Encyclopedia of Geomagnetism and Paleomagnetism

ENCYCLOPEDIA of GEOMAGNETISM AND PALEOMAGNETISM

Encyclopedia of Earth Sciences Series ENCYCLOPEDIA OF GEOMAGNETISM AND PALEOMAGNETISM Volume Editors David Gubbins is Research Professor of Earth Sciences in the School of Earth and Environment, University of Leeds, UK. He did his PhD on geomagnetic dynamos in Cambridge, supervised by Sir Edward Bullard and has worked in the USA, and in Cambridge before moving to Leeds in 1989. His work has included dynamo theory and its connection with the Earth's thermal history, modeling the Earth's magnetic field from historical measurements, and recently the interpretation of paleomagnetic data. He is a fellow of the Royal Society and has been awarded the gold medal of the Royal Astronomical Society and the John Adam Fleming Medal of the American Geophysical Union for original research and leadership in geomagnetism. Emilio Herrero-Bervera is Research Professor of Geophysics at the School of Ocean and Earth Science and Technology (SOEST) within the Hawaii Institute of Geophysics and Planetology (HIGP) of the University of Hawaii at Manoa, where he is the head of the Paleomagnetics and Petrofabrics Laboratory. During his career he has published over 90 papers in professional journals including Nature, JGR, EPSL, JVGR. He has worked in such diverse fields as volcanology, sedimentology, plate tectonics, and has done field work on five continents.

Aim of the Series The Encyclopedia of Earth Sciences Series provides comprehensive and authoritative coverage of all the main areas in the earth sciences. Each volume comprises a focused and carefully chosen collection of contributions from leading names in the subject, with copious illustrations and reference lists. These books represent one of the world's leading resources for the Earth sciences community. Previous volumes are being updated and new works published so that the volumes will continue to be essential reading for all professional Earth scientists, geologists, geophysicists, climatologists, and oceanographers as well as for teachers and students. See the back of this volume for a current list of titles in the Encyclopedia of Earth Sciences Series. Go to http://www.springerlink.com/referenceworks/ to visit the “Earth Sciences Series” on-line.

About the Editors

Professor Rhodes W. Fairbridge{ has edited 24 encyclopedias in the Earth Sciences Series. During his career he has worked as a petroleum geologist in the Middle East, been a World War II intelligence officer in the SW Pacific, and led expeditions to the Sahara, Arctic Canada, Arctic Scandinavia, Brazil, and New Guinea. He was Emeritus Professor of Geology at Columbia University and was affiliated with the Goddard Institute for Space Studies. Professor Michael Rampino has published more than 100 papers in professional journals including Science, Nature, and Scientific American. He has worked in such diverse fields as volcanology, planetary science, sedimentology, and climate studies, and has done field work on six continents. He is currently Associate Professor of Earth and Environmental Sciences at New York University and a consultant at NASA's Goddard Institute for Space Studies.

ENCYCLOPEDIA OF EARTH SCIENCES SERIES

ENCYCLOPEDIA of GEOMAGNETISM AND PALEOMAGNETISM edited by

DAVID GUBBINS

University of Leeds and

EMILIO HERREROBERVERA

University of Hawaii at Manoa

A C.I.P. Catalogue record for this book is available from the Library of Congress.

ISBN-13: 978-1-4020-3992-8 This publication is available also as Electronic publication under ISBN 978-1-4020-4423-6 and Print and electronic bundle under ISBN 978-1-4020-4866-1

Published by Springer PO Box 17, 3300 AA Dordrecht, The Netherlands

Printed on acid‐free paper

Cover photo: Part of “A Digital Age Map of the Ocean Floor”, by Mueller, R.D., Roest, W.R., Royer, J.‐Y., Gahagan, L.M., and Sclater, J.G., SIO Reference Series 93-30, Scripps Institution of Oceanography (map downloaded courtesy of NGDC).

Every effort has been made to contact the copyright holders of the figures and tables which have been reproduced from other sources. Anyone who has not been properly credited is requested to contact the publishers, so that due acknowledgment may be made in subsequent editions.

All Rights Reserved © 2007 Springer No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work.

Dedication Jack A. Jacobs (1916–2003) This encyclopedia is dedicated to the memory of Jack Jacobs. He made contributions across the whole spectrum of geomagnetism and paleomagnetism throughout a long and productive career. His books Micropulsations, Geonomy, The Earth’s Core and Geomagnetism, Reversals of the Earth’s Magnetic Field, and the four volumes of

Geomagnetism intended to replace Chapman & Bartels’ work of the same name, cover the field. For this encyclopedia he completed articles for both editors, on disc dynamo and geomagnetic excursions, and was working on superchrons and changes in reversal frequency at the end.

Contents

List of Contributors Preface

xv xxv

Bemmelen, Willem van (1868–1941) Art R.T. Jonkers

44

Benton, E. R. David Loper

44

Bingham Statistics Jeffrey J. Love

45

Biomagnetism Michael D. Fuller and Jon Dobson

48

Aeromagnetic Surveying Mark Pilkington

1

Agricola, Georgius (1494–1555) Allan Chapman

3

Alfvén Waves Christopher Finlay

3

Alfvén, Hannes Olof Gösta (1908–1995) Carl-Gunne Fälthammar and David Gubbins

6

Blackett, Patrick Maynard Stuart, Baron of Chelsea (1887–1974) Michael D. Fuller

53

Alfvén's Theorem and the Frozen Flux Approximation Paul H. Roberts

7

Bullard, Edward Crisp (1907–1980) David Gubbins

54

Carnegie Institution of Washington, Department of Terrestrial Magnetism Gregory A. Good

56

Anelastic and Boussinesq Approximations Stanislav I. Braginsky and Paul H. Roberts

11

Anisotropy, Electrical Karsten Bahr

20

Carnegie, Research Vessel Gregory A. Good

58

Antidynamo and Bounding Theorems Friedrich Busse and Michael Proctor

21

Champ Stefan Maus

59

Archeology, Magnetic Methods Armin Schmidt

23

Chapman, Sydney (1888–1970) Stuart R.C. Malin

61

Archeomagnetism Donald D. Tarling

31

Coast Effect of Induced Currents Ted Lilley

61

Auroral Oval Stephen Milan

33

Compass Art R.T. Jonkers

66

Baked Contact Test Kenneth L. Buchan

35

Conductivity Geothermometer Ted Lilley

69

Bangui Anomaly Patrick T. Taylor

39

Conductivity, Ocean Floor Measurements Steven Constable

71

Barlow, Peter (1776–1862) Emmanuel Dormy

40

Convection, Chemical David Loper

73

Bartels, Julius (1899–1964) Karl-Heinz Glaßmeier and Manfred Siebert

42

Convection, Nonmagnetic Rotating Andrew Soward

74

Bauer, Louis Agricola (1865–1932) Gregory A. Good

42

Core Composition William F. McDonough

77

viii

CONTENTS

Core Convection Keke Zhang

80

D00 as a Boundary Layer David Loper

145

Core Density Guy Masters

82

D00 , Anisotropy Michael Kendall

146

Core Motions Kathryn A. Whaler

84

D00 , Composition Quentin Williams

149

Core Origin Francis Nimmo

89

D00 , Seismic Properties Thorne Lay

151

Core Properties, Physical Frank D. Stacey

91

Della Porta, Giambattista (1535–1615) Allan Chapman

156

Core Properties, Theoretical Determination David Price

94

Demagnetization Jaime Urrutia-Fucugauchi

156

Core Temperature David Price

98

Depth to Curie Temperature Mita Rajaram

157

Core Turbulence Bruce Buffett and Hiroaki Matsui

101

Dipole Moment Variation Catherine Constable

159

Core Viscosity Lidunka Vočadlo

104

Dynamo Waves Graeme R. Sarson

161

Core, Adiabatic Gradient Orson L. Anderson

106

Dynamo, Backus Ashley P. Willis

163

Core, Boundary Layers Emmanuel Dormy, Paul H. Roberts, and Andrew Soward

111

Dynamo, Braginsky Graeme R. Sarson

164

Core, Electrical Conductivity Frank D. Stacey

116

Dynamo, Bullard-Gellman Graeme R. Sarson

166

Core, Magnetic Instabilities David R. Fearn

117

Dynamo, Disk Jack A. Jacobs

167

Core, Thermal Conduction Frank D. Stacey

120

Dynamo, Gailitis Agris Gailitis

169

Core-Based Inversions for the Main Geomagnetic Field David Gubbins

122

Dynamo, Herzenberg Paul H. Roberts

170

Core-Mantle Boundary Topography, Implications for Dynamics Andrew Soward

124

Dynamo, Lowes-Wilkinson Frank Lowes

173

Core-Mantle Boundary Topography, Seismology Andrea Morelli

125

Dynamo, Model-Z Rainer Hollerbach

174

Core-Mantle Boundary, Heat Flow Across Stéphane Labrosse

127

Dynamo, Ponomarenko Paul H. Roberts

175

Core-Mantle Coupling, Electromagnetic Richard Holme

130

Dynamo, Solar Eugene N. Parker

178

Core-Mantle Coupling, Thermal Jeremy Bloxham

132

Dynamos, Experimental Andreas Tilgner

183

Core-Mantle Coupling, Topographic Dominique Jault

135

Dynamos, Fast Michael Proctor

186

Cowling, Thomas George (1906–1990) Leon Mestel

137

Dynamos, Kinematic Philip W. Livermore

188

Cowling's Theorem Friedrich Busse

138

Dynamos, Mean-Field Karl-Heinz Raedler

192

Cox, Allan V. (1926–1987) Kenneth P. Kodama

139

Dynamos, Periodic David Gubbins

200

Crustal Magnetic Field Dhananjay Ravat

140

Dynamos, Planetary and Satellite David J. Stevenson

203

D00 and F-Layers David Gubbins

145

Earth Structure, Major Divisions Brian Kennett

208

CONTENTS

ix

Elsasser, Walter M. (1904–1991) Eugene N. Parker

214

Geodynamo, Symmetry Properties David Gubbins

306

EM Modeling, Forward Dmitry B. Avdeev

215

Geomagnetic Deep Sounding Roger Banks

307

EM Modeling, Inverse Gary D. Egbert

219

Geomagnetic Dipole Field Frank Lowes

310

EM, Industrial Uses Graham Heinson

223

Geomagnetic Excursion Jack A. Jacobs

311

EM, Lake-Bottom Measurements Adam Schultz

227

Geomagnetic Field, Asymmetries Phillip L. McFadden and Ronald T. Merrill

313

EM, Land Uses Louise Pellerin

228

Geomagnetic Hazards Alan W.P. Thomson

316

EM, Marine Controlled Source Nigel Edwards

231

Geomagnetic Jerks Susan Macmillan

319

EM, Regional Studies Oliver Ritter

242

Geomagnetic Polarity Reversals Alain Mazaud

320

EM, Tectonic Interpretations Malcolm Ingham

245

Geomagnetic Polarity Reversals, Observations Bradford M. Clement

324

Environmental Magnetism Barbara A. Maher

248

Geomagnetic Polarity Timescales William Lowrie

328

Environmental Magnetism, Paleomagnetic Applications Andrew P. Roberts

256

Geomagnetic Pulsations Karl-Heinz Glaßmeier

333

Equilibration of Magnetic Field, Weak- and Strong-Field Dynamos Keke Zhang

262

Geomagnetic Reversal Sequence, Statistical Structure Phillip L. McFadden

335

Euler Deconvolution Alan B. Reid

263

Geomagnetic Reversals, Archives Jean-Pierre Valet and Emilio Herrero-Bervera

339

First-Order Reversal Curve (FORC) Diagrams Adrian R. Muxworthy and Andrew P. Roberts

266

Geomagnetic Secular Variation Ingo Wardinski

346

Fisher Statistics Jeffrey J. Love

272

Geomagnetic Spectrum, Spatial Frank Lowes

350

Fleming, John Adam (1877–1956) Shaun J. Hardy

273

Geomagnetic Spectrum, Temporal Catherine Constable

353

Fluid Dynamics Experiments Jonathan M. Aurnou

274

Geomagnetism, History of Art R.T. Jonkers

355

Galvanic Distortion Karsten Bahr

277

Gilbert, William (1544–1603) Allan Chapman

360

Gauss’ Determination of Absolute Intensity Stuart R.C. Malin

278

Gravitational Torque Jean-Louis Le Mouel

362

Gauss, Carl Friedrich (1777–1855) Karl-Heinz Glaßmeier

279

Gravity-Inertio Waves and Inertial Oscillations Keith Aldridge

364

Gellibrand, Henry (1597–1636) Stuart R.C. Malin

280

Grüneisen's Parameter for Iron and Earth's Core Orson L. Anderson

366

Geocentric Axial Dipole Hypothesis Michael W. McElhinny

281

Halley, Edmond (1656–1742) Sir Alan Cook

375

Geodynamo Chris Jones

287

Hansteen, Christopher (1784–1873) Johannes M. Hansteen

376

Geodynamo, Dimensional Analysis and Timescales David Gubbins

297

Harmonics, Spherical Denis Winch

377

Geodynamo, Energy Sources Stéphane Labrosse

300

Harmonics, Spherical Cap G.V. Haines

395

Geodynamo: Numerical Simulations Gary A. Glatzmaier

302

Hartmann, Georg (1489–1564) Allan Chapman

397

x

CONTENTS

Helioseismology Michael J. Thompson

398

Laplace's Equation, Uniqueness of Solutions David Gubbins

466

Higgins-Kennedy Paradox Friedrich Busse

401

Larmor, Joseph (1857–1942) David Gubbins

468

Humboldt, Alexander Von (1759–1859) Friedrich Busse

402

Lehmann, Inge (1888–1993) David Gubbins

468

Humboldt, Alexander Von and Magnetic Storms G.S. Lakhina, B.T. Tsurutani, W.D. Gonzalez, and S. Alex

404

Length of Day Variations, Decadal Richard Holme

469

IAGA, International Association of Geomagnetism and Aeronomy David Kerridge

407

Length of Day Variations, Long-Term L.V. Morrison and F.R. Stephenson

471

Ideal Solution Theory Dario Alfè

408

Lloyd, Humphrey (1808–1881) Deanis Weaire and J.M.D. Coey

472

IGRF, International Geomagnetic Reference Field Susan Macmillan

411

Magnetic Anisotropy, Sedimentary Rocks and Strain Alteration Peter D. Weiler

475

Induction Arrows Oliver Ritter

412

Magnetic Anomalies for Geology and Resources Colin Reeves and Juha V. Korhonen

477

Induction from Satellite Data Steven Constable

413

Magnetic Anomalies, Long Wavelength Michael E. Purucker

481

Inhomogeneous Boundary Conditions and the Dynamo Keke Zhang

416

Magnetic Anomalies, Marine James R. Heirtzler

483

Inner Core Anisotropy Xiaodong Song

418

Magnetic Anomalies, Modeling Jafar Arkani-Hamed

485

Inner Core Composition Lidunka Vočadlo

420

Magnetic Domains Susan L. Halgedahl

490

Inner Core Oscillation Keith Aldridge

422

Magnetic Field of Mars Jafar Arkani-Hamed

502

Inner Core Rotation Paul G. Richards and Anyi Li

423

Magnetic Field of Sun Steven M. Tobias

505

Inner Core Rotational Dynamics Michael G. Rochester

425

Magnetic Indices Jeffrey J. Love and K.J. Remick

509

Inner Core Seismic Velocities Annie Souriau

427

512

Inner Core Tangent Cylinder Rainer Hollerbach and David Gubbins

430

Magnetic Mineralogy, Changes due to Heating Bernard Henry

515

Inner Core, PKJKP Hanneke Paulssen

433

Magnetic Properties, Low-Temperature Andrei Kosterov

525

Instrumentation, History of Gregory A. Good

434

Magnetic Proxy Parameters Mark J. Dekkers Magnetic Remanence, Anisotropy Ann M. Hirt

535

Interiors of Planets and Satellites Gerald Schubert

439

Magnetic Shielding Gary R. Scott

540

Internal External Field Separation Denis Winch

448

Magnetic Surveys, Marine Maurice A. Tivey

542

Ionosphere Arthur D. Richmond

452

Magnetic Susceptibility, Anisotropy František Hrouda

546

Iron Sulfides Leonardo Sagnotti

454

Magnetic Susceptibility, Anisotropy, Effects of Heating Jaime Urrutia-Fucugauchi

560

Jesuits, Role in Geomagnetism Agustín Udías

460

Magnetic Susceptibility, Anisotropy, Rock Fabric Edgardo Cañón-Tapia

564

Kircher, Athanasius (1602–1680) Oriol Cardus

463

Magnetic Susceptibility (MS), Low-Field Brooks B. Ellwood

566

Langel, Robert A. (1937–2000) Michael E. Purucker

465

Magnetization, Anhysteretic Remanent Bruce M. Moskowitz

572

CONTENTS

xi

Magnetization, Chemical Remanent (CRM) Shaul Levi

580

Melting Temperature of Iron in the Core, Theory David Price

692

Magnetization, Depositional Remanent Jaime Urrutia-Fucugauchi

588

Microwave Paleomagnetic Technique John Shaw

694

Magnetization, Isothermal Remanent Mike Jackson

589

Nagata, Takesi (1913–1991) Masaru Kono

696

Magnetization, Natural Remanent (NRM) Mimi J. Hill

594

Natural Sources for Electromagnetic Induction Studies Nils Olsen

696

Magnetization, Oceanic Crust Julie Carlut and Hélène Horen

596

Néel, Louis (1904–2000) Pierre Rochette

700

Magnetization, Piezoremanence and Stress Demagnetization Stuart Alan Gilder

599

Nondipole Field Catherine Constable

701

Magnetization, Remanent, Ambient Temperature and Burial Depth from Dyke Contact Zones Kenneth L. Buchan

603

Nondynamo Theories David J. Stevenson

704

Magnetization, Remanent, Fold Test Jaime Urrutia-Fucugauchi

607

Norman, Robert (flourished 1560–1585) Allan Chapman

707

Magnetization, Thermoremanent Özden Özdemir

609

Observatories, Overview Susan Macmillan

708

Magnetization, Thermoremanent, in Minerals Gunther Kletetschka

616

Observatories, Instrumentation Jean L. Rasson

711

Magnetization, Viscous Remanent (VRM) David J. Dunlop

621

Observatories, Automation Lawrence R. Newitt

713

Magnetoconvection Keke Zhang and Xinhao Liao

630

Observatories, Intermagnet Jean L. Rasson

715

Magnetohydrodynamic Waves Christopher Finlay

632

Observatories, Program in Australia Peter A. Hopgood

717

Magnetohydrodynamics Paul H. Roberts

639

Observatories, Program in the British Isles David Kerridge

720

Magnetometers, Laboratory Wyn Williams

654

Observatory Program in France Mioara Mandea

721

Magnetosphere of the Earth Stanley W.H. Cowley

656

Observatories, Program in USA Jeffrey J. Love and J.B. Townshend

722

Magnetostratigraphy William Lowrie

664

Observatories in Antarctica Jean-Jacques Schott and Jean L. Rasson

723

Magnetotellurics Martyn Unsworth

670

Observatories in Benelux Countries Jean L. Rasson

725

Magsat Michael E. Purucker

673

Observatories in Canada Lawrence R. Newitt and Richard Coles

726

Main Field Maps Mioara Mandea

674

Observatories in China Dongmei Yang

727

Main Field Modeling Mioara Mandea

679

Observatories in East and Central Europe Pavel Hejda

728

Main Field, Ellipticity Correction Stuart R.C. Malin

683

Observatories in Germany Hans-Joachim Linthe

729

Mantle, Electrical Conductivity, Mineralogy Tomoo Katsura

684

Observatories in India Gurbax S. Lakhina and S. Alex

731

Mantle, Thermal Conductivity Frank D. Stacey

688

Observatories in Italy Massimo Chiappini

733

Matuyama, Motonori (1884–1958) Masaru Kono

689

Observatories in Japan and Asia Toshihiko Iyemori and Heather McCreadie

733

Melting Temperature of Iron in the Core, Experimental Guoyin Shen

689

Observatories in Latin America Luiz Muniz Barreto

734

xii

CONTENTS

Observatories in New Zealand and the South Pacific Lester A. Tomlinson

735

Potential Vorticity and Potential Magnetic Field Theorems Raymond Hide

840

Observatories in Nordic Countries Truls Lynne Hansen

736

Precession and Core Dynamics Philippe Cardin

842

Observatories in Russia Oleg Troshichev

737

Price, Albert Thomas (1903–1978) Bruce A. Hobbs

844

Observatories in Southern Africa Pieter Kotzé

739

Principal Component Analysis in Paleomagnetism Jeffrey J. Love

845

Observatories in Spain Miquel Torta and Josep Batlló

739

Project Magnet David G. McMillan

850

Ocean, Electromagnetic Effects Stefan Maus

740

Proudman-Taylor Theorem Raymond Hide

852

Oldham, Richard Dixon (1858–1936) Johannes Schweitzer

742

Radioactive Isotopes, Their Decay in Mantle and Core V. Rama Murthy

854

Ørsted Nils Olsen

743

Reduction to Pole Dhananjay Ravat

856

Oscillations, Torsional Mathieu Dumberry

746

Repeat Stations Susan Macmillan

858

Reversals, Theory Graeme R. Sarson

859

749

Paleointensity, Absolute, Techniques Jean-Pierre Valet

753

Rikitake, Tsuneji (1921–2004) Y. Honkura

862

Paleointensity, Relative, in Sediments Stefanie Brachfeld

758

Ring Current Thomas E. Moore

863

Paleomagnetic Field Collection Methods Edgardo Cañón-Tapia

765

Robust Electromagnetic Transfer Functions Estimates Gary D. Egbert

866

Paleomagnetic Secular Variation Steve P. Lund

766

Rock Magnetism Ronald T. Merrill

870

Paleomagnetism Ronald T. Merrill and Phillip L. McFadden

776

Rock Magnetism, Hysteresis Measurements David Krása and Karl Fabian

874

Paleomagnetism, Deep-Sea Sediments James E.T. Channell

781

Rock Magnetometer, Superconducting William S. Goree

883

Paleomagnetism, Extraterrestrial Michael D. Fuller

788

Runcorn, S. Keith (1922–1995) Neil Opdyke

886

Paleomagnetism, Orogenic Belts John D.A. Piper

801

Runcorn's Theorem Andrew Jackson

888

Parkinson, Wilfred Dudley Ted Lilley

807

Sabine, Edward (1788–1883) David Gubbins

890

Peregrinus, Petrus (flourished 1269) Allan Chapman

808

Seamount Magnetism James R. Heirtzler and K.A. Nazarova

891

Periodic External Fields Denis Winch

809

Secular Variation Model Christopher G.A. Harrison

892

Plate Tectonics, China Xixi Zhao and Robert S. Coe

816

Sedi David Loper

902

Pogo (OGO-2, -4 and -6 Spacecraft) Joseph C. Cain

828

Seismic Phases Brian Kennett

903

Polarity Transition, Paleomagnetic Record Kenneth A. Hoffman

829

Seismo-Electromagnetic Effects Malcolm J.S. Johnston

908

Polarity Transitions: Radioisotopic Dating Brad S. Singer

834

Shaw and Microwave Methods, Absolute Paleointensity Determination John Shaw

910

Pole, Key Paleomagnetic Kenneth L. Buchan

839

Shock Wave Experiments Thomas J. Ahrens

912

Paleointensity: Absolute Determinations Using Single Plagioclase Crystals John A. Tarduno, Rory D. Cottrell, and Alexei V. Smirnov

CONTENTS

xiii

Spinner Magnetometer Jiří Pokorný

920

Transient Em Induction Maxwell A. Meju and Mark E. Everett

954

Statistical Methods for Paleovector Analysis Jeffrey J. Love

922

True Polar Wander Vincent Courtillot

956

Storms and Substorms, Magnetic Mark Lester

926

ULVZ, Ultra-Low Velocity Zone Ed J. Garnero and M. Thorne

970

Superchrons, Changes in Reversal Frequency Jack A. Jacobs

928

Units David Gubbins

973

Susceptibility Eduard Petrovsky

931

Upward and Downward Continuation Dhananjay Ravat

974

Susceptibility, Measurements of Solids Z.S. Teweldemedhin, R.L. Fuller, and M. Greenblatt

933

Variable Field Translation Balance David Krása, Klaus Petersen, and Nikolai Petersen

977

Susceptibility, Parameters, Anisotropy Edgardo Cañón-Tapia

937

Verhoogen, John (1912–1993) Peter Olson

979

Taylor's Condition Rainer Hollerbach

940

Vine-Matthews-Morley Hypothesis Maurice A. Tivey

980

Thellier, Émile (1904–1987) David J. Dunlop

942

Volcano-Electromagnetic Effects Malcolm J.S. Johnston

984

Thermal Wind Peter Olson

945

Voyages Making Geomagnetic Measurements David R. Barraclough

987

Time-Averaged Paleomagnetic Field David Gubbins

947

Watkins, Norman David (1934–1977) Brooks B. Ellwood

992

Time-Dependent Models of the Geomagnetic Field Andrew Jackson

948

Westward Drift Richard Holme

993

Color Plates

997

Transfer Functions Martyn Unsworth

953

Subject Index

1013

Contributors

Thomas J. Ahrens CALTECH MS 252-21 Pasadena, CA 91125, USA email: [email protected] Keith Aldridge Department of Earth & Atmospheric Sciences York University 4700 Keele Street Toronto, ON M3J 1P3, Canada email: [email protected] Sobhana Alex Indian Institute of Geomagnetism New Panvel (W) Navi Mumbai-410 218, India email: [email protected] Dario Alfè Department of Earth Sciences University College London Gower Street London, WC1E 6BT, UK email: [email protected] Orson L. Anderson Institute of Geophysics and Planetary Physics F83 Department of Earth and Space Sciences University of California, Los Angeles, CA 90095, USA email: [email protected] Jafar Arkani-Hamed Earth & Planetary Sciences McGill University 3450 University St Montreal, QC H3A 2A7, Canada email: [email protected] Jonathan M. Aurnou Department of Earth and Space Sciences University of California, Los Angeles 595 Charles Young Drive East Los Angeles, CA 90095-1567, USA email: [email protected]/[email protected] Dmitry B. Avdeev School of Cosmic Physics Dublin Institute for Advanced Studies 5 Merrion Square Dublin 2, Ireland email: [email protected]

Karsten Bahr Geophysical Institute Universität Göttingen Herzberger Landstr. 180 37075 Göttingen, Germany email: [email protected] Roger Banks Fernwood, Rogerground Hawkshead, Ambleside Cumbria, LA22 0QG, UK email: [email protected] David Barraclough 49 Liberton Drive Edinburgh, EH16 6NL, UK email: [email protected] Luiz Muniz Barreto Observatorio Nacional Rua general Jose Cristino, 77 Rio de Janeiro, Brazil email: [email protected] Josep Batlló Department Matematica Aplicada 1 Universitat Polytecnica de Catalunya Spain email: [email protected] Jeremy Bloxham Department of Earth and Planetary Sciences Harvard University 20 Oxford Street Cambridge, MA 02138, USA email: [email protected]/[email protected] Stefanie Brachfeld Department of Earth and Environmental Studies Montclair State University Montclair, NJ 07043, USA email: [email protected] Stanislav I. Braginsky Institute of Geophysics and Planetary Physics UCLA 405 Hilgard Avenue Los Angeles, CA 90024-1567, USA email: [email protected] Kenneth L. Buchan Geological Survey of Canada Natural Resources Canada 601 Booth Street Ottawa, Ontario, K1A 0E8, Canada email: [email protected]

xvi

CONTRIBUTORS

Bruce Buffett Department of Geophysical Sciences University of Chicago 5734 S. Ellis Avenue Chicago, IL 60637, USA email: [email protected]

J. Michael D. Coey Physics Dept. Trinity College College Green Dublin 2, Ireland email: [email protected]

Friedrich Busse Institute of Physics University of Bayreuth 95440 Bayreuth, Germany email: [email protected]

Richard Coles Geomagnetism Laboratory Natural Resources Canada 7 Observatory Crescent Ottawa, Ontario K1A0Y3, Canada

Joseph C. Cain Department of Geology Florida State University Tallahassee, FL 32306-3026, USA email: [email protected]

Catherine Constable Institute of Geophysics and Planetary Physics Scripps Institution of Oceanography University of California at San Diego La Jolla, CA 92093 0225, USA email: [email protected]

Edgardo Cañón-Tapia Department of Geology CICESE P.O. Box 434843 San Diego, CA 92143, USA email: [email protected] Philippe Cardin Universite Joseph Fourier de Grenoble Laboratoire de Geophysique interne et Tectonophysique 1381 Rue de la Piscine, BP 53 Grenoble Cedex 9, 38041, France email: [email protected] Oriol Cardus Observatori de l'Ebre Roquetes Tarragona, 43520, Spain email: [email protected] Julie Carlut Laboratoire de Geologie Ecole Normale Superieure 24 rue Lhomond Paris, 75235, France email: [email protected] James E.T. Channell University of Florida Department of Geological Sciences P.O. Box 112120 Gainesville, FL 32611-2120, USA email: [email protected] Allan Chapman Modern History Faculty Office University of Oxford Broad St. Oxford, OX1 3BD, UK email: [email protected] Massimo Chiappini Instituto Nazionale di Geofisica e Vulcanologia Vigna Murata 605 Rome, 00143, Italy email: [email protected] Bradford M. Clement Florida International University Department of Earth Science SW 8th St & 107th Ave Miami, FL 33199, USA email: [email protected] Robert S. Coe Institute of Geophysics and Planetary Physics University of California Santa Cruz 1156 High Street Santa Cruz, CA 95064, USA email: [email protected]

Steven Constable Scripps Institution of Oceanography La Jolla, CA 920930225, USA email: [email protected] Sir Alan Cook (deceased) Rory D. Cottrell Department of Earth and Environmental Sciences University of Rochester Hutchison Hall 227 Rochester, NY 14627, USA email: [email protected] Vincent Courtillot Institut de Physique du Globe de Paris 4 place Jussieu Paris Cedex 05, 75252, France email: [email protected] Stanley W.H. Cowley Department of Physics & Astronomy University of Leicester Leicester, LE1 7RH, UK email: [email protected] Mark J. Dekkers Department of Earth Sciences Utrecht University Budapestlaan 17 Utrecht, 3584 CD, The Netherlands email: [email protected] Jon Dobson Centre for Science & Technology in Medicine Keele University Thornburrow Drive Hartshill, Stoke-on-Trent ST4 7QB, UK Emmanuel Dormy C.N.R.S./I.P.G.P./E.N.S. Département de Physique Ecole Normale Supérieure 24, rue Lhomond 75231 Paris Cedex 05, France email: [email protected] Mathieu Dumberry School of Earth and Environment University of Leeds Leeds, LS2 9JT, UK email: [email protected] David J. Dunlop Department of Physics University of Toronto Mississauga, Ontario L5L 1C6, Canada email: [email protected]

CONTRIBUTORS

Nigel Edwards Department of Physics University of Toronto 60 St George Street Toronto, Ontario M5S 1A7, Canada email: [email protected]

Karl-Heinz Glaßmeier Institute of Geophysics and Extraterrestrial Physics Technical University of Braunschweig Mendelssohnstr. 3 38106 Braunschweig, Germany email: [email protected]

Gary D. Egbert College Oceanography Oregon State University Oceanography Admin Bldg 104 Corvallis, OR 97331-5503, USA email: [email protected]

Gary A. Glatzmaier Department of Earth Sciences University of California Santa Cruz, CA 95064, USA email: [email protected]

Brooks B. Ellwood Department of Geology and Geophysics Louisiana State University Baton Rouge, LA 70803, USA email: [email protected]

Walter Demétrio Gonzalez INPE-Caixa Postal 515 2200 Sao Jose Dos Campos Sao Paulo, Brazil

Mark E. Everett Department of Geology & Geophysics Texas A & M University College Station, TX 77843-3114, USA email: [email protected]/[email protected]

Gregory A. Good History Department West Virginia University Morgantown, WV 26506-6303, USA email: [email protected]

Karl Fabian Department of Earth and Environmental Sciences University of Munich Theresienstr. 41 80333 München, Germany email: [email protected]

William S. Goree Inc. and 2 G Enterprises 2040 Sunset Drive Pacific Grove, CA 93950, USA email: [email protected] [email protected]

Carl-Gunne Fälthammar Dept. of Plasma Physics Royal Institute of Technology Stockholm, SE-10044, Sweden email: [email protected] David R. Fearn Department of Mathematics University of Glasgow Glasgow G12 8QW, UK email: [email protected] Christopher Finlay ETH-Hönggerberg Institute of Geophysics Schaftmattstrasse 30 CH-8093 Zürich, Switzerland email: [email protected] Michael D. Fuller Paleomagnetics and Petrofabrics Laboratory 1680 East West Rd Honolulu, Hawaii, 96822, USA email: [email protected] Robert L. Fuller 909 River Rd. Colgate Palmolive co. Piscataway, NJ 08854, USA Agris Gailitis Institute of Physics University of Latvia Miera iela 32 Salaspils, LV 2169, Latvia email: [email protected]

Martha Greenblatt Department of Chemistry and Chemical Biology Rutgers University 610 Taylor Road Piscataway, NJ 08854-8087, USA email: [email protected] David Gubbins School of Earth and Environment University of Leeds Leeds LS2 9JT, UK email: [email protected] G.V. Haines 69 Amberwood Cr Ottawa, ON K2E 7C2, Canada email: [email protected] [email protected] Susan L. Halgedahl Department of Geology and Geophysics University of Utah Salt Lake City, Utah 84112, USA email: [email protected] Truls Lynne Hansen Tromso Geophysical Laboratory University of Tromso Tromso, N-9037, Norway email: [email protected] Johannes M. Hansteen (deceased)

Edward J. Garnero Dept Geological Sciences Arizona State University Box 871404 Tempe, AZ 85287-1404, USA email: [email protected]

Shaun J. Hardy Carnegie Institution of Washington 5241 Broad Branch Rd., N.W. Washington, DC 20015, USA email: [email protected]

Stuart Alan Gilder Geophysics Section Ludwig Maximillians University Theresienstrasse 41 80333 München, Germany email: [email protected]

Christopher G.A. Harrison Rosenstiel School of Marine and Atmospheric Science University of Miami 4600 Rickenbacker Causeway Miami, FL 33149, USA email: [email protected]

xvii

xviii

Graham Heinson Department of Geology and Geophysics Adelaide University Adelaide, SA 5005, Australia email: [email protected] [email protected]

CONTRIBUTORS

Peter A. Hopgood Geoscience Australia GPO Box 378 Canberra, ACT, 2601, Australia email: [email protected]

James R. Heirtzler NASA, Goddard Space Flight Center MC 920 Greenbelt, MD 20771-0001, USA email: [email protected]

Hélène Horen Laboratoire de Geologie Ecole Normale Superieure 24 rue Lhomond, Paris 75235, France email: [email protected]

Pavel Hejda Geophysical Institute Academy of Sciences of the Czech Republic Bocni II/1401 Prague 4, 141 31, Czech Republic email: [email protected]

František Hrouda AGICO, Inc. Advanced Geoscience Instruments Company Jecna 29a Brno, CZ 621 00, Czech Republic email: [email protected]

Bernard Henry Géomagnétisme et Paléomagnétisme IPGP and CNRS 4 avenue de Neptune Saint-Maur Cedex, 94107, France email: [email protected] Emilio Herrero-Bervera Paleomagnetics and Petrofabrics Laboratory University of Hawaii at Manoa Honolulu, Hawaii 96822, USA email: [email protected] Raymond Hide 17 Clinton Ave East Molesey, Surrey, KT8 0HS, UK email: [email protected] Mimi J. Hill Department of Earth and Ocean Sciences University of Liverpool Oxford Street Liverpool, L69 7ZE, UK email: [email protected] [email protected] Ann M. Hirt ETH-Hönggerberg Institute of Geophysics Zürich, 8093, Switzerland email: [email protected] Bruce A. Hobbs Department of Geology and Geophysics University of Edinburgh West Main Road Edinburgh, EH9 3JW, UK email: [email protected] Kenneth A. Hoffman Physics Department California Polytechnic State University San Luis Obispo, CA 93407, USA email: [email protected] Rainer Hollerbach School of Mathematics University of Leeds Leeds, LS2 9JT, UK email: [email protected] Richard Holme Department of Earth and Ocean Sciences University of Liverpool 4 Brownlow Street Liverpool, L69 3GP, UK email: [email protected] Yoshimori Honkura Tokyo Institute of Technology Department of Earth & Planetary Sciences 2-12-1-I2-6 Ookayama Meguro-ku, Tokyo, 152-8551, Japan email: [email protected]

Malcolm Ingham Department of Physics University of Victoria PO Box 600 Wellington, New Zealand email: [email protected] Toshihiko Iyemori Graduate School Science Kyoto University Data Analysis Center Kyoto, 606-8502, Japan email: [email protected] Andrew Jackson School of Earth and Environment University of Leeds Leeds LS2 9JT, UK email: [email protected] Mike Jackson Department of Geology and Geophysics Institute for Rock Magnetism University of Minnesota 100 Union Street SE Minneapolis, MN 55455, USA email: [email protected] Jack A. Jacobs (deceased) Dominique Jault LGIT, CNRS and University Joseph-Fourier BP 53 38041Grenoble Cedex9, France email: [email protected]. Malcolm J.S. Johnston USGS 345 Middlefield Rd MS 977 Menlo Park, CA 94025, USA email: [email protected] Chris Jones Department of Applied Mathematics University of Leeds Leeds LS2 9JT, UK email: [email protected] Art R.T. Jonkers Department of Earth and Ocean Sciences The Jane Herdman Laboratories University of Liverpool 4 Brownlow St Liverpool, L69 3GP, UK email: [email protected] Tomoo Katsura Institute for Study of the Earth's Interior Okayama University Misasa, Tottori-ken, 682-0193, Japan email: [email protected]

CONTRIBUTORS

Michael Kendall Dept of Earth Sciences University of Bristol Queen's Rd. Bristol, BS8 1RJ, UK email: [email protected] Brian Kennett Research School of Earth Sciences Australian National University GPO Box 4 Canberra, ACT 0200, Australia email: [email protected] David Kerridge Geomagnetism Group British Geological Survey West Mains Road Edinburgh, EH9 3LA, UK email: [email protected] Gunther Kletetschka NASA Goddard Space Flight Center Greenbelt, Maryland, 20771, USA email: [email protected] Kenneth P. Kodama Department of Earth and Environmental Sciences Lehigh University 31 Williams Drive, Bethlehem, PA 18015-3188, USA email: [email protected] Masaru Kono Institute for Study of the Earth's Interior Okayama, University of Misasa Yamada 827 Misasa, Tottori Prefecture, 682 0193, Japan email: [email protected] Juha V. Korhonen Geological Survey of Finland POB 96 Espo, 02151, Finland email: [email protected] Andrei Kosterov Nikolaeva 5-56, Kiev, Ukraine email: [email protected]/[email protected] Pieter Kotzé Geomagnetism Group Hermanus Magnetic Observatory PO Box 32, Hermanus, 7200, South Africa email: [email protected] David Krása School of GeoSciences University of Edinburgh King's Buildings Edinburgh, EH9 3JW, UK email: [email protected] Stéphane Labrosse Departement des Geomateriaux Institut de Physique du Globe de Paris 4 place Jussieu Paris Cedex 05, 75252, France email: [email protected]

Jean-Louis Le Mouel Institut de Physique du Globe de Paris 4 place Jussieu Paris Cedex 05, 75252, France email: [email protected] Mark Lester Dept. of Physics and Astronomy University of Leicester Leicester, LE1 7RH, UK email: [email protected] Shaul Levi College of Oceanic and Atmospheric Sciences Oregon State University Corvallis, OR 97331-5503, USA email: [email protected] Anyi Li Lamont-Doherty Earth Observatory and Department of Earth and Environmental Sciences Columbia University Palisades, NY 10964, USA email: [email protected] Xinhao Liao Shanghai Astronomical Observatory 80 Nandan Road Shanghai, 200030, China Ted Lilley Research School of Earth Sciences Australian National University GPO Box 4 Canberra, ACT 0200, Australia email: [email protected] Hans-Joachim Linthe Geomagnetic Adolf Schmidt Observatory Niemegk Geoforschungszentrum Potsdam Lindenstr. 7 14823 Niemegk, Germany email: [email protected] Philip W. Livermore School of Mathematics University of Leeds Leeds, LS2 9JT, UK email: [email protected] David Loper Florida State University GFDI-4360 Tallahassee, FL 32306-0000, USA email: [email protected] Jeffrey J. Love USGS Golden Box 25045 MS966 DFC Denver, CO 80227, USA email: [email protected] Frank Lowes Department of Physics The University of Newcastle-upon-Tyne Newcastle upon Tyne, NE1 7RU, UK email: [email protected]

Gurbax S. Lakhina Indian Institute of Geomagnetism New Panvel, Navi Mumbai, 410218, India email: [email protected]

William Lowrie Institute of Geophysics ETH-Hönggerberg CH-8093 Zürich, Switzerland email: [email protected]

Thorne Lay Earth Sciences Dept. University California Santa Cruz Santa Cruz, CA 95064-1077, USA email: [email protected]/[email protected]

Steve P. Lund Department of Earth Sciences University of Southern California Los Angeles, CA 90089-0740, USA email: [email protected]

xix

xx

CONTRIBUTORS

Susan Macmillan British Geological Survey West Mains Road Edinburgh, EH9 3LA, UK email: [email protected] Barbara A. Maher Centre for Environmental Magnetism and Palaeomagnetism Lancaster University Lancaster, LA1 4YB, UK email: [email protected] Stuart R.C. Malin 30 Wemyss Road Blackheath London, SE3 0TG, UK email: [email protected] Mioara Mandea GeoForschungZentrum Telegrafenberg 14473 Potsdam, F269, Germany email: [email protected] Guy Masters IGPP Scripps Institute of Oceanography University of California, San Diego 9500 Gilman Drive La Jolla, CA 92093-0225, USA email: [email protected] Hiroaki Matsui Dept. Geophysical Sciences University of Chicago 5734 S. Ellis Ave. Chicago, IL 60637, USA email: [email protected] Stefan Maus National Geophysical Data Center NOAA E/GC1 325 Broadway Boulder, CO 80305-3328, USA email: [email protected] Alain Mazaud Laboratoire des Sciences du Climat et de l'Environnement (LSCE) CEA-CNRS Avenue de la Terrasse Gif-sur-Yvette Cedex, 91198, France email: [email protected] Heather McCreadie World Data Centre for Geomagnetism Kyoto University Kyoto, 606-8502, Japan William F. McDonough Department of Geology University of Maryland College Park, Maryland, 20742, USA email: [email protected]

Maxwell A. Meju Dept Environmental Sci. Lancaster University University Rd. Bailrigg, LA1 4YQ, UK email: [email protected] Ronald T. Merrill Geophysics Program, AK50 University of Washington P.O. Box 433934 Seattle, Washington 98195, USA email: [email protected] Leon Mestel Astronomy Centre University of Sussex Brighton BN1 9QH, UK email: [email protected] Stephen Milan Radio and Space Plasma Physics Group Department of Physics and Astronomy University of Leicester Leicester, LE1 7RH, UK email: [email protected] Thomas E. Moore Laboratory for Solar and Space Physics Mail Code 612 Greenbelt, MD 20771, USA email: [email protected] Andrea Morelli Istituto Nazionale di Geofisica e Vulcanologia Via Donato Creti 12 40128 Bologna, Italy email: [email protected] Leslie V. Morrison 28 Pevensley Park Road Westham, Pevensley East Sussex, BN24 5HW, UK email: [email protected] Bruce M. Moskowitz Department of Geology and Geophysics University of Minnesota 310 Pillsbury Dr. SE Minneapolis, MN 55455, USA email: [email protected] [email protected] V. Rama Murthy Department of Geology and Geophysics University of Minnesota 310 Pillsbury Drive SE Minneapolis, MN55455, USA email: [email protected]

Michael W. McElhinny Gondwana Consultants 31 Laguna Place Port Macquarie, NSW 2444, Australia email: [email protected] [email protected]

Adrian R. Muxworthy National Oceanography Centre School of Ocean and Earth Science University of Southampton Southampton, SO14 3ZH, UK email: [email protected] [email protected]

Phillip L. McFadden Geoscience Australia GPO Box 378 Canberra, ACT 2601, Australia email: [email protected]

Katherine A. Nazarova ITSS/NASA Goddard Space Flight Center Greenbelt, MD 20771, USA email: [email protected]

David G. McMillan Department of Earth & Space Science and Engineering York University Toronto, Ontario, Canada email: [email protected]

Lawrence R. Newitt 1 Observatory Crescent Geological Survey Canada Ottawa, ON K1A 0Y3, Canada email: [email protected]

CONTRIBUTORS

Francis Nimmo Dept. Earth Sciences University of California Santa Cruz, CA 95064, USA email: [email protected] Nils Olsen Danish National Space Center Juliane Maries Vej Copenhagen, 2100, Denmark email: [email protected] Peter Olson Earth & Planetary Sciences The Johns Hopkins University Baltimore, MD 21218-2681, USA email: [email protected] Neil Opdyke Department of Geology University of Florida 1112 Turlington Hall Gainsville, FL 32611, USA email: [email protected] Özden Özdemir Department of Physics University of Toronto Mississauga, Ontario L5L 1C6, Canada email: [email protected] Eugene N. Parker 1323 Evergreen Rd Homewood, IL 60430, USA email: [email protected] Hanneke Paulssen Universiteit Utrecht Institute of Earth Sciences P O Box 80021 Utrecht, 3508 TA, The Netherlands email: [email protected]

Jiří Pokorný AGICO, Inc. Advanced Geoscience Instruments Company Jecna 29a Brno, CZ 621 00, Czech Republic email: [email protected] David Price Department of Geological Science University College London Gower Street London, WC1E 6BT, UK email: [email protected] Michael Proctor University of Cambridge D.A.M.T.P, F1.07 CMS Wilberforce Rd. Cambridge, CB3 0WA, UK email: [email protected] [email protected] Michael E. Purucker Goddard Space Flight Centre Geodynamics Branch Hughes-STX Greenbelt, MD 20771, USA email: [email protected] Karl-Heinz Raedler Astrophysikalisches Institut Potsdam Andersternwarte 16 14482 Potsdam, Germany email: [email protected] [email protected] Mita Rajaram Indian Institute of Geomagnetism New Panvel, Navi Mumbai, 410218, India email: [email protected]

Louise Pellerin Green Engineering, Inc. 6543 Brayton Drive Anchorage, AK 99507, USA email: [email protected]

Jean L. Rasson Centre de Physique du Globe Institut Royal Météorologique Dourbes, 5670, Belgium email: [email protected]

Klaus Petersen Petersen Instruments Torstr. 173 10115 Berlin, Germany email: [email protected]

Dhananjay Ravat Department of Geology 4324 Southern Illinois University Carbondale Carbondale, IL 62901-4324, USA email: [email protected] [email protected]

Nikolai Petersen Department of Earth and Environmental Sciences University of Munich Theresienstr. 41 80333 München, Germany email: [email protected]

Colin Reeves Earthworks Achterom 41a Delft, 2611PL, The Netherlands email: [email protected]

Eduard Petrovsky Geophysical Institute Bocni II/1401 Prague 4, 141 31, Czech Republic email: [email protected]

Alan B. Reid 49 Carr Bridge Drive Leeds, LS16 7LB, UK email: [email protected]

Mark Pilkington Geological Survey of Canada 615 Booth Street Ottawa, ON, Canada, K1A 0E9 email: [email protected]

Karen J. Remick USGS Golden Box 25045 MS966 DFC Denver, CO 80227, USA email: [email protected]

John D.A. Piper Department of Earth Sciences Geomagnetism Laboratory University of Liverpool Liverpool, L69 7ZEm, UK email: [email protected]

Paul G. Richards Lamont-Doherty Earth Observatory Columbia University 61 Route 9W Palisades, NY 10964-1000, USA email: [email protected]

xxi

xxii

CONTRIBUTORS

Arthur D. Richmond NCAR High Altitude Observatory POB 3000 Boulder, CO 80307-3000, USA email: [email protected]

Johannes Schweitzer NORSAR Instituttveien 25 POB 53 Kjeller, 2027, Norway email: [email protected]

Oliver Ritter GeoForschungsZentrum Telegrafenberg A45 14473 Potsdam, Germany email: [email protected]

Gary R. Scott Berkeley Geochronology Center 2455 Ridge Road Berkeley, CA 95709, USA email: [email protected]

Andrew P. Roberts National Oceanography Centre School of Ocean and Earth Science University of Southampton Southampton, SO14 3ZH, UK email: [email protected] [email protected]

John Shaw Department of Earth Sciences University of Liverpool Oxford Street, P.O. Box 147 Liverpool, L69 3BX, UK email: [email protected]

Paul H. Roberts Institute of Geophysics and Planetary Physics UCLA 405 Hilgard Avenue Los Angeles, CA 90095, USA email: [email protected] Michael G. Rochester Dept of Earth Sciences Memorial University of Newfoundland St. John's, N.L., A1B 3X5, Canada email: [email protected] Pierre Rochette CNRS-Université d'Aix-Marseille 3 CEREGE BP80 Europole de l'Arbois Aix en Provence Cedex 4, 13545, France email: [email protected] Leonardo Sagnotti Istituto Nazionale di Geofisica e Vulcanologia Via di Vigna Murata 605 Roma, 00143, Italy email: [email protected]

Guoyin Shen Center for Advanced Radiation Sources University of Chicago Chicago, IL 60439, USA email: [email protected] Manfred Siebert Institute of Geophysics University of Göttingen Friedrich-Hund-Platz 1 37077 Göttingen, Germany Brad S. Singer Department of Geology and Geophysics University of Wisconsin-Madison 1215 West Dayton Street Madison, WI 53706, USA email: [email protected] Alexei V. Smirnov Department of Earth and Environmental Sciences University of Rochester Hutchison Hall 227 Rochester, NY 14627, USA email: [email protected]

Graeme R. Sarson School of Mathematics and Statistics University of Newcastle Newcastle upon Tyne, NE1 7RU, UK email: [email protected]

Xiaodong Song Dept. of Geology University of Illinois 1301 W. Green St. 245NHB Urbana, IL 61801, USA email: [email protected]

Armin Schmidt Department of Archaeological Sciences University of Bradford BD7 1DP, UK email: [email protected]

Annie Souriau CNRS Observatoire Midi-Pyrenees 14 Ave. Edouard Belin Toulouse, 31400, France email: [email protected]

Jean-Jacques Schott Ecole et Observatoire des Sciences de la Terre 5, rue Descartes Strasbourg Cedex, 67084, France email: [email protected] Gerald Schubert Department of Earth & Space Sciences University of California 2707 Geology Building Los Angeles, CA 90024-1567, USA email: [email protected] [email protected] Adam Schultz College of Oceanic and Atmospheric Sciences Oregon State University Corvallis, OR 97331-5503, USA email: [email protected]

Andrew Soward School of Mathematical Sciences University of Exeter Exeter, EX4 4QE, UK email: [email protected] Frank D. Stacey CSIRO Exploration and Mining PO Box 883 Kenmore, Queensland 4069, Australia email: [email protected] F. Richard Stephenson Dept. of Physics University of Durham South Road Durham, DH1 3LE, UK email: [email protected]

CONTRIBUTORS

David J. Stevenson CALTECH Div Geology & Planetary Sci, 150-21 Pasadena, CA 91125, USA email: [email protected]

Oleg Troshichev Arctic and Antarctic Research Institute 38 Bering St. St. Petersburg, 199397, RUSSIA email: [email protected]

John A. Tarduno Department of Earth and Environmental Sciences University of Rochester Hutchison Hall 227 Rochester, NY 14627, USA email: [email protected] [email protected]

Bruce T. Tsurutani Jet Propulsion Laboratory California Institute of Technology 4800 Oak Grove Drive Pasadena, CA 91009, USA email: [email protected]

Donald D. Tarling Department of Geological Sciences Plymouth Polytechnic Drake Circus Plymouth, Devon, PL4 8AA, UK email: [email protected]

Agustín Udías Facultad de Ciencias Físicas Departamento de Geofísica Universidad Complutense Ciudad Universitaria Madrid, 28040, Spain email: [email protected]

Patrick T. Taylor Laboratory for Planetary Geodynamics NASA/Goddard Space Flight Center Greenbelt, MD 20771, USA email: [email protected]

Martyn Unsworth University of Alberta Edmonton, Alberta, T6G 2J1, Canada email: [email protected]

Z.S. Teweldemedhin (no address) Michael J. Thompson Dept. of Applied Mathematics University of Sheffield Sheffield, S3 7RH, UK email: [email protected] Alan W.P. Thomson British Geological Survey West Mains Road Edinburgh, EH9 3LA, UK email: [email protected] Michael Thorne Department of Geological Sciences Arizona State University Tempe, AZ 85287-1404, USA Andreas Tilgner Institute of Geophysics University of Gottingen Herzberger Landstr. 180 37075 Göttingen, Germany email: [email protected] Maurice A. Tivey Dept Geology & Geophysics WHOI 360 Woods Hole Rd Woods Hole, MA 02543-1542, USA email: [email protected] Steven M. Tobias Department of Applied Mathematics University of Leeds Leeds, LS2 9JT, UK email: [email protected] Lester A. Tomlinson Geoscience, Electronics & Data Services 30 Kirner St. Christchurch, 8009, New Zealand email: [email protected] Miquel Torta Observatori de l'Ebre Roquetes (Tarragona), 43520, Spain email: [email protected] John B. Townshend USGS Box 25046 MS 966 Golden, CO 80225, USA

Jaime Urrutia-Fucugauchi Instituto de Geofisica, Laboratorio de Paleomagnetismo y Paleoambientes Universidad Nacional Autonoma de Mexico Mexico D.F., 04510, Mexico email: [email protected] Jean-Pierre Valet Institut de Physique du Globe de Paris 4 place Jussieu Paris Cedex 05, 75252, France email: [email protected] Lidunka Vočadlo Dept. Earth Sciences University College London Gower St. London, WC1E 6BT, UK email: [email protected] Ingo Wardinski GeoForschungsZentrum Potsdam Sektion 2.3 Geomagnetische Felder Telegrafenberg 14473 Potsdam, Germany email: [email protected] Deanis Weaire Department of Physics Trinity College College Green Dublin 2, Ireland email: [email protected] Peter D. Weiler Baseline Environmental Consulting 5900 Hollis Street Emeryville, CA 94608, USA email: [email protected] Kathryn A. Whaler Grant Institute of Earth Science The University of Edinburgh West Mains Road Edinburgh, EH9 3JW, UK email: [email protected] Quentin Williams Earth Sciences UCSC Santa Cruz, CA 95064, USA email: [email protected]

xxiii

xxiv

Wyn Williams Department of Geology & Geophysics University of Edinburgh West Mains Road Edinburgh, EH9 3JW, UK email: [email protected] Ashley P. Willis Dept of Mathematics University of Bristol Bristol, BS8 1TW, UK email: [email protected] Denis Winch School of Mathematics & Statistics F07 University of Sydney Sydney, NSW 2006, Australia email: [email protected] [email protected]

CONTRIBUTORS

Dongmei Yang Institute of Geophysics China Earthquake Administration, No. 5 Minzudaxuenanlu Haidan District, Beijing, 100081, China email: [email protected] Keke Zhang Department of Mathematical Sciences University of Exeter Exeter, EX4 4QE, UK email: [email protected] Xixi Zhao Institute of Geophysics and Planetary Physics University of California Santa Cruz 1156 High Street Santa Cruz, CA 95064, USA email: [email protected] [email protected]

Preface

Geomagnetism is the study of the earth's magnetic field: its measurement, variation in time and space, origins, and its use in helping us to understand more about our Earth. Paleomagnetism is the study of the record left in the rocks; it has contributed much to our understanding of the geomagnetic field's past behavior and many other aspects of geology and earth history. Both have applications, pure and applied: in navigation, in the search for minerals and hydrocarbons, in dating rock sequences, and in unraveling past geologic movements such as plate motions. The entire subject is a small subdiscipline of earth science, and our goal has been to cover it in fine detail at a level accessible to anyone with a general scientific education. We envisage the encyclopedia to be of greatest use to those starting in the subject and those needing to know something of the field for their own application, but the topic is broad and demanding—as we have become increasingly aware while editing the huge variety of contributions— and we also expect it to be of use to experts in geomagnetism or paleomagnetism who need to stray outside their own area of expertise, for nobody is an expert in the whole field. The scope of the encyclopedia is defined by the “GP” section of the American Geophysical Union: the magnetic field of internal origin. Over 25% of the membership of GP has contributed to this book. External sources of magnetic field are included insofar as they are used in solid earth geomagnetism—for example periodic external fields, because they induce electric currents in the earth that are useful in mapping deep electrically conducting regions—and articles are included on the ionosphere, magnetosphere, Sun, and planets. External geomagnetism as such is a separate discipline in most research establishments as well as the AGU, and is therefore not treated. Geomagnetism is the oldest earth science, having its origins in simple human curiosity in the lodestone's ability to point north. It claims what most believe to be the first scientific treatise, William Gillbert's (q.v.) De Magnete published in 1601, the claim being founded on its use of deduction from experimental measurement. These innocent beginnings were soon to give way to the intensely practical business of finding one's way at sea, and during the European age of discovery understanding the geomagnetic field and using it for navigation became a burning challenge for early scientists. A century after the publication of De Magnete saw Edmond Halley (q.v.) in charge of a Royal Navy vessel making measurements throughout the Atlantic Ocean. Halley's plans to fix position more accurately by using the departures of magnetic north from geographic north were dashed by the geomagnetic field's rapid changes in time, and the longitude problem was of course finally solved by Harrison and his accurate clock, but the compass remains an essential aid to navigation to this day. Almost a century after Halley's voyages James Cook was making even more accurate measurements throughout the oceans, and in the 19th century, Alexander von Humboldt (q.v.) and Carl Friedrich Gauss (q.v.) set up a network of magnetic observatories, the first example of international cooperation in a scientific endeavor. The data compilation continued throughout the 19th century, with typical Victorian tenacity, led by Edward Sabine (q.v.), and detailed magnetic measurements were made by James Clark Ross’ expedition to the poles and the voyage of HMS Challenger. Impressive though these data collections were, with

hindsight they yielded rather little in the way of pure scientific discovery or useful application. True, Sabine was to identify the source of magnetic storms with activity on the Sun and they left us a wonderful record of the geomagnetic field in the 19th century, laying the foundation for modern surveying, but the real prize of discovering the geomagnetic field's origin eluded them. Developments in the early 20th century were to catapult geomagnetism into the limelight yet again, this time in the quest for minerals. Metal ores, base and noble, are concentrated in rocks rich in magnetites that are intensely magnetic. Geomagnetism provided a cheap and simple prospecting tool for exploration, and magnetic surveys proliferated on land as never before. Geomagnetism provides the cheapest geophysical exploration tool, and while it may lack the precision of seismic methods it continues to produce economic returns—a year's profit from one of the larger mines would probably pay for all the mapping in the last century. The discovery of electromagnetic induction provided yet another technique for exploring the earth's interior and even more significantly it changed prevailing views on the origin of geomagnetic fluctuations and the earth's main dipole field. The many theories for the origin proposed around the turn of the 20th century are reviewed here by David Stevenson (see Nondynamo theories). The only one to survive the test of time was Joseph Larmor's (q.v.) self-exciting dynamo theory, but even this was to suffer a half-century setback from Thomas Cowling's (q.v.) proof that no dynamo could sustain a magnetic field with symmetry about an axis, which the earth's dipole has to a good approximation. Spectacular progress was being made at about this time by French physicists such as Bernard Brunhes and Motonori Matuyama (q.v.) from Japan trying to understand the magnetic properties of rocks. In founding the science of paleomagnetism they discovered they were able to determine the direction of the earth's magnetic field at the time of the rock's formation, and made the astonishing discovery that the magnetic field had reversed direction in the past. This discovery, like the dynamo theory, suffered a setback when, in the late 1950s, Seiya Uyeda and Takesi Nagata from Japan found that some minerals reverse spontaneously: this providing an alternative but rather mundane explanation that appealed to some in a skeptical scientific community. Evidence for polarity reversals mounted, thanks in great part to the efforts of Keith Runcorn (q.v.) and colleagues in England, but it required precise radiometric dating and access to a suite of rocks younger than 5 million years to establish a complete chronology and put the question beyond any doubt: this was finally achieved by Allan Cox (q.v.) and colleagues in the USA in 1960, on the eve of the plate tectonic revolution. It is hard to comprehend the rapidity of scientific developments in earth science in the 1960s and impossible to underestimate the importance of the role played by paleomagnetism and geomagnetism in the development of plate tectonics. The establishment of polarity reversals came together with H. Hess’ ideas on seafloor spreading and the discovery of magnetic stripes on the ocean floor to provide a means to map the age of the oceans (see Vine-Matthews-Morley hypothesis) and confirm once and for all Wegener's ancient ideas of continental drift. Even today, almost all our quantitative knowledge of plate movements in the geological past comes from paleomagnetism and

xxvi

PREFACE

geomagnetism and the development of a reversal timescale (q.v. see also Bill Lowrie's article Geomagnetic polarity timescales). 1958 saw at last the removal of Cowling's objection to the dynamo theory (see Dynamo, Backus and Dynamo, Herzenberg) and work done in Eastern Europe and the former Soviet Union provided the mathematical foundation and physical insight needed to understand how the geomagnetic field could be generated in the earth's liquid iron core (see Dynamos, mean-field and Dynamo, Braginsky). Improvements in instrumentation accelerated magnetic surveys: electromagnetic surveys became routine, proton magnetometers could be towed behind ships, and Sputnik ushered in the satellite era with a magnetometer on board. So where does geomagnetism and paleomagnetism stand now? The dynamo theory of the origin of the main field still presents one of the most difficult challenges to classical physics, but computers are now fast enough to solve the equations of magnetohydrodynamics (q.v., see also Geodynamo, numerical simulations) in sufficient complexity to reproduce many of the observed phenomena; we have just entered a decade of magnetic observation with two satellites operational at the time of writing and two more launches planned in the near future; there are more aircraft devoted to industrial magnetic surveying than ever before; electromagnetic methods have found application in the search for hydrocarbon reserves and have moved into the marine environment (see EM, marine controlled source). Paleomagnetists have begun to map details of the magnetic field during polarity transition (see Geomagnetic polarity reversals), discover many examples of excursions (aborted reversals), are mapping systematic departures from the simple dipole structure, and have automated laboratory techniques to the point where, in a single day, they can make more measurements of the absolute paleointensity of magnetic materials such as ceramics and basaltic rocks than pioneer Emile Thellier (q.v.) could do in a lifetime (see articles on Absolute paleointensity). The subject divides naturally into the studies of magnetic fields with different origins—indeed these differences often make it difficult even for experts to understand other branches of their own subject! Those studying the permanent magnetization of the earth's crustal rocks deal

with magnetic fields that owe their origin to permanent magnetism at the molecular or crystal grain level; those in electromagnetic induction study magnetic fields caused by electric currents induced in solid rocks deep inside the Earth; while dynamo theorists deal with induction by a fluid, and have to deal with the additional complexity of advection by a moving conductor. Paleomagnetism naturally separates into studies of the magnetism of rocks, or rock magnetism (q.v.), laboratory methods for determining the ancient field, and the history of the ancient field itself. This classification dictated our choice of topics. Special effort has been made to represent the activities of the global network of permanent magnetic observatories. These rarely feature in scientific papers and most practicing scientists are unaware of the meticulous nature of the work and the dedication of those unsung heroes charged with maintaining standards over decades—a persistence rarely experienced in modern science. The observatory section represents, to our knowledge, the first attempt to draw together into one place this rather loosely connected international endeavor. Our subject relates to many other disciplines, either because the geomagnetic field is a vital part of our environment and provides a surprising range of useful techniques to others, from stratigraphy through navigation to radio communication. Partly because of this, and partly in an effort to provide a self-contained volume, we have strayed outside the strict remit of GP. We have included articles on earth structure, particularly those esoteric regions (see for example articles on D”) important for geomagnetism, and have covered the fascinating magnetic fields of other planets and satellites. Our main thanks must go to our contributors, who have so willingly and energetically contributed to make this a truly community effort: we have received very few refusals to our requests to contribute. Alan Jones and Kathryn Whaler advised on electromagnetic induction, a difficult subject for both editors. Thanks go to Stella Gubbins for her unstinting work in organizing the geomagnetism articles and presenting them to the publishers in good order. March 2007 David Gubbins and Emilio Herrero-Bervera

A

AEROMAGNETIC SURVEYING Introduction Magnetic surveying is one of the earliest geophysical methods ever used, with a magnetic compass survey being used in Sweden in 1640 to detect magnetic iron ores. Once instruments were developed in the 1880s to measure the magnitude of the Earth’s magnetic field, magnetic surveys applied to mineral exploration became widespread (Hanna, 1990). These early surveys, comprising magnetometer measurements taken on or close to the Earth’s surface, were limited in extent and only small areas could be covered in any great detail. With the advent of an aircraft-mounted magnetometer system (Muffly, 1946), developed primarily for submarine detection during World War II, the number and areal coverage of magnetic surveys expanded rapidly. The first airborne magnetic or aeromagnetic survey for geological purposes was flown in 1945 in Alaska by the U.S. Geological Survey and the U.S. Navy. By the end of the 1940s, aeromagnetic surveys were being flown worldwide.

Survey objectives Aeromagnetic surveys are flown for a variety of reasons: geologic mapping, mineral and oil exploration, and environmental and groundwater investigations. Since variations in the measured magnetic field reflect the distribution of magnetic minerals (mainly magnetite) in the Earth’s crust and human-made objects, surveys can be used to detect, locate, and characterize these magnetic sources. Most surveys are flown to aid in surface geologic mapping, where the magnetic effects of geologic bodies and structures can be detected even in areas where rock outcrop is scarce or absent, and bedrock is covered by glacial overburden, bodies of water, sand, or vegetation. Magnetic surveys can also detect magnetic sources at great depth (tens of kilometers) within the Earth’s crust, being limited only by the depth at which magnetic minerals reach their Curie point and cease to be ferrimagnetic. Broad correlations can be made between rock type and magnetic properties, but the relationship is often complicated by temperature, pressure, and chemical changes that rocks are exposed to (Grant, 1985). Nevertheless, determining the location, shape, and attitude of magnetic sources and combining this with available geologic information can result in a meaningful geological interpretation of a given area. Certain kinds of ore bodies may produce magnetic anomalies that are desirable targets for mineral exploration surveys. Although hydrocarbon reservoirs are not directly detectable by aeromagnetic surveys, magnetic data can be used to locate geologic structures that

provide favorable conditions for oil/gas production and accumulation (Gibson and Millegan, 1998). Similarly, mapping the magnetic signatures of faults and fractures within water-bearing sedimentary rocks provides valuable constraints on the geometry of aquifers and the framework of groundwater systems.

Survey design Aeromagnetic surveys are usually flown in a regular pattern of equally spaced parallel lines (flightlines). A series of control or tie-lines is also flown perpendicular to the flightline direction to assist in the processing of the magnetic field data set. The ratio of control to flightline spacing generally ranges from 3 to 10 depending on the desired data quality. Flightline spacing is dependent on the primary aim of the survey and governs the amount of detail in the measured magnetic field. For reconnaissance mapping of large regions (e.g., states, countries), where little or no knowledge of the magnetic field to be mapped is available, typical line spacings are 1.5 km in Australia and 0.8 km in Canada. Reconnaissance survey lines are usually oriented north-south or east-west. For smaller areas with similar financial resources available, these spacings can be reduced (e.g., 0.2 km in Sweden and Finland). For smaller regions, the line spacing is governed by the size of the target (geological structure, oil prospect, ore body) being investigated. For example, mineral exploration surveys generally have spacings in the range 50–200 m and are flown perpendicular to the dominant geologic strike direction, although directions no more than 45o from magnetic north are preferred because N-S wavelengths are shorter than E-W wavelengths at low and intermediate latitudes. For areas with distinct regions of differing geologic strike, costs may permit splitting the survey into several flight directions or a single compromise direction must be assigned. The survey flying height is intimately related to flightline spacing with lower heights being more appropriate for closer line spacings. This is due to the rapid decrease in magnetic field intensity and increase in wavelength as a function of the distance from a magnetic source. Theoretical arguments suggest a line spacing to height ratio of 2 or less is desirable to accurately sample variations in the magnetic field (Reid, 1980). The majority of surveys, however, have ratios higher than this, i.e., 2.5 to 8. The minimum flight height is limited by government restrictions, safety considerations, and ruggedness of the terrain. Topographic variations also affect the mode of flying used in a survey. For geologic mapping and mineral exploration surveys, measurements are desirable at levels as close to the magnetic sources as possible, hence surveys are flown at a constant height (mean terrain clearance) above the ground surface. Prior to the availability of

2

AEROMAGNETIC SURVEYING

real-time global positioning system (GPS) navigation, surveys over mountainous regions were flown at a constant altitude. Postsurvey processing of the collected magnetic field data could then be applied to artificially “drape” the measurements onto a surface at some specified mean terrain clearance. This can improve the resolution of magnetic anomalies that is degraded by flying at the higher constant altitude. However, current usage of more accurate navigational systems (GPS) now permits fixed-wing surveys to be flown along a preplanned artificial drape surface in regions of rugged topography. The drape surface is set to a constant height above ground level but is modified to take into consideration the climb and descent rate of the survey aircraft. Helicopters can also offer an alternative (usually costlier) to fixed-wing aircraft in mountainous areas.

Along with measurements of the magnetic field, information relating to the accurate location and time of these recorded values is gathered. A modern data acquisition system (usually computer-based) records and archives magnetic, navigational, temporal, altitude, and perhaps aircraft attitude data. In addition, video recordings of the flight-path of the aircraft are made that may be used later to identify magnetic effects of human-made sources and to provide checks on navigation. Accurate synchronization of all the data streams is crucial for subsequent processing of the raw magnetometer data. Magnetometer measuring rates of 10 samples per second and navigational information at one sample per second are current norms.

currents. The permanent magnetization of the aircraft results in a heading error which is a function of the flightline direction. Induced magnetizations occur due to the motion of the aircraft in the Earth’s magnetic field. These effects are partially reduced by mounting the magnetometer as far away as possible from the source of the noise, either on a boom mounted on the aircraft’s tail or in a towed “bird” attached by a cable. Although the bird can be located several tens of meters below the aircraft, this may introduce orientation and positional errors if the sensor does not maintain a fixed location relative to the aircraft. For magnetometers mounted on the aircraft, a technique called compensation is needed to reduce remaining magnetic noise effects to an acceptable level. Modern aircraft are compensated by establishing and modeling the magnetic response of the aircraft to changes in its orientation. This is usually done prior to surveying by flying lines in each of the four local survey directions in an area of low magnetic field gradient and at high altitude, while carrying out several roll, pitch, and yaw maneuvers. The compensation may be done digitally in real time through the data acquisition system, or later as part of data processing. Information on the aircraft’s orientation and direction is used to predict the resultant magnetic effect, which is then removed as the survey is flown, or later. The figure of merit, consisting of the sum of the absolute values of the magnetic effects of the roll, pitch, and yaw maneuvers for each of the four headings (12 values in all), can be as low as one nanotesla. Once a survey is started, the aircraft’s noise level should be assessed repeatedly to ensure that the same level of data quality is maintained.

Magnetometers

Positioning

Compared to the first magnetometers used in the 1940s, the resolution and accuracy of magnetic field measurements have increased significantly. The early fluxgate magnetometers had resolutions of about 1 nT and noise envelopes of 2 nT (Horsfall, 1997). These instruments only produced relative measurements of the field and suffered from appreciable drift, possibly 10 nT/hr. Proton precession magnetometers followed with a resolution of 0.1 nT and noise envelope of 1 nT. These instruments measured the magnetic field intensity and had minimal drift. These have been superseded by the optically pumped, generally caesium-vapor magnetometers which have a resolution of 0.001 nT and a noise envelope of 0.005 nT. Such levels of measurement accuracy are not met in practice since errors arise through unremoved aircraft magnetic effects, navigation effects, and postsurvey data processing. Standard sampling rates of 10 samples per second coupled with aircraft speeds of 220–280 km/hr result in measurements at an interval of 6–8 meters. Single magnetometer aircraft configurations simply produce measurements of the magnetic field intensity in the direction of the Earth’s magnetic field. By adding extra sensors, various other quantities can be measured. Two vertically separated sensors can provide vertical magnetic gradient information while two wing-tip sensors allow measurement of the transverse horizontal magnetic gradient. Vertical gradient measurements have the advantage in effectively suppressing longer wavelength magnetic anomalies and providing a higher resolution definition of shorter wavelength features. This is crucial in surveys flown for mapping of near-surface geology where the longer wavelength effects of deep sources are of secondary interest. In theory, having two sensors also implies that a simple subtraction of the two measurements removes any temporal effects in the data. However, in practical applications, temporal effects may still be present in the data. Transverse (perpendicular to the flight path) gradient measurements are not generally used as a final interpretation product but can be exploited when the magnetic data are interpolated between the flight lines to produce a regular grid of magnetic field values (Hogg, 1989).

No matter what accuracies are achieved in the magnetic field measurements, the value of the final survey data is dependent on accurate location of the measurement points. Traditional methods of navigation relied heavily on visual tracking using aerial photographs. The actual location of the flight path would be recovered manually by comparing these photographs with images from onboard video cameras. Synchronizing the data recording with the cameras would allow positioning of the magnetic field data. Electronic navigation systems were also used either to interpolate between visually located points or provide positioning information for offshore surveys. Positional errors of the order of hundreds of meters would not be unexpected. These techniques have been superseded by the introduction in the 1990s of GPS. Navigation using GPS relies on the information sent from an array of satellites whose locations are known precisely. Signals from a number of satellites are used to triangulate the position of the receiver in the aircraft so that its position is known for navigational purposes and to locate the magnetic field measurements. GPS systems also provide highly accurate time information that forms the basis for synchronizing each component of the data acquisition system. Positional accuracies of
5 m horizontally and
10 m vertically can be achieved with a single GPS receiver. These values can be improved upon by using another receiver and refining positions by making differential corrections. The added receiver is set up at the survey base and simultaneously collects data from the satellites as the survey is flown. Postsurvey processing of all the positional data reduces errors in information provided by the satellites and accuracies of
2–4 m can be achieved. Current surveys generally use real-time differential GPS navigation where the raw positional information is corrected as the data is being collected. The GPS location and altitude are also used to calculate (and subtract) the expected normal field for the location, as predicted by the International Geomagnetic Reference Field (IGRF). This generally removes an unhelpful regional component arising from the Earth’s core field.

Survey data acquisition

Temporal effects Aircraft noise Magnetic noise caused by the survey aircraft arises from permanent and induced magnetization effects and from the flow of electrical

Since the Earth’s magnetic field varies temporally as well as spatially, time variations that occur over the period of surveying must be determined and removed from the raw measurements. The dominant effect

AGRICOLA, GEORGIUS (1494–1555)

for airborne surveys is the diurnal variation, which generally has an amplitude of tens of nanotesla. Shorter time-period variations due to magnetic storms can be much larger (hundreds of nanotesla) and severe enough to prevent survey data collection. Data can also be degraded my micropulsations with short periods and amplitudes of several nanotesla. Monitoring of the Earth’s field is an essential component of a survey in order to mitigate these time-varying effects. One or more base station magnetometers is used to track changes in the field during survey operations. When variations are unacceptably large, surveying is suspended and any flightlines recording during disturbed periods are reflown. The smoothly changing diurnal variation is removed from the data using tie-line leveling. Simply subtracting this variation from the measured data is not sufficient since diurnal changes may vary significantly over the survey area. Nonetheless, the recorded diurnal can be used as a guide in the leveling process. Tie-line leveling is based on the differences in the measured field at the intersection of flightlines and tie-lines. If the distance and hence the time taken to fly between these intersection points is small enough then it can be assumed that the diurnal varies approximately linearly and can be corrected for (Luyendyk, 1997).

Data display and interpretation The final product resulting from an aeromagnetic survey is a set of leveled flightline data that are interpolated onto a regular grid of magnetic field intensity values covering the survey region. These values can be displayed in a variety of ways, the most common being a color map or image, where the magnetic field values, based on their magnitude, are assigned a specific color. Similarly, the values can be represented as a simple line contour map. Both kinds of representation can be used in a qualitative fashion to divide the survey area into subregions of high and low magnetizations. Since the data are available digitally, it is straightforward to use computer-based algorithms to modify and enhance the magnetic field image for the specific purpose of the survey. Transformation and filtering allows certain attributes of the data to be enhanced, such as the effects due to magnetic sources at shallow or deep levels, or occurring along a specified strike direction. More sophisticated methods may estimate the depths, locations, attitudes, and the magnetic properties of magnetic sources. Mark Pilkington

Bibliography Gibson, R.I., and Millegan, P.S. (eds.), 1998. Geologic Applications of Gravity and Magnetics: Case Histories. Tulsa, OK, U.S.A.: Society of Exploration Geophysicists and American Association of Petroleum Geologists. Grant, F.S., 1985. Aeromagnetics, geology and ore environments, I. Magnetite in igneous, sedimentary and metamorphic rocks: An overview. Geoexploration, 23: 303–333. Hanna, W.F., 1990. Some historical notes on early magnetic surveying in the U.S. Geological Survey. In Hanna, W.F., (ed.), Geologic Applications of Modern Aeromagnetic Surveys. United States Geological Survey Bulletin 1924, pp. 63–73. Hogg, R.L.S., 1989. Recent advances in high sensitivity and high resolution aeromagnetics. In Garland, G.D. (ed.), Proceedings of Exploration ’87, Third Decennial International Conference on Geophysical and Geochemical Exploration for Minerals and Groundwater. Ontario Geological Survey, Special Volume 3, pp. 153–169. Horsfall, K.R., 1997. Airborne magnetic and gamma-ray data acquisition. Australian Geological Survey Organization Journal of Australian Geology and Geophysics, 17: 23–30. Luyendyk, A.P.J., 1997. Processing of airborne magnetic data. Australian Geological Survey Organization Journal of Australian Geology and Geophysics, 17: 31–38.

3

Muffly, G., 1946. The airborne magnetometer. Geophysics, 11: 321–334. Reid, A.B., 1980. Aeromagnetic survey design. Geophysics, 45: 973–976.

Cross-references Compass Crustal Magnetic Field Depth to Curie Temperature IGRF, International Geomagnetic Reference Field Magnetic Anomalies for Geology and Resources Magnetic Anomalies, Modeling

AGRICOLA, GEORGIUS (1494–1555) Agricola, which seems to have been the classical academic pen-name for Georg Bauer, was born at Glauchau, Germany, on March 24, 1494 as the son of a dyer and draper. He studied at Leipzig University and trained as a doctor, and became Town Physician to the Saxon mining community of Chemnitz. His significance derives from his great treatise on metal mining, De Re Metallica (1556), or “On Things of Metal.” This sumptuously illustrated work was a masterly study of all aspects of mining, including geology, stratigraphy, pictures of mineshafts, machinery, smelting foundries, and even the diseases suffered by miners. Book III of De Re Metallica contains a description and details of how to use the miner’s compass which, by 1556, seems to have been a well-established technical aid to the industry. Agricola drew a parallel between the miner’s and the mariner’s compass, saying that the direction cards of both were divided into equidistant divisions in accordance with a system of “winds.” Agricola’s compass is divided into four quadrants, with each quadrant subdivided into six: 24 equidistant divisions around the circle. And just as a mariner named the “quarters” of his compass from the prevailing winds, such as “Septentrio” for north and “Auster” for south, so the miner likewise named his underground metal veins, depending on the directions in which they ran. Agricola also tells us that when prospecting for ore, a mining engineer would use his compass to detect the veins of metal underground and use it to plot their direction and the likely location of their strata across the landscape. Agricola, it was said, always enjoyed robust health and vigor. He served as Burgomaster of Chemnitz on several occasions and worked tirelessly for the sick during the Bubonic Plague epidemic that ravaged Saxony during 1552–1553. Then, on November 21, 1555, he suddenly died from a “four days” fever. Allan Chapman

Bibliography Agricola, G., De Re Metallica (Basilae, 1556), translated by Hoover, H.C., and Hoover, L.H. (1912; Dover Edition, New York, 1950).

ALFVE´N WAVES Introduction and historical details Alfvén waves are transverse magnetic tension waves that travel along magnetic field lines and can be excited in any electrically conducting fluid permeated by a magnetic field. Hannes Alfvén (q.v.) deduced their existence from the equations of electromagnetism and hydrodynamics (Alfvén, 1942). Experimental confirmation of his prediction was found seven years later in studies of waves in liquid mercury (Lundquist, 1949). Alfvén waves are now known to be an important mechanism

4

ALFVE´N WAVES

for transporting energy and momentum in many geophysical and astrophysical hydromagnetic systems. They have been observed in Earth’s magnetosphere (Voigt, 2002), in interplanetary plasmas (Tsurutani and Ho, 1999), and in the solar photosphere (Nakariakov et al., 1999). The ubiquitous nature of Alfvén waves and their role in communicating the effects of changes in electric currents and magnetic fields has ensured that they remain the focus of increasingly detailed laboratory investigations (Gekelman, 1999). In the context of geomagnetism, it has been suggested that Alfvén waves could be a crucial aspect of the dynamics of Earth’s liquid outer core and they have been proposed as the origin of geomagnetic jerks (q.v.) (Bloxham et al., 2002). In this article a description is given of the Alfvén wave mechanism, the Alfvén wave equation is derived and the consequences of Alfvén waves for geomagnetic observations are discussed. Alternative introductory perspectives on Alfvén waves can be found in the books by Alfvén and Fälthammar (1963), Moffatt (1978), or Davidson (2001). More technical details concerning Alfvén waves in Earth’s core can be found in the review article of Jault (2003).

The Alfve´n wave mechanism The restoring force responsible for Alfvén waves follows from two simple physical principles: 1. Lenz’s law applied to conducting fluids: “Electrical currents induced by the motion of a conducting fluid through a magnetic field give rise to electromagnetic forces acting to oppose that fluid motion.” 2. Newton’s second law for fluids: “A force applied to a fluid will result in a change in the momentum of the fluid proportional to the magnitude of the force and in the same direction.” The oscillation underlying Alfvén waves is best understood via a simple thought experiment (Davidson, 2001). Imagine a uniform magnetic field permeating a perfectly conducting fluid, with a uniform flow initially normal to the magnetic field lines. The fluid flow will distort the magnetic field lines (see Alfvén’s theorem and the frozen flux approximation) so they become curved as shown in Figure A1 (part (b)). The curvature of magnetic field lines produces a magnetic (Lorentz) force on the fluid, which opposes further curvature as predicted by Lenz’s law. By Newton’s second law, the Lorentz force changes the momentum of the fluid, pushing it (and consequently the magnetic field lines) in an attempt to minimize field line distortion and restore the system toward its equilibrium state. This restoring force provides the basis for transverse oscillations of magnetic fields in conducting fluids and therefore for Alfvén waves. As the curvature of the magnetic field lines increases, so does the strength of the restoring force. Eventually the Lorentz force becomes strong enough to reverse the direction of the fluid flow.

Magnetic field lines are pushed back to their undistorted configuration and the Lorentz force associated with their curvature weakens until the field lines become straight again. The sequence of flow causing field line distortion and field line distortion exerting a force on the fluid now repeats, but with the initial flow (a consequence of fluid inertia) now in the opposite direction. In the absence of dissipation this cycle will continue indefinitely. Figure A1 shows one complete cycle resulting from the push and pull between inertial acceleration and acceleration due to the Lorentz force. Consideration of typical scales of physical quantities involved in this inertial-magnetic (Alfvén) oscillation shows that the strength of the magnetic field will determine the frequency of Alfvén waves. Balancing inertial accelerations and accelerations caused by magnetic field curvature, we find that U=TA ¼ B2 =LA rm where U is a typical scale of the fluid velocity, TA is the time scale of the inertial-magnetic (Alfvén) oscillation, LA is the length scale associated with the oscillation, B is the scale of the magnetic field strength, r is the fluid density, and m is the magnetic permeability of the medium. For highly electrically conducting fluids, magnetic field changes occur primarily through advection (see Alfvén’s theorem and the frozen flux approximation) so we have the additional constraint that B=TA ¼ UB=LA or U ¼ LA =TA. Consequently L2A =TA2 ¼ B2 =rm or vA ¼ B=ðrmÞ1=2. This is a characteristic velocity scale associated with Alfvén waves and is referred to as the Alfvén velocity. The Alfvén velocity will be derived in a more rigorous manner and its implications discussed further in the next section. Physical intuition concerning Alfvén waves can be obtained through an analogy between the response of a magnetic field line distorted by fluid flow across it and the response of an elastic string when plucked. Both rely on tension as a restoring force, elastic tension in the case of the string and magnetic tension in the case of the magnetic field line and both result in transverse waves propagating in directions perpendicular to their displacement. When visualizing Alfvén waves it can be helpful to think of a fluid being endowed with a pseudoelastic nature by the presence of a magnetic field, and consequently supporting transverse waves. Nonuniform magnetic fields have similar consequences for Alfvén waves as nonuniform elasticity of solids has for elastic shear waves.

The Alfve´n wave equation To determine the properties of Alfvén waves in a quantitative manner, we employ the classical technique of deriving a wave equation and then proceed to find the relationship between frequency and wavelength necessary for plane waves to be solutions. Consider a uniform, steady, magnetic field B0 in an infinite, homogeneous, incompressible, electrically conducting fluid of density r, kinematic viscosity n, and magnetic diffusivity
¼ 1=sm where s is the electrical conductivity.

Figure A1 The Alfve´n wave mechanism. In (a) an initial fluid velocity normal to the uniform field lines distorts them into the curved lines shown in (b) giving rise to a Lorentz force which retards and eventually reverses the fluid velocity, returning the field lines to their undisturbed position as shown in (c). The process of field line distortion is then reversed in (d) until the cycle is completed with the return to the initial configuration in (e).

ALFVE´N WAVES

We imagine that the fluid is perturbed by an infinitesimally small flow u inducing a perturbation magnetic field b. Ignoring terms that are quadratic in small quantities, the equations describing Newton’s second law for fluids and the evolution of the magnetic fields encompassing Lenz’s law are ]u ]t |{z}

1  rp r |fflfflffl{zfflfflffl}

¼

Inertial acceleration

þ

Combined mechanical and magnetic pressure gradient

1 ðB0  rÞb rm |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl}

þ nr |fflffl{zfflfflu} ;

Lorentz acceleration due to magnetic tension

2

Viscous diffusion

5

Equation 8 is the dispersion relation, which specifies the relationship between the angular frequency o and the wavenumber k of Alfvén waves. It is a complex quadratic equation in o, so we can use the well known formula to find explicit solutions for o, which are iðn þ
Þk 2
o¼ 2

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ^ 0 Þ2 ðn 
Þ2 k 4 B20 ðk  B  : rm 4

(Eq. 9)

In an idealized medium with n ¼
¼ 0 there is no dissipation and the dispersion relation simplifies to

(Eq. 1) ]b ]t |{z} Change in the magnetic field

¼

ðB0  rÞu |fflfflfflfflfflffl{zfflfflfflfflfflffl}

þ
r2 b |fflffl{zfflffl}

Stretching of magnetic field by fluid motion

(Eq. 2)

Taking the curl ðrÞ of equation 1, we obtain an equation describing how the fluid vorticity x ¼ r  u evolves (Eq. 3)

r  equation 2 gives r

]b ¼ ðB0  rÞx þ
r2 ðr  bÞ: ]t

(Eq. 4)

To find the wave equation, we take a further time derivative of equation 3 so that     ]2 x 1 ]b ]x ðB0  rÞ r  þ n r2 ; ¼ 2 ]t rm ]t ]t

(Eq. 5)

1 and then eliminate b using an expression for rm ðB0  rÞðr  ]b=]tÞ in 1 terms of x obtained by operating with rm ðB0  rÞ on (4) and substitut1 ing for rm ðB0  rÞðr  bÞ from equation 3 which gives

    1 ]b 1 ]x ðB0  rÞ r  ¼ ðB0  rÞ2 x  n
r4 x þ ðn þ
Þr2 ; rm ]t rm ]t (Eq. 6) which when substituted into equation 5 leaves the Alfvén wave equation   ]2 x 1 2 4 2 ]x ðB : ¼  rÞ x  n
r x þ ðn þ
Þr 0 ]t 2 rm ]t

(Eq. 7)

The first term on the right hand side is the restoring force which arises from the stretching of magnetic field lines. The second term is the correction to the restoring force caused by the presence of viscous and ohmic diffusion, while the final term expresses the dissipation of energy from the system due to these finite diffusivities.

Dispersion relation and properties of Alfve´n waves

Substituting a simple plane wave solution of the form x ¼ Ref^ xeiðkrotÞ g x þ y^y þ z^z into the Alfvén wave where k ¼ kx ^x þ ky^y þ kz^z and r ¼ x^ equation 7, we find that valid solutions are possible provided that ! ^ 0 Þ2 B20 ðk  B o2 ¼  n
k 4  iðn þ
Þk 2 o: (Eq. 8) rm ^ 0 ¼ B0 =jB0 j. where B

(Eq. 10)

where vA is the Alfvén velocity

Magnetic diffusion

]x 1 ¼ ðB0  rÞðr  bÞ þ nr2 x: ]t rm

^ 0 Þ; o ¼
vA ðk  B

vA ¼

B0 ðrmÞ1=2

:

(Eq. 11)

This derivation illustrates that the Alfvén velocity is the speed at which an Alfvén wave propagates along magnetic field lines. Alfvén waves are nondispersive because their angular frequency is independent of jkj and the phase velocity and the group velocity (at which energy and information are transported by the wave) are equal. Alfvén waves are however anisotropic, with their properties dependent on the angle between the applied magnetic field and the wave propagation direction. An idea of Alfvén wave speeds in Earth’s core can be obtained by inserting into equation 11 the seismologically determined density of the outer core fluid r ¼ 1  104 kg, the magnetic permeability for a metal above its Curie temperature m ¼ 4p  107 T2 mkg1 s2 and a plausible value for the strength of the magnetic field in the core (which we take to be the typical amplitude of the radial field strength observed at the core surface B0 ¼ 5  104 T) giving an Alfvén velocity of vA ¼ 0:004 m s1 or 140 km yr1. The time taken for such a wave to travel a distance of order of the core radius is around 25 years. It should also be noted that Alfvén waves in the magnetosphere (where they also play an important role in dynamics) travel much faster because the density of the electrically conducting is very much smaller. Considering Alfvén waves in Earth’s core, Ohmic dissipation is expected to dominate viscous dissipation but for large scale waves will still be a small effect so that nk 2 
k 2  v2A . Given this assumption the dispersion relation reduces to o¼

i
k 2 ^ 0Þ
vA ðk  B 2

(Eq. 12)

and the wave solutions have the form of simple Alfvén waves, damped on the Ohmic diffusion timescale of Tohm ¼ 2=
k 2 n o t=T ^ x ¼ Re ^ xeiðkr
vA ðkB0 ÞtÞe ohm

(Eq. 13)

Smaller scale waves are rapidly damped out by Ohmic diffusion, while large scale waves will be the longer lived, so we expect these to have the most important impact on both dynamics of the core and the observable magnetic field.

Observations of Alfve´n waves and their relevance to geomagnetism The theory of Alfvén waves outlined above is attractive in its simplicity, but can we really expect such waves to be present in Earth’s core? In the outer core, rotation will have a strong influence on the fluid dynamics (see Proudmann-Taylor theorem). Alfvén waves cannot exist when the Coriolis force plays an important role in the force balance; in this case, more complex wave motions arise (see Magnetohydrodynamic waves). In addition convection is occurring (see Core convection)

6

¨ STA (1908–1995) ALFVE´N, HANNES OLOF GO

and gives rise to a dynamo generated magnetic field (see Geodynamo), which is both time dependent and spatially nonuniform, while the boundary conditions imposed by the mantle are heterogeneous—these factors combine to produce a formidably complex system. Braginsky (1970) recognized that, despite all these complications, a special class of Alfvén waves is likely to be the mechanism by which angular momentum is redistributed on short (decadal) timescales in Earth’s core. He showed that when Coriolis forces are balanced by pressure forces, Alfvén waves involving only the component of the magnetic field normal to the rotation axis can exist. The fluid motions in this case consist of motions of cylindrical surfaces aligned with the rotation axis, with the Alfvén waves propagating along field lines threading these cylinders and being associated with east-west oscillations of the cylinders. Similarities to torsional motions familiar from classical mechanics led Braginsky to christen these geophysically important Alfvén waves torsional oscillations (see Oscillations, torsional). Although the simple Alfvén wave model captures the essence of torsional oscillations and leads to a correct order of magnitude estimate of their periods, coupling to the mantle and the nonaxisymmetry of the background magnetic field should be taken into account and lead to modifications of the dispersion relation given in equation 9. A detailed discussion of such refinements can be found in Jault (2003). The last 15 years have seen a rapid accumulation of evidence suggesting that Alfvén waves in the form of torsional oscillations are indeed present in Earth’s core. The transfer of angular momentum between the mantle and torsional oscillations in the outer core is capable of explaining decadal changes in the rotation rate of Earth (see Length of day variations, decadal). Furthermore, core flows determined from the inversion of global magnetic and secular variation data show oscillations in time of axisymmetric, equatorially symmetric flows which can be accounted for by a small number of spherical harmonic modes with periodic time dependence (Zatman and Bloxham, 1997). The superposition of such modes can produce abrupt changes in the second time derivative of the magnetic field observed at Earth’s surface, similar to geomagnetic jerks (Bloxham et al., 2002). Interpreting axisymmetric, equatorially symmetric core motions with a periodic time dependence as the signature of torsional oscillations leads to the suggestion that geomagnetic jerks are caused by Alfvén waves in Earth’s core. Further evidence for the wave-like nature of the redistribution of zonally averaged angular momentum derived from core flow inversions has been found by Hide et al. (2000), with disturbances propagating from the equator towards the poles. The mechanism exciting torsional oscillations in Earth’s core is presently unknown, though one suggestion is that the time dependent, nonaxisymmetric magnetic field could give rise to a suitable fluctuating Lorentz torque on geostrophic cylinders (Dumberry and Bloxham, 2003). Future progress in interpreting and understanding Alfvén waves in Earth’s core will require the incorporation of more complete dynamical models of torsional oscillations (see, for example, Buffett and Mound, 2005) into the inversion of geomagnetic observations for core motions. Christopher Finlay

Bibliography Alfvén, H., 1942. Existence of electromagnetic-hydrodynamic waves. Nature, 150: 405–406. Alfvén, H., and Fälthammar, C.-G., 1963. Cosmical Electrodynamics, Fundamental Principles. Oxford: Oxford University Press. Bloxham, J., Zatman, S., and Dumberry, M., 2002. The origin of geomagnetic jerks. Nature, 420: 65–68. Braginsky, S.I., 1970. Torsional magnetohydrodynamic vibrations in the Earth’s core and variations in day length. Geomagnetism and Aeronomy, 10: 1–10.

Buffett, B.A., and Mound, J.E., 2005. A Green’s function for the excitation of torsional oscillations in Earth’s core. Journal of Geophysical Research, Vol. 110, B08104, doi: 10.1029/2004JB003495. Davidson, P.A., 2001. An introduction to Magnetohydrodynamics. Cambridge: Cambridge University Press. Dumberry, M., and Bloxham, J., 2003. Torque balance, Taylor’s constraint and torsional oscillations in a numerical model of the geodynamo. Physics of Earth and Planetary. Interiors, 140: 29–51. Gekelman, W., 1999. Review of laboratory experiments on Alfvén waves and their relationship to space observations. Journal of Geophysical Research, 104: 14417–14435. Hide, R., Boggs, D.H., and Dickey, J.O., 2000. Angular momentum fluctuations within the Earth’s liquid core and solid mantle. Geophysical Journal International, 125: 777–786. Jault, D., 2003. Electromagnetic and topographic coupling, and LOD variations. In Jones, C.A., Soward, A.M., and Zhang, K., (eds.), Earth’s core and lower mantle. The Fluid Mechanics of Astrophysics and Geophysics, 11: 56–76. Lundquist, S., 1949. Experimental investigations of magnetohydrodynamic waves. Physical Review, 107: 1805–1809. Moffatt, H.K., 1978. Magnetic Field Generation in Electrically Conducting Fluids. Cambridge: Cambridge University Press. Nakariakov, V.M., Ofman, L., DeLuca, E.E., Roberts, B., and Davila, J.M., 1999. TRACE observation of damped coronal loop oscillations: Implications for coronal heating. Science, 285: 862–864. Tsurutani, B.T., and Ho, C.M., 1999. A review of discontinuities and Alfvén waves in interplanetary space: Ulysses results. Reviews of Geophysics, 37: 517–541. Voigt, J., 2002. Alfvén wave coupling in the auroral current circuit. Surveys in Geophysics, 23: 335–377. Zatman, S., and Bloxham, J., 1997. Torsional oscillations and the magnetic field within the Earth’s core. Nature, 388: 760–763.

Cross-references Alfvén’s Theorem and the Frozen Flux Approximation Alfvén, Hannes Olof Gösta (1908–1995) Core Convection Geodynamo Length of Day Variations, Decadal Magnetohydrodynamic Waves Oscillations, Torsional Proudman-Taylor Theorem

¨ STA (1908–1995) ALFVE´N, HANNES OLOF GO Hannes Alfvén is best known in geomagnetism for the “frozen flux” theorem that bears his name and for the discovery of magnetohydrodynamic waves. He started research in the physics department at the University of Uppsala, where he studied radiation in triodes. His early work on electronics and instrumentation was sound grounding for his later discoveries in cosmic physics. When his book Cosmical Electrodynamics (Alfvén, 1950) was published, the author was referred to by one of the reviewers—T.G. Cowling (q.v.)—as “an electrical engineer in Stockholm.” All of Hannes Alfvén’s scientific work reveals a profound physical insight and an intuition that allowed him to derive results of great generality from specific problems. Hannes Alfvén is most widely known for his discovery (Alfvén, 1942) of a new kind of waves now generally referred to as Alfvén waves (q.v.). These are a transverse mode of magnetohydrodynamic waves, and propagate with the Alfvén velocity, B=ðm0 rÞ1=2 . In the Earth’s core they occur as torsional oscillations as well as other Alfvén-type modes that are altered by the Coriolis force and have quite a different character (see Magnetohydrodynamic waves). Before Alfvén, electromagnetic theory and hydrodynamics were well developed but as separate

ALFVE´N’S THEOREM AND THE FROZEN FLUX APPROXIMATION

scientific disciplines. By combining them, Alfvén founded the new discipline of magnetohydrodynamics (q.v.). Magnetohydrodynamics is of fundamental importance for the physics of the fluid core of the Earth and other planets but has also much wider significance. It is indispensable in modern plasma physics and its applications, both in the laboratory (e.g., in fusion research) and in space (ionospheres, magnetospheres, the sun, stars, stellar winds and interstellar plasma). A famous theorem in magnetohydrodynamics, sometimes called the Alfvén theorem (q.v.), is that of frozen flux, which says that if the conductivity of a magnetized fluid is high enough, magnetic field and fluid motion are coupled in such a way that the fluid appears to be frozen to the magnetic field lines. This is a powerful tool in many applications, for example in studying motions in the Earth’s core (see Core motions). The degree to which the frozen condition holds in a fluid is characterized by a dimensionless parameter called the Lundquist number, m0 sBl=r1=2 (see Geodynamo, dimensional analysis and timescales). It was derived by Alfvén’s student Stig Lundqvist (1952), who also was the first to prove the existence of magnetohydrodynamic waves experimentally in liquid metal (Lundqvist, 1949). In plasmas, especially space plasmas, there are important exceptions to the validity of the frozen field concept, and Alfvén himself vigorously warned against its uncritical use. The second fundamental contribution by Alfvén was the guiding center theory, in which the average motion of a charged particle gyrating in a magnetic field is represented by the motion of its center of gyration. This theory dramatically simplified many plasma physics problems and laid the foundation of the adiabatic theory of charged particle motion. Specific fields where Hannes Alfvén contributed were the theory of aurora and magnetic storms (q.v.) (Alfvén, 1939), evolution of the solar system, and cosmology (he was a vocal opponent to the Big Bang theory). His auroral theory was disputed, in particular by S. Chapman (q.v.), and was generally disregarded. But when direct measurements in space became possible, many of Hannes Alfvén’s ideas, especially about the auroral acceleration process, were vindicated. A similar fate was shared by many of Hannes Alfvén’s ideas. Even his discovery of the magnetohydrodynamic waves was not taken seriously until Enrico Fermi acknowledged the possibility of their existence. His early work in cosmic ray physics led Alfvén to predict the existence of a galactic magnetic field, but his prediction was universally rejected, and by the time the galactic field was discovered, his prediction was long forgotten. Hannes Alfvén was active in public affairs throughout his life, particularly in opposition to nuclear proliferation. He was president of the Pugwash Conference from 1970 to 1975. He wrote several popular science books, including a work of science fiction under the pseudonym O. Johannesson. Further details of his life and a complete list of publications may be found in Pease and Lindqvist (1998); more details of his life are on http://public.lanl.gov/alp/plasma/people/alfven.html. Carl-Gunne Fälthammar and David Gubbins

Bibliography Alfvén, H., 1939. A theory of magnetic storms and of the aurorae (I), Kungliga Svenska Vetenskapsakademiens Handlingar, Tredje Serien, 18: 1–39; Partial reprint EOS Transactions of the American Geophysical Union, 1970, 51: 181–193. Alfvén, H., 1942. Existence of electromagnetic-hydrodynamic waves. Nature, 150: 405–406. Alfvén, H., 1950. Cosmical Electrodynamics, Oxford: Clarendon Press. Lundqvist, S., 1949. Experimental demonstrations of magneto-hydrodynamic waves. Nature, 164: 145–146. Lundqvist, S., 1952. Studies in magnetohydrodynamics. Arkiv för Fysik, 5: 297–347. Pease, R.S., and Lindqvist, S., 1998. Hannes Olof Gösta Alfvén. Biographical Memoirs of the Royal Society, 44: 3–19.

7

Cross-references Alfvén’s Theorem and the Frozen Flux Approximation Alfvén Waves Chapman, Sydney (1888–1970) Core Motions Cowling, Thomas George (1906–1990) Geodynamo, Dimensional Analysis and Timescales Magnetohydrodynamics Magnetohydrodynamic Waves Oscillations, Torsional Storms and Substorms, Magnetic

ALFVE´N’S THEOREM AND THE FROZEN FLUX APPROXIMATION History In 1942, Hannes Alfvén (q.v.) published a paper that announced the discovery of the wave that now bears his name (Alfvén, 1942; see Alfvén waves), and that is now often regarded as marking the birth of magnetohydrodynamics or “MHD” for short (see Magnetohydrodynamics). In interpreting the waves in an associated paper (Alfvén, 1943, }4), he enunciated a result that has become known as “Alfvén’s theorem”: Suppose that we have a homogeneous magnetic field in a perfectly conducting fluid. . . . In view of the infinite conductivity, every motion (perpendicular to the field) of the liquid in relation to the lines of force is forbidden because it would give infinite eddy currents. Thus the matter of the liquid is “fastened” to the lines of force. . . Here, one understands “in relation” to mean “relative.” Also the term “frozen” instead of “fastened” has been thought more appealing, and the theorem is frequently referred to as “the frozen flux theorem.” It is now more often stated in terms such as Magnetic flux tubes move with a perfectly conducting fluid as though frozen to it. The reason why it is a little more accurate to refer to “magnetic flux tubes” rather than “magnetic field lines” will be explained in section “Formal demonstrations of the theorem.” According to Alfvén’s theorem, a perfect conductor cannot gain or lose magnetic flux. The theorem therefore has no bearing on field generation by dynamo action although, to understand how the conductor initially acquired the magnetic flux threading it, one must recognize the imperfect conductivity of the fluid and reopen the dynamo question. Similarly, the theorem shows that it is impossible to change the topology of the field lines passing through the conductor in any way whatever and, to understand how a real fluid allows the field lines to sever and reconnect, one must recognize that its resistivity is finite.

Underlying physics Although MHD phenomena usually require the coupled equations of fluid dynamics and electromagnetism (em) to be solved together, all that is needed to establish Alfvén’s theorem is the “kinematic” part of the relationship; the fluid velocity V is regarded as given and the magnetic field B is found by solving rB¼0

(Eq. 1)

and the em induction equation: ]t B ¼ r  ðV  BÞ þ
r2 B:

(Eq. 2a)

ALFVE´N’S THEOREM AND THE FROZEN FLUX APPROXIMATION

8

Here

where we have used (Eq. 1) and a vector identity. The result is
¼ 1=m0 s

(Eq. 2b)

is the “magnetic diffusivity” (assumed uniform), s being the electrical conductivity, and m0 the permeability of free space: SI units are used here. In kinematic dynamo theory, V is sought for which (Eq. 2a) has self-sustaining solutions B (satisfying the appropriate boundary conditions). There is also considerable geophysical interest in the inverse problem of inferring V from the B observed at the Earth’s surface; see section “The inverse problem.” Alfvén’s theorem follows from (Eq. 2a) in the zero diffusion limit: ]t B ¼ r  ðV  BÞ

for

s¼1

(Eq. 3)

A solution to this equation can be regarded as the first term in an expansion of the field in inverse powers of a magnetic Reynolds number. Such an expansion is, in the parlance of perturbation theory, “singular” since the term
r2 B involving the highest spatial derivatives of B has been ejected from (Eq. 2a). The highest derivatives reassert themselves in the structure of boundary layers; see Core, boundary layers and the section “The inverse problem”. Equation (3) has the same form as the equation that governs the vorticity v ¼ r  V in an inviscid fluid of uniform density driven by conservative forces: ]t v ¼ r  ðV  vÞ:

(Eq. 4)

Vortex tubes move with an inviscid fluid of uniform density as though frozen to it. Clearly, Alfvén’s theorem is the MHD analog of Kelvin’s theorem. There is however a significant difference: (Eq. 3) is a linear equation that determines the evolution of B from an initial state for given V; (Eq. 4) is a nonlinear relationship between r  V and V.

Formal demonstrations of the theorem Some readers, with as firm a grasp of em theory as Alfvén, will find his explanation of the theorem, given in the section “History”, sufficiently persuasive. Others, who have a grounding in classical fluid mechanics, may be satisfied by the analogy with Kelvin’s theorem, although that theorem too requires proof. Yet others, with a more mathematical bent, may prefer a direct demonstration, such as the one given below. Let Gðt Þ be an arbitrary curve “frozen” to the fluid as it moves, and let ds(t) be an infinitesimal element of G whose ends are situated at s(t) and sðt Þ þ dsðt Þ. The fluid velocities at these points are V(s(t),t) and Vðsðt Þ þ dsðt Þ, t Þ ¼ Vðsðt Þ, t Þ þ dsðt Þ  rVðsðt Þ, t Þ. At a time dt later, the ends are therefore situated at s þ Vdt and s þ ds þ ½V þ ds  rVdt . It follows that

B  ds ¼ 0

(Eq. 6b)

t¼0

at

)

B  ds ¼ 0

for all t : (Eq. 6c)

A magnetic field line is a curve every element ds of which is parallel to B. According to (Eq. 6c), if G is initially a field line, it is always a field line. This establishes that magnetic field lines move with a perfectly conducting fluid, the weak form of Alfvén’s theorem. The conclusion would still follow if a source term parallel to B were added to the right-hand side of (Eq. 3). The absence of this term leads to the stronger form of the theorem given below. For the second application of (Eq. 5a), let Sðt Þ be an arbitrary (open) surface “frozen” to the fluid as it moves and let dS(t) be an infinitesimal element of S having the shape of a parallelogram, two adjacent sides being along curves Gð1Þ and Gð2Þ , so that dS ¼ dsð1Þ dsð2Þ ;

(Eq. 7a)

where ds(1) and ds(2) are elements of Gð1Þ and Gð2Þ to which (Eq. 5a) applies, leading to the result

(Eq. 5b)

dt ðB  dSÞ ¼ 0;

for

s ¼ 1;

t¼0

)

(Eq. 8a)

and therefore B  dS ¼ 0

at

B  dS ¼ 0 for all t :

(Eq. 8b)

A magnetic surface, M(t), is a surface composed of field lines, so that every surface element, dS, is perpendicular to B. According to (Eq. 8b), if M is initially a magnetic surface, it is always a magnetic surface. This conclusion also follows from (Eq. 6c). To obtain the stronger result, let M be a magnetic flux tube. This is a bundle of magnetic field lines and is therefore bounded by a magnetic surface for all t. Let Sðt Þ be a cross-section of the tube, i.e., a surface that cuts across every magnetic field line within the tube. The net magnetic flux through S, Z B  dS; (Eq. 9) m¼ SðtÞ

is called the “strength” of the tube. It is easily seen from (Eq. 1) that m is the same for every cross-section S. Equation (8b) shows further that m is time independent: Z Sðt Þ

B  dS ¼ 0;

for

s ¼ 1:

(Eq. 10a)

This conservation of flux is the strong form of Alfvén’s theorem. It is worth observing that, according to Eqs. (1), (3), and Stokes’ theorem, (Eq. 10a) may also be written in Eulerian terms as Z SðtÞ

(Eq. 6a)

(Eq. 7b)

It follows from (Eq. 6a) and (Eq. 7b) that

dt

is the “motional” or “Lagrangian” derivative, i.e., the time derivative following the fluid motion. In the first application of this kinematic result, we combine it with (Eq. 3), written in the form dt B ¼ B  rV  Br  V;

s ¼ 1;

where the summation convention applies to the repeated suffix j. It follows that

(Eq. 5a)

where dt ¼ ]t þ V  r

for

dt ðdSÞ ¼ dSðr  VÞ  dSj rVj :

Kelvin’s theorem follows from (Eq. 4):

dt ðdsÞ ¼ ds  rV;

dt ðB  dsÞ ¼ ðB  dsÞj rVj ;

I ð]t BÞ  dS þ

GðtÞ

B  ðV  dsÞ ¼ 0;

for

s ¼ 1; (Eq. 10b)

ALFVE´N’S THEOREM AND THE FROZEN FLUX APPROXIMATION

where Gðt Þ is the perimeter of Sðt Þ. If S were fixed in space, the first term on the left-hand side of (Eq. 10b) would be the rate of increase of the flux through S. The V  ds in the second term is the rate at which the vector area of S increases as ds, on the perimeter G of S, is advected by the fluid motion V. The associated rate at which S gains magnetic flux is therefore B  ðV  dsÞ.

The inverse problem Even though no material, apart from superconductors, can transmit electricity without ohmic loss, Alfvén’s theorem is often useful in visualizing MHD processes. For example, in rapidly rotating convective systems such as the Earth’s core, the Coriolis force deflects the rising and falling buoyant streams partially into the zonal directions, and creates zonal field. This “o-effect” is readily pictured with the help of Alfvén’s theorem: The field lines of the axisymmetric part of B, for example, are dragged along lines of latitude by the zonal shear, i.e., the shear adds a zonal component to B. Alfvén’s theorem is also helpful in attacking the problem of inferring unobservable fluid motions from observed magnetic field behavior, and it is this application that will be considered now: B and ]t B will be assumed known and V will be sought. The electrical conductivity of the mantle is so low that the toroidal electric currents induced in the mantle by the changing MHD state of the core are small. If they are neglected, the magnetic field created by the core is, everywhere in the mantle and above, a potential field that can be expressed as a sum of internal spherical harmonics with time-varying Gauss coefficients whose values are obtained by analyzing observatory and satellite data (see Time-dependent models of the geomagnetic field ). This sum can be used to compute B at the coremantle boundary (CMB) but the larger the order n of the spherical harmonic the more rapidly it increases with depth into mantle. The energy spectrum, which is dominated by the dipolar components at the Earth’s surface, is almost flat at the CMB (see Geomagnetic spatial spectrum). Clearly, small errors that arise in the large n coefficients when the data at the Earth’s surface is analyzed can have serious consequences when B is extrapolated to the CMB. Although B is continuous across the CMB, its radial derivative is not. Extrapolation cannot therefore provide information about r2 B in the core, even at its surface. If the unknown
r2 B in (Eq. 2a) is significant, it is impossible to learn anything about V from the observed B. If however
r2 B is small enough, (Eq. 3) may be a good first approximation to (Eq. 2a), and the extrapolated B and ]t B may yield information about V (Roberts and Scott, 1965). The first task is to assess the importance of
r2 B in (Eq. 2a), restricting attention to the large scales L of B corresponding to the Gauss coefficients ðn  12Þ that are accessible. The three terms appearing in (Eq. 2a) are respectively of order B/T, VSB/L, and
B=L2 , where T is the timescale of the large-scale B. The third term is small compared with one or both of the other terms if VS >
=L  5  106 m s1 and=or T pl=r0 this criterion is more severe than Childress’ criterion mentioned above in connection with relationship (Eq. 6). Antidynamo theorems have also been derived for velocity fields with only a radial component (Namikawa and Matsushita, 1970) in which case the property r  v ¼ 0 must be dropped, of course. For a discussion of antidynamo theorems for a combination of a toroidal velocity field and a purely radial velocity we refer to Ivers (1995) and earlier papers mentioned therein.

Magnetic fields that cannot be generated by the dynamo process We have already mentioned the fact that equation 1 does not permit growing axisymmetric magnetic fields as solutions as stated by Cowling’s theorem. But it is also generally believed that neither a purely toroidal nor a purely poloidal magnetic field can be generated by the dynamo process in a sphere. The assumption of spherical symmetry is essential since the boundary conditions for poloidal and toroidal components separate on a spherical surface. The only other configuration with this property appears to be the planar layer, which may be regarded as the limit of a thin spherical shell. Proofs in the spherical and planar cases are thus essentially identical. A proof that any physically reasonable purely toroidal field must decay has been obtained by

ARCHEOLOGY, MAGNETIC METHODS

23

Cross-references

Table A1 Existence of homogeneous dynamos Properties

Of magnetic field

Of velocity field

Axisymmetry Purely toroidal Purely poloidal Helical symmetry

No No No (?) Yes

Yes No Yes Yes

Cowling’s Theorem Cowling, Thomas George (1906–1990) Dynamo, Backus Dynamo, Gailitis Dynamo, Herzenberg Elsasser, Walter M. (1904–1991) Larmor, Joseph (1857–1942)

Kaiser et al. (1994). In the poloidal case a proof depending on a physical plausibility argument has been given by Kaiser (1995).

ARCHEOLOGY, MAGNETIC METHODS

Concluding remarks The search for antidynamo theorems has led to some interesting mathematical theorems in the past decades. Just as important are the physical insights into the dynamo process that have been gained. The interactions of poloidal and toroidal components of the magnetic field are evidently essential for the operation of a dynamo. To summarize the main results of antidynamo theorems the Table A1 has been composed. In it we have included the case of helical symmetry, which is known to be dynamo friendly. Indeed Lortz (1968) was able to derive a dynamo with helical symmetry. Friedrich Busse and Michael Proctor

Bibliography Backus, G., 1958. A class of self-sustaining dissipative spherical dynamos, Annals of Physics, 4: 372–447. Bullard, E., and Gellman, H., 1954. Homogeneous dynamos and terrestrial magnetism. Philosophical Transactions of the Royal Society of London, A247: 213–278. Busse, F.H., 1975. A necessary condition for the geodynamo. Journal of Geophysical Research, 80: 278–280. Childress, S., 1969. Théorie magnétohydrodynamique de l’effet dynamo, Lecture Notes. Département Méchanique de la Faculté des Sciences, Paris. Cowling, T.G., 1934. The magnetic field of sunspots. Monthly Notices of the Royal Astronomical Society, 34: 39–48. Elsasser, W.M., 1946. Induction effects in terrestrial magnetism. Physical Review, 69: 106–116. Gailitis, A., 1970. Magnetic field excitation by a pair of ring vortices. Magnetohydrodynamics (N.Y.), 6: 14–17. Herzenberg, A., 1958. Geomagnetic Dynamos. Philosophical Transactions of the Royal Society of London, A250: 543–585. Ivers, D.J., 1995. On the antidynamo theorem for partly symmetric flows. Geophysical and Astrophysical Fluid Dynamics, 80: 121–128. Kaiser, R., 1995. Towards a poloidal magnetic field theorem. Geophysical and Astrophysical Fluid Dynamics, 80: 129–144. Kaiser, R., Schmitt, B.J., and Busse, F.H., 1994. On the invisible dynamo. Geophysical and Astrophysical Fluid Dynamics, 77: 91–109. Larmor, J., 1919. How could a rotating body such as the sun become a magnet? Reports of the British Association for the Advancement of Science, 159–160. Lortz, D., 1968. Exact solutions of the hydromagnetic dynamo problem. Plasma Physics, 10: 967–972. Namikawa, T., and Matsushita, S., 1970. Kinematic dynamo problem. Geophysical Journal of the Royal Astronomical Society, 19: 319–415. Proctor, M.R.E., 1977. On Backus’ necessary condition for dynamo action in a conducting sphere. Geophysical and Astrophysical Fluid Dynamics, 9: 89–93. Proctor, M.R.E., 1979. Necessary conditions for the magnetohydrodynamic dynamo. Geophysical and Astrophysical Fluid Dynamics, 14: 127–145. Proctor, M.R.E., 2004. An extension of the toroidal theorem. Geophysical and Astrophysical Fluid Dynamics, 98: 235–240.

Magnetism and archeology Introduction Magnetic methods have become important tools for the scientific investigation of archeological sites, with magnetic prospection surveys and archeomagnetic dating being the most prominent ones. The principles behind these techniques were initially applied to larger and older features, for example prospecting for ore deposits (see Magnetic anomalies for geology and resources) or paleomagnetic dating (see Paleomagnetism). When these techniques were adapted for archeological targets it was soon established that very different methodologies were required. Archeological features are relatively small and buried at shallow depth and the required dating accuracy is in the order of tens of years. More importantly, the relationship between archeological features and magnetism is often difficult to predict and the planning of investigations can hence be complicated. Related is the problem of interpretation. Geophysical results on their own are only of limited use to resolve an archeological problem. It is the archeological interpretation of the results using all possible background information (site conditions, archeological background knowledge, results from other investigations, etc.), which provides useful new insights. If the relationship between magnetic properties and their archeological formation is unknown, such interpretation may become speculative. All magnetic investigations depend on the contrast in a magnetic property between the feature of interest and its surrounding environment, for example the enclosing soil matrix. The most important magnetic properties for archeological studies are magnetization and magnetic susceptibility.

Remanent magnetism Thermoremanent magnetization is probably the best understood magnetic effect caused by past human habitation. If materials that are rich in iron oxides are heated above their Curie temperature and then allowed to cool in the ambient Earth’s magnetic field they have the potential to acquire a considerable thermoremanence that is fixed in the material until further heating. Typical archeological examples are kilns and furnaces, often built of clay, which during their heating cycles often exceed the Curie temperatures of magnetite (Fe3O4) and maghaemite (g-Fe2O3) (578 C and 578–675 C, respectively). Such iron oxides are commonly found in the clay deposits that were used for the construction of these features. Even if the clays only contained weakly magnetic haematite or goethite, the heating and cooling cycles may have converted these into ferrimagnetic iron oxides (see Magnetic mineralogy, changes due to heating). Similarly, fired bricks and pottery can exhibit thermoremanence but when the finished bricks are used as building material their individual vectors of magnetization will point into many different directions producing an overall weakened magnetic signature (Bevan, 1994). The same applies to heaps of pottery shards. The strong magnetic remanence of kilns led to their discovery with magnetometers in 1958 (Clark, 1990), which triggered the widespread use of archeological magnetometer surveys today.

24

ARCHEOLOGY, MAGNETIC METHODS

Kilns also helped to establish the archeomagnetic dating technique as a new tool for chronological studies in archeology (Clark et al., 1988). Detrital remanence is caused when the Earth’s magnetic field aligns magnetized particles that are suspended in solution and gradually settle (see Magnetization, depositional remanent). An example is stagnant water loaded with “magnetic sediments”. The resulting deposits can exhibit a weak but noticeable remanence, which could be used for dating and prospection. It is suspected that similar effects have produced a small remanence and hence a noticeable positive signal in Egyptian mud-bricks, which were made from wet clay pushed into moulds and dried in the sun (Herbich, 2003). However, other magnetometer surveys over mud-brick structures (Becker and Fassbinder, 1999) have shown a negative magnetic contrast of these features against the surrounding soil. Although it is possible to consider burning events that could lead to such results during the demolition of buildings, it is more likely that these bricks were made from clay that has lower magnetic susceptibility than the soil on the building site.

Induced magnetism Any material with a magnetic susceptibility will acquire an induced magnetization in the Earth’s magnetic field (see Magnetic susceptibility). Hence, if past human habitation has led to enhanced levels of magnetic susceptibility in the soils, magnetic measurements can be used for their detection. The relationship between human activities and the enhancement of magnetic susceptibility was investigated by Le Borgne (1955, 1960), distinguishing thermal and bacterial enhancement. When soil is heated in the presence of organic material (for example during bush- or camp-fires), oxygen is excluded and the resulting reducing conditions lead to a conversion of the soil’s hematite (a-Fe2O3, antiferromagnetic) to magnetite (Fe3O4, ferrimagnetic) with a strong increase of magnetic susceptibility. On cooling in air, some of the magnetite may be reoxidized to maghemite (g-Fe2O3, ferrimagnetic), thereby preserving the elevated magnetic susceptibility. In contrast, the “fermentation effect” refers to the reduction of hematite to magnetite in the presence of anerobic bacteria that grew in decomposing organic material left by human habitation, either in the form of rubbish pits (“middens”) or wooden building material. This latter effect requires further research but it is reported that changes in pH/Eh conditions as well as the bacteria’s use of iron as electron source are responsible for the increase of magnetic susceptibility (Linford, 2004). The level of magnetic susceptibility that can be reached through anthropogenic enhancement also depends on the amount of iron oxides initially available in the soil for conversion. The level of enhancement can hence be quantified by relating a soil’s current magnetic susceptibility to the maximum achievable value. This ratio is referred to as “fractional conversion” and is determined by heating a sample to about 700 C to enhance its magnetic susceptibility as far as possible (Crowther and Barker, 1995; Graham and Scollar, 1976). Whether the initial magnetic susceptibility was enhanced by pedogenic or anthropogenic effects can however not be distinguished with this method. It is also worth remembering that magnetite and maghemite have the highest magnetic susceptibility of the iron oxides commonly found in soils, and in the absence of elemental iron, a sample’s magnetic susceptibility is hence a measure for the concentration of these two minerals. More recent investigations have indicated additional avenues for the enhancement of magnetic susceptibility. One of the most interesting is a magnetotactic bacterium that thrives in organic material and grows magnetite crystals within its bodies (Fassbinder et al., 1990) (see also Biomagnetism). Their accumulation in the decayed remains of wooden postholes led to measurable magnetic anomalies and is probably responsible for the detection of palisade walls in magnetometer surveys (Fassbinder and Irlinger, 1994). Other causes include the low-temperature thermal dehydration of lepidocrocite to maghemite (e.g., Özdemir and Banerjee, 1984) and the physical alteration of the constituent magnetic minerals, especially their grain size (Linford

and Canti, 2001; Weston, 2004). Magnetic susceptibility can also be enhanced by the creation of iron sulfides in perimarine environments with stagnant waters (Kattenberg and Aalbersberg, 2004). These may fill geomorphological features, like creeks, that were used for settlements and can therefore be indirect evidence for potential human activity.

Archeological prospection Archeological prospection refers to the noninvasive investigation of archeological sites and landscapes for the discovery of buried archeological features. To understand past societies it is of great importance to analyze the way people lived and interacted and the layout of archeological sites gives vital clues; for example the structure of a Roman villa’s foundations or the location of an Iron-aged ditched enclosure within the wider landscape. Such information can often be revealed without excavation by magnetic surveys. These techniques have therefore become a vital part of site investigation strategies. Buried archeological features with magnetic contrast will produce small anomalies in measurements on the surface and detailed interpretation of recorded data can often lead to meaningful archeological interpretations. The techniques are not normally used to “treasure-hunt” for individual ferrous objects but rather for features like foundations, ditches, pits, or kilns (Sutherland and Schmidt, 2003).

Magnetic susceptibility surveys Since human habitation can lead to increased magnetic susceptibility (see above), measurements of this soil property are used for the identification of areas of activity. Such surveys can either be carried out in situ (i.e., with nonintrusive field measurements) or by collecting soil samples for measurements in a laboratory. These two methods are often distinguished as being volume- and mass-specific, respectively, although such labeling only vaguely reflects the measured properties. Most instruments available for the measurement of low field magnetic susceptibility internally measure the “total magnetic susceptibility” (kt with units of m3), which is proportional to the amount of magnetic material within the sensitive volume of the detector: the more material there is, the higher will be the reading. For field measurements, the amount of investigated material is usually estimated by identifying a “volume of sensitivity” (V ) for the employed sensor (e.g., a hemisphere with the sensor’s diameter for the Bartington MS2D field coil). The “volume specific magnetic susceptibility” is then defined as k ¼ kt/V (dimensionless). In contrast, laboratory measurements normally relate instrument readings to the weight of a sample (m), which can be determined more accurately. The “mass specific magnetic susceptibility” is then w ¼ kt/m (with units of m3 kg1). Accordingly, it is possible to calculate one of these quantities from the other using the material’s bulk density (r): w ¼ k/r. The main difference between field and laboratory measurements, however, is the treatment of samples. It is common practice (Linford, 1994) to dry and sieve soil samples prior to measuring their magnetic susceptibility in the laboratory. Drying eliminates the dependency of mass specific magnetic susceptibility on moisture content, which affects the bulk density. Sieving removes coarse inclusions (e.g., pebbles) that are magnetically insignificant. In this way, laboratory measurements represent the magnetic susceptibility of a sample’s soil component and can therefore be compared to standard tables. For field measurements, however, results can be influenced by nonsoil inclusions and the conversion of volume-specific measurements to mass-specific values is affected by changes in environmental factors (e.g., soil moisture content). The measured magnetic susceptibility depends on the amount of iron oxides available prior to its alteration by humans (mostly related to a soil’s parent geology), and also on the extent of conversion due to the anthropogenic influences. As a consequence, the absolute value

ARCHEOLOGY, MAGNETIC METHODS

of magnetic susceptibility can vary widely between different sites and “enhancement” can only be identified as a contrast between areas of higher susceptibility compared to background measurements. There is no predetermined threshold for this contrast and it is therefore justifiable to use the more qualitative measurements of field-based magnetic volume susceptibility (see above) for archeological prospection. The instrument most commonly used for field measurements of volume specific magnetic susceptibility is the Bartington MS2D field coil consisting of a 0.2 m diameter “loop,” which derives 95% of its signal from the top 0.1 m of soil (Schmidt et al., 2005). Since most archeological features are buried deeper than this sensitivity range, the method relies on the mixing of soil throughout the profile, mostly by ploughing. Magnetic susceptibility surveys, either performed as in situ field measurements or as laboratory measurements of soil samples, can be used in three different ways: (i) as primary prospection method to obtain information about individual buried features, (ii) to complement magnetometer surveys and help with their interpretation by providing data on underlying magnetic susceptibility variations, or (iii) for a quick and coarse “reconnaissance” survey using large sampling intervals to indicate areas of enhancement instead of outlining individual features. Figure A4a shows the magnetic susceptibility survey (MS2D) of a medieval charcoal burning area. Since a detailed study of this feature was required, the data were recorded with a spatial resolution of 1 m in both x- and y-directions. Corresponding magnetometer data (FM36) are displayed in Figure A4b. The magnetic susceptibility results outline the burnt area more clearly and are on this site well suited for the delineation of features. On former settlement sites where ploughing has brought magnetically enhanced material to the surface and spread it across the area, magnetic susceptibility measurements at coarse intervals of 5, 10, or 20 m can be used to identify areas of enhancement that can later be investigated with more detailed sampling, for example with a magnetometer (see below). Even where ploughing has mixed the soil, magnetic susceptibility measurements can vary considerably over short distances of about 2 m. It is hence not normally possible to interpolate coarsely sampled data and

25

settlement sites can only be identified if several adjacent measurements have consistently high levels compared to the surrounding area. Figure A5 shows data from a survey at Kirkby Overblow, North Yorkshire, in search for a lost medieval village. The dots mark the individual measurements and their size represents the strength of the magnetic susceptibility, highlighting the singularity of each measurement. Nevertheless, areas of overall enhancement can be identified and these were later investigated with magnetometer surveys. The gray shading in this diagram visualizes the data in a contiguous way, coloring the Voronoi cell around each individual measurement according to its magnetic susceptibility. The limitations of such diagrams for widely separated measurements should be considered carefully.

Magnetometer surveys In contrast to surveys that measure magnetic susceptibility directly, magnetometer surveys record the magnetic fields produced by a contrast in magnetization, whether it is induced as a result of a magnetic susceptibility contrast, or remanent, for example from thermoremanent magnetization. If the shape and magnetic properties of a buried archeological feature were known, the resulting magnetic anomaly could be calculated (Schmidt, 2001) (see also Magnetic anomalies, modeling). The inverse process, however, of reconstructing the archeological feature from its measured anomaly, is usually not possible due to the nonuniqueness of the magnetic problem and the complex shape and heterogeneous composition of such features. Some successful inversions were achieved when archeologically informed assumptions were made about the expected feature shapes (e.g., the steepness of ditches) and magnetic soil properties (for example from similar sites) (Neubauer and Eder-Hinterleitner, 1997; Herwanger et al., 2000). If surveys are conducted with sufficiently high spatial resolution, the mapped data often already provide very clear outlines of the buried features (Figure A6), allowing their archeological interpretation even without data inversion. However, when interpreting measured data directly, the typical characteristics of magnetic anomalies have to be taken into consideration. For example, in the northern hemisphere

Figure A4 Medieval charcoal production site in Eskdale, Cumbria. (a) Magnetic susceptibility survey with Bartington MS2D field coil. (b) Fluxgate gradiometer survey with Geoscan FM36. Both surveys were conducted over an area of 40 m  40 m with a spatial resolution of 1 m  1 m.

26

ARCHEOLOGY, MAGNETIC METHODS

Figure A5 Kirkby Overblow, North Yorkshire. To search for a deserted medieval village, a magnetic susceptibility survey was undertaken with the Bartington MS2D field coil. The measurements are represented as scaled dots and as shaded Voronoi cells. Basemap from 1st edition Ordnance Survey data.

anomalies created by soil features with induced magnetization have a negative trough to the north of the archeological structure and a slight shift of the positive magnetic peak to the south of its center (Figure A7). To estimate the strength of a typical archeological response it is possible to approximate the anomaly with a simplified dipole field: B ¼ m0m/r3 ¼ DkVBEarth/r3, where m0 is the magnetic permeability of free space (4p  10–7 Tm A1), m is the total magnetic moment of the feature, r the distance between measurement position and sample, Dk its volume-specific magnetic susceptibility contrast, V its volume, and BEarth the Earth’s magnetic flux density. For a buried pit, the following values can be used: Dk ¼ 10  10–5, V ¼ 1 m3, BEarth ¼ 48,000 nT, and r ¼ 1 m. This yields an anomaly strength of only 4.8 nT, which is typical for archeological soil features (e.g., pits or ditches). Anomalies created by the magnetic enrichment of soils through magnetotactic bacteria, for example in palisade ditches, can be as low as 0.3 nT (Fassbinder and Irlinger, 1994) and are therefore only detectable with very sensitive instruments and on sites where the signals caused by small variations in the undisturbed soil’s magnetic properties are very low (low “soil noise”). Peak values higher than approximately 50 nT are normally only measured over ferrous features with very high magnetic susceptibility, or over features with thermoremanent magnetization, like furnaces or kilns. As shown by the numerical approximation above, the anomaly strength depends both on the magnetization and the depth of a buried feature and can therefore not be used for the unambiguous characterization of

that feature. More indicative is the spatial variation of an anomaly (see Figure A7) since deeper features tend to create broader anomalies. Surveys are normally undertaken with magnetometers on a regular grid. The required spatial resolution obviously depends on the size of the investigated features, but since these are often unknown at the outset, a high resolution is advisable. Recommendations by English Heritage have suggested a minimum resolution of 0.25 m along lines with a traverse spacing of 1 m or less (0.25 m  1 m) (David, 1995). However, to improve the definition of small magnetic anomalies even denser sampling is required. Such high-definition data can even show peaks and troughs of bipolar anomalies from small, shallowly buried iron debris (e.g., farm implements) and therefore help to distinguish these from archeological features with wider anomaly footprints. More recently, it has become possible to collect randomly sampled data with high accuracy (Schmidt, 2003) that can either be gridded to a predefined resolution or visualized directly with Delaunay triangulation (Sauerländer et al., 1999). Due to the often small anomaly strength caused by archeological features, very sensitive magnetometers are required for the surveys. The first investigations (Aitken et al., 1958) were made with proton-free precession magnetometers (see also Observatories, instrumentation) but due to the slow speed of operation and their relative insensitivity, these are rarely used for modern surveys. In Britain, the most commonly used magnetometers use fluxgate sensors (see also Observatories, instrumentation), which can achieve noise levels as low as 0.3 nT if sensors and electronics

ARCHEOLOGY, MAGNETIC METHODS

Figure A6 Fluxgate gradiometer data from Ramagrama, Nepal. The outlines of a small Buddhist temple complex are clearly visible, including the outer and inner walls of the courtyard building (1). In addition, the foundations of a small shrine (2) within an enclosure wall (3) can be discerned.

are carefully adjusted. In Austria and Germany, some groups use cesium vapor sensors that are built to the highest possible specifications and have reported sensitivities of 0.001 nT (Becker, 1995). On loess soils, which produce very low magnetic background variations, the weak anomalies caused by magnetotactic bacteria in wood (see above) were detected with cesium magnetometers, revealing palisade trenches of Neolithic enclosures (Neubauer and Eder-Hinterleitner, 1997). Magnetometer sensors measure the combination of the archeological anomaly and the Earth’s magnetic field. Hence, to reveal the archeological anomalies, readings have to be corrected carefully for changes in the ambient field, caused by diurnal variations (see Geomagnetic secular variation) or magnetic storms (see Storms and substorms, magnetic). Proton free-precession instruments are often used in a differential arrangement (“variometer”) by placing a reference sensor in a fixed location to monitor the Earth’s magnetic field. The survey is then carried out with an additional sensor, which is affected by the same ambient field. Subtracting the data from both sensors cancels out the Earth’s field. The same can be achieved with a gradiometer arrangement where the second sensor is usually rigidly mounted 0.5–1 m above the first so that both are carried together. Their measurements can be subtracted instantly to form the gradiometer reading. Despite the heavier weight of such an arrangement, it allows for improved instrument design. Especially for fluxgate sensors a signal feedback system, linked to the upper sensor, can be used to enhance the sensitivity of the instrument. Archeological features that can be detected with magnetometers are often buried at shallow depth and a gradiometer’s sensor separation is then similar to the distance between the feature and the instrument. Therefore, the gradiometer reading is not an approximation for the field gradient and is better recorded as the difference in magnetic flux density (nT) between the sensors and not as a gradient (nT m1). The magnetic field created

27

by the feature of interest and the ambient magnetic field combine at the sensor and the recorded signal depends on the vector component measured with the particular instrument. For example, fluxgate sensors measure a single component of the magnetic field (usually the vertical component) and a gradiometer therefore records exactly this component of the archeological anomaly. In contrast, gradiometers built from field intensity sensors, like proton free-precession and cesium vapor sensors, measure the component of the anomaly in the direction of the ambient Earth’s magnetic field, since this is much larger than the anomaly. Its direction governs the vector addition of the two contributions and the gradiometer reading can be approximated as  BEarth þ Banomaly   jBEarth j  Banomaly  eEarth where eEarth is the unitvector in the direction of the Earth’s magnetic field (Blakely, 1996). As a consequence, vertical fluxgate sensors record weaker signals in areas of low magnetic latitude (i.e., near the equator) and data from the two sensor types cannot be compared directly. The “detection range” of archeological geophysical surveys depends on the magnetization and depth of the features (see above) as well as the sensitivity of the used magnetometers and can therefore not easily be specified. Based on practical experience with common instruments, it is estimated that typical soil features, like pits or ditches, can be detected at depths of up to 1–2 m, while ferrous and thermoremanent features can be identified even deeper. Weak responses were recorded from paleochannels that are buried by more than 3 m of alluvium (Kattenberg and Aalbersberg, 2004). Prior to their final interpretation, magnetometer data often have to be treated with computer software for the improvement of survey deficiencies and the processing of resulting data maps (Schmidt, 2002). Data improvement can reduce some of the errors introduced during the course of a survey, such as stripes and shearing between adjacent survey lines. To reduce the time of an investigation, adjacent lines of a magnetometer survey are often recorded walking up and down a field (“zigzag” recording). However, since most magnetometers have at least a small heading error, a change in sensor alignment resulting from this data acquisition method can lead to slightly different offsets for adjacent lines, which are then visible as stripes in the resulting data (Figure A8a, middle). A common remedy for this effect is the subtraction of the individual mean or median value from each survey line. This helps to balance the overall appearance but also removes anomalies running parallel to the survey lines. If the sensor positions for the forward and backward survey direction are systematically offset from the desired recording position (e.g., always 0.1 m “ahead”) anomalies will be sheared and data will appear “staggered” (Figure A8a, left). If this defect is sufficiently consistent, it can be removed (Figure A8b) by fixed or adaptive shifting of every other data line (Ciminale and Loddo, 2001). Another common problem found with some instruments is the “drift” of their offset value, mostly due to temperature effects. If regular measurements are made over dedicated reference points the effects of drift can be reduced numerically. All these methods of data improvement require detailed survey information, for example about the length of each survey line, the direction of the lines, the size of data blocks between reference measurements, etc. It is hence essential that metadata are comprehensively recorded (see below). Once all data have been corrected for common problems and all survey blocks balanced against each other, they can be assembled into a larger unit (often referred to as “composite”) and processed further. Typical processing steps may include low- and high-pass filtering and reduction-to-thepole. However, most processing can introduce new artifacts into the data (Schmidt, 2003) and should therefore only be used if the results help with the archeological interpretation. Many archeological magnetometer surveys are commissioned to resolve a clearly defined question, for example to find archeological remains in a field prior to its development for housing. However, most data are also of potential benefit beyond their initial intended use and should therefore be archived. For example, the removal of

28

ARCHEOLOGY, MAGNETIC METHODS

Figure A7 Calculated shape of the magnetic anomaly caused by the induced magnetization of a buried cubic feature with 3 m side length. The anomaly is measured with a vertical fluxgate gradiometer (0.5 m sensor separation), 0.5 m above the cube’s top surface. The strength of the anomaly was calculated for an inclination of 70 and a magnetic susceptibility contrast of k ¼ 1  10–8.

Figure A8 Fluxgate gradiometer data from Adel Roman Fort, West Yorkshire. (a) Field measurements with staggered data (left) and stripes (center). (b) After their improvement the data clearly show a ditch (1) and the soldiers’ barracks (2) to the north of the road through the Fort (3).

all archeological remains during the development of a building site may mean that the collected geophysical data are the most important record of an ancient settlement. It is therefore essential that data archiving is undertaken according to recognized standards. In particular, detailed information describing data collection procedures and the layout of site

and survey is important. Such information is usually referred to as “metadata” (Schmidt, 2002) and complements the numerical instrument readings as well as processed results. Related to the archiving of data from geophysical surveys is the recommendation that at least a brief report should be provided and archived, whenever possible.

ARCHEOLOGY, MAGNETIC METHODS

Archeomagnetic dating As with paleomagnetic dating (see Paleomagnetism), the magnetic remanence preserved in archeological structures can be used for their dating. Although some research has been undertaken on depositional (“detrital”) remanence in sediments (Batt and Noel, 1991), archeomagnetic dating is mainly applied to thermoremanent magnetization. By firing archeological structures that are rich in iron oxides above their Curie temperature (ca 650–700 C) they become easily magnetized in the direction of the ambient field, which is usually the Earth’s magnetic field. On subsequent cooling below the blocking temperature, this acquired magnetization will form a magnetic remanence. When archeological features that were exposed to such heating and cooling are excavated, oriented samples can be recovered and their thermoremanent magnetization measured. By comparing these data with an archeomagnetic calibration curve that charts the variation of magnetic parameters with time, a date can be determined for the last firing of the archeological feature. Most archeomagnetic dating methods use two or three components of the remanent magnetization vector (inclination, declination, and sometimes intensity). It is therefore a pre-requisite that samples are collected from structures that have not changed their orientation since the last firing. Typical features include kilns, hearths, baked floors, and furnaces. Unfortunately, it is not always possible to assess whether an archeological feature is found undisturbed and in situ. For example, due to instabilities following the abandonment of a kiln, the walls may have moved slightly or the area around a fire place may have been disturbed by modern agricultural activities. Only the final statistical analysis (see below) can ascertain the validity of results. In recent years, advances in archeointensity dating have been made using only the magnitude of the magnetization for age determinations (see Shaw and microwave methods, absolute paleointensity determination). It is therefore possible to magnetically date materials that are no longer in their original position, like fired bricks that were used in buildings or even nonoriented pottery fragments (Shaw et al., 1999; Sternberg, 2001). A variety of different sampling methods exist, which all have their respective benefits. Some groups extract samples with corers, others encase the selected samples in Plaster of Paris before lifting them together with the plaster block, and in Britain plastic disks are commonly glued to the samples before extraction. As it is important to accurately record the orientation of the sample while still in situ, plaster and disks are usually leveled horizontally and the north direction is marked with a compass (either conventional, digital, or sunbased). Determining the right sampling locations within a feature is important as the effect of magnetic refraction often causes magnetic field lines to follow the shape of heated features similar to the demagnetization effects observed in grains and elongated objects. Soffel (1991) reports an approximately sinusoidal dependence of both inclination and declination from the azimuthal angle in a hollow cylindrical feature while Abrahamsen et al. (2003) have found that sample declinations from a hemisphere of solid iron slag with 0.5 m diameter vary throughout 360 . It is therefore essential that several samples from different parts of a feature are compared to statistically assess whether a consistent magnetization vector can be determined. After the last firing of an archeological structure, magnetically soft materials may have acquired a viscous remanent magnetization that gradually followed the changing direction of the Earth’s magnetic field (see Geomagnetic secular variation). Its contribution to a sample’s overall magnetization can lead to wrong estimates for the age, and it therefore has to be assessed with a Thellier experiment and then removed. This removal can either be accomplished through stepwise thermal demagnetization (see Demagnetization) or with stepwise alternating field (AF) demagnetization (Hus et al., 2003). The subsequent measurement of a sample’s magnetization vector, for example in a spinner magnetometer, is then a good approximation of its thermoremanence. Fisher statistics (see Fisher statistics) is used to assess the distribution of all the measured magnetization vectors of an

29

archeological feature by calculating a mean value for the direction together with its angular spread a95. The latter describes the distribution of all measured directions around the mean and is half of the opening angle of a cone (hence appearing as a “radius” on a stereographic plot) that contains the mean vector direction with a probability of 95%. The spread of the individual vector directions can be due to errors in sample marking, measurement errors, and the distribution of different directions within a single feature (see above). It has therefore become common practice to expect a95 to be less than 5 for a reliable investigation (Batt, 1998). Once the magnetization of an archeological feature has been established it can be compared to a calibration curve to derive the archeological age. The construction and use of such calibration data has been a matter of recent research. In Britain, a calibration curve was compiled by Clark et al. (1988) using 200 direct observations (since 1576 AD) and over 100 archeomagnetic measurements from features that were dated by other means, as far back as 1000 BC. All data were corrected for regional variations of the Earth’s magnetic field and converted to apparent values for Meriden (52.43 N, 1.62 W). After plotting results on a stereographic projection, the authors manually drew a connection line that was annotated with the respective dates. Measurements from any new feature could be drawn on the same diagram and the archeological date was determined by visual comparison. This approach is compatible with the accuracy of the initial calibration curve but has clear limitations (Batt, 1997). Similar calibration curves exist for other countries and due to short-scale variations of the Earth’s magnetic field’s nondipole component (see Nondipole field), they are all slightly different and have to be constructed from individually dated archeological materials. The reference curve for Bulgaria, for example, now extends back to nearly 6000 BC (Kovacheva et al., 2004). Although the calibration curve by Clark et al. (1988) has been a useful tool for archeomagnetic dating, improvements are now being made. Batt (1997) used a running average to derive the calibration curve more consistently from the existing British data. Kovacheva et al. (2004) calculated confidence limits for archeological dates using Bayesian statistics (Lanos, 2004) for the combination of inclination, declination, and paleointensity of measured samples. To improve the accuracy and reliability of the archeomagnetic method, more dated archeological samples are required and comprehensive international databases are currently being compiled. Some researchers have attempted to use magnetometer surveys over furnaces to derive archeomagnetic dates for their last firing. For this, the magnetization causing the recorded magnetic anomaly has to be estimated and can then be used with a calibration curve for the dating of the buried archeological feature. To accommodate the complex shape and inhomogeneous fill of partly demolished iron furnaces in Wales, Crew (2002) had to build models with up to five dipole sources to approximate the measured magnetic anomaly maps. The dipole parameters were chosen to achieve the best possible fit between measured and modeled data and their relationship with the magnetization of the furnaces’ individual components is not entirely clear. In addition to the sought after thermoremanence, this magnetization also has contributions from acquired viscous remanence and from the induced magnetization in the current Earth’s magnetic field. Even after estimates for these two sources have been taken into consideration, the results were in poor agreement with the archeomagnetic calibration curve. The magnetic anomalies from slag-pit furnaces in Denmark were approximated with individual single dipole sources by Abrahamsen et al. (2003). They found that the distribution of 32 adjacent furnaces produced an unacceptably high a95 value of 18 and concluded that the method is therefore unsuitable for the dating of these features.

Conclusion There are many ways in which magnetic methods can be used in archeological research and this application has made them popular with the

30

ARCHEOLOGY, MAGNETIC METHODS

public. Magnetometer surveys have become a tool for archeologists, nearly as important as a trowel. In an excavation, many archeological remains are only revealed by their contrast in color or texture compared with the surrounding soil. Searching for a contrast in magnetic properties is therefore only an extension of a familiar archeological concept. The science explaining the magnetic properties of buried features may be complex but the application of the techniques has become user-friendly. Similarly, archeomagnetic dating is an important part of an integrated archeological dating strategy. For archeological sites, dates are often derived with many different methods simultaneously, ranging from conventional archeological typological determinations over radiocarbon dating to luminescence methods. Combining these different data, for example with Bayesian statistics, allows a significant reduction of each method’s errors and leads to improved results. The wealth of information stored in the magnetic record has certainly made an important contribution to modern archeology. Armin Schmidt

Bibliography Abrahamsen, N., Jacobsen, B.H., Koppelt, U., De Lasson, P., Smekalova, T., and Voss, O., 2003. Archaeomagnetic investigations of iron age slags in Denmark. Archaeological Prospection, 10: 91–100. Aitken, M.J., Webster, G., and Rees, A., 1958. Magnetic prospecting. Antiquity, 32: 270–271. Batt, C.M., and Noel, M., 1991. Magnetic studies of archaeological sediments. In Budd et al. (eds.), Archaeological Science 1989, pp. 234–241. Batt, C.M., 1997. The British archaeomagnetic calibration curve: an objective treatment. Archaeometry, 39: 153–168. Batt, C.M., 1998. Where to draw the line? The calibration of archaeomagnetic dates. Physics and Chemistry of the Earth, 23: 991–995. Becker, H., 1995. From Nanotessla to Picotessla—a new window for magnetic prospecting in archaeology. Archaeological Prospection, 2: 217–228. Becker, H., and Fassbinder, J.W.E., 1999. In search for piramesses — the lost capital of Ramses II. the Nile delta (Egypt) by caesium magnetometry. In Fassbinder, J.W.E., and Irlinger, W.E. (eds.), Archaeological Prospection. München: Bayerisches Landesamt für Denkmalpflege, 146–150. Bevan, B.W., 1994. The magnetic anomaly of a brick foundation. Archaeological Prospection, 1: 93–104. Blakely, R.J., 1996. Potential Theory in Gravity and Magnetic Applications. Cambridge: Cambridge University Press. Ciminale, M., and Loddo, M., 2001. Aspects of magnetic data processing. Archaeological Prospection, 8: 239–246. Clark, A., 1990. Seeing Beneath the Soil. London: Batsford. Clark, A.J., Tarling, D.H., and Noel, M., 1988. Developments in archaeomagnetic dating in Britain. Journal of Archaeological Science, 15: 645–667. Crew, P., 2002. Magnetic mapping and dating of prehistoric and medieval iron-working sites in Northwest Wales. Archaeological Prospection, 9: 163–182. Crowther, J., and Barker, P., 1995. Magnetic susceptibility: distinguishing anthropogenic effects from the natural. Archaeological Prospection, 2: 207–216. David, A., 1995. Geophysical survey in archaeological field evaluation, English Heritage Research and Professional Services Guideline, Vol. 1. Fassbinder, J.W.E., and Irlinger, W.E., 1994. Aerial and magnetic prospection of an eleventh to thirteenth century Motte in Bavaria. Archaeological Prospection, 1: 65–70. Fassbinder, J.W.E., Stanjek, H., and Vali, H., 1990. Occurrence of magnetic bacteria in soil. Nature, 343: 161–163.

Graham, I., and Scollar, I., 1976. Limitations on magnetic prospection in archaeology imposed by soil properties. Archaeo-Pysika, 6: 1–125. Herbich, T., 2003. Archaeological geophysics in Egypt: the Polish contribution. Archaeologia Polona, 41: 13–56. Herwanger, J., Maurer, H., Green, A.G., and Leckebusch, J., 2000. 3-D inversions of magnetic gradiometer data in archeological prospecting: Possibilities and limitations. Geophysics, 65: 849–860. Hus, J., Ech-Chakrouni, S., Jordanova, D., and Geeraerts, R., 2003. Archaeomagnetic investigation of two mediaeval brick constructions in north Belgium and the magnetic anisotropy of bricks. Geoarchaeology, 18: 225–253. Kattenberg, A.E., and Aalbersberg, G., 2004. Archaeological prospection of the Dutch perimarine landscape by means of magnetic methods. Archaeological Prospection, 11: 227–235. Kovacheva, M., Hedley, I., Jordanova, N., Kostadinova, M., and Gigov, V., 2004. Archaeomagnetic dating of archaeological sites from Switzerland and Bulgaria. Journal of Archaeological Science, 31: 1463–1479. Lanos, P., 2004. Bayesian inference of calibration curves: application to archaeomagnetism. In Buck, C.E., and Millard, A.R. (eds.), Tools for Constructing Chronologies: Crossing Discipline Boundaries. Lecture Notes in Statistics. Berlin: Springer-Verlag, pp. 43–82. Le Borgne, E., 1955. Susceptibilité magnétique anormale du sol superficiel, Annales de Géophysique, 11: 399–419. Le Borgne, E., 1960. Influence du feu sur les propriétés magnétiques du sol et sur celles du schiste et du granite. Annales de Géophysique, 16: 159–195. Linford, N., 1994. Mineral magnetic profiling of archaeological sediments. Archaeological Prospection, 1: 37–52. Linford, N.T., 2004. Magnetic ghosts: mineral magnetic measurements on Roman and Anglo-Saxon graves. Archaeological Prospection, 11: 167–180. Linford, N.T., and Canti, M.G., 2001. Geophysical evidence for fires in antiquity: preliminary results from an experimental study. Archaeological Prospection, 8: 211–225. Neubauer, W., and Eder-Hinterleitner, A., 1997. 3D-Interpretation of postprocessed archaeological magnetic prospection data. Archaeological Prospection, 4: 191–205. Özdemir, O., and Banerjee, S.K., 1984. High temperature stability of maghemite. Geophysical Research Letters, 11: 161–164. Sauerländer, S., Kätker, J., Räkers, E., Rüter, H., and Dresen, L., 1999. Using random walk for on-line magnetic surveys, European Journal of Environmental and Engineering Geophysics, 3: 91–102. Schmidt, A., 2001. Visualisation of multi-source archaeological geophysics data. In Cucarzi, M., and Conti, P. (eds.), Filtering, Optimisation and Modelling of Geophysical Data in Archaeological Prospecting. Rome: Fondazione Ing. Carlo M. Lerici, pp. 149–160. Schmidt, A., 2002. Geophysical Data in Archaeology: A Guide to Good Practice, ADS series of Guides to Good Practice. Oxford: Oxbow Books. Schmidt, A., 2003. Remote Sensing and Geophysical Prospection. Internet Archaeology, 15 (http://intarch.ac.uk/journal/issue15/ schmidt_index.html). Schmidt, A., Yarnold, R., Hill, M., and Ashmore, M., 2005. Magnetic Susceptibility as Proxy for Heavy Metal Pollution: A Site Study. Geochemical Exploration, 85: 109–117. Shaw, J., Yang, S., Rolph, T.C., and Sun, F.Y., 1999. A comparison of archaeointensity results from Chinese ceramics using microwave and conventional Thellier’s and Shaw’s methods. Geophysical Journal International, 136: 714–718. Soffel, H. Chr., 1991. Paläomagnetismus und Achäomagnetismus. Berlin, Heidelberg, New York: Springer-Verlag.

ARCHEOMAGNETISM

Sternberg, R.S., 2001. Magnetic properties and archaeomagnetism. In Brothwell, D.R., and Pollard, A.M. (eds.), Handbook of Archaeological Sciences, Chichester: John Wiley & Sons, Ltd. Sutherland, T., and Schmidt, A. 2003. Towton, 1461: An Integrated Approach to Battlefield Archaeology. Landscapes, 4: 15–25. Weston, D.G., 2004. The influence of waterlogging and variations in pedology and ignition upon resultant susceptibilities: A series of laboratory reconstructions. Archaeological Prospection, 11: 107–120.

Cross-references Biomagnetism Demagnetization Fisher Statistics Geomagnetic Secular Variation Magnetic Anomalies for Geology and Resources Magnetic Anomalies, Modeling Magnetic Mineralogy, Changes Due to Heating Magnetic Susceptibility Magnetization, Depositional Remanent (DRM) Magnetization, Thermoremanent (TRM) Nondipole Field Observatories, Instrumentation Paleomagnetism Shaw and Microwave Methods, Absolute Paleointensity Determination Storms and Substorms, Magnetic

ARCHEOMAGNETISM The science and utilization of the magnetization of objects associated with archeological sites and ages. In practice, it is mostly concerned with the magnetic dating of archeological materials, but can also be used for reconstruction, magnetic sourcing, environmental analysis, and exploration. It depends on (1) the property of particular archeological materials containing magnetic impurities (particularly, the iron oxides, magnetite, and hematite) that can retain a record of the past direction and strength of the Earth’s magnetic field, usually at the time that they cooled after being heated, but also when deposited, fluidized, or chemically altered; (2) the Earth’s magnetic field gradually changes in both direction and intensity (secular variation, q.v.). As archeological materials normally record events, such as firing, at a specific point and time, most archeomagnetic records are intermittent in both space and time. The establishment of regional records of directions and intensities therefore require spatial corrections to be applied to individual determinations. Such “Master Curves” for directional studies are now mostly based on the assumption that the geomagnetic field, over an area of some 106 km2, can be represented by an inclined geocentric dipole model of the field and have been constructed for several areas, particularly in Europe, the Middle East, Japan, and Central and southwestern North America. Such corrections appear justified by studies of the regional variation of the present geomagnetic field but may not be valid for all areas and times. Spatial variations in intensity are usually modeled using an axial geocentric dipole model of the geomagnetic field.

History Bricks were shown to be magnetic by Boyle, in 1691, and that they lose their magnetization when heated and acquire it while cooling. Volcanic rocks used in the amphitheatre of Pompeii were demonstrated to have retained their original magnetization for at least 2000 years by Melloni in 1853. Folgerhaiter examined the magnetization of pottery

31

in 1894. Chevallier, in 1925, constructed a record of historical changes in the direction of the geomagnetic field using lavas from Mt Etna. Actual dating of archeological fired clays, involving incremental thermal demagnetization of their direction and intensity of magnetization, were initiated by Thellier in 1936.

Procedures (a) Directional Studies. Oriented samples are selected from part of the structure or object considered to have been magnetized at the same time. Their initial directions of remanence are measured and the samples subjected to demagnetization in incremental steps. This demagnetization can by heating, alternating magnetic fields or tuned microwaves. The vectors at successive steps are then analyzed to determine the characteristic vector associated with the time when the sample had originally cooled. These vectors are then combined, giving each determination equal weight, to determine the mean direction for the site. When the age of the firing is already known, then this direction is incorporated into a “Master Curve” for that region. When the age is unknown, the direction is compared with the appropriate “Master Curve” to assess the likely age of firing. (b) Paleointensity Studies. As the intensity of magnetization is a scalar quantity, it is unnecessary that the sample should be unoriented. This also enables the technique to be used for isolated objects, such as pottery shards, that are no longer in their original firing position. It is therefore a far more widely applicable method. Until 2000, most paleointensity determinations were based on comparing the observed intensity of magnetization of a sample with the intensity it acquired in a known external magnetic field strength. Assuming no chemical changes occurred, i.e., the bulk magnetic susceptibility (K) remains constant, there is a simple relationship in which the ancient geomagnetic field strength (A) equals the intensity of natural remanence (NRM) acquired in that field multiplied by the strength of laboratory field (F ) divided by the intensity of thermal remanence (TRM) acquired in that field (F ), i.e., K ¼ NRM/A ¼ TRM/F. The problems are to ensure there is no change in susceptibility and that the NRM thermal component that was actually acquired during the original cooling has been isolated from all other NRM components. Conventionally, various tests, such as monitoring the susceptibility, repeat readings of the NRM component at successive temperatures and testing for the linear NRM/TRM relationship during incremental heating are all used. However, even when these are all satisfied, there can still be discrepancies between paleointensity determinations on specimens taken from the same sample. While such discrepancies are few, they suggest that thermally induced changes in the magnetic properties are occurring during laboratory heating even when all such tests are satisfied. The origin of these effects remains unclear and possibly relates to thermally induced physical changes as well as chemical changes in the magnetic properties. While other tests are still being developed, the more recent microwave method offers a radically different technique of demagnetizing both the natural and laboratory remanences. The microwaves are tuned to preferentially affect the quanta of spin wave energy in magnetic materials, magnons. The microwave energy is imposed briefly at different power levels, resulting in incremental demagnetization at successive power increments. In the current systems, little of the microwave energy leaks into the thermal band and the consequent rise in temperature is less than 120 C at the current maximum power of 120 W at 14 GHz. Such a temperature rise could still affect some minerals, but the hydroxide minerals that are most likely to be affected will only be present in weathered samples that are, in any case, not suitable for paleointensity determinations. This form of energy could also affect the magnetic domain structures but, as for thermally induced chemical changes, the brevity of the energy application appears to inhibit all such changes. A major advantage is that it can be applied rapidly, currently enabling paleointensity estimates to be made in a fraction of the

32

ARCHEOMAGNETISM

Figure A10 The “Master Curve” of geomagnetic secular variation in SW USA. This shows only the best-dated, well defined declinations and inclinations, after spatial correction. Modified from J.Eighmy, University of Colorado State University (personal communication).

Figure A9 The “Master Curve” of geomagnetic secular variation in Bulgaria. The curves are discontinuous where there is little or no data. The 95% probability error is shaded. Modified from M.Kovacheva, Bulgarian Academy of Sciences (personal communication).

time needed for standard thermal paleointensity methods. Nonetheless, several aspects of this method must still be considered experimental and it is mostly designed for materials containing magnetite rather than hematite.

Errors In directional studies, the largest, non-Gaussian errors in regional “Master Curves” arise from original age inaccuracies. Individual determinations can be affected by magnetic anisotropy, magnetic refraction, but mainly from physical disturbances of the site since firing. The precision of measurement and inhomogeneity remain problems in paleointensity studies, although the former is reducing. Present Situation: The longest, continuous, and most systematic record that is available is for Bulgaria (FigureA9) and includes both directional and intensity data. The most readily accessible data are for England and Wales, France, southwestern USA (FigureA10), and potentially for Japan. Standard methods of curve fitting are being applied to local sequences of directional and intensity determinations, mostly assuming the errors to be Gaussian. This assumption may not always be valid and can result in smoothing fluctuations in the readings that may be of geomagnetic origin. Paleointensities using

Figure A11 The paleointensity secular variation in Peru. These data are determined using microwave demagnetization techniques. Modified from J. Shaw, University of Liverpool (personal communication). microwave techniques are now available for certain periods in Egypt and Peru (FigureA11). Donald D. Tarling

Cross-references Geomagnetic Secular Variation Paleomagnetism Secular Variation Model Shaw and Microwave Methods, Absolute Paleointensity Determination

AURORAL OVAL

AURORAL OVAL The auroras have fascinated humans since prehistory, but it was not until 1770 that Captain James Cook reported that the aurora borealis in the Northern Hemisphere had a counterpart in the Southern Hemisphere, the aurora australis. It was later noted that the frequency of observation of the aurora did not increase all the way to the poles, but maximized near latitudes of 70 and decreased again as the poles were approached. This allowed Elias Loomis in 1860 to draw a map of greatest auroral frequency as an irregular oval encircling the northern pole, though with a centroid somewhat displaced toward the northern coast of Greenland, such that it ran through the northern reaches of Scandinavia, Canada, and Siberia. Subsequently, photographic surveys have shown that the auroral observations are organized by the geomagnetic field, the ovals being roughly centered on the geomagnetic poles, though displaced away from the Sun, such that a magnetic latitude and magnetic local time coordinate system is the most appropriate for describing auroral morphology. The auroras are located at higher magnetic latitudes ( 75 ) on the dayside than on the nightside ( 70 ); the oval has a typical radius of 1500–2000 km and a latitudinal width of some 200–1000 km (Feldstein and Starkov, 1967). The auroras tend to be most luminous and of greatest latitudinal extent near local midnight. Before the advent of the Space Age, the ovals could only be considered as a statistical phenomenon. However, with the launch of auroral imagers onboard spacecraft such as Dynamics Explorer 1, it was found that the ovals were visible as continuous rings surrounding the pole (Frank and Craven, 1988). In recent years, near-continuous, high-temporal resolution auroral imagery from the ground and space has provided a wealth of information regarding the nature of the auroral oval and its relationship to the dynamic magnetosphere (q.v.) (e.g., Milan et al., 2006).

Auroral processes The auroras are formed by precipitation of charged particles, electrons, and protons, following magnetic field lines from the magnetosphere into the atmosphere. Here, collisions with neutral atmospheric atoms or molecules can lead to the promotion of electrons from their ground state or, in cases where the collisional energy exceeds the ionization potential, produce ion-electron pairs to supplement the pre-existing ionosphere (q.v.) produced mainly by solar photoionization (e.g., Vallance Jones, 1974). Excited atoms and molecules emit photons as they relax back to their ground states, to form the luminous aurora; approximately 1% of the incoming kinetic energy is converted to visible light, the power output of a typical 10  1000 km auroral arc being of the order of 1 GW. The initial kinetic energy of the precipitating particles determines the altitude at which most collisions occur, higher energy particles penetrating deeper into the atmosphere before being slowed significantly (Rees, 1989). Early triangulation of auroral altitudes (St
rmer, 1955) showed that peak auroral emission occurred at 100–150 km altitude (the ionospheric E region), though significant luminosity could be observed higher, even above 250 km (F region). The auroral wavelengths produced depend on the atoms or molecules involved in the collisions, the atmospheric concentrations of which themselves depend on altitude. Although many spectral lines can be identified in the aurora, a few dominate. In the lower altitude regime, the main spectral line observed is the forbidden green oxygen line at 557.7 nm (1S to 1D), produced by incoming electrons with energies of 10 keV. At higher altitudes, the red doublet at 630 and 636.4 nm (1D to ground state) dominates, generated by 1 keV electrons. Weak hydrogen lines are also observed, as incoming protons are excited by their collisions with the atmosphere. In addition, emission outside of the visible spectrum, the N2 Lyman-Birge-Hopfield lines in the ultraviolet for instance, are important in auroral imaging from space. Other emission mechanisms include bremsstrahlung X-ray emission from highly energetic electrons penetrating to altitudes below 90 km.

33

The forms of aurora vary greatly, from auroral curtains or arcs, to curls, corona, and rayed aurora. Auroras can also be separated into diffuse and discrete forms, the latter often appearing embedded within the former. Discrete auroras are thought to arise due to acceleration processes taking place at relatively low altitudes (below 1 Earth radius). Low-Earth orbit observations of the characteristics of precipitating particles are shedding new light on these accelerating mechanisms (e.g., Carlson et al., 1998). Spatial scales also vary from the global scale of the auroral ovals themselves, to meso- and microscale features within individual arcs; indeed, the closer the auroras are examined, the smaller the spatial scales that become apparent. One of the many open questions within auroral physics is the mechanism by which such diverse features, some as small as a few meters across, arise within the oval (Borovsky, 1993).

Auroral characteristics and the magnetosphere The ultimate source of the precipitating electrons and protons is the solar wind plasma constantly streaming away from the Sun. Observing the morphology and dynamics of the auroral oval provides a great deal of information about the mechanisms by which solar wind particles gain entry to the magnetosphere, are distributed within it, and are subsequently accelerated to impact the atmosphere. The auroral ovals are not static, uniform features but display local time asymmetries and temporal variations on timescales ranging from subsecond, through minutes, hours, the seasons, and up to the 11-year solar cycle. In general, shorter timescale variations reflect internal magnetospheric dynamics such as magnetohydrodynamic wave activity, whereas longer changes are caused by magnetospheric responses to external stimuli, such as interactions between the terrestrial field with the Sun’s magnetic field (q.v.) carried frozen-in (Alfvén’s Theorem and the frozen flux approximation, q.v.) to the solar wind, where it is known as the interplanetary or heliospheric magnetic field. Solar cycle variations arise due to long-term changes in the characteristics of the solar wind and its magnetic field. Magnetic reconnection between the interplanetary field and the magnetospheric field at the dayside magnetopause causes the two to become interconnected, such that the magnetosphere is no longer closed. The open field lines are eventually disconnected from the solar wind again by reconnection occurring in the magnetotail. It is this constant topological reconfiguration of the magnetospheric field that results in a general circulation and structuring of the magnetospheric plasma, as first proposed by James Dungey (Dungey, 1961), now known as the Dungey cycle. The open field lines allow solar wind plasma to enter the magnetosphere, in general streaming towards the Earth, funneled towards auroral latitudes in the noon sector by the dipolar nature of the geomagnetic dipole field (q.v.). Particles, which penetrate to sufficiently low altitudes that they can collide with atmospheric neutrals, give rise to cusp aurora in the dayside auroral oval, so-called after the magnetospheric cusp topology through which particles enter the magnetosphere. These particles tend to be relatively unaccelerated and so excite red line aurora. Those which do not collide are said to mirror on entering the higher field strength at low altitudes, return upwards along the magnetic field and are carried antisunward to populate the magnetotail. Such particles come eventually to reside near the equatorial plane where they are heated to form a dense hot plasma sheet. Confined in general to move along the near-dipolar magnetic field lines, these trapped particles approach the Earth most closely at auroral latitudes as they mirror backwards and forwards between conjugate hemispheres. At each bounce there is a possibility of an atmospheric collision and the generation of aurora; plasma sheet particles, having been accelerated in the magnetotail, give rise to green line aurora. The plasma sheet extends from the magnetotail around the Earth to the dayside, forming the familiar rings of aurora around the poles. The lower latitude boundary of the auroral oval is governed by the distance of the inner edge of the plasma sheet from the Earth in the equatorial plane, usually of order 7 Earth radii. The higher latitude

34

AURORAL OVAL

boundary of the oval is closely related to the transition from closed dipolar field lines to open field lines which map deep into the magnetotail and out into the solar wind. Such nondipolar field lines do not provide the trapping geometry that requires particles to constantly return to the atmosphere, and so auroral luminosities tend to be negligible poleward of 80 latitude. On occasions, Sun-aligned arcs can be observed at high latitudes inside the oval, also known as transpolar arcs or theta aurora, in the latter case due to the characteristic shape they form together with the oval. There is still much debate regarding the source of these auroras and the magnetospheric configuration that gives rise to them. Seasonal variations have been found in the distribution of auroral luminosity within the ovals (Newell et al., 1998) and the overall level of auroral activity (Russell and McPherron, 1973). The latter is in part a consequence of the differing reconnection geometries available at the magnetopause as the tilt of the rotational and magnetic axes of the Earth change the orientation of the magnetosphere with respect to the Sun-Earth line. However, variations in auroral distribution controlled by the extent of solar illumination of the polar regions point toward complex feedback mechanisms between ionosphere and magnetosphere.

The substorm cycle The auroral oval is not static with time, but undergoes a 2–4 hr cycle of growth and decay known as the substorm cycle (intimately related to the Dungey cycle), first adequately described by Akasofu (1968). During periods of strong reconnection at the dayside magnetopause, generally when the interplanetary magnetic field points southwards, an increasing proportion of the terrestrial magnetic flux becomes opened. As a consequence, the auroral oval moves to lower latitudes as the open field line region expands, during what is known as the growth phase, which can last several tens of minutes. At some point the magnetosphere can no longer support continued opening of flux and reconnection is initiated in the magnetotail to close flux once again. This flux closure is associated with vivid nightside auroral displays, termed substorm break-up aurora, as plasma is accelerated earthwards into the midnight sector atmosphere. The nightside aurora expands rapidly polewards to form a substorm auroral bulge as the dim open flux region contracts, leading to this phase of the substorm cycle being known as expansion phase, which can last from 30 min to a few hours. The auroral bulge expands not only polewards but also rapidly westwards, spear-headed by the most luminous feature of the substorm, the westwards-travelling surge. Eventually, the substorm enters the recovery phase as the magnetosphere relaxes back to a more quiescent state. This phase is accompanied by large-scale wave-like perturbations of the poleward border of the postmidnight oval, termed omega bands after their characteristic appearance. Associated with substorms are elevated levels of geomagnetic activity caused by currents flowing horizontally in the E region ionosphere, facilitated by enhanced ionospheric conductivity associated with the auroral ionization. These currents flow to close pairs of upward and downward field-aligned currents (Iijima and Potemra, 1978), also known as Birkeland currents, which transmit stress from the outer magnetosphere into the ionosphere and drive ionospheric convection, the counterpart of the magnetospheric circulation described by the Dungey/substorm cycle. The main components of the ionospheric current systems are the westward and eastward electrojets flowing along

the auroral ovals in the dawn and dusk sectors, respectively, and the development of the westward substorm electrojet in the midnight sector during expansion phase. These currents are largely responsible for the magnetic deflections detected at the ground and compiled to form various geomagnetic indices that measure the general level of geomagnetic disturbance, such as the planetary KP index devised by Julius Bartels (q.v.) (Bartels et al., 1939) and auroral electrojet index AE (Davis and Sugiura, 1966). Stephen Milan

Bibliography Akasou, S.-I., 1968. Polar and magnetospheric substorms. Dordrecht: Reidel. Bartels, J., Heck, N.H., and Johnston, H.F., 1939. The three-hourrange index measuring geomagnetic activity. Journal of Geophysical Research, 44: 411. Borovsky, J.E., 1993. Auroral arc thickness as predicted by various theories. Journal of Geophysical Research, 9: 6101. Carlson, C.W., Pfaff, R.F., and Watzin, J.G., 1998. The Fast Auroral SnapshoT (FAST) mission. Geophysical Research Letters, 25: 2013. Davis, T.N., and Sugiura, M., 1966. Auroral electrojet activity index AE and its universal time variations. Journal of Geophysical Research, 71: 785. Dungey, J.W., 1961. Interplanetary magnetic field and the auroral zones. Physics Review Letters, 6: 47. Feldstein, Y.I., and Starkov, G.V., 1967. Dynamics of the auroral belt and polar geomagnetic disturbances. Planetary and Space Science, 15: 209. Frank, L.A., and Craven, J.D., 1988. Imaging results from Dynamics Explorer 1. Reviews of Geophysics and Space Physics, 26: 249. Iijima, T., and Potemra, T.A., 1978. Large-scale characteristics of field-aligned currents associated with substorms. Journal of Geophysical Research, 83: 599. Milan, S.E., Wild, J.A., Grocott, A., and Draper, N.C., 2006. Spaceand ground-based investigations of solar wind-magnetosphereionosphere coupling. Advances in Space Research, 38: 1671–1677. Newell, P.T., Meng, C.-I., and Wing, S., 1998. Relation to solar activity of intense aurorae in sunlight and darkness. Nature, 393: 342. Rees, M.H., 1989. Physics and Chemistry of the Upper Atmosphere. Cambridge: Cambridge University Press. Russell, C.T., and McPherron, R.L., 1973. Semi-annual variation of geomagnetic activity. Journal of Geophysical Research, 78: 92. St
rmer, C., 1955. The Polar Aurora. Oxford: Clarendon Press. Vallance Jones, A., 1974. Aurora. Dordrecht: Reidel.

Cross-references Alfvén’s Theorem and the Frozen Flux Approximation Bartels, Julius (1899–1964) Geomagnetic Dipole Field Ionosphere Magnetic Field of Sun Magnetosphere of the Earth Storms and Substorms, Magnetic

B

BAKED CONTACT TEST Introduction At the time of emplacement, an igneous unit (intrusion or volcanic flow) heats adjacent host rocks. Upon cooling in the Earth’s magnetic field the igneous unit and adjacent host rocks acquire similar directions of remanent magnetization. The simple case of thermal overprinting of host rocks that have a consistent composition and magnetic mineralogy is illustrated in Figure B1. The host rock magnetization is completely reset in a “baked” zone adjacent to the contact, partially reset in a zone of “hybrid” remanence farther away, and unaffected in the “unbaked” zone at even greater distances.1 On the other hand, a later regional metamorphic event would, if sufficiently intense, produce a consistent remanence direction throughout the profile. Two types of baked contact tests utilize the paleomagnetic remanence across the contact of igneous units in order to establish whether the remanence in the igneous unit is primary. The standard baked contact test (or contact test) compares remanence directions in the igneous unit, the baked zone, and the unbaked zone (Figure B1). Although relatively robust, difficulties can arise with the interpretation of this test if chemical changes have occurred in the host rocks. A more rigorous test, herein called the baked contact profile test, includes a detailed study of overprinting in the hybrid zone (Figure B1). It can demonstrate conclusively that the overprint remanence was acquired as a thermoremanent magnetization (TRM) at the time of emplacement of the igneous unit. Both the standard baked contact and the more detailed baked contact profile tests are described and discussed below.

Importance of baked contact tests Well-dated paleomagnetic poles (see Pole, key paleomagnetic) are a prerequisite to defining reliable apparent polar wander paths, tracking the drift of continents, and establishing continental reconstructions. In general, key paleopoles should be demonstrated primary and the rock unit from which they are derived should be precisely and accurately dated (Buchan et al., 2001). In the Precambrian most key paleopoles are derived from igneous rocks because fossil evidence for the age of sedimentary rocks is usually lacking. Although there are several field tests of the primary nature of the magnetic remanence of igneous rocks (e.g., Buchan and Halls, 1990), the baked contact test is the most widely used.

1 In some publications the “baked”, “hybrid” and “unbaked” zones are referred to as the “contact”, “hybrid” and “host (magnetization)” zones, respectively.

In a recent review of the worldwide database of paleomagnetic poles for the 1700–500 Ma period, Buchan et al. (2001) concluded that only 18 paleopoles, of many hundreds that are available, are sufficiently well-dated to be used for robust continental reconstructions. All 18 are from well-dated igneous rocks. Ten of the 18 have a baked contact test or a baked contact profile test demonstrating that the remanence is primary, whereas eight are shown primary using other types of tests. This emphasizes the importance of igneous rocks and baked contact tests in paleomagnetic studies, especially in the Precambrian.

Baked contact test Brunhes (1906) first proposed a comparison of the remanence of an igneous unit with that of adjacent baked host rocks, concluding that if the remanence direction is similar in the two units it is stable and primary. Everitt and Clegg (1962) pointed out that the Brunhes test was incomplete because regional metamorphism will result in simultaneous remagnetization of both igneous and baked rocks. Therefore, they proposed an extension of the Brunhes test in which unbaked host rocks farther from the intrusion are also sampled. Everitt and Clegg used their test to look for a difference in remanence characteristics between baked and unbaked sedimentary host rocks. Today, the Everitt and Clegg (1962) test is commonly referred to as the baked contact test (or contact test) and applied mainly to a comparison of the direction of remanent magnetization in an igneous unit and its igneous or sedimentary host rocks.

Method The standard baked contact test requires collection of oriented paleomagnetic samples from the igneous unit, from baked host rocks and from unbaked host rocks in relatively close proximity to the igneous unit. In addition, the baked and unbaked host rocks should be of similar composition so that a direct comparison can be made of their rock properties. The baked contact test is “positive” (Figure B2a) and the remanence considered primary if the igneous unit and baked host rocks carry a consistent, stable direction of magnetization, whereas the unbaked host rocks have a stable but different direction. It is important to reiterate that the unbaked host rock must be stably magnetized. An example of a positive baked contact test is shown in Figure B3. Here a Marathon dyke crosscuts an older Matachewan dyke in the Superior Province of the Canadian Shield. The Matachewan dyke far from the Marathon contact carries a stable SSW remanence direction, whereas the Marathon dyke and adjacent baked Matachewan dyke have similar SE remanence

36

BAKED CONTACT TEST

Figure B1 Direction of remanent magnetization with distance from the contact of an igneous intrusion. It is assumed that the magnetic mineralogy of the host rock is similar throughout the profile and that overprinting in the baked and hybrid zones is a thermoremanent magnetization.

Figure B2 Baked contact tests. (a) Positive test. (b) Negative test. (c-h) Examples of inconclusive tests.

directions. This positive test indicates that the Marathon dyke carries a primary remanence from the time of its intrusion (Buchan et al., 1996). Comparing the magnetic properties of the unbaked and baked host rocks may provide information on whether chemical alteration has occurred in the baked rocks at the time of emplacement of the igneous unit. The presence of chemical changes can make the test less reliable. For example, consider the case of an igneous unit with unblocking temperatures that are lower than those of host rocks into which it is intruded. Suppose that the unblocking temperatures of host rocks adjacent to the igneous unit are lowered as a result of chemical changes

Figure B3 Example of a positive baked contact test for a ca. 2.10 Ga Marathon dyke which crosscuts a ca. 2.45 Ga Matachewan dyke in a roadcut at a locality in the Superior Province of the Canadian Shield. (a) Patterns indicate area of outcrop. Open and closed squares refer to samples for which typical Marathon and Matachewan remanence directions were obtained, respectively. (b) Paleomagnetic directions are plotted on an equal-area net. All magnetizations are directed up. Data are discussed in the text and in Buchan et al. (1996).

BAKED CONTACT TEST

that occur due to heating by the igneous unit. Then a later mild regional reheating event can reset the remanence of the igneous unit and adjacent host rocks without resetting the remanence of distant host rocks that have retained their higher unblocking temperatures. This situation would yield a positive baked contact test even though the remanence of the igneous unit is secondary. (Note that the more detailed baked contact profile test described below can eliminate any uncertainty about chemical changes by clearly establishing that the remanence in the baked zone is a TRM and that it was acquired at the time when the igneous unit was emplaced ). A positive test also demonstrates that the unbaked host rocks carry a remanence that significantly predates the emplacement of the igneous unit. The baked contact test is “negative” (Figure B2b) if the igneous unit and the baked and unbaked host all carry a similar stable direction of magnetization. A negative test usually indicates that the remanence of the igneous unit is secondary, having been acquired significantly after emplacement. However, there are specific situations in which the igneous unit carries a primary remanence, but the baked contact test is negative. These include cases where (a) the igneous unit and its host rocks are of similar age, (b) the host rocks acquired a regional overprint shortly before emplacement of the igneous unit, and (c) the direction of the Earth’s magnetic field was similar at the time of emplacement of the igneous unit and at some earlier time when the host rocks were magnetized. Therefore, although a remanence can usually be established as primary with a positive baked contact test, it cannot be demonstrated as secondary with the same degree of certainty by a negative baked contact test. Many baked contact tests do not meet the criteria described above for a positive or a negative test. In a few specific situations, some limited information concerning the age of the remanence in the baked zone can still be obtained. For example, if the igneous unit is unstably magnetized, but the baked and unbaked host rocks are stably magnetized and can be shown to have similar magnetic properties, similar remanence directions in the baked and unbaked host rocks likely indicate a secondary overprint, whereas dissimilar directions likely indicate that the remanence of the baked zone was acquired at the time of emplacement of the igneous unit. However, in most cases in which the criteria for a positive or negative test are not achieved, the test is “inconclusive” (e.g., Figure B2c-h) and does not provide information concerning the primary or secondary nature of the remanence.

Incorrect application of the baked contact test The baked contact test is often applied incorrectly or misinterpreted in the literature. In addition, terminology applied to the test is often confusing or misleading. Examples of these problems are discussed below. In some studies only the igneous unit and its baked zone are sampled and the remanence is interpreted to be primary based on similar remanence directions in these two units. However, as noted above, Everitt and Clegg (1962) described how this result would also be obtained if both units were overprinted at some later date. Such tests are incomplete and should not be referred to as baked contact tests because all three elements (igneous, baked host, and unbaked host rocks) that are necessary for a baked contact test have not been sampled. There are also many examples in the literature where an igneous unit and baked host rocks carry a consistent direction of magnetization but unbaked host is unstably magnetized (Figure B2c). Terms such as “semipositive test,” “not fully positive test,” or “partial test” are sometime used in such cases. However, these terms are misleading and should not be used, because they imply incorrectly that the test gives some information on whether the remanence of the igneous unit is primary. For example, host rocks that are unstably magnetized before emplacement of the igneous unit can acquire a stable remanence in the baked zone at the time of emplacement as a result of chemical alterations that occur only in the baked zone. During a subsequent regional metamorphism the igneous unit and baked host will be

37

remagnetized (with similar remanence directions), while the unbaked host remains unstably magnetized. This is especially likely if temperatures during the metamorphic event remain below that necessary to induce chemical changes in the unbaked host rocks. Thus, when unbaked host rocks carry an unstable remanence, the baked contact test is inconclusive and the age of the remanence in the igneous unit is uncertain. In other studies, the unbaked host rocks are collected at great distance from the igneous unit and its baked host rocks, often as a result of a lack of suitable outcrop in the intervening area. This can cause serious problems in the interpretation of the results. For example, overprinting of the local area near the igneous unit may have occurred after its emplacement, resulting in remagnetization of the igneous unit and its baked zone, whereas the distant unbaked site may have escaped overprinting at that time. In addition, it is more likely that magnetic mineralogy will differ between such widely separated baked and unbaked host rocks, making the comparison of the magnetic directions between the two locations more problematic.

Baked contact profile test Although the baked contact test described above is generally considered to be a robust test of primary remanence, there are some circumstances involving chemical alterations in the host rocks when it can be misleading. Buchan et al. (1993) and Hyodo and Dunlop (1993) pointed out that analyzing magnetic overprinting along a profile that includes the hybrid zone (Figure B1), following the procedure of Schwarz (1977), yields a more rigorous test of primary remanence than the standard baked contact test. In particular, such a baked contact profile test can demonstrate that the overprint component is a TRM and that it was acquired at the time of emplacement of the igneous unit (Buchan and Schwarz, 1987; Schwarz and Buchan, 1989; Hyodo and Dunlop, 1993). The baked contact profile test is the most powerful method for establishing that remanence in igneous rocks is primary. However, it is not widely used because of the necessity of sampling a continuous profile, the difficulty in locating the often-narrow hybrid zone, and the difficulty in analyzing the hybrid zone magnetizations. To date it has only been applied in a few instances to the study of dyke contacts (e.g., Schwarz, 1977; Buchan et al., 1980; Symons et al., 1980; Buchan and Schwarz, 1981; McClelland Brown, 1981; Schwarz and Buchan, 1982; Schwarz et al., 1985; Hyodo and Dunlop, 1993; Oveisy, 1998; Wingate and Giddings, 2000), usually as part of studies to determine ambient host rock temperatures and depth of burial (see Magnetization, Remanent, Ambient Temperature and Burial Depth from Dyke Contact Zones).

Method A continuous profile is sampled from the igneous unit, through the baked and hybrid zones into the unbaked host zone (see Figure B1). The distance of each sample from the contact is recorded, as are the dimensions of the igneous unit. Samples are thermally demagnetized in stepwise fashion in order to determine the maximum temperature (Tmax) to which each hybrid sample was reheated at the time of emplacement of the igneous unit. If the overprint component is a partial TRM (pTRM) resulting from heating by the igneous unit, the unblocking temperature spectra of the overprint component will occupy a range immediately below that of the earlier host component (Dunlop, 1979; McClelland Brown, 1982; Schwarz and Buchan, 1989; Dunlop and Özdemir, 1997). Any overlap of the two unblocking spectra should be explainable in terms of viscous pTRM. Tmax is given by the maximum unblocking temperature of the overprint component, most easily determined using an orthogonal component (or Zijderveld) plot of the horizontal and vertical component of the thermal demagnetization data. Tmax must be corrected for the effects of magnetic viscosity using published curves

38

BAKED CONTACT TEST

Figure B4 Positive baked contact profile test. Tmax, the maximum temperature that is obtained at a given location in the host rocks as the result of emplacement of the igneous unit, is determined from orthogonal component projections in the hybrid zone. A systematic decrease in the Tmax through the hybrid zone with increasing distance from the contact demonstrates that the overprint is a TRM.

for the appropriate magnetic mineral (e.g., Pullaiah et al., 1975). The procedure for analyzing magnetic components in the hybrid zone and determining Tmax is described in more detail in the article on Magnetization, remanent, application (q.v.). If the overprint component is a chemical remanent magnetization (CRM), Tmax cannot be determined. Schwarz and Buchan (1989) discuss how CRM components can be distinguished in hybrid zones. The baked contact profile test is “positive” (Figure B4) if the criteria for a positive baked contact test are satisfied (i.e., the igneous unit and baked host carry a consistent, stable direction of magnetization, whereas the unbaked host has a stable but different direction) and if Tmax values in the hybrid zone decrease systematically away from the contact, as predicted by heat conduction theory (Jaeger, 1964). The critical aspect of the test is that Tmax values decrease systematically across the hybrid zone. The most detailed comparisons of experimental and theoretical Tmax values across hybrid zones have been described by McClelland Brown (1981). A positive baked contact profile test demonstrates that the overprint remanence of the hybrid zone is a pTRM and that it was acquired as the igneous unit cooled following emplacement. The similarity between the direction of the overprint in the hybrid zone and that of the remanence in the igneous unit itself indicates that the latter is primary. The baked contact profile test is “negative” if the igneous unit and all elements of the host profile yield consistent remanence directions. Interpretation of the negative baked contact profile test is similar to that of the negative baked contact test described above. The test is “inconclusive” if the criteria for “positive” or “negative” test are not met. For example, if Tmax values cannot be obtained from the hybrid zone or if they do not decrease systematically with distance from the contact, then it cannot be concluded that the overprint is a TRM from the time of emplacement of the igneous unit. Kenneth L. Buchan

Bibliography Brunhes, B., 1906. Recherches sur la direction d’aimantation des roches volcaniques. Journal de Physique, 5: 705–724. Buchan, K.L., and Halls, H.C., 1990. Paleomagnetism of Proterozoic mafic dyke swarms of the Canadian Shield. In Parker, A.J.,

Rickwood, P.C., and Tucker, D.H., (eds.), Mafic Dykes and Emplacement Mechanisms. Balkema: Rotterdam, 209–230. Buchan, K.L., Mortensen, J.K., and Card, K.D., 1993. Northeasttrending Early Proterozoic dykes of southern Superior Province: multiple episodes of emplacement recognized from integrated paleomagnetism and U-Pb geochronology. Canadian Journal of Earth Sciences, 30: 1286–1296. Buchan, K.L., Halls, H.C., and Mortensen, J.K., 1996. Paleomagnetism, U-Pb geochronology, and geochemistry of Marathon dykes, Superior province, and comparison with the Fort Frances swarm. Canadian Journal of Earth Sciences, 30: 1286–1296. Buchan, K.L., and Schwarz, E.J., 1981. Uplift estimated from remanent magnetization: Munro area of Superior Province since 2150 Ma. Canadian Journal of Earth Sciences, 18: 1164–1173. Buchan, K.L., and Schwarz, E.J., 1987. Determination of the maximum temperature profile across dyke contacts using remanent magnetization and its application. In Halls, H.C., and Fahrig, W.F. (eds.), Mafic Dyke Swarms. Geological Association of Canada, Special Paper 34, pp. 221–227. Buchan, K.L., Ernst, R.E., Hamilton, M.A., Mertanen, S., Pesonen, L.J., and Elming, S.-Å., 2001. Rodinia: the evidence from integrated palaeomagnetism and U-Pb geochronology. Precambrian Research, 110: 9–32. Buchan, K.L., Schwarz, E.J., Symons, D.T.A., and Stupavsky, M., 1980. Remanent magnetization in the contact zone between Columbia Plateau flows and feeder dykes: evidence for groundwater layer at time of intrusion. Journal of Geophysical Research, 85: 1888–1898. Dunlop, D.J., 1979. On the use of Zijderveld vector diagrams in multicomponent paleomagnetic studies. Physics of the Earth and Planetary Interiors, 20: 12–24. Dunlop, D.J., and Özdemir, Ö., 1997. Rock magnetism: fundamentals and frontiers. Cambridge: Cambridge University Press, 573 pp. Everitt, C.W.F., and Clegg, J.A., 1962. A field test of paleomagnetic stability. Journal of the Royal Astronomical Society, 6: 312–319. Hyodo, H., and Dunlop, D.J., 1993. Effect of anisotropy on the paleomagnetic contact test for a Grenville dike. Journal of Geophysical Research, 98: 7997–8017. Jaeger, J.C., 1964. Thermal effects of intrusions. Reviews of Geophysics, 2(3): 711–716. McClelland Brown, E., 1981. Paleomagnetic estimates of temperatures reached in contact metamorphism. Geology, 9: 112–116. McClelland Brown, E., 1982. Discrimination of TRM and CRM by blocking-temperature spectrum analysis. Physics of the Earth and Planetary Interiors, 30: 405–411. Oveisy, M.M., 1998. Rapakivi granite and basic dykes in the Fennoscandian Shield; a palaeomagnetic analysis. Ph.D. thesis, Luleå University of Technology, Luleå, Sweden. Pullaiah, G., Irving, E., Buchan, K.L., and Dunlop, D.J., 1975. Magnetization changes caused by burial and uplift. Earth and Planetary Science Letters, 28: 133–143. Schwarz, E.J., 1977. Depth of burial from remanent magnetization: the Sudbury Irruptive at the time of diabase intrusion (1250 Ma). Canadian Journal of Earth Sciences, 14: 82–88. Schwarz, E.J. and Buchan, K.L., 1982. Uplift deduced from remanent magnetization: Sudbury area since 1250 Ma ago. Earth and Planetary Science Letters, 58: 65–74. Schwarz, E.J. and Buchan, K.L., 1989. Identifying types of remanent magnetization in igneous contact zones. Physics of the Earth and Planetary Interiors, 68: 155–162. Schwarz, E.J., Buchan, K.L., and Cazavant, A., 1985. Post-Aphebian uplift deduced from remanent magnetization, Yellowknife area of Slave Province. Canadian Journal of Earth Sciences, 22: 1793–1802. Symons, D.T.A., Hutcheson, H.I., and Stupavsky, M., 1980. Positive test of the paleomagnetic method for estimating burial depth using a dike contact. Canadian Journal of Earth Sciences, 17: 690–697.

BANGUI ANOMALY

Wingate, M.T.D., and Giddings, J.W., 2000. Age and paleomagnetism of the Mundine Well dyke swarm, Western Australia: implications for an Australia-Laurentia connection at 755 Ma. Precambrian Research, 100: 335–357.

Cross-references Magnetization, Remanent, Ambient Temperature and Burial Depth from Dyke Contact Zones Magnetization, Thermoremanent (TRM) Paleomagnetism Pole, Key Paleomagnetic

BANGUI ANOMALY “Bangui anomaly” is the name given to one of the Earth’s largest crustal magnetic anomalies and the largest over the African continent. It covers two-thirds of the Central African Republic and the name derives from the capital city Bangui that is near the center of this feature. From surface magnetic survey data, Godivier and Le Donche (1962) were the first to describe this anomaly. Subsequently high-altitude world magnetic surveying (see Aeromagnetic surveying) by the US Naval Oceanographic Office (Project Magnet) recorded a

39

greater than 1000 nT dipolar, peak-to-trough anomaly with the major portion being negative (Figure B5). Satellite observations (Cosmos 49) were first reported in 1964 (Benkova et al., 1973); these revealed a –40nT anomaly at 350 km altitude. Subsequently the higher altitude (417–499km) Polar Orbiting Geomagnetic Observatory (POGO) satellites data recorded peak-to-trough anomalies of 20 nT. These data were added to Cosmos 49 measurements by Regan et al. (1973) for a regional satellite altitude map. In October 1979, with the launch of Magsat (see Magsat), a satellite designed to measure crustal magnetic anomalies, a more uniform satellite altitude magnetic map was obtained (Girdler et al., 1992). From the more recent CHAMP (see CHAMP) satellite mission a map was computed at 400 km altitude, a greater than 16 nT anomaly was recorded (Figure B6/Plate 6c). The Bangui anomaly is elliptically shaped and is approximately 760 by 1000 km centered at 6
N, 18
E (see Magnetic anomalies, long wavelength). It is composed of three segments, with positive anomalies north and south of a large central negative anomaly. This displays the classic pattern of a magnetic anomalous body being magnetized by induction in a zero inclination field. This is not surprising since the magnetic equator passes near the center of the body. While the existence and description of the Bangui anomaly is wellknown, what is less established and controversial is the origin or cause that produced this large magnetic feature. It is not possible to discuss its origin without mentioning the other associated geophysical and geologic information. There is a 120 mGal Bouguer gravity anomaly coincident with the magnetic anomaly (Boukeke, 1994) and a putative

Figure B5 Total field aeromagnetic anomaly profile data over the Central African Republic from Project MAGNET, US Naval Oceanographic Office (from Regan and Marsh, 1982, their figure 3, Reproduced by permission of American Geophysical Union).

40

BARLOW, PETER (1776–1862)

Figure B6/Plate 6c Satellite altitude (400 km) scalar magnetic anomaly map of the Central African Republic region from CHAMP mission data. Anomaly maximum, minimum, standard deviation and contour interval are given on the figure (Hyung Rae Kim, UMBC and NASA/GSFC).

topographic ring some 810 km diameter associated with this feature (Girdler et al., 1992). In Rollin’s (1995) recently compiled tectonic/ geologic map of the Central African Republic, Late Archean and Early Proterozoic rocks are exposed beneath the central part of the anomaly. Lithologically the area is dominated by granulites and charnockites (a high temperature/pressure granite believed to be part of the lower crust). There are, in addition, significant exposures of greenstone belts and metamorphosed basalts with itabrite (a metamorphosed iron formation). There are several theories for the origin of the anomaly. Regan and Marsh (1982) proposed that a large igneous intrusion into the upper crust became denser on cooling and sank into the lower crust with the resulting flexure producing the overlying large basins of this region (see Magnetic anomalies for geology and resources). The intrusion the source of the magnetic anomaly; the sedimentary basin fill the source of the gravity anomaly. Another hypothesis is that it is the result of a large extraterrestrial impact (Green, 1976; Girdler et al., 1992). Ravat et al. (2002) applied modified Euler deconvolution techniques to the Magsat data and their analysis supports the impact model of Girdler et al. (1992). Unfortunately, it is not possible to discriminate between these theories based solely on geophysical data. However, the key to the solution may lie in the origin of carbonados (microcrystalline diamond aggregates). Carbonados are restricted to the Bahia Province, Brazil and the Central African Republic, with the latter having a greater number. Smith and Dawson (1985) proposed that a meteor impacting into carbon-rich sediment produced these microdiamonds. More recently De et al. (1998) and Magee (2001) have failed to confirm this hypothesis. The origin of this large crustal anomaly remains uncertain. Patrick T. Taylor

Bibliography Benkova, N.P., Dolginow, S.S., and Simonenko, T.N., 1973. Residual magnetic field from the satellite Cosmos 49. Journal of Geophysical Research, 78: 798–803. Boukeke, D.B., 1994. Structures crustales D’Afrique Centrale Déduites des Anomalies Gravimétriques et magnétiques: Le domaine précambrien de la République Centrafricaine et du SudCameroun. ORSTOM TDM 129. De, S., Heaney, P.J., Hargraves, R.B., Vicenzi, E.P., and Taylor, P.T., 1998. Microstructural observations of polycrystalline diamond: a

contribution to the carbonado conundrum. Earth and Planetary Science Letters, 164: 421–433. Godivier, R., and Le Donche, L., 1962. Réseau magnétique ramené au 1er Janvier 1956: République Centrafricaine, Tchad Méridonial. 19 pages: 6 maps 1:2,500,000, Cahiers ORSTOM/Geophysique, No. 1. Girdler, R.W., Taylor, P.T., and Frawley, J.J., 1992. A possible impact origin for the Bangui magnetic anomaly (Central Africa). Tectonophysics, 212: 45–58. Green, A.G., 1976. Interpretation of Project MAGNET aeromagnetic profiles across Africa. Journal of the Royal Astronomical Society, 44: 203–208. Magee, C.W. Jr., 2001. Constraints on the origin and history of carbonado diamond. PhD thesis, The Australian National University, Canberra. Ravat, D., Wang, B., Widermuth, E., and Taylor, P.T., 2002. Gradients in the interpretation of satellite-altitude magnetic data: and example from central Africa. Journal of Geodynamics, 33: 131–142. Regan, R.D., Davis, W.M., and Cain, J.C., 1973. The detection of “intermediate” size magnetic anomalies in Cosmos49 and 0602.4.6 data. Space Research, 13: 619–623. Regan, R.D., and Marsh, B.D., 1982. The Bangui magnetic anomaly: Its geological origin. Journal of Geophysical Research, 87: 1107–1120. Rollin, P., 1995. Carte Tectonique de République Centrafricaine. Université de Besançon. Smith, J.V., and Dawson, J.B., 1985. Carbonado: diamond aggregates from early impacts of crustal rocks? Geology, 13: 342–343.

Cross-references Aeromagnetic Surveying CHAMP Crustal Magnetic Field Magnetic Anomalies for Geology and Resources Magnetic Anomalies, Long Wavelength Magsat

BARLOW, PETER (1776–1862) A British mathematician and physicist, born at Norwich, England, Peter Barlow is now remembered for his mathematical tables, the Barlow wheel and Barlow lens. His contributions to science in general and magnetism in particular are most impressive. We will concentrate here chiefly on his contributions in direct relation with geomagnetism, which are too often not given the attention they deserve. Despite lacking formal education, Peter Barlow became assistant mathematical master at the Royal Military Academy in Woolwich in 1801. He was promoted to a professorship in 1806 and worked in Woolwich until retiring in 1847. His first researches were mainly focused on pure mathematics (his “Theory of Numbers” appeared in 1811), but in 1819 he began to work on magnetism. In May 1823, Peter Barlow was elected fellow of the Royal Society. He later also became a member of several of the leading overseas societies (including correspondant of the French Académie des Sciences in 1828). He worked on problems associated with magnetic mesurements and the issue of deviation in ship compasses caused by iron pieces in the hull. In 1825, he was awarded the Royal Society Copley Medal for his method of correcting the deviation by juxtaposing the compass with a suitably shaped piece of iron used as neutralizing plate. Guided by a suggestion from John Herschel, Peter Barlow conducted experiments on the influence of rotation upon magnetic and non-magnetic bodies. In a letter to Major Colby dated December 20, 1824, he relates: “Having been lately speculating on the probable causes of the earths magnetic polarity. It occured to me that it might possibly be due to the rotation, and if so the same ought to be the case

41

BARLOW, PETER (1776–1862)

with any revolving mas of iron. I therefore fixed one of our 13 inch shells upon one of the turning lathes in the arsenal driven by the steam engine, and the very few trials were most conclusive and satisfactory.”

magnetism, than does this artificial globe (. . .) I may therefore, I trust, be allowed to say, that I have proved the existence of a force competent to produce all the phenomena of terrestrial magnetism, without the aid of any body usually called magnetic.”

The next year, in the Philosophical Transactions, Peter Barlow describes how the experimental mesurements were made extremely difficult because of the disturbing influence of the lathe and other machinery on the needle. After careful investigations, he reports negative conclusions:

This interpretation of the principal geomagnetic field clearly represents the premise of present dynamo theory. Barlow’s globe was originally constructed in 1824; this experiment yielded a teaching apparatus still preserved in some universities around the world (see Figure B7). Peter Barlow also did work with geomagnetic observations, in 1833 he constructed a new declination chart (then called “variation” chart) in which he embraced earlier magnetic observations. This chart is illustrated and described in the Philoposphical Transactions in 1833. Barlow notes that the lines of equal variation (following the terminology introduced by E. Halley in his original 1701 chart) are very regular, denoting the deep origin of these structures. Barlow also discusses the evolution of these lines in time by comparison to previous charts (i.e., the secular variation). Barlow concludes his opus by noting that he shall be most happy if this

“I have certainly found a stronger effect produced by rotation than I anticipated, yet it does not appear to be of a kind to throw any new light upon the difficult subject of terrestrial magnetism. I think there are strong reasons for assuming, that the magnetism of the earth is of that kind which we call induced magnetism; but at present we have no knowledge of the inductive principle, (. . .)” Years later, Lord Blackett (q.v.) revisited this possibility with similar conclusions (Blackett, 1952). Following on Öersted’s discovery of the magnetism associated with electrical current (Experimenta circa effectum Conflictus Electrici in Acum Magneticam, 1820), the French physicist André-Marie Ampère proposed (Annales de chimie et de physique, 1820) that electrical currents within the Earth could account for the geomagnetic field (these currents were then assumed to be of galvanic origin). Barlow was the first to test the practicability of Ampère’s proposal and designed a remarkable experiment to that end. This experiment is presented in the Philoposphical Transactions for 1831. Barlow built a wooden hollow globe 16 in. in diameter and cut grooves in it. A copper wire was placed around the sphere along the grooves in the manner of a solenoid. When this globe is connected to a powerful galvanic battery, current passing through the coils sets up a dipolar magnetic field. Barlow describes how, if one turns “(. . .) the globe so as to make the pole approach the zenith, the dip will increase, till at the pole itself the needle will become perfectly vertical. Making now this pole recede, the dip will decrease, till at the equator it vanishes, the needle becoming horizontal. (. . .) Nothing can be expected nor desired to represent more exactly on so small a scale all the phenomena of terrestrial

Figure B7 Teaching instrument, based on Barlow’s sphere, used to demonstrate how a current passing through a coil produces a dipolar field similar to that of the Earth. [The Physics Museum of the University of Coimbra, CAT. 1851: 25.O.III, 39  25.8  41, wood, brass, and copper. Photography: Joa˜o Pessoa-Divisa˜o de Documentac¸a˜o Fotogra´fica do Instituto Portugueˆs de Museus].

“labour should furnish the requisite data for either a present or future development of those mysterious laws which govern the magnetism of the terrestrial globe, an object as interesting in philosophy as it is important in navigation.” Peter Barlow died in March 1862 in Kent, England.

Acknowledgments Figure reproduced by permission of João Pessoa (Divisão de Documentação Fotográfica do Instituto Português de Museus) and the Physics Museum of the University of Coimbra. Emmanuel Dormy

Bibliography Barlow, P., 1824. Letter to Major Colby at the Royal Military Academy dated dec. 20th 1824. Archives of the Royal Society, HS.3. 287. Barlow, P., 1825. On the temporary magnetic effect induced in iron bodies by rotation, In a Letter to J.F.W. Herschel. Philosophical Transactions, 115: 317–327. Barlow, P., 1831. On the probable electric origin of all the phenomena of terrestrial magnetism; with an illustrative experiment. Philosophical Transactions, 121: 99–108. Barlow, P., 1833. On the present situation of the magnetic lines of equal variation, and their changes on the terrestrial surface. Philosophical Transactions, 123: 667–673. Blackett, P.M.S., 1952. A Negative Experiment Relating to Magnetism and the Earth’s Rotation. Philosophical Transactions, A245: 309–370. Ingenuity and Art, 1997. A collection of Instruments of the Real “Gabinete de Física”. Catalogue of the Physics Museum of the University of Coimbra (Portugal). Mottelay, P.F., 1922. Biographical History of Electricity and Magnetism. London: Charles Griffin & Company Limited. Obituary notices of fellows deceased, Proceedings of the Royal Society, 1862–1863, 12: xxxiii–xxxiv.

Cross-references Blackett, Patrick Maynard Stuart, Baron of Chelsea (1897–1974) Geodynamo Geomagnetic Secular Variation Halley, Edmond (1656–1742)

42

BARTELS, JULIUS (1899–1964)

BARTELS, JULIUS (1899–1964) Amongst the 20th century scientists working in the field of geomagnetism, Julius Bartels was certainly one of the outstanding contributors. He is known to most geophysicists as the author, along with Sydney Chapman (q.v.), of the two-volume monograph Geomagnetism, which used to be the bible of geomagnetism for several decades. Born on August 17, 1899 in Magdeburg (Germany), his life and scientific career was intimately related with the development of the field of geophysics to become a scientific discipline of its own. Julius Bartels studied mathematics and physics at the University of Göttingen as a student of Emil Wiechert as well as David Hilbert, Max Born, and James Franck. In 1923 he received his Ph.D. with a dissertation on “New methods of the calculation and display of daily pressure variations, especially during strong nonperiodic oscillations,” which was supervised by Wilhelm Meinardus. Also his habilitation thesis “On atmospheric tides,” which he submitted to the University of Berlin, dealt with the determination of tidal-like oscillations of the atmosphere using long-term pressure observations. Trained as a mathematician Julius Bartels always had a keen interest in developing and applying statistical methods to geophysical problems. He became interested in geomagnetism when studying daily variations of the geomagnetic field caused by tidal oscillations of the ionosphere. His teacher in this new field was Adolf Schmidt whose assistant he became in 1923 after he graduated from Göttingen. In 1929 he was offered professorship in meteorology and geophysics at the Forestry College in Eberswalde, close to Berlin. After several years of teaching there he took over the chair of geophysics at the University of Berlin and became the director of the Geomagnetic Observatory in Potsdam, which later became the Observatory of Niemegk. During his years in Potsdam, Julius Bartels laid the foundation for our current quantitative knowledge about geomagnetic variations and their relation to solar activity. With these statistical methods, he was able to disprove many widely accepted geophysical models on the one hand and develop some very useful and easy to apply geophysical indices on the other. In particular, the K-index and its planetary counterpart, KP, have been developed in these years. The KP index was, and partially still is, the key index to describe geomagnetic activity. Eliminating regular variations of the Earth’s magnetic field, Bartels demonstrated the impact of solar wave and particle radiation on the geomagnetic field. He recognized that seizing this concept in numbers would form a powerful tool for scientists in the research of solarterrestrial relations. Bartels realized that his KP tables would only be accepted by the scientific community if they were presented in a clearly arranged manner. The striking success of the KP index is also due to its appealing representation as a kind of musical diagram. Using only geomagnetic data and statistical methods, Bartels claimed the existence of M-regions on the Sun that are responsible for geomagnetic activity. The claimed M-regions were later identified as coronal holes, which are the source of the fast solar wind that intensifies geomagnetic activity. Between 1931 and 1940 Bartels also worked as a research associate at the Department of Terrestrial Magnetism at the Carnegie Institution (Washington, D.C.) for longer periods of time, where he closely cooperated with his friend Sydney Chapman. In 1940, the first edition of their book Geomagnetism was printed. After the Second World War, Bartels accepted an offer from the University of Göttingen to take over the chair of geophysics there, succeeding Gustav Angenheister. In 1956 Julius Bartels also became the director of the Institute for Stratospheric Research of the Max-Planck-Institut für Aeronomie, now the Max-Planck-Institute for Solar System Research. Following the tradition of Carl-Friedrich Gauss (q.v.), Wilhelm Weber, and Alexander von Humboldt (q.v.), who initiated the “Göttinger Magnetischer Verein,” Julius Bartels strongly favored international cooperation. He was one of the initiators of the International Geophysical Year 1957/1958, served as the president of the International Association of Geomagnetism and Aeronomy (q.v.) from 1954 to 1957, and became

the vice-president of the International Union of Geodesy and Geophysics in 1961. Julius Bartels was an elected member of the Academies in Berlin and Göttingen, the Deutsche Akademie der Naturforscher und Ärzte Leopoldina in Halle, the Royal Astronomical Society in London, and the International Academy of Astronautics in Paris. His outstanding scientific contribution to the field of geomagnetism was honored by the Emil Wiechert Medal of the Deutsche Geophysikalische Gesellschaft, the Charles Chree Medal of the British Physical Society, and the William Bowie Medal of the American Geophysical Union. His published work spans 23 monographs and 143 scientific papers. Julius Bartels died on March 6, 1964 in Göttingen. Karl-Heinz Glaßmeier and Manfred Siebert

Bibliography Kertz, W., 1971. Einführung in die Geophysik II, Mannheim: Bibliographisches Institut. Siebert, M., 1964. Julius Bartels, Gauss-Gesellschaft e. V., Göttingen.

Cross-references Chapman, Sydney (1888–1970) Gauss, CarlFriedrich (1777–1855) Humboldt, Alexander Von (1759–1859) IAGA, International Association of Geomagnetism and Aeronomy Magnetosphere of the Earth Periodic External Fields

BAUER, LOUIS AGRICOLA (1865–1932) Louis Agricola Bauer (January 26, 1865 to April 12, 1932) founded and directed the Department of Terrestrial Magnetism (DTM) of the Carnegie Institution of Washington from 1904 until the 1920s. (See Carnegie Institution of Washington, Department of Terrestrial Magnetism). Seizing the opportunity provided by a large bequest by Andrew Carnegie, Bauer conceived of a tightly structured, worldwide magnetic survey that would be completed in a single generation. This goal was largely achieved by the time he was replaced as director of the DTM by John Adam Fleming (q.v.). Bauer was born in Cincinnati, Ohio, as the son of German immigrants Ludwig Bauer and Wilhelmina Buehler. Bauer was the sixth of nine children. He studied civil engineering at the University of Cincinnati, graduating in 1888 and moving directly into a position as an aide in the US Coast and Geodetic Survey’s computing division. Here he worked under Charles Anthony Schott, a German immigrant of 1848 who had dedicated his career at the Survey to geomagnetism. Under Schott, Bauer learned how to use magnetic instruments and how to reduce magnetic data. After four years at the Survey, Bauer enrolled in 1892 at the University of Berlin, Germany, to earn a Ph.D. in physics and learn the theoretical background to geomagnetic research. There he studied under Max Planck, Wilhelm von Bezold, and Wilhelm Foerster, and in 1895 successfully defended a dissertation on geomagnetic secular variation. Many geomagnetic researchers are still familiar with “Bauer plots” of declination versus inclination, which he introduced in 1895. These appear to indicate a cyclical behavior of the magnetic field (Bauer, 1895; Good, 1999). While at the University of Berlin, Bauer worked as an observer at the magnetic observatory at Telegraphenberg, Potsdam. While a student, he met Adolf Schmidt, one of the most important geomagnetic researchers of the time, with whom he collaborated throughout his career. Bauer returned to the United States as one of the country’s first selfproclaimed geophysicists, with a prestigious Ph.D. from Germany. In 1896 he founded the international journal Terrestrial Magnetism, which became the Journal of Geophysical Research in 1948.

43

BAUER, LOUIS AGRICOLA (1865–1932)

He taught physics and mathematics at the University of Chicago and the University of Cincinnati in the late 1890s and then returned to the US Coast and Geodetic Survey in 1899 to direct its new division of terrestrial magnetism. Although he periodically taught a course on geomagnetism at the Johns Hopkins University in Baltimore, his move to the Survey marked a permanent transition away from academia and toward a life dedicated to researching geomagnetism full time. He was following, as he saw it, the tradition of Edmond Halley (q.v.), Alexander von Humboldt (q.v.), and Carl Friedrich Gauss (q.v.). His goal was to understand Earth’s magnetism in all its facets, including the origin of the main field, secular variation, and shorter term variations due to “cosmic” influences. The Coast and Geodetic Survey gave him the opportunity to develop an ambitious magnetic survey across the broad expanse of the United States and also the opportunity to elaborate a chain of magnetic observatories, which would allow him to investigate the variation of the field on different timescales. Here he also learned to manage an instrument shop, a computational division, field operations, cartographers, and publications. He soon deployed these skills on a global scale. In 1902, Andrew Carnegie endowed the Carnegie Institution of Washington to encourage scientific research beyond the established disciplines. Bauer saw an opportunity in this to move beyond the borders of academic and government science and to address for the first time the global nature of geomagnetism. He established the Department of Terrestrial Magnetism (q.v.) at the Carnegie Institution in 1904. Bauer intended the DTM to observe Earth’s magnetism everywhere that others were not. This meant Asia, Africa, South America, and the polar and ocean regions. The DTM was also to act as a coordinating and standardizing bureau to guarantee that data obtained by all observers would be intercomparable. The DTM under Bauer completed the first “World Magnetic Survey,” sending out 200 land-based survey teams and circling the globe numerous times with the oceangoing magnetic observatories, the ships Galilee and Carnegie (q.v.) (see Figures B8 and B9). He also established magnetic observatories

Figure B9 Louis Agricola Bauer, portrait presented to the crew of the Carnegie on its maiden voyage in 1909. Bauer was noted for his sense of duty. Photo courtesy of Carnegie Institution of Washington. at Huancayo, Peru, and Watheroo, Western Australia. The survey was nearly complete by 1929, when the Carnegie burned at Apia, Samoa (Good, 1994). Bauer saw the need to professionalize and coordinate geomagnetic research and wanted to see progress made both in observational programs and in theory. Toward the end of his career, he participated extensively in the formation of the International Union of Geodesy and Geophysics, and served as general secretary (1919–1927) and as president (1927–1930) of the Section of Terrestrial Magnetism and Electricity (now the International Association of Geomagnetism and Aeronomy or IAGA, q.v.). His retirement from the DTM was gradual, as he experienced increasingly intense periods of depression from 1924 on. Fleming was “acting director” from 1927 until 1932. In 1932, the Washington Post reported Bauer’s death (by suicide) on the front page. Bauer’s influence on geomagnetic research in the early 20th century was immeasurable. Gregory A. Good

Bibliography Bauer, L.A., 1895. On the secular motion of a free magnetic needle. Physical Review, 2: 456–465; 3: 34–48. Good, Gregory A., 1994. Vision of a global physics: the Carnegie Institution and the first world magnetic survey. History of Geophysics, 5: 29–36. Good, Gregory A., 1999. Louis Agricola Bauer. In John A. Garraty and Mark C. Carnes (eds.), American National Biography. 24 vols. New York: Oxford University Press, Vol. 2, pp. 349–351.

Cross-references

Figure B8 Louis Agricola Bauer, observing the horizontal intensity of Earth’s magnetic field onboard the DTM vessel Galilee in 1906. Courtesy of Carnegie Institution of Washington, Department of Terrestrial Magnetism Archives. Photo 0.4.

Carnegie Institution of Washington, Department of Terrestrial Magnetism Carnegie, Research Vessel Fleming, John Adam (1877–1956) Gauss, Carl Friedrich (1777–1855) Halley, Edmond (1656–1742) Humboldt, Alexander von (1759–1859) IAGA, International Association of Geomagnetism and Aeronomy

44

BEMMELEN, WILLEM VAN (1868–1941)

BEMMELEN, WILLEM VAN (1868–1941) Dutch geophysicist and meteorologist Willem van Bemmelen was born on August 26, 1868 in Groningen (the Netherlands) as the son of Prof. Dr. Jacob Maarten van Bemmelen and Maria Boeke. As a student of physics and mathematics in his early twenties, he became interested in the Earth’s magnetic field through the work of Christopher Hansteen (q.v.) (Hansteen, 1819), who had derived isogonic charts and a geomagnetic hypothesis from historical magnetic measurements gathered from ship’s logbooks. Aware of Hansteen’s limited coverage of the seventeenth century and the period before 1600 and having encountered an abundance of original manuscript sources in various Dutch archives during an attempt to construct a secular variation curve for the city of Utrecht, Van Bemmelen decided to fill the gap with all useful data he could find from the period 1540–1690. The results he initially laid down in his Ph.D. thesis (Bemmelen, 1893), which tabulated the original declination observations gleaned from 38 ships’ logbooks (169 on land, about 1000 at sea). The thesis moreover included secular variation curves for 19 cities and six isogonic reconstructions, at 1540, 1580, 1610, 1640, 1665, and 1680. After receiving his doctorate, he extended this research during the next three years, covering additional archives in the Netherlands (1893–1896), London (early 1896), and Paris (summer 1897). Later that year he sailed to the East Indies, where he was appointed director of the Dutch Royal Magnetical and Meteorological Observatory at Batavia (Djakarta). There he met his future wife, Soetje Hermanna de Iongh (November 6, 1876, Pankal Pinang to July 5, 1969, Driebergen), whom he married on January 10, 1899. That same year appeared Van Bemmelen’s “Die Abweichung der Magnetnadel” (Bemmelen, 1899), a substantial extension of the earlier compilation, which contained 388 observations on land, 5276 at sea (about 60% from original sources), 20 small isogonic charts for the period 1492–1740, and six larger ones on Mercator projection for 1500, 1550, 1600, 1650, and 1700. A decade later, he published a geomagnetic survey of the Dutch East Indies (1903–1907), which included the readings obtained at the Batavia observatory (Bemmelen, 1909). In the next decade, he directed his attention increasingly to meteorological investigations in the Dutch colony (notably rain and wind). He eventually returned to his native country, where he died at The Hague on January 28, 1841. His collected declinations have since become incorporated in the world’s largest compilation of historical geomagnetic data, held at the World Data Center C1 at Edinburgh, United Kingdom. Art R.T. Jonkers

Bibliography Bemmelen, W. van, 1893. De Isogonen in de XVIde en XVIIe Eeuw. Ph.D. thesis, Leyden University. Utrecht: De Industrie. Bemmelen, W. van, 1893. Über Ältere Erdmagnetische Beobachtungen in der Niederlanden, Meteorologische Zeitschrift, 10 (February 1893), 49–53. Bemmelen, W. van, 1899. Die Abweichung der Magnetnadel: Beobachtungen, Säcular-Variation, Wert- und Isogonensysteme bis zur Mitte des XVIIIten Jahrhunderts. Supplement to Observations of the Royal Magnetical and Meteorological Observatory, Batavia, 21: 1–109. Batavia: Landsdrukkerij (Government Printing Office). Bemmelen, W. van, 1909. Magnetic Survey of the Dutch East Indies, 1903–1907. Observations made at the Royal Magnetical and Meteorological Observatory at Batavia, 30, appendix 1. Batavia: Landsdrukkerij. Hansteen, C., 1819. Untersuchungen über den Magnetimus der Erde. Christiania: Lehmann & Gröndahl.

Cross-references Geomagnetism, History of Hansteen, Christopher (1784–1873) Voyages Making Geomagnetic Measurements

BENTON, E. R. Edward R. Benton, known to all as “Ned”, is remembered for both his geophysical research and administration. He began his career as an applied mathematician, receiving the PhD degree in that discipline from Harvard University in 1961 under the supervision of Professor Arthur E. Bryson, Jr. Ned’s early research focused on aerodynamics, beginning with a dissertation on “Aerodynamic origin of the magnus effect on a finned missile”. Following a year as lecturer in mathematics at the University of Manchester, Ned moved to Boulder, Colorado, taking a position as a staff scientist at the National Center for Atmospheric Research (NCAR) in the fall of 1963. In 1964 he became a lecturer in the Department of AstroGeophysics (now the Department of Astrophysical, Planetary and Atmospheric Sciences) at the University of Colorado (UC), thus beginning an employment oscillation between NCAR and UC that lasted through 1977. Milestones along the way included academic appointments at UC to assistant professor in 1965, to associate professor in 1967, and Full Professor in 1971, and serving as assistant director of the NCAR Advanced-Study Program from 1967 to 1969, departmental chairman from 1969 to 1974, special assistant to the president for university relations of the University Corporation for Atmospheric Research (the parent organization for NCAR) from 1975 to 1977 and associate director of the Office of Space Science and Technology of UC from 1986 to 1987. Ned’s transition from engineer to geophysicist began with the publication in 1964 of a study of zonal flow inside an impulsively started rotating sphere. This work blossomed into a series of fundamental studies of the combined effects of rotation and magnetic fields on confined fluids, culminating in 1974 with a review of spin up (Benton, 1974). During this period, Ned also had a secondary interest in solutions of Burghers’ equation. The next geophysically related area in which Ned took an interest was “kinematic dynamo theory,” with a series of three papers published in 1975–1979 on Lortz-type dynamos. These papers developed a systematic categorization of these simple kinematic dynamos and provided some useful insights into their possible structures. This work was not continued as Ned’s interest was soon drawn in other directions. The work up until 1979 was prelude to his major scientific contribution in the area of inversion of the geomagnetic field. In that year he published a burst of papers on this topic, highlighted by papers on “Magnetic probing of planetary interiors” and “Magnetic contour maps at the core-mantle boundary” (Benton 1979a, b). In the mid-60s, workers had peered through this geomagnetic window on the Earth’s core, but when it was shown that the view was both incomplete and flawed, interest in this subject had languished. Following Ned’s revitalization of the subject, it has flourished, with contour maps of magnetic field and velocity at the top of the core being published by a number of groups. Now, the accurate determination of these features is a vital part of solving the dynamo problem. Many know Ned best for his service as chairman of Study of the Earth’s Deep Interior (SEDI) from 1987 to 1991. SEDI began as an idea at the IAGA Scientific Assembly in Prague, Czechoslovakia, in August, 1985, when IAGA Working Group I-2 (on theory of planetary magnetic fields and geomagnetic secular variation) called for the creation of a project to study the Earth’s core and lower mantle. Ned cochaired the approach to IAGA and IUGG which led to SEDI’s adoption, and served as its first chairman from 1987 to 1991. During his term as chairman, SEDI held symposia at Blanes, Spain, in June 1988, and Santa Fe, New Mexico in August 1990, as well as a large number of special sessions at various meetings of AGU, EGS, IAGA and IASPEI. Also, a number of national SEDI groups were formed, including those in Britain, Canada,

45

BINGHAM STATISTICS

France, Japan and the United States. Several of these groups were successful in instituting national scientific projects with the help of SEDI. Also, SEDI endorsed several new scientific projects, including ISOP, INTERMAGNET, and the Canadian GGP. All these activities were a great boon for the deep-earth geoscientific community and helped these studies maintain their visibility and viability. This success was due in large part to Ned’s steady hand at the helm during SEDI’s crucial formative years. Ned received a number of honors for his scientific research and service, including NASA’s Group Achievement Award in 1983, a scholarship from the Cecil H. and Ida M. Green Foundation for Earth Sciences, and election as Fellow of the American Geophysical Union shortly before his untimely death. The loss of his scientific talents, leadership ability, and companionship is still felt strongly by his many colleagues. David Loper

Bibliography Benton E.R. 1974. Spin-up. Annual Review on Fluid Mechanics, 6: 257–280. Benton E.R. 1979a. Magnetic probing of planetary interiors. Physics of the Earth and Planetary Interiors, 20: 111–118. Benton E.R. 1979b. Magnetic contour maps at the core-mantle boundary. Journal of Geomagnetism and Geoelectricity, 31: 615–626.

Let us now examine some of the properties of the Bingham distribution. The density function (Eq. 2) is clearly antipodally symmetric: xÞ ¼ pb ð^ xÞ: pb ð^

(Eq. 5)

The distribution is also invariant for any change in the sum of the concentration parameters, X X km ! km þ k0 ; (Eq. 6) m

m 0

where k is an arbitrary real number. Therefore, for specificity, we set the largest parameter equal to zero, and we choose an order for the other two, so that k1  k2  k3 ¼ 0:

(Eq. 7)

Thus, the Bingham distribution is a two-parameter distribution. For all possible values of k1 and k2 , the density (Eq. 2) describes a wide range of distributions on the sphere (see Figure B10). In spherical coordinates of inclination I and declination D the Bingham density function reduces to xjk1 ; k2 ; e1 ; e2 Þ ¼ pb ð^

Cross-references

1 F ðk1 ; k2 Þ h i exp k1 ð^ x  e1 Þ2 þ k2 ð^ x  e2 Þ2 cos I ;

IAGA, International Association of Geomagnetism and Aeronomy SEDI, Study of the Earth’s Deep Interior

(Eq. 8) where the unit data vectors are given by ^x1 ¼ cos I cos D;

BINGHAM STATISTICS When describing the dispersion of paleomagnetic directions expected to have antipodal symmetry, it is a standard practice within paleomagnetism to employ Bingham (1964, 1974) statistics. The Bingham distribution that forms the basis of the theory is derived from the intersection of a zero-mean, trivariate Gaussian distribution with the unit sphere. For full-vector Cartesian data x ¼ ðx1 ; x2 ; x3 Þ the Gaussian density function is
 1 1 T 1 x pg ðxjCÞ ¼ exp  C x ; (Eq. 1) 2 ð2pÞ3=2 jCj1=2

^x2 ¼ cos I sin D;

^x3 ¼ sin I ;

(Eq. 9)

where the unit eigenvectors are given by m m ^em 1 ¼ cos I cos D ;

m m ^em 2 ¼ cos I sin D ;

m ^em 3 ¼ sin I ; (Eq. 10)

and where the normalization function is Fðk1 ; k2 Þ ¼

1 pffiffiffi X Gði þ 12ÞGð j þ 12Þ ki1 ki2 : p i!j! Gði þ j þ 32Þ i;j¼0

(Eq. 11)

where C is a covariance matrix. But for directional data the Cartesian vectors ^x ¼ ð^x1 ; ^x2 ; ^x3 Þ are of unit length, and the corresponding density function is Bingham’s distribution   pb x^jEKET ¼

 T  1 exp ^ x EKET ^ x : F ðk1 ; k2 ; k3 Þ

(Eq. 2)

The matrix E is defined by the eigenvectors em of the covariance matrix C (see Principal component analysis for paleomagnetism). The diagonal concentration matrix 0 1 k1 0 0 (Eq. 3) K ¼ @ 0 k2 0 A; 0 0 k3 is formed from the Bingham concentration parameters km . Normalization of (Eq. 2) is obtained through the confluent hypergeometric function, represented here as a triple sum (see Abramowitz and Stegan, 1965):       1 X G i þ 12 G j þ 12 G k þ 12 ki1 kj2 kk3   ; F ðk1 ; k2 ; k3 Þ ¼ i!j!k ! G i þ j þ k þ 32 i;j;k ¼0 (Eq. 4)

Figure B10 Bingham density function, with representative contours for a (a) uniform density k1 ¼ k2 ¼ 0, (b) symmetric bipolar density k1 < k2  0, (c) asymmetric bipolar density k1 < k2  0, (d) asymmetric girdle k1  k2 < 0, and (e) symmetric girdle density k1  k2 ¼ 0. (After Collins and Weiss, 1990).

46

BINGHAM STATISTICS

Three important special cases are worthy of attention, which are most clearly illustrated in the coordinate system determined by the eigenvectors em. First, for k1 ¼ k2 we have a axially symmetric bipolar distribution, with density function p3 ðyjkÞ ¼

  1 exp k cos2 y sin y; for k  0; F ðkÞ

(Eq. 12)

(see (Eq. 19) of Statistical methods for paleomagnetic vector analysis.) The angle y is defined by the directional datum ^x and the principal axis determined by the eigenvector with the largest eigenvalue e3 cos y ¼ ^x  e3 ;

(Eq. 13)

normalization is given by F ðkÞ ¼

1 X Gði þ 1Þ ki 2

i¼0

Gði þ 32Þ i!

:

(Eq. 14)

Second, for k1 < k2 ¼ 0 the distribution is a axially symmetric girdle, with density function (Dimroth 1962; Watson 1965) p1 ðyjkÞ ¼

  1 exp k cos2 y sin y; F ðkÞ

for

k0

(Eq. 15)

The angle y is defined by the directional datum ^x and the principal axis determined by the eigenvector with the smallest eigenvalue e1 cos y ¼ ^x  e1 :

(Eq. 16)

These two special cases of the more general Bingham distribution are useful for describing the dispersion (Eq. 12) of bipolar data about some mean pole, and the dispersion (Eq. 15) of bipolar data about some mean plane. And, finally, the third symmetric distribution is uniform, obtained for k1 ¼ k2 ¼ 0. In spherical coordinates this is just p0 ðyjkÞ ¼ sin y;

for

k ¼ 0:

(Eq. 17)

The uniform distribution has no preferred direction and so y can be measured from an arbitrary axis. Of course, if we now allow k to be positive or negative (or zero), then the full range of axially symmetric density functions is available (Mardia, 1972, p. 234): p1;3 ðyjkÞ ¼

  1 exp k cos2 y sin y; F ðkÞ

for

 1  k  1: (Eq. 18)

The off-axis angle y is then defined by the axis of symmetry of the distribution. Symmetric density functions, ranging from bipolar to girdle, obtained for a variety of values of k, are illustrated in Figure B11.

Maximum-likelihood estimation In fitting a Bingham distribution to paleomagnetic directional data a convenient method is that of maximum-likelihood; for a general review see Stuart et al. (1999). With this formalism, the likelihood function is constructed from the joint probability-density function for the existing data set. Using the general form of the Bingham density function (Eq. 8) the likelihood for N data is just Lðk1 ; k2 ; e ; e Þ ¼ 1

2

Figure B11 Examples of the axially symmetric Bingham probability density function p1;3 ðyÞ, (Eq. 18) for a variety of k concentration parameters: 0; 1; 4; 16. Note that as jkj is increased the dispersion decreases.

N Y j¼1

Maximizing L is accomplished numerically (Press et al., 1992), an exercise yielding a pair of eigenvectors e1 and e2 and their corresponding concentration parameters k1 and k2 . The third eigenvector e3 is determined by orthogonality and its concentration parameter by convention (Eq. 7) is zero. Some investigators (e.g., Onstott, 1980; Tanaka, 1999) prefer a two-step estimation method, where the eigenvectors are determined by principal component analysis, but these vectors are identical to those found by maximizing (Eq. 19). In any case, obtaining the eigenvalues of the data set through principal component analysis is still required for establishing confidence limits.

Confidence limits It is unfortunate that the relationship between the concentration parameters km , determined (usually) by maximum likelihood, and the eigenvalues of the covariance matrix lm , determined (usually) by principal component analysis, is very complicated. This fact makes the establishment of confidence limits on the eigenvectors difficult. However, Bingham (1974, p. 1220) has discovered an approximate formula for the confidence limit valid under certain circumstances. The confidence ellipse, within which a specified percentage (%) of estimated eigenvectors em can be expected to be realized from a statistically identical data set, is given by the elliptical axes
amn % ¼

x2% 2N Dmn

1=2 ;

(Eq. 20)

for   pb Ij ; Dj jk1 ; k2 ; e1 ; e2 :

(Eq. 19)

Dmn ¼ ðkm  kn Þðlm  ln Þ  1;

and N ! 1;

(Eq. 21)

47

BINGHAM STATISTICS

Table B1 Principal component and maximum likelihood analysis of the Hawaiian paleomagnetic directional data recording a mixture of normal and reverse polarities over the past 5 Ma Eigenvalue

0.0304 0.0601 0.9088

Bingham κm

Eigen direction I(o)

D(o)

14.6 58.9 31.1

87.6 48.4 1.5

Confidence axes am1 95

9.8749 9.3490 0.0000

>1 0.0765

am2 95

am3 95

>1

0.0765 0.0801

0.0801

and where w2% is the usual chi-squared value for two degrees of freedom. Alternative methods have been proposed for establishing confidence limits, most notably the bootstrap method popularized within paleomagnetism by Tauxe (1998). Using the mixed-polarities directional from Hawaii covering the past 5 Ma (see Principal component analysis), in Table B1 we summarize the statistical parameters, and in Figure B12 we show the 95% confidence limit a3n 95 about the eigenvector defining the mean bipolar direction (e3). Note that the confidence limit is much smaller than the variance of the data. Jeffrey J. Love

Bibliography

Figure B12 Equal-area projection of Hawaiian directional data, defined in (a) geographic coordinates and (b) eigen coordinates. Also shown are the projections of the variance minor ellipse, defined by l1 and l2 , and, inside of that, the a3n 95 confidence ellipse. As is conventional, the azimuthal coordinate is declination (clockwise positive, 0
to 360
), and the radial coordinate is inclination (from 90
in the center to 0
on the circular edge).

Abramowitz, M., and Stegun, I.A., 1965. Handbook of Mathematical Functions. New York: Dover. Bingham, C., 1964. Distributions on the sphere and on the projective plane, PhD Dissertation, Yale University, New Haven, CT. Bingham, C., 1974. An antipodally symmetric distribution on the sphere. Annals of Statistics, 2: 1201–1225. Collins, R., and Weiss, R., 1990. Vanishing point calculation as a statistical inference on the unit sphere. International Conference on Computer Vision, December, pp. 400–403. Dimroth, E., 1962. Untersuchungen zum Mechanismus von Blastesis und syntexis in Phylliten und Hornfelsen des südwestlichen Fichtelgebirges I. Die statistische Auswertung einfacher Gürteldiagramme. Tscherm. Min. Petr. Mitt., 8: 248–274. Mardia, K.V., 1972. Statistics of Directional Data. New York: Academic Press. Onstott, T.C., 1980. Application of the Bingham distribution function in paleomagnetic studies. Journal of Geophysical Research, 85: 1500–1510. Press, W.H., Teukolsky, S.A., Vetterling, W.T., and Flannery, B.P., 1992. Numerical Recipes. Cambridge: Cambridge University Press. Stuart, A., Ord, K., and Arnold, S., 1999. Kendall’s advanced theory of statistics, Classical Inference and the Linear Model. Volume 2A, London: Arnold. Tanaka, H., 1999. Circular asymmetry of the paleomagnetic directions observed at low latitude volcanic sites. Earth, Planets, and Space, 51: 1279–1286. Tauxe, L., 1998. Paleomagnetic Principles and Practice. Dordrecht: Kluwer Academic. Watson, G.S., 1965. Equatorial distributions on a sphere. Biometrika, 52: 193–201.

Cross-references Fisher Statistics Principal Component Analysis in Paleomagnetism Statistical Methods for Paleo Vector Analysis

48

BIOMAGNETISM

Use of the geomagnetic field as a navigational cue

changes in firing rates in response to small magnetic field variations have been reported for some time (Figure B13) (for example, see Semm and Beason, 1990; Schiff, 1991). Magnetite in the bee abdomen has been implicated in the orientation of honeybees during their “dance” with a sensitivity to local magnetic field variations many times smaller than the geomagnetic field (Kirschvink et al., 1997). Having established that certain animals can sense the geomagnetic field, the next question is “how is the sensing used in navigation?” One of the most extensively investigated topics is avian navigation, which has recently been reviewed (Wiltschkow and Wiltschkow, 2003). Here we restrict the discussion to the pigeon (for which some of the most convincing evidence of a magnetic compass is available), although as noted above, the topic remains controversial (e.g., Walcott, 1977, 1978, 1996; Walker, 1998, 1999; Walraff, 1999; Reilly, 2002). Kramer’s map and compass model (e.g., Kramer, 1961) is the starting point of most interpretations. In such magnetic navigational systems the two necessary requirements are: (1) some means whereby the animal can locate itself in relation to its loft and (2) some form of compass whereby it can set and maintain a course for its target. These require sensing of the magnetic field and of relatively small changes in the field, as we noted above. How exactly the ability to sense the field is used to provide a map or compass, remains obscure and may well differ from species to species. A particularly important clue has emerged in the case of the homing pigeon from studies of the direction flown immediately after release (Windsor, 1975; Walker, 1998). This is illustrated in Figure B14, in which we see that initial departure directions are symmetrical about the local field gradient, which is indicated by the heavy solid NW/SE line. If birds are released close to this line, the errors are small, but they increase as release sites depart from it. Moreover, if the birds are released to the SW of the gradient line, the starting direction errs to the SW, and conversely if they are released from the site to the NE they err to the NE. Walker (1998) proposed a vector summation model to account for the directions and final return to the loft. It is clear that the chosen directions for the most part take the bird up or down the local gradient in the appropriate sense to move toward the loft. When the release site is close to the local gradient leading to the loft, errors are small, but when the site is away from this line flying down the gradient gives rise to large errors.

A wide variety of animals have been shown to sense the geomagnetic field and to use it as a navigational cue, e.g., salmon (Quin, 1980) honeybees (Gould et al. 1978), robins (Wiltschkow and Wiltschkow, 1972), homing pigeons (Walcott, 1977, 1978, 1996), field mice (Mather and Baker, 1981) and it has been suggested that human beings can also sense the geomagnetic field (Baker, 1980). There is no longer any doubt that certain animals can sense the magnetic field and indeed detect small changes in that field (e.g., Walker and Bitterman, 1989). The homing pigeons appear to use the gradient of the field (Walker, 1998, 1999), although this work remains controversial. The geomagnetic field intensity is of the order of tens of microteslas and typical gradients are a few nanoteslas per kilometer. To detect the necessary changes in the field to establish the gradients over distances of kilometers would therefore require sensitivity to field differences of the order of nanoteslas. Yet, pigeons appear to be able to do this (Windsor, 1975; Walker, 1998). It has long been thought that a magnetite-based sensory receptor makes this possible. The nature of the magnetic sense organ has proved elusive, but an example of a magnetite based sensor has been exquisitely demonstrated in the rainbow trout by Walker et al. (1997). Behavioral studies established that the animal could discriminate between the presence and the absence of a field comparable to the geomagnetic field. Intracellular magnetite was discovered in its nose. Magnetic force microscopy observations of the magnetite were consistent with the magnetic behavior of single domain magnetite particles (Diebel et al., 2000). Finally, neural signal responses to field changes were recorded in nerves from this region. Other detailed demonstrations of magneto-reception are likely to emerge as various animals, known to be capable of sensing the field, are examined by similar techniques. Indeed demonstrations of

Figure B13 Observation of enhanced firing rates in response to magnetic field changes in the Bobolink. The response of a single ganglion cell to field intensity changes—(a) spontaneous activity, (b) response to 200 nT change, (c) response to 5000 nT change, (d) response to 15000 nT change, (e) response to 25000 nT change, and (f) response to 100000 nT change. Note the geomagnetic field is of the order of 10000 nT, or gammas, or tens of microteslas. The stimulus is indicated by the horizontal heavy line. The vertical bar indicates 2 mV and the horizontal bar 50 ms (From Semm and Beason. 1990).

BIOMAGNETISM Biomagnetism has emerged from a somewhat checkered earlier history. Indeed it has the dubious honor through Dr. Mesmer of contributing the verb mesmerize to the language and to have been the butt of Da Ponte and Mozart’s humor in Così fan Tutte. However, it has survived to become a mature science covering a wide range of areas including animal navigation and orientation, possible harmful effects of magnetic fields, and searches for mechanisms explaining these phenomena are underway. In addition, electromagnetic instrumentation plays a major role in medicine and synthetic magnetic particles have been used to measure viscosity in cells and to transport chemicals within the human body, Here we discuss biological sensitivity to magnetic fields, the biogenic synthesis of magnetite, and its possible role in neurodegenerative diseases in humans. We leave the major areas of electromagnetically based medical instrumentation techniques, such as magnetocardiagrams, magnetoencephalograms, and the uses of nanoparticles of magnetite for other reviews.

Sensitivity to magnetic fields The most readily demonstrated sensitivity to magnetic fields is that of magnetotactic bacteria. Several discussions of the phenomena are given in “Magnetite Biomineralization and Magnetoreception in Organisms” by Kirschvink et al. (1985), which provides an excellent source for work in this research area up to that time. As is well known, the bacteria are flagellum driven and the presence of a linear chain of magnetite particles results in their motion being constrained to move along, but not across magnetic lines of force. However, the sense of motion with respect to the field is more complicated than originally recognized and can in some cases change during the course of the day. The simple chain enables the bacterium to navigate between oxygenated and anoxic conditions at different depths in the water column as it swims.

BIOMAGNETISM

49

(Baker, 1980), but the results have not been replicated. A review of human magnetoreception is given in Kirschvink et al. (1985). In addition to useful magnetic field sensing in relation to navigation, there is a broader issue of field sensitivity including possible harmful effects of magnetic fields. Such effects are not likely to be related to the normal field strength of the geomagnetic field because animals have evolved in the presence of this field. Rather, it is the absence of the geomagnetic field, or the presence of much stronger fields that may prove harmful. During the period of low field intensity associated with field reversals, when the simple geometry of the field is also lost, animals relying on the field for navigation will presumably experience difficulty. In a more direct test of the importance of the geomagnetic field, Shibab et al. (1987) reported impairment of axonal ensheathment and myelination of peripheral nerves of newborn rats kept in 10 nT and 0.5 mT environments compared with controls.

Sensitivity to ac fields and dc fields larger than the geomagnetic field

Figure B14 Initial departure directions of pigeons released at various distance and azimuthal direction from their loft. Note that the departure directions are symmetrical about the local field gradient indicated by the heavy NW/SE line (From Windsor, 1975 and Walker, 1998).

Indeed, some of the birds released due west of the loft appear to set off in completely the wrong direction. The overall pattern of directions demonstrates that the birds are able to determine the local field gradient and have a memory of whether the field is higher or lower at the release site than at the loft. The birds appear to use the gradient of the field in their navigation, which is a long standing idea in homing noted by Wiltschkow and Wiltschkow (2003), but it is not clear whether they use nonmagnetic clues or a more sophisticated vector summation magnetic model, such as that suggested by Walker (1998) to complete their return to the loft. These ideas have been strongly criticized by Wallraff (1999) and Reilly (2002), but given the observation of the relation of initial flight directions to the field gradient (Windsor, 1975; Walker, 1998), it is hard to avoid the interpretation that the pigeon is able to sense and follow the field gradient. Many other animals are able to sense the geomagnetic field. There are examples demonstrating determination of the direction of the horizontal component and of the inclination of the total field vector field. Beason (1989) demonstrated sensitivity to the sign of the vertical component of a magnetic field in the Bobolink (Dolichonyx orzivorous), which is a nocturnal migratory species. The experiments were carried out in a planetarium and the initial departure directions were found to reverse with the reversal of the vertical component of the field. The experiment provided convincing evidence of an inclination based compass because the direction of horizontal component of the field was maintained when the vertical component was reversed and yet the initial departure directions reversed. Particularly convincing evidence of the ability to sense the direction of the horizontal component of the field comes from species which orient themselves, e.g., resting termites (Roonwal, 1958) or structures they build, e.g., termite galleries (Becker, 1975), and honey bee hives (Lindauer and Martin, 1972). These and other examples are presented in an excellent review of orientation in Arthropods (Lehrer, 1997). With all the many examples of sensitivity to the geomagnetic field in animals, it is natural to ask whether human beings are sensitive to the Earth’s field and claims that we are indeed able to sense the field have been made

Studies of effects of alternating magnetic fields and static fields that are stronger than the geomagnetic field have also tested sensitivity of the nervous system to these fields, but the main emphasis of the studies is on possible harmful effects, such as carcinogenesis, mutagenesis, and developmental effects. Much of this work has been stimulated by epidemiological studies such as that of Feychting and Ahlbom (1995), which suggested links between power line exposure and cancer in children. This has generated an enormous literature, but no conclusive evidence of cause and effect has emerged. As an example of an effect of a static field only marginally larger than the geomagnetic field, Bell et al. (1992) reported changes in electrical activity in the human brain on exposure to fields of 0.78 G, or 78 mT. Power spectra of electroencephalograms (EEG) were found to be modified in all but one case, and in most cases an increase was observed. Interictal (between seizure) firing rates in the hippocampus of epileptic patients are increased by fields of the order of milliteslas. Such fields are some 20 times larger than the geomagnetic field (Dobson et al., 1995; Fuller et al., 1995, 2003). The procedure had some clinical value in the preoperative evaluation of patients with drug resistant epilepsy. In contrast to the generally observed increases in activity, activity was inhibited when the field was applied during periods of strong activity (Dobson et al., 1998). Work on hippocampal slices by Wieraszko (2000) showed a complicated response pattern, with initial inhibition of firing followed by an amplification phase after the field was removed. The amplification was modulated by dantrolene, which is an inhibitor of intracellular calcium channels. In contrast, to the observations of neural responses to field changes observed in animals that use the field sensitivity for navigation, these responses were on a relatively long timescale of seconds and increased firing rates persisted in some case for tens of minutes. This suggests that these phenomena are very different from the magnetite sensor-mediated responses discussed above in the trout and the Bobolink. In those cases, the response took place within milliseconds of the field change and was completed within seconds. Wikswo and Barach (1980) showed that very large fields of 24 T would be needed to bring about a 10% difference in conduction, in the discussion of the direct effect of magnetic fields on nerve transmission. In this, they noted that they had specifically ignored “microscopic chemical effects as well as those due to induced fields.” It is evidently through these more subtle effects that weak magnetic fields affect biological systems. The direct effect of magnetic fields on the rate of recombination of free radicals is an example of a mechanism that could play a role in field sensitivity. Moreover, given the importance of free radicals in biological systems this could be a significant effect. It arises because the probability of recombination of free radicals depends upon their spin state, which in turn can be affected by a dc field (McLauchlan, 1989) and possibly by ac fields (Hamilton et al., 1988).

50

BIOMAGNETISM

Whereas the sensitivity of the trout to the geomagnetic field is clearly through the intermediary of a magnetite based sensory receptor, it may well be that there are also direct effects of magnetic fields on the nervous system. Indeed, it seems that the effects on the epileptic patients are very different from the process in the trout and may result from direct effects on cell processes.

Biogenic magnetite Biogenic magnetite clearly plays a central role in the sensitivity of the trout to magnetic fields and probably in many other animals. Biogenic magnetite was first discovered in organisms as a capping on the radula (teeth) of the chiton—a marine mollusk (Lowenstam, 1962). Since that time it has been found in many organisms ranging from bacteria to humans. In some species it has been demonstrated to form a part of a geomagnetic field receptor system (e.g., Frankel, 1984; Walker et al., 1997; Wiltschko and Wiltschko, 2002; Kirschvink and Walker, 1995; Lohmann and Lohmann, 1996), but in most it is not obviously linked to magnetic orientation/navigation and its role is unclear. Magnetite is ferrimagnetic because of the imbalance of Fe2þ, which is confined to one of the two opposed sub-lattices, i.e., on the octahedral sites. It has a saturation magnetization of 4.71  105 A m1 at room temperature, a Curie (Néel) point of 578
C and a high density of 5.2 g cm3. It is presumably the high density and hardness that account for its presence in the radula of Chiton. In magnetotactic bacteria and in species that sense the magnetic field, it plays a role in detecting the field. However, magnetite is also a source of ferrous iron, which can generate potentially harmful free radicals. Here we will discuss the synthesis of magnetite, its occurrence in organisms including human beings and its possible role in neurodegenerative diseases.

Synthesis and occurrence of biogenic magnetite Perhaps the best understood example of biogenic magnetite occurs in the various species of magnetotactic bacteria. These motile, gramnegative bacteria synthesize chains of highly pure and crystallographically perfect magnetite nanoparticles for use as a navigational aid—a phenomenon known as “magnetotaxis” (Blakemore, 1975; Frankel, 1984). The morphology of these particles is characteristic of the various species, however, in all cases, the size of the particles is generally just above the superparamagnetic limit such that they are magnetically blocked. This enables the chain to experience a torque when the geomagnetic field is at an angle to the long axis of the chain. Recent work from Bertani and others (2001) has led to the identification of genes, which are involved in the synthesis of magnetite in magnetotactic bacteria. Though this gives an indication of the processes involved in magnetite synthesis, the mechanism employed by the bacteria to form perfect crystals of very specific size has remained difficult to identify. Work published recently, however, characterized proteins associated with biogenic magnetite crystals from the bacterium M. magnetotacticum, which are believed to regulate crystal growth (Arakakai et al., 2003). The authors were able to use these proteins to direct the synthesis of magnetite nanoparticles in the laboratory with sizes and shapes very similar to those found in M. magnetotacticum.

of studies on tissue removed from the human hippocampus was undertaken. The analyses were performed on tissue resected during amygdalohippocampectomies (a surgical procedure in which the damaged hippocampus of focal epilepsy patients is removed) as well as from cadaver tissue. Thus it was possible to control post mortem artifacts, which may have complicated the interpretation of Kirschvink’s earlier results. These studies demonstrated clearly that biogenic magnetite is present in human brain tissue and confirm the results of the Cal Tech group (Dunn et al., 1995; Dobson and Grassi, 1996; SchultheissGrassi and Dobson, 1999). In addition, recent results appear to demonstrate that biogenic magnetite concentration increases with age in males; however, this relationship is not seen in female subjects (Dobson, 2002). While initial analysis of human brain tissue focused on magnetometry studies and the identification of biogenic magnetite by proxy, particles of this material also have been extracted, imaged, and characterized using transmission electron microscopy (TEM) (Kirschvink et al., 1992; Schultheiss-Grassi et al., 1999) (Figure B15). The particles are generally smaller than 200 nm and, in most cases, are on the order of a few tens of nanometers. While some particles exhibit dissolution edges, others preserve pristine crystal faces, and all particles examined thus far are chemically pure (this is common in biogenic magnetite). Morphologically, many of the particles are similar to those observed in magnetotactic bacteria (Schultheiss-Grassi et al., 1999) though other particles are more irregularly shaped. A particularly intriguing transmission electron image has suggested that the iron oxide present in some hippocampal tissue may be concentrated in or near the cell membrane (Dobson, et al., 2001) (Figure B16). Whether this location is related to cell membrane transport, or simply iron storage, is not known. However, the high concentration of iron oxide in this cell suggests a degree of specialization because such a concentration in all hippocampal tissue would give an unrealistically high concentration in this tissue. Unfortunately, magnetic particles have, for the most part, only been observed in tissue extracts, and work is currently underway to develop new techniques for imaging the particles in tissue slices and mapping their distribution to tissue structures. Biogenic magnetite also has been found in the human heart, spleen, and liver. Again, this was determined by magnetometry studies, and calculation of magnetite concentrations showed that the heart has the highest levels of all organs examined thus far (Schultheiss-Grassi et al., 1997). These investigations are continuing with the aim of determining the origin and role of biogenic magnetite in the human body.

Biogenic magnetite in organisms Biogenic magnetite is found in many organisms; however, its physiological role in most of these is not well understood (Webb et al., 1990). The mineral is also found in human tissue. Biogenic magnetite, along with maghemite (gFe2O3—an oxidation product of magnetite), was first reported in humans in 1992 by a group at the California Institute of Technology led by Joseph Kirschvink (Kirschvink et al., 1992). In that study, the group examined human brain tissue samples taken from cadavers and this work proved somewhat controversial. In order to allay this controversy and to examine the possibility of contamination and postmortem changes in brain iron chemistry, a series

Figure B15 Transmission electron micrograph of biogenic magnetite extracted from the human hippocampus. Inset shows an electron diffraction pattern confirming the presence of magnetite. (From Schultheiss-Grassi et al., 1999).

BIOMAGNETISM

51

Conclusion

Figure B16 Electron microscope image of cell with iron oxide particles. (a) scale bar 200 nm and (b) scale bar 100 nm (From Dobson et al., 2001).

Biomagnetism is now a major research area with a healthy mix of biologists, physicists, and chemists involved. We can safely conclude that many organisms sense the geomagnetic field and use it as a navigational cue, or for other purposes, such as nest alignment. Again it is clear that strong magnetic fields can produce mutagenic effects and produce harmful developmental effects in the several animals investigated. It is much less clear how much damage the relatively weak fields of power lines, mobile phones, and other electrical devices of modern society actually harm human beings. Magnetite is synthesized by many organisms and is involved in field detection in some. It is however present in many organisms in which its role is unclear. Indeed, as we noted, magnetite may be harmful as a source of free radicals. Presently, the possibility that magnetite may play a role in neurodegenerative diseases, such as Alzheimer’s, Parkinson’s, or Huntington’s, is being investigated and may explain the association of iron enrichment and these diseases. Mike Fuller and Jon Dobson

Biogenic magnetite and neurological disorders Though the physiological role of magnetite in humans is not yet known, recent studies have demonstrated a tentative link between magnetite biogenesis and neurodegenerative disorders such as Alzheimer’s disease (AD) (Dobson, 2001, 2004; Hautot et al., 2003). It has been known for 50 years that neurodegenerative diseases are associated with elevated levels of iron (Goodman, 1953). However, since that discovery, very little progress has been made in identifying the iron compounds present, their role, and their origin until this recent work. Magnetite was first reported in AD tissue in the original study of human brain tissue by Kirschvink et al. (1992). These results did not demonstrate any correlation between elevated levels of biogenic magnetite and AD. Although the results were negative, it appears that this may have been due to the particular measurement technique rather than the absence of a correlation altogether. The isothermal remanent magnetization (IRM) of the tissue was measured using the superconducting quantum interference device (SQUID) magnetometry. One of the drawbacks of this method is that the contribution of very fine (superparamagnetic) particles (less than 50 nm) is lost from the signal, particularly when measured at 0
C as was the case in this first study. In this case, any concentration which is calculated from the data depends on making some assumptions—one assumption being that the signal is due to particles which are large enough to contribute to the remanent magnetization (i.e., blocked). In this study, that may not have been a true picture of all the biogenic magnetite in the tissue samples. Recently, Hautot et al. (2003) have developed methods for measuring the total biogenic magnetite signal. This is done by measuring the magnetization vs. applied field and then modeling the contributions from material such as the diamagnetic tissue, ferritin, and heme iron. In this way, the total concentration of magnetite—regardless of grain size—can be determined. Using this method, preliminary evidence of a correlation between biogenic magnetite concentration has been demonstrated in female AD patients. By measuring other parameters, such as magnetic interactions between groups of magnetite particles, it has also been shown that there is preliminary evidence of a correlation between particle packing geometry and focal epilepsy. There was no demonstrable correlation between biogenic magnetite concentration and epileptic tissue, but again, these results are from measurements of remanent magnetization. New studies using microfocused synchroton x-ray sources have now enabled iron compounds in brain tissue to be located and characterized in situ with subcellular resolution (Mikhaylova et al., 2005; Collingwood et al., 2005). This breakthrough allows, for the first time, the mapping of specific iron compounds including magnetite, to structures and cells within neurodegenerative tissue. Magnetite-rich iron anomalies have been unequivocally identified in AD tissue using this method, and it shows great promise for furthering our understanding of the role of magnetite and other iron compounds in neurological disorders.

Bibliography Arakaki, A., Webb, J., and Matsunaga, T., 2003. A novel protein tightly bound to bacterial magnetic particles in Magnetospirillum magneticum strain AMB-1. Journal of Biological Chemistry, 278: 8745–8750. Baker, R.R., 1980. Goal orientation by blindfolded humans after long distance displacement: possible involvement of a magnetic sense. Science, 210: 555–557. Beason, R.C., 1989. Use of an inclination compass during migratory orientation by the Bobolink (Dolichonyx orzivorous). Journal of Experimental Biology, 125: 49–56. Becker, G., 1975. Einfluss von mangetischen, elektrischen und Schwerefeldern auf den Galeriebau von Termiten, Umschau in Wissenschaft und Technik, 75: 183–185. Bell, G.B., Marno, A.A., and Chesson, A.L., 1992. Alterations in electrical activity caused by magnetic fields, detecting the detection process. Electroencephalography and clinical Neurophysiology, 83: 389–392. Bertani, L.E., Weko, J., Phillips, K.V., Gray, R.F., and Kirschvink, J.L., 2001. Physical and genetic characterization of the genome of Magnetospirillum magnetotacticum, strain MS-1. Gene, 264: 257–263. Blakemore, R.P., 1975. Magnetotactic bacteria. Science, 190: 377–379. Collingwood, J.F., Mikhailova, A., Davidson, M., Batich, C., Streit, W.J., Terry, J., and Dobson, J., 2005. In-situ characterization and mapping of iron compounds in Alzheimer’s tissue. Journal of Alzheimer’s Disease, 7: 267–272. Diebel, C.E., Proksch, R., Green, C.R., Neilson, P., and Walker, M.M., 2000. Magnetite defines a magnetoreceptor. Nature, 406: 299–302. Dobson, J., 2001. Nannoscale biogenic iron oxides and neurodegenerative disease. FEBS Letters, 496: 1–5. Dobson, J., 2002. Investigation of age-related variations in biogenic magnetite levels in the human hippocampus. Experimental Brain Research, 144: 122–126. Dobson, J., 2004. Magnetic iron compounds in neurological disorders. Annals of the New York Academy of Sciences, 1012: 183–192. Dobson, J., Fuller, M., Moser, S., Dunn, J.R., and Zoeger, J., (1995) Evocation of epileptiform activity by weak DC magnetic fields and iron biomineralization in the human brain. In Decke, L., Baumgartner, C., Stroink, G., Williamson, S.J. Dobson, J., Weiser, H.G., and Fuller, M., (eds.), Advances in Biomagnetism, 16–19, 1995. Dobson, J., St. Pierre, T., Gregor Wieser, H., and Fuller, M., 2000. Changes in paroxysmal brainwave patters of epileptics by weakfield magnetic stimulation, Bioelectromagnetics, 21: 94–96. Dobson, J., and Grassi, P., 1996. Magnetic properties of human hippocampal tissue: Evaluation of artefact and contamination sources. Brain Research Bulletin, 39: 255–259.

52

BIOMAGNETISM

Dobson, J., Gross, B., Exley, C., Mickhailova, A., Batich, C., and Pardoe, H., 2001. Biomedical applications of biogenic and biocompatible magnetic nanoparticles. In Ayrapetyan, S., and North, A. (eds.) Modern Problems of Cellular and Molecular Biophysics, Yerevan: Noyan Tapan/UNESCO, 121–130. Dunn, J.R., Fuller, M., Zoeger, J., Dobson, J., Heller, F., Caine, E., and Moskowitz, B.M., 1995. Magnetic material in the human hippocampus. Brain Research Bulletin, 36: 149–153. Feychting, M., and Ahlbom, A., 1995. Magnetic fields and cancer in children residing near Swedish high-voltage power lines. American Journal of Epidemiology, 138: 7, 467–481. Frankel, R.B., 1984. Magnetic guidance of organisms. Annual Review of Biophysics and Bioengineering, 13: 85–103. Fuller, M., Dobson, J., Wieser, H.G., and Moser, S., 1995. On the sensitivity of the human brain to magnetic fields: Evocation of epileptiform activity. Brain Research Bulletin, 36(2): 155–159. Fuller, M., Wilson, C.L., Velasco, A.L., Dunn, J.R., and Zoeger, J., 2003. On the confirmation of an effect of magnetic fields on the interictal firing rate of epileptic patients. Brain Research Bulletin, 611: 1–10. Goodman, L., 1953. Alzheimer’s disease—a clinicopathologic analysis of 23 cases with a theory on pathogenesis. Journal of Nervous and Mental Diseases, 118: 97–130. Gould, J.L., Kirschvink, J.L., and Deffeyes, K.S., 1978. Bees have magnetic remanence. Science, 201: 1026–1028. Hautot, D., Pankhurst, Q.A., Kahn, N., and Dobson, J., 2003. Preliminary evaluation of nanoscale biogenic magnetite and Alzheimer’s disease. Proceedings of the Royal Society B: Biology Letters, 270: S62–S64. Kirschvink, J.L., Jones, D.S., and MacFadden, B., 1985. Magnetite Biomineralization and Magnetoreception in Organisms: A New Biomagnetism. New York: Plenum Press. Kirschvink, J.L., Kobayashi-Kirschvink, A., and Woodford, B.J., 1992. Magnetite biomineralization in the human brain. Proceedings of the National Academy of Science USA, 89: 7683–7687. Kirschvink, J.L., and Walker, M.M., 1995. Magnetoreception in honeybees. Science, 269: 1889–1889. Kirschvink, J.L., Padmanabha, S., Boyce, C.K., and Oglesby, J., 1997. Measurement of the threshold sensitivity of honeybees to weak, extremely low frequency magnetic fields. Journal of Experimental Biology, 200: 1363–1368. Kramer, G., 1961. Long distance orientation, In Marshall, E.G. (ed.), Biology and Comparative Physiology of Birds, London: Academic Press, pp. 341–371. Lehrer, M. (ed.), 1997. Orientation and Communication in Arthropods. Berlin: Birkhäuser. Lindauer, M., and Martin, M., 1972. Magnetic effects on dancing bees, In Galler, S.R., Schmidt-Koenig, K., and Belleville, R.E. (eds.), Animal Orientatioin and Navigation. Washington, D.C.: US Government Printing Office, pp. 559–567. Lohmann, K.J, and Lohmann, C.M.F., 1996. Detection of magnetic field intensity by sea turtles. Nature, 380: 59–61. Lowenstam, H.A., 1962. Magnetite in denticle capping in recent chitons (Polyplacophora). Geological Society of America Bulletin, 73: 435–438. Mather, J.G., and Baker, R.R., 1981. A magnetic sense of direction in woodmice for route based navigation, Nature, 291: 152–155. McLauchlan, K.A., 1992. Are environmental magnetic fields dangerous? Physics World, January: 41–45. Mikhailova, A., Davidson, M., Channel, J.E.T., Guyodo, Y., Batich, C., and Dobson, J., 2005. Detection, identification and mapping of iron anomalies in brain tissue using x-ray absorption spectroscopy. Journal of the Royal Society Interface, 2: 33–37. Quin, T.P., 1980. Evidence for celestial and magnetic compass orientation in lake migrating sockeye salmon fry, Journal of Comparative Physiology, 137: 243–248. Reilly, WI., 2002. Magnetic position determination by homing pigeons, Journal of Theoretical Biology, 218: 47–54.

Roonwal, M.L., 1958. Recent work on termite research in India (1947–1957) Transactions of the Bose Research Institute, 22: 77–100. Schiff, H., 1991. Modulation of spike frequencies by varying the ambient magnetic field and magnetite candidates in bees (Apis mellifera). Comparative Biochemistry and Physiology, 100A: 975–985. Schultheiss-Grassi, P.P., Heller, F., and Dobson, J., 1997. Analysis of magnetic material in the human heart, spleen and liver. Biometals, 10: 351–355. Schultheiss-Grassi, P.P. and Dobson, J., 1999. Magnetic analysis of human brain tissue. Biometals, 12: 67–72. Schultheiss-Grassi, P.P., Wessiken, R., and Dobson, J., 1999. TEM observation of biogenic magnetite extracted from the human hippocampus. Biochimica et Biophysica Acta, 1426/1: 212–216. Semm, P., and Beason, R.C., 1990. Responses to small Magnetic Variations by the trigeminal system of the Bobolink, Brain Research Bulletin, 25: 735–740. Shibab, K., Brock, M., Gosztonyi, G., Erne, S.E., Hahlbohm, H.D., and Schoknecht, G., 1987. The geomagnetic field: a factor in cellular interactions. Neurological Research, 9: 225–235. Walcott, C., 1977. Magnetic fields and the orientation of homing pigeons under sun, Journal of Experimental Biology, 70: 105–123. Walcott, C., 1978. Anomalies in the earth’s magnetic field increase the scatter of pigeon’s vanishing bearings, In Schmidt-Koenig, K., and Keeton, W.T., (eds.), Animal Migration, Navigation and Homing, Berlin: Springer, pp. 143–151. Walcott, C., 1996. Pigeon homing observations, experiments and confusions, Journal of Experimental Biology, 199: 21–27. Walker, M.M., 1998. On a wing and a vector: a model for magnetic navigation by homing pigeons. Journal of Theoretical Biology, 192: 341–349. Walker, M.M., 1999. Magnetic position determining by homing positions, Journal of Theoretical Biology, 197: 271–276. Walker, M.M., and Bitterman, M.E., 1989., Bees can be trained to respond to very small changes in geomagnetic field intensity, Journal of Experimental Biology, 141: 447–451. Walker, M.M., Diebel, C.E., Haugh, C.V., Pankhurst, P.M., Montgomery, J.C., and Green, C.R., 1997. Structure and function of the vertebrate magnetic sense, Nature, 390: 371–376. Wallraff, H.G., 1999. The magnetic map of the homing pigeon, an evergreen phantom, Journal of Theoretical Biology, 197: 265–269. Webb, J., St. Pierre, T.G., Macey, D.J., 1990. Iron biomineralization in invertebrates. In Frankel, R.B., and Blakemore, R., (eds.), Iron Biominerals, New York: R.P. Plenum Publishing Corp., 193–220. Wieraszko, A., 2000. Dantrolene modulates the influence of steady magnetic fields on hippocampal evoked potentials in vitro, Bioelectromagnetics, 21: 175–182. Wiltschkow, W., and Wiltschkow, R., 1972. Magnetic compass of European Robins, Science, 176: 62–64. Wiltschko, W., and Wiltschko, R., 2003. Magnetic compass orientation in birds and its physiological basis. Naturwissenschaften, 89: 445–452. Windsor, D.M., 1975. Regional expression of directional preferences by experienced homing pigeons, Animal Behaviour, 23: 335–343. Winklhofer, M., Holtkamp-Rotzler, E., Hanzlik, M., Fleissner, G., and Petersen, N., 2001. Clusters of superparamagnetic magnetite particles in the upper-beak skin of homing pigeons: evidence of a magnetoreceptor? European Journal of Mineralogy, 13: 659–669.

Cross-references Magnetization, Isothermal Remanent Rock Magnetism

BLACKETT, PATRICK MAYNARD STUART, BARON OF CHELSEA (1887–1974)

53

BLACKETT, PATRICK MAYNARD STUART, BARON OF CHELSEA (1887–1974) Blackett (Figure B17) came to geophysics late in a remarkable career. He had been educated at Osborne Naval College and Dartmouth and joined the Royal Navy, seeing action at the Battles of Jutland and of the Falklands in the Great War, as it was then known. After the war, he resigned from the navy, went to Cambridge to Magdalene College, and read for the Natural Science Tripos. He joined Lord Rutherford’s group at the Cavendish as a graduate student in 1921, where he worked on transmutation of atoms using cloud chambers, carrying out in the succeeding two decades much of the work for which he was eventually awarded a Nobel Prize in Physics in 1948. In 1933 he became professor of physics at Birkbeck College, London and in 1937 he moved to Manchester University as the Langworthy Professor of Physics. Prior to the Second World War, he had been brought into the Air Defense Committee and with the advent of the war was involved in many technical projects and made notable contributions in operations research. After the Second World War, he returned to Manchester and to research and was instrumental in establishing the first chair in radioastronomy and in the development of the Jodrell Bank Research Station that would lead to the famous steerable radio telescope. Meanwhile he became interested in magnetic fields in stars and planets, which led to his research in geophysics. In 1953, he was appointed professor and head of the physics department at Imperial College and built up an influential group in rock magnetism and paleomagnetism. Professor Blackett was elected president of the Royal Society in 1965 and made a life peer in 1969. He died in 1974. Blackett was a member of the heroic generation of physicists, most of whom were European, who revolutionized physics between the two world wars. As one of Lord Rutherford’s students, he was assigned to study the disintegration of elements using the cloud chamber, which had been invented by C.T.R. Wilson at Cambridge nearly a quarter of a century earlier. With it, Wilson had demonstrated the tracks of ionizing alpha particles emitted from a small radioactive source. In his initial studies of thousands of photographs using such a source and with the cloud chamber containing nitrogen, Blackett found just eight, in which something remarkable was recorded. As Figure B18 shows, most of the alpha particles from the radioactive source below the chamber pass through it, not striking the atoms of nitrogen gas in the chamber. However, on the left a track shows that an alpha particle has hit a nitrogen atom. This has yielded a heavy oxygen isotope and a proton that leaves to the right of the original track. The heavy oxygen leaves to the left with a stronger track and is seen to collide again. Blackett had trapped a nuclear transmutation on film! He went

Figure B17 P.M.S. Blackett.

Figure B18 Cloud chamber photograph showing alpha particles traversing the chamber and an example on the left of a collision between an alpha particle and a nitrogen atom, yielding a heavy oxygen isotope and a proton. The proton leaves to the right of the initial track (after Close et al., 1987).

on to establish the nature of the scattering of alpha particles from a number of nuclei of varying mass. A major problem with the cloud chamber used in Blackett’s early experiments was that thousands of photographs were needed to have a good chance of seeing an interesting transmutation. This problem became all the more daunting when the cloud chambers were turned to the analysis of cosmic rays. It was then a matter of chance, whether a cosmic ray crossed the chamber close to the time of expansion. The problem was solved by Blackett and a young Italian physicist Giuseppe Occhialini. Occhialini had studied with Rossi at Florence and brought with him knowledge of the work on coincident signals from Geiger counters. Blackett and Occhialini’s solution was to place a Geiger counter above and below the cloud chamber so that a simultaneous signal from the two could be used to detect a cosmic ray particle and fire the cloud chamber. Of course by that time, the cosmic ray particle was long gone, but the ionization trail was still there and produced the necessary condensation for a track to be formed when the chamber was expanded. Although Anderson was the first to detect Dirac’s antielectron, or positron as it has become known, Blackett and Occhialini confirmed its existence with their technique and demonstrated the production of positrons. To do this they had placed a copper plate above the cloud chamber, which produced gamma rays from the interaction of energetic cosmic ray electrons with the copper. Within the chamber numerous symmetrically divergent tracks were formed, as the particles with opposite electric charge formed and responded to the strong magnetic field imposed. These divergent tracks recorded the generation of matter, in the form of electrons and positrons, from energy, in the form of gamma rays, as predicted by Einstein. The citation for Blackett’s 1948 Noble Prize was “for his development of the Wilson cloud chamber method and his discoveries therewith in the fields of nuclear physics and cosmic radiation.” Blackett’s Nobel acceptance speech in 1948 reveals his social sensitivity. After expressing his personal feelings of satisfaction in the award, he remarked that he saw the Nobel Prize “as a tribute to the vital school of European Experimental Physics.” He then went on to address the problem of avoiding the catastrophe of nuclear war, noting that “technological progress and pure science are but different facets of the same growing mastery by man over the force of nature. It is

54

BULLARD, EDWARD CRISP (1907–1980)

our task as scientists and citizens to ensure these forces are used for the good of men and not their destruction.” In 1948 he published more on these ideas in the book Military and Political Consequences of Atomic Energy. His socialist political thought caused difficulty with the America of that time, but his recognition of the dangers of a world divided into very poor and very rich countries remains all too relevant to this day. Blackett was clearly an excellent experimentalist and it was this ability that led him into the development of a sensitive magnetometer that opened up the field of paleomagnetism of the weakly magnetized sediments. Blackett had very skillfully constructed a sensitive magnetometer to test a theory of his about magnetic fields and the rotation of massive bodies. The idea was in relation to the origin of the magnetic fields of stars and of the Earth as a fundamental property of their rotation. The work developing this system and the negative result were described in a famous paper by Blackett entitled “A Negative Experiment Relating to Magnetism and the Earth’s Rotation” and published in the Transactions of the Royal Society. There used to be a reprint of this paper in the geophysics library at Madingly Rise, Cambridge, with a note from Blackett, apologizing for the mistaken theoretical basis for the experiment. However, his instrument provided the means of measuring the weak magnetization of sediments, which was essential to test the idea of continental drift. This was one mistake that had a substantial silver lining! Blackett’s rock magnetism group worked initially at Manchester and then moved with him to Imperial College. His laboratory at Imperial College, with its orderly arrangement of instruments and polished wooden floors, was reminiscent of the quarterdeck of a Royal Navy battleship, presumably a legacy of his naval days. Important work was done both in fundamental rock magnetism and in paleomagnetism. Leng and Wilson demonstrated the reality of geomagnetic field reversals. Haigh made critical contributions to the understanding of magnetization acquired by magnetic materials as they grow in a magnetic field. Everitt studied the acquisition of magnetization as magnetic particles cooled in a magnetic field and developed the baked contact test—one of the standard paleomagnetic field tests. The paper by Blackett, Clegg, and Stubbs (1961) was one of the most convincing early demonstrations by paleomagnetism of the veracity of the idea of continental drift. Blackett’s lectures on rock magnetism taken from his Second Weizman Memorial Lectures of 1954 is a little gem, combining insight on the grand scale, but also reflecting the pleasure of experimental work (Blackett 1956). Blackett’s main contributions were outside of geophysics. One immediately thinks of his experimental skills and the insights that came from the cloud chamber work; his leadership during World War II; his contributions to the development of radioastronomy. In addition to all this, his experimental skills played an important role in the development of rock magnetism and paleomagnetism. The way in which his group operated was in a direct lineage from Rutherford’s Cavendish, which was brought to our field. We were lucky that our subject was touched by Blackett.

BULLARD, EDWARD CRISP (1907–1980) Edward Crisp Bullard, known to everyone as Teddy, was the greatest geophysicist working in the United Kingdom in the second half of the 20th century (Figure B19). He made seminal contributions to the theory of the origin of the magnetic field, to the analysis of the westward drift and secular variation, and to studies of electromagnetic induction. He championed a number of developments in the theory of plate tectonics, and placed his department at the centre of the plate tectonic revolution of the 1960s. His full biography is in McKenzie (1987). Teddy was born in Norwich into a wealthy family of brewers. The building that housed the brewery still stands in an attractive part of Norwich, but the business was taken over in the 1960s and Bullard’s beer can no longer be found; the name enjoyed a revival in the early days of the campaign for Real Ale and won a prize at the 1974 Cambridge beer festival. The young Bullard was educated at Repton School where his physics teacher, A.W. Barton, initiated his interest in physics. He studied physics in Cambridge, where he found the lectures dull with the exception of J. Larmor (q.v.), Rutherford, and Pars, and proceeded to research in Rutherford’s laboratory. He was in fact supervised by P.M.S. Blackett (q.v.), although Rutherford was probably the greater influence. His Ph.D. project was on electron scattering, but on taking up a position in the Department of Geodesy and Geophysics he began work on the measurement of gravity with pendulums and made observations with them in East Africa. His Ph.D. thesis, only 26 pages long, includes both the electron scattering and gravity work. He was elected a fellow of the Royal Society in 1941; the citation includes work on seismics, heat flow, and gravity: everything, in fact, except geomagnetism. On the outbreak of the Second World War he joined the Admiralty Research Laboratory and set to work on reducing the magnetic fields around ships. The Germans had laid magnetic mines in shallow waters around Britain, which sank 60 ships in the “phoney war” in December 1939. The “degaussing” was successful (Bullard, 1946). He subsequently joined Blackett to work on what later became known as operational research. The degaussing of ships was his first work on magnetism, the only previous reference being to the effect of the Earth’s magnetic field on the period of invar pendulums (Bullard, 1933), but during his war work he read Chapman and Bartels (1940), and this probably stimulated him to work on geomagnetism after the war was over. Postwar Cambridge offered little for an ambitious researcher, particularly in geophysics, and Teddy moved to Toronto. There followed the most productive period of his career. He used his knowledge of

Michael D. Fuller

Bibliography Blackett, P.M.S., 1952. A negative experiment relating to magnetism and the Earth’s rotation, Philosophical Transactions of the Royal Society of London, Series A, A245: 309–370. Blackett, P.M.S., 1954. Lectures on Rock Magnetism. Jerusalem: The Weizmann Science Press of Israel. Blackett, P.M.S., Clegg, J.A., and Stubbs, P.H.S., 1960. An analysis of rock magnetic data, Proceedings of the Royal Society of London, 256: 291–322. Close, F., Marten, M., and Sutton, C., 1987. The Particle Explosion. Cambridge: Oxford University Press.

Figure B19 “Teddy” as the author knew him in about 1970.

55

BULLARD, EDWARD CRISP (1907–1980)

heat flow to develop the convection theory for the dynamo (Bullard, 1950) and analyzed global observations for secular variation and the westward drift (Bullard et al., 1950). Both these studies remain substantially correct today, although his theory for westward drift, that the magnetic field acted on the lower mantle to drive it eastwards by an electric motor effect, is no longer believed because the electrical conductivity of the mantle is thought to be too small. W.M. Elsasser (q.v.) also began working on the origin of the Earth’s magnetic field at this time, but he concentrated on the mechanism by which a magnetic field could be generated by flow in the core, whereas Bullard had worked with the observations and on the heat sources driving the convection. Elsasser derived some fundamental results in dynamo theory and constructed a formalism for solving the equations, but did not attempt to solve them. This was a very formidable problem for that time—indeed, it is still a very difficult problem with today’s computers. Bullard and a research student at Toronto, H. Gellman, attempted to find a solution with the tools at hand (see Bullard-Gellman dynamo). In 1950 Bullard moved back to the United Kingdom as director of the National Physical Laboratory, where he had access to one of the most powerful computers of the day. Although Bullard and Gellman (1954) is one of the most influential papers ever published in dynamo theory, their solutions had not converged and their claim to have found a working dynamo was subsequently proved incorrect (Gibson and Roberts, 1969). Nonetheless, it was a brave and very early effort to carry out a numerical solution of a partial differential equation governing a problem in fluid mechanics. The dynamo failed because they chose a fluid flow that lacked helicity. It is clear that they intended to design a flow with helicity because their paper contains a discussion of magnetic field lines being pulled and twisted to reinforce the dipole field, but they were restricted by numerical considerations to a very simple flow. Bullard was undoubtedly influenced by Elsasser’s ideas, which were later developed by E.N. Parker into what is now known as an a-effect dynamo (Parker, 1955). Kumar and Roberts (1975) eventually found a working dynamo using the Bullard-Gellman formalism and a more complicated fluid flow. Bullard desperately wanted to return to Cambridge. Eventually a post became available when S.K. Runcorn (q.v.) moved to Newcastle, and Bullard accepted a very junior position, lower than the one he occupied before the war. He was eventually promoted as reader then professor in 1964, over 20 years since his election to fellowship of the Royal Society and 10 years after his knighthood. On the question of polarity reversals he sat on the fence, much to Runcorn’s annoyance, remaining sceptical because of the discovery of self-reversal in certain minerals. He was finally won over by the classic work of Cox et al. (1963), and when Harry Hess visited Cambridge in 1962 and talked about his ideas on seafloor spreading he encouraged Fred Vine, then a research student, to pursue his interpretation of reversely magnetized seafloor in terms of Hess’ model. The result is now known as the Vine-Matthews-Morley hypothesis (q.v.). In later years he had a succession of successful research students working on electromagnetic induction, among them Bob Parker, Nigel Edwards, and Dick Bailey. He put his name to very few scientific papers written during this period, but his influence was everywhere, both

on the group as a whole and on individual students. Bullard worked to the end, finishing his last paper on historical measurements at London with Stuart Malin (Malin and Bullard, 1981) shortly before he died. David Gubbins

Bibliography Bullard, E.C., 1933. The effect of a magnetic field on relative gravity determinations by means of an invar pendulum. Proceedings of the Cambridge Philosopjical Society, 29: 288–296. Bullard, E.C., 1946. The protection of ships from magnetic mines. Proceedings of the Royal Institution of Great Britain, 33: 554–566. Bullard, E.C., 1950. The transfer of heat from the core of the Earth. Monthly Notices of the Royal Astronomical Society, 6: 36–41. Bullard, E.C., Freedman, C., Gellman, H., and Nixon, J., 1950. The westward drift of the Earth’s magnetic field. Philosophical Transactions of the Royal Society of London, Series A, A243: 67–92. Bullard, E.C., and Gellman, H., 1954. Homogeneous dynamos and terrestrial magnetism. Philosophical Transactions of the Royal Society of London, Series A, 247: 213–278. Chapman, S., and Bartels, J., 1940. Geomagnetism. London: Oxford University Press. Cox, A., Doell, R.R., and Dalrymple, G.B., 1963. Geomagnetic polarity epochs and Pleistocene geochronometry. Nature, 198: 1049. Gibson, R.D., and Roberts, P.H., 1969. The Bullard and Gellman dynamo. In Runcorn, S.K. (ed.), The Application of Modern Physics to the Earth and Planetary Interiors. New York: Wiley Interscience, pp. 577–602. Kumar, S., and Roberts, P.H., 1975. A three-dimensional kinematic dynamo. Proceedings of the Royal Society, 344: 235–258. Malin, S.R.C., and Bullard, E.C., 1981. The direction of the Earth’s magnetic field at London, 1570–1975. Philophical Transactions of the Royal Society of London, Series A, A299: 357–423. McKenzie, D.P., 1987. Edward Crisp Bullard. Biographical Memoirs of Fellows of the Royal Society, 33: 67–98. Parker, E.N., 1955. Hydromagnetic dynamo models. Astrophysical Journal, 122: 293–314.

Cross-references Blackett, Patrick Maynard Stuart, Baron of Chelsea (1897–1974) Bullard-Gellman dynamo Dynamo, Bullard-Gellman Dynamos, Kinematic Elsasser, Walter M. (?–1991) Geomagnetic Polarity Reversals, Observations Geomagnetic Secular Variation Larmor, Joseph (1857–1942) Mantle, Electrical Conductivity, Mineralogy Runcorn, S. Keith (1922–1995) Transient EM Induction Vine-Matthews-Morley Hypothesis Westward Drift

C

CARNEGIE INSTITUTION OF WASHINGTON, DEPARTMENT OF TERRESTRIAL MAGNETISM When the Carnegie Institution of Washington (CIW) was established in 1902 with a large bequest from Andrew Carnegie, one of its goals was to encourage research in the “borderlands” between the scientific disciplines. In its first years, the CIW established two departments that became stages for research in the geosciences: the Geophysical Laboratory and the Department of Terrestrial Magnetism (DTM). While the Geophysical Laboratory moved quickly in the direction of geochemical investigation and toward developing laboratory techniques to study how rocks behave at great depth within the Earth, the DTM focused on a series of problems related to geomagnetism and geo-electricity during its first several decades. The DTM branched out starting in the 1920s to investigate the atomic nucleus, cosmic rays, and even cosmology. These two departments have now existed for over a century and today are located on a single campus where their researchers can more easily interact (Trefil and Hazen, 2002). The geophysicist Louis Agricola Bauer (q.v.), who had been engaged in geomagnetic research for over a decade, established the DTM in 1904 with the general goal of studying Earth’s magnetic and electric phenomena, but with a definite project to prosecute “quickly” a magnetic survey of the planet. This project was to be completed within a generation in order to provide data which could be more easily reduced to a common epoch than the data then available. Magnetic data had only been gathered since the mid-19th century with any degree of consistency, and even then, it was concentrated in just a few regions and was spread over decades. Entire continents were uninvestigated and charts available for magnetic declination in the world’s oceans were frequently in error by 1
and 2
, while inclination measurements were 1
–3
off. Moreover, outside the main shipping lanes very few readings had been made at all (Good, 1994b). Bauer proposed a World Magnetic Survey to the Carnegie Institution of Washington because, as a nongovernmental agency, it would enjoy greater credibility for mounting expeditions in Africa, Asia, South America, and other places where the institutions of major governments might be looked upon with suspicion. Moreover, free of any government connection, the DTM could directly approach scientific officials in other countries without going through their supervising bureaucracies. As he put it in his mission statement for the new department, the DTM would survey the magnetism of areas not being studied by others. This was most of the globe. The DTM was to have a second purpose too: to coordinate and standardize magnetic measurements then being undertaken by dozens of

governments and agencies around the world. That is, it would be the international standardizing bureau. It would carefully compare the great variety of geomagnetic instruments (see History of Instrumentation) and promote the best methods of investigation and measurement. To this end, for decades the DTM had its traveling “magneticians” visit major magnetic observatories around the world to standardize instruments and to learn and teach different working methods. DTM promoted data exchange and uniform methods of reduction. In these regards the DTM acted in the way that various commissions of the International Association for Geomagnetism and Aeronomy or IAGA (q.v.) and Intermagnet do today. Bauer set the DTM’s first task as evaluation of existing data for secular variation studies and he contracted Adolph Schmidt to determine the status of data on magnetic storms. The second over-arching goal was to “establish the facts” and to convince theoreticians that the data and mathematical tools could become available for decisive testing among different theories. Bauer also brought his journal Terrestrial Magnetism and Atmospheric Electricity (since 1948 the Journal of Geophysical Research) to the DTM, where he and his associate director John A. Fleming (q.v.) used it to provide a disciplinary and international identity for geomagnetic researchers. In its pages, Bauer published critical studies of various instruments, theoretical discussions, and “news of the profession,” including capsule biographies of geomagnetic role models like Edmond Halley (q.v.), Alexander von Humboldt (q.v.), and Carl Friedrich Gauss (q.v.). Bauer’s most dramatic project, however, was the global magnetic survey. Between 1905 and World War II, the DTM employed two hundred magnetic observers on land expeditions. Counting the number of expeditions is a little difficult, since many of them split and rejoined and they varied from a few weeks to many months in duration. The DTM reported in 1928 that 178 land expeditions had thus far occupied 5685 magnetic stations. This averaged 271 stations per year. By 1928, when Fleming was taking over supervisory duties for the DTM, the main land survey was completed. Dwindling numbers of expeditions from then until the mid-1940s concentrated on secular variation measurements at a smaller number of repeat stations (q.v.). These troops of magneticians were a tightly organized, well-trained army of observers. They learned the physics of electricity and magnetism, the use of magnetometers, induction coils, and sine galvanometers (see History of Instrumentation) and the techniques of data reduction. They also were resourceful, independent travelers. Unlike other expeditionaries, these scientists often traveled alone and rarely in groups of more than three. They transported hundreds of kilograms

CARNEGIE INSTITUTION OF WASHINGTON, DEPARTMENT OF TERRESTRIAL MAGNETISM

of instrumentation and other gear and frequently traveled by camel or canoe, as well as by steamer or locomotive. The magnetic data was hard won and surprisingly, not one observer died of disease or accident in all of those travels into the Polar Regions, Amazonia, and banditridden China (Figure C1).

Figure C1 Magnetic observer Frederick Brown traveled around Asia by camel and sedan chair. This photo is from April 1915. Photo courtesy of Carnegie Institution of Washington. Photo DTM 5769.

57

Meanwhile, Bauer carefully executed a parallel set of oceanic expeditions. He consulted in 1904 and 1905 with experts in geomagnetism (e.g., Arthur Schuster) and hydrographers (e.g., George W. Littlehales and Ettrick W. Creak) (Good, 1994b). In 1906, 1907, and 1908 the brig Galilee followed three, spiraling loops in the northern Pacific Ocean, providing magnetic measurements within each 5
quadrangle in the region, or one station every 200 miles. The DTM constantly improved sea-going magnetic instruments, so that by 1908, magnetic observers read declination to within 0.5
, compared with 0.1
for land observers. By this last cruise, errors due to iron on board ship was a larger source of error than the instruments. Hence, Bauer obtained funding from the CIW for construction of a nonmagnetic yacht, the Carnegie (q.v.), which was launched in 1909. This yacht was magnificent, if peculiar. Its dominant peculiarity was the deck house, a glass enclosed galley with a glass observing dome at each end. Protected within, magnetic observers could make measurements even in quite rough seas. From 1909 until its demise in 1929, the Carnegie conducted seven cruises, including a celebrated fastest-ever circumnavigation of Antarctica for a sailing vessel in 118 days; all the while the observers continued their scientific work. A map published by the DTM in 1929 shows the globe enmeshed in the observations of Bauer’s remarkable World Magnetic Survey (Figure C2). Bauer had imagined an ambitious plan to make thousands of measurements around the globe in a single generation and he found the funding, personnel, and instruments to do it. The effort required to produce this first geomagnetic “snap-shot” was nothing short of remarkable.

Figure C2 A map showing all of the expeditions undertaken by the DTM for the first World Magnetic Survey, 1904 to 1929. Each dot is a discreet station. The lines on the oceans indicate the tracks of the Galilee and the Carnegie between 1905 and 1929. Map courtesy of Carnegie Institution of Washington.

58

CARNEGIE, RESEARCH VESSEL

Although the DTM still exists, its focus on expeditionary, geomagnetic work ended with World War II. The department published two volumes in 1947, coordinated by E.H. Vestine, in which the data of the enterprise were summarized and analyzed (Vestine et al., 1947a,b). Although the DTM has continued to conduct geophysical research since then, its character has been much more varied, including research in seismology and nuclear physics. Gregory A. Good

Bibliography Good, Gregory A. (ed.), 1994a. The Earth, the Heavens and the Carnegie Institution of Washington. History of Geophysics, volume 5. Washington, D.C.: American Geophysical Union, 252 pp. Good, Gregory A., 1994b. Vision of a global physics: The Carnegie Institution and the first world magnetic survey. History of Geophysics, 5: 29–36. Trefil, J., and Hazen, M.H., 2002. Good Seeing: a Century of Science at the Carnegie Institution of Washington. Washington, D.C.: Joseph Henry Press. Vestine, E.H., Laporte, L., Cooper, C., et al.,1947a. Description of the earth’s main magnetic field and its secular change, 1905–1945. Carnegie Institution of Washington, Washington, D.C., Publication No. 578. 532 p. Vestine, E.H., Laporte, L., Lange, I., and Scott, W.E., 1947b. The Geomagnetic Field, its Description and Analysis. Carnegie Institution of Washington, Washington, D.C., Publication No. 580.

Figure C3 The Carnegie under full sail. Photo courtesy of Carnegie Institution of Washington. Photo H-112r.

Cross-references Bauer, Louis Agricola (1865–1932) Carnegie, Research Vessel Fleming, John Adam (1877–1956) Gauss, Carl Friedrich (1777–1855) Halley, Edmond (1656–1742) Humboldt, Alexander von (1759–1859) IAGA, International Association of Geomagnetism and Aeronomy Repeat Stations

CARNEGIE, RESEARCH VESSEL When Louis Agricola Bauer (q.v.) established the Department of Terrestrial Magnetism (DTM) at the Carnegie Institution of Washington (cv) in 1904, he intended to focus the DTM’s efforts strongly on an oceanic magnetic survey. The oceans represented both the largest and the most accessible gap in geomagnetic data. Bauer felt that most of the oceanic data available was not of high enough quality and that it was mainly restricted to coastal waters and trade routes. From 1905 to 1908, the DTM leased a wooden sailing vessel, the brig Galilee. Bauer had the steel rigging replaced with hemp and stripped away as much other iron as possible, but bolts and nails had to stay. The Galilee sailed over 70000 miles and established 442 magnetic stations, but its most important result was this: The Galilee demonstrated that a completely nonmagnetic ship was necessary to carry on this research. Bauer convinced the Carnegie Institution’s Board (and Andrew Carnegie) that the DTM needed its own yacht, specially designed for magnetic research. The shipyard in Brooklyn, New York that had built Kaiser Wilhelm’s Meteor (and the winner of the 1906 Atlantic cup race) designed and built a most peculiar ship: the Carnegie (Figure C3). It was 155 feet long and carried 13000 square feet of sails. Bauer stationed a magnetic investigator at the shipyard for 7 months to guarantee that no magnetic materials were used in its construction. The planking was all wood, the nails were of locust tree, and the bolts were copper or bronze. Its bronze anchors each weighed 5500 lb and instead of anchor chains,

Figure C4 A full complement of Carnegie observers in the two observing domes. Captain W.J. Peters is on the deck above. The photo was taken in the Indian Ocean in 1911. Courtesy of Carnegie Institution of Washington. Photo H-118. the ship used 11-in. hemp cable. At $115000 (perhaps ten million dollars today), the Carnegie cost more than any other scientific instrument in history. The most important part of the ship for geomagnetic research was the observation deck, a special glass-enclosed room with a circular observation dome at each end (Figure C4). The domes were of glass panels with a brass framework. Magnetic observations were made in foul weather and fair, in order to obtain the density of measurements that was required. A crowd of 3500 people turned out to launch the ship in 1909. It sailed the world’s oceans for 20 years, collecting many thousands of magnetic measurements, and sailed on seven cruises, the equivalent distance of circumnavigating the globe twelve times. It met its demise in 1929 in the harbor at Apia, Samoa, when a gasoline engine exploded during refueling. The Carnegie Institution never again operated its own research vessels. Although the British government proposed a new “nonmagnetic” research vessel in the 1930s, the Research, World War Two

CHAMP

prevented it being placed in service. The Soviet government did launch the nonmagnetic schooner Zarya in the 1950s and continued to use it for several decades. (The name Zarya is now attached to the Russian component of the International Space Station.) Simultaneously with the oceanic surveys of the Galilee and the Carnegie, the DTM (see Carnegie Institution of Washington, Department of Terrestrial Magnetism) conducted an ambitious magnetic land campaign. The goal was to gather enough magnetic data in a generation to provide a much better “snapshot” of Earth’s magnetism than was then available. While this survey was completed by about 1930, the DTM continued to send observers to selected “repeat stations” (q.v.) to obtain data useful for studying secular variation. Gregory A. Good

Bibliography Good, Gregory A., 1994. Vision of a Global Physics: The Carnegie Institution and the First World Magnetic Survey. History of Geophysics, 5: 29–36.

Cross-references Bauer, Louis Agricola (1865–1932) Carnegie Institution of Washington, Department of Terrestrial Magnetism Repeat Stations

CHAMP Challenging Minisatellite Payload (Figure C5) is a satellite mission dedicated to improving gravity and magnetic field models of the Earth. CHAMP was proposed in 1994 by Christoph Reigber of GeoForschungsZentrum Potsdam in response to an initiative of the German Space Agency (DLR) to support the space industry in the “New States” of the united Germany by financing a small satellite mission. The magnetic part of the mission is lead by Hermann Lühr. CHAMP was launched with a Russian COSMOS vehicle on July 15, 2000 onto a low Earth orbit. Initially planned to last 5 years, the mission is now projected to extend to 2009 (Figure C6). The official CHAMP website is at http://op.gfz-potsdam.de/champ/.

Satellite and orbit A limiting factor for low Earth satellite missions is the considerable drag of the atmospheric neutral gas below 600 km altitude. This

59

brought down Magsat (q.v.) within 7 months, despite of its elliptical orbit, and necessitated the choice of a higher altitude orbit for Ørsted (q.v.). To achieve long mission duration on a low orbit, CHAMP was given high weight (522 kg), a small cross section (Figure C5), and a stable attitude. It was launched onto an almost circular, near polar (i ¼ 87.3
) orbit with an initial altitude of 454 km. While Magsat was on a strictly sun synchronous dawn/dusk orbit, CHAMP advances one hour in local time within eleven days. It takes approximately 90 min to complete one revolution at a speed of about 8 km s1.

Magnetic mission instrumentation Magnetometers At the tip of the 4 m-long boom, a proton precession Overhauser magnetometer (q.v.), measures the total intensity of the magnetic field (q.v.) once per second. This instrument, which was developed by LETI, Grenoble, has an absolute accuracy of 0. In the core, C quantifies the nonmetallic constituent and b is of unit order.

74

CONVECTION, NONMAGNETIC ROTATING

If the surfaces of constant density are not perpendicular to the local acceleration of gravity, convection invariably occurs. On the other hand, if these surfaces are everywhere perpendicular to gravity, then a quiescent state is always possible. An important question is whether convection can and will occur in such a fluid body. The fluid is stable if all perturbations of arbitrary form and amplitude decay with time. On the other hand, if any small perturbation of the quiescent state leads to motions which increase in amplitude with time, the fluid is said to be (convectively) unstable. The tendency for a fluid body to convect is measured by the Rayleigh number: Ra ¼

ðDrÞgh3 r0 nk

where g is the acceleration of gravity, h is the vertical extent of the body, Dr is a measure of the density perturbations (excluding those due to pressure), n is the kinematic viscosity of the fluid, and k is the diffusivity of the factor causing density variations (either temperature or composition). Often the symbol D is used in place of k when variations are due to composition. Commonly Dr is defined such that a positive value denotes lighter fluid beneath denser and a negative value denotes denser fluid beneath lighter. There is a dynamic similarity between the situation in which the density of the fluid depends only on the relative proportions of two chemical constituents of the fluid (with Dr/r0 ¼ bDC) and that in which density depends only on the temperature (with Dr/r0 ¼ aDT ). In either case the fluid is convectively unstable when the Rayleigh number exceeds a critical value, the magnitude of which depends on the shape and nature of the boundaries of the fluid body (see Chandrasekhar, 1961). If the density is a function of both temperature and composition, then the stability of the fluid body depends on two Rayleigh numbers, measuring the density changes due to these two variables, and is significantly more complicated (see Turner, 1974). The existence of Earth’s magnetic field is very strong evidence that convection occurs in the outer core, as there is no plausible explanation for the origin of this field other than dynamo action driven by convective motions (see Geodynamo). The outer core convects because it is cooled and the principal sources of buoyancy are the latent heat and light material released as the inner core grows by solidification of the denser component of the liquid outer core (see Geodynamo, energy sources and Core convection). That is, convection is likely due to both thermal and chemical (or compositional) differences; both the thermal and compositional Rayleigh numbers are likely to be positive, particularly near the inner-core boundary. Chemical convection of another sort can occur within the uppermost portion of the inner core. It is likely that the inner core is not completely solid, but rather a mixture of solid and liquid, sometimes called a mush. The solid and liquid phases are in phase equilibrium, placing a constraint on the temperature and composition of the liquid phase. Ignoring pressure effects, this may be characterized by a simple linear liquidus relation: T ¼ T0  G½C  C0 ; where G is a constant quantifying the rate at which the temperature of pure iron is decreased by addition of (nonmetallic) impurities in the core. Using this constraint to eliminate composition from the equation for density,   b r ¼ r0  r0 a  ½T  T0 : G For almost all materials, the change of density with composition is sufficiently large that b > aG. In this case, the normal density-temperature relation is reversed; in a mush, cold fluid is less dense than warm.

In the core, pressure effects are important; and in the above formulas, one must replace the temperature by the potential temperature, or equivalently, normalize the temperature with the adiabatic temperature (see Core, adiabatic gradient). While the actual temperature in the core increases with depth, the normalized temperature decreases with depth. (This is why the outer core solidifies at the bottom, rather than the top.) The point to be emphasized is that this situation prevails in the mush at the top of the inner core and the liquid phase is prone to convective instability. Within the mush, thermal and chemical effects are no longer independent and it is not proper to consider thermal and chemical convections separately. However, given the dominance of density changes due to chemical differences, convection is properly characterized as chemical. Chemical convection within mushy zones is commonly observed in the casting of metallic alloys (e.g., see Worster, 1997). The situation in Earth’s core is simulated when an alloy is cooled from below and the denser phase freezes first. In this situation, convection within the mush has a curious structure. Upward motion induces melting of the crystals of the solid phase, thereby reducing the resistance to flow. This provides a positive feedback leading to the formation of discrete chimneys in the mush. The cold, chemically buoyant fluid in the mush convects by moving laterally to and rising up these chimneys; mass is conserved by a downward return flow of warm, chemically denser fluid into the mush away from the chimneys. It is uncertain whether chimney convection occurs in the core, where both Coriolis and Lorentz forces act on the moving fluid. David Loper

Bibliography Chandrasekhar, S., 1961. Hydrodynamic and Hydromagnetic Stability, London: Oxford University Press. Turner, J.S., 1974. Double-diffusive phenomena, Annual Review of Fluid Mechanics, 6: 37–56. Worster, M.G., 1997. Convection in mushy layers, Annual Review of Fluid Mechanics, 29: 91–122.

Cross-references Core Convection Core, Adiabatic Gradient D00 and F layers of the Earth Geodynamo Geodynamo, Energy Sources

CONVECTION, NONMAGNETIC ROTATING Various physical processes are responsible for maintaining motion, possibly turbulent, in the Earth’s fluid core (see Core motions), which is ultimately responsible for the geodynamo (q.v.). Here, however, we focus attention on thermal convection (see Core convection) driven by an unstable temperature gradient. Compressibility will be ignored except where density variations lead to buoyancy forces, the so-called Boussinesq approximation (q.v.). The basic model (Chandrasekhar, 1961) generally adopted in theoretical studies concerns a self-gravitating sphere, gravitational acceleration gr, in which fluid is confined to a spherical shell, inner radius ri, and outer radius ro. Relative to cylindrical polar coordinates ðs; f; zÞ, the system rotates rapidly about the z-axis with constant angular V ¼ O^z  ð0; 0; OÞ. The Boussinesq fluid, density r, kinematic viscosity n, has coefficient of thermal expansion a and thermal diffusivity k. The fluid is assumed to contain a uniform distribution of heat sources with thermal boundary conditions, which leads to a spherically symmetric temperature exhibiting an adverse radial temperature gradient br.

75

CONVECTION, NONMAGNETIC ROTATING

The basic static equilibrium state just described may be buoyantly unstable. The ensuing fluid motion, velocity u, causes temperature perturbations T of the basic state (see Core temperature), which satisfy the heat conduction equation Dt T ¼ br  u þ kr2 T ;

(Eq. 1a)

where Dt  ]=]t þ u  r is the material derivative: here b is a constant for the case of heating by a uniform distribution of heat sources in a full sphere but may be a function of the radius r for more general situations. In turn, the ensuing density variations arT drive the motion itself, which is governed by Dt u þ 2V  u ¼ rðp=rÞ þ gaT r þ nr u

apply to the representation of z. Within the framework of these approximations, the linearized form of the z-components of the vorticity Eq. (1c) and the equation of motion (1b) are ð]t  nr2? Þr2? C þ 2O]z W ¼ g a]f T

(Eq. 4a)

ð]t  nr2? ÞW  2O]z C ¼ gazT ;

(Eq. 4b)

respectively, where ]t  ]=]t etc. The corresponding form of the heat conduction Eq. (1a) is ð]t  kr2? ÞT ¼ bð]f C þ zW Þ:

(Eq. 4c)

2

with

r  u ¼ 0;

(Eq. 1b)

where p is the fluid pressure, and both r and g are constants. The vorticity equation obtained by taking its curl is Dt z  ð2V þ zÞ  ru ¼ gar  rT þ nr2 z;

(Eq. 1c)

where z  r  u is the relative vorticity (2V þ z is the absolute vorticity). Generally the inner and outer spherical boundaries are regarded as isothermal ðT ¼ 0Þ and impermeable ðr  u ¼ 0Þ, whereas both the cases of stress-free and no-slip boundaries are often considered separately. The physical system is characterized by four dimensionless numbers, namely the aspect ratio
, the Ekman number, the Prandtl number, and the Rayleigh number:
¼ ri =ro ;

E ¼ n=2Oro2 ;

P ¼ n=k;

R ¼ gabro6 =kn

(Eq. 2)

These parameters arise naturally for the linear convection that occurs at the onset of instability. For a given system such as the Earth’s core, we must regard
, E, and P as given. Interest then focuses on what happens with increasing R. The initial bifurcation occurs at the socalled critical Rayleigh number Rc. The strength of the finite amplitude that ensues is measured by some convenient parameter such as the Rossby number U0 =ro O based on a typical velocity U0.

Correct to leading order it is sufficient to solve the system (4) subject to the impermeable boundary condition r  u  ]f C þ zW ¼ 0 on

r ¼ ri and ro ;

(Eq. 5)

namely the inner and outer boundaries. In the early solutions of Eq. (4) for a full sphere, it was assumed that motion is localized in the vicinity of some cylinder s ¼ sc on a short radial scale, yet long compared to the azimuthal length scale Oðsc E 1=3 Þ. Accordingly, the approximation r2? ¼ s2 ]2f was also made (Roberts, 1968; Busse, 1970). The localized convection has a columnar structure, for which the evolution of its axial vorticity r2? C is governed by Eq. (4a). The buoyancy torques ga]f T that drive motion are most effective near the equator, where the radial component of gravity is strongest. Nevertheless, the large tilt of the boundaries at the ends of the column necessarily break the constraints of the Proudman-Taylor theorem and inhibit motion. This effect is quantified by the term 2O]z W , which describes the stretching of absolute vorticity and is minimized at the poles where the boundary tilt vanishes. In consequence, the preferred location of the convection is at mid latitudes, where sc/ro is roughly a half. A heuristic appreciation of the solution is obtained by making the geostrophic ansatz that C and T are independent of z (i.e., assume Eq. (4b) is simply 2O]z C ¼ 0) and that W is linear in z, which according to the boundary condition (5) implies W ¼ ðz=h2 Þ]f C, where h ¼ ðr02  s2 Þ1=2 is the column half height. Linked to these heuristic assumptions is the neglect of W in Eq. (4c), which then leaves the model equations (Busse, 1970)

The onset of instability The linear problem for the onset of instability has a long history. The early studies (Chandrasekhar, 1961) focused on axisymmetric convection. Later it was realized (Roberts, 1968) that, in the geophysically interesting limit of small Ekman number ðE  1Þ, the onset is characterized by nonaxisymmetric wave-like modes proportional to exp iðmf  ot Þ with m ¼ OðE 1=3 Þ and ðro2 =nÞo ¼ OðE2=3 Þ; they occur when R ¼ OðE 4=3 Þ. Since the rotation (as measured by the smallness of E ) is so rapid, the flow is almost geostrophic; 2V  u  rðp=rÞ (see Eq. (1b)). True geostrophic flow satisfies the Proudman-Taylor theorem (q.v.) and is independent of z, i.e., 2V  ru ¼ 0 (see Eq. (1c)). In a spherical shell, geostrophic flow is purely azimuthal and not convective. The system overcomes this difficulty by occurring on the small OðE 1=3 ro Þ azimuthal length scale upon which it is quasigeostrophic (i.e., independent of z on the OðE1=3 ro Þ length scale of the convection but dependent on z on the Oðro Þ length scale of the core radius): p  2rOC;

u  r  C^z þ W ^z;

z  ðr2? CÞ^z þ r  W ^z; (Eq. 3)

where r2?  r2  ]2 =]z2 . Note that r  u ¼ 0 does not imply that ]W =]z ¼ 0. Instead it simply means that there are small extra components of u, orthogonal to ^z, of order E1/3W ignored in Eq. (3). Similar remarks

ð]t  nr2? Þr2? C  2ðO=h2 Þ]f C ¼ ga]f T ;

(Eq. 6a)

ð]t  kr2? ÞT ¼ b]f C

(Eq. 6b)

In the absence of viscosity n ¼ 0 and buoyancy forces gab ¼ 0, Eq. (6a) possesses eastward propagating Rossby wave solutions, which under the long radial length assumption r2? ¼ s2 ]2f have frequency o ¼ 2Os2 =h2 m. When dissipation and buoyancy are reinstated, the dispersion relation becomes ~ þ ik2 Þ  E 1~sð1  ~s2 Þ1 =k þ ~s2 R ¼ 0; ~ þ ik2 Þ½ðo ðPo

(Eq. 7)

~ ¼ ro2 o=n, where we have introduced the dimensionless parameters o k ¼ ro m=s and ~s ¼ s=ro . The onset of convection occurs at the ~ is minimum value of R over ~s and k subject to the constraint that o real. This recovers our earlier critical value estimates ~sc  0:5, ~ ~ c ¼ OðE 2=3 Þ, and Rc ¼ OðE 4=3 Þ (Busse, 1970). kc ¼ OðE 1=3 Þ, o The marginal waves, like the Rossby waves travel eastward, and indeed have the Rossby wave character (not to be confused with an alternative class of solutions, namely the equatorially trapped inertial waves discussed later) in the small Prandtl number limit. For that reason, the waves that occur at finite P are referred to as thermal Rossby waves.

76

CONVECTION, NONMAGNETIC ROTATING

Though the system (6) only constitutes model equations in relation to the rotating self-gravitating sphere, they are correct for an ingenious rapidly rotating annulus model (Busse, 1970). The idea is that the annulus rotates so fast that the centrifugal acceleration produces an effective gravity in the radial s-direction. The top and bottom boundaries are slightly tilted to produce small axial velocities of exactly the form proposed in the heuristic model. This configuration provides the basis of many laboratory investigations (e.g., Busse and Carrigan, 1976; Carrigan and Busse, 1983; but see also Fluid dynamics experiments). The local solutions of both the full-sphere system (4) and annulus system (5) certainly illuminate the physical processes involved but they fail to capture the radial structure, whose resolution is essential for the proper solution of the eigenvalue problem. The correct asymptotic theory was first developed for a model system of annulus type (Yano, 1992) and later for the full-sphere Eqs. (4) and (5) (Jones et al., 2000). The essential difficulty lies in the character of the local dispersion relation o ¼ oðsÞ at given m with all other physical parameters fixed. Since o, now complex in general, varies with position, the angular phase velocity o=m varies with radius s. As a result the convection rolls, which the theory assumes are elongated in the radial s-direction (to obtain the relatively long radial length scale), are in fact twisted in a prograde sense because of their tendency to propagate eastward faster with increasing s (Busse and Hood, 1982; Carrigan and Busse, 1983). A balance is in fact achieved at a finite angle of twist which asymptotic theory can predict (Jones et al., 2000) together with the true critical Rayleigh number, which is larger than that obtained by simply minimizing over local values as in the Roberts-Busse theory. In the case of a spherical shell, the Jones et al. asymptotic theory continues to apply provided the convection is localized outside the tangent cylinder s ¼ ri of the inner sphere, which it will be for parameter values of the Earth’s core. For planets with large inner cores, however, that will no longer be true. Then convection generally occurs outside but adjacent to the inner sphere tangent cylinder at critical values predicted by local theory (Busse and Cuong, 1977). A systematic theory of this inner core tangent cylinder (q.v.) convection (Dormy et al., 2004) also shows that, with alternative heating profiles exhibiting r-dependent b, such convection may occur irrespective of the inner core size. The asymptotic theories described effectively apply to stress-free boundaries for which there are no strong additional boundary layers (see Core, boundary layers). When the boundaries are rigid, an Ekman layer forms on the boundary and the analysis of the system (4) applies outside the Ekman layer. The Ekman pumping condition on the outer sphere that replaces the impermeable condition is r  u  ðEr0 =2hÞ1=2 r  z on r ¼ ro (Greenspan, 1968) and this leads to a correction to the Rayleigh number smaller by a factor OðE1=6 Þ. Numerical results (Zhang and Jones, 1993) suggest that this correction is positive (negative) for small (large) Prandtl number, with a switch over when P is roughly unity. There are other even smaller corrections arising from weak thermal boundary layers. The validity of the asymptotic theories presented is not so clear in the limit of small Prandtl number and the double limit E ! 0, P ! 0 is not yet properly resolved. Nevertheless, numerical studies suggest that at fixed small E the onset of instability in the limit P ! 0 occurs in the form of equatorially trapped inertial waves with m ¼ Oð1Þ rather than as finite latitude Rossby waves with m 1. Asymptotic theory indicates that, for k=2Oro ¼ P1 E Oð1Þ, the critical Rayleigh number Rc for this inertial wave convection is sensitive to the kinematic boundary conditions: for stress-free conditions Rc ¼ Oð1Þ (Zhang, 1994); for no-slip conditions Rc ¼ OðE 1=2 Þ (Zhang, 1995).

The nonlinear development An early attempt at a nonlinear asymptotic theory (Soward, 1977) suggested that finite amplitude solutions existed close to the local critical Rayleigh number, which is somewhat smaller than the true (or global) critical value. There is recent evidence that such solutions exist but are

unstable. The asymptotic theory depends strongly on the large azimuthal wave number m of the convective mode. As a consequence, the nonlinearity only generates large axisymmetric perturbations of T and u, manifest dynamically in large azimuthal geostrophic flows and thermal winds (q.v.). These interact with the assumed convective mode and the generation of higher harmonics is ignored. Recent investigations, similar in spirit, taken well into the nonlinear regime have adopted Busse’s annulus model with constant tilt boundaries (Abdulrahman et al., 2000), as well as nonlinear versions of Eq. (6) adopting spherical boundaries (Morin and Dormy, 2004). Most of the earlier numerical work on nonlinear convection at small but finite E (as opposed to the perceived asymptotic limit E ! 0 of the previous paragraph) was based on Busse’s annulus model. The bifurcation sequence has been traced in a series of papers (see Schnaubelt and Busse, 1992, which should be contrasted with Abdulrahman et al., 2000). An important primary instability is the so-called mean flow instability discussed recently in the review (Busse, 2002). The corresponding studies in spherical shells following the pioneering linear investigations (see, for example, Zhang and Busse, 1987; Zhang, 1992) are now extensive and described in review articles (Busse, 1994, 2002). As the Rayleigh number R is increased, a typical scenario is that vacillating convection develops, followed by spatial modulation and a breakdown towards chaos. Surprisingly, with further increase of R more orderly relaxation oscillations are identified (Grote and Busse, 2001). It would seem that the strong shear produced by the convection actually suppresses the convection. There is then a long period over which the shear decays slowly due to viscous damping. After this relaxation, convection begins again becoming vigorous in a relatively short time and the cycle is repeated. There is recent evidence that, for annulus type model systems (Morin and Dormy, 2003), this relaxation oscillation can occur at Rayleigh numbers close to critical in the asymptotic limit E ! 0. Andrew Soward

Bibliography Abdulrahman, A., Jones, C.A., Proctor, M.R.E., and Julien, K., 2000. Large wavenumber convection in the rotating annulus. Geophysical and Astrophysical Fluid Dynamics, 93: 227–252. Busse, F.H., 1970. Thermal instabilities in rapidly rotating systems. Journal of Fluid Mechanics, 44: 441–460. Busse, F.H., 1994. Convection driven zonal flows and vortices in the major planets. Chaos, 4(2): 123–134. Busse, F.H., 2002. Convective flows in rapidly rotating spheres and their dynamo action. Physics of Fluids, 14(4): 1301–1314. Busse, F.H., and Carrigan, C.R., 1976. Laboratory simulation of thermal convection in rotating planets and stars. Science, 191: 81–83. Busse, F.H., and Cuong, P.G., 1977. Convection in rapidly rotating spherical fluid shells. Geophysical and Astrophysical Fluid Dynamics, 8: 17–44. Busse, F.H., and Hood, L.L., 1982. Differential rotation driven by convection in a rotating annulus. Geophysical and Astrophysical Fluid Dynamics, 21: 59–74. Carrigan, C.R., and Busse, F.H., 1983. An experimental and theoretical investigation of the onset of convection in rotating spherical shells. Journal of Fluid Mechanics, 126: 287–305. Chandrasekhar, S., 1961. Hydrodynamic and Hydromagnetic Stability. Oxford: Clarendon Press. Dormy, E., Soward, A.M., Jones, C.A., Jault, D., and Cardin, P., 2004. The onset of thermal convection in rotating spherical shells. Journal of Fluid Mechanics, 501: 43–70. Greenspan, H.P., 1968. The Theory of Rotating Fluids. Cambridge: Cambridge University Press. Grote, E., and Busse, F.H., 2001. Dynamics of convection and dynamos in rotating spherical fluid shells. Fluid Dynamics Research, 28: 349–368.

CORE COMPOSITION

Jones, C.A., Soward, A.M., and Mussa, A.I., 2000. The onset of thermal convection in a rapidly rotating sphere. Journal of Fluid Mechanics, 405: 157–179. Morin, V., and Dormy, E., 2004. Time dependent b-convection in rapidly rotating spherical shells. Physics of Fluids, 16: 1603–1609. Roberts, P.H., 1968. On the thermal instability of a rotating-fluid sphere containing heat sources. Philosophical Transactions of Royal Society of London, A263: 93–117. Schnaubelt, M., and Busse, F.H., 1992. Convection in a rotating cylindrical annulus. Part 3 Vacillating spatially modulated flow. Journal of Fluid Mechanics, 245: 155–173. Soward, A.M., 1977. On the finite amplitude thermal instability of a rapidly rotating fluid sphere. Geophysical and Astrophysical Fluid Dynamics, 9: 19–74. Yano, J.-I., 1992. Asymptotic theory of thermal convection in rapidly rotating systems. Journal of Fluid Mechanics, 243: 103–131. Zhang, K., 1992. Spiralling columnar convection in rapidly rotating spherical fluid shells. Journal of Fluid Mechanics, 236: 535–556. Zhang, K., 1994. On coupling between the Poincaré equation and the heat equation. Journal of Fluid Mechanics, 268: 211–229. Zhang, K., 1995. On coupling between the Poincaré equation and the heat equation: Non-slip boundary conditions. Journal of Fluid Mechanics, 284: 239–256. Zhang, K.-K., and Busse, F.H., 1987. On the onset of convection in rotating spherical shells. Geophysical and Astrophysical Fluid Dynamics, 39: 119–147. Zhang, K., and Jones, C.A., 1993. The influence of Ekman boundary layers on rotating convection in spherical fluid shells. Geophysical and Astrophysical Fluid Dynamics, 71: 145–162.

Cross-references Anelastic and Boussinesq Approximations Core Convection Core Motions Core Temperature Core, Boundary Layers Core, Magnetic Instabilities Fluid Dynamics Experiments Geodynamo Inner Core Tangent Cylinder Proudman–Taylor Theorem Thermal Wind

CORE COMPOSITION Primary planetary differentiation produced a metallic core and silicate shell surrounded by a thin hydrous and gaseous envelope. Emil Wiechert proposed this simple first order picture of the Earth at the end of the 19th century, while in 1914 Beno Gutenberg, Wiechert’s former PhD student, determined that the depth to the core-mantle boundary at 2900 km (c.f., the present day value is 2895  5 km depth, Masters and Shearer, 1995). Establishing a more detailed picture of the Earth’s core has been a considerable intellectual and technological challenge, given the core’s remote setting. The composition of the Earth’s core is determined by integrating observations and constraints from geophysics, cosmochemistry, and mantle geochemistry; a unilateral approach from any of these perspectives cannot produce a significant compositional model. Geophysical methods provide the only direct measurements of the properties of the Earth’s core. The presence and size of the core and its material properties are revealed by such studies. Foremost among these observations include (1) its seismic wave velocity and the free oscillation frequencies, (2) the moment of inertia (both of these observations plus the Earth’s mass collectively define a density profile for the core and mantle that is mutually and internally consistent

77

(Dziewonski and Anderson, 1981), (3) the distribution and secular variation in magnetic field, and (4) laboratory data on mineral physics (e.g., equation of state (EOS) for materials at core appropriate conditions). When combined with the element abundance curve for the solar system and compositional models for the Earth’s mantle (McDonough and Sun, 1995) these observations give us constraints on the mineralogical and chemical constituents of the core and mantle. Washington (1925), Birch (1952), and more recent studies (see McDonough (2004) for a recent update and literature review) have used these constraints to develop compositional models for the Earth and the core. These models consistently converge on the result that the core contains approximately 85% Fe, 5% Ni, and 10% of minor lighter components (in weight %, or about 77% Fe, 4% Ni, and 19% other in atomic proportions). The minor component in the core is an alloy of lower atomic mass that accounts for the core’s lower density when compared with that of liquid Fe at core conditions. Washington (1925), drawing upon analogies with phases in iron meteorites, recognized that the core contained a minor amount of an atomically light component (e.g., sulfide, carbide, phosphide). Birch (1964) suggested that this light component represented some 10% of the core’s mass and offered a suite of candidate elements (e.g., H, C, O, Si, or S). Anderson and Isaak (2002) more recently reviewed the relevant literature and concluded that only

5% of this light component is needed. More recently, however, Masters and Gubbins (2003) show that the density increase for the liquid outer core to the solid inner core is much greater than previously considered and Lin et al. (2005) found that Birch’s law (a linear relationship between sound velocity and density) does not hold at core pressures. Both findings have implications for the bulk core composition and imply potentially greater amounts of a light component in the outer core. The nature and proportion of the elements that make up this alloy are controlled by three main factors: (1) the behavior of elements during metal-silicate segregation, (2) the integrated pressures and temperatures experienced during core formation, and (3) whether or not there is (or has been) mass transfer across the core-mantle boundary since core formation. Studies of meteorites identify the behavior of the elements in the early solar nebula and during planetismal formation, thus identifying elements that are likely concentrated in the core. Analyses of mantle samples constrain the composition of the Earth’s primitive mantle (the combined crust plus mantle) and from this one ascertains the volatile element inventory for the planet. Studies of the secular variation of the mantle composition define the extent of core–mantle mass exchange. However, there is the proviso that we must sample this change; chemical changes occurring at the core-mantle boundary that remain isolated at the base of the mantle can only be speculated upon, but not demonstrated. Collectively, these data establish a bulk planetary composition; subtracting the primitive silicate mantle composition from this reveals the core composition. The compositional diversity of the planets in the solar system and that of chondritic meteorites (primitive, undifferentiated meteorites) provide a guide to the bulk Earth composition. However, this diversity of samples presents a problem in that there is no unique meteorite composition that characterizes the Earth. The solar system is compositionally zoned from volatile-poor planets closer to the sun to volatilerich gas giants further out. Relative to the other planets, the Earth has a relatively intermediate size-density relationship and volatile element inventory and is more depleted in volatile components than CI chondrites, the most primitive of all of the meteorites. The bulk Earth’s composition is more similar to that of some carbonaceous chondrites and less so the ordinary or enstatite chondrites (Figure C21), especially in regard to the four most abundant elements (Fe, O, Si, and Mg; in terms of atomic proportions these elements represent 95% of the inventory of elements in chondrites and the Earth) and their ratios. Thus, we need to establish the absolute abundances of the refractory elements in the Earth and the signature of the volatile element depletion pattern. The silicate Earth, or primitive mantle, encompasses the solid Earth minus the core. There is considerable agreement at the major and minor element level for the composition of the primitive mantle. The relative abundances of the lithophile elements (e.g., Ca, Al, Ti, REE

78

CORE COMPOSITION

(rare earth elements), Li, Na, Rb, B, F, Zn, etc) in the primitive mantle establish both the absolute abundances of the refractory elements in the Earth and the planetary signature of the volatile element depletion pattern. The volatile lithophile elements, those with half-mass condensation temperatures Times > 5 Ma For times older than 5 Ma, it has been proposed that significant octupole (g30 terms) are present in the time-averaged paleomagnetic field. Axial octupole terms can arise in a variety of ways as a result of poor data quality or data artifacts. A full discussion of these effects is given by McElhinny et al. (1996) and McElhinny (2004). When the

GEOCENTRIC AXIAL DIPOLE HYPOTHESIS

285

observed paleomagnetic inclinations appear to be consistently shallower worldwide than those predicted by the GAD field, spherical harmonic analyses interpret this as an axial octupole term. Such inclination shallowing can arise in the following ways: 1. Inclination errors in sediments arising from detrital remanent magnetization (DRM) processes (King and Rees, 1966), or compaction effects (Blow and Hamilton, 1978). 2. Inclination errors in lavas arising from the bulk demagnetizing effect (shape anisotropy) (Coe, 1979). This is unlikely to be a significant effect except in very thin lavas. 3. Creer (1983) has shown that the use of unit vectors in paleomagnetism can cause an artificial shallowing of the observed inclinations. 4. McElhinny et al. (1996) have shown that the incomplete magnetic cleaning (demagnetization) of Brunhes-age overprints in reversely magnetized rocks can also create an artificial axial octupole term. Viscous components that would average to the axial dipole field during the Brunhes epoch would have longer relaxation times and would require more careful cleaning procedures. 5. In the least-squares method, the mapping of the inclination anomaly curve DI leads to nonorthogonality of the spherical harmonic terms such that the estimates of the coefficients are not entirely independent. However, for small values of the quadrupole and octupole terms, the effect is small. 6. Poor data distribution can cause aliazing between the various axial multipole terms. This problem is exacerbated in the least-squares method where an infinite spherical harmonic representation is truncated to a finite series with only 1, 2, or 3 zonal harmonics. Full spherical harmonic analyses helps overcome this problem, but in this case the individual coefficient estimates should not be overinterpreted as they could arise purely from the poor quality of the data.

Random paleogeography test The random paleogeography test proposed by Evans (1976) has been much used in recent times (e.g., Kent and Smethurst, 1998), purporting to show that significant axial octupole terms were present in the timeaveraged field in pre-Cenozoic times. McElhinny and McFadden (2000) surmised that the basic assumption of random paleogeographic sampling required by the Evans (1976) test has not been fulfilled. Meert et al. (2003) and McFadden (2004) have both demonstrated that this is indeed the case. On a GAD Earth, sampling over 600 Ma will produce a GAD-like distribution of inclinations as required by the Evans (1976) test only 30% of the time. Inadequate sampling can produce false quadrupole and octupole effects. With the present global paleomagnetic data set, it now appears unlikely that the GAD hypothesis can be tested in this way even using data covering the age of the Earth (McFadden, 2004).

Figure G10 Global paleointensities plotted as a function of paleomagnetic latitude using the mean values averaged over 20 latitude bands calculated by Tanaka et al. (1995) for 0–10 Ma (dashed curve) and Perrin and Shcherbakov (1997) for 0–400 Ma (solid curve). The number of units used in each average is indicated with 95% confidence limits for each mean (crosses with dashed error bars for 0–10 Ma, solid circles with solid error bars for 0–400 Ma). The curves represent the best fits for a geocentric axial dipole field.

GAD field. This provides further confirmation that the GAD model is valid, at least to first-order, for the past 400 Ma.

Conclusions At the present time, the GAD model is a reasonable first-order approximation for the time-averaged field at least for the past 400 Ma and probably for the whole of geological time. A persistent geocentric axial quadrupole term of about 4% of the axial dipole term is present for the interval 0–5 Ma and, with possible variations, can be expected to be a permanent feature of the time-averaged field through time. More precise evaluation of the time-averaged field for 0–5 Ma will become possible when results from the time-averaged field initiative become available and are fully analyzed. Previous analyses were based on poor quality data and new results from the acquisition of new highquality data worldwide appear to indicate that the time-averaged field is much simpler than was previously thought.

Paleointensities and the GAD hypothesis The intensity (F) of the GAD field has twice the value at the poles as it does at the equator and varies with latitude (l) according to the relation F ¼ F0 ð1 þ 3sin2 lÞ1=2 ;

(Eq. 13)

where F0 is the intensity of the GAD field at the equator. Tanaka et al. (1995) and Perrin and Shcherbakov (1997) have summarized global paleointensity values for the time intervals 0–10 Ma and 0–400 Ma, respectively and calculated mean values over 20 latitude bands. These values should conform with the expected variation in Eq. (13) from the GAD hypothesis. Figure G10 shows these mean values plotted as a function of paleomagnetic latitude and the best-fit curves for a GAD field are drawn through each data set. A chi-square test indicates that the data are consistent with the latitude variation expected for a

Michael W. McElhinny

Bibliography Blow, R.A., and Hamilton, N., 1978. Effect of compaction on the acquisition of a detrital remanent magnetization in fine-grained sediments. Geophysical Journal of the Royal Astronomical Society, 52: 13–23. Briden, J.C., and Irving, E., 1964. Paleolatitude spectra of sedimentary paleoclimatic indicators. In Nairn, A.E.M. (ed.) Problems in Paleoclimatology. New York: Wiley-Interscience, pp. 199–250. Brown, L., 2002. Paleosecular variation from Easter Island revisited: modern demagnetization of a 1970s data set. Physics of the Earth and Planetary Interiors, 133: 73–81. Camps, P., Henry, B., Prevot, M., and Faynot, L., 2001. Geomagnetic paleosecular variation recorded in plio-Pleistocene volcanic rocks

286

GEOCENTRIC AXIAL DIPOLE HYPOTHESIS

from Possession Island (Crozet Archipelago), southern Indian Ocean. Journal of Geophysical Research, 106: 1961–1971. Carlut, J., and Courtillot, V., 1998. How complex is the time-averaged geomagnetic field over the past 5 million years? Geophysical Journal International, 134: 527–544. Carlut, J., Quidelleur, X., Courtillot, V., and Boudon, G., 2000. Paleomagnetic directions and K-Ar dating of 0 to 1 Ma lava flows from La Guadeloupe Island (French West Indies): implications for timeaveraged field models. Journal of Geophysical Research, 105: 835–849. Coe, R.S., 1979. The effect of shape anisotropy on TRM direction. Geophysical Journal of the Royal Astronomical Society, 56: 369–383. Cook, P.J., and McElhinny, M.W., 1979. A reevaluation of the spatial and temporal distribution of sedimentary phosphate deposits in the light of plate tectonics. Economic Geology, 74: 315–330. Cox, A., and Doell, R.R., 1960. Review of paleomagnetism. Geological Society of America Bulletin, 71: 645–768. Creer, K.M., 1983. Computer synthesis of geomagnetic palaeosecular variation. Nature, 304: 695–699. Evans, M.E., 1976. Test of the nondipolar nature of the geomagnetic field throughout Phanerozoic time. Nature, 262: 676–677. Hatakeyama, T., and Kono, M., 2002. Geomagnetic field models for the last 5 Myr time-averaged field and secular variation. Physics of the Earth and Planetary Interiors, 133: 181–201. Herrero-Bervera, E., and Valet, J-P., 2002. Paleomagnetic secular variation of the Honolulu Volcanic Series, (33–700 ka), Oahu (Hawaii). Physics of the Earth and Planetary Interiors, 133: 83–97. Hospers, J., 1954. Rock magnetism and polar wandering. Nature, 173: 1183. Irving, E., 1956. Palaeomagnetic and palaeoclimatological aspects of polar wandering. Geofisica Pura et Applicata, 33: 23–41. Irving, E., 1964. Paleomagnetism and Its Application to Geological and Geophysical Problems. New York: John Wiley. Johnson, C.L., and Constable, C.G., 1995. The time-averaged geomagnetic field as recorded in lava flows over the past 5 Myr. Geophysical Journal International, 122: 489–519. Johnson, C.L., and Constable, C.G., 1997. The time-averaged geomagnetic field: global and regional biases for 0–5 Ma. Geophysical Journal International, 131: 643–666. Johnson, C.L., Wijbrans, J.R., Constable, C.G., Gee, J., Staudigal, H., Tauxe, L., Forjaz, V-H., and Salguiero, M., 1998. 40Ar-39Ar ages and paleomagnetism of San Miguel lavas, Azores. Earth and Planetary Science Letters, 160: 637–649. Kelly, P., and Gubbins, D., 1997. The geomagnetic field over the past 5 million years. Geophysical Journal International, 128: 315–330. Kent, D.V., and Smethurst, M.A., 1998. Shallow bias of paleomagnetic inclinations in the Paleozoic and Precambrian. Earth and Planetary Science Letters, 160: 391–402. King, R.F., and Rees, A.I., 1966. Detrital magnetism in sediments: an examination of some theoretical models. Journal of Geophysical Research, 71: 561–571. Laj, C., Guillou, H., Szeremeta, N., and Coe, R.S., 1999. Geomagnetic secular variation at Hawaii around 3 Ma from a sequence of 107 lavas at Kaena Point (Oahu). Earth and Planetary Science Letters, 170: 365–376. McElhinny, M.W., 1973. Palaeomagnetism and Plate Tectonics. Cambridge: Cambridge University Press. McElhinny, M.W., 2004. The geocentric axial dipole hypothesis: a least squares perspective. In Channell, J.E.T., Kent, D.V., and Lowrie, W. (eds.), Timescales of the Internal Geomagnetic Field. American Geophysical Union Monograph, 145: 1–12. McElhinny, M.W., and Brock, A., 1978. A new palaeomagnetic result from East Africa and estimates of the Mesozoic palaeoradius. Earth and Planetary Science Letters, 27: 321–328.

McElhinny, M.W., and McFadden, P.L., 1997. Palaeosecular variation over the past 5 Myr based on a new generalized database. Geophysical Journal International, 131: 240–252. McElhinny, M.W., and McFadden, P.L., 2000. Paleomagnetism: Continents and Oceans. San Diego, CA: Academic Press. McElhinny, M.W., McFadden, P.L., and Merrill, R.T., 1996. The timeaveraged geomagnetic field 0–5 Ma. Journal of Geophysical Research, 101: 25007–25027. McFadden, P.L., 2004. Is 600 Myr long enough for the random palaeogeographic test of the geomagnetic axial dipole assumption? Geophysical Journal International, 158: 443–445. Meert, J.G., Tamrat, E., and Spearman, J., 2003. Nondipole fields and inclination bias: insights from a random walk analysis. Earth and Planetary Science Letters, 214: 395–408. Mejia, V., Barendregt, R.W., and Opdyke, N., 2002. Paleosecular variation of Brunhes age lava flows from British Columbia. Geochemistry, Geophysics, Geosystems, 3(12): 8801, doi:10.1029/ 2002GC000353. Mejia, V., Opdyke, N.D., Vilas, J.F., Singer, B.S., and Stoner, J.S., 2004. Plio-Pleistocene time-averaged field in southern Patagonia recorded in lava flows. Geochemistry, Geophysics, Geosystems, 5(3): Q03H08, doi:10.1029/2003GC000633. Merrill, R.T., and McElhinny, M.W., 1977. Anomalies in the timeaveraged paleomagnetic field and their implications for the lower mantle. Reviews of Geophysics and Space Physics, 15: 309–323. Merrill, R.T., and McElhinny, M.W., 1983. The Earth’s Magnetic Field: Its History, Origin and Planetary Perspective. London: Academic Press. Merrill, R.T., McFadden, P.L., and McElhinny, M.W., 1990. Paleomagnetic tomography of the core-mantle boundary. Physics of the Earth and Planetary Interiors, 64: 87–101. Miki, M., Inokuchi, H., Yamaguchi, S., Matsuda, J., Nagao, K., Isazaki, N., and Yaskawa, K., 1998. Geomagnetic secular variation in Easter Island, southeast Pacific. Physics of the Earth and Planetary Interiors, 106: 93–101. Morinaga, H., Matsumoto, T., Okimura, Y., and Matsuda, T., 2000. Paleomagnetism of Pliocene to Pleistocene lava flows in the northern part of Hyogo prefecture, northwest Japan and Brunhes chron paleosecular variation in Japan. Earth, Planets and Space, 52: 437–443. Opdyke, N.D., and Henry, K.W., 1969. A test of the dipole hypothesis. Earth and Planetary Science Letters, 6: 139–151. Opdyke, N.D., and Musgrave, R., 2004. Paleomagnetic results from the Newer Volcanics of Victoria: contribution to the time averaged field initiative. Geochemistry, Geophysics, Geosystems, 5(3): Q03H09, doi:10.1029/2003GC000632. Perrin, M., and Shcherbakov, V., 1997. Paleointensity of the Earth’s magnetic field for the past 400 Ma: evidence for a dipole structure during the Mesozoic low. Journal of Geomagnetism and Geoelectricity, 49: 601–614. Schneider, D.A., and Kent, D.V., 1990. The time-averaged paleomagnetic field. Reviews of Geophysics, 28: 71–96. Tanaka, H., Kono, M., and Uchimura, H., 1995. Some global features of paleointensity in geological time. Geophysical Journal International, 120: 97–102. Tauxe, L., Staudigal, H., and Wijbrans, J.R., 2000. Paleomagnetism and 40Ar-39Ar ages from La Palma in the Canary Islands. Geochemistry, Geophysics, Geosystems, (9):doi:10.1029/2000GC000063. Tauxe, L., Constable, C., Johnson, C.L., Koppers, A.A.P., Miller, W.R., and Staudigal, H., 2003. Paleomagnetism of the southwestern U.S.A. recorded by 0–5 Ma igneous rocks. Geochemistry, Geophysics, Geosystems, 4(4): 8802, doi:10.1029/ 2002GC000343. Wilson, R.L., 1970. Permanent aspects of the Earth’s non-dipole magnetic field over Upper Tertiary times. Geophysical Journal of the Royal Astronomical Society, 19: 417–437.

GEODYNAMO

Wilson, R.L., 1971. Dipole offset: the time-averaged palaeomagnetic field over the past 25 million years. Geophysical Journal of the Royal Astronomical Society, 22: 491–504. Zanella, E., 1998. Paleomagnetism of Pleistocene rocks from Pantelleria Island, (Sicily Channel), Italy. Physics of the Earth and Planetary Interiors, 108: 291–303.

Cross-references Geocentric Axial Dipole Hypothesis Pole, Paleomagnetic Time-averaged Paleomagnetic Field

GEODYNAMO Introduction The discovery of the magnetic compass (q.v.) by the Chinese dates back at least to the A.D. 1st century A.D., and it was known in Europe during the A.D. 12th century. An early idea was that the magnetic needle was attracted to the pole star, but by the year 1600, William Gilbert (q.v.) had realized that the source of the Earth’s magnetic field came from within the Earth rather than outside it. Final confirmation of this was not made until 1838, when Carl Friedrich Gauss (q.v.) used a spherical harmonic (q.v.) decomposition of the geomagnetic field to establish that the main field is internal. The important discovery that the field changes with time was made in 1634 by Henry Gellibrand (q.v.). This showed that the Earth’s field cannot be a permanent magnet, as William Gilbert had envisaged. Edmond Halley (q.v.) investigated the changes in the Earth’s field and showed that some magnetic features were drifting westward, and on this basis suggested that the interior of the Earth might be liquid. Evidence from the study of rock magnetism shows that the Earth’s magnetic field is certainly not static, but varies dramatically over long periods of time. Indeed, the Earth has undergone many complete magnetic reversals throughout its history. The most widely accepted theory is that the magnetic field is continually being created and destroyed by fluid motions in the interior of the Earth, as suggested by Joseph Larmor (q.v.) (1919). Since electricity and magnetism are commonly generated by means of dynamos, the mechanism by which the Earth’s magnetic field is created is known as the geodynamo. Permanent magnetism does occur in the crustal magnetic field (q.v.) of the Earth, and contributes a small and relatively static contribution to the main internally generated magnetic field, the core field. There are also external components of the magnetic field measured at the Earth’s surface. They can be distinguished from the internal core field partly because they increase upward rather than decrease upward, but also because they vary on a much shorter timescale. The origin of these external fields is in the Earth’s ionosphere (q.v.), where charged particles in the solar wind interact with the upper atmosphere. Since solar magnetic activity changes on a timescale of a few days, short bursts of activity known as magnetic storms and substorms (q.v.) can be detected in magnetic observatories. The external components of the geomagnetic field will not be discussed further here as they are not part of the geodynamo.

Magnetic fields on other planets and stars The Earth is by no means the only planet to have a strong internal magnetic field. Jupiter has a surface field 10 times larger than the geomagnetic field, and Saturn, Uranus, and Neptune have surface fields at least as strong. Internal fields exist on Mercury (though this field is quite weak) and on Ganymede, one of Jupiter’s moons. There are planetary and satellite dynamos (q.v.) as well as a dynamo in the

287

Earth. Planetary magnetic fields have been reviewed by Stevenson (1982, 2003) and Jones (2003). Not all the planetary magnetic fields are dominated by an axial dipole, as is the Earth. Uranus and Neptune have more complex fields. Our moon, and the planet Mars, have crustal magnetic fields (q.v.) which are very likely to have been created by internal dynamos, which operated early in their history, but have now ceased to function. The issue of whether a planet or satellite has an internally generated magnetic field depends on the physical conditions in the interiors of the planets and satellites (q.v.). The magnetic field of Sun (q.v.) is much stronger than the geomagnetic field, and some stars have fields that are stronger still. The solar magnetic field is believed to be generated by dynamo processes occurring in the Sun’s deep interior, (see the Solar dynamo).

The dynamo process According to dynamo theory, the magnetic field in the Earth is maintained by a system of electrical currents flowing through its liquid metal core. There are four key laws from electromagnetic theory that describe how the magnetic field and the currents behave in terms of concepts from elementary vector calculus. These are discussed in detail in books on magnetohydrodynamics (q.v.), or MHD for short, such as Roberts (1967), Moffatt (1978), and Davidson (2001). The first is Ampere’s law, which can be written r  B ¼ mj;

(Eq. 1)

where B is the magnetic field, j is the current density (the current flowing through a wire is j times its cross-sectional area), and m is the permeability of free space (the high core temperature makes the free space value 4p  10
7 Tm A
1 appropriate). Maxwell showed that a displacement current term should be added to this equation, but for the Earth’s core this extra term can be ignored, so Eq. (1) is sometimes called the pre-Maxwell equation. The second equation is Faraday’s law of electromagnetic induction, which is ]B ¼
r  E; ]t

(Eq. 2)

and says that if a magnetic field is varied in time, an electric field E is created. The third result from electromagnetic theory is Ohm’s law in a moving conductor, which can be written j ¼ sðE þ u  BÞ:

(Eq. 3)

Here the conductor (which may be fluid or solid) is moving with velocity u. s is the electrical conductivity of the core (q.v.), a physical quantity of fundamental importance for the geodynamo. It is estimated to be around 5  105 S m
1 . Dividing by s and taking the curl of Eq. (3) allows us to eliminate the electric field E, and dividing Eq. (1) by m and taking the curl allows us to eliminate j. The fourth law is simply r  B ¼ 0;

(Eq. 4)

which is equivalent to saying there are no magnetic monopoles. If m and s are constants, we obtain (using the vector identity curl curl ¼ grad div
r2 ) the induction equation ]B ¼ r  ðu  BÞ þ
r2 B; ]t

(Eq. 5)

where
¼ 1=ms is called the magnetic diffusivity, and its typical value in the core is
 2 m2 s
1 . The last diffusive term in Eq. (5) arises because all metals have some electrical resistance.

288

GEODYNAMO

Seismology has shown that the Earth consists of a solid inner core, the inner core radius being approximately 1100 km (see Core properties, physical ), surrounded by a fluid outer core of radius approximately 3500 km, surrounded in turn by the mantle. The electrical conductivity of the mantle is very low compared with that of the outer core, so the liquid metal outer core is the most natural seat of the geodynamo. We first consider what happens if there is no motion, u ¼ 0, so that Eq. (5) becomes the diffusion equation. When the power flowing into a laboratory electromagnet is switched off, the magnetic field usually collapses quite quickly, but the Earth’s core is very large and this slows down the decay. Equation (5) with u ¼ 0 can be solved for a spherical conducting core surrounded by an insulating mantle (see e.g., Moffatt, 1978), and the field is found to decay in time as expð
p2
t =a2 Þ, where a is the core radius. Putting in the estimates given above, the field reduces by a factor e (the e-folding time) in 20 ka. The core of the Earth is so large that even if the source of the driving were suddenly removed, the field would decay only on this very long timescale. However, the age of the Earth is around 4:5  109 y, and paleomagnetic studies show that the Earth’s magnetic field has existed for at least 3  109 y. This is much longer than the 20 ka magnetic decay time, so an energy source for the geodynamo (q.v.) is required to maintain the field. The generation mechanism for the geodynamo is that the field is maintained by core motions (q.v.) inside the liquid metal outer core of the Earth, so u is nonzero and the electromagnetic induction term in Eq. (5), r  ðu  BÞ, is important. This mechanism has now been demonstrated to work in laboratory experimental dynamos (q.v.). The same process is used in power stations to generate virtually all our electricity, though there the conducting material is an array of copper wires rather than a core of liquid metal. The velocity required to generate the magnetic field can be estimated from the size of the induction term relative to the diffusion term. This ratio is approximately U a=
¼ Rm, the magnetic Reynolds number, where U is a typical value of the velocity of the fluid inside the core relative to the frame rotating with the mantle. This can be estimated by observing the westward drift (q.v.) velocity of the magnetic field itself. Since the inhomogeneities in the field are to a large extent carried along by the fluid motion in the core, this provides a rough estimate of approximately U  2  10
4 m s
1 for the core flow, giving Rm  300 (note though that in some places in the core the velocity can be as much as 8  10
4 m s
1 , Bloxham and Jackson, 1991). Unfortunately, it is not possible to get a complete picture of the flow below the core-mantle boundary (CMB), but core-based inversions (q.v.) can extract much useful information about core velocities. Since Rm is significantly larger than unity, this ensures that the induction term, which generates the magnetic field, makes good the losses arising from magnetic diffusion. This large value of Rm arising from these natural estimates provides strong support for the dynamo hypothesis in the Earth’s core. It is of interest to seek the minimum value of Rm required to overcome diffusion. A number of bounding theorems have been proved to answer this question. The actual flow u inside the core that is driving the dynamo will be time-dependent and have a complex structure. However, some idea of the type of flow inside the core suggested by the theory of the dynamics of the core (see section below on “Core dynamics”) is shown in Figure G11.

Nondynamo generation mechanisms Although the dynamo theory is a very reasonable hypothesis, it is necessary to consider the other nondynamo theories (q.v.) which have been suggested for driving the Earth’s magnetic field. Most of these are capable of generating some magnetic field, but are not up to the task of generating the rather strong magnetic field observed. Permanent magnets are demagnetized when raised to temperatures above the Curie point temperature, which for the materials which constitute

Figure G11 A possible configuration for the fluid flow and the field inside the core (after Bloxham and Gubbins, 1989).

the core is well below any reasonable estimate of core temperature. Admittedly, the Curie point might be affected by the high pressure in the Earth’s core (see Depth to the Curie temperature). The strong variability of the geomagnetic field is the best argument against a permanent magnetism explanation. Thermoelectric effects can create currents and hence magnetic fields; indeed Stevenson (1987) has proposed this mechanism for Mercury’s field, but it cannot produce enough field to explain geomagnetism. Other possible mechanisms, with references, are listed by Merrill et al. (1996), but none has gained any widespread acceptance.

How is the geodynamo driven? A more controversial issue is the energy source of the fluid flow inside the core. The diffusion of the magnetic field is accompanied by ohmic heating, sometimes called Joule dissipation. This loss of energy requires a source to replace it. Core convection (q.v.) is the mostly widely accepted source, and the two most developed driving mechanisms are thermal convection and compositional convection. Other possibilities are precession and tidal interaction. For a more detailed discussion on these mechanisms (see Energy source for the geodynamo). The fundamental source of energy for precession and tidal interactions is the rotational energy of the Earth. If these contribute to driving the geodynamo, they would lead to a slowing down of the Earth’s rotation, i.e., to a lengthening of the day (see Decadal length of day variations). Earth tides and oceanic tides are also slowing down the rotation rate of the Earth, so a precession-driven dynamo would give additional slowing. Thermal convection derives its energy from the cooling down of the Earth, although if there is radioactive heating in the core, this could contribute too. The issue of whether there is radioactivity in the core has a long history and remains highly controversial (see Radioactive isotopes and their decay). There is no doubt that radioactivity is important in the mantle, but whether radioactive elements were carried into the core during formation is a very challenging problem for geochemists. The primordial heating of the Earth occurred as part of the core formation process through gravitational collapse, so the thermal energy of the core may be thought of as originating from gravitational energy.

GEODYNAMO

289

Another source of energy for driving the dynamo is compositional or chemical convection (q.v.) (Braginsky, 1963; J. Verhoogen (q.v.), 1961). The solid inner core of the Earth contains a higher fraction of iron than the fluid outer core (see Inner core composition), and so is more dense than the fluid outer core. As the inner core grows, due to core cooling, iron is deposited on the inner core, and light material is released at the moment of freezing. This light material rises due to buoyancy, and stirs up the outer core. Since the net effect of this process is to move heavier material toward the center, gravitational energy is liberated. It is slightly surprising that the core first freezes near the center of the Earth, where the temperature is highest. This happens because the melting temperature of iron in the core increases strongly with pressure. The question of whether tidal forcing or precessional forcing (see Precession and core dynamics) is large enough to drive the geodynamo has been discussed by Malkus (1994). The typical height of an Earth tide inside the core is expected to be about h  0:15 m, and precession gives a similar displacement. The typical radial velocity is then Oh  10
5 m s
1 . The azimuthal velocity might be a little larger, so a value not that far off the convective velocity U might be achieved. There is a difficulty because this velocity is fluctuating on a timescale of a day, whereas a flow varying only on a much longer timescale (a 1 ka) is needed to sustain a dynamo. Some mechanism for converting oscillatory flows into steady flows is required, and some suggestions for overcoming this difficulty have been proposed (Kerswell, 2002; Aldridge, 2003). A recent example of a precessiondriven dynamo has been given by Tilgner (2005), though the conditions under which these dynamos operate are still very far from Earth-like.

a nonadiabatic stably stratified region where the temperature gradient is significantly subadiabatic, thus reducing the conducted flux, or if compositional convection stirs this subadiabatic region the heat flux produced by compositional stirring is negative (back into the interior). A further possibility is that light material released at the ICB as the inner core forms accumulates just below the CMB. This possibility has been called by Braginsky (1993) the “inverted ocean,” since it would consist of a sea of relatively light material floating up against the CMB. Unfortunately, our knowledge of the heat flow across the Core-Mantle Boundary is imprecise, and we cannot yet be certain whether an inverted ocean exists or not. We know that at the present time there are 44 TW of heat coming through the Earth’s surface, and that most of that originates in the mantle. Estimates of the CMB heat flux range from 3 TW (Sleep, 1990) to 15 TW (Roberts et al., 2003), and are highly dependent on whether there is core radioactivity or not. In principle, mantle convection simulations could tell us the heat flux across the CMB, since this is one of the bottom boundary conditions to such simulations, and so will affect the structure of mantle convection. However, to date the uncertainties surrounding mantle convection modeling (see e.g., Schubert et al., 2001) preclude this possibility. Numerical simulations of the geodynamo (q.v.), such as those of Glatzmaier and Roberts (1995, 1997), have typically adopted a compromise value of around 7 TW for the CMB heat flux, which just makes the core fully convective. The overall energy balance in the core, ignoring any contributions from precession or tides, can be written

Energy balance for a convective dynamo

where QS is the rate of cooling, QCMB is the heat flux passing through the CMB, QICB is the heat flux passing through the ICB, QL is the latent heat released by the freezing process on the inner core, Verhoogen (q.v.), 1961 QG is the gravitational energy released by the same process, and QR is the heat produced by radioactivity (if any). For the sake of definiteness, we give reasonable estimates for each of these quantities (see e.g., Braginsky and Roberts, 1995; Roberts et al., 2003), but it cannot be emphasized too strongly that all of these estimates are uncertain, and it would be very surprising if improvements in our understanding of high pressure physics did not lead to radical revision of these estimates over the next decades. Discussions of some of the physics that goes into these estimates can be found in the articles on core composition (q.v.), core density (q.v.), and core properties, physical (q.v.). With this proviso, we take the cooling rate of the core to be QS  2:3 TW, the latent heat released at the ICB to be QL  4:0 TW, the gravitational energy released at the ICB QG  0:5 TW, and QICB  0:25 TW. This gives QCMB  7 TW, the value used by Glatzmaier and Roberts (1997). QR , the heat released by core radioactivity is extremely uncertain, as mentioned above. Perhaps the best hope is that an improved understanding of the geodynamo might enable us to constrain QR , as might a better understanding of the thermal history of planetary interiors. It might seem surprising that the ohmic dissipation generated by the magnetic field does not appear in the energy balance. This is because it is balanced by the work done by buoyant convection. Some of the heat energy flowing through the core is extracted to drive the fluid motions, only to be returned in full by the ohmic and viscous dissipation.

This naturally leads on to the question of how efficient the geodynamo actually is. In the solar dynamo (q.v.), only a tiny fraction of the heat energy pouring out of the Sun is converted into the solar magnetic energy: is the Earth more efficient in this respect? To investigate this we need to examine the overall energy and entropy budget in the Earth’s core, and to do this we need a model for the Core temperature of the Earth (q.v.). Since we believe the fluid outer core is stirred it is reasonable to assume it is approximately adiabatically stratified, so the temperature gradient is given by the adiabatic gradient in the core (q.v.). This is dependent on a quantity called the Grüneisen parameter (q.v.) which itself varies somewhat between the inner core boundary (ICB) and the Core-Mantle boundary but if we adopt the currently accepted estimates, we find that the temperature drop from the ICB to the CMB is about 1100 K (Braginsky and Roberts, 1995; Roberts et al., 2003). In principle, the melting temperature of iron at high pressure (see melting temperature of iron in the core), should determine the ICB temperature, but unfortunately, the impurities expected in the core significantly depress the melting temperature, so this is somewhat uncertain. A reasonable guess at the present time (Roberts et al., 2003) is TICB ¼ 5100 K, giving TCMB ¼ 4000 K. An important issue is the amount of thermal conduction in the core (q.v.). Again there is uncertainty due to the difficulty of the very high core pressures but if we take  40 W m
1 K
1 as our estimate, the conducted flux down the adiabat near the CMB is Qcond  6 TW, and only  0:25 TW near the ICB. There are two reasons for the big difference; first the adiabatic temperature gradient decreases somewhat with depth, so the conducted heat flux per square meter decreases with depth, but secondly the geometry of the much larger surface area near the CMB means that a much larger number of terawatts is conducted there. The critical question is whether the actual heat flux out of the core is greater or less than this conducted flux. If the actual flux exceeds the conducted flux Qcond at any point in the fluid outer core, the excess flux is carried by convection. If however the actual heat flux is less than Qcond then there will be a region where there is a subadiabatic temperature gradient. This is most likely to occur just below the CMB. Here there is either

QS ¼ QCMB
QICB
QL
QG ð
QR Þ:

(Eq. 6)

Entropy balance in the core To get an estimate which involves the magnetic field we need to consider the rate of entropy production (Hewitt et al., 1975; Gubbins, 1977; Gubbins et al., 1979; Braginsky and Roberts, 1995; Roberts et al., 2003). As magnetic field diffuses, it generates heat through ohmic dissipation, Z (Eq. 7) QD ¼ m
j2 dv

290

GEODYNAMO

Viscous dissipation is believed to be negligible in comparison with ohmic dissipation, because the magnetic energy is much larger than the kinetic energy in the core, and we obtain (Roberts et al., 2003)       TD  TCMB QD ¼ þ QS þ QR QICB þ QL 1
TCMB TICB    X TCMB G 1
Q
(Eq. 8) T CMB TM where TD is the average temperature where dissipation occurs, and TM is the average temperature of the fluid outer core. As in a Carnot heat engine, there are “efficiency factors” which mean that thermal convection cannot extract more than ð1
TCMB =TICB Þ  20% of the heat flux passing through the core, and for sources which are distributed throughout the core, such as cooling and radioactivity, the efficiency factor may be as low as 10%. Using the estimates above, we find that in the absence of radioactivity the dissipation is around 1 TW, with about 0.5 TW being from compositional convection QG , and 0.5 TW being from thermal convection (see also Christensen and Tilgner, 2004). Most current numerical simulations of the geodynamo (q.v.) have an ohmic dissipation that is less than 1 TW, but this is most probably because they are not yet in the right parameter regime for the geodynamo. Roberts and Glatzmaier (2000) have shown that as the resolution is increased the dissipation rises (see also Kono and Roberts, 2002). Again some geodynamo simulations show a viscous dissipation rate as large as the ohmic dissipation rate, but this again is likely to be a consequence of the numerical difficulties in achieving the right parameter regime.

component of flow in the longitudinal direction is called the azimuthal flow.) The time dependence of a kinematic dynamo is proportional to expðst Þ, and s is in general complex. If s happens to real, we say that the dynamo is steady, but if s has an imaginary part we say it is oscillatory. The solar dynamo (q.v.) reverses continually in an approximately periodic 22-year cycle, and is therefore naturally modeled as an oscillatory dynamo, but the geodynamo is comparatively steady, although it does reverse (irregularly) occasionally (see Reversals, theory). Kinematic dynamo calculations (Roberts, 1972) showed that meridional circulation helps to make a dynamo steady, indicating that meridional circulation could play an important role in the geodynamo. Recent work on kinematic dynamos relevant to the Earth can be found in e.g., Gubbins et al. (2000a,b) and Sarson (2003).

Mean field dynamo models In the 1960s, mean field dynamos (q.v.) were developed (Steenbeck et al., 1966) see also Moffatt (1978). The basic idea is that small-scale core turbulence (q.v.) could help to generate the large-scale field. If the magnetic field is B þ b, and the velocity is U þ u where b and u are the small-scale turbulent parts, and B and U are the mean parts averaged over the small scales, then the induction equation averaged over the small scales becomes ]B ¼ r  ðU  BÞ þ r  ðu  bÞ þ r2 B: ]t

(Eq. 9)

Provided certain criteria are satisfied the new term can be written r  ðu  bÞ ¼ r  aB

(Eq. 10)

Kinematic dynamo models A major difficulty in constructing geodynamo models was noted by Cowling (1934). He showed that it is impossible to maintain an axisymmetric magnetic field by means of dynamo action, a result now known as Cowling’s theorem (q.v.). This was the first of a class of antidynamo theorems (q.v.) showing that fields with too much symmetry cannot be created by dynamo action. The earliest geodynamo models based on electromagnetic induction appeared in the 1950s (Bullard (q.v.) and Gellman, 1954), thus showing that although Cowling’s theorem (q.v.) is a mathematical inconvenience (because symmetric solutions of partial differential equations are much easier to find) it is not a fatal objection to the dynamo idea. These models only considered the induction equation (5), the velocity field of the fluid flow being imposed. This is known as the kinematic dynamo problem, the significance of the word kinematic being that the velocity field is simply prescribed, rather than taking the flow to be a solution of an equation of motion. In practice fairly simple flows are chosen, but despite the induction equation (5) being linear in B, the nonaxisymmetric three-dimensional nature of the solutions makes it hard to solve, even with today’s fast computers. Because of the linearity, the induction equation (5) has either exponentially growing or decaying solutions. We say that a flow is a kinematic dynamo if the solutions grow rather than decay. Fast dynamos, that is dynamos whose growth rate remains finite in the limit of small magnetic diffusion (Rm ! 1) are usually studied in the kinematic dynamo approximation. It was noticed early on in the study of kinematic dynamos that quadrupolar dynamos, that is dynamos with a dominant quadrupole field (see symmetry and the geodynamo) rather than a dipole field can also be obtained, depending on the chosen flow. The issue of what makes some flows give dipolar dynamos and others quadrupolar dynamos is not yet fully resolved, so we cannot yet give a definite answer to the question why is the Earth dipole dominated rather than quadrupole dominated. Numerical simulations of the geodynamo (q.v.) usually, but not always, give dipolar fields. Another issue to arise out of kinematic dynamo studies was the role of meridional circulation, which is the axisymmetric component of flow in the radial and latitudinal direction. (The axisymmetric

where a is in general a function of position in the core. More generally, a is a tensor, but it is usually taken to be isotropic so that the comparatively simple form of Eq. (10) can be used. This term in the induction equation is known as the a-effect. The types of turbulent flow which give a nonzero a are those with nonzero mean helicity. The helicity of a flow is u  v where v ¼ r  u is the vorticity. Flows with a “screw-type” motion have helicity. Parker (1955) pointed out that a blob of hot fluid, rising because of its buoyancy, would twist as it rises due to the action of Coriolis force in a rotating body like the Earth or the Sun. This gives a natural source of helicity. The great advantage of mean field models is that axisymmetric mean fields are now possible solutions of Eqs. (9) and (10). The nonaxisymmetric components of the field which must be there because of Cowling’s theorem (q.v.), do not need to be calculated explicitly. This simplification made it feasible to include dynamics into dynamo models, to obtain what are now known as intermediate dynamo models (see section below on “Intermediate dynamo models”). Mean field models have been extensively used in the solar dynamo (q.v.) problem, because they give rise to dynamo waves (q.v.), which can explain many features of the solar dynamo (q.v.). The main disadvantage of mean field models is that the distribution of a over the core depends on the form of the turbulence in the Earth’s core. This cannot be observed, and indeed it is not even known whether such turbulence satisfies the criteria required for a mean field dynamo to be valid. If it were the case that the spatial form of a was not critical, this would be less important, but unfortunately numerical calculations indicated that the a-distribution can make a very big difference to the types of dynamo generated (Hollerbach et al., 1992).

Core dynamics Some of the most interesting properties of the geodynamo are concerned with the dynamics that is the force balance in the core. This is described by the Navier-Stokes equation with additional forces which are important in the geodynamo. A simplification that is often

GEODYNAMO

used is the Boussinesq approximation (see the article on Boussinesq and anelastic approximations), in which density variations are ignored except where they are multiplied by g, so that buoyancy forces can be included. This approximation is not strictly valid in the core, as density variations of order 20% can occur, but it is nevertheless a useful simplification of equations which are difficult to solve, and we adopt it here. The equation of motion for a rotating fluid in which Lorentz (magnetic) forces, buoyancy, and viscosity act can be written   Du þ 2V  u ¼
rp þ j  B þ rnr2 u þ rgaT ^r: r Dt (Eq. 11) Here r is the density (assumed constant), u is the velocity, p is the pressure, j is the current density, B is the magnetic field, n is the kinematic viscosity, g is gravity, a is the coefficient of expansion, T is the temperature, and ^r is the unit vector in the direction of gravity, the radial direction. Not surprisingly, such a complex equation with so many different forces leads to many different physical effects. Note that the Coriolis acceleration, 2V  u is often thought of as a force (when multiplied by r) in a rotating frame. D/Dt is the convective derivative, ]=]t þ u  r. Since the temperature occurs in this equation, another equation determining T is needed, ]T ¼ kr2 T
u  rT þ Q: ]t

(Eq. 12)

Here Q is the heat source and k is the thermal diffusivity. A number of dimensionless parameters can be formed from these equations. The Ekman number, E ¼ n=2Od 2 , measures the relative importance of the viscous force to the Coriolis force. To estimate this we need to know the core viscosity (q.v.), but E is certainly very small, probably of order 10
15 in the core. There are two diffusion coefficients, n and k, and a third,
, occurs in the induction equation (5), so we have two dimensionless ratios, Pr ¼ n=k, known as the Prandtl number, and Pm ¼ n=
known as the magnetic Prandtl number. Another useful combination involving the magnetic field is the Elsasser (q.v.) number L ¼ B20 =2Omr
, where B0 is a typical value of the magnetic field. The Elsasser number at the CMB, where the field is typically 0.5 mT, is about 0.25, but inside the core the field is likely to be stronger, and the Elsasser number is believed to be of order unity or a little more.

Rapidly rotating convection Much of our understanding of convection in rapidly rotating systems comes from the problem of thermally driven nonmagnetic rotating convection (q.v.), and the problem of convection in an imposed magnetic field (usually either a uniform magnetic field or one with a simple geometry) which is called magnetoconvection (q.v.). The articles on these topics give a more detailed discussion, but there is a very large literature on both these topics, and some understanding is essential for an appreciation of the geodynamo. Our knowledge has been enhanced by fluid dynamic experiments (q.v.) and experimental dynamos (q.v.). It turns out that magnetic field affects the convection considerably, but at least the nonmagnetic case gives us a framework on which to build understanding of the magnetic case. The onset of instability in a rapidly rotating sphere is now fairly well understood (Roberts, 1968; Busse, 1970; Jones et al., 2000). Rapid rotation means the Ekman number, E ¼ n=2Od 2 is small, and in this case convection occurs in a columnar form, somewhat as illustrated in Figure G11. However, in nonmagnetic convection the columns are tall but very thin, so that instead of the three columns shown in Figure G11, a much larger number of order E
1=3 fit into the sphere. These columns are known as Busse columns (Busse, 1970;

291

Busse and Carrigan, 1976), and their basic structure can be understood in terms of the Proudman-Taylor theorem (q.v.). The onset of convection in a uniformly heated sphere with stress-free boundaries in the absence of rotation occurs when the Rayleigh number Ra ¼ g abra6 =kn exceeds Ra crit ¼ 1 546 (Chandrasekhar, 1961), where b ¼ Q=3k, so the static gradient is br. Gravity is taken to be
gr, as in a uniform sphere gravity increases linearly from the center. In the small Ekman number limit, i.e., in rapid rotation, the critical Rayleigh number for the onset of convection approaches 4:117E
4=3 at Prandtl number Pr ¼ 1 (Jones et al., 2000), which gets very large at small E. Although the onset of instability is strongly delayed by the rotation as the Rayleigh number is increased above critical, once the critical value for convection is achieved the flow becomes time-dependent, and then aperiodic, rather quickly. Abdulrahman et al. (2000) showed that the bifurcations to chaotic convection occur in a relatively small window in Rayleigh number space Racrit < Ra < Rcrit þ OðE
2=3 Þ. Convection continues to occur in tall thin columns in the fully nonlinear regime provided the Rossby number Ro ¼ U =aO remains small, although the thickness of the columns does increase somewhat over its very thin linear value, and the columns tend to become rather transient, and occur fairly randomly throughout the sphere. Experiments suggest that in the nonlinear regime, inertial effects take over from viscous effects, and the leading order balance is between inertia, Coriolis force, and buoyancy (for details see Aubert et al., 2001). When a magnetic field is applied, the Proudman-Taylor theorem (q.v.) no longer controls the pattern of convection, as the Lorentz force becomes important. An important effect of magnetic field is to increase the thickness of the Busse columns (see e.g., Jones et al., 2003), which occurs even at small Elsasser number L  OðE1=3 Þ field strengths. If the Elsasser number is further increased to O(1) values, the columns are still visible, but the number fitting into the sphere is much reduced. Because magnetic field can break the ProudmanTaylor constraint, the critical Rayleigh number is reduced in the presence of a L  Oð1Þ field, so magnetic field can help the system convect, see e.g., Fearn et al. (1988).

Dynamical regime in the core In the magnetic case it is not unreasonable as a first approximation to assume that the large-scale flows and fields have dominant length scales of the size of the core radius. It is then possible to do a useful scale analysis of the dynamics of the Earth’s core using Eq. (11) (see e.g., Braginsky and Roberts, 1995; Starchenko and Jones, 2002), although we must keep in mind this can only give order of magnitude estimates, and more exact results require that the geometry of the convection and the spherical domain be taken properly into account. Some of the physical quantities in the core are quite accurately known, others somewhat less so, but here we only need order of magnitude estimates, to see which terms are large and which small in Eq. (11). We take as a typical value for the velocity in the core U ¼ 2  10
4 m s
1 , see section “The dynamo process.” For D/Dt we take 1/300 years, or 10
10 s
1. Significant changes to the field occur on this timescale, though a full reversal of the field might take ten times as long. The density r  104 kg m
3 in the core. The Earth’s angular velocity O ¼ 7  10
5, and already we can see that the magnitude of the inertial term jDu=Dtj is very small compared to the Coriolis term j2V  uj. We turn next to the Lorentz force term, j  B. The field at the CMB is about 5  10
4 T. The current is r  B=m, and since the curl is a space derivative, we crudely estimate it at jBp=amj where a is the core radius, 3:5  106 m. The Coriolis term j2rV  uj  3  10
4 N m
3 while the Lorentz term jj  Bj has magnitude 2:5  10
7 N m
3 . This is smaller than the Coriolis term, but it is generally believed that the invisible field inside the core is at least 10 times larger than the visible field, which leaves the core, increasing the magnitude of the Lorentz term. Furthermore, the magnetic field pattern at the CMB suggests that it varies on a shorter length scale than the core radius, which will

292

GEODYNAMO

enhance j. It is therefore likely that the Lorentz force will be comparable to the other forces at least in some places. The main argument for a significant Lorentz force is that the induction equation (5) is linear in B and therefore cannot determine the field strength. This must be limited by the Lorentz force term, so j  B must be significant at least somewhere in the core, and most dynamo models suggest that equilibration of the magnetic field takes place when the Lorentz and Coriolis forces are comparable. One of the difficulties of numerical simulations is that they require a substantial amount of dissipation, in particular viscous dissipation, for numerical stability. Thus the balance in the models is often between Coriolis force and viscosity, with the Lorentz force playing only a small role in limiting the field strength without greatly affecting the flow. This is known as the weak field regime, in contrast to the strong field regime in which the Lorentz force is comparable to the other large forces in the equation of motion. This is the essential difference between the strong and weak field dynamo regimes. We address this issue further in the section “Taylor’s constraint.” The buoyancy force drives the flow, and so we would expect this force to be comparable to the Coriolis and Lorentz force terms. If we estimate g at 10 ms
2 , and the coefficient of expansion a at 10
5 K
1 , we find that the typical temperature fluctuation is in the range 10
3 –10
4 K. It is rather remarkable that such tiny temperature fluctuations in the core are driving the whole dynamo process, but we should remember that core flows are very slow; even a lethargic snail could go faster! We also have a consistency check here, because the convective heat flux Z (Eq. 13) Qconv ¼ rcp ur T dS ; S

can be estimated, and when we remember that T here is the very small superadiabatic temperature, which is 10
3 –10
4 K only, we reassuringly discover that this flux is comparable to that suggested by the energy arguments of the section above “Energy balance for a convective dynamo.” The viscous term in the equation of motion turns out to be very small, and so is not in the primary force balance, except in thin boundary layers in the core (q.v.). It is however necessary to include the viscous term in the force balance when doing numerical simulations, to ensure numerical stability. Pressure forces are always important in convecting fluids; when buoyant fluid rises, the pressure forces push the ambient fluid aside to make room for the hot rising fluid. The main balance of forces in the core is therefore between Lorentz force, Coriolis force, buoyancy force, and pressure force. This is known as MAC balance, M standing for magnetic, A for Archimedean force (Buoyancy), and C for Coriolis force (Braginsky, 1967). The word “magnetostrophic” is sometimes used as a synonym for MAC balance, though more commonly magnetostrophic balance refers to situations where buoyancy is absent. Pressure can usually be eliminated by taking the curl of the equation of motion, which converts it into a vorticity equation. If compositional convection is present, an additional equation of similar form to the temperature equation (12) is required for x, the fraction of light material, and the resulting buoyancy must be included in the equation of motion (see e.g., Braginsky and Roberts, 1995). Equations (5), (11), and (12) can be solved numerically in dynamo simulations, but an important issue is whether laminar or turbulent values should be used for the diffusivities. Laminar values of k and n are so small it is hard to believe they can be relevant to the core. Turbulent values are therefore generally used, and this raises the interesting question of the nature of turbulence in the Earth’s core. The inertial terms, which are responsible for cascading energy down to small scales in “normal” fluids are only effective on tiny length scales in the core, so core turbulence may be quite different from normal turbulence. In particular, the diffusion processes may be quite anisotropic, as suggested by Braginsky and Meytlis (1990).

Waves and instabilities Calculating how the field and the flow evolves in the core can only be done using sophisticated numerical dynamo models. However, insight can be obtained into the rather complicated nature of the dynamics of the core by considering simple equilibrium flows and fields, and analyzing the magnetohydrodynamic waves (q.v.) and instabilities that these simple equilibria undergo when they are slightly perturbed. This leads to linear problems which can be investigated mathematically. Perhaps the most fundamental problem is the case of a uniform magnetic field with fluid at rest (or in uniform motion) with a uniform temperature gradient imposed. Linearizing Eqs. (5), (11), and (12) about a uniform magnetic field B0 and constant temperature gradient b, so that B ¼ B0 þ b0 , the velocity is u0 and the temperature is T ¼ bz þ T 0 . The primed quantities are assumed small so that squares and products of primed quantities are neglected, and then wave solutions with u0 ; b0 , and T 0 proportional to expiðk  x
ot Þ can be found. Equations (5), (11), and (12) then give the dispersion relation between k and o. If the temperature gradient is stabilizing (subadiabatic) then real values of o are found if diffusion is neglected. If diffusion is retained, o is complex with a negative imaginary part corresponding to damped oscillations. If the temperature gradient is superadiabatic, growing waves (instabilities) are found provided the diffusion is not too large. The dispersion relation is actually very complicated, but it can be simplified when as in the Earth’s core, different types of waves have very different frequencies (Fearn et al., 1988). The fastest waves are inertial waves (see the article on gravity-inertio waves and inertial oscillations), which balance the Coriolis and inertial accelerations, and the inertial wave frequency is given by oC ¼ 2ðV  kÞ=jkj. The typical period is therefore of the order of a day. These waves have not yet been observed in the core, but they are expected to be driven by tidal forcing. Alfvén waves (q.v.) result from a balance of inertia and Lorentz force in the equation of motion, when combined with the induction equation. These waves have frequency oM ¼ ðB0  kÞ=ðmrÞ1=2, and travel at the Alfvén speed, B0 =ðmrÞ1=2 , which for a moderate 1 mT core field is around 10
2 m s
1 , giving around 60 years for the wave to travel round the core. An important class of Alfvén waves are those corresponding to azimuthal motion constant on cylinders, which are called torsional oscillations (q.v.). These are believed to be important in the core; see “Taylor’s constraint” below. Another timescale comes from the temperature gradient, from the balance of buoyancy and inertia in the equation of motion, combined with the temperature equation (12). The frequency of internal gravity waves is oA ¼ ðgabÞ1=2 kH =kj, where b is the subadiabatic temperature gradient, and kH is the component of k perpendicular to gravity. In a convectively unstable region, the temperature gradient is superadiabatic and b is negative. Then oA is imaginary, which corresponds to an exponentially growing unstable mode, with joA j being the growth rate. With the estimate of 10
4 K for a typical superadiabatic temperature fluctuation, b  10
4 =a, giving a typical growth rate of about 1 year. When all the terms in the equation of motion are present, the dispersion relation gives a fast inertial wave and a slow wave in which only the time-derivative terms in the induction and temperature equations are important, inertia being negligible. These slow waves are known as MAC waves and have frequency oMAC ¼ oMC ð1 þ o2A =o2M Þ1=2 ;

where oMC ¼ o2M =oC : (Eq. 14)

When joA j > joM j, oMAC is imaginary, corresponding to a growing convective mode. The rate of growth is slow, since tMC ¼ 2p=oMC is of the order of some thousands of years. It is comparable to the magnetic diffusion time, since the ratio tdiff =tMC ¼ B20 =2mrO
¼ L and the Elsasser number L has a value of O(1) in the core. The MAC growth

GEODYNAMO

rate will be a little larger than 1=tMC because joA j > oM , but these slow growth times are consistent with the time taken for the typical convective velocity to take fluid across the core, tconv 1010 s, so the dynamical picture does seem to be self-consistent. The main role of the Coriolis acceleration and the Lorentz force is to constrain the convection and slow it down from an unimpeded growth rate of about a year down to tconv . There are, however, some difficulties. Although for general wave vectors k the MAC wave timescale is slow, it is much faster if k and V are perpendicular. This is the case for motions which are independent of z, the coordinate parallel to the rotation axis. Then oMC is infinite, which means we have to restore inertia, and we then get the much faster torsional wave frequency. It is also possible for k to be perpendicular to both V and B0 . Such waves have motion only along “plates,” planes containing the rotation vector and the magnetic field vector. For these waves Eq. (14) is degenerate, with both oM and oC zero. Such motions are unaffected by the field and the rotation and so will grow on the much shorter oA timescale. We therefore expect these modes to dominate small scale convection in the core (Braginsky and Meytlis, 1990). The above analysis assumes uniform magnetic fields. Actually the field inside the core is likely to have a complicated structure. Such fields often become unstable, and allow magnetic energy to be converted into kinetic energy. In the Sun, this can happen on a very short timescale, and solar flares are the result. Nothing quite as dramatic is expected in the Earth’s core, because magnetic diffusion is much more important there. Nevertheless magnetic instabilities could be important in driving shorter timescale motions and in controlling the strength of the magnetic field. Magnetic instabilities generally become important when the Elsasser number is O(1) (Fearn, 1998), (see also Core magnetic instabilities) so this may well be an important mechanism in the core.

Taylor’s constraint One interesting consequence of the MAC balance in the core pointed out by J.B. Taylor (1963) was that the geomagnetic field must satisfy a special condition, known as Taylor’s condition (q.v.). Azimuthal flows that are constant on cylinders coaxial with the rotation axis play a rather special role in rotating flows, because the Coriolis acceleration can be balanced entirely by the pressure force. Such flows are called geostrophic flows. The special feature of the cylinders is that a column of fluid moving along such a cylinder has constant height, so no vortex stretching occurs. J.B. Taylor noted that if we integrate the net azimuthal force over a cylinder, the contribution from the Coriolis term is zero, as is the pressure term, and since buoyancy acts radially, not azimuthally, the only force left is Lorentz force. It follows that Z ðj  BÞf ds ¼ 0; (Eq. 15) S

where S is any cylindrical surface not intersecting the inner core. This is known as Taylor’s constraint, and provided viscosity is negligible it must be satisfied in the core. Since the current is related to the field by Ampere’s law, (1), Taylor’s condition is a constraint on the magnetic field. An interesting question is what happens if we solve an initial value problem in which Taylor’s constraint is not initially satisfied? We must then restore part of the inertial term, r]u=]t , and the system responds with a fast torsional oscillation (q.v.), in which Lorentz forces directly accelerate the fluid. These oscillations have a period of some tens of years, and there is evidence that they occur in the core, as they may explain the decadal variations of the length of day (q.v.) (Jault et al., 1988; Jackson, 1997; Jault, 2003), and possibly shorter period elements of the geomagnetic secular variation (q.v.), such as the so-called geomagnetic jerks (q.v.). On the much longer dynamo timescale, these oscillations would then decay away, restoring a field

293

satisfying Taylor’s condition. We can view the MAC balance state satisfying Taylor’s constraint as a slowly evolving equilibrium state, with the magnetic field slowly changing as the convection moves the core fluid, on a timescale of hundreds of years. Short-term variations of the field on small length scales can be explained by convection, as variations on a short length-scale ‘ would occur on short timescale ‘=U , but the short-term variations on long length scales, such as give rise to the decadal variations of the length of day, could be explained if torsional oscillations were continually being excited, so that the core is always vibrating around its equilibrium MAC state. How these torsional oscillations are excited is not yet understood.

Intermediate dynamo models The full time-dependent three-dimensional equations (5), (11), and (12) are difficult to solve numerically, so it is only recently that computers have become fast enough to make it feasible to study them. Numerical simulations can also be hard to interpret. A compromise position has been to still retain the a-effect, thus allowing axisymmetric solutions to be found, but to include some dynamics, normally the meridional circulation and the azimuthal flow driven by the magnetic field. Intermediate dynamo models have often been used to investigate how Taylor’s constraint could be met. It was suggested by Malkus and Proctor (1974) that the magnetic field would drive flows of exactly the right type to generate fields satisfying Taylor’s constraint. Soward and Jones (1983) looked at an intermediate plane layer model which did indeed lead to solutions satisfying Taylor’s constraint, in accord with the Malkus-Proctor scenario. With values of a close to critical, the magnetic field did not satisfy Taylor’s constraint. The azimuthal force balance was therefore between the Lorentz force and the Ekman friction in the boundary layer which is OðE 1=2 Þ, so that the magnetic field was limited to a value with Elsasser number OðE 1=2 Þ. This balance is known as an Ekman state. However, as a was increased, a value was reached at which the field did satisfy Eq. (15), and at that point the field dramatically increased, leaving the Ekman state and having a strength with Elsasser number O(1). However, Braginsky (1976, 1994) proposed an almost axisymmetric model-Z dynamo (q.v.) in which the Taylor constraint was met in an unusual way. In an axisymmetric configuration, Eq. (15) can be written  Z  d 2 s BBs dz ¼ 0; (Eq. 16) ds where Bs is the component of magnetic field pointing away from the rotation axis. Taylor’s constraint could be satisfied if the meridional part of B was purely in the z-direction, so Bs ¼ 0, hence the name model Z dynamo. Braginsky proposed a specific distribution of a concentrated near the equatorial region and the CMB, which led to a steady rather than an oscillatory dynamo, as part of the model Z scenario. A more detailed analysis by Jault (1995) suggested however that if the viscosity is reduced to very low values, “Taylorization” (that is the approach to a state in which Eq. (15) holds) occurs in the conventional Malkus-Proctor way, with Bs being small but finite. Fully three-dimensional simulations usually have to be run with a viscosity too large to achieve “Taylorization” in order to maintain numerical stability. However, Rotvig and Jones (2002) have run fully three-dimensional simulations in plane geometry, which is less demanding computationally, and allows smaller E to be reached. They found that “Taylorization” did occur at the lowest values of E they could reach. Another way of simplifying the full three-dimensional problem, is the so-called “212-dimensional” approach. Here full resolution is used in the r and y directions, but in the azimuthal fdirection a severe truncation is imposed; in the most draconian approximation only the axisymmetric and one nonaxisymmetric expðimfÞ modes are retained

294

GEODYNAMO

(e.g., Jones et al., 1995). This still allows nonaxisymmetric convection to occur, so no a-mechanism is needed, and the results are generally qualitatively similar to those obtained from fully three-dimensional simulations, but at much reduced cost. The disadvantage is that unless results are checked by fully three-dimensional simulations, one cannot be certain that the truncation is not affecting the behavior; nevertheless “212-dimensional” models are very useful for testing out new ideas in dynamo theory.

Role of the inner core As mentioned in the energy balance section, the inner core makes an important contribution to the driving of convection in the core. It also significantly affects the dynamics of core convection. The importance of geostrophic motions in rapidly rotating fluids suggests that the solid inner core could play an important role in the dynamics of the Earth’s interior. The cylinder, which just touches the inner core, is called the inner core tangent cylinder (q.v.). Whereas the volume of the inner core itself is only about 4% of the volume of the outer core, the fraction inside the tangent cylinder is much larger. The flow outside the tangent cylinder can convect heat out in tall thin columns, which although not exactly geostrophic, because some vortex stretching occurs as the columns rotate due to the sloping boundaries, nevertheless minimizes the effect of rotation. Inside the tangent cylinder the convection needs to get the heat out in a direction almost parallel to the rotation axis, and tall thin columns run into the inner core, causing additional friction. In consequence, most models show that convection is more efficient outside the tangent cylinder than inside it. This difference produces a latitudinal temperature gradient, with the poles being slightly hotter than the equator on a sphere of constant radius. Neglecting magnetic forces, the azimuthal component of the curl of Eq. (11), then gives the thermal wind (q.v.) equation, 2O

]u f g a ]T ¼ r ]y ]z

(Eq. 17)

where z is the coordinate parallel to the rotation axis and y is the meridional direction in spherical polars ðr; y; fÞ. This shows that uf decreases with z in the northern hemisphere. A similar effect drives the jet stream in the Earth’s atmosphere, though the sign is different (the jet stream velocities increase with height, because the pole is colder than the equator, and so the jet stream goes from west to east relative to the surface) and of course temperature gradients and velocities are much smaller in the core. Nevertheless, the thermal wind is sufficient to give a rotation of the inner core (q.v.) relative to the mantle. The strength of the thermal wind can also be affected by magnetic torques. It has been suggested (Buffett and Glatzmaier, 2000) that the acceleration produced by the thermal wind may be counteracted by a gravitational coupling of the inner core to the mantle, which may be expected if there are slight departures from axisymmetry in the figure of the mantle and the inner core. The rotation of the inner core relative to the mantle can in principle be measured by seismologists, but the actual value of the inner core rotation rate is still controversial (e.g., Collier and Helffrich, 2001).

Reversals and field morphology Two obvious questions about the geodynamo are (i) why is the field dominated by an axial dipole component? (ii) why does the field periodically reverse its polarity? Our lack of a full understanding of the geodynamo is perhaps highlighted by the fact that neither of these questions can be answered unambiguously. The majority of dynamo simulations do show a dipole dominated field, but by no means all do (e.g., Christensen et al., 1999; Busse, 2002), and the Karlsruhe experimental dynamo was dominated by an equatorial rather than an axial dipole. Quadrupolar dynamos, equatorial dipoles, and axial dipoles can all be produced from flows, which are apparently not that

dissimilar. A number of trends are apparent from the simulations; for example, a strong azimuthal flow makes an equatorial dipole less likely, as the shearing motion associated with differential rotation tends to disrupt an equatorial dipole field. Dipolar fields tend to be more common when the convection is driven by a flux of heat from the inner core, whereas quadrupolar dynamos are more common in uniformly heated models, and there is a tendency (Busse, 2002) for dipoles to be preferred over quadrupoles at higher Rayleigh number. However, all these observations are model dependent, and it should not be forgotten that it is not currently possible to run geodynamo models in the correct parameter regime. A large number of papers have addressed the issue of the theory of reversals (q.v.) (see e.g., Sarson, 2000). One simple point is that the dynamo equations are invariant under a reversal of the field direction, so if a solution with “normal” polarity exists, an exactly similar one with reversed polarity also exists. To see this note that the transformation B !
B changes the sign of both B and j, so the Lorentz force j  B is unchanged, and the induction equation is linear in B so reversing the sign of B leaves this equation unchanged. This observation shows that it is reasonable to expect the geodynamo to reverse occasionally, but it doesn’t explain why reversals are comparatively infrequent (there are only a few reversals per million years on average) and why they happen relatively quickly (around 5 ka or less) when they occur. The magnetic field is continually fluctuating, and reversals appear to be part of this process. Magnetic excursions, events where the axis of the dipolar component moves rapidly away from the geographic poles occur much more frequently than full reversals, so it is possible that excursions are in some sense failed reversals, where the fluctuation is large but not large enough to cross a threshold leading to full reversal. Hollerbach and Jones (1993, 1995) suggested that the threshold necessary was that the field in the solid inner core had to be reversed. Since this can only occur by magnetic diffusion, rather than on the somewhat more rapid convective turnover time, only the larger, longer, excursions would allow the inner core field to reverse, and hence establish a full reversal. Numerical simulations of the geodynamo also suggest that the core is in a continually fluctuating state, and that some of these fluctuations occasionally lead to reversals. Glatzmaier et al. (1999) found that a spatially inhomogeneous heat flux at the core-mantle boundary could affect reversal frequency (see also Inhomogeneous boundary conditions and the dynamo). This is an interesting possibility, as mantle convection models suggest that such inhomogeneities are likely, and indeed seismic measurements of the CMB region also suggest there is such inhomogeneity. This would provide a link between mantle convection and the dynamo, and since mantle convection evolves on a very long million year timescale, this could explain why reversals are very infrequent compared to excursions. For example, on this picture during the 60-million-year Cretaceous superchron, during which there were no significant reversals, mantle convection was in a pattern, which affected the core flow in such a way as to make reversals unlikely. Sarson and Jones (1999) suggested that this might be connected with the meridional circulation, which would be associated with an inhomogeneous CMB heat flux, as in their model the strength of the meridional circulation played an important role in the reversal process. There is much that we do not yet understand about the dynamics of the Earth’s core and the reversal process, but progress is being made on a number of different fronts: better numerical simulations, and deeper understanding of the processes involved; more detailed and more accurate paleomagnetic studies giving information about the past behavior of the geomagnetic field; new results from seismology, probing the structure of the Earth’s deep interior ever more thoroughly; a new and better understanding of the physical properties of matter at very high pressure. With all these stimuli, our understanding of the geodynamo will surely radically improve over the coming decades. Chris Jones

GEODYNAMO

Bibliography Abdulrahman, A., Jones, C.A., Proctor, M.R.E., and Julien, K., 2000. Large wavenumber convection in the rotating annulus. Geophysical and Astrophysical Fluid Dynamics, 93: 227–252. Aldridge, K.D., 2003. Dynamics of the core at short periods. In Jones, C.A., Soward, A.M., and Zhang, K. (eds.) Earth’s Core and Lower Mantle. London: Taylor & Francis, pp. 180–210. Aubert, J., Brito, D., Nataf, H.C. et al., 2001. A systematic experimental study of rapidly rotating spherical convection in water and liquid gallium. Physics of the Earth and Planetary Interiors, 128: 51–74. Bloxham, J., and Gubbins, D., 1989. The evolution of the Earth’s magnetic field. Scientific American, 261: 30–37. Bloxham, J., and Jackson, A., 1991. Fluid flow near the surface of the Earth’s outer core. Reviews of Geophysics, 29: 97–120. Braginsky, S.I., 1963. Structure of the F layer and reasons for convection in the Earth’s core. Soviet Physics Doklady, 149: 8–10. Braginsky, S.I., 1967. Magnetic waves in the Earth’s core. Geomagnetism and Aeronomy, 7: 851–859. Braginsky, S.I., 1976. On the nearly axially-symmetrical model of the hydro-magnetic dynamo of the Earth. Physics of the Earth and Planetary Interiors, 11: 191–199. Braginsky, S.I., 1993. MAC-oscillations of the hidden ocean of the core. Journal of Geomagnetism and Geoelectricity, 45: 1517–1538. Braginsky, S.I., 1994. The nonlinear dynamo and model-Z. In Proctor, M.R.E., and Gilbert, A.D. (eds.) Lectures on Solar and Planetary Dynamos. Cambridge: Cambridge University Press, pp. 267–304. Braginsky, S.I., and Meytlis, V.P., 1990. Local turbulence in the Earth’s core. Geophysical and Astrophysical Fluid Dynamics, 55: 71–87. Braginsky, S.I., and Roberts, P.H., 1995. Equations governing convection in the Earth’s core and the geodynamo. Geophysical and Astrophysical Fluid Dynamics, 79: 1–97. Buffett, B.A., and Glatzmaier, G.A., 2000. Gravitational braking of inner-core rotation in geodynamo simulations. Geophysical Research Letters, 27: 3125–3128. Bullard, E.C., and Gellman, H., 1954. Homogeneous dynamos and terrestrial magnetism. Philosophical Transactions of the Royal Society of London A, 247: 213–255. Busse, F.H., 1970. Thermal instabilities in rapidly rotating systems. Journal of Fluid Mechanics, 44: 441–460. Busse, F.H., 2002. Convective flows in rapidly rotating spheres and their dynamo action. Physics of Fluids, 14: 1301–1314. Busse, F.H., and Carrigan, C.R., 1976. Laboratory simulation of thermal convection in rotating planets and stars. Science, 191: 81–83. Chandrasekhar, S., 1961. Hydrodynamic and Hydromagnetic Stability. Oxford: Clarendon Press. Christensen, U., and Tilgner, A., 2004. Power requirement of the geodynamo from ohmic losses in numerical and laboratory dynamos. Nature, 429: 169–171. Christensen, U., Olson, P., and Glatzmaier, G.A., 1999. Numerical modelling of the geodynamo: a systematic parameter study. Geophysical Journal International, 138: 393–409. Collier, J.D., and Helffrich, G., 2001. Estimate of inner core rotation rate from United Kingdom regional seismic network data and consequences for inner core dynamical behaviour. Earth and Planetary Science Letters, 193: 523–537. Cowling, T.G., 1934. The magnetic field of sunspots. Monthly Notices of the Royal Astronomical Society, 94: 39–48. Davidson, P.A., 2001. An Introduction to Magnetohydrodynamics. Cambridge: Cambridge University Press. Fearn, D.R., 1998. Hydromagnetic flow in planetary cores. Reports on Progress in Physics, 61: 175–235. Fearn, D.R., Roberts, P.H., and Soward, A.M., 1988. Convection, stability and the dynamo. In Galdi, G.P., and Straughan, B. (eds.) Energy, Stability and Convection. New York: Longmans, pp. 60–324.

295

Glatzmaier, G.A., and Roberts, P.H., 1995. A three-dimensional selfconsistent computer simulation of a geomagnetic field reversal. Nature, 377: 203–209. Glatzmaier, G.A., and Roberts, P.H., 1997. Simulating the geodynamo. Contemporary Physics, 38: 269–288. Glatzmaier, G.A., Coe, R.S., Hongre, L., and Roberts, P.H., 1999. The role of the Earth’s mantle in controlling the frequency of geomagnetic reversals. Nature, 401: 885–890. Gubbins, D., 1977. Energetics of the Earth’s core. Journal of Geophysics, 43: 453–464. Gubbins, D., Masters, T.G., and Jacobs, J.A., 1979. Thermal evolution of the Earth’s core. Geophysical Journal of the Royal Astronomical Society, 59: 57–99. Gubbins, D., Barber, C.N., Gibbons, S. et al., 2000a. Kinematic dynamo action in a sphere. I. Effects of differential rotation and meridional circulation on solutions with axial dipole symmetry. Proceedings of the Royal Society of London A, 456: 1333–1353. Gubbins, D., Barber, C.N., Gibbons, S. et al., 2000b. Kinematic dynamo action in a sphere. II. Symmetry selection. Proceedings of the Royal Society of London A, 456: 1669–1683. Hewitt, J.M., McKenzie, D.P., and Weiss, N.O., 1975. Dissipative heating in convective flows. Journal of Fluid Mechanics, 68: 721–738. Hollerbach, R., and Jones, C.A., 1993. Influence of the Earth’s inner core on geomagnetic fluctuations and reversals. Nature, 365: 541–543. Hollerbach, R., and Jones, C.A., 1995. On the magnetically stabilizing role of the Earth’s inner core. Physics of the Earth and Planetary Interiors, 87: 171–181. Hollerbach, R., Barenghi, C.F., and Jones, C.A., 1992. Taylor’s constraint in a spherical ao-dynamo. Geophysical and Astrophysical Fluid Dynamics, 67: 3–25. Jackson, A., 1997. Time-dependency of tangentially geostrophic core surface motions. Physics of the Earth and Planetary Interiors, 103: 293–311. Jault, D., 1995. Model Z by computation and Taylor’s condition. Geophysical and Astrophysical Fluid Dynamics, 79: 99–124. Jault, D., 2003. Electromagnetic and topographic coupling, and LOD variations. In Jones, C.A., Soward, A.M., and Zhang, K. (eds.) Earth’s Core and Lower Mantle. London: Taylor & Francis, pp. 56–76. Jault, D., Gire, C., and Lemouel, J.L., 1988. Westward drift, core motions and exchanges of angular-momentum between core and mantle. Nature, 333: 353–356. Jones, C.A., 2003. Dynamos in planets. In Thompson, M.J., and Christensen-Dalsgaard, J.C. (eds.) Stellar Astrophysical Fluid Dynamics. Cambridge: Cambridge University Press, pp. 159–176. Jones, C.A., Longbottom, A.W., and Hollerbach, R., 1995. A selfconsistent convection driven geodynamo model, using a mean field approximation. Physics of the Earth and Planetary Interiors, 92: 119–141. Jones, C.A., Soward, A.M., and Mussa, A.I., 2000. The onset of convection in a rapidly rotating sphere. Journal of Fluid Mechanics, 405: 157–179. Jones, C.A., Mussa, A.I., and Worland, S.J., 2003. Magnetoconvection in a rapidly rotating sphere: the weak-field case. Proceedings of the Royal Society of London A, 459: 773–797. Kerswell, R.R., 2002. Elliptical instability. Annual Review of Fluid Mechanics, 34: 83–113. Kono, M., and Roberts, P.H., 2002. Recent geodynamo simulations and observations of the geomagnetic field. Reviews of Geophysics, 40(4): 1013. Malkus, W.V.R., 1994. Energy sources for planetary dynamos. In Proctor, M.R.E., and Gilbert, A.D. (eds.) Lectures on Solar and Planetary Dynamos. Cambridge: Cambridge University Press, pp. 161–179.

296

GEODYNAMO

Malkus, W.V.R., and Proctor, M.R.E., 1974. The macrodynamics of a-effect dynamos in rotating fluids. Journal of Fluid Mechanics, 67: 417–443. Merrill, R.T., McElhinny, M.W., and McFadden, P.L., 1996. The Magnetic Field of the Earth. San Diego, CA: Academic Press. Moffatt, H.K., 1978. Magnetic Field Generation in Electrically Conducting Fluids. Cambridge: Cambridge University Press. Parker, E.N., 1955. The formation of sunspots from the solar toroidal field. Astrophysics Journal, 121: 491–507. Roberts, P.H., 1967. An Introduction to Magnetohydrodynamics. London: Longmans. Roberts, P.H., 1968. On the thermal instability of a rotating fluid sphere containing heat sources. Philosophical Transactions of the Royal Society of London A, 263: 93–117. Roberts, P.H., 1972. Kinematic dynamo models. Philosophical Transactions of the Royal Society of London A, 272: 663–703. Roberts, P.H., and Glatzmaier, G.A., 2000. A test of the frozen flux approximation using geodynamo simulations. Philosophical Transactions of the Royal Society of London A, 358: 1109–1121. Roberts, P.H, Jones, C.A., and Calderwood, A., 2003. Energy fluxes and ohmic dissipation. In Jones, C.A., Soward, A.M., and Zhang, K. (eds.) Earth’s Core and Lower Mantle. London: Taylor & Francis, pp. 100–129. Rotvig, J., and Jones, C.A., 2002. Rotating convection-driven dynamos at low Ekman number. Physical Review E, 66: 056308-1-15. Sarson, G.R., 2000. Reversal models from dynamo calculations. Philosophical Transactions of the Royal Society of London A, 358: 921–942. Sarson, G.R., 2003. Kinematic dynamos driven by thermal-wind flows. Proceedings of the Royal Society of London A, 459: 1241–1259. Sarson, G.R., and Jones, C.A., 1999. A convection driven geodynamo reversal model. Physics of the Earth and Planetary Interiors, 111: 3–20. Schubert, G., Turcotte, D.L., and Olson, P., 2001. Mantle Convection in the Earth and Planets. Cambridge: Cambridge University Press. Sleep, N.H., 1990. Hot spots and mantle plumes: some phenomenology. Journal of Geophysical Research, 95: 6715–6736. Soward, A.M., and Jones, C.A., 1983. a2 -dynamos and Taylor’s constraint. Geophysical and Astrophysical Fluid Dynamics, 27: 87–122. Starchenko, S., and Jones, C.A., 2002. Typical velocities and magnetic field strengths in planetary interiors. Icarus, 157: 426–435. Steenbeck, M., Krause, F., and Rädler, K-H., 1966. A calculation of the mean electromotive force in an electrically conducting fluid in turbulent motion, under the influence of Coriolis forces. Zeitschrift für Naturforschung, 21a: 369–376. Stevenson, D.J., 1982. Interiors of the giant planets. Annual Review of Earth and Planetary Sciences, 10: 257–295. Stevenson, D.J., 1987. Mercury magnetic field—a thermoelectric dynamo. Earth and Planetary Science Letters, 82: 114–120. Stevenson, D.J., 2003. Planetary magnetic fields. Earth and Planetary Science Letters, 208: 1–11. Taylor, J.B., 1963. The magneto-hydrodynamics of a rotating fluid and the Earth’s dynamo problem. Proceedings of the Royal Society of London A, 274: 274–283. Tilgner, A., 2005. Precession driven dynamos. Physics of Fluids, 17: 034104-1. Verhoogen, J., 1961. Heat balance in the Earth’s core. Geophysical Journal, 4: 276–281.

Cross-references Alfvén Waves Antidynamo and Bounding Theorems

Boussinesq and Anelastic Approximations Bullard, Edward Crisp (1907–1980) Compass Convection, Chemical Convection, Nonmagnetic Rotating Core Composition Core Convection Core Density Core Magnetic Instabilities Core Motions Core Properties, Physical Core Temperature Core Turbulence Core Viscosity Core, Adiabatic Gradient Core, Boundary Layers Core, Electrical Conductivity Core, Thermal Conduction Core-based Inversions for the Main Geomagnetic Field Core-Mantle Boundary, Heat Flow Across Cowling’s Theorem Crustal Magnetic Field Depth to the Curie Temperature Dynamo Waves Dynamo, Model-Z Dynamo, Solar Dynamos, Experimental Dynamos, Mean Field Dynamos, Planetary and Satellite Elsasser, Walter M. (1904–1991) Fluid Dynamics Experiments Gauss, Carl Friedrich (1777–1855) Gellibrand, Henry (1597–1636) Geodynamo, Energy Sources Geodynamo, Numerical Simulations Geomagnetic Jerks Geomagnetic Secular Variation Gilbert William (1544–1603) Gravito-inertio Waves and Inertial Oscillations Grüneisen’s Parameter for Iron and Earth’s Core Halley, Edmond (1656–1742) Harmonics, Spherical Inhomogeneous Boundary Conditions and the Dynamo Inner Core Composition Inner Core Rotation Inner Core Tangent Cylinder Interiors of Planets and Satellites Ionosphere Larmor, Joseph (1857–1942) Length of Day Variations, Decadal Magnetic Field of Sun Magnetoconvection Magnetohydrodynamic Waves Magnetohydrodynamics Melting Temperature of Iron in the Core Nondynamo Theories Oscillations, Torsional Precession and Core Dynamics Proudman-Taylor Theorem Radioactive Isotopes and their Decay in Core and Mantle Reversals, Theory Storms and Substorms, Magnetic Taylor’s Condition Thermal Wind Verhoogen, John (1912–1993) Westward Drift

297

GEODYNAMO, DIMENSIONAL ANALYSIS AND TIMESCALES

GEODYNAMO, DIMENSIONAL ANALYSIS AND TIMESCALES It is common to put the equations governing a complex physical system like the geodynamo into dimensionless form. This clarifies the importance of each term in the equations and can reduce the number of input parameters to a minimum. Consider the very simple example of the kinematic dynamo (q.v.), which is governed by a single partial differential equation, the induction equation: ]B 1 ¼ r  ðv  BÞ þ r2 B: ]t m0 s

(Eq. 1)

The fluid velocity is specified within a sphere and the induction equation solved for a growing magnetic field. It depends on just three input parameters: the amplitude of the velocity V, the radius of the sphere c, and the electrical conductivity s. In dimensionless form this reduces to just one input parameter, the magnetic Reynolds number Rm ¼ m0 sVc, where m0 is the permeability of free space (see Tables G2 and G3). Kinematic dynamo behavior therefore depends on the product of the three dimensional input parameter but not on all three independently. This is a great help in exploring the behavior of the system as a function of the input parameters. Incidentally, it means that scaling the Earth down to a laboratory-sized experiment would require a great increase in fluid velocity to compensate for the reduction in

Table G2 Illustrative numerical values of core parameters used Property

Symbol

Molecular

Density Gravity Core radius Inner core radius Outer core depth Angular velocity Kinematic viscosity

r g c ri d ¼ c
ri O n

104 kg m
3 0–10 m s
2 3 484 km 1 215 km 2 269 km 7:272  10
5 10
6 m2 s
1

Electrical conductivity Thermal conductivity Specific heat Magnetic diffusivity Thermal diffusivity Molecular diffusion constant Thermal expansion Typical core velocity Typical magnetic field Adiabatic gradient Core heat flux Temperature gradient

s

5  105 S m
1

k

50 W m
1 K
1

Cp
¼ ðm0 sÞ
1

Turbulent

1 at room temperature


1

700 J kg K
1 1:6 m2 s
1

1:6 m2 s
1

k ¼ k ðrCp Þ
1 7  10
7 m2 s
1 1:6 m2 s
1 D 10
6 m2 s
1 1:6 m2 s
1 a V

5  10
6 K
1 10
4 m
1

B

1 mT

Ta0 Q T0

0:1 K km
1 5TW 0:5 K km
1

Those above the line are measured directly; those below the line are inferred from putative core composition, temperature, and pressure T 0 is the excess temperature gradient over the adiabat Ta0 , as defined in Eq. (4). Current dynamo models usually assume some form of turbulence that brings the small diffusivities of heat, momentum (viscosity), and mass up to the larger value of the electrical diffusivity. For illustration I have taken these values equal to 1.6 when calculating turbulent values in Tables G3 and G4.

length scale (increasing the conductivity is impossible since there are no materials at room temperature with electrical conductivity significantly higher than that of iron in the core). Nondimensionalization is by no means a unique process: there are many ways in which to scale the dimensional variables, each giving different versions of the same equations. Consider once again the kinematic dynamo problem. The usual approach is to scale length with the radius c and time with the magnetic diffusion time, m0 sc2 . This leads to the form ]B ¼ Rm r  ðv  BÞ þ r2 B: ]t

(Eq. 2)

so that Rm can be regarded as a dimensionless measure of the fluid velocity. A second way to nondimensionalize the induction equation is to scale time with the overturn, or advection, time c/V, the time it takes for fluid to cross the sphere. This leads to the form ]B 1 2 ¼ r  ðv  BÞ þ R
m r B: ]t

(Eq. 3)

Now the inverse magnetic Reynolds number is a dimensionless measure of magnetic diffusion. At first sight, Eqs. (2) and (3) appear to contradict each other; the difference arises from the definition of time, t. In this article, I shall consider the fairly general case of a geodynamo driven by a combination of heat and compositional buoyancy, governed by the equations of momentum, induction, heat, and mass transfer. The full geodynamo problem is usually put into a nondimensional form that leaves just four independent input parameters, the Rayleigh number Ra, Ekman number E, the Prandtl number Pr , and the magnetic Prandtl number Pm. These are defined and estimated in Table G3. Other nondimensional numbers are calculated from the solution of the equation, although they can also be independent input parameters for subproblems. Two examples are the magnetic Reynolds number, which depends on the fluid velocity and is therefore calculated from the convective flow from the full dynamo problem but is

Table G3 Dimensionless numbers Name

Definition

Molecular Turbulent

Rayleigh Modified Rayleigh Buoyancy Ekman Prandtl Magnetic Prandtl Roberts Schmidt Magnetic Schmidt Lewis Magnetic Reynolds Elsasser Rossby Nusselt Thermal Peclet Mass Peclet

Ra ¼ gDr0 d 3 =kn 0 Rm a ¼ Ra E ¼ gDr d=kO 2 B 0 Ra ¼ gDr =O d E ¼ n=Oc2 Pr ¼ n=k Pm ¼ m0 sn ¼ n=
Pq ¼ k=
Ps ¼ n=D PD ¼
=D PL ¼ D=k Rm ¼ m0 sVc L ¼ B2 s=rO Ro ¼ V =2Oc Q=ð4pc2 Ta0 Þ Pe ¼ Vc=k PD ¼ Vc=D

1030 1015 3 10
15 1.4 6  10
7 4  10
7 1 1:6  106 1.4 200 1 2  10
7 7 5  108 3  108

7  1017 8  108 3 10
9 1 1 1 1 1 1 200 1 2  10
7 7 200 200

Those above the line appear explicitly in formulations of the geodynamo equations; those below are derived from solutions to the geodynamo problem; those in column 4 are for nominal turbulent values of the diffusivities, k ¼ n ¼ D ¼
¼ 1:6 m2 s
1 . Rayleigh and buoyancy numbers are calculated for thermal convection by replacing Dr0 with aDT , where r0 is the departure from the adiabatic density gradient, and DT is the temperature difference from the adiabat. Values for compositional convection are likely to be higher. Qa is the heat conducted down the adiabat.

298

GEODYNAMO, DIMENSIONAL ANALYSIS AND TIMESCALES

the sole input parameter for the kinematic dynamo as already mentioned; and the Elsasser number (see below), which depends on the magnetic field strength but is the main input parameter to the magnetoconvection problem, in which a magnetic field is imposed rather than being self-generated. The nondimensional numbers represent ratios of terms appearing in the governing equations: they are ratios of forces in the momentum equation, of magnetic induction in the induction equation, of heat transfer in the heat equation, and of mass transfer in the material diffusion equation. Further physical insight may be obtained by expressing them as ratios of timescales. The Earth’s magnetic field varies on an enormous range of timescales, from less than a year for geomagnetic jerks (q.v.) to a million years between reversals, and its age is comparable with that of the Earth. Timescales contained within the geodynamo equations span an even wider spectrum, from a single day, the timescale of the Coriolis force, to the molecular viscous and heat diffusion times, which exceed the age of the Earth itself (Table G4). This enormous disparity of timescales makes numerical simulation of the geodynamo very difficult, more so than the range of length scales involved, because the computer must resolve details on the shortest timescale, then integrate the equations for sufficient time to demonstrate dynamo generation in order to follow the evolution. The very long viscous and thermal diffusion times are impossible to simulate and are usually shortened by assumed turbulent enhancement of the diffusivities (see Core turbulence), often by simply equalizing the diffusivities to the largest, electrical, diffusivity. A summary of relevant timescales is given in Table G4. Diffusion times are quoted by the formula shown; true diffusion times in a sphere for magnetic, viscous, thermal, and compositional diffusion contain the factor p2 to give the time taken for a dipole field to fall by a factor of e. Geometrical factors and wavenumbers have been omitted from the formulae for dimensionless numbers in Table G5. The periods of MAC waves depend on the wavenumber and are considerably shorter than the time quoted in the table. Each group of nondimensional parameters is now discussed in turn.

Rayleigh, modified Rayleigh, buoyancy numbers— buoyancy: dissipation The Rayleigh number multiplies the buoyancy force in the dimensionless equation of motion and as such measures the driving force of the convective dynamo. This discussion is restricted to thermal convection but the same remarks apply to compositional convection with concentration replacing temperature, D replacing k, and a compositional expansion coefficient replacing a. I do not discuss complications involving both thermal and compositional buoyancy since there has been very little discussion of it in the literature to date, and no direct observations. Ra is a complicated combination of dimensional parameters and is very difficult to estimate numerically. For free convection heated from below, a critical value of the Rayleigh number must be exceeded before convection starts. This is because motion is opposed by viscous forces and heat, the source of the buoyancy, can dissipate away by conduction. Consider the balance of buoyancy and viscous forces for a rising, hot, light blob: raDTg ¼ rnVS =d 2 . The blob falls with terminal (Stokes) velocity when buoyancy equals viscous drag, of order VS ¼ agDTd 2 =n. The time to rise through the core is tB ¼ d=VS ¼ n=aDTgd. Buoyancy is lost by diffusion of heat into the surrounding fluid, restricting the power of the convection. The Rayleigh number is the ratio of the thermal diffusion time tk ¼ d 2 =k to the viscous buoyant rise time: Ra ¼ gaDTd 3 =kn. Vigor of convection is measured by the speed with which fluid rises compared to the speed with which heat is lost by conduction. There are two major difficulties with estimating Ra in the core. First, it is virtually impossible to estimate the temperature difference DT driving the convection. The correct boundary condition for core

Table G4 Timescales for the Earth’s core, definition, and numerical estimates in years Time

Definition

Molecular

Turbulent

Magnetic diffusion (core) Magnetic diffusion (inner core) Thermal diffusion Viscous diffusion Mass diffusion Overturn Buoyant rise Coriolis rise Day MAC wave

t
¼ c2 =
p2

25000

25000

ti ¼ ri2 =
p2

3000

tk ¼ c2 =kp2 tn ¼ c2 =np2 tD ¼ c2 =Dp2 tV ¼ d=V tB ¼ n=aDTgd tO ¼ cO=gaDT t tMAC ¼ Orm0 c2 =B2

6  1010 4  1010 4  1010 700 2  10
19 10
4 3  10
3 3  105

25000 25000 25000 3  10
13

Table G5 Dimensionless numbers and their relationship with timescales Rayleigh Modified Rayleigh Buoyancy Ekman Prandtl Magnetic Prandtl Roberts Schmidt Magnetic Schmidt Lewis Magnetic Reynolds Elsasser Rossby Thermal Péclet Mass Péclet

tk =tB tk =tO t=tO t=tn tk =tn t
=tn t
=tk tD =tn tD =t
tk =tD tV =t
t
=tMAC t=tn tV =tk tV =tD

convection is constant heat flux imposed by the convecting and cooling mantle. This gives the temperature gradient at the core surface, from which we must subtract the adiabatic gradient, which would hold in the absence of any convection: T0 ¼

Q
Ta0 : 4pc2 k

(Eq. 4)

We could then take DT to be the product of the temperature gradient T 0 and the outer core depth d. The problem is that Q is very uncertain and Ta0 poorly known. We do know, however, that the core must convect in order to produce a dynamo, and that T 0 must exceed the adiabat by enough to power the dynamo. Gubbins (2001) estimates Ra in this way to be about 1000 times the critical value for nonmagnetic convection. Jones (2000) gives an independent argument based on the speed of the flow in the core and arrives at a similar value. The critical Rayleigh number depends strongly on both rotation, which increases it, and magnetic field, which in combination with rotation decreases it; Ra in the core seems to be close to the critical value in the absence of any magnetic field (Gubbins, 2001). The Rayleigh number for compositional convection could be much higher. Studies of magnetoconvection have shown that, at high rotation rates, the critical Rayleigh number varies as E
1 , prompting some authors to use the modified Rayleigh number RaE. This takes account, to some extent, the effect of rotation on the stability of convection. Ra E

GEODYNAMO, DIMENSIONAL ANALYSIS AND TIMESCALES

contains only the thermal diffusivity and not the fluid viscosity, which plays a negligible role in resisting the flow compared to the magnetic force. The presumed force balance is between Coriolis and buoyancy forces, leading to the scaling gaDT  OV ; the relevant timescale is then tO ¼ c=V ¼

cO : gaDT

The modified Rayleigh number is then seen as the ratio of thermal diffusion time to Coriolis rise time tk =tO . Others have used a buoyancy parameter that involves no diffusivity. This is the ratio of the day to the Coriolis rise time. Further details may be found in, for example, Gubbins and Roberts (1987).

Nusselt number—convected heat: conducted heat

299

for which there are lower bounds (see Antidynamo and bounding theorems). Estimates of fluid flow in the core, based on inferences from secular variation, yields Rm 200, low but well above the critical values required for dynamo action.

Pe´clet numbers—advection: diffusion Like the magnetic Reynolds numbers, Pe and PD measure the importance of advection in the heat and mass transfer equations, respectively. They are the ratios of the day to viscous diffusion time, and day to compositional diffusion times, respectively.

Elsasser number—magnetic: Coriolis force L is a dimensionless measure of the Lorentz force and an essential input parameter for magnetoconvection. The balance between magnetic, buoyancy, and Coriolis forces (MAC) leads to magnetohydrodynamic waves with periods of centuries to millennia (see Magnetohydrodynamic waves). They are highly dispersive and under certain simple conditions have wavespeeds given by VMAC ¼ kB2 =Orm0, where k is the wavenumber. A “MAC” timescale can then be defined as the time taken for a MAC wave with wavelength c to cross the core,

The Nusselt number is a different dimensionless measure of the heat driving the convection. It is usually quoted as the ratio of convected heat flux to conducted heat flux for convection between boundaries with fixed temperatures. The situation in the Earth’s outer core is somewhat different. First, the boundary conditions are fixed temperature at the bottom (melting temperature) and fixed heat flux at the top (as dictated by mantle convection). The Nusselt number is therefore, in some sense, an input parameter for the geodynamo problem. Second, the conduction profile includes the adiabatic gradient, which is steep and responsible for conduction of a large amount of heat in the context of the Earth’s thermal history. It is not really an estimate of the vigor of the convection, which depends on the superadiabatic temperature gradient rather than the absolute value. Estimates of Nu in the core are inevitably restricted to low values, from 1 to 10, because of limits on the heat coming from the core.

The Elsasser number is then L ¼ t
=tMAC . Small L is the condition for large-scale MAC waves to pass through the core with little dissipation. The numerical value for tMAC in Table G4 is rather long: a typical MAC wave with wavelength 1000 km would have timescale closer to 1 ka; a stronger core field of 10 mT, the value usually taken in MAC wave studies, would reduce it by a factor of 100.

Ekman number—viscous force: Coriolis force

Less common dimensionless numbers

E is tiny, whether we use molecular or turbulent values of the viscosity, reflecting the enormous disparity between the viscous and diurnal timescales. Smallness of E causes the greatest difficulty in numerical simulations of the geodynamo, even when turbulent diffusivities are used. The smallest values of E achieved in numerical simulations so far are 10
5 –10
7 , compared with E ¼ 10
9 for the Earth’s core assuming a turbulent viscosity.

The above list includes those in common usage in geodynamo theory. Other numbers are used occasionally, and some have different names. The Taylor number is the inverse square of the Ekman number, the ratio of centrifugal to viscous force (and for the core is one of the biggest numbers one is ever likely to come across!). The Chandrasekhar number B2 sd 2 is a useful measure of the magnetic force in place of the Elsasser number when rotation is unimportant; it measures the relative strengths of magnetic and diffusive forces. The Hartmann 1=2 number, Bs1=2 d=m0 , is important in boundary layer theory; it measures the ratio of magnetic to viscous forces. The Alfvén (q.v.) number, V ðrm0 Þ1=2 =B, is a magnetic Mach number, the ratio of flow speed to the speed of Alfvén waves. Its inverse is the Cowling number. Neither are in common use in geomagnetism, although Merrill and McElhinny (1996) define the Alfvén number as the Cowling number. The degree to which magnetic fields are frozen-in is usefully measured by the Lundqvist number, m0 sB
=r1=2 , the time for Alfvén waves to cross the core divided by the magnetic diffusion time.

Rossby number—inertial: Coriolis force Ro is a measure of the importance of inertial forces Dv/Dt in the equation of motion. It is small in the core and many studies have dropped the inertial terms altogether. This may not be appropriate, since inertial terms may play a role in restoring the balance between magnetic, buoyancy, and rotational forces. Inertia also plays an essential role in torsional oscillations (q.v.).

tMAC ¼

c VMAC

Prandtl numbers The Prandtl number measures the ratio of viscous to thermal diffusion. Liquid metals generally have small Prandtl numbers of order 0.1, and many studies have focused on this. Others have taken Pr ¼ 1, consistent with the assumption of turbulence equalizing the diffusivities. Pm and Pq also have extremely low values that cause problems. Similar remarks apply to compositional diffusion.

Magnetic Reynolds number—induction: magnetic diffusion Rm is a dimensionless measure of the fluid velocity, the input parameter for kinematic dynamos, and is the ratio of the overturn and magnetic diffusion times. It must exceed a critical value for dynamo action,

¼

Orm0 c2 : B2

David Gubbins

Bibliography Gubbins, D., 2001. The Rayleigh number for convection in the Earth’s core. Physics of the Earth and Planetary Interiors, 128: 3–12. Gubbins, D., and Roberts, P.H., 1987. Magnetohydrodynamics of the Earth’s core. In Jacobs, J.A. (ed.), Geomagnetism, Vol. II, Chapter 1. London: Academic Press, pp. 1–183. Jones, C.A., 2000. Convection-driven geodynamo models. Philosophical Transactions of the Royal Society of London, Series A, 873: 873–897. Merrill, R.T., and McElhinny, M.W., 1996. The Magnetic Field of the Earth. San Diego, CA: Academic Press.

300

GEODYNAMO, ENERGY SOURCES

Cross-references Alfvén, Hannes Olof Gösta (1908–1995) Antidynamo and Bounding Theorems Convection, Chemical Convection, Nonmagnetic Rotating Core, Adiabatic Gradient Core Convection Core Motions Core Turbulence Dynamos, Kinematic Geodynamo Geodynamo, Numerical Simulations Geomagnetic Jerks Geomagnetic Spectrum, Temporal Magnetoconvection Magnetohydrodynamic Waves Magnetohydrodynamics Oscillations, Torsional

GEODYNAMO, ENERGY SOURCES The magnetic field of the Earth is maintained against ohmic losses by dynamo action in the fluid core which takes its energy from several sources of different natures and amplitudes, all being consequences of the thermal evolution of the core. Moreover, each source has a different efficiency in maintaining ohmic dissipation compared to the others. The study of these problems is then of major importance for understanding dynamo theory as well as the thermal evolution of the Earth.

Evolving reference state The core of the Earth is composed of iron, nickel, and some lighter elements, with concentrations that are still matter of lively debates (see Core composition, and Poirier, 1994). The light elements play a very important dynamical role in maintaining the geodynamo since their rejection upon inner core crystallization drives compositional convection in the core (see Convection, chemical). The core is therefore usually modeled as a binary alloy of Fe and some light element X. Including a more realistic chemistry in thermal evolution models on the core would add complexity (although no real difficulty) in the derivations without really improving the understanding, unless not yet documented coupled effects occur. The Earth’s core is different from a well-controlled convection experiment in a laboratory, in the fact that it is continually evolving on geological timescales and that the energy for the motion comes from that evolution. Fortunately, the timescales relevant for this evolution and that for the dynamics are very different and can be separated. To get expressions for the different energy sources in the balance, we need to know in what state is the core when averaged on a timescale long compared to the one relevant to the dynamics but short compared to the one associated with the thermal evolution. The short-time dynamics is responsible for maintaining the core close to that reference state whereas the evolution of the reference state provides the energy sources needed by the dynamics. Convection in the liquid core is assumed to be very efficient so that, outside of tiny boundary layers (see Core, boundary layers), all extensive quantities responsible for the movement (entropy s and mass fraction of light elements x) are uniform: rx ¼ 0 and rs ¼ 0. In addition, the momentum equation is assumed to average to the hydrostatic balance: rp ¼ rg, with r the density and g the acceleration due to gravity. It is often useful to know what the temperature is in the reference state (see Core temperature) and isentropy and isochemistry implies it to be adiabatic (see Core, adiabatic gradient)

rT ¼ agT =Cp , with a the coefficient of thermal expansion and Cp the heat capacity at constant pressure. The set of partial differential equations defining the reference state must be supplemented by an equation of state relating the density to the pressure, for example at constant entropy, and by boundary conditions. In terms of temperature, the equilibrium between solid and liquid at the inner core boundary (ICB) provides the required condition, the liquidus of the core alloy being given (see Melting temperature of iron in the core, Theory and melting temperature of iron in the core, experimental ) as a function of radius in the core. Depending on the choice of equation of state, one can get different expressions for the different profiles in the reference state, usually in the form of some series expansion of powers of the radius. It should be noted that the actual averaged state cannot be exactly hydrostatic, owing to its being compressible and compressing with time, but the corrections to this balance are negligible for the energetics of the core (Braginsky and Roberts, 1995). The deviations from the isentropic, well mixed state can also be estimated from the amplitude of fluid velocity at the top of the core which also allows an estimate of the typical scale for the size of boundary layers. These are found to be a negligible contribution to averaged quantities (e.g., Braginsky and Roberts, 1995), although they are very important for the dynamics of the core.

Energy equation The equation for total (internal, kinetic, magnetic, and gravitational) energy conservation can be integrated over the volume of the core and it states that the total heat loss of the core, the heat flow across the core-mantle boundary (CMB), is balanced by the sum of four terms, the secular cooling QC associated with the heat capacity of the core, the latent heat QL associated with the gradual freezing of the inner core, the gravitational energy EG associated with the rearrangement in the outer core of the light elements that are released at the ICB by the freezing of the inner core, and possibly the radioactive heating QR , QCMB ¼ QC þ QL þ EG þ QR :

(Eq. 1)

The first three terms are related to the growth of the inner core and can be written as a function of the radius of the inner core c times its time derivative. The exact expression of these three functions depends on the choice of equation of state and parameterization and the expressions of Labrosse (2003) can be written to first order as:   8p 2 c dc ; (Eq. 2) QC ¼ b3 r0 CP TL ðcÞ 1
3 3g L2T dt QL ¼ 4pc2 rðcÞTL ðcÞDS

EG ¼

dc ; dt

  8p2 3 c2 dc
2 ; GDrr0 c2 b2 5 b dt 3

(Eq. 3)

(Eq. 4)

with b and c the radii of the outer and inner core respectively, r0 the central density, CP the heat capacity, TL the liquidus temperature, g the Grüneisen parameter (see Grüneisen’s parameter for iron and Earth’s core), LT the adiabatic length scale, DS the entropy of crystallization, Dr the chemical density jump across the ICB (see Core density), and G the gravitational constant. All these physical parameters can be estimated with more or less accuracy using combinations of geophysical data (mostly seismology) and mineral physics (see Core properties, physical; Core properties, theoretical determination). Convective mixing can be assumed to be sufficient to ensure a uniform mass rate hðtÞ of radioactive heating in the core. Therefore,

301

GEODYNAMO, ENERGY SOURCES

radioactive heating is not related to inner core growth and is simply equal to MN hðt Þ, with MN the mass of the core. This energy source can then be easily computed, provided one knows the concentration in radioactive elements in the core. Among all possible heat producing isotopes, 40 K has always been the most popular candidate, owing to its apparent depletion in the mantle compared to Earth forming meteorites and its predicted metalization at high pressure that would allow it to enter the core. However, potassium is also somewhat volatile and its budget in the Earth is influenced by accretion processes. In addition, the concentration of potassium in the core depends strongly on the scenario of core formation, a process still far from perfectly understood (e.g., Stevenson, 1990). The most recent experiments devoted to the partitioning of potassium between iron and silicates (e.g., Gessmann and Wood, 2002; Lee and Jeanloz, 2003; Rama Murthy et al., 2003) favor a concentration of potassium in the core of O(100) ppm, producing less than 1 TW at present but exponentially more in the past. Such a value is too small to affect importantly the thermal evolution of the core (Labrosse, 2003) and radioactivity will not be considered further. The gravitational energy actually comes in the equations as a compositional energy, due to a change of composition in a gradient of chemical potential (Braginsky and Roberts, 1995; Lister and Buffett, 1995). It is equal to the change of gravitational energy due only to chemical stratification of the core. Other sources of change of gravitational energy do not contribute significantly to this balance and are mostly stored as strain energy. As can be seen on Eqs. (2–4), for the different energy sources to be estimated, one needs to know the growth rate of the inner core. Since there is no direct way of measuring this number, one usually uses an estimate of the heat flow across the CMB to get this number from the energy balance (1). The heat flow across the CMB is not very well-known (see Core-mantle boundary, heat flow across) but for a value of 10 TW (say) and no radioactive elements, one gets approximately (Labrosse, 2003) QC ¼ 5:5 TW; QL ¼ 2:8 TW; EG ¼ 1:7 TW. The energy balance can be used for any time in the history of the Earth to give the growth history of the inner core and the evolution of all energy sources in the balance, provided the heat flow across the CMB is known as a function of time. Moreover, this equation can be integrated between the onset time of inner core crystallization and the present to compute the age of the inner core (Labrosse et al., 2001). A typical example of time evolution of the energy balance is shown on Figure G12, where the onset of inner core crystallization, at an age of 1 Ga, is marked by a qualitative change in the balance: before, only secular cooling is balancing QCMB (in absence

of radioactivity) but this term is greatly decreased when the inner core starts to crystallize and both latent heat and compositional energy come in the balance.

Entropy equation The energy equation does not involve the magnetic field directly and contains no contribution from the dissipative heating. This is a wellknown characteristic of convective engines: in contrary to Carnot engines, dissipation occurs inside the system and is then not lost. In order to relate the energy sources to the magnetic field generation, an entropy balance equation must be written. This equation comes from a combination of the momentum balance equation and the energy balance equation written above (e.g., Braginsky and Roberts, 1995) and states that the entropy that flows out through the CMB is balanced by the sum of the entropy that flows in due to the different heat sources and the entropy that is produced by nonreversible processes, mostly ohmic dissipation and conduction along the adiabatic temperature profile. This equation does not directly involve the gravitational energy, since it is not a heat source. It can however, be brought back into the equation by use of the energy balance to suppress the heat flow across the CMB, giving an efficiency equation, 

Z F þ TD

k OC

rT T

2 dV ¼

 TD TICB
TCMB EG þ ðQICB þ QL Þ TCMB TICB  TR
TCMB TC
TCMB þ QR þ QC ; TR TC (Eq. 5)

where it can be seen that each energy source contributes in maintaining both the total ohmic dissipation F and the conduction along the adiabatic gradient, but with different efficiency factors. In particular, this equation shows that the gravitational energy has an efficiency factor that is the ratio of the effective temperature TD at which ohmic dissipation occurs to the temperature at the CMB, TCMB , whereas all heat sources efficiency factors have a contribution from the classical Carnot engine efficiency factor, ðTX
TCMB Þ=TX , with TX the temperature at which the heat source X is provided. This shows that compositional energy is more efficiently transformed in dissipation than all heat sources and that the efficiency of each heat source depends on the temperature at which it is provided.

Figure G12 Energy (left) and entropy (right) balances of the core as a function of time for a typical evolution model without radioactivity. (After Labrosse, 2003).

302

GEODYNAMO: NUMERICAL SIMULATIONS

The energy sources on the right-hand side of Eq. (5) are the same as that appearing in the energy balance Eq. (1) and, except for TD, their efficiency factors can be expressed using the parameters characterizing the reference state of the core. It can then be proved that all terms, except the radioactive heating one, is a function of the radius of the inner core and is proportional to its growth rate. This means that, if the heat flow across the CMB is known, this growth rate can be computed from the energy equation (1), and the entropy equation (5) then gives the ohmic dissipation that is maintained. Alternatively, one can take the opposite view and compute the growth rate of the inner core that is required to maintain a given ohmic dissipation, the energy balance being then used to get the heat flow across the CMB that makes this growth rate happen. Unfortunately, the ohmic dissipation in the core is no better known than the heat flow across the CMB, since it is dominated by small-scales of the magnetic field and possibly by the invisible toroidal part of it (Gubbins and Roberts, 1987; Roberts et al., 2003). However, the value of QCMB ¼ 10 TW used above gives a contribution of ohmic dissipation to the entropy balance F=TD ¼ 500 MWK
1 . The temperature TD is not well-known, but is bounded by the temperature at the inner core boundary and the CMB and this gives F ’ 2 TW. The evolution with time of the entropy balance associated with a given heat flow evolution can be computed and the example shown above gives the result of Figure G12. An interesting feature is the sharp increase of the ohmic dissipation in the core when the inner core starts crystallizing, latent heat and, even more so, gravitational energy being more efficient than secular cooling. Unfortunately, the link between this ohmic dissipation and the magnetic field observed at the surface of the Earth is far from obvious and the detection of such an increase in the paleomagnetic record is unlikely (Labrosse and Macouin, 2003). Some alternative models for the average structure of the core involving some stratification have been proposed. In particular, the heat conducted along the adiabatic temperature gradient can be rather large (about 7 TW) and might be larger than the heat flow across the CMB (see Core-mantle boundary, heat flow across). In this case, two different models have been proposed. In the first one, the adiabatic temperature profile is maintained by compositional convection against thermal stratification, except in a still very thin boundary layer, and this means that the entropy flow across the CMB is less than that required to maintain the conduction along the average temperature profile. In other words, the compositional convection has to fight against thermal stratification to maintain the adiabatic temperature profile in addition to maintaining the dynamo. In the second model (see Labrosse et al., (1997); Lister and Buffett (1998)), a subadiabatic layer of several hundreds of kilometers is allowed to develop at the top of the core and the entropy flow out of the core balances the conduction along the average temperature gradient. In this case, the compositional energy is entirely used for the dynamo. Which of these two options would be chosen by the core is a dynamical question that cannot be addressed by simple thermodynamic arguments as used here. Stéphane Labrosse

Bibliography Braginsky, S.I., and Roberts, P.H., 1995. Equations governing convection in Earth’s core and the geodynamo. Geophysical Astrophysical Fluid Dynamics, 79: 1–97. Gessmann, C.K., and Wood, B.J., 2002. Potassium in the Earth’s core? Earth and Planetary Science Letters, 200: 63–78. Gubbins, D., and Roberts, P.H., 1987. Magnetohydrodynamics of the Earth’s core. In Jacobs, J.A., (ed.), Geomagnetism, Vol. 2. London: Academic Press, pp. 1–183. Labrosse, S., 2003. Thermal and magnetic evolution of the Earth’s core. Physics of the Earth and Planetary Interiors, 140: 127–143. Labrosse, S., and Macouin, M., 2003. The inner core and the geodynamo. Comptes Rendus Geosciences, 335: 37–50.

Labrosse, S., Poirier, J.-P., and Le Mouël, J.-L., 1997. On cooling of the Earth’s core. Physics of the Earth and Planetary Interiors, 99: 1–17. Labrosse, S., Poirier, J.-P., and Le Mouël, J.-L., 2001. The age of the inner core. Earth and Planetary Science Letters, 190: 111–123. Lee, K.K.M., and Jeanloz, R., 2003. High-pressure alloying of potassium and iron: radioactivity in the Earth’s core? Geophysical Research Letters, 30: 2212, doi:10.1029/2003GL018515. Lister, J.R., and Buffett, B.A., 1995. The strength and efficiency of the thermal and compositional convection in the geodynamo. Physics of the Earth and Planetary Interiors, 91: 17–30. Lister, J.R., and Buffett, B.A., 1998. Stratification of the outer core at the core-mantle boundary. Physics of the Earth and Planetary Interiors, 105: 5–19. Poirier, J.-P., 1994. Light elements in the Earth’s core: a critical review. Physics of the Earth and Planetary Interiors, 85: 319–337. Rama Murthy, V., van Westrenen, W., and Fei, Y., 2003. Radioactive heat sources in planetary cores: experimental evidence for potassium. Nature, 423: 163–165. Roberts, P.H., Jones, C.A., and Calderwood, A.R., 2003. Energy fluxes and ohmic dissipation in the Earth’s core. In Jones, C A., Soward, A.M., and Zhang, K. (eds.) Earth’s Core and Lower Mantle. London: Taylor & Francis, pp. 100–129. Stevenson, D.J., 1990. Fluid dynamics of core formation. In Newsom, H.E., and Jones, J.H. (eds.) Origin of the Earth. New York: Oxford University Press, pp. 231–249.

Cross-references Convection, Chemical Core Composition Core Density Core Properties, Physical Core Properties, Theoretical Determination Core Temperature Core, Adiabatic Gradient Core, Boundary Layers Core-Mantle Boundary, Heat Flow Across Grüneisen’s Parameter for Iron and Earth’s Core Melting Temperature of Iron in the Core, Experimental Melting Temperature of Iron in the Core, Theory

GEODYNAMO: NUMERICAL SIMULATIONS Introduction The geodynamo is the name given to the mechanism in the Earth’s core that maintains the Earth’s magnetic field (see Geodynamo). The current consensus is that flow of the liquid iron alloy within the outer core, driven by buoyancy forces and influenced by the Earth’s rotation, generates large electric currents that induce magnetic field, compensating for the natural decay of the field. The details of how this produces a slowly changing magnetic field that is mainly dipolar in structure at the Earth’s surface, with occasional dipole reversals, has been the subject of considerable research by many people for many years. The fundamental theory, put forward in the 1950s, is that differential rotation within the fluid core shears poloidal (north-south and radial) magnetic field lines into toroidal (east-west) magnetic field; and three-dimensional (3D) helical fluid flow twists toroidal field lines into poloidal field. The more sheared and twisted the field structure the faster it decays away; that is, magnetic diffusion (reconnection) continually smooths out the field. The field is self-sustaining if, on average, the generation of field is balanced by its decay. Discovering and understanding the details of how rotating convection in Earth’s fluid outer

GEODYNAMO: NUMERICAL SIMULATIONS

core maintains the observed intensity, structure, and time dependencies requires 3D computer models of the geodynamo. Magnetohydrodynamic (MHD) dynamo simulations are numerical solutions of a coupled set of nonlinear differential equations that describe the 3D evolution of the thermodynamic variables, the fluid velocity, and the magnetic field. Because so little can be detected about the geodynamo, other than the poloidal magnetic field at the surface (today’s field in detail and the paleomagnetic field in much less detail) and what can be inferred from seismic measurements and variations in the length of the day and possibly in the gravitational field, models of the geodynamo are used as much to predict what has not been observed as they are used to explain what has. When such a model generates a magnetic field that, at the model’s surface, looks qualitatively similar to the Earth’s surface field in terms of structure, intensity, and time-dependence, then it is plausible that the 3D flows and fields inside the model core are qualitatively similar to those in the Earth’s core. Analyzing this detailed simulated data provides a physical description and explanation of the model’s dynamo mechanism and, by assumption, of the geodynamo. The first 3D global convective dynamo simulations were developed in the 1980s to study the solar dynamo. Gilman and Miller (1981) pioneered this style of research by constructing the first 3D MHD dynamo model. However, they simplified the problem by specifying a constant background density, i.e., they used the Boussinesq approximation of the equations of motion. Glatzmaier (1984) developed a 3D MHD dynamo model using the anelastic approximation, which accounts for the stratification of density within the sun. Zhang and Busse (1988) used a 3D model to study the onset of dynamo action within the Boussinesq approximation. However, the first MHD models of the Earth’s dynamo that successfully produced a time-dependent and dominantly dipolar field at the model’s surface were not published until 1995 (Glatzmaier and Roberts, 1995; Jones et al., 1995; Kageyama et al., 1995). Since then, several groups around the world have developed dynamo models and several others are currently being designed. Some features of the various simulated fields are robust, like the dominance of the dipolar part of the field outside the core. Other features, like the 3D structure and time-dependence of the temperature, flow, and field inside the core, depend on the chosen boundary conditions, parameter space, and numerical resolution. Many review articles have been written that describe and compare these models (e.g., Hollerbach, 1996; Glatzmaier and Roberts, 1997; Fearn, 1998; Busse, 2000; Dormy et al., 2000; Roberts and Glatzmaier, 2000; Christensen et al., 2001; Busse, 2002; Glatzmaier, 2002; Kono and Roberts, 2002).

303

included that account for perturbations in composition and gravitational potential and their effects on buoyancy. This set of coupled nonlinear differential equations, with a set of prescribed boundary conditions, is solved each numerical time step to obtain the evolution in 3D of the fluid flow, magnetic field, and thermodynamic perturbations. Most geodynamo models have employed spherical harmonic expansions in the horizontal directions and either Chebyshev polynomial expansions or finite differences in radius. The equations are integrated in time typically by treating the linear terms implicitly and the nonlinear terms explicitly. The review articles mentioned above describe the variations on the equations, boundary conditions, and numerical methods employed in the various models of the geodynamo.

Current results Since the mid-1990s, 3D computer simulations have advanced our understanding of the geodynamo. The simulations show that a dominantly dipolar magnetic field, not unlike the Earth’s, can be maintained by convection driven by an Earth-like heat flux. A typical snapshot of the simulated magnetic field from a geodynamo model is illustrated in Figure G13 with a set of field lines. In the fluid outer core, where the field is generated, field lines are twisted and sheared by the flow. The field that extends beyond the core is significantly weaker and dominantly dipolar at the model’s surface, not unlike the geomagnetic field. For most geodynamo simulations, the nondipolar part of the surface field, at certain locations and times, propagates westward at about 0.2 y
1 as has been observed in the geomagnetic field over the past couple hundred years. Several dynamo models have electrically conducting inner cores that on average drift eastward relative to the mantle (e.g., Glatzmaier and Roberts, 1995; Sakuraba and Kono, 1999; Christensen et al., 2001), opposite to the propagation direction of the surface magnetic

Model description Models are based on equations that describe fluid dynamics and magnetic field generation. The equation of mass conservation is used with the very good assumption that the fluid flow velocity in the Earth’s outer core is small relative to the local sound speed. The anelastic version of mass conservation accounts for a depth-dependent background density; the density at the bottom of the Earth’s fluid core is about 20% greater than that at the top. The Boussinesq approximation simplifies the equations further by neglecting this density stratification, i.e., by assuming a constant background density. An equation of state relates perturbations in temperature and pressure to density perturbations, which are used to compute the buoyancy forces, which drive convection. Newton’s second law of motion (conservation of momentum) determines how the local fluid velocity changes with time due to buoyancy, pressure gradient, viscous, rotational (Coriolis), and magnetic (Lorentz) forces. The MHD equations (i.e., Maxwell’s equations and Ohm’s law with the extremely good assumption that the fluid velocity is small relative to the speed of light) describe how the local magnetic field changes with time due to induction by the flow and diffusion due to finite conductivity. The second law of thermodynamics dictates how thermal diffusion and Joule and viscous heating determine the local time rate of change of entropy (or temperature). Additional equations are sometimes

Figure G13 A snapshot of the 3D magnetic field simulated with the Glatzmaier-Roberts geodynamo model and illustrated with a set of magnetic field lines. The axis of rotation is vertical and centered in the image. The field is complicated and intense inside the fluid core where it is generated by the flow; outside the core it is a smooth, dipole-dominated, potential field. (From Glatzmaier, 2002.)

304

GEODYNAMO: NUMERICAL SIMULATIONS

field. Inside the fluid core the simulated flow has a “thermal wind” component that, near the inner core, is predominantly eastward relative to the mantle. Magnetic field in these models that permeates both this flow and the inner core tries to drag the inner core in the direction of the flow. This magnetic torque is resisted by a gravitational torque between the mantle and the topography on the inner core surface. The amplitude of the superrotation rate predicted by geodynamo models depends on the model’s prescribed parameters and assumptions and on the very poorly constrained viscosity assumed for the inner core’s deformable surface layer, which by definition, is near the melting temperature. The original prediction was an average of about 2 longitude per year faster than the surface. Since then, the superrotation rate of the Earth’s inner core (today) has been inferred from several seismic analyses, but is still controversial. There is a spread in the inferred values, from the initial estimates of 1 to 3 eastward per year (relative to the Earth’s surface) to some that are zero to within an uncertainty of 0.2 per year. More recent geodynamo models that include an inhibiting gravitational torque also predict smaller superrotation rates. On a much longer timescale, the dipolar part of the Earth’s field occasionally reverses (see Reversals, theory). The reversals seen in the paleomagnetic record are nonperiodic. The times between reversals are measured in hundreds of thousands of years; whereas the time to complete a reversal is typically a few thousand years, less than a magnetic dipole decay time. Several dynamo simulations have produced spontaneous nonperiodic magnetic dipole reversals (Glatzmaier and Roberts, 1995; Glatzmaier et al., 1999; Kageyama et al., 1999; Sarson and Jones, 1999; Kutzner and Christensen, 2002). Regular (periodic) reversals, like the dynamo-wave (q.v.) reversals seen in early solar dynamo simulations, have also occurred in recent dynamo simulations. One of the simulated reversals is portrayed in Figure G14/Plate 16 with four snapshots spanning about 9 ka. The radial component of

the field is shown at both the core-mantle boundary (CMB) and the surface of the model Earth. The reversal, as viewed in these surfaces, begins with reversed magnetic flux patches in both the northern and southern hemispheres. The longitudinally averaged poloidal and toroidal parts of the field inside the core are also illustrated at these times. Although when viewed at the model’s surface, the reversal appears complete by the third snapshot, another 3 ka is required for the original field polarity to decay out of the inner core and the new polarity to diffuse in. Small changes in the local flow structure continually occur in this highly nonlinear chaotic system. These can generate local magnetic anomalies that are reversed relative to the direction of the global dipolar field structure. If the thermal and compositional perturbations continue to drive the fluid flow in a way that amplifies this reversed field polarity while destroying the original polarity, the entire global field structure would eventually reverse. However, more often, the local reversed polarity is not able to survive and the original polarity fully recovers because it takes a couple of thousand years for the original polarity to decay out of the solid inner core. This is a plausible explanation for “events,” which occur when the paleomagnetic field (as measured at the Earth’s surface) reverses and then reverses back, all within about 10 ka. On an even longer timescale, the frequency of reversals seen in the paleomagnetic record varies. The frequency of nonperiodic reversals in geodynamo simulations has been found to depend on the pattern of outward heat flux imposed over the CMB (presumably controlled in the Earth by mantle convection) and on the magnitude of the convective driving relative to the effect of rotation. Many studies have been conducted via dynamo simulations to, for example, assess the effects of the size and conductivity of the solid inner core, of a stably stratified layer at the top of the core, of

Figure G14/Plate 16 A sequence of snapshots of the longitudinally averaged magnetic field through the interior of the core and of the radial component of the field at the core-mantle boundary and at what would be the surface of the Earth, displayed at roughly 3 ka intervals spanning a dipole reversal from a geodynamo simulation. In the plots of the average field, the small circle represents the inner core boundary and the large circle is the core-mantle boundary. The poloidal field is shown as magnetic field lines on the left-hand sides of these plots (blue is clockwise and red is counterclockwise). The toroidal field direction and intensity are represented as contours (not magnetic field lines) on the right-hand sides (red is eastward and blue is westward). Aitoff-Hammer projections of the entire core-mantle boundary and surface are used to display the radial component of the field (with the two different surfaces displayed as the same size). Reds represent outward directed field and blues represent inward field; the surface field, which is typically an order of magnitude weaker, was multiplied by 10 to enhance the color contrast. (From Glatzmaier et al., 1999.)

GEODYNAMO: NUMERICAL SIMULATIONS

heterogeneous thermal boundary conditions, of different velocity boundary conditions, and of computing with different parameters. These models differ in several respects. For example, the Boussinesq instead of the anelastic approximation may be used, compositional buoyancy and perturbations in the gravitational field may be neglected, different boundary conditions, and spatial resolutions may be chosen, the inner core may be treated as an insulator instead of a conductor or may not be free to rotate. As a result, the simulated flow and field structures inside the core differ among the various simulations. For example, the strength of the shear flow on the “tangent cylinder ” (the imaginary cylinder tangent to the inner core equator; Figure G13), which depends on the relative dominance of the Coriolis forces, is not the same for all simulations. Likewise, the vigor of the convection and the resulting magnetic field generation tends to be greater outside this tangent cylinder for some models and inside for others. But all the solutions have a westward zonal flow in the upper part of the fluid core and a dominantly dipolar magnetic field outside the core. When assuming Earth values for the radius and rotation rate of the core, all models of the geodynamo have been forced (due to computational limitations) to use a viscous diffusivity that is at least three to four orders of magnitude larger than estimates of what a turbulent (or eddy) viscosity should be (about 2 m2 s
1 ) for the spatial resolutions that have been employed. In addition to this enhanced viscosity, one must decide how to prescribe the thermal, compositional, and magnetic diffusivities. One of two extremes has typically been chosen. These diffusivities could be set equal to the Earth’s actual magnetic diffusivity (2 m2 s
1 ), making these much smaller than the specified viscous diffusivity; this was the choice for most of the Glatzmaier-Roberts simulations. Alternatively, they could be set equal to the enhanced viscous diffusivity, making all (turbulent) diffusivities too large, but at least equal; this was the choice of most of the other models. Neither choice is satisfactory.

Future challenges Because of the large turbulent diffusion coefficients, all geodynamo simulations have produced large-scale laminar convection. That is, convective cells and plumes of the simulated flow typically span the entire depth of the fluid outer core, unlike the small-scale turbulence that likely exists in the Earth’s core. The fundamental question about geodynamo models is how well do they simulate the actual dynamo mechanism of the Earth’s core? Some geodynamo modelers have argued, or at least suggested, that the large (global) scales of the temperature, flow, and field seen in these simulations should be fairly realistic because the prescribed viscous and thermal diffusivities may be asymptotically small enough. For example, in most simulations, viscous forces (away from the boundaries) tend to be 104 times smaller than Coriolis and Lorentz forces. Other modelers are less confident that current simulations are realistic even at the large-scales because the model diffusivities are so large. Only when computing resources improve to the point where we can further reduce the turbulent diffusivities by several orders of magnitude and produce strongly turbulent simulations will we be able to answer this fundamental question. In the mean time, we may be able to get some insight from very highly resolved 2D simulations of magnetoconvection. These simulations can use diffusivities a thousand times smaller than those of the current 3D simulations. They demonstrate that strongly turbulent 2D rotating magnetoconvection has significantly different spatial structure and time-dependence than the corresponding 2D laminar simulations obtained with much larger diffusivities. These findings suggest that current 3D laminar dynamo simulations may be missing critical dynamical phenomena. Therefore, it is important to strive for much greater spatial resolution in 3D models in order to significantly reduce the enhanced diffusion coefficients and actually simulate turbulence. This will require faster parallel computers and improved numerical methods and hopefully will happen within

305

the next decade or two. In addition, subgrid scale models need to be added to geodynamo models to better represent the heterogeneous anisotropic transport of heat, composition, momentum, and possibly also magnetic field by the part of the turbulence spectrum that remains unresolved. Gary A. Glatzmaier

Bibliography Busse, F.H., 2000. Homogeneous dynamos in planetary cores and in the laboratory. Annual Review of Fluid Mechanics, 32: 383–408. Busse, F.H., 2002. Convective flows in rapidly rotating spheres and their dynamo action. Physics of Fluids, 14: 1301–1314. Christensen, U.R., Aubert, J., Cardin, P., Dormy, E., Gibbons, S. et al., 2001. A numerical dynamo benchmark. Physics of the Earth and Planetary Interiors, 128: 5–34. Dormy, E., Valet, J.-P., and Courtillot, V., 2000. Numerical models of the geodynamo and observational constraints. Geochemistry, Geophysics, Geosystems, 1: 1–42, paper 2000GC000062. Fearn, D.R., 1998. Hydromagnetic flow in planetary cores. Reports on Progress in Physics, 61: 175–235. Gilman, P.A., and Miller, J., 1981. Dynamically consistent nonlinear dynamos driven by convection in a rotating spherical shell. Astrophysical Journal, Supplement Series, 46: 211–238. Glatzmaier, G.A., 1984. Numerical simulations of stellar convective dynamos. I. The model and the method. Journal of Computational Physics, 55: 461–484. Glatzmaier, G.A., 2002. Geodynamo simulations—how realistic are they? Annual Review of Earth and Planetary Sciences, 30: 237–257. Glatzmaier, G.A., and Roberts, P.H., 1995. A three-dimensional selfconsistent computer simulation of a geomagnetic field reversal. Nature, 377: 203–209. Glatzmaier, G.A., and Roberts, P.H., 1997. Simulating the geodynamo. Contemporary Physics, 38: 269–288. Glatzmaier, G.A., Coe, R.S., Hongre, L., and Roberts, P.H., 1999. The role of the Earth’s mantle in controlling the frequency of geomagnetic reversals. Nature, 401: 885–890. Hollerbach, R., 1996. On the theory of the geodynamo. Physics of the Earth and Planetary Interiors, 98: 163–185. Jones, C.A., Longbottom, A., and Hollerbach, R., 1995. A selfconsistent convection driven geodynamo model, using a mean field approximation. Physics of the Earth and Planetary Interiors, 92: 119–141. Kageyama, A., Ochi, M., and Sato, T., 1999. Flip-flop transitions of the magnetic intensity and polarity reversals in the magnetohydrodynamic dynamo. Physics Review Letters, 82: 5409–5412. Kageyama, A., Sato, T., Watanabe, K., Horiuchi, R., Hayashi, T. et al., 1995. Computer simulation of a magnetohydrodynamic dynamo. II. Physics of Plasmas, 2: 1421–1431. Kono, M., and Roberts, P.H., 2002. Recent geodynamo simulations and observations of the geomagnetic field. Review of Geophysics, 40: 41–53. Kutzner, C., and Christensen, U.R., 2002. From stable dipolar towards reversing numerical dynamos. Physics of the Earth and Planetary Interiors, 131: 29–45. Roberts, P.H., and Glatzmaier, G.A., 2000. Geodynamo theory and simulations. Reviews of Modern Physics, 72: 1081–1123. Sakuraba, A., and Kono, M., 1999. Effect of the inner core on the numerical solution of the magnetohydrodynamic dynamo. Physics of the Earth and Planetary Interiors, 111: 105–121. Sarson, G.R., and Jones, C.A., 1999. A convection driven dynamo reversal model. Physics of the Earth and Planetary Interiors, 111: 3–20. Zhang, K., and Busse, F.H., 1988. Finite amplitude convection and magnetic field generation in a rotating spherical shell. Geophysical and Astrophysical Fluid Dynamics, 41: 33–53.

306

GEODYNAMO, SYMMETRY PROPERTIES

Cross-references Core Convection Core Turbulence Core Viscosity Core-Mantle Boundary Topography, Implications for Dynamics Core-Mantle Coupling, Electromagnetic Core-Mantle Coupling, Thermal Core-Mantle Coupling, Topographic Dynamo, Solar Geodynamo Geomagnetic Dipole Field Harmonics, Spherical Inner Core Rotation Inner Core Seismic Velocities Inner Core Tangent Cylinder Magnetohydrodynamics Reversals, Theory Thermal Wind Westward Drift

GEODYNAMO, SYMMETRY PROPERTIES The behavior of any physical system is determined in part by its symmetry properties. For the geodynamo this means the geometry of a spinning sphere and the symmetry properties of the equations of magnetohydrodynamics. Solutions have symmetry that is the same as, or lower than, the symmetry of the governing equations and boundary conditions. By “symmetry” here we mean a transformation T that takes the system into itself. Given one solution with lower symmetry, we can construct a second solution by applying the transformation T to it. Solutions with different symmetry can evolve independently and are said to be separable. If the governing equations are linear separable solutions are also linearly independent: they may be combined to form a more general solution. If the governing equations are nonlinear they may not be combined or coexist but they remain separable. The full geodynamo (q.v.) problem is nonlinear and separable solutions exist; fluid velocities and magnetic fields with the same symmetry are linearly independent solutions of the linear kinematic dynamo (q.v.) problem. Symmetry considerations are important for both theory and observation. For example, solutions with high symmetry are easier to compute than those with lower symmetry and are often chosen for that reason. Time-dependent behavior of nonlinear systems (geomagnetic reversals for example) may be analyzed in terms of one separable solution becoming unstable to one with different symmetry (“symmetry breaking”). Observational applications include detection of symmetries in the geomagnetic field. The axial dipole field has very high symmetry but is not a separable solution of the geodynamo; it does, however, belong to a separable solution with a particular symmetry about the equator. Paleomagnetic data rarely have sufficient global coverage to allow a proper assessment of the spatial pattern of the geomagnetic field, but they can sometimes be used to discriminate between separable solutions with different symmetries. The sphere is symmetric under any rotation about its centre while rotation is symmetric under any rotation about the spin axis. The symmetries of the spinning sphere are therefore any rotation about the spin axis and reflection in the equatorial plane (Figure G15). This conflict of spherical and cylindrical geometry lies at the heart of many of the properties of rotating convection and the geodynamo. The equations of magnetohydrodynamics (q.v.) are also invariant under rotation. They are invariant under change of sign of magnetic field B (but not other dependent variables) because the induction equation is linear in B and the magnetic force and ohmic heating are quadratic in B. They are also invariant under time translation.

Figure G15 Reflection of a rotating sphere in a plane parallel to the equator. A0 , B0 are the reflections of the points A, B. The reflected sphere turns in the same direction as the original sphere. A vector is equatorial-symmetric (ES) if its value at C0 appears as a reflection as shown: it is EA if it appears with a change of sign.

The group of symmetry operations is Abelian because of the infinite number of allowed rotations about the spin axis and time translations. The full set of symmetry operations is found by constructing the group table and using the closure property. The group table, including rotation of p about the spin axis but no higher rotations, is shown in Table G6. Note the additional symmetry operations O; these are combinations of reflection in the equatorial plane and rotation about the spin axis; they amount to reflection through the origin. Note also the subgroups formed by (I,i) and (I,i,E S,E A). These are fundamental to some analyses of paleomagnetic data. Arbitrary time translation can be applied to any symmetry to produce steady solutions that are invariant under translation, drifting solutions that are steady in a corotating frame, more complicated time-periodic solutions that may vascillate or have reversing magnetic fields, and solutions that change continually and are sometimes loosely called “chaotic.” A word is needed about the behavior of vectors under reflection. A vector is usually defined by its transformation law under rotation. An axial or pseudovector (or tensor) changes sign on reflection whereas a polar or true vector (or tensor) does not. A true scalar is invariant under reflection, a pseudoscalar changes sign. Examples of true vectors are fluid velocity and electric current. Examples of axial vectors are angular velocity and magnetic field. The cross product changes sign under reflection (to see this consider the simple case of the cross product of two polar vectors); the curl also changes sign under reflection. Vectors v satisfying r  v ¼ 0 are often represented in terms of their toroidal and poloidal parts:

307

GEOMAGNETIC DEEP SOUNDING

Table G6 Multiplication table for a finite subgroup of spatial symmetry operations for a buoyancy-driven dynamo in a rotating sphere I

i

ES

EA

P2S

P2A

OS

OA

i ES EA P2S P2A OS OA

I EA ES P2A P2S OA OS

EA I i OS OA P2S P2A

ES i I OA OS P2A P2S

P2A OS OA I i ES EA

P2S OA OS i I EA ES

OA P2S P2A ES EA I i

OS P2A P2S EA ES i I

Note: I, identity; i, field reversal; E S, reflection in equatorial plane; E A, reflection in equatorial plane with change of sign of magnetic field; P2, rotation by p about spin axis; and O, reflection in origin.

v ¼ r  Tr þ r  r  Pr; if the toroidal part is a true vector the poloidal part will be a pseudovector and vice versa because of the extra curl involved. Helicity, v  r  v, is a pseudoscalar because vorticity changes sign under reflection but velocity does not. Properties of true and pseudovectors are used to determine the symmetry of individual terms in the governing equations and to find separable solutions. The symmetry of solutions is reflected in their spherical harmonic expansions. Potential fields with E A symmetry involve only harmonics Ylm with l
m odd; E S fields have l
m even. The symmetries are usually referred to as “dipole” and “quadrupole” families because of their leading terms Y10 (dipole) and Y20 (quadrupole). The terminology is somewhat unsatisfactory for two reasons: first, the equatorial dipole Y11 is a member of the quadrupole family, and second the internal, toroidal, field of each family has the opposite series to that of the external, poloidal, field. Fields with P2S symmetry have spherical harmonic series with m even, P2A have m odd. Solutions with higher symmetry have series containing m differing by larger integers, for example P4S has m multiples of 4. The main use of symmetry properties in theoretical studies has so far been restricted to reducing the complexity of the solution in order to effect numerical solutions. For example, allowing only a “dipole family,” or E S, solution halves the number of spherical harmonics required to represent the solution (or, equivalently, the solution only need be found in one hemisphere); P2S halves it again. In practice solutions with high symmetry may be poor dynamos and be more difficult to compute, despite their apparent lack of spatial complexity, because they require strong driving and involve small-scale magnetic fields. The existence of a separable solution is no guarantee of maintaining a magnetic field: B may still decay to zero. Thus axial symmetry is an allowed symmetry but Cowling’s theorem (q.v.) shows that no axisymmetric magnetic field can be sustained by dynamo action. Kinematic dynamos (q.v.) with axisymmetric fluid velocities have axisymmetric solutions that decay by Cowling’s theorem, but solutions with lower symmetry, each proportional to expimf, can grow with time. They do not form separable solutions of the full, dynamical, dynamo problem because nonlinear terms in the equations couple the modes to include many values of m. Bullard and Gellman (1954) investigated an E S P2S flow that also possessed a meridional plane of symmetry, again to reduce the computational effort required for the very small, early computer at their disposal. This last symmetry is not an allowed separable solution of the full dynamo problem. The lack of helicity imposed by this symmetry was later found to be the reason for the failure of the Bullard-Gellman dynamo (q.v.). The future may see further studies of dynamo behavior making more use of symmetries. For example, reversals may be understood in terms of the “dipole” E A solution becoming unstable to a “quadrupole” E S or an oscillatory solution. It was once suggested that reversals

may involve only the observed poloidal field, the larger internal toroidal field retaining the same polarity, but this is unlikely because it violates the symmetry properties of the solution. Symmetry properties have received more attention in observational studies, particularly with paleomagnetic data (see Geomagnetic field, asymmetries). The basic observation is that of an E A field, the dipole, but equatorial symmetry appears to go beyond that of the dipole: the main concentrations of magnetic field on the core-mantle boundary form four lobes, two in the northern hemisphere and two in the southern hemisphere, on closely similar longitudes (see Plate 10c). This basic pattern is close to P2S symmetry, but a nonaxisymmetric pattern would only be of interest if it were long term. With homogeneous boundary conditions we would expect the solution to drift without reference to any longitude, averaging to axial symmetry. This has been assumed in many studies in the past; departures from axial symmetry would imply an effect of inhomogeneity on the boundary, such as variable heat flux from the core (see Core-mantle coupling, thermal). The lower mantle seismic velocities suggest a P2S pattern associated with subduction around the Pacific rim, which could favor P2S magnetic fields. Further discussion of the theory is in Gubbins and Zhang (1993) and of the data analysis in Merrill et al. (1996). David Gubbins

Bibliography Bullard, E.C., and Gellman, H., 1954. Homogeneous dynamos and terrestrial magnetism. Philosophical Transactions of the Royal Society of London, Series A, 247: 213–278. Gubbins, D., and Zhang, K., 1993. Symmetry properties of the dynamo equations for paleomagnetism and geomagnetism. Physics of the Earth and Planetary Interiors, 75: 225–241. Merrill, R.T., McElhinny, M.W., and McFadden, P.L., 1996. The Magnetic Field of the Earth. San Diego, CA: Academic Press.

Cross-references Cowling’s Theorem Core-Mantle Boundary Topography, Implications for Dynamics Core-Mantle Coupling, Thermal Dynamo, Bullard-Gellman Dynamos, Kinematic Geomagnetic Field, Asymmetries Magnetohydrodynamics Paleomagnetic Secular Variation

GEOMAGNETIC DEEP SOUNDING Geomagnetic deep sounding (GDS) is the use of electromagnetic induction methods to determine the electrical conductivity within the Earth, working from observations of natural geomagnetic variations. It is differentiated from the magnetotelluric method (q.v.) in employing only the magnetic, and not the electric field. The term GDS is applied both to global and to regional studies. The aim of global investigations is to determine the variation of electrical conductivity with depth. That of regional studies is to map lateral differences in the conductivity of the crust and upper mantle. The book by Rokityansky (1982) covers all aspects of the subject. Weaver (1994) gives a detailed account of the theory.

Sounding the earth using natural geomagnetic variations A slowly varying magnetic field inside a uniform conductor (conductivity s and relative magnetic permeability m) satisfies the induction equation:

308

GEOMAGNETIC DEEP SOUNDING

r2 B ¼ mm0 s

]B ]t

The time-varying field induces eddy currents in the conductor which flow so as to exclude the field from the deeper parts. The amplitude of a spatially uniform field of frequency o falls to 1/e of its surface value at the “skin-depth”: z0 ¼

p

represented by spherical harmonics up to degree n ¼ 4. The continuum from a few days to a few minutes period originates in the current systems produced by geomagnetic storms. Their spatial structure is complex, and in the auroral regions is localized as electrojets in the ionosphere, and field-aligned currents connecting with the outer magnetosphere. Micropulsations are vibrations of geomagnetic field lines, while Schumann resonances are oscillations of the Earth-ionosphere waveguide excited by large-scale thunderstorm activity.

2=omm0 s:

This expression provides a rough guide to the “sounding depth” which might be expected of a particular frequency. However, the geometry of the external source also restricts the depth or volume sampled by the field. When induction effects are negligible, a field with spatial wavelength l falls to 1/e of its surface value at depth l/2p. The basis of the sounding method is to measure the Earth response at a range of frequencies and/or source wavelengths. If the response of a one-dimensional earth is known precisely either at all frequencies or all spatial scales, the radial variation of conductivity is uniquely defined.

The geomagnetic variation spectrum The frequency range of the externally generated electromagnetic spectrum (Figure G16) is extremely wide. The longest periods available are associated with the solar cycle (11 or 22 years), and penetrate into the lower mantle. Unfortunately, they are difficult to separate from the purely internal variations with periods longer than 3 years generated by the secular behavior of the dynamo. Much of the spectrum between 2 years and 2 days period comes from fluctuations in the total energy of particles in the radiation belts, which drift in the geomagnetic field, creating the ring current (q.v.). Because the current is located between 3 and 5 Earth radii, the field it creates at the Earth is relatively uniform, and its spatial structure can be represented by a small number of zonal spherical harmonics (q.v.), of which the first (n ¼ 1) is much the most important. Variations with this structure include the semiannual line, the quasiperiodic harmonics at 27, 13.5, 9 days, etc., driven by the Sun’s rotation and the persistence both of solar sources and sector structure in the solar wind, and the continuum. The annual variation, however, has a distinct spatial structure that is antisymmetric about the equator, suggesting a seasonal driving force. The daily variation and its harmonics are created by dynamo action in the ionosphere, where thermal and gravitational tides move plasma through the magnetic field. Their spatial structure can be adequately

GDS—the global problem Only magnetic observatories provide the record lengths required for global sounding to depths of hundreds of kilometers. Because of their poor distribution and insufficient numbers, only the smoother fields are defined adequately. Temporary arrays of magnetometers are deployed to map the more complex fields, and, in the absence of conductivity anomalies, the field gradients can be used for local soundings. The determination of the vertical variation of conductivity is conveniently divided into two steps. The first is the measurement of the response or transfer function (q.v.) which links the input—the external part of the magnetic field—to the output—the internal part created by the induced currents. The second is the inversion of the response for the conductivity—discovering what can be inferred about s(r) from the response and its associated errors.

Definition and determination of the response function In global GDS, spherical harmonic functions (q.v.) are commonly used to define the spatial structure of the field. The response Qnm ðoÞ is the ratio of the internal and external parts of the field at frequency o for a spherical harmonic component of degree n and order m. An alternative is the C response: C¼

Br ]Br =]r

The radial gradient of the vertical field is replaced by the horizontal gradients of the horizontal components using the condition div B ¼ 0. For plane earth geometry, and a smooth external field, C¼

Bz ]Bx =]x þ ]By =]y

The C response has been favored in recent investigations because of its intercomparability between global and local studies, and physical significance as a penetration scale (its dimension is length). The magnetic observatory network is inadequate for determining all but the largest scale spatial structures. Instead of a full spherical harmonic analysis, a simple spherical harmonic model is usually adopted, based on what is known of the source. For periods between 2 years and 2 days, a single zonal harmonic has been used, with the advantage that the response can be computed from vertical (Z) and horizontal (H) component records at a single observatory. The total potential of internal (gi) and external (ge) sources for a spherical harmonic degree n ¼ 1 and order m ¼ 0, is O¼

Figure G16 Schematic representation of the natural geomagnetic spectrum. SV, secular variation of the geodynamo; AV, annual variation; RS, recurrent storms; C, continuum; D, daily variation; S, storms and substorms; MP, micropulsations; SR, Schumann resonance.

 2 a a r cos y gi 2 þ ge m0 a r

The corresponding components of the magnetic induction are Z ¼
Br ðr ¼ aÞ ¼ f
2gi þ ge g cos y H ¼
By ðr ¼ aÞ ¼
fgi þ ge g sin y

GEOMAGNETIC DEEP SOUNDING

309

The ratio of the vertical and horizontal components of the field, multiplied by tany, is itself a response (W10 ), which can be computed from the component records by standard response estimation techniques:

 1
2Q 01 Z  W10 ¼ tan y ¼

H
1 þ Q 01 The major factors limiting the precision of response estimates are the inadequacy of the source model and the influence of lateral variations in conductivity. Attempts to incorporate these effects into response determination and modeling are hampered by the limitations of the observatory network.

Determination of the conductivity The second step is the transformation of the electromagnetic response data into a conductivity model. In the forward modeling approach (q.v.), a conductivity distribution is selected either on the basis of independent geophysical constraints, or for mathematical convenience. Its response is computed and compared with the observations, and the model parameters adjusted until a satisfactory fit is achieved. In the inverse modeling method (q.v.), parameters which define the structure are determined directly from the data. In practice, the two approaches are more alike than at first appears. The class of structure, which is to be the target of an inversion must be selected, and preferences about the smoothness of models incorporated. A preliminary exploration of simple models, for which the relationship between data and structure is well understood, is always worthwhile. Consider a model in which a perfectly conducting “core,” radius r ¼ Ra, is surrounded by an insulating shell. The Q01 response (to excitation by ring current-generated fields) can be interpreted using the spherical harmonic model to downward continue the field. The radial component of the field in any source-free region is: 
2a3 Br ¼
gi 3 þ ge cos y; r and it must be zero at the surface of the perfect conductor. If Br ¼ 0 at r ¼ Ra, Q01 ¼ gi =ge ¼ R3=2 At 27 days period, the value of Q01 is 0.3, which implies R ¼ 0.84, corresponding to a depth of 1000 km. This is a strong indication that the conductivity rises steeply in the upper mantle.

The global electromagnetic response and global conductivity distribution The earliest determinations of the global response are summarized by Chapman and Bartels (1940). Schuster, Chapman, Price, and Lahiri used the daily variation and time-domain analyses of magnetic storms. Conductivity models were restricted to those with analytical solutions—a uniform core surrounded by a uniform insulator, and a power law increase. With the arrival of the digital computer, Fourier transform-based spectral analysis methods were introduced (Currie, 1966). The response could be determined at a continuous range of frequencies, and it was possible to calculate the theoretical response of arbitrary models (Banks, 1969). Recognition of the problems posed by source complexity and lateral variations in conductivity led to more robust response estimation and regionalization of the models (see Constable, 1993). Weidelt and Parker (see Parker, 1983) clarified the inverse problem and demonstrated that the best-fitting model for any set of data was a set of thin conducting sheets. They also showed how to construct more realistic models, which would, however, fit the

Figure G17 Representative electrical conductivity models. LP, Lahiri and Price model d; B, Banks; C, Constable; O, Olsen.

data less well. Constable applied their techniques to collated response data. Olsen (1998, 1999) further refined both the response and models for the European area. There was early recognition of a steep rise in conductivity in the 400–800 km depth range to a value of 2 Sm–1 (Figure G17). Later work has made only minor differences. The major factors inhibiting improvement are the inadequacy of the source model and the influence of lateral variations in conductivity. Attempts to incorporate these effects into response determination and modeling (Schultz and Zhang, 1994) are hampered by the nature of the observatory network. Satellite observations may be one route to future progress.

GDS—mapping lateral variations in conductivity Conductivity anomalies and their response Outside the auroral zones, the externally generated part of the timevarying magnetic field is uniform over hundreds of kilometers. If the electrical conductivity were similarly uniform, the induced currents would double the horizontal component of the external magnetic field but cancel the vertical component. Such a conductivity structure, and the fields associated with it, are referred to as “normal.” What additional “anomalous” fields are created when a region of different conductivity—a conductivity “anomaly”—is embedded within the normal structure? The anomaly’s response depends on how its characteristic size (L) relates to the length scale of the normal field, and with a uniform field what matters is its skin-depth z0, which increases with period. When z0 L (at high frequencies), the characteristics of the induced currents are controlled by the local structure. When z0  L (at low frequencies), they are determined by the host body. The pattern of the “normal” induced currents is modified by electric charges set up on the boundaries of the anomaly. A dipolar current system is created which enhances (a conductive anomaly) or opposes (a resistive anomaly) the normal current flow, but which is in phase with it. Considered as an input/output problem, the input is the normal, spatially uniform horizontal field; the output is the anomalous magnetic field. This is either the entire measured vertical component (since the normal vertical field is zero), or the difference between the local and normal horizontal fields. The output is the sum of the response to two independent inputs—orthogonal directions of the normal horizontal field: Bz ðoÞ ¼ Zzx ðoÞB nx ðoÞ þ Zzy ðoÞB ny ðoÞ; where Bz is the vertical field at frequency o at a site influenced by the anomaly, and Bnx , Bny the north and east components at a normal site.

310

GEOMAGNETIC DIPOLE FIELD

In practice, recordings at a single site are often used, and the vertical field related to the horizontal field (Bx, By) at the same site: Bz ðoÞ ¼ Tzx ðoÞBx ðoÞ þ Tzy ðoÞBy ðoÞ: Z and T are complex quantities, reflecting the phase shifts between the anomalous currents and the normal fields.

Magnetic variation mapping experiments An ideal magnetic variation (MV) mapping experiment requires a large array of simultaneously recording magnetometers, spaced sufficiently closely as to avoid aliasing the structure, with at least one instrument at a normal site. Between 1965 and 1985, arrays of up to 50 magnetometers were constructed and deployed (Gough, 1989). They provided valuable initial information on the conductivity structure of the crust and upper mantle in North America, East and Southern Africa, Australia, India, and Europe. However, their limitations were soon clear. Even relatively low frequencies were affected by structures in the upper crust, so spacings less than 10 km were required, limiting the coverage. The analog recording method limited the frequency response and volume of data which could be interpreted. In particular, sampling at 0.1 to 1 Hz for a few weeks restricted the response to the range for which scattering by crustal anomalies was the dominant process. Later experiments with broader-band digital recording were restricted to a small number of instruments, and were forced to revert to the transfer function techniques described above. The first task in interpretation is to determine the spatial pattern of conductivity without actually modeling the response. With small numbers of irregularly distributed magnetometers, a useful technique is to plot induction arrows (q.v.). For the T response, the vector is plotted with lengths in the north and east directions proportional to Tzx and Tzy , respectively. Near two-dimensional bodies, the arrows point toward or away from the structure. Arrows are harder to interpret when the structure is three-dimensional. Once the responses at a network of sites have been determined, the defining equations can be used to predict the spatial pattern of the vertical fields for a selected horizontal field—the “hypothetical event.” However, the T response includes the effect of local anomalous horizontal fields, so the predicted vertical field anomalies do not correspond to induction by a uniform horizontal field. Banks (1986) devised a method of determining the Z transfer functions from the spatial variation of T, together with the constraint that the field derives from a potential that satisfies the Laplace equation. Egbert (2002) reviews methods of organizing and displaying MV array data. These include the further step of inverting the MV data for a map of the conductance in a thin sheet. The final step is to apply forward and inverse modeling techniques (q.v.) to combined MV and MT responses. The latter provide the vital constraints on absolute conductivity values.

General remarks GDS was the most popular natural source electromagnetic method between 1950 and 1980. The magnetotelluric method was viewed with some suspicion because of the limitations of technology (difficulties in measuring electric fields, the need for high capacity data storage facilities, infield processing to evaluate data quality, etc.—all this came along after 1980 with microprocessor technology), lack of understanding of the effects of very local distortion on the electric field, and inability to compute the electromagnetic fields associated with twoand three-dimensional structures. But it had the huge advantage that measurements at a single site had the potential to define both the vertical and horizontal structure. With the advent of more powerful computing facilities, MT took over from GDS, which was then somewhat ignored. Now there is a realization that measuring and analyzing both electric and magnetic fields adds enormously to the information that can be derived. Roger Banks

Bibliography Banks, R.J., 1969. Geomagnetic variations and the electrical conductivity of the upper mantle. Geophysical Journal of the Royal Astronomical Society, 17: 457–487. Banks, R.J., 1986. The interpretation of the Northumberland trough geomagnetic variation anomaly using two-dimensional current models. Geophysical Journal of the Royal Astronomical Society, 87: 595–616. Chapman, S., and Bartels, J., 1940. Geomagnetism. London: Oxford University Press. Constable, S., 1993. Constraints on mantle electrical conductivity from field and laboratory measurements. Journal of Geomagnetism and Geoelectricity, 45: 1–22. Currie, R.G., 1966. The geomagnetic spectrum—40 days to 5.5 years. Journal of Geophysical Research, 71: 4579–4598. Egbert, G.D., 2002. Processing and interpretation of electromagnetic induction array data. Surveys in Geophysics, 23: 207–249. Gough, D.I., 1989. Magnetometer array studies, Earth Structure and tectonic processes. Review of Geophysics, 27: 141–157. Olsen, N., 1998. The electrical conductivity of the mantle beneath Europe derived from C-responses from 3 to 720 h. Geophysical Journal International, 133: 298–308. Olsen, N., 1999. Long-period (30 days–1 year) electromagnetic sounding and the electrical conductivity of the lower mantle beneath Europe. Geophysical Journal International, 138: 179–187. Parker, R.L., 1983. The magnetotelluric inverse problem. Geophysical Surveys, 6: 5–25. Rokityansky, I.I., 1982. Geoelectromagnetic Investigation of the Earth’s Crust and Mantle. Berlin: Springer-Verlag. Schultz, A., and Zhang, T.S., 1994. Regularized spherical harmonic analysis and the 3-D electromagnetic response of the earth. Geophysical Journal International, 116: 141–156. Weaver, J.T., 1994. Mathematical Methods for Geoelectromagnetic Induction. Taunton: Research Studies Press.

Cross-references EM Modeling, Forward EM Modeling, Inverse EM, Regional Studies Harmonics, Spherical Induction Arrows Induction from Satellite Data Internal External Field Separation Magnetotellurics Mantle, Electrical Conductivity, Mineralogy Ring Current Robust Electromagnetic Transfer Functions Estimates Transfer Functions

GEOMAGNETIC DIPOLE FIELD A long thin bar magnet gives a magnetic field, the lines of force of which (in the usual sign convention) leave the magnet near its north magnetic pole, and reenter near its south magnetic pole. If we think of the magnet being physically reduced in size, but keeping the same magnetic moment (see below), then in the limit of infinitesimal size we have, what we call, a dipole field. Near the Earth, its magnetic field resembles that of a magnetic dipole situated at the geocenter; formally, if we represent the field as a series of spherical harmonics (see Harmonics, spherical), then the field given by the n ¼ 1, dipole, terms dominates. (Note that these are the fields of fictitious dipoles; the real source is electric currents

GEOMAGNETIC EXCURSION

distributed throughout the core.) For many purposes it is adequate to approximate the geomagnetic field as that of a dipole; however, there are several possible definitions of such an approximating dipole. When averaged over thousands of years, the field is very nearly that of a central axial dipole; i.e., the dipole is at the center of the Earth, and directed along the geographical axis, the spin axis. (This alignment is almost certainly due to the very strong influence the Earth’s rotation has on the motions in the liquid core which produce the electric currents—see Geodynamo.) At present, the dipole points from north to south (in the sense that the north pole of the fictitious magnet is nearer the south geographic pole), but the direction has reversed many times during geological time (see Geomagnetic polarity reversals, observations). This axial dipole corresponds to the coefficient g10 in a spherical harmonic analysis. Physically, an axial dipole field is produced by a suitable axially symmetric distribution of electric currents, and its magnitude (or strength), called the dipole moment, is given in units of Am2 (current multiplied by “area turns”); at present its value is about 8  1022 A m2, a value probably rather larger than its average over the last 109 y (see Dipole moment variations). For the geomagnetic axial dipole, the north-to-south horizontal magnetic field on the equator at radius r is given by B ¼ ðm0 =4pr3 Þ

ðDipole momentÞ:

The spherical harmonic Gauss coefficient g10 gives this value for the Earth’s surface, r ¼ a, and at present it is about –30000 nT; the negative sign is there because the field is actually directed south-to-north. But while on average the dipole is axial, at any one time the bestfitting dipole is usually inclined to the spin axis, by an angle of about 10 . A general central dipole can be resolved into three orthogonal components: one along the spin axis (corresponding to the Gauss coefficient g10 ), plus two in the equatorial plane—one (g11 ) in the direction of zero longitude (the Greenwich meridian), and the other (h11 ) in the direction of 90 longitude. The total dipole is called the inclined dipole, and its axis is called the geomagnetic axis; for phenomena (such as the ionosphere) which are controlled by the geometry of the geomagnetic field, it is often convenient to work in a coordinate system which is based on this geomagnetic axis, rather than on the geographic axis. While this dipole field dominates, the remaining nondipole field (q.v.) is still significant. This nondipole field corresponds to all the n > 1 terms in a spherical harmonic analysis, and at the Earth’s surface is typically about 25% that of the dipole field. (At the core-mantle boundary the nondipole field is larger than the dipole field, as its smaller scale fields increase downward more rapidly than the large-scale dipole field.) (See Harmonics, spherical and Geomagnetic spectrum, spatial.) Several other planets of the solar system, and many other astronomical bodies, have dipole-like magnetic fields. For a given body, and at a given time, the magnitude and orientation of the dipole are unique. In fact the magnitude and direction of the vector dipole moment are invariant; whatever coordinate system we choose to measure from, we will always get the same dipole moment, i.e., magnitude and direction in space. The (central) inclined dipole is a reasonable approximation to the observed field. However, if we move the position of this inclined dipole away from the center this introduces three more parameters, so it is possible to get a slightly better fit to the observed field; the displacement reduces the magnitude of what we have called the nondipole field. Conventionally, such a fit is not made to the whole of the nondipole field, but only to the n ¼ 2, quadrupole, part of it; while keeping the moment and direction of the inclined dipole constant, the dipole is moved away from the geocenter in such a direction, and by such an amount, as to minimize (in a least-squares sense) the quadrupole field as seen from the displaced origin. Such a displaced dipole is called the eccentric dipole. It should be noted that (like the

311

other dipoles) it is a simply a convenient mathematical fiction, but using it can be a useful arithmetic simplification in, for example, the study of the deflection of cosmic rays. At present the displacement is about 550 km, toward Japan; the displacement is increasing with time, because the dipole field is reducing in magnitude compared with the quadrupole field. Various authors have suggested other definitions for the “best-fit” dipole (see Lowes, 1994), but those discussed above are the ones currently used. Frank Lowes

Bibliography Lowes, F.J., 1994. The geomagnetic eccentric dipole; facts and fallacies. Geophysical Journal International, 118: 671–679.

Cross-references Dipole Moment Variations Geodynamo Geomagnetic Polarity Reversals, Observations Geomagnetic Spectrum, Spatial Harmonics, Spherical Nondipole Field

GEOMAGNETIC EXCURSION Records of the Earth’s magnetic field have shown that on occasions it has reversed its (see geomagnetic polarity reversals). Intervals during which the field is predominantly of the same polarity (>1 Ma) have been called chrons. Occasionally within a chron, the magnetic field reverses its polarity for a short time ( 1 the probability of a future reversal drops to zero as soon as a reversal occurs and then gradually recovers to its undisturbed value. Thus, the process has a memory of the previous event and this memory temporarily depresses the probability of occurrence of future events. Hence, the parameter k can be interpreted in terms of the stability of the field (against a further reversal) immediately after a reversal. Early studies suggested that k is different for the two polarities, indicating that the relative stabilities of the polarity states are different. A concern with these early studies was that the sense of this asymmetry was sensitive to minor details in the polarity chronology. A major problem is that it is easy to miss a short interval in the polarity record, and this has major consequences for interpretation. Consider the situation if a short reverse interval is missed. The three-interval polarity segment normal-reverse-normal appears as a single-interval-normal polarity segment: the three original intervals have been incorrectly concatenated into a single interval. It is simple to show that if three intervals are drawn at random from a Poisson distribution (k ¼ 1) and concatenated into a single interval, then the result is the same as drawing a random interval from a gamma distribution with k ¼ 3. Hence, when short intervals are missed it erroneously increases the estimated value of k, and this was not appreciated in the early studies. With this in mind, and with improved data, it is now apparent that the data are consistent with the two polarities having the same value of k. Indeed, for the reversal process, k is close to 1 and so the process is nearly Poisson, if not actually Poisson.

Intensity of the field Reliable determination of the paleointensity of the geomagnetic field is notoriously difficult. Consequently, conclusions based on paleointensities are generally less robust than those based on paleodirections or on the reversal chronology. Nevertheless, paleointensities are an important source of information. During the 1980s several studies indicated that there were minor but statistically significant differences in the distributions of paleointensity for the normal and reverse polarity fields. However, none of these observations was robust and each suffered from structural problems such as dependence on one or two extreme values or poor spatial distribution of observations. Further work has not supported these apparent differences and there now seems no reason to reject the simple view that the two polarity states have a common time-average paleointensity.

Structure of the normal and reverse polarity fields By far the most common way to analyze the geomagnetic field is through the use of spherical harmonics. The lower-degree harmonics are probably best known by the terms dipole (degree 2), quadrupole (degree 3), and octupole (degree 4). Zonal harmonics are harmonics that are symmetrical about the spin axis and dominate the structure of the time-average field. In the late 1970s and into the 1980s, analyses of data from the past 5 Ma indicated that the structure of the timeaverage reverse polarity field has been discernibly different from that of the time-average normal polarity field. Specifically, it appeared that the time-average reverse polarity field had a proportionately larger quadrupole and octupole content than the time-average normal polarity field. However, there were (and remain) significant problems with the data. The data came primarily from lava flows on continents and islands and from deep-sea sedimentary cores, which generally provided only values of the paleomagnetic inclination. The spatial distribution of the

GEOMAGNETIC FIELD, ASYMMETRIES

data is poor, particularly in the southern hemisphere, and the data are poorly distributed in time. Also, rock magnetic and other effects can produce spurious estimates of odd-degree harmonics, particularly the octupole term. Later work has shown that these early conclusions were not robust and were mainly due to data artifacts.

Polarity transition asymmetry Our understanding of the structure of the field during transition from one polarity to the other is, at best, rudimentary. Despite the fact that the field at the Earth’s surface is unlikely to be dominantly dipolar during a transition, it is standard practice to use the field direction to calculate the position on the Earth’s surface where the pole would be if the field were dipolar. This is referred to as a virtual geomagnetic pole, or VGP. In the absence of a dominant dipolar field structure it would be expected that, for a single reversal transition, the VGP transition path would be quite different for observations taken from different locations. Of particular interest here is the fact that in the absence of a persistent asymmetry in the Earth, there should be no consistency in the VGP transition paths from one reversal transition to another. Several investigators have noted the existence of apparently preferred VGP polarity transition paths, both within individual transitions and persisting across several different transitions. Indeed, there is also some suggestion of similar preferential paths in VGP movement as a consequence of normal secular variation. There are also proponents for periods of VGP stasis during transitions with preferred positions for the clustering of VGPs. Appeals have been made to the existence of persistent regions of relatively high electrical conductivity in D00 , anomalous regions of heat flux through D00 , or topography at the core-mantle boundary to influence the dynamo behavior and produce these preferred paths. Conversely, there are investigators who feel the interpretation of preferred paths is a consequence of grossly inadequate data. This is currently one of the more controversial topics in paleomagnetism.

Structure of the time-average field Over the years, several attempts have been made to determine the structure of the time-average field by undertaking spherical harmonic analyses of paleomagnetic data for the past 5 Ma. Of particular interest here is the question of whether any nonzonal harmonic terms are genuinely present in the time-average field. Most of the modelers who have undertaken these investigations do claim the presence of nonzonal terms, but the agreement between different models is poor and this suggests that the conclusions are as yet not robust. As already discussed, there is no reliable evidence for polarity asymmetry and so a comparison of normal and reverse polarity results can be used to assess errors in the estimation of individual harmonic terms. Such a comparison suggests that the actual errors exceed the formal errors assigned by modelers. Once again there are questions regarding reliability of the data. The data are not well distributed either in space or in time, and it is difficult to detect small rotations in the rocks providing the data. Consequently, our knowledge about the structure of the time-average field remains inconclusive and there is inadequate robust evidence to identify any specific asymmetries.

315

symmetrically on either side of the equator at 60 latitude and at 120 W and 120 E longitude. Most of the nondipole field drifts westward, but these flux lobes appear to have been fixed for the time interval investigated. A third north-south pair of lobes is required to produce symmetry, and the speculation is that such a third pair has been disrupted by fluid flow near the surface of the outer core. Bloxham and Gubbins hypothesized that the stationary lobes reflect convection rolls found much earlier in weak-field dynamo models by Fritz Busse. (“Weak-field” means that the magnetic field has a negligible effect on fluid motions in the core. Most theory today involves strong-field models.) However, dynamo models indicate that the convection rolls should drift westward or eastward depending on the details of the model. Thus, it was posited that there are thermal anomalies that pin the flux lobes to certain locations in the lowermost mantle. If true, this would lead to long-term asymmetry in the time-average magnetic field. Gubbins and Bloxham also emphasize that the data show a persistent low secular variation in the Pacific hemisphere and, as already discussed, this would also lead to a manifestation of asymmetry in the secular variation data.

Conclusion Dynamo theory is notoriously complex and even with the computing power available today a working model, based on parameters that actually match those in the Earth, remains elusive. Hence, it is not currently possible to predict that the particular asymmetry in the Earth will manifest as a particular asymmetry in the geomagnetic field. Thus, although there are several mechanisms that it is speculated might lead to geomagnetic asymmetry, such as anisotropy in the inner core, asymmetry in any initial field, thermal-electric and battery effects originating in the mantle, or thermal and chemical heterogeneities in D00 , there is no robust evidence that this would actually occur. Certainly, it is not currently possible to invert from an observed asymmetry in the geomagnetic field to the underlying cause of that asymmetry. However, dynamo models do indicate that thermal structure at the base of a mantle does influence the structure of the field generated by the geodynamo. The case for or against time-average asymmetry in the magnetic field and/or its secular variation remains inconclusive. The case against asymmetry is simple to state: all evidence put forth to advocate asymmetry contains errors, inadequacies, or debatable assumptions. The case for asymmetry is more subtle. Proponents recognize the problems, but point to a variety of data types that each weakly suggest a common location for an asymmetry of some form in the lowermost mantle. For example, the stationary flux lobes evidenced in the direct observations, claimed biases of VGP polarity transition data, and some secular variation data, have all been used to argue for anomalous mantle near the eastern margin of the Pacific hemisphere. This is also a region that appears to exhibit different seismic properties. Although a scientific cliché, we conclude that more data and analyses are required.

Acknowledgments This article is published with the permission of the Chief Executive Officer, Geoscience Australia. Phillip L. McFadden and Ronald T. Merrill

Field structure from direct observations Spherical harmonic models have been created using data from magnetic observatories, satellites, and (corrected) ancient mariner logs. There are now models based on these data that extend from approximately 400 years ago to the present. While this time span is well short of that required to obtain a valid time-average magnetic field, some of the results combined with theory have stimulated research on possible long-term magnetic field asymmetry. In particular, Jeremy Bloxham and David Gubbins found four lobes (in two pairs) in the structure of the field that are placed approximately

Bibliography Bloxham, J., and Gubbins, D., 1985. The secular variation of the Earth’s magnetic field. Nature, 317: 777–781. Bloxham, J., and Gubbins, D., 1986. Geomagnetic field analysis. IV. Testing the frozen-flux hypothesis. Geophysical Journal of the Royal Astronomical Society, 84: 139–152. Constable, C., and Johnson, C., 1999. Anisotropic paleosecular variation models: implications for geomagnetic field observables. Physics of the Earth and Planetary Interiors, 115: 35–51.

316

GEOMAGNETIC HAZARDS

Cowling, T.G., 1934. The magnetic field of sunspots. Monthly Notices of the Royal Astronomical Society, 94: 39–48. Merrill, R.T., and McFadden, P.L., 1999. Geomagnetic polarity transitions. Reviews of Geophysics, 37: 201–226. Merrill, R.T., McElhinny, M.W., and McFadden, P.L., 1996. The Magnetic Field of the Earth: Paleomagnetism, the Core, and the Deep Mantle. San Diego, CA: Academic Press.

in the polar regions and at airline altitudes anywhere may solar radiation storms be a health issue, and major airline operators and aviation authorities are considering the risks (Getley, 2004). In terms of hazard to health, modeling and predicting the changing morphology of the geomagnetic field is important in determining where charged particles may enter into the lower atmosphere (see Main field modeling; Geomagnetic secular variation).

Cross-references

Conducting networks and geomagnetically induced currents

Cowling’s Theorem D00 Geocentric Axial Dipole Hypothesis Geomagnetic Polarity Timescales Geomagnetic Secular Variation Harmonics, Spherical Inner Core Anisotropy Nondipole Field Reversals, Theory Westward Drift

GEOMAGNETIC HAZARDS Introduction Geomagnetic variations take place over a wide range of timescales. The longer-term variations, typically those occurring over decades to millennia, are predominantly the result of dynamo action in the Earth’s core (see Geodynamo). However, geomagnetic variations on timescales of seconds to many years also occur due to dynamic processes in the ionosphere (see Ionosphere), magnetosphere (see Magnetosphere of the Earth), and heliosphere (see Magnetic field of Sun). These changes are ultimately tied to variations associated with the solar activity cycle. To a lesser extent geomagnetic variations are also caused by changes in the solar ultraviolet emission that controls the ionization of the Earth’s ionosphere (see Periodic external fields). However, the largest geomagnetic variations are due to sporadic solar activity in the form of solar coronal mass ejections (CMEs) and the number of CMEs varies with the phase of the solar cycle. In a CME, the rapid reconfiguration, through magnetic reconnection processes, of the fields in and above solar sunspot regions results in the release of magnetic flux and plasma into the solar wind. Traveling toward Earth, these plasma “bubbles” can interact with the Earth’s magnetic field, depositing particles and energy into the magnetosphere, and driving geomagnetic storms (see Storms and substorms, magnetic). Solar activity, and hence geomagnetic storm frequency, varies on a cycle of between about 9 and 14 years (see Dynamo, solar). A largescale dipolar solar magnetic field exists during times of sunspot minimum. This dipolar field progressively diminishes through solar (sunspot) maximum, becoming reestablished with the opposite polarity around the time of the next cycle minimum. Sunspots, flares, and associated CMEs are the means by which the large-scale field is changed and magnetic flux is expelled into the solar wind (see Magnetic field of Sun). The fact that the geomagnetic field does respond to solar conditions can be useful, for example, in investigating Earth structure using magnetotellurics (see Magnetotellurics), but it also creates a hazard. This geomagnetic hazard is a risk to technology, rather than to health (for recent reviews see, e.g., Lanzerotti, 2001; Pirjola et al., 2005). Astronauts are certainly at risk from bursts of ionizing solar energetic particle radiation that follow solar flares (Turner, 2001). Astronaut protection involves appropriate spacecraft shielding and positioning relative to the Sun. However, at the Earth’s surface, the atmosphere acts as a protective layer equivalent to several meters of concrete. Only

A time-varying magnetic field external to the Earth induces a secondary magnetic field, internal to the Earth, as a consequence of Faraday’s law. Associated with time variations in the induced field is an electric field, which is measurable at the surface of the Earth. The surface electric field causes electrical currents, known as geomagnetically induced currents (GIC), to flow in any conducting structure, for example, a power grid grounded in the Earth. This electric field, measured in V m
1, acts as a voltage source across networks, with the different grounding points at different electrical potentials. Examples of conducting networks are electrical power transmission grids, oil and gas pipelines, undersea communication cables, telephone and telegraph networks, and railways. GIC are often described as being quasi-direct current (DC), although the variation frequency of GIC is governed by the time variation of the electric field. For GIC to be a hazard to technology, the current has to be of a magnitude and frequency that makes the equipment susceptible to either immediate or cumulative damage. The size of the GIC in any network is partly governed by the electrical resistance of the network, relative to the resistance of the underlying Earth. The largest external magnetic field and magnetospheric current variations occur during geomagnetic storms and it is then that the largest GIC occur. Significant variation periods are typically from seconds to about an hour, so the induction process involves the upper mantle and lithosphere (see Magnetotellurics; Geomagnetic deep sounding). Since the largest magnetic field variations are observed at higher magnetic latitudes, GIC have been regularly measured in Canadian, Finnish, and Scandinavian power grids and pipelines since the 1970s. GIC of tens to hundreds of Amps have been recorded. However, GIC and effects have also been recorded in countries at midlatitudes during major storms. There may even be a risk to low latitude nations during a storm sudden commencement (see Storms and substorms, magnetic) because of the high, short-period, rate of change of the field that occurs everywhere on the dayside of the Earth. GIC have been known since the mid-1800s when it was noted that telegraph systems could run without power during geomagnetic storms, described at the time as operating by means of the “celestial battery” (Boteler et al., 1998). However, technological change and the growth of conducting networks have made the significance of GIC greater and more pervasive in modern society. We therefore describe the GIC hazard in detail in two of the best-studied networks, power grids and pipelines. The technical considerations for undersea cables, telephone and telegraph networks, and railways are similar. However, fewer problems are known, or have been reported in the open literature, about these systems. This suggests that the hazard is less, or that there are reliable methods for equipment protection. Modern electrical transmission systems consist of generating plant interconnected by electrical circuits that operate at fixed transmission voltages controlled at transformer substations. The grid voltages employed are largely dependent on the path length between these substations and 200–700 kV system voltages are common. There is a trend toward higher voltages and lower line resistances to reduce transmission losses over longer and longer path lengths. However, low line resistances produce a situation favorable to the flow of GIC. Power transformers have a magnetic circuit that is disrupted by the quasi-DC GIC: the field produced by the GIC offsets the operating point of the magnetic circuit and the transformer may go into half-cycle

GEOMAGNETIC HAZARDS

saturation. This produces a harmonic-rich AC waveform, localized heating, and leads to high reactive power demands, inefficient power transmission and possible misoperation of protective measures. Balancing the network in such situations requires significant additional reactive power capacity (Erinmez et al., 2002). The magnitude of GIC that will cause significant problems to transformers varies with transformer type. Modern industry practice is to specify GIC tolerance levels on new transformer purchases. On 13 March 1989 a severe geomagnetic storm caused the collapse of the Hydro-Quebec power grid in a matter of seconds as equipment protection relays tripped in a cascading sequence of events (Bolduc, 2002). Six million people were left without power for 9 hours, with significant economic loss. Since 1989 power companies in North America, the UK, Northern Europe, and elsewhere have invested time and effort in evaluating the GIC risk and in developing mitigation strategies. GIC risk can, to some extent, be reduced by capacitor blocking systems, maintenance schedule changes, additional on-demand generating capacity, and, ultimately, shedding of load. However, these options are expensive and sometimes impractical. The continued growth of high voltage power networks, for example, in North America and in mainland Europe, is leading to a higher risk. This is partly due to the increase in the interconnectedness at higher voltages; connections to grids in the auroral zone, and commercial considerations that see grids run closer to capacity than was the case historically. To understand the flow of GIC in power grids and therefore to advise on GIC risk, analysis of the quasi-DC properties of the grid is necessary. This must be coupled with a geophysical model of the Earth that provides the driving surface electric field, determined by combining time-varying ionospheric source fields and a conductivity model of the Earth. Such analyses have been performed for North America, the UK, and in Northern Europe. However, the complexity of power grids, the source ionospheric current systems, and the 3D ground conductivity makes an accurate analysis difficult. By being able to analyze, postevent, major storms, and their consequences we can build a picture of the weak spots in a given transmission system and even run hypothetical event scenarios. An example of a postevent grid analysis for Central Scotland and Northern England during the peak of the 30 October 2003 magnetic storm is shown in Figure G20 (Thomson et al., 2005). At this time the measured GIC was 42 A, a significant, though in this case, manageable level and comparable with the model results. Grid management is also aided by space weather forecasts of major geomagnetic storms. This allows for mitigation strategies to be implemented. Solar observations provide a 1–3 day warning of an Earthbound CME, depending on CME speed. Following this, detection of the solar wind shock that precedes the CME in the solar wind, by spacecraft at the Lagrangian L1 point, gives a definite 20–60 min warning of a geomagnetic storm (again depending on local solar wind speed). However, the magnitude and accurate time of arrival of a CME prior to shock detection is unknown, although there is much research and model development within the space weather community. Given the demands on managing power grids more accurate information would have high value. Major pipeline networks exist at all latitudes and many systems are on a continental scale. Pipeline networks are constructed from steel to contain high-pressure liquid or gas and are covered with special coatings to resist corrosion. Weathering and other damage to the pipeline coating can result in the steel being exposed to moist air or to the ground, causing localized corrosion problems. Cathodic protection rectifiers are used to maintain pipelines at a negative potential with respect to the ground. This minimizes corrosion without allowing any chemical decomposition of the pipe coating and the operating potential is determined from the electrochemical properties of the soil and Earth in the vicinity of the pipeline. The GIC hazard to pipelines is that GIC cause swings in the pipe-to-soil potential, increasing the rate of corrosion during major geomagnetic storms (Gummow, 2002).

317

Figure G20 Estimated GIC in a network model of the Central Scotland and Northern England high voltage power grid, at the peak of the 30 October 2003, storm at 21:20 UT. Circle shading denotes GIC flowing to/from earth (dark/light) at transformers in the grid and the arrows denote the instantaneous E and B fields at Eskdalemuir magnetic observatory. The E-field is calculated from a radially varying Earth conductivity model using measured Eskdalemuir geomagnetic variations at this time. GIC amplitudes are proportional to spot size, with 40 A shown for scale. White spots show the locations of permanent GIC measurement sites in the grid. GIC risk is not, therefore, a risk of catastrophic failure, rather the reduced service lifetime of the pipeline, or parts of it. Pipeline networks are modeled in a similar manner to power grids, for example, through distributed source transmission line models that provide the pipe-to-soil potential at any point along the pipe (Boteler, 1997). These models need to take into account complicated pipeline topologies that include bends as well as electrical insulators, or flanges, that electrically isolate different sections of the network. From a detailed knowledge of the pipeline response to GIC, pipeline engineers can understand the behavior of the cathodic protection system even during a geomagnetic storm, when pipeline surveying and maintenance may often be suspended.

Satellite operation, navigation, and radio communication The ionosphere plays a significant role in very low frequency (VLF) through to high-frequency (HF) radio communication, and in navigation systems such as Loran-C and Omega (Lanzerotti, 2001; Schunk, 2001). Ionospheric conductivity is partly affected by geomagnetic storms but more significantly by solar ultraviolet and x-ray control of the ionospheric D, E, and F layers (see Ionosphere). Solar flares cause signal-phase anomalies and amplitude variations to occur (fades and enhancements) and conditions can persist for minutes to hours. Solar flares and CMEs are also important as sources of solar energetic particles, affecting radio communications at high latitudes. Disturbed conditions can persist for days to weeks in the Polar Regions during periods of high particle flux into the polar caps. Ultrahigh-frequency (UHF) radio signals are central to the Global Positioning System (GPS) that utilizes satellites in Earth orbit for precise ground position determination. UHF waves pass largely unattenuated through the

318

GEOMAGNETIC HAZARDS

ionosphere but the system accuracy is sensitive to variations in the total electron content (TEC) in the path between ground and satellite. The TEC determines the signal propagation delay. TEC variations occur during geomagnetic storms and these particularly degrade the accuracy of single-frequency GPS equipment. Geomagnetic storms also produce ionospheric irregularities and scintillations that occur both on the dayside and nightside of the Earth. In practical terms, radio navigational and communication systems and operators adjust to the prevailing conditions, although accurate forewarning of solar and geomagnetic activity may be useful for planning purposes. Low Earth orbit satellites and space stations (up to around 1000 km altitude) experience increased air drag during geomagnetic storms. Enhanced ionospheric currents deposit heat in the atmosphere. This causes the atmosphere to expand outward and, at a given altitude, atmospheric density increases. This leads to heightened drag forces, slowing of satellite velocities, and lowering of orbit altitudes. The cumulative effect of geomagnetic storms is therefore to reduce the operational lifetime of satellites, particularly those in low initial orbits, where air density is higher. High-altitude atmospheric density models often parameterize geomagnetic heating effects by geomagnetic activity indices (e.g., Roble, 2001). By predicting geomagnetic indices, on day-to-day and solar cycle timescales, more efficient use can be made of satellite fuel supplies, with judicious orbit reboosts used to extend mission lifetimes. More sophisticated models of the upper atmosphere are being developed, involving near real-time calibration data from many orbiting satellites, as well as a better understanding and modeling of the physics of the atmosphere and its response to solar and geomagnetic forcing. Satellites also suffer increased surface and internal electrical charging from ionized particles during geomagnetic storms. This can result in system malfunctions as electronic components experience physical damage and logic errors. Satellite manufactures take the solar cycle varying radiation environment into account when designing components, using statistical models of radiation dosages over component lifetimes. Modeling of the geomagnetic field morphology and predicting changes in the field (see Main field modeling; Geomagnetic secular variation) help to map the radiation environment of the Earth and the response of satellites to that environment. In operation, satellites can be temporarily powered down or placed in an appropriate “safe mode” following warnings of solar and geomagnetic activity.

Geophysical exploration and geomagnetic variations Aeromagnetic surveys are affected by geomagnetic storm variations (see Crustal magnetic field; Magnetic anomalies for geology and resources; Aeromagnetic surveying). These cause data interpretation problems where external field amplitudes are similar to those of the crustal field in the survey area. Accurate geomagnetic storm warnings, including an assessment of the magnitude and duration of the storm, would allow for an economic use of survey equipment. For economic and other reasons, oil and gas exploration often involves the directional drilling of well paths many kilometers from a single wellhead in both the horizontal and vertical directions. Target reservoirs may only be a few tens to hundreds of meters across and accurate surveying by gyroscopic methods is expensive since it can involve the cessation of drilling for a number of hours. An alternative is to use magnetic referencing while drilling (Clark and Clarke, 2001). Near real-time magnetic data are used to correct the drilling direction and nearby magnetic observatories prove vital. There is no drilling “down-time” during a magnetic storm and storm forecasts are not normally seen as being important.

Summary and outlook The geomagnetic hazard to technology results from the strengthening of magnetospheric and ionospheric current systems by the solar wind

and by CMEs. These electrical current enhancements cause rapid and high amplitude magnetic variations during geomagnetic storms. Processes internal to the magnetosphere also drive variations, particularly after CME-driven activity. A common theme that emerges from a study of geomagnetic hazards is a need for accurate geomagnetic storm forecasting, in terms of onset time and duration, maximum amplitude, and variation periods. The close connection of geomagnetic hazard with solar activity and space weather is also clear. Some practical applications of geomagnetic variations require only monitoring data, but other applications (will) clearly benefit from a thorough physical understanding of the Sun-Earth magnetic interaction and, in particular, accurate prediction of geomagnetic variations. Alan W.P. Thomson

Bibliography Bolduc, L., 2002. GIC observations and studies in the Hydro-Quebec power system. Journal of Atmospheric and Solar-Terrestrial Physics, 64(16): 1793–1802. Boteler, D.H., 1997. Distributed source transmission line theory for electromagnetic induction studies. In Supplement of the Proceedings of the 12th International Zurich Symposium and Technical Exhibition on Electromagnetic Compatibility, February 18 to 20, 1997 at the Swiss Federal Institute of Technology in Zurich, Switzerland, pp. 401–408. Boteler, D.H., Pirjola, R.J., and Nevanlinna, H., 1998. The effects of geomagnetic disturbances on electrical systems at the Earth’s surface. Advances in Space Research, 22(1): 17–27. Clark, T.D.G., and Clarke, E., 2001. In Space Weather Workshop: Space weather services for the offshore drilling industry. Looking Towards a Future European Space Weather Programme. ESTEC, ESA WPP-194. Erinmez, I.A., Kappenman, J.G., and Radasky, W.A., 2002. Management of the geomagnetically induced current risks on the national grid company’s electric power transmission system. Journal of Atmospheric and Solar-Terrestrial Physics, 64(5–6): 743–756. Getley, I.L., 2004. Observation of solar particle event on board a commercial flight from Los Angeles to New York on October 29, 2003. AGU Space Weather, 2: S05002, doi:10.1029/2003SW000058. Gummow, R.A., 2002. GIC effects on pipeline corrosion and corrosion-control systems. Journal of Atmospheric and Solar-Terrestrial Physics, 64(16): 1755–1764. Lanzerotti, L.J., 2001. Space weather effects on technologies. In Song, P., Singer, H.J., and Siscoe, G.L. (eds.) Space Weather. Geophysical Monograph 125. Washington, DC: American Geophysical Union, pp. 11–22. Pirjola, R., Kauristie, K., Lappalainen, H., and Viljanen, A., 2005. Space weather risk. AGU Space Weather, 3: S02A02, doi:10.1029/ 2004SW000112. Roble, R.G., 2001. On forecasting thermospheric and ionospheric disturbances in space weather events. In Song, P., Singer, H.J., and Siscoe, G.L. (eds.) Space Weather. Geophysical Monograph 125. Washington, DC: American Geophysical Union, pp. 369–376. Schunk, R.W., 2001. Ionospheric climatology and weather disturbances: a tutorial. In Song, P., Singer, H.J., and Siscoe, G.L. (eds.) Space Weather. Geophysical Monograph 125. Washington, DC: American Geophysical Union, pp. 359–368. Thomson, A.W.P., McKay, A.J., Clarke, E., and Reay, S.J., 2005. Surface electric fields and geomagnetically induced currents in the Scottish power grid during the October 30, 2003, geomagnetic storm. AGU Space Weather, 3: S11002, doi:10.1029/2005SW 000156. Turner, R., 2001. What we must know about solar particle events to reduce the risk to astronauts. In Song, P., Singer, H.J., and Siscoe, G.L. (eds.) Space Weather. Geophysical Monograph 125. Washington, DC: American Geophysical Union, pp. 39–44.

GEOMAGNETIC JERKS

Cross-references Aeromagnetic Surveying Crustal Magnetic Field Dynamo, Solar Geodynamo Geomagnetic Deep Sounding Geomagnetic Secular Variation Ionosphere Magnetic Anomalies for Geology and Resources Magnetic Field of Sun Magnetosphere of the Earth Magnetotellurics Main Field Modeling Periodic External Fields Storms and Substorms, Magnetic

GEOMAGNETIC JERKS Observations Geomagnetic jerks are abrupt changes in the second-time derivative, or secular acceleration, of the magnetic field that arises from sources inside the Earth. They delineate intervals of oppositely signed, nearconstant secular acceleration. The first observed geomagnetic jerk was that around 1969 (Courtillot et al., 1978), and since the late 19th century when direct and continuous measurements of the Earth’s magnetic field have been made at a number of observatories around the world, geomagnetic jerks have also been observed to occur around 1925, 1978, 1991, and 1999 and possibly also around 1901 and 1913 (e.g., Malin and Hodder, 1982; Courtillot and Le Mouël, 1984; Macmillan, 1996; Alexandrescu et al., 1996; Mandea et al., 2000). These jerks are most readily observed in the first-time derivative of the east component at European observatories (Figure G21). Understanding their origin is important, not only because they result from interesting dynamical processes in the core and may help determine the conductivity of the mantle, but also for improving time-dependent models of

319

the geomagnetic field (q.v.) and for the strictly practical purpose of forecasting its future behavior, for example, as used in navigation.

Analysis of jerks Many studies have been undertaken to assess jerk characteristics, such as origin, times of occurrence, and spatial patterns. At observatories the measured magnetic field is the vector sum of fields arising from three primary sources, namely the main field generated in the Earth’s core, the crustal field from local rocks, and a combined disturbance field from electrical currents flowing in the upper atmosphere and magnetosphere, which also induce electrical currents in the sea and the ground. As the long-period variations of the disturbance field associated with the 11-year solar cycle are in the same frequency band as variations arising from sources inside the Earth, the separation of the sources is an important part of any analysis of geomagnetic jerks. Spherical harmonic analysis applied to monthly or yearly mean geomagnetic observatory data has shown that jerks are internal in origin and, where sufficient data exist, are global phenomena (e.g., Malin and Hodder, 1982, McLeod, 1985). Various analysis techniques have been applied to jerks to investigate specific aspects of their temporal and spatial characteristics. Methods such as optimal piecewise regression analysis and wavelet analysis have established the exact dates of occurrence at different observatories without any a priori assumptions. Using wavelets, the 1969 and 1978 jerks have been shown to have different arrival times at the Earth’s surface, with the northern hemisphere leading the southern hemisphere by about 2 years (Alexandrescu et al., 1996).

Implications for studies of the Earth’s deep interior The shortest variations in the Earth’s internal field that can be seen at the Earth’s surface are determined by the rate of change of the magnetic field at the core-mantle boundary (CMB) and by the electrical conductivity of the mantle through which the magnetic signal from the core must pass. From analyses of surface data we know that the duration of individual jerks, as indicated by the width of impulses in the third-time derivative, is no more than about 2 years. However, it

Figure G21 Geomagnetic jerks as seen in the secular variation of the east component of the Earth’s magnetic field observed at European observatories. Time of jerks are shown by arrows.

320

GEOMAGNETIC POLARITY REVERSALS

is difficult to make any estimate of the rate of change of the magnetic field at the CMB, or the electrical conductivity of the mantle, from geomagnetic field observations at the Earth’s surface. As in most geophysical problems, assumptions have to be made or constraints have to be applied using information from other data sources. In the case of geomagnetic jerks one useful independent dataset is length of day (LOD) observations. Correlation of the occurrences of jerks with marked changes in the deficit LOD (the former appearing to lead the latter by a few years) indicates some form of coupling between the core and the mantle. It has recently been shown that inflexion in the time derivative of splines fitted to filtered LOD data from 1960 onwards coincide remarkably well with geomagnetic jerks, including the late arrivals reported for the southern hemisphere (Holme and de Viron, 2005). Jerks are most likely to result from dynamic processes inside the Earth that produce changes in the magnetic field near the top of the fluid core. These include the changes caused by advection and shear within the flow. Diffusion is not likely to be an important factor because of the short timescales involved. Core flows can be estimated directly from observatory secular variation data or indirectly from spherical harmonic models of secular variation. Again, assumptions have to be made or constraints applied so that a unique solution is found. One common assumption applied when determining core flows from magnetic data over time intervals of a few decades is the frozenflux hypothesis with tangentially geostrophic time-varying flows. With this assumption the fine detail of the secular variation, including jerks, can be well fitted (Jackson, 1997) but the models offer little physical explanation of the flows in terms of core processes. Another assumption that also gives good fits to data is steady flow in a core reference frame that is allowed to rotate about the Earth’s rotation axis with respect to the mantle (Holme and Whaler, 2001). This, not surprisingly, provides results which are consistent with LOD data. A time-dependent flow model that comprises only torsional oscillations, recovers jerks very well while not producing spurious jerks where none are observed (Bloxham et al., 2002). This suggests that the origin of jerks is related to the action of torsional oscillations on the local magnetic field at the CMB. If this is correct, then the torsional oscillations provide a physical representation for the origin of jerks, for they are not only a part of the mathematical representation of core flows but, more importantly, are an actual component of flow predicted to occur in the geodynamo. They also provide the angular momentum changes required to explain the LOD changes. However, the specific details of jerk generation in the core are still to be resolved, in particular the mechanism driving torsional oscillations. Susan Macmillan

Bibliography Alexandrescu, M., Gibert, D., Hulot, G., Le Mouël, J.-L., and Saracco, G., 1996. Worldwide wavelet analysis of geomagnetic jerks. Journal of Geophysical Research, 101(B10): 21975–21994. Bloxham, J., Zatman, S., and Dumberry, M., 2002. The origin of geomagnetic jerks. Nature, 420: 65–68. Courtillot, V., Ducruix, J., and Le Mouël, J.-L., 1978. Sure une accéleration récente de la variation seculaire du champ magnétique terrestre. Comptes rendus des séances de l’ Académie des Sciences, 287, D: 1095–1098. Courtillot, V., and Le Mouël, J.-L., 1984. Geomagnetic secular variation impulses. Nature, 311: 709–716. Holme, R., and de Viron, O., 2005. Geomagnetic jerks and a high-resolution length-of-day profile for core studies. Geophysical Journal International, 160: 413–413. doi: 10.1111/j.1365246X.2005.02547.x. Holme, R., and Whaler, K.A., 2001. Steady core flow in an azimuthally drifting reference frame. Geophysical Journal International, 145: 560–569.

Jackson, A., 1997. Time-dependency of tangentially geostrophic core surface motions. Physics of the Earth and Planetary Interiors, 103: 293–312. Macmillan, S., 1996. A geomagnetic jerk for the early 1990’s. Earth and Planetary Science Letters, 137: 189–192. Malin, S.R.C., and Hodder, B.M., 1982. Was the 1970 geomagnetic jerk of internal or external origin? Nature, 296: 726–728. Mandea, M., Bellanger, E., and Le Mouël, J.-L., 2000. A geomagnetic jerk for the end of the 20th century? Earth and Planetary Science Letters, 183: 369–373. McLeod, M.G., 1985. On the geomagnetic jerk of 1969. Journal of Geophysical Research, 90(B6): 4597–4610.

Cross-references Decade Variations in LOD Main Field Modelling Spherical Harmonics Time-dependent Models of the Main Magnetic Field Torsional Oscillations

GEOMAGNETIC POLARITY REVERSALS Early records of reversals Natural magnetization of rocks opposed to that of the present day field was observed by David and Brunhes about one century ago. Brunhes concluded that “at a certain moment of the Miocene epoch in the neighborhood of Saint-Flour the north pole was directed upward: it was the south pole which was the closest to central France” (see Laj et al., 2002). Then, Matuyama observed in 1929 other reverse polarities in the magnetic directions of lava flows from Japan and Manchuria. However, early studies did not provide definite evidence that the geomagnetic field had reversed its polarity, in particular because self-reversal in some rocks can produce thermoremanent magnetization antiparallel to the geomagnetic field at cooling time. Existence of polarity reversals was definitely established when K-Ar dating demonstrated that all lavas from the same age have a similar polarity, whatever their geographical position at the Earth surface. First Geomagnetic Polarity Timescales (GTPS) were constructed in the 1960s (Cox, 1969). Then, discovery of marine magnetic anomalies (Figure G22) confirmed seafloor spreading (Vine and Matthews, 1963), and the GTPS was extended to older times (Vine, 1966; Heirtzler et al., 1968; Lowrie and Kent, 1981). Since then, succession of polarity intervals has been extensively studied and used to construct magnetostratigraphic timescales linking biostratigraphies, isotope stratigraphies, and absolute ages (see Opdyke and Channell, 1996, for a review).

Reversal frequency The geomagnetic polarity timescale constructed from seafloor magnetic anomalies (Figure G23) revealed that polarity reversals were common in Earth’s history. Their occurrence, however, is not constant through time. Reversal frequency has increased during the past 85 Ma, since the long Cretaceous interval of normal polarity. Before this superchron, reversal frequency decreased since the Late Jurassic. Changes in reversal frequency occur with time constant of tens of millions of years or more, of the same order of magnitude of those involved in mantle overturning. It was therefore suggested that reversal frequency is linked to the structure of the mantle, an idea that was reinforced by 3D computations of the geodynamo (Glatzmaier et al., 1999). The question of the existence of a 15 Ma periodicity in reversal frequency (Mazaud et al., 1983) has been largely debated (see Jacobs, 1994).

GEOMAGNETIC POLARITY REVERSALS

321

Figure G22 Magnetic structure of seafloor near the Reyjkanes Ridges (after Phinney, 1968).

different views of the phenomenon. Lava flows acquire magnetic memory during their cooling, and provide instantaneous pictures of the Earth magnetic field. Polarity transitions are recorded only if eruptions occurred at the time of the transition, and if no alteration or secondary heating affected the recording rocks. Lava may provide absolute paleointensities, by using Thellier, Shaw, or Thellier and Coe determination methods. In contrast, sediments provide continuous records of geomagnetic field changes, but acquisition process may smooth out fastest field changes. Only relative changes of past geomagnetic intensity are obtained, and only with sediment exhibiting limited downcore variations in magnetic mineralogy and magnetic particle grain size. Despite difficulties to obtain paleomagnetic records of polarity transitions, several characteristics have emerged in the last few decades. The geomagnetic intensity decreases a few thousands of years before directional changes. Then, the magnetic vector exhibit directional changes while intensity remains low. When the direction reaches opposite polarity, the geomagnetic intensity rises to normal values. The total process takes place in few thousand years, and may vary for different reversals. It was also suggested that the geomagnetic field intensity progressively decreases during periods of stable polarity and recovers high values immediately after a transition (Valet and Meynadier, 1993).

The standing field hypothesis and the flooding models

Figure G23 Geomagnetic polarity timescale with magnetic anomaly numbers for the past 160 Ma. (Figure from Merrill et al., 1996).

Geometry and dynamics of the transitional field Polarity transitions occur so quickly on a geological timescale that it is difficult to find rocks that have preserved in detail variations of the transitional field. Also, sediment and lava may sometimes provide

The geometry and the dynamics of the geomagnetic field during polarity transitions have been subject to large debate. Hillhouse and Cox (1976) suggested that if the usual nondipole drift (secular variation) was unchanged during a polarity reversal, which is theoretically envisaged (Le Mouël, 1984), then one should observe large longitudinal swings in reversal records. Their absence in the first records obtained from sediments led these authors to suggest the standing field hypothesis, in which an invariant component dominates the transitional field when the axial dipole component has vanished. The standing field hypothesis, however, was not confirmed. A classical tool used for investigating the morphology of the transitional field is the virtual geomagnetic pole (VGP), defined as the pole of the dipolar field that gives the observed direction of magnetization at studied site. If VGPs obtained at several sites for a given instant in time coincide, then the field has a dipolar structure. If different VGPs are obtained at different sites, then the field was not dipolar (the word “virtual” indicates that VGPs can be calculated for any field, not necessarily dipolar).

322

GEOMAGNETIC POLARITY REVERSALS

Sedimentary records also indicated different VGP paths for the same transition studied at different sites at the Earth’s surface, with transitional VGPs moving progressively along longitudinal great circles. This led Hoffman and Fuller (1978) to develop flooding models, in which reversals originate in a localized region of the core and then progressively propagate into other regions.

Stop and go behavior Higher resolution records were obtained from lava and sedimentary series. They suggested some repetition over successive reversals (Valet and Laj, 1984), and also stop-and-go behavior, with alternating phases of rapid change and stationarity during transitions. This is seen in the Miocene reversal record obtained at Steens Mountain, Oregon (Figure G24) (Prévot et al., 1985), and also in Miocene sedimentary records from Northwest Greece (Laj et al., 1988).

The two preferred bands An intriguing point was issued by Laj and colleagues (1991), examining the longitudinal distribution of the VGP paths of sedimentary records of reversals over the past 10 Ma. They found that VGP paths tend to follow paths located into two preferred bands, over Americas and eastern Asia. These two preferred bands coincide with regions of fast seismic wave propagation in the lower mantle, in the downward continuation of subducted slabs in the mantle (Figure G25). These two bands also coincide with the main patches of radial magnetic flux at the core-mantle boundary in the present-day field and the historical field (Bloxham and Gubbins, 1985), and in paleofield reconstructions (Constable, 1992; Gubbins and Kelly, 1993). Thus, Laj and colleagues (1991) argued that the persistence of these two preferred bands over time of the order of magnitude of mantle convection was experimental evidence that the geodynamo was constrained by temperature patterns at the core-mantle boundary, as suggested by present day

Figure G24 The Steens Mountain directional record. Stereographic projection of field directions after rotation about the east-west horizontal axis, so dipole field directions coincide with the pole of the projection sphere (from Pre´vot et al., 1985).

GEOMAGNETIC POLARITY REVERSALS

323

Figure G25 The two preferred bands for VGP paths of sedimentary records of reversals (from Laj et al., 1991). Zones of fast seismic velocity in the lower mantle are indicated in dark (blue in online version of encyclopedy).

Figure G26 Clusters of VGPs (from Hoffman, 1992).

and historical magnetic observations. The existence of the two preferred bands was and is still debated, in particular because they were not seen in the volcanic compilation of Prévot and Camps (1993). However, another analysis of volcanic databases suggests that the two bands also exist in compilations of volcanic reversals records (Love, 1998). Influence of variable conditions at the core-mantle boundary is suggested in 3D computations of the geodynamo (Glatzmaier et al., 1999).

Toward a consistent picture of polarity reversals? Progressively, a consistent picture emerges from high-resolution sediment and lava records. In 1992, Hoffman observed long-lived transitional states of the geomagnetic field, with VGPs of volcanic reversal records clustering in the two preferred bands (Figure G26). More recently, several high-resolution sedimentary records were obtained that exhibited complex field behavior, with VGP loops and clusters reminiscent of volcanic records. Some clusters, but not all, lie in the two preferred bands (Figure G27). Transitional precursors are sometimes observed.

Conclusion The situation is still complex, because different reversals may document different behavior and the question of the two preferred longitudinal bands is still open. The relation between polarity reversals and excursions is under investigation. Geomagnetic excursions are seen as aborted reversals in which the field may reverse in the liquid outer core, which has timescales of 500 years or less, but not in the solid inner core, where field must change by diffusion with a timescale of 3 ka. This disparity of dynamical timescales between the inner and outer core is consistent with the presence of several excursions between full reversals (Gubbins, 1999). Overall, the mechanisms that trigger reversals and excursions have to be better understood. Whether or not the present-day axial dipole field decrease corresponds to a reversal or excursion onset, or to a field fluctuation during stable polarity, is not yet known. Combination of supercomputer 3D simulations (see for instance Glatzmaier and Olson, 2005, for a review) and new high-resolution records from both sediments and lava flows should lead to a better understanding of the mechanisms involved in field generation and

324

GEOMAGNETIC POLARITY REVERSALS, OBSERVATIONS

Figure G27 Examples of VGP paths for the Bunhes-Matuyama (a) and upper Jaramillo (b) transitions obtained from high deposition rate sediments in the North Atlantic (after Channell and Lehman, 1997). polarity transitions, and ultimately of the global dynamics of the Earth’s interior. LSCE contribution n 2407. Alain Mazaud

Bibliography Bloxham, J., and Gubbins, D., 1985. The secular variation of the Earth’s magnetic field. Nature, 317: 777–781. Channell, J.E.T., and Lehman, B., 1997. The last two geomagnetic polarity reversals recorded in high-deposition-rate sediment drifts. Nature, 389: 712–715. Constable, C., 1992. Link between geomagnetic reversal paths and secular variation of the field over the past 5 Ma. Nature, 358: 230–233. Cox, A., 1969. Geomagnetic reversals. Science, 263: 237–245. Glatzmaier, G.A., and Olson, P., 2005. Probing the geodynamo. Scientific American, 292: 51–57. Glatzmaier, G.A., Coe, R., Hongre, L., and Roberts, P.H., 1999. The role of the Earth’s mantle in controlling the frequency of geomagnetic reversals. Nature, 401: 885–890. Gubbins, D., 1999. The distinction between geomagnetic excursions and reversals. Geophysical Journal International, 137: F1–F3. Gubbins, D., and Kelly, P., 1993. Persistent patterns in the geomagnetic field over the past 2.5 Ma. Nature, 365: 829–832. Heirtzler, J.R., Dickson, G.O., Herron, E.M., Pitman W.C. II, and Le Pichon, X., 1968. Marine magnetic anomalies, geomagnetic field reversals, and motions of the ocean floor and continents. Journal of Geophysical Research, 73: 2119–2136. Hillhouse, J.W., and Cox, A., 1976. Brunhes-Matuyama polarity transition. Earth and Planetary Science Letters, 29: 51–64. Hoffman, K.A., and Fuller, M., 1978. Polarity transition records and the geomagnetic dynamo. Nature, 273: 715–718. Hoffman, K.A., 1992. Dipolar reversal states of the geomagnetic field and core mantle dynamics. Nature, 359: 789–794. Jacobs, J.A., 1994. Reversals of the Earth’s Magnetic Field, 2nd edn. Cambridge: Cambridge University Press, 339 pp. Laj, C., Guitton, S., Kissel, C., and Mazaud, A., 1988. Complex behavior of the geomagnetic field during three successive polarity reversals, 11–12 Ma BP. Journal of Geophysical Research, 93: 11655–11666. Laj, C., Mazaud, A., Weeks, R., Fuller, M., and Herrero-Bervera, H., 1991. Geomagnetic reversals paths. Nature, 351: 447. Laj, C., Kissel, C., and Guillou, H., 2002. Brunhes research revisited: magnetization of volcanic flows and backed clays. EOS Transactions: American Geophysical Union, 83(35): 381, 386–387. le Mouël, J.L., 1984. Outer core geostrophic flow and secular variation of Earth’s magnetic field. Nature, 311: 734–735. Love, J.J., 1998. Paleomagnetic volcanic date and geometric regularity of reversals and excursions. Journal of Geophysical Research, 103: 12435–12452.

Lowrie, W., and Kent, D.V., 1981. One hundred million years of geomagnetic polarity history. Geology, 9: 392–397. Mazaud, A., Laj, C., De Seze, L., and Verosub, K.L., 1983. 15 Ma periodicity in the frequency of geomagnetic reversals since 100 Ma Reply to McFadden. Nature, 304: 328–330. Merrill, R.T., McElhinny, M.W., and McFadden, P.L., 1996. The Magnetic Field of the Earth. International Geophysics Series 63. San Diego, CA: Academic Press, 527 pp. Opdyke, N., and Channell, J.E.T, 1996. Magnetic Stratigraphy. International Geophysics Series 64. San Deigo, CA: Academic Press, 346 pp. Phinney, R.A., 1968. The History of the Earth’s Crust. Princeton, NJ: Princeton University Press, 244 pp. Prévot, M., and Camps, P., 1993. Absence of preferred longitudinal sectors from pole from volcanic records of geomagnetic reversals. Nature, 366: 53–57. Prévot, M., Mankinen, E.A., Grommé, C.S., and Coe, R., 1985. How the geomagnetic field vector reverses polarity. Nature, 316: 230–234. Valet, J.P., and Laj, C., 1984. Invariant and changing transitional field in a sequence of geomagnetic reversals. Nature, 311: 552–555. Valet, J.P., and Meynadier, L., 1993. Geomagnetic intensity and reversals during the past four million years. Nature, 366: 234–238. Vine, F.J., 1966. Spreading of the ocean floor: new evidence. Science, 154: 1405. Vine, F.J., and Matthews, D.H., 1963. Magnetic anomalies over oceanic ridges. Nature, 199: 947–949.

Cross-references Paleointensity, Absolute, Determination Polarity Transition, Paleomagnetic Record

GEOMAGNETIC POLARITY REVERSALS, OBSERVATIONS Earth’s magnetic field exhibits the remarkable property of undergoing a 180 change in directions at geologically frequent, but irregular, intervals. As a result, a compass needle that had been pointing to one geographic pole (such as North), would point to the opposite geographic pole. Why the field reverses polarity remains a mystery and solving this mystery is important for understanding the processes in Earth’s fluid outer core responsible for generating the geomagnetic field. The best record of how frequently the geomagnetic field reverses polarity comes from marine magnetic anomalies (Cande and Kent, 1995). Neither the marine anomaly record nor long stratigraphic sequences of polarity history, often record field directions that fall in between the two stable polarity directions. Given the seafloor spreading and sedimentation rates, this indicates that polarity reversals are rapid events,

GEOMAGNETIC POLARITY REVERSALS, OBSERVATIONS

at least on geological timescales. For example the marine magnetic anomaly record limits the time for a reversal to occur to less than 20 ka. The first step in attempting to solve the mystery of why Earth’s magnetic field reverses is to determine what happens as the field reverses. Paleomagnetic records of polarity transitions are the only source of information about how the geomagnetic field behaves during a reversal. Because polarity reversals are very short events, polarity transition records are difficult to obtain. Not only is it necessary to find a paleomagnetic recorder that provides great enough time resolution to catch the field in the act of reversing, but also it is necessary for that recorder to provide a high fidelity recording of the field, even when the field was weak. Two very different types of paleomagnetic recorders have repeatedly yielded polarity transition records. Volcanic sequences created by rapid extrusion periods that coincide with a polarity reversal, record the changing field as successive lava flows cool and become permanently magnetized in the transitional field. Sequences of sedimentary rocks or unconsolidated sediments have also proven useful in transition studies. If sedimentation rates are rapid and continuous enough, a record of a polarity transition may result as layers of sediment become magnetized as they accumulate during the reversal. These two types of paleomagnetic recorders document transitional fields in fundamentally different ways. Most importantly, the way each type of recorder becomes magnetized is very different, and these differences must be considered carefully when interpreting each type of record. A second important difference is the two kinds of recorders provide different types of temporal information about reversing fields. The processes by which lavas become magnetized are well founded in the theory of thermoremanent magnetization (TRM). This theory explains how grains of magnetic minerals become permanently magnetized as they cool through their magnetic blocking temperatures in the presence of an external field. Because the cooling times for typical lava flows are rapid, it may be assumed that the lava flows provide a spot reading in time of the magnetic field direction and intensity at the time of cooling. TRM acquisition theory also provides a basis for obtaining measures of the strength or intensity of the field, known as absolute paleointensities. On the other hand, the processes by which sediments become magnetized are less well understood. Although it is clear that many types of sediments can record the geomagnetic field accurately, as evidenced by recording the full polarity directions predicted for their site location, it is not clear how these magnetizations are acquired. A number of studies indicate that a postdepositional remanent magnetization process that locks-in the magnetization at some depth beneath the seafloor magnetizes marine sediments. Of particular interest in polarity transition studies is not so much the depth offset of remanence acquisition, but rather over what depth range the acquisition process occurs simultaneously. If the remanence becomes fixed over a relatively thick depth interval, the sediment in that layer will average out the changes in the field that occur. The resulting magnetization therefore would be an integration of the field during that time, and would provide an average of those changes in the field. As the thickness of sediment over which the remanence becomes blocked in becomes thinner, the magnetization approaches more of a spot reading of the changing field. Unfortunately, the remanence lock-in thicknesses in sediments are not known, and this presents an important difficulty in interpreting sediment records of polarity transitions. Unlike the theory of TRM, theories of how sediments become magnetized do not provide a basis for determining absolute field intensities. But, if conditions are favorable, it is possible to obtain records of relative changes in field strength (Tauxe, 1993). In other words, sediments are capable of providing information on the increasing or decreasing field strength, but as of yet, it is not possible to calibrate those relative changes to absolute magnitudes of field strength. In addition to the different ways these two types of recorders become magnetized, there is also a fundamental difference in the temporal resolution provided by each type of recorder. On the timescales

325

over which polarity reversals occur, it is not possible to determine the rates at which lavas are extruded from a volcano, or how much time has elapsed between successive lava flows. This means, that while lavas may provide high fidelity recordings of the field, the timing of the sequence of magnetizations of the flows is unknown. Recent advances in radiometric dating techniques using 40Ar/39Ar and K/Ar methods (Singer and Pringle, 1996) are approaching the precision needed to provide bounding limits on the durations of the youngest reversals but cannot yet provide information of timing with reversals. Sediment records on the other hand, generally provide a much more continuous recording of the field. Correlation of @ 18O records with the Marine Isotopic Stages can provide high-resolution age control through a reversal. Such records provide important information regarding the age, timing, and duration of polarity transitions. Paleomagnetic studies of polarity transitions result in a sequence of magnetizations that were acquired as a reversal occurred. These results are generally presented as magnetization declination, inclination, and intensity or virtual geomagnetic pole (VGP) latitude as a function of stratigraphic position (either as positions in a sequence of sediments or lava flows). In order to compare records from different geographic locations it is necessary to take into account the fact that even during intervals of full polarity, different field directions are expected at different sites. For full polarity intervals, paleomagnetists do this by using the Geocentric Axial Dipole hypothesis to calculate the equivalent geomagnetic pole for the given magnetization directions (where the site results would place Earth’s magnetic north pole). If the magnetization is thought to represent a spot reading (nearly instantaneous) of the field in time, the pole is called a VGP. Polarity transition data are often presented as VGP positions as a function of stratigraphic position (Figure G28). This method of viewing the data is a convenient way to show the polarity reversal by illustrating how the apparent north magnetic pole (VGP), calculated from the observed directions at a site, moves from one high latitude position to the other. An additional way of presenting a transition record is to plot the path of the VGP on a world map as it moved from one geographic pole to the other (Figure G29). This method makes it possible to compare records from distant sites and to test the hypothesis that the transitional field was dipolar.

Figure G28 A record of the Lower Jaramillo polarity transition (Gee et al., 1991) plotted as virtual geomagnetic pole position versus stratigraphic position. The full polarity intervals exhibit VGPs, which cluster close to the geographic poles. The dotted intervals represent a measure of the scatter of the VGPs about the full polarity average direction. The progression of VGP positions from one geographic pole to the other defines the polarity transition (shown by dashed interval). In this case, the transitional interval is defined as the zone in which the VGPs fall a statistically significantly distance from the geographic poles.

326

GEOMAGNETIC POLARITY REVERSALS, OBSERVATIONS

Figure G29 The Matuyama-Brunhes polarity transition obtained from North Atlantic deep-sea sediments (Channell and Lehman, 1997) shown as the path the VGP positions track along as the field gradually changes from reverse polarity (VGPs near the south geographic pole) to normal polarity (VGPs near the north geographic pole).

If the transitional fields were not dipolar, then it is not strictly appropriate to calculate VGPs from the transitional magnetizations, because this calculation assumes a dipolar field. But then some other method is needed for comparing records from distant sites. One method that has been proposed is to plot the changing directions on an equal-area stereographic projection; a method commonly used by paleomagnetists. However, before plotting the directions, the vectors are rotated so that the plot is constructed looking down the full polarity direction (Hoffman, 1984). This method provides a way of comparing the directional behavior observed from different locations without invoking the assumption of dipolar field geometries. This method, however does not illustrate absolute or relative paleointensities. Although the details of transitional field behavior remain uncertain, the number of paleomagnetic records of polarity transitions now available makes it possible to address the major features of reversals with greater certainty. It is generally agreed that during a reversal, the strength of the geomagnetic field drops to low levels, close to that of the strength of the present-day nondipole field (Merrill and McFadden, 1999). This represents about an 80% drop in intensity. Both volcanic and sedimentary transition records record intensity lows. The timing of the directional change relative to the intensity change is not tightly constrained; however, the majority of sediment records and many volcanic records show that the directional change occurs while the field is weak. In other words, the directional change begins after the field has weakened and finishes before the field strength increases to full polarity values. Simple geometrical models of reversals generally suggest that the directional change occurs at slightly different times during the low intensity depending on the site location. For example, the directional change may occur earlier at low latitudes than at high latitudes. Unfortunately, the records available to date do not have enough time control to rigorously test for this result. The time it takes for a reversal to occur provides a major constraint on the geodynamo process. The reversal durations obtained from the available sediment records indicate that the time it takes for the directional change to occur is 7  1 ka. The intensity change takes longer, perhaps up to 11 ka. These duration estimates are less than estimates of the free-decay time of the geodynamo (20 ka), indicating that reversals do not result from a passive decay of the field (Clement, 2004). The reversal durations are also dependent on site latitude, with the directional change occurring faster at low latitude than at high latitude (Clement, 2004). Again, this observation is expected based on simple geometrical models of reversals and provides constraints on three-dimensional numerical simulations of the dynamo.

While the features described above are generally agreed upon, there remain several controversial interpretations of transitional field behavior, most of which involve interpreting the transitional field geometries (based on the directional data). Some of the earliest comparisons of records of the most recent reversal showed that VGP paths from widely separated sites do not coincide (Hillhouse and Cox, 1976). This means that the field geometry was not dipolar, but was more complex. In other words, the reversal did not occur by a simple rotation of a dipole from one geographic pole to the other. This interpretation has held for the majority of reversals for which multiple records are available. Based on this result, it has traditionally been held that the simplest hypothesis is that, because the field is weak during a reversal, the field geometry would likely be very complex: much like that of the presentday, time-varying, nondipole field. This interpretation is supported by dynamo theory and what is known about the properties of the core. This hypothesis predicts that there should be no systematic variation in intermediate polarity field directions observed at distant sites. Perhaps the most controversial issue regarding polarity transitions is that several polarity reversals do, in fact, exhibit systematic variations in directions. For the most recent and several older reversals, it has been shown that the distribution of transitional VGPs from multiple sites fall into two, nearly antipodal, preferred longitudinal bands, one passing through the Americas and the other through eastern Asia (Clement, 1991; Laj et al., 1991). The observation that multiple records of the same reversal exhibit VGPs in both longitudinal bands means that the transitional fields were not dipolar, but instead, some other, relatively simple transitional field geometry gave rise to the VGP distribution. Because there is no known intrinsic property of Earth’s outer core or the geodynamo that should give rise to this distribution of VGPs or the recurrence of the pattern in multiple reversals, it has been suggested that lateral variations in the lowermost mantle affect the dynamo and influence the geometry of the transitional fields. However, this interpretation remains controversial because the grouping of longitudinal bands of transitional VGPs comes primarily from sediment records, and because the geographic distribution of available transition records is not wide enough to rigorously demonstrate the grouping. Some volcanic transition records have been obtained that exhibit clusters of VGPs that fall within one or the other of the preferred longitudinal bands, suggesting that this distribution may not be an artifact of the remanence acquisition process in sediments (Love and Mazaud, 1997). Because clusters of VGPs from multiple lava flows occur, it has been suggested that the transitional field may get temporarily locked into a geometry that produces these VGPs (Hoffman, 1992). If so, this could explain the longitudinal grouping of VGPs from the sediment records by assuming that the sediment records have averaged the dominant field geometry over the reversal. This process would produce a great circle VGP path over the cluster of VGPs from the lava records. An additional controversy centers over the observation of recurring VGP positions that occur within individual transition records. Several records, some volcanic and some sedimentary, exhibit VGP paths remarkable in that the VGPs return to a position that had occurred previously during the reversal. This observation suggests that the reversal process possesses a memory, at least over the timescales of a single reversal. A few lava records have also been interpreted as exhibiting VGPs that return to similar positions during different reversals, suggesting a memory in the dynamo process that exists over much longer timescales. In both cases, lateral variations in the lowermost mantle are the likeliest candidate for providing such a memory (Hoffman, 1991). This interpretation has been questioned by suggesting that the recurrent VGP positions may be an artifact of the remanence acquisition processes. If such an artifact is present or a magnetic overprint was acquired at a later age, it is possible that the observed sequence of transitional directions does not correspond to the actual temporal sequence that occurred during the reversal. So far, however, only one example

GEOMAGNETIC POLARITY REVERSALS, OBSERVATIONS

327

Figure G30 A record of a polarity reversal recorded in lava flows at Steens Mountain, Oregon (Mankinen et al., 1985; Prevot et al., 1985). The changes in magnetization through the reversal are shown here plotted as declination, inclination, and paleointensity in stratigraphic sequence of similar magnetization vectors. The data are plotted versus vector group number because it is assumed that successive flows exhibiting the same magnetizations were extruded rapidly and represent multiple records of the same field and do not necessarily represent times when the field was not changing.

has been found for a remagnetization of intermediate directions in a sediment record of an excursion (Coe and Liddicoat, 1994). Yet another controversy regards how fast the magnetic field can change during a reversal. The reversal recorded by lava flows at Steens Mountain, Oregon provides evidence that the transitional field may change as fast as degrees per day (Coe and Prevot, 1989; Coe et al., 1995). This rate is extremely fast and in fact is thought to be too fast (Figure G30). This is because the electrical conductivity of Earth’s mantle filters field changes produced by the dynamo and should limit just how fast the dynamo-produced field can change at Earth’s surface. The evidence for the rapid changes comes from magnetizations recorded within a single lava flow. The magnetizations differ with position in the flow. Using cooling rate estimates for the flow, the rate at which the field was changing can be estimated. Despite an intensive effort, no evidence has been found to suggest that the different magnetizations result from differences in the magnetic minerals that record the field. These controversies will likely be resolved as additional polarity transition records are obtained from a greater geographic distribution and as records of older reversals are obtained. These records will help solve the mystery of what happens during a polarity reversal, and that knowledge will in turn help us understand what it is about Earth that gives rise to this fascinating feature of our magnetic field. Bradford M. Clement

Bibliography Cande, S.C., and Kent, D.V., 1995. Revised calibration of the geomagnetic polarity timescale for the Late Cretaceous and Cenozoic. Journal of Geophysical Research, 100: 6093–6095. Channell, J.E.T., and Lehman, B., 1997. The last two geomagnetic polarity reversals recorded in high-deposition-rate sediment drifts. Nature, 389: 712–715. Clement, B.M., 1991. Geographical distribution of transitional VGPs: evidence for non-zonal equatorial symmetry during the Matuyama-Brunhes geomagnetic reversal. Earth and Planetary Science Letters, 104: 48–58. Clement, B.M., 2004. Dependence of the duration of geomagnetic polarity reversal on site latitude. Nature, 428(6983): 608–609. Coe, R.S., and Liddicoat, J.C., 1994. Overprinting of natural magnetic remanence in lake sediments by a subsequent high intensity field. Nature, 367: 57–59. Coe, R.S., and Prevot, M., 1989. Evidence suggesting extremely rapid field variation during a geomagnetic reversal. Earth and Planetary Science Letters, 92: 292–298. Coe, R.S., Prevot, M., and Camps, P., 1995. New evidence for extraordinarily rapid change of the geomagnetic field during a reversal. Nature, 374: 687–692. Gee, J.S., Tauxe, L., Barge, E., Peirce, J.W., Weissel, J.K., Taylor, E., Dehn, J., Driscoll, N.W., Farrell, J.W., Fourtanier, E., Frey, F.A.,

328

GEOMAGNETIC POLARITY TIMESCALES

Gamson, P.D., Gibson, I.L., Janecek, T.R., Klootwijk, C.T., Lawrence, J.R., Littke, R., Newman, J.S., Nomura, R., Owen, R.M., Pospichal, J.J., Rea, D.K., Resiwati, P., Saunders, A.D., Smit, J., Smith, G.M., Tamaki, K., Weis, D., and Wilkinson, C., 1991. Lower Jaramillo polarity transition records from the equatorial Atlantic and Indian oceans: Proceedings of the Ocean Drilling Program, Scientific Results, 121: 377–391. Hillhouse, J.A., and Cox, A., 1976. Brunhes-Matuyama polarity transition. Earth and Planetary Science Letters, 29: 51–64. Hoffman, K.A., 1984. A method for the display and analysis of transitional paleomagnetic data. Journal of Geophysical Research, 137: 6285–6292. Hoffman, K.A., 1991. Long-lived transitional states of the geomagnetic field and the two dynamo families. Nature, 354: 273–277. Hoffman, K.A., 1992. Dipolar reversal states of the geomagnetic field and core-mantle dynamics. Nature, 359: 789–794. Laj, C., Mazaud, A., Weeks, R., Fuller, M., and Herrero-Bervera, E., 1991. Geomagnetic reversal paths. Nature, 351: 447. Love, J.J., and Mazaud, A., 1997. A database for the MatuyamaBrunhes magnetic reversal. Physics of the Earth and Planetary Interiors, 103: 207–245. Mankinen, E.A., Prevot, M., Gromme, C.S., and Coe, R.S., 1985. The Steens Mountain (Oregon) geomagnetic polarity transition. I. Directional history, duration of episodes, and rock magnetism. Journal of Geophysical Research, 90: 10393–10417. Merrill, R.T., and McFadden, P.L., 1999. Geomagnetic polarity transitions. Reviews of Geophysics, 37: 201–226. Prevot, M., Mankinen, E.A., Coe, R.S., and Gromme, C.S., 1985. The Steens Mountain (Oregon) geomagnetic polarity transition. 2. Field intensity variations and discussion of reversal models. Journal of Geophysical Research, 90: 10417–10448. Singer, B.S., and Pringle, M.S., 1996. Age and duration of the Matuyama-Brunhes geomagnetic polarity reversal for 40Ar/39Ar incremental heating analyses of lavas. Earth and Planetary Science Letters, 139: 47–61. Tauxe, L., 1993. Sedimentary records of relative paleointensity of the geomagnetic field: theory and practice. Reviews of Geophysics, 31: 319–354.

Cross-references Core-Mantle Boundary Topography, Implications for Dynamics Core-Mantle Boundary Topography, Seismology Core-Mantle Boundary, Heat Flow Across Geodynamo Geodynamo, Dimensional Analysis and Timescales Geodynamo, Numerical Simulations Magnetic Anomalies, Marine Nondipole Field Reversals, Theory

GEOMAGNETIC POLARITY TIMESCALES Marine magnetic anomaly record It is well established that Earth’s magnetic field has alternated frequently but irregularly between two opposing polarity states for at least most of the Phanerozoic. Older rocks of Proterozoic age that display both polarity states are also known, but whether they correspond to reversals of a dipole field is less firmly established. The time interval during which geomagnetic polarity remains constant is called a polarity chron; long episodes of continuous reversal behavior are called superchrons. The polarity equivalent to the present state is referred to as “normal” and the opposite state as “reversed.”

The history of geomagnetic polarity is derived from two sources: the interpretation of lineated marine magnetic anomalies, and magnetic polarity stratigraphy in continuous sedimentary sequences and radiometrically dated igneous rocks. Since the Late Jurassic, when the current phase of seafloor spreading began, these records support and confirm each other. The geomagnetic polarity timescale (GPTS) is most reliable for this time. It is divided into two sequences of alternating polarity, the younger covering the Late Cretaceous and Cenozoic, and the older corresponding to Early Cretaceous and Late Jurassic time. The reversal sequences are referred to as the C-sequence and M-sequence, respectively. They are separated in the oceanic record by the Cretaceous Quiet Zone in which lineated magnetic anomalies are absent. It appears that Earth’s magnetic field did not reverse polarity during this time interval, which is referred to as the Cretaceous normal polarity superchron (CNPS). Prior to the Late Jurassic only the polarity record preserved in rocks is available. Knowledge of older geomagnetic polarity history is patchy and, despite some excellent magnetostratigraphic results, largely unconfirmed.

Construction and identification of a polarity chron sequence Although the pioneering studies of geomagnetic polarity were carried out on radiometrically dated lavas, the marine magnetic anomaly record provides the most extensive, detailed and continuous record of reversal history and forms the basis of all GPTS covering the last 160 Ma of the Earth history. The appearance of lineated large amplitude, long wavelength marine magnetic anomalies (Figure G31) depends on the latitude and direction of the corresponding measurement profile. The first step in constructing a GPTS thus consists of interpreting the anomalies as a block model of alternating magnetization of the oceanic crust responsible for the anomalies. The polarity pattern is used to correlate coeval segments of different profiles and to obtain an optimized model that minimizes local variations in spreading rate on any given profile. The ensuing composite block model constitutes a polarity sequence in which the distances between the block boundaries are proportional to the relative lengths of the polarity chrons. No consideration is usually given to the finite duration of a polarity transition, which is thought to last about 4–6 ka (Clement and Kent, 1984). This time is included in the lengths of polarity chrons, which are measured between the midpoints of polarity transitions. This simplification is generally acceptable but it may be problematic for accurately describing short polarity chrons for which the transitional time may be an appreciable fraction of the duration of the chron. The anomalies are identified by numbering them in increasing order away from the spreading axis. In the C-sequence the positive magnetic anomalies, corresponding to normal polarity, are numbered and preceded by the letter “C,” as anomalies C18, C29, etc. The M-sequence oceanic crust in the North Pacific Ocean was magnetized south of the equator but plate motion has brought it into the northern hemisphere. The positive anomalies, identified by the letter “M” as anomalies M0, M13, etc., correspond to reversely magnetized crust. To reconcile the two numbering schemes, adjacent normally and reversely polarity chrons are paired, so that the younger member has normal polarity. Thus marine magnetic anomaly C15 is associated with polarity chrons C15n and C15r. To obtain an optimized global model of polarity, which then becomes the polarity sequence for a GPTS, block models for different spreading centers must be compared, stretched, and squeezed differentially. The matching of block models may be visual or more sophisticated. To form an optimized record of the C-sequence of polarity Cande and Kent (1992a) used nine rotation poles to stack polarity records on contemporaneous profiles and determine relative widths of crustal blocks for the modeled reversal sequence. Before a timescale can be generated, the optimized polarity sequence must be dated. Biostratigraphic stage boundaries or other dated levels are correlated by magnetostratigraphy to the polarity

GEOMAGNETIC POLARITY TIMESCALES

329

Figure G31 Marine magnetic profiles, correlation tie-lines, and block model of the polarity of oceanic crustal magnetization (black, normal; white, reverse) for M-sequence anomalies in the North Pacific (after Larson and Hilde, 1975).

sequence. Intervening reversal boundaries between the calibration levels are dated by linear interpolation and older boundaries by extrapolation. The largest sources of error in this dating are the “absolute” ages associated with the tie-levels. The relative lengths of the polarity chrons, derived from a crustal block model that assumes constant spreading rate, are known more accurately.

Cenozoic and Late Cretaceous GPTS The pioneering efforts to determine a GPTS were carried out on marine magnetic anomalies at actively spreading oceanic ridge systems. Heirtzler et al. (1968), hereafter referred to as HDHPL68, derived a GPTS from the present time to the Late Cretaceous by matching profiles in different oceans to a reference profile in the South Atlantic, which was judged to be the most likely to have a constant spreading rate. The spreading rate at the South Atlantic Ridge was estimated by correlating distances to young reversals with radiometrically dated reversals from lava sequences. By assuming this spreading rate to be constant, the distance of a polarity reversal from the ridge could be converted to age. However, it was later found that the magnetic anomalies composing anomaly C14 in HDHPL68 were not reproducible in other oceans and thus did not correspond to polarity intervals; moreover, the HDHPL68 timescale did not cover the full range of older anomalies following the Cretaceous Quiet Interval. Improved, extended, and modified, the HDHPL68 timescale served as the basis for several subsequent versions of the GPTS for this time interval (Figure G32). Magnetostratigraphic correlation of the Cretaceous– Tertiary boundary to the upper part of chron C29r in a continental exposure of marine limestones (Alvarez et al., 1977) provided a way of associating age with the older end of the reversal sequence and resulted in improved calibration of the C-sequence GPTS. The magnetostratigraphy also confirmed that the C-sequence began at C33r, as termination of the CNPS. Together with more detailed analysis of the oceanic record, this led to a more complete and accurate GPTS (LaBrecque et al., 1977), hereafter LKC77. The number of stage boundaries now tied to the polarity record led Lowrie and Alvarez (1981) to propose a modification of the GPTS. They assumed the polarity sequence in LKC77 and best estimates of

the “absolute” ages for 11 tie-levels that correlated the stratigraphic and marine magnetic polarity records. Disregarding the effects on seafloor spreading rates they stretched and squeezed the polarity record between the tie-points and obtained a new GPTS. The ensuing timescale resulted in a history of seafloor spreading characterized by sudden large changes in spreading rate. To avoid this, Harland et al. (1982) modified the ages of tie-points and obtained a GPTS that gave a smoother seafloor spreading record. One of the most important uses of a GPTS is now recognized to be the ability to attach numeric ages to faunal appearances and extinctions. Berggren et al. (1985) and Harland et al. (1990) produced GPTS versions in which the biostratigraphy was well tied to the magnetic polarity record, which was derived from that of LKC77. Cande and Kent (1992a) carried out a detailed reevaluation of the C-sequence marine magnetic anomalies and improved the definitions of the corresponding oceanic block models, resulting in adjustments of the relative lengths of polarity chrons. They associated ages with the improved polarity sequence by fitting a cubic spline curve to nine dated correlation points. In so doing they used a nonstandard age for the Cretaceous-Tertiary boundary. An updated version of their timescale (Cande and Kent, 1995), hereafter referred to as CK95, is the current reference GPTS for the Late Cretaceous and Cenozoic. An archive of Cenozoic-Late Cretaceous timescales has been assembled by Mead (1996).

The Cretaceous normal polarity superchron The C-sequence and M-sequence polarity chrons are separated by the CNPS. Using the nomenclature for labeling chrons, this might be called C34n or M0n, but it is customary to designate it separately from the adjacent sequences. In the marine magnetic anomaly record the CNPS corresponds to regions of oceanic crust referred to as the Cretaceous Quiet Zone, in which magnetic anomalies, although present, are not lineated. It is believed to represent an interval of time, from about 121 to 83 Ma ago, in which the geomagnetic field had a consistent normal polarity. Despite magnetostratigraphic and deep-tow investigations to find polarity reversals within this 38 Ma interval, there is no incontrovertible evidence of them. The absence of polarity changes

330

GEOMAGNETIC POLARITY TIMESCALES

Figure G32 Evolution of the geomagnetic polarity timescale for the Cenozoic and Upper Cretaceous. The authors of individual C-sequence models are as follows: HDHPL68, Heirtzler et al. (1968); LKC77, LaBreque et al. (1977); LA81, Lowrie and Alvarez (1981); GTS82, Harland et al. (1982); BKFV85, Berggren et al. (1985); GTS89, Harland et al. (1990); CK95, Cande and Kent (1995). in the CNPS may imply a special behavior of the geodynamo. A few very long polarity chrons, lasting several million years each, are found adjacent to the CNPS.

Early Cretaceous and Late Jurassic GPTS The oldest regions of oceanic crust characterized by lineated magnetic anomalies were formed during the Late Jurassic. The Phoenix, Japanese, and Hawaiian lineations are found in the Western Pacific, and the Keathley lineations in the western North Atlantic. Larson and Pitman (1972) identified and correlated these sets of anomalies, numbering them from M1 to M22 in order of increasing age. They realized that the three sets of Pacific lineations were inverted with respect to the Keathley sequence. The North Pacific oceanic crust had been magnetized south of the magnetic equator, so that positive anomalies are now found over reversely magnetized crust. Larson and Hilde (1975)

refined the reversal record for the Hawaiian lineations, resolving additional anomalies, adding a younger anomaly M0, and extending the older anomaly record to M25 (Figure G31). They dated their timescale (LH75) by estimating the age of magnetic basement at drillholes of the Deep Sea Drilling Project from the paleontological ages of the oldest calcareous fossils found in the holes. The magnetic reversal block model was derived for the Hawaiian lineations, but the ages were determined for sites on other lineation sets and correlated by the magnetic polarity pattern to the Hawaiian set. The LH75 polarity sequence has formed the basis for subsequent modifications and improvements to the M-sequence GPTS (Figure G33). Later versions (Kent and Gradstein, 1985; Harland et al., 1990) are somewhat better dated, but still rely on bottom ages in groups of drillholes near the ends of the sequence. A new Hawaiian block model with an optimized approximation to a constant spreading rate was derived by Channell et al. (1995) after critical comparison of block models for the Hawaiian,

GEOMAGNETIC POLARITY TIMESCALES

331

Figure G33 Evolution of the geomagnetic polarity timescale for the Early Cretaceous and Late Jurassic. The authors of individual M-sequence models are as follows: LH75, Larson and Hilde (1975); KG85, Kent and Gradstein (1985); GTS89, Harland et al. (1990); CENT94, Channell et al. (1995).

Japanese, Phoenix, and Keathley lineations. This model (CENT94), covering magnetic polarity chrons CM0r to CM29r, is probably the optimum current GPTS for the M-sequence anomalies. Oceanic crust older than chron CM25r was thought to be free of lineated magnetic anomalies and was labeled the Jurassic Quiet Zone by analogy to the Cretaceous equivalent. However, magnetic lineations with low amplitude were subsequently identified in the youngest part of this Quiet Zone, extending the polarity sequence to CM29r (Cande et al., 1978). These weaker old anomalies are related to oceanic crust whose magnetization decreases with increasing age. Even older anomalies, with short wavelengths and low amplitudes, have been detected within the Jurassic Quiet Zone, both from aeromagnetic profiles (Handschumacher et al., 1988) and from deep-towed magnetometer surveys (Sager et al., 1998). Their origin is as yet uncertain. They may have been formed during an episode of high reversal

frequency, in which case they would extend the M-sequence from CM29r to CM41r. This would imply that the Jurassic Quiet Zone is different in origin from the Cretaceous Quiet Zone, in which no reversals are thought to have occurred. However, the polarity pattern corresponding to these anomalies has not been established definitively and the reversal sequence has not been confirmed independently by magnetostratigraphy. It is possible that the anomalies may be due to fluctuations of paleomagnetic field intensity, as suggested to explain low amplitude, short wavelength anomalies in the Cenozoic (Cande and Kent, 1992b).

Early Jurassic and Triassic GPTS The record of geomagnetic polarity prior to the onset of seafloor spreading is much less well known. In the absence of a marine

332

GEOMAGNETIC POLARITY TIMESCALES

Bibliography

Figure G34 Construction of a GPTS for the Late Triassic based on overlapping magnetostratigraphies from drillholes in the Newark Basin, USA (after Kent et al., 1995). The GPTS is that of Kent and Olsen (1999), calibrated by astrochronological dating of the magnetozones.

magnetic anomaly record, the GPTS must be pieced together from magnetostratigraphic results. This is a large undertaking. Changes of sedimentation rate within a given depositional basin or from one basin to another can strongly modify the “fingerprint” pattern of reversals that is essential for correlation. Investigations in marine limestones of Middle and Early Jurassic age have identified magnetozones, but these have only rarely been confirmed by other magnetostratigraphic results. Kent et al. (1995) determined detailed magnetostratigraphies in overlapping drillholes through continental redbeds of Late Triassic age in the Newark basin. They matched the individual records stratigraphically to produce a composite geomagnetic polarity record (Figure G34). The sediments displayed lithological variations related to cyclical changes in Earth’s orbital parameters, of which the 400 ka fluctuation of orbital eccentricity was prominent. Kent and Olsen (1999) used this Milankovitch cycle to convert the polarity sequence to a GPTS (Figure G34), assuming an “absolute” age of 202 Ma for the Jurassic-Triassic boundary. The Newark section currently serves as the standard of reference for the history of geomagnetic polarity in the Late Triassic. William Lowrie

Alvarez, W., Arthur, M.A., Fischer, A.G., Lowrie, W., Napoleone, G., Premoli Silva, I., and Roggenthen, W.M., 1977. Upper CretaceousPaleocene magnetic stratigraphy at Gubbio, Italy. V. Type section for the Late Cretaceous-Paleocene geomagnetic reversal time scale. Geological Society of America Bulletin, 88: 383–389. Berggren, W.A., Kent, D.V., Flynn, J.J., and Van Couvering, J.A., 1985. Cenozoic geochronology. Geological Society of America Bulletin, 96: 1407–1418. Cande, S.C., and Kent, D.V., 1992a. A new geomagnetic polarity time scale for the Late Cretaceous and Cenozoic. Journal of Geophysical Research, 97: 13917–13951. Cande, S.C., and Kent, D.V., 1992b. Ultra-high resolution marine magnetic anomaly-profiles: a record of continuous paleointensity variations? Journal of Geophysical Research, 97: 15075–15083. Cande, S.C., and Kent, D.V., 1995. Revised calibration of the geomagnetic polarity timescale for the Late Cretaceous and Cenozoic. Journal of Geophysical Research, 100: 6093–6095. Cande, S., Larson, R.L., and LaBrecque, J.L., 1978. Magnetic lineations in the Pacific Jurassic quiet zone. Earth and Planetary Science Letters, 41: 434–440. Channell, J.E.T., Erba, E., Nakanishi, M., and Tamaki, K., 1995. Late Jurassic-Early Cretaceous time scales and oceanic magnetic anomaly block models. In Berggren, W.A., Kent D.V., Aubry, M., and Hardenbol, J. (eds.), Geochronology, Timescales, and Global Stratigraphic Correlation. Tulsa, Oklahoma: SEPM Special Publication, pp. 51–64. Clement, B.M., and Kent, D.V., 1984. Latitudinal dependency of geomagnetic polarity transition durations. Nature, 310: 488–491. Handschumacher, D.W., Sager, W.W., Hilde, T.W.C., and Bracey, D.R., 1988. Pre-Cretaceous evolution of the Pacific plate and extension of the geomagnetic polarity reversal time scale with implications for the origin of the Jurassic “Quiet Zone”. Tectonophysics, 155: 365–380. Harland, W.B., Cox, A.V., Llewellyn, P.G., Pickton, C.A.G., Smith, A.G., and Walters, R., 1982. A Geologic Time Scale. Cambridge: Cambridge University Press, 131 pp. Harland, W.B., Armstrong, R.L., Cox, A.V., Craig, L.E., Smith, A.G., and Smith, D.G., 1990. A Geologic Time Scale 1989. Cambridge: Cambridge University Press, 263 pp. Heirtzler, J.R., Dickson, G.O., Herron, E.M., Pitman, W.C. III, and Le Pichon, X., 1968. Marine magnetic anomalies, geomagnetic field reversals and motions of the ocean floor and continents. Journal of Geophysical Research, 73: 2119–2136. Kent, D.V., and Gradstein, F.M., 1985. A Cretaceous and Jurassic geochronology. Geological Society of America Bulletin, 96: 1419–1427. Kent, D.C., and Olsen, P.E., 1999. Astronomically tuned geomagnetic polarity timescale for the Late Triassic. Journal of Geophysical Research, 104: 12,831–12,841. Kent, D.V., Olsen, P.E., and Witte, W.K., 1995. Late Triassic-Earliest Jurassic geomagnetic polarity sequence and paleolatitudes from drill cores in the Newark rift basin, eastern North America. Journal of Geophysical Research, 100: 14965–14998. LaBrecque, J.L., Kent, D.V., and Cande, S.C., 1977. Revised magnetic polarity timescale for Late Cretaceous and Cenozoic time. Geology, 5: 330–335. Larson, R.L., and Hilde, T.W.C., 1975. A revised time scale of magnetic reversals for the Early Cretaceous and Late Jurassic. Journal of Geophysical Research, 80: 2586–2594. Larson, R.L., and Pitman, W.C. III, 1972. World-wide correlation of Mesozoic magnetic anomalies, and its implications. Geological Society of America Bulletin, 83: 3645–3662. Lowrie, W., and Alvarez, W., 1981. One hundred million years of geomagnetic polarity history. Geology, 9: 392–397.

333

GEOMAGNETIC PULSATIONS

Mead, G.A., 1996. Correlation of Cenozoic-Late Cretaceous geomagnetic polarity timescales: an Internet archive. Journal of Geophysical Research, 101: 8107–8109. Sager, W.W., Weiss, C.J., Tivey, M.A., and Johnson, H.P., 1998. Geomagnetic polarity reversal model of deep-tow profiles from the Pacific Jurassic Quiet Zone. Journal of Geophysical Research, 103: 5269–5286.

Cross-references Crustal Magnetic Field Geomagnetic Polarity Reversals Geomagnetic Polarity Timescales Magnetic Anomalies, Marine Magnetic Surveys, Marine Magnetostratigraphy Paleomagnetism Reversals, Theory

GEOMAGNETIC PULSATIONS Geomagnetic pulsations or mircopulsations are ultralow frequency (ULF) plasma waves in the Earth’s magnetosphere. These waves have frequencies in the range 1 mHz to greater than 10 Hz and appear as more or less regular oscillations in records of the geomagnetic field (see Figure G35). Geomagnetic oscillations, or ULF pulsations as they are also called, can also be identified in electric field measurements in the ionosphere as well as observations of the electromagnetic field of the magnetosphere as made onboard spacecraft. The lower frequency pulsations have wavelengths comparable to typical scale lengths of the entire magnetosphere. They may also be interpreted as eigenoscillations of or standing waves in the magnetospheric systems. The higher frequency waves are usually identifiable as proton ion-cyclotron waves in the magnetospheric plasma. The amplitudes of the lower frequency pulsations can reach several tens up to hundreds of nanotesla in the auroral zone region while the higher frequency waves reach amplitudes of the order of a few nanotesla. The first observation of a geomagnetic pulsation was published in 1861 by the Scottish scientist Balfour Stewart who identified quasisinusoidal variations of the geomagnetic field in records of the Kew observatory after the great magnetic storm that occurred in 1859. The International Geophysical Year (1958–1959) with its large

number of coordinated geomagnetic field observations made available a multitude of studies on ULF pulsations, stimulated the interest in this type of geomagnetic field variation and created an active area of research. The classification scheme (Table G7) of the International Association of Geomagnetism and Aeronomy (IAGA) distinguishes seven different types of geomagnetic pulsations based on their oscillation period and appearance in magnetograms as almost continuous and more irregular pulsations. The two classes, continuous pulsations (Pc) and irregular pulsations (Pi), are usually divided into subclasses. A more detailed overview of this morphological classification was given by Jacobs (1970). Though this classification is still widely in use it is somewhat outdated as the increased understanding of the physical nature and properties of geomagnetic pulsations allows a more indepth classification based on physical processes and generating mechanisms. The long-period pulsations are at present interpreted as magnetohydrodynamic waves while the short-period pulsations are related to ion-cyclotron waves propagating in the magnetosphere. As for many magnetospheric phenomena the solar wind provides the energy for geomagnetic pulsations, partly directly, partly indirectly. A direct energy source is plasma waves generated in the solar wind and penetrating the magnetopause. A major source of these waves are plasma instabilities in the upstream region of the near-Earth solar wind where, for example, protons reflected at the magnetospheric bow shock constitute an unstable particle distribution, generating a variety of upstream waves. These waves are convected downstream toward the magnetopause and couple through it into the magnetosphere. Detailed investigations, however, indicate that the transmission of hydromagnetic waves through the magnetopause is a rather inefficient process. Only a small percentage of the energy of the upstream solar wind waves couples to oscillations of the magnetosphere. A more efficient, solar wind-driven process is the impulsive excitation of plasma waves by sudden impulses from the solar wind. The magnetosphere constitutes a kind of body capable of eigenoscillations. It can be excited much like a bell. The British scientist Jim Dungey in 1954 was the first to analyze such eigenoscillations in more detail. Assuming that the magnetospheric magnetic field is of dipole nature only, he studied the magnetohydrodynamic equations of such a dipole-magnetosphere filled with an ionized gas of spatially depending mass density. The boundaries of this magnetosphere are the magnetopause and the ionosphere, where the geomagnetic field lines are anchored much as strings of a violin are anchored between the pegs and the tailpiece. He derived the so-called Dungey equations, a set of partial differential equations describing the coupling between toroidal oscillations of the fluid velocity field and the toroidal (poloidal) component of the electric (magnetic) field oscillations in this dipole-magnetosphere. If the excitation of the dipole-magnetosphere is axisymmetric, then the toroidal components of these two fields are decoupled. This in turn

Table G7 The IAGA classification of geomagnetic pulsations Name

Figure G35 Geomagnetic pulsation of the Pc4 type, recorded at a magnetic observatory in North Scandinavia. The y- or east-west component of the geomagnetic field is displayed relative to a quiet day record.

Continuous Pc1 Pc2 Pc3 Pc4 Pc5 Irregular Pi1 Pi2

Period range (s)

0.2–5 5–10 10–45 45–150 150–600 1–40 40–150

334

GEOMAGNETIC PULSATIONS

implies that individual field line shells are oscillating independent from each other, much as different violin strings oscillate independently. The oscillation period depends on Alfvén wave velocity along the considered field line. Therefore geomagnetic pulsations can also be used as a diagnostic tool for magnetospheric physics. This is very much in accord with the observational finding that the periods of geomagnetic pulsations in the Pc4–5 range decrease with decreasing geomagnetic latitude. At lower latitudes the length of field lines anchored between the northern and southern ionosphere is much shorter than at higher latitudes, which causes a smaller oscillation period at low-latitudes. Geomagnetic oscillations can thus be viewed at as standing field line oscillations with fundamental and higher harmonic waves being generated (Figure G36) . Besides an impulsive excitation of geomagnetic pulsations the socalled Kelvin-Helmholtz instability can drive magnetospheric magnetohydrodynamic waves. At the interface between the solar wind plasma and the magnetosphere plasma, the so-called magnetopause, strong shear flows exists. The solar wind plasma has to flow around the magnetosphere along the magnetopause. Much as atmospheric wind flow over a water surface can cause water waves the velocity shear at the magnetopause destabilizes this boundary and causes surface waves which are coupled into the magnetosphere. The energy for these waves is drained out of the solar wind flow and maximum instability of the magnetopause is expected at the flanks of the magnetosphere, which is in the dawn and dusk hours. Furthermore, the unstable waves should propagate tailward that is they propagate east at dusk and west at dawn. This is indeed observed in ground and satellite observations of geomagnetic pulsations. As the plasma density distribution is not uniform wave propagation in the magnetosphere may lead to a very interesting physical effect, field line resonance or resonant mode coupling. A Kelvin-Helmholtz instability generated wave is a compressional magnetohydrodynamic wave, which couples to a transverse oscillation, an Alfvén wave, at a region in the magnetosphere where the surface wave’s period equals the local eigenperiod of the toroidal field oscillation or Alfvén wave. At this resonance point the oscillation magnitude maximizes and the wave phase changes by 180 when crossing the resonant field line in the radial direction. The physics of this resonant mode coupling between a surface wave and a local eigenmodes is actually a tunneling process (Southwood and Hughes, 1983). Internal to the magnetosphere a variety of wave sources exist. Most important is the ring current region with its energetic protons. Proton distributions are usually nonthermal and tend to thermalize via interaction with electromagnetic waves. These waves in turn can be generated by plasma instabilities such as the so-called bounce-resonance or driftmirror instabilities (Samson, 1991; Walker, 2004). The ring current region is also the source region of many of the Pc1Pc2 geomagnetic pulsations. During the expansive phase of magnetospheric substorms large numbers of energetic ions are injected from the magnetotail into the inner magnetosphere where they drift westward to create the substorm-enhanced ring current. These energetic particle populations are highly unstable against the ion-cyclotron instability and cause the generation of short-period geomagnetic pulsations (Kangas et al., 1998). Once geomagnetic pulsations have been generated in the magnetosphere their energy must be dissipated somewhere. Most of this dissipation occurs in the ionosphere where the pulsation-associated electric fields cause current flow, which in turn leads to significant Joule heating of the ionosphere. Local kinetic temperature increases of several thousand Kelvin have been observed. Part of the wave energy is also used to accelerate magnetospheric particles. Such high-energy particles may subsequently hit the atmosphere where they can cause aurora. Much as the terrestrial magnetosphere also magnetospheres of other planets, in particular those of Mercury, Jupiter, and Saturn exhibit magnetic field oscillations comparable to geomagnetic pulsations

Figure G36 Schematic representation of standing geomagnetic field line oscillations. The perturbed field line is the dashed-dotted line. The electric field always has a node in the ionosphere as large electrical conductivity there shortcuts all electric potential differences.

(Glassmeier et al., 1999). Magnetic pulsations of these magnetospheric systems have different properties than those at Earth due to, for example, different spatial scales of the oscillating system. Karl-Heinz Glaßmeier

Bibliography Glassmeier, K.H., Othmer, C., Cramm, R., Stellmacher, M., and Engebretson, M., 1999. Magnetospheric field line resonances: a comparative planetology approach. Surveys in Geophysics, 20: 61–109. Jacobs, J.A., 1970. Geomagnetic Micropulsations. Berlin: SpringerVerlag. Kangas, J., Guglielmi, A., and Pokhotelov, O., 1998. Morphology and physics of short-period magnetic pulsations. Space Science Reviews, 83: 435–512. Samson, J.C., 1991. Geomagnetic pulsations and plasma waves in the Earth’s magnetosphere. In Jacobs, J.A. (ed.), Geomagnetism, Vol. 4. London: Academic Press, pp. 481–592. Southwood, D.J., and Hughes, W.J., 1983. Theory of hydromagnetic waves in the magnetosphere. Space Science Reviews, 35: 301–366. Walker, A.D.M. (ed.), 2004. Magnetohydrodynamic Waves in Geospace: The Theory of ULF Waves and Their Interaction with Energetic Particles in the Solar-Terrestrial Environment. Philadelphia: Institute of Physics Publishing.

Cross-references Alfvén Waves Ionosphere Magnetohydrodynamic Waves Magnetosphere of the Earth Ring Current

GEOMAGNETIC REVERSAL SEQUENCE, STATISTICAL STRUCTURE

GEOMAGNETIC REVERSAL SEQUENCE, STATISTICAL STRUCTURE Introduction It is now well recognized that the geomagnetic field is produced by dynamo action within the molten iron in the outer core. This is a dynamic process intimately linked to cooling of the core and the rapid spin of the Earth. The equations governing the geodynamo, which have to be solved jointly to obtain a full model of the process, are Maxwell’s equations, Ohm’s law, the Navier-Stokes’ equation, the continuity equation, Poisson’s equation, the generalized heat equation, and the equation of state for the material in the outer core. This is a complex, nonlinear set of equations, making it extraordinarily difficult to obtain a full solution. However, the equations are even in H, the magnetic field. That is, the equations are insensitive to the sign of H, and so if H is a solution then so also is –H. We know from present-day observations that the geomagnetic field can exist in a relatively stable state in which the field at the Earth’s surface is approximately that of a dipole with its axis almost parallel to the Earth’s spin axis. Consequently we should expect that there is a similar relatively stable solution, with the same statistical properties as the field we observe today, that simply has the opposite polarity, that is, the north and south poles are swapped. Hence, if there is a mechanism for the field to move from one solution to the other, we should expect to see reversals of the geomagnetic field. The solar magnetic dynamo reverses regularly with a full period of about 22 years (see Magnetic field of Sun), so we know that reversal is possible in other self-sustaining dynamos. This then leaves open the question of how we can tell whether the field has reversed polarity in geological time, well before humans started to observe the field and its behavior. By at least the late 18th century it was recognized that deviation of magnetic compasses could occur because of nearby strongly magnetized rocks. The first observations that the magnetization in certain rocks was actually parallel to the Earth’s magnetic field were made independently by Delesse and Melloni. Folgerhaiter extended their work but also studied the magnetization of bricks and pottery. He argued that when a brick or pot was fired in the kiln then the remanent magnetization it acquired on cooling provided a record of the direction of the Earth’s magnetic field. With the wisdom of hindsight it is fairly obvious that this would be the case. Volcanic rocks are heated well above the Curie point so the magnetization is free to align with the external magnetic field and becomes locked in as the rock cools. This is known as a thermoremanent magnetization (TRM) (q.v.) and, in extrusive volcanics, provides a record of the direction of the magnetic field at that locality at a specific point in time. There are several processes by which a rock will lock in a fossil record of the ancient (or paleo) magnetic field. The fossil magnetism naturally present is termed the natural remanent magnetization (NRM) (q.v.) and its existence provides us with an opportunity to discover the direction of the geomagnetic field over geological time. David in 1904 and Brunhes in 1906 reported the first discovery of NRM that was roughly opposite in direction to that of the present field and this led to the speculation that the Earth’s magnetic field had reversed its polarity in the past. At that time it was not recognized that the field was generated by a self-sustaining dynamo in the outer core, so the possibility of reversal was more exciting than it may seem from our perspective today. In 1926, Mercanton pointed out that if the Earth’s field had in fact reversed itself in the past then reversely magnetized rocks should be found in all parts of the world. He demonstrated that this was indeed the case for Quaternary-aged rocks around the world. The speculation gained further support when Matuyama in 1929 observed reversely magnetized lava flows from the past 1 or 2 Ma in Japan and Manchuria. However, doubts about the validity of the field reversal hypothesis surfaced during the 1950s after Néel presented theory that showed it was possible for samples to acquire a magnetization antiparallel to the external field during cooling, a

335

process referred to as self-reversal. Shortly thereafter Nagata and Uyeda found the first laboratory-reproducible self-reversing rock, the Haruna dacite. Subsequently, it was recognized that self-reversal is relatively rare and by the early 1960s it was accepted that the Earth’s magnetic field has indeed reversed and that the phenomenon of field reversal has occurred many times. An excellent history of this subject was given by Glen (1982). A critical component in our understanding of reversals was development in the early 1960s of the K-Ar dating method, which made it possible to date young volcanic rocks with reasonable precision. Consequently, it was possible to undertake systematic studies attempting to define the geomagnetic polarity timescale (GPTS) using joint magnetic polarity and K-Ar age determinations on young lavas. As data rapidly became available, it was established that rocks of the same age had the same polarity of magnetization, helping to confirm that the observed reversals of magnetization were indeed due to reversals of the geomagnetic field itself. A few of the earliest compilations, covering the years 1959–1966, of the GPTS for the past 4 Ma are shown in Figure G37. Not surprisingly, when the field has the polarity that we observe today, it is referred to as normal polarity, and the opposite polarity is referred to as reverse polarity. As already noted, the solar magnetic field has a periodic reversal, and the first timescale put forward by Cox et al. (1963) appeared to be consistent with geomagnetic reversals having a periodicity of about 1 Ma intervals. However, as new data appeared it rapidly became apparent that there was no simple periodicity; some of the observed polarity intervals were nearly a million years in length and some were as short as 0.1 Ma. Furthermore, there did not appear to be any regular pattern to these different lengths. This led to the suggestion that there is a random component to the reversal process and, therefore, to interest in the statistical structure of the geomagnetic reversal sequence. It is extremely difficult to solve the geodynamo problem, that is, to determine just how the geodynamo operates. Determination and understanding of the statistical structure of the geomagnetic reversal sequence provides insight into the long-term dynamic process and in so doing can provide powerful constraints on geodynamo models. For example, the observation that the solar dynamo reverses with a clear periodicity but the geodynamo reversal process appears random, is probably a reflection of the different boundary conditions on the two dynamos.

Figure G37 Early compilations of the GPTS. Black represents normal polarity, white represents reversed polarity, and grey indicates uncertain polarity. Abstracted with permission from Cox (1969). ã American Association for the Advancement of Science.

336

GEOMAGNETIC REVERSAL SEQUENCE, STATISTICAL STRUCTURE

When examining a time sequence such as the GPTS it is crucial that the observed events be ordered correctly. This, together with other difficulties, has led to severe problems with the land-based polarity sequence. As one goes further back in time, the absolute error in dating a rock soon becomes larger than the length of the shorter polarity intervals. Because the land-based information comes from combined observations of magnetic polarity from rocks in widely spaced localities worldwide and not from a single continuous sequence, the ordering relies on the accuracy of the K-Ar dates. Hence, the ordering of events is effectively indeterminate for events closer together than the dating error. Consequently the length of the reliable land-based polarity timescale has been too short for reliable estimation of the statistical structure of the GPTS. Recently however, Kent and coworkers (e.g., Kent and Olson, 1999) have shown the potential for GPTS extension in thick, complete continental sedimentary sections. The most useful data source for development of the GPTS has been marine magnetic anomalies, and the reversal chronology is now quite well determined for the past 165 Ma. The absolute error in the age assigned to individual reversals will still often be greater than the length of the shorter intervals, but the continuous nature of the data source means that the ordering of events will be correct. Furthermore, because of the way ages are assigned in the marine magnetic anomaly timescales, the lengths of the intervals between reversals will typically be about right. Naturally though, there are difficulties and problems with the marine magnetic anomaly timescales. As we go further back in time the record is more degraded, so the timescale from 120 Ma back to about 165 Ma is less reliable than that from the present back to 80 Ma. Furthermore, because of subduction of the seafloor, any record prior to about 170 Ma has been destroyed. A major problem with marine magnetic anomalies is that it is fairly difficult to identify all of the very short intervals, particularly if a short interval appears between two relatively long intervals of the opposite polarity. This had led to significant problems in interpreting the GPTS. Despite the problems in the marine magnetic anomaly timescales, these provide us with our best GPTS and so analyses of the statistical structure of the geomagnetic reversal sequence have typically been performed on these timescales.

Relevant probability distributions It has been established that individual reversals take only a few thousand years to complete, which is a short time relative to the average interval between reversals. Thus it is a reasonable approximation to assume that the reversals themselves are instantaneous relative to the time constants of interest. It has already been noted that there appears to be a random component in the reversal process, and so it is sensible to look for a probabilistic description. Consider an interval of time D x short enough that the probability of having two reversals in D x is negligible, but with probability of lD x of having a single reversal in D x. This is then a Poisson process, and it is a simple matter to show that the probability density p(x) of interval lengths x between reversals is given by pðxÞdx ¼ le
lx dx;

1 ðk LÞk xðk
1Þ e
k Lx dx Gðk Þ 1 k ðk
1Þ
lx l x ¼ e dx; l ¼ k L Gðk Þ

pðxÞdx ¼

(Eq: 2)

where the mean interval length is now given by m ¼ 1/L ¼ k/l. G(k) is the gamma function of k, given by Z Gðk Þ ¼

1

z k
1 e
z dz

(Eq. 3)

0

If k is an integer, then this is just the factorial function of k–1, i.e., G(k) ¼ (k–1)!. From Eqs. (1) and (2) it is apparent that the gamma process leads to a family of distributions depending on the value of k, and that the Poisson process is simply the special case of a gamma process with k ¼ 1. Figure G38 shows appropriately scaled probability densities for this family of distributions. It is immediately apparent that a gamma process with k > 1 has far fewer very short intervals than a Poisson process. Statistical tools for analyzing the reversal sequence in terms of a renewal process are given by McFadden (1984a) and McFadden and Merrill (1986, 1993).

Implications of a gamma or Poisson process Consider the general probability of a reversal occurring at an interval D x. Let x be the interval of time since the most recent reversal (i.e., no reversals have occurred in the interval 0 to x), and let f (x)D x be the probability that a reversal will then occur in the interval x to (x þ D x). Hence the function f (x) describes how the instantaneous probability for the occurrence of another reversal varies with the time since the last reversal and is given by f ðxÞ ¼

1


pðxÞ Rx 0 pðtÞdt

(Eq. 4)

Figure G39 shows f (x)/l plotted as a function of (xl). For a Poisson process the occurrence of a reversal has no impact on the probability of a future reversal. That is, the system has no memory. For k > 1, the probability for a future reversal drops to 0 as soon as a reversal

(Eq. 1)

where l is the rate of the process and m ¼ 1/l is the mean interval length. It is possible to test if the observed distribution of interval lengths is compatible with the Poisson distribution. Initially it was concluded that there was indeed compatibility but after further testing it was suggested that there were too few short intervals in the observed record. Naidu (1971) showed that a gamma distribution provided a good fit to the then observed intervals of the Cenozoic timescale and Phillips (1977) confirmed this in an extensive study of geomagnetic reversal sequences. For a gamma process the probability density p(x) of interval lengths x is given by

Figure G38 Probability densities of gamma distributions plotted against the interval length scaled to the mean length. The k ¼ 1 curve represents a Poisson distribution, which has a relatively large number of shorter intervals compared with the other distributions.

GEOMAGNETIC REVERSAL SEQUENCE, STATISTICAL STRUCTURE

Figure G39 Illustration of the memory (inhibition) in a gamma process and the absence of a memory in the special case of a Poisson process (k ¼ 1). After Merrill et al. (1996).

occurs and then gradually rebuilds to its undisturbed value. That is, the system has a memory of the previous reversal and this memory causes inhibition of future events by depressing their probability of occurrence.

Is the process gamma or Poisson? As noted above, there may be too few very short intervals in the observed timescale for a Poisson process, but a gamma distribution provides a good fit. Thus it may initially appear as a clear case in favor of a general gamma process, and indeed this was felt to be so for some time. Analysis of early timescales showed that k was quite different for the normal and reverse polarity sequences: at times it was about 4 for the normal polarity sequence while it was close to unity for the reverse polarity sequence (from about 40 to about 25 Ma); and in recent times (the past 15 Ma) it was about 2½ for the reverse sequence while close to unity for the normal sequence. This suggested a substantial asymmetry between normal and reverse polarity, which was surprising when theoretical considerations strongly implied that there should be no asymmetry (see above and Geomagnetic field, asymmetries). However, McFadden (1984a) showed that these estimates of k were not robust and that the actual reversal process is much more likely to be nearly Poisson. Obtaining a reliable GPTS from the marine magnetic anomaly record is not a trivial matter: the major problems relate to accurate dating of the individual events and reliable recognition of the shorter intervals. A small error in the dating of an individual reversal has small consequences. By contrast, there are significant consequences when a short polarity interval is actually missed. When a short interval is not identified it is, in effect, combined with the preceding and succeeding intervals of the opposite polarity. This means that the short interval is missed from its own polarity sequence thereby tending to increase the apparent value of k for that polarity sequence. At the same time, it incorrectly produces a long interval of the opposite polarity that is the sum of the interval preceding the missed short interval, the short interval itself, and the succeeding interval. McFadden (1984a) has shown that if n intervals from a gamma process with index k are joined together then the resulting interval appears to have been drawn from a gamma process with index nk. Therefore, if the process is Poisson and a short interval is missed, the resulting long interval of the opposite polarity appears to be an observation drawn from a gamma process with k ¼ 3. Hence the parameter k can be a fairly sensitive indicator of polarity intervals that have been missed. McFadden and Merrill (1984) concluded that k for the observed sequence was about 1.25 for the period from about 80 Ma to the

337

Figure G40 Estimated reversal rate l for the past 160 Ma. Constructed from the timescales of Kent and Gradstein (1986) and Cande and Kent (1995), following the methods of McFadden (1984a). From Merrill et al. (1996). present. This suggested that several short intervals had not been identified in the GPTS. McFadden and Merrill (1993) discussed the possible alternative that the reversal process is actually gamma and showed that this would imply that following a reversal the probability for another reversal would be depressed for about 50 ka. The observed sequence is statistically similar to a sequence that would be obtained by taking a Poisson process with the appropriate rate l and filtering the sequence so that any interval less than 30 ka in duration is incorporated into the surrounding intervals of opposite polarity. This is consistent with the conclusion of Parker (1997) that the maximum resolution of the GPTS from marine magnetic anomalies is about 36 ka. Overall it is probably safe to conclude that the reversal process is either Poisson or nearly Poisson, that there is no asymmetry between the normal and reverse sequences, and that the polarity sequence can be considered as a single sequence without regard to the actual polarity of the individual intervals.

Nonstationarity in the reversal process Recently the reversal rate has been about 4.5 Ma–1, around 40 Ma it was about 2 Ma–1, and from about 83 Ma back to about 119 Ma there were no reversals, the field having normal polarity. Clearly the rate at which the reversal process occurs has not been constant through time. This is of central geophysical interest because the existence of nonstationarity implies a change in the properties of the origin of the process, typically in the boundary conditions. Estimates of the reversal rate l, using the reversal chronology of Cande and Kent (1995) for the interval 0 to 118Ma and that of Kent and Gradstein (1986) for the interval 118–160 Ma, are shown in Figure G40. These estimates have been obtained using a sliding window containing 50 polarity intervals. The Cretaceous Superchron, an interval of about 36 Ma from about 118 to 82 Ma when the polarity was normal and there were no reversals, is very obvious in the sequence and requires explanation. The characteristic time of these changes is the same as that associated with changes in boundary conditions imposed by the mantle on the outer core (e.g., Jones, 1977; McFadden and Merrill, 1984, 1993, 1995, 1997; Courtillot and Besse, 1987). Suggestions that spatial variations in the core-mantle boundary conditions can affect the reversal rate are now supported by some dynamo theory (e.g., Glatzmaier et al., 1999). The variation in l shown in Figure G40 led to the interpretation first articulated by McFadden and Merrill (1984), that changes in the core-mantle boundary conditions gradually slowed the reversal process

338

GEOMAGNETIC REVERSAL SEQUENCE, STATISTICAL STRUCTURE

until eventually the process ceased, creating the superchron without any reversals. Gradually the boundary conditions became more favorable for reversals until eventually the reversal process restarted and gradually sped up to its current rate. Gallet and Hulot (1997), using the same timescales, proposed an alternative nonstationarity. They suggested that the reversal rate had been essentially constant from about 158 to 130 Ma and from about 25 Ma to the present, with an intermediate nonstationary segment including the Cretateous Superchron. However, McFadden and Merrill (2000) developed a statistical test that showed that the data demanded the trends identified in Figure G40. Subsequently, using the methodology developed by McFadden and Merrill (2000), Hulot and Gallet (2003) analyzed the Gradstein et al. (1994) and Channell et al. (1995) timescales. These timescales update the pre-superchron information. Hulot and Gallet (2003) concluded that these newer timescales show no long-term trend in reversal rate leading into the superchron. Indeed, they suggest that the only precursor to the superchron was perhaps a single unusually long interval (CM1n) just before the superchron. If this is confirmed by future timescales then the interpretation of the causes of the superchron will naturally change. There is some fine structure observable in Figure G40, and the question arises as to whether this structure is meaningful. Several authors (Mazaud et al., 1983; Negi and Tiwari, 1983; Raup, 1985; Stothers, 1986; Marzocchi and Mulargia, 1990; Mazaud and Laj, 1991; Rampino and Caldeira, 1993) have suggested either a 15- or a 30 Ma periodicity in the reversal chronology record. Hulot and Gallet (2003) have recently suggested a 20 Ma periodicity. Clearly the perceived periodicity is not robust. McFadden (1984b) showed that similar apparent periodicities are produced when using fixed-length sliding windows to analyze a Poisson process with a linear trend in the reversal rate, but that such periodicities are not observed when using sliding windows with a fixed number of intervals (McFadden and Merrill, 1984). Lutz (1985), Stigler (1987), McFadden (1987), and Lutz and Watson (1988) all showed that the perceived periodicities are more likely an artifact of the methods of analysis than real geophysical phenomena.

Acknowledgment This paper is published with the permission of the Chief Executive Officer, Geoscience Australia. Phillip L. McFadden

Bibliography Cande, S., and Kent, D.V., 1995. Revised calibration of the geomagnetic polarity timescale for the Late Cretaceous and Cenozoic. Journal of Geophysical Research, 100: 6093–6095. Channell, J., Erba, E., Nakanishi, M., and Tamaki, K., 1995. Late JurassicEarly Cretaceous time scales and oceanic magnetic anomaly block models. In Berggren, W., Kent, D., Aubry, M., and Hardenbol, J. (eds.), Geochronology, Timescales and Global Stratigraphic Correlation. Society of Economic Paleontologists and Mineralogists Special Publications, 54: 51–63. Courtillot, V., and Besse, J., 1987. Magnetic field reversals, polar wander, and core-mantle coupling. Science, 237: 1140–1147. Cox, A., 1969. Geomagnetic reversals. Science, 163: 237–245. Cox, A., Doell, R.R., and Dalrymple, G.B., 1963. Geomagnetic polarity epochs and Pleistocene geochronometry. Nature, 198: 1049–1051. Gallet, Y., and Hulot, G., 1997. Stationary and nonstationary behavior within the geomagnetic polarity timescale. Geophysical Research Letters, 24: 1875–1878. Glatzmaier, G.A., Coe, R.S., Hongre, L., and Roberts, P.H., 1999. The role of the Earth’s mantle in controlling the frequency of geomagnetic reversals. Nature, 401: 885–890.

Glen, W. 1982. The Road to Jaramillo. Critical Years of the Revolution in Earth Science. Stanford: Stanford University Press. Gradstein, F.M., Agterberg, F.P., Ogg, J.G., Hardenbol, J., van Veen, P., Thierry, J., and Huang, Z., 1994. A Mesozoic timescale. Journal of Geophysical Research, 99: 24051–24074. Hulot, G., and Gallet, Y., 2003. Do superchrons occur without any palaeomagnetic warning? Earth and the Planetary Science Letters, 210: 191–201. Jones, G.M., 1977. Thermal interaction of the core and the mantle and long term behaviour of the geomagnetic field. Journal of Geophysical Research, 82: 1703–1709. Kent, D.V., and Gradstein, F.M., 1986. A Jurassic to recent chronology. In Vogt, P.R., and Tucholke, B.E. (eds.), The Geology of North America, Vol. M, The Western North Atlantic Region. Boulder: Geological Society of America. Kent, D.V., and Olsen, P.E., 1999. Astronomically tuned geomagnetic polarity timescale for the Late Triassic. Journal of Geophysical Research, 104: 12831–12842. Lutz, T.M., 1985. The magnetic reversal record is not periodic. Nature, 317: 404–407. Lutz, T.M., and Watson, G.S., 1988. Effects of long-term variation on the frequency spectrum of the geomagnetic reversal record. Nature, 334: 240–242. Marzocchi, W., and Mulargia, F., 1990. Statistical analysis of the geomagnetic reversal sequences. Physics of the Earth and Planetary Interiors, 61: 149–164. Mazaud A., and Laj, C., 1991. The 15 Ma geomagnetic reversal periodicity: a quantitative test. Earth and the Planetary Science Letters, 107: 689–696. Mazaud A., Laj, C., de Seze, L., and Verosub, K.L., 1983. 15 Ma periodicity in the reversal frequency of geomagnetic reversals since 100 Ma. Nature, 304: 328–330. McFadden, P.L., 1984a. Statistical tools for the analysis of geomagnetic reversal sequences. Journal of Geophysical Research, 89: 3363–3372. McFadden, P.L., 1984b. 15 Ma periodicity in the frequency of geomagnetic reversals since 100 Ma. Nature, 311: 396. McFadden, P.L., 1987. Comment on “A periodicity of magnetic reversals?” Nature, 330: 27. McFadden, P.L., and Merrill, R.T., 1984. Lower mantle convection and geomagnetism. Journal of Geophysical Research, 89: 3354–3362. McFadden, P.L., and Merrill, R.T., 1986. Geodynamo energy source constraints from paleomagnetic data. Physics of the Earth and Planetary Interiors, 43: 22–33. McFadden, P.L., and Merrill, R.T., 1993. Inhibition and geomagnetic field reversals. Journal of Geophysical Research, 98: 6189–6199. McFadden, P.L., and Merrill, R.T., 1995. History of Earth’s magnetic field and possible connections to core-mantle boundary processes. Journal of Geophysical Research, 100: 317–316. McFadden, P.L., and Merrill, R.T., 1997. Asymmetry in the reversal rate before and after the Cretaceous normal polarity superchron. Earth and the Planetary Science Letters, 149: 43–47. McFadden, P.L., and Merrill, R.T., 2000. Evolution of the geomagnetic reversal rate since 160 Ma: Is the process continuous? Journal of Geophysical Research, 105: 28455–28460. Merrill, R.T., McElhinny, M.W., and McFadden, P.L., 1996. The Magnetic Field of the Earth: Paleomagnetism, the Core, and the Deep Mantle. San Diego, CA: Academic Press. Naidu, P.S., 1971. Statistical structure of geomagnetic field reversals. Journal of Geophysical Research, 76: 2649–2662. Negi, J.G., and Tiwari, R.K., 1983. Matching long-term periodicities of geomagnetic reversals and galactic motions of the solar system. Geophysical Research Letters, 10: 713–716. Parker, R.L., 1997. Coherence of signals from magnetometers on parallel paths. Journal of Geophysical Research, 102: 5111–5117.

GEOMAGNETIC REVERSALS, ARCHIVES

Phillips, J.D., 1977. Time-variation and asymmetry in the statistics of geomagnetic reversal sequences. Journal of Geophysical Research, 82: 835–843. Rampino, M.R., and Caldeira, K., 1993. Major episodes of geologic change: correlations, time structure and possible causes. Earth and the Planetary Science Letters, 114: 215–227. Raup, D.M., 1985. Magnetic reversals and mass extinctions. Nature, 314: 341–343. Stigler, S., 1987. A periodicity of magnetic reversals? Nature, 330: 26–27. Stothers, R.B., 1986. Periodicity of the Earth’s magnetic reversals. Nature, 332: 444–446.

Cross-references Core-Mantle Boundary Core-Mantle Boundary Topography, Implications for Dynamics Geodynamo, Numerical Simulations Geomagnetic Field, Asymmetries Geomagnetic Polarity Timescales Magnetic Field of Sun Reversals, Theory Superchrons, Changes in Reversal Frequency

GEOMAGNETIC REVERSALS, ARCHIVES It is now widely accepted that the Earth’s magnetic field is generated by electric currents in the iron-rich liquid outer core. A dynamo process converts the energy associated with fluid convection within the Earth’s core into magnetic energy. At the surface of the Earth, the field varies on timescales that range over more than 18 orders of magnitude, from less than a millisecond to more than 100 Ma. The most dramatic field variations are reversals. Almost exactly one century ago, Bernard Brunhes (1903, 1905) and his colleague David measured the magnetization of a lava flow, which was magnetized in the opposite direction to the present field. After investigating several possibilities, they convinced themselves that this resulted from a magnetic field with its magnetic north pole close to the south geographic pole. In other words the field would have been flipped in the opposite configuration to the present field with its south magnetic pole (in contrast to the current belief the pole referred as the north magnetic pole is actually a south pole which attracts the northern edge of the magnet) close to the north geographic pole. It took about 50 years before the existence of the geomagnetic reversals was established. Between 1925 and 1935 the discovery of new reversely magnetized rocks from different continents convinced the Japanese Matuyama (1926, 1929) and the French scientist Mercanton (1931) of their existence. However Louis Néel (1955) and his colleagues reported that reversely magnetized rocks could result from specific arrangements of the atoms lattices and the Japanese Uyeda (1958) observed this mechanism in a dacite from Mount Haruna. Further work demonstrated that this process was actually associated with specific chemical configurations and thus very rarely met in natural rocks. In the meantime, dating using new radiometric techniques indicated that rocks of the same age had the same polarity.

339

Hence, it was obvious that the presence of reversed magnetization was a worldwide phenomenon, not caused by self-reversing processes. The discovery of the geomagnetic reversals allowed also understanding why the basaltic seafloor magnetization was organized in alternances of positive and negative giant anomalies parallel to the ridge axis. This pattern actually reflects the succession of the two polarities of the geomagnetic field, which were initially recorded at the ridge axis and then pushed away by the spreading seafloor.

Geomagnetic polarity timescale and the duration of reversals The first reversals were dated using the ages derived from land sequences. This yielded the construction of the first geomagnetic polarity timescale (GPTS), which covered the past 5 Ma (Cox et al., 1963). Using the ages of these individual reversals, it was then possible to interpolate or to extrapolate on the basis of the width of the marine magnetic anomalies, which led to extend the timescale over the past 160 Ma (Figure G41). Beyond this period, we are faced to the absence of magnetic anomalies. During the past 20 years much activity has been devoted at studying long sequences of exposed sediments or lava flows in order to build up the polarity timescale for the older periods. The succession of polarity intervals during the past 160 Ma shows the existence of periods of high reversal frequency, which alternate with periods of low frequency. The mean reversal frequency is of the order of 1 reversal per million years and the maximum value does not seem to exceed 6 reversals per million years. During the period extending from 118 to 83 Ma, the field remained in a virtually uninterrupted normal state. These 35 Ma in the upper Cretaceous are known as the “Cretaceous normal superchron (CNS)” (between 120 and 83 Ma). Before the Kiaman reversed superchron stretches undisturbed for 50 Ma from the late Carboniferous to the middle Permian (between 310 and 260 Ma). The period between these two superchrons looks very much like the magnetic record that has followed the CNS. A third superchron, although apparently shorter in duration (20 ka), has been proposed from magnetostratigraphic studies (Pavlov and Gallet, 1998). Thus, in the present state of knowledge superchrons can be seen as a constant characteristic of the polarity timescale. It was frequently claimed that the slow decrease of reversal frequency before the Cretaceous superchron (McFadden and Merrill, 1995) would led to the superchron, thus resulting from a long-term influence of the mantle dynamics on core convection. Recent analysis (Hulot and Gallet, 2003) has shown that no long-term behavior over the 40 Ma preceding the Cretaceous superchron can be seen in the reversal rate that could be invoked as announcing its occurrence 120 Ma ago. Superchrons might correspond to times when the process responsible for geomagnetic reversals passed below a certain critical threshold, and reversals could start again when that threshold was exceeded again. Other analyses suggest that the superchrons indicate the existence of a nonreversing state of the geodynamo in contrast to its reversing state. Alternatively, changes between the two processes may depend on factors directly connected to the boundary conditions, such as the distribution of heterogeneities within the “D” layer (just above the core-mantle boundary) that affect the flow pattern within the core and therefore may not be related to the intrinsic time constants of the geodynamo.

Figure G41 Geomagnetic polarity timescale for the past 160 Ma. Black (white) bars indicate periods of normal (reverse) polarity. Note the existence of the long Cretaceous period without any reversal.

340

GEOMAGNETIC REVERSALS, ARCHIVES

The time span between two successive reversals appears to be random. Many statistical analyses were aimed at detecting periodicities or recurrent features, which would be hidden behind the succession of reversals. The succession of polarity intervals is closely approximated by a Poissonian distribution or at least by a gamma process if one assumes that short events have been missed in the reversal sequence. One can always consider that subsequent refinement of the reversal sequence can change this view. A critical aspect is the presence of field excursions which large deviations from the geocentric axial dipole during periods of very-low-field intensity. Excursions and reversals share common characteristics, which suggest that they could be treated as manifestations of similar processes within the core. In fact many authors proposed that excursions must be seen as aborted reversals whereas others regard them as intrinsic secular variation in presence of a weak dipolar field. In the first case, they should be treated at the same level as reversals in the statistics, which would provide a very different picture of the field. A critical point to answer these problems would be to determine whether excursions existed during the superchrons.

Dating reversals and estimating their duration The age of the successive reversals is critical for analyzing their succession. As described earlier most reversals were dated by extrapolating the ages of the recent reversals after combining the spreading rates. Very significant progress was accomplished during the past 20 years with the discovery that carbonated sediments revealed the succession of the orbital cyclicities of the Earth (23 ka). The ages of the Pliocene and Pleistocene reversals were refined by using this independent method (Shackleton et al., 1990; Tauxe et al., 1992, 1996; Channel and Kleiven, 2000) which relies on correlating climate proxies (such as oxygen isotope ratios, susceptibility or density variations . . . or simply counting the astronomical cycles recorded in continuous sedimentary sequences) with calculated variations of the Earth’s orbit. Using seafloor spreading rates for five plate pairs, Wilson (1993) has shown that the errors in the astronomical calibration are not greater than 0.02 Ma (which corresponds to a precessional cycle of the Earth’s orbit) and also that spreading rate can remain constant for several million years. A critical question that comes to mind is how long it takes for the field to reverse from one polarity to the other. Before dealing with this aspect, it is important to determine when a reversal starts and when it ends. There are many observations showing successions of large oscillations (Hartl and Tauxe, 1996; Dormy et al., 2000) preceding or following polarity changes, and it is not clear whether or not they should be incorporated in the reversal process. Such oscillations can be linked to enhanced secular variation in presence of low dipole field. Alternatively, they can be considered as successive attempts by the field to reverse. Important also is the definition of a transitional direction, which must exceed the normal range of secular variation (i.e., the range of the field variations during periods of polarity). Limits on the virtual geomagnetic pole (VGP) latitudes have been mostly used since it is rare that VGPs reach latitudes lower than 60 , although a strict definition should be restrained to positions lower than 45 (episodes of large secular variation can occasionally reach these latitudes). The sharp transition between magnetic anomalies of opposite polarities, the very narrow thickness of the intervals recording sedimentary columns as well as the small number of lava flows with directions being unambiguously identified as “intermediate” between the two polarities were rapidly convincing and strong evidences that reversals are short phenomena on a geological timescale. The data recorded from the best continuous and documented sedimentary sequences indicate that the jump between the two polarities is shorter than one precessional cycle (23 ka). If we refer to radiometric dating (e.g., K-Ar or Ar-Ar techniques) of lava flows, the problem is delicate because of the sporadic succession of the flows characterized by pulses of eruptions occurring over a short period and long intervals without any magmatic event. A direct consequence is that sequences of lavas

provide only very partial records of one or several phases of the reversal but not the entire process. Finally and even more critical is the fact that uncertainties on the ages cannot be ruled out as this is clearly illustrated by the large number of studies performed on the last reversal (Brunhes-Matuyama). The compilation of K-Ar and/or Ar-Ar dating (Quidelleur et al., 2003) for 23 volcanic records indicates an age of 789  8 ka (total error). The tuning of the d18O records from sedimentary sequences to orbital forcing models gives an age of 779  2 ka (Tauxe et al., 1996). Recently, Singer et al. (2005) mentioned that the astronomical determination is close to the age of the lavas from Maui (Hawaii) (776  2 ka), and therefore that the other volcanic records are related to the onset of the transitional process. Consequently they deduce that a field reversal would require a significant period. One can oppose that there is no reason to consider that the onset of the reversal was initiated at the same time everywhere but above all if we consider the total uncertainties on the ages, there is no significant difference. Actually this uncertainty leaves doubts as to the possibility of constraining the duration of the transition with precision. There is also no reason to consider that all reversals have the same duration. Most studies converge to estimates between 5 and 10 ka but durations as short as 1 ka or as long as 20–30 ka have been proposed also. Clement (2004) recently analyzed the four most recent reversals recorded in sediments with various deposition rates and found a mean average duration of 7 ka for the directional changes (Figure G42). An interesting characteristic is that the mean duration seems to vary with latitude, as expected from simple geometrical models in which the nondipole fields are allowed to persist while the axial dipole decays through zero and then builds in the opposite direction.

The reversing field Recorded by sediments There has been a great deal of speculations concerning the processes governing field reversals. In order to decipher the mechanisms it was important to focus on the morphology of the field during the transition from one polarity state to the other. The first major step was to determine whether this “transitional” field would keep its dipolar dominant character. To achieve this goal, the paleomagnetists took advantage of

Figure G42 (a) Geographical distribution of Matuyama-Brunhes transitional VGPs derived from a selection of sedimentary records of the last reversal (from Clement, 1991). (b) Longitudinal distribution of transitional VGPs plotted as the number of transitional VGPs in a sliding 30 wide longitude window. Note the presence of two peaks over the American and Asian continents.

GEOMAGNETIC REVERSALS, ARCHIVES

the concept of the VGP, which defines the position of the pole assuming that the field is dominantly dipolar. The idea was very simple. If the field remained dipolar, then any vector at the surface of the Earth would point toward the same pole following the rotation of the dipole with its north (south) pole passing to the (south), while in the opposite situation the poles would be different at any site. Using the first records from sediments, it was rapidly shown that the field was not dipolar during reversals (Dagley and Lawley, 1974; Hillhouse and Cox, 1976). In the meantime the first records of paleointensity established that the field intensity was systematically very low during the transitions. This decrease in dipole intensity is necessary before directions depart significantly from dipolar ones (Mary and Courtillot, 1993) and therefore consistent with the concept of a nondipolar field. The nature of this nondipolar field is one of the major questions to elucidate. Two objectives were pursued in order to provide some answers to this question—the first one relied on the acquisition of multiple records of the same transition. It was proposed that because the Earth’s rotation keeps a major role in field regeneration the transitions would be dominated by axisymmetrical components (quadropolar or octupolar). This suggestion could be tested at least at the first order by referring again to the useful concept of the VGPs. Indeed to satisfy the axial symmetry, the VGPs always follow the great circle passing through the observation site (or its antipode), which implies to study the same reversals at different sites. Records from sediments were appropriate because the transitions can be identified without ambiguity and thus correlated from widely separated sites. In the meantime the development of cryogenic magnetometers was very helpful as they provided the possibility of measuring weakly magnetized sediments. The results established that the VGP trajectories were effectively constrained in longitude but in many cases 90 away from the site meridian rather than centered above it. The second objective was to investigate whether successive reversals were characterized by some recurrent or persistent features. This required studying sequences of reversals at the same site. Again long sequences of marine sediments were appropriate for this kind of study as well as the use of VGP paths, despite the dominance of nondipolar components. The most appealing observation emerged from a selection of sedimentary records (Clement, 1991; Tric et al., 1991), which were all showing VGP paths within two preferred longitudinal bands, over the Americas and eastern Asia. Going one step further Laj et al. (1992) noted that these areas coincide with the cold circum Pacific regions in the lower mantle (outlined by seismic tomography), thereby suggesting that density or temperature conditions in the lower mantle could control the geometry of the reversing field. However, this observation was controversial because it relied on a selection of records. Another intriguing characteristic (Valet et al., 1992; McFadden et al., 1993) was the fact that these VGP paths were also found 90 away from the longitude of their sites despite their relatively wide geographic distribution (Figure G43), which could suggest some artifacts in the recording processes. Several studies effectively questioned the fidelity of sediments as recorders of the field variations, particularly during periods of low field intensity. Many factors (compaction, alignment of the elongated particles) reduce the inclination of the magnetization, a process that moves the VGPs paths far away from the longitude of the observation sites (Rochette, 1990; Langereis et al., 1992; Quidelleur and Valet, 1994, 1995; Barton and McFadden, 1996,). There is also some indication that for some sediments the magnetic torque generated by a weak field is too low to provide accurate orientation of the magnetic grains, leaving in this case a prominent role to the hydrodynamic forces. These problems suggest that sediments could not be as appropriate as it was originally thought to study reversals.

Recorded by volcanic lava flows One must thus turn toward volcanic records keeping in mind that in this case we are faced to the very discontinuous character of the

341

Figure G43 Distance between the mean longitudes of the virtual geomagnetic poles (VGPs) and their site meridian. Note that most lie 90 away from their site longitude.

eruption rates. Following the presentation of preferred longitudinal bands, Prévot and Camps (1993) compiled all volcanic records with VGPs latitudes lower than 60 , considering only one position for poles that were identical or very close to each other. They observed that no preferred longitudinal band emerged from this database. Love (1998) questioned this interpretation, arguing that similar directions from successive flows should not be averaged, because they do not necessarily result from a very rapid succession of eruptions. Instead he treated each individual flow as a single time event (implying no correlation between successive flows (Figure G44), or identically that the duration between flows was larger than the typical correlation times of secular variation). Because VGPs obtained from volcanics can reach latitudes as low as 45–50 during episodes of large secular changes (and of course excursions), it is also important to restrain the analysis to the most transitional directions, i.e., those with VGP latitudes less than 45 . Unfortunately in this case the number of points becomes too small to perform any robust analysis. This illustrates the difficulties of finding detailed records of reversals but also implies and confirms that indeed transitions occur very rapidly. Using another selection of records, Hoffman (1991, 1992, 1996) pointed out the existence of clusters of VGPs in the vicinity of South America and above western Australia. Because of the apparent longevity of these directions they were interpreted as indicating the existence of a persistent inclined dipolar field configuration during the reversal process. This is an attractive suggestion, which would establish some link between the sedimentary and the volcanic records but limited to a selection of data. It is striking that these interpretations depend on the chronology of the lava flows. Volcanism is mostly governed by short periods of intense eruptions alternating with quiet intervals. It is usually admitted that the active periods can be very short with respect to the intervals of quiescence. An indirect “magnetostratigraphic” indication has been given by three parallel sections of Hawaii (Herrero-Bervera and Valet, 1999, 2005), which are not distant, by more than a few kilometers. They all recorded the same reversal but do not show the same successions of transitional directions. Clusters of similar directions can be present in one section without being recorded 2 km away. Similarly

342

GEOMAGNETIC REVERSALS, ARCHIVES

apparent rapid changes can be observed in detail in one case and be absent 2 km away. In the first case, the field remained in the same position during a period short enough for not being recorded in the

Figure G44 (a) Map of gray-scale histogram of the VGPs from volcanic lava flows with latitudes that are low enough for being considered as transitional (from Love et al., 1998). (b) Histogram showing their longitudinal distribution. All lava flows were considered and thus not necessarily incorporated within a sequence of overlying flows. The similar directions that could be linked to a phase of rapid volcanic eruptions were considered as a result of a random process and thus were all taken in consideration. The database is strongly dominated by Icelandic lava flows.

nearby section. In the second case, the field variations occurred either over a short time interval or the field moved very rapidly and was thus recorded only at a single location. Thus we cannot rule out that the apparent concentrations of VGPs close to South America and western Australia can be purely coincidental so that it cannot be considered yet as certain that long-lived transitional states represent an actual characteristic of the reversing field. Another approach is to rely exclusively on volcanic records with well-defined pre- and posttransitional directions and a sufficient number of intermediate directions. These records are more significant in terms of transitional field characteristics because they are not associated with uncertainties about the origin of the directions and their stratigraphic relation. The distribution of poles extracted from this limited database does not display any preferred location, nor does it show any evidence for systematics in the reversal process (Figure G45). Note also the presence of clusters at various longitudes and latitudes. Despite the difficulties inherent to the interpretation of sedimentary and volcanic transitional directions, the two kinds of data may share some common characteristics. We already noted a possible link between the volcanic clusters and the preferred VGP paths in sediments but outlined some difficulties in reconciling these two aspects. Another important issue is that the two kinds of records are associated with very different timing in the acquisition of their magnetization, the almost instantaneous cooling of the lavas being almost opposite to the slow processes governing the lock-in of magnetization in sediments. It is thus more justified to attempt a comparison by considering sediments with very high deposition rates in order to reach a better resolution (for magnetization acquisition) thus closer to the volcanic characteristics. A few detailed sedimentary records with deposition rates exceeding 5 cm per 1 ka have been published (Valet and Laj, 1984; Clement and Kent, 1991; Channell and Lehman, 1997). A dominant and common characteristic is that they display a complex structure with large directional variations preceding and/or following the transition which reminds the features seen in the volcanics (Mankinen et al., 1985; Chauvin et al., 1990; Herrero-Bervera and Valet, 1999). These large loops share similarities with the secular variation of the present field. This observation reinforces the simplest model initially suggested by Dagley and Lawley (1974) of a rather complex transitional field which would be dominated by nondipole components following the large drop of the dipole field (Valet et al., 1989; Courtillot et al., 1992).

Figure G45 Positions of the VGPs derived from the most detailed volcanic records of reversals published so far. In contrast with Figure G44 no Icelandic sequence of superimposed flows met the selection criteria. Note also the large number of clusters (surrounded by circles) at various locations of the globe.

GEOMAGNETIC REVERSALS, ARCHIVES

Fast impulses during reversals? The existence of fast impulses during the 16 Ma old reversal recorded at the Steens Mountain in Oregon (Mankinen et al., 1985) was suggested from a puzzling progressive evolution of the paleomagnetic directions in the interiors of two transitional lava flows (Coe and Prévot, 1989; Camps et al., 1999). Each lava unit recorded a complete sequence of directions going all the way from that of the underlying flow to the direction of the overlying flow. In the absence of any clear evidence for anomalous rock magnetic properties, these features have been interpreted in terms of very fast geomagnetic changes, which would have reached 10 and 1000 nT per day. For comparison values typical of the present-day secular variation of the field (of internal origin) are of the order of 0.1 and 50 nT per year, i.e., some 104 times slower. In this specific case the timing of these fast changes can be constrained by estimates of the cooling times of individual flows. However, such rapid changes do not seem to be compatible with accepted values of mantle conductivity (Ultré et al., 1995). As a consequence this interpretation of the magnetization generated exciting controversy. Additional detailed investigations have been conducted in order to see whether this situation could not have arisen because of remagnetization of the flows. Remagnetizations of lava flows, yielding complex or unusual directions, have been detected at several locations where reversals have been recorded. A first interesting example was given by Hoffman (1984) from Oligocene basaltic rocks. Valet et al. (1998) observed the coexistence of both polarities (with similar characteristics as for the directions recorded at Steens Mountain) within flows marking the last reversal boundary (0.78 Ma) at the Canary Islands, and also in a lava flow associated with the onset of the upper Réunion subchron (2.13 Ma) in Ethiopia. In these cases, a purely geomagnetic interpretation would imply that a full reversal took place in only a few days. Similarly to the Steens Mountain, there is no striking difference between the rock magnetic properties of these units and the rest of the sequence, but a scenario involving thermochemical remagnetization is not incompatible with the results. Thermochemical magnetic overprinting can be particularly serious when it affects a flow emplaced at a time of very low field intensity, sandwiched between flows emplaced at a time of full (stable polarity) intensity. Recent investigation of additional flows at Steens did not shed more light on this problem but rather casts doubts on a geomagnetic interpretation of the paleomagnetic directions (Camps et al., 1999). Therefore, the existence of very large and rapid changes that have been documented from a single site by a unique team remains controversial. In the meantime no alternative explanation has been completely accepted yet.

343

cosmogenic isotopic records (Frank et al., 1997; Baumgartner et al., 1998; Carcaillet et al., 2003, 2004; Thouveny et al., 2004). Apart many other interesting observations a dominant feature is the existence of a large drop of intensity about every 100 ka, which coincide with excursions reported from various sequences in the world. The most recent curve of relative paleointensity was extended to the past 2 Ma (Valet et al., 2005) and is in good agreement with the absolute dipole moments derived from volcanic lavas, which were used for calibration. It shows that the time-averaged field was higher during periods without reversals but the amplitude of the short-term oscillations remained the same. As a consequence, few intervals of very low intensity and thus less instability are expected during periods with a strong average dipole moment, whereas more excursions and reversals are produced during periods of weak field intensity. Prior to reversals, the axial dipole decays during 60 to 80 ka, but rebuilds itself in the opposite direction in a few thousand years at most (Figure G46). The most complete volcanic records confirm that recovery following a transition is short and culminates to very high values. The detailed volcanic records including determinations of absolute paleointensity provide support for such an asymmetry. Strong posttransitional field values have been reported in a Pliocene reversal recorded at Kauai (Bogue and Paul, 1993) and in the upper Jaramillo (0.99 Ma) subchron recorded from Tahiti (Chauvin et al., 1990) as well as for the lower Mammoth reversal (3.33 Ma) from Hawaii (Herrero-Bervera and Valet, 2005). The same characteristics emerge also from the 60 Ma oldest record obtained so far in Greenland (Riisager and Abrahamsen, 2000), from the 15 Ma old Steens Mountain reversal (Prévot et al., 1985) and from the last reversal (0.78 Ma) recorded from La Palma in the Canary Islands (Valet et al., 1999).

Conclusion and perspectives Several hundreds reversals have been documented from geological records and it is not unlikely, yet not demonstrated, that reversals always accompanied the existence of the geomagnetic field. Their internal origin neither makes any doubt. Many numerical models for the Earth’s dynamo were produced during the last decade, including three-dimensional self-consistent dynamos that exhibit magnetic reversals. However, mostly for computation difficulties the parameters used remain far away from the Earth. This demonstrates the importance of

Field intensity variations across reversals Variations in field intensity accompanying the reversals provide important and unique information concerning the transition itself but also the evolution of the dipole prior and after the reversals. We mentioned earlier that a significant decrease of the dipole was reported since the earliest studies. It is well established, based on all sedimentary as well as volcanic records that field intensity drops significantly and in most cases these changes last longer than directional changes (see e.g., Lin et al., 1994; Merrill and Mc Fadden, 1999). Note that similar drops have been mentioned in all records of excursions with one exception (Leonhardt et al., 2000). Initially, the records were restrained to the transitional interval or to a few thousand years preceding and following the reversal. Several long records of relative paleointensity have now been obtained using sequences of deep-sea sediments, which make possible to observe the evolution of the field over a long period. These independent records from sediment cores in different areas of the world can be stacked together to extract the evolution of the geomagnetic dipole moment (Guyodo and Valet, 1996, 1999; Laj et al., 2000). There is a remarkable consistency between the first stacks published for the past 200 and 800 ka and a similar approach performed from field measurements immediately above bottom seafloor magnetic anomalies (Gee et al., 2000). There is also a good agreement with the

Figure G46 Field intensity variations across the five reversals occurring during the past 2 Ma. In this figure we superimposed the changes in dipole moment during the 80 and 20 ka time intervals, respectively preceding and following each reversal. Note the 60–80 ka long decrease preceding the reversals, and the rapid recovery following the transitions.

344

GEOMAGNETIC REVERSALS, ARCHIVES

accumulating data. For about 30 years the paleomagnetists attempted to acquire as many detailed records as possible using the magnetic memory of sediments and lava flows. One of the first objectives was to determine whether the field keeps its dipolar character when reversing. The complexity of the directional changes shown by the detailed records and the large decrease of the field intensity indicates that the dipolar component strongly decreases by at least 80%, if not vanishing completely. One of the major constraints is the rapidity of the reversal process. There is no clear estimate for reversal duration, which may vary. Indeed it seems easier to isolate transitional directions for some reversals than for some others. There is no estimate for a lower limit of the duration of a transition which could be as short as a few hundreds years, if not less. It is reasonable to consider that the upper limit does not exceed 20 ka. After many years the suitability of sedimentary records (which have the advantage of preserving continuous information on field evolution) has been heavily questioned because their direction of magnetization can be affected by other factors (climate, alignment of the magnetic grains, postdepositional reorientations), particularly in presence of low field intensity. It is thus wise to turn also toward volcanic records despite their intrinsic limits in terms of resolution and dating. If we refer to the existing volcanic database, different views are presently defended regarding the field configuration during the short transitional period. Some claim that there is a dominance of the pole positions within preferred longitudinal bands, particularly within the American and Australian sectors, while others oppose rock magnetic artifacts and defend that the distribution of the transitional directions is typical of a nondipole field that would be similar to the present one. These two views have different implications and impose different constrains. The first one assumes that the lower mantle exerts some control on the reversal processes while the alternative interpretation defends that the transitional field would result from intrinsic processes linked to the dynamic of the core fluid. Another aspect is the existence of precursory events. The complexity of the field evolution prior to reversals depicts some “excursions” of the directions that are interpreted as precursory events. This observation can be linked to the long-term decay of the dipole component prior to the reversal, which is responsible for the complexity of the directional changes observed at the surface. Under this scenario the “precursory” excursions simply reflect the dominance of the nondipole part of the field, which will then prevail during the transition. Finally, a fast and strong recovery takes place immediately after the transition. The amplitude of this restoration phase is certainly critical as recent observations suggest that the dipole field strength could be a dominant factor controlling the frequency of reversals. Jean-Pierre Valet and Emilio Herrero-Bervera

Bibliography Barton, C.E., and McFadden, P.L., 1996. Inclination shallowing and preferred transitional VGP paths. Earth and the Planetary Science Letters, 140: 147–157. Baumgartner, S., Beer, J., Masarik, J., Wagner, G., Meynadier, L., and Synal, H.-A., 1998. Geomagnetic modulation of the 36Cl flux in the Grip ice core, Greenland. Science, 279: 1330–1332. Bogue, S.W., and Paul, H.A., 1993. Distinctive field behaviour following geomagnetic reversals. Geophysical Research Letters, 20: 2399–2402. Brunhes, B., 1903. Sur la direction de l’aimantation permanente dans diversees roches volcaniques. Comptes Rendus De l Academie Des Sciences Paris, 137: 975–977. Brunhes, B., 1905. Sur la direction de l’aimantation dans une argile de Pontfarein. Comptes Rendus De l Academie Des Sciences Paris, 141: 567–568. Camps, P., Coe, R.S., and Prévot, M., 1999. Transitional geomagnetic impulses hypothesis: geomagnetic fact or rock magnetic artifact? Journal of Geophysical Research, 104(B8): 17747–17758.

Carcaillet, J., Thouveny, N., and Bourlès, D., 2003. Geomagnetic moment instability between 0.6 and 1.3 Ma from cosmonuclide evidence. Geophysical Research Letters, 30(15): 1792. Carcaillet, J., Bourlès, D.L., Thouveny, N., and Arnold, M., 2004. A high resolution authigenic 10Be/9Be record of geomagnetic moment variations over the last 300 ka from sedimentary cores of the Portuguese margin. Earth and Planetary Science Letters, 219: 397–412. Channell, J.E., and Kleiven, H.F., 2000. Geomagnetic paleointensity and astronological ages for the Matuyama-Brunhes boundary and the boundaries of the Jaramillo subchron: paleomagnetic and oxygen isotope records from ODP site 983. Philosophical Transactions of the Royal Society of London A, 358: 1027–1047. Channell, J.E.T., and Lehman, B., 1997. The last two geomagnetic polarity reversals recorded in high deposition-rate sediments drifts. Nature, 389: 712–715. Chauvin, A., Roperch, P., and Duncan, R.A., 1990. Records of geomagnetic reversals from volcanic islands of French Polynesia. Journal of Geophysical Research, 95: 2727–2752. Clement, B.M., 1991. Geographical distribution of transitional VGPs: evidence for non zonal equatorial symmetry during the Matuyama-Brunhes geomagnetic reversal. Earth and Planetary Science Letters, 104: 48–58. Clement, B.M., 2004. Dependence of the duration of geomagnetic polarity reversal. Nature, 428: 637–640. Coe, R.S., and Prévot, M., 1989. Evidence suggesting extremely rapid field variation during a geomagnetic reversal. Earth and Planetary Science Letters, 92: 292–298. Courtillot, V., Valet, J.-P., Hulot, G., and Le Mouël, J.-L., 1992. The Earth’s magnetic field: which geometry? EOS Transactions, American Geophysical Union, 73(32): 337. Cox, A., Doell, D.R., and Dalrymple, G.B., 1963. Geomagnetic polarity epochs and Pleistocene geochronometry. Nature, 198: 1049. Dagley, P., and Lawley, E., 1974. Paleomagnetic evidence for the transitional behaviour of the geomagnetic field. Geophysical Journal of the Royal Astronomical Society, 36: 577–598. Dormy, E., Valet, J.-P., and Courtillot, V., 2000. Observational constraints and numerical dynamos. Geochemistry, Geophysics and Geosystems, 1: Vol1, N 10, doi:10,1029/2000GCD00062. Frank, M., 1997. A 200 ka record of cosmogenic radionuclide production rate and geomagnetic field intensity from 10Be in globally stacked deep-sea sediments. Earth and Planetary Science Letters, 149: 121–129. Gee, J., Cande, S.C., Hildebrand, J.A., Donnelly, J.A., and Parker, R.L., 2000. Geomagnetic intensity variations over the past 780 ka obtained from near-seafloor magnetic anomalies. Nature, 408: 827–832. Guyodo, Y., and Valet, J.P., 1996. Relative variations in geomagnetic intensity from sedimentary records: the past 200 thousand years. Earth and the Planetary Science Letters, 143: 23–26. Guyodo, Y., and Valet, J.P., 1999. Global changes in intensity of the Earth’s magnetic field during the past 800 ka. Nature, 399: 249–252. Hartl, P., and Tauxe, L., 1996. A precursor to the Matuyama/Brunhes transition-field instability as recorded in pelagic sediments. Earth and the Planetary Science Letters, 138: 121–135. Herrero-Bervera, E., and Valet, J.-P., 1999. Paleosecular variation during sequential geomagnetic reversals from Hawaii. Earth and the Planetary Science Letters, 171: 139–148. Herrero-Berrvera, E., and Valet, J.P., 2005. Absolute paleointensity from the Waianae volcanics (Oahu, Hawaii) between the GilbertGauss and the upper Mammoth reversals. Earth and the Planetary Science Letters, 234: 279–296. Hillhouse, J., and Cox, A., 1976. Brunhes-Matuyama polarity transition. Earth and the Planetary Science Letters, 29: 51–64. Hoffman, K.A., 1984. Late acquisition of “primary” remanence in some fresh basalts: a case of spurious paleomagnetic results. Geophysical Research Letters, 11: 681–684.

GEOMAGNETIC REVERSALS, ARCHIVES

Hoffman, K.A., 1991. Long-lived transitional states of the geomagnetic field and the two dynamo families. Nature, 354: 273–277. Hoffman, K.A., 1992. Dipolar reversal states of the geomagnetic field and core-mantle dynamics. Nature, 359: 789–794. Hoffman, K.A., 1996. Transitional paleomagnetic field behavior: preferred paths or patches? Surveys in Geophysics, 17: 207–211. Hulot, G., and Gallet, Y., 2003. Do superchrons occur without any paleomagnetic warming? Earth and the Planetary Science Letters, 210: 191–201. Laj, C., Mazaud, A., Weeks, R., Fuller, M., and Herrero-Bervera, E., 1992. Geomagnetic reversal paths. Nature, 351: 447. Laj, C., Kissel, C., Mazaud, M., Channell, J.E.T., and Beer, J., 2000. North Atlantic paleointensity stack since 75 ka (NAPIS 75) and the duration of the Laschamp event. Philosophical Transactions of the Royal Society of London A, 358: 1009–1025. Langereis, C.G., van Hoof, A.A.M., and Rochette, P., 1992. Longitudinal confinement of geomagnetic reversal paths. Sedimentary artefact or true field behavior? Nature, 358: 226–230. Love, J.J., 1998. Paleomagnetic volcanic data and geometric regularity of reversals and excursions. Journal of Geophysical Research, 103(B6): 12435–12452. Leonhardt, R., Hufenbecher, F., Heider, F., and Soffel, H.C., 2000. High absolute paleointensity during a mid Miocene excursion of the Earth's magnetic field. Earth and the Planetary Science Letters, 184(1): 141–154. Lin, J.L., Verosub, K.L., and Roberts, P.A., 1994. Decay of the virtual dipole moment during polarity transitions and geomagnetic excursions. Geophysical Research Letters, 21: 525–528. Mankinen, E.A., Prévot, M., Grommé, C.S., and Coe, R.S., 1985. The Steens Mountain (Oregon) geomagnetic polarity transition. 1. Directional history, duration of episodes and rock magnetism. Journal of Geophysical Research, 90: 10393–10416. Mary, C., and Courtillot, V., 1993. A three-dimensional representation of geomagnetic field reversal records. Journal of Geophysical Research, 98: 22461–22475. Matuyama, M., 1929. On the direction of magnetisation of basalt in Japan, Tyosen and Manchuria. Proceedings of the Imperial Academy of Japan, 5: 203. McFadden, P., and Merrill, R., 1995. Fundamental transitions in the geodynamo as suggested by paleomagnetic data. Physics of the Earth and Planetary Interiors, 91: 253–260. McFadden, P.L., Barton, C.E., and Merrill, R.T., 1993. Do virtual geomagnetic poles follow preferred paths during geomagnetic reversals? Nature, 361: 342–344. Mercanton, P.L., 1926. Inversion de l’inclinaison magnétique aux âges géologiques. Nouvelles constattations. Comptes Rendus De l Academie Des Sciences. Paris, 192: 978–980. Merrill, R.T., and McFadden, P.L., 1999. Geomagnetic polarity transitions. Reviews of Geophysics, 37: 201–226. Néel, L., 1955. Some theoretical aspects of rock magnetism. Advances in Physics, 4: 191. Pavlov, V., and Gallet, Y., 1998. Upper Cambrian to Middle Ordovician magnetostratigraphy from the Kulumbe river section (northwestern Siberia). Physics of the Earth and Planetary Interiors, 108(1): 49–59. Prévot, M., and Camps, P., 1993. Absence of longitudinal confinement of poles in volcanic records of geomagnetic reversals. Nature, 366: 53–57. Prévot, M., Mankinen, E.A., Coe, R.S., and Grommé, C.S., 1985. The Steens Mountain (Oregon) geomagnetic polarity transition. Field intensity variations and discussion of reversal models. Journal of Geophysical Research, 90: 10417–10448. Quidelleur, X., and Valet, J.-P., 1994. Paleomagnetic records of excursions and reversals: possible biases caused by magnetization artefacts? Physics of the Earth and Planetary Interiors, 82: 27–48.

345

Quidelleur, X., Valet, J.-P., Le Goff, M., and Bouldoire, X., 1995. Field dependence on magnetization of laboratory redeposited deep-sea sediments: first results. Earth and the Planetary Science Letters, 133: 311–325. Quidelleur, X., Carlut, J., Soler, V., Valet, J.-P., and Gillot, P.Y., 2003. The age and duration of the Matuyama-Brunhes transition from new K-Ar data from LaPalma (Canary Islands) and revisited 40 Ar/39Ar ages. Earth and the Planetary Science Letters, 208: 149–163. Riisager, P., and Abrahamsen, N., 2000. Paleointensity of west Greenland Paleocene basalts: asymmetric intensity around the C57nC26r transition. Physics of the Earth and Planetary Interiors, 118: 53–64. Rochette, P., 1990. Rationale of geomagnetic reversal versus remanence recording process in rocks: a critical review. Earth and the Planetary Science Letters, 98: 33–39. Shackleton, N.J., Berger, A., and Peltier, W.R., 1990. An alternative astronomical calibration of the lower Pleistocene timescale based on ODP Site 677. Transactions of the Royal Society of Edinburg Earth Science, 81: 251–261. Singer, B.S., Hoffman, K.A., Coe, R.S., Brown, L.L., Jicha, B.R., Pringle, M.S., and Chauvin, A., 2005. Structural and temporal requirements for geomagnetic reversal deduced from lava flows. Nature, 434: 633–636. Tauxe, L., Deino, A.D., Behrensmeyer, A.K., and Potts, R., 1992. Pinning down the Brunhes/Matuyama and Upper Jaramillo boundaries: a reconciliation of orbital and isotopic timescales. Earth and the Planetary Science Letters, 109: 561–572. Tauxe, L., Herbert, T., Shackleton, N.J., and Kok, Y.S., 1996. Astronomical calibration of the Matuyama Brunhes Boundary: consequences for magnetic remanence acquisition in marine carbonates and the Asian loess sequences. Earth and the Planetary Science Letters, 140: 133–146. Thouveny, N., Carcaillet, J., Moreno, E., Leduc, G., and Nérini, D., 2004. Geomagnetic moment variation and paleomagnetic excursions since 400 ka BP: a stacked record of sedimentary sequences of the Portuguese margin. Earth and Planetary Science Letters, 219: 377–396. Tric, E., Laj, C., Jehanno, C., Valet, J.-P., Mazaud, A., Kissel, C., and Iaccarino, S., 1991. A detailed record of the upper Olduvai polarity transition from high sedimentation rate marine deposits of the Po valley. Physics of the Earth and Planetary Interiors, 65: 319–336. Ultré-Guérard, P., and Achache, J., 1995. Core flow instabilities and geomagnetic storms during reversals: the Steens mountain impulsive field variation revisited. Earth and the Planetary Science Letters, 139: 91–99. Uyeda, S., 1958. Thermoremanent magnetism as a medium of paleomagnetism with special references to reverse thermoremanent magnetism. Japanese Journal of Geophysics, 2: 1–123. Valet, J.-P., and Laj, C., 1984. Invariant and changing transitional field configurations in a sequence of geomagnetic reversals. Nature, 311(5986): 552–555. Valet, J.-P., Tauxe, L., and Clement, B.M., 1989. Equatorial and midlatitudes records of the last geomagnetic reversal from the Atlantic Ocean. Earth and the Planetary Science Letters, 94: 371–384. Valet, J.P., Tucholka, P., Courtillot, V., and Meynadier, L., 1992. Paleomagnetic constraints on the geometry of the geomagnetic field during reversals. Nature, 356: 400–407. Valet, J.-P., Kidane, T., Soler, V., Brassart, J., Courtillot, V., and Meynadier, L., 1998. Remagnetization in lava flows recording pretransitional directions. Journal of Geophysical Research, 103(B5): 9755–9775. Valet, J.P., Meynadier, L., and Guyodo, Y., 2005. Geomagnetic field strength and reversal rate over the past 2 Million years. Nature, 435: 802–805.

346

GEOMAGNETIC SECULAR VARIATION

Wilson, D., 1993. Confirmation of the astronomical calibration of the magnetic polarity timescale from seafloor spreading rates. Nature, 364: 788–790.

Cross-references Core motions Geomagnetic dipole field Geomagnetic excursion Geomagnetic hazards Magnetization, chemical remanent (CRM) Magnetization, depositional remanent (DRM) Magnetization, natural remanent (NRM) Non-dipole field Paleomagnetic secular variation Paleomagnetism, deep-sea sediments Polarity transitions: Radioisotopic Dating

seven jerks have been reported (1912, 1925, 1969, 1978, 1983, 1991, and 1999), some of them of global extent. The 1969 event (first described by Courtillot et al., 1978) was widely investigated; on the basis of observatory records. Courtillot et al. (1978) and Malin and Hodder (1982) showed its global extent, although it was not evident in all field components. This fact and the coincidental occurrence of jerks and sunspot maxima set off a lively discussion between Alldredge and McLeod in the 1980s (Alldredge, 1984; McLeod, 1985; Backus et al., 1987) about the causative processes of jerks. The general understanding is that jerks are of internal origin, mainly because of two reasons: First, the potential of the solar cycle has approximately the form of zonal spherical harmonics, therefore any contribution to the East component would be small. Jerks are most clearly visible in the East component of European observatories. The second argument is based on a comparison of the strength of the solar maxima adjacent to the 1969 jerk, 1958 and 1980. At both epochs the solar maxima were more pronounced than in 1969, so we would expect two jerks at those epochs, but nothing obvious happened in 1958 and the jerks around 1978 and 1983 do not fit well to the solar maximum in 1980 (see also entry on Geomagnetic jerks).

GEOMAGNETIC SECULAR VARIATION Determination of secular variation models Introduction The term secular comes from the Latin Seculum which means the duration of the influence of a powerful family becoming steadily less. The Romans accounted 100 years for that, why it also was synonymous for century. In context of geomagnetism it implies a long-term variation of the Earth’s magnetic field. The observed temporal variations of the Earth’s magnetic field cover timescales from milliseconds to a few million years and originate in two distinct (external and internal) source regions with respect to the Earth’s surface. With this respect the fluctuations of the external field ranges from milliseconds to a few decades, where the longer periods are related to variations of the solar magnetic field, e.g., the turnover of solar magnetic field (about 22 years). Changes of the internal field are of the order of a few years to millions of years. This variation results from the effect of magnetic induction in the fluid outer core and from effects of magnetic diffusion in the core and the mantle. Here, we distinguish geomagnetic secular variation and paleomagnetic secular variation, where the latter includes temporal changes longer than several hundreds of years, such as reversals (see Reversals, theory). The overlap of periods of internal and external sources in the range of a few years to decades can be separated in internal and external contributions applying spherical harmonic analysis (see Internal external field separation). The geomagnetic secular variation was first noticed by Gunter and “Gellibrand” in 1635, who collected measurements of magnetic declination made at Limehouse near London between 1580 and 1634. This component gradually changed over the period of 350 years from 11 E to 24 W in 1820, before turning eastward again. Figure G47 shows the declination measured in London and Danzig (Gdansk) for about 350 years. Whereas the main field is dominated by its dipolar nature, the secular variation is clearly nondipolar, which is reflected in regions of different magnitudes of secular variation. For example, in the pacific region the secular variation appears to be crestless. However, the (geomagnetic) secular variation has been observed to be the feature of the main field and not of the local field. One other prominent feature of the secular variation is the tendency of isoporic foci (areas of maximum secular variation) to drift westward. Analyzes by Vestine et al. (1947) and Bloxham et al. (1989) found an averaged drift rate of about 0.3 y
1 . In addition to the slowly varying secular variation event-like features appear, the so-called geomagnetic jerks. Such jerks show up as a change of sign in the slope of the secular variation, a discontinuity in the second time derivative of the field, most clearly seen in the east (Y ) component of the geomagnetic field. For the last 100 years at least

Before considering the generation of secular variation and its link to other observables of processes in the Earth’s core, we shall first describe a modeling approach to map the secular variation at the source region, the core-mantle boundary (CMB). In principle, the magnetic field and its variation can be separated in parts due to external and internal sources by spherical harmonic analysis (Gauss, 1839). The geomagnetic field is then represented by the so called Gauss coefficients (see entries on Internal external field separation and Main field modeling). The variations of the internal field can be modeled by expanding the internal Gauss coefficients in a Taylor series in time about some epoch, te, e.g., glm ðtÞ ¼ glm ðte Þ þ g_ lm ðte Þðt
te Þ þ g€lm

ðt
te Þ2 þ  2!

(Eq. 1)

where the first time derivative g_ lm is the secular variation and the second time derivative g€lm the secular acceleration. The determinations usually have been truncated after some derivatives. The GSFC(12/ 66) model (Cain et al., 1967) included second time derivatives and the GSFC(9/80) model of the magnetic field between 1960 and 1980 by Langel et al. (1982) included third time derivatives. These

Figure G47 The declination at London and Danzig (Gdansk).

GEOMAGNETIC SECULAR VARIATION

representations are only adequate to represent the temporal variation over short periods, e.g., the GSFC(9/80) is only sufficient for the period 1955–1980. Outside this period the misfit increases drastically. An alternative attempt to model the secular variation was put forward by Langel et al. (1986). The methodology is to model the secular variation directly from first time derivatives of observatory annual means. For example, using the differences between annual means for different years, i.e., X_ ¼ DX =Dt. Then performing with those X_ ; Y_ ; Z_ spherical harmonic analysis. Langel et al. (1986) achieved a continuous representation of the secular variation model for 1903 to 1982 by fitting each coefficient with cubic B-splines. In a series of publications Bloxham (1987); Bloxham and Jackson (1989); Bloxham and Jackson (1992) developed a method which gives the most favorable description of the secular variation. This method bases on the simultaneous construction of a time-dependent model of the secular variation and main field. Their description of the timedependent geomagnetic potential at the CMB follows: V ðr; y; fÞ ¼

L X l X N lþ2 X a

ðl þ 1Þ  ðglmn c l¼1 m¼0 n¼1 m þ hmn l sinðmfÞÞPl ðcos yÞMn ðtÞ;

cosðmfÞ (Eq. 2)

where a is the Earth’ radius, c is the radius of the Earth’s core, Mn ðtÞ are the temporal basis functions, i.e. cubic B-splines. The expansion (2) involves coefficients fglmn ; hmn l g which are related to the standard Gauss coefficients fglm ; hm l g by glm ¼

N X

glmn Mn ðtÞ

(Eq. 3)

n¼1

and for the hm l likewise. The model is derived to meet some constraints, first it should be spatially smooth. Spatial smoothness, for instance could be controlled by the minimum ohmic dissipation based on the ohmic heating bound of Gubbins (1975). The second constraint to meet is the temporal smoothness of the model, which can be controlled by minimizing the second time derivative of the radial component of the field at the CMB. Further constraints, invoking satellite field models for certain epochs are conceivable as shown by Wardinski (2005) (see entry on Time-dependent models of the geomagnetic field).

347

which it is present. Further, Langel et al. (1986) argued for an internal origin, because it is only present in a few secular variation coefficients, namely g_ 12 ; g_ 22 ; h_ 33. The picture is less clear when considering longer periods such as the 60 years period. Slaucitajis and Winch (1965) provided evidence of this period by an analysis of five observatory data sets. They found an averaged period of 61  6 years. These results were confirmed by Jin and Jin (1989) with a slightly different period of 60  12 years, where the uncertainty is due to an average of the results from an analysis of inclination and declination data. But its true nature is not clear since the match with a period of about 60 years in geomagnetic activity and sunspot numbers suggests an external origin which may cause an inductive 60 years period. On the other hand, there also may be an internal origin of this signal related to the variation of the westward drift and torsional oscillations of the fluid outer core. A comparison of the variation of the “westward drift” of the eccentric dipole (which is part of the secular variation) with the variation of the “length of day” reveals a significant correlation between both variations at about 60 years (Vestine, 1953; Vestine and Kahle, 1968). Braginsky (1970) developed a theory which is capable of characterizing both observables as two aspects of one phenomenon related to torsional magnetohydrodynamic oscillations in the Earth’s core. These oscillations can have periods of about 60, 30, and 20 years, and via coupling processes between mantle and core they should produce fluctuation of the Earth’s rotation. (see Core-mantle coupling, Length of day variations and Torsional oscillations (q.v.)). Bloxham et al. (2002) suggested that the occurrence of geomagnetic jerks should be linked to “torsional oscillation,” therefore also to the length of day variations. Indeed, recent analyzes by Holme and de Viron (2005) and Shirai et al. (2005) reveal that “geomagnetic jerks” are directly linked to the processes responsible for changes in the core angular momentum. Further long periods may exist, such as 18.6 and 9.3 years resulting from the periodic gravitational action of celestial bodies on the equatorial bulge of the Earth. However, their spectral peaks are fairly close to those of the solar activity. Another was predicted by Yukutake (1972) to be the free modes of the electromagnetically coupled coremantle system, which has a length of about 6.7 years. Currie (1973) may have found this period. For other periods, e.g., 70, 55, 32, 29, and 13 years (Langel et al., 1986) a theory is only vaguely outlined and needs further investigation.

Generation of secular variation Periodicities in the geomagnetic secular variation and related phenomena An analysis of the geomagnetic activity and the solar activity reveals periods which can be attributed to the effect of the sunspot cycle. These periodicities are linked to external field variations, such as fluctuation of the strength and position of the ring current and variations of the strengths of current systems in the magnetosphere. Therefore, geomagnetic field variations exhibit nearly the same periods. If we assume that the effect of sunspot cycle and its harmonics is independent of observatory longitudes, then the effect can be approximated by a series of zonal harmonics. We would expect that these contributions dominantly map into the first degree Gauss coefficients of the internal field. Langel et al. (1986) found evidence that some of the periods also exist in higher degree secular variation coefficients ðg_ lm ; h_ m l Þ. It is most likely, that this is due to induction in the mantle. Beside the periods related to solar variations there exists a bundle of periods which are supposed to be inherent features of the geomagnetic secular variation. One prominent long period is the 23 (22.9) years period. It is found in the data of geomagnetic observatories (Alldredge, 1977b) as well as in secular variation coefficients (Langel et al., 1986). Although the closeness of this period with the double solar cycle period may refer to a common cause, the origin of the 23 years period seems to be internal. Alldredge (1977b) showed that it does not appear in all analyzed observatories and is not in phase at those observatories at

Secular variation results from the effect of magnetic induction in the fluid outer core and from effects of magnetic diffusion in the core and the mantle. These processes are constituted by the induction equation, which follows from Faraday’s induction law ]B ¼
r  E; ]t

(Eq. 4)

where E is the electric field, showing that a spatially varying electric field can induce a magnetic field. And further, Ampere’s law 1 1 j¼EþuB¼ r  B; s sm

(Eq. 5)

where s is the electric conductivity, j the current density, and m the magnetic permeability. Combining both and after some algebra we get ]B ¼ r  ðu  BÞ þ
r2 B: ]t

(Eq. 6)

This is the induction equation, where the first term on the right-hand side displays the advection of the magnetic field due to the fluid motion in the liquid outer core, the second represents the action of magnetic diffusion.

348

GEOMAGNETIC SECULAR VARIATION

It is generally assumed that the diffusive timescale is factor 300 to 500 to the advective timescale. This implies that on short timescales, i.e., less than 100 years, the secular variation is entirely caused by the rigidly coupled movement of the magnetic field lines with the fluid motion in the liquid outer core and therefore diffusion can be neglected. ]B ¼ r  ðu  BÞ ]t

(Eq. 7)

That is the so-called frozen flux hypothesis (see Alfvén’s Theorem and the frozen flux approximation (q.v.)). The “frozen flux assumption” simplifies the inversion for the fluid motion u of the liquid outer core (see Core motions), but it imparts an incomplete description of the mechanism of the secular variation generation, for two reasons. First, it reduces the algebraic order of the induction equation by one degree, therefore the solution for the fluid flow may not be expected to be complete or correct (Love, 1999). Second, there is evidence that the frozen flux assumption is violated in the last 40 or 50 years. Bloxham and Gubbins (1985); Bloxham (1986) evaluated the magnetic flux through individual flux patches at the CMB. A necessary condition for the “frozen flux approximation” to apply is that the flux F through a patch S on the core surface bounded by a contour of zero radial field must be constant with respect to time Z Br dS ¼ const: (Eq. 8) F¼ S

Patches located in the southern hemisphere show a significant change of the flux during the period 1960–1980. On a global scale the unsigned flux integral I jBr jdS ¼ const:; (Eq. 9) jFj ¼ CMB

must also be constant with respect to time. Recent analyzes by Holme and Olsen (2005) and Wardinski (2005) suggest that this is not the case for the period 1980–2000. Figure G48 shows the differences of this integral for successive years averaged over the period 1980 to 2000. The given error bars are the sample variance of the integral

for a specific truncation degree. For truncation degrees less than 9 a conservation of unsigned flux is not achieved, for higher degree this seems to be achieved within error margins, but it should be mentioned that the higher degrees of the model more and more reflect the a priori beliefs, i.e., spatial and temporal damping applied in the computation of the secular variation model. In a related discussion about the origin of the “westward drift” it has been suggested that waves rather than a bulk advective motion account for the “westward drift” and most of the secular variation. The possible set of wave types are: inner core oscillation, which can have very short periods of about a year (Gubbins, 1981; Glatzmaier and Roberts, 1996). And, “torsional oscillations” of coaxial cylindrical shells oscillating in a solid-body rotation, for which the force balance is between Lorentz force and inertia. The periods are of the order of decades (Braginsky, 1970; Zatman and Bloxham, 1997, 1999). On longer timescales ( 300 years) magnetohydrodynamic waves which are dependent on Magnetic, Archimedean (buoyancy), and Coriolis force could play a role. The difference between torsional oscillation and MAC-waves is the additional acting of buoyancy in MAC-waves. An interaction of these waves, core-surface flows and the morphology of the magnetic field on a timescale shorter than 300 years is likely. However, it should be mentioned that our distance to core surface and the mantle low conductivity attenuate any details in time and space.

The influence of the mantle on the secular variation The secular variation as observed at the Earth’s surface undergoes two principal processes when it permeates from the core through the mantle. First a geometrical attenuation determined by the factor a lþ2 ; c where a=c 1:8. The higher the harmonic number l, the stronger is the geometrical attenuation. The second effect concerns the magnetic diffusion. Here the temporal change of the field is entirely given by the diffusion of the magnetic field through the mantle implicitly assuming that there is no magnetic field generation in the mantle. The shortest temporal variation of the core’s magnetic field that can be seen at the Earth’s surface is determined by the conductivity of the mantle and in particular boundary between core and mantle. This region of compositional variation, partial melting and heterogeneous high conductivity, known as D00 , may have a thickness of about 200 km and should have a significant impact to the permeability of the magnetic field (see D00 ). However, our understanding of the electrical conductivity of the mantle and its lower part is very poor. All method, which have been employed to deduce the mantle conductivity from the observed secular variation are based on the assumption that the mantle conductivity s varies as a power law of the radius r s ¼ s0

Figure G48 The mean change of the unsigned flux integral for 1980–2000.

c a r

:

(Eq. 10)

This model permits high conductivity in the boundary layer as well as low values of conductivity in the upper mantle, well in agreement with “magnetotellurics.” The values of s0 ranges from 223 S m
1 (McDonald, 1957) and 100 000 S m
1 (Alldredge, 1977a) depending on the method of analysis (see Electrical conductivity of the mantle). A value between 600 and 3000 S m
1 (the later preferred by Backus, 1983) would be in good agreement with length of day variation and electromagnetic core-mantle coupling (Stix and Roberts, 1984; Paulus and Stix, 1989; Stewart et al., 1995). Our knowledge of the mantle conductivity is far from being complete and a rigorous analysis, along theoretical lines as set by Backus (1983) remains to be done. This requires a consolidated knowledge

GEOMAGNETIC SECULAR VARIATION

about the secular variation. In order to achieve this the observation of the Earth’s magnetic field have to be improved, either in terms of more geomagnetic observatories or satellite missions such as MAGSAT, ØRSTED, and CHAMP. Ingo Wardinski

Bibliography Alldredge, L.R., 1977a. Deep mantle conductivity. Journal of Geophysical Research, 82: 5427–5431. Alldredge, L.R., 1977b. Geomagnetic variations with periods from 13 to 30 years. Journal of Geomagnetism and Geoelectricity, 29: 123–135. Alldredge, L.R., 1984. A discussion of impulses and jerks in the geomagnetic field. Journal of Geophysical Research, 89: 4403–4412. Backus, G.E., 1983. Application of mantle filter theory to the magnetic jerk of 1969. Geophysical Journal of the Royal Astronomical Society, 74: 713–746. Backus, G.E., Estes, R.H., Chinn, D., and Langel, R.A., 1987. Comparing the jerk with other global models of the geomagnetic field from 1960 to 1978. Journal of Geophysical Research, 92: 3615–3622. Bloxham, J., 1986. The expulsion of magnetic flux from the Earth’s core. Geophysical Journal of the Royal Astronomical Society, 87: 669–678. Bloxham, J., 1987. Simultaneous stochastic inversion for geomagnetic main field and secular variation. I. A large-scale inverse problem. Journal of Geophysical Research, 92: 11597–11608. Bloxham, J., and Gubbins, D., 1985. The secular variation of earth’s magnetic field. Nature, 317: 777–781. Bloxham, J., and Jackson, A., 1989. Simultaneous stochastic inversion for geomagnetic main field and secular variation II: 1820–1980. Journal of Geophysical Research, 94: 15753–15769. Bloxham, J., and Jackson, A., 1992. Time-dependent mapping of the magnetic field at the core-mantle boundary. Journal of Geophysical Research, 97: 19537–19563. Bloxham, J., Gubbins, D., and Jackson, A., 1989. Geomagnetic secular variation. Philosophical Transactions of the Royal Society of London A, 329: 415–502. Bloxham, J., Dumberry, M., and Zatman, S., 2002. The origin of geomagnetic jerks. Nature, 420: 65–68. Braginsky, S.I., 1970. Torsional magnetohydrodynamic vibrations in the Earth’s core and variations in day length. Geomagnetism and Aeronomy (English translation), 10: 1–8. Cain, J.C., Hendricks, S.J., Langel, R.A., and Hudson, W.V., 1967. A proposed model for the International Geomagnetic Reference Field—1965. Journal of Geomagnetism and Geoelectricity, 19: 335–355. Courtillot, V., Ducruix, J., and Le Mouël, J.-L., 1978. Sur une accélération récente de la variation séculaire du champ magnétique terrestre. Comptes Rendus de l’ Academie des Sciences Paris-Series D, 287: 1095–1098. Currie, R.G., 1973. Geomagnetic line spectra—2 to 70 years. Astrophysics and Space Science, 21: 425–438. Gauss, C.F., 1839. Allgemeine Theorie des Erdmagnetismus. In Gauss, C.F., and Weber, W. (eds.), Resultate aus den Beobachtungen des magnetischen Vereins im Jahre 1838. Leipzig, pp. 1–57. Glatzmaier, G.A., and Roberts, P.H., 1996. On the magnetic sounding of planetary interiors. Physics of the Earth and Planetary Interiors, 98: 207–220. Gubbins, D., 1975. Can the Earth’s magnetic field be sustained by core oscillations? Geophysical Research Letters, 2: 409–412. Gubbins, D., 1981. Rotation of the inner core. Journal of Geophysical Research, 86: 11695–11699.

349

Holme, R., and de Viron, O., 2005. Geomagnetic jerks and a high resolution length-of-day profile for core studies. Geophysical Journal International, 160: 435–440. Holme, R., and Olsen, N., 2005. Core-surface flow modelling from high resolution secular variation. Geophysical Journal International, submitted. Jin, R.S., and Jin, S., 1989. The approximately 60-year power spectral peak of the magnetic variations around London and the earth’s rotation rate fluctuations. Journal of Geophysical Research, 94: 13673–13679. Langel, R.A., Estes, R.H., and Mead, G.D., 1982. Some new methods in geomagnetic field modeling applied to the 1960–1980 epoch. Journal of Geomagnetism and Geoelectricity, 34: 327–349. Langel, R.A., Kerridge, D.J., Barraclough, D.R., and Malin, S.R.C., 1986. Geomagnetic temporal change: 1903–1982—a spline representation. Journal of Geomagnetism and Geoelectricity, 38: 573–597. Love, J.J., 1999. A critique of frozen-flux inverse modelling of a nearly steady geodynamo. Geophysical Journal International, 138: 353–365. Malin, S.R.C., and Hodder, B.M., 1982. Was the 1970 geomagnetic jerk of internal or external origin. Nature, 296: 726–728. McDonald, K.L., 1957. Penetration of the geomagnetic secular field through a mantle with variable conductivity. Journal of Geophysical Research, 62: 117–141. McLeod, M.G., 1985. On the geomagnetic jerk of 1969. Journal of Geophysical Research, 90: 4597–4610. Paulus, M., and Stix, M., 1989. Electromagnetic core-mantle coupling: the Fourier method for the solution of the induction equation. Geophysical Astrophysical Fluid Dynamics, 47: 237–249. Shirai, T., Fukushima, T., and Malkin, Z., 2005. Detection of phase disturbances of free core nutation of the Earth and their concurrence with geomagnetic jerks. Earth, Planets and Space, 57: 151–155. Slaucitajis, L., and Winch, D.E., 1965. Some morphological aspects of geomagnetic secular variation. Planetary and Space Science, 13: 1097–1110. Stewart, D.N., Busse, F.H., Whaler, K.A., and Gubbins, D., 1995. Geomagnetism, Earth rotation and the electrical conductivity of the lower mantle. Physics of the Earth and Planetary Interiors, 92: 199–214. Stix, M., and Roberts, P.H., 1984. Time-dependent electromagnetic core-mantle coupling. Physics of the Earth and Planetary Interiors, 36: 49–60. Vestine, E.H., 1953. On the variations of the geomagnetic field, fluid motions, and the rate of the Earth’s rotation. Journal of Geophysical Research, 58: 127–145. Vestine, E.H., and Kahle, A.B., 1968. The westward drift and geomagnetic secular change. Geophysical Journal of the Royal Astronomical Society, 15: 29–37. Vestine, E.H., Laporte, L., Lange, I., and Scott, W.E., 1947. The geomagnetic field, its description and analysis, Technical Report Publication 580, Carnegie Institution of Washington. Wardinski, I., 2005. Core surface flow models from decadal and subdecadal secular variation of the main geomagnetic field, PhD thesis, Freie Universität Berlin (http://www.gfz-potsdam.de/bib/pub/ str0507/0507.htm). Yukutake, T., 1972. The effect of change in the geomagnetic dipole moment on the rate of the Earth’s rotation. Journal of Geomagnetism and Geoelectricity, 24: 19–48. Zatman, S., and Bloxham, J., 1997. Torsional oscillations and the magnetic field within the Earth’s core. Nature, 388: 760–763. Zatman, S., and Bloxham, J., 1999. On the dynamical implications of models of Bs in the Earth’s core. Geophysical Journal International, 138: 679–686.

350

GEOMAGNETIC SPECTRUM, SPATIAL



Cross-references Alfvén’s Theorem and the Frozen Flux Approximation CHAMP Core-Mantle Coupling, Electromagnetic Core Motions Core-Mantle Coupling, Electromagnetic Core-Mantle Coupling, Thermal Core-Mantle Coupling, Topographic D00 and F Layers of the Earth D00 as a Boundary Layer D00 , Anisotropy D00 , Composition D00 , Seismic Properties Gauss, Carl Friedrich (1777–1855) Gellibrand, Henry (1559–1636) Geomagnetic Jerks Internal External Field Separation Length of Day Variations, Decadal Length of Day Variations, Long Term Magnetotellurics Main Field Modeling Mantle, Electrical Conductivity, Mineralogy Oscillations, Torsional Reversals, Theory Time-dependent Models of the Geomagnetic Field Westward Drift

Application to the geomagnetic field The case of the geomagnetic field is more complicated for two reasons. The first complication is that the field is a vector (at each point in space it has direction as well as magnitude) rather than a scalar (which has only magnitude). However, in regions where there is no significant electric current (in practice this means throughout the upper mantle and the lower atmosphere) we can represent the vector magnetic field B as the gradient of a scalar potential, V, B ¼
grad V ;

Introduction Just as for a function defined round a circle it is useful to separate the contributions of different wavelengths/spatial frequencies, so for a function defined on the surface of a sphere it is useful to separate the contributions of different (two-dimensional) wavelengths/spatial frequencies. In the context of geomagnetism it is useful to apply this concept of a spatial spectrum to the main magnetic field (coming from the core) and its time variation (the secular variation), and to the field from the magnetization of the crust. In the one-dimensional case we call this process Fourier analysis; if we have a variable F, which is known round a circle (and which is therefore a periodic function round the circle), we can express it in the form X n

ðam cos ml þ bm sin mlÞ;

(Eq. 3)

the factor of (1/2) is the mean-square (ms) value of cos ml or sin ml round the circle. (For m ¼ 0 the mean-square value is in fact 1, but I will ignore this minor complication.) We see that the two harmonics of a given spatial frequency m contribute to the overall mean-square value independently of all the other harmonics. We can write Fm(l) ¼ (am cos ml þ bm sin ml) alternatively as cm cos(ml
e); it is a sinusoidal variation of amplitude cm ¼ ða2m þ b2m Þ1=2 , and phase e. Our choice of the origin of the angle l is arbitrary; while the numerical values of am and bm, and the phase e, will depend on this choice of origin, the variation of Fm round the real circle, and its amplitude cm, is independent of this choice. A plot of c2m against m is called the (power) spectrum of F. (The notation arises because for a wave-motion periodic in time, c2m is proportional to the average power (over the cycle) carried by the wave at that frequency.)

GEOMAGNETIC SPECTRUM, SPATIAL

FðlÞ ¼ Sm Fm ðlÞ ¼

    X  2 am cos2 ml þ b2m sin2 ml F 2 ¼ Sm Fm2 ¼ m X ¼ ða2m þ b2m Þ=2; m

(Eq. 1)

where l is the angle round the circle, measured from some origin. am and bm are numerical coefficients; the integers m, which start from zero, can be thought of as spatial frequencies, or wave numbers. Each term cos ml or sin ml is called a harmonic, though this name is sometimes also applied to the numerical values of am cos ml and bm sin ml. The concepts of frequency and harmonic come from the analogous situation of a variable which is a periodic function in time; this function of time can be separated into harmonics specified by frequencies (in time) which are multiples of the fundamental frequency. These harmonics are orthogonal; in the sense that the average value round the circle of the product of any two different harmonics is zero:   (Eq. 2) Fm Fm ¼ 0 unless m ¼ m (and both harmonics use only either cos ml or sin ml), where <x> represents the average of x round the circle. Therefore, if we expand an arbitrary F in terms of its harmonic components Fm, and then square it, all the cross-terms vanish when we take the average round the circle. So the mean-square value of F round the circle, , becomes

(Eq. 4)

and can then work in terms of this scalar variable V. The second complication is that the relevant geometry is threedimensional space rather than a one-dimensional circle. As we are concerned with a roughly spherical Earth, it is convenient to use a spherical polar coordinate system (r, y, l) based at the center of the Earth, with y ¼ 0 being the direction of the north geographic pole, and l ¼ 0 the direction of the Greenwich meridian. (The angle l is the conventional longitude, but because the Earth is a slightly oblate ellipsoid the (geocentric) y is not quite the same as the (geodetic) colatitude.) Provided that there is also no significant magnetization in the region of interest, it is convenient to approximate the scalar potential V as a finite sum of “spherical harmonics” (see Harmonics, spherical). (In practice, although the rocks of the crust are magnetized, even in the crust they produce a field which is small compared with that of the electric currents in the core, and we can often ignore this minor complication. See the last section below for the relative magnitude of the fields.) We then have X V m ðr; y; lÞ V ðr; y; lÞ ¼ n;m n X nþ1 Pm ðcos yÞðgnm cos ml þ hm ¼a n sin mlÞða=rÞ n;m n (Eq. 5) for that part of the field having sources internal to the region of interest. The Pnm ðcos yÞ are (scaled versions of ) the mathematical functions of y called Legendre polynomials. The different harmonics are labeled by two integers, the degree n and the order m, with m  n. (Note that some authors use the symbol l rather than n for the degree.) The factor a, the mean radius of the Earth, is incorporated so that the numerical coefficients gnm and hm n have the dimensions of magnetic field, conventionally expressed in units of nanotesla. The particular variation with radius in Eq. (5) is valid only for internal sources, and only in a region free from sources of magnetic field. But on any given spherical surface, whatever the source

351

GEOMAGNETIC SPECTRUM, SPATIAL

distribution, a scalar such as V can be represented uniquely as a function of (y, l) given by the sum of a set of surface harmonics, where any radial variation is incorporated numerically into the coefficients. For simplicity, considering a field of internal origin on the surface r ¼ a (approximately the surface of the Earth), we have X X V m ðy; lÞ ¼ V ðy; lÞ (Eq. 6) V ðy; lÞ ¼ n;m n n n where Vnm ðy; lÞ ¼ aPnm ðcos yÞðgnm cos ml þ hm n sin mlÞ:

(Eq. 7)

We can think of these “surface harmonics,” Pnm ðcos yÞ cos ml and Pnm ðcos yÞ sin ml, as being the two-dimensional analogs of the onedimensional Fourier harmonics on a circle. For a given physical field, the numerical values of the coefficients, gnm and hm n will depend on the origin of our coordinate system. If, for example, we used the meridian of Paris, rather than that of Greenwich, to define l ¼ 0, we would find that the coefficients would be different. However, for a given pair of gnm and hm n we find that the sum of their 2 Þ is a constant, independent of our choice of the squares, ðgnm Þ2 þ ðhm n origin l ¼ 0 (exactly as in one-dimensional Fourier analysis). More generally, even if we also move the y ¼ 0 axis to another point on the sphere, it turns out that we have X 2 ½ðgnm Þ2 þ ðhm (Eq. 8) Cn2 ¼ n Þ ¼ constant; m where the sum is over m ¼ 0,1,2,. . .,n. (This particularly simple result is true provided that we use the Schmidt seminormalized definitions for the Legendre polynomials Pnm , as is conventional in geomagnetism. It is also true if we use fully normalized Pnm (as do workers in gravity), but not if we use the basic mathematical definitions (sometimes denoted by Pnm) which give unnormalized functions, the mean-square of which varies with m as well as with n.) In fact any one of the Vn of Eq. (6) is physically the same, whatever our choice of coordinate system; the processes producing the fields do not “know” of our arbitrary choice of coordinate system. We can think of all the contributions of a given degree n as having essentially the same minimum (surface) wavelength on the sphere, of value (approximately) the circumference divided by degree n. Mathematically, the three terms for n ¼ 1 correspond to the potentials given by three dipoles at the center, the n ¼ 2 terms to five quadrupoles at the center, and so on, but of course physically the sources are distributed current systems. Just as in Fourier analysis the various harmonics are orthogonal round the circle, these surface harmonics are orthogonal over the spherical surface; the average value over the sphere of the product of two different surface harmonics is zero. Now using for the average over the sphere r ¼ a, the mean-square value of V over the surface can be expressed as the sum 

 X  2   Vn where Vn2 V2 ¼ n X 2 ¼ a2 ½ðgnm Þ2 þ ðhm n Þ =ð2n þ 1Þ; m

(Eq: 9Þ

the factor 1/(2nþ1) is the mean-square value of each harmonic over the surface for our Schmidt seminormalized Pnm . However we are using the scalar potential V only as a convenient mathematical simplification. What we are really interested in is the magnetic field B itself. It turns out that if we represent each harmonic Pnm ðcos yÞ cos ml or Pnm ðcos yÞ sin ml by (say) Wnm , and put m Bm n ¼
gradWn ;

(Eq. 10)

then the vector fields Bm n are themselves also orthogonal over the sphere: using the scalar product for the multiplication, we have



 m Bm n  Bn ¼ 0

unless n ¼ n, m ¼ m (and both harmonics use only either cos ml or sin ml); 

 m Bm n  Bn ¼ ðn þ 1Þ:

(Eq. 11)

The factor (n þ 1) is the mean-square value over the sphere of the vector fields corresponding to each of the individual harmonics of the degree n part of the scalar potential V (Lowes, 1966). (This factor (n þ 1) is valid only if the sources of the field are internal to the sphere. For external sources, e.g., ionospheric or magnetospheric currents, the factor is n.) So if B is the observed field (X, Y, Z), where X, Y, Z are Cartesian components of the vector, the mean-square value of the field vector over the sphere r ¼ a becomes   X R (Eq. 12) < B  B >¼ jBj2 ¼ X 2 þ Y 2 þ Z 2 ¼ n n Where Rn ¼ ðn þ 1Þ

X h  2   2 i : gnm þ hm n m

(Eq. 13)

If we plot Rn as a function of n, we have the geomagnetic spatial spectrum. An alternative interpretation of Eq. (13) is that if it is multiplied by 1/2m0 (and the coefficients are expressed in units of tesla) it gives the stored magnetic energy density B·H/2 (the energy stored per unit volume in the magnetic field, in J m
3) averaged over the surface; so this spatial spectrum (of the vector field) is sometimes called the geomagnetic power spectrum, or the geomagnetic energy spectrum. (Rn is roughly analogous to the “degree variance” used in gravity. But note that while Rn is zero for n ¼ 0 in geomagnetism, as there are no magnetic monopoles, the n ¼ 0 term completely dominates in gravity.)

Results for the geomagnetic field If we apply Eq. (13) to the results of a spherical harmonic analysis of the geomagnetic field of internal origin, as observed globally by satellites such as Ørsted and CHAMP, and plot Rn on a logarithmic scale against n, we obtain the spectrum of Figure G49. Formally, given observation of the field only at and above the Earth’s surface, we cannot tell whereabouts inside the surface are the currents producing the field. But if we look at the plot, we see that to a first approximation it can be represented as the sum of two straight lines (two power-law functions on a linear scale). Although there is argument about the exact shape of the best-fitting curves, it is now accepted that (almost all of) the field from harmonics up to about degree n ¼ 14 comes from electric currents in the Earth’s core, while (almost all of) the field of higher degrees comes from the magnetization of crustal rocks. As is well known, at the Earth’s surface the geomagnetic field is dominated by its dipole component, and indeed the n ¼ 1, dipole, point lies well above the first line. But the line is a good fit for values of n from 2 to about 12. If we go back to the definition of Eq. (5), and remember that taking the gradient in effect introduces another power of (a/r), we see that to draw the equivalent plot for a radius r different from a, we have to multiply each Rn by the factor (a/r)2nþ4. This has more effect on the points for larger n, so that for r < a the line becomes less steep. In fact, if we go down to the radius of the core-mantle boundary (CMB) the line becomes nearly horizontal, and if we continued the extrapolation a little further we would find that just below the CMB the line becomes horizontal; each degree would make about the same contribution to the total ms field. Strictly, because there will be electric currents in this region, this further extrapolation is not allowable, but this result is a convincing argument that these fields do have their origin in the core.

352

GEOMAGNETIC SPECTRUM, SPATIAL

Figure G49 Spatial spectrum of the internal part of the geomagnetic field at 2000.0. The circles are the values of Rn, the mean-square vector field produced by the harmonics of degree n over the surface r ¼ a, plotted on a logarithmic scale against n; right-hand scale. The squares are the corresponding values for the secular variation; left-hand scale. The coefficients used are those of CM4, Sabaka et al. (2005).

The other line, for higher spatial frequencies/shorter wavelengths, is already sloping upward, so we cannot use such a simple argument to indicate the source depth. The results for even higher frequencies (see below) show that the curve does in fact eventually turn downward, as it must for the crust to contribute only a finite field, and the overall shape is consistent with reasonable models of crustal magnetization. So there is no doubt that this separation into core and crustal origin must be largely correct; the division is usually put at about n ¼ 14. But note that the spectrum of the core field will continue above this value, and that of the crustal field will continue below this value. We do not know the detailed physics that lies behind each of the two spectra, so we can only guess at how they are to be extrapolated. But if we simply extrapolate the straight lines, then we find that at the Earth’s surface the core field (of total root-mean-square (rms) magnitude about 45000 nT) would contribute about 10 nT rms from degrees beyond 13, while the crust (total rms about 300 nT) would contribute about 10 nT from degrees below n ¼ 14. While the spectrum of the main field itself is now quite well determined (by measurements from polar-orbiting satellites such as Magsat, Ørsted, and CHAMP), the spectrum of the crustal field (see Crustal magnetic field) is much more uncertain. All the points shown on Figure G49 are obtained from satellite measurements. However at satellite altitudes the crustal field is small, and for individual measurements is often not much larger than the instrument noise. Probably more importantly, there are also comparable or larger time-varying fields of external origin, and the separation of the spatial-variation and time-variation of the field seen by a moving satellite is very difficult. In fact, as techniques improve there has been a tendency for the derived crustal spectrum to become lower on a graph such as that of Figure G49! For degrees above about 30–40 it is at present difficult

to do a formal spherical harmonic analysis of the satellite data, though it is sometimes possible to do a Fourier, one-dimensional, analysis along individual tracks. Similarly we can use the long tracks of the Project MAGNET airborne vector magnetometer (see Project MAGNET ), and a much larger number of tracks from oceanographic vessels towing a scalar magnetometer; these approaches can extend the data out to about n ¼ 1000. However these are only onedimensional cross sections of the field variation over the two-dimensional sphere, so some assumptions have to be made (see, e.g., Korte et al,, 2002); also, the near-surface tracks are almost all over the oceans, where the crust is systematically thinner than under the continents. Such results however do suggest that the mean-square field per degree does in fact start to decrease at higher degrees. In theory, for the very high degrees, we could analyze data from the large number of detailed aeromagnetic surveys which have been carried out (though mostly over land), but there are many problems, and this has not yet been attempted. This concept of a spatial spectrum can obviously be extended to the secular variation, the time variation of the main field. A complication is that there are many external sources of field variations, of period of a year or less, the effects of which are most easily removed by taking averages over one year. This averaging is most easily done with the long-term surface observations at magnetic observatories and repeat stations; unfortunately these have only a very poor global coverage. Satellites have much better global coverage, and are now also giving continuous coverage in time, though unfortunately their measurements are subject to more interfering effects. However, at present it looks as though we can estimate the secular variation over the last year or so, and hence its spectrum, up to about n ¼ 10 (see, e.g., Langlais et al., 2003). The spectrum for 2000.0 is shown on Figure G49; it is much flatter than that of the main field, so that (in this range) at the CMB the shorter wavelengths are varying more rapidly than the longer wavelengths.

History Lowes (1966) was the first to introduce to the English-speaking community the expression for mean-square field, Eq. (11) of this article, in the context of geomagnetism; he also produced the first power spectrum, analogous to Figure G49, in Lowes (1974). However, essentially the same expression had been introduced earlier in Germany, by Lucke in a colloquium in 1955, referred to by Fanselau and Lucke (l956). Lucke worked in terms of the energy per unit volume stored in a magnetic field, which (in modern notation) is E ¼ ð1=2ÞB  H ¼ ð1=2m0 ÞB  B:

(Eq. 14)

So, when he averaged over the sphere r ¼ a, he obtained h    i X 2 2 ðn þ 1Þ gnm þ hm h E i ¼ ð1=2m0 ÞhB:Bi ¼ ð1=2m0 Þ n n X R ; (Eq. 15) ¼ ð1=2pm0 Þ n n or h E i ¼ ð1=2m0 Þ

X n

Rn ;

(Eq. 16)

in the notation of this article. Lucke’s lecture was also referred to by Mauersberger (1956), who produced the same expression for the mean energy density, but using a different derivation. Hence graphs such as those of Figure G49 are sometimes referred to as a “Lowes spectrum,” “Lowes-Mauersberger spectrum,” or “Mauersberger-Lowes spectrum.” Frank Lowes

GEOMAGNETIC SPECTRUM, TEMPORAL

Bibliography Fanselau, von G., and Lucke, O., 1956. Über die Veränderlichkeit des erdmagnetischen Hauptfeldes und seine Theorien. Zeitshrift für Geophysik, 22: 121–216. Korte, K., Constable, C.G., and Parker, R.L., 2002. Revised magnetic power spectrum of the oceanic crust. Journal of Geophysical Research, 107(B9): doi:10.1029/2001JB1389. Langlais, B., Mandea, M., and Ultrée-Guérard, P., 2003. High-resolution magnetic field modeling: application to MAGSAT and Ørsted data. Physics of the Earth and Planetary Interiors, 135: 77–79. Lowes, F.J., 1966. Mean-square values on sphere of spherical harmonic vector fields. Journal of Geophysical Research, 71: 2179. Lowes, F.J., 1974. Spatial power spectrum of the main geomagnetic field, and extrapolation to the core. Geophysical Journal of the Royal Astronomical Society, 36: 717–730. Mauersberger, P., 1956. Das Mittel der Energiedichte des geomagnetischen Hauptfeldes an der Erdoberfläche und seine säkulare Änderung, Gerlands Beitrage Geophysik, 65: 207–215. Sabaka, T.J., Olsen, N., and Purucker, M., 2005. Extending comprehensive models of the Earth’s magnetic field with Ørsted and CHAMP data. Geophysical Journal International, 159: 521–547.

Cross-references Crustal Magnetic Field Harmonics, Spherical IGRF, International Geomagnetic Reference Field Main Field Modeling Nondipole Field Project Magnet

GEOMAGNETIC SPECTRUM, TEMPORAL The geomagnetic field varies on a huge range of timescales, and one way to study these variations is by analyzing how changes in the geomagnetic field are distributed as a function of frequency. This can be done by estimating the spectrum of geomagnetic variations. The power spectral density S( f ) is a measure of the power in geomagnetic field variations at frequency f. When integrated over all frequencies it measures the total variance in the geomagnetic field. Figure G50/Plate 2 shows a schematic of the various processes that contribute to the geomagnetic field, and these can be roughly divided according to the frequency range in which they operate. The bulk of Earth’s magnetic field is generated in the liquid outer core, where fluid flow is influenced by Earth rotation and the geometry of the inner core. Core flow produces a secular variation in the magnetic field, which propagates upward through the relatively electrically insulating mantle and crust. Short-term changes in core field are attenuated by their passage through the mantle so that at periods less than a few months most of the changes are of external origin. The crust makes a small static contribution to the overall field, which only changes detectably on geological timescales making an insignificant contribution to the long-period spectrum. Above the insulating atmosphere the electrically conductive ionosphere (q.v.) supports Sq currents with a diurnal variation as a result of dayside solar heating. Lightning generates high-frequency Schumann resonances in the Earth/ionosphere cavity. Outside the solid Earth the magnetosphere (q.v.), the manifestation of the core dynamo, is deformed and modulated by the solar wind, compressed on the dayside and elongated on the nightside. At a distance of about 3 earth radii, the magnetospheric ring current acts to oppose the main field and is also modulated by solar activity. Although changes in solar activity probably occur on all timescales the associated magnetic variations are much smaller than the changes in

353

the core field at long periods, and only make a very minor contribution to the power spectrum. With an adequate physical theory to describe each of the above processes one could predict the power in geomagnetic field variations as a function of frequency. The inverse problem is to use direct observations or paleomagnetic measurements of the geomagnetic field to estimate the power spectrum. Power spectral estimation is usually carried out using variants of one of the following well-known techniques: (1) direct spectral estimation based on the fast Fourier transform, and using extensions and improvements to the time-honored periodogram method introduced by Schuster in 1898 to search for hidden periodicities in meteorological data; (2) the autocovariance method which exploits the Fourier transform relationship between the time-domain autocovariance and frequency-dependent power spectral density of a process; (3) parametric modeling schemes like the maximum entropy method based on a discrete autoregressive process. The relative merits of these techniques have been widely discussed (see Constable and Johnson, 2005: or Barton, 1983, for the geomagnetic context), while all have been used in analyzes of geomagnetic intensity and directional variations. It is generally acknowledged (e.g., Percival and Walden, 1993) that direct spectral estimation combined with tapering and averaging of nearly independent spectral estimates can provide highresolution estimates and be used to optimize the unavoidable trade-offs between variance and bias. The parameter often chosen to represent the geomagnetic spectrum is the field strength at midlatitudes, or a proxy form for times where it is not possible to obtain a direct measurement. This is the case for paleomagnetic time series derived from lacustrine or marine sediments which provide only directional information and/or relative intensity variations. Barton (1982) combined spectra from lake sediment directional paleomagnetic records with those from full vector data recorded at magnetic observatories and used periodogram analysis in the first attempt to provide an integrated power spectrum for periods ranging from less than a year to 105 years. Courtillot and Le Mouël (1988) merged Barton’s result with other spectral estimates at longer and shorter periods extending the timescales from seconds to millions of years. They debated whether the result was compatible with an overall 1/f 2 spectrum, and concluded that it was too early to make such an inference. A recent version of such a composite spectrum (Constable and Constable, 2004) uses spectral estimates from relative paleointensity variations (Constable et al., 1998) at long periods and is shown in Figure G51 (note that this is an amplitude rather than a power spectrum, that is the square root of power spectral density). Between 10
10 and 1 Hz, the spectrum is from Filloux (1987). Above 1 Hz, the results are those of Nichols et al. (1988). Internal variations reflecting motions of the fluid core dominate at periods longer than a few months, and the spectrum generally rises toward longer period (low frequency) with reversals (q.v.) of the dipolar part of the field the dominant influence on 105 to 106 year timescales. The 11-year sunspot cycle, solar rotation, and Earth’s orbit modulate the distortions of the field associated with geomagnetic storms, which themselves have energy in the several hour to several second band. Energy at the daily variation and harmonics comes from diurnal heating of the ionosphere. Lightning creates high frequency energy in the Earth/ionosphere cavity, which resonates at 7–8 s and associated harmonics. At the highest frequencies there is presumed to be a continued fall-off in the natural spectrum. The upturn seen in Figure G51 reflects the dominant influence of human-made sources. The spectrum in Figure G51 lacks information at very long periods where the spectrum is dominated by the intensity variations associated with changing geomagnetic reversal rate, and between about 10 ka and 10-year periods, where the timescales for processes of internal and external origin overlap. Figure G52/Plate 4c provides a range of estimates at centennial to 50 Ma periods for the spectrum of the geomagnetic dipole moment (q.v.). The longest period power spectrum is estimated from reversal times given by the magnetostratigraphic

354

GEOMAGNETIC SPECTRUM, TEMPORAL

Figure G51 Composite amplitude spectrum of geomagnetic variations as a function of frequency (Constable and Constable, 2004): annotations indicate the predominant physical processes at the various timescales. Copyright American Geophysical Union, 2004, reproduced with their and the authors’ permission. timescale and shown in black. Further information comes from various sedimentary relative paleointensity records with varying accumulation rates and the dipole moment estimate of a time varying global paleomagnetic field model for the past 7 ka.

In constructing a paleomagnetic power spectrum like that in Figure G52/Plate 4c there are a number of challenges. A basic requirement for spectral analysis is a time series of observations, but there is no single record that covers the time span of interest. The magnetostratigraphic record appears to be nonstationary with long-term changes in reversal rate, and provides no information about intensity variation on long timescales. The relative paleointensity records from sediments not only lack an absolute scale, but are usually unevenly sampled in time so that some stable calibration and interpolation scheme is required before using the standard analysis techniques. It is likely that some sediments record a smoothed version of the geomagnetic signal because of low sedimentation rates, while in others it may be necessary to consider the possibility of aliasing. Nongeomagnetic signals may be inadvertently interpreted as arising from geomagnetic variations with time. The choice of dipole moment (as opposed to some other geomagnetic field parameter) is motivated in large part by the dominance of the geocentric axial dipole when the field is averaged over long time intervals: the strength of the axial dipole is representative of the global field, and may be related to the amount of energy required by the geodynamo or to the geomagnetic reversal rate. Although it is possible that other properties of the field (such as nondipole field contributions) directly reflect particular physical processes controlling the secular variation, resolving such variations in paleomagnetic time series remains controversial. Overall, it is likely that the power in geomagnetic field variations is underestimated. There is substantial scope for improving the spectra in both Figures G51 and G52/Plate 4c. Relative intensity records from sediments are steadily improving, and new modeling techniques may extend time-varying geomagnetic field models providing better dipole moment estimates on million year timescales. This may give new insight into what controls very long-period secular variation. Many newer observations could be used for direct spectral estimates to replace the current schematic spectrum at periods from decades down to 1 s. Although the general form of the spectrum is quite well understood in this region, detailed analyzes could provide further insight

GEOMAGNETISM, HISTORY OF

355

Courtillot, V., and Le Mouël, J.-L., 1988. Time variations of the Earth’s magnetic field: from daily to secular. Annual Review of Earth and Planetary Science, 16: 389–476. Filloux, J.H., 1987. Instrumentation and experimental methods for oceanic studies. In Jacobs, J.A. (ed.), Geomagnetism. London: Academic Press, pp. 143–248. Guyodo, Y., Gaillot, P., and Channell, J.E.T., 2000. Wavelet analysis of relative geomagnetic paleointensity at ODP Site 983. Earth and Planetary Science Letters, 184: 109–123. Nichols, E.A., Morrison, H.F., and Clarke, J., 1988. Signals and noise in measurements of low-frequency geomagnetic fields. Journal of Geophysical Research, 93: 13743–13754. Percival, D.B., and Walden, A.T., 1993. Spectral Analysis for Physical Applications: Multitaper and Conventional Univariate Techniques. Cambridge: Cambridge University Press.

Cross-references Dipole Moment Variation Harmonics, Spherical Ionosphere Magnetosphere of the Earth Mantle Nondipole Field Reversals, Theory Secular Variation Model

Figure G52/Plate 4c Composite spectrum for the geomagnetic dipole moment constructed from the magnetostratigraphic reversal record with and without cryptochrons (frequencies 10
2–20 Ma
1), various sedimentary records of relative paleointensity (100–103 Ma
1), and from the dipole moment of a 0–7 ka paleomagnetic field model (103–104 Ma
1). Figure redrawn from Constable and Johnson (2005).

into the underlying physical processes. Techniques such as wavelet analysis, that can take account of nonstationarity in the underlying geomagnetic processes, have yet to be fully exploited for geomagnetic data (Guyodo et al., 2000) and may prove useful. However, despite the relatively crude nature of existing spectral estimates it is apparent that the geomagnetic power spectrum does not follow a simple power law fall-off with increasing frequency. The form is instead influenced by the characteristic timescales that reflect the distinct physical processes contributing to the geomagnetic field. Catherine Constable

Bibliography Barton, C.E., 1982. Spectral analysis of palaeomagnetic time series and the geomagnetic spectrum. Philosophical Transactions of the Royal Society of London A, 306: 203–209. Barton, C.E., 1983. Analysis of paleomagnetic time series-techniques and applications. Geophysical Survey, 5: 335–368. Constable, C.G., and Constable, S.C., 2004. Satellite magnetic field measurements: applications in studying the deep earth. In Sparks, R.S.J., and Hawkesworth, C.J., (eds.), The State of the Planet: Frontiers and Challenges in Geophysics. Washington, DC: American Geophysical Union, doi: 10.1029/150GM13, pp. 147–160. Constable, C.G., and Johnson, C.L., 2005. A paleomagnetic power spectrum. Physics of the Earth and Planetary Interiors, 153: 61–63. Constable, C.G., Tauxe, L., and Parker, R.L., 1998. Analysis of 11 Ma of geomagnetic intensity variation. Journal of Geophysical Research, 103: 17735–17748.

GEOMAGNETISM, HISTORY OF In its present form, the geophysical discipline of geomagnetism is of relatively recent origin (the term “geomagnetism” was coined in 1938 by Sydney Chapman, (q.v.)), yet interpretations of the Earth’s magnetic field have been propounded by scholars from the Middle Ages onward, in the broader context of cosmologies, natural philosophies, and oceanic navigation. In this historical overview, geomagnetism will be considered as the scientific study of the Earth’s internal field and its secular change, encompassing both descriptive and causal hypotheses; paleomagnetism and the external field are treated elsewhere. Furthermore, since most of this volume is dedicated to current issues, stress will here be placed on earlier times.

Main developmental stages The history of geomagnetism can be subdivided into three main periods: firstly, a proto-scientific stage (up to the 16th century), during which awareness slowly grew of the existence of a global magnetic property worthy of investigation. At this time, causal hypotheses almost exclusively identified the heavens as the seat of magnetic attraction. Secondly, an early-modern stage took place (16th to early 19th century), during which directional data (magnetic declination and inclination) were increasingly measured, compiled, and mapped. These efforts led to the discovery of secular variation and causal models involving one to three crustal or nuclear dipoles (terrestrial polar attraction). Thirdly, a modern stage emerged (from the 1830s), characterized by measurement of the full magnetic vector (direction and intensity) in dedicated scientific surveys, observatories, and satellites; by the description of the field as a whole, both at the surface and at the top of the source region; and by the mid-20th century introduction of geodynamo theory (q.v.), leading to a multitude of numerical and laboratory magnetohydrodynamic simulations. These developments are summarized per century in Table G8, which pays separate attention to empirical (data and mapping) and theoretical aspects (hypotheses). As is apparent, the main watershed between the early-modern and modern stage pervades both. The most influential scientist in bringing about this change was Carl Friedrich Gauss

356

GEOMAGNETISM, HISTORY OF

Table G8 Empirical and theoretical stages in geomagnetism Century

Data

Mapping

Hypotheses

16th 17th 18th 19th 20th

Maritime Maritime; site series Maritime; site series; surveys Surveys; observatories Surveys, repeat stations, Observatories, satellites

D; interpolation D; interpolation D, I; interpolation D, I, H, F; spherical harmonics (surface) All components; spherical harmonics (surface, CMB)

Polar attraction (celestial, crustal) Polar attraction (crustal, core) Polar attraction (core) Geomagnetic field Geodynamo (fluid outer core)

Notes: D, declination; I, inclination; H, horizontal intensity; F, total intensity; CMB, core-mantle boundary.

(1777–1855) (q.v.), who not only invented the first absolute magnetometer (1832) but, moreover, developed for geomagnetic application the mathematical techniques of spherical harmonic expansion (1839) and least-squares analysis (independently discovered by Adrien-Marie Legendre, 1752–1833). Furthermore, around the same time occurred a shift in the realm of data acquisition, up till then performed mostly by navigational practitioners and hydrographers, thereafter predominantly by professional astronomers and physicists. This article will therefore apply the same temporal caesura in discussing geomagnetic data processing and theory formation before, and after Gauss’s contributions.

The era of polar attraction Although the phenomenon of magnetic attraction had been known since antiquity, it was in the Middle Ages that the notions of polarity and orientation were first discovered, and several cultures recorded its potential (or actual) utility. The Chinese record goes back furthest in this respect, with magnetized objects being employed in geomancy from at least the A.D. 6th century, and a wire-suspended magnetic needle first described in 1096. Moreover, in the next few decades appeared the first Chinese references to mariners relying for direction on needles made to float in a bowl of water (Needham, 1962), a technique independently discovered in Europe. By the 13th century, the magnetic needle was commonly relied upon in both Asian and European navigation at sea, in particular when overcast skies made an astronomical fix impossible. In subsequent centuries, the dry-pivoted compass (q.v.) was moreover increasingly relied upon on land, in surveying and to meridionally align portable instruments and some newly planned official buildings (churches, temples, and palaces). The difference between magnetic and true north was as of yet unappreciated; in keeping with Aristotelian cosmology, the needle was still thought to respect the imagined immutable perfection of the supralunary spheres (Smith, 1992). This view is epitomized in the “Epistle on the Magnet” (1269) in which engineer Pierre de Maricourt (Petrus Peregrinus) related his experiments with a spherical lodestone. This magnes rotundus represented the firmament, its two magnetic poles coinciding with the celestial poles (the only fixed points in a diurnally rotating field of stars). Given this arrangement, magnetic and true meridian would always coincide (declination was zero everywhere), and inclination would change as a function of geographical latitude. The concept of a celestial axial dipole was reiterated over the next two centuries by many medieval and Renaissance authors. Some, however, attributed the magnetic force to the nearby Polestar, relying on the Classical concept of sympathy to establish an occult bond between compass and star. These tenets would be challenged when magnetic declination was finally acknowledged as a real phenomenon, rather than being ascribed to an error in instrument or measurement. The first tacit evidence thereof can be found on German portable sundials from the second quarter of the 15th century. Craftsmen in Augsburg and Nuremberg then started to mark local declination in the base of the instrument, adjacent to a small inset compass, used to enable proper orientation in the geographical meridian. Around that

time, similar markings also started to appear next to the wind rose on some German maps (e.g., by cartographers Etzlaub, Waldseemüller, Ziegler, and Murer). But it would take another century before the spatial variability of the Earth’s magnetic field became overwhelmingly clear, and with it the need for global descriptive and causal geomagnetic hypotheses.

The 16th century The advent of oceanic navigation, without the benefit of land sightings, soundings, or a practical method with which to measure longitude, meant that mariners had to rely on compass and astronomical observations to a hitherto unequalled degree. Off the continental shelves, steering, dead reckoning, log keeping, charting, and sailing instructions consequently came to incorporate true compass directions. Any local magnetic declination had to be measured regularly, and compensated for in courses and calculations. Fortunately, establishing this so-called compass allowance was a fairly simple calculation, based on sighting the Sun or other stars at rise and/or set. As Portuguese and Spanish explorers traversed increasing parts of the Atlantic and Indian Oceans, the peculiar spatial distribution of needle behavior was slowly recorded, collated, and compared, both by navigators at sea and hydrographers at home. In large parts of the Atlantic, the compass northeasted, whereas in the Indian Ocean, variable northwesting was the rule; near-zero declination was furthermore associated with the mid-Atlantic, South Africa, Southeast Asia, and Middle America. These early data sets seemed to suggest that the Earth’s dipole was tilted relative to the Earth’s rotation axis, since local declination was assumed to have a direct bearing on the position of the dipole. By applying spherical trigonometry, two observations of declination at places far apart in longitude could provide a cross bearing of two great circles, each through one of the points and at the given angle to true north. Where the two circles intersected, the geomagnetic poles were supposed to lie. Once the position of the dipole was known, a similar calculation at sea would yield the longitude of a place where declination and latitude were measured. In other words, given a tilted dipole at a fixed position, an observer traveling around the globe at the same latitude would see his compass needle diverge from true north as a function of longitude. At the meridian of the dipole’s longitude and on the antimeridian (180 E) the great circle through the place of the observer and the dipole would cut the geographical pole as well, resulting in zero degrees declination; deflection from true north would reach a maximum near 90 longitude east and west from either meridian. Actual measurements of zero declination near the Azores led to the mistaken belief that such a meridional agonic line was found, forming a “natural” indicator of a prime meridian to reckon longitude from. Geomagnetic considerations have thereby greatly affected early-modern cartography. The combination of reigning northeasting in Western Europe and northwesting on the American east coast furthermore led to the conclusion that the dipole was situated on longitude 180 E (of the Azorean prime meridian). The notion of the tilted dipole neatly concurred with a more ancient conjecture, that of the “magnetic mountain.” This lodestone rock,

GEOMAGNETISM, HISTORY OF

mountain, or island, often located in Arctic regions, was not only supposed to affect compass needles all over the globe, but, according to some legends, could even draw, capture, or destroy iron-bearing vessels that sailed too close by (Balmer, 1956). Several scholars have repeatedly tried to calculate its exact coordinates, among them cartographer Gerard Mercator (1512–1594) and astronomer Johannes Kepler (1571–1631). Others instead interpreted the postulated tilted dipole as evidence that the points of magnetic attraction were identical with the poles of the ecliptic. Nevertheless, by this time, the majority opinion tended to favor magnetic poles on the Earth’s surface. This shift from celestial to terrestrial magnetism was reinforced by the discovery of magnetic inclination. In a qualitative sense, it was made by German mathematician and instrument maker Georg Hartmann (reported in a letter to his patron in 1544). It would take until about 1580 before London compass maker Robert Norman constructed the first inclinometer (a magnetized needle rotating in the vertical plane on a horizontal axis), with which the first quantified readings were taken. Inclination too was briefly considered as a navigational aid; in the 1590s, English mathematician Henry Briggs (1561–1630) assumed an axial dipole and produced a table of dip for each degree of latitude. Its lasting impact, however, was to strengthen the growing conviction that the origin of the Earth’s field was located deep inside the planet, instead of on, or above its surface. The consideration of additional sites of observed zero declination meanwhile led Iberian and Dutch cosmographers to postulate a more complex arrangement of two tilted dipoles, resulting in two perpendicular great circles that quartered the world into alternating sections of northeasting and northwesting. In the Dutch Republic, cartographer Petrus Plancius (1552–1622) turned such a hypothesis into a practical method to find longitude, which was used at sea for about three decades. His compatriot Simon Stevin (1548–1620) instead put forward the first sextupole hypothesis. Other schemes developed by European navigators proposed a locally valid ratio between distance traveled along a parallel of latitude and change in needle stance. The striking diversity of explanations formulated during the 16th century bears evidence, not just of the numerical paucity of geomagnetic data, but also of poor charts (longitudinal uncertainty) and lack of standardization in instruments and measurement, generating error margins large enough to tentatively confirm the witnessed variety of (one or multiple) tilted dipole arrangements and other explanations.

The 17th century With the arrival of overseas trading companies, missionary networks, and better instruments, both the quantity and quality of geomagnetic data substantially increased in the 17th century. Moreover, this period witnessed several official efforts to process and publish these measurements, such as by the newly founded scientific societies (Royal Society, Académie Royale des Sciences), and the hydrographic office of some East India Companies. Observations of declination and inclination also appeared in printed compilations, sometimes ordered by latitude or longitude, both for mariners’ and scholars’ benefit. In addition, the first manuscript chart (now lost) depicting curved isogonics (lines connecting all points with equal declination) was produced in the 1620s by the Italian Jesuit Christoforo Borro (or Bruno), a teacher of navigation in Portugal. Although isogonic charts would never attain prominence as a tool in navigational practice, they did serve to underline that geomagnetism implied more than a collection of isolated data points, enabling global visualization of both a priori and empirically derived patterns. The continuing stream of readings from places near and far furthermore stressed the inadequacy and simplification of imposing a tilted dipole model on observed reality. The nascent geomagnetic discipline therefore avidly sought more appropriate explanations, while being increasingly influenced by competing paradigms of natural philosophy. The stage was set with the publication of De Magnete in 1600, by Elizabethan court physician William Gilbert (1544–1603) (q.v.). This work contained the famous

357

conclusion that “the Earth itself is a great magnet.” Gilbert posited an axial dipole field distorted by the attraction of iron-rich continental landmasses (and more localized sources); a single Arctic magnetic pole was thus supplemented by global crustal heterogeneity, dispensing with any perceived regularity. Regarding cosmology, the Englishman maintained that the geomagnetic force was likewise responsible for the Earth’s orientation and diurnal rotation, and the orbit of other planets, a view that brought him into conflict with the religious order of the Jesuits (q.v.). To the likes of Nicolo Cabeo (1629), Athanasius Kircher (1639), and Jacques Grandamy (1645), who supported geocentrism, Gilbert’s universal “magnetic philosophy” was anathema; they claimed instead that it was geomagnetism which held the Earth firmly fixed at the center of creation. However, upon closer examination, Jesuit interpretations of the Earth’s (seemingly irregular) field proved far from consistent themselves, attributing its source to quasiorganic magnetic “fibers,” mines of iron ore, subterranean heat, chemical processes, and the Earth’s heterogeneous crustal constitution (Daujat, 1945). Outside of religious circles, the composition of the deep Earth also evoked much speculation and debate; Galileo Galilei postulated intense pressures at the core, while others imagined an infernal furnace there, driving hot sulfurous gases through a huge system of caverns. The notion of a hollow Earth would become even more important after the discovery of secular variation. The first sustained series of measurements at a single site, which gave rise to the realization that the geomagnetic field was subject to time-dependent change, was compiled in east London (foremost by naval commander William Borough, instrument maker John Marr, and Gresham astronomers Edmund Gunter and Henry Gellibrand. It was Gellibrand (q.v.) who eventually published all findings in 1635, concluding “a sensible diminution” at London of about seven degrees westerly over the period 1580–1634 (Chapman and Bartels, 1940). It was one of the earliest scientific conclusions based on averaging sets of measurements, a procedure through which observational error was substantially reduced. The notion of geomagnetic change over time was accepted by scholars across Europe within two decades, forcing a reassessment of all earlier work. Compiled undated measurements became useless, as did all time-invariant geomagnetic hypotheses put forward so far. The acceptance of secular variation thus inaugurated the third phase of geomagnetic models, which introduced a temporal parameter. Nonetheless, this variable was not necessarily always quantified. Kircher, for example, interpreted time-variance as resulting from the slow “generation” and “deterioration” of iron mines in the crust, a view shared by the philosopher René Descartes (1596–1650), who additionally blamed atmospheric circulation as affecting the vortices of tiny magnetic particles he postulated as the essence of the magnetic force (1644). A more atomistic (but equally qualitative) micromechanistic hypothesis was formulated by his compatriot Pierre Gassendi (1592–1655). Once (geo)magnetic corpuscularism reached English soil, research into magnetic particle circulation was initially promoted by the Royal Society, but by the 1680s, “magnetic philosophy” lost much of its former identity, becoming subsumed in broader theories involving the ether and effluvial emanations of matter in general. Although these speculations would eventually lead to the formulation of the laws of electromagnetism, none of these concepts made much impact on English geomagnetic hypotheses throughout the 17th century. The path followed here led scholars instead to reinstate the notion of polar attraction in dynamic terms, through one or more dipoles precessing around the Earth’s rotation axis in hundreds of years. Such solutions were launched by Henry Bond (1639), Henry Phillippes (1659), Robert Hooke (1674), Peter Perkins (1680), Edmond Halley (1683 and 1692) (q.v.), and Edward Harrison (1696). In addition to an array of parameters to define these geomagnetic models (precessional period and direction, dipole colatitude and longitude at a given year), a physical rationale was added by some, for example, Halley’s magnetic core rotating relative to the crust, separated from the latter by a fluid or gaseous intermediary layer. Even more parameters were required

358

GEOMAGNETISM, HISTORY OF

in the 18th century multipole variant (several hollow spheres nested around a common kernel), which allowed dipoles at different depths to precess with distinct speeds. This greatly expanded the scope for accommodating the observed asymmetrical global distribution of surface declination, as visualized on the first printed isogonic charts. These were based on the first scientific naval surveys, by Edmond Halley in 1698–1700.

The 18th century Halley’s declination charts were revised twice, by mathematics teachers William Mountaine and James Dodson (1744, 1756). Isogonics were likewise drawn on near-global-scale by engineer Frezier (1717), physicist Van Musschenbroeck (1729, 1744), Captain Nicolaas van Ewyk (1752), cartographer Samuel Dunn (1775), natural philosopher Johann Heinrich Lambert (1777), and astronomer Le Monnier (1778). Physicist Johann Karl Wilcke published the first world chart depicting isoclinics (lines of equal dip) in 1768, based on inclinations observed on board Swedish Eastindiamen. In the last decade of the century, surveyor John Churchman moreover published several editions of his “Magnetic Atlas,” which delineated both isogonics and isoclinics worldwide. These efforts were based on unprecedented data compilations, mostly derived from maritime sources. Mountaine and Dodson’s, for instance, used over 50000 geomagnetic observations for their 1756 world chart. More accurate compasses with enlarged sights, a greater volume of world shipping than ever before, and improved infrastructures for processing and disseminating magnetic data, allowed increased data accuracy at higher resolution over larger parts of the globe. Strenuous regimes of daily or hourly observations at fixed locations furthermore offered workers the chance to explore diurnal variation, secular acceleration, and the link between erratic needle behavior and the aurora polaris. In this century, major cities such as London and Paris maintained almost uninterrupted series of annual observations at astronomical observatories. Eighteenth century geomagnetic hypotheses displayed the full gamut of existing explanations, with proponents of static and dynamic tilted dipoles and multipoles in numerous countries. Nevertheless, data and theory still continued to disagree to an appreciable extent. In 1732, Royal Society Fellow Servington Savery was the first to propose that irregular surface topography of the magnetic core might be to blame. Mathematical work by Leonhard Euler (1757) furthermore laid the groundwork for the abandonment of the assumption of pairs of poles in diametrical opposition (Fleury Mottelay, 1922). Implicitly, some earlier multipole solutions had featured nonantipodal configurations, but only after Euler did the fourth phase of geomagnetic hypothesis receive explicit attention, with detailed discussion of each pole’s position and movements inside the Earth. In the most extreme cases (from the 1790s), the two disjointed poles of a dipole were assigned different coordinates, directions, and velocities. And like in earlier phases, when a single dipole was no longer deemed sufficient to account for the observations, another one was added. The disjointed quadrupole solution (1819) developed by Norwegian mathematician Christopher Hansteen (1784–1873) is a case in point, incorporating compass readings from 74 sea voyages. But eventually, even four independent magnetic poles proved insufficient, heralding the end of the belief that magnetized needles were solely governed by the attraction of a few, distant, all-powerful geomagnetic poles. The four main phases of geomagnetic hypotheses up till then are illustrated in Figure G53.

The era of the global field The 19th century Relative geomagnetic intensity data began to be compiled in earnest from the 1790s, by comparing the time it took a magnetized needle (horizontally or vertically) displaced by a standard distance from its

Figure G53 The four phases of geomagnetic hypotheses based on polar attraction. Top left, axial dipole (static); top right, tilted dipole (static); bottom left, precessing dipole (dynamic, tilted, antipodal); bottom right, disjointed dipole (dynamic, tilted, nonantipodal). (From Jonkers, A.R.T., Earth’s Magnetism in the Age of Sail, p. 36, Fig. 2.2. ã 2003 Johns Hopkins University Press. Reprinted with permission of The Johns Hopkins University Press.)

preferred orientation to return to it, or by timing a given number of such oscillations. Admiral de Rossel, on d’Entrecasteaux’s voyage in search of the lost expedition of La Pérouse, obtained such readings (1791–1794) relative to Brest. Naturalist Alexander von Humboldt (1768–1859), on his south American explorations (1799–1804), instead chose the number of dip oscillations in 10 min on the magnetic equator (where vertical geomagnetic intensity is lowest) at Micuipampa, Peru as his reference value. Other readings obtained at intermediate latitudes led him to postulate the “law” of decreasing (vertical) magnetic force from the magnetic poles to the magnetic equator. After Humboldt’s return to Paris, this initial standard of relative intensity soon became eclipsed by the Paris unit, whereas similar local units based on swing times at London, Christiania (Oslo), and St. Petersburg found more limited application elsewhere. Global isodynamics (lines of equal intensity) were drawn from 1825. Two periods in the 19th century stand out as of particular significance for geomagnetism: the first of these was 1820–1840. After Hans Christian Ørsted’s (1777–1851) discovery in 1820 of electromagnetism, and André Marie Ampère’s (1775–1836) “Theorie Mathématique des Phénomènes Électro-dynamiques” (1826), it was Michael Faraday (1791–1867) who discovered electromagnetic induction in 1831, and who applied it to build the first dynamo (a copper disk rotating in a strong magnetic field, with an electrical circuit connecting rim and center). The year thereafter, Carl Friedrich Gauss built the first absolute magnetometer in his Göttingen laboratory (21 May 1832). Units of absolute intensity soon replaced all relative scales, and apart from an increasing number of surveys on land, networks of fixed observatories were set up in numerous countries. Geomagnetism gradually

GEOMAGNETISM, HISTORY OF

became separated from traditional ferromagnetic studies, championed instead by professional astronomers as one of a range of interconnected “telluric” forces (including heat, volcanism, and atmospheric electricity), in a framework of terrestrial or cosmic physics. Such phenomena obviously warranted globally coordinated data collection. Already from 1828, von Humboldt had tirelessly advocated international cooperation in geomagnetism, personally establishing a string of observatories in Europe and Asia that made simultaneous measurements at specific intervals. In 1834, Gauss and his colleague Wilhelm Weber (1804–1891) joined this scheme, expanding it to about 50 stations, known as the “Göttingen Magnetic Union” (1836– 1841). At the time, the three recorded elements were declination, inclination, and intensity. When these had all been mapped on a global scale for (approximately) the same era (declination 1833, inclination 1836, intensity 1837), it became possible to interpolate the complete magnetic vector for any point on Earth. This is what Gauss set out to do in 1839. He promulgated the idea of representing the magnetic field by the gradient of a potential function, to be written as a linear combination of a series of spherical harmonic coefficients (or Gauss coefficients), to be derived by least-squares analysis. Assuming a spherical Earth, and converting the charted data into X, Y, and Z components at fixed intervals of colatitude and longitude, Gauss obtained the first 24 coefficients (equivalent to an expansion up to degree and order 4). The results, laid down in his 1839 “Allgemeine Theorie des Erdmagnetismus” showed that the strength of the internal field overwhelmingly dominated any possible external sources affecting the surface field. This conclusion contrasted starkly with the “cosmological” interpretation, which emphasized the importance of the external field. While many workers employed, expanded, and refined Gauss’s method in subsequent decades (over 300 such field models have been computed since, Jacobs, 1987), army officer Edward Sabine (1788–1883) (q.v.), rejected Gauss’s theory, reverting instead to Hansteen’s disjointed quadrupole as a working hypothesis, the verification of which culminated in the British movement known as the “Magnetic Crusade” of the 1840s. This consisted mainly of two Antarctic exploration voyages (1839–1842) under James Clark Ross (1800–1862), and a network of colonial observatories (1840–1848) run by the British Admiralty, the War Office, and the English East India Company, supplemented by over twenty, mostly European, Russian, and American stations (Cawood, 1979). At a new geomagnetic observatory at Kew (est. 1841), Sabine and his staff processed the network data, plus those from ship’s logbooks, land surveys, and earlier compilations, to be published in 15 “Contributions to Terrestrial Magnetism” (1840–1877) in the Royal Society’s “Philosophical Transactions.” The most notable results were Sabine’s distinction of diurnal variation of internal and external origin, and his 1851 discovery that the periodicity of magnetic storms was correlated with the 11-year sunspot cycle. Nevertheless, some observatories proved less permanent than originally envisaged, and geomagnetic land surveys in the second half of the century often had to rely on more established disciplines for funding, as part of grand empirical programmes. Whereas France and Russia were the main sponsors of oceanic circumnavigations up to the midcentury (acquiring substantially better coverage in geomagnetic field measurements, not least in the Pacific Ocean), a plethora of nations (Germany, Sweden, Austria-Hungary, Britain, Norway, the United States, Denmark, and others) partook in the Arctic exploration frenzy characteristic of the second peak period of geomagnetism in the 19th century, from about 1870 to 1885. These ventures, partly stimulated by the First International Polar Year (1882– 1883), followed the policy adopted on naval surveys of carrying geomagnetic instruments as standard equipment. Other significant empirical contributions were made on the British “Challenger” (1872–1875), the German “Gazelle” (1874–1876), and the Swedish “Vanadis” (1883–1885) expedition.

359

On land, networks of repeat stations rose to prominence where observatory coverage was sparse (eventually reaching a total of about 3000). These carefully chosen, fixed and marked locations were to be revisited for a few days at regular intervals (1, 2, 5, or 10 years) to study secular and diurnal variation. Time-dependent change on the historical timescale was meanwhile studied in the 1890s by physicist Willem van Bemmelen (1868–1941) (q.v.), who reconstructed 16and 17th century isogonics from nautical data in 165 historical ship’s logbooks (based in part on original manuscripts). Arctic and observatory data were also reexamined time and again, forming the basis for new spherical harmonic expansions. By this time, geomagnetism (and related geoelectricity) had become almost exclusively the fieldtheoretical domain of physicists.

The 20th century Empirical geomagnetism continued to prosper in the 20th century. Besides the European colonial powers, the United States, Russia, and Japan likewise established new observatories and organized new surveys to track the field’s appearance and change. Moreover, Antarctica became a novel focal point of investigation, for example, on Robert Falcon Scott’s (1868–1912) first Antarctic expedition (1901–1904), and later during the Second International Polar Year (1932–1933). In 1904, Louis Agricola Bauer (1865–1932) (q.v.), at the Carnegie Institution in Washington (q.v.), established the “Department of International Research in Terrestrial Magnetism,” which employed the freighter “Galilee” (1905–1908) and the nonmagnetic vessel Carnegie (1909–1929) (q.v.), for oceanic geomagnetic surveys. A similar initiative, with the “Zarya,” was launched by the USSR in 1956, just before the International Geophysical Year (1957–1958) renewed interest in geomagnetic observations through its “World Magnetic Survey.” From 1953, novel magnetometers were towed at distance behind a ship to avoid deviation effects, a practice still routinely carried out on recent ocean-bound scientific missions. With the advent of new technology, land observatories no longer recorded the local vector in terms of angles and total intensity, but adopted the three orthogonal vector components (X, Y, and Z ). Airborne magnetic surveys (in particular to study the crustal field) became increasingly common in the second half of the century, for example, in the “World Magnetic Modelling and Charting” programme of the US. Defence Mapping Agency. Satellite measurements of the Earth’s magnetic field started with the Soviet “Sputnik 3” (1958), maintained in subsequent decades by their “Vostoks” and the American “OGO” series (see POGO). In 1979 the United States launched Magsat (q.v.), the first geomagnetic vector satellite, followed by the European “Ørsted” (1999) (q.v.), and “CHAMP” (2000) (q.v.). Once workers were able to compare the near-global, uniform coverage provided by satellites with earlier field maps and historical datasets, the continuity of some static and dynamic features provided theoreticians new food for thought. Important strides forward were also made in the theoretical domain before that time. At the turn of the 20th century, a great number of nondynamo theories (q.v.) were launched, based on thermoelectricity and (gravitational or geochemical) charge separation. Another hypothesis explored by several workers is the one often associated with the work of Patrick M.S. Blackett (1897–1974) (q.v.), which posited that the Earth’s magnetic field could be due to its rotation (Good, 1998). Two of Blackett’s former students explored this idea: Edward C. Bullard (1907–1980) (q.v.) suggested that such a distributed source would cause the field to be weaker at greater depth, whereas actual measurements made in deep mine shafts by S. Keith Runcorn (1922–1995) (q.v.), ultimately concurred with Blackett’s own laboratory research, in supporting a negative conclusion. A different hypothesis, more in keeping with a growing body of seismological evidence that delineated a fluid outer core and solid inner core, originated with the supposition of Joseph Larmor in 1919 that the Sun’s magnetic field might operate as a self-exciting dynamo.

360

GILBERT, WILLIAM (1544–1603)

Building on the mathematical foundations of electromagnetism laid down by James Clark Maxwell (1831–1879) in 1861–1864, and armed with new methods of analysis, workers in the 1940s and 1950s developed the theory of the fluid geodynamo, which still forms the basis of current understanding regarding geomagnetism. In 1939, Walter Elsasser published “On the Origin of the Earth’s Magnetic Field,” in which turbulent convective motions were deemed to create inhomogeneities in the liquid metallic interior of the Earth’s core, giving rise to thermoelectric currents. After World War II, Elsasser focused on kinematic instabilities, positing in 1946 that a toroidal core field could play a vital role in generating and sustaining a self-exciting dynamo process. In the late 1940s and 1950s, the theory was elaborated further by Bullard and George E. Backus (q.v.), among others. Despite initial skepticism, first after publication of Cowling’s theorem (1934, q.v.), and later, after the demise of the Bullard-Gellman numerical solution, dynamo theory has since gained, in broad outline, near universal acceptance. Nevertheless, important areas of controversy remain (for example, regarding the magnetohydrodynamic wave and frozen flux hypotheses), and to date, no causal geomagnetic model exists that simultaneously and accurately mimics physics and behavior of the Earth’s field on all relevant timescales. By the end of the 20th century, the discipline of geomagnetism, which in the 1930s had split into the three subdisciplines of paleomagnetism, internal field modeling, and investigations of the external field, had initiated a move toward reintegration, combining data, theories, and constraints from all. In addition, field maps no longer only captured the surface field, but likewise displayed geomagnetic features at the top of the source region, at the Core-Mantle Boundary (q.v.), and increasing attention was being devoted to possible coupling mechanisms at work across this interface. Advances in mathematics, instruments, and computing power likewise opened up vast new horizons to explore. Together, these developments promise geomagnetists in the 21st century many exciting, more comprehensive insights into the myriad fascinating aspects of their chosen field of study.

Cross-references Bauer, Louis Agricola (1865–1932) Bemmelen, Willem van (1868–1941) Blackett, Patrick Maynard Stuart, Baron of Chelsea (1897–1974) Bullard, Edward Crisp (1907–1980) Carnegie Institution of Washington, Department of Terrestrial Magnetism Carnegie, Research Vessel CHAMP Chapman, Sydney (1888–1970) Compass Core-Mantle Boundary Cowling’s Theorem Elsasser, Walter M. (1904–1991) Gauss, Carl Friedrich (1777–1855) Gellibrand, Henry (1597–1636) Geodynamo Gilbert William (1544–1603) Halley, Edmond (1656–1742) Harmonics, Spherical Humboldt, Alexander von (1759–1859) Instrumentation, History of Jesuits, Role in Geomagnetism Kircher, Athanasius (1602–1680) Larmor, Joseph (1857–1942) Magsat Nondynamo Theories Norman, Robert (1560–1585) Ørsted POGO (OGO-2, -4, and -6 Spacecraft) Runcorn, S. Keith (1922–1995) Sabine, Edward (1788–1883) Voyages Making Geomagnetic Measurements

Art R.T. Jonkers

Bibliography Balmer, H., 1956. Beiträge zur Geschichte der Erkenntnis des Erdmagnetismus. Veröffentlichungen der Schweizerischen Gesellschaft für die Geschichte der Medizin und Naturwissenschaft 20. Aarau: H. R. Sauerländer. Bullard, E.C., 1954. Homogeneous dynamos and terrestrial magnetism. Philosophical Transactions of the Royal Society of London, A, 247: 213–278. Cawood, J., 1979. The magnetic crusade: science and politics in Early Victorian Britain. Isis, 70(254): 493–518. Chapman, S., and Bartels, J., 1940. Geomagnetism, 2 vols. Oxford: Clarendon Press. Daujat, J., 1945. Origines et Formation de la Théorie des Phénomènes Électriques et Magnétiques. Exposés d’Histoire et Philosophie des Sciences vol. 989–991. Paris: Hermann. Fleury Mottelay, P., 1922. Bibliographical History of Electricity and Magnetism Chronologically Arranged. London: Charles Griffin. Good, G.A. (ed.), 1998. Sciences of the Earth: An Encyclopedia of Events, People, and Phenomena. New York, London: Garland. Jacobs, J.A. (ed.), 1987. Geomagnetism, 4 vols. London: Academic Press. Jonkers, A.R.T., 2003. Earth’s Magnetism in the Age of Sail. Baltimore, MD: Johns Hopkins University Press. Needham, J., 1962. Science and Civilisation in China, vol. 4. Cambridge: Cambridge University Press. Smith, J.A., 1992. Precursors to Peregrinus: the early history of magnetism and the Mariner’s compass in Europe. Journal of Medieval History, 18: 21–74. Still, A., 1946. Soul of Lodestone: The Background of Magnetical Science. Toronto: Murray Hill.

GILBERT, WILLIAM (1544–1603) William Gilbert (sometimes spelled “Gilberd”) was born in Colchester, England, in 1544, the son of Jerome, a successful Burgess and Recorder of the town, and his wife Elizabeth. From St John’s College, Cambridge, where he graduated M.D. in 1569, and of which foundation he became a Senior Fellow, serving the College in a variety of offices, William Gilbert began to practice medicine in London. Sometime before 1581, he was admitted into the prestigious Royal College of Physicians, and over the years occupied several of its senior offices. His professional career was crowned in 1600 when he was appointed Physician to Queen Elizabeth I, though as that monarch wisely avoided medical attention, one suspects that this distinguished appointment was, in practical terms, a sinecure. When King James I ascended to the throne in March 1603, Gilbert continued as Royal doctor, only to die later that year on November 30, 1603. Gilbert never married. Gilbert’s enduring fame, however, lies not in his medical achievements, which were conventional, but in his pioneering and privately conducted researches into magnetism, geomagnetism, and electrostatics, which he published in De Magnete, magneticisque corporibus, et de magno magnete tellure; physiologia nova, plurimis et argumentis, et experimentis demonstrata (London, 1600). For as this long title tells us, Gilbert’s book was far more than just a book on magnets, but developed an argument, based on abundant experimental evidences, that magnetism was a property of the terrestrial globe itself. In short, it was the first coherent treatise on geomagnetism, and in this respect, is a foundation text of modern experimental physics, having a profound impact on all subsequent researches, and influencing figures such as Kepler, Galileo, Hooke, Newton, Halley, and the early Fellows of the Royal Society.

GILBERT, WILLIAM (1544–1603)

At the heart of Gilbert’s scientific rationale was his skepticism about aspects of the philosophy of Aristotle, which was enshrined in university curricula across Europe, and which saw the earth as essentially passive and made of a fundamentally different stuff from the dynamic and shining heavens. Magnetism, however, tended to challenge such an attitude, suggesting that the earth possessed a lively dynamic force of its own: a force, moreover, that was amenable to experimental investigation. For it is very clear that Gilbert was part of a tradition of emerging experimental practice in Elizabethan England. A practice sometimes seen as closer in principle to the mechanic practice of artisans than to the refined philosophy of university men, and which Gilbert’s younger contemporary, Sir Francis Bacon, would develop into a coherent model of how to investigate nature by means of experiments in his Novum Organon (1620) (q.v.). Central to the argument of De Magnete is that the earth itself is a spherical magnet, with a north and south pole. Moving between these poles in what we might call a curved field was the invisible magnetic force deriving from the very being, or “soul,” of the earth itself. This force was actively present in all magnetite stone and ferrous metals, and Gilbert knew, from writers going back to Petrus Peregrinus in 1269 and before, that suspended iron needles would orientate themselves along it. He also knew from his Elizabethan contemporary, the compass-maker and artisan-scientist, Robert Norman, in 1581, that this force not only acted in a lateral plane, but also in a vertical one, to produce the magnetic dip. Central to Gilbert’s geomagnetic ideas, and probably picked up initially from Peter Peregrinus’ Epistola, were his experiments conducted upon what he called “terrellae,” or little earths. These were spheres of magnetic material, probably lumps of spherically chiseled magnetite, all of which were found to have magnetic poles, equators, and contours, just like the earth. These characteristics were discovered with an instrument, which Gilbert sometimes called his magnetized “Versorium” (presumably from Latin versare, “to turn around”), which was a delicately poised magnetic needle, capable of moving both horizontally and vertically when held near the terrella. Over the terrella’s magnetic equator, for instance, the needle tended to position itself north-south horizontally, or in a tangent to the equator. But as it moved forever closer to a pole, it dipped in its angle, and stood vertically upon reaching one or other pole. This suggested a continuous force field emanating from and connecting the poles in a series of invisible arches, which were at their flattest over the equator. Gilbert also found that the slight irregularities in the terrella produced in turn local force fields irregularities, which he saw as analogous to local variations in the magnetic field of the earth itself. The major aspect of Gilbert’s genius as an experimental physicist was his realization that one could study and model phenomena with the terrella in the laboratory that were physically identical to the phenomena exhibited by the globe of the earth itself as it hung in space. A concept, indeed, so familiar to modern scientific practice as to be taken for granted, but outrageous for the 16th century, for Aristotle taught not only that the heavens were fundamentally different from the earth, but that beneath the fire, air, and water that surrounded our planet, there was a primary element of Earth. Yet how could this Earth be homogeneous or at one with itself, if parts of it were magnetic and other parts were not? Gilbert’s terrella experiments enabled him to develop a coherent and verifiable model for the Earth’s magnetic field, explaining the northfinding properties of compass needles, local irregularities, and the dip. It was arguably the first experimentally based comprehensive theory in the history of physics, and it is hardly surprising that several subsequent generations of scientists found inspiration in his work. From his laboratory and terrestrial studies, Gilbert then took the portentous step of developing a magnetic cosmology in Book VI of De Magnete. For one thing, he argued that the Earth’s magnetic field suggested that our planet rotated on its axis, contra Aristotle, Ptolemy, and the classical philosophers, who said that it was stationary, with the universe rotating around us. And while he never formally proclaimed

361

himself a Copernican, all of Gilbert’s cosmological arguments presumed the Sun, and not the Earth, to be at the center of the solar system. He also argued that, instead of being made of a unique cosmological fifth element, as the ancients had thought, the planets themselves were probably made of magnetic material, and moved through space under the influence of magnetic force fields. Very important in this context, moreover, was his abandonment of Aristotle’s cosmological divide of the Moon’s orbit, which was believed to separate the terrestrial from the celestial realms. Substantiated in part from Tycho Brahe’s recent astronomical discoveries, And in his posthumously published De Mundo (Amsterdam, 1651), Gilbert suggested that space was, in effect, homogeneous and empty: without qualitative divisions, yet traversed by magnetic forces which were themselves the sources of all motion. He also speculated that the diffuse light of the Milky Way might be occasioned by masses of very distant stars, no individual star of which we could see from Earth, though here, in some respects, Gilbert was in keeping with earlier medieval writers such as Jean Buridan, Nicolas Oresme, and Simon Tunsted, to name but a few. There is no evidence to suggest that Gilbert’s failure to openly embrace the Copernican theory derived from a fear of religious persecution. Copernicanism only became a contentious issue for the Roman Catholic Church after Galileo used it for his own highly adversarial purposes after 1612, while the Church of England never had any official policies on scientific issues one way or the other. One suspects that his reluctance comes from covering his back professionally, as an eminent physician. Academic medicine was a deeply conservative art in Gilbert’s time, and learned physicians risked professional suicide if they openly proclaimed novel ideas, which cast doubt on the timehonored wisdom of the ancients. After all, Gilbert’s Royal doctor and Physicians’ College colleague of the next generation, William Harvey, found that his published discovery of the circulation of the blood in 1628 badly damaged his practice as a society doctor. Writing a book about magnetism was one thing, but openly espousing a theory, so contradictory to common sense as Copernicanism then seemed in 1600 was risking being branded an unsound man. Not a good trait for a Royal doctor to have attributed to him, indeed! In addition to his experimental and speculative cosmological work, De Magnete also aspired to present a history of and devise a taxonomy for magnetic phenomena. And very significantly, in Chapter 2 of Book II (out of the six books into which De Magnete is divided) Gilbert set out his researches into the properties of amber, jet, and other substances, which displayed what he called Electric characteristics. (The term, which introduced the words electric and electrical into the modern world, was derived by Gilbert from the Greek word for amber: electrum.) From his experiments, he differentiated between magnetic phenomena proper, which he saw as innate and permanent properties of the stuff from which God had created the world, and friction-generated electric phenomena, which were a short-lasting product of the residual moisture or effluvium of once-fluid substances, such as those resins which solidified into amber, and which could be temporarily excited, and draw things to themselves by rubbing. Though by modern standards Gilbert’s explanations were wrong, his recognition that magnetism and electrical phenomena were two quite different forms of attraction was correct. Gilbert’s De Magnete was one of those milestone books in the history of science which turned a hitherto vague and confused collection of observations that was magnetics into a coherent discipline, the phenomena of which could be tested at leisure by its readers and applied to new situations and monitored with refined instruments. Its taxonomy of phenomena, moreover, introduced new terms, such as magnetic polarity and electric into general usage. And very portentously, it began that transformation away from the Aristotelian doctrine of motion to something much more dynamic, and, by its exploration through a series of tests and hypotheses, laid the foundation for modern experimental physics. Allan Chapman

362

GRAVITATIONAL TORQUE

Bibliography Stephen, P., ‘William Gilbert’, 2004. New Dictionary of National Biography. Oxford: Oxford University Press. Taylor, E.G.R., 1954, 1968. The Mathematical Practitioners of Tudor and Stuart England 1485–1714. Cambridge: Cambridge University Press, p. 174, no. 31. William Gilbert, 1600. De Magnete. London: Chiswick press. See P. Fleury Mottelay, William Gilbert of Colchester...on the Great Magnet of the Earth (Ann Arbor, 1893). Re-issued in facsimile as De Magnete, New York: Dover, 1958.

GRAVITATIONAL TORQUE The expression can be given a wide definition. We will restrict the presentation to the torques mutually exerted one on the other by the three envelopes of the internal Earth (inner core, outer core, and mantle) through the perturbative gravitational force field superimposed on the gravitational field of an equilibrium state—hydrostatic in the fluid core.

The equilibrium state For the sake of simplicity we consider the inner core and mantle to be rigid. In fact the mantle is convecting, and density anomalies exist, as well as bumps at the core-mantle boundary (CMB). A disturbed gravitational potential results in the mantle, liquid outer core (Wahr and de Vries, 1989), and inner core. Our hydrostatic state is the state in which the mantle, outer core and inner core are at rest with respect to one another under self-gravitation potential C0 ð^rÞ (^r is the current ! point), the whole Earth rotating with uniform angular velocity O0 around its polar axis of inertia fixed in space (we ignore precession, although an external gravitational force is responsible for driving it). Let r0 ð^rÞ and p0 ð^rÞ be the corresponding density and pressure fields in the liquid outer core. We have: ! ~ ðC0 þ U0 Þ ¼ rp ~ 0 ; rC ~ 0 ¼
! O0 ^ ð O0 ^~ r0 r rÞ r2 U0 ¼
4p Gr0 G is the gravitational constant, C0 the rotational potential. Level surfaces of U0 þ C0 , r0 , p0 are parallel. It is essential in the following that U0 be nonaxisymmetric. In fact r0 changes with time, but we consider time constants much shorter than the time constant of the convection in the mantle. This (pseudo) static equilibrium state corresponds to a minimum gravitational energy, and no mutual torque exists. We will now consider perturbations from this state and the resulting gravitational torque. We separate different problems for the sake of clarity.

The mantle—inner core gravitational coupling and l.o.d. Let us ignore in a first step the convection in the core (density heterogeneities responsible for this convection are of the order of 10
8 r0 , r0 being the mean core density, whereas those generated by mass anomalies in the mantle are of the order of 10
4 r0 . Consider that r0 ð^rÞ is not axisymmetric, for example r0 ð^rÞ ¼ r0 ðrÞ ð1 þ D22 ðrÞ P22 ðcos yÞ cos 2ðj
j0 ÞÞ, y and j being colatitude and longitude; (this quadrupole term is evidenced by seismic studies) and that a difference (AS–BS) results between the equatorial moments of inertia of the inner core. Let the mantle and inner core be rotated by angles jm and js , respectively,

from their equilibrium positions. A restoring gravitational torque is exerted by the mantle on the inner core (and vice versa), proportional to the relative rotation (Buffet, 1996a):
m
js Þ Gi ¼ Gðj

(Eq. 1)


(the action of the outer core is just to contribute to the value of G). That means that the inner core is gravitationally locked to—aligned
can be computed from with—the mantle in the equilibrium figure. G the gravitational energy, which is minimum in the equilibrium state. Small oscillations of the inner core and mantle around the rotation ! axis O0 —i.e., oscillations in the length of the day—arise whose period is proportional to the polar momentum of inertia of the inner core
Periods of few years (approximately) and inversely proportional to G. are obtained with current models of density anomalies in the mantle. The liquid outer core does not play an important role in the former mechanism, but it can be the seat of the so-called torsional oscillations (Braginsky, 1970; Zatman and Bloxham, 1998; Hide et al., 2000; Dumberry and Bloxham, 2003; Jault and Legaut, 2005) made of rotations o ðsÞ of the geostrophic cylinders C(s) of radius s (whose axis is ! the rotation axis O0 ). The adjacent cylinders are coupled through electromagnetic forces due to the radial cylindrical component of the main field, and the inner ones are coupled with the solid inner core through magnetic friction. The inner core and mantle are as before coupled by gravitational forces. Again, if the inner core and mantle happen to be moved with respect one to the other, restoring gravitational torques will generate small oscillations around the equilibrium configuration. Their periods are increased up to several decades due to the involvement of geostrophic cylinders (Buffet, 1996b). As for the excitation of these necessarily damped oscillations, it could be due to the dynamo process itself.

The mantle—inner core gravitational coupling and nutations The system made of the rotating mantle, outer core and inner core displays small motions of the rotation axes of the three envelopes, called nutations. Forced nutations are generated by the lunisolar tidal potential. Free nutations are rotational eigenmodes of the system. Two modes are ! nearly diurnal in a frame corotating with the mantle ð O0 Þ, the free core rotation (FCN) and the free inner core nutation (FICN). Two modes have periods much longer than a day in this same frame, the Chandler wobble and the inner core wobble (Mathews et al., 1991). The gravitational torque exerted by the mantle and outer core on the ellipsoidal inner core has a profound effect on the inner core wobble and FICN. This torque results from the tilt of the inner core (due to the equatorial component of its rotation ! os ) or misalignment of the figure axes of the inner core and the mantle (Xu and Zseto, 1996). In other words, as before in the axial problem, the restoring gravitational torque tends to lock the inner core with respect to the mantle (here the flattening of an axisymmetric earth is to be invoked). If the inner core is rigid, or purely elastic, the locking is strong. If it is viscous, it yields to the gravitational forces and tends to cancel the torque. In the limit of a liquid inner core, the inner core wobble disappears. As for the period of the FICN (in an absolute frame), it may vary, in the absence of magnetic field, from 75 days (liquid inner core) to 485 days (elastic inner core) (Greff-Lefftz et al., 2000).

Gravitational forces due to the convective density heterogeneity in the core Consider now the independent of (r0 ) density heterogeneity r0, thermal or compositional, generating the convective flow ~ u driving the dynamo. Extra gravitational and rotational potentials appear: ! ! ! r0 ! r0 þ r0 ; U0 ! U0 þ U 0 ; O0 ! O0 þ O0 ; C0 ! C0 þ C0

GRAVITATIONAL TORQUE

A gravitational torque is exerted on the mantle by this density distribution (in this section we ignore the inner core): ZZZ

! ~ 0 dv (Eq. 2) r0 ~ r ^ rU Gg ¼ M

(we ignore the effect of C0 for the sake of simplicity); of course an equal and opposite torque is applied on the core by the mantle. (a) The axial problem ! ^k being the unit vector of ! O (or O0 ) ZZZ ZZZ ]U 0 ]r ! ! dv ¼ Ggz ¼ Gg  k ¼
r0 U 0 0 dv ]j ]j M

M

that makes it clear again that only the non axisymmetric part of r0 intervenes in the axial torque. If IM ! is the polar moment of inertia of ! the mantle and oz the component of O0 along O0 , we have IM

do0z dt

¼ Ggz

 Z ]U 0 ]r
U 0 0 dz dj Ggz ðsÞ ¼ ds r0 ]j ]j s, j, z being the cylindrical coordinates. An accelerated rotation of ! ^, governed by CðsÞ results, with linear velocity t1 ðs; t Þ ¼ t1 ðs; t Þ j the equations   ]t1 ðs; t Þ do0z
¼ Gg ðsÞ I ðsÞ (Eq. 3) ]t dt Z a do0z ¼
Gg ðsÞ ds ¼ Ggz dt 0

(Eq. 4)

I(s) is the moment of inertia of cylinder CðsÞ, a is the radius of the core. The reason why the geotrophic flow t1 ðs; t Þ is accelerated is that it is the only one for which the growing Coriolis force can be balanced by a growing pressure torque. Equations (3) and (4) would govern changes in o0z , i.e., changes in l.o.d generated by gravitational ! forces linked to r0 , should Gg be only acting. In fact it is rather artificial to separate here the gravitational torque from the topographic torque (action of the pressure field linked to flow ~ u on the CMB bumps). Furthermore the electromagnetic forces which couple the adjacent cylinders (and the outer core with the inner core and the poorly conducting mantle) provide the dissipative mechanism, which prevents rotation of the core relative to the mantle to grow indefinitely. Unfortunately the evaluation of the gravitational torque, which requires the knowledge of both the anomalous (nonaxisymmetric) part of r0 and the convective density heterogeneity r‘, is far from being easy. It could generate significant variations in l.o.d. ! (b) The gravitational torque Gg and the pole motion Let us consider the equatorial component of the gravitational torque Eq. (2) and the Euler equations for the mantle (i.e., the equations of the equatorial components of the angular momentum of the mantle ! in a frame rotating with angular velocity O0 ): AM

dO0
i ðCE
AE Þ O0 O0 þ i O20 c ¼ Gg dt

with the usual notations: AM is the equatorial moment of inertia of the mantle, CE and AE the polar and equatorial moments of inertia of the whole Earth (taken axisymmetric), O0 ¼ o01 þ io02 , Gg ¼ Gg1 þ i Gg2 , o01 and o02 being the two equatorial orthogonal compffiffiffiffiffiffiffi ! ! ponents of O0 , Gg1 and Gg2 the same components of Gg ; i ¼
1. And c ¼ c0E1 þ i c0E2 is the change in the Earth equatorial moment of inertia due to the perturbations r0 . Again, it is artificial to separate here the gravitational torque from the pressure torque (Hulot and Le Mouel, 1996). The gravitational torque linked to a time varying r0 —i.e., a flow—might be invoked to excite the Chandler wobble; but short time constants ( 435 days) are requested; or to generate the so-called Markovitz wobble; but balances with other torques have to be considered (Hulot and Le Mouel, 1996). Note that dynamo modeling could now provide estimates of the gravitational torque due to the corresponding core flow. Jean-Louis Le Mouel

Bibliography

! if Gg is only acting. The effect of the gravitational forces on the fluid core itself is more subtle since it does not react like a rigid body. Let us again consider the cylinders CðsÞ of radius s and thickness ds. The axial gravitational torque acting on C ðsÞ is (Jault and le Mouel, 1989):

IM

363

Braginski, S.I., 1970. Torsional magnetohydrodynamic vibrations in the Earth’s core and variations in day length. Geomagnetism and Aeronomy, English Translation, 10: 1–8. Buffet, B., 1996a. Gravitational oscillations in the length of day. Geophysical Research Letters, 23: 2279–2282. Buffet, B., 1996b. A mechanism for decade fluctuations in the length of day. Geophysical Research Letters, 25: 3803–3806. Dumberry, M., and Bloxham, J., 2003. Torque balance, Taylor’s constraint and torsional oscillations in a numerical model of the geodynamo. Physics of the Earth and Planetary Interiors, 140: 29–51. Greff-Lefftz, M., Legros H., and Dehant, V., 2000. Influence on the inner core viscosity on the rotational modes of the Earth. Physics of the Earth and Planetary Interiors, 122: 187–204. Hide, R., Boggs, D.H., and Dickey, J.O., 2000. Angular momentum fluctuations within the Earth’s liquid core and torsional oscillations of the core-mantle system. Geophysical Journal International, 143: 777–786. Hulot, G., Le Huy, M., and Le Mouel, J.L., 1996. Influence of core flows on the decade variations of the polar motion. Geophysical and Astrophysical Fluid dynamics, 82: 35–67. Jault, D., and Le Mouel, J.-L., 1989. The topographic torque associated with a tangentially geostrophic motion at the core surface and inferences on the flow inside the core. Geophysical and Astrophysical Fluid Dynamics, 48: 273–296. Jault, D., and Legaut, G., 2005. Fluid dynamics and dynamos in a strophysics and geophysics. In Soward, A.M., Jones, C.A., Hughes, D.W., and Weiss, N.O., (eds.) The Fluid Mechanics of Astrophysics and Geophysics Series, vol. 12, chap. 9. London: Commodity Resource Corporation Press, pp. 277–293. Mathews, P.M., Buffett, B.A., Herring, T.A., and Shapiro, I.L., 1991. Forced nutations of the Earth: influence of inner core dynamics. I. Theory. Journal of Geophysical Research, 96(B5): 8219–8242. Wahr, J., and de Vries, D., 1989. The possibility of lateral structure inside the core and its implications for rotation and Earth tide observations. Geophysical Journal International, 99: 511–519. Xu, S., and Szeto, A.M.K., 1996. Gravitational coupling within the Earth: computation and reconciliation. Physics of the Earth and Planetary Interiors, 97: 95–107. Zatman, S., and Bloxham, J., 1998. A one-dimensional map of Bs from torsional oscillations of the Earth’s core. In Gurnis, M., Wysession, M.E., Knittle, E., and Buffett, B.A., (eds.), The Core-mantle Boundary Region. Geophysical Monograph, 28, Geodynamics series. Washington, DC: American Geophysical Union, pp. 183–196.

364

GRAVITY-INERTIO WAVES AND INERTIAL OSCILLATIONS

GRAVITY-INERTIO WAVES AND INERTIAL OSCILLATIONS Earth’s outer core is a predominantly iron, rotating liquid body, bounded by a silicate mantle, that is quasispherical. Thus an understanding of the global dynamics of the core can be had by considering it to be an incompressible fluid contained by an approximately spherical, rigid boundary. If the core were rotating at a constant rate as though it were a solid, individual parcels of fluid would be at rest relative to Earth’s surface. A parcel of fluid displaced from equilibrium would experience a buoyancy force if the density were not uniform, and a coriolis force due to Earth’s rotation, resulting in what are called core undertones, their periods being large compared to Earth’s free (elastic) oscillations. The combined result of these two restoring forces would produce gravity-inertio waves or gravito-inertial waves as they are sometimes termed. In Earth’s atmosphere they are usually termed inertio-gravity waves while in the Sun they are called g-modes; more generally they are described as internal gravity waves modified by rotation or simply gravity waves. If there were no density effects, just inertial waves result. The local motion of a fluid parcel in this latter case is an inertial oscillation. A member of the triply infinite set of inertial modes can be excited in a contained rotating fluid like the core if there is a global source of excitation at an appropriate frequency and will decay by viscous dissipation in the boundary layers and interior. The physics of these modes is discussed below where the range of frequencies for each type of response is found heuristically from basic physical principles.

like the core, a doubly infinite set of axially symmetric inertial oscillations can exist, usually called inertial modes. Surfaces on which disturbance pressure vanishes form nodal lines which are ellipses in meridional planes. Some examples of these lines which divide the fluid into cells are given in Figure G55 for a unit sphere. The small ticks at the surface show what is known as the critical colatitude for the mode where y satisfies Eq. (1). Tangents to the surface at this location define a direction within the fluid, called a characteristic, along which small disturbance will propagate as illustrated in Figure G54.

Inertial oscillations Starting with the simplest case, assume that there are no density differences from place to place in the core and consider a uniform core rotating at the diurnal rate. A ring of fluid encircling the axis of rotation is held in equilibrium by the balance between centrifugal and radial pressure gradient forces on the ring. Expanding the ring slows it down to conserve angular momentum so that the centrifugal force in the expanded position is now less than the local, axially directed radial pressure gradient. Accordingly, the ring is forced back to its starting position just as it would be if the ring had been initially contracted toward the axis of rotation. Thus the ring of fluid is stable with respect to radial perturbations so it oscillates.

Figure G54 Characteristic lines for an inertial wave of frequency o ¼ O propagating in a sphere. The critical colatitude y ¼ 60 is shown corresponding to the point where a tangent to the sphere parallels a characteristic line.

Frequency of oscillations A ring of fluid displaced in a direction perpendicular to the axis of rotation will oscillate at frequency o ¼ 2O where O is the assumed steady rotation rate of Earth. If instead of a radial displacement of the fluid ring, it were moved axially with its motion parallel to the rotation axis, no restoring force would exist so that o ¼ 0. Displacement of the ring in any other direction would produce a component of restoring force only in the direction perpendicular to the rotation axis giving o ¼ 2O cosy

(Eq. 1)

where y is the colatitude of the ring as shown in Figure G54. Thus any frequency of oscillation o in the range 0 < o < 2O is allowed so that for the idealized constant density core considered here (see also adiabatic gradient in the core), the shortest period of oscillation would be approximately 12 h. Individual particle motions follow elliptical paths that are the same in every meridional plane. These are axisymmetric inertial oscillations (Aldridge and Toomre, 1969; Greenspan, 1969) that have particle motions predominantly parallel to the rotation axis when o ! 0 and perpendicular to this axis when o ! 2O. In a bounded rotating fluid

Figure G55 Nodal surfaces of pressure for some axisymmetric inertial oscillations with velocity shown by sketched curves. Index (n, m) corresponds to the mth root of the Legendre polynomial P2nþ2 ðo=2OÞ ¼ 0. (From Aldridge and Toomre, 1969. With permission.)

GRAVITY-INERTIO WAVES AND INERTIAL OSCILLATIONS

The source of these characteristics is that the differential equation describing the gravity waves is hyperbolic in space and it must satisfy a no-slip boundary condition at the surface of the core-mantle boundary (CMB). A boundary value problem which is hyperbolic (rather than elliptic) is said to be ill-posed. No continuous solutions are guaranteed for such problems so that approximate and laboratory methods must be used to solve them. A notable example is the limiting case o ¼ 0 (steady motion) where y ¼ 90 so the tangent is parallel to the rotation axis; this is a Taylor column, the axial column seen in a rotating fluid when an object is towed steadily through the fluid. These disturbances are inertial waves and are discussed below.

Amplitude of oscillations Inertial oscillations could be excited in Earth’s core through coupling of the interior fluid at the boundary by viscous forces. If the boundary motion is sufficiently well-coupled spatially to a mode—e.g., the semidiurnal tide has very similar spatial structure to a mode with azimuthal dependence cos2f where f is geographic longitude. There are several modes with periods near the semidiurnal tide (Aldridge et al., 1988). Modes coupled to the CMB by viscosity will be excited on a timescale of spin-up or E
1=2 O
1 where E is the Ekman number for the core. For laminar flow this turns out to be about 5 ka. This is also the timescale for free decay, which is mostly caused by boundary layer dissipation (see also Boundary layers in the core) with additional internal dissipation which increases dramatically for modes of smaller spatial structure. Theorems have been developed (Zhang et al., 2001) that imply no dissipation for inertial modes in a sphere; though elegant they are not relevant for real fluids since they only apply to ideal (inviscid) fluids.

Inertial waves At frequencies o ! 2O the waves travel in the direction of the axis of rotation while at intervening frequencies they travel in directions given by y in Eq. (1). The particle motion of the waves is perpendicular to the propagation direction since it is the Coriolis force that is acting on the particle. Thus the inertial waves are transverse. The waves are reflected at boundaries such that they maintain equal angles of reflection and incidence with the direction of the rotation axis rather than a normal to the reflecting boundary. Thus these waves will have multiple reflections in a closed container. In a spherical shell of fluid, multiple reflections lead to repeated paths (Hollerbach and Kerswell, 1995; Tilgner, 1999) which are known as attractors and have been studied extensively (Rieutord et al., 2001). The assumption of axial symmetry can be relaxed by introducing azimuthal dependence and this reveals that there is a triple infinity of inertial modes. The critical directions become cones that define the propagation of waves in the fluid. Inertial waves have been excited in laboratory experiments (Noir et al., 2001) on precession of a spheroidal shell of fluid (see also Fluid dynamic experiments, Precession and core dynamics). Since an observer in the rotating frame of reference with the container’s spin will see the precession as a diurnal disturbance, y ¼ arccosð1=2Þ ¼ 60 which corresponds to the characteristics illustrated earlier in Figure G54. It is estimated that the precession will excite inertial waves of amplitude 6  10
6 m s
1 in a band 20 km wide in Earth’s core, thus setting a level for their detectability. At semidiurnal periods, inertial modes with azimuthal dependence cos2f that are viscously coupled to the CMB will produce a semidiurnal wobble as angular momentum of Earth must be conserved. In principle this wobble should be detectable using both VLBI and supergravimetry as changes in latitude of observatories but so far only limited success has been achieved in the search for evidence of these modes in the core (Aldridge and Cannon, 1993).

365

Gravity-inertio waves The previously ignored buoyancy force can be returned by assuming the fluid is stratified. Stratification is characterized by N2 ¼


g dr r dr

where N is the Brunt-Väisälä frequency, g is acceleration due to gravity, and rðrÞ is the fluid density assumed to be a function of spherical radius r. (The
sign is included to ensure N 2 > 0 as rðrÞ is a decreasing function of r.) With rotation both coriolis and buoyancy forces act on a fluid parcel. If buoyancy dominates coriolis in the sense that N > 2O then wave gravity wave frequencies are such that 2O  o  N while if stratification is very weak so that N < 2O, then 2O o N : In the presence of stratification, the characteristics shown in Figure G54 become curved (Dintrans et al., 1999). In principle gravity waves should be detectable at Earth’s surface since there must be a redistribution of mass when fluid parcels oscillate (Crossley et al., 1991). Changes in gravity are due to this redistribution of mass directly as well as the movement of Earth’s surface through the gradient of the gravity field. Amplitudes are extremely small, estimated at 10
11 m s
2 , and are only likely to be detectable by using a global array of superconducting gravity meters as organized under the Global Geodynamics Project (GGP). Successful detection of gravity waves in Earth’s core would constrain the core’s density (q.v.) distribution as has already been demonstrated for the Sun (see also Helioseismology).

Instability of inertial modes The stable inertial modes described above can be destabilized by straining of the fluid streamlines (Kerswell, 1993, 2002). This fact has significant implications for Earth’s core as it provides for a mechanism to pump energy parametrically into the core at a diurnal rate and cause a disturbance to grow at a rate determined by the strain. For Earth, this strain is very small leading timescales of instability growth of several thousand years (Aldridge and Baker, 2003) (see also Turbulence in the core). The strain comes from two sources, Earth’s precession (q.v.) that introduces a predominately shearing of the core fluid’s streamlines and Earth’s semidiurnal tide which deforms the streamlines into ellipses. The former is small because of the relatively long period of the precession compared to the diurnal rotation while the latter owes its small size to the amplitude of tidal deformation in the core. At present there is a small amount of evidence for the existence of parametric instability in Earth’s fluid core. Based on laboratory observations of rotating parametric instabilities (Aldridge, 2003), a search has been made for a signature of these instabilities in records of relative paleointensity (q.v.) obtained from seafloor sediments. Initial results of this work (Aldridge and Baker, 2003) confirmed that a 400 ka record of relative paleointensities yielded a sequence of geophysically plausible growths and decays of magnetic field intensity over several thousand years that is consistent with what would be expected for rotating parametric instabilities. On longer timescales, tidal deformation and precessional forcing have been identified in paleomagnetic records. For example, variation in relative paleomagnetic intensity corresponding to Earth’s obliquity has been reported (Kent and Opdyke, 1977). Although the origin of these variations has been often considered to be climatic, recent paleomagnetic records have revealed robust

366

¨ NEISEN’S PARAMETER FOR IRON AND EARTH’S CORE GRU

evidence of external forcing that is independent of climatological variations. Spectral analysis of the relative paleointensity record from ocean sediments (Channell et al., 1997) show modulation of intensity corresponding to orbital obliquity and eccentricity periods. The inclination error found in a 2 Ma long record from the west Caroline basin (Yamazaki and Oda, 2002) correlates to orbital eccentricity. Although gravitational energy released through compositional convection is considered to drive the core’s geodynamo that maintains the geomagnetic field, other phenomena like rotational parametric instabilities may prove to play a significant role in maintaining the geodynamo. Keith Aldridge

Bibliography Aldridge, K.D., 2003. Dynamics of the core at short periods: theory, experiments and observations. In Jones, C.A., Soward, A.M., and Zhang, K. (eds.), Earth’s Core and Lower Mantle, The Fluid Mechanics of Astrophysics and Geophysics. London: Taylor & Francis. Aldridge, K.D., and Baker, R.E., 2003. Paleomagnetic intensity data: a window on the dynamics of Earth’s fluid core? Physics of the Earth and Planetary Interiors, 140: 91–100. Aldridge, K.D., and Cannon, W.H., 1993. A search for evidence of short period polar motion in VLBI and supergravimetry observations. Proceedings of the IUGG XX Assembly, Symposium U6: Dynamics of the Earth’s Deep Interior and Earth Rotation, American Geophysical Union Geophysical Monograph 72, pp. 17–24. Aldridge, K.D., and Toomre, A., 1969. Axisymmetric inertial oscillations of a fluid in a rotating spherical container. Journal of Fluid Mechanics, 37: 307–323. Aldridge, K.D., Lumb, L.I., and Henderson, G., 1988. Inertial modes in the Earth’s fluid outer core. Proceedings of the International Union of Geodesy and Geophysics XIX Assembly, Symposium U2: Instability within the Earth and core dynamics, American Geophysical Union Monograph 46, pp. 13–21. Channell, J.E.T., Hodell, D.A., and Lehman, B., 1997. Relative geomagnetic paleointensity and d18 O at ODP Site 983 (Garder Drift, North Atlantic) since 350 ka. Earth and Planetary Science Letters, 153: 103–118. Crossley, D., Hinderer, J., and Legros, H., 1991. On the excitation, detection and damping of core modes. Physics of the Earth and Planetary Interiors, 116: 68–97. Dintrans, B., Rieutord, M., and Valdettaro, L., 1999. Gravito-inertial waves in a rotating stratified spherical shell. Journal of Fluid Mechanics, 398: 271–297. Greenspan, H., 1969. The Theory of Rotating Fluids. Cambridge: Cambridge University Press. Hollerbach, R., and Kerswell, R.R., 1995. Oscillatory internal shear layers in rotating and precessing flows. Journal of Fluid Mechanics, 298: 327–339. Kent, D.V., and Opdyke, N., 1977. Paleomagnetic field intensity variations recorded in a Brunhes epoch deep-sea sediment core. Nature, 266: 156–159. Kerswell, R.R., 1993. The instability of precessing flow. Geophysical and Astrophysical Fluid Dynamics, 72: 107–114. Kerswell, R.R., 2002. Elliptical instability. Annual Review of Fluid Mechanics, 34: 83–113. Noir, J., Brito, D., Aldridge, K., and Cardin, P., 2001. Experimental evidence of inertial waves in a precessing spheroidal cavity. Geophysical Research Letters, 19: 3785–3788. Rieutord, M., Georgeot, B., and Valdettaro, L., 2001. Inertial waves in a rotating spherical shell: attractors and asymptotic spectrum. Journal of Fluid Mechanics, 435: 103–144.

Tilgner, A., 1999. Driven inertial oscillations in spherical shells. Physical Review E, 59: 1789–1794. Yamazaki, T., and Oda, H., 2002. Orbital influence on Earth’s magnetic field: 100000 year periodicity in inclination. Science, 295: 2435–2438. Zhang, K., Earnshaw, P., Liao, X., and Busse, F.H., 2001. On inertial waves in a rotating fluid sphere. Journal of Fluid Mechanics, 437: 103–119.

Cross-references Core Density Core Turbulence Core, Adiabatic Gradient Core, Boundary Layers Fluid Dynamics Experiments Helioseismology Paleointensity, Relative, in Sediments Precession and Core Dynamics

¨ NEISEN’S PARAMETER FOR IRON AND GRU EARTH’S CORE Introduction The Grüneisen parameter g is a necessary tool for assessing Earth’s thermal properties. Here we emphasize aspects of g that are pertinent to Earth’s core. g has various definitions, many of which are derivable one from another by thermodynamic identities. The following equation for g quantifies the relationship between the thermal and elastic properties of a solid. The parameter g can be considered as a measure of the change in pressure P resulting from an increase in energy density at constant volume V. It is dimensionless, since pressure and DU=V have the same units  g¼V

]P ]U

 (Eq. 1) V

where U is the internal energy. From Eq. (1) it is seen that g connects pressure and energy, and therefore, pressure and temperature. Elasticity properties of Earth’s core (especially pressure) come from seismological data. Gamma is useful to obtain energy and temperature from these seismological data. Another example of the importance of g is in the expression used for thermal pressure at high temperature PTH ¼

g ETH V

(Eq. 2)

the so-called Mie-Grüneisen relationship (ETH is the thermal energy and PTH is the thermal pressure). This equation is often used with an isothermal equation of state, P(V,T0). Equation (1) is the general statement of the relationship between pressure and energy of which Eq. (2) is a special case. Equation (2) can be considered as the historical presentation of g, attributed to Grüneisen (1926), who derived the equation for pressure as a function of volume and temperature in an early version of lattice dynamics. For many physical properties of the Earth’s interior, we need to know the value of g of a solid at the pressure and temperature of the Earth’s interior, not just at ambient conditions. We need to know how g changes with T, especially at high T, if at all, and how g changes with pressure (or rather, volume), especially at high compression.

¨ NEISEN’S PARAMETER FOR IRON AND EARTH’S CORE GRU

Thermodynamic derivations of gamma In order to evaluate Eq. (1) in parameters representing measurable physical properties, start with the following equation 

]P ]U



 ¼

V

]P ]T

   ]U ]T V V

(Eq. 3)

The definition of specific heat is ð]U=]T ÞV ¼ CV at unit mass (all quantities are per unit mass). The numerator on the right side of Eq. (3) is found from calculus: ð]P=]T ÞV ¼
ð]V =]T ÞP =ð]V =]PÞT ¼ aKT , where a is the volume thermal expansivity, and KT is the isothermal bulk modulus. Equation (3) is therefore equivalent to 

]P ]U

 ¼ V

aKT CV

(Eq. 4)



]P ]U

 ¼ V

aKT V CV

(Eq. 5)

g as given by Eq. (5) is composed of individual measurable physical properties, each of which varies significantly with temperature. The ratio of these properties as given by Eq. (5), however, does not vary greatly with temperature, and often not at all. There are many approximations to Eq. (5) for g. The few approximations we will use here will be given special subscripts and names. Another method of finding g involves adiabatic compression. Start with one of Maxwell’s relationships: 

]T ]V



 ¼


S

]P ]S

3N X

V

Expand the right side of Eq. (6): ð]P=]SÞV ¼ ð]P=]T ÞV ð]T =]SÞV . By using ð]P=]T ÞP ¼ aKT and ð]S=]T ÞV ¼ CV , the right side of this equation becomes
T aKT =CV . Using Eq. (5), the right side becomes T g=V , so that Eq. (6) can also be written as     ] ln T ] ln T ¼ g¼
] ln V S ] ln r S

(Eq. 7)

Equation (7) is the thermodynamic basis for finding the adiabatic thermal gradient in the core and mantle. Equations (1), (2), (5), and (7) are thermodynamically equivalent definitions of g. The choice of the equation to use depends on the parameters at hand and the result desired.

Lattice dynamic derivations of gamma The lattice dynamic view of a solid is that of statistical mechanics: a solid is composed of N atoms (where N is Avogadro’s number); each atom is an oscillator having three degrees of freedom and connected to neighboring atoms by a spring. The solid’s thermodynamic properties are found from the dynamics of 3N vibrations with modal frequencies n1 , n2 , n3 ; . . . ; n3N . Each modal vibration arises from a simple harmonic oscillation, the energy of which is given by an Einstein function, and the frequency of which is classically related to the atomic mass and the spring constant. For a monatomic solid, such as iron, all the masses are equal. A standard treatment of this subject shows that the Helmholtz free energy for a monatomic solid of N degrees of freedom is (Slater, 1939)

(Eq. 8)

where k is Boltzmann’s constant and
h is h=2p, where h itself is Planck’s constant, and where the quantity under the summation sign is the Einstein function. The isothermal equation of state P(V ) is found by differentiating Eq. (8) with respect to V at constant T, giving   3N hnj
]U 0 1X  P¼
þ g  ]V T V j j e
hnj =kT
1

(Eq. 9)

Attention is directed to gj , called the mode gamma, which arises from the V derivative gj ¼


] ln nj ] ln V

 (Eq. 10) T

Although gj is dimensionless, its value is influenced by the rate of change of mode frequency with volume. Thus, in order for gj to be nonzero, it must change with volume, and for it to be positive, it must decrease with volume. It is customary to make an assumption (called the quasiharmonic assumption) that the mode gammas depend on volume but are independent of temperature. For core physics, where the temperature is high, the high-temperature limit of the above equations is needed. The Debye temperature YD marks the division between the high temperature region and the quantum state region of a solid. For a monatomic solid, YD is (Anderson, 1995)

 (Eq. 6)



kT ln 1
e

hni =kT

j



Using Eq. (4) in Eq. (1), the most useful definition of g is found: g¼V

F ¼ U0 þ

367

YD ¼ 251:2

 1=3 r vm k m

(Eq. 11)

where r is density (in g cm
3 ), m is the atomic mass number (55.85 for iron), and vm is the mean sound velocity (in km s
1 ), given in terms of the longitudinal and shear velocities as "  3 # vp 3 1 ¼ 2þ v3m v3s vs

(Eq. 12)

The numerical factor in Eq. (11) is composed of the atomic constants h, k, and N.
Body-centered cubic (bcc) iron is the phase of iron that exists at ambient conditions. There is much data on this well-known phase. We use the properties of bcc iron to explain principles such as the evaluation of Eq. (11), but our chief interest lies in hexagonal closepacked (hcp) iron, which is the most likely pure iron phase at core conditions. In the evaluation of Eq. (11), properties of ambient bcc iron are r, 7:87 g cm
3 ; vp , 5:9 km s
1 ; vs , 3:25 km s
1 ; vm , 3:62 km s
1 (note that vm is only slightly larger than vs). This gives YD ðT ¼ 300Þ ¼ 415 K at ambient conditions. Thus, above 415 K, bcc iron is a classical solid, and below, it is a quantum solid. Since the core is in the 5000–6000 K temperature range, its properties are in the high T regime. Since the value of YD =T at core temperatures is of the order of 0.1, the exponential term in Eq. (8) is of the order of 10
5 , insignificant compared to unity. To find the lattice dynamical equation for g, divide Eq. (5) into two factors as follows: g¼

ðaKT Þ CV =V

(Eq. 13)

¨ NEISEN’S PARAMETER FOR IRON AND EARTH’S CORE GRU

368

and evaluate each of the two parts separately from Eq. (8). From calculus, aKT ¼ ð]P=]T ÞV . CV ¼ ð]U=]T ÞV , where U is the internal energy. Using Eq. (9), ( )   ]P
h X 2 e yi ¼ (Eq. 14) yj gj ]T V V ðe yi
1Þ2 where yj ¼
hnj =kT . Since U ¼ F
ð]F =]T ÞV , then U ¼ E0 þ kT

3N X j¼1

yj ðe yj
1Þ

(Eq. 15)

and CV ¼ k

3N X j¼1

y2j e yj ðe yj
1Þ

2

¼

3N X

CVj

(Eq. 16)

j¼1

  1 V ]vs gs ¼
3 vs ]V T

Sumino and Anderson (1984) evaluated g for bcc iron using Eq. (5) and experimental values of a, KT, and CP. They found that g ¼ 1:81. They also evaluated g, using the acoustic approximation to g (Eq. (19)) and the measured elastic constant data under pressure and found g ¼ 1:81. Since the derivation of Eq. (19) starts with Eq. (5), it is concluded that the quasiharmonic assumption and the assumptions of the Debye solid do not modify the value of g for bcc iron. This is a way of demonstrating that bcc iron is a Debye solid. Debye’s theory of a solid assumes that the whole vibrational spectrum can be represented by the long wave limit where frequency is proportional to the wave number ðn ¼ vkÞ. All modes are acoustic, each with the same average velocity. Equation (19) can also be written in terms of the derivatives of the bulk modulus and the shear modulus, G gac ¼

Following Eq. (13), there are major cancellations, leaving P3N j gj CVj g ¼ P3N j CVj

(Eq. 17)

where the g means that this Grüneisen parameter may be affected by the approximation of the quasiharmonic assumption. Note that all factors containing y ¼
hnj =kT have cancelled out, leaving the temperature dependence alone in CV, but since there is input of CV in both the numerator and the denominator, there is very little temperature dependence in either the low- or high-temperature regimes. In the very hightemperature regime (such as found in the core), all CVj are equal to k, so the denominator in Eq. (17) is equal to 3Nk, while the numerator P is equal to k 3N gj . Thus, the high-temperature limit of the lattice dynamical Grüneisen parameter is the arithmetic average of all mode gammas (Barron, 1957). g¼

3N 1 X gj 3N

KT ð]KS =]PÞT þ ð4=3Þð]G=]PÞT KT ð]G=]PÞT 1 þ
6 6 KS þ ð4=3ÞG 3 G (Eq. 22)

(Stacey and Davis, 2004). Stacey and Davis (2004) stated, “The acoustic formula (for g) withstands critical scrutiny well. For application to the lower mantle it has no serious rival.” Thus, for the Debye solid, the 3N modal frequencies are replaced by one frequency, nD , and one gamma,   ] ln nD gD ¼
] ln V

(Eq. 23)

Since
hn0 ¼ kYD , the above can be replaced by   ] ln YD gD ¼
] ln V T

(Eq. 24)

A variant of Eq. (24) is (Eq. 18)

Simplifications made by assuming a Debye solid Evaluation of Eq. (18) is complicated by the fact that N (Avogadro’s number) is quite large. We wish to reduce the summation in this equation from a limit of 3N to a lower value. We accomplish this by invoking the assumptions of a Debye solid (Debye, 1912), which are: the solid is monatomic and isotropic; its frequency spectrum (modal frequency versus wave number k) is quadratic with a sharp cutoff at the maximum frequency called the Debye frequency, vD; the slope of the longitudinal and shear nj versus k curves is constant; and there are no optic modes. Barron (1957) showed that invoking the properties of a Debye solid reduces the sum over 3N modes in Eq. (18) to a sum of 3 modes. 1 gac ¼ ðgp þ 2gs Þ 3

(Eq. 21)

(Eq. 19)

where gp is related to the longitudinal sound velocity and gs to the shear sound velocity. In Eq. (19), gac has replaced g from Eq. (18) because acoustic information is the sole input to gamma. The mode gammas can be expressed in terms of velocity derivatives (Anderson, 1995),   1 V ]vp gp ¼
(Eq. 20) 3 vp ]V T



YD YD0

 ¼

 g D r r0

(Eq. 25)

which is true only for a Debye solid. Equation (24) is used as follows. Measure YD as a function of V, and thus determine g as a function of V. Equation (25) is used to find a shift in YD corresponding to a shift in density. A stringent test of the applicability of the theory of Debye solid to properties of bcc iron is that the frequency spectra of the real solid (called the phonon density of states, PDOS) has a sharp cutoff at a frequency corresponding to the Debye temperature.

Proving epsilon (hcp) iron is a Debye solid In the last section, bcc iron was proven to be a Debye solid. That is only one of many proofs that have been made for this solid. Early in the 20th century, a popular proof was to measure the entropy versus T data for temperatures below the Debye temperature and show that there was good agreement with the entropy calculated from the Debye theory. Unquestionably, bcc iron is taken as a Debye solid. It is not evident however, that hcp iron should be a Debye solid because this phase of iron has hexagonal symmetry, which requires two atoms in the lattice dynamic cell. Consequently, there are vibrational modes in which both atoms move in the cell, but the center of mass remains fixed. Such a vibration is an optical mode, and thus hcp iron has an optical branch in its frequency spectrum, which usually implies that there are modes with frequencies much higher than nD .

¨ NEISEN’S PARAMETER FOR IRON AND EARTH’S CORE GRU

For hcp iron, however, the two masses in the cell have equal values, and lattice dynamic theory shows that in this case the frequencies found in the optic branch are contained within a narrow band anchored close to the maximum acoustic frequency. This means that within the frequency spectrum f ðnÞ, called the phonon density of states (PDOS), there is a clustering of modes near the maximum acoustic value of f ðnÞ. A criterion for the validity of the Debye solid is that the high-frequency edge of the PDOS is the same (or nearly the same) as vD, which is determined by YD. Debye (1965) said, “it is not too important to know the details of the frequency spectrum because at high temperature one only has to know the number of degrees of freedom [see Eq. (18)] and at low temperature the high frequency modes carry less and less energy.” It is the maximum frequency of the PDOS that is significant. The crucial test for hcp iron is to show that the PDOS has a high-frequency, sharp cutoff close in value to vD. A recent paper by Giefers et al. (2002) reported the measured PDOS of hcp and bcc iron by nuclear inelastic scattering. Since hcp iron is a high-pressure phase and does not exist at room pressure, the experiment was done at P ¼ 28 GPa. An experiment was also done for bcc iron at P ¼ 0 GPa (resulting data for both shown in Figure G56). Their plots of the density of states, G(E), are reported in units of energy (meV) instead of the usual units of frequency, but they report the cutoff in terms of temperature, so the comparison with the Debye temperature is eased. Note that for both phases of iron there is a sharp cutoff of the spectrum with only a very small percentage of modes seen above the cutoff. This is proof that both phases are Debye solids. Further proofs come from the value of YD . The authors report that the value of the temperature corresponding to the cutoff is 511 K for hcp iron (at 29 GPa) and 417 K for bcc iron (at P ¼ 0). The value for hcp iron is to be compared with the measurement of YD reported by Anderson et al. (2001), 521 K at 29 GPa and discussed in the next section. These values are sufficiently close to conclude that hcp iron is a Debye solid. The value of YD ¼ 417 K for bcc iron reported by Giefers et al. (2002) is to be compared with YD ¼ 415 K found from Eq. (11). Anderson et al. (2001) also report that for hcp iron, the value of YD at P ¼ 0 is 446 K. This value can also be used to predict the cutoff of bcc iron by the bcc-hcp phase change using Eq. (25), and the ratio of the uncompressed density of bcc iron to hcp iron is (7.87/8.28) with g ¼ 1:7. The predicted YD is 410 K, to be compared with 417 K from the cutoff of the PDOS for bcc iron.

Figure G56 Experimental phonon density of states g(E ) of bcc iron and hcp iron by Giefers et al. (2002). For hcp iron, the pressure is 28 GPa. The cutoff of bcc iron at P ¼ 0 is 40 meV (with a corresponding temperature of 417 K). For hcp iron, the cutoff is 45 meV (with a corresponding temperature of 511 K). The Debye temperature of bcc iron is 415 K, according to Eq. (11), and the Debye temperature of hcp iron is 520 K according to Anderson et al. (2001) and as seen in Figure G57.

369

Experimental determination of lattice g(V) for iron from Eq. (24) The state of the art in diamond anvil pressure cells at high pressure has improved so that by using a controlled intense x-ray beam from a synchrotron radiation facility, determination of high quality powder x-ray diffraction data is possible. (The intensity of the diffracted x-ray beam is a measurable function of pressure.) For a Debye solid, the intensity of the beam is given by the mean-square amplitude of atomic displacements < u2 >. Classical x-ray theory relates < u2 > to the Debye temperature, which Gilvarry (1956) gives as < u2 > ¼

c
h2 T mkY2D

(Eq. 26)

where c is a numerical constant, and m is the atomic mass. Equation (26) with c ¼ 3 is found in Willis and Pryor (1975). Using this equation and the measured values of < u2 > versus V, Anderson et al. (2001) measured the experimental values of Y versus V for hcp iron from ambient pressure to P ¼ 300 GPa; these values are plotted in Figure G57. The value of g versus V was then found from Eq. (24). The data are plotted in g
P space as the lower curve in Figure G58. It is seen that at 330 GPa, gvib decreases with pressure from the zero pressure value of 1.7 to 1.2, the normal trend for nonmetallic solids. But metals have an additional contribution to gamma not found in nonmetallic solids arising from conduction electrons (often called free electrons). This contribution to g is especially important at high temperature. Since this electron contribution adds to the Helmholtz energy, g as defined by Eq. (1) will be sensitive to the electronic contribution. This means that for metals, the gamma found in this section is only a part of that defined by Eq. (1) at core temperatures. We therefore give the gamma shown in the lower curve of Figure G58 a special name, gvib , indicating vibrational energy, or that arising from the PDOS.

Figure G57 The experimental variation of Debye temperature YD for hcp iron (Anderson et al., 2001). For P ¼ 28 GPa, V ¼ 5:95 cm3 mol
1 with a corresponding YD ¼ 519 K.

¨ NEISEN’S PARAMETER FOR IRON AND EARTH’S CORE GRU

370

Some of the detailed calculations for hcp iron are given in Table G9. Values in Table G9 were found for T increasing with P along the solidus of hcp iron, as calculated by Anderson et al. (2003). The value of g descends from 1.62 at P ¼ 135 GPa to 1.53 at 330 GPa, a gradual decrease. The values of g in Table G9 are for the solid state edge of melting, called the solidus. It is seen that g for the ICB pressure (330 GPa) is 1.53, and for the CMB (135 GPa) it is 1.62. Figure G58 shows g (the top curve), as well as gvib , versus P.

First principles calculations of gamma for liquid iron

Figure G58 Variation of gvib and g for pure hcp iron with pressure (Anderson et al., 2003). For the core range of pressure, g varies between 1.62 at 135 GPa and 1.51 at 330 GPa. The calculations were made with temperatures from the hcp solidus, so the values of g represent the solid state edge of melting.

Accounting for the electronic contribution to gamma for iron The electronic contribution to the specific heat in a metal has a dominant effect at low T and sometimes at high T. The electronic contribution to specific heat, for example, is given by Kittel (1956) as 1 CVe ¼ p2 DðeF Þk 2 T 3

(Eq. 27)

where DðeF Þ is the electronic density of states at the Fermi energy level.1 The important point is that CVe increases steadily with T at high T, whereas the lattice specific heat levels out for T > YD and remains independent of T (neglecting anharmonic terms). At 6000 K, the specific heat of iron is 37% greater than the classical value of 3k per atom, due to Eq. (27). From Eq. (27), the electronic Grüneisen parameter is ge ¼

] ln DðeF Þ ] ln r

(Eq. 28)

Numerical evaluation of Eq. (28) (Bukowinski, 1977) gives ge ¼ 1:5. Thus, for iron the thermodynamic gamma of Sections and consists of gvib , suitably modified by ge. The formalism for finding g for metals is found in Bukowinski (1977); it requires the electronic specific heat found from the electronic density of states (EDOS). Anderson (2002a) found gc and g from gvib by using the PDOS and CVe , as presented by Stixrude et al. (1997). The theory of Bukowinski (1977) results in the useful equation, g¼

CVvib CV g þ eg CV vib CV e

(Eq. 29)

Core’s gamma: Consideration of impurities in iron Structural changes

where CV ¼ CVvib þ CVe

The value of g is likely to be different for the liquid state than for the solid state. Verhoogen (1980) first emphasized that the liquid state gl should be used for calculations of properties of Earth’s core. The difference between the values of the liquid gamma (gl ) and the solid gamma ðgs Þ for iron is quite large at low pressure but small at high pressure. The value of gl is 2.44 at P ¼ 0 (Stevenson, 1981), but that of gs is 1.66 at P ¼ 0 (Boehler and Ramakrishnan, 1980) for bcc iron. The value of gl decreases with pressure, becoming 1.63 at 30 GPa (Chen and Ahrens, 1997), and then changes much more slowly with further pressure, as we shall see. Measurements and/or calculations of gl at core pressures have long been needed. Important progress has been made in the theory of thermodynamic properties of condensed matter by calculations of parameter-free ab initio techniques using quantum mechanics. The use of ab initio techniques for several decades by many physicists has resulted in considerable advances in understanding of solid state properties. Recently, ab initio methods have advanced to the point that successful calculations of properties of the liquid state at high pressure and high temperature have been made. Using ab initio methods, Alfè et al. (2002a) calculated 1.51 for gl at P ¼ 330 GPa for iron along with the liquidus, giving Tm ¼ 6350 K. This is quite close to the value of gs (1.53) at P ¼ 330 GPa for hcp iron reported by Anderson et al. (2003), who found Tm ¼ 6050 K for the solidus (see Table G9). Thus, Dg at the ICB pressure is small, substantially smaller than at the CMB pressure, where gl ¼ 1:52 (Alfè et al., 2002a), while gs ¼ 1:62 (Anderson et al., 2003). For the solid state, Stixrude et al. (1997) reported gs ¼ 1:5 for hcp iron at ICB pressure from their ab initio calculations. Alfè et al. (2001) reported gs ¼ 1:52 for hcp iron at P ¼ 330 GPa, where the solidus is found to be 6250 K, as compared to 1.53, as obtained by Anderson et al. (2003). Thus, three laboratories have found very close values of gs at 330 GPa, 1:5 < gs < 1:53. The various values for gamma are plotted on the phase diagram of iron in Figure G59 (in the diagram, T is versus V with isobars as solid lines). We see that as the pressure increases, at core conditions, the difference between liquidus and solidus temperature values decreases and Dg decreases, as well. This is interpreted to mean that as pressure increases, the volume of crystallization diminishes. Indeed, Vocadlo et al. (2003) report DVm ¼ 0:77 cm mol
1 at 330 GPa, while Anderson (2002b) report DVm ¼ 0:55 also at 330 GPa from an analysis of shock wave data. The values of gl and gs approach each other at deep core pressures.

(Eq. 30)

1 For further information on the Fermi energy level for metals, see Kittel, Introduction to Solid State Physics, 2nd edn. New York: John Wiley & Sons, 1956.

One approach for finding the properties of the core is to learn all that is possible of pure iron at core pressure and then insert a certain quantity of light elements into the iron structure and determine the resulting changes. A number of problems arise, one of which is that the crystallographic structure is reported to change from hcp when impurities are added. The consensus is that hcp is the stable phase of pure iron, but there is strong evidence that for the light impurities in iron, light silicon, sulfur, and oxygen, the high-pressure stable structure is bcc.

¨ NEISEN’S PARAMETER FOR IRON AND EARTH’S CORE GRU

371

Table G9 Heat capacity, gvib and g along the calculated solidus of hcp iron V (cm3 mol
1)

Tm (K)a

P (GPa)

CVe a

gvib a

ga

6.060 6.000 5.800 5.600 5.500 5.300 5.181 5.100 5.000 4.900 4.800 4.700 4.620 4.600 4.500 4.400 4.316 4.300 4.200 4.100

2790 2840 3040 3270 3400 3690 3880 4030 4230 4440 4660 4910 5120 4170 5460 5770 6050 6100 6450 6830

55.0 58.1 71.6 88.0 97.5 119.5 135.0 146.5 162.2 179.7 199.2 220.8 240.0 244.9 271.7 301.7 330.0 335.3 372.8 414.9

1.331 1.336 1.360 1.390 1.407 1.445 1.472 1.491 1.517 1.545 1.575 1.607 1.633 1.640 1.675 1.712 1.743 1.749 1.788 1.827

1.62 1.61 1.57 1.53 1.51 1.47 1.44 1.43 1.41 1.35 1.37 1.35 1.32 1.32 1.30 1.28 1.25 1.25 1.23 1.20

1.74 1.73 1.70 1.68 1.67 1.64 1.62 1.62 1.61 1.60 1.59 1.58 1.56 1.56 1.55 1.54 1.53 1.53 1.52 1.50

a

Anderson et al. (2003).

calculations by Alfè et al. (2002b) incorporating as a boundary condition the seismically determined density jump at the inner core-outer core boundary found that substantial oxygen was required in the outer core because silicon and sulfur could not account for the size of the jump. From their conclusions of the amount and type of impurities in the core, Alfè et al. (2002c) found the freezing point depression at DT ¼
700 K  100 K. In a subsequent paper, Alfè et al. (2002d) reported the freezing point depression to be DT ¼
800 K, a value they have used in subsequent work. They estimated the core density deficit, the isobaric density of pure hcp iron, less the corresponding density of the core, to be 6.6% (Alfè et al., 2002c). The core temperature at the ICB was found by subtracting 800 K from the value 6350 K (the ab initio calculation of Tm ð330 GPaÞ for hcp iron), giving 5550 K. As seen in the previous paragraph, it is concluded from ab initio studies that

Figure G59 Values of gs (solid state) and g‘ (liquid state) plotted on the phase diagram of iron (T versus V with isobars as shown). Vocadlo et al. (2003) suggest that a mole of 4% silicon or sulfur, either in combination or separately, may change the structure of hcp iron to the bcc phase above 3000 K at high pressure. When light impurities are placed in iron, its density decreases. The percentage change in density, called the “core density deficit” (cdd), is reported to be 4% to 7% for the outer core. The cdd is used to find the DT drop at the ICB, giving the value of the liquidus temperature at P ¼ 330 GPa for the core. This DT is called the “freezing point depression.”

The effects of Si, S, and O impurities on the value of gl Alfè et al. (2002a) found from ab initio calculations that the core density deficit due to O, Si, and S, is 6.6%. The concentrations of these light elements were found in a series of papers in which chemical equilibrium between coexisting solid and liquid was treated. Ab initio

1. The difference between gl for pure iron and gl for the core at the ICB is only  0:02 despite the addition of 9% oxygen and 9% Si/Si to the hcp structure. This difference is trivial. 2. The difference between gl and gS at the ICB for both the core and hcp iron is  0:01; again, it is trivial. 3. The value of gl at the ICB of the core found by Alfè et al. (2002b) is so close to 1.5 that it has been taken to be 1.5 in the paper by Gubbins et al. (2003) in which D. Alfè and G.D. Price were coauthors. 4. The value of gl ¼ 1:5 at the ICB is associated with a cdd of 6.6% and a freezing point depression of
800 K. 5. The value of gl is virtually unchanged from the CMB pressure to the ICB pressure.

The effect of Ni in the core on the value of gc The relative abundances of Fe and Ni in the core are probably about 10
1 (McDonough and Sun, 1995). Thus, we consider the effect of Fe0.90-Ni0.10 in place of Fe. Mao et al. (1990) found that r for Fe0.80-Ni0.20 at 330 GPa and 300 K is about 2% higher than for pure Fe. Therefore, the alloy Fe0.90-Ni0.10 at the ICB is 1% denser than hcp iron.

372

¨ NEISEN’S PARAMETER FOR IRON AND EARTH’S CORE GRU

Table G10 Values of g, Tm, and a for pure hcp iron and Earth’s core Parameters Pure hcp iron, solidus, 330 GPa gs Tm (K) að10
5 K
1 Þ Pure hcp iron, solidus 135 GPa gs Tm (K) að10
5 K
1 Þ Pure hcp iron, liquidus 330 GPa gl Tm (K) að10
5 K
1 Þ Pure hcp iron, liquidus 135 GPa g‘ Tm (K) að10
5 K
1 Þ Core, liquidus, 330 GPa gl Tm (K) að10
5 K
1 Þ Core, liquidus, 135 GPa gl Tm (K) að10
5 K
1 Þ

Ab initio calculationsa

Thermal physics measurements and theoryb

1.50 6250 1.07

1.53 6050 0.7

1.519 4734 1.78

1.62 4062 2.5

K-primed EoS and seismic datac

1.518 6350 1.072 1.520 4734 1.78 1.50 5550 0.99

5100

1.391 5001 0.971

1.53 4111 1.70

4100

1.443 3739 1.804

a

Alfè et al. (2001, 2002a-d), Vocadlo et al. (2003). Anderson (2002b); Anderson et al. (2003), Isaak and Anderson (2003). Stacey and Davis (2004).

b c

A 1% increase in density would have virtually no effect on gl or gs because g is nearly constant with pressure in the vicinity of the ICB (Figures G58 and G59). From the data plotted in both figures, a 1% change in density changes g only slightly in the third significant figure. Thus, the value of gl is insensitive to the addition of nickel to iron. In the ab initio approach, all physical properties are found from derivatives of the Helmholtz free energy, F, which is made to account for all atoms (Fe and impurities, if any) in their respective lattice sites. The physical properties of the core are found from the core’s free energy. These properties are different from those found from the free energy of pure iron, except for the value of gl , which is virtually the same for both cases. How can this be explained? The value of gl changes with volume, but in the case of iron, the value of gamma changes very slowly with volume at core pressures, as shown in Figure G58. Consider the drop in temperature called the freezing point depression, DTm ¼
800 K. The relative change in volume due to freezing point depression is DV =V ¼
Dr=r ¼ aDT at the ICB. The value of a varies from 1:07  10
5 at 6350 K and 330 GPa (for pure iron) to 0:99  10
5 and 330 GPa for the core (D. Alfè, personal communication). If we know a and DT, then we can calculate Dr=r, the percent density change. The average a  DT gives Dr=r ¼
0:85%, less than but opposite to the 1% increase arising from placing 10% Ni in the Fe-Ni alloy.

Finding core properties independent of pure iron properties Stacey (2000) introduced a new equation of state called the K-primed approach because he found all EoSs used in geophysical treatment of Earth’s mantle to be in error with regard to the higher derivatives, especially the derivative of KS with respect to V. Stacey successfully

applied his K-primed equation of state to the Earth’s mantle and core. In Stacey and Davis (2004), thermal core properties were extracted directly from seismic data of Earth without the intermediate step of using properties of pure iron. The values of gamma for the core they report, to three significant figures, are 1.39 at the CMB pressure and 1.44 at the ICB pressure. They also report that the values of Tm for the liquidus are 5000 K and 3739 K, for the ICB and CMB, respectively.

Summary and conclusions Our prime focus is on the value of g at pressures of 135 GPa (CMB) and 330 GPa (ICB) for the liquidus and the solidus of hcp iron and for the liquidus of the outer core and the solidus of the inner core. The values of g and associated properties of thermal expansivity a and melting temperature Tm described in the previous sections are summarized in Table G10. It is noted that the values of gs at the ICB from the two approaches (ab initio and thermal physics) agree quite well. There is good agreement on the value of gl , but the value reported from the K-primed EoS approach is about 0.1 less than that reported from the ab initio approach. The ab initio approach yields the largest range of results, giving values for both gl and gs and for both hcp iron and the core, whereas the thermal physics approach is limited to gs for hcp iron and the inner core. The K-primed EoS approach is limited to gl for the outer core and gs for the inner core. An advantage of the K-primed EoS approach is that information about the chemistry—in particular, the concentrations of the various impurity elements—is not required to obtain gl of the core. However, this is also a disadvantage because no information can be given to geochemists about impurities. In contrast, the

¨ NEISEN’S PARAMETER FOR IRON AND EARTH’S CORE GRU

ab initio approach can give valuable results about core impurities: it was shown that sulfur and silicon as impurities in iron cannot account for the seismic density jump at the ICB, but about 8 mol% oxygen in the outer core is consistent with the seismic jump. From the results of the ab initio calculations, the value of gl is virtually constant from the ICB pressure to the CMB pressure, varying by 1.3%. The variation using the K-primed EoS approach is somewhat larger, 5.7%. If gl is assumed to be constant over this range, the relationship between Tm and density r given by Eq. (7) is simplified, 

Tm1 Tm0

 ¼

 g r1 r0

(Eq. 31)

Using the ratio of the seismic density of the core at the CMB to that at the ICB (12166.34/9903.49), Eq. (31) becomes   Tm1 ¼ 0:7344 for gl ¼ 1:5 Tm0 ¼ 0:7466

for

gl ¼ 1:42

Thus, if Eq. (31) is used to find Tm (CMB) from Tm (ICB), the answer will differ by only 1.2% if one uses the ab initio gamma in comparison with the K-primed EoS gamma (starting out with the same value of Tm0 ). If one uses the ab initio data or the thermal physics data and is satisfied that the value of Tm(P) is rounded to three significant figures, then the value of gl can be rounded to three significant figures for the entire outer core region. The value of gl ¼ 1:5 was used by Gubbins et al. (2003) throughout the core in their analysis of the geodynamo. The value of gs for the inner core decreases very slowly with the depth from its value at the ICB pressure. Table G9 shows that gs (listed in the table as gamma) drops from 1.53 at the ICB (330 GPa) to 1.52 at a pressure of 372.8 GPa; by interpolation gamma is 1.525 at the Earth’s center, which has a pressure of 363.9 GPa. One may as well use gs ¼ 1:53 for calculation of Tm at the Earth’s center. We need to know the temperature of the core; this requires DTm , the freezing point depression from pure iron. Alfè et al. (2002c) report DTm ¼
800 K. Thus, for the temperature at the ICB pressure, one finds Tm ðcoreÞ ¼ 5 550 K from the solidus at ICB (6250 K) for the ab initio approach, and Tm ðcoreÞ ¼ 5 250 K from the thermal properties approach. The K-primed EoS approach leads to Tm ðcoreÞ ¼ 5000 K. This calculation using Eq. (31) is Tm ðE centerÞ= Tm ðICBÞ ¼ 1:1213, giving, for Earth’s center, 6111 K for the ab initio ICB value (5450 K) and 5456 K for the thermal physics ICB value (5250 K). The K-primed EoS approach gives 5030 K for Earth’s center.

Acknowledgments The author gratefully acknowledges the data on g, Tm ðKÞ, and a (listed in Table G10), from Dario Alfè’s work (some unpublished), which were sent to him on request. Orson L. Anderson

Bibliography Alfè, D., Price, G.D., and Gillan, M.J., 2001. Thermodynamics of hexagonal close packed iron under Earth’s core conditions. Physical Review B, 64: 1–16.04123, Alfè, D., Price, G.D., and Gillan, M.J., 2002a. Iron under Earth’s core conditions: thermodynamics and high pressure melting from abinitio calculations. Physical Review B, 65(118): 1–11. Alfè, D., Gillan, M.J., and Price, G.D., 2002b. Composition and Earth’s core constrained by combining ab initio calculations and seismic data. Earth and Planetary Science Letters, 195: 91–98.

373

Alfè, D., Gillan, M.J., and Price, G.D., 2002c. Ab-initio chemical potentials of solid and liquid solutions and the chemistry of the Earth’s core. Journal of Chemical Physics, 116: 7127–7136. Alfè, D., Gillan, M.J., and Price, G.D., 2002d. Complementary approach to ab-initio calculations of melting properties. Journal of Chemical Physics, 116: 6170–6177. Anderson, O.L., 1995. Equations of State for Geophysics and Ceramic Science. New York: Oxford University Press, 405 pp. Anderson, O.L., 2002a. The power balance at the core-mantle boundary. Physics of the Earth and Planetary Interiors, 131: 1–17. Anderson, O.L., 2002b. The three dimensional phase diagram of iron. In Karato, S., Dehant, V., and Zatman, S. (eds.), Core Structure and Rotation. Washington, DC: American Geophysical Union. Anderson, O.L., Dubrovinsky, L., Saxena, S.K., and Le Bihan, T., 2001. Experimental vibrational Grüneisen ratio value for e-iron up to 330 GPa at 3000 K. Geophysical Research Letters, 28: 399–402. Anderson, O.L., Isaak, D.G., and Nelson, V.E., 2003. The high-pressure melting temperature of hexagonal close-packed iron determined from thermal physics. Journal of Physics and Chemistry of Solids, 64: 2125–2131. Barron, T.H.K., 1957. Grüneisen parameters for the equation of state of solids. Annals of Physics, 1: 77–89. Boehler, R., and Ramakrishnan, J., 1980. Experimental results on the pressure dependence of the Grüneisen parameter. A review. Journal of Geophysical Research, 85: 6996–7002. Bukowinski, M.S.T., 1977. A theoretical equation of state for the inner core. Physics of the Earth and Planetary Interiors, 14: 333–339. Chen, G.Q., and Ahrens, T.J., 1977. Sound velocities of liquid g and liquid iron under dynamic compression (abstract). EOS Transactions of the American Geophysical Union, 78: P757. Debye, P., 1912. Theorie der spezifischen wärmen. Annals of Physics (Berlin), 39: 789–839. Debye, P., 1965. The early days of lattice dynamics. In Wallis, R.I. (ed.) Lattice Dynamics: Proceedings of an International Conference. Oxford, UK: Pergamon Press, pp. 9–17. Giefers, H., Lubbers, P., Rupprecht, K., Workman, G., Alfè, D., and Chumakov, A.I., 2002. Phonon spectroscopy of oriented hcp iron. High Pressure Research, 22: 501–506. Gilvarry, J.J., 1956. The Lindemann and Grüneisen laws. Physical Review, 102: 308–316. Grüneisen, E., 1926. The state of a solid body. In Handbuch der Physik, vol. 10, Berlin: Springer-Verlag, pp. 1–52, (English translation, NASA RE 2-18-59W, 1959). Gubbins, D., Alfè, D., Masters, G., Price, G.D., and Gillan, M.J., 2003. Can the Earth’s dynamo run on heat alone? Geophysical Journal International, 155(2): 609–622, doi: 10.1046/j.1365246X.2003.02064.x. Gubbins, D., Alfè, D., Masters, G., Price, G.D., and Gillan, M., 2004. Gross thermodynamics of 2-component core convection. Geophysical Journal International, 157: 1407–1414. Isaak, D.G., and Anderson, O.L., 2003. Thermal expansivity of hcp iron at very high pressure and temperature. Physica B, 328: 345–354. Kittel, C., 1956. Introduction to Solid State Physics, 2nd edn. New York: Wiley, 617 pp. Mao, H.K., Wu, Y., Chen, C.C., Shu, J.F., and Jephcoat, A.P., 1990. Static compression of iron to 300 GPa and Fe0.8Ni0.2 alloy to 200 GPa. Journal of Geophysical Research, 95: 21691–21693. McDonough, W.F., and Sun, S., 1995. The composition of the Earth. Chemical Geology, 120: 228–253. Slater, J.C., 1939. Introduction to Chemical Physics, 1st edn. New York: McGraw-Hill. Stacey, F.D., 2000. The K-primed approach to high pressure equations of state. Geophysical Journal International, 128: 179–193. Stacey, F.D., and Davis, P.M., 2004. High pressure equations of state with applications to the lower mantle and core. Physics of the Earth and Planetary Interiors, 142: 137–184.

374

¨ NEISEN’S PARAMETER FOR IRON AND EARTH’S CORE GRU

Stevenson, D.J., 1981. Models of the Earth’s core. Science, 214: 611–619. Stixrude, L., Wasserman, E., and Cohen, R.E., 1997. Composition and temperature of the Earth’s inner core. Journal of Geophysical Research, 102: 24729–24739. Sumino, Y., and Anderson, O.L., 1984. Elastic constants of minerals. In Carmichael, R.S. (ed.) CRC Handbook of Physical Properties of Rocks. Boca Raton, FL: CRC Press, pp. 39–138. Verhoogen, J., 1980. Energetics of the Earth. Washington, DC: National Academy of Sciences, 139 pp. Vocadlo, L., Alfè, D., Gillan, M.J., Wood, I.G., Brodholt, J.P., and Price, G.D., 2003. Possible thermal and chemical stabilization of body-centred-cubic iron in the Earth’s core. Nature, 424: 536–538.

Willis, B.T.M., and Pryor, A.W., 1975. Thermal Vibrations in Crystallography, 1st edn. London, UK: Cambridge University Press.

Cross-references Core Composition Core Density Core Properties, Theoretical Determination Core, Adiabatic Gradient Core-Mantle Boundary, Heat Flow Across Higgins-Kennedy Paradox Shock Wave Experiments

H

HALLEY, EDMOND (1656–1742) Edmond Halley (Figure H1) was born in London in 1656 and educated at St. Paul’s School in London and the Queen’s College, Oxford, which he left in 1676 without a degree to catalog southern stars and observe a transit of Mercury on the island of St. Helena. He also measured magnetic inclination on this voyage, in the Cape Verdes and on St Helena. On his return the Royal Society elected him a fellow. He visited Johannes Hevelius in Danzig and G.D. Cassini in Paris and observed with them. He became clerk to the Royal Society in 1686. He prompted Newton to write the Philosophiae Naturalis Principia Mathematica and published it himself. He was often at sea, surveying and diving for salvage until he was elected Savilian Professor of Geometry in Oxford in 1704. He became the second Astronomer Royal in 1720.

The Westward Drift (q.v.) In 1680, it was known that the major deviations of the Earth’s field from that of a uniformly magnetized sphere appeared to move westward. Halley selected representative observations of the variation (magnetic declination) from 1587 to 1680, and from New Zealand (170
E) to Baffin Bay (80
W). They came from mariners’ reports, and perhaps from collections not to be found now. He found that the westward drift was not the same everywhere (Halley, 1683). He argued that the extensive anomalies had much deeper sources than iron near the surface that Descartes (Descartes, 1644) and Gilbert (q.v.) proposed. He suggested that the Earth had a shell with one pair of poles and a core with another pair, the two in relative rotation (Halley, 1692). Would there be life in the space between the core and the shell? Other worlds in other places and at other times were then seriously contemplated.

noon. Longitude at sea was much more difficult, and sometimes had gross errors. Halley had delays, bad weather, and difficult officers, and was too late in the season to go into the South Atlantic. His second cruise from September 1699 to September 1700 accomplished much more, and he went further into the Antarctic ice than anyone before. Halley’s chart of isogonic lines over the Atlantic hung in the Royal Society building for many years but is now lost. Many printed copies were published, the first isarhythmic chart of any variable to have been published. There may already have been private manuscript charts and Athanasius Kircher (q.v.) had written of one in his Magnes. (Kircher, 1643) Halley claimed, justly, that his chart was the first with isogonic lines to have been printed and published: He was a founder of modern cartography. His lines were known as Halleyan lines, and were soon used for other data such as temperature and depth of water. About two hundred years later Christopher Hansteen (q.v.) and W. van Bemmelen (q.v.), having access to the logs of Dutch ships, compiled worldwide isogonic charts at regular epochs. His chart for 1700 is generally close to Halley’s where they overlap.

The Atlantic cruises In October 1698, Halley sailed as a captain in the Royal Navy in a ship built for him by the government, to cruise around the Atlantic and observe the magnetic variation for navigation, the first-ever scientific naval surveys (Thrower, 1981). He measured the angle between compass north and geographic north as established by the Sun’s position at rise or set. He found his latitude from the elevation of the Sun at

Professor Sir Alan Cook died in August 2004 after completing this article. Final minor changes were made by the editors, who take full responsibility for any omissions or errors.

Figure H1 Edmond Halley.

376

HANSTEEN, CHRISTOPHER (1784–1873)

Aurorae In 1716 Halley observed an intense auroral display over London (Halley, 1716). He collected reports from distant places and plotted the forms of the auroral arcs. They followed the lines of the Earth’s magnetic field and were most intense around the magnetic, not the geographic, pole. Halley argued that matter circulating around the field lines produced the aurorae. He thought the matter leaked out of hollow spaces in the Earth, perhaps between the core and the mantle. He did not explain the colors. As there were then no instruments to detect the daily variation of the geomagnetic field or magnetic storms, Halley could not relate aurorae to events on the Sun. Halley established entirely new lines of enquiry in his three principal contributions to geomagnetism. He was buried with his wife in the churchyard of St Margaret’s Lee, not far from the Royal Observatory where a wall now carries his gravestone with its memorial inscription. There is a tablet in the cloisters of Westminster Abbey, and the observatory at Halley Bay on Antarctica recalls his magnetic and auroral observations and the cruise that went so close to the Antarctic continent. A recent biography is in Cook (1998). Sir Alan Cook

Bibliography Cook, A., 1998. Edmond Halley: Charting the heavens and the seas. Oxford: Oxford University Press. Descartes, René, 1644. Principia Philosophiae. Amsterdam. Halley, E., 1683. A theory of the variation of the magnetical compass. Philosophical Transactions of the Royal Society, 13, 208–221. Halley, E., 1692. An account of the cause of the change of the variation of the magnetic needle, with an hypothesis of the structure of the internal parts of the Earth. Philosophical Transactions of the Royal Society, 17, 563–578. Halley, E., 1716. An account of the late surprising appearance of lights seen in the air, on the sixth of March last, with an attempt to explain the principal phenomena thereof, as it was laid before the Royal Society by Edmund Halley, J.V.D., Savilian Professor of Geometry, Oxon, and Reg. Soc. Secr. Philosophical Transactions of the Royal Society, 29, 406–428. Kircher, A., 1643. Magnes, sive de arte magnetica, opus tripartium. Rome. Thrower, N.J.W. (ed.), 1981. The three voyages of Edmond Halley in the “Paramour”, 1698–1701., 2nd series, vol. 156, 157. London: Hakluyt Society Publications.

University of Oslo. His pioneering achievements in terrestrial magnetism and northern light research are today widely appreciated (Brekke, 1984; Josefowicz, 2002), though it has not always been the case earlier. Hansteen was selected as university teacher because of his successful participation in a prize competition, answering the question posed by the Royal Danish Academy of Sciences in 1811: “Can one explain all the magnetic peculiarities of the Earth from one single magnetic axis or is one forced to assume several?” The prize, a gold medal, was won by Hansteen. In his treatise he meant to demonstrate the necessity to assume that the Earth possesses two magnetic axes, implying our globe to be a magnetic quadrupole. The terrestrial magnetism became his main scientific interest through the rest of his life. For economic reasons Hansteen’s one-volume treatise was not published until 1819, but then in a considerably extended form. The book has the title Untersuchungen über den Magnetismus der Erde (Hansteen, 1819). It is still quoted in the literature. This work appeared in print only one year ahead of the discovery of the connection between electricity and magnetism by his Danish friend and colleague H.C. Ørsted. With his well-formulated treatise, Hansteen thus obtained a central position in the development of the geophysical sciences taking place in that period (see Figure H2). A Norwegian expedition under the leadership of Hansteen operated in Siberia in the years 1828–1830, traveling to the Baikal Sea and crossing the border into China (Hansteen, 1859). One measured the numerical values of the total magnetic field strength, the inclination, and the declination. Hansteen found no evidence of any additional magnetic pole in Siberia. To him this was an enormous disappointment. Nevertheless, in a letter to H.C. Ørsted of June 21, 1841, he proudly relates the written statement of Gauss, that to a large extent, it was the measurements of Hansteen, that had made Gauss devote himself to the study of magnetism. Furthermore, Hansteen also made contributions to the investigation of northern light phenomena (Hansteen, 1825, 1827) as indicated in Figure H3. Related to his extensive studies of the Earth’s magnetic

Cross-references Auroral Oval Bemmelen, Willem van (1868–1941) Geomagnetism, History of Gilbert, William (1544–1603) Hansteen, Christopher (1784–1873) Humboldt, Alexander von (1759–1859) Humboldt, Alexander von and magnetic storms Jesuits, Role in Geomagnetism Kircher, Athanasius (1602–1680) Storms and Substorms, Magnetic Voyages Making Geomagnetic Measurements Westward Drift

HANSTEEN, CHRISTOPHER (1784–1873) Christopher Hansteen (1784–1873) was born in Christiania (now Oslo), Norway. In the years 1816–1861 he was professor of applied mathematics and astronomy at the institution today denoted as the

Figure H2 Christopher Hansteen, drawn by C.W. Eckersberg ca. 1828. Shown is also an instrument constructed by Hansteen for the determination of the magnetic intensity. It exploits the fact that a magnetic needle suspended in a magnetic field, when set in motion; its movement in time will among other factors depend on the field strength. This device received international acclaim and was used during the so-called Magnetic Crusade (Cawood, 1979).

377

HARMONICS, SPHERICAL

HARMONICS, SPHERICAL Introduction Spherical harmonics are solutions of Laplace’s equation ]2 V ]2 V ]2 V þ 2 þ 2 ¼0 ]x 2 ]y ]z

Figure H3 Picture from work of Hansteen (1827) on the concept of the aurora, showing an auroral ring encircling the polar cap. The figure is presumably one of the first drawings of the auroral ring. The illustration also demonstrates how to measure the height of the aurora from one point only.

field he claimed that the center of the auroral ring to be situated somewhere north of Hudson’s bay (Brekke, 1984). At the meeting of the Scandinavian natural scientists in Stockholm in 1842, Christopher Hansteen gave a talk on the development of the theory of terrestrial magnetism (Hansteen, 1842). He points to the discovery of Ørsted of the connection between magnetism and electricity as a possible way to explain the interplay between the processes in Earth’s interior and the terrestrial magnetic phenomena. The modern dynamo theory is a development of this highly intuitive idea. The scientific results from the Siberian expedition were published as late as 1863 (Hansteen and Due, 1863). It represents a dignified finish to Christopher Hansteen’s contributions to this part of geoscience. It deserves attention that Christopher Hansteen was also active as a nation builder in the first 50 years after 1814 of Norwegian independence. Important subjects of his activities were the mapping of the Norwegian costal area together with inland triangulation, exact time determination, and many years of almanac edition, to mention just a few of his undertakings. Johannes M. Hansteen

Bibliography Brekke, A., 1984. On the evolution in history of the concept of the auroral oval. Eos, Transactions of the American Geophysical Union, 65: 705–707. Cawood, J., 1979. The magnetic crusade: science and politics in the early Victorian Britain. Isis, 70: pp. 493–518. Hansteen, C., 1819. Untersuchungen über den Magnetismus der Erde. Erster Teil, Christiania: J. Lehman and Chr. Gröndal. Hansteen, C., 1825. Fors
g til et magnetisk Holdningskart. Mag. Naturvidensk, 2: pp. 203–212. Hansteen, C., 1827. On the polar light, or aurora borealis and australis. Philos. Mag. Ann. Philos, New Ser., 2: 333–334. Hansteen, C., 1842. Historisk Fremstilling af hvad den fra det forloebne Seculums Begyndelse til vor Tid er udrettet for Jordmagnetismens Theorie. In Proceedings of the Third Meeting of the Scandinavian Natural Scientists, Stockholm, pp. 68–80. Hansteen, C., 1859. Reiseerindringer, Chr. T
nsbergs Forlag, Christiania. Hansteen, C., and Due, C. 1863. Resultate Magnetischer, Astronomischer und Meteorologischer Beobachtungen auf einer Reise nach dem Östlichen Sibirien in den Jahren 1828–1830. Christiania Academy of Sciences. Josefowicz, J.G., 2002. Mitteilungen der Gauss-Gesellschaft 39: pp. 73–86.

(Eq. 1)

in three dimensions, and they are collected together as homogeneous polynomials of degree l, with 2l þ 1 in each group. The simplest spherical harmonics are the three Cartesian coordinates x, y, z, which are three homogeneous polynomials of degree 1. Inverse distance 1=r satisfies Laplace’s equation everywhere, except at the point r ¼ 0, forming a spherical harmonic of degree –1, and the Cartesian derivatives of inverse distance generate spherical harmonics of higher degree (and order). Because inverse distance is the potential function for gravitation, electric, and magnetic fields of force, the theory of spherical harmonics has many important applications. It is a simple matter to observe that on taking the gradient of Laplace’s equation (1), that if V is a solution, then so also is the vector rV , and also the tensor rðrV Þ, showing clearly the need to consider vector and tensor spherical harmonics. As a simple example, the gradients of the Cartesian coordinates x; y; z, lead to i; j; k; as vector spherical harmonics, and gradients of xi; xj; xk; yi; . . . ; lead to Cartesian tensors ii; ij; ik; ji; . . . as tensor spherical harmonics. With increasing interest in atomic structure and electron spin, the study of Laplace’s equation in four dimensions, ]2 V ]2 V ]2 V ]2 V þ 2 þ 2 þ 2 ¼ 0; ]p2 ]q ]r ]s

(Eq. 2)


which is satisfied by 1 ðp2 þ q2 þ r2 þ s2 Þ and its derivatives, is important. The solution uses surface spherical harmonics from Eq. (1), and “spin weighted” associated Legendre functions.

Spherical polar coordinates In three dimensions, the spherical polar coordinates of a point P on the surface of the sphere are r; y; f; where r is the radius of the sphere, y is the colatitude of the point measured from the point chosen as the north pole of the coordinate system, and f is the east longitude of the point measured from the meridian of longitude chosen as the prime meridian, x ¼ r sin y cos f; y ¼ r sin y sin f; z ¼ r cos y:

(Eq: 3)

If a spherical harmonic of degree l is denoted Vl , then in spherical polars, by virtue of being a homogeneous polynomial of degree l, we may write Vl ðr; y; fÞ ¼ rl Sl ðy; fÞ

(Eq. 4)

and the function Sl ðy; fÞ is called a surface spherical harmonic of degree l. From the theory of homogeneous functions, ðr  rÞVl ¼ r

]Vl ]Vl ]Vl ]Vl ¼x þy þz ¼ l Vl : ]r ]x ]y ]z

(Eq. 5)

Conversely, it can be shown that if a function Vl satisfies Eq. (5), then it is a homogeneous function of degree l in x; y; z. The four-dimensional case arises when considering rotations through and angle w, about an axis of rotation with which is in the direction that

378

HARMONICS, SPHERICAL

has colatitude u and east longitude v. For rotations, the hypersphere has unit radius, for we include a value r here, and the four-dimensional Cartesian coordinates p; q; r; s; are 1 p ¼ r sin u cos v sin w; 2 1 q ¼ r sin u sin v sin w; 2 1 r ¼ r cos u sin w; 2 1 s ¼ r cos w: 2

eiV ! eiðV þaÞ ¼ eiV eia ; iV

e

In spherical polar coordinates, the Laplacian r2 V is       1 ] ]V 1 ] ]V 1 ]2 V r2 þ sin y þ 2 : 2 2 r ]r ]r sin y ]y ]y sin y ]f (Eq. 7) The method of separation of variables is used to solve Laplace’s equation in spherical polars. With V ðr; y; fÞ ¼ RðrÞSl ðy; fÞ; and with l ðl þ 1Þ as a constant of separation, then   d dR r2  l ðl þ 1ÞR ¼ 0; dr dr

(Eq. 8)

  1 ] ]S l 1 ]2 S l sin y þ 2 þ l ðl þ 1ÞSl ¼ 0: sin y ]y ]y sin y ]f2

(Eq. 9)

and

Note that the constant of separation l ðl þ 1Þ remains unchanged if l is replaced by  l  1. Radial dependence of the spherical harmonics is given by (8), B : rlþ1

! eiðV þaÞ ¼ eiV eia ;

(Eq. 10)

Orthogonality of surface spherical harmonics A number of properties of spherical harmonics can be established using only the definition, namely, that they are homogeneous functions of degree l that satisfy Laplace’s equation, and specific mathematical expressions are not required. Let Sl ðx; y; zÞ and SL ðx; y; zÞ be two surface spherical harmonics of degree l and L respectively. A vector F is defined by F ¼ ðrSl ÞSL  Sl ðrSL Þ;

 1 l ðl þ 1ÞSl SL þ LðL þ 1ÞSl SL ; r2 1 ¼ 2 ðL  l ÞðL þ l þ 1ÞSl SL : r

rF¼

(Eq. 11)

spherical volume

spherical surface

where, in this case, dS ¼ r2 er sin y dy df; and er is the unit radial vector. The vector F has no radial component, and therefore the spherical surface integral is zero, and the volume integral, after integrating with respect to radius, reduces to Z rZ

2p Z p 0

0

    1 d dY m2 sin y þ l ð l þ 1Þ  2 Y ¼ 0: sin y dy dy sin y

l ðl þ 1Þ Sl ðy; fÞ r2

0

0

and therefore, when l 6¼ L or when l 6¼ L  1, then

0

2p

Z

p

Sl ðy; fÞSL ðy; fÞ sin y dy df ¼ 0:

(Eq. 19)

0

(Eq. 12)

Legendre polynomials

From Eqs. (7) and (9), the Laplacian of the surface spherical harmonic Sl ðy; fÞ is r2 S l ð y ; f Þ ¼ 

r  F dv ¼ ðL  l ÞðL þ l þ 1Þ Z r Z 2p Z p dr Sl ðy; fÞSL ðy; fÞ sin ydydf; ¼ 0; 0

Z

and

(Eq: 17)

Gauss’s theorem, applied to a spherical surface and the enclosed spherical volume, is ZZZ ZZ r  F dv ¼ F  dS ; (Eq. 18)

Equation 9 can be solved by further separation of variables, using Sl ðy; fÞ ¼ YðyÞFðfÞ, with m2 as the constant of separation, 2

(Eq. 16)

when, by Eq. (13), the divergence of F is

0

d F þ m2 F ¼ 0; df2

(Eq: 15)

(Eq: 6)

Separation of variables

RðrÞ ¼ Al rl þ

(Eq: 14)

and are “reducible”, whereas the complex exponential functions do not become mixed, and are therefore “irreducible.”

With r ¼ 1, the coordinates p; q; r; s; are better known as quaternions.

r2 V ¼

cos V ! cosðV þ aÞ ¼ cos V cos a  sin V sin a; sin V ! sinðV þ aÞ ¼ sin V cos a þ cos V sin a;

(Eq. 13)

Solutions of Eq. (11) are the trigonometric functions cos V and sin V , and the complex exponential functions eiV and eiV . The concept of “irreducibility” favors the use of the complex exponential expressions, because, with the transformation V ! V þ a, the trigonometric functions become mixed,

In the case m ¼ 0, the spherical harmonics are independent of longitude f and are said to be “zonal”. The differential equation is obtained from Eq. (12) with m ¼ 0, namely ð1  m2 Þ

d2 Y dY þ l ðl þ 1ÞY ¼ 0;  2m dm2 dm

(Eq. 20)

The factor ð1  m2 Þ of the second derivative shows that the solution has singularities at m ¼ 1, corresponding to colatitudes y ¼ 0 and y ¼ p, at the north and south poles respectively of the chosen reference frame. The two independent solutions of Eq. (20) are

379

HARMONICS, SPHERICAL

Legendre functions of the first and second kind, Pl ðmÞ and Ql ðmÞ, respectively, where 

Pl ðmÞ ¼

The function

l

1 d 2l l ! dm

Ql ðmÞ ¼ Pl ðmÞ ln 

Generating function

ðm2  1Þl ;

1 X 1 hl Pl ðcos yÞ V ðr; mÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 2 1  2mh þ h l ¼0

(Eq. 21)

sffiffiffiffiffiffiffiffiffiffiffi 1 þ m 2l  1 2l  5  Pl1 ðmÞ  Pl3 ðmÞ 1m 1l 3ðl  1Þ

2l  9 Pl5 ðmÞ þ . . . ; jmj < 1: 5ðl  2Þ

(Eq. 22)

is the generating function for the Legendre polynomials. If a gravitating particle of mass m is moved from the origin a distance d along the z-axis, regarded as the pole of a coordinate system, then, using the cosine rule of trigonometry, the gravitational potential of the particle (in a region free of gravitating material) is

Legendre polynomials are orthogonal over the range  1  m  1, and the normalization of Pl ðmÞ has been chosen so that 1 2

Z

1

1

Pl ðmÞPL ðmÞdm ¼

1 dL ; 2l þ 1 l

Gm V ðr; yÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; r2  2r d cos y þ d 2 1  l þ 1 GmX d Pl ðcos yÞ; when r > d ; V ðr; yÞ ¼ d l¼0 r 1 l GmX r V ðr; yÞ ¼ Pl ðcos yÞ; when r < d : d l¼0 d

(Eq. 23)

where dLl is the Kronecker delta. The descending power series expansion for the Legendre polynomials is  ð2l Þ! l l ðl  1Þ l2 m m  Pl ðmÞ ¼ l 2 l !l ! 2ð2l  1Þ  l ðl  1Þðl  2Þðl  3Þ l4 m  ... ; þ 2  4ð2l  1Þð2l  3Þ

P1 ¼ m;

1 P3 ¼ ð5m3  3mÞ; 2

1 P2 ¼ ð3m2  1Þ; 2 1 P4 ¼ ð35m4  30m2 þ 3Þ: 8

(Eq: 24) V ðzÞ ¼

Ql ðmÞ ¼

 lþ1 1 GX d al d l¼0 z

for z > d;

and V ðzÞ ¼

(Eq: 25)

Note that Pl ð1Þ ¼ 1 and that Pl ð1Þ ¼ ð1Þl for all values of l. A series expansion for Legendre functions of the second kind, Ql ðmÞ is

1 z l GX al d l¼0 d

for z < d;

(Eq. 30)

then, from the continuity of the potential, the potential in regions away from the z-axis is



2 l !l ! 1 ðl þ 1Þðl þ 2Þ 1 þ ð2l þ 1Þ! mlþ1 2ð2l þ 3Þ mlþ3  ðl þ 1Þðl þ 2Þðl þ 3Þðl þ 4Þ 1 þ þ . . . : 2  4ð2l þ 3Þð2l þ 5Þ mlþ5 l

(Eq: 29)

If the potential of a distribution of matter is independent of azimuth or east longitude, it is said to be zonal. In the case where the potential of the distribution in a source-free region along the z-axis is known to be

and the first few are P0 ¼ 1;

(Eq. 28)

V ðr; yÞ ¼

 lþ1 1 GX d al Pl ðcos yÞ d l¼0 r

V ðr; yÞ ¼

1 r l GX al Pl ðcos yÞ d l¼0 d

for r > d;

and for r < d :

(Eq. 31)

Neumann’s formula Neumann’s formula, given by 1 X l ¼0

1 ; ð2l þ 1ÞPl ðzÞQl ðmÞ ¼ mz

Recurrence relations jmj > jzj;

(Eq. 26)

can be used to show that the recurrence relations for Ql ðmÞ are the same as those for the Legendre polynomials Pl ðmÞ. Therefore, rounding errors, regarded as proportional to Ql ðmÞ, in the generation of Legendre polynomials using recurrence relations, are likely to become large at or near the poles. Neumann’s formula (26) can also be written in the form Ql ðmÞ ¼

1 2

Z

Pl ðzÞ dz; m 1  z

Differentiation of the generating function (28) and the Rodrigues formula (21) can be used to derive the recurrence relations for Legendre polynomials. The more important ones are Bonnet's formula

ðl þ 1ÞPlþ1  ð2l þ 1ÞmPl þ l Pl1 ¼ 0; (Eq. 32)

dPlþ1 dPl ¼m þ ðl þ 1ÞPl ; dm dm

(Eq. 33)

1

(Eq. 27)

which can be used to derive the Christoffel formula (22) for Ql ðmÞ.

and ð2l þ 1ÞPl ¼

dPlþ1 dPl1  : dm dm

(Eq. 34)

380

HARMONICS, SPHERICAL

 d m Pl ðmÞ; dm  l þ m 1 d ðm2  1Þl ; ¼ l ð1  m2 Þm=2 2 l! dm

Multiple derivatives of inverse distance

Pl;m ðmÞ ¼ ð1  m2 Þm=2

Parameters x and
are required, and they are defined by 1 1 x ¼  pffiffiffi ðx þ iyÞ ¼  pffiffiffi r sin yeif ; 2 2 1 1
¼ pffiffiffi ðx  iyÞ ¼ pffiffiffi r sin yeif : 2 2

(Eq. 42) See Table H1 for a list of these functions. ’s theorem for multiple derivatives of h Applying Leibnitz i ðm  1Þl ðm þ 1Þl in the function Plm ðmÞ and then re-arranging

(Eq. 36)

terms, gives the expression in terms of Plm ðmÞ,  lm 1 d 2 m=2 ð 1  m Þ ðm2  1Þl 2l l ! dm ðl  mÞ! Pl;m ðmÞ: ¼ ð1Þm ðl þ mÞ!

Pl;m ðmÞ ¼

The solid spherical harmonic, inverse distance, is 1 1 1 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; r x 2 þ y 2 þ z2 z2  2x


(Eq. 37)

 m ] 1
m ¼ 1  3  5 . . . ð2m  1Þ 2mþ1 ; ]x r r  m ] 1 xm ¼ 1  3  5 . . . ð2m  1Þ 2mþ1 : ]
r r

(Eq: 38)


The ðl  mÞth derivative of 1 r2mþ1 with respect to z is  l  m ] 1 ¼ ð2m þ 1Þð2m þ 3Þ . . . ð2l  1Þ ð1Þlm ]z r2mþ1  l m z ðl  mÞðl  m  1Þ zlm2  2ð2l  1Þ r 2 l þ1 r2l1  ðl  mÞðl  m  1Þðl  m  2Þðl  m  3Þ z lm4 þ  . . . 2  4ð2l  1Þð2l  3Þ r2l3 (Eq. 39) The required 2l þ 1 independent spherical harmonics of degree l are obtained by differentiating (38) partially with respect to z some l  m times, for m ¼ l ; l þ 1; . . .  1; 0; 1; . . . ; l  1; l, when, with the substitutions z ¼ r cos y and m ¼ cos y,  m  lm ] ] 1 1 ðl  mÞ! m ¼ x ð1Þlm ]
]z r rlþmþ1 2l l !  ð2l Þ! lm l ð2l  2Þ! lm2 m m  ðl  mÞ! ðl  m  2Þ!  l ðl  1Þð2l  4Þ! lm4 m þ  ... : 2ðl  m  4Þ!

(Eq: 40)

The series in (40) is the ðl þ mÞth derivative of a series which can be summed by the binomial theorem, and hence ð1Þ

] ] þi ]x ]y

¼ ð1Þm

m  lm ] 1 ]z r

1 2

Z

1 1

Pl;m ðmÞPL;m ðmÞdm ¼

1 ðl þ mÞ! L d : 2l þ 1 ðl  mÞ! l

(Eq. 44)

Surface spherical harmonics Surface spherical harmonics Ylm ðy; fÞ are defined in terms of the Ferrers normalized functions Pl;m ðcos yÞ Ylm ðy; fÞ

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðl  mÞ! ð2l þ 1Þ ¼ ð1Þ e Pl;m ðcos yÞ; ðl þ mÞ! sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  lþm ðl  mÞ! 1 d ¼ ð1Þm eimf ð2l þ 1Þ ðm2  1Þl : ð1  m2 Þm=2 l ðl þ mÞ! 2 l! dm m imf

(Eq. 45) The initial factor ð1Þm is now used following the influential work of Condon and Shortley (1935). See Table H1 for a list of the functions Ylm ðy; fÞ, and Figure H4 for tesseral, sectorial, and zonal harmonics. Therefore, in terms of multiple derivatives of inverse distance, from Eqs. (41) and (45), 1 m Y ð y; f Þ rlþ1 l sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi    2l þ 1 ] ] m ] l m 1 l m þi ¼ ð1Þ ðl  mÞ!ðl þ mÞ! ]x ]y ]z r (Eq. 46)



lþm

e ðl  mÞ! d ð1  m2 Þm=2 dm rlþ1 2l l ! imf

(Eq. 43)

Both Eqs. (42) and (43) can be used with positive or negative values of m. Caution is needed with definitions based on jmj. Equations (42) and (43) can be used to derive the normalization integral for associated Legendre functions with Ferrers normalization,

with multiple derivatives



for l  jmj

(Eq: 35)

Partial derivatives with respect to x and
are     ]f 1 ]f ]f ]f 1 ]f ]f ¼  pffiffiffi i ; ¼ pffiffiffi þi : ]x ]y ]
]y 2 ]x 2 ]x

l m



The series expression for surface spherical harmonics is ðm2  1Þl : (Eq. 41)

Ferrers normalized functions Associated Legendre functions are solutions of Eq. (12), and are given in the first instance as the Ferrers normalized functions Pl;m ðmÞ, (Ferrers, 1897), defined by

Ylm ðy; fÞ 

ð2l Þ! ¼ l eimf 2 l!

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2l þ1 ð1  m2 Þm=2 ðl  mÞ!ðl þ mÞ!

ðl  mÞðl  m  1Þ lm2 m 2ð2l  1Þ  ðlmÞðlm1Þðlm2Þðl  m  3Þ nm4 m þ  ... : 2  4ð2l  1Þð2l  3Þ (Eq. 47) mlm 

381

HARMONICS, SPHERICAL

Table H1 List of associated Legendre functions Plm ð cos yÞ in Ferrers normalization and Schmidt normalization to degree and order six, and the corresponding surface spherical harmonics Ylm ðy; fÞ, normalized after Condon and Shortley l

m

Ferrers Pl;m ðcos yÞ

Schmidt Plm ðcos yÞ

Ylm ðy; fÞ

0 1 1

0 0 1

1 c s

1 c s

2

0

1 2 2ð3c

1 pffiffiffi 3pcffiffi  26seif pffiffi 5 2  1Þ 2 ð3c pffiffiffiffi 30  sceif pffiffiffiffi 2 30 2 2if 4 s e pffiffi 7 3  3cÞ 2 ð5c pffiffiffiffi 21  4 sð5c2  1Þeif pffiffiffiffiffiffi 210 2 2if 4 s ce pffiffiffiffi  435s3 e3if 3 4 2 8ð35c  30c þ 3Þ pffiffi  3 4 5sð7c3  3cÞeif pffiffiffiffi 3 10 2 2 2if 8 s ð7c  1Þe pffiffiffiffi 3 35 3 3if 4 s ce pffiffiffiffi 3 70 4 4if 16 s e pffiffiffiffi 11 5 3 8 ð63c  70c þ 15cÞ pffiffiffiffiffiffi 330  16 sð21c4  14c2 þ 1Þeif pffiffiffiffiffiffiffi 2310 2 3 2if 8 s ð3c  cÞe pffiffiffiffiffiffi 385 3  16 s ð9c2  1Þe3if pffiffiffiffiffiffi 3 770 4 4if 16 s ce pffiffiffiffi  3 1677s5 e5if pffiffiffiffi 6 4 13 16 ð231c  315c

 1Þ

1 2 2ð3c  1Þ p ffiffiffi 3 sc pffiffi 3 2 2 s

2

1

3 sc

2

2

3s2

3

0

3

1

1 3 2ð5c  3cÞ 3 2 2sð5c  1Þ

3

2

15 s2 c

3

3

15 s3

4

0

4

1

1 4 2 8ð35c  30c 5 3 2sð7c  3cÞ

4

2

4

3

105 s c

4

4

105 s4

5

0

1 5 8ð63c

5

1

15 4 8 sð21c

5

2

105 2 3 2 s ð3c

 cÞ

5

3

105 3 2 2 s ð9c

 1Þ

5

4

945 s4 c

5

5

945 s5

6

0

6 1 16ð231c

1

21 5 8 sð33c

6

2

105 2 4 8 s ð33c

 18c2 þ 1Þ

6

3

315 3 3 2 s ð11c

 3cÞ

6

4

945 4 2 2 s ð11c

 1Þ

6

5

10395 s5 c

6

6

10395 s6

15 2 2 2 s ð7c

1 3 2ð5c  3cÞ pffiffi 6 2 4 sð5c  1Þ pffiffiffiffi 15 2 2 s c pffiffiffiffi 10 3 4 s 1 4 2 8ð35c  30c þ 3Þ pffiffiffiffi 10 3 4 sð7c  3cÞ pffiffi 5 2 2 4 s ð7c  1Þ pffiffiffiffi 70 3 4 s c pffiffiffiffi 35 4 8 s 1 5 3 8ð63c  70c þ 15cÞ pffiffiffiffi 15 4 2 8 sð21c  14c þ 1Þ pffiffiffiffiffiffi 105 2 3 4 s ð3c  cÞ pffiffiffiffi 70 3 2 16 s ð9c  1Þ pffiffiffiffi 3 35 4 8 s c pffiffiffiffi 3 14 5 16 s 6 4 1 16ð231c  315c 2

þ 3Þ

 1Þ

3

 70c3 þ 15cÞ  14c2 þ 1Þ

 315c4

þ 105c2  5Þ 6

þ 105c  5Þ

pffiffiffiffi 21 5 3 8 sð33c  30c þ 5cÞ pffiffiffiffiffiffi 210 2 4 2 32 s ð33c  18c þ 1Þ pffiffiffiffiffiffi 210 3 3 16 s ð11c  3cÞ pffiffi 3 7 4 2 16 s ð11c  1Þ pffiffiffiffiffiffi 3 154 5 16 s c pffiffiffiffiffiffi 462 6 32 s

 30c þ 5cÞ 3

In the special case m ¼ 0; from Eq. (45), Yl0 ðy; fÞ ¼

 l pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d ð2l þ 1Þ l ðm2  1Þl ¼ ð2l þ 1ÞPl ðmÞ; 2 l ! dm (Eq. 48)

Changing the sign of m in Eq. (45) and making use of Eq. (43), gives Ylm ðy; fÞ

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðl þ mÞ! Pl;m ðcos yÞ; ð2l þ 1Þ ¼ ð1Þ e ðl  mÞ! sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðl  mÞ! Pl;m ðcos yÞ; ¼ eimf ð2l þ 1Þ ðl þ mÞ!

pffiffiffi pffiffiffi 6 3 sin yeif ¼ x; 2 r pffiffiffi pffiffiffi 3 Y10 ðy; fÞ ¼ 3 cos y z; ¼ r pffiffiffi pffiffiffi 6 3 Y11 ðy; fÞ ¼ sin yeif ¼
: 2 r Y11 ðy; fÞ ¼ 

(Eq. 49)

Note that x; z;
are related to the surface spherical harmonics of the first degree by

(Eq: 50)

The normalization and orthogonality integral is

m imf

¼ ð1Þm Ylm ðy; fÞ:

þ 105c2  5Þ pffiffiffiffiffiffi 546  16 sð33c5  30c3 þ 5cÞeif pffiffiffiffiffiffiffi 1365 2 4 2 2if 32 s ð33c  18c þ 1Þe pffiffiffiffiffiffiffi 1365 3  16 s ð11c3  3cÞe3if pffiffiffiffiffiffi 3 182 4 2 4if 32 s ð11c  1Þe pffiffiffiffiffiffiffi 1001 5 5if  3 16 s ce pffiffiffiffiffiffiffi 3003 6 6if 32 s e

1 4p

Z 0

2p

Z

p 0

Ylm ðy; fÞYLM ðy; fÞ sin ydydf ¼ dLl dM m:

(Eq. 51)

In theoretical physics texts, it is common practice to replace the initial pffiffiffiffiffiffiffiffiffiffi factor 1=4p in Eq. (51) by applying a factor 1=4p to the surface spherical harmonic Ylm ðy; fÞ. This factor is sometimes broken up into pffiffiffiffiffiffiffiffiffiffi a factor 1=2p applied to the associated Legendre function and a facpffiffiffiffiffiffiffiffi tor 1=2 applied to the complex exponential part.

382

HARMONICS, SPHERICAL

They are normalized to have the same value as the Legendre polynomials and they are widely used in geomagnetism in accordance with a resolution of the Association of Terrestrial Magnetism and Electricity of the International Union of Geodesy and Geophysics (Goldie and Joyce, 1940). Following Schmidt (1899), it is convenient to write the two formulae of Eq. (55) in the form Plm ðcos yÞ

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðl  mÞ! Pl;m ðcos yÞ ¼ em ðl þ mÞ!

(Eq. 56)

in which the parameter em is defined by e0 ¼ 1;

e1 ¼ e2 ¼ e3 ¼ . . . ¼ 2;

(Eq. 57)

or alternatively, in terms of the Kronecker delta function em ¼ 2  d0m :

(Eq. 58)

For expressions f ðy; fÞ involving real variables only, we may write it as a linear combination of Schmidt normalized functions, f ð y; f Þ ¼

N  X

m glm cos mf þ hm l sin mf Pl ðcos yÞ

(Eq. 59)

l ¼0

where the constant coefficient where the constants gnm and hm n are determined using Z Z 1 2p p f ðy; fÞPlm ðcos yÞ cos mf sin ydydf 4p 0 0 Z Z 1 2p p ¼ ð2 l þ 1 Þ f ðy; fÞPlm ðcos yÞ sin mf sin ydydf: hm l 4p 0 0 (Eq. 60) glm ¼ ð2l þ 1Þ

Figure H4 Zonal surface spherical harmonics are of the form Pl0 ðcos yÞ; sectorial surface spherical harmonics are of the form Pll ðcos yÞcos lf and Pll ðcos yÞsin lf. Tesseral surface spherical harmonics are those that are neither zonal nor sectorial. The ð2l þ 1Þ independent solutions of degree l of Laplace’s equation, with no logarithmic singularity at the poles, are the ð2l þ 1Þ solid spherical harmonics, rl Ylm ðy; fÞ;

where m ¼ l ; l þ 1; . . . l  1; l :

(Eq. 52)

Given a sufficiently differentiable function f ðy; fÞ over the surface of a sphere, it can be written as a finite linear combination of surface spherical harmonics f ðy; fÞ ¼

L X n X

flm Ylm ðy; fÞ;

(Eq. 53)

l¼0 m¼n

where the complex constant coefficients flm are determined using flm

1 ¼ 4p

Z

2p 0

Z 0

p

f ðy; fÞYlm ðy; fÞ sin ydydf:

Schmidt normalized functions The Schmidt normalized associated Legendre functions Plm ðcos yÞ are defined by

m 6¼ 0:

Recurrence relations for spherical harmonics are derived using derivatives of Eq. (46) and derivatives of the recurrence relations for Legendre polynomials. The spherical polar components for the derivatives used in Eq. (46) defining Ynm ðy; fÞ, are   ]f ]f ]f 1 ]f i ]f if þi ¼ sin y þ cos y þ e ; (Eq. 61) ]x ]y ]r r ]y r sin y ]f ]f ]f 1 ]f ¼ cos y  sin y; ]z ]r r ]y

(Eq. 62)

  ]f ]f ]f 1 ]f i ]f if i ¼ sin y þ cos y  e : ]x ]y ]r r ]y r sin y ]f

(Eq. 63)

(Eq. 54)

Vector spherical harmonics are required for the representation of vector fields over the surface of a sphere.

Pl0 ðcos yÞ ¼ Pl;0 ðcos yÞ; sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðl  mÞ! m Pl;m ðcos yÞ; Pl ðcos yÞ ¼ 2 ðl þ mÞ!

Recurrence relations

(Eq: 55)

The basic set of recurrence relations in which the degree n and order m on the left hand side are changed by –1, 0, or þ1 on the right-hand side are ]Y m m m Y  ðl þ 1Þ sin yYlm þ cos y l  sin y l ffi ]y rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2l þ 1 mþ1 ðl þ m þ 1Þðl þ m þ 2ÞYlþ1 ¼ eif ; 2l þ 3 ]Y m  ðl þ 1Þ cos yYlm  sin y l ]y rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2l þ 1 m ðl  m þ 1Þðl þ m þ 1ÞYlþ1 ¼ ; 2l þ 3

(Eq: 64)

(Eq: 65)

383

HARMONICS, SPHERICAL

]Y m m m Y  ðl þ 1Þ sin yYlm þ cos y l þ sin y l ffi ] y rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2l þ 1 ðl  m þ 1Þðl  m þ 2ÞYlmþ1 ¼ eif 1 ; 2l þ 3 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ]Ylm  m cot yYlm ¼ eif ðl  mÞðl þ m þ 1ÞYlmþ1 ; ]y pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ]Ylm þ m cot yYlm ¼ eif ðl þ mÞðl  m þ 1ÞYlm1 ; ]y ]Y m m m Y l sin yYlm þ cos y l  sin y l ] y rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2l þ 1 ðl  mÞðl  m  1ÞYlmþ1 ¼ eif 1 ; 2l  1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ]Ylm 2l þ 1 m ðl þ mÞðl  mÞYlm1 ; l cos yYl  sin y ¼ 2l  1 ]y ]Y m m m Y l sin yYlm þ cos y l þ sin y l ] y rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2l þ 1 1 ðl þ mÞðl þ m  1ÞYlm ¼ eif 1 : 2l  1

From the transformation formula for spinors Eq. (73), it follows that (Eq: 66)

ul2 Þlþm Ll1m Ll2þm ¼ ðul1 þ vl2 Þlm ðvl1 þ 

(Eq. 76)

and after expanding by the binomial theorem (Eq. 67) Ll1m Ll2þm ¼

l þm X l m X j¼0 k ¼0

(Eq. 68)

ðl  mÞ!ðl þ mÞ! ðl  m  k Þ!k !ðl þ m  jÞ! j! :

(Eq. 77)

k l þmj lmk 2l jk

u  u

v

v

Replace the summing index k by M, where k ¼ l  M  j, and with the substitutions (Eq: 69)

(Eq. 70)

1 u ¼ ðcos bÞeiðgþaÞ=2 ; 2

Most recurrence relations can be derived from this basic set. Schuster (1903) gives a list of recurrence relations useful in ionospheric dynamo theory. Chapman and Bartels (1940) also derive some recurrence relations.

(Eq. 78)

we find that Ll1m Ll2þm ¼

(Eq: 71)

1 v ¼ ðsin bÞeiðgaÞ=2; 2

XX j

M

ð1Þj ðM  m þ jÞ!ðl  M  jÞ!ðl þ m  jÞ! j!

1 1  ðcos bÞ2lM þm2j ðsin bÞM mþ2j eiðM aþmgÞ ll1M ll2þM 2 2 (Eq. 79) The range of summing indices M and j, is such that none of the factorial expressions or powers of trigonometrical functions become negative.

Transformation of spherical harmonics On using the spinor forms of derivatives (see Rotations; Eq. (45)) ] ¼ l21 ; ]x

1 ] pffiffiffi ¼ l1 l2 ; 2 ]z

It follows from Eq. (79) that

]f ¼ l22 ; ]


it follows that solid spherical harmonics can also be written using spinors, in the more symmetric form 1

Y m ðy; fÞ r l þ1 l

¼ ð1Þ

l m

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2l þ 1 1 llm llþm : ðl  mÞ!ðl þ mÞ! 1 2 r (Eq. 72)

The transformation law for spherical harmonics under rotation of the reference frame follows from Eq. (72), using the transformation law (see Rotations; (190)) L1 ¼ ul1 þ vl2 ; L2 ¼ vl1 þ ul2 :

(Eq: 73)

where the Cayley-Klein parameters u and v are given in terms of Euler angles ða; b; gÞ, by

¼ ð1Þ

where the functions DlMm ða; b; gÞ are called “rotation matrix elements” and are written l DlMm ða; b; gÞ ¼ dMm ðbÞeiðM aþmgÞ ;

(Eq. 80)

l ðbÞ is and the purely real function dMm l ðbÞ ¼ ð1ÞM m dMm

X

ð1Þj

j

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðl  M Þ!ðl þ M Þ!ðl  mÞ!ðl þ mÞ! ðM  m þ jÞ!ðl  M  jÞ!ðl þ m  jÞ! j !

1 1  ðcos bÞ2lM þm2j ðsin bÞM mþ2j : 2 2

(Eq: 74) l ðbÞ ¼ dMm

If a point P has spherical polar coordinates, colatitude and east longitude ðy; fÞ, and if after a rotation through Euler angles ða; b; gÞ the coordinates are ðY; FÞ, then from Eq. (72) Y m ðY; FÞ r l þ1 l

l X Ll1m Ll2þm ffi¼ ð1Þlm pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi DlMm ða; b; gÞ ðl  mÞ!ðl þ mÞ! M ¼l " # ll1M ll2þm ð1ÞlM pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ðL  mÞ!ðL þ mÞ!

(Eq. 81)

See Table H2 for a list of these functions. The form often given for l the functions dMm ðbÞ is obtained with the substitution j ¼ m  M þ t, when

1 u ¼ cos beiðgþaÞ=2 ; 2 1 v ¼ sin beiðgaÞ=2 : 2

1

Rotation matrix elements

l m

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2l þ 1 1 Llm Ll2þm : ðl  mÞ!ðl þ mÞ! 1 r (Eq. 75)

X t

ð1Þt

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðl  M Þ!ðl þ M Þ!ðl  mÞ!ðl þ mÞ! ðm  M þ tÞ!ðl þ M  tÞ!ðl  m  tÞ!t!

1 1  ðcos bÞ2lmþM 2t ðsin bÞmM þ2t ; 2 2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi    lþm lm ðl  M Þ!ðl þ M Þ! X ð1Þt ¼ ðl  mÞ!ðl þ mÞ! t lþM t t 1 2lmþM 2t 1 mM þ2t  ðcos bÞ ðsin bÞ ; (Eq. 82) 2 2

384

HARMONICS, SPHERICAL

Table H2 List of rotation matrix elements to degree and order 3. Symmetry properties are required to complete the ð2l þ 1Þ  ð2l þ 1Þ table of rotation matrix elements for degree l l

M

m

l Rotation matrix element dM ;m ðbÞ

l

M

m

l Rotation matrix element dM ;m ðbÞ

0

0

0

1

3

1

–1

1

0

0

cos b

3

1

0

1

1

–1

3

1

1

1

1

0

sin2 b2 pffiffiffi  2 cos b2 sin b2

3

2

–2

2b 1 2 4 sin 2ð15 cos b þ 10 cos b  1Þ pffiffi b b 3 2 2 cos 2 sin 2ð1  5 cos bÞ b 1 2 2 4 cos 2ð15 cos b  10 cos b  1Þ 4b sin 2ð3 cos b þ 2Þ pffiffiffiffi  210 cos b2 sin3 b2ð1 þ 3 cos bÞ pffiffiffiffiffi 2 b 2 b 30 cos 2 sin 2 cos b pffiffiffiffi b 10 3b 2 cos 2 sin 2ð1  3 cos bÞ cos4 b2ð3 cos b  2Þ sin6 b2 pffiffiffi  6 cos b sin5 b pffiffiffiffiffi 2 b2 4 b2 15 cos 2 sin 2 pffiffiffi 2 5 cos3 b2 sin3 b2 pffiffiffiffiffi 4 b 2 b 15 cos 2 sin 2 pffiffiffi  6 cos5 b2 sin b2 cos6 b2

1

1

1

2

0

0

2

1

–1

2

1

0

2

1

1

2

2

–2

2

2

–1

2

2

0

2

2

1

2 3

2 0

2 0

cos2 b2 1 2 2ð3 cos b  1Þ 2b sin 2ð2 cos b þ 1Þ pffiffiffi  6 cos b2 sin b2 cos b cos2 b2ð2 cos b  1Þ sin4 b2 2 cos b sin3 b pffiffiffi 2 2b 2 2b 6 cos 2 sin 2 2 cos3 b2 sin b2 cos4 b2 1 3 2ð5 cos b  3 cos bÞ

Ylm ðY; FÞ ¼

DlMm ða; b; gÞYlM ðy; fÞ:

2

–1

2

0

3

2

1

3

2

2

3

3

–3

3

3

–2

3

3

–1

3

3

0

3

3

1

3 3

3 3

2 3

Closure

The transformation law for spherical harmonics is therefore l X

3 3

(Eq. 83)

M ¼l

When a rotation through Euler angles ða1 ; b1 ; g1 Þ is followed by a second rotation through Euler angles ða2 ; b2 ; a2 Þ, then l X

0

Ylm ðY; FÞ ¼ Equation (82) is valid for half-odd integer values of the parameters l, M, and m. In particular, 1

1 ða; b; gÞ 2;2 1 D21 1 ða; b; gÞ 2;2

D2 1

1 ¼ cos beiðgþaÞ=2 ¼ u; 2 1 ¼ sin beiðgaÞ=2 ¼ v: 2

DlMm0 ða1 ; b1 ; g1 Þ YlM ðy; fÞ;

M ¼l

YlM ðY0 ; F0 Þ ¼

l X

0

m0 ¼l

Dlm0 m ða2 ; b2 ; g2 Þ Ylm ðY; FÞ;

and therefore, (Eq: 84)

"

l X

Ylm ðY0 ; F0 Þ ¼

M ¼l

l X

m0 ¼l

# DlMm0 ða1 ; b1 ; g1 ÞDlm0 m ða2 ; b2 ; g2 Þ YlM ðy; fÞ:

The transformation law for spinors, Eq. (73), becomes 0 

L1 L2



1

2 B D1;1 ða; b; gÞ B 2 2 ¼B 1 @ D2 1 1 ða; b; gÞ 2;2

However, this is equivalent to a rotation through Euler angles ða; b; gÞ where

1 ða ; b ; g Þ 1 C  C l1 2;2 : C 1 A l2 2 D 1 1 ða; b; gÞ 1

D21

2;2

DlMm ða; b; gÞ YlM ðy; fÞ;

M ¼l

(Eq. 85) The transformation law for surface spherical harmonics of degree one, is 1 0 1 Y 1 ð Y; F Þ C B 0 @ Y 1 ð Y; F Þ A Y11 ðY; FÞ 1 0 pffiffiffi 1 1 1 1 cos2 b eiðaþgÞ 2 sin b cos beig sin2 beiðaþgÞ C B 2 2 2 2 C B pffiffiffi pffiffiffi 1 1 1 ia 1 ia C B ¼ B  2 sin b cos b e cos b 2 sin b cos be C C B 2 2 2 2 A @ pffiffiffi 1 1 ig 21 iðagÞ 2 1 iðaþgÞ sin b e  2 sin b cos be cos be 2 2 2 1 0 1 2 Y 1 ð y; f Þ C B 0 (Eq. 86) @ Y 1 ð y; f Þ A Y11 ðy; fÞ

l X

Ylm ðY0 ; F0 Þ ¼

from which it follows that DlMm ða; b; gÞ ¼

l X m0 ¼l

DlMm0 ða1 ; b1 ; g1 ÞDlm0 m ða2 ; b2 ; g2 Þ:

(Eq. 87)

Equation (87) is an important result, with special cases giving the sum rule, the addition theorems of trigonometry, and formulae of spherical trigonometry, including the analogies of Napier and Delambre, as well as the various haversine formulae. For example, in the special case that a1 ¼ a2 ¼ a3 ¼ 0 and g1 ¼ g2 ¼ g3 ¼ 0; then b3 ¼ b1 þ b2 ; and l dMm ðb1 þ b2 Þ ¼

l X m0 ¼l

l l dMm 0 ðb1 Þdm0 m ðb2 Þ;

(Eq. 88)

which is a generalization of all of the sum formulae of trigonometry.

385

HARMONICS, SPHERICAL

Rodrigues formula

with symmetries for general values of b

The rotation matrix elements DlMm ðbÞ have two types of orthogonality. Firstly, they are orthogonal under integration over the range of the Euler angles, 0  a < 2p; 0  b  p, and 0  g < 2p, and secondly, for a fixed value of l, regarded as a ð2l þ 1Þ  ð2l þ 1Þ matrix array, they have matrix orthogonality properties. These properties are most easily derived using the Rodrigues formula, substituting z ¼ cos b, when l ðzÞ ¼ dMm

ð1ÞlþM 2l

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðl  M Þ! ðl þ mÞ!ðl  mÞ!ðl þ M Þ!

 ð1  zÞðM mÞ=2 ð1 þ zÞðM þmÞ=2  lþM d ½ð1  zÞlþm ð1 þ zÞlm : dz

The sum rule

Orthogonality

ð1Þlm ¼ 2l

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðL þ M Þ! ðl þ mÞ!ðl  mÞ!ðL  M Þ!

 ð1  zÞðM þmÞ=2 ð1 þ zÞðM mÞ=2  lM h i d ð1  zÞlm ð1 þ zÞlþm : dz

¼

0

l iM a ; DlM ;0 ða; b; gÞ ¼ dM ;0 ðbÞe sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðl  M Þ! Pl;M ðmÞeiM a ; ¼ ð1ÞM ðl þ M Þ!

1 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi YlM ðb; aÞ: 2l þ 1

Z

0

2p

Therefore, in the special case m ¼ 0, the transformation formula (83) becomes Yl0 ðY; FÞ ¼

l X

DlM 0 ða; b; gÞ YlM ðy; fÞ :

0

Pl ðcos YÞ ¼

l 1 X Y M ðb; aÞ YlM ðy; fÞ : 2l þ 1 M ¼l l

(Eq. 99)

(Eq: 91)

In terms of Schmidt normalized functions, the sum rule takes the wellknown form,

(Eq. 92)

Special values At the particular values 0; p; 2p of b, the rotation matrix elements are m l dM ;m ð0Þ ¼ dM ; l þM m l dM dM ¼ ð1Þlm dm ;m ðpÞ ¼ ð1Þ M ;

ð1Þ2l dm M;

(Eq. 98)

l X ðl  M Þ! Pl;M ðcos bÞPl;M ðcos yÞeiM ðfaÞ : ðl þ M Þ! M ¼l

Pl ðcos YÞ ¼

l X M ¼0

PlM ðcos bÞPlM ðcos yÞ cos mðf  aÞ:

(Eq. 100)

The case l ¼ 1 of Eq. (100) is the cosine rule of spherical trigonometry, cos Y ¼ cos b cos y þ sin b sin y cosðf  aÞ;

¼

(Eq. 97)

M ¼l

Pl ðcos YÞ ¼

DlMm ða; b; gÞDlM 0 m0 ða; b; gÞ sin bdadbdg

0 0 0 1 dl dM dm : 2l þ 1 l M m

l dM ;m ð2pÞ

(Eq: 96)

and the sum rule in the form Eq. (98) becomes

The required property is that p

An important special case of Eq. (89) is that for which m ¼ 0; when, from Eq. (75) defining Ylm ðy; fÞ, and with m ¼ cos b, we obtain

From Eqs. (48) and (96), Eq. (97) becomes

The D-functions form unitary matrices, with orthonormal rows and columns for each fixed degree l. They also have orthogonality properties under integration over all three Eulerian angles. The derivation l ðzÞ, namely makes use of a second equivalent form for dMm

Z

mM l l l dM ;m ðbÞ ¼ dm; dM ;m ðbÞ ¼ ð1Þ M ðbÞ;

(Eq: 89)

A full description of the properties of the rotation matrix elements, including contour integral formulae is given in Vilenkin (1968).

2p

and

(Eq. 95)

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  lþM ð1Þ ðl  M Þ! d l ð1  z2 ÞM =2 ðz2  1Þl ; dM ;0 ðzÞ ¼ l ðl þ M Þ! dz 2 l! sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðl  M Þ! M Pl;M ðzÞ; ¼ ð1Þ ðl þ M Þ! sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðl þ M Þ! Pl;M ðzÞ: ¼ (Eq. 90) ðl  M Þ!

Z

(Eq: 94)

l M b l dM dM ;m ðbÞ ¼ ð1Þlþm dl M ;m ðbÞ: ;m ðp  bÞ ¼ ð1Þ

M

1 8p2

l dml ;M ðbÞ ¼ ð1ÞmM dm;M ðbÞ;

l m l l dM dM ;m ðbÞ; ;m ðp þ bÞ ¼ ð1Þ

l l The “symmetry”, dM ;m ðbÞ ¼ dm;M ðbÞ, indicates that there are two other equivalent formulae (Schendel, 1877). Equation (89), in the case m ¼ 0, using Eq. (43), gives

l ðzÞ dMm

l l dM ;m ðbÞ ¼ dm;M ðbÞ;

(Eq. 101)

and the case l ¼ 2, is the basic rule in the theory of tides, with the longitudinal terms showing the dependence on M ¼ 0 long-period terms, M ¼ 1 lunar diurnal terms and M ¼ 2 lunar semidiurnal terms, P2 ðcos yÞ ¼ P2 ðcos bÞP2 ðcos yÞ þ P21 ðcos bÞP21 ðcos yÞ cosðf  aÞ

(Eq: 93)

þ P22 ðcos bÞP22 ðcos yÞ cos 2ðf  aÞ:

(Eq. 102)

386

HARMONICS, SPHERICAL

In the case that Y ¼ 0, and the spherical triangle collapses to a straight line so that b ¼ y and f ¼ a, then since Pl ð1Þ ¼ 1 for all values of l, then Eq. (98) becomes l X M ¼l

YlM ðy; fÞ YlM ðy; fÞ ¼ 2l þ 1 :

M ¼0

PlM ðcos yÞ

2

¼ 1;

M ¼l

b

c



00

DlM 0 M ða; b; gÞ DlM 00 M ða; b; gÞ ¼ dM M0 :

(Eq. 109) Conditions on the values of the parameters require that the nine elements of the following 3  3 array, called a Racah square, are all positive, a þ b þ c aa aþa

(Eq. 105)

 

Schmidt (1899) derived the transformation formula (83) for spherical harmonics well before the later derivations in the context of the theory of atomic spectra. Schmidt gave the result in terms Schmidt normalized functions, using real variables only, as



l m

(Eq. 110)

l m l m

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 9  > ðl þ m þ 1Þðl þ m þ 2Þ > lþ1 1 l m > ; > ¼ ð1Þ > m  1 1 ð2l þ 1Þð2l þ 2Þð2l þ 3Þ > > > > s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi >  = 2ðl  m þ 1Þðl þ m þ 1Þ lþ1 1 lm1 ; ¼ ð1Þ m 0 ð2l þ 1Þð2l þ 2Þð2l þ 3Þ > > sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > >  > > ð l  m þ 1 Þðl  m þ 2Þ lþ1 1 > l m ;> ¼ ð1Þ > m þ 1 1 ð2l þ 1Þð2l þ 2Þð2l þ 3Þ ;



l m  l m  l m

(Eq: 106)

Because m and M ¼ 1; 2; 3; . . . only in the expression for imaginary component, therefore, in this case, em ¼ 2 and eM ¼ 2, and Plm ðcos YÞ sin m½p  ðF þ gÞ l h i X l l M ð1ÞM dM ¼ ;m ðbÞ  dM ;m ðbÞ Pl ðcos yÞ sin M ðf  aÞ

9 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  > > 2ðl  mÞðl þ m þ 1Þ l 1 > > ; ¼ ð1Þlm > > m  1 1 2ð2l þ 1Þð2l þ 2Þ > >  > = 2m l 1 l m ¼ ð1Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi; m 0 > 2l ð2l þ Þð2l þ 2Þ > s1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > >  > > 2ð l þ mÞðl  m þ 1Þ > l 1 l þm1 > ;> ¼ ð1Þ m þ 1 1 2l ð2l þ 1Þðl þ 2Þ ; (Eq. 112)



M ¼1

(Eq. 107)



The expression Schmidt developed for the rotation matrix elements was given in a slightly different, but nevertheless equivalent form to that of Eq. (89) where z ¼ cos b. In Schmidt’s result, c ¼ cos b and s ¼ sin b, and sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðl  mÞ!ðl  M Þ! ð1 þ cÞm sM m ¼ ð1Þ ðl þ mÞ!ðl þ M Þ!  M   m  d d ð1  cÞm Pl ðcÞ : dc dc

aþbc cg cþg

(Eq. 111)

Plm ðcos YÞ cos m½p  ðF þ gÞ l i pffiffiffiffiffiffiffiffiffiffih 1X l l ¼ em eM ð1ÞM dM ;m ðbÞ þ dM ;m ðbÞ 2 M ¼0  PlM ðcos yÞ cos M ðf  aÞ:

abþc bb bþb

and that a þ b þ g ¼ 0. The following special cases of the 3-j coefficients are determined from the series definition of Eq. (109),

Schmidt’s analysis

l ðbÞ dMm

¼ ð1Þabg

(Eq. 104)

which can be used to provide a useful check on numerical work. The result of Eq. (103) is equivalent to the conservation of “lengths” under rotation, and therefore requires that the ð2l þ 1Þ  ð2l þ 1Þ rotation matrices DlMm ða; b; gÞ be unitary, i.e., l X

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ða  aÞ!ða þ aÞ!ðb  bÞ!ðb þ bÞ!ðc  gÞ!ðc þ gÞ! ða þ b þ cÞ!ða  b þ cÞ!ða þ b  cÞ! a b g     X abþc a þ b þ c aþbc 1 ð1Þt  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi aat bþbt t ða þ b þ c þ 1Þ! t a

(Eq. 103)

and Eq. (100) in terms of Schmidt normalized polynomials gives l  X





l m l m l m

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 9  > ðl  m  1Þðl  mÞ > l1 1 l m > ; > ¼ ð1Þ > m  1 1 ð2l  1Þ2l ð2l þ 1Þ > > > > s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi >  = 2ðl  mÞðl þ mÞ l1 1 l m : ; ¼ ð1Þ > m 0 ð2l  1Þ2l ð2l þ 1Þ > sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > >  > > ð l þ m  1 Þðl þ mÞ l1 1 > l m > ¼ ð1Þ > m þ 1 1 ð2l  1Þ2l ð2l þ 1Þ ; (Eq. 113)

M

(Eq. 108)

Leibnitz’s theorem is used to show the equivalence of Eqs. (108) and (89), where z ¼ cos b.

Wigner 3-j coefficients The Wigner 3-j coefficients (Wigner, 1931) are just generalizations of the factors that appear in recurrence relations for associated Legendre polynomials, and rotation matrix elements. They have an interesting structure, and are best defined as a sum of products of three combinatorial coefficients. Thus

An integration formula The following integration formula can be used to derive the properties of 3-j coefficients directly from their definitions, Z

 l2 þl2 l3 h i d ðm  1Þl2 m2 ðm þ 1Þl2 þm2 ðm  1Þl1 m1 ðm þ 1Þl1 þm1 dm dm 1   l2 l3 l1 ¼ ð1Þ2l1 ðl1 þ l2  l3 Þ! m1 m2 m3   ðl1  m1 Þ!ðl1 þ m1 Þ!ðl2  m2 Þ!ðl2 þ m2 Þ!ðl3  m3 Þ!ðl3 þ m3 Þ! :  ðl1 þ l2  l3 Þ!ðl1  l2 þ l3 Þ!ðl1 þ l2 þ l3 Þ!ðl1 þ l2 þ l3 Þ! (Eq. 114) 1

387

HARMONICS, SPHERICAL

The 3-j coefficients have two important orthogonality properties   0  X 1 l1 l2 l3 l2 l3 l1 l0 dl11 ; ¼ m m m m m m 2l1 þ 1 1 2 3 1 2 3 m2 ;m3 m1 þm2 þm3 ¼0

l3

(Eq. 115) and Eq. (119) given below.

1 8p2

0



¼

Z 0

l1 M1

p

Z

2p 0

l2 M2

DlM1 1 m1 DlM2 2 m2 DlM3 3 m3 sin bdadbdg   l3 l2 l3 l1 M3 m1 m2 m3

(Eq: 116)

and therefore, from the orthogonality of rotation matrix elements, it follows that the product of two rotation matrix elements can be written as a sum of rotation matrix elements, DlM1 1 m1 ða; b; gÞDlM2 2 m2 ða; b; gÞ  X l2 l1 ¼ ð2l3 þ 1Þ M1 M2 l3

l3 M3



l1 m1

l2 m2

 l3 DlM3 3 m3 ða; b; gÞ : m3

(Eq. 117) Equation (117) is an important equation because it effectively contains all the recurrence relations for rotation matrix elements, and therefore for surface spherical harmonics. The first orthogonality property of 3-j coefficients applied to the product formula (117) gives    X  l1 l1 l2 l3 l2 l3 DlM1 1 ;m1 DlM2 2 ;m2 ¼ DlM3 3 ;m3 : M1 M2 M3 m1 m2 m3 M ;M 1

M3 Cll13mm13l2 m2 DlM1 1 ;m1 DlM2 2 ;m2 ¼ Cll13M DlM3 3 ;m3 ; 1 l2 M2

M1 þ M2 ¼ M3 ;

DlM ;m ða; b; gÞ ¼ ð1ÞM m DlM ;m ða; b; gÞ :

3

(Eq. 119)

Ylm1 1 ðY1 ; F1 Þ ¼ Ylm2 2 ðY2 ; F2 Þ ¼

X m1 ;m2

l1 X

m1 þm2 ¼m3

l0 m Cll13mm13l2 m2 Cl13m13l2 m2

¼

l0 dl33 ;

(Eq. 121) X l3

0

0

1 m2 Cll13mm13l2 m2 Cll13mm03l2 m0 ¼ dm m1 dm2 ; 2

2

with m1 þ m2 ¼ m3 ; m01 þ m02 ¼ m3 :

(Eq. 122)

M1 ¼l1

l2 X

"

2 DlM2 2 ;m2 ða; b; gÞYlM ðy2 ; f2 Þ: 2

X

m1 ;m2

M1 ¼l1 M2 ¼l2

(Eq: 126)

# Cll13mm13l2 m2 DlM1 1 m1 DlM2 2 m2 (Eq: 127)

The sums on the right of Eq. (127) can be rearranged and the expressions in square brackets replaced by means of Eq. (124), giving the results that X lm Cl13m13l2 m2 Ylm1 1 ðY1 ; F1 ÞYlm2 2 ðY2 ; F2 Þ X

" DlM3 3 m3

X M1 ;M2

# M3 2 Cll13M Y M1 ðy1 ; f1 ÞYlM ðy2 ; f2 Þ 1 l 2 M2 l 1 2

; (Eq: 128)

showing that the coupled expression in square brackets in Eq. (128) 3 ðY; FÞunder obeys the transformation law for spherical harmonics YlM 3 rotation of the reference frame; that is 3 YlM ðY; FÞ ¼ 3

with m1 þ m2 ¼ m3 ;

l1 X

 Ylm1 1 ðy1 ; f1 ÞYlm2 2 ðy2 ; f2 Þ :

M3

when the orthogonality properties, Eqs. (115) and (119) are

M1 ¼l1

1 DlM1 1 ;m1 ða; b; gÞYlM ðy1 ; f1 Þ; 1

Cll13mm13l2 m2 Ylm1 1 ðY1 ; F1 ÞYlm2 2 ðY2 ; F2 Þ

¼

¼

In applications, it is more convenient to use a coupling coefficient, (Varshalovich et al., 1988),  pffiffiffiffiffiffiffiffiffiffiffiffiffiffi l1 l2 l3 l1 l2 m3 3 Cll13m;m ; (Eq. 120) ¼ ð1 Þ þ 1 2 l 3 1 l2 m2 m1 m2 m3

l1 X

When the coupled spherical harmonic expression is formed

m1 ;m2

Vector coupling coefficients

(Eq. 125)

We can now use vector-coupling coefficient to couple together two quantities that transform like spherical harmonics. Consider

(Eq. 118) Setting a ¼ b ¼ g ¼ 0 in Eq. (117) gives the second orthogonality property of the 3-j coefficients,    X l2 l3 l2 l3 l1 l 1 m2 ¼ dm ð2l3 þ 1Þ 1 M1 d M2 : M1 M2 M3 m1 m2 m3 l

and m1 þ m2 ¼ m3

Note that the complex conjugate expressions required on the right-hand sides of Eqs. (117) and (118) are not required in the corresponding vector coupling coefficient formulae. This occurs because of the result that

2

X

(Eq. 124)

respectively, and in both of which

The integral of a product of three rotation matrix elements is 2p

X m1 ;m2

Products of rotation matrix elements Z

The product formulae (117) and (118), for rotation matrix elements become X lM Cl13M13L2 M2 Cll13mm13l2 m2 DlM3 3 ;m3 ; (Eq. 123) DlM1 1 ;m1 DlM2 2 ;m2 ¼

X M1 ;M2

M3 2 Cll13M Y M1 ðy1 ; f1 ÞYlM ð y2 ; f 2 Þ ; 1 l2 M2 l1 2

(Eq. 129)

so that coupled spherical harmonics on the right of Eq. (129) will transform under rotation of the reference frame like a spherical harmonics.

Vector spherical harmonics If the unit vectors in the x-, y- and z-directions are denoted ex ; ey ; ez ; respectively, the complex reference vectors e1 ; e0 ; e1 ; are defined by

388

HARMONICS, SPHERICAL

1 e1 ¼  pffiffiffi ðex þ iey Þ ¼ rx; 2 e0 ¼ ez ¼ rz; 1 e1 ¼ pffiffiffi ðex  iey Þ ¼ r
: 2

(Eq: 130)

If em denotes the complex conjugate of em ; m ¼ 1; 0; 1; then the complex reference vectors satisfy em  en ¼ dnm ; and em  en ¼ ð1Þmn dn m ;

where er ; ey ; ef ; are unit vectors in the direction of r; y; f; increasing, respectively. However, from the recurrence relations given in Eqs. (64)–(71), and from the spherical polar forms of the vector operators given in Eqs. (133) and (134), it follows immediately that rffiffiffiffiffiffiffiffiffiffiffiffiffi 2l þ 1 2l þ 3  1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi mþ1  pffiffiffi ðl þ m þ 1Þðl þ m þ 2ÞYlþ1 e1 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m  ðl  m þ 1Þðl þ m þ 1ÞYlþ1 e0  1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m þ pffiffiffi ðl  m þ 1Þðl  m þ 2ÞYlþ1 e1 ; 2 (Eq. 137)

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m ðl þ 1Þð2l þ 1ÞYl;lþ1 ¼

for m; n ¼ 1; 0; 1: (Eq. 131)

Therefore, a vector B, with Cartesian components Bx ; By ; Bz ; will have complex reference components B1 ; B0 ; B1 ; such that B ¼ B1 e1 þ B0 e0  B1 e1 ; and 1 B1 ¼  pffiffiffi ðBx þ iBy Þ; 2 B0 ¼ Bz ; 1 B1 ¼ pffiffiffi ðBx  iBy Þ: 2

(Eq: 132)

Complex reference components of the gradient operator r and the angular momentum operator L ¼ i r  r, are obtained, and can be expressed in terms of spherical polar coordinates, r; y; f; as follows:   1 ] cos y ] i ] þ ; r1 ¼  pffiffiffi eif sin y þ ]r r ]y r sin y ]f 2 ] sin y ] ; r0 ¼ cos y  ]r  r ]y  1 ] cos y ] i ] r1 ¼ pffiffiffi eif sin y þ  ; (Eq: 133) ]r r ]y r sin y ]f 2 and   1 ] ] L1 ¼  pffiffiffi eif þ i cot y ; ]y ]f 2 ] L0 ¼ i ; ]f   1 ] ]  i cot y : L1 ¼  pffiffiffi eif ]y ]f 2

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi lðl þ 1ÞYl;l ¼ pffiffiffi ðl þ m þ 1Þðl  mÞYlmþ1 e1 þ mYlm e0 2 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  pffiffiffi ðl þ mÞðl  m þ 1ÞYlm1 e1 ; 2 (Eq. 138) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m lð2l þ 1ÞYl;l1 ¼

rffiffiffiffiffiffiffiffiffiffiffiffiffi 2l þ 1 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi mþ1 pffiffiffi ðl  m  1Þðl  mÞYl1 e1 2l  1 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m þ ðl þ mÞðl  mÞYl1 e0  p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi m1 1 þ pffiffiffi ðl þ m  1Þðl þ mÞYl1 e1 : 2 (Eq. 139)

Using the 3-j coefficients listed in Eqs. (111)–(113), it follows that all three Eqs. (137)–(139) can be written as a single expression 1  pffiffiffiffiffiffiffiffiffiffiffiffiffi X lþn lþn1þm Ym 2l þ 1 ðy; fÞ ¼ ð1Þ l;lþn mm m¼1

1

l

m

m

 mm Ylþn em :

(Eq. 140) In terms of the coupling coefficient of Eq. (120) (Eq: 134) Ym l;lþn ðy; fÞ ¼

We now define three vector spherical harmonics m m Ym l ;l þ1 ðy; fÞ; Yl ;l ðy; fÞ; Yl;l1 ðy; fÞ; in terms of complex reference vectors, and also present them in the better known and widely used forms in terms of spherical polars. Thus   1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m 1 ðl þ 1Þð2l þ 1ÞYl;lþ1 ðy; fÞ ¼ r lþ1 Ylm ðy; fÞ ; l þ 2 r r pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m l ðl þ 1ÞYl;l ðy; fÞ ¼ LYlm ðy; fÞ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  l m  rl1 l ð2l þ 1ÞYm l ;l 1 ðy; fÞ ¼ r r Yl ðy; fÞ :

1 X

mm l;m Clþn;mm;1;m Ylþn em :

(Eq. 141)

m¼1

Because of the orthogonality properties of complex reference vectors and surface spherical harmonics, it follows that the vector spherical harmonics are orthogonal under integration over the surface of a sphere: 1 4p

Z

2p

0

Z

p 0

m2 l2 m2 m 1 Ym l1 ;l1 þm ðy; fÞ  Yl2 ;l2 þn ðy; fÞ sin ydydf ¼ dl1 dm1 dn :

(Eq. 142)

(Eq. 135) The spherical polar components of the vector spherical harmonics are pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m ]Y m 1 ]Ynm ef ; ðl þ 1Þð2l þ 1ÞYl;lþ1 ðy; fÞ ¼ ðl þ 1ÞYlm er þ n ey þ sin y ]f ]y pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m i ]Ynm ]Y m ey  i n ef ; lðl þ 1ÞYl;l ðy; fÞ ¼ sin y ]f ]y m pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m ]Y 1 ]Ynm lð2l þ 1ÞYl;l1 ðy; fÞ ¼ l Ylm er þ n ey þ ef ; sin y ]f ]y (Eq. 136)

Appendix Rotations Introduction The word “rotation” has related, but nevertheless, different meanings in different disciplines, such as agriculture, medicine, psychology, and mechanics. This article is intended to deal with the theory of rotation about an axis, and to bring together the relationships between the

389

HARMONICS, SPHERICAL

different methods used to describe rotations, including the angle and axis of rotation, the Euler-Rodrigues parameters, quaternions, the Cayley-Klein parameters, spinors, and Euler angles.

Rotation about an axis Consider a positive rotation of a system of particles without deformation, through an angle V about an axis n, in a reference frame that remains fixed. An origin O is chosen along the axis of rotation, and r is the position vector of a general particle that becomes the vector R. From Figures H5 and H6, it can be seen that

! ! ! ! R ¼ OQ ¼ OP þ PN þ NQ ; ¼ r þ ½ðn  rÞ  nðcos V  1Þ þ ðn  rÞ sin V :

The equation for the transformation of the coordinates of P, under rotation of the reference frame, follows directly from Eq. (144) by changing the sign of the angle V. Thus R ¼ r cos V þ nðn  rÞð1  cos V Þ  ðn  rÞ sin V :

(Eq: 143)

(Eq. 144)

(Eq. 145)

Writing out the Cartesian components of Eq. (145) gives a matrix form R ¼ A  r;

where A ¼ Aðn; V Þ; and

0

where N is a point along PM, such that PN is perpendicular to NQ. Therefore, R ¼ r cos V þ nðn  rÞð1  cos V Þ þ ðn  rÞ sin V :

Rotation of the reference frame

cos V B þa2 ð1  cos V Þ B B B B abð1  cos V Þ B Aðn; V Þ ¼ B B c sin V B B B @ acð1  cos V Þ þb sin V

abð1  cos V Þ þc sin V cos V þb2 ð1  cos V Þ bcð1  cos V Þ a sin V

(Eq. 146) acð1  cos V Þ

1

C C C C bcð1  cos V Þ C C C: C þa sin V C C C A cos V b sin V

þc2 ð1  cos V Þ

(Eq. 147)

The angle and axis of rotation The angle of rotation and the axis of rotation can be determined from the trace and the antisymmetric components of the transformation matrix A. The trace can be expressed in a number of different ways, traceAðn; V Þ ¼ 3 cos V þ ða2 þ b2 þ c2 Þð1  cos V Þ 3 sin V 21 2 : ¼ 1 þ 2 cos V ¼ 4 cos V  1 ¼ 1 2 sin V 2 (Eq. 148) The trace of Aðn; V Þ is a single valued function of V in the range 0 < V < p, and for rotations V in the range p < V < 2p, the axis of rotation n is reversed. In the case of no rotation when V ¼ 0; and cos V ¼ 1, the rotation matrix reduces to Aðn; 0Þ ¼ 1, and there is no axis of rotation. Having determined the angle of rotation, the Cartesian components ða; b; cÞ of the axis of rotation are given by

Figure H5 The point P with position vector r relative to the origin O, is carried to the point Q, with the position vector R, by rotation through an angle V about the axis n.

1 1 2a sin V ¼ A23  A32 ¼ 4a sin V cos V ; 2 2 1 1 2b sin V ¼ A31  A13 ¼ 4b sin V cos V ; 2 2 1 1 2c sin V ¼ A12  A21 ¼ 4c sin V cos V : 2 2

(Eq: 149)

The eigenvalues of the rotation matrix Aðn; V Þ depend only on the angle of rotation V, l1 ¼ eiV ;

l2 ¼ eiV ;

l3 ¼ 1;

(Eq. 150)

and their sum is equal to the trace of the rotation matrix, 1 þ 2 cos V .

Infinitesimal rotations When the angle of rotation V about the axis n, is written as a differential, or infinitesimal dV, then Eq. (144) for the new position vector R of a particle with original position vector dr, of a system of particles, rotating as a rigid body, reduces to R ¼ r þ ðn  rÞdV : Figure H6 Showing the vector geometry of the rotation through an angle V about an axis of rotation n.

(Eq. 151)

The actual displacement during the rotation is dr ¼ R  r, and therefore,

390

HARMONICS, SPHERICAL

dr ¼ ðn  rÞdV :

(Eq. 152)

If the change takes place over an infinitesimal interval of time, dt, then dr dV ¼ ðn  rÞ ; dt dt ¼ V  r;

Euler Rodrigues parameters (Eq: 153)

where V is said to be the angular velocity of the particle. The time rate of change of a scalar function of position f ðrÞ relative to a position vector r rotating with angle velocity V is therefore df ]f d x ]f d y ]f d z ¼ þ þ ; dt ]x d t ]y d t ]z d t dr ¼ rf  ; dt ¼ rf  ðV  rÞ;

(Eq. 155)

At this point it is convenient to introduce the operator L, known as the angular momentum operator in quantum physics, and Lf ¼ ir  rf ¼ ir  ðrf Þ;

(Eq. 156)

when the expression Eq. (154) for the time rate of change of f ðrÞ becomes  n df d ¼ iO  Lf ; and f ¼ ðiO  rLÞn f : dt dt

(Eq. 157)

The Cartesian components of the angular momentum operator are   ]f ]f Lx f ¼ i z  y ; ]y ]z   ]f ]f Ly f ¼ i x  z ; ]z ]x   ]f ]f Lz f ¼ i y  x : (Eq: 158) ]x ]y

1 p ¼ a sin V ; 2 1 q ¼ b sin V ; 2 1 r ¼ c sin V ; 2 1 s ¼ cos V ; 2

(Eq: 161)

and are spherical polar coordinates in a four-dimensional space. Greek symbols x;
; z; w; are also used (Whittaker, 1904), as well as l; m; n; r (Kendall and Moran, 1962). Note that ða; b; cÞ, regarded as ðx; y; zÞ, are spherical harmonics of the first degree, and those of higher degree being generated by multiple derivatives inverse distance. The Euler-Rodrigues parameters ðp; q; r; sÞ are spin-weighted spherical harmonics, with terms of higher degree generated by derivatives of the inverse square of distance.

Resultant of two rotations When a reference frame is rotated through angle V1 about an axis n1 , and then rotated through an angle V2 about an axis n2 , the resulting configuration is equivalent to a single rotation through an angle V3 about an axis n3 . In terms of rotation matrices Aðn3 ; V3 Þ ¼ Aðn2 ; V2 ÞAðn1 ; V1 Þ:

(Eq. 162)

The trace of the product matrix is found to have the form 1 þ 2 cos V3 ¼ ðn1  n2 Þ2 ð1  cos V2 Þð1  cos V2 Þ  2ðn1  n2 Þ sin V2 sin V1 þ ð1 þ cos V2 Þð1 þ cos V1 Þ leads to a perfect square for cos 12V3, the positive square root of which gives

In spherical polar coordinates, the Cartesian derivatives are ]f ]f cos y cos f ]f sin f ]f ¼ sin y cos f þ  ; ]x ]r r ]y r sin y ]f ]f ]f cos y sin f ]f cos f ]f ¼ sin y sin f þ þ ; ]y ]r r ]y r sin y ]f ]f ]f sin y ]f ¼ cos y  : ]z ]r r ]y

The Euler-Rodrigues parameters p,q,r,s, are defined by

(Eq: 154)

where it is assumed that there is no “local” time derivative, usually denoted ]f =]t . By the rules for scalar triple products, we may write df ¼ O  ðr  rf Þ: dt

succession of a large number of small rotations, which can be used to show that the knowledge of the infinitesimal transformation amounts implicitly to a knowledge of the entire transformation.

1 1 1 1 1 cos V3 ¼ cos V2 cos V1  ðn1  n2 Þ sin V2 sin V1 : 2 2 2 2 2 (Eq. 163) (Eq: 159)

Similarly, the antisymmetric parts of the product matrix, after some algebra, lead to

The spherical polar forms of the angular momentum operators are therefore

1 1 1 1 1 a3 sin V3 ¼ ðb1 c2  c1 b2 Þ sin V2 sin V1 þ a1 cos V2 sin V1 2 2 2 2 2 1 1 þ a2 sin V2 cos V1 ; 2 2 1 1 1 1 1 b3 sin V3 ¼ ðc1 a2  a1 c2 Þ sin V2 sin V1 þ b1 cos V2 sin V1 2 2 2 2 2 1 1 þ b2 sin V2 cos V1 ; 2 2 1 1 1 1 1 c3 sin V3 ¼ ða1 b2  b1 a2 Þ sin V2 sin V1 þ c1 cos V2 sin V1 2 2 2 2 2 1 1 þ c2 sin V2 cos V1 : (Eq. 164) 2 2

  ] ] Lx ¼ i sin f þ cot y cos f ; ]y ]f   ] ] Ly ¼ i  cos f þ cot y sin f ; ]y ]f ] Lz ¼ i : ]f

(Eq: 160)

The Eq. (160) are used in determining recurrence relations for surface spherical harmonics. Note also that a finite rotation can be done by a

HARMONICS, SPHERICAL

Using p,q,r,s, as defined in Eq. (161), with suitable subscripts, Eq. (164) leads to the following formulae for the combination of rotations of the reference frame, p3 q3 r3 s3

¼ s2 p1 þ r2 q1  q2 r1 þ p2 s1 ¼  r2 p1 þ s2 q1 þ p2 r1 þ q2 s1 ¼ q2 p1  p2 q1 þ s2 r1 þ r2 s1 ; ¼  p2 p1  q2 q1  r2 r1 þ s2 s1 :

Without using half-angles From Eqs. (163) and (164), the trace and the antisymmetric part of the resultant rotation matrix Aðn3 ; V3 Þ can be expressed as the product of the Euler Rodrigues parameters, 1 1 2a3 sin V3 ¼ 4a3 sin V3 cos V3 ¼ 4p3 s3 ; 2 2 1 1 2b3 sin V3 ¼ 4b3 sin V3 cos V3 ¼ 4q3 s3 ; 2 2 1 1 2c3 sin V3 ¼ 4c3 sin V3 cos V3 ¼ 4r3 s3 ; 2 2 2 þ 2 cos V3 ¼ 4s23 :

(Eq. 165)

The beginnings of vector algebra The product formula of Eq. (165) contains within it as a special case, the vector product of the axes of rotation, namely n1 and n2, and also their scalar product. By choosing the rotation angles V1 and V2 to be p radians, equivalent to 180
, then s1 ¼ s2 ¼ cos 12p ¼ 0; and the remaining Euler-Rodrigues parameters reduce to the Cartesian components of the rotation axes,

391

Thus, starting with s3 ¼

1pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi trace½Aðn3 ; V3 Þ þ 1; 2

then ðp1 ; q1 ; r1 ÞjV1 ¼p ¼ ða1 ; b1 ; c1 Þ ¼ n1


p3 ¼ ½Aðn3 ; V3 Þ23  Aðn3 ; V3 Þ32  2s3 ;
q3 ¼ ½Aðn3 ; V3 Þ31  Aðn3 ; V3 Þ13  2s3 ;
r3 ¼ ½Aðn3 ; V3 Þ12  Aðn3 ; V3 Þ21  2s3 :

and ðp2 ; q2 ; r2 ÞjV2 ¼p ¼ ða2 ; b2 ; c2 Þ ¼ n2 The product formula (165) reduces to 9 a3 ¼ c2 b1  b2 c1 = b3 ¼ a2 c1  c2 a1 so that ða3 ; b3 ; c3 Þ n1  n2 ; ; c3 ¼ b2 a1  a2 b1

(Eq. 166)

Rigid body rotations (Eq. 167)

while the expression for s3 reduces to the scalar product times  1; s3 jV1 ;V2 ¼p ¼ a1 a2  b1 b2  c1 c2 so that s3 jV1 ;V2 ¼p ¼ n1  n2 : (Eq. 168) The results of Eqs. (167) and (168) would be written Vn1 n2 and Sn1 n2 respectively, where V and S refer to “vector” and “scalar,” respectively. Maxwell’s equations were given in Cartesian form (Maxwell, 1881), and also as equivalent “quaternion expressions” using V and S as in Eqs. (167) and (168). It was the fashion, when using Hamilton’s quaternions, to write vectors, such as the electromagnetic momentum A and the magnetic induction B in Gothic upper case script, A and B, respectively.

Hyperspherical trigonometry If the rotation axes n1 and n2 subtend an angle Y between them, and n1 ¼ ða1 ; b1 ; c1 Þ ¼ ðsin y1 cos f1 ; sin y1 sin f1 ; cos y1 Þ; n2 ¼ ða2 ; b2 ; c2 Þ ¼ ðsin y2 cos f2 ; sin y2 sin f2 ; cos y2 Þ;

then n1  n2 ¼ cos Y, and hence Eq. (163) for s3 becomes (Eq. 170)

where cos Y is the scalar product of n1 and n2 , cosY ¼ cosy1 cosy2 þ sin y1 sin y2 cosðf1  f2 Þ:

p3 ¼ s2 p1  r2 q1 þ q2 r1 þ p2 s1 ; q3 ¼ r2 p1 þ s2 q1  p2 r1 þ q2 s1 ; r3 ¼ q2 p1 þ p2 q1 þ s2 r1 þ r2 s1 ; s3 ¼ p2 p1  q2 q1  r2 r1 þ s2 s1 :

(Eq: 172)

It is important to distinguish carefully between Eq. (165) for rotation of the reference frame and Eq. (172) for a rigid-body rotation.

Quaternions The product formula Eq. (172) for combining rigid-body rotations, corresponds exactly to the law for the product Q2 Q1 of quaternions Q1 and Q2 , where Q1 ¼ 1s1 þ ip1 þ jq1 þ kr1 ; Q2 ¼ 1s2 þ ip2 þ jq2 þ kr2 ;

(Eq: 173)

subject to the rules that 1 is the identity and (Eq. 169)

1 1 1 1 1 cos V3 ¼ cos V1 cos V2  sin V1 sin V2 cos Y; 2 2 2 2 2

The formulae for rigid body rotations are obtained by replacing ðV1 ; V2 ; V3 Þ by ðV1 ; V2 ; V3 Þ, when the required formulae are

(Eq. 171)

Thus Eq. (170) uses half-angles of rotation, and Eq. (171) uses polar coordinates of rotation axes.

i2 ¼ j2 ¼ k 2 ¼ 1; jk ¼ i; kj ¼ i; ki ¼ j; ik ¼ j; ij ¼ k; ji ¼ k; ijk ¼ 1:

(Eq: 174)

It is difficult to see why such a fuss is made over the “rule” that ijk ¼ 1, when it is an immediate consequence of the rules ij ¼ k and k 2 ¼ 1. Quaternions are useful for the occasional calculation, but if any quantity of such rigid-body rotation calculations has to be done, then clearly the matrix form of Eq. (172) is easier to deal with.

392

HARMONICS, SPHERICAL

Cayley-Klein parameters The Cayley-Klein parameters u and v are defined by 1 1 u ¼ s þ ir ¼ cos V þ ic sin V ; 2 2 1 v ¼ q þ ip ¼ ðb þ iaÞ sin V : 2

(Eq: 175)

With this substitution, the quaternion product formula (165) for the combining of reference frame rotations can be written s3 þ ir3 ¼ ðs2 þ ir2 Þðs1 þ ir1 Þ  ðq2 þ ip2 Þðq1  ip1 Þ; q3 þ ip3 ¼ ðq2  ip2 Þðs1 þ ir1 Þ  ðs2  ir2 Þðq1  ip1 Þ; (Eq. 176)

u3 ¼ u2 u1  v2v1 ; v3 ¼  v2 u1   u2v1 ;

(Eq: 177)

and, in matrix form, Eq. (177) becomes u3 v3



 ¼

(Eq: 181)

Using the Cayley-Klein parameters u and v defined in Eq. (175), the Eq. (181) become ðX þ iY Þ ¼ ðx þ iyÞ u2 þ 2z u v þ ðx  iyÞv2 ; Z ¼ ðx þ iyÞ uv þ zðu u  vvÞ þ ðx  iyÞuv; ðX  iY Þ ¼ ðx þ iyÞv2  2zuv þ ðx  iyÞu2 :

(Eq: 182)

Introducing the parameters x and
, defined by

which becomes



  1 2 X  iY ¼ ðx þ iyÞ ða  ibÞ sin V 2 1 1 1  2zðb þ iaÞ sin V ðcos V þ ic sin V Þ 2 2 2 1 1 2 þ ðx  iyÞðcos V þ ic sin V Þ : 2 2

u2 v2

v2  u2



 u1 : v1

(Eq. 178)

Equation (178) can be regarded as the transformation law for the Cayley-Klein parameters u and v, and, more importantly, can be used to derive the transformation law for homogeneous polynomials of u and v of degree n.

Spinors Using the elementary results that 1 ax þ by ¼ ½ða þ ibÞðx  iyÞ þ ða  ibÞðx þ iyÞ; 2 1 1 1 cos V þ ic sin V þ ða2 þ b2 Þð1  cos V Þ ¼ ðcos V þ ic sin V Þ2 ; 2 2 2 (Eq. 179) and the vector relationships 10 1 1 0 0 X 1 i 0 X þ iY CB C C B B @ Z A ¼ @ 0 0 1 A@ Y A; and Z 1 i 0 X  iY 0 1 0 1 1 10 1 x x þ iy 0 2 2 B C B 1 C CB @ y A ¼ @ 2i 0 12i A@ z A; z x  iy 0 1 0

(Eq: 183)

the Eq. (182) become pffiffiffi x0 ¼ x  u2 þ 2 z u v þ
v2 ; pffiffiffi pffiffiffi Z ¼  2x uv þ zðu u  vvÞ þ 2
uv; p ffiffi ffi
0 ¼ x v2  2 zuv þ
u2

(Eq: 184)

These equations are easily solved with V ! V ; when by Eq. (175), u! u and v ! v giving pffiffiffi x ¼ x0 u2  2 Z uv þ
0v2 ; pffiffiffi pffiffiffi u  vvÞ  2
0  u v; z ¼ 2 x0 uv þ Zðu pffiffiffi 0 2 0 2 uv þ
 u :
¼x v þ 2Z 

(Eq: 185)

The required partial derivatives then follow from Eq. (185) and the chain rule of partial differentiation, 1 1 0 ]f ]f 0 C B ]x C 0 1B C B ]x C B u2 2uv v2 B 1 ]f C B 1 ]f C C B B pffiffiffi C @  v AB pffiffiffi C: u  v v u C ¼ u v u  B B 2 ]z C B 2 ]Z C 2 2  v 2 u v u C C B B @ ]f A @ ]f A 0

(Eq: 180)

one obtains from the basic Eq. (147) for reference frame rotation, that 1 1 X þ iY ¼ ðx þ iyÞðcos V  ic sin V Þ2 2 2 1 1 1  2zðb  iaÞ sin V ðcos V  ic sin V Þ 2 2 2   1 2  ðx  iyÞ ðb  iaÞ sin V ; 2 1 1 1 Z ¼ ðx þ iyÞðb þ iaÞ sin V ðcos V  ic sin V Þ 2 2 2   1 21 2 21 2 þ z cos V þ c sin V  ð1  c Þ sin2 V 2 2 2 1 1 1 þ ðx  iyÞðb  iaÞ sin V ðcos V þ ic sin V Þ; 2 2 2

1 1 x ¼  pffiffiffi ðx þ iyÞ; x0 ¼  pffiffiffi ðX þ iY Þ; 2 2 1 1 0
¼ pffiffiffi ðx  iyÞ;
¼ pffiffiffi ðX  iY Þ; 2 2

(Eq. 186)

]


]
0 The substitutions ]f ¼ L21 ; ]x0 1 ]f pffiffiffi ¼ L1 L2 ; 2 ]Z ]f ¼ L22 ; ]
0

]f ¼ l21 ; ]x 1 ]f pffiffiffi ¼ l1 l2 ; 2 ]z ]f ¼ l22 ; ]


(Eq: 187)

are valid only for spherical harmonic functions, since ]2 f ]2 f ]2 f þ þ ; ]x2 ]y2 ]z2 ] ]f ]2 f þ ¼2 ; ]x ]
]z2

r2 f ¼

¼  2l21 l22 þ 2l21 l22 ; ¼ 0:

(Eq: 188)

393

HARMONICS, SPHERICAL

where the matrix Aða; b; gÞ has the form

With the substitutions of Eqs. (187), the Eq. (186) becomes

0

L21 ¼ ðul1 þ vl2 Þ2 ; L1 L2 ¼ ðul1 þ vl2 Þðvl1 þ  ul2 Þ; L22

2

¼ ðvl1 þ  ul2 Þ ;

(Eq: 189)

and are equivalent to the 2  2 form L1 ¼ ul1 þ vl2 ; L2 ¼  vl1 þ ul2 ;

sin a cos b cos g þ cos a sin g

 sin b cos g

 sin a cos b sin g þ cos a cos g sin a sin b

1

C C C sin b sin g C : C A cos b

(Eq. 193) (Eq: 190)

where the positive square roots are required. In the case when there is no rotation and the angle V ¼ 0, for which u ¼ 1 and v ¼ 0, the Eq. (190) reduce correctly to L1 ¼ l1 and L2 ¼ l2 . In matrix form, Eq. (190) can be written      L1 u v l1 ¼ : (Eq. 191) L2 l2 v  u The parameters L1 ; L2 ; and l1 ; l2 are called spinors, and although from their basic definition given in Eq. (187), they appear to be the square roots of differential operators, they are in fact, apart from the transformation formula Eq. (191) (and representations of “spin weighted” spherical harmonics), used only in combinations of integer powers of differential operators. The 2  2 matrix in Eq. (191) has already been derived in Eq. (178) from the product formula for the Euler-Rodrigues parameters p,q,r,s, based on the complex forms (the Cayley-Klein parameters) u and v, arising solely from the need for a formula for the resultant of two successive rotations.

Euler angles ða; b; gÞ In this system theðx; y; zÞreference frame is rotated through an angle a about the zaxis to form the ðx0 ; y0 ; z0 Þ reference frame, which is rotated though an angle b about the y0 -axis to form the ðx00 ; y00 ; z00 Þ reference frame, which is rotated through an angle g about the z00 axis to form the final ðX ; Y ; Z Þ frame (see Figure H7). The point with coordinates of r becomes a point with coordinates R in the rotated reference frame, and the relationship can be written R ¼ Aða; b; gÞr;

cos a cos b cos g B  sin a sin g B B Aða; b; gÞ ¼ B  cos a cos b sin g B  sin a cos g @ cos a sin b

For the reverse rotation, the Euler angles ða; b; gÞ are replaced by ðg; b; gÞ, r ¼ Aðg; b; aÞR:

(Eq. 194)

The matrix can be written in terms of its factors, which derive from the three rotations: 1 10 0 1 0 cos b 0  sin b cos g sin g 0 X C CB B C B 1 0 A @ Y A ¼ @  sin g cos g 0 A@ 0 sin b 0 cos b 0 0 1 Z 10 1 0 cos a sin a 0 x CB C B  @  sin a cos a 0 A@ y A: 0 0 1 z (Eq. 195) Comparing the off-diagonal terms of Aðn; V Þ of Eq. (147), and Aða; b; gÞ of Eq. (193), gives sin a sin b ¼ cbð1  cos V Þ  a sin V ; sin b sin g ¼ cbð1  cos V Þ þ a sin V ;  sin b cos g ¼ cað1  cos V Þ  b sin V ; cos a sin b ¼ acð1  cos V Þ þ b sin V ;  cos a cos b sin g  sin a cos g ¼ abð1  cos V Þ  c sin V ; sin a cos b cos g þ cos a sin g ¼ bað1  cos V Þ þ c sin V ; (Eq. 196) and from Eq. (196) it follows that

(Eq. 192) 1 sin bðsin g  sin aÞ; 2 1 b sin V ¼ sin bðcos g þ cos aÞ; 2 1 c sin V ¼ ð1 þ cos bÞ sinðg þ aÞ: 2

a sin V ¼

(Eq: 197)

The right-hand sides of Eq. (197) can be written as a product of two terms, 1 1 1 1 a sin V ¼ 2 sin b sin ðg  aÞ  cos b cos ðg þ aÞ; 2 2 2 2 1 1 1 1 b sin V ¼ 2 sin b cos ðg  aÞ  cos b cos ðg þ aÞ; 2 2 2 2 1 1 1 1 c sin V ¼ 2 cos b sin ðg þ aÞ  cos b cos ðg þ aÞ; 2 2 2 2

(Eq: 198)

from which we obtain 1 1 1 ða2 þ b2 Þ sin2 V ¼ 4 sin2 b cos2 b cos2 ðg þ aÞ: 2 2 2 Figure H7 The Oxyz reference frame is rotated through Euler angles ða; b; gÞ to become the OXYZ reference frame.

(Eq. 199)

Comparing the diagonal terms of Aðn; V Þ of Eq. (147), and Aða; b; gÞ of Eq. (193), gives

394

HARMONICS, SPHERICAL

cos a cos b cos g  sin a sin g ¼ cos V þ a2 ð1  cos V Þ;  sin a cos b sin g þ cos a cos g ¼ cos V þ b2 ð1  cos V Þ;

ð1Þlm

cos b ¼ cos V þ c2 ð1  cos V Þ; (Eq. 200) from the third of Eq. (200) we obtain 1 1 1 1 sin2 b ¼ sin2 V  c2 sin2 V ¼ ða2 þ b2 Þ sin2 V 2 2 2 2

(Eq. 201)

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  m  lm ] ] 1 1 ðl þ mÞ!ðl  mÞ! m ¼ lþ1 Yl ðy; fÞ: ]
]z r r 2l þ 1 (Eq. 206)

and on using the spinor forms of derivatives given in Eqs. (187)–(206), it follows that solid spherical harmonics can also be written in a more symmetric, spinor form, 1

Y m ðy; fÞ r lþ1 l

From Eqs. (199) and (201)

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2l þ 1 1 llm llþm : ðl  mÞ!ðl þ mÞ! 1 2 r (Eq. 207)

1 1 1 cos2 V ¼ cos2 b cos2 ðg þ aÞ; 2 2 2 and the positive square root is required, because in the case in which a ¼ g ¼ 0, the angle of rotation V ¼ b. Therefore 1 1 1 cos V ¼ cos b cos ða þ gÞ: 2 2 2

¼ ð1Þ

lm

(Eq. 202)

Using the transformation law (191) for spinors, under rotation of the reference frame, the transformation law for spherical harmonics is easily derived. This leads, for example, to formulae in spherical (and hyperspherical) trigonometry, and to the identification of vector and tensor quantities, which are formed into “irreducible” parts, with profound physical implications, on account of their identification as spherical harmonics in their own right. Denis Winch

Euler-Rodrigues parameters and Euler angles Combining the results of Eq. (161) for rotation about n axis, with Eqs. (198) and (202), 1 p ¼ a sin V 2 1 q ¼ b sin V 2 1 r ¼ c sin V 2 1 s ¼ cos V 2

1 1 ¼ sin b sin ðg  aÞ; 2 2 1 1 ¼ sin b cos ðg  aÞ; 2 2 1 1 ¼ cos b sin ðg þ aÞ; 2 2 1 1 ¼ cos b cos ðg þ aÞ; 2 2

(Eq: 203)

and these important equations relate the axis and angle of rotation with the Euler angle formulation of the same rotation.

Cayley-Klein parameters and Euler angles Using the Cayley-Klein parameters, u,v, defined in Eq. (175), and the results of Eq. (203), 1 1 1 u ¼ s þ ir ¼ cos V þ ic sin V ¼ cos beiðgþaÞ=2 ; 2 2 2 1 1 iðgaÞ=2 v ¼ q þ ip ¼ ðb þ iaÞ sin V ¼ sin be : 2 2

(Eq: 204)

In terms of the rotation matrix elements DlMm ða; b; gÞ, (see Spherical l harmonics), and the generalized Legendre functions dM ;m ðbÞ, the parameters u and v are 1

1 iðaþgÞ=2 2 ; 1 ða; b; gÞ ¼ d 1 1 ðbÞe 2;2 2;2 1 1 D21 1 ða; b; gÞ ¼ d12 1 ðbÞeiðaþgÞ=2 : 2;2 2;2

u ¼ D2 1 v¼

(Eq: 205)

Spherical harmonics Briefly, spherical harmonics are multiple derivatives of inverse distance,

Bibliography Chapman, S., and Bartels, J., 1940. Geomagnetism. London: Oxford Clarendon Press. Condon, E.U., and Shortley, G.H., 1935. The Theory of Atomic Spectra. Cambridge: Cambridge University Press. [7th printing 1967.] Ferrers, Rev. N.M., 1897. An Elementary Treatise on Spherical Harmonics and Subjects Connected with them. London: Macmillan and Co. Goldie, A.H.R., and Joyce, J.W., editors, 1940. Proceedings of the 1939. Washington Assembly of the Association of Terrestrial Magnetism and Electricity of the International Union of Geodesy and Geophysics. International Union of Geodesy and Geophysics (IUGG), Edinburgh: Neill & Co., Bulletin 11, part 6, 550. Kendall, M.E., and Moran, P.A.P., 1962. Geometrical Probability. London: Griffin. Maxwell, J.C., 1881. A Treatise on Electricity and Magnetism. 2nd edition. London: Oxford Clarendon Press. Schendel, L., (1877). Zusatz zu der Abhandlung über Kugelfunktionen S. 86 des 80. Bandes. Crelle’s Journal, 82: 158–164. Schmidt, A., 1899. Formeln zur Transformation der Kugelfunktionen bei linearer Änderung des Koordinatensystems..Zeitschrift für Mathematik, 44: 327–338. Schuster, A., 1903. On some definite integrals, and a new method of reducing a function of spherical coordinates to a series of spherical harmonics. Philosophical Transactions of the Royal Society of London, 200: 181–223. Varshalovich, D.A., Moskalev, A.N., and Khersonskii, V.K., 1988. Quantum Theory of Angular Momentum. Irreducible Tensors, Spherical Harmonics, Vector Coupling Coefficients, 3nj Symbols. Singapore: World Scientific Publishing Co. Vilenkin, N.J., 1968. Special Functions and the Theory of Group Representations, American Mathematical Society, Second printing 1978. Translated from 1965 Russian original by V.N. Singh. Wigner, E., 1931. Gruppentheorie und ihre Anwendungen auf die Quantenmechanik und Atomsspektren, Braunschweig. Translation: Griffin, J.J. (ed.) (1959). Group Theory and its Application to the Quantum Mechanics of Atomic Spectra. New York and London: Academic Press. Whittaker, E.T., 1904. Treatise on the Analytical Dynamics of Particles and Rigid Bodies. Cambridge: Cambridge University Press.

395

HARMONICS, SPHERICAL CAP

HARMONICS, SPHERICAL CAP Spherical cap harmonics are used to model data over a small region of the Earth, either because data are only available over this region or because interest is confined to this region. The three-dimensional region considered in this article is that of a spherical cap (Figure H8), although one can also consider only the two-dimensional surface of a spherical cap. A three-dimensional model is used where the data are vector data that represent a field with zero curl and divergence, whereas a two-dimensional model is used for general fields with no such constraints. In a source free region of the Earth, the vector magnetic field B can be expressed as the gradient of a scalar harmonic potential V. (This scalar potential is “harmonic” since its Laplacian is zero.) The potential, and therefore the field, can then be expressed mathematically as a series of basis functions, each term of the series being harmonic by design. This harmonic solution is usually identified by the coordinate system used. For example, when the coordinates chosen are rectangular coordinates, we say the field is expressed in terms of rectangular harmonics. These would be useful when dealing with a local or very small portion of the Earth’s surface, such as in mineral exploration. When the coordinate system is the whole sphere, we speak of spherical harmonics (see Harmonics, spherical). These are used for investigating global features of the field. This article will discuss the harmonic solution for the case of a spherical cap coordinate system, which involves wavelengths intermediate between the local and the global solutions.

Basis functions The basis functions for the series expansion of the potential over a spherical cap region are found in the usual way by separating the variables in the given differential equations (that the curl and divergence of the field are zero) and solving the individual eigenvalue problems subject to the appropriate boundary conditions (e.g., Smythe, 1950; Sections 5.12 and 5.14). The boundary conditions include continuity in longitude, regularity at the spherical cap pole, and the appropriate Sturm-Liouville conditions on the basis functions and their derivatives at the boundary of the cap (e.g., Davis, 1963, Section 2.4). The details have been given by Haines (1985a), and computer programs in Fortran by Haines (1988). Let r denote the radius, y the colatitude, and l the east longitude of a given spherical cap coordinate. This coordinate system is, of course, identical to the usual spherical or polar coordinate system, except that the colatitude y must be less than y0 , the half angle of the spherical cap (Figure H8). Also, the spherical cap pole is not usually the geographic

Figure H8 Three-dimensional spherical cap region: colatitude y  y0 . Thickness of cap indicates radial coverage of data.

North Pole, in which case the geographic coordinate system is rotated to the new spherical cap pole giving new spherical cap colatitudes and longitudes. In both spherical and spherical cap systems, the radius r must lie between the outer radius of any current sources within the Earth and the inner radius of any current sources within the ionosphere, and the longitude l takes on the full 360
range. The harmonic solution of the potential V ðr; y; lÞ applicable to this three-dimensional spherical cap region is then given by: V ðr; y; lÞ ¼ a

Ki X k X

ða=rÞnk ðmÞþ1 Pnmk ðmÞ ðcos yÞ

k ¼0 m¼0  ½gkm;i cosðmlÞ Ke X k X

þa 

þ hm;i k sinðmlÞ

ðr=aÞnk ðmÞ Pnmk ðmÞ ðcos yÞ

k ¼1 m¼0 ½gkm;e cosðmlÞ

e þ hm; k sinðmlÞ

ðEq: 1Þ

where a is some reference radius, usually taken as the radius of the Earth, and Pnmk ðmÞ ðcos yÞ is the associated Legendre function of the first kind. It is usual in geomagnetism for the Legendre functions P to be Schmidt-normalized (Chapman and Bartels, 1940; Sections 17.3 and 17.4). The subscript of P is known as the degree of the Legendre function and the superscript is known as the order; k is referred to as the index and simply orders the real (usually nonintegral) degrees nk ðmÞ. The gkm and hm k are the coefficients, which are each further identified with the superscript i or e to denote internal or external sources, respectively. The internal source terms involve powers of (a/r) while the external source terms involve powers of (r/a). If one intended to fit only internal (or external) sources, only the internal (or external) terms of the expansion would be used. The truncation indices Ki and Ke are the maximum indices for the internal and external series, respectively. The potential can easily be transformed into a function of time t as well as space, V ðr; y; l; t Þ, by simply making the coefficients functions of time, gkm ðt Þ and hm k ðt Þ, and expanding these coefficients in terms of some temporal basis functions, such as cosine functions or Fourier functions or whatever is appropriate. The degree nk ðmÞ is chosen so that dPnmk ðmÞ ðcos y0 Þ dy

¼ 0 when k  m ¼ even

(Eq. 2)

and Pnmk ðmÞ ðcos y0 Þ ¼ 0 when k  m ¼ odd

(Eq. 3)

where the Legendre functions Pnm ðcos y0 Þ are here considered to be functions of n, given the order m and the cap half-angle y0 . The index k simply starts at m and is incremented by 1 each time a root is found to one of the Eqs. (2) or (3). This choice of k is analogous to the case of ordinary spherical harmonics (y0 ¼ 180
), and in fact for that case the degree nk(m) is simply the integer k. Table H3 gives the roots nk ðmÞ for y0 ¼ 30
, up to k ¼ 8. Figure H9 shows how Pnmk ðmÞ ðcos yÞ varies over a 30
cap, up to index 4, both for k  m ¼ even and for k  m ¼ odd. We can see how the Pnmk ðmÞ ðcosyÞ, 0  y  y0 , for k  m ¼ even, are analogous to the cosine functions cosðnyÞ, 0  y  p, in that each has zero slope at the upper boundary (y0 or p, respectively). In fact, we can think of Pnmk ðmÞ as being defined on the interval ½y0 ; þy0 , just as cosðnyÞ is defined on ½p; þp, so that an expansion over the spherical cap using Pnmk ðmÞ ðcosyÞ, k  m ¼ even, as basis functions is analogous to an expansion in Cartesian coordinates over ½p; þp using cosðnyÞ as basis functions. The extension of Pnmk ðmÞ to the ½y0 ; 0 interval of course, takes place on the meridian 180
away from the meridian on which the ½0; þy0  interval lies, which is why the Pnmk ðmÞ ðcosyÞ must be zero at y ¼ 0 when m 6¼ 0 (Figure H9a).

396

HARMONICS, SPHERICAL CAP

Table H3 nk (m) for y0 = 30˚ m k

0

1

2

3

4

0 1 2 3 4 5 6 7 8

0.00 4.08 6.84 10.04 12.91 16.02 18.94 22.02 24.95

3.12 6.84 9.71 12.91 15.82 18.94 21.87 24.95

5.49 9.37 12.37 15.62 18.58 21.72 24.69

7.75 11.81 14.92 18.22 21.25 24.42

9.96 14.18 17.39 20.76 23.84

5

6

7

8

12.13 16.50 14.29 19.81 18.80 16.42 23.24 22.19 21.07 18.55

square (or “in the mean”). So to fit an arbitrary function over the region, one only needs to include the basis functions with zero slope at the boundary (like the cosine functions). On the other hand, if one wishes to fit the derivative of the arbitrary function, these basis functions would not do (unless that arbitrary function has zero slope at the boundary). However, the basis functions that are zero (and whose derivatives are therefore not zero) at the boundary, will be able to fit uniformly the derivative of that arbitrary function. Including basis functions both with zero slope and zero function at the boundary thus permits the uniform expansion of functions with both arbitrary values and arbitrary derivatives at the boundary of the expansion region. The Legendre functions thus defined comprise two sets of separately orthogonal functions, with respect to the weight function siny. That is, Pnmj ðmÞ ðcos yÞ and Pnmk ðmÞ ðcos yÞ are orthogonal when j 6¼ k and when j  m and k  m are either both odd or both even. However, they are not orthogonal when j  m ¼ even and k  m ¼ odd (or vice versa).

The vector magnetic field B

Figure H9 Legendre functions Pnmk ðmÞ ðcos yÞ of nonintegral degree nk ðmÞ (given in Table H3) and integral order m, up to index k ¼ 4: (a) those that have zero slope at the cap boundary y0 ¼ 30
, and (b) those that are zero at the cap boundary.

Similarly, the Pnmk ðmÞ ðcosyÞ, for k  m ¼ odd, are analogous to the sine functions sinðnyÞ, 0  y  p, in that each is zero at the upper boundary. Here, of course, Pnmk ðmÞ ðcosyÞ must be nonzero at y ¼ 0 when m ¼ 0 (Figure H9b). So an expansion over the spherical cap using Pnmk ðmÞ ðcosyÞ, k  m ¼ odd, as basis functions is analogous to an expansion in Cartesian coordinates using sinðnyÞ as basis functions. Finding the Legendre functions with zero slope (Eq. (2)) or zero value (Eq. (3)) at the boundary of the expansion region is therefore analogous to finding the Fourier cosine and sine basis functions (on the interval ½0; p) applicable to a given expansion region in Cartesian coordinates. Those with zero slope at the boundary are able to fit nonzero functions there, while those that are zero at the boundary are only able to fit functions that go to zero there. Of course, we are considering Eq. (1) to be uniformly convergent, not merely convergent in mean

The vector magnetic field B is given by the negative gradient of Eq. (1). That is, we compute the vertical and the horizontal spherical cap north and east components of B by differentiating Eq. (1) with respect to the spherical cap coordinates r, y, and l, respectively (with appropriate signs and of course including the metrical coefficients 1, r, and rsiny). Note that only when the spherical cap pole is the geographic North Pole, are these components the usual geographic components. We can see now why we need the spherical cap basis functions with k  m ¼ odd. It is because the north component of the field involves a differentiation with respect to y. If the north component of the data happened to be nonzero at y ¼ y0 , the north component of the model would not be uniformly convergent if only basis functions with dPðcosy0 Þ=dy ¼ 0 were included in the model. Of course, if the north component is not being modeled, the second set of basis functions are not strictly required (the vertical anomaly field from Magsat data, e.g., was modeled by Haines, 1985b). In this case, the constraint on the derivative of P results in the field having zero slope, with respect to y, at the cap boundary. Even if one did not need to fit a derivative, it can still be an advantage to include basis functions that allow for a nonzero derivative at the boundary of the analysis region. That advantage is in faster convergence (a smaller number of coefficients for a given truncation error), as discussed by Haines (1990). Mean square and cross product values for the internal and external harmonics of the vector field B have been derived by Lowes (1999), in terms of the harmonics of the potential, as well as for their horizontal and radial components. Power spectra have been defined and expressed by Haines (1991), and the relationship of spherical cap harmonics to ordinary spherical harmonics has been described by De Santis et al. (1999).

Subtraction of a reference field The price paid for having uniform convergence over a spherical cap by including both sets of basis functions is that some of these functions are mutually nonorthogonal. Although most are mutually orthogonal (those within each of the two sets and all those of different order by virtue of the orthogonality of the longitude functions), even this sparse nonorthogonality can have an adverse effect on the coefficient solution, particularly at high truncation levels. This is because the nonorthogonality results in an ill-conditioned least-squares matrix, which will have some nondiagonal terms. The smaller the computer word size and the larger the field values being fitted, the more serious is this problem. It is therefore advantageous to subtract a spherical harmonic reference field from the data, do the spherical cap harmonic fit on the resulting residuals, and simply add the reference field back on when

HARTMANN, GEORG (1489–1564)

the full field is required. Which particular harmonic reference field is used for this purpose is unimportant; the idea is simply to have smaller numbers in the modeling process so as not to lose too much numerical accuracy for the given computer word size. A second difficulty with spherical cap and other regional models, is the accumulation of errors in upward or downward continuation of the field (see Upward and downward continuation). The field at the continuation point, of course depends on field values outside the cap, as well as inside, and so unless the effect of these outside values could somehow be compensated for by more complicated boundary conditions, there will be an error in continuing a model based only on field values inside the cap. Here again, we can get considerable relief from subtraction of global reference fields, which are models of data from the whole Earth. Haines (1985a, Figures 3–6) shows the kinds of errors to expect in upward continuing fields to 300 and 600 km, and the lower errors when subtracting an International Geomagnetic Reference Field (IGRF) (q.v.) prior to analysis.

General fields over a cap surface Previously, we have discussed fitting a harmonic function in a threedimensional region of space over a spherical cap, as portrayed in Figure H8. However, the spherical cap basis functions can also be used to expand a general function over a (two-dimensional) spherical cap surface (y  y0 ; r ¼ r0 ). That is, there is no constraint for this surface field to be harmonic since the radial field is not being defined; it can represent, on the surface, fields that are very complex in three-dimensional space. This is analogous to the expansion theorem for the whole sphere (e.g., Courant and Hilbert, 1953, Chapter VII, Section 5.3). The vertical field is able to play the role of the general function by considering only the internal field (i.e., putting Ke ¼ 0), and setting r ¼ a. The factor  ½nk ðmÞ þ 1 arising from the differentiation with respect to r can also be set to unity. Similarly, the north and east components can be made to play the role of north and east derivatives of a general function by similar modifications (see Haines, 1988, p. 422 for details). This latter aspect can be used to model electric fields (horizontal derivatives of an electric potential) over a surface in the ionosphere. Although an electric potential satisfies Poisson’s equation in three-dimensional space, it can be treated as a general function on the spherical surface. This surface expansion is in fact the solution of the two-dimensional eigenvalue problem in the two variables (colatitude and longitude) whose functions (Legendre and trigonometric) were orthogonalized in the solution of the three-dimensional differential equation. This is a common technique in expansion methods. In Cartesian coordinates, for example, the solution to Laplace’s equation in three-dimensional space (“Rectangular Harmonics”) gives rise to a surface expansion (two-dimensional “Fourier Series”) simply by finding the eigensolutions in the coordinates whose (trigonometric) functions were orthogonalized there. Again, these Fourier series allow the expansion of a very large class of functions, certainly functions that are in no way constrained as are the rectangular harmonic functions of three-dimensional space. Of course, we can go down another dimension and expand onedimensional functions in either of the surface variables. This gives an expansion, on the spherical cap, in trigonometric functions for longitude and in Legendre functions for colatitude, analogously to the one-dimensional trigonometric Fourier series in Cartesian coordinates.

Example application Regional magnetic field models over Canada have been produced every 5 years since 1985 using spherical cap harmonics. For each model, the IGRF at an appropriate epoch is subtracted from the data, and a spherical cap harmonic model of the residuals is then determined. (This also provides an estimate of the “error” or wavelength limitations of the IGRF that was subtracted.) The final model, obtained by adding the IGRF back on to the spherical cap model, is referred to as the Canadian Geomagnetic Reference Field (CGRF). Details of the

397

data and processing methods used for the 1995 CGRF have been given by Haines and Newitt (1997). G.V. Haines

Bibliography Chapman, S., and Bartels, J., 1940. Geomagnetism, Vol. II. New York: Oxford University Press. Courant, R. and Hilbert, D., 1953. Methods of Mathematical Physics, translated and revised from the German original, Vol. I. New York: Interscience Publishers. Davis, H.F., 1963. Fourier Series and Orthogonal Functions. Boston: Allyn and Bacon. De Santis, A., Torta, J.M., and Lowes, F.J., 1999. Spherical cap harmonics revisited and their relationship to ordinary spherical harmonics. Physics and Chemistry of the Earth (A), 24: 935–941. Haines, G.V., 1985a. Spherical cap harmonic analysis. Journal of Geophysical Research, 90: 2583–2591. Haines, G.V., 1985b. Magsat vertical field anomalies above 40
N from spherical cap harmonic analysis. Journal of Geophysical Research, 90: 2593–2598. Haines, G.V., 1988. Computer programs for spherical cap harmonic analysis of potential and general fields. Computers and Geosciences, 14: 413–447. Haines, G.V., 1990. Modelling by series expansions: a discussion. Journal of Geomagnetism and Geoelectricity, 42: 1037–1049. Haines, G.V., 1991. Power spectra of subperiodic functions. Physics of the Earth and Planetary Interiors, 65: 231–247. Haines, G.V., and Newitt, L.R., 1997. The Canadian Geomagnetic Reference Field 1995. Journal of Geomagnetism and Geoelectricity, 49: 317–336. Lowes, F.J., 1999. Orthogonality and mean squares of the vector fields given by spherical cap harmonic potentials. Geophysical Journal International, 136: 781–783. Smythe, W.R., 1950. Static and Dynamic Electricity, 2nd ed. New York: McGraw-Hill.

Cross-references Harmonics, Spherical IGRF, International Geomagnetic Reference Field Upward and Downward Continuation

HARTMANN, GEORG (1489–1564) Hartmann was born on February 9, 1489, at Eggolsheim, Germany. After studying theology and mathematics at Cologne around 1510, he spent some time in Rome, where he ranked Andreas, the brother of Nicholas Copernicus, amongst his friends. Hartmann belonged to that class of Renaissance scholar who, while the recipient of a learned education, also had a fascination with mechanics, horology, instrumentation, and natural phenomena. His principal claim to fame as a student of magnetism lay in his discovery that in addition to the compass needle pointing north, it also had a dip, or inclination, out of the horizontal. He claims to have noticed, for instance, that in Rome, the needle of a magnetic compass dipped by 6
towards the north. Although his numerical quantification of this phenomenon was wrong, the compass does, indeed, dip in Rome, as it does in most nonequatorial locations. Although Hartmann never published his discovery, he did communicate it in a letter of 4 March 1544 to Duke Albert of Prussia. Unfortunately, this remained unknown to the wider world for almost the next three centuries, until it was finally printed in 1831. Hence, Hartmann’s discovery had no influence on other early magnetical researchers, and credit for the discovery of the “dip” went to the Englishman Robert

398

HELIOSEISMOLOGY

Norman (q.v.), who published his own independent discovery in A Newe Attractive (London, 1581). Hartmann was a priest by vocation, and held several important benefices. He settled in Nuremberg in 1518, where he was no doubt in his element, as that city was one of Europe’s great centers for the manufacture of clocks, watches, instruments, ingenious firearms, and other precision metal objects. He collected astronomical and scientific manuscripts and was associated with the Nuremberg observational astronomer Johann Schöner, as well as Joachim Rheticus, who saw Copernicus’ De Revolutionibus through the press at Nuremberg in 1543. Hartmann died in Nuremberg on April 9, 1564. Allan Chapman

Bibliography Heger, K., 1924. Georg Hartmann von Eggolsheim. Der Fränkische Schatzgräber 2: 25–29. Hellmann, G. (ed.), Rara Magnetica 1269–1599, Neudrucke von Schriften und Karten über Meteorologie und Erdmagnetismus no. 10 (Berlin, 1898). Reprints Hartmann’s letter to Duke Albert of Prussia. Letter first noticed in Prussian Royal Archives, Königsberg; see J. Voigt, in Raumer’s Historisches Taschenbuch II (Leipzig, 1831), 253–366. Ritvo, L.B., ‘Georg Hartmann’, Dictionary of Scientific Biography (Scribner’s, New York, 1972–). Zinner, Ernst, Deutsche und niederländische astronomische Instrumente des 11–18 Jahrhunderts (Munich, 1956).

HELIOSEISMOLOGY Helioseismology is the study of the interior of the Sun using observations of waves on the Sun’s surface. It was discovered in the 1960s by Leighton, Noyes, and Simon that patches of the Sun’s surface were oscillating with a period of about 5 min. Initially these were thought to be a manifestation of convective motions, but by the 1970s it was understood from theoretical works by Ulrich and Leibacher and Stein and from further observational work by Deubner that the observed motions are the superposition of many global resonant modes of oscillation of the Sun (e.g., Deubner and Gough, 1984, Christensen-Dalsgaard, 2002). The Sun is a gaseous sphere, generating heat through nuclear fusion reactions in its central region (the core) and held together by self-gravity. It can support various wave motions, notably acoustic waves and gravitational waves; these set up global resonant modes. The modes in which the Sun is observed to oscillate are predominantly acoustic modes, though the acoustic wave propagation is modified by gravity and the Sun’s internal stratification, and by bulk motions and magnetic fields. The observed properties of the oscillations, especially their frequencies, can be used to make inferences about the physical state of the solar interior. The excitation mechanism is generally believed to be turbulent convective motions in subsurface layers of the Sun, which generate acoustic noise. This is a broadband source, but only those waves that satisfy the appropriate resonance conditions constructively interfere to give rise to resonant modes. There are many reasons to study the Sun. It is our closest star and the only one that can be observed in great detail, so it provides an important input to our understanding of stellar structure and evolution. The Sun greatly influences the Earth and near-Earth environment, particularly through its outputs of radiation and particles, so studying it is important for understanding solar-terrestrial relations. And the Sun also provides a unique laboratory for studying some fundamental physical processes in conditions that cannot be realized on Earth. From the 1980s up to the present day, much progress has been made in helioseismology through analysis of the Sun’s global modes of

oscillation. Some of the results, on the Sun’s internal structure and its rotation, are given below. Though not discussed here, similar studies are now also beginning to be made in more limited fashion for some other stars, in a field known as asteroseismology. In the past decade, global mode studies of the Sun have been complemented by socalled local helioseismology, techniques such as tomography which have been used to study flows in the near-surface layers and flows and structures under sunspots and active regions of sunspot complexes. Local helioseismology is also discussed below.

Solar oscillations The Sun’s oscillations are observed in line-of-sight Doppler velocity measurements of the visible solar disk, and in measurements of variations of the continuum intensity of radiation from the surface. The latter are caused by compression of the radiating gas by the waves. Spatially resolved measurements are obtained by observing separately different portions of the visible solar disk; but the motions with the largest horizontal scales are also detectable in observations of the Sun as a star, in which light from the whole disk is collected and analyzed as a single time series. To measure the frequencies very precisely, long, uninterrupted series of observations are desirable. Hence, observations are made from networks of dedicated small solar telescopes distributed in longitude around the Earth (networks such as the Global Oscillation Network Group [GONG] making 1024  1024-pixel resolved observations and the Birmingham Solar Oscillation Network [BiSON] making Sun-as-a-star observations) or from space (for instance, the Solar and Heliospheric Orbiter [SOHO] satellite has three dedicated helioseismology instruments onboard, in decreasing order of resolution MDI, VIRGO and GOLF). In Doppler velocity, the amplitudes of individual modes are of order 10 cm s1 or smaller, their superposition giving a total oscillatory signal of the order of a few kilometers per second. The highest amplitude modes have frequencies n of around 3mHz. The outer 30% of the Sun comprises a convectively unstable region called the convection zone (e.g., Christensen-Dalsgaard et al., 1996). The solar oscillations are stochastically excited by turbulent convection in the upper part of the convection zone. The modes are both excited and damped by their interactions with the convection. Although large excitation events such as solar flares may occasionally contribute, the dominant excitation is probably much smaller-scale and probably takes place in downward plumes where material previously brought to the surface by convection has cooled and flows back into the solar interior. These form a very frequent and widespread set of small-scale excitation sources. The fluid dynamics of the solar interior is described by the fluid dynamical momentum equation and continuity equation, an energy equation and Poisson’s equation for the gravitational potential. On the timescales of the solar oscillations of interest here, the bulk of the solar interior can be considered to be in thermal equilibrium and providing a static large-scale equilibrium background state in which the waves propagate. Treating the departures from the timeindependent equilibrium state as small perturbations with harmonic time dependence expðiotÞ, where o 2pnpffiffiffiffiffiffi is ffian (as yet) unknown resonant angular frequency, t is time and i ¼ 1, the above mentioned equations can be linearized in perturbation quantities and constitute a coupled set of linear equations describing the waves. The timescales of the solar oscillations are sufficiently short that the energy equation can be replaced by the condition that the perturbations of pressure ( p) and density ðrÞ are related by an adiabatic relation. At high frequencies, pressure forces provide the dominant restoring force for the perturbed motions, giving rise to acoustic waves. A key quantity for such acoustic waves is the adiabatic sound speed c, which varies with position inside the Sun: c2 ¼ G1 p=r, where G1 is the first adiabatic exponent (G1 ¼ 5=3 for a perfect monatomic gas). To an excellent approximation, p / rT =m, where T is temperature and m is the mean molecular weight of the gas, so

HELIOSEISMOLOGY

c / T 1=2 inside the Sun. Except for wave propagating exactly vertically, inward-propagating waves get refracted back to the surface at some depth, because the temperature and hence sound speed increase with depth. Near the surface, outward-propagating waves also get deflected, by the sharply changing stratification in the near-surface layers. Hence the acoustic waves get trapped in a resonant cavity and form modes, with discrete frequencies o. To a good approximation, the Sun’s structure is spherically symmetric, hence, the horizontal structure of the eigenfunctions of the modes is given by spherical harmonics Ylm , where integers lðl  0Þ and mðl  m  lÞ are respectively called the degree and azimuthal order of the mode. The structure of the eigenfunction in the radial direction into the Sun is described by a third quantum number n, called the order of the mode: the absolute value of n is essentially the number of nodes in the (say) pressure perturbation eigenfunction between the center and the surface of the Sun. Hence, each resonant frequency can be labeled with three quantum numbers thus: onlm . The labeling is such that at fixed l and m, the frequency onlm is a monotonic increasing function of n. In a wholly spherically symmetrical situation the frequencies would be independent of m. However, the Sun’s rotation, as well as any other large-scale motions, thermal asphericities, and magnetic fields break this degeneracy and introduce a dependence on m in the eigenfrequencies. The modes of different m but with the same values of n and l are called a multiplet: it is convenient also to introduce the mean multiplet frequency onl. High-frequency modes, with positive values of the order n, are essentially acoustic modes (p modes): for them, the dominant restoring force is pressure. Low-frequency modes, with negative values of n, are essentially gravity modes (g modes) set up by gravity waves, which can propagate where there is a stable stratification. There is an intermediate mode with n ¼ 0: this is the so-called fundamental or f mode. For large values of l, the f mode has the physical character of a surface gravity mode. The observed global modes of the Sun are p modes and f modes. Modes of low degree (mostly l ¼ 0; 1; 2; 3) are detectable in observations of the Sun as a star: low-degree p modes are sensitive to conditions throughout the Sun, including the energy-generating core. Spatially resolved observations have detected p and f modes up to degrees of several thousand: such modes are no longer global in character, but nonetheless measuring their properties conveys information about the Sun’s outer subsurface layers. Internal g modes have not unambiguously been observed to date, and they are expected to have small amplitudes at the surface: this is because their region of propagation is the stably stratified radiative interior and they are evanescent through the intervening convection zone. The computed mean multiplet frequencies for modes of a current model of the Sun are shown in Figure H10. Points on each branch of the dispersion relation (each corresponding to a single value of n) have been joined with continuous curves.

399

Figure H10 Mean multiplet cyclic frequencies nnl ð¼ onl =2pÞ of p, f, and g modes of a solar model, as a function of the mode degree l. The low-order p modes are labelled according to the value of the mode order n. The g modes are only illustrated up to l ¼ 7 and for jnj  20.

Results from global-mode helioseismology The frequencies of the Sun’s global modes are estimated by making suitable spatial projections and temporal Fourier transformations of the observed Doppler or intensity fluctuations. From these, using a variety of fitting or inversion techniques (similar to those used in other areas of, e.g., geophysics and astrophysics), properties of the solar interior can be deduced. Indeed, helioseismology has borrowed and adapted various approaches and ideas on inversion from geophysics, notably those of G. Backus and F. Gilbert (Backus and Gilbert, 1968, 1970). One of the principal deductions from helioseismology has been the sound speed as a function of position in the Sun (e.g., Gough et al., 1996). The value of this deduction lies not in the precise value of the sound speed but in the inferences that follow concerning the physics that determines the sound speed. The sound speed in a solar model that is closely in agreement with the helioseismic data is illustrated in Figure H11. As discussed above, the adiabatic sound

Figure H11 Square of the adiabatic sound speed c inside a model of the present Sun, as a function of fractional radius (center at 0, photospheric surface at 1.0).

speed c is related to temperature T, mean molecular weight m, and adiabatic exponent G1 by c2 / G1 T =m, where the constant of proportionality is the gas constant. The general increase in sound speed with depth reflects the increase in temperature from the surface to the center of the

400

HELIOSEISMOLOGY

Sun. The gradient of the temperature, and hence of the sound speed, is related to the physics by which heat is transported from the center to the surface. In the bulk of the Sun, that transport is by radiation; but in the outer envelope the transport is by convection. A break in the second derivative of the sound speed near a fractional radius of 0.7 indicates the transition between the two regimes and has enabled helioseismology to determine the location of the base of the convection zone: this is important because it delineates the region in which chemical elements observed at the surface are mixed and homogenized by convective motions. Beneath the convection zone, the gradients of temperature and sound speed are influenced by the opacity of the material to radiation: it has thus been possible to use the seismically determined sound speed to find errors in the theoretical estimates of the opacity, which is one of the main microphysical inputs for modeling the interiors of stars. Not readily apparent on the scale of Figure H11 is the spatial variation of G1 and hence of sound speed in the regions of partial ionization of helium and hydrogen in the outer 2% of the Sun. This variation depends on the equation of state of the material and on the abundances of the elements: it has been used to determine that the fractional helium abundance by mass in the convection zone, which is poorly determined from surface spectroscopic observations, is about 0.25. This is significantly lower than the value of 0.28 believed from stellar evolutionary models to have been the initial helium abundance of the Sun when it formed. The deficiency of helium in the present Sun’s convection zone is now understood to arise from gravitational settling of helium and heavier elements out of the convection zone and into the radiative interior over the 4.7 Ba that the Sun has existed as a star. This inference is confirmed by an associated slight modification to the sound speed profile beneath the convection zone. The dip in the sound speed at the center of the Sun is the signature of the fusion of hydrogen to helium that has taken place over the Sun’s lifetime: the presence of helium increases the mean molecular weight m, decreasing the sound speed. Thus the amount of build-up of helium in the core is an indicator of the Sun’s age. Apart from inversions, a sensitive indicator of the sound speed in the core is the difference nnl  nn1;lþ2 between frequencies of p modes of low degree ðl ¼ 0, 1, 2, 3Þ differing by two in their degree. Such pairs of modes penetrate into the core and have similar frequencies and hence very similar sensitivity to radial structure in the outer part of the Sun, but have different sensitivities in the core. Another major deduction of global-mode helioseismology has been the rotation as a function of position through much of the solar interior (Figure H12) (e.g., Thompson et al., 1996, 2003). In the convection zone, it has been discovered that the rotation varies principally with latitude and rather little with depth: at low solar latitudes the rotation is fastest, with a rotation period of about 25 days; while at high latitudes the rotation periods in the convection zone are in excess of 30 days. These rates are consistent with deductions of the surface rotation from spectroscopic observations and from measurements of motions of magnetic features such as sunspots. The finding was at variance with some theoretical expectations that the rotation in the convection zone would be constant on Taylor columns. At low- and mid-latitudes there is a near-surface layer of rotational shear, which may account for the different rotational speeds at which small and large magnetic features are observed to move. Near the base of the convection zone, the latitudinally differential rotation makes a transition to latitudinally independent rotation. This gives rise to a layer of rotational shear at low and high latitudes, which is called the tachocline. It is widely believed that the tachocline is where the Sun’s large-scale magnetic field is generated by dynamo action, leading to the 11-year solar cycle of sunspots and the large-scale dipole field. Deeper still the rotation appears to be consistent with solid-body rotation, presumably caused by the presence of a magnetic field. In the core, there is some hint of a slower rotation, but the uncertainties on the deductions are quite large: nonetheless, some earlier theoretical predictions that the core would rotate much faster than the surface, a relic of the Sun’s faster rotation as a

Figure H12 (a) Contour plot of the rotation rate inside the Sun inferred from MDI data. The rotation axis is up the y-axis, the solar equator is along the x-axis. Contour spacings are 10 nHz; contours at 450, 400, and 350 nHz are thicker. The shaded region indicates where a localized solution has not been possible with these data. (b) The rotation rate deeper in the interior, on three radial cuts at solar latitudes 0
; 30
, and 60
, using data from Sun-as-a-star observations by BiSON and spatially resolved observations by the LOWL instrument. young star, are strongly ruled out by the seismic observations. Superimposed on the rotation of the convection zone are weak but coherent migrating bands of faster and slower rotation (of amplitude only a few meters per second, compared with the surface equatorial rotation rate of about 2 km s1 ), which have been called torsional oscillations: the causal connection between the migrating zonal flows and the sunspot active latitudes, which also migrate during the solar cycle, is as yet uncertain. There have also been reported weak variations, with periodicities around 1.3 years, in the rotation rate in the deep convection zone and in the vicinity of the tachocline. To date there has been no direct helioseismic detection of magnetic field in the region of the tachocline: indeed, because the pressure increases rapidly with depth, a field there would have to have a strength of order 106 Gauss to have a significant direct influence on the mode frequencies. Thus detecting

HIGGINS-KENNEDY PARADOX

the effect of a temporally varying magnetic field on the angular momentum may be the most likely way to infer seismically the presence of such a field in the deep interior. On the other hand, nearsurface magnetic fields can influence mode frequencies in a detectable way, because the pressure is much lower there; and there is strong evidence that mode frequencies vary over the solar cycle in a manner that is highly correlated with the temporal and spatial variation of photospheric magnetic field.

Local helioseismology Analysis of the Sun’s global mode frequencies has provided an unprecedented look at the interior of a star, but such an approach has limitations. In particular, the frequencies sense only a longitudinal average of the internal structure. To make more localized inferences, the complementary approaches of local helioseismology are used. One such technique is to analyze the power spectrum of oscillations as a function of frequency and the two horizontal components of the wavenumber in localized patches. To obtain good wavenumber resolution, the patches are usually quite large: square tiles of up to about 2  105 km on the side. The technique is known as ring analysis, because at fixed frequency the p-mode power lies on near-circular rings in the horizontal wavenumber plane. By performing inversions for the depth dependence under each tile, maps with horizontal resolution similar to the size of the tiles can be obtained of structures and particularly of flows in the outer few per cent of the solar interior (e.g., Toomre, 2003). As well as the zonal flows, the meridional (i.e., northward and southward) flow components have also been measured. Beneath the surface, these are generally found to be poleward in both hemispheres down to the depth at which the ring analyzes lose resolution: this depth is of order 104 km. However, the analyses indicate that the flow patterns in the northern but not the southern hemispheres changed markedly in 1998–2000 as the Sun approached the maximum of its 11-year magnetic activity cycle. Other local techniques include time-distance helioseismology and acoustic holography. So for example, in time-distance helioseismology, the travel-time of waves between different points on the surface of the Sun are used to infer wavespeed and flows under the surface (e. g., Kosovichev, 2003). This can be achieved on much smaller horizontal scales than have yet been achieved by ring analysis, down to just a few thousand kilometers. Since individual excitation events are rarely if ever seen, the travel-times are isolated from the measured Doppler velocities across the solar disk by cross-correlating pairs of points, or sets of points. Using high-resolution observations such as those from the MDI instrument onboard SOHO, travel-times can be measured between many different locations on the solar surface, and with different spatial separations which in turn provide different depth sensitivities to subsurface conditions. By comparing the measured traveltimes with those of a solar model, inversion techniques similar to those used in global mode helioseismology have been used to map conditions in the convection zone using travel-time data. In particular, wavespeed anomalies and flows under sunspots and active regions of strong magnetic fields on the surface have been mapped. An unexpected finding has been that the wavespeed beneath sunspots (which in such regions is not just the sound speed but is modified by the magnetic field) is increased relative to the spot’s surroundings, except in a shallow layer of up to 5000km depth where the wavespeed is decreased. Such studies are providing much-needed constraints on models of sunspot structure and theories of their origin, and on models of the emergence of magnetic flux from the solar interior. Michael J. Thompson

Bibliography Backus, G. and Gilbert, F., 1968. The resolving power of gross Earth data. Geophysical Journal, 16: 169–205.

401

Backus, G. and Gilbert, F., 1970. Uniqueness in the inversion of inaccurate gross Earth data. Philosophical Transactions of the Royal Society of London, Series A, 266: 123–192. Christensen-Dalsgaard, J., 2002. Helioseismology. Reviews of Modern Physics, 74: 1073–1129. Christensen-Dalsgaard, J., et al.,1996. The current state of solar modeling. Science, 272: 1286–1292. Deubner, F.-L., and Gough, D.O., 1984. Helioseismology: oscillations as a diagnostic of the solar interior. Annual Review of Astronomy and Astrophysics, 22: 593–619. Gough, D.O., et al.,1996. The seismic structure of the Sun. Science, 272: 1296–1300. Kosovichev, A., 2003. Telechronohelioseismology. In Thompson, M.J., and Christensen-Dalsgaard, J. (eds.), Stellar Astrophysical Fluid Dynamics. Cambridge: Cambridge University Press, pp. 279–296. Thompson, M.J., Christensen-Dalsgaard, J., Miesch, M.S., and Toomre, J., 2003. The internal rotation of the Sun. Annual Review on Astronomy and Astrophysics, 41: 599–643. Thompson, M.J., et al.,1996. Differential rotation and dynamics of the solar interior. Science, 272: 1300–1305. Toomre, J., 2003. Bridges between helioseismology and models of convection zone dynamics. In Thompson, M.J., and ChristensenDalsgaard, J., (eds.), Stellar Astrophysical Fluid Dynamics. Cambridge: Cambridge University Press, pp. 299–314.

Cross-references Dynamo, Solar Harmonics, Spherical Magnetic Field of Sun Proudman-Taylor Theorem

HIGGINS-KENNEDY PARADOX For the student learning about properties of the Earth’s core it may be surprising that the core is solid at its center where its temperature is highest, while it becomes liquid at a distance of about 1220 km from the center where the temperature is lower. This property is caused by the dependence of the melting temperature Tm on the pressure p. That the melting temperature Tm of nearly all materials increases with pressure has been a well-known property for a long time and finds its most simple expression in Lindemann’s law   1 dTm 1 (Eq. 1) ¼ 2 g  =k Tm dp 3 where k is the compressibility along the melting curve Tm(p) and g is the Grüneisen parameter which, in thermodynamics, is defined by g¼

akT akS ¼ : rcv rcp

(Eq. 2)

Here, r denotes the density, a is the coefficient of thermal expansion, and kT and kS are the isothermal and adiabatic compressibilities, while cv and cp are the specific heats at constant volume and constant pressure, respectively. The Grüneisen parameter g plays a prominent role in studies of the thermal state of planetary interiors since it assumes a value of the order unity for all condensed materials and does not vary much with pressure or temperature in contrast to the material properties on the right hand sides of Eq. (2). Using thermodynamic relationships one finds (see Grüneisen’s parameter for iron and Earth’s core)   T ]V  ] ln V  ¼ (Eq. 3) g¼  V ]T S ] ln T S

402

HUMBOLDT, ALEXANDER VON (1759–1859)

which indicates that g describes the negative slope in logarithmic plots of volume versus temperature under adiabatic compression. The g used in relationship (1) is not necessarily the thermodynamic one, since the melting temperature may be influenced by electronic contributions and other effects which are not taken into account in Eqs. (2) and (3). The simple approximate relationship (1) called Lindemann’s law is far from a rigorous law in any mathematical sense and is mentioned here only as an example of many similar “laws” that have been derived in the literature. For a review and a recent paper, see Jacobs (1987) and Anderson et al. (2003). The adiabatic temperature gradient is defined by (see Core, adiabatic gradient)  ]T  aT gT ¼ ¼ (Eq. 4) ]p S rcp kS Here, the isentropic compressibility kS is well determined in the Earth’s core by seismic data, since the relationship kS ¼ rðVp2  43Vs2 Þ holds where Vp and Vs are the velocities of propagation of p (compressional)- and s (shear)-waves. The adiabatic temperature gradient defines a state where the exchange of fluid parcels from different radii can be accomplished without gain or loss of energy when dissipative effects such as those connected with viscous friction can be neglected. When the increase of temperature with pressure is less than the adiabatic gradient, a stably stratified state is obtained which requires an input of energy for the exchange of fluid parcels from different radii. When the temperature increases more strongly by a finite amount e than the adiabatic gradient, the static state becomes unstable and convection sets in. In fact, for dimensions as large as those of the core, the amount e is minute, such that a state of convection corresponds essentially to an adiabatic temperature distribution. Since convection flows of sufficient strength are needed to generate the geomagnetic field (Busse, 1975) it has always been assumed that the temperature field of the liquid outer core must be close to an adiabatic one except, perhaps, close to the core-mantle boundary. In their paper of 1971, Higgins and Kennedy shattered this confidence in the adiabatic temperature distribution. On the basis of the extrapolation of numerous experimental results measured at relatively low pressures, they suggested that the melting temperature Tm in the core depends only weakly on the pressure and that the adiabatic temperature would exhibit a much steeper dependence. Accordingly they claimed that the latter temperature distribution could only be compatible with a frozen outer core in contradiction to all seismic evidence, while any temperature distribution at the melting temperature or above would imply a stably stratified core in contradiction to the dynamo hypothesis of the origin of geomagnetism. This is the HigginsKennedy core paradox. The publication of the paper by Higgins and Kennedy (1971) stimulated numerous attempts to circumvent the paradoxial situation. It turns out that the melting temperature Tm(p) can coincide with an isentropic temperature distribution if a suspension is assumed of small solid particles; the melting and freezing of which contributes the correct energies for a neutral exchange of fluid parcels from different radii (Busse, 1972; Malkus, 1973). Stacey (1972) pointed out that the adiabat corresponding to the temperature at the inner core-outer core boundary may well lie above the melting temperature of the outer core since the latter corresponds to that of iron alloyed with a significant amount of light elements. It is thus considerably lower than the temperature at the boundary of the inner core, which corresponds to the melting of nearly pure iron or an iron-nickel alloy. In later years rather convincing arguments have been put forward which cast doubts on the validity of the extrapolation of experimental data carried out by Higgins and Kennedy. Their proposed melting temperature together with their assumed value of g is far removed from Lindemann’s law (1). On the other hand, modern theoretical studies support relationships similar to Eq. (1). Stevenson (1980) and others argued that a

weak pressure dependence of Tm( p) is compatible only with an unphysically low value of the Grüneisen parameter g. Indeed, comparing Eqs. (1) and (4) one finds that  dTm ]T  <  (Eq. 5) ]p S dp can be satisfied only for g < 23, which contrasts with the value of about 1.5 of g found for most liquid metals at high pressures. For details on the various theoretical arguments we refer to the comprehensive review given by Jacobs (1987). For a recent review of experimental measurements see Boehler (2000). Although it is unlikely today that a situation as imagined by Higgins and Kennedy exists in the Earth’s core, this possibility cannot be excluded entirely. In view of our ignorance about properties of the core, it seems advisable to keep the core paradox in the back of one’s mind in thinking about problems of planetary interiors (see Dynamos, planetary and satellites). Friedrich Busse

Bibliography Anderson, O.L., Isaak, D.G., and Nelson, V.E., 2003. The highpressure melting temperature of hexagonal close-packed iron determined from thermal physics. Journal of Physics and Chemistry of Solids, 64: 2125–2131. Boehler, R., 2000. High-pressure experiments and the phase diagram of lower mantle and core materials. Reviews of Geophysics, 38: 221–245. Busse, F.H., 1972. Comments on paper by G. Higgins and G.C. Kennedy, The adiabatic gradient and the melting point gradient in the core of the Earth, Journal of Geophysical Research, 77: 1589–1590. Busse, F.H., 1975. A necessary condition for the geodynamo. Journal of Geophysical Research, 80: 278–280. Higgins, G. and Kennedy, G.C., 1971. The adiabatic gradient and the melting point gradient in the core of the Earth. Journal of Geophysical Research, 76: 1870–1878. Jacobs, J.A., 1987. The Earth’s Core, 2nd edn, London: Academic Press. Malkus, W.V.R., 1973. Convection at the melting point: a thermal history of the Earth’s core. Geophysical Fluid Dynamics, 4: 267–278. Stacey, F.D., 1972. Physical Properties of the Earth’s Core. Geophysical Surveys, 1: 99–119. Stevenson, D.J., 1980. Applications of liquid state physics to the Earth’s core. Physics of the Earth and Planetary Interiors, 22: 42–52.

Cross-references Core, Adiabatic Gradient Dynamos, Planetary and Satellite Grüneisen’s Parameter for Iron and Earth’s Core

HUMBOLDT, ALEXANDER VON (1759–1859) Alexander von Humboldt (Figure H13) was born in Berlin in 1769 as the son of a nobleman and former officer of the Prussian Army. He studied at the universities of Frankfurt/Oder and Göttingen, at the trade academy of Hamburg and at the mining academy of Freiberg in Saxonia. In his studies he was interested in all aspects of nature from botany and zoology to geography and astronomy. His friendship with Georg Forster who had participated in Cook’s second voyage and whom he met in Göttingen had a strong influence on him and

HUMBOLDT, ALEXANDER VON (1759–1859)

Figure H13 Alexander von Humboldt in 1832. (Lithography of F.S. Delpech after a drawing of Francois Ge´rard.)

motivated him to see his vocation as a scientific explorer. First he entered, however, the career as a mining inspector in the service of the Prussian state. But when his mother died in 1796—he had lost his father already when he was only nine years old—and he became the heir of a considerable fortune, he declined the offer of the directorship of the Silesian mines and started the realization of his long held plan for a scientific expedition. In 1798, he embarked on his six years voyage to South and North America from which he returned in 1804 with huge collections of scientific data and materials. It was in the preparation for this expedition that he was instructed by the French mathematician and nautical officer Jean-Charles de Borda in the measurements of the components of the Earth’s magnetic field and throughout the years of his voyage he collected magnetic data in addition to geodetic, meteorological, and other ones. One of his findings was the decrease of magnetic intensity with latitude which, apparently, had not been clearly recognized until that time since measurements had been focused on the direction of the field. A. von Humboldt continued his magnetic measurements on his voyage with his friend Gay-Lussac to Italy in 1805 and back to Berlin. Here he was offered a wooden cabin, which allowed him to take readings of the declination each night for several months, a task he shared with the astronomer Oltmann. In December 1805 he was lucky in observing strong fluctuations of the magnetic field while an aurora borealis occurred. Humboldt coined the term “Magnetischer Sturm” for the period of strong oscillations of the magnetometer needle. Nowadays the term “magnetic storm” is generally accepted for this phenomenon. Because of the defeat of Prussia by Napoleon’s army von Humboldt had to stay longer in Berlin than he had planned and returned to Paris only in late 1807. Here he was occupied with the edition of the scientific results of his American voyage until 1827. It is worth noting that during a shorter journey to Berlin in 1826, he stopped in Göttingen to meet Carl Friedrich Gauss (see Gauss, Carl Friedrich) for the first time in person. A. von Humboldt and Gauss were the most prominent German scientists of their time and they had been in contact by letters for quite a while. They had high regards for each other even though they were opposites in their styles of research. A. von Humboldt was one of the last universally educated scientists interested in the descriptive comprehension of all natural phenomena, while the mathematician and physicist Gauss used primarily deductive analysis in his research.

403

The study of the Earth’s magnetic field was one of their common interests. Their interaction intensified after von Humboldt had invited Gauss to participate in the 7th Assembly of the Society of German Scientists and Physicians, which took place in the fall of 1828 in Berlin. During that time Gauss stayed at von Humboldt’s house. The latter had left Paris reluctantly in 1827 to follow a call from the Prussian King Friedrich Wilhelm III to assume the position of a chamberlain at the Berlin court. In his free time, von Humboldt continued his scientific work among which magnetic measurements had a high priority. For this purpose his friend A. Mendelson-Bartholdy (father of the famous composer) had provided a place in his garden where von Humboldt built an iron free wooden cabin for his magnetometers. Gauss pursued his studies of the Earth’s magnetic field quite independently from those of von Humboldt, which caused the latter some irritation. Von Humboldt had thought that he had motivated Gauss to do magnetic measurements when the latter visited him in Berlin in 1828. But Gauss’ interests in geomagnetism went back to a time 40 years earlier, as he mentions in a letter of 1833, and in 1806 he had already contemplated a description of the field in terms of spherical harmonics. At the Berlin Assembly, Gauss had met the promising physicist Wilhelm Weber and had arranged that this young man got a professorship in Göttingen in 1831. In the following six years, an intense and highly productive collaboration between Gauss and Weber ensued on all kinds of electromagnetic problems. In the course of this research Gauss developed his method of the absolute determination of magnetic intensity (see Gauss’ determination of absolute intensity) and built together with Weber appropriate instruments. It now became possible to calibrate instruments locally independently from any other. For the purpose of absolute measurements of the magnetic field and its variations in time at different places on the Earth, Gauss and Weber initiated the “Göttinger Magnetischer Verein.” Alexander von Humboldt was also interested in simultaneous measurements of the geomagnetic field at different geographic locations in order to determine, for instance, whether magnetic storms are of terrestrial origin or depended on the position of the Sun. Already in 1828 he had arranged for coincident measurements in Paris and inside a mine in Freiberg (Saxonia). When he received a glorious reception at the Russian court at St. Petersburg in 1829 at the end of his expedition to Siberia, von Humboldt used the opportunity to suggest the creation of a network of stations throughout the Russian empire for the collection of magnetic and meteorological data. In the following years such stations were indeed installed including one in Sitka, Alaska, which at that time belonged to Russia. Von Humboldt realized that he had to persuade British authorities in order to achieve his goal of a nearly worldwide distribution of stations. In 1836 he wrote to the Duke of Sussex whom he had gotten to know during his student days at the University of Göttingen and who was now the president of the Royal Society. For an English translation of this letter see the paper by Malin and Barraclough (1991). Von Humboldt’s recommendations for the establishment of permanent magnetic observatories in Canada, St. Helena, Cape of Good Hope, Ceylon, Jamaica, and Australia were well received. Besides realizing these proposals, the British Government went a step further and organized an expedition to Antarctica under the direction of Sir James Clark Ross with magnetic measurements as one of its main tasks. Based on the observatory data, Sir Edward Sabine (see Sabine, Edward) who supervised the network of stations could later establish a connection between magnetic storms and sunspots and demonstrate in particular the correlation between the 11-year sunspot cycle and a corresponding periodicity of magnetic storms. In the course of his continuing research on geomagnetism, von Humboldt realized the superiority of Gauss’ method of measurement and he also often expressed his admiration for Gauss’ theoretical work on the representation of the geomagnetic field in terms of potentials and the separation of internal and external sources in particular. From today’s point of view it is obvious that Gauss made the more fundamental contributions to the field of geomagnetism. But Alexander

404

HUMBOLDT, ALEXANDER VON AND MAGNETIC STORMS

von Humboldt’s lasting legacy has been the organization of a worldwide cooperation in the gathering of geophysical data with simultaneous measurements at prearranged dates; the influence of which can still be felt today. A. von Humboldt died in 1859 at the age of nearly 90 years. For further details on von Humboldt’s scientific endeavours the reader is referred to the book by Botting (1973). Friedrich Busse

Bibliography Botting, D., 1973. Humboldt and the Cosmos, George Rainbird Ltd., London. Malin, S.R.C. and Barraclough, D.R., 1991. Humboldt and the Earth’s magnetic field Quarterly Journal of the Royal Astronomical Society 32: 279–293.

Cross-references Gauss, Carl Friedrich (1777–1855) Gauss’ Determination of Absolute Intensity Sabine, Edward (1788–1883)

HUMBOLDT, ALEXANDER VON AND MAGNETIC STORMS Baron Alexander von Humboldt (September 14, 1769–May 6, 1859) was a great German naturalist and explorer, universal genius and cosmopolitan, scientist, and patron. Application of his experience and knowledge gained through his travels and experiments transformed contemporary western science. He is widely acknowledged as the founder of modern geography, climatology, ecology, and oceanography. This article focuses on Alexander von Humboldt’s contribution to the science of geomagnetism (Busse, 2004), which is not so widely known.

Historical background on geomagnetism With the publication of De Magnete by William Gilbert in A.D. 1600, the Earth itself was seen as a great magnet, and a new branch of physics, geomagnetism, was born. The first map of magnetic field declination was made by Edmund Halley in the beginning of 18th century. Alexander von Humboldt prepared a chart of “isodynamic zones” in 1804, based on his measurements made during his voyages through the Americas (1799–1805). He swung a dip needle in the magnetic meridian plane, and observed the number of oscillations during 10 min. He noticed that the number of oscillations were at maximum near the magnetic equator, and decreased to the north and to the south. This indicated a regular decrease of the total magnetic intensity from the poles to the equator. The publication of this result stimulated further investigations on the Earth’s magnetic field (Chapman and Bartels, 1940). Von Humboldt, together with Guy Lussac, improvised the method to measure the horizontal intensity by observing the time of oscillation of a compass needle in the horizontal plane. They took the first measurements of the relative horizontal intensity of the Earth’s magnetic field on a journey to Italy in 1807. From May 1806 to June 1807 in Berlin, Humboldt and a colleague observed the local magnetic declination every half hour from midnight to morning. On December 21, 1806, for six consecutive hours, von Humboldt observed strong magnetic deflections and noted the presence of correlated northern lights (aurora) overhead. When the aurora disappeared at dawn, the magnetic perturbations disappeared as well. Von Humboldt concluded that the magnetic disturbances on the ground and the auroras in the polar sky were two manifestation of the same phenomenon (Schröder, 1997). He gave this phenomenon involving

large-scale magnetic disturbances (possibly already observed by George Graham) the name “Magnetische Ungewitter,” or magnetic storms (von Humboldt, 1808). The worldwide net of magnetic observatories later confirmed that such “storms” were indeed worldwide phenomena. Alexander von Humboldt organized the first simultaneous observations of the geomagnetic field at various locations throughout the world after his return from his South American journey. Wilhelm Weber, Karl F. Gauss, and he organized the Göttingen Magnetische Verein (Magnetic Union); and from 1836–1841, simultaneous observations of the Earth’s magnetic field were made in nonmagnetic huts at up to 50 different locations, marking the beginning of the magnetic observatory system. Through his diplomatic contacts, Alexander von Humboldt was instrumental in the establishment of a number of magnetic observatories around the world, especially in Britain, Russia, and in countries then under the British and Russian rule, e.g., cities such as Toronto, Sitka, and Bombay, etc. He constructed his own iron-free magnetic observatory in Berlin in 1832. These observatories have played an important role in the development of the science of geomagnetism. With the beginning of the space age, our knowledge about the near-Earth space environment, the magnetosphere-ionosphere system in general, and geomagnetic storms in particular, has improved dramatically.

Intense geomagnetic storms and their causes A geomagnetic storm is characterized by a main phase during which the horizontal component of the Earth’s low-latitude magnetic fields are significantly depressed over a time span of one to a few hours. This is followed by a recovery phase, which may extend for 10 h or more (Rostoker, 1997). The intensity of a geomagnetic storm is measured in terms of the disturbance storm-time index (Dst). Magnetic storms with Dst < 100 nT are called intense and those with Dst < 500 nT are called superintense. Geomagnetic storms occur when solar wind-magnetosphere coupling becomes intensified during the arrival of fast moving solar ejecta like interplanetary coronal mass ejections (ICMEs) and fast streams from the coronal holes (Gonzalez et al., 1994) accompanied by long intervals of southward interplanetary magnetic field (IMF) as in a “magnetic cloud” (Klein and Burlaga, 1982). It is now well established that the major mechanism of energy transfer from the solar wind to the Earth’s magnetosphere is magnetic reconnection (Dungey, 1961). The efficiency of the reconnection process is considerably enhanced during southward IMF intervals (Tsurutani and Gonzalez, 1997), leading to strong plasma injection from the magnetotail towards the inner magnetosphere causing intense auroras at highlatitude nightside regions. Further, as the magnetotail plasma gets injected into the nightside magnetosphere, the energetic protons drift to the west and electrons to the east, forming a ring of current around the Earth. This current, called the “ring current,” causes a diamagnetic decrease in the Earth’s magnetic field measured at near-equatorial magnetic stations (see Figure H14 for the results from one such station). The decrease in the equatorial magnetic field strength, measured by the Dst index, is directly related to the total kinetic energy of the ring current particles (Dessler and Parker, 1959; Sckopke, 1966); thus the Dst index is a good measure of the energetics of the magnetic storm. The Dst index itself is influenced by the interplanetary parameters (Burton et al., 1975). One obviously cannot directly determine the solar/interplanetary causes of storm events prior to the space age. However, based on the recently gained knowledge on solar, interplanetary, and magnetospheric physics, one can make these determinations by a process of elimination. For example, by examining the profile of magnetic storms using ground magnetic field data, storm generation mechanisms can be identified (Tsurutani et al., 1999). From the ground magnetometer data shown in Figure H14, reports on the related solar flare event (Carrington, 1859; Hodgson, 1859), and reports on the concomitant aurora (taken from newspapers and private correspondence, Kimball,

HUMBOLDT, ALEXANDER VON AND MAGNETIC STORMS

405

Figure H14 The Colaba (Bombay) magnetogram for the September 1–2, 1859 geomagnetic storm. The peak near 0400 UT September 2, is due to the storm sudden commencement (SSC) caused probably by the shock ahead of the magnetic cloud. This was followed by the storm “main phase” which lasted for about one hour and a half. Taken from Tsurutani et al. (2003). 1960), Tsurutani et al. (2003) were able to deduce that an exceptionally fast (and intense) magnetic cloud was the interplanetary cause of the superintense geomagnetic storm of September 1–2, 1859 with a Dst –1760 nT. This large value of Dst is consistent with the decrease of DH ¼ 1600  10 nT recorded at Colaba (Bombay), India (Figure H14). The supposition that the intense southward IMF was due to a magnetic cloud was surmised by the simplicity and short duration of the storm in the ground magnetic field data. Main phase compound events or “double storms” (Kamide et al., 1998) can be ruled out by the (simple) storm profile. Compound stream events (Burlaga et al., 1987) can also be eliminated by the storm profile. The only other possibility that might be the cause of the storm is sheath fields associated with an ICME. This can be ruled out because the compression factor of magnetic fields following fast shocks is only approximately four times (Kennel et al., 1985). Thus with quiet interplanetary fields being typically 3 to 10 nT, the compressed fields would be too low to generate the inferred interplanetary and magnetospheric electric fields for the storm. Thus by a process of elimination the interplanetary fields that caused this superintense storm have been determined to be part of a fast, intense magnetic cloud. Geomagnetic storms produce severe disturbances in Earth’s magnetosphere and ionosphere, creating so-called adverse space weather conditions. They pose major threats to space- and ground-based technological systems on which modern society is becoming increasingly dependent. The intense magnetic storms during October 29–31, 2003, the so-called “Halloween storms”, associated with the largest X-ray solar flare of the solar cycle 23, caused damage to 28 satellites, ending the operational life of two, disturbed flight routes of some airlines, telecommunications problems, and power outage in Sweden. These Halloween storms with Dst –400 nT were about four times less intense than the superintense storm of 1859. One can imagine the loss to society if a magnetic storm similar to the superintense storm of 1859 were to occur today!

Acknowledgments GSL would like to thank Prof. Y. Kamide for the kind hospitality during his stay at STEL, Nagoya University, Japan. Portions of the research for this work were performed at the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. G.S. Lakhina, B.T. Tsurutani, W.D. Gonzalez, and S. Alex

Bibliography Burlaga, L.F., Behannon, K.W., and Klein, L.W. 1987. Compound streams, magnetic clouds, and major geomagnetic storms. Journal of Geophysical Research, 92: 5725. Burton, R.K., McPherron, R.L., and Russell, C.T. 1975. An empirical relationship between interplanetary conditions and Dst. Journal of Geophysical Research, 80: 4204. Busse, F., 2004. Alexander von Humboldt, this volume. Carrington, R.C., 1859. Description of a singular appearance seen in the Sun on September 1, 1859. Monthly Notices of the Royal Astronomical Society, XX, 13. Chapman, S. and Bartels, J., 1940. Geomagnetism, vol. II, Oxford University Press, New York, pp. 913–933. Dessler, A.J., and Parker, E.N., 1959. Hydromagnetic theory of magnetic storms. Journal of Geophysical Research, 64: 2239. Dungey, J.W., 1961. Interplanetary magnetic field and the auroral zones. Physical Research Letters, 6: 47. Gonzalez, W.D., Joselyn, J.A., Kamide, Y., Kroehl, H.W., Rostoker, G., Tsurutani, B.T., and Vasyliunas, V.M., 1994. What is a geomagnetic storm? Journal of Geophysical Research, 99: 5771. Hodgson, R.,1859. On a curious appearance seen in the Sun. Monthly Notices of the Royal Astronomical Society London, XX, 15. Kamide, Y., Yokoyama, N., Gonzalez, W., Tsurutani, B.T., Daglis, I.A., Brekke, A., and Masuda, S., 1998. Two-step development of geomagnetic storms. Journal of Geophysical Research, 103: 6917. Kennel, C.F., Edmiston, J.P., and Hada, T. 1985. A quarter century of collisionless shock research. In Stone, R.G. and Tsurutani, B.T. (eds.), Collisionless Shocks in the Heliosphere: A Tutorial Review. Washington, DC: American. Geophysical Union, Vol. 34, p. 1. Kimball, D.S., 1960. A study of the aurora of 1859. Sci. Rpt. 6, UAGR109, University of Alaska. Klein, L.W. and Burlaga, L.F., 1982. Magnetic clouds at 1 AU. Journal of Geophysical Research, 87: 613. Rostoker, G., 1997. Physics of magnetic storms. In Tsurutani, B.T., Gonzalez, W.D., Kamide, Y., and Arballo, J.K. (eds.), Magnetic Storms. Geophysical Monograph 98, AGU, Washington DC, p. 149. Schröder, W., 1997. Some aspectss of the earlier history of solarterrestrial physics. Planetary and Space Science, 45: 395. Sckopke, N., 1966. A general relation between the energy of trapped particles and the disturbance field near the Earth. Journal of Geophysical Research, 71: 3125. Tsurutani, B.T. and Gonzalez, W.D., 1997. Interplanetary causes of magnetic storms: a review. In Tsurutani, B.T., Gonzalez, W.D.,

406

HUMBOLDT, ALEXANDER VON AND MAGNETIC STORMS

Kamide, Y., and Arballo, J.K. (eds.), Magnetic Storms. Geophysical Monograph 98, AGU, Washington DC, p. 77. Tsurutani, B.T., Kamide, Y., Arballo, J.K., Gonzalez, W.D., and Lepping, R.P., 1999. Interplanetary causes of great and superintense magnetic storms, Physics and Chemistry of the Earth, 24: 101. Tsurutani, B.T., Gonzalez, W.D., Lakhina, G.S., and Alex, S., 2003. The extreme magnetic storm of September 1–2, 1859.

Journal of Geophysical Research, 108, A7, 1268, doi:10.1029/ 2002JA009504. von Humboldt, A., 1808. Annalen der Physik, 29: 425.

Cross-references Gilbert, William (1544–1603) Halley, Edmond (1656–1742)

I

IAGA, INTERNATIONAL ASSOCIATION OF GEOMAGNETISM AND AERONOMY History The origin of the International Association of Geomagnetism and Aeronomy (IAGA) can be traced to the Commission for Terrestrial Magnetism and Atmospheric Electricity, part of the International Meteorological Organization, which was established in 1873. The Commission planned the geomagnetic observation campaign for the First International Polar Year, 1882–1883. Following the end of World War I, the International Research Council (IRC) was established to promote science through international cooperation. At a meeting of the IRC in July 1919, the International Geodetic and Geophysical Union was formed, with Terrestrial Magnetism and Electricity as Section D, and with its leadership provided by the International Meteorological Organization. In 1930, the International Geodetic and Geophysical Union agreed to cooperate with the International Meteorological Organization to organize a Second International Polar Year, 50 years after the first. The Union changed its name to the International Union of Geodesy and Geophysics (IUGG), its Sections were renamed Associations, and the International Association of Terrestrial Magnetism and Electricity (IATME) came into existence. At the Brussels IUGG General Assembly in 1951, upper atmosphere scientists lobbied to have their interests recognized within IATME. Following the Brussels Assembly, Sydney Chapman (q.v.), who had previously suggested replacing the term “Terrestrial Magnetism” by “Geomagnetism” coined the term “Aeronomy” and suggested adoption of the name International Association of Geomagnetism and Aeronomy. The expansion of the scope of IATME was ratified at the General Assembly in Rome in 1954, the new title IAGA was agreed, and aeronomy was defined as “the science of the upper atmospheric regions where dissociation and ionization are important.” In 1950, Lloyd Berkner proposed a Third International Polar Year to provide motivation to re-equip geophysical observatories, many of which had been damaged or destroyed during World War II. The International Council of Scientific Unions adopted and broadened the scope of the idea, and designated July 1957 to December 1958 the International Geophysical Year (IGY). Observations were made in a number of IUGG discipline areas, including geomagnetism. The IGY resulted in a leap forward in geophysics through coordinated observational campaigns and through the establishment of new observatories and the World Data Center system. It was the beginning of the space age, and marked the start of the modern era for IAGA science, which has expanded to include solar-terrestrial interactions and studies of the Sun and the planets.

Present structure and organization In 2005, IAGA is one of seven scientific associations of the IUGG. It is an international nongovernmental organization deriving the majority of its funding from the IUGG member nations. An Executive Committee, elected by member countries, runs the Association, and the scientific work is organized through a structure defined in the Association’s Statutes and By-Laws: Division I: Internal Magnetic Fields; Division II: Aeronomic Phenomena; Division III: Magnetospheric Phenomena; Division IV: Solar Wind and Interplanetary Field; Division V: Geomagnetic Observatories, Surveys, and Analyses. Interdivisional Commission on History Interdivisional Commission on Developing Countries Several of the Divisions have Working Groups in specialist topic areas and establish Task Groups to deal with specific issues.

IAGA’s purpose and linkages IAGA’s “mission” is defined by the Association’s first Statute, which states that IAGA should promote scientific studies of international interest and facilitate international coordination and discussion of research. An important defining characteristic of IAGA is, therefore, to encourage inclusiveness in the scientific community, making excellent science accessible to scientists worldwide. There is a particular commitment to the less developed countries, and to the free exchange of scientific data and information. One way in which IAGA achieves its objectives is through the organization of meetings. IAGA Scientific Assemblies are held every 2 years, and in conjunction with IUGG General Assemblies every 4 years. Smaller scale meetings and specialized workshops make IAGA science accessible to a wider audience and help younger scientists and scientists from developing countries to accelerate their learning. IAGA sponsors several such meetings each year. They are held in all parts of the world, an important factor in enabling attendance by scientists with limited resources. Meetings enable scientific results to be presented and debated, new ideas generated, and collaborations to be established. The major Assemblies are also used to conduct divisional and working group business meetings, and matters of interest to IAGA are presented at open meetings convened for discussions between the IAGA Executive, the official delegates from member countries, and individual scientists. Resolutions are adopted as a formal means for IAGA to express views on scientific

408

IDEAL SOLUTION THEORY

matters. Support is often given to scientific initiatives under consideration by national or international agencies that will benefit IAGA science. For example, a series of IAGA Resolutions supported the initiation of and extensions to the International Equatorial Electrojet Year (IEEY) project (1991–1994). The project not only produced good science, but also resulted in investment in observational facilities at low-latitudes, benefiting scientists in the less developed world. Although the IEEY has finished, the impetus it gave to research in the equatorial regions continues. Similarly, IAGA has supported the Decade of Geopotential Research, which appears likely to achieve its goal of securing uninterrupted geomagnetic field satellite survey measurements spanning a decade (including the Ørsted and CHAMP missions). IAGA science is also promoted through collaboration with other bodies with similar interests. There are links with the other IUGG Associations and with Inter-Association bodies including Studies of the Earth’s Deep Interior (SEDI) (q.v.) and the Working Group on Electric and Magnetic Studies on Earthquakes and Volcanoes (EMSEV). There are formal contacts for liaison with the International Lithosphere Program (ILP), the Scientific Committee on Antarctic Research (SCAR), the Committee on Space Research (COSPAR), and the Scientific Committee on Solar Terrestrial Physics (SCOSTEP). IAGA cooperates with the World Data Center system on the definition of geomagnetic data exchange formats and management and preservation of analog and digital databases. The Association has provided strong support to INTERMAGNET (q.v.), the international program promoting the modernization of magnetic observatory practice and the distribution of data in near real time. IAGA advises bodies such as the International Organization for Standardization (ISO).

IAGA science in the 21st century IAGA science, because of the pervasiveness of the geomagnetic field and its interactions with charged particles and electrically conducting materials, is useful for studies of properties and processes in practically all parts of the solid Earth, the atmosphere, and the surrounding space environment. As well as covering a vast range of length scales, IAGA science covers timescales from seconds to billions of years. Modern-day observatories record rapid variations during magnetic storms caused by the interaction of the solar wind with the magnetosphere; the imprinting of the paleomagnetic field in rocks provides records of geodynamic changes on geological timescales. While IAGA provides an international focus for fundamental research resulting in advances in understanding in specialist areas, national and international funding for research often focuses on issues of societal concern. IAGA science is providing answers to many important questions, and through its links to other bodies and projects the Association is able to foster the building of the interdisciplinary teams required to address complex problems of interest to society. For instance, a natural goal for IAGA scientists is to be able to understand and model the whole Sun-Earth system including the complex interactions and feedbacks controlling the transfer of energy momentum and matter between parts of the system. Research in this area is proving relevant to the problem of how to disentangle natural from anthropogenic causes of climate change. This area of science also underpins the understanding of how “space weather” conditions affect the risk to technological systems and human activities on the ground and in space. For example, during magnetic storms, electrical power distribution grids, radio communications, GPS accuracy, and satellite operations can be adversely affected. Also in the geohazards area, EMSEV is charged with establishing firm scientific understanding of the generation mechanisms of any signals that may help to mitigate the effects of earthquakes and volcanoes. IAGA is responsible for the production of the International Geomagnetic Reference Field, used in a variety of scientific and “real world” applications, including navigation, and hydrocarbons exploration and production. The Association is responsible for the definition of the most widely used magnetic activity indices, and works closely

with the International Service for Geomagnetic Indices, the body responsible for their production and distribution. IAGA has a long history, and its science remains vigorous and relevant. Rapid advances in scientific understanding are resulting from improved instruments, better observations and data analysis techniques, the wealth of satellite data now routinely available, and the power of modern computer technology. As the 50th anniversary of the IGY approaches, IAGA is promoting the concept of an “Electronic Geophysical Year” (eGY), for 2007–2008, taking advantage of the modern capability to link distributed computing resources to multiple remote sources of data and modeling codes to address scientific problems. This initiative is in line with the Association’s mission to promote international scientific cooperation and collaboration, and has the potential to advance the ability of scientists in developing countries to participate in leading-edge research. (The principal point of contact with the Association is the IAGA Secretary General, and IAGA communicates with its members and the public through its Web site and through issues of IAGA News.) David Kerridge

Bibliography Naoshi Fukushima, 1995. History of the International Association of Geomagnetism and Aeronomy (IAGA). IUGG Chronicle, 226: 73–87. The IAGA Web site: http://www.iugg.org/IAGA/

Cross-references CHAMP Chapman, Sydney (1888–1970) Ørsted SEDI

IDEAL SOLUTION THEORY Consider two different substances; mix them together and in general they will form a solution, like sugar and coffee, for example. We call solvent the substance present in the largest quantity (coffee), and solute the other (sugar). In general solutions may have more than one solute, and/or more than one solvent, but for simplicity we will focus here only on binary mixtures. The behavior of solutions can be understood in terms of the chemical potential mi , which represents the constant of proportionality between the energy of the system and the amount of the specie i (Wannier, 1966):
 ]E mi ¼ (Eq. 1) ]Ni S;V where E is the internal energy of the system, S is the entropy, V is the volume, and Ni is the number of particles of the specie i. Alternative equivalent definitions of the chemical potential are (Wannier, 1966; Mandl, 1997):
mi ¼

]F ]Ni




¼

T;V

]G ]Ni




¼
T

T; p

]S ]Ni

 (Eq. 2) E;V

where F and G are the Helmholtz and Gibbs free energies of the system, T is the temperature, and p is the pressure. We recall the statistical mechanics definition of the Helmholtz free energy for a classical system (Frenkel, 1996): Z Z 1 dR1 ... dRN e
U ðR1 ;...;RN ;TÞ=kB T ; (Eq. 3) F ¼
kB T ln 3N L N! V V

409

IDEAL SOLUTION THEORY

where U ðR1 ; . . . ; RN ; T Þ is the potential energy which depends on the positions ðR1 ; . . . ; RN Þ of all the particles in the system and on T, kB is the Boltzmann constant and L ¼ h=ð2pMkB T Þ1=2 is the thermal wavelength, with M being the nuclear mass and h the Plank’s constant. Consider now a solution with NA particles of solvent A and NX particles of solute X, with N ¼ NA þ NX . The Helmholtz free energy of this system is: Z Z 1 F ¼
kB T ln 3NA 3NX dR1 ... dRN e
U ðR1 ;...;RN ;T Þ=kB T : LA LX mA ! NX ! V V (Eq. 4)

X

Ni dmi ¼ 0:

(Eq. 12)

i

In particular, in our two-components system the Gibbs-Duhem equation implies: cA dmA þ cX dmX ¼ 0; which gives (Alfè et al., 2002a): mA ð p; T ; cX Þ ¼ m0A ð p; T Þ þ ðkB T þ lX ð p; T ÞÞlnð1
cX Þ þ lX ð p; T ÞcX þ Oðc2X Þ;

According to (Eq. 2), we have:
 ]F mX ¼ ¼ F ðNA ; NX þ 1Þ
F ðNA ; NX Þ; ]NX T ; V

(Eq. 5)

1 L3X ðNX þ 1Þ R R R . . . V dRN V dRN þ1 e
U ðR1 ;...;RN ;RN þ1 ;T Þ=kB T V dR 1 R R :
U ðR1 ;...;RN ;T Þ=kB T V dR 1 . . . V dR N e

(Eq: 14)

where m0A is the chemical potential of the pure solvent. To linear order in cX, this gives: mA ðp; T ; cX Þ ¼ m0A ðp; T Þ
kB TcX þ Oðc2X Þ:

which can be evaluated using (Eq. 4)

(Eq. 13)

(Eq. 15)

Notice that this expression for mA is not restricted to ideal solutions. Though m0A is the chemical potential of the pure solvent, observe that m0X is not the chemical potential of the pure solute, unless the validity of (Eq. 11) extends all the way up to cX ¼ 1.

mX ¼
kB T ln

(Eq. 6) The ratio of the two integrals is an extensive quantity, but mX is an intensive quantity; therefore it is useful to rewrite the expression as follows: N mX ¼
kB T ln ðNX þ 1Þ ( ) R R R 1 V dR1 . . . V dRN V dRN þ1 e
U ðR1 ;...;RN ;RN þ1 ;T Þ=kB T R R
U ðR1 ;...;RN ;T Þ=kB T N L3X V dR1 . . . V dR N e (Eq. 7) so that the value in curly brackets is now independent on system size. By setting cX ¼ NX =N (which in the limit of large N and NX is the same asðNX þ 1Þ=N ÞÞ and (

1 ~X ¼
kB T ln m N L3E

R V

) R dRN V dRN þ1 e
U ðR1 ;...;RN ;RN þ1 ;T Þ=kB T R
U ðR1 ;...;RN ;T Þ=kB T V dR1 ... V dRN e

dR1 ... R

R

V

(Eq. 8)

we can rewrite the chemical potential in our final expression: ~X ðp; T ; cX Þ: mX ðp; T ; cX Þ ¼ kB T ln cX þ m

(Eq. 9)

The first term of (Eq. 9) depends only on the number of particles of solute present in the solution, while the second term is also responsible for all possible chemical interactions. For small concentration of solute ~X : we can make a Taylor expansion of m ~X ¼ m0X þ lcX þ oðc2X Þ; m

(Eq. 10)

~X =]cX Þp;T . If the solution is so dilute that the particles where l ¼ ð]m of the solute do not interact with each other, we can stop the expansion to the first term: mX ð p; T ; cX Þ ¼ kB T ln cX þ m0X ð p; T Þ:

(Eq. 11)

We define ideal solution a system in which (Eq. 11) is strictly satisfied. To find an expression for the chemical potential of the solvent we employ the Gibbs-Duhem equation, which for a system at constant pressure and constant temperature reads (Wannier, 1966):

Volume of mixing It is often interesting to study the change of volume of a solution as a function of the concentration of the solute. To this end, it is useful to express the volume of the system as the partial derivative of the Gibbs free energy with respect to pressure, taken at constant temperature and number of particles: V ¼ ð]G=]pÞT;NX ;NA :

(Eq. 16)

If we now add to the system one particle of solvent at constant pressure, the total volume changes by vA, and becomes V þ vA . We call vA the partial molar volume of the solvent. The total Gibbs free energy is G þ mA , so that according to (Eq. 16) vA ¼ ð]mA =]pÞT;NX ;NA ¼ ð]mA =]pÞT;cX , where the last equality stems from the fact that mA only depends on NX and NA through the molar fraction cX (we assume here that cX does not change when we add one particle of solvent to the system, this is obviously true if the number of atoms of solvent NA is already very large). The partial volume in general depends on cX, p, and T, but under the assumption of ideality vA ¼ ð]m0A =]pÞT;cX , and it depends only on p and T. In an ideal solution vA is the same as in the pure solvent. Similarly, the partial molar volume of the solute is: vX ¼ ð]mX =]pÞT;cX , which becomes independent on cX under the assumption of ideality. Notice that this is not in general equal to the partial volume of the pure solute. As an illustration of the applicability of the ideal solution approximation, I mention the recent first-principles calculations of the density of the Earth’s liquid outer core as a function of the concentration of light impurities like sulfur, silicon, and oxygen in liquid iron. As reported in (Gubbins et al., 2004), explicit first-principles calculations of the density of the core were not able to resolve any departure from the prediction of ideal solution theory. However, we shall see below that for other properties ideal solution theory is not necessarily a good working hypothesis.

Solid-liquid equilibrium We want to study now the conditions that determine the equilibrium between solid and liquid, and in particular how the solute partitions between the two phases. Thermodynamic equilibrium is reached when the Gibbs free energy of the system is at its minimum (Wannier, 1966; Mandl, 1997), and therefore, 0 ¼ dG ¼ dðGl þ Gs Þ, where

410

IDEAL SOLUTION THEORY

superscripts s and l indicate quantities in the solid and in the liquid, respectively. In a multicomponent system, the Gibbs free energy can be expressed in terms of the chemical potentials of the species present in the system (Wannier, 1966; Mandl, 1997): X Ni mi : (Eq. 17) G¼ i

Using (Eq. 17) and the Gibbs-Duhem equation (12), we obtain: X dG ¼ mi dNi : (Eq. 18) i

If the system is isolated, particles can only flow between the solid and the liquid, and we have dNis ¼
dNil , which implies: X dG ¼ dNi ðmli
msi Þ: (Eq. 19) i

If mli < msi there will be a flow of particles from the solid to the liquid region ðdNi > 0Þ, so that the Gibbs free energy of the system is lowered. The opposite will happen if mli > msi . The flow stops at equilibrium, which is therefore reached when mli ¼ msi . In particular, in our two-components system, the equilibrium between solid and liquid implies that the chemical potentials of both solvent and solute are equal in the solid and liquid phases: msX ð p; Tm ; c sX Þ ¼ mlX ð p; Tm ; clX Þ; msA ð p; Tm ; c sX Þ ¼ mlA ð p; Tm ; clX Þ;

(Eq. 20)

where Tm is the melting temperature of the solution at pressure p. Using (Eq. 9), we can rewrite the first of the two equations above as: ~lX ðp; Tm Þ þ kTm lnclX ; ~sX ðp; Tm Þ þ kTm lncsX ¼ m m

(Eq. 21)

from which we obtain an expression for the ratio of concentrations of solute between the solid and the liquid: ~sX ð p; Tm Þ=kTm g: mlX ð p; Tm Þ
m csX =clX ¼ expf½~

(Eq. 22)

~sX , because the greater mobility of the liquid can ~lX < m In general m usually better accommodate particles of solute, and therefore their energy (chemical potential) is lower. This means that the concentration of the solute is usually smaller in the solid. Equation (22) was used by Alfè et al. (2000, 2002a,b) to put constraints on the composition of the Earth’s core. The constraints came from a comparison of the calculated density contrast at inner core boundary, and that obtained from seismology, which is between 4.5%  0.5% (Shearer and Masters, 1990) and 6.7%  1.5% (Masters and Gubbins, 2003). This density contrast is significantly higher than that due to the crystallization of pure iron, and therefore must be due to the partitioning of light elements between solid and liquid. This partitioning for some candidate impurities can be obtained by calculating ~sX . Alfè et al. (2000, 2002a,b) considered sulfur, silicon, and ~lX and m m oxygen as possible impurities, and using first-principles simulations, in which the interactions between particles were treated using quantum mechanics, obtained the partitions for each impurity. The calculations ~sX are very similar, ~lX and m showed that for both sulfur and silicon m which means that csX and clX are also very similar, according to (Eq. 22). As a result, the density contrast of a Fe/S or a Fe/Si system is not much different from that of pure Fe, and still too low when com~sX ~le and m pared with the seismological data. By contrast, for oxygen m are very different, and the partitioning between solid and liquid is very large. This results in a much too large density contrast, which also does not agree with the seismological data. The conclusion from these calculations was that none of these binary mixtures can be viable for

the core. The density contrast can of course be explained by ternary or quaternary mixtures. Assuming no cross-correlated effects between the chemical potentials of different impurities, Alfè et al. (2002a,b) proposed an inner core containing about 8.5% of sulfur and/or silicon and almost no oxygen, and an outer core containing about 10% of sulfur and/or Si, and an additional 8% of oxygen. It is worth noting that these calculations (Alfè et al., 2000, 2002a,b) ~lX on csX and clX was ~sX and m also showed that the dependence of m significant, and therefore departure from ideal solution behavior. If nonideal effects were ignored, the calculations would result in an outer core containing about 12% of sulfur and/or silicon, and about 6% of oxygen.

Shift of freezing point The partitioning of the solute between the solid and the liquid is generally responsible for a change in the melting temperature of the mixture with respect to that of the pure solvent. To evaluate this, we expand the chemical potential of the solvent around the melting temperature of the pure system, Tm0 : mA ð p; Tm ; cX Þ ¼ mA ð p; Tm0 ; cX Þ
s0A dT þ   

(Eq. 23)

where dT ¼ ðTm
Tm0 Þ, and s0A ¼
ð]mA =]T ÞT ¼Tm0 is the entropy of the pure solvent at Tm0 . We now impose continuity across the solid-liquid boundary: 0 0s s 0l 0 0l l m0s A ð p; Tm Þ
sA dT
kTm cX ¼ mA ð p; Tm Þ
sA dT
kTm cX ; (Eq. 24)

where we have considered only the linear dependence of mA on cX 0 0l 0 (See Eq. 15). Noting that m0s A ð p; Tm Þ ¼ mA ð p; Tm Þ we have dT ¼

kTm ðcs
clX Þ: 0s X s0l A
sA

(Eq. 25)

Since usually csX < clX, there is generally a depression of the freezing point of the solution. Using (Eq. 25), Alfè et al. (2002a,b) estimated a depression of about 600–700 K of the melting temperature of the core mixture with respect to the melting temperature of pure Fe, and they suggested an inner core boundary temperature of about 5600 K. Dario Alfè

Bibliography Alfè, D., Gillan, M.J., and Price, G.D., 2000. Constraints on the composition of the Earth’s core from ab initio calculations. Nature, 405: 172–175. Alfè, D., Gillan, M.J., and Price, G.D., 2002a. Ab initio chemical potentials of solid and liquid solutions and the chemistry of the Earth’s core. Journal of Chemical Physics, 116: 7127–7136. Alfè, D., Gillan, M.J., and Price, G.D., 2002b. Composition and temperature of the Earth’s core constrained by combining ab initio calculations and seismic data. Earth and Planetary Science Letters, 195: 91–98. Frenkel, D., and Smit, B., 1996. Understanding Molecular Simulation. San Diego, CA: Academic Press. Gubbins, D. et al., 2004. Gross thermodynamics of two-component core convection. Geophysical Journal International, 157: 1407–1414. Mandl, F., 1997. Statistical Physics, 2nd edn. New York: John Wiley & Sons.

IGRF, INTERNATIONAL GEOMAGNETIC REFERENCE FIELD

Masters, T.G., and Gubbins, D., 2003. On the resolution of the density within the Earth. Physics of the Earth and Planetary Interiors, 140: 159–167. Shearer, P., and Masters, T.G., 1990. The density and shear velocity contrast at the inner core boundary. Geophysical Journal International, 102: 491–498. Wannier, G.H., 1966. Statistical Physics. New York: Dover Publications.

IGRF, INTERNATIONAL GEOMAGNETIC REFERENCE FIELD History of the IGRF The IGRF is an internationally agreed global spherical harmonic model of the Earth’s magnetic field whose sources are in the Earth’s core (see Harmonics, spherical and Main field modeling). It is revised every 5 years under the auspices of the International Association of Geomagnetism and Aeronomy (see IAGA, International Association of Geomagnetism and Aeronomy). The concept of an IGRF grew out of discussions concerning the presentation of the results of the World Magnetic Survey (WMS). The WMS was a deferred element in the program of the International Geophysical Year, which, during 1957–1969, encouraged magnetic surveys on land, at sea, in the air, and from satellites and organized the collection and analysis of the results. At a meeting in 1960, the Committee on World Magnetic Survey and Magnetic Charts of IAGA recommended that, as part of the WMS program, a spherical harmonic analysis be made using the results of the WMS, and this proposal was accepted. Another 8 years of argument and discussion followed this decision and a summary of this, together with a detailed description of the WMS program, is given by Zmuda (1971). The first IGRF was ratified by IAGA in 1969. The original idea of an IGRF had come from global modelers, including those who produced such models in association with the production of navigational charts. However, the IGRF as it was first formulated was not considered to be accurate or detailed enough for navigational purposes. The majority of potential users of the IGRF at this time consisted of geophysicists interested in the geological interpretation of regional magnetic surveys. An initial stage in such work is the removal of a background field from the observations that approximates the field whose sources are in the Earth’s core. With different background fields being used for different surveys, difficulties arose when adjacent surveys had to be combined. An internationally agreed global model, accurately representing the field from the core, eased this problem considerably.

411

Another group of researchers who were becoming increasingly interested in descriptions of the geomagnetic field at this time were those studying the ionosphere and magnetosphere and behavior of cosmic rays in the vicinity of the Earth. This remains an important user community today.

Development of the IGRF The IGRF has been revised and updated many times since 1969 and a summary of the revision history is given in Table I1 (see also Barton, 1997, and references therein). Each generation of the IGRF comprises several constituent models at 5-year intervals, some of which are designated definitive. Once a constituent model is designated definitive it is called a Definitive Geomagnetic Reference Field (DGRF) and it is not revised in subsequent generations of the IGRF. New constituent models are carefully produced and widely documented. The IAGA Working Group charged with the production of the IGRF invites submissions of candidate models several months in advance of decision dates. Detailed evaluations are then made of all submitted models, and the final decision is usually made at an IAGA Assembly if it occurs in the appropriate year, otherwise by the IAGA Working Group. The evaluations are also widely documented. The coefficients of the new constituent models are derived by taking means (sometimes weighted) of the coefficients of selected candidate models. This method of combining several candidate models has been used in almost all generations as, not only are different selections of available data made by the teams submitting models, there are many different methods for dealing with the fields which are not modeled by the IGRF, for example the ionospheric and magnetospheric fields and crustal fields. The constituent main field models of the most recent generation of the IGRF (IAGA, 2005) extend to spherical harmonic degree 10 up to and including epoch 1995.0, thereafter they extend to degree 13 to take advantage of the excellent coverage and quality of satellite data provided by Ørsted and CHAMP (see Ørsted and CHAMP). The predictive secular variation model extends to degree 8.

Future of the IGRF Firstly, no model of the geomagnetic field can be better than the data on which it is based. An assured supply of high-quality data distributed evenly over the Earth’s surface is therefore a fundamental prerequisite for a continuing and acceptably accurate IGRF. Data from magnetic observatories (see Observatories, overview) continue to be the most important source of information about time-varying fields. However their spatial distribution is poor and although data from other sources such as repeat stations (see Repeat stations), Project MAGNET (see Project MAGNET), and marine magnetic surveys (see Magnetic surveys, marine) have all helped to fill in the gaps,

Table I1 Summary of IGRF history Full name

Short name

Valid for

Definitive for

IGRF IGRF IGRF IGRF IGRF IGRF IGRF IGRF IGRF IGRF

IGRF-10 IGRF-9 IGRF-8 IGRF-7 IGRF-6 IGRF-5 IGRF-4 IGRF-3 IGRF-2 IGRF-1

1900.0
2010.0 1900.0
2005.0 1900.0
2005.0 1900.0
2000.0 1945.0
1995.0 1945.0
1990.0 1945.0
1990.0 1965.0
1985.0 1955.0
1980.0 1955.0
1975.0

1945.0
2000.0 1945.0
2000.0 1945.0
1990.0 1945.0
1990.0 1945.0
1985.0 1945.0
1980.0 1965.0
1980.0 1965.0
1975.0



10th generation (revised 2004) 9th generation (revised 2003) 8th generation (revised 1999) 7th generation (revised 1995) 6th generation (revised 1991) 5th generation (revised 1987) 4th generation (revised 1985) 3rd generation (revised 1981) 2nd generation (revised 1975) 1st generation (revised 1969)

412

INDUCTION ARROWS

the best spatial coverage is provided by near-polar satellites. Measurements made by the POGO satellites (1965–1971) (see POGO (OGO-2, -4, and -6 spacecraft)), Magsat (1979–1980), POGS (1990–1993), Ørsted (1999–), and CHAMP (2000–) have all been utilized in the production of the IGRF. Secondly, the future of the IGRF depends on the continuing ability of the groups who have contributed candidate models to the IGRF revision process to produce global geomagnetic field models. This ability is dependent on the willingness of the relevant funding authorities to continue to support this type of work. Thirdly, the continued interest of IAGA is a necessary requirement for the future of the IGRF. This is assured as long as there is, as at present, a large and diverse group of IGRF-users worldwide. One reason why the IGRF has gained the reputation it has is because it is endorsed and recommended by IAGA, the recognized international organization for geomagnetism. A topic still under discussion is how best to extend the IGRF backward in time. The current generation includes non-definitive models at 5-year intervals covering the interval 1900.0–1940.0, and, more importantly, some of the earlier DGRFs are of questionable quality. Constructing an internationally acceptable model that describes the time variation better than the IGRF using splines is one way forward (see Time-dependent models of the geomagnetic field). Susan Macmillan

Bibliography Barton, C.E., 1997. International Geomagnetic Reference Field: the seventh generation. Journal Geomagnetism and Geoelectricity, 49: 123–148. International Association of Geomagnetism and Aeronomy (IAGA), Division V, Working Group VMOD: Geomagnetic Field Modeling, 2005. The 10th-Generation International Geomagnetic Reference Field. Geophysical Journal International, 161, 561–565. Zmuda, A.J., 1971. The International Geomagnetic Reference Field: Introduction, Bulletin International Association of Geomagnetism and Aeronomy, 28: 148–152.

Cross-references CHAMP Harmonics, Spherical IAGA, International Association of Geomagnetism and Aeronomy Magnetic Surveys, Marine Magsat Main Field Modeling Observatories, Overview Ørsted POGO (OGO-2, -4 and -6 spacecraft) Project MAGNET Repeat Stations Time-dependent Models of the Geomagnetic Field

INDUCTION ARROWS Introduction The history of continuous magnetic field observations in Germany dates back to the 19th century (see Observatories in Germany). Modern observatory work began in 1930, with the foundation of the geomagnetic observatory in Niemegk, south of Potsdam. When a second observatory started operation in the late 1940s in Wingst, northern Germany, scientists soon noticed reversed signal amplitudes in the vertical field components between the two locations.

This observation was unexpected because magnetic field disturbances were known to be generated by large-scale processes in the Earth’s atmosphere, far away from the surface of the Earth (see Natural sources for EM induction studies). Hence, for external sources, the magnetic field variations should have been very similar, given the relatively short distance between the two observatories. However, it was soon speculated that this peculiar behavior may be caused by electromagnetic induction of currents into an electrically conducting region deep in the earth’s interior. When portable magnetometers became available, this phenomenon was investigated more systematically. With measurements following a N-S oriented profile at selected field sites, Schmucker (1959) could not only confirm the observatory results but could also confine the width of this North German anomaly to approximately 100 km. Northern Germany was not the only place where magnetic field variations were studied; similar conductivity anomalies were reported from many other parts of the world. Over the years, some of the largest conductivity anomalies on Earth were identified with studies relating vertical magnetic fields to horizontal magnetic fields (see Geomagnetic deep sounding).

The induction arrow Wiese (1962) and Parkinson (1962) independently developed graphical methods to locate the observed “induction effect,” thereby inventing the induction arrow (IA). Both original definitions of the IA are based on a time-domain approach which is not in use any more. However, we still draw our IAs in either the Wiese or the Parkinson (q.v.) convention according to their original designs (see Hobbs, 1992). Functions that interrelate vertical or horizontal magnetic fields are known as transfer functions (see Transfer functions). If lateral conductivity variations exist within the subsurface on length scales equal to or greater than the penetration depth of the induced horizontal magnetic fields, a vertical magnetic field component is generated. The frequency-domain, geomagnetic transfer functions Tx ðoÞ; Ty ðoÞ are defined as (dependence on frequency assumed): Bz ¼ Tx Bx þ Ty By

(Eq. 1)

x, y, and z denote north, east, and vertical directions, respectively. B is the magnetic field component in [T]. The IA is a graphical representation of the complex vertical magnetic field transfer function T. The real and imaginary parts of Tx and Ty can be combined to form ðTxr ; Tyr Þ and ðTxi ; Tyi Þ. The visual display of the IAs in maps is a comprehensive presentation of changes in vertical magnetic field anomalies, both, as a function of frequency and location. In Parkinson convention the ð
Txr ;
Tyr Þ is plotted, which tends to point toward regions of higher conductivity. The IA in the Wiese convention (Txr, Tyr) tends to point away from areas of higher conductivity. Table I2 gives the definitions of the IAs in the Wiese and Parkinson conventions. Figure I1 shows an example. The conductivity contrast associated with a fault zone can be either due to a juxtaposition of different lithologies (upper panel of Figure I1) or the damaged rocks within a fault zone can become conductive to form a conductive channel (bottom panel of Figure I1). If the conductivity contrast is assumed to extend indefinitely, both cases resemble a two-dimensional (2D) problem. Table I2 The definition of induction arrows in Parkinson and Wiese conventions Real arrow Convention

Length

Parkinson

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Txr2 þ Tyr2

Wiese

Imaginary arrow Angle   xr arctan
T
Tyr   arctan TTxryr

Length qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Txi2 þ Tyi2

Angle   T arctan Tyixi

INDUCTION FROM SATELLITE DATA

413

Figure I1 Expected and observed behavior of induction arrows (IA) for two different fault structures. In Wiese convention real (black) arrows point away from good conductors; imaginary (gray) arrows are parallel or antiparallel to the real arrows, given 2D Earth structure. Across a conductivity contrast (top panel) real IAs point away from the more conductive side and are largest over the less conductive side. Measurements from the Dead Sea Transform, locally known as the Arava Fault, are representative of this type of fault structure. A conductive fault zone sandwiched between more resistive units is characterized by IAs pointing away from the fault on both sides and small IAs over the fault zone conductor. The West Fault in Chile is representative of this type of fault structure. (Figure after Ritter et al., 2005.)

The expected behavior for the real IA is indicated on the left-hand side of the figure. In 2D, the real and imaginary IAs point parallel or antiparallel to each other, and perpendicular to the strike of the structure. The largest real IAs are observed close to a conductivity contrast, they become smaller with increasing distance from that boundary. A larger conductivity contrast will also cause a longer arrow. Although vertical magnetic field response functions do not contain direct information about the underlying conductivity structure, they are extremely valuable in detecting the lateral extension and the strike of conductivity anomalies. In the absence of lateral resistivity gradients IAs vanish. Real and imaginary IAs pointing obliquely indicate the influence of off-profile features of a more complicated three-dimensional Earth. The observed IAs on the right-hand side of Figure I1 show this influence to some extend but overall, they are still in good agreement with the simple 2D scenarios on the left. If IAs show great variation, both between sites and between real and imaginary arrows then a simple interpretation in terms of geoelectric strike direction is not possible. Many scientists, including the author, prefer the term induction vector over IA because we construct and treat IAs like vectors, i.e., their length and the direction have a physical meaning. They are however, not vectors in a strict mathematical sense. IAs caused by a superposition of two different lateral conductivity contrasts which are linked by inductive coupling will generally not be the same as the vector constructed by adding the individual IAs. Furthermore, IAs are very sensitive to electrical anisotropy (q.v.) (Pek and Verner, 1997).

Ritter, O., Hoffmann-Rothe, A., Bedrosian, P.A., Weckmann, U., and Haak, V., 2005. Electrical conductivity images of active and fossil fault zones. In Bruhn, D., and Burlini, L., (eds.), High-Strain Zones: Structure and Physical Properties, Vol. 245. London: Geological Society of London Special Publications, pp. 165–186. Schmucker, U., 1959. Erdmagnetische Tiefensondierung in Deutschland 1957/59. Magnetogramme und erste Auswertung. Abh. Akad. Wiss.Götingen. Math.-Phys. Klasse: Beiträge zum Int. geophys. Jahr 5. Wiese, H., 1962. Geomagnetische Tiefentellurik Teil II: die Streichrichtung der Untergrundstrukturen des elektrischen Widerstandes, erschlossen aus geomagnetischen Variationen. Geofisica Pura e Applicata, 52: 83–103.

Cross-references Anisotropy, Electrical Geomagnetic Deep Sounding Natural Sources for EM Induction Studies Observatories in Germany Parkinson, Wilfred Dudley (1919–2001) Transfer Functions

INDUCTION FROM SATELLITE DATA

Oliver Ritter

Bibliography Hobbs, B.A., 1992. Terminology and symbols for use in studies of electromagnetic induction in the Earth. Surveys of Geophysics, 13: 489–515. Parkinson, W., 1962. The influence of continents and oceans on geomagnetic variations. Geophysical Journal, 2: 441–449. Pek, J., and Verner, T., 1997. Finite-difference modelling of magnetotelluric fields in two-dimensional anisotropic media. Geophysical Journal International, 132: 535–548.

In addition to sources of internal origin, magnetic satellites record the variations in Earth’s external magnetic field caused by interaction of the ionosphere and magnetosphere with the solar wind and radiation, along with the secondary magnetic fields induced in Earth. The ratio of the internal to external field may be used to probe the electrical conductivity structure of the planet. Satellite induction studies date from Didwall’s (1984) early work with the POGO satellite data. Olsen (1999a) and Constable and Constable (2004a) provide reviews of satellite induction methods. Table I3 presents a list of past, present, and future magnetic satellite missions that are relevant to induction studies (see MAGSAT, Ørsted, POGO).

414

INDUCTION FROM SATELLITE DATA

Table I3 Satellite missions relevant to magnetic induction studies, as of July, 2005 Satellite

Type

Launch date

Data to

Inclination

Altitude (km)

POGO 2 POGO 4 POGO 6 MAGSAT Ørsted SAC-C CHAMP SWARM

Scalar Scalar Scalar Vector Vector Vector Vector Three vector sats

October 14, 1965 July 28, 1967 June 5, 1969 October 30, 1979 February 23, 1999 November 18, 2000 July 15, 2000 Proposed 2008

October 2, 1967 January 19, 1969 April 26, 1971 May 6, 1980 Present Present Present

87.3 86.0 82.0 96.8 96.6 98.2 87.3

410–1510 410–910 400–1100 352–561 500–850 702 200–454 450–550

Longer period EM variations penetrate deeper into Earth, because EM fields decay exponentially with a characteristic length given by the skin depth, or pffiffiffiffiffiffiffiffiffi zs  500 T =s m where s is electrical conductivity (S m
1) and T is the period of the field variations (s). Traditionally, studies of deep Earth conductivity have relied on either the magnetotelluric (MT) method (q.v.) or the geomagnetic depth sounding (GDS) method (q.v.). For MT studies, time series measurements of horizontal magnetic and electric fields are transformed into frequency domain Earth impedance (see Transfer functions for EM). In GDS, observatory records of three-component magnetic fields (or data from portable arrays of instruments), along with an assumption about the geometry of the inducing field, are transformed into a similar impedance. MT studies have the advantage that they are largely independent of the geometry of the external fields, but because induced electric fields decay with increasing period, MT studies are limited to periods of less than a few days. GDS studies, on the other hand, have been extended out to the period of the 11-year sunspot cycle, the longest wherein field variations are external to Earth’s core. The usual assumption about the geometry of the external field variations is that it is dominated by the fundamental spherical harmonic (q.v.) aligned with Earth’s internal dipole field called P10 . This is based both on observations of field geometry (e.g., Banks, 1969) and the morphology of the equatorial ring current (q.v.), which is composed mainly of oxygen ions circulating at a distance of 2–9 Earth radii (a review of the ring current is given by Daglis et al., 1999). Within the satellite orbit, the magnetic field B can be expressed as the gradient of a scalar potential F B ¼
m0 rF

Figure I2 The equatorial ring current generates an external magnetic field of uniform P10 geometry (grey arrows). The external field variations induce an internal field (black arrows) which is also described by P10 geometry but that decays with altitude. The different altitude dependence and different directional behavior with latitude allows a separation of the internal and external fields using satellite measurements made during pole to pole passes.

(Eq. 1)

(m0 is the magnetic permeability of space) where for P10 geometry F is described by a spherical harmonic representation in radial distance from Earth’s center r, Earth radius a0, and geomagnetic colatitude y: (
2
) a r F01 ðr; yÞ ¼ a0 i01 ðtÞ 0 þ e01 ðtÞ P10 ðcos yÞ: a0 r

(Eq. 2)

In GDS studies, knowledge of the field geometry allows observatory time series measurements of vertical and horizontal components of the magnetic field to be separated into the internal ði10 Þ and external ðe01 Þ coefficients, which may be Fourier transformed into the frequency domain to derive a complex geomagnetic response function of period: Q01 ðT Þ ¼ i10 ðT Þ=e01 ðT Þ:

(Eq. 3)

Figure I3 Amplitude spectrum taken from 13 years of the hourly Dst geomagnetic index. A broad peak in energy is seen around the 27-day solar rotation period, and narrow peaks at periods of 1 day and harmonics. The daily variations are generated mainly in the ionosphere, and are not P10 geometry. Power at frequencies lower than 10
7 Hz is probably dominated by secular variation in the internal magnetic field.

INDUCTION FROM SATELLITE DATA

The more familiar MT apparent resistivity and phase can be derived from the real and imaginary components of Q01 ðT Þ, as can the inductive scale length C-response (see transfer functions for EM). Magnetic satellites, collecting either scalar (total field magnitude only) or vector (three component) data, offer an opportunity to take GDS induction studies beyond the global observatory network. There are numerous advantages to using satellites for these types of studies: 1. Global coverage can be achieved using satellites, while the magnetic observatory network is sparse, nonuniform, and restricted to land. 2. During each mission magnetic measurements are made using a single, high-quality instrument having a single calibration and which is removed from the distorting effects of crustal magnetic anomalies. Satellite magnetic sensors include scalar proton precession magnetometers, which are often augmented by vector fluxgate magnetometers oriented by star cameras (see Magnetometers). 3. Extensive spatial coverage means that the geometry of the inducing magnetic fields can be estimated, instead of being assumed. Point (3) is illustrated in Figure I2. As a satellite makes a pass across the equator from pole to pole, the geometry of the magnetic field variations is different for the internal and external components. Variations in satellite altitude provide additional discrimination to fit the separate geometries of the internal and external fields. Induction is powered by variations in the ring current intensity caused by interactions with the solar wind. During magnetic storms (q.v.), these may amount to a few hundred nanoteslas. A proxy for ring current intensity is provided by the Dst (disturbed storm time) magnetic index, a weighted average of horizontal fields at four low latitude magnetic observatories. Ring current variations are observed at periods of a few hours to many months or years, but are strongest at about 1 month because the solar wind is modulated by solar rotation.

415

Figure I3 shows an amplitude spectrum of the Dst index; power at the daily harmonics is caused by contamination from the daily variation in the magnetic field, generated in the ionosphere and by the Chapman-Ferraro currents outside the magnetosphere, and is more complicated than simple P10 geometry. Power at periods greater than 1 month is contaminated by secular variation of the main magnetic field (q.v.), of internal origin and unsuitable for induction studies. For GDS observatory studies, it is difficult to reject these contaminations and these periods have to be avoided, but satellite data have some power to reject non-P10 geometries. The above discussion ignores the contributions to the magnetic field that come from the main (internal) field and the crustal field (q.v.). These parts of the field must be removed before an induction analysis can be carried out, and therein lies one of the great difficulties of satellite induction. Without additional information, it is impossible to

Figure I4 Without additional information, it is impossible to distinguish between temporal variations in the external magnetic field and spatial variations in the internal magnetic field sampled as the magnetic satellite flies though its orbit.

Figure I5 Satellite responses from the MAGSAT (circles, from Constable and Constable, 2004b) and Ørsted (triangles, from Olsen et al., 2003) missions, compared with Olsen’s (1999b) long period observatory response for Europe (squares). The real components of C are positive, the imaginary are negative; error bars are one standard deviation.

416

INHOMOGENEOUS BOUNDARY CONDITIONS AND THE DYNAMO

distinguish between time variations in the ring current source field and spatial variations in the main and crustal fields sampled as the spacecraft flies over Earth. This is illustrated in Figure I4. A model of the main field that includes secular variation, along with a model of the crustal fields and possibly the daily variation, electrojets, and field-aligned currents must be used to remove all these elements before the induction analysis can be carried out. A convenient tool for removing these contributions is the Comprehensive Field Model of Sabaka et al. (2004). Figure I5 shows GDS response functions from the MAGSAT and Ørsted satellite missions, compared with a response function generated using magnetic observatories. The agreement is generally good, and radial conductivity models generated from observatory data also fit the satellite responses fairly well. However, differences do exist. At short periods, the satellite response is influenced by oceans (Constable and Constable, 2004b). At long periods, there may be contamination of the source-field geometry, and some differences are undoubtedly due to the various analysis techniques. The subject of magnetic satellite induction is relatively young and many issues remain to be addressed. For example, there are complications to the ring current geometry that have largely been ignored (e.g., Balasis et al., 2004). The daily variation in the magnetic field has variously been considered a source of noise (Constable and Constable, 2004b) and a signal for 3D induction studies (Velimsk and Everett, 2005). Ultimately, the goal of satellite induction is to move beyond global response functions and radial conductivity models toward true 3D conductivity structure.

Velimsk, J., and Everett, M.E., 2005. Electromagnetic induction by Sq ionospheric currents in heterogeneous Earth: modeling using ground-based and satellite measurements. In Reigber, Ch., Luehr, H., Schwintzer, P., and Wickert, J. (eds.) Earth Observation with CHAMP Results from Three Years in Orbit. Berlin: Springer-Verlag, pp. 341–346.

Cross-references CHAMP Crustal Magnetic Field EM, Marine Controlled Source Geomagnetic Deep Sounding Geomagnetic Spectrum, Temporal Harmonics, Spherical Magnetometers, Laboratory Magnetosphere of the Earth Magnetotellurics Magsat Ørsted POGO (OGO-2, -4, and -6 spacecraft) Ring Current Secular Variation Model Storms and Substorms, Magnetic Transfer Functions

Steven Constable

Bibliography Balasis, G., Egbert, G.D., and Maus, S., 2004. Local time effects in satellite estimates of electromagnetic induction transfer functions. Geophysical Research Letters, 31: L16610, doi:10.1029/2004GL020147. Banks, R.J., 1969. Geomagnetic variations and the conductivity of the upper mantle. Geophysical Journal of the Royal Astronomical Society, 17: 457–487. Constable, C.G., and Constable, S.C., 2004a. Satellite magnetic field measurements: applications in studying the deep Earth. In Sparks, R.S.J., and Hawkesworth, C.T. (eds.) The State of the Planet: Frontiers and Challenges in Geophysics, Geophysical Monograph 150. Washington, DC: American Geophysical Union, pp. 147–159. Constable, S., and Constable, C., 2004b. Observing geomagnetic induction in magnetic satellite measurements and associated implications for mantle conductivity. Geochemistry Geophysics Geosystems, 5: Q01006, doi:10.1029/2003GC000634. Daglis, I.A., Thorne, R.M., Baumjohann, W., and Orsini, S., 1999. The terrestrial ring current: origin, formation, and decay. Reviews of Geophysics, 37: 407–438. Didwall, E.M., 1984. The electrical conductivity of the upper mantle as estimated from satellite magnetic field data. Journal of Geophysical Research, 89: 537–542. Olsen, N., 1999a. Induction studies with satellite data. Surveys in Geophysics, 20: 309–340. Olsen, N., 1999b. Long-period (30 days–1 year) electromagnetic sounding and the electrical conductivity of the lower mantle beneath Europe. Geophysical Journal International, 138: 179–187. Olsen, N., Vennerstr
m, S., and Friis-Christensen, E., 2003. Monitoring magnetospheric contributions using ground-based and satellite magnetic data. In Reigber, Ch., Luehr, H., and Schwintzer, P. (eds.), First CHAMP Mission Results for Gravity, Magnetic and Atmospheric Studies. Berlin: Springer-Verlag, pp. 245–250. Sabaka, T.J., Olsen, N., and Purucker, M.E., 2004. Extending comprehensive models of the Earth’s magnetic field with Orsted and CHAMP data. Geophysical Journal International, 159: 521–547.

INHOMOGENEOUS BOUNDARY CONDITIONS AND THE DYNAMO Thermal core-mantle coupling and geodynamo Modern observations indicate that the Earth’s magnetic field takes the form of an approximate geocentric axial dipole, i.e., a dipole positioned at the Earth’s center and aligned with the axis of rotation. Paleomagnetic measurements indicate that, when it is averaged over a sufficiently long time, this has been the case for the observable past. This suggests that the observed Earth’s magnetic field is primarily the result of a geodynamo consistent with turbulent flows and rapid rotation (Moffatt, 1978). In consequence, the geocentric axial dipole is widely employed to provide a reference state for the Earth’s magnetic field. Any persistent departure from the axial dipole is indicative of the external influence such as inhomogeneous boundary conditions on the geodynamo imposed by the overlying mantle (Hide, 1967; Bloxham and Gubbins, 1987; Bloxham, 2000). Convection is occurring in both the Earth’s outer core and mantle. However, there exists a huge difference between the timescales of mantle and core convection. Core convection has an overturn time of several hundred years while that of mantle convection is a few hundred million years. As far as the Earth’s mantle is concerned, the core-mantle boundary is a fixed temperature interface. As far as the Earth’s fluid core is concerned, the core-mantle boundary is rigid and imposes a nearly unchanging heat-flux heterogeneity on the core convection and dynamo. The structure of the inhomogeneous boundary condition can be inferred from the seismic tomography of the Earth’s lower mantle (for example, Masters et al., 1996; van der Hilst et al., 1997). By assuming that the hotter regions of mantle correspond to the lower seismic velocity while the cold regions are related to the anomalously high seismic velocity, the inhomogeneous boundary condition at the core-mantle boundary for the core convection and dynamo can be determined. The temperature variations above the core-mantle boundary impose a nearly unchanging heat-flux heterogeneity on the core dynamo.

INHOMOGENEOUS BOUNDARY CONDITIONS AND THE DYNAMO

Paleomagnetic and historical magnetic field measurements suggest persistent distinct patterns of variation of the geomagnetic field taking place in different regions of the Earth. For example, secular variation under the Pacific region is much slower than that under the Atlantic region (Bloxham and Gubbins, 1985; McElhinny et al., 1996). This longitudinal asymmetry of the geomagnetic field, the so-called Pacific dipole window, may be explained by thermal heterogeneity at the lower mantle. Bloxham and Gubbins (1987) proposed that thermal interaction whereby lateral variations in heat flux across the coremantle boundary drive core flows directly and influence the preexisting deep convection. Furthermore, comparison of the geomagnetic field at the core surface with lower mantle temperature or seismic velocity suggests a close connection between them (Gubbins and Richards, 1986; Gubbins and Bloxham, 1987) and order-of-magnitude estimates also suggest the variations are large enough to drive the core flow required for secular variation of the Earth’s magnetic field (Bloxham and Gubbins, 1987). The inhomogeneous boundary condition in heat flux or temperature at the core-mantle boundary, together with the strong effect of rotation, drive fluid motions directly in the outer core which influence the generation of geomagnetic field in the core. Moreover, an inhomogeneous boundary condition may also influence the core convection being driven from below by locking it to the nonuniform boundary condition (Zhang and Gubbins, 1993; Sarson et al., 1997).

Inhomogeneous boundary conditions: models With a uniform thermal boundary, convection takes place when and only when a sufficient radial temperature gradient, measured by the size of the Rayleigh number, exists to drive it. With a nonuniform thermal boundary, there are two driving parameters: one measures the radial buoyancy and the other the strength of the lateral heating. It follows that how an inhomogeneous boundary condition affects the core convection and dynamo can be illustrated by a simple nonlinear amplitude equation in the form A_ ¼ ðm þ iÞA
AjAj2 þ E;

(Eq. 1)

pffiffiffiffiffiffiffi where the flow amplitude A is a complex variable, i ¼
1; E is a positive parameter measuring the strength of the imposed inhomogeneous boundary condition (thermal or nonthermal) at the base of mantle, and m is a positive parameter like the Rayleigh number in connection with the deep core convection. The amplitude equation is simple but provides a helpful mathematical framework in the understanding of complicated nonlinear dynamics affected by an imposed inhomogeneous boundary condition. Without lateral heterogeneity of the lower mantle, E ¼ 0, convection is always in the form of an azimuthally traveling wave, such as convection rolls aligned with the axis of rotation. With the influence of the lateral heterogeneity E 6¼ 0, a steady equilibrium solution A0 becomes possible and can be obtained by setting A_ 0 ¼ 0. Denoting A0 ¼ X0 þ iY0 and Z ¼ jA0 j2 ¼ X02 þ Y02 , we obtain the cubic equation Z 3
2mZ 2 þ Zðm2 þ 1Þ
E2 ¼ 0

(Eq. 2)

for steady solutions of convection, which can be solved analytically using a standard formula. The stability of a steady solution A0 can be investigated by linearizing the nonlinear equation by letting A ¼ ðX0 þ xÞ þ iðY0 þ yÞ;

(Eq. 3)

where x and y are small perturbations of X0 and Y0. Stability of the nonlinear equilibrium A0 is then related to the two linear equations x_ ¼ ðm
3X02
Y02 Þx
ð1 þ 2X0 Y0 Þy;

(Eq. 4)

y_ ¼ ð1
2X0 Y0 Þx þ ðm
3Y02
X02 Þy:

417

(Eq. 5)

The corresponding growth rate s of the perturbations, or stability of the boundary-driven steady solution, is then described by s ¼ ðm
2jA0 j2 Þ  ðjA0 j4
1Þ1=2 :

(Eq. 6)

Two important features are evident from the stability of the boundarydriven steady solution: (i) a Hopf bifurcation (traveling wave convection) cannot occur if the amplitude is sufficiently large, jA0 j > 1; (ii) pffiffiffia necessary condition for the occurrence of a Hopf bifurcation is E < 2. In other words, the core convection and dynamo in the presence of an inhomogeneous core-mantle boundary can be lockedpinto ffiffiffi the boundary provided that the effect is sufficiently strong, E > 2. It follows that the Pacific dipole window may reflect the thermal structure of mantle convection under the Pacific region. Complex three-dimensional numerical simulations in rotating spherical geometry show a similar behavior revealed by the above simple model, demonstrating that an inhomogeneous boundary condition imposed by the lower mantle can alter the properties of core convection and dynamo in a fundamental way (Zhang and Gubbins, 1992, 1993; Sarson et al., 1997; Olson and Christesen, 2002). The mantle convection imposes a different length scale upon the system of the core convection and dynamo. When a geodynamo model uses the inhomogeneous boundary condition derived from seismic tomography, it can produce an anomalous magnetic field and westward fluid velocity which are largely consistent with the observed features of the Earth’s magnetic field (Olson and Christensen, 2002). Keke Zhang

Bibliography Bloxham, J., 2000. Sensitivity of the geomagnetic axial dipole to thermal core-mantle interactions. Nature, 405(6782): 63–65. Bloxham, J., and Gubbins, D., 1985. The secular variation of the Earth’s magnetic field. Nature, 317: 777–781. Bloxham, J., and Gubbins, D., 1987. Thermal core-mantle interactions. Nature, 325: 511–513. Gubbins, D., and Bloxham, J., 1987. Morphology of the geomagnetic field and implications for the geodynamo. Nature, 325: 509–511. Gubbins, D., and Richards, M., 1986. Coupling of the core dynamo and mantle: thermal or topographic? Geophysical Research Letters, 13: 1521–1524. Hide, R., 1967. Motions of the earth’s core and mantle, and variations of the main geomagnetic field. Science, 157: 55–56. Masters, G., Johnson S., Laske, G., and Bolton, H., 1996. A shearvelocity of the mantle. Philosophical Transactions of the Royal Society of London, Series A, 354: 1385–1411. McElhinny, M.W., McFadden, P.L., and Merrill, P.T., 1996. The myth of the Pacific dipole window. Earth and Planetary Science Letters, 143: 13–22. Moffatt, H.K., 1978. Magnetic Field Generation in Electrically Conducting Fluids. Cambridge, England: Cambridge University Press. Olson, P., and Christensen U.R., 2002. The time-averaged magnetic field in numerical dynamos with nonuniform boundary heat flow. Geophysical Journal International, 151(3): 809–823. Sarson, G.R., Jones, C.A., and Longbottom, A.W., 1997. The influence of boundary region heterogeneities on the geodynamo. Physics of the Earth and Planetary Interiors, 104: 13–32. van der Hilst, R.D., Widiyantoro S., and Engdahl E.R., 1997. Evidence for deep mantle circulation from global tomography. Nature, 386: 578–584. Zhang, K., and Gubbins, D., 1992. On convection in the Earth’s core forced by lateral temperature variations in the lower mantle. Geophysical Journal International, 108: 247–255.

418

INNER CORE ANISOTROPY

Zhang, K., and Gubbins, D., 1993. Convection in a rotating spherical fluid shell with an inhomogeneous temperature boundary condition at infinite Prandtl number. Journal of Fluid Mechanics, 250: 209–232.

Cross-references Core, Magnetic Instabilities Core Motions Magnetoconvection Magnetohydrodynamic Waves Proudman-Taylor Theorem

INNER CORE ANISOTROPY Evidence for anisotropy of inner core Anisotropy is the general term used to describe a medium whose properties (i.e., elastic properties in the case of seismic anisotropy) depend on orientation. Seismic waves in an anisotropic medium travel with different speeds depending on both their particle motion and propagation directions. The Earth’s solid inner core was found to possess significant anisotropy in seismic velocity in the 1980s and early 1990s. Before that time, the inner core was known to us as a featureless small solid ball (at a radius of 1220 km) of iron-nickel alloy with some light elements. The evidence for the anisotropy of the inner core came from two different kinds of observations: directional variations of travel-times

of seismic body waves that go through the inner core and anomalous splitting, splitting not explainable simply by Coriolis and ellipticity effects, of the Earth’s normal modes that are sensitive to the structure of the Earth’s core. Anomalies in inner core arrival times and in normal-mode splitting were observed in early 1980s (Masters and Gilbert, 1981; Poupinet et al., 1983) and the hypothesis that the inner core is anisotropic was first proposed in 1986 (Morelli et al., 1986). Subsequent studies using travel-time data (e.g., Shearer et al., 1988) generally favored the existence of inner core anisotropy. However, the interpretation that anomalous splitting of core-sensitive modes is primarily caused by the anisotropy of the inner core was controversial (e.g., Widmer et al., 1992). Strong support for the inner core anisotropy was provided in the early 1990s when new sets of information started to emerge. Measurements of high quality differential travel-times of PKP waves (waves that pass through the Earth’s core) show large travel-time anomalies (2–6 s) for rays traveling through the inner core nearly parallel to the Earth’s spin axis (polar paths) (Creager, 1992; Song and Helmberger, 1993; Vinnik et al., 1994). Reanalysis of arrival time data also confirmed large travel-time anomalies (Shearer, 1994; Su and Dziewonski, 1995). At the same time, Tromp (1993) demonstrated that a simple transversely isotropic inner core explained the anomalously split core-sensitive modes and the larger, newly observed PKP differential travel-time anomalies reasonably well. An example of the evidence for the inner core anisotropy from travel-time observations is shown in Figure I6. In analyzing travel-time data, residuals of travel-times that seismic waves (phases) take from an earthquake to arrive at recording stations are often formed by subtracting the times predicted for a standard Earth model from measured travel-times; similarly, residuals of differential travel-times are formed

Figure I6 (a) Ray paths of PKP(AB) and PKP(DF) phases at near antipodal distances. The PKP(AB) phase turns at mid-outer core and the PKP(DF) phase passes through the inner core, sampling the bulk of the inner core at near antipodal distances as shown in this example. (b) Residuals of differential PKP(AB)-PKP(DF) travel-times for near antipodal paths (modified from Sun and Song, 2002). Assuming uniform cylindrical anisotropy in the inner core, the curve is the best fit to the data.

INNER CORE ANISOTROPY

by subtracting the time differences between two phases concerned predicted for a standard Earth model from measured differential traveltimes. Figure I6b shows the residuals of differential between AB and DF branches of PKP waves (Figure I6a). Here the PKP(AB) is used as a reference phase to form the differential travel times, which eliminates bias from uncertainties in earthquake origin times and reduces the biases from earthquake mislocations and heterogeneous upper mantle; heterogeneity in the outer core is assumed to be small because of efficient mixing in the fluid core (Stevenson, 1987). The residuals for the polar paths are systematically larger than those of the equatorial paths, suggesting faster wave speed along the NS direction through the inner core than along the equatorial plane. Assuming cylindrical anisotropy with fast axis parallel to the spin axis, this data set indicates an average anisotropy of about 2.5% in P-velocity in the inner core.

Three-dimensional structure of inner core anisotropy and heterogeneity The presence of significant anisotropy in the inner core is now well accepted. The anisotropy appears to be dominantly cylindrical, with the axis of symmetry aligned approximately with the NS spin axis of the Earth. The detailed structure of the inner core anisotropy is still being mapped out. Some studies suggest the inner core has a simple constant anisotropy except for the innermost inner core (Ishii and Dziewonski, 2002). Other studies, however, suggest that the inner core anisotropy is more complex, varying both laterally and in depth, although some of the complexity may be due to contamination from the lowermost mantle heterogeneity (Breger et al., 2000; Tromp, 2001). The outermost part of the inner core appears nearly isotropic (Shearer, 1994). The thickness of the weak anisotropy layer varies from upper 100–250 km in the western inner core to upper 400 km or more in the eastern inner core (Creager, 2000; Souriau et al., 2003). The transition from isotropy in the upper inner core to strong anisotropy in lower inner core under Central America appears to be sharp enough to cause multipathing of seismic waves (Song and Helmberger, 1998). There is also growing evidence for strong lateral variation in inner core structure. The inner core appears to vary on all scales, from the scale of half a hemisphere to the scale of a few kilometers (Vidale and Earle, 2000; Cormier and Li, 2002). At the uppermost 100 km of the inner core, the P-velocity in the quasi-eastern hemisphere (40 E–180 E) is isotropically faster than the quasi-western hemisphere (180 W–40 E) by about 0.8% (Niu and Wen, 2001). At intermediate depth, 100–400 km below the inner core boundary (ICB), the inner core possesses a hemispherical pattern of a different form. The quasi-western hemisphere is strongly anisotropic but the quasi-eastern hemisphere is nearly isotropic (Tanaka and Hamaguchi, 1997). Deeper into the inner core, significant anisotropy (3%) seems to exist in both hemispheres (Creager, 2000; Souriau et al., 2003); however, in the innermost 300–400 km of inner core, it is suggested that the form of the anisotropy is distinctly different (Ishii and Dziewonski, 2002; Beghein and Trampert, 2003).

Sources of inner core anisotropy Apparent seismic velocity anisotropy can, in general, arise from preferred orientation of anisotropic crystals or from lamination of a solid. The properties of the inner core are widely believed to be consistent with those of anisotropic iron crystals in the hexagonal close-packed (hcp) phase (e.g., Brown and McQueen, 1986). Thus, the anisotropy is believed to be due to a preferred orientation of the hcp iron (e.g., Stixrude and Cohen, 1995). However, the mechanisms responsible for creating such a preferred alignment are under debate. One category of the proposed models involves texturing established during the solidification of iron crystals at the surface of the inner core. Possible mechanisms of solidification texturing include dendritic growth of iron crystals as they solidify (Bergman, 1997) and the

419

development of anisotropic polycrystalline solid due to the presence of thermal gradient and initial nucleation in the melt (Brito et al., 2002). However, solidification texturing alone cannot explain the observed depth dependence of the anisotropy. The other category of the proposed models involves the alignment of iron crystals from plastic deformation in the inner core after solidification. Proposed mechanisms of plastic deformation in the inner core include solid state thermal convection driven by internal heating (Jeanloz and Wenk, 1988); flow induced by Maxwell stress (stress due to magnetic field) (Karato, 1999; Buffett and Wenk, 2001); and flow induced by a differential stress field created by preferential growth of the inner core in the equatorial belt, which results from more efficient heat transport in the equator than near the polar regions (Yoshida et al., 1996).

Implications of inner core anisotropy The inner core anisotropy has important implications for improving our understanding of the structure, composition, and dynamics of the Earth’s deep interior. It has been suggested that the inner core is rotating relative to the mantle (Song and Richards, 1996). Detailed mapping of the lateral variations of the inner core anisotropic structure is crucial for quantification of the rotation rate (Creager, 1997), which would contribute to our understanding of the geodynamo (e.g., Glatzmaier and Roberts, 1995; Kuang and Bloxham, 1997) and could influence our interpretation of decadal variations in the length of day (Buffett and Glatzmaier, 2000). The understanding of the source of the inner core anisotropy is expected to improve our understanding of the inner core evolution and its interactions with the outer core (e.g., Romanowicz et al., 1996; Buffett, 2000). Viable models for the origin of anisotropy must be compatible with the estimates of symmetry, depth-dependence and lateral variations. The basic cylindrical symmetry of the inner core anisotropy probably reflects the strong influence of rotation on the dynamics of the fluid core. The absence of anisotropy from the uppermost region of the inner core suggests that any texture acquired during solidification is weak. However, a weak initial texture could be subsequently amplified by grain growth and plastic deformation. In this case, the rate of increase in anisotropy with depth is closely related to the relative rates of inner core growth, grain growth, and strain accumulation. The existence of sharp increase with depth and strong lateral variations of inner core anisotropy poses serious restrictions of the mechanisms responsible for the anisotropy. Improved models of the anisotropic structure of the inner core from seismic imaging will be vital in addressing these issues. Xiaodong Song

Bibliography Beghein, C., and Trampert, J., 2003. Robust normal mode constraints on inner-core anisotropy from model space search. Science, 299: 552–555. Bergman, M.I., 1997. Measurements of elastic anisotropy due to solidification texturing and the implications for the Earth’s inner core. Nature, 389: 60–63. Breger, L., Tkalcic, H., and Romanowicz, B., 2000. The effect of D” on PKP(AB-DF) travel time residuals and possible implications for inner core structure. Earth and Planetary Science Letters, 175: 133–143. Brito, D., Elbert, D., and Olson, P., 2002. Experimental crystallization of gallium: ultrasonic measurement of elastic anisotropy and implications for the inner core. Physics of the Earth and Planetary Interiors, 129: 325–346. Brown, J.M., and McQueen, R.G., 1986. Phase-transitions, Gruneisenparameter, and elasticity for shocked iron between 77-GPa and 400-GPa. Geophysical Journal of the Royal Astronomical Society, 91: 7485–7494. Buffett, B.A., 2000. Dynamics of the Earth’s core. In Karato, S., Forte, A.M., Liebermann, R.C., Masters, G., and Stixrude, L. (eds.),

420

INNER CORE COMPOSITION

Earth’s Deep Interior: Mineral Physics and Tomography from the Atomic to the Global Scale, American Geophysical Union Monograph 117. Washington, DC: American Geophysical Union, pp. 37–62. Buffett, B.A., and Glatzmaier, G.A., 2000. Gravitational braking of inner-core rotation in geodynamo simulations. Geophysical Research Letters, 27: 3125–3128. Buffett, B.A., and Wenk, H.R., 2001. Texturing of the inner core by Maxwell stresses. Nature, 413: 60–63. Cormier, V.F, and Li, X., 2002. Frequency-dependent seismic attenuation in the inner core. 2. A scattering and fabric interpretation. Journal of Geophysical Research, 107(B12): doi:10.1029/2002 JB001796. Creager, K.C., 1992. Anisotropy of the inner core from differential travel times of the phases PKP and PKIKP. Nature, 356: 309–314. Creager, K.C., 1997. Inner core rotation rate from small-scale heterogeneity and time-varying travel times. Science, 278: 1284–1288. Creager, K.C., 2000. Inner core anisotropy and rotation. In Karato, S., Forte, A.M., Liebermann, R.C., Masters, G., and Stixrude, L. (eds.), Earth’s Deep Interior: Mineral Physics and Tomography from the Atomic to the Global Scale, American Geophysical Union Monograph 117. Washington, DC: American Geophysical Union, pp. 89–114. Glatzmaier, G.A., and Roberts, P.H., 1995. A three-dimensional convective dynamo solution with rotating and finitely conducting inner core and mantle. Physics of the Earth and Planetary Interiors, 91: 63–75. Ishii, M., and Dziewonski, A.M., 2002. The innermost inner core of the earth: evidence for a change in anisotropic behavior at the radius of about 300 km. PNAS, 99(22): 14026–14030. Jeanloz, R., and Wenk, H.R., 1988. Convection and anisotropy of the inner core. Geophysical Research Letters, 15: 72–75. Karato, S., 1999. Seismic anisotropy of the Earth’s inner core resulting from flow induced by Maxwell stresses. Nature, 402: 871–873. Kuang, W.J., and. Bloxham, J., 1997. An earth-like numerical dynamo model. Nature, 389: 371–374. Masters, G., and Gilbert, F., 1981. Structure of the inner core inferred from observations of its spheroidal shear modes. Geophysical Research Letters, 8: 569–571. Morelli, A., Dziewonski, A.M., and Woodhouse, J.H., 1986. Anisotropy of the inner core inferred from PKIKP travel-times. Geophysical Research Letters, 13: 1545–1548. Niu, F.L., and Wen, L.X., 2001. Hemispherical variations in seismic velocity at the top of the Earth’s inner core. Nature, 410: 1081–1084. Poupinet, G., Pillet, R., and. Souriau, A., 1983. Possible heterogeneity of the Earth’s core deduced from PKIKP travel-times. Nature, 305: 204–206. Romanowicz, B., Li, X.D., and Durek, J., 1996. Anisotropy in the inner core: could it be due to low-order convection? Science, 274(5289): 963–966. Shearer, P.M., 1994. Constraints on inner core anisotropy from PKP(DF) travel-times. Journal of Geophysical Research, 99: 19,647–19,659. Shearer, P.M., Toy, K.M., and Orcutt, J.A., 1988. Axi-symmetric Earth models and inner-core anisotropy. Nature, 333: 228–232. Song, X.D., and Helmberger, D.V., 1988. Seismic evidence for an inner core transition zone. Science, 282: 924–927. Song, X.D., and Helmberger, D.V., 1993. Anisotropy of Earth’s inner core. Geophysical Research Letters, 20: 2591–2594. Song, X.D., and Richards, P.G., 1996. Observational evidence for differential rotation of the Earth’s inner core. Nature, 382: 221–224. Souriau, A., Garcia, R., and Poupinet, G., 2003. The seismological picture of the inner core: structure and rotation. Comptes Rendus Geoscience, 335(1): 51–63. Su, W.J., and Dziewonski, A.M., 1995. Inner core anisotropy in three dimensions. Journal of Geophysical Research, 100: 9831–9852. Sun, X.L., and Song, X.D., 2002. PKP travel times at near antipodal distances: implications for inner core anisotropy and lowermost

mantle structure. Earth and Planetary Science Letters, 199: 429–445. Stevenson, D.J., 1987. Limits on lateral density and velocity variations in the Earth’s outer core. Geophysical Journal of the Royal Astronomical Society, 88: 311–319. Stixrude, L., and Cohen, R.E., 1995. High-pressure elasticity of iron and anisotropy of Earth’s inner core. Science, 267: 1972–1975. Tanaka, S., and Hamaguchi, H., 1997. Degree one heterogeneity and hemispherical variation of anisotropy in the inner core from PKP (BC)-PKP(DF) times. Journal of Geophysical Research, 102: 2925–2938. Tromp, J., 1993. Support for anisotropy of the Earth’s inner core from free oscillations. Nature, 366: 678–681. Tromp, J., 2001. Inner-core anisotropy and rotation. Annual Review of Earth and Planetary Sciences, 29: 47–69. Vidale, J.E., and Earle, P.S., 2000. Fine-scale heterogeneity in the Earth’s inner core. Nature, 404(6775): 273–275. Vinnik, L., Romanowicz, B., and Breger, L., 1994. Anisotropy in the center of the inner-core. Geophysical Research Letters, 21(16): 1671–1674. Widmer, R.W., Masters, G., and Gilbert, F., 1992. Observably split multiplets—data analysis and interpretation in terms of large-scale aspherical structure. Geophysical Journal International, 111: 559–576. Yoshida, S., Sumita, I., and. Kumazawa, M., 1996. Growth-model of the inner core coupled with the outer core dynamics and the resulting elastic anisotropy. Journal of Geophysical Research, 101: 28085–28103.

Cross-references Core Convection Core Properties, Theoretical Determination Geodynamo, Energy Sources Geodynamo, Numerical Simulations Grüneisen’s Parameter for Iron and Earth’s Core Inner Core Composition Inner Core Oscillation Inner Core Rotation Inner Core Rotational Dynamics Inner Core Seismic Velocities Lehmann, Inge (1888–1993)

INNER CORE COMPOSITION Why is the inner core important? The solid inner core is complex and not yet fully understood. This is hardly surprising given that the temperature of the Earth’s core (q.v.) is in the range 5000–6000 K and inner core pressures are 330– 360 GPa. Knowledge of the exact composition and structure of the Earth’s inner core would enable a better understanding of the internal structure and dynamics of the Earth as a whole; in particular, better constraints on core composition (q.v.) would not only have fundamental implications for models of the formation, differentiation, and evolution of the Earth, but would also enable successful interpretation of seismic observations which have revealed inner core anisotropy (q.v.), layering and heterogeneity (e.g., Creager, 1992; Song, 1997; Beghein and Trampert, 2003; Cao and Romanowicz, 2004; Koper et al., 2004). The elastic anisotropy of the inner core is well established: seismic velocities of the inner core (q.v.) show P-wave velocities 3% faster along the polar axis than in the equatorial plane (e.g., Creager, 1992). However, more recent seismic observations suggest further complexity. The evidence is for a seismically isotropic upper layer, with lateral variations in thickness of 100–400 km,

INNER CORE COMPOSITION

overlaying an irregular, nonspherical transition region to an anisotropic lower layer (Song and Helmberger, 1998; Ouzounis and Creager, 2001; Song and Xu, 2002). The existence of an isotropic upper layer implies that the magnitude of the seismic anisotropy in the lower inner core must be significantly greater than previously thought, possibly as much as 5%–10%. The observed layering also implies that the upper and lower inner core are compositionally or structurally different. Small-scale heterogeneities in the Earth’s inner core have also been observed, possibly associated with phase changes, a “mushy layer,” melt inclusions and/or compositional differences (Cao and Romanowicz, 2004; Koper et al., 2004). The question now arises as to the mechanisms by which such complexity can occur, all of which could be highly compositionally dependent. Therefore, we cannot test any hypotheses for the observed features without knowing the inner core composition.

421

elements is included (Lin et al., 2003). It now seems very possible that a bcc phase might be formed (Vočadlo et al., 2003). Previously, the bcc phase of iron was considered an unlikely candidate for an inner coreforming phase because athermal calculations show it to be elastically unstable at high pressures, with an enthalpy considerably higher than that of hcp iron (Stixrude and Cohen, 1995; Söderlind et al., 1996; Vočadlo et al., 2000). However, more recent ab initio molecular dynamics calculations at core pressures and temperatures have suggested that the bcc phase of iron becomes entropically stabilized at core temperatures. Although for pure iron the thermodynamically most stable phase is still the hcp phase, the free energy difference is so very slight that a small amount of light element impurity could stabilize the bcc phase at the expense of the hcp phase (Vočadlo et al., 2003).

Alloying elements The inner core is not just made of iron The exact composition of the Earth’s inner core is not very well known. On the basis of cosmochemical and geochemical arguments, it has been suggested that the core is an iron alloy with possibly as much as 5 wt% Ni and very small amounts (only fractions of a wt% to trace) of other siderophile elements such as Cr, Mn, P, and Co (McDonough and Sun, 1995). On the basis of materials-density/ sound-wave velocity systematics, Birch (1964) further concluded that the core is composed of iron that is alloyed with a small fraction of lighter elements. The light alloying elements most commonly suggested include S, O, Si, H, and C, although minor amounts of other elements, such K, could also be present (e.g., Poirier, 1994; Gessmann and Wood, 2002). From seismology it is known that the density jump across the inner core boundary is between 4.5% and 6.7% (Shearer and Masters, 1990; Masters and Gubbins, 2003) indicating that there is more lighter element alloying in the outer core. The evidence thus suggests that the outer core contains 5%–10% light elements, while the inner core has 2%–3% light elements. Our present understanding is that the Earth’s solid inner core is crystallizing from the outer core as the Earth slowly cools and the partitioning of the light elements between the solid and liquid is therefore crucial to understanding the evolution and dynamics of the core.

Which phase of iron exists in the inner core? Before tackling the more detailed problem of inner core composition (q.v.) in terms of alloying elements, it is essential to know what crystalline structure(s) are present and also their seismic properties, as these will greatly affect both interpretations made from seismic data and also the possible alloying mechanism. Many experimentalists have put an enormous effort over the last 10–15 years into obtaining a phase diagram of pure iron under core conditions, but above relatively modest pressures and temperatures there is still much uncertainty. Experimental techniques have evolved rapidly in recent years, and today, using diamond anvil cells or shock wave experiments (q.v.), the study of minerals at pressures up to 200 GPa and temperatures of a few thousand Kelvin are possible. These studies, however, are still far from routine and results from different groups are often in conflict. At low pressures and temperatures the phase diagram of iron is well understood: the body-centered-cubic (bcc) phase is stable at ambient conditions, transforming to the hexagonal close-packed phase (hcp) at high pressure (>10–15 GPa) and to the face-centered-cubic phase at high temperature (>1200 K). Both experiments and also theoretical calculations of the static, zero-Kelvin solid (Stixrude et al., 1997; Vočadlo et al., 2000), have suggested that the hcp phase has a wide stability field at high pressures and temperatures right to the conditions of the Earth’s inner core. However, this assumption that iron must have the hcp structure at core conditions has recently been challenged (Brown, 2001; Beghein and Trampert, 2003), especially if the presence of lighter

It is generally assumed that the small amount of nickel alloyed to iron in the inner core is unlikely to have any significant affect on core properties (q.v.) as nickel and iron have sufficiently similar densities to be seismically indistinguishable, and addition of small amounts of nickel are unlikely to appreciably change the physical properties of iron. However, the presence of light elements in the core does have an affect on core properties. The light element impurities most often suggested are sulfur, oxygen, and silicon. Although these alloying systems have been experimentally studied up to pressures of around 100 GPa (e.g., Li and Agee, 2001; Lin et al., 2003; Rubie et al., 2004) there seems little prospect of obtaining experimental data for iron alloys at the highly elevated pressures and temperatures of the Earth’s inner core. An alternative approach to understanding inner core composition (q.v.) is to simulate the behavior of these iron alloys with ab initio calculations which are readily able to access the pressures and temperatures of the inner core. Alfè et al. (2000, 2002) calculated the chemical potentials of iron alloyed with sulfur, oxygen, and silicon. They developed a strategy for constraining both the impurity fractions and the temperature at the inner core boundary (ICB) based on the supposition that the solid inner core and liquid outer core are in thermodynamic equilibrium at the ICB. For thermodynamic equilibrium the chemical potentials of each species must be equal both sides of the ICB, which fixes the ratio of the concentrations of the elements in the liquid and in the solid, which in turn fixes the densities. If the core consisted of pure iron, equality of the chemical potential (the Gibbs free energy in this case) would tell us only that the temperature at the ICB is equal to the melting temperature of iron (q.v.) at the ICB pressure of 330 GPa. With impurities present, the ab initio results reveal a major qualitative difference between oxygen and the other two impurities: oxygen partitions strongly into the liquid, but sulfur and silicon, both partition equally in the solid and liquid. Having established the partitioning coefficients, Alfè et al. (2002) then investigated whether the known densities of the outer and inner core, estimated from seismology, could be matched by one of their calculated binary systems. For sulfur and silicon, their ICB density discontinuities were considerably smaller than the known seismological value at that time of 4.5%  0.5% (Shearer and Masters, 1990); for oxygen, the discontinuity was markedly greater than that from seismology. Therefore none of these binary systems were plausible, i.e., the core cannot be made solely of Fe/S, Fe/Si, or Fe/O. However, the seismic data can clearly be matched by a ternary/quaternary system of iron and oxygen together with sulfur and/or silicon. A system consistent with seismic data could contain 8 mol% oxygen and 10 mol% sulfur and/or silicon in the outer core, and 0.2 mol% oxygen and 8.5 mol% sulfur and/or silicon in the inner core (Alfè et al., 2002). However, it should be remembered that it is likely that several other light elements could exist in the inner core and would therefore have to be considered before a true description of inner core composition could be claimed. Lidunka Vočadlo

422

INNER CORE OSCILLATION

Bibliography Alfè, D., Gillan, M.J., and Price, G.P., 2000. Constraints on the composition of the Earth’s core from ab initio calculations. Nature, 405: 172–175. Alfè, D., Gillan, M.J., and Price, G.P., 2002. Ab initio chemical potentials of solid and liquid solutions and the chemistry of the Earth’s core. Journal of Chemical Physics, 116: 7127–7136. Beghein, C., and Trampert, J., 2003. Robust normal mode constraints on inner-core anisotropy from model space search. Science, 299: 552–555. Birch, F., 1964. Density and composition of the mantle and core. Journal of Geophysical Research, 69: 4377–4388. Brown, J.M., 2001. The equation of state of iron to 450 GPa: another high pressure solid phase? Geophysical Research Letters, 28: 4339–4342. Cao, A., and Romanowicz, B., 2004. Hemispherical transition of seismic attenuation at the top of the Earth’s inner core. Earth and Planetary Science Letters, 228: 243–253. Creager, K.C., 1992. Anisotropy of the inner core from differential travel-times of the phases PKP and PKIKP. Nature, 356: 309–314. Gessmann, C.K., and Wood, B.J., 2002. Potassium in the Earth’s core? Earth and Planetary Science Letters, 200: 63–78. Koper, K.D., Franks, J.M., and Dombrovskaya, M., 2004. Evidence for small-scale heterogeneity in Earth’s inner core from a global study of PkiKP coda waves. Earth and Planetary Science Letters, 228: 227–241. Li, J., and Agee, C.B., 2001. Element partitioning constraints on the light element composition of the Earth’s core. Geophysical Research Letters, 28: 81–84. Lin, J-F., et al., 2003. Sound velocities of iron-nickel and iron-silicon alloys at high pressures. Geophysical Research Letters, 30: doi:10.1029/2003GL018405. Masters, G., and Gubbins, D., 2003. On the resolution of density within the Earth. Physics of the Earth and Planetary Interiors, 140: 159–167. McDonough, W.F., and Sun, S-S., 1995. The composition of the Earth. Chemical Geology, 120: 223–253. Ouzounis, A., and Creager, K.C., 2001. Isotropy overlying and isotropy at the top of the inner core. Geophysical Research Letters, 28: 4331–4334. Poirier, J-P., 1994. Light elements in the Earth’s outer core: a critical review. Physics of the Earth and Planetary Interiors, 85: 319–337. Rubie, D.C., Gessmann, C.K., and Frost, D.J., 2004. Partitioning of oxygen during core formation of the Earth and Mars. Nature, 429: 58–61. Shearer, P., and Masters, G., 1990. The density and shear velocity contrast at the inner core boundary. Geophysical Journal International, 102: 491–498. Söderlind, P., Moriarty, J.A., and Wills, J.M., 1996. First-principles theory of iron up to Earth’s core pressures: structural, vibrational and elastic properties. Physical Review B, 53: 14,063–14,072. Song, X., 1997. Anisotropy of the Earth’s inner core. Reviews in Geophysics, 35(3): 297–313. Song X., and Helmberger, D.V., 1998. Seismic evidence for an inner core transition zone. Science, 282: 924–927. Song, X., and Xu, X., 2002. Inner core transition zone and anomalous PKP(DF) waveforms from polar paths. Geophysical Research Letters, 29: 1–4. Stixrude, L., and Cohen, R.E., 1995. Constraints on the crystalline structure of the inner core—mechanical instability of bcc iron at high pressure. Geophysical Research Letters, 22: 125–128. Stixrude, L., Wasserman, E., and Cohen, R.E., 1997. Composition and temperature of the Earth’s inner core. Journal of Geophysical Research, 102: 24729–24739.

Vočadlo. L., et al., 2000. Ab initio free energy calculations on the polymorphs of iron at core conditions. Physics of the Earth and Planetary Interiors, 117: 123–127. Vočadlo. L., et al., 2003. Possible thermal and chemical stabilisation of body-centred-cubic iron in the Earth’s core. Nature, 424: 536–539.

Cross-references Core Composition Core Properties, Physical Core Properties, Theoretical Determination Core Temperature Inner Core Anisotropy Inner Core Seismic Velocities Melting Temperature of Iron in the Core, Experimental Melting Temperature of Iron in the Core, Theory Shock Wave Experiments

INNER CORE OSCILLATION Earth’s inner core, identified in relatively recent time (Lehmann, 1936) (q.v.), can be considered to be an elastic solid on timescales of seismic waves. With a radius of 1220 km and hence containing less than 1% of Earth’s volume, the inner core’s importance to our understanding of Earth structure far exceeds both its relative size and our knowledge of its properties. Decadal changes in Earth’s magnetic field, linked to torsional oscillations (q.v.) of the fluid outer core, can produce corresponding oscillations of the inner core through electromagnetic coupling. Since the inner core is well coupled gravitationally to the mantle, both changes in LOD (q.v.) and polar motion of the mantle on a decade timescale could be used to constrain torsional oscillations (Dumbery and Bloxham, 2003) (see also Inner core rotation). Gravitational force is responsible for holding the slightly denser inner core at Earth’s center in the surrounding fluid core. Thus the inner core will return to this central equilibrium position if displaced axially and subsequently oscillate at periods of a few hours. This fact was initially recognized by Slichter (1961) who also realized that Earth’s rotation would lead to two additional circular modes, one prograde and one retrograde if the inner core were displaced in a direction perpendicular to the rotation axis. Theoretical values of three modes of oscillation, often referred to as the Slichter triplet, depend on the physical properties of the core described by the Earth model chosen, as well as the method used to calculate their periods. In the case of the Earth model 1066A (Gilbert and Dziewonski, 1975), for example, the buoyancy oscillation period without Earth rotation or degenerate period, is reported as 4.599 h by Rogister (2003), 4.309 h by Rieutord (2002), and 4.45471 h by Smylie et al. (2001). The central period of the triplet has been calculated using normal mode theory (Rogister, 2003), yielding estimates near 5.309 h for the Earth model PREM (Dziewonski and Anderson, 1981) and 4.529 h for 1066A which closely agree with those from perturbation methods (e.g., Dahlen and Sailor, 1979; Crossley et al., 1992) that ignored ellipticity and centrifugal effects. A central period of 4.255 h for 1066A has been found by Rieutord (2002) who reminded us that the buoyancy of the inner core is determined by the difference between the mean density of the inner core and its surrounding fluid rather than the density jump at the inner core-outer core boundary. An even shorter central period of 3.7926 h is predicted (Smylie et al., 2001) for the Cal8 Earth model (Bullen and Bolt, 1985) with a higher density inner core that experiences a larger restoring force when displaced.

INNER CORE ROTATION

The Slichter triplet has been the subject of search in gravimetric records since a movement of the inner core will produce a small change in Earth’s gravitational field. Following the Chilean earthquake of May 22, 1960, a peak in the gravimetric record near 86 min was tentatively identified as an inner core oscillation, implying a much denser inner core than would be consistent with existing Earth models. Significant attempts had been made to detect the Slichter triplet by locating gravimeters near the South Pole where the diurnal and semidiurnal tidal effects are small (Rydelek and Knopoff, 1984) but there was no identification of the triplet. In recent years, stable, low noise, superconducting gravimeters have allowed for the measurement of changes in gravity down to 2  10
11 m s
2 in Earth’s gravity field of about 9.8 ms
2. Thus, long records can be analyzed in the frequency domain for three spectral peaks separated by precisely the amount predicted by the Earth model chosen. Even with the above sensitivity, however, the signalto-noise ratios are so small that no one has yet observed the Slichter triplet in a single gravimetric record. By combining several superconducting gravimetric records with a wide geographical distribution, through a method known as product spectral analysis (Smylie et al., 1993), a claim has been made (Smylie et al., 2001) to have observed the Slichter triplet at periods 3.5822, 3.7656, and 4.0150 h with a statistical error of 0.04%. These observed values are reported to correspond to periods of 3.5168, 3.7926, and 4.1118 h predicted from the Cal8 model assuming the fluid core is inviscid. This identification and the method of analysis reported by Smylie et al. (2001) are yet to be widely accepted. Other groups have either failed to find the Slichter triplet (Hinderer et al., 1995) or, more recently, have reported several possible candidate signals for the Slichter triplet depending on the Earth model chosen (Rosat et al., 2004). Still others have cautioned (Florsch et al., 1995) that there is a high likelihood in finding significant fits to a triplet of spectral peaks even if one chooses the frequencies at random. The rotational splitting of the Slichter triplet should include a contribution from viscosity of the liquid outer core as noted by Smylie (1999). Using boundary layer theory, he calculated the viscosity of the outer core fluid in a layer next to the inner core boundary, based on the small discrepancy between the observed and predicted triplet periods. He found a very high kinematic viscosity, close to 107 m2 s
1 , that was interpreted as confirming the semifluid nature of the layer at the base of the fluid core as the source of energy for the geodynamo. Such a large viscosity, however, brings into question the validity of results obtained by using only the leading term of the asymptotic series, as pointed out by Rieutord (2002). Future models that include dissipation due to turbulence (see Core turbulence) and partial solidification in the boundary layer around the inner core would decrease estimates of viscosity of the fluid core. Periods of the Slichter triplet are extremely sensitive to the mean density of the inner core (q.v.), about 3 h for each gm cm
3 for the axial mode (Smylie et al., 2001), and thus would provide an independent estimate of that important quantity. Confirmation of the measured periods of the Slichter triplet will provide the basis for an estimation of the density jump at the inner core-outer core boundary, that is central to estimation of the inner core’s age and its role in a geodynamo driven by compositional convection. Further improvement in signalto-noise ratio of gravity measurements and a larger excitation of the inner core, whose source remains unknown, are needed to validate the detection of the Slichter triplet. Keith Aldridge

423

Dahlen, F.A., and Sailor, R.V., 1979. Rotational and elliptical splitting of the free oscillations of the Earth. Geophysical Journal of the Royal Astronomical Society, 58: 609–623. Dumbery, M., and Bloxham, J., 2003. Torque balance, Taylor’s constraint and torsional oscillations in a numerical model of the geodynamo. Physics of the Earth and Planetary Interiors, 140: 29–51. Dziewonski, A.M., and Anderson, D.L., 1981. Preliminary Reference Earth Model (PREM). Physics of the Earth and Planetary Interiors, 25: 297–356. Florsch, N., Legros, H., and Hinderer, J., 1995. The search for weak harmonic signals in a spectrum with applications to gravity data. Physics of the Earth and Planetary Interiors, 90: 197–210. Gilbert, F., and Dziewonski, A.M., 1975. An application of normal mode theory to the retrieval of structural parameters and source mechanisms from seismic spectra. Philosophical Transactions of the Royal Society of London, Series A, 278: 187–269. Hinderer, J., Crossley, D., and Jensen, O., 1995. A search for the Slichter triplet in superconducting gravimeter data. Physics of the Earth and Planetary Interiors, 90: 221–241. Lehmann, I., 1936. P0. Publication Bureau Central Seismologique International, Series A, 14: 87–115. Rieutord, M., 2002. Slichter modes of the Earth revisited. Physics of the Earth and Planetary Interiors, 131: 269–278. Rogister, Y., 2003. Splitting of seismic-free oscillations and of the Slichter triplet using the normal mode theory of a rotating, ellipsoidal earth. Physics of the Earth and Planetary Interiors, 140: 169–182. Rosat, S., Hinderer, J., Crossley, D., and Boy, J.P., 2004. Performance of superconducting gravimeters from long-period seismology to tides. Journal of Geodynamics, 38: 461–476. Rydelek, P., and Knopoff, L., 1984. Spectral analysis of gapped data: search for the mode 1S1 at the South Pole. Journal of Geophysical Research, 89: 1899–1911. Slichter, L.B., 1961. The fundamental free mode of the Earth’s inner core. Proceeding of the National Academy of Sciences USA, 47: 186–190. Smylie, D.E., 1999. Viscosity near earth’s solid inner core. Science, 284: 461–463. Smylie D.E., Hinderer, J., Richter, B., and Ducarme, B., 1993. The product spectra of gravity and barometric pressure in Europe. Physics of the Earth and Planetary Interiors, 80: 135–157. Smylie, D.E., Francis, O., and Merriam, J.B., 2001. Beyond tides— determination of core properties from superconducting gravimeter observations. Journal of the Geodetic Society of Japan, 47: 364–372.

Cross-references Core Density Core Turbulence Inner Core Rotation Lehmann, Inge (1888–1993) Length of Day Variations, Decadal Oscillations, Torsional

INNER CORE ROTATION Observational evidence of inner core rotation is based on changes in the travel-time of seismic waves. The rotation rate appears to be a few tenths of a degree per year eastward with respect to the mantle.

Bibliography Bullen, K.E., and Bolt, B.A., 1985. An Introduction to the Theory of Seismology. Cambridge: Cambridge University Press. Crossley, D., Rochester, M., and Peng, Z., 1992. Slichter modes and Love numbers. Geophysical Research Letters, 19: 1679–1682.

Early speculations The inner core with radius of about 1215 km resides concentrically within the much larger fluid outer core, which has a low viscosity (e.g., Poirier, 1998). Patterns of convection within the fluid core

424

INNER CORE ROTATION

associated with the geodynamo are presumed to undergo temporal variations because the Earth’s magnetic field is changing on timescales ranging over several orders of magnitude. It is therefore reasonable to speculate (Gubbins, 1981; Anderson, 1983) that the inner core might have a rotation rate somewhat different from that of the rest of the solid Earth, which is dominated by the daily rotation. If such relative rotation could be detected, it would provide information on the energy of convection patterns that maintain the geomagnetic field. The study of inner core motion relative to the mantle and crust is difficult, because of the remoteness of the inner core (more than 5150 km from the Earth’s surface where the nearest observations can be made) and its small size (about 0.7% of the Earth’s volume). Also there are intrinsic difficulties in telling whether a spherical object is rotating, unless a marker on or within the object can be identified and tracked if it moves.

First claims, based on seismological observations and implications The first published claim of observational evidence for inner core rotation, relative to the mantle and crust, was given by Song and Richards (1996). They noted that seismic waves originating in the South Sandwich Islands (in the southernmost South Atlantic) and recorded in Alaska, which had traveled through the inner core, appeared to have traveled systematically faster for earthquakes in the later years during the period from 1967 to 1995, compared to observations of seismic waves from earlier earthquakes in this time period. The rate of travel-time decrease amounted to about 0.01 s y
1— a value that was close to the precision of measurement. More detailed studies by Creager (1997), Song (2000), and Li and Richards (2003), have provided additional support for travel-time change of about this same value, for seismic P-waves crossing the

inner core. Figure I7 shows a clear example of PKIKP waves (which pass through the inner core) showing a faster arrival for the later earthquake, when the seismograms are aligned on PKP waves (which avoid the inner core). Song and Richards (1996) interpreted the travel-time change as an effect of anisotropy, which causes P-waves to travel with speeds that depend on direction relative to a crystalline axis that rotates with the inner core. Creager (1997) persuasively argued for a more important marker of inner core rotation, namely a lateral gradient of the P wavespeed (increasing from east to west) in the part of the inner core traversed by PKIKP waves. He and Song (2000), Richards (2000), and Li and Richards (2003) concluded the observed travel-time changes indicate an eastward rotation of the inner core amounting to a few tenths of a degree per year. The concept of an inner core rotating fast enough to be detected on a human timescale has attracted numerous investigators since 1996. Dehant et al. (2003) describe inner core research in mineral physics, seismology, geomagnetism, and geodesy. The rate of inner core rotation is an indication of the vigor of convection in the outer core, associated with the geodynamo. A nonzero rotation rate can be used to place limits on the outer core’s viscosity (Buffett, 1997).

Counterclaims, and additional methods and evidence Several papers since 1996 have argued that the seismological evidence for inner core rotation is equivocal. Thus, Souriau and Poupinet (in Dehant et al., 2003) claim the reported travel-time changes of PKIKP waves on the path between the South Sandwich Islands and Alaska are an artifact of mislocated earthquakes. Early reports of purported changes in the absolute arrival times (not differential times) of PKIKP waves were later dismissed as based on inadequate evidence.

Figure I7 One minute segments of short-period seismograms recorded at College, Alaska, for two earthquakes in the South Sandwich Islands (March 28, 1987 and August 14, 1995). For 30 s following the PKP arrival, and for an additional 3 min (not shown here), these seismograms (passed in the band from 0.6 to 3 Hz) show excellent waveform agreement for signals that have traversed the Earth but not via the inner core. For the PKIKP waveform, from time 250 to 255 s, an insert shows an expanded view of the narrowband filtered version of the two arrivals (in the passband from 0.8 to 1.5 Hz) that have traversed the inner core. With the two seismograms aligned on the PKP phase, it is seen that the PKIKP phase of the later event (shown in gray) traveled slightly faster. An explanation is that the inner core rotated during the 8-year period between the earthquakes, in a manner that provided a faster path for the later PKIKP signal through the inner core.

INNER CORE ROTATIONAL DYNAMICS

Laske and Masters have used normal mode data to study inner core inhomogeneities. Some modes appear to indicate eastward rotation, others westward, and their paper in Dehant et al. (2003) concludes the rate is only marginally indicative of a small eastward rotation, about 0.15 y
1 (but alternatively estimated as 0.34  0.13 y
1 if the normal modes likely to be most contaminated by upper mantle structure are excluded). Vidale and Earle (2000) used backscatter from within the inner core, following PKIKP waves, to find an eastward rotation of the inner core amounting to a few tenths of a degree per year. It appears that the strongest claims of evidence for inner core rotation derive from differential traveltimes, for earthquakes separated by several years and which occur at essentially the same location, generating very similar waveforms. The evidence for inner core rotation is still under debate, and consensus on inner core rotation will likely depend on whether examples such as that given in Figure I7 can be accumulated, since waveform doublets avoid artifacts of event mislocation. A report on 18 high-quality doublets with time separation of up to 35 years in the South Sandwich Islands region, observed at up to 58 stations in and near Alaska, provides such an accumulation (see Zhang et al., 2005). Paul G. Richards and Anyi Li

Bibliography Anderson, D.L., 1983. A new look at the inner core of the Earth. Nature, 302: 660. Buffett, B.A., 1997. Geodynamic estimates of the viscosity of the Earth’s inner core. Nature, 388: 571–573. Creager, K.C., 1997. Inner core rotation rate from small-scale heterogeneity and time-varying travel times. Science, 278: 1284–1288. Dehant, V., Creager, K.C., Karato, S., and Zatman, S. (eds.), 2003. Earth’s Core: Dynamics, Structure, Rotation, Geodynamics series 31. Washington, DC: American Geophysical Union, p. 279. Gubbins, D., 1981. Rotation of the inner core. Journal of Geophysical Research, 86: 11695–11699. Li, A., and Richards, P.G., 2003. Using doublets to study inner core rotation and seismicity catalog precision. G-Cubed, 4: 1072, doi:10.1029/2002GC000379. Poirier, J.P., 1998. Transport properties of liquid metals and viscosity of the Earth’s core. Geophysical Journal of the Royal Astronomical Society, 92: 99–105. Richards, P.G., 2000. Earth’s inner core—discoveries and conjectures. Astronomy and Geophysics, 41: 20–24. Song, X., 2000. Joint inversion for inner core rotation, inner core anisotropy, and mantle heterogeneity. Journal of Geophysical Research, 105: 7931–7943. Song, X., and Richards, P.G., 1996. Seismological evidence for differential rotation of the Earth’s inner core. Nature, 382: 221–224. Vidale, J.E., and Earle, P.S., 2000. Slow differential rotation of the Earth’s inner core indicated by temporal changes in scattering. Nature, 405: 445–448. Zhang, J., Song, X., Li, Y., Richards, P.G., Sun, X., and Waldhauser, F., 2005. Inner core differential motion confirmed by earthquake waveform doublets. Science, 310(5752): 1279.

Cross-references Core Motions Geodynamo Geodynamo, Energy Sources Geomagnetic Spectrum, Temporal Inner Core Anisotropy Inner Core Composition Inner Core Seismic Velocities Lehmann, Inge (1888–1993) Length of Day Variations, Long Term

425

INNER CORE ROTATIONAL DYNAMICS For several decades after its discovery in 1936, the Earth’s solid inner core (IC) played only a passive role in geodynamics. Now the ICs material properties, internal structure, and rotational dynamics are lively concerns of geomagnetism, seismology, and geodesy. Suppose the mantle spins about its figure axis e3 at rate O relative to the stars, i.e., 2p/O ¼ 1 sidereal day (sd). In principle the IC can exhibit: a. differential rotation, i.e., spin relative to the mantle at angular rate Do about the IC figure axis i3 (where i3 is possibly inclined at angle E to e3) and b. small periodic changes in orientation of i3 relative to e3 (wobble) and relative to the stars (nutation). The spinning IC can be regarded as a gyroscope bathed in the surrounding liquid outer core (OC), and will orient itself relative to the mantle in response to the torques exerted on it. Changes in OC flow will cause fluid pressure torques at the inner core boundary (ICB). The dominantly iron IC (see Inner core composition) is expected to have an electrical conductivity comparable to that of the OC, allowing it to be penetrated by changing magnetic fields generated by dynamo action in the OC, and experience electromagnetic torques as a result of Lenz’s law. The OC viscosity is probably so small near the ICB that viscous torques are negligible (see Core viscosity). The vital importance of gravitational torques, exerted on the body of the IC not only by the OC but also by the mantle, has been realized only in the past two decades. Braginsky (1964) first noted that one boundary condition on the geomagnetic dynamo is a (possibly nonzero) value of Do fixed by the requirement of zero net electromagnetic torque on the IC. Gubbins (1981) pointed out that any equatorial component of Do would be shielded by the electrically conducting lower mantle, and (by modeling the OC as a solid spherical shell rotating with the mantle) showed that a steady axial component could result from the balance of two electromagnetic torques: one exerted by the dynamo-generated field in the OC, and the other induced by the nonzero Do itself. Such a torque balance was incorporated in their 2-D numerical hydromagnetic dynamo model by Hollerbach and Jones (1993), who found that the IC (by virtue of its finite electrical conductivity and solidity) played a crucial role in stabilizing dynamo operation. This was confirmed by the numerical 3-D model of a self-sustaining geodynamo created by Glatzmaier and Roberts (1995). This model predicted that, under the balance of both electromagnetic and viscous torques (assuming an artificially large value for OC viscosity for computational reasons), the IC rotates dominantly eastward relative to the mantle at a few degrees per year, changing on a timescale of 500 years. However Kuang and Bloxham (1997) found that a much smaller, more realistic, value of OC viscosity leads to an alternative dynamo in which Do fluctuates, on a timescale of several thousand years, between eastward and westward values. (See Geodynamo, numerical simulations for details.) Buffett and Glatzmaier (2000) showed that dynamo models incorporating gravitational torques between the mantle and IC yield much smaller values of Do, on average about 0.17 or 0.02 y
1 eastward according as viscous torques do or do not act on the ICB. Seismological evidence that the IC departed from isotropy and radial stratification began to emerge nearly two decades ago (see Inner core anisotropy and Inner core composition for details). Using such nonuniformities to directly determine IC rotation became an exciting new branch of seismology when Song and Richards (1996) realized that any tilt of the anisotropy axis away from the rotation axis would cause a systematic change with time in the travel-times of P-waves through the IC from the same source location to the same seismograph. They used 30 years of data to estimate a tilt of about 10 and Do ffi 1 y
1 eastward. Su et al. (1996), analyzing a different but equally long data set, found Do ffi 3 y
1. However most subsequent studies, involving the passage

426

INNER CORE ROTATIONAL DYNAMICS

of a particular IC feature (anisotropy or lateral heterogeneity) across a particular ray path, led to smaller values of Do, of order 0.1 y
1 (e.g., Creager, 1997; Poupinet et al., 2000; Vidale et al., 2000; Isse and Nakanishi, 2002). A very different seismological approach (Laske and Masters, 1999) examined the effects of such nonuniformities on the spectra of nine free oscillations sampling the IC, and yielded Do ¼ 0.01  0.21 y
1. Because such vibrations are comparatively unaffected by errors in locating seismic events and by local IC structure, and are independent of the earthquake source mechanism, this result seems particularly robust and provides strong evidence that Do is likely to be small, possibly not significantly different from zero. The latter result is consistent with the inference that the IC is likely to be gravitationally locked to the mantle, because the gravitational torque exerted on the IC by mantle heterogeneities, when the IC and mantle are not in their equilibrium orientation, greatly exceeds any other (e.g., electromagnetic) torque that might act to maintain such a misalignment (Buffett, 1996). A more extensive analysis of gravitational coupling between IC and mantle by Xu et al. (2000) showed that IC superrotation cannot be supported by such torques alone, and that unless E < 3 the corresponding values of Do far exceed those inferred from seismology. However, Buffett (1996) pointed out that a nonzero Do could be sustained if the IC viscosity is low enough ( a;

r < a;

The field of internal origin can be represented by a system of surface currents flowing over the surface of a sphere, concentric with and within the minimum sphere over which observations are made. A stream function Cðy; fÞ, with units of amperes, called the current function, gives the surface current density Kðy; fÞ (in amperes per meter) in the form Kðy; fÞ ¼

1 ]C 1 ]C ey
ef ¼
er  rC ¼ r  ðer CÞ R sin y ]f R ]y (Eq. 2)

The vector rC is in the direction of Cðy; fÞ increasing, that is, directed toward maxima, and from (Eq. 2) the electrical current flow is counterclockwise around maxima when seen from above. Note however, that the maxima of Cðy; fÞ can be negative and the minima positive. The current function is denoted Cðy; fÞ, with units of amperes, and has a spherical harmonic representation Cðy; fÞ ¼

N X n X

m m ðJnc cos mf þ Jns sin mfÞPnm ðcos yÞ;

(Eq. 3)

n¼1 m¼0

Subscripts c and s have been added to the coefficients Jnm to indicate application to cos mf and sin mf respectively. For a current function in the ionosphere, of radius R, to represent the magnetic daily variations, whose coefficients have been determined relative to a sphere of radius a, the current function coefficients in amperes are given by m Jnc ¼
ðaÞkm

10 2n þ 1 Rn m 10 2n þ 1 Rn m m ðg Þ ; Jns ¼
ðaÞkm ðh Þ : 4p n þ 1 an ne nT 4p n þ 1 an ne nT (Eq. 4)

For a current function within the Earth to represent main field coefficients that have been determined relative to a sphere of radius a, the current function coefficients in amperes are given by m Jnc ¼ ðaÞkm

10 2n þ 1 anþ1 m 10 2n þ 1 anþ1 m m ðg Þ ; Jns ¼ ðaÞkm ðh Þ : 4p n Rnþ1 ni nT 4p n Rnþ1 ni nT (Eq. 5)

Global magnetic field separation

r m gne cos mf þ hm ne sin mf n a n¼1 m¼0

 Pnm ðcos yÞ;

The current function representation

(Eq. 1)

are the potential functions for sources internal to the reference sphere r ¼ a and for sources external to the reference sphere, respectively. Subscripts i and e have been applied to coefficients gnm and hm n to indicate internal or external fields. The initial factor a in (Eq. 1) has been included so that the g and h coefficients will have the dimensions of magnetic flux density. Gauss’s first analysis of the main field to degree and order N ¼ 4 used only the term Vint ðr; y; fÞ, on the assumption that the geomagnetic main field is a term of internal origin only. Analysis of the field gives no indication as to the location of the internal and external sources. Therefore, the potential of the main field, or its variations, is represented as a linear combination of the two types given in (Eq. 1), and applied to field measurements made in the region between the ground and the ionosphere, or, in the case of satellite magnetic data, between the ionosphere and the magnetopause.

In studies of the geomagnetic daily variations, current systems in the ionosphere are the basic source, and eddy currents are induced in the Earth by the ionospheric currents, with only a very small contribution from the magnetospheric current system. Therefore, for the study of the geomagnetic daily variations, the potential function must include terms of both internal and external origin, with the expectation that the terms of external origin will be the greater of the two, being the basic source of the phenomenon. The two types of potential can be separated by numerical calculations as follows, the process being the title of this article, “internal-external field separation.”

Magnetic flux density field components The three components of the geomagnetic field are denoted X, Y, and Z. The northward component is X, the eastward component Y, and the vertically downward component Z. The components, X, Y, and Z in that order, form a right-handed orthogonal system. The mathematical expressions for the field components over the reference sphere, r ¼ a, are



1 ]V

1 ]V

]V

; Y ¼
; Z ¼ ; (Eq. 6) X ¼ a ]y r¼a a sin y ]f r¼a ]r r¼a

450

INTERNAL EXTERNAL FIELD SEPARATION

and therefore in a region free of magnetic sources and electric currents, the ambient magnetic field can be derived from fields of internal and external origin, and the components will have the mathematical form X ða; y; fÞ ¼

N X n  X n¼1 m¼0

Y ða; y; fÞ ¼ Z ða; y; fÞ ¼

N X n  X

 m m m þ gne Þ cos mf þ ðhm ðgni ni þ hne Þ sin mf

 m m m ðgni þ gne Þ sin mf
ðhm ni þ hne Þ cos mf

n¼1 m¼0 N X n  X n¼1 m¼0

m m P ðcos yÞ; sin y n

(Eq: 7)

It will be seen from (Eq. 7), that the coefficients from the northward m m þ gne Þ and component of the field X, obtained in the form ðgni m þ h Þ, can be used to estimate the values of the eastward compoðhm ni ne nent Y. Should the estimated values of Y not be in agreement with the observed values of Y, then to resolve the situation, a nonpotential field m and hm is introduced, as shown below by coefficients gnv nv . The radial dependence required for the internal and external fields is quite definite, but the radial dependence of the nonpotential field depends upon the hypothesis chosen for the origin of the fields; for example, Earthair currents, or field-aligned currents. The nonpotential field Bv ðr; y; fÞ is B v ð r ; y; f Þ ¼

(Eq: 8)

and has no radial component. The scalar function Vv ðr; y; fÞ (with units of magnetic flux density) is given by N X n X qm ðrÞ  n m ðaÞ q n n¼1 m¼0

N X n  X

m m gnv cos mf þ hm nv sin mf Pn ðcos yÞ:

The electrical current system comes only from the nonpotential field as 1 r  Bv ðr; y; fÞ m0  N X n 1 X 1 d m rq ðrÞ ¼ qm ðrÞnðn þ 1Þ er þ m m0 r n¼0 m¼0 qn ðaÞ n dr n
  m ef ]  m ] gnv cos mf þ hm  ey þ sin mf P ðcos yÞ nv n ]y sin y ]f (Eq. 13)

Jv ðr; y; fÞ ¼

The hypothesis of Earth-air currents requires qm n ðrÞ ¼ 1=r when the current system of (Eq. 13) reduces to a radial term only. An expression for a field of internal origin, along whose field lines the electrical current system travels, can also be found (Winch et al., 2005). The representation of (Eqs. 10–12) can be interpreted as a representation of the magnetic field over a sphere in terms of the three independent and orthogonal vector spherical harmonics, usually given in m m complex form as Ym n;nþ1 ðy; fÞ; Yn;n ðy; fÞ; Yn;n
1 ðy; fÞ.

Magnetic daily variations The magnetic daily variations of solar and lunar type originate in the ionosphere by a dynamo process, so that in contrast to the main magnetic field in which the internal coefficients are almost the only significant terms, the magnetic daily variations are dominantly of external origin. The extra complication of dependence on time adds an extra dimension to the complexity of the analysis. A very basic assumption of local time t* dependence on time, where t* ¼ t þ f, and t is universal time, can reduce the complexity, but ignores significant nonlocal time terms that arise because of the inclination of the Earth’s geomagnetic and geographic axes. A less restrictive assumption is that of westward movement only, since the solar and lunar magnetic variations are driven by the Sun, one can expect that eastward moving variations will be mostly insignificant. The analysis begins with Fourier analysis of the elements X, Y, Z, at each observatory. Thus X ða; y; f; tÞ ¼

4 X

½AXM ða; y; fÞ cos Mt þ BXM ða; y; fÞ sin Mt ;

M ¼1

(Eq. 9) Y ða; y; f; tÞ ¼ The function qm n ðrÞ cannot be determined from analysis of theory or data alone, but requires a hypothesis to be made on the form of the electrical current system. The field components of the magnetic field over the reference sphere, r ¼ a, are X ða; y; fÞ ¼

m m
ðn þ 1Þgni þ ngne cos mf



m þ
ðn þ 1Þhm þ nh sin mf Pnm ðcos yÞ: (Eq: 12) ni ne

The nonpotential field

Vv ðr; y; fÞ ¼

Zða; y; fÞ ¼

m m dPnm m Pn
ðgnv ; cos mf þ hm nv sin mfÞ sin y dy (Eq. 11)

n¼1 m¼0

Because the radial dependence of the internal and external fields is known, (Eq. 7) can be easily adapted for calculations using magnetic field measurements which are not made over the surface of a sphere, and where the coordinates of the point at which each measurement is made is given in terms of geocentric coordinates, r; y; f. From the equations for the horizontal components of the field, namely X and m m þ gne Þ and Y, it will be seen that the coefficients involve only ðgni m þ h Þ, so that from the X component only, it is possible to deterðhm ni ne mine the Y components. However, from these two elements only, it is not possible to separate the coefficients into internal and external parts, m m and gne separately. How- meaning it is not possible to determine g ni m m m ever, the coefficients
ðn þ 1Þgni þ ngne and
ðn þ 1Þhm ni þ n hne as determined from the radial or vertical component Z, will allow the separation of the coefficients and therefore allow separation of the field into parts of internal and external origin, respectively.

1 ]Vv ]Vv ey
ef sin y ]f ]y ¼
r  rVv ðr; y; fÞ; ¼ r  ½rVv ðr; y; fÞ;

n¼1 m¼0



dPnm ðcos yÞ ; dy

m m
ðn þ 1Þgni þ ngne cos mf



m m þ
ðn þ 1Þhm ni þ n hne sin mf Pn ðcos yÞ:

N X n X

m m m ðgni þ gne Þ sin mf
ðhm ni þ hne Þ cos mf

Y ða; y; fÞ ¼

N X n X

dPnm m m m ðgni þ gne Þ cos mf þ ðhm ni þ hne Þ sin mf dy n¼1 m¼0 m m m m P ; þ ðgnv sin mf
hnv cos mfÞ (Eq: 10) sin y n

4 X

½AYM ða; y; fÞ cos Mt þ BYM ða; y; fÞ sin Mt ;

M ¼1

Zða; y; f; tÞ ¼

4 X

½AZM ða; y; fÞ cos Mt þ BZM ða; y; fÞ sin Mt :

M ¼1

(Eq. 14) It is the standard practice to do separate spherical harmonic analyses of the A and B coefficients, to obtain internal and external coefficients mM mM mM ; hmM gnAi nAi and gnAe ; hnAe ;

mM mM mM gnBi ; hmM nBi and gnBe ; hnBe :

451

INTERNAL EXTERNAL FIELD SEPARATION

of the potential functions VAM ðr; y; fÞ and VBM ðr; y; fÞ, where N X n  nþ1  X a mM gnAi cos mf þ hmM nAi sin mf r n¼1 m¼0  r n  i m mM gnAe cos mf þ hmM þ nAe sin mf Pn ðcos yÞ; a (Eq. 15)

VAM ðr; y; fÞ ¼ a

P¼

N þ1 X n  X

m m am n cos mf þ bn sin mf Pn ðcos yÞ;

n¼0 m¼0

G¼

N þ1 X n  X

m m cm n cos mf þ dn sin mf Pn ðcos yÞ;

n¼0 m¼0

X¼

N þ1 X n  X

m m em n cos mf þ fn sin mf Pn ðcos yÞ:

(Eq: 20)

n¼0 m¼0 N X m  nþ1  X a

VBM ðr; y; fÞ ¼ a

m¼1 n¼0  r n 

þ

a

r

mM gnBi cos mf þ hmM nBi sin mf

i m mM gnBe cos mf þ hmM nBe sin mf Pn ðcos yÞ; (Eq. 16)

where only a limited range of m is used, for example, m ¼ M
1; M ; and M þ 1. However, if one were to include the following expression for VBM ðr; y; fÞ, which contains only the g and h coefficients appearing in VAM ðr; y; fÞ, along with the least-squares analysis of VAM ðr; y; fÞ, X X anþ1  mM hmM nAi cos mf
gnAi sin mf r n n  r n  i mM mM hnAe cos mf
gnAe sin mf Pnm ðcos yÞ; þ a (Eq. 17)

VBM ðr; y; fÞ ¼ a

then the combination VAM ðr; y; fÞ cos Mt þ VBM ðr; y; fÞ sin Mt N X n  nþ1 X

m a mM ¼a gnAi cosðmf þ Mt Þ þ hmM nAi sinðmf þ Mt Þ Pn ðcos yÞ r n¼1 m¼0 N X n  n X

m r mM gnAe cosðmf þ MtÞ þ hmM þa nAe sinðmf þ Mt Þ Pn ðcos yÞ: a n¼1 m¼0

(Eq. 18) consists only of terms with argument mf þ Mt , which are westward moving. In the special case that m ¼ M , the terms with argument mf þ Mt reduce to local time t *, and mt * ¼ mðt þ fÞ. The use of (Eqs. 15 and 17) together doubles the number of equations of condition for the westward moving terms. Because the magnetic variation “follows the Sun” and moves westward, the eastward moving terms are therefore much smaller, and in the first instance, can be regarded as random noise. Analysis of residuals after removing the westward moving part can be done by using an expression for
VBM ðr; y; fÞ in place of (Eq. 17) for VBM ðr; y; fÞ.

By forming specified orthogonal combinations of the computed coefficients, the internal, external, and nonpotential coefficients are determined independently, so that, for example, if one decided to have only internal terms in the model, there is no need to repeat the calculation (see Winch, 1968 for the necessary details).

Surface integral method Kahle and Vestine (1963) put forward a method of surface integrals for separating a potential function V ðy; fÞ, calculated as a set of numerical values over the globe, into its terms of internal and external origin. The method first requires the determination of the numerical values of V ðy; fÞ by integration of the given X values with respect to colatitude y and integration of Y values with respect to east longitude f. The reconciliation of values of potential from the two different elements is discussed by Price and Wilkins (1963), and the method applied to the analysis of Sq variations for the International Polar Year. Using the known potential function V, the internal and external components of V are determined from a form of Green’s third identity, Z 1 dS (Eq. 21) ðV þ 2aZ Þ : Vext
Vint ¼ 2p s r The separation into internal and external parts is done using the numerical arrays determined by (Eq. 21) and the known array V ðy; fÞ ¼ Vext ðy; fÞ þ Vint ðy; fÞ:

(Eq. 22)

Weaver (1964) describes a method for separating a local geomagnetic field into its external and internal parts using two-dimensional Fourier analyses of the X, Y, and Z components. The formulae for separating the X field are 1 Xe ¼ ðX þ M1 Z Þ; 2

1 Xi ¼ ðX
M1 Z Þ; 2

where M1 Z ¼

1 2p

Z

1

Z


1

1
1

Z ðu; vÞ h

x
u 2

ðx
uÞ þ ðy
vÞ2

i3=2 dudv:

Separation using Cartesian components

A simple method

For a Cartesian coordinate system at the center of the Earth, the x-axis directed toward the intersection of the equator and the Greenwich meridian, the y-axis at the intersection of the equator and the 90 E meridian, and the z-axis directed toward the North Pole. If P; G; X; denote field components parallel to the x-, y- and z- axes, then

In order to determine the direction and strength of the external field in magnetic daily variation studies, without the need for any analysis at all, the horizontal components can be rotated clockwise through 90 . For the Sq field, it can be shown that the method is accurate to within 10% of the maximum overhead current circulation, with largest errors occurring in middle and equatorial latitudes, e.g., Winch (1966). The method is useful for checking results of other more complicated forms of analysis and interpolation.

P ¼
Z sin y cos f
X cos y cos f
Y sin f; G ¼
Z sin y sin f
X cos y sin f þ Y cos f; X ¼
Z cos y þ X sin y:

Denis Winch

(Eq: 19)

The method requires the surface spherical harmonic analysis of the m m m m m Cartesian field elements, P; G; X. Coefficients am n ; bn ; cn ; dn ; en ; fn ; are determined using

Bibliography Gauss, C.F., 1838. Allgemeine Theorie des Erdmagnetismus, Resultate aus den Beobachtungen des magnetischen Vereins im Jahre 1838.

452

IONOSPHERE

(Reprinted in Werke, Bd. 5, 121, see also the translation by Mrs. Sabine and Sir John Herschel, Bart, Taylor, Richard, 1841. Scientific Memoirs Selected from the Transactions of Foreign Academies of Science and Learned Societies and from Foreign Journals, 2: 184–251.) Kahle, A.B., and Vestine, E.H., 1963. Analysis of surface magnetic fields by integrals. Journal of Geophysical Research, 68: 5505–5515. Price, A.T., and Wilkins, G.A., 1963. New methods for the analysis of geomagnetic fields and their application to the Sq field of 1932– 1933. Philosophical Transactions of the Royal Society of London, Series A, 256: 31–98. Weaver, J.T., 1964. On the separation of local geomagnetic fields into external and internal parts. Zeitschrift für Geophysik, 30: 29–36. Winch, D.E., 1966. The Sq overhead current system approximation. Planetary and Space Science, 14: 163–172. Winch, D.E., 1968. Analysis of the geomagnetic field by means of Cartesian components. Physics of the Earth and Planetary Interiors, 1: 347–360. Winch, D.E., Ivers, D.J., Turner, J.P.R., and Stening, R.J., 2005. Geomagnetism and Schmidt quasi-normalization. Geophysical Journal International, 160: 487–504.

Cross-references Harmonics, Spherical Harmonics, Spherical Cap

IONOSPHERE The ionosphere is the weakly ionized region of the upper atmosphere above 60 km altitude where free electrons and ions form a plasma that influences radio wave propagation and conducts electrical currents. The plasma is created by solar extreme-ultraviolet and X-ray radiation impinging on the atmosphere and at high latitudes in the northern and southern auroral ovals (q.v.), by energetic particles precipitating from the magnetosphere (q.v.). Plasma is destroyed through chemical reactions that lead to electron-ion recombination. The ionizing solar radiation varies by about a factor of two over the 11-year solar sunspot cycle, which results in significant solar-cycle variations in the plasma density. Figure I27 shows typical altitude profiles of the mid-latitude electron density at day and night for years of minimum and maximum

solar activity. For historical reasons relating to radio sounding of the ionosphere, the region of maximum density above 150 km is called the F-region (or layer), the region around the secondary maximum density at about 110 km is called the E region, and the region below about 90 km is called the D region. The ionosphere refracts, reflects, retards, scatters, and absorbs radio waves in a manner that depends on the radio wave frequency. Aroundthe-Earth communications are possible by utilizing ionospheric and ground reflections of waves at frequencies below about 3–30 MHz (100 to 10 m wavelength), depending on the peak electron density. However, frequent collisions between electrons and air molecules in the D region remove energy from the radio waves, leading to partial or even complete absorption, depending on the electron density. At higher frequencies, radio waves penetrate entirely through the ionosphere, allowing radio astronomy and communications with spacecraft. Nevertheless, such signals can still be degraded by refraction and scattering off of small-scale density irregularities. In the case of geolocation signals like those of the Global Positioning System (GPS), variable and uncertain ranging errors can be introduced by ionospheric signal retardation. The retardation is proportional to the total electron content (TEC), or height-integrated electron density. A typical global pattern of TEC is shown in Figure I28. The TEC tends to be larger in winter than in summer, because of slower chemical loss in winter. The ionospheric electrons and ions are strongly influenced by the geomagnetic field, and tend to move in spirals along geomagnetic-field lines. In the F-region, where collisions with air molecules are infrequent, the plasma diffuses rapidly along the magnetic field, but not across it. Winds in the upper atmosphere can also push the ions along the field lines. Plasma motion across magnetic field lines, on the other hand, is produced by electric fields. In the presence of an electric field E and a magnetic field B, electrons and ions drift perpendicular to both fields at the velocity E  B/B2. In the upper F-region, where ion lifetimes are long, the ions and electrons can be transported vertically and horizontally over significant distances between the times they are produced and lost, giving rise to a pattern of plasma density that depends on the geomagnetic-field configuration. As seen in Figure I28, a relative minimum in TEC tends to form along the magnetic equator in the afternoon and evening. It is created by an upward E  B/B2 plasma drift during the day, with subsequent rapid plasma diffusion along the magnetic field, driven downward and away from the magnetic equator by gravity and plasma pressure gradients. Maxima in

Figure I27 Altitude profiles of electron density at 18 N, 67 W, September equinox, representative of noon and midnight, solar minimum (solid lines) and solar maximum (dashed lines), with the F, E, and D regions indicated.

IONOSPHERE

453

Figure I28 Global map of TEC at 12 UT, December solstice, solar maximum. Local time increases with longitude as shown on the bottom scale. Contours are spaced at intervals of 10  1016 electrons per square metre. The thick solid line is the magnetic equator.

the F-region ion density build up about 1500 km on either side of the magnetic equator. The geomagnetic influence on ion and electron mobility makes the electrical conductivity of the ionosphere highly anisotropic. The relatively free motion of electrons along the magnetic-field direction gives rise to large conductivity in that direction. In fact, the large parallel conductivity extends far into space, out to many Earth radii, and permits electric currents to flow relatively easily along geomagnetic-field lines from one hemisphere of the Earth to the opposite hemisphere. In the direction perpendicular to the geomagnetic field, current can flow in response to an electric field only when the electrons and positive ions move at different velocities. Their common drift at the velocity E  B/B2 in the upper F-region does not produce current. However, at lower altitudes, ion-neutral collisions deflect the ion motion away from the velocity E  B/B2, while the electrons continue to drift at that velocity so that current flows perpendicular to the magnetic field. Part of this current, called Pedersen current, is in the direction of E, and is associated with a Pedersen conductivity. Another part of the current, called Hall current, flows in the B  E direction, opposite to the electron velocity, and is associated with a Hall conductivity. Figure I29 shows typical daytime altitude profiles of the parallel, Hall, and Pedersen conductivities. In the E-region the conductivities vary by about 40% over the solar cycle, but in the F-region the percentage variation is much greater. Unlike the TEC, which represents primarily F-region plasma, the height-integrated Pedersen and Hall conductivities vary mainly with the E-region plasma densities, which have short lifetimes and therefore depend directly on the solar and auroral ionization. They maximize during daylight, and at night in the auroral zone. The electric field that drives current in the conductive medium is the field present in the reference frame moving with the air, E þ v  B, where v is the wind velocity. Upper atmospheric winds drive electric current by moving the conductive medium through the geomagnetic field, thereby driving an ionospheric dynamo. Convergent or divergent wind-driven currents create space charge and an electrostatic field E that drives offsetting divergent or convergent current. The ionospheric dynamo is the main source of quiet-day ionospheric electric fields

Figure I29 Solid lines: altitude profiles of the noon-time parallel (s||), Pedersen (sP), and Hall (sH) conductivities at 18 N, 67 W, September equinox, solar minimum. Dashed line: Pedersen conductivity at solar maximum.

and currents, and of their associated geomagnetic daily variations, at middle and low latitudes (see Periodic external fields). At high magnetic latitudes, electric fields and currents connect along geomagnetic-field lines with sources in the outer magnetosphere. They intensify greatly during magnetic storms (see Storms and substorms), and can produce strong heating and dynamical changes in the upper atmosphere and ionosphere. For further information about the ionosphere and ionospheric processes, see Kelley (1989) and Schunk and Nagy (2000). Arthur D. Richmond

454

IRON SULFIDES

Bibliography Kelley, M.C., 1989. The Earth’s Ionosphere. San Diego: Academic Press. Schunk, R.W., and Nagy, A.F., 2000. Ionospheres: Physics, Plasma Physics and Chemistry, Cambridge University Press.

Cross-references Auroral Oval Magnetosphere Periodic External Fields Storms and Substorms, Magnetic

IRON SULFIDES General Iron sulfides are generally quoted as minor magnetic minerals and the interest of paleomagnetists for this family of minerals progressively developed only during the last 10–15 years. This was due partly to the fact that their occurrence was originally believed to be restricted to peculiar geological environments (i.e., sulfidic ores, anoxic sulfatereducing sedimentary environments) and partly to their metastability with respect to pyrite (FeS2), which is paramagnetic. Magnetic iron sulfides were therefore not expected to carry a stable remanent magnetization and to survive over long periods of geological time in sedimentary environments. However, their occurrence as main carriers of a remanent magnetization stable through geological times has been increasingly reported in recent years from a large variety of rock types, primarily as a result of more frequent application of magnetic methods to characterize the magnetic mineralogy in paleomagnetic samples. The recognition of the widespread occurrence of magnetic iron sulfides as stable carriers of natural remanent magnetizations (NRMs) in rocks propelled specific researches on their fundamental magnetic properties.

Pyrrhotite: magnetic properties Among magnetic iron sulfides, the pyrrhotite solid solution series (Fe1
xS; with x varying between 0 and 0.13) exhibits a wide range of magnetic behaviors. Stochiometric pyrrhotite FeS (troilite) shows

a hexagonal crystal structure based upon the NiAs structure, with alternating c-planes of Fe and S (Figure I30). In each Fe-layer the Fe2þ ions are ferromagnetically coupled, while neighboring Fe-layers are coupled antiferromagnetically to each other via intervening S2
ions. The nonstoichiometric pyrrhotites are cation deficient. The increase of Fe deficiency affects both the crystallographic and magnetic structures: ordering of Fe vacancies leads to an alternation of partially and fully filled Fe layers, the hexagonal structure distorts to monoclinic and the magnetic ordering turns from antiferromagnetic to ferrimagnetic. Pyrrhotite shows strong magnetocrystalline anisotropy, with the easy directions of magnetization confined in the crystallographic basal plane (the magnetic crystalline anisotropy constant K1 is positive and estimated at ca. 104 J m
3, see Dunlop and Özdemir, 1997). The crystallographic c-axis perpendicular to the basal plane is the axis of very hard magnetization. The more Fe-rich, hexagonal, pyrrhotites (F9S10, and possible F10S11, F11S12) are antiferromagnetic at room temperature and are characterized by a l transition at temperatures between ca. 180  C and 220  C, depending on composition, above which they exhibit ferrimagnetism. The l transition represents a change in the vacancyordering pattern of the crystal structure and is distinctive and diagnostic of hexagonal (Fe-rich) pyrrhotites. The Curie temperature for the different compositions of hexagonal pyrrhotites varies between 210  C and 270  C. The most Fe-deficient pyrrhotite (F7S8) is monoclinic and ferrimagnetic at room temperature, with a Curie temperature of ca. 325  C. It has no l transition, but it shows a diagnostic low-temperature transition in remanence and coercivity at 30–35 K (Rochette et al., 1990). Natural pyrrhotite usually occurs as a mixture of superstructures of monoclinic and hexagonal types, which results in an intermediate overall composition. Magnetic properties of pyrrhotite and their dependence from grain size and temperature have been investigated in detail in a number of specific studies (e.g., Clark, 1984; Dekkers, 1988, 1989, 1990; Menyeh and O’Reilly, 1991, 1995, 1996; Worm et al., 1993; O’Reilly et al., 2000). Monoclinic ferrimagnetic pyrrhotite (F7S8) has a saturation magnetization (Js) of 18 A m2 kg
1 at room temperature. The magnetic susceptibility of pyrrhotites is field independent in grains smaller than 30 mm, however for larger grains the magnetic susceptibility (w) and its field dependence increase with increasing grain size (1  10–5–7  10–5 m3 kg
1). Conversely, the coercivity parameters all increase with decreasing grain size. The room temperature coercive force (Hc) ranges from 10 to 70 kA m
1 (i.e., 12–88 mT for coercive force expressed as magnetic induction Bc)

Figure I30 Sketch of the crystalline structure of stoichiometric troilite (FeS). The structure is basically hexagonal, with alternating layers of Fe2þ and S2
ions. The c-axis of hexagonal symmetry is the axis of very hard magnetization and is perpendicular to the basal planes. Elemental magnetic moments are parallel within a particular cation basal plane. The alternating Fe2þ layers define the two magnetic sublattices with oppositely directed magnetic moments. In nonstoichiometric monoclinic pyrrhotite Fe7S8, the cation vacancies are preferentially located on one of the two magnetic sublattices, giving rise to ferrimagnetism.

IRON SULFIDES

455

and for synthetic powders with grain size in the range 1–24 mm, it could be fitted by a power-law dependence of the form Hc / Ln, where L is the particle size and n ¼
0.38 (O’Reilly et al., 2000). During hysteresis measurements and isothermal remanent magnetization (IRM) acquisition experiments pyrrhotite powders approach complete saturation only in fields greater than 1 T, with the magnetic hardness increasing with decreasing grain sizes (O’Reilly et al., 2000). In synthetic monoclinic pyrrhotites the critical size for the transition from the single-domain (SD) to the multidomain (MD) state, with typical lamellar domains normal to the c-axis, is estimated at a mean value of ca. 1 mm. The micromagnetic structures of hexagonal synthetic pyrrhotites are far more complex than that of monoclinic pyrrhotites, with typical wavy walls and a significantly smaller size of the individual magnetic domains (O’Reilly et al., 2000). The remanent magnetization carried by pyrrhotites can be of prime importance for paleomagnetic studies, but various factors may complicate the paleomagnetic interpretation of the data obtained from the classical demagnetization treatments. During alternating field (AF) demagnetization pyrrhotite-bearing samples were reported to acquire significant gyromagnetic and rotational remanent magnetization (RRM) in fields higher than 20 mT (Thomson, 1990) and upon heating at temperatures greater than 500  C during thermal demagnetization pyrrhotite transforms irreversibly to magnetite and, via subsequent further oxidation and heating, to hematite (Dekkers, 1990), producing new magnetic phases that may acquire new remanent magnetizations in the laboratory. Furthermore, self-reversal phenomena of pyrrhotite have been earlier reported under various laboratory experiments (Everitt, 1962; Bhimasankaram, 1964; Bhimasankaram and Lewis, 1966). Recent studies linked such phenomena to crystal twinning (Zapletal, 1992) and/or to a close coexistence of pyrrhotite and magnetite crystals, the latter being nucleated from direct oxidation of the pyrrhotite grains during heating (Bina and Daly, 1994).

Greigite: magnetic properties The iron sulfide greigite (Fe3S4) is the ferrimagnetic inverse thiospinel of iron and its crystalline structure is comparable to that of magnetite (Fe3O4) in which sulfur replace oxygen atoms. In the cubic closepacked crystal structure of greigite, the tetrahedral A-sites are filled by Fe3þ ions and the octahedral B-sites are filled half by Fe3þ ions and half by Fe2þ ions (Figure I31). Greigite is characterized by a high magnetocrystalline anisotropy (with a positive magnetic crystalline anisotropy constant K1 estimated at ca. 103 J m
3, see Diaz Ricci and Kirschvink, 1992; Dunlop and Özdemir, 1997) and its magnetic easy axis is aligned along the crystallographic direction. Basic magnetic properties at room temperature in synthetic greigite powders were systematically investigated by Dekkers and Schoonen (1996), indicating a minimum lower bound for the saturation magnetization (Js) of 29 A m2 kg
1, a magnetic susceptibility (w) between ca. 5  10–5 and 20  10–5 m3 kg
1 and intermediate coercivities (i.e., IRM acquisition curves saturate after application of fields 0.7–1 T). Greigite is unstable during heating to temperature higher than ca. 200  C. As a consequence of such instability the magnetic susceptibility and the magnetization in greigite-bearing sediments both undergo significant changes during heating (see Krs et al., 1990, 1992; Reynolds et al., 1994; Roberts, 1995; Horng et al., 1998; Sagnotti and Winkler, 1999; Dekkers et al., 2000): a major drop is generally observed between ca. 250  C and 350  C, reflecting decomposition of greigite in nonmagnetic sulfur, pyrite, and marcasite, followed by a dramatic increase above 350–380  C and a subsequent decrease above 400–450  C, indicating progressive production of new magnetic phases (pyrrhotite, then magnetite/maghemite, and finally hematite). Thermal decomposition of greigite precludes direct determination of its Curie temperature. Greigite does not show any low temperature (5–300 K) phase transition (Roberts, 1995; Torii et al., 1996; Dekkers et al., 2000), though

Figure I31 Sketch of the ½ of a unit cell of greigite (Fe3S4). The crystalline structure of greigite is basically the same inverse spinel structure typical for magnetite (Fe3O4) in which S2
ions substitute O2
ions. Cations are both in tetrahedral (A-site) and octahedral (B-site) coordination with S2
ions. In greigite the easy axis of magnetization is the crystallographic axis.

a broad Js maximum peak value was observed at 10 K during cooling from 300 to 4 K (Dekkers et al., 2000). Typical for greigite are: low to intermediate coercivities, with spectra that partly overlap those of magnetite and pyrrhotite (i.e., for natural greigite-bearing sediments and synthetics greigite the published values for the coercivity of remanence (Bcr) ranges from 20 to 100 mT, while for the coercivity (Bc) varies from 13 to 67 mT), maximum unblocking temperatures in the range 270–380  C, the lack of lowtemperature phase transitions and the presence of distinct stable SD properties (i.e., Roberts, 1995). With regards to these latter properties, in particular, greigite-bearing sediments show: (a) SD-like hysteresis ratios, with Mrs/Ms often exceeding 0.5 (where Mrs is the saturation remanent magnetization and Ms is the saturation magnetization) and Bcr/Bc often lower than 1.5, (b) high values of the SIRM/k ratio (where SIRM is the saturation IRM and k the low-field magnetic susceptibility), (c) a sensitivity to field impressed anisotropy and (d) a marked tendency for acquisition of gyromagnetic remanent magnetization (GRM) and RRM. GRM and RRM acquisition in greigite-bearing samples has been investigated in the detail in several specific studies (Snowball, 1997a,b; Hu et al., 1998; Sagnotti and Winkler, 1999; Stephenson and Snowball, 2001; Hu et al., 2002) that have shown that greigite has the highest effective gyrofield (Bg) reported so far for all magnetic minerals (of the order of several hundred microteslas for a peak AF of 80 mT) and that gyromagnetic effects are powerful indicators for the presence of greigite in sediments.

456

IRON SULFIDES

GRM and RRM are effects due to the application of an AF on SD grains. They are produced whenever there is an asymmetry in the number of magnetic moments that flip in a particular sense during the AF treatment. In other words, such remanences appear in any system where a particular sense of flip predominates (Stephenson, 1980a,b). RRM was explained in terms of a gyromagnetic effect associated with the irreversible flip of SD particles during rotation of the sample in an AF (Stephenson, 1985). The typical SD properties of natural greigite grains may be explained by its intrinsic magnetic and crystalline structure, with the magnetocrystalline anisotropy dominating the magnetization process. The theoretically estimated size range of stability for prismatic greigite SD grains extends well beyond that of magnetite, suggesting that elongated greigite crystals may be in a SD state even for very large sizes (up to several micrometers) (Diaz Ricci and Kirschvink, 1992). Moreover, direct magnetic optical observations indicated that the single- to two-domain transition in greigite may occur for grain sizes of 0.7–0.8 mm (Hoffmann, 1992) and that greigite usually occurs in framboidal aggregates of grains individually smaller than 1 mm (i.e., Jiang et al., 2001). The maximum value for the Mrs/Ms ratio in SD grains with shape anisotropy is 0.5. Conversely, if magnetocrystalline anisotropy controls the hysteresis behavior and the axis is the easy axis of magnetization, it would be expected that Mrs/Ms would approach a value of 0.832 as Bcr/Bc approach unity. Under the same circumstances if is the easy axis of magnetization Mrs/Ms is predicted to approach 0.866 (O’Reilly, 1984; Dunlop and Özdemir, 1997). A magnetic method for discriminating between greigite and pyrrhotite in paleomagnetic samples has been proposed by Torii et al. (1996), based on the thermal demagnetization of a composite IRM and relying upon the instability and alteration of greigite during thermal heating at temperatures above 200  C. Notwithstanding its metastable properties and magnetic instability, the importance of greigite for paleomagnetism and magnetostratigraphy has been particularly stressed in recent years since it has been widely recognized as a carrier of stable chemical remanent magnetization (CRM) in lacustrine and marine sediments, with ages ranging from the Cretaceous to the Present (e.g., Snowball and Thomson, 1990; Snowball, 1991; Hoffmann, 1992; Krs et al., 1990, 1992; Roberts and Turner, 1993; Hallam and Maher, 1994; Reynolds et al., 1994; Roberts, 1995; Roberts et al., 1996; Sagnotti and Winkler, 1999), as well as in soils (Fassbinder and Stanjek, 1994).

Pyrrhotite and greigite occur also as authigenic phases in geologically young or recent sediments deposited under anoxic, sulfatereducing conditions (Figure I32). The presence of magnetic iron sulfides in sediments is of special interest, since they have important implication for magnetostratigraphy and environmental magnetism. Reduction diagenesis and authigenesis can significantly affect the magnetic mineralogy of sediments (Karlin and Levi, 1983; Karlin, 1990a,b; Leslie et al., 1990a,b). In particular, magnetic iron sulfides can be produced through a number of different processes: (1) bacterially mediated synthesis of single-domain greigite, in the form of magnetosomes produced by magnetotactic bacteria (Mann et al., 1990; Bazylinski et al., 1993), and/or (2) precipitation from pore waters, and/or (3) dissolution of detrital iron oxides and subsequent precipitation of iron sulfides. Authigenic growth of iron sulfides is a common and relatively wellunderstood process in anoxic sedimentary environments with relatively high organic carbon contents (e.g., Berner, 1984; Leslie et al., 1990a,b; Roberts and Turner, 1993). Authigenic magnetic iron sulfides

Fe-Ni sulfides Smythite (Fe, Ni)9S11, pentlandite (Fe, Ni)9S8, and other complex Fe-Ni sulfides were occasionally reported in sediments (i.e., Krs et al., 1992; Van Velzen et al., 1993) in association with pure Fe sulfides, but their magnetic properties have not been studied in the detail so far.

Occurrences: formation, preservation, problems, and potential for use in paleomagnetism Troilite is common in meteorites and lunar rocks, but not on Earth. Pyrrhotites are ordinary magnetic carriers in magmatic, hydrothermal, and metamorphic rocks. Pyrrhotite in metamorphic rocks has been shown to acquire postmetamorphic partial thermomagnetic remanent magnetization (pTRM) during the uplift of mountain belts and has been used as a thermometer for postmetamorphic cooling (thermopaleomagnetism) and for dating and evaluation of exhumation rates through various reversals of the Earth magnetic field (i.e., Crouzet et al., 1999, 2001a,b). Pyrrhotite has also been recognized as the main remanence carrier in Shergotty-Nakhla-Cassigny type (SNC) Martian meteorites, and inferred as a main magnetic phase in the crust of Mars with significant implication for a proper evaluation of magnetic anomalies on such planet (Rochette et al., 2001).

Figure I32 (a) Backscattered electron micrograph of an iron sulfide nodule from the Valle Ricca section, Italy (see Florindo and Sagnotti, 1995). The nodule shows evidence for multiphase sulfidization and contains framboidal pyrite (Py) and greigite (G), which always occurs with finer crystal sizes than the co-occurring pyrite, and intergrown plates of hexagonal pyrrhotite (Pyrr). (Courtesy of Andrew P. Roberts and Wei-Teh Jiang.) (b) Backscattered electron micrograph of a Trubi marl sample from Sicily (see Dinare`s-Turell and Dekkers, 1999) showing framboidal pyrite (Py) filling foraminifera shells and pyrrhotite (Pyrr) intergrown plates dispersed in the rock matrix. (Courtesy of Jaume Dinare`s-Turell.)

IRON SULFIDES

(pyrrhotite and greigite) are intermediate mineral phases within the chemical iron reduction series that eventually forms stable paramagnetic pyrite (FeS2; Berner, 1969, 1970, 1984; Canfield and Berner, 1987). The process can be mediated by bacterial activity that reduces pore-water sulfate (SO42þ) to sulfide (S
) associated with consumption of organic carbon during burial. Hydrogen sulfide (H2S) then progressively reacts with detrital iron minerals to ultimately produce pyrite. Intermediate magnetic iron sulfides are metastable with respect to pyrite in the presence of excess H2S. The major factors controlling pyrite formation and the preservation of the (magnetic) intermediate phases in marine sediments are the amounts of dissolved sulfate, reactive iron detrital minerals, and decomposable organic matter (Berner, 1970, 1984). Intermediate magnetic iron sulfides may be preserved in all cases when the limited availability of sulfide, reactive iron and/ or organic matter prevents completion of the processes that result in formation of pyrite. In normal marine environments, sulfate and reactive detrital iron minerals are practically unlimited and the major controlling factor on the pyritization process appears to be the availability of detrital organic matter (Kao et al., 2004). In this case, the small amounts of sulfide produced will rapidly react with dissolved iron; the rapid consumption of sulfide means that formation of intermediate greigite is favored over pyrite. However, in some settings, organic carbon may have different sources and greigite has also been reported to form during later burial as a consequence of the diffusion of hydrocarbons and gas hydrates through permeable strata (Reynolds et al., 1994; Thompson and Cameron, 1995; Housen and Mousgrave, 1996). Greigite is the more common magnetic iron sulfide in fine-grained sediments; pyrrhotite, however, has also been reported in various fine-grained sediments, often alongside greigite (Linssen, 1988; Mary et al., 1993; Roberts and Turner, 1993; Dinarès-Turell and Dekkers, 1999; Horng et al., 1998; Sagnotti et al., 2001; Weaver et al., 2002) (Figure I32). The relative abundance of reactive iron versus organic matter appears to be the controlling factor for the transformation pathway of initial amorphous FeS into greigite or into pyrrhotite (Kao et al., 2004). Compared to greigite, pyrrhotite is favored by more reducing environments (i.e., lower Eh) and higher concentration of H2S, both implying a higher consumption of organic carbon (Kao et al., 2004). Even under appropriate diagenetic conditions, however, monoclinic pyrrhotite (Fe7S8) formation will be extremely slow below 180  C, which makes it a highly unlikely carrier of early diagenetic remanences in sediments (Horng and Roberts, 2006). The abundance of monoclinic pyrrhotite in regional metamorphic belts makes it a likely detrital rather than authigenic magnetic mineral in marginal basins in such settings (Horng and Roberts, 2006). During nucleation and crystal growth, authigenic greigite and pyrrhotite acquire a CRM, which contributes to the total NRM of the sediments. Authigenic formation of magnetic iron sulfides is generally believed to occur at the very first stages of diagenesis and NRM carried by magnetic iron sulfides were used for detailed paleomagnetic studies, assuming they reflect primary components of magnetization (e.g., Tric et al., 1991). However, many studies documented a significant delay between the deposition of the sediment and the formation of magnetic iron sulfides, implicating late diagenetic magnetizations. In particular, sediments bearing magnetic iron sulfides were often reported to carry CRM that are antiparallel either to those carried by coexisting detrital magnetic minerals and of opposite polarity with respect to the polarity expected for the age of the rock unit (e.g., Florindo and Sagnotti, 1995; Horng et al., 1998; Dinarès-Turell and Dekkers, 1999; Jiang et al., 2001; Weaver et al., 2002; Oms et al., 2003; Roberts and Weaver, 2005; Sagnotti et al., 2005). Such studies demonstrate that magnetic iron sulfides can carry stable magnetizations with a wide range of ages and that a syndepositional age should not be automatically assumed. Leonardo Sagnotti

457

Bibliography Bazylinski, D.A., Heywood, B.R., Mann, S., and Frankel, R.B., 1993. Fe3O4 and Fe3S4 in a bacterium. Nature, 366: 218. Berner, R.A., 1969. Migration of iron and sulfur within anaerobic sediments during early diagenesis. American Journal of Science, 267: 19–42. Berner, R.A., 1970. Sedimentary pyrite formation. American Journal of Science, 268: 1–23. Berner, R.A., 1984. Sedimentary pyrite formation: an update. Geochimica et Cosmochimica Acta, 48: 605–615. Bhimasankaram, V.L.S., 1964. Partial magnetic self-reversal of pyrrhotite. Nature, 202: 478–480. Bhimasankaram, V.L.S., and Lewis, M., 1966. Magnetic reversal phenomena in pyrrhotite. Geophysical Journal of the Royal Astronomical Society, 11: 485–497. Bina, M., and Daly, L., 1994. Mineralogical change and self-reversal magnetizations in pyrrhotite resulting from partial oxidation; geophysical implications. Physics of the Earth and Planetary Interiors, 85: 83–99. Canfield, D.E., and Berner, R.A., 1987. Dissolution and pyritization of magnetite in anoxic marine sediments. Geochimica et Cosmochimica Acta, 51: 645–659. Clark, D.A., 1984. Hysteresis properties of sized dispersed monoclinic pyrrhotite grains. Geophysical Research Letters, 11: 173–176. Crouzet, C., Ménard, G., and Rochette, P., 1999. High-precision threedimensional paleothermometry derived from paleomagnetic data in an alpine metamorphic unit. Geology, 27: 503–506. Crouzet, C., Rochette, P., and Ménard, G., 2001a. Experimental evaluation of thermal recording of successive polarities during metasediments uplift. Geophysical Journal International, 145: 771–785. Crouzet, C, Stang, H., Appel, E., Schill, E., and Gautam, P., 2001b. Detailed analysis of successive pTRMs carried by pyrrhotite in Himalayan metacarbonates: an example from Hidden Valley, Central Nepal. Geophysical Journal International, 146: 607–618. Dekkers, M.J., 1988. Magnetic properties of natural pyrrhotite. I. Behavior of initial susceptibility and saturation-magnetisationrelated rock-magnetic parameters in a grain-size dependent framework. Physics of the Earth and Planetary Interiors, 52: 376–394. Dekkers, M.J., 1989. Magnetic properties of natural pyrrhotite. II. High- and low-temperature behavior of Jrs and TRM as a function of grain size. Physics of the Earth and Planetary Interiors, 57: 266–283. Dekkers, M.J., 1990. Magnetic monitoring of pyrrhotite alteration during thermal demagnetization. Geophysical Research Letters, 17(6): 779–782. Dekkers, M.J., and Schoonen, M.A.A., 1996. Magnetic properties of hydrothermally synthesized greigite (Fe3S4). I. Rock magnetic parameters at room temperature. Geophysical Journal International, 126: 360–368. Dekkers, M.J., Passier, H.F., and Schoonen, M.A.A., 2000. Magnetic properties of hydrothermally synthesized greigite (Fe3S4). II. Highand low-temperature characteristics. Geophysical Journal International, 141: 809–819. Diaz Ricci, J. C., and Kirschvink, J.L., 1992. Magnetic domain state and coercivity prediction for biogenic greigite (Fe3S4): a comparison of theory with magnetosome observations. Journal of Geophysical Research, 97: 17309–17315. Dinarès-Turell, J., and Dekkers, M.J., 1999. Diagenesis and remanence acquisition in the Lower Pliocene Trubi marls at Punta di Maiata (southern Sicily): palaeomagnetic and rock magnetic observations. In Tarling, D.H., and Turner, P. (eds.), Palaeomagnetism and Diagenesis in Sediments. Geological Society of London, Special Publication, 151: 53–69. Dunlop, D.J., and Özdemir, Ö., 1997. Rock Magnetism: Fundamentals and Frontiers. New York: Cambridge University Press.

458

IRON SULFIDES

Everitt, C.W.F., 1962. Self-reversal of magnetization in a shale containing pyrrhotite. Philosophical Magazine Letters, 7: 831–842. Fassbinder, J.W.E., and Stanjek, H., 1994. Magnetic properties of biogenic soil greigite (Fe3S4). Geophysical Research Letters, 21: 2349–2352. Florindo, F., and Sagnotti, L., 1995. Palaeomagnetism and rock magnetism in the upper Pliocene Valle Ricca (Rome, Italy) section. Geophysical Journal International, 123: 340–354. Hallam, D.F., and Maher, B.A., 1994. A record of reversed polarity carried by the iron sulphide greigite in British early Pleistocene sediments. Earth and Planetary Science Letters, 121: 71–80. Hoffmann, V., 1992. Greigite (Fe3S4): magnetic properties and first domain observations. Physics of the Earth and Planetary Interiors, 70: 288–301. Horng, C.-S., and Roberts, A.P., 2006. Authigenic or detrital origin of pyrrhotite in sediments? Resolving a paleomagnetic conundrum. Earth and Planetary Science Letters, 241: 750–762. Horng, C.-S., Torii, M., Shea, K.-S., and Kao, S.-J., 1998. Inconsistent magnetic polarities between greigite- and pyrrhotite/magnetitebearing marine sediments from the Tsailiao-chi section, southwestern Taiwan. Earth and Planetary Science Letters, 164: 467–481. Housen, B.A., and Musgrave, R.J., 1996. Rock-magnetic signature of gas hydrates in accretionary prism sediments. Earth and Planetary Science Letters, 139: 509–519. Hu, S., Appel, E., Hoffmann, V., Schmahl, W.W., and Wang, S., 1998. Gyromagnetic remanence acquired by greigite (Fe3S4) during static three-axis alternating field demagnetization. Geophysical Journal International, 134: 831–842. Hu, S., Stephenson, A., and Appel, E., 2002. A study of gyroremanent magnetisation (GRM) and rotational remanent magnetisation (RRM) carried by greigite from lake sediments. Geophysical Journal International, 151: 469–474. Jiang, W.T., Horng, C.S., Roberts, A.P., and Peacor, D.R., 2001. Contradictory magnetic polarities in sediments and variable timing of neoformation of authigenic greigite. Earth and Planetary Science Letters, 193: 1–12. Kao, S.J., Horng, C.S., Liu, K.K., and Roberts, A.P., 2004. Carbonsulfur-iron relationships in sedimentary rocks from southwestern Taiwan: influence of geochemical environment on greigite and pyrrhotite formation. Chemical Geology, 203: 153–168. Karlin, R., 1990a. Magnetic mineral diagenesis in suboxic sediments at Bettis site WN, NE Pacific Ocean. Journal of Geophysical Research, 95: 4421–4436. Karlin, R., 1990b. Magnetite diagenesis in marine sediments from the Oregon continental margin. Journal of Geophysical Research, 95: 4405–4419. Karlin, R., and Levi, S., 1983. Diagenesis of magnetic minerals in recent hemipelagic sediments. Nature, 303: 327–330. Krs, M., Krsova, M., Pruner, P., Zeman, A., Novak, F., and Jansa, J., 1990. A petromagnetic study of Miocene rocks bearing microorganic material and the magnetic mineral greigite (Solokov and Cheb basins, Czechoslovakia). Physics of the Earth and Planetary Interiors, 63: 98–112. Krs, M., Novak, F., Krsova, M., Pruner, P., Kouklíková, L., and Jansa, J., 1992. Magnetic properties and metastability of greigitesmythite mineralization in brown-coal basins of the Krusné hory Piedmont, Bohemia. Physics of the Earth and Planetary Interiors, 70: 273–287. Leslie, B.W., Hammond, D.E., Berelson, W.M., and Lund, S.P., 1990a. Diagenesis in anoxic sediments from the California continental borderland and its influence on iron, sulfur, and magnetite behavior. Journal of Geophysical Research, 95: 4453–4470. Leslie, B.W., Lund, S.P., and Hammond, D.E., 1990b. Rock magnetic evidence for the dissolution and authigenic growth of magnetic minerals within anoxic marine sediments of the California continental borderland. Journal of Geophysical Research, 95: 4437–4452.

Linssen, J.H., 1988. Preliminary results of a study of four successive sedimentary geomagnetic reversal records from the Mediterranean (Upper Thvera, Lower and Upper Sidufjall, and Lower Nunivak). Physics of the Earth and Planetary Interiors, 52: 207–231. Mann, S., Sparks, N.H.C., Frankel, R.B., Bazylinski, B.A., and Jannasch, H.W., 1990. Biomineralization of ferrimagnetic greigite (Fe3S4) and iron pyrite (FeS2) in a magnetotactic bacterium. Nature, 343: 258–261. Mary, C., Iaccarino, S., Courtillot, V., Besse, J., and Aissaoui, D.M., 1993. Magnetostratigraphy of Pliocene sediments from the Stirone River (Po Valley). Geophysical Journal International, 112: 359–380. Menyeh, A., and O’Reilly W., 1991. The magnetization process in monoclinic pyrrhotite (Fe7S8) particles containing few domains. Geophysical Journal International, 104: 387–399. Menyeh, A., and O’Reilly, W., 1995. The coercive force of fine particles of monoclinic pyrrhotite (Fe7S8) studied at elevated temperature. Physics of the Earth and Planetary Interiors, 89: 51–62. Menyeh, A., and O’Reilly, W., 1996. Thermoremanent magnetization in monodomain pyrrhotite Fe7S8. Journal of Geophysical Research, 101: 25045–25052. Oms, O., Dinarès-Turell, J., and Remacha, E., 2003. Paleomagnetic results on clastic turbidite systems in compressional settings: example from the Eocene Hecho Group (Southern Pyrenees, Spain). Studia Geophysica et Geodaetica, 47: 275–288. O’Reilly, W., 1984. Rock and Mineral Magnetism. Glasgow: Blackie. O’Reilly, W., Hoffmann, V., Chouker, A.C., Soffel, H.C., and Menyeh, A., 2000. Magnetic properties of synthetic analogues of pyrrhotite ore in the grain size range 1–24 mm. Geophysical Journal International, 142: 669–683. Reynolds, R.L., Tuttle, M.L., Rice, C.A., Fishman, N.S., Karachewski, J.A., and Sherman, D.M., 1994. Magnetization and geochemistry of greigite-bearing Cretaceous strata, North Slope Basin, Alaska. American Journal of Science, 294: 485–528. Roberts, A.P., 1995. Magnetic properties of sedimentary greigite (Fe3S4). Earth and Planetary Science Letters, 134: 227–236. Roberts, A.P., and Turner, G.M., 1993. Diagenetic formation of ferrimagnetic iron sulphide minerals in rapidly deposited marine sediments, South Island, New Zealand. Earth and Planetary Science Letters, 115: 257–273. Roberts, A.P., and Weaver, R., 2005. Multiple mechanisms of remagnetization involving sedimentary greigite (Fe3S4). Earth and Planetary Science Letters, 231: 263–277. Roberts, A.P., Reynolds, R.L., Verosub, K.L., and Adam, D.P., 1996. Environmental magnetic implications of greigite (Fe3S4) formation in a 3 Ma lake sediment record from Butte Valley, northern California. Geophysical Research Letters, 23: 2859–2862. Rochette, P., Fillion, G., Matteí, J.L., and Dekkers M.J., 1990. Magnetic transition at 30–34 K in Fe7S8: insight into a widespread occurrence of pyrrhotite in rocks. Earth and Planetary Science Letters, 98: 319–328. Rochette, P., Lorand, J.P., Fillion, G., and Sautter, V., 2001. Pyrrhotite and the remanent magnetization of SNC meteorites: a changing perspective on Martian magnetism. Earth and Planetary Science Letters, 190: 1–12. Sagnotti, L., and Winkler, A., 1999. Rock magnetism and palaeomagnetism of greigite-bearing mudstones in the Italian peninsula. Earth and Planetary Science Letters, 165: 67–80. Sagnotti, L., Macrí, P., Camerlenghi, A., and Rebesco, M., 2001. Environmental magnetism of Antarctic Late Pleistocene sediments and interhemispheric correlation of climatic events. Earth and Planetary Science Letters, 192: 65–80. Sagnotti, L., Roberts, A.P., Weaver, R., Verosub, K.L., Florindo, F., Pike, C.R., Clayton, T., and Wilson, G.S., 2005. Apparent magnetic polarity reversals due to remagnetization resulting from late diagenetic growth of greigite from siderite. Geophysical Journal International, 160: 89–100.

IRON SULFIDES

Snowball, I.F., 1991. Magnetic hysteresis properties of greigite (Fe3S4) and a new occurrence in Holocene sediments from Swedish Lapland. Physics of the Earth and Planetary Interiors, 68: 32–40. Snowball, I.F., 1997a. Gyroremanent magnetization and the magnetic properties of greigite-bearing clays in southern Sweden. Geophysical Journal International, 129: 624–636. Snowball, I.F., 1997b. The detection of single-domain greigite (Fe3S4) using rotational remanent magnetization (RRM) and the effective gyro field (Bg): mineral magnetic and palaeomagnetic applications. Geophysical Journal International, 130: 704–716. Snowball, I.F., and Thompson, R., 1990. A stable chemical remanence in Holocene sediments. Journal of Geophysical Research, 95: 4471–4479. Stephenson, A., 1980a. Gyromagnetism and the remanence acquired by a rotating rock in an alternating field. Nature, 284: 48–49. Stephenson, A., 1980b. A gyroremanent magnetization in anisotropic magnetic material. Nature, 284: 49–51. Stephenson, A., 1985. The angular dependence of rotational and anhysteretic remanent magnetization in rotating rock samples. Geophysical Journal of the Royal Astronomical Society, 83: 787–796. Stephenson, A., and Snowball, I.F., 2001. A large gyromagnetic effect in greigite. Geophysical Journal International, 145: 570–575. Thomson, G.F., 1990. The anomalous demagnetization of pyrrhotite. Geophysical Journal International, 103: 425–430. Thompson, R., and Cameron, T.D.J., 1995. Palaeomagnetic study of Cenozoic sediments in North Sea boreholes: an example of a magnetostratigraphic conundrum in a hydrocarbon producing area.

459

In Turner P., and Turner, A. (eds.), Palaeomagnetic Applications in Hydrocarbon Exploration and Production. Geological Society of London, Special Publication, 98: 223–236. Torii, M., Fukuma, K., Horng, C.S., and Lee, T.Q., 1996. Magnetic discrimination of pyrrhotite- and greigite-bearing sediment samples. Geophysical Research Letters, 23: 1813–1816. Tric, E., Laj, C., Jehanno, C., Valet, J.P., Kissel, C., Mazaud, A., and Iaccarino, S., 1991. High-resolution record of the upper Olduvai polarity transition from Po Valley (Italy) sediments: support for dipolar transition geometry? Physics of the Earth and Planetary Interiors, 65: 319–336. Van Velzen, A.J., Dekkers, M.J., and Zijderveld, J.D.A., 1993. Magnetic iron-nickel sulphides in the Pliocene and Pleistocene marine marls from the Vrica section (Calabria, Italy). Earth and Planetary Science Letters, 115: 43–55. Weaver, R., Roberts, A.P., and Barker, A.J., 2002. A late diagenetic (synfolding) magnetization carried by pyrrhotite: implications for paleomagnetic studies from magnetic iron sulphide-bearing sediments. Earth and Planetary Science Letters, 200: 371–386. Zapletal, K., 1992. Self-reversal of isothermal remanent magnetization in a pyrrhotite (Fe7S8) crystal. Physics of the Earth and Planetary Interiors, 70: 302–311.

Cross-references Demagnetization Magnetization, Chemical Remanent (CRM)

J

JESUITS, ROLE IN GEOMAGNETISM The Jesuits are members of a religious order of the Catholic Church, the Society of Jesus, founded in 1540 by Ignatius of Loyola. From 1548, when Jesuits established their first college, their educational work expanded rapidly and in the 18th century, in Europe alone, there were 645 colleges and universities and others in Asia and America. As an innovation in these colleges, special attention was given to teaching of mathematics, astronomy, and the natural sciences. This tradition has been continued in modern times in the many Jesuit colleges and universities and this tradition thus spread throughout the world. Jesuits’ interest in geomagnetism derived from teaching in these colleges and universities. In many of these colleges and universities observatories were established, where astronomical and geophysical observations were made (Udías, 2003). Missionary work in Asia, Africa, and America, where scientific observations were also made and some observatories established, was another factor in the role of Jesuits in geomagnetism. In this work we must distinguish two periods. The first, from 1540 to 1773, ended with the suppression of the Jesuit order. The second began in 1814 with its restoration and lasts to our times.

Early work on magnetism Terrestrial magnetism attracted Jesuits’ attention from very early times (Vregille, 1905). José de Acosta, a missionary in America, in 1590 described the variation of the magnetic declination across the Atlantic and the places where its value was zero, one of them at the Azores islands. His work is quoted by William Gilbert (q.v.). In 1629, Nicolò Cabbeo (1586–1650) was the first Jesuit to write a book dedicated to magnetism, Philosophia Magnetica, where he collected all that was known in his time, together with his own experiments and observations. Some of his work is based on that of a previous Jesuit Leonardo Garzoni (1567–1592), professor at the college of Venice, who could well be the first Jesuit interested in such matters. Cabbeo was opposed to Gilbert on the origin of terrestrial magnetism. The best-known early work on magnetism by a Jesuit is that of Athanasius Kircher (1601–1680) (q.v.), Magnes sive de arte magnetica, published in 1641. Martin Martini (1614–1661), who sent many magnetic observations from China to Kircher, proposed to him in 1640 to draw a map with lines for the magnetic declination. Had his suggestion been followed this would have been the first magnetic chart (before that of Edmund Halley (q.v.)). A curious work is that of Jacques Grandami (1588–1672), Nova demonstratio inmobilitatis terrae petita ex virtute magnetica (1645),

in which, in order to defend the geocentric system, he tried to show that the Earth does not rotate because of its magnetic field. Among the best observations made in China are those of Antoine Gaubil (1689–1759), who mentioned that the line of zero declination has with time a movement from east to west. His observations and those of other Jesuits in China were published in France in three volumes between 1729 and 1732. In 1727, Nicolas Sarrabat (1698–1739) published Nouvelle hypothèse sur les variations de l’aiguille aimantée, which was given an award by the Académie des Sciences of Paris. In 1769, Maximilian Hell (1720–1792), director of the observatory in Vienna, made observations of the magnetic declination during his journey to the island of Vardö in Lapland, at a latitude of 70 N, where he observed the transit of Venus over the solar disk.

Magnetic observatories The Jesuit order was suppressed in 1773 and restored in 1814. From this time work on geomagnetism was taken up again at the new Jesuit observatories (Udías, 2000, 2003). A total of 72 observatories were founded throughout the world. Magnetic stations were installed in 15 of them: five in Europe, one in North America, four in Central and South America, and five in Asia, Africa, and the Middle East. Some of those magnetic stations in Europe were among the first to be in operation. Observatories in Central and South America, Asia, and Africa provided for some time the only magnetic observations in those regions. Details of these observatories can be found in Udías (2003). At present, most of them have been either closed or transferred to other administration. The first of these observatories was established in 1824 at the Collegio Romano (Rome). There, in 1858, Angelo Secchi (1818– 1878) began magnetic observations, using a set of magnetometers, a declinometer, and an inclinometer. He studied the characteristics of the periodic variations of the different components of the magnetic field and tried to relate magnetic variations with solar activity, considering the Sun to be a giant magnet at a great distance. Relations between geomagnetism and solar activity were to become a favorite subject among Jesuits. In the same year, 1858, magnetic observations begun at Stonyhurst College Observatory (Great Britain). Stephen J. Perry (1833–1889, Figure J1) began his work on geomagnetism, carrying out three magnetic surveys: two in France in 1868 and 1869 and the third in Belgium in 1871 (Cortie, 1890). In each of these surveys, at each station, careful measurements were made of the horizontal component of the magnetic field, magnetic declination, and inclination or dip (Perry, 1870). In Belgium, Perry found large magnetic anomalies related to coal mines. In order to study the relation

JESUITS, ROLE IN GEOMAGNETISM

461

Figure J2 Ebro Observatory, Roquetas, Tarragona, Spain.

Figure J1 Stephen J. Perry (1833–1889), Director of Stonyhurst Observatory.

between solar and terrestrial magnetic activity, which was still a controversial subject, Perry at Stonyhurst began a series of observations of sunspots, faculae, and prominences in 1881. For this purpose he installed direct-vision spectroscopes and photographic-grating spectrometers and made large drawings of the solar disk (27 cm diameter). Perry collaborated with Edward Sabine (q.v.) in this work. Perry participated in several scientific expeditions. The most important was to Kerguelen Island in 1874 to observe the transit of Venus; there he carried out a very comprehensive program of magnetic observations. His project of collecting and comparing all his magnetic and solar observations was never completed due to his untimely death during a scientific expedition to the Lesser Antilles to observe a solar eclipse. In 1874 he was elected a Fellow of the Royal Society for his work in terrestrial magnetism. Perry’s successor Walter Sidgreaves (1837–1919) completed the work and showed the correlation between magnetic storms and the maxima of sunspots (Sidgreaves, 1899– 1901). The continuous magnetic observations from 1858 to 1974 at Stonyhurst may be one of the longest series at the same site. Solar-terrestrial relations were also the main subject of Haynald Observatory, founded in 1879 by Jesuits in Kalocsa (Hungary). Between 1885 and 1917, Gyula Fényi (1845–1927) carried out a long program of magnetic and solar observations. He concentrated his efforts on the study of sunspots and prominences, making detailed observations (about 40000 of them) and proposing some interesting, for that time, new ideas about their nature. Magnetic observations began in 1879. The Manila Observatory (Philippines) was founded by Spanish Jesuits in 1865. Martín Juan (1850–1888) was trained in geomagnetism by Perry in Stonyhurst. Juan brought new magnetic instruments to Manila where he took charge of the magnetic section in 1887. In 1888, he carried out a magnetic field survey on various islands of the archipelago. His death did not allow him to finish the work; this was done in 1893 by Ricardo Cirera (1864–1932), who extended the survey to the coasts of China and Japan (Cirera, 1893). In the observatory of Zikawei, founded in 1872 near Shanghai, magnetic instruments for absolute measurements and variations were installed in 1877; instruments were moved to two nearby sites, first Lukiapang in 1908 and then Zose in 1933. Results were published in

nine volumes between 1908 and 1932 (Études sur le magnetisme terrestre, pp. 1–39). Joseph de Moidrey (1858–1936) carried out some early work on the secular variation in the Far East region (de Moidrey and Lou, 1932). These observations continued until 1950 when Jesuits were expelled from China by the Communist government. After returning to Spain, Cirera founded in 1904 the Observatorio del Ebro, Roquetas (Tarragona; Figure J2), dedicated specifically to the study of the relations of solar activity and terrestrial magnetism and electricity. The observatory was equipped from the beginning with the most up-to-date instrumentation for this purpose and has kept updating its instrumentation since then. Luis Rodés (1881–1939) studied the influence of various forms of solar activity, mainly sunspots and prominences on the terrestrial magnetic and electric fields (Rodés, 1927). After the Spanish civil war, work was resumed by Antonio Romañá (1900–1981) and José O. Cardús (1914–). Since 1958, Ebro has been the base of the International Association of Geomagnetism and Aeronomy (IAGA) (q.v.) Commission for rapid magnetic variations. Romañá and Cardús held office in IAGA for many years. Maintaining magnetic observatories in Africa was a difficult task, which Jesuits undertook early. One of the earliest observatories was established in 1889 in Antananarive, Madagascar. Continuous magnetic observations were made until 1967 when the observatory was transferred to the University of Madagascar. A magnetic station was established in the observatory founded in 1903 in Bulawayo, Zimbabwe. Edmund Goetz (1865–1933), its director for 23 years, in 1909 and 1914 carried out two magnetic field surveys, the first with a profile from Broken Hill (Kabwe, Zambia) to the “Star of the Congo Mine” (Congo) and the second in Barotseland (Zambia) covering a distance of more than 200 miles from Kazungula to Lealui (Goetz, 1920). The third observatory was a state observatory entrusted to and directed by Jesuits in Ethiopia. This observatory arose from a recommendation of the Scientific Committee of the International Geophysical Year in 1955 for a magnetic station near the magnetic equator. The station was established in Addis Ababa in 1958 and directed by Pierre Gouin (1917–) for 20 years. Gouin with the collaboration of Pierre Noel Mayaud (1923–), a Jesuit working at the Institute de Physique du Globe in Paris, studied temporal magnetic variations and magnetic storms. Mayaud participated in French expeditions to the Antarctica, studied the magnetic activity in the polar regions and worked mainly on the nature and practical use of geomagnetic indices (Mayaud, 1980). Early magnetic observations were also made by Jesuits in Central and South America. These observations began in 1862 in Belen Observatory (Havana, Cuba) and continued to about 1920. In Puebla (Mexico), magnetic observations were made between 1877 and 1914.

462

JESUITS, ROLE IN GEOMAGNETISM

Summary As an important part of their scientific tradition Jesuits dedicated much of their efforts to the study of terrestrial magnetism. Jesuits’ dedication to science can be explained by their peculiar spirituality, which unites prayer and work and finds no activity too secular be turned into prayer. Teaching mathematics and observing the magnetic field of the Earth or of the Sun are activities, as has been shown, that a Jesuit finds perfectly compatible with his religious vocation and through which he can find God in his life. Early work on magnetism was carried out in the 17th and 18th centuries. Jesuit missionaries during those centuries carried out magnetic observations, which were analyzed and published in Europe. Modern work was done through the establishment of observatories. In at least 15 of these observatories magnetic observations were made. Jesuits contributed to the earliest magnetic observations in Africa, Asia, and Central and South America. Agustín Udías

Bibliography Cirera, R., 1893. El magnetismo terrestre en Filipinas. Manila: Observatorio de Manila, 160 pp. Cortie, A.L., 1890. Father Perry, F.R.S.: The Jesuit Astronomer. London: Catholic Truth Society, 113 pp. Goetz, E., 1920. Magnetic observations in Rhodesia. Transactions of the Royal Society of South Africa, 8(4): 297–302. Mayaud, P.N., 1980. Derivation, Meaning, and Use of Geomagnetic Indices. Geophysical Monograph 22. Washington, D.C.: American Geophysical Union.

Moidrey, J. de and Lou, F. 1932. Variation séculaire des éléments magnétiques en extreme orient. Études sur le magnetisme terrestre 39, Observatoire de Zikawei, 9: 1–21. Perry, S., 1870. Magnetic survey of the west of France in 1868. Philosophical Transactions Royal Society of London, 160: 33–50. Rodés, L., 1927. Some new remarks on the cause and propagation of magnetic storms. Terrestrial Magnetism and Atmospheric Electricity, 32: 127–131. Sidgreaves, W., 1899–1901. On the connection between solar spots and Earth magnetic storms. Memoirs of the Royal Astronomical Society, 54: 85–96. Udías, A., 2000. Observatories of the Society of Jesus, 1814–1998. Archivum Historicum Societatis Iesu, 69: 151–178. Udías, A., 2003. Searching the Heavens and the Earth: The History of Jesuit Observatories. Dordrecht: Kluwer Academic, 369 pp. Vregille, P. de, 1905. Les jésuites et l’étude du magnétisme terrestre. Études, 104: 492–511.

Cross-references Geomagnetism, History of Gilbert, William (1544–1603) Halley, Edmond (1656–1742) IAGA, International Association of Geomagnetism and Aeronomy Kircher, Athanasius (1602–1680) Observatories, Overview Sabine, Edward (1788–1883)

K

KIRCHER, ATHANASIUS (1602–1680) Athanasius Kircher was born in Geisa near Fulda (Germany) on May 2, 1601 (or 1602—there is a disagreement in his biographies). After his primary education at the Jesuit school in Fulda, he joined (October 1618) the Jesuit order in Paderborn, where he began to study Latin and Greek. As a consequence of the Thirty Years War he went to Münster and Cologne, where he studied philosophy. He taught for a short time at the secondary school in Coblenz and afterward in Heiligenstadt. Later he went to Mainz to study theology, where he was ordained priest in 1628 and at the end of his Jesuit training was appointed professor of ethics and mathematics at the Jesuit University in Würzburg. He again migrated as a result of the war and went to Avignon (France) and was finally appointed professor of mathematics and oriental languages at the Roman College, which is now the well-known Gregorian University in Rome, where he died on November 27, 1680. One cannot use modern standards of scientific evaluation to understand his real value, as Kircher’s thinking and his methods are typical for the 17th century. First of all, he was a polymath interested in almost every kind of knowledge. His books (over 30 printed volumes) range from oriental languages—Arabic grammar; translation of old Coptic books and an extensive research to decipher Egyptian hieroglyphics: Lingua Egyptiaca restituta (1643), Oedypus Egyptiacus (1652), Obeliscus Pamphilius (1650)—to physics—optics: Ars Magna Lucis et Umbra (1646); magnetism: Ars Magnesia (1631), Magnes, sive de Arte Magnetica (1641, 1643, 1654), Magneticum Naturae Rerum (1667); music: Musurgia Universalis sive Ars Magna consoni and disoni (1650), Phonurgia Nova (1673); geophysics: Mundus Subterraneus (1664); and also medicine: Scrutinium physico-medicum contagiosae luis quae pestis dicitur (1657). He also wrote on biblical subjects such as Arca Noe (1675) and Turris Baabel (1679), in which he combines science and archeological imagination. This imagination together with his knowledge of many different subjects inclined him to encyclopedism as in his works Itinerarium Extaticum (1656) and Iter Extaticum II (1657), in which he discusses theories of the solar system and beyond while telling the fictitious story of an imaginary journey through Earth and celestial bodies. Kircher tries to find some sort of unity among the many different subjects with which he dealt, not only in written language—Polygraphia nova universalis (1663)—but also in the much more universal concept of knowledge itself as in Ars Magna Sciendi (1669). Although mathematical combinations in Kircher’s book are far removed from the cabalistic usage, which was so frequent in his time, there is a

misleading use of the word “magnetism” in his writings: for him magnetism with its two polarities is a metaphorical symbol of the dualism in things and forces acting in nature: attraction and repulsion; love and hatred; and light and darkness. This use allows him to talk about the magnetism of celestial bodies, music, medicine, love, and to print, perhaps for the first time in a scientific book, the word electromagnetism (Kircher, 1654, p. 451), although this word has nothing to do with the concept introduced many years afterward by Maxwell. He was certainly dominated by Baroque culture, redundant in many aspects of his expression, overlengthy in his writings, and not easy to read in his Latin, which is far removed from the classical language. That Kircher was accepted during his lifetime as one of the best scholars in Europe is proven by the fact that the Imperial Court in Vienna invited Kircher to be professor of astronomy in the University of Vienna as successor of Kepler, who died in 1632. It was only by the influence of Cardinal Barberini and of Pope Urban VIII that he was prevented from accepting the post, and was called to Rome. They accepted the request of the French gentleman Nicolaus Claude de Peiresc, who wanted Kircher to continue his work on the interpretation of Egyptian hieroglyphics. Since his death importance of Kircher as a scientist has been much debated although now his value seems to be accepted again. Of the three books that Kircher wrote on magnetism the first Ars Magnesia (his first printed book) is a short booklet of 48 pages and seems to be the printed edition of a talk he gave some years earlier when he was teaching at Heiligenstadt. That can be inferred from the explanation that follows the title where he says he will deal with the nature, force, and effects of the magnet together with several applications of it. In the third book, Magneticum Naturae Rerum, the word “magneticum” is taken in the metaphorical sense that includes all forces that act on all things in nature. The second book Magnes sive de Arte Magnetica Opus Tripartitum (Kircher, 1654) (a copy of its 3rd edition is in the library of the Ebro Observatory) contains the scientific contribution of Kircher to geomagnetism. It is divided into three “books:” 1. On the Nature and Properties of the Magnet 2. The Use of the Magnet 3. The Magnetic World In the first book Kircher tries to investigate what magnetism is, using as a starting point what was known in his time: the natural existence of loadstones, the fact that iron may become magnetic with two poles acting in opposite senses on other magnetic bodies. He also

464

KIRCHER, ATHANASIUS (1602–1680)

accepts the hypothesis of the Earth being a magnet after repeating Gilbert’s experiments with the terrella, but it is easy to see that the data is too poor to explain what magnetism is. In the second book, Kircher shows with the help of several experiments how magnetic force is distributed in magnetic bodies of different shapes and how to find what he calls the barycentre. He tries to show that in a spherical body magnetism propagates linearly to the poles in its interior and spherically to the poles in the exterior. He finds that both poles have equal force and explains how to make magnetic needles to measure inclination and declination accurately. For him it is a well-established fact that “in the Boreal Hemisphere there is a distribution of inclination in accordance with latitude, whether it exists also past the equator I do not know, as to my knowledge nobody has studied these concordances” (Kircher, 1654, p. 295). At this point he accepts the idea already given by Gilbert and prints a table of magnetic inclinations for each degree of northern latitudes. He includes his own calculations and some of Gilbert’s values (Kircher 1654, p. 306). Declination is treated quite differently in Kircher’s works: he accepts that its value is related with geographical position but he does not believe that it depends only on longitude because it deviates from the local meridian either eastward or westward and also because it changes values at different latitudes on the same meridian. The solution of this point is of greatest importance in geography and navigation. He also realized the errors of many of these values: “there is no doubt about declination, the only doubt is in its quantity (¼value)” and a little afterward he adds “the only way to get out of this problem is to make all around the world as many reliable observations as possible” (Kircher, 1654, p. 313) and for Kircher it was easy to see that his Jesuit order, with many colleges in Europe and missionaries traveling to America and Asia, could be a reliable instrument to perform this task. After talks with his colleagues in Rome and with the approval of his superior general, he prepared a Geographical Consilium (an instruction manual) about declination and inclination, what kind of instruments were needed, how to use them, and how to report the results. On the occasion of one of the regular meetings of Jesuits from around the world in Rome in 1639, Kircher collected names of Jesuits who could do the work and wrote letters to all of them. Three long tables are the result: the first one (Table K1) “Magnetic declination on the oceans observed by Jesuits” is a long list of nearly 200 geographical points with their declination and latitude. The latitudes extend as far as from 81
00 N to 54
270 S, and the declinations from 25
00 E to 35
300 W. Table K2 “Longitudes and latitudes of several places with their declination” seems to be arranged in increasing longitude from 0
170 to 359
400 , but longitudes from 347
250 to 359
400 are included between 41
100 and 42
70 . Declinations go from 0
0’ to 33
. There is no indication whether they are to the east or to the west, but the fact that longitudes from 42
to 346
are listed at the end of the table may suggest some elaboration of the data, which are not explained by Kircher. Table K3 is the list of declinations observed in

Europe following Kircher’s request. The names of some 57 authors and of 66 cities in Europe with their declination are mentioned in it. Values in the list extend from Lisbon to Constantinople and from Vilna to Malta. Kircher adds six other cities: Aleppo (Syria), Alexandria (Egypt), Goa and Narfinga (India), and Canton and Macau (China). Behind this third table Kircher adds the copy of a letter from Francis Brescianus, who observed in his travel westward to Canada the following declination changes: from Europe to a place near the Azores the declination was around 2
E or 3
E, at the Azores it was about 0
, from there for some 300 leagues it increased regularly up to around 21
or 22
and from there it decreased again but more quickly as one approached America. In Quebec it was 16
and further west, in the land of the Huron Indians, some 200 leagues from Quebec, it was only 13
. Kircher adds also a letter from his pupil in Rome, P. Martin s.I. with 13 declinations but their geographical positions are too vague as “in the meridian of river Laurentij Marques” (Kircher, 1654, p. 348). In this letter Martin suggested a method for drawing a world chart of declination. This suggestion was not followed by Kircher and the first isogonic chart of the Atlantic was not published until 1701 by Edmund Halley (Chapman and Bartels, 1940). Unfortunately, the final result of all this observational work was never completed. As Kircher says: “When I was keeping the work, composed with no small effort, in my Museum and waiting for the right moment to publish it for the good of the Republic of Letters, it was secretly removed by one of those people who came to me almost every day from all around the world to see my Museum” (Kircher, 1654, p. 294). This combined and centralized effort is probably the first one of that kind in magnetism and must be considered as the great contribution of Kircher to the advancement of geomagnetism. It is certainly not a milestone on the long way of progress in science but it is an important step in it. Oriol Cardus

Bibliography Chapman, S., and Bartels J., 1940. Geomagnetism. Oxford: The Clarendon Press. Diccionario Histórico de la Compañía de Jesús, 2001. Comillas (4 volumes). Kircher, A., 1654. Magnes sive de Arte Magnetica. 3rd edn. Rome. Virgille P. de, 1905. Les Jesuites et l’étude du magnetisme terrestre. Etudes, 492–511.

Cross-references Gilbert, William (1544–1603) Halley, Edmond (1656–1742) Observatories, Overview

L

LANGEL, ROBERT A. (1937–2000) A NASA (UnitedStates) scientist who served as the project scientist for the Magsat mission. Trained as an applied mathematician and physicist, Langel was born in Pittsburgh, Pennsylvania. He was a strong advocate of high-accuracy vector magnetic measurements from satellite. These measurements were used to demonstrate the break in the power spectra at about spherical harmonic degree 13 that represents the transition from core-dominated processes to lithosphericdominated processes (Figure L1; Langel and Estes, 1982). Tantalizing

suggestions of such a break had previously been reported from total field-intensity measurements acquired on an around-the-world magnetic profile (Alldredge et al., 1963). Langel also made use of the high-accuracy vector measurements, and magnetic-observatory measurements, for the coestimation of internal (core and lithosphere), external (ionosphere and magnetosphere), and induced magnetic fields. These comprehensive models of the geomagnetic field were developed in collaboration with T. Sabaka (NASA) and N. Olsen (Denmark). Earlier workers had relied on a sequential approach, beginning with the largest (core) fields. Langel also pioneered mathematical techniques for the merging of a priori long-wavelength lithospheric magnetic field information with shorter wavelength observations (Figure L1; Purucker et al., 1998) using inverse techniques. This is necessary because the longest-wavelength lithospheric magnetic fields are inaccessible to direct observation as they overlap with the much-larger magnetic fields originating in the Earth’s outer core. Significant earlier developments with forward models came from Cohen (1989) and Hahn et al. (1984). Langel authored or coauthored in excess of 94 peer-reviewed publications and helped train a generation of geomagnetists by virtue of his work on Magsat and his teaching at Purdue University (UnitedStates) and Copenhagen University (Denmark). Bob was devoted to his family and church. His religiosity was well known among his colleagues, and he undertook missionary work, lead Bible study groups, and worked with teenagers. Michael E. Purucker

Bibliography

Figure L1 Comparison of the Lowes-Mauersberger spectra at the Earth’s surface for a recent comprehensive model (line) of the core and lithospheric fields by Sabaka et al. (2004) and a lithospheric field model (symbols) by Fox Maule et al. (2005). Rn is the mean square amplitude of the magnetic field over a sphere produced by harmonics of degree n.

Alldredge, L.R., Vanvoorhis, G.D., and Davis, T.M., 1963. A magnetic profile around the world. Journal of Geophysical Research, 68: 3679–3692. Cohen, Y., 1989. Traitements et interpretations de donnees spatiales in geomagnetisme: etude des variations laterales d’aimantation de la lithosphere terrestre. Docteur es sciences physiques these, Paris: Institut de Physique du Globe de Paris, 95 pp. Fox Maule, C., Purucker, M., Olsen, N., and Mosegaard, K., 2005. Heat flux anomalies in Antarctica revealed by satellite magnetic data. Science, 10.1126/science.1106888. Hahn, A., Ahrendt, H., Meyer, J., and Hufen, J.H., 1984. A model of magnetic sources within the Earth’s crust compatible with the field measured by the satellite Magsat. Geologischer Jahrbuch A, 75: 125–156. Langel, R.A., 1987. The main geomagnetic field. In Jacobs, J. (ed.), Geomagnetism. New York: Academic Press, pp. 249–512.

466

LAPLACE’S EQUATION, UNIQUENESS OF SOLUTIONS

Langel, R.A., and Estes, R.H., 1982. A geomagnetic field spectrum. Geophysical Research Letters, 9: 250–253. Langel, R.A., and Hinze, W.J., 1998. The Magnetic Field of the Earth’s Lithosphere. Cambridge: Cambridge University Press. Lowes, F.J., 1974. Spatial power spectrum of the main geomagnetic field, and extrapolation to the core. Geophysical Journal of Royal Astronomical Society, 36: 717–730. Purucker, M., Langel, R., Rajaram, M., and Raymond, C., 1998. Global magnetization models with a priori information. Journal of Geophysical Research, 103: 2563–2584. Sabaka, T.J., Olsen, N., and Langel, R.A., 2002. A comprehensive model of the quiet-time, near-Earth magnetic field: phase 3. Geophysical Journal International, 151: 32–68. Sabaka, T.J., Olsen, N., and Purucker, M., 2004, Extending comprehensive models of the Earth’s magnetic field with Oersted and CHAMP data. Geophysical Journal International, 159: 521–547. Taylor, P., and Purucker, M., 2000. Robert A. Langel III (1937–2000). EOS. Transactions of the American Geophysical Union, 81(15): 159.

Cross-references Harmonics, Spherical IAGA, International Association of Geomagnetism and Aeronomy Magsat

LAPLACE’S EQUATION, UNIQUENESS OF SOLUTIONS Potential theory lies at the heart of geomagnetic field analysis. In regions where no currents flow the magnetic field is the gradient of a potential that satisfies Laplace’s equation: B ¼
rV

(Eq. 1)

r2 V ¼ 0

(Eq. 2)

We can find the geomagnetic potential V everywhere in the source-free region by solving Eq. (2) subject to appropriate boundary conditions. Equation (1) means that we are dealing with a single scalar, the geomagnetic potential, and Eq. (2) means that we only have to know that scalar on surfaces rather than throughout the entire region. This reduces the number of measurements enormously. I shall discuss here two idealized situations appropriate to geomagnetism and paleomagnetism: measurements on a plane in Cartesian coordinates (x, y, z) with z down, continued periodically in x and y, appropriate for a small survey, and measurements on a spherical surface r ¼ a in spherical coordinates ðr; y; fÞ, where a is Earth’s radius. Simple generalizations are possible to more complex surfaces. It is well known that solutions of Laplace’s equation within a volume O are unique when V is prescribed on the bounding surface ]O, the so-called Dirichlet conditions. This is of no use to us because we cannot measure the geomagnetic potential directly. We can, however, measure the component of magnetic field normal to the surface, which also guarantees uniqueness (the so-called Neumann conditions). Thus measurement of a single component of magnetic field on a surface can be used to reconstruct the magnetic field throughout the current-free region, provided the sources are entirely inside or outside the measurement surface. When both internal and external sources are present two components need to be measured on a single surface, when it is also possible to separate the parts (see Internal external field separation). Here I shall focus on the case of purely internal sources. Components other than the vertical also provide uniqueness. As illustration consider the general solution of Laplace’s equation in Cartesian coordinates:

V ðx; y; zÞ ¼

X

Cm expðmzÞ cosðkx þ ‘y þ ak ;‘ Þ;

(Eq. 3)

k ;l ;m

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where m ¼ k 2 þ ‘2 is positive to ensure a finite solution as z !
1 and the fak ;‘ g are phase factors. If Bz ¼ ]V =]z is known on z ¼ 0 we can differentiate Eq. (3) with respect to z, set z ¼ 0, and determine the arbitrary coefficients fCk ;‘ ; ak ;‘ g from the usual rules for Fourier series. This is the case of Neumann boundary conditions. Of course, solutions of Eq. (3) are periodic in x and y with periods 2p=k and 2p=l , respectively. Bx and By, however, do not give unique solutions. Differentiating Eq. (3) with respect to x and putting z ¼ 0, for example, allows determination of all the Fourier coefficients except those with kP¼ 0. The ambiguity in the solution takes the form V0 ð y; zÞ ¼ l Cl expð
lzÞ cosðly þ al Þ: Similarly, measurement only of By leaves an ambiguity independent of y. Knowing Bx and By overdetermines the problem. The spherical case is slightly different. Br (
Z in conventional geomagnetic notation) provides Neumann boundary conditions and so will give a unique solution. Bf (east) allows an axisymmetric ambiguity similar to that for the cartesian Bx and By cases. Surprisingly,
By (north) gives a unique solution; the condition that V be finite at the poles provides the necessary constraint to remove the arbitrary parts of the solution. Proof involves the general solution of Laplace’s equation as a spherical harmonic expansion (see Main field modeling) in similar fashion to the use of the Fourier series solution in the cartesian case (see Magnetic anomalies, modeling). More elegant formal proofs of these standard results making use of Green’s theorem are to be found in texts on potential theory (e.g., Kellogg (1953)). Formal mathematical results dictate the type of measurements we must make in order to obtain meaningful estimates of the geomagnetic field in the region around the observation surface. Clearly in the real situation other complications arise from finite resolution and errors, but with a uniqueness theorem we can at least be confident that the derived field will converge toward the correct field as the data are improved; for this to happen we must measure the right components of magnetic field, but this is not always possible. Instrumentation has changed over the years, dictating changes in the type of measurement available. Table L1 gives a summary of the bias in available datasets. Backus (1970) addressed the problem of uniqueness when only F is known on the surface. He showed that, for certain magnetic field configurations, there exist pairs of magnetic fields B1 and B2, which satisfy r  B ¼ 0 and r  B ¼ 0 outside ]O and such that jB1 j ¼ jB2 j everywhere on ]O, but jB1 j 6¼ jB2 j elsewhere. This result led to the decision to make vector measurements on the MAGSAT satellite. In practice B obtained from intensity measurements alone can differ by 1000 nT from the correct vector measurements. The difference takes the form of a spherical harmonic series with terms dominated by l ¼ m, now called the Backus ambiguity or series. Additional information does provide uniqueness (e.g., locating the dip equator (Khokhlov et al., 1997)). Backus (1968, 1970) gives details of the proof in both two and three dimensions. The 2D proof uses complex variable theory and is particularly simple and illuminating. We use the complex potential w(z), where z ¼ x þ iy. The field is given by the real and imaginary parts of dw/dz, and jdw=dzj is known on the bounding circle. Focus attention on zðzÞ ¼ ilogdw=dz ¼ x þ i
¼
argjdw=dzj þ ilogjdw=dzj:

(Eq. 4)

Both x and
satisfy Laplace’s equation as the real and imaginary parts of an analytic function, except where dw=dz ¼ 0. When such singularities exist, taking a circuit enclosing the source region gives an increase of 2p in the argument of the logarithm. This defines the ambiguity. The number of singularities determines the number of different solutions; singularities exist if the field has more than two dip poles on the surface. Paleomagnetic and older geomagnetic data sets are dominated by directional measurements. Early models made the reasonable but

467

LAPLACE’S EQUATION, UNIQUENESS OF SOLUTIONS

Table L1 Datasets are dominated by the components shown for each epoch Date

Dominant component

Comment

–1700 1700–1840 1840–1955 1955–1980 1980–

D D, I D, I, F F X, Y, Z

Compass only, I from 1586 restricted to Europe No absolute intensity until Gauss’ method (q.v.) Magnetic surveys generally include all components Proton magnetometer gives absolute F, satellites not orientated MAGSAT starts global vector measurements

Paleomagnetism Archeomagnetism Borehole samples

D, I F I, F

Paleointensity is a difficult and inaccurate measurement Sample orientation usually not available Declination usually not logged in boreholes

Note: Relative horizontal intensities were measured by von Humboldt (q.v.) before 1840, but in a global model these give no more information than does direction.

Table L2 Summary of uniqueness results Component

Ambiguity

Comment

Z (down) X (north) Y (east) F (total intensity) F þ location of dip equator D, I (direction) D þ partial H (horizontal)

Unique Unique Any axisymmetric solution Usually nonunique Unique Arbitrary multiplicative constant Unique

Neumann boundary condition Y ¼ 0 for any axisymmetric field Backus effect n constants in general H needed on line joining dip poles

Note: 2n is the number of dip poles on the measurement surface.

unproven assumption that direction determined the magnetic field up to a single multiplicative factor. Proctor and Gubbins (1990) found a result in 2D using Backus’ method above, focussing instead on the potential zðzÞ ¼ logdw=dz ¼ logjdw=dzj þ iargjdw=dzj:

(Eq. 5)

The above argument then leads to a general solution with one arbitrary multiplicative constant plus n
1 additional arbitrary constants, where 2n is the number of dip poles. As with the intensity problem, extension of the 2D proof into 3D is not straightforward. Proctor and Gubbins (1990) conjectured a similar result in 3D and provided an axisymmetric example. They also gave a practical recipe for determining the ambiguity using standard methods of inverse theory. The problem is homogeneous in the sense that, if B1 and B2 are fields derived from harmonic potentials and are parallel on ]O, then any linear combination of the form a1 B1 þ a2 B2 is also a solution. Thus, having found one solution, B1 say, a linear error analysis around B1 will yield a direction in solution space along B1
B2 that defines an annihilator for the problem. Homogeneity then guarantees that this linearized analysis will be valid globally. Application of this procedure to the axisymmetric example did yield both solutions, but only when a very large number of spherical harmonics were used in their representation, confirming Backus’ view that uniqueness demonstrations on solutions restricted to finite spherical harmonic series could be misleading. Hulot et al. (1997) finally found a formal proof in 3D using the homogeneity of the problem. The key is to show that the solution space is linear; the proof follows using relatively recent general results from potential theory. They show that, if n is the number of loci where the field is known to be either zero or normal to the surface then the dimension of the solution space (number of independent arbitrary constants) is n
1. This result holds whether sources are outside or inside the surface. For the related problem of gravity, which allows monopoles, the dimension is n. It is rather easy to prove that directional measurements throughout a volume provide uniqueness to within a single arbitrary constant (Blox-

ham, 1985). Given one solution B others must have the form aB and must be solenoidal and curl-free: r  ðaBÞ ¼ B  ra ¼ 0

(Eq. 6)

r  ðaBÞ ¼
B  ra ¼ 0:

(Eq. 7)

Thus ra ¼ 0 and a must be a constant. Knowing the field everywhere inside the volume of measurement enables us to construct the field component normal to the boundary and solve Laplace’s equation outside the volume of measurement, all within a single-multiplicative constant. The earliest historical data sets are dominated by declination. In this case we know the direction of the horizontal field and location of the dip poles. Suppose the potential field Br þ aBh fits the data; Bh is known but Br and a are unknown. The radial component of r  B does not involve Br or ]=]r, so rh  ðaBh Þ ¼ 0 and ra  Bh ¼ 0:

(Eq. 8)

The unknown a is therefore invariant along contours of Bh; measurement of horizontal intensity along any line joining the dip poles is therefore sufficient to guarantee uniqueness of Bh, which in turn would allow us to determine the potential and B. The coverage of additional data required is therefore reduced enormously from the whole surface to a line. Unfortunately horizontal intensity is not available for the early historical epoch, and paleoinclinations are too inaccurate to be of any use (Hutcheson and Gubbins, 1990), but inclination did become available from the early 18th century on voyages from Europe round Cape Horn and through the Magellan Straits into the Pacific, which represents a fair approximation to the ideal case of horizontal intensity between the north and south dip poles. Uniqueness results are summarized in Table L2. Practical aspects of the problem are discussed further by Lowes et al. (1995). David Gubbins

468

LARMOR, JOSEPH (1857–1942)

Bibliography Backus, G.E., 1968. Application of a nonlinear boundary value problem for Laplace’s equation to gravity and geomagnetic intensity surveys. Quarterly Journal of Mechanics and Applied Mathematics, 21: 195–221. Backus, G.E., 1970. Nonuniqueness of the external geomagnetic field determined by surface intensity measurements. Journal of Geophysical Research, 75: 6339–6341. Bloxham, J., 1985. Geomagnetic secular variation. PhD thesis, Cambridge University, Cambridge. Hulot, G., Khokhlov, A., and Mouël, J.L.L., 1997. Uniqueness of mainly dipolar magnetic fields recovered from directional data. Geophysical Journal International, 129: 347–354. Hutcheson, K., and Gubbins, D., 1990. A model of the geomagnetic field for the 17th century. Journal of Geophysical Research, 95: 10,769–10,781. Kellogg, O.D., 1953. Foundations of Potential Theory. New York: Dover. Khokhlov, A., Hulot, G., and Mouël, J.L.L., 1997. On the Backus effect—I. Geophysical Journal International, 130: 701–703. Lowes, F.J., Santis, A.D., and Duka, B., 1995. A discussion of the uniqueness of a Laplacian potential when given only partial field information on a sphere. Geophysical Journal International, 121: 579–584. Proctor, M.R.E., and Gubbins, D., 1990. Analysis of geomagnetic directional data. Geophysical Journal International, 100: 69–77.

Cross-references Gauss, Carl Friedrich (1777–1855) Internal External Field Separation Magnetic Anomalies, Modeling Magsat Main Field Modeling

leading authority on geomagnetism. This was a time when theories of the origin of the magnetic fields of the Earth and of the Sun abounded. His 1919 paper, presented at the British Association for the Advancement of Science meeting in Bournemouth (Larmor, 1919) is only two pages long; he dismisses permanent magnetization and convection of electric charge as candidates for both bodies. For the Sun he considers three theories: the dynamo theory, electric polarization by gravity or centrifugal force, and intrinsic polarization of crystals. He dismisses the last two in the case of the Earth because of the secular variation, arguing that they predict proportionality between the magnetic field and spin rate in contrast to the observation of a changing dipole moment and constant spin rate. His version of the dynamo theory is quite explicit and requires fluid motion in meridian planes. It was this very specific theory that was tested by Cowling (1934) in establishing his famous antidynamo theorem (see Cowling’s theorem). According to Eddington (1942–1944) he was a retiring and complicated man, refusing a celebration in his honor organized by Cambridge University. His lectures were “ill-ordered and obscure but. . . even the examination-obsessed student could perceive that here he was coming to an advanced post of thought, which made all his previous teaching seem behind the times.” He was one of only three lecturers E. C. Bullard (q.v.) bothered with during his undergraduate days. Darrigol (2000) says “Larmor’s physics was freer and broader than conceptual rigor and practical efficiency commanded.” In similar vein, Bullard once described him to me as a lazy man, which I took to refer to his rather casual outline of the dynamo theory and failure to follow it up with any mathematical analysis, of which he was eminently capable, leaving Cowling to pick it up a decade later. But how right he was! How much better to take the credit for a correct theory and avoid the disappointments of the next half-century, from the antidynamo theorems to the numerical failure of Bullard and Gellman’s attempt at solution, and the graft of the subsequent 30 years of mathematical and computational effort it has taken to put the dynamo theory onto a secure footing!

LARMOR, JOSEPH (1857–1942) Joseph Larmor is known in geomagnetism for the precession frequency of the proton that now bears his name and his seminal paper of 1919 proposing the dynamo theory for the first time. His main purpose was to explain the solar magnetic field, but he also had the geomagnetic field in mind. Born in 1857 in Magherhall, County Antrim, and educated at the Royal Belfast Academical Institution and Queen’s College, Belfast, Larmor showed an early ability in mathematics and classics. He was senior wrangler in the Mathematics Tripos at Cambridge, beating J.J. Thomson to second place. He succeeded George Stokes as Lucasian Professor of Mathematics in 1903, and was knighted in 1909. Larmor worked at the dawn of modern physics and was concerned with the aether and with unifying electricity and matter. His greatest work, Aether and Matter, was revolutionary and inspiring. His contributions are often compared with those of Lorentz. His discovery of what we now call Larmor precession was incidental: he showed that in a magnetic field, an electron orbit precesses with angular velocity proportional to the magnetic field strength while remaining unchanged in form and inclination to the magnetic field. The same principle applies to a particle, such as a proton, that possesses a magnetic moment, and the frequency of precession of a proton in an applied magnetic field is called the Larmor frequency. The proton magnetometer, developed 30 years after his death, determines the absolute intensity of a magnetic field by measuring the precession frequency. Later in his career he contributed to many of the geophysical problems of the day, publishing on the Earth’s precession and irregular axial motion, and on the effects of viscosity on free precession, which Harold Jeffreys later used in his argument against mantle convection. Larmor published on the origin of sunspots, which G.E. Hale had shown were associated with strong magnetic fields, and became a

David Gubbins

Bibliography Cowling, T.G., 1934. The magnetic field of sunspots. Monthly Notices of Royal Astronomical Society, 94: 39–48. Darrigol, O., 2000. Electrodynamics from Ampère to Einstein. Oxford: Oxford University Press. Eddington, A.S., 1942–1944. Joseph Larmor. Obituary Notices of Fellows of the Royal Society, 4: 197–207. Larmor, J., 1919. How could a rotating body such as the Sun become a magnet?. Reports of the British Association, 87: 159–160.

Cross-references Bullard, Edward Crisp (1907–1980) Cowling, Thomas George (1906–1990) Cowling’s Theorem Dynamo, Bullard-Gellman Magnetometers, Laboratory Nondynamo Theories

LEHMANN, INGE (1888–1993) In 1936, Miss Inge Lehmann (Figure L2) proposed a new seismic shell of radius 1400 km at the center of the Earth, which she called the “inner core.” It was already known that low seismic velocities in the outer core created a shadow at angular distances between 105 and 142 and that seismic waves diffracted some considerable distance into the shadow. Wiechert also attributed seismic arrivals before 142 to diffraction, but Gutenberg and Richter (1934) had already found their

469

LENGTH OF DAY VARIATIONS, DECADAL

held until retirement in 1953. In retirement she enjoyed frequent visits to Lamont and other institutions in the United States and Canada. It seems her international reputation was strong before her work was recognized at home in Denmark. She was elected Foreign Member of the Royal Society in 1969, was awarded the Bowie Medal (the American Geophysical Union highest honor) in 1971, and now has an AGU medal named in her honor. Further details of her life may be found in Bolt (1997). David Gubbins

Bibliography

Figure L2 Inge Lehmann (Reprinted with permission from the Royal Society).

Bolt, B.A., 1997. Inge Lehmann. Biographical Memoirs of Fellows of the Royal Society, 43: 28285–28301. Bullen, K.E., 1947. An Introduction to the Theory of Seismology. London: Cambridge University Press. Gutenberg, B., and Richter, C.F., 1938. P’ and the Earth’s core. Monthly Notices of Royal Astronomical Society, Geophysical Supplement, 4: 363–372. Haddon, R.A.W., and Cleary, J.R., 1974. Evidence for scattering of seismic PKP waves near the mantle-core boundary. Physics of the Earth and Planetary Interiors, 8: 211–234. Jeffreys, H., 1939. The times of the core waves. Monthly Notices of Royal Astronomical Society, Geophysical Supplement, 4: 548–561. Lehmann, I., 1936. Publication’s Bureau Centrale Seismologique Internationale Series A, 14: 87–115.

Cross-reference Inner Core Tangent Cylinder amplitude too large and frequency too high to fit with the diffraction theory. Measurements of the vertical component of motion clinched the interpretation as waves reflected or refracted through a large angle. Lehmann’s paper remains a classic example of lateral thinking, careful interpretation, and cautious conclusions. The proposed radius was close to the currently accepted one of 1215 km. The inner core was rapidly accepted into the new Earth models being constructed by Gutenberg and Richter (1938) in the United States and Jeffreys (1939) in the U.K., although the scattered nature of the arrivals in the shadow zone demanded a thick transition zone at the bottom of the outer core, where the seismic velocity increased gradually. In the first edition of his book on seismology, Bullen (1947) labeled the Earth’s layers A-G, with A as the crust and G the inner core, with F denoting the inner core transition. The need for this layer was largely removed when seismic arrays showed that much of the scattered energy came not from the inner-core boundary but from the core-mantle boundary (Haddon and Cleary, 1974). The inner core is vital in the theory of geomagnetism because its slow accretion, a consequence of the Earth’s secular cooling, powers the geodynamo and influences core convection (see Inner core tangent cylinder). The F-layer lives on in name at least, despite the removal of seismological evidence, in the form of the required boundary layer between the inner and outer cores. Although the inner core was probably the most spectacular of Lehmann’s discoveries, she enjoyed a long and distinguished career in seismology, remaining active throughout her long retirement. Born in Copenhagen into a distinguished family, she benefited from an exceptionally enlightened early education and read mathematics at the University of Copenhagen from 1908. A spell in Cambridge in 1910 came as a shock because of the restrictions placed on women there (Cambridge did not grant women degrees until much later). Lehmann’s career as a seismologist began in 1925 when she was appointed assistant to Professor N.E. Norland, who was engaged in establishing a network of seismographs in Denmark. She met Beno Gutenberg at this time. In 1928 she was appointed chief of the seismological department of the Royal Danish Geodetic Institute, a post she

LENGTH OF DAY VARIATIONS, DECADAL It is a truism to say that the Earth rotates once per day. However, the length of each day is not constant, but varies over timescales from everyday to millions of years. Perhaps the best-known variation is the gradual slowing of the rotation rate due to the tidal interaction between the Earth and the moon (see Length of day variations, longterm). Historical observations are well fit by an increase in length of day (LOD) of about 1.4 ms per century, most clearly seen in the timing of eclipses recorded by ancient civilizations. (Note that there are 86 400 seconds in a day, so these millisecond changes are of order 1 part in 108 of the basic signal.) Geological evidence also supports increase in LOD over Earth history (Williams, 2000). At the other end of the scale, modern measurements from the global positioning system (GPS) or very-long baseline interferometry (VLBI) demonstrate a signal on yearly and sub-yearly timescales, again of a few milliseconds in magnitude. These variations are dominated by angular momentum exchange between the atmosphere and the solid Earth; put crudely, winds blowing against mountain topography (particularly north-south ranges such as the Rockies in America) push the solid Earth, speeding up or slowing down the rate of rotation. Atmospheric angular momentum calculated from models of the global circulation can explain almost all the short-period signals. The small remaining residual (of amplitude 0.1 ms) shows strong coherence with estimates of oceanic angular momentum (Marcus et al., 1998), and the hydrological cycle is also probably important, by causing small variations in the Earth’s moment of inertia. However, surface observations are unable to explain a decadal fluctuation, again of a few milliseconds in LOD. This signal was first recognized in the early 1950s, and because the core is the only fluid reservoir in the Earth capable of taking up the requisite amount of angular momentum and due to its correlation with various geomagnetic data (in particular records of magnetic intensity and declination at certain magnetic observatories), it was assumed to result from exchange

470

LENGTH OF DAY VARIATIONS, DECADAL

of angular momentum between the fluid core and solid mantle. More robust evidence was provided by Vestine (1953), who demonstrated a good correlation between the change in length of day (DLOD) signal and core angular momentum (CAM) estimated from observations of westward drift (q.v.). The connection between core processes and DLOD was put on a rigorous footing by Jault et al. (1988). They realized that models of surface core flow reflect motions throughout the core and can be used to estimate changes in angular momentum of the core, matching changes in angular momentum of the solid Earth required to explain the observed decadal DLOD. Most processes in the core have characteristic timescales which are either of centuries or longer (for example the planetary waves that have been proposed as an explanation for westward drift (q.v.)), or which are short, of order days (Alfvén waves (q.v.) or inertial waves). Torsional oscillations (q.v.), however, are thought to have characteristic periods of decades. These are solid-body rotations of fluid cylinders concentric with the Earth’s spin axis, about that axis. Such motions carry angular momentum about the spin axis, and extend to the core surface. Therefore, changes in CAM can be calculated using surface flow models. Such a calculation ignores the inner core, and assumes uniform density for the liquid core; however, the value calculated is not sensitive to these assumptions. Most commonly, a surface flow is used that is assumed to be tangentially geostrophic (the force balance in the horizontal direction is assumed to be between pressure gradients and Coriolis force). The core surface flow is expanded in a poloidal-toroidal decomposition in spherical coordinates (r, y, j) (r is radius, y is colatitude, and j is longitude), such that v¼

1 X l X

    m rH rsm þ rH  rtlm Ylm l Yl

(Eq. 1)

l¼1 m¼0 m where rH is the horizontal part of the gradient operator, and sm l and tl are poloidal and toroidal scalars, respectively, corresponding to surface spherical harmonics (q.v.) Ylm ð; jÞ of degree l and order m. Then, core angular momentum depends only on two large-scale toroidal, zonally symmetric, surface flow harmonics. Using standard parameters for the Earth, the relation   12 (Eq. 2) T ¼ 1:138 t10 þ t30 7

is obtained, where dT is the change in LOD in milliseconds and the changes in the toroidal flow coefficients are measured in km y
1. The t10 harmonic corresponds to a uniform rotation of the core surface about the Earth’s rotation axis. Figure L3 shows a calculation of Jackson et al. (1993) using the ufm time-dependent model of the main field (q.v.) to calculate the predicted LOD variations, compared with observations of McCarthy and Babcock (1986). Despite the highly idealized theory, the agreement is striking, especially over the last century. The correlation post-1900 between LOD and calculated coreangular momentum is remarkably robust, being evident for many different strengths of flow (explaining more or less of the secular variation) and for different physical assumptions about the flow (for example, assuming that the flow is toroidal, or steady but within a drifting reference frame). By 1900 the network of magnetic observatories (q.v.) was globally distributed; these stations provide by far the most sensitive and accurate measurements of the global magnetic secular variation from which the flows are calculated. Before 1900, as more observatory data become available, the time variation of the field models reflects changes in data quality as well as any physical processes; these changes map into the flow models, giving unphysical results and poorer correlation with observed DLOD. There is some evidence that this problem can be overcome by calculating simpler flow models; a notably better correlation is provided by the original analysis (Vestine, 1953) of westward drift!

Figure L3 Comparison of observed decadal variations in length of day LOD (with a long-term trend of 1.4 ms/century removed) with the prediction from a model of surface core flow (following Jackson et al., 1993). More recent work has considered whether the core could have a role to play in angular momentum exchange at shorter periods. It has been argued that the core may well change the phase of the observed LOD signal compared with forcing from the atmosphere and oceans, but there is no strong evidence in subdecadal DLOD of a signal from the core. The study of LOD variations as described here is essentially phenomenological—there is no need to assume any particular mechanism for the exchange of angular momentum between the core and the mantle. How the exchange of angular momentum may be effected, and the influence of the dynamics of the torsional oscillations (q.v.), are discussed elsewhere. Richard Holme

Bibliography Jackson, A., Bloxham, J., and Gubbins, D., 1993. Time-dependent flow at the core surface and conservation of angular momentum in the coupled core-mantle system. In Le Mouël, J.-L., Smylie, D.E., and Herring T. (eds.), Dynamics of the Earth’s Deep Interior and Earth Rotation, pp. 97–107, AGU/IUGG. Jault, D., Gire, C., and Le Mouël, J.L., 1988. Westward drift, core motions, and exchanges of angular momentum between core and mantle. Nature, 333: 353–356. Marcus, S.L., Chao, Y., Dickey, J.O., and Gregout, P., 1998. Detection and modeling of nontidal oceanic effects on Earth’s rotation rate. Science, 281: 1656–1659. Vestine, E.H., 1953. On variations of the geomagnetic field, fluid motions and the rate of the Earths rotation. Journal of Geophysical Research, 38: 37–59. Williams, G.E., 2000. Geological constraints on the Precambrian history of earth’s rotation and the moon’s orbit. Reviews of Geophysics, 38: 37–59.

Cross-references Alfvén Waves Core-Mantle Coupling, Electromagnetic Core-Mantle Coupling, Topographic Harmonics, Spherical Length of Day Variations, Long-Term Observatories, Overview Oscillations, Torsional Time-Dependent Models of the Geomagnetic Field Westward Drift

LENGTH OF DAY VARIATIONS, LONG-TERM

LENGTH OF DAY VARIATIONS, LONG-TERM Tidal friction The Moon raises tides in the oceans and solid body of the Earth. The Sun does likewise, but to a lesser extent. Let us first consider the action of the Moon. Due to the anelastic response of the Earth’s tides, the tidal bulges can be thought of as being carried ahead of the sublunar point by the Earth’s rotation. This misalignment of the tidal bulges produces a torque on the Moon, which increases its orbital angular momentum and drives it away from the Earth, while reducing the Earth’s rotational angular momentum by the opposite amount. The rate of rotation of the Earth is reduced through the action of tidal friction, which occurs very largely in the oceans. By analogy, the solar tides make a smaller contribution to slowing down the Earth, but the reciprocal effect on the Sun is negligible. As the Earth slows down the length of the day (LOD) increases. The change in the rate of rotation of the Earth under this mechanism is usually termed its tidal acceleration. Beside this long-term change, there are relative changes between the rotation of the mantle and the core on shorter timescales. The transfer of angular momentum, which produces these changes, is caused by core-mantle coupling and the astronomical results discussed here shed light on the timescale of the geomagnetic processes involved. The tidal acceleration of the Earth has been measured reliably in the following way. Analysis of the perturbations of near-Earth satellites produced by lunar and solar tides, together with the requirement that angular momentum be conserved in the Earth-Moon system, leads to an empirical relation between the retardation of the Earth’s spin and the observed tidal acceleration of the Moon (see, for example, Christodoulidis et al. (1988)). Lunar laser ranging gives an accurate value for the Moon’s tidal acceleration and inserting this in the relation gives the result –6.1  0.4  10
22 rad s
2 for the total tidal acceleration of the Earth (Stephenson and Morrison, 1995). This result is equivalent to a rate of increase in the LOD of þ2.3  0.1 ms per century (ms/100 y). For the interconversion of units see Table 2 of Stephenson and Morrison (1984). These satellite and lunar laser ranging measurements were obtained from data collected over the past 30 years or so. However, they can be applied to the past few millennia because the mechanism of tidal friction has not changed significantly during this period (see, for example, Lambeck, 1980, Section 10.5). While the tidal component of the Earth’s acceleration can be derived from recent high-precision observations, the actual long-term acceleration, which is the sum of the tidal and nontidal components, cannot be measured directly because it is masked by the relatively large decade fluctuations. Instead, observations from the historical past, albeit crude by modern standards, have to be used. By far the most accurate data for measuring the Earth’s rotation before the advent of telescopic observations (ca. A.D. 1620) are records of eclipses. Useful records of eclipses extend back to about 700 B.C.

471

Stephenson and Morrison (1995) analyzed all the available reliable records of solar and lunar eclipses during the period 700 B.C. to A.D. 1600 from ancient Babylon, China, Arab, and European sources in order to measure DT at epochs in the past. For full translations of the various historical records see Stephenson (1997). The untimed data consist of unambiguous descriptions of total solar eclipses from known places on particular dates, but without timing. The measurement of time is not necessary because of the narrowness of the belt of totality (a few minutes of time) relative to the quantity being measured (DT), which amounts to several hours in the era B.c. The timed data, on the other hand, include a timing of the occurrence of one or more of the contacts during solar and lunar eclipses. Their results for DT from the untimed data are shown in Figure L4. Some less-critical data have been omitted here from the figure for the sake of clarity. A single parabola does not adequately represent all the data, so Stephenson and Morrison fitted a smooth curve to the data using cubic splines, with economical use of knots in order to follow the degree of smoothness in the observed record after A.D. 1600. The details of the fitting procedure can be found in their 1995 paper. Their results from the independent timed data are very close to the untimed data, and are not reproduced here.

Long-term accelerations in rotation The parabola in Figure L4 corresponds physically to an acceleration of –4.8  0.2  10
22 rad s
2 in the rotation of the Earth. This is significantly less algebraic than the acceleration expected on the basis of tidal friction alone, –6.1  0.4  10
22 rad s
2. Some other mechanism, or mechanisms, must account for the accelerative component of þ1.3  0.4  10
22 rad s
2. This nontidal acceleration may be associated with the rate of change in the Earth’s oblateness attributed to viscous rebound of the solid Earth from the decrease in load on the polar caps following the last deglaciation about 10000 years ago (Peltier and Wu, 1983; Pirazzoli, 1991). From an analysis of the acceleration of the motion of the node of near-Earth satellites, a present-day fractional

Eclipses and the Earth’s rotation Early eclipse observations fall into two main independent categories: untimed reports of total solar eclipses; and timed measurements of solar and lunar eclipse contacts. The tracks of total solar eclipses on the Earth’s surface are narrow and distinct. The retrospective calculation of their occurrence is made by running back the Sun and Moon in time along their apparent orbits around the Earth, the Moon’s motion having been corrected for the lunar tidal interaction with the Earth. Unless special provision is made, this computation presupposes the uniform progression of time and hence the uniform rotation of the Earth on its axis. For each observation, the displacement in longitude between the computed position of the track of totality and the actual observed track measures the cumulative correction to the Earth’s rotational phase due to variations in its rate of rotation over the intervening period. This displacement in degrees, divided by 15, gives the correction in hours to the Earth’s “clock,” usually designated by DT.

Figure L4 Corrections to the Earth’s clock derived from the differences between the observed and computed paths of (untimed) solar eclipses. The vertical lines are not error bars, but solution space, anywhere in which the actual value is equally likely to lie. Arrowheads denote that the solution space extends several hours in that direction. The solid line has been fitted by cubic splines. The dashed line is the best-fitting parabola.

472

LLOYD, HUMPHREY (1808–1881)

Figure L5 Changes in LOD –500 to þ2 000 obtained by taking the first time-derivative along the curves in Figure L4. The change due to tidal friction is þ2.3 ms/100 y. The high-frequency changes after þ1700 are taken from Stephenson and Morrison (1984). rate of change of the Earth’s second zonal harmonic J2 of –2.5  0.3  10
11 per year has been derived, which implies an acceleration in the Earth’s rotation of þ1.2  0.1  10
22 rad s
2. This is consistent to within the errors of measurement with the result from eclipses, assuming an exponential rate of decay of J2 with a relaxation time of not less than 4000 years. Other causes may include long-term core-mantle coupling and small variations in sea level associated with climate changes.

Lambeck, K., 1980. The Earth’s Variable Rotation. Cambridge UK: Cambridge University Press. Peltier, W.R., and Wu, P., 1983. History of the Earth’s rotation. Geophysical Research Letters, 10: 181–184. Pirazzoli, P.A., 1991. In Sabadini, R., Lambeck, K., and Boschi, E. (eds.), Glacial Isostacy, Sea-level and Mantle Rheology. Dordrecht: Kluwer, pp. 259–270. Ponsar, S., Dehant, V., Holme, R., Jault, D., Pais, A., and Van Hoolst, T., 2003. The core and fluctuations in the Earth’s Rotation. In Dehant, V., Creager, K.C., Karato, S., and Zatman, S. (eds.), Earth’s Core: Dynamics, Structure, Rotation Vol 31. Washington, DC: American Geophysical Union Geodynamics Series, pp. 251–261. Stephenson, F.R., 1997. Historical Eclipses and Earth’s Rotation. Cambridge UK: Cambridge University Press. Stephenson, F.R., and Morrison, L.V., 1984. Long-term changes in the rotation of the Earth: 700 B.C. to A.D. 1980. In Hide, R. (ed.), Rotation in the Solar System. The Royal Society, London. Reprinted in Philosophical Transactions of the Royal Society of London, Series A, 313: 47–70. Stephenson, F.R., and Morrison, L.V., 1995. Long-term fluctuations in the Earth’s rotation: 700 B.C. to A.D 1990. Philosophical Transactions of the Royal Society of London, Series A, 351: 165–202.

Cross-references Core-Mantle Coupling-Electromagnetic Geomagnetic Spectrum, Temporal Halley, Edmond (1656–1742) Length of Day Variations, Decadal Mantle, Electrical Conductivity, Mineralogy Westward Drift

Long-term changes in the length of the day The first time-derivative along the curves in Figure L4 gives the change in the rate of rotation of the Earth from the adopted standard. This standard is equivalent to a LOD of 86400 SI s. The change in the LOD is plotted in Figure L5. The wavy line is the derivative along the cubic splines in the period 500 B.C. to A.D. 1600. After A.D. 1600, the higher resolution afforded by telescopic observations reveals the decade fluctuations (see Length of day variations, decadal), which are usually attributed to core-mantle coupling (see Core-mantle coupling) (see Ponsar et al., 2003). Similar fluctuations are undoubtedly present throughout the whole historical record, but the accuracy of the data is not capable of resolving them. All that can be resolved is a long-term oscillation with a period of about 1500 years and amplitude comparable to the decade fluctuations. The observed acceleration of –4.8  0.2  10
22 rad s
2 in Figure L4 is equivalent to a trend of þ1.8  0.1 ms/100 y in the LOD. In our considered opinion, no reliable eclipse data suitable for LOD determinations are available before ca. 700 B.C.

LLOYD, HUMPHREY (1808–1881) Humphrey Lloyd (Figure L6) was a distinguished and influential experimental physicist whose entire career was pursued at Trinity College, Dublin (O’Hara, 2003).

Paleorotation The rotation of the Earth about 400 Ma ago can, in principle, be measured from the seasonal variations in growth patterns of fossilized corals and bivalves. This leads to estimates of the number of days in the year at that time, and hence the LOD (the year changes very little). Lambeck (1980) reviews the data and arrives at the value –5.2  0.2  10
22 rad s
2 (¼ þ1.9  0.1 ms/100 y in LOD) for the tidal acceleration over the past 4  108 years, which implies that tidal friction was less in the past. L.V. Morrison and F.R. Stephenson

Bibliography Christodoulidis, D.C., Smith, D.E., Williamson, R.G., and Klosko, S.M., 1988. Observed tidal braking in the Earth/Moon/Sun system. Journal of Geophysical Research, 93: 6216–6236.

Figure L6 Humphrey Lloyd.

LLOYD, HUMPHREY (1808–1881)

473

Figure L7 Lloyd’s magnetic survey of Ireland, undertaken with Ross and Sabine. In optics, he is remembered for his experimental confirmation of Hamilton’s prediction of conical refraction, his demonstration of interference fringes with a pair of mirrors, and his authoritative report on the wave theory of light for the British Association. He had a lifelong interest in terrestrial magnetism and played an important role in the establishment of magnetic observatories, beginning in the 1830s. His first venture into geomagnetism was a survey of Ireland (Figure L7) (Lloyd et al., 1836) undertaken, at the instigation of the British Association, with Ross and Sabine (q.v.) in 1834 and 1835. In addition to dip and the horizontal component of the Earth’s field, measured following the oscillation method of Hansteen (q.v.), simultaneous measurements of dip and intensity were made by a static balance method of his own devising. Following the lead of von Humboldt (q.v.) and Gauss (q.v.) and with the detailed advice of the latter, Lloyd proceeded to have his Magnetical Observatory built in the grounds of the college. This charming edifice can still be viewed, but in the grounds of University College Dublin, some miles distant from its original home. He conceived and designed many of the instruments that were to be used in the observatory: most were made by Thomas Grubb of Dublin. These included his balance magnetometer and a bifilar magnetometer, which he had developed to measure the horizontal component.

Lloyd participated enthusiastically in Gauss’s Europe-wide Magnetic Union, and when such observatories came to be established throughout the British colonies, at the urging of Sabine (q.v.) Lloyd was heavily involved. The Dublin facility served as a model for those far-flung stations and new observers were trained in the college. The association of European, Russian, and British observatories represented what Lloyd called “a spirit unparalleled in the history of science.” They constituted the first global network devoted to a scientific project. (see Observatories, overview). The specific idea of a network for magnetic observations lives on today in Intermagnet. Even his honeymoon in 1840 was subsumed in the busy program of overseas observatory visits that followed. Meanwhile his inventiveness continued to show itself in new and refined designs for instrumentation: for example an induction magnetometer for the vertical component (1841). The current notion that terrestrial magnetism was linked to meteorological phenomena led him to study the latter, with a characteristically systematic approach. He also explored the conjecture that internal electric currents were responsible for the Earth’s field, tried to relate this to the diurnal variation, and proposed new experiments to test this theory.

474

LLOYD, HUMPHREY (1808–1881)

Lloyd’s father Bartholomew had been an inspiring leader of the college as a reforming mathematician and provost. Humphrey followed his example in becoming provost, and served as president of the Royal Irish Academy and the British Association. He was recognized with such awards as the German “Pour le Mérite” and Fellowship of the Royal Society. At home, he was awarded the Royal Irish Academy’s Cunningham Medal. He wrote 8 books and 64 papers and reports that record the achievements and insights of a remarkable experimentalist, organizer, and educator. A bust and a number of portraits survive in his college. There his father had created a new impetus in mathematics; the son accomplished the same for experimental science. On his death, the Erasmus Smith Chair of Experimental and Natural Philosophy passed to George Francis Fitzgerald, together with much of Lloyd’s energy and ideals. He had “struck an almost twentieth century note in his emphasis on the importance to a university of the cultivation of scientific research” (McDowell and Webb, 1982). Deanis Weaire and J.M.D. Coey

Bibliography Lloyd, H., Sabine, E., and Ross, J.C., 1836. Observations on the Direction and Intensity of the Terrestrial Magnetic Force in Ireland. In the Fifth Report of the British Association for the Advancemment of Science. London: John Murray, pp. 44–51. McDowell, R.B., and Webb, D.A., 1982. Trinity College Dublin, 1592–1952: An Academic History. London: Cambridge University Press. O’Hara, J.G., 2003. Humphrey Lloyd 1800–1881. In McCartney, M., and Whitaker, A. Physicists of Ireland. Bristol: IOP Publishing, pp. 44–51.

Cross-references Gauss, Carl Friedrich (1777–1855) Hansteen, Christopher (1784–1873) Humboldt, Alexander von (1759–1859) Observatories, Overview Sabine, Edward (1788–1883)

M

MAGNETIC ANISOTROPY, SEDIMENTARY ROCKS AND STRAIN ALTERATION The relationship between anisotropy of magnetic susceptibility (AMS) fabrics (the shape and orientation of the AMS ellipsoid) and rock fabrics has been studied for over 40 years. Many studies have employed magnetic fabrics as an independent data set to compare with remanent magnetization directional data. Magnetic fabrics can also be very sensitive indicators of low-intensity strain, though deformation exceeding 10% to 20% often modifies or destroys the fabric to the point of making a functional interpretation impossible. This is in part due to diagenetic processes; for example, reduction, pressure solution, recrystallization, fluid migrations, and growth of new mineral constituents occurring with greater degrees of strain. Thus, AMS techniques are most often successful in investigating tectonic strains on weakly deformed sedimentary rocks. While AMS fabrics are most often easily interpreted in sedimentary rocks, the method is nonetheless complicated by the great range of primary sedimentary fabrics requiring correct interpretation before any modification to that fabric can be understood. It is important to note that AMS ellipsoids and grain-shape often do not correspond. Recent studies have pointed out the strong dependence of the AMS ellipsoid shape on mineral composition, for example, which can have a much larger effect than strain modification. Slight variations in ferromagnetic (or ferrimagnetic) trace minerals, for example, can strongly influence the shape and magnitude of the AMS ellipsoid, to the extent that composition-dependent variation may completely obscure the effects of strain. Finally, modeling and experimental studies have demonstrated that the relationship between incremental strain and the change in degree of anisotropy is highly complex. Some studies have determined power-law relationships, but with a great deal of variability reflecting differences in deformational style, intensity, and minerals that carry the anisotropy. Despite such complications, AMS methods as fabric indicators in sedimentary rocks have the advantages of being rapid, nondestructive, and result from the averaging of a susceptibility tensor from a very large population of magnetic grains. Weak deformation distinguished by AMS techniques may often be undetectable by nonmagnetic methods.

Depositional (primary) sedimentary fabrics and processes AMS studies have been directed at analyzing deep-sea current flow, bioturbation, fluvial and lacustrine fabrics, wind-deposited sand and loess fabrics, to name a few, as interpreted from AMS ellipsoids. Since

sedimentary rocks should nearly always have a primary magnetic fabric, any fabric acquired as a result of later deformation will be superimposed upon a preexisting depositional fabric. Therefore, interpretation of altered AMS fabrics must begin with interpretation of the earlier (primary) fabric, which controls any subsequent alteration. Studies of depositional AMS fabrics in sediments show that the shape and orientation of the susceptibility ellipsoid can be affected by a great variety of sedimentological processes predominantly formed during the deposition of particles from suspension in water or air (for a review, see Ellwood, 1980). Initial sedimentary fabrics are generally determined by gravity and hydrodynamics. Thus, to a first approximation, the shape and orientation of the AMS ellipsoid (which is the sum of the susceptibilities of a large number of grains present in the measured sample) is determined by both the characteristics of the grains themselves, and by the velocity and direction of transport of the medium through which they are carried. In general, sedimentary depositional fabrics are comparatively low in both susceptibility ( 0 or (b) Ms is perpendicular to s and ls < 0. Thus in case (a) the application of stress will cause Ms to rotate toward the stress axis, whereas in case (b) Ms will rotate toward the direction perpendicular to the axis of stress.

Magnetostatic energy The work, per unit volume, required to assemble a population of magnetic “free poles” into a particular configuration is called the magnetostatic energy. Magnetostatic calculations even for the simplest domain configurations are complex and usually require numerical methods. One of the simplest examples was addressed by Kittel (1949), who analyzed a semi-infinite plate of thickness L that contained lamellar domains of uniform width D, with spontaneous magnetizations normal to the plate surface (Figure M14). Walls were

E ¼ K0 þ K1 sin2 y þ K2 sin4 y þ    higher order terms: Here, K0, K1, K2 are the material’s magnetocrystalline anisotropy constants (units of erg cm–3 in cgs). When K1 > 0 and K2 > K1, the c-axis is the easy direction of magnetization, as in cobalt. In a cubic material, the magnetocrystalline anisotropy energy density is given by: E ¼ K0 þ K1 ða21 a22 þ a22 a23 þ a21 a23 Þ þ K2 ða21 a22 a23 Þ þ    higher order terms:

Figure M14 Illustration of the semi-infinite, magnetized plate of thickness L, whose magnetostatic energy was calculated by Kittel (1949). The plate contains domains of identical width D, with magnetizations perpendicular to the plate’s surface. Walls are assumed to be infinitely thin.

492

MAGNETIC DOMAINS

assumed to be of negligible thickness with respect to the domains’ widths and, therefore, the magnetostatic energies due to the walls’ magnetic moments were not taken into consideration. Using the magnetic potential and a Fourier series approach, Kittel obtained Em ¼ 1.705 Ms2 D, where Em is the magnetostatic energy, per unit area of plate surface (erg cm–2, in cgs). Note, however, that the expression above is an approximation, because Kittel did not account for magnetostatic interactions among all combinations of polarized slabs on the top and bottom surfaces of the plate. Subsequently, these extra energies were included in calculations by Rhodes and Rowlands (1954). Also, they relaxed the assumption of semi-infinite geometry and addressed finite, rectangular grains. Their numerical calculations yielded “Rhodes and Rowlands” functions, with which one may calculate the total magnetostatic energy of rectangular grains of specified relative dimensions.

Energy and width of the domain wall In the classical sense, a domain wall is the transition region where the spontaneous magnetization changes direction from one domain to the next. According to classical theories, each domain is spontaneously magnetized in one direction and is clearly distinct from the wall. (Micromagnetic theories relax this assumption and will be discussed in a later section.) The two most important energies that affect the domain wall’s energy and width are (1) exchange energy and (2) anisotropy energy, the latter being due either to magnetocrystalline anisotropy or stress. The magnetostatic energy of the wall itself also plays a role; this energy was first analyzed by Amar (1958) and grows especially important when the wall width approaches that of the particle. Were magnetocrystalline energy acting alone, the spin vectors would change directions abruptly from one domain to the next. For example, if a substance possessed very strong, uniaxial crystalline anisotropy and exchange energy was very much weaker, then the lowest energy “transition” between two adjacent domains would virtually consist of two adjacent spins pointing 180
apart. This kind of abrupt transition usually involves a large amount of exchange energy, however, because exchange energy is minimized when adjacent spins are parallel. Because minimization of anisotropy energy alone would produce an infinitely thin wall, while minimization of exchange energy alone would produce an infinitely broad wall, the spins in a wall rotate gradually from one domain to the next (Figure M15). This produces a wall with finite width. By this model, spins in a wall reach an equilibrium configuration when magnetocrystalline and exchange energies are balanced. For the 180
Bloch wall shown in Figure M15 in which the spins rotate through 180
, the wall energy per unit area of wall surface is Ew ¼ 2ðJex S 2 p2 K=aÞ1=2 ¼ 2pðAKÞ1=2

Figure M15 Illustration of spins in a 180
Bloch wall. y is the angle between a spin and the easy axis of magnetization; f is the angle between adjacent spins.

where Jex is exchange integral, S is spin, K is anisotropy constant due to magnetocrystalline anisotropy energy and/or stress energy, A is exchange constant (equal to JexS2/a), and a is lattice constant. The width of a 180
wall is dw ¼ ðJS 2 p2 =KaÞ1=2 ¼ pðA=KÞ1=2 In magnetite, values predicted for Ew and dw are approximately 0.9 erg cm–2 and 0.3 mm, respectively (e.g., see Dunlop and Özdemir, 1997).

Domain width versus grain size For the simple, planar domain structure illustrated in Figure M14, domain width D can be calculated for the lowest energy state by minimizing the sum of magnetostatic and wall energies per unit volume of material. One obtains the familiar half-power law derived originally by Kittel (1949): D ¼ ð1=Ms ÞðEw L=1:705Þ1=2 By this model, domain width increases with the half-power of crystal thickness, so long as the crystal occupies the state of absolute minimum energy. Experimental determinations of domain width versus grain thickness will be discussed in a subsequent section. Rocks, however, rarely contain magnetic minerals in the shape of thin platelets that might be fair approximations to the semi-infinite plate of Kittel’s model. Moskowitz and Halgedahl (1987) calculated the number of domains versus grain size in rectangular particles of x ¼ 0.6 titanomagnetite (“TM60”: Fe2.4Ti6O4) containing planar domains separated by 180
walls. Magnetostatic energy, including that due to the walls’ moments, was determined with the method developed by Amar (1958), based on Rhodes and Rowlands’ method (1954). It was assumed that grains occupied states of absolute minimum energy. The effects of two dominant anisotropies were investigated: magnetocrystalline anisotropy (zero stress) and a uniaxial stress (s ¼ 100 MPa) which was strong enough to completely outweigh the crystalline term. Their calculations yielded two principal results: (1) particles encompassing a wide range of grain sizes can contain the same number of domains, and (2) a plot of N (number of domains) versus L (grain thickness) is fitted well by a power law N / L1/2. It follows that domain width D also follows a half-power law in L. Thus, the general form of functional dependence of D on L is the same for finite, rectangular grains, and semi-infinite plates.

Single-domain/two-domain transition size Grains containing only two or three domains can rival the remanences and coercivities of SD grains, and such particles are referred to as being “pseudosingle-domain,” or PSD. PSD grains can be common in many rocks and, in terms of interpreting rock magnetic behavior, it is important to know their size ranges in different magnetic minerals. The onset of PSD behavior begins at the single-domain–two-domain transition size, d0. In general, a particle will favor a two-domain over a SD state because the magnetostatic energy associated with two domains is much lower than that of a saturated particle. However, below d0 the energy price of adding a domain wall is too great to produce a state of minimum energy. At d0 the total energy of the two-domain state (E2D) and the SD state are equal; above d0, E2D < ESD, while E2D > ESD below it. In zero applied field this transition size depends on the material’s magnetic properties, its state of stress, the particle shape, and temperature. Moskowitz and Banerjee (1979) calculated total energies of SD, two-domain, and three-domain cubes of stress-free magnetite containing lamellar domains at room temperature. Magnetostatic energy of the walls was included in their calculations (Amar, 1958). They obtained d0  0.08 mm.

MAGNETIC DOMAINS

Similar calculations for TM60 by Moskowitz and Halgedahl (1987) yielded d0  0.5 mm for unstressed particles at room temperature. Raising the stress level to 100 MPa shifted d0 upward to 1 mm.

Domains and domain walls at crystal surfaces In relatively thick crystals, domains and domain walls may change their geometric styles near and at crystal surfaces, in order to lower the total magnetostatic energy with respect to that of the “open” structure shown in Figure M14. The particular style depends largely on the dominant kind of anisotropy, as well as on the relative strengths of magnetostatic and anisotropy energies. When 2pMs2 /K  1 in a uniaxial material, prism-shaped closure domains bounded by 90
walls, in which spins rotate through 90
from one domain to the next, may completely close off magnetic flux at the crystal surface. Lamellar domains separated by 180
walls may fill the crystal’s volume (Figure M16a). In this case, the two main energies originate from the walls and from magnetoelastic energy due to magnetostrictive strain where closure

493

domains abut body domains. Closure domain structures such as these can subdivide further into elaborate arrays of smaller closure and nested spike domains, if the crystal is sufficiently thick. Because closure domains can either greatly reduce or completely eliminate magnetostatic energy, the body domains can be several times broader than predicted for the Kittel-like, “open” structure. When 2pMs2 /K  1 in a uniaxial substance like barium ferrite, a large amount of anisotropy energy results when Ms is perpendicular to the “easy” axis. Therefore, the style of surface closure shown in Figure M16a is energetically unfavorable. Instead, a very different style of surface domain structure may evolve in thick crystals. Walls which are planar within the body of the crystal can become wavy at the surface (Figure M16b). In extremely thick crystals, wavy walls can alternate with rows of reverse spikes. These elaborate surface domain structures lower magnetostatic energy by achieving a closer mixture of “positive” and “negative” free magnetic poles (e.g., see Szymczak, 1968). Large crystals governed by cubic magnetocrystalline anisotropy reduce surface flux through prism-shaped closure domains at the surface. When are easy directions, as in iron, closure domains are bounded by 90
walls (Figure M16a). When are easy directions, as in magnetite, closure domains are bounded by 71
and 109
walls and, within the closure domains, Ms is canted with respect to the crystal surface (Figure M16c).

Temperature dependence of domain structure

Figure M16 Three styles in which domains and domain walls may terminate at a crystal surface. (a) Illustration of prism-shaped surface closure domains at the surface of a material which is either uniaxial, with 2pM2s /K  1.0, or cubic, such as iron, whose easy axes are along . Here, 90
walls separate closure domains from the principal “body” domains that fill most of the crystal. Arrows indicate the sense of spontaneous magnetization within the domains. (b) Illustration of wavy walls at the surface of a uniaxial material with 2pM2s /K < 1.0. Waviness dies out with increasing distance from the surface. (c) Prism-shaped closure domains at the surface of a cubic material, such as magnetite, whose easy directions of magnetization are along . Closure domains and body domains are separated by 71
and 109
walls.

Understanding how domain structure evolves during both heating to and cooling from the Curie point is crucial to understanding the acquisition and thermal stability of thermal remanent magnetization (TRM). If the number of domains changes significantly during cooling from the Curie point, it is reasonable to hypothesize that TRM will not become blocked until the overall domain structure reaches a stable configuration. According to Kittel’s original model (Figure M14), grains will nucleate (add) domain walls and domains with heating in zero field, if the wall energy drops more rapidly with increasing temperature than does the magnetostatic term. Energywise, in this first case a particle can “afford” to add domains with heating. Conversely, during cooling from the Curie point a grain will denucleate (lose) domains and domain walls if wall energy rises more quickly than does Ms2 with decreasing temperature. If wall energy drops less rapidly with increasing temperature than does Ms2 , then the opposite scenarios apply. Of course, such behavior relies on the assumptions that the particle is able to maintain a global energy minimum (GEM) domain state at all temperatures and that the total magnetostatic energy of the walls themselves can be ignored. Using Amar’s (1958) model, Moskowitz and Halgedahl (1987) calculated the number of domains between room temperature and the Curie point in parallelepipeds of TM60. As discussed earlier, they investigated two cases: dominant crystalline anisotropy (zero stress) and high stress (s ¼ 100 MPa). Magnetostatic energy from wall moments was included in the calculations. At all temperatures they assumed that particles occupied GEM domain states. Because the temperature dependences of the material constants A (exchange constant) and l (magnetostriction constant) for TM60 had not been constrained well by experiments, they ran six models to bracket the least rapid and most rapid drops in wall energy with heating. In TM60 grains larger than a few micrometers, most of their results gave an increase in the number of domains with heating. Exceptions to this overall pattern were cases in which walls broadened so dramatically with increasing temperature that they nearly filled the particle and rendered nucleation unfavorable. During cooling from the Curie point in zero field, the domain “blocking temperature”—that is, the temperature below which the number of domains remained constant—increased both with decreasing grain size and with internal stress.

494

MAGNETIC DOMAINS

Micromagnetic models In contrast to classical models of magnetic domain structure, micromagnetic models do not assume the presence of discrete domains— that is, relatively large volumes in a crystal where the spontaneous magnetization points in a single direction. Instead, micromagnetic models allow the orientations of Ms-vectors to vary among extremely small subvolumes, into which a grain is divided. A stable configuration is obtained numerically when the sum of exchange, anisotropy, and magnetostatic energies is minimum. To reduce computation time, all micromagnetic models for magnetite run to date assume unstressed, defect-free crystals. Moon and Merrill (1984, 1985) were the first in rock magnetism to construct one-dimensional (1D) micromagnetic models for defect-free magnetite cubes. To simplify calculations, they assumed that magnetite was uniaxial. In their models, they subdivided a cube into K thin, rectangular lamellae. Within a Kth lamella, the spin vectors were parallel to the lamella’s largest surface but oriented at angle yk with respect to the easy axis of anisotropy. The distribution of angles was varied throughout the cube until a state of minimum energy was obtained. Because the yk’s varied continuously across the grain, there was no sharp demarcation between domains and domain walls. For practical purposes, a wall’s effective width was defined as the region in which the moments rotated most rapidly. Nucleation was modeled by “marching” a fresh wall from a particle’s edge into the interior. A nucleation barrier was determined by calculating the maximum rise of total energy as the wall moved to its final location and as preexisting walls adjusted their positions to accommodate the new wall. The denucleation barrier was determined by reversing the nucleation process. Moon and Merill’s major breakthrough was the discovery that a particle can occupy a range of local energy minimum (LEM) domain states. Each LEM state is characterized by a unique number of domains and is separated from adjacent states by energy barriers. As illustrated in Figure M17, the GEM state is the configuration of lowest energy but, owing to the energy barriers between states, a LEM state can be quite stable as well. Any given LEM can be stable over a broad range of grain sizes.

Figure M17 Diagram illustrating the relative energies and energy barriers associated with local energy minimum (LEM) domain states. Each LEM state is characterized by a unique number of domains and is separated from adjacent states by nucleation and denucleation energy barriers. In this diagram, the global energy minimum, or GEM, domain state has five domains.

Several authors have extended micromagnetic calculations for magnetite to two- and three-dimensions (e.g., Williams and Dunlop, 1989, 1990; Newell et al., 1993; Xu et al., 1994; Fabian et al., 1996; Fukuma and Dunlop, 1998; Williams and Wright, 1998). In 2D models, grains are subdivided into rods. In 3D models, the crystal is subdivided into a multiplicity of extremely small, cubic cells (e.g.,  0.01 mm on a side), within each of which Ms represents the average magnetization over several hundred atomic dipole moments. Each cell’s Ms-vector is oriented at angle y with respect to the easy axis, and the y’s are varied independently with respect to their neighbors until an energy minimum is achieved. Owing to the extremely large number of cells and the even larger number of computations, energy calculations begin with an initial guess of how the final, minimum energy structure might appear. To date, the largest magnetite cubes addressed by 3D models are only a few micrometers in size, due to limitations of computing time (e.g., Williams and Wright, 1998). 2D and 3D micromagnetic models yield a variety of exotic, nonuniform configurations of magnetization, such as “flower” and “vortex” states. Analogous to Moon and Merrill’s results, these models yield both LEM and GEM states, although very different in their Ms-structures from those of 1D models. For example, Figure M18 illustrates a “flower” state in a cube magnetized parallel to the z-axis. The flower state is reminiscent of a classical SD state of uniform magnetization, except that the Ms-vectors are canted at and near the crystal surface. Increasing the cube size makes other nonuniform states energetically favorable (e.g., Fabian et al., 1996). For example, according to 3D models of magnetite cubes between 0.01 and 1.0 mm in size, the flower state is the lowest energy state between about 0.05 and 0.07 mm (Williams and Wright, 1998). Further increase of cube size yields a vortex state, in which the Ms-vectors circulate around a closed loop within the grain. By Williams and Wright’s calculations, both flower and vortex states are stable between about 0.07 and 0.22 mm, although the vortex state has the lower energy. Raising the cube size to the 0.22–1.0 mm range results in the flower state becoming unstable, so that the vortex state is the only possible state. In relatively large cubes of magnetite—e.g., 4 or 5 mm—2D and 3D models yield Ms-configurations closely approaching those of

Figure M18 Illustration of a “flower” state obtained through three-dimensional micromagnetic modeling of a cube largely magnetized along the cube’s z-axis.

MAGNETIC DOMAINS

classical domain structures expected for magnetite; some models predict “body” domains separated by domain walls, with closure domains at the surface (e.g., Xu et al., 1994; Williams and Wright, 1998). According to several micromagnetic models of hysteresis in submicron magnetite, magnetization reversal can occur through LEM-LEM transitions (e.g., flower to vortex state). Reversal can take place through almost independent reversals of the particle’s core and outer shell (e.g., Williams and Dunlop, 1995). Dunlop et al. (1994) used 1D micromagnetic models to investigate transdomain TRM. Transdomain TRM—that is, acquisition of TRM through LEM-LEM transitions—had been proposed earlier by Halgedahl (1991), who observed denucleation of walls and domains in titanomagnetite during cooling (see below for a discussion of these results). In particular, Halgedahl’s observations strongly suggested that denucleation could give rise to SD-like TRMs in grains that, after other magnetic treatments, contained domain walls. Furthermore, Halgedahl (1991) found that a grain could “arrive” in a range of LEM states after replicate TRM acquisitions. To determine whether transdomain TRM could be acquired by stress-free, submicron magnetite particles free of defects, Dunlop et al. (1994) calculated the energy barriers for all combinations among single domain-two-domain-three-domain transitions with decreasing temperatures from the Curie point of magnetite in a weak external field. They assumed that LEM-LEM transitions were driven by thermal fluctuations across LEM-LEM energy barriers and that thermalequilibrium populations of LEM states were governed by Boltzmann statistics. According to their results, after acquiring TRM most populations would be overwhelmingly biased toward GEM domain states, and an individual particle should not exhibit a range of LEM states after several identical TRM runs. Using renormalization group theory, Ye and Merrill (1995) arrived at a very different conclusion. According to their calculations, shortrange ordering of spins just below the Curie point could give rise to a variety of LEM states in the same particle after replicate coolings. These conflicting theoretical results are discussed below in the context of experiments.

Domain observations Methods of imaging domains and domain walls Rock magnetists have mainly used three methods to image domains and domain walls: the Bitter pattern method, the magneto-optical Kerr effect (MOKE), and magnetic force microscopy (MFM). Two other methods—transmission electron microscopy (TEM) and off-axis electron holography—have been used in a very limited number of studies on magnetite. The Bitter method images domain walls through application of a magnetic colloid to a smooth, polished surface carefully prepared to eliminate residual strain from grinding (e.g., see details in Halgedahl and Fuller, 1983). Colloid particles are attracted by the magnetic field gradients around walls, giving rise to Bitter patterns. When viewed under reflected light, patterns appear as dark lines against a grain’s bright, polished surface. With liquid colloid, this method can resolve details as small as about 1 mm. If colloid is dried on the sample surface and the resultant pattern is viewed in a scanning electron microscope (SEM), features as small as a few tenths of one micrometer can be resolved (Moskowitz et al., 1988; Soffel et al., 1990). The MOKE is based on the rotation of the polarization plane of incident light by the magnetization at a grain’s surface. Unlike the Bitter method, the MOKE images domains themselves, rather than domain walls. Domains appear as areas of dark and light, the result of their contrasting magnetic polarities (Hoffmann et al., 1987; Worm et al., 1991; Heider and Hoffmann, 1992; Ambatiello et al., 1999). The MOKE has the same resolving power as the Bitter method. In the MFM, a magnetized, needle-shaped tip is vibrated above the highly polished surface of a magnetic sample. Voltage is induced in the

495

tip by the magnetic force gradients resulting from domains and domain walls. The effects of any surface topography are removed by scanning the surface with the nonmagnetic tip of an atomic force microscope. The MFM can resolve magnetic features as small as 0.01 mm (Williams et al., 1992; Proksch et al., 1994; Moloni et al., 1996; Pokhil and Moskowitz, 1996, 1997; Frandson et al., 2004).

Styles of domains observed in magnetic minerals of paleomagnetic significance In rock magnetism, the majority of domain observation studies have focused on four magnetic minerals, all important to paleomagnetism: pyrrhotite (Fe7S8), titanomagnetite of roughly intermediate composition (near Fe2.4Ti6O4, or TM60), magnetite (Fe3O4), and hematite (Fe2O3). Owing to its high magnetocrystalline anisotropy constant and relatively weak magnetostriction constant, pyrrhotite behaves magnetically as a uniaxial material. When studied with the Bitter method, pyrrhotite often exhibits fairly simple domain patterns, which suggest lamellar domains separated by 180
walls (Figure M19a) (Soffel, 1977; Halgedahl and Fuller, 1983). Despite being cubic, intermediate titanomagnetites rarely, if ever, exhibit the arrays of closure domains, 71
, and 109
walls that one would predict. Instead, these minerals usually display very complex patterns of densely spaced, curved walls. Possibly, these complex structures result from varying amounts of strain within the particle (Appel and Soffel, 1984, 1985). Occasionally, grains exhibit simple arrays of parallel walls, such as that shown in Figure M20 (e.g., Halgedahl and Fuller, 1980, 1981), but it is not unusual to observe wavy walls alternating with rows of reverse spikes (Halgedahl, 1987; Moskowitz et al., 1988). Both simple and wavy patterns suggest a dominant, internal stress that yields a uniaxial anisotropy, although the origin of this stress is still unclear. On one hand, it could originate from the mechanical polishing required to prepare samples for domain studies. Alternatively, in igneous rock samples stress could be generated during cooling, due to differences in the coefficients of thermal expansion among the various minerals. Small magnetite grains randomly dispersed in a rock or a synthetic rock-like matrix generally display simple arrays of straight domain walls, if the domains’ magnetizations are sufficiently close to being parallel to the surface of view (e.g., Worm et al., 1991; Geiß et al., 1996) (Figure M21). In such samples, however, there appears to be a paucity of closure domains, perhaps the result of observation surfaces being other than {100} planes. In an early study by Bogdanov and Vlasov (1965), Bitter patterns of 71
and 109
walls were observed on cleavage planes of a few small magnetite particles obtained by crushing a natural crystal. Similarly, from Bitter patterns Boyd et al. (1984) found both closure domains and networks of 180
, 71
, and 109
walls in several magnetite particles in a granodiorite. Body domains accompanied by closure domains were also reported by Smith (1980), who applied the TEM method to a

Figure M19 Bitter patterns on a particle of natural pyrrhotite (a) after demagnetization in an alternating field of 1000 Oe and (b) in an apparently SD-like state after acquiring saturation remanence in 15 kOe.

496

MAGNETIC DOMAINS

Figure M20 Bitter pattern on a grain of intermediate titanomagnetite (x  0.6) in oceanic basalt drilled near the Mid-Atlantic Ridge.

Figure M22 Bitter pattern of a wall on a very large (approximately 150 mm width, 1 mm diameter) platelet of natural hematite from Elba, Italy. At most this platelet exhibits only 1–2 principal walls, although small edge domains are often observed. Apparently, the wall is bowing around a defect near the upper right-hand edge of the photograph. Only a small part of the crystal surface is shown.

contain very few walls and a fairly simple domain structure, owing to hematite’s weak spontaneous magnetization (about 2 emu cm–3) and low magnetostatic energy (Figure M22) (Halgedahl, 1995, 1998).

Observed number of domains versus grain size

Figure M21 Bitter pattern on a grain of magnetite synthesized with the glass-ceramic method. In this particular state of magnetization, the grain contains four walls, whose lengths are commensurate with the particle’s length. Note that one wall is pinned very near the particle’s extreme left-hand edge. The small triangular patterns at the lower edge of the particle represent walls which enclose small reverse spike domains.

few small magnetite crystals in rock. Note that this method images domains through deflection of the electron beam by the magnetizations within domains. Consequently, the TEM method reveals volume domains, not just their surface manifestations. Similarly, large magnetite crystals cut and polished on {100} planes almost invariably exhibit the closure domains bounded by 71
and 109
walls expected for truly multidomain magnetite (e.g., Bogdanov and Vlasov, 1966; Özdemir and Dunlop, 1993, 1997; Özdemir et al., 1995). To date, domain studies on hematite have been limited to large (e.g., 100 mm–1 mm) platelets. Even large crystals such as these

Both Kittel’s original model of domains in a semi-infinite platelet and calculations for finite grains by (e.g., Moskowitz and Halgedahl, 1987) lead to the prediction that domain width D / L1/2, where L is plate or particle thickness. These predictions are supported by domain studies of natural magnetic minerals, which generally yield a powerlaw dependence of D on L, although the power may differ somewhat from 0.5. In rock magnetism, Soffel (1971) was the first to study the grainsize dependence of the number of domains in a paleomagnetically important magnetic mineral. From Bitter patterns on grains of natural, intermediate (x  0.55) titanomagnetite in a basalt, Soffel determined that, on average, N (number of domains) / L1/2, where L is the average particle size. This relation translates to D (domain width) / L1/2. Extrapolation of these data to N ¼ 1 (or, equivalently, to D ¼ L) yielded a single domain-two domain transition size of 0.6 mm for this composition. By assuming that Kittel’s model could be applied to roughly equidimensional grains without serious errors and that Ms ¼ 100 emu cm–3 for x  0.55, Soffel obtained a wall energy density of about 1 erg cm–2. Bitter patterns on natural pyrrhotite in the Bad Berneck diabase from Bavaria were studied both by Soffel (1977) and by Halgedahl and Fuller (1983). Halgedahl and Fuller determined the dependence of domain width on grain size for three states of magnetization: NRM, after alternating field demagnetization (AFD) in a peak field of 1000 Oe, and saturation remanence imparted in a field of 15 kOe. Particles that contained walls in these three different states yielded similar power-law dependences of D on L: D / L0.43 (NRM), D / L0.40 (AFD), and D / L0.45 (saturation remanence). The single domain-two-domain boundary sizes estimated for the three states also

MAGNETIC DOMAINS

were similar, falling between 1.5 and 2 mm. These results were consistent with those of Soffel (1977). Likewise, Geiß et al. (1996) obtained D / L0.45 from Bitter patterns on synthetic magnetite particles grown in a glass-ceramic matrix (Worm and Markert, 1987). They obtained a single domain-twodomain transition size for magnetite at approximately 0.25 mm. In contrast to the results discussed above for titanomagnetite, pyrrhotite, and glass-ceramic magnetite, one of Dunlop and Özdemir’s (1997) data compilations for magnetite samples from several very different provenances yielded D / L0.25. The cause of the discrepancy between this result and those obtained from other samples is not understood.

Domain wall widths When studied in the SEM, patterns of dried magnetic colloid afford high-resolution views of domain walls—or, more precisely, the colloid accumulations around walls. Moskowitz et al. (1988) applied this method to polycrystalline pellets of synthetic titanomagnetite substituted with aluminum and magnesium (Fe2.2Al0.1Mg0.1Ti0.6O4). Henceforth, this composition is referred to as “AMTM60.” Dried Bitter patterns on unpolished surfaces were virtually identical in style to those common to materials controlled by strong uniaxial anisotropy, such as barium ferrite. Patterns indicated closely spaced stripe domains, sinusoidally wavy walls, and elaborate wavy walls, which alternated with nested arrays of reverse spikes. Direct measurements from SEM photographs yielded wall widths between 0.170 and 0.400 mm. In the same study, Moskowitz et al. (1988) also measured the wavelengths, amplitudes, and domain widths of the sinusoidally wavy patterns. Using these parameters and the experimental value of spontaneous magnetization for AMTM60, they applied Szymczak’s (1968) model to estimate the exchange constant and the uniaxial anisotropy constant, the latter presumably due to residual stress generated, while the pellet cooled from the sintering temperature. Very high-resolution images of domains and domain walls have been obtained with MFM on magnetite. The first images were reported by Williams et al. (1992) from a {110} surface on a large crystal of natural magnetite. They recorded a magnetic force profile across a 180
wall. This record indicated that the spins within the wall reversed their polarity of rotation along the length of the wall, demonstrating that walls in real materials can be much more complex than those portrayed by simple models. In a study of glass-ceramic magnetite particles with MFM, Pokhil and Moskowitz (1996, 1997) found that, along its length, an individual wall can be subdivided into several segments of opposite polarity. The segments are separated by Bloch lines, the transition regions where polarity changes sense. Rather than being linear, subdivided walls zigzag across a grain. Analogous to wavy walls, the zigzags help to reduce a grain’s total magnetostatic energy. The number of Bloch lines within any specific wall was found to vary with repeated AF demagnetization treatments. Thus walls, like particles, can occupy LEM states. Owing to its high-resolution capabilities, the MFM can provide estimates of wall width. Proksch et al. (1994) obtained MFM profiles across a 180
wall on a {110} surface of natural magnetite. The deconvolved signal yielded a half-width, assumed to be commensurate with wall width, of about 0.21 mm.

Experimental evidence for local energy minimum domain states In their study of Bitter patterns on natural pyrrhotite, Halgedahl and Fuller (1983) noted that grains of virtually identical size could contain very different numbers of domains, despite these same particles having undergone the same magnetic treatments. Moreover, they found that an individual particle could arrive in very different domain states—i.e., with different numbers of walls—after different cycles of minor

497

hysteresis. Clearly, such particles did not always occupy a domain state of absolute minimum energy. These observations led Halgedahl and Fuller (1983) to the conclusion that a particle could occupy domain states other than the ground state. Shortly thereafter, Moon and Merrill (1984, 1985) dubbed these states as LEM states, on the basis on their one-dimensional micromagnetic results for magnetite. Subsequent domain observation studies by Geiß et al. (1996) provided strong evidence for LEM states in magnetite as well.

LEM states and thermomagnetic treatments: hysteresis A particularly unexpected type of LEM state in pyrrhotite and intermediate titanomagnetite is a SD-like state that certain particles can occupy after being saturated in a strong external field, even though these same particles readily accommodate walls in other states of magnetization. Bitter patterns on intermediate (x  0.6) titanomagnetite in oceanic basalt and on natural pyrrhotite in diabase were studied during hysteresis by Halgedahl and Fuller (1980, 1983). An electromagnet was built around the microscope stage for these experiments, permitting one to track the evolution of walls continuously, while varying the applied field. In states of saturation remanence, most particles in a large population contained one or more walls, as expected on the basis of previous theories. However, a significant percentage—nearly 40%—of the finer (5–15 mm) particles observed appeared saturated after the maximum field was shut off (Figure M19a and b). Within a large population of such grains it was found that the number of domains was described well by a Poisson distribution. Halgedahl and Fuller (1980, 1983) proposed that grains, which failed to nucleate walls, could make a substantial contribution to saturation remanence. Furthermore, such grains might explain much of the known decrease in the ratio of saturation remanent magnetization to saturation magnetization with increasing particle size. This hypothesis was extended to the case of weak-field TRM. Particles in SD-like states remained saturated, until nucleation was accomplished by applying a back-field of sufficient magnitude. In some cases, the nucleating field was strong enough to sweep a freshly nucleated wall across the particle and thereby result in a state of saturation (or near-saturation) in the opposite sense. In other cases, the new wall moved until it was stopped by a pinning site. In the smaller grains studied (e.g., smaller than 20 mm) the nucleation field dropped off with increasing grain size L according to a power law in L1/2. Boyd et al. (1984) reported Bitter patterns on natural magnetite grains carrying saturation remanence, and these patterns suggested SD-like states. These results were surprising, in view of magnetite’s strong tendency to self-demagnetize. Subsequent domain observations and hysteresis studies of hematite platelets from Elba, Italy demonstrated that, after exposure to strong fields, even large crystals could arrive in states suggesting near-saturation. Often, however, small, residual domains clung to the platelet’s surface in these states. Back-fields were necessary to nucleate principal domains, and these nucleation fields also followed a power-law in L1/2 (Halgedahl, 1995, 1998). Results from titanomagnetite and pyrrhotite were interpreted by Halgedahl and Fuller (1980, 1983) in light of previous theoretical and experimental work on high-anisotropy, industrial materials synthesized to produce magnetically hard, permanent magnets. In such materials, nucleation of walls is an energetically difficult process (Brown, 1963; Becker, 1969, 1971a,b, 1976). According to Brown’s original theory, the nucleation of a fresh wall requires the following condition to be fulfilled: * * 2K=Ms < jH appl ¼ Hd j Here, 2K/Ms is the local anisotropy field at the nucleation site; Hd is the local demagnetizing field at the site, assumed to be opposite in sense to Ms; and Happl is the field applied to the sample. In samples

498

MAGNETIC DOMAINS

where back-fields are necessary to accomplish nucleation, Happl is opposite to the sample’s original saturation magnetization. Wall nucleation may occur either at crystal defects where 2K/Ms is anomalously low or where the local demagnetizing field Hd is anomalously high, as at shape defects on a crystal’s surface. Note that Brown’s theory of nucleation differs from the kind of “global” nucleation resulting from micromagnetic models. Brown-type nucleation may depend on the presence of defects, which fulfill the above condition. By contrast, micromagnetic models assume that particles are defect-free. However, SD-like states also can result from submicroscopic walls being trapped by surface and volume defects (Becker, 1969, 1971a,b, 1976). There are two cases. First, when a saturating field is shut off, a submicroscopic, virgin wall can nucleate but remain pinned at a local defect. In the second case, a strong applied field may be insufficient to completely saturate a particle. When this occurs, preexisting domain walls may be driven into surface or volume defects and become trapped. In both cases, the particle appears to be saturated at the top of a hysteresis loop. Experimentally, the saturation remanent state appears to be SD-like. A back-field, opposite in sense to the maximum field, is required to break free these residual walls and the submicroscopic domains which they enclose. These two phenomena can be distinguished by observing both the position where a reverse nucleus first appears in a grain and the field required for its appearance. If complete saturation of a grain, followed by nucleation of a fresh wall, has occurred, then the nucleus will always appear at the same location and in the same back-field, regardless of the maximum field’s polarity. Of course, the submicroscopic nucleus could have been trapped by a defect just after it was nucleated initially. If the grain was not saturated completely, then both the location where the nucleus appears and the field required to expand this nucleus and make it visible depend on the direction in which a wall was driven in the first place and on the back-field’s magnitude— that is, on the specific trapping site (Becker, 1969, 1971a,b, 1976; Halgedahl and Fuller, 1983). The style of domain structure and the range of LEM states that a particle can occupy can depend on thermomagnetic history. In a study of Bitter patterns on natural, polycrystalline pyrrhotite, Halgedahl and Fuller (1981) discovered that crystallites displayed arrays of undulating walls on their surfaces after acquiring TRM in a weak external field. By contrast, after AF demagnetization in a strong peak field the same crystallites exhibited planar walls. Evidently, cooling through the TRM blocking temperature locked in a high-temperature configuration of walls, whose curved shapes promoted lower magnetostatic energy than did a planar geometry. As indicated by Moon and Merrill’s theoretical results for magnetite, a particular LEM state can be stable across a broad range of grain size. This prediction is born out by experiments and calculations by Halgedahl and Ye (2000) and Ye and Halgedahl (2000), who investigated the effects of mechanical thinning on domain states in natural pyrrhotite particles. In their experiments, several individual pyrrhotite grains in a diabase were mechanically thinned and their Bitter patterns observed after each thinning step. Despite some grains being thinned to about one-fourth of their initial diameter, the widths of surviving domains and positions of surviving walls remained unaffected (Figure M23). Neither nucleations nor denucleations were observed, although calculations indicated that thinning would cause significant changes in GEM domain states. Bitter patterns on crystallites of polycrystalline AMTM60 were studied after many replicate TRM acquisition and AF demagnetization experiments (Halgedahl, 1991). In each particle, the number of domains varied from one experiment to the next, describing a distribution of TRM states. For states of weak-field TRM, this distribution could be broad and could include SD-like states (Figure M24). After replicate AF demagnetizations, however, typical distributions were narrow and clustered about a most probable state, possibly the GEM state.

Figure M23 Bitter patterns on a grain of natural pyrrhotite in a diabase before and after mechanical thinning. (a) Initial state, before thinning, (b) after thinning the particle to about one-half of its original length along a direction parallel to the trends of the Bitter lines, and (c) final state, after the particle has been thinned to about one-fourth of its original length.

Thermal evolution of magnetic domain structures observed at elevated temperatures The manner in which domain structure evolves with temperature carries clear implications for TRM. In very early models of TRM, such changes were either ignored or were assumed not to occur. Instead, some models assumed that the number of domains remained constant from the Curie point to room temperature in a weak external field, and that the blocking of TRM was controlled solely by the growth of energy barriers associated with defects which, eventually, locked walls in place. Yet changes in the number of domains with temperature could profoundly affect how pseudosingle-domain and multidomain particles acquire TRM. Temperature-induced nucleations or denucleations of walls should trigger sudden changes of the internal demagnetizing field. As a result, preexisting walls, which survive a domain transition, could be dislodged from imperfections where they were pinned initially at higher temperatures. In rock magnetism, domain observation experiments to study the thermal response of domain structure have focused on two magnetic minerals: (a) magnetite, owing to its great importance to the paleomagnetic

MAGNETIC DOMAINS

Figure M24 Bitter patterns on a crystallite of titanomagnetite (“AMTM60”: see text) after each of eight replicate TRM acquisition runs in the Earth’s field.

signal carried by many rocks, and (b) intermediate titanomagnetite, owing to its significance to marine magnetic anomalies and to its moderately low-Curie temperature, which renders study of Bitter patterns feasible at moderate temperatures. High-temperature experiments have yielded highly variable results. The first domain observations on magnetite above room temperature in the Earth’s field were made by Heider et al. (1988), using particles of hydrothermally recrystallized magnetite embedded in an epoxy matrix. Depending on the particle, Bitter patterns could be followed to approximately 200
C, above which temperature the patterns grew too faint to distinguish against a grain’s bright background. Surprisingly, heating to very moderate temperatures drove certain walls across much of a particle; in some cases, denucleation occurred. Upon cooling, walls reassembled in a similar, though not identical, arrangement to that observed initially at room temperature. In some cases repeated thermal cycling between room temperature and about 200
C produced different numbers of domains in the same particle. Ambatiello et al. (1995) used the MOKE to study domain widths versus temperature in several large (>5 mm) crystals of natural magnetite cut and polished on {110} planes. At room temperature, these planes were dominated by broad (40–90 mm), lamellar domains separated by 180
walls, terminating in closure domains at crystal edges. On {111} planes they found complex, nested arrays of very small closure domains, which finely subdivided the main closure structure. Unlike the Bitter technique, the MOKE is applicable to the Curie point of magnetite (580
C), at least in theory. In practice, the amount

499

of Kerr rotation in magnetite is very small even at room temperature; this rotation decreases progressively as Ms drops with heating. As the Curie point is approached, the contrast between adjacent domains grows exceedingly faint. For this reason, Ambatiello and colleagues successfully imaged domains to 555
C, but no higher. According to their observations, domain widths generally increased with heating, and such changes were thermally reversible. However, the temperature that triggered changes depended on the provenance of the sample. In some samples, domain widths remained nearly constant until significant broadening of domains occurred above 400
C. In other samples, domains began to broaden at far lower temperatures. To explain the overall character of their results, Ambatiello et al. (1995) calculated the thermal dependence of domain widths on the basis of the closed, “Landau-Lifshitz”-type structure shown in Figure M16c. Their calculations included the magnetoelastic energy associated with closure domains, as well as the “m*” correction to magnetostatic energy, originating from small deviations of spins away from the easy axis near a crystal surface. There were several discrepancies between experimental and model results. One mechanism which they put forth to explain these discrepancies was that heating promoted an even finer subdivision into small domains at the crystal surface. This would cause a greater collapse of magnetostatic energy than predicted from their model and thus a more pronounced broadening of body domains than expected. Bitter patterns observed at elevated temperatures on natural, intermediate titanomagnetites have been reported by Soffel (1977), Metcalf and Fuller (1987a,b, 1988), and Halgedahl (1987). Because the magnetic force gradients around a wall weaken rapidly with heating, walls attract increasingly less magnetic colloid as the Curie point is approached. Consequently, these authors could follow patterns from room temperature to about 10
to 20
below their samples’ Curie points (Soffel, 1977: Tc ¼ 105
C; Metcalf and Fuller, 1987a,b, 1988, Halgedahl, 1987: Tc approximately 150
C). During heating, patterns gradually faded to obscurity, with few significant changes in domain structure. Remarkably, some titanomagnetite particles studied by Metcalf and Fuller (1987a,b, 1988) displayed no Bitter lines after cooling from the Curie point in the Earth’s field. This observation provided supporting evidence for Halgedahl and Fuller’s (1983) proposal that weak-field TRM acquired by populations of pseudosingle-domain grains could, in part, be attributable to particles which failed to nucleate walls during cooling. The possible importance of LEM states to TRM acquisition was raised by work on synthetic, polycrystalline AMTM60 (Halgedahl, 1991). Observations focused on grains that displayed simple “Kittellike” patterns suggesting lamellar domains (Figure M14). The sample was cycled repeatedly between room temperature and the Curie point of 75
C in the Earth’s field, although patterns lost definition near 70
C. Bitter patterns were observed continuously throughout heatings and coolings. As discussed above and shown in Figure M24, replicate TRM experiments often produced different numbers of domains in the same particle. Some of these domain states suggested that the particle was entirely saturated, with no visible Bitter lines, or nearly saturated, with only small spike domains at grain boundaries. During heating, few changes in domain patterns were observed, except for the usual fading as the Curie temperature was approached. SD-like states were quite thermally stable, exhibiting no changes during thermal cycling until the sample was heated nearly to, or above, the Curie point. As a result of heating above a certain critical temperature, followed by cooling, a SD-like state could be transformed to a LEM state with several domain walls. Continuous observation of Bitter patterns on AMTM60 during cooling revealed that denucleation was the mechanism by which a particle arrived in a final LEM state. Walls which initially nucleated just below Tc grew visible after the sample cooled slightly through just a few degrees. With further cooling, however, the number of domains often

500

MAGNETIC DOMAINS

decreased either through contraction of large, preexisting spike domains, or by straight walls moving together and coalescing into spikes. In both cases, the spikes often would collapse altogether. Nucleations were never observed during cooling, once Bitter patterns grew visible. In some cases, denucleation left behind large volumes which, apparently, contained no walls, although walls were present elsewhere in the grain. In other cases, denucleation left behind a grain that appeared nearly saturated, but for small edge domains. In both cases denucleation appeared to produce anomalously large moments in particles that contained several walls after other TRM runs (Halgedahl, 1991). Results of the experiments described above on AMTM60 lend further support to the hypothesis that particles in metastable SD states can make significant contributions to stable TRM acquired in weak fields. Furthermore, particles which do contain domain walls can carry anomalously large TRM moments, if denucleation leaves behind large volumes in which walls are absent. These experimental results are at odds with theoretical results by Dunlop et al. (1994) for LEM-LEM transitions in magnetite at high temperatures. At present this disagreement has not been explained fully. However, it is important to note that the model of Dunlop et al. (1994) assumes that (1) such transitions are driven by thermal fluctuations, and that (2) occupation frequencies of LEM states follow Boltzmann statistics for thermal equilibrium, before domains are blocked in. It is reasonable to expect that thermally activated jumps among LEM states occur very rapidly and discontinuously during times on the order of 1 s or less; rapid transitions such as these are impossible to track under a microscope. By contrast, many of the denucleations observed by Halgedahl (1991) occurred via continuous wall motions easily tracked over laboratory time scales of several seconds. It is questionable, therefore, whether the changes exhibited by AMTM60 were driven by the thermal fluctuation mechanism modeled by Dunlop et al. (1994). Perhaps the denucleations observed are not stochastic processes. Additional analyses and experiments are needed to determine the controlling mechanisms.

Observed temperature dependence of magnetic domain structure: low temperatures Magnetite undergoes two types of transitions at low temperatures, which can profoundly affect domain structure and remanence (e.g., see detailed discussions in Stacey and Banerjee, 1974; Dunlop and Özdemir, 1997). First, at the isotropic point (approximately 130 K) the first magnetocrystalline anisotropy constant, K1, passes through zero as it changes sign from negative at temperatures above the transition to positive below it. Second, at the Verwey transition, Tv (approximately 120 K), magnetite undergoes a crystallographic transition from cubic to monoclinic. At the isotropic point domain walls should broaden dramatically, because wall width is proportional to K1/2, K being the crystalline anisotropy constant. It follows that, by cooling through the isotropic point, walls may break free of narrow defects, which pinned them at higher temperatures. At the Verwey transition the easy axis of magnetization changes direction. In multidomain and pseudosingle-domain magnetite, this thermal passage should cause domain structure and domain magnetizations to reorganize completely, so that much of an initial remanence acquired at room temperature would be demagnetized by cooling below Tv. Low-temperature demagnetization has proved useful in removing certain spurious components of NRM, which are surprisingly resistant to thermal demagnetization above room temperature. Thus, it is important to determine how these transitions affect domain structure. Using an MFM specially adapted to operate at low temperature, Moloni et al. (1996) were the first to image domains in magnetite at temperatures near the two transitions. Their sample was a synthetic magnetite crystal cut and polished on a {110} surface. At room

temperature, the crystal displayed 180
, 71
, and 109
walls on {110}, as expected for multidomain magnetite. At 77 K—that is, below the Verwey transition—they observed both straight, 180
walls separating broad, lamellar domains, and wavy walls accompanied by reverse spikes. Both types of domain structure were consistent with a uniaxial anisotropy arising from monoclinic crystal structure and for which 2pMs2 /K < 1. They interpreted the broad, “body” domains as being magnetized along easy axes within {110} planes. The wavy walls were interpreted to indicate domains with magnetizations directed along an easy axis that lay out-of-plane. This mixture of domain styles was thought to reflect lateral variations in the easy axis, due to c-axis twinning below Tv. As the crystal was warmed to a few degrees below Tv the domain structure disappeared entirely below the instrument’s noise level, evidently undergoing a complete reorganization near the crystallographic transition.

Ultrahigh-resolution imaging of micromagnetic structures Very recently, off-axis electron holography in the TEM has been used to image magnetic microstructures in submicron magnetite intergrown with ulvospinel (Harrison et al., 2002). This method enables one to image the magnetization vectors within very small, single particles, at a resolution approaching nanometers. Harrison et al. (2002) discovered that clusters of magnetically interacting blocks of magnetite could assume both vortex and multidomain-like states. This promising method should prove highly fruitful in future experimental tests of micromagnetic models. Susan L. Halgedahl

Bibliography Amar, H., 1858. Magnetization mechanism and domain structure of multidomain particles. Physical Review, 111: 149–153. Ambatiello, A., Fabian, K., and Hoffmann, V., 1999. Magnetic domain structure of multidomain magnetite as a function of temperature: observation by Kerr microscopy. Physics of the Earth and Planetary Interiors, 112: 55–80. Appel, E., and Soffel, H.C., 1984. Model for the domain state of Ti-rich titanomagnetites. Geophysical Research Letters, 11: 189–192. Appel, E., and Soffel, H.C., 1985. Domain state of Ti-rich titanomagnetites deduced from domain structure observations and susceptibility measurements. Journal of Geophysics, 56: 121–132. Becker, J.J., 1969. Observations of magnetization reversal in cobaltrare-earth particles. IEEE Transactions on Magnetics, MAG-5: 211–214. Becker, J.J., 1971a. Magnetization discontinuities in cobalt-rare-earth particles. Journal of Applied Physics, 42: 1537–1538. Becker, J.J., 1971b. Interpretation of hysteresis loops of cobaltrare-earths. IEEE Transactions on Magnetics, MAG-7: 644–647. Becker, J.J., 1976. Reversal mechanism in copper-modified cobaltrare-earths. IEEE Transactions on Magnetics, MAG-12: 965–967. Bogdanov, A.K., and Ya. Vlasov, A., 1965. Domain structure in a single crystal of magnetite. (English trans.). Izvestiya Akademii Nauk, SSSR, Earth Physics, series no. 1., 28–32. Bogdanov, A.A., and Ya Vlasov, A., 1966. The domain structure of magnetite particles. (English trans.), Izvestiya Akademii Nauk, SSSR, Physics, Solid Earth, 9: 577–581. Boyd, J.R., Fuller, M., and Halgedahl, S., 1984. Domain wall nucleation as a controlling factor in the behaviour of fine magnetic particles in rocks. Geophysical Research Letters, 11: 193–196. Brown, W.F., 1963. Micromagnetics. New York: John Wiley, 143 pp. Chikazumi, S., 1964. Physics of Magnetism. New York: John Wiley, 664 pp.

MAGNETIC DOMAINS

Cullity, B.D., 1972. Introduction to Magnetic Materials. AddisonWesley, Reading, MA, 666 pp. Dunlop, D.J., and Özdemir, O., 1997. Rock Magnetism: Fundamentals and Frontiers. Cambridge: Cambridge University Press, UK, 573 pp. Dunlop, D.J., Newell, A.J., and Enkin, R.J., 1994. Transdomain thermoremanent magnetization. Journal of Geophysical Research, 99: 19,741–19,755. Fabian, K., Kirchner, A.,Williams, W., Heider, F., and Leibl, T., Threedimensional micromagnetic calculations for magnetite using FFT. Geophysical Journal International, 124: 89–104. Foss, S., Moskowitz, B., and Walsh, B., 1996. Localized micromagnetic perturbation of domain walls in magnetite using a magnetic force microscope. Applied Physics Letters, 69: 3426–3428. Foss, S., Moskowitz, B.M., Proksch, R., and Dahlberg, E.D., Domain wall structures in single-crystal magnetite investigated by magnetic force microscopy. Journal of Geophysical Research, 103: 30,551–30,560. Frandson, C., Stipp, S.L.S., McEnroe, S.A., Madsen, M.B., and Knudsen, J.M., 2004. Magnetic domain structures and stray fields of individual elongated magnetite grains revealed by magnetic force microscopy (MFM). Physics of the Earth and Planetary Interiors, 141: 121–129. Fukuma, K., and Dunlop, D.J., Grain-size dependence of twodimensional micromagnetic structures for pseudo-single-domain magnetite (0.2–2.5 mm). Geophysical Journal International, 134: 843–848. Geiß, C.E., Heider, F., and Soffel, H.C., Magnetic domain observations on magnetite and titanomaghemite grains (0.5–10 mm). Geophysical Journal International, 124: 75–88. Halgedahl, S.L., 1987. Domain pattern observations in rock magnetism: progress and problems. Physics of the Earth and Planetary Interiors, 46: 127–163. Halgedahl, S.L., 1991. Magnetic domain patterns observed on synthetic Ti-rich titanomagnetite as a function of temperature and in states of thermoremanent magnetization. Journal of Geophysical Research, 96: 3943–3972. Halgedahl, S.L., 1995. Bitter patterns versus hysteresis behavior in small single particles of hematite. Journal of Geophysical Research, 100: 353–364. Halgedahl, S.L., 1998. Barkhausen jumps in larger versus small platelets of natural hematite. Journal of Geophysical Research, 103: 30,575–30,589. Halgedahl, S., and Fuller, M., 1980. Magnetic domain observations of nucleation processes in fine particles of intermediate titanomagnetite. Nature, 288: 70–72. Halgedahl, S.L., and Fuller, M., 1981. The dependence of magnetic domain structure upon magnetization state in polycrystalline pyrrhotite. Physics of the Earth and Planetary Interiors, 26: 93–97. Halgedahl, S., and Fuller, M., 1983. The dependence of magnetic domain structure upon magnetization state with emphasis upon nucleation as a mechanism for pseudo-single domain behavior. Journal of Geophysical Research, 88: 6505–6522. Halgedahl, S.L., and Ye, J., 2000. Observed effects of mechanical grain-size reduction on the domain structure of pyrrhotite. Earth and Planetary Science Letters, 176(3): 457–467. Harrison, T.J., Dunin-Borkowski, R.E., and Putnis, A., 2002. Direct imaging of nanoscale magnetic interactions in minerals. Proceedings of the National Academic Sciences, 99: 16,556–16,561. Heider, F., 1990. Temperature dependence of domain structure in natural magnetite and its significance for multi-domain TRM models. Physics of the Earth and Planetary Interiors, 65: 54–61. Heider, F., and Hoffmann, V., 1992. Magneto-optical Kerr effect on magnetite crystals with externally applied fields. Earth and Planetary Science Letters, 108: 131–138.

501

Heider, F., Halgedahl, S.L., and Dunlop, D.J., 1988. Temperature dependence of magnetic domains in magnetite crystals. Geophysical Research Letters, 15: 499–502. Heisenberg, W., 1928. Zur Theorie des Ferromagnetismus. Zeitschrift fuer Physik, 49: 619–636. Hoffmann, V., Schafer, R., Appel, E., Hubert, A., and Soffel, H., 1987. First domain observations with the magneto-optical Kerr effect on Ti-ferrites in rocks and their synthetic equivalents. Journal of Magnetism and Magnetic Materials, 71: 90–94. Kittel, C., 1949. Physical theory of ferromagnetic domains. Reviews of Modern Physics, 21: 541–583. Metcalf, M., and Fuller, M., 1987a. Magnetic remanence measurements of single particles and the nature of domain patterns in titanomagnetites. Geophysical Research Letters, 14: 1207–1210. Metcalf, M., and Fuller, M., 1987b. Domain observations of titanomagnetites during hysteresis at elevated temperatures and thermal cycling. Physics of the Earth and Planetary Interiors, 46: 120–126. Metcalf, M., and Fuller, M., 1988. A synthetic TRM induction curve for fine particles generated from domain observations. Geophysical Research Letters, 15: 503–506. Moloni, K., Moskowitz, B.M., and Dahlberg, E.D., 1996. Domain structures in single crystal magnetite below the Verwey transition as observed with a low-temperature magnetic force microscope. Geophysical Research Letters, 23: 2851–2854. Moon, T., and Merrill, R.T., 1984. The magnetic moments of nonuniformly magnetized grains. Physics of the Earth and Planetary Interiors, 34: 186–194. Moon, T.S., and Merrill, R.T., 1985. Nucleation theory and domain states in multidomain magnetic material. Physics of the Earth and Planetary Interiors, 37: 214–222. Moskowitz, B.M., and Banerjee, S.K., 1979. Grain size limits for pseudosingle domain behavior in magnetite: implications for paleomagnetism. IEEE Transactions on Magnetics, MAG-15: 1241–1246. Moskowitz, B.M., and Halgedahl, S.L., Theoretical temperature and grain-size dependence of domain state in x ¼ 0.6 titanomagnetite. Journal of Geophysical Research, 92: 10,667–10,682. Moskowitz, B.M., Halgedahl, S.L., and Lawson, C.A., 1988. Magnetic domains on unpolished and polished surfaces of titaniumrich titanomagnetite. Journal of Geophysical Research, 93: 3372–3386. Muxworthy, A.R., and Williams, W., Micromagnetic calculations of hysteresis as a function of temperature in pseudo-single domain magnetite. Geophysical Research Letters, 26: 1065–1068. Newell, A.J., Dunlop, D.J., and Williams, W., 1993. A two-dimensional micromagnetic model of magnetization and fields in magnetite. Journal of Geophysical Research, 98: 9533–9549. Özdemir, O., and Dunlop, D.J., 1993. Magnetic domain structures on a natural single crystal of magnetite. Geophysical Research Letters, 20: 1835–1838. Özdemir, O., and Dunlop, D.J., 1997. Effect of crystal defects and internal stress on the domain structure and magnetic properties of magnetite. Journal of Geophysical Research, 102: 20, 211–20,224. Özdemir, O., Xu, S., and Dunlop, D.J., 1995. Closure domains in magnetite. Journal of Geophysical Research, 100: 2193–2209. Pokhil, T.G., and Moskowitz, B.M., 1996. Magnetic force microscope study of domain wall structures in magnetite. Journal of Applied Physics, 79: 6064–6066. Pokhil, T.G., and Moskowitz, B.M., 1997. Magnetic domains and domain walls in pseudo-single-domain magnetite studied with magnetic force microscopy. Journal of Geophysical Research, 102: 22,681–22,694. Proksch, R.B., Foss, S., and Dahlberg, E.D., 1994. High resolution magnetic force microscopy of domain wall fine structures. IEEE Transactions on Magnetics, 30: 4467–4472.

502

MAGNETIC FIELD OF MARS

Rhodes, P., and Rowlands, G., 1954. Demagnetizing energies of uniformly magnetised rectangular blocks. Proceedings of the Leeds Philosophical and Literary Society, Science Section, 6: 191–210. Smith, P.P.K., 1980. The application of Lorentz electron microscopy to the study of rock magnetism. Institute of Physics Conference Series, 52: 125–128. Soffel, H., 1971. The single-domain-multidomain transition in natural intermediate titanomagnetites. Zeitschrift fuer Geophysik, 37: 451–470. Soffel, H.C., 1977. Domain structure of titanomagnetites and its variation with temperature. Journal of Geomagnetism and Geoelectricity, 29: 277–284. Soffel, H., 1977. Pseudo-single-domain effects and single-domain multidomain transition in natural pyrrhotite deduced from domain structure observations. Journal of Geophysics, 42: 351–359. Soffel, H.C., Aumuller, C., Hoffmann, V., and Appel, E., 1990. Threedimensional domain observations of magnetite and titanomagnetites using the dried colloid SEM method. Physics of the Earth and Planetary Interiors, 65: 43–53. Stacey, F.D., and Banerjee, S.K., 1974. The Physical Principles of Rock Magnetism. Amsterdam: Elsevier, 195 pp. Szymczak, R., 1968. The magnetic structure of ferromagnetic materials of uniaxial structure. Electronics Technology, 1: 5–43. Weiss, P., 1907. L’hypothèse du champ moléculaire et la propriété ferromagnétique. Journal of Physique, 6: 661–690. Williams, W., and Dunlop, D.J., 1989. Three-dimensional micromagnetic modelling of ferromagnetic domain structure. Nature, 337: 634–637. Williams, W., and Dunlop, D.J., 1990. Some effects of grain shape and varying external magnetic fields on the magnetic structure of small grains of magnetite. Physics of the Earth and Planetary Interiors, 65: 1–14. Williams, W., and Dunlop, D.J., 1995. Simulation of magnetic hysteresis in pseudo-single-domain grains of magnetite. Journal of Geophysical Research, 100: 3859–3871. Williams, W., and Wright, T.M., 1998. High-resolution micromagnetic models of fine grains of magnetite. Journal of Geophysical Research, 103: 30,537–30,550. Williams, W., Hoffmann, V., Heider, F., Goddenhenreich, T., and Heiden, C., 1992. Magnetic force microscopy imaging of domain walls in magnetite. Geophysical Journal International, 111: 417–423. Worm, H.-U., and Markert, H., 1987. The preparation of dispersed titanomagnetite particles by the glass-ceramic method. Physics of the Earth and Planetary Interiors, 46: 263–270. Worm, H.-U., Ryan, P.J., and Banerjee, S.K., 1991. Domain size, closure domains, and the importance of magnetostriction in magnetite. Earth and Planetary Science Letters, 102: 71–78. Xu, S., Dunlop, D.J., and Newell, A.J., 1994. Micromagnetic modelling of two-dimensional domain structures in magnetite. Journal of Geophysical Research, 99: 9035–9044. Ye, J., and Halgedahl, S.L., 2000. Theoretical effects of mechanical grain-size reduction on GEM domain states in pyrrhotite. Earth and Planetary Science Letters, 178: 73–85. Ye, J., and Merrill, R.T., 1995. The use of renormalization group theory to explain the large variation of domain states observed in titanomagnetites and implications for paleomagnetism. Journal of Geophysical Research, 100: 17,899–17,907.

Cross-references Magnetic Properties, Low-Temperature Magnetization, Isothermal Remanent (IRM) Magnetization, Thermal Remanent (TRM) Paleomagnetism

MAGNETIC FIELD OF MARS Introduction Strong magnetic anomalies have been detected over the south hemisphere of Mars. The Mars missions prior to Mars Global Surveyor (MGS) detected no appreciable magnetic field around the planet, which led to the conclusion that the Martian core field is weaker than Earth’s by more than an order of magnitude. However, orbiting at elevations as low as 100–200 km during the science phase and aerobreaking phase, MGS detected a very strong crustal magnetic field, as strong as 200 nT, over the ancient southern highlands (Acuna et al., 1999), indicating that the Martian crust is more magnetic than Earth’s by more than an order of magnitude. There is evidence from the Martian meteorites that a magnetic field as strong as 3000 nT has existed on the surface of Mars. The oldest Martian meteorite (ALH84001) formed before 4 Ga (e.g., Collinson, 1997; Kirschvink et al., 1997; Weiss, et al., 2002; Antretter et al., 2003) and the young Martian meteorites that have crystallization ages ranging from 0.16 to 1.3 Ga (Nyquist et al., 2001) are magnetized in a weak field of less than 3000 nT (e.g., Cisowski, 1986; Collinson, 1997; McSween and Treiman, 1998). Whether the meteorites were magnetized by a weak core field or by the local crustal field is still debated. There is good evidence that the strong anomalies of the crust have existed for the last 4 Ga (Arkani-Hamed, 2004a). The magnetometer on board of MGS has provided immense amount of magnetic data since it resumed its mapping phase when its highly elliptical orbit was reduced to an almost circular polar orbit at a mean elevation of 400 km. Besides the magnetometer, MGS carried an electron reflectometer, which has also indirectly provided data on the magnetic field of Mars (e.g., Mitchell et al., 2001). This article is concerned with the magnetometer data. The first section presents a spherical harmonic model of the magnetic field of Mars that is derived from the mappingphase magnetic data. The second section is concerned with the firstorder interpretation of the anomalies. The final section summarizes the major points of the article.

The spherical harmonic model of the magnetic field The first magnetic anomaly map of Mars was derived by Acuna et al. (1999) from the low-altitude MGS data acquired at 100–200 km elevations during the science phase and aerobreaking phase. It was presented without altitude corrections. Subsequently, Purucker et al. (2000) reduced the radial component anomalies to a constant elevation of 200 km, and Arkani-Hamed (2001a) derived a 50
spherical harmonic model of the magnetic potential at 120 km altitude using all three vector components of the magnetic field. The later magnetic field models of Mars by Arkani-Hamed (2002), Cain et al. (2003), and Langlais et al. (2004) were obtained using all three components of the magnetic field acquired at the low- and high-altitude phases. A highly repeatable and reliable magnetic anomaly map of Mars is essential for the investigation of the relationship between the tectonic features and the magnetization of the Martian crust. MGS has provided a huge amount of high-altitude magnetic data acquired within 360–420 km altitudes. Although the high altitude of the spacecraft limits the resolution of the data (e.g., Connerney et al., 2001; ArkaniHamed, 2002), the huge amount of the data provides an opportunity to derive a highly repeatable and accurate magnetic anomaly map at that altitude. For this purpose Arkani-Hamed (2004b) used the nighttime radial component of the high-altitude data, which is least contaminated by noncrustal sources, and selected the most common features on the basis of covariance analysis. The vast amount of the data allowed him to divide the entire data into two almost equal sets, acquired during two different periods separated by more than a year, and to derive two separate spherical harmonics models of the magnetic field. The spherical harmonic model of the radial component of the magnetic field R is expressed as

MAGNETIC FIELD OF MARS

Rðr; q; jÞ ¼

N X

ðn þ 1Þða=rÞnþ2

n¼1

n X

ðCnm cos mj þ Snm sin mjÞPmn ðcos yÞ

m¼0

where a is the mean radius of Mars (3393 km); r, y, and j are distance from the center, colatitude, and east longitude (the coordinate system is centered at the center of mass of Mars); Pmn ðcos yÞ is the Schmidtnormalized associated Legendre function of degree n and order m; Cnm and Snm are the Gauss coefficients; and N denotes the highest degree harmonics retained in the model. The covariance analysis of the two models showed that the covarying coefficients of the models are almost identical over harmonics of degree up to 50, with correlation coefficients greater than 0.95. These harmonics are highly repeatable and hence reliable. Figure M25/Plate 13a shows the radial component of the magnetic field of Mars at 100 km altitude, derived by averaging the covarying harmonic coefficients of the two models up to degree 50 and downward continuing from 400 to 100 km altitude. The color bar is saturated to better illustrate the weak anomalies.

Interpretation of the magnetic anomalies Figure M25/Plate 13a shows strong magnetic anomalies arising from the remanent magnetization of the crust. This section provides firstorder interpretation of the anomalies. The magnetic anomalies in the northern lowlands are very weak compared to those in the south. Many buried impact craters in the lowlands suggest similar ages for the crust in the north and south (Frey et al., 2001). There is no striking evidence that the composition of the Martian crust had a distinct north-south dichotomy before the formation of the lowlands that would have prevented the formation of strong magnetic bodies in the north. The weak anomalies in the lowlands are most likely the relicts of stronger ones that existed before the formation of the lowlands. Zuber et al. (2000) disputed the impact origin of the lowlands and suggested crustal thinning due to a giant mantle plume. According to this hypothesis, the magnetic source bodies must have been partially demagnetized by the thermal effects

503

of the plume, and to a lesser extent by near-surface low-temperature hydration (e.g., Arkani-Hamed, 2004a). Figure M25/Plate 13a shows that the giant impact basins Hellas (41
S, 70
E), Argyre (50
S, 316
E), and Isidis (13
N, 88
E) are almost devoid of magnetic anomalies. These basins were formed by large impacts 4 Ga, which resulted in strong shock pressures and high temperatures, capable of demagnetizing the crust beneath the basins (Hood et al., 2003; Mohit and Arkani-Hamed, 2004; Kletetschka et al., 2004). The crust beneath these basins is completely demagnetized to a distance of 0.8 basin radius, where the shock pressure exceeded 3 GPa, and partially demagnetized to 1.4 radius where the pressure exceeded 2 GPa. This observation together with the fact that many intermediate-size impacts that created craters of 200–500 km diameter have little demagnetization effect led Mohit and Arkani-Hamed (2004) to suggest that the magnetic carriers of Martian crust have high coercivity. The resulting demagnetization depends on the coercivity of the magnetic minerals, magnetite, hematite, and pyrrhotite, suggested for the Martian crust (e.g., Kletetschka et al., 2000, 2004; Hargraves et al., 2001). Cisowski and Fuller (1978) found that remanence with a coercivity of 70 mT was 20% demagnetized after a shock of 1 GPa and 70% after a shock of 4 GPa. Under a shock of 1 GPa, the remanence of the multidomain hematite and magnetite are reduced by 20% and 68%, respectively (Kletetschka et al., 2002, 2004). Shock experiments on high coercivity (300 mT) single-domain pyrrhotite samples showed that a shock of 1 GPa removed 50% of the magnetization at room temperature, and they were completely demagnetized by shocks exceeding 2.75 GPa, undergoing a transition to a paramagnetic phase (Rochette et al., 2003). The high-pressure magnetic measurements in a diamond anvil cell showed that magnetite behaved as single domain with high coercivity at high pressures (Gilder et al., 2002). The survival of the magnetic anomalies beneath intermediate-size craters can be partly due to the increase of the coercivity of already high-coercive magnetic carriers of the Martian crust during the passage of the shock wave. The magnetic anomalies over Tharsis bulge are much weaker than those in the south. The bulge formed through major volcanic activities in Noachian and Early Hesperian, although minor volcanism likely continued to the recent past (e.g., Hartmann and Neukum, 2001).

Figure M25/Plate 13a The radial component magnetic map of Mars at 100 km altitude derived from the high-altitude MGS data using the spherical harmonic of degree up to 50. The color bar is saturated to better illustrate the weak anomalies. Units are in nanotesla.

504

MAGNETIC FIELD OF MARS

There are many lines of evidence that the volcanic layers of Tharsis are not significantly magnetized, they are formed either in the absence of a core dynamo or in the waning period of the dynamo (Arkani-Hamed, 2004b). Many places of Tharsis have been punctured by tectonic processes, which could have created detectable magnetic anomalies if the Tharsis plains were appreciably magnetized. Here two major features, Valles Marineris and shield volcanoes, are discussed in some detail. Figure M25/Plate 13a shows no distinct magnetic edge effects associated with Valles Marineris, a canyon of over 5 km mean depth, 100– 400 km width, and 3500 km length. The models proposed for the formation of this giant canyon (e.g., Schultz, 1997; Tanaka, 1997; Wilkins and Schultz, 2003) indicate a vast amount of mass wasting. Many of the proposed models imply that upper strata of the canyon floor are porous. The porous floor and possible long-existing water make the rocks more susceptible to demagnetization by low-temperature hydration. The magnetization of mid-ocean ridge basalt on Earth declines from 20 to 5 A m–1 in the first 30 Ma, likely a result of low-temperature hydration by circulating pore water (e.g., Bliel and Petersen, 1983). Also a positive Bouguer anomaly of 150 mGal is associated with the central part of the canyon, implying crustal thinning and mantle uplift of 15 km. If the Tharsis crust were strongly magnetized before the formation of the canyon, it is to be expected that the crust in the canyon would be partly demagnetized, giving rise to detectable magnetic edge effects. The absence of the expected edge effects indicates that the preexisting Tharsis plains were not significantly magnetized. Figure M25/Plate 13a shows that the formation of the canyon has ruptured the magnetic anomalies in the eastern part of the canyon between 290
E and 320
E. The anomalies must have existed before the formation of Valles Marineris that occurred during Late Noachian to Early Hesperian (e.g., Anderson et al., 2001). Figure M25/Plate 13a also shows no magnetic signature associated with the shield volcanoes Olympus, Arsia, Pavonis, and Ascreaus, suggesting that the underlying preexisting Tharsis plains had not been appreciably magnetized. Otherwise, the thermal effects of the shield volcanoes could have demagnetized part of them and created low-magnetic patches in the magnetized plains, giving rise to detectable magnetic anomalies. There was no active core dynamo during the formation of the shield volcanoes, from Early Hesperian to Late Amazonian. Hood and Hartdegen (1997) estimated the magnetic anomaly that would have been produced by the entire volcanic structure if were magnetized by a core field an order of magnitude weaker than the Earth’s core field. The lack of such a magnetic anomaly indicates that no core dynamo existed during the formation of shield volcanoes to magnetize the volcanic flows. Many lines of evidence suggest that the core dynamo decayed during Early Noachian. The lack of magnetic anomalies associated with giant impact basins, the absence of magnetic edge effect of Valles Marineris, and the fact that the huge amount of volcanic flows of Syria Planum, occurred from Noachian to Early Amazonian (e.g., Anderson et al., 2001) and those of Olympus and Tharsis mounts, occurred from Early Hesperian to Late Amazonian, show no magnetic signatures strongly argue against an active core dynamo since Early Noachian (Arkani-Hamed, 2004b). Attempts have been made to estimate the paleomagnetic pole position of Mars. Arkani-Hamed (2001b) modeled 10 widely distributed relatively isolated small magnetic anomalies. Assuming that the source bodies were magnetized by a dipole core field, he found that most of the pole positions were clustered within a 30
circle centered at 25
N, 230
E. Moreover, both north and south magnetic poles were found in the cluster, suggesting reversal of the core field polarity. Using the two weak magnetic anomalies near the geographic north pole, Hood and Zacharian (2001) found paleomagnetic poles in general agreement with Arkani-Hamed (2001b). Arkani-Hamed and Boutin (2004) used magnetic profiles from the low- and high-altitude data to model nine anomalies. The magnetic pole positions determined from their models showed clustering of the poles at the same general region, but not as tightly as those of Arkani-Hamed (2001b).

The magnetic field of Mars provides information about the rotational dynamics of the planet. None of the paleomagnetic poles are close to the present rotation pole. If the axis of dipole core field were close to the axis of rotation at the time the magnetic sources acquired magnetization, which is the case for the terrestrial planets at present, the paleomagnetic pole positions suggest appreciable, 20
–60
, polar wander since the magnetic bodies were magnetized. Polar wander of 20
to 120
were proposed on the basis of geological features (Murray and Malin, 1973; Schultz and Lutz-Garihan, 1982; Schultz and LutzGarihan, 1988). Approximating the topographic mass of Tharsis bulge by a surface mass, Melosh (1980) predicted polar wander of up to 25
from the effects of the bulge on the moment of inertia of the planet. Willeman (1984) suggested that compensation of Tharsis bulge would limit the amount of polar wander to less than 10
. The theoretical studies of polar wander by Spada et al. (1996) showed that Mars could have undergone polar wander by as much as 70
, in response to surface loads such as Olympus and Tharsis mounts.

Summary Figure M25/Plate 13a presents the most recent and highly accurate spherical harmonic model of the radial component of the magnetic field of Mars derived from the high-altitude mapping-phase magnetic data from MGS, using harmonics of degree up to 50. It shows that the northern lowland formation and the large impacts that created the giant basins Hellas, Isidis, and Argyre have significantly demagnetized the crust and there was no active core dynamo to remagnetize the crust. The absence of magnetic edge effects associated with Valles Marineris, and the lack of magnetic signatures of the large shield volcanoes Olympus and Tharsis mounts imply that core dynamo did not exist from Late Noachian to Late Amazonian, and likely to the present. The fact that intermediate-size craters, with diameters 200–500 km, show no sign of demagnetization of the underlying crust indicates that the magnetic carriers of the Martian crust have high coercivity. The clustering of the paleomagnetic poles far from the present rotation axis implies polar wander of Mars by about 20
–50
since the magnetic source bodies were magnetized. Finally, the presence of both north and south paleomagnetic poles in the cluster suggests that the core field of Mars had polarity reversals.

Acknowledgment This research was supported by the Natural Sciences and Engineering Research Council (NSERC) of Canada. Jafar Arkani-Hamed

Bibliography Acuna, M.H. et al., 1999. Global distribution of crustal magnetization discovered by the Mars Global Surveyor MAG/ER experiment. Science, 284: 790–793. Anderson, R.C. et al., Primary centers and secondary concentrations of tectonic activity through time in the western hemisphere of Mars. Journal of Geophysical Research, 106: 20,563–20,585. Antretter, M., Fuller, M., Scott, E., Jackson, M., Moskowitz, B., and Soleid, P., 2003. Paleomagnetic record of Martian meteorite ALH84001. Journal of Geophysical Research, 108(E6): 5049, doi:10, 1029/2002JE001979. Arkani-Hamed J., 2001a. A 50 degree spherical harmonic model of the magnetic field of Mars. Journal of Geophysical Research, 106: 23,197–23,208. Arkani-Hamed, J., 2001b. Paleomagnetic pole positions and poles reversals of Mars. Geophysical Research Letters, 28: 3409–3412. Arkani-Hamed, J., 2002. An improved 50-degree spherical harmonic model of the magnetic field of Mars, derived from both highaltitude and low-altitude observations. Journal of Geophysical Research, 107: 10.1029/2001JE001835.

MAGNETIC FIELD OF SUN

Arkani-Hamed, J., 2004a. Timing of the Martian core dynamo. Journal of Geophysical Research, 109(E3): E03006, doi:10.1029/ 2003JE002195. Arkani-Hamed J., 2004b. A coherent model of the crustal magnetic field of Mars. Journal Of Geophysical Research, 109: E09005, doi:10.1029/2004JE002265. Arkani-Hamed, J., and Boutin, D., 2004. Paleomagnetic poles of Mars: Revisited. Journal of Geophysical Research, 109: doi:10.1029/ 2003JE0029. Bliel, U., and Petersen, N., 1983. Variations in magnetization intensity and low-temperature titano-magnetite oxidation of ocean floor basalts. Nature, 301: 384–388. Cain, J.C., Ferguson, B., and Mozzoni, D., 2003. An n ¼ 90 model of the Martian magnetic field. Journal of Geophysical Research, 108: 10.1029/2000JE001487. Cisowski, S.M., 1986. Magnetic studies on Shergotty and other SNC meteorites. Geochemica Cosmochemica Acta, 50: 1043–1048. Cisowski, S., and Fuller, M., 1978. The effect of shock on the magnetism of terrestrial rocks. Journal of Geophysical Research, 83: 3441–3458. Collinson, D.W., 1997. Magnetic properties of Martian meteorites: implications for an ancient Martian magnetic field. Planetary Science, 32: 803–811. Connerney, J.E.P., Acuna, M.H., Wasilewski, P.J., Kletetschka, G., Ness, N.F., Remes, H., Lin, R.P., and Mitchell, D.L., 2001. The global magnetic field of Mars and implications for crustal evolution. Geophysical Research Letters, 28: 4015–4018. Frey, H., Shockey, K.M., Frey, E.L., Roark, J. H., and Sakimoto, S.E.H., 2001. A very large population of likely buried impact basins in the northern lowlands of Mars revealed by MOLA data. Lunar and Planetary Science Conference XXXII, Abstr. 1680. Gilder, S.A., Le Goff, M., Peyronneau, J., and Chervin, J., 2002. Novel high pressure magnetic measurements with application to magnetite. Geophysical Research Letters, 29: 10,1029/ 2001GL014227, 2002. Hargraves, R.B., Knudsen, J.M., Madsen, M.B., and Bertelsen, P., 2001. Finding the right rocks on Mars. EOS: Transactions, American Geophysical Union, 82: 292–293. Hartmann, W.K., and. Neukum, G., 2001. Cratering chronology and the evolution of Mars. Space Science Reviews, 96: 165–194. Hood, L.L., and Hartdegen, K., 1997. A crustal magnetization model for the magnetic field of Mars: a preliminary study of the Tharsis region. Geophysical Research Letters, 24: 727–730. Hood, L.L., and Zacharian, A., 2001. Mapping and modeling of magnetic anomalies in the northern polar region of Mars. Journal of Geophysical Research, 106: 14601–14619. Hood, L.L., Richmond, N.C., Pierazzo, E., and Rochette, P., Distribution of crustal magnetic fields on Mars: Shock effects of basinforming impacts. Geophysical Research Letters, 30(6): 1281, doi:10.1029/2002GL016657. Kletetschka, G., Wasilewski, P.J., and Taylor, P.T., 2000. Mineralogy of the sources for magnetic anomalies on Mars. Meteoritics and Planetary Science, 35: 895–899. Kletetschka, G., Wasilewski, P.J., and Taylor, P.T., 2002. The role of hematite-ilmenite solid solution in the production of magnetic anomalies in ground- and satellite-based data. Tectonophysics, 347: 167–177. Kletetschka, G., Connerney, J.E.P., Ness, N.F., and Acuna, M.H., 2004. Pressure effects on Martian crustal magnetization near large impact basins. Meteoritics and Planetary Science, 39: 1839–1848. Kirschvink, J.L., Maine, A.T., and Vali, H., 1997. Paleomagnetic evidence of a low-temperature origin of carbonate in the Martian meteorite ALH84001. Science, 275: 1629–1633. Langlais, B., Purucker, M.E., and Mandea, M., 2004. Crustal magnetic field of Mars. Journal of Geophysical Research, 109: E002008, doi:10.1029/2003JE002048.

505

McSween, H.Y., and Treiman, A.H., 1998. Martian meteorites, Chapter 6 in Planetary materials. Reviews in Mineralogy, 36: 53. Melosh, H.J., 1980. Tectonic patterns on a reoriented planet: Mars. Icarus, 44: 745–751. Mitchell, D.L. et al., 2001. Probing Mars’ crustal magnetic field and ionosphere with the MGS electron reflectometer. Journal of Geophysical Research, 106: 23,419–23,427. Mohit, P.S., and Arkani-Hamed, J., 2004. Impact demagnetization of the Martian crust. Icarus, 168: 305–317. Murray, B.C., and Malin, M.C., 1973. Polar wandering on Mars. Science, 179: 997–1000. Nyquist, L.E. 2001. Ages and geological history of Martian meteorites. In Kallenbach, R., Geiss, J., and Hartmann, W.K. (eds.), Chronology and Evolution of Mars. Dordrecht, the Netherlands: Kluwer Academic Publishers. Purucker, M., Ravat, D., Frey, H., Voorhies, C., Sabaka, T., and Acuna, M., 2000. An altitude-normalized magnetic map of Mars and its interpretation. Geophysical Research Letters, 27: 2449–2452. Rochette, P., Fillion, G., Ballou, R., Brunet, F., Ouladdiaf, B., and Hood, L., 2003. High pressure magnetic transition in pyrrhotite and impact demagnetization on Mars. Geophysical Research Letters, 30(13): 1683, doi:10.1029/2003GL017359. Schultz, P.H., and Lutz-Garihan, A.B., 1982. Grazing impacts on Mars: a record of lost satellites. Journal Geophysical Research, 87: A84–A96. Schultz, P.H., and Lutz-Garihan, A.B., 1988. Polar wandering of Mars. Icarus, 73: 91–141. Schultz, R.A., 1997. Dual-process genesis for Valles Marineris and troughs on Mars, presented at the XXVIII Lunar and Planetary Science Conference, Houston, Texas. Spada, G. 1996. Long-term rotation and mantle dynamics of the Earth, Mars and Venus. Journal of Geophysical Research, 101: 2253–2266. Tanaka, K.L., 1997. Origin of Valles Marineris and Noctis Labyrinthus, Mars, by structurally controlled collapses and erosion of crustal materials, presented at the XXXVIII Lunar and Planetary Science Conference, Houston, Texas. Weiss, B.P., Vali, H., Baudenbacher, F.J., Kirschvink, J.L., Stewart, S.T., and Schuster, D.L., 2002. Records of an ancient Martian field in ALH84001. Earth and Planetary Science Letters, 201: 449–463. Wilkins, S.J., and Schultz, R.A., 2003. Cross faults in extensional settings: stress triggering, displacement localization, and implications for the origin of blunt troughs at Valles Marineris, Mars. Journal of Geophysical Research, 108: E6, 5056, doi:1029/2002JE001968. Willeman, R.J., 1984. Reorientation of planets with elastic lithospheres. Icarus, 60: 701–709. Zuber, M.T. 2000. Internal structure and early thermal evolution of Mars from Mars Global Surveyor topography and gravity. Science, 287: 1788–1793.

MAGNETIC FIELD OF SUN The Sun has been observed to exhibit a breathtaking variety of magnetic phenomena on a vast range of spatial and temporal scales. These vary in spatial scale from the solar radius down to the limit of present resolution of the most powerful satellite instrumentation, and with durations varying from minutes to hundreds of years, encompassing the famous 11-year sunspot cycle. This dynamic and active field, which is visible in extreme ultraviolet wavelengths as shown in Figure M26/Plate 14a, is responsible for all solar magnetic phenomena, such as sunspots, solar flares, coronal mass ejections and the solar wind, and also heats the solar corona to extremely high temperatures. These have important terrestrial consequences, causing severe magnetic storms and major disruption to satellites, as well as having a possible impact on

506

MAGNETIC FIELD OF SUN

Figure M27 Hale’s polarity law: Magnetogram of the solar surface as a function of longitude and latitude. White indicates strong radial field pointing out of the solar surface, while black indicates strong radial field pointing into the solar surface. Notice that the sense of the active regions is opposite in the northern and southern hemispheres.

Figure M26/Plate 14a Compound extreme ultraviolet (171 A˚) full-disk image of the Sun taken by the Transition Region and Coronal Explorer (TRACE) satellite. The two bands on either side of the Sun’s equator contain bright features indicating magnetic activity.

the terrestrial climate. The magnetic field also plays a major role in the evolution of the Sun as a star, with magnetic braking drastically reducing the angular momentum of the star as it evolves over its lifetime (Mestel, 1999). This article will review the observations of the solar magnetic field and briefly indicate the features that need to be understood in order to formulate a theory for the generation and evolution of such a field.

The large-scale solar field Although magnetic fields in the Sun are observed on a whole spectrum of length-scales, it is usual to classify magnetic phenomena either as large- or small-scale events. Here the large-scale magnetic field is defined to be that which is observed on scales comparable to the solar radius, with systematic properties, spatial organization, and temporal coherence. The most obvious manifestation of the large-scale solar magnetic field is the appearance of sunspots at the solar surface. Sunspots have been systematically observed in the West since the early 17th century when Galileo utilized the newly invented telescope to document the passage of dark spots across the solar surface. Although the 11-year period for the cycle of sunspot activity was discovered by Schwabe in 1843, the connection between sunspots and the magnetic field of the Sun was only determined when Hale in 1908 used the Zeeman splitting of spectral lines to establish the presence of a strong magnetic field in sunspots (Hale, 1908). By then it could clearly be seen that sunspots appear in bipolar pairs, with a leading and trailing spot nearly aligned with the solar equator. Magnetic observations of sunspots indicate that they have systematic properties—often described as Hale’s polarity laws. These state that the magnetic polarities of the leading and trailing spots are opposite, with the sense of the polarity being opposite in each hemisphere of the Sun (see Figure M27). Moreover, the polarities of the bipolar sunspot pairs that are observed at the solar surface reverse after each minimum in sunspot activity. Finally, Joy’s law states that there is a systematic tilt (of the order of 4
) in the alignment of sunspot pairs with respect to the solar equator, with the leading spot being closer to the equator. Taken together, these indicate that

sunspots are the surface manifestation of an underlying magnetic field with an opposite sense in each hemisphere. This magnetic field is cyclic, with a mean period of 22 years. Sunspot pairs are interpreted as an indicator for an azimuthal (toroidal) Bf field. A radial (poloidal) component of the magnetic field, Br , may also be detected. The radial field is particularly visible at the solar poles with a distinct and opposite polarity at each pole. This polar field reverses at the maximum of sunspot activity and the polarity is such that between solar maximum and solar minimum the radial field has the same polarity as the trailing spots of the sunspot pairs in the same hemisphere. Hence Bf and Br fields are nearly in antiphase with Bf Br < 0 for most of the time (Stix, 1976). Also particularly visible at high latitudes are ephemeral active regions. These are small regions that do not systematically obey Hale’s polarity laws (although they do show a preference for the same orientation as sunspots) and have a lifetime of a few days. It is now believed that these ephemeral active regions are either reprocessed flux from active regions or the result of local regeneration in the solar convection zone. An important measure of the topology of the large-scale solar magnetic field is given by the current helicity of the magnetic field Hj ¼ hj  Bi, where j ¼ 1=m0 r  B is the electric current. This is a difficult quantity to measure, although some information about the radial contribution to the current helicity h jr Br i can be estimated from vector magnetograms of active regions. The measurements are consistent with the premise that magnetic structures possess a preferred orientation corresponding to a left-hand screw in the northern hemisphere (with an opposite sense in the southern hemisphere). Another crucial topological measure of the magnetic field is the magnetic helicity H ¼ hA  Bi (where B ¼ r  A). Although some estimates have been made of the magnetic helicity from vector magnetograms, it is almost impossible to measure this important property of the solar magnetic field with any great certainty.

Spatiotemporal variability In addition to the systematic properties of the solar magnetic field indicated by Hale’s and Joy’s laws, these large-scale features follow a systematic spatiotemporal pattern. At the beginning, of a cycle spots appear on the Sun at latitudes of around 27
; as the cycle progresses the location of activity drifts toward the equator and the spots then die out (at a latitude of around 8
) in the next minimum. It is important to note that it is the location of magnetic activity rather than individual sunspot pairs—which have a lifetime of at most a few months—that migrates toward the solar equator. This migration of activity produces the familiar butterfly diagram first exhibited by Maunder (1913). An up-to-date version showing the incidence of sunspots as a function

MAGNETIC FIELD OF SUN

of colatitude is in Figure M28a. It is clear from this figure that the level of sunspot activity has been approximately the same in each hemisphere, and that the solar magnetic field has a high degree of symmetry. The above observations indicate that the solar field is primarily axisymmetric and dipolar, although a small nonaxisymmetric component of the solar field has been observed. In addition to the basic solar cycle, sunspot activity undergoes further significant temporal variability. By using the (arbitrarily defined) sunspot number (e.g., Eddy, 1976), the variation in magnetic activity over the past 400 years may be obtained. An up-to-date record of sunspot numbers is shown in Figure 28b. This figure shows that an average cycle is temporally asymmetric, with a sharp rise from minimum to maximum of duration 3–6 years being followed by a gradual decay lasting between 5 and 8 years. The amplitude of the cycle has been shown to be anticorrelated with the length of the rise time and that of the solar cycle itself. The solar cycle appears to be chaotic with an amplitude that is modulated aperiodically. Despite the shortness of the time sequence, the existence of a characteristic timescale for the modulation of approximately 90 years (the Gleissberg cycle) has been postulated—although this is open to debate. What is clear, however, is the presence in the record of an episode with a dearth of sunspots, which lasted for about 70 years at the end of the 17th century. This interruption is known as the Maunder minimum. There is no doubt that this is a real phenomenon, and this is not due to the lack of awareness on the part of the observers (see Ribes and Nesme-Ribes, 1993). It is also interesting to note that at the end of the Maunder minimum large-scale solar magnetic activity started again primarily in the southern hemisphere of the Sun and hence the field was not dipolar for a cycle. Although small asymmetries between sunspot activity in the northern and southern hemispheres have been observed since the Maunder minimum, when the field is strong the field is largely dipolar, with only a small quadrupole component as noted above. It is only when the field is weak and the dipole component is small that a comparable quadrupolar contribution can be seen. More useful information about the temporal variability of the solar magnetic activity can be gained by the use of proxy data (see e.g., Beer, 2000). Magnetic fields in the solar wind modulate the cosmic ray flux entering the Earth’s atmosphere. These high-energy particles lead to the production of radioisotopes, including 10Be and 14C,

507

and so the abundance of these isotopes can act as a measure of solar magnetic activity—the abundance is anticorrelated with magnetic activity. The isotope 10Be is preserved in polar ice-cores and its production rate, together with the sunspot number, is shown in Figure M29. The figure clearly shows the presence of the Maunder minimum and that, although sunspot activity is largely shut off during this period, cyclic magnetic activity continued with a period of approximately 9 years. The 10Be record extends back over 50 ka and analysis of this record clearly shows both continued presence of the 11-year solar cycle. Moreover analysis indicates that the Maunder minimum is not an isolated event, with regularly spaced minima (termed grand minima) interrupting the record of activity with a significant recurrent timescale of 205 years (Wagner et al., 2001). The variations in 14C production confirm this pattern of recurrent grand minima with a timescale of about 200 years. Moreover, both of these radioisotope records show significant power at a frequency that corresponds to roughly 2100 years. It appears as though grand minima occur in bursts, with this as a well-defined period of recurrence (J. Beer, private communication). Taken together, these data indicate that the solar magnetic field undergoes a considerable amount of temporal variability on scales varying from days to hundreds of years.

The small-scale solar magnetic field In addition to the large-scale magnetic field with systematic properties described above, magnetic field is also observed on scales comparable with the solar convective scales and smaller. The magnetic network is seen to coincide largely with the network forming the boundaries of supergranular convection. The network has a different form at solar maximum and minimum. At solar minimum it consists of mixed polarity, vertically oriented bundles of magnetic flux, while at solar maximum the network is dominated by the presence of large unipolar regions in close proximity to the bipolar active regions. The magnetic network is therefore only weakly linked to the solar magnetic cycle. The flux bundles in the magnetic network are dynamic and are observed to move randomly across the solar surface, with a motion that is consistent with two-dimensional diffusion. The magnetic network is also interspersed with weak stochastic magnetic fields of random orientations. This intranetwork field is

Figure M28 (a) Solar butterfly diagram: Location of sunspots as a function of time (horizontal axis) and latitude (vertical axis). As the cycle progresses sunspots migrate from midlatitudes to the equator. (Courtesy of D.N. Hathaway). (b) Time series for yearly sunspot group numbers. Notice the Maunder minimum at the end of the 17th century.

508

MAGNETIC FIELD OF SUN

Figure M29 Comparison of the proxy 10Be data from the dye 3 ice core (measured in 10000 atoms g–1), filtered using a low pass (6 year) filter, with the filtered (6 year) sunspot group number as determined by Hoyt and Schatten (1998). Note that the 10Be is anticorrelated with sunspot activity.

weaker than the network field and has a significantly shorter dynamical timescale associated with it (on the scale of days as opposed to months). There appears to be little or no correlation of this field with the solar cycle and the behavior of this field is consistent with it being generated by the action of a local dynamo (see Dynamo, solar) situated in the upper reaches of the solar convection zone. All these small-scale magnetic features visible at the solar surface migrate slowly toward the solar pole with a velocity of between 1 and 10 m s–1. This migration is due to the presence of a systematic large-scale meridional flow at the solar surface. The meridional flow has also been detected by helioseismic inversions of the solar interior and is believed to continue poleward to a depth of 0:9R , where R is the solar radius.

Solar cycle indices—irradiance, flows and the shape of the corona The solar magnetic cycle can not only be detected in the large-scale magnetic features such as sunspots. Many other solar indices follow the 11-year activity cycle. Of particular interest terrestrially is the variation in the solar luminosity (the so-called solar constant) that occurs over the solar cycle. Total solar irradiance does vary systematically in phase with the solar cycle. In the visible spectrum the modulation varies only weakly, with the effect of bright magnetic faculae nearly exactly canceling out (but just overcoming) that of dark sunspots. However in the far-ultraviolet and X-ray part of the spectrum the modulation is large and systematic (see e.g., Frölich, 2000). The shape of the solar corona and strength of the solar wind is also correlated with the solar cycle, and these in turn interact with the geomagnetic field and are measurable in the AA and AP geomagnetic indices and visible in the record of aurorae. In addition, the presence of the magnetic field is also responsible for driving flows that are correlated with the solar activity cycle. This pattern of parallel belts of faster slower rotation was initially discovered by Doppler measurements of the solar surface and dubbed torsional oscillations. It is now known from helioseismic inversions for the internal solar rotation (Vorontsov et al., 2002) that these oscillations have an amplitude of approximately 5–10 m s–1 and that they extend deep into the solar interior—reaching at least halfway into the solar convective zone (see Figure M30/Plate 14b). These bands of zonal flows, which have an equatorward and poleward propagating branch, have a period of 11 years, which is consistent with them being driven by the Lorentz force associated with the magnetic field.

Figure M30/Plate 14b Top panel: The rotational variation as a function of time and latitude at radius r ¼ 0:98R are shown for the first nearly 6 years, and thereafter the time series is continued by exhibiting the 11-year harmonic fit. Shown to the right are the residuals from the fit, on the same color scale. Bottom panel: As for above, but showing the rotation variation as a function of depth instead of latitude, at latitude 20
. (From Vorontsov et al., 2002).

Stellar magnetic fields The Sun is just one example of a moderately rotating star whose magnetic properties can be observed in detail. Much can be learned about the magnetic behavior of the Sun, by examining the properties of nearby magnetically active stars that have similar properties (e.g., age and rotation rate). The most important results indicating magnetic activity in stars are obtained by measuring the level of Caþ, H, and K emission. The rate of Caþ emission in the Sun is well-known to be correlated with solar magnetic fields. Results from the Mount Wilson Survey indicate the level of magnetic activity in a relatively large sample of slow and moderate rotators. It is now clear that, for stars of a fixed spectral type, there is a range of activity, but that this scatter is a function of age. It can be shown that the level of activity in a star can be represented as a function of the inverse Rossby (or Coriolis) number s ¼ Otc, where O is the angular velocity of the star and tc is a suitable convective timescale (Brandenburg et al., 1998). For slow rotators, cyclic magnetic activity may be deduced from variations in the Caþ emission and it is clear that the cycle period of activity decreases with increasing rotation rate. Moreover as the rotation rate is increased the magnetic activity of the stars becomes more disordered, with a transition between cyclic, doubly periodic and chaotic activity occurring as the rotation rate is increased. On studying this relationship in greater detail it becomes apparent that the stars fall into two distinct groups, an active and an inactive branch (Brandenburg et al., 1998). This is enough to discourage the extension of the properties of slowly rotating stars to cover those with a greater angular momentum. Direct measurement of magnetic fields in stars is also possible using a variety of techniques, including broadband polarimetry and photometry (see e.g., Rosner, 2000). The luminosity of a highly active star may vary by up to 30%, whereas the known variation in solar luminosity is only 0.2%. Whether an increase in magnetic activity in a star is positively or negatively correlated with an increase in luminosity depends upon the absolute level of activity in the star. It is also found

MAGNETIC INDICES

that the level of magnetic activity is proportional to the inverse Rossby number of the star—a result that is consistent with the emission data. Clearly, as there is a small sample of stars for which we have a detailed record of magnetic activity and this record is comparatively short, there is a need to continue these observations as they give us great understanding of the magnetic properties of our nearest star—the Sun. S. Tobias

Bibliography Beer, J., 2000. Long-term indirect indices of solar variability. Space Science Reviews, 94: 53–66. Brandenburg, A., Saar, S.H., and Turpin, C.R., 1998. Time evolution of the magnetic activity cycle period. The Astrophysical Journal, 498: L51–L54. Eddy, J.A., 1976. The Maunder minimum. Science, 192: 1189–1202. Fröhlich, C., 2000. Observations of irradiance variations. Space Science Reviews, 94: 15–24. Hale, G.E., 1908. On the probable existence of a magnetic field in sunspots. Astrophysics Journal, 28: 315–343. Hoyt, D.V., and Schatten, K.H., 1998. The Role of the Sun in Climate Change. New York: Oxford University Press. Maunder, E.W., 1913. Note on the distribution of sunspots in heliographic latitude. Monthly Notices of the Royal Astronomical Society, 64: 747–761. Mestel, L., 1999. Stellar Magnetism. Oxford: Clarendon Press. Ribes, J.C., and Nesme-Ribes, E., 1993. The solar sunspot cycle in the Maunder minimum AD 1645–AD 1715. Astronomy and Astrophysics, 276: 549–563. Rosner, R., 2000. Magnetic fields of stars: using stars as tools for understanding the origins of cosmic magnetic fields. Philosophical Transactions of the Royal Society of London, Series. A, 358: 689–708. Stix, M., 1976. Differential rotation and the solar dynamo. Astronomy and Astrophysics, 47: 243–254. Wagner, G. et al., 2001. Presence of the solar de vres cycle (205 years) during the last ice age. Geophysical Research Letters, 28: 303. Vorontsov, S.V., Christensen-Dalsgaard, J., Schou, J., Strakhov, V.N., and Thompson, M.J., 2002. Helioseismic measurement of solar torsional oscillations. Science, 296: 101–103.

Cross-references Dynamo, Solar

MAGNETIC INDICES Magnetic indices are simple measures of magnetic activity that occurs, typically, over periods of time of less than a few hours and which is recorded by magnetometers at ground-based observatories (Mayaud, 1980; Rangarajan, 1989; McPherron, 1995). The variations that indices measure have their origin in the Earth’s ionosphere and magnetosphere. Some indices having been designed specifically to quantify idealized physical processes, while others function as more generic measures of magnetic activity. Indices are routinely used across the many subdisciplines in geomagnetism, including direct studies of the physics of the upper atmosphere and space, for induction studies of the Earth’s crust and mantle, and for removal of disturbed-time magnetic data in studies of the Earth’s deep interior and core. Here we summarize the most commonly used magnetic indices, using data from a worldwide distribution of observatories, those shown in Figure M31 and whose sponsoring agencies are given in Table M1.

509

Range indices K and Kp The 3-h K integer index was introduced by Bartels (1938) as a measure of the range of irregular and rapid, storm-time magnetic activity. It is designed to be insensitive to the longer term components of magnetic variation, including those associated with the overall evolution of a magnetic storm, the normal quiet-time diurnal variation, and the very much longer term geomagnetic secular variation arising from core convection. The K index is calculated separately for each observatory, and, therefore, with an ensemble of K indices from different observatory sites, the geography of rapid, ground level magnetic activity can be quantified. When it was first implemented, the calculation of K relied on the direct measurement of an analog trace on a photographic record. Today, in order to preserve continuity with historical records, computer programs using digital data mimic the original procedure. First, the diurnal and secular variations are removed by fitting a smooth curve to 1-min horizontal component (H ) observatory data. The range of the remaining data occurring over a 3-h period is measured. This is then converted to a quasilogarithmic K integer, 0, 1, 2, . . . , 9, according to a scale that is specific to each observatory and which is designed to normalize the occurrence frequency of individual K values among the many observatories and over many years. A qualitative understanding of the K index and its calculation can be obtained from Figure M32. There we show a trace, Figure M32a, of the horizontal intensity at the Fredericksburg observatory recording magnetically quiet conditions during days 299–301 of 2003, followed by the sudden commencement in day 302 and the subsequent development of the main and recovery phases of the so-called great Halloween Storm. In Figure M32b we show, on a logarithmic scale, the range of the Fredericksburg data over discrete 3-h intervals, and in Figure M32c we show the K index values themselves. Note the close correspondence between the magnetogram, the log of the range and the K index. This storm is one of the 10 largest in the past 70 years since continuous measurements of storm size have been routinely undertaken. For more information on this particular storm, see the special issue of the Journal of Geophysical Research, A9, 110, 2005. Planetary-scale magnetic activity is measured by the Kp index (Menvielle and Berthelier, 1991). This is derived from the average of fractional K indices at 13 subauroral observatories (Table M1) in such a way as to compensate for diurnal and seasonal differences between the individual observatory K values. The final Kp index has values 0, 0:3, 0:7, 1:0, 1:3, . . . etc. For illustration, in Figure M33 we show magnetograms from the 13 observatories contributing to Kp, recording the Halloween Storm of 2003, along with the Kp index itself. The distribution of observatories is far from uniform, with a predominant representation from North America and Europe, and very little representation from the southern hemisphere. In fact, in Figure M33, it is easy to see differences during the storm in the magnetograms among the different regional groupings of observatories. Although geographic bias is an obvious concern for any index intended as a measure of planetary-scale magnetic activity, Kp has proven to be very useful for scientific study (e.g., Thomsen, 2004). And, since it has been continuously calculated since 1932, Kp lends itself to studies of magnetic disturbances occurring over many solar cycles. There are several other indices related to the K and Kp. Ak and Ap are linear versions of K and Kp. Kn, An, Ks, and As are similar to Kp and Ap except that they use, respectively, northern and southern hemisphere observatories; their global averages are Km and Am. The aa index is like the Kp except that it utilizes only two, roughly antipodal, observatories, one in the northern hemisphere and one in the southern hemisphere. aa has been continuously calculated since 1868, making it one of the longest historical time series in geophysics.

Auroral electrojet indices AU, AL, AE, AO During magnetic storms, particularly during substorms, magnetospheric electric currents are often diverted along field lines, with

510

MAGNETIC INDICES

Figure M31 Map showing geographic distribution of magnetic index observatories.

Table M1 Summary of index observatories used here Agency

Country

Observatory

Observatory

Index

Geoscience Australia Geological Survey of Canada Geological Survey of Canada Geological Survey of Canada Geological Survey of Canada Geological Survey of Canada Danish Meteorological Institute Danish Meteorological Institute GeoForschungsZentrum Potsdam GeoForschungsZentrum Potsdam University of Iceland Japan Meteorological Agency Geological and Nuclear Science National Research Foundation Swedish Geological Survey Swedish Geological Survey British Geological Survey British Geological Survey British Geological Survey US Geological Survey US Geological Survey US Geological Survey US Geological Survey US Geological Survey US Geological Survey

Australia Canada Canada Canada Canada Canada Denmark Denmark Germany Germany Iceland Japan New Zealand South Africa Sweden Sweden United Kingdom United Kingdom United Kingdom United States United States United States United States United States United States

Canberra Fort Churchill Meanook Ottawa Poste-de-la-Baleine Yellowknife Brorfelde Narsarsuaq Niemegk Wingst Leirvogur Kakioka Eyerewell Hermanus Abisko Lovoe Eskdalemuir Hartland Lerwick Barrow College Fredericksburg Honolulu San Juan Sitka

CNB FCC MEA OTT PBQ YKC BFE NAQ NGK WNG LRV KAK EYR HER ABK LOV ESK HAD LER BRW CMO FRD HON SJG SIT

Kp AE Kp Kp AE AE Kp AE Kp Kp AE Dst Kp Dst AE Kp Kp Kp Kp AE AE Kp Dst Dst Kp

current closure through the ionosphere. To measure the auroral zone component of this circuit, Davis and Sugiura (1966) defined the auroral electrojet index AE. Ideally, the index would be derived from data collected from an equally spaced set of observatories forming a necklace situated underneath the northern and southern auroral ovals.

Unfortunately, the southern hemispheric distribution of observatories is far too sparse for reasonable utility in calculating AE, and the northern hemispheric observatories only form a partial necklace, due to the present shortage of reliable observatory operations in northern Russia. Progress is continuing, of course, to remedy this shortcoming, but for

MAGNETIC INDICES

Figure M32 Example of (a) magnetometer data, horizontal intensity (H) from the Fredericksburg observatory recording the Halloween Storm of 2003, (b) the maximum range of H during discrete 3 h intervals, and (c) the K index for Fredericksburg. now the partial necklace of northern hemisphere observatories is used to calculate an approximate AE. The calculation of AE is relatively straightforward. One-min resolution data from auroral observatories are used, and the average horizontal intensity during the five magnetically quietest days is subtracted. The total range of the data from among the various AE observatories for each minute is measured, with AU being the highest value and AL being the lowest value. The difference is defined as AE ¼ AU  AL, and for completeness the average is also defined as AO ¼ 1=2ðAU  ALÞ. For illustration, in Figure M34, we show magnetograms from the eight auroral observatories contributing to AE, during the Halloween Storm of 2003, along with the AE and its attendant relatives.

Equatorial storm indices Dst and Asym One of the most systematic effects seen in ground-based magnetometer data is a general depression of the horizontal magnetic field as recorded at near-equatorial observatories (Moos, 1910). This is often interpreted as an enhancement of a westward magnetospheric equatorial ring current, whose magnetic field at the Earth’s surface partially cancels the predominantly northerly component of the main field. The storm-time disturbance index Dst (Sugiura, 1964) is designed to measure this phenomenon. Dst is one of the most widely used indices in academic research on the magnetosphere, in part because it is well motivated by a specific physical theory. The calculation of Dst is generally similar to that of AE, but it is more refined, since the magnetic signal of interest is quite a bit smaller.

511

Figure M33 Example of (a) magnetometer data, horizontal intensity (H), from the observatories used in the calculation of the Kp index, together with (b) the corresponding Kp index. The observatories have been grouped into North American, European, and southern hemisphere regions in order to highlight similarities of the data within each region and differences in the data across the globe.

One-min resolution horizontal intensity data from low-latitude observatories are used, and diurnal and secular variation baselines are subtracted. A geometric adjustment is made to the resulting data from each observatory so that they are all normalized to the magnetic equator. The average, then, is the Dst index. It is worth noting that, unlike the other indices summarized here, Dst is not a range index. Its relative Asym is a range index, however, determined by the difference between the largest and smallest disturbance field among the four contributing observatories. In Figure M35 we show magnetograms from the four observatories contributing to Dst and Asym, for the Halloween storm of 2003, along with the indices themselves. The commencement of the storm is easily identified, and although the magnetic field is very disturbed during the first hour or so of the storm, the disturbance shows pronounced longitudinal difference, and hence a dramatically enhanced Asym. With the subsequent worldwide depression of H through to the beginning of day 303 the storm is at its main phase of development. During this time Dst becomes increasingly negative. It is of interest to note that it is during this main phase that AE is also rapidly variable, signally the occurrence of substorms with the closure of magnetospheric electric currents through the ionosphere. AE diminishes during the recovery period of the storm as Dst also pulls back for its most negative values and Asym is diminished. Toward the end of day 303 the second act of this complicated storm begins, with a repeat of the observed relationships of the various indices.

512

MAGNETIC MINERALOGY, CHANGES DUE TO HEATING

Availability Magnetic indices are routinely calculated by a number of different agencies. Intermagnet agencies routinely calculate K indices for their observatories (www.intermagnet.org). The GeoForschungsZentrum in Potsdam calculates Kp (www.gfz-potsdam.de). The Kyoto World Data Center calculates AE and Dst (swdcwww.kugi.kyoto-u.ac.jp). Other agencies supporting the archiving and distribution of the indices include the World Data Centers in Copenhagen (web.dmi.dk/fsweb/ projects/wdcc1) and Boulder (www.ngdc.noaa.gov), as well as the International Service of Geomagnetic Indices in Paris (www.cetp. ipsl.fr). Jeffrey J. Love and K.J. Remick

Bibliography

Figure M34 Example of (a) magnetometer data, horizontal intensity (H ), from the observatories used in the calculation of the AE indices, together with (b) the corresponding AU and AL indices and the (c) AE and AO indices.

Bartels, J., 1938. Potsdamer erdmagnetische Kennziffern, 1 Mitteilung. Zeitschrift f ür Geophysik, 14: 68–78, 699–718. Davis, T.N., and Sugiura, M., 1966. Auroral electrojet activity index AE and its universal time variations. Journal of Geophysical Research, 71: 785–801. Mayaud, P.N., 1980. Derivation, Meaning, and Use of Geomagnetic Indices, Geophysical Monograph 22. Washington, DC: American Geophysical Union. McPherron, R.L., 1995. Standard indices of geomagnetic activity. In Kivelson, M.G., and Russell, C.T. (eds.), Introduction to Space Physics. Cambridge, UK: Cambridge University Press, pp. 451–458. Menvielle, M., and Berthelier, A., 1991. The K-derived planetary indices—description and availability. Reviews of Geophysics, 29: 415–432. Moos, N.A.F., 1910. Colaba Magnetic Data, 1846 to 1905. 2. The Phenomenon and its Discussion. Bombay, India: Central Government Press. Rangarajan, G.K., 1989. Indices of geomagnetic activity. In Jacobs, J.A. (ed.), Geomagnetism , Vol. 2. London, UK: Academic Press, pp. 323–384. Sugiura, M., 1964. Hourly values of equatorial Dst for the IGY. Annals of the International Geophysical Year, 35: 945–948. Thomsen, M.F., 2004. Why Kp is such a good measure of magnetospheric convection. Space Weather, 2: S11004, doi:10.1029/ 2004SW000089.

Cross-references IAGA, International Association of Geomagnetism and Aeronomy Ionosphere Magnetosphere of the Earth

MAGNETIC MINERALOGY, CHANGES DUE TO HEATING

Figure M35 Example of (a) magnetometer data, horizontal intensity (H ), from the observatories used in the calculation of the Dst indices, together with (b) the corresponding Dst plotted as the center trace and the maximum and minimum disturbance values, the difference of which is Asym.

Mineralogical alterations occur very often in rocks subjected to thermal treatment. Laboratory heating may cause, in many cases, not only magnetic phase transformations, but also changes in the effective magnetic grain sizes, the internal stress, and the oxidation state. The presence or absence of such alterations is crucial to the validity and success of numerous magnetic studies. The basic assumption in paleointensity determinations in the measurement of anisotropy of thermoremanent magnetization is that the rock is not modified during the different successively applied heating treatments. For simple thermal demagnetization, the occurrence of mineralogical alteration can introduce errors in the determination of the magnetic carrier if the latter had undergone transformation a at lower

MAGNETIC MINERALOGY, CHANGES DUE TO HEATING

temperature than its unblocking temperature. Acquisition of parasitic chemical remanent magnetization is also possible if the heating was not made in a perfectly zero magnetic field. Formation of magnetic grains with very short relaxation times can lead to a large variation in the magnetic viscosity, making the stable components of the remanent magnetization sometimes immeasurable during thermal demagnetization. The interpretation of the results obtained by the Lowrie’s method (1990) of identification of magnetic minerals by their coercivity and unblocking temperatures can be biased by mineralogical alteration, because rock can undergo mineralogical alteration instead of thermal demagnetization. Analysis of the Curie curve (the measurement of the susceptibility in the low- or high magnetic field as a function of the temperature) is a key method for the identification of magnetic minerals, because each mineral has its Curie temperature (corresponding to a change from a ferrimagnetic to a paramagnetic state). On a Curie curve with increasing temperature, the Curie temperature is shown as a strong decrease in the susceptibility, sometimes preceded by an increase in susceptibility (a Hopkinson peak). However, mineralogical alteration can also increase or decrease susceptibility so it is not possible to discriminate the Curie temperature or the Hopkinson peak from mineralogical alteration from the shape of a simple curve directly until the highest temperature is reached, except if the heating and cooling curves are identical.

Example of mineralogical transformations due to heating The increase of susceptibility during heating is mainly due to the formation of iron oxides. A decrease in susceptibility is often related to transformation of these oxides, such as in the oxidation of magnetite to hematite. Magnetite growth between 500 and 725
C (susceptibility increase) and the hematization of the magnetite at a higher temperature (susceptibility decrease) have been pointed out in a study of biotite granites (Trindade et al., 2001). Furthermore, magnetic oxides can be formed from iron sulfides (pyrite, pyrrhotite, greigite, troilite), carbonates (siderite, ankerite), silicates, other iron oxides or hydroxides (e.g., Schwartz and Vaughan, 1972; Dekkers, 1990a,b). During heating, hexagonal pyrrhotite (antiferromagnetic) can be transformed first, to monoclinic pyrrhotite (ferrimagnetic) by partial oxidation to magnetite (Bina et al., 1991). Pyrrhotite oxidizes mostly to magnetite at higher temperatures. Siderite oxidizes to magnetite or maghemite when exposed to air, even at room temperature but the oxidation is faster during heating (Ellwood et al., 1986; Hirt and Gehring, 1991). Maghemite converts to hematite in the temperature range from 250 to 750
C, depending on the grain size, degree of oxidation, and the presence of defects in the crystallographic lattice (Verwey, 1935; Özdemir, 1990). In the range 250–370
C, goethite is transformed generally to very fine grains of hematite (Gehring and Heller, 1989; Dekkers, 1990b). Lepidocrocite starts to transform into superparamagnetic (SP) maghemite at 175
C, with further conversion of this maghemite into hematite at around 300
C (Gehring and Hofmeister, 1994). In an oxidative atmosphere, ferrihydrite transforms into hematite (Weidler and Stanjek, 1998). But, ferrihydrite heated in the presence of an organic reductant forms magnetite and/or maghemite, or magnetitemaghemite intermediates (Campbell et al., 1997). Exsolved magnetite has been found in plagioclases, pyroxene, and micas. With increasing temperature phyllosilicates undergo different reactions (Murad and Wagner, 1998). Thus, new phases developed at high temperatures depend on the composition of the original clay minerals, the firing temperature, and the atmosphere around the samples (Osipov, 1978). During thermal treatment, ferrimagnetic minerals can also be affected by changes in the magnetic grain size (alteration, grain growth, or grain breaking due to fast heating and cooling). Therefore, grain size dependence of magnetic susceptibility is weak for pseudosingle-domain (PSD) and multidomain (MD) magnetite grains (Hartstra, 1982), but becomes important if the grain size variation is a transformation into SP—SingleDomain (SD) or a SD—PSD transformation. The susceptibility of the SD grains is lower than for the other grain sizes (Stacey and Banerjee, 1974).

513

Homogenization of the distribution of crystal defects can also have significant effects, leading to an increase in the susceptibility of large grains and in the redistribution of the domain wall. Inversion by filling up by new cations of the vacancy sites within the crystal lattice (Bina and Henry, 1990) can give significant susceptibility variations. This last mechanism and grain breaking could be at the origin of susceptibility variation since the heating takes place at relatively low temperatures.

Methods of analysis A change of color of the sample is an indication of mineralogical alteration, but not all mineralogical transformations of magnetic minerals do lead to such color changes. Moreover, such change can be limited to a very fine surface layer and concern only to paramagnetic or diamagnetic minerals. Different investigations are therefore necessary to determine the magnetic mineralogy characteristics. To investigate the mineralogical alteration of magnetic minerals due to heating, as a first possibility the simple observation of a thin section or conduct a microprobe analysis. In practice, such an approach is often difficult since alteration can have affected a limited part of the magnetic minerals, which can be moreover of a very small size. It is therefore easy to miss the modified minerals. However, when it works, this method yields precise indications about the affected minerals. Transformation of a nonferrimagnetic mineral to another nonferrimagnetic mineral does not introduce a significant variation in susceptibility. Since the mineralogical change of the ferrimagnetic phase mostly corresponds on the contrary to the susceptibility change, measurement of susceptibility is generally a simple, fast, and efficient method to point out the alteration of the ferrimagnetic part. The least time consuming and common approach uses the monitoring of the variations of bulk magnetic susceptibility measured at room temperature and then after subsequent thermal steps, for example during thermal demagnetization. The different parts of Curie curve measured during heating and cooling respectively, are similar if no alterations have occurred. The nonreversibility of the curve, on the contrary, indicates mineralogical changes. Hrouda et al. (2003) proposed to make several Curie curves for the same sample, with increasing maximum temperature, until obtaining a nonreversible curve. A faster approach is to apply cooling as soon as a significant susceptibility change occurs during heating. If the cooling curve is similar to the heating curve, the sample can be then heated again until the next significant susceptibility variation, etc. (Figure M36). Comparison of the heating and cooling curves can yield key information about the affected and altered minerals. If a Curie point occurs only on the heating curve, the corresponding mineral not disappeared. On the other hand, if this Curie point appears on the heating curve at a higher temperature than a transformation, it is

Figure M36 Thermomagnetic curve (low field normalized susceptibility K/K0 as a function of the temperature T in
C) of a dolerite sample during heating (thick line) and cooling (thin line). Partial loop “a” corresponds to an absence of mineralogical alteration (reversible curve) contrary to the final part “b” of the loop (irreversible curve).

514

MAGNETIC MINERALOGY, CHANGES DUE TO HEATING

often difficult or even impossible to determine if this Curie temperature was associated with the preexisting or the newly formed mineral. A Curie temperature occuring only on the cooling curve is related to a newly formed mineral. Van Velzen and Zijderveld (1992) proposed a complementary method for monitoring mineralogical alteration by using not only the susceptibility but several remanent magnetization characteristics. This timeconsuming method allows a much more precise study of the effect of heating. However, if several different alterations had occurred during the same thermal treatment, they cannot be distinguished using these approaches. Hysteresis loop analysis is another classical method in rock magnetism, giving, in particular, data on grain size and coercivities of the magnetic minerals. The sample is first subjected to a strong magnetic field to obtain the saturation of the magnetization. The total magnetization is then measured during progressive decrease of the field followed by the application of an increasing field in the opposite direction until obtaining the saturation of the magnetization in this opposite direction (“descending” curve d, Figure M37). The “ascending” curve a is then obtained by measurement of the total magnetization during a decrease of the field in the opposite direction followed by an increase in the field in the direction of the initial field. To further investigate the mineralogical alteration, hysteresis loops at room temperature are measured before and after heating (Figure M37a). Difference hysteresis loops (Jb–a) are obtained by subtraction for a same field (H) value of the measured magnetization value before (Jb) and after (Ja) heating in curves d and a, respectively (Figure M37b). They are sometimes relatively complicated curves, but using the separation method proposed by Von Dobeneck (1996) in two curves (half-difference and half-sum), the interpretation becomes much easier (Figure M37c). In particular, it is possible to discriminate characteristics (coercivity, magnetic field for saturation) of disappearing and occurring magnetic components, even during the same thermal treatment (Henry et al., 2005).

Applications If heating at a temperature T applied in laboratory introduces a mineralogical change, previous heating at the same temperature T should have given the same transformation. The studied rocks in their present state were therefore not subjected to this temperature before being sampled. The minimum temperature to develop mineralogical change, therefore, corresponds to the maximum temperature possibly undergone by the rock in the past (Hrouda et al., 2003). This is therefore the maximum paleotemperature indicator. However, that means obviously, the maximum temperature since the rock has the present mineralogical composition, i.e. sometimes relatively recently in case of weathering and low-temperature oxidation. Modification of the magnetic fabric as a result of heating has been applied to different types of rocks (see Magnetic susceptibility, anisotropy, effects of heating). The outcome was sometimes a simple enhancement of the fabric, but significant changes of the fabric have also been obtained. It is moreover possible to determine the anisotropy of magnetic susceptibility of the ferrimagnetic minerals formed or that have disappeared during successive heatings. To this aim, the tensor resulting from the subtraction of the tensors measured before and after heating is used (Henry et al., 2003). When a same magnetic fabric is obtained from several following thermal steps, it cannot be related to the randomly oriented ferrimagnetic minerals. Instead, the newly formed fabric must be related to characteristics of the preexisting rock. By comparing this ferrimagnetic mineral fabric with the initial whole rock fabric, it is possible to distinguish cases where heating simply enhances of preexisting fabric from those where thermal treatment induces a different fabric. Relative to the preheated fabric, this different fabric may simply be an inverse fabric or one whose principal susceptibility axes are oriented in a different direction, relative to petrostructural elements other than those defining the initial magnetic fabric. Important applications concerning all the methods, assumes that no transformation occurred during heating. It is fundamentally important to

Figure M37 Example of hysteresis loops (ascending—a and descending—d curves) measured at room temperature on dolerite sample after heating at 400 and at 550
C (a) and corresponding difference loop (b). From curves (c) obtained from half-sum and half-difference of the curves b, transformation of a magnetic component with low coercivity and saturation field to another magnetic phase with higher coercivity and saturation field can be inferred (Henry et al., 2004): Part of the curves pointing out disappearance (in grey) of component with low coercivity (1) and moderate saturation field (2), and occurrence (in black) of component with higher coercivity (3) and high saturation field (4). Field H in T, Magnetization in A m2 kg–1. verify this assumption when paleointensity or anisotropy of thermoremanent magnetization studies are carried out. It is important to point out that mineralogical alteration can also have important implications for all the studies using heating, such as Lowrie’s (1990) method of identification of magnetic minerals (using corercivity and unblocking temperatures) or even simple thermal demagnetization. Bernard Henry

Bibliography Bina, M., Corpel, J., Daly, L., and Debeglia, N., 1991. Transformation de la pyrrhotite en magnetite sous l’effet de la temperature: une source potentielle d’anomalies magnétiques. Comptes Rendus de I’Académie des Sciences, Paris. 313(II): 487–494. Bina, M., and Henry, B., 1990. Magnetic properties, opaque mineralogy and magnetic anisotropies of serpentinized peridotites from ODP hole 670A near Mid-Atlantic ridge. Physics of the Earth and Planetary Interiors, 65: 88–103.

MAGNETIC PROPERTIES, LOW-TEMPERATURE

Campbell, A.S., Schwertmann, U., and Campbell, P.A., 1997. Formation of cubic phases on heating ferrihydrite. Clay Minerals, 32: 615–622. Dekkers, M.J., 1990a. Magnetic monitoring of pyrrhotite alteration during thermal demagnetisation. Geophysical Research Letters, 17: 779–782. Dekkers, M.J., 1990b. Magnetic properties of natural goethite—III. Magnetic behaviour and properties of minerals originating from goethite dehydration during thermal demagnetisation. Geophysical Journal International, 103: 233–250. Ellwood, B.B., Balsam, W., Burkart, B., Long, G.J., and Buhl, M.L., 1986. Anomalous magnetic properties in rocks containing the mineral siderite: paleomagnetic implications. Journal of Geophysical Research, 91: 12779–12790. Gehring, A.U., Heller, F., and 1989. Timing of natural remanent magnetization in ferriferous limestones from the Swiss Jura Mountains. Earth and Planetary Science Letters, 93: 261–272. Gehring, A.U., and Hofmeister, A.M., 1994. The transformation of lepidocrocite during heating: a magnetic and spectroscopic study. Clays and Clay Minerals, 42(4): 409–415. Hartstra, R.L., 1982, Grain size dependence of initial susceptibility and saturation magnetization-related parameters of four natural magnetites in the PSD-MD range. Geophysical Journal of the Royal Astronomical Society, 71: 477–495. Henry, B., Jordanova, D., Jordanova, N., Souque, C., and Robion, P., 2003. Anisotropy of magnetic susceptibility of heated rocks. Tectonophysics, 366: 241–258. Henry, B., Jordanova, D., Jordanova, N., and Le Goff, M., 2005. Transformations of magnetic mineralogy in rocks revealed by difference of hysteresis loops measured after stepwise heating: Theory and cases study. Geophysical journal International, 162: 64–78. Hirt, A.M., and Gehring, A.U., 1991. Thermal alteration of the magnetic mineralogy in ferruginous rocks. Journal of Geophysical Research, 96: 9947–9953. Hrouda, F., Müller, P., and Hanak, J., 2003. Repeated progressive heating in susceptibility vs. temperature investigation: a new palaeotemperature indicator? Physics and Chemistry of the Earth, 28: 653–657. Lowrie, W., 1990. Identification of ferromagnetic minerals in a rock by coercivity and unblocking temperature properties. Geophysical Research Letters, 17: 159–162. Murad, E., and Wagner, U., 1998. Clays and clay minerals: The firing process. Hyperfine Interactions, 117: 337–356. Osipov, J., 1978. Magnetism of Clay Soils (in Russian). Moscow: Nedra. Özdemir, Ö., 1990. High temperature hysteresis and thermoremanence of single-domain maghemite. Physics of the Earth and Planetary Interiors, 65: 125–136. Schwartz, E.J., and Vaughan, D.J., 1972. Magnetic phase relations of pyrrhotite. Journal of Geomagnetism and Geoelectricity, 24: 441–458. Stacey, F.D., and Banerjee, S.K., 1974. The Physical Principles in Rock Magnetism. Amsterdam: Elsevier, 195 pp. Trindade, R.I.F., Mintsa Mi Nguema, T., and Bouchez, J.L., 2001. Thermally enhanced mimetic fabric of magnetite in a biotite granite. Geophysical Research Letters, 28: 2687–2690. Van Velzen, A.J., and Zijderveld, J.D.A., 1992. A method to study alterations of magnetic minerals during thermal demagnetization applied to a fine-grained marine marl (Trubi formation, Sicily). Geophysical Journal International, 110: 79–90. Verwey, 1935. The crystal structure of gFe2O3 and gAl2O3. Z. Krist., Zeitschrift für Kristallographie 91: 65–69. Von Dobeneck, T., 1996. A systematic analysis of natural magnetic mineral assemblages based on modeling hysteresis loops with coercivity-related hyperbolic basis functions. Geophysical Journal International, 124: 675–694.

515

Weidler, P., and Stanjek, H., 1998. The effect of dry heating of synthetic 2-line and 6-line ferrihydrite. II. Surface area, porosity and fractal dimension. Clay Minerals, 33: 277–284.

Cross-references Chemical Remanent Magnetization Depth to Curie Temperature Magnetic Remanence, Anisotropy Magnetic Susceptibility Magnetic Susceptibility, Anisotropy Magnetic Susceptibility, Anisotropy, Effects of Heating Paleointensity: Absolute Determination Using Single Plagioclase Creptals

MAGNETIC PROPERTIES, LOW-TEMPERATURE Introduction Use of magnetic measurements at cryogenic temperatures for characterizing magnetic mineralogy of rocks was initiated in the early 1960s, when it was realized that several minerals capable to carry natural remanent magnetization (NRM), e.g., magnetite and hematite, show distinctive magnetic phase transitions below room temperature. In the last decade, low-temperature magnetometry of rocks and minerals has seen a new boost due to increasing availability of commercial systems capable to carry out magnetic measurements down to and below 4.2 K. Low-temperature magnetometry has the potential to complement conventional high-temperature methods of magnetic mineralogy while offering an advantage of avoiding chemical alteration due to heating. This is especially important in the case of sedimentary rocks, which alter much more readily upon heating. However, additional complications may arise because of a possible presence in a rock of mineral phases showing ferrimagnetic or antiferromagnetic ordering below room temperature. On the other hand, these minerals are often of a diagnostic value by themselves, being the signature of various rock-forming processes. In all, low-temperature magnetometry is a valuable new tool in rock and environmental magnetism. Several factors control low-temperature behavior of remanent magnetization and low-field susceptibility of minerals and rocks. Phase transitions, which may occur below room temperature, have the most profound effect. Also of importance is the temperature variation of the intrinsic material properties such as magnetocrystalline anisotropy and magnetostriction. Low-temperature magnetic properties of minerals are also affected by their stoichiometry and degree of crystallinity. Last but not least, low-temperature variation of remanence and magnetic susceptibility is generally grain-size dependent. In particular, ultrafine (say, 0.5. Compositions close to both hematite and ilmenite are quite common in igneous rocks, while truly single-phase titanohematites of an intermediate composition (0.2 < y < 0.8) can be obtained only in the laboratory by quenching from high temperatures. Natural intermediate titanohematites of nominal composition 0.5  y  0.7, found in rapidly chilled dacitic pyroclastic rocks, received much interest in rock magnetism due to the ability to acquire a self-reversed TRM (Uyeda, 1958; Hoffman, 1992). Magnetic properties of titanohematites are determined by their composition: (i) for 0  y < 0.5 an overall ordering is antiferromagnetic with a weak parasitic ferromagnetism due to a spin canting (Dzyaloshinsky, 1958; Moriya, 1960); (ii) compositions with 0.5 < y < 0.7 are ferrimagnetic; (iii) titanohematites with 0.7 < y  0.93 show rather complex magnetic structures, which will be discussed in more detail below; (iv) compositions with y greater than about 0.93 show antiferromagnetism. Curie temperatures of titanohematites decrease approximately linearly with increasing ilmenite content up to y  0.93, reaching the values below room temperature at y  0.75 (Nagata, 1961). Hematite is the most stable iron oxide in oxidizing conditions and, as such, is found in many sedimentary and in some highly oxidized volcanic rocks. Its small net saturation moment of about 0.4 A m2 kg1 (Morrish, 1994) results primarily from a spin canting. Another mechanism that can produce a magnetic moment is a presence

MAGNETIC PROPERTIES, LOW-TEMPERATURE

of defects or impurity atoms in the hematite crystalline lattice (Bucur, 1978). Since hematite is much less strongly magnetic than magnetite or maghemite, the two latter minerals dominate the NRM in the case when concentrations of these minerals are comparable. However, in an important class of sedimentary rocks called red beds, NRM is almost entirely carried by hematite. Below room temperature, bulk hematite exhibits the so-called Morin transition (Morin, 1950). At the Morin transition spin directions change from the basal plane to the rhombohedral c-axis, and the spin canting, and the magnetic moment associated with it, seems to vanish. The defect moment, however, remains largely intact. This explains, at least qualitatively, the peculiar behavior of room temperature remanence being cycled in a zero field through the Morin transition (Haigh, 1957; Gallon, 1968; Bucur, 1978): magnetization lost on cooling is largely, but not completely, restored on warming (Figure M41). Still unresolved is an existence of the transition temperature hysteresis during magnetization cycling in a zero field. Transition temperatures determined from cooling curves are often lower than those determined from warming curves, sometimes by as much as 20
. Alike the Verwey transition in magnetite, the Morin transition is sensitive to impurities. Titanium is known to suppress the transition in concentrations as low as less than 1% (Kaye, 1962), while the transition exists until 10% of Al substitution (da Costa et al., 2002). The transition becomes more gradual for highly aluminous hematites, in the sense that low- and high-temperature phases appear to coexist over a considerable range of temperatures, and the effective transition temperature is lowered to about 180 K for hematite with 10% Al. The transition temperature is lowered and the transition itself appears considerably suppressed in fine grains (Bando et al., 1965; Schroeer and Nininger, 1967; Nininger and Schroeer, 1978; Amin and Arajs, 1987). This may be the reason why the Morin transition is rarely found in hematite-bearing rocks (Dekkers and Linssen, 1989). Titanohematites with y up to about 0.7 do not show any distinctive low-temperature magnetic properties (Ishikawa and Akimoto, 1957).

Figure M41 Variation of IRM (0.6 T), acquired at room temperature in a dispersed hematite powder, during a zero-field cycle to 193 K (–80
C). (Modified from Haigh (1957), with the permission of the publisher, Taylor & Francis Ltd., http://www.tandf.co.uk/journals.)

519

On the contrary, more Ti-rich compositions show fairly complex behavior (Figure M42). At very low temperatures, a spin glass structure seems to arise (Arai and Ishikawa, 1985; Arai et al., 1985a,b; Ishikawa et al., 1985), the spin glass transition temperatures Tsg being considerably lower than the respective Curie temperatures. Between these two transition points compositions with y  0.8–0.9 show superparamagnetic behavior (Ishikawa et al., 1985). The spin glass phase is capable to carry a relatively strong remanence, which disappears at the Tsg. The coercive force also increases sharply below this temperature reaching 300–600 mT at 4.2 K. It must be noted, however, that the above studies have been carried out using the synthetic materials. Whether their natural analogs would show similar magnetic properties remains largely unknown. The second end member of the titanohematite series, ilmenite, orders antiferromagnetically below its Néel temperature of 57 2 K (Senftle et al., 1975). Antiferromagnetism in the low-temperature phase persists up to about 6.6% of Fe3þ substitution, while Néel temperatures shift to below 50 K (Thorpe et al., 1977). Ilmenite therefore contributes only to induced magnetization. A susceptibility peak at 45 K, which may be due to ilmenite, has been reported for a basalt sample from the Hawaiian deep drill hole SOH-1 (Moskowitz et al., 1998).

Iron hydroxides Hydrous iron oxides are produced commonly by weathering of iron-bearing rocks in ambient conditions. They are also ubiquitous in

Figure M42 Low-temperature magnetic-phase diagram of ilmenite-rich part of the FeTiO3-Fe2O3 system. Down triangles indicate the temperature where TRM disappears, up triangles the temperature of the susceptibility peak. Closed symbols refer to single crystals, open symbols to polycrystalline samples. Note that x 1–y. (After Ishikawa et al. (1985), with the permission of the publisher, the Physical Society of Japan.)

520

MAGNETIC PROPERTIES, LOW-TEMPERATURE

marine sediments (Murray, 1979). Of various iron hydroxides, the orthorhombic form, goethite (a-FeOOH), is the most stable. Goethite is an antiferromagnet but, like hematite, often carries a small magnetic moment believed to be of defect origin (Strangway et al., 1968; Hedley, 1971). Below its Néel temperature of 120
C (in wellcrystalline samples) goethite is extremely magnetically hard, requiring fields in excess of 10 T to approach the magnetic saturation. An intense TRM, nearly equal to the spontaneous magnetization Js, can be acquired by cooling from TN in a strong field. Upon warming in a zero field it is demagnetized almost linearly with temperature until fully vanishing somewhat below TN, but on the consequent cooling in a zero field a considerable part of TRM is restored (Figure M43a, Rochette and Fillion, 1989). TRM is far greater than an IRM that could be acquired by application of the same field at a low temperature. Therefore, FC and ZFC warming curves of goethite are very strongly different for both bulk and nanocrystalline goethite (Figure M43b, Guyodo et al., 2003). Both features can be used as a discriminatory criterion for this mineral. Other iron hydroxides often found in sediments and soils include lepidocrocite (g-FeOOH) and ferrihydrite (5Fe2O39H2O). These minerals are paramagnetic at room temperature and do not contribute to NRM; however, both can transform to ferrimagnetic phases (maghemite) on moderate heating. Recent studies of the lowtemperature magnetic properties (Zergenyi et al., 2000; Hirt et al., 2002) have revealed that below their ordering temperatures (100 K for ferrihydrite and 50–60 K for lepidocrocite) these minerals can acquire a considerable, on the order of several tenths of A m2 kg1, remanence which decays nearly exponentially on heating in zero field, mimicking the superparamagnetic behavior of ultrafine particles of strongly magnetic minerals, e.g., magnetite or maghemite (Figure M44).

Iron sulfides Of a variety of iron sulfides occurring in nature, monoclinic 4C pyrrhotite (Fe7S8) is the most extensively studied. The magnetic transition at 30–34 K, initially discovered in early 1960s (Besnus and Meyer, 1964), has been since found in a number of natural and synthetic pyrrhotites (Rochette, 1988; Dekkers, 1989; Dekkers et al., 1989; Rochette et al., 1990). Phenomenologically, the behavior of the SIRM and the coercive force on crossing the transition has much in common with the much more extensively studied Verwey transition in magnetite. Part of a SIRM given at some temperature below the transition is demagnetized at the latter. A SIRM acquired above the transition, when cycled through the transition in zero magnetic field is also partly demagnetized, and a rebound of magnetization at the transition is observed during warming (Figure M45). The latter feature is grain-size dependent, becoming progressively less pronounced in finer grains. Pyrrhotite grains several tens of microns in size show an irreversible loss of remanence even during a zero-field cycle to 77 K (Dekkers, 1989). The Mrs/Ms ratio and the coercive force increase sharply below the transition, indicating that the low-temperature phase is in the single-domain magnetic state. These properties can serve as discriminatory criteria for monoclinic pyrrhotite in the case when heating the sample above room temperature is impossible or undesirable. Thiospinel of iron, mineral greigite (Fe3S4) once thought to be rare, is being identified in growing number of environments, contributing to NRM of sediments and soils (see e.g., Snowball and Torii, 1999 and references therein). Its low-temperature magnetic properties have been the subject of several recent studies (Roberts, 1995; Snowball and Torii, 1999; Dekkers et al., 2000); however, no distinctive features between 4.2 K and room temperature have been observed. Generally, SIRM given at a low temperature decreases monotonously on warming in zero-field albeit at different rates (Roberts, 1995), while roomtemperature SIRM hardly show any changes on cooling to 4 K (Dekkers et al., 2000, Figure M46). The latter study also reported on the magnetic hysteresis measurements on synthetic greigite. A peak

Figure M43 (a) Saturation magnetization Js (crosses) in 101 A m2 kg1 and high-field susceptibility wl (open circles) in 109 m3 kg1 vs temperature for two goethite samples. Js and saturation remanence Jrs are equal for the G70 sample. The upper curve of Jrs (triangles) is obtained by heating peak TRM at 4.2 K and the lower one (dots) by cooling peak TRM from 290 K. (b) Field-cooled (FC) and zero-field-cooled (ZFC) remanence curves for 350-nm (left vertical axis) and 3.5-nm goethite (right vertical axis) samples. (After Rochette and Fillion (1989) and Guyodo et al. (2003) ã American Geophysical Union, with the permission of the authors and the publisher.)

in Mrs was found at 10 K, and both the coercive force and coercivity of remanence increased significantly below at about 30 K. Reasons for this behavior are not clear. Pyrite (FeS2) is often found in sediments affected by diagenesis. Theoretical calculations show that pyrite and its polymorph marcasite are low-spin paramagnets whose magnetic susceptibility is temperature independent (Hobbs and Hafner, 1999). However, in practice an increase of susceptibility is often observed in both compounds below

MAGNETIC PROPERTIES, LOW-TEMPERATURE

521

Figure M45 Examples of the 34-K transition in samples of pyrrhotite with contrasting grain size. Crosses correspond to cooling and circles to rewarming. (Modified from Dekkers et al. (1989) ã American Geophysical Union, with the permission of the author and the publisher.)

Siderite and rhodochrosite

Figure M44 (a) Thermal demagnetization of IRM acquired at 5 K in a 2.5 T field during warming to 300 K after cooling the sample in zero-field cooling (ZFC) or in a 2.5 T field (FC) for two lepidocrocite samples with different crystallite size. (b) Thermal demagnetization of IRM acquired at 10 K in a 2.5 T field during warming to 300 K for a ferrihydrite sample. (After Hirt et al. (2002) and Zergenyi et al. (2000) ã American Geophysical Union, with the permission of the authors and the publisher.)

10 K (Burgardt and Seehra, 1977; Seehra and Jagadeesh, 1979). Most likely it is due to otherwise undetectable impurities showing usual Curie-Weiss paramagnetism. Pyrrhotites richer in iron than Fe7S8 are antiferromagnetic at room temperature, but transform to a ferrimagnetically ordered phase at 200
C, and eventually to monoclinic pyrrhotite on further heating. Other Fe-S minerals (mackinawite, smythite) have been often considered metastable in ambient conditions. These minerals received so far a limited interest in rock magnetism, and their magnetic properties at low temperatures are largely unknown.

Siderite (iron carbonate, FeCO3) and rhodochrosite (manganese carbonate, MnCO3) usually precipitate jointly in anoxic sedimentary environments when iron and manganese are present in soluble form. The two compounds may form solid solutions spanning the entire range of intermediate compositions. Although paramagnetic at room temperature, siderite has some importance in paleomagnetism, since secondary maghemite is produced by weathering siderite-bearing rocks, and also because siderite easily converts to magnetite at heating to 400
C, thereby putting an effective limit to thermal demagnetization (Ellwood et al., 1986, 1989). Siderite orders antiferromagnetically below the Néel temperature of 37–40 K (natural samples of varying purity, cf. Jacobs, 1963; Robie et al., 1984). MnCO3 is a canted antiferromagnet with slightly lower Néel temperature of 32 K (synthetic crystal, Borovik-Romanov et al., 1981). Both compounds can acquire remanence below TN, which is lost quite sharply on warming in zero field. These minerals could be thus confused with pyrrhotite, which shows the magnetic transition in the same temperature range. However, siderite and rhodochrosite can still be distinguished from pyrrhotite since both minerals show a large (over one order of magnitude for siderite) difference between SIRMs acquired after FC and ZFC, respectively (Housen et al., 1996; Frederichs et al., 2003), not observed for pyrrhotite (Figure M47). It might be expected that cycling of room-temperature SIRM in siderite- and pyrrhotite-bearing samples will also distinguish between the two minerals.

Vivianite Vivianite, iron hydrophosphate (Fe3(PO4)28H2O), usually precipitates in moderately to highly productive, iron-rich sedimentary environments. At low temperatures, vivianite shows antiferromagnetic ordering in each of two sublattices formed by Fe2þ ions occupying nonequivalent positions (van der Lugt and Poulis, 1961). The two sublattices are canted by approximately 42
(Forsyth et al., 1970). However, conflicting results have been reported on changes of the antiferromagnetic order with temperature. Forstat et al. (1965) have

522

MAGNETIC PROPERTIES, LOW-TEMPERATURE

Figure M46 (a) Thermal demagnetization of IRM acquired at 5 K in a 2.5 T field for four greigite samples. (b) Behavior of IRM (1.2 T) acquired at room temperature for synthetic greigite during zero-field cooling to 4 K. Sample EXP31 was stored for 22 months before the measurements, whereas sample G932 was a “fresh” sample. (After Roberts (1995) and Dekkers et al. (2000), with the permission of the authors.) found two sharp anomalies in the specific heat at 9.60 and 12.40 K, respectively, which were attributed to two distinct antiferromagneticparamagnetic transitions. On the other hand, in a neutron diffraction experiment Forsyth et al. (1970) found only one antiferromagneticparamagnetic transition at 8.84 0.05 K. A recent rock magnetic study (Frederichs et al., 2003) has confirmed previous data on the magnetic susceptibility, and found that below 10 K vivianite can acquire only a small remanence on the order of 103 A m2 kg1.

Figure M47 Warming curves of IRM acquired at 2 K in a 5 T magnetic field after cooling in zero field (open symbols) and in a 5 T field (solid symbols) for well-crystalline siderite (a) and rhodochrosite (b). (After Frederichs et al. (2003), with the permission of the author and the publisher, Elsevier.) state, can exhibit the magnetic hysteresis and acquire remanent magnetization at low temperatures. An example is mineral minnesotaite, or ferrous talc, of chemical formula Fe3Si4O10(OH)2, which acquires an IRM(2 T) of about 40 A m2 kg1 at 4.2 K (Ballet et al., 1985). The underlying mechanism of this behavior is that in minnesotaite the metamagnetic transition from an antiferromagnetic to a ferromagnetic state occurs in an anomalously low field of about 0.5 T. After once being subjected to fields higher than the metamagnetic threshold, the sample follow hysteresis loop never to return to the antiferromagnetic state (Coey and Ghose, 1988). In summary, silicate minerals may have certain potential to contribute to the low-temperature magnetic signal of rocks, particularly to the magnetic susceptibility; whether and how often this is really the case remains to be seen. Andrei Kosterov

Iron silicates Silicates are the most abundant minerals in the Earth’s crust. They often contain iron in either ferrous (Fe2þ) or ferric (Fe3þ) form. At low temperatures, iron-rich silicates typically order antiferromagnetically, and the observed Néel temperatures range from 10–15 mm) contain many domains (>10); they are paleomagnetically less stable. The number of domains is a function of the grain’s size and shape, and of the saturation magnetization of the magnetic material. In essence, a grain of a material with a high-saturation magnetization like magnetite will contain (much) more domains than a grain with the same size but of material with a low-saturation magnetization like hematite (see Table M3). Therefore we need to know the magnetic mineralogy before we can make a proper grain-size estimate with

The concept of using magnetic properties as proxy parameters and correlation tool in a geoscientific context was put forward in the late 1970s mostly by Oldfield and his coworkers; the article in Science (Thompson et al., 1980) is often considered as the formal definition of “environmental magnetism” as a subdiscipline. Before briefly highlighting some magnetic proxies, we need to introduce two other important mineral-magnetic parameters, the low-field or initial susceptibility (win or kin) and the anhysteretic remanent magnetization (ARM). The initial susceptibility is the magnetization of a sample in a small applied field (up to a few times the intensity of the geomagnetic field, 30–60 mT) divided by that field. Measurement takes a few seconds and requires no specific sample preparation; instrumentation is sensitive. This makes initial susceptibility an attractive proxy parameter, especially for correlation purposes. All materials have a magnetic susceptibility, also paramagnets and diamagnets. The initial susceptibility

528

MAGNETIC PROXY PARAMETERS

Table M3 Saturation magnetization, typical coercive forces, remanent coercive forces and field required to reach saturation of some magnetic minerals (at room temperature) Mineral

Saturation magnetization (A m2 kg1)

Coercive forcea (mT)

Remanent coercive forcea (mT)

Saturation field (T)

Magnetite Maghemite Hematite Iron Goethite Pyrrhotite Greigite

92b 65–74c 0.1–0.4 217 0.01–1 18 30e

5–80 5–80 100–500 1000 8–100 15–40

15–100 15–100 200–800 5–15 >2000 9–115 35–70

E3 corresponds to a girdle-type pattern (Figure M77). In a perfect cluster, in which all linear elements are parallel, E1 ¼ 1, E2 ¼ 0, E3 ¼ 0. In a perfect girdle, E1 ¼ 0.5, E2 ¼ 0.5, E3 ¼ 0. A straightforward relationship exists between the orientation tensor and the susceptibility tensor (for a summary see Ježek and Hrouda, 2000). For a rock, whose AMS is carried by a single magnetic mineral, the relationship is: k ¼ KI þ DE where k is the rock susceptibility tensor, I is the identity matrix. For magnetically rotational oblate grains, K ¼ K2 ¼ K3, D ¼ K1–K (K1, K2, K3 are the grain principal susceptibilities) and E is the orientation tensor of the grain minimum susceptibility axes. For magnetically rotational oblate grains, K ¼ K1 ¼ K2, D ¼ K–K3, and E is the orientation tensor of the grain maximum susceptibility axes. For the grains displaying so-called perfectly triaxial AMS (K1–K2 ¼ K2–K3), K ¼ K2, D ¼ K1–K2 ¼ K2–K3 and E ¼ Ex–Ez is the Lisle orientation tensor, where Ex and Ez are the Scheidegger orientation tensors of the maximum and minimum axes, respectively. Since the rock susceptibility is affected by all minerals present in the rock, the theoretical rock susceptibility can be calculated if the susceptibility tensors of all minerals and their relative contributions are known. Unfortunately, this is never known in practice and studies of the quantitative relationship between the AMS and preferred orientation are made only in such rocks in which the contribution of one mineral to the rock susceptibility is dominant. In this case, the model rock susceptibility is:

550

MAGNETIC SUSCEPTIBILITY, ANISOTROPY

Figure M75 Mineral contribution to the susceptibility of a rock. Note that 1% of magnetite contributes more than 100% of phyllosilicates. (Adapted from Hrouda and Kahan, 1991.)

Figure M76 The dependence of the degree of AMS of a model rock (Ps) on the grain degree of AMS (Pg) and on the parameter C characterizing the intensity of the preferred orientation of the axes of magnetically uniaxial grains. (Adapted from Hrouda, 1980.)

universal stage methods), through determining the orientation tensor using X-ray pole figure goniometry, or electron backscatter diffraction with orientation contrast. Studies of this type were made for hematite (deformed hematite ore, Minas Gerais, Brazil, Hrouda et al., 1985), biotite (Bíteš orthogneiss, Hrouda and Schulmann, 1990), chlorites (Lüneburg et al., 1999; Chadima et al., 2004), and calcite (Carrara marble, Owens and Rutter, 1978). Results from these studies show that the principal susceptibilities of the whole rock measured are oriented in the same way as those calculated from the crystallographic axis pattern. The degree of AMS and the shape parameter are not always in agreement with that calculated from the crystallographic axis pattern. The differences may reflect the situation that the AMS and X-ray measurements are often not executed on exactly the same specimens, and the respective specimen volumes differ substantially (1 cm3 vs thin section). In addition, minor effect of other magnetic minerals cannot be excluded. Nevertheless, the agreement between the theoretical and measured AMS can be considered to be satisfactory. Some of these results are illustrated in Figure M78.

Resolution of AMS into paramagnetic and ferromagnetic components k ¼ pSK ¼ pSðOKd O0 Þ; where k is the rock susceptibility tensor, K is the magnetic grain susceptibility tensor, Kd the diagonal from the grain susceptibility tensor, O the matrix specifying the mineral orientation (O0 is the transposed matrix of O), and p the percentage of a magnetic mineral in a rock. The orientation of magnetic minerals can be investigated either through direct measurement of individual grains (microscope and

In rocks with the bulk susceptibility between 5  10–4 and 5  10–3, the AMS is, in general, controlled by both ferromagnetic and paramagnetic minerals. As these mineral groups may behave differently in various geological situations, it is desirable to resolve the rock AMS into its ferromagnetic and paramagnetic components. This resolution is usually made through measuring the AMS in various strong magnetic fields of the order of tesla in which the ferromagnetic and paramagnetic minerals behave in a different way (described in an another part

MAGNETIC SUSCEPTIBILITY, ANISOTROPY

551

Figure M77 Relationship between preferred orientation of a linear element and principal values of the orientation tensor. (Adapted from Woodcock, 1977.)

Figure M78 Relationship between orientation tensor calculated from mineral preferred orientation (c-axes of chlorite) and from AMS. Constructed from Chadima et al. (2004) data.

552

MAGNETIC SUSCEPTIBILITY, ANISOTROPY

The first equation is that of a hyperbola offset along the susceptibility axis, and the second is a combination of a hyperbola and a straight line. By fitting the hyperbola and straight line either horizontal (first equation), or sloped (second equation) to k vs T curve using the least-squares method, one obtains the paramagnetic susceptibility contribution ( ppC/T ) and the ferromagnetic susceptibility contribution ( pf kf or pf (bT þ a)). These methods were developed by Hrouda (1994) and Hrouda et al. (1997), respectively, and are broadly used in resolving the mean bulk susceptibility measured on powder specimens into paramagnetic and ferromagnetic components. Richter and van der Pluijm (1994) developed a technique for the AMS resolution into the paramagnetic and ferromagnetic components based on susceptibility measurements at low temperatures (between the temperature of liquid nitrogen and room temperature). At these temperatures, the contribution of paramagnetic fraction to the rock susceptibility is amplified because of the hyperbolic course. By plotting the reciprocal susceptibility against temperature one can easily visualize the linear paramagnetic susceptibility contribution and separate it from the ferromagnetic contribution. If this measurement is made in six directions, one can obtains six directional paramagnetic and ferromagnetic susceptibilities from which the paramagnetic and ferromagnetic susceptibility tensor can be calculated in a standard way (Figure M80).

Thermal enhancement of the magnetic fabric

Figure M79 Susceptibility vs temperature relationship in the temperature interval between room temperature and 200 oC for biotite (a) and magnetite (b).

of this volume) or at variable temperatures in which the ferromagnetic and paramagnetic minerals also behave differently. Variation of the susceptibility of paramagnetic minerals with temperature is represented by a hyperbola, while in ferromagnetic minerals it is a complex curve in case of magnetite characterized by acute susceptibility decreases at about 155
C, the Verwey transition, and at about 580
C, the Curie temperature. Fortunately, in some temperature intervals, typically between room temperature and 200
C, the ferromagnetic susceptibility is either constant or follows only a mildly sloped straight line (Figure M79). The susceptibility vs temperature curve of a rock containing both ferromagnetic and paramagnetic minerals for the case in which the ferromagnetic susceptibility is constant within the considered temperature interval, is: k ¼ pp C=T þ pf kf ; and for the case in which the ferromagnetic susceptibility is represented by only a mildly sloped straight line, it is: k ¼ pp C=T þ pf ðbT þ aÞ; where k is the rock susceptibility, kf the ferromagnetic susceptibility, pp and pf the percentages of paramagnetic and ferromagnetic fractions, respectively, C the paramagnetic constant and T the absolute temperature, a and b are constants.

The AMS is often measured using the instrumentation of a paleomagnetic laboratory. Paleomagnetic techniques, such as AF demagnetization and thermal demagnetization, have then been logically tested as to whether they are also applicable to the AMS (Urrutia-Fucugauchi, 1981). It has been shown that the AF demagnetization has only weak effect on the AMS and its use ceased. On the other hand, the effect of thermal demagnetization on the AMS can be strong. However, thermal demagnetization mechanisms are different in AMS than in remanent magnetization. Thermal demagnetization affects the induced magnetization very weakly, probably negligibly, but may induce the creation of new magnetic phases or even new minerals. If this creation is made in a zero magnetic field, it would have virtually no effect on the remanent magnetization, but would have a very strong effect on the AMS. Often, the newly created phases are more magnetic than the original phases and may be coaxial to the old phases. Thermal demagnetization, in fact, enhances the original magnetic fabric. Newly created phases can also be noncoaxial, and their creation may obscure the original magnetic fabric. If the new creation is confined to a phase of different origin than the original phase, thermal treatment may “visualize” the cryptic fabric (Henry et al., 2003). Even though positive results may be obtained in this way in some cases, this technique cannot be recommended to be used in a routine way because the process of the creation of new magnetic phases due to specimen heating is very complex, and our knowledge about this is far from complete.

Statistical evaluation of the AMS data The AMS technique is so fast that several specimens are usually measured in a site. The site data or the data from geological bodies should be evaluated statistically. The simplest approach is to consider the individual principal directions as vectors and to process them using a vector statistics, for example the Fisher statistics. Unfortunately, the principal directions do not have a Fisherian distribution, being orthogonal and bipolar, so various modifications of the Fisher statistics have been proposed (Ellwood and Whitney, 1980; Park et al., 1988). To avoid these problems Jelínek (1978) has developed a statistical method that is based on averaging out the individual components of the susceptibility tensors in one coordinate system (e.g., geographical) after they have been normalized against their mean susceptibility. In this way the mean tensor (and the orientations of its principal directions) as well as the elliptical confidence regions (in terms of confidence angles) about each principal direction are computed from the

MAGNETIC SUSCEPTIBILITY, ANISOTROPY

553

Figure M80 Principle of the resolution of AMS into paramagnetic and ferromagnetic components using measurement of low-temperature susceptibility. (a) Six orientations of a specimen used to construct the susceptibility tensor, (b) the relative amount of paramagnetic and ferromagnetic susceptibilities is determined in six orientations from heating curves, the upper curve being the original heating curve and the lower curve being the paramagnetic remnant, and (c) the resolved paramagnetic and ferromagnetic subtensors. (Adapted from Richter and van der Pluijm, 1994.) (1990) developed a bootstrap method which delineates the confidence regions using Monte Carlo probability modeling. Statistical evaluation of the AMS data is a complex problem and further investigations are required to evaluate the effectiveness of the individual methods (for discussion see Ernst and Pearce, 1989; Tarling and Hrouda, 1993; Borradaile, 2001).

Geological applications of AMS Magnetic fabric in sedimentary rocks

Figure M81 Results of statistical evaluation of the AMS data from a locality of the Cˇista´ granodiorite made by the Jelı´nek (1978) method. Small symbols indicate individual specimens, and large symbols indicate the mean tensor. The confidence regions around the mean principal directions are elliptic on a sphere, being slightly deformed due to the equal area projection.

variance-covariance matrix (Figure M81). This technique seems to be the most frequently used at present. In data sets with widely scattered principal directions, neither analytical method provides an adequate visual representation of the observed data. To overcome these disadvantages, Constable and Tauxe

The AMS in sedimentary rocks provides information on the deposition and compaction processes (for a summary, see Hamilton and Rees, 1970; Rees and Woodall, 1975; Rees, 1983; Taira, 1989; Tarling and Hrouda, 1993; Hrouda and Ježek, 1999). In natural sedimentary rocks unaffected by later deformation, the relationship that is primarily investigated is the agreement between the magnetic foliation and lineation and the sedimentary external structures (sole markings, flute casts, groove casts) and internal structures (cross-bedding, current-ripple lamination, symmetric and asymmetric ripples). It has been found that the magnetic foliation is always oriented near the bedding/compaction plane, while the magnetic lineation is mostly roughly parallel to the near-bottom water current directions determined using sedimentological techniques (see Figure M82). Less frequently, the magnetic lineation may be perpendicular to the current direction, which is typical of the flysch sediments of the lowermost A member of the Bouma sequence. The degree of AMS is relatively low and the AMS ellipsoid is, in general, oblate. In artificial sediments deposited through grain by grain (or from thin suspension) deposition from still or running water onto a flat or sloping bottom under controlled conditions in tanks and flumes in laboratories, the AMS ellipsoid is clearly oblate; the magnetic foliation dips less than 15
from the bedding toward the origin of flow and the magnetic lineation is parallel to the direction of flow, slightly plunging toward the origin of flow. During deposition from medium-concentrated suspensions, the AMS ellipsoids can be clearly triaxial to slightly prolate, in which the

554

MAGNETIC SUSCEPTIBILITY, ANISOTROPY

found to be near or even parallel to the flow plane in lava flows, sills, and dikes (Ernst and Baragar, 1992; Ca
non-Tapia et al., 1994; Raposo and Ernesto, 1995; Hrouda et al., 2002a). The magnetic lineation is mostly parallel to the lava flow directions (Figure M83), even though perpendicular or oblique relationships can also be rarely found. AMS enables not only the lava flow process to be investigated, but also the rheomorphic flow in ignimbrites and the motions of tuffs and tuffites (MacDonald and Palmer, 1990).

Magnetic fabric in plutonic rocks

Figure M82 Magnetic and sedimentary fabrics in sediments of the Rosario Formation, La Jolla, California (crosses, magnetic foliation poles; open circles, magnetic lineation). Equal-area projection on lower hemisphere (projection plane ¼ bedding plane). (Adapted from von Rad, 1971.)

magnetic foliation dips less than 15
toward the origin of flow, and the magnetic lineation is perpendicular to the direction of flow and to the dip of magnetic foliation. During deposition from very concentrated grain dispersions onto a sloping bottom, the AMS ellipsoids are triaxial. The magnetic foliation dips 25
–30
toward the origin of the flow and the magnetic lineation is parallel both to the flow and to the dip of magnetic foliation, and plunges slightly toward the origin of flow. During the process of diagenesis, the originally sedimentary magnetic fabric may be slightly modified due to ductile deformation accompanying this process. If the ductile deformation is represented by vertical shortening due to the loading by the weight of the overlying strata, the degree of AMS and the oblateness of the AMS ellipsoid increase, whereas the magnetic foliation and lineation retain their orientations (e.g., Lowrie and Hirt, 1987). If the ductile deformation is represented by the bedding parallel shortening or by the bedding parallel simple shears or by both, the degree of AMS initially decreases and only later increases when the deformation is strong enough to overcome the initial compaction. The magnetic fabric initially becomes more planar, and only later it does become more triaxial or even linear. The magnetic lineation deviates gradually from the direction of flow toward that of maximum strain, often creating a bimodal pattern. The magnetic foliation remains initially near the bedding. After a stronger strain, it deviates from it, creating a girdle in the magnetic foliation poles that are perpendicular to the magnetic lineation.

Magnetic fabric in volcanic rocks The AMS of volcanic rocks is, in general, very weak, reflecting a very poor dimensional orientation of the magnetic minerals (mostly titanomagnetites) in these rocks. In spite of this, their AMS can, in general, be measured precisely because the volcanic rocks are often strongly magnetic, and the AMS seems to be the quickest method to be able to reliably investigate the weakly preferred orientation of the minerals. Since the first investigations of the AMS of volcanic rocks it has been clear that it reflects the dimensional orientation of magnetic minerals created during a lava flow. The magnetic foliation is often

In plutonic rocks, the AMS is one of the most powerful tools of structural analysis, because it can efficiently measure the magnetic fabric even in massive granites that are isotropic at the first sight. The AMS of plutonic rocks varies from very weak (typical of volcanic rocks) to extraordinarily strong (typical of metamorphic and/or strongly deformed rocks). Granites, the most important representatives of plutonic rocks, usually display a bimodal distribution of their magnetic susceptibility. One mode corresponds to susceptibilities in the order of 10–5 to 10–4 SI units and are due to weakly magnetic (Dortman, 1984) or paramagnetic (Bouchez, 2000) granites. These are equated with the S (sedimental) type granites, such as the ilmenite-bearing granites. The other mode has susceptibilities in the order of 10–3–10–2 SI units, and this is due to magnetic or ferromagnetic granites. These are equated with the I (igneous) type granites such as magnetite-bearing granites. In magnetic granites, the AMS investigates the preferred orientation of magnetite by grain shape. In weakly magnetic granites, the AMS reflects the preferred orientation of mafic silicates (mainly biotite, and less frequently amphibole) by crystal lattice. The plutonic rocks that have suffered no postintrusive deformation show the magnetic fabric created during the process of magma emplacement. The characteristic features of this magnetic fabric are as follows: the degree of AMS is relatively low, indicating only weak preferred orientation of magnetic minerals created during the liquid flow of magma; the magnetic fabric ranges from oblate to prolate according to the local character of the magma flow; the magnetic foliation is parallel to the flow plane and the magnetic lineation is parallel to the flow direction; the magnetic foliations are steep in stocks and upright sheetlike bodies in which the magma flowed vertically (Figure M84). On the other hand, it is oblique or horizontal in the bodies where the magma could not ascend vertically and moved in a more complex way. The magnetic lineation can be vertical, horizontal, or oblique according to the local direction of the magma flow. The magnetic fabric elements usually show a close relationship to the shapes of magmatic bodies and to magmatic structural elements, if observable. Many examples are shown in the book by Bouchez et al. (1997). Some plutonic rocks exhibit a very high degree of AMS and a very high mean magnetic susceptibility. The magnetic fabric in these rocks usually indicates a preferred orientation of magnetite by grain shape that is younger than the fabric of the surrounding silicates and takes its shape from highly anisometric intergranular (interfoliation) spaces. This magnetic fabric is usually termed as the mimetic magnetic fabric (e.g., Hrouda et al., 1971). Some granites have suffered tectonic ductile deformation after their emplacement. In this process, the originally intrusive magnetic fabric is overprinted or even obliterated by the deformational magnetic fabric. The degree of AMS of such rocks is often much higher, because the ductile deformation is a relatively efficient mechanism for the reorientation of magnetic minerals. The magnetic foliations and lineations deviate from the directions of the intrusive fabric elements toward the directions of the principal strains. A good example includes granites of the West Carpathians whose magnetic fabrics are coaxial with those of surrounding metamorphic rocks and covering sedimentary rocks as result of ductile deformation associated with retrogressive deformation acting during creation and motion of the West Carpathian nappes (Hrouda et al., 2002b).

MAGNETIC SUSCEPTIBILITY, ANISOTROPY

555

Figure M83 Magnetic fabric in basalt of the Chrˇibsky´ les lava flow. (a) Synoptic geological map of the Chrˇibsky´ les lava flow (A) and its environs (Legend: 1, basalt tuff; 2, boundaries of lava bodies), (b) orientations of principal susceptibilities in the quarry 1 (Legend: triangle, magnetic lineation; square, intermediate susceptibility; circle, magnetic foliation pole). Equal-area projection on lower hemisphere. (Adapted from Kolofı´kova, 1976.)

AMS of metamorphic and deformed rocks The AMS of metamorphic rocks in general and even of low-grade metamorphic rocks is considerably higher than the AMS in undeformed sedimentary and volcanic rocks. Hence, the mechanism orienting magnetic grains during low-grade metamorphism is very effective. It probably involves ductile deformation or recrystallization in an anisotropic stress field operating during metamorphism. The AMS therefore enables these processes to be studied. AMS is an extremely powerful tool in investigating the progressive modification of the sedimentary magnetic fabric by ductile deformation that takes place in accretionary prisms (Figure M85). After sediment deposition, the AMS ellipsoid is usually oblate with the magnetic foliation near the bedding, and if a magnetic lineation is present, it is near the current direction. In initial stages of deformation, represented by

shortening along the bedding, the AMS ellipsoid changes from oblate to triaxial; the degree of AMS may slightly decrease, the magnetic foliation remains parallel to bedding, but the magnetic lineation reorients perpendicular to the shortening direction. In progressing deformation, spaced cleavage develops and the magnetic lineation becomes parallel to the bedding/cleavage intersection line. The AMS ellipsoid becomes prolate. If the deformation continues, giving rise to the development of slaty cleavage, the AMS ellipsoid becomes oblate again, the degree of AMS considerably increases, the magnetic foliation pole reorientates parallel to the cleavage and the magnetic lineation remains parallel to the bedding/cleavage intersection line. If the deformation is very strong, the magnetic lineation may be dip-parallel in the cleavage plane. AMS can be used in revealing the origin of folds (Hrouda, 1978; Hrouda et al., 2000). Various types of folds differing in terms of the

556

MAGNETIC SUSCEPTIBILITY, ANISOTROPY

Figure M84 Magnetic fabric in the Flamanville granite, Normandy, France. Note steep magnetic foliations conforming the magmatic body shape. (Adapted from Cogne´ and Perroud, 1988.)

relation of the magnetic fabric to the fold curve were found, and simple techniques for the recognition of unfoldable and homogeneous folds were elaborated. AMS can also be used in deciphering the sense of a shear movement (sinistral or dextral) (Rathore and Becke, 1980). This method is based on the assumption that the minimum susceptibility direction rotates during a simple shear movement in a manner similar to that of the minimum strain direction. AMS can also be used as a strain indicator of deformed igneous rocks. During the process of deformation, the degree of AMS, in general, increases and the principal directions reorientate into the directions parallel to the strain directions. Studies of this type were made in dikes, volcanic bodies, and granitic rocks (e.g., Henry, 1977). Strain analysis is one of the most laborious techniques of structural analysis and it is confined to rocks containing convenient strain indicators (oolites, concretions, reduction spots, lapilli, fossils). For this reason, many attempts have been made to use the AMS as a strain indicator. After revealing a close correlation between the directions of principal susceptibilities and principal strains (for examples, see Tarling and Hrouda, 1993), the quantitative relationship between the AMS and strain has been investigated theoretically through mathematical modeling (for review, see Hrouda, 1993), empirically through examining natural rocks with known strain (for review, see Borradaile, 1991), and experimentally through deforming rocks and rock analogs in the laboratory (for review, see Borradaile and Henry, 1997).

The first set of formulae describing a quantitative relationship between the AMS and strain was obtained through empirical studies of the AMS and strain (determined through investigation of natural strain markers—reduction spots) in Welsh slates (Figure M86) (Kneen, 1976; Wood et al., 1976): L ¼ ðk1 =k2 Þ ¼ ðs1 =s2 Þa F ¼ ðk2 =k3 Þ ¼ ðs2 =s3 Þa P ¼ ðk1 =k3 Þ ¼ ðs1 =s3 Þa or in terms of natural strain: ln L ¼ aðe1  e2 Þ ln F ¼ aðe2  e3 Þ ln P ¼ aðe1  e3 Þ where k1 k2 k3 are the principal susceptibilities, s1 s2 s3 are the principal strains, and e1 ¼ lns1, e2 ¼ lns2, e3 ¼ lns3, are the principal natural strains. In later empirical studies, several other formulae were suggested, but as shown by Hrouda (1982), all of them can be converted, within an error of a few percent, into the above formulae. In theoretical studies pioneered by Owens (1974), no simple formula for the AMS-to-strain relationship has been found. However, the curves

MAGNETIC SUSCEPTIBILITY, ANISOTROPY

557

Figure M85 Scheme of the AMS development in progressively deformed accretionary prisms sediments. (Adapted from Pares et al., 1999.) representing the lnP vs (e1–e3) relationship (for summary, see Hrouda, 1993), considering various AMS carriers (magnetite, hematite, pyrrhotite, and paramagnetic mafic silicates), are represented by monotonously rising and only gently curved lines (Figure M87). In small strain ranges, mainly in their initial parts, the lines do not differ very much from straight lines. Consequently, using a reasonable simplification, the theoretical AMS-to-strain relationship can be described in the same way as the empirical AMS-to-strain relationship. In experimental studies, the AMS of rock analogs deformed in laboratory was investigated (Borradaile and Alford, 1987, 1988). The rock analogs were relatively strongly magnetically anisotropic before

deformation, and to describe the AMS to strain relationship the following formula respecting the predeformational AMS was used: P 0  P 00 ¼ Bðe1  e3 Þ; where P 00 and P 0 are the corrected degrees of AMS before and after deformation, respectively, and B is a constant. Analysis by Hrouda (1991) shows that this relationship can also be converted into the relationship presented for empirical data. It is obvious from the above analysis that the AMS-to-strain relationship obtained through empirical, theoretical, and experimental studies

558

MAGNETIC SUSCEPTIBILITY, ANISOTROPY

Figure M86 AMS to strain relationship in Welsh slates. (Adapted from Kneen, 1976.)

Figure M87 An example of theoretical AMS to strain relationship. Pure shear strain, March model, prolate ellipsoids magnetite grains with variable aspect ratios (a/c). (Adapted from Hrouda, 1993.)

can, with a reasonable simplification, be represented by the formulae first derived for the empirical relationship. Unfortunately, the above studies have also shown that the a values are not the same for all rock types, but vary according to the mineral carrying the AMS (Tarling and Hrouda, 1993) and also probably by the lithology of the rock investigated. Even though a great majority of the empirical studies had revealed a relatively close correlation between AMS and strain, some studies had obtained results suggesting no correlation or even inverse correlation between the AMS and strain (Borradaile and Mothersill, 1984;

Borradaile and Tarling, 1984). This implies that some rocks may have had complex deformation histories in which various rock components responded in different ways to overall strain and such rocks should not be used in the strain determination via AMS. Nevertheless, it is believed that in homogeneously deformed rocks, the AMS can serve as a quantitative strain indicator, which is evident from the great majority of empirical and experimental studies. František Hrouda

MAGNETIC SUSCEPTIBILITY, ANISOTROPY

Bibliography Banerjee, S.K., and Stacey, F.D., 1967. The high-field torque-meter method of measuring magnetic anisotropy of rocks. In Collinson, D.W., Creer, K.M., and Runcorn, S.K. (eds.) Methods in Paleomagnetism. Amsterdam: Elsevier, pp. 470–476. Borradaile, G.J., 1991. Correlation of strain with anisotropy of magnetic susceptibility (AMS). PAGEOPH, 135: 15–29. Borradaile, G.J., 2001. Magnetic fabrics and petrofabrics: their orientation distribution and anisotropies. Journal of Structural Geology, 23: 1581–1596. Borradaile, G., and Alford, C., 1987. Relationship between magnetic susceptibility and strain in laboratory experiments. Tectonophysics, 133: 121–135. Borradaile, G.J., and Alford, C., 1988. Experimental shear zones and magnetic fabrics. Journal of Structural Geology, 10: 895–904. Borradaile, G.J., and Henry, B., 1997. Tectonic applications of magnetic susceptibility and its anisotropy. Earth-Science Reviews, 42: 49–93. Borradaile, G.J., and Mothersill, J.S., 1984. Coaxial deformed and magnetic fabrics without simply correlated magnitudes of principal values. Physics of the Earth and Planetary Interiors, 35: 294–300. Borradaile, G.J., and Tarling, D., 1984. Strain partitioning and magnetic fabrics in particulate flow. Canadian Journal of Earth Sciences, 21: 694–697. Bouchez, J.-L., 2000. Anisotropie de susceptibilitée magnétique et fabrique des granites. Comptes Rendus Académie des Sciences Paris, Sciences de la Terre et des planétes, 330: 1–14. Bouchez, J.-L., Hutton, D.W.H., and Stephens, W.E. (eds.), 1997. Granite: From Segregation of Melt to Emplacement Fabric. Dordrecht: Kluwer Academic Publishers, 358 pp. Canon-Tapia, E., Walker, G.P.L., and Herrero-Bervera, E., 1994. Magnetic fabric and flow direction in basaltic Pahoehoe lava of Xitle Volcano, Mexico. Journal of Volcanology and Geothermal Research, 65: 249–263. Chadima, M., Hansen, A., Hirt, A.M., Hrouda, F., and Siemes, H., 2004. Phyllosilicate preferred orientation as a control of magnetic fabric: evidence from neutron texture goniometry and low and high-field magnetic anisotropy (SE Rhenohercynian Zone of Bohemian Massif ). In: Martín-Hernández, F., Lüneburg, C.M., Aubourg, C., and Jackson, M. (eds.). Magnetic Fabric: Methods and Applications. Geological Society, London, Special Publications, 238: 361–380. Constable, C., and Tauxe, L., 1990. The bootstrap for magnetic susceptibility tensors. Journal of Geophysical Research, 95: 8383–8395. de Wall, H., 2000. The field dependence of AC susceptibility in titanomagnetites: implications for the anisotropy of magnetic susceptibility. Geophysical Research Letters, 27: 2409–2411. Dortman, N.B. (ed.), 1984. Physical Properties of Rocks and Mineral Deposits (in Russian). Moscow: Nedra, 455 pp. Ellwood, B.B., and Whitney, J.A., 1980. Magnetic fabric of the Elberton granite, Northeast Georgia. Journal of Geophysical Research, 85: 1481–1486. Ernst, R.E., and Baragar, W.R.A., 1992. Evidence from magnetic fabric for the flow pattern of magma in the Mackenzie giant radiating dyke swarm. Nature, 356: 511–513. Ernst, R.E., and Pearce, G.W., 1989. Averaging of anisotropy of magnetic susceptibility data. In Agterberg, F.P., and Bonham-Carter, G. F. (eds.), Statistical Applications in the Earth Sciences, Geological Survey of Canada Paper 89-9, pp. 297–305. Hamilton, N., and Rees, A.I., 1970. The use of magnetic fabric in palaeocurrent estimation. In Runcorn, S.K. (ed.) Palaeogeophysics. London: Academic Press, pp. 445–463. Henry, B., 1977. Relations entre deformations et propriétés magnétiques dans des roches volcaniques des Alpes francaises. Mémoires du B.R.G.M. 91: 79–86. Henry, B., Jordanova, D., Jordanova, N., Souque, C., and Robion, P., 2003. Anisotropy of magnetic susceptibility of heated rocks. Tectonophysics, 366: 241–258.

559

Hrouda, F., 1978. The magnetic fabric in some folds. Physics of the Earth and Planetary Interiors, 17: 89–97. Hrouda, F., 1980. Magnetocrystalline anisotropy of rocks and massive ores: a mathematical model study and its fabric implications. Journal of Structural Geology, 2: 459–462. Hrouda, F., 1982. Magnetic anisotropy of rocks and its application in geology and geophysics. Geophysical Surveys, 5: 37–82. Hrouda, F., 1991. Models of magnetic anisotropy variation in sedimentary sheets. Tectonophysics, 186: 203–210. Hrouda, F., 1993. Theoretical models of magnetic anisotropy to strain relationship revisited. Physics of the Earth and Planetary Interiors, 77: 237–249. Hrouda, F., 1994. A technique for the measurement of thermal changes of magnetic susceptibility of weakly magnetic rocks by the CS-2 apparatus and KLY-2 Kappabridge. Geophysical Journal International, 118: 604–612. Hrouda, F., 2002. Low-field variation of magnetic susceptibility and its effect on the anisotropy of magnetic susceptibility of rocks. Geophysical Journal International, 150: 715–723. Hrouda, F., and Ježek, J., 1999. Magnetic anisotropy indications of deformations associated with diagenesis. In Tarling, D.H., and Turner, P. (eds.), Palaeomagnetism and Diagenesis in Sediments. London: Geological Society, Special Publications 151, pp. 127–137. Hrouda, F., and Kahan, S., 1991. The magnetic fabric relationship between sedimentary and basement nappes in the High Tatra Mts. (N Slovakia). Journal of Structural Geology, 13: 431–442. Hrouda, F., and Schulmann, K., 1990. Conversion of magnetic susceptibility tensor into orientation tensor in some rocks. Physics of the Earth and Planetary Interiors, 63: 71–77. Hrouda, F., Chlupacova, M., and Rejl, L., 1971. The mimetic fabric of magnetite in some foliated granodiorites, as indicated by magnetic anisotropy. Earth Science and Planetary Interiors, 11: 381–384. Hrouda, F., Siemes, H., Herres, N., and Hennig-Michaeli, C., 1985. The relation between the magnetic anisotropy and the c-axis fabric in a massive hematite ore. Journal of Geophysics, 56: 174–182. Hrouda, F., Jelínek, V., and Zapletal, K., 1997. Refined technique for susceptibility resolution into ferromagnetic and paramagnetic components based on susceptibility temperature-variation measurement. Geophysical Journal International, 129: 715–719. Hrouda, F., Krejčí, O., and Otava, J., 2000. Magnetic fabric in folds of the Eastern Rheno-Hercynian Zone. Physics and Chemistry of the Earth (A), 25: 505–510. Hrouda, F., Chlupáčová, M., and Novák, J.K., 2002a. Variations in magnetic anisotropy and opaque mineralogy along a kilometer deep profile within a vertical dyke of the syenogranite porphyry at Cínovec (Czech Republic). Journal of Volcanology and Geothermal Research, 113: 37–47. Hrouda, F., Putiš, M., and Madarás, J., 2002b. The Alpine overprints of the magnetic fabrics in the basement and cover rocks of the Veporic Unit (Western Carpathians, Slovakia). Tectonophysics, 359: 271–288. Jelínek, V., 1977. The statistical theory of measuring anisotropy of magnetic susceptibility of rocks and its application. Geofyzika, n.p. Brno, 88 pp. Jelinek, V., 1978. Statistical processing of anisotropy of magnetic susceptibility measured on groups of specimens. Studia Geophysica et Geodaetica, 22: 50–62. Ježek, J., and Hrouda, F., 2000. The relationship between the Lisle orientation tensor and the susceptibility tensor. Physics and Chemistry of the Earth (A), 25: 469–474. Kolofikova, O., 1976. Geological interpretation of measurement of magnetic properties of basalts on example of the Chribsky les lava flow of the Velky Roudny volcano (Nizky Jesenik Mts.) (in Czech). Časopis pro mineralogii a geologii, 21: 387–396. Kneen, S.J., 1976. The relationship between the magnetic and strain fabrics of some haematite-bearing Welsh slates. Earth and Planetary Science Letters, 31: 413–416.

560

MAGNETIC SUSCEPTIBILITY, ANISOTROPY, EFFECTS OF HEATING

Lowrie, W., and Hirt, A.M., 1987. Anisotropy of magnetic susceptibility in the Scaglia Rossa pelagic limestone. Earth and Planetary Science Letters, 82: 349–356. Lüneburg, C.M., Lampert, S.A., Lebit, H.D., Hirt, A.M., Casey, M., and Lowrie, W., 1999. Magnetic anisotropy, rock fabrics and finite strain in deformed sediments of SW Sardinia (Italy). Tectonophysics, 307: 51–74. MacDonald, W.D., and Palmer, H.C., 1990. Flow directions in ashflow tuffs: a comparison of geological and magnetic susceptibility measurements, Tshirege member (upper Bandelier Tuff ), Valles caldera, New Mexico, USA. Bulletin of Volcanology, 53: 45–59. Nagata, T., 1961. Rock Magnetism. Tokyo: Maruzen. Nye, J.F., 1957. Physical Properties of Crystals. Oxford: Clarendon Press. Owens, W.H., 1974. Mathematical model studies on factors affecting the magnetic anisotropy of deformed rocks. Tectonophysics, 24: 115–131. Owens, W.H., and Rutter, E.H., 1978. The development of magnetic susceptibility anisotropy through crystallographic preferred orientation in a calcite rock. Physics of the Earth and Planetary Interiors, 16: 215–222. Pares, J.M., van der Pluijm, B.A., and Dinares-Turell, J., 1999. Evolution of magnetic fabrics during incipient deformation of mudrock (Pyrenees, northern Spain). Tectonophysics, 307: 1–14. Park, J.K., Tanczyk, E.I., and Desbarats, A., 1988. Magnetic fabric and its significance in the 1400 Ma Mealy diabase dykes of Labrador, Canada. Journal of Geophysical Research, 93: 13689–13704. von Rad, U., 1971. Comparison between “magnetic” and sedimentary fabric in graded and cross-laminated sand layers, Southern California. Geologische Rundschau, 60: 331–354. Raposo, M.I.B., and Ernesto, M., 1995. Anisotropy of magnetic susceptibility in the Ponta Grossa dyke swarm (Brazil) and its relationship with magma flow direction. Physics of the Earth and Planetary Interiors, 87: 183–196. Rathore, J.S., and Becke, M., 1980. Magnetic fabric analyses in the Gail Valley (Carinthia, Austria) for the determination of the sense of movements along this region of the Periadriatic Line. Tectonophysics, 69: 349–368. Rees, A.I., 1983. Experiments on the production of transverse grain alignment in a sheared dispersion. Sedimentology, 30: 437–448. Rees, A.I., and Woodall, W.A., 1975. The magnetic fabric of some laboratory-deposited sediments. Earth and Planetary Science Letters, 25: 121–130. Richter, C., and van der Pluijm, B.A., 1994. Separation of paramagnetic and ferrimagnetic susceptibilities using low temperature magnetic susceptibilities and comparison with high field methods. Physics of the Earth and Planetary Interiors, 82: 111–121. Scheidegger, A.E., 1965. On the statistics of the orientation of bedding planes, grain axes, and similar sedimentological data. US Geological Survey Professional Paper, 525-C: 164–167. Siegesmund, S., Ullemeyer, K., and Dahms, M., 1995. Control of magnetic rock fabrics by mica preferred orientation: a quantitative approach. Journal of Structural Geology, 17: 1601–1613. Taira, A., 1989. Magnetic fabrics and depositional processes. In Taira, A., and Masuda, F., (eds.), Sedimentary Facies in the Active Plate Margin. Tokyo: Terra Publications, pp. 43–77. Tarling, D.H., and Hrouda, F., 1993. The magnetic anisotropy of rocks. London: Chapman & Hall, 217 pp. Urrutia-Fucugauchi, J., 1981. Preliminary results on the effects of heating on the magnetic susceptibility anisotropy of rocks. Journal of Geomagnetism and Geoelectricity, 33: 411–419. Uyeda, S., Fuller, M.D., Belshe, J.C., and Girdler, R.W., 1963. Anisotropy of magnetic susceptibility of rocks and minerals. Journal of Geophysical Research, 68: 279–292. Wood, D.S., Oertel, G., Singh, J., and Bennet, H.G., 1976. Strain and anisotropy in rocks. Philosophical Transactions of the Royal Society of London, Series A, 283: 27–42.

Woodcock, N.H., 1977. Specification of fabric shapes using an eigenvalue method. Geological Society of America Bulletin, 88: 1231–1236.

Cross-references Fisher Statistics Magnetic Remanence, Anisotropy Magnetic Susceptibility

MAGNETIC SUSCEPTIBILITY, ANISOTROPY, EFFECTS OF HEATING Study of the anisotropic properties of magnetic susceptibility and remanent magnetizations is an active field of research in paleomagnetism. These studies have important applications in petrofabrics, structural geology, metamorphism, rock magnetism, volcanology, and tectonics. Here we concentrate on the anisotropy of magnetic susceptibility (AMS) of rocks measured at low magnetic fields, and in particular, on the effects of laboratory heating on the AMS of rocks. Mineralogical changes resulting from heating samples in the laboratory have long been recognized and studied (see Table M6). The potential use of temperature-induced effects to investigate composite magnetic fabrics needs further development.

Introduction The magnetic susceptibility of rocks given by the ratio between magnetization (M) and applied magnetic field (H) exhibits anisotropic properties. These properties have been used to investigate the characteristics of rock fabric (Tarling and Hrouda, 1993). Mi ¼ ki; j Hj The susceptibility ki, j of anisotropic samples is a second-order tensor (Nye, 1957), which is usually represented in terms of the principal susceptibility axes (maximum k1, intermediate k2, and minimum k3). The Table M6 Laboratory data on heating-induced thermochemical reactions From

To




C

Igneous Impure titanomagnetites Magnetite Olivines Pyrite Maghemite Magnetite Pyroxenes

Magnetite Maghemite Magnetite Magnetite Hematite Hematite Magnetite

>300 150–250 >300 350–500 350–450 >500 >600

Sediments Siderite Lepidocrocite Goethite Maghemite Pyrite Magnetite Hematite

Magnetite Maghemite Hematite Hematite Magnetite Hematite Magnetite

>200 220–270 110–120 350–450 350–500 >500 >550

Note: The actual temperatures are strongly dependent on grain size and shape, redox conditions, rate of heating, etc., and are only preliminary and indicative. Similarly, the magnetic product resulting from heating is dependent on the specific redox conditions at that temperature and it is likely that many of these reactions take place slowly at lower temperatures under natural conditions—for example, goethite may change to magnetite at 100
C–120
C under low oxidation conditions (after Tarling, 1983; Tarling and Hrouda, 1993).

MAGNETIC SUSCEPTIBILITY, ANISOTROPY, EFFECTS OF HEATING

AMS is generally measured at low magnetic fields, and several instrumental systems have been developed and commercially produced in the last decades (e.g., Kappabridge, Saphire, Bartington, Digico, Molspin, and PAR-SM2). AMS is also determined from measurements in cryogenic magnetometers. AMS is usually analyzed in terms of the orientation of the principal susceptibility axes and the AMS parameters. The parameters based on the magnitudes of principal susceptibilities have been proposed to quantify the anisotropy degree, lineation, foliation, and shape of susceptibility ellipsoid (Tarling and Hrouda, 1993). The magnitude and shape of susceptibility ellipsoid have been determined from various plots, which include the Flinn diagram of lineation as a function of foliation and the plot of shape parameter as a function of anisotropy degree (Jelinek, 1981). They permit distinction of susceptibility ellipsoid, with prolate, oblate, and triaxial (neutral) ellipsoids. In paleomagnetic and rock magnetic studies, several methods have been developed to investigate the temperature dependence of magnetic properties and remanent magnetizations (Tarling, 1983; Dunlop and Özdemir, 1997). These studies use laboratory heating or thermal treatment on samples under controlled conditions (e.g., thermal demagnetization, thermomagnetic measurements, temperature-dependent susceptibility, and paleointensity experiments). Understanding the effects of heating on the magnetic properties and magnetic mineralogy is then an active research field. This note is concerned with the effects of laboratory heating on the AMS of rocks. Thermal treatment has been proposed to enhance AMS in rocks, and used to investigate on temperature-induced fabric changes. The low-field magnetic susceptibility changes as a result of growth/ destruction/transformation of iron oxides, sulfides and carbonates, and hydroxides. Oxidation and reduction processes occur at different temperature ranges during thermal treatment. Magnetic oxides can result from transformation of iron sulfides (pyrite, pyrrhotite). Several methods have been implemented to study mineral alteration during laboratory heating (e.g., Dekkers, 1990; Van Velzen and Zijderveld, 1992), and to promote, reduce, or avoid oxidation/reduction of magnetic minerals (for instance, use of inert or reducing atmospheres in heating/ cooling chambers of ovens). Susceptibility is a bulk property, resulting from relative contributions of the diamagnetic, paramagnetic, and ferromagnetic minerals. It depends on its mineralogy, grain size and shape, mineral distribution, crystallographic preferred orientation, layering, etc. Interpretation of AMS in terms of rock fabric can often be complex, particularly for composite or multicomponent fabrics. Thermal treatment has been proposed to enhance and quantify given components of AMS fabric (Urrutia-Fucugauchi, 1979, 1981), where heating produces new magnetic minerals from the phyllosilicates and ferromagnetic minerals, which mimic the crystallographic structure in micas and clays.

Temperature-dependent mineral transformations Mineralogical changes resulting from heating samples in the laboratory have long been identified and studied (e.g., Abouzakhm and Tarling, 1975; Urrutia-Fucugauchi, 1979, 1981). Susceptibility increases and decreases during heating results mainly from formation of iron-titanium oxides (titanomagnetites and magnetite) and transformation of oxides (hematization of magnetite), respectively. Magnetic oxides can be produced from a range of minerals and processes, such as iron sulfides, carbonates and silicate minerals, hydroxides, and other iron oxides (e.g., Tarling, 1983; Borradaile et al., 1991; Tarling and Hrouda, 1993; Henry et al., 2003). Lithologies in which iron oxides are not present, other minerals play an important role and constitute special cases that require further investigation. Pyrrhotite-bearing rocks can display changes due to transformation of antiferromagnetic hexagonal pyrrhotite to ferrimagnetic monoclinic pyrrhotite (Bina et al., 1991). Most often, complex series of mineralogical changes occur at different temperature ranges. Oxidation of pyrrhotite at high temperature results in further susceptibility increase

561

(Dekkers, 1990). Magnetite formation between 500 and 725
C result in susceptibility increase, which can be followed by decrease at higher temperature as a result of hematization (biotite granites; Trinidade et al., 2001). Mineral transformations depend not only on mineral type, but also on grain size, mineral impurities, time, atmosphere conditions (inert, oxidizing, or reducing conditions), and water content. Temperature-induced changes in the bulk susceptibility and AMS have often been related to the formation of magnetite. Perarnau and Tarling (1985) proposed magnetite formation from vermiculite at 450
C and from illite at >850
C.

Thermal enhancement of AMS Early studies of AMS thermal enhancement used samples from Late Precambrian tillites, Late Jurassic red beds, and Late Tertiary volcanics (Urrutia-Fucugauchi, 1981). Thermal treatment resulted in enhancing given components of the AMS fabric in the first two cases, and no significant changes in the AMS axis pattern for the volcanic rocks. Thermal enhancement was proposed as a method to study composite fabrics (Urrutia-Fucugauchi, 1979). Subsequent studies applying thermal treatment to glacial deposits, slates, sandstones, and red beds supported that heating enhanced the primary AMS fabric and pointed out the complexity of heating-induced effects on magnetic mineralogy (e.g., Urrutia-Fucugauchi and Tarling, 1983; Perarnau and Tarling, 1985; Borradaile et al., 1991; Borradaile and Lagroix, 2000). An example of simple thermal enhancement of AMS is illustrated in Figure M88. Samples studied are Cretaceous sandstones sampled near a fault system in Venezuela (Perarnau and Tarling, 1985). The initial fabric is poorly defined with mixed maximum, intermediate, and maximum axes (Figure M88a). Maximum and minimum axes are oriented normal to bedding, and maximum, intermediate, and minimum axes conform a girdle distribution on the bedding plane. After heating to 350
C, the AMS axes pattern changes, resulting in minimum axes preferentially aligned normal to bedding, and maximum axes on the bedding plane and oriented to the north (Figure M88b). The thermal enhanced fabric is interpreted as a depositional sedimentary fabric, resulting from new magnetic minerals that mimic the crystallographic structure and shape of micas and clays. Samples from sites closer to the fault show a fault-parallel oblateness effect (Perarnau and Tarling, 1985). Composite AMS fabrics are in general difficult to interpret, and several methods have been proposed in an attempt to separate the different mineral contributions (e.g., Hrouda and Jelinek, 1990). Other methods using measurements at high and low fields, low-temperature observations, etc. have also been used contributions (e.g., Rochette and Fillion, 1988; Pares and Van der Pluijm, 2002). An example of magnetic fabric enhancement of a given component of a composite fabric in glacial sediments induced by laboratory heating is summarized in Figures M89 and M90. Samples used come from a Late Precambrian Port Askaig Formation from southwestern Scotland (Urrutia-Fucugauchi, 1981; Urrutia-Fucugauchi and Tarling, 1983). The mean susceptibility showed little change up to 300–400
C and increased markedly after heating at 400 and 500
C (Figure M90). The orientation of principal susceptibility axes before thermal treatment shows a pattern with vertical minimum axes and maximum axes on the bedding plane (with some scatter; Figure M89a). This pattern conforms that expected for a depositional sedimentary fabric in sediments deposited under the influence of dominant currents (indicated by the lineation). With thermal treatment up to 500
C, there are marked changes in bulk susceptibility and AMS axes pattern (Figure M89b). The maximum axes are grouped close to paleovertical and minimum axes lie in the paleohorizontal plane. Thus, heating-induced changes result in an inverse fabric. The scatter in AMS axes is reduced, which gives a well-defined NW-SE lineation (Figure M89a,b). This new fabric was interpreted in terms of compressional stresses acting parallel to bedding on strata free to extend in directions normal to bedding and compression. Deformation results on lineation increase eventually exceeding foliation, with predominantly prolate-shaped ellipsoids.

562

MAGNETIC SUSCEPTIBILITY, ANISOTROPY, EFFECTS OF HEATING

Figure M88 Example of thermal enhancement of AMS (for explanation see text).

Figure M89 Example of magnetic fabric enhancement induced by laboratory heating. Orientation of principal susceptibility axes before and after thermal treatment (for explanation see text). The initial fabric can be totally obliterated, resulting in oblate ellipsoids but with different orientation to bedding. Maximum axes align normal to bedding and minimum axes parallel to compression (Graham, 1966). The AMS after heating to 500
C then suggests a magnetic fabric that is strain-controlled with compressional stresses locally NW-SE oriented, in agreement with geologic and petrofabric studies which indicate strong deformation of this region during the Caledonian orogeny (Borradaile, 1979). AF and thermal demagnetization of remanence suggests that magnetic carriers are slightly oxidized titanomagnetites.

Thermal demagnetization to 300
C–400
C produces further oxidation with some hematization accompanied by increase of coercive force and Curie temperature. Heating to higher temperatures results in the formation of new magnetite from paramagnetic minerals, which accounts for the marked increase of mean susceptibility (Figure M90). Formation of new magnetic minerals occurs preferentially controlled by existing tectonic fabrics. Borradaile and Lagroix (2000) presented data for high-grade gneisses from the Kapuskasing granulite zone. They used measurements of AMS

MAGNETIC SUSCEPTIBILITY, ANISOTROPY, EFFECTS OF HEATING

563

Figure M90 Example of magnetic fabric enhancement induced by laboratory heating: mean susceptibility (for explanation see text).

and anisotropy of anhysteretic remanent magnetization (AARM) to define the magnetic fabric. AMS was mainly due to the contributions of magnetite and silicate minerals. Thermal treatment does not result in increase of mean susceptibility, but definition of AMS and AARM axial orientation is improved. Heating results in enhancement of the magnetite subfabric, which can then be quantified. This magnetite subfabric reveled by the thermal treatment is also shown in the heated AARM fabric, which is also preferentially enhanced by the laboratory heating. Other examples of successful thermal enhancement of component of composite fabrics have been observed in granites, particularly in S-type granites in which fabrics arise from biotite and tourmaline minerals (Trinidade et al., 2001; Mintsa et al., 2002). The AMS of paramagnetic phyllosilicates in granites generally correlate well with the crystallographic fabric (e.g., Borradaile and Werner, 1994). Mintsa et al. (2002) presented results for the Carnmenellis granite, where thermal treatment above 500
C results in enhancement of the biotite fabric overcoming the tourmaline contribution. Secondary magnetite produced by laboratory heating within biotite crystals mimics the phyllosilicate cleavage, which in turn corresponds to magmatic (linear) flow structures in the granitic body. Comparison of the AMS fabric lineations before and after the laboratory thermal treatment shows clear differences between the composite initial fabric produced by the biotite fabric and the (inverse) tourmaline fabric (Mintsa et al., 2002).

Discussion Measurement of magnetic susceptibility has long been used to monitor changes in magnetic minerals produced by heating of rocks in the laboratory. The effects of heating on the AMS and the enhancement of components of AMS fabric open new exciting possibilities in petrofabric studies. In the simple case, heating produces new magnetic minerals, which mimic the crystallographic structure in micas and clays (Urrutia-Fucugauchi, 1979, 1981).

The initial studies showed, however, that heating-induced changes in AMS are more complex than simple enhancement of the magnetic fabric. Recent studies investigating different lithologies (e.g., granites, gabbros, diorites, ignimbrites, volcanoclastic sediments, loess, and limestones) confirm that heating does not always result in simple AMS fabric enhancement (Henry et al., 2003). Henry et al. (2003) distinguish three different cases when comparing the heating-induced fabric with the preexisting fabric. First case is simple enhancement of preexisting fabric or masked poorly defined initial fabric. Second case corresponds to formation of an inverse fabric. Third case is the formation of a different fabric, which shows principal susceptibility axes related to petrostructural elements, but differing from the initial fabric. In the cases where thermal treatment results in increasing the bulk susceptibility and anisotropy degree, heating-induced changes facilitate the measurement of the susceptibility tensor and principal susceptibility axes. Susceptibility increase during heating mainly results from the formation of iron-titanium oxides, principally titanomagnetites and magnetite. Susceptibility decrease results from transformation of oxides, like hematization of magnetite. Iron-titanium oxides could be produced from a range of minerals and processes; most often including iron sulfides, iron carbonates, iron silicate minerals, hydroxides, and other iron oxides (Tarling, 1983; Henry et al., 2003). Lithologies in which iron oxides do not play a dominant role present special cases that are still not fully investigated for magnetic properties. Study of AMS changes induced by progressive thermal treatment appears as a useful method to investigate the fabric of rocks. Laboratory stepwise heating has potential for enhancement of AMS and revealing masked or cryptic fabrics. Further investigation of heatinginduced effects in mineralogy, grain size, and textural changes, applied systematically to different lithologies, is clearly required. The methodology permits studying composite fabrics and may give further insight on the relationship of AMS to the different petrofabrics arising from stratification, depositional conditions, current-induced features, strain effects, cleavage, microfracturing, etc.

564

MAGNETIC SUSCEPTIBILITY, ANISOTROPY, ROCK FABRIC

Acknowledgments Partial support for the magnetic fabric studies has been provided by DGAPA projects IN-116201. Thanks are due to Editor E. Herrero-Bervera for the invitation to contribute, encouragement, and useful comments. Jaime Urrutia-Fucugauchi

Bibliography Abouzakhm, A.G., and Tarling, D.H., 1975. Magnetic anisotropy and susceptibility from northwestern Scotland. Journal of the Geological Society London, 131: 983–994. Bina, M., Corpel, J., Daly, L., and Debeglia, N., 1991. Transformation de la pyrrhotite en magnetite sous l’effet de la temperature: une source potentielle d’anomalies magnetiques. Comptes Rendus de l’Académie des Sciences de Paris, 313: 487–494. Borradaile, G.J., 1979. Strain study of the Caledonides in the Islay region, S.W. Scotland: implications for strain histories and deformation mechanisms in greenschists. Journal of the Geological Society London, 136: 77–88. Borradaile, G.J., and Henry, B., 1997. Tectonic applications of magnetic susceptibility and its anisotropy. Earth-Science Reviews, 42: 49–93. Borradaile, G.J., and Lagroix, F., 2000. Thermal enhancement of magnetic fabrics in high grade gneisses. Geophysical Research Letters, 27: 2413–2416. Borradaile, G.J., and Werner, T., 1994. Magnetic anisotropy of some phyllosilicates. Tectonophysics, 235: 223–248. Borradaile, G.J., Mac Kenzie, A., and Jensen, E., 1991. A study of colour changes in purple-green slate by petrological and rockmagnetic methods. Tectonophysics, 200: 157–172. Dekkers, M.J., 1990. Magnetic monitoring of pyrrhotite alteration during thermal demagnetization. Geophysical Research Letters, 17: 779–782. Dunlop, D.J., and Özdemir, O., 1997. Rock Magnetism: Fundamentals and Frontiers. Cambridge: Cambridge University Press, 573 pp. Graham, J.W., 1966. Significance of magnetic anisotropy in Appalachian sedimentary rocks. In Steinhart, J.S., and Smith, T.J. (eds.), The Earth Beneath the Continents. Geophysical Monograph Series 10. Washington, DC: American Geophysical Union, pp. 627–648. Jelinek, V., 1981. Characterization of the magnetic fabric of rocks. Tectonophysics, 79: 63–67. Henry, B., Jordanova, D., Jordanova, N., Souque, C., and Robion, P., 2003. Anisotropy of magnetic susceptibility of heated rocks. Tectonophysics, 366: 241–258. Hrouda, F., and Jelinek, V., 1990. Resolution of ferrimagnetic and paramagnetic anisotropies in rocks, using combined low-field and highfield measurements. Geophysical Journal International, 103: 75–84. Mintsa Mi Nguema, T., Trinidade, R.I.F., Bouchez, J.L., and Launeau, P., 2002. Selective thermal enhancement of magnetic fabrics from the Carnmenellis granite (British Cornwall). Physics and Chemistry of Earth, 27: 1281–1287. Nye, J.F., 1957. Physical Properties of Crystals. London: Oxford University Press, 322 pp. Pares, J.M., and Van der Pluijm, B.A., 2002. Phyllosilicate fabric characterization by low-temperature anisotropy of magnetic susceptibility (LT-AMS). Geophysical Research Letters, 29: 2215, doi:10.1029/ 2002GL015459. Perarnau, A., and Tarling, D.H., 1985. Thermal enhancement of magnetic fabric in Cretaceous sandstone. Journal of Geological Society London, 142: 1029–1034. Rochette, P., and Fillion, C., 1988. Identification of multicomponent anisotropies in rocks using various field and temperature values in a cryogenic magnetometer. Physics of the Earth and Planetary Interiors, 51: 379–386. Tarling, D.H., 1983. Palaeomagnetism. London: Chapman & Hall, 379 pp. Tarling, D.H., and Hrouda, F., 1993. The Magnetic Anisotropy of Rocks. London: Chapman & Hall, London, 217 pp.

Trinidade, R.I.F., Mintsa Mi Nguema, T., and Bouchez, J.L., 2001. Thermally enhanced mimetic fabric of magnetite in a biotite granite. Geophysical Research Letters, 28: 2687–2690. Urrutia-Fucugauchi, J., 1979. Variation of magnetic susceptibility anisotropy versus temperature. Thermal cleaning for magnetic anisotropy studies? European Geophysical Union Meeting, Vienna, Austria. Urrutia-Fucugauchi, J., 1981. Preliminary results on the effects of heating on the magnetic susceptibility anisotropy of rocks. Journal of Geomagnetism and Geoelectricity, 33: 411–419. Urrutia-Fucugauchi, J., and Tarling, D.H., 1983. Palaeomagnetic properties of Eaocambrian sediments in northwestern Scotland: implications for world-wide glaciation in the Late Precambiran. Paleogeography Paleoclimatology Paleoecology, 1: 325–344. Van Velzen, A.J., and Zijderveld, J.D.A., 1992. A method to study alterations of magnetic minerals during thermal demagnetization applied to a fine-grained marine marl (Trubi formation, Sicily). Geophysics Journal International, 110: 79–90.

MAGNETIC SUSCEPTIBILITY, ANISOTROPY, ROCK FABRIC Although the anisotropy of magnetic susceptibility (AMS) of rocks first called the attention of scientists because of its possible influence on the direction of the remanent magnetization, it was relatively soon realized that the AMS of a rock is in general too weak to exert a noticeable influence in the paleomagnetic record (e.g., Uyeda et al., 1963). On the other hand, due to the close relationship that exists between the crystalline structure of minerals and their AMS (Nye, 1960), it was well established that the AMS should bear a very close relationship with the mineral fabric of the rock. It is therefore not surprising that the modern interest in the study of the AMS of rocks resides mainly in its value as a petrofabric indicator rather than as a paleomagnetic tool. There are several sources for the AMS of rock. The most common of these sources is the anisotropy arising from the crystalline structure of the minerals (Nye, 1960). Almost all minerals will have an AMS that is controlled by these sources, although their bulk susceptibility will be generally small. A second source of AMS is found only in some minerals with a strong permeability. These minerals will show an anisotropy that arises from the overall shape of the grain rather than from its crystalline structure. For this reason, this source of AMS is called shape anisotropy. Finally, when two or more ferromagnetic minerals are in close proximity to each other, they can interact magnetically and such interaction becomes the source of a magnetic anisotropy. If the distance between the grains is too small ( 1 mm magnetites, TRM/ARM ratios between 1 and 2 suggest that similar micromagnetic states, probably with many domains, are produced during either field-cooling or field-cycling.

ARM theories in single- and multidomain particles For an assembly of noninteracting SD particles undergoing only fieldassisted switching (i.e., no thermal fluctuations or time-dependent switching), the intensity of ARM is theoretically independent of HDC (and will equal the saturation remanence) and the initial ARM susceptibility is infinite (Wohlfarth, 1964). However, these theoretical predictions are not observed in synthetic or natural materials. For instance, Figure M97 clearly demonstrates that the shape of the ARM acquisition curve is strongly affected by particle concentration, where high concentrations of particles result in stronger dipolar (magnetostatic) interactions between magnetic particles. The effect of increasing interactions is to lower the efficiency of ARM acquisition. Dunlop and West (1969) used the Néel-Preisach model for interactions to explain weak-field ARM and TRM behavior in several synthetic and natural samples containing

MAGNETIZATION, ANHYSTERETIC REMANENT

575

parm ¼ tanhðwarm HDC =Mrs Þ
3=2 VMs pffiffiffiffiffiffiffiffiffiffiffi warm ¼ 1:797m0 Mrs kT m0 HK "
 # 0:35F kT 3=2 0 1=3 pffiffiffiffiffiffiffiffiffiffiffi ln fac DHAF m0 HK VMs

Figure M97 ARM acquisition curves for samples containing dispersed grains of magnetite in different concentrations. The curves are normalized by saturation remanence and the magnetic field is the DC bias field applied during AF demagnetization. The magnetite sample contains a mixture of SD and PSD grains. (Redrawn from Sugiura, 1979).

SD particles of iron oxides. In the Néel-Preisach model, interactions are accounted for in particle assemblies by assuming each particle has an asymmetric rectangular hysteresis loop, with the amount of asymmetry being proportional to the strength of the interaction field. A distribution of interaction fields, determined by separate experiments, is then used to model the ARM acquisition curves. A shortcoming to this type of model is that moment switching occurs instantaneously, while it is known that time-dependent switching via thermal fluctuations is important in SD and small PSD particles according to the Néel theory of thermally activated magnetization (Dunlop and Özdemir, 1997). ARM theories for SD particles incorporating interparticle, dipolar interactions, and thermal fluctuations have been proposed by Jeap (1971), Walton (1990), and Egli and Lowrie (2002). The following equation was derived by Jeap (1971) for ARM for interacting SD grains: parm ¼ tanhðm0 VMs =kT ðBH0  lparm Þ B  ðMsb =Ms0 ÞðT0 =Tb Þ1=2 where parm ¼ Marm/Mrs, Ms is the saturation magnetization, Mrs is the saturation remanence, k is Boltzmann’s constant, V is grain volume, T is absolute temperature, l is the mean interaction field, and the subscripts 0 and b refer to room temperature (T0) and the blocking temperature (Tb). A field-blocking condition, derived from Néel’s relaxation theory, is assumed whereby the alternating field necessary to reverse the magnetization in a given time interval corresponds to a relaxation time that makes ln( f0t) ¼ 25, where f0 is the frequency factor usually taken as 109 Hz. Note, that when B ¼ 1, the equation reduces to the SD equation for TRM with interactions. In the most recent theoretical work, Egli and Lowrie (2002) show that even in noninteracting systems, ARM can be described solely by thermal fluctuation effects. Therefore, in dilute systems, such as in many rocks and sediments containing low concentrations of magnetic carriers, ARM properties are controlled by intrinsic parameters (coercivity and grain volume) rather than by interactions. The SD ARM equation derived by Egli and Lowrie (2002) is

where Hk is the anisotropy field, ƒac is the AF frequency, and DHAF is the decay rate of AF. This model predicts an increase in ARM intensity proportional to d2 within the SD size range (d < 60 nm). ARM intensity is also predicted to be weakly dependent on the characteristics of the AF. Experimental results on natural and synthetic samples gave results compatible with theoretical predictions. For non-single-domain grains, Gillingham and Stacey (1971) proposed a multidomain theory of ARM incorporating domain wall translation under the influence of the self-demagnetizing field. The self-demagnetizing field itself can be thought of as an internal interaction field. Theory predicts that for DC fields in which domain wall translation occurs (HDC < Hc), Marm / HDC. Reasonable agreement has been obtained between theory and experiment for magnetite grains with d > 40 mm, which most likely contain true multidomain states. However, experimental ARM intensities were significantly larger than predicted for d < 40 mm or for particle sizes within the pseudosingledomain size range (Gillingham and Stacey, 1971). Dunlop and Argyle (1997) developed a PSD theory for ARM by combining theories incorporating a wall displacement term linearly dependent on HDC (an MD process) and a wall moment term proportional to the hyperbolic tangent function (an SD process). Reasonable fits to experimental ARM acquisition data for magnetite particles in the 0.2–0.5 mm size range were obtained with this theory.

Applications Absolute paleointensity Shaw (1974) proposed an alternative paleointensity method (now called the Shaw method) to the conventional Thellier-Thellier method, employing ARM and a single-step laboratory TRM heating. First, the NRM of the sample is stepwise AF demagnetized. Then an ARM (designated ARM1) is imparted and subsequently AF is demagnetized. Next, the sample is given a total TRM by heating the sample once above its Curie temperature and cooling in the same DC field as used in the ARM experiment. The TRM is then AF demagnetized using the same step as the one used for ARM. Finally, a second ARM (designated ARM2) is given and AF is demagnetized. If the sample does not experience any chemical alteration during the TRM step then a plot of ARM1 vs ARM2 using the peak AF as the common parameter should be linear with a slope of 1.0. In practice, a portion of the coercivity spectrum is selected that gives the best fit to a line of slope 1.0. Within this coercivity window it is assumed that the TRM is likewise unchanged and a plot of NRM vs TRM using the peak AF as the common parameter yields an estimate of the paleofield. Idealized Shaw diagrams are shown in Figure M98. This method is considerably quicker than the Thellier-Thellier method because less heating is required. Furthermore, samples are heated only once above the Curie temperature, which reduces the potential for chemical alteration that can occur during the multiple heatings used in the Thellier-Thellier method. Kono (1978) showed experimentally that the Shaw method could be as reliable as the Thellier-Thellier method for samples that undergo little or no chemical change during heating. However, a disadvantage of the Shaw method is that the entire experiment is unusable if alteration takes place during the single-step heating to the Curie temperature. Thermochemical alteration of the sample can lead to changes in its TRM capacity and produce erroneous paleointensity results. Modifications of the original Shaw method have been proposed to correct for limited amounts of thermochemical alteration

576

MAGNETIZATION, ANHYSTERETIC REMANENT

Figure M98 Idealized Shaw diagrams for absolute paleointensity determination. (a) ARM1/ARM2 diagram is used to test samples for thermochemical alteration during heating. ARM1 (before heating) and ARM2 (after heating) are AF-demagnetized in identical steps (5, 10, 20,. . .,100 mT). The solid line represents the slope where ARM1/ARM2 ¼ 1.0. Points that fall off the line (0, 5, 10 mT) may indicate part of the coercivity spectrum that was altered during heating. (b) NRM/TRM diagram is used to estimate the paleointensity. The line with slope ¼ 1.0 from part (a) between 20 and 100 mT delimits the range of identical AF demagnetization characteristics of NRM and TRM within which the TRM and NRM may be compared with each other for estimating paleointensity.

during the TRM heating step (Kono, 1978; Rolph and Shaw, 1985). In these methods, a “slope correction” is (1) applied to the NRM/TRM slope using slopes of linear segments of the ARM1/ARM2 plot over “unaltered” coercivity windows or (2) applied individually to TRM data points using ARM1/ARM2 ratios measured at the same AF level as the corresponding TRM point. Besides the Shaw method and its modified forms, several ARM techniques were developed earlier, which made use of the theoretical

Figure M99 Pseudo-Thellier method for relative paleointensity in sediments. (a) Normalized remanence curves for stepwise AF-demagnetization of NRM and the subsequent acquisition of partial ARM in the identical steps as the NRM demagnetization. (b) NRM/ARM plot (also called an Arai plot) for the data in (a). The Arai plot shows ARM gained during magnetization in specified AF fields vs NRM left after demagnetization in the same AF fields. The solid line represents the best-fit line through the data and the slop gives the relative paleointensity. (Data taken from Tauxe et al., 1995).

relationships between the intensities of ARM and TRM (Banerjee and Mellama, 1974; Collinson, 1983). Here, the laboratory heating step was eliminated and replaced entirely by room-temperature ARM. However, the main shortcoming of such an approach was the difficulty in determining the correct calibration factor, R ¼ ARM/ TRM, for particular samples. Here R is known to be a function of grain size, interactions, and applied field (Figures M96 and M97) and would vary from sample to sample. These methods were used primarily to estimate paleointensities from lunar samples containing iron or ironnickel grains, which are highly susceptible to chemical alteration even when heated or slightly heated above room temperature (see review by Collinson, 1983).

MAGNETIZATION, ANHYSTERETIC REMANENT

Relative paleointensity For marine and lacustrine sediments in which NRM is a detrital remanent magnetization (DRM), ARM is used as a normalizing parameter to remove the effects on the NRM intensity signal due to depth variations in concentration, grain size, and composition of the magnetic carriers of remanence (e.g., Tauxe, 1993; Valet, 2003 for comprehensive reviews). According to this hypothesis, down-core variations in NRM/ARM values should theoretically reflect relative variations in paleointensity of the Earth’s field rather than variations in physical properties (grain size, concentration) of the magnetic particles in the sediment. The idea was originally proposed by Johnson et al. (1975a,b) and Levi and Banerjee (1976). These authors argued that the magnetic particles responsible for the stable portion of the NRM signal, namely, SD- and PSD-sized grains, will be the same grain size fraction that is most efficient in acquiring ARM (see Figure M95). King et al. (1983) propose three uniformity criteria for sediments under which NRM/ARM normalization provides a reliable measure of relative paleointensity: (1) magnetite is the predominate remanence carrier (uniformity in composition); (2) grain size varies between 1 and 15 mm (uniformity in grain size); and (3) variability in magnetite concentration is less than 20–30 times the minimum concentration (uniformity in concentration). Another approach, which uses ARM to determine relative paleointensity in sediments, is the pseudo-Thellier method of Tauxe et al. (1995). In this procedure, NRM lost in successive AF demagnetization steps is compared to partial ARMs gained in matching AF steps, similar to the pNRM-pTRM steps in the Thellier-Thellier method. When NRM lost is plotted against the pARM gained for each matching AF step, the slope of the resulting line is proportional to the relative paleointensity (Figure M99). This method has two advantages over the conventional NRM/ARM normalization. (1) It allows identification of possible overprints due to viscous remanent magnetization (VRM), particularly at low AF demagnetization steps ( TR, where TR denotes room temperature, is in particles that have grown beyond the blocking volume, vAB; chemically precipitated particles with volumes v < vAB do not contribute to the CRM. On cooling from TA to TR, a PTRM will be produced in particles with blocking temperatures TA > TB > TR. Also, ubiquitous time effects might contribute significant VRM superimposed on the CRM. In addition, the resultant remanence is likely to grow on cooling from TA to TR, due to the increase in MS on cooling for most magnetic minerals. Such complexities make it difficult to uniquely isolate CRM from other remanences, hence it is probable that CRM occurrences in the paleomagnetic record are more numerous than is usually recognized. As long as the magnetic ensemble consists of non-interacting, homogeneously magnetized SD particles, t and the magnetic stability increase exponentially with particle volume. For multi-domain (MD) particles, t decreases with increasing volume, principally because it becomes progressively easier to alter the remanence by domain wall movements, as opposed to rotating the magnetic moments of SD particles. These considerations of remanence stability and relaxation times were demonstrated experimentally by coercivity measurements of dispersed fine particles of ferromagnetic metals (Fe, Ni, Co), as the grain sizes increased due to progressive heat treatments (e.g., Meiklejohn, 1953; Becker, 1957). Zero initial coercivity in the SP region is followed by increasing coercivity with particle sizes, presumably in the SD size range; the coercivity then decreases for larger presumably MD particles (Figure M104). Haigh (1958) first applied Néel’s theory and the grain size dependence of the coercivity to rock magnetism to explain CRM properties in growing particles of magnetite obtained during laboratory reduction of hematite. Kobayashi (1959) also examined CRM in magnetite obtained from hematite reduction and showed that CRM stability with respect to both alternating fields and thermal demagnetization was much greater than for IRM (isothermal remanence) and very similar to the stability of TRM (Figure M105). Kobayashi (1961) produced CRM in cobalt grains precipitated from Cu-Co alloy and showed that the specific CRM intensity had a similar bell-shaped grain size dependence as the coercivity, increasing from zero for SP particles, attaining a maximum value and then decreasing for inhomogeneously magnetized MD particles (Figure M106). Grain growth CRM can explain many examples in paleomagnetism, where the natural remanence (NRM) is predominantly a secondary remanence in chemically-altered rock and where new particles of a chemicallynucleated magnetic phase have grown beyond the SP size range.

CRM in Igneous Rocks General Considerations In igneous rocks, CRM can be produced by phase transformations or nucleation and growth of new magnetic minerals at temperatures

582

MAGNETIZATION, CHEMICAL REMANENT (CRM)

Figure M104 (a) Intrinsic coercive force of iron and cobalt as a function of particle size at liquid nitrogen temperature (modified from Meiklejohn, 1953). (b) Change in coercive force of Cu(98%)Co(2%) alloy as a function of annealing time (particle size) at 700
C, measured at 300 K (modified from Becker, 1957).

Figure M105 (a) Thermal, (b) alternating fields (AF) demagnetization of remanence in magnetite. All remanence measurements were made at room temperature. CRM was produced at 340
C with external fields of (a) 0.3 mT, (b) 1 mT. Total TRM was produced in (a) 0.3 mT, (b) 0.05 mT. IRM(T0) at room temperature was produced in fields of (a) 20 mT, (b) 3 mT. IRM(T) was produced at 340
C in fields of (a) 2 mT, (b) 1 mT. CRM (T ¼ 340
C) and IRM (T ¼ 340
C) were cooled to T0 in zero field prior to being demagnetized (modified from Kobayashi, 1959).

Figure M106 Total, remanent, and reversible magnetization of Cu-Co alloy as a function of annealing time (particle size) at 750
C, measured at 750
C. The reversible magnetization is the difference between the total magnetization (in presence of an applied field) and the remanence (modified from Kobayashi, 1961).

MAGNETIZATION, CHEMICAL REMANENT (CRM)

below the Curie point of the new magnetic species. The effective magnetic field at the site of the new magnetic particles determines the direction and intensity of the CRM. In igneous rocks, magnetic interactions among multiple phases, such as during exsolution of the iron-titanium solid solution series, might significantly modify the external field through magnetostatic or exchange interactions with phase(s) having higher blocking temperatures. Occasionally, such interactions might produce CRM with oblique directions or opposite polarity to the external magnetic field and the existing magnetic phase (negative magnetic interactions), and in rare cases a self-reversal may result (Néel, 1955).

CRM Origin of Marine Magnetic Anomalies The first remanence of freshly extruded submarine basalts is TRM in stoichiometric titanomagnetites, xFe2TiO4(1-x)Fe3O4, with x 0.6 0.1, that is, about 60% molar ulvöspinel, with Curie points between 100
–150
C (Readman and O’Reilly, 1972). The initial magnetic phase is transformed at the sea floor by topotactic low temperature oxidation to cation deficient titanomaghemites. The oxidation is thought to proceed by a net cation migration out of the crystal lattice to accommodate a higher Fe3þ/Fe2þ ratio and accompanying changing proportions of Fe:Ti. The resulting cation deficient phases retain the cubic crystal structure with smaller lattice dimensions, higher Curie points, approaching 500
C, and lower saturation magnetizations, responsible for the rapid diminution of the amplitudes of the marine magnetic anomalies away from spreading centers (e.g., Klitgord, 1976). Therefore, marine magnetic anomalies over the world’s oceans can be considered to be preserved predominantly as CRM in oxidized Fe-Ti oxides. The secondary CRM in submarine basalts usually retains the same polarity as the initial TRM recorded upon extrusion. This is indicated by the agreement, for overlapping time intervals, between the polarity time scales from marine magnetic anomalies, continental lavas and marine sediments, as well as the magnetic polarity of oriented dredged and drilled submarine basalts. The agreement of CRM and TRM directions is further supported by laboratory low temperature oxidation experiments of predominantly SD titanomagnetites (e.g., Marshall and Cox, 1971; Johnson and Merrill, 1974; Özdemir and Dunlop, 1985). The superexchange interactions vary with changes in cation distribution and lattice dimensions, as indicated by the reduced saturation magnetization and higher Curie points for the more oxidized titanomaghemites. However, it is possible that the orientations of the sub-lattice magnetic moments remain intact during low temperature oxidation of the titanomagnetite minerals, so that the ensuing CRM retains or inherits the original TRM direction. Evidence from magnetic anomalies and magnetic properties of drilled oceanic basalts suggests an increase of the magnetization of extrusive submarine basalts of oceanic crust older than about 40 Ma (e.g., Johnson and Pariso, 1993). At present, the data are too sparse for one to be confident of the generality of this phenomenon or to select from several mechanisms that might be responsible.

583

However, because reaction rates increase with temperature, such a procedure tends to maximize chemical alterations. The Thelliers’ double heating method (Thellier and Thellier, 1959) was developed to diminish this problem by gradually heating the samples in steps from room temperature, TR, to the highest blocking temperature. At each temperature step, TI > TR, the samples are heated twice; to determine both the thermally demagnetized partial-NRM (PNRM) and the acquired partial-TRM (PTRM) between TI and TR. The Thellier method depends on the additivity and independence of PTRMs acquired in different temperature intervals; that is, totalTRM ¼ SPTRM. Also, it is assumed that the unknown paleointensity, HU, was constant throughout the remanence acquisition process and that HU and HL are sufficiently small that both TRMs are linearly proportional to the imposed field. When these conditions are satisfied and in the absence of chemical changes on heating, HU can be calculated from the ratio PNRM(TI,TR)/PTRM(TI,TR) ¼ HU/HL and should be the same for each temperature interval. Every temperature step gives an independent paleointensity value. These consistency checks, provided by several independent paleointensity estimates at the different temperatures, are the primary asset of the Thellier method. The data can be displayed on a PNRM versus PTRM plot (an Arai diagram), with data corresponding to the different temperature steps (Figure M107). Ideal behavior in the Thellier sense implies linear data with a slope equal to -HU/HL (line A, Figure M107). The Thelliers’ procedure is well suited to detect the onset of chemical alterations, which are more common at higher temperatures, and are often expressed as deviations from the ideal straight line. Another feature of the Thelliers’ procedure is the PTRM check, where a PTRM is repeated at a lower temperature TP < TI. The PTRM check provides information about changes in the PTRM capacity of magnetic particles with TB  TP . For these reasons, the Thelliers’ procedure is usually considered to be the most reliable paleointensity method. High-temperature chemical alterations during the Thelliers’ paleointensity procedure is one of the more common causes for failed or abbreviated paleointensity experiments. If the CRM is expressed as a

CRM Influence on Paleointensity Studies At present, only TRM can be used for obtaining absolute paleointensities of the Earth’s magnetic field (see Paleointensity from TRM). Hence, all paleointensity methods require that the NRM be essentially pure TRM, or that the TRM can be readily isolated from the NRM. The paleointensity methods compare the NRM of each specimen to a new laboratory TRM produced in a known laboratory field, HL. Because chemical and mineralogical alterations of specimens during laboratory heatings are common and often preclude reliable paleointensity determinations, the different paleointensity methods apply various pre- and post-heating tests to assess the extent of chemical changes on the remanence and paleointensity experiments. Several paleointensity methods use a single heating to above the specimen’s highest Curie temperature to produce a total laboratory TRM.

Figure M107 PNRM-PTRM diagrams for five hypothetical Thellier paleointensity experiments, A-E, discussed in the text. Like symbols indicate identical temperatures. Dashed lines 1, 2, and 3 indicate PTRM checks between the designated temperatures for experiments A, B, and D, respectively. Solid diamonds refer to unsuccessful PTRM checks.

584

MAGNETIZATION, CHEMICAL REMANENT (CRM)

greater PTRM capacity, which increases with temperature, the data will form a concave-up PNRM-PTRM plot (Figure M107, curve B). Provided the lower temperature data are linear and the PTRM checks show no evidence of alteration, then the lower temperature data can be used to calculate a paleointensity (Figure M107 curve B, points T1-T3). Recent studies suggest that at least 50% of the NRM should be used to obtain reliable results (e.g., Chauvin et al., 2005). CRM production that increases the PTRM capacity may result from precipitation of new magnetic particles or from unmixing of titanomagnetite grains to a more Fe-rich phase with higher saturation magnetization. Chemical modifications, which lead to higher PTRM, usually cause the PNRM-PTRM points to lie above the ideal line, and the calculated paleointensity will be lower than the actual paleofield. Alternatively, if the chemical alterations decrease the PTRM potential by destroying magnetic particles or by transforming them to a phase with lower intrinsic magnetic moments, then the PNRMPTRM points will plot below the ideal line, with higher apparent paleointensities than the actual values (e.g., Figure M107, curve D and PTRM check 3). For a special case, where CRM acquisition grows linearly with temperature, the PNRM-PTRM plot might be linear (Figure M107, line C); however, these data plot above the ideal line (Figure M107, line A), and the calculated paleointensity would be lower than its actual value. This result emphasizes that linear PNRM-PTRM data are a necessary but not sufficient condition for obtaining reliable paleointensities. The adverse effects of chemical alterations and CRM on paleointensity studies do not always arise from heatings in the laboratory. It is also possible that the NRM is not a pure TRM but contains a significant CRM component. Yamamoto et al. (2003) suggested that high temperature CRM contributes to the NRM of the Hawaiian 1960 lava, which results in higher than expected paleointensities. Alternatively, lowtemperature hydrothermal alteration might produce CRM in new magnetic particles that contribute to the NRM. If these particles have not grown significantly beyond their blocking volumes, vAB, they would be demagnetized at T TA, leading to rapid decrease of the NRM. This scenario might explain the precipitous diminution of the NRM observed for some basalts, with decreases on the order of 20% to more than 50% in the first few temperature steps of the Thellier experiment. When this decrease in NRM cannot be attributed to viscous remanence, it is possible that the NRM is augmented by CRM. It is no longer pure TRM.

CRM in Sedimentary Rocks Oxidized Red Sediments Red beds are a broad and loosely defined category of highly oxidized sediments with colors ranging from brown to purple, usually resulting from secondary fine particles of hematite, maghemite and/or ferric oxyhydroxide. The color is a complex function of the mineralogy, chemical composition and particle sizes of the iron oxides, as well as the impurity cations and their concentrations; however, for paleomagnetism, color is unimportant. Red sediments have been used extensively for paleomagnetism since the late 1940s, because they are widely distributed geographically and with respect to geologic time. In addition, the remanence of red sediments is often stable and sufficiently intense for paleomagnetic measurements, even with early-generation magnetometers. Already in the 1950s, it was deduced that low temperature oxidation was responsible for transforming the original magnetite to fine particles of hematite, maghemite, and/or goethite, which provide the pigment and CRM of red beds (Blackett, 1956). Larson and Walker (1975) studied CRM development during early stages of red bed formation in late Cenozoic sediments; they showed that in their samples CRM occurred in several authigenic phases including hematite and goethite. The CRM, which obscured the original DRM, had formed over multiple polarity intervals, as indicated by different polarities in several generations of authigenic minerals. Complex multi-generation patterns of CRM, with several polarities within

single specimens, have also been observed in Paleozoic and Mesozoic red beds. The influence of the secondary CRM on the primary remanence depends on the relative stability and intensity of the CRM carriers as compared with the primary DRM. In many cases the DRM may have been entirely obliterated by diagenetic processes, and the CRM is the dominant characteristic remanence. However, there are examples of red beds, where the primary DRM in specularite hematite remains the characteristic remanence with respect to CRM (e.g., Collinson, 1974). Sometimes distinct CRM components can be isolated by thermal demagnetization, selective leaching in acids (e.g., Collinson, 1967) and removal of altered phases of sediment by selective destructive demagnetization (Larson, 1981). During the past more than five decades, paleomagnetic studies of red sediments have contributed significantly to magnetostratigraphy, plate tectonics and rock magnetism. Many data of apparent polarwander paths are from red beds, where it is assumed that the CRM was produced soon after deposition, so that the paleomagnetic pole accurately represents the depositional age of the sediments. Moreover, CRM is not subject to inclination shallowing, which often affects the primary DRM. The utility of red sediments for high resolution studies of the geomagnetic field and paleosecular variation is limited and depends on how pervasive the CRM is as compared to the primary DRM, the time lag between the CRM and initial DRM, and the duration of CRM production.

Non-red Sediments Here we discuss CRM in a subset of non-red mostly carbonate sediments, whose remanence is usually much weaker than for red beds. The low remanence intensity of these sediments was a key reason that they were generally excluded from paleomagnetic investigations until the introduction in the early 1970s of cryogenic magnetometers, capable of measuring minute signals, down to the 109–1010 Gauss range. Since then, there has been an explosion of studies of weakly magnetized non-red sediments. For example, it has been shown that the characteristic remanence of some early Paleozoic carbonate sequences was acquired in the late Paleozoic, hundreds of millions of years younger than their biostratigraphic ages (e.g., McCabe et al., 1983). Magnetic extracts from these diagenetically altered sediments contained essentially pure magnetite particles, whose botryoidal and spheroidal forms have been used to infer their secondary origin, and they are thought to be responsible for the secondary characteristic remanence of these sediments. This conclusion has been buttressed by electron microscope observations (Figure M108) of in-situ authigenic magnetites in Paleozoic limestones (Suk et al., 1993). In the absence of evidence of significant heating of these sediments, their remanence has been attributed to low temperature CRM in secondary magnetite, which might have been produced by “diagenetic alteration of preexisting iron sulfides (e.g., framboidal pyrites)” (McCabe et al., 1983). For Miocene dolomites and limestones of the Monterey Formation, Hornafius (1984) concluded that the secondary remanence, presumably a low temperature CRM, resides in diagenetic magnetite produced by partial oxidation of pyrite upon the introduction of oxygenated meteoric groundwaters to the formation. CRM in some Paleozoic carbonates resides in hematite particles (e.g., Elmore et al., 1985) produced by diagenetic dedolomitization, where oxidizing fluids with high calcium contents cause calcite replacing dolomite (McCabe and Elmore, 1989). The presence of magnetite (and siderite, FeCO3) in oil impregnated sediments was discovered by Bagin and Malumyan (1976), and Donovan et al. (1979) reported correspondence of near surface magnetic anomalies over an oil field with a higher concentration of magnetic minerals in the sediments. Paleomagnetic studies of remagnetized hydrocarbon-impregnated Paleozoic sediments (McCabe et al., 1987; Benthien and Elmore, 1987) indicate a relationship between hydrocarbon migration and the precipitation of authigenic magnetite particles, carrying the secondary CRM. This scenario is supported by extracted magnetite spherules up to several tens of microns in diameter and

MAGNETIZATION, CHEMICAL REMANENT (CRM)

585

Figure M108 Scanning electron microscope images of pseudoframboidal magnetite in the New York carbonates. Symbols are MGT, magnetite; PF, pseudoframboid; F, framboid; P, pyrite; C, calcite; D, dolomite; Q, quartz; and H, hole. (a) Densely distributed framboids and pseudoframboids in a calcite matrix with occasional occurrence of dolomite and quartz; backscattered electron image (BEI). (b) Cross section of a pseudoframboid in a microcrack showing individual octahedral/cubo-octahedral crystals; secondary electron image (SEI). (c) A pseudoframboid in a microcrack showing almost perfect spherical shape (SEI). (d) An imperfectly spherical magnetite pseudoframboid in a void showing pyrite cores or voids within originally homogeneous pyrite crystals. Layered iron-rich clay minerals surround the grain (SEI) (from Suk et al., 1993).

the presence of other authigenic textures in the sediments. All these studies suggest a net reduction of ferric ions in oxides, hydroxides and silicates, caused by the biodegradation (oxidation) of the hydrocarbons and the precipitation of more reduced iron oxide phases such as magnetite (Fe3O4), siderite (FeCO3) and wustite (FeO). Of these, only magnetite is ferrimagnetic; hence it is responsible for the CRM and for being preferentially extracted during magnetic separations. Rapidly deposited marine and lacustrine sediments are increasingly being used to study high-resolution behavior of the Earth’s magnetic field, including secular variation and relative paleointensities. However, to accurately interpret the sedimentary record, geochemical processes that influence the magnetic signal must be understood. In anoxic and suboxic environments, bacterial sulfate reduction produces H2S, which reacts with the detrital iron oxides to precipitate sulfide minerals (Berner, 1970, 1984). An abundance of sulfate favors reactions that produce relatively more stable pyrite (FeS2), which does not carry remanence. When the sulfate supply is more limited, the formation of ferrimagnetic pyrrhotite (Fe7S8) and/or greigite (Fe3S4) is preferred. Pyrrhotite formation is less common in sediments because it is thought to require pH > 11 (Garrels and Christ, 1965), which is outside the range of values measured in sedimentary pore waters. However, for extremely low sulfur activity, it is possible for pyrrhotite to form in sediments. For anoxic sediments from the Gulf of California and suboxic hemipelagic muds from the Oregon continental slope with sedimentation rates exceeding 1m/kyr, Karlin and Levi (1983, 1985) documented very rapid, dramatic decreases with depth of the intensity of NRM

and artificial remanences, paralleled by downcore decrease in porewater sulfate and systematic growth in solid sulfur, mainly as pyrite (Figure M109). In both environments, the remanence resides in finegrain nearly pure magnetite. These data suggest that early oxidative decomposition of organic matter leads to chemical reduction of the ferrimagnetic minerals and other iron oxides, which are subsequently sulfidized and pyritized with depth. Changes in the remanence intensity and stability are consistent with selective dissolution of the smaller particles, causing downcore coarsening of the magnetic fraction. In these environments, there was no evidence for the formation of authigenic magnetitic minerals. In this example, there is no CRM formation; rather, the sediments experience chemical demagnetization via dissolution. The chemical processes cause substantial reduction of the remanence intensity, while the directions appear to be unaffected. In other suboxic marine environments, characterized by lower sedimentation rates, on the order of centimeters/kyrs, CRM in authigenic magnetite particles accompanies oxidative decomposition of organic matter immediately above the Fe-reducing zone (Karlin et al., 1987; Karlin, 1990). Some of the smaller authigenic magnetite particles are subsequently dissolved downcore on entering the zone of Fe-reduction (Figure M110). In the past approximately fifteen years, an increasing number of paleomagnetic studies have identified CRM in ferrimagnetic iron sulfides, pyrrhotite and greigite, in a variety of marine and lacustrine settings (e.g., Roberts and Turner, 1993; Reynolds et al., 1999; Weaver et al., 2002; Sagnotti et al., 2005). While in many cases the Fe-sulfides are formed during early diagenesis upon initial burial, they can also result from later diagenesis, deeper in the sections.

586

MAGNETIZATION, CHEMICAL REMANENT (CRM)

Figure M109 Downcore profiles of magnetic intensities and solid sulfur for Kasten core W7710-28, Oregon continental slope. The magnetization intensities were partially AF demagnetized at 15 mT. Solid sulfur concentrations of total (circles) and acid-insoluble (triangles) fractions on a sulfate-free basis, measured by X-ray fluorescence (modified from Karlin and Levi, 1983). might be produced in magnetite due to diagenesis of iron sulfides or, possibly, in hematite due to dedolomitization in an oxidizing environment, depending on the sediment composition and prevailing geochemistry.

Concluding Remarks

Figure M110 Downcore profiles of the NRM intensity, partially AF demagnetized at 20 mT, and the saturation magnetization for core TT11 from NE Pacific Ocean (modified from Karlin, 1990).

Recently, Roberts and Weaver (2005) described multiple mechanisms for CRM involving sedimentary greigite. The resolution of paleomagnetic time series would be compromised in sediments where CRM occurs in authigenic magnetic minerals, formed below the remanence lock-in depths. In light of this analysis t, it is possible that sediments discussed earlier in this section experienced multiple episodes of CRM. The first might arise from diagenesis of magnetite and other iron oxides to iron sulfides during initial burial. A later CRM

This report on CRM is not exhaustive and reflects the interests, biases and limitations of the author. It is an update of a similar article written over fifteen years ago (Levi, 1989). In the future, as paleomagnetists address more difficult tectonic and geomagnetic questions, requiring data from structurally more complex, metamorphosed, and older formations, it will be increasingly likely that CRM will contribute to the NRM. Paleomagnetists have become more adept at isolating different remanence components, using detailed and varied demagnetization procedures. It is usually assumed that the most resistant remanence, whether with respect to increasing temperatures, alternating fields, or a particular leaching agent is also the primary component. However, CRM stabilities are highly variable, and this assumption is unlikely to be satisfied universally. During the past fifteen years, there has been progress in understanding several aspects of CRM, including (a) the recognition that even for some very young subaerial lavas the NRM may comprise a low-temperature CRM component, and (b) that in some active sedimentary environments, diagenesis leads to CRM in ferrimagnetic iron sulfides. A more comprehensive understanding of CRM is needed to assist paleomagnetists to interpret complex, often multicomponent, NRMs with probable CRM overprints. This goal would be advanced by conducting controlled field and laboratory CRM experiments to (1) recognize the varied geochemical environments that produce different magnetic minerals and their associated CRMs; (2) determine the ranges of magnetic and mineralogical stabilities with respect to different demagnetization procedures and for isolating different CRM components; and (3) develop procedures for identifying the timing and sequencing of multi-component CRMs. Shaul Levi

MAGNETIZATION, CHEMICAL REMANENT (CRM)

Bibliography Bagin, V.I., and Malumyan, L.M., 1976. Iron containing minerals in oil-impregnated sedimentary rocks from a producing rock mass of Azerbaidzhan. Izvestiya, Academy of Sciences, USSR. Physics of the solid earth (English translation) 12: 273–277. Becker, J.J., 1957. Magnetic method for the measurement of precipitate particle size in a Cu-Co alloy. Journal of Metals, 9: 59–63. Beetz, W., 1860. Über die inneren Vorgänge, welche die Magnetisierung bedingen. Annalen der Physik und Chemie, 111: 107–121. Benthien, R.H., and Elmore, R.D., 1987. Origin of magnetization in the phosphoria formation at Sheep Mountain, Wyoming: a possible relationship with hydrocarbons. Geophysical Research Letters, 14: 323–326. Berner, R.A., 1970. Sedimentary pyrite formation. American Journal of Science, 268: 1–23. Berner, R.A., 1984. Sedimentary pyrite formation: an update. Geochimica et Cosmochimica Acta, 48: 605–615. Blackett, P.M.S., 1956. Lectures on Rock Magnetism. Jerusalem: The Weizmann Science Press of Israel, 131 pp. Chauvin, A., Roperch, P., and Levi, S., 2005. Reliability of geomagnetic paleointensity data: the effects of the NRM fraction and concave-up behavior on paleointensity determinations by the Thellier method. Physics of the Earth and Planetary Interiors, 150(4): 265–286, doi:10.1016/j.pepi.2004.11.008. Collinson, D.W., 1967. Chemical demagnetization. In Creer, K.M., and Runcorn, S.K. (eds.) Methods in Paleomagnetism. Amsterdam: Elsevier, pp. 306–310. Collinson, D.W., 1974. The role of pigment and specularite in the remanent magnetism of red sandstone. Journal of the Royal Astronomical Society, 38: 253–264. Donovan, T.J., Forgey, R.L., and Roberts, A.A., 1979. Aeromagnetic detection of diagenetic magnetite over oil fields. American Association of Petroleum Geologists Bulletin, 63: 245–248. Elmore, R.D., Dunn, W., and Peck, C., 1985. Absolute dating of a diagenetic event using paleomagnetic analysis. Geology, 13: 558–561. Garrels, R.M., and Christ, C.L., 1965. Solutions, Minerals and Equilibria. New York: Harper & Row, 450 pp. Haigh, G., 1958. The process of magnetization by chemical change. Philosophical Magazine, 3: 267–286. Hornafius, J.S., 1984. Origin of remanent magnetization in dolomite from the Monterey Formation. In Garrison, R.E., Kastner, M., and Zenger, K.H. (eds.), Dolomites of the Monterey Formation and Other Organic-Rich Units. Society of Economic Paleontologists and Mineralogists pp. 195–212. Pacific Section Publication No. 41. Johnson, H.P., and Merrill, R.T., 1973. Low-temperature oxidation of a titanomagnetite and the implication for paleomagnetism. Journal of Geophysical Research, 78: 4938–4949. Johnson, H.P., and Pariso, J.E., 1993. Variations in oceanic crustal magnetization: systematic changes in the last 160 million years. Journal of Geophysical Research, 98: 435–445. Karlin, R., 1990. Magnetic mineral diagenesis in suboxic sediments at Bettis site W-N, NE Pacific Ocean. Journal of Geophysical Research, 95: 4421–4436. Karlin, R., and Levi, S., 1983. Diagenesis of magnetic minerals in recent hemipelagic sediments. Nature, 303: 327–330. Karlin, R., and Levi, S., 1985. Geochemical and sedimentological control of the magnetic properties of hemipelagic sediments. Journal of Geophysical Research, 90: 10373–10392. Karlin, R., Lyle, M., and Heath, G.R., 1987. Authigenic magnetite formation in suboxic marine sediments. Nature, 326: 490–493. Klitgord, K.D., 1976. Sea-floor spreading: the central anomaly magnetization high. Earth and Planetary Science Letters, 29: 201–209. Kobayashi, K., 1959. Chemical remanent magnetization of ferromagnetic minerals and its application to rock magnetism. Journal of Geomagnetism and Geoelectricity, 10: 99–117.

587

Kobayashi, K., 1961. An experimental demonstration of the production of chemical remanent magnetization with Cu-Co alloy. Journal of Geomagnetism and Geoelectricity, 12: 148–164. Koenigsberger, J.G., 1938. Natural residual magnetism of eruptive rocks. Terrestrial Magnetism and Atmospheric Electricity, 43, 119–130, part 1: part 2: 299–320. Larson, E.E., 1981. Selective destructive demagnetization, another microanalytic technique in rock magnetism. Geology, 9: 350–355. Larson, E.E., and Walker, T.R., 1975. Development of CRM during early stages of red bed formation in late Cenozoic sediments, Baja, California, Geological Society of America Bulletin, 86: 639–650. Levi, S., 1989. Chemical remanent magnetization. In James, D.E. (ed.) The Encyclopedia of Solid Earth Geophysics. London, UK: Van Nostrand Reinhold Ltd., pp. 49–58. Marshall, M., and Cox, A., 1971. Effect of oxidation on the natural remanent magnetization of titanomagnetite in suboceanic basalt. Nature, 230: 28–31. Maurain, Ch. 1901. Propriétés des dépots électolytiques de fer obtenus dans un champ magnétique. Journal of Physique, 3(10): 123–135. Maurain, Ch. 1902. Sur les propriétés magnétiques de lames trés minces de fer et de nickel. Journal of Physique, 4(1): 90–151. McCabe, C., and Elmore, R.D., 1989. The occurrence and origin of late Paleozoic remagnetization in the sedimentary rocks of North America. Reviews of Geophysics, 27: 471–494. McCabe, C., Van der Voo, R., Peacor, D.R., Soctese, R., and Freeman, R., 1983. Diagenetic magnetite carries ancient yet secondary remanence in some Paleozoic sedimentary carbonates. Geology, 11: 221–223. McCabe, C., Sassen, R., and Saffer, B., 1987. Occurrence of secondary magnetite within biodegraded oil. Geology, 15: 7–10. Meiklejohn, W.H., 1953. Experimental study of coercive force of fine particles. Reviews of Modern Physics, 25: 302–306. Néel, L., 1949. Théorie du traînage magnétique des ferromagnétiques en grains fins avec applications aux terres cuites. Annales de Géophysique, 5: 99–136. Néel, L., 1955. Some theoretical aspects of rock magnetism. Advances in Physics, 4: 191–243. Özdemir, Ö., and Dunlop, D.J., 1985. An experimental study of chemical remanent magnetizations of synthetic monodomain titanomaghemites with initial thermoremanent magnetizations. Journal of Geophysical Research, 90: 11513–11523. Readman, P.W., and O'Reilly, W., 1972. Magnetic properties of oxidized (cation deficient) titanomagnetites. Journal of Geomagnetism and Geoelectricity, 24: 69–90. Reynolds, R.L., Rosenbaum, J.G., van Metre, P., Tuttle, M., Callender, E., and Goldin Alan 1999. Greigite (Fe3S4) as an indicator of drought—the 1912–1994 sediment magnetic record from White Rock Lake, Dallas, Texas, USA. Journal of Paleolimnology, 21: 193–206. Roberts, A.P., and Turner, G.M., 1993. Diagenetic formation of ferromagnetic iron sulphide minerals in rapidly deposited marine sediments, South Island, New Zealand. Earth and Planetary Science Letters, 115: 257–273. Roberts, A.P., and Weaver, R., 2005. Multiple mechanisms of remagnetization involving sedimentary greigite Fe3S4. Earth and Planetary Science Letters, 231(3–4): 263–277, doi:10.1016/j.epsl. 2004.11.024. Sagnotti, S., Roberts, A.P., Weaver, R., Verosub, K.L., Florindo, F., Pike, C.R., Clayton, T., and Wilson, G.S., 2005. Apparent magnetic polarity reversals due to remagnetization resulting from late diagenetic growth of greigite from siderite. Geophysical Journal International, 160: 89–100. Suk, D., Van der Voo, R., and Peacor, D.R., 1993. Origin of magnetite responsible for remagnetization of early Paleozoic limestones of New York State. Journal of Geophysical Research, 98: 419–434.

588

MAGNETIZATION, DEPOSITIONAL REMANENT

Thellier, E., and Thellier, O., 1959. Sur l'intensité du champ magnétique terrestre dans le passé historique et géologique. Annales de Géophysique, 15: 285–376. Weaver, R., Roberts, A.P., and Barker, A.J., 2002. A late diagenetic (syn-folding) magnetization carried by pyrrhotite: implications for paleomagnetic studies from magnetic iron sulphide-bearing sediments. Earth and Planetary Science Letters, 200: 371–386. Yamamoto, Y, Tsunakawa, H., and Shibuya, H., 2003. Palaeointensity study of the Hawaiian 1960 lava: implications for possible causes of erroneously high intensities. Geophysical Journal International, 153: 263–276.

MAGNETIZATION, DEPOSITIONAL REMANENT The magnetization acquisition processes in unconsolidated sediments have been long studied (e.g., Johnson et al., 1948; King, 1955; Granar, 1958). The early studies showed that magnetic minerals in the sediments align along the ambient magnetic field during deposition through the water column. The magnetization resulting from the sedimentation process has been referred as depositional or detrital remanent magnetization (DRM). The magnetization acquisition process is still not well understood, and the role of the complex interplay of processes occurring during deposition, water-sediment interface processes, burial, and compaction, etc., require further analyses. The characteristics and stability of the remanent magnetization of unconsolidated sediments are determined by the composition, grain size, and shape of individual grains. During deposition in aqueous media, the magnetic particles are subject to the aligning force of the ambient magnetic field, plus the gravitational and dynamic forces. In tranquil conditions alignment of magnetic grains is relatively effective in depositional timescales, which are affected by Brownian forces. Nagata (1961) showed that equilibrium with the ambient magnetic field is attained in a scale of 1 s. Therefore, saturation magnetization should be expected in natural depositional systems, with the DRM intensity independent of the ambient magnetic field intensity. However, this is not the case, and the magnetization intensity is related to the intensity of the ambient magnetic field (Johnson et al., 1948). The DRM intensities lower than saturation values have been related to misalignment effects of Brownian motion of submicron ferrimagnetic grains (Collinson, 1965; Stacey, 1972). Near the water-sediment interface, flow conditions may become relatively stable and simple by having laminar flow (Granar, 1958) but still the deposition process is complex. The interplay and characteristics of the bottom sediments result in a variety of fabrics in the deposited sediments. In general, reorientation of magnetic minerals occurring after deposition and before consolidation of the sediment is referred as postdepositional DRM. Perhaps, the most notable distinction between depositional DRM and postdepositional DRM is the occurrence of the so-called inclination error present in depositional DRM (Johnson et al., 1948; King, 1955; Granar, 1958). If IH is the inclination of the ambient Earth’s magnetic field at the time of sediment deposition, then the inclination of magnetization IS can be expressed in terms of IS ¼ arctanðf tan IH Þ where f is a factor that is determined experimentally. The inclination error has been ascribed to deposition of elongated grains with alongaxis magnetizations tending to lie parallel to the sediment interface and deflecting the magnetization toward the horizontal plane. Laboratory experiments have been conducted to evaluate effects of the intensity of the ambient magnetic field, size and shapes of the magnetic grains, deposition on horizontal, inclined, and irregular surfaces, bottom currents, etc. (e.g., Johnson et al., 1948; King, 1955; Rees, 1961). The inclination error has been observed in laboratory experiments, where

the angular difference can be as high as 20
(King, 1955), but it is smaller (5–10
) or absent in natural sediments. Barton et al. (1980) studied the change with time in the DRM acquisition process of laboratory-deposited sediments and found that in less than 2 days, there was no appreciable inclination error. In natural conditions, postdepositional DRM presents no significant inclination error. One of the major differences between laboratory experiments and natural conditions is the deposition rate. The time taken for realignment has been estimated in a few years and is apparently related to the water content of the sediments. Verosub et al. (1979) experimentally reexamined the role of water content in acquisition of postdepositional DRM and suggested that small-scale shear-induced liquefaction is the main magnetization process. There are also several additional factors involved; for instance bottom water currents, changing water levels, presence of organic matter, biological activity (bioturbation), particle flocculation, floccule disaggregation, dewatering, etc. In general, it appears that rapidly deposited sediments show inclination and bedding inclination errors, similar to those observed in laboratory experiments. Slowly deposited or high-porosity sediments show small or no inclination error. Postdepositional DRM will realign the magnetization direction; this process may occur in short timescales of days or months, but may occur in periods of years or decades following deposition (Tarling, 1983). In addition to studies of secular variation and magnetostratigraphy in sedimentary sequences, there has also been much interest in determining relative paleointensities from sedimentary records (Tauxe, 1993). Long records of relative paleointensities have been derived from marine and lake sedimentary sequences, and results have been compared with volcanic records and other records. There have been also several attempts to examine the effects of depositional factors in the DRM intensity, including for instance the effects of clay mineralogy, electrical conductivity of sediments, pH and salinity (Lu et al., 1990; Van Vreumingen, 1993; Katari and Tauxe, 2000). Katari and Bloxham (2001) examined the effects of sediment aggregate sizes on the DRM intensities, and proposed that intensity is related to viscous drag that produces misalignment of magnetic particle aggregates. They argue that interparticle attractions arising from electrostatic or van der Waals forces and/or biologically mediate flocculation results in formation of aggregates (which present a log-normal size distribution) preventing settling of individual smaller grains. The depositional and postdepositional DRM in laboratory experiments and naturally deposited sediments have been intensively studied; nevertheless, further work is required to understand the complex interplay of processes and then develop magnetization acquisition models (e.g., Verosub, 1977; Tarling, 1983; Tauxe, 1993; Katari and Bloxham, 2001). Jaime Urrutia-Fucugauchi

Bibliography Barton, C., McElhinny, M., and Edwards, D., 1980. Laboratory studies of depositional DRM. Geophysical Journal of the Royal Astronomical Society, 61: 355–377. Collinson, D., 1965. Depositional remanent magnetization in sediments. Journal of Geophysical Research, 70: 4663–4668. Granar, 1958. Magnetic measurements on Swedish varved sediments. Arkiv for Geofysik, 3: 1–40. Irving, E., 1964. Paleomagnetism and its Application to Geological and Geophysical Problems. New York: Wiley. Johnson, E., Murphy, T., and Wilson, O., 1948. Pre-history of the Earth’s magnetic field. Terrestrial Magnetism and Atmospheric Electricity, 53: 349. Katari, K., and Bloxham, J., 2001. Effects of sediment aggregate size on DRM intensity: a new theory. Earth and Planetary Science Letters, 186: 113–122.

MAGNETIZATION, ISOTHERMAL REMANENT

Katari, K., and Tauxe, L., 2000. Effects of pH and salinity on the intensity of magnetization in redeposited sediments. Earth and Planetary Science Letters, 181: 489–496. King, R.F., 1955. The remanent magnetism of artificially deposited sediments. Monthly Notices of the Royal Astronomical Society, 7: 115–134. Lu, R., Banerjee, S., and Marvin, J., 1990. Effects of clay mineralogy and the electrical conductivity of water in the acquisition of depositional remanent magnetism in sediments. Journal of Geophysical Research, B95: 4531–4538. McElhinny, M.W., 1973. Palaeomagnetism and Plate Tectonics. Cambridge Earth Science Series. Cambridge: Cambridge University Press, 358 pp. Nagata, T., 1961. Rock Magnetism. Tokyo, Japan: Mazuren, 350 pp. Rees, A.I., 1961. The effect of water currents on the magnetic remanence and anisotropy of susceptibility of some sediments. Geophysical Journal of the Royal Astronomical Society, 5: 235–251. Stacey, F., 1972. On the role of Brownian motion in the control of detrital remanent magnetization of sediments. Pure and Applied Geophysics, 98: 139–145. Tarling, D.H., 1983. Palaeomagnetism, Principles and Applications in Geology, Geophysics and Archaeology. London: Chapman & Hall, 379 pp. Tauxe, L., 1993. Sedimentary records of relative paleointensity of the geomagnetic field: theory and practice. Reviews of Geophysics, 31: 319–354. Van Vreumingen, M., 1993. The influence of salinity and flocculation upon the acquisition of remanent magnetization in some artificial sediments. Geophysical Journal International, 114: 607–614. Verosub, K.L., 1977. Depositional and postdepositional processes in the magnetization of sediments. Reviews of Geophysics and Space Physics, 15: 129–143. Verosub, K.L., Ensley, R.A., and Ulrick, J.S., 1979. The role of water content in the magnetization of sediments. Geophysical Research Letters, 6: 226–228.

MAGNETIZATION, ISOTHERMAL REMANENT Introduction As indicated by the name, isothermal remanent magnetization (IRM) is a remanent magnetization (RM) acquired without the aid of changes in temperature. There are actually many processes that can produce RMs isothermally (see, e.g., CRM, VRM, DRM, and ARM ), but by convention we restrict the use of the term IRM to denote a remanence resulting from the application and subsequent removal of an applied DC field. IRMs are thus intimately related to hysteresis. Conventional nomenclature uses MR for the IRM determined by measurement of a hysteresis loop (and MRS if the cycle reaches saturation), whereas IRM and SIRM are typically used for remanence measurements made with a zero-field magnetometer after magnetization on a separate instrument (electromagnet, pulse magnetizer, etc.). Weak fields, such as the geomagnetic field, are generally very ineffective in producing IRMs, and therefore IRMs are rarely significant components of NRM. (A notable exception is found in rocks struck by lightning, as discussed below.) IRMs are therefore primarily of interest as laboratory-produced remanences, and are used in sample characterization, in studying the mechanisms affecting stable remanence, and as a normalizing parameter for estimation of relative paleointensity. The attributes of principal interest are: the intensity (dipole moment per unit mass or volume); the coercivity (range of fields over which significant acquisition occurs) or stability (resistance to demagnetization by thermal, alternating-field, and/or low-temperature methods); and the anisotropy (dependence on orientation of the applied field).

589

Grain-scale processes and remanent states of individual particles In a sufficiently strong applied field, the magnetic structure inside a ferromagnetic particle is simple: uniform magnetization parallel to the field, i.e., saturation magnetization (MS). With decreasing applied field intensity, interesting complexities develop in response to the internal demagnetizing field, magnetocrystalline anisotropy and stress: magnetization may rotate away from the applied field direction toward an easy axis determined by grain shape, crystal orientation or stress; the magnetization may become nonuniform, with spins arranged in “flower” or “vortex” structures; domains may nucleate and grow, with magnetizations oriented at high angles to those of neighboring domains. Upon complete removal of the saturating applied field, the particle reaches its saturated remanent state, with a net intensity MRS  MS, oriented in general with a positive component parallel to the applied field direction, i.e., MRS// > 0 (although self-reversal of IRM has been reported in rare cases, e.g., Zapletal, 1992). The ratio MRS/MS for an individual particle thus ranges from 0 to 1, depending on the saturated remanent domain state, and MRS///MS depends additionally on easy-axis orientation(s) with respect to the applied field direction. For an ideal single-domain (uniformly magnetized) particle, MRS is equal in magnitude to MS, and the ratio MRS///MS is thus determined by the particle anisotropy (number and symmetry of easy axes) and orientation (Gans, 1932; Wohlfarth and Tonge, 1957). For an ideal SD particle with uniaxial (e.g., shape) anisotropy, oriented at an arbitrary angle f to the applied field, MRS///MS ¼ cos(f) and MRS⊥/MS ¼ sin(f); MRS///MS thus ranges from 0 for an applied field perpendicular to the easy axis (i.e., the moment rotates 90
as the field is removed) to 1 for the parallel case. Magnetocrystalline anisotropy often involves multiple symmetrically equivalent easy axes (e.g., the cubic body diagonals in magnetite, and the cube edges in Ti-rich titanomagnetites). For a multiaxial SD particle, the maximum angle (fmax) that may separate the applied field from the nearest easy axis is reduced (Figure M111), and the minimum ratio MR///MS is accordingly increased. For an individual cubic SD magnetite grain, fmax ¼ 54.7
and thus 0.577  MRS///MS. Above a threshold grain size, uniformly magnetized remanent states become energetically unfavorable, and vortex (Williams and Wright, 1998) or multidomain structures consequently develop, lowering the total energy by reducing the net remanence of the particle. In a two-domain particle with antiparallel domain magnetizations, the net remanence is primarily due to domain imbalance; in asymmetrically shaped grains the minimum-energy state may yield a significant net (imbalance) moment (Fabian and Hubert, 1999). Even when domain moments exactly cancel in two-domain grains, the domain-wall moment (“psark,” Dunlop, 1977) provides a small but stable net particle moment. Increasing domain multiplicity generally results in more perfect mutual cancellation of both the domain moments and the wall moments (which have been observed to alternate in polarity across magnetite grains with several domains, Pokhil and Moskowitz, 1997), and there is consequently a strong reduction in MRS///MS with increasing particle size.

Remanent states: ensembles of particles Except in very coarse-grained rocks, even a centimeter-sized cubic or cylindrical specimen contains a very large number of ferrimagnetic particles. The net IRM of a population of grains is simply the vector sum of the individual grain remanent moments, and thus its magnitude depends on the number of remanence-carrying particles, their size (domain-multiplicity) distribution, particle-level anisotropy, and orientation distribution (i.e., population-level anisotropy) relative to the applied-field direction. In interacting assemblages, each particle is influenced not only by the externally applied field but also by the fields of neighboring particles, so the spatial distribution of grains is an additional important factor (Muxworthy et al., 2003).

590

MAGNETIZATION, ISOTHERMAL REMANENT

Figure M111 Cube edge directions (diamonds) are the magnetocrystalline easy axes for high-Ti titanomagnetites. Due to the cubic symmetry, an applied field can be oriented no more than ymax ¼ 54.7
from the nearest easy axis of an individual grain. For a randomly oriented assemblage of ideal SD grains with easy axes, MRS/MS ¼ 0.832 (Gans, 1932). This is most easily calculated in the particle coordinate system, by noting that an IRM parallel to the polar easy axis results from application of a sufficiently strong field oriented anywhere within the “capture area” comprised of the eight adjacent congruent spherical triangles (outlined in bold). For each particle, MR///MS ¼ cos(f), and for the assemblage MRS /MS is obtained by integrating cos(f) over the “capture area.” Body diagonal directions (triangles) are the magnetocrystalline easy axes for magnetite at temperatures above the 130 K isotropic point; the “capture area” for these consists of six congruent spherical triangles, and integration of cos(f) over this area yields MR/MS ¼ 0.866 for a randomly oriented assemblage of ideal SD grains with easy axes.

Orientation distribution and bulk anisotropy The simplest case to consider is that of a perfectly aligned population of identical single-domain particles, for which the ratio MRS///MS of the assemblage equals that of each individual particle: MRS///MS ¼ cos(f), where f is the angle between the applied field and the nearest easy-axis direction in the aligned assemblage. Note that in this case, as for an individual particle, the net IRM generally has both parallel and transverse components. A perfectly aligned population represents an extreme not often approached in natural materials, most of which are considerably closer to the opposite limiting case: isotropic (random or quasiuniform) orientation distribution (OD). For an ensemble of SD particles with an isotropic OD, MRS///MS is determined entirely by the particle-level anisotropy (Gans, 1932; Stoner and Wohlfarth, 1948; Wohlfarth and Tonge, 1957), and the net transverse component is effectively zero. For uniaxial anisotropy, grain moments rotate on average 60
to the nearest easy axis when the applied field is removed, yielding MRS/// MS ¼ MRS/MS ¼ cos(60
) ¼ 0.5. (This result is obtained by integrating parallel components over the assemblage, equivalent to integrating cos(f) over the hemisphere surrounding the applied field orientation, within which the proximal easy axes are distributed.) With multiaxial anisotropies the average rotation angle is significantly reduced and MRS/MS is concomitantly increased. For an ensemble of cubic SSD particles with easy axes (e.g., TM60) and a uniform OD, the expected MRS///MS ratio is calculated by integrating cos(f) over the “capture area” surrounding a cube edge orientation (Figure M111),

yielding MRS///MS ¼ MRS/MS ¼ 0. 832. For easy axes (e.g., magnetite), integration over the region surrounding a body diagonal orientation yields MRS///MS ¼ MRS/MS ¼ 0.866 (Gans, 1932; Stoner and Wohlfarth, 1948). For the imperfect antiferromagnets hematite and goethite, as well as the hard ferrimagnet pyrrhotite, the applied fields typically used in rock magnetism (1 T produced by electromagnets; 5 T produced by typical superconducting magnets) are insufficient to achieve saturation. When these minerals occur with fine-grain sizes and/or impure compositions, applied fields exceeding 60 T may be required to saturate the remanence (Mathé et al., 2005). A hard magnetocrystalline anisotropy largely confines the remanence to the basal planes of hematite and pyrrhotite, and to the c-axis for goethite. Hematite’s basal-plane anisotropy has both a weak triaxial magnetocrystalline component and a uniaxial magnetoelastic component (Banerjee, 1963; Dunlop, 1971). When the uniaxial anisotropy dominates MRS/ MS ¼ 0.637. In the absence of strong uniaxial anisotropy, the hexagonal symmetry results in much higher ratios, up to 0.955 (Wohlfarth and Tonge, 1957; Dunlop, 1971). When particle moments are diminished by the development of nonuniform remanent states, the ratio MRS/MS of the population decreases proportionally. For magnetite there is a more or less continuous decrease in the remanence ratio with increasing grain size. Data compiled by Dunlop (1995) for the size range 40 nm  d  400 mm show that MRS/MS varies in proportion to d –n (0.5  n  0.65), with two separate size-dependent trends: stress-free hydrothermally grown grains have significantly lower remanence ratios than similarly sized crushed grains or grains grown by the glass ceramic method. Bulk IRM anisotropy is controlled both by particle-level anisotropy and by the orientation distribution of the population. Like other bulk magnetic anisotropies, the anisotropy of IRM (AIRM) has been used to characterize rock fabric for geological applications and for evaluating directional fidelity of NRM (Fuller, 1963; Cox and Doell, 1967; Daly and Zinsser, 1973; Stephenson et al., 1986; Jackson, 1991; Kodama, 1995). IRM differs from weak-field induced and remanent magnetizations (e.g., susceptibility, ARM, and TRM) in that it is generally a nonlinear function of applied field, and consequently the tensor mathematics used to describe linear anisotropic magnetizations are not valid for AIRM, except when very weak magnetizing fields ( a few millitesla) are used. Nevertheless it is a relatively rapid and very sensitive method of characterizing magnetic fabric (Potter, 2004).

“Partial IRMs” and coercivity spectrum analysis Partial TRMs (pTRMs) and partial ARMs (pARMs) are generated when a weak but steady bias field is applied during some part, but not all, of the cooling or AF decay. Clearly no direct analog can exist for IRM, but “pIRM” equivalents can be produced physically (by partial demagnetization of an IRM), or calculated mathematically (by subtraction of IRMs acquired in different fields). Partial IRMs and their ratios are very widely used for sediment characterization in environmental magnetism (Thompson and Oldfield, 1986; King and Channell, 1991; Oldfield, 1991; Verosub and Roberts, 1995; Dekkers, 1997; Maher and Thompson, 1999). For example the hard fraction, HIRM, is determined by subtracting the IRM acquired in 300 mT (IRM300) from SIRM, to estimate the contribution of antiferromagnetic minerals (e.g., hematite and goethite) to the saturation remanence. HIRM, as conventionally defined, is an absolute, concentration-dependent parameter. The soft fraction is more often quantified in relative terms through the so-called S-ratios, calculated from measurements of SIRM and of the IRM subsequently acquired in an oppositely directed field of 100 mT (IRM–100) or 300 mT (IRM300): S100 ¼ IRM–100/SIRM, and S300 ¼ IRM–300/SIRM. These ratios range from 1 (for samples containing only hard antiferromagnets) to þ1 (for samples dominated by soft ferrimagnets). A more thorough method of deconstructing an SIRM uses application of two successive orthogonal applied fields of decreasing strength to reset the remanence of the intermediate- and low-coercivity fractions;

MAGNETIZATION, ISOTHERMAL REMANENT

the resultant “triaxial” IRM is then thermally demagnetized to determine the unblocking temperature distribution associated with each coercivity fraction (Lowrie, 1990). The threshold fields are selected for isolation of different mineralogical and grain-size fractions; Lowrie (1990) used 5 T for the initial SIRM (sufficient to magnetize hematite and goethite at least partially); 0.4 T for the first orthogonal overprint (to reorient the remanence of the ferrimagnets magnetite and pyrrhotite); and 0.12 T for the second overprint (to realign the remanence of soft MD carriers). For many geological applications, recognition and quantification of a few discrete IRM-carrying fractions is adequate, but finer subdivision of SIRM into a quasicontinuous coercivity distribution can be made by differentiation of the IRM acquisition curve (Dunlop, 1972). Starting from a demagnetized state, application of successively stronger fields results in growth of IRM as domain structures change and single-domain moments are reoriented, each at a critical field determined by particle anisotropy and orientation. The resulting coercivity spectrum can be analytically decomposed if it assumed that the distributions each follow some prescribed form, typically lognormal (Figure M112) (Robertson and France, 1994; Stockhausen, 1998; Heslop et al., 2002; Egli, 2003, 2004 a,b). Starting from the SIRM state, application of successively larger opposite-polarity “backfields” causes the net remanence first to decrease and then to grow in the reverse direction, eventually reaching the negative SIRM state; this process is often termed “DC demagnetization” but it is perhaps more useful to think of it as a part of the isothermal remanent hysteresis cycle. The remanent coercivity HCR is defined as the field-axis intercept of the remanent hysteresis loop, exactly as the bulk coercivity HC is defined for the in-field loop.

Interaction effects In the absence of interparticle interactions, the IRM acquisition curve for SD ensembles has a simple and direct relationship to the “DC demagnetization” curve (Wohlfarth, 1958). In each case, a particular applied field activates the same set of particles; in the former case randomly oriented moments (in the demagnetized ensemble) are replaced with a net IRM, whereas in the backfield case a net IRM is replaced with

591

its opposite-polarity equivalent. The change in the latter case is exactly twice that in the former, and thus the relationship between the initial acquisition curve MIRA(H ) and the backfield curve MIRB(H ) is: MIRB(H ) ¼ SIRM2MIRA(H ). For AF demagnetization, each peak AF erases the remanence imprinted by the corresponding DC applied ~ ¼ SIRMMIRA(HDC), i.e., there is a mirror symmetry field, so MIR(H) of the IRM acquisition and AF demagnetization curves for noninteracting SD populations. There are two commonly used graphical illustrations of these relationships. The Henkel plot graphs MIRB(H ) as a function of MIRA(H ) (Henkel, 1964). For noninteracting SD ensembles, the relationship is linear, with a slope of 2 (Figure M113a). The Cisowski plot (Cisowski, 1981) or crossover plot (Symons and Cioppa, 2000) simultaneously graphs the acquisition curve MIRA(HDC) together with the ~ ) and/or the appropriately rescaled AF-demagnetization curve MIR(H 0 (H ) ¼ 0.5(SIRM þ MIRB(H )), as functions DC-backfield curve MIRB of the AC and DC fields (Figure M113b). On this plot, noninteracting SD populations are indicated by mirror symmetry, with an intersection at the point whose field coordinate is equal to both the median destruc~ 1=2 , Dankers, 1981) and the median tive field (MDF; also referred to as H 0 (Dankers, 1981)), and acquisition field (MAF; also referred to as HCR whose magnetization coordinate is equal to 0.5*SIRM. A crossover ratio R < 0.5 (equivalent to a trajectory on the Henkel plot that “sags” below the line of slope 2) indicates that MAF > MDF, in other words that the ensemble is harder to magnetize than to demagnetize. Such behavior is a hallmark of either a negatively interacting SD population or MD carriers (for which self-demagnetization produces the equivalent effect); these two possibilities often cannot be definitively distinguished on the basis of IRM data alone (Dunlop and Özdemir, 1997). Crossover ratios R > 0.5 are virtually never observed, in part due to the extreme rarity of MD-free magnetic assemblages in natural materials, but further suggesting that interactions in SD assemblages, when they occur, are primarily negative. Theoretical treatment of interactions in multidomain assemblages has been hindered by the analytical and numerical intractability of modeling interacting nonuniformly magnetized grains. However, recent results obtained at the resolution limit of micromagnetic modeling (Muxworthy et al., 2003) shows that in general, interaction begins

Figure M112 Coercivity spectrum analysis (methods of Heslop et al., 2002), modeling the measured IRM acquisition (solid circles) as the sum of log-Gaussian distributions, in this case with three components (open symbols). Integrating and summing yields the model IRM curve (solid black). The individual components can be interpreted as different mineral and/or grain-size populations.

592

MAGNETIZATION, ISOTHERMAL REMANENT

SD behavior occurs under these conditions at approximately 20–30 nm. A highly sensitive method of probing the SP-SSD threshold uses IRM acquisition with varying exposure times (Worm, 1999). At any fixed absolute temperature T, the initial magnetization M0 of an assemblage of single-domain grains grows or decays in time (t), approaching an equilibrium value in a constant applied field H: M ðtÞ ¼ M0 et=t þ Meq ð1  et=t Þ

(Eq. 1)

where the equilibrium magnetization Meq is:
 m VMS H Meq ðT ; HÞ ¼ MS tanh 0 kT

(Eq. 2)

and the relaxation time t is: VMS ðT ÞHK ðT Þð1  H=HK ðT ÞÞ2 tðT ; HÞ ¼ t0 exp 2kT

! (Eq. 3)

In these equations V is grain volume, HK is microscopic coercivity, k is Boltzmann’s constant (1.38  10–23 J K–1), m0 is the permeability of free space (4p  10–7 H m–1), and t0 is approximately 10–9 s. Very fine (small V ) and/or very soft (low HK) grains have shorter relaxation times, and equilibrate more quickly, than do larger and/or magnetically harder SD grains. Note that relaxation time drops very sharply as the applied field H approaches HK. IRM is thus acquired quickly, and then becomes stabilized (t increasing sharply) when the applied field is removed. Timedependent IRM is significant in populations of SD particles with narrow distributions of V and HK, i.e., with a narrow distribution of relaxation times in the applied field H. Exposure times less than the mean relaxation time are too short for significant acquisition of remanence, but exposure times one or two orders of magnitude larger may allow substantial equilibration with the applied field. The timedependence of IRM in such assemblages is much stronger than the frequency-dependence of susceptibility (wfd). For example, ash-flow tuffs from Yucca Mountain Nevada have a wfd of up to 35% per decade, the highest known for any geological material; in contrast their IRM intensities increase by as much as 800% with a tenfold increase in exposure time (Worm, 1999). The time-dependence is effectively amplified for IRM by the strong asymmetry between acquisition and decay rates. Figure M113 Top: Henkel (1964) plot for a sample with ideal noninteracting SD behavior. Bottom: Cisowski (1981) plot of the same data (backfield data rescaled and reflected). After Worm and Jackson (1999), with permission of the authors and publisher.

to have significant effects for SD and PSD grains (30–250 nm) when the separation between particles is less than or equal to twice the particle diameter; this is comparable to the threshold for susceptibility interaction effects (Hargraves et al., 1991; Stephenson, 1994; CañonTapia, 1996). Interestingly, the model results show that IRM interactions are primarily negative (decreasing MRS/MS) for single-domain ensembles, but positive (increasing MRS/MS) for grains >100 nm.

Kinetic effects: time-dependent IRM (TDIRM) Néel (1949a,b, 1955) theory provides the basis for understanding phenomena such as magnetic viscosity, thermoremanence, and frequencydependent susceptibility, by specifying how magnetization kinetics depend on factors including temperature, applied field, and particle size. At room temperature, and for typical experimental timescales of milliseconds to hours, thermal fluctuations are most significant for nanometric particles; for magnetite the transition from superparamagnetic to stable

Lightning strikes and IRM in nature Lightning has long been recognized as a significant mechanism for overprinting natural remanence (Matsuzaki et al., 1954; Cox, 1961; Graham, 1961), due primarily to the strong magnetic fields locally generated by the intense currents involved in lightning strikes. These currents, typically 104–105 A, are dominantly upward-directed (i.e., electrons flow down from the cloud to the ground along the ionized flow path), and most lightning flashes include three or four separate upward current pulses (“return strokes”), spaced about 50 ms apart (e. g., Krider and Roble, 1986). The magnetic field B generated by a straight filamentary current I of effectively infinite length, at a point located at a perpendicular distance r, has a magnitude m0I/(2pr). For example, a 30 kA current produces a field of 60 mT at r ¼ 10 cm, sufficient to impart a significant IRM in most materials. The orientation of the field is perpendicular to both I and r, i.e., the field lines are circles, centered on and perpendicular to the filamentary current. Both of these characteristics, concentric directional geometry and 1/r field dependence, have been observed by Verrier and Rochette (2002) in IRM overprints in samples collected around known lightning strikes (Figure M114). However in most cases the patterns are not so clear. Although the current in a lightning bolt flows along an effectively infinite (typically

MAGNETIZATION, ISOTHERMAL REMANENT

593

Bibliography

Figure M114 Orientation and intensity of lighting-induced IRM in samples collected around a tree struck by lightning. Modified from Verrier and Rochette (2002), with kind permission of the authors and publisher.

5 km) and relatively straight vertical path between cloud and ground, the path along and below the ground surface is much shorter and more irregular, as shown by the geometry of fulgurites, which are formed when rocks or sediments are melted by lightning. (Temperatures in the ionized channel typically reach 30000 K, five times the temperature of the sun’s surface). This irregularity may be expected to disrupt the simple cylindrical symmetry of the induced magnetic field. In the absence of simple directional symmetry, the diagnostic feature for identification of lightning-produced IRM is a combination of high NRM intensity (or more specifically, high NRM/SIRM ratios) and moderate-to-low resistance to AF demagnetization (Cox, 1961; Graham, 1961); thermal demagnetization is generally much less effective at separating lightning overprints (e.g., Tauxe et al., 2003). Quantitative estimates of lightning currents (Wasilewski and Kletetschka, 1999; Verrier and Rochette, 2002; Tauxe et al., 2003) require both a paleointensity determination and an estimate of the separation between sample location and lightning strike. Wasilewski and Kletetschka (1999) and Tauxe et al. (2003) estimate lightning field intensities by finding the lab field that produces an IRM of the same intensity as the lightning overprint. Because lightning remanences may decay with time, this approach yields a minimum estimate of the magnetizing field associated with a strike. Verrier and Rochette (2002) use an alternative approach based not on the overprint intensity but on the AF amplitude required to erase it. Mike Jackson

Banerjee, S.K., 1963. An attempt to observe the basal plane anisotropy of hematite. Philosophical Magazine, 8: 2119–2120. Cañon-Tapia, E., 1996. Single-grain versus distribution anisotropy: a simple three-dimensional model. Physics of the Earth and Planetary Interiors, 94: 149–158. Cisowski, S., 1981. Interacting vs. non-interacting single-domain behavior in natural and synthetic samples. Physics of the Earth and Planetary Interiors, 26: 77–83. Cox, A., 1961. Anomalous remanent magnetization of basalt. US Geological Survey Bulletin, 1083-E: 131–160. Cox, A., and Doell, R.R., 1967. Measurement of high-coercivity anisotropy. In Collinson, D.W., Creer, K.M., and Runcorn, S.K. (eds.) Methods in Palaeomagnetism. Amsterdam: Elsevier, pp. 477–482. Daly, L., and Zinsser, H., 1973. Étude comparative des anisotropies de susceptibilité et d’aimantation rémanente isotherme: conséquences pour l’analyse structurale et le paléomagnétisme. Annales de Géophysique, 29: 189–200. Dankers, P.H.M., 1981. Relationship between median destructive field and coercive forces for dispersed natural magnetite, titanomagnetite, and hematite. Geophysical Journal of the Royal Astronomical Society, 64: 447–461. Dekkers, M.J., 1997. Environmental magnetism: an introduction. Geologie en Mijnbouw, 76: 163–182. Dunlop, D.J., 1971. Magnetic properties of fine-particle hematite. Annales de Géophysique, 27: 269–293. Dunlop, D.J., 1972. Magnetic mineralogy of unheated and heated red sediments by coercivity spectrum analysis. Geophysical Journal of the Royal Astronomical Society, 27: 37–55. Dunlop, D.J., 1977. The hunting of the ‘psark’. Journal of Geomagnetism and Geoelectricity, 29: 293–318. Dunlop, D.J., 1995. Magnetism in rocks. Journal of Geophysical Research B: Solid Earth, 100: 2161–2174. Dunlop, D.J., and Özdemir, Ö., 1997. Rock Magnetism: Fundamentals and Frontiers. Cambridge: Cambridge University Press, 573 pp. Egli, R., 2003. Analysis of the field dependence of remanent magnetization curves. Journal of Geophysical Research—Solid Earth, 108(B2): 2081, doi:10.1029/2002JB002023. Egli, R., 2004a. Characterization of individual rock magnetic components by analysis of remanence curves. 2. Fundamental properties of coercivity distributions. Physics and Chemistry of the Earth, 29: 851–867. Egli, R., 2004b. Characterization of individual rock magnetic components by analysis of remanence curves. 3. Bacterial magnetite and natural processes in lakes. Physics and Chemistry of the Earth, 29: 869–884. Fabian, K., and Hubert, A., 1999. Shape-induced pseudo-single-domain remanence. Geophysical Journal International, 138: 717–726. Fuller, M., 1963. Magnetic anisotropy and paleomagnetism. Journal of Geophysical Research, 68: 293–309. Gans, R., 1932. Über das magnetische Verhalten isotroper Ferromagnetika. Annalen der Physik, 15: 28–44. Graham, K.W.T., 1961. The remagnetization of a surface outcrop by lightning currents. Geophysical Journal of the Royal Astronomical Society, 6: 85–102. Hargraves, R.B., Johnson, D., and Chan, C.Y., 1991. Distribution anisotropy: the cause of AMS in igneous rocks?. Geophysical Research Letters, 18: 2193–2196. Henkel, O., 1964. Remanenzverhalten und Wechselwirkungen in hartmagnetischen Teilchenkollektiven. Physica Status Solidi, 7: 919–924. Heslop, D., Dekkers, M.J., Kruiver, P.P., and van Oorschot, I.H.M., 2002. Analysis of isothermal remanent magnetization acquisition curves using the expectation-maximization algorithm. Geophysical Journal International, 148: 58–64. Jackson, M.J., 1991. Anisotropy of magnetic remanence: a brief review of mineralogical sources, physical origins, and geological

594

MAGNETIZATION, NATURAL REMANENT (NRM)

applications, and comparison with susceptibility anisotropy. Pure and Applied Geophysics, 136: 1–28. King, J.W., and Channell, J.E.T., 1991. Sedimentary magnetism, environmental magnetism, and magnetostratigraphy. Reviews of Geophysics Suppl. (IUGG Report—Contributions in Geomagnetism and Paleomagnetism), 358–370. Kodama, K.P., 1995. Remanence anisotropy as a correction for inclination shallowing: a case study of the Nacimiento Formation. Eos, Transactions of the American Geophysical Union, 76: F160–F161. Krider, E., and Roble, R.E., 1986. The Earth’s Electrical Environment. Washington, DC: National Academy Press. Lowrie, W., 1990. Identification of ferromagnetic minerals in a rock by coercivity and unblocking temperature properties. Geophysical Research Letters, 17: 159–162. Maher, B.A., and Thompson, R., 1999. Quaternary Climates, Environments and Magnetism. Cambridge: Cambridge University Press, 390 pp. Matsuzaki, H., Kobayashi, K., and Momose, K., 1954. On the anomalously strong natural remanent magnetism of the lava of Mount Utsukushi-ga-hara. Journal of Geomagnetism and Geoelectricity, 6: 53–56. Muxworthy, A.R., Williams, W., and Virdee, D., 2003. Effect of magnetostatic interactions on the hysteresis parameters of singledomain and pseudo-single-domain grains. Journal of Geophysical Research, 108: 2517. Néel, L., 1949a. Influence des fluctuations thermiques sur l’aimantation de grains ferromagnétiques très fins. Comptes rendus hebdomadaires des séances de l’Académie des Sciences (Paris), Série B, 228: 664–666. Néel, L., 1949b. Théorie du traînage magnétique des ferromagnétiques en grains fins avec applications aux terres cuites. Annales de Géophysique, 5: 99–136. Néel, L., 1955. Some theoretical aspects of rock magnetism. Advances in Physics, 4: 191–243. Oldfield, F., 1991. Environmental magnetism: a personal perspective. Quaternary Science Reviews, 10: 73–85. Pokhil, T.G., and Moskowitz, B.M., 1997. Magnetic domains and domain walls in pseudo-single-domain magnetite studied with magnetic force microscopy. Journal of Geophysical Research B: Solid Earth, 102: 22681–22694. Potter, D.K., 2004. A comparison of anisotropy of magnetic remanence methods—a user’s guide for application to palaeomagnetism and magnetic fabric studies. In Martin-Hernández, F., Lüneburg, C.M., Aubourg, C., and Jackson, M. (eds.), Magnetic Fabric: Methods and Applications, Vol. 238. London: The Geological Society of London. Geological Society Special Publications, pp. 21–36. Robertson, D.J., and France, D.E., 1994. Discrimination of remanencecarrying minerals in mixtures, using isothermal remanent magnetization acquisition curves. Physics of the Earth and Planetary Interiors, 82: 223–234. Rochette, P., Mathé, P.-E., Esteban, L., Rakoto, H., Bouchez, J.-L., Liu, Q., and Torrent, J., 2005. Non-saturation of the defect moment of goethite and fine-grained hematite up to 57 Teslas, Geophysical Research Letters, 32, doi:10.1029/2005GL024196. Stephenson, A., 1994. Distribution anisotropy: two simple models for magnetic lineation and foliation. Physics of the Earth and Planetary Interiors, 82: 49–53. Stephenson, A., Sadikun, S., and Potter, D.K., 1986. A theoretical and experimental comparison of the anisotropies of magnetic susceptibility and remanence in rocks and minerals. Geophysical Journal of the Royal Astronomical Society, 84: 185–200. Stockhausen, H., 1998. Some new aspects for the modelling of isothermal remanent magnetization acquisition curves by cumulative log Gaussian functions. Geophysical Research Letters, 25: 2217–2220. Stoner, E.C., and Wohlfarth, E.P., 1948. A mechanism of magnetic hysteresis in heterogeneous alloys. Philosophical Transactions of the Royal Society of London, Series A, 240: 599–602.

Symons, D.T.A., and Cioppa, M.T., 2000. Crossover plots: a useful method for plotting SIRM data in paleomagnetism. Geophysical Research Letters, 27: 1779–1782. Tauxe, L., Constable, C., Johnson, C.L., Koppers, A.A.P., Miller, W.R., and Staudigel, H., 2003. Paleomagnetism of the southwestern USA recorded by 0–5 Ma igneous rocks. Geochemistry, Geophysics, Geosystems, 4(4): 8802, doi:10.1029/2002GC000343. Thompson, R., and Oldfield, F., 1986. Environmental Magnetism. London: Allen & Unwin, 227 pp. Verosub, K.L., and Roberts, A.P., 1995. Environmental magnetism: past, present, and future. Journal of Geophysical Research B: Solid Earth, 100: 2175–2192. Verrier, V., and Rochette, P., 2002. Estimating peak currents at ground lightning impacts using remanent magnetization. Geophysical Research Letters, 29: 1867, doi:10.1029/2002GL015207. Wasilewski, P., and Kletetschka, G., 1999. Lodestone: nature’s only permanent magnet—what it is and how it gets charged. Geophysical Research Letters, 26: 2275–2278. Williams, W., and Wright, T.M., 1998. High-resolution micromagnetic models of fine grains of magnetite. Journal of Geophysical Research, 103: 30537–30550. Wohlfarth, E.P., 1958. Relations between different modes of acquisition of the remanent magnetization of ferromagnetic particles. Journal of Applied Physics, 29: 595–596. Wohlfarth, E.P., and Tonge, D.G., 1957. The remanent magnetization of single-domain ferromagnetic particles. Philosophical Magazine, 2: 1333–1344. Worm, H.U., 1999. Time-dependent IRM: a new technique for magnetic granulometry. Geophysical Research Letters, 26: 2557–2560. Worm, H.-U., and Jackson, M., 1999. The superparamagnetism of Yucca Mountain Tuff. Journal of Geophysical Research B: Solid Earth, 104: 25,415–25,425. Zapletal, K., 1992. Self-reversal of isothermal remanent magnetization in a pyrrhotite (Fe7S8) crystal. Physics of the Earth and Planetary Interiors, 70: 302–311.

Cross-references Magnetic Anisotropy, Sedimentary Rocks and Strain Alteration Magnetic Domain Magnetization, Anhysteretic Remanent (ARM) Magnetization, Chemical Remanent (CRM) Magnetization, Depositional Remanent (DRM) Magnetization, Natural Remanent (NRM) Magnetization, Thermoremanent (TRM) Magnetization, Viscous Remanent (VRM)

MAGNETIZATION, NATURAL REMANENT (NRM) Natural remanent magnetization (NRM) is remanent magnetization that has been acquired naturally (i.e., not artificially acquired in a laboratory). It is the remanent magnetization of a sample (such as rock or baked archaeological material) that is present before any laboratory experiments are carried out. Magnetization is usually measured normalized to either the sample volume (units of A m1) or sample mass (units of A m2 kg1). NRM can consist of one or more types of magnetization depending on the history of the sample. Rock samples that have been exposed to the Earth’s magnetic field for many millions of years may have experienced several different processes of magnetization during that time, producing multiple components of magnetization. Components acquired at the time of rock formation are termed primary

MAGNETIZATION, NATURAL REMANENT (NRM)

and later components are termed secondary. The NRM is then the vector sum of all the naturally acquired components of magnetization: X NRM ¼ Primarycomponent þ Secondarycomponentsi i

i ¼ 0, 1, 2, . . . , n, where n is the number of secondary components. Both primary and secondary components in a rock can record geological

595

events (for example, the time of formation, metamorphism, or an impact event) and are of interest in unraveling the geological history of the rock. Generally, however, it is the primary remanence acquired on formation that is of most interest to the paleomagnetist. Depending on the intensity and duration of the events responsible for secondary magnetizations (also known as overprints), the latter may partially or completely replace the primary magnetization.

Figure M115 Example of thermal and AF demagnetization. Solid (open) symbols on the orthogonal vector plot (OVP) represent vertical (horizontal) components of magnetization. (a) Thermal demagnetization of Australian Tertiary basalt. There is a viscous secondary component that is removed by 200
C to leave the primary component of remanence. (b) AF demagnetization of recent Turkish basalt. There is a small viscous secondary component that is removed by 20 mT to leave the primary component of remanence.

596

MAGNETIZATION, OCEANIC CRUST

Primary and secondary remanence

Cross-references

When igneous rocks and baked archaeological materials cool on formation, they acquire a primary thermoremanent magnetization (TRM) (q.v.) in the ambient geomagnetic field as the temperature falls below the Curie temperature of the magnetic minerals. A later thermal event below the Curie point will unblock part of the TRM and replace it with a later partial thermal remanent magnetization (pTRM) secondary remanence. If magnetic minerals grow in the presence of a magnetic field, a chemical remanent magnetization (CRM) (q.v.) is formed when a critical blocking volume is attained. The remanence in sedimentary rocks is often a CRM. CRMs can also be produced in igneous and metamorphic rocks. In sediments, the primary remanence is usually a depositional remanent magnetization (DRM) (q.v.) acquired when ferromagnetic particles are aligned in the geomagnetic field during deposition. In sedimentary rocks, dewatering during the prolonged process of lithification will usually have enhanced the alignment of these grains with the geomagnetic field to produce a postdepositional detrital magnetization (PDRM). Samples may acquire a magnetization at ambient temperatures over prolonged lengths of time in a weak magnetic field such as the Earth’s. This is called viscous remanent magnetization (VRM) (q.v.) and its importance is a function of time spent in the field, the strength of the field, and the grain size and composition of the material. VRM is the most common form of secondary remanence and is present, to varying degrees, in all palaeomagnetic samples. Lightning reaching the ground is capable of inducing magnetization in rocks in a very strong, essentially instantaneous magnetic field. This is called an isothermal remanent magnetization (IRM) (q.v.) and can often completely obscure the primary remanence. Meteorites are extraterrestrial samples that can acquire a secondary shock remanent magnetization (SRM), both in the event that excavated the material from the parent body and during impact at arrival on Earth. Provided the rock has isotropic properties, each of these types of magnetization is in the direction of the magnetic field at the time of formation. The intensity of magnetization is dependent on many things (for example, the number, composition, and grain size of magnetic minerals) and also depends on the strength of the magnetic field. It is only possible to determine the absolute intensity of the field from samples containing a TRM (see Palaeointensity).

Magnetization, Magnetization, Magnetization, Magnetization, Magnetization, Paleointensity

Isolation of NRM components To isolate the different components of NRM, partial demagnetization techniques are used, as the different components will normally have different stabilities to the demagnetization procedures. The techniques used to resolve the components are thermal and alternating field (AF) demagnetization. Components acquired by differing mechanisms will usually have contrasting blocking temperature and coercivity spectra. During thermal demagnetization, samples are heated to successively higher temperatures in a magnetic field-free space so that magnetization below the treatment temperature is unblocked and randomized. As the temperature increases, more and more of the remanence will be removed until the Curie temperature is reached and the sample is completely demagnetized. With AF demagnetization, the peak AF field is increased until magnetic minerals with the highest coercivity are demagnetized (or maximum laboratory peak AF field is reached). Figure M115 shows an example of thermal and AF demagnetization. The characteristic remanent magnetization (ChRM) is the earliest acquired component of magnetization that can be isolated. So for both examples in Figure M115 this is the primary TRM that the basalt acquired on cooling. Mimi J. Hill

Bibliography Butler, R.F., 1992. Paleomagnetism: Magnetic Domains to Geologic Terranes. Boston, MA: Blackwell Science. Tarling, D.H., 1983. Palaeomagnetism Principles and Applications in Geology, Geophysics and Archaeology. London: Chapman & Hall.

Chemical Remanent (CRM) Depositional Remanent (DRM) Isothermal Remanent (IRM) Thermoremanent (TRM) Viscous Remanent (VRM)

MAGNETIZATION, OCEANIC CRUST In the late 1950s, the first magnetometers adapted to sea surface measurements became available for the scientific community, leading to the discovery of magnetic lineations on the oceanic crust. Since this discovery, first described by Fred Vine and Drummond Matthews in 1963 (the work of Morley and La Rochelle at the same time should also be mentioned), the nature and thickness of the magnetized crust has long been a subject of debate. The depth of the Curie point isotherm for magnetite (580
C) is on the order of a few hundred meters to tens of kilometers below the seafloor (depending on crustal age) and represents a depth limit for remanent magnetization. Below this isotherm, all rocks from the oceanic crust could give rise to a remanent magnetic anomaly signal. However, several decades of magnetic studies on the oceanic crust showed that magnetization is complex and varies by several orders of magnitude depending on the structure of oceanic crust, types of oceanic rocks, and alteration processes. Magnetic properties of the oceanic crust can be studied by the analysis of magnetic surveys made using sensing instruments mounted on submersibles, towed by ships, or incorporated in satellites. Depending on the distance between the survey and the crustal sources, the focus of remote magnetic analysis can be at very different scales. Submersible surveys can be used to map, for example, the meter scale contrast of magnetization above hydrothermal structures or between flows along ridge axes. By contrast, satellite surveys will give information on the properties of wide oceanic areas and the deepest magnetic sources. In between these extreme scales, sea surface magnetic surveys adapted to the kilometer scale are the reference. Local information can also be collected using downhole magnetometer tools in holes drilled within the crust. When available, the magnetic characterization of drilled or dredged oceanic rocks is a way to directly access magnetization of different rock types and is complementary to remote sensing measurements. Ophiolites, ancient ocean-crust fragments on continents, offer a way to easily access properties of deeper lithologic units. However, the obduction process probably affects mineralogical properties of rocks in such material, and it has been observed that less intense magnetizations are found compared to drilled or dredged oceanic samples. Altogether the fragmented information collected since the early 60’s allows a global pattern of magnetization of the oceanic crust to be retreived. In the following, we present the global structure of the oceanic crust and its variability, then we review the magnetic properties of the extrusive and intrusive basalts, the gabbroic sections, and peridotites.

Global structure of the oceanic crust In 1972, a model for “typical” oceanic crust structure was derived from ophiolite observation and seismic velocity analysis. Often referred to as “Penrose structure,” following the name of the international conference on ophiolites where it was established, this layered oceanic crust is as follows: 500 m of basalts, 1500 m of dikes, and 5000 m of gabbros, which together give an average thickness of 7 km of oceanic crust upon unaltered mantle. Since the establishment of this basic model, drilling and dredging show that deep materials, consisting of altered mantle rocks and gabbros, can be seen at outcrops. This more complex threedimensional structure of the crust is due to spatial and temporal variations of the magmatic and tectonic processes and is more likely to be

MAGNETIZATION, OCEANIC CRUST

Figure M116 Sketch illustrating (a) the uniform oceanic crust following the model of a Penrose layered crust and (b) an example of idealized portion of crust with a discontinuity in its center allowing exposure of ultramafic outcrops (modified after Cannat et al., 1995). UM, ultramafic rocks; G, Gabbros, D, Dikes, E, Extrusives.

adapted to slow spreading ridges. Ultramafic rocks such as serpentinized peridotites, also called serpentinites, are notably reported in these new crustal models (Cannat et al., 1995). Two “idealized” views of ocean crust are represented in Figure M116, showing a Penrose “layer cake” crust and a discontinuous crust with serpentinite emplaced at the surface. Serpentinized peridotites are altered mantle rocks but will be considered as part of the magnetic oceanic crust, which is described below.

Basalt layer The extrusive basalt layer, also referred to as seismic layer 2A (layer 1 being the sediment cover), consists of pillow and sheeted lava erupted at the ridge axes, pillows being more widely associated with slow spreading ridges and sheet flows with fast spreading ridges. The surface of flows often consists of a few centimeters of rapidly quenched glasses. Magnetic properties of extrusive basalts of the oceanic crust were the subject of numerous studies as a result of the important amount of dredged and drilled samples made available to the scientific community. The magnetic minerals in oceanic basalts are predominantly titanomagnetites associated with various amounts of titanomaghemites, an alteration product of titanomagnetites. The primary natural remanent magnetization (NRM) of basalts is acquired during the extrusion process and is a thermal remanent magnetization. Magnetic properties can show significant variations between flows depending, in particular, on iron and titanium content but also at the within-flow scale depending on local crystallization conditions and alteration progression. On average, NRM of young basalts is in the range of 10–20 A m1, but can reach extreme values on the order of 80 A m1 or higher for young flows with high iron content. The Koenigsberger parameter, which is defined as a ratio of remanent magnetization to induced magnetization, is almost always greater than one in basalts and can often reach 100 confirming that magnetization of the basalt layer is dominantly of remanent origin. This property favors the magnetic recording of past inversions of the polarity of Earth’s magnetic field leading to the pattern of lineated magnetic anomalies. Microscopic observations and Curie temperatures dominantly within 150–250
C for young unoxidized basalts characterize a primary population of titanomagnetic grains with 60% ulvospinel (called TM60). In addition to this dominant population within the bulk part of the lava, fine-grained titanomagnetite with a broader composition (between 0% and 50% ulvospinel) is observed in the rapidly quenched glassy part (Zhou et al., 2000). As a consequence, higher Curie temperatures and lower NRMs are commonly found at the pillow margin. However, magnetic anomalies are caused by basalts rather than the thin glassy part and the high magnetization of young basalts led some authors during the late 1960s to suggest that rocks from only the first few hundred meters belonging to the extrusive basaltic layers could actually be responsible for most, if not all, of the magnetic signal observed by Vine and Mathews (e.g. Klitgord et al., 1975). This led to the first magnetic models of the oceanic crust as a uniformly highly magnetized basalt

597

layer. However, extensive magnetic measurements on oceanic rocks made available, in particular by the Deep Sea Drilling Project (DSDP) program, in the late 1970s and early 1980s showed that magnetization of the basaltic extrusive layers was not homogeneous. This is mostly a consequence of the low temperature alteration of titanomagnetites to titanomaghemite, caused by circulation of seawater in the rocks pores and cracks (O'Reilly, 1984). During the process, Curie temperature progressively increases, whereas NRM decreases typically down to values on the order of a few A m1. A secondary, less intense remanent magnetization replace part of the original thermal remanent magnetization. The magnetic record of Earth’s magnetic field inversions in the upper oceanic crust is nevertheless well preserved, because this secondary magnetization remains parallel to the original direction despite its chemical origin (for a review on this topic see Dunlop and Ozdemir, 1997). Depending on the alteration environment, secondary minerals can also be created at the expense of magnetite, iron sulfides being an example often associated to hydrothermal alteration. The common occurrence of secondary viscous magnetization contributes to lowering of the magnetic signal. These findings corroborate the idea that to account for the magnetic anomaly signal, deeper seated rocks are more highly magnetized than first expected.

Dike layer Below the extrusive basalt layer, dike complexes are the first intrusive bodies. Oceanic intrusive rocks have been the subject of fewer studies than basalts due to increasing difficulties in sample accessibility. Rocks from escarpments, tectonized areas, and ophiolites are thus overrepresented and might not represent the norm. From what we know, the sheeted dike layer (also called layer 2B) shows contrasting magnetizations. The much slower cooling rate of intrusive rocks results in an initial magnetic population of large multidomain titanomagnetite grains. Curie temperatures are reported to be quite homogeneous and close to 580
C indicating magnetite as the main carrier of the magnetization. This is the result of widespread oxidation-exsolution of titanomagnetite grains. This process by which the multidomain grains exsolved to a solid solution with pure magnetite could take place during the slow initial cooling (the so-called deuteric oxidation common in subaerial lavas) but the presence of the characteristic lamellar structures associated with such a process is not always observed in submarine dikes. It is likely that low temperature and hydrothermal alteration make a significant contribution to exsolution in this context. Hydrothermal alteration also plays an important role in the loss of iron from grains, recrystallization of magnetite, and replacement of titanomagnetite crystals by silicate minerals (Smith and Banerjee, 1986). Typical remanent magnetization in dikes is typically 10–1–10–2 A m1, but stronger magnetizations on the order of 1 A m1 are found in the freshest samples. Koenigsberger ratios are lower than in basalts but usually above one (ophiolites dikes have lower Koenigsberger ratio) showing that magnetization of submarine dikes is for its most part of remanent origin. This remanent magnetization is mainly acquired during alteration and its relationship with the geomagnetic field at the time of emplacement is not yet clear.

Gabbros Gabbros are found below oceanic basalts and are coarse-grained igneous rocks (seismic layer 3). The dominant magnetic mineral in gabbros is magnetite, occurring as a primary mineral produced by the exsolution of large titanomagnetite crystals at high temperature and as a secondary mineral typically as recrystallization along cracks. Curie temperatures in gabbros are consequently dominantly found in the range 550
C–590
C. The intensity of magnetization is quite variable spanning several orders of magnitude and depends on the concentration of magnetic minerals, which is often rather low, giving on average NRM in the range 10–1 A m1. In a few sites, ferrogabbro and gabbros rich in olivine were reported and give rather high NRM

598

MAGNETIZATION, OCEANIC CRUST

values due to their high concentration of magnetite of primary origin (in the case of ferrogabbro) or resulting from serpentinization (in the case of olivine-rich gabbros) (Dick et al., 2000). The Koenigsberger ratio is often close to one, but usually shows that remanent magnetization is dominant over induced magnetization. Remanent magnetization is often reported to be quite resistant to alternating field demagnetization showing good stability, this magnetic behavior, characteristic of fine-grained population, arises from the subdivision of magnetite in lamellae structures within exsolved coarse titanomagnetite crystals, secondary magnetite is often observed as fine-grained population as well. The contribution of secondary magnetization of chemical origin (CRM) is difficult to isolate and seems relatively low. Presumably gabbros can preserve a faithful record of the magnetic field acquired during the cooling of the units shortly after intrusion at the spreading axis. Gabbros could contribute to the magnetic anomalies but their low NRM and depth usually make them marginal contributors. Gabbros show important chemical variability and the effect of alteration on magnetic properties is difficult to single out; studies made on sets of metamorphosed gabbros show NRMs and Curie temperatures in the same range as unaltered gabbros (Kent et al., 1978).

Contribution from deeper material: Peridotites In “typical” Penrose crust, the lower crust is restricted to the gabbroic layer and no contribution from deeper material is reported. However, as the occurrence of serpentinized peridotites (also called serpentinites) within the crust was more and more observed (see Figure M116), their contribution to magnetic signal became obvious. Unaltered peridotites in the mantle have a paramagnetic behavior. Magnetization of peridotites is acquired during the serpentinization process, which is the reaction of the silicate minerals of peridotites (mostly olivine) with water producing serpentine and pure magnetite. The amount of magnetite

produced during the serpentinization process is not proportional to the serpentinization progression resulting in complex iron remobilization history (Oufi et al., 2002). Even in fully serpentinized peridotites, a very different behaviors are described with, in particular, Koenigsberger ratios well below or above unity. Curie temperatures are clustering around 580
C as expected for magnetite. Remanent magnetization is variable in the range of 10–1 –10 A m1, peak values of 25 A m1 have been reported, which makes these serpentinite samples as magnetized as young basalts (Oufi et al., 2002). From microscopic observation, magnetite grains can be large and are concentrated along the veins of serpentines, which is believed to indicate the preferential direction of fluid circulation. In this case, large grains contribute to a low NRM and Konigsberger ratio and a strong induced magnetization. Magnetite is also found as small single-domain grains disseminated in the serpentinized peridotite matrix, higher NRM and larger Koenigsberger ratios are then observed. Serpentinites with such a magnetic mineralogy could be important contributors to magnetic anomalies. Nevertheless no clear evidence of serpentinized peridotites recording a reverse magnetization have yet been established.

Conclusion The typical range in two of the most used magnetic parameters, NRM and Curie temperature, are synthesized on Figure M117 for different rocks within the crust. The effect of serpentinization and low temperature alteration is represented. As shown and discussed earlier, postemplacement alteration has a significant effect on magnetic properties of rocks and contributes to the magnetic heterogeneity of the crust. Due to their shallow depth, strong magnetic intensity and remanent behavior oceanic basalts are thought to be the most important contributors to marine magnetic anomalies allowing the establishment of the sequence of reversals of the Earth’s magnetic field. In addition,

Figure M117 NRM and Curie temperatures frequently observed in oceanic rocks from drilled or dredged samples. Bold lines represent the average values for “fresh” rocks, the black arrow represents the evolution with serpentinization and small grey arrows show the evolution of these magnetic properties during low temperature alteration process. Scale bar for photo is 5mm.

MAGNETIZATION, PIEZOREMANENCE AND STRESS DEMAGNETIZATION

although this is still a matter of debate, oceanic basalts could provide a record of the magnetic field intensity variations within polarity intervals, as suggested by the worldwide occurrence of small scale magnetic variations (called “tiny wiggles”) within the main magnetic anomaly pattern (Cande and Kent, 1992). In any case, it should be mentioned again that the local crustal architecture is probably complex (Karson et al., 2002) and is a major factor to take into account in the analysis of marine magnetic signals. Julie Carlut and Helene Horen

Bibliography Cannat, M.C. et al., 1995. Thin crust, ultramafic exposures, and rugged faulting patterns at the Mid-Atlantic Ridge (22
N–24
N). Geology, 23: 149–152. Cande, S.C., and Kent, D.V., 1992. Ultrahigh resolution marine magnetic anomaly profiles—a record of continuous paleointensity variations. Journal of Geophysical Research, 97(B11): 15075–15083. Dick, H.J.B. et al., 2000. A long in situ section of the lower ocean crust: results of ODP Leg 176 drilling at the Southwest Indian Ridge. Earth and Planetary Science Letters, 179: 31–51. Dunlop, D., and Ozdemir, O., 1997. Rock Magnetism. Cambridge: Cambridge University Press, 573 pp. Kent, D.V. et al., 1978. Magnetic properties of dredged oceanic gabbros and the source of marine magnetic anomalies. Geophysical Journal of the Royal Astronomical Society, 55: 513–537. Karson, J.A. et al., 2002. Structure of the uppermost fast spread oceanic crust exposed at the Hess Deep Rift: implications for subaxial processes at the East Pacific Rise. Geochemistry, Geophysics, Geosystems, 3: DOI:10.1029/2001GC000155. Klitgord, K.D. et al., 1975. An analysis of near bottom magnetic anomalies: seafloor spreading and the magnetized layer. Geophysical Journal of the Royal Astronomical Society, 43: 387–424. Morley, L., and La Rochelle, A., 1963. Paleomagnetism and the dating of geological events. Transactions of the Royal Society of Canada, 1: App 31. O’Reilly, W., 1984. Rock and Mineral Magnetism. London: Blackie, 220 pp. Oufi, O. et al., 2002. Magnetic properties of variably serpentinized abyssal peridotites. Journal of Geophysical Research, 107(B5): 10.1029/2001JB000549. Smith, G.M., and Banerjee, S.K., 1986. Magnetic structure of the upper kilometer of the marine crust at deep sea drilling project hole 504B, Eastern Pacific Ocean. Journal of Geophysical Research, 91(B10): 10337–10354. Vine, F.J., and Matthews, D.H., 1963. Magnetic anomalies over oceanic ridges. Nature, 199: 947–949. Zhou, W. et al., 2000. Variable Ti-content and grain size of titanomagnetite as a function of cooling rate in very young MORB. Earth and Planetary Science Letters, 179: 9–20.

MAGNETIZATION, PIEZOREMANENCE AND STRESS DEMAGNETIZATION With the advent of paleomagnetic studies in the 1940s, scientists began to wonder how the effect of stress, from burial, folding, etc., would influence the magnetic remanence of rocks. By the mid1950s, an intense research effort was underway aimed at understanding piezoremanence, which is the remanent magnetization produced by stress, as well as the opposite effect of stress demagnetization. The laboratory experiments suggested that the magnetic signals of rocks were sensitive to stresses typically found around fault zones; and it was calculated that such stress-induced changes would in turn

599

modify the local magnetic field. Scientists thought that earthquakes could be predicted by monitoring magnetic-field variations.

How pressure (stress) affects magnetic remanence The origin of a spontaneous magnetic remanence is commonly associated with electron exchange between iron (or other transition metals such as Cr, Ni, etc.) atoms. For the iron oxide magnetite (Fe3O4), the most abundant magnetic mineral in the Earth’s crust, the exchange is indirect, passing through an oxygen atom. The spontaneous magnetization of magnetite is generated in magnetic lattices established by the crystallographic arrangement of the atoms. The sum of the magnetic moments from the lattices comprises the net spontaneous magnetization of the mineral. The bond lengths and bond angles between iron and oxygen atoms of a particular lattice control the magnetic intensities and directions of that lattice. Thus, when a stress acts on a material, the bond lengths and angles change, which will in turn modify the lattices’ magnetic moments. This is why the nature of the stress, be it uniaxial, hydrostatic (equal on all sides), compressive, or tensile, can have important consequences on the magnetic properties of materials. If a material compresses isotropically under a pure hydrostatic stress, bond lengths will decrease, yet bond angles will remain unchanged; whereas, uniaxial or shear stress will change the bond angles as well as the bond lengths. Such changes depend on the volumetric compressibility of the magnetic mineral. For example, the volume of magnetite decreases by about 2% per gigapascal (GPa) (note that 1 GPa ¼ 10 kilobars (kbar); for the Earth, 1 kbar corresponds to 3.5 km depth). This is why one needs very large stresses to significantly modify the electronic configuration and the magnetic lattice networks. Size and shape are other factors influencing the magnetic properties of materials. In extremely small grains, electron exchange exists, but the spontaneous moment does not remain in a fixed direction within the crystal due to thermal fluctuations, even at room temperature. These grains are called superparamagnetic because they have no spontaneous moment in the absence of an external magnetic field, yet in the presence of even a small external field, the randomizing thermal energy is overcome, and the grains display magnet-like behavior. At a critical size, about 0.05 mm for needle-shaped magnetite grains, the spontaneous magnetic moment becomes fixed within the crystal, thus resembling a dipolar magnet with a positive and a negative pole. Such grains are called single domain. At even greater volumes, the magnetic energy of the grain becomes too great and new magnetic domains grow such that the magnetic vectors of the neighboring domains are oriented in directions that diminish the overall magnetization of the crystal. These are called multidomain grains. Multidomain grains often have a net moment because the sum of the magnetic vectors of all the domains does not exactly equal zero. The magnetizations of single- and multidomain grains react differently to an imposed stress, just as they do with temperature or applied fields. Hydrostatic stress may cause the net spontaneous moment in a single-domain grain to reorient to a new position depending on the grain’s shape, presence of crystal defects, etc. Under subhydrostatic stress, the magnetization in a single-domain grain will rotate and change in intensity because different crystallographic axes will experience different stresses, which will deform the exchange network. Under any stress condition, the individual domains in a multidomain grain may grow or shrink to compensate for the deformation, which will modify the magnetic intensity and direction of the grain. Uniaxial stresses are more efficient than hydrostatic stresses at causing domain reorganization. In sum, two processes contribute to the piezoremanent behavior of materials, both of which can act independently or simultaneously at a given stress condition. The first is a microscopic effect that modifies the electron exchange couple, and the second is more of a macroscopic effect where the magnetization becomes reoriented within a grain. Each type of magnetic mineral and the domain states of each mineral have their own particular piezoremanent and stress demagnetization

600

MAGNETIZATION, PIEZOREMANENCE AND STRESS DEMAGNETIZATION

characteristics. Because rocks are usually composed of a spectrum of magnetic grain sizes, and sometimes more than one type of magnetic mineral, in order to understand how and why the magnetic properties of a rock change in response to an imposed stress, one must know the composition of the magnetic phases and their size distributions. Understanding how stress affects a rock must be viewed in a statistical sense by summing the changes in magnetization of each grain. The type of imposed stress must also be specified.

Details surrounding magnetic measurements at high pressure Magnetic measurements at high pressures are difficult for several reasons. For one, the pressure vessel apparatus must be non-, or only slightly, magnetic. This is a tall order because most materials strong enough to transmit high pressures are steel alloys—most of which are magnetic or interfere with the electronics of the detector. Second, the sample’s magnetic strength should be detectable by the measuring system and be significantly higher than the pressure vessel. In order to maximize the magnetic signal, one wants as many magnetic grains as possible in the sample. It follows that larger samples are more magnetic, and more easily measured than smaller ones—not because the density of magnetic grains is greater, but because of a greater number (or mass) of magnetic grains. As pressure equals force divided by area, a trade-off exists between pressure and sample size, which explains why most experimental work on this problem has focused on large, rock samples. The majority of experiments have been performed using uniaxial stresses and pressures of less than 1 GPa. Indirect measurements of the magnetic states of minerals at much higher pressure cells have been employed since the 1960s using Mössbauer spectroscopy, which measures magnetic interactions between electrons and the nucleus. Although Mössbauer data yields important information about the existence and type of magnetic network in materials, it does not quantify the strength or direction of the moment or their stress dependencies and thus is beyond the scope of this summary. Direct measurements of the full magnetic vector to significantly higher pressures (to 40 GPa) are now becoming possible with the development of diamond anvil cell technology. This new method also allows one to experiment on magnetic minerals whose composition and domain state are well characterized.

An important observation of the high-pressure experiments preformed under uniaxial stress is that magnetic properties change in relation to the maximum stress axis direction. For stress demagnetization (i.e., stress applied in the absence of an applied magnetic field), the magnetization intensity decreases faster parallel to the maximum stress direction rather than perpendicular to it. When pressure is applied in the presence of a magnetic field, magnetic vectors in the grains rotate toward the direction perpendicular to the maximum stress direction. Both magnetic susceptibility and magnetization increase perpendicular to the maximum stress direction and decrease parallel to the maximum stress direction (Figure M118). This has important implications because several experimental apparatuses can only measure magnetic parameters in one direction. Obviously this will lead to a bias of the interpretations of the data. A special case of stress demagnetization and piezoremanent magnetization is called shock demagnetization and shock magnetization. The term shock is added when the application and removal of a stress occurs in a brief instant in time, such as during a meteorite impact or by firing a projectile at a magnetic sample. To date no study has established that either stress demagnetization or piezoremanence is a time-dependent process, so the difference in terminology should be considered semantic.

Stress demagnetization Despite the experimental restrictions, several things are known about the magnetic properties of rocks and minerals under stress. The effect of stress in the absence of an external magnetic field has a permanent demagnetizing effect on a rock composed of several magnetic grains. In this case, permanent means that once the pressure is released, the magnetic moment does not grow back to its original, precompressed value because the magnetic vector directions of the grains have been randomized, thus lowering the net magnetization of the rock. It does not mean that the individual magnetic grains have lost their capacity to be magnets. This makes stress demagnetization similar to alternating field demagnetization. Place a rock in a decaying alternating field in a null or very weak external field and the rock will demagnetize. However, the magnetic intensity of a rock placed in a decaying alternating field in the presence of an external field will often increase to levels

Figure M118 Relative change in magnetic susceptibility (w) measured perpendicular and parallel to the maximum compression direction in two titanomagnetite samples (Curie temperatures of 250
C and 200
C, respectively) (after Kean et al., 1976).

MAGNETIZATION, PIEZOREMANENCE AND STRESS DEMAGNETIZATION

higher than its starting intensity. The same is true for pressure (stress) demagnetization and its converse, piezoremanent magnetization. Figure M119 shows a good example of the differences between hydrostatic and uniaxial stress demagnetization of a diabase rock containing large (20–300 mm) titanomagnetite (Curie temperature ¼ 535
C) grains. The pressures employed in these experiments are relevant to the Earth’s crust, as 100 MPa (which equals 1 kbar or about 3.5 km depth) is well within the seismogenic zone, i.e., the upper region of the Earth’s crust where earthquakes are generated. One quickly sees that uniaxial stress demagnetizes this material about two times faster than hydrostatic stress; e.g., at 100 MPa, hydrostatic and uniaxial pressures demagnetize roughly 20% and 40% of the original, noncompressed magnetization intensity, respectively. This is true whether the material is under pressure, or if it has been decompressed. Upon decompression, a demagnetization effect also exists, which is greater under uniaxial than hydrostatic stress. It should also be noted that during pressure cycling, once a rock has been compressed and then decompressed from a given pressure, it resists further stress demagnetization up until the maximum pressure that the rock has previously experienced. Once that pressure is exceeded, the rock continues demagnetizing in a manner coherent with the previously defined path. The pressure history dependence can be likened to a work or magnetic hardening effect owing to the strain state

Figure M119 An example of the differences between hydrostatic and uniaxial stress demagnetization for a diabase sample containing multidomain titanomagnetite (after Martin and Noel, 1988).

601

of the grains. Multidomain grains are more sensitive to stress than single-domain grains because the magnetic remanence of multidomain grains is controlled by domain wall migration, which is highly sensitive to dislocations and other imperfections in the crystal.

Piezoremanent magnetization Imagine a rock that has acquired its magnetization by cooling in the presence of the Earth’s magnetic field. The magnetic intensity of the rock will be proportional to the strength of the field in which it cooled, the number and type of magnetic grains, their sizes and shapes, and the way the grains are geometrically distributed in space. For a rock whose grains are randomly distributed (no preferred orientation), the magnetic moment of each nonspherical grain will tend to lie parallel to its long-axis, favoring the long-axis direction pointing toward the magnetic field direction. If a relatively low magnetic field is then applied, say three times that of the present Earth’s field, the rock’s magnetization will slightly increase, proportional to the amount of magnetic material in the rock (as defined by its magnetic susceptibility). After the field is removed, the magnetization will decay reversibly back to its original value. When relatively high fields, say 100 times the Earth’s field, are applied, the magnetization directions of the grains rotate parallel to the field, leading to a greater net magnetization of the rock. Upon removal of the field, the rock’s magnetic intensity will decrease, but will still remain above the value prior to the application of the strong field. This is due to an irreversible reorganization of the individual moments in each grain. As successively higher fields are applied and then removed, the magnetic moments of the grains become more and more aligned parallel to one another, again due to irreversible rotation, leading to greater and greater magnetic intensities. At some applied field value, the moments of all the grains become aligned to the extent that successively higher fields can no longer cause irreversible rotations and the rock’s magnetic intensity remains constant. This point is called the saturation remanence of isothermal remanent magnetization (SIRM) and the plot of magnetic intensity as a function of applied field is an isothermal remanent magnetization (IRM) curve (Figure M120). When a magnetic material is subject to stress in the presence of a magnetic field, the material will acquire a remanence proportional to the strength of the field and the level and type of stress. Figure M120 shows four IRM curves obtained from the same titanium-free, multidomain magnetite sample (a) in the stress-free state (had never been subjected to any pressure), (b) when the sample was compressed and held at a hydrostatic pressure of 3.13 GPa, (c) when the sample was further compressed and held at a hydrostatic pressure of 5.96 GPa, and (d) after decompression (P ¼ 0) from P ¼ 5.96 GPa. Piezoremanence refers to the phenomenon where the application of stress in the presence of a magnetic field increases the material’s magnetization above the level that it would have acquired if the same field had been applied in the absence of stress; e.g., by definition, a piezoremanent magnetization lies anywhere in the gray region above the IRM curve (a) in Figure M120. For this multidomain magnetite example, one observes that the magnetization gained at any applied field increases with increasing pressure (Figure M120, curves b and c). This means that pressure (stress) enhances irreversible rotations of the grains’ moments, by domain reorganization, more than an applied field does in the absence of stress. After pressure is released, the domain structures of the grains are even more disposed to irreversible rotation than before (Figure M120, curve d). The sequence that stress and magnetic fields are applied in will modify a rock’s magnetism in different ways and is again dependent on the species of magnetic mineral and its domain state. For the case in Figure M120, one observes that the net magnetization of a multidomain magnetite sample having been (1) brought up in pressure, (2) subject to an applied field, and then (3) removed from the field (curves b and c) is less than having been (1) brought up in pressure, (2) removed from pressure, (3) subject to an applied field, and then

602

MAGNETIZATION, PIEZOREMANENCE AND STRESS DEMAGNETIZATION

Figure M120 Isothermal remanent magnetization (IRM) curves for the same sample of titanium-free, multidomain magnetite at (a) initial (zero pressure), (b) and (c) under hydrostatic pressures, and (d) at zero pressure after decompressing from 5.96 GPa. Any point lying in the gray area is a piezoremanent magnetization. (S. Gilder, M. LeGoff, J.C. Chervin and J. Peyronneau, unpublished data).

(4) removed from the field (curve d). The opposite is generally true for single-domain magnetite grains. The reason for this is partly due to the way domains facilitate irreversible rotation. Other contributions to irreversibility may come from grain-size reduction (e.g., breaking multidomain grains into single-domain grains) or a permanent modification of the exchange couple. Note that the magnetization of a sample (1) brought up in pressure, (2) subject to an applied field, (3) removed from the field and then (4) removed from pressure may or may not exhibit piezoremanence. This depends on whether compressive stress produces more irreversible rotation than the demagnetization effect that accompanies decompression.

Piezoremanence and stress demagnetization in nature On Earth, piezoremanent magnetization and stress demagnetization are relevant anywhere transient or permanent stresses provoke a measurable change either in a short-term sense (e.g., during an earthquake) or in a long-term sense (e.g., during the build up or relaxation of stress in the crust due to tectonic motion). Mathematical models of fault motion during earthquakes estimate local deviations in the magnetic field intensity and may reach 1–10 nT (the present Earth’s field intensity ranges from 30000 to 60000 nT), which is well within the measurable limit of about 0.1 nT. The size of the seismomagnetic effect depends on several parameters including: magnetic mineralogy, domain state, magnetization direction and intensity of the country rock, direction and intensity of the Earth’s field, strike and dip of the fault, sense and amount of motion on the fault, and earthquake magnitude and depth. An important unknown is the stress history of the rock. As shown above in Figure M119, rocks undergo a magnetic hardening during stress demagnetization. So, for a significant change in the rock’s magnetization to occur, the rock must be stressed to higher levels than it has previously seen. This could be why piezoremanent effects during an earthquake have never been witnessed. On the other hand, it is difficult to set up and maintain an array of magnetometers over very long time-periods. As earthquakes cannot be predicted, the chances of being at the right place at the right time are extremely low.

Long-term changes in the local magnetic field have been documented around faults and are thought to coincide with nonseismic stress loading due to plate motion. Moreover, long-term changes in the local magnetic environment were observed when large reservoirs were filled with water. The changes were attributed to ground loading and the piezomagnetic effect. Magnetic field variations around volcanoes have been observed during or preceding volcanic eruptions. Although the changes were originally attributed to stress-field changes during the charge or withdraw of magma, closer examination of the data suggests that the magnetic variations were induced by fluid circulation. A similar debate exists for certain subduction zones that exhibit long wavelength magnetic anomalies. Subduction of cold lithosphere depresses the Curie isotherm to depths down to 100 km. Thus, magnetic minerals can be displaced to, or form at, great depths and still be above the Curie isotherm. Note that the Curie temperature of titanomagnetite increases by 2
kbar–1. Some workers attribute the anomalies to piezoremanence while others believe it is related to fluid flow in the subduction zone. Meteorite craters are among the most likely place to find the effects of piezoremanence or stress demagnetization. The presence of shatter cones, pseudotachylite, microscopic textures such as planar deformation features, and large volumes of melted rocks, all suggest that pressures during impact range from about 5 to more than 30 GPa, for reasonably sized (>20 km diameter) craters. However, finding clearcut evidence that the magnetism of the country rocks has been affected by shock is not so easy because most craters are old (>1 Ga) and have been severely eroded, deeply buried, or the thermal effects of the impact have not been eroded deeply enough to expose the shocked rocks. Moreover, after several hundreds of millions of years, the original magnetic mineralogy of the shocked rocks may become altered by oxidation, or new magnetic minerals may grow. However, the fact that aeromagnetic anomalies are characteristic features of meteorite impacts, there must be some long-term memory of the country rock’s magnetization. Some scientists have found that the shocked rocks are remagnetized in the direction of the Earth’s field at the time of impact. If true, one can perform an impact test by collecting samples along a

MAGNETIZATION, REMANENT, AMBIENT TEMPERATURE AND BURIAL DEPTH FROM DYKE CONTACT ZONES

radial transect away from the center. The remagnetization direction should be most prevalent near the center and eventually die out with distance. An intriguing case for shock demagnetization has been proposed for Mars. The southern hemisphere of the Martian crust is highly magnetic except for two very large (>500 km diameter) circular regions that have significantly lower magnetic intensities. Some scientists believe these areas of low magnetic intensity are due to stress demagnetization that occurred during meteorite impacts. Although the magnetic mineral in Martian rocks is presently thought to be magnetite, little is known about the piezomagnetic and stress demagnetization behavior of this substance at the cold (9
C to –128
C) temperatures of the red planet’s surface. Despite some 50 years of research, our knowledge of piezoremanence and stress demagnetization remains limited. A renaissance in this research field, partially due to recent space exploration, will surely lead to exciting discoveries in the future. Stuart Alan Gilder

603

Figure M121 Directions of remanent magnetization with distance from the contact of an igneous intrusion. It is assumed that the magnetic mineralogy of the host rock is similar throughout the profile and that overprinting in the baked and hybrid zones is a thermoremanent magnetization.

Bibliography Kean, W., Day, R., Fuller M., and Schmidt, V., 1976. The effect of uniaxial compression on the initial susceptibility of rocks as a function of grain size and composition of their constituent titanomagnetites. Journal of Geophysical Research, 81: 861–872. Martin, R., and Noel, J., 1988. The influence of stress path on thermoremanent magnetization. Geophysical Research Letters, 15: 507–510.

Cross-references Magnetic Remanence, Anisotropy Magnetization, Thermoremanent (TRM)

MAGNETIZATION, REMANENT, AMBIENT TEMPERATURE AND BURIAL DEPTH FROM DYKE CONTACT ZONES Introduction Nagata (1961, p. 320) proposed using the directions of remanent magnetization in the contact zone of an igneous intrusion as a geothermometer. Carslaw and Jaeger (1959) and Jaeger (1964) applied heat conduction theory to calculate the variation in maximum reheating temperature with distance from a cooling intrusion of simple geometry. Schwarz (1976, 1977) carried out the first study in which the paleomagnetism across a dike contact and theoretical maximum temperature curves were used to estimate the ambient temperature of the host rocks and the depth of burial of the present erosion surface at the time of dike intrusion. In principle, the depth of burial method can be applied to the contacts of a variety of intrusions with different shapes. However, calculating the maximum reheating temperature in the vicinity of an intrusion with complicated geometry is difficult. To date, depth of burial studies that use remanent magnetization have usually been confined to dike contacts. The case of a dike contact is used to illustrate the method in this article.

Magnetic overprinting in the vicinity of a dike The depth of burial method is based on the thermal overprinting of remanent magnetization in host rocks near dike contacts. It has been reviewed by Buchan and Schwarz (1987). When a dike intrudes, temperatures in adjacent host rocks rise above the highest magnetic unblocking temperatures (Tub). During subsequent cooling in the Earth’s magnetic field, the dike and this adjacent

“baked” zone acquire magnetic remanences with similar directions (Figure M121). Far from the dike contact, in the “unbaked” zone, peak temperatures are insufficient to thermally reset even the lowest blocking temperature component (see Baked contact test). Between the baked and unbaked zones lies a region of hybrid magnetization where reheating is only sufficient to result in thermal resetting of the lower blocking temperature portion of the host rock magnetization. If, in this “hybrid” zone,1 the older and younger components of magnetization can be separated on the basis of their blocking temperature spectra, then the maximum temperature (Tmax) reached following dike intrusion can be determined as a function of distance from the contact (Schwarz, 1977). This maximum temperature may be corrected for the effects of magnetic viscosity during prolonged heating in the field. Using heat conduction theory (Carslaw and Jaeger, 1959; Jaeger, 1964), the ambient temperature of the host rock just prior to dike intrusion can then be calculated. Depth of burial of the present erosion surface is determined by dividing the ambient temperature of the host rock by the estimated geothermal gradient at the time of intrusion. It should be noted that the study of the magnetic hybrid zone according to the procedure of Schwarz (1977) also yields the most rigorous type of baked contact test (the baked contact profile test), a field test for establishing the primary nature of the remanent magnetization of an intrusion (see Baked contact test).

Assumptions Several assumptions are made in the following discussion of the depth of burial method applied to the remanent magnetization across a dike contact (Buchan and Schwarz, 1987). 1. 2. 3. 4.

The dike can be approximated by an infinite sheet. The dike was formed by a single magma pulse of short duration. Heat transfer occurs solely by conduction. Dike intrusion results in a (partial) thermal resetting of the remanence of the host rock, without chemical changes. 5. Magnetic mineralogy in the hybrid zone is relatively simple, so that the effects of magnetic viscosity can be estimated from published time-temperature curves. Complicating factors such as prolonged duration of magma flow in the dike, multiple magma pulses, groundwater, volatiles escaping from the dike, and chemical resetting of remanence in the hybrid zone are 1 In some publications the baked, hybrid, and unbaked zones are referred to as the contact, hybrid, and host (magnetization) zones, respectively.

604

MAGNETIZATION, REMANENT, AMBIENT TEMPERATURE AND BURIAL DEPTH FROM DYKE CONTACT ZONES

usually difficult to take into account (see Buchan et al., 1980; Delaney, 1982, 1987; Delaney and Pollard, 1982; Schwarz and Buchan, 1989). They are not addressed in this article.

Sampling Oriented paleomagnetic samples are collected along a continuous profile that includes the dike, and the baked, hybrid, and unbaked zones of the host rocks (e.g., Figure M122). The horizontal distance of each sample from the contact is recorded, along with the horizontal width of the dike. If the dike is tilted from vertical, the perpendicular distance of each sample from the contact and the thickness of the dike should be calculated.

Determining maximum reheating temperature (Tmax) in the hybrid zone Each hybrid sample is thermally demagnetized in a stepwise fashion. In general, purely viscous components are eliminated at low temperatures to reveal magnetization that is the sum of a host component and a component acquired at the time of emplacement of the igneous unit (Figure M121). Upon thermal demagnetization to higher temperatures, the remanence direction of each hybrid sample moves progressively along a great circle path to the host direction (e.g., Figure M123). This reflects the fact that the Tub spectra of the overprint and host magnetization components are essentially discrete, with the overprint Tub spectrum occupying a range immediately below that of the host component. Tmax, the maximum temperature attained at a given locality in the hybrid zone following dike intrusion is given by the highest Tub of the overprint component. It can most easily be obtained using an

orthogonal component (or Zijderveld) plot of the horizontal and vertical projection of the thermal demagnetization data (e.g., Figure M124). Tmax is given by the temperature at which the straight-line segments through the overprint and host components intersect (Dunlop, 1979; McClelland Brown, 1982; Schwarz and Buchan, 1989; Dunlop and Özdemir, 1997). In the example that is shown in Figure M124 the intersections are fairly sharp so that Tmax values are readily determined. Note that the interpretation of more complicated orthogonal component plots involving multiple magnetic minerals and chemical overprinting are discussed in Schwarz and Buchan (1989). The Tmax values determined from the orthogonal component plots of Figure M124 decrease progressively across the hybrid zone with increasing distance from the dike contact as expected for thermal overprinting due to the dike emplacement.

Correcting Tmax for magnetic viscosity The maximum blocking temperatures (Tmax) that are reset in a given thermal event are dependent upon the length of time over which the elevated temperatures are maintained (Dodson, 1973; Pullaiah et al., 1975). Therefore, Tmax must be corrected using published timetemperature curves for the appropriate magnetic mineral (e.g., Pullaiah et al., 1975).

Determining ambient temperature of host rock at time of dike emplacement (Tamb) The maximum temperature (Tmax) attained at a particular locality in the host rock following dike intrusion equals the sum of the maximum temperature increase (DTmax) due to heat from the dike and the ambient temperature (Tamb) of the present surface just before the dike was emplaced. Therefore, in the hybrid zone, where Tmax falls within the overall unblocking spectrum, Tamb is calculated from the equation: Tamb ¼ Tmax  DTmax

(Eq. 1)

Different heat conduction models can be utilized to determine the maximum increase in temperature (DTmax) in the contact zone of a dike (Carslaw and Jaeger, 1959; Jaeger, 1964; Delaney, 1987). The simplest model that accounts for the latent heat of crystallization of the dike magma is one in which the heat released during solidification is added to the initial magma temperature as an equivalent temperature increase (Jaeger, 1964). This model gives a reasonable approximation to more rigorous models for distances greater than a quarter of a dike width from the contact, and has been employed in a number of studies of dike contacts where the hybrid zone is relatively far from the contact (e.g., Schwarz, 1977; Buchan and Schwarz, 1981; Schwarz and Buchan, 1982; Schwarz et al., 1985; Hyodo et al., 1993; Oveisy, 1998). Using this simple model, the maximum temperature increase at a distance x from the center of a dike of thickness 2d is calculated using the following equations from Jaeger (1964), modified slightly to allow for nonzero Tamb: DTmax ¼ ½V þ ðL=cÞ  ðTmax  DTmax Þ 1  ½erf fðx þ dÞ=2dt1=2 g  erf fðx  dÞ=2dt1=2 g 2 (Eq. 2)

Figure M122 Sampling profile perpendicular to contact of a ca. 2200–2100 Ma Indin diabase dike of the Slave Province of the Canadian Shield (modified after Schwarz et al., 1985). The dike intrudes Archean metavolcanic rocks.

where t ¼ kt/d2, k ¼ k/(rc), and erf (u) is the error function. V and L are respectively the intrusion temperature and latent heat of crystallization of the dike magma, k, k, r, and c are respectively the diffusivity, conductivity, density, and specific heat of the host rock, t is the time elapsed since intrusion of the dike, and t is dimensionless time.

MAGNETIZATION, REMANENT, AMBIENT TEMPERATURE AND BURIAL DEPTH FROM DYKE CONTACT ZONES

605

Figure M123 Thermal demagnetization characteristics of selected samples from the Indin dike profile of Figure M122 (after Schwarz et al., 1985). Distance to the contact is given for each sample. Directions are plotted on equal area nets with closed (open) symbols indicating positive (negative) inclinations. Heat-treatment temperatures are indicated in degree Celsius. Data points that represent the great circle between dike and host magnetization directions are connected by a dashed line. Tmax values, estimated from the experimental data before correction for magnetic viscosity, are given for hybrid zone and host zone samples.

tmax is determined from the equation: tmax ¼ ðx=dÞ= ln½ðx þ dÞ=ðx  dÞ

(Eq. 3)

which is satisfied when the temperature reaches its maximum value, Tmax. In this model, the required thermal parameters are assumed to be constant. Typical values are: V ¼ 1150
C and L ¼ 3.77  105 J kg1 for mafic dike rocks; c ¼ 1.26  103 J kg1
C1 for mafic host rocks; and c ¼ 1.09  103 J kg1
C1 for granitic host rocks (e.g., Touloukian et al., 1981). The simple heat conduction model described above is not valid within a quarter of a dike width of the contact (Jaeger, 1964), because it does not adequately account for the release of the latent heat of crystallization during cooling of the dike or the temperature dependence of thermal conductivity and diffusivity (Delaney, 1987). In some magnetic studies, hybrid samples have been obtained close to the dike (Buchan et al., 1980; Symons et al., 1980; McClelland Brown, 1981), so that the simple model is inappropriate. The authors in these studies applied more rigorous analytical or numerical models described by Carslaw and Jaeger (1959). The most sophisticated analysis of heat conduction models for a dike cooling by conduction has been described by Delaney (1987, 1988). He demonstrated the advantage of numerical solutions over analytical solutions, especially at and close to the dike contact. Delaney

(1988) published computer programs to calculate the temperature in a dike and host rocks as they cool conductively, in which the temperature dependence of thermal properties and the latent heat of crystallization are taken into account. These programs have been applied to the case of remanent magnetization in a dike contact by Adam (1990). Finally it should be noted that Tamb may be determined in one of two ways. It can be calculated from individual Tmax values (e.g., Schwarz and Buchan, 1982). Alternatively, when there is a sufficiently wide hybrid zone, Tamb can be determined by comparing the whole experimentally determined profile of Tmax values with the theoretical Tmax curves of maximum temperature based on conductive heating of the dike (McClelland Brown, 1981; Buchan and Schwarz, 1987).

Determining depth of burial of the present erosion surface at the time of dike emplacement The depth of burial of the present erosion surface can be determined by dividing Tamb (determined from Eq. (1)) by the paleogeothermal gradient for the sampling area at the time of dike intrusion. To estimate the paleogeothermal gradient, the present-day geothermal gradient must be corrected for the decrease in heat generation resulting from the decay of radiogenic isotopes since the time of dike intrusion (e.g., Jessop and Lewis, 1978), and in some areas for the effects of

606

MAGNETIZATION, REMANENT, AMBIENT TEMPERATURE AND BURIAL DEPTH FROM DYKE CONTACT ZONES

Figure M124 Orthogonal component plots showing thermal demagnetization results for selected samples from the Indin dike profile of Figure M122 (after Schwarz et al., 1985). Peak temperatures are indicated in degree Celsius. Closed and open symbols indicate projections onto the horizontal and N-S vertical planes, respectively.

glacial perturbation (e.g., Jessop, 1971). The uncertainties involved in estimating the paleogeothermal gradient are significant and can lead to relatively large uncertainties in the depth of burial, even if Tamb is well constrained (see Schwarz et al., 1985; Hyodo et al., 1993). Kenneth L. Buchan

Bibliography Adam, E., 1990. Temperature ambiante de la roche hôte deduite de l’histoire thermique d’un dyke: apport du paleomagnetisme. B.Eng. thesis, École Polytechnique, Montreal: Université de Montréal, 62 pp.

MAGNETIZATION, REMANENT, FOLD TEST

Buchan, K.L., and Schwarz, E.J., 1981. Uplift estimated from remanent magnetization: Munro area of Superior Province since 2150 Ma. Canadian Journal of Earth Sciences, 18: 1164–1173. Buchan, K.L., and Schwarz, E.J., 1987. Determination of the maximum temperature profile across dyke contacts using remanent magnetization and its application. In Halls, H.C., and Fahrig, W.F., (eds.), Mafic Dyke Swarms. Geological Association of Canada Special Paper 34, pp. 221–227. Buchan, K.L., Schwarz, E.J., Symons, D.T.A., and Stupavsky, M., 1980. Remanent magnetization in the contact zone between Columbia Plateau flows and feeder dykes: evidence for groundwater layer at time of intrusion. Journal of Geophysical Research, 85: 1888–1898. Carslaw, H.S., and Jaeger, J.C., 1959. Conduction of Heat in Solids, 2nd edn. New York: Oxford University Press, 510 pp. Delaney, P.T., 1982. Rapid intrusion of magma into wet rock: groundwater flow due to pore pressure increases. Journal of Geophysical Research, 87: 7739–7756. Delaney, P.T., 1987. Heat transfer during emplacement and cooling of mafic dykes. In Halls, H.C., and Fahrig, W.F. (eds.), Mafic Dyke Swarms. Geological Association of Canada Special Paper 34, pp. 31–46. Delaney, P.T., 1988. Fortran 77 programs for conductive cooling of dikes with temperature-dependent thermal properties and heat of crystallization. Computers and Geosciences, 14: 181–212. Delaney, P.T., and Pollard, D.D., 1982. Solidification of basaltic magma during flow in a dike. American Journal of Science, 282: 856–885. Dodson, M.H., 1973. Closure temperature in cooling geochronological and petrological systems. Contributions to Mineralogy and Petrology, 40: 259–274. Dunlop, D.J., 1979. On the use of Zijderveld vector diagrams in multicomponent paleomagnetic studies. Physics of the Earth and Planetary Interiors, 20: 12–24. Dunlop, D.J., and Özdemir, Ö., 1997. Rock magnetism: fundamentals and frontiers. Cambridge: Cambridge University Press, 573 pp. Halls, H.C., 1986. Paleomagnetism, structure and longitudinal correlation of Middle Precambrian dykes from northwest Ontario and Minnesota. Canadian Journal of Earth Sciences, 23: 142–157. Hyodo, H., York, D., and Dunlop, D., 1993. Tectonothermal history in the Mattawa area, Ontario, Canada, deduced from paleomagnetism and 40Ar/39Ar dating of a Grenville dike. Journal of Geophysical Research, 98: 18001–18010. Jaeger, J.C., 1964. Thermal effects of intrusions. Reviews of Geophysics, 2(3): 711–716. Jessop, A.M., 1971. The distribution of glacial perturbation of heat flow. Canadian Journal of Earth Sciences, 8: 162–166. Jessop, A.M., and Lewis, T., 1978. Heat flow and heat generation in the Superior Province of the Canadian Shield. Tectonophysics, 50: 55–77. McClelland Brown, E., 1981. Paleomagnetic estimates of temperatures reached in contact metamorphism. Geology, 9: 112–116. McClelland Brown, E., 1982. Discrimination of TRM and CRM by blocking-temperature spectrum analysis. Physics of the Earth and Planetary Interiors, 30: 405–411. Nagata, T., 1961. Rock Magnetism. Tokyo, Japan: Maruzen Company Ltd., 350 pp. Oveisy, M.M., 1998. Rapakivi granite and basic dykes in the Fennoscandian Shield: a palaeomagnetic analysis, PhD thesis, Luleå, Sweden: Luleå University of Technology. Pullaiah, G., Irving, E., Buchan, K.L., and Dunlop, D.J., 1975. Magnetization changes caused by burial and uplift. Earth and Planetary Science Letters, 28: 133–143. Schwarz, E.J., 1976. Vertical motion of the Precambrian Shield from magnetic overprinting. Bulletin of the Canadian Association of Physicists, 32: 3.

607

Schwarz, E.J., 1977. Depth of burial from remanent magnetization: the Sudbury Irruptive at the time of diabase intrusion (1250 Ma). Canadian Journal of Earth Sciences, 14: 82–88. Schwarz, E.J., and Buchan, K.L., 1982. Uplift deduced from remanent magnetization: Sudbury area since 1250 Ma ago. Earth and Planetary Science Letters, 58: 65–74. Schwarz, E.J., and Buchan, K.L., 1989. Identifying types of remanent magnetization in igneous contact zones. Physics of the Earth and Planetary Interiors, 68: 155–162. Schwarz, E.J., Buchan, K.L., and Cazavant, A., 1985. Post-Aphebian uplift deduced from remanent magnetization, Yellowknife area of Slave Province. Canadian Journal of Earth Sciences, 22: 1793–1802. Symons, D.T.A., Hutcheson, H.I., and Stupavsky, M., 1980. Positive test of the paleomagnetic method for estimating burial depth using a dike contact. Canadian Journal of Earth Sciences, 17: 690–697. Touloukian, Y.S., Judd, W.R., and Roy, R.F., (eds.), 1981. Physical Properties of Rocks and Minerals. New York: McGraw-Hill, 548 pp.

Cross-reference Baked Contact Test

MAGNETIZATION, REMANENT, FOLD TEST Establishing the age of the remanence acquisition with respect to the origin of the rock unit or age of structural deformation events is critical in the interpretation of paleomagnetic data. It is also important to establish that the magnetic minerals carry a stable magnetization over geological timescales. These two related questions remain difficult problems, which often affect the application of paleomagnetism to tectonic, stratigraphic, and paleogeographic problems (Cox and Doell, 1960; Irving, 1964). It is therefore not surprising that early in the development of the paleomagnetic method, several field and laboratory tests were developed and applied to a range of geological contexts. Among them, an elegant simple answer was put forward to constrain the age of remanence acquisition by using field information by Graham (1949). He realized that in folded rocks a simple test based on rotation of the magnetization directions about the local strike could determine if the remanence was acquired before the deformation event —if magnetization directions were dispersed in present-day coordinates and clustered upon tilt rotation (Figure M125a). Magnetization directions that cluster in present-day coordinates and disperse upon tilt rotation indicate that remanence acquisition occurred after the deformation event (Figure M125b). At the time Graham (1949) developed this test, the multivectorial nature of natural remanent magnetization (NRM) had not been properly recognized, and the demagnetization techniques for isolating remanent components had not been developed. The fold test nevertheless worked well as applied to the deformed sedimentary strata of the Rose Hill Formation from Maryland, mainly because the NRMs are univectorial and the strata are tilted to large angles without major structural complexities (French and Van der Voo, 1979). The Graham fold test can be applied to study single deformed structures, including a fold within an undeformed sequence, or over wide deformed areas with sampling at sites with varying bedding tilts. The impact of paleomagnetic data in the study of regional tectonic processes, particularly in the establishment of the continental drift theory, and the increasing use of demagnetization and statistical techniques in investigating multi-vectorial remanences made the use of field tests fundamental tools for paleomagnetists (Irving, 1964; McElhinny, 1973). Field tests to investigate the magnetization stability and the timing and mode of remanent magnetization acquisition include

608

MAGNETIZATION, REMANENT, FOLD TEST

Figure M125 Paleomagnetic fold test.

the (a) fold test, (b) conglomerate test, (c) baked contact test, (d) reversal test, and (e) consistency test. Looking at the many applications developed over the years can easily assess the importance of the field tests in paleomagnetic research. Development of the fold test in paleomagnetism has involved incorporation of statistical criteria to evaluate the significance of the test, advances in separating multicomponent magnetizations, advances in paleomagnetic instrumentation providing more sensitive instrumentation and methods to measure weak magnetizations, and application to a wide range of deformed rock units as well as local and regional tectonic structures.

Statistical criteria Application of statistical criteria in evaluating the test significance was first attempted by McElhinny (1964), by using the concentration parameter k of the Fisher (1953) statistics to distinguish between pre- and postfolding magnetization data sets. The criterion is an application of the statistical test for comparing the precisions using the Fisherian concentration parameter of two separate groups of sample directions with the precision of unfolded directions (ka) compared with the precision for in situ directions (kb). The ratio ka/kb is referred to F-ratio tables at given confidence limits and with equal degrees of freedom (2N  1), where N is the number of sample directions studied. The application of statistical criteria permitted the use of the fold test in a wide range of deformed structures, and McElhinny (1964) fold test is still used in paleomagnetic studies. The test is applied in spite of the problem noticed by McFadden and Jones (1981) that the use of Fisher statistics for directional data sets, which are not Fisher distributed, is not valid. This occurs either way if magnetizations are pre- or postfolding. For instance, if directions are Fisher distributed in in situ coordinates, then they are not tilt-corrected coordinates.

McFadden and Jones (1981) proposed a test based on comparison of Fisher distributed data sets derived from opposite limbs of a fold in in situ and tilt-corrected coordinates. The new test again implies no internal distortion of magnetization and more intensive field sampling, but it can be applied under less stringent conditions than the McElhinny (1964) test. Two further statistical significance tests based on comparison of distributions of magnetization directions, which require less demanding sampling strategies than the test by McFadden and Jones (1981), have been proposed by McFadden (1990) and Bazhenov and Shipunov (1991). The test of Bazhenov and Shipunov (1991) considers different structural attitudes by making division of bedding plane data into groups, which gives greater flexibility in the sampling strategy. Nevertheless, the tests have been little used in paleomagnetic studies mainly because their statistical significance is relatively weak (e.g., Weil and Van der Voo, 2002). Evaluation of fold test data sets has also been examined in terms of a parameter estimation problem, by applying simple bootstrap or Monte Carlo approaches. In these tests, large numbers of sample data are generated to determine confidence limits and significance of the maximum clustered directional distribution (Fisher and Hall, 1990; Tauxe et al., 1991; Watson and Enkin, 1993; Tauxe and Watson, 1994). Tauxe and Watson (1994) developed a test using eigen analysis of directional data sets that estimates the degree to which directions from different structural attitude sites are parallel. In this way, directional data sets do not need to conform to a Fisher distribution, and confidence limits are derived from bootstrap and parametric bootstrap techniques according to data set size. McFadden (1998) discussed the underlying assumption of parametric estimation fold tests that the magnetization directions show maximum clustering with the strata in the position magnetization was acquired. This is not the case when folding does not conform to

MAGNETIZATION, THERMOREMANENT

simple cylindrical horizontal axis geometry and additional factors are involved like plunging folds, unaccounted overprints, etc. Jaime Urrutia-Fucugauchi

Bibliography Bazhenov, M.L., and Shipunov, S.V., 1991. Fold test in paleomagnetism: new approaches and reappraisal of data. Earth and Planetary Science Letters, 104: 16–24. Chan, L.S., 1988. Apparent tectonic rotations, declination anomaly equations, and declination anomaly charts. Journal of Geophysical Research, 93: 12151–12158. Cogné, J.P., and Perroud, H., 1985. Strain removal applied to paleomagnetic directions in an orogenic belt; the Permian red slates of the Alpes Maritimes, France. Earth and Planetary Science Letters, 72: 125–140. Cox, A., and Doell, R.R., 1960. Review of paleomagnetism. Bulletin of the Geological Society of America, 71: 647–768. Facer, R.A., 1983. Folding, Graham’s fold tests in paleomagnetic investigations. Geophysical Journal of the Royal Astronomical Society, 72: 165–171. Fisher, N.I., and Hall, P., 1990. New statistical methods for directional data—I. Bootstrap comparison of mean directions and the fold test in paleomagnetism. Geophysical Journal International, 101: 305–313. Fisher, R.A., 1953. Dispersion on a sphere. Proceedings of the Royal Society of London Series A, 217: 295–305. French, A.N., and Van der Voo, R., 1979. The magnetization of the Rose Hill Formation at the classic site of Graham’s fold test. Journal of Geophysical Research, 48: 7688–7696. Graham, J.W., 1949. The stability and significance of magnetism in sedimentary rocks. Journal of Geophysical Research, 54: 131–167. Irving, E., 1964. Paleomagnetism and Its Application to Geological and Geophysical Problems. New York: Wiley. Kodama, K.P., 1988. Remanence rotation due to rock strain during folding and the stepwise application of the fold test. Journal of Geophysical Research, 93: 3357–3371. MacDonald, W.D., 1980. Net rotation, apparent tectonic rotation, and the structural tilt correction in paleomagnetic studies. Journal of Geophysical Research, 85: 3659–3669. McElhinny, M.W., 1964. Statistical significance of the fold test in palaeomagnetism. Geophysical Journal of the Royal Astronomical Society, 8: 338–340. McElhinny, M.W., 1973. Paleomagnetism and Plate Tectonics. Cambridge Earth Science Series, Cambridge University Press, 358 pp. McFadden, P.L., 1990. A new fold test for paleomagnetic studies. Geophysical Journal International, 103: 163–169. McFadden, P.L., 1998. The fold test as an analytical tool. Geophysical Journal International, 135: 329–338. McFadden, P.L., and Jones, D.L., 1981. The fold test in paleomagnetism. Geophysical Journal of the Royal Astronomical Society, 67: 53–58. Stamatakos, J., and Kodama, K.P., 1991a. Flexural flow folding and the paleomagnetic fold test: an example of strain reorientation of remanence in the Mauch Chunk Formation. Tectonics, 10: 807–819. Stamatakos, J., and Kodama, K.P., 1991b. The effects of grain-scale deformation on the Bloomsburg Formation pole. Journal of Geophysical Research, 96: 17919–17933. Stewart, S., 1995. Paleomagnetic analysis of plunging fold structures: errors and a simple fold test. Earth and Planetary Science Letters, 130: 57–67. Tauxe, L., and Watson, G.S., 1994. The fold test: an eigen analysis approach. Earth and Planetary Science Letters, 122: 331–341. Tauxe, L., Kylstra, N., and Constable, C., 1991. Bootstrap statistics for paleomagnetic data. Journal of Geophysical Research, 96: 11723–11740.

609

Van der Plujim, B.A., 1987. Grain-scale deformation and fold testevaluation of syn-folding remagnetization. Geophysical Research Letters, 14: 155–157. Watson, G.S., and Enkin, R.J., 1993. The fold test in paleomagnetism as a parameter estimation problem. Geophysical Research Letters, 20: 2135–2137. Weil, A.B., and Van der Voo, R., 2002. The evolution of the paleomagnetic fold test as applied to complex geologic situations, illustrated by a case study from northern Spain. Physics and Chemistry of the Earth, 27: 1223–1235.

MAGNETIZATION, THERMOREMANENT Introduction Thermoremanent magnetization (TRM) is acquired when magnetic minerals cool in a weak magnetic field H from above their Curie temperatures. TRM is the most important remanent magnetization used in paleomagnetism. It is almost always close to parallel to the field which produced it, and its intensity is proportional to the strength of the field for weak fields like the Earth’s. The TRM of rocks is therefore a vast storehouse of recorded information about past movements of the Earth’s lithospheric plates and the history of the geomagnetic field. The primary natural remanent magnetization of an igneous rock or a high-grade metamorphic rock is a TRM. Newly erupted seafloor lavas at mid-ocean ridges acquire an intense TRM on cooling below the Curie temperature TC. Rapid cooling also results in fine grain size, which makes the TRM highly stable, so that oceanic basalts are excellent recorders of the paleomagnetic field. However, the TRM is largely replaced within at most a million years of formation by chemical remanent magnetization (CRM) of reduced intensity. The magnetic signal recorded in archeological materials such as pottery, bricks, and the walls and floors of the ovens in which they were fired at high temperature is also a pure TRM residing in single-domain (SD) and pseudosingle-domain (PSD) grains. These materials should be ideal for determinations of the paleointensity of the Earth’s field at the time of firing because they are strongly magnetized and the minerals have been stabilized physically and chemically well above their Curie temperatures before their initial cooling. TRM results from thermally excited changes in magnetization. In the case of SD grains, it is a frozen high-temperature equilibrium distribution between two microstates in which the spins of all atoms are either parallel or antiparallel to an applied field H. At temperatures below TC, there are large perturbations of a crystal’s spin structure, leading to transitions between structures of different types. With cooling, transitions become more difficult, as evidenced by a rapidly increasing relaxation time t. Eventually, decreasing thermal energy and increasing energy barriers EB between states prevent further transitions and TRM is frozen in very abruptly at a blocking temperature TB. The TRM of multidomain (MD) grains results from the blocking of domain walls at positions determined by the externally applied field H, the internal demagnetizing field HD of the grain, and the pinning effect of lattice defects. Theories of TRM and reviews of experimental data have been given by Néel (1949, 1955), Everitt (1961, 1962), Stacey (1958), Schmidt (1973), Day (1977), and Dunlop and Özdemir (1997).

Relaxation time and single-domain TRM Néel’s (1949, 1955) theory of relaxation time deals with noninteracting uniaxial SD grains with saturation magnetization MS and volume V. MS results from the parallel exchange coupling of atomic spins and is a strong function of temperature T near TC, but is weakly dependent on T at low temperatures. Néel’s SD theory makes use of

610

MAGNETIZATION, THERMOREMANENT

Stoner and Wohlfarth’s (1948) results for coherent rotation in spheroidal SD grains aligned with a weak field H. The magnetic moment VMS has a choice of orientations controlled by the easy axes of anisotropy. Uniaxial shape anisotropy is dominant in minerals such as magnetite with high MS. Shape anisotropy produces two energy minima, corresponding to spins in one or the other direction along the longest axis of the grain. These two minima define the two SD microstates. At 0 K, spins would be exchange coupled exactly parallel; but at ordinary temperatures, thermal excitations perturb the spin lattice, resulting in a steady decrease in MS(T ). At high temperatures, reversals of grain moments are also excited as MS(T ) drops rapidly toward zero at TC. At ordinary temperatures, spontaneous reversals of an SD grain are unlikely because the energy barrier between microstates due to shape anisotropy is much larger than the available thermal energy (25kT for experimental times of a few minutes, where k is Boltzmann’s constant). Close to TC, where EB is small, the energy barrier becomes comparable to 25kT and SD moments can reverse with the help of thermal excitations. This thermally excited condition is called superparamagnetism. The transition from a superparamagnetic state to a stable magnetic state is quite sharp and defines the blocking temperature TB. At any time, SD particles are either in state 1, in which moments are aligned with H, or state 2, in which moments are antiparallel to H. The transition probability set by the energy barrier DEB between states leads to a relaxation time for exponential magnetization decay 1=t ¼1=t0 expðEB =kT Þ ¼ 1=t0 exp½ðm0 VMS HK =2kT Þ ð1  H=HK Þ2 

(Eq: 1)

where t0 ¼ 109–1010 s is the atomic reorganization time for transitions between microstates and microcoercivity HK ¼ (Nb  Na)MS for shape anisotropy. Na and Nb are the demagnetizing factors when MS is directed parallel or perpendicular, respectively, to the long axis. The thermally activated transitions between states 1 and 2 cause the average moment of a grain ensemble to relax to a thermal equilibrium value M ðtÞ ¼ M ð0Þ expðt=tÞ þ M ð1Þ½1  expðt=tÞ:

where MRS is the saturation value of TRM. Single-domain MTRM should be relatively intense because tanh(a) saturates rapidly as a ¼ m0VMSBH/kTB increases and TRM then approaches the SD saturation remanence MRS. The high thermal stability of TRM is also accounted for, because MTRM can only be unfrozen and reset to zero by reheating to the unblocking temperature TUB in H ¼ 0. According to Eq. (4), if H  HK, then TUB  TB. TRM can only be demagnetized by reheating to its original high blocking temperature.

Experimental single-domain TRM According to Eq. (5), the intensity of weak-field TRM is proportional to the applied field strength H, since tanh(a) / a for small a. The proportionality between MTRM and H in weak fields (Figure M126) has been verified by measuring TRM acquisition curves in the weak-field region for synthetic oxidized SD titanomagnetites Fe2.3Al0.1Ti0.6O4 over a range of oxidation parameters 0.15 < z < 0.41 (Özdemir and O’Reilly, 1982). Figure M126 also shows a decrease in the intensity of TRM with increasing degree of maghemitization. This fall in intensity is about the same as that reported to take place in submarine basalts with increasing distance from mid-ocean ridges. Although Eq. (5) gives a reasonable match to observed absolute TRM intensity in the weak-field region, the Néel SD theory is less successful in explaining TRM acquisition for higher fields, H 01 Oe or 0.1 mT. The TRM intensity predicted by Eq. (5) reaches saturation at quite low fields. Experimentally, SD TRM usually saturates in much larger fields, on the order of 100 Oe or 10 mT (Figure M127). This discrepancy has been attributed to either angular dispersion of particle axes (Stacey and Banerjee, 1974) or particle interactions. Dunlop and West (1969) modified Eq. (5) by introducing an interaction field distribution that was estimated from experimental Preisach diagrams.

(Eq. 2)

The thermal equilibrium magnetization above and at the blocking temperature TB is Meq ¼ M ð1Þ ¼ MS tanhðm0 VMS H=kT Þ:

(Eq. 3)

The dependence of t on temperature by Eq. (1) is very significant. As T decreases, the energy barriers increase and t grows exponentially, changing from a few seconds to millions of years over a narrow temperature interval. This rapid change leads to the concept of the blocking temperature at which t ¼ t (experimental time). TB ¼ ½m0 VMSB HKB =2k lnðt=t0 Þð1  H=HKB Þ2 ;

(Eq. 4)

where MSB and HKB are the values of MS and HK at the blocking temperature. Equation (1) also predicts a strong dependence of t on grain size. SD grains only slightly above superparamagnetic size have relaxation times greater than the age of the earth, but relaxation times of only a few minutes at TB of a few hundred degrees Celsius. Thus, on cooling below TB in an applied field, all of the orientations of the SD moments remain fixed as their MS increases and TRM is frozen in: MTRM ¼ MRS ðT Þ tanhðm0 VMSB H=kTB Þ;

(Eq. 5)

Figure M126 The acquisition of TRM in weak fields for single domain titanomaghemites with composition Fe2.5Al0.1Ti0.4O4 (ATM/60). TRMs are linear with H and decrease with increasing ¨ zdemir and O’Reilly, 1982). oxidation parameter z (after O

MAGNETIZATION, THERMOREMANENT

611

Figure M127 Experimental TRM acquisition curves for a number of synthetic magnetites with mean grain sizes from 0.04 to 0.22 mm compared to the prediction of Ne´el’s (1949) theory of noninteracting SD grains. The theory cannot explain the rapid initial rise of the experimental curves combined with their gradual approach to saturation (after Dunlop and West, 1969).

TRM intensity varies strongly with grain size d. Eq. (5) predicts an increase in MTRM in proportion to increasing grain volume V. Figure M128 shows the grain-size dependence of TRM intensity for SD hematites. MTRM increases in almost exact proportion to grain size d, not d3. A similar dependence is observed for magnetite grains smaller than the critical SD size d0  0.1 mm, the threshold between SD and nonSD states (Figure M129). Above d0, MTRM decreases with increasing grain size, as d –1 below 1 mm and as d –0.55 above 1 mm. Thus, the Néel SD theory does not explain experimental TRM data for magnetite over most of its natural grain-size range. However, Néel’s SD theory is more relevant to TRM in SD hematites because hematite has a small MS (2 kA m–1) and a large d0. One of the important properties of SD TRM is the equality of the blocking temperature TB during field cooling and the unblocking temperature TUB during zero-field reheating. Experimentally, as in Figure M126, and theoretically according to Eq. (5), the intensity of a weak-field TRM, and the intensities of all the partial TRMs with different TB values that compose it, are proportional to the applied field strength. The equality of TB and TUB for weak fields and the proportionality of TRM and pTRM intensities to field H form the basis for determining paleofield intensity by the Thellier and Thellier (1959) method. Field dependence of blocking temperature has been examined by Sugiura (1980) and Clauter and Schmidt (1981). TB(H) as predicted by Eq. (4) has been verified directly by measuring pTRMs covering the entire blocking-temperature spectrum for SD magnetites. Sugiura (1980) obtained a close match of theoretical and experimental TB spectra by assuming a single value of V and subdividing the HK spectrum into 11 fractions (Figure M130). The blocking-temperature spectrum shifts to lower temperature as H increases. Unblocking temperatures during zero-field heating are therefore always higher than blocking temperatures observed during in-field cooling (Dunlop and West, 1969). Equation (4) predicts that blocking and unblocking temperatures vary with grain volume V. Experimentally determined blocking temperatures (Dunlop, 1973) agree well with the theoretical blocking temperatures calculated with a 3D micromagnetic model (Winklhofer et al., 1997). Small SD grains have lower blocking and unblocking

Figure M128 Grain-size dependence of TRM intensity for the single-domain hematites. The dashed line has slope 1.04 showing that MTRM is very nearly proportional to d. Open circles: experimental data points; crosses: theoretical TRM values from ¨ zdemir and Dunlop, 2002). Ne´el (1949) SD theory (after O temperatures than large SD grains. The predicted quadratic dependence of TB on the applied field is confirmed experimentally. The dependence of TRM intensity on the rate of cooling is of practical interest to paleomagnetists studying slowly cooled orogens.

612

MAGNETIZATION, THERMOREMANENT

Figure M129 Weak-field TRM as a function of magnetite grain size d. MTRM decreases approximately as d –1 between 0.1 and 1 mm and less strongly as d–0.55 above 1 mm (after Dunlop and Argyle, 1997).

Partial TRM and paleofield intensity determination

Figure M130 Continuous cumulative TRM spectra measured during cooling of single domain magnetite (points and solid curves) compared with theoretical stepwise spectra for H ¼ 2, 100, and 400 Oe (after Sugiura, 1980). Dodson and McClelland-Brown (1980) and Halgedahl et al. (1980) predicted that the intensity of TRM in SD grains should increase for longer cooling times, possibly by as much as 40% between laboratory and geological settings. Experiments by Fox and Aitken (1980) for baked clay samples containing SD magnetites showed that there was about a 7% decrease in the intensity of TRM when the cooling time changed from 2.5 h to 3 min.

According to Eq. (4), TB depends on grain volume V and microscopic coercivity HK. Any real rock has a distribution f (V, HK) of both V and HK, and as a result will have a spectrum of blocking temperatures. Therefore, the total TRM of a rock will consist of a spectrum of partial TRMs, each carried by grains with similar (V, HK). When a rock cools from T1 to T2 in a field H, SD ensembles with T2  TB  T1 pass from the unblocked to the blocked condition, acquiring a partial TRM, MPTRM (T1, T2, H). Thellier (1938) showed that SD partial TRMs follow experimental laws of additivity, reciprocity, and independence. Independence: Partial TRMs acquired in different temperature intervals are mutually independent in direction and intensity. Each partial TRM disappears over its own blocking-temperature interval. Additivity: Partial TRMs produced by the same H have intensities that are additive. This is expected theoretically because the blocking-temperature spectrum can be decomposed into nonoverlapping fractions, each associated with one of the partial TRMs. The total TRM is the sum of partial TRMs covering the entire blocking-temperature interval from TC to room temperature because each partial TRM contains a unique part of the TB spectrum of the total TRM. Reciprocity: The partial TRM acquired between T1 and T2 during cooling in H is thermally demagnetized over the interval (T2, T1) when heated in zero field. In other words, the blocking and unblocking temperatures are identical for weak fields. Since partial TRM acquired at TB during cooling is erased at TB during heating, one can replace the natural remanent magnetization (NRM) MNRM of a rock in a stepwise fashion with a laboratory TRM. This is done in practice by a series of double heatings, the first in zero field and the second in a known laboratory field HLAB. If the NRM of a rock was acquired as a TRM in an ancient geomagnetic field HA, MNRM should be proportional to HA. The intensity of the ancient geomagnetic field is then given by HA ¼ MNRM HLAB =MTRM ;

613

MAGNETIZATION, THERMOREMANENT

each pair of heating steps giving an independent estimate of HA. This is the Thellier and Thellier (1959) method of paleointensity determination as modified by Coe (1967).

Multidomain TRM In large grains containing many domain walls, TRM acquisition results from several processes: domain wall pinning, domain nucleation or denucleation, and nucleation failure. A multidomain grain contains crystal imperfections such as inclusions, voids, and line defects such as dislocations, all of which create surrounding stress fields. These defects tend to pin domain walls. Voids and inclusions reduce the volume of a wall and create magnetic poles, affecting both the wall energy and demagnetizing energy. Dislocations pin walls by the magnetoelastic interaction between their stress fields and the spins in the wall, which are rotated out of easy axes (Özdemir and Dunlop, 1997). Pinned domain walls can be thermally activated just as SD moments can. Wall displacements in response to a weak field applied at high temperatures, like reversals of SD grains, can be frozen in by cooling to room temperature, resulting in TRM. The Néel (1955), Everitt (1962), and Schmidt (1973) multidomain TRM theories consider the wall-defect interaction and the field dependence of the blocking temperature. These theories follow Néel (1949) SD theory fairly closely except that V becomes the volume of one Barkhausen jump of a domain wall and HK becomes the critical field for such a Barkhausen jump. McClelland and Sugiura (1987) observed that during cooling below the blocking range, after H had been zeroed, multidomain partial TRM did not remain constant but decreased in real terms, i.e., M(T)/ MS(T). This observation compromises the concept of the blocking temperature, because evidently, magnetization can partially relax during cooling below TB. Later Shcherbakov et al. (1993) developed a kinetic model that invokes nucleation or denucleation of domain walls as a way of explaining changes in M(T ) below TB. The only direct observations of domain nucleation during zero-field warming or cooling are by Heider et al. (1988) and Ambatiello et al. (1999) for magnetite and by Metcalf and Fuller (1987) and Halgedahl and Fuller (1983) for titanomagnetite. In each case, any changes in the number or widths of domains occurred either close to TC or in a range of T where anisotropy changes rapidly, just above room temperature in magnetite. In intermediate T ranges, where most observations of partial TRM relaxation have been made, there is no evidence for nucleation. A related phenomenon is nucleation failure. Titanomagnetite grains field-cooled from TC sometimes nucleate fewer than the equilibrium number of domain walls, and occasionally none at all (Halgedahl and Fuller, 1980). The latter structure is of particular interest because single-domain TRM is so intense that a small fraction of such grains in metastable SD states could account for multidomain TRM. In particular, the size dependence of TRM in grains larger than SD size is then explicable. There are fundamental differences between SD and most MD TRM models. The internal demagnetizing field HD ¼ –NM favors a configuration of domain walls with minimum net moment. Loosely pinned domain walls may undergo a series of Barkhausen jumps during heating and cooling. Therefore, there is no single blocking temperature for a given domain wall or for a grain containing many walls since a jump by one wall alters the internal field at the location of every other wall. This is an alternative way of explaining TRM relaxation below TB.

Ne´el and Schmidt two-domain TRM theories Néel’s (1955) 2D theory is based on the temperature dependence of the wall-defect interaction and the wall displacement driven by the internal demagnetizing field HD ¼ N M where M is the local magnetization vector. A domain wall, which has been displaced by H, will reequilibrate its position at each new temperature during cooling under the influences of HD and of local potential wells created by interaction

Figure M131 Total energy, EW þ ED þ EH, as a function of wall displacement x for various fields (after Schmidt, 1973). Downward arrows show successive local energy minima in which the wall is trapped as H increases from 0 to 50 Oe. Upward arrows indicate minima where the wall is pinned as H decreases from 50 to 0 Oe. The wall occupies different minima in increasing and decreasing fields of the same strength, giving rise to magnetic hysteresis and remanence.

between the domain walls and lattice defects (Dunlop and Özdemir, 1997). The pinning strength is measured by the microcoercivity HC and determined by the barrier height between adjacent local energy minimums (LEMs). Figure M131 shows the total energy in a twodomain grain with many identical wall energy (EW) barriers, each with the same coercivity HC for positive and negative jumps. The central position of the wall is favored by the parabolic energy well due to the demagnetizing energy ED. As the applied H increases, the parabolas are tilted to the right by the field energy EH and the wall jumps from one LEM to another in a series of Barkhausen jumps. Repeated jumps of the wall generate the ascending part of the hysteresis loop, which has a slope 1/N. The barriers to wall motion are not symmetric in increasing and decreasing H. As H decreases, the wall is pinned in a different set of LEMs and generates a descending loop with the same slope 1/N. When H becomes zero, the grain is left with a displaced wall and a net moment, which is an isothermal remanent magnetization. As the temperature decreases, the demagnetizing field and the barriers between wells grow. At the blocking temperature TB, the potential barriers EW begin to grow more rapidly than the demagnetizing field HD pushing the wall back toward the demagnetized state and the wall is trapped. TRM is frozen in. The basic result of Néel’s 2D theory is that 1=2

MTRM ¼ 2H 1=2 HC =N :

(Eq. 6)

Equation (6) predicts absolute TRM intensity at room temperature. The TRM is blocked by the growth of energy barriers to wall motion

614

MAGNETIZATION, THERMOREMANENT

and thermal fluctuations are ignored. This is referred to as fieldblocked TRM. Néel (1955) pointed out that field blocking of TRM occurs very close to TC if H is small. At such high temperatures, thermal fluctuations can unpin a domain wall below its blocking temperature TB. Néel introduced a thermal fluctuation field HF such that a wall cannot be blocked until HC ðTB Þ ¼ HF ;

observed properties of multidomain partial TRMs and to justify the kinematic equation for partial TRM relaxation during cooling or heating (Shcherbakov et al., 1993). Although not a theory of TRM acquisition in the fundamental sense, the Fabian model is useful in describing and connecting a body of experimental data and making predictions about the TRM properties of a particular sample based on its measured w(TB, TUB).

(Eq. 7)

Experimental multidomain TRM where HC(TB) is the thermal fluctuation blocking temperature. TRM intensity in the presence of thermal fluctuations is given by 1=2

1=2

MTRM ¼ HF HC H=N :

(Eq. 8)

Thermally blocked TRM should be proportional to H as observed experimentally for small fields. Schmidt’s (1973) MD model is based on the temperature variation of total energy in a two-domain grain with a single wall (Figure M131). The model has the same physical basis as Néel’s 2D theory, but the coercivity appears only as a derived quantity. The total energy has contributions due to the interactions between the domain wall and the applied field, the demagnetizing field, and lattice defects. The temperature dependence of wall-defect interaction is contained in a term mp, where m is the reduced domain magnetization m(T) ¼ M/MS(T). The index p takes a value between 2 and 10 depending on the relative importance of magnetostriction or magnetocrystalline anisotropy in the wall pinning. The basic results of the Schmidt model are that HC ðT Þ / mðT Þp1 ;

(Eq. 9)

MTRM / H 11=ðp1Þ ;

(Eq. 10)

1=ðp1Þ

MTRM / HC

:

There are relatively few testable predictions of multidomain TRM theories. The prediction that TRM intensity should increase with increasing HC (Eqs. (6), (8), (11)) was verified in a general way in Néel’s original paper and the test has not been improved upon since. The predicted dependence on applied field H has received more attention. Tucker and O’Reilly (1980) showed that in the low-field region (H  1 kA m–1, where 80 A m–1  1 Oe ¼ 0.1 mT), the TRM of large titanomagnetite crystals, as predicted by Eq. (8), was more or less proportional to H (Figure M132). For intermediate field strengths, the field dependence was as a power of H < 1 (Eqs. (6) or (10)), and saturation occurred around 10 kA m–1. Similar results for magnetite and magnetite-bearing rocks were reported by Dunlop and Waddington (1975) and for pyrrhotite by Menyeh and O’Reilly (1998).

(Eq. 11)

These are identical to Néel’s predictions (Eq. (6)) when p ¼ 3.

Stacey multidomain TRM theory Stacey (1958) considered the blocking temperature TB to be independent of the applied field and neglected the interaction between the domain wall and lattice defects. At high temperatures, TB < T < TC, the walls are highly mobile and take up positions in which the internal field HI ¼ H  NM is zero. Above TB, the magnetization is given by M ¼ H/N. During cooling below TB, the walls’ displacements are frozen, so that M changes only by the reversible increase in MS(T ). At room temperature, the TRM is given by MTRM ¼ H MS =½NMSB ð1 þ N w0 Þ;

(Eq. 12)

where w0 is the initial volume susceptibility. Equation (12) seemingly predicts the linear dependence of TRM on the applied field observed experimentally for weak fields. In reality, as Néel (1955) showed, blocking temperature TB is itself dependent on H. TRM as given by Eq. (12) actually varies approximately as H1/2, a dependence observed experimentally only when H > 1 mT.

Fabian phenomenological TRM model Fabian (2000) has proposed a phenomenological approach in which a sample is described in terms of a distribution function w(TB, TUB). This formulation establishes no direct link between (TB, TUB) and physical properties like grain size and coercivity, but recognizes that for MD grains TB 6¼ TUB in general, i.e., Thellier reciprocity does not hold. With this simple approach, it is possible to explain many

Figure M132 TRM acquisition curves for millimeter-size single crystals of titanomagnetite with composition Fe2.4Ti0.6O4 (TM60), plotted bilogarithmically (after Tucker and O’Reilly, 1980). The experimental data have been fitted by a series of linear segments corresponding to power law dependences of TRM on applied field. TRM intensity is proportional to H for fields below about 2 mT, then increases approximately as H1/2 until it saturates at the value MRS ¼ H/N, in agreement with Ne´el’s (1955) theory of TRM due to domain wall displacement limited by self-demagnetization.

MAGNETIZATION, THERMOREMANENT

615

An interesting sidelight on multidomain TRM behavior is the fact that for H of the order of the Earth’s magnetic field, weakly magnetic hematite (MS  2 kA/m) in a multidomain state has a much stronger TRM than MD magnetite (MS ¼ 480 kA m–1) or SD hematite (Figure M133). The reason is that the weak MS of hematite results in a negligible demagnetizing field HD, so that even a small H is sufficient to drive walls to their limiting positions and saturate the TRM (Dunlop and Kletetschka, 2001). MD magnetite does not approach saturation TRM except in fields 3 orders of magnitude larger because of its correspondingly greater HD. The model of Fabian (2000) was a response to recent experimental work on multidomain partial TRMs. The thermal demagnetization data of Figure M134 are one example. Partial TRMs produced by applying H over a narrow blocking interval TB ¼ 370–350
C are demagnetized over increasingly broad intervals of unblocking temperature TUB as the grain size increases. While partial TRMs of 1 and 6 mm magnetites demagnetized mainly over the original TB interval, partial TRMs of 110 and 135 mm grains demagnetized almost entirely outside the 370
C–350
C interval. This almost total violation of reciprocity led Fabian (2000) to treat TB and TUB as independent variables. Özden Özdemir

Bibliography

Figure M133 Comparison of experimental (points and solid curves) and theoretical (dashed curves) TRM values for SD and MD hematites and magnetites (Dunlop and Kletetschka, 2001). The contrast in TRM intensities for two minerals is due to the internal demagnetizing field HD ¼ NM. At saturation, HD  200 mT for magnetite, but for hematite HD  1 mT, making it much easier for a small field like the Earth’s to push the magnetic domain walls to their limiting positions.

Figure M134 Stepwise thermal demagnetization of partial TRMs produced by a small field applied during cooling from 370 to 350
C in magnetites ranging from small PSD to MD size. About 50%–90% of the remanence unblocks at temperatures below or above the PTRM blocking temperature range. Tailing of thermal demagnetization curves is not confined to large MD grains but is significant even for fairly small PSD grains (after Dunlop and ¨ zdemir, 2001). O

Ambatiello, A., Fabian, K., and Hoffmann, V., 1999. Magnetic domain structure of multidomain magnetite as a function of temperature: observations by Kerr microscopy. Physics of the Earth and Planetary Interiors, 112: 55–80. Coe, R.S., 1967. The determination of paleointensities of the earth’s magnetic field with emphasis on mechanisms which could cause non-ideal behaviour in Thellier’s method. Journal of Geomagnetism and Geoelectricity, 19: 157–179. Clauter, D.A., and Schmidt, V.A., 1981. Shifts in blocking temperature spectra for magnetite powders as a function of grain size and applied magnetic field. Physics of the Earth and Planetary Interiors, 26: 81–92. Day, D., 1977. TRM and its variation with grain size. Journal of Geomagnetism and Geoelectricity, 29: 233–265. Dodson, M.H., and McClelland-Brown, E., 1980. Magnetic blocking temperatures of single-domain grains during slow cooling. Journal of Geophysical Research, 85: 2625–2637. Dunlop, D.J., 1973. Thermoremanent magnetization in submicroscopic magnetite. Journal of Geophysical Research, 78: 7602–7613. Dunlop, D.J., and Argyle, K.S., 1997. Thermoremanence, anhysteretic remanence and susceptibility of submicron magnetites: nonlinear field dependence and variation with grain size. Journal of Geophysical Research, 102: 20199–20210. Dunlop, D.J., and Kletetschka, G., 2001. Multidomain hematite: a source of planetary magnetic anomalies? Geophysical Research Letters, 28: 3345–3348. Dunlop, D.J., and Özdemir, Ö., 1997. Rock Magnetism: Fundamentals and Frontiers. Cambridge and New York: Cambridge University Press. Dunlop, D.J., and Özdemir, Ö., 2001. Beyond Néel theories: thermal demagnetization of narrow-band partial thermoremanent magnetizations. Physics of the Earth and Planetary Interiors, 126: 43–57. Dunlop, D.J., and Waddington, E.D., 1975. The field dependence of thermoremanent magnetization in igneous rocks. Earth and Planetary Science Letters, 25: 11–25. Dunlop, D.J., and West, G.F., 1969. An experimental evaluation of single domain theories. Reviews of Geophysics, 7: 709–757. Everitt, C.W.F., 1961. Thermoremanent magnetization. I. Experiments on single domain grains. Philosophical Magazine, 6: 713–726. Everitt, C.W.F., 1962. Thermoremanent magnetization. III. Theory of multidomain grains. Philosophical Magazine, 7: 599–616.

616

MAGNETIZATION, THERMOREMANENT, IN MINERALS

Fabian, K., 2000. Acquisition of thermoremanent magnetization in weak magnetic fields. Geophysical Journal International, 142: 478–486. Fox, J.M.W., and Aitken, M.J., 1980. Cooling-rate dependence of thermoremanent magnetisation. Nature, 283: 462–463. Halgedahl, S.L., and Fuller, M., 1980. Magnetic domain observations of nucleation processes in fine particles of intermediate titanomagnetite. Nature, 288: 70–72. Halgedahl, S.L., and Fuller, M., 1983. The dependence of magnetic domain structure upon magnetization state with emphasis on nucleation as a mechanisms for pseudo-single-domain behaviour. Journal of Geophysical Research, 88: 6506–6522. Halgedahl, S.L., Day, R., and Fuller, M., 1980. The effect of cooling rate on the intensity of weak-field TRM in single domain magnetite. Journal of Geophysical Research, 85: 3690–3698. Heider, F., Halgedahl, S.L., and Dunlop, D.J., 1988. Temperature dependence of magnetic domains in magnetite crystals. Geophysical Research Letters, 15: 499–502. McClelland, E., and Sugiura, N., 1987. A kinematic model of TRM acquisition in multidomain magnetite. Physics of the Earth and Planetary Interiors, 46: 9–23. Menyeh, A., and O’Reilly, W., 1998. Thermoremanence in monoclinic pyrrhotite particles containing few domains. Geophysical Research Letters, 25(18): 3461–3464. Metcalf, M., and Fuller, M., 1987. Domain observations of titanomagnetites during hysteresis at elevated temperatures and thermal cycling. Physics of the Earth and Planetary Interiors, 46: 120–126. Néel, L., 1949. Théorie du traînage magnétique des ferromagnétiques en grains fins avec applications aux terres cuites. Annales de Geophysique, 5: 99–136. Néel, L., 1955. Some theoretical aspects of rock magnetism. Advances in Physics, 4: 191–243. Özdemir, Ö., and Dunlop, D.J., 1997. Effect of crystal defects and internal stress on the domain structure and magnetic properties of magnetite. Journal of Geophysical Research, 102: 20211–20224. Özdemir, Ö., and Dunlop, D.J., 2002. Thermoremanence and stable memory of single-domain hematite. Geophysical Research Letters, 29:doi:10.1029/2002GL015597. Özdemir, Ö., and O’Reilly, W., 1982. An experimental study of thermoremanent magnetization acquired by synthetic monodomain titanomaghemites. Journal of Geomagnetism and Geoelectricity, 34: 467–478. Schmidt, V.A., 1973. A multidomain model of thermoremanence. Earth and Planetary Science Letters, 20: 440–446. Shcherbakov, V.P., McClelland, E., and Shcherbakova, V.V., 1993. A model of multidomain thermoremanent magnetization incorporating temperature-variable domain structure. Journal of Geophysical Research, 98: 6201–6216. Stacey, F., 1958. Thermoremanent magnetization (TRM) of multidomain grains in igneous rocks. Philosophical Magazine, 3: 1391–1401. Stacey, F., and Banerjee, S.K., 1974. The Physical Principles of Rock Magnetism. Elsevier, Amsterdam. Stoner, E.C., and Wohlfarth, E.P., 1948. A mechanism of magnetic hysteresis in heterogeneous alloys. Philosophical Transactions of the Royal Society of London, A240: 599–642. Sugiura, N., 1980. Field dependence of blocking temperature of single-domain magnetite. Earth and Planetary Science Letters, 46: 438–442. Thellier, E., 1938. Sur l’aimantation des terres cuites et ses applications géophysiques. Annales de. l’Institut de Physique du Globe, Université de Paris, 16: 157–302. Thellier, E., and Thellier, O., 1959. Sur l’intensité du champ magnétique terrestre dans le passé historique et géologique. Annales de Géophysique, 15: 285–376.

Tucker, P., and O’Reilly, W., 1980. The acquisition of thermoremanent magnetization by multidomain single-crystal titanomagnetite. Geophysical Journal of the Royal Astronomical Society, 60: 21–36. Winklhofer, M., Fabian, K., and Heider, F., 1997. Magnetic blocking temperatures of magnetite calculated with a three-dimensional micromagnetic model. Journal of Geophysical Research, 102: 22695–22709.

Cross-references Archeomagnetism Magnetic Anisotropy, Sedimentary Rocks and Strain Alteration Magnetic Domain Magnetic Susceptibility Magnetization, Chemical Remanent (CRM) Magnetization, Isothermal Remanent (IRM) Magnetization, Natural Remanent (NRM) Magnetization, Viscous Remanent (VRM)

MAGNETIZATION, THERMOREMANENT, IN MINERALS Thermoremanent magnetization Perhaps the best understood of the primary magnetizations, of natural rocks and specimens, is thermal remanent magnetization (TRM). Most of the natural rocks are magnetized primarily by the geomagnetic field (30000 nT) and acquire natural remanent magnetization (NRM). Magnetic minerals acquire TRM when they are contained within the rock that is cooled in an external magnetic field from temperatures above the minerals’ blocking temperatures. Blocking of remanent magnetization at a specific temperature results in locking of a specific direction and intensity of magnetization as it becomes stable on the timescale of the TRM acquisition. The generally recognized first-order theory of TRM can be applied only to small uniformly magnetized grains (Néel, 1949) and it provides a reasonable explanation for the intensity of TRM vs inducing field. The theory explains the changes of stability of TRM with temperature and with the inducing field, explaining how a rock can maintain a TRM record for billions of years. This theory can also provide an explanation on how the secondary component of magnetization of a rock can be removed from the primary one. However, besides being useful only for small single-domain (SD) grains it is also less successful in explaining TRM acquisition for more intense ambient fields (1 mT). According to Neel’s theory, the TRM reaches saturation, in lower fields than actually measured on specimens (10 mT). This effect was assigned to particle interaction and long axis dispersions (Stacey and Banerjee, 1974). In general, the TRM is not carried only by a small single-domain (SD) fraction of magnetic mineral grains. When the volume of the grain is larger (>0.5 mm for magnetite mineral) the demagnetizing field (caused by magnetic sources distributed on the surface of the grain) has a slightly different geometry than the pattern of the uniform magnetization (Figure M135). This geometry causes inhomogeneity of magnetization of the larger grains that is replaced by more energetically favorable state containing domain walls bounding volume with reversed magnetic moments. Neel first attempted to construct the theory for multidomain materials (Néel, 1955). Stacey (1963) and Everitt (1962) applied a concept of Barkhausen discontinuities, small jumps of the domain walls into the new position, believed to be due to crystal imperfections. The resistance of the domain wall motion was thought to be responsible for remanent magnetism in MD grains. As the smaller MD grains were

MAGNETIZATION, THERMOREMANENT, IN MINERALS

617

Figure M135 Intensities of the ambient magnetic field B (tesla) against thermoremanent efficiencies e of hematite (Fe2O3), magnetite (Fe3O4), iron-nickel alloy (FeNi) and iron (Fe). The pyrrhotite data (Fe7S8) are from Dekkers (1989). Straight lines are drawn according to B ¼ aJe, where a is a dimensionless constant equal to 0.0046 (see Figure M137a) at 300 K, and J ¼ m0 Js where m0 is permeability of vacuum and Js is saturation magnetization at 300 K. Magnetization efficiency e is defined as e ¼ Mtr =Jsr , the ratio of thermoremanence to saturation remanence. Single-domain (SD) and acicular (elongated crystal parallel to the applied field) magnetite data are redrawn from Dunlop and Argyle (1997) and Dunlop and West (1969), respectively.

more stable than larger MD grains, Stacey invented the term pseudosingle-domain (PSD) grains for grains slightly larger than SD grains (Stacey, 1963). Because, in general, small grains of magnetite have 2–3 orders of magnitude larger TRM intensity than larger MD grains. For some time it was thought that the TRM of MD grains is negligible (Hargraves and Young, 1969; Hoye and Evans, 1975). More small grains are present in rocks than apparently visible, resulting from the formation of the iron oxides through oxidation inside the silicates. The intensity of the remanent magnetization acquired by rocks is determined by an unknown strength of the ambient magnetic field, an unknown magnetic mineral composition, and an unknown temperature history of the sample. Stacey pointed out in his theory of multidomain TRM (Stacey, 1958) that because the demagnetizing energy falls off more slowly with temperature than any other, the condition under which TRM is first acquired is simply the minimization of the internal field. This guarantees that at least at this temperature the TRM is related only to the magnetostatic energy and the demagnetizing energy. Néel’s theory of MD TRM is incomplete since it fails to describe many aspects of pTRM (partial TRM) behavior (Néel, 1955; Shcherbakova et al., 2000). There the blocking occurs at temperature Tb when magnetic coercivity increases high enough to pin domain walls against the demagnetizing field. For TRM of SD grains at room temperature, Mtr(Tr), magnetic remanence was frozen in high temperature equilibrium distribution achieved by thermally excited transitions among the different magnetic states. Transitions cease below the Tb, because in the course of cooling, the energy barriers between different magnetization states grow larger than the available thermal energy. For both SD and MD states the resulting magnetization, composed of many magnetic moments, is in the direction of and for mineral specific field range (Kletetschka et al., 2004) proportional to the applied magnetic field B. Efficiency e(Tr) of Mtr(Tr) of SD grains of saturation remanence Jsr(Tr), volume V, and saturation magnetization Js(Tb) is (Néel, 1949):

EðTr Þ ¼


 Mtr ðTr Þ mVJs ðTb ÞB ¼ tanh ; Jsr ðTr Þ kTb

(Eq. 1)

with m0 ¼ 4p  10–7, and k ¼ 1.38  10–23 J K–1. On the timescale 50–100 s:
 m0 VJs ðTb ÞBc ðTb Þ ¼ 2 lnð f0 tÞ  50; kTb where Bc(Tb) is a critical field for moment rotations in the absence of thermal energy (microcoercivity), and frequency of moment fluctuation f0  109 s1. Therefore, from Eq. (1) one can derive for small fields (E  1): EðTr Þ ¼

50B Bc ðTb Þ

(Eq. 2)

In most fine-grained magnetic material, a typical efficiency E(Tr) of thermoremanent magnetization Mtr(Tr) acquired in the geomagnetic field is about 1% (Wasilewski, 1977, 1981; Cisowski and Fuller, 1986; Kletetschka et al., 2000a). This small efficiency is consistent with the Mtr(Tr) acquisition curves for magnetite (Dunlop and Waddington, 1975; Tucker and O’Reilly, 1980; Özdemir and O’Reilly, 1982) with grain sizes covering the range from the single-domain (SD) to multidomain (MD) magnetic states. However, Mtr(Tr) experiments with hematite (Kletetschka et al., 2000b,c; Dunlop and Kletetschka, 2001; Kletetschka et al., 2002) showed e >10%.

Experimental TRM constrains for theory The following part is largely reiteration of the results from Kletetschka et al. (2004). There in an attempt to reconcile the contrast between TRM acquisition of hematite and magnetite series of magnetic Mtr

618

MAGNETIZATION, THERMOREMANENT, IN MINERALS

acquisitions performed using distinct magnetic materials (Kletetschka et al., 2004): iron (Fe), iron-nickel (FeNi), magnetite (Fe3O4), hematite (a-Fe2O3), and resistance wires MWS-294R and ALLOY52. Figure M135 shows the field B required to reach efficiency E(Tr) of the Mtr(Tr) acquisition for equidimensional samples and a literature sample of acicular magnetite (Dunlop and Argyle, 1997) in which the crystals are highly elongated parallel to the applied field. Data near and at saturation, where the simple power law breaks down (see Figure M136), were excluded from the data set. The data set includes literature data for pyrrhotite (Dekkers, 1989). Remarkably, each mineral is restricted to its own line with the unit slope in the logB-logE space. Another important feature reported in Kletetschka et al. (2004) is that the larger the Js(Tr) (see legend) the larger the field B required to achieve a predefined efficiency level E. An increase of the minerals’ Js(Tr) is equivalent to a similar increase in opposing demagnetizing field Hd(Tr) (Dunlop and Özdemir, 1997; Kletetschka et al., 2000b, c; Dunlop and Kletetschka, 2001) as well as critical fields B requiring material to reach the saturation magnetization at Tb. The demagnetizing field at saturation relates to Bc(Tb), above which the energy minima become unstable causing the magnetic moment to irreversibly rotate in an absence of thermal fluctuations. Low Js(Tr) value is associated with low value of both Bc(Tb) and Hd and leads to a large critical SD size and large magnetic domain wall spacing while large Js(Tr) implies large Bc(Tb) (or Hd) causing a fine scale of the individual domains. Other effects, like magnetostriction, anisotropy, and exchange constants, may also cause changes in the overall magnetic domain size. Data in Figure M135 are representative of multidomain (MD) magnetic materials (1 mm grain size). However, the Mtr(Tr) varies with the grain size according to the domain type (Kletetschka et al., 2004). For example, MD magnetite has Mtr(Tr) that increases with decreasing grain size (Dunlop, 1990). Similar grain size dependence has been observed at saturation for Jsr(Tr) (Dunlop, 1990) (E ¼ 1). Thus, efficiency E(Tr) of Mtr(Tr) rather than just Mtr(Tr) reduces the grain size dependence to a minimum, and a line separation in the logB-logE plot can be used to identify the magnetic mineralogy in ideal circumstances.

The insensitivity of the logB-logE(Tr) plot to various grain sizes is illustrated in Figure M135, where the Mtr(Tr) efficiency for SD magnetite (literature data; Dunlop and Argyle, 1997) correlates with MD magnetites; this breaks down when mineral size becomes so small that it is near or in the superparamagnetic size range (Dunlop and Argyle, 1997). The grain size independence is resolved in Figure M136 where we have literature data of various Mtr(Tr) acquisitions of titanomagnetite with disparate magnetic domain states identified by the specific grain sizes (Dunlop and Waddington, 1975; Tucker and O’Reilly, 1980; Özdemir and O’Reilly, 1982). Despite much stronger Mtr(Tr) of fine vs large magnetic grains, all sizes appear to have identical acquisitions when normalized by saturation remanence Jsr(Tr). When neglecting effects near saturation (E < 0.3), this linear dependence predicts approximate maximum values of the magnetic field that can be recorded by a specific material (Figure M135). Near the saturation, the magnetization is not linear with the applied field (Figure M136) owing to Eq. (1) when E  1. Thus, the magnetic fields at which E  1 should be the magnetic fields that define the values of intrinsic Bc(Tb) for the specific mineral. Knowledge of Bc(Tb) fields in principle can be used for dating of the magnetization according to known magnetization viscous decay curves (Heller and Markert, 1973; Borradaile, 1996). Bc 1 Ms for shape anisotropy, Bc / l Msn for magnetoelastic anisotropy (Syono and Ishikawa, 1963b; Moskowitz, 1993) (for n > 2) and Bc1K Msn for crystalline anisotropy (Syono and Ishikawa, 1963a; Fletcher and O’Reilly, 1974) (for n > 8) where n is the experimentally determined exponent. Both K and l go to zero much faster than Ms when approaching Curie temperature Tc. Energy minimum, related to magnetic ordering, becomes shallower when approaching Tb due to thermal fluctuations (Stacey, 1958). Thus, for the purpose of magnetic remanence blocking near Tc we may consider only the shape anisotropy: Bk 1 Ms. The distribution of demagnetization field vectors (tensors in general; Dunlop and Özdemir, 1997) relates to Bc(Tb), the nature of the resulting Mtr(Tr) and the E (Tr) dependencies. The Bc(Tb) fields are small in minerals with low Js(Tr) causing them to reach saturation (E(Tr) ¼ 1) in much lower applied fields B

Figure M136 Acquisition fields are plotted against thermoremanent efficiency for contrasting domain states of titanomagnetite. Data for multidomain and pseudosingle-domain mineral are from Tucker and O’Reilly (1980). Data for single-domain minerals are from ¨ zdemir and O’Reilly (1982). O

MAGNETIZATION, THERMOREMANENT, IN MINERALS

619

Figure M137 Magnetic acquisition fields are plotted against JE, which is the saturation magnetization J ¼ m0Js multiplied by efficiency E of various materials at 300 K. (a) Equidimensional grains of iron, iron-nickel, magnetite, pyrrhotite, and hematite define a straight line that is a result of a linear fit to all of the data. For Js at 300 K this fit has the form of B ¼ (4.6 0.3)  10–3JE. The linear regression coefficient is R ¼ 0.97. (b) Effect of shape (nonequidimensional crystals) on thermoremanent acquisition fields for wire materials (MWS-294R and ALLOY52) with small (1/5) and large (1/14) predefined length to diameter ratios compared with the predicted acquisition (solid line) for equidimensional materials. Measurements were made with wires aligned parallel to the applied field.

(Kletetschka et al., 2000c; Dunlop and Kletetschka, 2001). Larger Bc(Tb) in minerals with large Js(Tr) creates larger resistance against acquisition of Mtr(Tr) and requires larger magnetizing fields to achieve the saturation (E(Tr) ¼ 1). For example, because magnetoelastic and crystalline constants go to zero much faster than Ms close to TC, hematite has low Bc(Tb) (due to shape anisotropy) at the point at which TRM is acquired in contrast to its high Bc at room temperature caused by high magnetoelastic anisotropy. The fundamental role of Js(Tr) in mineral specific Mtr(Tr) acquisition can be crystal clear by taking the data from Figure M135, multiplying the magnetic efficiency E(Tr) by J(Tr) ¼ m0Js(Tr) (Figure M137a). Remarkably the resulting data set completely eliminates the effect of the demagnetizing field during the Mtr(Tr) acquisition. Figure M137a

suggests that mineral Mtr(Tr) acquisitions can be in general approximated (linear regression coefficient R ¼ 0.97) by following a linear fit: B ¼ aðT ÞJ ðT ÞEðTr Þ;

(Eq. 3)

where a ¼ (4.6 0.3)  103 with 95% confidence level for T ¼ Tr ¼ 300 K. The product Js(Tr)E(Tr) is essentially the Mtr(Tr) normalized by the squareness ratio Jsr(Tr)/Js(Tr) of the hysteresis loop. This linear behavior (Figure M137a) indicates that all magnetic minerals should contribute to a planetary thermoremanent magnetic anomaly (e.g., intense magnetic anomalies detected on Mars (Acuña et al., 1999) with the same, squareness Jsr(Tr)/Js(Tr)-normalized, Mtr(Tr) intensity. Because Js(Tr)

620

MAGNETIZATION, THERMOREMANENT, IN MINERALS

eliminates the mineral dependence observed in Figure M135 produced byvariation in Bc(Tb) in different minerals, Bc(Tb) ¼ 0.23Js(Tr) in Eq. (2) leads to the empirically observed relationship equation (3). Equation (3) breaks down if the Curie temperature of the magnetic material gets near or below 300 K. With decreasing temperature, Js increases and reaches a maximum at absolute zero temperature unless the material undergoes a phase transition (e.g. Verwey transition for magnetite). Extending the trend of the published Js(T) curves (Dunlop and Özdemir, 1997) into 0 K (ignoring any phase transitions) results in increase of Js by a factor of 1.05 for iron, 1.00 for hematite, 1.10 for magnetite, and 1.25 for pyrrhotite. This change of Js values has a negligible effect on Eq. (3) and still results in a near perfect linear relationship where a ¼ (4.2 0.3)  10–3 with 95% confidence level for T¼T0¼0 K and Bc(Tb) ¼ 0.21J(T0). Using magnetic constants at absolute zero temperature, the problem of Curie temperature is eliminated and Eq. (3) can be applied for any magnetic material. As the microcoercivity Bc(Tb) modifies the Mtr(Tr) acquisition, the shape, magnetostriction, and crystalline anisotropy of the carriers should have significant influence on the Mtr(Tr) acquisition curves. For example, the length vs diameter ratio of the carrier should reduce or increase the effect of Bc(Tb) or the demagnetizing field (Dunlop and Özdemir, 1997) for sample lengths parallel or perpendicular to the field and thus shift the Mtr(Tr) acquisitions into lower or higher field intensities, respectively. In Figure M135, Mtr(Tr) acquisition for acicular (elongated crystals parallel to the applied field) magnetite (Dunlop and West, 1969) with diameter to length ratio 1:7 violates the equidimensionality assumption. Although the data are for magnetic fields near saturation of the magnetite, the demagnetizing field due to elongation causes these grains to acquire magnetization at lower fields than equidimensional magnetite grains. This shape effect was verified experimentally (Kletetschka et al., 2004) by measuring Mtr(Tr) acquisition in industrial wires (MWS-294R and ALLOY52) with length to diameter ratios 1:5 and 1:14 (Figure M137b), where the longer wires, parallel to the field, required lower fields to acquire the predicted intensity of magnetization. The effect of placing wires perpendicular to the applied field should be equal and opposite to placing them parallel. Consequently for a large number of randomly oriented, elongated grains (such as is frequently the case in igneous rocks) E should follow the same relationships as for the single equidimensional grains used in this study. This makes the TRM relationship far more applicable to those who study natural materials. However, this information would have to be accompanied by the caveat that, for the relationship to hold for multiple grains, the grains would have to be identical in size and composition (equal Js, Jsr, and Mtr). It is important to emphasize that a substitution of E to Eq. (3) B ¼ aðT0 ÞJs ðT0 Þ

Mtr ðT Þ Jsr ðT Þ

(Eq. 4)

represents the first ever means to obtain a paleointensity determination using measurable quantities that does not involve the comparison of a TRM imparted in the lab with that acquired in nature. Practical considerations may pose a serious hindrance to it ever being used as such because the above equation would not be satisfied by bulk values and natural grains, capable of retaining a remanence over geological time, would be too small to be measured individually. Possible solutions involve isolating and amassing grains with sufficiently similar properties, decomposition of bulk values using FORC diagrams, and so on to satisfy the requirements of Eq. (4). Gunther Kletetschka

Bibliography Acuña, M.H., Connerney, J.E.P., Ness, N.F., Lin, R.P., Mitchell, D., Carlson, C.W., McFadden, J., Anderson, K.A., Rème, H.,

Mazelle, C., Vignes, D., Wasilewski, P., and Cloutier, P., 1999. Global distribution of crustal magnetization discovered by the Mars global surveyor MAG/ER experiment. Science, 284: 790–793. Borradaile, G.J., 1996. An 1800-year archeological experiment in remagnetization. Geophysical Research Letters, 23(13): 1585–1588. Cisowski, S., and Fuller, M., 1986. Lunar paleointensities via the IRMs normalization method and the early magnetic history of the Moon. In Hartmann, W.K., Phillips, R.J., and Taylor, G.J. (eds.), Origin of the Moon. Houston: Lunar and Planetary Institute, pp. 411–424. Dekkers, M.J., 1989. Magnetic properties of natural pyrrhotite. II. Highand low-temperature behavior of Jrs and TRM as a function of grain size. Physics of the Earth and Planetary Interiors, 57: 266–283. Dunlop, D.J., 1990. Developments in rock magnetism. Reports on Progress in Physics, 53: 707–792. Dunlop, D.J., and Argyle, K.S., 1997. Thermoremanence, anhysteretic remanence and susceptibility of submicron magnetites: nonlinear field dependence and variation with grain size. Journal of Geophysical Research-Solid Earth, 102(B9): 20,199–20,210. Dunlop, D.J., and Kletetschka, G., 2001. Multidomain hematite: a source of planetary magnetic anomalies? Geophysical Research Letters, 28(17): 3345–3348. Dunlop, D.J., and Özdemir, Ö., 1997. Rock magnetism: fundamentals and frontiers. In Edwards, D. (ed.), Cambridge Studies in Magnetism, Vol. 3. Cambridge: Cambridge University Press, 573 pp. Dunlop, D.J., and Waddington, E.D., 1975. Field-dependence of thermoremanent magnetization in igneous rocks. Earth and Planetary Science Letters, 25(1): 11–25. Dunlop, D., and West, G., 1969. An experimental evaluation of singledomain theories. Reviews of Geophysics, 7: 709–757. Everitt, C.W.F., 1962. Thermoremanent magnetization II: experiments on multidomain grains. Philosophical Magazine, 7: 583–597. Fletcher, E.J., and O’Reilly, W., 1974. Contribution of Fe2þ ions to the magnetocrystalline anisotropy constant K1 of Fe3–xTixO4 (0 < x < 0.1). Journal of Physics C, 7: 171–178. Hargraves, R.B., and Young, W.M., 1969. Source of stable remanent magnetism in Lambertville diabase. American Journal of Science, 267: 1161–1177. Heller, F., and Markert, H., 1973. Age of viscous remanent magnetization of Hadrians wall (Northern-England). Geophysical Journal of the Royal Astronomical Society, 31(4): 395–406. Hoye, G.S., and Evans, M.E., Remanent magnetizations in oxidized olivine. Geophysical Journal of the Royal Astronomical Society, 41: 139–151. Kletetschka, G., Taylor, P.T., Wasilewski, P.J., and Hill, H.G.M., 2000a. The magnetic properties of aggregate polycrystalline diamond: implication for carbonado petrogenesis. Earth and Planetary Science Letters, 181(3): 279–290. Kletetschka, G., Wasilewski, P.J., and Taylor, P.T., 2000b. Hematite vs. magnetite as the signature for planetary magnetic anomalies? Physics of the Earth and Planetary Interiors, 119(3–4): 259–267. Kletetschka, G., Wasilewski, P.J., and Taylor, P.T., 2000c. Unique thermoremanent magnetization of multidomain sized hematite: implications for magnetic anomalies. Earth and Planetary Science Letters, 176(3–4): 469–479. Kletetschka, G., Wasilewski, P.J., and Taylor, P.T., 2002. The role of hematite-ilmenite solid solution in the production of magnetic anomalies in ground and satellite based data. Tectonophysics, 347 (1–3): 166–177. Kletetschka, G., Acuna, M.H., Kohout, T., Wasilewski, P.J., and Connerney, J.E.P., 2004. An empirical scaling law for acquisition of thermoremanent magnetization. Earth and Planetary Science Letters, 226(3–4): 521–528. Moskowitz, B.M., 1993. High-temperature magnetostriction of magnetite and titanomagnetites. Journal of Geophysical Research, 98: 359–371.

MAGNETIZATION, VISCOUS REMANENT (VRM)

Néel, L., 1949. Théorie du traînage magnétique des ferromagnétiques en grains fins avec applications aux terres cuites. Annales de Géophysique, 5: 99–136. Néel, L., 1955. Some theoretical aspects of rock magnetism. Advances in Physics, 4: 191–243. Özdemir, Ö., and O’Reilly, W., 1982. An experimental study of the intensity and stability of thermoremanent magnetization acquired by synthetic monodomain titanomagnetite substituted by aluminium. Geophysical Journal of the Royal Astronomical Society, 70: 141–154. Shcherbakova, V.V., Shcherbakov, V.P., and Heider, F., 2000. Properties of partial thermoremanent magnetization in pseudosingle domain and multidomain magnetite grains. Journal of Geophysical Research-Solid Earth, 105(B1): 767–781. Stacey, F.D., 1958. Thermoremanent magnetization (TRM) of multidomain grains in igneous rocks. Philosophical Magazine, 3: 1391–1401. Stacey, F.D., 1963. The physical theory of rock magnetism. Advances in Physics, 12: 45–133. Stacey, F.D., and Banerjee, S.K., 1974. The Physical Principles of Rock Magnetism. Amsterdam: Elsevier, 195 pp. Syono, Y., and Ishikawa, Y., 1963a. Magnetocrystalline anisotropy of xFe2TiO4.(1–x)Fe3O4. Journal of the Physical Society of Japan, 18: 1230–1231. Syono, Y., and Ishikawa, Y., 1963b. Magnetostriction constants of xFe2TiO4.(1–x)Fe3O4. Journal of the Physical Society of Japan, 18: 1231–1232. Tucker, P., and O’Reilly, W., 1980. The acquisition of thermoremanent magnetization by multidomain single-crystal titanomagnetite. Geophysical Journal of the Royal Astronomical Society, 63: 21–36. Wasilewski, P.J., 1977. Magnetic and microstructural properties of some lodestones. Physics of the Earth and Planetary Interiors, 15: 349–362. Wasilewski, P.J., 1981. Magnetization of small iron-nickel spheres. Physics of the Earth and Planetary Interiors, 26: 149–161.

Cross-references Blocking Temperature Crystalline Anisotropy Curie Temperature Demagnetization Field Demagnetizing Energy Empirical Law Hematite Iron Iron-Nickel Magnetic Ordering Magnetite Magnetoelastic Anisotropy Microcoercivity Multidomains (MD) Mysterisis Loop Natural Remanent Magnetization (NRM) Néel Theory Phase Transition Pseudo Single Domain (PSD) Pyrrhotite Remanent Efficiency of Magnetization (REM) Saturation Magnetization Saturation Remanence Shape Anisotropy Single Domains (SD) Squareness Ratio Super Paramagnetic Thermal Remanent Magnetization (TRM) Titanomagnetite Verwey Transition Viscous Decay

621

MAGNETIZATION, VISCOUS REMANENT (VRM) Introduction Viscous magnetization is the gradual change of magnetization with time in an applied magnetic field H. Brief exposure of a ferromagnetic material to a field results in isothermal remanent magnetization (IRM). The additional remanence produced by a longer field exposure is viscous remanent magnetization (VRM). The longer the exposure time t, the stronger is the VRM. Viscous remagnetization is the time-dependent change of VRM or other remanences, such as thermoremanent magnetization (TRM), depositional remanent magnetization (DRM), or chemical remanent magnetization (CRM), in response to a change in the direction or strength of H. In nature, such field changes are due to secular variation, excursions, polarity transitions, or plate motion. In laboratory experiments, but never in nature, samples may be exposed to zero field and the viscous decay of their magnetization measured. The natural remanent magnetization (NRM) of rocks, sediments, and soils usually includes a VRM produced at ambient temperature by exposure to the weak (≲100 mT) Earth’s magnetic field during the Brunhes normal polarity epoch. This VRM obscures the useful paleomagnetic information residing in older components of the NRM. Removing the VRM is the purpose of standard “cleaning” procedures, such as alternating field (AF) and thermal demagnetization. Brunhes-epoch VRM is distinctive because it is roughly parallel to the present-day local geomagnetic field. It is usually weaker than TRM and DRM and also “softer” or more easily cleaned than these older NRMs. However, magnetically “hard” or high-coercivity minerals like hematite, goethite, pyrrhotite, and some compositions of titanomagnetites and titanohematites have hard VRM that is not easy to AF demagnetize. Rocks that have been exposed to the Earth’s field at elevated temperatures for long times acquire thermoviscous magnetization (TVRM), which may replace part or all of their primary NRM. Usually these rocks are slowly cooling plutons, or have been deeply buried in sedimentary basins, volcanic piles, or mountain belts. TVRM blurs the distinction between thermal processes like TRM and partial TRMs and time-dependent processes like VRM and viscous remagnetization. In reality, even ambient temperature is sufficiently high compared to 0 K that time t and temperature T are interwoven in VRM and viscous overprinting. The fundamental mechanism of all viscous processes is the slow, continuous approach of the magnetization M to its thermal equilibrium value in field H at T, aided by thermal fluctuations of the magnetization in each crystal. Fortunately, the approach to equilibrium is usually extremely sluggish (hence viscous) at ordinary temperatures. If this were not so, there would be no surviving NRM of ancient origin, and no paleomagnetism or seafloor magnetic anomalies to track plate tectonic movements. Although most viscous magnetization changes are driven by thermal fluctuations, another possible source of viscous effects is slow diffusion of lattice defects that pin domain walls in multidomain crystals. This diffusion after-effect is most significant in titanomagnetites over geologically short times (Moskowitz, 1985).

Theory of single-domain VRM The Néel (1949, 1955) theory of time- and temperature-dependent magnetization deals for simplicity with ensembles of N identical uniaxial single-domain grains, all mutually aligned and having volume V, anisotropy constant K(T), and spontaneous magnetization Ms(T). At time t ¼ 0, magnetic field H is applied parallel to the anisotropy axis. At time t, nþ grains have their moments þVMs in the direction of H and n grains have their moments –VMs opposite to H. The distribution function n(t)/N ¼ (nþ–n)/N, which equals M(t)/Ms, obeys the kinetic equation

622

MAGNETIZATION, VISCOUS REMANENT (VRM)

dn=dt ¼ ½nðtÞ  neq =t;

(Eq. 1)

where the equilibrium value neq/N ¼ tanh(VMsH/kT) and the relaxation time t, due to thermal activation of moments over the energy barrier VK due to anisotropy, is 1=t ¼ 2f0 exp½VKðT Þ=kT ;

(Eq. 2)

with f 0  1010 s1. At constant temperature, the solution to Eq. (1) is nðtÞ  neq ¼ ½nð0Þ  neq  expðt=tÞ;

(Eq. 3)

an exponential decay of the out-of-equilibrium distribution function and magnetization. Néel introduced an approximation that replaces the real exponential decay by a step function, nðtÞ ¼ nð0Þ nðtÞ ¼ neq

if t  t if t > t:

(Eq: 4)

In the case of TRM, where t changes rapidly with small changes in T, Néel’s approximation leads to the definition of the blocking temperature TB, at which the ensemble rapidly passes from a thermal equilibrium state to an out-of-equilibrium state during cooling, or the reverse during heating, e.g., in thermal demagnetization. The blocking concept is less useful in isothermal viscous magnetization because t remains constant and all changes are a result of changing t. Any real sample contains many ensembles with different values of V and K. The grain distribution f (V, K)dV dK is generally poorly determined, but the formal solution of Eq. (3) for acquisition of magnetic moment m(t) by all ensembles, starting from a demagnetized state, is RR (Eq. 5) mðtÞ ¼ VMs neq ½1  expðt=tÞ f ðV ; KÞdV dK: Numerical solutions can be found for specified forms of f (V, K).

Time and temperature dependence of VRM If f (V, K) is constant or nearly constant over a range of (V, K), the convolution of the exponential time dependence for a single ensemble (Eq. (3)) with f (V, K), as expressed by Eq. (5), leads to a logarithmic dependence of m(t) on t (see Dunlop, 1973, or Dunlop and Özdemir, 1997, Chapter 10 for details). Experimentally most ferromagnetic materials on a laboratory timescale exhibit viscous changes that are indeed approximately proportional to logt. Figure M138 illustrates

the measured time dependence of VRM for oxidized and unoxidized pyroclastics on a t scale from 15 min to 28 days (about 4 decades of t). Only a limited range of (V, K) is activated in a few decades of t and the VRM increases more or less linearly on a logt scale. According to Eq. (2), for T constant, VK is proportional to log2f 0t. Using Néel’s blocking approximation (4), it is then possible to transform f (V, K) directly into the corresponding distribution g(logt). In this way, one can predict the viscosity coefficient S ¼ @M /@logt for a sample whose grain distribution f (V, K) is known. Another approach (Walton, 1980) is to calculate S directly from Eq. (5) without explicitly invoking a blocking approximation. If we replace K by a suitable average value and approximate neq/N by VMsH/kT, since H is small, we find S¼

RR

ðMs2 H=kT Þðt=tÞ½1  expðt=tÞV 2 N f ðV ÞdV :

(Eq. 6)

The integrand is of the form x(1–ex), with x itself exponential. It peaks sharply at x ¼ 1 or t ¼ t, which is in fact the blocking condition. All nonexponential factors can be assigned their blocking values, e.g., V ! VB ¼ (kT/K) log2f 0t, and taken outside the integral, giving S ¼ Ik 2 N ðMs2 T 2 =K 3 ÞHðlog 2f0 tÞ2 f ½kT =KÞ log 2f0 t;

(Eq. 7)

the integral I being approximately 1. Assuming that the anisotropy is due to grain shape, so that K(T) / Ms2 ðT Þ, and that the grain distribution is uniform over the range of ensembles affected in time t, we obtain S ¼ C HðT =Ms2 Þ2 ðlog 2f0 tÞ2 ;

(Eq. 8)

whereas if f (V) / 1/V, as for the larger grains in a lognormal distribution, S ¼ C 0 HðT =Ms2 Þ2 log 2f0 t;

(Eq. 9)

0

C, C being constants. Either Eq. (8) or (9) predicts a logt dependence for VRM intensity over times short enough that logt  log2f 0  23.7, i.e., for a few decades of t, as observed. Deviations from logt dependence will become obvious even at short times for very viscous samples, which scan larger fractions of the grain distribution, and at longer times for all samples. Deviations will appear earlier if Eq. (8) applies rather than Eq. (9).

VRM of multidomain and interacting grains Viscous magnetization due to thermal activation of domain walls has been treated theoretically in the weak-field limit by Néel and in the

Figure M138 Viscous remanent magnetization measured after exposure times of 15 min to 4 weeks to a 200 mT field (data: Saito et al., 2003). Upper curve, unoxidized pyroclastic sample containing large grains of titanomagnetite. Lower curve, oxidized pyroclastic containing hematite and titanohematite.

MAGNETIZATION, VISCOUS REMANENT (VRM)

presence of self-demagnetizing fields by Stacey (see Dunlop, 1973, for a review). All theories predict a logt dependence of VRM for geologically short times. The viscosity coefficient S varies with temperature as T/Ms or (T/Ms)1/2. The theories resemble single-domain formulations with V in Eq. (2) replaced by the volume Vact activated in a single Barkhausen jump. The frequency constant f 0 is 1010 s1, as for single-domain grains. Viscous magnetization due to activation of entire walls should be negligible because Vact  V of a singledomain grain. It must be that segments of walls are activated past single pinning sites. In nature, titanomagnetite and pyrrhotite crystals sometimes fail to renucleate equilibrium domain structures following saturation, resulting from heating to the Curie temperature. A possible mechanism for multidomain viscous magnetization is thermally activated nucleation of domains in parts of the crystal where the internal field is close to the critical nucleation field. This process has been termed transdomain VRM. Nucleation events have been observed in magnetite during small changes of T (Heider et al., 1988) but are not yet documented for changing t at constant T. Interacting single-domain grains have been approached in two different ways. One method is to treat them collectively by increasing V from the single-grain value. This approach preserves the experimentally observed logt behavior but it is not known at what level of interaction the picture breaks down. Another approach is to deal with individual particles under the influence of a randomly varying interaction field. Walton and Dunlop (1985) predicted deviation from logt behavior when the interactions are strong. Their theory gave a good fit to viscous magnetization data for an interacting single-domain assemblage with known f (V).

Experimental results Grain size dependence of viscosity coefficients Room-temperature viscosity coefficients S have their highest values in single-domain grains with volumes V just above the critical superparamagnetic volume VB. Figure M139 illustrates the data for magnetite. Data sets for other minerals are even smaller. Between the magnetite critical superparamagnetic size (0.025–0.03 mm) and the maximum size for single-domain behavior (0.07–0.08 mm), S decreases by at least a factor 4. The decrease may actually be larger and more precipitous than shown. Even in carefully sized samples, there is always a fraction of very fine grains. This fine fraction probably controls the short-term viscous behavior, because, according to Eqs. (2) and (4), only grains with very small V will be activated in the short times used in laboratory experiments. Similar “contaminating” ultrafine grains may explain the constant baseline value of S in grains larger than single-domain size.

623

The twofold increase in S over the upper pseudosingle-domain region (2–15 mm) is probably real and due to the increasing ease with which segments of walls can escape from their pins as the scale of the domains grows. On the other hand, large multidomain grains 80–100 mm in mean size have significantly lower S values. In summary, strong viscous magnetization is found mainly in the finest single-domain grains. However, small multidomain (so-called pseudosingle-domain) grains are also significantly viscous. Observations on lunar rocks and soils (see Dunlop, 1973 and Dunlop and Özdemir, 1997, Chapter 17 for summaries) confirm this trend. Lunar soils and loosely welded soil breccias containing nearly superparamagnetic iron particles are among the most viscous materials known, whereas crystalline rocks and mature breccias containing multidomain iron exhibit a consistent but small viscous magnetization. The viscous nature of lunar samples greatly complicated the study of their other magnetic properties during the Apollo missions.

Temperature dependence of viscous magnetization Viscous magnetization data for a single-domain magnetite sample (mean grain size 0.037 mm) at and above room temperature appear in Figure M140. Zeroing the field H for each measurement would have changed the initial state for subsequent measurements. Therefore viscous induced magnetization in the presence of H was measured. Over the short time-spans examined (2.4 decades of t), Mvr is linear in logt, as predicted by Eqs. (2), (8), and (9). The viscosity coefficient S, the slope of each data run, increases steadily with increasing T. This too is as predicted by theory. However, the inherent temperature dependence of S, in Eqs. (8) and (9) for example, tends to be obscured by the variability of the grain distribution f (V, K), which is scanned with a narrow thermal “window,” as expressed by the factor f (kT/K)log2f0t] in Eq. (7). Very different temperature dependences of viscous magnetization have been reported for magnetite and titanomagnetites of various grain sizes by different authors. In many cases, the higher-T results are distinctly nonlinear in logt. However, a good theoretical match is evident in Figure M140 between the single-domain magnetite data (Dunlop, 1983) and the theory of equations (6)–(9), using the measured f (V) of this sample (Walton, 1983).

Field dependence of viscous magnetization Experiments confirm the theoretical prediction (e.g., Eqs. (6)–(9)) that VRM intensity is proportional to field strength H for weak fields. The most detailed studies are those of Creer (1957) on single-domain hematite and Le Borgne (1960) on soils containing single-domain magnetite and maghemite. Linear behavior was observed for 0.05 mT  H  1 mT.

Figure M139 Viscosity coefficients S for sized grains of magnetite. Data: triangles, Shimizu (1960); circles, Dunlop (1983); squares, Tivey and Johnson (1984).

624

MAGNETIZATION, VISCOUS REMANENT (VRM)

Figure M140 Viscous magnetization of a single-domain magnetite sample (mean grain size 0.037 mm) at various temperatures. Dots: measured results, Dunlop (1983); lines: theoretical fits, Walton (1983).

Viscous magnetization at short times: frequency dependent and quadrature susceptibility Viscous changes in magnetization can be detected over very short times by measuring initial susceptibility w (short-term induced M, normalized by H) as a function of the frequency f of an alternating field ~ In effect, each cycle of the AF restarts the viscous magnetiza(AF) H. tion experiment. A second method is to measure the quadrature (90
out-of-phase) component of susceptibility wq, which represents a ~ time-delayed response to H. In the classic experiments of Mullins and Tite (1973) on soils containing single-domain size magnetite and maghemite, @w/@logf and wq were constant over the tested range of f (66–900 Hz, 1.4 decades of t). No frequency dependence or quadrature susceptibility was detected for multidomain magnetite. A pair of measurements of w at two standard frequencies is routinely used in environmental magnetic studies as a test for the presence of nearly superparamagnetic magnetite or maghemite. Because the frequencies are preset, there is no attempt to tune the viscosity measurement to other minerals or to domain states other than single-domain. Much more granulometric information could be obtained with a little extra effort by scanning a wider range of f and by extending the

measurements to temperatures (e.g., Jackson and Worm, 2001).

other

than

room

temperature

Demagnetizing VRM AF demagnetization ~ used in frequency-dependent susceptibility measurements The AFs H have small amplitudes, usually 0.1 mT, and do not seriously modify preexisting remanences. If larger AFs are applied, they entrain singledomain moments or domain walls, causing them to oscillate and ~ To destroy VRM (or any other remanence, for sufficiently large H). ~ is slowly decreased demagnetize a previously magnetized sample, H from its maximum value at a rate f, leaving about equal numbers of moments or domains in one or the other polarity. AF demagnetization is carried out in a stepwise fashion. The maximum field is increased in steps, starting from small values, so as to gradually erase remanence components of increasing AF coercivity. VRM is often the NRM component of lowest coercivity and therefore erased first. Indeed the strategy of AF cleaning in paleomagnetism is firmly rooted in this assumption.

MAGNETIZATION, VISCOUS REMANENT (VRM)

Strong fields lower the energy barrier VK to rotation of single-domain moments or the corresponding “barrier” (pinning energy) for domain wall motion. In the single-domain case, Eq. (2) is modified to 1=tH ¼ f0 exp½ðVKðT Þ=kT Þð1  Ms H=2KÞ2 :

(Eq. 10)

625

One consequence is hard VRM in minerals with high K/Ms, like hematite, goethite, pyrrhotite, elongated single-domain iron, and many compositions of titanomagnetite, titanomaghemite, and titanohematite. VRM is produced by a weak field and Eq. (2) applies, but it ~ for which Eq. (10) is is demagnetized by a strong field H

Figure M141 AF demagnetization of some of the VRMs of Figure M138 (data: Saito et al., 2003). VRM of the unoxidized titanomagnetite is erased by 2.5 mT. VRM of the oxidized sample containing titanohematite is much more resistant.

Figure M142 AF demagnetization of VRMs produced by 2.5 h exposure to a 50 mT field at the temperatures shown (after Dunlop and ¨ zdemir, 1990). VRMs produced at higher temperatures are increasingly resistant to AF cleaning. O

626

MAGNETIZATION, VISCOUS REMANENT (VRM)

appropriate. Using the blocking approximations t ¼ t for the last ~ for AF cleaning ensemble to acquire VRM and tH ¼ 1/f when H ¼ H of this same ensemble, we have by combining (2) and (10), ~ ¼ ð2K=Ms Þ ½1  ðlog f0 =f Þ1=2 =ðlog 2f0 tÞ1=2 : H

(Eq. 11)

The resistance of VRM to AF cleaning is thus proportional to K/Ms, the other factor being independent of mineral properties. Figure M141 confirms this prediction. One sample is an unoxidized pyroclastic containing iron-rich titanomagnetite with low coercivity K/Ms. VRMs produced over all ranges of t are very soft, demagnetizing ~ ¼ 2.5 mT. The other sample is oxidized and almost completely for H its VRM is carried by titanohematite of much higher coercivity. VRMs for this sample are more resistant to AF demagnetization, surviving to ~ ¼ 10–20 mT. This is only a moderately hard VRM, because in fact H both these samples contain multidomain-size grains, not single-domain material to which Eq. (11) strictly applies. However, practically any rock whose NRM is due mainly to hematite has VRM that cannot be ~ ¼ 100 mT (Biquand and Prévot, 1971). entirely removed by H VRM produced in nature at above-ambient temperature and subsequently AF cleaned at room temperature is also relatively hard. Figure M142 shows an example for a single-domain magnetite sample that was given VRMs in 2.5 h runs at temperatures as high as 552
C (the Curie point is 580
C). The effect of T is to shift the window of f (V, K) affected viscously in t ¼ 2.5 h to grains with increasingly high K (see Dunlop and Özdemir, 1997, Chapter 10). Ambient-temperature VRM in magnetically hard minerals and elevated-temperature VRM in both hard and soft minerals are both difficult to remove completely or cleanly (without removing part of the primary NRM as well) by AF demagnetization. A more satisfactory method is thermal demagnetization.

Thermal demagnetization Because VRM is a thermoviscous relaxation phenomenon, thermal demagnetization should be the most efficient way of erasing VRM. Randomizing single-domain moments or domain wall positions is accomplished by random thermal excitations, resulting in a truly demagnetized state, not the polarized condition that follows AF cleaning. From Eq. (2), the laboratory temperature TL to which a single-domain sample must be heated (in zero field) on a timescale tL (typically an hour or less) to erase VRM produced over a long time tN in nature at temperature TN is given by TL log 2f0 tL =Ms2 ðTL Þ ¼ TN log 2f0 tN =Ms2 ðTN Þ;

(Eq. 12)

(Pullaiah et al., 1975), assuming K(T) / Ms2 ðT Þ as for shape anisotropy. Substituting tN ¼ 0.78 Ma for VRM acquired over the Brunhes epoch at surface temperatures, one obtains for magnetite TL  250
C. This is indeed the typical reheating temperature needed to completely erase magnetite VRM, although AF cleaning often serves very well in this case. For pyrrhotite, with a different Curie temperature (320
C) and Ms(T) variation, the corresponding TL is  130
C. Figure M143 illustrates how complete erasure of VRM is detected. The NRM vector of this sample (Milton Monzonite, SE Australia), plotted in vector projection, reveals four components whose removal requires successively higher heatings. The VRM, produced by the Earth’s field over 100 ka at  165
C (Dunlop et al., 2000), and the primary TRM are carried by single-domain pyrrhotite, the two CRMs by magnetite and hematite. The junction between the VRM and TRM vectors is sharp because single-domain thermoviscous remanences separate cleanly in heating. The vector direction changes abruptly at the junction temperature because the TRM was produced by a reverse-polarity field and the VRM by a normal-polarity field. The junction temperature TL is >130
C because TN was above ambient while the rock was buried and VRM was being produced. Using

Figure M143 Thermal demagnetization of NRM of a sample of the Milton Monzonite (after Dunlop et al., 1997). The pyrrhotite primary TRM and VRM separate cleanly in thermal demagnetization to 222
C. Triangles and circles: vertical- and horizontal-plane vector projections, respectively.

tN ¼ 100 ka, TN ¼ 165
C, and Ms(T) for pyrrhotite in Eq. (12) yields TL ¼ 222
C, as observed. Contours of constant (V, K) are often plotted on a time-temperature diagram (Pullaiah et al., 1975), which can then be used as a nomogram for locating matching (T, t) pairs. Figure M144 is such a diagram, with contours for magnetite based on Eq. (12). It is clear from the data plotted, some of which are from the Milton Monzonite, that singledomain magnetite samples (0.04 mm, A) obey Eq. (12) but samples with pseudosingle-domain (20 mm, B) and multidomain (135 mm, C) magnetites follow contours with smaller slopes. That is, these latter grains require more heating to remove their VRMs than the single-domain theory predicts. Observations of “anomalously high” unblocking temperatures in thermal demagnetization of VRM are well documented in the literature. Initially it seemed possible to reconcile the data by using the Walton (1980) single-domain theory outlined in an earlier section. From Eq. (9),

MAGNETIZATION, VISCOUS REMANENT (VRM)

627

Figure M144 Theoretical time-temperature contours based on Eq. (12) compared to data from thermal demagnetization of ¨ zdemir, VRMs produced at elevated temperatures for synthetic magnetites and samples of the Milton Monzonite (after Dunlop and O 2000). Single-domain samples agree well with the contours. Larger pseudosingle-domain and multidomain samples are more difficult to demagnetize than predicted.

TL ðlog 2f0 tL Þ2 =Ms2 ðTL Þ ¼ TN ðlog 2f0 tN Þ2 =Ms2 ðTN Þ;

(Eq. 13)

which produces shallower contours resembling the trend of some of the deviant data (Middleton and Schmidt, 1982). However, on reflection it became clear that Eqs. (12) and (13) actually answer different questions (Enkin and Dunlop, 1988). Equation (13) tells us what exposure time tL to field H at temperature TL will produce the same VRM intensity as the original exposure to H for time tN at TN. The question that is relevant to erasure of VRM by thermal demagnetization, and is answered by Eq. (12), is what exposure tL to H at TL will reactivate the same ensembles (V, K) as the exposure in nature to H for time tN at TN. Because VRM intensity increases with rising T even if f (V, K) is constant (e.g., Eq. (7)), the two answers will always be different. This has now been shown beyond doubt by the experiments of Jackson and Worm (2001), shown in Figure M145. They carried out experiments on the magnetite-bearing Trenton Limestone below room temperature at fixed, very short times t and varied T to either achieve unblocking of the same (V, K) ensembles (upper graph) or produce the same viscous magnetization intensity (lower). The former data sets agree well with the Pullaiah et al. (1975) contours from Eq. (12), while the latter are close to the predicted Walton-Middleton-Schmidt contours from Eq. (13). Discrepancies between the predictions of Eq. (12) and thermal demagnetization results for rocks are mainly attributable to VRMs of grains larger than single-domain size. The larger the grain size, the greater the discrepancy, as illustrated by Figure M144. It is well known that the Thellier (1938) law of independence of partial TRMs is increasingly violated as grain size increases. One manifestation of this violation is that partial TRMs produced over narrow T intervals do not demagnetize over the same intervals, so that a clean separation of partial TRMs is no longer possible using thermal demagnetization. VRMs behave analogously. The average demagnetization temperature of both VRMs and partial TRMs is close to that predicted by Eq. (12) (Dunlop and Özdemir, 2000) but domain walls continue to

move toward a demagnetized state at higher T, producing a “tail” of unblocking temperatures. Unfortunately only the ultimate highest unblocking temperature is easily detected experimentally, and it is not predictable by any current theory. To further complicate matters, the initial state of a multidomain grain can strongly affect its magnetic behavior, including thermal demagnetization. Figure M146 demonstrates that VRMs produced in an AF demagnetized sample or in one thermally demagnetized and then heated in a zero field to the VRM production temperature are subsequently erased by further heating of 100
C. But a sample cooled in a zero field directly from the Curie point to the VRM temperature acquires a VRM that requires 250
C of further heating to remove. These experiments use continuous thermal demagnetization, which differs from standard stepwise demagnetization by not revisiting room temperature for each measurement. Although these results may not be directly applicable to standard paleomagnetic practice, they do make the point that any multidomain sample retains a memory of its entire past history and this may influence its later behavior in as yet unpredictable ways.

Applications of viscous magnetization Magnetic granulometry Viscous magnetization is a sensitive probe of narrow (V, K) bands of the grain distribution. If K has a narrow spread compared with V, inversion of Eq. (5) by Laplace transforms yields f (t), i.e., f (V). A cruder estimate, which is adequate in most cases, is to use a blocking approximation as in Eq. (7) to obtain f (VB) ¼ f (kTlog2f0t/K). Small VB can be accessed by using the frequency dependence of w. Large VB can be activated in VRM experiments above room temperature.

Estimating Brunhes-chron VRM In very viscous rocks, particularly unoriented or partially oriented seafloor samples, one would like to know if a large part of the NRM is

628

MAGNETIZATION, VISCOUS REMANENT (VRM)

Figure M145 Comparison of equivalent (t, T) sets determined from frequency-dependent susceptibility data at low temperatures for the magnetite-bearing Trenton Limestone (after Jackson and Worm, 2001). The upper data compare reactivation of equivalent grain ensembles (V, K) and are in good agreement with Eq. (12). The lower data compare equal viscosity coefficients at different T and agree reasonably well with Eq. (13).

VRM produced during the Brunhes chron (the past 0.78 Ma). For samples magnetized during an earlier normal-polarity chron, there is not much difference between the directions of primary TRM and recent VRM. No sharp junctions like those of Figure M143 appear in thermal demagnetization trajectories. Since relaxation times of 1 Ma are inaccessible in laboratory experiments, two different methods have been used. First is simple extrapolation of laboratory data, assuming S remains constant over many decades of t beyond the laboratory scale. This is certainly unjustified with the more viscous shallow oceanic rocks, e.g., doleritic flows (Lowrie and Kent, 1978). With less viscous rocks, like those of Figure M138, simple extrapolation implies that a substantial part of the NRM could be VRM, but a precise quantitative estimate is not possible (Saito et al., 2003). The second method is to activate long relaxation times in the laboratory in mild heatings above room temperature. For titanomagnetite of mid-ocean ridge compositions, the heatings must remain below the

Curie point (150–200
C). Another criterion is that chemical alteration must be prevented. Thermal demagnetization of the NRM, using Eq. (12) or Figure M144 as a guide to the (TL, tL) combination needed, is a better approach than production of a new VRM. In the latter experiment it is not possible to reactivate at TL all the ensembles (V, K) that carried the room-temperature VRM.

Paleothermometry If the approximate residence time tN over which VRM was acquired can be estimated (within about an order of magnitude, since log tN is involved), but the burial temperature during VRM production is unknown, Eq. (12) or Figure M144 can again be used to analyze laboratory thermal demagnetization data and estimate TN. This method is useful for estimating depth of burial in a sedimentary basin as a guide to hydrocarbon potential. It of course only works well if the VRM has single-domain carriers. A recent variant, which is capable

MAGNETIZATION, VISCOUS REMANENT (VRM)

629

approach depends on constructing a calibration (tN, TN) curve using samples from buildings of known age. It is specific to a certain area and a particular building stone, and requires substantial labor in establishing the calibration curve, but is undeniably useful to archeologists thereafter.

VRM in rocks

Figure M146 Continuous thermal demagnetization of VRMs produced at 225
C in a synthetic multidomain magnetite sample with different initial states: crosses, AF demagnetized; solid circles and triangles, thermally demagnetized and then heated to 225
C; open circles, zero-field cooled from the Curie point to 225
C (after Halgedahl, 1993).

in principle of estimating both tN and TN independently, is to analyze thermal demagnetization data for VRMs carried by two minerals, e.g., magnetite and pyrrhotite (Dunlop et al., 2000).

Cooling-rate dependence of TRM Another thermoviscous effect is the variation of TRM intensity depending on cooling rate. For single-domain grains, about 5% increase in intensity is predicted for the slowest versus the fastest practical laboratory cooling rates. The theory is not simple because T and t vary simultaneously but the few experimental data that exist agree quite well with predictions (Fox and Aitken, 1980). There is some evidence that for multidomain grains, TRM intensity may decrease for slower cooling (Perrin, 1998). The enormous range of cooling rates present in nature when metamorphic terrains are uplifted over millions or tens of millions of years by erosional unroofing may lead to much larger changes than can be measured in the laboratory. So far no method of simulating or estimating changes over these very long timescales has been proposed.

VRM as an archeological dating tool When blocks of stone were reoriented in building historical or more ancient structures, viscous remagnetization of their NRMs began. This “clock” is less than ideal because the changes in M are roughly in proportion to logt, so that the resolution dM/dt ¼ t1 dM/dlogt ¼ S/t decreases with increasing age. The classic study by Heller and Markert (1973) found reasonable values, 1.6–1.8 ka, for the age of Hadrian’s Wall (Roman, northern England) from the viscous behavior of two of three blocks tested. This success is somewhat unanticipated because it implies that S is constant for times of historical length, contrary to most laboratory observations. Multidomain grains, with their lower S values, may perhaps succeed where single-domain grains would fail. More recently, Borradaile has successfully used ad hoc relations between tN and TN for specific limestones used in the construction of buildings and monuments in England and Israel to interpret VRM thermal demagnetization data of samples from other structures of unknown age (see Borradaile, 2003 and references therein). The success of this

Lunar rocks owe their magnetic properties to metallic iron. Lunar soils and low-grade breccias contain ultrafine (0.02 mm) single-domain grains which are potently viscous. High-grade breccias, basalts, and anorthosites have weaker viscous magnetizations originating in multidomain iron. For a full discussion, see Dunlop (1973) or Dunlop and Özdemir (1997, Chapter 17). Terrestrial submarine basalts are rather weakly viscous. Coarsegrained massive flows, on the other hand, can be extremely viscous (Lowrie and Kent, 1978). The VRM is likely due to large homogeneous titanomagnetite grains with easily moved walls. VRM of titanomaghemites is very dependent on the degree of oxidation (see Dunlop and Özdemir, 1997, Chapter10). For Tertiary and early Quaternary subaerial basalts, Brunhes epoch VRM averages about one-quarter of total NRM intensity (Prévot, 1981). Soft VRM is due to homogeneous multidomain titanomagnetite grains (cf. Figure M141). Relatively hard VRM is carried by singledomain size magnetite-ilmenite intergrowths formed by oxyexsolution. Thermal cleaning to 300
C erases all Brunhes epoch VRM, as Eq. (12) suggests. Red sediments of all types have relatively strong and hard VRMs. Creer’s (1957) study remains the classic. Among magnetite-bearing sedimentary rocks, limestones have been the most thoroughly studied (e.g., Borradaile, 2003). Plutonic rocks, especially intermediate and felsic ones, tend to be quite viscous. The VRM is generally due to multidomain magnetite in ~