Physics of the Space Environment (Cambridge Atmospheric and Space Science Series)

Physics of the Space Environment This book provides a comprehensive introduction to the physical phenomena that result from the interaction of the Sun and the planets - often termed space weather. Physics of the Space Environment explores the basic processes in the Sun, in the interplanetary medium, in the near-Earth space, and down into the atmosphere. The first part of the book summarizes fundamental elements of transport theory relevant for the atmosphere, ionosphere, and the magnetosphere. In the rest of the book this theory is applied to physical phenomena in the space environment. The fundamental physical processes are emphasized throughout, and basic concepts and methods are derived from first principles. This book is unique in its balanced treatment of space plasma and aeronomical phenomena. Students and researchers with a basic math and physics background will find this book invaluable in the study of phenomena in the space environment.

Cambridge Atmospheric and Space Science Series Editors: J. T. Houghton, M. J. Rycroft, and A. J. Dessler This series of upper-level texts and research monographs covers the physics and chemistry of different regions of the Earth's atmosphere, from the troposphere and stratosphere, up through the ionosphere and magnetosphere, and out to the interplanetary medium.

A native of Hungary, Professor Gombosi carried our postdoctoral research at the Space Research Institute in Moscow before moving to the United States to participate in theoretical work related to NASA's Venus exploration. He is currently Professor of Space Science and Professor of Aerospace Engineering at the University of Michigan. He has served as Senior Editor of the Journal of Geophysical Research - Space Physics and is the author of Gaskinetic Theory, published in 1994.

Cambridge Atmospheric and Space Science Series

EDITORS Alexander J. Dessler John T. Houghton Michael J. Rycroft

TITLES I N PRINT I N THIS SERIES M. H. Rees Physics and chemistry of the upper atmosphere Roger Daley Atmosphere data analysis Ya. L. Al'pert Space plasma, Volumes 1 and 2 J. R. Garratt The atmospheric boundary layer J. K. Hargreaves The solar—terrestrial environment Sergei Sazhin Whistler-mode waves in a hot plasma S. Peter Gary Theory of space plasma microinstabilities Martin Walt Introduction to geomagnetically trapped radiation

Tamas I. Gombosi Gaskinetic theory Boris A. Kagan Ocean—atmosphere interaction and climate modelling Ian N. James Introduction to circulating atmospheres J. C. King and J. Turner Antarctic meteorology and climatology J. F. Lemaire and K. I. Gringauz The Earth's plasmasphere Daniel Hastings and Henry Garrett Spacecraft—environment interactions Thomas £. Cravens Physics of solar system plasmas John Green Atmospheric dynamics

Physics of the Space Environment Tamas I. Gombosi University of Michigan

CAMBRIDGE UNIVERSITY PRESS

PUBLISHED BY THE PRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIDGE The Pitt Building, Trumpington Street, Cambridge, United Kingdom CAMBRIDGE UNIVERSITY PRESS The Edinburgh Building, Cambridge CB2 2RU, UK 40 West 20th Street, New York NY 10011-4211, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia Ruiz de Alarcon 13, 28014 Madrid, Spain Dock House, The Waterfront, Cape Town 8001, South Africa http://www.cambridge.org © Tamas I. Gombosi 1998 This book is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1998 First paperback edition 2004 Typeset in Times 10/4/13/4 pt. and Joanna in E T E X 2 £ [TB] A catalogue record for this book is available from the British Library Library of Congress Cataloguing in Publication Data Gombosi, Tamas I. Physics of the space environment / Tamas I. Gombosi. p. cm. ISBN 0 521 59264 X hardback 1. Atmospheric physics. 2. Space environment. 3. Atmosphere, Upper. 4. Solar wind. 5. Magnetosphere. I. Title. QC861.2.G64 1998 551.51'4-dc21 97-51318 CIP ISBN 0 521 59264 X hardback ISBN 0 521 60768 X paperback

To the memory of my mentor Konstantin Iosifovich Gringauz, a dedicated pioneer of space exploration

Contents

Preface xvii Parti

Theoretical Description of Gases and Plasmas 1

Chapter 1 Particle Orbit Theory 3 1.1

Electromagnetic Fields 3

1.1.1

Maxwell's Equations in a Vacuum 3

1.1.2

Lorentz Transformation 5

1.1.3

Lorentz Force 6

1.2

Particles in Constant External Fields 6

1.2.1

Neutral Particles 6

1.2.2

Uniform Electric Field 1

1.2.3

Uniform Magnetic Field 1

1.2.4

Guiding Center Drifts 9

1.3

Nonuniform Magnetic Fields 11

1.3.1

Magnetic Mirror Force 11

1.3.2

Magnetic Moment 13

1.3.3

Magnetic Mirroring 14

1.3.4

Gradient and Curvature Drifts 16

1.4

Adiabatic Invariants 19

1.4.1

The First Adiabatic Invariant 21

1.4.2

The Second Adiabatic Invariant 21

Contents

1.4.3

The Third Adiabatic Invariant 23

1.5

Problems 28

Chapter 2 Kinetic Theory 30 2.1

Collisions 30

2.1.1

Mean Free Path 30

2.1.2

Collision Cross Section 32

2.1.3

Collision Frequency, Collision Rate 32

2.2

The Boltzmann Equation 34

2.2.1

Phase-Space Distribution Function 34

2.2.2

Evolution of the Phase-Space Distribution Function 36

2.2.3

Local Equilibrium Distribution 40

2.2.4

Collision Terms 43

2.3

Multispecies Gases 48

2.4

Elastic Binary Collisions 49

2.5

Problems 51

Chapter 3 Basic Plasma Phenomena 53 3.1

Debye Shielding 54

3.2

Plasma Parameter 56

3.3

Plasma Frequency 58

3.4

Problems 60

Chapter 4 Fluid and MHD Theory 61 4.1

Moment Equations 61

4.1.1

Velocity Moments 61

4.1.2

The Euler and Navier-Stokes Equations 63

4.2

Transport Equations for Multispecies Gases 67

4.3

MHD Equations for Conducting Fluids 69

4.3.1

Single-Fluid Equations 70

4.3.2

Generalized Ohm's Law 72

4.4

Ideal MHD 75

4.5

Problems 78

Chapter 5 Waves and Oscillations 79 5.1

Linearized Fluid Equations 79

5.2

Soundwaves 80

Contents

5.3

Alfven Waves 82

5.4

Plane Waves 83

5.5

Internal Gravity Waves 84

5.6

Waves in Field-Free Plasmas 86

5.6.1

Langrnuir Waves 89

5.6.2

Electromagnetic Waves 90

5.7

MHD Waves 90

5.8

Plasma Waves in a Cold Magnetized Plasma 93

5.9

Parallel Plasma Waves 97

5.9.1

Circularly Polarized Waves 98

5.9.2

Whistler Waves 100

5.9.3

Low-Frequency Waves 100

5.10

Problems 101

Chapter 6 Shocks and Discontinuities 103 6.1

Normal Shock Waves in Perfect Gases 103

6.2

MHD Shocks and Discontinuities 106

6.2.1

Contact and Tangential Discontinuities 108

6.2.2

MHD Shocks 108

6.3

Problems 111

Chapter 7 Transport of Superthermal Particles 113 7.1

Transport of Energetic Particles 113

7.2

Guiding Center Transport 118

7.3

Problems 121

Part II

The Upper Atmosphere 123

Chapter 8 The Terrestrial Upper Atmosphere 125 8.1

Hydrostatic Equilibrium 126

8.2

Stability of the Atmosphere 128

8.3

Winds and Waves 129

8.3.1

Acceleration Due to Planetary Rotation 130

8.3.2

Linearized Equations 131

8.3.3

Geostrophic and Thermal Winds 132

8.3.4

Acoustic-Gravity Waves 133

Contents

8.4

Diffusion 136

8.4.1

Molecular Diffusion 136

8.4.2

Eddy Diffusion 138

8.4.3

Diffusive Equilibrium 139

8.4.4

Maximum Diffusion Velocities 140

8.5

Thermal Structure 141

8.6

TheExosphere 143

8.7

Some Concepts of Atmospheric Chemistry 145

8.7.1

Thermodynamics 145

8.7.2

Chemical Kinetics 147

8.8

Atmospheric Composition and Chemistry 149

8.8.1

Stratosphers and Mesosphere 149

8.8.2

Thermo sphere 155

8.9

Problems 156

Chapter 9 Airglow and Aurora 158 9.1

Measuring Atmospheric Emissions: The Rayleigh 159

9.2

Atomic and Molecular Spectra 161

9.2.1

Ground States of Atoms 162

9.2.2

Atomic Excited States 163

9.2.3

Molecular Structure 165

9.3

Airglow 167

9.4

Aurora 169

9.5

Auroral Electrons 172

9.6

Problems 174

Chapter 10 The Ionosphere 176 10.1

Ionization Profile 177

10.2

Ion Composition and Chemistry 181

10.2.1

The D Region 182

10.2.2

The E Region 183

10.2.3

TheF\ Region 183

10.2.4

The F2 Peak Region 184

10.3

Gyration-Dominated Plasma Transport 185

10.4

Ambipolar Electric Field and Diffusion 188

10.5

Diffusive Equilibrium in the F2 Region 190

Contents

10.6

The Topside Ionosphere and Plasmasphere 193

10.7

The Polar Wind 195

10.8

Ionospheric Energetics 199

10.9

Ionospheric Conductivities and Currents 203

10.10

Problems 206

Fart III

Sun-Earth Connection 209

Chapter II The Sun 211 11.1

Thermonuclear Energy Generation in the Core 213

11.2

Internal Structure 214

11.2.1

Pressure Balance 214

11.2.2

Energy Transport in the Solar Interior 216

11.2.3

Radial Structure 111

11.3

Solar Oscillations 217

11.4

Generation of Solar Magnetic Fields 219

11.5

The Sunspot Number and the Solar Cycle 221

11.6

The Solar Atmosphere 225

11.6.1

The Photosphere 226

11.6.2

The Chromosphere 226

11.6.3

The Transition Region 227

11.6.4

The Corona 227

11.7

Radiative Transfer in the Solar Atmosphere 230

11.7.1

Local Thermodynamic Equilibrium Approximation 231

11.7.2

The Gray Atmosphere 232

11.8

Flares and Coronal Mass Ejections 233

11.9

Problems 235

Chapter 12 The Solar Wind 236 12.1

Hydrostatic Equilibrium - It Does Not Work 236

12.2

Coronal Expansion 238

12.3

Interplanetary Magnetic Field (IMF) 241

12.3.1

Field Lines and Streamlines 241

12.3.2

Magnetic Field Lines 243

12.4

Coronal Structure and Magnetic Field 245

12.5

Solar Wind Stream Structure 248

12.6

Nonrecurring Disturbances in the Solar Wind 251

Contents

12.7

The Heliosphere 252

12.8

Problems 255

Chapter 13 Cosmic Rays and Energetic Particles 257 13.1

Galactic Cosmic Rays 258

13.1.1

Solar Cycle Modulation of Galactic Cosmic Rays 258

13.1.2

Diffusion Theory of Cosmic Ray Modulation 260

13.1.3

Modulation Due to Diffusion and Cosmic Ray Drift 262

13.2

Solar Energetic Particles 265

13.3

Interstellar Pickup Particles and Anomalous Cosmic Rays 269

13.3.1

Interstellar Pickup Ions 269

13.3.2

Anomalous Cosmic Rays 271

13.4

Energetic Particle Acceleration in the Heliosphere 272

13.4.1

Stochastic Acceleration 273

13.4.2

Shock Acceleration 21A

13.5

Problems 277

Chapter i 4 The Terrestrial Magnetosphere 278 14.1

The Intrinsic Magnetic Field 278

14.2

Interaction of the Solar Wind with the Terrestrial Magnetic Field 280

14.2.1

The Chapman-Ferraro Model 280

14.2.2

The Bow Shock and the Magnetopause 281

14.2.3

The Magneto spheric Cavity 284

14.3

Magnetospheric Current Systems 286

14.3.1

The Magnetopause Current 286

14.3.2

The Tail Current 288

14.3.3

The Ring Current 289

14.3.4

Field-Aligned Currents 291

14.4

Plasma Convection in the Magnetosphere 292

14.4.1

The Axford-Hines and the Dungey Models 292

14.4.2

Magnetic Diffusion 295

14.4.3

Magnetic Reconnection 297

14.4.4

Convection Electric Field 300

14.5

High-Latitude Electrodynamics 303

Contents

14.5.1

Polar Cap Convection for Southward IMF 303

14.5.2

Ionospheric Convection Velocities 304

14.6

Magnetic Activity and Substorms 306

14.6.1

Sq Current and the Equatorial Electro]et 306

14.6.2

Magnetic Storms 308

14.6.3

Substorms 309

14.6.4

Geomagnetic Activity Indices 311

14.7

Problems 312 Appendices

A

Physical Constants 315

Table A.I Universal Constants 315 Table A.2 Conversion Factors 316 Table A.3 Solar Physics Parameters 316 Table A.4 Heliosphere Parameters at 1 AU 317 Table A.5 Terrestrial Parameters 317 Table A.6 Planetary Parameters 318 Table A.7 Atmospheric Composition 319 Table A.8 Thermospheric Composition 319 Table A.9 A Model Atmosphere 320 B

Vector and Tensor Identities and Operators 321

B.I

Vector and Tensor Identities 321

B.2

Differential Operators in Curvilinear Coordinates 322

B.2.1

Spherical Coordinates 322

B.2.2

Cylindrical Coordinates 323

C

Some Important Integrals 325

D

Some Useful Special Functions 327

D.I

The Dirac Delta Function 327

D.2

The Heaviside Step Function 328

D.3

The Error Function 329

Preface

This book provides a comprehensive introduction to the physics of the space environment for graduate students and interested researchers. The text is based on graduate level courses I taught in the Department of Aerospace Engineering and in the Department of Atmospheric, Oceanic, and Space Sciences of the University of Michigan College of Engineering. These courses were intended to provide a broad introduction to the physics of solar-planetary relations (or space weather, as we have started to call this discipline more recently). The courses on the upper atmosphere and on the solar wind and magnetosphere have been taught for a long period of time by many of my friends and colleagues here at Michigan before I was fortunate enough to teach them. I greatly benefited from discussions with Drs. Thomas M. Donahue, Lennard A. Fisk, and Andrew F. Nagy here at the University of Michigan and Drs. Thomas E. Cravens (University of Kansas), Jack T. Gosling (Los Alamos National Laboratory), and Jozsef Kota (University of Arizona). I am grateful for their advise, criticism, and physical insight. I would also like to acknowledge the constructive criticism of Konstantin Kabin, my graduate student here at the University of Michigan. His mathematical rigor and helpful suggestions greatly helped me in producing the final version of the manuscript. This book was intended to provide a comprehensive introduction to students with very different backgrounds and interests. Over the years my students came from physics, aerospace and electrical engineering, computational fluid dynamics, meterology, aeronomy, planetary science, astronomy, and astrophysics. In the presentation of the material my approach was to emphasize the fundamental physical processes and not the morphology of the various phenomena. This means that, in some respect, the book is a survey of transport theory applied to the space environment. This approach is also reflected in the organization of the book: Part I

Preface

summarizes fundamental elements of transport theory relevant for the space environment, whereas Part II and III apply the basic concepts to physical phenomena in the Sun, the helisphere, the upper atmosphere, the ionosphere, and the magnetosphere. The reader is expected to have some familiarity with integral and differential calculus, vector and tensor algebra, complex variables, statistics, classical mechanics, and electricity and magnetism. It was my intent to produce a more or less self-contained text and introduce all important concepts and definitions used in the book. In the preparation of this text I have consulted other textbooks, monographs, review articles, and research papers. When I used research papers in the preparation of this text, I usually referenced the source in a footnote. Textbooks and monographs, however, are listed in the bibliography at the end of the book. During my scientific career I greatly benefited from many exceptional individuals. However, I would like to dedicate this book to the memory of my first scientific mentor, Konstantin Iosifovich Gringauz. Gringauz was a true pioneer of space science: He designed devices and instruments for all generations of space vehicles from Sputnik-1 to the first missions of the Moon, to the first planetary probes, and finally, to the spacecraft that intercepted Halley's comet. He not only introduced me to worldclass space science, but he also stood by me during difficult periods of my life. I greatly benefited from his scientific insight, from his no-nonsense approach to science, and from his personal friendship. Ann Arbor, Michigan

Tamas I. Gombosi

Parti Theoretical Description of Gases and Plasmas

Chapter 1 Particle Orbit Theory

In this chapter we investigate how single particles (charged or neutral) behave in gravitational, electric, and magnetic fields. It is assumed that these fields are externally prescribed and that they are not affected by the particles themselves. This approach is usually referred to as the "test particle" method.

l. l

Electromagnetic Fields

ill

Maxwells Equations in a Vacuum

In vacuum, electromagnetic fields are generated by electric charges and currents. These quantities are characterized by the net electric charge density, p, and by the electric current density vector, j . The relation between the source quantities, p and j , and the resulting electric and magnetic field vectors (E and B) are described by Maxwell's equations. Maxwell's equations can be written in differential or in integral form. For the sake of completeness, we present here both forms. Maxwell's equations for vacuum contain three universal constants: the speed of light in vacuum (c = 2.9979 x 108 m/s), the magnetic permeability of vacuum (/x0 = 471 x 10~7 henry/m), and the electric permittivity of vacuum (e 0 = 8.8542 x 10~12 farad/m). These constants are not independent of each other: They are related by the /zO£o = 1/c2 relation. Differential Form of Maxwell's Equations are:

The four equations in differential form

Particle Orbit Theory

Poisson ys equation V-E=—,

(1.1)

Absence of magnetic monopoles V B = 0,

(.1.2)

Faraday's law VxE =

,

(1.3)

dt

and Ampere's law l dE V x B = — —+/zoj(1.4) cl at Integral Form of Maxwell's Equations The integral form of Maxwell's equations can be obtained by integrating Eqs. (1.1) and (1.2) over a finite volume, V, bounded by a closed surface, A. One can now use Stokes's theorem to obtain the following forms of the two source equations: Gauss's law

J)E.ndA= — f pdV, A

(1.5)

v

Absence of magnetic monopoles B-ndA = 0.

(1.6)

A

In these integrals dA is an infinitesimal element of area on A, and n is the unit normal vector to the surface element dA pointing outward from the enclosed volume. The integral form of Faraday's and Ampere's laws can be obtained by integrating Eqs. (1.3) and (1.4) over an open surface area, T, bounded by the closed curve, C. After invoking Gauss's theorem, one obtains the following relations: Faraday ys law

J c

ds

= - / J ^t ^t

ri dT,

(1.7)

Ampere '±? law -xidT. at C

T

(1.8)

j T

In these integrals n' is the unit normal vector to the surface element dT in the direction given by the right-hand rule from the sense of integration around C.

1.1

1.1.2

Electromagnetic Fields

5

Lorentz Transformation

Consider two inertial reference frames K and K' moving with respect to each other with a constant relative velocity, u. The time and space coordinates are (t,x,y,z) and (tf, x\ yf, z') in the frames of reference K and K\ respectively. For the sake of simplicity, we choose the x axis of our coordinate systems (in both frames) to be parallel to the relative velocity. In this case u = (u, 0, 0). The time and space coordinates in K' are related to those in K by the Lorentz transformation'. ct = y [ct x'

x ],

= y[X--Ct),

(1.9)

y = y, where the Lorentz factor, y, is defined as y = ,

(1.10)

The inverse Lorentz transformation is

i

u

ct = y I ct H—j

y = y', z = z f.

The relativistic momentum and energy of a particle moving with velocity v are the following: p = ym\ and £ = yrac 2 ,

(1.12)

where m is the particle's rest mass and s is its total energy. It can be readily seen that e2

(1.13)

6

Particle Orbit Theory

The transformation of the electric and magnetic field vectors between K and K' are given by

E' = y(E + u x B) -

y +1

(E]

y+

(1.14) ( l\c

The inverse Lorentz transformation can be found by interchanging the primed and unprimed quantities and replacing u by — u. In this book we will restrict ourselves to nonrelativistic situations, when u /c