The earth's core

The Earth's Core

This is Volume 37 in INTERNATIONAL GEOPHYSICS SERIES A series of monographs and textbooks Edited by WILLIAM L. D O N N A complete list of the books in this series appears at the end of this volume.

The Earth's Core

J . A. JACOBS Department of Earth Sciences University of Cambridge. England

SECOND EDITION

1987

ACADEMIC PRESS San Diego

New York

Harcourt Brace Jovanovich, Publishers London Orlando Austin Boston Sydney Tokyo Toronto

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W'

United States Edition published by ACADEMIC PRESS INC. Orlando, Florida 32887

Copyright @ 1987 by ACADEMIC PRESS INC. (LONDON) LTD All rights reserved. No part of this book may be reproduced in any form by photostat, microfilm, or any other means, without written permission from the publishers

British Library Cataloguing in Publication Data

Jacobs, J. A. The earth's core.-2nd ed.(International geophysics series, ISSN 0074-6142; v. 37). 1. Earth-Core I. Title II. Series 551 .I '1 2 QE509 ISBN 0 - 1 2-378951 - 6

Filrnset by Eta Services (Typesetters) Ltd, Beccles, Suffolk and printed in Great Britain by St Edrnundsbury Press Ltd, Bury St Edmunds, Suffolk

Preface t o the First Edition

The innermost regions of the Earth are inaccessible to man and no direct measurements of any of its physical properties can be made. Much attention has been given in the last few years to the “inverse problem in geophysicsthat of determining some physical parameter from a set of observations made at the surface of the Earth. Our knowledge of the core of the Earth comes from many different fields, of which seismology and geomagnetism are the most important and a discussion of these disciplines form a large part of this book. The role of the Earth’s core is essential to our understanding of many geophysical phenomena-it is the seat of the Earth’s magnetic field and coupling between the core and mantle is responsible for some of the variations in the length of the day. The advent of satellites and spacecraft to the moon and terrestrial planets has given an added interest to the internal properties of these bodies-measurements of their magnetic field has already given us some clues on their constitution and possible cores. The question of cores in the other planets is discussed in the last chapter. J. A. JACOBS J u l y 1975 ”

V

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Preface t o the Second Edition

It is just over ten years since the first edition of The Earth‘s Core was published It has been said that the typical doubling period for the accumulation of scientific knowledge during the last two centuries is about fifteen years, and this is certainly the case for the Earth’s core. The second edition has been largely rewritten and expanded, but follows the same layout as the first edition. The first chapter on the general physical properties of the Earth now includes a long section on the core-mantle boundary (the so-called D” layer). The section on free oscillations of the Earth is extended and a section on attenuation (the Q of the Earth) added. In Chapter 2, the accretion mechanism is treated in much more detail and the sections on heat sources in the early Earth and the time of core formation more thoroughly discussed. Chapter 3 now contains a section on Griineisen’s parameter and its role in the thermal regime of the Earth. Melting-point depth curves are discussed in more detail in the light of more recent highpressure shock wave data and a new section is added on the thermal consequences of an iron-alloy core. In Chapter 4 the section on reversals of the Earth’s magnetic field is greatly expanded and the sections on the energetics of the Earth’s core and on variations in the length of the day largely rewritten. A new section on the secular variation is added. Chapter 5 has been considerably modified. I t now contains a more detailed discussion of equations of state and a new section on experimental methods (both static and dynamic). The constitution of the Earth’s core is reviewed in more detail and the case for the presence of silicon, sulphur, oxygen and potassium in the core assessed. Chapter 6 is essentially rewritten. It contains much more information on the cores of the Moon and vii

viii

Preface to the Second Edition

other planets-mainly as a result of increased data from space probes and more detailed numerical modelling. In general SI units have been used. However, in some cases where the older system seems to persist in much of the literature, their usage has been retained, e.g. densities are expressed in g/cm3 rather than in kg/m3. This also has the advantage that their magnitude is reduced by a factor of three. Again the ST unit of pressure is the pascal (Pa), but much experimental work at high pressures still quotes results in kilobars (kb). The conversion is 1 giga pascal (GPa) = 10 k bar. Pressures in the Earth’s core exceed 100 G P a = 1 M bar. J. A. JACOBS

Contents

Preface to the First Edition Preface to the Second Edition

Chapter O n e

V

vii

General Physical Properties of t h e Earth

I. I I .2 1.3 I .4

Introduction Travel-Time and Velocity-Depth Curves The Core-Mantle Boundary (MCB) Free Oscillations of the Earth 1.5 Attenuation in the Earth 1.6 Variation of Density and Other Physical Properties within the Earth 1.7 Models of the Earth’s Interior

Chapter Two 2.1 2.2 2.3 2.4 2.5 2.6

3. I 3.2 3.3 3.4 3.5 3.6 3.7 3.8

30 39 48 56

The Origin of t h e Core

Introduction The Accretion Mechanism Heat Sources for an Earth Accreting Cold Time of Core Formation Inhomogeneous Models of the Earth Variation of the Gravitational Constant G with Time

Chapter Three

i

3 18

81

82 93 107 116 126

The Thermal Regime of t h e Earth’s Core

Introduction Gruneisen’s Parameter The Earths Inner Core Melting-Point-Depth Curves Adiabatic Temperatures The Earth’s Inner Core Reconsidered Thermal Consequences of an Iron-Alloy Core The Core and the Thermal History of the Earth ix

137 137 145 147 159 164 169 175

Contents

X

Chapter Four 4. I 4.2 4.3 4.4 4.5 4.6 4.7 4.8

The Earth’s Magnetic Field

Introduction The Origin of the Earth’s Magnetic Field The Homogeneous Dynamo Equations Mean-Field Electrodynamics Reversals of the Earth’s Magnetic Field Energetics of the Earth’s Core The Secular Variation Variations in the Length of the Day

191 20 I 207 213 217 236 247 267

Chapter Five The Constitution of t h e Core 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8

Equations of State The Birch--Murnaghan Equation of State Experimental Methods Ramsey’s Hypothesis Bullen’s (k,p ) Hypothesis Bullen’s Fe,O Hypothesis The Constitution of the Core The Possibility of Potassium in the Core

Chapter Six 6.1 6.2 6.3 6.4 6.5 6.6 6.7

The Cores of Other Planets

Introduction Mercury Mars Venus TheMoon Summary The Great Planets

Appendix A Index

291 299 302 308 310 314 315 336

347 349 359 371 377 39 1 394

407 409

Chapter One

Genera I Phys ica I Properties of the Earth

1.1 Introduction

This book is concerned with the Earth’s core-the innermost regions of the Earth. No direct measurements can be made of any of its physical properties and in some ways we know more about the distant stars than we do about the deep interior of the Earth. However, the core plays a key role in many geophysical studies-amongst other things it is believed that the Earth’s magnetic field is a result of motions in the fluid outer core ($4.2). The core, whose radius is just over one-half of that of the Earth, consists of an outer core (OC) that is fluid and an inner core (IC) that is most probably solid-there may also be one or more transition zones between these two regions. The radius of the IC is about one-fifth that of the Earth-thus, although its linear dimensions are quite large, its volume is only about 0.007 that of the whole Earth. Geophysics is an observational science and our objective is to try to explain our observations with theories based on sound scientific principles. To do this we set up models that approximate conditions in the real Earth and try to solve the problem for this idealized situation. A whole series of models of ever-increasing sophistication can be considered, but it must not be forgotten that they are only models and that the real Earth may behave quite differently. The assumptions and limitations of any model must be clearly stated and always borne in mind when assessing the success of any theory. 1

2

The Earth‘s Core

The mathematical equations that describe most physical phenomena are often non-linear and extremely complex so that analytical solutions are in general not possible. In many cases order-of-magnitude arguments are used to draw some preliminary conclusions, but one must be very wary of much of this type of “arithmetic”, since it is all too easy to be misled by “geophysical numerology” (Jacobs, 1970). In such cases intuition is a notoriously bad guide. Moreover, order-of-magnitude calculations can often lead to contradictions resulting from over simplification of the original equations. Our knowledge of the deep interior of the Earth is pieced together from information obtained from a number of different disciplines-physics, chemistry, astronomy and geology. A fundamental problem in geophysics is the determination of some physical parameter (e.g. density) from a set of observations made at the surface of the Earth. This “inverse” problem as it is called has received much attention during the last few years. It has been formulated using sophisticated mathematics, and with the aid of modern computers it is also possible to obtain information about the uncertainty and non-uniqueness of Earth models. The resulting proliferation of Earth models has had two consequences. Firstly, there is the difficulty of choice of an adequate Earth model for researches that depend on the structure of the Earth; and secondly, several researchers have adopted some properties from one model and other properties from another, with the result that their models are not self-consistent. At the XV General Assembly of the International Union of Geodesy and Geophysics (IUGG), held in Moscow in 1971, a committee was set up to advise on a Standard Earth Model (SEM) covering the distribution of various physical properties of the Earth’s interior. A necessary requirement of a reference Earth model is that it must fit data on the Earth’s mean radius R, mass M , and z where I = z M R 2 and I is the Earth’s mean moment of inertia. Account must be taken of data derived from records of seismic body waves, seismic surface waves, and free Earth oscillations. In addition there is a large body of information from other sources, including data on Earth tides, finitestrain and solid-state theory, laboratory experiments on rocks (including shock-wave experiments at pressures up to 4 million atmospheres), and evidence from other disciplines such as planetary physics, geology and geochemistry. The SEM committee set up a number of subcommittees to examine particular aspects of the SEM project. At a meeting of the Symposium on Mathematical Geophysics held at Banff in 1972, three research groups, headed by D. L. Anderson, F. Gilbert and F. Press, presented Earth models giving the main parameters. The results were published in the Geophysical Journal 35, 1973. At a meeting of the International Association of Seismology and Physics of the Earth’s Interior (IASPEI) in Lima in 1973, there was agree-

General Physical Properties of the Earth

3

ment about the need for a parameterization of an Earth model and Dziewonski et al. (1975) later published parametrically simple Earth models (PEM) consistent with geophysical data. It was also decided in Lima to publish some of the interim reports prepared by the sub-committees (Physics of the Earth & Planetary Interiors 9, 1 4 4 , 1974). By the time of the IASPEI meeting in Durham in 1977 it had become evident that it was possible to construct Earth models taking account of damping. The effects of attenuation were considered to be important, but since such effects are not well-determined it was decided to produce two Earth models-one including attenuation and one not. It was further decided to entrust to D. L. Anderson and A. M. Dziewonski the task of constructing suitable models. They presented a Preliminary Reference Earth Model (PREM) at the XVII General Assembly of the IUGG in Canberra in 1979 that was subsequently published in Physics of the Earth & Planetary Interiors 25,297-356,1981. Although a definitive SEM has not yet been produced, the SEM committee was finally dissolved at the XVIII General Assembly of the IUGG in Hamburg in 1983. It was realized, however, that refinement of the Preliminary Reference Earth Model is necessary, and further work is continuing, Details of recent Earth models are given in $1.7.

1.2 Travel-Time and Velocity-Depth Curves

The major source of our information about the Earth’s interior comes from the field of seismology. Following an earthquake, elastic waves travel throughout the Earth and may be observed at the large number of seismological stations distributed across the world. The theory of the propagation of elastic waves may be found in any standard text on seismology (see, for example, Bullen, 1963) and will not be developed here. There are two distinct types of elastic waves-P waves and S waves. A P wave is a condensation- . rarefaction wave involving change of volume. Motion of the medium is longitudinal, so that there is no polarization of a P wave. An S wave is a shear wave in which there is distortion without change of volume. S waves are transverse waves and thus exhibit polarization. In addition surface waves are set up: these have yielded valuable information about the crustal layers of the Earth. Recently, long-period surface waves have been used to obtain additional information about the deeper parts of the Earth. It can be shown that the velocities of P and S waves are given by

4

The Earths Core

and r

V, =

J’P

where k, is the bulk modulus or adiabatic incompressibility, ,u the modulus of rigidity and p the density. Thus V p and V, depend only on the elastic parameters and density of the medium. In particular, if the rigidity is zero, V, = 0, i.e. shear waves cannot be transmitted through a liquid. It follows from (1.1) and (1.2) that r#~ =

k,/p = V;

-

$Vg

(1.3)

When an elastic wave meets a sharp boundary between two media of different properties, part of it will be reflected and part refracted, and laws of reflection and refraction analogous to those of geometrical optics apply. The case of elastic waves is more complicated, however, since waves of both P and S type may be reflected and refracted. The‘ theory is based on Fermat’s principle, according to which an elastic wave takes the quickest path between any two points. This does not imply that there is only one path-there may be a number of alternative paths-but each path must involve a minimum transit time relative to small deviations in the path. Figure 1.1 illustrates some of the many possible reflections and refractions of elastic waves at discontinuities within the Earth. The figure also illustrates the terminology used to designate some of the different phases. A P wave reflected from the Earth’s surface can give rise to both a P wave and an S wave (called PP and PS waves respectively). Likewise an S wave reflected from the surface can give both P and S waves (called SP and SS waves respectively). The letters c and i denote reflections from the boundaries of the OC and IC respectively, and h denotes reflection from the surface of the F shell (the transition layer between the OC and IC). The letters Kand I are used for P waves in the O C and IC. Thus SKP is an S wave that has been refracted into the OC (necessarily as a P wave) and refracted back into the mantle as a P wave. S waves have never been observed in the OC, which is thus considered to be liquid. On the other hand, it has often been conjectured that the IC is solid and the symbol J was proposed for paths of S waves (if they exist) in the IC. It was not until 1972 that Julian et al. claimed to have identified the phase P K J K P on seismograms4onfirmation of their work would establish the existence of rigidity in the IC directly. They obtained a value of V, N 2.95 km s - ’ in the IC which is difficult to reconcile with that of -3.6 km s - ’ obtained by Gilbert et al. (1973) from an analysis of free Earth oscillation data (see $1.4). A value of -3.6 km s- also follows from Bullen’s ( k , p ) hypothesis (see 55.5). There is a possibility that Julian et al. observed the phase (1974) has suggested that the SKJKP, rather than PKJKP-Doornbos

’

5

General Physical Properties of the Earth

----s rays Fig. 1 .l.Representative seismic rays through the Earth (After Bullen, 1954.)

phase P K J K P is too small to be observed. Multiple ( n ) internal refections from the mantle-core boundary (MCB) into the O C are indicated by nK. Thus P7KP is a P wave that has been refracted into the OC, suffering 7 internal reflections in the core before being refracted back into the mantle as a P wave (see Fig. 1.2). The times of arrival of the different seismic waves may be determined from the records at a number of stations, so that it is possible to construct traveltime curves, i.e. plots of arrival times T against distance A (measured in degrees along the surface of the Earth) between the source and the seismic observatory. The construction of travel-time curves has had a long history of successive approximation and increasing accuracy. Revision of the early travel-time curves was undertaken by Jeffreys in 1931 (see Jeffreys, 1936, 1961 for complete details) using a least-squares technique. In collaboration with Bullen he produced the first J.B. Tables in 1935. Substantial refinements were incorporated in a new set of J.B. Tables first published in 1940. These gave travel-times, not only of P and S waves, but also of reflected and refracted

6

The Earths Core

Fig. 1.2. Diagram showing seven internal reflections from the M C B into the OC (P7KP). (After C. H . Chapman.)

waves. Details of the production of these tables and of the early history of the subject may be found in Bullen (1963). Improvements in the records and the use of large artificial explosions have led to corrections to the J.B. Tables (see e.g. Herrin, 1968). However, the production of the J.B. Tables, before highspeed computers were avilable, was a monumental achievement. From the travel-time curves, it is possible to calculate the velocities of P and S waves at any depth within the Earth. The details of such an inversion will not be given here (see e.g. Bullen (1963) for details). Figure 1.3 shows the gross features of the velocity-depth curves in the Earth according to Jeffreys and Gutenberg. During 1940-42, Bullen divided the Earth into a number of regions based on such curves. His nomenclature

7

General Physical Properties of the Earth

Depth ( k m ) Fig. 1.3. Seismic velocities of P and S waves as a function of depth. (After Birch, 1952.)

has since been widely used, and, in spite of uncertainties in the positions (and realities) of the boundaries between the different regions, continues to serve as a useful basis in discussing the Earth’s interior. The upper mantle consists of region B extending from the base of the crust (region A) to a depth of about 400 km, and region C which is a transition zone between depths of about 400 and 1000 km. The lower mantle, below a depth of about 1000km, is called region D. The core is divided into an outer core E, a transition region F and an inner core G. In recent years much finer detail has been obtained, particularly in the upper mantle (regions B and C, see Fig. 1.4), and in the vicinity of the boundary of the IC (region F). In the following discussion of the physical properties of the Earth, attention will be confined to the core and very deep mantle: no account will be given of the crust or the low-velocity layer in the upper mantle, nor of possible phase changes in region C. Sengupta and Julian (1978) have investigated the variation of P and S wave velocities in the lower mantle using data from deep-focus earthquakes. The chief advantage of using deep-focus earthquakes is the reduction of anomalies caused by complex structures near the source. Their standard deviations of average travel times are as much as a factor of two smaller than those determined from the data of shallow-focus earthquakes (see e.g., Herrin et al., 1968; Hales and Roberts, 1970). No pronounced decrease in either the P or S

8

The Earth's Core

Pressure ( k bar )

I. I. I

I' 1.

I:p I:

1:

I:

I: I: I'

i

c IL

I' 6.

LOWER

MANTLE

:?

CORE

L

t

;z .........../*'

..'

......

1: I:..

I I

fNIA1; MOHO. I I

i

I

I

Depth ( k m 1 Fig. 1.4. Seismic velocity distribution in the mantle for P waves, solid line (Johnson, 1967, 1969); S waves, solid line (Nuttli, 1969); broken lines P and S waves (Jeffreys, 1937, 1 9 3 9 ~ ) (After . Ringwood, 1972.)

velocity just above the MCB is required by their data, though a small velocity decrease could not be ruled out (see also $1.3). However, anomalous rapid changes in the gradient of the P velocity were found around depths of 2400 km and 2600 km. Their model for P velocities in the lower mantle agrees best with that of Herrin et al. (1968), and for S velocities with the 1066B model of Gilbert and Dziewonski (1975). Lee and Johnson (1984a) used travel-time data from the ISC Bulletin to

9

General Physical Properties of the Earth

0

1000 Y v

f a cu

a 2000

3000

4

6

a

10 Velocity ( k m s-')

12

14

Fig. 1.5. P a n d S velocitydepth bounds computed with upper mantle 7 ( p ) envelope. (After Lee and

Johnson, 1984a.)

determine confidence limits for the seismic parameter z(p) = T ( p ) - pA(p) where p is the ray parameter for mantle P and S waves. One year of P data (- 108,000 travel-times) and four years of S data (- 75,000 travel-times) were used. Based on these results, they obtained (Lee and Johnson, 1984b) extrema1 bounds on seismic velocities in the mantle (see Fig. 1.5). Bounds on the lower mantle P velocity have an average resolution of 70 km in the depth range -800-2900 km. The average bound width is -0.14 km sC1 at the 97% confidence level. Bounds on the lower mantle S velocity have an average resolution of 85 km in the depth range -850-2900 km and an average width of 0.13 km s-l. These confidence limits were computed with an assumed upper mantle velocity distribution. It should be noted that linearized inverse techniques (e.g. Backus and Gilbert, 1970; Johnson and Gilbert, 1972a,b) provide uncertainties in velocity and depth at a particular point in a model, whereas the method of Lee and Johnson establishes a confidence region for the entire model. The inversion methods of obtaining velocity-depth curves from travel-time curves break down if there is a region in the Earth in which the velocity decreases with depth. In such a case there will be a range of A in which arrivals from an earthquake are not observed (a shadow zone). There is an abrupt, discontinuous drop in the velocity of P waves across the MCB with a corresponding shadow zone in the range 105" < A < 143". Some waves are recorded in this range, however, so that it is not a true shadow zone (see Fig. 1.6).The

10

The Earths Core

Fig. 1.6. P, PcP, PKP, PKIKP, and diffracted P waves. (After Bullen, 1954)

amplitudes of such waves are much reduced and for many years their presence was attributed to diffraction round the boundary of the core. Miss I. Lehmann suggested in 1936 that such waves had passed through an IC in which the P wave velocity is significantly greater than that in the O C and later work has corroborated her hypothesis. Our present knowledge of seismic velocities in the core comes chiefly from the interpretation of travel-times of short-period P waves. There is a wide variation in the estimated velocities of P waves just below the MCB with likely values ranging from 7.9 k m s - ' (Hales and Roberts, 1971) to 8.26 k m s - l (Randall, 1970). Inversion studies favour a value fairly close to Jeffreys'( 1939d) value of 8.10 km s-'. At depths in the O C greater than about 200 km, most velocity models converge to values that are generally higher than those of the Jeffreys (1939d) model, although within 0.1 km s-' of it. A detailed discussion of the results of different authors has been given by Engdahl(1968). Nearly all authors of travel-time studies agree that there is a gradual increase in P velocity in the O C down to a depth of about 4600 km. Most disagreements arise in the transition zone F between the OC and IC. Calculations by Gutenberg and Richter (1938, 1939) and by Jeffreys (1939c,d) corroborated Miss Lehmann's hypothesis of a rapid or discontinuous increase in the velocity of P waves in the Earth's core, but differed in the interpretation of the velocity distribution near the inner core boundary (ICB). Jeffreys assumed that the wave velocity decreased with depth just above the IC, and then increased discontinuously at the ICB (see Fig. 1.3). Gutenberg, on the other hand, favoured a gradual increase in velocity in a transition zone without a preceding decrease. Neither of these models, however, can explain (using ray theory alone) the observed short-period precursors to P K P waves at distances less than 140". Jeffreys (1939~)considered

General Physical Properties of the Earth

11

diffraction near a caustic and showed that appreciable amplitudes for a diffracted P K P wave could not exist more than about 3" from the P K P caustic at 143" for periods of 1 s, or 14" for periods of 10 s. Denson (1952) and Gutenberg (1957, 1958a,b, 1959) attributed these early arrivals to dispersion in the transition zone. The lack of definitive observational evidence, however, is a significant argument against accepting this as the mechanism that is operative in the Earth. An explanation for the early onsets in terms of magnetoelastic coupling was also shown to be unlikely by Knopoff and Macdonald (1958). Bolt (1962,1964) proposed that two discontinuous increases in the velocity distribution of P waves in the Earth's core could explain these arrivals as rays refracted through an intermediate shell. He used observations from a study of P K P readings in the range 11O0 0 everywhere the core is gravitationally unstable. It should be noted that if the Adams-Williamson equation holds, i.e. if p ( r ) = 0, the liquid core is neutrally stable. Smylie showed that in the static case for spheroidal displacements of degree n 2 1, either the core must deform in such a way that the individual fluid elements, though displaced, suffer no dilatation, or the equilibrium density in

36

The Earths Core

the core must satisfy the Adams-Williamson equation. Smylie and Mansinha (1971) had earlier pointed out that, for fluids in static equilibrium, equipotential, isobaric and isopycnic (i.e. equal density) surfaces are parallel and that individual fluid elements may be displaced on such surfaces without resistance. Fluid elements are free to move aside on such surfaces to permit penetration of the solid IC and mantle into the liquid OC. Moreover, such surfaces not only remain parallel but are carried radially with the fluid element. There has been much confusion in the literature on the proper boundary conditions for the motions coupling the mantle with the liquid core, and different authors have come to different conclusions. Wunsch (1974) has shown that the essential difficulties and most of the physical phenomena can be reproduced in a much simplified model-the deformation in the static limit of an elastic half-space overlying a liquid half-space. He showed that, as is often the case in fluid dynamical problems, the problem is one of singular perturbation which leads to paradoxes if not handled correctly. Wunsch showed that in the case of the coupling of the motion of an elastic mantle to a stratified fluid core, the static limit is singular in a perfect fluid and can lead to physically meaningless results. Including rotation, however, makes the static limit a regular one, since Coriolis forces can support the deformed isopycnic surfaces in a steady flow. In a later paper (1975), Wunsch extended his analyses to include the effects of viscosity and density diffusion, and showed that paradoxes and contradictions that arise by treating only perfect fluids may be resolved by taking dissipation into account. Wunsch also considered the effect of a large ambient magnetic field-depending upon its orientation, its effect is similar to that of stratification or rotation. In particular, a toroidal field introduces a static limit singularity similar to that of density stratification. When a non-rotating spherical Earth model is used (Alterman et al., 1959), each spheroidal and toroidal component can be separated and the solution of the dynamic equation of equilibrium presents no major problems. Once rotation and ellipticity are taken into account, however, the spheroidal and toroidal fields become infinitely coupled (Smith, 1974; Shen and Mansinha, 1976). To circumvent the problem of infinite coupling, the effects of rotation and ellipticity can be considered as perturbations to a spherical, non-rotating Earth. This is feasible when the frequencies of the normal modes are much greater than the diurnal frequency. For elastic oscillations with periods of less than a sidereal hour, adequate results can be obtained with a first-order perturbation approximation (e.g. Backus and Gilbert, 1961; Pekeris et al., 1961; Dahlen, 1968; Luh, 1974). The approach fails, however, when long-period oscillations such as those confined mainly to the liquid OC are considered. Because the eigenperiods for these oscillations are comparable to or greater

37

General Physical Properties of the Earth

than the diurnal period, the full effects of rotation, and in some cases of ellipticity, must be taken into account. We then have an infinitely coupled system of ordinary differential equations. Traditionally, solutions of this system have been attempted by straightforward numerical integration (e.g. Smylie, 1974; Crossley, 1975; Shen and Mansinha, 1976; Johnson and Smylie, 1977; Crossley and Rochester, 1980). This approach, however, runs into the unavoidable problem of having to truncate the coupling chain severely and it is also difficult to interpret the numerical results. Shen (1983) has shown that by invoking appropriate approximations, analytic solutions can be developed for free oscillation of the Earth’s fluid OC. Crossley and Rochester (1980) attempted to justify certain approximations to their Earth model which permit more terms to be included in the expansion without increasing computer time. By considering simplified models of the Earth’s core they showed that, at least for the case of a non-rotating Earth, the effects of mantle structure and boundary rigidity on the undertone periods are minimal. Further, the combined effects of core compressibility and self-gravitation in a real Earth model can, to a good approximation, be replaced by a single parameter, the Brunt-Vaisala buoyancy frequency N.* Applying these simplifications to a rotating Earth model, Crossley and Rochester solved the resulting equations of motion with a longer coupling chain than any used previously. This approach, based on the chain beginning with the spheroidal term S;, supports the hypothesis that the periods of core undertones for a realistic rotating Earth model are crowded into a narrow band around 12 h. A different approach has been taken by Johnson and Smylie (1977) who used a variational procedure and considerable computational manipulation to include three spheroidal terms in the displacement field throughout the Earth. Figure 1.13 shows their results for the periods of the first few undertones of the spheroidal chain of azimuthal number two compared with earlier results obtained by direct integration. Moon (1982) has also used a variational procedure to derive seismic free oscillations and core modes for a rotating, slightly elliptical Earth model. He obtained numerical solutions for two Earth models (1066B and B156) minimizing the energy functionals for the terrestrial spectral range longer than the lowest-order free oscillation. For model B156, with stably stratified OC, the periods of core modes range from about 4-13 h with most of the modes crowded around periods of 6 h and 12 h. (Crossley and Rochester (1980) had shown that the periods for B156 are expected to be limited to the range * N is defined by the equation N2 =

-gdP-&

P d.r k, If the fluid is incompressible, the second term vanishes.

38

The Earths Core

24

0

t

I

2

3

I

1

1

4

5

6

UNDERTONE ORDER Fig. 1.13. Periods of the first few undertones of the spheroidal chain of azimuthal number 2. The curves show the effect of successive truncations. The case of no rotational coupling is due to Pekeris and Accad (1972) as modified by Smylie (1 974) to include a solid IC; the self-coupling case is due to Smylie (1974). who considered only rotational coupling of the first member of the spheroidal chain back on itself; coupling to T: was considered by Crossley (1975) in a direct integration of the equations of motion; and the case of coupling up to and including T; is due to Johnson and Smylie (1977). After Johnson and Smylie, 1977.)

between 6 and 12 h.) The core-mode periods for model 1066B are much more spread out without clustering around periods of 6 h and 12 h. Moon concluded that an Earth model can support long-period oscillations even when it is not stably stratified. According to the nature of the fluid motions and the primary driving force, free oscillations of the Earth’s core may be divided into two classes: gravitational oscillations and inertial oscillations. Gravitational oscillations are those that arise from density stratification in the core and are characterized by fluid flows in which the pressure variation due to compression is approxi-

General Physical Properties of the Earth

39

mately balanced by that due to transport (Shen, 1978; Smylie and Rochester, 1981). Inertial oscillations, on the other hand, are characterized by a predominantly solenoidal flow in the fluid and result from the rotation of the Earth (Greenspan, 1968). Shen (1983) argued that the equation of motion for inertial oscillations of the Earth’s core is identical in form to that of a nongravitating, homogeneous, incompressible fluid, density stratification coming in only when the change in gravitational potential is taken into account. Thus, the existence of inertial oscillations is independent of the density stratification in the liquid OC, in disagreement with the findings of Olson (1977) and Smylie and Rochester (1981). However, Shen agrees with Smylie and Rochester that the flow will be non-solenoidal for gravitational oscillations of the O C (undertones) in which density stratification is the principal governing factor. Smylie and Rochester showed that for sufficiently small (subacoustic) frequencies, an approximation that neglects the effect of flow pressure on the density is valid (they called this the “sub-seismic approximation”). This leads to a second-order partial differential equation in a single scalar variable, a great simplification from the original fourth-order partial differential vector equation. The validity of the sub-seismic approximation was first shown by Pekeris and Accad (1972) for gravitational oscillations of a fluid core in a non-rotating Earth using explicit solutions, and by Smylie and Rochester (1981) for a rotating Earth using scaling analysis. Shen (1983) showed that, despite Coriolis coupling, the sub-seismic equation can still be solved analytically for some core models. In a non-rotating Earth, gravitational oscillations are possible only for a stably stratified fluid core (Pekeris and Accad 1972). Shen showed, however, that with rotation gravitational oscillations become possible for any density stratification in the OC, the eigenfrequencies being divided into alternating allowed and forbidden zones. Furthermore, in each allowed zone, there exist an infinite number of such normal modes with their eigenfrequencies approaching asymptotically the eigenfrequency of a corresponding inertial oscillation. Both the solenoidal flow approximation (for inertial oscillations) and the sub-seismic approximation (for gravitational oscillations) ignore the fluid flow arising from variations in gravitational potential. Shen developed an asymptotic formulation to allow for this as well as taking into account deformation of the solid part of the Earth. He showed that the eigenfrequencies can be determined without explicitly solving for the correction terms.

1.5 Attenuation in the Earth

Most models based on free-oscillation periods assume that the elastic proper-

40

The Earth’s Core

ties of the Earth are independent of frequency. It is well known that absorption decreases the intrinsic elastic properties and causes velocity dispersion (Zener, 1948). This means that laboratory, body wave, surface wave, and freeoscillation data cannot be directly compared unless the absorptiondispersion correction is applied. The effect of absorption on free oscillation periods was initially ignored, however. Serious discrepancies were found with body wave results, which were attributed to a “baseline” effect, a presumed difference between the average Earth and that sampled by body wave data. Later, the effects of absorption on dispersion were re-examined and found to be of the first-order as Jeffreys (1965) had all along maintained, and not second-order as was initially assumed. The damping of free oscillations changes, for example, the eigenfrequencies of oT2 and oSz by about 0.5 and 0.25% (13 and 8 s) respectively (Randall, 1976; Hart et al., 1977). Jeffreys (1967) pointed out that for any form of imperfection of elasticity, except elasticoviscosity, there must be a systematic lengthening of free periods that may be about one part in 160. Thus, Earth models obtained by inversion of uncorrected normal mode data will yield lower velocities than models based on body wave data. The method for correcting seismic data has been given by Liu et al. (1976) and consistent Earth models have been determined by Hart et al. (1976). The physical dispersion accompanying absorption also complicates comparisons of laboratory and seismic data since they are taken at quite different frequencies. This must be taken into account in attempting to infer the composition of the Earth by comparing laboratory and seismic data. A large number of body wave observations are consistent with a low-Q zone at the bottom of the mantle.* Mikumo and Kurita (1968) estimated a 300-km thick layer at the base with QP 100. Teng (1968) determined the variation of Q with depth from the spectra of P phases from deep earthquakes. His data indicate a 200-km thick zone with QP 200. The observed amplitude ratios of ScS/S of Mitchell and Helmberger (1973) can be interpreted in terms of a thin low-Q zone (approximately 150 km thick with Qs about 100). The amplitudes of grazing P K P phases (Sacks and Snoke, 1976) also indicate a low-Q zone at the base of the mantle, Kuster (1972) proposed a low-Q zone (QP = 300, thickness 150 km) on the basis of P K P and P K K P spectral amplitude ratios. Berzon, et al. (1974) estimated the attenuation parameter QP in the mantle based on changes in the form of P-wave amplitude spectra as a function of distance. They used two independent sets of data from narrow-band and wide-band seismometers. They found the average value of QP in the depth in-

-

-

* Q has already been defined on page 16. Q for P waves is denoted by QP and for S waves by Q-in some papers by Q,. Often the subscript is omitted if there is no confusion about which Q is being discussed.

41

General Physical Properties of the Earth

TABLE 1.3 Comparison of Qpvalues in the upper and lower mantle. (After Berzon etal., 1974.)

0,according to: Depth interval (km)

Velocity Values used (krns-’)

Berzon eta/. (1974)

Anderson

~- et al.

lstset

2ndset

100-760

9.04

760-2900

12.48

710 k 150 + 730 1330- 360 1200:

100-2900

11.45

+ 420 1080- 285

0-100 0-760

7.62 8.91

0-2900

11.29

220 5 3 0 k 150 + 420 845 - 260

,”7”,”

(1965)

Kanamori (1967a) (1967b)

380

165

3210

1210

990

420

230 345

100 150

850

375

0-900 900-2900

410-630

Mikumo and Kurita (1968) M-5 M-14

1690

4900

120

274

292

702

180-240 1600-6000

terval 100-760 km to be 710 and in the interval 76@2900 km to be between 1200 and 1330. Table 1.3 compares their values with those obtained by other investigators using different methods. The differences are most likely due to the different data used (particularly the different frequency ranges) and the methods of analysis. Canas and Bolt (1983) showed that the ratio of the observed amplitudes of PcP and PcS phases at a station from the same earthquake is directly related to the density ratio at the MCB and the average attenuation parameter Qs for the mantle. Later Bolt and Canas (1985) obtained clear wave pulses at Berkeley and Jamestown for three earthquakes giving reflections at the MCB under both continental and oceanic areas. Comparison with plausible Earth models confirms a density jump at the MCB of 1.7. The average mantle Qsvalues under continental and oceanic regions are 705 f 365 and 585 ? 305, giving a mean of 650 k 335, in general agreement with results from surface waves. Agreement between observations and theoretical models is best if it is assumed that Qs increases with frequency o as oawhere c1 also increases with frequency. The Q of the O C is extremely high. High-frequency PnKP phases ( P K P , P K K P , P K K K P , etc.) travel with little change in shape or amplitude for long distances. Cormier and Richards (1976) concluded that QP in the OC is at least lo4 and probably much higher. A high Q for the O C is also supported by studies by Buchbinder (1971), Sacks (1971a, 1972), Adams (1972), Miiller (1973) and Qamar and Eisenberg (1974). Most of these studies infer a Q greater than 4000. It must be pointed out, however, that part of these differences in Q-values may be due to differences in the period ranges upon which

-

42

The Earths Core

the estimates were based. Qamar and Eisenberg’s investigation used spectral ratios of seismic core waves. For seismic waves of frequencyf2 1 Hz they found a high value of Q( 2 5000) in the OC. This was based on the fact that PnKP phases were observed for large values of n (at least n = 7): the P7KP/ P4KP ratios suggest a Q in the range 400CL8000, while the average slope of the P7KPIP4KP spectral ratio yields Q 11 10,000. Qamar and Eisenberg’s analysis, which assumed the applicability of classical ray theory and plane wave reflection coefficients, could not explain the observation that P7KP has consistently greater high-frequency content than P4KP. They suggested that frequency dependence of the MCB reflectiontransmission coefficients (Richards, 1973) may be important even in the highfrequency range of their study (0.2-2.0 Hz). Cormier and Richards (1976) later showed that this was in fact the case, and put a lower bound to Q in the oc of 10,000. There have been a number of body wave measurements of dissipation in the IC. Buchbinder (1971), using P K P amplitude ratios, found an average value of Q in the IC of about 400. Spectral ratios of IC and OC phases (Sacks, 1971b) indicate an average value of Q in the IC of 600. Sacks’ data suggest that QP is not constant through the IC but ranges from 100 to 200 in the outer 300 km to 1000 at greater depths. Kuster (1972) found a value of 300 for QP from spectral ratios of core phases. Julian et al. (1972) estimated that Q, in the IC is between 500 and 1000. A high value for Q, in this region is also implied by the fact that IC shear modes (e.g. 2S2) are observed (Buland and Gilbert, 1978). The Q of 2 S 2 implies a Q, for the IC of about 550. For the outer 450 km of the IC, Qamar and Eisenberg (1974) using the P K I K P I P K P ratio, obtained a value of Q in the range 120-1400, more than an order of magnitude less than that in the OC. Doornbos (1974) studied the anelasticity of the IC using spectral ratios of short-period core phases with common source and receiver and with nearly the same ray paths in the mantle. For 1 Hz compressional waves, he found that Q rises from a value of about 200 near the ICB to about 600 at a depth of 400km inside the IC. Below 450km from the boundary, the Q structure cannot be determined with any precision but is probably less than 2000. From observations of high-Q normal modes with a high energy concentration in the IC, Doornbos found a suggestion that, over a wider frequency range, Q is frequency-dependent. A frequency-dependent Q with low Q values around 1 Hz is compatible with partial melting in the IC. He suggested that at the ICB the temperature is close to the melting point and that in the IC the temperature gradient only slightly exceeds the melting point gradient (see also 63.6). From a spectral analysis of the phases PKZZKP and P K i K P from the same record, Bolt (1977) obtained a mean QP of 450 f (100) in the IC. Akopyan et al. (1975, 1976), Liu et al. (1976) and Hart et al. (1977) tackled

General Physical Properties of the Earth

43

the problem of dispersion caused by the Earth’s anelasticity by “correcting” the observed eigenperiods for attenuation before performing inversion to find an Earth model. This procedure can eliminate the “base-line” problem mentioned above-that travel times predicted by Earth models derived from free oscillations are generally slower than those predicted by Earth models derived from body waves. Lee and Solomon (1978), in a study of surface waves, showed that it is strictly more correct to perform simultaneous inversion for elastic and anelastic properties, rather than “correcting” some of the observations. Sailor and Dziewonski (1978) attempted to obtain the anelastic properties of the average Earth. As a first approximation, they considered the anelastic properties separately, without taking into account the coupling to elastic properties. In spite of the known lateral heterogeneity of both elastic and anelastic properties of the Earth, it is still useful to consider the concept of a spherically symmetric average Earth. The average Earth can serve as a standard of comparison for seismic observations made in more limited regions. Sailor and Dziewonski derived models of the gross Earth attenuation and found no indication of frequency dependence of Q in the period range from 100 to 3000 s. They obtained more than 230 Q measurements, using two different techniques-individual LaCoste-Romberg gravimeter recordings of three large earthquakes and 211 WWSSN recordings of two deep earthquakes. They inverted these data to obtain models of the distribution of Q in the mantle. The data-space inversion gave no indication of large variations of Q in the lower mantle-the data being consistent with very simple Q models. The data for modes dominated by shear energy can be satisfied by models with constant Q,-’ (attenuation of shear energy) in two regions: the upper mantle in which Q, is about 110, and the lower mantle below 670 km, in which Q, is a little below 400. The average Q, of the mantle is about 210. Models in which only shear energy is dissipated do not satisfy the observations of Q for radial modes. They also found that Q K (bulk dissipation) is finite in the upper mantle and possibly in the IC. Simultaneous inversion for Q, and QK showed that the data can be satisfied if QK is infinite in the IC and lower mantle, but finite in the upper mantle. Anderson and Hart (1978a) revised their earlier Q model MM8 for the Earth (Anderson et al., 1965). Their data set included fundamental and overtone observations of radial, spheroidal and toroidal modes, ScS observations and amplitudes of body waves as a function of distance. Their preferred model includes a low-Q zone at both the top and bottom of the mantle. Absorption in the mantle is predominantly due to losses in shear, although in the IC compressional absorption may be important as well. Most of the attenuation data can be satisfied by a frequency-independent Q versus depth model. Although Q is probably frequency-dependent in the high-Q regions of

-

44

The Earths Core

the Earth (the lithosphere and mid-mantle, (200-2000 km)), it is the low-Q regions of the Earth, where Q is only weakly dependent on frequency, that dominate the absorption of teleseismic body waves and normal modes. Anderson and Hart interpreted their Q model in terms of a thermally activated relaxation process that involves a distribution of activation energies and a small activation volume. A grain boundary relaxation model explains the dominance of shear over compressional dissipation, the roughly frequency-independent average value for Q and the variation of Q with depth. In a later paper Anderson and Hart (197%) made further estimates of the variation of Q with depth in the Earth. Their approach was to construct models based on body wave data and then to test and modify these models in order to satisfy the normal-mode data. Most of the attenuation data are consistent with frequency-independent Q versus depth models. They found that Q increases smoothly with depth over most of the lower mantle with a low-Q zone at its base. Normal-mode and body wave data indicate that there is bulk and shear dissipation in the IC, although no bulk dissipation is required in the mantle. Table 1.4 and Fig. 1.14 show the values of Q for their preferred model. Cormier (1981) synthesized short-period (0.2-2 Hz) seismograms for P K P waves interacting with the IC in anelastic Earth models. The agreement he obtained between predicted and observed amplitudes and spectral ratios requires neither a low-Q zone nor a low- or negative-P velocity gradient at the bottom of the OC. Thin low-Q, zones beneath the ICB fit the spectral data that sample the upper 200 km of the IC but fail to fit data that sample the lower IC. Only models having Q p 280, nearly constant with depth in the IC, satisfy all the spectral ratio and amplitude data. This suggests nearly constant temperature, pressure and phase composition with depth in the IC. The closeness to the melting temperature of some thermal models of the IC (see $3.6) suggests that the anelasticity may be due to partial melting. The amplitude of a mode at its resonant frequency is proportional to its Q. Thus even a weakly excited core mode (i.e. a spheroidal mode of elasticgravitational oscillation that has an energy partition dominated by shear energy in the IC) may be observable if Qs in the IC is high. Masters and Gilbert (1981), using recordings from the IDA gravimeter network of an earthquake in the Tonga region, obtained the first unambiguous observations of core modes. At low frequencies, the IC has a very high Q (about 3500) Core modes are extremely sensitive to the travel-time of shear waves in the IC. Masters and Gilbert computed the frequencies of the identified core modes using a constant shear velocity in the IC and a variable IC radius. The calculations are insensitive to structure above the ICB and frequencies computed using models 1066A and 1066B (Gilbert and Dziewonski, 1975) differ by less than 1 pHz. The observations are consistent with a low-frequency

-

TABLE 1.4 Q model SL8. (After Anderson and Hart, 1978b ) Thickness

Qs

Q P

(km) ~~

6371 00 6365 50 6360 00 6325 79 6291 62 6257 46 6223 29 618913 6154 97 61 20 80 6086 64 6052 17 601 8 31 5984 15 5950 00 5908 30 5866 64 5824 97 5783 31 5741 65 5700 00 561 3 93 5527 00 5440 09 5353 15 5266 23 51 79 39 5092 39 5005 47 491 8 54 4831 62 4744 70 4657 78 4570 86 4483 93 4397 02 431 0 09 4223 18 41 36 25 4049 33 3962 41 3875 48 3788 55 3701 62 361 4 70 3527 76 3484 30 2326 40 1229 50 61475

5 50 5 50 34 20 34 16 34 16 34 16 34 16 34 16 34 16 34 16 34 16 34 16 34 16 34 16 41 68 41 66 41 66 41 66 41 66 41 66 86 06 86 92 86 92 86 92 86 92 86 92 86 92 86 92 86 92 86 92 86 92 86 92 86 92 86 92 86 92 86 92 86 92 86 92 86 92 86 93 86 93 86 93 86 93 86 93 86 93 43 46 115787 1096 92 614 75 614 75

00 5 50 1 1 00 45 20 79 37 11353 147 70 181 86 21 6 02 250 19 284 35 31 8 52 352 68 386 84 421 00 462 69 504 35 546 02 587 68 629 34 671 00 757 07 843 99 930 91 101784 110476 1191 68 1278 60 1365 52 1452 45 1539 37 1626 29 171321 180013 1887 06 1973 98 2060 90 2147 82 2234 74 2321 67 2408 59 2495 52 2582 45 2669 38 2756 30 2843 24 2886 70 4044 60 5141 50 5756 25

Read 0.1OOE - 05 as 0.100

X

1 O-’.

500 00 500 00 500 00 11000 90 00 90 00 105 00 105 00 105 00 105 00 105 00 105 00 120 00 124 00 140 00 150 00 166 00 180 00 196 00 21 5 00 230 00 260 00 287 00 31 0 00 338 00 360 00 370 00 390 00 41 0 00 425 00 440 00 455 00 465 00 477 00 485 00 495 00 505 00 510 00 51500 51 5 00 515 00 51 0 00 500 00 300 00 200 00 100 00 0 00 0 00 425 00 850 00

1242 22 124411 1047 31 240 20 207 14 218 50 265 92 273 85 279 38 282 06 282 18 280 41 317 59 324 98 363 50 386 40 424 59 457 15 494 09 537 10 566 31 631 43 691 66 744 03 81218 871 42 905 02 963 85 1022 26 1067 57 111251 115709 118794 1221 83 1242 57 1266 60 1291 80 1308 10 1329 50 1342 44 1357 89 1359 37 1343 27 809 24 539 77 268 99 1 000,00000 1,000,00000 425 00 850 00

00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00

00 00 0 100E - 05 0 lOOE - 05 0 235E - 02 0 118E- 02

46

The Earth's Core

0

0

100

200

300

QP 400 500

600

700

800

J

,-

500 -

Model SL8 I000 -

1500-

2000 -

E

1

-

v

2500-

a

Q,

D

Outer Core

4871 -

Q,= co

5371 -

Fig. 1.14. 0 model SL8. (After Anderson and Hart, 1978b.)

travel-time of shear waves (radius/mean shear velocity) in the IC of 345 f 1.5 s. Stiller et al. (1980) have given two possible explanations for the distribution of the attenuation coefficient Q-' in the Earth's core. An appreciable asymmetric peak occurs at the ICB (see Fig. 1.15). It is somewhat surprising that the attenuation is higher in the solid IC than in the liquid OC, since one would expect sound waves to be generally damped less in solids than in their melts. Doornbos (1974) suggested that this is due to partial melting in the IC. Stiller et al. discuss two possible mechanisms that could explain the large anelasticity in the IC: one takes into account slow relaxation processes near

47

General Physical Properties of the Earth

4 5-

4-

3c

I

Q

m

0 2-

1 -

--

1

-.-

, tl200

ICB

I

I 1000

I

-------- - - _ _ _ _ _ _ I 800

I

1

I

600

I 400

I

I 200

I

3

I 0

Radius (km )

Fig. 1.15. Attenuation coefficient of seismic P waves in the Earth’s core. 1, data of Doornbos (1 974). 2, relaxation model of Stiller e t a / . (1980) with temperature distribution of Higgins and Kennedy (1 971). 3,relaxation model of Stiller et a/. (1980) with temperature distribution of Stacey (1 977); 4, contribution from attenuation in inhomogeneous matter (second model). (In the paper of Higgins and Kennedy the ICB is located at a different depth from that in the papers of Doornbos and Stacey). (After Stiller, Franck and Schmidt, 1980.)

the melting transition at the ICB, and the other is a quasi-polycrystalline structure for the IC. The first mechanism via slow order parameter relaxation can account for the shape of the Q - curve for the Earth’s core if Higgins and Kennedy’s (1971) temperature distribution is used (see Fig. 1.15). This explanation is a direct result of the condition that the whole IC, although solid, is very close to its melting temperature or solidus. Stiller et al. prefer this mechanism. The second mechanism would be more likely if the temperature distribution in the IC is like that of Stacey (1977), in which case the order parameter relaxation effect cannot account for the high energy loss in the lower 1000 km of the core. Karato (1980) has estimated temperatures in the lower mantle from the distribution of Q. He found a thermal boundary layer at the base of the mantle corresponding to the low-Q zone found by Anderson and Hart (1978a,b). This is not really surprising, but the accuracy of such temperature determinations is very dependent on that of Q, which is not at all well-known.

48

The Earths Core

1.6 Variation of Density and Other Physical Properties within the Earth

The density p will depend on the pressure p , the temperature T, and an indefinite number of parameter nispecifying the chemical composition, i.e. P = P ( P , T,ni)

(1.5)

If m is the mass of material within a sphere of radius r, then, since the stress in the Earth's interior is essentially equivalent to a hydrostatic pressure,* dP dr

- = -9P

where g=-

Gm

(1.7)

r2

and G is the gravitational constant. Considering first a chemically homogeneous layer in which the temperature variation is adiabatic, it follows from (1.5) that nP-----dPdP - - 9 P 2 dr - d p dr k, where k, is the adiabatic incompressibility defined by the equation

where s is entropy. A homogeneous region is here defined as one in which there are no significant changes of either phase or chemical composition. It follows from (1.7), (1.8) and (1.3) that dp

dr

-

-Gmp2 - -Gmp -~

ksr2

rz 4

(1.10)

The distribution of 4 throughout the Earth is known from the velocity-depth curves (cf. 1.3). Since dmldr = 47cpr2, a second-order differential equation for p = p ( r ) can be written down by differentiating (1.10). This equation, first obtained by

* In the deeper interior of the Earth, strains are fairly large (e.g. near the MCB, the mean of the principal strains has magnitude about 0.14). The mean (+pix) of the principal stresses is correspondingly large. In contrast the strength of the materials in the Earth, and therefore the greatest deviatoric stresses that can be attained, is small, i.e. the ratio of any pi, to fp,.. (which corresponds to pressure) is small at depths greater than -50 km and diminishes steadily as the depth increases. We can thus neglect the p,, compared with j p k k in the Earth's deep interior and treat the internal stresses of the Earth, when in its equilibrium state, as hydrostatic.

General Physical Properties of the Earth

49

Adams and Williamson in 1923 (Williamson and Adams, 1923), may be integrated numerically to obtain the density distribution in those regions of the Earth where chemical and non-adiabatic temperature variations may be neglected. Any density distribution must satisfy two conditions-it must yield the correct total mass of the Earth and the moment of inertia about its rotational axis. Using these two conditions and a value p1 of 3.32 g cm-3 for p at the top of layer B of the mantle (assumed to be at a depth of 33 km), Bullen applied (1.10) throughout the regions B, C and D. He then found that this led to a value of the moment of inertia I , of the core greater than that of a uniform sphere of the same size and mass. This would entail the density to decrease with depth in the core and would be an unstable state in a fluid. Birch (1952) showed that a non-adiabatic temperature gradient only worsened the situation. A reasonable value for I , could be obtained by increasing the initial value of pl, but only if an impossibly high value (at least 3.7 gem-' were chosen. Thus the assumption of chemical homogeneity must be in error; the region where this assumption is most likely to be invalid is region C where there are large changes in the slope of the velocity-depth curves. In his original Earth Model A, Bullen thus used the Adams-Williamson equation (1.10) in regions B and D while in region C he fitted a quadratic expression in r for p = p(r). In the O C (region E) (1.10) is likely to apply, and values of p down to a depth of about 5000 km can be obtained with some confidence. In the core one boundary condition is m = 0 at Y = 0, but lack of evidence on the value of the density p o at the centre of the Earth leads to some indeterminacy in the density distribution in regions F and G. Bullen initially derived density distributions on two fairly extreme hypotheses: (i) p o = 12.3 g cmP3and (ii) p o = 22.3 g cm-3 (this value being taken quite arbitrarily). A model with density values midway between those of these two hypotheses has been called Model A. More recent evidence indicates that p o is probably much nearer its minimum value and that a model based on po = 12.3 (Model A-i) is more likely to be correct.* These earlier determinations of the density distribution only made use of the information contained in the velocity-depth curves. The biggest source of additional information has come from analyses of the free vibrations of the Earth excited by major earthquakes. None of the existing velocity-depth curves combined with Bullen’s density distribution were consistent with the longer-period free-oscillation data obtained from the great Chilean earthquake of May 22,1960, and the great Alaska earthquake of March 28,1964. Using this additional data Landisman et al. (1965) investigated a number * A good account of this early work can be found in Bullen (1963,1975).

50

The Earth's Core

of Earth models without assuming homogeneity and an adiabatic temperature gradient (except in the OC). A feature of their models is constant density between depths of about 1600 and 2800 km. Bullen and Haddon (see later), were able to avoid this very implausible conclusion by treating the radius of the core rc as a free parameter-a normal value of the density gradient in the lower mantle could be obtained by increasing r, by about 15 km. Dorman et al. had earlier (1965) proposed an Earth model in which rc was increased by 10 km. Pekeris (1966) also obtained density distributions in the Earth without assuming homogeneity and an adiabatic temperature gradient in any region of the Earth. His density distribution p ( r ) was represented by 50 pivotal values p(rk) with linear variations in between and with discontinuities at the base of the crust and at the MCB. Pekeris varied the p k by the method of steepest descent so as to minimize the sum of the squares of the residuals of all the observed periods of the free oscillations of the Earth. As might be expected the density distribution in the IC has little effect on the spectrum as a whole. Other workers have supplemented the travel-time data with experimental empirical data. Birch (1960, 1961a,b, 1964) used an approximate linear relationship between density and the velocity of compressional waves that he had observed for silicates and oxides of about the same iron content, viz. p = a(m)

+ bVp

(1.11)

where m is the mean atomic weight. Birch only used this equation in the upper mantle (where there are high thermal gradients) and the transition zone (where there are phase changes). In the rest of the Earth he used the Adams-Williamson equation. A relationship between p and V , was proposed by Simmons (1964) and verified by Christensen (1966). McQueen et al. (1964) suggested that a better estimate of the density in the upper mantle might be obtained by using an empirical relationship between the bulk sound velocity c = (V: - SVi)* and density p rather than Birch's relation ( 1 . 1 1) between V p and p. This was verified by Wang (1968,1970). Birch's law implies a unique relationship between V pand p for materials of common atomic weight irrespective of whether density changes are due to crystallographic structure, pressure, or temperature. Liebermann and Ringwood (1973) obtained data for low- and high-pressure polymorphs for a wide variety of crystallographic phase transformations likely to be important in the Earth's interior. Their data indicate that V p - p relationships across such polymorphic transformations are not always equivalent to V p- p changes caused by varying pressure, temperature, or composition at constant mean atomic weight.

General Ph ysical Properties of the Earth

51

Chung (1974) obtained ultrasonic data on some crustal structures important in mantle minerology, using a powder-matrix method (Simmons and Chung, 1968) in which the elastic parameters of the unknown material are inferred from the measured elastic parameters of a composite made of the powdered material imbedded in a vacuum hot-pressed matrix of AgCI. Chung showed that a power law describes the velocitydensity (V,, ,J relationship for high-pressure polymorphs better than Birch’s linear law. In particular he found that V , varied quite linearly with density over very wide density (and thus wide pressure) ranges for materials of lower mean atomic weight, the linearity even being preserved through solid-solid phase transformations. For materials of higher mean atomic weight, on the other hand, evidence for linearity is weak. In a later paper Shankland and Chung (1974) obtained a power law equation for the dependence of V, upon m and p . Birch’s linear law and Anderson’s (1967) seismic equation of state may be deduced from it as special cases. Anderson (1973) has reviewed the literature on the power law representation of Birch’s law. Clark and Ringwood (1964) and Wang (1972) estimated densities in the Earth using petrological models for the upper mantle. Mizutani and Abe (1972) obtained an Earth model consistent with all the data using a trial-and-error method with the help of an equation of state of some rock-forming minerals. It must be emphasized that the overall density distribution within the Earth is not changed drastically by taking into account the additional observational data on its free vibrations and the revised estimate of its moment of inertia about its axis of rotation as determined from analyses of the orbits of artificial satellites (Cook, 1963). Inside the mantle, the largest difference in p between models A-i and HB, (to be discussed later) is only 0.15 gcm-3, while inside the core the values of p in model HB, exceed those in model A-i at all depths by amounts between 0.2 and 0.3 g ~ m - ~ . Buchbinder (1968) carried out a detailed study of PcP phases from eight explosions and three earthquakes. He reported cases of reversed polarity of PcP phases relative to P for epicentral distances < 32” at which distance the amplitudes of PcP pass through a minimum. Buchbinder suggested that these results reflect properties at the MCB that cannot be satisfied by any of the ‘‘conventional’’ Earth models. He found that the velocity at the top of the core is some 7-8% lower than values usually quoted and that there is no density discontinuity across the MCB. Buchbinder interpreted his results to indicate that the bottom of the mantle is inhomogeneous-the inhomogeneity being caused by an increase in iron (or other heavy metal) content with depth that increases the mean atomic weight and density with but little change in P velocity. The drop in P velocity across the MCB may then be explained by a discontinuous increase in mean atomic weight with but little change in density.

52

The Earth's Core

Various authors have objected to Buchbinder's interpretation. Berzon et al. (1972) noted that PcP phases are observed near 30" as frequently as at other epicentral distances, thus ruling out a zero in amplitude. Kogan (1972) demonstrated, by correlation between P and PcP phases, that PcP phases do not invert their polarity. Chowdhury and Frasier (1973) carried out an analysis of short-period P and PcP phases from earthquakes in the range 26" < A < 40" from the Large-Aperture Seismic Array (LASA) in Montana. No reversals of PcP polarity for A < 32" and no minimum in amplitudes at 32" were observed-again ruling out Buchbinder's model and favouring the more conventional models with a density ratio at the MCB of about 1.7. Buchbinder himself has now abandoned his original interpretation. A different approach to the inverse problem is the Monte Carlo method described in its geophysical context by Keilis-Borok and Yanovskaya (1967). It was used by Press (1968a,b) to obtain a number of Earth models using as data 97 eigenperiods, travel-times of P and S waves and the mass and moment of inertia of the Earth. The Monte Carlo method uses random selection to generate large numbers of models in a computer, subjecting each model to a test against geophysical data. Only those models whose properties fit the data within prescribed limits are retained. This procedure has the advantage of finding models without bias from preconceived or over-simplified ideas of Earth structure. Monte Carlo methods also offer the advantage of exploring the range of possible solutions and indicate the degree of uniqueness obtainable with currently available geophysical data. Press was able later (1970a,b) considerably to speed up his Monte Carlo procedures. Using new and more accurate data he was able to find a large number of successful models and also confirmed that the Earth's core is inhomogeneous and that the density at the top of the core is constrained to the narrow range 9.9-10.2 g cmP3.Wiggins (1969) also used Monte Carlo techniques to investigate the nature of the non-uniqueness inherent in the interpretation of body wave data. Haddon and Bullen (1969) constructed a series of Earth models (HB) using free oscillation data consisting of the observed periods of fundamental spheroidal and toroidal oscillations for 0 Q n Q 48 and 2 Q n Q 44 respectively and certain spheroidal overtones. Data from the records of both the Chilean and Alaskan earthquakes were used. Their procedure was to start from models derived independently of the oscillation data and to produce a sequence of models showing improved agreement with all the available data. In passing from one model to the next, a guiding principle was to introduce and vary one or more of the parameters in the model description at any stage in order to satisfy the oscillation data. They thus tried to establish models described in terms of the minimum number of parameters demanded by the data. A major difference in principle between their method and the Monte Carlo procedure

General Physical Properties of the Earth

53

of Press is the comparatively large number of parameters that Press permits to be randomly varied. Haddon and Bullen pointed out that the predominance of complex models found by Press is inherent in his method, since a simple random walk would automatically have a low probability. Haddon and Bullen also stressed that the “average” Earth to which average periods of free-oscillation modes relate is not necessarily the same as an Earth model to which the currently available average seismic body wave travel-times apply, since earthquake epicentres and recording stations are not randomly distributed over the Earth’s surface. In 1970, Bullen and Haddon used additional evidence to derive an improved Earth model (HB,). Model HB2 incorporated the newer P travel-time data of Herrin (1968) and takes into account the abnormalities in the body wave observations in the lower 200 km of the mantle and the detailed structure of the lower core (Bullen, 1965). In their earlier model HB1 a simplified core structure was assumed (the whole core being considered fluid), since free Earth oscillation data are incapable at present of resolving fine detail in the lower core. Derr (1969) also developed a series of Earth models using freeoscillation data. Anderssen and Seneta (1971, 1972) examined in detail the Monte Carlo method of inversion of geophysical data: in particular they developed a statistical procedure for estimating the reliability of non-uniqueness bounds defined by a family of randomly generated models. Anderssen et a / . (1972) applied these techniques to the problem of obtaining the density distribution within the Earth, and in a later paper (Worthington et al., 1972) attempted to resolve the major discrepancies between the density models of Bullen and Haddon and those of Press. They showed that these differences cannot be due to the different techniques employed to derive the models, and concluded that they are predominantly due to differences in assumptions about the permissible range of values of the shear velocity in the upper mantle. An upper bound to the density increase at the ICB was obtained by Bolt and Qamar (1970) from measured amplitudes of P K i K P and PcP phases recorded at LASA in Montana. The P K i K P waves were reflected at steep angles from the ICB (epicentral distance A (40”) and demonstrate the sharpness of that boundary (Engdahl et a!., 1970). Bolt and Qamar’s analysis indicated a minimum value of 0.875 for the ratio of the densities at the ICB. This would give a maximum density jump Apic at the ICB of 1.8 g cm-3 if the density in the O C at the boundary is 12.35 g cm-3 (see also Bolt, 1972). This result has been obtained on seismological evidence alone and does not depend on any assumptions concerning the chemical composition of the core. It is also in agreement with recent core models that indicate that the density in the IC is much lower than was at one time supposed. Dziewonski (1971) has stressed that only observations of overtones will

54

The Earths Core

allow the density distribution to be determined with sufficient detail to enable meaningful estimates of the structure and composition of the Earth’s deep interior. The diversity of models that satisfy travel-time and fundamental mode data indicates the insufficiency of the constraints provided by the more limited set of data. Dziewonski was able to identify overtones with periods greater than 250-300 s on records of the March 1964 Alaska earthquake from stations belonging to the WWSSN. Using this additional information, he constructed a number of Earth models. His final models indicate a change in composition in the bottom 500 km of the mantle. His density distributions show systematic differences from the HB1 model and Press’s models in the lower mantle, although all models lie within Press’s band of solutionsexcept that lower shear velocities are found in the depth range 480-650 km. Although agreement with these other models is good for the fundamental modes and the first two spheroidal overtones, it is poor for the third and fourth spheroidal and torsional overtones. A solid IC is also demanded by Dziewonski’s data-if the shear velocity in the IC is 3.5 km s - ’ Apic = 0.81 gcm-3. Details of the most recent Earth models are given in the next section. From (1.6) and (1.7). it follows that (1.12)

Hence by numerical integration, the pressure distribution may be obtained once the density distribution has been determined. Since the density is used only to determine the pressure gradient, the pressure distribution is insensitive to small changes in the density distribution and may be determined quite accurately. The variation of g can be calculated from (1.7). From a knowledge of the density distribution it is easy to compute values of the elastic constants. Thus (1.2) and (1.3) give p and ks directly. Poisson’s ratio r~ can be expressed in terms of V, and V, or ks and p; viz. o=

V; - 2V: 2(V; - Vg)

3k.y - 2 p 6ks + 2p

=-

(1.13)

Estimates of the IC radius compatible with differential P K i K p travel-times (Engdahl et al., 1974) correspond to a shear velocity of 3.55 ? .03 kms-’. This estimate is very close to the value found by Dziewonski and Gilbert (1971) from an analysis of other modes sensitive to IC properties. The low shear velocity results in a high Poisson’s ratio (- 0.45) for IC material and it has been suggested that this is indicative of a material near its melting point. The high value of CJ in the IC of many Earth models has often been used to argue that it cannot be a crystalline solid. However Al’tshuler and Sharip-

55

General Physical Properties of the Earth

dzhanov (1971) have reported shock wave measurements of 0 for copper and iron up to pressures of 1 and 2 Mbar. They obtained values of 0.43 at about 2 Mbar and the data extrapolate smoothly to the value found for the IC. Unlike other elastic properties, 0 increases with both temperature and pressure, although some metals have extremely high values of 0 even at low temperatures and pressures (Simmons and Wang, 1971). Falzone and Stacey (1980) have shown that it is not merely normal but necessary that 0 should be very high in any crystal at very high pressures. They point out that terms in conventional elasticity theory that are normally neglected become important when the ambient pressure is not negligible in comparison with the elastic moduli. These terms give a pressure dependence to the elastic moduli in addition to that resulting from the pressure dependence of atomic forces. The net effect is to reduce the modulus of rigidity and to increase the incompressibility, resulting in an increase in 0 (see (1.13)). The high value of 0 implies a state which allows large lateral contraction compared with longitudinal extension, suggesting a soft solid that is perhaps near its melting point. Under these conditions some variation of physical properties throughout the IC might be expected. The viscosity of the Earth's core is one of the least known physical parameters of the Earth. Estimates in the literature of the kinematic viscosity v differ by many orders of magnitude, ranging from to 10" cm2 s-'. High values of the viscosity have been deduced from the observation that compressional waves traverse the core without suffering appreciable attenuation (see e.g. Rochester, 1970). Such results lead to a value of v of the order of 109-f011 cm2 s-'. Gans (1972a) has shown that the effect of the Earth's magnetic field at the MCB cannot reconcile seismic data for S H waves with a very low viscosity. However, he offered the very interesting speculation that a highly 10" c m2 s -l could viscous region 5 10km thick at the MCB with v satisfy the seismic data and still permit a very low value of the viscosity in the OC as a whole. Hide (1971) has estimated the value of v at the MCB. He argued that if "bumps" on the boundary strongly influence the flow in the core, as suggested by himself and Malin (see §4.7), their height must exceed the viscous boundary layer thickness by a certain factor. Using the estimate for the height of these bumps given by himself and Horai (1968), he obtained an upper limit for the effective kinematic viscosity (i.e. eddy plus molecular) at the MCB of 1o6cm2s-'. Gans (1972b) has estimated the value of v at the ICB, assuming that this boundary is a melting transition and using Andrade's formula for the viscosity of a substance at its melting point. The estimates for pure iron give

-

-

9.0 x

< v < 1.7

x

10-2cm2s-'

56

The Earth’s Core

The core, however, is not pure iron, but contains some 10% lighter alloying element-silicon, oxygen and sulphur have been proposed (see $5.7). Both substances would lower the liquidus temperature, density and average atomic weight. The experimental data on the effect of alloying on the viscosity, however, are very sparse, although it seems probable that it would be lowered. It is not possible to estimate with any certainty the amount of the decrease. Gans suggests that in the core 2.8 x

< v < 1.5 x lo-’ cm2 s - ’

with a typical value of 6 x cm2 s-’. This range of values applies at the ICB. However if the melting point gradient is very shallow throughout the core (see $3.4), this range will not change much. With these values, the precessional mechanism proposed by Malkus (1968) to generate the Earth’s magnetic field (see $4.6) would be unstable. Whether it would be sufficiently unstable to overcome a highly sub-adiabatic temperature gradient in the core as proposed by Higgins and Kennedy (1971) cannot be determined. Crossley and Smylie (1975) have considered in detail the rate at which energy is dissipated in the liquid OC of the Earth. There are two possible dissipative mechanisms, ohmic and viscous, and they showed that oscillations in the core are virtually unaffected by either damping mechanism. Most of the ohmic dissipation takes place within a few “skin-depths’’ of the boundariesCrossley and Smylie expect that a similar result would be obtained for viscous dissipation, although their calculations were confined to estimating laminar dissipation in the body of the liquid core. The conclusion that the Earth’s core appears to be nearly free of dissipation for small harmonic oscillations, such as those which occur in the natural elastic oscillations of the Earth or those which may occur as gravitational oscillations if the temperature profile in the core is subadiabatic (see 991.4 and 3.6), suggests that modes of oscillation that are largely confined to the core may persist for unexpectedly long times-perhaps long enough to be important in driving the geodynamo (see $4.6). The viscosity of the lower mantle is discussed in $3.8.

1.7 Models of the Earth‘s Interior

The mathematical formulation of the inverse problem in geophysics characterizes possible variations of physical parameters as entities in an abstract function space, each entity representing an Earth model. In particular, a spherically symmetric, non-rotating, linearly elastic, isotropic Earth can be

General Physical Properties of the Earth

57

described by specifying the compressional velocity, the shear velocity, and the density as functions of the radius. An observation is the value of a functional defined in this space of Earth models+xamples are the Earth’s mass, moment of inertia, the measured travel-times of seismic waves, and the observed periods of free oscillation. Since the distributions of physical parameters are continuous in some interval and the number of data obtainable is necessarily finite, the inverse problem generally has no unique solution. Furthermore, the observations used as data are invariably contaminated by errors; only estimates of the values of data functionals for the Earth are available. A general discussion of the inverse problem in geophysics has been given in a series of papers by Backus and Gilbert. They investigated (1967) the extent to which a finite set of data functionals (called by them gross Earth data) can be used to determine the Earth’s internal structure. In a later paper (1968) they showed how to determine the shortest length-scale the given data can resolve at any particular depth. The principal result of their work is that it is possible to draw rigorous conclusions about the internal structure of the Earth from a finite set of gross Earth data. Infinite resolution can never be achieved, yet rigorous answers can nevertheless be given to a number of qualitative questions, such as whether there are low-velocity zones or density inversions in the mantle. In these first two papers, observational errors were neglected. In a later paper, Backus and Gilbert (1970) considered the effect of such errors and investigated the inversion of a finite set G of inaccurate gross Earth data-they showed that from some sets G it is possible to determine the structure of the Earth (except for fine-scale detail) within certain limits of error. They also showed how to determine whether a given set G will permit the construction of such localized averages of Earth structure, and how to find the shortest length-scale over which G gives a local average structure at a given depth if the variance of the error is to be less than a given amount. Detailed accounts of their work have been given by Backus (1971) and Parker (1970,1972a). In practice all data will be subject to some error, so that there are two measures of imprecision (an error of the estimate and its resolution) both of which one would like to make as small as possible. It is not possible, however, to minimize both at once. The error estimate of a property can be improved, but only at the expense of the resolving power and vice versa. Thus there exists a tradeoff between error and resolution at every radius, and some compromise must be made in choosing the best model. Parker (1972b) has also considered the inverse problem with grossly inadequate data. Such is the case, not only when the number of observations is small, but also if the inverse problem is intrinsically non-unique as is the case of attempting to determine the density inside a body from gravity obser-

58

The Earths Core

vations outside the body. Parker showed that although inadequate data cannot yield detailed structure, they can nevertheless be used to rule out certain classes of structures and provide bounds on acceptable models. The interpretation of inaccurate, insufficient and inconsistent data has also been investigated by Jackson (1972). The question of uniqueness has been further discussed by Dziewonski (1970) and the general linear inverse problem has been reviewed by Wiggins (1972). Finally, it must be emphasized that most inverse problems in geophysics are non-linear. In such cases Backus and Gilbert linearize the problem. Basically, their method employs an iterative perturbation algorithm that approximates the difference between the sought representation of the Earth and some initial model as a particular solution to the finite system of linear, inhomogeneous, integral equations relating changes in the data. The data functionals are computed for the starting model and subtracted from the observed data; the system of perturbation equations is solved, and the calculated perturbation added to the starting model. This process is iterated until the data are satisfied. Because the inverse problem is non-linear and has no unique solution, interpretation of any numerical results needs special care. A common mistake is to infer that because a certain model satisfies the data, some feature of that model actually exists in the Earth, when in reality the data do not require this feature. Moreover there is no guarantee that other starting models do not exist which are outside the scope of a linear description. Wiggins et al. (1973) have shown that body wave behaviour is too strongly non-linear for linearized schemes to be effective in predicting uncertainties. McMechan and Wiggins (1972) have developed a direct method for inverting seismic travel-time (T,A) data to determine velocity-depth profiles. They showed how a set of data points in the ( T , A ) plane, with estimates of the uncertainties, may be converted into an equivalent envelope in the velocitydepth plane. They refer to the problem of finding such envelopes of all possible models consistent with the observations as “extremal inversion”. They believe that the extremal inversion technique makes the usual Monte Carlo inversion method obsolete. In a later paper, Wiggins et al. (1973) examined the range of Earth structures that can be obtained from a set of body wave observations. Figure 1.16 shows the envelope of all possible models consistent with the set of data they used. In particular they found that there is an uncertainty of f 4 0 km in the radius of the ICB and f 18 km at the MCB. The velocity uncertainty is about f0.08 km s-’ for P and S waves in the lower mantle and about kO.10 km s-’ in the core. An interesting result of their investigation is that quite crude observations of SKKS-SKS traveltimes restrict the range of possible models far more than do the most precise estimates of PnKP travel-times.

..

c

'm lo-

**+**

E

Fixed crust" Fixed upper mantle

Y

v

,9

L .-

x

I

3

@

-

76-

5-

6!500

6000

5500

5ooo

4500

4000

: Radius (km

Fig. 1.16. Envelopes of all possible models of Earth structure from a given set of body wave observations. The envelope for the core was found under the assumption of 8 fixed core radius uf 3481 km. The heavy middle line is the standard model. (After Wiggins et a / . ,1973.)

60

The Earth's Core

The basic concept in the theory of Backus and Gilbert is that, although the exact solution cannot be computed because the information provided by the data is insufficient, it is possible to estimate accurately linear averages of the desired model. The aim of their theory is the construction of an optimal inverse filter from the constraints imposed by the observations, through which the correct solution may be viewed. As already mentioned, there is a tradeoff between the ability to resolve detail and the accuracy with which this detail can be estimated. Jordan (1972) has used a variation of the Backus-Gilbert theory incorporating the stochastic inverse theory of Franklin (1970). A particular, unique solution to the linear system is obtained by minimizing a specified quadratic measure of error. This quadratic form is the sum of two terms, a measure of the resolution of the estimate and a measure of its accuracy, parameterized to yield a Backus-Gilbert-type tradeoff curve. Jordan showed that the generalized inverse of Penrose (1955) and Moore (1920) and the stochastic inverse of Franklin (1970) lie on this tradeoff curve, the stochastic inverse being, in one sense, an optimal point. Any particular solution computed by selecting a point on the tradeoff curve is shown to be an estimate of the correct solution convolved with a projection-like smoothing operator. Dziewonski and Gilbert (1973a) observed an IC mode (llSz) in 11 spectra of seismograms with large lag times-over two days after the origin time in some cases (see Fig. 1.17). This represents direct proof of the solidity of the

r

Period ( s 1 Fig. 1.17. Histograms of a number of spectral peaks read from the spectra of vertical component readings. The histogram for seismograms with a lag-time of less than 18 h, broken line; over 18 h, solid line. (After Dziewonski and Gilbert, 1973a.)

General Physical Properties of the Earth

61

IC-the mode lSz could not exist if the IC were liquid. The IC mode lSz is coupled with a compressional mode of nearly identical phase velocity (loSz)-the observed periods are 246.89 s and 247.74 s respectively. The coincidence of loSzand llSz implies that the average shear velocity in the IC must be 3.6 km s - l , and that dissipation in the IC is low. Because of the large wavelengths of the normal modes, their eigenfrequencies are likely to be more closely related to the properties of a spherically symmetric, radially stratified “average” Earth than the nearly exclusive land-based travel-time observations. The resolving power of the normal mode data set shows that the average velocities and densities are known in the mantle to 1% for averaging lengths of -200 km (Gilbert et al., 1973). For the same precision the averaging lengths in the OC are -400 km. The structure of the region D” has been discussed in 51.3. There is no question about the sharpness of the MCB, although there may be minor lateral variations in some of the physical properties there. Dziewonski and Haddon (1974) have reviewed all data on the radius rc of the core. The first close estimate was made by Gutenberg (1913), who obtained a value of 3471 km from a determination of the distance at which P waves become diffracted. Much later, Jeffreys (1939a) proposed a method using the travel-times of core reflections and obtained a value of 3473 ( f 4.2) km, which he revised later (1939b) to 3473.1 (k2.5) km. His method (with some modifications) is still being used and it is only in recent years that additional data have forced revision of his estimate. Using travel-times of PcP phases from nuclear explosions, Taggart and Engdahl (1968) obtained a value of 3477 (k2.0) km for rc. Bolt (1972) inferred a 2% decrease in V , through the lowest 150 km of the mantle leading to a value for rc of 3475 (k2.0) km. Hales and Roberts (1970) obtained values of I , of 3490 (f4.7) km and 3486 (k4.6) km for two possible mantle models, basing their calculations on the differences in travel-times between ScS and S in the distance range 48” < A < 70”. Engdahl and Johnson (1972) analysed the differential travel-times Pep-P from three nuclear explosions in the Aleutian Islands and concluded that yC should be increased by 5-15 km over the value of 3477 km obtained by Taggart and Engdahl(l968). Their preferred estimation is 3482.2 ( 2.9) km. The free oscillation data demanded an increase in I , over earlier estimates. Dorman et al. (1965) proposed a model in which r, was increased by 10 km and a “soft” layer introduced at the base of the mantle. Haddon and Bullen (1969) showed that the travel-time and free oscillation data could be reconciled with an Adams-Williamson density gradient (see 51.6) in the lower mantle by increasing r, by 15-20 km over Jeffreys’ (1939b) value of 3473 km. Press used a Monte Carlo method to generate a large number of Earth models (see 51.6). The successful models (1970b) did not show a strong prefer-

-

62

The Earths Core

ence for any particular value of the core radius within the bounds from 3463 to 3483 km. Dziewonski (1970) showed that only additional overtone data could narrow the range of permissible solutions based on fundamental mode data alone. Using the eigenperiods of 70 spheroidal and toroidal overtones from the spectra of 84 recordings of the Alaska earthquake of March 28, 1964 (identified and measured by Dziewonski and Gilbert, 1972), he was able to narrow the range of r, to 3486-3491 km. Many more overtone data have now been obtained (Dziewonski and Gilbert, 1973b) and have been used by a number of authors in constructing Earth models. Jordan and Anderson (1974) obtained a value for r, of 3845 km and Gilbert et al. (1973) values of 3482.6 and 3484.9 km for two different models. Engdahl and Johnson (1974) using differential P C P travel-times, obtained values in the range 34843486 km. The best available estimate at present for I , is 3485 ( IfI 3) km. A radius of 1250 km postulated for the IC in the Jeffreys-Bullen model now appears to be somewhat too great. Bolt’s (1962) model used a radius of 1216 km and later work by Bolt (1964) supported by wide-angle reflections of P K i K P (Bolt et al., 1968) gave values near 1220 km. The models of Buchbinder (1971) and Qamar (1973) incorporate IC radii of 1226 and 1213 km respectively. Engdahl et al. (1974) have estimated the radius of the IC from differential P K i K P (i.e. P K i K P minus PcP) arrival time data for a number of different Earth models. The data indicate an IC radius in the range 12201230 km. For a model incorporating a decreased velocity gradient just above the ICB, their preferred value for the radius of the IC is 1227.4 & 0.6 km. Comparison of pulse shapes and pulse durations for P , PcP and P K i K P phases indicates that the ICB is very sharp. A value in the range 12151235 km is now preferred. Bolt and Urhammer (1975) carried out least-squares iterations with penalty functions on Earth model CAL5* based on observations of eigenvibrations, surface waves and body waves. In the penalty function method, linear equality constraints are weighted so that the least-squares estimator suffers a specified penalty for progressive violation of the physical restrictions. They also derived a procedure for calculating the trade-off between spread and variance for density estimates, and then assessed the resolvability of density and index of homogeneity at a number of levels in the Earth. The density in the lower mantle in their final model is a little closer to homogeneous, adiabatic conditions. Density values are about 0.15 g cm-3 less than those of B1 (Jordan and Anderson, 1974) in the region 20W2500 km. As a consequence, density values in the core are significantly greater than those of

* Details of these earlier models will not be given-they required.

can be found in the references if

63

General Physical Properties of the Earth

-

B1 (the mean density at the bottom of the OC is 1 2 . 3 4 g ~ m -compared ~ with 12.11 g ~ m -in ~ Model Bl). In the IC, V , N 3.55 kms,-l, p N 13.30g cm-3 and Poisson's ratio ~ 0 . 4 5 . Hales et al. (1974) suggested the construction of a spherically symmetric Earth model in which the radial variations of seismic parameters are expressed by piece-wise continuous analytical functions such as low-order polynomials in the radius. This idea was taken up by Dziewonski et al. (1975) who used polynomials of order not higher than three to construct a parametric Earth model (PEM). Their starting model was based on model 1066B (Gilbert and Dziewonski, 1975) with some modifications. The data used in the inversion consisted of observations of eigenperiods for 1064 normal modes, 246 travel-times of body waves for five different phases and regional surface-wave dispersion data. They presented three Earth models, all identical below a depth of 420 km; models P E M - 0 and PEM-C reflecting the different properties of the oceanic and continental upper mantles and model PEM-A representing an average Earth. The densities and velocities in the lower mantle (depths greater than 670 km) and core are consistent with the Adams-Williamson equation to less than 0.2%. Thus, any departures from homogeneity and adiabaticity within these regions must be very small. Dziewonski et al. (1975) also showed that the velocities in the lower mantle were consistent with the complete first-order finite-strain theory to within 0.2% for V p and 0.4% for Vs. In the IC, Vs varies from 3.44 to 3.56 km s-', confirming the earlier estimate of 3.5 km s-' by Dziewonski and Gilbert (1971). Buchbinder (1976) computed PcP and PmKP travel times for three simple or parametric Earth models based on free oscillation and travel-time dataH B , (Haddon and Bullen, 1969), B1 (Jordan and Anderson, 1974) and PEMA (Dziewonski et al., 1975)-and compared them with PCP and PmKP travel-times from different sources. The comparison was made only for the region above and below the MCB. He found that only model B1 does not need a correction for its PcP travel-times. For the PmKP travel-times all three models needed corrections. Normal-mode data are not adequate to resolve details of Earth structure having wavelengths of the order of 1 W 2 0 0 km. To resolve these features, body wave data, which are of higher resolving power, must be used. Such data are by their nature more path-dependent than normal modes-the role of free oscillations is to determine differences between the average Earth and the more path-specific body wave structures. Anderson and Hart (1976) thus used a starting model based on high-resolution body wave data and perturbed it to fit the normal mode data set. The resulting model retains the features found by body wave studies, but the average properties are suitably adjusted to correspond to average Earth properties, as is required by the nor-

64

The Earths Core

4000

3000

2000

1000

0

Radius (km) Fig. 1.18. A comparison of densities p and seismic velocities Vp and V, estimated in three Earth models, CALG, PEM and QM2. (After Bolt and Uhrhammer 1981 .)

ma1 mode data set. Their final model (C2) shows slight inhomogeneity at the base of the mantle, the top of the core and the regions on both sides of the ICB. Both the V p and density jump at the ICB are small. Hart et al. (1977) later examined the effect of correcting Earth model C2 for attenuation. The result was generally to increase seismic velocities, particularly shear velocities, throughout the model. The largest increases occur in those regions where Q is lowest. In spite of recent refinements, the resolution of P and S velocity gradients places considerable limitations on the sharpness of inferences that can be drawn about dynamical properties, density stratification and deviations from adiabatic conditions. Recent core models have been based on limited sets of data on travel-times and amplitudes of seismic core waves, and measurements of the eigenfrequencies of the Earth. A comparison shows that the differences between independently constructed models such as CAL6 (Bolt and Urhammer, 1975), PEM (Dziewonski et al., 1975) and QM2 (Hart et al., 1977) are almost everywhere less than 1% (see Fig. 1.18). The differences depend on weighting and selections of data and on assumptions on smoothing. Figure 1.19 gives the distribution of the seismic parameter 4

6ot

50 4000

1

I

1

3Ooo

2000

Radius

Fig. 1 .IS.A comparison of the seismic parameter and Q M 2 . (After Bolt & Urhammer, 1981 .)

-0 -0

3

1

I

0

loo0

(km)

4 calculated from three Earth models, CALG, PEM,

-

-

2 -

-

v

c I

1

I

I

I

I

Fig. 1.20. The PREM model. Dashed lines are the horizontal components of velocity. Where q is 1 the model is isotropic. The core is isotropic. (After Dziewonski and Anderson, 1981 ).

TABLE 1.5 Earth model PREM and its funcrionals evaluated at a reference period of 1 s. Above 220 km the mantle is transversely isotropic;the parameters given are ”equivalent” isotropic moduli and velocities. (After Dziewonski and Anderson, 1981 .) Radius (km)

Depth (km)

Density (gcmP)

VP

v5

(kms-’)

(kms-’)

Q,

0. 100.0 200.0 300.0 400.0 500.0 600.0 700.0 800.0 900.0 1000.0 1 1 00.0 1200.0 1221.5 1221.5 1300.0 1400.0 1500.0 1600 0 1700.0 1800.0 1900.0 2000.0 21 00.0 2200.0 2300.0 2400.0 2500.0 2600.0

6371 0 6271 0 6171 0 6071 0 5971 0 5871 0 5771 0 5671 0 5571 0 5471 0 6371 0 5271 0 5171 0 5149 5 51 49 5 5071 0 4971 0 4871 0 4771 0 4671 0 4571 0 4471 0 4371 0 4271 0 4171 0 4071 0 3971 0 3871 0 3771 0

13.08848 13.08630 13.07977 13.06888 13.05364 13.03404 13.01009 12.98178 12.94912 12-3121 1 12.87073 12.82501 12.77493 12.76360 12.16634 12.12500 12.06924 12.00989 11.94682 11.87990 1 1 -80900 11.73401 11.65478 1 1 -57119 1 1 -48311 11.39042 1 1 29298 11,19067 11.08335

11.26220 11.26064 11.25593 11.24809 11.23712 11.22301 11.20576 11,18538 1 1 ,I6186 11,13521 1 1.1 0542 11.07249 11.03643 1 1.02827 10.35568 10.30971 10.24959 10.18743 10.12291 10.05572 9.98554 9.91206 9.83496 9.75393 9.66865 9.57881 9.48409 9.38418 9.27876

3.66780 3.66670 3.66342 3.65794 3.65027 3.64041 3.62835 3.61411

85 85 85 85

3.59767 3.57905 3.55823 3.53522 3.51002 3.50432

0. 0.

0. 0. 0. 0. 0.

0. 0. 0. 0. 0.

0. 0. 0.

0

328 328 328 328 328 328 328 1328 1328 1328 1328 1328 1328 1328 57822 57822 57822 57822 57822 57822 57822 57822 57822 57822 57822 57822 57822 57822

0

57822

85 85 a5

85 a5 85

85 a5 85 85

0 0 0 0 0 0 0 0 0 0 0 0

0

108.90 108.88 108.80 108.68 108.51 108.29 108.02 107 70 107.33 106.91 106.45 105.94 105.38 105.25 107.24 106.29 105.05 103.78 102.47 101.12 99.71 98.25 96.73 95.14 93.48 91 7 5 89.95 88 06 86.10

ti

P

(kbar)

(kbar)

14253 14248 14231 14203 14164 14114 14053 13981 13898 13805 13701 13586 13462 13434 13047 12888 12679 12464 12242 1201 3 1 1 775 1 1 529 1 1 273 1 1009 10735 10451 101 58 9855 9542

1761 1759 1755 1749 1739 1727 1713 1696 1676 1654 1630 1603 1574 1567

0 0 0 0 0 0 0

0 0 0

0 0 0

0 0

IS

0.4407 0.4407 0.4408 0.4409 0.4410 0.4412 0.4414 0.6417 0.4420 0.4424 0.4428 0.4432 0.4437 0.4438 0.5000 0.5000 0.5000 0.5000 0.5000 0.5000 0.5000 0.5000 0.5000 0.5000 0.5000 0.5000 0.5000 0.5000 0.5000

Gravity

Pressure (kbar)

dkldp

3638.524 3636.131 3628.956 361 7.011 3600.315 3578.894 3552.783 3522.024 3486.665 3446.764 3402.383 3353.596 3300.480 3288.513 3288.502 3245.423 31 87.493 31 26.159 3061.461 2993.457 2922.221 2847.839 2770.407 2690.035 2606.838 2520.942 2432.484 2341.603 2248.453

2 3360 2 3363 2 3365 2 3369 2 3375 2 3382 2 3391 2 3402 23414 2 3428 2 3443 2 3460 2 3480 2 3486 3 7545 3 6539 3 5478 3 4649 3 401 7 3 3552 3 3230 3 3028 3 2927 3 291 1 3 2966 3 3080 3 3242 3 3441 3 3670

(crnr-2)

0. 36.56 73.11 109.61 146.04 182.39 21 8.62 254.73 290.68 326.45 362.03 397.39 432.51 440.02 440.03 463.68 494.13 524.77 555.48 586.14 61 6.69 647.04 677.15 706.57 736.45 765.56 794.25 822.48 850.23

2700 0 2800 0 2900 0 3000 0 31000 3200 0 3300 0 3400 0 3480 0 3480 0 3500 0 3600 0 3630 0 3630 0 3700 0 3800 0 3900 0 4000 0 4100 0 4200 0 4300 0 4400 0 4500 0 4600 0 4700 0 4800 0 4900 0 5000 0 51 00 0 5200 0 5300 0 5400 0 5500 0 5600 0

3671 .O 3571 .O 3471 .O 3371 .O 3271 .O 31 71 .O 3071 .O 2971 .O 2891 .O 2891 .O 2871 .O 2771 .O 2741 .O 2741 .O 2671 .O 2571 .O 2471 .O 2371 .O 2271 .O 21 71 .O 2071 .0 1971 0 1871 .O 1771.O 1671 .O 1571 0 1471 .O 1371 .O 1271 .O 1171 0 1071 .O 971 .O 871 .O 771 0

10.97091 10.85321 10.73012 10.60152 10.46727 10.32726 10.181 34 10.02940 9 90349 5.56645 5.55641 5.50642 5.49145 5.49145 5.45657 5.40681 5.35706 5.30724 5.25729 5.20713 5 15669 5.10590 5.05469 5.00299 4.95073 4.89783 4.84422 4.78983 4.73160 4.67844 4.62129 4.56307 4.50372 4.44317

9.16752 9.05015 8.92632 8.79573 8.65805 8.51298 8.36019 8.19939 8.06482 13.71660 13.71168 13.68753 13.68041 13.68041 13.59597 13.47742 13 36074 13.24532 13.13055 13.01579 12.90045 12.78389 12.66550 12.54466 12.42075 12.29316 12.161 26 12.02445 11.88209 11.73357 11.57828 11.41560 11.24490 11.06557

0 0

0 0 0 0

0 0

0 726466 726486 726575 7 26597 7 26597 723403 718892 714423 709974 705525 701053 696538 691957 687289 682512 6 77606 672548 667317 661891 656250 650370 644232 637813 6 31091 624046

0

0 0 0

0 0 0

0 0

312 312 312 312 312 312 312 312 312 312 312 312 312 312 312 312 312 312 312 312 312 312 312 312 312

57822 57822 57822 57822 57822 57822 57822 57822 57822 57822 57822 57822 57822 57822 57822 57822 57822 57822 57822 57822 57822 57822 57822 57822 !m22 57822 57822 57822 57822 57822 57822 57822 57822 57822

84 04 81 91 79 68 77 36 74 96 ?2 47 69 89 67 23 65 04 17 78 17 64 16 96 16 76 16 76 1508 11273 11046 708 23 10604 103 88 101 73 99 59 97 43 95 26 93 06 90 81 88 52 86 17 83 76 81 28 78 72 76 08 73 34 70 52

9220 8889 8550 8202 7846 7484 7116 6743 6441 6556 6537 6440 641 2 641 2 6279 6095 591 7 5744 5575 5409 5246 5085 4925 4766 4607 4448 4288 41 za 3966 3803 3638 3471 3303 31 33

0 05000 0 05000 0 05000 0 05000 0 05000 0 05000 0 05000 0 05000 0 05000 2938 03051 2933 03049 2907 03038 2899 03035 2899 03035 2855 03026 2794 03012 2734 02998 2675 02984 2617 02971 2559 02957 2502 0 2943 2445 02928 2388 02913 2331 02898 2273 02881 2215 02864 2157 02846 2098 02826 2039 02805 1979 02183 1918 02758 1856 02731 1794 02701 1730 02668

21 53 189 2055 978 1956.991 1856.409 1754418 1651.209 1546.982 441 941 357 51 0 357 509 345 61 9 287 067 269 742 269 741 229 71 9 1173 465 1 1 18 207 1063 864 101 0 363 957 641 905 646 854 332 803 660 753 598 704 119 655 202 606 830 558 991 511 676 464 882 41 8 606 372 852 321 673 282 928

33919 3 41a0 3 4448 3 471 4 3 4972 3 5215 3 5437 3 5629 3 5769 16435 7 6434 16424 16420 3 3344 3 2957 3 2443 3 2029 31716 3 1503 3 1393 3 1383 3 1472 3 1657 3 1935 3 2302 3 2750 3 3276 3 3871 3 4527 3 5236 3 5989 3 6775 3 7582 3 8403

877 46 904 14 930 23 955 70 980 51 1004 64 1028 04 1050 65 1068 23 1068 23 1065 32 1052 04 1048 44 1048 44 1040 66 1030 95 1022 72 1015 80 101006 1005 35 1001 56 998 59 996 35 994 74 993 69 993 14 993 01 993 26 993 83 994 67 995 73 996 98 998 36 999 85

T A B L E l . 5 continued overleaf

TABLE 1.5-contmued

5600 0 5650 0 5701 0 5701 0 5736 0 5771 0 5771 0 5821 0 5871 0 5921 0 5971 0 59Ji 0 60160 6061 0 6106 0 6151 0 6151 0 6186 0 6221 0 6256 0 6291 0 6291 0 631 1 0 6331 0 6346 6 6346 6 6356 0 6356 0 6368 0 6368 o 6371 0

771 0 721 0 670 0 670 0 635 0 600 0 600 0 550 0 500 0 450 0 400 0 400 0 355 0 3100 265 0 220 0 220 0 1850 1500 1150 80 0 80 0 60 0 40 0 24 4 24 4 150 150 30 30 0

4 4431 6 4 41 241 4 38071 3 9921 4 3 98399 3 97584 3 97584 3 91 282 3 84980 3 78678 3 72378 3 54325 3 51 639 3 48951 3 46264 3 43578 3 35950 3 36330 3 36710 3 37091 3 37471 3 37471 3 37688 3 37906 3 38076 2 90000 2 90000 2 60000 2 60000 102000 102M)o

VP

vs

(kms-')

(kms-')

11 06556

6 24046 6 0941 8 5 94508 5 57020 5 5431 1 5 51 602 5 51 600 5 3701 4 5 22428 5 07842 4 93259 4 76989 4 73840 4 70690 4 67540 4 64391 4 41 885 4 431 08 4 44361 4 45643 4 46953 4 46954 4 4771 5 4 48486 4 49094 3 90000 3 90000 3 20000 3 20000

10 91 005

I075131 10 26622 10 21 203 10 15782 10 15782 9 901 85 9 64588 9 38990 9 13397 8 90522 8 81 867 8 73209 8 64552 a 55896 7 98970 801180 8 03370 8 0554Q 8 07688 8 07689 8 08907 810119 811061 6 80000 6 80000 5 80000 5 80000 145000 145000

0 0

# QP

31 2 31 2 31 2 143 t 43 143 143 143 143 143 143 143 143 143 143 143 80 80 80 80 80 600 600 600 600 600 600

600 600 0 0

QA

57822 57822 57822 57822 57822 57822 57822 578.22 57822 57822 57822 57822 57822 57822 57822 57822 57822 57822 57822 57822 57822 57822 57822 57822 57822 57822 57822 57822 57822 57822 57822

L:

( k m 2 K z ) (kbar)

70.52 69.51 68.47 64.03 63.32 62.61 62.61 59.60 56.65 53.78 50.99 48.97 47.83 46.71 45.60 44.50 37.80 38.01 38.21 38.41 38.60 38.60 38.71 38.81 38.89 25 96 25.96 i9.99 19.99 210 2.10

P

(kbar)

u

21

02668 02732 02798 02914 02911 02909 02909 02917 02924 02933 02942 02988 02971 02952 02933 0 2914 02797 0.2797 0.2796 02795 02793 02793 02792 02790 02789 02549 02549 02812 02812 0 05000

21

0 05000

31 33 3067 2999 2556 2523 2489 2489 2332 21 81 2037 1899 1735 1682 1630 1579 1529 1270 1278 1287 1295 1303 1303 1307 1311 1315 753 753 520 520

1730 1639 1548 1239 1224 1210 1210 1128 1051 977 906 806 790 773 757 741 656 660 665 669 674 674 677 680 682 441 441 266 266

Pressure (kbar)

282 927 260 783 238 342 238 334 224 364 21 0 426 21 0 425 190 703 171 311 152 251 133 527 133 520 117 702 102 027 86 497 71 115 71 108 59 466 47 824 36 183 24 546 24 539 17 891 1 1 239 6 043 6 040 3 370 3 364 0 303 0 299 -0 000

dk/dp

2 981 9 3 0086 3 0358 2 4000 2 3868 2 3734 8 091 0 7 8833 7 6761 7 4695 7 2633 33718 3 3369 3 301 7 3 2662 3 2305 - 0 7364 - 0 7200 -0 7035 - 0 6868 - 0 6700 - 0 6700 - 0 6603 -0 6505 - 0 6428 - 0 0000 0 0000 0 0000 - 0 0000 - 0 0000

0 000

Gravity (ems-')

999 85 1000 63

1001 43 1001 43

1000 88 1000 38 1000 38 999 65 998 83 997 90 996 86 996 86 995 22 993 61 992 03 990 48 990 48 989 1 1 987 83 986 64 985 53 985 53 984 93 984 37 983 94 983 94 983 32 983 31 982 22 982 22 gal 56

General Physical Properties of the Earth

69

(equation 1.3) for the above three models and shows that the gradients of this parameter are not well-resolved. (It is the gradient of 4 that is important for solid state, temperature, compositional and convection studies.) Recent seismological evidence has favoured a simple two-layer core with a liquid OC, and a sharply-bounded solid IC. As discussed in 51.2, the hypothesis of a sharply-bounded transition region F has fallen from favour because of the explanation of precursors to P K I K P in terms of scattering sources near the MCB. Nevertheless, velocity jumps in F need be as small as only 0.01 km s-' to give rise to back-scattered waves with the observed amplitudes (Bolt and Urhammer, 1981). The P wave velocity distribution in the O C shows that the P velocity between 1600km and about 1200km changes by only a few percent, i.e. there is near-independence of sound speed over a pressure change of 400 kbar. Thus, a region F between E and G does exist. The density distributions in E (see Fig. 1.18) determined from eigenvibration and travel-time inversions, without using the assumptions of Adams and Williamson, confirm a density gradient close to the Adams and Williamson value (at least, in the central part of E). The conclusion is that zero deviations from homogeneity and adiabaticity throughout most of E are sufficient but not necessary to satisfy the data. The largest differences in the estimates of density occur in the IC. However, both CAL6 and PEM have about the same jump in density (0.6 g cm-3) at the ICB. Models PEM and QM2 have essentially equal density distributions throughout the liquid OC. Observations of the amplitude of P K i K P and P K I I K P imply that there must be a sharp discontinuity in density, or in compressibility, or both, at the ICB. A jump in rigidity will not yield either of these reflections with the amplitudes observed. The central densities of CAL6, PEM and QM2 are 13.35, 13.01 and 12.57 g cm-3 respectively. In 1981 Dziewonski and Anderson published their Preliminary Earth Model (PREM) which they had presented at the 1979 IUGG General Assembly in Canberra. In order to satisfy the large amount of normal mode, surface wave and body wave data, they found it necessary to introduce anelastic dispersion and anisotropy for the mantle above 220 km. Their model is thus frequency-dependent. Like the earlier model PEM, PREM is parametric in nature, the various physical parameters being expressed as loworder polynomials in the radius, of order between zero and three in the various regions of the Earth. They used about 1000 normal mode periods, 500 summary travel-time observations and 100 normal mode Q values. This data set was supplemented with a special study of 12 years of ISC phase data which yielded an additional 1.75 x lo6 travel-time observations for P and S waves. To obtain reasonably good control over velocities in the OC, they com-

70

The Earths Core

bined data from four travel-time branches: S K K S , SKS, P K P ( A B ) and P K P ( B C ) . Differential travel-times, PCP - P (Engdahl and Johnson, 1974) and ScS- S (Jordan and Anderson, 1974) are important for the control of the radius of the OC. The P K i K P - PcP differential travel-times of Engdahl et al. (1974) give the best control of the radius of the IC. Their final model is shown in Fig. 1.20 and values of various physical parameters in Table 1.5. This table includes values of the parameter r] of Bullen (see (5.32)) which represents a measure of the deviation of a model from the Adams-Williamson equation (for which r] = 1). For the most part r] in the core and lower mantle is very close to unity, the small deviations, in some cases, being an artifact of the polynomial representation. In the lower 150 km of the mantle (region D”), the velocity is nearly constant; the radius of the OC is 3480 km. Dziewonski and Anderson suggest that the core radius may be 1.7 km larger, which would satisfy the observations that slower velocities may exist in D”.

References

Adams, R. D. (1972). Multiple inner core reflections from a Novaya Zemlya explosion. Bull. Seism. SOC.Amer. 62, 1063. Adams, R. D. and Randall, M. J. (1963). Observed triplication of PKP. Nature 200,744. Adams, R. D. and Randall, M. J., (1964). The fine structure of the Earth’s core. Bull. Seism. Soc. Amer. 54, 1299. Akopyan, S. Ts., Zharkov, V. N. and Lyubimov, V. M. (1975). The dynamic shear modulus in the interior of the Earth. Dokl. Earth Sci. Ser. 223, 1. Akopyan, S. Ts., Zharkov, V. N. and Lyubimov, V. M. (1976). Corrections to the eigenfrequencies of the Earth due to the dynamic shear modulus. Izo. Acad. Sci. U S S R Phys. Solid Earth 12,625. Alder, A. J. and Trigueros, M. (1977). Suggestion of a eutectic region between the liquid and solid core of the Earth. J . Geophys Res. 82,2535. Alterman, Z., Jarosch H. and Pekeris, C. L. (1959). Oscillations of the Earth. Proc. Roy. SOC. London A252,80. Al’tshuler, L. and Sharipdzhanov, L. (1971). Distribution of iron in the Earth and its chemical differentiations. Izv. Acad. Sci. USSR. Phys. Solid Earth 4 3 . Anderson, D. L. (1967). A seismic equation of state. Geophys. J . 13,9. Anderson, D. L. and Hanks, T. C. (1972). Formation of the Earth’s core. Nature 237,387. Anderson, D. L. and Hart, R. S. (1976). An Earth model based on free oscillations and body waves. J . Geophys. Rex 81, 1461. Anderson, D. L. and Hart, R. S. (1978a). Attenuation models of the Earth. Phys. Earth Planet. Int. 16, 289. Anderson, D. L. and Hart, R. S.(1978b). Q of the Earth. J . Geophys. Res. 83,5869. Anderson, D. L., Ben-Menahem, A. and Archambeau, C. B. (1965). Attenuation of seismic energy in the upper mantle. J . Geophys. Res. 70,505. Anderson, 0. L. (1973). Comments on the power law representation of Birch’s law. J . Geophys. Res. 78,4901. Anderssen, R. S. and Seneta, E. (1971). A simple statistical estimation procedure for Monte Carlo inversion in geophysics. Pure Appl. Geophys. 91,5014.

General Physical Properties of the €arrh

71

Anderssen, R. S. and Seneta, E. (1972). A simple statistical estimation procedure for Monte Carlo inversion in geophysics. 11: Efficiency and Hempel's paradox. Pure Appl. Geophys. 96, 5. Anderssen, R. S., Worthington, M. H. and Cleary, J. R. (1972). Density modelling by Monte Carlo inversion. I: Methodology. Geophys. J . 29,433. Backus, G. (1971). Inference from inadequate and inaccurate data. In Mathematical Problems in the Geophysical Sciences (W. H. Reid, ed.). Lectures in Applied Mathematics, Vol. 14, Amer. Math. SOC. Backus, G. E. and Gilbert, J. F. (1961). The rotational splitting of the free oscillations of tfie Earth. Proc. Nat. Acad. Sci. U S A 47, 362. Backus, G. E. and Gilbert, J. F. (1967). Numerical applications of a formalism for geophysical inverse problems. Geophys. J . 13,247. Backus, G. E. and Gilbert, J. F. (1968). The resolving power of gross Earth data. Geophys. J . 16, 169. Backus, G. and Gilbert F. (1970). Uniqueness in the inversion of inaccurate gross Earth data. Ehil. Trans. Roy. Soc. London A266, 123. Balakina, L. M. and Vvedenskaya, A. V. (1962). Izv. Akad. Nauk SSR. Ser. Geojiz 11,909. Bertrand, A. E. S. and Clowes, R. M. (1974). Seismic array evidence for a two-layer core transition zone. Phys. Earth Planet. Int. 8,251. Berzon, I. S., Kogan, S. D. and Pessechnik, I. P. (1972). The character of the mantle-core boundary from observations of PCP waves. Earth Planet. Sci. Lett. 16, 166. Berzon, I. S., Pessechnik, I. P. and Polikarpov, A. M. (1974). The determination of P wave attenuation values in the Earth's mantle. Geophys. J . 39,603. Birch, F. (1952). Elasticity and constitution of the Earth's interior. J . Geophys. Res. 57,227. Birch, F. (1960). The velocity of compressional waves in rocks to 10 kilobars, 1. J . Geophys. Res. 65, 1083. Birch, F. (1961a). The velocity ofcompressional waves in rocks to 10 kilobars, 2. J . Geophys. Res. 66,2199. Birch, F. (1961b). Composition of the Earth's mantle. Geophys. J . 4,295. Birch, F. (1964). Density and composition of mantle and core. J . Geophys. Res. 69,4377. Bolt, B. A. (1959). Travel times of PKP up to 145".Geophys. J . 2,190. Bolt, B. A. (1962). Gutenberg's early PKP observations. Nature 196, 122. Bolt, B. A. (1964). The velocity of seismic waves near the Earth's centre. Bull. Seism. SOC.Amer. 54,191. Bolt, B. A. (1972). The density distribution near the base of the mantle and near the Earth's centre. Phys. Earth Planet. Int. 5,301. Bolt. B. A. (1977). The detection of PKIIKP and damping in the inner core. Ann. Geojis. 30,507. Bolt, B. A. and Canas, J. A. (1985). Constraints from core reflections on mantle Q and density at the core boundary. Phys. Earth Planet. Int. 38,l. Bolt, B. A. and Niazi, M. (1984). S velocities in D" from diffracted SH-waves at the core boundary. Geophys. J . 79,825. Bolt, B. A. and Qamar, A. (1970). Upper bound to the density jump at the boundary of the Earth's inner core. Nature 228, 148. Bolt, B. A. and Urhammer, R. (1975). Resolution techniques for density and heterogeneity in the Earth. Geophys. J . 42,419. Bolt, B. A. and Urhammer, R. (1981). The structure, density and homogeneity of the Earths core. In Evolution ofthe Earth (R. J. OConnell, W. S. Fyfe eds). Geodynamics Series Vol. 5, Amer. Geophys. Union. Bolt, B. A,, ONeill, M. and Qamar, A. (1968). Seismic waves near 110": is structure in core or upper mantle responsible? Geophys. J . 16,475.

72

The Earth’s Core

Bolt, B. A,, Niazi, M. and Somerville, M. (1970). Diffracted SCS and the shear velocity of the core boundary. Geophys. J . 19,299. Buchbinder, G. G. R. (1968). Properties of the core-mantle boundary and observations of PCP. J . Geophys. Res. 73,5901. Buchbinder, G. G. R. (1971). A velocity structure of the Earth’s core. Bull. Seism. Soc. Amer. 61, 429. Buchbinder, G. G. R. (1976). A test of new Earth models against PCP and PmKP travel times. Phys. Earth Planet. Int. 11,13. Buchbinder, G. G. R. and Poupinet, G. (1973). Problems related to PCP and the core-mantle boundary illustrated by two nuclear events. Bull. Seism. SOC.Amer. 63,2047. Buland, R. and Gilbert, F. (1978). Improved resolution of complex eigenfrequencies in analytically continued seismic spectra. Geophys. J . 52,457. Bullen, K. E. (1954). Seismology. Methuen, London. Bullen, K. E. (1963). An Introduction to the Theory of Seismology. Cambridge University Press, London. Bullen, K. E. (1965). Models for the density and elasticity of the Earths lower core. Geophys. J . 9, 233. Bullen, K. E. (1975). The Earths Density. Chapman and Hall. Bullen, K. E. and Haddon, R. A. W. (1970). Evidence from seismology and related sources on the Earth’s present internal structure. Phys. Earth Planet. Int. 2,342. Busse, F. H. (1974). On the free oscillations of the Earth’s inner core. J . Geophys. Res. 79,753. Caloi, P. (1961). Seismic waves from the outer core and the inner core. Geophys. J . 4,139. Canas, J. A. and Bolt, B. A. (1983). Nuevo metodo para la obtencion simultanea del factor medio de calidad especifica en el manto y litosfera terrestre y el salto de densidades en el limite. Nucleo-manto. Teoria Anales de Fisica Serie A 79, 57. Cathles, L. M. (1975). The oiscosity of the Earth’s mantle. Princeton University Press, Princeton, MA. Chang, A. C. and Cleary, J. R. (1978). Precursors to PKKP. Bull. Seism. Soc. Amer. 68,1059. Chang, A. C. and Cleary, J. R. (1981). Scattered P K K P further evidence for scattering at a rough core-mantle boundary. Phys. Earth Planet. Int. 24, 15. Chowdhury, D. K. and Frasier, C. W. (1973). Observations of PCP and P phases at LASA at distances from 26” to 40”. J . Geophys. Res. 78,6021. Choy, G. L. and Cormier, V. F. (1983). The structure of the inner core inferred from short-period and broadband GDSN data. Geophys. J . 7 2 , l . Christensen, N. I. (1966). Elasticity of some ultrabasic rocks. J . Geophys. Res. 71,5921. Christensen, U. (1984). Instability of a hot boundary layer and initiation of thermo-chemical plumes. Ann. Geophys. 2,311. Chung, D. H. (1974). General relationship among sound speeds. I: New experimental information. Phys. Earth Planet. Int. 8, 113. Clark, R. A. and Pearce, R. G. (1981). Comments on “Lateral heterogeneity at the base of the mantle revealed by observations of amplitudes of PKP phases” by I. S. Sacks, J. A. Snoke and L. Beach. Geophys. J . 66,741. Clark, S. P. Jr. and Ringwood, A. E. (1964). Density distribution and constitution of the mantle. Rev. Geophys. 2,35. Clark, S . P. Jr. and Turekian, K. K. (1979). Thermal constraints in the distribution of long-lived radioactive elements in the Earth. Phil. Trans. Roy. Soc. London A291,269. Cleary, J. R. (1969). The S-velocity at the core-mantle boundary from observations of diffracted S . Bull. Seism. Soc. Amer. 59, 1399. Cleary, J. R. (1974). The D region. Phys. Earth Planet. Int. 9, 13.

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Smith, M. L. (1976). Translational inner core oscillations of a rotating, slightly elliptical Earth. J . Geophys. Res. 81,3055. Smith, M. L. (1977). Wobble and nutation of the Earth. Geophys. J . 50, 103. Smylie, D. E. (1973). Dynamics of the outer core. Proc. 2nd Int. Symp. Geod. Phys. Earth, Potsdam. Smylie, D. E. (1974). Dynamics of the outer core. Veroff Zentralinst. Phys. Erde. Akad. Wiss. DDR 30,91. Smylie, D. E. and Mansinha, L. (1971). The elasticity theory of dislocation in real Earth models and changes in the rotation of the Earth. Geophys. J . 23,329. Smylie, D. E. and Rochester, M. G. (1981). Compressibility, core dynamics and the sub-seismic wave equation. Phys. Earth Planet. Int. 24,308. Stacey, F. D. (1977). A thermal model of the Earth. Phys. Earth Planet. Int. 15,341. Stacey, F. D. and Loper, D. E. (1983). The thermal boundary layer interpretation of D and its role as a plume source. Phys. Earth Planet. Int. 33,45. Stiller, H., Franck, S. and Schmidt, U. (1980). On the attenuation of seismic waves in the Earth’s core. Phys. Earth Planet. Int. 22,221. Subiza, G. P. and Bath, M. (1964). Core phases and the inner core boundary. Geophys. J . 8,496. Taggart, J. and Engdahl, E. R. (1968). Estimation of PCP travel times and depth to the core. Bull. Seism. SOC.Amer. 58,1293. Teng, T. L. (1968). Attenuation of body waves and the Q structure of the mantle. J . Geophys. Res. 73,2195. Wang, C.-Y. (1968). Constitution of the lower mantle as evidenced from shock wave data for some rocks. J . Geophys. Res. 73,6459. Wang, C.-Y. (1970). Density and constitution of the mantle. J . Geophys. Res. 75,3264. Wang, C.-Y. (1972). A simple Earth model. J . Geophys. Res. 77,4318. Wiggins, R. A. (1969). Monte Carlo inversions of body wave observations. J . Geophys. Res. 74, 3171. Wiggins, R. A. (1972). The general linear inverse problem: implication of surface waves and free oscillations for Earth structure. Reu. Geophys. Space Phys. 10,251. Wiggins, R. A,, McMechan, G . A. and Toksoz, M. N. (1973). Range of Earth structure nonuniqueness implied by body wave observations. Rev. Geophys. Space Phys. 11,87. Williamson, E. D. and Adams, L. H. (1923). Density distribution in the Earth. J . Wash. Acad. Sci. 13,413. Won, I. J. and Kuo, J. T. (1973). Oscillation of the Earth’s inner core and its relation to the generation of geomagnetic field. J . Geophys. Res. 78,905. Worthington, M. H., Cleary, J. R. and Anderssen, R. S. (1972). Density modelling by Monte Carlo inversion. 11: Comparison of recent Earth models. Geophys. J . 29,445. Wright, C. (1973). Array studies of P phases and the structure of the D region of the mantle. 1. Geophys. Res. 78,4965. Wunsch, C. (1974). Simple models of the deformation of an Earth with a fluid core, I. Geophys. J . 39,413. Wunsch, C. (1975). Simple models of the deformation of an Earth with a fluid core, 11. Dissipation and magnetohydrodynamic effects. Geophys. J . 41,165. Zener, C. (1948). Elasticity and Anelasticity ofMetals. Chicago University Press, Chicago. Zharkov, V. N., Karpov, P. B. and Leontjev, V. V. (1985). On the thermal regime of the boundary layer at the bottom of the mantle. Phys. Earth Planet. Int. 41,138.

Chapter Two

The Origin of the Core

2.1

Introduction

A fundamental question in any discussion of the Earth’s core is its origin. Has the Earth always had a core, or has its present structure evolved over geological time? Such a question cannot be divorced from the much broader issue of the origin of the Earth itself. There is one constraint, however, that can be imposed upon possible evolutions. Most theories of the origin of the Earth’s magnetic field ascribe it to motions in the liquid OC (see $4.2). Rocks as old as 3500 Ma have been found which possess remanent magnetization, so that it is extremely probable that the Earth had a molten OC, comparable in size to that at present at least that long ago. No similar deductions can be made about the state of the IC. With regard to the broader question of the origin and evolution of the solar system, it must not be forgotten that it is by no means certain a priori that the problem can be given a definite answer. It may well be that all memory has been lost of the circumstances under which our solar system was born. All that we can do is to attempt to derive its present state from an assumed event or series of events that occurred in the distant past. Thus, in a sense, the method is one of trial and error. It is more reasonable to assume that planets are normally present in the vicinity of certain stars than to suppose that our planetary system is unique or at least very rare-there are about 200,000 million stars in our galaxy alone. The origin of the solar system is thus part of a very much larger problem-the evolution of the sun and stars in general. Theories of the origin of the Earth and other members of the solar system 81

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The Earth's Core

may be classified in a number of ways. The Earth may have cooled from a hot gas or accreted cold from dust particles, probably of composition similar to that of chondritic meteorites. In the past, most theories of the origin of the Earth have assumed that the proto-Earth was homogeneous and that the present differentiation into a core, mantle and crust occurred later. There are, however, a number of difficulties with such theories, and there has been some discussion in recent years of non-homogeneous models. In such models, the core of the Earth is formed first and the mantle is deposited upon it later. Non-homogeneous models can also have either a hot or a cold origin. A hot origin involves a consideration of the order of condensation of the elements in the primitive solar nebula; a cold origin involves a study of the mechanics of the accretion process. 2.2

The Accretion Mechanism

Regardless of how stars like the sun form, at some time matter in the form of grains or small bodies must have aggregated to form the terrestrial planets. There are two possible mechanisms by which this may have been accomplished-gravitational collapse and accumulation. In the first case, matter aggregates into larger objects under the influence of a gravitational field following a gravitational instability. In the second case, bodies grow by the gradual addition of matter following collisions and coherence. These two processes may occur with or without the presence of a dynamically significant gas phase. During the initial stages of the evolution of the solar system, large amounts of gas must have been present and the beginning of planetary growth must have involved a gas phase. However, it is possible that the gas was removed by solar ultraviolet or corpusular radiation on a time-scale less than that required for the growth by collisions of the initial solid bodies that formed from gravitational instabilities in the dust layer of the solar nebula. A large body of literature exists on gas-free accumulation models of the evolution of the terrestrial planets: see, for example, Safronov (1969). The evolution of terrestrial planetary material from dust to planets can be considered in roughly three main stages. The early stage, producing kilometre-size planetesimals formed by smallscale gravitational instabilities of the dust layer alone. The intermediate stage, during which kilometre-sized planetesimals grow to 1000-km-sized bodies. The dynamical evolution of this stage is governed by two competing mechanisms: collisional fragmentation and gravitational accumulation. The late stage, or final stage, during which 1000-km-sized embryos grow to full-sized planets, with the right orbital spacing, eccentricities, inclina-

The Origin of the Core

83

tions, and rotations. Wetherill (1980a) has reviewed in detail these different phases of the formation of the terrestrial planets. For the smallest particle sizes, accumulation cannot be very rapid. The particles will collide, partly as a result of their Brownian motion in the gas, and partly as a result of acceleration by very weak electric fields that can be expected to be produced in the nebula through the interaction of convection with ionization produced by natural radioactivity. This state of accumulation may be greatly accelerated as a result of the natural magnetism of the interstellar grains. Purcell and Spitzer (1971) believe that the interstellar grains most probably would be ferromagnetic or super-paramagnetic, so that they could be aligned by the very weak interstellar magnetic field. If this is the case, and it appears to be consistent with evidence from meteorites (Brecher, 1971), then the interstellar grains would be able to come together to form considerably larger units, both during the late stages of the collapse that formed the primitive solar nebula and during the early history of the solar nebula itself. The cross-section for magnetic capture of one particle by another would be very much greater than the geometric cross-sections involved in ordinary non-magnetic collisions. The initial step is the presence of interstellar grains in the gas which, after some dregree of gas absorption, find themselves concentrated in the nebular plane and somehow manage to accrete into larger bodies. The main problem is the fact that the number of these bodies must eventually be considerably reduced: from swarms of small bodies, eventually a few large planets must be produced. This reduction can only take place by collisions. The model of Safronov (1969, 1972) starts with a quiet nebula in which dust particles rotate and begin to sink slowly (under gravity) towards the equatorial plane. The assumption is made that all the particles such a dust particle meets on the way down adhere to it, and, upon arriving at mid-plane (after 2: 103-104 years) it has reached a size of N 1 cm. The critical assumption is the so called “cold welding” of matter which represents an upper limit to the efficiency of the process. When the density of the dust layer exceeds a certain value, condensation of matter would be induced by gravitational instabilities. In those regions close to the sun, gas motions would have been too fast for gravitational instabilities to take place and growth of bodies could only occur through aggregation during collisions. Safronov argued that the relative velocities of the bodies would be determined by their gravitational perturbations at encounters and should be less than 1 m s - so long as the objects were less than 5 km in diameter. Alfvkn and Arrhenius (1970a,b, 1973) solved the problem of reducing the relative velocities of colliding bodies so as to facilitate coalescence by the concept of “jet” streams. They argued that collisions

a4

The Earth‘s Core

between particles in Keplerian orbits do not lead to a spreading but to an equalization of the orbits of the particles when the collisional frequency is smaller than the orbital frequency. This state, in any theory of this type, either prevails originally or develops by accretion. It results in a focusing of particles into jet streams with increasingly similar orbital elements and velocities of the individual particles. The partially inelastic collisions will lead to the growth of embryo bodies of which some eventually reach the size where gravitational accretion becomes increasingly important. Alfven and Arrhenius believe that the maximum possible velocity for accretion is 0.5 km s- It must be pointed out that the jet stream hypothesis of Alfvtn and Arrhenius has not been universally accepted (see e.g. Reeves (1972), p. 86-87, for a discussion on this point). Cameron (1972) has also recognized that, during the lifetime of the nebula, collision probabilities were too low for accretion to take place in any reasonable time. He suggested, however, that during the collapse phase from the interstellar cloud to the nebula the situation was much more favourable, since violent gas motions (in the form of turbulent eddies) would have considerably increased the collision probabilities of gas and grains. Cameron estimated typical grain velocities to be N 10 m s-’. At the end of the collapse the nebula is formed and turbulence rapidly decays out, embedding a set of bodies up to 20 cm in radius. Goldreich and Ward (1973) investigated the process whereby dust particles can accrete to form gravitationally active objects. They concluded that gravitational instabilities can account for the growth of objects up to several km in radius. These instabilities develop, not in the gaseous solar nebula, but in a thin disc of particulate matter that forms in the central plane during an initial condensation phase. A first generation of planetesimals whose radii range up to 0.1 km form from this dust disc by gravitational collapse in a time-scale of order 1 year. The resulting disc of first-generation planetesimals is still gravitationally unstable and the planetesimals group into clusters. The clusters then contract on a time-scale of a few thousand years to form a second generation of planetesimals having radii of order 5 km. The contraction is controlled by the rate at which gas drag damps their internal rotation and random kinetic energy. In Goldreich and Ward’s model planetary accretion does not depend on the if the dust grains tend to “stickiness” of the surface of the dust particles-ven stick together upon impact, the growth of solid bodies by this process would be much slower than by gravitational instabilities. However, Weidenschilling (1980) pointed out that non-Keplerian rotation of the solar nebula results in shear between the gas and a dust layer. This shear produces turbulence within the layer that inhibits gravitational instability unless the mean particles are larger than a critical size (- 1 cm at 1 AU). Weidenschilling believes that

’.

85

The Origin of the Core

some coagulation of grains is possible in the solar nebula; Van der Waals forces may be sufficient to allow formation of centimetre-sized aggregates during settling to the central plane. Growth is primarily due to the sweeping up of small particles by larger ones: settling times are lo3 years in the inner nebula and lo4 years in the outer regions. Drag-induced relative velocities then result in collisions and larger bodies can accrete if reasonable impact lo3cm, relative velocities decrease strength is assumed. After sizes reach and the growth rate declines. Damping of out-of-plane motions by gas drag concentrates these intermediate-sized bodies into a thin layer that is subject to gravitational instability. Kilometre-sized planetesimals are then formed. In Weidenschilling’s model, the formation of planetesimals is a composite process involving, first, a stage of collisional accretion, followed by gravitational instability. Similar conclusions were reached by Nakagawa et al. (1981) using a different numerical method and different assumptions in their model. In the evolution of a growing swarm of planetesimals, relative velocities will be increased by mutual gravitational perturbations, and decreased by dissipative collisions. If the relative velocity is too high (say, several times the gravitational escape velocity of the growing planet), fragmentation rather than accumulation will be dominant and planets will not grow. On the other hand, if relative velocities are too low, the system of planetesimals will be in nearly circular concentric orbits, and the collisions required for growth will not take place. Safronov (1962) showed that for plausible assumptions regarding dissipation of energy in collisions, and size distribution of the bodies, the mutual gravitational perturbations of the bodies cause their mean relative velocity to be only somewhat less than the escape velocity of the larger bodies. Thus, throughout the entire course of planetary growth, from 1-km planetesimals to Earth-sized planets, the system regulated itself in such a way that the larger bodies could always grow, whereas smaller objects would fragment. In its initial, non-gravitational stage, the accretion process may be regarded as a series of collisions between grains in intersecting orbits under conditions that result in a net transfer of mass to some of the grains. An important parameter for accretion theories is the range of impact velocities over which accretion occurs. Kerridge and Vedder (1 972) experimentally investigated this velocity range by observing the impact at different velocities of particles similar to material thought to be present in the early solar system. The projectile material used was kaolinite (a hydrated aluminium silicate) in the form of thin micrometer-sized laminated flakes with a density of 2.63 g cm-3. Such material structurally resembles the major mineral component of the primitive type I carbonaceous chondrites and may well have been widespread in the early solar system. For target material, thick ( - 5 mm) sections of clinochrysotile (a magnesium silicate) with a density of 2.50 g cm- were

-

-

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86

The Earths Core

used. They found that in any velocity range, impacts in which a significant transfer of projectile material occurred were outnumbered by those in which little or no transfer took place. Moreover, such impacts, at least those above 2 km s-’ were accompanied by the formation of craters with dimensions comparable to or greater than those of the projectile. Thus no accretion could take place for impacts in the velocity range studied (1.5-9.5 km s-l). By contrast, Neukum (1968) showed that for micrometer-sized iron particles impacted into a variety of metallic targets, almost the entire projectile mass is transferred to the target in the velocity range 0.5-13 km s - I . Below 0.5 km s - ’ iron particles rebounded from the metallic targets. There is thus a striking difference between the accretion behaviour of silicates and that of metals-over at least part of the velocity range in which silicate accretion was efficient, metal particles would have rebounded without accreting. Simple accretion theory cannot, therefore, be applied to silicate particles in markedly dissimilar orbits but must be restricted to situations characterized by generally low interparticle velocities. Whether or not impacts at lower velocities can actually lead to accretion of material is not known. No obvious sticking mechanism exists for bonding normal silicates together during lowvelocity impact, although a number of mechanisms have been proposed based on assumed properties of primordial dust grains. Possibly the sticking process was aided by the presence of the volatile coatings on the grains. Alternatively, adhesion may have been prompted by electrostatic charge asymmetries produced in the surface regions of circumsolar grains by charged particle irradiation, as proposed by Arrhenius et al. (1972) A third possible mechanism, suggested by Maurette and Bibring (1972), is the release of stored energy from the irradiation-damaged amorphous rims of grains exposed to the solar wind. Kerridge and Vedder (1972) suggested that metal-silicate fractionation in the solar system may have been affected by differences in the accretionary behaviour of metal and silicate particles. In particular, at the low velocities that must have prevailed during most of the accretionary process, silicates would have preferentially accreted. Similarly, if some regions of the solar system were characterized by higher interparticle velocities, bodies enriched in metal may have been produced. In this respect it may be significant that among the terrestrial planets there is a rough tendency for density, and therefore assumed metal content, to increase with orbital velocity. Dole (1970) developed a computer program to simulate the formation of planetary systems. Planets are assumed to form by aggregation of particles within a cloud of dust and gas surrounding the newly-formed sun. In his model nuclei are “injected” into the cloud one at a time, on elliptical orbits. The dimensions of the semimajor axis and the eccentricity of the orbit of each nucleus are determined by using random numbers. As the nuclei orbit within

87

The Origin of the Core

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Fig.2.1. Planetary systems generated by Dole (1 970). Solid circles represent terrestrial planets, while open circles indicate Jovian planets that have accreted gas as well as dust. The radius of each circle is scaled solely by the cube root of the planet's mass, given in the figures in units of Earth masses. The positions of Jovian planets are given by the centres of the circles. All planets produced are well separated, as the relative positions of their centres indicate. (After lsaacman and Sagan. 1977.)

the cloud they grow by aggregation and gradually sweep out dust-free annular lanes. If they grow larger than a specified critical mass they can begin to accumulate gas as well from the cloud. If the orbit of a planet comes inside a certain interaction distance from a planet that was formed earlier, or if the orbits cross one another, the two bodies coalesce to form a single, more massive, planet which may then continue to grow by aggregation. The process of injecting nuclei is continued until all the dust has been swept from the system. Each planetary system produced by using a different random number sequence is unique. However, all the systems so produced share the major regular features of our solar system (see Fig. 2.1.) The orbital spacings have patterns of regularity suggestive of "Bode's Law". The innermost planets are small rocky bodies; the mid-range planets are large gaseous bodies; the outermost planets are generally small. The general pattern of planetary mass distribution is similar to that in our solar system with masses ranging from less than that of Mercury to greater than Jupiter's. Dole was able to produce planetary systems of recognizable characteristics

a8

The Earth’s Core

only with a certain choice of input parameters and assumed structure of the solar nebula. Isaacman and Sagan (1977) investigated the consequences of varying these parameters and assumptions. They found that only for a small subset of conceivable cases are planetary systems generated that are closely like our own. Many models have tendencies toward one of two preferred configurations: multiple-star systems, or planetary systems in which Jovian planets either have substantially smaller masses than in our system or are absent altogether. However, for a wide range of cases recognizable planetary systems are generated, ranging from multiple-star systems with accompanying planets, to systems with Jovian planets at several hundred astronomical units, to single stars surrounded only by asteroids. Many systems exhibit planets like Pluto and objects of asteroidal mass, in addition to usual terrestrial and Jovian planets. No terrestrial planets were generated more massive than five Earth masses. The number of planets per system is for most cases of the order 10, and, roughly, inversely proportional to the orbital eccentricity of the accreting grains. All systems generated obey a relation of the TitiusBode variety for relative planetary spacing. Isaacman and Sagan comment that it is rather remarkable that such a simple and incomplete model, based merely on Newtonian physics and accretion with unit sticking efficiency, should generate plausible planetary systems for a wide variety of initial conditions. A problem with these simulations is the neglect of mutual gravitational perturbations, which could play a major role during the latest stage of planetary growth. Also, velocities are assumed that are impossibly high in the gaseous interplanetary medium. In his models, Hills (1970) includes gravitational perturbations but ignores collisions. Wetherill (1978) has discussed the shortcomings of many of these earlier attempts at numerical simulations of multi-planet growth and pointed out that in the real solar system accumulation of the terrestrial planets proceeded more or less simultaneously. Greenberg et al. (1978a) carried out a numerical simulation of collisional and gravitational interactions in a possible early solar system. Their results show that “planets” 500 km in diameter can be generated from an initial swarm of kilometre-sized planetesimals, such as might have resulted from gravitational instabilities in the solar nebula. Their models are based on experimental (see Hartmann, 1978) and theoretical impact results (such as rebound, cratering and catastrophic fragmentation) for a reasonable range of parameters and initial conditions. The experimental work of Hartmann indicated that by the time bodies of multi-metre size have formed, they should have regolith surfaces or granular structure throughout. Regolith strongly favours accretion by inhibiting rebound, so that such bodies would have grown very efficiently by collisions with neighbouring bodies. The small planets grow fast on time-scales of a few tens of thousands of years, during which time most of the mass of the system continues to remain in particles

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The Origin of the Core

near the original size. Relative random velocities remain of the order of a kilometre-sized body’s escape velocity, those of the largest objects being somewhat lower because of damping by the bulk of the material. Greenberg et al. suggested that the few 500-km-sized planets, in a swarm still dominated by kilometre-scale planetesimals, may act as seeds for the subsequent, gradual accretional growth into full-sized planets. While there are uncertainties about many stages of the origin of the solar system and planetary accretion, one of the most serious questions concerns the intermediate phase of accretion, i.e. growth after the hypothetical formation of planetesimals by gravitational instability (Safronov, 1972; Goldreich and Ward, 1973) but before the later stages of accretion when the largest bodies have substantial gravitational cross-sections. It is not clear how planetesimals could have efficiently accreted one another. However, the modelling of Greenberg et al. (1978a), based on detailed physical experiments involving low-velocity collisions among rocky bodies, demonstrates that accretion through this intermediate size range is efficient and rapid-it is a natural result of low-velocity rebound phenomena. Greenberg et al. point out that additional physical processes might be important in this intermediate stage, such as resonant phenomena that might accelerate or retard growth in certain zones, and gas-drag. A potentially disrupting influence on accretion would be high-velocity bodies, perhaps scattered into the zone of interest by an early-formed Jupiter. In this regard, Hartmann (1978) suggested that highvelocity planetesimals passing through the asteroid belt could have stopped further growth of Ceres, rather than a lack of accretable material. Greenberg (1979) later considered the problem of the size of the largest planetesimals that grew near, and later impacted, those that became full-sized planets. The question is important since the final stages of the growth of a terrestrial planet would significantly affect many of its fundamental properties, such as its thermal history and petrology. Greenberg showed that given the range of plausible mechanisms, parameters and initial conditions, possible results for the radius of the second largest planetesimal in the Earth’s zone range from only a few hundred kilometres to 2500 km, with corresponding accretion times of 7 x lo6 and lo8 years respectively. Cox (1978) and Cox and Lewis (1980) carried out numerical simulations to determine whether terrestrial-like planetary systems could be produced as a result of gravitationally-induced collisions and accretions occurring in a swarm of protoplanets assumed to have already reached approximately lunar mass. In their calculations the mass and relative velocity distributions of the bodies are free to evolve simultaneously in response to close gravitational encounters and occasional collisions between bodies. Thus, collisions between bodies arise in a natural way and assumptions concerning the relative velocity distribution and the gravitational collision cross-section are

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The Earth‘s Core

unnecessary. The initial eccentricities of the orbits were randomly chosen from a uniform distribution ranging from zero to emax.Their model was twodimensional: the inclinations remain zero throughout the evolutionary ~ 0 . 1 0resulted in process. Cox and Lewis found that simulations with emax final-state configurations having many bodies of nearly equal mass, none of which were comparable to that of the Earth or Venus. Simulations with emax >0.20 resulted in a substantial number of catastrophically disruptive collisions between bodies. On the other hand, simulations with emaxN 0.15 gave six planets, surprisingly close to the actual number (4), with spacings between orbits resembling those of the terrestrial planets. This rather large value of emaxis mainly the consequence of assuming coplanar motions. Cox and Lewis estimated the length of time for the stage of accretion considered in their simulations to be ~2 x lo7 years. Wetherill (1978) and Greenberg et al. (1978b) have questioned whether a self-contained swarm can avoid premature orbital isolation of embryo planets, with the consequent production of too many small bodies. Wetherill (1980b) also investigated to what extent the simulations of Cox (1978) and Cox and Lewis (1980) depend on the calculations being two-dimensional. Wetherill used a different (less exact) method of calculation, but reproduced = 0.05 and 0.10 for the two-dimensional case in which 8 Cox’s results for emax to 10 bodies finally remain. In the three-dimensional case, the final number of bodies is reduced to 3-5 for a low value of emax= 0.05 (see Fig. 2.2) The difference between the three- and two-dimensional results is a consequence of the lesser importance of collisions relative to gravitational perturbations in three dimensions. The time-scale for accretion is lo8 years, most of it occurring during the first 20 Ma. The three-dimensional treatment allows accumulation to proceed for a longer time prior to the separation of embryo planets into isolated orbits*. Spaute et al. (1985) have studied the dynamical evolution of an initial population of kilometre-sized planetesimals in the early solar system subject to collisions, taking into account accretion, rebound, cratering and catastrophic fragmentation. They used a Monte-Carlo statistical method which yielded the mass and velocity distribution of the planetesimals as a function of time and heliocentric distance. They carried out a number of computations simulating the accretional growth of planetesimals (initial mass N 10l6g) in the absence (or presence) of gas-drag, with (or without) one larger embryo among them and with (or without) a size gradient. The work of Spaute et al.

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* Wetherill has since carried out more sophisticated similar calculations (in Origin of the Moon, W. K. Hartmann, R. J. Phillips, G. J. Taylor eds, Lunar & Planetary Institute, Houston, 1986). He concluded that for a wide range of initial conditions, terrestrial planet accumulation was characterized by giant impacts, ranging in mass up to 3 times that of Mars, at typical impact velocities of 9 km s - I .

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The Origin of the Core

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is an extension of that of Greenberg et al. (1978a) by considering the effect of gas-drag. Gas-drag can cause some planetesimals to move inside the feeding zone of an embryo, allowing their further growth to reach planetary size. This could help to overcome the difficulty of embryos becoming isolated from the remaining population. Nakagawa et al. (1983) also considered the effect of gas in their study of the evolution of intermediate-sized planetesimals. However, the effect of gas-drag is more important in the simulations of Spaute et al. than in those of Nakagawa et al., since the initial mass of the planetesimals in Spaute et al.3 models is one-hundredth of the mass assumed by Nakagawa et al. Specifically, Spaute et al. showed that, assuming an initial negative size gradient as the heliocentric distance increases, local accumulation of planetesimals can take place by gas-drag alone leading to embryos a few hundred kilometres in radius. Moreover, when an embryo is present among a population of planetesimals, its growth is about twice as rapid in the presence of gas-drag since gas-drag allows a continuous feeding of the embryo by planetesimals moving inwards towards the sun. When an embryo has become massive enough, it becomes more and more efficient in inhibiting the inward diffusion of planetesimals, limiting the growth of the next inner embryo. Such a process may have acted to prevent the formation of many small planets in contrast to the few large existing ones. Cazenave et a!. (1982) have also carried out numerical simulations of the latest stages of planet growth. They considered the motions of a small number of large protoplanets of unequal mass in orbit around the sun.

92

The Earth's Core

Mutual gravitational perturbations, accretion and collisional fragmentation are all taken into account. Their model is three-dimensional, gravitational encounters are treated exactly by numerical integration of the N-body problem, and the outcome of collisional fragmentation is based on the model of Greenberg et al. (1978a). They found that motion during close encounters departs significantly from that predicted by the two-body approximation. For large protoplanets (mass loz5g), accretion always prevails against fragmentation as long as relative velocities are not too great. The mean eccentricities and inclinations evolve towards a steady state, the average value of the inclination (in radians) being about one-half that of the eccentricity. Also, the population evolves towards a quasi-equilibrium relative velocity distribution whatever the initial conditions. Wetherill later (1985) extended his earlier (1980a, 1980b) work on the late stages of growth of the terrestrial planets. He considered the simultaneous growth of several planets from planetesimals in the absence of gas-drag using three-dimensional numerical calculations. He supposed that as most of the mass began to be concentrated in larger bodies, further evolution would be dominated by perturbations by large bodies in orbits of low eccentricity in neighbouring zones rather than by collisions of small bodies. This would cause the eccentricities to increase gradually and Wetherill estimated that there would be -500 bodies of mass -2.5 x when they reached -8 x g (i.e. - 3 the lunar mass). During the first 1.8 Ma, mutual perturbations increased eccentricities to values as high has 0.2. After 9.4 Ma two bodies with masses greater than 3 x loz7g and with relatively low eccentricities (about 0.03 or less) and inclinations ( < 3") formed, in addition to a large number of g (the size of Mercury or Mars). bodies with masses in the range loz6to At this stage, 71% of the final mass of the Earth accumulated. During the last stages of accumulation, requiring up to 2 x lo8 years, almost all these intermediate-mass bodies, including failed planets with masses greater than 3 x loz6g, collided and merged with the two bodies in orbits similar to those of the present Earth and Venus. The accumulation process was accompanied by the impact of very large bodies-as large as three times the mass of Mars at velocities of -9 km s-'. The kinetic energy released in these largest impacts would be several times greater than that required to melt the entire Earth. The Earth, as it grew, would have been at least partially melted throughout, leading to core formation by gravitational segregation of iron. A problem common to all models of cold origin is the means by which the Earth would have heated up sufficiently to lead to a (predominantly iron) molten OC at least 3500 Ma ago. This problem will be discussed in the next section.

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The Origin of the Core

2.3 Heat Sources for an Earth Accreting Cold 2.3.1

Long-lived Radioactive Isotopes

The radioactive isotopes that contribute significantly to the present heatproduction within the Earth are 238U,235U,232Thand 40K,all of which have half-lives comparable to the age of the Earth. The temperature increase due to the radioactive decay of these long-lived isotopes is thus small, of the order of 150 K after 100 Ma (MacDonald, 1959). During the first 1000 Ma, the temperature increase would only be about 700 K (assuming no heat escape) while the total heat produced over the life-time of the Earth (-4500 Ma), if trapped within the Earth, would raise the temperature by 1800 K. Thus long-lived radioactive isotopes can account for part of the initial heating of the Earth, but other sources are necessary as well. It is usually assumed that there is no uranium in the Earth’s core, since uranium (and potassium) exist as siderophobic compounds, such as silicates and oxides in the Earth’s mantle, and were thus thought to be immiscible with the metal core (Urey, 1952). However Hodkin and Potter 1980) have carried out experiments on the binary system of steel and UO, and shown that above 3120 K the system is a two-phase liquid, one rich in UOz and the other poor in UOz. Their phase diagram predicts that there must be a temperature above which there is total miscibility between UO, and steel. Feber et al. (1984) thus suggested that uranium is steadily dissolving into the core and supplying radioactive heat, sufficient to power the geodynamo (see $4.6). The power required is IOl3 W (Gubbins 1981). Murrell et al. (1983), Murrell and Burnett (1985,1986) have carried out experimental work on the partitioning of K, U and Th between sulphide and silicate liquids. Contrary to what has usually been assumed, they found U and Th partitioning into Fe FeS liquids much more important than K partitioning. Of course, uranium could not act as a heat source in the core to power the geodynamo until the core was already molten. If the power required (- 1013 W) is supplied solely by the decay of 40K, the K concentration currently required in the core is 1500 ppm. On the other hand, the actinide core concentrations’ currently required are about 50 ppb U or 180 ppb Th. The question of the possibility of K in the Earth’s core is discussed further in Section 5.8.

-

-

-

2.3.2 Short-lived Radioactive Isotopes

Short-lived radioactive isotopes could have contributed to the initial heat of the Earth if the time between the formation of the elements and the aggregation of the Earth was short compared with the half-lives of the isotopes. The

94

The Earfhs Core

most important short-lived isotopes are 236U,I4%m, 244Puand 247Cm,all of which have half-lives sufficiently long to have heated up the Earth for some tens of million years after the initial formation. These four isotopes would, in fact, have contributed about 20 times the heat produced by 40K during this period. The decay of three shorter-lived radionuclides 26A1, 36Cl and 60Fe would have significantly heated up accreting planetary bodies for a period of about 5-15 Ma after the termination of nucleosynthesis in the primitive solar system (Fish et al., 1960). Of these, 26A1 is the most important. It decays to 26Mgwith a half-life of 0.74 Ma and would remain as a significant source of heat for about 10 Ma. If the Earth accreted within 20 Ma of the termination of nucleosynthesis, the heat released through the decay of 26A1 could be the main cause of its high internal temperature. If, on the other hand, the time of accretion was of the order of 100 Ma, the decay of 26A1would have had a negligible effect on the Earth's thermal history. In this regard, no trace of any anomaly in the 26Mg/24Mgratio had, until recently, been found in meteorites or in lunar and terrestrial samples (Schramm et al., 1970). Gray and Compston (1974), however, found an anomalous value of the 26Mg/24Mg ratio in a melilite-bearing chondrule in the Allende meteorite. They suggested that the anomaly results entirely from excess 26Mg,implying an origin by the radioactive decay of 26Al formed during intense proton irradiation in the early stages of the solar system. Later, Typhoon Lee et al. (1976, 1977) measured the isotopic composition of Mg in different phases of a Ca-Al-rich inclusion in the Allende meteorite and found large excesses of 26Mg of up to 10% that correlated with the 27A1/24Mgratio for four coexisting phases with distinctive chemical compositions. They concluded that the most plausible cause of the 26Mganomaly is the decay in situ of 26A1,providing definite evidence for the presence of 26A1in the early solar system. This would require either injection of freshly synthesized nucleosynthetic material into the solar system immediately before condensation and planet formation, or local production within the solar system by intense activity of the early sun. One suggestion to account for the excess 26Mg in the aluminium-rich minerals from the Allende meteorite is that a supernova exploded beside the forming solar system, peppering the solar cloud from which Allende formed with 26A1. This conclusion has now been questioned by the detection of surprisingly large amounts of 26A1in the interstellar medium (Mahoney et al. 1984; Mahoney 1985) by the third HighEnergy Astrophysical Observatory (HEAO 3). The concentration is approximately 1 atom of 26A1 on the average for every 100,000 atoms of 27A1. This discovery has been confirmed by a pray spectrometer on board the Solar Maximum Mission Satellite (Share et al., 1985) The prays coming from the general direction of the galactic centre are a product of the P-decay of the 26A1 nucleus. Clayton (1984) showed that if the 26A1 is spread uniformly

The Origin of the Core

95

throughout the interstellar gas, its concentration (- 10 ppm of Al) is close to that in the Allende minerals, in which case it is unnecessary to invoke the injection of 26A1into the forming solar system by a supernova. In any case, supernova explosions are not adequate to maintain the average level of observed interstellar radioactivity. The excess of 26A1 in Allende minerals, however, is too large to be the result of the average interstellar concentration. If 26A1decayed in the Allende minerals, there must have been some source of 26A1enhancement in the solar cloud as it collapsed to form the solar system. Clayton (1984) now expresses some doubts that 26A1 was ever "alive" in Allende. Some information on the time of nucleosynthesis has come from a study of xenon isotopes. "'I decays through p-emission to "'Xe with a half-life of 16.4 Ma. This half-life is so short that no "'I now exists in nature. However, this isotope should be produced at the time of formation of the other elements. It can be expected that the condensation of planetary material would incorporate any of the iodine present, but would be unlikely to incorporate much xenon. "'Xe formed within the condensed materials would be trapped and might be observed today-very small amounts of excess "'Xe have in fact been found in a number of meteorites. Absolute formation intervals calculated by the 129Xe-1291method generally give values between 40 and 300 Ma depending on the nucleosynthesis model assumed. Variations in relative formation intervals are much less, but are difficult to assess because of different methods of data treatment between different laboratories. Podosek (1970) has subjected all of the Berkeley data to a common method of analysis and found 15 different meteorites possessing formation intervals within a period of 15 Ma. Other noble-gas components in meteorites that arise from an extinct radionuclide and have the potential for the determination of formation intervals are the heavy xenon isotopes 131-136Xethat arise from the fission of 244Pu. These fission Xe components are much more difficult to identify than the excess lZ9Xe and were not discovered until precise isotopic data were obtained on the calcium-rich achondrites, a class of meteorites high in uranium (supposedly chemically similar to plutonium) and very low in trapped Xe (Rowe and Kuroda, 1965). Rao and Gopalan (1973) have also considered the fission Xe components in different types of meteorites and in particular have investigated whether some actinide isotopes could produce the excess fission Xe found in primitive chondrites. They concluded that the I-Xe and Cm-Xe systematics relate to different events in the early history of the solar system. The former refers to the time when I-Xe clocks are re-set during metamorphism of the ordinary chondrites while the latter refers to the time of primary condensation of the refractory materials from the cooling solar nebula (Hoyle and Wickrama-

96

The Earth's Core

singhe, 1968). The occurrence of the short-lived nuclide 248Cm ( Q T , ,=~ 0.4 Ma) in primitive meteorites has important consequences for the time-scales of the earliest events in the evolution of the solar system. As the solar nebula cools from about 2000 to 1400 K, i.e. to the temperature at which iron and magnesium silicates condense, the high-temperature minerals (or phases) form as refractory condensates (Fireman et al., 1970; Podosek and Lewis, 1972). Curium, being a refractory element similar to uranium, plutonium and the rare-earths, is most likely to be incorporated into these minerals, which would preserve the spontaneous fission record of 248Cmif they cooled fast to the Xe-retention temperature. Hence this actinide isotope could fix the timescale for a fast cooling stage, i.e. if the time interval for condensation was about 2 Ma, the 248Cmrecord would be preserved in these early condensations, but if it lasted for tens of millions of years, 248Cm would have completely decayed before it could be incorporated into these minerals. Ozima et al. (1985) have re-examined xenon isotopes produced from the decay of '291 and from 244Puand considered the constraints they impose on the early history of the Earth. The accretion time of planetesimals into the terrestrial planets is of the order of lo7- lo8 years. This is a significant time for the short-lived "'I and 244Pu.While condensation and accretion result in closure of solid elements such as Pu and I, Xe closure might be established only after collisions among planetesimals, since impact shock could result in the effective degassing of volatiles (Lange and Ahrens, 1982). Thus it would be possible for early- and late-accreting material to have significantly different Xe closure times. The excess '29Xe in some terrestrial materials indicates that the region from which it was derived must have been isolated from the atmosphere for nearly the whole history of the Earth. The commonly held view is that this excess lZ9Xeis a result of early mantle degassing (see e.g. Staudacher and AIlCgre, 1982; Allhgre et al., 1983; Hart et al., 1983). Ozima et al. (1985) offer an alternative explanation, viz. that the Earth's inner regions accreted a few tens of millions of years earlier than the outer regions from which the atmosphere evolved. The minimum time interval between the formation of the two regions is of order 1.4 x lo7 years.

2.3.3 Superheavy Elements

Runcorn et al. (1977) and Runcorn (1978) have suggested that superheavy elements (SHE) with atomic numbers between 114 and 126 may be relatively stable and may have provided the early heat source that melted the moon in its first few 100 Ma. They further suggested that these elements may be soluble in iron and hence provide the driving mechanism for the generation of the ancient lunar magnetic field. In a later paper, Libby et al. (1979) suggested

The Origin of the Core

97

that the heavy elements in iron meteorites may be partly the fission products of SHE. Flerov et al. (1977) claim to have observed in carbonaceous chondritic meteorites spontaneous fission of nuclei that they argue are SHE nuclei. However, there is no firm evidence as yet for the existence of relatively stable SHE. Kaiser et al. (1981) carried out a search for any records of fission events in the metal phase of IA and IVB meteorites, and found no evidence in support of a fissionable, siderophile superheavy element; limits for fission events are too low for fission to be a significant heat source in planetary cores. The highest possible value for heat production deduced from their Nd-Sm data in iron meteorites is one order of magnitude smaller than that required to drive the geodynamo by heat release due to radioactive decay (see 94.6). The lower values obtained from Kr, Xe and fission track data are two to three orders of magnitude smaller than the required energy production. Kaiser et al. conclude that nuclear fission could not have contributed significantly as a heat source to drive an early lunar, or a terrestrial, core dynamo.

2.3.4 Adiabatic Compression

The temperature of the material within the aggregating Earth would also increase because of adiabatic compression. Although data (particularly on the variation with pressure of the coefficient of thermal expansion) are rather uncertain, a rise in temperature of several hundred degrees from this source seems possible.

2.3.5 Potential Energy due to the Mutual Gravitational Attraction of the Particles of the Dust Cloud

The kinetic energy of the aggregating particles is either converted into internal energy or radiated away. It is extremely difficult to estimate the contribution from this source because of our lack of knowledge of the physical processes of accretion. The result depends quite critically on the temperature attained at the surface of the aggregating Earth and on the transparency of the surrounding atmosphere to radiation. Comparatively low surface temperatures (of the order of a few hundred degrees) have been suggested, mainly because the atmosphere of the primitive Earth was assumed transparent so that the thermal energy of the impinging particles was immediately re-radiated into space. Ringwood (1960) has argued, however, that during these early years the primitive Earth would have had a large reducing atmosphere. In the presence of these reducing

98

The Earth’s Core

agents (chiefly carbon and methane), the accreting material would be reduced to metallic alloys-principally of iron, nickel and silicon. The outer regions of the Earth would thus be metal-rich and dense (referred to zero pressure) compared with the interior. Such a state is gravitationally unstable and convective overturn would follow, leading to a sinking of the metal-rich outer regions to the centre. This would release further heat due to the energy of gravitational rearrangement. Ringwood believes the whole process is likely to be catastrophic, since the overturn would be accelerated as the initial temperature rose. The relative importance of the gravitational potential energy of the dust cloud as a means of heating the proto-Earth depends critically on the duration of the accretion process-rapid accretion is necessary to produce high temperatures and melt the OC. The rate of growth of mass of the embryo Earth would increase in the beginning of the accretion process (because of the increase in the capture cross-section of the embryo), pass through a maximum and then decrease (because of the exhaustion of accretable material). The law governing the flux, h, of falling matter determines the ultimate thermal profile, since losses by radiation depend upon how fast the recent fall is covered by new layers thus insulating the former from radiative loss. The energy balance is given by the equation T 3

+ CP[T(t)- T&l

for infalling matter having the velocity of escape at impact, where G is the universal constant of gravitation, M ( t ) the mass of the Earth, m the mass flux, r(t) the instantaneous radius of the Earth, p the mass density, E, the emissivity, c the Stefan-Boltzmann constant, T ( t )the instantaneous temperature of the surface of the proto-Earth at time t, To the background temperature of free space of the circumsolar nebula into which radiative losses take place, and cp the specific heat at constant pressure, In (2.1) the term on the left-hand side represents the gravitational energy input, the first term on the right-hand side the radiative loss rate, and the second term the rate of heat takeup by the infalling matter in coming to the temperature T(t).Heat conduction to the interior from the hot surface regions is neglected since it is vanishingly small for time-scales of the order of 103-106years. Heating due to adiabatic compression is also ignored. The mass flux, m(t), is the dominant parameter in calculating the thermal profile resulting purely from accretion, although To has an important effect under certain conditions. In general m = m(t) but, as will be shown later, whether m = m ( t ) or is constant has little bearing upon the main conclusions. It is easy to generalize a set of laws for m that include gravity and slower rates

99

The Origin of the Core

of accretion as well. It can be shown that the exact form of the law has little effect upon the time-limit for sufficient energy to be stored to melt surface or subsurface regions. Write m = c,/r"

where r

= r(t) and c

(2.2)

is a constant with index n. Since M = 47cpr2i, m = h / 4 n r 2 = pi.

Hence from (2.2) and (2.3),

except for the case n = - 1. In this case

For n = 0, m is constant and the radius grows linearly with time while for 1 the growth is slowed. None of these assumptions is especially plausible physically, as the accretion rate should depend upon the gravitational force (n = - 1) or potential (n = - 2), assuming that the escape velocity is not exceeded and an infinite source reservoir is available. For the case n = - 2, (2.1) becomes la>

$ c p G c - 2r4 =

T4 - T:)

+ c pc

-

2r2(T - To)

(2.7)

where the constant c - ~= mf/R: where mf is the mass flux at the termination of growth, r = R,. From (2.5), it then follows that

and the time taken for the Earth to grow from radius, ro, to final radius, Rmis given by

where Ar = r

-

ro. When r

= R,,

(2.10)

100

The Earths Core

For R,/ro = 2, t = p R , / m f which is the same as for the case m = co. Thus the time taken for the Earth to grow from half-radius to its full value under gravitational accumulation is the same as for riz = co. Although the radiative losses depend upon the accretion time, there is not a great difference in the final temperature for these two cases. The effect of heat being admitted to the incoming matter tends to diminish the differences between the different mass laws for k. Without this term in (2.1), the temperature would vary as r for gravitational accretion and be less for cases where n> -2. If all the gravitational energy of accumulation were trapped in the Earth, the first term on the right-hand side of (2.1) vanishes and the resulting increase in temperature AT can be estimated from the equation (2.1 1)

where R , is the final radius. This gives a temperature of the order of 50,000 K. This is the maximum thermal effect of the accretion process-such a hot origin of the Earth would lead to core formation and large-scale differentiation upon accretion (Ringwood, 1966). The minimum thermal effect of the accretion process may be obtained by assuming that the energy released by inelastic collisions of matter falling in from the primordial gas/dust cloud is partially compensated by energy reradiated to space. The equilibrium condition is then given by (2.1) with the last term omitted. If we assume black-body radiation, E, = 1 and the resulting temperatures are minimal. The unknown quantity is i., the rate of accretion. Hanks and Anderson (1969) have considered a number of models. The simplest assumed a constant accretion rate: more plausible models were obtained with accretion rates i = A sin at and i = Ct2 sin y t , where i is zero at r = 0 and r = Rm. The constants A , a, C and y are determined by the final radius R , and t,,,, the total time of accretion. Figure 2.3 shows the radial growth as a function of time for this last model, when t,,, = 1 Ma. It can be seen that it takes 0.3 Ma for the Earth to reach asteroidal size-the rapid acceleration of the accretion process between 0.4 and 0.9 Ma is responsible for the major part of the temperature rise. The accretion rate begins to decelerate significantly only in the last 200 km. Safronov (see, e.g. 1969) and Urey (1962) prefer an accretion time of the order of lo8 years, rather than the very much shorter times advocated by Ringwood (1959, 1966), Hanks and Anderson (1969) and Turekian and Clark (1969). The above calculation ignores the details of each accretion event-an approximation that can only be justified if all the accreting particles are very small compared to the planet. If the terrestrial planets were .formed by accumulation, there must have been impacts involving proto-planetary

101

The Origin of the Core

I

I

I

I

7

5000

"0

0.2 0.4 0.6 0.8 Time of ter accretion begins ( M a1

1.0

Fig. 2.3. Radial growth of the Earth. (After Hanks and Anderson, 1969.)

bodies of many different sizes: the above theory would not simulate the heating from such events at all. Again, this simple theory does not take into account any mass loss by the growing planets during accretion: high speed impacts could cause considerable heating, at the same time ejecting more mass from the forming body than they brought to it. The heating effect of impacts of large falling bodies is considered in more detail in $2.3.8. The accretional heating of terrestrial planets has been reviewed by Ransford (1982).

2.3.6 Dissipation of the Earth's Rotational Energy

As the Earth slows down through tidal interaction with the moon (and to a lesser extent with the sun), part of its rotational energy is dissipated by tides in the oceans and part in the interior of the Earth by Earth tides. It is difficult to estimate how the energy loss is divided between ocean and Earth tidesprobably most of it is dissipated in the shallow seas. If the heat generated by tides were evenly distributed over the whole volume of the Earth, the increase

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The Earths Core

in temperature would be unimportant (probably less than 100 K). However, if there is a region of low viscosity in the upper mantle, most of the energy of tidal deformation should be dissipated there, resulting in a greater increase of temperature. This would have occurred when the Earth and moon were relatively close together and the Earth’s rotation was much faster than it is now; but even if the moon was once close to the Roche limit, it would recede so rapidly that the effect of tidal heating would be negligible.

2.3.7 The Formation of the Core

If the Earth formed by accretion from approximately homogeneous material and later differentiated into crust, mantle and core, the formation of the core would release a large amount of gravitational energy as a result of the concentration of the high density nickel-iron in the centre of the Earth. Tozer (1965) estimated the increase of heat arising from core formation from an originally undifferentiated Earth to be 470 cal/g. About 6% of this would melt the Ni-Fe phase, while the rest would raise the mean temperature of the Earth about 1500 K. Birch (1965) has made a similar estimate and obtained essentially the same result (400 cal/g) for the energy available for heating. In a later paper, Flaser and Birch (1973) corrected an error in Birch’s original paper. Their revised estimate of the energy release due to core formation is 590 cal/g which would be sufficient to heat the Earth by about 2000 K. Thus the formation of the Earth’s core may well have played a dominant role in the thermal history of the Earth.

2.3.8 Impacts of Falling Bodies

Safronov (1969, 1972, 1978) is of the opinion that one of the main sources of heat for an Earth accreting cold was the impacts of falling bodies. Estimates of the initial temperature from this source depend largely on the body sizes assumed. Calculations based on the assumption that these bodies were small give a low initial temperature-from 300 K near the surface to 800-900 K at the centre. The impact energy of small bodies and particles would be liberated near the surface of the growing Earth, practically all of it being radiated back into space. Safronov believes, however, that relatively large bodies were involved in the formation of the planets, the largest bodies falling on the Earth having diameters of several hundred kilometres. The larger the incident body, the greater the depth at which its impact energy would be released and hence the greater the fraction of this energy that would be trapped inside the Earth, unable to escape into space. Moreover, larger bodies would produce

The Origin of the Core

103

deeper craters, inducing more intensive mixing of the material on impact. Heat transfer by mixing of material during the impact of large bodies is far more efficient than heat transfer by ordinary thermal conduction. Numerical estimates show that the maximum initial temperature of the Earth occurred in the region of the upper mantle (at a depth -500 km) and probably exceeded 1500 K. The largest bodies striking the Earth would also cause primary inhomogeneities in the mantle. Apart from inhomogeneities arising from differences in chemical composition of the impacting bodies, there would also be temperature inhomogeneities. Regions of impact of the largest bodies with sizes up to 1000 km would be additionally heated some hundreds of degrees-in such large areas the temperature would not be equalized even after 1000 Ma. With the additional heat due to radioactive decay, the central zones of these regions would be the first to reach melting temperatures. When about half of the material had melted, local gravitational differentiation would take place-heavy materials would settle down to the lower part of the zone, and light materials rise to the upper part. During such global differentiation large amounts of thermal energy would be released. As a result, a greatly heated vertical column of lower viscosity would be formed along which masses of heavier materials would sink downwards to form the core. Safronov also suggested that it was above such regions excessively heated by impacts that the formation of the continents took place. Safronov’s conclusions were based on an extremely simple model, Kaula (1979) refined the model by treating the thermal evolution of the Earth to be time-varying in a spherical geometry. The energy partitioning at impact was taken from the work of O’Keefe (1977) and O’Keefe and Ahrens (1977), the distribution of impact energy declining with depth. The greatest uncertainty in Kaula’s models is that portion of the impact energy partitioned to internal energy that is retained after material falls back in great impacts. Kaula found that models of the Earth with growth times of the order of 50 Ma became hot enough for vaporization to occur if more than about 12% of the heat energy is retained upon impact. He also showed that early planetary heating is insensitive to planetary growth time-rapid formation is not required. In a later paper, Kaula (1980) re-examined his earlier model and reaffirmed his conclusion that the impact of falling bodies would have led to significant heating, more than enough to trigger core formation. The principal modification from the conclusions of his 1979 paper is that the temperature would never have risen much above melting, since the viscosity of molten material would have been low enough to remove heat as rapidly as it was generated by impacts. The effects of lateral heterogeneity are mainly to promote convection and thus heat removal. Core formation probaby started before the Earth was 10% formed, since melting temperatures would have been reached by

104

The Earths Core

that time. Kaula estimated that the peak energy release was near the MCB, and that about 60% went to the core. The resulting temperature gradient in the mantle was strongly super-adiabatic. The energy released by differentiation of subsequently infalling material would have gone mainly into the mantle, where it would have further contributed to vigorous convection, and not into the core. The calculations of the accretion heating effect by Safronov (1972, 1978) and Kaula (1979, 1980) assumed that there was no overlap between the growth zones of the Earth and Venus during their formation. Wetherill (1976) considered this possibility and also the effect of slightly eccentric orbits for the proto-Earth and proto-Venus, and obtained much higher relative velocities in both the Earth and Venus zones and much larger masses for the planetesimals relative to the proto-Earth and proto-Venus. In this model, both the Earth and Venus and the Moon would have melted completely during accretion. Davies (1985) has discussed in detail the heating of a planet due to the impact of falling bodies. In a growing planet, the heating rate will be greatest at the surface and will decrease with depth. Heat transport back to the surface will also be affected by mixing accompanying impacts and this effect will also decrease with depth. Moreover, the surface of the planet will be moving upwards as new material is added. Relative to the accreting surface, the material is moving downwards. Thus, heat that is deposited too deeply will not be conducted back to the surface before it is carried into the low conductivity interior and there trapped. Davies thus identifies three depth-scales that affect the fraction of deposited heat that is retained in the growing planetthe depth-scales of heat deposition, impact mixing and the ratio u/v where u is the thermal diffusivity and u the upward velocity of the accreting surface. He showed that the efficiency of heat retention is independent of planetary growth rate, and confirmed Safronov’s (1969, 1972, 1978) conclusion that a significant amount of impact energy will be retained in a slowly growing planet. Assuming a heat deposition efficiency of 20%, a terrestrial planet will reach melting temperatures in its outer layers at a radius of between 2000 and 3000 km (see Fig. 2.4). Davies stresses the importance of the correct scaling for impact effects, and questions some of the results of Kaula (1979, 1980) that are inconsistent with the general scaling forms deduced by Holsapple & Schmidt (1982).

2.3.9 Blanketing Effect of a Primordial Dense Atmosphere

Hayashi et al. (1979) considered the heating due to the blanketing effect of a primordial dense atmosphere surrounding the proto-Earth. When the mass

105

The Origin of the Core

1

I

5

I

6 log R (rn)

Fig. 2.4. Temperature profiles for growing terrestrial (rock-iron) planets. Solid lines bound the preferred range corresponding to impactor heating, while short-dashed lines correspond to crater heating. M and E denote momentum and energy scaling, respectively. Long-dashed sloping lines are the limits assuming lOO%-efficient heat retention ( T A ) and 100%-efficient deposition plus retention ( T T ) . Deposition efficiency of 20% is assumed; a different value would shift TAand lower profiles vertically relative to TT. Horizontal lines bound estimated melting ranges of relevant silicate and metal phases. Temperature profiles near and above the melting temperatures are not realistic, since heat transport by convection and melting would then dominate. (After Davies, 1985.)

of the proto-Earth was 20.1 of the mass of the present Earth, an appreciable amount of gas from the solar nebula would form an optically thick, dense atmosphere. They showed that the temperature at the bottom of the atmosphere, i.e. at the surface of the proto-Earth, increases greatly with the mass of the proto-Earth and is 1500 K when its mass is a quarter that of the present Earth, and 4000 K when the proto-Earth reached its present size. However, their results depend rather critically on the boundary conditions of the Earth’s proto-atmosphere, and are difficult to assess. Later Matsui and Abe (1986) considered the blanketing effect of an impactinduced atmosphere. The impact of large bodies-planetesimals-would result in the ejection of enormous amounts of fine dust and the release of

-

106

The Earrh’s Core

volatile gases such as HzO and COz. Lange and Ahrens (1982a,b) had earlier suggested that the primordial H 2 0 atmosphere was formed on the Earth by impact dehydration of H,O-bearing minerals. Since grains, C O , and HzO are all good absorbers of infrared radiation, Matusi and Abe suggested that the impact energy released at the surface of a growing planet could not have escaped directly into space and was thus expended in heating a surface layer. They estimated that the surface temperature due to the blanketing effect of an impact-induced H 2 0 atmosphere would exceed the melting temperature of the surface materials leading to a “magma ocean” covering the entire surface of the Earth (and Moon) just after their formation.

2.3.10 Electrical Induction

Electrical induction as a means of joule heating of planets has been studied extensively by Sonett and coworkers (see e.g. Sonett and Colburn, 1967,1968; Sonett et al. 1968, 1970; Sonett and Herbert, 1977). Electromagnetic induction depends upon certain properties of the pre-main-sequence Sun-that it passed through a T Tauri phase and at that time had a high spin and surface magnetic field of about 10-100 G. Under these conditions a strong electromagnetic field would be set up in the interplanetary cavity. Planetary objects in such an environment would be subject to electrical induction: the Apollo magnetometer detected a very large transverse electric (TE) mode in the Moon excited by the forcing field of the solar wind (Sonett et al., 1972). As seen by the planetary object, the induction takes the form of a unipolar generator of linear geometry using the solar wind motion and the interplanetary magnetic field, with the conductive interior of the planet acting as the armature. Sonnett and coworkers have shown that joule heating can cause the melting of objects in the inner solar system, the amount depending on the solar distance, and the bulk electrical conductivity of the rocky matter. Other parameters important for excitation by the electric field are the temperature in the interplanetary cavity into which the object will radiate, the strength of the magnetic field, accretion time, and the solar wind ram-pressure. Sonnett and coworkers initially suggested electrical induction as a means of heating small planetary objects (meteorites and asteroids) and possibly the Moon and Mercury. Application to the other terrestrial planets-Earth, Mars and Venus is much more complex. These planets have an atmosphere and, in the case of the Earth, a strong intrinsic magnetic field. For a body whose temperature and conductivity decrease towards the surface, the bulk of the electric field will occur near the surface. The deep interior of a hot object will effectively be a short-circuit and will show little electric field.

The Origin of the Core

2.4

107

Time of Core Formation

Views of the time required to form the Earth’s core have changed considerably over the past two decades. Runcorn (1962a,b) suggested that the core has continued to grow throughout geological time and developed a theory of continental drift based on its growth. Vening Meinesz (1952) had argued that the positions of the continents today could result from a large-scale, regular pattern of convective motions in the mantle, continental material tending to congregate at places where the currents are descending. Using the results of Chandrasekhar’s (1961) studies on the convection of a fluid in a spherical shell, Runcorn showed that the present radius of the core is only just greater than the value at which the fourth harmonic convection pattern in the mantle should become less readily excited than the fifth. During a transition from one harmonic to another, the continents will be under much stress as the convection pattern changes to a new form. The relatively recent time (geologically speaking) at which continental drift took place is thus understandable. Runcorn also predicted other epochs of large continental displacements when the core radius reached successively the values at which the first degree convection pattern in the mantle, initially present when the core was only beginning to form, gave place to the second and the second to the third, etc. These other epochs of continental drift occurred early in the early Precambrian. In addition to continental displacements, one would expect a series of orogenies during such transition periods. Radioactive age determinations have mainly been obtained from igneous rocks that come from depth or metamorphic rocks formed in the deeper parts of the crust as the result of orogenic forces (Gastil, 1960). It is interesting that the ages obtained for such rocks group in broad peaks around the dates 2600, 1800, 1000, and 300 Ma ago. The most recent peak covers much of the geological record since the middle Palaeozoic and this is associated with continental drift and the transition from a fourth to a fifth degree convection pattern in the mantle. Runcorn suggested that it is naturai to identify the 1000 Ma peak with the transition from the third to the fourth degree convection pattern, the 1800 Ma peak with the transition from tne second to the third and the 2600 Ma peak with the transition from the first to the second. In this way the dates at which the core radius reached these critical values can be determined-see Fig. 2.5 which also shows that, on this theory, the core only started its growth a little over 3000 Ma ago. Elsasser (1963) has considered in some detail the model of an Earth accreted cold with the material uniformly distributed. The main feature of such an Earth model is that the melting point curve of the silicates rises much more steeply with depth than the actual temperature. This implies that the viscosity of the silicates should increase appreciably with increasing depth. As

108

The Earth’s Core

0.60

0-55 0.50 r 0-45

0.40

-

1

0-35

3

7

-

Final radius of core

\

n.4-

n=5/

n=3-

n.4’

/ n~2-n-3

Present

0.30

day

f

2000 Origin

of Earth

1000

I

Age ( M a )

Fig. 2.5. Growth of the Earth’s core compared with radioactive age determination peaks. (After Runcorn. 1962b.)

the original Earth is heated by radioactivity, the outer layers are then the first to become soft enough to permit iron to sink towards the centre-farther down the fall of iron is slowed by increased viscosity. It then forms a coherent layer which, however, is gravitationally unstable and results in the formation of quite large drops. The latter fall rapidly to the centre giving rise to a protocore. Elsasser estimated that the formation of a proto-core slightly smaller than the present core probably took no more than several hundred million years. The fall of iron is controlled by the elasto-viscous properties of the silicate matrix. Below the melting point, the viscosity will increase exponentially with decreasing temperature. Thus, instead of a uniform rain of iron towards the centre we have a far more complicated process in which the fall of iron is largely controlled by the variation in temperature at any given depth. As iron fell through the almost fluid silicate material, the general flow pattern of the latter would at first be nearly streamline, but, since the viscosity of the mantle increases with depth, the inner portion would not have time to flow but would be pushed outwards by the falling drop of iron. Elsasser suggested that the composition of the mantle might thus become asymmetrical, the material displaced to the antipodal point containing iron, while that above the falling drop lost most of its iron. A difficulty with Elsasser’s model of core formation is that the centre of the

The Origin of the Core

109

proto-Earth is likely to be initially cold and thus possibly strong enough to resist penetration of iron droplets. Davis (1982) pointed out that the work of Kinsland and Bassett (1977) and Kinsland (1978) indicates that the ultimate strengths of silicates are only of the order of 1-2 G P a even under extreme pressures, Deviatoric stresses of this magnitude can easily be generated in a planet with large density contrasts, such as between rock and iron. Davies showed that the minimum size of a density heterogeneity that would exceed the strength of the mantle and sink towards the centre of the planet is a few tens of kilometres in radius. He further investigated the possibility that such heterogeneities might arise by the accretion of differentiated planetesimals, in which case core segregation could be initiated by heterogeneous impact heating well before extensive near-surface melting occurred. In the most favourable circumstances segregation could begin in a proto-planet when its radius is only 700 km. This increases the likelihood that core segregation was essentially contemporaneous with accretion. Tozer (1965) has reconsidered the question of the kinetics of core formation and concluded that the simple theory of falling iron masses in silicate material is untenable. He suggested as an alternative a mechanism based on the flow of iron along channels in the silicate phase-the acceptability of this theory, however, depends quite critically on whether iron is able to flow over distances of the order of a kilometre under such conditions. Tozer concluded that in any case core formation is proceeding today much more slowly than in the past and that it was virtually complete very early in the Earth’s history. Tolland (1973) has also considered core formation following accretion from an initially homogeneous mixture (82% by volume of silicate and 18% of iron plus sulphide). His model differs from other models (such as that of Elsasser, 1963) in that he considers the role of convection during core formation, taking into account the interaction between infall and convection processes. Tolland used Tozer’s (1972) model of the thermal history of the Earth (see $3.8). Tozer showed that for small departures from hydrostatic equilibrium, the rheology of the Earth can be described in terms of a Newtonian viscosity. If the convecting region is of the order of hundreds of kilometres thick, the viscosity is lo2*poise for any reasonable value of heat generation. This answers the objection to infall theories that the mantle viscosity is too high. Tolland assumed the present core to be an Fe-S mix (see §5.7.2)-the first melt in the proto-Earth would then be well below the melting point of iron. Rapid convection would prevent the Earth from ever having been extensively melted-a fact which Urey (1952) deduced on geochemical grounds. Tolland estimated that, for a mantle viscosity of 10’’ poise, to form the core in about lo8 years requires an initial “seed” of about 1 m in size, which does not seem unreasonable. Vityazev (1973) calculated the radial distribution of gravitational energy

-

110

The Earth’s Core

Model# 15

Model#30

Mode l#O

ModeW60

0@ @ a

a

b

b

Fig. 2.6. Earth models at various stages of core separation according to (a) the sinking layer model, ( 6 ) Elsasser’s model. Model 15 has = + o f the total mass differentiated: model 30 shows differentiation to a depth = t. Earth’s radius. (After Shaw, 1978.)

resulting from core formation. His solution necessitated a number of simplifying assumptions-incompressible Earth, no change in g as core formation proceeds and no consideration of the question of energy transfer between the silicate and iron phases. His model gave maximum energy release in the region of the MCB. A feature of Vityazev’s model was that the iron would separate out as a discrete layer falling to the centre of the Earth. This is different from the earlier approach of Elsasser (1963), who envisaged core formation resulting from the fall of a series of drops. Shaw (1978) has investigated a series of Earth models at various stages of core formation. The initial state is a homogeneous mixture of “silicate” and “iron”. The iron separates from the outside inwards, leaving an outer silicate layer as it moves towards the centre as in Vityazev’s model: for comparison he considered as well a set of models based on Elsasser’s mode of core formation (see Fig. 2.6). Shaw calculated the density, pressure and gravitational acceleration at various stages of core formation and hence the energy of assembly as differentiation proceeds. He was thus able to obtain the energy release as a function of degree of differentiation, in particular as a function of depth to which iron separation has taken place. Probably the most serious assumption in Shaw’s model is that the calculations are isothermal rather than adiabatic, so that the temperature rise is ignored. Shaw found that for the sinking-layer model the energy of assembly is approximately a linear function of depth of separation, following a slow change near the surface. Shaw then calculated the energy release on the assumption that it takes place primarily within the iron layer, as silicate rises through that layer. At each stage of the separation process, most of the Earth is remote from the site of energy release and thus does not receive any signifi-

The Origin of the Core

111

cant fraction of the energy. The energy release is small (- 100 J g- ') for the separation of the 100-km layer near the Earth's surface. As separation proceeds and the iron layer thickens, the energy release quickly rises to about 400 J g-', remaining at about this value for the separation of most of the mass, and then gradually drops to zero as separation is completed. By contrast, in Elsasser-type models the energy release is largest during the early stages of separation and then rapidly decreases with very little energy released by the separation of the mass which was originally within one-half Earth-radius from the centre. For the sinking-layer model, Shaw found that the core receives a substantial amount of the energy released, about one-half. This concentration of energy in the iron is due to its participation in the energy release throughout the process. This contrasts with the results of Vityazev and Mayeva (1976), who found that the mantle would receive -83% of the energy and the core 17%. This difference is due to the details of energy exchange between silicate and iron and the effect of isolation of silicate from the site of energy release, which was not taken into account in Vityazev and Mayeva's calculations*. Melting of disseminated iron would occur before melting of the major silicate fraction, especially if any alloying constituent is present to depress the melting point. Liquid iron would have readily penetrated downward through the silicate matrix, but since deeper levels were probably colder, the iron could not have immediately migrated to the centre of the planet. Rather, it would have formed a spheroidal layer, the bottom of which would necessarily have been at its freezing point. Below this layer there would have been a zone consisting of an undifferentiated, cold primordial mixture of materials, and the mantle above the iron-rich layer would have been iron-depleted and wellstirred by thermal convection and Rayleigh-Taylor instabilities caused by incoming iron-bearing planetesimals. Stevenson (1 981) believes that both the Elsasser blob model (1963) and the sinking layer model of Vityazev and Mayeva (1976) are feasible, given enough time. The problem with both models is the length of time needed for core formation-several hundred million years-mainly due to the slow thermal diffusion. Stevenson has suggested an alternative model. Figure 2.7(1) shows a cold, undifferentiated primordial core overlain by the denser iron that has accumulated from the partially molten mantle above. In Fig. 2.7(11), a spontaneous asymmetry develops. Large, non-hydrostatic stresses on the resulting primordial core lead

* Shaw (personal communication) has pointed out that the model depicted in Fig. 2.6 is highly oversimplified. The highly symmetrical process shown is also highly unstable. Shaw believes, however, that it does give a limiting result in that it favours maximum energy transfer to the material of the mantle. Any more realistic model would result in more energy (higher temperature) in the core, with a consequent cooler mantle. This implies even steeper thermal gradients immediately following core separation.

112

The Earth’s Core

PR IMOR D IA L

I

IRON LAYER

II

COLD ‘ROCKBERG’

m

Fig. 2.7. A hypothetical scenario for the early stages of core formation. At this stage, the proto-earth may be much smaller than the final earth. In I, a cold, undifferentiated primordial core is overlain by the more dense iron that has accumulated from the partially molten mantle above. In II, a spontaneous asymmetry develops. Large, non-hydrostatic stresses on the resulting primordial core lead to deformation and even fracturing. In 111, the debris from this is distributed in the form of ‘rockbergs‘ around the newly formed core. Thermal equilibration of these ‘rockbergs’ with the surroundings is by thermal diffusion and takes longer than Earth accretion. (After Stevenson, 1981 .)

to deformation and even fracturing. In Fig. 2.7(111), the debris from this is distributed in the form of “rockbergs” around the newly formed core. Thermal equilibration of these rockbergs with the surroundings is by thermal diffusion and takes longer than Earth accretion. Stevenson had earlier (1980) suggested that the nearside-farside asymmetry of the Moon was a natural result for a rigid, low-density core overlain by an inviscid high-density layer. Although the initiation of the core-forming process may be difficult because of the cold central part of the Earth, Stevenson believes that the subsequent migration of iron-rich material would be quite rapid (- 1 Ma), so that core formation was essentially contemporaneous with the accretion of the Earth. As pointed out in $2.3, a rapid rate of accretion of the Earth is essential if the gravitational potential energy of the dust cloud is to play a major role in the heating of the embryo Earth. There are a number of lines of evidence which indicate that the time-scale for accretion was quite short. It is generally believed that the Earth’s magnetic field arises from dynamo action in the fluid, electrically conducting OC (see 54.2). Remanent magnetization has been found in rocks as old as 3500 Ma so that the OC must have been molten at least that long ago. Hanks and Anderson (1969) have carried out an investigation of the thermal history of the Earth with this additional constraint and come to the conclusion that the Earth must have accreted in a period less than 0.5 Ma. There would not have been time for the decay of long-lived radioactive isotopes to have had sufficient heating effect and the authors discount the importance of the decay of short-lived radionuclides. A short accretion time is thus essential so that most of the gravitational potential energy of the dust cloud is trapped and not lost to space. The accretion

The Origin of the Core

113

time could be lengthened somewhat, since the core is not pure iron. The addition of silicon or sulphur (see 55.7) would lower the melting point of ironHall and Murthy (1972) have estimated that at core pressures the melting temperature of the Fe-FeS eutectic is about 1600°C lower than that of pure iron. Wetherill (1972) pointed out that measurements of the concentration of Rb and Sr and the Sr isotopic composition of lunar samples indicate that a major fractionation of Rb relative to Sr took place on the moon 4600 Ma ago. Similar studies of the highly differentiated lunar breccias provide additional evidence for a very early lunar differentiation. Wetherill concluded that the most likely heat source for this initial differentiation is the gravitational energy of lunar accretion and that, for a sufficiently high temperature to be reached during the accretion period, the time-scale for accretion must be of the order of 1000 years. By analogy with the moon, Wetherill suggested that the entire Earth was initially melted and fractionated, and that the age of the oldest rocks ( N 3500 Ma) indicates that it was not until then that the Earth cooled sufficiently to enable the formation of extensive areas of stable crust. Oversby and Ringwood (1971) have also produced evidence for an early formation of the Earth’s core. Ringwood had argued in 1960 that iron descending during core formation would take with it substantial amounts of lead but not of uranium, i.e. core formation would change the Pb/U ratio in the upper crust and mantle. He thus concluded that the 4550 Ma terrestrial event recorded by Pb/U geochronology referred to the time of core formation, which must have taken place very soon after the formation of the Earth. The main argument for Ringwood’s conclusion is that during core formation the Pb/U ratio of the metal phase would be higher than that in silicates; the opposite point of view was taken by Patterson and Tatsumoto (1 964). Oversby and Ringwood (1967) carried out experimental measurements of the distribution coefficient of lead between relevant metal and silicate systems to settle this question. Since the core is believed to contain some 1&30% of one or more light elements in addition to iron (see 95.7), they examined a number of core models with different synthetic metal phase compositions (Fea3SlsC2, FesgSi, 1, Fe3,S30-weight per cent). The silicate phase used in the experiments consisted of simplified synthetic basaltic compositions, doped with 1000 ppm of lead (as oxide) in some runs and 100 ppm in others. Finely powdered samples of metal phase were mixed with silicate in the ratio 1 part (by weight) of metal to 2 parts silicate. The mixture was heated to a temperature at which both metal and silicate phases were known to be molten (the experiments were carried out at high pressures under closed-system conditions to prevent volatilization of lead). Their results showed that lead ends up preferentially in the (i.e. the ratio of Pb metal phase and values of the distribution coefficient Kmis

114

The Earths Core

A

5 (Ma)

Fig. 2.8. Relationship of the time delay for core formation after accretion of the Earth, ATc, with the distribution coefficient for lead between metal and silicate phases, I?"'. Time of accretion is taken as 4550 Ma. Isotopic composition of upper mantle lead is taken as that from tholeiites dredged from the floor of the Pacific Ocean. (After Oversby and Ringwood, 1971 .)

concentration in the metal to that in the silicate phase) were obtained for each run. A relationship exists between K"/" and the time interval ATc between accretion of the Earth and segregation of the core. The exact relationship depends on certain "boundary" conditions-Fig. 2.8 shows one of the curves produced by Oversby and Ringwood. The distribution coefficient for a model with a metal phase composition Fea3SI5Czis 2.5 and Fig. 2.8 shows that core segregation must have occurred within about 20 Ma after accretion. For the Fe70S30 and Fe&i compositions, K"Is values are larger-however, because the ATc curve is very steep above K"/" = 1.5, the ATc intervals are only slightly shorter. Oversby and Ringwood thus concluded that the Earth's core formed either during accretion or very soon after. Reasonable variations in the boundary conditions do not affect this basic result. These conclusions strongly favour models in which much of the accretion of the Earth occurred rapidly under sustained high-temperature conditions, rather than those in which the Earth accreted in a cool, unmelted state, followed by subsequent internal heating and core formation at a much later stage. Vollmer (1 977) repeated Oversby and Ringwood's calculations using the

115

The Origin of the Core

revised U decay constants of Jaffey et al. (1971), which somewhat alters Oversby and Ringwood’s conclusions. Vollmer found that core formation cannot have occurred before 200 Ma and probably not later than 500 Ma after accretion. Oversby and Ringwood concluded that accretion must have resulted in a hot initial Earth, since the time interval between accretion and formation of the core in their calculation was less than 100 Ma. On the other hand, Vollmer’s results are compatible with an initially solid Earth being heated by radioactive decay until the metallic phase suddenly segregated into a core. Vollmer also considered another model compatible with hot accretion resulting in a completely or partially molten Earth and came to the same conclusion, that core formation was not quasi-simultaneous with accretion but occurred after or during a period of several hundred million years following accretion. Vollmer appreciates that his models are over-simplifications and do not take into account the non-uniform temperature distribution. Thus, they cannot give any details of the process of core formation but do demonstrate that there was probably a maximum time of some 100Ma between accretion and the completion of core formation. Gancarz and Wasserburg (1977) also showed that major differentiation of the Earth, accompanied by the transport of most of the Earth’s lead into the core, must have taken place in the first few hundred million years. From an analysis of Sr and P b isotopic data, Vidal and Dosso (1978) concluded that, although the core most probably formed early and quickly in the Earth‘s history, core growth has nevertheless continued but at a much slower rate for the following 3 x lo9 years. Allegre et al. (1982) used data on lead isotope ratios in mantle material to calculate growth curves for the core. Neglecting the continental crust, the chemical balance equation can be written as PE = Pnd1 - a )

(2.12)

where p = 238U/204Pband the subscripts E and rn stand for the bulk Earth C is the concentration of 204Pband the suband the mantle; a = Ccmc/CEmE, script c stands for the core. The core is assumed to contain no uranium so that pc = 0. Equation (2.12) can then be written (2.13)

p,,,(t) is determined from isotopic data, and equation (2.13) then gives mc(t)for fixed values of the parameter C,/C, and p E . Growth curves for the core for various models and values of p E are shown in Fig. 2.9. For their preferred model, Allegre et al. found that 85% of the core would have formed early in

116

The Earth's Core

0.5

0

I

4

I

2 9 Age (10 years)

Fig. 2.9. Growth of the mass of the core throughout geological time according to various models and two values of p~ = 2"U/2MPb for the whole Earth. (After Allegre e t a / . , 1982.)

the Earth's history (during the first 5Ck200 Ma) and 15% subsequently, its growth still continuing.

2.5

lnhomogeneous Models of the Earth

Orowan (1969) was one of the first to suggest that the Earth may have accreted inhomogeneously. He pointed out that iron is plastic ductile, even at low temperatures, provided that it does not contain far more carbon than is found in meteorites. As a result, metallic particles would be expected to stick together when they collide because they can absorb kinetic energy by plastic deformation. They can therefore combine by cold or hot welding. Silicates, on the other hand, are brittle and break up on collision, except within a narrow temperature range near their melting point. The accretion of the planets may thus have started with metallic particles. Once sufficiently large a body could easily collect non-metallic particles by embedding them in ductile metal, and

The Origin of the Core

117

later by gravitational attraction. Orowan thus suggested that planets may arise cold in this way with a metal core already partially differentiated. The problem of heat sources to provide subsequent melting of the core remains. Harris and Tozer (1967) pointed out that in the preplanetary dust cloud, particle adhesion and aggregation would have been most effective for magnetic grains. They thus suggested that the magnetostatic attraction of ferromagnetic dust particles could lead to enhanced capture or “sticking” crosssections. They showed that, with a Curie point for iron or an iron-nickel alloy of more than 700°C, and an average time interval between dust particle collisions (based on the optical cross-section) of the order of one year, there would have been time for a very large number of such collisions before oxidation of the iron occurred to any significant degree. Makalkin (1980) also investigated the possibility of the formation of an originally inhomogeneous Earth. His model depends on differences in the physical properties of dust grains in the early stages of evolution of the protoplanetary nebula. These differences are expected to produce difference in the sizes of iron-rich and silicate-rich grains leading to a segregation of the two types of grain. Makalkin believes that segregation would occur if the silicon-rich grains were smaller than the iron-rich grains, the latter being about lo-’ cm in size, or larger. Thus provided that gas was present in the solar nebula during accumulation of the present core-sized embryo, the Earth could form as an inhomogeneous planet with an original iron-rich core and a silicate mantle. Wesson and Lermann (1978) have examined the interaction of dust grains with each other in the solar nebula, taking into account that such grains would carry net steady-state charges like those of grains in interstellar clouds. This would provide a screening mechanism during accretion, resulting in bodies of different compositions depending on the local temperature in the nebula. They found that planetesimals of 0.1-10’ cm in size would form in a time 106-107 years and in the inner solar system would be of iron, stone or mixed composition. Such a mechanism could explain the different types of meteorites. Wesson and Lermann also found that iron grains would have accreted fastest in the inner regions of the primitive nebula. If such iron planestesimals were able to accumulate gravitationally to form larger bodies on a time-scale much shorter than the silicate accretion time-scale, then a planet could evolve with an iron core formed first, the silicate mantle being deposited later. Eucken (1944), Anders (1968), Turekian and Clark (1969) and Clark et al. (1972) have also proposed a model Earth in which the Earth accreted inhomogeneously-not a cold accretion model like that of Orowan (1969), but as a result of condensations from the solar nebula. The observed decrease in density of solid bodies with distance from the sun may then be causally N

118

The Earths Core

related to the radial temperature gradient present in the solar nebula during the condensation of the component materials. The sequence of chemical compositions of solar system bodies may be pictured as paralleling the sequence of condensation reactions undergone by a gas of solar composition during cooling from >2000 K to perhaps 30 K. More specifically, it is possible in principle to calculate the equilibrium chemical composition of solar material as a general function of pressure and temperature, and specify the exact sequence of reactions and the composition of the condensate along any pressure-temperature profile. Larimer (1967) calculated that the order of condensation would be Fe and Ni, magnesium and iron silicates, alkali silicates, metals such as Ag, Ga, Cu, iron sulphide, and finally metals such as Hg, TI, Pb, In and Bi. Such an order of condensation is that grossly inferred in the Earth and usually attributed to differentiation. Turekian and Clark (1969) suggested that the Earth’s core formed by accumulation of the condensed FeNi in the vicinity of its orbit which, as in Orowan’s model, then became the nucleus upon which the silicate mantle was deposited. However, Blander and Katz (1967) and Blander and Abdel-Gawad (1969) maintain that, at the pressures expected to have existed in the solar nebula, the silicates would have condensed before iron. Larimer later revised his estimates and concluded also that iron condensed slightly later than the silicates (Anders, 1968; Larimer and Anders, 1970). Lewis (1972a,b) has developed a model for the origin of the solid bodies in the solar system beginning with a solar nebula in which there is a steep radial temperature gradient. He assumed that the bulk composition of condensates in the nebula is determined by chemical equilibium between the condensates and gases in a system of solar composition. He then calculated the bulk density of condensate as a function of temperature (over the range 0-2200 K) and pressures from lo-’ to loc’ atm. He found that the plot of density of condensate versus temperature was quite insensitive to total pressure over a wide range of pressures, the sequence of reactions and compositions of solid phases formed at equilibrium being essentially pressure-invariant. The condensation and reaction sequence for inner solar system material of the most abundant elements is given in Table 2.1. Lewis obtained a specific density versus temperature curve (Fig. 2.10) that shows large density decreases due to oxidation of iron, hydration of silicates, condensation of water ice, and formation of solid gas hydrates. A notable exception to the trend of decreasing density with falling temperature is the formation of FeS, the mineral troilite, at 680 K. Retention of the volatile element sulphur leads to a density increase because of the high atomic weight of sulphur. Lewis’ model also permits specific correlation of the total volatile content of the bodies with their bulk densities. Thus density, oxidation state, and volatile content are all causally interrelated.

119

The Origin of the Core

TABLE 2.1 Condensationsequence for solar material. (After Lewis, 1973.)

Reaction Refractory oxide condensation Metallic iron condensation MgSiO, condensation Alkali aluminosilicateformation FeS formation Tremoliteformation End of Fe oxrdation to FeO Talclserpentine formation

Temperature (at lO-’bar) ( K) 1720 1460 1420 1250 680 540‘ 490

400

Cumulative bulk density of condensate (gem ’) -3 -7 4 4 4 -4

5 0 40 40 46 3’

3 a5 32

* Tremolite formations occur while the density of the condensate IS already falling rapidly due to Fe oxidation A small uncertainty in the free energy of formation of tremolite thus corresponds to a large uncertainty in the bulk condensate density at the temperatureof tremolite formation

The observed density trends of the terrestrial planets thus need not be the result of special fractionation processes, but a consequence of physical and chemical restraints on the structure of the solar nebula, particularly the variation of temperature with heliocentric distance. The densities of Venus, Mars and the Earth are then due to different degrees of retention of S, 0 and H as FeS, FeO and hydrous silicates produced in chemical equilibrium between condensates and solar-composition gases. Only the Earth is likely to have differentiated so as to extract the heavy alkali metals into the core-thus the very large heat source ( N 10L3Js-’), possible within the OC of the Earth, is impossible on Mercury or Venus and unlikely on Mars. Lewis envisions the planets and smaller bodies to be formed by slow, low-temperature accretion of condensate particles, initially by grain-interaction forces (“stickiness”, magnetic moments, electrostatic forces, etc.) but eventually, after growth of some protoplanets to multikilometre dimensions, by gravitational attraction. Hoyle (1972) has also discussed the origin of the solar system from the viewpoint of condensations in the primitive solar nebula. The temperature of the gases would fall off with increasing distance d from the centre, and certain solid and liquid materials become thermodynamically stable as the temperature falls. As the temperature fell below -1500K a group of refractory materials would condense first, iron in metallic form and certain metal oxides, particularly SiOz, MgO and CaO. This would happen for d - 2 x 1013cm, i.e. at about the radius of the Earth’s orbit. If such condensing materials formed into large enough bodies they would fall out of the gas and continue moving in more or less Keplerian orbits with d remaining of the same order of magnitude. The relevant size for this to happen is of the order of a few metres. Smaller particles would be swept outward by the gas. Solid materials forming

120

The Earths Core

3

6

I

~

/

P

Mercury Venus

d

Earth

d

Mars

MgSi03 condensation

Fe S

formotipn a Fe /

FeO ond

Fe/Si = 0.96

24

1600 1400 1200 1000 800 600 400 Temperature ( K ) Fig.2.10. Density of condensed material in equilibrium with a solar-composition gas, 400-1 600 K at 10-3 bar. A simplified chemical system (the 20 most abundant elements) is employed for three different values of the Fe:Si ratio. The densities of some of the planets are in excellent agreement with an Fe:Si ratio of 1.08, but the omission of rare elements and uncertainties in the abundance of major elements could have displaced the entire manifold of curves slightly. (After Lewis, 1972b.)

in a slowly cooling gas might be expected to be quite pure chemically. Also, for slow cooling the condensation of any one substance should be essentially completed before that of another starts, unless the two substances happen to become thermodynamically stable at much the same temperature. Hoyle thus predicted that iron would be likely to condense independently of MgO and SO2. At a temperature of 1500 K the iron would be metallic, not FeS as usually assumed by geochemists. Hoyle suggested that "chunks of iron would aggregate together more readily at 1500 K than chunks of silica, because the

-

The Origin of the Core

121

iron would be more sticky. Hence it does not seem unreasonable that the first substantial condensations, with sizes of some kilometres, would be balls of comparatively pure iron. I would imagine that from such a beginning the core of the Earth was formed first and that the rock-forming materials were added later”. This general conclusion has been substantiated by the detailed investigations of Grossman (1972a,b) on the condensation sequence of the elements as the primitive solar nebula cooled. Grossman assumed the vapour to be in chemical equilibrium with each condensed phase over the entire temperature range between its condensation point and the temperature at which it was consumed by reaction to form new phases. Thus, the derived sequence of condensation is an accurate description of the changing distribution of the elements between vapour and solid phase and between solid phases themselves, since the effects of high temperature condensates on the composition of the gas were considered in determining the condensation temperatures of lowertemperature species. The temperature of condensation and disappearance of all phases (at 1O-j atm total pressure) are given in Table 2.2. Metallic iron first appears at 1473 K and contains 12.1 mole% Ni. As the gas cools and more alloy condenses, its equilibrium Ni content decreases, reaching 4.9 mole %, corresponding to the Ni/Fe ratio of the solar system (Cameron, 1968), by 1350K. Forsterite first appears at 1444 K, at which temperature 46% of the total iron has already condensed. The pressure dependence of the condensation temperatures of iron, forsterite and enstatite are shown in Fig. 2.1 1, from which it can be seen that iron has a higher condensation temperature than forsterite and enstatite at pressures above about 7.1 x atm respectively. The differand 2.5 x ence between the temperatures of appearance of iron and forsterite gradually increases with increasing pressure, reaching approximately 80” at 10- atm. Forsterite condenses just before enstatite at all pressures above at least atm, with the temperature gap between them also gradually increasing with increasing pressure. At 10-2atm, forsterite condenses at 1528 K and enstatite at 1511 K. Equilibrium condensation of iron-nickel alloy proceeds in the same way over the entire pressure range investigated. The first alloy to condense is relatively nickel-rich but its nickel content rapidly decreases on cooling. Table 2.3 gives the compositions of the first-condensing alloys and their condensation points at several different total pressures. The initial nickel content of the alloy appears to increase very slowly with decreasing pressure, rising 1-1.5 mole % per ten-fold decrease in the total pressure. In Grossman’s inhomogeneous model, in a cooling gas of solar composition at atm, total pressure, 46% of the iron condenses before the forsterite appears. As accretion continues, the release of gravitational poten-

122

The Earth's Core

Stability fields of equilibrium condensates at

TABLE 2.2 atmospheres total pressure. (After Grossman. 1972a.) Condensation temperature

Temperature of disappearance

( K)

(K)

1758 1647 1625 1513 1473 1450 1444 1393 1362 1349 1294 1274 1139 1125' 1000 700 405 Q 200

1513 1393 1450 1362

Phase Corundum Perovskite Melilite Spinel Metallic Iron Diopside Forsterite Anorthite Enstatite Eskolaite Metallic Cobalt Alabandite Rutile Alkali Feldspar Troilite Magnetite Ice

A1203 CaTiO, Ca,AI,SiO,-Ca,MgSi,O, MSAW, (Fe, Ni) CaMgSi,O, Mg,SiO, Ti,O, CaAI,Si,O, MgSiO, CrA co MnS TiO, (Na, K)AlSi,O, FeS Fe,O, Hk-0

1125

-

' Below this temperature, calculations were performed manually using extrapolated high-temperature vapour composition data. In some cases, gaseous species that had been very rare assumed major importance at low temperature (CH,).

tial energy causes melting of the core and the less dense components of the early condensate such as Al2O3,perovskite and melilite float to the surface. Below 1444 K, iron and magnesium silicates condense together and accrete upon the Ni-Fe planetary nucleus. Melting of silicates and iron takes place as a result of the impacting of infalling material and 5-10% Si may enter the sinking metallic liquid by reactions such as those proposed by Brett (1971). In Grossman's model, a large fraction of the total iron is buried inside the Earth at temperatures far above 700 K, below which it would have reacted with solar gases to form FeS. Although high temperature olivine and pyroxene are nearly Fe-free, severe loss of Hz relative to HzO by Jeans escape from the atmosphere of the planet allows 15-20 mole % of the iron end members to enter these phases by the time the temperature has fallen to 1000-900 K. In the later stages of accretion, the accumulation rate decreases and the surface temperature falls, allowing more volatile and oxidized condensates to be retained by the Earth. Lower surface temperatures lead to a convective stage during which both the core and that part of the mantle already accreted are each internally homogenized. A difficulty with the Turekian-Clark and Grossman models is that there does not appear to be enough energy available to melt the OC either during

h

E

c

0

Fig. 2,11. The variation with pressure of the Condensation temperatures of Fe. forsterito and enstatite The deprtvsion of the condensation point of enstatite by the crystallization of forsterite has not been considered in this calculation. At any given pressure, the alloying of N i in the metal will widen slightly the temperature difference between the appearance of Fe and forsterite. Although this temperature gap is small at lo-’ atm, 46% of the Fe will have condensed before forsterite appears. (After Grossman, 1972b.)

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The Earth’s Core

TABLE 2.3 Composition of the first condensing alloy, its condensation temperature and the fraction of the total iron condensed before the appearance of forsterite as a function of pressure. (After Grossman, 1972a.)

Pressure (am)

10-2

10-3 10-4 10-5 10-6

Condensation temperature of alloy (K)

Initial nickel content of alloy (mole %)

Percentage of total iron condensed at condensation point of forsterite

1584 1473 1377 1292 1218

10.9 12.1 13.5 14.9 16.4

66 46 13 0 0

or after accretion-it is even more difficult to do so for a cold accretion model. It was for this reason that Hanks and Anderson (1969) postulated a very short accretion time-less than 0.5 Ma. This difficulty can be alleviated to some extent if the lighter component of the core is sulphur. In an Earth of meteoritic composition, a sulphur-rich iron liquid would be the first melt to form (see 95.7.2). Core formation could proceed under these conditions at a temperature some 600°C lower than would be required to initiate melting in pure iron. In the vicinity of the core, the eutectic temperature is probably some 1600°C lower than that for pure iron (Hall and Murthy, 1972). Anderson and Hanks (1972) reconsidered the inhomogeneous accretion model and concluded that it could account for the early melting of the core. In their model the proto-Earth consists of a uranium/thorium rich central nucleus, composed chiefly of Ca-, A l - and Ti-rich silicates; a shell of Fe-Ni containing some of the earlier condensates that had not fully condensed or accreted when the Fe and Ni condensed; a shell of less refractory silicates, mainly pyroxene and olivine; a shell of potassium- and sodium-rich silicates; and finally a shell of hydrated minerals and volatile-rich condensates. The proto-core is composed of the refractory nucleus and the Fe-Ni shell. It is the presence of a radioactive nucleus that provides the mechanism for melting the metal shell. In the model of Anderson and Hanks, melting of Fe commences at about 0.4 x lo9 years. The nucleus and the proto-core are not in gravitational equilibrium because Fe is denser than the nucleus. As melting of Fe commences, the nucleus will attempt to rise-the rise of the nucleus to the base of the mantle will not be symmetric because of the convective pattern in the core which is controlled by the rotation of the Earth. When the nucleus leaves the centre of the Earth it will be replaced by nickel-iron which, before its descent, is close to its melting point. Because of the steep slope of the melting curve relative to the adiabatic curve, it will refreeze, explaining the seismic evidence of the solidity and composition of the IC. Their model also suggests

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The Origin of the Core

that part of the extra-light alloying material in the molten O C may be residue from the nucleus, i.e. Ca-Al-rich oxides and silicates either in solution or in suspension. In this case, the high radioactivity of the material could provide part of the energy for driving the geomagnetic dynamo (see w.7). Ruff and Anderson (1980) have refined this model using additional data from the Allende meteorite. Although their model predicts long-lived heat sources within the initial core, the early thermal history of the Earth is dominated by the short-lived nuclide 26A1 which is responsible for melting the core. In addition, another significant heat source is the potential energy release associated with the rising refractories-their calculations indicate that this heat source is capable of melting at least some fraction of the core. They further suggested that the emplacement of refractory material (including U and Th) in the lowermost mantle is the cause of motions in the core (and could also drive convection in the mantle). They believe that U and Th can provide enough heat energy to maintain the Earth’s magnetic field against dissipative losses. Further experimental work is highly desirable. In this regard Lange and Ahrens (1984) used shock-wave experiments to estimate the strength of impact necessary to free water from minerals such as brucite and serpentine on the surface of an accreting planet. They found that when the Earth was about half its present radius, the impact of further planetesimals would have freed all the water. They then examined how the water would react with iron as the Earth continued to accrete. Water, iron and enstatite react to form forsterite, fayalite and hydrogen; the hydrogen gas could escape, depleting the Earth of its water content. Lange and Ahrens circumvent this difficulty by proposing an inhomogeneous accretion, in which the first planetesimals include more iron than the later arrivals. In order to obtain loz5g of atmospheric water by the end of accretion, slightly heterogeneous accretion is required-the first planetesimals being 36% iron by mass, the last arrivals having almost no iron, as compared to a homogeneous value of 34%. Such models yield final FeO budgets that either require a higher FeO content of the mantle (17 weight %) or oxygen as the light element in the OC of the Earth (see $57.3). Some meteorites contain chondrules, small ( - 1 mm) bodies of crystalline material which may have formed initially as molten droplets by direct condensation in the solar nebula. Other meteorite materials, lacking chondrules, may have crystallized directly from a gas phase, i.e. may have solidified without passing through a liquid state. Workers at the Geophysical Laboratory of the Carnegie Institution, Washington DC, have studied the triple-point conditions for the major meteoritic minerals. High-temperature experiments bar. The triple-point for diopside were carried out at pressures as low as (CaMgSi,O,) was found to be 4 x lo-* bar at 1553 K, and that for enstatite

-

126

The Earths Core

(MgSiO,) somewhat lower. Preliminary work has been carried out at pressures as low as lo-’’ bar. Many of the questions raised in this chapter will be discussed again in Chapter 5 on the constitution of the Earth’s core. Grossman and Larimer (1974) have reviewed the literature on chemical fractionations during the condensation of the solar system, and their consequences on the establishment of chemical differences between the different classes of chondrites and between the planets. As already mentioned, the temperatures for condensation of iron and the major silicate phases from the solar nebula are within a few tens of degrees of one another. Stevenson (1981) thus believes that efficient separation would be very difficult and that, in the absence of an efficient physical or chemical mechanism for large-scale separation, the Earth is more likely to have accreted in an approximately homogeneous fashion with incoming planetesimals containing both iron and silicates in roughly solar (chondritic) abundances.

2.6 Variation of the Gravitational Constant G with Time

Another physical process that could affect the evolution of the Earth is the possibility that the gravitational constant G varies with time. A number of geophysicists (e.g. Egyed, 1956; Carey, 1958; Heezen, 1959; Wilson, 1960) have suggested that the Earth has been expanding with time. In Carey’s hypothesis the expansion took place mainly during the past 500 Ma, leading to an average rate of increase in the radius of the Earth during this period of about 5 mm per year; Egyed inferred a rate of increase of the Earth’s radius of 0.4-0.8 mm per year. Creer (1965) noticed that all the continents fit closely together on a globe about half its present size. Expansion from a completely sial-covered sphere of about 3700km radius at a constant rate of about 0.6 mm per year over 4.5 x lo9 years would give continents the present-day configuration, perhaps modified by continental drift (Wesson 1973). One of the mechanisms that has been suggested to account for such an expansion is Dirac’s (1938) speculation that the gravitational constant G varies inversely with time. With a gradual decrease of G, the pressure would decrease inside the Earth and the volume increase. Dicke (1957) showed that, to account for the expansion demanded by Carey by this mechanism, it would be necessary to assume that G has been decreasing at a rate of roughly 1 part in lo8 per year: to meet the expansion of Egyed, the rate of decrease of G would have to be 1 part in lo9 per year. Egyed (1960) first suggested that palaeomagnetic data might be used to calculate the radius of the Earth at various times in the past. On the expansion hypothesis of Carey, Heezen, and Egyed, the continents do not increase

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127

in area, so that the distance between any two points on a stable part of one continent remains the same. Thus if the Earth’s radius increases, the geocentric angle between the two points decreases. Assuming the Earth’s ancient magnetic field to be dipolar, the Earth’s ancient radius may then be found from the measured inclinations of contemporaneous rocks from two localities on the same stable continental block. Cox and Doell (1961) used this method on Permian data from Europe and Siberia to obtain an estimated Permian radius of 0.99 times the present radius. Ward (1963) generalized Egyed’s method of calculation and applied it to Devonian and Triassic data as well as Permian data from Europe and Siberia. He obtained estimated Earth radii for these periods of 1.12, 0.94 and 0.99 times the present radius, respectively, and considered none of these estimates to be significantly different from the present radius. McElhinny et al. (1978) have used palaeomagnetic data for the Earth, and a study of the morphology of the planetary surfaces in other cases, to obtain limits to possible expansion for the Earth, Moon, Mars and Mercury. They concluded that the Earth has not increased its radius by more than 0.8% of its present value over the last 4 x lo8 years, and that the Moon has not expanded by more than 0.06% over the last 4 x lo9 years. Mars and Mercury may have changed their radii slightly over their histories, but not by much. The Mercury data have also been used to set the limit lG/Gl 5 8 x 10- l 2 a for variable-G cosmologies without continuous creation, and IG/Gl 5 2.5 x lo-” a - in cosmologies like Dirac’s (X)-model where there is continuous creation. The palaeomagnetic limit for Earth expansion is equivalent to a limit on the expansion rate of about 0.13 mm per year. The palaeomagnetic method of estimating the radius of the Earth in the past has been criticized by Carey (1976), Wesson (1978) and Smith (1978). Wesson summarizes the arguments and concludes that there is no evidence that rules out slow expansion; but on the other hand there is no evidence that proves the reality of expansion. The expanding Earth hypothesis has also been reviewed by Dearnley (1966), Creer (1967), Carey (1976) and Wesson (1973, 1978). Using high-pressure shock wave data, Birch (1968) showed that the increase in radius of an Earth having a chemically distinct mantle and core would only be about 370 km for a decrease in the gravitational constant from 2G to its present value of G. A larger increase in radius would be possible if Ramsey’s hypothesis (see 95.4) were true-Birch showed, however, that Ramsey’s hypothesis is extremely unlikely to hold. He concluded that if the mass of the Earth has remained constant, changes in the Earth’s radius are unlikely to exceed 100 km. Thus, the sum total of evidence indicates that any large expansion of the Earth has not taken place and that the upper limit to any rate of change of G is about 1 part in 10’O per year.

’

128

The Earths Core

If G has been decreasing with time, the rate of radiation of the sun would have been higher in the past and hence asteroids and meteorite bodies would have been warmer, possibly leading to loss of argon from the material of the meteorites. From the observed K-Ar ages of meteorites, Peebles and Dicke (1962) have shown that G cannot have been decreasing by more than about 1 part in 10" per year. This rules out the possibility that a decrease in G could lead to the large expansions required by Carey and Egyed: the limit of 1 part in 10" per year in the variation in G leads to an upper limit in the rate of increase in the Earth's radius of about 0.05 mm per year. Further deductions by Dicke (1966) on various geophysical effects indicate that only a very small decrease in G is possible. On the other hand, Hoyle (1972) is in favour of a decrease in G with time. One result of this would be higher temperatures in the past-the mean sea-level temperature of the Earth being about 70°C 2000 Ma ago. The most serious objection to a variable G is, in Hoyle's opinion, the Precambrian glaciation of the Canadian shield, which has been estimated to have occurred 2500 Ma ago. Recently there has been a fairly rapid increase in the number of observations of "discrepant redshifts". Hoyle and Narlikar (1971) have given a possible explanation in a theory that incorporates a gravitational constant G that is decreasing with time. In a later paper (1972) they showed that their theory implies that the radius of the Earth has increased at a rate of about 0.1 mm per year. Shapiro ef al. (1971) have placed an observational upper limit of 4 x lo-'' per year on G/G. Taking Hubble's constant as 5 x lo-" year, the variation expected would be lo-'' per year, which is in close agreement with Dicke's (1962) estimate. If G decreases by a sufficient amount a cluster of particles will expand. In some cases the particles may have sufficient velocity to escape from the cluster as the gravitational binding decreases. Since it is known that clusters of galaxies and globular clusters currently exist with finite dimensions, Dearborn and Schramm (1974) were able to set limits on the magnitude of the variation of G , by investigating numerically the dynamical effects on an isolated cluster of galaxies caused by non-zero values of G/G. For a range of initial conditions, the maximum rate of change in G was determined that would still allow the existence of clusters of galaxies at the current epoch. A similar study was made for globular clusters and the results were found to be comparable. They found an upper limit for G/G of 4-10 x lo-" per year. This limit is much stronger than that of Shapiro et al. (1971). However, the Dirac cosmology is not consistent with this limit. Also the Hoyle-Narlikar (1972) cosmology with G/G K l/t would seem to be inconsistent with the observations. On the other hand, Morrison (1973) obtained a limit of similar magnitude to that given by Dearborn and Schramm. He used occultations timed on atomic scales to determine the slowing of the Earth's rotation due to

The Origin of the Core

129

tidal and non-tidal (i.e. depending on 6)forces. Previous determinations were based on data timed on a gravitational basis on which the assumed effects of G/G would not be apparent, leaving only tidal slowing. The close agreement of the deceleration rates then implies G/G = 0, yielding an upper limit of 2 x lo-'' per year on G/C from the uncertainty in the data. This result is approximately the same as that of Dearborn and Schramm. There have been a number of suggestions for obtaining information about the strength of the Earth's gravitational field in the past. To obtain mean rates of change over millions of years a gravity-sensitive geological system is required. There is no shortage of gravity-controlled phenomena, but gravity seldom leaves a permanent record in the rocks and the effects of gravity are usually small compared with those of other often unpredictable variables. Stewart (1970) has suggested a number of phenomena, both geological and biological, that could possibly be used for this purpose. These include the gravitational compaction of clays, the compaction of clays beneath glacial ice, palaeobarometry, the size of flying animals, the depth of animals' footprints, the dimensions of the frames of land animals, and the growth of diapirs. Later Stewart (1972), developed a method that, while not determining accurate values of palaeogravity, has been able to define a limit to the decrease of g with time. The limit, though broad, is not inconsistent with theoretical predictions. Stewart argued that if gravity in the past had been higher than today it is conceivable that some fine-grained sedimentary deposits would be over-consolidated, i.e. compacted more by the smaller sedimentary column above them in the past than by the larger one existing today. Since compaction is relatively rapid and largely irreversible, the requisite evidence could be preserved. Stewart measured the degree of overconsolidation in sediments in the London basin. He found that the London clay now exposed has been consolidated by higher pressures than would have been produced by what is now the greatest thickness of sedimentary overburden to be found anywhere in the London basin. It is of course possible that this extra pressure was derived from sediments younger than any now observed, additional overburden that has since been eroded away. However, if the additional pressure was caused solely by a higher value of g in the past (about 26 Ma ago), Stewart's results indicate that gravity at that time could not have been more than twice its present value. This implies that the maximum possible decrease in g over the past 26 Ma is 4 parts in lo8 per year. Stewart later (1978) set a limit to changes in palaeogravity since the late Precambrian by analysing the pressure-temperature relation of upper-mantle kimberlite nodules, finding that g (now) 5 g (Precambrian) 5 29 (now). This limit confirms his earlier conclusion that g (now) 5 g (Phanerozoic) 5 1.49 (now), obtained from the non-occurrence of lawsonite in deep sedimentary basins (Stewart, 1977). These limits rule out the fast-expansion hypothesis for the

130

The Earths Core

Earth if G and the mass of the Earth have not changed significantly over the last lo9 years; and while the slow-expansion hypothesis is not ruled out, there is no evidence to suggest that g has changed significantly during the last lo9 years. A bibliography on measurements of G, including the question of its possible time-dependence, has been given by Gillies (1982). References

Alfven, H. and Arrhenius, G . (1970a). Structure and evolutionary history of the solar system. I. Astrophys. Space Sci. 8,338. Alfven, H. and Arrhenius, G. (1970b). Origin and evolution of the solar system, 11. Astrophys. Space Sci. 9,3. Alfven, H. and Arrhenius, G. (1973). Structure and evolutionary history of the solar system, 111. Astrophys. Space Sci. 21, 117. Allegre, C. J., DuprB, B. and Brevart, 0. (1982). Chemical aspects of the formation of the core. Phil. Trans. Roy. SOC.London A M , 49. Allkgre, C. J., Staudacher, T. L., Sarda, P. and Kurtz, M. (1983). Constraints on evolution of Earth’s mantle from rare gas systematics. Nature 303,762. Anders, E. (1968). Chemical processes in the early solar system, as inferred from meteorites. Acc. Chem. Res. 1,289. Anderson, D. L. and Hanks, T. C. (1972). Formation of the Earth’s core. Nature 237,387. Arrhenius, G., Asunmaa, S . K. and Fitzgerald, R. W. (1972). In Lunar Sci. Inst. (C. Watkins, ed.). Contribution No. 88 in Lunar Science 111. Birch, F. (1965). Energetics ofcore formation. J . Geophys. Res. 70.6217. Birch, F. (1968). On the possibility of large changes in the Earth‘s volume. Phys. Earth Planet. Int. 1, 141. Blander, M. and Katz, J. L. (1967). Condensation of primordial dust. Geochim. Cosmochim. Acta 31, 1025. Blander, M. and Abdel-Gawad, M. (1969). The origin of meteorites and the constrained equilibrium condensation theory. Geochim. Cosmochim. Acta 33,701. Brecher, H. (1971). On the primordial condensation and accretion environment and the remanent magnetization of meteorites. Proceedings of the IAU Symposium on the Evolutionary and Physical Properties of Meteoroids, Albany, June, 1971. Brett, R. (1971). The Earth’s core; speculations on its chemical equilibrium with the mantle. Geochim. Cosmochim. Acta 35,203. Cameron, A. G. W. (1968). A new table of abundances of the elements of the solar system. In Origin and Distribution of the Ekments (L. H. Ahrens, ed.). Pergamon Press, Oxford. Cameron, A. G. W. (1972). Models of the primitive solar nebula. Symposium on the Origin ofthe Solar System, Nice 1972 (H. Reeves, ed.) Cent. Nat. Rech. Scient., Paris. Carey, S. W. (1958). A tectonic approach to continental drift. In Continental Drijt, a Symposium. University of Tasmania, Hobart. Carey, S. W. (1976). The Expanding Earth. Elsevier, Amsterdam. Cazenave, A., Lago, B. and Dominh, K. (1982). Numerical experiment applicable to the latest stage of planet growth. Icarus 51,113. Chandrasekhar, S . (1961). Hydrodynamic and Hydromagnetic Stability. Oxford University Press, London. Clark, S. P. Jr., Turekian, K. K. and Grossman, L. (1972). Model for the early history of the Earth. In The Nature ofthe Solid Earth (E. C. Robertson, ed.). McGraw-Hill, New York.

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Clayton, D. (1984). 26A1 in the interstellar medium. Astrophys. J . 280, 144. Coradini, A., Fedorico, C. and Lanciano, P. (1983). Earth and Mars; early thermal profiles. Phys. Earth Planet. Int. 31, 145. Cox, L. P. (1978). Numerical simulation of the final stages of terrestrial planet ,formation. PhD Thesis, MIT. Cambridge, MA. Cox, A. and Doell, R. R. (1961). Palaeomagnetic evidence relevant to a change in the Earth’s radius. Nature 189,45. Cox, L. P. and Lewis, J. S. (1980). Numerical simulation of the final stages of terrestrial planet formation. Icarus 44,706. Creer, K. M. (1965).An expanding Earth? Nature 205,539. Creer, K. M. (1967).Earth, possible expansion of. In International Dictionary ofGeophysics (S. K. Runcorn, ed.). Pergamon Press, Oxford. Davies, G. F. (1982). Ultimate strength of solids and formation of planetary cores. Geophys. Res. Lett. 9, 1267. Davies, G. F. (1985). Heat deposition and retention in a solid planet growing by impacts. Icarus 63,45. Dearnley, R. (1966). Orogenic belts and a hypothesis of Earth evolution. Phys. Chem. Earth 7 , l . Dearborn, D. S. and Schramm, D. N. (1974). Limits on variation of G from clusters of galaxies. Nature 247,441. Dicke, R. H. (1957). Principle of equivalence and the weak interactions. Rev. Mod. Phys. 29,355. Dicke, R. H. (1962). Implication for cosmology of stellar and galactic evolution rates. Rev. Mod. Phys. 34, 110. Dicke, R. H. (1966). The secular acceleration of the Earth’s rotation and cosmology. In The Earth-Moon System (B. G. Marsden and A. G. W. Cameron, eds.). Plenum Press, New York. Dirac, P. A. M. (1938).A new basis for cosmology. Proc. Roy. Soc. London A165 199. Dole, S. H. (1970).Computer simulation of the formation of planetary systems. Icarus 13,494. Egyed, L. (1956). A new theory on the internal constitution of the Earth and its geologicalgeophysical consequences. Acta. Geol. Hung, 4,43. Egyed, L. (1960). Some remarks on continental drift. Geofis. Pura Appl. 45,115. Elsasser, W. M. (1963). Early history of the Earth. In Earth Science and Meteoritics (J. Geiss and E. D. Goldberg, eds.). North-Holland, Amsterdam. Eucken, A. (1944). Physikalisch-chemische Betrachtungen uber die fruheste Entwicklungsgeschichte der Erde. Nachr Akad. Wiss. Gottingen, Math-Phys. K I . , Heft 1, 1. Feber, R. C., Wallace, T. C. and Libby, L. M. (1984). Uranium in the Earth’s core. Trans. Amer. Geophys. Union. 65,785. Flerov, G. N., Ter Akopjan, G. M., Popecko, A. G., Fefilov, B. V. and Subbotin, V. G. (1977).Sou. J . Nucl. Phys. 26,449. Fireman, E. L., DeFelice J. and Norton, E. (1970). Ages of the Allende meteorite. Geochim. Cosmochim. Acta 34,873. Fish, R. A,, Goles, G. G. and Anders, E. (1960). The record in the meteorites. 111: On the development of meteorites in asteroidal bodies. Astrophys. J . 132,243. Flasar, F. M. and Birch, F. (1973). Energetics of core formation: a correction. J . Geophys. Res. 78, 6101. Gancarz, A. J. and Wasserburg, G. J. (1977). Initial P b of the Amitsoq gneiss, West Greenland and implications for the age of the Earth. Geochim. Cosmochim. Acta 41,1283. Gastil, G. (1960).The distribution of mineral dates in time and space. Amer. J . Sci. 258, 1 . Gillies, G. T. (1982).The Newtonian gravitational constant: an index of measurements. Rapport Bur. Int. Poids Mesures 8219. Sevres, France. Goldreich, P. and Ward, W. R. (1973).The formation of planetesimals. Astrophys. J . 183, 1051.

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Kieffer, S. W. and Simonds, C. M. (1980). The role of volatiles and lithology in the impact cratering process. Rev. Geophys. Space Phys. 18,143. Kinsland, G. L. (1978). The effect of the strength of materials on the interpretation of data from opposed-anvil high-pressure devices. High Temp. High Press. 10,627. Kinsland, G. L. and Bassett, W. A. (1977). Strength of MgO and NaCl polycrystals to confining pressures of 250 kbar at 25°C. J. Appl. Phys. 48,978. Lange, M. A. and Ahrens, T. J. (1982a). Impact induced dehydration of serpentine and the evolution of planetary atmospheres. J. Geophys. Res. 87, Suppl. A, 451. Lange, M. A. and Ahrens, T. J. (1982b). The evolution of an impact generated atmosphere. Icarus 51, 96. Lange, M. A. and Ahrens, T. J. (1984). FeO and HzO and the homogeneous accretion of the Earth. Earth Planet. Sci. Lett. 71, 111 . Larimer, J. W. (1967). Chemical fractionation in meteorites. I: Condensation of the elements. Geochim. Cosmochim. Acta 31,1215. Larimer, J. W. and Anders, E. (1970). Chemical fractionation in meteorites. 111: Major element fractionations in chondrites. Geochim. Cosmochim. Acta 34,367. Typhoon Lee, Papanastassiou, D. A. and Wasserburg, G. J. (1976). Demonstration of 26Mg excess in Allende and evidence for 26A1.Geophys. Res. Lett. 3, 109. Typhoon Lee, Papanastassiou, D. A. and Wasserburg, G. J. (1977). Aluminium-26 in the early solar system: fossil or fuel? Astrophys. J. 211, L107. Lewis, J. S. (1972a). Low temperature condensation from the solar nebula. Icarus 16,241. Lewis, J. S. (1972b). Metal/silicate fractionation in the solar system. Earth Planet. Sci. Lett. 15, 286. Lewis, J. S. (1973). Chemistry of the planets. Annu. Rev. Phys. Chem. 24,339. Libby, L. M., Libby, W. F. and Runcorn, S. K. (1979). The possibility of superheavy elements in iron meteorites. Nature 278,613. MacDonald, G. J. F. (1959). Calculations on the thermal history of the Earth. J . Geophys. Res. 64,1967. Mahoney, W. A. (1985). 165th Meeting Amer. Astr. SOC.,Tucson, Arizona, Jan 13-16. Mahoney, W. A., Ling, J. C., Wheaton, Wm. A. and Jacobson, A. S. (1984). HEAO 3 discovery of 26A1in the interstellar medium. Astrophys. J . 286,578. Makalkin, A. B. (1980). Possibility of formation of an originally inhomogeneous Earth. Phys. Earth Planet. Int. 22, 302. Matsui, T. and Abe, Y. (1986). Formation of a “magma ocean” on the terrestrial planets due to the blanketing effect of an impact induced atmosphere. Earth, Moon, Planets 34,223. Maurette, M. and Bibring, J. P. (1972). Stellar wind radiation damage in cosmic dust grains: implications for the history of early accretion in the solar nebula. Symposium on the Origin of the Solar System, Nice 1972 (H. Reeves, ed.). Cent. Nat. Rech. Scient., Paris. McElhinny, M. W., Taylor, S. R. and Stephenson, D. J. (1978). Limits to expansion of Earth, Moon, Mars and Mercury and to changes in the gravitational constant. Nature 271,316. Morrison, L. V. (1973). Rotation of the Earth from A.D. 1663-1972 and the constancy of G. Nature 241,519. Murrell, M. T. and Burnett, D. S. (1985). K, U and Ca partitioning between sulfide and silicate liquids: heating of planetary cores. Lunar Planet. Sci. XVI, 603. Murrell, M. T. and Burnett, D. S. (1986). Partitioning of K, U, and Th between sulfide and silicate liquids: implications for radioactive heating of planetary cores. J . Geophys. Res. 91, 8 126. Murrell, M. T., Heuser, W. R. and Burnett, D. S. (1983). Partitioning of potassium and uranium between Fe- FeS and silicate liquids. Lunar Planet. Sci. XIV, 536.

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Safronov, V. S. (1972). Accumulation of the planets. In Symposium on the Origin of’the Solar System, Nice 1972 (H. Reeves, ed.). Cent. Nat. Rech. Scient., Paris. Safronov, V. S. (1978). The heating of the Earth during its formation. Icarus 3 3 , l . Schramm, D. M., Tera, F. and Wasserburg, G. J. (1970). The isotope abundance of 26Mg and limits on 26A1 in the early solar system. Earth Planet. Sci. Lett. l O , 4 4 . Shapiro, I. I., Smith, W. B., Ash, M. B., Ingalls, R. P. and Pettengill, G. H. (1971). Gravitational constant: experiment bound on its time variation. Phys. Rev. Lett. 26,27. Share, G. H., Kinzer, R. L., Kurfess, J. D., Forrest, D. J., Chupp, E. L. and Rieger, E. (1985). Detection of galactic 26A1gamma radiation by the SMM spectrometer. Astrophys. J . Lett. 292,2 Pt 2, L6l. Shaw, G. H. (1978). Effects ofcore formation. Phys. Earth Planet. Int. 16,361. Smith, P. J. (1978). The end of the expanding Earth hypothesis? Nature 271,301. Sonett, C. P. and Colburn, D. S. (1967). Establishment of a lunar unipolar generator and associated shock and wake. Nature 216, 340. Sonett, C. P. and Colburn, D. S. (1968). The principle of solar wind induced planetary dynamos. Phys. Earth Planet. Int. 1, 326. Sonett, C. P. and Herbert, F. L. (1977). Pre-main sequence heating of planetoids. In Proc. IAU Coll. No. 39, Relationships Between Comets, Asteroids and Meteorites (A. H. Delsemme, ed.). University of Toledo. Sonett, C. P., Colburn, D. S. and Schwartz, K. (1968). Electrical heating of meteorite parent bodies and planets by dynamo induction from a pre-main sequence T-Tauri solar wind. Nature 219,924. Sonett, C. P., Colburn, D. S., Schwartz, K. and Keil, K. (1970). The melting of asteroidal sized bodies by unipolar dynamo induction from a primordial T-Tauri sun. Astrophys. Space Sci. 7,446. Sonett, C. P., Smith, B. F., Colburn, D. S., Schubert, G, and Schwartz, K. (1972), in Proc. Third Lunar Sci. Conf. 3,2309 (Geochim. Cosmochim Acta Suppl. 3). Spaute, D., Lago, B. and Cazenave, A. (1985). Gaseous drag and planetary formation by accretion. Icarus 64,139. Staudacher, T. R. and Allegre, C. J. (1982). Terrestrial xenology. Earth Planet. Sci. Lett. 60,389. Stevenson, D. J. (1980). Lunar asymmetry and palaeomagnetisrn. Nature 287,520. Stevenson, D. J. (1981). Models of the Earth’s core. Science 214,61 I. Stewart, A. D. (1970). Palaeogravity. In Palaeogeophysics (S. K. Runcorn, ed.). Academic Press. London and Orlando. Stewart, A. D. (1972). Palaeogravity from the compaction of fine-grained sediments. Nature 235, 322. Stewart, A. D. (1977). Quantitative limits to palaeogravity, J . Geol. SOC.London 133,281. Stewart, A. D. (1978). Limits to palaeogravity since the late Precambrian. Nature 271, 153. Tolland, H. G. (1973). Formation of the Earth’s core. Nature Phys. Sci. 243, 141. Tozer, D. C. (1965). Thermal history of the Earth. 1: The formation of the core. Geophys. J . 9,95. Tozer, D. C. (1972). The present thermal state of the terrestrial planets. Phys. Earth Planet. Int. 6, 182. Turekian, K. K. and Clark, S. P. Jr. (1969). Inhomogeneous accumulation of the Earth from the primitive solar nebula. Earth Planet. Sci. Lett. 6, 346. Urey, H. C. (1952). The Planets. Yale University Press. Urey, H. C. (1962). The origin of the moon and its relationship to the origin of the solar system. In The Moon ( Z . Kopal and Z. K. Mikhailov, eds.). Academic Press, London and Orlando. Vening Meinesz, F. A. (1952). Convection currents in the Earth and the origin of the continents, I. Kon. Ned. Akad. Weten. 55, 527. Vidal, P. and Dosso, L. (1978). Core formation+atastrophic or continuous-Sr and Pb isotope geochemistry constraints. Geophys. Res. Lett. 5, 169.

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Chapter Three

The Thermal Regime of the Earth’s Core

3.1

Introduction

Early work on the thermal history of the Earth was based on the supposition that the thermal regime was controlled by conduction alone. This raised two difficulties. First, the time z required to reach equilibrium from some initial temperature distribution N R2/k,where R is the radius of the Earth and k the thermal conductivity. z is of the order of 10“ s, longer than the age of the Earth, so that most of the original heat would not have been dissipated. The second difficulty is the unknown initiai boundary conditions, which depend on the circumstances surrounding the origin of the solar system. Allowing for convection resolves these difficulties. z can be considerably reduced and, if the thermal regime of the deep interior is controlled by a fixed viscosity as maintained by Tozer (1972) and discussed in detail in $3.8, then the physical properties of the interior have only a “fading memory” of the initial conditions at the time of the formation of the solar system and do not depend on a prescribed cosmological theory. To obtain some idea of the thermal structure of the deep interior of the Earth, it is necessary to obtain estimates of the melting point and adiabatic gradients: these are considered in &3.4 and 3.5. Since both these gradients depend on Gruneisen’s parameter, this will be discussed first. 3.2

Griineisen’s Parameter

Many basic physical relationships that are important in a study of the Earth‘s 137

138

The Earths Core

deep interior can be expressed in terms of the dimensionless Gruneisen parameter y. These include the adiabatic gradient, the melting-point curve, the Hugoniot curve and the equation of state for solids and liquids. y can be defined as

where CI is the volume coefficient of expansion, kT and ks the isothermal and adiabatic incompressibilities, p the density and cv, c p the specific heats at constant volume and constant pressure respectively. Numerical values of y are close to unity for most condensed substances, including liquids and gases, and y is thus much more nearly constant over the range of physical conditions in the Earth’s deep interior than its component parameters a, k,, ks or p. For solids above about half of their Debye temperature, y is virtually independent of temperature, but decreases slightly with pressure. The definition of y in (3.1) is sometimes referred to as the thermodynamic Gruneisen parameter ?, to distinguish it from related parameters with different definitions. Using standard mathematical identities of thermodynamics, alternative definitions of y are (3.2)

where p is pressure, V volume and T temperature. This last expression shows that y is the negative slope of a log-log plot of temperature against volume in a material undergoing adiabatic compression. In his studies of crystal lattice dynamics, Gruneisen (1912, 1926) defined a parameter yi as y. =

- d (In vi)

d(In V )

(3.3)

where v i is the frequency of the ith mode of lattice vibration. If all vi have the same volume dependence and are independent of temperature, yi defined by (3.3) can be shown to be equivalent to y defined by (3.1)-see e.g., Knopoff (1963); If a material is heated from zero temperature and then recompressed to its original volume V, the total pressure p at temperature T is given by the Mie-Gruneisen equation of state yEth

P = P o + v

(3.4)

where E t h is the thermal energy. This assumes that the thermal pressure can be separated from the static pressure. In the absence of an electron contribution, y in (3.4) is the same as that defined in (3.1).

139

The Thermal Regime of the Earth‘s Core

Neglecting the electron contribution, (3.4) has been used by many authors to obtain a relationship between y and the pressure dependence of the incompressibility k. In 1939, Slater obtained the result 1 dk 2dp

y=----

1 6

(3.5)

In his analysis, he appealed to the Debye theory of lattice vibrations, in which 2/3 of the modes are transverse. Later, Dugdale and MacDonald (1953) obtained the expression

Irvine and Stacey (1975) showed that the main defect in this relationship is the assumption that the forces due to atomic vibrations in mutually perpendicular directions are independent. By considering classical (hightemperature) three-dimensional anharmonic thermal oscillations of atoms in simple crystal structures with purely central interatomic forces, Irvine and Stacey obtained the equation

Vashchenko and Zubarev (1963) obtained the same result by a completely different approach, and Irving and Stacey have since called it yvz. It reduces to Dugdale and MacDonald’s expression (3.6) if the vibrations are assumed to be one-dimensional. The greatest departures of (3.7) from laboratory values of y are for strongly covalent materials in which inherent bond rigidity introduces an additional constraint on thermal vibrations. Irving and Stacey allowed for non-central forces by adding a termfto the expression for interatomic force constants to account for the constraint imposed by bond flexure. This leads to a modification of equation (3.7): Y=

It must be emphasized that (3.7) has been derived on the assumption that y is independent of temperature. Since k(p) is fairly well known from seismic data, y can be estimated from Earth model data. It is assumed that the data on k(p) for a mixture of phases can be used to estimate y for the same mixture. This is unlikely to introduce serious errors, since finite-strain theories indicate a common approximate k ( p ) relationship for all materials at very high pressures; see also Bullen’s (k, p ) hypothesis (95.5). Application of (3.7) to the fluid

140

The Earths Core

OC may be justified by the fact that, at core pressures, the material must approximate a close-packed atomic arrangement despite being completely dislocated in the fluid state, and that the frequency of thermal atomic oscillations is much greater than the frequency of diffusive transitions. For the upper mantle, however, non-central forces are important and the additional correction factorfin (3.8) must be used. For plausible materials in the upper . the core and lower mantle, laboratory tests for y indicate a value ~ 0 . 8 For mantle,fN 0, and seismic data indicate values of y in the lower mantle N 1.0 and in the core N 1.4. It is interesting to note that this is not the trend of the pressure dependence of y for any particular material at constant phase, which shows a decrease with pressure. Anderson (1979a) has preferred to use (3.3) to define what he has called the high-temperature acoustic y:

(3.10)

(3.1 1)

The rationale of the derivation of (3.9) from equation (3.3) is given in Anderson (1968). Essentially all modes are lumped into two kinds-shear and longitudinal-and since all shear modes are equal, they are equal to ys. Anderson prefers his approach to Stacey’s (1977), since it involves the seismic data direct rather than derived values of k and dkldp which smooth out fluctuations in up and us. He found that yo decreases exponentially from 1.3 to 1.0 in the lower mantle and is about 1.5 in the core. In a later paper, Falzone and Stacey (1980) developed, for a face-centred cubic element, an atomic potential formulation for the second-order theory of elasticity, in which terms to second order in strain are retained in calculating atomic bond length changes and elastic moduli. They used the same approach in a further paper (1981) to calculate the energy of thermal vibrations. The analysis differs from that of Irvine and Stacey (1975) by allowing for the mutual dependence of atomic bonds. In the 1975 paper it was assumed that the only elastic forces that were significant were due to nearest-neighbour central force atomic interactions and that all the bonds in a crystal structure were independent. Falzone and Stacey (1980) showed that, although the central force approximation is good for close-packed structures, the assumption of bond independence is not. Allowing for this mutual dependence, they obtained (1981) a new formulation of their expression for the thermal

141

The Thermal Regime of the Earth’s Core

Gruneisen parameter y. Equation (3.7) again has additional terms infin the numerator and denominator, leading to the result

The casef= 1, representing independent bonds, reduces to (3.7). It should be noted that the factorfin (3.12) is not the same as the factorfin (3.8), where f = 0 reduces to (3.7). Falzone and Stacey (1980) believe that second-order elasticity theory provides the best explanation of the elasticity of the IC and that (3.12) is appropriate there. However, it cannot be applied to the lower mantle where atomic bond angle rigidity, not considered in the derivation of (3.12), may be appreciable. Jamieson et al. (1978) considered all possible contributions to y Iikely to be important in the core-vibrational, electronic, magnetic ordering (ferromagnetic) and Schottky anomaly (high-spin-low-spin). The thermodynamic Gruneisen parameter is then given by (3.13)

where yi are the partial Griineisen parameters and ci the specific heats. The vibrational contribution is estimated from (3.9) in the solid IC. In the OC another formula must be used that does not explicitly include the effect of shear waves, for example the Slater (3.9, Dugdale-MacDonald (3.6) or the Irvin Stacey (3.7) formulation. Jamieson et al. prefer the Irvine-Stacey formulation and estimated yvibfor a number of Earth models. They estimated the electronic contribution y e from the equation d In D ( Ye

=

d In p

E ~ )

(3.14)

where D(+) is the electronic density of states at the Fermi energy. They obtained the same range of values for ye as for yvib viz. 1.2-1.8. Finally, Jamieson et al. considered possible contributions from magnetic effects to c, = Ciciand y in the high-pressure form of iron. For the Schottky anomaly effect on the specific heat, they concluded that it might have a small effect under core conditions; if so, it would lead to a slight increase in y. For ferromagnetic ordering to be important under core conditions, the Curie temperature would have to increase fairly rapidly with pressure. This would lead to a further increase in y. Figure 3.1 shows their estimated contributions to the specific heat of iron in the core as a function of temperature. It can be seen that both the vibrational and electronic specific heats are significant under core conditions. Magnetic contributions may cause a small perturbation.

142

The Earth’s Core

I

3

I

I

I

I

V i brat ionol

-

cV

N k 2

I

I 1

200

500

1000 2000 5000 I( 100

Fig. 3.1. Contributions to Cv in iron under core pressures and temperatures (After Jamieson et a / , 1978 )

Table 3.1 summarizes their conclusions on the different contributions to c, and y. Fazio et al. (1978) also estimated y as a function of temperature and volume for iron. They considered various crystal structures and several kinds of intermolecular potential functions. They concluded that it was not possible to obtain sufficiently accurate estimates of y in the Earth’s core, mainly because experimental data cannot help choose between possible potential functions. Jeanloz (1979) has used shock-wave (Hugoniot) data for iron to estimate values of y at high pressures and temperatures. He found that they satisfy the empirical relation (Anderson, 1968, 1979a) Y = Yo (PO/PY

(3.15)

TABLE 3.1 Contributionsto Cvand y in core. (After Jamieson eial., 1978.) Term Vibrational Electronic Schottky Anomaly Ferromagnetic Total

CvlNk

Y

3 1-2 0-0.4

1.2-1.9 1 .z-1 .a 4

0 4-5.4

0 1.2-2.0

~

143

The Thermal Regime of the Earth‘s Core

Using the initial density of E-Fe as a centering volume, the best values for y o and q are 2.2(+0.5) and 1.62( f0.37) respectively. The value of y o is not very well constrained by the data because of greater uncertainties at low densities. However, the value of q is significantly different from 1, implying that the assumption often used (e.g. McQueen et al., 1970) py = const

(3.16)

is not correct for Fe at high pressures. Using only seismic data, Anderson (1979b) found that the empirical relation (3.15) was also applicable to the lower mantle. A value of q in the range, 0.8 TL or TL, or both, since otherwise the O C would be solid. If TL > Tj4 heat is removed from the core by thermal conduction and the core is stably stratified thermally with Tc = TLat the ICB. There are six possible variations of the relative magnitudes of Tj4, TL and T;7.It is easy to see that the cases Tk < T i < Th and T i < TL < T‘’ are not possible, since they would both result in a solid OC. This leaves four possible thermal regimes, which have been discussed in detail by Loper (1978) and summarized below.

Regime A: Tj4 < TL < T i The solid freezes directly onto the IC, and the latent heat released there is removed by a thin conductive layer in which the temperature remains above the liquidus; see Fig. 3.16. The fluid is buoyant both compositionally and thermally. This regime represents the conventional picture of a thermally driven dynamo.

171

The Thermal Regime of the Earth's Core

I

TL ( P , )

T = la

/ ' /

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Fig. 3.17. A schematic representation of thermal regime B: 16 < 1; < 1:. A slurry layer must form at the bottom of the OC. The IC grows by sedimentation of solid particles. (After Loper, 1978.)

Regime B: T i < TL < TL In this regime the solid cannot freeze directly onto the IC because a conductive layer to remove the latent heat cannot be constructed. Since TL < TL in such a layer, it would be frozen solid. This is prevented by the formation of a slurry of solid particles suspended in the liquid phase directly above the ICB. The latent heat released by the freezing of particles raises the temperature from the adiabat to the liquidus as shown in Fig. 3.17. Since the particles are heavier than the fluid because of their compositional difference, they tend to fall radially inward onto the surface of the solid IC. The deficit of particles in the slurry layer, together with the excess of light material released at the boundary makes the fluid compositionally buoyant, setting up convection. With some of the latent heat released throughout the layer, rather than directly at the ICB as in regime A, the conductive gradient at the boundary is the same as the liquidus gradient. Regime C: TL < T i < Ti In this case the thermal conductivity of the fluid is sufficiently large that the thermal component of the density gradient tends to stabilize the fluid. If the compositional component of the density gradient is destabilizing and sufficiently strong to overcome the stabilizing effect of the thermal component, the core fluid will overturn. Again a thin conductive layer exists close to the ICB as shown in Fig. 3.18 The opposing effects of conduction and mixing keep T between TL and TL. The temperature is everywhere above the liquidus, so that no slurry forms. Since T i < T , more heat is conducted down the temperature gradient with motion than in the static case. The excess of heat

172

The Earth's Core

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conducted radially outward cannot be transferred to the mantle, and must be carried radially inward by the convective motions. This is in contrast to motions driven by thermal buoyancy in which the convective heat transfer is invariably radially outward.

RegimeD: TL < TL < TL There are two possibilities depending upon the relative magnitude of the effects of conduction and convection. Regime Df:High thermal conductivity If the thermal conductivity is sufficiently high that

the situation is essentially the same as regime C (see Fig. 3.19). Convection is driven by compositional buoyancy despite the stabilizing effect of the thermal gradients.

Regime 02: Low thermal conductivity If the thermal conductivity is too small to maintain T < TL, then the regime

173

The ThermalRegime of the Earth’s Core

7; (P,

T=Tc

-

+‘0

I I

1

I I

7

I I

’TL

0

,/

,I

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I

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I

I

I

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I

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I

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Fig. 3.19. A schematic representation of thermal regime D1: T ; < T’ < T ; < T i . This regime is very similar to C. (After Loper, 1978.)

holds for the bulk of the O C as shown in Fig. 3.20. A thin conductive layer occurs near the ICB, and a similarly thin convective layer occurs immediately above it, with the bulk of the core being filled with a slurry as envisaged by Busse (1972) and Malkus (1973). Thermal conduction leads to a stable thermal gradient, and gravitational settling of the dense, metal-rich particles also creates a stable compositional gradient. Thus the slurry would be very stable and any convective motions restricted to a very thin layer near the ICB, making any dynamo action very improbable. Fearn and Loper (198 1) have estimated the relative importance of thermal and compositional buoyancy in the core. They calculated the density profile in the core on the assumption that, as the IC grows, the lighter material and the latent heat released are redistributed by diffusion. They concluded that compositional effects dominate thermal effects in determining the stratification of the core. The diffusive flux of light material has two components-the flux down the compositional gradient acting to make the composition homogeneous (minimizing the chemical potential energy) and the baro-diffusive flux acting to separate out the light component from the heavy component (minimizing the gravitational potential energy). Using estimates of parameters relevant to the core, Fearn and Loper found that the diffusive density profile is unstable throughout the O C except for a layer some 70 km thick adjacent to the MCB. This would imply that, except for this thin layer just below the MCB, compositional convection acts throughout the OC. Loper and Roberts (1981, 1983) have examined in more detail conditions in the core if it is in regime B of Loper (1978). i.e. if T i < Ti, < TL, which implies that the liquid above the IC is frozen (constitutionally supercooled).

174

The Earth’s Core

P

PI

Fig.3.20. A schematic representation of thermal regime D2: T: < T’ = T; < T i . A slurry occurs in the OC. Irreversible processes generate stable gradients that inhibit radial motion. (After Loper, 1978.)

The resolution of this apparent paradox lies in reducing the amount of freezing directly on the growing IC and instituting freezing above it in a slurry. Copley et ul. (1970) studied the freezing of a 30 wt% solution of ammonium chloride in water as it is cooled from below by liquid nitrogen. The cooling causes crystals of ammonium chloride to freeze into a dendritic interface at the bottom of the sdution. The experiments of Copley et ul. have convinced Loper and Roberts that the slurry is more likely to be a mushy zone of dendrites between the liquid O C and the solid IC. Such dendrites are tree-like structures and are likely to be easily broken-in Copley et d ’ s experiments they were broken by natural convection flow creating isolated “chimneys” free of solid. The flow pattern was highly non-linear with all the upward flow occurring within these chimneys. Fearn et at. (1981) showed that the thickness of the mushy zone is strongly dependent on the difference in composition between the O C and the corresponding liquid with the lowest melting point. Unless the two compositions are very nearly equal, the mushy zone will be very thick, possibly extending to the centre of the Earth. Loper and Roberts (1983) thus suggested that the entire IC may be a mixture of liquid and solid. Seismic evidence has given some hints that the IC may be in a partially molten state. Doornbos (1974), Stiller et al. (1980) and Cormier (1981) have suggested that the attenuation of seismic waves in the IC may be due to the presence of fluid inclusions. In a later paper, Loper and Fearn (1983) constructed a model of a partially molten IC. They developed a model for the bulk anelasticity of a solid containing liquid inclusions nearly in melting-freezing equilibrium that is quan-

The Thermal Regime of the Earth's Core

175

titatively capable of explaining the observed anelasticity. However, the mass fraction of liquid must not be too small and there is a restriction on the size distribution of the liquid inclusions. Braginsky (1963) originally suggested that the geodynamo (see also $4.6) is driven through the downward settling of a heavy iron-rich component in the fluid OC to form a growing solid IC. He proposed a silicon-rich core in order to explain the F layer. However, the existence of the F layer is now in some doubt (see 51.2). t o p e r (1978) discussed the thermal regime of a fluid whose composition is less metallic than the eutectic, as proposed by Braginsky (1963). One difficulty with this model is that the solid that forms initially is less metallic and hence less dense than the liquid. This light solid cannot freeze directly onto the IC, since it would result in an IC less dense than the surrounding liquid. The metal-poor particles must float upwards, leaving a thin metal-rich layer at the bottom of the OC. Since the metal is the more dense constituent, this layer is stably stratified, and of variable composition, and thus heat can only be removed by conduction. Because of the difficulties in removing heat from this layer, toper concluded that a metal-poor composition for the core is very unlikely. Fearn and Loper (1983) examined in more detail how an IC of eutectic composition might grow in an iron-poor core. They investigated the evolution of a completely molten core as it cooled and concluded that the IC cannot be explained by the secular cooling of an iron-poor core because of the difficulty of removing the latent heat released. McCammon et al. (1983) presented a model of core formation in which a solid IC grows by pressureinduced freezing as the Earth grows by accretion, i.e. the cause of the freezing of the IC is not a decrease in temperature but an increase in the central pressure as the mass of the Earth grows. Fearn and Loper (1985) later showed that, as in the case of secular cooling, there is no difficulty in pressure-freezing a solid IC from an iron-rich core, but there are serious problems if the core is iron-poor. Their general conclusion is that the core has an iron-rich composition and was essentially entirely molten immediately following its formation. 3.8 The Core and the Thermal History of the Earth

The classical approach to the Earth's thermal history is to formulate it as an initial boundary-value problem with calculations based on the theory of heat conduction in a solid. Although calculations based on the theory of heat conduction are relatively straightforward, the results are not applicable to the real Earth. The data required are poorly known and conduction does not describe all the processes of heat and mass transfer within the Earth. Largescale convection is extremely efficient in transporting heat and such convec-

176

The Earth's Core

tive heat transfer will dominate thermal lattice and radiative heat conduction even for small velocities. Goto et al. (1980) have carried out shock wave measurements of the optical absorption spectrum from 41G580 nm of 0.14 and 0.26 mole% FeZf-bearing MgO (synthetic periclase) to pressures of 42 GPa. They found considerably lower opacities at short wavelengths than those inferred from static highpressure'measurements on more Fe2+-rich samples, implying that significant radiative thermal conductivity of the lower mantle may take place in Fe z + depleted minerals under highly-reducing conditions. However, although the effective conductivity may be enhanced by radiative transfer at high temperatures, it is unlikely to be a major factor in heat transport in the Earth. A difficult feature of any calculation of the thermal regime of the Earth based on conduction theory alone is that the thermal time constant is much greater than the age of the Earth, with the result that temperatures in the deep interior depend on the thermal conditions existing at the time of formation of the Earth. This difficulty is largely overcome by convection theory. A stabilization temperature of less than about 2500 K, and the great increase in heat transport above it, result in the present thermal conditions being virtually independent of initial conditions, and make the temperature distribution far less sensitive to details of the heat source distribution. An investigation of the thermal history of the Earth taking into account convection and fractionation of radioactive elements is extremely difficult. The difficulties are two-fold-mathematical and physical. A mathematical treatment of the problem entails the formulation and solution of the complete field equations of a multicomponent, multiphase and radioactive continuum of varying properties. The physical difficulties arise mainly from our lack of understanding of the Earth's rheological behaviour and fractionation processes. Lee (1967, 1968) developed mathematical techniques to treat the Earth's thermal history beyond simple heat-conduction theory, taking into account latent heat, convection, and fractionation of radioactive elements. He showed that large-scale convection within the Earth is unlikely and that heat transfer by small-scale penetrative convection is also unimportant. Such convection is of great importance, however, as a means of moving the radiogenic heat sources upward. Very little work has been done on convection as a mode of heat transfer in the mantle, although it has been realized that convection may play a dominant role in the thermal regime in the Earth. Tozer (1967, 1970a,b) has made some significant contributions in this regard, although the value of his work has unfortunately not been fully recognized or appreciated. As a result of both experimental work and theoretical considerations, Tozer proposed that convective motions in the Earth are such as to minimize the mean temperature across any spherical shell in which the conduction solution is unstable

177

The Thermal Regime of the Earth’s Core

and that it is possible to use data from laboratory model experiments to find that minimum. Tozer also showed that it is possible to estimate mean temperatures on level surfaces without knowing the exact details of the velocity distribution-such temperatures below a depth of about 800 km are controlled by the viscosity dependence on temperature. For all plausible viscositytemperature relationships, the temperature is always that which gives a viscosity N 1020-1021poise. Thus, the prevalent idea that any convection theory for the mantle must be very imprecise because of uncertainty in the viscositytemperature relationship is not true; rather the viscosity itself is constrained to lie within very narrow limits. In a later paper, Tozer (1972) showed that, for a very large range of physically plausible values of the material parameters, the temperature distribution in bodies larger than about 800 km in radius is very different from that predicted by conduction theory. In any body the average temperature rises with depth according to the conduction or state of rest solution until either the centre of the body is reached or the kinematic viscosity has fallen to a value -1020cm2s-’-whichever is reached first. Once a body is large enough for its central viscosity to be incapable of stabilizing a state of rest solution, the steady central temperature becomes comparatively independent of the radius and surprisingly low (see Fig. 3.21). Thus, of all bodies in the solar system, small objects (radius 5 800 km) best preserve conditions existing at their birth. The magnitude of the velocity of internal motions, the nonhydrostatic stresses and the viscosity in the convective zones are all relatively insensitive to the choice of values of the material parameters. Most planetary thermal evolution models now use the “parameterized” convection approach. Instead of solving the complete set of equations which govern convection, parameterized convection uses a relationship between the Nusselt number and the Rayleigh number independent of the actual details of the convective structure. The Nusselt number Nu describes the efficiency of convection in transporting heat: it is the ratio of the convective heat flux to the heat flux that would obtain if heat were conducted across the layer. In the case of constant viscosity and with isothermal boundaries, the Rayleigh number is defined as R

= (xyd3AT)/(Kv)

(3.33)

where a is the coefficient of thermal expansion, g the acceleration due to gravity, d the depth of the fluid layer, AT the temperature drop across it, K the thermal diffusivity and v the kinematic viscosity. The Rayleigh number describes the tendency of a fluid to convect: the larger the number, the more vigorous the convection. The relationship Nu

=

u(R/R,)O

(3.34)

178

The Earths Core

External radius (1000 k r n ) Fig. 3.21. The central temperature T, as a function of the external radius R for two values of the heat source density H. Curve 1 : H = 1.6 x 10-’4calcm-’s ’. Curve 2: H = 5 x 1 0 - ‘ 5 c a l c m - 3 s ‘, On the left, the steeply rising curves are steady state of rest solutions-unstable where dotted. Note the relative independence of T, on R when R > 800 km and the surprisingly low values of T,. (After T o m , 1972.)

+

where , ! is Ibetween and +,and R, is the critical Rayleigh number for the onset of convection, gives good results for a number of conditions (see, e.g. Turcotte and Oxburgh, 1967; Sharpe and Peltier, 1978; Schubert, 1979). Nataf and Richter (1982) point out that the “existence of such a simple relationship is basically due to the way convection ‘works’: convective motions are organized in cells whose typical dimensions are of the same order as the depth of the convective layer; heat is conducted through a boundary layer at the bottom, advected in plumes around an isothermal core to the upper boundary layer which conducts it to the top surface. However, when there is a very large viscosity variation across the layer, one expects a thick rigid lid to develop on top of the convective region where the Nu-R relationship applies. The problem is to determine the thickness of the ‘lid’ or the temperature drop across it. In other words, one has to define where the cut-off between a more or less rigid and conducting lid and the convective regime takes place.” Different authors have used different assumptions to define the cut-off. In his original model, Tozer (1972) took the viscosity at the base of the lid to calculate the Rayleigh number and the parameterized Nusselt number for the convective region beneath the lid, and then assumed that the lid thickness is that which minimizes the bottom temperature (i.e. maximizes the overall

The Thermal Regime of the Earth’s Core

179

Nusselt number). Most authors have assumed a constant viscosity-that applicable to the interior of the convective region-although with very different values. Ellsworth et al. (1985) have estimated the variation of viscosity q across the lower mantle based on models of the Gibb’s free energy of activation G* and the adiabatic temperature. The variation of G* with depth is estimated in two ways: using an elastic strain energy model, in which G* is related to the seismic velocities, and a model assuming G* proportional to the melting temperature (obtained from Lindemann’s law-see 93.4). The adiabatic temperature gradient is estimated from (3.2) and (3.9) which assume that the acoustic Griineisen parameter is the same as the thermodynamic Griineisen parameter. Finally estimates of q depend on the rheology of the lower mantle, whether it can be considered as a Newtonian or power law fluid. Ellsworth et al. obtained a wide range of values of q. For G* based on the melting temperature, increases in y~ with depth vary from a factor of about 100 for Newtonian deformation or power-law flow with constant stress to about 5 for non-Newtonian deformation with constant strain rate. (See Fig. 3.22.) For G* based on the elastic strain energy model, increases in q with depth range from a factor of about 1500 for Newtonian deformation or power-law flow with constant stress to about 10 for non-Newtonian deformation with constant strain rate. Only a non-Newtonian lower mantle convecting with constant strain rate or constant strain energy dissipation rate is consistent with recent estimates of mantle viscosity obtained from post-glacial rebound and true polar-wander data (Nakiboglu and Lambeck, 1980; Peltier, 1981; Sabadini and Peltier, 1981; Yuen et al., 1982), which indicate that the viscosity of the mantle is essentially constant with depth. Verhoogen (1973) has shown that it is possible to construct models of a radioactive core (due to the decay of 40K) that agree with seismic data and account for convective instability, the existence of a solid IC and other details of core structure (particularly the F layer) He assumed that the OC consists of Fe, S and a small amount of K (see 55.8) which generates heat by the radioactive decay of 40K. He considered two cases, one corresponding to a high rate of heat production h (8 x 10’’ W) which would require about 0.1% K by weight. This value of h corresponds to a heat flux at the surface of the Earth of about 25% of the average measured value. Verhoogen took as the other case a lower limit for h that was 10 times smaller. The temperature in the lower mantle has been estimated by Wang (1972) to be about 3300 K 800 at a depth of 2800 km. Bolt (1 972) has shown that the decrease with depth of the seismic velocities V pand Vs in the lowermost 100-150 km of the mantle (region D”) may imply a sharp density gradient.*

*

The D” layer is discussed in detail in $1.3.

I

I

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I

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I

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1

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f a U

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-

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Radius (km)

Fig 3 22 Variation o f effective viscosity '1 across the lower mantle calculated (a) from the elastic strain energy model for G* { b j from the melting temperdturc model for G' Solid line-constant stress Dashed Ilne--constant strain energy dissipation rate Dotted line-constant strain rate Calculations are based on a power law rheology w i t h n = 3 (After Ellsworth e t a / , 1985 )

The Thermal Regime of the Earth’s Core

181

This density gradient, caused perhaps by an admixture of core material may be sufficiently steep to prevent convection and maintain a high (conductive) temperature gradient, which accounts in part for the decrease in the seismic velocities. The layer D” may then be thought of as a non-convecting thermal boundary layer separating the convecting thermal regime of the core from that of the lower mantle above D”. If this interpretation is correct the temperature T, at the MCB may be considerably higher than the temperature at the top of D”. Verhoogen thus took T, to be about 4500-5000K for h = 8 x 10” W and perhaps only 3500 K for the lower limit of h or if the layer D” participates in mantle convection. Instability in the OC probably requires a temperature gradient at least as great as the adiabatic. Verhoogen thus first determined the gradient that would be maintained by the heat sources in the absence of convection. For distributed heat sources (40K) the temperature difference across the OC is approximately 1600 K for high h and only 160 K for low h. With this lower value of h the core is essentially isothermal. The heat drop across the core could be raised to 850 degrees even for this low value of h if it was released by crystallization at the ICB (Verhoogen, 1961), rather than by the radioactive decay of 40K. Verhoogen estimated the adiabatic gradient from (3.30) and showed that the lower rate of heat production from distributed heat sources leads to a conductive gradient that is everywhere less than the adiabatic and is thus incompatible with convection. Stacey and Loper (1984) have discussed the thermal histories of the core and mantle. Essential to their model is the existence of a thermal boundary layer at the base of the mantle (the D” layer). Core cooling is not effective until this layer develops by cooling of the bulk of the mantle. The core cooling rate is strongly influenced by the temperature increment across D that developed within the first few hundred million years and is still increasing (at present it is N 800 K). These results are shown in Figs 3.23 and 3.24. It can be seen that mantle cooling is non-linear with time. On the other hand, the core cooling rate has been essentially steady for more than 3 x lo9 years. The figures also show that the energy available to drive the geodynamo has been sensibly constant for a long period. This is consistent with the long-term average geomagnetic field strength that has been more or less steady for the entire palaeomagnetic record. In Stacey and Loper’s model, the core does not begin to cool apppreciably until a temperature difference has been established across the D” layer by cooling of the mantle. This places a limit on the heat flux from the core and makes it difficult to have a thermally driven dynamo (see $4.6) early in the Earth’s history as proposed by Stevenson (1983). The addition of radioactive 40K in the core would not help-in fact it would create a further difficulty because of its relatively short half-life-as the early core would be heating rather than cooling.

182

The Earths Core A

E

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(D

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v

i- 1.0QG

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.-

6 core

U 0

L

2 0.5 -

- 1.0

0 U

QD

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1

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I

4.5 4.0

l

3.0

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1.0

Time before present (lOgyears) Fig. 3.23.Preferred thermal history for a Newtonian mantle. The lower plot gives the mantle solidus temperature, Tu, core temperature T,. and lower mantle temperature, Ts, as functions of time The upper plot gives IC radius, r,, total core heat, d,,,,. the rate of gravitational energy release by rejection from the IC of the light alloying ingredient of the OC, QG and the net energy for the geomagnetic dynamo, do. The total mantle-derived heat flux is 27.5 x 10l2W and the mantle radiogenic heat is 22.78 x 10l2W at the present time (After Stacey and Loper, 1984.)

There is still much argument as to whether convection in the mantle is single-stage or two-stage: the question will not be reviewed here. It is important, however, for the constitution of the mantle, since whole-mantle convection would be incompatible with a difference in the bulk chemical composition between the upper and lower mantle: the question of whether there is such a compositional difference is controversial. Jeanloz and Richter (1979) constructed a thermal model for the lower mantle based on petrologically-derived estimates of the temperature in the transition zone (see Fig. 1.4) and from an adiabat based on the thermal properties of MgO and S i 0 2 measured at high pressures. They found that a thermal boundary layer ( - 100 km thick) is required at the base of the mantle (the D” region) in order to satisfy their estimate of the lowest possible temperature in the core ( - 2800 K). Their preferred model (with more reasonable core temperatures of about 320C3500 K) requires a second boundary layer in the lower mantle, most likely close to its base or at the transition zone. A thermal structure with only one boundary layer at the base of the mantle is acceptable only if the OC

183

The Thermal Regime of the Earth's Core

2.0

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74000

TM ,-,

Y

v

2

-3500 2

!i

' E

3000 ,

I

I

4.5 4.0

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3.0

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2.0

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Time before present (lOgyears) Fig. 3.24. Thermal history for non-Newtonian mantle rheology. In this case, the mantle radiogenic heat is 18.6 x 10" W. (After Stacey and Loper, 1984.)

is at a lower temperature, or if the heat flux from the core is significantly larger than has been considered reasonable ( > 50 m Wm-'). Davies (1980) has constructed thermal histories for a number of Earth models in which heat is transported by convection alone. The only heat source considered is that due to radioactive decay: the calculations thus only apply to the time since core segregation (or accretion) ceased to be a significant heat source. Transient conductive near-surface cooling is also ignored. Davies used an empirical relationship between the rate of convective heat transport and the temperature difference across a convecting fluid, and an approximate proportionality between effective mantle viscosity and T-" ( n N 30 throughout the mantle). He found that assuming the present surface heat flux to be entirely primordial (i.e. non-radiogenic) leads to unrealistically high temperatures about 1700 Ma ago. The inclusion of radiogenic heat sources moves these high temperatures further back in time. Davies concluded that there is a significant imbalance between present heat production and heat loss in the Earth, the present heat production/heat loss ratio being between about 0.45 and 0.65. The internal structure of the Earth consists essentially of a silicate material overlying an iron core. The fluidity of the OC then depends primarily on its

184

The Earth's Core

T

Fig. 3.25. A likely tempetaturexomposition phase diagram for a terrestrial core alloy. X represents the light alloying constituent (e.g sulphur, oxygen). S refers to the solid (almost pure iron), L is the liquid phase, OC refers to the liquid outer core and I C refers to a solid inner core. The full vertical line represents a cooling evolution and the broken line represents the present OC composition after some I C growth. (After Stevenson, 1983.)

composition and thermal evolution. This in turn implies the existence of alloying constituents in the OC capable of substantially reducing the melting point of iron. This follows from the fact that the mantle undergoes solid-state convection and self-regulates at a temperature considerably lower than the melting point of its major mineral phases. Moreover, the melting points of iron and of the major silicate and oxide phases are very similar even at high pressures. The question of the composition of the core and the likely alloying element(s) is discussed in $5.7. Figure 3.25, after Stev-enson (1983), shows a likely temperature-composition phase diagram for the Earth's core, where X represents the light alloying constituent. Consider a thermal evolution in which the core starts out hotter than the melting curve and then gradually cools. The evolution trajectory of the centre would follow the vertical line shown in the figure until it intercepted the phase boundary. At that point, solid iron would form and the coexisting fluid remnant would try to evolve towards being richer in the alloying constituent X. However, this fluid is intrinsically less dense than the overlying fluid (which has not yet undergone any freezing) and must therefore rise. This compositionally driven convection will homogenize the liquid OC to a composition that is in phase equilibrium with the solid IC. Eventually, the eutectic composition is reached and com-

The Thermal Regime of the Earth’s Core

185

plete freeze-out of the liquid is possible. The rate at which the core can cool over geological time is determined by the heat-transporting properties of the overlying mantle. However, the core cannot cool faster than the mantle, and the mantle cools rather slowly because its viscosity is very strongly temperature dependent. The existence of the Earth’s magnetic field for at least the last 3500 Ma demands a continued energy source and this provides a constraint on the thermal evolution of the core. The amount of power required depends on the nature of the source. Gubbins et al. (1979) used the equations of global energy and entropy balance and showed that for radioactive heating alone, 1013W are required, which is just within the limits set by the observed surface heat flux ( ~ x 4 l o i 3W). Cooling itself cannot release enough heat to power the dynamo, because the required cooling rate is so high that the IC would be a very recent feature of the Earth. The release of gravitational energy can produce a magnetic field of 1-2 x T with the IC growing slowly to its present size over 4 x lo9 years and a heat release of 2.5 x 1 O I 2 W (a lower heat flux is required because of the greater efficiency of conversion of gravitational energy into magnetic field than heat). For most of their calculations, the heat flowing from the core is less than that conducted down the adiabatic gradient, suggesting that convection is driven gravitationally against thermal stratification. These questions are discussed in more detail in $4.6.

References

Abelson, R. S. (1981). P h D Thesis, University ofCalifornia, Los Angeles. Alder, B. J. (1966). Is the mantle soluble in the core? J . Geophys. Re.?.71,4973. Anderson, 0. L. (1968). Some remarks o n the volume dependence of the Gruneisen parameter. J . Geophys. Rex 73, 5187. Anderson, 0. L. (1979a). The high-temperature acoustic Gruneisen parameter in the Earth’s interior. Phys. Earth Planet. Int. 18, 221. Anderson, 0. L. (1979b). Evidence supporting the approximation yp = const. for the Griineisen parameter of the Earths lower mantle. J . Geophys. Res. 84,3537. Anderson, 0. L. (1981a). Temperature profiles in the Earth. In Evolution of the Eurth (R. J. O’Connell and W. S. Fyfe, eds) Geodynamics Series 5, Amer. Geophys. Union. Anderson, 0. L. (1981b). A decade of progress in Earth’s internal properties and processes. Science 213, 76. Anderson, 0. L. (1982). The Earth’s core and the phase diagram of iron. Phil. Trans. Roy, Soc. London A306,21. Anderson, 0.L. (1986a). The phase diagram ofiron to 300 GPa. J . Phys. Chem. Solids. (in press). Anderson, 0. L. (1986b). Properties of iron at the Earths core condition. Geophys. J . 84,561. Baumgardner, J. R. and Anderson, 0. L. (1981). Using the thermal pressure to compute the physical properties of terrestrial planets. Adu. Space Res. 1,159. Birch, F. (1972). The melting relations of iron and temperatures in the Earth’s core. Geophys. J . 29,373.

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The Earth’s Core

Bolt, B. A. (1972). The density distributions near the base of the mantle and near the Earth’s centre. Phys. Earth Planet. Int. 5, 301. Boschi, E., (1974). Melting of iron. Geophys. J . 38,327. Boschi, E. (1975). The melting relation of iron and temperatures in the Earth’s core. Riu. Nuovo. Cim. 5,501. Boschi, E. and Mulargia, F. (1977). Equation of state of liquid iron at the Earth’s core conditions. J . Geophys. 43,467. Boschi, E., Mulargia, F. and Bonafede, M. (1979). The dependence of the melting temperatures of iron upon the choice of the interatomic potential. Geophys. J . 58,201. Braginsky, S. I. (1963). Structure of the F layer and reasons for convection in the Earth’s core. Dokl. Akad., Nauk, SSSR, 149,8. Brown, J. M. and McQueen, R. G. (1980). Melting of iron under core conditions. Geophys. Res. Lett. 7, 533. Brown, J. M. and Shankland, T. J. (1981). Thermodynamic parameters in the Earth as determined from seismic profiles. Geophys. J . 66,579. Brown, J. M. and McQueen, R. G. (1982). The equation of state for iron and the Earth’s core. In High Pressure Research in Geophysics (S. Akimoto and M. H. Manghnani, eds), pp. 61 1623. Center Acad. Publ., Tokyo. Brown, J. M. and McQueen, R. G. (1986). Phase transitions, Griineisen parameter, and elasticity for shocked iron between 77 G P a and 400 GPa. J . Geophys. Res. 91,7485. Bukowinski, M. S. T. (1977). A theoretical equation of state for the inner core. Phys. Earth Planet. Int. 14,333. Bullard, E. C. and Gubbins, D. (1971). Geomagnetic dynamos in a stable core. Nature 232,548. Bundy, F. P. and Strong, H. M. (1962). In Solid State Physics (F. Seitz and D. Turnbull, eds), Vol. 13, p. 81. Academic Press, London and Orlando. Busse, F. H. (1972). Comment on the paper “The adiabatic gradient and the melting point gradient in the core of the Earth” by G. Higgins and G. C. Kennedy, J . Geophys. Res. 77, 1589. Clark, S. P. Jr. and Turekian, K. K. (1979). Thermal constraints on the distribution of long lived radioactive elements in the Earth. Phil. Trans. Roy. Soc. London A291,269. Copley, S. M., Giamei, A. F., Johnson, S. M. and Hornbecker, M. F. (1970). The origin of freckles in unidirectionally solidified castings. Metall. Trans 1,2193. Cormier, V. F. (1981). Short period P K P and the anelastic mechanism of the inner core. Phys. Earth Planet. Int. 24, 291. Croxton, C. (1975). Introduction to Liquid State Physics. Wiley, New York. Davies, G. F. (1980). Thermal histories of convective Earth models and constraints on radiogenic heat production in the Earth. J . Geophys. Rex 85,2517. Doornbos, D. J. (1974). The anelasticity of the IC. Geophys. J . 38,397. Dugdale, J. S. and MacDonald, D. K. C. (1953). Thermal expansion of solids. Phys. Rev. 89,832. Dziewonski, A. M. and Anderson, D. L. (1981). Preliminary reference Earth model. Phys. Earth Planet. Int. 25,297. Egelstaff, P. A. (1967). A n Introduction to the Liquid State. Academic Press, London and Orlando. Ellsworth, K., Schubert, G. and Sammis, C. G. (1985). Viscosity profile of the lower mantle. Geophys. J . 83, 199. Elsasser, W. M. (1972). Thermal stratification and core convection. Trans. Amer. Geophys. Union 53, 605. Falzone, A. J. and Stacey, F. D. (1980). Second order elasticity theory: explanation for the high Poisson’s ratio of the inner core. Phys. Earth Planet. Int. 21,371. Falzone, A. J. and Stacey, F. D. (1981). Second-order elasticity theory: an improved formulation of the Griineisen parameter at high pressure. Phys. Earth Planet. Int. 24,284.

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Fazio, D., Mulargia, F. and Boschi, E. (1978). Thermodynamical Griincisen’s function for pure iron at the Earth’s core conditions. Geophys. J . 53,531. Fearn, D. R. and Loper, D. E. (1981). Compositional convection and stratification of Earth’s core. Nature 289,393. Fearn, D. R. and Loper, D. E. (1983). The evolution of an iron-poor core 1. Constraints on the growth of the inner core. In Stellar and Planetary Magnetism (A. M. Soward, ed.). Gordon & Breach, London. Fearn, D. R., Loper, D. E. and Roberts, P. H. (1981). Structure of the Earth’s inner core. Nature 292,232, Fearn, D. R. and Loper, D. E. (1985). Pressure freezing of the Earth’s inner core. Phys. Earth Planet. Int. 39, 5. Frazer, M. C. (1973). Temperature gradients and the convective velocity in the Earth’s core. Geophys. J . 34, 193. 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. Phil. Trans. Roy. Soc. London A278,187. Gilvarry, J. J. (1956a). The Lindemann and Griinciscn laws. Phys. Rev. 102,308. Gilvarry, J. J. (1956b).Gruneisen’s law and the fusion curve at high pressures. Phys. Rev. 102,317. Gilvarry, J. J. (1956~).Equation of the fusion curve. Phys. Rev. 102,325. Gilvarry, J. J. (1957). Temperatures of the Earth’s interior. J . Atmos. Terr. Phys. 10,84. Goto, T., Ahrens, T. J., Rossman, G. R. and Syono, Y. (1980). Absorption spectra of shockcompressed Fezf-bearing MgO and the radiative conductivity of the lower mantle. Phys. Earth Planet. Int. 22,277. Gruneisen, E. (1912). Theorie des festen zustandes einatomiger elemente. Ann. Phys. 39,257. Griineisen, E. (1926). The state of a solid body. Handhuch Phys. 10, 1. [NASA translation RE218-59W (1959).] Gubbins, D., Masters, T. G., and Jacobs, J. A. (1979). Thermal evolution of the Earth’s core. Geophys. J . 59,57. Hall, H. T. and Murthy, V. R. (1972). Comments on the chemical structure of an Fe-Ni-S core of the Earth. Trans Amer. Geophys. Union 53,602. Hide, R. (1969). Interaction between the Earth’s liquid core and solid mantle. Nature 222, 1055. Higgins, G. H. and Kennedy, G. C. (1971). The adiabatic gradient add the melting point gradient in the core of the Earth. J . Geophys. Res. 76,1870. Hiwatari, Y. and Matsuda, H. (1972a). Ideal three-phase model and the melting of molecular crystals and metals. Prog. Theor. Phys. 47,741. Hiwatari, Y. and Matsuda, H. (1972b). “Ideal three-phase model” and the melting of molecular crystals and metals. In The Properties of Liquid Metals ( S . Takeuchi, cd.). Taylor and Francis, London. Hoover, W. G., Ross, M., Johnson, K. W., Henderson, D., Barker, J. A. and Brown, B. C. (1970). Soft sphere equation of state. J . Chem, Phys. 52,4931. Irvine, R. D. and Stacey, F. D. (1975). Pressure dependence of the thermal Griineisen parameter, with application to the Earth’s lower mantle and outer core. Phys. Earth Planet. Int. 11,157. Jacobs, J. A. (1953). The Earth’s inner core. Nature 172,297. Jacobs, J. A. (1971a). Boundaries of the Earth’s core. Nature Phys. Sci. 231,170. Jacobs, J. A. (1971b). The thermal regime in the Earth’s core. Comm. Earth Sci. Geophys. 2,61. Jacobs, J. A. (1973). Physical state of the Earth’s core. Nature Phys. Sci. 243, 113. Jamieson, J. C., Demarest, H. H. and Schiferl, D. (1978). A re-evaluation of the Gruneisen parameter for the Earth’s core. J . Geophys. Res. 83,5929. Jayaraman, A,, Newton, R. C. and McDonough, J. M. (1967). Phase relations, resistivity and electronic structure of cesium at high pressures. Phys. Rev. 159,527.

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Jeanloz, R. (1979). Properties of iron at high pressures and the state of the core. J . Geophys. Res. 84, 6059. Jeanloz, R. and Richter, F. M. (1979). Convection, composition and the thermal state of the lower mantle. J . Geophys. Res. 84, 5497. John, M. S. and Eyring, H. (1971). The significant structure theory of liquids. In Physical Chemisty, A n Advanced Treutise (D. Henderson, ed.). Vol. VIIIa. Academic Press, London and Orlando. Kennedy, G. C. and Higgins, G. H. (1973a). The core paradox. J . Geophys. Res. 78,900. Kennedy, G. C. and Higgins, G. H. (1973b). Temperature gradients at the core-mantle interface. The Moon 7, 14. Kennedy, G. C. and Vaidya, S. N. (1 970). The effect of pressure on the melting points of solids. J . Geophys. Res. 75, 1019. Kim, Ki-Tae, Vaidya, S. N. and Kennedy, G. C. (1972). Effect of pressure on the temperature of the eutectic minimums in two binary systems: NaF-NaCl and CsCI-NaCI. J . Geophys. Res. 77,6984. Knopoff, L. (1963). Equations of state of matter at moderately high pressure. In High Pressure Physics und Chemistry (R. S. Bradley, ed.). Vol. 1, p. 227. Academic Press, London and Orlando. Knopoff, L. and Shapiro, J. N. (1969). Comments on the interrelationship between Griineisen’s parameter and shock and isothermal equations of state. J . Geophys. R e x 7, 1439. Kraut, E. A. and Kennedy, G. C. (1966). New melting law at high pressures. Phys. Rev. 151,668. Lee, W. H. K. (1967). The thermal history ofthe Earth. P h D Thesis, University of California, Los Angela. Lee, W. H. K. (1968). Effects of selective fusion in the thermal history of the Earth’s mantle. Earth Planet. Sci. Lett. 4,270. Leppaluoto, D. A. (1972). Melting ofiron by significant structure theory. Phys. Earth Planet. Int. 6, 175. Lindemann, F. A. (1910). Uber die Berechnung Molecular Eigenfrequenzen. Phys. Z.1 I , 609. Liu, L. G. (1975). On the ( y . E , I ) triple point of iron and the Earth’s core. Geophys. J . 43,697. Liu, L. G. ( 1 982a). The F zone. Geophys. Res. Lett. 9,808. Liu, L. G. (1982b). Phase transformation in MgSiO, compounds at high pressures and their geophysical implications. Earth Planet. Sci. Lett. 57, 110. Liu, L. and Bassett, W. A. (1975). The melting of iron up to 200 kbar. J . Geophys. Res. 80,3777. Loper, D. E. (1978). Some thermal consequences of a gravitationally powered dynamo. J , Geophys. Res. 83,5961, Loper, D. E. and Fearn, D. R. (1983). A seismic model of a partially molten inner core. J . Geophys. Res. 88,1235. Loper, D. E. and Roberts, P. H. (1978). O n the motion of an iron-alloy core containing a slurry 1. General theory. Geophys. Asrrophys. Fluid Dyn. 9,289. Loper, D. E. and Roberts, P. H. (1981). A study of conditions at the inner core boundary of the Earth. Phys. Earth Planet. Int. 24,302. Loper, D. E. and Roberts P. H. (1983). Compositional convection and the gravitationally powered dynamo. In Stellar and Planetary Magnetism (A. M. Soward, ed.). Gordon and Breach, London. Malkus, W. V. R. (1973). Convection at the melting point, a thermal history of the Earth’s core. Geophys. Fluid Dyn. 4, 267. Mao, H. K., Bassett, W. A. and Takahashi, T. (1967). Effect of crystal structure and lattice parameters of iron up to 300 kbar. J. Appl. Phys. 38,272. Masters, G. (1979). Observational constraints o n the chemical and thermal structure of the Earth‘s deep interior. Geophys. J. 57,507. McCammon, C. A., Ringwood, A. E. and Jackson, I. (1983). Thermodynamics of the system

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Fe-FeO-MgO at high pressure and temperature and a model for the formation of the Earths core. Geophys. J . 72,577. McLachlan, D. and Ehlers, E. G. (1971). EKect of pressure on the melting temperature of metals. J . Geophys. Res. 76,2780. McQueen, R.G., Marsh, S. P., Taylor, J. W., Fritz, J. N. and Carter, W. J. (1970).The equation of state of solids from shock wave studies. In High Velocity Impact Phenomena (R.Kinslow, ed.). Academic Press, London and Orlando. Mulargia, F. (1977). Is the common definition of the Mie-Griineisen equation of state inconsistent? Geophys. Res. Lett. 4, 590. Mulgaria, F. and Boschi, E. (1978). The generalization of the Mie-Griineisen equation of state. Geophys. J . 55,263. Mulargia, F. and Boschi, E. (1980). The problem of the equation of state in the Earth’s interior. In Physics ofthe Earth‘s Interior (Proc. Int. School Phys. Course LXXVII). North Holland, Amsterdam. Murthy, V. Rama and Hall, H. T. (1972). The origin and chemical composition of the Earth’s core. Phys. Earth Planet. Int. 6, 123. Nakiboglu, S. M. and Lambeck, K. (1980). Deglaciation effects on the rotation of the Earth. Geophys. J . 62,49. Nataf, H. C. and Richter, F. M. (1982). Convection experiments in fluids with highly temperature-dependent viscosity and the thermal evolution of the planets. Phys. Earth Planet. Int. 29, 320. Ninomiya, T. (1978). Theory of melting, dislocation model. J . Phys. Soc. Japun 44,263. ONions, R. K., Evensen, N. M., Hamilton, N. J. and Carter, S. R. (1978). Melting of the mantle past and present: isotope and trace element evidence. Phil. Trans. Roy. SOC.London A258,547. Peltier, W. R.(1981). Ice age geodynamics. Ann. Rev. Earth Planet. Sci. 9, 199. Poirier, J. P. (1986). Dislocation mediated melting of iron and the temperature of the Earths core. Geophys. J . 85,315. Ross, M. (1969). Generalized Lindemann melting law. Phys. Rev. 184,233. Sabadini, R. and Peltier, W. R. (1981). Pleistocene deglaciation and the Earth’s rotation: implications for mantle viscosity. Geophys. J . 66, 553. Salter, L. (1954). The Simon melting equation. Phil.M a g . 45, 369. Schloessin, H. H. (1974). Corrugations on the core boundary interfaces due to constitutional supercooling and effects on motion in a predominantly stratified liquid core. Phys. Earth Planet. Int. 9, 147. Schubert, G. (1979). Subsolidus convection in the mantles of terrestrial planets. Ann. Rev. Earth Planet. Sci. 7, 289. Sharpe, H. N. and Peltier, W. R. (1978). Parameterized mantle convection and the Earth’s thermal history. Geophys. Res. Lett. 5, 737. Simon, F. E. (1937). On the range of stability of the fluid state. Trans. Faraday S o t . 33,65. Slater, J. C. (1939). Introduction to Chemical Physics. McGraw-Hill, New York. Spiliopoulos, S. and Stacey, F. D. (1984). The Earth’s thermal profile: is there a mid-mantle thermal boundary layer? J . Geodyn. 1,61. Stacey, F. D. (1972). Physical properties of the Earth’s core. Geophys. Surv. 1,99. Stacey, F. D. (1977). A thermal model of the Earth. Phys. Earth Planet. Int. 15,341. Stacey, F. D. and Irvine, R. D. (1977). Theory of melting: thermodynamic basis of Lindemann law. Aust. J . Phys. 30,631. Stacey, F. D. and Loper, D. E. (1984). Thermal histories of the core and mantle. Phys. Earth Plunet. Int. 36,99. Stacey, F. D., Brennan, B. J. and Irvine, R.D. (1981). Finite strain theories and comparisons with seismological data. Geophys. Surv. 4, 189.

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Chapter Four

The Earth‘s Magnetic Field

4.1

Introduction

At its strongest near the poles the Earth’s magnetic field is several hundred times weaker than that between the poles of a toy horseshoe magnet-being Thus, in geomagnetic studies we are measuring extremely less than a gauss (r). small magnetic fields and a more convenient unit is the gamma (y), defined as lO-’r. Strictly speaking the unit of magnetic field strength is the oersted, the gauss being the unit of magnetic induction. The distinction is somewhat pedantic in geophysical applications since the permeability of air is virtually Wbm-2 unity in cgs units. In SI units, which will be used here, l r = T (tesla). Thus, l y = 10-9T = In T. (Weber/m2) = The magnetic field at any point on the Earth’s surface may be specified by three parameters, e.g. the total intensity F , declination D and inclination I or the two horizontal components X and Y and the vertical component 2. Simple relationships exist between these different magnetic elements (see e.g. Jacobs, 1966). The variation of the magnetic field over the Earth’s surface is best illustrated by isomagnetic charts, i.e. maps on which lines are drawn through points at which a given magnetic element has the same value. Contours of equal intensity in any of the elements X , Y, Z , H or F are called isodynamics. Figures 4.1 and 4.2 are world maps showing contours of equal declination (isogonics) and equal inclination (isoclinics) for the year 1985. It is remarkable that a phenomenon (the Earth’s magnetic field) whose origin, as we shall see later, lies within the Earth should show so little relation to the broad features of geology and geography. 191

192

The Earths Core

Not only do the intensity and direction of magnetization vary from place to place across the Earth, but they also show a time variation. There are two distinct types of temporal changes: transient fluctuations and long-term secular changes. Transient variations produce no large or enduring changes in the Earth's field and arise from causes outside the Earth. Secular changes, on the other hand are due to causes within the Earth and over a long period of time the net effect may be considerable. If successive annual mean values of a rnagnetic element are obtained from a particular station, it is found that over a long period of time the changes are in the same sense, although the rate of change does not usually remain constant. Figure 4.3 shows the change in declination and inclination at London, Boston and Baltimore. The declination at London was 113"E in 1580 and 24i0W in 1819, a change of almost 36" in 240 years. Lines of equal secular change (isopors) in an element form sets of ovals centring on points of local maximum change (isoporic foci). Figure 4.4 shows the secular change in Z for the year 1987.5. If the pattern of secular change is compared at different epochs it can be seen that the secular variation is a regional, rather than a planetary, phenomenon and that considerable changes can take place in the general distribution of isopors even within 20 years. The secular variation is discussed in more detail in 94.7. In 1839 Gauss showed that the field of a uniformly magnetized sphere, which is the same as that of a dipole at its centre, is an excellent first approximation to the Earth's magnetic field. Gauss further analysed the irregular part of the Earth's field, i.e. the difference between the actual observed field, and that due to a uniformly magnetized sphere, and showed that both the regular and irregular components of the Earth's field are of internal origin. The geomagnetic poles, i.e. the points where the axis of the geocentric dipole which best approximates the Earth's field meets the surface of the Earth, are situated at approximately 79"N, 70"W, and 79"S, f 10"E. The geomagnetic axis is thus inclined at about 11" to the Earth's geographical axis. If greater accuracy is needed, the potential of the Earth's magnetic field may be expanded in a series of spherical harmonics. It can also be shown that a better approximation to the Earth's field is an eccentric dipole obtained by displacing the centre of the equivalent dipole by about 300 km in the direction of Indonesia. The non-dipole components of the Earth's field, though much weaker than the dipole component, show more rapid changes. The time-scale of the nondipole changes is measured in decades and that of the dipole in centuries. The isoporic foci also drift westward at a fraction of a degree per year. The drift of the dipole field is slower than that of the non-dipole field, at least by a factor of three. Both eastward and westward drifts of the non-dipole field have been inferred from palaeomagnetic studies. Observatory records from Sitka, Alaska, indicate an eastward drift during the past 60 years, in contrast to the

Fig. 4 1 World map showtng contours of equal declination (tsoyonlcs) in dcgrccs for 1985 0 Contour interval 5' (Providsd by 0 R Barraclough Courtesy of the Geomagnetism Unit, Brlttsh Geological Survey )

I

60

90

120

Fig. 4.2. World map showing contours of equal inclination (isoclinics) rn degrees for 1985.0. Contour interval 70' for -40" (Provided by D. R. Barraclough. Courtesy of the Geomagnetism Unit, British Geological Survey.)

150

I80

< / < 40". and 5" for l / l z 40"

195

The Earths Magnetic Field CJECllNATlON

Fig. 4.3. Secular change of declination and inclination a t London, Boston and Baltimore (After Nelson ef a / , 1962 )

predominance of a westward direction of drift observed over most areas of the world for the past several hundred years (Skiles, 1970; Yukutake, 1962). In addition to the westward drift, the pattern of the secular variation field may alter appreciably in a few decades. In some areas the amplitude of the secular change is anomalously large or small. At present it is larger than average in the Antarctic and smaller than average in the Pacific hemisphere. A number of workers have given an alternative description of the Earth's magnetic field in terms of a n'umber of dipoles on, and inside, the MCB (see e.g. McNish (1940), Lowes and Runcorn (1951), Alldredge and Hurwitz (1964), Alldredge and Stearns (1969). More recently Mayhew and Estes (1983) have modelled the field by a number of dipoles at fixed radii inside the OC. Fixing the radius avoids problems of convergence that result from the extreme non-linearity of the problem when dipole positions are allowed to vary. The dipoles are centred on equal-area blocks and the algorithm Mayhew and Estes developed can be solved for dipole magnitudes only (fixed orientations), or have full freedom of orientation and be solved for vector components. A model consisting of 93 dipoles arranged 21" apart was computed, based on observatory annual means from 1974 through 1977 and a selected set of Magsat data. This was compared with models of the International Geomagnetic Reference Field for 1975 and their 1980 secular variation obtained by standard spherical harmonic analysis. The equivalent dipole

3

3

3'

3

31

6

2 :

16C13C10656

F i g . 4.4. World map showing contours of rate of change i n intensity (isopors) of the vertical component 2 for 1987.5. Contour interval 20 nT per year. (Provided by D. R. Barraclough. Courtesy of the Geomagnetism Unit, British Geological Survey.)

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The Earths Magnetic Field

source was shown to be comparable in accuracy. It must be stressed, however, that the representation of the geomagnetic field by a spherical harmonic analysis or by a number of dipoles is simply a mathematical procedure and has no physical basis. The geomagnetic field (of internal origin) varies with time on other timescales besides that of the secular variation. Braginskii (1970a,b, 1971, 1972) divided the oscillation “spectrum” of geomagnetic variations into three major categories. (i) The fundamental frequency, characteristic of dipole field strength oscillations, with a period =9000 years. (ii) Medium frequency oscillations, with periods in the range 1O(F-5000 years (typically 1000 years), with peaks in the spectrum close to the period of the westward drift of the non-dipole field. (iii) Highfrequency oscillations, with periods < 100 years (Braginskii, 1984). The theoretical treatment of these oscillations is highly complicated. Braginskii suggested that the existence of a fundamental frequency is a consequence of the two-stage nature of the dynamo process, in which a weak poloidal* field leads to the generation of a strong toroidal field, the toroidal field being responsible for the regeneration of the poloidal field. Cox (1968) had earlier suggested that the intensity of the geomagnetic dipole undergoes steady oscillations with a period of 8000 years. This suggestion was based on palaeomagnetic determinations of the ancient field intensity for the past 9000 years. However, Kono (1972) and McFadden and McElhinny (1982), in an analysis of VDMst for the past 5 Ma, found no support for Cox’s model. McElhinny and Senanayake (1982), in an analysis of archaeomagnetic intensity data for the past 50,000 years also found no evidence for any sinusoidal variations. McFadden (1984a) has looked into the question again, and suggested that deviations of the dipole moment from a short-term mean dipole moment may be modelled as a linear sum of previous, random effects. Using archaeomagnetic data he showed that if the

-

* For the definition and a discussion of poloidal and toroidal vector fields see Appendix A.

t The virtual geomagnetic pole (VGP) is defined as the pole of the dipolar field which gives the observed direction of magnetization at the site under consideration. It is calculated from any spot-reading of the field direction, the word ‘virtual’ meaning that no implication about the position of an average dipole is being made. To compare data from different sampling sites at different latitudes it is convenient to calculate the equivalent dipole moment that would have produced the measured intensity at the calculated palaeolatitude (assuming a dipolar field) of the sample. Such a calculated dipole moment is called a virtual dipole moment (VDM) and has the advantage that no scatter is introduced by any wobble of the main dipole, since the determined magnetic palaeolatitude is independent of the orientation of the dipole relative to the Earth’s rotational axis. However, it is not possible to determine that portion of the observed intensity (and therefore of the VDM) that arises from non-dipole components. Hence a true dipole moment cannot be obtained directly from palaeointensity studies.

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The Earths Core

dipole moment is observed at intervals up to 1000 years, such deviations are serially connected. However, with observations 2000 years or more apart, he found no significant serial correlation nor any indication of an oscillation. He concluded that if there is a deterministic time of -2000 years, observations taken at shorter time-intervals will give a visual impression of quasioscillations dominated by quasi-periods of about 4000 and 8000 years. Such a model explains the apparent oscillatory behaviour seen by Cox (1968). Medium frequency oscillations are associated with the so-called “MACwaves” (i.e. Magnetic-Archimedean-Coriolis waves) in the Earth’s fluid core. High-frequency oscillations are linked to torsional magnetohydrodynamic oscillations and turbulent pulsations in the core. Currie (1973a) has used the maximum entropy method to obtain improved knowledge of the geomagnetic spectrum in the period range 2-70 years. His work provides the first successful spectrum detection of the solar cycle (SC) and double solar cycle (DSC) variations. The structure consists of band spectra centred at 10.5 and 21.4 years. In addition, he was also able to identify the first four harmonics of the SC and, except for one, the first nine harmonics of the DSC. Currie had earlier (1966) shown that the geomagnetic continuum from 40 days to at least 3.7 years is coherent and in phase over the entire Earth and deduced that it was generated by physical processes external to the Earth. The more detailed analysis of his later paper indicates that fluctuations over two years consist of a number of band spectra harmonics extending all the way to the SC which are a result of SC and DSC sunspot variations on the sun’s surface. Currie concluded, however, that one cannot dismiss the possibility that some of the spectral lines are due to internal signals generated in the Earth’s core. In particular tlie band at 6.07 years in Fig. 4.5 cannot be ascribed to a harmonic of either fundamental band and thus may be evidence for internal signals of this period. In this respect it is interesting to note the tendency for secular variation impulses to recur approximately every 5 years. Yukutake (1972) has also shown that one of the free modes of the electrically coupled core mantle Earth system has an oscillation period of 6.7 years. Alldredge (1977) has carried out a harmonic analysis of the H and Z components from a number of observatories. For the pass band 13-30 years, several cases were found in which a group of observatories showed similar variations up to some tens of nanoTeslas (see Fig. 4.6). The largest variations occurred at periods of 25 years. Alldredge concluded that the variations are of internal origin and not external as maintained by Currie (1973a). If the variations were caused by an external source, all observatories should show the effect (not just 17 out of 49 in Currie’s analysis) and all should have a common period. Courtillot and Le Moue1 (1976) also do not believe that Currie’s histogram is meaningful, since his 22-year peak is not significantly

-

-

199

The Earth’s Magnetic Field

I

0

1

004 000 0.12 0.16 0 20 0.24 0.20 0.32 0.36 0.40 0 44 0.48 0.50 Cycles per year

Fig 4.5. HandZsmoothed histograms showing the distribution of a number of interpolated linesas a function of frequency that occurred in 97 spectra The “linds” were chosen by the computer without regard to signal-to-noise ratio (After Currie. 1973a )

above the histogram noise level. Alldredge showed that the variations cannot be explained by detailed fine structure in the westward drift of a core dynamo, and suggested that hydromagnetic waves in the core, as proposed by Hide (1966) to account for the secular variation, may be the cause of these shorterperiod fluctuations. Currie also obtained evidence for a 60 year line in the geomagnetic spectrum. The estimated amplitudes, although rather crude and variable, are too large to be associated with the sun and thus this “line” must represent a signal from the core. In a later paper (1973b), Currie showed that the line is global in character; however the amplitudes of the signal are anomalously low in and around the Pacific basin region. This is also true of the present secular variation (Cox and Doell, 1964; Doell and Cox 1971, 1972). A physical explanation for these lows is not clear. The existence of the Pacific (magnetically) quiet zone suggests that generation of irregularities in the core dynamo may be partly controlled by lateral inhomogeneities at the MCB. Such inhomogeneities might be topographic “bumps” at the base of the mantle (see $4.7) and could be related to the non-drifting components of the non-dipole field (Yukutake and Tachinaka, 1969; Yukutake, 1970). Golovkov and Kolomiitseva (1970), Jin and Thomas (1977) and Rotanova et al. (1985) have also found 60- and 30-year periodicities in the geomagnetic spectrum of internal origin (see $4.8). Evidence of a 50-60 year magnetic variation had been made earlier in con-

-

I

I-

APIA p.b. (13.827.6 years

p.b. (13.1-28.8 years)

-

p.b. (13.8 -24.8 years)

0

1920

1940 Year

1960

0

1920

1940

1960

. Year

Fig. 4.6. Vertical component variation synthesized from Fourier terms representing periods between 13 and 30 years: p.b. indicates the actual period pass band used. (After Alldredge, 1977.)

201

The Earth's Magnetic Field

(i 3

"

0

57.5 y e a r s

1 n

0.05

0.1

Frequency ( c y c l e s per y e a r ) Fig. 4.7. Amplitude spectrum in seconds of time for yearly fluctuations in the length of the day from 1824 to 1950. Two hundred estimates were computed with a bandwidth of 0.0025 cycles per year, but only estimates up to 0.25 the Nyquist limit are plotted. Also shown is the transfer function of the highpass filter applied to the source data. (After Currie, 1973b.)

nection with variations in the length of the day (I.o.d.)--e.g. Vestine and Kahle (1968) (see g4.8). The curves of Orlov (1965), based on data from four observatories, clearly show a 5&60 year cycle. However, until the more detailed and conclusive analysis of Currie (1973b), such a variation does not appear to have been widely known or accepted. In a further paper (1973c), Currie applied the maximum entropy method to 19th and 20th century data on variations in the 1.o.d. The spectrum is dominated by a strong line at 57.5 years (see Fig. 4.7), which is close to the estimate of 57.8 years in the magnetic spectrum (Fig. 4.5). Currie suggested that 60 year lines in the geomagnetic and astronomical data are causally related and that these results are the spectral analogues of the correlation previously established between 1.o.d. variations and the westward drift of the eccentric dipole (Vestine and Kahle, 1968; Kahle et al., 1969).

-

4.2

The Origin of the Earth's Magnetic Field

The problem of the origin of the Earth's magnetic field (and secular variation)

202

The Earths Core

is one of the oldest in geophysics and one to which no completely satisfactory answer has as yet been found. A number of possible sources for the field have been suggested-such as permanent magnetization or theories involving the rotation of the Earth-but most of them have proved to be inadequate.* The only possible means seems to be some form of electromagnetic induction: electric currents flowing in the Earth‘s fluid, electrically conducting core. This still poses the problem of how such currents were initiated-perhaps they arose from chemical irregularities that separated charges and set up a battery action, generating weak currents. Palaeomagnetic measurements have shown that the Earth’s main field has existed throughout geological time and that its strength has never differed significantly from its present value. In a bounded, stationary, electrically conducting body, any system of electric currents will decay. The field or the current may be analysed into normal modes, each of which decays exponentially with its own time constant. The time constant is proportional to 01’ where o is the electrical conductivity and I a characteristic length representing the distance in which the field changes by an appreciable amount. For a sphere the size of the Earth the most slowly decaying mode is reduced to lie of its initial strength in a time of the order of 100,000 years. Since the age of the Earth is more than 4000 Ma, the geomagnetic field cannot be a relic of the past, and a mechanism must be found for generating and maintaining electric currents to sustain the field. The most likely source of the electromotive force needed to maintain these currents is the motion of core material across the geomagnetic lines of force. The study of this process, in which the currents generated reinforce the magnetic field that gives rise to the driving e.m.f. is known as the homogeneous dynamo problem. The dynamo theory of the Earth’s magnetic field was due originally to Sir Joseph Larmor, who in 1919 suggested that the magnetic field of the sun might be maintained by a mechanism analogous to that of a self-exciting dynamo. The pioneering work in dynamo theory was later carried out by Elsasser (1946a,b, 1947) and Bullard (1949a,b). The Earth’s core is a good conductor of electricity and a fluid in which motions can take place, i.e. it permits both mechanical motion and the flow of electric current, the interaction of which could generate a self-sustaining magnetic field. It has not been possible to demonstrate the existence in the laboratory of such a dynamo action. If a bowl of mercury is heated from below, thermal convection will be set up-but no electric currents or magnetism will be detected in the bowl. Such a model experiment fails because electrical and mechanical processes do not scale down in the same way. An electric current in a bowl of mercury 30 cm in diameter would have a decay time of about one hundredth of a second. The decay time, however, increases as the square of the diameter-an electric

* A review and discussion of theories that have been shown to fail has been given by Stevenson (1974).

The Earth‘s Magnetic Field

203

current in the Earth’s core would persist for about 10,000 years before it decayed. This time is more than sufficient for the current and its associated magnetic field to be altered and amplified by motions in the fluid, however slow. The development of dynamo theory for the Earth has had to be based on theoretical models, because the materials available for any laboratory experiments are not sufficiently good conductors for the models to be of a reasonable size. Even if energy sources exist within the Earth’s core sufficient to maintain the field, there remains the critical problem of sign, i.e. it must also be shown that the inductive reaction to an initial field is regenerative. An engineering dynamo is exceedingly non-homogeneous, containing rotors, stators, wires, bearings, etc., whereas the fluid core of the Earth is simply-connected, and virtually homogeneous and isotropic. It might be suspected therefore that any dynamo trying to operate within such a fluid would be, in some sense, short-circuited; it is thus not at all clear that such a homogeneous dynamo can work under any conditions at all. In fact, in 1934 Cowling showed that a magnetic field that possesses an axis of symmetry could not be produced by dynamo action. For many years, this important result was generally misinterpreted as implying also that an axisymmetric motion could not generate a magnetic field, but this has since been shown to be untrue under certain circumstances. The limitations imposed by Cowling’s theorem (and its incorrect extension) led at one time to fears that dynamo action might in fact not be possible under any conditions: perhaps some “super-Cowling theorem” existed that prohibited it. Fortunately this has not proved to be the case. Cowling’s result has been extended by many authors, e.g. Backus and Chandrasekhar (1956), and it now appears that homogeneous dynamos must possess a low degree of symmetry. A number of other “non-dynamo” theorems that prohibit particular types of motion in a sphere from acting as dynamos have also been obtained. For a discussion of Cowling’s theorem, see James et al. (1980), and Hide and Palmer (1982). The interesting generalization that growing axisymmetric poloidal fields do not exist even if timedependent and compressible fluid flows with arbitrary spatial distributions of the magnetic diffusivity, v, are admitted has recently been proven by Lortz and Meyer-Spasche (1982a) and by Hide and Palmer (1982) using different mathematical methods. More recently, Lortz and Meyer-Spasche (1982b) have extended their analysis and obtained a proof that the toroidal component of an axisymmetric or two-dimensional field decays as well. Cowling’s theorem thus appears to hold under the most general conditions. Elsasser (1964a) noted that any magnetic field is bound to decay if the velocity field is toroidal and has no radial component. More specifically Bullard and Gellman (1954) showed that velocity fields of the form u = V x r$ cannot generate a magnetic field if v, is a function of IrI only, where Y is the position

204

The Earths Core

vector. The important geophysical implication of this theorem is that a radial component of the velocity must be maintained over periods of the order of the magnetic decay time. A lower bound on the radial velocity in terms of the ratio between poloidal and toroidal field strengths has been derived by Busse (1975a). It is clear that the dynamics of the core is governed by Lorentz and Coriolis forces. It is usually supposed that the fluid motion contains some differential rotation that winds up any dipole field present into a toroidal field. Such a mechanism is very efficient and well understood, and presents a simple way in which the fluid may amplify the field-the field may be increased indefinitely simply by increasing the fluid motion. This does not solve the dynamo problem-to do that the dipole field must in some way be produced from the toroidal field, thus completing a cycle by which energy may be put into the field. This second stage of the cycle is much more difficult to account for. It is often assumed that a principal action of the Lorentz force is to limit the magnitude of differential rotation, rather than suppressing other motions, as supposed in some dynamical theories. It is now known that small-scale velocities can lead to dynamo action (as periodic motions or turbulence) and the smallscale magnetic fields produced by them may contribute to a large Lorentz force. Busse (1975b) has developed a model for the geodynamo on the assumption that the Lorentz force is small compared with the Coriolis force. Only that part of the Coriolis force that cannot be balanced by the pressure gradient is available for a balance with the Lorentz force. Busse solves the complete hydromagnetic problem in the case of a cylindrical geometry-because of the dominant effect of rotation Busse maintains this configuration will incorporate the essential features of the geodynamo. In his model, the toroidal field in the core is of the same order of magnitude as the poloidal field, unlike most other models in which the toroidal field is substantially greater than the poloidal field (see $4.4). Hide and Roberts (1979) have summarized the methods that have been used to estimate the magnitude of the toroidal field B, in the core. They concluded that BT is probably a good deal stronger than the poloidal field Bp.BT is probably about lo-’ T, might be as low as T, but is unlikely to exceed 5 x 10-2T. Too high a value of B T would imply a rate of ohmic heating incompatible with geophysically reasonable models of heat flow in the deep interior of the Earth. Several successful dynamos have been developed that involve axially symmetric motions, the earliest of which are those of Tverskoy (1966) and Gailitis (1970). Their models are interesting in that the velocity field is axisymmetric, although the resulting magnetic fields are not: the symmetry restrictions of Cowling’s theorem are thus avoided.

The Earths Magnetic Field

205

In 1958 two dynamo models were found. However, though rigorous mathematical solutions were obtained, in each case the motions were physically very improbable. The first model was due to Backus (1958), who made use of the fact that the natural decay time of field components in a stationary conductor depends upon the length-scale of the component. He proposed a model in which short periods of vigorous motion within a fluid sphere are followed by longer periods with no motions. During these longer periods, all field harmonics except that of the lowest degree, and therefore greatest characteristic length, diffuse away. Another period of non-zero motion strengthens this harmonic and twists it into a different direction, and after several repetitions of this process the original field is regenerated. The second model, due to Herzenberg (1958), consists of two spheres in the core, each of which rotates as a rigid body at a constant angular velocity about a fixed axis. The axially-symmetric component of the magnetic field of one of the spheres is twisted by rotation, resulting in a toroidal field that is strong enough to give rise to a magnetic field in the other sphere. The axial component of this field is twisted as well and fed to the first sphere. If the rotation of the spheres is sufficiently rapid, a steady state may be reached. Lowes and Wilkinson (1963) built a working model of what is effectively a homogeneous self-maintaining dynamo based on Herzenberg’s theory. For mechanical convenience they used, instead of spheres, two cylinders placed side by side with their axes at right angles to one another so that the induced field of each is directed along the axis of the other. If the directions of rotation are chosen correctly, any applied field along one axis will lead, after two stages of induction, to a parallel induced field. If the velocities are large enough the induced field will be larger than the applied field, which is no longer needed, i.e. the system would be self-sustaining. Rikitake and Hagiwara (1966) investigated the stability of a Herzenberg dynamo and concluded that it is unstable for small disturbances about its steady state. Their analysis is not applicable to the case of the Earth, however, since numerical integrations could only be performed for parameters very different from those in the Earth’s core. Kerridge and Wilkinson (1982) have since constructed a laboratory homogeneous dynamo consisting of several identical metal blocks. Four of the blocks contain cylindrical cavities, inside which there are cylindrical rotors of the same material with a radial and end clearance of 0.1 mm. The annular gap is filled with mercury to maintain good electrical contact. By stacking the blocks together in different configurations and ensuring electrical contact between them through a mercury layer, they obtained 2-, 3- and 4-rotor selfexciting dynamos that displayed time-varying magnetic fields with different characteristics-including reversals. Wilkinson (1984) has reviewed the results of laboratory dynamo experiments.

206

The Earths Core

/

Fig. 4.8. The production of toroidal field components from poloidal field components by differential rotation: the poloidal field line ADG is stretched by differential rotation to produce the toroidal components BC and EF. (After Elsasser, 1950.)

A different type of model was suggested by Parker (1955). His velocity field has two components-rotation (producing a toroidal field from a poloidal field) and convection (producing a poloidal field from a toroidal field). He supposed that a rapidly rotating fluid sphere is heated from within, leading to large-scale, asymmetric convection cells. As fluid in these cells rises and falls, the conservation of angular momentum results in non-uniform rotation of the fluid, with the areas further away from the axis rotating more slowly than those nearer. This rotational shear will draw out magnetic lines of force that are in the meridonal plane (ADG in Fig. 4.8), to produce an azimuthal component of the field (BC and EF). Eventually the magnetic stresses that develop will become large enough to reduce the differential rotation, and the stretching of the field lines will cease. The powerful toroidal field that results from such a mechanism can never be observed at the surface of the sphere, as no portion of it can escape from the fluid (i.e. it is a “contained” field). Parker next suggested that the action of the Coriolis force on the convection cells produces a cyclonic or twisting motion within these cells, rather like the circular motions in atmospheric weather systems. The convective lifting and rotational twisting in the cells between them turn the toroidal lines of force into a loop of flux (see Fig. 4.9, right-hand portion). This loop includes the new poloidal field components AB and CD in its projection on to the meridional plane (Fig. 4.9, left-hand portion). With diffusion outward of the component C D and inward of the component, AB, and continuous coales-

The Earth’s Magnetic Field

207

Fig. 4.9. The production of poloidal field components from toroidal field components by cyclonic twisting: the convective lifting and rotational twisting motion turns the toroidal field line into a loop (right-hand half of the figure), which includes the new poloidal field components AB and CD in its projection on the meridional plane (left-hand half of the figure). (After Frazer, 1973b.)

cence of such flux lines produced in many of these cyclonic convection cells, Parker suggested that it is possible that the original poloidal field could be regenerated and maintained. Parker’s model has been further developed by Levy (1972a,b).

4.3 The Homogeneous Dynamo Equations

A number of detailed accounts of dynamo theory have been given (see e.g. Roberts, 1967a; Gubbins, 1974; Busse, 1978; Moffatt, 1978; P. H. Roberts and Soward, 1978; Krause and Radler, 1980) and no attempt will be made to review the subject in this book. The homogeneous dynamo problem involves the solution of a highly complicated system of coupled partial differential equations. The equations fall into four major groups.

(i) The electrodynamic equations. These include Maxwell’s equations, the constitutive relations among the various electric and magnetic fields, Ohm’s law, and the transformation relating the fields observed in one reference frame to those observed in another in relative motion. (ii) The hydrodynamic equations. These include the equations of the conser-

208

The Earth‘s Core

vation of mass and the conservation of momentum, and the constitutive equation for the total stress tensor. (iii) The thermodynamic equations. These include the postulate of local thermodynamic equilibrium, the equation of heat conduction, and the constitutive law for the heat conduction vector. When combined with (i) and (ii) above, these equations lead to a detailed description of the energy flow within the system considered. (iv) The boundary and initial conditions. Elsasser (1954) has shown by a dimensional analysis that in geophysical and astrophysical problems the displacement current and all purely electrostatic effects are negligible, as are all relativistic effects of order higher than Ujc where U is the fluid velocity. Thus the electromagnetic field equations are the usual Maxwell equations

v x E = - asjat VxB=y,j V.B=O

(4.3)

where B and E are the magnetic and electric fields respectively and j the electric current density. The electromotive forces that give rise to j are due both to electric charges and to motional inductions, so that the total current j is given by j = c r ( E + U X B) (4.4) Assuming the electrical conductivity to be constant, taking the curl of (4.2) and using (4.4) and (4.1), E can be eliminated, leading to the equation V x (V x B) = you

(4.5)

Since V x (V x B) = V ( V -B) - V 2 B= - V2B, on using (4.3), we finally obtain

7

aB/at

=

v x ( ux B) + V , V Z B

(4.6)

where vm = ~ / P O O

(4.7)

is the magnetic diffusivity. Equations (4.3) and (4.6) give the relationships between B and U that have to be satisfied from electromagnetic considerations. The term V x (U x B) in (4.6) is the source term, and represents the physical process by which magnetic induction is “created” through the flow of fluid across lines of force. The term v,V*B represents the tendency for the field to decay through ohmic dissipation by the electric currents supporting

209

The Earth’s Magnetic Field

the field. The balance between these two terms, at a particular point, determines how the magnetic field changes with time at that point. If the material is at rest, (4.6) reduces to

aslat = V , V ~ B

(4.8)

This has the form of a diffusion equation, and indicates that the field leaks through the material from point to point. Dimensional arguments indicate a decay time of the order Lz/v, where L is a length representative of the dimensions of the region in which current flows. For conductors in the laboratory this decay time is very small--even for a copper sphere of radius 1 m it is less than 10 s. For cosmic conductors, on the other hand, because of their enormous size, it can be very large. As an alternative limiting case, suppose that the material is in motion but has negligible electrical resistance. Equation (4.6) then becomes

aslat = v

x

(u x

B)

(4.9)

This equation is identical to that satisfied by the vorticity in the hydrodynamic theory of the flow of a non-viscous fluid, where it is shown that vortex lines move with the fluid. Thus (4.9) implies that the field changes are the same as if the magnetic lines of force were “frozen” into the material. When neither term on the right-hand side of (4.6) is negligible, both the above effects are observed, i.e. the lines of force tend to be carried about with the moving fluid and at the same time leak through it. If L, T, V represent the order of magnitude of a length, time and velocity respectively, transport dominates leak if LV % v,. The condition for the onset of turbulence in a fluid is that the non-dimensional Reynolds number Re = LV/v be numerically large. By analogy, a magnetic Reynolds number R , may be defined as R , = LV/V,

(4.10)

Thus the condition for transport to dominate leak is that R , >> 1. This condition is only rarely satisfied in the laboratory-in cosmic masses, however, it is easily satisfied because of the enormous size of L. Thus, under laboratory conditions, lines of force slip readily through the material-in cosmic masses, on the other hand, the leak is very slow and the lines of force can be regarded as very nearly frozen into the material. The generation of magnetic fields in astrophysical bodies relies on fluid motion having a large R,-in the case of a dynamo operating in the Earth’s core, it has been estimated that R , 2 10. This is necessary so that the magnetic field can be distorted enough by the fluid motion to reinforce the largescale field. Additionally, however, it is necessary that magnetic field lines diffuse sufficiently through the fluid and dissipate rapidly enough to keep the

210

The Earth's Core

field topology simple. Otherwise the main effect of the fluid motion would simply be to tangle the field lines without producing efficient regeneration. It has been shown (Backus 1958; Childress 1968) that a necessary condition for a dynamo in a sphere of radius ro with constant diffusivity v, is that (4.11)

where Vo is the maximum velocity inside the fluid sphere. Proctor (1977) has improved this bound but his results d o not change the fact that the estimate for the critical magnetic Reynolds number R , is far below those required for realistic dynamos. A physical interpretation of the condition (4.11) is that the time required by a fluid parcel to traverse the distance ro/n must be shorter than the magnetic decay time r i / h c 2 .This time is about 25 x lo3 years for the Earth's core. It has been shown (e.g. Roberts, 1978; Hide, 1982) that a magnetic field B cannot be maintained or amplified by fluid motions U against the effects of Ohmic decay unless R , is sufficiently large and the patterns of B and U are sufficiently complicated. Thus the main features of the geomagnetic field can be reproduced by a variety of very different velocity fields. Dynamo action is stimulated by Coriolis forces, which promote departures from axial symmetry in the pattern of U when B is weak, and is opposed by Lorentz forces, which increase in influence as B grows in strength. Axisymmetric magnetic fields will always decay, which suggests (Hide, 1981) that palaeomagnetic and archaeomagnetic data might show evidence that departures from axial symmetry in the geomagnetic field are systematically less during the decay phase of a polarity reversal or excursion than during the recovery phase. To the electromagnetic equations must be added the hydrodynamical equation of fluid motion in the Earth's core (the Navier-Stokes equation) together with the equation of continuity, which, for an incompressible fluid (the speed of flow is much less than the speed of sound in the Earth's core) reduces to

v.u=o

(4.12)

The Navier-Stokes equation is -Vp+pVW

(4.13)

where U is the velocity relative to a system rotating with angular velocity a,p the pressure, W the gravitational potential (in which is absorbed the centrifugal force) and p and v the density and kinematic viscosity, respectively.

The Earth’s Magnetic Field

211

Equations (4.6) and (4.13) contain only the vectors U and Band are the basic equations of field motion. Because of the complexity of the equations describing the hydromagnetic conditions in the Earth’s core, most effort has been directed to seeking solutions of Maxwell’s equations for a given velocity distribution. This approach, known as the kinematic dynamo problem is linear and has been the subject of much investigation. Expansion in spherical harmonics reduces (4.6) to an infinite set of differential equations containing the radial functions for each harmonic, their first and second radial derivatives and their first time derivatives. For a given set of initial conditions these could in theory be integrated in time steps. Little progress has been made in this direction, however, and most workers have looked for steady-state solutions, putting dB/dt = 0. In 1954, Bullard and Gellman looked for steady-state solutions using first-order finite differences for the radial variation. When the series are truncated a generalized algebraic eigenvalue problem in R , remains. The difficulty has been in obtaining a convergent solution when the truncation level is raised. Bullard and Gellman (1954) chose as their velocity function differential rotation TI and a poloidal convection P:‘ and obtained steady-state solutions that appeared satisfactory. However, with the advent of large computers Gibson and Roberts (1969) were able to show that in fact their solution did not converge. A clue to the reason for the failure of the BullardGellman dynamo can be found in the work of Braginskii (1965a,b). Braginskii considered the generation of magnetic fields by a fixed fluid velocity that is mainly toroidal and axisymmetric, but which also has a small asymmetric component. Under certain limiting conditions (primarily a high value of the magnetic Reynolds number) he showed that a necessary condition for dynamo action was the simultaneous existence of both sin rn4 and cos m 4 terms, with the same value of m in each, in the Fourier expansion of the asymmetric velocity component (4 being the aximuthal angle in spherical polar coordinates r, 0,4).Unless both of these terms occur, dynamo action is not possible. The velocity distribution used by Bullard and Gellman contained only a cos 24 term, with no sin 24 term, and this may be the reason for its failure. Lilley (1970) added a term in sin 24 to their velocity distribution, i.e. he assumed a velocity field of the form

and demonstrated convergence which appeared to be much better than Bullard and Gellman’s. Later work (P. H. Roberts, 1972; Gubbins 1973), however, has shown that the convergence is still not satisfactory. Gubbins (1973) has since attempted to find solutions using modifications of Lilley’s velocity field, but without success. A possible reason for the failure of Lilley’s

212

The Earth’s Core

model may be that steady-state solutions do not exist. His velocity field contains a non-zero angular rotation and the field may rotate-such a rotating field would not yield a solution to the steady-state eigenvalue problem. The first satisfactory numerical demonstration of dynamo action in a sphere was given by G. 0. Roberts (see P. H. Roberts, 1971b). A solution was possible because he chose an axially symmetric velocity, for which solutions of the induction equation proportional to e’”4 decouple. G. 0. Roberts considered the case rn = 1, the calculation being essentially two-dimensional. In his analysis he used a finite difference method rather than spherical harmonics. Flows similar to those of Roberts have also been investigated by Gubbins (1972, 1973) and Frazer (1973a) using the Bullard-Gellman expansion method. Gubbins (1973) in particular obtained dynamo action in which the numerical convergence is convincing. P. H. Roberts and Kumar reported in Moffatt (1973) apparent convergence for the velocity field

G. 0. Roberts (1970, 1972) investigated dynamos in infinite bodies of fluids. In particular he showed that “almost all” spatially periodic motions in an unbounded conductor will lead to dynamo action. These dynamos are remarkable in that, although the motion extends to infinity, the field does not. Also the field, unlike the motion, is not spatially periodic. The real significance of these interesting and unexpected results is not clear. Childress (1967, 1968,1970) has been able to show that such a periodic motion giving dynamo action in an unbounded conductor retains this property when fitted into a finite spherical volume by means of a “cut off” function. In discussions of the dynamo mechanism, it is usually assumed that the fluid O C has uniform electrical resistivity. Cowan (1980) has investigated the consequences of removing this restriction. The variations in resistivity appear as a large-scale average effect resulting from small-scale differences in the resistivity of the materials in ascending and descending convection currents in the liquid O C that are deflected in opposite tangential directions by the Coriolis acceleration. Cowan carried out a numerical analysis and showed that a dipole field can be generated, sustained and even be reversed. If the anisotropic layer is near the top of the core, the differential rotation for regeneration must be from east to west (the actual direction of the Earth’s non-dipole field). If the anisotropic layer is near the bottom of the OC, the differential rotation must be from west to east. A critical question is whether there is sufficient anisotropy in the core to support the mechanism. Differences in resistivity between ascending and descending currents could result from either a difference in temperature or a difference in chemical composition. Cowan showed that the changes in resistivity due to temperature differences are

The Earths Magnetic Field

21 3

many orders of magnitude too small for the required anisotropy. This leaves the possibility that the required degree of anisotropy could arise from differences in chemical composition that may occur as the convecting fluid reaches the liquid-solid interface. Although the dynamo mechanism could operate with the anisotropy in a localized region, that total energy dissipation required for a given dipole moment increases as the region of anisotropy is made smaller. Quantitative calculations are difficult because of lack of information about the constitution and turbulence in the core. Although Cowan’s suggestion is possible, it seems on balance rather unlikely. Hibberd (1979) proposed a different model for a homogeneous axisymmetric generator of the geomagnetic field. He suggested that the driving force for the electric currents is provided by the Nernst effect, in which a flow of heat across a magnetic field in an electrical conductor gives rise to an electromotive force that acts mutually perpendicularly to the directions of the temperature gradient and the magnetic field. In the case of the Earth, the Nernst effect is associated with a radially outward flow of heat from heat sources within the core across an initial meridional magnetic field. The thermomagnetic e.m.f. drives a system of two azimuthal current shells in the core region, one nested inside the other, with the currents flowing in opposite directions. The current shells slowly expand radially. As the outer shell decays a new current shell develops inside the inner shell. The resultant magnetic field near and beyond the Earth’s surface approximates to a dipole field that undergoes repeated reversals. Hibberd estimates that the magnitude of the Nernst effect is large enough to drive the generator, although this seems very doubtful. Certainly the electrical conductivity of the core would have to be several times larger than the generally accepted value.

4.4

Mean-field Electrodynamics

Two approaches have in recent years been made to obtain a better understanding of dynamo processes. In both cases the magnetic and velocity fields are split into mean and residual fields. The mean field is defined by an averaging procedure and has a relatively simple geometrical structure, whereas the residual field shows a more complicated structure. The first approach is that of Braginskii’s (1965a,b) “nearly symmetric” dynamo. In his model, the mean fields are the axisymmetric parts of the original fields. The theory can explain the axisymmetric dipole character of the Earth’s mean magnetic field, which is symmetric with respect to the rotational axis. A differential rotation and a meridional circulation, both symmetric, and special non-axisymmetric motions have to be assumed. Braginskii (1976) later developed a dynamo model for the Earth that he

214

The Earths Core

called Model Z. It is a nearly axially symmetrical model, with laminar flow in the core and the toroidal field being much greater than the poloidal field. His model is characterized by a large geostrophic velocity in the core, a boundary layer, large gradients in the fields B4 and Be and a large velocity U, near the MCB. A characteristic of the model is the strong dependence on the value of the (weak) friction between the core and the mantle. This friction depends essentially on the value of B, at the MCB. Changes of stochastic fields B, at the boundary can arise from external causes (e.g. local fluctuations in the supply of denser mantle material to the MCB) and from internal causes connected with the development of local instabilities at the boundary. Changes in magnetic friction cause large perturbations in the couple acting on the mantle leading to observable sharp changes in the angular velocity of the mantle and hence length of the day. Braginskii showed that changes in the field B, at the MCB can cause a large change in magnetic friction. Moreover, changes in the non-symmetrical fields at the MCB influence the behaviour of the field in the current layer, radically changing the whole pattern of the fields in the dynamo, even on occasion producing a reversal. Most dynamo models use large-scale, highly ordered fluid motions, i.e. motions in which the characteristic length of the velocity field is not much less than the radius of the Earth. In the early 1950s several attempts were made to produce models in which turbulent (i.e. random and small-scale) velocities might act as dynamos. The theory, which has been called mean-field electrodynamics, has been developed independently by Moffatt (1 970) in Britain and by Krause, Radler and Steenbeck in Germany. An account of the German work has been given in a book by Krause and Radler (1980). In mean-field dynamo models the velocity U and magnetic field B are each represented as the sum of a statistical average and a fluctuating part. We thus write

u=uof u

(u>=O

(4.14)

B= Bo+b

(b)=O

(4.15)

The average fields U, and Bo are assumed to vary on a length-scale L, while the fluctuating fields u and b (with zero statistical average) are assumed to vary on a length-scale 1 ( I 6 L). This separates the velocity and magnetic fields into mean, slowly varying and fluctuating parts. The induction equation may then be divided into its mean and fluctuating parts

(u, x B,) + v x E + v , ~ z ~ o ablat = v x (u, x b) + v x (u x B,) x v x G + v,,,vZb aBolat = v x

(4.16) (4.17)

where E = ( u x ~ )

and

G=uxb-(uxb)

(4.18)

21 5

The Earttis Magnetic Field

can be regarded as an extra mean electric force arising from the interaction of the turbulent motion and magnetic field. If the velocity field is isotropic, it can be shown that E = uBO - PV x Bo (4.19) E

where CI and p depend on the local structure of the velocity field. The induction equation 4.15 satisfied by the mean field is

The term aBo represents an electric field parallel to Bo. The quantity p is an eddy diffusivity similar in its effects to the ohmic diffusivity v,. It operates by mixing magnetic fields transported from neighbouring regions-its effect is to 8. Parker (1955) first drew attenreplace v, by a total diffusivity vT = v, tion to the possibility that E = aBo, and Steenbeck and Krause (1966) christened this the CI effect. A key concept in the theory is the helicity defined as u-(V x u). Parker (1955,1970) showed that convective fluid motions having non-zero helicity could distort lines of magnetic force in such a way as to produce a regeneration mechanism. A non-vanishing helicity indicates that the vorticity V x u tends to turn predominantly either in a clockwise or anticlockwise sense about the direction of the velocity (for further details see Moffatt 1978; Krause and Radler 1980). The helicity of fluid motions is a consequence of the action of the Coriolis force on convection in rotating bodies. This is the reason for the importance of rotation in the dynamo mechanism. It must be remembered, however, that the Coriolis force can only change the direction of flow: it cannot change the speed since it acts at right angles to the direction of flow, and so cannot drive it against the retarding effects of other forces. We can picture the meaning of helicity by representing a turbulent velocity flow by a box containing a large number of randomly directed screws. If the spiral edge of each screw represents the short-term path of a moving element of fluid, then the fluid over-all will have zero helicity if there are equal numbers of right-handed (i.e. normal) thread screws and left-handed (i.e. reversed) thread screws. In such a case, viewing the fluid-or the box of screws-through a mirror produces no visible change in the average properties: turbulence with zero helicity has reflectional symmetry. On the other hand, if more screws in the box are right-handed than are left-handed (or vice versa), the helicity is non-zero, and viewing the system in a mirror clearly produces a different average picture, interchanging a surplus of right-handed screws for a surplus of left-handed ones (or vice versa). Thus helicity measures the lack of reflectional symmetry in the turbulence, and it is turbulence that lacks such symmetry that can give rise to dynamo action.

+

21 6

The Earth's Core

Much effort has gone into solving (4.20) for various choices of -a and U,. Analytical solutions are very rare, and most of the work involves numerical calculation. Helicity in the Earth's core is due to rotation and -a is expected to have opposite signs in the northern and southern hemispheres. -a = a. cos 8, where ct0 is a constant and 8 the colatitude, is a common choice. U, includes large-scale motions, the most important of which is differential rotation of the core liquid, which can produce a large toroidal field from the dipole. There are two possible types of dynamo using the -a effect--a2 dynamos and aw dynamos. In an a2 dynamo the a effect generates poloidal field from toroidal field and generates toroidal field from poloidal field. The toroidal field has lines of force that lie on spherical surfaces and has no component external to the core. The poloidal field has a radial component in general, and joins continuously with the external, observed field. The -a effect can also be used in conjunction with a large-scale shear flow (the o effect) to produce an ao dynamo. Parker (1955, 1971, 1979) is primarily responsible for the development of dynamo models of this type, in which an -a effect from cyclonic turbulence generates poloidal field from toroidal field, and differential rotation creates toroidal field from poloidal field thereby completing the cycle. A differential rotation by itself cannot produce a poloidal field from a toroidal field, only a toroidal from a poloidal one. However, if there is an -a effect as well as a strong differential rotation, the latter may dominate in producing a toroidal from a poloidal field. For a dynamo of the -a2 type, the poloidal and toroidal fields are of the same order of magnitude, whereas in the case of an a o type dynamo, the toroidal field is much stronger than the poloidal one. Watanabe (1977) has estimated bounds on the fluid velocity and magnetic field in the core if the field is produced by an -am dynamo. He concluded that a toroidal magnetic field greater than 30r is unlikely, and that the poloidal field in the core is then less than lor. He also estimated that the scale of turbulence in the core must be larger than 10 km and the intensity smaller than 5 x l o p 3cm s-' if the energy dissipated due to Joule heat does not exceed the energy available to drive the dynamo. His results, however, depend on the assumption that the turbulence in the core is steady, homogeneous and isotropic and on assumed values for physical parameters in the core. Singer and Olson (1984) have shown that short (high-wavenumber) inertial gravity waves propagating through rotating, stably stratified, electrically conducting fluid can support dynamo action. The effect of the Coriolis force is to give the waves an elliptical polarization and a net helicity. The helicity can give rise to an -a effect and thus a field of waves can generate a dynamo of the a2 type. Stable stratification does not prevent dynamo action-indeed Singer and Olson showed that the minimum energy density is small so that only weak energy sources are necessary to drive the dynamo. However, increased stratification requires increased wave energy (see also 53.8).

The Earths Magnetic Field

4.5

217

Reversals of the Earth‘s Magnetic Field

One of the most interesting results of palaeomagnetic studies is that many igneous rocks show a permanent magnetization approximately opposite in direction to the present field. Reverse magnetization was first discovered in 1906 by Brunhes in a lava from the Massif Central mountain range in France-since then examples have been found in almost every part of the world. About one-half of all rocks measured are found to be normally magnetized and one-half reversely. Dagley et al. (1967) carried out an extensive palaeomagnetic survey of Eastern Iceland sampling some 900 separate lava flows lying on top of each other. The direction of magnetization of more than 2000 samples representative of individual lava flows was determined covering a time interval of 20 Ma. At least 61 polarity zones, or 60 complete changes of polarity were found giving an average rate of at least 3 inversions per Ma. There is no a priori reason why the Earth’s magnetic field should have a particular polarity, and there is no fundamental reason why its polarity should not change. It is easy to see that dynamos can produce a field in either direction. Equation (4.6) is linear and homogeneous in the field and (4.13) inhomogeneous and quadratic. Thus, if a given velocity field will support either a steady or a varying magnetic field, then it will also support the reversed field and the same forces will drive it. This, however, merely shows that the reversed field satisfies the equations-it does not prove that reversal will take place. It is possible that some physical or chemical processes exist whereby a material could acquire a magnetization opposite in direction to that of the ambient field. In fact Nee1 (1951, 1955) suggested, on theoretical grounds, four possible mechanisms-and within two years two of them had been verified, one by Gorter for a synthetic substance in the laboratory (although no naturally occurring rock has been found to behave in the required manner), and one by Nagata for a dacite pumice from Haruna in Japan. However, the great majority of igneous rocks that show reversal in the field, unlike the Haruna pumice, d o not exhibit this property in the laboratory. To prove that a reversed rock sample has become magnetized by a reversal of the Earth’s field, it is necessary to show that it cannot have been reversed by any physico-chemical process. This is almost impossible to do, since physical changes may have occurred since the initial magnetization or may occur during laboratory tests. More positive results can only come from the correlation of data from rocks of varying types at different sites and by statistical analyses of the relation between the polarity and other chemical and physical properties of the rock sample. There have been many cases where reversely magnetized lava flows cross sedimentary layers. Where the sediments have been baked by the heat of the

218

The Earths Core

cooling lava flow, they are also found to be strongly magnetized in the same reverse direction as the flow. It seems improbable that the adjacent rocks as well as the lavas themselves should possess a self-reversal property, and such results seem difficult to interpret in any other way than by a reversal of the Earth’s field. The same pattern of reversals observed in igneous rocks has also been found in deep sea sediments (see Fig. 4.10). No two substances could be more different or have more different histories than the lavas of California and the pelagic sediments of the Pacific. The lavas were poured out, hot and molten, by volcanoes and magnetized by cooling in the Earth’s field; the ocean sediments on the other hand accumulated grain-by-grain by slow sedimentation and by chemical deposition in the cold depths of the ocean. If these two materials show the same pattern of reversals then it must be the result of an external influence working on both and not due to a recurrent synchronous change in the two materials. Examples have been found of time-ordered sequences of lavas that apparently record the change of the ancient magnetic field from one polarity to the other, with directionally intermediate steps. Self-reversal cannot explain a gradual swing of direction of magnetization from one lava flow to another. A more reasonable explanation is that these examples record field reversal actually taking place. Finally, it is found that the range and mean value of the ancient field intensity that is deduced from normally magnetized rocks is the same as that deduced from reversely magnetized rocks. It is unlikely that a self-reversal mechanism would produce the same values as occur in normally magnetized rocks and this suggests, again, that the Earth’s field has reversed. Thus, although it has been established that self-reversal does occur in some rocks, the total evidence is overwhelmingly in favour of field reversal in the great majority of cases. Reversals occurred during the Precambrian and have been observed in all subsequent periods. There is no evidence that periods of either polarity are systematically of longer or shorter duration. However during the Kiaman-a period of about 60 Ma within the upper Carboniferous and Permian (about 235-290 Ma ago) the polarity of the Earth’s field appears to have been almost always reversed-until quite recently no normal intervals at all were known within this period. Again from about 85-107 Ma ago, to Earth’s field was almost always normal (the Cretaceous normal polarity interval). During the last few million years reversals have occurred at intervals of roughly 20&300 thousand years. Four major normal and reversed sequences have been found during the past 3.6 Ma. These major groupings have been called geomagnetic polarity epochs and have been named by Cox et al. (1964) after people who have made significant contributions to geomagnetism. Superimposed on these polarity epochs are brief fluctuations in magnetic polarity

Fig. 4.10. Correlation of magnetic stratigraphy in seven cores from the Antarctic. Minus signs indicate normally magnetized specimens; plus signs. reversely magnetized. Greek letters denote faunal zones. Inset source of cores (after Opdyke er a/.. 1966 )

220

The Earths Core

with a duration that is an order of magnitude shorter. These have been called polarity events and have been named after the localities where they were first recognized (see Fig. 4.11). The reality of such a distinction has since been questioned.* The time-scale of reversals was originally determined by radiometric dating and palaeomagnetic measurements on volcanic rocks. It was then extended using the palaeomagnetic record of reversals seen in the oceanic crust. (For a discussion of marine magnetic anomaly time-scales, see e.g. Ness et al. (1980); Lowrie and Alvarez, 1981; Butler and Coney, 1981; Cox, 1983). A polarity transition takes place so quickly (on a geological time-scale) that it is difficult to find rocks that have preserved a complete and accurate record. Observational evidence on changes in intensity during a reversal is conflicting. Although there seems to be general agreement that there is a marked reduction in intensity associated with the reversal, there is some disagreement as to whether the onset of changes in field direction coincides with the reduction in intensity. Several workers have now succeeded in obtaining a record of the geomagnetic field during a polarity transition. It appears that during a reversal the intensity of the field first decreases by a factor of 3 or 4 for several thousand years while maintaining its direction. The magnetic vector then usually executes several swings of about 30", before moving along an irregular path to the opposite polarity, the intensity still being reduced, rising to its normal value later. It is not certain whether the field is dipolar during a transition. There do not seem to be any precursors of a reversal or any indication later that a reversal has occurred. A detailed record of a field reversal in the Tatoosh intrusion in Mount Rainier National Park has been described by Dunn et al. (1971) and Dodson et al. (1978). They found that the field intensity decreased by a factor of 10 before any change in field direction occurred, and did not return to normal until after the directional change was completed. The directional change was estimated to have taken 1000-4000 years, while the intensity change took 10,000 years. Although cases of simultaneous changes in intensity and direction have been recorded, the majority of cases seem to follow the above pattern. A surprising result has been reported by Shaw (1975). During a well documented transition in a lava flow in western Iceland, he found the geometric field was large and stable when the VGP was close to the equator (see Fig. 4.12). Shaw later (1977) found another case of a strong intermediate palaeofield-in the Lousetown Creek, Nevada transition. The fact that the intermediate palaeofield can remain fixed in one direction for a considerable period of time (43 lavas recorded in western Iceland and 34 lavas in Lousetown * The terms epoch and event have now been superseded by a system of magnetostratigraphic polarity units (Cox, 1983).

. BRUNHES NOR- POLARITY MAL EPOCH EPOCH MATUYAMA REVERSED EPOCH GAUSS NORMAL EPOCH G I L B E R T REVERSED EPOCH

I Ill I I II lllll I I I I IIIIII

I (I I I I

Fig. 4.11. Time-scale for geomagnetic reversals. Each short horizontal line shows the age as determined by potassium-argon dating and the magnetic polarity (normal or reversed) of one volcanic cooling unit. Normal polarity intervals are shown by the solid portions of ?he"field normal" column, and reversed polarity intervals by the solid portions of the "field reversal" column. The duration of events is based in part on palaeomagnetic data from sediments and magnetic profiles. (After Cox, 1969.)

222

The Earths Core 2;

x 10

h

N

E Q

v

50

L

C

5E a 5

.-Dn

5

-13

3

L L

.-

> 0

20'

40'

60'

80'

100'

120'

140' 160'

180'

Co-lahrude 6

Fig. 4.12 Graph of V D M against co-latitude for the A, to N, transition zone of western Iceland The error shown is the standard deviation-in most cases it IS less than the black circles Arrows indicate the time progression and show that the large intermediate VDMs grow and decay smoothly (After Shaw, 1975 )

Creek), and also that it can change in magnitude without changing direction, indicates that the Earth's magnetic field may have a third metastable state that occurs much less frequently than the normal and reversed states. Another unresolved question is whether the magnetic field vector at a given locality tends to move along the same path during successive polarity transitions. If only the non-dipole field were present over much of a transition, the VGP from many reversals would be expected to be randomly scattered through all longitudes. Some analyses have suggested, however, that there may be preferred meridional bands within which most transition pole paths lie. From an analysis of Tertiary polarity transitions, Creer and Ispir (1970) concluded that during a transition a significant equatorial dipolar component remains that can reverse independently of the axial dipole and on a shorter time-scale. The movement of the dipole from one hemisphere to the other, corresponding to transition field directions recorded at widely separated sites, follows the same path, passing through the Indian Ocean. Most analyses, however, indicate no strong preference for transitional poles to be found in any one particular longitudinal interval (see e.g. Dagley and Lawley, 1974; Dodson et al. 1978). Hoffman (1977) noted that there appears to be a tendency for several paths to be controlled in part by the site location. In this regard Dodson et al. (1978) found two reversal paths from the same general area but differing in age by about 10 Ma to be essentially the same. Hoffman

The Earths Magnetic Field

223

pointed out that for R -+ N reversals observed a t mid latitudes in the northern hemisphere, the poles tend to lie predominantly in the hemisphere centred about the site meridian. These he termed near poles. Conversely there is some indication that N -+ R transitional poles tend to be found in the hemisphere centred about the antimeridian (far-sided poles). Williams and Fuller (1981a) later obtained an unambiguousfar-sided VGP path for an R -+ N reversal recorded in the Agno batholith (a 14-15 Ma old quartz diorite) in Luzon Island, the Philippines. This result is contrary to those obtained from nearly all other R -+ N northern hemisphere reversals, which have near-sided VGP paths, and lessens the claim of a general sitedependence. In fact Williams and Fuller (1981b) later pointed out that there is no guarantee that each reversal has the same harmonic content. Hoffman (1982) suggested that such an overall characteristic, if it does exist, may be time-dependent or determinable only in a statistical manner. Steinhauser and Vincenz (1973) investigated the latitudinal distribution of transitional palaeopoles. They found a U-shaped distribution, there being a decrease in the number of observed poles with decreasing latitude. They interpreted this result as reflecting an acceleration in the movement of the dipole axis as it approaches the equator-they estimated that it is moving 3.4 times faster in equatorial latitudes than in latitudes around 40". In contrast the record of a non-dipole field would give a random latitudinal distribution of poles. They further estimated that the dipole moment is reduced by about one order of magnitude for only about 12% of the transition time, while for two-thirds of the time its magnitude is comparatively high-with field intensities considerably greater than the intensity of the non-dipole field. Kristjansson and McDougall (1982) have also studied the latitude distribution of VGP, using data from late Tertiary lava flows from Iceland. They found that the latitude range where the VGP was most commonly found was around 70'" or S, which is unexpectedly low when compared to the presentday geomagnetic field. The chance that the VGP will be found within 10" of the geographic pole was only 11%. The VGP spent on average 50% of its time below latitude 64.5", 9% below 35", and about 2% below 10". Clement and Kent (1984) also found that the duration of a transition is dependent on the site latitude, with durations at mid latitudes being more than a factor of two longer than at equatorial latitudes. Mankinen et al. (1985) and Prtvot et al. (1985) carried out a detailed investigation of the Miocene R -+ N polarity transition recorded in lava flows from Steens Mountain in southeastern Oregon. The record begins with an estimated several thousand years of reversed polarity with an average intensity about one-third less than the expected Miocene intensity. The polarity transition consists of two phases. At the onset of the first phase, a further onethird decrease in intensity may have preceded directional changes by about

224

The Earth’s Core

600 years. Changes in field direction are confined to within 60” of the northsouth vertical plane and occurred very rapidly with normal polarity being attained in possibly 550 f 150 years. Field intensities rose to their pretransitional values at the same time that normal polarity directions were recorded. A quasi-normal behaviour lasted for only 1 W 3 0 0 years, after which field intensities dropped to very low values with changes in direction describing a long counter-clockwise loop, in contrast to the earlier narrowly constrained changes. The second part of the transition lasted 2900 f 300 years and both normal field directions and intensities were attained at the same time. The total time from the initial onset of directional changes until a normal field configuration was “permanently” re-established was 3600 5 700 years, while the period of marked intensity decrease lasted 4400 f 900 years. Three periods of extremely rapid changes were found-all when the intensity of the field was very low. In the best-documented example, the directional change was 58“ k 21” per year. The intensity change during the same period was 6700 f 2700 nT per year, which is about 15-50 times larger than the maximum rate of change of the non-dipole field observed in historical records. The record ends with several thousand years of normal polarity, with an average intensity in accord with the expected Miocene value. However, the occurrence of rather large and apparently rapid intensity fluctuations with little directional changes suggests that the nearly re-established dipole was still somewhat unstable. Hoffman (1986) examined the transitional behaviour observed in lavas from three mid-southern-hemisphere Cenozoic shield volcanoes. The data suggest that a geomagnetic reversal is accomplished in two steps, separated by an interval of relative quiescence. At its onset the geomagnetic reversal process is characterized by an essentially stationary intermediate field geometry associated with vector directions less than about 60” from that of the immediate preceding axial dipole field. The data indicate that several aborted attempts involving the same intermediate field geometry may precede an actual polarity transition. This is in accord with the results of the detailed study of the Steens Mountain R + N transition (Mankinen et al., 1985; Prevot et a!., 1985), which show a large and regular change in direction at the onset of the reversal to an intermediate orientation that remains essentially unchanged during the extrusion of several flows. There follows a rapid change to full normal polarity and then, before the completion of the reversal, a return to the same intermediate field direction. In addition to polarity changes, the Earth’s magnetic field has often departed for brief periods from its usual near-axial configuration, without establishing, and perhaps not even instantaneously approaching, a reversed direction (See Fig. 4.13). This type of behaviour has been called a geomagnetic excursion. Geomagnetic excursions have been reported in lava flows of vari-

The Earths Magnetic Field

Core Treatment

225

26 - 0 4 150 Oe Indinahon

-90' 0

-60"

-30"

Oo

30"

60"

90'

50C

h

E e

f CL

1000

cl

150C

2000 Fig. 4.13. Inclination of short-period events seen in a deep-sea core from the Southern Ocean. Polarity log at right: black is normal, clear is reversed. (After Watkins, 1968.)

ous ages in different parts of the world and from some deep-sea and lake sediments. Excursions are generally observed to commence with a sudden and often fairly smooth movement of the VGP towards equatorial latitudes. The VGP may .then return almost immediately, or it may cross the equator and move through latitudes in the opposite hemisphere before swinging back again to resume a near-axial position. It is possible that excursions represent aborted reversals. Although records have been obtained from a wide variety of environments and places, in no case has a proposed excursion been recorded with the same palaeomagnetic signature in two nearby but distinct environments. Again, in no case has it been shown that two independently dated proposed excursions at widely separated sites are of precisely the same age. Demonstration of such a temporal consistency is critical in establishing a

226

The Earth's Core

global as opposed to a regional nature of an excursion. Most geomagnetic excursions have been recorded in lake or deep-sea sediments and it is extremely difficult to assess the reliability of ages assigned to them. The horizons are usually dated by assuming uniform sedimentation rates between or beyond 14C-dated horizons. A number of models have been suggested to describe the behaviour of the geomagnetic field during a polarity transition. In some models the non-dipole field remains a small part of the total field but the dipole is considered to have two or three components, the strength, polarity and possible orientation of which change during the transition. Bochev (1969) suggested such a model with three dipoles-for epoch 1960 he found that two dipoles are approximately parallel to the Earth's axis of rotation while the third is aligned obliquely to the axis. Creer and Ispir (1970) suggested that these three dipoles correspond to different dynamo processes and that each dynamo might drift or undergo periodic changes in position--each might also fluctuate in strength and even reverse its polarity independently of the others. Verosub (1975) made a similar suggestion to account for reversals. In his model the field arises from two separate sources, the observed dipole component of the Earth's field being the vector sum of the dipole components of each source, i.e. Mo = Mi

+

M2

(4.21)

If the dipole components of each source are oppositely directed, one pointing essentially towards geographical north and the other south, then Mo

N

MI - Mz.

(4.22)

The prevailing magnetic polarity reflects the polarity of the dominant source component: a magnetic reversal, which corresponds to a change in the sign of M,, represents a shift in the relative sizes of M I and M2. If both M , and M 2 are very large compared with M,, then only a small fluctuation in M , , M2 or both will result in a reversal. Verosub suggested that the two sources are located in the 1C and OC. The source field in the liquid OC could be generated by MHD action, but it is much more difficult to postulate a source mechanism in the solid IC. Cox (1968) developed a probabilistic model in which it is assumed that polarity changes occur as the result of an interaction between steady oscillations and random processes. The steady oscillator is the dipole component of the field and the random variations are the components of the non-dipole field. The random variations serve as a triggering mechanism that produces a reversal whenever the ratio of the non-dipole to dipole fields exceeds a critical value (Fig. 4.14). Nagata (1969) suggested that the main dipole field is steadily maintained only as long as the convection pattern in the core is

227

The Earths Magnetic Field

I

0

0.01

I

0.02

0.03

0.04

I

r

0.05

0.06

Time ( M a ) Fig. 4.14. Probabilistic model for reversals. T,, is the period of the dipole field and ‘I the length of a changes sign. where Ma IS polarity interval. A polarity change occurs whenever the quantity (MA+ Mao) the axial moment of the dipole field and MA‘is a measure of the non-dipole field (After Cox, 1968.)

asymmetric, as is the case in the Bullard-Gellman-Lilley dynamo, but collapses when the convection pattern becomes symmetric as in the BullardGellman model. Parker (1969) developed a different model for reversals. He showed that a fluctuation in the distribution of the cyclonic convective cells in the core can produce an abrupt reversal of the geomagnetic field. The simplest fluctuation leading to a reversal is a general absence of cyclones below about latitude 25“ for a time comparable to the lifetime (- 1000 years) of an individual cell. In both Cox’s and Parker’s models, cyclonic convection cells in the core produce reversal by a two-step mechanism. At any instant they are randomly distributed throughout the core: reversals occur when, through random processes, they arrive at certain critical configurations. However, there is a fundamental difference between the two models. In that of Parker (and also that of Nagata) the occurrence of a reversal depends only on the spatial distribution of the cyclones and not on the intensity of the dipole field. In the model of Cox the occurrence of reversals depends both upon the distribution of the cyclones and on the field strength of the cyclone disturbances (i.e. the nondipole field) relative to the dipole field. Parker’s model has been further developed by Levy (1972a,b,c), and Hoffman (1977, 1979). Laj et al. (1979) pointed out that there is a mechanical difficulty in Cox’s model arising from the different time-constants of the random fluctuations of the non-dipole field and the steady osci!lations of the dipole field. Oscillations of the dipole field typically have a time-constant of the order of 2 x lo4 years. Characteristic times for the non-dipole field range up to about lo3 years; however, most of the power is concentrated in the range 2 4 x lo2 years. It is quite difficult to couple two systems with very different time constants. Laj et al. maintain that, if Cox’s model is correct, the field would reverse but would

228

The Earth‘s Core

maintain the opposite polarity for a very short time-of the order of the time constant of the non-dipole field. This problem of the differences between the time constants of the dipole and non-dipole fields also occurs in other models. In Parker’s (1969) model the quantitative nature of the cyclones is not sufficiently well known to be able to calculate the minimum amount of time during which they have to be absent from low latitudes in order to induce a reversal. Parker believes that reversals occurring as often as lo5 years are not likely. There are essentially two classes of models that have been suggested to account for reversals. In the first (the “standing field model of Hillhouse and Cox, 1976) the main dipole field diminishes while at least parts of the nondipole field do not. The transition field at any site is thus the sum of a steady non-dipole component and an axial dipole component that changes only in magnitude and sign. Unless the direction of the standing non-dipole field everywhere lies on the north-south vertical plane, there is no site dependence of path longitude. The second model (the “flooding” model of Hoffman, 1977, 1979) is based on the idea that the reversal process floods through the fluid core from a localized zone or point of initiation. The entire dipole source remains active, but normal and reverse flux is produced simultaneously in different regions. The VGP moves along the great circle defined by the geographical poles and the observer’s site. Depending on the sense of the reversal, where in the core source region the reversal process begins and the hemisphere of the observer, the VGP path for the reversal will be “near-sided” or “far-sided”. It is not known whether the non-dipole field is predominantly quadrupolar or octupolar. Figure 4.15 shows schematically a sequence of five field configurations for an R + N reversal. Figure 4.15a is for a pure octupolar transition field geometry and 4.15b for a pure quadrupolar field. In the upper section of each figure, the lines of force of the poloidal fields are shown-the shaded area represents the region of new polarity and illustrates its growth throughout the reversal. In the lower part of each diagram schematic dipoles are used to illustrate the sources of the observed fields. The shading in the lower section of the diagrams indicates the region of the core in which the reversal was initiated. Olson (1983) investigated the behaviour of the geomagnetic field during a polarity transition based on a ‘M dynamo model. He showed that rapid reversals of the dipole field occur when the helicity changes sign, and that two classes of polarity transitions are possible. In the first class (which he termed component reversals), the dipole field reverses but the toroidal field does not. In the second class (termed full reversals), both dipole and toroidal fields reverse. Component reversals result from long-term fluctuations in core helicity; full reversals result from short-term fluctuations. Olson suggested that the cause of reversals is the result of fluctuations in the core’s net helicity

229

The Earths Magnetic Field

stage 1

stage 2

stage 3

stogr

c

normal

atupole

reverse

stage 5

la I

stage 1

stage 2

reverse

stoge c

stage 3

quadrupde

-

i

stage 5

normal 161

N reversal with octupolar transition field (after Fuller eta/., 1979). ( b ) Sequence of field configurations for R -+ N reversal with quadrupolar transition Fig. 4.15. ( a ) Sequence of field configurations for R

field (after Fuller eta/., 1979).

in response to fluctuations in the level of turbulence produced by two competing energy sources-thermal convection and growth of the IC. Helicity generated by heat loss at the MCB may have the opposite sign to that generated by energy release at the ICB. There exists a wide class of systems of ordinary differential equations that represent forced dissipative hydrodynamic systems such as geodynamos, which have non-periodic solutions. Such systems may oscillate randomly between two states of fixed points; i.e. they may have two unstable polarity

230

The Earths Core

states. Such systems could display non-periodic reversals in the absence of any triggering mechanism. Lorenz (1963) first pointed out such behaviour in a discussion of the feasibility of very-long-range weather forecasting. The phenomenon (since named “chaos”) has been found in many other fields, e.g. models of population dynamics, physiological control systems and chemical reactions. The equations describing simple dynamo models are known to exhibit instabilities with respect to starting conditions and integration method and it has been shown that, in common with other systems (e.g. Sparrow, 1982), the existence of strange attractors and chaotic behaviour is inherent in the equations. Cook and Roberts (1970) and Ito (1980) showed that the Rikitake (1958) double-disk dynamo model shows all the attributes of chaotic behaviour associated with non-linear systems. However, in all physically realistic situations, the addition of some noise process (random or otherwise) must intrude to influence the evolution of the system. For fluid motions in the Earth’s core, irregularities either in material properties, physical or chemical processes or fluid dynamical turbulence will provide an adequate source of noise. Crossley et al. (1986) examined whether the essential behaviour of simple dynamo models can be attributed to inherent chaos or to external stimulation by a random component. They showed that the addition of three varieties of stochastic processes (Gaussian, flicker and brown noise) into the equations enriched the field evolution. Although it was not possible to decide whether the palaeomagnetic results favour a chaotic or a “noisy” dynamo (or some combination of the two), they favoured a physically derived stochastic process to trigger reversals-reminiscent of the empirical models of Cox (1981). Two features of the palaeomagnetic record that favour the presence of noise in the dynamo process are the absence of a linear oscillation in field intensity between reversals and the absence of an increase in amplitude of any oscillations prior to a reversal. Merrill et d.(1979) suggested that hydromagnetic dynamos might exist in which there are at least two different acceptable velocity fields and in which changes in polarity occur only when there is a change from one velocity field to the other. The existence of two velocity states suggests bifurcating solutions that are frequently encountered in fluid dynamical systems (e.g. Chandrasekhar, 1961; Huppert and Moore, 1976; Roberts, 1978). A characteristic of many finite-amplitude systems is that more than one state is stable or metastable for a given set of externally imposed conditions. It has been speculated (for example by Roberts, 1978) that reversals could be associated with “catastrophic” transitions between primarily geostrophic and primarily magnetostrophic states. McElhinny (1971) and Irving and Pullaiah (1976) have analysed palaeomagnetic data for different geological periods in the Phanerozoic in order to find the proportion of normal to reversed polarities (see Fig 4.16, which

231

The Earths Magnetic Field

-.

.

h

0

v

v

lJ

3 50

50

:

a a

z

(L

100

0

100

200

300

400

500

600

Age (Ma) Fig 4.16. Polarity bias of the geomagnetic field during the Phanerozoic. Overlapping 50 M a averages of polarity ratios as observed in palaeomagnetic results are shown together with the limits of the standard errors. (After Irving and Pullaiah, 1976).

shows the percentage of normal polarity plotted as a function of time). Figure 4.16 clearly illustrates polarity “bias”. There is a marked difference between the Lower and Middle Devonian, which have roughly equal occurrences of normal and reversed data, and the Upper Devonian which is largely reversed and marks the beginning of a predominantly reversed era lasting until the close of the Palaeozoic. For the mid-Carboniferous to the Lower Permian, only a few results show normal polarity. Apart from the Lower Triassic, where frequent reversals have been observed in several formations, the Mesozoic has predominantly normal polarity (- 75%); measurements in the Upper Triassic indicate that the field was normal as much as 83% of the time. Thus the assumption that the reversal process is symmetrical in the two polarity states, with no bias with regard to the lengths of normal and reversed intervals, i s not justified. Another interesting question is the variation in the average frequency of reversals. This was first examined by Cox (1975). Figure 4.17 (after Cox, 1975) shows the average frequency of reversals over the past 150 Ma based on the time-scale of Heirtzler et al. (1968) as seen through a sliding window 10 Ma long. It appears that the mean frequency of reversals shows no statistically significant changes from about 75 to 45 Ma ago and then increases (rather abruptly) by a factor of more than two. It has then remained approximately constant until the present. The geomagnetic dynamo appears to have been stationary for the 30 Ma prior to this 35 Ma “discontinuity” and to have been stationary since then. Thus 45 Ma seems to mark a boundary between two intervals during which the statistical properties of the dynamo were distinctly different. An even more striking discontinuity appears at 85 Ma: for the preceding 22 Ma the frequency of reversals was zero or nearly so, the polarity of the field remaining normal from about 107 to 85 Ma ago. During the past 108 years the frequency of reversals appears to have increased from zero to the present frequency of nearly 5 per Ma in two steps, one at 85 Ma and one at 45 Ma.

232

The Earths Core

a >.

5

.-

.-

3-

E 2all -

0

b

6

1-

r x

I 0

100 150 Time before present ( M a )

50

Fig. 4.17. Variation in the average frequency of reversals as seen through a sliding window 10 M a long. From 7 6 M a to the present the rates are from Heirtzler eta/. (1 968) and prior to 7 6 M a from Larson and Pitman (1 972). (After Cox, 1975.)

Phillips et al. (1975) have estimated the mean and variance of interval lengths as a function of time for the Cenozoic reversal timescale (G76 Ma ago) of Hertzler et al. (1968). Their analysis shows a discontinuity around 45 Ma. However, both curves are consistent with stationary behaviour between 0 and 45 Ma and between 45 and 76 Ma. Lowrie (1982) later estimated the number of reversals per million years in the Cenozoic and late Cretaceous averaged over intervals of 2, 5 and 10 Ma duration based on the time-scale of Lowrie and Alvarez (1981). The 2 Ma averages show large, apparently irregular, fluctuations in reversal frequency; averaging over 5 Ma and especially over 10 Ma intervals indicates that these fluctuations are superposed on an almost linear trend (see Fig. 4.18). This suggests that the average reversal frequency of the geomagnetic field has been steadily increasing since the late Cretaceous. Lowrie and Kent (f983) repeated the calculations using different polarity time-scales and an 8 Ma sliding window and arrived at similar conclusions. They thus disagree with the analyses of Cox (1975) and Phillips (1977), which indicated a fairly abrupt increase in the frequency of reversals around 45 Ma ago. They claim that subdivision into two periods of stationary behaviour is no longer warranted. They base their conclusions on the newer magnetic polarity time-scales of LaBrecque et al. (1977), Ness et al. (1980) and Lowrie and Alvarez ( 1 9 8 1 F the analyses of Cox (1975) and Phillips (1977) used the older magnetic timescale of Heirtzler et al. (1968). Mazaud et al. (1983) also analysed the structure of the reversal frequency curve using the time-scale of Lowrie and Alvarez (1981) and that of LaBrecque et al. (1977). The frequency of reversals was studied using a moving rectangular window, with window widths ranging from 2 to 10 Ma.

233

The Earths Magnetic Field

h

4k I I

0

IC)

I 1

I

I

30

I

I

I

60 A g e (Ma)

I

90

Fig. 4.18. The frequency of geomagnetic polarity reversals since the late Cretaceous, averaged over intervals of ( a ) 2. ( b )5 and ( c ) 10 M a respectively. (After Lowrie, 1982.)

The reversal frequency curves were then analysed in terms of a monotonically increasing component and an oscillating component. The monotonic component was modelled by a least-squares fit to a Lorentzian function. To analyse the oscillating part they subtracted the corresponding Lorentzian from the reversal frequency curve and performed a time autocorrelation on the remainder and found a periodicity of about 15 Ma. They then fitted the reversal frequency curve simultaneously with a Lorentzian function and a sine function. For the 4 Ma window and the LaBrecque et al. time-scale, this gave a 31 Ma half-width for the Lorentzian and a perioa of 15.1 Ma for the sine function. Similar results were found for windows ranging from 2 to 8 Ma and for the time-scale of Lowrie and Alvarez. McFadden (1984b), however, does not believe that the fluctuations seen by Lowrie and Kent (1983) or those of Mazaud et al. (1983) are real, but that the frequency of reversals has increased smoothly and linearly with time. He maintains that the superimposed fluctuations seen by Lowrie and Kent are due to their choice of smoothing filter (8 Ma) and those of Mazaud et al. are

234

The Earth’s Core

an artefact of their analysis, the filter generating the fluctuations. It is just by chance that the period of the fluctuations seen by these two sets of authors are approximately the same. In a later paper McFadden and Merrill (1984) found that there is strong evidence that the rate of reversals decreased in an approximately linear manner with time from 165Ma ago until it reached zero, resulting in the Cretaceous Normal Polarity Interval. At about 86 Ma ago the process of reversals began again, and the rate of reversals increased linearly with time. There is a slight indication that the rate of reversals reached a maximum about 10 Ma ago and is again decreasing. Two basic models have been suggested to describe the behaviour of the geomagnetic field during a polarity transition. In one, the main dipole field remains axial but decreases in strength during the transition, so that the nondipole field becomes dominant in the intermediate stages (see e.g. Larson et af., 1971). In the other the non-dipole field remains a small part of the total field but the dipole is considered to have two or three components, the strength, polarity and possible orientation of which change during the transition. Cox (1968, 1969) suggested that the variations in the length z of polarity intervals could be approximated by a Poisson distribution P ( T ) = 1 exp (- 17)

(4.23)

where the parameter l characterizes the observed variations in the length of polarity intervals. For time intervals longer than some minimum time zmin, (4.23) will be valid if and only if the probability of a reversal in any time interval z > T,,,~,, is independent of whether a reversal occurred during any prior interval. Thus, if reversals are truly Poisson, the core has no memory longer than tmin, which implies that the dynamo processes responsible for reversals have time constants no greater than zmi,. T,,,~,, is probably comparable to the longer time constants of the secular variation (- lo4 years). The observed sequence of reversals for the past 7 0 M a satisfies a Poisson distribution reasonably well except that the observed number of short polarity intervals is somewhat smaller than predicted (Cox, 1969; Naidu, 1971). Naidu (1971) suggested that the probability density function for reversals is better described by a gamma distribution P(T) = l(1~)~-exp (-

h)/r(k)

(4.24)

for which the mean value p = k / l . This reduces to (4.23) if the lengths of polarity intervals are Poisson distributed ( k = 1). The parameter k measures both the proximity of the observed distribution to an exponential distribution and the dispersion of the intervals about their mean. Phillips ( 1 977) examined the Cenozoic reversal sequence and found that the means of normal

The Earth’s Magnetic Field

235

and reversed intervals behaved in a similar way; both show a discontinuity at 45 Ma, and both are consistent with stationary behaviour on either side of this discontinuity. Statistical tests established that there is no significant difference between the mean lengths of normal and reversed intervals during the Cenozoic. However, the average value of k for normal intervals is 2.84 i 0.41, whereas for reversed intervals it is 1.31 0.18. This difference implies that normal and reversed polarity intervals have different distributions. An analysis of the Miocene (8-23 Ma) reversal time-scale of Blakely (1974) confirms these earlier findings-the mean lengths of normal and reversed intervals are virtually the same and the value of k determined for normal polarity intervals is greater than that determined for reversed polarity intervals. This difference in k for normal and reversed intervals, if confirmed, is evidence that the geodynamo is more stable after a transition to normal polarity than it is after a transition to reversed polarity. Lowrie and Kent (1983) have discussed the effect of adding short events to the magnetic polarity time-scale-this has a dramatic effect on the polarity interval distribution between 0-40 Ma. The number of polarity intervals increases from 126 to 226 and the mean length of a polarity interval is reduced from 0.314 Ma to 0.175 Ma. The exact duration of such short polarity events cannot be determined. If they are 20,000 years long, the polarity interval length distribution differs significantly from a Poisson distribution-if they are about 30,000 years long, there is no significant difference. The effect of additional short polarity events also dramatically affects the distribution of normal and reversed polarity interval lengths. If the short events are about 30,000 years long, the gamma index k for normal and reversed distributions is 1.14 and 2.01 respectively. Although different k values for normal and reversed distributions had been found before (e.g. Phillips, 1977), the addition of these short polarity events alters the sense of asymmetry between normal and reversed states-previous analysis had suggested k was larger for normal polarity. This change results from the unequal distribution of the short polarity events which are predominantly of positive polarity and concentrated in the Late Cenozoic. McFadden and Merrill (1984) have more recently analysed the statistical properties of recent marine magnetic anomaly time-scales covering the period 165 Ma ago to the present using methods developed by McFadden (1984~).As mentioned earlier, their analysis no longer supports the hypothesis of a discontinuity at 45 Ma ago. In addition, they found no evidence for a different value of k for normal and reversed polarities and hence no difference in the relative stabilities of the two polarity states. They showed that the parameter k is extremely sensitive to the number of intervals in a sequencethe addition of one normal interval in a total of 198 intervals resulted in a striking difference in the distribution of k against time for normal polarity.

236

The Earths Core

The overall structure of non-stationarity from 165 Ma ago to the present found by McFadden and Merrill explains the existence of extremely long intervals of time without any reversals, such as the Cretaceous Normal Polarity Interval and the Permo-Carboniferous Reverse Polarity Interval. From 165 Ma ago onwards the rate of reversals decreased in an essentially linear manner until 106 Ma ago, when the process ceased, until about 87 Ma ago and then gradually increased again. In other words, the process causing instabilities, leading to reversals, gradually slowed down until no more instabilities were produced. The field then remained in the polarity it happened to be in at the time that instabilities ceased to occur. Thus it is merely a matter of chance whether a long quiet interval happens to have normal or reverse polarity. There is no requirement for the relative stabilities of the two polarity states to alter between the two long quiet intervals. The process causing instabilities then begins again and gradually speeds up, reversals occurring with an essentially linear increase in the number of reversals with time. This work of McFadden and Merrill is comforting since asymmetries between normal and reversed states cannot be reconciled within the domain of dynamo theory. A detailed account of reversals of the Earth’s magnetic field has been given by Jacobs (1984).

4.6

Energetics of the Earth’s Core

Although it is now known that kinematic dynamos exist, solutions in which Maxwell’s equations are solved for specified velocities are of limited geophysical interest since there is no guarantee that there exist forces in the Earth’s core that can sustain them. Without a satisfactory theory to account for the driving force, the problem is not realistic and has only been “pushed one stage further back”. In a dynamical theory, the velocities would be calculated from assumed forces-almost no work has been done on this more difficult non-linear problem. From a consideration of the order of magnitude of the terms in the Navier-Stokes equation (4.13) it can be shown that the inertial terms may be neglected and most probably the viscous forces. We must then have a balance between the Coriolis forces, the pressure gradient, the electromagnetic forces and the applied force. It can easily be verified that if there is no applied force, the electromagnetic force cannot be balanced by the other two terms, since neither can supply energy. The pressure term can be removed by taking the curl of the resulting equation-this leads to an expression from which the curl of the force can be found for any dynamo for which the velocity and the field are known, i.e. if we had a solution to Maxwell’s equations for a specified velocity field, it is possible, in theory, to calculate the forces needed to drive

237

The Earths Magnetic Field

the system. To solve the inverse problem with specified forces is, as already mentioned, very much more difficult. There are three distinct types of magnetic fields making up a measurement at the Earth‘s surface. These are the main geomagnetic field generated by processes within the Earth’s core, field perturbations due to the crust of the Earth, and externally induced fields. Generally speaking the main field is several orders of magnitude larger than the other types of fields. Lowes (1974a) estimated the logarithmic spatial “power” spectrum of the Earth’s internal magnetic field for harmonics up to n = 500 using the equation, n

r

(4.25)

He showed that it consists of two components, long wavelengths being dominated by fields originating in the core, and short wavelengths by fields originating in the crust; the cross-over occurs at n 2 11, a wavelength < 3600 km. Both the long-wavelength and short-wavelength fields may be fitted well by straight lines. If these lines are extrapolated and combined, an idealized surface power spectrum may be obtained. Lowes estimated that it probably represents the spectrum of the present field to better than about 30%. McLeod and Coleman (1980) also obtained the spatial power spectra of crustal and core geomagnetic fields. Their equations are based on a statistical model using the principle of equipartition of energy. The two spectra are approximately equal for n = 15. Their equations are in good agreement with data for the core field given by Peddie and Fabiano (1976) and with data for the crustal field given by Alldredge et al. (1963). Their equation for the crustal field spectrum is, however, significantly different from that of Lowes (1974a), but is in better agreement with data obtained from the POGO spacecraft. Since their model predicts infinite energy for the geomagnetic field, it becomes invalid as n becomes increasingly large. They suggest a breakdown above n = 500 for the crustal field and above n = 12 for the core field. Langel and Estes (1982) used 26,500 Magsat scalar and vector data taken on 14 selected magnetically quiet days to obtain a spherical harmonic model of the Earth’s internal magnetic field of degree and order 23. They computed the power spectrum and found a distinct change of slope at about n = 14 that they interpreted to mean that the field from the core dominates for n d 13 and the field from the crust for n 2 15 (see Fig. 4.19), in agreement with earlier results of Cain ct al. (1974) and Cain (1976). This means that the presence of crustal fields places a limitation on our ability to estimate the field from the Earth’s core. In the Earth’s core, the energy needed to drive the geodynamo must have been available at a more or less constant rate for several billion years. This

238

The Earth's Core

t![rs,"; + ( h P j 2 J

~ , , = ( n + l )rn-0 10' - @

0

-

0

n Fig. 4.19. Geomagnetic field spectrum. R, is the total mean-square contribution to the vector field by all harmonics of degree n. The curves are fitted to the surface result and extrapolated to the core-mantle boundary. (After Langel and Estes, 1982).

energy could originate in a variety of forms (gravitational, chemical, thermal), ultimately being converted into heat that flows out into the mantle. The energy source must not be too great, otherwise it would melt the mantle and produce more heat at the Earth's surface than is observed. The average heat flux through the Earth's surface is 4 x l O I 3 W (Sclater et al., 1980). Most of this comes from the decay of radioactive elements in the crust, leaving l O I 3 W to come from the core.

-

The Earth‘s Magnetic Field

239

Estimates of the work done by the fluid motions in the Earth’s core in replacing the energy lost as heat by electric currents (Ohmic heating) range from about 108W (Parker, 1972) to 5 x 10” W (Braginskii 1964), the larger figure being the more likely. The actual power requirements for the Earth’s dynamo are, however, much greater than just Ohmic heating. If the liquid core is stirred by thermal convection, then most of the heat will be convected away without any magnetic field being generated at all. Moreover, heat in the core can be dissipated not only by convection but also by conduction. When heat is transmitted by conduction, the fluid does not move, so that the conducted heat flow does not contribute to dynamo action. The probable efficiency of any heat-driven dynamo is about 5%. There is not a great variety of forces that can produce motions in the Earth’s core. Most naturally occurring fluid motions are due ultimately to the action of gravity. The gravitational potential near the Earth consists of two components-that due to the Earth itself and that due to extraterrestrial sources (the sun and the moon). Consider first gravitational fields of extraterrestrial origin. The Earth‘s mantle undergoes complicated accelerations due to a number of causes, such as the bodily tide of the Earth, tidal friction in the oceans, sudden changes in the rate of rotation of the Earth and precession and nutation. It can be shown (see e.g. Hide, 1956) that of these only precession could have any appreciable effect on motions in the Earth’s core, and experiments by Malkus (1968) have indicated that precession may produce turbulent motion in the core and hence drive the dynamo. Malkus (1963) had earlier suggested that precessional torques might drive the Earth’s dynamo, but unfortunately there are errors in his paper. A detailed theory of a dynamo in a precessing turbulent core is difficult and no full treatment has as yet been given. However, Rochester et al. (1975) have pointed out the mathematical and physical errors in Malkus’ arguments and shown that precession fails by at least two orders of magnitude to satisfy the power requirements to drive the dynamo. Toomre (1966) has shown that the coupling between the core and mantle is provided primarily by inertial, or pressure, torques arising from the distribution of the fluid pressure at the spheroidal MCB. However, this coupling mechanism is conservative and thus incapable of supplying energy to the geodynamo. Rotational kinetic energy can possibly be supplied though only by dissipative torques. Processional power input to the core (at the expense of the obliquity) can be estimated in terms of the dissipative part of the core-mantle coupling and the tiltover angle, i.e. the inclination of the core angular momentum vector to that of the mantle, Malkus failed to allow for the electrical conductivity contrast at the MCB and neglected the dependence of the coupling strength on the diurnal frequency of the precession-induced core flow relative to the mantle. Stacey (1973) had also argued that precessional torques can power

240

The Earths Care

the geodynamo. He also neglected the above dependence of the coupling strength and in addition estimated the tiltover angle by a non-rigorous kinematic argument. A detailed discussion of motions within the Earth’s core has been given by Busse (1971). In an analysis of the flow in the core of a precessing Earth, he found that the angle between the axis of rotation of the core and that of the mantle in only radians. The fact that the essentially toroidal velocity field obtained in his solution cannot by itself act as a dynamo led him to the conclusion that precession cannot be the driving mechanism. The more detailed analysis of Rochester et al. (1975) indicates that the possibility of a precession-driven dynamo is extremely remote. Loper (1975) pointed out that both Malkus and Stacey assumed that all the rotational kinetic energy is available to drive the dynamo. However, since the energy is transmitted by dissipative couples, transmission losses must be taken into account. Loper showed that all of the energy from rotation can be dissipated in boundary layers at the MCB leaving none available to drive the geodynamo. In addition, he showed that the dissipative torques were weaker by a factor of than those estimated by Malkus and Stacey. Order of magnitude arguments on the feasibility of precession to drive the geodynamo (e.g. Malkus, 1968) also depend quite critically on the value of the electrical conductivity r~ of the core. There are two problems-the functional dependence on d of various critical parameters such as the ohmic dissipation Q and the correct value of CJ in the core. Rochester et al. (1975) have corrected errors . the in the literature and shown that Q in the core is proportional to o - ~For value of 0,Malkus assigned an uncertainty of a factor of 3 in either direction to the value he assumed in his calculations (7 x lo5 S/m). In 1967 Stacey suggested that allowance for the effect of impurities in the iron in the Earth’s core would increase its electrical resistivity by a factor of 10. However, in a later paper, Gardiner and Stacey (1971) withdrew this suggestion of such a large increase and preferred Bullard’s (1949a) original estimate of (3 x 10’) for the conductivity. They suggested that a plausible range of values of r~ is (1-6 x lo5). In this respect it is interesting to note that Braginskii (1964b) estimated the conductivity of a core alloyed with 30% Si to be just twice Bullard’s figure. The normalization Gardiner and Stacey used to allow for metallic impurities in the iron is reasonable, as liquid silicon is itself metallic. However, their extrapolation of Bridgman’s (1957) data on resistivities of solid iron alloys up to 100 kbar to liquid iron at Mbar pressures is not justified. Thus, although the conclusions of Gardiner and Stacey may well be right, some of their arguments in rebuttal of Stacey’s (1967) earlier suggestion of a large increase in resistivity in the core appear to be incorrect (see Jacobs et al., 1972). Jain and Evans (1972) have carried out a calculation of the resistivity of the Earth’s core based on a model for the electrical transport properties of simple

The Earths Magnetic Field

241

liquid metals proposed by Ziman (1961, 1971). Their estimate of the resistiOm), which is in the range of plausible values vity is between (1 and 2 x according to Gardiner and Stacey. It is also of interest to note the results of the experimental work of Kawai and Mochizuki (1971) on the metallic states in three 3d transition metal oxides (including Fe,O,) under very high pressures. They found a very sudden drop in resistivity of from four to six orders of magnitude at a pressure corresponding to that in the liquid core of the Earth. Whether the resistivity of the Earth’s core is of the same order of magnitude as that of pure iron at its melting temperature at atmospheric pressure will depend on the alloying material (see $5.7). If it is mainly silicon, the answer is probably yes since liquid silicon is a normal metal. If it is sulphur, the answer would be very different, since liquid sulphur is almost a perfect insulator. However, if the core material is in the form of the compound FeS, the resistivity can still be quite similar to that of pure iron since liquid FeS is a semi-conductor under normal temperature and pressure conditions and there may be a Mott transition to metal at very high pressures. Estimates of the conductivity obtained by Jain and Evans (1972) are probably the best to date and are in general agreement with experimental results obtained by Johnston and Strens (1973) on a molten Fe-Ni-S-C core mix. It must not be forgotten that the ohmic losses remain inside the liquid core as an additional heat source. The global expression for conservation of energy does not involve the magnetic field in any way but is simply an equality (in steady state) between the sum of the heat sources and the heat flowing out into the mantle. Gubbins (1981) has stressed that arguments involving the energy required to replace that lost to ohmic heating are false. The magnetic field does, however, enter into the entropy balance (see e.g. Backus, 1975) and one can argue on the basis of entropy balance. Dissipative effects such as electrical or viscous heating always lead to positive entropy contributions. It is a consequence of the second law of thermodynamics that the diffusion constants are positive, which ensures that the corresponding entropy changes are positive-definite. These gains must be offset by entropy losses arising from the various heat sources. The entropy depends not only on the quantity of heat supplied but also on the temperature at which it is supplied. Backus (1975) and Hewitt et al. (1975) obtained the inequality: (4.26)

where Q is the total heat flowing out of the core and 4 the ohmic heating. Gubbins (1976) has used (4.26) to obtain estimates of the heat required for the Earth’s dynamo. Using a lower bound for 4, based on Parker’s (1972) approach, he found that the heat flux Q exceeded the lower bound of

242

The Earths Core

5 x 10” W. In fact, for reasonable estimates of core parameters, this lower bound is greatly exceeded and Q is probably nearer 1013W (Gubbins et al., 1979). It was this high value that led to a search for alternative energy sources. Verhoogen (1961) suggested that the latent heat released by the solidification of the IC could power the dynamo. Gubbins et al. (1979) showed this to be untenable, since the power required would freeze the liquid core more rapidly than the present size of the IC would suggest. The differences between the calculations of Gubbins et al. and those of Verhoogen are due to the fuller use of the entropy equation and to the fact that they related the growth rate of the IC to the dzflerence between the melting point and adiabatic temperature gradients rather than to the melting point gradient, as did Verhoogen. The melting and adiabatic temperature gradients are probably quite similar in the core (see #3.4,3.5) and this could lead to a substantial numerical difference. Braginskii (1963) extended Verhoogen’s idea by suggesting that freezing of material at the ICB would separate a heavy fraction (mainly Fe) leaving behind a lighter liquid fraction in the OC that would be buoyant, leading to “compositionally driven” convection. Gravitational energy released in this way contributes a disproportionally large factor to the entropy balance when compared to heat, i.e. gravitational energy is much more effective than heat in generating magnetic fields. Light material rising up will drive fluid motions directly and all of this potential energy must be converted to heat via one of the dissipative processes before the heat escapes to the mantle. Electrical heating is the major dissipative process in the core, viscosity and the diffusion rate of the light material in the core being so small. Thus, a large part of the gravitational energy released will go into electrical heating. With heat-driven convection, however, most of the energy is simply carried away by hot fluid rising to the top of the core. Gubbins (1977) extended Backus’s (1975) thermodynamic treatment to include gravitational energy, latent heat and specific heat, and obtained an inequality analogous to (4.26) 4-(G-P)

Q

AT

1 where m is the harmonic coefficient. When Iml = 1, the principal oscillations can drift westwards, but this is unlikely to be the case. For higher modes of oscillation, waves which drift both eastward, and westward, can occur. This question has been discussed in more detail by Hide and Stewartson (1972) in a review paper on hydromagnetic oscillations of the Earth’s core, and by Acheson and Hide (1973). Much further work has been done on this question (see e.g. Malkus, 1967; Rickard, 1973; Negi and Singh, 1974; Acheson, 1975; Wood, 1977; and London, 1981). There does not seem to be any consensus of opinion, London concluding that the phase velocity has no preference for either eastward or westward drift. The coupling of the core to the mantle, which is mainly due to magnetic forces, may influence the secular variation. In this regard Hide (1969) sug-

250

The Earths Core

gested that the MCB may not be smooth.* Analysis of travel-times of compressional waves reflected at this boundary shows that any topographic feature there cannot exceed a few kilometres in height-this being the limit of resolution of present-day seismic techniques. Hide suggested that irregular features at the MCB might provide “topographic” coupling between the core and mantle and so account for the decade fluctuations in the length of the day (neither viscous nor electromagnetic coupling seems completely adequatesee $4.8). Topographical “bumps” on the MCB may also be the cause of horizontal density variations responsible for regional gravity anomalies. It can readily be shown (e.g. Hide and Horai, 1968) that, because of the density contrast at the MCB, bumps with horizontal dimensions up to thousands of kilometres and a kilometre or so in height would make a significant (although not dominant) contribution to the observed distortion of the gravitational field at the Earth’s surface. Hide (1967,1969,1970) also suggested that bumps on the MCB might affect the flow pattern in the core and thus influence the detailed configuration of the geomagnetic field and its time variations. While the liquid core of the Earth is the only likely location of electric currents responsible for the main geomagnetic field, it is the most unlikely place to find density variations of sufficient magnitude to cause the observed distortions of the gravitational field-these must arise largely in the mantle. Thus, any correlation between gravity and magnetic anomalies should reflect processes at the MCB. Hide and Malin (1970) argued that if both gravity and magnetic anomalies are the result of the same topographical features, it should be possible to find a statistically significant correlation between them. In fact they found, for spherical harmonic coefficients up to degree 4,a correlation coefficient of 0.84 between large-scale features of the Earth’s non-dipole magnetic field (for epoch 1965) and the gravitational field, provided the magnetic field is displaced 160”eastward in longitude 1.Hide and Malin also showed that A has increased linearly with time since 1835 (see Fig. 4.21), the date of the earliest reliable spherical harmonic analysis (by Gauss) of the geomagnetic field. Their result is I

=

(126.2 & 0.2)”

+ (0.273

0005) ( t - 1835

10)”

(4.28)

where t is the epoch (year AD). This dependence of 1on t is associated with the westward drift of the geomagnetic field. There has been some controversy over the above correlation, particularly concerning the statistical procedures used (see e.g. Khan, 1971; Lowes, 1971, and reply by Hide and Malin, 1971). It must be stressed that even if the correlation does exist it does not by itself * The suggestion that the MCB may become “warped” as a result of stresses developed by currents in the outer part of the core was suggested earlier by Garland (1957).

251

The Earths Magnetic Field

d I

160'

1

1

150'

x 140°

130'

1

1

I

I

1850

1900

1950

TIME Fig. 4.21. The variation with time of the eastward displacement in longitude I between the Earth's magnetic and gravitational fields. (After Hide and Malin, 1970.)

prove the existence of bumps on the MCB. It is possible that quite small temperature variations over the MCB could, through their effects on core motions, produce measurable distortions of the geomagnetic field. If these temperature variations in turn reflect the density structure of the lower mantle, then there would be a correlation between gravity and geomagnetic anomalies. Baranova et al. (1973) have extended Hide and Malin's computation back to 1600 using spherical harmonic analyses of the geomagnetic field for six earlier epochs (1600, 1650 . . . 1850), Izmiran model D-1 (Ben'kova et al., 1972). They found that a relationship between gravity and magnetic fields existed over the entire time interval, although the relationship is more complex than that obtained from an analysis of data for a short time interval ( 100 years). The westward drift, which was so clearly seen in the A(t) pattern for 1829-1950 is distinct only in the non-dipole field and is practically absent in the quadrupole field. Robinson (1974) has developed a boundary-layer model of thermal convection throughout the Earth's mantle and been able to estimate the distortion of the MCB due to such convective motion. By equating the additional gravitational force of the heavier descending plume with the hydrostatic force due to the distortion of the MCB, he estimated the displacement to be N

* There is now evidence (Hager et al., 1985) for large-scale dynamically maintained topography at the MCB.

252

The Earth’s Core

-

1.5 km, which is of the order of that required by Hide (1969, 1970) to account for core-mantle coupling. As the IC is a good electrical conductor, any ambient magnetic field would diffuse into it on a time-scale that was long compared to several thousand years, and at the same time be frozen there on shorter time-scales. From the observations that the dipole component of the Earth’s magnetic field has been inclined persistently to the spin axis over hundreds of thousands of years, and that the dipole drifts and decays significantly more slowly than the non-dipole field, Szeto and Smylie (1984) suggested that the external dipole is simply a manifestation of a field frozen in an inclined IC. They showed that the gravitational restoring torque can be significant for an inclined IC, so much so that its motion is in the main determined by gravity, with electromagnetic and inertial coupling effects being of secondary importance. They found a regular precession of the IC to be possible where its spin axis drifts westward relative to the mantle with a period of 7000 years. Bullard et al. (1950) had suggested much earlier that the westward drift of the non-dipole field was due to a magnetic field being carried bodily by the OC, and that the dipole drift, because of its slower rate, might have a deeper source. Further progress in understanding the secular variation has been made in recent years by considering the kinematics of the problem and ignoring the dynamics. If the core is treated as a perfect conductor, current sheets may form so that tangential components of B would be discontinuous across the MCB. Taking the radial component of B in (4.6) without ohmic dissipation gives

a

-B, at

+ V.(UB,)

-

V*(BU,)= 0

(4.29)

Diffusion will be negligible for sufficiently short periods of time (much shorter than the diffusion time-scale of 10,000 years) and for sufficiently long wavelengths (Roberts and Scott, 1965). In order to extrapolate the magnetic field and its variation in time as observed at the Earth’s surface down to the MCB, an insulating mantle must be assumed. The finite conductivity of the mantle filters out magnetic variations on a time-scale of a few years and precludes any information about short-period fluctuations that may be important for the understanding of the dynamics of the core. At the MCB, U, vanishes and (4.29) can be replaced by the equation

a ~

at

B,

= - V,. ( U,B,)

(4.30)

where the subscript s refers to the horizontal component of the respective vector quantity. Of particular interest are the contours at the MCB on which B,

253

The Earth's Magnetic Field

vanishes (called null flux curves). The flux F, enclosed by these contours obeys the equation (Backus, 1968): (4.3I )

where the integral is over the part of the spherical surface enclosed by these contours. Curves on which B, vanishes are topologically invariant, i.e. they are permanent features. They cannot merge, disappear or form, although they can move about. There appear to be about five null flux curves on the MCB at present (Whaler, 1984). On null flux curves, (4.30) reduces to

a

-B, iUs*V,B, = 0 at

V; U measures convergence or divergence of flow at a point, and for an incompressible flow this indicates upwelling or downwelling of fluid beneath the surface. Rewriting (4.30) in the form (4.32)

it can be seen that at extreme or saddle points of B,, the second term vanishes and the secular variation is determined solely by upwelling. Whaler (1980) plotted the zero contour of secular variation on the core surface and found that it passed close to all the extrema, suggesting that upwelling was small compared with other properties of the flow (see Fig. 4.22). Whaler (1982) concluded that secular variation data can give information about fluid motions at the core surface. She found significant flows in both east-west and north-south directions. However, she obtained no evidence to suggest that the westward drift of the geomagnetic field is associated with azimuthal flow, although the well-resolved longitudinal flow tends to be westward. The frozen flux approximation has been applied to recent magnetic observatory and satellite data (Booker, 1969; Shure et al., 1982). Gubbins and Roberts (1983) have used it to help in the interpretation of archaeomagnetic and palaeomagnetic data. They showed that it places a stringent restriction on possible field changes. It gives bounds on the size of the components of the magnetic field at a point, of the Gauss coefficients, and, if valid over such long time periods, limits the shape of the field during polarity transitions. The maximum intensity at a point, consistent with the present flux, is 281 pT, i.e. four times the maximum field observed today. The present dipole is about 50% of its upper bound. In a series of papers Gubbins (1983, 1984), Gubbins and Bloxham (1985)

Fig. 4.22 Caritours of the field strength (vertical component) at the core surface for 1965 0 Dotted hne$ are negatlve values of 2 The dashed curvo IS thc contour of zero secular change for the same epoch If there were no upwelling the dashed curve would pass through all rhe extrema. inarked with crosses on the map Whaler ( 1 980) suggests that this isevidence for a lack of upwelllny of core fluld (Aitcr Whaler, 1980)

The Earth’s Magnetic Field

255

and Bloxham and Gubbins (1985a) applied the method of stochastic inversion (Franklin, 1970) to the analysis of geomagnetic data. They used an extension of the stochastic inversion method given by Jackson (1979) that uses the concept of a priori data. Good a priori information about the magnetic field is available since it originates in the core. The great distance between the Earth’s surface and the MCB means that short-wavelength fields are strongly attenuated at the observation point, so that restrictions on the short-wavelength fields at the MCB become very severe restrictions at the surface. In the first paper, Gubbins (1983) explains the method in detail and uses it to construct core fields. One great advantage of stochastic inversion compared to the alternative methods of analysis-spherical harmonic analysis and harmonic splines-is that it enables a proper error estimate to be made. In his second paper, Gubbins (1984) used the method to find models of the secular variation that are both smooth on the MCB and are constrained to satisfy the conditions of frozen flux. Both constrained and unconstrained models that fit the observations from 1959 onwards equally well are very similar, showing that it is possible to account for 15 years of secular variation (from 1959 to 1974) by fluid motions of a perfectly conducting core. Gubbins also found evidence of a complex change in core motions associated with the secular acceleration impulse (jerk) that occurred in 1969. Gubbins and Bloxham (1985) later applied the method of stochastic inversion to map the main field at the MCB. They had to adapt the method to use non-linear as well as linear data (total intensity data depend non-linearly on the model), and to make allowance for crustal components in the observatory data. They used data from epochs 1969.5 and 1980.0. The resulting field models are much more complex than other models (such as IGRF models extrapolated to the core), and show considerable small-scale detail (see Figs 4.23 and 4.24). Their error analysis indicates that most of these features are real. Gubbins and Bloxham (1985) also obtained null flux curves. Core fields based on truncated spherical harmonic series all give similar pictures for the null flux curves, regardless of epoch, with a magnetic equator, one under South America, one under South Africa, and two other small curves in the North Pacific and Arctic (Shure et al., 1983, Fig. 1). The fields obtained by Gubbins and Bloxham are quite different. Figure 4.25 shows their null flux curves for 1980 together with the error corridor for one standard deviation. The magnetic equator now has sharp bends under Indonesia, there is one large null flux curve in the South Atlantic and a second small one to the north. There are four more small curves, most of which they believe to be significant. Current work on the determination of the velocity of the core fluid depends on the assumption that the core behaves as a perfect conductor, so that the

Fig. 4.23. Contour map of the radial field at the M C E for model D8011 'I. epoch 1980. Contour interval is l 0 0 p T : solid contours represent flux into the core, dashed contours flux out of the core. The thicker lines are null flux curves. The projection is Lamben equal area. (After Gubbins and Bloxham. 1985.)

Fig. 4.24. Contour map of the radial field at the M C B for model D69111, epoch 1969 5 (After Gubbins and Bloxham, 1985)

L

0

m

Fig. 4.25. One standard deviation error bounds on the null flux contours for model D80111, epoch 1980 (After Gubbins and Bloxham. 1985 )

The Earth's Magnetic Field

259

field lines remain frozen to the fluid at the core surface. This frozen-flux condition requires that the integrated flux over patches of the core surface bounded by contours of zero radial field remain constant in time. Bloxham and Gubbins (1986) presented a new method for constructing core fields that satisfies these frozen-flux constraints and used it to test the frozen-flux hypothesis by comparing the changes in the flux integrals between 1980.01969.5, 1969.5-1959.5 and 1980.0-1959.5 with the estimated errors. They concluded that the hypothesis can be rejected with 95% confidence. The main evidence for flux diffusion is in the South Atlantic region, where a new null flux curve appears between 1959.5 and 1969.5 and continues to grow at a rapid rate from 1969.5 to 1980.0. However, the statistical result depends critically on error estimates for the field at the MCB, which are difficult to assess with any certainty. Bloxham and Gubbins (1985) have obtained models of the magnetic field at the MCB at selected epochs from 1715.0 to 1980.0 using in all cases the original magnetic field observations. The data were analysed using the method of stochastic inversion (Jackson, 1979; Gubbins and Bloxham, 1985). The radial component of the magnetic field at the MCB is plotted for each of these models in Fig. 4.26. The resolution of small-scale features is remarkable even in the 1715.0 field model. Bloxham and Gubbins (1985b) identified a number of features: (i) static flux bundles (permanent regions of intense flux observed under Arctic Canada, Siberia and Antarctica, the central Pacific Ocean and the Persian Gulf,); (ii) static zero-flux patches (permanent regions of very low flux observed at the North Pole, under Easter Island, in the northern Pacific Ocean, and in many models near the South Pole); (iii) rapidly drifting flux spots (observed in the southern hemisphere from around 90"E, drifting westward towards South America with changes in intensity. Four such spots labelled A-D in the plot for 1980 in Fig. 4.26 can be traced through many epochs); (iv) localized field oscillations (such as that under Indonesia). This picture of the field at the MCB is very different from what has previously been inferred from maps of the surface field. Westward drift occurs only in certain well-defined regions of the core-in the northern Pacific Ocean there is slow eastward drift. It is evident that these models are not consistent with the frozen-flux hypothesis, so that the effects of magnetic diffusion must be taken into consideration in estimating fluid flow in the Earth's core. The intensification of flux through patches under the southern Atlantic Ocean clearly shows that the flux through patches bounded by null flux curves is not conserved. This confirms the earlier tentative conclusion (Bloxham and Gubbins 1986) based on only 20 years of data. The question of whether there is upwelling at the MCB has an important bearing on the thermodynamics of the core, no upwelling indicating that the core is stably stratified near the MCB (Gubbins

Fig. 4.26. Contour plots of the radial field at the M C B for a, 1715.0; 6 . 1377.5; c. 1842 5; d, 1905.5; e, 1969.5; and f, 1980.0 The contour interval is IOOpT, solid contours represent flux into thecore, broken contoursflux out of thecore The bold contours rcpresenrreruradial fleld. (After Gubbinsand Btoxham. 1985 )

Fig. 4.26.-contInoed

Fig. 4.26.-continued

Fig. 4.26.-continued

Fig. 4.26.-continued

Fig. 4.26.-continued

266

The Earth's Core

et al., 1982). Under the no-upwelling hypothesis, flux should be conserved through patches on the MCB bounded by any contour of radial field instead of for just zero contours. Whaler (1980, 1982) found these flux integrals changed less with time than integrals over arbitrary patches on the MCB, lending some support to the no-upwelling hypothesis. However, the validity of this test depends critically on the validity of the frozen flux hypothesis, and her conclusions must now be questioned. Whaler (1984, 1986) later attempted to use geomagnetic data to try and settle the question of whether there is fluid upwelling at the MCB. In her 1984 paper she calculated patch integrals of the radial secular variation at the MCB which should vanish if there is no fluid upwelling. The error bounds on the integrals from the rather small amount of data treated were sufficiently large that none of the integrals was significantly non-zero, but the range of possible values was so large that, like her 1980, 1982 papers, no firm conclusions could be reached. In her 1986 paper she used a different method (Gubbins, 1983) to invert secular variation data measured at the Earth's surface on the assumption that the velocity field at the MCB is purely toroidal. Two different main field models were used to assess the effect of uncertainty in its radial component at the MCB. In both cases purely toroidal motions provided a poor fit, i.e. there must be fluid upwelling and downwelling at the MCB. She does not expect the assumptions of an insulating mantle and perfectly conducting core to change this conclusion. However, no physical significance can be attached to the flows she obtained because of the nonuniqueness of the velocity determined using the frozen-flux form of the radial component of the induction equation. Thus no information can be obtained as to where upwelling and downwelling occurs. Gire et d.(1986) have also estimated core motions derived from secular variation data. They assumed the frozen-flux approximation and that the motion is of low degree with the spectrum decreasing with wavenumber. Some low-degree modes they obtained appear to be stable, i.e. they show little change with different inversion schemes. A feature of the stable flows is the westward drift. However, both poloidal and toroidal components are necessary, which would imply upwelling and downwelling at the core surface. From- 1970 to 1980 the geometry of the computed toroidal motion changes little, although its intensity increases. On the other hand, the geometry of the poloidal motion is different at the two epochs. Gire et a/. stress that they make no claim that the motions they obtained are unique-they are only examples of simple regular motions that could generate the observed secular variation.

The Earth’s Magnetic Field

267

4.8 Variations in the Length of the Day

As a result of the attraction of the Sun and Moon on the Earth’s equatorial

bulge and of movements of mass within the Earth, the angular velocity of the Earth is not constant. There are in fact fluctuations, not only in the rate of spin (i.e. changes in the length of the day, l.o.d.), but also in the direction of the axis of rotation, i.e. the Earth “wobbles”. There are a number of peaks in the frequency spectrum of the Earth’s rate of spin, covering a very long timescale. These peaks are believed to arise from different causes: Three distinct components have been recognized-a steady increase in the 1.o.d. by about 2x s a century, seasonal fluctuations of about l o p 3s, and less regular variations up to about 5 x l o p 3s having time-scales of the order of years (the so-called decade fluctuations). The seasonal variations in the 1.o.d. are chiefly the result of torques on the mantle exerted by oceanic currents and atmospheric winds. The rapid irregular variations in the 1.o.d.over a decade, however, cannot be explained by surface phenomena. No transport of mass at the surface could alter the Earth’s moment of inertia by a sufficient amount to account for such large changes as are observed. It has been suggested that these “decade” fluctuations are caused by the transfer of angular momentum between the Earth’s solid mantle and liquid core. This in turn implies some form of core-mantle coupling. There are a number of ways in which angular momentum could be transferred between the core and the mantle; the principal possible mechanisms are inertial, electromagnetic and topographic coupling. Inertial coupling could arise from hydrodynamic pressure forces acting over the ellipsoidal MCB when internal flow is induced in the liquid core by a shift in the earth’s rotation axis (Toomre, 1966). Electromagnetic coupling could arise from leakage of the secular variation into the electrically conducting lower mantle (Rochester, 1960, 1968). It is difficult, however, to make quantitative estimates of the horizontal stresses at the MCB. Calculations indicate that neither viscous coupling nor electromagnetic coupling is really adequate to account for the decade fluctuations (see e.g. Rochester, 1970, 1973). Hide (1969) has suggested the possibility of topographic coupling as a result of irregular features (bumps) at the MCB (see $4.7). It would be interesting to try to detect any small departures in uniformity in the rate of rotation of other planets such as Mars and Venus and of planetary satellites with rigid surfaces, since any such departures might yield information on a possible liquid convecting core and on planetary magnetic fields. Changes over geological time are predominantly a constant deceleration as a result of tidal friction. It is difficult to estimate the rate at which this deceleration has taken place, since our knowledge of palaeogeography is scant and the result depends critically on a few shallow seas. Urey (1952) has

268

The Earth‘s Core

suggested, on the other hand, that, because of differentiation of the materials of the Earth and the growth of the core, the moment of inertia of the Earth about its axis of rotation may have been reduced and as a result the 1.o.d.decreased from about 30 to 24 h. The changes caused by a growing core are considerably smaller (and of opposite sign) to those due to tidal friction (Runcorn, 1964,1970a). In both the atmosphere and the oceans the role of rotation is of fundamental importance. This is also true-with some qualifications-for the Earth’s fluid core. With regard to rotational effects, we must distinguish between the core and oceanic-atmospheric layers in the following manner. The core should only be divided up into thin layers if it is stably stratified (see $3.6). In such a case dynamo action, though still possible, would be significantly constrained to a predominantly two-dimensional motion in concentric spherical shells. If, on the other hand, the core is not stratified, any perturbation of the otherwise steady rotation would lead to a predominantly two-dimensional motion in planes perpendicular to ‘the rotation axis as predicted by the Proudman-Taylor theorem. In both cases these statements must be modified to include the effects of the Lorentz force. In certain special cases the effects of the Lorentz force can be included without too much difficulty. For example, we can consider the possibility of interpreting the westward drift of the non-dipole field as the propagation of hydromagnetic waves in the fluid core around the axis of rotation in the presence of a dominant toroidal magnetic field (Hide, 1966). If the fluid is not stably stratified, the Lorentz force enters into the equations of motion for hydromagnetic wave solutions in the same manner as the Coriolis force. In fact, the flow in this case is described by the PoincarC equation-the same equation that describes the flow in the absence of a magnetic field, But even in this rather simple example there is an added difficulty. We are looking for solutions to a hyperbolic differential equation with certain specified boundary conditions. This is an ill-posed mathematical problem and there are serious difficulties in attempting to find a solution. A major source of the difficulty is the presence of an inner boundary to the fluid, viz. the solid IC. It has been suggested that a change in the dipole field could be the cause of the irregularities in the 1.o.d. (e.g. Braginskii, 1970; Yukutake, 1972). Yukutake (1973a) investigated the possibility that the observed changes in the dipole field with periods of 8000,400 and 65 years were sufficient to produce the observed fluctuations in the 1.o.d. for these periods. He used archaeomagnetic data, observations of the variation of the Moon’s longitude and observatory data. He found that such a relationship does exist, but that it is highly dependent on the period, the magnitude of the change increasing as the period decreases. He was able to account for the dependence of the excitation of the change in rotation rate upon period by electromagnetic coupling

The Earths Magnetic Field

269

between the core and mantle, provided the electrical conductivity of the lower mantle is as large as lo3 S/m-', that toroidal fields were generated only at the MCB by the changing dipole, and that the sources of the change in the dipole field lay near the surface of the core. His model was very simple, the mantle consisting of two spherical shells with different conductivity. He also assumed uniform rotation of the OC. In a later paper, Watanabe and Yukutake ( I 975) refined the model, taking into account differential rotation of the OC and a multi-layered mantle. They obtained essentially the same results as Yukutake (1973a) provided that the dipole change in the core is limiied to a surface layer less than a few hundred kilometres thick and that the electrical conductivity of the lower 500km of the mantle is -102-103Sm-'. The mean motion of the surface layers of the core is compatible with the observed variations in the rate of the westward drift. From ancient observations of eclipses, the Earth's rotation is known to have been accelerated during the past few thousand years in addition to the steady retardation due to tidal friction. Archeomagnetic studies have shown that the dipole field has also been changing, approximately periodically, with a large amplitude ( N 50% of the present dipole moment) and with a period -8000 years (Bucha, 1967, 1970; Cox, 1968; Kitazawa, 1970). Since the last maximum of the dipole moment (sometime between 0 and AD 500), the electromagnetic coupling between the mantle and core has been diminishing and acceleration of the Earth's rotation is thus to be expected during the past 2000 years. Observation confirms the theoretical prediction (Yukutake, 1972) of a phase difference of about 71 between rotation and dipole moment change, for periods 8000 years. Yukutake (1971) has also shown that the magnitude of the gauss coefficient gy increased during the seventeenth and eighteenth centuries and then began to decrease from the early nineteenth century-this variation being superposed on the general trend discussed above (the gradual decrease since about 2000 years ago). During this period there was also a large fluctuation in the observed longitude of the Moon which has been ascribed to a change in the Earth's rate of rotation (see Fig. 4.27). The curve showing fluctuations in the length of the year leads the dipole curve by about 742 for these periods ( N 400 years). The non-tidal variations in the 1.o.d. must conserve the total angular momentum of the Earth: for variations of the order of a decade, the core of the Earth appears to be the only reasonable source. Yukutake (1973a, see Fig. 4.28) found a variation in the dipole field in phase with length of the year fluctuations ( N65 year period). Variations in the westward drift of the secular variation, however, are very different from those obtained earlier by Vestine (1953). Figure 4.29 (by Vestine and co-workers) shows a comparison between the motion of the eccentric dipole and the deviation in the 1.o.d. Vestine and

-

270

The Earth's Core

-

3 -0.32 U m

c 0

$

0

-20

-

I

I

I

Fig. 4.27. Fluctuations in various features of the geomagnetic field and the Earth's rotation. From the top, fluctuations in the gauss coefficient & (the dipole term), the Moon's longitude (curve 6). and the length of the year. (After Yukutake, 1973a.)

Kahle (1 968) have interpreted these curves as showing that the angular velocity of the eccentric dipole is related to changes in the angular momentum of the outer 200 km of the core. Figure 4.29 also shows that there is an apparent phase lag between variations in the rate of rotation of the mantle and in the velocity of the eccentric dipole. Ball et al. (1969) suggested that this is the effect of diffusion of the magnetic signal through the conducting mantle. Yukutake (1973b), however, has questioned whether the movement of the eccentric dipole really represents that of the geomagnetic field as a whole. He showed that the westward movement of the eccentric dipole over the last 150 years is determined almost entirely by the westward drift arising from only one term ( n = 2, m = 1) of the geomagnetic potential. Cain et al. (1985) recalculated the rate of westward drift of the eccentric geomagnetic dipole since 1900 and confirmed its correlation with irregularities in the 1.o.d. as originally reported by Vestine and co-workers. They also confirmed a slower rate of westward drift for the past decades first shown by Kahle et al. (1 969) from an analysis of POGO data. However, the time lag of the magnetic response noted by Vestine for the fluctuations near the beginning of the century was not confirmed. On the other hand Cain et al. found

-

271

The Earth's Magnetic Field

-

1

Eccentric dipole dipole Eccentric

...*'-.,.. ,. . ...). 0.

I

I

.*-

'

......,, 0..

--0.4 .rue.

'.

0%.

-

*p*-a2

-

b

. p Q,

>.

U

-0 I

I

I

Fig. 4.28. Comparison of fluctuations in the length of the year with those of various features of the geomagnetic field. From the top: variations in the dipole term (gd)-curve A from an analysis of 21 observatories, curve B from an analysis of 6 observatories; fluctuations in the length of the year; and the drift rate of the eccentric dipole. (After Yukutake. 1973a.)

that the data for the drift of the eccentric dipole did not support magnetic leading of the 1.o.d. changes as suggested by the work of Le Moue1 and Courtillot (1982). The close correlation between fluctuation in the 1.o.d. and the westward drift of the eccentric dipole as noted by Vestine and Kahle (1968) was not found by Jin (1974). Jin and Thomas (1977) carried out a power spectral density analysis using Burg's maximum entropy method (MEM) of the geomagnetic dipole field and its rate of change for the years 1901-1969. Both spectra indicate relative maxima at 0.015 cycle/year and its harmonics (see Figs 4.30,4.31). These maxima correspond approximately to 66-, 33-, 22-, 17-, 13-, 1 1-, and 9-year spectral lines. The application of the same analysis techniques to the 1.o.d.fluctuations for the period 1865-1961 reveals similar spectral characteristics (see Fig. 4.32). The existence of the common spectral peaks with periods of 66 and 33 years in the 1.o.d. fluctuations and the geomagnetic dipole field is clearly established, although the existence of the higher harmonics with periods of 22, 17, 13, 11, and 9 years is somewhat uncertain because of the line-splitting problem in the MEM spectral analysis.

272

The Earths Core

h

L

. 0

aJ

x

03

$ 0.4r)

v

.-5 0.3 E a, 0 0.2 0 . .U U

.r 0.1 L

-

4.0

c

d Y O

1880

1

1

1

'

"

1900

1

'

"

1

'

'

1920

1

1

'

1

L'i

- 0.006

50-

0.008

1940

1960 AD

Year Fig. 4.29. Comparison of astronomical and geomagnetic data on the speed of the Earth's rotation The solid line represents the westward motion of the eccentric dipole relative to the mantle, the dashed line the eastward angular velocity of the mantle (with scales appropriate to deviations from both standard angular velocity and standard length of day (After Ball e f a / , 1969 )

Jin and Thomas point out that the 22- and 11-year spectral lines cannot be attributed to the solar magnetic cycle and the solar cycle unambiguously, because they are the higher harmonics of the 66-year period. They suggest that the spectral line similarity in the 1.o.d. fluctuations and the dipole field variations is related to motions within the Earth's fluid core during the past 100 years. Braginskii (1970a, 1980) showed that torsional oscillations can exist in the Earth's core with eigenperiods of about 60, 30 and 20 years. Golovkov and Kolomiitzeva (1971) suggested that the 60-year variation was caused by a localized pulse of the radial field on the MCB. This would excite torsional oscillations, and also 1.o.d. variations, because of the action of the variable magnetic field on the conducting mantle. It should be noted, however, that variations produced by torsional oscillations are global, while the pulse mechanism is local both in space and time. In Braginskii's (1976) model Z dynamo, large gradients of the main magnetic field and velocity are present near the MCB that could lead to local hydromagnetic instabilities. Braginskii later (1984) suggested an alternative mechanism for generating motions near the MCB that are needed in addition to torsional oscillations to account for the secular variation-wave motions in a thin layer H of light fluid at the MCB. He estimated the thickness of the layer to be 20 km but with a density difference of only g cm-3, so that it could not be detected by seismic methods.

-

1

2

273

The Earths Magnetic Field

9.0

8.0

L

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Fig. 4.30 Power spectra of the geomagnetic dipole field for the period 1901-1969 The number ot PEF coefficients used is 68, and the number of spectral estimates is 300 The unit for power spectral den sity is (nanotesla)’ (After Jin and Thomas, 1977 )

Anufriyev and Braginskii (1977) showed that small bumps on the MCB have little effect on fluid motions in the core (see Hide and Malin (1970) and 54.7). However, distortion of the flow by a bump in the H layer could be sufficient for it to influence the secular variation. Braginskii (1984) suggested that the interaction of torsional oscillations with such bumps could excite motions in the H layer with the frequencies of the oscillations and thus contribute to the

274

The Earth’s Core

c

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I

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Fig. 4.31. Power spectra of the time rate of change of the geomagnetic dipole field for the period 1905-1 965. The number of PEF coefficients used is 61, and the number of spectral estimates is 300 The unit for power spectral density is (nanotesla/year)2. (After Jin and Thomas, 1977.)

secular variation. Braginskii further suggested that excursions (see $4.5) of the magnetic field could be thought of as “storms” in the “ocean” of light fluid in the H layer. Large perturbations could thus be generated near the MCB without any substantial change in the main body of the core. These large variations in the H layer would be accompanied by significant 1.o.d. variations.

275

The Earth’s Magnetic Field

2.E

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Fig. 4.32. Power spectra of the time rate of change ofA7 for the period 1865-1965. The number of PEF coefficients used is 60, and the number of spectral estimates is 300. The unit for power spectral density is (seconds/year)2. (After Jin and Thomas, 1977.)

If the core were a perfect conductor, both the changing field at the Earth’s surface and the electric currents induced in the mantle would be determined completely by the rearrangement of the lines of force emerging from the core. To each flow at the core surface consistent with the observed geomagnetic field, a corresponding electromagnetic couple on the mantle could be com-

276

The Earth's Core

-21

I

I

I

1880

1900

,

I

1920 Year

I

1940

I

,

1960

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1980AD

Fig. 4.33.Decade fluctuations in the length of the day (After Rochester, 1984 )

puted. Stix and Roberts (1984) showed how this could be carried out on the geophysically reasonable hypothesis that the electromagnetic time-constant of the mantle is short compared with the time-scale of the secular variation. They initially associated toroidal fields in the mantle with motions at the core surface that create the observed geomagnetic field by flux rearrangement and computed the resulting couple parallel to the geographic axis. Using only zonal core motions and a depth-dependent electrical conductivity of the form CT = C T ~(r/a)-' with CT, = 3 x lo3 S m p l , LY = 30, they found that the toroidal induced fields create a couple rTthat over most of this century has been roughly ten times greater than the poloidal part rs and has the same sign. The total couple r has fluctuations of the order 10'' N m as required to explain the observed decade fluctuations in the 1.o.d. Its average value, however, is - 1.5 x lo'* N m-too large to remain unbalanced. Stix and Roberts suggest that a couple in the opposite sense is created by flux leakage from the core, thus reducing the value of the mean electromagnetic couple. Morrison (1979) has estimated that the maximum torque required to act across the MCB to account for the decade fluctuations in the 1.o.d. is 10l8N m. This corresponds to an average tangential stress of 2 x N m-' on the MCB. Changes in torque of this order must take place in a decade in order to account for the change in the 1.o.d. that occurred around 1900 (see Fig. 4.33). Rochester (1984) has reviewed the various possible mechanisms for core-mantle coupling. It is difficult to assess the role of viscous friction mainly because of the uncertainty in the value of the kinematic m2 s viscosity of the core. However, even with the highest estimate of (Bukowinski and Knopoff, 1976), viscous coupling seems to be inadequate. Electromagnetic coupling arises as the lower mantle (a semiconductor at the temperatures there) is penetrated by changing magnetic fields from the liquid core. Such coupling is both direct (because of leakage of the secular

The Earth’s Magnetic Field

277

variation into the lower mantle) and passive (because of the generation of toroidal magnetic field as the O C accelerates past the mantle). The critical coupling parameter is the radial distribution of electrical conductivity cr in the conducting part (the lower 2000 km) of the mantle. Estimates of cr are based on the secular variation spectrum observed at the Earth’s surface after being filtered by diffusion through the conducting lower mantle. The inversion is not unique, however, because little is known about the input spectrum at the MCB. Recent estimates of cr have used the 1969-1970 ‘?jerk” in the magnetic field. Achache et al. (1980) obtained an upper bound on the average value (Q) of 150 S m a more detailed analysis by Backus (1983) led to a value of 400 S m - I . These later estimates allow significantly tighter coupling than earlier estimates of the 1960s which favoured ( Q ) N 100s m-’. If Backus’ estimate of the lower mantle conductivity distribution is accepted, an unbalanced electromagnetic torque of N 10’‘ N m can be achieved within a decade by the secular variation. Hide (1977) suggested that irregularities in topography at the MCB could provide the necessary coupling: as the core fluid flows past such bumps, dynamic pressure would produce a tangential stress. Hassan and Eltayeb (1982) estimated that bumps a few kilometres in height could produce a mean tangential stress of the order of N m-’, provided that below the MCB the ratio of toroidal to poloidal field strength does not greatly exceed unity. It is not easy, however, to estimate the drag coefficient and it is not clear that such stresses would integrate to give a torque of the magnitude needed to account for the decade fluctuations in the 1.o.d. A step change in the second time derivative (a “jerk”) in the geomagnetic field in about 1969 was first reported by Courtillot et al. (1978). After decreasing during most of this century, the energy of the secular variation held and the rate of westward drift began increasing in 1969. Ducruix et al. (1980) later used annual mean values from observatories all over the world for the period 1947-1977 and found convincing evidence for a large secular variation impulse to have occurred in the late 1960s. They were able to isolate this from the external field due to variations in the solar cycle. They concluded that the electrical conductivity of the mantle cannot exceed 100 S m-’. Courtillot et al. (1984) later concluded that the electrical conductivity is likely to be less than 300 S m - ’ in more than 97% of the volume of the mantle. This is still orders of magnitude smaller than previous estimates for the conductivity of the lower mantle. Malin and Hodder (1982) carried out a spherical harmonic analysis of the 1961-1978 records from 83 observatories and confirmed that most of the jerk is of internal origin, the mean-squared value of the internal part exceeding that of the external part by a factor of 3.4 k 0.8. However, Alldredge (1984) is not convinced; he maintains that the method of analysis of piecemeal fitting of parabolas to the data will always create a discontinuity in

’;

278

The Earths Core

the secular acceleration where the parabolas join and that the place where they join is an a priori assumption and not a result of the analysis. Courtillot and Le Moue1 (1984) have replied to Alldredge’s criticisms and later (1985) reiterated their belief that their analysis does show that the impulse was quite short (1-2 years), occurred around 1969.5, is entirely internal in origin, and is of worldwide extent. However, Alldredge (1985a,b) is still not convinced and maintains that the method of analysis used by proponents of the jerk “will yield a jerk at any preselected date, and, if this condition is imposed on all observatories, then the effect will appear to be a worldwide instantaneous effect”. His point is that any piecemeal polynomial approximation of degree n will lead to discontinuities in the nth derivative of the function. McLeod (1985) also questioned Alldredge’s conclusions and showed that internal sources can give rise to changes in the secular acceleration on timescales as short as 1 or 2 years for some field components at some geographical positions. He estimated the average centroid of the jerk to be 1969.5. However, the centroid of the jerk differs for different spherical harmonic degrees. The centroid is significantly later (possibly by as much as 4 years) for the degree-one spherical harmonic than for the next three spherical harmonic degrees which all occur around 1969. McLeod calculated the spherical harmonic coefficients of the change of secular accelerations that occurred around 1969 using field model GSFC (9/80) of Langel and Estes (1982). He estimated that the Malin and Hodder (1982) internal Coefficients contain about 25% mean-squared error due to “noise” from external current systems not related to the jerk. Although this is fairly large, McLeod believes that it does not invalidate any of their conclusions. McLeod’s coefficients are in good qualitative agreement with those of Gubbins (1984) which are, however, systematically lower. Malin et al. (1983) attempted to quantify the jerk, assuming it to be of internal origin, by calculating its mean-square value over the Earth’s surface. Kerridge and Barraclough (1985) showed that the mean-square value of the jerk field is not a reliable indicator of the existence of a jerk when considered alone, since noise contributes to it additively, giving rise to “apparent” jerks. However, contour maps derived from global models show where the power in a jerk is concentrated and corroboration can be sought from observatory data. They thus claim that real and spurious jerks can be distinguished. Kerridge and Barraclough carried out their analysis at 2-year intervals from 1931.5 to 1971.5, making no assumptions about either the existence or timing of a jerk, and found only one jerk-that at 1970. Gire et al. (1984) have since suggested that a second jerk took place in 1912, when there was a sudden change from an increase to a decrease in the energy of the secular variation field and in the westward drift rate.

The Earth's Magnetic Field

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Courtillot et al. (1978) noted that periods of intense secular acceleration changes occurred around about the time of extrema in 1.o.d. variations. Le M o d 1 et al. (1981) found a correlation between the secular variation of D in Europe and 1.o.d. over the past 120 years, the 1.o.d. variations lagging the geomagnetic variations by 1&15 years. They further suggested that secular variation impulses near 1900(?), 1913 and 1969 should belinked with extrema of the 1.o.d. curve near 1910, 1930 and 1980(?)-increased (decreased) westward drift being followed by increased (decreased) rotational velocity of the Earth. This correlation is opposite to that proposed by Vestine (1952), Runcorn (1982) and Backus (1983). If we accept the correlation suggested by Backus and Vestine, the fact that changes in westward drift rate follow 1.o.d. changes and are of the same sign can be understood if we assume convective mass-exchange between a thin rigid shell, forming the top of the core, and the rest of the liquid core below it. This model is similar to that originally proposed by Bullard (1948) and later extended by Rochester (1960). The radial mass-exchange exerts a torque on the OC and (in the case of the 1956 1.o.d. event studied by Backus) decelerates the mantle. The changes in rotational velocity of the mantle would immediately show as 1.o.d. changes (a secondorder impulse in mantle angular velocity in 1956), and be followed by geomagnetic changes, observed with a 13-year mean delay due to diffusion of westward drift related signals through a rather strongly conducting mantle (Backus, 1983). If, on the other hand, we accept the correlation found by Le Mouel et al. (198I), we have to explain the discrepancy between the short time-scale over which the geomagnetic impulse is established (1 year) and the long delay after which the rotation of the mantle changes (10-15 years). Le Mouel and Courtillot (1981, 1982) have suggested a model that, although it gives a reasonable account of the observations, does not consider a quantitative treatment of magnetic field transport. They interpret secular acceleration impulses of the geomagnetic field as secondary motions in the outer layers of the core. They estimate that such motions have dimensions of the order of 5000 km and velocities of 3 x m s- and occur in less than a year. The correlation with minima in the Earth's rotation rate (around 1840, 1905 and 1970) is attributed to electromagnetic coupling of an OC layer (lOCb-200km thick) to the rest of the core and to the weakly conducting mantle. The radial velocity of the fluid at the base of the layer is of the order of 1 0 - 6 m s- ' , about two orders of magnitude less than the horizontal velocity in the layer. Courtillot et al. (1982) have also suggested that decade fluctuations in climate and 1.o.d. are both correlated with changes in the Earth's magnetic field (see Fig. 4.34), implying a possible long-term influence of core motions on climate. Courtillot and Le Mouel (1984) have reviewed the published work on geomagnetic secular variation impulses.

280

The Earth‘s Core

C

4-

-12

m >.

‘Eu ‘9

. U

0.5 0.4 0.3

v

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_-

_1860AD

;

(0, 1880

I-

.-

l ’ q w\ v- fi 1900

I

I

1920

1940

1960

0.2 01 1940

Fig. 4.34. Plot of the secular variation of geomagnetic declination D (solid curve), of the excess length of day d ( A T ) / d t (dashed curve) and of successive 5-year averages of the temperature over the whole Earth (expressed as departures from the means for 1880-84. after Lamb (1977). dotted curve) (After Courtillot e t a / , 1 9 8 2 )

In addition to fluctuations in the Earth’s rate of spin, there are also variations in the direction of the axis of rotation, a “wobble”. In 1891, Chandler isolated a component with a 14-month period and r.m.s. amplitude ~ 0 . 1 arc 5 seconds (since called the Chandler wobble). For a rigid Earth it can be shown that the period of its free nutation is about 10 months; the effect of the Earth’s deformation is thus to increase the period by approximately 40%. Smith and Dahlen (1981) have shown that the discrepancy between the observed Chandler period and that for a rigid Earth can be reduced to about 8 days by the combined effects of the fluid core (largely decoupling the core from the mantle and thus reducing the period), mantle elasticity, and a non-global ocean. Smith and Dahien further suggested that the remaining 8-day discrepancy arises from dispersion, the decrease in mantle rigidity with increasing period due to anelasticity. Observations show that the Chandler wobble is damped, so that it must be constantly re-excited. Rochester and Smylie (1965) showed that electromagnetic core-mantle coupling fails by four orders of magnitude to provide the necessary damping. The excitation mechanism of the Chandler wobble has been the subject of much debate. Changes in the annual variation of the mass distribution of the atmosphere fail by about an order of magnitude (Munk and Hassan, 1961). However, Wahr (1982, 1983) has re-examined the meteorological and oceanographic contributions to the wobble from 1900 to 1973 and concluded that on average 20-25% of the required power is avail-

The Earth’s Magnetic Field

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able from the redistribution of atmospheric masses and the accompanying pressure-driven ocean loading of the Earth. A more recent analysis by Barnes et al. (1983) and Hide (1984) is even more optimistic: they suggest that the observed polar motion might be accounted for without the need for any significant contributions from non-meteorological processes. A possible connection with seismic activity has often been proposed. Estimates of the contribution of earthquakes to the excitation of the Chandler wobble have until recently been several orders of magnitude too small, mainly because the displacement fields of even the largest earthquakes were thought to extend no more than a few hundred kilometres from the focus. Following the work of Press (1965), which indicated that for great earthquakes a measurable displacement field may extend several thousand kilometres from the epicentre, Mansinha and Smylie (1967, 1968), using dislocation theory, estimated the changes in the products of inertia of the Earth arising from several large faults associated with major earthquakes. They found that the cumulative effect (based on earthquake statistics) could account for both the excitation of the Chandler wobble and a slow secular shift of the mean pole of rotation. However, Mansinha and Smylie (1970) and Dahlen (1971) disagree on whether the cumulative effect of all earthquakes is enough to sustain the Chandler wobble. Independent formulations of the theory for a model of a self-gravitating Earth with a liquid core and realistic distributions of density and elastic properties in the mantle have been given by Smylie and Mansinha (1971), Dahlen (1971, 1973, 1974), Israel et al. (1973), Saito (1974) and Crossley and Gubbins (1975). Their theoretical treatments differ in detail and have given rise to some controversy over the physical principles governing static deformation of the liquid core. The subject has been well reviewed by Mansinha et al. (1979). The effect of different boundary conditions at the MCB is likely to be small, and the authors are in general agreement that a major earthquake can produce a polar shift of the order of 0.1 arc seconds. However, very large earthquakes are sufficiently infrequent that their cumulative effect in maintaining the Chandler wobble against damping seems inadequate, although Mansinha et al. (1 979) believe that earthquakes do indeed provide the mechanism. O’Connell and Dziewonski (1976) estimated the cumulative seismic excitation for 234 large earthquakes between 1901 and 1970. Their synthetic Chandler wobble reproduced reasonably well the variations observed during that period, lending support to the idea that earthquakes represent the major factor in the excitation of the Chandler wobble. Kanamori (1977) agreed that there is a correlation between the amplitude of the Chandler wobble and the energy released in great earthquakes, but argued (1976) that O’Connell and Dziewonski had over-estimated the magnitude of the seismic moments. Souriau and Cazenave (1985) re-examined the problem using more recent

282

The Earth’s Core

high-quality data on the determination of the seismic moments of great earthquakes and polar motion data obtained from space techniques. Using 1287 strong and moderate earthquakes that occurred between 1977 and 1983, they found that seismic excitation is far too small (by about two orders of magnitude) to explain the amplitude of the Chandler wobble, re-affirming Kanamori’s (1976) conclusions that in the past the magnitudes of the seismic moments of major earthquakes have been over-estimated. Although Rochester and Smylie (1965) showed that electromagnetic coupling at the MCB is completely inadequate to excite the Chandler wobble, Runcorn (1970b, 1982) believes that high-frequency components of the secular variation at the MCB are the core equivalent of sunspots and can supply an impulsive torque to the mantle that can transfer angular momentum rapidly enough to sustain the Chandler wobble. Earthquakes leave the instantaneous rotation pole unchanged but shift the axis of figure, so that the pole path experiences a discontinuous change in direction. Impulsive torques, on the other hand, leave the axis of figure unchanged and shift the rotation pole so that the radius of the pole path is changed discontinuously. The details of electromagnetic coupling on such a short time-scale have not been fully worked out, partly because high-frequency components of the secular variation are screened from observation by the electrical conductivity of the lower mantle that provides the coupling. Kakuta (1965) concluded that magnetohydrodynamic oscillations in the core could not excite a detectable wobble. The main difficulty of Runcorn’s suggestion is that excitation of the Chandler wobble by impulsive torques requires that most of the power in these torques be at periods much shorter than the resonant period (1.2 years). Lambeck and Hopgood (1982) have shown that the spectrum of changes in the 1.o.d. from 1958-1980 has significant power only at periods over 5 years once the excitation by zonal winds has been removed. Longer-period wobbles have also been suggested. Markowitz (1970) has adduced empirical evidence for a 24-year-period wobble, which Busse (1970a,b) has suggested may represent the response of the mantle to a wobble of the solid IC inertially coupled to the mantle via the liquid OC. Rykhlova (1969), using a longer but less homogeneous record, has found evidence for a 40-year period. McCarthy (1972) also found, from latitude observations at Washington, a “period” somewhat longer than Markowitz’s. If it is real, the phenomenon may well be the only observable manifestation in the entire spectrum of changes in the Earth’s rotation of the presence of the solid IC. References

Achache, J., Courtillot, V., Ducruix, J. and Le Mouel, J-L. (1980). The late 1960s secular varia-

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tion impulse: further constraints on deep mantle conductivity. Phys. Earth Planet. In/. 23, 72. Acheson, D. (1975). O n the theory of the geomagnetic westward drift and rapid fluctuations in the Earth’s rotation rate. Pure Appl. Geophys. 113,611. Acheson, D. J. and Hide, R. (1973). Hydromagnetics of rotating fluids. Rep. Prog. Phys. ( U K ) 36, 159. Alldredge, L. R. (1977). Geomagnetic variations with periods from 13 to 30 years. J . Geomag. Geoelect. 29, 123. Alldredge, L. R. (1984). A discussion of impulses and jerks in the geomagnetic field. J . Geophys. Res. 89,4403. Alldredge, L. R. (1985a). Reply to Comments on “A discussion of impulses and jerks in the geomagnetic field” by V. Courtillot and J. L. Mouel. J . Geophys. Res. 90,6899. Alldredge, L. R. (1985b). More on the alleged 1970 geomagnetic jerk. Phys. Earth Planet. Int. 39, 255. Alldredge, L. R. and Hurwitz, L. (1964). Radial dipoles as the sources of the Earth’s main magnetic field. J . Geophys. Res. 69,2631. Alldredge, L. R. and Steams, C. 0. (1969). Dipole model of the sources of the Earth’s magnetic field and secular change. J . Geophys. Res. 74,6583. Alldredge, L. R., Van Voorhis, G. D. and Davids, T. M. (1963). A magnetic profile around the world. J . Geophys. Res. 68,3679. Anufriyev, A. P. and Braginskii, S. I. (1977). Effect of irregularities of the boundary of the Earth’s core on the speed of the fluid and on the magnetic field 111. Geomayn. Aeronom. XVII, 492. Backus, G. E. (1957). The ax[-symmetric self-excited fluid dynamo. Astrophys. J . 125, 500. Backus, G. E. (1958). A class of self-sustaining dissipative spherical dynamos. Ann. Phys. 4, 372. Backus, G . E. (1968). Kinematics of geomagnetic secular variation in a perfectly conducting core. Phil. Trans. R o y . Soc. London A263,239. Backus, G. E. (1975). Gross thermodynamics of heat engines in deep interior or Earth. Proc. Nut. Acad. Sci. U S A 72, 1555. Backus, G. E. (1983). Application of mantle filter theory to the magneticjerk of 1969. Geophys. J . 74, 713. Backus, G. E. and Chandrasekhar, S. (1956). O n Cowling’s theorem on the impossibility of selfmaintained axi-symmetric homogeneous dynamos. Proc. Nut. Acad. Sci. U S A 42, 105. Ball, R. H., Kahle, A. B. and Vestine, E. H. (1969). Determination of surface motions of the Earth’s core. J . Geophys. Res. 74,3659. Baranova, T. N., Ben’kova, N. P. and Freyzon, A. A. (1973). Correlation of geomagnetic gravity fields. Geomagn. Aeronom. XIII, 644. Barnes, R. T. H., Hide, R., White, A. A. and Wilson, C. A. (1983). Atmospheric angular moment u n fluctuations, length of day changes, and polar motion. Proc. Roy. Soc. London A387.31. Ben’kova, N. P., Cherevko, T. N. and Adam, N. V. (1972). Izmiran, Preprint No. 35. Benton, E. R. (1979a). O n fluid circulation around null-flux curves at Earth’s core-mantle boundary. Geophys. Astrophys. Fluid Dyn. 11,323. Benton, E. R. (1979b). Magnetic probing of planetary interiors. Phys. Earth Planet. Int. 20, 1 1 1 . Benton, E. R. (1981). A simple method for determining the vertical growth rate of vertical motion at the top of Earth’s outer core. Phys. Earth Planet. Int. 24,242. Benton, E. R. and Muth, L. A. (1979). O n the strength of electric currents and zonal magnetic fields at the top of the Earth’s core: methodology and preliminary estimates. Phys. Earth Planet. Int. 20, 127. Blakely, R. J. (1974). Geomagnetic reversals and crustal spreading rates during the Miocene. J . Geophys. Res. 79,2919.

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Opdyke, N. D., Glass, B., Hays, J. D. and Foster, J. (1966). Paleomagnetic study of Antarctic deep-sea cores. Science 154, 349. Orlov, V. P. (1965). The leading trends of the secular variation investigation. J . Geomagn. Geoelect. 17,277. Parker, E. N. (1955). Hydromagnetic dynamo models. Astrophys. J . 122,293. Parker, E. N. (1969). The occasional reversal of the geomagnetic field. Astrophys. J . 158,815. Parker, E. N. (1970). The generation of magnetic fields in astrophysical bodies I. The dynamo equations. Asrrophys. J . 162,665. Parker, E. N. (1971). The generation of magnetic fields in astrophysical bodies IV. The solar and terrestrial dynamos. Astrophys. J . 164,491. Parker, E. N. (1979). C o m i c a l Magnetic Fields, their Origin and their Activity. Clarendon Press, Oxford. Parker, R. L. (1972). Inverse theory with grossly inadequate data. Geophys. J . 29, 123. Peddie, N. W. and Fabiano, E. B. (1976). A model of the geomagnetic field for 1975. J . Geophys. Res. 81, 2539. Phillips, J. D. (1977). Time variation and asymmetry in the statistics of geomagnetic reversal sequences. J . Geophys. Res. 82,835. Phillips, J. D., Blakely, R. J. and Cox, A. (1975). Independence of geomagnetic polarity intervals. Geophys. J . 43,747. Pitt, G. D. and Tozer, D. C. (1970a). Optical absorption measurements on natural and synthetic ferromagnesian minerals subjected to high pressures. Phys. Earth Planet. Int. 2, 179. Pitt, G. D. and Tozer, D. C. (1970b). Radiative heat transfer in dense media and its magnitude in olivines and some other ferromagnesian minerals under typical upper mantle conditions. Phys. Earth Planet Int. 2, 189. Press, F. (1965). Displacements, strains and tilts at teleseismic distances. J . Geophys. Res. 70, 2395. 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. J . Geophys. Res. 90, 10417. Proctor, M. R. E. (1977). On Backus’ necessary condition for dynamo action in a conducting sphere. Geophys. Astrophys. Fluid Dyn. 9,89. Rickard, J . A. (1973). Free oscillations of a rotating fluid contained between two spheroidal surfaces. Geophys. Fluid Dyn. 5, 369. Rikitake, T. (1958). Oscillations of a system of disk dynamos. Proc. Camh. Phil. Soc. 54,89. Rikitake, T. (1967). Non-dipole field and fluid motions in the Earth’s core. J . Geomayn. Geoelecr. 19, 129. Rikitake, T. and Hagiwara, Y. (1966). Non-steady state of a Herzenberg dynamo. J . Geomagn. Geoelect. 18,393. Roberts, G. 0. (1970). Spatially periodic dynamos. Phil. Trans. Roy. Soc. London A266,535. Roberts, G. 0. (1972). Dynamo action of fluid motions with two-dimensional periodicity. Phil. Trans. R o y . Soc. London A271,411. Roberts, P. H. (1967a). A n Introduction t o Maynetohydrodynamics. Longman, London. Roberts, P. H. (1967b). The Dynamo Problem. Woods Hole Ocean. Inst. Rept. 67-54, Vol. 1, pp. 51 and 178. Roberts, P. H. (1971). Dynamo theory of geomagnetism. In World Magnetic Survey IY57-196Y (A. J. Zmuda, ed.). IAGA Bull, 28, IUGG Publ. Off., Paris. Roberts, P. H. (1972). Kinematic dynamo models. Phil. Trans. Roy. Sac. London A272,663. Roberts, P. H. (1978). Diffusive instabilities in magnetohydrodynamic convection. Les instabilites hydrodynamiques en convection libre, forces et mixte. Lecture N o f e s in Physics (J. C . Legras and J. K . Platten eds), Vol. 72.

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Chapter Five

The Constitution of the Core

5.1

Equations of S t a t e

By an equation of state (EOS) we mean a relationship between the pressure, specific volume and temperature of a material. An equation of state cannot involve the history of the material and thus non-hydrostatic stresses are excluded-large non-hydrostatic stresses in general lead to irreversible (plastic) deformation. The pressures and temperatures in the deep interior of the Earth are on the one hand sufficiently high to make them difficult to reproduce in the laboratory, and on the other sufficiently low to make the quantum statistical models of Thomas-Fermi-Dirac not applicable. However, temperatures are generally above the characteristic Debye temperatures of the materials involved, so that classical analysis can be used to describe their vibrational properties. The free energy F is related to the internal energy U , temperature T and entropy S by the equation F=U-TS

(5.1)

From the first law of thermodynamics, Hence and

dU = - p d V + T d S

(5.2)

dF= -pdV-SdT

(5.3)

P7 )= ;(-

297

298

The Earth's Core

Since

P

+ T(%)"

= -(%)I

(5.5)

Thus, at a temperature of absolute zero,

P=

):(-

7'= 0

and depends only on the volume. The equation p = - d U / d V when U = U ( V) has often been used to determine a barotropic EOS. Such equations may be used if the temperature is low enough for thermal vibrations to be neglected in comparison with elastic effects. Anderson (e.g. 1979a,b, 1980) has shown that in a number of cases, the EOS may be approximated by expressing the pressure as the sum of two functions, one independent of T and one independent of V, i.e.

The general class of solutions given by (5.7) is known as the Hildebrand EOS. Hildebrand (1931) showed that if the internal energy U is a function of V only, the pressure is given by (5.5). Born and Meyer (1932) pointed out that this is equivalent to assuming that

where a is the (volume) coefficient of thermal expansion and k , the isothermal bulk modulus. This is similar to the Mie-Gruneisen EOS

where y is the Gruneisen parameter and Eththe thermal energy. As already noted in $3.2, the difficulty in using (5.9) is the uncertain dependence of y on V and T at high temperatures and pressures. This can to a certain extent be overcome if (5.8) is used instead of (5.9) since a and kT are measurable quantities. To do so it is necessary to show that a k , is independent of V (or nearly

299

The Constitution of the Core

so). This can be done by showing that (dp/dT), is independent of V in which case the isotherms on a p , Vplot are parallel. Since (5.10)

an alternative method is to show that ( d k T / d T ) , is small (i.e. kT at constant V is parallel to the Taxis). Anderson (1979a) showed that Pth is independent of V for a number of minerals including MgO, A120, andMg,SiO,. The data used were for polycrystalline solids; Anderson later (1982) made measurements on single-crystal specimens. Swenson (1966) had earlier found that the thermal pressure Pth was independent of V and proportional to T for lithium, sodium and potassium and solid argon. The later work of Anderson (1982) indicates that for the lower mantle Pth is probably linear with T a t high T ( > the Debye temperature) for all volumes and that anharmonicity corrections to the thermal pressure may not be necessary. Further theoretical work has been carried out in an effort to obtain better EOS at high pressures that might be more applicable to the Earth's deep interior (Walzer et al., 1979; Ullmann and Pan'kov, 1980; Walzer, 1982).

5.2 The Birch-Murnaghan Equation of State

In the ordinary theory of elasticity (infinitesimal strain), the Helmholtz free energy is equated to the strain energy and expressed as a quadratic function of the strain components, higher powers being neglected. For finite strain, suppose that xi are the coordinates of a point after deformation and y i those before, and define a strain tensor (Murnaghan, 1944, 1951) as (5.1 1)

where djk in the Kronecker delta. For the special case of the deformation of an isotropic body under hydrostatic pressure, xi =

(1

+ xjy,

(1

+

so that dXi

-= aYj

(5.12)

300

The Earth's Core

There is thus only one component of strain and the corresponding changes in volume and density are

vo/v= P/Po = (1 + 2 4 3 ' 2

(5.13)

It should be noted that while the above definition of strain (5.1 1) seems a natural extension of infinitesimal strain, Knopoff (1963) has pointed out that it is not unique and that there are other tensors of rank 2 that reduce to the infinitesimal strain form. Birch (1947, 1952) supposed that the free energy F could be written as a power series I

1 a,,?

F =

(5.14)

n=2

where the coefficients a, are in general functions of temperature (there is no term n = 1, since p( V o )= 0). Since

it can be written as a function of p / p o , viz. 513

P = iko(;)

c ")

n = 2 x"[(;)"3

-

(5.15)

17-

where k o is the bulk modulus at zero pressure. If the free energy contains just the quadratic term u2 2,this reduces to p =tk,

[(:)'I3

I">?!(

(5.16)

-

which is known as the Birch-Murnaghan equation and has been widely used to extrapolate experimental data. If terms up to the third order are retained, it can be shown that p = ~ k o [ ( ' ) "Po 3 - ( ' ) 5 ' 3 ] { lP o

-:(k&4)[(i)z'3-

I])

(5.17)

If fourth-order terms are included, it is necessary to introduce the second derivative kg in the expression for p ( p ) , which cannot be measured directly. Murnaghan (1944) assumed that the isothermal bulk modulus k is a linear function of pressure, i.e. (5.18)

301

The Constitution of the Core

where

This may be integrated to give (5.19)

where po is the density when p = 0. The parameters ko, go can be measured in the laboratory or determined from seismic observations. (Strictly speaking, seismic observations yield the adiabatic, not the isothermal bulk modulus). Bullen (see 95.5) has used the relation k = k ( p ) for many of his Earth models. By differentiating (5.19), we obtain (5.20) which is identical with the seismic EOS proposal by Anderson ( I 967). If the mutual forces between atoms at any separation in a crystal are known, the density is a known function of pressure. The forces between atoms can be represented by one (or more) potential functions that may be combined to an average effective potential function between nearest neighbours. Every potential function leads to a finite-strain relationship. The inter-atomic potential of Mie and Born is U ( r ) = r""

-

rh/'",

n

> rn; a, b > O

(5.21)

In the particular case n = 2m, the number of parameters is reduced to three, which can be determined from the conditions

It can be shown that (5.22)

This reduces to the Birch-Murnaghan equation (5.16) when n = 4 and m = 2. This form of the equation of state was obtained by Ullmann and Pan'kov (1976). They derived their equation not from the Born-Mie potential but from the expansion in a Taylor series of the free energy F ( p o / p , To) or the internal energy U(po/p, So) around the initial state (po, To)or (Po, So). Their equation describes the relation p = p ( p ) under isothermal or adiabatic conditions and is an approximate formulation, although it is an exact relation corresponding to a Born-Mie potential. Stacey et al. (1981) have given a

302

The Earth's Core

detailed review of finite-strain equations and listed those of geophysical interest.* The majority of the equations are entirely empirical-in the sense of having plausible analytical forms that appear to follow the trend of observations, with constants that can be adjusted to suit different materials. Comparison of finite-strain equations with terrestrial data requires some assumptions about temperatures and thermodynamic properties of Earth materials. In this regard an important parameter is Gruneisen's ratio y (see $3.2 and Stacey, 1977). A simple empirical form commonly used is (5.23)

with the special case q

=

1 strongly favoured by Anderson (1979b).

5.3 Experimental Methods

Studies of wave propagation measurements in rocks show that the velocity of compressional waves depends principally upon density and mean atomic weight. Since most common rccks have mean atomic weights close to 21 or 22 regardless of composition, rocks and minerals of very different composition may have the same densities and seismic velocities. Thus it is not easy to infer chemical composition from seismic data alone, and laboratory experiments at the conditions of pressure and temperature that exist deep within the Earth are necessary. There are two principal experimental methods in high-pressure physics: static and dynamic, In static studies, the pressure and temperature can be changed independently; in shock wave methods, pressure and temperature are related by the Hugoniot conditions. Further, a static experiment can be carried out for almost any length of time, whereas shock wave experiments are of the nature of an impulse with a duration of, at most, the order of a few microseconds. The pioneering experimental work of P. W. Bridgman up to pressures of 100 kbar corresponds to a depth of only 300 km within the Earth. However, in the last few years static measurements have been made up to pressures greater than 1 Mbar and temperatures greater than 3000°C. Dynamic determinations of the compressibility of minerals and rocks have been made by a number of workers up to pressures in excess of those at the centre of the Earth.

* A recent review of the reduction of mantle and core properties to zero pressure by adiabatic decompression has been given by R. Jeanloz and E. Knittle (in Chemistry and Physics of Terrestrial Planets, S. K. Saxona ed., Springer-Verlag, 1986).

The Constitution of the Core

5.3.1

303

Static Measurements

Static measurements have been achieved in a diamond-anvil high-pressure cell devised by Mao and Bell at the Carnegie Institution, Washington. A description of the apparatus with continuing improvements together with results have been published regularly in the Annual Reports of the Director of the Geophysical Laboratory from the year 1974-1975 onwards. Pressures of 1.2 Mbar have been reached and held constant for two weeks. This enables the physical chemistry of the region near the MCB to be investigated experimentally. Heating of the diamond-anvil, high-pressure cell is achieved with a neodymium-doped YAG (yttrium aluminium garnet) continuous laser beam. The diamond is transparent to the wavelength of laser radiation, and therefore only the sample to be investigated is heated, avoiding excessive heating of the diamonds and supporting blocks. The specimen can be viewed during the experiment, and its temperature measured with an optical pyrometer. The determination of the stress in the specimen is more difficult, since it is not hydrostatic (the material tends to be extruded from between the anvils). Pressure is measured by the shift of the R, strong-line ruby fluorescence spectrum excited with an He-Cd laser. The pressure is also calibrated with the BILB2 transition in NaCl at 300 kbar. The first observations (Carnegie Institution of Washington, Year Book 74) indicated that major reactions and transitions predicted to occur in the Earth’s mantle may proceed rapidly. The olivinespinel transition was observed visually. The ruby R , scale was calibrated to 195 kbar by the National Bureau of Standards; a linear extrapolation was used above that pressure. However, it is necessary to calibrate the R 1 scale directly in the higher range of pressures now attainable and this can be obtained only from independently determined pressure-volume relationships of suitable materials. Mao and Bell (1 976) used four metals-Cu, Mo, Pd and Ag-for which accurate pressure-volume relationships from shock wave measurements up to 1 Mbar were available. The metals were selected because data on their Hugoniot EOS are among the most accurate and because corrections to the isothermal conditions of the static experiments are small. The phases MgO and &Fehave been studied by shock wave methods, and their EOS determined (McQueen and Marsh, 1960; Carter et ul., 1971). Compressibility of the two phases has been measured in the diamond-anvil highpressure cell to 0.9 Mbar by Mao and Bell (1976), who found that no phase changes occurred above 130 kbar. The data serve to set volume--pressure (and density-pressure) constraints on the potential mineral properties to an equivalent depth in the Earth of 2000-2500 km. Figure 5.1 shows the compressibility of &Feas obtained by static measurements compared with shock

304

The Earth’s Core

0.6 0

I .o

0.5

1.5

Pressure (Mbar) Fig. 5.1. Compressibility of r:Fe (After Mao and Bell, 1976.)

Hugoniot data (uncorrected). Mao and Bell (1977) later reached pressures of 1.5 1 Mbar, at which pressure the diamonds failed. Mao and Bell ( I 978) have since achieved pressures of 1.72 Mbar for a period of 18 hours. At this pressure, macroscopic flow in one of the diamond anvils was observed. In later experiments, Bell and Mao (1979) made measurements of the force per unit area to compare with the accurate ruby-fluorescence pressure-calibration scale to 1 Mbar and to provide calibration above 1 Mbar. In these experiments, ruby crystals located throughout the surface of a stainless-steel sample were used with the fluorescence technique to determine the pressure distribution and the mean pressure. The force-per-unit-area measurements are less precise than the ruby measurements, but are considered absolute pressure measurements. The two independent calibrations were done simultaneously and yielded the same results, confirming the ruby pressure scale to 1 Mbar. Mao et al. (1984) later redesigned the diamond cell by increasing the rigidity of the cell’s steel structure and redistributing force loadings within the diamonds by reducing the flat area where the two diamonds faced, and by changing the angle of bevelling at the sides where shattering most often occurred. Ruby-fluorescence is not observed above about 1.85 Mbar and pressures had to be determined by extrapolation from lower values obtained by the ruby fluorescence method at a number of points at different distances from the centre of the sample chamber (see Fig. 5.2). Pressures up to 2.8 Mbar at the centre of the diamond face were reported (Ma0 et al., 1984). This pressure is equivalent to that well inside the Earth’s IC. More recently, Xu, Mao and Bell (1986) have reached pressures of 5.5 Mbar, well in excess of that at

305

The Constitution of the Core

0 I

I

I /

/

30 p m

f

2

4

6

Force ( lo3 N )

1.0 Pressure 30 pm from centre (Mbar)

0

Fig. 5.2. Determination of 2 8 Mbar static pressure achieved in experiments at the Geophysical Laboratory, Carnegie Institution, Washington. ( a ) Plot of pressure vs. applied force as a function of distance from the centre of the sample chamber of the diamond-cell high-pressure apparatus. The dashed line is the linear extrapolation from the 1 8 Mbar measurable by the ruby fluorescence method ( b ) Plot of pressure at the centre of the sample chamber vs. pressure 30 pm from the centre. The line defined by the four lower readings is extended to intercept the 1 8-Mbar value measured 30pm from the centre. this method is independent of applied-force measurements. (After Mao e t a / , 1984.)

the centre of the Earth (3.5Mbar). In the previous experiment up to 2.8 Mbar, the pressure calibration had to be done indirectly from load calculations. The shift of the fluorescent line of ruby crystals placed in the sample could not be used because of strong interference from diamond-anvil fluorescence at pressures above 2.7 Mbar. In the new experiments, the overlapping diamond emission was found to disappear at pressures above 3 Mbar, and the ruby pressure calibration scale could be employed once again.

5.3.2 Dynamic Measurements

If a slab of material is struck on one side by a blow, strong enough that the resulting strain cannot be considered to be infinitesimal, a pressure wave travels through the slab at a speed greater than the speed of sound. The pressure rises in a thin layer of material to the value set up by the blow and, behind the thin layer (the shock front), the material as a whole is set in motion. When the shock reaches the far side of the slab it is reflected as a rarefaction wave satisfying the condition that the pressure on that side is atmospheric (i.e. 2: zero). The whole slab is then moving at the speed of the material behind the shock. Shock waves are generated by driving a metal plate at high

306

The Earth's Core

velocity against specimens of the material. The plate is driven by an explosive charge, the composition of which determines the speed of the plate and hence the strength of the shock. Details of the experimental procedures may be found in Rice et a/. (1 9%) and in Duvall and Fowles (1 967). An EOS obtained from shock wave experiments usually takes the form of a relationship between shock pressure p , shock-induced density p , and internal energy U along a curve called the Hugoniot. The Hugoniot curves, upon reduction, yield pressure-temperature states for different materials. One form of the EOS consists of pressure-density isentropes (constant entropy curves). Upon differentiation the isentropes yield the seismic parameter $ = (dp/dp),. Direct comparison is thus possible between values of 4 obtained from the seismic velocities V p and Vs (see (1.3)) and values of $ measured for rocks and minerals in the laboratory under similar conditions of temperature and pressure. Two assumptions are made in shock wave experiments-that the measured ( p , p, U ) states are in thermodynamic equilibrium and that the compression for a given pressure is the same as that which would be produced by a hydrostatic pressure of the same magnitude. The first condition is satisfied if thermodynamic equilibrium is attained in about s or less. This implies a shock front with a thickness of a few tenths of a millimetre or less. Carter et al. (1971) have discussed the second point and concluded that the Hugoniot curve is equivalent to that obtained under hydrostatic conditions if a small correction is made for the effects of the strength of the material. The thermodynamic states that are produced behind shock waves are determined from the Rankine-Hugoniot equations which express the conservation of mass, momentum, and internal energy or enthalpy across the pressure discontinuity or shock front. Consider a disturbance (corresponding to a shock front) propagating with a velocity us into an undisturbed state defined by pressure po, density p o ( = l/Vo) and mass velocity zero (see Fig. 5.3). The shock front is assumed to consist of a time-independent pressure profile. If p , , p1 and u p are the pressure, density and mass velocity behind the front, the condition that the mass flux in and out of the shock front is equal is POUS

= PI(% - up)

(5.24)

The net force on a unit cross-section of the material between x = A and x = B (see Fig. 5.3) is p1 - po. This must equal the time rate of change of momentum for this material, viz. pous through the shock multiplied by the associated velocity change up,i.e. PI - P o = PoUsUp

(5.25)

Finally, the power input to a unit cross-section of material between A and B,

307

The Constitution of the Core

I

I

1 PI vl

I I I I

I 1 I I I

I

I

I

x SB

X=B

P

I

I

xFig. 5.3. Shock wave pressure profile The surface x 1958 )

=

8 moves with the fluid (After Rice ef a / ,

viz. plu,, must equal the time rate of change of energy for the enclosed material, i.e. PlU, =

Pou,(:u;)

+ POUs(U,

-

Ud

(5.26)

where U o and U1 are the specific internal energies ahead of and behind the shock front respectively. From (5.24) and (5.25) (5.27)

and (5.28)

Hence UI

-

Uo =

:(PI

+ POl(V0 - V1).

(5.29)

These equations were first derived by Rankine and Hugoniot. From measurements of the velocity of the shock u, and the velocity of the material behind the shock up, (5.26) and (5.27) enable the change of pressure and change of specific volume to be found. Since U 1 and U o are different, the ( p , V) relationship is not one of constant internal cnergy-it is thus neither adiabatic nor isothermal. Since the internal energy of a material is a function of its pressure and density, (5.28) may be regarded as the locus of the (pl, V 1 ) states attainable by propagating a shock wave into a fixed initial state ( p o , Vo).This locus is defined as the Hugoniot curve centred at (po, V o ) . Each shock of a given strength gives one point specified by p l and V1 in the shock. If the material is given a series of shocks of different strengths, a curve of density against pressure (i.e. the Hugoniot) can be built up.

308

The Earth’s Core

In 1955 Walsh and Christian carried out such calculations for the states produced by 500 kbar shock waves in metals. Since then a large number of papers have been published giving shock wave EOS data for many metallic elements to pressures, in some cases, up to 9 Mbar (see e.g. Walsh et ul., 1957; McQueen and Marsh, 1960; Al’tshuler et ul., 1958a,b, 1962). A considerable body of data for compounds as well as for rocks and minerals has been collected in the Compendium of Shock Wave Data (van Thiel, 1967). The use of shock wave data to constrain the thermal and chemical state of the Earth’s core has been extensively discussed (see e.g. McQueen and Marsh, 1966; Al’tshuler et al., 1968; Jeanloz, 1979; Ahrens, 1979) and will be reviewed in 45.7. it must be stressed that in order to interpret the results, the EOS as determined from shock wave data, which is neither adiabatic nor isothermal, must be reduced to a reference temperature. Temperatures in the shock front are not generally known and additional measurements or assumptions must be made to reduce the pressure-density data to those at absolute zero. Moreover the shock-produced states are characterized by pressure and internal energy-thus a U ( p , T ) EOS is required before the data can effectively be used. Lyttleton (1973) has also pointed out that in the seismic case, harmonic waves of small amplitude are propagated through material already under static pressures in excess of one megabar (and at high temperature), whereas in laboratory experiments a shock wave moves into hitherto unstressed material and what is propagated through it is a non-oscillating disturbance in which the peak pressure may attain this same order of magnitude.

5.4 Ramsey’s Hypothesis

In 1949 Ramsey proposed that the lower mantle and core have the same chemical composition, the discontinuity at the MCB resulting from a change of mantle silicates to a high-pressure liquid metallic form. This suggestion met with two main difficulties-that of reconciling the large jump in density (by a factor of 1.7) at the MCB with geochemical theory, and the failure to find positive evidence of such a transformation in shock wave experiments at the relevant pressures. The materials in the lower mantle are already tightly packed and transformation to a metallic form is unlikely to increase the density. Also, as Birch (1968) pointed out, at one atmosphere the mean atomic volumes of oxides and silicates are less than the mean atomic volumes of the pure metals of which they are made. The transformation to metallic form of mantle material, composed mainly of light elements, must result in a light metal, and metals lighter than chromium ( Z = 24) are all too light for the

-

The Constitution of the Core

309

-

h

u)

E

Y

v

Fig. 5.4. Hydrodynamical sound velocity ( ~ ? p / ? p ) ;versus ’~ density along the Hugoniot compression curves for metals up to cobalt and several rocks. For a few materials, the shock wave velocity has been plotted instead of the adiabatic sound velocity Atomic numbers or representative atomic numbers (in parentheses) are attached to each curve. The line marked ( I 1.2) shows values for Twin Sisters dunite, a rock composed mainly of olivine (92%) of composition about Fa,,; the line marked (14.3) shows values for hortonolite dunite, composed mainly of olivine (90%) of composition about Fa,; the line marked (20.1) is for magnetite; the line marked (23.2) is for an iron-silicon alloy having nearly the composition Fe,Si. Similar data for a large number of other rocks fall between the lines for the dunites. but have been omitted for the sake of clarity The areas in which the corresponding quantities for the Earth’s mantle and core must lie are indicated by the pairs of dotted lines Several oxides, silica, periclase and corundum have been compressed to Mbar pressures and also fall in the “mant1e”area. (After Birch, 1968.)

Earth’s core. Figure 5.4 is a plot of the hydrodynamical sound velocity (dp/dp):’* against density along the Hugoniot compression curves for metals up to cobalt and for several rocks. The areas in which the corresponding quantities for the Earth’s mantle and core must lie are indicated by the pairs of dotted lines. O n the diagram, Ramsey’s hypothesis corresponds to a transition from a point such as A to a point such as B. The experimental evidence clearly shows that this corresponds to a change of atomic number, i.e. of chemical composition, and by a large amount (roughly from 12 to 23). For a given atomic number and given velocity, the rocks are both denser and less compressible than metals. Moreover all experimental data have shown that the densities of rocks when compressed well beyond the pressure at the MCB,

310

The Earth’s Core

fall on a smooth continuation of the curves for lower pressures and give no indication of transforming to core densities. Anderson, (1985,1986) has recently summarized all the data on the properties of iron at high pressures and temperatures and concluded that the evidence is overwhelmingly in favour of an iron core. Shock wave experiments of improved resolution (Ahrens, 1980) have shown that the data for pure iron at core conditions correspond to the seismically determined density of the IC. Jeanloz (1979) has also shown that the mechanical properties of iron (4, k, p ) agree with the known properties of the IC. The main issue now is what the light alloying element in the O C is (see $5.7). Liu (1 974) has constructed a more detailed velocity-density plot than Birch’s figure (Fig. 5.4) for all solid elements having densities in the range 8 g cm113.Without considering information on the chemical abundances of the elements in the solar system, Liu found that a “Birch” diagram does not unambiguously distinguish the elements of the iron group from other heavy elements as candidates for the core. Elements Ga (31), Ge (32), As (33), Y (39), Zr (40) and Ba (56) are all possible candidates. Birch himself (personal communication) further commented that elements lighter (i.e. of lower atomic weight or number) than V (23) are not likely candidates for the core. 5.5

Bullen’s ( k ,p ) Hypothesis

Bullen found for his Earth Model A that there was no noticeable difference in the gradient of the incompressibility, dkldp, between the base of the mantle and the top of the core. Moreover there was only a 5% difference in the value of k across the MCB. These features are in marked contrast to the large changes in density and rigidity at the boundary. Because of the smallness in the change in k across the MCB and because this change (a diminution) is opposite in sign to that predicted theoretically, Bullen (1 949, 1950) suggested another Earth Model B in which he assumed that k and d k / d p are smoothly varying functions throughout the Earth below a depth of about 1000 km. This hypothesis, called the compressibility-pressure ( k , p) hypothesis, implies that at high pressures the compressibility of a substance is independent of its chemical composition. O n the basis of this hypothesis, Bullen found that there must be a concentration of more dense material near the base of the mantle (region D”). This material could be a, mixture of metallic iron with silicates near the MCB or an iron sulphide phase at the base of the mantle. Again if the entire core is liquid so that V, = 0, then from (1.3), V pis given by V: = k / p

(5.30)

311

The Constitution of the Core

Bullen first pointed out in 1946 that if the IC (region G) is solid, and thus capable of transmitting S waves, (5.30) is replaced by (1.3) in G and the increase in V p can be accounted for without violating his (k, p ) hypothesis. Bullen and Haddon (1 967) and Haddon and Bullen (1 969) have since revised Bullen’s original model B and constructed a series of new Earth models based on the ( k , p ) hypothesis (see 41.6). The high-pressure work of Ahrens et al. (1969) on magnesium and aluminium oxides indicates that along an adiabat over the pressure range in the lower mantle (dkldp), is not constant but decreases markedly. Lattice dynamical calculations also predict that (dkldp), should decrease as the pressure increases. Moreover such a prediction is given by quantum mechanical calculations of EOS (such as the Thomas-Fermi-Dirac equation) and general theories of finite deformation (such as the Birch-Murnaghan equation). The results of shock wave experiments and theoretical considerations also indicate quite clearly that k and (dkldp), d o not depend on pressure in the same way for all materials. The behaviour of solids in general is thus inconsistent with Bullen’s (k, p ) hypothesis, although it is a fairly good approximation in the case of the Earth. This is in part fortuitous-the compressibilities of iron oxide, stishovite, aluminium oxide and magnesium oxide are similar at pressures comparable to those at the MCB. Again the gradients of the bulk modulus of such substances are not constant, although the average values are not so very different from the mean terrestrial value. If the mantle covered a greater range of pressures, the agreement would become less close. Bullen (1963) also investigated the chemical inhomogeneity of the Earth, where chemical inhomogeneity is used here to include also inhomogeneity arising from phase changes. Assuming hydrostatic pressure (1.6) and an adiabatic temperature gradient, we have (by definition of 4 ) k

= P4

so that

For a chemically homogeneous region, d p l d p

=p/k

(by definition), and thus (5.31)

At any point of a region, whether chemically homogeneous or not, at which dpldr can be assumed to exist, we can write (5.32)

31 2

The Earth’s Core

i.e.

where (5.33)

When y = 1, (5.32) reduces to the Adams-Williamson equation (1 .8), so that ‘Iis a measure of the departure from chemical homogeneity-it is the ratio of the actual density gradient to the gradient that would be obtained if the composition were uniform. An excess temperature gradient normally reduces, while chemical inhomogeneity increases, the value of y. From (5.33) it can be seen that y depends on dkldp, g and d 4 l d r . On Bullen’s compressibility-pressure hypothesis, d k / d p is slowly varying and lies between about 3 and 6 throughout most of the Earth’s deep interior. Uncertainties in estimates of g are not large, while values of d 4 / d r are immediately obtainable from the P and S wave velocity distributions. It follows from (5.33) that 11 can be estimated in most parts of the Earth’s deep interior within limits that can be assigned. Thus it is possible to estimate the degree of departure from chemical homogeneity in any given region and to assess density gradients where the Adams-Williamson equation cannot be used. In a later paper, Bullen (1 965a) refined (5.33)-in particular he investigated the implications of the variation of k with composition and of the deviation of d k l d p from (dkl8p)constdnt composltlon. In the lower 200 km of the mantle (region D”), the seismic velocity distributions of both Jeffreys and Gutenberg indicate that d d l d r ‘v 0, so that y 2i d k l d p . Bullen’s Model A gives d k j d p N 3 in D” indicating that the lower 10&200 km of the mantle is inhomogeneous. The inhomogeneity is not too great, however: with y = 3 it contributes only an increase of 0.2 g c m - 3 in density across D”. Using more recent values of the seismic velocities (including a decrease in D”), Bolt (1972) obtained a value of 4.7 for y leading to an increase in density of 0.33 g cm-3 through D”. y = 3.1 for model B1 of Jordan and Anderson (1974) and 2.3 for model UTD 124A‘ of Dziewonski and Gilbert (1972). The density gradients in both these models correspond approximately to adiabatic compression (y = 1). A very slight increase in the gradient of V pwould result in a value of y closer to unity. Using the data of Taggart and Engdahl (1968) on PcP times, Muirhead and Cleary (1969) devised a variant of the Haddon and Bullen model HB1 with a core radius of 3478 km and with V, decreasing linearly from the Jeffreys value of 7.25 km s-’ at 2700 km to 6.8 km s-’ at the MCB. The resultant model, designated ANUl, was found to fit the free-oscillation modes

The Constitution of the Core

31 3

reasonably well. Model ANU2 is a variation of model ANUl in which the thickness of D” is reduced to 100 km and the values of V pand Vs at the top of the layer are taken from the models of Hales and Herrin (1972) and Hales and Roberts (1970) respectively while the values at the bottom are those given by “diffracted” wave studies for a core radius of 3482 km. The calculation gives y = 4.5. Jordan’s (1972) model B2 has an unjustifiably low value of Vs at the MCB leading to a change of sign in the density gradient between that of the model and that calculated from (5.32) and a negative value (- 10.0) for y. The structure of D” has been discussed in more detail in $1.3. In the transition region F between the OC and IC, Jeffreys’ velocity distribution (characterized by a large negative P velocity gradient, i.e. d b / d r 9 0) leads to a value of q of 38 entailing a density increase of the order of 3 g cm-3 through F. On the other hand, Gutenberg’s velocity distribution gives large negative values of d b / d r so that y is significantly less than unity (actually negative) implying an unstable distribution of mass. It would seem that seismic velocity gradients much in excess of those in regions D and E cannot exist in the Earth’s deep interior. An infinite gradient (i.e. a velocity discontinuity) on the other hand, is not impossible since then the range of depth of any instability would be zero. Birch (1963), using shock wave data at pressures of the order of lo6 bar, inferred that the density p o at the centre of the Earth does not exceed 13 g ~ m - Bullen ~ . (1965b) investigated the consequences of this limiting value of po using Bolt’s (1964) distribution of V p in the core. Since p is not likely to decrease with depth in the core, Bullen’s ( k , p ) hypothesis implies, through (1.1) that departures from smooth variations of V pwith r are accompanied by similar departures in the variation of p rather than of k. Bullen showed that it is impossible for p o to be as low as 13 unless there is substantial rigidity in both regions F and G. In addition Bullen found that d p / d r > 0 over a significant range of depth in the lower core, i.e. the rigidity must decrease with increase in depth. This is further evidence that the IC is solid, since a region cannot be entirely fluid if the rigidity is significantly changing inside it. If F and G are both fluid (i.e. complete absence of rigidity), po must be at least 14.7 g ~ m - This ~ . value is sufficiently in excess of 13 g c m p 3to give additional support to the conclusion that the IC is solid. Using a more recent P velocity distribution in the central core (due to Qamar, 1971), Bolt (1972) has shown that p o probably lies between 13.0 and 14.0 g ~ m - the ~ , lower value being more compatible with shock wave data for iron. His preferred model is p o = 13.0 g ~ m - ko ~ ,= 15.05 Mbar and po = 1.25 Mbar. None of Bolt’s models, however, showed a decrease in p with depth in the IC as was found by Bullen (1965b). In a later paper, Bullen (1969) re-examined conditions in the OC. He found a strong suggestion of slight inhomogeneity inside the outermost 700 km of

31 4

The Earth’s Core

the core. It thus appears that the whole depth range from about 270& 3600 km, which embraces regions on both sides of the MCB, is somewhat abnormal. If the indicated increase of d k l d p (from 3.1 to 3.6 across the outermost 700 km of the core) is due solely to compositional changes, such changes might be enough to inhibit convection currents and confine convection in the core to the depth range 3600-4500 km. A range of depth of the order of only 1000 km might well be insufficient to satisfy the requirements of the dynamo theory of the Earth’s magnetic field ($4.2). Bullen thus tentatively suggested that there may be a continuous phase change near the top of the core, the rate of change diminishing with depth-the lower part of the OC (below a depth of about 3600 km) being probably nearly uniform in composition and phase ( q N 1 and d k l d p N 3.6). Using values of d 6 l d z computed for his model KOR5 (see Fig. 1.7), Qamar (1973) found, from (5.32) that q ru 1 in region E. In F, the small velocity gradient leads to a value of ‘1in the range 3-4 indicating a possible change in phase or chemical composition. The large velocity gradient at the top of region G in model KOR5 has interesting consequences. Assuming that d k l d p has about the same value as in the OC, the large value of dVp/dr leads to negative values of q if Vs = 0. With ‘1 = 1, d k l d p = 4 in G (Bullen, 1963) and Vs N 3.5 km s-’ near the Earth’s centre (Dziewonski and Gilbert, 1972), Qamar found that V , must increase rapidly with depth by about 0.3 km s-’ from the ICB to about r = 950 km and then slowly decrease by about 0.1 5 km s-‘ toward’s the Earth’s centre.

5.6

Bullen‘s FetO Hypothesis

Bullen (1973a,b) has developed an alternative model of the cores of the terrestrial planets that avoids the main difficulties of the phase-transition theory while retaining the important feature that the pressure pc at the MCB is critically involved in the change of properties there. His theory is based on a suggestion by Sorokhtin (1971) that the Earth’s O C consists of Fe,O. Sorokhtin calculated that this oxide, which is unstable at ordinary pressures, becomes stable at the pressures in the Earth’s core and has a density-pressure relationship in agreement with that in the Earth’s OC. However, whereas Sorokhtin attributes the occurrence of Fe,O in the OC to a breakdown of FeO into FezO and oxygen, Bullen associates it with the equation FeO + Fe. Bullen considers a model family of planets with the folFe,O lowing properties. All planets of the family are composed of two primary materials-a basic mantle material X, and Fe,O. (The composition of X is not specified, but it is likely to contain some FeO.) In all the planets, the ratio of the mass of X to the mass M of the planet is the same. In those planets that

+

31 5

The Constitution of the Core

@@ RC

H

J

K

F i g . 5.5. Materials in the interiors of the three sub-sets ( H ,J. K ) of the terrestrial planets The outermost zones, all containing the material X, are mantles. The Fe,O zones are referred t o as “outer cores”, the Fe zones as ”inner cores”. (After Bullen, 1973a.)

contain Fe,O, the Fe,O occurs as a distinct zone (OC), throughout which p 3 pc. In those planets where p < pc in an Fe,O zone, some or all of the

Fe,O has broken down into FeO and Fe. This FeO (which Bullen calls Y) forms part of the mantle and is additional to any FeO that may be part of X. The Fe falls to form an IC. Bullen’s family of models has three sub-sets which he calls H , J , K (see Fig. 5.5). The subset H includes the smallest planets that have no Fe,O zones and thus no OCs: they have mantles composed of X and Y, and ICs of Fe. The sub-set K includes the largest planets that have mantles composed purely of X, OCs of Fe,O, and no ICs. The subset J consists of intermediate planets that have mantles composed of X and some Y, OCs of Fe,O, and ICs of Fe. Earth and Venus correspond to members of J , Mars of H . No known planets correspond to K . Table 5.1 gives the results which Bullen has computed for a number of members of the subset J . For a first approximation, compressibility effects have been neglected (and variations of density inside mantles, ICs and OCs) as well as possible volume changes and/or chemical interactions in the mixing of the materials X and Y and all free iron has been assumed to have dropped to the innermost regions. The members of the subset J in Table 5.1 range from the extreme (top) member, which has no IC (no free Fe) and for which p = p c at the MCB, to the other extreme (bottom) member which has no OC (no Fe,O) and for which p = pc at the mantle ICB. The second member (with M = 6.03 x loz4kg) corresponds to the Earth model. The member with M = 4.86 x kg was constructed for comparison with Venus. 5.7 The Constitution of the Core

A review of our ideas on the composition of the core up to 1976 has been

316

The Earth's Core

TABLE 5.1 Structuresof model planets oftheJsubset (possessing both OC and ICs) (After Bullen 1973b )

Mass ( M ) (10'"kg) 6.29 6.03 5.96 5.65 5.34 5.04 4.86 4.75 4.46 4.36 *

Radius (R) (IO'krn) 6.44 6.37 6.35 6.26 6.16 6.07 6.01 5.97 5.87 5.83

Mean Mantle thickness density ( p ) ( g ~ m - ~ ) (103km) 5 62 5 57 5 57 5 50 5 45 5 38 5 34 5 32 5 26 5 22

2 83 2 89 2 91 3 00 3 09 3 21 3 29 3 34 3 52 3 54

Mantle density (gcrn ")

Outer-core thickness (IO3km)

Inner-core radius (103km)

4 47 4 50 4 51 4 55 4 60 4 65 4 68 4 69 4 73 4 74

3 61 2 28 2 13 162 121 0 83 0 60 0 46 0 07 0

0 120 131 164 186 2 03 212 2 17 2 28 2 29

Radius.

given by Brett (1976). In a detailed study of the properties of iron at high pressures using shockwave data, Jeanloz (1979) concluded that the gradients of density, sound speed and bulk modulus in the OC are grossly consistent with a homogeneous composition, the actual values of the velocity being close to, but probably less than, those of Fe under equivalent conditions. These results are contrary to the conclusions of other analyses (e.g. Davies and Dziewonski, 1975; Anderson 1977) that did not properly take account of the large differences in temperature between states on the Hugoniot and those within the core. Seismic data are also consistent with chemical homogeneity on a gross scale for the bulk of the OC (Dziewonski et al., 1975; Butler and Anderson, 1978). Both densities and bulk moduli in the OC are less than those of Fe under equivalent conditions (by about 10 and 12% respectively). However, both densities and bulk moduli for the IC are compatible with those of Fe. This suggests that the ICB represents not only a phase boundary, but a compositional one as well. If this is the case, then the boundary cannot readily be used as a fixed temperature point in the melting curve of Fe (or a related compound). The reason for the OC being molten is then simply that it is alloyed with some lighter component such that not only its density but also its melting point is lowered below that of pure Fe. Jeanloz (1979) also showed how data on bulk modulus and density provide constraints on the composition of the OC. Fig. 5.6 gives the unique combination of density and bulk modulus as a function of mass fraction of the core that the lighter compound X must possess to satisfy the seismic data for the OC as well as the shock wave data on pure Fe. When more data are available on the densities and bulk moduli of other materials, Fig. 5.6 can be used to see what compounds (along with Fe) may be present in the OC and in what proportions.

31 7

The Constitutionof the Core

12.0

-

P (Fe)

-

p(core)

-

10.0-

-

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E

- 400 g AT 2 0 0 G P a IN OUTER CORE

3

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m

- 200 I

0

I

I

I

I

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0.2 0.4 0.6 Mass fraci-ion of X

I

I

0

I

0.0

1.0

Fig. 5.6. Core diagram showing the values of density and bulk modulus that must simultaneously be satisfied by a candidate compound (X) coexisting in a given mass fraction with Fe in the OC, at a pressure of 200 GPa (radius -2860 km). Errors in density may be as much as 3-5%; those in bulk modulus may be up to *5-10% The densities and bulk moduli of the core and of iron (under the same conditions) used to construct this figure are also shown. Ideal mixing is assumed. (After Jeanloz, 1979.)

*

Bukowinski (1976a) has carried out a quantum-mechanical calculation of the effect of pressure on the electronic structure of iron. Elsasser and Isenberg (1949) had suggested that, at core pressures, iron might undergo a transition to the 3d8 state. The suggestion was quickly forgotten, but was revived by McLachlan and Ehlers (1971), who pointed out that the presence of an electronic transition would invalidate extrapolations of the melting curve of iron. Bukowinski and Knopoff (1976) constructed a model of the core in which an electronic transition in iron is responsible for the IC-OC transition, and showed that it is possible that a highly viscous, dense iron IC may transmit shear waves at seismic frequencies. Moreover, they found that Poisson's ratio of a liquid IC is a slowly varying function of frequency with values close to the extremely high observed value of 0.45. The high Poisson's ration (0.45) of the IC has been used to argue that it cannot be a crystalline solid. However, Al'tshuler and Sharpdzhanov (1971a,b) have carried out shock wave measurements of Poisson's ratio of copper and iron to pressures of 1 and 2 Mbar. They obtained 0.43 at about 2Mbar and the data extrapolate smoothly to the value found for the IC. Unlike other elastic properties, Poisson's ratio increases with both tempera-

-

318

The Earth’s Core

ture and pressure. Some metals have extremely high Poisson’s ratios even at low temperatures and pressures. It does not seem necessary, therefore, to invoke a residual fluid character or an electronic phase transition to explain the properties of the IC. The rapid increase of V , at the outer part of the IC, however, may indicate the elimination of a melt fraction with depth in this region. Bukowinski’s (1976a) calculations on the electronic structure of iron showed that it is very stable at least to two-fold compression. However, the collapse to a 3d8 state occurs at four-fold compression-the predicted transition density being 2.5 times as large as the IC density. This effectively rules out the possibility of any electronic transitions within the Earth’s core. Limitations on possible choices for the light alloying element in the OC are that the element be reasonably abundant, miscible with liquid iron, and possess chemical properties that would allow it to enter the core. Possible candidates are H, He, C, 0,N, Mg, Si and S. Ringwood (1966a) has pointed out that H, He, C, 0 and N may be rejected since they are known to form interstitial solid solutions with iron.* Additions of these elements would not significantly decrease the density of iron, since they occupy holes already present in the lattice. Magnesium is unlikely to be present in any amount since it has a much greater affinity for oxygen than has silicon, i.e. any chemical conditions that may have led to the incorporation of magnesium in the core would have caused the incorporation of much larger amounts of silicon.

5.7.1

Silicon in the Outer Core

Ringwood (1966a,b) favoured silicon as the most likely extra component of the Earth’s core. He pointed out that the presence of substantial quantities of silicon in the metal phase of enstatite chondrites shows that chemical conditions during the formation of the solar system were, at least in some regions, favourable for the reduction of silicates. Ringwood also showed that an Earth model constructed from the abundances of non-volatile elements in Type I carbonaceous chondrites also requires the presence of silicon in the core. In Ringwood’s model the terrestrial planets formed in an initially cold gas and dust cloud. The Earth is believed to have accreted from primitive oxidized dust similar in composition to the material of Type 1 carbonaceous chondrites. In this model, reduction of the iron oxide by organic matter in the dust begins to occur inside the growing Earth owing to the heat liberated by the release of gravitational potential energy. As the temperature of the Earth in-

* Ringwood has since changed his opinion and believes that oxygen may be the light alloying element in the OC. The case for oxygen is discussed in 55.7.3.

The Constitution of the Core

319

creases, the metallic iron thus produced melts and sinks to the centre of the Earth to form the core. As a consequence of this high-temperature reduction process, some Si enters the metal phase, accounting for the low density of the core, and some S is lost from the Earth by degassing along with other volatiles that are believed to be depleted in the Earth relative to Type 1 carbonaceous chondrites. In the final stages of the accretion of the Earth, a hot, dense silicate atmosphere builds up that later escapes from the Earth and condenses into a sediment-ring from which, Ringwood believes, the Moon accreted. Ringwood's model has encountered a number of difficulties. If the core of the Earth is formed in situ by reduction, the reaction products, H2 and CO, will form an enormous atmosphere totalling perhaps half the mass of the core. No trace of such an atmosphere remains and an efficient dissipation mechanism must be postulated. Again the H20/H2 and CO2/CO ratios in the gases that degas from the mantle far exceed those that would be in equilibrium with metallic iron and ferromagnesian silicates. The presence of silicon in the core, whereas the mantle contains substantial quantities of oxidized iron, implies that the mantle is not in chemical equilibrium with the core. Ringwood regards this as an important constraint on any process of core formation. In particular it is incompatible with the theory that the material from which the Earth accreted was composed of an intimate mixture of silicate and metal particles similar to ordinary chondrites. The disequilibrium between mantle and core has been discussed in more detail in earlier papers by Ringwood (1959, 1961). The deep interior of the Earth is initially highly oxidized and rich in volatiles, whereas the outer regions are progressively more reduced and poor in volatile components. After melting near the surface, the metal phase, consisting of an iron-nickel-silicon alloy collects into bodies that are large enough to sink into the core. In Ringwood's model, core-mantle separation took place so rapidly that chemical equilibrium was not attained. In addition, the Fe2+/Fe3+ratio of the mantle is a factor of 3 W O below that which would be in equilibrium with metallic iron at 25OO0C,the mean temperature believed by Ringwood for the interior of the Earth during the metal-silicate separation. Again the abundance of Ni in the upper mantle, as measured in ultramafic rocks, is nearly two orders of magnitude higher than would be expected if equilibrium partitioning of Ni between metal and silicate phases had occurred during core formation. Ringwood thus believes that the mean oxidation state of the mantle is such that it is not now, nor has it ever been, in equilibrium with metallic iron. Clark, Jr. et al. (1972) have shown, however, that the diffusion coefficients for Ni in olivine (Clark and Long, 1971) imply that complete removal of Ni from 2 m m olivine grains is possible in about 10 years at the zeropressure melting point of iron, 1805K. This, coupled with the high Ni content of mantle rocks, places severe limits on the time-scale of the coalescence of

320

The Earth’s Core

small metallic droplets into the larger pools that sink to the centre of the Earth. Equilibrium between metal and silicate can only be obtained by diffusion across the interface of the sinking bodies of metal. If the rate of sinking of metal is high compared to the rate of attainment of equilibrium by diffusion (which Ringwood believes to be the case), the core that separates will not be in equilibrium with the mantle. After the core has separated and the mantle is molten, both regions will become homogenized by convection, but they will remain out of equilibrium with one another. The composition of the Earth’s core depends very much on the mechanism of the accretion of the Earth. Ringwood (1978) later showed that homogeneous accretion would permit Si, but not C, to enter the core by means of reduction of silicates to metallic Fe-Si material during the late stages of the accumulation of the Earth. As already pointed out, if the light element in the OC is silicon, there must be gross chemical disequilibrium between the core and mantle. This difficulty remains if carbon is also present in the OC. Sat0 (1980) pointed out that the presence of carbon in the core would lead to a depletion of FeO in the lower mantle. Larimer (1975) had shown that if the Earth had accreted inhomogeneously in a region of the solar nebula in which the C : O (atomic) ratio was 1 as compared with a C : O ratio of 0.6 for the present solar photosphere, the initial condensates would be FeNi, minor amounts of CaS and TIN and moderate amounts of C, Fe3C and Sic. If the Earth’s core formed in this way, it would be composed of FeNi and contain high-temperature condensates including silicon and have an appreciable carbon content (up to a minimum of 11%). Herndon (1979) has also suggested the possibility of such compounds as Ni,Si and Ni3Si being in the core. Lyzenga et al. (1983) have carried out shock wave experiments on SiOz that have relevance for Si in the mantle. Shock temperatures were measured by an optical pyrometry method (Lyzenga and Ahrens, 1979) for samples of single-crystal a quartz and fused quartz for pressures between 60 and 140 GPa. Shock-reduced phase transitions were observed at 70 GPa and 50 G P a along the a and fused quartz Hugoniots respectively. They suggested that this transformation is the melting of shock-synthesized stishovite, the onset of melting being delayed by metastable superheating of the crystalline phase. They were thus able to construct the stishovite-liquid phase boundaries. The melting temperature of stishovite near 100 G Pa (1 Mbar) suggests an upper limit of -3500K for the melting temperature of S O 2 bearing solid mantle mineral assemblages, even though, as an end-member in the MgO-FeO-SiOz system, it is unlikely that stishovite would be present as a pure phase. However, their work indicates that in order for a silica-bearing mantle to be solid at pressures near the MCB, the temperature may have to be 3500 K. Lyzenga et al. (1983) also estimated the increase of melting point with pressure and, assuming creep is diffusion-controlled in the lower

-

-

-

-

321

The Constitution of the Core

mantle, concluded that the viscosity across the mantle would not increase by more than a factor of lo3.

-

5.7.2

Sulohur in the Outer Core

Some of the difficulties of Ringwood's (1966a,b) model have been removed by Murthy and Hall (1970), who proposed a homogeneous accretion model in which the composition of the Earth is 40% carbonaceous chondrite, 50% ordinary chondrite and 10% iron meteorite. Since all the iron that eventually enters the core already exists in the reduced state in this model, no enormous atmosphere of reaction products is built up and there is, therefore, no difficulty encountered in its dissipation. Again, since this mixture of meteorite types contains abundant Fe and FeS grains, melting first begins at a temperature close to the Fe-FeS eutectic, 550 K lower than the melting-point of pure Fe at zero pressure. Because the Fe-FeS eutectic temperature is relatively insensitive to pressure (Brett and Bell, 1969), temperatures in the Earth keep the resulting Fe-S melt liquid until it reaches a depth well within the present ICB, thus avoiding the refreezing difficulty in Ringwood's model. The discovery of '29Xe produced by the decay of the short-lived extinct radioactive 1291 (tt = 17 x lo6 years) by Buolos and Manuel (1971) in deep mantle gases sets limits, not only on the time of formation of the Earth relative to nucleosynthesis, but also on the degassing history of the outer parts of the Earth. Specifically, this precludes a sustained high-temperature regime for the mantle after a few mean life-times ( z80 Ma) of this extinct radionuclide. If the Earth accreted rapidly (lo3-lo4 years), it is doubtful whether it would cool to low enough temperatures to retain Iz9Xe in this short a time. Hot accretion models for the Earth (Ringwood, 1966a,b; Anderson and Hanks, 1972) in which an efficient trapping of the gravitational potential energy occurs, seem untenable in the light of this discovery (but see also 92.4). Traces of the fission-produced Xe isotopes 131,132,134, and 136 were also found by Buolos and Manuel. The relative isotopic compositions of these isotopes strongly suggest that a mixture of spontaneous fission by 238Uand 244Puhas been responsible for their production. These observations give direct evidence that the formation of the Earth did not appreciably postdate the formation of the meteorites. If it did, isotopic anomalies from the decay of extinct radioactivities should almost certainly have been obliterated in the formation of the planet. Murthy and Hall (1972) thus maintain that models for the early history of the Earth must simultaneously be in accord with an essentially lowtemperature accretion and yet lead to the formation of the core nearly simul-

322

The Earth’s Core

taneously with, or very soon after, accretion. In any evolutionary model of core formation, this restriction effectively rules out pure Fe-Ni melting and sinking into the interior. Murthy and Hall therefore investigated low-melting components (in particular sulphur) that would satisfy the geophysical data of the core as well as a number of geochemical considerations of the mantle and core. The lowest-melting composition for any initial Earth is probably in the binary eutectic system Fe-FeS. The eutectic melting temperature of this system at zero pressure is 988°C. Additional components such as FeO, C, Ni would almost certainly reduce this value. Brett and Bell (1969) have shown that the Fe-FeS eutectic is little affected by pressure. This has been confirmed by Ryzhenko and Kennedy (1973) who carried out experimental work on the system Fe-FeS up to pressures of 60 kbar. They found that the temperature of the eutectic rises to a broad maximum of 1015°C in the 30-40 kbar region and then decreases to 1005°C at 60 kbar. The composition of the eutectic shifts with pressure in the iron-rich direction. If this trend continues, the melting temperatures of the eutectic must begin to rise again to approach that of pure iron. The curve that best fits the experimental data of Ryzhenko and Kennedy, which shows a broad maximum in the 30-40 kbar region, should also show a broad minimum at some pressure above 60 kbar before it begins to rise again. The authors offer no explanation for this strange behaviour. In general, the effect of pressure on the eutectic temperature is less than on the melting point of the end members (Newton et al., 1962) because of the large entropy of mixing. Thus it can be expected that in the interior of the Earth, the difference between the melting point of iron and an Fe-FeS melt would be even larger than that at zero pressure. At 1 atm, the difference in melting temperatures of pure Fe and the eutectic Fe-FeS is 550°C. From a consideration of the behaviour of this eutectic at higher pressures, Hall and Murthy have shown that this difference may be as much as three times higher in the core. The initial temperatures at zero time for a number of time-scales of accretion are shown in Fig. 5.7 (after Murthy and Hall, 1972).It can be seen that in all the models, melting temperatures of the Fe-FeS eutectic are exceeded for the bulk of the Earth’s mass but are well below the melting point of Fe. If core formation practically simultaneous with accretion is a significant constraint, it appears that the only way is to segregate the Fe-FeS eutectic liquids into the core as discussed earlier by Murthy and Hall (1970). Sulphur is highly depleted both in carbonaceous and ordinary chondrites as compared to its abundance in the silicate fraction of the Earth-it is depleted more than other volatiles such as HZO, the halogens and the rare gases. Murthy and Hall state that “we cannot conceive of any volatilization processes in a hot accretion model to lead to a sulphur loss more than of

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323

The Constitution of the Core

0

0

1

0.2

I

0.4

0.6

01)

ID

Radius (fraction of Earth radius) Fig. 5.7. Temperature profiles for the Earth for time-scales of accretion ranging from 5 x 10' to 1O6 years, schematically shown from Hanks and Anderson (1969). The lower three profiles are for rapid accretion of material maintaining low surface temperatures (300 K). The uppermost profile assumes an initial high nebular condensation temperature and radiative equilibrium during accretion and adiabatic compression in the interior for a short time-scale (5 x 10' years) accretion. (After Murthy and Hall, 1972.)

other volatiles". They concluded that either this pattern of depletion was characteristic of solid material that formed the Earth, in which case the fractionation must have taken place in the solar nebula during condensation processes, or that sulphur was segregated into the interior by Fe-FeS melting in the early Earth. The first possibility--cosmochemical fractionation during the condensation of matter in the solar nebula-is unlikely, and Murthy and Hall are forced to accept the second. It must be pointed out, however, that Ringwood (1966a,b) is of the opinion that most of the cosmic abundance of sulphur was not retained by the Earth and that the core contains only a small fraction of the primordial sulphur, which must have been lost from the Earth before or during accretion. This question will be discussed again in 95.8. As soon as the temperature in the accreting Earth exceeds 990°C, the FeFeS liquid will form and, because of its higher density, will sink down to form the core. Once the process has started the gravitational energy liberated will be available practically simultaneously with the accretion process, thus satisfying the constraint that the Earth's core be formed at the time of, or very soon after, accretion. The energetics of core formation by iron sinking to the centre has been discussed in #2.3 and 2.4. Birch (1965) has estimated that the

324

The Earth’s Core

energy available for heating is about 400calg-’.* The density of Fe is 7.9 g cm-3 and of the Fe-FeS eutectic about 5.0 g cmP3. Keeping other factors the same, the thermal energy released in Murthy and Hall’s model would be approximately 250 cal g-’ and would result in a temperature increase of z 1000°C. This temperature increase is probably an upper limit because segregation of the core during accretion will entail radiative losses. It can be seen from Fig. 5.7 that, even with this additional heat, metallic iron is not likely to melt in a substantial fraction of the Earth and large-scale silicate melting and degassing will certainly not occur. This is in keeping with the retention of radiogenic ‘29Xe formed in the mantle during the first few million years of the Earth’s lifetime. Murthy and Hall thus interpret the mode of formation of the solid IC as due to segregation of iron from a Fe-Ni-S melt. Since the liquid in equilibrium with a solid in a multi-component system cannot have the same composition as the solid, there must be a compositional difference between the IC and OC. The presently accepted density of the IC of about 13.5 g cmP 3(Bolt, 1972) is compatible with an Fe-Ni alloy. A consequence of this mode of formation of the IC is that the ICB be on the liquidus surface of the Fe Ni-S ternary system. The temperature at the ICB should thus be less than the melting point of iron. Not only sulphur but substantial amounts of carbon will also be incorporated into the core. The low density of the core is then due to the presence of about 15% of these elements in an Fe-Ni core. A composition of the Earth based on a core containing about 15% S and C and a “pyrolite” (Green and Ringwood, 1963) type of mantle would not correspond to any single class of known meteorites. Appropriate initial compositions can only be represented by a mixture of materials condensed over a range of temperatures in the solar nebula. A satisfactory mixture for the Earth’s composition with respect to major oxides, metal phases and volatiles can be obtained from about 40% carbonaceous chondrites, 45% ordinary chondrites and 15% iron meteorites (Murthy and Hall, 1970). These authors do not suggest that these proportions of meteorites actually accreted to form the Earth. The mixture is simply an approximate representation of bulk Earth material of solar nebular condensates in terms of the major classes of meteorites. Usselman (1 975a) has investigated the liquidus relations of the Fe-rich portions of the Fe-Ni-S system at pressures from 30 to 100 kbar. He found that up to 6.5 wt% Ni had very little effect on the melting relations of the FeFeS system at and below 80 kbar. He thus extrapolated the Fe-FeS eutectic to higher pressures in order to estimate eutectic temperatures at core pressures, using the compressibility of Fe and the calculated compressibility of

* In a later paper, Flaser and Birch (1973) revised this value to 590 cal g-

The Constitution of the Core

325

FeS in the Kraut and Kennedy (1 966) melting relationship. The eutectic temperatures and compositions are about 1800°C and 17.5 wt% S at the MCB, and about 2100°C and 15 wt% S at the ICB. In a later paper, Usselman (1975b) used these results to obtain a model of the formation, composition and temperature of the core. As the core-forming liquid coalesced, it would become gravitationally unstable and sink while the silicate material would be displaced. This process would lead to an initially homogeneous core of Fe-FeS eutectic composition. The high-pressure Fe-FeS eutectic conditions closely approximate the firstmelting liquid in the Fe-Ni-S system with up to 6.5 wt% Ni (Usselman, 1975a). Because of the loss of gravitational potential energy resulting from core formation there would be an increase of temperature. This increase in temperature would cause the incorporation of much of the available metallic Fe into the Fe-FeS liquid, resulting in a liquid richer in Fe than the eutectic composition at the ambient pressure. As the initial thermal profile approaches the present thermal profile, Fe will crystallize out along the Feeutectic liquidus and sink gravitationally to begin to form the solid IC. As the Earth cools, additional Fe will crystallize out of the liquid and enlarge the solid IC. The present composition of the OC may be approximated by the intersection of the temperature gradient with the liquidus surface. Usselman (1975b) pointed out that the ICB is probably a chemical boundary as well as a solidliquid boundary. This is supported by the earlier conclusions of Derr (1 969) and Press (1968) that the IC is essentially pure Fe or Fe-Ni. Bolt (1972) estimated that there is a density increase of about 1.8 g cm-3 at the ICB---such an increase is greater than would be associated with a solid-liquid transition of a homogeneous material. For the Fe-Ni-S core models (Murthy and Hall, 1970 Anderson et al., 1971) the minimum temperature of the O C is essentially the Fe-FeS eutectic. If temperatures are extrapolated to the pressure at the ICB, a temperature of approximately 2200°C is obtained (Usselman, 1975a). Using Higgins and Kennedy’s (1971) extrapolation of the melting temperature of pure iron (about 4250°C at the ICB), the possible range of temperatures at the ICB lies between 2200°C and 4250°C. For the Fe-FeS core model, Fig. 5.8 shows the sulphur content of the liquidus surface between Fe and the Fe-FeS eutectic, contoured with depth and temperature. Usselman (1975b) has estimated the densities of various Fe-FeS compositions on the liquidus surface at the MCB and at the ICB. It is then possible to select the liquidus compositions at these two boundaries that agree with the seismically-determined densities of the liquid OC. If the seismic densities of Haddon and Bullen (1969) are used, the composition of the O C at the MCB is about 10 wt% S, and at the ICB about 7 wt% S. The

326

The Earth's Core

5000

I

1

I

I

1

I Fe MELTING CURVE

Y

4000

-

3000

-

v

W

e

5

Qc

W

n

5

I

'

I

2000 0.1

I

I

14

I

1

I

1

0.2

0.3

0.4

0.6

0.5

FRACTION OF RADIUS INNER

CORE

I

OUTER CORE

I

MANTLE

Fig. 5.8. Schematic sketch of the sulphur content of the Fe-eutectic liquidus surface (contoured at constant sulphur content). The sulphur content of the core is determined by the intersection of the contours with the thermal gradient (After Usselman, 1975b.)

average sulphur content of the liquid O C is about 9 wtx. If the densities are chosen from the precision bands given by Bolt (1972), the composition of the liquid OC at the MCB lies between 13 and 9 wt% S, while at the ICB it lies between 8 and 5 wt% S. The average sulphur content of the liquid OC lies between 11 and 8 wt x. These results may be compared with shock wave data (to 400 kbar) of iron sulphide (King and Ahrens, 1973): when extrapolated to core conditions and compared to shock wave data for Fe, they satisfy the pressure-density profile for the OC with 10-12 wt% S. The compositions of the liquids indicate that the temperature at the MCB is between 2850 K and 3300 K (depending on the seismic densities used), and at the ICB between 3750 K and 4050 K. The thermal gradient derived by Usselman (1975b) implies that the liquid O C is stable against thermal convection, as the thermal gradient is the melting-point gradient (the liquidus surface). In such a case, fluid motions in the core, necessary for the generation

327

The Constitution of the Core l6 14

-

I

'

'FeS,,

I

I

'

'

I

3500 K (SWEOS)

I

I

Feo,S,

,

20

I

I60

1

OUTER CORE I 1 I I 200 240

-

INMR I

280

1

I

C

320

F

3500 KEWEOS) 5000 K (SMOS). 3500 K(SWE0S)

l

360

I

I

400

Pressure (GPO) F i g 5.9 Sulphur content (weight %) versus pressure in the IC and OC as inferred from constraining shock wave data t o seismologically determined (Hart e t a / 1977) pressure-density distributions for the earth SWEOS indicates shock wave equation of state Unlabelled curves are calculated using the fourthorder finite-strain equation (After Ahrens 1979 )

of the Earth's magnetic field would have to be driven by other means (see $4.6). More recently Ahrens (1979) has carried out shock wave experiments on pyrrhotite (Feo.$S) over the pressure range 0.03-1.58 Mbar. His results are consistent with the assumption that the Hugoniot curve in this pressure range indicates a denser polymorph that may be similar to the local bonding of sulphur in the liquid core of the Earth. The raw Hugoniot data for Feo.9S,FeS, and Fe, when constrained to the seismic density-pressure curve for the OC, indicate a systematic decrease of sulphur content with depth from 10% to 6.5% (see Fig. 5.9). If the shock wave data are reduced to isotherms, a nearly constant sulphur content in the range 9-12% results. These bounds on the sulphur content of the core are consistent with the Earth being depleted in S by factors of 1.8-3.2 with respect to the abundances of Si or 2.8-7.5 with respect to Fe abundances in carbonaceous chondrites. Ahrens thus concludes that although the shock wave data permit the major light element in the core to be sulphur, the Earth can be modelled as being depleted in sulphur along with other volatile elements. In a later paper, Brown et al. (1984) extended the Hugoniot data for pyrrhotite to a pressure of 2.74 Mbar (nearly two-fold compression). They found a minor discontinuity in the Hugoniot between 1 and 1.5 Mbar which they interpreted as the melting transition. At the MCB ( p = 1.35 Mbar, T = 4000 K), they estimated the density of liquid pyrrhotite to be . is the light alloying element of the core, their 7.80 & 0.20 g ~ m - If~ sulphur data are consistent with a homogeneous mix containing 10 5 4 wt% sulphur.

328

The Earth’s Core

There is some uncertainty in the values of auxiliary thermodynamic parameters needed to infer the equation of state (such as Griineisen’s parameter and the heat capacity). In spite of all these difficulties, the limited amount of data, and possible uncertainties in the interpretations as to which high-pressure-high-temperature phases are represented by the Hugoniot data, valuable information has been obtained from shock wave experiments on the possible constitution of the core. Using experimental, static high-pressure data, Kraft et al. (1982) calculated an isothermal EOS for various compositions of the monosulphid solid solution (MSS) in the Fe-Ni-S system. They used the method developed by Ullmann et al. (1976, 1977), which enabled them to extrapolate their results to conditions in the OC. A model of the O C composed of (Fe, Ni) and MSS, requires a MSS content of 30-35 wt% at the MCB and 13-17 wt% at the ICB. This corresponds to a sulphur content of 10.8-13.3 wt% at the MCB and 4.96.5 wt% at the ICB, consistent with.the results obtained by Ahrens (1979) from shock wave data. Kraft et al. also found that the sulphur content increases with increasing nickel content. Franck (1982) has investigated the gravitationally powered dynamo in the Earth’s core (see 94.6) under the assumption of liquid immiscibility in the Fe-S system. Experiments at normal pressures have shown that immiscibility develops in the Fe-Ni-S system after the introduction of a small percentage of either P, Si or C (Vogel, 1963, 1964). Whether such immiscibility persists at pressures corresponding to those in the O C is not known. If so, it would be interesting to know what constraints this would impose on convection in the OC. Verhoogen (1973) suggested that the Fe-S system at high temperature and pressure might resemble that of the S-Sb system at low pressures, which exhibits liquid immiscibility between Sb and the eutectic. If this were so in the Earth’s core, there would be two liquids above the liquidus: a heavy liquid with a low S content (about 2.5% by mass), representing the composition of layer F in equilibrium with the solid IC, and a lighter liquid with a higher S content (about 15% by mass), representing the composition of layer E. We do not know what the effect of pressure would be on such phase diagrams. If this is true for the Earth, then, because of liquid immiscibility above the IC boundary, the upward flux of S at the IC boundary could not alloy with the heavy liquid. This would lead to the nucleation of liquid droplets with a sulphur content of about 15% by mass. Franck (1982) suggested that such droplets could grow to such a size that buoyancy forces become strong enough to enable them to rise, perhaps releasing enough gravitational potential energy to drive the geodynamo. Another possibility is that the liquid immiscibility is confined to a small region above the liquidus. If this were so, the droplets would dissolve in the OC, leaving only compositional convection in the rest of the OC.

329

The Constitution of the Core

In a later paper, Stiller et al. (1986) used this model to infer a stably stratified layer about 1 1 km in thickness below the MCB. They then interpreted travel-time data from the seismic station MOX in the GDR by scattering at the base of this stably stratified layer.

5.7.3

Oxygen in the Outer Core

The idea that oxygen might be the principal light alloying element in the Earth's OC was revived by Ringwood (1978), who argued that the solubility of FeO in liquid iron might increase rapidly with both temperature and pressure. There has thus been increased interest in the high-pressure behaviour of the iron oxides. Shock wave data (McQueen and Marsh, 1966; Simakov et al., 1974) for haemetite (Fe,O,) and magnetite (Fe,O,) have since been supplemented by shock compression measurements by Jeanloz and Ahrens ( 1 980) on wiistite (Feo.940).They made measurements on FeO and CaO to pressures in excess of 1.5 Mbar (see Fig. 5.10). Phase transitions in both these oxides were discovered at about 0.7 Mbar. Because of the resulting density changes, considerable amounts of Ca O and FeO may be present within the Earth. Their data show that the density of the O C is equal to that of an equal mix (by weight) of Fe and FeO ( N 10 wt% oxygen) which is consistent with geochemical arguments for oxygen in the core (Ringwood, 1978). The most interesting aspect of the high-pressure behaviour of haematite and magnetite is the very large increases of density ( - 10% at 100 Gpa) associated with shock-induced polymorphism. The data for wustite also reveal a major shock-induced phase change that Jeanloz and Ahrens interpret as B1 =+B2 polymorphism. Jackson and Ringwood (1981) have reassessed the wiistite data and found that the zero-pressure density increase associated with shock-induced polymorphism is at least 1 G l 6 % and possibly as great as 18-28%. They thus conclude that the density increases for all three iron oxides is too large to be explained in terms of geometric re-arrangement (e.g. B1 S B2) of the usual Fe 2+,Fe 3+ and 0 2 -ions. Since the extent to which FeO is soluble in molten iron is determined primarily by the nature of the bonding in the competing crystalline FeO phase, Jackson and Ringwood (1981) examined the relative plausibility of a wide range of candidate structures, electronic configurations and bond types for high-pressure FeO. They proposed that the more covalently bonded nickel arsenide (NiAs) and derivative structures might accommodate very dense iron oxide phases of all stoichiometrics at high pressures-calculations indicate a B1 NiAs transformation pressure of -31 Gpa for FeO, with an estimated zero-pressure density increase of 11%. An additional mechanism by which FeO bonds

+

-

P

6

7

8

9

I0

II

12

13

Densify (106g ~ n - ~ ) Fig. 5.10. Shock-wave data for FeO (Jeanloz and Ahrens, 1980). Fe (McQueen e t a l . , 1970). Fe,O, and Fe,O, (McQueen and Marsh, quoted in Birch, 1966; not corrected for porosity)-heavy curvescompared with the seismologically determined compression curve of the core (Dziewonski etal.. 1975; Anderson and Hart 1976). The Fe and FeO data, corrected to core temperatures, are shown as light dashed curves. Hugoniots of mixtures of oxides and Fe corresponding to FeO are given as thin curves: and (Fe,O, - Fe,O,). are plotted but overlap; open squares are from f(Fe + Fe,O,). t(Fe + Fe,O,) Al'tshuler and Sharipdzhanov (1971 b). Hugoniots for only the high-pressure phase are shown, and the wustite data are corrected for initial porosity and non-stoichiornetry. (After Jeanloz and Ahrens, 1980.)

331

The Constitution of the Core

might be significantly shortened involves spin-pairing of 3d electrons. Using structural data for Co3+ oxides, Jackson and Ringwood estimated a spinpairing pressure of 50 Gpa for FeO with a density increase of 9% at the transition. The NaCl e NiAs polymorphism in FeO, possibly coupled with spin-pairing in Fe2+,provides an attractive mechanism for the incorporation of large amounts of oxygen into the Earth’s core. Liu et al. (1982) carried out a theoretical investigation of FeO at high pressures and concluded that stoichiometric FeO cannot be formed at temperatures above 600°C, regardless of pressure. Liu (1976) had shown earlier that the stoichiometry of Fe,O quenched from static high-pressure-hightemperature experiments decreases with increasing pressure in the range 100300 kbar. Liu et al. (1982) thus re-examined the Hugoniot data of Feo.940of Jeanloz and Ahrens (1980) and found that the stoichiometry for Feo 940 does not remain constant during the shock wave experiments, but decreases with increasing pressure as observed in the static experiments at lower pressures, the rate of decrease decreasing with increasing pressure. The value of x approaches 0.75 at pressures greater than 2000 kbar (Fe, 750 is equivalent to the stoichiometry of magnetite, Fe304). If oxygen is assumed to be the light component of the OC, constant stoichiometry would imply that the oxygen content decreases by a factor of 2 from the top (6 wt%) to the bottom (3 wt%) of the OC. On the other hand, if Fe,O is the stable phase at high pressures, the 0 content is approximately constant ( - 11-12 wt%) throughout the OC, implying that it may be chemically homogeneous. It is interesting to note that the bulk chemistry for the O C in the latter model ( Fe 2 , 4 0 Fe2.,0) is in close agreement with the ideal formula of F e 2 0 for the OC suggested by Sorokhtin (1971) and Bullen (1973a,b) (see also $5.6). Jackson et al. (1984) take issue with Liu et al.’s (1982) interpretation of the Hugoniot of Fe0.940of Jeanloz and Ahrens (1980). They claim that Liu et al.’s analysis was based on highly inappropriate equations of state for the hypothetically grossly non-stoichiometric wiistites. Thus, although further pressure- and temperature-induced variation in wustite, non-stoichiometry beyond 30 G P a is almost certain to occur. Jackson et al. believe that such variation in stoichiometry is not expected to account for the high densities of wiistite shocked beyond 70 GPa. They still prefer their interpretation of the shock wave data in terms of a pressure-induced phase transformation involving a change in crystal structure, electronic configuration or some combination of both (Jeanloz and Ahrens, 1980; Jackson and Ringwood, 1981). Liu and Bassett (1984) d o not accept the criticisms of Jackson et al. and find the above interpretation of Jackson and Ringwood (1981) inadequate. Yagi et al. (1985) have since carried out X-ray diffraction studies on wiistite up to 120 G P a at room temperature using a diamond-anvil cell high-pressure apparatus. No discontinuous volume reduction was found, in disagreement

-

-

332

The Earth’s Core

with the large density increase at 70GPa observed in shock wave experiments. It is clear that further work on this question is necessary. McCammon et al. (1983) have examined the process by which FeO may be segregated from a silicateeoxide mixture into core liquid and how the present core-mantle structure of the Earth may have evolved. They begin their investigation with a reassessment of the shock wave data of Jeanloz and Ahrens (1980), who found a phase transformation of Feo.940 at about 70 GPa. Although the structure of the high-pressure phases has not yet been determined, McCammon et al. believe that the large inferred density increase suggests that significant shortening of Fe-0 or Fe-Fe bond lengths, or perhaps both, is involved in the phase transformation. They therefore disagree with the suggestion of Liu et ai. (1982) that it can be explained by a progressive change in wustite stoichiometry resulting from ex-solution of Fe, and believe that the phase transition in FeO involves a change in crystal structure, possibly accompanied by a substantial change in electronic structure. They then calculated the FeO-MgO phase diagram and suggested that an increase in pressure in the system FeO-MgO will lead to a gradual ex-solution of an almost pure FeO high-pressure phase (h.p.p.), leaving an iron-depleted (Fe, Mg)O rocksalt (Bl) phase. They find that ex-solution of FeO (h.p.p.) from the rocksalt phase in the present-day lower mantle occurs only in the region immediately overlying the core. Before core segregation took place, however, the increased iron content of the rocksalt phase could place the exsolution region at lesser depths, perhaps 2000 km. They further showed that extrapolation of the solubility data of oxygen in liquid iron indicates that the liquid miscibility gap in the system Fe-FeO at atmospheric pressure should close at approximately 4250 K. At high pressure ( p 90 GPa) the phase diagram of Fe-FeO becomes qualitatively similar to the system Fe-FeS, displaying complete liquid miscibility and a eutectic temperature well below the melting point of pure iron. Ohtani and Ringwood (1984a) have carried out experiments on the solubility of crystalline FeO(c) in molten iron at temperatures between 2100 and 2550°C (Fig. 5.1 1). Their results agree well with the extrapolation of Fischer and Schumacher’s (1978) curve for the solubility of liquid FeO in molten iron in the 1523-2050°C temperature range. Together, the combined results indicate that at temperatures above 2500°C the solubility of FeO(1) in molten iron exceeds 35 mol% and that complete miscibility should be reached at temperatures around 2800°C. Ohtani and Ringwood have used the data to construct a phase diagram of Fe-FeO at ambient pressure (Fig. 5.12) and to estimate the solubility of FeO in molten iron in equilibrium with crystalline magnesiowustite (Mgo.sFeo.z)O(Fig. 5.13). Magnesiowustite is believed to be an important mineral in the lower mantle and would dissolve about 14mol% of FeO at 2500°C and 40mol% at 2800°C. Jeanloz and Ahrens

-

333

The Constitution of the Core

Temperahre

(OC)

1500

Experimental solubility of FeO(c) Calculated solubility of FeO(l)

0.2

Fig. 5.1 1. Lower limits to the solubility of FeO in molten iron in equilibrium with metastable crystalline FeO and with liquid ionic FeO. as determined in experiments by Ohtani and Ringwood (1984a). The solubility of liquid FeO in molten Fe determined by Fischer and Schumacher (1 978) is also shown (solid curve). The broken line represents an extrapolation of the high-temperature data of Fischer and Schumacher. (After Ohtani and Ringwood, 1984a.)

(1980) had earlier concluded that the density of the O C could be explained if it contained about 40 mol% FeO. The work of Ohtani and Ringwood indicates that if the core were in equilibrium with the mantle it should contain -4Omolx of FeO even if the temperature at the MCB were as low as 2800°C. In a companion paper, Ohtani et al. (1984b) carried out an experimental and theoretical investigation of the effect of pressure on the solubility of FeO in molten iron. When FeO dissolves in molten iron, it is believed to form a metallic solution (Ringwood, 1978). Ringwood considers the metallic Fe-0 solutions as mixtures of pure Fe and of a hypothetical FeO end-member possessing metallic bonding, which he denotes as FeO* that may be either liquid FeO(l)* or crystalline FeO(c)*. McCammon et al. (1983) had shown that the high-pressure phase (FeO(c)*) formed under shock wave compression

334

The Earth's Core

3ooo

ALLIC LIQUID

'

'\

\ \ I

.

I

I

'

I

I

REGION OF LIQUID IMMISCIBILITY LIQUID (Fe+FeO)

+

LIQUID FeO

Fe

Fe

20

'

.

IONIC LIQUID'~

+

LiQUlD FeO 40 60

80

FeO

FeO mol% Fig. 5 12. Phase diagram of Fe-FeO at ambient pressure based on the results of Fischer and Schumacher (1978) and Ohtani and Ringwood (1984a) Note that complete miscibility between (Fe-FeO) liquid and ionic FeO liquid will occur around 2800°C (After Ohtani and Ringwood, 1984a )

(Jeanloz and Ahrens, 1980) would be ex-solved from magnesiowustite at pressures exceeding 70 GPa. This phase is expected to dissolve in molten iron, forming a simple eutectic system. McCammon et al. developed a model of core formation that was characterized by the formation and segregation of Fe-FeO* liquid alloys only at considerable depths in the Earth, exceeding those corresponding to a pressure of 70 GPa. The results obtained by Ohtani et al. (1984b) imply that solution of large amounts of FeO* in molten iron could occur at much smaller pressures than were considered by McCammon et al. They carried out an experimental investigation of the effect of pressure on the position of the Fe-FeO eutectic. Thermodynamic calculations based on these experiments gave an estimate for the partial molar volume p* of FeO* that is in reasonable agreement with the theoretical estimates. They used the experimental value of P* to calculate the effect of high pressure on the Fe-FeO phase diagram (Fig. ;.14).* Solubility of FeO in molten iron in-

* Knittle and Jeanloz (Geophys. Res. Letters, December 1986) have determined experimentally the phase diagram of F e O to pressures of 155 G P a and temperatures of 4000 K. They found a metallic phase of FeO at pressures greater than 70 G P a and temperatures exceeding 1000 K implying that oxygen can be present as the light alloying element in the Earth's OC. The high pressures necessary for metallization suggest that the composition of the core was a late process during the Earth's accretion

335

The Constitution of the Core

0.1

-1.0.

v

0

0 -

, .

:

2./

. /

a

1.0.

/

!/'

/

LL

I'

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/

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20 50

creases sharply with pressure; the liquid immiscibility region contracts and disappears around 20 GPa, and they predict that the Fe-FeO phase diagram should resemble a simple eutectic system above about 20 GPa. Analogous calculations predict that the solubility of FeO in molten iron in equilibrium with magnesiowiistite (Mg,.,Feo.2)0 at 2500°C should increase from 14 mole% ( p = 0)to above 25 mole% at 20 GPa.*

* M. Brown (personal communication) has looked at recent velocity data for FeS and Fe,O, and found that both iron-oxygen and iron-sulphur mixtures can be made which match Earth model PREM in the OC to better than 5% in velocity. However there is a systematic increase in misfit with depth. He is of the opinion that these velocity data constrain the allowable core chemistry to a much greater extent than before, although it appears not possible to distinguish between alloying elements in the OC on the basis of seismic velocities.

336

300 The Earth's Core

1

I

5 GPa

Ll+L2

L1 + L 2

2000

FeO+L1

FC

30 01

Fe*L2

Fa0

I

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i

15GPa

Ll*LZ

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1 . Fe

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. . . . . . . .

I

1

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3 0 GPa

FeO+Ll

F.

20

Fe*FeO 40 60

80

f

I0

Composition (mol o/o FeO) Fig. 5.14. Schematic diagrams depicting phase relationships in the Fe-FeO system at pressures up to 30 GPa. The liquid immiscibility field is expected to disappear at pressures around 20 GPa (After Ohtani e t a / . 1984b.)

5.8 The Possibility of Potassium in the Core

It has been suggested that in the deep interior of the Earth, under the strongly reducing conditions prevailing at the time of Fe-FeS liquid segregation, the alkali elements K, Rb and Cs would show strongly chalcophilic tendencies in possible reactions between silicates and Fe-FeS liquids (Lewis, 1971; Hall and Murthy, 1971). Thermodynamic calculations by these authors have shown that in a mixture of silicates and Fe-FeS, equilibrations will result in

337

The Constitution of the Core

an incorporation of potassium into the sulphide melt in reasonable concentrations, at temperatures where the silicates remain solid but Fe-FeS exists as a liquid. This suggests that during core formation significant amounts of potassium can be withdrawn into the core. The chalcophilic behaviour of potassium under conditions appropriate to the primitive Earth has been verified by laboratory experiments (Goettel, 1972). Moreover, the incorporation of potassium into the core can proceed under less strongly reducing conditions than those required to introduce elemental silicon into the core. Hall and Murthy (1971) showed that during core formation it is possible to separate K from U and Th. Differential segregation of K into the core, leaving behind the initial complement of U and Th in the mantle and the crust would also be consistent with the low K/U ratio of terrestrial rocks ( 1 x lo4) relative to chondritic meteorites ( 7 x lo4) (see Wasserburg et al., 1964). The presence of 40K in the core has many implications for the thermal history of the core as well as the lower mantle. For the mixture chosen by Murthy and Hall (1972), the total K content would be about 500ppm. Assuming that about three-quarters of this has been segregated into the core, the K content, about 375 ppm, would lead to a present-day heat generation in the core of about 2 x 10” W. Thus, a significant amount of thermal energy due to radioactivity should exist in the core. If the solid IC was formed by the segregation of pure metal from an Fe-FeS melt, and is in chemical equilibrium with the liquid OC, the O C will be the site of the 40K. The heat produced by 40K in the liquid O C would thus be available to set up convective motions. Estimates of the total ohmic dissipation by currents in the core to maintain the geomagnetic field are of the order of 5 x lo9 W. Thus the present estimate of heat generation due to 40K in the core (2 x 10” W) is more than sufficient for the generation of the geomagnetic field even at very low thermodynamic efficiencies ( ~ 0 . 0 1 ) . Experimental work by Murrell & Burnett (1986) on the partitioning of K, U and Th between sulphide and silicate liquids indicates that U and Th partitioning into Fe-FeS liquids is much more important than K partitioning. Their experiments were carried out at 1.5 GPa and at 1 atmosphere. At 1 atmosphere measurabie U and Th partitioning into sulphide was nearly always observed, but K partitioning never. They do not believe the higher pressures and temperatures in the core would seriously affect their main conclusions. Pressure effects near the MCB would have to increase the partition coefficient for potassium by a factor of lo3 with a much smaller increase in that for uranium in order to have terrestrial K and U abundances at chondritic levels. However, as discussed in 44.6, the geodynamo may be driven by other means than thermal convection. The hypothesis that a large fraction of the Earth’s potassium is in an Fe-FeS core has been disputed by Oversby and Ringwood (1972). They first

-

N

-

338

The Earth’s Core

presented data on the distribution of K in meteorites. The distribution of K in the Abee enstatite and in two ordinary chondrites has been investigated by Shima and Honda (1967). On a strict analogy between Abee and the Earth, a maximum of 2.5% of the Earth’s K could be in the core-the ordinary chondrites analysed by Shima and Honda had less than 2% K in non-silicate phases. Oversby and Ringwood also measured the distribution of K between a synthetic basalt and an Fe, FeS metallic phase containing 28% S. In their first run, they found a minimum distribution coefficient for K between silicate and metal of 25, limiting the amount of K in the Earth’s core to less than 2% of the total amount available in the whole Earth. In their second run, they obtained a minimum distribution coefficient of SO, corresponding to a maximum of 1% of K in the Earth being in the core. The minimum distribution coefficients result from their inability to detect any K in the metal phase. Moreover, their experiments contained a much higher FeS/Fe ratio than would be possible for the Earth-this should have favoured entry of K into the metal phase. Their work points out the dangers of making predictions based upon thermodynamic calculations for which pure oxides are used as analogues of the actual silicate phases involved in the reaction of interest. The paper by Oversby and Ringwood (1972) brought forth objections from Goettel and Lewis (1973) to which Oversby and Ringwood (1973) replied. One point that Goettel and Lewis make is that the chemical and physical conditions that are relevant to the partitioning of K between the Fe-FeS core and the silicate mantle and crust are the conditions that existed during the primary differentiation of the chondritic Earth into core and mantle and not the conditions in the present crust and mantle. They therefore argue that the partitioning experiments of Oversby and Ringwood using highlydifferentiated basaltic material are not necessarily relevant to the distribution of K in a primitive, differentiating, chondritic Earth. Oversby and Ringwood mention further experiments they have carried out that answer some of the objections of Goettel and Lewis-these additional experiments confirm their earlier results, ruling out the possibility of any significant amount of K in the core. Both sets of authors agree, however, that up to 1.5% of the Earth’s K may be in the core. This may just be sufficient to supply sufficient energy through the decay of 40K to set up thermal convection in the core. Seitz and Kushiro (1974) have carried out a series of experiments up to pressures of 2.5 G P a on the melting relations of a bulk sample of the Allende Type 3 carbonaceous chondrite to try to establish the initial evolutionary sequence of a planet of chondritic composition. Their experiments suggest that ferrobasaltic liquid can be generated by partial melting of materials similar in bulk composition to the Allende meteorite under anhydrous conditions and in the presence of less than 10%metal. Under more reducing conditions, more metallic phase would precipitate, making the silicate melt more mag-

The Constitution of the Core

339

nesian. The latter case would be expected in planets like the Earth, with a large metal core. If the core formed after accretion, it would be expected to contain a large proportion of the nickel and sulphur of the system. In these experiments, two immiscible liquid phases were obtained, the iron-nickel sulphide melt containing no detectable potassium. Goettel (1976) later carried out experiments in which iron sulphide melts were equilibrated with K feldspar at 1030°C and 1070°C and found that the solubility of potassium in the melt increased rapidly with temperature. He calculated the potassium contents of iron sulphide melts equilibrated with plagioclase of chondritic composition to be 10% of the total potassium at 1500°Cand 62% at 2000°C. Ganguly and Kennedy (1977) carried out a series of partitioning experiments to 3.0 G P a and 1150°C with an omphacitic pyroxene as the solid phase and with oxygen fugacity in the field of metallic iron. They found that the solid silicate phase directly in contact with the potassium-bearing sulphide melt became enriched in K by a factor of about 40 in relation to the sulphide-in general agreement with the results of Oversby and Ringwood (1 972) and Seitz and Kushiro (1974). Bukowinski (1976b) obtained augmented plane-wave solutions to Schroedinger’s equation for the electronic state of potassium in the static (0 K) b.c.c. lattice. He found a series of pressure-induced electronic phase changes brought about by the sequential filling of initially unoccupied d-like electronic states. He concluded that at pressures as low as 500 kbar the ionic radius of potassium becomes comparable to that of iron and its electronic structure becomes similar to that of a transition metal. These electronic transitions could (in principle) greatly increase the miscibility of potassium in iron or iron-sulphide melts. However, Bukowinski did not consider the question of whether the electronic transitions in potassium would occur at comparable pressures when it is in the environment of an ionic or covalent phase. Somerville and Ahrens (1980) carried out shock wave experiments on KFeS, at pressures up to 110 GPa. The Hugoniot data indicate a phase change at 13 GPa. Comparing the inferred isentrope of KFeS, (h.p.p.) with those of Fe, Feo.9Sand FeS,, they found that the atomic volume of potassium in KFeS, is twice that of iron at 75 GPa. There is no indication of a marked phase transition or even an anomalous region of compression that might be due to a gradual 4s-3d type electronic transition in K at high temperatures, as suggested by Bukowinski (1976b). Somerville and Ahrens also carried out thermochemical calculations of the partitioning of K between a sulphide and silicate phase that showed that pressure does not have a pronounced effect on the relative stability of solid KFeS, and potassium aluminosilicate h.p.p. The calculations suggest that the high-pressure phase of KFeS2 would not be stable in relation to KAISiO, in the upper mantle or in relation to KAISi308

340

The Earths Core

in the lower mantle. Somerville and Ahrens conclude that “although the results cannot absolutely rule out the hypothesis that a large fraction of the terrestrial potassium budget has dissolved into a molten iron-sulphidebearing core, the analysis of the pressure-volume relation for potassium, iron, iron sulphides, potassium aluminosilicate and potassium iron sulphide yields no evidence in support of the hypothesis.”

References

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The Constitution of the Core

341

Bell, P. M. and Mao, H. K. (1979). Absolute pressure measurements and their comparison with the ruby fluorescence ( R , ) pressure scale to 1.5 Mbar. Carnegie Inst. Washington Year Book 78,665. Birch, F. (1947). Finite elastic strain of cubic crystals. Phys. Rev. 71,809. Birch, F. (1952). Elasticity and constitution of the Earth’s interior. J . Geophys. Res. 57,227. Birch, F. (1963). Some geophysical applications of high pressure research. In Solids Under Pres.sure (W. Paul and D. M . Warschauer, eds). McGraw-Hill, New York. Birch, F. (1965). Energetics of core formation. J . Geophys. Res. 70,6217. Birch, F. (1966). Compressibility; elastic constants. In Handbook of Physical Constants ( S . P. Clark, Jr., ed.) p. 97. Geol. Soc. Amer. Mem. 97. Birch, F. (1968). On the possibility of large changes in the Earth’s volume. Phys. Earth Planet. Int. 1, 141. Bolt, B. A. (1962). Gutenberg’s early P K P observations. Nature 196,122. Bolt, B. A. (1964). The velocity of seismic waves near the Earth’s core. Bull. Seism. Soc. Amer. 54. 191. Bolt, B. A. (1972). The density distribution near the base of the mantle and near the Earth’s centre. Phys. Earth Planet. Int. 5, 301. Born, M. and Meyer, J. B. (1982). Zur Gitter der Ionenkristalle. Z . Phys. 75, 1. Brett, R. (1976). The current status of speculations on the composition of the core of the Earth. Rev. Geophys. Space Phys. 14,375. Brett, R. (1984). Chemical equilibrium of the Earth’s core and upper mantle. Geochim. Cosmochim. Actu 48, 1183. Brett, R. and Bell, P. M. (1969). Melting relations in the Fe-rich portion of the system Fe-FeS at 30 k b pressure. Earth Planet. Sci. Lett. 6,479. Brown, J. M. and McQueen, R. M. (1982). The equation of state for iron and the Earth’s core. Adv. Earth Planet. Sci. 12,611. Brown, J. M., Ahrens, T. J. and Shampine, D. L. (1984). Hugoniot data for pyrrhotite and the Earth’s core. J . Geophys. Res. 89,6041. Bukowinski, M. S. T. (1976a). On the electronic structure of iron at core pressures. Phys. Earth Planet. Int. 13, 57. Bukowinski, M. S. T. (1976b). The effect of pressure on the physics and chemistry of potassium. Ceophys. Res. Lett. 3,49 I , Bukowinski, M. S. T. and Knopoff, L. (1976). Electronic structure of iron and models of the Earth’s core. Geophys. Res. Lett. 3,45. Bullen, K. E. (1946). A hypothesis of compressibility at pressures of the order of a million atmospheres. Nature 157,405. Bullen, K. E. (1949). Compressibility-pressure hypothesis and the Earth’s interior. Mon. Not. Roy. Astron. Soc. Geophys. Suppl. 5,355. Bullen, K. E. (1950). An Earth model based on a compressibility-pressure hypothesis. M o n . Not. Roy. Astron. Soc. Geophys. Suppl. 6, 50. Bullen, K. E. (1963). An index of degree of chemical inhomogeneity in the Earth. Geophys. J . 7, 584. Bullen, K. E. (1965a). O n compressibility and chemical inhomogeneity in the Earth’s core. Geophys. J . 9, 195. Bullen, K. E. (1965b). Models for the density and elasticity of the Earth’s lower core. Geophys. J . 9,233. Bullen, K. E. (1969). Compressibility-pressure gradient and the constitution of the Earth’s outer core. Geophys. J . 18,73. Bullen, K. E. (1973a). Cores of the terrestrial planets. Nature 243,68. Bullen, K. E. (1973b). O n planetary cores. The Moon 7,384.

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The Earth's Core

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Hanks, T. C. and Anderson, D. L. (1969). The early thermal history of the Earth. Phys. Earlh Planet. Int. 2, 19. Hart, R. S., Anderson, D. L. and Kanamori, H. (1977). The effect of attenuation on gross earth models. J . Geophys. Res. 82, 1647. Herndon, J. M. (1979). The nickel silicide inner core of the Earth. Proc. Roy. Soc. London A368, 495. Higgins, G. and Kennedy, G. C. (1971). The adiabatic gradient and the melting point gradient in the core of the Earth. J . Geophys. Res. 76,1870. Hildebrand, J. H. (1931). Gitterenergien von Thermodynamischen. Standpunkt Aus., Z.Phys. 67, 124. Jackson, I. and Ringwood, A. E. (1981). High-pressure polymorphism of the iron oxides. Geophys. J . 64, 767. Jackson, I., Ringwood, A. E. and McCammon, C. A. (1984). Comment on “High-pressure polymorphism of FeO? An alternative interpretation and its implications for the Earth’s core” by L. Liu, P. Shen and W. A. Bassett. Geophys. J . 77,279 Jeanloz, R. (1979). Properties of iron at high pressures and the state of the core. J . Geophys. Res. 84, 6059. Jeanloz, R . and Ahrens, T. J. (1980). Equations of state of FeO and CaO. Geophys. J . 62,505. Jordan, T. H. (1972). Estimation of the radial variation of seismic velocities and density in the Earth. P h D Thesis, California Institute of Technology. Jordan, T. H. and Anderson, D. L. (1974). Earth structure from free oscillations and travel-times. Geophys. J . 36,411. King, D. A. and Ahrens, T. J. (1973). Shock compression of iron sulphide and the possible sulphur content of the Earth’s core. Nature 243,82. Knopoff, L. (1963). Solids: Equations of state of solids at moderately high pressures In: High Pressure Physics and Chemistry ( R . S . Bradley, ed.), Vol. 1, p. 227. Academic Press, London and Orlando. Kraft, A., Stiller, H. and Vollstadt, H. (1981). The monosulfid solution in the Fe-Ni-S system: relationship to the Earth’s core o n the basis of experimental high-pressure investigations. Phys. Earth Planet. Int. 27,255. Kraut, E. A. and Kennedy, G. C. (1966). New melting law at high pressures. Phys. Rev. 151,668. Lange, M. A. and Ahrens, T. J. (1984). FeO and H,O and the homogeneous accretion of the Earth. Earth Planet. Sci. Lett. 71, 11 1 . Larimer, J. W. (1975). The effect of C / O ratio on the condensation of planetary material. Geochirn. Cosmochirn. Actn 39,389. Lewis, J. S. (1971). Consequences of the presence of sulphur in the core of the Earth. Earth Planet. Sci. Lett. 11, 130. Liu, L-G. (1974). Birch’s diagram: some new observations. Phys. Earth Planet. Int. 8,56. Liu, L-G. (1976). The high pressure phases of FeSiO, with implications for Fe2Si04 and FeO. Earth Planet. Sci. Lett. 33, 101. Liu, L. and Bassett, W. A. (1984). Reply to comment by Jackson, Ringwood and McCammon with further observations on wiistite and magnetite. Geophys. J . 77,283. Liu, L-G., Shen, P. and Bassett, W. A. (1982). High-pressure polymorphism of FeO? An alternative interpretation and its implication for the Earth’s core. Geophys. J . 70,57. Lyttleton, R. A. (1973). The end of the iron-core age. The Moon 7,422. Lyzenga, G. A. and Ahrens, T. J. (1979). A multi-wavelength optical pyrometer for shock compression experiments. Rev. Sci. Instrum. 50, 142 I. Lyzenga, G. A., Ahrens, T. J. and Mitchell, A. C. (1983). Shock temperatures of SiO, and their geophysical implications. J . Geophys. Res. 88,2431. McCammon, C. A,, Ringwood, A. E. and Jackson, I. (1983). Thermodynamics of the system

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Fe-FeO-MgO at high-pressure and temperature and a model for the formation of the Earth’s core. Geophys. J . 72,577. McLachlan, D. and Ehlers, E. G. (1971). Effect of pressure on the melting temperature of metals. J . Geophys. Res. 76,2780. McQueen, R. G. and Marsh, S. P. (1960). Equation of state for nineteen metallic elements from shock-wave experiments to two megabars. J . Appl. Phys. 31,1253. McQueen, R. G. and Marsh, S. P. (1966). Shock wave compression of iron-nickel alloys and the Earth’s core. J . Geophys. Res. 71,1751. McQueen, R. G., Marsh, S. P., Taylor, J. W., Fritz, J. N. and Carter, W. J. (1970). The equation of state of solids from shock wave studies. In Nigh Velocity Impact Phenomena (R. Kinslow, ed.), p. 294. Academic Press, London and Orlando. Mao, H. K. and Bell, P. M. (1976). Compressibility and X-ray diffraction of the epsilon phase of metallic iron (8-Fe) and periclase (MgO) to 0.9 Mbar pressure with bearing on the Earth’s mantle-core boundary. Carnegie Inst. Washington Year Book 75,509. Mao, H. K. and Bell, P. M. (1977). Generation of static pressures t o 1.5 Mbar. C a r n q i e Inst. Washington Year Book 76,644. Mao, H. K. and Bell, P. M. (1978). Static generation of 1.72 Megabars pressure. Carnegie Inst. Washington Year Book 77,908. Mao, H. K., Bassett, W. A. and Takahashi, T. (1967). Effect of pressure on crystal structure and lattice parameters of iron up to 300 kbar. J . Appl. Phys. 38,272. Mao, H. K., Goettel, K. A. and Bell, P. M. (1984). Ultrahigh-pressure experiments at pressures exceeding 2 megabars. In Proc. Int. Symp. Solid State Phys. Under Pressure. Muirhead, K. J. and Cleary, J. R. (1969). Free oscillations of the Earth and the D” layer. Nature 223,1146. Murnaghan, F. D. (1944). The compressibility of media under extreme pressures. Proc. Nut Acad. Sci. U S A 30,244. Murnaghan, F. D. (1951). Finite deformation o f a n elastic solid. Wiley, New York. Murrell, M. T. and Burnett, D. S. (1986). Partitioning of K, U, and T h between sulfide and silicate liquids: implications for radioactive heating of planetary cores. J . Geophys. Res. 91, 8 126. Murthy, V. R. and Hall, H. T. (1970). The chemical composition of the Earth’s core; possibility of sulphur in the core. Phys. Earth Planet. Int. 2,276. Murthy, V. R. and Hall, H. T. (1972). The origin and chemical composition of the Earth’s core. Phys. Earth Planet. Int. 6, 123. Newton, R. C., Jayaraman, A. and Kennedy, G. C. (1962). The fusion curves of the alkali metals up to 50 kilobars. J . Geophys. Res. 67,2559. Ohtani, E. and Ringwood, A. E. (1984a). Composition of the core I: solubility of oxygen in molten iron at high temperatures. Earth Planet. Sci. Lett. 71,85. Ohtani, E., Ringwood, A. E. and Hibberson, W. (1984b). Composition of the core 11: effect of high pressure on solubility of FeO in molten iron. Earth Planet Sci. Lett. 71,94. Oversby, V. M. and Ringwood, A. E. (1972). Potassium distribution between metal and silicate and its bearing on the occurrence of potassium in the Earth’s core. Earth Planet Sci. Lett. 14, 345. Oversby, V. M. and Ringwood, A. E. (1973). Reply to comments by K. A. Goettel and J. S. Lewis. Earth Planet. Sci. Lett. 18, 151. Press, F. (1968). Density distribution in the Earth. Science 160, 1218. Qamar, A. (1971). Seismic wave oelocity in the Earth‘s core; a study o f P K P and P K K P . PhD Thesis, University of California, Berkeley. Qamar, A. (1973). Revised velocities in the Earth’s core. Bull. Seism. Soc. Amer. 63, 1073.

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Ramsey, W. H. (1949). On the nature of the Earth’s core. Mon. Not. ROJJ.Astron. Soc Geophys. Suppl. 5,409. Rice, M. H., McQueen, R. C. and Walsh, J. M. (1958). Compression of solids by strong shock waves. In Solid State Physics (F. Seitz and D. Turnbull, eds), Vol. 6. Academic Press, London and Orlando. Ringwood, A. E. (1959). O n the chemical evolution and densities of the planets. Geochim. Cosmochim. Acta 15,257. Ringwood, A. E. (1961). Silicon in the metal phase of enstatite chondrites and some geochemical implications. Geochim. Cosmochim. Acta 25, 1. Ringwood, A. E. (1966a). Chemical evolution of the terrestrial planets. Geochim. Cosmochim. Acta 30,41. Ringwood, A. E. (1966b). The chemical composition and origin of the Earth. In Advances in Earth Science (P, M. Hurley, ed.). M I T Press, Cambridge, Mass. Ringwood, A. E. (1978). Composition of the core and implications for origin of the Earth. Geochem. J . 11, 1 1 1. Ryzhenko, B. and Kennedy, G. C. (1973). The effect of pressure on the eutectic in the system Fe-FeS. Amrr. J . Sci. 273, 803. Sato, M. (1980). In Abstracts, X I Lunar & Planetary Science Conference, p. 974. Seitz, M. G. and Kushiro, I. (1974). Melting relations of the Allende meteorite. Science 183,954. Shima, M. and Honda, M. (1967). Distribution of alkali, alkaline earth and rare elements in component minerals of chondrites. Geochim. Cosmochim. Acta 31, 1995. Simakov, G. V., Pavlovskiy, M. N., Kalashnikov, N. G. and Trunin, R. F. (1974). Shock compressibility of twelve minerals. Izu. Earth Phys. 8, 1 1. Somerville, M. R. and Ahrens, T. J. (1980). Shock compression of KFeS, and the question of potassium in the core. J . Geophys. Res. 85,701 6. Sorokhtin, 0.G. (1971). Possible physicochemical process involved in the formation of the core of the Earth. Dokl. Akad. Nauk. S S S R 198,29. Stacey, F. D. (1 977). Applications of thermodynamics to fundamental Earth physics. Geophys. Surv. 3, 175. Stacey, F. D., Brennan, B. J . and Irvine, R. D. (1981). Finite strain theories and comparisons with seismological data. Geophys. Suru. 4,89. Stiller, H., Franck, S. and Kowalle, G. (1986). O n the formation of a stably stratified layer near core-mantle boundary. J . Geodyn. 5,303. Swenson, C-A. (1966). Lithium metal: an experimental equation of state. J . Phys. Chem. Solids 27, 33. Taggart, J. and Engdahl, E. R. (1968). Estimation of PCP traveltimes and depth to the core. Bull. Seism. Soc. Amer. 58, 1293. Ullmann, W. and Pan’kov, V. L. (1976). A new structure of the equation of state and its application in high-pressure physics and geophysics. VerqfJ:Z e n t r a h s t . Phys. Erde, Potsdam 41, 1. Ullmann, W. and Pan’kov, V. L. (1980). Application of the equation of state to the Earth’s lower mantle. Phys Earth Planet. I n t . 22, 194. Usselman, T. M. (1975a). Experimental approach to the state of the core: Pt I The liquidus relations of the Fe-rich portion of the Fe-Ni-S system from 30 to 100 kb. Amer. J . Sci. 275,278. Usselman, T. M. (197%). Experimental approach to the state of the core; Pt I1 Composition and thermal regime. Amer. J . Sci. 275,291. Van Thiel, M. (1967). Compendium of Shock Ware Data. Univ. Calif., Radiation Lab. Rept. UCRL 50108, 1967. Verhoogen, J. (1973). Thermal regime of the Earth’s core. Phys. Earlh Planet. Int. 7,47. Vogel, R. (1963). Arch. Eisenhiitt Wes. E3, 21 1.

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Chapter Six

The Cores of Other Planets

6.1 introduction

In this chapter the question of whether the other terrestrial planets (Venus, Mars and Mercury) and the Moon have cores will be discussed. Since it is believed that the Earth’s magnetic field arises from motions in the fluid, predominantly iron OC, the question of whether these other bodies possess a magnetic field is of crucial importance to whether they also have cores. In this respect, of all the known planetary bodies, only Jupiter, Saturn and Uranus, besides the Earth have been found to have a significant magnetic field. It must not be forgotten, however, that in the case of the Earth the poloidal field is probably only about 1% of the toroidal field. Thus, failure to observe a poloidal field in another planet does not in itself imply that the planet has no fluid core containing a toroidal field. In a series of papers, Anderson and Baumgardner (Anderson and Baumgardner, 1980; Baumgardner and Anderson, 1981) estimated the physical properties of the interiors of the terrestrial planets-particularly their temperature distribution. They used a very simple equation of state in which the thermodynamic variables are separated (see (5.7) and (5.9) and discussion in 45.1):

dV?

= P(v,o)

where the thermal pressure pthis given by

347

+ Pth

(6.1)

34%

The Earth's Core

and a and b are independent of depth, and are taken from experimental data appropriate to some assumed chemical model. The term a is found from experimental data on a,the thermal coefficient of expansion, and kT, the isothermal bulk modulus ( a = ak, at high T ) . The term b is a correction factor allowing for the fact that kT becomes small at low T (Anderson 1979, 1980). Anderson (1982) examined the experimental and theoretical data appropriate to mantle materials to justify the use of (6.1) and (6.2) for planetary interiors and obtained density and temperature profiles for the Earth, Mars, Mercury, Venus and the Moon (see Fig. 6.1). Shaw (1979) has calculated the radial distribution of energy released in core formation for all the terrestrial planets and the Moon-using the method he employed (1978) in the case of the Earth (see $2.4). In all cases, core formation results in the core receiving a disproportionately large amount of the gravitational energy release, with the result that the core is initially quite hot while the mantle is comparatively cool. Shaw also showed that it is necessary for a certain amount of core phase to separate and accumulate before the energy released by gravitational settling is sufficient to supply the latent heat of fusion of the core phase. The amount of melting required to reach this point varies according to the total mass of the planet and mass fraction of core, and is < 5% in the case of the Earth and 37% for the Moon. O n the basis of mean energy release, core formation is likely to be a runaway process in the case of the Earth and Venus, but not for the Moon. Shaw concluded that for Mercury and Mars it may be minimal. However, core separation for all the terrestrial planets appears to be a self-sustaining process provided that a small amount of a dense potential core phase is present. The radial non-uniformity of energy release may require an induction period before the process can begin-even for the larger planets. Figure 6.2 shows the temperature profiles produced by core separation. Schubert (1 979) has discussed in detail the importance of sub-solidus convection in the mantles of the terrestrial planets, and Schubert et al. (1979a) have traced their thermal evolution taking this into account. They showed that sub-solidus convection can quickly remove large amounts of heat from the deep interior of a planet. Even for Rayleigh numbers only 50-100 times the critical value, the time-scale for the removal of this heat is comparable to the age of the Earth (the Rayleigh numbers for the mantles of the terrestrial planets were probably greater than lo4 times the critical value). Schubert et al. (1979b) showed that vigorous mantle convection removes several times the latent heat of fusion from the core during the thermal evolution of a planet, and then posed the question of why the core is not completely frozen. Their reason, in the case of the Earth, lies in the large difference between the core and mantle melting temperatures at the MCB (cf. Fricker et al., 1976). This difference in melting points puts a limit on the reservoir of internal

-

The Cores of Other Planets

349

energy in the core that can be drawn upon by mantle convection before solidification occurs. At the lower pressures in the interiors of the Moon, Mercury and Venus, the differences in the melting temperatures of iron and silicates will be much less. 6.2

Mercury

Because of its small size, its proximity to the sun and its low reflectivity, it is very difficult to observe telescopically and to photograph the markings on Mercury. However, since the 1880s most astronomers believed that Mercury was rotating slowly with a period equal to its orbital period of 88 days. It was not until 1965 (Pettengill and Dyce) that delay Doppler maps of the surface with radar showed the rotation period to be 59 k 3 days. Following these radar observations, several groups have independently re-examined the visual determinations of the rotation period and concluded that the optical data are consistent with a rotation period of about 59 days-in fact the most accurate rotation rate for Mercury (58.66 days) has been obtained from visual data (Smith and Rees, 1968; Chapman, 1968). Colombo (1965) first noticed that the observed sidereal spin period T, was nearly two-thirds of the 88-day orbital period To and suggested that the axial rotation might be “locked” to the orbital motion in a three-halves resonance state by the additional solar torque exerted on an axial asymmetry in Mercury’s inertia ellipsoid. Colombo and Shapiro (1966) investigated a two-dimensional model of Sun-Mercury interaction and showed that resonances occur when a planet makes an integral number of half rotations during one orbital revolution, i.e. when T, = 2T0/k (k an integer) (6.3) Thus, the solar torque exerted on the permanent asymmetry could cause Mercury to be trapped into a k = 3 resonance spin state (see also Goldreich and Peale, 1966). In order to explain why Mercury is trapped in a resonant state of rotation with a period two-thirds of its orbital period, Runcorn (1977a) proposed that Mercury is not in a state of hydrostatic equilibrium, and showed that if this were the case its gravity field must include a second-degree harmonic term. Runcorn further argued that this must be caused by solid-state creep driven by thermal convection, rather than be due to a distortion produced earlier in the planet’s history and retained by the finite strength of its interior. It is not easy, however, to explain a second-degree harmonic convection pattern in the silicate mantle of Mercury if it has a large iron core. Runcorn suggested that perhaps Mercury has only a small iron core (0.1-0.3 of the planet’s radius). If this is the case, differentiation of Mercury is still very incomplete.

a-

MARS

/-----=2000

-

m

'5

6-

rn

/

v

/

/

/

- 1500p v

E3

I

- 1000

U

!g

E

c

I

7 I J

01

Lit-hosphere thickness = 200 krn - 500 Cold uncompressed mantle density = 3.44 g ~ r n - ~ Cold uncompressed core density = 5.7 g ct7-1;~ ' 0

Fig. 6.1. Density (solid line) and temperature (broken line) distributions in Mars, Mercury, Venus and the Moon. (After Anderson, 1982.)

-- ---

2000

- 150O0S

6-

h

0

v

5

f

s

I

-1 1

/ 0

L

- 1000

I I ,

t u) :

n

t3

I

m 1 4 -

v

U

E

P

2

1 I M R = 0.346 Lithosphere thickness = 300 km Cold uncompressed mantle density = 3.5 g ~ r n - ~ Cold uncompressed core,density = 7.0 g cm-:

- 500

'

0

1500

h

U 0 v

1000

g L

O

L Q)

unto::

E"

Core radius = 412 km

4*

21,,/''

f

Lithosphere thickness =lo00 km- 500 dcore ; density = 6.2.g cm-3

,

Cold uncompressed -3 mantle density = 3.40 g cm 0

0 0

400

800

1200

Dept-h ( k m ) Fig. 6.1. continued

1600

352

The Earths Core

8000

I

I

I

Earth

W

-

6000

0 v

LI

L

al

-

4000

f

F -

2000

0

0

2000

4000 Depth

6000 (km)

8000

Fig. 6.2. Temperature profiles in the terrestrial planets produced by core formation. (After Shaw,

1979.)

Information about the magnetic field of Mercury has been obtained from two triaxial fluxgate magnetometers carried by Mariner 10, which made three passes by the planet from March 1974 to March 1975; only the first and third were suitable for studying its magnetic field. O n the first encounter, a maximum field strength of 98y was observed at closest approach, 723 km (Ness et al., 1975a). A well-developed detached bow shock was observed behind which a magnetosphere-like region was crossed with boundaries similar to the terrestrial magnetopause. The observed magnetic field is a factor of 5 greater than the interplanetary magnetic field strength ( 1 87) measured outside the bow shock. The features of the third pass were similar to those observed in the first. This pass was closer to the planet (327 km) and at higher latitudes and a magnetic field of over 400y was measured. External current systems strongly affected measurements made by Mariner 10 and there has been much controversy over the strength of the magnetic moment of Mercury. The first estimate of the moment was given by Ness et a/. (1974b) who obtained a value of 5.5 x 10’’ gauss cm3. Ogilvie et al. (1977) later revised this estimate to 3.1 x 10” gauss cm3 using additional solar wind ion data deduced from electron measurements. Ness et al. (1 974b) also calculated the coordinates of a simple model of an offset, tilted dipole. This dipole is orientated within 20” of the ecliptic pole, i.e. it is almost aligned with the axis of rotation of the planet and is offset 0.47 Mercury radii. Ness et al. (1975b) later added two external sources to their inversion (a uniform field

The Cores of Other Planets

353

and the first harmonic) and obtained a value of 5.1 x 10” gauss cm3 for the planetary moment. Using data from Mercury 111, Ness et al. (1975a) obtained a value of 4.8 x loz2 gauss cm3 and combining Mercury I and I11 data a value of 5 x loz2gauss cm3, assuming one external (uniform) source and one internal source. Whang (1977) employed a completely different method using an image dipole and a two-dimensional current sheet to fit the Mercury I and 111 data and computed quadrupole and octupole (axial) terms as well. He obtained a value for the moment of 2.4 x 10” gauss cm3: the ratio of dipole to quadrupole to octupole fields at the surface was 1.0:0.45:0.29. Jackson and Beard (1977) and Ng and Beard (1979) used a different approach to estimate the magnetic moment by fitting the data to a magnetospheric model field. They calculated the ratio of Mercury’s compressed dipole field to that of the Earth for each encounter separately. For Mercury I, Jackson and Beard found that the dipole moment is not very sensitive to the addition of a quadrupole moment, although the opposite was true for Mercury 111-presumably because it came much closer to the planet. Their best overall dipole moment was 2.5 x loz2 gauss cm3, the axial quadrupole moment being about two-thirds this value. Ng and Beard (1979) repeated the analysis of Jackson and Beard but used an offset dipole rather than a dipoleplus-quadrupole; they obtained a value of 2.75 x 10” gauss cm3 with a 0.19 Mercury radii northward displacement. Slavin and Holzer (1979a,b) believe that the above analyses have underestimated the dynamic pressure of the solar wind during the encounters and did not take into account the tangential stress on Mercury’s magnetopause. gauss Their estimate of Mercury’s moment is much higher at 6 _+ 2 x cm3. The magnetic field of Mercury is almost certainly of internal origin, although it is not easy to account for it. Sharpe and Strangway (1976) and Stephenson (1976a) have suggested a magnetized crust. For a global moment to have arisen, the crust must have cooled through the Curie point of some carrier of magnetic remanence in an applied field. If the magnetizing field were external, there must have been an extremely high (- 10 gauss) steady primordial interplanetary field. If the source were an internal dipole, Runcorn (1975a,b) showed that an overall dipole moment would not arise as the crust cooled, since a dipolar magnetized shell produces no external field. Stephenson (1976a) showed that it is possible to produce a field as large as Mercury’s present field by an internal dipole source if second-order effects (such as crustal asymmetries and finite cooling time) are present, but the necessary conditions are rather extreme. In any case, as Srnka and Mendenhall (1979) have pointed out, if the magnetic field of Mercury reverses (as does the Earth’s), the amount of crustal magnetization in any one direction would be

354

The Earth's Core

severely limited. Russell (1980) concludes that Mercury most probably has a presently active internal dynamo. Constraints on models of the interior of a planet are given by astronomical data such as its mass and mean diameter. Recent determinations of these parameters by radar techniques have a much greater accuracy than that attainable by ordinary optical means. The mean density of Mercury is about the same as that of the Earth and the problem is to account for this for a small planet with a mass intermediate between that of Mars and the Moon. If the composition of Mercury is anything like that of the other terrestrial planets, it must have a very large iron core surrounded by a thin rocky mantle. Plagemann (1965) estimated the core radius to be about 0.86 that of the planet. He also estimated the temperature distribution in the planet and concluded that neither the core nor the mantle is molten. This, together with its slow rate of rotation, would seem to prohibit any planetary magnetic field by dynamo action. However, his thermal calculations indicate that, inside Mercury, heat flows continuously through the MCB, which could be considered a natural semi-conducting junction. Potential differences may then be generated by the Seebeck effect. If a complete circuit exists, it is possible that a small magnetic field may be set up. Lewis (1972), in a discussion of the chemistry of the solar system based on the sequence of condensations in the cooling solar nebula, concluded that Mercury has a massive core of Fe-Ni alloy surmounted by a small mantle of Fe2+-free magnesium silicates. The high density of Mercury is thus attributed to accretion at temperatures so high that MgSi03 is only partially retained but Fe metal is condensed. Siegfried and Solomon (1974) carried out a series of calculations on the thermal evolution of Mercury using the gross physical properties of the planet and the cosmochemical model of Lewis (1972). Their computations are based on the heat conduction equation. They used the finite-difference method described by Toksoz et al. (1 972), which allows for melting, differentiation and simulated convection. When the melting temperature of Fe is reached at a grid point, and an additional amount of heat corresponding to the heat of fusion of iron has been produced, the material at that point is considered to have melted. Within the melted region, complete differentiation of silicate from metal is assumed. The lower grid points in the region are assigned physical property values corresponding to iron, and the upper grid points values appropriate to a silicate. All radioactive heat sources are assumed to go into the silicate portion, since U and Th can be accommodated in silicate structures fairly readily, Convection is simulated in the molten silicate layer by holding the temperature to the melting curve and shifting any excess heat upwards. Solid-state convection is not considered. Siegfried and Solomon argued that because of the high thermal conductivity of the metallic fraction of Mercury, solid-state convection is unlikely to postpone or prevent

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355

differentiation of an originally homogeneous planet-its principal effect would probably be to lower temperatures in the mantle by a few hundred degrees for some time after differentiation until the present. A fullydifferentiated core would have a radius equal to 75% of the planetary radius and a mass equal to 66% of Mercury’s total mass. Their models favour a solid core at present. Fricker et al. (1976) have investigated the thermal evolution of Mercury and believe that a liquid O C 500 km thick may persist if account is taken of a thermal barrier at a liquid iron-solid silicate MCB. The thermal barrier is the result of the higher melting point and lower thermal conductivity of the silicates. The barrier permits temperatures in the OC to rise above the melting point of iron near the MCB thereby decreasing the core temperature gradient. The models of Siegfried and Solomon (1974) neglected the discontinuity in melting temperatures and thus predicted solidification of the core and a large molten region in the mantle. Whether a differentiated model leads to a molten core persisting to the present depends critically on initial conditions, the values of the physical parameters assumed, and the possibility of solidstate convection in the mantle. Cassen et al. (1976) have shown that if solidstate convection in the mantle is permitted, it must contain a density of heat sources comparable to the Earth’s mantle-wide average if the core is to remain molten. From the general similarity of the surfaces of the Moon and Mercury, particularly the extensive system of smooth plains inferred to be of volcanic origin (Strom et al. 1975; Trask and Guest, 1975), Murray et al. (1975) have argued that there must be at least local melting and hence silicate-metal differentiations in at least the outer regions of Mercury. The surface features of Mercury have been interpreted as thrust faults indicative of planet-wide compressive horizontal stress. The faults are most abundant in the oldest geological units-those older than, or contemporaneous with, heavy bombardment (Trask and Guest, 1975)-but also transect the youngest units, the smooth plains (Strom et al. 1975). However, Solomon (1976) has pointed out that differentiation of a core from an originally homogeneous planet would have involved an increase in planetary radius and large tensional horizontal stresses at Mercury’s surface. (See Fig. 6.4 which shows the change of radius with time for the thermal history model for Mercury shown in Fig. 6.3. This model was obtained by Solomon (1976) following essentially the procedures of Siegfried and Solomon (1974).) Solomon (1977) thus argues that core formation must predate the oldest surface terrain on Mercury, in particular the terminal phase of heavy surface bombardment. If this time is placed at -4 x lo9 years ago, by analogy with the Moon, high initial temperatures are required throughout much of Mercury’s interior in order that global differentiation be essentially complete at that time. Tozer (1985) is

356

The Earth's Core

3000

I

I

I

I

2500-

-v

-

2000 -

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I

1.8 Ga

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OL

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I

400

800

1200

1600

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I1

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Depth ( k m ) Fig. 6.3.A model for the thermal evolution of Mercury. Initial temperature is 1400 K (after Lewis, 1972) plus a contribution from adiabatic compression. Time in billions of years (Ga) since planetary formation is shown adjacent to the corresponding temperature profile. The temperature of metal-silicate differentiation is taken to be the Fe melting curve, labelled solidus. (After Solomon, 1976 )

critical of the use of surface features as justification for the use of heat conduction theory of heat transfer, arguing that viscosity controls the temperature distribution in a planet through efficient solid-state convection (see $3.8). In the model of Fig. 6.3, core formation begins 1.2 x lo9 years after planetary origin and is complete by 1.8 x lo9 years.* The total heat gained during differentiation is equivalent to a mean temperature rise of less than 700K. Because the thermal conductivity of the iron core is high, and because all radioactive heat sources are presumed to remain in the silicate phase during differentiation, the core cools rapidly in this model and is solid 1.5 x lo9 years after core infall is complete. Solomon (1976) also estimated what fraction of U and Th would have to be trapped in the core to keep the iron moll ten at present for a thermal model otherwise similar to that of Fig. 6.3. He showed that if 10% of the total U and Th were uniformly distributed in the core, it would retard core solidification by lo9 years. More than 20% of Mercury's U and Th would need to be distributed in the core to keep it at least partly melted at present-such fractionation seems very unlikely. Toksoz and Johnson (1977), using similar models to those of Siegfried and

* A total duration of about 500 M a for core segregation with most of the differentiation occurring within 200 M a is typical of thermal models with a nearly flat initial temperature profile.

357

The Cores of Other Planets

5

I

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0

I

1

Differentiated

p

fP, T )

Core segregation h

E -5

Y

v

Dt

-' Q

-10

CI

L

Q

-1 5

-20

0

1

2

t (10'

3

4

years)

Fig. 6.4. Radius of Mercury versus time for the thermal model of Fig. 6.3. The finite-strain equation of state prior to core differentiation is that of a homogeneous mixture of 66.5wt% F e N i and 33.5wt% silicate (Siegfried and Solomon, 1974). After differentiation, separate equations of state are used for mantle and core. For comparison, a curve for R(t) after core infall using the equation of state for a homogeneous mixture is shown as a broken line. The rapid expansion prior to completion of differentiation is accompanied by large horizontal tensional stresses at Mercury's surface; the subsequent contraction by 2 km in radius (cf. Strom eta/., 1975) is accompanied by horizontal compressive stresses at the surface. (After Solomon, 1979.)

Solomon (1974) but with earlier core formation, estimated that 156 ppb 40K in the core would be needed for the iron to be at least partly molten at present. This corresponds to a total potassium abundance in Mercury of about 0.1% by weight, or a bulk K/U ratio of about 20,000. Such a high ratio is not in accord with cosmochemical models of the terrestrial planets (Lewis, 1972; Grossman, 1972). Solomon (1977) found that for differentiation of an initially homogeneous Mercury to be essentially complete by 4 x lo9 years, extensive early heating is necessary. Figure 6.5 shows a thermal history model with a very hot initial temperature and early core segregation; differentiation is complete after 0.4 x lo9 years. Because the heat sources (U and Th) are uniformly distributed in the mantle, planetary cooling is very slow. The main differences between 0.4 and 4.6 x lo9 years are a slightly cooler and thicker lithosphere and a small solid IC for the present state.

358

The Earth’s Core

3000

I

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2 500 h

u

0

v

2000

t 41

2

1500

Q,

CI.

5

I-

1000

500

OO

400

800

1200 1600 Depth ( k m )

2000

2400

Fig. 6.5. A thermal history model for Mercury with hot initial temperature (central temperature 1800 K) and early core formation. Time in billions of years (Ga) since planetary formation is shown next to the appropriate temperature profile. The Fe melting curve is from Liu and Bassett (1 975). and the silicate melting curve from Siegried and Solomon (1 974). (After Solomon, 1977.)

Gubbins (1977) has shown that, provided part of the core is liquid, the observed magnetic field of Mercury can be maintained by a dynamo driven by the heat released by cooling of the whole planet. If heating is required at a rate of 10” W, a drop in temperature of 10 K in lo9 years would suffice. The IC would grow in radius by 30 km. Cassen et al. (1976) have considered the implications for the thermal history of Mercury of assuming that its magnetic field is driven by an internal dynamo, thereby demanding the existence of a metallic liquid core or shell within the planet. They showed that, in the absence of internal heating of the mantle, Mercury’s core would solidify in less than 3 x lo9 years. To obviate this, they considered a number of possibilities-high mantle viscosity, an iron-sulphide component in the core and heat production in the core. Geochemical considerations and radioactive abundances argue against the retention of significant quantities of radioactives in the core (see 92.3). If the kinematic viscosity of Mercury’s mantle were considerably higher than cm2 s-l, the mantle would be stable against solid convection. In such a case, Fricker et al. (1976) have shown that a molten iron core would not completely solidify. If Mercury had a core containing FeS, its melting point would

The Cores of Other Planets

359

be considerably lowered and Mercury’s mantle could be at low enough temperatures to be stable against thermal convection. However, if heat sources have already been removed from the mantle, Cassen et al. showed that conductive cooling might then be sufficient for solidification of the iron sulphide. Cassen et al. conclude that, in order to avoid complete core solidification, a minimum heat source density at least comparable to the global average for the Earth’s mantle must be retained in Mercury’s mantle. Toksoz et al. (1978) have considered the thermal evolution of the terrestrial planets taking into account conduction, solid-state convection and differentiation. Geological, geochemical and geophysical data indicate that the planets were heated during, or shortly after formation, and that they all started their differentiation early in their history. Initial temperatures and core formation play the most important roles in early differentiation, while the size of the planet is the primary factor in determining its present-day thermal state. The initial temperatures chosen by Toksoz et al. show a peak towards the melting curve near the surface. The details of conductive models can be found in Toksoz et al. (1972a,b), the effects of melting in Reynolds et a / . (1966), and of solid-state convection in Young (1974). Figure 6.6 after Toksoz et al. (1978) shows the temperature profile of Mercury as a function of time. The solidus of the mantle was taken as that of diopside and the initial temperature calculated with a base temperature of 1400°C. After initial formation, Mercury starts to heat up, and when the core is formed a substantial amount of potential energy is released-sufficient to raise the temperature by 600-700 degrees. The whole planet might melt, leading to extensive differentiation. The planet then starts to cool rapidly. Solid state creep in the mantle takes place up to about 2.5 x lo9 years after formation, when the mantle probably becomes rigid. Figure 6.6 assumes a heat generation of 1.5 x J s - l in the core, and shows that the model has a molten core at the present time. Figure 6.7 shows the present-day thermal profiles for Mercury for various heat source densities in the core. If there are no heat sources, the present-day core temperature is below the Fe solidus and the core is solid. The minimum heat source density necessary to maintain a molten core is about 1.5 x lo-’’ J c ~ s-’. - ~This is equivalent to about 2 p p b of the concentration within the core. Figure 6.8 shows the evolution of Mercury in space and time. Core separation is complete 1.5 x lo9 years after formation, and large-scale partial melting ceased about 2 x lo9 years ago. On this model, if Mercury had an iron core its radius would be about three-quarters that of the whole planet. 6.3 Mars

The Mariner 4 spacecraft passed within about 13,200km of Mars on July 14-

3000

I

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400

800

1

I

I

1

2000

2440

2400 h

1200 1600 Depfh (km)

Fig. 6.6. A thermal evolution model for Mercury (details described in the text). Small amounts of heat sources are included in the core. In this and in Figs. 6.1 1, 6.1 4 and 6.1 7, time in billions of years (Ga) since planetary origin is indicated by the number adjacent to each temperature profile. The planet is partially or completely molten at those depths where the temperature profile lies along the solidos. (After Toksoz e t a / . , 1978.)

3000 r

0

I

400

I

800

I

I

1200 1600 Depth ( k m )

I

2000

24.40

Fig. 6.7. Present-time thermal profile of Mercury as a function of depth for three different heat source strengths 0 (in J cm-'s-') in the core. (After Toksoz eta/., 1978.)

361

The Cores of Other Planets

0

- 500 1

--

-.

-

2 Time

3

I3UU-

4

4.6

(lo9 years)

Fig. 6.8. Thermal evolution of the interior of Mercury as a function of time based on the temperature calculations shown in Fig. 6.6. Shading above the core region indicates melting. Isotherms are in “C. (After Toksoz et a/., 1978.)

15, 1965. During the close encounter with the planet, measurements were made of the magnetic field and various particle fluxes, all of which indicated that Mars had at most a very weak magnetic field. We do not have a quantitative theory of the origin of the Earth’s radiation belts and it is not possible to predict the nature of such belts for a planet at arbitrary distance from the Sun. However, the planet must have a sufficiently strong magnetic field and be exposed to the solar wind. Since the distance of Mars from the Sun is intermediate between that of the Earth and Jupiter, both of which have intense radiation belts, it is reasonable to assume that Mars also would have radiation belts provided that it is a sufficiently magnetized body. A system of sensitive particle detectors on Mariner 4 indicated the presence of electrons of energy >40 keV out to a radial distance of 165,000 km on the morning side of the Earth, yet failed to detect any such electrons during the close encounter with Mars (Van Allen et al., 1965). This implies that the magnetic dipole moment of Mars is less than 0.001 that of the Earth, i.e. the upper limit on the equatorial magnetic field at the surface of Mars is about 200y. Similar results were reported by O’Gallagher and Simpson (1 965). The Mariner 4 carried a solid-state charged particle telescope capable of detecting electrons with energies greater than 40 keV and protons with energies greater than 1 MeV. The trajectory of Mariner 4 would have carried it through a bow shock, transition region and magnetospheric boundary had these existed. No

362

The Earth’s Core

evidence of charged-particle radiation was found in any of these regions. Again, a planet with even a very small magnetic field might be expected to produce a wake in the anti-solar direction. Mariner 4 passed sufficiently close to Mars to have detected such a wake, had it existed-no escape of electrons, as would be expected in such a wake, was observed. O’Gallagher and Simpson placed the same upper limit on any Martian magnetic field, viz. 0.1% of that of the Earth. Mariner 4 also carried a magnetometer during the close encounter with Mars; no magnetic effects were observed that could be definitely associated with a Martian magnetic field. Smith et al. (1965a) put an upper limit on a Martian magnetic moment of 3 x that of the Earth. The next observations of a possible Martian magnetic field came from the Soviet Mars 2 and 3 spacecraft in November and December 1971, which passed within 1100 km of the planet. Dolginov et al. (1972, 1973) interpreted the data from one pass of Mars 3 as indicating that it had crossed a magnetosphere. They claimed that the magnetograms indicated a field intensity some 7-10 times greater than that of the interplanetary magnetic field at the distance of the orbit of Mars, and concluded that Mars has an intrinsic dipole moment of 2.4 x gausscm3 and an intensity of -6Oy at the magnetic equator. This interpretation has been criticized by Wallis (1975) and Russell (1978a). Wallis suggested that the behaviour of the field and plasma that Dolginov et al. (1973) claim to be the magnetosheath could be disturbed solar wind. Russell pointed out that the direction of the field in the postulated magnetosphere looks like draped magnetosheath field lines. Dolginov (1978a), however, does not accept this interpretation and reaffirms his belief in his original analysis. Bogdanov and Vaisberg (1975) suggested that the observed magnetic field variations are a consequence of a solar-planetary ionosphere interaction rather than that of an internal planetary dipole magnetic field. The latest magnetic observations of Mars were made in February 1974 by Mars 5, whose nearest approach to the planet was 1800 km. Dolginov et at. (1976b) identified a region behind the planet as the magnetotail of Mars. Russell (1978b), on the other hand, again believes that the postulated magnetotail encounters are due to draping of magnetosheath field lines and that there is no conclusive evidence that the spacecraft ever entered a Martian magnetosphere. He estimated the upper limit of a Martian magnetic moment to be 2 x lo2’ gaus cm3, i.e. only 2 x lo-’ of the Earth’s magnetic moment. Dolginov (1978b), as before, does not accept this conclusion. Although dynamo theory is still very incomplete, one might expect that, since the Earth and Mars have similar rotation rates, and since Mercury and Mars are expected to have similar core sizes, the Martian magnetic moment should be intermediate between those of the Earth and Mercury. If the dependence on rotation rate were linear, the Martian moment as scaled from

The Cores of Other Planets

363

Mercury would be far larger even than that reported by Dolginov et al. Russell (1978~)concludes that Mars no longer has an active dynamo, if it ever had one. On the other hand, the Viking retarding potential analyser data have been interpreted as suggesting a small permanent field (Intriligator and Smith, 1979; Cragin et al., 1982). The much weaker magnetic moment of Mars would suggest that it has at most a very small fluid electrically conducting core. The density and moment of inertia coefficient of Mars show that it possesses a ratio of iron to silicate smaller than that of the Earth but larger than that of the Moon. The radius of an iron core should not exceed a few hundred kilometers. Such a core could have sustained, through dynamo action, a stronger magnetic field in earlier times than any now present. Runcorn (1972) is of the opinion that the field of 60y, of apparent internal origin, detected by the Mars 2 orbiter is more likely to be due to such a dynamo now acting weakly than to permanent magnetization, though the latter may be present in crustal rocks as it is on the Moon. O n Bullen’s FezO hypothesis (see $5.6), Mars would have an iron core of 1400 km, but no Fe,O zone and therefore, presumably, no fluid radius zone. This would be compatible with the failure to observe any significant magnetic field. Lewis (1972) concluded that Mars is essentially devoid of free iron. It may possess a small core of FeS with or without a small amount of Feo-its mantle is rich in FeO, with a FeO/FeO + MgO ratio 2~0.5.His conclusions are based on his model of an initial nebula of solar composition in which there is a steep radial temperature gradient. Studies of the internal structure of Mars have been hampered by uncertainty in its radius, values of which have ranged from 3310 to 3423 km. However, an oscillation experiment on board Mariner 4 led to more precise estimates (Fjeldbo et al., 1966). The mean equatorial radius of Mars, as determined from the Mariner 4, 6 and 7 missions and ground-based radar observations, is 3394 km. This value assumes that the surface of Mars is an oblate spheroid whose flattening is equal to the dynamical flattening, 1/190-i.e. Mars is in complete hydrostatic equilibrium. However, preliminary analysis of the Mariner 9 occultation data (Cain et al., 1972) indicates that the surface of Mars may best be represented by a triaxial ellipsoid with equatorial radii of 3401 and 3395 km an a polar radius of 3372 km. The radius of a sphere whose volume is equal to that of this ellipsoid is 3389 km. The mean density of Mars, 3.933 & 0.002 g cm-3 (Bills and Ferrari, 1978) is well-determined from the observed mass and mean radius. However, the dimensionless moment of inertia factor C / M R 2 has to be estimated from J 2 (the second degree zonal component in the spherical harmonic expansion of the gravitational potential) and is complicated by significant departures from hydrostatic equilibrium. If non-hydrostatic contributions are ignored, the predicted value of C / M R 2 is 0.377 (Binder, 1969). Reasenberg (1977) and Kaula (1979) attri-

-

364

The Earth’s Core

buted all non-hydrostatic contributions to the Tharsis plateau region and obtained a value of 0.365, a value about 3% lower. Models of the internal structure of Mars constructed before 1977 have used the older value of C I M R 2 ,viz. 0.377. Binder (1969) has constructed a number of models of the internal structure of Mars using the Mariner 4 data and studies of Anderson (1967) on the composition and structure of the Earth’s upper mantle, assuming that there is no radical difference between mantle materials of Mars and mantle materials of the Earth. He found that Mars has an iron core with a radius in the range 790-950 km and a mass in the range 2.74.9% of the total mass of the planet. The percentage of free iron in Mars is considerably lower than that in the Earth, whose core contains about 31% of the Earth’s mass. Binder also concluded that in its early history the iron core of Mars was molten, adding support to his earlier (1966) suggestion that initially the internal conditions of Mars satisfied the requirements of dynamo theory, Mars possessing an early magnetic field. Models (e.g. Anderson and Kovach, 1967) in which Mars has a smaller, less dense core than the Earth with a mantle more dense than the Earth’s have often been interpreted as suggesting that Mars has not differentiated completely, so that both core and mantle are cross-contaminated with each other’s material. The data can, however, be simply explained if the core of Mars is pure FeS and the mantle very-FeO-rich silicates as suggested by Lewis (1972). Binder and Davis (1973) later constructed a number of other models of the internal structure of Mars taking into account the work of Anderson (1972) and the new data returned by Mariner 6, 7 and 9. They still used the older value of 0.376 for C / M R 2 , although they did consider one model with a slightly lower value (0.372). They assumed a solid core, which in their opinion is necessary to account for the absence of a substantial magnetic field. If Mars has a liquid core, then it must have had a hot thermal history and the models of Anderson (1972) and Johnson et al. (1974), to be discussed later, show that the core has to include an alloying element, probably sulphur, to account for its low density. In either case the value of J 2 (and hence C / M R 2 )determined from the Mariner 9 mission is incompatible with an undifferentiated planet, i.e. Mars must have a core, be it solid or liquid. Ringwood (1959, 1966) had proposed an alternative model for the constitution of Mars, viz. that the relative abundances of the common metals Fe, Mg, Si, Ca, Al, etc., in Mars are similar to those in the Earth, Venus, the Sun and Type 1 carbonaceous chondrites and that the differing densities between the planets are caused primarily by the varying amounts of oxygen present. Mars is believed to be completely oxidized, containing no free Fe. In a later paper, Ringwood and Clark (1971) investigated this hypothesis in some detail-this was made possible by the more recent accurate determinations of

365

The Cores of Orher Planets

UPPER

MANTLE

Olivine and Pyroxene

SDinel and Garnet

Fig. 6.9. Hypothetical cross-sectionthrough Mars. (After Ringwood and Clark, 1971 .)

the mean density and moment of inertia of Mars, together with advances in high-pressure research enabling the role of phase transitions within Mars to be evaluated. In their preferred model (see Fig. 6.9), magnetite has segregated to form a core with a radius of 1638 km (21% of the mass of Mars), the magnetite being present as the high-pressure polymorph. If differentiation has proceeded sufficiently to form a core, it is likely that a crust is also present. It is also likely that a small metallic Fe core would be possible within the constraints laid down by their models. However, an Fe core is not readily reconcilable with the inferred oxidized state of surface rocks and atmospheric composition. Ringwood and Clark's final model is also in agreement with the probable thermal history of Mars. (MacDonald (1962) had shown that if Mars has the chondritic abundances of U, Th and K, the radioactive heat generated over 4500 Ma would have caused extensive internal melting and differentiation.) The absence of a large, electrically conducting convective liquid core in their model is also consistent with the failure to observe a significant magnetic field on Mars. Anderson (1972) has also constructed a number of models of the possible internal structure of Mars (see Fig. 6.10). He found that Mars cannot be homogeneous but must have a core, the size depending on its density. Taking

366

The Earth's Core

4-

----_

------____

Earth's Core

-

-

-

--

Eufeclic mix.{::?

CORE

Lkz FeS

I

-

MANTLE

L

I

I

1

I

I

1

I

Earlh's Mantle

-

-

-

Fig. 6.10. Density versus radius for some representative models of Mars that have the correct mean density and moment of inertia. Models with chondritic compositions fall within the hatched region All models have approximately the same mantle c'ensity. (After Anderson, 1972.)

the density of pure Fe and pure troilite (FeS) as upper and lower bounds for the density of the core, he found that its radius must lie between 0.36 and 0.60 of that of Mars (i.e. between about 1220 and 2035 km). A meteoritic model for Mars leads to an Fe-S-Ni core, of 12% of the mass of the planet. Mars has an Fe content of 25 wt%, which is significantly less than that of the other terrestrial planets but is close to the total Fe content of ordinary and carbonaceous chondrites. Anderson obtained a satisfactory model for Mars on the assumption that ordinary chondritic material was subjected to relatively moderate temperatures. Core formation would begin when temperatures exceeded the eutectic temperature in the system Fe-FeS ( -990°C), but would not be completed unless temperatures exceeded the liquidus throughout most of the planet. Anderson estimated that approximately 63% of the potential coreforming material (Fe-S-Ni) has entered the core. Thus Mars, in contrast to the Earth, is an incompletely differentiated planet, and its core is considerably richer in S than is the Earth's core. The absence of a significant magnetic field may be due to the smaller size of its core, as compared with that of the Earth, and its higher resistivity (due to large amounts of sulphur in the core). Alternatively, if the dynamo is driven by precessional torques, the absence of a magnetic field may be due to the lack of a significant lunar torque. Okal and Anderson (1978) later showed that the use of Reasenberg's more accurate value of 0.365 for C / M R 2 instead of the older value (0.377) does not alter Anderson's (1972) conclusions concerning the composition of Mars viz.

The Cores o f Other Planets

367

an incompletely differentiated, ordinary (high-iron) chondrite, or a mixture of ordinary and carbonaceous chondrites. Goettel(l981) has estimated that cores ranging from Ni to FeS would constitute from 13 to 36wt”/, of Mars, with core radii ranging from 1320 to 1890 km. In contrast to earlier studies, Young and Schubert (1974) considered solidstate convection as a significant means of heat transport. They investigated two models-one with an iron core (radius and mass 0.4 and 0.1 that of Mars) and the other with a pure FeS core (radius and mass 0.6 and 0.26 that of Mars). These core models are the maximum density-minimum size and maximum size-minimum density cores consistent with the observed mass, radius and moment of inertia of Mars. The radioactive heat source concentration in the crust-mantle was taken to be the same as that in the terrestrial J cm-3 s - I ) . Young and Schubert found convection in mantle (2.61 x the mantle to be very effective in keeping temperatures relatively low. Thus, for a pure iron core a temperature difference of 8000°C between the MCB and the surface when there is no convection is reduced to about 2300°C when convection occurs at a Rayleigh number 100 times the critical value for the onset of convection. (The actual Rayleigh number is uncertain because viscosities are not known.) It is clear from their work, however, that for pure conduction, temperatures exceed typical silicate melting temperatures, implying that convection in the Martian mantle is actually necessary to prevent large-scale melting there. Young and Schubert also found that whether Mars has a liquid or solid core depends on the value of the kinematic viscosity v of its mantle at temperatures above about 1500°C. If v S 1022-1023cm2 s - ’ at such temperatures, and evidence suggests that this is probably the case, then convection is so efficient as to cool nearly the entire mantle, and in particular the MCB, to temperatures below those at which Fe or Fe-FeS cores would be liquid. If such were the case, on the assumption that a planetary magnetic field could only originate through dynamo action in a liquid core, Mars would not have an intrinsic magnetic field. Johnston et al. (1974) have also re-examined the thermal history and evolution of Mars, and come to somewhat different conclusions. They obtained a number of models of the Martian interior that had to satisfy certain constraints (mean density, moment of inertia and recent volcanism as inferred from the Mariner 9 photographs). Assuming an Fe-FeS core, they found that core formation and differentiation of a crust took place within the first 1000 Ma of the planet’s history. At the present time the radius of the core is between about 1300 and 1800 km depending upon the exact composition, and the core is liquid even if its composition is not that of the eutectic. Photographs obtained by Mariner 9 have revealed evidence of extensive volcanism (including a series of prominent shield volcanoes), mildly cratered

368

The Earths Core

plains similar to the lunar maria, and circular features such as domes and craters; see e.g. Masursky (1973), Mutch and Saunders (1976). This suggests that at some time in the past Mars must have been hot enough for partial melting to have occurred at depth, but gives no indication of present conditions. Evidence for tectonic and volcanic activity was confirmed by the 1976 Viking missions (Arvidson et al., 1980). No compressional features comparable in scale or in extent to those on Mercury have been observed. Solomon (1 979), therefore, concluded that Mars must have experienced a thermal history involving global expansion and surface extension for most of its history. Toksoz and Hsui (1978) considered the thermal history and evolution of Mars drawing upon the results and constraints of earlier models but using the new data provided by the Viking missions. Their calculations include conduction and sub-solidus mantle convection. Earlier models, although not unique, favour the presence of an Fe-FeS core and FeO enrichment of the mantle. This is consistent with the enrichment of Fe and S in the surface samples at the Viking lander sites (see e.g. Baird et al., 1976; Toulmin et al., 1977). Toksoz and Hsui assumed a differentiated planet with an Fe-FeS core of 1900 km radius. The mean core density is 6 g cm-3 and that of the silicate mantle 3.5gcmP3 (Johnston and Toksoz, 1977). From a study of the Mariner 9 and Viking photographs, it appears that Mars may have undergone a period of intense bombardment about 4 x lo9 years ago (Tera and Wasserberg, 1976). This would imply that a crust had formed prior to this time, requiring early differentiation and hence relatively high initial temperatures. The later stages of Martian evolution show episodes of tectonic and volcanic activity. Tharsis volcanism probably occurred about lo9 years ago (Soderblom et al., 1974) and the Martian channels, whatever their origin, indicate differentiation of the planet prior to Tharsis volcanism. The gravity field of Mars has been determined from the Mariner and Viking orbiters; analysis of the results indicates that large areas of Mars may not be isostatically compensated. Maintaining large stresses, such as those around the Tharsis bulge, requires either a thick lithosphere (Thurber and Toksoz, 1977) or strong convection. The relatively low seismicity (Anderson et al., 1977), and the lack of evidence for active plate tectonics, favours a thick lithosphere independent of convection. In their thermal model calculations, Toksoz and Hsui (1978) assume that long-half-life radioactive isotopes are the major heat sources. Since some potassium is likely to partition with the sulphide phase, 10% of 40K is assumed to be in the Fe-FeS core. Figure 6.11 shows a typical theoretical model. The initial temperature profile is calculated assuming a constant base temperature of 1000°C (this is an ad hoc representation of all heating mechanisms besides accretion) plus a contribution due to accretion (assumed to last for lo5 years). Figure 6.1 1 also shows the Fe-FeS eutectic and liquidus curves

369

The Cores of Other Planets

500

-

I Mantle -Core

I

I 0-

I

I

) .

I I

I

I

I

-

Fig. 6.11. A thermal evolution model for Mars (details described in the text). U = 1 5 p p b with K/U = 50,000 Numbers adjacent to curves indicate time in billions of years (Ga) since planetary origin. (After Toksor eta/., 1978.)

based on the compositional models of Johnston and Toksoz (1977). The mantle solidus is an extrapolated peridotite solidus (Green and Ringwood, 1969). Figure 6.12 presents the evolution of Mars in the form of a space-time diagram, and shows that it can be approximately characterized by four different stages. The first stage lasts from formation of the planet for about lo9 years. During this period, a Martian core has formed and an initial crust differentiated. The second stage lasts from about lo9 to 3 x lo9 years. First, mantle temperatures rise and reach partial melting. Between 2 x lo9 and 3 x lo9 years extensive melting, differentiation and outgassing occur. The radius of the planet increases and extensional features observed at the surface of Mars most likely occur. The third stage, between 3 x lo9 and 4 x lo9 years, sees slow but continued evolution. The lithosphere thickens and the partial melting zone deepens. The final stage is one of cooling. The partial melting zone shrinks and volcanic activity diminishes. At present, the Martian lithosphere is about 200 km thick, the mantle convecting slowly, and the core molten. These four stages in the thermal evolution of Mars are seen in other models developed by Toksoz and Hsui (see e.g. Toksoz et al., 1978) that satisfy the available constraints. Changing the initial conditions can change the time-scale of evolution. Thus, hotter initial models move the period of extensive differentiation to an earlier time; colder initial modes1 do the opposite. However, the relative sequence of events is not radically changed.

370

The Earth's Core

I

I

a

n

6 Fig. 6.12 Thermal evolution of the Martian interior as a function of time, based on the temperature calculations shown in Fig 6 11 Isotherms are in "C (After Toksoz e t a / , 1 9 7 8 )

3

0

I

1000

2000

I

3000

Deprh (km)

Fig. 6.13. Density profiles for Mars satisfying the mean density and moment of inertla for varying core compositions using temperatures based on a convection model (After Johnston and Toksoz, 1977 )

Figure 6.13 (Johnston and Toksoz, 1977) shows density profiles for Mars satisfying both the mean density and revised moment of inertia (Reasenberg, 1977) for different core compositions, and using a thermal model based on convection. It can be seen that changes in core density are reflected primarily in core radius, with relatively small changes in mantle density, setting rather narrow bounds on mantle composition. Also in the upper mantle, thermal expansion is dominant over compression, resulting in a slight density inver-

371

The Cores of Other Planets

sion of about 2% at a depth of about 250-300 km, a region where a small partial pressure of water could induce local partial melting. Coradini et al. (1983) have also considered the thermal evolution of Mars with particular emphasis on early heating due to the impact of large bodies (see also $2.3). The thermodynamic results of an accreting Mars are complicated by the likely presence of a sizeable proto-Jupiter in a nearby zone during its formation (Weidenshilling, 1976). A sizeable influx of bodies from Jupiter’s zone could have caused complete melting of Mars early on, in which case core formation would have been contemporaneous with accretion. Coradini et al. estimated that the melting temperature of silicates would be reached in a shell 360 km thick with a maximum temperature of 1850 K at a depth of 1300 km-assuming quite a long accumulation time (130 Ma). It is thus not necessary to have a very short time of formation. After accumulation, the melted shell grows due to the decay of long-lived radioactive isotopes and after 460 Ma it is 1500 km thick. At this time, core formation is presumably over. In their model, Coradini et al. neglected the thermal consequences of core formation. Coradini et al. (1983) have also considered the heating of the embryo Earth and Mars (see also 52.3.8) caused by the impact of large bodies. They point out that Safronov (1978) and Kaula (1979) neglected the depth of penetration in their models and thus underestimated the different heating efficiency of small and large planetesimals. Coradini et al. showed that, when massive bodies reach depths of several kilometres, they are able to deposit significant amounts of heat below the surface. Even if a long accumulation time is assumed (> 100 Ma), a large melted shell is possible with temperatures sufficiently high to initiate differentiation of an iron alloy to form a core. With an accumulation time of 130 Ma for the size of the present Earth, a maximum temperature of 3300 K is reached at a depth of 3600 km, deeper than the present MCB. Vigorous convection is set up in a melted layer 3300 km thick, and inhibits a rise in temperature beyond the melting point. In the case of Mars, even if an accumulation time as long as the Earth is assumed, the melting temperature of silicate is reached in a shell 360 km wide at a depth of 1300 km. A good account of both theoretical and experimental work on impact processes has been given by Kieffer and Simonds (1980).

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6.4 Venus

The first measurements of the environment of Venus were made by the spacecraft Mariner 2, which on December 14, 1962, passed within about 41,000 km of the centre of the planet. No disturbances of the interplanetary medium that could be attributed to the planet were observed (Frank et a/., 1963; Smith et

372

The Earth’s Core

a/., 1963, 1965b). A search for charged particles magnetically trapped around Venus, and measurements made by the magnetometer carried by Mariner 2, gave no indication that Venus possesses a magnetic field: an upper limit to the magnetic dipole moment of Venus of about 1/1&1/20 that of the Earth was deduced. Later, Venera 4 (Dolginov et a/., 1968, 1969; Gringauz et al., 1968) and Mariner 5 (Bridge et al., 1967) carried out magnetic field and plasma measurements one day apart in October 1967. When Venera 4 was 200 km above the surface of Venus, it indicated a magnetic moment < l o z 2 gauss cm3, i.e. < that of the Earth (equivalent to a surface equatorial field strength < 4y) (Dolginov et al., 1969). Such a low value would be insufficient to deflect the flow of the solar wind as does the Earth’s magnetic field. Mariner 5 passed about 10,150 km from the centre of Venus on October 19, 1967 and observed abrupt changes in the amplitude of magnetic fluctuations, in field strength and in plasma properties. Data from both the magnetometer and plasma probe showed clear evidence for the existence of a bow shock around Venus, similar to, but much smaller than, that near the Earth. The solar wind appears to flow around the planet without striking it, in contrast to the case of the Moon: it appears to be deflected by a dense ionosphere on the sunlit side of Venus. Its high electrical conductivity prevents the passage of the incoming magnetic field, the “pile-up’’ of the field altering the plasma flow. Strong support for this interpretation comes from the observation of the Mariner Stanford Group (1967) that the upper boundary of the daylight ionosphere is very sharp and is pushed down to within about 500 km of the planet by the momentum of the solar wind. In the case of the Moon, the plasma ions are absorbed by the lunar surface and no shock develops (Lyon et a/., 1967):the Moon appears to be a sufficiently good insulator to allow the interplanetary field to be convected through it essentially unaltered (Colburn et al., 1967; Ness et al., 1967). Although the shock around Venus resembles that around the Earth (except in scale), conditions inside it are quite different: in the case of Venus it is the ionosphere that deflects the solar wind, in the case of the Earth, it is its magnetic field. The above observations also place an upper bound on the magnetic dipole moment of Venus: it cannot exceed 0.001 that of the Earth (within a factor of two). Similar conclusions were drawn by Van Allen et al. (1967) from the absence of energetic electrons (>45 keV) and protons (> 320 keV) during the flyby of Mariner 5-they concluded that the dipole moment of Venus is almost certainly less than 0.01 and probably less than 0.001 that of the Earth. Russell (1976a) has re-examined the Venera 4 data and concluded that when allowance is made for the external current systems, the “planetary” field in the data appears to increase with decreasing distance according to an inverse-cube law. Moreover, the Mariner 5 data in the wake have the direc-

373

The Cores of Other Planets

tions expected for tail field lines (Russell, 1976b). Russell concluded that Venus may have an intrinsic magnetic dipole moment of about 6.5 x 10” gauss cm3 (corresponding to a surface field of 30y). The next spacecraft to study the plasma field environment of Venus were Venera 6, which impacted the planet in May 1969 (Gringauz et al., 1970), and Mariner 10 (Ness et a/., 1974a), which passed within 11,900km from the centre of Venus on February 5, 1974. Because of the trajectory of Mariner 10, data obtained from its dual magnetometer system cannot further reduce the upper limit to the magnetic moment of Venus obtained by Venera 5 that of the Earth’s). The following spacecraft missions carrying magnetometers to Venus were Venera 9 and 10, which went into orbit on October 22 and 25, 1975. Their closest approach to Venus was 1500 km, but they did pass directly through the wake of the planet, out through the magnetosheath, and into the solar wind (Dolginov et al. 1976a). The magnetic field stretches out from the planet in the southern lobe and is consistent with Russell’s (1976~)inference of an intrinsic field based on Mariner 5 and Venera 4 data. The Russian investigators disagree on the interpretation of the Venera 9 and 10 data (Dolginov et al. 1977). However, in a later paper Dolginov et al. (1978) support the existence of an intrinsic magnetic moment of -2.5 x lo2’ gauss cm3. Dolginov (unpublished) claims that the variable magnetic fields seen near periapsis are spacecraft fields and not the result of induction of the planetary ionosphere as claimed by Yeroshenko (minority report in Dolginov et a/. 1977). The most recent spacecraft to probe the magnetic field of Venus was the Pioneer Venus orbiter, which went into orbit on December 4, 1978; the nearest approach to the planet was 150 km. The data from more than 120 orbits in the centre of the solar wind wake behind Venus show that the field behind Venus can be quite strong, reaching many tens of gammas for periods of the order of seconds, but is irregular and changes from oribt to orbit. Russell et al. (1980) obtained an upper limit for the moment of 3 x loz1 gausscm3. Venus rotates only slightly slower than Mercury and is only slightly smaller than the Earth. Thus almost any scaling law for planetary dynamos will give a sizeable magnetic moment for Venus: Busse (1976) predicted 1.6 x gausscm3, which is over 50 times that observed. The implication is that motions in the core of Venus are not strong enough to drive a self-sustaining dynamo. Vector measurements made by the Pioneer-Venus retarding potential analyser within the ionosphere of Venus show a global ion velocity field with the ion velocity near the terminator generally directed across the terminator and towards the anti-solar region (Knudsen et al., 1980a,b). Knudsen et al. (1982) showed that the magnetohydrodynamic conditions of the ionosphere near the terminator favour convection of field rather than diffusion. Thus,

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374

The E a r t h s Core

any planetary magnetic field that Venus might possess will be strongly affected by the global plasma flow pattern they found. The magnetic flux from an internal magnetic field will accumulate in the night hemisphere. Although the structure and dynamics of such accumulations will depend on details of the magnetic field source, Knudsen et ul. (1982) interpreted the observational data to imply a dipole moment for Venus of 7 x lo2' gausscm3 directed along the planet spin vector. Because of its dense cloud cover, no reliable estimates of the rotation rate of Venus were possible prior to radar measurements in 1962 (Carpenter, 1964; Goldstein, 1964).These led to the surprising result that the rotation rate is slow ( - 250 days) and retrograde. More recent determinations indicate the period to be within a day or two of 243.15 days, for which period Venus would present the same face to the Earth at every inferior conjunction-to an observer on Earth Venus would appear to rotate exactly 4 times between close approaches. The spin of Venus may thus be controlled by the Earth. Determination of the rotation period (242.98 days) by Carpenter (1970) from radar tracking of features and Doppler broadening differs from the Earth's synchronous value, and it is now uncertain whether Venus is in synodic spin resonance. That Venus is not in a hydrostatic state follows from measurements of the equatorial figure, which show a difference of the equatorial axes of 1.1 4 km. The oblateness of the equatorial figure is sufficient to provide the required difference in the equatorial gravitational moments for spin-orbit coupling of the Earth and Venus. It is not possible to decide whether this non-hydrostatic state is due to convection within the planet. Goldreich and Peale (1970) have investigated the dynamical implications of the anomalous obliquity* of Venus. They found that solar gravitational tides acting on Venus tend to reduce the planet's obliquity to a value less than 90". This result is true whether or not Venus is locked in synodic spin resonance with the Earth. They discussed two possible mechanisms that may stabilize the obliquity near 180"-atmospheric torques due to thermal tides and frictional dissipation of energy at the boundary between a rigid mantle and a differentially rotating liquid core. The atmospheric torque model is speculative because of large uncertainties in the properties of the Venusian atmosphere and Goldreich and Peale thus prefer the second possibility. Core-mantle interaction, which is induced by precession, must produce a reduction of the component of angular momentum in the orbital plane. Goldreich and Peale showed that such a mechanism is capable of driving the

* A planet's obliquity is defined as the angle between its spin and orbital angular momenta. Uranus and Venus are anomalous among solar system bodies in that their spin angular momenta have components that are antiparallel to their orbital angular momenta. The obliquity of Uranus is 97" and that of Venus close to 180".

375

The Cores o f Other Planets

obliquity to 180" from values greater than 90" for a wide range of reasonable core viscosities and spin angular velocities. They had previously (1 967) invoked coupling between a rigid mantle and liquid core as the means of capturing Venus into synodic spin resonance with the Earth. Thus, both the surprising and unexpected features of the rotation of Venus-the possibility of synodic spin resonance with the Earth and obliquity close to 180O-can be accounted for if it possesses a fluid core similar to that of the Earth. Since Venus has about the same size and average density as the Earth, it is quite probable that it does possess a metallic core with internal temperatures similar to those in the Earth. On Bullen's Fe,O hypothesis (see §5.6), the OC of Venus is only about 600 km in thickness (Bullen, 1973). Bullen suggested that allowance for compressibility may reduce this estimate still further and thatathis may be the reason for the failure to detect any significant magnetic field. Lewis (1972) concluded that Venus has a massive core of Fe-Ni alloy surrounded by a mantle of Fez +-free magnesium silicates. His conclusions are based on his assumption that the bulk composition of condensates in the solar nebula is determined by chemical equilibrium between the condensates and gases in a system of solar composition. Assuming that Venus and the Earth have similar moments of inertig, Toksoz and Johnston (1977) estimated that Venus has a completely separated Fe-Ni core of radius -2900 km. Figure 6.14 shows a thermal evolution model for Venus by Toksoz et al. (1978)-their method was the same as they had used for Mercury and Mars. The initial temperature was estimated from an accretion model with an accretion time of 0.25 x lo5 years. Heat source

-0

2000

4000

6000

Depfh ( km) Fig. 6 14. A thermal evolution model for Venus (details described in the text) Numbers adjacent to curves indicate time in billions of years (Ga) since planetary origin (After Toksoz et a / , 1978 )

376

The Earths Core

O r

h

E 2000

1

r

Y

v

fp. QI

n

4000

Origin

Time (109years)

Present

Fig. 6.15. Thermal evolution of Venus as a function of time, based on the temperature calculations shown in Fig 6.14, showing core formation and mantle melting (shaded region). Isotherms are in "C. (After Toksoz et a/., 1978.)

strengths were assumed to be similar to that of the Earth. The model predicts that at present Venus has a thin lithosphere, a partially molten asthenosphere and a solid but convecting lower mantle. The O C may be molten or partly molten; with no heat sources in the core, it would cool sufficiently to form a solid IC. Figure 6.15 shows the evolution of Venus in a space-time diagram. Core formation is assumed to have taken place early, lo9 years after origin. Radar maps of the surface of Venus indicate that it may be tectonically active (Campbell et al., 1972; Malin and Saunders, 1977). Arkani-Hamed and Toksoz (1984) later considered the thermal evolution of Venus using solid-state convection theory (Arkani-Hamed, 1979). They based their study on finite-amplitude three-dimensional convection in the mantle of Venus, whose physical parameters are both temperature- and pressure-dependent. Because of the similarity in size and total mass between the Earth and Venus, they used values of the physical parameters of the Earth to guide them in their choice of values for Venus. The core was assumed to be uniform with an adiabatic tempeiature distribution. They found that a significant proportion of the present temperature in the mantle and heat flux at the surface of Venus is probably due to the decay of high temperatures set up in the planet as the result of the formation of its core. Their models indicate that Venus had been highly convective until about 500 Ma ago. The rate of cooling was so high that its core solidified-in fact, they conclude that the planet is now completely solid with the exception of the upper part of its mantle where partial melting may still occur. By contrast, the model of

The Cores of Other Planets

377

Stevenson et al. (1983) (see $6.7) predicts a stably stratified, completely fluid core as the most likely present state for Venus.

6.5 TheMoon

In 1959 a magnetometer aboard Luna 2 passed within 55 km of the Moon's surface and detected no magnetic field. Taking into account the sensitivity of the magnetometer, this put an upper limit of 6 x lo2' gausscm3 on the Moon's global magnetic moment-a value some four orders of magnitude lower than the present dipole moment of the Earth (Dolginov et al., 1961). This lack of any significant lunar magnetic field was confirmed later by Luna 10 (Oolginov et al., 1966) and Explorer 35 (Sonett et al., 1967), which showed that the upper limit of any lunar dipole moment was even lower being 6 x 10'' gauss cm3. A more detailed analysis of the Explorer 35 data carried out by Behannon (1968) yielded a still lower value of 1 x lozogauss cm3, and later estimates of the maximum lunar dipole moment (based on Apollo subsatellite magnetometer results) gives an upper limit of 6 x loL9gausscm3 (Russell et al., 1975). The results of the Apollo programme were very surprising. Stable components of natural remanent magnetization (NRM) of lunar origin were found in Apollo 11 samples by a number of workers (Geochim. Cosmochim. Acta 34, Suppl. 1, 1970, contains a number of papers on this topic). Later, lunar surface magnetometers at the Apollo 12, 14, 15 and 16 landing sites revealed surprisingly high local surface fields of tens and hundreds of y (Sonett et al., 1971). Finally, magnetometer measuremenets made with Apollo 15 and 16 subsatellites orbiting the moon at a height ~ 1 1 0 k m detected sharp magnetic anomalies up to several y on both the front and far side of the moon. These anomalies are apparently associated with topographical and/or geological features, and imply a degree of homogeneous magnetization of the lunar crust on a scale of tens or hundreds of km (Coleman et al., 1972). Some large-scale magnetic anomalies appear to be associated with thin, highly magnetized layers deposited during the impact of a large meteorite (Hood, 1981). It is also possible that a globally magnetized crust, disrupted by impacts, could provide the observed magnetic anomalies (Runcorn, 1982). The principal difference between lunar and terrestrial samples is that the remanent magnetization of the lunar samples is carried by iron with small amounts of nickel (see e.g. Strangway et al., 1970; Pearce et al., 1971; Collinson et al., 1973). The magnetic behaviour of these materials is very different from that of the familiar titanomagnetites and other ferromagnetics in terrestrial rocks. The interpretation of the NRM of the returned lunar samples

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The Earth’s Core

ascribes it to three possible sources: magnetic contamination, secondary effects on the lunar surface, and primary magnetization associated with the formation of the rock. The primary NRM of igneous rocks is of the thermoremanent (TRM) type. In the case of the breccias, the primary NRM will be thermoremanent if the Curie point is exceeded during formation but not if they have a relatively low temperature of origin. Among the secondary processes, impact-related shock is likely to be important. It can readily account for the observed inhomogeneity of some of the NRM, and clearly much of the lunar surface material has been shocked. No evidence is available on the direction of the ancient lunar field, as the lunar samples are unorientated. All effort has thus been expended in trying to determine the strength of the lunar field. Early results suggested that the field decreased from about 100 pT 3900 Ma ago by about two orders of magnitude some 3200 Ma ago (Stephenson et al., 1975). Cisowski et al. (1977) have questioned this finding, claiming that some of the variation observed is due to shock effects. However, using a much larger sampling, Cisowski et al. (1983) later obtained evidence for a rapid growth of the field 3800 Ma ago decaying 200 Ma later to about a tenth of the value. Such a time variation is important if substantiated, since it would favour an intrinsic magnetic field as the cause of the rock magnetization. Such a variation would be very unlikely if the dominant magnetizing process is shock magnetization in a transient impact-generated field. Many theories have been advanced to explain the remanent magnetization of the lunar crust (reviewed by Dyal et al., 1974; Fuller, 1974; Daily and Dyal, 1979; none of them is completely satisfactory). Possible processes on a global scale that have been suggested are magnetization by an external magnetic field of solar or terrestrial origin, an ancient lunar core, and a regenerative lunar dynamo. A large solar dipole field out to lunar orbit, present during lunar crustal formation, would cause the lunar crust to be uniformly magnetized, resulting in a dipole field external to the Moon. The present Moon cannot have a uniformly magnetized shell, since it would yield a global remanent dipole moment 2-3 orders of magnitude larger than the upper limit of Russell et al. (1974). An objection to a solar origin is that the magnetizing field must remain constant during the cooling of a large volume of crust if it is to acquire a uniform magnetization. The cooling time of large crustal areas is likely to be much longer than fluctuations in an early solar field. If the magnetization of lunar rocks were due to the Earth’s field, the Moon would have had to be much closer to the Earth or the Earth’s field would have had to have been much greater than at present. If the Moon were within 3 4 Earth radii (dangerously close to the Roche limit*), the present terrestrial

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* At the Roche limit, the self-gravitational field of the Moon would just balance the disruptive force of the Earth’s gravitational field.

The Cores of Other Planers

379

R,/R Fig. 6.16 . The effect of a heavy core of radius R on the moment of inertia CIMR’and the mantle density pm in t w o models of the Moon with homogeneous crust and mantle. The two sets of models differ only in the density of the assumed core material. The adopted temperature distribution is taken from a thermal history calculation and satisfies the surface heat flow; the compressibility of aFe and the volume thermal expansion of iron (Clark, 1966) were used for both core materials. The mass M, of the core as a fraction of the lunar mass M, t w o recent estimates of CIMRZ (Sjogren, 1971; Williams eta/., 1973), and the range in radii of a low-Q “core” inferred by Nakamura e t a / . (1973) from S wave amplitudes are shown. (After Toksoz eta/., 1975.)

field could account for lunar fields of -1OOOy, but not for the high fields present in the Moon’s early history. Moreover the Moon would not remain this close to the Earth but would recede from it, since tidal effects would rapidly transfer spin angular momentum from the Earth to the Moon. To explain the lunar magnetic field with the present separation of the Earth would require a terrestrial dipole at least two orders of magnitude greater than its present value. There is no evidence that the Earth’s magnetic field has been stronger in the past. Rocks more than 3000 Ma old have a remanence comparable to that of young rocks: the strength of the Earth’s field does not appear to have changed by more than a factor of 2 over geological time (Strangway, 1978). Strangway et al. (1973) and Runcorn and Urey (1973) suggested that the crustal field may have resulted from a lunar core magnetized during, or shortly after, its formation in an intense solar field. The lunar crust is sup-

380

The Earth's Core

posed to have acquired a TRM as it cooled, while subsequent radioactive heating of the interior of the Moon raised the temperature of the core above the Curie point, thus destroying the internal magnetizing field. Strangway and Sharpe (1974, 1975) believe that the thermal evolution of the Moon is compatible with this model. Runcorn (1 975a,b) showed that if the lunar crust were magnetized by the field of an internal dipole, there would be no external field after the internal field was removed, except at holes in the magnetization as might be caused by craters. Unfortunately, the observed magnetic anomalies are not correlated with craters (Russell et al., 1977; Lin 1978; Hood et ul., 1978). Stephenson (1976b) later considered the case where the permeability of the crust and core are different, and showed that the dipole moment of a crust magnetized to a depth of 35 km is of the same order of magnitude as the upper limit of Russell et al. (1974). A regenerative dynamo in a lunar core has often been proposed as the origin of the magnetic field (see e.g. Runcorn et al., 1971; Strangway et ul., 1971). One real difficulty is that there would have to have been a lunar core at least 4000 Ma ago, since remanence in lunar rocks that old has been measured, and this would seem to require melting of the whole Moon to form an iron or iron-sulphide core (Runcorn, 1976). As a possible heat source, Runcorn (1977b, 1978) has suggested the decay of super-heavy elements (see also 42.3.3). Nozette and Boynton (1979) estimated the abundance of '52Sm in the Santa Clara iron meteorite. If this is attributed to fission of a super-heavy element with atomic number 107-110, they estimated that the amount of super-heavy elements in the early solar system would be about three orders of magnitude less than that needed by Runcorn. Some evidence for the existence of a lunar core has come from orbital magnetometer data (Goldstein et al., 1976): such a core would exclude magnetic field lines on the time-scale of the Moon's passage through the geomagnetic tail. Observations limit the size of the core to 400-600 km in radius. The moment of inertia of the Moon is consistent with a core of radius 400 km or less (Gapcynski et al., 1975), and Nakamura et al. (1974) have tentatively inferred a small core based on greatly reduced P wave velocities within about 350 km of the lunar centre. They put limits on the radius of a core of 170360 km. A number of local processes where small regions have been magnetized have also been invoked. A dynamo driven by thermoelectric currents in the cooling lunar crust has been proposed by Dyal et al. (1 973, 1977). In their model, meteorite impacts are postulated to penetrate the solid crust that has solidified from a primordial magma ocean, forming lava-filled basins. Two nearby basins formed at about the same time would be connected below the crust by subsurface magma and above by the solar plasma. Different crustal conditions could have resulted in different cooling rates leading to a temperature difference between the two basins. Such a model is essentially a thermo-

Thecores of Other Planets

381

electric circuit, two dissimilar conductors (plasma and lava) joined at two junctions (the surfaces of the lava basins) that are at different temperatures. It is difficult to quantify this model. It would seem that extremely large thermoelectric coefficients would be needed to generate fields as large as 100 FT. Magnetization of localized regions by impact cratering was one of the first suggestions made to account for the Moon’s magnetic field (Nagata et al., 1970, 1971). Some evidence for this has been reported by Cisowski et al. (1975) for certain regolith breccias. Magnetic fields have not been detected during hyper-velocity impact experiments, but have been observed in some laser-target experiments (Stamper et al. 1971; Tidman, 1974; Stamper and Ripin, 1975). However, the magnetization process in a transient field is not well-understood. Even if rocks in the vicinity of the impact are heated to high temperatures, the field is unlikely to persist long enough for conventional TRM to be acquired. Direct evidence of the generation of plasmas necessary for any field-generation process during impact events on the Moon has come from detectors at the Apollo 12 and 14 sites, which recorded increased ionic activity that could be correlated with the impact of the Apollo 15 lunar module carrier rocket on the Moon (Hills et al., 1972). The Apollo 15 surface magnetometer also recorded fluctuations in the vertical field that coincided with the arrival of plasma from the Apollo 15 lunar module impact 93 km away. Hide (1 972) suggested an alternative mechanism-amplification of the existing interplanetary or lunar surface field. This could arise by compression of the ambient field by an expanding plasma cloud. Some of the strongest evidence against shock-induced magnetization in a transient field as the dominant magnetization process in the early Moon is the decay of the field with time. This would not be expected, since transient fields are likely to be random in intensity and independent of time. Many lunar samples show no evidence of shock. When the field was strongest, it seems most likely that TRM in a global field was the dominant magnetizing process. When the field decayed, TRM was still important, although in some samples (such as regolith microbreccias and glasses) shock magnetization is the main cause. Coilinson (1984) has recently reviewed the whole question of the existence of a lunar magnetic field throughout geological time. The origin of the NRM of the returned Apollo samples-particularly the origin of the fields in which the magnetization was acquired-is still unresolved. If it is necessary to invoke a lunar dynamo during the early history of the moon, this would place severe constraints on geochemical models of the moon’s evolution. It would require that the moon differentiated early in its history in order to have a metallic core. It is also not clear whether a single explanation will account for all aspects of the observed magnetic phenomena-the NRM of the lunar samples, the surface fields observed with lunar surface magnetometers and the fields observed by the subsatellites.

382

The Earth's Core

The NRM of the samples argues strongly for the existence of ancient lunar fields in which they acquired their magnetization. Unfortunately, there are no samples of demonstrably undisturbed bedrock that could give the configuration of the magnetic fields. The stable magnetization carried by the rocks suggests that many of them were magnetized in fields of lo3 or lo4?. However, the range of fields implied by the natural remanence of the returned lunar rocks is remarkable, being from hundreds of y to the order of one gauss (G). Local magnetic fields, typically two orders of magnitude greater than would be produced by a global dipole field, have been observed by the subsatellites placed in orbit by Apollo 15 and 16. Some large-scale magnetic anomalies appear to be associated with thin, highly magnetized layers deposited during the impact of a large meteorite (Hood, 1981). It is also possible that a globally magnetized crust, disrupted by impacts, could provide the observed magnetic anomalies (Runcorn, 1982). As already mentioned, there are models that assume that the whole Moon or a substantial part of it was magnetized early in its history giving rise to a primitive remanent field. Subsequently the rocks that are observed to be magnetized and those that give rise to present lunar remanent fields were magnetized in this primitive remanent field. Later still this field is postulated to have been thermally demagnetized; this hypothesis is consistent with some of the proposed thermal histories of the Moon (e.g. Solomon and Toksoz, 1973). Such models may conveniently be referred to as fossil field models, since the magnetization that is now observed is held to have been acquired in the primitive remanent field of the Moon and not directly in a solar wind field, a terrestrial field, or a lunar dynamo field. The lunar dynamo theory has been most strongly advocated by Runcorn et a / . (1971) and Strangway et al. (1971). It is held to have operated from some time before the oldest rocks studied were formed until some period after the origination of the last volcanic rocks and breccias that have been investigated. At this point it then shut off. Much of the support for a lunar dynamo comes from the suggestion that lunar rocks for the most part exhibit NRM appropriate for TRM acquired in a field of about lo3?. Fuller (1974) has pointed out, however, that this is not very well substantiated-it appears that it does not apply to certain of the rocks that have the most convincing TRM-like natural remanence. It may well be that the dynamo will have to account for a wide range of ancient lunar fields (including values as high as 1 G). Levy (1972) has also concluded that the Moon would have to rotate faster than its strength would allow before it could give rise to a selfgenerating dynamo (his dynamo model is based on that of E. N. Parker-see $4.2). Formation of a molten lunar core or pockets of pure Fe or FeeNi would

383

The Cores of Other Planets

require temperatures above the melting point of iron in the lunar interior. Extrapolating the data of Sterrett et a / . (1 965), the melting point of metallic iron at the centre of the Moon is about 1660°C. The presence of Ni would lower this temperature only slightly. To produce a magnetic field 4000 Ma ago, such a core would have had to have been formed early. O n the other hand, temperatures of the order of 1660°C at the centre of the Moon, early in lunar history, are unacceptable according to many models of the Moon’s thermal history that call for a cool lunar interior (e.g. Hanks and Anderson, 1969; Wood, 1972; McConnell and Gast, 1972; Toksoz et a/., 1972a,b). Moreover, unreasonably high accretional temperatures would be required to heat the centre of the Moon to 1660°C in the first 500 Ma of its history. The presence of S in the core of the Moon would allow a molten lunar core at considerably lower temperatures ( 1000°C). The eutectic temperature of 988°C (at 1 atm)* would also be lowered somewhat by the presence of Ni. Anderson (1972) has calculated that the density of an Fe-FeS eutectic mix at 30 kbar pressure is about 5.3 g cm-3. This value is significantly less than the density of pure Fe which Runcorn et al. (1970) assumed in their calculations on the maximum permissible size of a lunar core. A core of lower density than Fe could be larger than one of pure Fe without violating the constraints imposed by the Moon’s moments of inertia. The Moon does not have an internally generated magnetic field today, so that, if the field was caused by a convecting core, either the core froze or the magnetic Reynolds number fell below the critical value (Runcorn, 1972). Failure of the lunar dynamo by solidification of a core or of deep pockets of Fe-FeS is inconsistent with most models of lunar thermal history, nearly all of which require the temperature of the deep interior to be considerably above 1200°C today. On the other hand, such thermal models are consistent with the present model for a core provided that the lunar dynamo ceased because the magnetic Reynolds number fell below the critical value. If the dynamo failed by solidification, it is still possible to modify most thermal models by postulating convective cooling. (Tozer (1972) believes that the Moon is likely to undergo solid-state convective motion as soon as the temperature of the lunar interior rises above about 1000°C, and that the central temperature adjusts to a steady-state temperature of between 600 and 1000°C.) Convection would also aid in core formation. If the dynamo failed by solidification and the dynamo was a central core rather than a pocket or a number of pockets, it is unlikely that temperatures within the Moon could have exceeded the solidification temperature of the core at any time after-

-

* Usselman (1972) has extended the earlier work of Brett and Bell (1969) on the effect of pressure on the Fe-FeS eutectic temperature. At a pressure of 50 kbar (the approximate pressure at the centre of the Moon), the eutectic temperature is 1000°C.

-

384

The Earth’s Core

wards. Brett (1973) has re-examined the geochemical arguments in favour of a lunar dynamo and produced an ad hoc model that attempts to reconcile, as much as possible, some of these divergent opinions. If we assume that the Moon accreted cold but was heated from the outside during the final stage of accretion, this would lead to a molten shell very early on in lunar history, which later formed the crust as indicated by seismic studies. The interior of the Moon, however, would have been cool during its early history and would have warmed up because of radioactive heating only relatively recently, thus forming a molten or partially molten core, which is also indicated by seismic data (Nakamura et al., 1973). Strangway and Sharpe (1 974) have considered whether an early hot-outside, cold-inside Moon, which evolved into the present cold-outside, hot-inside Moon, could quantitatively account for the evidence of an ancient magnetic field. Two questions need to be answered. First, what is the origin of the magnetizing field? Secondly, once the Moon had become magnetized by an isothermal process, could the memory have been retained until the termination of igneous activity about 3200 Ma ago? There is no definitive answer to the first question, because there are no clear records of planetary fields predating those frozen into the lunar rocks. Strangway and Sharpe, however, showed that provided the moon contained a few per cent of metallic Fe and was exposed to an extra-lunar field of about 1&20 G while much of it was still below the Curie point of Fe, it is possible to derive a restricted class of thermal evolution models that satisfy the known constraints. The question of the time-decay of isothermal remanent magnetization (IRM) is difficult to answer, but Gose et al. (1972) have shown, by extrapolating the decay rate of IRM acquired in the laboratory in multi-domain, iron-bearing rocks, that the decay coefficient is very small. The best value for the lunar radius comes from the Apollo laser altimetry experiment (Kaula et al. 1972, 1974). The weighted average of the three mean lunar radii determined in the Apollo 15, 16 and 17 orbital planes is 1737.7 f 0.4 km (Kuala et al., 1974). If this value is assumed to be the radius of a spherical Moon equal in volume to the actual Moon, the mean density of the Moon is 3.343 0.003 g cm-3. The best estimate of the moment of inertia factor C / M R 2 is 0.3950 -t 0.0050 (Sjogren, 1971; Williams et al., 1973). Toksoz et al. (1975) considered the effect on C / M R 2and on the density of the mantle of a dense metallic core (see Fig. 6.16). Two extremes were chosenpure Fe and pure FeS (troilite). Although the existence of a lunar core cannot be decided on the basis of the value of C / M R 2 ,limits can be placed on its size for given assumptions about the structure of the crust and mantle. If C / M R 2 = 0.395 and if the Moon has a 60 km crust and a chemically and mineralogically homogeneous mantle, Toksoz et al. (1975) showed that the radius of a core cannot exceed 305 km if the core is Fe, or 530 km if the core is

The Cores of Other Planets

385

FeS. If C / M R 2 can be as low as 0.390, these limits are raised to 470 km and 720 km respectively. Goins et al. (1977, 1981) have analysed the arrival times of direct P and S waves measured on seismograms recorded from natural lunar seismic events. The complete Apollo seismic network consisting of long- and short-period instruments at each of four stations was operative from April 1972 to September 1977. The signals from roughly 400 seismic events have been used to determine the structure of the lunar interior. Because of the distribution of events, arrival-time analyses are not very useful in studying the structure below a depth of about 1100 km. Nakamura et al. (1973) had earlier suggested that there was an attenuating zone in the deep lunar mantle beginning at a depth of about 1100 km. Goins et al. infer from their analysis that the attenuating zone should extend to at least 1300 km and possibly deeper. The attenuation is more pronounced for shear waves and they suggest that it is possibly the result of temperatures approaching the solidus producing a small amount of partial melting. Nakamura et al. (1973) reported a meteorite impact event, and determined the location and origin time from the three nearest stations and used this data to predict the arrival-time for the fourth P wave. They found that the observed arrival-time was delayed about 50 s and tentatively proposed the existence of a lunar core with a radius 17c360 km. Goins et al. pointed out that the event was very weak and the signal-to-noise ratios were very low, and suggested that the uncertainty in the P wave arrivals could explain the proposed arrival-time delay. They thus find the seismic evidence for a core inconclusive. They conclude that “all lunar data (moment of inertia, density, seismic, electrical conductivity) allow, but do not require, a core.” The lunar crust and lithosphere have supported mascons for over 3 x lo9 years. This implies that temperatures in the outer few hundred kilometres of the Moon must have been significantly lower than the melting temperature for this length of time in order that the lithosphere be thick enough to support such loading. Moreover creation of a 60 km thick crust requires initial or partial melting and differentiation of a significant fraction of the Moon (Wood et al., 1970). Workers at M.I.T. have constructed many thermal models of the Moon taking into account all the constraints imposed by other disciplines (see e.g. Toksoz et al., 1972a,b; Solomon and Toksoz, 1973; Toksoz and Solomon, 1973). Figure 6.17 shows their latest model (1978) in which they employed the same method as they used for Mercury. The solidus curve shown is that of basalt determined by Ringwood and Essene (1970). The initial profile was chosen to have a partial melt region about 500 km thick near the surface and base temperature of 500°C. In time the Moon starts to cool from the surface while the interior warms up. At about 2 x lo9 years, planet-wide convection takes place and the central regions of the Moon cool.

The Earth's Core

2000

1600

h

V

-

0

1200

P, L

3

L

n L

P)

E 800

400

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500

1000 Oepbh ( k m )

1500

Fig. 6.17. A thermal evolution model for the M o o n (details described in text). Numbers adjacent t o curves indicate time in billions of years since planetary origin. (After Toksoz eta/., 1978 )

However, these regions are more viscous (temperatures are further away from the solidus), so that convective heat transport becomes less effective and they subsequently warm up again. The present temperature profile below 1000 km is sufficiently close to the solidus (given the uncertainties involved in their determination) to explain the shear wave attenuation within this region. If there is a small core in the Moon, then in this model it is totally molten. Figure 6.18 shows the thermal evolution of the Moon with depth plotted against time. The absence of large-scale compressional and extensional tectonic features on the Moon implies that the lunar volume cannot have varied greatly in time. Soiomon and Chaiken (1976) have put a limit on the change in radius in the last 3.8 x lo9 years of 1 km. The thermal consequences of this imply initial temperature profiles with near-melting temperatures to 20CL300 km in depth and relatively cold temperatures (&50OoC) in the deep interior. There is evidence of extensive heating of the outer layers of the Moon during at least two epochs, that of crustal formation 4.64.4 x lo9 years ago (Wasserburg et al., 1977) and that of mare flooding 4.0-2.5 x lo9 years ago (Boyce, 1976). It

The Cores of Other Planets

387

Time ( IO'years) Fig.6.18. Thermal evolution of the lunar interior as a function of time, based on the temperature calculations shown in Fig. 6.1 7 . Coarse and fine shading denote regions of partial melting and solid state convection respectively. Isotherms are in "C (After Toksoz etal., 1978.)

is possible that there was one long period of decreasing magmatic activity, the record between 4.4 and 4.0 x lo9 years being obliterated by bombardment. Binder (1982) and Binder and Gunga (1985), however, favour an initially totally molten Moon, They argue that their thermoelastic stress calculations show that if only the outer few hundred kilometres of the Moon was initially molten, and if it had a cool interior (as proposed in the model of Solomon and Chaiken (1 976), Solomon and Head (1 979)), then the highlands should not have any young, compressional tectonic features. On the other hand, if the Moon was initially totally molten, the highlands should have 10 km-scale thrust faults less than 0.5-1.0 x lo9 years old. Schultz (1972), using lunar orbiter images, and Binder and Gunga (1985), using Apollo high-resolution imagery, showed that young thrust faults do exist in the highlands. Cassen et al. (1979) have investigated the thermal history of the Moon using a modified version of the programme devised by Reynolds ef al. (1 966), taking into account sub-solidus convection based on an empirical relationship between the Nusselt number and the Rayleigh number (cf. Schubert, 1979; Schubert et al. 1979a). Their results are shown in Fig. 6.19, which compares their thermal histories with those of Toksoz et al. (1978), which were based on finite-difference solutions of the Navier-Stokes equation. Both solutions assume the same overall heat source abundances, material properties and initial conditions. However, Cassen et al. assumed constant viscosity, while Toksoz et al. allowed for a variable viscosity. The onset of convection,

2000

I

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I

2000

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900 1200 Radius ( k m )

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Time= 0

Melting curve

I

1500

T i m e - 0.5 Go

I

1

1800

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600 900 1200 Radius ( k m )

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1800

F i g . 6.19. Cornpartson of thermal histories of the Moon calculated by different methods. Solid line: emplrical slrnulation (Cassen e f a / , , 1979) Broken Ilne: Navier-Stokes solution (Toksoz e t a / . . 1978). In (d). curve 1. radioactives have been moved according to the scheme of Fricker e t a / . (1 967), which moves less radioactives than the method of Toksoz e t a / . ( 1 978) Curve 2 results from enhanced differentiation. (After Cassen e t a / . 1979 ) 1 Ga = 1 billlon years.

389

The Cores of Other Planets

1600

Melting curve -------- -J

2.5 Go

-----

10

Radius (km) Fig. 6.20. A thermal history model of the Moon satisfying the following constraints. The outer layers are initially molten; convection commences about 0.5 billion years (Ga) after formation; the temperature remains close enough to melting in the outer 300 km ro produce magma at least until 2 5 Ga ago; the average temperature is presently below melting to at least a depth of 900 km; the change in radius since 3.5 Ga ago is less than 1 krn; and the present day heat flow is between 14 x 10.’ and 21 x lo-’ J cm‘2s-’. The activation energy in this model varies from 2.9 eV for a convecting region near the surface, to 4.0eV for one extending ro the centre. The central model kinematic viscosity and heat flow at c m z s - ’ and 16.5 x lo-’ J cm - z s - ’ . (After Cassen e l a/.. 1979 ) present are, respectively. 5 x 102’

early evolution, final central temperature, and final surface heat flow agree well for the two calculations. Cassen et al. also calculated a series of lunar thermal history models for a range of viscosities and initial temperature distributions and for different assumptions regarding the mobility of heat sources. Two types of initial temperature distributions were used: hot (temperatures just below the melting curve for the whole Moon) and accretion (constant sub-solidus temperature in the interior, rising to melting in the outer few hundred kilometres). They showed that the initial temperature profile has little or no effect on the present thermal state of the deep interior, although it could affect the outer layers by determining the time of onset of convection and the extent of heat source differentiation. The present thermal state of the deep interior is determined primarily by the rheology and melting properties of the rocks. The central temperature is at least 1100°C and kinematic viscosities are in the range 1021-1022 cm2s-1. Cassen et al. also found that, as with the conduction models of Solomon (1977), convective models also demand that cooling of the outer part of the Moon be offset by heating of the inner part if a radius

390

The Earth's Core

_ _ _ _ _ _ _ _ _ _ _ -----------Fe melting

2000

h

Y

.-.-.-

v

._._

Fe-S eutectic

2 2 1000

F

?i

2

0

1000

500

1500

Depth (km) Fig. 6.21. Comparison of lunar internal temperatures (shaded region-see text for references) with the melting curve for pure Fe and the eutectic curve for Fe-S Since the lunar temperatures are intermediate, any lunar core must be at least partially fluid. (After Stevenson, 1983.)

change of less than 1 km is to be maintained over the last 3.5 x lo9 years. Figure 6.20 shows the evolution of a model of Cassen et al. satisfying the constraints given in the caption. Finally, Fig. 6.21 shows the estimated present temperature distribution in the Moon (after Stevenson, 1983) obtained from a number of considerations--electrical conductivity (Duba et al., 1976; Hood and Sonett, 1982), evolution models (Schubert et al., 1980), and gravity and rheological state (Lambeck and Pullen, 1980). The figure shows that the present internal temperature is well above the Fe-S eutectic, indicating that if the Moon has a core, then it must be partially fluid. Although it is possible that the lunar core contains sufficient sulphur still to be entirely fluid, Stevenson and Yoder (1981) suspect that its sulphur content is more likely to be comparable to or less than that in the Earth. They estimate a present-day OC thickness of 100 km for an initially fluid core of radius 400 km containing 5% sulphur by mass. Stevenson (1983) also showed that gravitational energy release from gradual cooling of this core is too small by about a factor of 3 to sustain compositionally driven convection, although it is possible that gravitational energy release during gradual core formation could provide dynamo generation (Stevenson, 1980). However, because the core is so small, Stevenson estimated that the energy available would be about two orders of magnitude too low.

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The Cores of Other Planets

6.6

391

Summary

It must be stressed that a determination of the constitution of the interior of the Moon and planets is not unique--even in the case of the Earth, for which much more knowledge is available. It is possible, however, to place bounds on some parameters. After the Earth we know most about the Moon, although a lunar core has only been indirectly inferred. Mars, unlike Venus and Mercury, has a well-determined moment of inertia that places some constraints on the nature and extent of its core. Knowledge of the moment of inertia of Venus and Mercury would greatly narrow speculation about their interiors. The preceding sections have discussed the Moon and terrestrial planets in turn, largely from a historical point of view, indicating advances that have been made, particularly from new data obtained by spacecraft. In this section, I give what I believe are the best predictions we can make at the moment, those of Stevenson et al. (1983). More details, particularly on planetary magnetism have been given by Stevenson (1983). Key points, I believe, are the power of solid-state mantle convection to remove substantial amounts of heat from the core, and the ability of light alloying materials to substantially lower the melting point of pure iron. In their study of the thermal evolution of the terrestrial planets, Stevenson et al. (1983) paid particular attention to the presence or absence of an intrinsic magnetic field. The basic assumptions of all their models are that core dynamos (if they exist) are driven by thermal and/or chemical convection, that radiogenic heat production is confined to the mantle, that mantle and core cool from initially hot states that are at the solidus and superliquidus respectively, and that any IC that may nucleate excludes the light alloying component (S or O?), which then mixes uniformly upward through the OC. The assumption that the cores contain no radioactive heat sources implies that convective motions necessary to drive a dynamo must be generated either by secular cooling of an entirely fluid core or by IC solidification. All models assume whole-mantle convection, which is parameterized by a Nusselt-Rayleigh-number relationship specifying the rate at which heat escapes from the core (see $3.8). Material parameters are chosen so that the correct present-day values of heat flow, upper mantle temperature and viscosity and IC radius are obtained from the Earth. Stevenson et al. (1983) found that small changes in model parameters can result in completely fluid, non-convecting cores (without dynamo action), convecting fluid OCs with IC growth (and with dynamo action), and almost solid cores with only thin O C fluid shells now remaining (and probably with no dynamo). In their models, as the core freezes the lighter constituent is concentrated into the remaining outer fluid shell and the liquidus is lowered, retarding IC growth. This is why

392

The Earth’s Core

6371

krn

3485

core ) Fe

Earth

Venus

Fig. 6.22. Schematic representation of probable present-day states for Earth and Venus The lower pressures and higher temperatures in Venus may prevent IC growth This would cause the core to be stably stratified and incapable of magnetic field generation (After Stevenson, 1983 )

more of their models lead to a completely solid core. Young and Schubert (1974) obtained lower temperatures and a completely solidified core for Mars by taking a constant (low) viscosity instead of the strongly temperaturedependent values adopted by Stevenson et al. Figure 6.22 shows the probable present-day states for the Earth and Venus and Fig. 6.23 those for Mars and Mercury. In their preferred model, th,: IC, in the case of the Earth, nucleates late in its history. Since the Earth’s magnetic field is at least 3.5 x lo9 years old (see §4.2), Stevenson et al. suggest that the mode of powering the dynamo may have changed during geological time. In the Earth’s early thermal history, the magnetic field was generated by thermal convection. After IC growth began (1.5-2.5 x f09 years ago), the release of gravitational energy became the dominant heat source for the geodynamo (Fig. 6.24). However, there is no palaeomagnetic evidence at present to indicate low field strength preceding nucleation of the IC as indicated by this model. Their models for the core of Venus admit a state similar to that of the Earth, but also a completely fluid, stably stratified core, and a core that is mostly frozen. They favour a completely fluid core-models with almost frozen cores require implausibly low amounts of light alloying constituents. The absence of a significant IC in Venus is probably due to its slightly higher temperature and lower central pressure relative to the Earth (290 GPa compared to 360 GPa). An interesting implication is that perhaps Venus once had a substantial magnetic field that died 1.5 x lo9 year ago and that it might eventually nucleate an IC, causing revival of the dynamo (see Fig. 6.24). Mars

393

The Cores of Other Planets

3390

2440 krn

1780 krn

Mars 6.23.Schematic

Mercury

representation of probable present-day states for Mars and Mercury Mars may have

no IC if the core fluid is sulphur rich, whereas Mercury's core may he mostly frozen because of very low sulphur abundance. The thin but vigorously convecting Mercurian fluid shell may be capable of magnetic field generation. (After Stevenson, 1983.)

again admits all possibilities for the state of its core. Stevenson et al. favour a completely fluid core, since it is predicted for a cosmochemically plausible sulphur content of 15% or more by weight and also provides an explanation for the absence of a substantial magnetic field. Their Mercury models predict

Fig. 6.24. Venus (After

the Earth and

394

The Earth's Core

TABLE 6.1 Observations and interpretations of planetary magnetic fields. (After Stevenson, 1983.) Observed surface field ( G ) Mercury Venus Earth Moon Mars Jupiter Saturn

- 2 x 10-3 5 2 x 10-5

-