Physics of the Earth and Planetary Interiors, 15 (1977) 341—348 341

© Elsevier Scientific Publishing Company, Amsterdam — Printed in The Netherlands

A THERMAL MODEL OF THE EARTH

FRANK D. STACEY
Physics Department, University of Queensland, Brisbane, QId. 4067 (Australia)

(Received January 21, 1977; revised and accepted May 17, 1977)

Stacey, F.D., 1977. A thermal model of the earth. Phys. Earth Planet, Inter., 15: 341—348.

A thermal model, consistent as far as possible with the parameterised earth model of Dziewonski et al. and with
thermodynamic principles and relevant equations of state, is tabulated. This is made more secure by two recent
developments, an experimental study of the Fe—S eutectic to 100 kbar by Usselman and a calculation by Bukowinski
which reveals an electronic phase collapse of potassium in the 200—300 kbar range andexplains the core heat source.
Use is made of the Vashchenko-Zubarev formulation of the Grüneisen ratio, and Lindemann’s melting law, both of
which have been shown recently to have particular relevance at very high pressures. Values of electronic specific heat
and the Grüneisen ratio, which contribute significantly to core properties are calculated from the electron equation
of state of Zharkov and Kalinin.

1. Summary of notation 2. Introduction

A coefficient of electron heat capacity There is no prospect that the thermal (or electrical)
(eqs. 7, 8) properties and behaviour of the earth’s deep interior
Ce, C
1 electron and lattice contributions to Cv will ever be known as precisely as the seismologically
Cji, Cp specific heat at constant volume, pressure determined parameters, elasticity and density. Never-
D density of electron states at Fermi level
Ethermal, Ee, El thermal energy, electron and lattice con- theless some of the major doubts about thermal behav-
tributions to Ethermal iour have recently been resolved and we are now in
g gravity a position to express the hope that subsequent im-
K, KT, KS incompressibility, isothermal and adia-
batic values provements in our understanding will be in the nature
L Wiedemann-Franz constant of refinements and not dramatic revisions of basic
m mean atomic weight principles. It therefore becomes relevant to consider
p pressure properly constructed thermal models of the earth,
r radius (distance from centre of earth) compatible with and similar in principle to the seis-
SM entropy of melting
T temperature mological models. By properly constructed I mean
TM melting point that the tabulated thermal parameters are precisely
V volume related by thermodynamic principles and vary with
a volume expansion coefficient depth according to specified equations of state, con-
7~Ye’ 71 thermal Grüneisen parameter, electron strained as far as possible by seismological data.
and lattice values For some time it has been supposed that the bound-
7w Vashchenko-Zubarev (integrated) 7
p density ary between the inner and outer cores marks the soli-
Pe electrical resistivity dus of core material and therefore that if the core
Debye temperature composition is known and its melting point extra-
thermodynamic efficiency of convection polated to 3.2 Mbar, there is a very deep fixed point
thermal conductivity on the temperature profile of the earth. In the past
342

decade, this approach has been subjected to a series of results of experiments on its solubility in Fe at labora-
doubts. The empirical melting law of Kennedy and co- tory pressures is explained.
workers (Kraut and Kennedy, 1966; Higgins and The assumption that the inner-core boundary is at
Kennedy, 1971)led to a suggestion that the melting- the normal solidus temperature of core material is
point gradient was less steep than the adiabatic gra- followed here and used to constrain the temperature
client, which the outer core is believed to follow very profile of the whole earth. This has become an effective
closely because it is quite vigorously stirred (whether approach since the measurements of Usselman (1975a)
by convection or not) in the process of dynamo action. on the Fe—S eutectic temperature at pressures up to
If this were correct the core should be solid outside 100 kbar. Up to 50 kbar Usselman found the eutectic
and liquid inside the “core paradox” (Higgins and
— temperature to be almost constant while the S content
Kennedy, 1971; Kennedy and Higgins, 1973). Although of the eutectic composition decreased, but above this
vigorously refuted by some workers (Boschi, 1974a, b; pressure the composition stabilised close to Fe2S (an
Leppaluoto, 1972) the “paradox” was treated serious- acceptable core composition in terms of its density)
ly and attempts to resolve it included a revival of the and the eutectic temperature increased, the increase
suggestion by Elsasser and Isenberg (1949) that the closely following Lindemann’s law. This law is used
inner core consisted
8) state of Fe1975),
(e.g., Stacey, in an aelectronically
treatment ofcollapsed
the core to
100extrapolate
kbar range.toIt 3.2 Mbar Usselman’s
is possible data in contrast
that the density the 50—
(3da slowly precipitating slurry (Malkus, 1973) and an
as at the inner-core boundary exceeds that associated
approach to the problem of a dynamo in a stably with solidification of core material of constant corn-
stratified (i.e. sub-adiabatic) core (Bullard and Gubbins, position (Stacey, 1977b); if so the eutectic assump-
197 1). The paradox was strengthened by the difficulty tion may be in error.
in explaining a core heat source. Experiments on the The estimate of the core—mantle boundary (CMB)
solubility of K in molten Fe (Goettel, 1972; Oversby temperature is as secure (or insecure) as the inner-core
and Ringwood, 1972; Seitz and Kushiro, 1974; temperature because the evidencethat the outer core
Ganguly and Kennedy, 1975) failed to give any indica- is adiabatic is quite convincing and the ratio of adia-
tion that a significant amount of K can enter the core. batically related temperatures is very simply calculated
All of these difficulties are now resolved. The if the GrUneisen ratio is known. Thus both the inner-
Lindemann melting law has a sound thermodynamic core (melting) temperature and CMB (adiabatically
basis (Stacey and Irvine, 1977a) as well as an atomistic extrapolated) temperature depend equally upon 7.
explanation which gives it particular relevance at high Model temperatures for the mantle are not as well
pressures (Stacey and Irvine, 1977b). The Kraut- constrained by physical observations. It now appears
Kennedy law can be viewed as a special consequence that the gradient everywhere exceeds the adiabatic
of the strong volume dependence of for alkali metals,
~‘ value, if very slightly, and that in the asthenosphere
which cannot be extrapolated to apply to Fe or Fe the temperature is very close to or even at the solidus.
alloys. Lindemann’s law gives a melting-point gradient The model presented here follows the author’s argu-
steeper than the adiabat provided Grüneisen ratio, y, ment that there is a steep rise in temperature in the
exceeds 2/3 [not the reverse as Kennedy and Higgins lowest 200 km of the mantle and that the CMB is also
(1973) suggest]. A new derivation and justification of at the solidus of mantle material (Stacey, 1975).
the Vashchenko-Zubarev formulation of 7 (Irvine and In view of the relevance of convection to current
Stacey, 1975) yields outer-core values in the range geophysical thinking the thermodynamic efficiency,
1.2—1 .4. There can be no electronic collapse of Fe in t~,of convective transfer of heat from each depth is
the core pressure range (Bukowinski, 1976a), but at tabulated. This gives the mechanical work which unit
pressures of a few hundred kilobars K undergoes an heat transfer performs, either in deforming the mantle
electronic collapse, losing its 4s-type electron to the against its strength or viscosity or in deforming and
3d shell, making it an element of the first transition amplifying the magnetic field in the core. This refers
series with an ionic radius comparable to that of to heat transfer to the surface from points in the man-
Fe (Bukowinski, 1976b). Thus the presence ofKin the tle, or to the CMB from points in the core. As shown
core must be expected, while the reason for negative by Stacey (1977a, b) the efficiency is determined
 343

from the integral of the adiabatic gradient over the value of y implied by the Mie-GrUneisen equation for
relevant depth range. thermal pressure at volume V:
As far as possible the tabulation presented here P(T) = P(T = 0) + 7Ether~l/ V (2)
has been made to conform to the earth model of
Dziewonski et al. (1975), by using their model densi- In the absence of an electron contribution to 7 this is
ties, p, pressures, P, and incompressibilities, K, and the same as the conventionally defined thermodynamic
listing thermal values at the same depths. However,
the parameterization of the earth model gives an im- — — .1~Ks
plausible curvature of the dK/dP vs. P relationship (3)
and this has been adjusted in calculating the GrUneisen — PcV —

ratio. Although it is clear that the seismological model and in the lower mantle 7 is taken to be identical to
is very much more precise and certain than the thermal 7vz’ However, curvature of the dK/dP vs. P relation-
model, the strong dependence of the thermal model ship of the earth model is implausible in terms of likely
upon the estimates of the Griineisen ratio, which for equations of state and y is adjusted to fit the values
much of the range is obtained from the fine details of dK/dP= 3.35 at p = 4,553 kg m3 by a simple empiri-
the K(P) tabulation, means that imprecision of the cal relationship:
seismological model may actually limit the accuracy
of the thermal model in this way. However, the present 7 = 68.08 p112 (lower mantle) (4)
thermal model goes further than earlier discussions of This is appropriate for close-packed crystal structures
the thermal regime of the earth in recognising the of the lower mantle, but for the tetrahedrally bonded
depth-dependences of thermodynamicvariables. Suffi- crystal structures in the upper mantle 7vz is map-
cient significant figures are given to show these varia- propriate and an estimate of the average 7 for upper-
tions, even where they greatly exceed the absolute ac- mantle minerals (7o = 0.850) is used with a stronger
curacy of the estimates, density dependence:
The crust is here neglected and the difference be-
tween sub-continental and sub-oceanic mantles is
recognised only in the tabulation of temperature.
Where the continental and oceanic models of
7 = 7~exp ~
I ~
~)
dP
(upper mantle) (5)

Dziewonski et al. (1975) differ, the continental data The integral is ln(p/p
0) only if there are no phase
have been followed, except at radius 6,360 km, which changes and so was tabulated by integration of model
would correspond to crust in the continental areas. data instead of using density ratios (which include in-
Upper-mantle temperatures can be inferred from the crements associated with phase changes). The density
petrological work of Boyd (1973) and MacGregor dependence of -y in eq. 5 is as p~,stronger than the
(1975). dependence assumed at higher pressures.
For the core ‘yvz was obtained from a similar rela-
tionship, constrained to fit the zero-pressure value for
3. Origin and parameterization of relationships laboratory Fe (~o= 1.59) but at a density reduced by
7%3)to and
correspond
an average
to value
core composition
for the outer(Po
core
= 7,364
(y = 1.23
kg
Where
values of rthey
haveare needed
been takenp,from
P, g, the
K atearth
tabulated
model of m p = 11,189 kg m3):
at
Dziewonski et a!. (1975). In principle this gives 7vz
directly from: 7vz = 330.51 p0’6 (core) (6)
1 dK 5 2P mis gives 7 as defined by eq. 1 but not eq. 3 because
—+ —
2 dP 6 9K there is an important electron contribution for which
7vz = 4P (1) thermal energy depends upon the square of tempera-
1 ture:

As discussed by Irvine and Stacey (1975) this gives the CeAT (7)
344

T required in the thermodynamic identities:

Ee = f CedT = ~ AT2 (8)
0
The value of Yvz accounts for both lattice and elec-
~\aT/~ =~
V
(16)

tron components of thermal pressure: 3T yT

(17)
(9) (~)s=~
and to Yvz as the integral ‘y, related to K(P) by eq. 1
so that: and to the component y’s by eq. 10. A summary of
thermodynamic identities in terms of y is given by
y
1E1 + YeEe (10) Stacey (1977b).
Yvz = ________

+ Ee The Debye temperature is tabulated from:

whereas:
________
_Y1C1+YeC’e
C
(11)
K
pm
(
3)1/6
(18)

1+Ce
which has the functional form required by Debye
Eqs. 10 and 11 were used to obtain ~ from Yvz and theory but a constant adjusted to give a better fit to
Ye using the Debye relationship to obtain C1 and E1 laboratory data. It is not suggested that Debye theory
and assuming a mean core atomic weight iii = 50. Ce is more than approximately valid but that it provides
and Ee were obtained from the tabulations of A and an adequate empirical correction of lattice specific
Ye’2,
taking
Ye as zero-pressure values A0 = 0.0959 1 kg~ heat Cv from the classical (Dulong-Petit) value, 3R
K 0 = 2.1, adjusted to the density of core material per atomic mole. For this purpose ill = 50 for the core
from values for iron (Kittel, 1971, p. 254) and as- and 20.2 for the mantle, and the Debye factor for Cv
suming: 2+ for the mantle, and C1 and E1 for the core were taken
alnD 1433(P)31 (12) from large-scale plots. For the core:
Ye3lV3 Po C~C1+Ce (19)

and C1 being virtually constant (497 J kg~K~),very

Ce = Ce0(D/Do) (13) close by
given to the
eq. classical
7. Cpandvalue
~ aresince T/OD>
obtained by 1, and Cethe
solving h

from the work of Zharkov and Kalinin (1971). These simultaneous eq. 3 and:
integrate to give:
(20)

A = 0.92115
r2 (___

expL_~lnk76l3S)
CpC~(1 +yo~T)

which give:
Cv
2pTCv (21)
—
~
7.6135) 1/3 1}~ (14) C~=1 7 K
4299C( 8
For the outer core the latent heat of melting: — ypCp (22)
L=TMSM (15)
C~lSnot tabulated, being less relevant than Cp.
where SM = 191.6 J kg’ K’, assumed to be con- The core melting point is obtained by integrating
stant, as well observed for close-packed metals numerically the differential Lindemann equation:
(Gschneider, 1964), is added to the Debye values of
E1ineq.10. I dTM2(7vz—~) (23)
We could,
1efined perhaps,
by eqs. 3 and 11refer to differential
as the ‘y, as tabulated and ‘y,
thermal TM clP KM
___________
 345

TABLE I
Thermal model of the core
7vz TM T 0D A Cp a 1<
r
(km) (K) (K) (K) (J kg~ (J kg~ (106 (W m~ K1)
K2) K~) K~)

0 1.186 1.124 4,337 4,286 1,3360.02780 637 6.9 — 36.0

100 1.186 1.124 4,336 4,286 1,335 0.02781 637 6.9 — 36.0
200 1.186 1.124 4,332 4,283 1,335 0.02784 637 6.9 — 36.0
300 1.186 1.124 4,326 4,279 1,334 0.02789 637 6.9 — 35.9
400 1.187 1.125 4,319 4,274 1,333 0.02796 638 6.9 — 35.9
500 1.188 1.126 4,308 4,266 1,332 0.02806 638 7.0 — 35.8
600 1.189 1.127 4,296 4,258 1,330 0.02818 638 7.0 — 35.8
700 1.191 1.129 4,281 4,247 1,328 002832 638 7.0 — 35.7
800 1.192 1.130 4,264 4,235 1,325 0.02848 639 7.0 — 35.6
900 1.194 1.132 4,245 4,221 1,323 0.02867 639 7.0 — 35.5
1,000 1.196 1.134 4,223 4,206 1,320 0.02888 639 7.1 — 35.3
1,100 1.199 1.137 4,198 4,199 1,316 0.02912 640 7.1 — 35.2
1,200 1.202 1.139 4,172 4,171 1,313 0.02938 641 7.2 — 35.1
1,217.1 1.203 1.140 4,168 4,168 1,312 0.02942 641 7.2 — 35.0
1,217.1 1.265 1.171 4,168 4,168 1,280 0.03275 659 7.9 0.244 35.0
1,300 1.267 1.174 4,143 4,149 1,276 0.03302 660 8.0 0.239 34.9
1,400 1.270 1.177 4,112 4,125 1,270 0.03337 661 8.1 0.235 34.7
1,500 1.274 1.180 4,078 4,100 1,263 0.03374 662 8.2 0.230 34.5
1,600 1.278 1.184 4,043 4,074 1,255 0.03416 663 8.3 0.225 34.3
1,700 1.282 1.188 4,005 4,045 1,246 0.03459 664 8.4 0.219 34.0
1,800 1.286 1.192 3,966 4,014 1,237 0.03506 665 8.5 0.213 33.8
1,900 1.290 1.196 3,924 3,981 1,226 0.03557 667 8.7 0.206 33.5
2,000 1.296 1.201 3,879 3,947 1,214 0.03612 668 8.9 0.200 33.2
2,100 1.300 1.206 3,832 3,910 1,202 0.03671 670 9,1 0.192 32.9
2,200 1.307 1.212 3,783 3,872 1,188 0.03734 671 9.3 0.184 32.6
2,300 1.312 1.217 3,731 3,831 1,174 0.03802 673 9.6 0.176 32.3
2,400 1.319 1.224 3,676 3,789 1,159 0.03876 675 9.8 0.166 31.9
2,500 1.325 1.230 3,618 3,745 1,142 0.03956 677 10.1 0.156 31.6
2,600 1.332 1.237 3,558 3,697 1,125 0.04042 679 10.5 0.146 31.2
2,700 1.340 1.244 3,494 3,648 1,107 0.04135 682 10.8 0.134 30.8
2,800 1.348 1.252 3,427 3,595 1,088 0.04236 684 11.3 0.121 30.4
2,900 1.357 1.261 3,357 3,540 1,068 0.04345 687 11.7 0.108 29.9
3,000 1.366 1.270 3,283 3,483 1,047 0.04464 690 12.2 0.093 29.4
3,100 1.375 1.279 3,206 3,422 1,025 0.04594 693 12.8 0.077 28.9
3,200 1.386 1.290 3,126 3,358 996 0.04733 697 13.6 0.060 28.4
3,300 1.397 1.301 3,043 3,290 979 0.04886 700 14.1 0.040 27.9
3,400 1.408 1.312 2,955 3,220 955 0.05053 704 14.9 0.020 27.3
3,485.7 1.419 1.323 2,878 3,157 933 0.05207 707 15.7 — 26.8

where liquid phase at this pressure, so that:

KM = KT + 2Y(Yvz — ~)Cv/V (24) TM = 1265 exp(1 1.366 — 1 102p~—~1np) (25)
is the incompressibility along the melting curve, with For calculation of inner-core values, p is multiplied by
Yvz given by eq. 6. The reference value TM
9 Pa (55 kbar) is from the 1data
= 1,265
of the factor 0.95553,
continuity which core.
with the outer adjusts the
It is model
the values 7,to
integrated
K at P1 = 5.5
Usselman i0 and Pt = 6,917 kg m3 for the
(1975a) Yvz which is appropriate for use in Lindemann’s law.
346

This approach differs from Usselman’s (1975b) inter- assumed to follow eq. 23 very nearly to the 5,701-km
pretation of his own data. transition from the bottom of the mantle, where the
The mantle TM refers to the solidus, which is also expression is normalised by the assumption that the

TABLE II
Thermal model of the mantle

r TM T Cp a
1 (106
(km) (K) (K) (K) (J kg
K1) K—1)

3,485.7 0.914 3,157 3,157 1,416 1,256 9.7 0.293

3,500 0.915 3,152 3,125 1,414 1,256 9.7 0.292
3,600 0.919 3,119 3,005 1,398 1,255 9.9 0.286
3,700 0.923 3.086 2,945 1,382 1,255 10.1 0.280
3,800 0.927 3,053 2,905 1,366 1,255 10.4 0.274
3,900 0.931 3,019 2,872 1,349 1,256 10.6 0.268
4,000 0.935 2,986 2,840 1,332 1,256 10.8 0.262
4,100 0.940 2,952 2,808 1,315 1,257 11.1 0.255
4,200 0.944 2,918 2,776 1,298 1,258 11.4 0.248
4,300 0.951 2,883 2,745 1,280 1,258 11.7 0.241
4,400 0.953 2,848 2,713 1,262 1,259 12.0 0.234
4,500 0.958 2,812 2,682 1,244 1,260 12.4 0.227
4,600 0.963 2,776 2,650 1,226 1,261 12.7 0.219
4,700 0.969 2,739 2,617 1,206 1,263 13.2 0.211
4,800 0.974 2,701 2,585 1,187 1,264 13.6 0.203
4,900 0.979 2,663 2,552 1,168 1,265 14.0 0.194
5,000 0.985 2,624 2,518 1,148 1,266 14.5 0.185
5,100 0.991 2,584 2,483 1,128 1,267 15.0 0.175
5,200 0.997 2,543 2,445 1,108 1,268 15.5 0.165
5,300 1.003 2,501 2,407 1,087 1,269 16.1 0.155
5,400 1.009 2,461 2,372 1,066 1,271 16.7 0.144
5,500 1.016 2,415 2,335 1,045 1,272 17.4 0.133
5,600 1.022 2,366 2,290 1,024 1,274 18.1 0.120
5,701 1.029 2,312 2,250 1,002 1,275 18.9 0.107
5,701 0.736 2,312 2,250 915 1,256 15.2 0.107
5,751 0.743 2,280 2,225 905 1,256 15.5 0.102
5,801 0.749 2,244 2,197 895 1,257 15.8 0.096
5,851 0.755 2,204 2,165 885 1,257 16.2 0.090
5,901 0.762 2,160 2,130 875 1,257 16.5 0.085
5,951 0.768 2,110 2,085 865 1,257 16.9 0.078
5,951 0.768 2,110 2,085 812 1,261 18.4 0.078
Conti- Oceans:
nents:

6,001 0.776 2,054 2,025 2,036 797 1,262 19.2 0.071

6,051 0.784 1,990 1,945 1,975 783 1,262 20.0 0.064
6,101 0.792 1,922 1,823 1,910 768 1,261 20.9 0.057
6,151 0.800 1,850 1,670 1,838 754 1,257 21.7 0.049
6,151 0.800 1,850 1,670 1,838 656 1,273 29,0 0.049
6,201 0.811 1,775 1,480 1,760 654 1,265 29.2 0.038
6,251 0.823 1,696 1,266 1,670 652 1,255 29.4 0.027
6,251 0.823 1,696 1,266 1,670 651 1,255 29.5 0.027
6,301 0.835 1,613 1,035 1,460 649 1,241 29.6 0.015
6,360 0.848 1,517 540 550 647 1,168 29.8 0.0016
 347

material is there atits solidus temperature. This allows areas. Petrological evidence (Boyd, 1973; MacGregor,
a simple numerical integration of model data to ob- 1975) is used as a constraint on the continental pro-
tam TM at any depth. In the upper-mantle equation file.
(23) becomes inapplicable because the bond rigidities The efficiency of thermal convection is determined
of tetrahedrally coordinated silicates make the values by adiabatic temperature differences between: (1) the
of irrelevant to the melting process. We must also level of a heat source; and (2) the surface in the case
expect some irregularity in solidus temperature of the mantle or the CMB in the case of the core
through the transition zone, although not as marked (Stacey, 1977a):
as the irregularity in liquidus temperature. However,
the lower-mantle curve is smoothly joined to an initial = 1 — L r’ ~
(surface) gradient, (dTM/dr)O = 1.6 deg km~from 17 exp ~ K8 ~ (27)
T~= 1 ,500 K, these being values suggested by labo- 2
ratory measurements on representative upper-mantle Thermal conductivity is important only in the case
minerals (Kennedy and Higgins, 1972). of the outer core, where it affects the heat balance. It
Outer-core temperatures are adiabatically extra- is believed that thermal conduction in the mantle is
polated from the inner-core boundary by eq. 17. Inte- ineffective except in the lithosphere, for which it
gration was carried out directly from the model data suffices to assume a constant value, ,~= 2.5 W m~
using the variable ‘y instead of the approximation of deg~.In the core there are both electron and lattice
constant -y (which leads to a simple relationship in contributions, the electron effect being dominant:
terms of density). It is this extrapolation to the CMB —
1<e + 28
which
profilesprovides
of both the
Tandlower
TM.fixed point
In the inneroncore,
the mantle
some as- The electron contribution is given by the Wiedemann-
—

sumption about heat sources must be made. The as- Franz relationship:
sumption is that they suffice to maintain an adiabatic ~ = L(T/ ) (29)
gradient against conductive loss at the inner-core e / Pe
boundary. Then, with K effectively constant over this where L 2.45
108 W ~2deg2 is the Wiedemann-
=

range (Table I) and assuming the heat source concen- Franz constant and Pe = 3 , l0~ ~2m is the electrical

tration proportional to density, temperature can be resistivity. The conclusion of Gardiner and Stacey
obtained directly in terms of gravity: (1971) that Pe is almost independent of depth in the
A
7i~iPi —1
core is accepted here, so that:
— dr g 1 (30)
1K81 g(r) = 0.0443 g deg km (26) K = (0.00817 T + 1) W m~ deg
for gin m s2. T is obtained by integration of the
model tabulation of g, with subscript 1 indicating inner- References
core boundary values.
Mantle temperatures are constrained by arguments, .
Boschi, E., 1974a. On the melting curve at high pressures.
used previously by the author (Stacey, 1975) that Geophys. J. R. Astron. Soc., 37: 45—50.
temperature approaches the solidus very closely in the Boschi, E., 1974b. The melting of iron. Geophys. J.R. Astron.
asthenosphere, more so and over a greater range of Soc., 38: 327—334.
depths for the sub-oceanic mantle than for the sub- Boyd, F.R., 1973. A pyroxene geotherm. Geochim. Cosmo-
continental mantle. Over most of the lower mantle the chirn. Acta,
Bukowinski, 37: 2533—2546.
M.S.T., 1976a. On the electronic structure of
temperature gradient exceeds slightly the adiabatic iron at core pressures. Phys. Earth Planet. Inter., 13: 57—
gradient, being extrapolated from the asthenosphere 66.
to within about 200 km of the CMB where it curves Bukowinski, M.S.T., 1976b. The effect of pressure on the
sharply upward, meeting the solidus at the boundary. physics and chemistry of potassium. Geophys. Res. Lett.,
In the upper mantle the different profiles are deter- 3. 49 E.
Bullard, 1—494.
and Gubbins, D., 1971. Geomagnetic dynamos in
mined by the assumptions of diffusive equilibrium in a stable core. Nature (London), 232: 548—549.
continental areas and lithospheric cooling in oceanic Dziewonski, A.M., Hales, A.L. and Lapwood, E.R., 1975.
348

Parametrically simple earth models consistent with geo- MacGregor, I.D., 1975. Petrological and thermal structure of
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