Download Some Aspects of the Thermal History of the Earth

Some Aspects of the Thermal History of the Earth
J. A. Jacobs
“
‘Where do you come from?’said the Red Queen, ‘and where are you going?’”Lxwis Carroll, Through the Looking Glass.
Summary
A number of thermal problems connected with the Earth are
discussed-the early thermal history of the Earth and some of its
consequences, the formation and state of the inner core, convection in
the core and its relation to the Earth’s magnetic field, and conditions
in the upper mantle.
I.
Introduction
Of all the problems connected with the interior of the Earth, that of the variation of its temperature, both with time and with depth, remains one of the most
difficult to settle. Verhoogen (1956) has given a table which lists various estimates
that have been made between 1915-1955 of the present temperature at depths of
Iookm, zgookm and at the centre of the Earth. To this table may now be added
some six or more further estimates. It is instructive, indeed, to plot the “secular
variation” of the Earth’s temperature over the last 45 years!
The temperature distribution is the key to a number of other problems that are
currently engaging the attention of many geophysicists. The question of possible
phase changes in the crust and upper mantle (MacDonald & Ness 1960) and any
attempt to discuss the physics of continental drift are but two examples. Rather
than attempt to estimate the actual temperature distribution, some aspects of
certain geothermal problems are discussed in the following sections.
2.
The early thermal history of the Earth
All thermal histories of the Earth are based on either a hot or on a cold origin ;
should a cold origin lead at some later stage in the evolution of the Earth to a
molten state, the subsequent thermal histories, whatever the initial origin, would be
the same. A cold origin is usually preferred now, and quite low initial temperatures (well below the melting point of the silicate rocks that form the mantle) are
generally postulated. Ringwood (I 960) however has given rather convincing reasons
for believing that an Earth formed by accretion would at an early stage in its
development become molton. A molten Earth also provides by fractional crystallization, the only reasonable explanation of the concentration of U and T h in its
outer regions.
Estimates of the time of accretion are of the order of 108 years. Possible heat
sources that could raise the temperature of the Earth during this period of accretion are the radioactive decay of both long-lived and short-lived isotopes, chemical
267
268
J. A. Jacobs
reactions, and the conversion of kinetic energy into thermal energy. Because of
the comparatively short time of accretion, the temperature increase due to the
radioactive decay of long-lived isotopes is small, of the order of 15oOC (MacDonald
I 959). 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 important
short-lived isotopes are U236, Sm146, Po244 and Cm247, all of which have half lives
sufficiently long to have heated up the Earth during the 107-108 years after the
initial formation. MacDonald (1959) estimates that if all this heat was retained by
the Earth, a temperature increase of the order of 2 0 0 0 3 ooooC may be possible.
The temperature of the material within the aggregating Earth will also increase
because of adiabatic compression. Although data (particularly on the variation of
the coefficient of thermal expansion) are rather uncertain, a rise in temperature
from this source of several hundred degrees seems likely. However the largest
source of available energy is the potential energy due to the mutual gravitational
attraction of the particles of the dust cloud. This energy, upon aggregation, is
either converted into internal energy or radiated away. It is difficult to estimate
the total contribution from this source because of the uncertainty of the physical
process of accretion. The result depends quite critically on the temperature attained
a t the surface of the aggregating Earth and on the transparency of the surrounding
atmosphere to radiation.
Comparativelylow surface temperatures (of the order of a few hundred degrees)
have been predicted, 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 shown, however, that
during these early years, the primitive Earth will have a large reducing atmosphere.
I n the presence of these reducing agents (chiefly carbon and methane), the accreting material will be reduced to metallic alloys-principally of iron, nickel and
silicon. T h e outer regions of the Earth will thus be metal rich and dense (referred
to zero pressure) compared to the interior. Such a state is gravitationally unstable,
and convective overturn will follow leading to a sinking of the metal rich outer
region into 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 will be accelerated as the initial temperature
rises.
Ringwood's (1960) theory of the early history of the Earth implies the production of an enormous atmosphere of C02, CO and H20 due to reduction processes
on the primitive Earth. A critical requirement of his theory is the dissipation of
this atmosphere at an early stage, and in his paper he discusses factors which may
have led to this. It is instructive to consider the effect on the Earth of the removal
of the surface pressure of this atmosphere. It would lead to an expansion of the
Earth and possible polymorphic phase changes in its interior. Suppose an expansion of 6km in the radius Y of the Earth took place, i.e. (dr/r)e 10-3. Then,
assuming hydrostatic equilibrium, the relief of pressure dp necessary for such an
expansion, is given by dp = k(dV/V), where K is the incompressibility and
(dV/V)e (3dr/r) is the volume change. Taking an average value of K for the
Earth (4x 1012 dyn/cm-2), this leads to a value of dp -h 1010 dyn/cm2 -rr 104
atmospheres.
Ringwood (personal communication) estimates the mass of this primitive
atmosphere to be approximately & that of the Earth, and the density at its base to
Some aspects of the thermal history of the Earth
269
be of the order of zg/cm3. This would imply a scale height of approximately
I ooo km and a pressure at the Earth’s surface of approximately 2.105 atmospheres.
Allowing for the increase of k with pressure, the removal of such a pressure would
lead to an expansion in the radius of the Earth of approximately 170km. Thus
an expansion of Iookm (due to the removal of an atmosphere with a pressure of
approximately 105 atmospheres at the Earth’s surface) in the early history of the
Earth seems reasonable. Such an expansion may well have been sufficient to rupture the Earth’s surface, and these initial “scars” may be the site of the mid-ocean
ridges and continents, which developed differently throughout geologic time
(cf. Wilson 1960). These surface expressions of the relief of pressure may be
accompanied by polymorphic phase changes occurring deeper within the Earth
(a pressure of 105 atmospheres is reached at a depth of approximately 300km).
3. The Earth’s inner core
In the last few years a number of estimates have been made of the variation
with depth of the melting point of the materials that compose the Earth. Using
solid-state theory and seismic data, Uffen (1952b) estimated the melting-point
depth curve in the mantle, while Simon (1953) used a semi-empirical equation to
estimate the melting point of iron and hence obtained melting-point depth curves
in the core. Bullard (1954)has also used Simon’s equation to estimate melting
points in both the mantle and the core. A feature of all these estimates is much
lower values for the melting point in the core than in the mantle. Recent experimental work by Strong (1959) on the fusion of iron up to a pressure of 96000
atmospheres, also leads by extrapolation to comparatively low values for the
melting point in the core.
Zharkov (1959) has estimated the melting point of the mantle on the theory
that fusion is obtained when the density of thermal defects in the crystalline solid
reaches a certain critical value. At depths greater than I ooo km, his results are about
I ooooC lower than those obtained by Uffen (1952b). On the other hand, his estimates of the melting point of iron at high pressures are greater than those obtained
by Simon (1953), his value of the melting point of iron at the core-mantle boundary
being approximately the same as that of the mantle at that depth. He thus finds an
almost smooth variation of the melting point-depth curve throughout the entire
Earth. Gilvarry (1956) under certain assumptions, has given a theoretical justification for Simon’s equation, and obtained fusion temperatures for both the mantle
and the core. Unlike other results, he obtains melting points for the inner core
in excess of those in the mantle (1957).
On the basis of Gilvarry’s curves, the explanation put forward by Jacobs (1953)
for the formation of a solid inner core as the Earth cooled from a completely molten
state, still holds. A solid inner core and solid mantle would form if the fusion
temperature for the inner core is higher than that for the mantle at its base (see
Figure I). In this case solidification would commence at the centre of the Earth.
A solid inner core would continue to grow until a curve representing the adiabatic
temperature intersected the melting point curve twice, once at A, the boundary
between the core and mantle, and again at B. As the Earth continued to cool, the
mantle would solidify from the bottom upwards, trapping a liquid layer between
A and B. The mantle would cool at a relatively rapid rate, leaving this liquid layer
essentially at its original temperature, insulated above by a rapidly thickening shell
of silicates and below by the already solid (iron) inner core.
J. A. Jacobs
Successive
adiabatic
\cooling curves
I
I
I
Mantle
(solid)
I Inner core I
1 (solid) I
Core
(liquid)
I
I
Depth
FIG.I .
Possible formation of the Earth's inner core (after Jacobs 1953).
Successive adiabatic
rooling r i x v e s
/
--
L
u
3
L
L1I
01
a
E
Melting point curve
c-"
I
I
I
Matitle
[solid)
I
I
I
Core
I Inner
I
core1
1
[solid) I
I
I
I
I
1
I
(liquid)
Depth
FIG.2.
Alternative formation of the Earth's inner core.
Some aspects of the thermal history of the Earth
27'
In view of the other evidence, particularly the experimental work of Strong
(1959),it is instructive to consider in some detail the (more probable) case when the
melting point-depth curve in the core never rises above its value at the bottom
of the mantle A. I n this case as the Earth cooled, the entire core would be left
liquid-at least initially. The critical point is whether the Earth could cool sufficiently for the temperature to fall below the melting point curve of the inner
core and so finally leave it solid (see Figure 2). Two lines of evidence show that this
is a possibility. Ringwood (1960)has pointed out that it is possible that the crystal
mush at the bottom of the mantle may undergo an independent convection, whilst
the overlying liquid is still crystallizing and convecting. This would greatly
reduce the temperature throughout the solidified zone. Thus the actual temperature would quickly follow the curves I + 2 -+ 3. When the mantle had completely solidified, the only mechanism for further cooling that has been considered
until quite recently, is conduction-which is an extremely slow process. However,
following a suggestion of Preston (1956)on the possibility of radiative heat transfer
in the mantle, Clark (1957),Lubimova (1958) and Lawson & Jamieson (1958)
have examined the possibility in some detail. Although differing in orders of
magnitude, all these workers agree that the effect of radiative transfer is to increase
considerably the effective thermal conductivity of the mantle. Lawson & Jamieson
in particular believe that in the lower part of the mantle, the effective conductivity
may be increased by a factor of a hundred. Moreover, although the ordinary
molecular conductivity will probably decrease in the outer layers of the Earth
due to the increase in temperature, Uffen (1952a)has shown that in the deeper parts
of the mantle, the effect of pressure may dominate that of temperature and cause a
considerable increase in the thermal conductivity. Finally Lubimova (I 960) has
considered one further process of heat transfer in the Earth's mantleenergy transfer by the excited states of atoms. At high temperatures, the contribution from the
higher excitation energies increases (in normal conditions practically only the
lowest excitation energy is important), and thus the rate of the excitation thermal
conductivity must increase. Thus all the recent evidence is for a considerable
increase in the effective conductivity with depth, and had the Earth cooled in its
early history to curve 3 (Figure 2), it is more than possible that it could have cooled
further throughout geologic time to curve 4,thus forming a solid inner core. The
inner core has thus been slowly growing.
The gradient of the fusion curve for iron at high pressures may be rather flat,
and it might therefore appear that if the further cooling suggested above actually
took place, the whole core might solidify-which, of course, is contrary to seismic
evidence. However present indications are that the outer core is likely to contain
a fair amount of silicon (of the order of 20 per cent) which would lower its melting
point compared to that of pure iron, thereby steepening the melting point gradient
across the outer core. Any tendency of the Earth to heat up at depth due to
long-lived radioactive isotopes may be offset by the increased values of the effective
thermal conductivity. In any case most of the radioactive elements are concentrated in the crust and upper mantle, and Gast (1960)has shown that the abundance
of potassium in the Earth may well be much lower than hitherto supposed. Taking
the older (higher) abundances of the radioactive isotopes and assuming that there
is no increase in the thermal conductivity with depth, Jacobs & Allan (1956)
showed that the increase in temperature at the core-mantle boundary throughout
geologic time was only about 300'C. This point will be considered again in the
next section.
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J. A. Jacobs
4. The Earth’s magnetic field and convection in the core
The only satisfactory explanation of the Earth’s main magnetic field is that it
arises from a self acting dynamo action set up by motions in the fluid electrically
conducting core. The nature and cause of these motions are unknown, although it
has been shown that thermal convection provides the only satisfactory mechanism
for their generation. Thermal convection will occur in the core if the transport of
heat radially outwards exceeds the heat transport by thermal conduction alone.
The material of the liquid (outer) core should be very nearly in a state of chemical
equilibrium, uniformity being maintained by the mixing action of the convective
motion. Thus it is unlikely that any motion is due to variations in the physical
properties of the liquid outer core-rather it must be determined by boundary
conditions. Three possible models have been proposed by Elsasser (1950). The
first is one in which heat supplied by the inner core to the outer liquid layer
exceeds the amount that can be carried away from there by conduction alone. ’I
he
second is one in which the heat flow in the mantle adjacent to the core exceeds the
purely conductive heat transport in the core itself. The third is a compromise
model in which the heat supplied by the inner core is carried to the core-mantle
boundary by convection and is then carried away by the mantle.
In the past it has been the first model that has been considered the most
plausible, the heat source being supplied by the radioactive content of the inner
core. It has been shown (Jacobs 1954) that very little radioactivity is neededprobably less than one per cent of the concentration in the Earth’s crust. But
even this small amount presents difficulties, since there are geochemical problems
in having any radioactivity in the inner core and neutron activation analyses have
failed to find any in iron meteorites (Reed & Turkevitch 1956;Bate & others
1958).
It is instructive therefore to consider Elsasser’s second model, particularly in
view of the large increase in the estimate of the effective thermal conductivity with
depth in the mantle due to radiative heat transfer. If the heat carried away in the
mantle exceeds that supplied by conduction alone in the core, the second model
will hold. Extra heat could be supplied in the mantle by radioactive material
distributed throughout the mantle, although such heat sources are not necessary.
The convective process will occur if the heat conduction in the mantle is large
enough, the net result being an average loss of heat from the core over geologic
time. It is extremely difficult to estimate the relative magnitudes of the heat
carried away by conduction in the mantle to that supplied by conduction in the
core. The temperature gradient in the liquid core will be approximately the
adiabatic, whilst that in the adjacent mantle should not be too different. Thus if
the (effective) thermal conductivity of the mantle at the core-mantle boundary is
greater than that of the core there, this model will hold. This may well be the case
since the effective thermal conductivity of the mantle at depth may be as much as a
hundred times that of surface rocks because of radiative transfer and thus greater
than that in the metal rich core where the conductivity will be essentially mclecular.
5. The upper mantle
In recent years there has accumulated a growing body of data which indicates
that there are very fundamental differences in the constitution of the upper mantle
beneath the oceans and beneath the continents. Jacobs (1960)has drawn attention
Some aspects of the thermal history of the Earth
273
to the implications of the evidence of the increased number of heat flow measurements. The rough equality of the heat flow through all parts of the Earth,
together with the fact that the oceanic crust, being relatively thin and composed
mainly of basaltic materials, can produce at most 10 per cent of the observed
values, whereas the whole of the heat flow could be produced by the radioactive
isotopes in about 25 km of granite, emphasizes the striking difference between
the oceanic and continental mantle. This difference is borne out by the results of
a recent study by Dorman, Ewing & Oliver (1960)of shear velocity distributicns
in the upper mantle by mantle Rayleigh waves. Two hundred and fifty-six Rayleigh
wave phase solutions were computed for I I models of the continental and oceanic
crust-mantle systems some with as many as 50 layers. The authors find a systematic
difference in the sub-oceanic and sub-continental mantle down to a depth of
qookm. They also confirm the existence of a low-velocity layer, which is however
shallower (beginning at about half the depth) under the oceans than under the
continents. This difference in the structure of the upper mantle has also been
reported by Miss Lehmann (1960)from an analysis of the travel times of seismic
P and S waves. Such differences (down to about qookm) must be considered in
any theory of continental drift, which would im.ply that the continents drag the
upper few hundred km of the mantle along with them, or fundamentally charge
the mantle as they drift. These fundamental differences between sub-oceanic and
sub-continental upper mantle are in accord with the idea that the whole of the
crust of the Earth has been derived from lavas extruded from the mantle during
geologic time (Wilson 1957,1959). On this theory, it is not surprising that there
are differences in upper mantle structure, in view cf the very different crustal
structure under continents and oceans. Recent studies (Russell 1956) of the
isotope ratios in lead ores can be interpreted to mean that the majority of these
ores have come directly from the mantle, giving further support to Wilson’s
thesis.
Lubimova (1958)has shown that the thermal conductivity of the upper layers
of the Earth is determined almost entirely by its molecular component and that it
has a minimum value (at a depth of 50-10okm) some 1*5-2*0times lower than
in normal surface rocks. The depth to the low velocity layer has been estimated
at about 50 km under the oceans and 120km under the continents. It is not unlikely that these two observations are related. Moreover because of the contribution from radiative transfer, the temperature gradient required to remove heat from
the Earth’s interior decreases as the effective thermal conductivity increases with
temperature. As a result melting temperatures are most likely to be approached
in the outer few hundred kilometres of the Earth. If the radioactivity is concentrated in the upper mantle, MacDonald (1959) estimates that the melting point
may be approached, or even exceeded in the range 200-600 km below the surface.
It is difficult to synthesize all these ideas and other possible relevant hypotheses
(such as the possibility that the Earth is expanding and not contracting, a hypothesis that has drawn three independent letters to Nature within three months
early in 1960)into a coherent whole-and even more difficult to distinguish between
cause and effect in the different phenomena. It is all too easy to concentrate (in
ignorance) on one point to the exclusion of another, while the first phenomenon
may turn out to be but quite a secondary result of the more dominating role
played by the other.
S
274
Acknowledgments
J. A. Jacobs
The author is grateful to Professor J. C. Jaeger, Australian National University,
Canberra, and Professor K. E. Bullen, University of Sydney, who made it possible for him to spend two months in Australia, where most of this paper was
written. Thanks are also due to Dr A. E. Ringwood for many stimulating discussions.
Geophysics Laboratory,
University of British Columbia,
Vancouver, Canada:
1960 September.
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