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OCR for page 67
A Dynamics
of the Core
We now turn to a closer study of the outer core. Our purpose is to
discover how much energy is generated in it, how much heat crosses
into the mantle, whether this amount of heat is consistent with our
earlier consideration of the lower mantle and of layer D" in particular
and whether it is sufficient to drive wholemantle convection.
Our principal clue is the geomagnetic field. It is now generally agreed
that this field is generated in the outer core by motion (flow) of the
electrically conducting fluid (molten iron and sulfur) that forms
the outer core. Since precession seems unlikely to be the main cause of
the flow (Rochester et al., 1975; Loper, 1975), convection is required.
Convection can be either thermal or chemical. Thermal convection
requires a source of heat. In chemical convection, the differences in
density that cause the motion are due to differences in chemical compo
sition (e.g., a difference in sulfur content). It is indeed conceivable that
slow cooling of the whole core might lead to crystallization of iron,
which accumulates to form a growing inner core; removal of iron from
an FeS liquid leaves a liquid richer in sulfur and presumably lighter
than the rest of the outer core; the lighter liquid rises by buoyancy.
Both the settling of the solid iron and the rising of the sulfurenriched
fluid release gravitational energy; hence the name gravitationally
powered dynamo given to generation of the magnetic field by this
process, which has long been advocated by Braginsky (1963), and more
recently by Gubbins (1977), by Loper (1978), and by Loper and Roberts
(1978~. The name is perhaps not quite appropriate, since a thermally
67
OCR for page 67
68 ENERGETICS OF THE EARTH
driven convecting system is also powered gravitationally; in addition,
it should be remembered that the assumed crystallization of iron and
formation of a light liquid enriched in sulfur result from cooling and are
therefore also thermal phenomena.
STABILITY OF THE CORE
Whether convection is thermal or chemical, the temperature gradient in
the outer core cannot, on the average, depart very much from being
adiabatic. If it could somehow be shown that the temperature gradient
is less than adiabatic, it would follow that convection does not occur
because the core is stably stratified.
One of the few things that seem certain for the core is that since the
inner core is solid and the outer core liquid, the temperature curve must
be below the melting point curve in the inner core and above it in the
outer core; at the inner core boundary (ICB), where r = ri, the tempera
ture gradient must be less, in absolute value, than the melting point
gradient. This is sketched out in Figure =1. If the core is unstable and
convecting, the temperature gradient must be steeper than the adiabat
and so also must the melting point curve. Higgins and Kennedy (1971)
have argued that this is not the case; but their conclusion is based on
an estimate of the melting point of iron at the inner core boundary that
is almost certainly too low (see Chapter 3~. As also mentioned in
Chapter 3, the adiabatic gradient in the outer core is uncertain by a
factor of perhaps as much as 2. Thus no firm conclusion can be reached
on those grounds.
Is there observational evidence to the effect that the core is stably
stratified or not? Olson (1977) has considered the problem of the in
ternal oscillations of a body consisting of a uniform solid elastic mantle
and a solid inner core bounding a stratified, rotating, inviscid, fluid
outer core. The interaction of buoyancy and rotation results in two
types of waves: (1) internal gravity waves that exist if N2 ~ 0 (N is the
BruntVaisala frequency, which describes the density stratification;
N ~ 0 in a stable core); and (2) inertial oscillations that exist if
N2 < 4Q2 (Q is the angular velocity of rotation). For a model with a
density stratification similar to that proposed by Higgins and Kennedy,
the internal gravity waves have eigenperiods of at least 8 hours. A
model with unstable stratification admits no gravity waves but admits
inertial oscillations whose eigenperiods have a lower bound of 12 hours.
There is unfortunately almost no observational evidence of longperiod
terrestrial oscillations; in any event, as Olson points out, their ampli
tude is confined predominantly to the outer core, so that their detection
OCR for page 67
Dynamics of the Core 69
` ~
'a '
111
it_
_
1 `, ~ _
1 ~e \~'/
ADIABAT
1 ~
I MELTING
I POINT
INNER CORE ~OUTER CORE
/
an.
TEMPERATURE
r'
RADIUS r
FIGURE =1 Temperature in the core. The temperature must lie
below the melting curve In the solid inner core, and above the melting
cuIve in the liquid outer core. For convection to occur throughout
the outer core, the temperature in it must lie below the adiabat
through the inner core boundary.
on the surface would be difficult. Since, furthermore, oscillations in the
real core are likely to be hydromagnetic rather than mechanical, ob
served periods of free oscillations of the earth cannot as yet be used to
discriminate between stable and unstable core.
In a gravitational field, a twocomponent fluid that is not stirred will
develop a compositional gradient, because the lighter constituent
(sulfur in the case of the core) will tend to concentrate at the top
(Guggenheim, 1933, pp. 103159~. Since sound velocity depends on
composition as well as on pressure and temperature, the sound velocity
profile through the outer core could be used, at least in principle and
given an adequate equation of state, to detect any such compositional
gradient. None has been clearly revealed to date. Too little is known,
however, of the diffusion coefficient of sulfur in an ironsulfur melt
under core conditions to predict whether the sulfur distribution could
reach its equilibrium value within the few billion years of the core's
existence.
OCR for page 67
70 ENERGETICS OF THE EARTH
Since, on the one hand, there is no observational evidence that the
core is stably stratified, and since on the other hand there is a geo
magnetic field, the generation of which does seem to require flow, we
shall assume henceforth that the outer core is indeed convecting, with
the implication that the temperature curve through the outer core is
only very slightly steeper than the adiabat through the ICB and lies
above the melting point curve. This will require the temperature at the
inner core boundary to be on the high side of the estimates given in
Chapter 3.
CAUSES FOR CONVECTION
Conceivably, the outer core could be convecting for two different
reasons:
1. The outer core contains distributed heat sources, for instance
radioactive potassium (see Chapter 21.
2. Cooling from the top. Convection can occur in a fluid cooled from
the top as well as in a fluid heated from below, or heated internally. The
mantle, losing heat, as it does at the surface of the earth, may be
cooling, with the temperature at the coremantle boundary (CMB) slowly
decreasing with time.
Two further consequences of the cooling hypothesis are interesting:
1. Since the temperature at the boundary of the inner core equals the
liquidus temperature (melting point), which increases downward with
increasing pressure, cooling moves the ICB outward. Crystallization of
iron releases some latent heat of melting, which slows down the cooling
and increases the average temperature gradient through the outer core.
The release of latent heat is, in effect, equivalent to a heat source.
The outer core is thus cooled from the top and heated from below
(Verhoogen, 19611.
2. Growth of the inner core releases gravitational energy. Crystalli
zation of iron enriches the remaining liquid in sulfur, prompting what
was described earlier as chemical convection.
ENERGY REQUIREMENTS OF THE GEOMAGNETIC
DYNAMO
Associated with a magnetic field B there is an energy density B218~,u.
In the electromagnetic units we shall be using, ,u = 1 in the core, which
is assumed not to be ferromagnetic.
OCR for page 67
Dynamics of the Core 71
The electric current density j = (1/4?T,U) curl B leads to ohmic dissipa
tion (=Joule heating) at the rate J~/`r per unit volume. Here ~ is the
electrical conductivity of the core, assumed to be uniform. The value
of a generally accepted at the moment is S x 108 s/cm2 = 5 x 105
mho/m (Gubbins, 1976~; it used to be 3 x 106. Reasons for the change
are not clear; in any event the value of or is uncertain, perhaps by as
much as a factor of 10. This is not surprising, considering that neither
composition nor temperature are exactly known.
Ohmic dissipation leads to gradual decay of the electric currents and
of the* magnetic field. In the absence of any source to maintain it, the
field decays with time according to the equation
SIB
at 4~
(4.1)
which shows that a characteristic decay time ~ is of the order of ,ucrL2,
where L is a characteristic length such that V2B B/L2. For the core,
if L is taken to be 3 x 108cm (approximately the radius of the core),
4.5 x 10~i s = 15,000 yr. Since paleomagnetic observations indicate
that the strength of the earth's field 2 billion years ago oscillated within
roughly the same limits as the field of the past few thousand years, it
follows that the core must be producing magnetic energy at least at the
rate Em /r, where
Em = ~d V
8rr,u
is the total magnetic energy, integration being over all space. Calcula
tion of Em is conveniently split into two parts. First we calculate the
energy En outside the core, where the field is a potential field charac
terized by the usual gaussian coefficients that describe the field at the
surface of the earth. The first term in the expansion (dipole field) gives
S x 1025 ergs. Other harmonic terms get progressively smaller as their
order increases. The grand total is unlikely to be greater than about
5 x 1026 ergs. Taking ~ _ S x 10~i s, as above? the dissipation rate
is ~10~5 ergs/e = 108 W. Gubbins (1975), by a different method, cal
culates a dissipation rate ~ for the poloidal field, ~ c 1.S x 108 W.
Inside the core, the situation is more complicated. In addition to the
poloidal field Bp that emerges from the core as the potential field we
measure at the surface, there may be a toroidal field B,,~, the field lines
OCR for page 67
72 ENERGETICS OF THE EARTH
of which remain within the core* and which cannot therefore be mea
sured directly. The existence of this field, which only has a component
B`p in the azimuthal direction, is made very likely by the ease with which
it can be induced by any differential rotation within the core. If two
parts of the fluid core rotate at different rates with angular velocities
that depend on r or ~ (r is the radial coordinate, ~ is colatitude), the field
lines of any poloidal field will be stretched and dragged by the differen
tial rotation and wrapped around the rotation axis, forming a toroidal
field that must necessarily vanish on the surface of the conducting fluid.
Since the strength of the toroidal field cannot be measured, it must
be guessed. Guessing customarily proceeds along the following lines.
The magnetohydrodynamic induction equation is
= 7,V2B + curl (u x B),
at
(4.2)
where u is the velocity of the fluid motion and ~ = 1/4~ r is the
magnetic diffusivity. In spherical coordinates (r, 8, ¢) the ~ component
of (4.2) is, for an axisymmetric field:
~~ (V2 ~ . 2 ~ BE =Br ~ ~ ¢) (4 3)
In the steady state, dB¢,/dt = 0. It may be reasonably assumed that
all remaining terms in equation (4. 3) are of the same magnitude, so that,
in particular,
71 ~Br ¢' or Be = BrRm,
r2 r
where the Reynolds magnetic number Rm is defined as
Rm = OR = 4~/l,~Rurp,
(4.4)
(4.5)
R being the radius of the core. It is also customary to suppose that u<,
may be about 4 x 102 cm/s. (The figure is obtained by interpreting the
*This assumes the lower mantle to be a perfect electric insulator, which it is not. The
toroidal field of the core does probably leak into the lower mantle with much diminished
intensity, but it still does not reach the earth's surface because of the very low conduc
tivity of the upper mantle.
OCR for page 67
Dynamics of the Core 73
rate of westward drift of the secular variation, 0.2°/yr, to represent the
differential rate of rotation of the outer layers of the core with respect
to the rest of it and to the mantle.) Then, for ~ = 2 x 104cm2/s, Rm
600. Since Be at the surface of the core is ~4 G (extrapolated from its
value, ~0.5 G. at the earth's surface), B., might be as large as 2400 G.
The argument is quite speculative. Equation (4.4) is obtained by
crudely equating terms on the right and left sides of (4.31; this procedure
amounts to saying that because 100  99 = 2  1, 100 _ 2 and 99 _ 1.
A different estimate of the total field B in the core is obtained by
assuming a rough balance between the Coriolis and Lorentz forces.
(The Lorentz force j x B is the force the magnetic field exerts on the
fluid.) This gives
B2
2pQu =
47r,uR '
(4.6)
since j = (1/4'r,u) curl B and curl B BIR. The density p of the core
being about 11 g/cm3 and the mean angular velocity Q = 7.29 x 1O5/s,
Equation (4.6), with R = 3 x 108 cm and u = 4 x 102cm/s as before,
gives B _ 300 G. The justification for equating Coriolis and Lorentz
forces comes mostly, it seems, from Chandrasekhar's (1961) calcula
tions on the onset of convection in a plane layer of viscous fluid that is
rotating in the presence of a uniform magnetic field. Chandrasekhar
showed indeed that under some special circumstances convection is
most easily started (minimum critical Rayleigh number) when the two
forces are approximately in balance (Acheson and Hide, 1973, p. 213).
But this view has been criticized by Busse (1975b) and also by Gubbins
(1976), who finds evidence against the existence of a toroidal field as
large as 100 G (10 mT). Gubbins' argument is, however, a bit circular;
he rejects a strong toroidal field because it leads to what he thinks is an
unreasonably high rate of ohmic heating. Busse, on the other hand,
reaches the conclusion that the toroidal field in the core is of the same
order of magnitude as the poloidal field by solving the complete hydro
magnetic problem (including the Lorentz force) in a cylindrical configu
ration that reproduces, he believes, the essential features of the core.
We are thus left guessing as to what the intensity of the magnetic field
might be inside the core. If the average field in the core is 100 G (.01T),
the corresponding magnetic energy is about 7 x 1028 ergs and the ohmic
dissipation rate is of the order of 10~° W. Braginsky, a proponent of the
"strong" toroidal field hypothesis, once calculated (Braginsky, 1965) a
dissipation rate of 3.8 x 10~2 W; he now suggests (Braginsky, 1976) a
"more realistic" value 10 times smaller, or 4 x 10~ W.
A tenuous clue may be provided by Bukowinski's (1977) determina
OCR for page 67
74 ENERGETICS OF THE EARTH
lion of the temperature gradient in the inner core, alluded to earlier
(Chapter 3~. By fitting "observed" properties of the inner core
(density, seismic velocities) to a quantummechanical equation of state
for iron, Bukowinski finds that the temperature at the center exceeds
the temperature at the ICB by an amount AT, which varies from 234° to
350° according to the density model chosen; his preferred model PEM
has /` T = 234°. In the steady state, a temperature gradient implies a heat
source in the inner core that can hardly be other than the ohmic heating
caused by electric currents diffusing from the outer into the inner core.
The total rate 4> of ohmic heating in the inner core with radius ri is then
of the order of 4aTrik AT _ 1 x 10~i W for k = 30 W/m deg (this is an
underestimate, since the temperature gradient at the ICBiS likely to be
steeper than the average gradient AT/ri used here). If, then, we boldly
assume the current densityj to be uniform throughout the core, the rate
of ohmic heating for the whole core comes out as ~ = Hi (rc /ri )3, where
rc is the radius of the CMB; this gives ~ _ 2.5 x 10~2 W. which seems
high but could be reduced by a factor of 10 if the current density were,
on the average, about 3 times greater in the inner core than in the outer
core. It is also possible that the temperature gradient in the inner core
also reflects the secular cooling postulated by the adherents of the
gravitational dynamo. Perhaps all that can be said at the moment is that
to balance ohmic dissipation, magnetic energy must be produced at a
rate of 10~°10~i W.
But how is magnetic energy created in a hydromagnetic dynamo? To
see this, start from the expression for the current density in an ohmic
conductor moving at velocity u relative to a magnetic field B.:
j = E + (u x B),
(4.7)
where E is the electric field such that curl E =dB/dt. Dot j into the
left side and the equivalent (1/4~,u) curl B into the right side. Using
standard vector identities one obtains, after some algebra,
(I j) = div (E x B) + ~ (B ~ _ u FL, (4 8)
i, 4=,u ~ t 2
where Fir stands for the Lorentz force (1/47r,u) [(curl B) x Bl. Multiply
both sides by the element of volume dV and integrate over all space to
infinity. The integral idiv (E x B) dV = r(E x B) · dS goes to zero at
infinity because both E and B decrease faster than r2. Both j and u are
zero outside the volume V of the conducting and flowing fluid. Thus we
obtain
OCR for page 67
Dynamics of the Core 75
[JEm= _l JdV ~ u FldV,
at v ~
Jv
(4.9)
where Em = .(v B2/8~,u dV is the total magnetic energy, inside and
outside the fluid. The first integral on the righthand side of (4.9) is of
course the ohmic dissipation. The last term on the right, with its minus
sign, is the rate at which the fluid does work against the Lorentz force.
Equation (4.9) shows that the rate of creation of magnetic energy
equals the rate at which the fluid does mechanical work against the
resistance offered by the Lorentz force, minus the rate of ohmic dissi
pation. In the steady state, bEm/6t = 0; all the work done by the fluid
is converted to heat by the electrical resistance of the conductor. Equa
tion (4.9) also shows that when the fluid does no work (as when, for
instance, the flow is parallel to B. or more generally when u lies in the
plane containing B and j), the magnetic field decays at a rate determined
by the ohmic dissipation.
Our problem now is to examine what conditions must pertain in the
core to enable the fluid to do mechanical work against the Lorentz force
at a rate at least equal to the ohmic dissipation (10910~t W).
We consider first the case of a dynamo activated by thermal convec
tion.
EFFICIENCY OF A STEADYSTATE THERMAL DYNAMO
Consider a convecting system receiving heat H at a rate Q = dH/dt. In
the case of the core, Q could represent the rate of radioactive heat
generation or the rate of release of latent heat by crystallization of the
inner core. The problem is to determine the rate W at which the system
can do mechanical work. The ratio 71 = W/Q is called the efficiency of
the system.
In classical thermodynamics, one usually considers a dissipationless
system receiving heat at rate A from a source at temperature To, losing
heat at rate Q0 to a sink at temperature To < T., and doing work on the
outside at rate W. In the steady state, conservation of energy requires
W = A  Qo.
(4. 10)
The system is considered dissipationless, so that there is no production
of entropy by irreversible processes (heat conduction, friction, etc.~.
The system receives entropy from the heat source at rate Q~/T~ and
OCR for page 67
76 ENERGETICS OF THE EARTH
loses entropy to the heat sink at rate Qo/To. In the steady state, entropy
remains constant, so that
To To
Combining (4.10) and (4.11) gives for the efficiency
W
= = 1
Q1 Q1 T
(4.11)
T1  To
(4.12)
which is necessarily smaller than 1 since To > 0. Since any real system
will be dissipative, (4.12) gives an upper bound to the efficiency.
Note that in (4.12) the work W must be done by the system on its
surroundings. The MHD dynamo in the core is different, in the sense that
in the steady state (constancy of B outside the core) no energy leaves
the core other than the heat GO transferred into the mantle ("the sink")
at the temperature To of the coremantle boundary. As explained above,
the work done by the fluid against Lorentz and viscous forces is con
verted back to heat inside the core by ohmic and viscous dissipation.
Backus (1975) has shown that under these circumstances
Wm ' ~ T 1) Q.
(4.13)
where Wm is the sum of the rates of production of magnetic energy
(which equals ohmic dissipation) and of viscous dissipation, To is, as
before, the temperature at the coremantle boundary, Tm is the maxi
mum temperature inside the core, and Q is the sum of the heat produced
by distributed radioactive sources and of the heat that enters at the
inner boundary (i.e., the innerouter core boundary). Since there is in
principle no reason why Tm cannot be greater than 2To, there is, again
in principle, no reason why the efficiency Wm/Q could not be greater
than 1.
This somewhat paradoxical result may perhaps be understood by
noting that since the ohmic and viscous heating occur within the con
vecting fluid, the heat generated by these dissipative processes could in
principle also be used to power convection. Imagine for instance a
system with a uniform distribution of radioactive sources in which
dissipative heating is also uniform; an element of fluid cannot dis
tinguish between the two sources, which therefore both contribute to
the motion. The answer to the paradox is that the dissipative heating
OCR for page 67
Dynamics of the Core 77
will not in general be uniform but will be distributed so as to oppose or
cancel the temperature gradient required to drive the convection, i.e.,
to raise To so that it approaches Tm. In a body with the low viscosity and
high Reynolds number characteristic of the outer core, flow inside the
core will be essentially inviscid, most of the viscous dissipation taking
place in a thin boundary layer at the coremantle interface. The same
sort of thing will happen for the ohmic heating because of the high
magnetic Reynolds number of the core. This may be seen for instance
in Braginsky's (1976) "model Z" dynamo, in which electric currents
flow mostly in a thin magnetic boundary layer where the magnetic field
changes from its rather uniform axial character in the interior to its
poloidal (mainly dipolar) form outside the core. In both cases, produc
tion of heat near the coremantle boundary will tend to raise the tem
perature there and therefore reduce the negative temperature gradient
that is needed for thermal convection. Clearly, if To rises so that To ~
Tm, the efficiency given by (4.13) goes to zero. The author knows of no
general theorem to prove that dissipation will occur so as to hinder the
motion, but he strongly suspects that there must be one.
Metchnik et al. (1974) have considered the efficiency of convection
in a layer of fluid heated from below. The flow is assumed to be isentro
pic so that the temperature T2 at the top lies on the adiabat through T.,
the temperature at the bottom. Similarly, T2 + AT2 lies on the adiabat
through To + ATE. They claim that the efficiency is
ToT2
A=
T1
(4.14)
as in (4.121. The result is erroneous. It is reached by assuming that since
the flow is isentropic, the difference in entropy between an ascending
column ED (Figure =2) and a descending column EA is the same at all
heights; thus the difference in entropy between A and B is the same as
between F and C. If this were true, it would also be the same as the
entropy difference between D and E. The entropy difference between
A and B arises from the heat received on the horizontal branch of the
flow, which is mcpAT~, where m is the mass of fluid and cp its specific
heat at constant pressure, and the entropy change ASH is mcp/iT~/T~
(assuming that /iT~ << Try. Similarly, the entropy difference ~S2 iS due
to cooling along the segment DE and is mcp/iT2/T2. Since the two are
assumed to be equal, i\T~/T~ =AT2/T2. But in the steady state no energy
leaves the system other than the heat lost at the top, which must
therefore equal the heat received at the bottom; hence mcPl`T~ =
mcp/`T2 and /iT, = IiT2. Then To = T2, which contradicts the assumption
OCR for page 67
Dynamics of the Core 89
result of crystallization of iron, a layer of liquid depleted in iron and
therefore enriched in sulfur forms at the ICB. This layer is assumed to
be less dense than the rest of the outer core. It rises by buoyancy to
produce the convective motion that generates electric currents.
GRAVITATIONAL ENERGY
Loper (1978) has calculated the amount of gravitational energy released
by comparing the gravitational energy of the earth as it is today to the
gravitational energy of the hotter earth just prior to beginning of crystal
lization. He constructs a model of a compressible earth in which the
mass of the inner core is an independent variable, using an approximate
equation of state based on present properties of the earth and making
no allowance for the higher temperatures prevailing before crystalliza
tion began. He calculates a total release to date of approximately
2.5 x 1029 J. If the inner core began to form 4.5 billion years ago, the
average rate of release is 1.76 x 10~2 W; if the age of the inner core is
only 3 x 109 yr, the rate is 2.64 x 10~2 W. These numbers are very
uncertain, since the gravitational energy (2.5 x 1029 J) is the difference
between two large numbers, both of the order of 1032 J. and both
uncertain by at least 10 percent, or perhaps even 50 percent. Central
condensation of matter in the core causes g to rise, so that the pressure
in the core also rises, and so does the temperature. Part of the gravita
tional energy thus goes into heat of adiabatic compression. Loper,
comparing results for compressible and incompressible earths, es
timates that not more than 27 percent of the power goes into internal
heating and elastic compression; this, Loper says, leaves 1.28 x 10~2 W
to drive the dynamo, which seems ample. This, however, neglects all
dissipative processes, other than ohmic heating, by which the gravita
tional energy could be converted to heat. That such dissipative process
es do exist is beyond doubt, as we can see by asking ourselves where
the gravitational energy would go if the outer core consisted of pure
iron, so that no buoyant layer could form, or if the electrical resistivity
of the outer core happened to be so large that electrical currents could
not flow. Where, for instance, did the much larger amount of gravita
tional energy released by separation of mantle and core (Chapter 2) go?
A more precise evaluation of the gravitational input to the dynamo
may be obtained by returning to the momentum equation (4.15), form
ing its dot product with velocity u and integrating over the volume of
the core. Assuming the magnetic and kinetic energies to be constant,
we obtain as before for the driving term D,
OCR for page 67
90 ENERGETICS OF THE EARTH
D = 4? =~ u · VP dV +
1
~ V
Jv
pu · V ~ d V , (4.45)
where ~ stands for the total dissipation, ohmic plus viscous plus what
ever frictional dissipation may occur. This is Gubbins' Equation (7)
(Gubbins, 19771. After some simple transformations, we get
D = ~ P div u~V ~ P u dS + ~ Peru dS
v
+~6PdV
(4.46)
where the surface integrals are over the CMB.
The last three terms on the right of (4.46) were dropped in (4.17)
because of the assumption Up/ = 0, which is no longer valid, and
because the normal component of velocity on the CMB was assumed to
be zero. This, however, would no longer be the case if, as a result of
crystallization of iron, the volume of the whole core were to change,
thereby moving the CMB outward if the core expands, or inward if the
core contracts (see below).
A slight simplification is obtained in (4.46) if, following Gubbins
(1977), we replace ~ by ¢~ + by, where As is the value of the potential
on the CMB, which is taken to be an equipotential surface. Then
v 8t iv Bi V ¢,' ~ div (pu) dV
=¢~ i p u dS =  ( ply u dS (4.47)
so that, finally,

D = ~ = J P div u dVJ P u · dS + J ~r ''~ dV
(4.48)
The first integral on the right of (4.48) is the same as before extent
. . . .
_ ~A it
that vacations in density are now caused by compositional differences
rather than by changes in temperature. The two other terms specifically
represent the gravitational contribution to the dynamo. If the whole
core contracts while it cools and crystallizes, so that u is directed
inward, the pressure term represents work done by the mantle falling
OCR for page 67
Dynamics of the Core 91
in, so to speak, on the shrinking core. If, on the other hand, the core
expands (see below), this same term represents work that must be done
to lift the mantle and is a negative contribution to the dynamo.
The last term on the right of (4.48) is estimated by Gubbins to be 1.7
x 10~ W. This evaluation is based on an assumed rate of crystallization
of 25 mats; at that rate it would take the inner core 101° yr to grow to its
present size, so that Gubbins' value may be an underestimate.
To estimate the second term on the righthand side of (4.48) we must
know the rate u at which the core boundary moves. This requires some
consideration of volumetric relations.
VOLUMETRIC RELATIONS
It is customary in calculations pertaining to the core to suppose that the
ironsulfur melt behaves as a perfect binary solution, in which, by
definition, the two liquid end members (e.g., iron and FeS) mix in all
proportions without change in volume. Unfortunately, the FeFeS
system is not a perfect situation, at least at low pressure. This is shown,
for instance, by the fact that below 50 kbar, pressure has very little
effect on the eutectic temperature, implying that the volume of the
eutectic liquid is very nearly equal to the volume of a mixture of solids
in the eutectic proportion. Since both pure iron and pure FeS melt with
an increase in volume, contraction of the liquid must occur when the
two pure liquids are mixed. Whether this effect persists at high pressure
is not known. It is known that at pressures greater than about 55 kbar
the eutectic temperature begins to rise with increasing pressure, but
this effect may be due to a phase change in solid FeS.
Consider now a melt containing n, mol of iron (molecular mass Mel
and n2 mol of FeS (molecular mass My. The volume VO of the melt is
V0 = n1 V1 + n2 V2,
and its density pO is
_ nlM1 + n2M2
go _ _
nlv1 + n2V2
(4.49)
(4.50)
where v1 and v2 are, respectively, the partial molar volumes of iron and
FeS in the melt.
Suppose now that n1 is changed by an amount &~1, corresponding for
instance to crystallization of And mol of iron that separate from the melt.
From (4.50),
OCR for page 67
92 ENERGETICS OF THE EARTH
~= V 2 (V2M1V1M2 )
(4.51)
assuming that &~1 is sufficiently small that vat and v2 do not change
appreciably. The formation of a buoyant layer lighter than the remain
ing liquid requires that p0 decrease when iron is taken out of the melt,
or dpO/6n1 ~ 0. This, by (4.51), requires
90
v2 . ~1.55 .
v1 M1 58
(4.52)
If en 1 mol of iron crystallize out of the liquid, the volume Vs of the solid
so formed is V ~  as At 1, v,, being the molar volume of the solid iron. The
volume V' of the remaining liquid is
V' = (n1  &~) V1 + n2V2,
and the total volume V of solid and liquid is
V = Vs + Vz = Vo + (vsV1) ~1 ~
(4.53)
The volume of the whole core increases if vs > v1. Contraction occurs
if v*
Dynamics of the Core 93
V
1
~ 
J
o
>
J
IP







N2=0 N
Nl = I
MOLAR FRACTION
V2
\12
N2=1
N I =0
FIGURE =3 Molar volume diagram for a binary solution with
negative volume of mixing. The tangent at P to the curve APB
intersects the two ordinate axes at points representing, respectively,
the partial molar volumes of the two components in a solution with
molar composition N.
core inside a liquid outer core) requires that the liquidus temperature
(i.e., the temperature Tm at which solid pure iron is in equilibrium with
a FeFeS melt) should increase with increasing pressure. Now, at con
stant composition
(6Tm: V1Vs
~ UP JN s1 So;
where s1 and so are, respectively, the partial molar entropy of iron in the
melt and the molar entropy of solid iron. If v1  v ~ < 0, (6Tm/3P)N can
be positive only if so  s' > 0, which implies that crystallization absorbs
heat; if so, crystallization could not be induced by cooling. It is much
more likely that vat > v,, the difference v1  vat being of the same order
as the difference TV = vlO  vat, between the molar volumes of pure
liquid and pure solid iron; Leppaluoto (1972b) estimates Av = 0.055
cm3/mol at the pressure of the inner core boundary.
OCR for page 67
94 ENERGETICS OF THE EARTH
The volume change sustained by the whole core since crystallization
of the inner core began some time At ago is then
AV = Evevs.) An,
where ~ n = MOMS, Mi being the mass of the inner core, approximately
9.8 x 1025 g. Thus /iV9. 3 x 1022 cm3.
A much larger contraction results from cooling, if our estimate of
250° for the average cooling AT of the core since crystallization began
is correct (see below). If the average coefficient of thermal expansion
~ is taken to be 1 x 105/deg (Stacey, 1977), the contraction amounts
to AV = TV AT =  4.3 x 1023 cm3. For At = 4 x 109 yr = 1.26 x 10~7
s, the velocity of the boundary u = da /aft = pa ~/3 At = 2.3 x 10~2
cm/s, where a is the radius of the core; this corresponds to an inward
displacement of the core boundary of 2.9 km over 4 billion years.
A further contraction results from the increase in pressure caused by
the central condensation of matter. Loper (1978) estimates that the
pressure at r = 0 may have risen by some 0.23 Mbar since the inner core
began to form, enough to raise locally the density by more than 1
percent. Thus our value of AV = 4.3 x 1023 cm3 may be seriously
underestimated.
A simple mechanism for converting gravitational energy into kinetic
energy is by formation on the ICB of a layer of fluid lighter than the rest
of the outer core liquid. This, by (4.52), requires
V2 M2
>
V1 M1
(4.54)
This condition is likely to be satisfied on the whole, since the density
of the outer core is assumed to be less than that of pure liquid iron
precisely because of the addition of sulfur. But recall from Figure ~3
that v1 and v2 are both likely to be sensitive to composition; small or
even negative values of v2 are not excluded at low FeS contents. Since
the shape of the curve APB in Figure =3 is not even approximately
known, the possibility cannot be excluded a priori that (4.54) not be
satisfied for certain compositions, including the actual composition of
the core. The formation of a buoyant layer is therefore not certain, even
though it appears likely. Formation of a buoyant layer also requires that
diffusion of sulfur (or FeS) be sufficiently slow to prevent equaliza
tion of composition before the buoyant layer has had time to rise.
Finally, we must choose to ignore the possibility, pointed out earlier by
OCR for page 67
Dynamics of the Core 95
Verhoogen (1973), that two immiscible liquids with different sulfur
content might form. Clearly, a lot of experimental work on the FeFeS
system at high pressure is needed.
We return to the evaluation of terms in Equation (4.481. The pressure
on the CMB being about 1.4 Mbar, the surface pressure integral amounts
to4.9 x 10~ W. ormoreifwe have underestimated end da/dt. This
term does not contribute directly to the dynamo; being a measure of the
work done in the core by its surroundings (i.e., the mantle), it goes into
internal energy and heat, slowing down the rate of cooling of the core,
so that it must be retained when we later consider the rate at which the
core is losing heat.
There remains to evaluate the first term on the righthand side of
(4.48~. Here div u = ~1/p) Splat  (1/p)U · alp iS a function of the
density variations induced by crystallization of the core, by formation
of a buoyant layer, and by the temperature gradient that must neces
sarily exist since the core is assumed to be cooling. There is no simple
way of evaluating the integral.
All that can be said at the moment is that the only gravitational
contribution to the dynamo that can be approximately evaluated is the
term iv ¢~ (6p/dt ~ dV, which is probably larger than Gubbins' estimate
of it (1.7 x 10~ W) and presumably sufficient to maintain the dynamo
if, as Gubbins claims, gravitational energy released by rearrangement
of matter in the core is completely converted to magnetic dissipation.
That claim, however, can hardly be sustained at the moment. Clearly,
the same release of gravitational energy by rearrangement of matter
could occur in a nonconducting fluid in which no current can flow and
no ohmic dissipation is permitted and in which other nonohmic dissipa
tive processes would necessarily occur; these might also be operative
in the earth's core. Evaluation of efficiency would, however, be even
more difficult than for the thermal dynamo, because of chemical diffu
sion. Just as irreversible entropy production by conduction of heat
turned out to be an important factor in the thermal dynamo, irreversible
entropy production by chemical diffusion in a fluid of varying composi
tion could well limit the efficiency of the chemical dynamo.
HEAT OUTPUT OF THE CORE
We now attempt to estimate the rate at which the core must be losing
heat for the gravitational dynamo to operate. The heat output will
consist of (1) the released gravitational energy transformed into heat by
ohmic heating and other forms of dissipation, and (2) the heat released
OCR for page 67
96 ENERGETICS OF THE EARTH
by cooling of the core and crystallization of the inner core. The first
source we have found to be greater than 6.6 x 10~ W; we now proceed
to evaluate the second.
We start at the moment when the temperature at the center of the
earth has cooled down to the solidus temperature appropriate to the
pressure and composition of the core. Suppose the core contains 10
percent sulfur by weight (= 28.1 percent FeS), the molar fraction x~ of
iron being 0.8. To make the calculation at all feasible, we must now
assume that the melt behaves as a perfect solution. The solidus tem
perature Tm at molar fraction x, is
1 Ah°
T =
(RT1O Inx~J
(4.55)
where Ah° is the latent heat of pure component 1 and TV is its melting
point. For pure iron at P = 3 .3 Mbar, Leppaluoto (1972) estimates T,°
= 7400°K, l~h° = Tl°As~° = 3560 cal/mol. For x~ = 0.8, (4.55) gives*
Tm = 4770°K.
The pressure coefficient of Tm, As = dTm/dP, is
Vl  V
s 51  S.
Av°
A5° Rln al '
since the partial molar entropy so in a perfect solution is soRln x~.
For liv,°, the volume change in melting of pure iron at 3.3 Mbar, we take
again Leppa~uoto's estimates, AvO = 0.55 cm3/mol and Ash = 0.81
cal/mol deg. Then A, ~ 1 x 103 °/bar. Since the pressure at the center
is presently greater than the pressure at the ICB by about 0.34 Mbar, the
solidus temperature at the center is Tmo = 4770 + 0.34 x 103 = 5110°K.
Assuming that prior to the start of crystallization the temperature
distribution was adiabatic, the temperature Ta at r = ri was initially
Ta = TmoPa AP, where
A" = _ If) = cYT
dP s mp
*The same calculation at room pressure gives Tm = 1474°K = 1200°C, whereas the
observed solidus temperature form = 0.8 is ~ 1380°C. This large discrepancy shows how
far FeFeS melts depart from being perfect solutions.
OCR for page 67
Dynamics of the Core 97
and AP = 0.34 Mbar. Taking ax = 5 x 106/deg, T = 4.9 x 103°K,
p = 12.5 g/cm3, and cp = 0.16 cal/g deg = 670 J/kg deg. we find
ha = 3 x 1040/bar and Ta = 5010°K. Thus at r = ri the temperature had
dropped since crystallization started by AT = TaTm250° (Figure
4~. If this figure is typical of the whole core, the corresponding
cooling rate is ~ 2 x 10~50/s, assuming the inner core began to form 4
billion years ago. The rate of heat loss by the core with mass M is Qc
= Mcp ~T/dt = 2.6 x 10~2 W.
This calculation omits consideration of the increase in sulfur content
of the liquid caused by crystallization of iron and the corresponding
lowering of the liquidus temperature. The initial sulfur content of the
liquid was slightly smaller before crystallization started than it is today,
and its liquidus may have been higher by some 20° or so; total cooling
since the inner core began to grow would then be 270° rather than 250°.
The calculation also ignores the fact that prior to separation of the inner
ADI ABAT
~ Tmo
or
on
£
t
MELTING
INNER CORE
To
m
l
i
ll
rj
RADIUS
FIGURE 4 4 The temperature drop Ta  Tm since crystallization
of the inner core began, when the melting temperature at the center
was Tmo. At that time the temperature Ta at the inner core boundary
r = ri was on the adiabat through Tmo. Tm is the present temperature
at the inner core boundary.
OCR for page 67
98 ENERGETICS OF THE EARTH
core, the pressure everywhere in the core was lower than it is today; as
mentioned above, Loper (1978) estimates that the pressure at the center
has risen by some 0.23 Mbar since crystallization began. The pressure
difference between r = 0 and r = ri is also likely to have increased
somewhat, since the density of the region between r = 0 and r = ri has
also risen. Thus AP may have been smaller than the present value (0.34
Mbar) used here.
Finally, the latent heat released by crystallization is Ah = T As = T
[Asp  Rlr~x~ ~ _ 106 cal/g = 4.45 105 J/kg. For the inner core, with mass
Mi = 9.8 x 1022 kg, AH = Mi Ah and the average rate of release I is,
for an inner core 4 x 109 yr old, 3.46 x 10~ W.
These figures are, of course, very uncertain. The assumption of a
perfect solution leads (see footnote, p. 96) to underestimating the
solidus temperature at zero pressure by some 15 percent; if the same
correction applied at core pressures, Tm would be about 5500°K, and all
other temperatures would rise in proportion. The rate of cooling bT/dt
might not be much changed, but /\h may have been overestimated, as
so is probably less than so  Rl71x~ due to the exothermic effect of
· ~
mlxmg.
These figures differ appreciably from earlier estimates (Verhoogen,
1961), mainly because estimates of the melting temperatures of pure
iron at the pressure of the ICB have greatly increased in recent years,
and also because we have now considered the crystallization of iron
from a FeSFe melt rather than from its own pure liquid.
The total rate of heat loss Go of the core, assuming the inner core
started to form 4 billion years ago, is
Qo = Qc + Qe + I
= 2.6 x 10~2 + 0.34 x 10~2 + 0.66 10~2
= 3.6 x 10~2W
where Qc represents cooling of the whole core, Qe is the latent heat of
crystallization, and the third term, Qg, comes from the gravitational
energy, the largest part of which is, as we have seen, the work done by
the mantle falling in on a shrinking core. The value of Q9 is, however,
quite uncertain and may have been underestimated by a factor of 2 or
more. The uncertainty stems mostly from our ignorance of the volu
metric properties of the FeFeS system. Our evaluation of Qc and Qe
was based on the perfectsolution assumption, which is almost certainly
wrong.
OCR for page 67
Dynamics of the Core 99
~ Recall our earlier result that the radioactive dynamo requires a heat
output in the range 4 x 10~21 x 10~3 W. not markedly greater than for
the gravitational dynamo. There is thus little basis for the claim that a
gravitational dynamo requires a much lower heat flow into the mantle
than a radiogenic one.
There is at the moment no compelling evidence to tell us that the core
is not cooling and the inner core not growing (nor, for that matter, is
there any evidence that the core is not heating and the inner core
shrinking). If it seems more plausible to assume that the core is cooling,
then surely there is a gravitational contribution to the dynamo. How
large this contribution may be still seems very uncertain, mainly be
cause of dffl~culties encountered in evaluating terms in Equation (4.481;
these difficulties stem mostly from our ignorance of the composition of
the core and of its physicochemical properties (liquidus temperature,
heat and volume of mixing, etc.~. The radiogenic thermal dynamo is
conceptually simpler. But who will tell us how much potassium there
is in the core?