Ceophys. J. R. astr. SOC.(1978) 54,389-404
The gravitationally powered dynamo
David E. Loper Department of Mathematics and Geophysical Fluid Dynamics
Institute, Florida State University, Tallahassee, Florida 32306, USA
Received 1977 December 8;in original form 1977 June 22
Summary. The energetics of the gravitationally powered dynamo have been
studied with the aid of a compressible-earth model which allows for the
growth of the solid inner core. The basic premise of this study is that as the
Earth gradually cooled over geological time the solid inner core continually
accreted dense material which crystallized from an outer core composed of a
molten binary alloy. This process requires a continual rearrangement of
matter which generates the fluid motions needed to sustain the dynamo.
These motions maintain the outer core in a well-mixed state, in apparent
contradiction to Higgins & Kennedy’s hypothesis that the outer core is stably
stratified. The vigour of these motions is dependent primarily upon the
composition of the solid inner core, but is surprisingly independent of the
density of the light constituent in the core. If the solid core is composed
entirely of heavy metal, then as much as 3.7 x 1OI2W may be transferred
from the core to the mantle as a result of cooling and gravitational settling.
This is roughly equal to estimates of the amount of heat conducted down the
adiabat in the core, but it is argued that there is no direct relation between
the amount of heat conducted down the adiabat and the amount transferred
to the mantle if the convection is driven non-thermally. The gravitational
energy released per unit mass of the solid inner core is remarkably constant
and may be as much as 2 x 106J/kg, roughly five times the value of the
latent heat of iron. These values are reduced if the solid inner core contains
some light constituents. It was found that the efficiency of the gravitationally
powered dynamo may exceed 50 per cent, a much higher figure than is
possible for either the thermally or precessionally driven dynamo. Also, the
amount of gravitational energy available to drive the dynamo in the future is
many times that expended so far. The size of the magnetic field sustained by
gravitational settling was related to the density jump at the inner-outer core
boundary and the field strength was estimated to lie between 390 and 685 G,
strongly suggesting that the dynamo is of the nearly-axisymmetric type developed by Braginsky.
390
D.E. Loper
1 Introduction
It is reasonably certain that the Earth's magnetic field is generated by dynamo action within
the liquid outer core but the power source for the fluid motion has been the subject of
controversy for some time. A popular candidate for the power source has been some form of
thermal convection (see, e.g. Verhoogen 1961 ;Busse 1975) but a serious objection is that the
efficiency of conversion of thermal energy into fluid motion is likely to be very low.
(Metchnik, Gladwin & Stacey 1974, have estimated the efficiency to be 4 to 7 per cent.)
Malkus (1963, 1968) has advanced the idea that rotational kinetic energy of the Earth is
supplied to fluid motions within the liquid outer core by hydromagnetic torques arising
from the precessional motion of the core and mantle. Gubbins (1977) incorrectly dismisses this mechanism by claiming that the deviatoric stress integral
I,
u .T1*dS
is zero for no-slip boundary conditions. The velocity appearing in this integral is that of the
bounding surface but the integral need not be zero if the surface is in motion. For example,
one can impart motion to a viscous fluid within a spherical container via torsional oscillations of the container. However, the power must be fed to the precessionally powered
dynamo via dissipative torques (see, e g . Loper 1975), making this mechanism inherently
inefficient and an unlikely candidate for the power source for the dynamo. A third possible
power source which was first advocated by Braginsky (1963) is the gravitational energy
released as the dense solid inner core forms and grows over geological time. (For definiteness we shall assume the inner core to be solid but the arguments and conclusions are similar
for an inner core composed of a dense, immiscible liquid phase.) Recent studies by Gubbins
(1977) and Loper & Roberts (1978) lend support to the idea that sufficient gravitational
power is available to meet the requirements of the dynamo. What is more, Gubbins (1977)
has shown that the energy released by gravitational settling within an incomprehensible
fluid is converted into heat via ohmic dissipation. That is, in contrast to the thermally and
precessionally powered dynamos, the gravitational dynamo is inherently efficient.
In this paper we shall investigate the energetics of the gravitationally powered dynamo
in order to estimate the size of the magnetic field which may be sustained and to elucidate
several important properties of the energy source. Specifically the effects of compressibility,
core composition and alloying within the solid shall be studied. These effects will be
discussed in some detail following a description of the basic operation of the gravitational
dynamo which we wish to model.
To simplify this description we shall assume that the separation of the core and mantle
is complete, thus ignoring the stirring mechanism advocated by Urey (1952) and Artyushkov
(1970, 1972). We shall consider the outer core to be homogeneous, molten binary alloy
composed of a heavy metal (iron with some nickel) and a light non-metal (sulphur, silicon
or possibly oxygen). It is known from metallurgy that the solid which crystallizes from a
binary melt does not in general have the same composition as the liquid. Following the
reasoning of Usselman (1975) we shall assume the c~mpositionof the melt to be more
metallic than the eutectic composition. This results in a solid more metallic and hence more
dense than the liquid even if we ignore the change of density upon solidification. As the
inner core grows by accretion of solid crystallizing from the molten outer core, a residue of
light material is left in the liquid near the inner core. Since diffusion of matter is ineffective
over length scales typical of the core, this excess of light material distributes itself throughout the outer core by means of convective motions driven by compositional buoyancy. It
is virtually certain that within the rotating core these convective motions are of proper form
Gravitationally powered dynamo
39 1
and sufficient vigour to generate a magnetic field by dynamo action. In fact, since the
magnetic diffusivity is much larger than the kinematic viscosity, the primary means of
dissipating the energy fed into these motions is via ohmic dissipation. This accounts for the
inherent efficiency of the gravitationally driven dynamo. It should be noted that if this
discussion is geophysically correct, then the core is close to a homogeneous adiabatic (i.e.
well-mixed) state, rather than stably stratified as proposed by Higgins & Kennedy (1971).
Also, since metallurgy dictates a density jump between the inner and outer core in general,
models which have none (e.g. Jordan & Anderson 1974) must be considered anomalous.
In the following section a simple earth model is developed in which the size of the solid
inner core is an independent variable. This model is used to calculate the change in gravitational potential energy of the Earth as the solid inner core grows. The gravitational energy
released due to rearrangement of matter and compressibility is included in the calculation
but the model excludes the less important effects of density change upon solidification and
density change resulting from temperature change. As Gubbins (1977) has noted, the energy
released by rearrangement of matter contributes directly to the motions which drive the
dynamo while that resulting from compressibility appears in part as internal heating and
is much less effective in driving the dynamo. The model is used to determine the relative
sizes of these two effects, giving a rough measure of the efficiency of the gravitationally
driven dynamo.
A central feature of the model developed in Section 2 is the treatment of the liquid outer
core as a binary alloy of heavy and light constituents. The most likely elements composing
the heavy constituent are iron and nickel while those composing the light constituent are
sulphur, silicon and/or oxygen (Brett 1976; Ringwood 1977). The composition of the light
constituent has been of vital concern to the advocates of the thermally driven dynamo
because of the expected occurrence of radioactive potassium 40 with sulphur, which
presumably would supply the necessary heat within the core. One would anticipate that in
the present study, the composition of the light constituent would be irrelevant, provided
the difference in density between heavy and light constituents is large (as in fact it is).
However, this presumption is incorrect; the amount of gravitational power released by the
continual growth of the solid inner core depends upon the composition of the light constituent, as we shall now explain.
The viability of the gravitationally driven dynamo rests on the assumption that the solid
phase which forms from the liquid outer core has a larger fraction of the heavy constituent
than the liquid. The difference in composition between liquid and solid phases depends,
among other things, upon the materials composing the alloy. In particular, if the light and
heavy materials have different atomic radii, they will not fit into a crystal lattice
conveniently and a large difference in composition between liquid and solid would be
expected. The atomic radii of iron, nickel, silicon, sulphur and oxygen are respectively 1.26,
1.24, 1.27, 1.06 and 0.74 8, at zero pressure. This indicates that iron, nickel and silicon can
coexist within a crystal fairly easily whereas the sulphur and oxygen atoms are too small.
This property is reflected in the position of the solidus curves in the phase diagrams of
Fe-S and Fe-Si (from Hansen 1958) shown in Fig. 1, with almost no solid alloying in the
Fe-S system and a high degree of solid alloying in the Fe-Si system. (The phase diagram
of Fe-0 is not given because the behaviour of this alloy at high pressure is likely to be much
different than that at low pressure, being substitutional at high pressure and interstitial at
low pressure (Ringwood 1977). Ringwood argues that the Fe-0 phase diagram will be
similar to that of Fe-S.) Assuming this property to hold at core pressures, the amount of
gravitational energy released by density differentiation depends upon the amount of solid
alloying which in turn is dependent upon the composition of the alloy. This point is
discussed further in Section 4.
392
D.E,Loper
WEIGHT PERCENT SULFUR
10
5
15
20
25
1600
e
1500
3
I-
a
u
1400
H
W
I-
t 300
10
20
30
40
ATOMIC PERCENT SULFUR
Figure 1. (a) The iron-rich portion of the Fe-S phase diagram (from Hansen 1958). The liquidus is
denoted by L and the solidus by S .
ATOMIC PERCENT SILICON
Figure 1. (b) The iron-rich portion of the Fe-Si phase diagram (from Hansen 1958).
2 A simple earth model
We wish to calculate the amount df gravitational energy released by the growth of the solid
inner core. This requires the construction of a model of a compressible earth in which the
size of the solid inner core is an independent variable. The basic equations for the model are
dpldr = - Gpm/r2,
dmldr = 47rpr2,
p =p(p),
dUfdr = - 4rrGpmr.
(2.1)
The notation is standard with U being the gravitational energy of the material below radius r .
The crucial step in modelling a compressible earth is proper formulation of the equation of
state p = p ( p ) . Normally the Adams-Williamson equation is used as the equation of state,
but this requires known seismic data. Since we wish to construct a model of the Earth valid
for other epochs for which seismic data is unavailable, this equation cannot be used. We
must instead construct an equation of state based upon the material properties of the Earth’s
interior.
Gravitationally powered dynamo
393
-.
,-
I
I
2
3
PRESSURE, MBAR
Figure 2. A plot of incompressibility k versus pressure p , taken from Table 1.3 of Jacobs (1975). This
figure demonstrates the continuity and linearity of the function k @). 1 Mbar = 10" Pa.
We shall use Bullen's k, p hypothesis (see, e.g. Jacobs 1975, p. 189) as the basis of our
equation of state. Cook (1972) has shown that this is not a general feature of materials at
high pressure but a fortuitous circumstance for the Earth. Since the mantle and the core as a
whole do not evolve in our model, the k , p hypothesis should remain valid as the size of the
solid inner core varies. A plot of k versus p derived from Table 1.3 of Jacobs (1975) is given
in Fig. 2. From this we see that the adiabatic incompressibility k is indeed a continuous
function of p throughout most of the Earth and as noted by Cook it is very nearly a linear
function. The most serious departures from continuity and linearity occur near and within
the solid inner core 0)> 3 x 10" Pa= 3 Mbar). It is very likely that these departures are
artifacts of Jordan's model rather than manifestations of actual conditions within the core
(see Cook 1972). (Jordan assumed that the density is continuous at the inner-outer core
boundary and that the shear velocity is constant at 3.5 km/s within the solid core: see
Jordan & Anderson (1974).) Therefore, we shall assume that k is a linear function of p
throughout the Earth;
k = (P + P x ) / a x
(2.2)
where p x and ax are parameters to be obtained from Fig. 2. Specifically, let a1= 0.304 and
pl = 0.68 x 10" Pa for p > 0.24 x 10" Pa which is all of the Earth below 671 km at
present (Cook prefers a1= 0.308, p1= 0.71), a2= 0.1306 and p z = 0.1 13 x 10" Pa in the
transition zone (0.24 > p > 0.14 x 10" Pa), a3= 0.14 and p 3 = 0.14 x 10" Pa in the upper
mantle (0.14 > p > 0.005 x 10" Pa) and a,= 0.0 in the crust.
The incompressibility k is defined as p d p / d p . Using (2.2), this may be integrated to yield
PX
=&(I
+P/PX>"X
(2.3)
where p: is the density of the material in a given layer at zero pressure. The densities in the
crust and mantle can be readily obtained from Table 1.3 of Jacobs (1975): py = 3.99 x
lo3kg m-3 in the lower mantle, p!: = 3.46 x lo3 kg m-3 in the transition zone, p:= 3.23 x
lo3 kg m-3 in the upper mantle and p: = 2.79 x lo3kg m-3 in the crust.
D. E. Loper
394
The equation of state for the core is more difficult to establish because in our model the
liquid outer core is composed of a binary alloy whose composition varies as the solid inner
core grows. Assuming the binary alloy to be an ideal mixture, its density is related to that of
its constituents by the formula
where 7 is the mass fraction of the light constituent and pc, PH and pL are the densities of
the core (either inner or outer), heavy constituent and light constituent. To maintain the
simple form of (2.4) for all core pressures, we shall assume that the incompressibilities of
the heavy and light constituents are identical, yielding
whereX=H,LorCand
7
- =1- 1 - 7 t o
.
p:
POH
PL
The justification for this assumption is given in Fig. 2 which shows that dk/dp i 3.3 for both
the silicate mantle and the iron core. Stacey (1972) had performed a similar analysis of
Haddon & Bullen's (1969) model HB1 and arrived at a value dk/dp i 3.6 for the entire
Earth. The present analysis which takes into account the greater incompressibility of the
upper mantle would appear to be more accurate. It should be remarked that the shock data
of Al'tshuler et al. (1958) and McQueen & Marsh (1960) can be correlated, using the k, p
hypothesis, by the equation
p = (8.57 x lo3kg m-3) (1 t p/0.43 x 10"
Their data yields dk/dp = 5.13, significantly higher than Jordan's or Haddon & Bullen's
models. The validity of this data for use in the Earth's core has been questioned by Stacey
(1972) who noted that the shockcompressed density of iron at 1.35 x 10" Pa (the mantlecore boundary) is greater than that of the core material at the same pressure. Also Stewart
(1973) has noted that the pressure-density relation along a Hugoniot may be quite different
from that in the Earth. Therefore we will not make use of the shock data in the construction
of our model.
Let us now estimate the densities p: and p i of the alloy constituents at zero pressure.
The heavy constituent is assumed to be primarily iron with a small amount of nickel. At low
pressure, iron has a density of 7.87 x lo3 kg md3 in a body-centred crystal (a)which is
not close packed. The shock data correlated by (2.7) indicates that iron transforms to a
close-packed crystal (E) at core pressures, giving an effective density at zero pressure of
8.57 x 103kg m-'. (The crystal-packing efficiency of a body-centred cubic is 0.6802 while
that of a close-packed crystal is 0.7505; (0.7405/0.6802) 7.87 = 8.57.) At low pressure,
nickel has a density of 8.9 x lo3 kg m-3 in a close-packed crystal; it has no important phase
changes as pressure increases. The value p; = 8.6 x lo3kg m-3 was chosen for the model,
corresponding roughly to 90 per cent iron and 10 per cent nickel.
The density of the light material in the core is particularly difficult to determine because
its composition is uncertain. We shall consider four possibilities: sulphur, silicon, oxygen and
lower-mantle material. At low pressure, sulphur has a low density, 2 x lo3 kg m-3, forming
covalent SB molecules. Allowing for phase change to a close-packed structure, a density of
p t = 2.5 x lo3kg m-3 was chosen as a rough lower limit for the density of the light con-
-
Gravitationally powered dynamo
395
stituent. Silicon compounds with iron to form FeSi which has a density of 6.1 x lo3kg m-3
at zero pressure. If the factor y in (2.4) is taken to represent the mass fraction of silicon,
rather than FeSi then the effective density of the light material, taking the compounding
into account, is 4.21 x lo3 kg m-3. This follows from the equation
where
-=-
&
PteSi
+q$-+]
WS
FeSi
PFe
and
W F ~= 55.85,
Wsi = 28.
A similar calculation for FeO with POL= 7.1 x 103kgm-3 (Ringwood 1977) yields an
effective density of 4.4 x lo3 kg m-3. If the light material in the core has .the same
composition as the lower mantle, the density at zero pressure is 3.99 x 103kg mA3. A
density of
= 4.0 x lo3 kg m-3 was chosen as a rough upper limit for the density of the
light constituent. The model was run with both densities, 2.5 and 4.0 x 103kgm-3, and it
was found that the gravitational energy released is remarkably (and fortunately) insensitive
to the choice of density for the light constituent. This point is discussed further in Section 4.
Equations of state (2.5)-(2.6) are valid for both the inner and outer core. To complete
the specification of them, we must determine how the mass fractions of light constituent
within the inner core yi and the outer core yo vary as the solid inner core grows. If the
total mass m, and the overall mass fraction yc for the entire core are held constant and yo
is assumed to be uniform in radius within the outer core, then conservation of mass of the
light constituent requires
Jo
where mi is the mass of the solid inner core. Let us assume that the composition of the
material being deposited on the solid inner core is related to that of the liquid by
yi(mi) = lo (mi)
(2.9)
where the alloying fraction p is a constant which measures the amount of alloying within the
solid: p = 0 means the solid is pure heavy material and p = 1 means the solid and liquid have
identical composition. From Fig. 1, we see that 1.1G 0.03 for sulphur and p A 0.82 for silicon;
p A 0 for oxygen. Assumption (2.9) is equivalent to assuming that the solidus curve is similar
in shape to the liquidus curve. Equations (2.8) and (2.9) may be easily solved to yield
(2.10)
(2.1 1)
These equations give the distribution of composition within the solid core as a function of
m(r) and the composition within the liquid as a function of mi(ri).
The equation of state described by (2.3), (2.5), (2.6), (2.10) and (2.11) contains a
number of discontinuities. Those which occur in the mantle are keyed by the pressure,
396
D. E. Loper
through (2.3), and are located automatically by the model. The remaining discontinuities
occur at the inner-outer core boundary ri, the mantle-core boundary ro and at the surface
r,. With a compressible earth, the mantle-core boundary and the surface move radially as
the inner core grows and the discontinuities in the equation of state cannot be located at
predetermined radii. To locate them we shall assume that the masses of the entire Earth
me,the entire core m, and the light material in the core m L remain constant as the solid
inner core forms and grows. me is known to be 5.977 x loz4kg while m, and m L can be
determined by requiring that ro & 3485 km and r, A 6371 km when ri I 1215 km, which
are the accepted current values. In the actual running of the model, the adjustment was
considered satisfactory if the discrepancy between the computed and actual values of r,
and r, was less than 10 km (< 0.3 per cent); typically the discrepancy was 2 to 5 km. The
value of the core mass obtained in this manner, 1.94 x
kg, agrees well with that
obtained from summing values given in Table 1.3 of Jacobs (1975). (As it transpired, r, and
rs are very weak functions of ri (Ar, G 4 km and Ar, Q 2.5 km for Ari 13000 km); the
mantle4ore boundary and the radius could have been located at the current values of r,
and r, with no loss in accuracy.)
Equations (2.1) were integrated numerically from r = 0 outward. This required initial
data on p , m and U . Obviously m(0) = 0 and U(0)= 0, but p ( 0 ) is not known beforehand;
it must be adjusted by an iteration scheme to achieve p G 0 at the Earth's surface. The
numerical integration was performed with a four-order Runge-Cutta scheme; a multistep scheme was not used because of the many discontinuities in the equations of state.
3 Computational results
The mathematical model developed in Section 2 was solved numerically for several choices
of density of light constituent and differing amounts of alloying within the solid inner core.
A given run consisted of the calculation of a sequence of earth models, one model for each
of a set of prescribed innercore radii. The values of the masses of the core m, and the
light constituent of the core m L were adjusted manually between trial runs to achieve
ro & 3485 km and r, G 637 1 km when ri & 1215 km. For example, the results of the run for
p t = 2.5 x lo3 kg m-3, p = 0 (no solid alloying), m, = 1.94 x loz4kg and m L = 0.17 x
10%kg are summarized in Table 1.
The entries in the second column of Table 1 give the change in gravitational potential
energy, from an initial value of U = -2.487 x lo3' J when ri = 0, as the solid inner core
grows. From this column we see that the gravitational energy released as the solid inner grew
from ri = 0 to 1215 km was approximately 2.5 x loz9J . Assuming that this energy has been
released uniformly over the past 4.5 x 109yr, this gives a continual power supply of
1.76 x 10" W. If we chose a smaller time for the age of the core such as 3 x lo9 yr
suggested by Verhoogen (1961), the power supply would be even larger: 2.64 x 10"W.
Looking at future earth models (with ri > 1215 km), we see that the energy release can be as
large as 30 x 1OZ9J , indicating that the gravitational power supply may be far from being
depleted.
The third column of Table 1 gives the gravitational potential energy released per unit
mass of the solid inner core. This is remarkably constant at roughly 2 x lo6J/kg. Intuitively
one would expect the energy released per unit mass to decrease as ri increases because the
mean distance travelled by the migrating masses decreases. However, the increase in the
strength of the local gravitational field as ri increases roughly offsets the decrease in the
travel distance, resulting in a fairly constant energy release per unit mass. This value is
approximately five times the value of the latent heat of iron quoted by Verhoogen (1961),
Gravitationally powered dynamo
397
Table 1. Model results for p i = 2.5 X lo3 kg m-3, p = 0, rnC = 1.94 X loz4kg and rnL = 0.17 x loz4kg.
Radius of solid inner
core
(106m)
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.o
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3 .O
3.1
Change in
gravitational energy
(1029 J )
0.0
- 0.001 3
-0.01 14
-0.0392
-0.0932
-0.1817
-0.3127
-0.4940
-0.7330
- 1.0366
- 1.41 13
- 1.8634
- 2.3982
- 3.0208
-3.7359
-4.5476
-5.4595
-6.4748
-7.5961
-8.8257
- 10.1654
- 11.6 166
- 13.1 801
- 14.8566
- 16.6462
- 18.5486
- 20.5634
-22.6896
- 23.8864
- 27.2316
- 29.6434
- 32.1996
Change in
gravitational energy
per unit mass of
solid inner core
(lo6J/kg)
Mass fraction of
light material in the
outer core
Central
pressure
(10" Pa)
-
0.08763
0.08763
0.08765
0.08771
0.08781
0.08798
0.08825
0.08861
0.0891 1
0.08974
0.09056
0.09156
0.09280
0.09430
0.096 11
0.09827
0.10085
0.10393
0.10761
0.11202
0.1 1732
0.12376
0.13163
0.14140
0.15375
0.16968
0.19089
0.22029
0.26344
0.33245
0.45957
0.76911
3.570
3.571
3.577
3.5 86
3.598
3.612
3.629
3.648
3.670
3.693
3.717
3.743
3.770
3.798
3.828
3.857
3.888
3.919
3.950
3.982
4.01 3
4.045
4.077
4.109
4.140
4.172
4.203
4.234
4.265
4.295
4.325
4.355
-2.0432
-2.2679
- 2.3 117
- 2.3 165
- 2.3 118
- 2.3033
- 2.2925
- 2.2797
-2.2659
-2.2509
-2.2352
-2.2185
-2.2012
-2.1833
-2.1650
-2.1464
-2.1275
-2.1085
- 2.0894
- 2.0703
-2.0512
-2.0324
-2.0137
- 1.9953
- 1.9772
- 1.9594
- 1.9421
- 1.9222
-1.9061
- 1.8880
- 1.8732
demonstrating that gravitational energy release can easily be larger than the thermal energy
once thought to drive the dynamo. This conclusion is in agreement with that of Braginsky
(1963).
It may be seen from the fourth column of Table 1 that the mass fraction of light
constituent in the entire core is 0.08763 for this run. The current mass fraction of light
material in the outer core is predicted by this model to be 0.0930, a change of less than
1 per cent. The mass fraction of light material in the outer core does not change rapidly until
the radius of the solid inner core roughly doubles its present value. In reality, the future
evolution of the core will change in character once the eutectic composition is reached in
the outer core. Subsequent growth of the solid inner core will not be accompanied by
radial separation of heavy and light material since the solid formed from an eutectic alloy
2 0results
,
has that same composition. For example, if the eutectic fraction is ~ ~ ~ 0 .the
listed in Table 1 predict that the gravitational dynamo will cease to function once the solid
inner core radius reaches 2633 km. However, the dynamo will have released 2.1 x lo3' J
D. E. Loper
by that time. If the current rate of power expenditure is 1.76 x 10" W, this event will not
occur until 37 x 109yr have elapsed. These numbers are of course not to be taken as
quantitatively accurate; the point to be made here is that gravitational settling has the
potential to power the dynamo for aeons to come. This is in contrast with Braginsky's
pessimistic conclusion that the energy source will be exhausted in less than 3 x 10' yr.
The central pressure and density predicted by this model are somewhat higher than those
obtained by Jordan (see Jacob's Table 1.3, 1975) because he made the overly restrictive
assumption that density is continuous at the inner-outer core boundary. (Although Jordan
8c Anderson (1974) claim to have assumed the density to be continuous at the inner-outer
core boundary, their model B1 curiously shows a density discontinuity of 0.17 x 103kg m-3
at the radius.) This assumption is contrary to metallurgical experience and automatically
precludes a gravitationally powered dynamo.
The gravitational energy release given in Table 1 arises from two separate effects: the
radial separation of light and heavy material and the radial contraction of the compressible
earth as the pressure increases. The former effect drives the dynamo efficiently via fluid
motion while the latter effect contributes only indirectly, through internal heating, 01 not at
all (some energy being stored in elastic compression). To estimate the relative magnitudes of
these two effects, the numerical programme was modified to model an incompressible earth.
It was found that the energy release in an incompressible earth is approximately 73 per cent
of that in a compressible earth. In other words, roughly 73 per cent of the 1.76 to 2.64 x
10" W of gravitational power released is fed through fluid motions into the magnetic field
while the remaining 27 per cent appears as internal heating and compression.
To test the importance of the choice of density of the light constituent, the model was
run with p i = 4.0 x lo3kg m-3, p = 0, m, = 1.92 x 10" kg and m L = 0.36 x 10%kg,
giving ro = 3479 km and rs = 6379 km at ri = 1215 km. The results obtained from this run
are almost identical to those given in Table 1, except that the mass fraction of the light
constituent is doubled. In particular, the total energy released as the solid inner core.grew
from ri = 0 to 1215 km is within 0.1 per cent of the value given in Table 1. The model is
indifferent to the choice of p t because, with no alloying, the density within the solid inner
core is independent of p: and the density within the outer core is virtually independent of
p: also. For example, from Table 1 we calculate, using (2.6), that & = 7.07 x lo3 kg m-3
while the data from the present case gives & = 7.05 x lo3 kg m-3. (Even this small
difference may be attributed to the disparities in adjustment of ro and rs between the two
cases.) Therefore, the amount of gravitational energy released is also independent of p t .
We shall elaborate on this point in Section 4.
We have stated previously that the composition of the solid is an important factor in
determining the amount of gravitational energy released by formation of the solid inner
core. This may be verified by noting that if the solid and liquid are of identical composition,
as is the case of solidification from an eutectic liquid, then no gravitational separation of
light and heavy material occurs and the gravitational dynamo fails. As an independent check
of this idea the run tabulated in Table 1 was repeated with an alloying fraction p = M (the
solid having M the mass fraction of light material that the liquid has) and it was found that
the resulting gravitational energy release was approximately 44 per cent of that given in
Table 1.
As a final check that the model we have constructed is realistic, the density as a function
of radius calculated by the present model is compared with that of Jordan's model B1, as
given by Jacobs (1975), in Fig. 3. Within the mantle and crust, the results are indistinguishable. Within the outer core, the present model calculates slightly lower densities with the
disparity being less than 2 per cent. The largest difference comes in the inner core because
398
Gravitationally powered dynamo
399
______-------
6
5
4
3
2
I
1,
RADIUS, lo6 M
Figure 3. A plot of density versus depth comparing the results of the present model with those of Jordan
& Anderson (1974) as given in Table 1.3 of Jacobs (1975). A solid line denotes Jordan & Anderson's
results; a dashed line denotes the results of the present model with no solid alloying (p = 0); a dotted line
denotes the results of the present model with p = %. The results are indistinguishable in the mantle.
Jordan assumed p to be continuous. This figure also shows the variation of density within
the inner core as the alloying fraction is increased from I.( = 0 to 34.
4 Discussion and conclusions
The energetics of a gravitationally powered dynamo have been studied with the aid of an
earth model which allows for the growth of the solid inner core. The molten outer core has
been modelled as a binary alloy composed of a heavy metal and a light non-metal. It is
assumed that the composition of the alloy is more metallic than the eutectic and that the
solid which crystallizes from the melt is more metallic and hence more dense than the liquid.
The basic premise of this study is that, as the Earth has gradually cooled over geological
time, the solid inner core has continually accreted the dense solid which crystallized from
the molten outer core. This process requires rearrangement of matter with the heavy constituent moving radially inward and the light moving outward. The result is a continual
release of gravitational energy which is capable of driving the dynamo efficiently since the
rearrangement of matter leads directly to fluid motions. In this discussion, we shall assume
the age of the solid inner core to be 4.5 x 109yr. If its age is only 3 x 109yr as Verhoogen
(1961) prefers, all figures concerning power release in the following discussion should be
increased by 50 per cent.
It was found in Section 3 that as much as 1.76 x 1OI2W may be released by the growth
of the solid inner core. Approximately 27 per cent of this power is a result of compressibility and appears as internal heating and elastic compression, primarily where the change in
the local acceleration of gravity produced by the change in density distribution is the largest,
that is, in and near the inner core. The remainder, roughly 1.28 x loL2W, represents an
400
D. E. Loper
upper bound on the power available to drive the dynamo. This power is fed into fluid
motions, thence by dynamo action into magnetic energy and finally by ohmic dissipation
into internal heating, primarily within the outer core. Therefore, most of the 1.76 x 1OI2 W
of gravitational energy must cross the mantle-core boundary (MCB) as heat. In addition the
heat of fusion liberated by crystallization and the heat cast off as the entire core cools must
cross the MCB. From Verhoogen (1961) these may be estimated to be roughly 0.7 x 10l2W,
making the total rate of heat transfer out of the core due to cooling and gravitational settling
2.5 x 10”W. (This does not include the effect of radioactive heating within the core (see,
t.g. Hall & Rama Murthy 1971).) This figure is remarkably close (i.e. within a factor of two)
to many other estimates of the heat flux (Bullard & Gellman 1954; Verhoogen 1961;
Braginsky 1963; Stacey 1972; Frazer 1973; Verhoogen 1973). In particular, Stacey (1972)
has estimated the heat conducted down the adiabatic gradient to be 3.9 x 10”W and
remarked that ’the only plausible source of this much heat is the radioactive decay of
potassium in the core’. The analysis of this paper shows that this statement is incorrect;
cooling and gravitational settling can supply more than half the figure quoted by Stacey.
A basic feature of the gravitationally driven dynamo is the fact that the outer core is
driven toward a well-mixed ,adiabatic state by motions driven by compositional buoyancy.
Assuming the above estimates to be correct, heat is conducted down the adiabat at a rate of
3.9 x 1OI2W but only 2.5 x 1OI2 W of this is transferred to the mantle. What becomes of
the remaining 1.4 x 10” W of heat? It must be carried radially inward by the convective
motions. In a sense, the core will be thermally stably stratified, although compositionally
unstably stratified. That is, if the rate of heat transfer to the MCB by conduction down the
adiabat is greater than that produced by cooling and gravitational settling, the excess heat is
returned to the core by the convective motions. On the other hand, if the heat transferred
by conduction is less than that produced by cooling and gravitational settling, the heat
deficit is made good by a radially outward transfer by the convective motions. In either case,
the thermal buoyancy is weaker than the convective buoyancy because of the inherent
inefficiency of the transfer of thermal energy to kinetic. From this argument, we see that the
magnitude of the heat conducted down the adiabat cannot be inferred from an estimate of
the heat transferred to mantle and, in fact, is irrelevant to the operation of the
gravitationally powered dynamo.
The efficiency of the gravitationally powered dynamo may be easily calculated by noting
that of the 2.5 x 10” W transferred to the mantle, 1.28 x 1OI2 W results from ohmic dissipation; hence the efficiency may exceed 50 per cent, a much higher figure than is possible for
either the thermally driven or the precessionally driven dynamo.
The rate of thermal evolution of the core is controlled by the rate of heat transfer
through the mantle as we shall now explain. The temperature at the inner-outer core
boundary (I-OCB) is dictated by the composition of the binary alloy and the phase diagram
and is related to the temperature at MCB by the adiabatic gradient in the outer core. Hence
the temperature at the MCB may be considered, to a first approximation, to be fixed and
independent of the rate of heat transfer. The mantle then is subject to a fixed bottom
temperature and a radiation boundary condition at the surface. The solution of this heattransfer problem will yield a definite value for the heat transfer from the core to the mantle;
this value controls the rate of thermal evolution of the core. Performing a calculation similar
to that of Verhoogen (1961), one finds that the change in temperature at the MCB as the
inner core evolves is very small (< 50°C/109yr). Thus the vigour of the mantle convection
and the gravitational dynamo is quite uniform in time, in contrast to a dynamo driven by
radioactive heating.
It was found in Section 3 that the amount of gravitational power released is independent
Gravitationally powered dynamo
40 1
of the density of the light constituent but does depend upon the fraction of light constituent alloyed within the solid. This finding, obtained by numerical solution of the model,
may be verified by the following simple analysis. If the mass of the solid inner core is small
compared with that of the entire core, then from (2.10) and (2.1 1) it follows that
7.1 I
PYo-
(4.1)
Using (2.4) and (4.1) we may write
1-YotY0
-1=
PH
Po
PL
and
1 - 1 - P Y o PYO
-=
tPi
PH
PL
(4 3)
where p o and pi are the densities of the outer and inner cores respectively. The densities PH
and p L are functions of pressure, hence (4.2) and (43) may be employed simultaneously
only at the ILOCB. At that radius, we may eliminate ro and obtain
-1= - 1 - P
PH
Pi
P
+ -.
(4-4)
PO
Note that PL is absent from this equation. The density jump at I-OCB may be expressed as
This density jump depends upon the density of the light constituent p~ only through its
dependence on p o . However, known geophysical data limits po to a narrow range
independent of pL. For example, it was noted in Section 3 that p : G7.06 f 0.01 x lo3
kg m-3, whether p t = 2.5 x lo3 kg m-3 or
4.0 x lo3 kg m-3, where a superscript zero
denotes density at zero pressure. Noting that & = 8.6 x lo3 kg m-3 and the pressure factor
in (2.5) is 1.71 forp = 3.3 x 10” Pa, we obtain the estimate
POL=
Ap =
1-P
2.63 x 103kgm-3.
1 t 0.22p
(4.6)
This formula predicts a maximum density jump, with no solid alloying, of 2.63 x lo3
kg m-3. This value is somewhat higher than Bolt & Qamar’s (1970) upper limit of 1.8 x lo3
kgm-3 obtained from seismic data. Using (4.6) we see that Bolt & Qamar’s figure
corresponds to an alloying fraction of p = 0.27. This value lies between the values quoted
for sulphur and silicon in Section 2 and hence appears geophysically plausible. However, it
is dangerous to attempt to deduce the relative amounts of sulphur and silicon composing the
light constituent from a known alloying fraction because the behaviour of a ternary alloy of
iron, sulphur and silicon can be quite different from that of a binary alloy.
To a first approximation, the density jump at the I-OCB is proportional to the power
available to drive the dynamo efficiently. Using (4.6) and the value quoted previously for
the maximum available power, we have
P=
14
1 -cc
1.28 x lO”W,
1 + 0.22p
(4.7)
D. E. Loper
402
where P is the power fed into fluid motions. This estimate is roughly three times larger than
Gubbins’ (1977) but in agreement with Braginsky’s (1963). The ohmic dissipation requires
a power supply of roughly
B2
4
p, =-n La2 3
Pi
0’
where L is the core radius, a! = L / l , I is the scale of variation of the toroidal component B
of the magnetic field, u is the conductivity and po = 4n x lo-’ i2 s/m. Kumar & Roberts
(1975) have argued that the level of turbulence in the core is such that a A 2.0. Letting
L = 3 x lo6m and u = 3 x 1OS/(Q m), (4.8) becomes
P,
= B ~ X
1014w~-2.
(4.9)
Equating the power available, given by (4.7), to the power dissipated, given by (4.9), we
have
BG(
1- P
)
0.1T
)
103G.
112
1 + 0.22p
or
B&(
1 -cc
112
(4.10)
1 + 0.22p
We may also combine (4.10) anc (4.6) to relate the toroidal magnetic field strength in the
core directly to the density jump at the I-OCB
AP
= 2.63 x lo3 kg m-’
.
(4.1 1)
Equation (4.10) predicts that the maximum strength of the magnetic field sustained by a
gravitational power source is 1000 G. However, if we accept Bolt & Qamar’s figure for A p ,
(4.1 1) gives a lower value of 685 G, still a very large figure. If p were as large as the value
quoted earlier for silicon, p = 0.82, giving a density jump of 0.40 x lo3kg m-3, the power
liberated would be sufficient to sustain a field of 390 G. (Even the value of Ap = 0.17 x
10’ kg m-3 in Jordan’s model B1 yields a field strength of 250 G!) These large values of
toroidal field strength are in agreement with estimates made by theorists using Braginsky’s
nearly-axisymmetric dynamo model (Braginsky 1964; Roberts 1971 ; Kumar & Roberts
1975). It is of interest to note that Roberts’ (1971) estimate of 500 G is based upon the
simplified induction equation
where BT is the-toroidal field strength, B M is the meridional field strength (5 G), u+ is the
observed westward drift
m/s) and L is a typical length scale (3 x lo6m).
Due to the uncertainties in the numerical values of various quantities such as the compressibility of iron at high pressure, the conductivity of the core and the level of turbulence,
the values in (4.7) and (4.9) may be in error, perhaps as much as a factor of two but
probably not as much as a factor of five. Consequently the estimate of the magnetic field
strength should not be in error more than a factor of two, say. This would give a minimum
Gravitationallypowered dynamo
403
estimate of the magnetic field strength 200 G. In contrast to estimates of the field strength in
a thermally or precessionally powered dynamo where it is difficult to justify a large toroidal
magnetic field, it is difficult to justify parameter values which yield a field strength less than
200 G for the gravitationally powered dynamo. This strongly suggests that the dynamo is
of the nearly axisymmetric type developed by Braginsky (1964) rather than the type
advocated by Busse (1975, 1976) with small toroidal field.
Acknowledgment
This work was supported by the National Science Foundation under grant No. EAR 7422249 and is contribution no. 136 of the Geophysical Fluid Dynamics Institute, Florida
State University.
References
Al’tshuler, L. V., Krupnikov, K. K., Ledenev, B. N., Zhuchikhin, V. I. & Brazhnik,M. I., 1958. Dynamic
compressibility and equations of iron under high pressure, Soviet Phys. JETP, 34,606-614.
Artyushkov, E. V., 1970. Density differentiation on the core-mantle interface and gravity convection,
Phys. Earth planet. Int., 2,318-325.
Artyushkov, E. V., 1972. Density differentiation of the Earth’s matter and processes at the core-mantle
interface, J. geophys. Res., 77,6454-6458.
Bolt, B. A. & Qamar, A., 1970. Upper bound to the density jump at the boundary of the Earth’s inner
core, Nature, 228,148-150.
Braginsky, S . I., 1963. Structure of the F-layer and reasons for convection in the Earth’s core, Dokl.
Akad. Nauk SSSR, 149,8-10.
Braginsky, S. I., 1964. Kinematic models of the Earth’s hydromagnetic dynamo, Geomag. Aeron., 4,
572-583.
Brett, R., 1976. The current status of speculations on the composition of the core of the Earth, Rev.
Geophys. Space Phys., 14,375-383.
Bullard, E. C . & Gellman, G., 1954. Homogeneous dynamos and terrestrial magnetism, Phil. Trans. R.
SOC.Lond. A , 247,213-278.
Busse, F. H., 1975. A model of the geodynamo, Geophys. J. R . astr. SOC..42,437-459.
Busse, F. H., 1976. Generation of planetary magnetism by convection,Phys. Earth. planet. Int., 12,350358.
Cook, A. H., 1972. The dynamical properties and internal structures of the Earth, the Moon and the
planets,Proc. R. SOC.Lond. A , 328,301-336.
Frazer, M. C., 1973. Temperature gradients and the convective velocity in the Earth’s core, Geophys. J. R.
astr. Soc., 34, 193-201.
Gubbins, D., 1977. Energetics of the Earth’s core, J. Geophys., 43,453-464.
Haddon, R. A. W. & Bullen, K. E., 1969. An Earth model incorporating free Earth oscillation data, Phys.
Earth planet. Int., 2,35-49.
Hall, H. T. & Rama Murthy, V., 1971. The early chemical history of the Earth: Some critical elemental
fractionations, Earth planet. Sci. Lett., 11,239-244.
Hansen, M., 1958. Constitution of binary alloys, McCraw-Hill, New York.
Higgins, G. & Kennedy, G. C., 1971. The adiabatic gradient and the melting point gradient in the core of
the Earth,J. geophys. Res., 76,1870-1878.
Jacobs, J. A., 1975. The Earth’s core, Academic Press, New York.
Jordan, T. H. & Anderson, D. L., 1974. Earth structure from free oscillations and travel times, Geophys.
J. R. astr. Soc.. 36,411-459.
Kumar, S . & Roberts, P. H., 1975. A three dimensional kinematic dynamo, Proc. R . SOC.Lond. A , 344,
235-258.
Loper, D. E., 1975. Torque balance and energy budget for the precessionally driven dynamo, Phys. Earth
planet. Int.. 11,43-60.
Loper, D. E. & Roberts, P. H., 1978. On the motion of an iron-alloy core containing a slurry. I. General
theory, Geophys. Astrophys. FI.Dyn. 9,289-321.
404
D.E. Loper
Malkus. W. V. R., 1963. Precessional torques as the cause of geomagnetism,J. geophys. Res., 68,28712886.
Malkus, W . V. R., 1968. Precession of the Earth as the cause of geomagnetism, Science, 160,259-264.
McQueen, R. G. & Marsh, S. P., 1960. Equation of state for nineteen metallic elements from shock-wave
measurements to two megabars,J. Appl. phys., 31,1253-1269.
Metchnik, V. I., Gladwin, M. T. & Stacey, F. D., 1974. Core convection as a power source for the geomagnetic dynamo - a thermodynamic argument,J. Geomag. Geoelect., 26,405-415.
Ringwood, A. E., 1977. Composition of the core and implications for origin of the Earth, Geochem. J.,
11,111-135.
Roberts, P. H., 1971. Dynamo theory,Lect. Appl. Math., 14,129-206.
Stacey, F. D., 1972. Physical properties of the Earth’s core, Geophys. Sum., 1,99-119.
Stewart, R. M., 1973. Composition and temperature of the outer core, J. geophys. Res., 78, 25862597.
Urey, H. C., 1952. The planets, University of Chicago Press.
Usselman. T. M., 1975. Experimental approach to the state of the core: part 11. Composition and thermal
regime,Am. J. Sci., 275,291-303.
Verhoogen, J., 1961. Heat balance of the Earth’s core, Geophys. J. R . astr. SOC.,4,276-281.
Verhoogen, J., 1973. Thermal regime of the Earth’s core,Phys. Earth planer. Int., 7,47-58.