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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 whole-mantle 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 Fe-S 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 sulfur-enriched 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
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 Brunt-Vaisala 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 long-period terrestrial oscillations; in any event, as Olson points out, their ampli- tude is confined predominantly to the outer core, so that their detection
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 two-component 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. 103-159~. 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 iron-sulfur 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.
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 core-mantle 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.
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 10-8 s/cm2 = 5 x 105 mho/m (Gubbins, 1976~; it used to be 3 x 10-6. 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
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 10-2 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.
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 1O-5/s, Equation (4.6), with R = 3 x 108 cm and u = 4 x 10-2cm/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
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 quantum-mechanical 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 ICB-iS 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
Dynamics of the Core 75 [JEm= _l J-dV- ~ 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 right-hand 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 (109-10~t W). We consider first the case of a dynamo activated by thermal convec- tion. EFFICIENCY OF A STEADY-STATE 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
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 core-mantle 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 core-mantle 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 inner-outer 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
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 core-mantle 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 core-mantle 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 To-T2 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
78 ENERGETICS OF THE EARTH T2+AT2 tout T2 _ . D E B A _ . Tl +AT' WIN Tl FIGURE =2 Temperatures in a rising limb (B to D) and a descending limb (E to A) of a convection cell In a layer of liquid heated from below. Met- chnik et al. (1974) assume erroneously that the dif- ference in specific entropy of the rising and falling fluids is the same at all levels (see text). of an adiabatic gradient between A and E or between B and D; further- more, if To = T2, the efficiency is zero by (4.14~. The error in (4.14) stems from the assumption of isentropic flow. Since the system does no work on its surroundings, any work done (e.g., by buoyancy forces) inside the system must be dissipated within the system as viscous or ohmic heating, with a corresponding irreversible production of en- tropy. There is also the unavoidable irreversible production of entropy by thermal conduction at the rate k(AT/T)2 per unit volume, where k is the thermal conductivity. Both vertical and horizontal temperature dif- ferences (i.e., T2 - Ti, ATi, AT2) contribute to this term. AN IMPROVED ESTIMATE OF EFFICIENCY Consider in particular the case of a core with distributed radioactive heat sources generating ~ watts per unit volume, so that the total heat generation is ~ ~ dV. In the steady state, the core must be losing heat to the mantle at the rate Q0 = ~ ~ dV, since no energy leaves the system under any form other than the heat flux through the core-mantle boundary, assumed to be held at the uniform temperature To. The appropriate momentum equation, written in a coordinate system rotating at the uniform rate Q. is
Dynamics of the Core 79 p [dU+(u V) u] + 2p (Q x u) = -VP + TV + FL + div a, (4. 15) where p is density, u is velocity, P is pressure, ~ is the sum of the gravitational and centrifugal potentials, Fir is the Lorentz force j x B. and ~ is the deviatoric stress tensor. Forming the scalar product of both sides of (4.15) with u and integrat- ing over the volume V of the fluid, one obtains after standard manipulations* (Hide, 1956), using (4.9), ~ (Ek + Em) = PdivudV+ | ~ PdV v v at -,J Emit-,( Add, V V (4.16) where Ek = 1/2 TV p U212 dV is the kinetic energy of the fluid, Em = j2 is the ohmic dissipation, and do is the viscous dissipation. Let ~ be the total dissipation rate, and let D = TV P div u dV. In the steady state, when bitt = 0, (4.16) reduces to the simple form D = fib = tPdivodV, (4.17) Jv which singles out D as the driving term for the dynamo. Since in the steady state rV Em dV = - rV u Fit dV by (4.9), the rate at which the fluid does work against the Lorentz force is measured by the same term D, and the efficiency 7' of the dynamo is, in the steady state, r1 = D/l ~ dV = D/QO = ~/QO (4.18) Equation (4.18) seems paradoxical since the efficiency increases with increasing rate of dissipation. The apparent paradox stems, as ex- plained above, from the fact that since dissipation takes place within the system, it must be counted as an additional heat source. *Repeated use is made of the identity div aB = a div B + B · V a, where a is any scalar quantity and B is any vector.
80 ENERGETICS OF THE EARTH Equation (4.17) also shows that a strictly divergenceless flow cannot maintain a dynamo. From the equation of continuity P+u·Vp+ pdivu=0, (4. 19) it follows that, in the steady state, div u = 0 implies Vp = 0, which cannot be true in a real fluid subjected to pressure and temperature gradients. It is nevertheless common practice to write div u = 0, as if the density were uniform, and to retain the density variation Bp only in the buoyancy term TV of the momentum equation. This, the Boussinesq approximation, is known to be valid provided that Bp/p << 1, or if the thickness d of the convecting layer of fluid is much smaller than the temperature scale height hT = cp/g~x (Hewitt et al., l975~. If we take for the earth's core cp ~ 0.6 J/g deg. ~ ~ 10-5/deg, and g _ 103 cm/s2, we find hT = 6 x 108 cm, and the ratio d/hT _ As, which is not small. Recall also that density varies by about 2 g/cm3 between the top and bottom of the outer core with a mean density of about 10 g/cm3, so that Bp/p-0.2, which is not small either. Thus the applicability of the Boussinesq approximation for the earth's core is questionable. It seems nevertheless that div u will be small if not neglible. Equation (4.18) then predicts that the efficiency of the dynamo will also be small, if not exactly zero. The important term D can be put under a variety of forms. We now spell out some of them. Since P div u = div (Pu) - u VP and the normal component of u vanishes on the core boundary, D = ~ P div u dV = - | u VP dV, (4.20) which shows that flow along isobaric surfaces contributes nothing to D or to A. Alternatively, since in the steady state div u = -(u V pJ/p, we also have D =-~-u · VpdV, Vp (4.21) which shows that flow along surfaces of equal density contributes nothing to D. Since Vp/p = -aVT + VP/KT, where KT is the isothermal bulk modulus, (4.21) can be further transformed to
Dynamics of the Core 81 D = J ~ P u VT dV-J P u ~ P dV (4.22) Still other forms can be obtained for D from the principle of local equilibrium (Glansdorff and Prigogine, 1971, p. 14), according to which the local values of the thermodynamic variables T. P. and v = Up can be defined as if the system were in thermodynamic equilibrium: T = fee ~ ~ -P = (be) ' If es Jv = P/T, where e is the specific internal energy and s is the specific entropy. Then Tds =de +Pdv (4.23) and since dv/dt = -(l/p2) dp/dt = (1/p) div u, (4.23) can be written as P div u = Tp AS _ p de (4.24) Making use of (4.19) it can also be shown that (Glansdorff and Prigogine, 1971, p. 4) d! pant + u Vs) = at (~) + Shiv four, dt P(dt +U · Ve) =~ (pe) +div (peak (4.25) Thus from (4.24) and (4.25) D = J Top dV-J p ~-dV = J T div (psu) dV v - ~ div (peu) d V. Jv The last integral on the right can be transformed to a surface integral
82 ENERGETICS OF THE EARTH that vanishes because the normal component of u is zero on the boundary. The other integral can also be transformed, noting that T div (psu) = div (Tpsu) - psu VT, to yield D =-my v or by the first of the equations (4.25) psu ~TdV D = ~ Tpu Vs dV, v (4.26) (4.27) which shows that isothermal or isentropic flows do not contribute to D. Finally, if we replace Vs in (4.27) by the equivalent (cp/T)VT (a/p)VP, we get D=l pcpu VTdV-~ tutu VPdV v Jv (4.28) If cp is constant, the Most integral vanishes because pu VT = div ~Tu) - T div (pu), div (pu) = 0 in the steady state, and Jv div (`pTu) dV = 0 because of the boundary condition on u. Thus D =-~ aTu VP dV, v \ (4.29) which is the form used by Hewitt et al. (1975) to estimate the efficiency of the core dynamo. They proceed as follows: First split P into a hydrostatic term PO such that VPO = pg and a dynamic term Pi, with P = PO + Pi. If the Reynolds number of the flow is small, VP~ << VPO and can be neglected. Let w be the radial velocity. Then D = ~ = ~ pgarTw dV= ~ g-F(r~dr, (4.30) v 0 cp where F(r) is the convective heat flux 4,rr2pcp<Tw> at radius r (the brackets denote averages over spherical surfaces) and a is the outer radius of the core. Suppose further that g (r) = rgO/a, where gO is the value of g on r = a. If heat is generated uniformly within the sphere and is carried by convection only, F(r) = (a ) Q° (4.31)
Dynamics of the Core 83 - where Q0 is the rate at which heat is supplied at the boundary. Substi- tuting in (4.30) gives <~ = ~ ego Ha ) Q0 and the efficiency q) 1 a 77 = = _ Qo ShT Since a 3.5 x 108 cm and hT-6 x 108 cm, ~ _ 0.12. (4.32) This value of a) is likely to be an overestimate, since the calculation leading to it ignores the conductive contribution to heat transport. We may note in the first place that since.{V u VPodV = 0 in the steady state, the integral (4.29) would be zero, and ~ would be zero, if c' and T were constant; but if T is not constant, VT is not zero everywhere, and heat cannot be carried by convection only. Clearly, Equation (4.31) is grossly wrong at r = a, where it requires F(r) = Q0; but on r = a the convective heat flux F(r) is necessarily zero since w vanishes there. It is precisely in the thermal boundary layer near r = a that VT is likely to be largest and the entropy production by conduction, which is pro- portional to (VT)2 is likely to be greatest. As we shall see in the next section, this conductive contribution to entropy production decreases the efficiency below the value it might have in the absence of conduc- tion and makes it impossible to determine the efficiency in the absence of detailed information on the temperature distribution. This can also be seen from the entropy balance equation, to which we now turn. THE ENTROPY BALANCE EQUATION Suppose that radioactivity is the only source of heat in the core. In the steady state, the entropy of the core must remain constant, even though it is losing entropy at the rate Qo/To, where To is the temperature in the CMB and Q0 = iv ~ dV, ~ being the rate of radiogenic heat production per unit volume. The entropy loss must be balanced by irreversible entropy production within the core. There are three internal sources of entropy, namely radiogenic heat production, dissipation (ohmic and viscous), and heat conduction. Thus entropy balance requires that Q°= 1 ~ edV To To v
84 ENERGETICS OF THE EARTH = J -dV +J ¢3m dV vT v T Jv T iv ( T ) , (4.33) where k is, as before, the thermal conductivity, ¢.m is the ohmic dissipa- tion rate, and ~v is the viscous dissipation rate. Thus J ~m dV v T = v (To T) dV JV k ( T ) dV _| ~VdV. v T (4.34) Clearly, a maximum value for ~mlT is obtained if the fluid is inviscid and ~v = 0, which we shall assume to be the case. Let Tm be the maximum temperature within the body, so that To ' T C Tm. Then, since t4m is necessarily positive everywhere, 1 J ~m dV ' J ~m dV _-J ~m dV Tm v v T To v or To I ¢T,m d V C J 4)m dV _ Tm J ¢T~m dV (4.35) Substituting the value of .rV(¢mlT) dV from (4.34) in (4.35) yields J ~dV = To J TdV-To J k ( T ) dV _ J ~m dV [ J ~ dV-To J ~ dV-To J k (VT) dV The eff~ciency ~ is defined as before, (4.36)
Dynamics of the Core 85 77=l~mdYlJ6dv=~DlQo Suppose that ~ is uniform, so that V edV = eV. Dividing (4.36) through- out by eV yields 1 A C c Tm (1 -A) o where (4.37) A = V° t; T +-~ k ~ T ~ dV] . (4.38) Equations (4.37) and (4.38) clearly show the role of the entropy production by conduction of heat, SC = k IV (VTIT)2 dV. Define an average temperature Ta as 1 r dV 1 , =. ~ Ta V Jv T Then (4.38) can be written as A = 0 + .c Ta 5 (4.39) (4.40) where S is the total rate of entropy production QJTo = eV/To. Substitut- ing (4.40) into (4.37) leads to an upper bound on 71 Ta Ta-To so T, ~ TaTo m Q ~ (4.41) which shows that the upper bound to ~ decreases as the conduction entropy production increases. This upper bound is, incidentally, al- ways below Backus' (1975) upper bound (Tm - To~lTo, since Sc _ O and Ta _ Tm Using the identity div ~-)= 1 V2T- 1 (VT)2 T T T2 and noting that
86 ENERGETICS OF THE EARTH -k div ~ T ~ dV = iv k To To To (4.38) can be rewritten as where J is defined as A= 1 +J, J= Tot (~+kV2~dv To eV v T Ta Then (4.37) becomes +- ~T dV. (4.42) T -Jan - - -J T 1 o (4.43) and since ~ is necessarily positive or zero (since ~ and ~ are positive or zero everywhere), J must be negative. If J = 0, ~ = 0; this happens when 1 +-V2T= 0, which describes the steady-state conductive temperature distribution in the absence of convection. Note that iv (e + kV2T) dV = 0 if k is constant. Indeed, Jv = kV2 T d V = J k div (VT) dV = J kVT · dS - ~qO.dS= -Go= -~ edV, JO JV where qO is the heat flux at the surface. Thus efficiency is proportional to iv [~e + kV2 T)/T] dV, with iv (e + kV2T) dV = 0. Clearly, J and limits on the efficiency cannot be calculated in the absence of precise knowl- edge of the temperature distribution and of v2 T. which in turn requires that the dynamo problem first be solved. No exact solutions are avail- able yet. If we try to estimate ~ by choosing a "likely" yet unproven tempera- ture distribution, we recall that ~ and 7' are zero if (1) the temperature
Dynamics of the Core 87 is a solution of the conduction equation, or if (2) the temperature profile is adiabatic so that Vs = 0 everywhere (see Equation (4.2711. ~ is also zero if the temperature is a function of r only. Indeed, in a steady state, constancy of mass inside a spherical surface S of radius r requires that the mass flux across the surface be zero, i.e., Js pu dS=0. But if T is constant on S. so is p, so that we must have ~ _ __ ~ J S v(r) u dS= | div udV=0, where the volume integral is over the volume V(r) inside S. Since this must be true for any r, it follows that div u = 0 everywhere, and D = = 0 (Equation (4.171~. The production of magnetic energy thus de- pends critically on the horizontal temperature gradients, without which there would obviously be no convection. The best we can do at the moment is to assume V2T =-T/l2 where I is a length characteristic of the scale of the temperature fluctuations. Then = To _ 1' Toll (kV'~ Ta ~QOJ Kl2J Substituting this in (4.43) and solving for QO gives Cal CT kV _m Ta T kV _ ~ Ta (4.44) for which the heat generation in the core, QO, could be estimated for any value of ~ if a reasonable choice could be made for 1. Take k = 30 W/m deg (Stacey, 1977), To = 4500°K, Ta = 5000°K, Tm = 6000°K, and = 4 x 10~i W. Then, for I = 106 m, 2.51 x 1013 ' QO-' 2.52 x 1013, implying an efficiency of less than 2 percent; but no real significance can be attached to this number, which depends on guessing at the value of 1. However, since I is unlikely to be much greater than 3 x 106 m (ro'~ahiv the radius of the outer core), QO is not likely to be much less ~_~ in,
88 ENERGETICS OF THE EARTH than 2.6 x 10~2 W. For comparison, we recall that 0.1 percent by weight of potassium in the outer core produces about 6.7 x 10~2 W (e-4 x 10-8 W/m31. To recapitulate: it is not possible to calculate the efficiency of the dynamo (i.e., the ratio of ohmic dissipation ~ to total heat output Q0 = eV) without a detailed knowledge of the temperature distribution in the convecting fluid. An upper limit of about 0.1 was calculated by Hewitt et al. (1975) by ignoring conduction and assuming that heat is carried by convection only. However, the effect of conduction, namely the en- tropy production associated with it, is not small if temperatures in the core are anywhere near the values previously determined (about 4500°K at the outer core-mantle boundary, about 6000°K at the inner core boundary), so that the actual efficiency may be an order of magni- tude smaller than the Hewitt et al. (1975) value. An approximate evaluation of the actual efficiency is given by Equation (4.44), from which it would be possible to calculate the heat output Q0, given the ohmic dissipation ~ and the length scale I of the horizontal temperature fluctuations defined by V2T =-Tl12. For I = 1 x 106-3 x 10; m, Q0 is in the range 2.8 x 10~2-2.5 x 10~3 W. THE GRAVITATIONAL DYNAMO Braginsky's early dynamo models required a very large toroidal field and had the correspondingly large dissipation rate of 3.8 x 10~2 W. Noting that the efficiency of a thermal dynamo would necessarily be low, he concluded that convection in the core could not be thermal, as it would require that more heat be generated in the core per unit time than escapes at the earth's surface. He suggested instead (Braginsky, 1963) that convection is powered gravitationally, either by the floating upward of the excess of the light component (silicon in those days) released as the inner core crystallizes, or by heavy material falling from the mantle into a growing core. Verhoogen (1961) had already sug- gested that the release of latent heat of crystallization could contribute significantly to maintenance of convection. The gravitational dynamo has been reconsidered lately by Gubbins (1977) and by Loper (19781. The problem is difficult. The general principle of the gravitational dynamo is as follows. Cool- ing of the core must cause an outward displacement of the inner core boundary; the inner core grows by crystallization of iron. Since the density of the inner core is greater than the density of the outer core, growth of the inner core entails a release of gravitational energy. As a
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,
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 dV-J 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
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 right-hand 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 iron-sulfur 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 Fe-FeS 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),
92 ENERGETICS OF THE EARTH ~= V 2 (V2M1-V1M2 ) (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 + (vs-V1) ~1 ~ (4.53) The volume of the whole core increases if vs > v1. Contraction occurs if v* <v1. To see what could happen in the core, it is instructive to plot molar volume against composition in a binary system (Figure =3). On the ordinate axis N2 = 0 (pure iron), we plot at A the molar volume v1O of pure liquid iron under the pressure and temperature conditions con- sidered; similarly at B we plot the molar volume v2O of pure liquid FeS. The dashed line AB represents the molar volume of a perfect solution in which neither contraction nor expansion occurs on mixing. If con- traction occurs, as it does in the Fe-FeS system at low pressure, the molar volume of the melt is represented by the curve APB, the exact shape of which is not known. It is a general property of such molar diagrams that the tangent at any point P to the curve cuts the two ordinate axes at points representing the two partial molar volumes vat and v2, respectively. It is clear from the graph that if departures from perfect behavior are serious, v2 could be negative at low FeS concentra- tion. It is also clear that at higher FeS concentrations say, near the minimum of the APB curve v, could be very much smaller than vie; and since at core pressures pure iron melts with relatively small change in volume (Leppaluoto, 1973), vat could conceivably be smaller than v,'. This, however, is unlikely. The configuration of the core (solid inner
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 Fe-FeS melt) should increase with increasing pressure. Now, at con- stant composition (6Tm: V1-Vs ~ 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.
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 = -Eve-vs.) An, where ~ n = MOMS, Mi being the mass of the inner core, approximately 9.8 x 1025 g. Thus /iV--9. 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 10-5/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
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 Fe-FeS 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 right-hand 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
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 so-Rln 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 10-3 °/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 = Tmo-Pa 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 Fe-FeS melts depart from being perfect solutions.
Dynamics of the Core 97 and AP = 0.34 Mbar. Taking ax = 5 x 10-6/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 10-40/bar and Ta = 5010°K. Thus at r = ri the temperature had dropped since crystallization started by AT = Ta-Tm-250° (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.
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 FeS-Fe 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 Fe-FeS system. Our evaluation of Qc and Qe was based on the perfect-solution assumption, which is almost certainly wrong.
Dynamics of the Core 99 ~ Recall our earlier result that the radioactive dynamo requires a heat output in the range 4 x 10~2-1 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 physico-chemical 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?
