Energetics of the Earth (1980) / Chapter Skim
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4 DYNAMICS OF THE CORE
4 DYNAMICS OF THE CORE
Pages 67-99
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From page 67... ...
. 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.
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From page 68... ...
Olson (1977) has considered the problem of the internal oscillations of a body consisting of a uniform solid elastic mantle and a solid inner core bounding a stratified, rotating, inviscid, fluid outer core.
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From page 69... ...
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.
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From page 70... ...
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.
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From page 71... ...
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*
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From page 72... ...
(4 3) In the steady state, dB¢,/dt = 0.
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From page 73... ...
, 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)
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From page 74... ...
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
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From page 75... ...
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 dissipation. 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.
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From page 76... ...
This somewhat paradoxical result may perhaps be understood by noting that since the ohmic and viscous heating occur within the convecting 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 distinguish between the two sources, which therefore both contribute to the motion.
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From page 77... ...
"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.
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From page 78... ...
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.
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From page 79... ...
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.
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From page 80... ...
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.
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From page 81... ...
' If es Jv = P/T, where e is the specific internal energy and s is the specific entropy.
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From page 82... ...
v 0 cp where F(r) is the convective heat flux 4,rr2pcp |
From page 83... ...
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.
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From page 84... ...
, (4.33) where k is, as before, the thermal conductivity, ¢.m is the ohmic dissipation rate, and ~v is the viscous dissipation rate.
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From page 85... ...
(4.40) where S is the total rate of entropy production QJTo = eV/To.
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From page 86... ...
If we try to estimate ~ by choosing a "likely" yet unproven temperature distribution, we recall that ~ and 7' are zero if (1) the temperature
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From page 87... ...
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 depends critically on the horizontal temperature gradients, without which there would obviously be no convection.
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From page 88... ...
Cooling 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.
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From page 89... ...
This, however, neglects all dissipative processes, other than ohmic heating, by which the gravitational energy could be converted to heat. That such dissipative processes 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.
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From page 90... ...
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
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From page 91... ...
(4.50) where v1 and v2 are, respectively, the partial molar volumes of iron and FeS in the melt.
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From page 92... ...
If contraction 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.
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From page 93... ...
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)
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From page 94... ...
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.
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From page 95... ...
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.
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From page 96... ...
The solidus temperature 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.
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From page 97... ...
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°.
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From page 98... ...
, 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.
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From page 99... ...
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)
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