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Are planetary dynamos driven by gravitational settling?
192
Physics o f the Earth and Planetary Interiors, 20 (1979) 192-193 © Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands
ARE PLANETARY DYNAMOS DRIVEN BY GRAVITATIONAL SETTLING?
DAVID E. LOPER 1 and PAUL H. ROBERTS 2 1 Geophysical Fluid Dynamics Institute, Florida State University, TaUahassee, FL (U.S.A.) 2 School o/Mathematics, University o f Newcastle upon Tyne, Newcastle upon Tyne (Great BritainJ
(Accepted for publication in revised form April 5, 1979)
Loper, D.E. and Roberts, P.H., 1979. Are planetary dynamos driven by gravitational settling?, Phys. Earth Planet. Inter., 20: 192-193. The basic features of gravitationally driven convection are summarized, and an expression for the power available to drive a hydromagnetie dynamo is given. Some thermal consequences of gravitationally driven convection are presented.
Several of the planets are known to possess magnetic fields. It is generally believed that these fields are sustained by dynamo action within the planetary interiors. One of the most challenging problems of planetary physics concerns the energy source for these dynamos. The two energy sources mentioned most often in the literature are thermal convection and precessionally induced motions, but each of these mechanisms encounters difficulties in efficiently converting the available energy into mechanical motion: in a metallic fluid such as the Earth's core, thermal convection may be short-circuited by thermal conduction, while the energy of precession appears to be dissipated in boundary layers. There is a third possible energy source which is not subject to these problems of efficiency: the energy released by gravitational settling. This idea was first proposed by Braginsky (1963) and has recently been discussed by Gubbins (1977) and Loper (1978a, b). A complete theory of dynamo operation driven by gravitational settling is currently being developed (Loper and Roberts, 1978). The basic features of gravitationally driven convection may be summarized as follows. Consider an initially homogeneous self-gravitating sphere of molten binary alloy representing a planetary core, the alloy being composed of a heavy metal (principally iron)
and a light non-metal in the case of a terrestrial planet, or of helium and metallic hydrogen in the case of a Jovian planet. As the planet cools over geological time, a phase change is likely to occur, either to a new liquid phase immiscible with the first phase, or to a solid phase. Assuming the adiabatic gradient to be smaller than the liquidus gradient, the phase change would first occur at the center of the sphere. Metallurgical studies have shown that the new phase which forms from a binary melt does not in general have the same composition as the liquid. To simplify the discussion we shall assume the melt to contain a larger fraction of the heavy component than does the eutectic. In this case, the new phase has a larger fraction of the heavy component and hence is more dense than the initial liquid, even in the absence of a density change associated with the phase change. The denser phase will accumulate at the center, forming an inner core. Associated with the growth of this inner core, there is a net motion of heavy material downward and light material upward, releasing gravitational potential energy. More specifically as the inner core grows by accretion of the denser phase, a residue of light material is left in the liquid near the inner core. Since diffusion of matter is ineffective over planetary lengthscales, this excess of light material must distribute
193 itself throughout the outer core by means of convective motions driven by compositional buoyancy. The fact that the material diffusion coefficient is typically much smaller than the thermal diffusion coefficient accounts for the inherent efficiency of the gravitationally driven dynamo, first pointed out by Bragin Braginsky (1963) and qualified by Gubbins (1977). The power available from gravitational settling depends upon the density contrast AO between the two phases and may be crudely estimated by the formula: P = AO VgL/t
where V is the inner core volume, L is the outer core radius, g is a typical value of the local acceleration of gravity and t is the time which has elapsed since the inner core began to form. Typical estimates for the Earth's core give P = 4 × 10 H W, which is sufficient to sustain a large field (see Loper, 1978a). In the light of the uncertainty in composition and structure of the planetary cores, it is difficult at present to estimate the viability of gravitational settling as a power source for the other planetary dynamos. If gravitationally-driven convection occurs, the outer core is in effect mechanically mixed. Loper (1978b) has shown that several interesting thermal regimes are possible. For example, if the liquidus graclient is less than the conduction gradient (i.e., the gradient which would occur in the absence of motion), a slurry must occur in the fluid at the bottom of the outer core. We have recently been able to show that the Earth's core must contain a slurry. A general thermodynamic theory for the motion of a binary-
alloy fluid containing a slurry has been developed by Loper and Roberts (1978). Also, compositionaUy driven convection can occur even if the actual temperature gradient is less than the adiabatic gradient, making the fluid thermally stably stratified. In this case the fluid motions convect heat radially inward up the temperature gradient. Hence the heat conducted down the adiabat cannor be taken as a lower limit on the heat transferred out of the core.
Acknowledgements
This work was supported by the National Science Foundation under grant No. EAR 78-00818 and is contribution No. 147 of the Geophysical Fluid Dynamics Institute, Florida State University, Tallahassee, Florida.
References 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. Gubbins, D., 1977. Energeties of the Earth's core. J. Geophys., 43: 453-464. Loper, D.E., 1978a. The gravitational powered dynamo Geophys. J. R. Astron. Sot., 54: 389-404. Loper, D.E., 1978b. Some thermal consequences of a gravitationally powered dynamo. J. Geophys. Res., 83: 59615970. Loper, D.E. and Roberts, P.H., 1978. On the motion of an iron-aUoy core containing a slurry. I. General theory. Geophys. Astrophys. Fluid. Dyn., 9: 289-321.
Physics o f the Earth and Planetary Interiors, 20 (1979) 192-193 © Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands
ARE PLANETARY DYNAMOS DRIVEN BY GRAVITATIONAL SETTLING?
DAVID E. LOPER 1 and PAUL H. ROBERTS 2 1 Geophysical Fluid Dynamics Institute, Florida State University, TaUahassee, FL (U.S.A.) 2 School o/Mathematics, University o f Newcastle upon Tyne, Newcastle upon Tyne (Great BritainJ
(Accepted for publication in revised form April 5, 1979)
Loper, D.E. and Roberts, P.H., 1979. Are planetary dynamos driven by gravitational settling?, Phys. Earth Planet. Inter., 20: 192-193. The basic features of gravitationally driven convection are summarized, and an expression for the power available to drive a hydromagnetie dynamo is given. Some thermal consequences of gravitationally driven convection are presented.
Several of the planets are known to possess magnetic fields. It is generally believed that these fields are sustained by dynamo action within the planetary interiors. One of the most challenging problems of planetary physics concerns the energy source for these dynamos. The two energy sources mentioned most often in the literature are thermal convection and precessionally induced motions, but each of these mechanisms encounters difficulties in efficiently converting the available energy into mechanical motion: in a metallic fluid such as the Earth's core, thermal convection may be short-circuited by thermal conduction, while the energy of precession appears to be dissipated in boundary layers. There is a third possible energy source which is not subject to these problems of efficiency: the energy released by gravitational settling. This idea was first proposed by Braginsky (1963) and has recently been discussed by Gubbins (1977) and Loper (1978a, b). A complete theory of dynamo operation driven by gravitational settling is currently being developed (Loper and Roberts, 1978). The basic features of gravitationally driven convection may be summarized as follows. Consider an initially homogeneous self-gravitating sphere of molten binary alloy representing a planetary core, the alloy being composed of a heavy metal (principally iron)
and a light non-metal in the case of a terrestrial planet, or of helium and metallic hydrogen in the case of a Jovian planet. As the planet cools over geological time, a phase change is likely to occur, either to a new liquid phase immiscible with the first phase, or to a solid phase. Assuming the adiabatic gradient to be smaller than the liquidus gradient, the phase change would first occur at the center of the sphere. Metallurgical studies have shown that the new phase which forms from a binary melt does not in general have the same composition as the liquid. To simplify the discussion we shall assume the melt to contain a larger fraction of the heavy component than does the eutectic. In this case, the new phase has a larger fraction of the heavy component and hence is more dense than the initial liquid, even in the absence of a density change associated with the phase change. The denser phase will accumulate at the center, forming an inner core. Associated with the growth of this inner core, there is a net motion of heavy material downward and light material upward, releasing gravitational potential energy. More specifically as the inner core grows by accretion of the denser phase, a residue of light material is left in the liquid near the inner core. Since diffusion of matter is ineffective over planetary lengthscales, this excess of light material must distribute
193 itself throughout the outer core by means of convective motions driven by compositional buoyancy. The fact that the material diffusion coefficient is typically much smaller than the thermal diffusion coefficient accounts for the inherent efficiency of the gravitationally driven dynamo, first pointed out by Bragin Braginsky (1963) and qualified by Gubbins (1977). The power available from gravitational settling depends upon the density contrast AO between the two phases and may be crudely estimated by the formula: P = AO VgL/t
where V is the inner core volume, L is the outer core radius, g is a typical value of the local acceleration of gravity and t is the time which has elapsed since the inner core began to form. Typical estimates for the Earth's core give P = 4 × 10 H W, which is sufficient to sustain a large field (see Loper, 1978a). In the light of the uncertainty in composition and structure of the planetary cores, it is difficult at present to estimate the viability of gravitational settling as a power source for the other planetary dynamos. If gravitationally-driven convection occurs, the outer core is in effect mechanically mixed. Loper (1978b) has shown that several interesting thermal regimes are possible. For example, if the liquidus graclient is less than the conduction gradient (i.e., the gradient which would occur in the absence of motion), a slurry must occur in the fluid at the bottom of the outer core. We have recently been able to show that the Earth's core must contain a slurry. A general thermodynamic theory for the motion of a binary-
alloy fluid containing a slurry has been developed by Loper and Roberts (1978). Also, compositionaUy driven convection can occur even if the actual temperature gradient is less than the adiabatic gradient, making the fluid thermally stably stratified. In this case the fluid motions convect heat radially inward up the temperature gradient. Hence the heat conducted down the adiabat cannor be taken as a lower limit on the heat transferred out of the core.
Acknowledgements
This work was supported by the National Science Foundation under grant No. EAR 78-00818 and is contribution No. 147 of the Geophysical Fluid Dynamics Institute, Florida State University, Tallahassee, Florida.
References 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. Gubbins, D., 1977. Energeties of the Earth's core. J. Geophys., 43: 453-464. Loper, D.E., 1978a. The gravitational powered dynamo Geophys. J. R. Astron. Sot., 54: 389-404. Loper, D.E., 1978b. Some thermal consequences of a gravitationally powered dynamo. J. Geophys. Res., 83: 59615970. Loper, D.E. and Roberts, P.H., 1978. On the motion of an iron-aUoy core containing a slurry. I. General theory. Geophys. Astrophys. Fluid. Dyn., 9: 289-321.