S. Labrosse $^{a}$, M. Macouin $^{a}$, J.-L. Le Mouël $^{a}$, J.-P. Poirier $^{a}$, J.-P. Valet $^{a}$ and J. Besse $^{a}$
$^{a}$ Institut de Physique du Globe de Paris, France
The thermal evolution of the Earth's core and the growth history of the inner core can be modeled using the global conservation equation for a convective core. However, this requires the knowledge of the heat flow at the core mantle boundary (CMB) as a function of time, a parameter that results from the dynamics and evolution of the mantle. Another limitation to that approach is that it cannot provide any information about the magnetic field that can be produced, since the dissipation is internally equilibrated by the work of buoyancy forces (Hewitt, McKenzie and Weiss, 1975). On the other hand, the entropy equation can be used to relate the Joule dissipation to the energy sources available to drive the dynamo (Gubbins, Masters and Jacobs, 1979; Roberts, Jones and Calderwood, 2002). All these sources, except the radioactive heating, can be parameterized by the inner core radius and growth rate that can then be derived for any given dissipation in the core. Another approach based on buoyancy fluxes has also been proposed (Lister and Buffett, 1995) and is compared with the one used by the authors. The present dissipation in the core is poorly known but taking its contribution to the entropy equation between $350$ and $700 MW\ K^{-1}$ gives a present heat flow across the CMB between 6 and 10 TW. To construct a thermal history of the core, the value of the Joule dissipation at all time is needed, and the published and some recently additional paleo-intensity data are used to constrain the Joule dissipation for the period before the inner core crystallization. The record is very scarce and displays variation of the largest amplitude on very short time scales that cannot be attributed to the change of buoyancy sources available but rather to exchange of energy between the dipole field and the smaller scales of the magnetic field, a process that can take place without change of the overall Joule dissipation. On the other hand, the virtual dipole moment (VDM) averaged for the ancient period (more than 1 Ga old) is about twice lower than the value averaged for the recent period. Consequently, computing the heat flow at the CMB that is necessary to maintain a total dissipation that is 4 times lower than the present estimates before the inner core, we get a value between 7 and 9~TW. Considering the large scatter of the record on short time-scales, one can consider that the dissipation is essentially constant during the whole time. In that hypothesis, the heat flow at the CMB has be between 14 and 24~TW before the crystallization of the inner core. The largest figure, when extrapolated backward, gives a temperature at the CMB of 10000 K at $t=-4.5\ Ga$ and should be excluded.
References. Gubbins, D., Masters, T.~G. and Jacobs, J.~A., 1979. Thermal evolution of the Earth's core. Geophys. J. R. astr. Soc., 59: 57--99. Hewitt, J.~M., McKenzie, D.~P. and Weiss, N.~O., 1975. Dissipative heating in convective flows. J. Fluid Mech., 68: 721--738. Lister, J.~R. and Buffett, B.~A., 1995. The strength and efficiency of the thermal and compositional convection in the geodynamo. Phys. Earth Planet. Inter., 91: 17--30. Roberts, P.~H., Jones, C.~A. and Calderwood, A.~R., 2002. Energy fluxes and ohmic dissipation in the Earth's core. In press.