S. Labrosse $^{a}$ and P. J. Tackley $^{b}$
$^{a}$ Institut de Physique du Globe de Paris, France $^{b}$ University of California, Los Angeles, USA
The cooling history of the Earth, controlled by mantle convection, is a central question in Earth's sciences and is still far from being fully understood. The complexity of that question lies in reconciling the geophysical and geochemical models of the mantle. The former generally produce a very fast initial cooling of the Earth and, to maintain the present heat flux at the surface of the Earth, tend to require more radioactive elements in the mantle than the latter would predict. However, the geophysical models are based on parameterization of heat transfer in convective systems that are rather different from the mantle and its rather peculiar plate tectonics regime. Recently, some mantle convection models have been built to allow the self consistent development of plate-like dynamics. We use the model proposed by Tackley (2000) of convection with a pseudo-plastic yielding and study heat transfer in this system as a function of the relevant parameters, the Rayleigh number $Ra$, the internal heating rate $h$ and yield stress $s_y$. Essentially two regimes are possible, depending on the values of these parameters: the stagnant lid regime and the plate-like regime. The stagnant lid regime is typically obtained in convection with strongly temperature dependent viscosity, or here when the yield stress is too large. The plate-like regime is characterized by an almost piecewise constant surface velocity, defining plates and plate boundaries. All other parameters being kept constant, the plate sizes increase with the internal heating rate and the system eventually undergoes a transition to stagnant lid convection if $h$ is too large. The reason for this behavior is that the mean temperature increases with the internal heating rate which makes the viscosity of the interior region decrease. Consequently, the viscous stress available from the mantle flow decreases making it more difficult to break the lid. Heat transfer in the plate-like regime follows the usual scaling $Q=A× Ra^{1/3} T^{4/3}$, but the coefficient $A$ strongly depends on the wavelength of the flow (twice the plate size) which is much larger than in typical Rayleigh-Bénard situations and, more importantly, increases with the internal heating rate. This suggests a possible scenario for the thermal evolution of the Earth: lithospheric plates might have been larger in the past, rendering heat transfer less efficient and helping to conserve the initial energy of Earth formation for a longer period. Reference. Tackley, P.~J., Self-consistent generation of tectonic plates in time-dependent, three-dimensional mantle convection simulations 1. pseudoplastic yielding, {\it Geochem. Geophys. Geosyst.\/}, {\it 1\/}, 2000, 2000Gc000041.