Three-dimensional thermal convection in an iso-viscous, infinite Prandtl number fluid heated from within and from below: applications to the transfer of heat through planetary mantles

Abstract

Numerical experiments have been carried out to explore the efficiency of heat transfer through a three-dimensional layer heated from both within and below as it is the case for the mantle of earth-like planets. A systematic study for Rayleigh numbers (Ra) between 10 5 and 10 7 and non-dimensional internal heating rate ( H s) between 0 and 40 allows us to investigate the pattern of convection and the thermal characteristics of the layer in a range of parameters relevant to mantle convection in earth-like planets. Inversion of the results for the mean temperature and non-dimensional heat flux at the top and the bottom boundaries yields simple parameterization of the heat transfer. It is shown that the mean temperature of the convective fluid ( θ) is the sum of the temperature that would exist with no internal heating and a contribution of the non-dimensional internal heating rate ( H s). As predicted by thermal boundary layer analysis, the non-dimensional heat flux at the upper boundary layer can be described by Q=[(Ra)/(Ra δ )] 1/3 θ 4/3 with θ=0.5+1.236[( H s) 3/4/(Ra) 1/4], and Ra δ being the thermal boundary layer Rayleigh number equal to 24.4. In agreement with laboratory experiments, this value slightly increases with the value of the Rayleigh number. This value is identical to that obtained for fluids heated from within only. In most cases, the hot plumes that form at the lower thermal boundary layer do not reach the upper boundary layer. No simple law has been found to describe the heat transfer through the lower thermal boundary layer, but the bottom heat flux can be determined using the global energy balance. The thermal boundary layer analysis performed in this study allows us to extrapolate our results to 3D spherical geometry and our predictions are in good agreement with numerical experiments described in the literature. A simple case of spherical 3D convection has been performed and provides the same thermal history of planetary mantles than that obtained from 3D numerical runs. Compared to previous parameterized analysis, this study shows that the behaviour of the thermal boundary layers is much different than that predicted by experiments for a fluid heated only from below: at similar Rayleigh numbers, the mean temperature is larger and the surface heat flux is much larger. It seems therefore necessary to reconsider previous models of the thermal evolution of planetary mantles.