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

Numerical experiments have been carried out to explore the efficiency of he at transfer through a three-dimensional layer heated from both within and b elow as it is the case for the mantle of earth-like planets. A systematic s tudy 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 p attern of convection and the thermal characteristics of the layer in a rang e of parameters relevant to mantle convection in earth-like planets. Invers ion of the results for the mean temperature and non-dimensional heat flux a t the top and the bottom boundaries yields simple parameterization of the h eat transfer. It is shown that the mean temperature of the convective fluid (theta) is the sum of the temperature that would exist with no internal he ating and a contribution of the non-dimensional internal heating rate (H-s) . As predicted by thermal boundary layer analysis, the non-dimensional heal flux at the upper boundary layer can be described by Q = [(Ra)/(Ra-delta)( 1/3)theta(4/3) with theta = 0.5 + 1.236[(H-s)(3/4)/(Ra)(1/4)], and Ra-delta being the thermal boundary layer Rayleigh number equal to 24.4. In agreeme nt with laboratory experiments, this value slightly increases with the valu e of the Rayleigh number. This value is identical to that obtained for flui ds 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 simp le law has been found to describe the heat transfer through the lower therm al boundary layer, but the bottom heat flux can be determined using the glo bal energy balance. The thermal boundary layer analysis performed in this s tudy allows us to extrapolate our results to 3D spherical geometry and our predictions are in good agreement with numerical experiments described in t he literature. A simple case of spherical 3D convection has been performed and provides the same thermal history of planetary mantles than that obtain ed from 3D numerical runs. Compared to previous parameterized analysis, thi s study shows that the behaviour of the thermal boundary layers is much dif ferent than that predicted by experiments for a fluid heated only from belo w: at similar Rayleigh numbers, the mean temperature is larger and the surf ace heat flux is much larger. It seems therefore necessary to reconsider pr evious models of the thermal evolution of planetary mantles. (C) 1999 Elsev ier Science B.V. All rights reserved.