Fully compressible convection for planetary mantles

Abstract

SUMMARY The numerical simulations of convection inside the mantle of the Earth or of terrestrial planets have been based on approximate equations of fluid dynamics. A common approximation is the neglect of the inertia term which is certainly reasonable as the Reynolds number of silicate mantles, or their inverse Prandtl number, are infinitesimally small. However various other simplifications are made which we discuss in this paper. The crudest approximation that can be done is the Boussinesq approximation (BA) where the various parameters are constant and the variations of density are only included in the buoyancy term and assumed to be proportional to temperature with a constant thermal expansivity. The variations of density with pressure and the related physical consequences (mostly the presence of an adiabatic temperature gradient and of dissipation) are usually accounted for by using an anelastic approximation (AA) initially developed for astrophysical and atmospheric situations. The BA and AA cases provide simplified but self-consistent systems of differential equations. Intermediate approximations are also common in the geophysical literature although they are invariably associated with theoretical inconsistencies (non-conservation of total energy, non-conservation of statistically steady state heat flow with depth, momentum and entropy equations implying inconsistent dissipations). We show that, in the infinite Prandtl number case, solving the fully compressible (FC) equations of convection with a realistic equation of state (EoS) is however not much more difficult or numerically challenging than solving the approximate cases. We compare various statistical properties of the Boussinesq, AA and FC simulations in 2-D simulations. We point to an inconsistency of the AA approximation when the two heat capacities are assumed constant. We suggest that at high Rayleigh number, the profile of dissipation in a convective mantle can be directly related to the surface heat flux. Our results are mostly discussed in the framework of mantle convection but the EoS we used is flexible enough to be applied for convection in icy planets or in the inner core.