Fully compressible convection for planetary mantles

Abstract

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 2D 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.

A playground for compressible natural convection with a nearly uniform density

“…For instance, the increasing value of thermal expansion as altitude increases is thought be be such an argument leading to the strengthening of ascending plumes and weakening of descending plumes. This seems to apply in mantle convection (Ricard et al 2022). However, we have the opposite effect in the present paper, although thermal expansion also increases with altitude.…”

“…This property seems to be related to the equation of state. In another paper (Ricard et al 2022), we consider an equation of state suitable for planetary mantles and in that case, the opposite is true: with compressible effects, ascending plumes are stronger.…”

Mass–Radius Relationships and Contraction of Condensed Planets by Cooling or Despinning

“…To get an estimate of the relation between the radius and temperature of a cooling planet, we must first choose an equation of state (EoS) to describe the thermodynamic relation between a planet's pressure P, temperature T, and density ρ. For a condensed planet (solid or liquid, silicate of metal), Murnaghan's EoS (Murnaghan 1937) gives a fairly simple and versatile expression that fits well with various high-pressure, high-temperature experiments on silicates and metals, as well as with the radius properties of the Earth (Ricard et al 2022;Ricard & Alboussière 2023):…”

“…In the remainder of this article, we take K 0 = 130 GPa which is typical incompressibility near the surface of the Earth, and ρ 0 = 4000 kg m −3 (for planets made of silicates and metal, we thought it reasonable to choose a reference density somewhat larger than that of crustal rocks or very shallow mantle-see also Ricard et al 2022-but this choice is not crucial). The length r 0 is therefore r 0 = 3113 km.…”

CaTSM: A Pseudo‐Spectral Thermodynamically Consistent Model of Compressible Flows

“…Under some circumstances, the fully compressible equations have been shown to be feasible to solve for such low-Mach number flows. In the limit of high Prandtl number, e.g., Ricard et al (2022) showed that fully compressible simulations can be performed at comparable computational cost to the sound-proof formulations. In addition, a number of studies have used implicit handling of the most challenging processes, typically the propagation of acoustic or Alfvén waves (Aydemir & Barnes, 1985;Erlebacher et al, 1990).…”