A playground for compressible natural convection with a nearly uniform density

Abstract

In the quest to understand the basic universal features of compressible convection, one would like to disentangle genuine consequences of compression from spatial variations of transport properties. For instance, one may choose to consider a fluid with uniform dynamic viscosity, but, then, compressible effects will generate a density gradient and consequently the kinematic viscosity will not be uniform. In the present work, we consider a very peculiar equation of state, whereby entropy is solely dependent on density, so that a nearly isentropic fluid domain is nearly isochoric. Within this class of equations of state, there is a thermal adiabatic gradient and a key property of compressible convection is still present, namely its capacity to viscously dissipate a large fraction of the thermal energy involved, of the order of the well-named dissipation number. In the anelastic approximation, under the assumption of an infinite Prandtl number, the number of governing parameters can be brought down to two, the Rayleigh number and the dissipation number. This framework is proposed as a playground for compressible convection, an opportunity to extend the vast corpus of theoretical analyses on the Oberbeck–Boussinesq equations regarding stability, bifurcations or the determination of upper bounds for the turbulent heat transfer. Here, in a two-dimensional geometry, we concentrate on the structure of numerical solutions. For all Rayleigh numbers, a change in the vertical temperature profile is observed in the range of dissipation number between $0$ and less than $0.4$ , associated with the weakening of ascending plumes. For larger dissipation numbers, the heat flux dependence on this number is found to be well predicted by Malkus's model of critical layers. For dissipation numbers of order unity, and large Rayleigh numbers, dissipation becomes related to the entropy heat flux at each depth, so that the vertical dissipation profile can be predicted, and so does the total ratio of dissipation to convective heat flux.

Fully compressible convection for planetary mantles

“…Here we enforce mass conservation by imposing the average values of P on the top and bottom boundaries to be equal. The Stokes equation integrated over the total volume when horizontal periodic conditions are imposed (see Alboussière et al, 2022), implies indeed that…”

“…No analytical results are available for the profile of dissipation with depth, Diss, however the equations (59) suggests a fundamental relation with the entropy transport in compressible convection that we will now emphasize. In a companion paper (Alboussière et al, 2022) and with very specific and different EoS, the role of entropy transport is also discussed.…”

Compressible convection in super-earths

“…7. When the dissipation number is large, it is difficult for cold plumes to continuously traverse the convective layer (Alboussière et al, 2022). As previously observed by Hansen et al (1993), hot instabilities gain buoyancy as they rise in the mantle as the thermal expansivity increases.…”

“…These anelastic codes often used a simplified equation of state (the relationship between density, temperature, and pressure) and implement some approximations that contradict the thermodynamic rules: this happens most often when some thermodynamic quantities are given an explicit dependence in depth rather than in pressure and temperature. We have shown in a series of papers that, when inertia can be neglected, which is the case in the mantle of planets, solving the exact mechanical and thermodynamical equations for a realistic equation is however not much more difficult than implementing an anelastic formalism (Alboussière and Ricard, 2017;Alboussière et al, 2022;Curbelo et al, 2019;Ricard et al, 2022).…”