The Breakdown of the Anelastic Approximation in Rotating Compressible Convection: Implications for Astrophysical Systems

TL;DR

This paper demonstrates that the anelastic approximation fails to accurately describe rapidly rotating, low Prandtl number compressible convection, revealing the importance of compressional oscillations and the limitations of traditional models in astrophysical contexts.

Contribution

It shows that the anelastic equations neglect critical compressional effects in rapidly rotating, low Prandtl number convection, leading to spurious eigenmodes and inaccurate instability predictions.

Findings

Anelastic equations cannot capture compressional quasi-geostrophic oscillations.

Rapidly rotating, low Prandtl number convection has intrinsically small Mach numbers.

Classical asymptotic laws are recovered at high rotation rates.

Abstract

The linear theory for rotating compressible convection in a plane layer geometry is presented for the astrophysically-relevant case of low Prandtl number gases. When the rotation rate of the system is large, the flow remains geostrophically balanced for all stratification levels investigated and the classical (i.e., incompressible) asymptotic scaling laws for the critical parameters are recovered. For sufficiently small Prandtl numbers, increasing stratification tends to further destabilise the fluid layer, decrease the critical wavenumber and increase the oscillation frequency of the convective instability. In combination, these effects increase the relative magnitude of the time derivative of the density perturbation contained in the conservation of mass equation to non-negligible levels; the resulting convective instabilities occur in the form of compressional quasi-geostrophic oscillations. We find that the anelastic equations, which neglect this term, cannot capture these instabilities and possess spuriously-growing eigenmodes in the rapidly rotating, low Prandtl number regime. It is shown that the Mach number for rapidly rotating compressible convection is intrinsically small for all background states, regardless of the departure from adiabaticity.