Oscillatory convection and limitations of the Boussinesq approximation

TL;DR

This paper investigates the validity of the Boussinesq approximation in oscillatory convection within rapidly rotating fluids, revealing its limitations in astrophysical regimes and identifying conditions where alternative approximations perform better.

Contribution

It establishes the asymptotic conditions for Boussinesq approximation validity in oscillatory convection, especially for small Prandtl numbers, and compares it with other sound-proof models.

Findings

Boussinesq approximation is valid only under very restrictive conditions in astrophysical regimes.

For an ideal gas, validity requires the domain height to be much smaller than the scale height, proportional to the Prandtl number.

A specific pseudo-incompressible approximation accurately predicts the instability threshold beyond Boussinesq's validity range.

Abstract

We determine the asymptotic conditions under which the Boussinesq approximation is valid for oscillatory convection in a rapidly rotating fluid. In the astrophysically relevant parameter regime of small Prandtl number, we show that the Boussinesq prediction for the onset of convection is valid only under much more restrictive conditions than those that are usually assumed. In the case of an ideal gas, we recover the Boussinesq results only if the ratio of the domain height to a typical scale height is much smaller than the Prandtl number. This requires an extremely shallow domain in the astrophysical parameter regime. Other commonly-used "sound-proof" approximations generally perform no better than the Boussinesq approximation. The exception is a particular implementation of the pseudo-incompressible approximation, which predicts the correct instability threshold beyond the range of validity of the Boussinesq approximation.