Validity of sound-proof approximations for magnetic buoyancy

TL;DR

This paper evaluates the accuracy of various sound-proof models like Boussinesq, anelastic, and pseudo-incompressible in simulating magnetic buoyancy within the solar interior, comparing them to fully compressible models across different regimes.

Contribution

It introduces a general sound-proof model, derives constraints for validity, and assesses which existing models satisfy these in different parameter regimes.

Findings

Certain sound-proof models accurately capture leading-order behavior under specific conditions.

The pseudo-incompressible MHD model has a Hamiltonian structure derived variationally.

Constraints for model validity depend on regime parameters.

Abstract

The presence of acoustic waves in models of compressible flows can present complications for analytical and numerical analysis. Therefore, several methods have been developed to filter out these waves, leading to various "sound-proof" models, including the Boussinesq, anelastic and pseudo-incompressible models. We assess the validity of each of these approximate models for describing magnetic buoyancy in the context of the solar interior. A general sound-proof model is introduced and compared to the fully compressible system in a number of asymptotic regimes, including both non-rotating and rotating cases. We obtain specific constraints that must be satisfied in order that the model captures the leading-order behaviour of the fully compressible system. We then discuss which of the existing sound-proof models satisfy these constraints, and in what parameter regimes. We also present a variational derivation of the pseudo-incompressible MHD model, demonstrating its underlying Hamiltonian structure.