  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.
