Magnetic buoyancy instability and the anelastic approximation: regime of validity and relationship with compressible and Boussinesq descriptions

Magnetic buoyancy instability, which is of astrophysical importance, results from the influence magnetic pressure variations on density a fluid in gravitational field. It inherently compressible phenomenon and is, as such, fully described by equations magnetohydrodynamics (MHD). For analytical computational reasons, it often convenient to study MHD within simpler, asymptotically consistent reduced systems; two most widely used result Boussinesq anelastic approximations. Within standard approximation MHD, leading magnetoconvection, excluded. can, however, be included rescaling basic-state variables making further assumptions about perturbation length scales. approximation, no special measures are taken incorporate buoyancy. priori unclear whether this neglect justified, particularly light results. Our aims here thus twofold. The first formulate relationship between descriptions compressible, systems. In so doing, we show that, under both approximations, can either through combination weak field strong gradient, or, conversely, gradient. Each has its own reduction, with dedicated governing equations. second aim address, linear stability analysis, conditions system provides faithful representation instability. completeness, also energy principle ideal framework, demonstrate relation counterpart.