The anelastic approximation is an asymptotically reduced form of the compressible Navier-Stokes equations (NSE) that has been in common use for studying the dynamics of both stably stratified fluid layers and unstably stratified convection. The primary advantage of the anelastic equations is the elimination of fast acoustic waves, that necessitate small time-steps for numerical simulations of the NSE. We show that the anelastic equations are incapable of capturing the linear instability of rotating compressible convection in low Prandtl number gases, despite the presence of an approximately adiabatic background state. The temporal derivative of the density fluctuation present in the conservation of mass equation remains important for this class of astrophysically-relevant fluids, and results in compressional inertial oscillations as the most unstable eigenmodes. Furthermore, a fully compressible quasi-geostrophic convection model will be presented that can capture both the linear stability and nonlinear evolution of low Rossby number compressible convection; some preliminary results for this new model will be discussed.
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