Validity of Sound-Proof Approaches in Rapidly-Rotating Compressible Convection: Marginal Stability vs. Turbulence

The validity of the anelastic approximation has recently been questioned in the regime of rapidly-rotating compressible convection in low Prandtl number fluids (Calkins et al. 2015). Given the broad usage and the high computational efficiency of sound-proof approaches in this astrophysically relevant regime, this paper clarifies the conditions for a safe application. The potential of the alternative pseudo-incompressible ap- proximation is investigated, which in contrast to the anelastic approximation is shown to never break down for predicting the point of marginal stability. Its accuracy, however, decreases as the temporal derivative of pressure term in the continuity equation becomes non-negligible. The magnitude of this pressure term is found to be controlled by the phase Mach number that we introduce as the ratio of the phase velocity (corresponding to the oscillatory instability) to the local sound speed. We find that although the anelastic approximation for compressible convection in the rapidly rotating low Prandtl number regime is inaccurate at marginal stability, it does not show unphysical behavior for supercritical convection. Growth rates com- puted with the linearized anelastic equations converge toward the corresponding fully compressible values as the Rayleigh number increases. Likewise, our fully nonlinear turbulent simulations, produced with our fully compressible and anelastic models and carried out in the regime in which Calkins et al. (2015) suspect the anelastic approximation to break down, show good agreement.