Convective heat transport in stratified atmospheres at low and high Mach number

We study fully compressible convection in the context of plane-parallel, polytropically stratified atmospheres. We perform a suite of 2D and 3D simulations in which we vary the initial superadiabaticity ($\epsilon$) and the Rayleigh number (Ra) while fixing the initial density stratification, aspect ratio, and Prandtl number. The evolved heat transport, quantified by the Nusselt number (Nu), follows scaling relationships similar to those found in the well-studied, incompressible Rayleigh-B\'{e}nard problem. This scaling holds up in both 2D and 3D and is not appreciably affected by the magnitude of $\epsilon$.