Weakly non-Boussinesq convection in a gaseous spherical shell

We examine the dynamics associated with weakly compressible convection in a spherical shell by running 3D direct numerical simulations using the Boussinesq formalism [1]. Motivated by problems in astrophysics, we assume the existence of a finite adiabatic temperature gradient $\nabla T_{\rm{ad}}$ and use mixed boundary conditions for the temperature with fixed flux at the inner boundary and fixed temperature at the outer boundary. This setup is intrinsically more asymmetric than the more standard case of Rayleigh-B\'{e}nard convection in liquids between parallel plates with fixed temperature boundary conditions. Conditions where there is substantial asymmetry can cause a dramatic change in the nature of convection and we demonstrate that this is the case here. The flows can become pressure- rather than buoyancy- dominated leading to anomalous heat transport by upflows. Counter-intuitively, the background temperature gradient $\nabla\bar{T}$ can develop a subadiabatic layer (where $\boldsymbol{g}\cdot\nabla\bar{T}<\boldsymbol{g}\cdot\nabla T_{\rm{ad}}$, where $\boldsymbol{g}$ is gravity) although convection remains vigorous at every point across the shell. This indicates a high degree of non-locality.