Turbulent Rayleigh-B\'enard convection in spherical shells

We simulate numerically Boussinesq convection in non-rotating spherical shells for a fluid with a unity Prandtl number and Rayleigh numbers up to ${10}^9$. In this geometry, curvature and radial variations of the gravitationnal acceleration yield asymmetric boundary layers. A systematic parameter study for various radius ratios (from $\eta =r_i\mathrm{/}{r}_o=0.2$ to $\eta =0.95$) and gravity profiles allows us to explore the dependence of the asymmetry on these parameters. We find that the average plume spacing is comparable between the spherical inner and outer bounding surfaces. An estimate of the average plume separation allows us to accurately predict the boundary layer asymmetry for the various spherical shell configurations explored here. The mean temperature and horizontal velocity profiles are in good agreement with classical Prandtl-Blasius laminar boundary layer profiles, provided the boundary layers are analysed in a dynamical frame, that fluctuates with the local and instantaneous boundary layer thicknesses. The scaling properties of the Nusselt and Reynolds numbers are investigated by separating the bulk and boundary layer contributions to the thermal and viscous dissipation rates using numerical models with $\eta =0.6$ and a gravity proportional to $1\mathrm{/}{r}^2$. We show that our spherical models are consistent with the predictions of Grossmann & Lohse’s (2000) theory and that $Nu(Ra)$ and $Re(Ra)$ scalings are in good agreement with plane layer results.