Scaling regimes in spherical shell rotating convection

Abstract

Rayleigh-B\'enard convection in rotating spherical shells can be considered as a simplified analogue of many astrophysical and geophysical fluid flows. Here, we use three-dimensional direct numerical simulations to study this physical process. We construct a dataset of more than 200 numerical models that cover a broad parameter range with Ekman numbers spanning $3\times 10^{-7} \leq E \leq 10^{-1}$, Rayleigh numbers within the range $10^3 < Ra < 2\times 10^{10}$ and a Prandtl number unity. We investigate the scaling behaviours of both local (length scales, boundary layers) and global (Nusselt and Reynolds numbers) properties across various physical regimes from onset of rotating convection to weakly-rotating convection. Close to critical, the convective flow is dominated by a triple force balance between viscosity, Coriolis force and buoyancy. For larger supercriticalities, a subset of our numerical data approaches the asymptotic diffusivity-free scaling of rotating convection $Nu\sim Ra^{3/2}E^{2}$ in a narrow fraction of the parameter space delimited by $6\,Ra_c \leq Ra \leq 0.4\,E^{-8/5}$. Using a decomposition of the viscous dissipation rate into bulk and boundary layer contributions, we establish a theoretical scaling of the flow velocity that accurately describes the numerical data. In rapidly-rotating turbulent convection, the fluid bulk is controlled by a triple force balance between Coriolis, inertia and buoyancy, while the remaining fraction of the dissipation can be attributed to the viscous friction in the Ekman layers. Beyond $Ra \simeq E^{-8/5}$, the rotational constraint on the convective flow is gradually lost and the flow properties vary to match the regime changes between rotation-dominated and non-rotating convection. The quantity $Ra E^{12/7}$ provides an accurate transition parameter to separate rotating and non-rotating convection.