Scaling behaviour in spherical shell rotating convection with fixed-flux thermal boundary conditions

Bottom-heated convection in rotating spherical shells provides a simple analogue for many astrophysical and geophysical fluid systems. We construct a database of 74 three-dimensional numerical convection models to investigate the scaling behaviour of seven diagnostics over a range of Ekman $(10^{-6}\leqslant E\leqslant 10^{-3})$ and Rayleigh $(15\leqslant \widetilde{Ra}\leqslant 18\,000)$ numbers while using a Prandtl number of unity. Our configuration is chosen to model Earth’s core as defined by the fixed flux thermal boundary conditions, radius ratio $r_{i}/r_{o}$ of $0.35$ and a gravity profile that varies linearly with radius. The quantities of interest are the viscous and thermal boundary layer thickness, mean temperature gradient, mean interior temperature, Nusselt number, horizontal flow length scale, and Reynolds number. We find four parameter regimes characterised by different scaling behaviour. For $E\leqslant 10^{-4}$ and low $Ra$ the weakly nonlinear regime is characterised by a balance between viscous, Archimedean and Coriolis forces and the heat transfer is described by weakly nonlinear theory. At low $E$ and moderate $Ra$ , the rapidly rotating regime sees inertia take over from viscosity in the global force balance. In this regime the heat transfer scaling has increasing exponent with decreasing Ekman number and shows no saturation to the diffusion free $Ra^{3/2}E^{2}$ scaling. At high $Ra$ and all $E$ the importance of the Coriolis force gradually decreases and all diagnostics continually change in the transitional regime before approaching the scaling behaviour of non-rotating convection.