Penetration of boundary-driven flows into a rotating spherical thermally stratified fluid

Motivated by the dynamics within terrestrial bodies, we consider a rotating, strongly thermally stratified fluid within a spherical shell subject to a prescribed laterally inhomogeneous heat-flux condition at the outer boundary. Using a numerical model, we explore a broad range of three key dimensionless numbers: a thermal stratification parameter (the relative size of boundary temperature gradients to imposed vertical temperature gradients), $10^{-3}\leqslant S\leqslant 10^{4}$ , a buoyancy parameter (the strength of applied boundary heat-flux anomalies), $10^{-2}\leqslant B\leqslant 10^{6}$ , and the Ekman number (ratio of viscous to Coriolis forces), $10^{-6}\leqslant E\leqslant 10^{-4}$ . We find both steady and time-dependent solutions and delineate the regime boundaries. We focus on steady-state solutions, for which a clear transition is found between a low $S$ regime, in which buoyancy dominates the dynamics, and a high $S$ regime, in which stratification dominates. For the low- $S$ regime, we find that the characteristic flow speed scales as $B^{2/3}$ , whereas for high- $S$ , the radial and horizontal velocities scale respectively as $u_{r}\sim S^{-1}$ , $u_{h}\sim S^{-3/4}B^{1/4}$ and are confined within a thin layer of depth $(SB)^{-1/4}$ at the outer edge of the domain. For the Earth, if lower mantle heterogeneous structure is due principally to chemical anomalies, we estimate that the core is in the high- $S$ regime and steady flows arising from strong outer boundary thermal anomalies cannot penetrate the stable layer. However, if the mantle heterogeneities are due to thermal anomalies and the heat-flux variation is large, the core will be in a low- $S$ regime in which the stable layer is likely penetrated by boundary-driven flows.