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 S 10 4 , a buoyancy parameter (the strength of applied boundary heat flux anomalies), 10 −3 B 10 6 , and the Ekman number (ratio of viscous to Coriolis forces), 10 −6 E 10 −4 . We find both steady and time-dependent solutions and delineate the temporal regime boundaries. We focus on steady-state solutions, for which a clear transition is found between a low S regime, in which buoyancy dominates dynamics, and a high S regime, in which stratification dominates. For the latter case, the radial and horizontal velocities scale respectively as u r ∼ S −1 , u h ∼ S − 3 4 B 1 4 and are confined to boundary-induced flow within a thin layer of depth (S B) − 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 † Email address for correspondence: gracecox@cp.dias.ie arXiv:1807.00310v2 [physics.flu-dyn]