  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), $10^{-3} \le
S \le 10^{4}$, a buoyancy parameter (the strength of applied boundary heat flux
anomalies), $ 10^{-3} \le B \le 10^{6}$, and the Ekman number (ratio of viscous
to Coriolis forces), $10^{-6} \le E \le 10^{-4}$. 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 $u_r \sim S^{-1}$, $u_h \sim S^{-\frac{3}{4}}\
B^{\frac{1}{4}}$ and are confined to boundary-induced flow within a thin layer
of depth $(S\ B)^{-\frac{1}{4}}$ 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.
