  The amount and spatial pattern of heat extracted from cores of terrestrial
planets is ultimately controlled by the thermal structure of the lower rocky
mantle. Using the most common model to tackle this problem, a rapidly rotating
and differentially cooled spherical shell containing an incompressible and
viscous liquid is numerically investigated. To gain the physical basics, we
consider a simple, equatorial symmetric perturbation of the CMB heat flux
shaped as a spherical harmonic $Y_{11}$. The thermodynamic properties of the
induced flows mainly depend on the degree of nonlinearity parametrised by a
horizontal Rayleigh number $Ra_h=q^\ast Ra$, where $q^\ast$ is the relative CMB
heat flux anomaly amplitude and $Ra$ is the Rayleigh number which controls
radial buoyancy-driven convection. Depending on $Ra_h$ we characterise three
flow regimes through their spatial patterns, heat transport and flow speed
scalings: in the linear conductive regime the radial inward flow is found to be
phase shifted $90^\circ$ eastwards from the maximal heat flux as predicted by a
linear quasi-geostrophic model for rapidly rotating spherical systems. The
advective regime is characterised by an increased $Ra_h$ where nonlinearities
become significant, but is still subcritical to radial convection. There the
upwelling is dispersed and the downwelling is compressed by the thermal
advection into a spiralling jet-like structure. As $Ra_h$ becomes large enough
for the radial convection to set in, the jet remains identifiable on
time-average and significantly alters the global heat budget in the convective
regime. Our results suggest, that the boundary forcing not only introduces a
net horizontal heat transport but also suppresses the convection locally to
such an extent, that the net Nusselt number is reduced by up to 50\%, even
though the mean CMB heat flux is conserved.
