We examine thermal convection in a rotating spherical shell with central gravity and a spatially non‐uniformly heated outer surface at two values of the Prandtl number:Pr=7.0, appropriate for water at room temperature, andPr=0.7, appropriate for air at standard temperature and pressure, by numerical calculation. Four calculations are performed in a sequence: the onset of convection with homogeneous temperature boundary condition, nonlinear boundary‐forced steady convection, stability of the forced steady convection to infinitesimal disturbances, and time stepping of subsequent secondary convection. Unlike our previous study of the infinite Prandtl number limit [J. Fluid Mech.250, 209 (1993)] inertial terms in the equation of motion for moderate Prandtl numbers play a key role in the dynamics. The effects of an inhomogeneous temperature boundary condition on nonlinear convection are illustrated by varying the wavelength and strength of the imposed boundary temperature. It is shown that even a slight inhomogeneity in the thermal boundary condition can lock azimuthally drifting convection and make it stationary, or modify the normal drifting convection rolls to a vacillating structure. In the infinite Prandtl number case, when inertial forces are absent from the equation of motion, resonance occurs when the wavelengths of boundary forcing and natural convection coincide. Fluid inertia destroys this resonance for finite Prandtl number fluids. The same effect reduces in size the stability region where steady convection is locked to the boundary, and steady convection becomes unstable to time‐dependent convection. The period of the secondary convection is close to that obtained with uniform temperature boundaries but the spatial structure is dramatically changed, exhibiting vacillations between the wavelength of the boundary temperature and that of the natural convection. ©1996 American Institute of Physics.