CONVECTION IN A ROTATING SPHERICAL FLUID SHELL WITH AN INHOMOGENEOUS TEMPERATURE BOUNDARY-CONDITION AT FINITE PRANDTL NUMBER

We examine thermal convection in a rotating spherical shell with centr al gravity and a spatially non-uniformly heated outer surface at two v alues of the Prandtl number: P-r=7.0, appropriate for water at room te mperature, and P-r=0.7, appropriate for air at standard temperature an d pressure, by numerical calculation. Four calculations are performed in a sequence: the onset of convection with homogeneous temperature bo undary condition, nonlinear boundary-forced steady convection, stabili ty 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 (1 993)] inertial terms in the equation of motion for moderate Prandtl nu mbers play a key role in the dynamics. The effects of an inhomogeneous temperature boundary condition on nonlinear convection are illustrate d by varying the wavelength and strength of the imposed boundary tempe rature. It is shown that even a slight inhomogeneity in the thermal bo undary condition can lock azimuthally drifting convection and make it stationary, or modify the normal drifting convection rolls to a vacill ating structure. In the infinite Prandtl number case, when inertial fo rces are absent from the equation of motion, resonance occurs when the wavelengths of boundary forcing and natural convection coincide. Flui d inertia destroys this resonance for finite Prandtl number fluids. Th e same effect reduces in size the stability region where steady convec tion 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 bet ween the wavelength of the boundary temperature and that of the natura l convection. (C) 1996 American Institute of Physics.