Effect of Robin boundary conditions on the onset of convective torsional flows in rotating fluid spheres

Torsional flows are preferred at the onset of thermal convection in fluid spheres with stress-free and perfectly conducting boundary conditions, in a narrow region of the parameter space for Prandtl numbers Pr <= 0.9 and ratios Pr / Ek = O ( 10 ), with Ek being the Ekman number. In this case, the transport of heat to the exterior is supposed instantaneous. When the thermal conductivity of the internal fluid is large, and the external convective heat transfer or radiative emissivity is low, the heat transmission is less efficient, and the thermal energy retained in the interior increases, enhancing the onset of convection. This study is devoted to analyzing the combined influence of the thermal conductivity and external conditions (temperature and resistance to heat transport) on the onset of the torsional convection by taking a Robin boundary condition for the temperature at the surface of the sphere. It is shown, by means of the numerical computation of the curves of simultaneous transitions to torsional flows and Rossby waves, that when the heat flux through the boundary decreases, the region where the axisymmetric flows are preferred shrinks, but it never strangles to an empty set. It has been found that with adequate scalings the curves delimiting the transition to torsional flows, and those of the critical Rayleigh number, Ra c, and the frequencies of the modes vs Ek become almost independent of the parameter of the Robin boundary condition.