An interpretation of the Nusselt-Rayleigh number relationship for convection in a spherical shell

Abstract

Recent studies on the relationship between the Nusselt (Nu) and Rayleigh (Ra) numbers for base-heated convection in a spherical shell with a constant viscosity show that the power-law index is around 1/4, which is different from the value of 1/3 predicted by a simple boundary layer theory. We show that such a difference may be caused by the flow pattern due to the geometry. The flow pattern of the convection in a spherical shell at relatively low Ra, at least, less than 106, is characterized by narrow upwelling and broad downwelling, which is similar to the opposite flow pattern of internally heated convection. Convection in the internally heated case predicts the power-law index of 1/4. We demonstrate this relationship based on the concept of 'local' Rayleigh (Ra1) and Nusselt (Nu1) numbers.