On the scaling of heat transfer for mixed heating convection in a spherical shell

Planetary mantles and solid shells of icy satellites potentially undergoing natural convection are subjected to a mixed heating configuration including basal (from thermal exchanges with a subjacent, possibly liquid, layer) and internal (from radioactive decay or tidal dissipation) sources. In the quasi-static approximation, the average cooling/heating of the layer is also considered as an instantaneous internal heat source to model transient evolutions. In a previous study (Choblet and Parmentier, 2009), we have proposed simple scaling relationships to describe heat transfer for an isoviscous fluid in such a mixed heating configuration in the case of a Cartesian geometry. Here, we extend this analysis to the case of a spherical shell. A framework based on a temperature scale associated with the global surface heat flux is introduced. This enables a simple description of the cold boundary layer, independent of the heating configuration and of the relative radius of the inner boundary of the shell. When free-slip mechanical boundaries are prescribed, numerical experiments present a significant departure from the prediction (up to ≃30%). We show that this is caused by the impact of hot plumes on the cold boundary layer when a large amount of basal heating is prescribed. The results of no-slip calculations are well predicted by the scaling which thus could be applied to planetary mantles where convection occurs beneath a rigid lithosphere. The lower hot boundary layer is included in our analysis through the ratio of the temperature differences across both boundary layers: the simple scaling indicates that this ratio is independent of the Rayleigh number, and varies only with the amount of basal heating and with the curvature of the layer. This is shown to be valid in the no-slip case. In the free-slip case, a departure from this scaling is observed in the calculations but for the range of values corresponding to planetary bodies, the agreement is good. We conduct transient numerical experiments and show that the quasi-static approximation is valid in the configuration investigated here. Implications for more complex planetary set-ups are discussed.