Convection in a spherical shell heated by an isothermal core and internal sources: Implications for the thermal state of planetary mantles

The parallelized three-dimensional spherical convection code, TERRA, is employed to study the mean temperatures and planforms of convecting planetary mantles in spherical shell geometries. We vary the factor f which controls the degree of curvature, defined as the ratio of the radii of the inner and outer bounding surfaces, the Bénard-Rayleigh number, RaB, and the dimensionless rate of internal heating, H. We develop parameterized expressions for predicting the mean temperature of convecting spherical shells which are heated partially from within and partially from below by a hot isothermal lower boundary. Our parameterization is fit to a data set of mean temperatures from 23 numerical model calculations for f=0.547 (appropriate to Earth's mantle). We then demonstrate that this parameterization of mean temperature in terms of f, RaB and H extends to other values of f as well. For all values of f, RaB and H considered in this study, our predicted mean temperatures agree with the model calculations to within 2.4%. The scaling analysis is extended to obtain an expression for surface heat flux in terms of RaB and H for f=0.547. In that case we obtain a predictive equation for surface heat flux that agrees to within 11% of the observed values. Our findings provide a useful tool for parameterizing the temperature and surface heat flux of planetary mantles of varying geometry and heating configurations.