Temperature and heat flux scalings for isoviscous thermal convection in spherical geometry

P>Parametrized convection, which has long been used to reconstruct the thermal history of planetary mantles, is based on scaling relationships between observables (including heat flux) and controlling parameters (the most important being the Rayleigh number, Ra). To explore the influence of spherical geometry on heat transfer, we have conducted two series of numerical experiments of thermal convection (one with bottom heating and the other with mixed heating) in an isoviscous spherical shell with various curvatures. Using these calculations and a generalized non-linear inversion, we then derive scaling laws for the average temperature and for the surface heat flux. In the case of bottom heating, we found that the non-dimensional average temperature is given by theta(m) = f2/(1 + f2), where f is the ratio between the core and total radii. The non-dimensional surface heat flux is fitted well by Nu(top) = 0.36f0.32 Ra(0.273+0.05f) theta 0.6(m). This scaling indicates that the available heating power decreases with increasing curvature (decreasing f). There exist strong trade-offs between the inverted parameters, that is, different sets of parameters explain our calculations well within error bars. For mixed heating, the non-dimensional average temperature and surface heat flux are well explained by theta(H) = theta(m) + (1.68 - 0.8f)[(1 + f + f2)/3]0.79 h0.79/Ra0.234, where h is the non-dimensional rate of internal heating, and Nu(top) = 0.59f0.05 Ra(0.300-0.003f) theta 1.23(H). Due to a competition between the radiogenic and convective powers, and for given values of h and Ra, there is a curvature for which the Urey ratio reaches a minimum. Applied to the Earth's mantle, the mixed heating scaling predicts a Urey ratio between 0.4 and 0.6, depending on the Rayleigh number. Additional parameters, including the thermal viscosity ratio, phase transitions, the presence of dense material in the deep mantle, and variability of the flow pattern in time, may enter an appropriate modelling of the Earth's mantle thermal history.