Effects of geometry on the convection with core-cooling

We study the dynamical (three-dimensional box, axisymmetric and spherical shell geometry) and parameterized models of the mantle convection with the core-cooling. The viscosity is constant in space and dependent on the volume averaged mantle temperature. Core is treated as a hot bath. To understand the process of cooling, we use the ‘local’ Rayleigh (Ra_l ) and Nusselt (Nu_l) numbers, which are defined in each thermal boundary layer. In the dynamical calculations, we check the various combinations of Ra_l and Nu_l, and find that the local Rayleigh number either at the top or bottom surface may control both the top and bottom local Nusselt numbers. This result suggests that the core-cooling in this case may be controlled by the flow either at top or bottom boundary layer. The least-square-fitting of Nu_l-Ra_l relationship shows that its power-law index is around 0.3, despite of the different geometry. Comparing the thermal history calculated by the dynamical and parameterized models, we find that the parameterized convection theory based on the local Ra-Nu relationship obtained by the dynamical calculation is useful for investigating the thermal history of the mantle and core. Applying the parameterized theory to the Earth, we find that the plausible Urey ratio is smaller than that obtained by the previous works which ignored the bottom thermal boundary layer.