Numerical modelling of convection interacting with a melting and solidification front: Application to the thermal evolution of the basal magma ocean

Melting and solidification are fundamental to geodynamical processes like inner core growth, magma chamber dynamics, and ice and lava lake evolution. Very often, the thermal history of these systems is controlled by convective motions in the melt. Computing the evolution of convection with a solid-liquid phase change requires specific numerical methods to track the phase boundary and resolve the heat transfer within and between the two separate phases. Here we present two classes of method to model the phase transition coupled with convection. The first, referred to as the moving boundary method, uses the finite element method and treats the liquid and the solid as two distinct grid domains. In the second approach, based on the enthalpy method, the governing equations are solved on a regular rectangular grid with the finite volume method. In this case, the solid and the liquid are regarded as one domain in which the phase change is incorporated implicitly by imposing the liquid fraction A as a function of temperature and a viscosity that varies strongly with f(L). We subject the two modelling frameworks to thorough evaluation by performing benchmarks, in order to ascertain their range of applicability. With these tools we perform a systematic study to infer heat transfer characteristics of a solidifying convecting layer. Parametrized relations are then used to estimate the super-isentropic temperature difference maintained across a basal magma ocean (BMO) (Labrosse et al.. 2007), which happens to be minute (<0.1 K), implying that the Earth's core must cool at the same pace as the BMO. (C) 2012 Elsevier B.V. All rights reserved.