A multiphase model of core formation

International audienceThe differentiation of solid planets with segregation of metal from silicates happens during the Hadean time while the planet is still growing by accretion. The separation of metal occurs when at least the metallic phase is liquid and proceeds by a combination of transport by diapiric instabilities and by more diffuse percolation flow. In this paper we develop a formalism derived from Bercovici et al. that can handle simultaneously two components, silicates and metal, and where the metal can be present both in solid and liquid states. The mechanical equations are non-Boussinesq as the lateral density variations are of the same order as the density itself. When the metal is solid, the metal and the silicates are locked together and we treat their mixture as a single-phase fluid where density is function of composition (iron-silicate proportions). When metal is liquid, it can separate from the silicates and the two phases interact through shear stress ( e. g. Darcy flow) and normal stress. The evolution of the volume proportion of liquid iron is controlled by the difference of pressure between the solid and liquid phases. The energy conservation equation takes into account the different mechanisms by which the gravitational energy is dissipated as heat. The 2-D Cartesian numerical code that we implemented to solve these equations makes use of numerical techniques that have not been previously used in geophysical two-phase modelling; we discuss the numerical aspects and benchmark the solutions. We present simulations of core-mantle differentiation showing that the first impact that melts the iron phase near the surface is potentially able to trigger the whole core-mantle segregation in a runaway phenomenon. The threshold of this instability in terms of the impactor and planetary size and the initial planetary temperature is investigated. The segregation of the metal occurs by a mechanism that was not suggested before and which is intermediate between the usual diapir instability and a porosity wave. Although we cannot explore the whole parameter space of our numerical model, we show various simulations that clarify the role of the most important parameters, such as the solid and metal viscosities or the depth dependence of gravit