Modelling of thermal stratification at the top of a planetary core: Application to the cores of Earth and Mercury and the thermal coupling with their mantles

Abstract

We present a new numerical scheme for solving one-dimensional conduction problems and in the radial direction of a spherically symmetric body or shell with variable spatial domain. This numerical scheme adopts a solution of the conduction equation in each interval of the chosen discretization which is valid if the fluxes at the interval boundaries are constant in time. This ‘piece-wise steady flux’ (PWSF) numerical scheme is continuous and differentiable in the space domain. These smoothness properties are convenient for implementing the numerical scheme in an energy-conserved thermal evolution approach for a terrestrial core in which a conductive thermally stratified layer is considered that develops below the core-mantle boundary when the heat flux drops below the adiabatic heat flux. The influence of a time-variable thermally stratified region on the general evolution of the planetary body is examined, in comparison to imposing an adiabatic temperature profile for the entire core. Also, the dependence of the model's accuracy to the applied grid size of the numerical scheme is studied. We showed that a very low numerical resolution of this approach suffices for obtaining a thermal evolution of a partially conductive planetary core with high accuracy.By considering thermal stratification in a planetary core where the heat flux is lower than the adiabatic heat flux, radial variations in the cooling rate are accounted for whereas otherwise the distribution of energy in the core is fixed by the imposed adiabat. During the growth of the thermally stratified region, the deep part of the core cools more rapidly than the outer part of the core. Therefore, the inner core grows to a larger size and the temperature and heat flux at the core-mantle boundary are higher and larger, respectively, if a thermally stratified region is considered. For the Earth, the implications are likely very minor and can be neglected in thermal evolution studies that are not specifically interested in the thermally stratified region itself. For Mercury, however, these implications are larger. For example, the age of Mercury's inner core can be underestimated by about a billion of years if thermal stratification is neglected. The consideration of thermal stratification in Mercury's core is also important for the evolution of Mercury's mantle. It increases the mantle temperature, leads to a higher Rayleigh number and therefore a larger heat flux into the lithosphere and prolongation of mantle convection.