Onset of thermal convection in a solid spherical shell with melting at either or both boundaries

Abstract

Thermal convection in planetary solid (rocky or icy) mantles sometimes occurs adjacent to liquid layers with a phase equilibrium at the boundary. The possibility of a solid-liquid phase change at the boundary has been shown to greatly help convection in the solid layer in spheres and plane layers and a similar study is performed here for a spherical shell with a radius-independent central gravity subject to a destabilising temperature difference. The solid-liquid phase change is considered as a mechanical boundary condition and applies at either or both horizontal boundaries. The boundary condition is controlled by a phase change number, Φ, that compares the time-scale for latent heat exchange in the liquid side to that necessary to build a topography at the boundary. We introduce a numerical tool, available at https://github.com/amorison/stablinrb, to carry out the linear stability analysis of the studied setup as well as other similar situations (cartesian geometry, arbitrary temperature and viscosity depth-dependent profiles). Decreasing Φ makes the phase change more efficient, which reduces the importance of viscous resistance associated to the boundary and makes the critical Rayleigh number for the onset of convection smaller and the wavelength of the critical mode larger, for all values of the radii ratio, γ. In particular, for a phase change boundary condition at the top or at both boundaries, the mode with a spherical harmonics degree of 1 is always favoured for Φ ≲ 10−1. Such a mode is also favoured for a phase change at the bottom boundary for small (γ ≲ 0.45) or large (γ ≳ 0.75) radii ratio. Such dynamics could help explaining the hemispherical dichotomy observed in the structure of many planetary objects.