Thermal convection in a spherical shell with melting/freezing at either or both of its boundaries

TL;DR

This study examines how melting and freezing at the boundaries of a spherical shell influence thermal convection patterns, revealing that boundary permeability significantly alters convective modes, with implications for planetary interiors like Earth's core and icy moons.

Contribution

It introduces a linear stability analysis of convection in spherical shells considering boundary melting/freezing, highlighting the impact of boundary permeability on convection modes and patterns.

Findings

Permeable boundaries lead to larger-scale convective modes.

Small P values cause boundary permeability effects to dominate.

Global translation mode occurs with fully permeable boundaries.

Abstract

In a number of geophysical or planetological settings (Earth's inner core, a silicate mantle crystallizing from a magma ocean, or an ice shell surrounding a deep water ocean) a convecting crystalline layer is in contact with a layer of its melt. Allowing for melting/freezing at one or both of the boundaries of the solid layer is likely to affect the pattern of convection in the layer. We study here the onset of thermal convection in a viscous spherical shell with dynamically induced melting/freezing at either or both of its boundaries. It is shown that the behavior of each interface depends on the value of a dimensional number P, which is the ratio of a melting/freezing timescale over a viscous relaxation timescale. A small value of P corresponds to permeable boundary conditions, while a large value of P corresponds to impermeable boundary conditions. The linear stability analysis predicts a significant effect of semi-permeable boundaries when the number P characterizing either of the boundary is small enough: allowing for melting/freezing at either of the boundary results in the emergence of larger scale convective modes. The effect is particularly drastic when the outer boundary is permeable, since the degree 1 mode remains the most unstable even in the case of thin spherical shells. In the case of a spherical shell with permeable inner and outer boundaries, the most unstable mode consists in a global translation of the solid shell, with no deformation. In the limit of a full sphere with permeable outer boundary, this corresponds to the "convective translation" mode recently proposed for Earth's inner core. As an example of possible application, we discuss the case of thermal convection in Enceladus' ice shell assuming the presence of a global subsurface ocean, and found that melting/freezing could have an important effect on the pattern of convection in the ice shell.