Rayleigh-B\'enard convection in a creeping solid with melting and freezing at either or both its horizontal boundaries

Solid state convection can take place in the rocky or icy mantles of planetary objects and these mantles can be surrounded above or below or both by molten layers of similar composition. A flow toward the interface can proceed through it by changing phase. This behaviour is modeled by a boundary condition taking into account the competition between viscous stress in the solid, that builds topography of the interface with a timescale $\tau_\eta$, and convective transfer of the latent heat in the liquid from places of the boundary where freezing occurs to places of melting, which acts to erase topography, with a timescale $\tau_\phi$. The ratio $\Phi=\tau_\phi/\tau_\eta$ controls whether the boundary condition is the classical non-penetrative one ($\Phi\rightarrow \infty$) or allows for a finite flow through the boundary (small $\Phi$). We study Rayleigh-B\'enard convection in a plane layer subject to this boundary condition at either or both its boundaries using linear and weakly non-linear analyses. When both boundaries are phase change interfaces with equal values of $\Phi$, a non-deforming translation mode is possible with a critical Rayleigh number equal to $24\Phi$. At small values of $\Phi$, this mode competes with a weakly deforming mode having a slightly lower critical Rayleigh number and a very long wavelength, $\lambda_c\sim 8\sqrt{2}\pi/ 3\sqrt{\Phi}$. Both modes lead to very efficient heat transfer, as expressed by the relationship between the Nusselt and Rayleigh numbers. When only one boundary is subject to a phase change condition, the critical Rayleigh number is $\Ray_c=153$ and the critical wavelength is $\lambda_c=5$. The Nusselt number increases about twice faster with Rayleigh number than in the classical case with non-penetrative conditions when the bottom boundary is a phase change interface.