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Inner core translation, with solidification on one hemisphere and melting onthe other, provides a promising basis for understanding the hemisphericaldichotomy of the inner core, as well as the anomalous stable layer observed atthe base of the outer core - the F-layer - which might be sustained bycontinuous melting of inner core material. In this paper, we study in detailsthe dynamics of inner core thermal convection when dynamically induced meltingand freezing of the inner core boundary (ICB) are taken into account. If theinner core is unstably stratified, linear stability analysis and numericalsimulations consistently show that the translation mode dominates only if theviscosity $\eta$ is large enough, with a critical viscosity value, of order $310^{18}$ Pas, depending on the ability of outer core convection to supply orremove the latent heat of melting or solidification. If $\eta$ is smaller, thedynamical effect of melting and freezing is small. Convection takes a moreclassical form, with a one-cell axisymmetric mode at the onset and chaoticplume convection at large Rayleigh number. [...] Thermal convection requiresthat a superadiabatic temperature profile is maintained in the inner core,which depends on a competition between extraction of the inner core internalheat by conduction and cooling at the ICB. Inner core thermal convectionappears very likely with the low thermal conductivity value proposed by Stacey& Davis (2007), but nearly impossible with the much higher thermal conductivityrecently put forward. We argue however that the formation of an iron-rich layerabove the ICB may have a positive feedback on inner core convection: it impliesthat the inner core crystallized from an increasingly iron-rich liquid,resulting in an unstable compositional stratification which could drive innercore convection, perhaps even if the inner core is subadiabatic.

Thermal convection in Earth's inner core with phase change at its boundary