Melting and solidification processes in the presence of convection are key to many geophysical problems. An essential question related to these phenomena concerns the estimation of the time-evolving melting rate, which is tightly connected to the turbulent convective dynamics in the bulk of the melt fluid and the heat transfer at the liquid-solid interface. In this talk, we consider a convective-melting model, constructed as a generalization of the Rayleigh-Bénard system, accounting for the basal melting of a solid. As the change of phase proceeds, an unstably-stratified fluid layer grows at the heated bottom of the system and eventually reaches a turbulent convection state. By means of extensive numerical simulations, we explore the model dynamics in two and three-dimensional configurations (fig. 84). The focus of our analysis is on the scaling of global quantities as well as on the interface morphology and the effects of space dimensionality. We find that independently of dimensionality the convective-melting system behavior shares strong resemblances with that of the Rayleigh-Bénard one, and that the heat flux is only weakly enhanced with respect to that case. Such similarities are understood, at least to some extent, considering the resulting slow motion of the melting front (with respect to the turbulent fluid velocity fluctuations) and its generally little roughness (compared to the height of the fluid layer). Varying the Stefan number, accounting for the thermodynamical properties of the material, also seems to have only a mild effect, which implies the possibility to extrapolate results in numerically delicate low-Stefan setups from more convenient high-Stefan ones [1]. Finally, we discuss possible extensions of the study to geophysically relevant problems such as the modeling of the evolution of melt-ponds at the surface of Arctic sea-ice.