Planetary ice shell

A planetary ice shell is a solid water-rich layer overlying an ocean or deeper planetary interior. Its thickness can be controlled by thermal conduction, basal heat flux, surface radiation, and solid-state mantle convection.

A planetary surface energy balance equates absorbed short-wave radiation and heat supplied from below with reflected short-wave radiation and emitted long-wave radiation. With incident flux $F_r$, Bond albedo $\alpha$, conductive upward flux $F_c$, and emissivity $ϵ$,

A Stefan problem is a moving-boundary problem in which a phase boundary moves as latent heat is released or absorbed. Temperature satisfies a heat equation in each phase and the Stefan condition supplies the interfacial energy balance.

The Stefan condition equates latent heat generated by motion of a phase boundary to the jump in conductive heat flux across it. With interface speed $V$ and normal directed from solid to liquid, one common sign convention is

A phase boundary is an interface separating distinct phases of matter. Its temperature, velocity, and flux jumps are constrained by phase equilibrium and conservation laws.

The Neumann solution places a planar phase boundary at $h(t)=2\lambda \sqrt{\kappa t}$. Similarity reduction of the heat equation gives temperature profiles in terms of the error function, while the Stefan condition determines $\lambda$ implicitly.

Constitutional supercooling occurs ahead of a solidification front when rejected solute raises the local liquidus above the actual liquid temperature. The planar front can then become morphologically unstable because a displaced piece of solid enters supercooled liquid.