 Large latent heat in a freezing and melting ice layer

For $S=L/(c_p\Delta T)\gg 1$, $ϵ=\sqrt{D/\kappa }\ll S^{-1}$ and $R=m{C}_0/\Delta T>1$, the leading thickness is the displayed expression. The basal position is $2A\sqrt{Dt}$, with $A$ determined byOnly when $ϵSA\ll 1$ can the basal translation be omitted from the upper position. Then $A{e}^{A^2}\simeq \sqrt{\pi }RS/4$, exposing the logarithmic refinement to the scale ordering.