= Large latent heat in a freezing and melting ice layer
{title2=$h\simeq2\sqrt{\kappa t}/(S\sqrt\pi)$}
For $S=L/(c_p\Delta T)\gg1$, $\epsilon=\sqrt{D/\kappa}\ll S^{-1}$ and $R=mC_0/\Delta T>1$, the leading thickness is the displayed expression. The basal position is $2A\sqrt{Dt}$, with $A$ determined by
$$
\frac{R}{1+\sqrt\pi A e^{A^2}\operatorname{erfc}(-A)}
\simeq\frac2{\pi S}+\frac{2\epsilon A}{\sqrt\pi}.
$$
Only when $\epsilon SA\ll1$ 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.
Back to article page
