Ice layer between freshwater and cold brine ($a(t)=2A\sqrt{Dt},b(t)=2B\sqrt{\kappa t}$)

In a planar no-flow model with pure ice, a stable cold brine below freshwater can support an ice layer whose upper phase boundary freezes freshwater while its lower boundary melts into diluted brine. Heat conduction and salt diffusion obey different similarity scales. Local liquidus equilibrium, salt conservation and two Stefan conditions determine both boundary positions. The heat lost into the brine supports net ice growth even while some basal ice melts.

For an ice layer between freshwater and cold brine, set $ϵ=\sqrt{D/\kappa }$ and use solutal similarity positions $a=2{\lambda }_a\sqrt{Dt}$, $b=2{\lambda }_b\sqrt{Dt}$. A linear ice temperature, two Stefan conditions and interfacial salt conservation determine both negative interface constants. As $ϵ\to 0$ with fixed latent-to-sensible heat ratio $S$, ${\lambda }_b=O(1)$ but ${\lambda }_a=O({ϵ}^{-1})$. Freezing above occurs over the thermal diffusion scale, while dilution-driven melting below has the much shorter solutal scale.

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.