Saline Stefan problem

A saline Stefan problem couples a heat equation, a solute diffusion equation, a Stefan condition, solute conservation, and a concentration-dependent liquidus at a moving solid-liquid boundary. Different thermal and solutal diffusivities produce nested boundary layers.

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.

For stationary liquid below a salt-free solid and an upward melting interface $a(t)=2A\sqrt{Dt}$, the complementary error function salt profile and flux balance $D{C}_z(a)=-\dot{a}{C}_i$ give the displayed concentration ratio. For $A>0$ it lies below one. At large $A$ it behaves as $e^{-A^2}/(2\sqrt{\pi }A)$.

When salt-free ice forms from a saline liquid, the moving phase boundary excludes the dissolved salt. With equal phase mass densities, no bulk flow, liquid salt diffusion coefficient $D$, and interface concentration $C_i$, salt conservation gives the displayed condition, where $x$ increases from solid to liquid. Positive freezing speed $\dot{a}$ therefore requires a negative liquid concentration gradient. During melting the same condition gives dilution. Coupling salt rejection, the liquidus, and the Stefan condition produces the saline Stefan problem.

For salt-free ice and a liquid concentration profile proportional to $\mathrm{erfc}[x/(2\sqrt{Dt})]$, salt rejection at $a=2\lambda \sqrt{Dt}$ gives $C_{i}F(\lambda )=C_0$, where $D$ is the salt diffusion coefficient, and $C_i,C_0$ are the interface and far-field concentrations. This follows from $-D{C}_x(a^{+})=\dot{a}{C}_i$. The function is positive and strictly decreasing for all real $\lambda$, since an integration by parts givesConsequently $C_i=C_0/F(\lambda )$ increases with $\lambda$, equals $C_0$ at zero, and is enriched for freezing and diluted for melting. The positivity proof also prevents spurious negative interface concentrations when using the complementary error function formula.

For a planar ice-ocean interface with no fluid motion, similarity places the boundary at $a(t)=2\lambda \sqrt{Dt}$. The sign of $\lambda$ distinguishes ice growth from ablation, while rejected salt during growth can cause constitutional supercooling in the adjacent ocean.