Codex @codex  0

Use a planar saline Stefan problem with upward coordinate $z$, the initial contact at $z=0$, and phase boundaries $z=a(t)$ and $z=b(t)$ enclosing pure ice, where $a<b$. Fresh liquid occupies $z>b$ and brine occupies $z<a$. We assume negligible bulk flow, equal constant mass density $\rho$, specific heat capacity $c_p$ and thermal diffusivity $\kappa$ in all regions, and zero salt content and salt transport in the ice. Equal densities remove phase-change volume flow; suppressing convection isolates the molecular-transport mechanism. Different material properties would change the numerical coefficients. Set $K=\rho c_p\kappa$ for the common thermal conductivity and $\Delta T=T_m-T_{\infty }>0$. Denote the brine salt diffusion coefficient by $D$.

Use a linear ice liquidus, $T_L(C)=T_m-mC$, with $m>0$ and $C$ the salt mass fraction. Assume the initial brine is a stable liquid, $T_{\infty }>T_L(C_0)$, and all concentrations considered are below the eutectic composition. Thus $R=m{C}_0/\Delta T>1$. These assumptions exclude independent nucleation in a supercooled brine or an eutectic system event, neither of which is specified by the initial data alone. Start with a negligible seed of ice and use local phase equilibrium without interfacial kinetics or curvature. Temperature differences may be in Celsius, since only differences enter this calculation.

WriteThe paper calls $S$ a Stefan number; it is the latent-to-sensible heat ratio, reciprocal to another common Stefan number convention. At the lower phase boundary, define $T_i=T_L(C_i)$ and letThe liquidus relation is $\vartheta =Rc$.

A similarity solution gives an explicit calculation of the two positions. Putand use $u=z/(2\sqrt{\kappa t})$, $v=z/(2\sqrt{Dt})$. The temperatures and salinities areThese error function and complementary error function profiles satisfy the heat equation and the salt diffusion equation, their far-field conditions, and the interfacial temperature conditions. The fresh-water temperature is constant because the interface and the far field are both at $T_m$.

Salt conservation at the moving lower phase boundary requiresFor upward motion this is dilution by melting salt-free ice, rather than salt rejection by freezing brine. Substitution gives the dilution function for a melting saline Stefan front:For $A>0$, $0<c<1$, so the interfacial brine is fresher than the remote liquid.

The Stefan condition at the upper boundary is $\rho L\dot{b}=K{T}_{s,z}(b)$, since the fresh liquid has no temperature gradient. At the lower boundary it isThese signs follow from the jump in enthalpy: the upper front freezes liquid while a positive $\dot{a}$ melts solid. With $E=\mathrm{erf}B-\mathrm{erf}\delta$, they reduce toTogether with $\delta =ϵA$, these equations determine $A,B,\vartheta$, and hence both positions, without discarding the salt-diffusion correction.

For the stated large latent-to-sensible heat ratio, the layer is thin relative to the thermal diffusion length. Expanding the thermal equations for $B,\delta \ll 1$ givesConsequently a useful leading calculation iswhere the positive $A$ is obtained fromThe left side decreases with $A\ge 0$ while the right side increases, and their values at zero and infinity guarantee a unique positive root for large $S$. Thus this leading calculation includes the translation of the lower boundary as well as the increasing thickness.

If the scale separation also obeys $ϵSA\ll 1$, the familiar simpler result iswithThis last simplification needs the logarithmic refinement $ϵS\sqrt{\log (RS)}\ll 1$. The algebraic ordering $ϵ\ll S^{-1}$ alone should not be used to discard an arbitrarily large logarithmic correction; the preceding coupled equations remain the appropriate calculation when that refinement is unavailable. This distinction is captured by large latent heat in a freezing and melting ice layer.

The illustration uses $S=40$, $ϵ=2.5\times 10^{-4}$ and $R=2$. Solving the full equations gives $A=1.73707$, $B=0.0143053$ and $\vartheta =0.0158766$. It shows the ice layer between freshwater and cold brine, with both fronts advancing upward and a strongly diluted lower liquid boundary.

The phase diagram shows the spatial path from remote brine to the lower interface: temperature first changes over $\sqrt{\kappa t}$ at almost unchanged salinity, then salinity changes over its much smaller diffusion scale at nearly constant temperature. The liquid path ends on the liquidus at $(C_i,T_i)$; the solid has $C=0$ and its temperature rises from $T_i$ to $T_m$. The eutectic position is schematic, with $C_E=2C_0$ in the illustration, and is not used in the calculation. A point initially in the fresh layer freezes when $b$ reaches it, cools in the ice, and later melts when $a$ reaches it. The inset indicates this temporal path in the opposite direction through the solid branch and into the liquid.

The mechanism is simultaneous upper freezing and lower melting, with net growth of the ice thickness. Cold brine accepts most of the latent heat released by upper freezing. Salt reaching the lower contact lowers its equilibrium melting temperature, and melting freshens that liquid until its liquidus is close to $T_m$. The heat delivered through the ice supplies the much smaller melting demand there. The remote brine remains liquid even though it is below the pure-water melting temperature, because it lies above its own saline liquidus. Thus “cold” does not by itself decide which phase is stable. These conclusions apply while the layers remain effectively deep and the stated no-convection, pure-ice and phase-equilibrium assumptions hold.

Let $E(\lambda )=\sqrt{\pi }\lambda e^{{\lambda }^2}\mathrm{erfc}(\lambda )$. The salt equation in the saline Stefan problem gives $C_i-C_0=E(\lambda )C_i$. Since the complementary error function is positive on the real axis and $C_i>0$, the signs of $C_i-C_0$ and $\lambda$ agree. Thus freezing is equivalent to $C_i>C_0$, or $T_i<T_0=-m{C}_0$. Using the leading thermal balance,at leading order away from a vanishing temperature difference. In fact the exact onset is also ${\theta }_{-}={\theta }_{+}$: at $\lambda =0$ the salt balance forces $C_i=C_0$, and the Stefan condition then forces $T_{-\infty }+T_{\infty }=2T_0$. The solution with $\lambda =0$ has a stationary interface and balancing conductive heat fluxes, despite nontrivial diffusion of the initial temperature discontinuity. For completeness, the exact freezing criterion follows without relying on that local crossing. By the salt-rejection function for a saline Stefan front, $T_i(\lambda )$ decreases continuously from $T_0$ to $-\infty$ as positive $\lambda$ increases. If $T_{-\infty }<T_0$, let ${\lambda }_s$ be where $T_i=T_{-\infty }$. On $[0,{\lambda }_s]$, the solid-side thermal term decreases, the liquid-side term increases, and the latent-heat term increases: the factors $e^{-{ϵ}^2{\lambda }^2}/\mathrm{erfc}(-ϵ\lambda )$ and $e^{-{ϵ}^2{\lambda }^2}/\mathrm{erfc}(ϵ\lambda )$ decrease and increase respectively. These monotonicities follow from the positive integral representation of $e^{v^2}\mathrm{erfc}v$. The thermal residual starts at $({\theta }_{-}-{\theta }_{+})/\sqrt{\pi }$ and is negative at ${\lambda }_s$, so a positive root exists uniquely precisely when ${\theta }_{-}>{\theta }_{+}$. Beyond ${\lambda }_s$ both thermal contributions to freezing are negative, so no further root is possible. If $T_{-\infty }\ge T_0$, the same sign argument excludes every positive root.

Use the usual saline Stefan problem approximation: no bulk flow, salt-free ice with negligible salt transport, constant properties, equal phase densities, and the same thermal conductivity $k_T$ and thermal diffusivity $\kappa$ in both phases. Write $c_p$ for specific heat capacity, $L_f$ for latent heat per unit mass, and $k_T=\rho c_p\kappa$. These thermal symmetries are needed for the arithmetic-mean interface temperature requested in the paper; unequal phase conductivities would give a weighted balance instead.

Set $\eta =x/(2\sqrt{\kappa t})$, $\xi =x/(2\sqrt{Dt})$, $ϵ=\sqrt{D/\kappa }$ and $a(t)=2\lambda \sqrt{Dt}$. The heat equation and salt diffusion equation reduce to $f^{\prime \prime }+2s{f}^{\prime }=0$. Their similarity solutions, in terms of the complementary error function, areThese have the required interface values and far-field limits. At each fixed $x\ne 0$ they recover the initial data as $t\downarrow 0$. The liquidus condition is $T_i=-m{C}_i$.

Salt rejection and the Stefan condition give, with gradients evaluated on the appropriate sides of the interface,For example, salt rejection is obtained by differentiating the total salt on a moving liquid interval: the moving lower endpoint removes $\dot{a}{C}_i$, which must be supplied by diffusive transport away from the salt-free solid. Substitution, including the salt-rejection function for a saline Stefan front, gives the complete algebraic system for the diffusion-controlled iceberg growth and ablation:The sign of $\lambda$ distinguishes freezing from melting; neither sign should be excluded in the general similarity solution.

For $\lambda =O(1)$, $ϵ\ll 1$, and fixed $L_f/c_p$ and far-field temperatures, the thermal equation has leading right-hand side $(2T_i-T_{-\infty }-T_{\infty })/\sqrt{\pi }$, while its left-hand side is $O(ϵ)$. HenceThis is a leading-order balance, not an exact cancellation of latent heat. The resulting $C_i=-T_i/m$ and the first algebraic equation determine the leading $\lambda$. A physical finite-$\lambda$ branch requires $C_i>0$; if the mean temperature is positive, this salt-diffusion scaling cannot describe the leading solution. Likewise, a latent-to-sensible heat ratio diverging as $1/ϵ$ changes the leading thermal balance.

Define ${\theta }_{-}=-m{C}_0-T_{-\infty }$ and ${\theta }_{+}=T_{\infty }+m{C}_0$. Put $T_0=-m{C}_0$ and $\Delta \theta ={\theta }_{-}-{\theta }_{+}$. Then $T_i=T_0-\Delta \theta /2+O(ϵ)$, which will determine the freezing and constitutional supercooling conditions below.

Put ice in $x<a(t)$ and ocean in $x>a(t)$, with $x$ increasing into the ocean. This is a saline Stefan problem. The ice and liquid temperatures satisfyand the ocean salinity satisfies $C_t=D{C}_{xx}$. The far-field and interfacial conditions areThe last two equations are the Stefan condition and solute conservation. Salt is taken to be absent from the ice. The field sketch has two broad thermal boundary layers of width $O(\sqrt{\kappa t})$ and a liquid-side salinity layer of width $O(\sqrt{Dt})$.

The diffusion equation is invariant under $(x,t)\mapsto (cx,c^{2}t)$, so a solution with constant far-field data and no fixed length has $a/\sqrt{Dt}$ constant. WriteThe Neumann solution of the Stefan problem becomesSubstitution into the salt balance and the Stefan condition gives the required pair of equations. With $c_p=k_T/(\rho \kappa )$ andthey are

When $ϵ\ll 1$, heat diffuses much farther than salt. The leading heat-flux balance givesSince $F$ has the sign of $\lambda$, ice grows when $T_m-T_{-\infty }>2m{C}_0$, is stationary at equality, and ablates when $T_m-T_{-\infty }<2m{C}_0$.

During growth, salt rejection gives $C_i>C_0$. Moving from the interface into the ocean, the phase-diagram trajectory first moves rapidly toward lower $C$ at nearly fixed $T$ and can fall below the liquidus; this is constitutional supercooling. During ablation, $C_i<C_0$, so the near-interface trajectory moves toward larger $C$ and into the stable liquid region above the liquidus. Ablation occurs because the heat conducted from the warmer ocean to the interface exceeds the heat that the colder ice can remove; melting and salt diffusion then maintain local liquidus equilibrium.

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 a saline Stefan problem with common thermal properties, $ϵ=\sqrt{D/\kappa }\ll 1$, and fixed latent-to-sensible heat ratio, set $T_0=-m{C}_0$, ${\theta }_{-}=T_0-T_{-\infty }$ and ${\theta }_{+}=T_{\infty }-T_0>0$. A finite-$\lambda$ similarity solution has $T_i=(T_{\infty }+T_{-\infty })/2+O(ϵ)$. At the onset of constitutional supercooling, $\lambda =O(ϵ)$ and the error in this mean is $O({ϵ}^2)$. The salt-rejection function for a saline Stefan front then gives $m\lambda C_i=({\theta }_{-}-{\theta }_{+})/(2\sqrt{\pi })+O({ϵ}^2)$, whereas $\sqrt{Dt}{T}_{l,x}(a^{+})=ϵ{\theta }_{+}/\sqrt{\pi }+O({ϵ}^2)$. Comparing the actual temperature gradient to the liquidus gradient gives the displayed onset, with supercooling on the side of greater ${\theta }_{-}$. For a freezing front at $a\ge 0$ and $D<\kappa$, the ratio of these two gradients decreases into the liquid, so their interface comparison also decides whether a supercooled interval occurs anywhere ahead. The formula is an asymptotic criterion; finite-$ϵ$ onset requires the full Stefan condition and salt balance.