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Measure $z$ upward from the heat exchanger and let the steady ice front be at $z=h>0$. In the exchanger frame, salt in the liquid satisfies the advection-diffusion equationThe decaying solution and the prescribed total salt mass aresoThe linear liquidus condition fixes the interface temperature asA solid layer between the exchanger and the interface can therefore exist only if

Write $\vartheta =T-T_{\infty }$. Heat advection, conduction, and environmental loss giveIts characteristic exponents areBelow the exchanger and above the ice front, boundedness givesIn $0<z<h$, the ice temperature iswhereSubstitution of these fields into the Stefan conditiongives one scalar equation for the steady height $h$, which can be solved numerically. The heat flux may jump at $z=0$ because the exchanger supplies the required localized cooling.

The local equilibrium freezing temperature ahead of the front isBecause $T_l(h)=T_L(h)$, constitutional supercooling begins when the actual liquid-temperature gradient at the interface is smaller than the liquidus gradient:Using the solutions above, this is

At criticality the inequality is an equality. If $F/k\ll V^2/{\kappa }^2$, then $q\sim V/\kappa$ and, for $\kappa >D$,The critical curve is proportional to $S^{-1}$. If $F/k\gg V^2/{\kappa }^2$, put $q\sim \sqrt{F/k}$ to obtainThis curve behaves as $S^{-1/2}$ for small $S$ and approaches $D\sqrt{F/k}$ for large $S$. Supercooling occurs above the corresponding critical curve, where solute rejection steepens the liquidus faster than heat transport raises the actual temperature.

In the frame of a planar interface moving at speed $V$, the steady liquid temperature obeys $-V{T}_z=\kappa T_{zz}$ and approaches $T_{\infty }$. ThusThe Stefan condition, with negligible solid-side heat flux, givesso $T_i-T_{\infty }=L/c_p=S\Delta T$. The kinetic undercooling law then giveswhich requires $S<1$ for advancing solidification.

Scale length with $\kappa /V$, time with $\kappa /V^2$, and temperature with $\Delta T$. The base liquid temperature is ${\Theta }_0=S{e}^{-z}$. For an interface displacement $\zeta e^{i\alpha x+\sigma t}$, write the thermal perturbation asThe heat equation givesLinearizing the Stefan condition giveswhile the kinetic law with stabilizing curvature undercooling givesEliminating $A$ gives the general implicit dispersion relation

At marginal stability $\sigma =0$,and the dispersion relation becomesSince ${\alpha }^2=\lambda (\lambda -1)$ on this curve, the right-hand side is simply $\Gamma {\lambda }^2$. A positive cutoff wavenumber therefore exists exactly whenThe unstable band is $0<\alpha <{\alpha }_c$, whereA sketch of $\Gamma {\lambda }^2$ against the constant $S/(1-S)$ shows no crossing above $\alpha =0$ below threshold and one cutoff above it. As $S\uparrow 1$, ${\lambda }_c$ and ${\alpha }_c$ diverge like $(1-S{)}^{-1/2}$, so the range of the morphological instability of a kinetically limited solidification front becomes unbounded.

Write the base-state gradients asThe base heat flux and Stefan condition relations are

Under the quasi-stationary approximation, and with advective heat transport neglected, each temperature perturbation satisfies the Laplace equation. The decaying normal-mode solutions areKeeping the displaced ice interface at $T_m$ omits the curvature correction described by the Gibbs--Thomson relation.
The fixed melting temperature at the displaced ice interface givesExpanding temperature and conductive heat flux continuity at the displaced outer interface gives

Put $r=k_a/k_w$. Since $G_a=G_w/r$, the temperature condition in the limits $r\ll 1$ and $\alpha h\ll 1$ givesAlthough $r$ is small, the product $rC$ is therefore order one and cannot be discarded. The flux condition then givesThis is why the stated asymptotic warning matters.

The perturbed Stefan condition at $x=0$ isUsing $B\sim {\eta }_2{G}_w$ and $-k_w{G}_w={\rho }_i{L}_{f}V$ yieldsSubstitution of the interface-amplitude ratio from part i produces the thin-film icicle-ripple instability dispersion relation

Set $b=\Gamma h{\alpha }^3$. Rationalizing this complex number givesso its growth rate and imaginary part areDifferentiating the growth rate with respect to the wavenumber shows that its positive stationary point satisfiesIt is the unique global maximum, and henceAt this wavenumber $b=1/\sqrt{5}$. With the convention $e^{i\alpha z+\sigma t}$, a constant phase travels with phase velocityThereforeBecause $z$ increases downward, the negative sign means that the icicle ripples migrate upward with speed $\sqrt{5}V/6$.

The unperturbed temperature is linear in each material. Continuity of the conductive heat flux at $x=h$ givesOn definingthis becomes $T_m-T_h=ϵ(T_h-T_a)$. Therefore

Let ${\rho }_i$ be the ice mass density and $L_f$ its latent heat of fusion per unit mass. The ice is isothermal in this model, so the Stefan condition equates latent-heat production to the heat flux conducted through the water:Consequently the unperturbed lateral solidification speed isThe material parameters are the thermal conductivities $k_w,k_a$, ice density ${\rho }_i$, and specific latent heat $L_f$; the geometric thermal length is $\delta$.

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.

Conservation of energy and Fourier's law giveAt steady state $k{T}_z=F_c$ is constant. Since $k=a/T$,The two temperature boundary conditions therefore giveAt the basal phase boundary, the Stefan condition isThe first term removes latent heat released by freezing, while the oceanic flux $F_w$ supplies heat to the interface. A steady shell has $\dot{h}=0$, hence $F_c=F_w$ andHere $T_s$ must simultaneously satisfy the surface balance from part a with $F_c=F_w$.

Take $z$ vertically downward from the cold surface. The ice occupies $0<z<h(t)$ and the water occupies $z>h(t)$. The temperature satisfies the heat equationin both phases, withThe remaining interfacial condition is the Stefan conditionA schematic should show the cold surface, the descending phase boundary, the ice temperature rising from $T_b$ to $T_m$, and the water temperature tending from $T_m$ to $T_{\infty }$.

The diffusion length suggests the similarity variableThe Neumann solution of the Stefan problem, written using the error function, isThese expressions satisfy all four thermal boundary conditions. At the interface,Since $\dot{h}=\lambda \sqrt{\kappa /t}$ and $k=\rho c_p\kappa$, the Stefan condition becomesThe second term represents heat supplied from the warm water and therefore reduces the freezing rate.

At times long compared with the encounter time at depth $H$, the fixed depth $H$ is negligible relative to the growing diffusion lengths. Salt obeys the diffusion equation with diffusivity $D$. Withthe self-similar concentration profile in the liquid isIt has $C(h,t)=C_i$ and tends to $C_0$ in the far field.

Because the solid contains no salt, conservation of solute at the moving boundary requires the rejected solute flux from Fick's first law to equal the rate at which the interface sweeps up salt:Substituting the similarity profile givesThis relation determines $C_i$ if $\lambda$ is already known. More generally it must be solved together with the thermal Stefan condition and the interfacial phase-equilibrium relation $T_i=-m{C}_i$ from the liquidus. For $ϵ\ll 1$, salt occupies a much thinner boundary layer than heat and $C_i$ can be much larger than $C_0$.

For a one-phase spherical Stefan problem with melting-temperature boundary value $T_s-\Gamma /a$ and supercooling $\Delta T=T_s-T_{\infty }>0$, use the latent-to-sensible heat ratio $S=\mathcal{L}/(c_p\Delta T)$. Large $S$ separates heat diffusion from slow interface motion. The exterior harmonic temperature is $T=T_{\infty }+(\Delta T-\Gamma /a)a/r$. The Stefan condition gives the displayed radius equation with $a_c=\Gamma /\Delta T$. This is the large inverse-Stefan number convention; the common sensible-to-latent Stefan number is $1/S$.

For a sublimating surface with exponentially absorbed volumetric heating, use a frame moving into the solid at speed $V$. Its steady heat equation is $k{T}^{\prime \prime }+\rho c_{p}V{T}^{\prime }+q_0{e}^{-\lambda z}=0$. The surface Stefan condition is $k{T}^{\prime }(0)=\rho LV$. Integrating over the half-space gives the displayed energy-limited speed. The surface gradient is positive, but the far field is colder, so the temperature has a subsurface maximum where melting can begin.

For a well-mixed lava lake of depth $H$ and a thin conductive crust of thickness $a$, put $\Delta T=T_m-T_s$, and use the convective flux $K(\overline{T}-T_m)$ with constant heat transfer coefficient $K>0$. The Stefan condition and liquid heat balance give, to leading order in $a/H$,Here latent heat $L_f$ is per unit mass and $k=\rho c_p\kappa$. Crust heating and changes of liquid depth are neglected. The linear conductive profile uses a quasi-steady approximation; thinness alone does not justify neglecting crust thermal storage. During initial growth, a small Stefan number supplies a sufficient rapid-adjustment regime. Define $h=a/H$, $\theta =(\overline{T}-T_m)/\Delta T$, and $\tau =\kappa t/H^2$. Scaling by the thermal diffusion time $H^2/\kappa$ and $\Delta T$ gives $h^{\prime }=S(1/h-B\theta )$ and ${\theta }^{\prime }=R-B\theta$, where $B=KH/k$, $R=Q{H}^2/(c_p\kappa \Delta T)$, and $S=c_p\Delta T/L_f$ is the Stefan number. Thus $\theta =R/B+({\theta }_0-R/B)e^{-B\tau }$ exactly. The reduction applies while $a/H\ll 1$, the conductive profile is approximately linear, and the thermal convection closure remains valid; it does not model complete freezing of a deep lake.

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.

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.

Sea ice is frozen ocean water. A local thermodynamic description uses thermal conduction, a surface energy balance and the Stefan condition; salinity, mechanical transport and fractional ice coverage add further physics.

A single-category model represents local ice by one thickness $h$ and one surface temperature $T_s$. With basal freezing temperature $T_m$, atmospheric temperature $T_A$, thermal conductivity $k$ and atmospheric heat transfer coefficient ${\lambda }_A$, quasistatic thermal conduction giveswhile $T_s<T_m$. If $F_O$ is upward ocean heat flux, the Stefan condition gives $\rho L\dot{h}={\lambda }_A(T_m-T_A)/(1+{\lambda }_{A}h/k)-F_O$. During surface melting, $T_s=T_m$ and $\rho L\dot{h}=-{\lambda }_A(T_A-T_m)-F_O$ when $T_A\ge T_m$. Thickness is constrained to be nonnegative, and the open-water energy balance applies after complete melt.

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.