Planetary ice shell

A planetary ice shell is a solid water-rich layer overlying an ocean or deeper planetary interior. Its thickness can be controlled by thermal conduction, basal heat flux, surface radiation, and solid-state mantle convection.

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.

The horizontal sea-ice drift momentum balance per unit area includes air and water drag forces, the Coriolis force, depth-integrated internal ice stress and the sea-surface pressure gradient. Here $m$ is ice-plus-snow mass per area and $\Sigma$ is a depth-integrated stress tensor.

// Target: geophysics.bigb

A sea-ice lead is a fracture or elongated opening in a sea-ice cover, containing open water or young thin ice. Divergence and shear create leads, while freezing and convergence close them.

// Target: geophysics.bigb

Sea-ice lead-width distributions depend on observational thresholds, resolution and whether leads are counted individually or weighted by area. Observations commonly exhibit a broad, approximately power law range, with many narrow leads and a relatively heavy tail.

// Target: geophysics.bigb

A sea-ice thickness distribution assigns fractional horizontal area to ice thickness intervals: $g(h)dh$ is the fraction between $h$ and $h+dh$. Include open water as a mass at $h=0$ when normalizing over the entire ocean area. The first moment is ice volume per ocean area. With growth rate $f$ and velocity $v$, its conservative evolution is ${\partial }_{t}g+\nabla \cdot (vg)+{\partial }_h(fg)=\psi$, with mechanical redistribution of sea ice represented by $\psi$.

// Target: geophysics.bigb

Mechanical redistribution of sea ice changes thickness-class area through ridging, rafting and lead formation. In the conservative thickness-distribution equation, local volume conservation gives $\int h\psi (h)dh=0$; area normalization including open water gives $\int \psi (h)dh=\nabla \cdot v$ when thermodynamic boundary fluxes are accounted for consistently.

// Target: geophysics.bigb

Snow ice forms when flooded snow or a seawater-snow mixture freezes. Snow loading of a thin floe can depress the ice surface below water level and initiate flooding; wave overwash is another route.

// Target: geophysics.bigb

Nilas is a thin, flexible, relatively continuous initial sea-ice sheet. Further freezing at its underside in quiet water produces congelation ice.

// Target: geophysics.bigb

The sea-ice crystal fabric describes grain shapes and crystallographic orientations. Calm-water congelation ice tends to develop vertical columns with predominantly horizontal crystal c-axes, whereas accumulated frazil ice produces a granular texture with much less systematic orientation.

// Target: geophysics.bigb

The Odden ice tongue is an intermittent extension of young sea ice into the Greenland Sea. Repeated formation and export of frazil ice and pancake ice can produce brine rejection without a steadily thickening local ice cover, increasing the salinity of the water left behind.

A melt pond is liquid water collected in a depression on melting ice. Its lower albedo than snow-covered ice can strengthen ice-albedo feedback. Pond drainage and connected pores can redistribute an oil spill under sea ice during warming.

Sea ice anchored to a coast, grounded feature or other fixed support is fast ice. It can be smooth or deformed, young or old. Currents can transport an oil spill under sea ice beneath it even though the ice itself remains stationary.

Multi-year sea ice has survived at least one summer melt season. Its age, thickness, salinity and deformation history affect how it responds to further melting and mechanical forcing. Ice age does not uniquely determine ridge shape.

First-year sea ice is ice that has formed since the preceding summer melt season and has not yet survived a summer. It is usually thinner than older ice, although deformation can create thick sea-ice pressure ridges in a young cover.

Congelation ice grows by freezing onto the underside of an existing sheet. Competitive growth selects grains whose fast basal-plane growth can advance downward, yielding columnar sea-ice crystal fabric with brine-bearing intergranular regions.

// Target: geophysics.bigb

Sea-ice draft is the depth of the ice underside below the local water surface. Total thickness also includes the part above the water. A sea-ice pressure ridge has a spatially variable draft, and its peak draft differs from its mean along a track.

A decline in Arctic sea ice can mean reduced extent, concentration, thickness, volume or survival of older ice. These measures differ: extent counts areas above a concentration threshold, whereas area weights actual coverage. Comparisons require consistent season, geographical domain and measurement method.

Loss of summer sea ice need not be irreversible at fixed external forcing. Open water can release excess heat in autumn, and thin ice grows faster than thick ice during winter because it offers less resistance to thermal conduction. Continued climate change can nevertheless shift the long-term ice cover downward. A one-summer perturbation and persistent forcing are different physical experiments.

A pressure ridge is a pile of fractured sea ice formed by horizontal convergence. Ice displaced upward forms a sail; ice displaced downward forms a keel. Newly formed ridges contain angular blocks and pores, and freezing, melting and further deformation alter their strength and shape.

A sonar ridge-identification rule can miss a shallow peak near a deeper peak because the intervening valley fails the required separation criterion. This apparent shadowing suppresses short observed spacings even without an acoustic blind zone. Finite beam width and genuine geometric occlusion can impose additional limitations. The resulting minimum resolvable spacing need not be a universal physical exclusion distance.

Here $n(H)dH$ is the expected number of ridge peaks per track length with sea-ice draft in $[H,H+dH]$. If total line density is $\mu$ and mean peak draft is $h_m>h_0$, integration gives $b=1/(h_m-h_0)$ and $B=\mu b{e}^{b{h}_0}$. The normalized peak draft has a shifted exponential distribution. This is a model over a specified draft range, not a universal description of every ridge.

For nonoverlapping triangular sea-ice pressure ridges with a common along-track side slope $\tan \delta$, a ridge whose peak exceeds $h$ contributes $2\cot \delta dh$ of track length in a draft interval $dh$. Summing over peaks proves the identity. Above the cutoff of an exponential ridge-draft model, $g(h)=A{e}^{-bh}$ with $A=2B/(b\tan \delta )$. This draft occupation is not the normalized peak probability density function; level ice and shallower parts supply the remaining probability.

Relative tangential motion across a fracture in sea ice can break and pile blocks along the fracture. Local convergence at rough contacts makes ridging possible even when the large-scale deformation is mainly shear.

Wave attenuation is a reduction of a chosen wave component as it travels through sea ice. An amplitude coefficient $\alpha$ corresponds to an energy coefficient $2\alpha$. Wave scattering at a planar interface can remove energy from the forward wave without dissipating total energy; viscous dissipation, collisions and internal ice losses irreversibly convert mechanical energy to heat.

Represent disconnected frazil ice and pancake ice as an upper layer with effective density, thickness and viscosity, rather than a continuous elastic plate. Linear Navier-Stokes equations in that layer are coupled to underlying water through velocity and stress conditions. A complex wavenumber describes both dispersion and wave attenuation in sea ice. A simpler conservative limit keeps ice mass loading but removes the elastic bending term; it does not predict viscous attenuation. The suspension model's effective viscosity is a fitted property, and a compact pancake cover need not behave like dilute frazil.

Interfaces between ice floes and open water reflect and redirect surface gravity waves. The effective attenuation depends on floe thickness, diameter, spatial arrangement and wave period. For very small diameter relative to wavelength, a floe responds nearly with the water and scatters weakly. Individual-floe interference and the number of interfaces per unit distance prevent a universal monotonic diameter law.

A polynya is a persistent or recurrent area of open water or relatively thin ice within surrounding sea ice. Wind can maintain it by transporting ice away; an oceanic heat flux can prevent freezing. Open-water width supplies an open-water fetch over which wind waves can grow.

In a uniformly wind-forced polynya with normalized heat loss $f$, let $H=L+c_{p}m{C}_{\infty }$, $K=f/H$, $J=3S/(4\rho )$ and $r=\rho -{\rho }_s$. The dilute equilibrium model gives $b{u}^2=Jx$, $\varphi bu=Kx$ and $rg\varphi b{b}^{\prime }=S/4$. Hence $u=(rgKx/\rho {)}^{1/3}$, $b=Jx/u^2$ and $\varphi =(K/J)u$. This similarity solution is valid only while the frazil ice volume fraction is small and no exported crystal flux has been neglected improperly.

The simplest Pease polynya model balances offshore ice export at speed $U$ against production of new ice at thickness rate $F$ within an open strip of width $L$. Newly formed ice is collected into an edge of prescribed thickness $H$, with negligible residence time. Per unit coastline length, conservation gives $H(U-\dot{L})=FL$. For constant positive parameters,Thus the equilibrium width is $HU/F$ and the relaxation time is $H/F$. If warming reduces $F$ while $U,H$ stay fixed, both increase. With $F\le 0$ the positive-freezing equilibrium ceases to apply. Finite frazil ice residence, variable atmospheric forcing and alongshore transport require extensions of the model.

The North Water Polynya, also called Pikialasorsuaq, lies between northern Greenland and the Canadian Arctic in the Smith Sound and northern Baffin Bay region. An upstream ice arch can restrict replenishment of drifting Arctic ice while winds and currents export newly formed ice. Ocean heat contributes in some parts of the polynya.

The Northeast Water Polynya is a recurrent open-water region off northeast Greenland. Ice barriers, local currents and winds organize ice divergence; summer opening and winter new-ice export need not have the same dominant forcing. Its regional northward coastal flow must not be confused with assuming that every current along east Greenland flows southward.

A sensible-heat polynya is maintained by ocean heat delivered through currents, mixing or upwelling, sufficient to inhibit local sea ice growth. Mechanical export and sensible heating may act together, so this classification identifies a dominant mechanism rather than an exclusive geographical class.

A latent-heat polynya remains relatively open because winds or currents remove newly formed sea ice. Sustained cooling can produce large amounts of ice even while local ice concentration remains low. Opening is maintained by export, not by latent heat warming the ocean enough to prevent freezing.

Pancake ice consists of approximately disk-shaped young ice floes, often with raised edges formed by collisions. Wave-driven collisions and motion in the surrounding frazil ice can dissipate energy.

Consolidated pancake ice forms when pancake ice packs, rafts and freezes together with interstitial frazil ice. Its heterogeneous granular structure can retain stacked layers and acquire later columnar basal growth.

// Target: geophysics.bigb

Frazil ice consists of small suspended ice crystals formed in supercooled turbulent water. A layer of crystals can damp surface gravity waves through viscous dissipation, even when individual crystals scatter very little wave energy.

Let $q$ be the depth-integrated volume flux, $E$ the entrainment rate, $R$ the upward export of pure ice, and $e=c_p(T-T_{\infty })-L\varphi$. Then $q^{\prime }=E-R$, $[\overline{C}q{]}^{\prime }=C_{\infty }E$ and $[eq{]}^{\prime }=-f-R[c_p(T-T_{\infty })-L]$. Export carries negative relative enthalpy, so it is not a heat-loss term with the same sign as atmospheric cooling. Total production is the increase in advected ice plus its exported amount. When both retained ice fraction and total processed fraction are small, equilibrium salinity and conservation of energy give production $f/(L+c_{p}m{C}_{\infty })$ even when retained and exported fractions differ.

The marginal ice zone is the transition between open ocean and compact sea ice, where interacting ice floes and surface gravity waves can dominate the mechanical evolution.

An ice-edge band is an elongated concentration of ice floes separated from neighbouring ice by open water. In one mechanism, an off-ice wind opens polynyas; short wind waves then exert wave radiation stress on their downwind floes. Wave forcing compacts floes into bands, opposed by incoming longer-period swell. Collisions and mergers can produce fewer, larger bands.

If forward-going surface-gravity-wave energy decays as $a^2(x)=a_0^2{e}^{-\kappa x}$ and local wave force is $F=C{\rho }_{w}g{R}^2{a}^2$, its spatial decrease is $-{\partial }_{x}F=C{\rho }_{w}g{R}^2\kappa a^2$. In a weak-reflection independent-layer approximation, $\kappa \simeq p{R}^2/d$. Standard deep-water wave radiation stress gives $C=1/2$, hence a compaction-force density ${\rho }_{w}gp{R}^4{a}^2/(2d)$. Different amplitude or force conventions change the coefficient; the force formula alone does not fix $\kappa$. This quantity is force per horizontal area, not automatically the three-dimensional stress in the ice skeleton.

An ice floe is a connected floating piece of sea ice. Its diameter relative to wavelength and its thickness affect the reflection, transmission and dissipation of surface gravity waves.

Sea-ice free drift neglects internal stress transmitted from neighboring ice. An isolated floe is governed by air and water drag forces, the Coriolis force and sea-surface tilt; steady drift also neglects acceleration.

// Target: geophysics.bigb

For stationary level water and wind stress approximated by $F={\rho }_a{C}_a{U}^2$ in the wind direction, equilibrium with quadratic water drag gives $F^2=({\rho }_w{C}_w{)}^2{q}^4+(mf{)}^2{q}^2$. The signed angle to the right of the wind is $\theta =\arctan [mf/({\rho }_w{C}_{w}q)]$. This neglects the floe velocity relative to the wind in the air drag; the exact relative-wind equation adds ${\rho }_a{C}_a|U-v|$ to the angle denominator.

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.

In a cycle starting at the atmospheric temperature maximum, positive thickness at the next maximum is a peak-temperature survival test. It is not sufficient for perennial ice: with sinusoidal forcing, surface melting continues for another quarter-cycle. If ocean heat was neglected during winter growth, the leading thickness at the next temperature maximum isLiteral year-round survival must be tested through the end of the subsequent melt season, rather than at the temperature maximum alone. For the specified forcing with annual mean atmospheric temperature $T_m$, the zero-ocean-heat upper bound on winter growth obeys $h_m^2/2+h_m/a=2H^2$, where $a={\lambda }_A/k$ and $H^2=\kappa \tau /\mathcal{S}$, with $\mathcal{S}$ the inverse Stefan number. Thus $h_m<2a{H}^2$, already less than the atmospheric melt over the full subsequent warm half-cycle. Positive ocean heat only strengthens the failure to produce literal perennial ice from an ice-free start.

For $T_A=T_m+\Delta T\cos (t/\tau )$ and $F_O={\lambda }_O(T_O-T_m)$, put $r=F_O/({\lambda }_A\Delta T)$. Open-water freezing requires $r<1$ and starts at $t_1=\tau \arccos (-r)$. Neglecting ocean heat during growth is a bulk approximation when $r(1+{\lambda }_A{h}_m/k)\ll 1$. It cannot hold pointwise at the onset or maximum, where the atmospheric removal and ocean input are comparable. The actual maximum satisfies $-\cos (t_2/\tau )=r[1+{\lambda }_{A}h(t_2)/k]$; neglecting $F_O$ gives $t_2=3\pi \tau /2$.

A planetary surface energy balance equates absorbed short-wave radiation and heat supplied from below with reflected short-wave radiation and emitted long-wave radiation. With incident flux $F_r$, Bond albedo $\alpha$, conductive upward flux $F_c$, and emissivity $ϵ$,

The solar constant is solar irradiance on a plane normal to incoming sunlight at a specified reference distance from the Sun, conventionally the mean Earth-Sun distance. Local horizontal irradiance includes solar geometry and atmospheric transmission.

At latitude $\varphi$ and solar declination $\delta$, integrate $S_0\max (0,\sin \varphi \sin \delta +\cos \varphi \cos \delta \cos H)$ over hour angle $H$. Sunset occurs at $H_0=\arccos (-\tan \varphi \tan \delta )$ when this lies in $[0,\pi ]$, with clipping for polar day or night. The displayed daily mean is at the top of the atmosphere; local absorbed heat flux also includes transmission and surface albedo.

Ocean heat content relative to a reference temperature $T_f$ is an integral through the water column. For a uniform column of depth $H$, $Q={\rho }_w{c}_{w}H(T-T_f)$ per horizontal area. Melting ice already at its fusion temperature requires its latent heat; warming cold ice first requires additional sensible heat. A surface temperature alone does not determine the column integral.

Surface albedo is the fraction of incident solar energy flux reflected by a surface, averaged over the specified spectrum and illumination geometry. The locally absorbed fraction is $1-\alpha$. It is distinct from geometric albedo, which measures brightness in a particular viewing configuration.

A Stefan problem is a moving-boundary problem in which a phase boundary moves as latent heat is released or absorbed. Temperature satisfies a heat equation in each phase and the Stefan condition supplies the interfacial energy balance.

A two-phase diffusion model with constant properties has similarity solutions built from the complementary error function in each half-line. Equilibrium gives $T_a=-m{C}_a$ and $C_s(a)=k_D{C}_a$. The Stefan condition and interfacial solute balance determine $\lambda$ and $C_a$; finite diffusion on both sides must be retained when constructing the narrow transition near $\lambda =0$.

For $r=-\lambda >0$ and vanishing solid-to-liquid diffusivity ratio, $H(r)=\sqrt{\pi }r{e}^{r^2}\mathrm{erfc}(-r)$ gives the interfacial liquid concentration. It increases from zero to $C_s$ as $r$ increases. Its derivative is $C_s{H}^{\prime }/(1+H{)}^2$, which decreases from $\sqrt{\pi }{C}_s$ to zero. For a direct proof, write $A=H/r=\sqrt{\pi }{e}^{r^2}\mathrm{erfc}(-r)$. Then $A^{\prime }=2rA+2$, so $A\ge \sqrt{\pi }+2r$. Differentiation gives $2H^{\prime 2}-(1+H)H^{\prime \prime }=2[(1+r^2+2r^4)A^2+r(4r^2-1)A+2r^2-2]>0$: the bracket increases with $A$, and substitution of its lower bound gives only positive polynomial coefficients, including constant $\pi -2$. Thus the concentration is strictly concave as a function of melting speed parameter $r$. Combining this with a linearized thermal balance produces one fold on the melting branch and, with the freezing branch, an interval of three possible interface speeds.

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 $1/S$.

The Gibbs--Thomson relation depresses a spherical interface's equilibrium temperature by $\Gamma /a$. For $a>a_c$, its interface is warmer than the supercooled ambient and heat loss drives growth. For $a<a_c$, its interface is colder than the ambient and heat gain drives melting. The equilibrium radius is unstable because the derivative of the quasistatic radius velocity at $a_c$ is positive.

In the quasistatic spherical model, ${\eta }_0=a_0/a_c\in (0,1)$. Separating the radius equation gives $t_e=(S/\kappa ){\int }_0^{a_0}{a}^2/(a_c-a)da$, which is finite and evaluates to the displayed expression. Near disappearance, $a^3\sim 3\kappa a_c(t_e-t)/S$. The time diverges logarithmically as the initial radius approaches the critical radius from below.

A spherical solid of radius $a$ growing inside a fixed cavity of radius $R$ expels volume at rate $4\pi (1-{\rho }_s/{\rho }_l)a^2\dot{a}$. The Darcy flux outside the cavity gives $p-p_m=\mu (1-{\rho }_s/{\rho }_l)a^2\dot{a}/(\Pi R)$. The common-pressure melting-point slope and quasi-steady spherical thermal conduction give ${\rho }_{s}La\dot{a}=k(T_i-T_{\infty })$. Define $x=a/R$, $\tau =tk(T_m-T_{\infty })/(R^2{\rho }_{s}L)$ and $K=L^2\Pi /[k{T}_m\mu (1/{\rho }_s-1/{\rho }_l{)}^2]$. Elimination yields $(x+x^2/K)x_{\tau }=1$ and the displayed zero-seed integral. The pressure rise is $p^{\ast }x/(x+K)$, with $p^{\ast }=L(T_m-T_{\infty })/[T_m(1/{\rho }_s-1/{\rho }_l)]$. Easy drainage gives square-root growth; low permeability gives an initial square-root regime followed by cube-root growth and pressure saturation. The model stops at $x=1$ and uses a uniform exterior conductivity.

A crystal dendrite is a branched crystal-growth morphology. Diffusive access to undercooling or supersaturation favours protruding tips, while interfacial capillarity regularizes fine scales and crystal interface anisotropy selects directions. Dendritic growth of a snowflake is a vapour-growth example.

For equal thermal properties and bounded travelling profiles on two deep half-spaces, the solid temperature is $T_m$ and the liquid profile is exponential. The Stefan condition requires $L=c_p\Delta T$. This balances the latent heat of solidification with warming of the initially supercooled liquid.

Retaining perturbation thermal diffusion givesThe displaced interfacial temperature and Stefan condition give the displayed implicit dispersion relation. In the capillarity-free limit its growing branch is exactly $\sigma =V\alpha$, which checks the limits of the small-wavelength thermal approximation for a solidification front.

Approximating perturbation temperatures by harmonic fields decaying as $e^{\pm \alpha \xi }$, while retaining displacement through the base gradient, produces this reduced dispersion relation. The Gibbs--Thomson relation supplies $\Gamma =\gamma T_m/(\rho L)$. This short-wave reduction suppresses perturbation advection and time dependence; its constant term and a threshold at $\alpha \kappa /V=O(1)$ are not controlled next-order results of the full diffusion problem.

The reduced dimensionless growth rate $g(q)=-1+q-2q^3/R_c$ has a positive maximum precisely when $R_c>27/2$. Its maximizing $q=\sqrt{R_c/6}$ follows by differentiation. At equality $q=3/2$, outside a strict $q\gg 1$ short-wave regime. The inequality is an exact criterion for the reduced cubic, not a quantitative onset theorem for full thermal stability of a travelling solidification front.

At zero liquid superheat, dense solid released by viscous overturn of a dense crust can sink and remain solid. For a thin crust on an asteroid of fixed radius $R$, efficient deposition and the equal-density volume approximation give $b^3-b_0^3=3R^2{J}_s(t-t_0)$. If surface gravity scales as $R$, the crustal flux scales as $R^{1/3}$ and the core radius as $R^{7/9}{t}^{1/3}$. The approximation ends when remaining liquid is exhausted or the renewed crust is no longer thin.

A growing ice-filled sphere expels liquid because ice is less dense than water. With porosity $\varphi$, $ϵ=1-{\rho }_s/\rho$, and pressure-induced freezing-point depression $T_i=T_m-\gamma p_i$, the Darcy velocity gives $p_i=\mu \varphi ϵa\dot{a}/\Pi$. Quasi-steady spherical thermal conduction gives $\varphi {\rho }_{s}La\dot{a}=k(T_i-T_{\infty })$. Hence $a\dot{a}=k(T_m-T_{\infty })/[\varphi {\rho }_{s}L(1+\Lambda )]$, where $\Lambda =k\gamma \mu ϵ/(\Pi {\rho }_{s}L)$. The interface approaches the far-field temperature at low permeability of a porous medium, while its pressure tends to a finite equilibrium value.

The Stefan condition equates latent heat generated by motion of a phase boundary to the jump in conductive heat flux across it. With interface speed $V$ and normal directed from solid to liquid, one common sign convention is

Kinetic undercooling makes the speed of a phase boundary proportional to the difference between its equilibrium and actual interface temperatures. A curvature correction raises the required undercooling for protrusions and therefore stabilizes short wavelengths.

The attachment mobility $\mathcal{M}>0$ is the coefficient converting interfacial undercooling into a solidification speed. In the linear isotropic law, $U=\mathcal{M}(T_{\mathrm{eq}}-T_i)$, and capillarity makes $T_{\mathrm{eq}}=T_m-{\gamma }_T\mathcal{K}$. Thus $T_i=T_m-U/\mathcal{M}-{\gamma }_T\mathcal{K}$. This kinetic coefficient refers to a moving phase boundary, rather than an order-parameter relaxation coefficient. The Gibbs--Thomson relation determines ${\gamma }_T=T_m\gamma /(\rho L)$ for isotropic solid-liquid surface energy $\gamma$.

Instantaneous interface kinetics is the limit of negligible kinetic undercooling at a moving solidification interface. Its temperature is set by local liquidus equilibrium and curvature-induced melting-temperature depression, without a further velocity-dependent attachment term. It does not imply that diffusion or heat transport is instantaneous.

A planar front growing into a supercooled melt can amplify corrugations because protrusions enter colder liquid. Interface curvature opposes this effect. Linearization couples the perturbation growth rate to the decaying thermal exponent ahead of the front.

Use undercooling $\Delta T=T_m-T_{\infty }$, solidification attachment mobility $\mathcal{M}$, velocity scale $U_0=\mathcal{M}\Delta T$, length $\ell =\kappa /U_0$ and time $\kappa /U_0^2$. Let $S=L/(c_p\Delta T)$ be the latent-to-sensible heat ratio, $\theta =(T-T_{\infty })/\Delta T$, and $\Gamma ={\gamma }_T/(\Delta T\ell )$. In the one-sided liquid-diffusion model, a travelling planar front has $\theta =S{e}^{-V\xi }$ and $V=1-S$. For a corrugation $\zeta e^{\sigma t+i\alpha y}$, the decaying thermal perturbation is $A_1{e}^{-p\xi }$, with $p^2-Vp=\sigma +{\alpha }^2$ and the branch $p=[V+\sqrt{V^2+4(\sigma +{\alpha }^2)}]/2$. The displaced kinetic condition gives $A_1=(SV-\sigma -\Gamma {\alpha }^2)\zeta$. The displaced Stefan condition gives $S\sigma \zeta =p{A}_1-S{V}^2\zeta$. Eliminating $A_1$ proves the displayed dispersion relation. Neglect of solid-side perturbation heat flux is part of this model.

The cubic follows by eliminating $\sigma =p^2-Vp-{\alpha }^2$ from one-sided kinetic solidification dispersion, with $V=1-S$. For $\Gamma =0$, the interface branch tends to $\sigma =SV-SV/\alpha +O({\alpha }^{-2})$. For fixed $0<\Gamma <1$, substituting $p=\sqrt{1-\Gamma }\alpha +d$ gives $2d-V=S\Gamma /(1-\Gamma )$, hence $\sigma =-\Gamma {\alpha }^2+[S\Gamma /\sqrt{1-\Gamma }]\alpha +O(1)$. At $\Gamma =1$ the balance is $p^3\sim S{\alpha }^2$, giving $\sigma =-{\alpha }^2+S^{2/3}{\alpha }^{4/3}+o({\alpha }^{4/3})$. For $1<\Gamma <1+2S/V$, $p\to S/(\Gamma -1)>V/2$ and $\sigma =-{\alpha }^2+p^2-Vp+o(1)$. Stronger capillarity can remove the real isolated interface mode from the chosen thermal branch; it does not create growth. These distinctions explain why replacing $p$ by $\alpha$ is not a controlled fixed-capillarity limit.

For $0<S<1$ and $V=1-S$, the translational branch of one-sided kinetic solidification dispersion has $\sigma =(S/V-\Gamma ){\alpha }^2+O({\alpha }^4)$. Expand $p=V+(\sigma +{\alpha }^2)/V+O({\alpha }^4)$ and substitute to prove this limit. At zero growth, $p_0=[V+\sqrt{V^2+4{\alpha }^2}]/2$, and the destabilizing right side has sign ${\alpha }^2(SV/p_0^2-\Gamma )$. For nonnegative real $\sigma$, the residual has derivative at least one, so there is one growing root precisely when $\Gamma <SV/p_0^2$. Thus instability exists exactly when $\Gamma <S/V$, with unstable band ${\alpha }^2<p_c(p_c-V)$, $p_c=\sqrt{SV/\Gamma }$ for $\Gamma >0$. At threshold the next term is $\sigma =-2S{\alpha }^4/V^3+O({\alpha }^6)$. Surface energy removes short-wave growth even when long waves remain unstable.

An icicle ripple is a corrugation of the ice-water interface produced while a thin flowing water film freezes. Coupling lubrication theory, surface tension, thermal conduction, and the Stefan condition determines its growth and migration.

For a water film of thickness $h$, lateral freezing speed $V$, ripple wavenumber $\alpha$, and $\Gamma =\gamma /(3\rho g)$, a long-wave model gives the dispersion relationThe fastest-growing mode has ${\alpha }_m=5^{-1/6}(\Gamma h{)}^{-1/3}$ and travels upward at speed $\sqrt{5}V/6$ when the vertical coordinate points downward.

A phase boundary is an interface separating distinct phases of matter. Its temperature, velocity, and flux jumps are constrained by phase equilibrium and conservation laws.

A planar freezing front can push a spherical particle steadily when repulsive sphere-plane dispersion force balances lubrication resistance. The thin parabolic gap gives hydraulic drag $6\pi \mu V{a}^2/d$, and balancing it against $Aa/(6d^2)$ gives the displayed speed. This balance assumes a nonretarded interaction, small gap and negligible inertial and other particle forces.

For a sphere partially below a planar melting isotherm, write $c=\cos {\theta }_0$ and depth $a(\cos \theta -c)$. Local premelting gives film thickness $d^3=K/(\cos \theta -c)$. Conservation in the thin spherical film gives flux $Va\sin \theta /2$ and pressure $p=3\mu V{a}^2(\cos \theta -c{)}^2/K$. Integrating vertical repulsive and hydraulic stresses over the wetted cap yields the displayed speed. The unknown thermodynamic constant $K$ cancels, but a narrow rim where the film ceases to be thin is excluded from the leading approximation.

The solid-liquid segregation coefficient is the ratio of the solute concentration in newly formed solid to that in the adjacent equilibrium liquid. A coefficient of zero means complete solute rejection: newly formed crystals are pure solvent. It does not mean the bulk solute in a mushy layer vanishes, because its pore liquid can remain concentrated.

A crystal's interfacial free energy and attachment kinetics depend on the interface normal $\mathbf{n}$. The orientation-dependent energy $\gamma$ and mobility $M$ select directions even when bulk diffusion is isotropic. This supplements the transport mechanism of crystal dendrites; isotropic transport alone does not specify the number of preferred arms.

Ordinary atmospheric snow crystals are predominantly hexagonal ice Ih. Rotating by sixty degrees around its crystallographic axis leaves the equilibrium orientation dependence unchanged, giving six equivalent basal-plane directions. Temperature and supersaturation determine whether plates, columns or crystal dendrites develop. Dendritic growth of a snowflake needs this crystal symmetry as well as the diffusion instability.

Premelting is the persistence of a liquid layer at a surface or interface below the bulk melting temperature. Molecular disjoining pressure and phase equilibrium can stabilize the film. Its hydraulic connection to a reservoir permits frost heave when water freezes onto the base of loaded ice.

Thermomolecular pressure is the temperature-dependent force per area transmitted through a thin liquid film between a solid and a substrate. For a repulsive ice-film interaction $p_T=A/(6\pi a^3)$, it can drive water through a porous material and support a load. Its thermodynamic relation to undercooling is the Clapeyron pressure relation for equal-density phases.

For equal solid and liquid mass densities, $d\mu =-sdT+dp/\rho$ and $s_l-s_s=L/T_m$ imply the displayed relation from equality of chemical potentials. $T_m$ must be absolute temperature. The phase pressures differ; this should not be confused with differentiating a coexistence curve along a common pressure.

Frost heave is displacement caused by segregated ice growth supplied by liquid flow, rather than solely by the expansion of water on freezing. A premelting film allows liquid access beneath the ice; thermomolecular pressure draws supply against viscous resistance and load.

In an equal-density, no-slip lubrication approximation, repulsive disjoining pressure $A/(6\pi d^3)$ sustains a premelting film of thickness $d$ beneath a rigid disk of radius $a$. With imposed radial undercooling $\Delta T(1-r^2/a^2)$, local thermomolecular pressure is $P_T=\rho L\Delta T(1-r^2/a^2)/T_m$, and $d^3=A/(6\pi P_T)$. A constant vertical lifting rate $U$ requires inward volume flux $q_r=-Ur/2$. Integrating the resulting lubrication pressure and disk force balance gives the displayed rate, where $W$ is the apparent load, $\mu$ is liquid dynamic viscosity, $L$ is specific latent heat, and $T_m$ is the absolute bulk melting temperature. Positive lifting requires $W<\pi a^2\rho L\Delta T/(2T_m)$. For fixed positive load, the rate rises from zero at that threshold towards $A/(6\pi \mu a^2)$ as undercooling increases. That plateau is an asymptotic limit within this constitutive model, not a claim that continuum-film physics persists at arbitrarily low temperatures.

A porous sheet of thickness $d$ and permeability of a porous medium $\Pi$ supplies a thin premelting film. The Darcy flow law equates water supply to ice growth for equal densities. With equal film and ice conductivities, negligible latent heat relative to the through-going conductive flux, and $a\ll h$, phase equilibrium gives $p_T=(\rho L\Delta T/T_m{)}^{3/4}(A/(6\pi h^3){)}^{1/4}$. The increasing gravitational load limits growth. A quasi-steady temperature field alone does not imply the negligible-latent-heat approximation.

For ${\eta }^{\prime }={\eta }^{-3/4}-\eta$, set $u={\eta }^{7/4}$ to get $u^{\prime }=(7/4)(1-u)$. The displayed solution approaches the stable equilibrium $\eta =1$. From the formal zero start it behaves as $(7\tau /4{)}^{4/7}$ at early times and $1-(4/7)e^{-7\tau /4}$ at late times. The zero-start asymptote is an intermediate reduced-model regime, since the physical thin-film approximation fails at vanishing ice thickness.