Sea ice thermodynamics

The thermodynamics of sea ice couples conductive heat transport, radiative and turbulent surface fluxes, basal ocean heat and latent heat of freezing and melting.

Melting a basal ice thickness $\Delta h$ consumes energy ${\rho }_i{L}_f\Delta h$ per unit area, ignoring sensible warming and salinity corrections. The corresponding time-averaged heat flux is this energy divided by $\Delta t$. The actual ocean heat flux also includes heat conducted into the ice.

// Target: geophysics.bigb

A freezing degree-day accumulates air temperature below a declared reference freezing temperature, with time in days and temperature difference in degrees Celsius or kelvins. The reference temperature and averaging convention must match the empirical sea ice thermodynamics formula using it. Warming by a constant $\Delta T$ subtracts $\Delta T$ times the number of days only if those days remain below the reference threshold.

This empirical sea ice growth relation gives thickness $H$ in metres from accumulated freezing degree-days $N$. Its coefficient includes the specified units and calibration. It is not a complete energy-balance model: snow, ocean heat and changing forcing can alter its applicability. Inversion gives $N=(H/0.0133{)}^{1/0.58}$.

The water-equivalent thickness expresses an ice mass per area as a depth of water at a declared reference density. With $m_i={\rho }_i{h}_i$ per area, $h_w=m_i/{\rho }_{w,\mathrm{ref}}$. Ice at $900\mathrm{kg}{\mathrm{m}}^{-3}$ and thickness $2.44\mathrm{m}$ has about $2.20\mathrm{m}$ water equivalent at a fresh-water reference density of $1000\mathrm{kg}{\mathrm{m}}^{-3}$. The depth of seawater consumed at $1025\mathrm{kg}{\mathrm{m}}^{-3}$ is instead about $2.14\mathrm{m}$. A reference depth must not replace ice thickness in an ice-volume calculation or the liquid mass removed in brine rejection.

When seawater freezes, most salt remains in liquid, increasing salinity and density and potentially driving convection.

A temperature gradient can move a brine inclusion by melting its warmer wall and freezing its colder wall. This slow process can redistribute salt, but is generally much less effective at net desalination than gravity drainage of sea-ice brine or meltwater flushing of sea ice.

// Target: geophysics.bigb

Freezing within a brine inclusion expands the solid fraction and can force concentrated liquid into neighboring pores or channels. Expulsion redistributes salt internally; net desalination requires a path crossing the ice boundary.

// Target: geophysics.bigb

Surface meltwater with a positive hydraulic head can percolate through permeable sea ice, displacing brine into the ocean. Flushing can efficiently desalinate warm summer ice, but requires meltwater and connected flow paths.

// Target: geophysics.bigb

Dense saline liquid in connected pore space can drain from sea ice while fresher seawater enters. The process is controlled by buoyancy, permeability of a porous medium and network connectivity, and is a major means of growing-ice desalination.

// Target: geophysics.bigb

If a fraction $f_i$ of initial liquid mass freezes into salt-free ice, all original salt remains in a liquid mass fraction $1-f_i$. Writing salinity as salt concentration per mass gives $S_1(1-f_i)=S_0$, hence $\Delta S=S_0{f}_i/(1-f_i)$. For equal initial cooling and subsequent freezing heat extraction, $f_i=c_{p,w}\Delta T/L_f$. The liquid specific heat capacity, rather than the ice value, determines the initial heat removal. If ice retains salt, the mass balance instead reads $S_1(1-f_i)+S_i{f}_i=S_0$.

Let liquid mass $M$ freeze incrementally, and let each new ice increment retain salinity $rS$ with constant $0\le r\le 1$, where $S$ is the instantaneous well-mixed liquid salinity. Conservation gives $d(MS)=rSdM$, hence $dS/S=(r-1)dM/M$. Integrating yields $S/S_0=(M/M_0{)}^{r-1}$. For frozen mass fraction $f$, $\Delta S=S_0[(1-f{)}^{-(1-r)}-1]\simeq (1-r)S_{0}f$. Specifying a fixed bulk ice salinity instead produces a different, though first-order compatible, finite-mass balance.

For equal heat extraction, the density increase caused by brine rejection can exceed that caused by cooling near the freezing point because the thermal density response is weak. Using conservation of salt during freezing, the ratio is $R=(\partial \rho /\partial S)S_0{f}_i/[(1-f_i)|\partial \rho /\partial T|\Delta T]$, with $f_i=c_{p,w}\Delta T/L_f$. This is a parcel comparison, not a guarantee that freezing overcomes any ambient density stratification.

For the conversion used in the exam, rejected salt increases density about one kilogram per cubic metre per practical salinity unit. The same heat loss can therefore produce much denser water through freezing than through modest sensible cooling.

The Maykut-Untersteiner model is a seasonally forced one-dimensional sea ice thermodynamics model with snow and saline ice layers. It couples radiative and turbulent surface fluxes, heat conduction, interior shortwave absorption, brine-dependent effective heat capacity and conductivity, snowfall, and a basal Stefan condition driven by prescribed ocean heat flux. Its annual sea-ice thermodynamic equilibrium is a periodic cycle with zero net yearly thickness change. The snow insulation and summer ice protection mechanisms compete, so additional snow need not reduce annual equilibrium thickness. The transient temperature profile need not be linear.

At fixed boundary temperatures, a snow cover decreases winter conductive heat loss through the resistance $h_s/k_s+h_i/k_i$ and inhibits sea ice growth. Its high albedo and persistence into summer reduce ablation. Therefore annual equilibrium thickness has competing snow sensitivities. The near cancellation at modest snow depth and thickening with persistent summer snow in the original Maykut-Untersteiner model are forcing-dependent responses, not a universal monotonically decreasing snow-thickness relation.

For annually repeated forcing, a thermodynamic equilibrium is a periodic temperature field and thickness cycle. The yearly integral of net ice growth is zero, although substantial winter growth and summer ablation can each occur. Equilibrium thickness must specify an annual mean or a phase of the cycle. A change in forcing can remove a perennial-ice solution without making every transient thickness decrease at the same rate.