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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.

Let $h_i,h_s$ denote ice and snow thickness, let $z$ increase downwards, and count external atmospheric fluxes as positive into the surface. The Maykut-Untersteiner model is a time-dependent column model, rather than a single thickness formula: heat conduction, heat storage, internal absorption of sunlight and moving freezing/melting boundaries must be solved together. In an ice-following description its energy equation can be writtenwhere $e_i$ is specific enthalpy, $S$ is the prescribed ice salinity, and $I$ is the downward penetrating shortwave flux. The material derivative includes coordinate motion if a moving numerical grid is used. A common attenuation form is $I=I_0{e}^{-\kappa z}$ within a homogeneous ice layer, with the origin taken at that layer's top. The corresponding source is $\kappa I$. Snow-covered cases have much less penetration; absorbed surface radiation must not also be counted as internal heating.

The liquid brine fraction contributes latent heat to the effective specific heat capacity: for the illustrative liquidus approximation ${\varphi }_b\simeq -\mu S/T$ with Celsius $T<0$ in the mixed-phase range $0<{\varphi }_b<1$, $c_{\mathrm{eff}}={\partial }_T{e}_i\simeq c_0+L_f\mu S/T^2$. The precise ice conductivity and heat capacity are salinity- and temperature-dependent. This is why the full transient temperature profile need not be linear. At the snow-ice interface impose continuity of temperature and conductive heat flux; at the ice bottom impose the seawater freezing temperature $T_f$.

Write $I_0=i_0(1-\alpha )F_{\mathrm{SW}}$ and define the atmospheric input absorbed at the surface byHere $\alpha$ is the surface albedo, $\sigma$ is the Stefan-Boltzmann constant, and $F_H,F_E$ are signed sensible and latent turbulent heat fluxes. With upward conductive flux $q_c=k{\partial }_{z}T$, the cold, nonmelting surface satisfies $Q_A+q_{c,s}=0$. Once the surface reaches its melting temperature, constrain it there and use excess energy for ablation:The mass-loss rate ${\dot{m}}_s$ first removes snow at thickness rate ${\dot{m}}_s/{\rho }_s$; after the snow is gone it removes ice at rate $m_i={\dot{m}}_s/{\rho }_i$. Snow thickness also changes through prescribed snowfall and any separately represented compaction or sublimation.

At the bottom, the Stefan condition equates latent heat released by growth to upward conductive removal minus upward ocean heat flux $F_o$:Here $g_b$ is positive for basal freezing and negative for basal melting, and $L_b$ is the appropriate effective latent heat. These equations, the incident fluxes and snowfall determine the seasonal thickness trajectory. Annual sea-ice thermodynamic equilibrium means a periodic temperature field and $h_i(t+\mathcal{T})=h_i(t)$, soWith constant $L_b$, this is equivalently $\int (q_{c,b}-F_o-{\rho }_i{L}_b{m}_i)dt=0$. An “equilibrium thickness” must refer to the mean or a specified phase of this seasonal cycle.

A useful winter reduction neglects heat storage and solar penetration and treats each conductivity as constant. The two layers act as thermal resistances in series:This displays the dependence on snow and ice thickness, but is not a replacement for the full seasonally forced Maykut-Untersteiner model.

Increasing ocean heat flux subtracts directly from basal growth and increases bottom ablation. On a stable perennial branch it lowers annual equilibrium thickness, and sufficiently large input can eliminate a year-round ice cover. With zero oceanic input, removing this heat sink can allow much thicker thermodynamic ice; it does not remove the seasonal surface energy balance.

Additional annual snowfall has competing effects. Larger $h_s/k_s$ suppresses winter heat conduction and basal growth, favoring thinner ice. But persistent bright snow raises albedo, consumes energy when it melts and delays exposure of dark bare ice, suppressing summer ice ablation. The original column calculations show near compensation for moderate snowfall and substantial thickening when much more snow survives into summer. Thus annual equilibrium thickness need not decrease with snowfall, even though greater snow thickness at fixed winter temperatures always reduces the conductive growth rate. The balance also depends on snowfall timing and the treatment of snow-to-ice conversion. These seasonal responses are illustrated by the original model calculations.

Reducing surface albedo increases absorbed shortwave radiation. It promotes surface ablation, internal heating and, where sunlight reaches the ocean, basal ablation; the ice-albedo feedback reinforces the reduction in equilibrium ice thickness. Increasing albedo has the opposite effect. This sensitivity is strongest when sunlight is available and dark surfaces are exposed, rather than during polar night.

For the observed bottom melt, use ${\rho }_i\simeq 917\mathrm{kg}{\mathrm{m}}^{-3}$, $L_f\simeq 3.34\times 10^5\mathrm{J}\mathrm{k}{\mathrm{g}}^{-1}$ and approximately 122 days. The basal melt heat flux equivalent isThe integrated latent heat is about $6.1\times 10^8\mathrm{J}{\mathrm{m}}^{-2}$. This is energy used for melting, not automatically the complete ocean-to-ice heat input: the basal balance gives $F_o={\rho }_i{L}_b(-g_b)+q_{c,b}$, with sensible heat and salinity corrections if needed.

A likely major source is solar heating of an unusually large open-water fraction. Dark water absorbs much more sunlight than snow-covered ice, and warm water transfers that stored heat to floe bottoms by mixing and lateral transport. Thin or ponded ice also transmits more sunlight. Wind-driven divergence and export expose additional water, while advection of warm Pacific or Atlantic water and mixing of subsurface heat can contribute regionally. The supplied melt amount alone cannot partition these sources. The 2007 Beaufort Sea heat-budget analysis found that locally absorbed solar heat was sufficient to account for the enhanced basal melt. The latent-heat equivalent is about $60\mathrm{W}{\mathrm{m}}^{-2}$; increased solar absorption by exposed ocean is a well-supported principal explanation.

The Maykut-Untersteiner model solves a seasonally forced, one-dimensional sea ice thermodynamics problem with a snow layer above saline ice. Let $z$ increase downward from the upper surface, with snow–ice interface $z=h_s$ and ice bottom $z=h_s+h_i$. Write $q=-k{T}_z$ for conductive heat flux, positive downward, and $I(z)$ for downward shortwave radiation penetrating the material. Local energy conservation givesFor $I=I_0{e}^{-\kappa z}$, the radiation term is $+\kappa I$: absorption heats the interior. Here $c_{\mathrm{eff}}$ is the derivative of specific enthalpy with temperature at prescribed ice salinity, and includes the latent heat of changing brine fraction. Using only pure-ice specific heat capacity would omit this storage. A usual approximation has ${\rho }_i{c}_{\mathrm{eff}}={\rho }_i{c}_{i,0}+a{S}_i/T_C^2$ and $k_i=k_{i,0}+b{S}_i/T_C$, with $T_C<0$ in degrees Celsius and positive, unit-dependent constants $a,b$. Thus warmer saline ice has larger effective heat capacity and lower thermal conductivity. Layerwise constant $k_i$ gives the usual finite-layer heat equation; full differentiation retains its spatial variation. Snow uses its own density and thermal conductivity, and its stronger extinction can make shortwave penetration negligible beneath it.

Define the net atmospheric input to the upper surface bywhere $\alpha$ is albedo, $H_{\uparrow }$ is turbulent sensible heat loss and $E_{\uparrow }$ is turbulent latent heat loss. Their signs can reverse. The outgoing longwave term uses the Stefan–Boltzmann law. Subtract the penetrating shortwave $I_0$, because it heats the interior rather than the infinitesimal surface. If $M_0\ge 0$ is surface ablation speed and ${\rho }_0$ the density of the material currently exposed, the upper boundary obeysBelow the melting point, $M_0=0$ and the flux balance determines $T_0$. Once it reaches $T_m$, the positive excess produces surface ablation; snow melts first, and only afterwards does that ablation remove ice. At the snow–ice interface, absent interfacial phase change,At the base, $T_i=T_f(S_w)$ and the Stefan condition isHere $G_b$ is signed bottom growth speed, and $F_o>0$ is ocean heat flux toward the ice. An upward conductive loss $-q_b>0$ freezes ice; an excess $F_o$ melts it. For a prescribed upward turbulent ocean heat flux, the model needs no separate ocean temperature solution. A water-side closure could write $F_o={\rho }_w{c}_{p,w}{K}_w{T}_{w,z}$ at the boundary in this downward coordinate. Snowfall is an independent mass input: with snowfall mass flux $P_s$, ${\dot{h}}_s=P_s/{\rho }_s-M_s$ in a constant-density approximation. The ice thickness obeys ${\dot{h}}_i=G_b-M_i$. Compaction or refreezing, if added, requires corresponding mass and energy terms rather than counting snowfall directly as ice growth. These boundary conditions and interior equations determine temperature, surface melt and basal growth together.

Under periodically repeated annual forcing, annual sea-ice thermodynamic equilibrium is a periodic annual state, not constant thickness:Winter accretion then balances net annual ablation. “Equilibrium thickness” normally denotes a mean or a specified point in this annual cycle. Different forcing can instead eliminate a perennial-ice solution.

Increasing ocean heat flux reduces bottom growth directly in the Stefan condition, lowering equilibrium thickness and eventually removing perennial ice. The stabilizing winter response is visible in the approximate conductive fluxwhen ice thins, conductive loss and winter regrowth increase. More snow lowers this flux and inhibits winter growth. However, snow also raises albedo and delays summer exposure of dark ice. The competing snow insulation and summer ice protection effects mean an annual snow sensitivity cannot be deduced from winter insulation alone. Under the original forcing, the Maykut-Untersteiner model gives an almost flat annual-thickness response below about $0.7\mathrm{m}$ annual snow depth, followed by strong thickening as snow persists into summer. This particular transition is not a universal threshold for other forcing or snowfall timing. Increasing summer albedo reduces absorbed solar energy, reduces surface and internal melting, and increases equilibrium thickness; sufficiently low albedo removes perennial ice. The original thermodynamic sensitivity calculations establish these three responses.

The three curves are original qualitative sketches, not a new numerical integration or digitization. The snow marker refers to that historical forcing; the unnumbered loss boundaries and ordinate indicate trends rather than calibrated universal thresholds.

For the summer retreat discussed around the time of this examination, the proposed mechanisms act as follows.

These processes overlap: circulation causes export, export changes thickness and open-water fraction, and those changes alter solar absorption. Without specified regional budgets there is no defensible unique percentage partition. For enhanced Beaufort summer basal melt, upper-ocean solar heating and western-Arctic heat pathways are more directly relevant than invoking unmodified heat from a deep, isolated Atlantic layer. ice-mass-balance observations of extreme Beaufort melt identify increased solar absorption in open water as an important source.

Take ${\rho }_i=900\mathrm{kg}{\mathrm{m}}^{-3}$ as an explicit ice-density approximation and use the supplied $L_f=336000\mathrm{J}\mathrm{k}{\mathrm{g}}^{-1}$. The latent heat for $1.5\mathrm{m}$ basal ablation isJune–September contains about $122$ days, soUsing $917\mathrm{kg}{\mathrm{m}}^{-3}$ would give $43.8\mathrm{W}{\mathrm{m}}^{-2}$. This is the net flux used for melting: the gross ocean heat flux can also supply conductive losses and changes in cold content. It greatly exceeds an annual central-pack background of a few $\mathrm{W}{\mathrm{m}}^{-2}$, but is compatible with a summer upper ocean absorbing solar radiation at tens to hundreds of $\mathrm{W}{\mathrm{m}}^{-2}$. For scale, an added open-water fraction $0.3$, a shortwave input $200\mathrm{W}{\mathrm{m}}^{-2}$ and an ice–water albedo contrast $0.6$ produce $0.3\times 200\times 0.6=36\mathrm{W}{\mathrm{m}}^{-2}$ additional area-mean absorption before losses. This is an illustrative budget, not a measurement. Wind mixing and circulation can deliver stored solar or advected heat to the underside.

The ice-albedo feedback closes the warming loop: retreat exposes darker water; extra solar absorption warms the upper ocean; basal and lateral melt increase retreat. The stored heat also delays autumn freeze-up, exposing water to atmospheric heat transfer for longer. Its later release warms the atmosphere, but must not be counted as a second source in addition to the solar energy that originally entered the ocean. In winter, greater heat loss from open water and rapid growth of thin ice partly oppose the summer positive feedback.