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The grounding line is the transition where an ice sheet detaches from its bed and begins to float as an ice shelf. Ice thickness there obeys the hydrostatic flotation condition.

An ice sheet is a large body of glacier ice covering land. Its mass budget combines accumulation, surface ablation, basal melting and discharge. Changes of floating ice shelves can influence discharge of grounded ice through changes in restraint, even though shelf melt itself has only a small direct sea-level contribution.

Ice-shelf buttressing is the restraint that a floating ice shelf, through its contact with sidewalls or pinning points, can exert on upstream grounded ice flow. Thinning or loss of the shelf can reduce this restraint and increase discharge, but the response depends on geometry and cannot be inferred from floating-ice area alone.

Ice-shelf water has acquired characteristic properties through interaction with the underside of an ice shelf. Melting at depth can cool and freshen the water, including cooling below the surface freezing temperature because pressure lowers the freezing point. Its eventual density and contribution to deep-water formation depend on mixing with surrounding shelf and ocean waters.

A steric sea level change is caused by a change of water volume at fixed ocean mass. Warming usually lowers seawater mass density, producing thermosteric sea level change; freshening also lowers density and produces a halosteric contribution. Both depend on the temperature, pressure and salinity dependence of the equation of state. Regional freshening or warming can therefore change local sea level even when the global ocean mass has not increased.

An eustatic sea level change concerns the ocean water amount or basin capacity. On the coming-century timescale, the principal positive mass contributions are melting mountain glaciers and ice caps, and loss of grounded ice from Greenland and Antarctica, including iceberg discharge. Terrestrial water storage also matters: groundwater depletion transfers water to the ocean, whereas reservoir filling retains it on land. Snowfall accumulation can partly compensate ice loss. Basin changes associated with geology matter over longer periods, and vertical land motion changes locally measured relative sea level. Melting floating sea ice has already largely displaced its eventual meltwater volume; its direct contribution is small compared with transfer of grounded ice into the ocean. Small density and salinity corrections do not alter that distinction.

For scale, a contemporary budget available in 2010 gave the following approximate contributions for 1993–2003, in $\mathrm{mm}\mathrm{y}{\mathrm{r}}^{-1}$:These are period-specific estimates with appreciable uncertainties, particularly for Antarctica, rather than exact closure of the budget. For comparison, over 1961–2003 the observed rate was about $1.8\mathrm{mm}\mathrm{y}{\mathrm{r}}^{-1}$ and the estimated thermal expansion contribution about $0.4\mathrm{mm}\mathrm{y}{\mathrm{r}}^{-1}$. The shorter interval need not define a permanent acceleration by itself, but the magnitudes show that thermal expansion and land-ice loss are both essential.

With continued warming, increasing ocean heat content sustains thermosteric sea level change, including after surface warming slows because the deep ocean equilibrates gradually. Increased ice ablation generally increases glacier runoff, although small glaciers eventually lose much of their available ice. Surface melt and changes in outlet-glacier flow can increase the ice-sheet contribution; ocean-driven loss of ice shelves can accelerate discharge of grounded ice. Greater snowfall provides some compensation but cannot be assumed to cancel these losses. Thus, under continued substantial greenhouse gas emissions, the global rate is likely to rise above its historical few millimetres per year, with an increasing contribution from the great ice sheets. The precise century-scale increase depends on emissions, ocean heat uptake and ice sheet dynamics; a single fixed rate extrapolated from one decade is inadequate.

For the idealized mixed-layer calculation, the stated density response givesAt fixed column mass, $\rho H=(\rho +\Delta \rho )(H+\Delta \eta )$, so to first orderThis is only the thermosteric sea level change from the prescribed layer and warming. The much larger total observed rise since the nineteenth century does not, by itself, prove deeper warming: part of that rise is eustatic sea level change. After independently allowing for the mass contribution, a larger remaining steric signal would require warming below $200\mathrm{m}$, a different temperature distribution, or corrections to the simplified equation of state. Direct temperature profiles do independently establish subsurface ocean warming. It reaches depth through winter mixing and open-ocean convection, especially in the North Atlantic and parts of the Southern Ocean; through subduction of surface waters into the interior; and through advection and turbulent mixing. Formation and spreading of intermediate and deep waters transmit surface properties far beyond the immediate region of heat uptake. Warm surface water need not simply sink locally: cooling, freshwater effects and density stratification govern the actual pathways.

Using $R=6.371\times 10^6\mathrm{m}$, the ocean area is $A_o=0.70(4\pi R^2)=3.57\times 10^{14}{\mathrm{m}}^2$. A liquid runoff volume $\Delta V=600\mathrm{k}{\mathrm{m}}^3=6.00\times 10^{11}{\mathrm{m}}^3$ givesThis is the contribution from that input, before allowing for other exchanges in the global ocean freshwater budget.

For the sea-ice dilution estimate, the ocean volume is $V_o=A_o(4000\mathrm{m})=1.428\times 10^{18}{\mathrm{m}}^3$. Let its initial mean salinity be $S_0$; the question does not specify it, so use $S_0\simeq 35$ psu for a numerical estimate. Interpret the $600\mathrm{k}{\mathrm{m}}^3$ as liquid freshwater-equivalent melt, with ${\rho }_f\simeq 1000\mathrm{kg}{\mathrm{m}}^{-3}$. Salt mass remains fixed while liquid water mass increases, givingConsequentlyApproximating the densities as equal gives $-1.47\times 10^{-5}$ psu per year. If the quoted volume instead denotes literal solid-ice volume, use ${\rho }_i\Delta V$ for the added mass; with ${\rho }_i=917\mathrm{kg}{\mathrm{m}}^{-3}$ the result is $-1.32\times 10^{-5}$ psu per year. Either convention gives the same order of magnitude. Floating-ice melt can change mean ocean salinity even though its direct sea-level effect is small: displaced volume and dissolved salt concentration are different budgets.

An observed net freshening is a statement about the sum of all freshwater exchanges, not a separate measurement of every positive contribution. Let $M$ be liquid ocean mass, $R$ net freshwater runoff from land, $I$ net global sea-ice melt minus formation, and $P-E$ net precipitation minus evaporation, all expressed as masses per unit time. The global ocean freshwater budget and salt conservation giveThe runoff term must include non-ice river input as well, or those other exchanges must already have been netted out. If the measured freshening equals that predicted from $I$ alone, positive runoff is still possible providedor an equivalent compensating term exists in the complete budget. For instance, excess evaporation can remove water at the same rate at which land-ice runoff supplies it; compensating net ice formation elsewhere must also be included if $I$ was initially only the Arctic contribution. Gross seasonal melt cannot be compared with an annual global trend without subtracting refreezing. Such compensation also means that the runoff-only sea-level increment need not equal the total retained increment. If runoff is additional, sea-ice melt is already net and global, and every other freshwater term is zero, the two contributions must add: under those stronger assumptions the alleged equality would be inconsistent. The apparent paradox is resolved by closing the complete freshwater and salt budgets.

A polynya is a persistent or recurrent area of open water or relatively thin ice within an otherwise ice-covered region. Persistence of the opening does not mean absence of ice formation: a latent-heat polynya can continuously make ice and export it. In contrast, a sensible-heat polynya remains open because ocean heat inhibits freezing. Both processes may occur at the same site.

At an Antarctic coast, cold air descending from the elevated continent produces strong katabatic winds. Their offshore component drives sea ice away from a fixed coastline, fast-ice edge or grounded-iceberg barrier. Open water exposed to the cold atmosphere loses heat rapidly; frazil ice forms, aggregates into pancake ice or consolidated new ice, and is exported offshore. The opening continues as long as export balances production. Headlands, the shelter provided by fast ice and the direction of the prevailing winds determine the location and size. Terra Nova Bay is a characteristic coastal example. Such latent-heat polynyas are ice factories rather than permanently warm holes in the ice pack.

At an ice shelf front the fixed boundary is floating shelf ice instead of land. Offshore winds blowing across the shelf can similarly remove newly formed sea ice, so an ice-front polynya need not have a different thermodynamic mechanism from a coastal one; the Ross Ice Shelf front provides an important wind-driven example. Shelf-front geometry, ice tongues, grounded icebergs and changes in calving alter the shelter and the supply of drifting pack ice. Some ice-front openings also receive significant sensible heat from currents, tidal mixing or upwelling, and heat carried out of an ice-shelf cavity can affect freezing. A sensible-heat polynya requires a sufficiently warm ocean supply, whereas the latent-heat mechanism requires continuing removal of the ice that does form. Consequently a strict identification of all coastal openings with one mechanism and all ice-shelf openings with the other would be incorrect.

The oceanic roles are substantial. Rapid heat loss and freezing produce brine rejection, raising shelf-water salinity and mass density. The resulting convection and circulation can carry dense shelf water into ice-shelf cavities; melting beneath an ice shelf modifies it into colder, fresher ice-shelf water. Dense shelf and cavity-modified waters can subsequently descend the continental slope, mix with adjacent waters and contribute to Antarctic bottom water. The outcome depends on bathymetry and the density after mixing: not all rejected brine reaches the abyss.

These localized regions can therefore have a disproportionate effect on deep-ocean ventilation, oxygen supply, carbon transport and the large-scale ocean circulation. They are also intense sites of ocean-to-atmosphere heat and moisture transfer. Seasonal sunlight arriving in open water supports high biological production where nutrients are available, with consequences for marine food webs and carbon cycling. Antarctic polynyas connect wind-driven ice export, concentrated ice production and dense-water formation; ice-shelf fronts add a floating boundary and possible cavity-related heat and freshwater exchanges.

The case for anthropogenic global warming rests on a physically consistent combination of forcing, energy storage and observed response, rather than on the claim that every unusual event has the same cause. For this examination, the observational comparison is the evidence available around 2010, and the two forecast horizons are approximately 2060 and 2110. Separating the robust mechanisms from uncertain regional details is essential to a coherent attribution of recent climate change.

The greenhouse effect follows from radiative transfer. Sunlight supplies energy, while the Earth loses energy mainly through long-wave radiation. Carbon dioxide, methane and other greenhouse gases absorb and emit in parts of that outgoing spectrum. Increasing their concentrations alters the levels and temperatures from which radiation escapes, initially reducing the net loss of energy for a given surface state. The surface and atmosphere must then warm until outgoing radiation again balances the input. Strong absorption near the centres of some bands does not eliminate the influence of additional gas: absorption in the wings and the changing height of emission still matter. For scale, the approximate carbon dioxide forcing law $\Delta F\simeq 5.35\log (C/C_0)\mathrm{W}{\mathrm{m}}^{-2}$ gives about $3.7\mathrm{W}{\mathrm{m}}^{-2}$ for doubling. It describes a forcing, not the ultimate temperature response, which additionally involves climate feedbacks and ocean heat uptake.

Measured increases in atmospheric carbon dioxide are consistent with fossil-fuel combustion and changes in land use. The carbon isotope signature and atmospheric oxygen consumption provide independent evidence of the combustion source, while the ocean and land absorb part of the emitted carbon. Methane and other climatically active gases add to the forcing. Aerosols exert partially compensating effects, so net human forcing is smaller than the sum of the warming gases alone. Uncertainty in that compensation affects the quantitative rate of warming without removing the greenhouse mechanism.

A sustained rise in global temperature is accompanied by increased ocean heat content, glacier retreat and sea level rise. This combination is harder to explain as an artefact of a few surface thermometers than a temperature series in isolation: the ocean is a large energy reservoir, and thermal expansion and melting have independent observational consequences. Internal variations of ocean circulation can move heat between regions and alter warming over particular decades, but cannot indefinitely supply a positive global energy accumulation without a change in the energy budget. Solar changes and volcanic eruptions also influence climate, and must be included in attribution; they do not reproduce the full recent warming pattern and sustained heat gain. Volcanic aerosol episodes chiefly produce temporary cooling. Broad troposphere warming together with stratosphere cooling is another useful fingerprint: additional greenhouse gases and ozone changes have a different vertical response from simply increasing solar input. No one fingerprint is unique to one cause, but their joint agreement with greenhouse forcing strengthens the attribution.

Polar observations are especially informative because they connect atmosphere, ocean and ice. In the Arctic, reduced summer sea ice extent, thinning and loss of multi-year sea ice indicate a diminished ice reservoir. Warmer air, earlier melt and delayed freeze-up contribute, while wind and current changes redistribute ice and export thick floes. Pacific heat inflow and changes in Atlantic-origin waters supply oceanic routes to melting. The protection afforded by the cold halocline means that warmer Atlantic water does not automatically heat the ice: the intervening mixing and density stratification must be considered. The 2007 retreat illustrates how long-term thinning, anomalous winds and greater solar absorption can combine. It neither proves that weather variability has ceased nor provides a reason to ignore the longer-term energy imbalance.

Loss of grounded ice has a different global implication. Increased surface melt and runoff on Greenland, and changes in outlet-glacier discharge, transfer water from land to the ocean. In Antarctica, warming of the Peninsula and disintegration of some ice shelves demonstrate strong regional change. Warm ocean water reaching shelf cavities can thin floating shelves from below; loss of ice-shelf buttressing can then speed grounded-ice discharge. The floating shelf's disappearance is not itself equivalent to adding its whole volume to sea level. The important subsequent response is increased loss from grounded ice sheets. Increased snowfall can compensate some losses, so a mass budget must distinguish accumulation, melting and ice flow.

The two polar regions should not be assumed to behave identically. Around 2010, Antarctic sea-ice changes included strong regional contrasts and an overall satellite-era behaviour quite different from Arctic summer retreat. Winds, ozone-related atmospheric changes, freshwater effects and the circulation of the Southern Ocean influence the Antarctic pattern. A large cold continent surrounded by an ocean is physically different from an ocean surrounded by land, and vigorous Southern Ocean heat uptake can delay surface warming. A regional increase in Antarctic sea ice is therefore compatible with global ocean heat gain and losses from some grounded Antarctic ice; it still needs its own dynamical and thermodynamic explanation. Credible attribution accounts for these differences instead of requiring uniform warming everywhere.

Several mechanisms produce polar amplification. The ice-albedo feedback is the most direct: retreating snow and ice expose darker surfaces that absorb more sunlight, causing further warming and melt. Much of the extra energy over newly open Arctic water is stored during summer and released to the atmosphere in autumn and winter, when an ice-covered ocean would be better insulated. Consequently the largest near-surface atmospheric response need not occur in the season of greatest solar input. Sea-ice loss changes both reflection and the timing of ocean heat release.

The atmospheric vertical structure also matters. In the tropics, convective adjustment often produces appreciable warming aloft, which increases radiative cooling. Stable polar boundary layers can concentrate warming near the surface, producing a different lapse-rate feedback and less compensating cooling from upper layers. Furthermore, the idealized Planck feedback scales as $4\sigma T^3$: colder emitting regions have a smaller increase of blackbody radiation per kelvin, although a quantitative atmospheric comparison needs full radiative transfer. Water-vapour and cloud changes modify both short-wave and long-wave fluxes, with cloud effects depending on season and surface properties. Changes in poleward heat and moisture transport provide further coupling. Thus polar amplification has several interacting causes; it is not simply the same local ice-albedo process repeated throughout the year.

Polar change affects the rest of the planet. Grounded-ice loss contributes to eustatic sea level change, while ocean warming contributes to thermosteric sea level change; together they increase coastal flooding and erosion risks. Redistribution of ice and freshwater changes density stratification and dense-water formation, potentially altering the Atlantic meridional overturning circulation and other oceanic heat and carbon transports. A weakening is physically plausible under warming and freshening, but an immediate complete shutdown is not a necessary prediction. Permafrost carbon feedback can release additional carbon dioxide and methane, increasing warming outside the Arctic. Habitat loss, changing primary production and altered food webs affect marine ecosystems and people, while a longer open-water season changes navigation, coastal exposure and resource access. These impacts are not exhausted by a single global-mean temperature number.

Forecasts require an emissions scenario as well as physical understanding. On the roughly fifty-year horizon, continued emissions imply further ocean and atmospheric warming, a longer Arctic melt season, less late-summer ice and a greater fraction of young seasonal ice. Mountain glaciers generally continue to lose mass, and sea level continues rising through thermal expansion and land-ice loss. Variability can interrupt a regional trend for several years, and the exact first nearly ice-free Arctic summer cannot be fixed from the 2007 minimum. Ocean heat storage and slow ice-sheet adjustment also mean that some further change is committed even if emissions are reduced.

By approximately 2110, sustained high emissions would imply a substantially warmer world, potentially several degrees above early twenty-first-century global temperatures, with stronger Arctic warming and likely recurrent nearly ice-free late summers. This is a seasonal prediction, not a claim that the entire Arctic Ocean must be ice-free in winter. Losses of grounded ice and thermal expansion would produce larger cumulative sea level rise, plausibly measured in several decimetres and potentially more if ice dynamics accelerate. The largest uncertainty is not whether adding heat expands water, but the future emissions trajectory and the response of major ice sheets. Complete loss of Greenland on this century-long horizon is not required by that prediction. With strong mitigation, substantially smaller late-century warming and ice loss are possible, although sea level can continue to rise beyond stabilization of surface temperature.

Anthropogenic forcing explains the broad warming signal; feedbacks amplify polar change, while emissions and ice dynamics govern future outcomes. This distinction permits both a firm physical conclusion and honest uncertainty about dates, regional details and the magnitude of rapid ice-sheet change.

Let $z$ increase upwards, so the bed is at $z=-b(x)$, the surface is at $z=h(x,t)$, and $H=h+b$. In the shallow-shelf approximation the incompressible flow has nearly depth-uniform horizontal speed $u$ and vertical strain $w_z=-u_x$. The Newtonian fluid stress tensor, together with the hydrostatic approximation, then givesThe factor four includes both horizontal extension and the pressure correction required by vertical compression. Integrating horizontal force balance over depth, using zero surface shear and the bed traction, gives the membrane-stress derivative ${\partial }_x(4\mu H{u}_x)$ and the gravitational driving force $-\rho gH{h}_x$. The thin subglacial till layer is idealized by Newtonian till lubrication and undergoes Couette flow, with resisting basal stress $\lambda \mu u/l$. Hence force balance and mass conservation yieldSince $b$ is fixed, $H_t=h_t$. No accumulation or ablation is included.

At the grounding line, the first boundary condition is ice-sheet flotation:For an unbuttressed ice shelf, the second is the extensional stress required to balance the difference between the integrated ice and seawater hydrostatic pressures. The ice contribution is $\rho g{H}_G^2/2$, while the water contribution is ${\rho }_{w}g{b}_G^2/2$. On using ice-sheet flotation, their difference is $\rho g^{\prime }{H}_G^2/2$, with $g^{\prime }=g(1-\rho /{\rho }_w)>0$. Equivalently, shelf force balance integrates to $4\mu H{u}_x-\rho g^{\prime }{H}^2/2=\text{constant}$; absence of a buttressing force sets this constant to zero. Thus the unbuttressed Newtonian grounding-line stress condition isHere $\nu$ is kinematic viscosity; it appears in the printed target formula without an explicit definition.

The ratio of longitudinal stress divergence to till drag is of order $4lH/(\lambda L^2)$. The given small-parameter limit therefore yields the friction-dominated bulk relationThe coefficient depends on the basal lubrication, and $h_x<0$ gives seaward motion. This is a bulk reduction: longitudinal stress has been neglected in the differential equation but its boundary traction remains specified. A complete uniformly valid approximation close to the grounding line could require a membrane-stress boundary layer. In the remainder, use the pointwise friction closure and retain the shelf stress condition as the boundary data, as requested in the paper.

Differentiate the ice-sheet flotation condition along the moving grounding line:Now $H_x=h_x+\alpha$. Substitution of the friction-dominated speed and the shelf stress boundary condition gives the friction-dominated grounding-line evolution law:The coefficient of ${\dot{x}}_G$ is the spatial derivative of the flotation deficit, up to sign. If it vanishes, the implicit equation remains the correct compatibility condition but division to determine a finite speed is invalid; tangential contact needs separate analysis.

In a steady state, volume flux per unit width is constant: $Hu=q_0$. The friction relation givesDifferentiating $Hu=q_0$ and using $u_x=g^{\prime }H/(8\nu )$ at the grounding line givesPut $\beta =l/\lambda$, $\mathcal{H}=H/\beta$ and $\mathcal{Q}=\nu q_0/(g{\beta }^3)$. Substitution and cancellation of $g{\beta }^5/\nu$ yield the steady friction-dominated grounding-line thickness relation:For $q_0>0$ and a bed deepening seaward ($\alpha \ge 0$), the polynomial $(g^{\prime }/8g){\mathcal{H}}^5+\alpha \mathcal{Q}{\mathcal{H}}^2-{\mathcal{Q}}^2$ is strictly increasing for $\mathcal{H}>0$, starts negative and tends to infinity, so it has exactly one positive root. At that root $\mathcal{Q}>\alpha {\mathcal{H}}^2$ and $H_x<0$, as required by steady outward extension. The relation belongs to the specified friction closure; it is not a flux law for arbitrary till rheology or arbitrary membrane-stress matching.

At the grounding line, hydrostatic flotation and continuity of ice flux giveThe normal stress and longitudinal membrane stress also match onto the floating ice shelf. For Newtonian ice this matching supplies the Newtonian grounding-line flux

The accumulation region still has $q=Ax$. In the ablation region, $q=A(2x_s-x)$, so $q(x_G)=q_G$ givesIntegrating the grounded flux law from $x_s$ to $x_G$ givesUnder the stated limit $A\ll g^{\prime }{h}_G^2/\nu$, the $h_G^4$ term is asymptotically smaller than the grounding-flux term, and henceFinally the accumulation-region profile gives $h_0^4=h_s^4+6\nu A{x}_s^2/g$, so

A shelfy stream is grounded ice with nearly depth-independent horizontal velocity. If the lubricating till has thickness $d$ and viscosity ${\mu }_t$, its simple-shear traction is approximately ${\mu }_{t}u/d$. Comparison with ${\tau }_b=\mu u/\lambda$ givesso $\lambda$ has dimensions of length.

Hydrostatic pressure contributes the depth-integrated longitudinal force $-\rho g{h}^2/2$, while the Newtonian extensional stress contributes $4\mu h{u}_x$ by the shallow-shelf approximation. Balancing the change of their sum against basal drag on a slice givesFor a freely floating ice shelf, hydrostatic flotation reduces the gravitational driving by $1-\rho /{\rho }_w$ and removes basal drag:Depth-integrated mass conservation isin the absence of accumulation or ablation.

In a thin stream with horizontal scale $L_x$, the ratio of extensional resistance to basal drag is $O(\lambda h/L_x^2)$, which is small when the thin-film condition $h/L_x\ll 1$ is combined with $\lambda \ll h_G$. For steady flux $hu=q$, neglecting extension givesAt the grounding line, flotation over a bed depth $b$ givesIntegration yieldsThe grounding-line slope is $|h_x(0)|=\mu q/(\rho g\lambda h_G^2)$. Thin-film theory therefore also requiresThis and $\lambda \ll h_G$ are compatible precisely when $\mu q/(\rho g{h}_G^3)\ll 1$.

The total horizontal force resultant in the stream isThe floating shelf equations and its ocean-front traction imply that the force it exerts at the grounding line is the ocean's hydrostatic force, soUsing flotation and the approximate stream solution,Consequentlyand the grounding-line flux-thickness relation is

A shelfy stream is a grounded, rapidly sliding ice stream whose internal resistance is dominated by longitudinal extension, as in an ice shelf, but which retains a basal drag. It is described by the shallow-shelf approximation.

For a two-dimensional Newtonian fluid ice sheet attached to an unbuttressed ice shelf, ice-sheet flotation gives $\rho H={\rho }_{w}b$. The difference between the depth-integrated ice and ocean hydrostatic pressures is $\rho g{H}^2/2-{\rho }_{w}g{b}^2/2=\rho g^{\prime }{H}^2/2$, where $g^{\prime }=g(1-\rho /{\rho }_w)$. Shelf force balance integrates $4\mu H{u}_x-\rho g^{\prime }{H}^2/2$ to a constant; absence of a buttressing force makes that constant zero. The shallow-shelf approximation therefore gives the displayed boundary condition, or $u_x=g^{\prime }H/(8\nu )$ with kinematic viscosity $\nu =\mu /\rho$.