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A glacier is a persistent mass of land-based ice that deforms and flows under gravity. Ice accumulation, ice ablation, internal deformation and basal sliding determine its geometry and motion.

Ice accumulation is the local gain of glacier or ice-sheet material, expressed in thickness per unit time after converting to ice-equivalent volume. The net accumulation rate subtracts ice ablation.

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.

Take $z$ upward from the horizontal bed and measure $x$ away from an ice divide. In the shallow-ice approximation, the hydrostatic pressure is $p=p_a+\rho g(h-z)$ and the horizontal balance for a Newtonian fluid isThe upper stress-free boundary condition gives $u_z(h)=0$. The resisting basal traction is $-\mu \beta u_b$, so the fluid-side shear stress satisfies $\mu u_z(0)=\mu \beta u_b$. This is a Navier slip boundary condition with slip length ${\ell }_s=1/\beta$. Integrating twice givesThe first term is the uniform basal sliding contribution, while the quadratic term is internal viscous deformation. Both are positive where the thickness decreases downstream.

The shallow-ice flux with linear basal drag isNo ice accumulation or ice ablation appears. For a symmetric glacier, work on one half, $0\le x\le x_N(t)$, with $q(0,t)=0$, $h(x_N,t)=0$ and no outgoing volume flux. Its conserved half-volume per unit width is $V_g={\int }_0^{x_N}hdx$; the full glacier has volume $2V_g$. This specifies the volume convention rather than silently supplying an unspecified value. For a general finite release on the whole line, the late similarity solution is centred on its conserved centre of mass and uses half its total volume for $V_g$.

The internal-shear and linear basal drag law mobilities are equal at $H_{\ast }=3/\beta$. Suitable vertical, horizontal and temporal scales areWrite $h=H_{\ast }\mathcal{H}$, $x=L_{\ast }X$, $t=T_{\ast }\tau$. ThenThe initial typical thickness is $\mathcal{H}\simeq 10$, the initial extent is of order $L_{\ast }/10$, and its shear-controlled spreading time is of order $T_{\ast }/10^5$. The shallow-ice approximation also requires small aspect ratio: in particular the initial thickness and extent must satisfy approximately $100H_{\ast }/L_{\ast }\ll 1$. The equations model the broad shallow bulk; the very tip can require physics beyond that approximation.

The two limiting equations are porous medium equations of the form ${\mathcal{H}}_{\tau }=({\mathcal{H}}^n{\mathcal{H}}_X{)}_X$, with $n=3$ for a thick shear-dominated glacier and $n=2$ for a thin sliding-dominated glacier. Volume conservation and balance of the time derivative give exponent $\alpha =1/(n+2)$. Set $\mathcal{H}={\tau }^{-\alpha }f(\eta )$, $\eta =X{\tau }^{-\alpha }$. One integration, using zero volume flux at the ice divide, gives $f^n{f}^{\prime }=-\alpha \eta f$, henceThe planar volume-conserving nonlinear-diffusion similarity is thereforeThe Beta function fixes its constants:Consequently the explicit early shear and late sliding limits areTheir centre depths are $A_3{\tau }^{-1/5}$ with $A_3\simeq 0.842252$, and $A_2{\tau }^{-1/4}$ with $A_2=\sqrt{2/\pi }$. Thus the early extent grows as $t^{1/5}$ and the late extent as $t^{1/4}$. These are asymptotic regime profiles; they solve the respective limiting equations, not the full sum of both mobilities. For arbitrary finite-width initial data they are appropriate after the corresponding spatial relaxation, with a virtual time origin depending on that data.

The shear-to-slip transition in a shallow ice current occurs at thickness of order $H_{\ast }$. Using equality at the centre of the early similarity solution givesIf the initial profile is already approximated by that early similarity solution and its centre depth is 10, the virtual age is ${\tau }_0=A_3^5/10^5\simeq 4.2385\times 10^{-6}$, and the elapsed estimate is ${\tau }_{\mathrm{tr}}-{\tau }_0$. Other reasonable definitions of typical thickness change the order-one coefficient; the initial thickness alone does not uniquely determine an exact transition time or an exact initial profile. The robust nondimensional estimate is a transition time of order one in the $T_{\ast }$ scale.

Finally, let a local nose translate steadily at positive speed $v$ and write $s=x_N-x\ge 0$. This is a local traveling nose of a glacier with linear basal drag; the speed of the globally spreading glacier changes slowly with time. The local mass conservation equation integrates to $q=vh$, since both $q$ and $h$ vanish at the front. Therefore

The right side is strictly increasing, giving a unique positive thickness at every $s>0$. With $S=X_N-X$ and $c=d{X}_N/d\tau$, the same relation reads $cS={\mathcal{H}}^3/3+{\mathcal{H}}^2/2$. Its limiting profiles areThe square-root tip reflects basal sliding; the cube-root thicker region reflects internal viscous deformation. Even during the early thick regime, the very nose is thin and has the square-root inner form. At late times that sliding balance governs nearly the whole profile. The dimensionless crossover thickness is $\mathcal{H}=1$ and its distance is $S=5/(6c)$; the early thick cube-root outer profile matches this smaller sliding region. The diverging geometric slope at the idealized tip also marks the local limit of the long-wave shallow-ice approximation.

Let $\ell =(H_m-H_0)/\sin \alpha$ be the along-slope distance to the snowline, so $H(x)=H_m-x\sin \alpha$ and $a(x)=A(1-x/\ell )$. Define $D=g\sin \alpha /(3\nu )$.

Treat ice as an incompressible Newtonian fluid, neglect inertia, and use lubrication theory with thickness measured normal to the slope. The bed has a no-slip boundary condition, the free surface has zero tangential stress, and normal pressure is hydrostatic. The small thickness slope allows its pressure-gradient contribution to be neglected against gravity along the mountain. With normal coordinate $y$, the tangential equation and boundary conditions areIntegration givesThe horizontal radius of a ring is $x\cos \alpha$, so mass conservation gives the gravity-driven ice flow on a conical slope equationNegative accumulation is ice ablation and applies only where ice exists; the ice-free region has $h=0$.

In a steady state, regularity and zero total flux at the apex require $xD{h}^3\to 0$ as $x\to 0$. Integrating givesRequiring a continuous zero-thickness steady terminus yields the steady conical ice cap with a linear accumulation gradient:The terminus lies below the snowline, allowing the ablation region to balance snowfall. The maximum thickness occurs at $x=3\ell /4$.

The volume follows from integrating the ring areas. With $s=x/\ell$,Substituting $D$ and $\ell =\Delta H/\sin \alpha$ givesThe ideal outer profile has steep slopes very close to the apex and terminus. Those small regions require local corrections to the assumed slope balance, while the bulk profile and leading volume follow from the stated approximation.

With both ice accumulation and ice ablation removed, the equation is $h_t+x^{-1}(xD{h}^3{)}_x=0$ and the conserved volume is $V_0=2\pi \cos \alpha M$, where $M={\int }_0^{x_N}xhdx$.

If the thickness and extent scales are $H(t)$ and $X(t)$, mass conservation gives $H{X}^2\sim M$, while the flow equation gives $H/t\sim D{H}^3/X$. Therefore $X\propto t^{1/5}$ and $H\propto t^{-2/5}$. Set $h={\tau }^{-2/5}f(\xi )$, $\xi =x{\tau }^{-1/5}$, with a possible virtual time origin $\tau =t+t_0$. The similarity solution satisfiesIntegration and regular zero total flux at the apex give $D\xi f^3={\xi }^{2}f/5$, hence the positive profile iswith dry bed beyond the front. Volume normalization givesso the volume-conserving conical ice-current similarity solution hasThe terminus has finite thickness and is a shock wave in the gravity-only kinematic wave equation. The Rankine-Hugoniot condition gives ${\dot{x}}_N=q_N/h_N=D{h}_N^2=x_N/(5\tau )$, exactly agreeing with the similarity extent. The characteristic speed behind the front is $3D{h}_N^2$, larger than its speed, while the dry-bed characteristic speed is zero, so the front is compressive and gives an entropy solution.

This solution describes the long-time spreading, rather than exactly matching the earlier steady profile at the instant snowfall stops. The initial transient can be described by the characteristic transformation for conical ice drainage: with starting point $s$, put $K(s)=s^{1/3}{h}_s(s)$; thenwhere characteristics remain smooth. Subsequent crossings are resolved by the same conservation and entropy conditions. Restoring the neglected local pressure gradient would smooth the idealized front.

On a horizontal bed, the shallow-ice approximation with a linear basal drag law gives $u=-(\rho g/\mu )h_x(h/\beta +hz-z^2/2)$ and $q=-(\rho g/\mu )(h^3/3+h^2/\beta )h_x$. With no ice accumulation or ice ablation, mass conservation is $h_t+q_x=0$. Internal shear supplies the cubic mobility and basal sliding the quadratic mobility.

For ice accumulation $a=A(1-x/\ell )$, regular zero flux at the apex givesThe steady zero-thickness terminus is at $3\ell /2$, below the snowline $x=\ell$. Accumulation above the snowline exactly balances ice ablation below it.

After ice accumulation and ice ablation cease, the similarity solution conserving $M=\int xhdx$ isIt is dry outside this interval. At its terminus the thickness $h_N$ is nonzero: the Rankine-Hugoniot condition gives ${\dot{x}}_N=q_N/h_N=D{h}_N^2=x_N/(5\tau )$. This is an entropy solution of the gravity-only kinematic wave model and a long-time approximation for general finite-volume initial profiles. Local pressure-gradient effects smooth the idealized discontinuous front.