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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 ablation removes glacier or ice-sheet material by melting, sublimation, or mechanical loss. A negative net ice accumulation rate removes ice only where ice is present; an initially bare region cannot acquire negative thickness.

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.

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.

Initially, away from the apex and snowline, transport is small and local ice accumulation dominates:The early ice volume is $V(t)\simeq \pi A{\ell }^2\cos \alpha t/3$. Balancing radial transport $D(At{)}^3/\ell$ with $A$ identifies the global transition scalesThere is also an earlier local adjustment near the apex: the factor $1/x$ makes the same flux significant at distance $x_a\sim D{A}^2{t}^3$. The apex filling layer of a conical ice cap is described byAt the apex, $F\sim (\eta /2{)}^{1/3}$ gives the local steady profile $h\sim (Ax/(2D){)}^{1/3}$. Thus gravity-driven redistribution reaches a fixed position $x$ on a time of order $[x/(D{A}^2){]}^{1/3}$, and its region grows out to the snowline when $t\sim t_{\ast }$.

At times comparable to $t_{\ast }$, ice transport becomes important across the mountain's accumulation region, carries ice below the snowline, and establishes an ablation zone. The cap then approaches the steady conical ice cap with a linear accumulation gradient, with $x_N\to 3\ell /2$. For a moving ice-covered disk, mass conservation and its front condition giveThe total input vanishes at $x_N=3\ell /2$, explaining the final extent. The narrow apex and front regions are governed by local adjustments to the bulk approximation; they do not create a further distinct large-scale time regime in this scaling description.

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.