Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 332 / 3

The shallow-ice approximation makes the pressure hydrostatic pressure,Horizontal Stokes flow balance and the stress-free boundary condition at $z=h$ givesoThe basal shear stress and the stated basal sliding law are thereforeIntegrating once more and imposing $u(0)=u_b$ yieldsFor the right half of an ice cap, $h_x<0$, so both basal sliding and internal deformation carry ice away from the ice divide.

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The depth-integrated ice flux isLocal mass conservation with accumulation rate $a$ saysSubstituting the flux givesas required. Equivalently, after using $u_b=-\beta h_x$,

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Use the dimensional analysisand define the dimensionless sliding parameterThe dimensionless flux and steady conservation law areOn the right half-cap, let the snowline be $X=X_s$ and the nose be $X=X_n$. The zero-flux condition at the ice divide gives $Q=X$ in the accumulation region. Continuity of $Q$ gives $Q=2X_s-X$ in the ablation region, and zero nose flux gives

Introduce the increasing functionSince ${\Phi }_X=-Q$, the ablation profile satisfying $H(X_n)=0$ isAt the snowline $H=1$, soIn the accumulation region,and the divide thickness $H_0$ is fixed byThese two implicit formulas give a continuous thickness and flux at the snowline. Reflection across $X=0$ gives the full two-dimensional ice cap.

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The ratio of sliding flux to deformation flux at thickness $h$ isThus basal sliding dominates throughout the dynamically important thick part of the cap whenwith the divide thickness giving a more conservative choice if it is known. In this limit $s\gg 1$, and the divide-to-nose extent from part c becomesThe full width of the symmetric cap is $2x_n$.

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When sliding dominates and the whole cap is below the snowline, its thickness obeysWith $\alpha =0$, this is a one-dimensional porous medium equation. Its mass-preserving Barenblatt solution isWithin its support,Now set $h=h_{ss}+ct$. Since $h_{ss}$ solves the unforced equation,Matching the uniform ablation term $-\alpha$ givesThe similarity solution has a virtual origin. Writing $s=t+t_0$ gives the exact decaying familyIf its initial centre thickness is $h^{\ast }$, then $C=(h^{\ast }+3\alpha t_0/4)t_0^{1/3}$ and the exact extinction time isIts value depends on the initial cap width through the virtual age $t_0$, but its dimensional scale is unambiguouslyThe additive melting correction alone has the corresponding time $4h^{\ast }/(3\alpha )$.

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