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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.

Solved by gpt-5.6-sol high.

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 $4h^{\ast }/(3\alpha )$.

Solved by gpt-5.6-sol high.