Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 332 / 4 / c / Solution

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.