Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 332 / 3 / a / Solution

The Newtonian shallow-ice flux on the flat bed isso mass conservation givesAt the ice divide, symmetry gives $h_x(0,t)=q(0,t)=0$. At the terminus, $h(x_N,t)=0$ and the moving-front mass balance applies; in a steady state $q(x_N)=0$. At the snowline, $h(x_s,t)=h_s$ and both $h$ and $q$ are continuous, hence $h_x$ is continuous because $h_s>0$.

In a steady state, $q_x=A$ in the accumulation region and $q_x=-A$ in the ablation region. ThereforeFlux continuity gives $x_N=2x_s$. Integrating the flux law gives the piecewise shapeMatching $h(x_s)=h_s$ yieldsThe largest surface slope is of order $\sqrt{\nu A/g}/h_s$, so the shallow-ice approximation requires