Past exam of the mathematics and the environment and mathematical physiology course of the University of Oxford / 2010 / A / 3 / iv / Solution

A typical net ice accumulation curve decreases downstream: snow gain at high elevation, zero at the equilibrium line, and negative values in the ice ablation zone. In a steady glacier with zero volume flux at its head, $Q(x)={\int }_0^{x}a(X)dX$. If its terminus also has zero volume flux, a natural glacier length $L$ is the positive root of ${\int }_0^{L}a(X)dX=0$, with the integral positive between the endpoints. Such a root requires a suitable given accumulation profile; it is not guaranteed by dimensional analysis alone.

Let $a_0$ be a representative accumulation rate and $K=A(\rho g\sin \alpha {)}^n$. ChooseUsing $x=L\hat{x}$, $H=H_{\ast }\hat{H}$, $Q=Q_{\ast }\hat{Q}$, $t=t_{\ast }\hat{t}$, $a=a_0\hat{a}$, both terms in mass conservation have scale $a_0$. The dimensionless flux isDropping hats gives the required partial differential equation. The parameter measures the surface-thickness slope correction relative to the bed slope. For the linear example $a=1-2x$, accumulation changes sign at $x=1/2$ and its integrated balance vanishes at $x=1$.