Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 65 / 3 / a

Let $h$ be the water-layer thickness and $z=0$ the horizontal bedrock. In the hydrostatic approximation, with constant ice-interface pressure,The prescribed two-wall stress balance givesIn this subglacial current with constant viscous wall layers, the uniform-pressure ice roof is treated as a movable confining boundary; the closure neglects inertia in comparison with pressure and wall stress. The bulk may be turbulent, while the specified wall viscous boundary layer supplies the linear stress relation. A different turbulent drag law would give a different transport coefficient and similarity exponent.

Without melting, the depth-integrated continuity equation gives $h_t+q_x=0$, henceTo write this in the PDF's displayed plus-sign form, its constant must beThe signed value is essential. If the printed $\gamma$ were interpreted as a positive diffusivity, its equation would drive water uphill and be backward parabolic. For example, linearizing around a uniform depth $H$ would give a Fourier perturbation growth rate $+\gamma H^2{k}^2$, whereas the derived physical equation damps it at $-D{H}^2{k}^2$. The plus-sign form is consistent only with the negative $\gamma$ above.