Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 345 / 2 / c / Solution

Use the gravity-current front condition $\dot{L}=\mathrm{Fr}\sqrt{d\varphi h}$, with a specified order-one front Froude number. In a triangular-channel gravity-current box model, the depth and concentration are uniform over $0<x<L(t)$. The volume isThe horizontal projected deposition area is ${\int }_0^{L}Tdx=\lambda h{L}^2$. Consequently the integral model isThis front speed is a closure for the integral model, rather than the assertion that its uniform interior is an exact solution of the spatially varying shallow water equations.

Eliminating time and integrating givesThe limiting runout length of a gravity current is thereforeThe additional distance beyond the release gate is $L_{\infty }-L_0$. The limit is approached asymptotically as $\varphi \to 0$ and $\dot{L}\to 0$; the idealized box model does not predict a finite-time stop. Its numerical runout depends on the adopted front Froude number.