Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 345 / 2 / c / Solution

Let $\mathcal{V}=H{L}_0$ be the layer volume per unit transverse width. The gravity-current box model imposes $hL=\mathcal{V}$; it approximates the front and scalar balances, rather than an exactly uniform solution of the local momentum equation. For a specified constant front Froude number $\mathrm{Fr}$, the heated particle-laden gravity current satisfiesThe initial data are $L=L_0$, $\varphi ={\varphi }_0$, and $\theta =0$. The gravity-current front condition applies only while $\gamma \varphi -\theta \ge 0$.

For heating without settling, $\varphi ={\varphi }_0$ and $\theta =\beta {\varphi }_{0}t$. Integration of the front equation givesHence the heated gravity-current runout isIt is reached at the neutral-buoyancy time $\gamma /\beta$ within this front model. A finite-momentum current could subsequently coast or lift off: the formula is the maximum predicted by the prescribed gravity-current front condition, which contains no independent front inertia.

For settling without heating, $\theta =0$. Set $K=\mathrm{Fr}\sqrt{g\gamma \mathcal{V}}$, so $\dot{L}=K\sqrt{\varphi }{L}^{-1/2}$. Eliminating time yieldsThus the runout length of a gravity current isThis length is approached as $t\to \infty$, because positive concentration cannot disappear at a finite time under the settling law. If distance means displacement from the removed barrier, the answer is $L_{\max }-L_0$ in either case.