Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 345 / 2

The shallow-water approximation requires current depth $h$ to be much smaller than its horizontal radius, hydrostatic vertical momentum balance, and horizontal velocity and temperature that are nearly depth-uniform. The radial scale and cooling time must also be long compared with the vertical adjustment scales.

The density contrast isso the inertia may use the common reference density while this small difference is retained in buoyancy: the flow is Boussinesq. Because the current is lighter, it occupies the top of the lake. Its free-surface displacement is smaller than its internal-interface displacement by the ratio of reduced gravity to gravity. A rigid-lid reduced-gravity model is therefore valid only when $g^{\prime }\ll g$; otherwise the external free-surface mode must be retained.

SetThe axisymmetric reduced-gravity equations areThe extra $h{g}_r^{\prime }/2$ in momentum is the depth average of the hydrostatic pressure gradient caused by horizontal density variation.

In variables $(u,c,\theta )$ these becomeThe three characteristic speeds areAlong $dr/dt=u$, temperature obeysAlong $dr/dt=u+sc$, $s=\pm 1$, a left-characteristic projection gives

The shallow-water equations cease to apply inside the narrow nose, so a gravity-current front condition is needed to relate its speed to the depth immediately behind it. Usewhere $Fr$ is the front Froude number; the ideal deep-ambient von Kármán condition gives $Fr=\sqrt{2}$.

In a uniform axisymmetric box model, conservation of contaminated volume givesThe total nondimensional heat content is $V\theta$, while cooling acts over area $\pi R^2$, soEliminating time and taking the initial release radius as negligible givesSpreading stops as $\theta \to 0$, atHence the maximum covered area is