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

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.