= 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 is
$$
\frac{\rho_0-\rho}{\rho_0}
=\beta(T-T_0)^2\ll1,
$$
so 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 $g'\ll g$; otherwise the external free-surface mode must be retained.
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