Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 71 / 1 / a / Solution

Write $b=g(\rho -{\rho }_0)/{\rho }_0>0$ for the reduced gravity. The Boussinesq approximation requires $|\rho -{\rho }_0|\ll {\rho }_0$: replace density by ${\rho }_0$ in inertia and retain its difference in the gravitational force. A shallow-water approximation additionally requires a small aspect ratio, negligible leading-order normal acceleration and approximately hydrostatic pressure relative to the ambient fluid:Incompressibility makes the normal velocity smaller than the downslope velocity by the aspect ratio.

Thinness alone does not force the downslope velocity to be uniform across the layer. The additional top-hat plume model assumes that turbulent mixing makes both $u(x,t)$ and $b(x,t)$ approximately uniform inside it, with a sharp transition to the ambient fluid. Then the depth-integrated volume and momentum fluxes are $hu$ and $h{u}^2$. This is the same integral closure used for a turbulent plume, applied to a shallow layer; the hydrostatic pressure term is retained because gravity has a component normal to the channel.