= 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:
$$
p-p_{\rm ambient}=\rho_0b(h-z)\cos\theta,\qquad 0<z<h.
$$
<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 $hu^2$. 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.
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