= Solution
The <Boussinesq approximation> requires $\delta=(\rho-\rho_0)/\rho_0=\phi(\rho_p/\rho_0-1)\ll1$. The reference <mass density> $\rho_0$ can then be used in inertia, while the small density excess is retained in <buoyancy>. Dilute <particle volume fraction> $\phi\ll1$ alone does not suffice if $\rho_p/\rho_0$ is exceptionally large. Define $G=g(\rho_p-\rho_0)/\rho_0$, so the <reduced gravity> is $g'=G\phi$.
The <shallow water equations> require depth small compared with the horizontal evolution scale, weak vertical acceleration and approximately <hydrostatic pressure>. Here the ambient is deep and quiescent, so its leading contribution is a hydrostatic reference pressure; the excess pressure in the layer is $\rho_0G\phi(h-z)$. The model also needs nearly uniform streamwise <velocity> across the section and the stated absence of secondary circulation. The advancing <gravity current> nose is a separate region where these assumptions can fail.
A nearly uniform <particle volume fraction> can be maintained by turbulent mixing with <eddy diffusivity> $K_z$, provided $h^2/K_z\ll h/w$ and the mixing time is short compared with horizontal advection and bulk evolution. Equivalently, $wh/K_z\ll1$ keeps the bulk settling-induced gradient small. The particles should respond rapidly enough to follow the mixing motions. A thin boundary layer adjacent to the absorbing bed or walls need not be well mixed.
\b[Uniform bulk <concentration> does not imply zero sedimentation.] The relative downward particle <velocity> still supplies a <particle deposition flux> of approximately $w\phi$ to an absorbing boundary. Mixing redistributes the remaining particles and maintains the bulk <concentration> profile while its overall level decreases; it need not suspend every particle indefinitely. Deposition requires negligible resuspension in the model.
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