= Solution
The depth $h$ is free-surface height $\eta$ minus bed elevation $s$. The first differential equation conserves water volume. Momentum balances local and advective acceleration against the free-surface <pressure> gradient, mean gravitational bed-slope forcing and the quadratic drag $fu^2/h$.
The <concentration> $c$ is suspended <sediment> mass per water volume. Multiplication of its material rate by depth gives the <sediment> change per bed area. Settling at speed $v_s$ removes mass at rate $v_sc$, while <entrainment> supplies $v_sE(u)$; increasing <equilibrium> <concentration> $E(u)$ represents stronger erosive capacity at greater speed. The same exchange changes the bed mass per area, whose solid volume fraction is $1-\phi$ and grain density is $\rho_s$. Hence net suspended gain equals $-\rho_s(1-\phi)s_t$: erosion lowers the bed and deposition raises it.
One derives these balances by depth integration with <hydrostatic pressure> and a sectionally averaged <velocity>, followed by closures for drag and <sediment> <entrainment>/settling. The model neglects additional bedload transport and uses the dilute-suspension approximation to keep the water-volume balance independent of <sediment> volume.
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