Past exam of the mathematics and the environment and mathematical physiology course of the University of Oxford / 2008 / A / 3 / i / 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 $f{u}^2/h$.

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 $v_{s}c$, while entrainment supplies $v_{s}E(u)$; 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 $1-\varphi$ and grain density is ${\rho }_s$. Hence net suspended gain equals $-{\rho }_s(1-\varphi )s_t$: 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.