Past exam of the mathematics and the environment and mathematical physiology course of the University of Oxford / 2008 / A / 3 / ii / Solution

Choose reference depth $h_0$, velocity $u_0=\sqrt{gS{h}_0/f}$ and concentration $c_0=E(u_0)$. Set the adaptation length $L=h_0{u}_0/v_s$ and the slow bed-evolution time $t_b={\rho }_s(1-\varphi )h_0/(v_s{c}_0)$. Scale both bed and surface elevation with $h_0$. ThenThese choices balance sediment advection with settling, bed mass change with exchange, and reference drag with downslope gravity. Direct substitution givesHere $E^{\ast }(1)=1$ by construction. The small parameter $ϵ$ is the suspended solid mass relative to bed solid mass at the chosen scales, explaining the separation between fast fluid motion and slow morphology.