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

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.

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.

The reduced mass equation gives $hu=1$ after normalizing the reference discharge. The reduced momentum equation integrates to $\eta +F^2{u}^2/2=1+F^2/2$ with the reference free-surface level one. Since $h=1/u$,Also $hu=1$ reduces the concentration equation toWith $E^{\ast }(1)=1$, a steady uniform reference state is $\boxed{u=h=c=\eta =1,s=0}$. The integration constants above specify this reference normalization; a different imposed discharge or head changes them.

Write $u=1+\tilde{u}$, $c=1+\tilde{c}$ and let $a_E=E^{\ast \prime }(1)>0$. To first order $\tilde{s}=(1-F^2)\tilde{u}$, and the other balances give ${\tilde{c}}_x=a_E\tilde{u}-\tilde{c}=-{\tilde{s}}_t$. Seek modes $e^{\sigma t+ikx}$. ThenThus every nonzero wavenumber grows when $F>1$, while it decays when $F<1$. The zero mode is neutral in this reduced model. The exceptional $F=1$ limit makes the algebraic depth-bed relation degenerate and requires neglected effects. If the entrainment derivative vanishes at the reference state, this first-order instability calculation is itself degenerate.

For $F>1$, the growth rate tends to the finite positive limit $a_E/(F^2-1)$ as $|k|\to \infty$. It does not diverge with wavenumber, although this reduced theory selects no finite fastest scale and should not be extrapolated beyond its long-wave regime. The phase speed isThe unstable bedforms migrate upstream against the water flow and represent antidunes arising from coupled erosion, deposition and supercritical free-surface flow. This is the relaxation-controlled antidune instability.