Past exam of the mathematics and the environment and mathematical physiology course of the University of Oxford / 2009 / A / 3

Here $h$ is water depth, $u$ the depth-averaged downstream speed, $s$ bed elevation relative to the reference slope, and $\eta =s+h$ free-surface elevation relative to that reference. The constants ${\rho }_w,{\rho }_s$ are water and grain densities, $S$ is the reference bed slope, $\tau$ bed shear stress, $\varphi$ bed porosity, $c$ suspended sediment mass concentration, $q_b$ bedload solid volume flux per unit width, and $v_E,v_D$ erosion and deposition velocities. The settling speed is denoted by $v_s$ in the constitutive laws.

Depth integration of incompressible water conservation gives the depth-storage plus discharge-divergence balance. Depth-averaged momentum has local and advective acceleration on the left, and gravity driven by the actual surface gradient together with bed drag on the right. Hydrostatic pressure and a shallow-water approximation underlie this Saint-Venant description.

Suspended sediment conservation first has the conservative form $(hc{)}_t+(huc{)}_x={\rho }_s(v_E-v_D)$; the water balance reduces it to the material-derivative form printed in the source. Solid bed volume per unit horizontal area is $(1-\varphi )s$. Its storage plus bedload divergence equals deposition minus erosion, giving the Exner bed balance. Adding the suspended and bed balances after converting bed volume to grain mass cancels exchange terms: erosion and deposition transfer sediment between the two reservoirs rather than create it.

Take an arbitrary reference depth $h_0>0$, and select $u_0=(gS{h}_0/f{)}^{1/2}$ so the uniform reference flow balances drag and slope. Work with positive downstream velocity, so $u_{\ast }=\sqrt{f}u$ in the dimensional constitutive laws. Introduce the erosion and bedload scalesMatching erosion with deposition, suspended advection with erosion, and bed storage with erosion gives respectivelyUse $h_0$ for the scales of $\eta$ and $s$ too, so $h=\eta -s$ survives unchanged. After dividing the balances by their matched scales, the parameters areFor example the momentum balance divided by $g{h}_0/x_0$ has inertial prefactor $F^2$, surface-gradient term $-{\eta }_x$, and equal slope/drag prefactors $S{x}_0/h_0=\delta$, because $f{u}_0^2=gS{h}_0$. The suspended balance has source $u^3-c/u$, and the bed balance has its negative with unit bed-time coefficient by the definition of $t_0$. These substitutions give the entire stated dimensionless system; $q_b=u^3$ is dimensionless bedload, with its dimensional scale already absorbed into $\beta$.

For quasistatic bed stability with suspended sediment, in the leading approximation, $(hu{)}_x=0$ and ${\eta }_x=0$. For spatially varying perturbations at fixed unit water flux and fixed surface level, write $u=1+U$, $h=1+H$, $c=1+C$, $s=s_{\ast }+S_1$. The constraints give $H=-U$ and $S_1=-H=U$. Linearizing the suspended and bed equations yieldsFor a Fourier mode $e^{\sigma t+i\kappa x}$, the first relation gives $C=4U/(1+i\kappa )$; substitution into the second givesThus every nonzero spatial frequency has negative growth rate, independently of $\beta$. The parameter changes propagation, not damping. The spatially uniform mode has $\sigma =0$ and is neutral, representing a change among uniform reference states compatible with the leading constraints. Hence there is no growing linear mode in this approximation; the claim is stability, not strict decay of every homogeneous perturbation. Restoring the small discarded terms or changing boundary conditions requires a separate calculation.