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

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.