 Bedload transport model with sediment inertia

Keeping sediment acceleration while neglecting water storage and a small gravity-friction residual gives a reduced model for the bed elevation $s$, water speed $u$ and sediment speed $v$. A prescribed water flux $q$ gives $h=q/u$; integrated water momentum gives $s=C-F^2{u}^2/2-q/u$. Around the normalized uniform state $h=u=q=1$, let $(1-v_0{)}^2=\kappa v_0$, $0<v_0<1$. A Fourier mode of wavenumber $k$ has growth-rate real partThus nonuniform modes decay for subcritical Froude number and grow for supercritical Froude number. The spatially constant mode is neutral, and the critical value needs terms neglected in this reduction. In a general uniform state the sign change is at the actual mean-flow Froude number equal to one, rather than at a velocity-scale parameter unrelated to that state.