Bedload transport

Bedload transport is motion of sediment grains along or close to the bed, represented in a depth-averaged model by a solid-volume flux per unit width. The Exner equation balances this flux divergence and changes in porous-bed solid storage against exchange with suspended sediment. Consistent mass units are necessary when adding this equation to a suspended mass-concentration balance.

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.