= Bedload transport model with sediment inertia
{title2=$s(u)_t+v_x=0,\quad vv_x=(u-v)^2-\kappa v$}
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 $s=C-F^2u^2/2-q/u$. 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 part
$$
-\frac{2(1-v_0)v_0k^2}{(1-F^2)([2(1-v_0)+\kappa]^2+k^2v_0^2)}.
$$
Thus 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.
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