Constant-flux river momentum relaxation ($u_t+[Q/(F^2{u}^2)-u]u_X=(1-u^3/Q)/(\epsilon F^2)$)

This scalar quasilinear partial differential equation results by imposing $h=Q/u$ in a frictional river momentum equation and measuring $X$ upstream. It retains a relaxation toward $u_0=Q^{1/3}$ and carries information at speed $c(u)=Q/(F^2{u}^2)-u$. Near equilibrium the relaxation time is $\epsilon F^2{u}_0/3$. A slowly moving downstream boundary can generate an attached boundary layer when $c(u)$ exceeds its speed throughout the relaxation. The model retains momentum dynamics only: exactly constant flux in both original Saint-Venant equations would instead force time-independent depth.