Past exam of the mathematics and the environment course of the University of Oxford / 2011 / 3 / i

These are the one-dimensional shallow water equations with bed slope and friction. The coordinate $s$ measures distance downstream, $t$ is time, $A(s,t)$ is the wetted cross-sectional area, $u(s,t)$ is section-averaged downstream velocity, and $h(s,t)$ is water water depth above the bed. Thus $Au$ is water volume flux and the first equation expresses mass conservation.

In the momentum equation, $u_t+u{u}_s$ is the material acceleration, $g$ is gravitational acceleration, and $S$ is the positive downward bed slope, so $gS$ drives the flow. The wetted perimeter is $l$ and $f$ is a dimensionless friction coefficient in the stated convention. The frictional deceleration is $fl{u}^2/A$ for downstream flow, while $-g{h}_s$ is the hydrostatic pressure contribution from surface-height variation relative to the sloping bed. The formula $u^2$ assumes the chosen positive-flow direction; a bidirectional friction model uses $u|u|$.