Zero-potential-vorticity rotating channel flow ($u=u_1+fy,h=h_1-f(u_{1}y+f{y}^2/2)/g$)

For a steady, slowly varying rotating channel, negligible transverse velocity and zero shallow-water potential vorticity give $u_y=f$. Cross-channel geostrophic balance gives $g{h}_y=-fu$, producing the displayed profiles. The specific energy $E=h+u^2/(2g)$ is constant across the channel, and $E+H$ is constant along it. Zero potential vorticity is not the value in an ordinary finite-depth reservoir at rest in the rotating frame; it requires vanishing absolute vorticity or a deep-reservoir approximation.

At a hydraulic control, the specific-energy function of a zero-potential-vorticity rotating channel flow is stationary with respect to wall depth at fixed discharge and width. Writing $v=u_1+fb/2$ gives $v(v+fb/2)=g{h}_1$ and $Q_c=b{v}^3/g$, hence the displayed discharge. The fully wetted positive-flow solution requires $fb<\sqrt{2g{h}_1}$; at equality the other wall dries and the first wall velocity vanishes. Beyond that limit the assumed two-wet-wall profile must be replaced.