 Zero-potential-vorticity rotating channel flow

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.