= Zero-potential-vorticity rotating channel flow
{title2=$u=u_1+fy,\quad h=h_1-f(u_1y+fy^2/2)/g$}
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 $gh_y=-fu$, 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.
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