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A hydraulic control of a zero-PV rotating channel is a critical section separating the upstream branch, where long-wave disturbances can propagate upstream, from the downstream branch, where they cannot. In the steady specific-energy description it is a stationary point of $E$ with respect to depth at fixed volume flux and local width. The control therefore fixes the discharge for a given upstream Bernoulli function and channel geometry; an arbitrary downstream condition cannot generally be imposed independently on the upstream reservoir.

Evaluate all quantities at $x=0$ and keep $Q,b$ fixed while differentiating. With $\Delta =fb$, the preceding branch givesAt the hydraulic control of a zero-PV rotating channel, this derivative vanishes. Using $h_1-h_2=\Delta v/g$ reduces it to $u_{1}v=g{h}_2$, where $v=u_1+\Delta /2$. Equivalently,Combining this with $Q=b{h}_{1}v-b\Delta v^2/(2g)$ cancels the quadratic term and leavesAt control $u_{1c}=\sqrt{g{h}_1+{\Delta }^2/16}-3\Delta /4$ and $h_{2c}=u_{1c}{v}_c/g$. Both walls must remain wet. Since $v_c>0$, the second depth is positive precisely when $u_{1c}>0$, givingEquality is the limiting case $u_{1c}=h_{2c}=0$, with the second wall just dry. For a wider channel, a dry lateral region requires a different solution and the two-wet-wall hydraulic control of a zero-PV rotating channel formula cannot be continued. The channel must also remain sufficiently long and slowly varying for the transverse geostrophic balance approximation to hold. As a check, $f\to 0$ gives $Q_c=b\sqrt{g}{h}_1^{3/2}$, the familiar rectangular-channel critical discharge.