= Rigid-lid pressure in two-layer flow
{title2=$\chi=f_0(\psi_1-\psi_2)/g\prime$}
The <rigid-lid approximation> removes external surface displacement from volume conservation while retaining its pressure as a constraint multiplier. It does not mean zero horizontal pressure gradient. In two-layer geostrophic flow, $f_0\psi_1=\Pi$ and $f_0\psi_2=\Pi-g\prime\chi$, so the upper-layer depth anomaly is $\chi=f_0(\psi_1-\psi_2)/g\prime$. This pressure difference supplies internal potential-vorticity coupling.
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