 Rigid-lid pressure in two-layer flow

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.