Rigid-lid approximation (${\nabla }_h\cdot \mathbf{u}=0(H=\mathrm{constant})$)

The rigid-lid approximation suppresses free surface displacement in a fluid layer while retaining a variable pressure that enforces the mass constraint. At constant depth it gives horizontally nondivergent motion. In a one-layer quasi-geostrophic approximation, removing free-surface stretching eliminates the $-\psi /R_D^2$ term from shallow-water quasi-geostrophic potential vorticity. This is also the small-horizontal-scale limit compared with the Rossby deformation radius, when that term is negligible rather than exactly absent.

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.