Nonrotating layerwise-two-dimensional vortex dynamics (${\partial }_t\zeta +J(\psi ,\zeta )=0,{\nabla }_h^2\psi =\zeta$)

In the limiting model of infinitely strong stable density stratification, small Froude number and fixed vertical scales suppress vertical displacement. Each horizontal level evolves by two-dimensional vortex dynamics; $z$ labels independent copies of the horizontal Poisson equation. The leading normalized Ertel potential vorticity is $Q/N^2=\zeta$. This is a constrained asymptotic model: finite buoyancy frequency restores vertical motion and coupling, and arbitrarily short vertical scales invalidate the independent-layer limit. McIntyre's discussion of potential-vorticity inversion explains the limiting independent-layer picture.