= Nonrotating layerwise-two-dimensional vortex dynamics
{title2=$\partial_t\zeta+J(\psi,\zeta)=0,\quad\nabla_h^2\psi=\zeta$}
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. https://pordlabs.ucsd.edu/wryoung/theorySeminar/pdf14/McIntyrePV.pdf[McIntyre's discussion of potential-vorticity inversion] explains the limiting independent-layer picture.
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