= Nonlinear exactness test for a stratified streamfunction mode
{title2=$J(\phi,a(z)\phi)=a'(z)\phi\phi_x$}
For two-dimensional flow under the <Boussinesq approximation> with $u=\psi_z$, $w=-\psi_x$, a linear mode whose <Laplacian> is proportional to its <streamfunction> has zero self-advection of <vorticity>. Its <buoyancy> anomaly is $\sigma=a(z)\phi$, with $a=N^2/(U-c)$. Self-advection of <buoyancy> vanishes only when the displayed Jacobian is zero. Constant stratification and constant flow make $a$ constant and yield finite-amplitude exact wave solutions. A varying polarization coefficient generally creates a second harmonic, even when the <vorticity> self-interaction vanishes.
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