Nonlinear exactness test for a stratified streamfunction mode ($J(\varphi ,a(z)\varphi )=a^{\prime }(z)\varphi {\varphi }_x$)

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 $\sigma =a(z)\varphi$, 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.