Barotropic vorticity equation

Remarks

Since $J(\psi, q) = u \, \partial_x q + v \, \partial_y q$, the equation says $D(\zeta + f)/Dt = 0$, where $D/Dt$ is the derivative along the flow: absolute vorticity is conserved by each fluid column. On the sphere the same form holds with $f = 2 \Omega \sin \varphi$ and the spherical Laplacian and Jacobian.

Example: $\psi = A \cos(kx + ly - \omega t)$ with $K^2 = k^2 + l^2 > 0$ solves the equation exactly when $\omega = -\beta k / K^2$. Indeed $\nabla^2 \psi = -K^2 \psi$, so $J(\psi, \nabla^2 \psi) = 0$, and the remaining terms give $(-K^2 \omega - \beta k) A \sin(kx + ly - \omega t) = 0$. This is the dispersion relation of Rossby waves; their phase moves westward.

Non-example: a flow with nonzero horizontal divergence $\partial_x u + \partial_y v \ne 0$, such as a flow with stretching of vortex columns, is not described by the equation, which assumes a stream function.

The integrals of the energy $\frac12 \int \abs{\nabla \psi}^2$ and of the enstrophy $\frac12 \int \zeta^2$ over a doubly periodic domain are conserved.