Charney–Hasegawa–Mima equation

Remarks

Equivalently, $q = \nabla^2 \psi - \psi / L_d^2 + \beta y$ satisfies $\partial_t q + J(\psi, q) = 0$, since $J(\psi, \psi) = 0$: the potential vorticity $q$ is carried by the flow. As $L_d \to \infty$ it becomes the barotropic vorticity equation. In plasma physics, after scaling, $\psi$ is the electrostatic potential and $L_d$ is the ion sound gyroradius. The $\beta$ term then comes from the background density gradient, and the sign conventions vary between texts.

Example: $\psi = A \cos(kx + ly - \omega t)$ with $K^2 = k^2 + l^2$ solves the equation exactly when $$\omega = -\frac{\beta k}{K^2 + L_d^{-2}}.$$ The nonlinear term vanishes because $\nabla^2 \psi = -K^2 \psi$, and the linear terms give $\left(-(K^2 + L_d^{-2}) \omega - \beta k\right) A \sin(kx + ly - \omega t) = 0$. Compared with the barotropic case, waves longer than $L_d$ are much slower.

Non-example: the shallow water equations on a beta plane are not of this form; they also contain gravity waves, which the equation filters out.

The energy $\frac12 \int \left( \abs{\nabla \psi}^2 + \psi^2 / L_d^2 \right)$ and the enstrophy $\frac12 \int (\nabla^2 \psi - \psi / L_d^2)^2$ are conserved on a doubly periodic domain.