Barotropic vorticity equation
This equation is a form of the Shallow-water equations, which are derived from Navier-Stokes in the incompressible limit and the vertical direction is integrated out.

One then takes the equation for the vorticity (which has only one component then) from the Shallow-water equations. The vorticity is split into planetary vorticity, as is the Coriolis term with the $\beta$-sheet-approximation.

Additionally it looks like they're using a velocity streamfunction.