Barotropic vorticity equation

I came across the barotropic vorticity equation (below) in Sakamoto (2002) and I cannot figure it out. The notation is not clear to me. How can it be derived from the Navier-Stokes equation?

$$\beta\psi_x=\frac{\tau_x^y-\tau_y^x}{\rho_0H}-r\zeta+A_H\nabla^2\zeta-\frac{f_0}{H}w_D$$

where

$$\psi$$ is the streamfunction

$$\nabla$$ is the horizontal gradient operator

$$\zeta=\nabla^2\psi$$ is the vertical component of relative vorticity

$$(\tau_x,\tau_y)$$ is the zonal wind stress

$$f=f_0+\beta y$$ is the Coriolis parameter

$$\rho_0$$ is the mean density

$$H$$ is the depth of the model ocean

$$r$$ is the inverse time scale for vorticity decay by bottom friction

$$A_H$$ is a horizontal eddy viscosity

$$w_D$$ is not explained in this paper.

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.