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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?

excerpt from paper

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

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.

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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.

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