Equations For Equivalent Barotropic QG Model

The equivalent barotropic quasigeostrophy evolution equations is

\[\partial_t\,{q} + \mathsf{J}\left(\psi\,, q\right) + \beta\, {\psi}_x = \text{ssd} \,.\]

The potential vorticity anomaly is

\[{q} = \nabla^2\psi - \kappa_d^2 \psi\,,\]

where \(\kappa_d^2\) is the deformation wavenumber. With \(\kappa_d = \beta = 0\) we recover the 2D vorticity equation.

The inversion relationship in Fourier space is

\[\widehat{q} = -\left(\kappa^2 + \kappa_d^2\right) \widehat{\psi}\,.\]

The system is marched forward in time similarly to the two-layer model.