Potential-vorticity gradients in a meridional two-layer current (${\nabla }_h{\bar{q}}_{1,2}=(\mp FV,\beta )$)

For ${\bar{\psi }}_1=Vx$ and ${\bar{\psi }}_2=0$, the two-layer quasi-geostrophic potential vorticity is ${\bar{q}}_1=f_0+\beta y-FVx$, ${\bar{q}}_2=f_0+\beta y+FVx$. A perturbation therefore feels both the northward gradient $\beta$ and opposite zonal gradients $\mp FV$. Dropping the zonal gradients removes the instability mechanism from the linearization.