Two-layer Sverdrup interior ($V=W/\beta$)

For a uniform normalized upper-layer wind stress curl $W$ and no lower-layer forcing, a uniform local Sverdrup balance isThe upper layer flows meridionally and the lower layer rests. Although the relative vorticity vanishes, the potential vorticity has the zonal gradients ${\bar{q}}_{1,x}=-FV$, ${\bar{q}}_{2,x}=FV$. These are essential for the perturbation stability and do not vanish merely because the background velocity is uniform.

Let $K^2=k^2+l^2$, $A=K^2+F$, $\mathcal{B}=K^2(K^2+2F)$ and use $e^{i(kx+ly-\sigma t)}$. The uniform upper-layer flow $V$ over a resting lower layer has the dispersion relationIts discriminant is $4F^2{\beta }^2{k}^2+\mathcal{B}{K}^2(K^2-2F)l^2{V}^2$. A negative value gives exponential baroclinic instability. This model includes advection across the interface-induced potential-vorticity gradients in a meridional two-layer current.

For purely meridional wavevectors, $k=0$, $l\ne 0$, the growth rate of a uniform meridional upper-layer flow $V$ over a resting lower layer isGrowth requires $V\ne 0$ and $0<l^2<2F$. The perturbation has no northward velocity, so planetary vorticity advection does not explicitly stabilize it. At fixed normalized wind stress curl, $V=W/\beta$ means stronger $|\beta |$ nevertheless reduces the shear and growth rate.

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.