 Solution

The potential-vorticity gradients in a meridional two-layer current are important here: besides ${\bar{q}}_{n,y}=\beta$, one has ${\bar{q}}_{1,x}=-FV$ and ${\bar{q}}_{2,x}=FV$. They must be retained when linearizing, even though the basic relative vorticity vanishes.

Let $K^2=k^2+l^2$ and $A=K^2+F$. For a normal mode the two-layer quasi-geostrophic potential vorticity amplitudes areBecause the basic velocities are northward $(V,0)$ in the two layers, their advection frequencies are $\sigma -lV$ and $\sigma$. The uniform forcing has no perturbation, $W^{\prime }=0$. The linear equations are thereforeIn particular, a perturbation's zonal velocity advects the basic interface-induced zonal potential-vorticity gradient. Substitution of the plane wave givesA nonzero disturbance exists exactly when the determinant vanishes. The requested relation for linear stability of a meridional two-layer current isEquivalently, with $\mathcal{B}=K^2(K^2+2F)$,Its discriminant isFor $K>0$, exponential baroclinic instability occurs when this discriminant is negative; otherwise the two frequencies are real. This also displays the stabilizing contribution of the planetary vorticity gradient when $k\ne 0$. As a check, $V=0$ gives the uncoupled barotropic mode and baroclinic mode frequencies $-\beta k/K^2$ and $-\beta k/(K^2+2F)$.