Linear stability of a meridional two-layer current ($\mathcal{B}{\sigma }^2+(2A\beta k-\mathcal{B}lV)\sigma +{\beta }^2{k}^2-A\beta klV+F{K}^2{l}^2{V}^2=0$)

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.