= Linear stability of a meridional two-layer current
{title2=$\mathcal B\sigma^2+(2A\beta k-\mathcal B lV)\sigma+\beta^2k^2-A\beta klV+FK^2l^2V^2=0$}
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 relation>
$$
\mathcal B\sigma^2+[2A\beta k-\mathcal B lV]\sigma+\beta^2k^2-A\beta klV+FK^2l^2V^2=0.
$$
Its discriminant is $4F^2\beta^2k^2+\mathcal B K^2(K^2-2F)l^2V^2$. A negative value gives exponential <baroclinic instability>. This model includes advection across the interface-induced <potential-vorticity gradients in a meridional two-layer current>.
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