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For $k=0$ and $l\ne 0$, divide the previous determinant equation by $l^2(l^2+2F)$ to obtainThusWith the chosen $e^{-i\sigma t}$ convention, a positive imaginary part of $\sigma$ is growth. For a nonzero basic shear $V$, the necessary condition for meridional-wave instability of a two-layer Sverdrup flow isFor these nondegenerate modes it is also sufficient. The growth rate isModes with $l^2>2k_I^2$ have real frequencies. Equality is the zero-growth coalescence of the two roots, while $l=0=k$ is a spatially uniform degeneracy and is not a growing finite-wavelength disturbance.

For $k=0$, the perturbation has $v^{\prime }={\psi }_x^{\prime }=0$, so it does not directly advect the northward planetary vorticity gradient. Consequently $\beta$ has no explicit restoring contribution to the fixed-$V$ instability criterion. It still sets the Sverdrup balance velocity: at fixed wind forcing, $V=W/\beta$, andIncreasing $|\beta |$ at fixed $W$ reduces the shear and the growth rate, without creating a finite-$\beta$ wavelength cutoff for these purely meridional modes. The limit $\beta \to 0$ at fixed $W$ is not a valid finite Sverdrup balance; its divergent velocity violates the weak-flow assumptions. Growth draws on the interfacial displacement and vertical shear, rather than on a time-dependent wind forcing, whose perturbation was set to zero.