The marginal stability is given by the critical value of $\alpha$ satisfying $$ \alpha_c H / 2 = \coth \, h \, (\alpha_c H / 2) $$ whose solution is $\alpha_c H = 2.4$

\[\alpha^{2}\equiv \frac{N^{2}}{f^{2}}(k^{2}+l^{2})\] The flow is unstable if $\alpha H < 2.4$ $$ \frac{HN}{f} < \frac{2.4}{\sqrt{k^2 + l^2}} $$ Since all values of k and l are allowed, a value of \(k^2 + l^2\) low enough to satisfy this inequality can always be found. The flow is therefore always unstable (to low wave number disturbances).

For a north-south wave number $l = 0$, instability is ensured if the east-west wave number $k$ is small enough such that $$ \frac{HN}{f} < \frac{2.4}{k} $$

In a continuously stratified ocean, the speed of a long internal wave for the $n = 1$ baroclinic mode is $c = NH/\pi$, so that the corresponding internal Rossby radius is $c/f = NH/\pi f$