= Solution
For nonzero meridional <wavenumber>,
$$
\eta=L_D\sqrt{k^2+l^2}<\eta_c,
\qquad k^2<\frac{\eta_c^2}{L_D^2}-l^2.
$$
At least one growing zonally varying mode can exist only if
$$
\boxed{|l|<l_{\sup}=\frac{2.39936}{L_D}}.
$$
The upper value is a supremum approached as $k\to0$, where the growth itself tends to zero. Thus not every permitted meridional <wavenumber> has the same fastest-growing zonal scale; nonzero $l$ restricts the available band of $k$ and weakens the growth factor $k/\sqrt{k^2+l^2}$ at fixed $\eta$.
Using $N=\sqrt{N^2}$, not the tabulated squared value itself, gives
$$
\boxed{\begin{array}{c|r|r|r}
&L_D\ (\mathrm{km})&l_{\sup}\ (\mathrm{m^{-1}})&2\pi/l_{\sup}\ (\mathrm{km})\\\hline
\text{Earth oceans}&54.8&4.38\times10^{-5}&143\\
\text{Jupiter}&1129&2.12\times10^{-6}&2958
\end{array}}
$$
The planetary radii put these into angular context: $l_{\sup}R$ is about 280 for Earth and 149 for Jupiter, corresponding to minimum meridional angular <wavelengths> about $1.28^\circ$ and $2.42^\circ$, respectively. They are reciprocal-length wave numbers; the dimensionless angular counts should not be substituted for $l$ in the Cartesian dispersion relation. If a channel admits a half-wave across its width, the corresponding minimum width is $\pi/l_{\sup}$ rather than a full <wavelength>.
The oceanic scale is mesoscale, while the much larger deformation length makes the Jovian modes thousands of kilometres across. Both are small fractions of the planetary radius, consistent with using a local patch as a first approximation. Real boundaries, planetary-vorticity gradients, varying stratification and jets can change these limits; this simple model alone does not predict the complete observed flow pattern.
Finally, the source's phrase $c^2<0$ must be interpreted in the mean-flow frame. The actual criterion is $D<0$, or $(c-\Lambda H/2)^2<0$. In the original frame $c$ has nonzero <real part>, so $c^2$ is generally complex, not an ordered negative <real number>. For example, at $\eta=1$ and $\Lambda H=1$, $c=0.5+0.2511i$ and $c^2\simeq0.1870+0.2511i$. This is an unstable wave even though the literal printed inequality is inapplicable.
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