It is usual to omit the factor $\pi$ and define the Rossby radius $\Lambda$ in a continuously stratified fluid as
$$
\Lambda \equiv \frac{HN}{f}
$$
The condition \(\frac{HN}{f} < \frac{2.4}{k}
\) for baroclinic instability is therefore that the east-west wavelength be large enough so that:
$$
\lambda > 2.6 \Lambda
$$
The wavelength $\lambda = 2.6 \Lambda$ does not grow at the fastest rate
\[c = \frac{U_0}{2} \pm \frac{U_0}{\alpha H}
\sqrt{\left(\frac{\alpha H}{2} - \tanh \frac{\alpha H}{2}\right)
\left(\frac{\alpha H}{2} - \coth \frac{\alpha H}{2}\right)}
\Rightarrow \lambda_{\max} = 3.9 \Lambda
\]
the wavelength with the largest growth rate \(\lambda_{\max}\) is 3.9\(\Lambda\)
This is therefore the wavelength that is observed when the instability develops.
Typical values for $f, N,$ and $H$ suggest that
- \(\lambda_{\max} \sim 4000 \, \mathrm{km}\) in the atmosphere
- \(\lambda_{\max} \sim 200 \, \mathrm{km}\) in the ocean
which agree with observation.
Waves much smaller than the Rossby radius do not grow, and those much larger than the Rossby radius grow very slowly.
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