Dispersion Relation for Rossby Waves
$$
\omega
= -\frac{
\overbrace{\beta k}^{\substack{\text{$\beta$-effect} \\ \text{(zonal wavenumber $k$)}}}
}{
\underbrace{k^{2} + l^{2} + \frac{f_{0}^{2}}{c^{2}}}_{\substack{\text{total wavenumber $K^2$ plus} \\ \text{deformation term $f_0^2/c^2$}}}
}
$$
This is the dispersion relation for Rossby waves
$$
\begin{aligned}
&\underbrace{\text{Only $k$ in numerator; $l$ only in $K^{2}$}}_{\text{$\beta$-effect anisotropy}}
\Longrightarrow
\text{not horizontally isotropic; sign of $k$ sets phase direction (for $\beta>0$)}
\\
&\text{For stratified flows, replace $c$ by:}\quad
\begin{cases}
c^{2} = g'H &\text{(reduced-gravity 1½-layer)}\\
c = \dfrac{NH}{n\pi} &\text{($n$th mode of continuously stratified model)}
\end{cases}
\\[8pt]
&\text{Barotropic mode: } c \ \text{very large} \ \Rightarrow \ f_{0}^{2}/c^{2} \approx 0
\quad\Longrightarrow\quad
\omega \approx -\frac{\beta k}{k^{2} + l^{2}}
\end{aligned}
$$
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