 Group velocity and polarization of an inertial wave

For $\omega =\sigma 2\Omega k_{\parallel }/k$, differentiating the dispersion relation gives the displayed group velocity. It is perpendicular to the wavevector. If $|k_{\parallel }|\ll k_{\perp }$, energy propagates almost along the rotation axis while the phase velocity is almost horizontal. In an orthonormal basis ${\mathbf{e}}_1,{\mathbf{e}}_2$ normal to $\mathbf{k}$, with ${\mathbf{e}}_2=\hat{\mathbf{k}}\times {\mathbf{e}}_1$, the complex velocity is $A({\mathbf{e}}_1-\sigma i{\mathbf{e}}_2)$ and its real oscillation is circular. In the anisotropic limit the vertical and transverse-horizontal velocity amplitudes are comparable, while the horizontal component parallel to ${\mathbf{k}}_{\perp }$ is smaller by $|k_{\parallel }|/k_{\perp }$.