Resonant Interaction of Rossby Waves; Helmholtz and Obukhov Singular Vortices; The Kirchhoff Equations

Abstract

The theory of wave processes tells us that the energy of waves of any nature propagates not with the phase velocity but with the group velocity

$$ \mathbf{C}_{gr}=\nabla_{\mathbf{k}}\omega,\quad \omega=\omega ( \mathbf{k} ), $$

where ω(k) is the dispersion relation for waves of this nature, and ∇ _k is the gradient operation in the k-space of wave numbers. The waves whose phase velocity does not coincide with the group velocity are called dispersion waves. So, for instance, are gravitational-gyroscopic waves with the dispersion relation (7.20), according to which they isotropically propagate in space.