On the impossibility of solitary Rossby waves in meridionally unbounded domains

Evolution of weakly nonlinear and slowly varying Rossby waves in planetary atmospheres and oceans is considered within the quasi-geostrophic equation on unbounded domains. When the mean flow profile has a jump in the ambient potential vorticity, localized eigenmodes are trapped by the mean flow with a non-resonant speed of propagation. We address amplitude equations for these modes. Whereas the linear problem is suggestive of a two-dimensional Zakharov-Kuznetsov equation, we found that the dynamics of Rossby waves is effectively linear and moreover confined to zonal waveguides of the mean flow. This eliminates even the ubiquitous Korteweg-de Vries equations as underlying models for spatially localized coherent structures in these geophysical flows.