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G.A. Gottwald and D.E. Pelinovsky

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On the impossibility of solitary Rossby waves in meridionally unbounded domains

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**Abstract:**

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.

**Keywords**:

quasi-geostrophic equation; Rayleigh–Kuo eigenvalue problem; Korteweg-de Vries equation; mod-
ified Korteweg-de Vries equation; Zakharov-Kuznetsov equation; Rossby waves