Asymptotic expansions and monopolar solitary rossby vortices in barotropic and two-layer models

Motivated by observational and experimental data on large-scale vortices close to the geostrophic balance we perform an asymptotic analysis of the barotropic and baroclinic rotating shallow-water systems in order to describe slowly-evolving monopolar coherent structures. The spherical and the paraboloidal geometries which are, respectively, relevant for planets and experiments with fast rotation are both studied. We analyse in detail the problem of long-living localised vortices of axisymmetric shape and find the anticyclonic solutions of a prescribed form on the sphere. In the paraboloidal geometry an arbitrary strongly non-linear axisymmetric structure may drift preserving its form for long enough times. We extend this study to the baroclinic two-layer model and find similar results with an important distinction that solitary vortices may be cyclones as well, their sign depending on the physical parameters of the model. The applications of this results in astrophysical, oceanographic and laboratory context are discussed.