The structure, stability and interaction of geophysical vortices

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Vortical structures or swirling masses of fluid abound in the Earth's atmosphere and oceans, environments strongly influenced by both the stable vertical density stratification and the planetary rotation. These structures, or vortices, exist over a wide range of spatial scales and generally at huge Reynolds numbers. Vortices are conspicuous features of planetary atmospheres in general, and they are believed to play a central role in shaping planetary-scale circulations. Their interactions can be extraordinarily complex, and to this day we have virtually no understanding how these interactions contribute toward the collective motion of the atmosphere and the oceans as a whole. The main objectives of this proposal is to provide a complete description of vortex stability and vortex interactions in geophysical flows.The planetary rotation is important when its associated vorticity is comparable to or larger than the relative vertical vorticity. This can be expressed by saying that the Rossby number Ro, the ratio of relative to planetary vorticity, is small compared to 1. On the other hand, the stratification is important when the buoyancy frequency is larger than the horizontal vorticity. Likewise this can be expressed by saying that the Froude number Fr, the ratio of horizontal vorticity to buoyancy frequency, is small. When the Froude number is smaller than or comparable to the Rossby number, itself small, the system of governing equation can be greatly simplified. Then, the flow is well modelled by the `quasi-geostrophic' (QG) equations. The QG model has enabled a comprehensive exploration of basic vortex dynamics, from isolated vortex equilibria and stability to 3D QG turbulence. We now understand why vortices in the QG model tend to be robust, how they react to external shear and strain, what precipitates strong interactions between both like-signed and opposite-signed vortices (potential vorticity anomalies), as well as general properties of vortex populations in turbulence. The important question is: how well do these results apply to finite Ro and Fr? Or, how are the QG results altered when using the full equations of motion? We intend to answer these fundamental questions.When Ro and Fr are not small, two new features arise. The first is the appearance of Inertia-Gravity Waves (IGWs) at super-inertial frequencies, i.e. at frequencies larger than those associated with the vortical motion. The IGWs often induce weak motions compared to those induced by the vortices, and therefore IGWs tend to be of secondary importance in many circumstances. Another feature arising from finite Ro and Fr is the added contribution of ageostrophic motions, which are missing in the QG model. Ageostrophic motions differ from the high frequency IGWs in that they are directly associated with vortices: they are generated in response to the instantaneous potential vorticity (PV) distribution and have a direct advective effect on the PV evolution. The retention of ageostrophic motion thus has a significant impact on the flow and implies potentially large departures from QG dynamics. It is this important difference between the full equations and the QG approximation that underlies the proposed work. We aim to understand and quantify these effects on vortex motions and vortex stability properties.