On the long-term evolution of an intense localized divergent vortex on the beta-plane

The evolution of an intense barotropic vortex on the beta -plane is analysed for the case of finite Rossby deformation radius. The analysis takes into account conservation of vortex energy and enstrophy, as well as some other quantities, and therefore makes it possible to gain insight into the vortex evolution for longer times than was done in previous studies on this subject. Three characteristic scales play an important role in the evolution: the advective time scale T-a (a typical time required for a fluid particle to move a distance of the order of the vortex size), the wave time scale T-w (the typical time it takes for the vortex to move through its own radius), and the distortion time scale T-d (a typical time required for the change in relative vorticity of the vortex to become of the order of the relative vorticity itself). For an intense vortex these scales are well separated, T-a much less than T-w much less than T-d, and therefore one can consider the vortex evolution as consisting of three different stages. The first one, t less than or equal to T-w, is dominated by the development of a near-field dipolar circulation (primary beta -gyres) accelerating the vortex. During the second stage, T-w less than or equal to t less than or equal to T-d, the quadrupole and secondary axisymmetric components are intensified; the vortex decelerates. During the last, third, stage the vortex decays and is destroyed. Our main attention is focused on exploration of the second stage, which has been studied much less than the first stage. To describe the second stage we develop an asymptotic theory for an intense vortex with initially piecewise-constant relative vorticity. The theory allows the calculation of the quadrupole and axisymmetric corrections, and the correction to the vortex translation speed. Using the conservation laws we estimate that the vortex lifetime is directly proportional to the vortex streamfunction amplitude and inversely proportional to the squared group velocity of Rossby waves. For open-ocean eddies a typical lifetime is about 130 days, and for oceanic rings up to 650 days. Analysis of the residual produced by the asymptotic solution explains why this solution is a good approximation for times much longer than the expected formal range of applicability. All our analytical results are in a good qualitative agreement with several numerical experiments carried out for various vortices.