The evolution of an intense barotropic vortex on the β-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 Ta (a typical time required for a fluid
particle to move a distance of the order of the vortex size), the wave time scale Tw
(the typical time it takes for the vortex to move through its own radius), and the distortion time scale
Td (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,
Ta [Lt ] Tw [Lt ] Td,
and therefore one can consider the vortex evolution as consisting of three different stages.
The first one, t [les ] Tw, is
dominated by the development of a near-field dipolar circulation (primary β-gyres)
accelerating the vortex. During the second stage,
Tw [les ] t [les ] Td, 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.