Resumen del producto

Group scattering of an initially compact two-dimensional configuration of N pointvortices with strengths of different signs in an unbounded ideal fluid is investigated. Thisphenomenon manifests itself in a permanent increase of the vortex cloud, accompaniedby mixing, clustering and emission of vortices. Possible mechanisms for such an evolutionare considered, and the necessary condition is formulated. An invariant function of relativemotion, F, is introduced, which is a combination of the first integrals of the system andcan be expressed through 2(N-2) independent phase variables. We use this functionto construct vortex equilibria and to investigate their nonlinear stability. Cases of N=3and N=5 (for vortex configurations which possess central symmetry), when F dependson two variables only, are examined in detail. In addition to the familiar self-similarscattering (or collapse), we found a regime of asymptotically self-similar scattering (butnot collapse) which occurs from any trajectory inside a certain finite domain in thephase space. A similar domain exists also for the case of dipole emission. A periodicabsolute motion consisting of alternating stages of expansion and convergence of vorticesis discussed. These mechanisms also work during group scattering. Simple examplesof this phenomenon are given for the case of five vortices. Finally, group scattering isdemonstrated for an initially compact cloud of 1009 vortices with strengths randomlydistributed in the range (-1, 1). The area of the main part of the vortex cloud grows withtime according to an almost linear law characteristic of self-similar motion.

Palabras clave: nonlinear instability; vortex dynamics

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