Large smoke rings with concentrated vorticity

In this paper we study an incompressible inviscid fluid when the initial vo rticity is sharply concentrated in N disjoint regions. This problem has bee n well studied when a planar symmetry is present, i. e., the fluid moves in R-2. In this case we know that, when the diameter sigma of each region sup porting the vorticity is very small, the time evolution of the fluid is qui te well described by a dynamical system with finite degrees of freedom call ed the "point vortex model.'' In particular the connection between this mod el and the Euler equation has been proved rigorously as sigma-->0. In the p resent paper we discuss the "stability'' of the point vortex model with res pect to a particular small perturbation of the planar symmetry. More precis ely we consider a fluid moving in R-3 with a cylindrical symmetry without s wirl in which each vortex is no longer a straight tube, but a vorticity rin g. We prove that large annuli of radii r approximate to sigma(-beta) b for any beta>0 remain "localized'' and hence we obtain the point vortex model a s sigma-->0. (C) 1999 American Institute of Physics. [S0022-2488(99)02701-2 ].