The evolution of Kirchhoff elliptic vortices

A Kirchhoff elliptic vortex is a two-dimensional elliptical region of uniform vorticity embedded in an inviscid, incompressible, and irrotational fluid. By using analytic theory and contour dynamics simulations, we describe the evolution of perturbed Kirchhoff vortices by decomposing solutions into constituent linear eigenmodes. With small amplitude perturbations, we find excellent agreement between the short time dynamics and the predictions of linear analytic theory. Elliptical vortices must have aspect ratios less than a /b=3 to be completely stable. At late times, unstable perturbations evolve to states consisting of filaments surrounding and connecting one or more separate vortex core regions. Even modes have two different evolution paths accessible to them, dependent on the initial phase. Ellipses can first fission into more than one separate region when a /b=6.046, from the negative branch m=4 mode. Increasing the perturbation amplitude can result in nonlinear instability, while the perturbation is still small relative to any vortex dimension. For the lowest m modes, we quantify the transition from linear to nonlinear behavior. © 2008 American Institute of Physics. DOI: 10.1063/1.2912991