Vortex Problems, Rotating Spiral Structures, and the Hannay-berry Phase

This paper describes the occurence of phase anholonomies in the context of point vortex problems for two-dimensional incompressible ows. After giving a brief description of anholo-nomic eeects in other contexts, we focus attention on the restricted three-vortex problem and a simpler modiied problem where thèHannay-Berry' phase can be computed using multi-scale asymptotic methods. Our main emphasis in this paper is to show how the Hannay-Berry phase arises as the leading term in an asymptotic expansion as the result of a non-uniform limit process. We show how it arises when computing the long time growth rate of passive scalar interfaces as they wrap around vortex cores in the presence of a slowly varying background eld due to other vortices, and discuss the results in the context of`spiral-vortex' models for 2D turbulence.