The Hannay-Berry Phase and Vortex Dynamics

Paul K. Newton, Department of Aerospace Engineering, University of Southern California, Los Angeles, CA 90089-1191

The Hannay-Berry phase (H-B phase) is a non-trivial phase change that can occur in Hamiltonian systems whose Hamiltonian $H(p,q;\vec{R}(\epsilon t))$ depends on a set of slowly changing parameters $\vec{R}$, where the parameters go around a closed circuit $\cal{C}$ in parameter space. It was discovered and subsequently studied in a paper by Berry in 1984 in the context of quantum mechanical problems where the equations are linear. The `semi-classical' limit was then studied and several classical examples were worked out by Hannay in 1985. Classically, the $O(1)$ phase discontinuity is analogous to the phase change that appears when a Foucault pendulum oscillates for a 24 hour period, as long as it is not located at the Equator or North pole. It is a {\it geometric} quantity as its value can be related to the geometric properties of the closed circuit $\cal{C}$, and not on such dynamical considerations as the time it takes to traverse the circuit.

This talk will focus on our applications of these ideas to point vortex dynamics in 2-D incompressible flow. The first part of the talk will focus on some basic canonical configurations and asymptotic methods to compute the H-B phase. We will emphasize the general point of view that the phase anholonomy arises from a non-uniform limiting procedure in which one quantity vanishes, the other goes to $\infty$, and the product exactly balances to leave an $O(1)$ term. The second part of the talk will focus specifically on the H-B phase which occurs during the vortex pairing stage of shear layer evolution in a two dimensional point vortex model. The talk will end with a description of how the methods can be used to derive estimates on the stretching rates of passive interfaces in flows modeled with point vortices. We will tie this discussion to some work on spiral structures in two dimensional turbulence discussed by other authors.