Vortex Motion and the Geometric Phase

B. N. Shashikanth, Department of Aerospace Engineering, University of Southern California

The existense of a non-trivial phase change in adiabatic evolutions of certain vortex configurations in 2-D incompressible, inviscid flows will be shown. The phase change is identified with the, by now well-documented, geometric phase (Berry's phase, Hannay angle) occuring in various classical and quantum systems. The calculation of the phase is performed using multi-scale asymptotics. In the geometric interpretation, the phase is shown to be the holonomy of a connection on an appropriately defined bundle. Three canonical point vortex configurations in which the phase appears will be first discussed. The planar configurations are a three-vortex problem, a particle and a vortex in a circle, and a particle in the model of a mixing layer flow in which an infinite number of vortices undergo subharmonic pairing. The phase appears as an $O(1)$ term in the relative angle variable of a point vortex or a particle in the flow at the end of one long time period of the vortex motion. The phase term is of the form $\theta_g=f(\Gamma_k,C) \cos 2\theta(0)$, where $f$ is a function of the vortex strengths $\Gamma_k$ and the periodic vortex orbit $C$, and $\theta(0)$ is the initial condition. With a view to applications, it will then be shown that the length formula for the long time growth of a passive interface in these flows inherits the geometric phase effect and shows the characteristic splitting into a `dynamic' part and a `geometric' part. The geometric part depends on the geometric phase $\theta_g$ for a particle in the flow and is given by $L_g=-\int_{\xi_A}^{\xi_B}d(\xi \theta_g)$, where $\xi$ parametrizes the interface curve joining particles $A$ and $B$ at $t=0$. Finally, the phase calculation for a system of two elliptical vortex patches will be presented. The phase appears in the orientation angle of each elliptical patch and is of the form $\theta_g=f(\Gamma_k,\lambda(0))\cos 2\theta(0)$, where $f$ depends on the patch strengths $\Gamma_k$ and initial aspect ratio $\lambda(0)$ of the patch.