Vortex Motion and the Geometric Phase. Part I. Basic Configurations and Asymptotics

Abstract

The geometric, or Hannay-Berry, phase is calculated for three canonical point vortex configurations in the plane. The simplest configuration is the three-vortex problem with arbitrary (like signed) circulations, where two of the vortices are near each other compared to the distance between them and a third vortex. We show that the third (distant) vortex induces a geometric phase in the relative angle variable between the two nearby vortices. The second configuration is a particle and vortex in a circular domain. In this problem, the geometric phase is induced on the particle by the presence of the boundary. The third configuration is an infinite row of point vortices undergoing subharmonic pairing. In this case, a geometric phase is induced on a particle orbiting one of the vortices as the vortex pairs orbit each other. In each case we derive the formula for the geometric phase using an asymptotic procedure, then we give it a geometric interpretation. For the asymptotic derivation, we show how the geometric phase can be interpreted as the product of two terms, one of which goes to zero, the other to infinity. Because they go at rates that balance each other, there is a residual O(1) term in the limit e → 0. In this way, we can see that the e → 0 problem is fundamentally different from the e = 0 problem. For the geometric interpretation, we introduce a 1-form γ defined on the plane and show that the phase θ g in the appropriate angle variable can be constructed by taking the contour integral of this 1-form over the closed vortex path. In each case this gives the simple formula θ g = ∮ γ.