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

Abstract

Summary.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 → 0. In this way, we can see that the → 0 problem is fundamentally different from the = 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 .

Geometric phases for corotating elliptical vortex patches

“…Using scaling arguments as in Ref. 22, one would expect that, as ⑀ →0, there is an O(1) contribution to the angle change of the patches at the end of the time period T l . The vector field considered is …”

“…18 Later, Marsden, Montgomery, and Ratiu 19 ͑see also Montgomery 20 and an introduction in Marsden and Ratiu 21 ͒ developed further the geometric theory in both adiabatic and nonadiabatic settings. We refer the reader to the references and discussion in Shashikanth and Newton 22,23 and Newton 24 for background as well as discussions of its relevance in the context of point vortex theory.…”

“…Geometric phases are then calculated explicitly in an adiabatic setting defined as in the point vortex configurations considered in the work of Newton 25 and Shashikanth and Newton. 22,23,26 The two length scales in the problem, due to the interpatch separation and the patch size, define two corresponding time scales which represent the ''slow'' point-vortex-like motion of the centroids and the ''fast'' rotation rates of the Kirchhoff-ellipse-like patches, respectively. The question we address is: what is the change in the orientation angle of each patch at the end of one appropriately defined time period of the centroid motion?…”

Vortex Motion and the Geometric Phase. Part II. Slowly Varying Spiral Structures

“…Assume that the interface is transversal at every point to the circular streamlines of the point vortex flow. 1 Consider now the evolution of an arbitrary particle (r 0 (t), θ 0 (t)) on the interface with r…”

“…The flowfields that we consider were introduced in Part I [1], [13] in the context of geometric phases in Hamiltonian systems with slowly varying parameters (see references therein). We identified three simple point-vortex configurations that, in an adiabatic setting, exhibit a geometric phase in the evolution of a "phase object".…”