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

Abstract

Summary.We derive formulas for the long time evolution of passive interfaces in three "canonical" incompressible, inviscid, two-dimensional flow models. The point vortex models, introduced in Part 1 [1] are (i) a "restricted" three-vortex problem, (ii) a vortex and a particle in a closed circular domain, and (iii) a particle in the flowfield of a mixing layer model undergoing a vortex pairing instability. In each configuration, it was shown in Part 1 that the passive particle exhibits a geometric or Hannay-Berry phase over long time periods induced by the slowly varying periodic background field. In this paper we show how the formula for the evolution of a passive interface driven by the dynamics of the vortices inherits this geometric phase effect. The interface wraps into a spiral formation around the "parent" vortex, with a slowly varying component induced by the farfield vorticity. The length formula for the long time growth of the slowly rotating spiral decomposes into a "dynamic" part and a "geometric" part. The dynamic part is the length in the "unperturbed" system-i.e., in the absence of the background field-and the geometric part is the contribution of the geometric phase θ g for a passive particle in the flow. We derive the following simple formula for an interface along a smooth curve joining two arbitrary particles labelled A and B. Define L(T ) as the difference in interface lengths between the "unperturbed" system and the "perturbed" slowly varying system at the end of the long time period T . In each case,where ξ is the radial coordinate parametrizing the interface at t = 0. * Current address: Control and Dynamical Systems,

Geometric phases for corotating elliptical vortex patches

“…16,17 In particular, to make direct contact with this work, one would average the initial orientations (0)Ϫ i (0) over the full circle and obtain zero for these averaged phases. In the context of point vortex models, we have shown 23 that these phases, despite averaging to zero, are important when computing quantities in the flow, such as interface stretching rates. We expect the same to be true when one replaces point vortices with vortex patches.…”

“…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?…”