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

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 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 . Received September 4, 1997; second revision received January 16, 1998; accepted January 27, 1998