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

Content Provider	Springer Nature Link
Author	Shashikanth, B. N. Newton, P. K.
Copyright Year	1999
Abstract	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 ΔL(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, $$\Delta L = \lim _{T \to \infty } \Delta L\left( T \right) = - \int_{\xi _A }^{\xi _B } {d\left( {\xi \theta _g } \right)}$$ , where ξ is the radial coordinate parametrizing the interface at t = 0.
Starting Page	233
Ending Page	254
Page Count	22
File Format	PDF
ISSN	09388974
Journal	Journal of Nonlinear Science
Volume Number	9
Issue Number	2
e-ISSN	14321467
Language	English
Publisher	Springer-Verlag
Publisher Date	1999-01-01
Publisher Place	Berlin, Heidelberg
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Analysis Theoretical, Mathematical and Computational Physics Mechanics ApplicationMathematics/Computational Methods of Engineering Economic Theory
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Engineering Modeling and Simulation

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