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Geometric phases for corotating elliptical vortex patches

Shashikanth, B. N. and Newton, P. K. (2000) Geometric phases for corotating elliptical vortex patches. Journal of Mathematical Physics, 41 (12). pp. 8148-8162. ISSN 0022-2488. doi:10.1063/1.1320855.

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We describe a geometric phase that arises when two elliptical vortex patches corotate. Using the Hamiltonian moment model of Melander, Zabusky, and Styczek [J. Fluid Mech. 167, 95–115 (1986)] we consider two corotating uniform elliptical patches evolving according to the second order truncated equations of the model. The phase is computed in the adiabatic setting of a slowly varying Hamiltonian as in the work of Hannay [J. Phys. A 18, 221–230 (1985)] and Berry [Proc. R. Soc. London, Ser. A 392, 45–57 (1984)]. We also discuss the geometry of the symplectic phase space of the model in the context of nonadiabatic phases. The adiabatic phase appears in the orientation angle of each patch—it is similiar in form and is calculated using a multiscale perturbation procedure as in the point vortex configuration of Newton [Physica D 79, 416–423 (1994)] and Shashikanth and Newton [J. Nonlinear Sci. 8, 183–214 (1998)], however, an extra factor due to the internal stucture of the patch is present. The final result depends on the initial orientation of the patches unlike the phases in the works of Hannay and Berry [J. Phys. A 18, 221–230 (1985)]; [Proc. R. Soc. London, Ser. A 392, 45–57 (1984)]. We then show that the adiabatic phase can be interpreted as the holonomy of a connection on the trivial principal fiber bundle pi:T2×S1-->S1, where T2 is identified with the product of the momentum level sets of two Kirchhoff vortex patches and S1 is diffeomorphic to the momentum level set of two point vortex motion. This two point vortex motion is the motion that the patch centroids approach in the adiabatic limit.

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Additional Information:©2000 American Institute of Physics. (Received 16 December 1999; accepted 6 September 2000) We would like to thank Fred Browand and Jerry Marsden for bringing the MZS model to our attention. P.K.N. would like to thank J. Hannay for discussions regarding the initial condition dependence on calculations of the phase. B.N.S. would also like to thank Matthew Perlmutter for assistance in computing the coadjoint isotropy subgroups, and Richard Murray for the support of a postdoctoral fellowship. P.K.N. was partially supported by NSF-DMS-9800797.
Subject Keywords:vortices; rotational flow; perturbation theory; geometry
Issue or Number:12
Record Number:CaltechAUTHORS:SHAjmp00
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:2567
Deposited By: Archive Administrator
Deposited On:10 Apr 2006
Last Modified:08 Nov 2021 19:49

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