Geometric phases for corotating elliptical vortex patches

Abstract

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 ͑1986͔͒ 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 ͑1985͔͒ and Berry ͓Proc. R. Soc. London, Ser. A 392, 45-57 ͑1984͔͒. 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 ͑1994͔͒ and Shashikanth and Newton ͓J. Nonlinear Sci. 8, 183-214 ͑1998͔͒, 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 ͑1985͔͒; ͓Proc. R. Soc. London, Ser. A 392, 45-57 ͑1984͔͒. We then show that the adiabatic phase can be interpreted as the holonomy of a connection on the trivial principal fiber bundle :T 2 ϫS 1 →S 1 , where T 2 is identified with the product of the momentum level sets of two Kirchhoff vortex patches and S 1 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.

Reconstruction phases in the planar three- and four-vortex problems

“…The cases N = 3 and N = 4 are considered. Adiabatic phases have been computed in this model previously by Newton [34] and by Shashikanth and Newton [42], and also in a model of interacting vortex patches by Shashikanth and Newton [43]. Pure reconstruction phases have been computed in an axisymmetric model of vortex rings in R 3 by Shashikanth and Marsden [44].…”

The Vortex Merger Factor in Aircraft Wake Turbulence

“…The prototype of a two-dimensional vortex is a vortex patch, defined as a connected, finite region of rotational fluid surrounded by fluid that is irrotational. The patch can be also viewed as the perpendicular section of an infinitely long rectilinear vortex tube whose vorticity distribution is invariant along the length of the tube [18]. If the region is circular of radius R and contains constant vorticity, ω = ω 0 , the vortex patch models a Rankine vortex, with the azimuthal velocity [19] v…”

Evaluation of the Biot–Savart Integral for Deformable Elliptical Gaussian Vortex Elements

“…recently studied the stability of a Gaussian vortex in an inhomogeneous fluid [36]. Similarly, a number of investigations have been conducted using interacting elliptical patches of constant vorticity (Kida vortices) to understand vortex mergers [25,33,44]. Elliptical Gaussian in linear flow field provides another family of base states for investigators to explore.…”