The evolution of Kirchhoff elliptic vortices

Abstract

A Kirchhoff elliptic vortex is a two-dimensional elliptical region of uniform vorticity embedded in an inviscid, incompressible, and irrotational fluid. By using analytic theory and contour dynamics simulations, we describe the evolution of perturbed Kirchhoff vortices by decomposing solutions into constituent linear eigenmodes. With small amplitude perturbations, we find excellent agreement between the short time dynamics and the predictions of linear analytic theory. Elliptical vortices must have aspect ratios less than a∕b=3 to be completely stable. At late times, unstable perturbations evolve to states consisting of filaments surrounding and connecting one or more separate vortex core regions. Even modes have two different evolution paths accessible to them, dependent on the initial phase. Ellipses can first fission into more than one separate region when a∕b=6.046, from the negative branch m=4 mode. Increasing the perturbation amplitude can result in nonlinear instability, while the perturbation is still small relative to any vortex dimension. For the lowest m modes, we quantify the transition from linear to nonlinear behavior.

Using Global Interpolation to Evaluate the Biot-Savart Integral for Deformable Elliptical Gaussian Vortex Elements

“…This particular example highlights the central role filamentation can play in asymmetrization when the core and filaments interact strongly to reduce the overall aspect ratio [22]. This process is ubiquitous in physical flows and with 2D elliptical vortices is often the dominant axisymmetrization process, although surface mode instabilities [24] and resonant wave-fluid interactions [37,19] can also play a role.…”

Explicit Equations to Transform from Cartesian to Elliptic Coordinates

“…It is the simplest example of non-smooth weak solutions to the Euler equations and has wide application in vortex dynamics [9][10][11]. It has a discontinuity of vorticity across its elliptical boundary …”

Parametric resonance in the dynamics of an elliptic vortex in a periodically strained environment

“…A large body of literature is devoted to the problem. Most of the papers consider spatial perturbations to the elliptic form in the case of con-stant linear deformation (Neu, 1984;Melander et al, 1986;Neu, 1990;Dritschel, 1990;Meacham et al, 1990;Legras and Dritschel, 1991;Kida and Takaoka, 1994;Miyazaki and Hanazaki, 1994;Bayly et al, 1996;Meacham et al, 1997;Mitchell and Rossi, 2008). A prominent result of this stability analysis is that an elliptic vortex is stable to linear perturbations of its form until its geometrical shape complies with the relation a/b ≤ 3, where a and b are the major and minor semi-axes of the ellipse.…”