Kirchhoff Vortex Contour Dynamics Simulation

A Kirchhoff elliptic vortex is a 2D elliptical region (or “patch”) of uniform vorticity embedded in an inviscid, incompressible and irrotational fluid. G. Kirchhoff showed that these are exact solutions to the nonlinear Euler equations in 1876.
Subsequently, A. E. H. Love analyzed the linear stability of Kirchhoff vortices, and established that at large aspect ratios they are unstable. He also obtained analytic expressions for the oscillation frequencies and growth rates. A transcription of his paper, which appeared in the Proceedings of the London Mathematical Society in 1893, is included in the readme file.

In 1979, N. J. Zabusky, M. H. Hughes and K. V. Roberts introduced a numerical scheme now commonly known as “Contour Dynamics.” This has been a popular tool for simulation of inviscid discrete patches of vorticity. It is numerically efficient because following the evolution of a uniform vorticity region only requires the tracking of its boundary.

We implemented the contour dynamics algorithm in Matlab in order to re-examine the evolution of the Kirchhoff vortex, with an emphasis on the modes of the system. Two fitting routines are included which decompose the solutions into constituent linear eigenmodes. Some results from these routines will appear in Physics of Fluids in April 2008, and a preprint of this paper is also included in the readme file.