Exact solutions for rotating vortex arrays with finite-area cores

Abstract

A class of explicit solutions of the two-dimensional Euler equations consisting of a finite-area patch of uniform vorticity surrounded by a finite distribution of co- rotating satellite line vortices is constructed. The results generalize the classic study of co-rotating vortex arrays by J. J. Thomson. For N satellite line vortices (N [ges ] 3) a continuous one-parameter family of rotating vortical equilibria is derived in which different values of the continuous parameter correspond to different shapes and areas of the central patch. In an appropriate limit, vortex patch equilibria with cusped boundaries are found. A study of the linear stability is performed and a wide range of the solutions found to be linearly stable. Contour dynamics methods are used to calculate the typical nonlinear evolution of the configurations. The results are believed to provide the only known exact solutions involving rotating vortex patches besides the classical Kirchhoff ellipse.