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.
Publisher
Cambridge University Press (CUP)
Subject
Mechanical Engineering,Mechanics of Materials,Condensed Matter Physics
Cited by
51 articles.
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