Nonlinear Analysis, Theory, Methods & Applicanonr. Vol. 9. No. 4, pp. 351469, 1985. 0362-546syigS53.00 + .oO Rited in Great Britain. @ 1985 Pergamon Press Ltd zyxwvutsrq COROTATING STEADY VORTEX FLOWS WITH N-FOLD SYMMETRY BRUCE TURKINGTON Department of Mathematics and Statistics, University of Massachusetts, Amherst, Massachusetts 01033. U.S.A. zyxwvutsrqp (Received 17 March 1984; received for publication 22 May 1984) Key words and phrases: Vortices, Euler fluid dynamical equations, variational method, free boundary problem, desingularization. IN THIS paper we study certain vertical flows of an ideal fluid in two dimensions which are steady relative to a uniformly rotating frame. In particular, we confine our analysis to those flows for which the vorticity relative to the rotating frame possesses N-fold rotational symmetry for some given N = 2,3, . . .; that is, we require that the relative vorticity be invariant under rotation through arbitrary multiples of 2zjN. As in our earlier related work [19,20], these (relative) steady solutions of the Euler fluid dynamical equations are obtained as extremals for a certain variational problem. The solutions then represent steady vortex flows in the sense defined in [19]-namely, each such flow has a prescribed distribution of relative vorticity and is separated into regions of zero and of positive relative vorticity. In 1883, Thomson [18] considered the flow due to a configuration of N (unit) point vortices spaced equally around the circumference of a circle. The relative vorticity of such a configur- ation on a circle of radius a is given by N-l o(x) = “ZO 6(x - z@)) (x E R2) with x = (T, 0) and z @)= (a, 2~m/N) in usual polar coordinates; 6(x) denotes the unit (Dirac) delta measure concentrated at x = 0. The induced flow is found to be steady relative to a rotating frame with angular velocity -(N - 1)/4~n2 2. Contemporary studies have been addressed to the numerical analysis of analogous configurations in which point vortices are replaced by vortex patches, the delta measures being replaced by the (scaled) characteristic functions of sets whose boundaries are to be determined. Zabusky et al. [21-231 and Saffman and Szeto [ 171 have computed a family of corotating (symmetric) vortex pairs-the case when N = 2-using the so-called contour dynamics method to solve the free boundary problem. Dritschel[8] has recently extended these computations to the case when 2 G N 6 6, remarking that tornadoes are often observed to consist of N corotating vortices with N > 2. The purpose of the present work is to provide existence theorems and certain other qualitative results for these N-fold symmetric corotating steady flows which complement the motivating numerical results. Accordingly, we focus our attention on vortex patches even though the variational method we employ can be applied to continuous distributions of vorticity; in doing so we maintain a connection with the known numerical results while, at the same time, we avoid some technical complications in the analysis. 351