TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 268, Number 1, November 1981 VORTEX RINGS: EXISTENCE AND ASYMPTOTIC ESTIMATES BY AVNER FRIEDMAN1 AND BRUCE TURKINGTON Abstract. The existence of a family of steady vortex rings is established by a variational principle. Further, the asymptotic behavior of the solutions is obtained for limiting values of an appropriate parameter X; as A —» oo the vortex ring tends to a torus whose cross-section is an infinitesimal disc. 0. Introduction. The study of steady vortex rings in an ideal fluid has been the subject of many investigations (see, for example, [3], [19] and the references given there). The classical examples are Helmholtz's rings of small cross-section [17] and Hill's spherical vortex [18]. A general existence theorem for vortex rings was first established by Fraenkel and Berger [13] (see also the very recent work [5], [20] with a similar approach); this paper also contains an excellent survey of the subject. The approach in [13] is based on a variational principle for the stream function. More recently Benjamin [4] developed a new approach based on a variational principle for the vorticity. This approach is more natural since (i) the vorticity has compact support (whereas the stream function does not) and (ii) the quantities involved in the variational principle have direct physical significance. In this paper we establish the existence of vortex rings by a new method. As in [4] we formulate the problem in a variational form for the kinetic energy as a functional of the vorticity. We take the admissible functions to vary in the set S^ of functions f(x) satisfying: f (x) = f (r, z) = f (r, - z) where x = (r, 0, z), (0.1) i , , - j r2$(x) dx = I, j$(x)dx<l, 0 <?(*)< A, i.e., an axisymmetric flow with prescribed impulse (= 1), circulation (< 1) and vortex strength (< X); in [4] f is taken to vary over all rearrangements of a given function f0(r, z). Our approach seems technically simpler; it has the further advantage that it leads to vortex rings with, essentially, any given vorticity func- tion, such as (0.2) fit) = cl{l>0] (c > 0), (0.3) fit) = c(t +)p (c>0,B>0). The method of solving our variational problem is in some sense an adaptation of the method of Auchmuty [1] and Auchmuty and Beals [2] (see also [14]-[16]) who Received by the editors May 22, 1980. AMS (MOS) subject classifications (1970). Primary 35J20, 76G05; Secondary31A15, 35J05. 1 The first author is partially supported by National Science Foundation Grant MCS-781 7204. © 1981 American Mathematical Society 0002-9947/81/0000-0500/$10.25 1 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use