A new family of uniform vortices related to vortex configurations before merging

Abstract

Stimulated by experimental observations of vortex merging, we compute a new family of uniform-vorticity steady solutions of the Euler equations in two dimensions. In experiments with two co-rotating vortices, one finds that, prior to the convective merging phase, and the formation of vortex filaments, the initial pair diffuses into a single structure (with two vorticity peaks) in the form of a symmetric ‘dumb-bell’. In the present computations, our exploration of the existence of vortex solutions has been guided by the streamline patterns of the co-rotating reference frame, and by the simple concept that the vortex boundary must be one of these streamlines. By varying the parameters which define the vortex patches, we find a family of vorticity structures which pass from the limiting case of point vortices, through the case of two equal co-rotating uniform vortices (as previously computed by Saffman & Szeto 1980; Overman & Zabusky 1982; Dritschel 1985), to the regime where the vortices touch in the form of a dumb-bell. Further exploration of this family of solutions leads to an elliptic vortex, which joins precisely to the local transcritical bifurcation from elliptic vortices with $n\,{=}\,4$ perturbation symmetry that was found by Kamm (1987) and Saffman (1988). Finally, one reaches a limiting ‘cat's-eye’ vortex patch of two-fold symmetry ($m\,{=}\,2$), which constitutes an extension to the limiting shapes of $m$-fold symmetry ($m \,{>}\, 2$) found by Wu, Overman & Zabusky (1984).