Two vortex rings produce chaos

Abstract. – We study the streamlines of the velocity field produced by two unlinked vortex rings. We find that two vortex rings can produce chaos and observe a route to chaos directly from periodic orbits. From the case study of numerous ring configurations we give conditions for integrability of streamlines. We also find that near the larger ring or the stronger ring there are more stable regular structures. Vortices are abundant in many fluid flows in nature. The most spectacular examples are tornadoes. Visualization of coherent vortical structures in turbulent fluid flows leads to the hope that complex turbulent flows might be modeled by vortex systems. Even relatively simple and static or slowly moving vortex structures could be capable of producing highly chaotic flow fields. There has been a significant amount of work on point vortices in two dimensions [1], but relatively little work on three-dimensional vortex flow. The frequently studied problem of coaxial vortex rings can also be reduced to a two-dimensional description. In the early 1990s, there have been studies on the superposition of spherical vortices in a steady flow and chaotic streamlines have been found in some cases [2–5]. A fundamental question is what vortex structures possess chaotic flow fields, in contrast to a periodic or quasi-periodic behavior [6]. In this letter, we consider the three-dimensional steady-state motion induced by two un-linked vortex rings. For a review on vortex rings, see ref. [7]. Our major objective is to investigate the circumstances that give rise to regular flow fields. This knowledge would be useful for illuminating and anticipating the flow properties of more general vortex structures. We use the well-known Helmholtz vortex ring solution, which is a circular vortex ring with infinitely small core. Its velocity field can be expressed in terms of a stream function ψ [8], ψ =