Chaotic phenomena in the interaction of vortex rings

Abstract

The problem of interaction of several coaxial vortex rings in an inviscid fluid is investigated numerically. It is assumed that the core shape of the vortex rings remain circular. At the initial time the rings are located at the same distance ρ0 from the center of the system. This distance is a control parameter of the problem. The cases of interaction of three, four, and five vortex rings are studied. It is shown, that in spite of the nonintegrability of the problem, there are certain domains of values ρ0 where the motion of the vortex rings is quasiperiodical. The transition from one such domain to another occurs always through a domain of chaotic interaction. The results of the interaction of the vortex rings are compared with the interaction of their plane analogies–vortex pairs.

Few-particle vortex cluster equilibria in Bose–Einstein condensates: existence and stability

“…In this context, it is also possible to write ordinary differential equations characterizing the interaction of the rings and their intrinsic translational dynamics as e.g. in [45]. However, we have not been able to identify simple ODEs that would describe the motion of such rings in a three-dimensional parabolic trap -a key ingredient for the system of ODEs, as we saw above for the case of vortices.…”

Single and multiple vortex rings in three-dimensional Bose-Einstein condensates: Existence, stability, and dynamics

“…For a set of co-axial VRs along the z-axis, a naïve approach to combine the above-mentioned VR-VR and VR-trap contributions would consist of simply adding the corresponding reduced dynamics at the level of the effective ordinary differential equations (ODEs) on the VR radii r i and positions z i . However, perhaps somewhat surprisingly, this approach turns out to produce nonHamiltonian ODEs because the two main contributions, namely the VR-VR interaction [36] and VR-trap interaction [44], originate from energy terms with different canonical variables (see below). Therefore, this approach -although capable of reasonably predicting the positions for stationary multi-VR configuration (results not shown here)-fails to describe the actual dynamics of trapped multi-VR configurations.…”

“…[44] and (ii) the VR-VR energy, denoted by E VR-VR , described in Ref. [36]. Importantly, both E VR-T and E VR-VR contain the VR self-induced velocity that is responsible for a single VR to always have an intrinsic velocity.…”

“…Here, the coherent waves can be thought of as individual "particles" that have kinematics and inter-particle dynamics that can be approximated by suitable particle models. Although attempts have been made to explore a single VR as such a particle, to describe its vibrational modes [32][33][34][35], as well as to explore multiple VRs in a homogeneous (untrapped) medium [36][37][38], to the best of our knowledge no such attempt has been made for the case of multiple trapped VRs. It is the purpose of this work to contribute to this direction by developing a systematic approach to study this problem.…”

“…(ii) the effect of a trap on a single VR [32][33][34]44], and the (iii) interaction of multiple VRs in the absence of a trap [36][37][38].…”