Dynamics and scattering of vortices and vortex rings

This project focuses on a classical subject that has found new relevance due to recent experimental results. Nineteenth century researchers including Helmholtz and Kirchhoff wrote down a system of equations describing how point vortices---infinitesimal whirlpools---induce motion in a fluid, which in turn affects the motion of the vortices. A similar system exists that describes the more complex problem of vortex rings. Both Bose-Einstein Condensates (BEC), an exotic phase of matter first produced experimentally in the 1990s, and superfluids including liquid helium, can support point vortices and vortex rings. Recent experimental work has imaged interacting point vortices in a BEC, showing strong agreement with mathematical predictions. This project will mathematically investigate a number of phenomena in the interactions of point vortices, and in the closely-related problem of interacting vortex rings. In particular, there is a motion called the leapfrogging orbit in which a pair of vortex rings propagate along a line, periodically expanding and contracting while moving through each other (this is a well-known trick performed by cigar smokers with smoke rings). This is well understood in the case of two vortex rings, and computer simulations have generalized this phenomenon to three or more rings, but have not performed a full mathematical study. This project will perform the first detailed mathematical study of the generalized problem. It is likely impossible to write down exact formulas for these solutions, so numerical techniques will be essential, especially numerical continuation and bifurcation methods and a recently-developed method called Lagrangian descriptors, which allows us to see structure in otherwise opaque chaotic systems. A second problem will study a phenomenon called chaotic scattering of point vortices: A pair of vortices can be made to propagate at a constant speed along a straight line. Collisions between such pairs can lead to extremely complex dynamics which will be investigated using dynamical systems methods.This project will apply techniques from dynamical systems to two classes of problems in the classical subject of the dynamics of interacting point vortices or vortex rings. The first is several generalizations of the problem of leapfrogging vortices, in which two pairs of vortices travel along a straight line while repeatedly and periodically widening, then narrowing, with one pair passing through the other, and the lead changing. This has been very well-studied for point vortices, but less well for vortex rings. The study aims to generalize this system to systems of three or more pairs of vortices (or three or more vortex rings), and to leapfrogging orbits in other geometries such as on the surface of a sphere. While limited numerical studies exist of such orbits, they remain unexplored mathematically. Hamiltonian reduction is the technique at the center of our understanding of the four-vortex system, leading two a two-degree-of-freedom Hamiltonian that is amenable to further analytical simplification. With larger sets of vortices, Hamiltonian reduction does not lead to systems of small enough dimension for the method to provide much insight, and other methods are required, including numerical continuation and bifurcation methods. The project will also make use of the newly-developed method of Lagrangian descriptors, although the size of the system means that choices need to be made about how to find surfaces in initial-condition space that will lead allow insight into the dynamics. The second class of problems is the chaotic scattering of colliding point vortex pairs. This system has some features in common with chaotic scattering studied in earlier NSF-sponsored research: a two degree-of-freedom system consisting of a slow dynamical system with a separatrix coupled to a fast dynamical system. However, the dynamics in this case are far more complex: there is no small parameter. Instead, the separation of time scales arises due to the dynamics but may not hold for all time. In the previous system, a separatrix describes how solutions escape to infinity, while the mechanism of escape here is still unknown.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.