Transition to instability of the leapfrogging vortex quartet

Roy Goodman and Brandon Behring. Mechanics Research Communications, 2023.
Feb 9, 2023·
Roy Goodman
Roy Goodman
Brandon Behring
Brandon Behring
· 0 min read
Abstract
The point-vortex system is a system of longstanding interest in nonlinear dynamics, describing the motion of a two-dimensional inviscid fluid that is irrotational except at a discrete set of moving point vortices, at which the vorticity diverges. The leapfrogging orbit consists of two rotating pairs of like-signed vortices which, taken as a quartet, propagate at constant velocity. It is known that if the two pairs are initially widely separated, the motion is stable, while if they are closer together it becomes unstable, with this relation represented by a dimensionless parameter \(\alpha\) defined in the text. We here demonstrate analytically that the transition from stability to instability happens at a critical value \(\alpha = \phi^{-2}\), where \(\phi\) is the golden ratio. This value had been hypothesized based on careful numerics by Tophøj and Aref, and by the present authors using a semi-analytic argument but not previously demonstrated through exact analysis.
Type
Publication
Mechanics Research Communications, 128, 104068
publication published
Vortices Bifurcation
Roy Goodman
Authors
Roy Goodman
Professor, Associate Chair for Graduate Studies, Department of Mathematical Sciences

I am a professor in the Department of Mathematical Sciences at NJIT. My research applies methods of dynamical systems to (conservative, dispersive) nonlinear wave equations arising, for example, in optics and fluid mechanics. I am interested in dynamical and bifurcation phenomena, and I use techniques from Hamiltonian mechanics and perturbation theory, as well as extensive computations. My current research in vortex dynamics is supported by the NSF under DMS-2206016.

I led the effort to start the SIAM New York-New Jersey-Pennsylvania Section (SIAM NNP) and was its first president in 2023-2024. I have handed over the reins to a new leadership group.

Brandon Behring
Authors
Brandon Behring
Data Scientist, Prudential Insurance

Brandon defended his dissertation on leapfrogging point vortices in August 2020. He spent about a year as a postdoc in the Dynamical Systems Laboratory at the NYU Tandon School of Engineering. He is now a data scientist at Prudential Insurance, and has a very cute baby girl.

Brandon completed his Ph.D. in Mathematical Sciences at NJIT. His dissertation project concerns the stability and associated dynamics of systems of leapfrogging vortices.