Stability of leapfrogging vortex pairs: A semi-analytic approach

Abstract

We investigate the stability of a one-parameter family of periodic solutions of the four-vortex problem known as 'leapfrogging' orbits. These solutions, which consists of two pairs of identical yet oppositely-signed vortices, were known to W. Gröbli (1877) and A. E. H. Love (1883), and can be parameterized by a dimensionless parameter α related to the geometry of the initial configuration. Simulations by Acheson (2000) and numerical Floquet analysis by Tophøj and Aref (2012) both indicate, to many digits, that the bifurcation occurs when 1/α = φ 2 , where φ is the golden ratio. This study aims to explain the origin of this remarkable value. Using a trick from the gravitational two-body problem, we change variables to render the Floquet problem in an explicit form that is more amenable to analysis. We then implement G. W. Hill's method of harmonic balance to high order using computer algebra to construct a rapidly-converging sequence of asymptotic approximations to the bifurcation value, confirming the value found earlier.

Transition to instability of the leapfrogging vortex quartet

“…They justify this with a formal argument, that the present authors have been unable to follow and believe to be incorrect. That α c takes such a fortiutous value seems like more than simple coincidence, and the authors' previous paper [7] documents their initial attempt to prove it. There, we devised a perturbative procedure which allowed Figure 2: Two perturbed leapfrogging orbits, the left one stable, the right one stable.…”

“…Fig. 3 shows the trajectories due to Hamiltonian (7). The orbit with ratio α corresponds to the (Q 1 , P 1 ) periodic orbit with h = (1−α) 2 8α .…”

“…Figure 3: Level sets of the reduced Hamiltonian function (7), showing periodic (leapfrogging) orbits in purple, unbounded orbits in red, and the critical orbit as a dashed blue line.…”

“…In Sec. 2 we set up the equations of motion and review the arguments from our earlier work [7] in which we transform the problem into a simplified form amenable to analysis. Sec.…”

Collinear interaction of vortex pairs with different strengths—Criteria for leapfrogging

“…We do not consider here any conditions of stability of the solutions presented, although we note that there is some computational consideration of this matter contained in Acheson [2]. The dynamical systems approach and stability issues are also pursued analytically in Berger [12], in Behring and Goodman [11], and in a numerical investigation in Whitchurch, Kevrekidis, and Koukouloyannis [33].…”