^{1,2,a)}and Hassan Aref

^{2,3}

### Abstract

Leapfrogging is a periodic solution of the four-vortex problem with two positive and two negative point vortices all of the same absolute circulation arranged as co-axial vortex pairs. The set of co-axial motions can be parameterized by the ratio 0 < α < 1 of vortex pair sizes at the time when one pair passes through the other. Leapfrogging occurs for α > σ2, where is the silver ratio. The motion is known in full analytical detail since the 1877 thesis of Gröbli and a well known 1894 paper by Love. Acheson [“Instability of vortex leapfrogging,” Eur. J. Phys.21, 269–273 (Year: 2000)]10.1088/0143-0807/21/3/310 determined by numerical experiments that leapfrogging is linearly unstable for σ2 < α < 0.382, but apparently stable for larger α. Here we derive a linear system of equations governing small perturbations of the leapfrogging motion. We show that symmetry-breaking perturbations are essentially governed by a 2D linear system with time-periodic coefficients and perform a Floquet analysis. We find transition from linearly unstable to stable leapfrogging at α = ϕ2 ≈ 0.381966, where is the golden ratio. Acheson also suggested that there was a sharp transition between a “disintegration” instability mode, where two pairs fly off to infinity, and a “walkabout” mode, where the vortices depart from leapfrogging but still remain within a finite distance of one another. We show numerically that this transition is more gradual, a result that we relate to earlier investigations of chaotic scattering of vortex pairs [L. Tophøj and H. Aref, “Chaotic scattering of two identical point vortex pairs revisited,” Phys. Fluids20, 093605 (Year: 2008)]10.1063/1.2974830. Both leapfrogging and “walkabout” motions can appear as intermediate states in chaotic scattering at the same values of linear impulse and energy.

The presented work was supported in part by a Niels Bohr Visiting Professorship at the Technical University of Denmark sponsored by the Danish National Research Foundation. This paper is dedicated to the memory of Hassan Aref, who passed away during its preparation. He was a friend and a great inspiration.

I. INTRODUCTION

II. EQUATIONS OF MOTION

III. LEAPFROGGING SOLUTIONS

IV. LINEAR STABILITY THEORY

V. FLOQUET ANALYSIS

VI. FLOQUET ANALYSIS APPLIED TO THE EQUATIONS OF SEC. IV

VII. THE α → 1 LIMIT

VIII. CONCLUSIONS

### Key Topics

- Rotating flows
- 38.0
- Eigenvalues
- 9.0
- Equations of motion
- 9.0
- Linear equations
- 5.0
- Numerical solutions
- 5.0

## Figures

Instability calculations of leapfrogging similar to those reported by Acheson. 6 Two different perturbations are shown. Vertical lines indicate the initial positions. (a) A “disintegration” instability for α = 0.25 perturbed by ξ− = η+ = −10−6. (b) A “walkabout” instability for α = 0.30 perturbed by ξ− = 0, η+ = 10−5 (see text for precise definitions).

Instability calculations of leapfrogging similar to those reported by Acheson. 6 Two different perturbations are shown. Vertical lines indicate the initial positions. (a) A “disintegration” instability for α = 0.25 perturbed by ξ− = η+ = −10−6. (b) A “walkabout” instability for α = 0.30 perturbed by ξ− = 0, η+ = 10−5 (see text for precise definitions).

Positive vortices are at in the complex plane, negative vortices at . Also shown are ζ±, Eqs. (2a) .

Positive vortices are at in the complex plane, negative vortices at . Also shown are ζ±, Eqs. (2a) .

Level curves of the Hamiltonian (9a) .

Floquet exponents as a function of α shown for α0 < α ⩽ α2. See the text for these values of α. The dashed curve shows 2|μ+|, the thick grey curve shows T/3, cf. (11) , and the solid curve shows |μ+|T. Note that μ+ is purely real for α < α2 and purely imaginary for α2 < α < 1, indicating that the monodromy matrix has complex conjugate eigenvalues of unit modulus in the latter range.

Floquet exponents as a function of α shown for α0 < α ⩽ α2. See the text for these values of α. The dashed curve shows 2|μ+|, the thick grey curve shows T/3, cf. (11) , and the solid curve shows |μ+|T. Note that μ+ is purely real for α < α2 and purely imaginary for α2 < α < 1, indicating that the monodromy matrix has complex conjugate eigenvalues of unit modulus in the latter range.

Detail of the Floquet exponent μ+ around α = α2 = ϕ2. For α < α2, where μ+ is imaginary, we show μ+/i. The dashed line at α = 0.382 shows Acheson's numerically determined value for the crossover from unstable to stable leapfrogging. 6 The text provides further discussion.

Detail of the Floquet exponent μ+ around α = α2 = ϕ2. For α < α2, where μ+ is imaginary, we show μ+/i. The dashed line at α = 0.382 shows Acheson's numerically determined value for the crossover from unstable to stable leapfrogging. 6 The text provides further discussion.

Scattering of two identical vortex pairs showing intermediate states consisting of leapfrogging and “walkabout” motions. The vertical bar marks the starting leapfrogging configuration, α = 0.25, (ξ−, η+) = (−1, 1) × 10−5, from which time is integrated forwards and backwards. The computation has been checked by reverse integration, and the variation of the integrals of motion is of negligible order (10−12).

Scattering of two identical vortex pairs showing intermediate states consisting of leapfrogging and “walkabout” motions. The vertical bar marks the starting leapfrogging configuration, α = 0.25, (ξ−, η+) = (−1, 1) × 10−5, from which time is integrated forwards and backwards. The computation has been checked by reverse integration, and the variation of the integrals of motion is of negligible order (10−12).

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