Instability of vortex pair leapfrogging

Laust Tophøj, Hassan Aref

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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 (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 (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.
Original languageEnglish
JournalPhysics of Fluids
Issue number1
Pages (from-to)014107
Publication statusPublished - 2013

Bibliographical note

© 2013 American Institute of Physics


  • Linear systems
  • Scattering
  • Vortex flow


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