Instability of vortex pair leapfrogging

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American Institute of Physics

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 < alpha < 1 of vortex pair sizes at the time when one pair passes through the other. Leapfrogging occurs for alpha > sigma(2), where sigma = root 2 - 1 is the silver ratio. The motion is known in full analytical detail since the 1877 thesis of Grobli and a well known 1894 paper by Love. Acheson ["Instability of vortex leapfrogging," Eur. J. Phys. 21, 269-273 (2000)] determined by numerical experiments that leapfrogging is linearly unstable for sigma(2) < alpha < 0.382, but apparently stable for larger alpha. 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 alpha = phi(2) approximate to 0.381966, where phi = 1/2 (root 5 - 1) 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 "walk-about" 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. Tophoj and H. Aref, "Chaotic scattering of two identical point vortex pairs revisited," Phys. Fluids 20, 093605 (2008)]. Both leapfrogging and "walkabout" motions can appear as intermediate states in chaotic scattering at the same values of linear impulse and energy.

Tophøj, Laust and Aref, Hassan, “Instability of vortex pair leapfrogging,”Phys. Fluids (1994-present), 25, 014107 (2013), DOI: