Instability of vortex pair leapfrogging
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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 < 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.