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Instability of vortex pair leapfrogging
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View: Figures


Image of FIG. 1.
FIG. 1.

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

Image of FIG. 2.
FIG. 2.

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

Image of FIG. 3.
FIG. 3.

Level curves of the Hamiltonian (9a) .

Image of FIG. 4.
FIG. 4.

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

Image of FIG. 5.
FIG. 5.

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

Image of FIG. 6.
FIG. 6.

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, 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).


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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Instability of vortex pair leapfrogging