Elliptical vortices in shear: Hamiltonian moment formulation and Melnikov analysis

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Abstract
The equations of motion for interacting elliptical vortices in a background shear flow are derived from a Hamiltonian moment formulation. The equations reduce to the sixth order system of Melander et al. [J. Fluid Mech. 167, 95 (1986)] when a pair of vortices is considered and shear is neglected. The equations for a pair of identical vortices are analyzed using a number of methods, with particular emphasis on the implications for vortex merger. The splitting distance between the stable and unstable manifolds connecting the hyperbolic fixed points of the intercentroidal motion—the separatrix splitting—is estimated with a Melnikov analysis. This analysis differs from the standard time‐periodic Melnikov analysis on two counts: (a) the ‘‘periodic’’ perturbation arises from a second degree of freedom in the system which is not wholly independent of the first degree of freedom, the intercentroidal motion; (b) this perturbation has a faster time scale than the intercentroidal motion. The resulting Melnikov integral appears to be exponentially small in the perturbation as the latter goes to zero. Numerical simulations, notably Poincaré sections, provide a global view of the dynamics and indicate that, as observed in previous studies, there are essentially two modes of merger. The effect of the shear on chaotic motion is also discussed.

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