Collinear interaction of vortex pairs with different strengths—Criteria for leapfrogging

Abstract

We formulate a system of equations that describes the motion of four vortices made up of two interacting vortex pairs, where the absolute strengths of the pairs are different. Each vortex pair moves along the same axis in the same sense. In much of the literature, the vortex pairs have equal strength. The vortex pairs can either escape to infinite separation or undergo a periodic leapfrogging motion. We determine an explicit criterion in terms of the initial horizontal separation of the vortex pairs given as a function of the ratio of their strengths, to describe a periodic leapfrogging motion when interacting along the line of symmetry. In an appendix we also contrast a special case of interaction of a vortex pair with a single vortex of the same strength in which a vortex exchange occurs.