Motion of Three Vortices near Collapse

A system of three point vortices in an unbounded plane has a special family of self-similarly contracting or expanding solutions: during the motion, vortex triangle remains similar to the original one, while its area decreases (grows) at a constant rate. A contracting configuration brings three vortices to a single point in a finite time; this phenomenon known as vortex collapse is of principal importance for many-vortex systems. Dynamics of close-to-collapse vortex configurations depends on the way the collapse conditions are violated. Using an effective potential representation, a detailed quantitative analysis of all the different types of near-collapse dynamics is performed when two of the vortices are identical. We discuss time and length scales, emerging in the problem, and their behavior as the initial vortex triangle is approaching to an exact collapse configuration. Different types of critical behaviors, such as logarithmic or power-law divergences are exhibited, which emphasizes the importance of the way the collapse is approached. Period asymptotics for all singular cases are presented as functions of the initial vortices configurations. Special features of passive particle mixing by a near-collapse flows are illustrated numerically.Comment: 45 pages, 22 figures Last version of the paper with all update

Jets, Stickiness and Anomalous Transport

Dynamical and statistical properties of the vortex and passive particle advection in chaotic flows generated by four and sixteen point vortices are investigated. General transport properties of these flows are found anomalous and exhibit a superdiffusive behavior with typical second moment exponent (\mu \sim 1.75). The origin of this anomaly is traced back to the presence of coherent structures within the flow, the vortex cores and the region far from where vortices are located. In the vicinity of these regions stickiness is observed and the motion of tracers is quasi-ballistic. The chaotic nature of the underlying flow dictates the choice for thorough analysis of transport properties. Passive tracer motion is analyzed by measuring the mutual relative evolution of two nearby tracers. Some tracers travel in each other vicinity for relatively large times. This is related to an hidden order for the tracers which we call jets. Jets are localized and found in sticky regions. Their structure is analyzed and found to be formed of a nested sets of jets within jets. The analysis of the jet trapping time statistics shows a quantitative agreement with the observed transport exponent.Comment: 17 pages, 17 figure