Motion of Three Vortices near Collapse

Abstract

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.