  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.
