Regular and Chaotic Motion of Two-Dimensional Point Vortices

Abstract

A point vortex is a model of a real vortex with δ –function-type vorticity configuration in space. The equations of motion for N-point vortices in an unbounded region are given as follows:

$$\frac{{d{z_j}}}{{dt}} = \frac{1}{{2\pi i}}\sum\limits_{m = 1}^N {\frac{{{\Gamma _m}}}{{{{\bar z}_j} - {{\bar z}_m}}}} (j = 1,2, \cdots ,N)$$

(1)

where Z_j (= x_j+iy_j) is a position of j-th vortex in the complex Z-plane, and Г_j is its strength. The bar on the variable means that we take complex conjugate, and the prime denotes that we omit the singular terms j=m from the sum.