4 VORTICES ON DOUBLY PERIODIC PATHS

Plane vortex configurations in ideal flow are considered for which the total ''mass'' of the vortex strengths, their ''moments,'' and their ''polar moments of inertia'' all vanish. These properties are conserved for all times. The simplest nontrivial realization ot such a configuration requires four vortices. For this case, which belongs to the more extended family of four-vortex problems that are known to be integrable [Phys. Fluids 31, 2796 (1989); Phys. Fluids A 2, 1477 (1990)], some simple closed-form results are given. The analysis shows that the paths are periodic in a ''configuration plane'' moving with the vortices as well as in the absolute fluid plane. A ''winding number'' is determined from the analysis, which gives the ratio of the two periods. Patterns of the vortex paths are determined by a program based on the step-by-step integration of the equations of motion, which is-beyond a certain level of the analysis--still the more practical method of solution. Results showing the typical behavior of the motion paths for different winding numbers are presented.