Rigid-Vortex Configurations of Four Point Vortices

In this work, we numerically investigate the distribution of initial vortex arrangements that result in the rigid vortex motion of four point vortices on the unbounded plane. A 6 × 4 configuration matrix can be assigned for each vortex configuration, whose null space determines the possibility of a rigid vortex motion. A systematic study of the rigid vortex configurations is provided based on the symmetry of inter-vortex distances. Furthermore, it is shown that rigid vortex configurations generally exist as one-dimensional continua with a bounded and an unbounded component, along which the associated circulations vary continuously.