Special Session 12: Complex and Chaotic Dynamics

Vortices are long-lived features of geophysical and laboratory flows. They emerge in free-decay turbulence and govern the long-term evolution of the flow. In 2D flows, two vortices can grow by merger when they are like-signed. The process of vortex merger in 2D flows is well known for a pair of isolated vortices. Nevertheless, in a turbulent field, the velocity field created by neighboring vortices can advect these two vortices away from each other, or exert a strain or shear flow on them. Furthermore, this external velocity field is most often time-varying since these neighboring vortices are not steady. Since vortex merger is strongly dependent on the distance between them, we first investigate the dynamics of a pair of point vortices, embedded in an external shear/strain field which mimics the influence of neighboring vortices. This external flow will first be considered as stationary, and then as time varying. The equilibrium positions of the two vortices in a steady external flow and their stability can easily be computed. We focus on neutral equilibria around which vortices rotate periodically. Then we add an unsteady component to the external strain and rotation, and tune its frequency to resonate with the vortex rotation. With a multiple time-scale expansion, we describe the vortex motion in this case, via an amplitude equation. We obtain the critical value of unsteady strain field, beyond which the vortex escapes its initial circular trajectory. With Poincaré sections of the dynamical system, we also show that chaos progressively fills the phase plane as this unsteady strain component increases. For finite amplitude of the unsteady strain, the vortex trajectory becomes chaotic and escapes the initial separatrices in the phase plane. This study is then generalized to finite-area vortices. It is shown that a properly oriented external strain is critical in bringing initially distant vortices closer to each other, and therefore facilitates vortex merger. Conclusions are drawn for vortices in 2D turbulence.