Regular and chaotic advection in the flow field of a three-vortex system

The dynamics of a passive particle in a two-dimensional incompressible flow generated by three point vortices advected by their mutual interaction is considered as a periodically forced Hamiltonian system. The geometry of the background vortex flow determines the degree of chaotization of the tracer motion. Two extreme regimes, of strong and weak chaos, are specified and investigated analytically. Mappings are derived for both cases, and the border between the chaotic and regular advection is found by applying the stochasticity criterion. In the case of strong chaos, there exist coherent regular structures around vortices (vortex cores), which correspond to domains with $\mathrm{KAM}$ curves. An expression for the radius of the cores is obtained. The robust nature of vortex cores, demonstrated numerically, is explained. In the near-integrable case of weak chaotization, a separatrix map is used to find the width of the stochastic layer. Numerical simulations reveal a variety of structures in the pattern of advection, such as a hierarchy of island chains and sticky bands around the vortex cores.