The vortex dynamics analogue of the restricted three-body problem: advection in the field of three identical point vortices

Jets, stickiness, and anomalous transport

Dynamical and statistical properties of the vortex and passive particle advection in chaotic flows generated by 4- and 16-point vortices are investigated. General transport properties of these flows are found to be anomalous and exhibit a superdiffusive behavior with typical second moment exponent mu approximately 1.75. The origin of this anomaly is traced to the presence of coherent structures within the flow, the vortex cores, and the region far from where vortices are located. In the vicinity of these regions stickiness is observed and the motion of tracers is quasiballistic. The chaotic nature of the underlying flow dictates the choice for thorough analysis of transport properties. Passive tracer motion is analyzed by measuring the mutual relative evolution of two nearby tracers. Some tracers travel in each other's vicinity for relatively long times. This is related to a hidden order for the tracers, which we call jets. Jets are localized and found in sticky regions. Their structure is analyzed and found to be formed of a nested set of jets within jets. The analysis of the jet trapping time statistics shows a quantitative agreement with the observed transport exponent.

Chaotic advection near a three-vortex collapse

Dynamical and statistical properties of tracer advection are studied in a family of flows produced by three point-vortices of different signs. Tracer dynamics is analyzed by numerical construction of Poincaré sections, and is found to be strongly chaotic: advection pattern in the region around the center of vorticity is dominated by a well developed stochastic sea, which grows as the vortex system's initial conditions are set closer to those leading to the collapse of the vortices; at the same time, the islands of regular motion around vortices, known as vortex cores, shrink. An estimation of the core's radii from the minimum distance of vortex approach to each other is obtained. Tracer transport was found to be anomalous: for all of the three numerically investigated cases, the variance of the tracer distribution grows faster than a linear function of time, corresponding to a superdiffusive regime. The transport exponent varies with time decades, implying the presence of multi-fractal transport features. Yet, its value is never too far from 3/2, indicating some kind of universality. Statistics of Poincaré recurrences is non-Poissonian: distributions have long power-law tails. The anomalous properties of tracer statistics are the result of the complex structure of the advection phase space, in particular, of strong stickiness on the boundaries between the regions of chaotic and regular motion. The role of the different phase space structures involved in this phenomenon is analyzed. Based on this analysis, a kinetic description is constructed, which takes into account different time and space scalings by using a fractional equation.

Passive particle transport in three-vortex flow

We study transport of tracer particles in a two-dimensional incompressible inviscid flow produced by three point vortices of equal strength. Time dependence of the flow caused by vortex motion gives rise to chaotic tracer trajectories, which fill parts of the flow plane referred to as mixing regions. For general vortex positions, a large connected mixing region (chaotic sea) is formed around vortices. It comprises a number of coherent fluid patches (islands), which do not mix with the rest of the chaotic sea, inside them particle motion is predominantly regular; three near-circular islands surrounding vortices are distinguished by their robust nature. Tracers in the chaotic sea rotate around the center of vorticity in an irregular way. Their trajectories are intermittent, long flights of almost regular motion are caused by trappings in the boundary regions of regular islands. The statistics of tracer rotation exhibits anomalous features, such as faster than linear growth of tracer ensemble variance and asymmetric probability distribution with long power tails. Exponent of the variance growth power law is different for different time ranges. Central part of the tracer distribution and its low (noninteger) moments evolve in a self-similar way, characterized by an exponent, which is different from that of the variance, and contrary to the latter is constant in time. Algebraic tails of the tracer distribution, controlling the behavior of the variance, are responsible for this effect. Long correlations in tracer motion lead to non-Poissonian distribution of Poincare recurrences in the mixing region. Analysis of long recurrences proves, that they are caused by tracer trappings inside boundary layers of islands of regular motion, which always exist inside the mixing region. Statistics of Poincare recurrences and trapping times exhibit power-law decay, indicating absence of a characteristic relaxation time. Values of the decay exponent for recurrences and for escape from the analyzed traps are very close to each other; long correlations are not dominated by a single trap, but are a cumulative effect of all of them, relative importance of a trap is determined by its size, and by its rotation frequency with respect to the background.

Chaotic advection, diffusion, and reactions in open flows

We review and generalize recent results on advection of particles in open time-periodic hydrodynamical flows. First, the problem of passive advection is considered, and its fractal and chaotic nature is pointed out. Next, we study the effect of weak molecular diffusion or randomness of the flow. Finally, we investigate the influence of passive advection on chemical or biological activity superimposed on open flows. The nondiffusive approach is shown to carry some features of a weak diffusion, due to the finiteness of the reaction range or reaction velocity. (c) 2000 American Institute of Physics.