Chaotic Advection near 3-Vortex Collapse

TL;DR

This paper investigates the chaotic advection of tracers near a three-vortex collapse, revealing super-diffusive transport, complex phase space structures, and a fractional kinetic model to describe the anomalous statistics.

Contribution

It provides a detailed analysis of tracer dynamics near vortex collapse, highlighting the chaotic structures and developing a fractional kinetic equation for anomalous transport.

Findings

Tracer advection is strongly chaotic with a stochastic sea dominating near collapse.

Tracer transport exhibits super-diffusive behavior with a near 3/2 exponent.

Poincaré recurrence statistics show long power-law tails indicating complex phase space.

Abstract

Dynamical and statistical properties of tracer advection are studied in a family of flows produced by three point-vortices of different signs. A collapse of all three vortices to a single point is then possible. Tracer dynamics is analyzed by numerical construction of Poincar\'{e} 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 approaches the collapse; 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 super-diffusive 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 Poincar\'{e} 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.