Braids of entangled particle trajectories

TL;DR

This paper introduces a topological approach using braid theory to analyze entanglement in particle trajectories, providing a global measure of flow complexity that surpasses traditional local dispersion metrics.

Contribution

The authors develop a novel method applying braid theory to estimate the topological entropy from trajectory data, capturing the global entanglement of fluid particles.

Findings

Method successfully applied to simple dynamical systems.

Applied to float data from the Labrador Sea.

Topological entropy correlates with flow complexity.

Abstract

In many applications, the two-dimensional trajectories of fluid particles are available, but little is known about the underlying flow. Oceanic floats are a clear example. To extract quantitative information from such data, one can measure single-particle dispersion coefficients, but this only uses one trajectory at a time, so much of the information on relative motion is lost. In some circumstances the trajectories happen to remain close long enough to measure finite-time Lyapunov exponents, but this is rare. We propose to use tools from braid theory and the topology of surface mappings to approximate the topological entropy of the underlying flow. The procedure uses all the trajectory data and is inherently global. The topological entropy is a measure of the entanglement of the trajectories, and converges to zero if they are not entangled in a complex manner (for instance, if the trajectories are all in a large vortex). We illustrate the techniques on some simple dynamical systems and on float data from the Labrador sea.