Braids of entangled particle trajectories

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 singleparticle 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. The method could eventually be used to identify Lagrangian coherent structures present in the flow. Keywords: topological chaos, dynamical systems, Lagrangian coherent structuresConsider particles floating on top of a fluid. We can follow their trajectories, either with a camera or by computer simulation. If we then plot their position in a three-dimensional graph, with time the vertical coordinate, we get a 'spaghetti plot,' which contains information about how entangled the trajectories are. We discuss how to measure the level of entanglements in terms of topological entropy, and the interpretation of the results. This provides a straightforward method of estimating the level of chaos present in a system. This approach could also be used to determine if some trajectories remain together for a long time, and are thus part of a Lagrangian coherent structure.

Heating of braided coronal loops

“…While the exact entropy depends on the full pattern of magnetic field lines, a good estimate may be obtained by ensemble averaging over finite sets of field lines. Applying the numerical method of Moussafir (2006), as implemented by Thiffeault (2010), and with sets of 40 field lines, we find T (E 3 ) ≈ 3.3 and T (S 3 ) ≈ 2.3.…”

“…It has the advantages both of a firm theoretical grounding and of being a robust quantity insensitive to small changes in the magnetic field. There are several equivalent definitions of the topological entropy, but a convenient one is the asymptotic growth rate (with z) of horizontal loops stretched around the magnetic field lines (Newhouse & Pignataro 1993;Thiffeault 2010). While the exact entropy depends on the full pattern of magnetic field lines, a good estimate may be obtained by ensemble averaging over finite sets of field lines.…”

Topological Chaos and Periodic Braiding of Almost-Cyclic Sets

“…1(d) over the length of time these particles remain in the ACS. The infinite-time braid of these particles is almost certainly aperiodic, but the entropy of the finite-time braid, particularly when shared by a number of different particles, provides a good representation of the actual entropy of the flow [15]. Thus, the entropy predicted by the braid of the ACS, h TN , gives an accurate lower bound on the actual entropy of the flow, h, as shown in Fig.…”

“…Recently, the analysis of ghost rod topology in a fluid has been extended to aperiodic orbits [14,15], relaxing a substantial restriction in the application of the TNCT. However, this approach introduces the complexity of needing to identify appropriate aperiodic orbits, as a random selection of trajectories generally leads to a poor estimate of the overall system behavior [14].…”

A study of mixing in coherent vortices using braiding factors

“…One of the latest techniques studied for such a purpose is that of particle braiding (Boyland et al, 2000;Kin and Sakajo, 2005;Thiffeault, 2005Thiffeault, , 2010. Braiding uses the trajectories of particles within the flow to give a measure of their entanglement, and hence how chaotic the flow is.…”