Finite-time braiding exponents

Topological braiding and virtual particles on the cell membrane

Braiding of topological structures in complex matter fields provides a robust framework for encoding and processing information, and it has been extensively studied in the context of topological quantum computation. In living systems, topological defects are crucial for the localization and organization of biochemical signaling waves, but their braiding dynamics remain unexplored. Here, we show that the spiral wave cores, which organize the Rho-GTP protein signaling dynamics and force generation on the membrane of starfish egg cells, undergo spontaneous braiding dynamics. Experimentally measured world line braiding exponents and topological entropy correlate with cellular activity and agree with predictions from a generic field theory. Our analysis further reveals the creation and annihilation of virtual quasi-particle excitations during defect scattering events, suggesting phenomenological parallels between quantum and living matter.

Material coherence from trajectories via Burau eigenanalysis of braids

In this paper, we provide a numerical tool to study a material's coherence from a set of 2D Lagrangian trajectories sampling a dynamical system, i.e., from the motion of passive tracers. We show that eigenvectors of the Burau representation of a topological braid derived from the trajectories have levelsets corresponding to components of the Nielsen-Thurston decomposition of the dynamical system. One can thus detect and identify clusters of space-time trajectories corresponding to coherent regions of the dynamical system by solving an eigenvalue problem. Unlike previous methods, the scalable computational complexity of our braid-based approach allows the analysis of large amounts of trajectories.

RePLaT-Chaos: A Simple Educational Application to Discover the Chaotic Nature of Atmospheric Advection

Large-scale atmospheric pollutant spreading via volcano eruptions and industrial accidents may have serious effects on our life. However, many students and non-experts are generally not aware of the fact that pollutant clouds do not disperse in the atmosphere like dye blobs on clothes. Rather, an initially compact pollutant cloud soon becomes strongly stretched with filamentary and folded structure. This is the result of the chaotic behaviour of advection of pollutants in 3-D flows, i.e., the advection dynamics of pollutants shows the typical characteristics such as sensitivity to the initial conditions, irregular motion, and complicated but well-organized (fractal) structures. This study presents possible applications of a software called RePLaT-Chaos by means of which the characteristics of the long-range atmospheric spreading of volcanic ash clouds and other pollutants can be investigated in an easy and interactive way. This application is also a suitable tool for studying the chaotic features of the advection and determines two quantities which describe the chaoticity of the advection processes: the stretching rate quantifies the strength of the exponential stretching of pollutant clouds; and the escape rate characterizes the rate of the rapidity by which the settling particles of a pollutant cloud leave the atmosphere.

Intricate features in the lifetime and deposition of atmospheric aerosol particles

The advection of particles emanated, e.g., from volcano eruptions or other pollution events exhibits chaotic behavior in the atmosphere. Due to gravity, the particles move downward on average and remain in the atmosphere for a finite time. The number of particles not yet deposited from the atmosphere decays exponentially after a while characteristic to transient chaos. The so-called escape rate describes the rapidity of the decrease, the reciprocal of which can be used to estimate the average lifetime of the particles. Based on measured wind field data, we follow aerosol particles and demonstrate that the geographical distribution of the individual lifetime of the particles distributed over the globe at different altitudes shows a filamentary, fractal distribution, typical for chaos: the lifetime of particles may be quite different at very nearby geographic locations. These maps can be considered as atlases for the potential fate of volcanic ash clouds or of particles distributed for geoengineering purposes. Particles with similar lifetime deposit also in filamentary structures, but the deposition pattern of extremely long-living particles covers more or less homogeneously the Earth. In general, particles emanated around the equator remain in the atmosphere for the longest time, even for years, e.g., for particles of 1μm radius. The escape rate does not show any considerable dependence on the particles' initial altitude, indicating that there exists a unique chaotic saddle in the atmosphere. We reconstruct this saddle and its stable and unstable manifolds on two planar slices and follow its time dependence.

Ensemble-based analysis of the pollutant spreading intensity induced by climate change

The intensity of the atmospheric large-scale spreading can be characterized by a measure of chaotic systems, called topological entropy. A pollutant cloud stretches in an exponential manner in time, and in the atmospheric context the topological entropy corresponds to the stretching rate of its length. To explore the plethora of possible climate evolutions, we investigate here pollutant spreading in climate realizations of two climate models to learn what the typical spreading behavior is over a climate change. An overall decrease in the areal mean of the stretching rate is found to be typical in the ensembles of both climate models. This results in larger pollutant concentrations for several geographical regions implying higher environmental risk. A strong correlation is found between the time series of the ensemble mean values of the stretching rate and of the absolute value of the relative vorticity. Here we show that, based on the obtained relationship, the typical intensity of the spreading in an arbitrary climate realization can be estimated by using only the ensemble means of the relative vorticity data of a climate model.

Ensemble-based topological entropy calculation (E-tec)

Topological entropy measures the number of distinguishable orbits in a dynamical system, thereby quantifying the complexity of chaotic dynamics. One approach to computing topological entropy in a two-dimensional space is to analyze the collective motion of an ensemble of system trajectories taking into account how trajectories "braid" around one another. In this spirit, we introduce the Ensemble-based Topological Entropy Calculation, or E-tec, a method to derive a lower-bound on topological entropy of two-dimensional systems by considering the evolution of a "rubber band" (piece-wise linear curve) wrapped around the data points and evolving with their trajectories. The topological entropy is bounded below by the exponential growth rate of this band. We use tools from computational geometry to track the evolution of the rubber band as data points strike and deform it. Because we maintain information about the configuration of trajectories with respect to one another, updating the band configuration is performed locally, which allows E-tec to be more computationally efficient than some competing methods. In this work, we validate and illustrate many features of E-tec on a chaotic lid-driven cavity flow. In particular, we demonstrate convergence of E-tec's approximation with respect to both the number of trajectories (ensemble size) and the duration of trajectories in time.

Influence of lateral boundaries on transport in quasi-two-dimensional flow

We assess the impact of lateral coastline-like boundaries on mixing and transport in a laboratory quasi-two-dimensional turbulent flow using a transfer-operator approach. We examine the most coherent sets in the flow, as defined by the singular vectors of the transfer operator, as a way to characterize its mixing properties. We study three model coastline shapes: a uniform boundary, a sharp embayment, and a sharp headland. Of these three, we show that the headland affects the mixing deep into the flow domain because it has a tendency to pin transport barriers to its tip. Our results may have implications for the siting of coastal facilities that discharge into the ocean.

Quantifying the tangling of trajectories using the topological entropy

We present a simple method to efficiently compute a lower limit of the topological entropy and its spatial distribution for two-dimensional mappings. These mappings could represent either two-dimensional time-periodic fluid flows or three-dimensional magnetic fields, which are periodic in one direction. This method is based on measuring the length of a material line in the flow. Depending on the nature of the flow, the fluid can be mixed very efficiently which causes the line to stretch. Here, we study a method that adaptively increases the resolution at locations along the line where folds lead to a high curvature. This reduces the computational cost greatly which allows us to study unprecedented parameter regimes. We demonstrate how this efficient implementation allows the computation of the variation of the finite-time topological entropy in the mapping. This measure quantifies spatial variations of the braiding efficiency, important in many practical applications.

Dynamics and transport properties of three surface quasigeostrophic point vortices

The surface quasi-geostrophic (SQG) equations are a model for low-Rossby number geophysical flows in which the dynamics are governed by potential temperature dynamics on the boundary. We examine point vortex solutions to this model as well as the chaotic flows induced by three point vortices. The chaotic transport induced by these flows is investigated using techniques of Poincaré maps and the Finite Time Braiding Exponent (FTBE). This chaotic transport is representative of the mixing in the flow, and these terms are used interchangeably in this work. Compared with point vortices in two-dimensional flow, the SQG vortices are found to produce flows with higher FTBE, indicating more mixing. Select results are presented for analyzing mixing for arbitrary vortex strengths.

Using heteroclinic orbits to quantify topological entropy in fluid flows

Topological approaches to mixing are important tools to understand chaotic fluid flows, ranging from oceanic transport to the design of micro-mixers. Typically, topological entropy, the exponential growth rate of material lines, is used to quantify topological mixing. Computing topological entropy from the direct stretching rate is computationally expensive and sheds little light on the source of the mixing. Earlier approaches emphasized that topological entropy could be viewed as generated by the braiding of virtual, or "ghost," rods stirring the fluid in a periodic manner. Here, we demonstrate that topological entropy can also be viewed as generated by the braiding of ghost rods following heteroclinic orbits instead. We use the machinery of homotopic lobe dynamics, which extracts symbolic dynamics from finite-length pieces of stable and unstable manifolds attached to fixed points of the fluid flow. As an example, we focus on the topological entropy of a bounded, chaotic, two-dimensional, double-vortex cavity flow. Over a certain parameter range, the topological entropy is primarily due to the braiding of a period-three orbit. However, this orbit does not explain the topological entropy for parameter values where it does not exist, nor does it explain the excess of topological entropy for the entire range of its existence. We show that braiding by heteroclinic orbits provides an accurate computation of topological entropy when the period-three orbit does not exist, and that it provides an explanation for some of the excess topological entropy when the period-three orbit does exist. Furthermore, the computation of symbolic dynamics using heteroclinic orbits has been automated and can be used to compute topological entropy for a general 2D fluid flow.

Braid Entropy of Two-Dimensional Turbulence

The evolving shape of material fluid lines in a flow underlies the quantitative prediction of the dissipation and material transport in many industrial and natural processes. However, collecting quantitative data on this dynamics remains an experimental challenge in particular in turbulent flows. Indeed the deformation of a fluid line, induced by its successive stretching and folding, can be difficult to determine because such description ultimately relies on often inaccessible multi-particle information. Here we report laboratory measurements in two-dimensional turbulence that offer an alternative topological viewpoint on this issue. This approach characterizes the dynamics of a braid of Lagrangian trajectories through a global measure of their entanglement. The topological length of material fluid lines can be derived from these braids. This length is found to grow exponentially with time, giving access to the braid topological entropy . The entropy increases as the square root of the turbulent kinetic energy and is directly related to the single-particle dispersion coefficient. At long times, the probability distribution of is positively skewed and shows strong exponential tails. Our results suggest that may serve as a measure of the irreversibility of turbulence based on minimal principles and sparse Lagrangian data.