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.

Scientific RepoRts | 5:18564 | DOI: 10.1038/srep18564 is capable of capturing the deformation of fluid elements using topological considerations and a limited number of Lagrangian trajectories. The method is suitable for studying two-dimensional (2D) flows. So far, the potential of the braid method has been rarely investigated experimentally 5,6,14,[29][30][31] .
Here, we report new experimental measurements of topological braids in 2D turbulent flows. Experiments have been carried out in a broad range of the turbulence kinetic energy by using both electromagnetically forced and Faraday wave driven 2D turbulence. The topological "length" N E of material fluid lines is derived from the behavior of Lagrangian trajectories, measured using high-resolution PTV techniques. After a transient period, the statistical average of N E grows exponentially with time and its probability density function (PDF) becomes positively skewed with strong exponential tails. The braid entropy S braid of the flow is measured. We show that S braid increases as the square root of the turbulent kinetic energy. This study also reveals that S braid is directly related to the single-particle diffusion coefficient D. Since quantifying the degree of irreversibility in turbulent flows [32][33][34][35] is still a matter of active debate, our results suggest that S braid could be a promising alternative measure based on topological considerations and sparse Lagrangian data.

Results
The experiments are carried out in two different experimental setups used to produce homogeneous two-dimensional (2D) turbulent flows. First, we take advantage of the remarkable similarity between the horizontal motion of particles on the surface of a fluid perturbed by Faraday waves and the fluid motion in 2D turbulence [17][18][19][20] . Though the fluid particle motion has a vertical component, these similarities stem from the ability of Faraday waves to generate lattices of horizontal vortices 17 . These vortices interact with each other and fuel the turbulent motion. In these experiments, the Faraday wave driven turbulence (FWT) is formed on the water surface in a vertically shaken container. The forcing is monochromatic with a frequency set to f 0 . Above a certain vertical acceleration threshold, parametrically forced Faraday waves appear with a dominant frequency of = / f f 2 0 and a wavelength λ. Tracer particles move erratically in the wave field. The forcing scale of the horizontal fluid motion is roughly λ/2. In the second set of experiments, we generate electromagnetically forced turbulence (EMT) in a layer of electrolyte by running an electric current J across the fluid cell 36,37 . A spatially periodic vertical magnetic field B is generated by placing a matrix of magnetic dipoles underneath the cell. The Lorenz J × B force produces local vortices at the forcing wave number k f which fuel the turbulent motion. An important aspect of both methods is that energy is injected at an intermediate scale (determined either by the distance between the magnets 37 or by the oscillon size 17 ) in the wave number spectrum, leaving it to the inverse energy cascade to spread energy over a broad range of scales.
Eulerian energy spectra. To visualize the horizontal fluid motion, the liquid-air interface is seeded with 50 μ m diameter particles. The Eulerian velocity field is measured by using particle image velocimetry (PIV) techniques. Figure 1 shows wave number spectra of the horizontal kinetic energy measured in both experiments for different parameters. The spectral scaling is consistent with the Kolmogorov-Kraichnan prediction of ∝ − / E k k 5 3 at wave numbers < k k f , revealing the presence of the inverse energy cascade 38 . At higher wave numbers, > k k f , some spectra follow the direct enstrophy cascade scaling ∝ − E k k 3 , while others are steeper, due to larger dissipation. The use of these two distinct methods allows us to study isotropic 2D turbulence in a broad range of kinetic energies, ∼ = (  Topological braids and topological fluid loops. The turbulent fluid motion is also characterized here by using PTV which allows us to measure simultaneously the Lagrangian trajectories of hundreds of particles in the horizontal x-y plane 17,18 . A few examples of the 2D trajectories are shown in Fig. 2(a,d). In these experiments, tracer particles are tracked with high resolution for long times ( > t T 10 L , where T L is the measured Lagrangian velocity autocorrelation time). We use tools from braid theory and the topology of surface mappings to characterize, in a topological sense, the deformation with time of fluid elements 6,14,[26][27][28][29] . In the following, we broadly refer to these different tools as the braid description. This method is built upon basic topological considerations and a limited number of Lagrangian trajectories. The connection between Lagrangian trajectories and the topological description of fluid lines is based on two minimal assumptions: i) particles act as local mixers for the surrounding fluid, and ii) fluid lines are impenetrable material objects. In physical terms, it emphasizes that the interaction of a fluid line with the stirring motion of surrounding particles determine completely its temporal evolution.
In this approach, 2D trajectories are viewed as 3D strands, with time t being the third coordinate, Fig. 2(b,e). The 3D x-y-t trajectories are projected onto the x-t (or y-t) plane, Fig. 2(c,f). In this plane, trajectories create a physical braid made of over-and under-crossings of strands. The crossings are the key topological information upon which the braid description hinges. The crossings of trajectories in 2D turbulence are qualitatively illustrated in Fig. 2(c,f). The braid approach then relies on two distinct objects ( Fig. 2(g)): -the topological braid which is really the sequence of crossings of the trajectories previously described.
-the topological loop which is like a fluid ribbon entangled within the braid. The topological braid is based only on the relative position of tracer particles and as such it does not require geometrical information such as the actual distance between strands ( Fig. 2(g)_left panel and ref. 14). The topological loop can neither intersect itself nor pass through the braid ( Fig. 2(g)_right panel). The degree of entanglement of the loop around the impenetrable strands of the braid can be quantified via a descriptor called the topological "length" N E which is also referred to as the braiding factor (see details on the computation of N E in Methods). In the course of time, each crossing along the braid distorts the loop and forces it to stretch or coil around the strands Fig. 2(g). If these deformations are irreversible, the degree of entanglement N E will increase. The time evolution of N E can be computed from the sequence of crossings in a given braid. The topological growth rate 3 m 2 s −2 and for various forcing scales L f . The probability distribution function (PDF) of N E and the statistical mean N E are estimated over at least 10 different braids and up to 100,000 initial topological loops. Figure 3 shows the PDF of / N N E E as a function of time at a flow kinetic energy = − u 10 2 3 m 2 s −2 . The initial loop is randomly entangled in the braid; as a consequence, at = t 0 s the PDF is a Gaussian function. After a transient period, the PDF becomes skewed and develops strong exponential tail at large values of N E . No saturation in the growth of the exponential tail could be observed in the temporal observation window (up to 30 T L for some runs). Figure 4 shows the temporal evolution of the statistical average N E in FWT and EMT as the flow energy is increased. After a transient state, N E grows exponentially with time and its growth rate increases with the flow energy. This behavior was observed in all our experiments as long as a sufficient number of trajectories compose the braid. The time evolution of N E reflects the non-trivial nature of braids made of Lagrangian trajectories in 2D turbulence.
To further characterize this complexity, we measure the braid entropy S braid as the growth rate of the logarithm of N E at long times: In these experiments, S braid increases as the square root of the flow kinetic energy Ẽ u 2 , as shown in Fig. 5(a). We observe no appreciable difference between data collected in different experiments, suggesting that S braid is independent on both the turbulence generation method and on the details of the energy injection. In particular, we detect no dependence of S braid on the energy injection scale L f . Figure 5

Discussion
Recently the concept of chaotic advection was further enriched by considering topological chaos 6 . The characterization of topological chaos hinges on the Thurston-Nielsen classification of surface mappings and on the concept of topological braids 6,14,26 . Our experimental work concerns topological chaos and explores the potential of the braid description to characterize 2D turbulence. For instance, Fig. 3 shows that the PDF of / N N E E presents growing exponential tails, a feature commonly associated with out-of-equilibrium systems.
The main results of this paper appear in Fig. 5(a), which shows that the braid entropy S braid is an increasing function of the flow kinetic energy Ẽ u 2 , independent of the forcing scale of the turbulent flows. Moreover S braid grows as E with no sign of saturation. It is expected that the more particles are included in the braid, the better S braid approximates the stretching rate of a "real" fluid line 14 . Batchelor showed that the exponential growth of fluid line in homogeneous turbulence is governed by the deformation of the small fluid elements of which it is comprised 26 . To further test the robustness of our results, we have measured the exponential stretching rate of small fluid elements in our turbulent flows. We have used the finite time Lyapunov exponent method 8 and found that the average exponential stretching rate Λ of fluid elements follows: Λ ∼ E, this result 40 supports strongly the behavior for S braid observed in Fig. 5(a). It is quite remarkable that S braid can record the actual behavior of fluid elements from scarce Lagrangian data (in our measurements, as low as 30 trajectories for which the average inter-particle distance is larger than L f ), while the computation of the Lyapunov exponents require high spatial resolution measurements of the entire velocity field.
A recent study 41 reported another type of entropy in 2D turbulent flows, namely the information entropy h Sh . This entropy quantifies the complexity of turbulence in terms of its predictability. Measurements were performed in turbulent flows in soap films; h Sh was computed from Eulerian velocity fluctuations. In these experiments, h Sh was a decreasing function of ≈ E u 2 . This is in sharp contrast with the behavior of S braid which increases with E in our work. This discrepancy highlights the fact that the relationship between the Eulerian and Lagrangian descriptions of turbulence remains an outstanding problem 3,41 . It also raises questions as to whether there are connections between different types of entropies in turbulent flows 39,42 .
S braid is a global topological quantity. Although it is connected to transport properties of the underlying flow, its connection to a "metric" descriptor of turbulent transport is not trivial. One of the most basic properties of Lagrangian trajectories is the single-particle dispersion δ = ( ) − ( )   r r t r 0 2 2 of a particle moving along the trajectory ( )  r t . In 2D turbulence, at long times, single-particle dispersion is similar to a Brownian motion, and it reads: δ = r D t 2 2 where D is the diffusion coefficient 18,35 . Recent experiments showed that ≈ D u L f 2 in 2D turbulence, where L f is the energy injection scale of turbulence 18 . The fact that ≈ S u braid 2 has therefore a remarkable consequence: the braid entropy S braid is a linear function of D (see Fig. 6). Quantitatively, we have measured: The forcing scale L f links a single-particle metric characteristic D, to a multi-particle topological descriptor S braid .
Although such a connection between single and multi-particle descriptors might sound surprising, it may originate from the uncorrelated motion of the particles that compose the braid. Indeed, we emphasize that the transition to the exponential growth regime of N E is observed for time scales > t T 6 L (see Fig. 4). At these time scales, both the single 18 and pair dispersion computed on the braided trajectories (with inter-particle distance being larger than L f ) show Brownian statistics. To our knowledge, there is as yet no theoretical understanding as to why the entanglement of independent Brownian trajectories results in an exponential growth of the topological length N E . We note that 2D turbulence plays an important role in this phenomenology since the r.m.s velocity Scientific RepoRts | 5:18564 | DOI: 10.1038/srep18564 u 2 depends on the kinetic energy accumulated in the inertial range. The relation linking S braid to D could be useful in oceanography to identify the energy injection scale L f from Lagrangian data 43 .
Much interest lies in determining Lagrangian tenets of turbulence irreversibility that would complement the Kolmogorov energy flux relations formulated in the Eulerian frame [32][33][34][35] . The braid entropy S braid is a promising topological measure of the irreversible deformation of fluid lines in 2D turbulence. On a practical note, the braid approach is particularly suitable for the analysis of natural flows in the ocean for which only sparse data are available. Much work is yet to be done to test the properties and potential applications of the braid entropy in fluid turbulence.

Methods
Turbulence generation. In these experiments, turbulence is generated using two different methods. In the first, 2D turbulence is generated electromagnetically in stratified layers of fluid 36 18,36,37 . The interaction between these vortices, through the inverse energy cascade process, provides the energy that drives the turbulent flow. The bottom layer reduces the bottom drag and makes the flow in the top layer two-dimensional.
In the second setup, Faraday surface waves are used to generate 2D turbulence 19,20 . The horizontal fluid motion on the surface of such parametrically excited waves shows strong similarities with the fluid motion in 2D turbulence. In these experiments, Faraday waves are formed in a circular container (178 mm diameter) filled with a liquid whose depth (30 mm) is larger than the wavelength of the perturbation at the surface (deep water approximation). An electrodynamic shaker is used to vertically vibrate the container. The forcing frequency f 0 is monochromatic and is set to 30, 45, 60 or 110 Hz. The wavelength λ of the sub-harmonic Faraday waves is a function of = / f f 2 0 . We have recently demonstrated that Faraday waves can generate lattice of horizontal vortices whose characteristic scale is roughly λ/2. The interaction between these vortices produces a turbulent flow. This method represents a versatile tool of laboratory modeling of 2D turbulence since Faraday wave turbulence can be produced in a broad range of kinetic energy level and forcing scales λ ≈ / surface. The use of surfactant ensures that particles do not aggregate on the surface and it facilitates the homogeneous spatial distribution of the particles. Videos are recorded at high frame rate (60 ~ 600 Hz) and a 16 bit resolution using the Andor Neo sCMOS camera. The flows are characterized using both particle image velocimetry (PIV) and particle tracking velocimetry (PTV) techniques. We use PIV to compute the Eulerian energy spectra of the flows shown in Fig. 1. The PIV velocity fields are computed on a 90 × 90 spatial grid (8 × 8 cm 2 (FWT), 10 × 10 cm 2 (EMT) field of view) with a time step of 0.008 s (FWT) or 0.033 s (EMT). The 2D Lagrangian trajectories used in the braid analysis are tracked by PTV techniques using a nearest neighbor algorithm 17,18 . In a highly turbulent flow (kinetic energy = − u 10 2 3 m 2 s −2 and integral characteristic timescale = − T 10 L 2 s), hundreds of particles can be tracked simultaneously for 4 s at 120 fps over a 8 × 8 cm 2 field of view.
The braid method. Topological fluid loops. In the course of time, each crossing along the braid distorts the topological loop and forces it to get more and more entangled in the impenetrable strands of the braid, see (Fig. 2g). It has recently been demonstrated that the level of entanglement of a loop can be described by a quantity N E called "topological length" or braiding factor 14 . N E is equal to the number of times the loop crosses a imagined line (horizontal dashed line in Fig. 2(g)_right panel) ) passing through all the particles that compose the braid at time t. Although it is named topological length, N E is a topological quantity that ultimately does not require the notion of "distance". Its time evolution is completely described by the sequence of crossings along a braid.
Experimental measurements. To compute the topological braids made of fluid tracers trajectories and their corresponding braiding factor N E , we use tools from the braidlab library 14,44 which have been modified to allow the computation to be carried out on a large number of trajectories. The analysis was performed over braids that are composed of = N 10 traj up to = N 100 traj Lagrangian trajectories for which the inter-particle distance is larger than energy injection scale L f . The experimental capacities allow the computation of the single particle diffusion coefficient D and the braiding factor N E over large statistical samples (~3000 trajectories).