** **

Special Session

**Complexity of spatio-temporal optical instabilities**

Chairman: **Marc SCIAMANNA**, CentraleSupélec (France)

Although nonlinear dynamics of optical systems have been studied since about forty years, recent years have seen an emerging new interest in applying concepts of complexity and time-series analysis to advanced photonic components. Examples include the analysis of entropy in a large class of laser systems with time-delayed optical feedback or coupling, statistical studies of extreme events in chaotic time-series, bifurcation of spatio-temporal dynamics in nonlinear light-matter interaction. This session will review recent works and identify perspectives that will make the most fruitful use of nonlinear science and nonlinear photonics.

Speakers (6) - Date: on May 23 or May 24, 2016

**Delphine Wolfersberger**, CentraleSupélec (France)

Dynamical interactions of counterpropagating Airy beams

Abstract: We analyse theoretically the spatiotemporal dynamics of two incoherent counterpropagating Airy beams interacting in a photorefractive crystal under focusing conditions. For a large enough nonlinearity strength the interaction between the two Airy beams leads to light-induced waveguiding. The stability of the waveguide is determined by the crystal length, the nonlinearity strength and the beam’s intensities and is improved when comparing to the situation using Gaussian beams. We further identify the threshold above which the waveguide is no longer static but evolves dynamically either time-periodically or even chaotically. Above the stability threshold, each Airy-soliton moves erratically between privileged output positions that correspond to the spatial positions of the lobes of the counterpropagating Airy beam. These results suggest new ways of creating dynamically varying waveguides, optical logic gates and chaos-based computing.

**Antonio Hurtado**, University of Strathclyde (UK)

Controllable activation and inhibition of spiking patterns in Vertical Cavity Surface Emitting Lasers

Abstract: Photonic techniques emulating the powerful computational capabilities of cortical neurons are attracting increasing research interest as these offer exciting prospects for novel ultrafast neuromorphic computing systems [1-4]. One of these approaches uses Semiconductor Lasers (SLs), as they can undergo a wide variety of nonlinear dynamical responses similar to those observed in neurons but several (up to 9) orders of magnitude faster [5-7]. Amongst SLs, Vertical Cavity Surface Emitting Lasers (VCSELs) are ideal devices for neuromorphic photonics given their inherent advantages, i.e. reduced manufacturing costs, high coupling efficiency to optical fibers, ease to integrate in 2D arrays, etc. However, it is only recently that VCSELs have started to attract attention for novel photonic neuronal models [1,6-8]. In this talk, we will review our recent work on the achievement of reproducible and controllable spiking patterns in VCSELs operating at important telecom wavelengths (e.g. 1300nm and 1550nm). Specifically, we will show that a wide variety of spiking regimes, e.g. single and multiple spiking and bursting patterns can be controllably produced in these devices in response to externally induced perturbations [8]. Moreover, these spiking regimes are obtained with sub-nanosecond speed resolution and reduced recovery time offering promise for ultrafast non-traditional information processing capabilities. During our talk, we will also introduce our very recent results on the controllable inhibition of spiking patterns in VCSELs with nanosecond speed resolution. The reproducible/controllable activation and inhibition of spiking responses at high speed rates in VCSELs operating at telecom wavelengths offer great potential for the use of these devices in novel ultrafast neuromorphic photonic information processing modules for applications in non-digital computing systems and future optical networks.

References

1. A. Hurtado et al, Appl. Phys. Letts. 100, 103703 (2012)

2. B. Garbin et al, Nat. Commun., 6, 5915 (2015)

3. R. Al-Seyab et al, IEEE International Semiconductor Laser Conference, 165-166 (2014)

4. A.N. Tait et al, Nanophotonic Information Physics, Springer, 183-222 (2014)

5. S. Wieczorek et al, Phys. Rev. Lett. 88, 063901 (2002)

6. K. Schires et al, Electron. Letts., 48, 872 (2012)

7. M Turconi et al, Phys. Rev. E 88, 022923 (2013)

8. A. Hurtado et al, Appl. Phys. Letts. 107, 241103 (2015)

**Sergei Turitsyn**, University of Aston (UK)

Practical applications of spatio-temporal instabilities in optical fibre systems

Abstract: to be defined

**Stefania Residori**, INLN, Université de Nice-Sophia Antipolis, CNRS (France)

Liquid crystals for dynamic control of optical phase and wavefront shaping

Abstract: Liquid crystals offer a unique versatile platform for optical phase control and wavefront shaping. In particular, liquid crystal textures such as planar and twisted nematics, homeotropic and their umbilical defects, allow manipulation of optical wavefronts in different configurations. Optical vortex beams are dynamically created either via two-beam interaction in an optically addressed spatial light modulator or directly by the anisotropy stabilized self-induction of vortex-like defects. I will present a review on optical phase manipulation in this kind of systems as well as more recent results on the control of light by means of geometric Berry phase.

**Claudio Conti**, Institute for Complex Systems CNR-ISC (Italy)

Landscape and Complexity of Nonlinear Waves

Abstract: We will review our work on the introduction of ideas of statistical mechanics of disordered systems in the study of nonlinear regimes, ranging from random lasers to solitons and rogue waves

**Andrea Fratalocchi**, KAUST (Saudi Arabia)

Evolutionary photonics: a review of recent results

Abstract: Evolutionary photonics takes inspiration from natural systems and phenomena such as chaos and unpredictability, creating new technologies in material science, energy harvesting and nanomedicine. In this invited talk, I summarize my research activity in the field, discussing recent results including chaotic energy harvesting (Nat Phot 7 474 2013), the control of subwavelength and ultrafast rogue waves (Nat Phys 11 358 2015), the development of disordered sensors for early-stage cancer detection (Science Advances, 2015), and finally a new generation of black-body “lasers” (Nat Nanotech 10 11 2016).

**Numerics of Chaos: Algorithms & Applications**

chaired by

**Péter Koltai & Florian Rupp**

The goal of this session is to bring together mathematicians who work in different areas of applied mathematics (numerics, bifurcation & chaos theory, random dynamical system, high performance computing etc.) and might thus not meet and exchange ideas and points of view. Consequently, the session program addresses a cross section of theoretical and computational developments and their applications to natural and social science, mechanics and life sciences. Areas of analytical interest include the theory of linear/ nonlinear deterministic/ random/ stochastic differential equations and chaos, the qualitative behavior of solutions and their bifurcation, stochastic stability and asymptotics, control-theoretic issues, efficient algorithms and related aspects. A key aspect of this session is its focus on the impact of theoretical results on the study of real-world problems.

Presentations in alphabetic order of the presenters:

• Ralf Banisch (University of Edinburgh, Scotland): Detecting coherent sets with spacetime diffusion maps

• Andreas Denner (Technische Universität München, Germany): Transfer Operator Families and Coherent Sets

• Peter Giesl (University of Sussex, England): Determination of the Basin of Attraction by Computing Contraction Metrics

• Sigurdur F. Hafstein (Reykjavik University, Reykjavik, Iceland): Lyapunov functions on finite time intervals: theory and a computational method

• Boumediene Hamzi (Imperial College London, England): Data-based methods for Lorenz-86: A simple atmospheric model.

• Erika Hausenblas (Montanuniversität Leoben, Austria):The Numerical approximation of the invariant measure of levy driven stochastic differential equations

• Péter Koltai (Freie Universität Berlin, Germany): Coherent Families: Spectral Theory for Transfer Operators in Continuous Time

• Marian Mrozek (Jagiellonian University, Kraków, Poland): Morse-Conley-Forman theory for combinatorial vector fields

• Florian Rupp (German University of Technology in Oman, Sultanate of Oman): Chaotic Attractors in Stochastic Hopf-Bifurcations

Detecting coherent sets with spacetime diffusion maps

**Ralf Banisch**

University of Edinburgh, Scotland

E-mail: ralf.banisch@ed.ac.uk

Abstract: Intuitively, coherent sets are subsets of the configuration space that stay together under the (possibly chaotic) dynamics. Many different approaches for making this notion precise exist in the literature. For example, one approach defines coherent sets via spectral properties of the transfer operator, and another defines coherent sets as tight bundles of trajectories by specifying a euclidean distance metric in spacetime. We show that these two approaches can be reconciled: By replacing the Euclidean distance in spacetime with an augmented version of the distance used in diffusion maps, one can make contact with the transfer operator notion of coherence in the infinite data limit. The resulting numerical method, which can be used to extract coherent sets directly from trajectory data, is related to similar methods that have been discussed in the past. We demonstrate its performance on several examples.

**Transfer Operator Families and Coherent Sets**

**Andreas Denner**

Technische Universität München, Germany

E-mail: andreas.denner@ma.tum.de

Abstract: The computation of sets in phase space of a time dependent dynamical system which are separated by transport barriers, so called coherent sets, is of interest for systems which often show chaotic behaviour, e.g. atmospheric flows or plasma physics. In this talk we present a way to compute those finite-time coherent structures via considering the system at all time instants. This is done by analysing a corresponding transfer operator family as a whole. We furthermore discuss different discretizations, some of them leading to recently developed, purely data-driven algorithms and so providing a set oriented justification for those.

**Determination of the Basin of Attraction by Computing Contraction Metrics**

**Peter Giesl**

University of Sussex, England

E-mail: p.a.giesl@sussex.ac.uk

Abstract: The determination of the basin of attraction of an equilibrium or periodic orbit can be achieved by different methods. In this talk, we discuss a local method using a contraction metric, i.e. a Riemannian metric with a local contraction property. It can be used to prove existence and uniqueness of an equilibrium (or periodic orbit) and determine a subset of its basin of attraction without requiring information about its position. We present a method to numerically construct such a contraction metric. The contraction metric is characterised as a matrix-valued function, satisfying a certain linear PDE. Then it will be approximated by meshless collocation, in particular Radial Basis Functions. We will present error estimates as well as numerical examples. This is joint work with Holger Wendland, University of Bayreuth, Germany.

Lyapunov functions on finite time intervals: theory and a computational method

**Sigurdur F. Hafstein**

Reykjavik University, Reykjavik, Iceland

E-mail: sigurdurh@ru.is

Abstract: Lyapunov functions for nonautonomous systems on finite time intervals can deliver useful information on the qualitative and quantitative behavior of the system trajectories on that interval. A novel method using linear programming to parameterize such Lyapunov functions is studied and some theoretical and numerical results are presented.

**Data-based methods for Lorenz-86: A simple atmospheric model.**

**Boumediene Hamzi**

Department of Mathematics, AlFaisal University, Riyadh, KSA

E-mail: hamzib110@gmail.com

Abstract: The model used in this study is that of Lorenz (1986), as modified by Wirosoetisno (2000), and will be referred to here as the extended Lorenz-1986 model or exL86. It has only 4 degrees of freedom, but admits both a fast gravity wave and a chaotic vortical mode, with an asymptotic, nonlinear balance between fast and slow variables. The advantage of models such as exL86 is that the balance between fast and slow variables is well understood, and the assimilated analysis can thus be easily interpreted in terms of the balanced and unbalanced components of the motion. The fact that this model is conservative does not pose a great difficulty, since the intention here is to use it to study kernel based algorithms in the context of the slow versus the fast variables. As pointed out by Lorenz (1986) and Wirosoetisno (2000), dissipation of gravity waves is not the cause of the existence of a slow manifold, and therefore models such as this one can be quite representative of realistic balanced dynamics. In this talk, we use kernel methods in (Bouvrie and Hamzi, 2011) for the data-based modelling of the Lorenz-86 model and show how we can detect the fast-slow dynamics.

**The Numerical approximation of the invariant measure of levy driven stochastic differential equations**

**Erika Hausenblas**

Montanuniversität Leoben, Austria

E-mail: erika.hausenblas@unileoben.ac.at

Abstract: A stochastic differential equation (SDE) is a differential equation in which one or more of the terms is a stochastic process, resulting in a solution which is itself a stochastic process. SDEs are used to model diverse phenomena such as fluctuating stock prices or physical systems subject to thermal fluctuations. However, most of the models are based on Gaussian noise, although, in recent years Levy randomness began to draw much attention. This is the case, e.g. if the random perturbation process should be used to model abrupt pulses or extreme events. A more natural mathematical framework for these phenomena takes into account perturbations other than purely Brownian, in particular Levy processes or general semimartingales with jumps. Levy randomness needs other techniques, is quite intricate and far from amenable to mathematical analysis. In case of monotonicity of the coefficients, one can show that there exists a unique invariant measure. This invariant measure characterize the long time behavior of such a solution. However, for most of the systems the invariant measure is not explicitly given and can only be found by numerical simulations. In our talk we will present some results concerning the numerical approximation of the invariant measure.

**Coherent Families: Spectral Theory for Transfer Operators in Continuous Time**

**Péter Koltai**

Freie Universität Berlin, Germany

E-mail: peter.koltai@fu-berlin.de

Abstract: The decomposition of the state space of a dynamical system into metastable or almost-invariant sets is important for understanding macroscopic behavior. This concept is well understood for autonomous dynamical systems, and has recently been generalized to non-autonomous systems via the notion of coherent sets. We elaborate here on the theory of coherent sets in continuous time for periodically-driven flows and describe a numerical method to find families of coherent sets without trajectory integration.

**Morse-Conley-Forman theory for combinatorial vector fields**

**Marian Mrozek**

Jagiellonian University, Kraków, Poland

E-mail: mrozek@ii.uj.edu.pl

Abstract: In late 90' R. Forman introduced the concept of a combinatorial vector field on a CW complex and presented a version of Morse theory for acyclic combinatorial vector fields. He also studied combinatorial vector fields without acyclicity assumption, introduced the concept of a chain recurrent set and proved Morse inequalities in this setting. In this talk we present the Morse-Conley theory for combinatorial vector fields and a certain generalization of combinatorial vector fields oriented on applications in sampled dynamics. In particular, we study Morse decompositions and Conley-Morse graph for such fields.

**Chaotic Attractors in Stochastic Hopf-Bifurcations**

**Florian Rupp**

German University of Technology in Oman, Sultanate of Oman

E-mail: florian.rupp@gutech.edu.om

Abstract: Unlike Hopf-bifurcations in deterministic systems were a pair of complex conjugate eigenvalues traverse the imaginary axis simultaneously, stochastic Hopf-bifurcations exhibit a sequential crossing of the eigenvalues. Thereby, instead of a limit cycle, a chaotic attractor is established that governs the dynamics during and after the change of stability. Exploiting novel simulation methods based on a cohomolgy between flows generated by stochastic and random ordinary differential equations we numerically study the generation and development this chaotic attractor and determine some of its key characteristics including subsets of its basin of mean-square stability.

**Special Session on "Lagrangian coherent structures in fluids and plasmas"**

Session chairs:

**Abraham Chian**, University of Adelaide - Australia, Institute for Aeronautical Technology (ITA) & National Institute for Space Research (INPE) - Brazil

**Rodrigo Miranda**, University of Brasilia (UnB) - Brazil

**Invited speakers**:

**Dario Borgogno**, University of Cote d'Azur - France: Lagrangian coherent structures in fusion plasmas.

**Abraham Chian**, University of Adelaide - Australia, ITA & INPE - Brazil: Magnetic coherent structures in space plasmas.

**Ana Mancho**, Instituto de Ciencias Matemáticas CSIC - Spain: Beautiful geometries underlying oceanic nonlinear processes.

**Rodrigo Miranda**, University of Brasilia - Brazil: Lagrangian and Eulerian coherent structures in complex fluids and plasmas.

**Maria Olascoaga**, University of Miami - USA: Lagrangian coherent structures in geophysical flows.

**Hua Xia**, The Australian National University- Australia: Lagrangian coherent structures in wave generated flows.

Session description: Lagrangian coherent structures (LCS) provide a new dynamical systems tool to determine the repelling and attracting material lines that organize the transport of fluid and plasma flows. The term “Lagrangian” refers to flows defined by the fluid motion instead of an instantaneous snapshot (Eulerian); the term “coherent” refers to the distinguished stability of these structures compared to other nearby material lines/surfaces. This special session will discuss the application of the LCS approach to improve the understanding of transport in complex flows in fluids and plasmas such as planetary atmosphere, oceans, and thermonuclear and astrophysical plasmas.

**Abstracts:**

**1.Lagrangian coherent structures in fusion plasmas**

**Dario Borgogno**

University of Cote d'Azur - France

E-mail: dario.borgogno@oca.eu

Magnetic field lines embedded in a plasma confinement system are often characterized by a chaotic motion. This weakens the confinement properties of any magnetic configuration. However, even in case of chaotic domains, magnetic barriers can emerge and limit the field line motion itself. In the context of the numerical simulation of a Reversed-Field Pinch (RFP) configuration a new magnetic topology analysis, borrowed from previous fluid dynamic studies, is discussed. This methodology relies on the behavior of the Finite Time Lyapunov Exponent (FTLE) associated with the magnetic field. By referring to a previous work in which the magnetic field is given in terms of analytical function the FTLE field shows the presence of ridges, special gradient lines normal to the direction of minimum curvature, forming magnetic barriers. These ridges can be recognized as Lagrangian Coherent Structures (LCSs) for the system, actually opposing the penetration of magnetic field lines across them. In this article a more general numerical scheme for the detection of the LCSs has been adopted that allows analysis of realistic cases in which the magnetic fields are numerically known on a discrete mesh. After a validation test performed on the analytical case, a first application to a numerical magnetohydrodynamics simulation of the RFP, characterized by a broad chaotic region, has been performed. A strong magnetic barrier has been observed that effectively limits the field lines motion inside the chaotic sea.

References:

1. G. Rubino, D. Borgogno, D., M. Veranda, et al., Plasma Phys. Contr. Fusion, 57, 085004 (2015)

2. D. Borgogno, D., D. Grasso, F. Pegoraro et al. Phys. Plasmas, 18, 102307 (2011)

3. D. Borgogno, D., D. Grasso, F. Pegoraro et al. Phys. Plasmas, 18, 102308 (2011)

4. D. Borgogno, D. Grasso, F. Pegoraro, F.; et al. Phys. Plasmas, 15, 102308 ( 2008)

**2. Magnetic coherent structures in space plasmas**

**Abraham Chian**

University of Adelaide - Australia, ITA & INPE - Brazil

E-mail: abraham.chian@gmail.com

We study coherent structures in solar photospheric flows in a plage in the vicinity of the active region AR10930 using the horizontal velocity data derived from Hinode/Solar Optical Telescope magnetograms. Eulerian and Lagrangian coherent structures (LCSs) are detected by computing the Q-criterion and the finite-time Lyapunov exponents of the velocity field, respectively. Our analysis indicates that, on average, the deformation Eulerian coherent structures dominate over the vortical Eulerian coherent structures in the plage region. We demonstrate the correspondence of the network of high magnetic flux concentration to the attracting Lagrangian coherent structures (aLCSs) in the photospheric velocity based on both observations and numerical simulations. In addition, the computation of aLCS provides a measure of the local rate of contraction/expansion of the flow.

References:

1. E. L. Rempel, A. C.-L. Chian, A. Brandenburg, Astrophysical Journal Letters, 735, L9 (2011)

2. E. L. Rempel, A. c.-L. Chian, A. Brandenburg, Physica Scripta 86, 018405 (2012)

3. E. L. Rempel, A. C.-L. Chian, A. Brandenburg, P. R. Munoz, S. C. Shadden, Journal of Fluid Mechanics, 729, 309 (2013)

4. A. C.-L. Chian ACL, E. L. Rempel, G. Aulanier, B. Schmieder, S. C. Shadden, B. T. Welsch, A. R. Yeates, Astrophysical Journal, 786, 51 (2014)

**3.Beautiful geometries underlying oceanic nonlinear processes**

**Ana Mancho**

Instituto de Ciencias Matemáticas CSIC - Spain

E-mail: a.m.mancho@icmat.es

Finding order in the apparent chaos that seems to govern ocean motions is a formidable task which has drawn the attention of scientists and oceanographers all over the world for the last decades. The endeavour of describing how different tracers (heat, salt, chlorophyll, contaminants) are transported in the ocean has become a global challenge, and its understanding is of vital importance for predicting and assessing their impact on global climate change or the distribution of natural marine resources.

In this talk I will provide an overview of recently developed tools, the so called Lagrangian Descriptors, which display beautiful geometries highlighting the always changing dynamical skeleton of the ocean. I will illustrate applications of these objects to the operational management of coastal emergencies such as the sinking and subsequent fuel spill by the Oleg Naydenov fishing ship in the Gran Canaria coast, Spain in April 2015.

References:

1. J. A. J. Madrid, Ana M. Mancho. Chaos 19, 013111-1-013111-18 (2009)

2. C. Mendoza, Ana M. Mancho. Physical Review Letters 105, 3, 038501-1-038501-4 (2010)

3. A. M. Mancho, S. Wiggins, J. Curbelo, C. Mendoza. Communications in Nonlinear Science

and Numerical Simulation. 18, 3530-3557 (2013)

4. C. Lopesino, F. Balibrea, S. Wiggins, A.M. Mancho. Communications in Nonlinear Science

and Numerical Simulation 27 (1-3), 40-51 (2015)

5. C. Mendoza, A. M. Mancho, S. Wiggins. Nonlinear Processes in Geophysics 21, 677-

689 (2014)

6. V. J. Garcia-Garrido, A. M. Mancho, S. Wiggins, C. Mendoza. Nonlinear Processes in

Geophysics 22 (6), 701-712 (2015)

7. V. J. Garcia-Garrido, A. Ramos, A. M. Mancho, J. Coca, S. Wiggins. Ocean Modelling

(submitted) (2015)

**4. Lagrangian and Eulerian coherent structures in complex fluids and plasmas**

**Rodrigo Miranda**

University of Brasilia - Brazil

E-mail: rmiracer@gmail.com

In this talk we present numerical simulations of a two-dimensional incompressible flow in a crisis-like transition to hyperchaos. We construct bifurcation diagrams and detect nonattracting chaotic sets known as chaotic saddles, using a fixed frame of reference in Fourier space (i.e., an Eulerian approach). We characterize the chaotic mixing properties of the fluid by detecting Lagrangian coherent structures and demonstrate that, prior to the transition, chaotic saddles can be used to predict the enhanced complexity of the spatiotemporal patterns observed in the hyperchaotic regime. In addition, we will characterize coherent structures in three-dimensional simulations of magnetized Keplerian shear flows in a regime of on-off intermittency. We demonstrate that large-scale coherent structures are characterized by high degrees of Fourier amplitude-phase synchronization by computing the Shannon entropy of amplitudes and phases in the three-dimensional spectral space. Finally, we will show the application of Lagrangian coherent structures and the Jensen-Shannon complexity-entropy technique to analyze numerical simulations of non-Newtonian flows in modelsof blood vessels.

References:

1. R. A. Miranda, E. L. Rempel, A. C.-L. Chian, N. Seehafer, B. A. Toledo and P. R. Muñoz. CHAOS, 23, 033107 (2013¬¬¬¬¬¬¬¬¬¬¬¬)

2. R. A. Miranda, E. L. Rempel and A. C.-L. Chian. Phys. Plasmas, 19, 112303 (2012)

3. R. A. Miranda, A. C.-L. Chian and E. L. Rempel. Adv. Space Res., 15, 1893 ( 2013)

4. R. A. Miranda, E. L. Rempel and A. C.-L.Chian. Mon. Not. Royal Astron. Soc.,448, 804 (2015)

**5. Lagrangian coherent structures in geophysical flows**

**Maria Olascoaga**

University of Miami – USA

E-mail: jolascoaga@rsmas.miami.edu

Lagrangian Coherent Structures (LCS) are key material surfaces that shape global transport. Thegeodesic LCS theory enables unified detection of all relevant LCS types in unsteady 2-d flows in a frame-independent manner. These are: hyperbolic LCS (generalized invariant manifolds ormaximally attracting/repelling material lines); elliptic LCS (generalized KAM tori or eddyboundaries); and parabolic LCS (generalized twistless KAM tori or jet-core barriers). In this talk I will review the basic elements of the theory and present results from several recent applications to oceanic flows which shed light into the nature of surface-ocean dispersion as well producedaccurate transport estimates and predicted sudden changes in the shape of pollutant distributions.

Work in collaboration with F. Beron-Vera, Y. Wang and G. Haller

References:

1. Y. Wang, M. J. Olascoaga, F. J. Beron-Vera. Geophys. Res. Lett. 42, 4072-4079 (2015).

2. M. J. Olascoaga, F. J. Beron-Vera, G. Haller, J. Trinanes, M. Iskandarani, E. F. Coelho, B. Haus, H. S. Huntley, G. Jacobs, A. D. Kirwan, Jr., B. L. Lipphardt, Jr., T. Ozgokmen, A. J. H. M. Reniers, A. Valle-Levinson, Geophys. Res. Lett., 40, 6171-6175 (2013).

3. M. J. Olascoaga, M. G. Brown, F. J. Beron-Vera, and H. Kocak.Nonlin. Processes Geophys., 19, 687-692 (2012).

4. M. J. Olascoaga, G. Haller. Proceedings of the National Academy of Sciences of the United States of America, 109, 4738-4743 (2012).

**6. Lagrangian coherent structures in wave generated flows**

**Francois,Hua Xia, N. Francois, H. Punzmann, and M. Shats**

Research School of Physics and Engineering,

The Australian National University, Canberra, ACT 0200 Australia

E-mail: hua.xia@anu.edu.au

Complex flows can be generated by both the Faraday waves on a vertical vibrating shaker [1] and by propagating waves using a solid plunger [2]. Under various flow conditions, we observed the vertical vorticity generation, which results in the generation of two-dimensional turbulence on the water surface [3] and water wave tractor beam [2].

The understanding of statistical properties of Lagrangian trajectories [4] in complex flows is crucial for problems such as spreading of plankton in the ocean, transport of pollutants in the atmosphere, or rain initiation in clouds. Our recent results show that fluid particle dispersion is diffusive and is determined by a single measurable Lagrangian scale related to the forcing scale [4]. Turbulence however is a state of a flow dominated by a hierarchy of scales, and it is not clear which of these scales mostly affect particle dispersion. Moreover, coherent structures often coexist with turbulence, such as the small scale forcing in laboratory experiments, or the self-generated spectral condensation [5]. How those coherent structures affect particle dispersion is not well understood. Recently developments in scientific imaging and computational power make it possible to tackle this problem experimentally.

In our experiments, flows are generated in a wide range of kinetic energy, forcing scales and boundary conditions. We analysed the Lagrangian properties of the flows using the single particle dispersion, pair separation, Finite Time Lyaponov Exponent (FTLE) and topological braids [6].

In this talk, I will give a brief overview of the chaotic flow generation in the laboratory using Faraday waves. Analysis of the Lagrangian properties of the flow will be presented. At last, I will report analysis of topological braids, which reveals the existence of coherent bundles in two-dimensional turbulence [7]. The existence of such bundles, whose width is determined by the turbulence forcing scale, substantially modifies the statistics of pair dispersion in 2D turbulence as compared with the expectations from the Richardson-Obukhov law. The results point to a single scale dynamics of the pair separation and the importance of extreme separation events.

References:

1. N. Francois, H. Xia, H. Punzmann, S. Ramsden and M. Shats, Physical Review X, 4, 021021

(2014).

2. H. Punzmann, N. Francois, H. Xia, G. Falkovich and M. Shats, Nature Physics, 10, 658 (2013)

3. N. Francois, H. Xia, H. Punzmann and M.Shats, Physical Review Letters, 110, 194501 (2013)

4. H. Xia, N. Francois, H. Punzmann, and M. Shats, Nature Communications, 4, 2013 (2013)

5. H. Xia, H. Punzmann, G. Falkovich and M. Shats, Physical Review Letters, 101, 194504

(2008)

6. N. Francois, H. Xia, H. Punzmann, B. Faber and M. Shats, Scientific Report, 5, 18564 (2015)

7. H. Xia, N. Francois, B. Faber, H. Punzmann and M. Shats, Physical Review Letters, submitted (2016)