Lorentz Center - Coherent Structures in Dynamical Systems from 16 May 2011 through 20 May 2011
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    Coherent Structures in Dynamical Systems
    from 16 May 2011 through 20 May 2011






Coherent structures and self-consistent chaotic transport in Hamiltonian systems
D. del-Castillo-Negrete


Self-consistent transport is the transport of a field F in which there is a feedback between F and the advection velocity. Two examples of interest to this talk are: vortex dynamics in shear flows, and charged particles in plasmas. In the first case, F is the vorticity, the advection equation is the 2-D Euler's equation, and the self-consistent constrain is the vorticity-streamfunction relation. In plasmas, F is the electron distribution function, the advection equation is the Vlasov equation, and the self-consistent constrain is the Poisson equation. We show that near marginal stability, the weakly nonlinear dynamics of these two systems can de described by a universal mean-field Hamiltonian model known as the Single Wave Model (SWM).  We present numerical and analytical results on the SWM in the N-->infinity kinetic limit and in the finite-N limit, where N is the number of degrees of freedom. Using this model, we study the formation and persistence of coherent structures in the presence of self-consistent chaos. Starting from the finite-N SWM we construct mean-field coupled symplectic map models of self-consistent transport.  We discuss how these maps open the possibility of studying chaos in Hamiltonian systems with a large number of degrees-of-freedom.



Formation of coherent structures by fluid inertia in 3D laminar flows

Herman Clercx


The formation and interaction of coherent structures that geometrically determine the transport properties of laminar 3D flow will be discussed. The impact of these structures on 3D laminar mixing will be demonstrated numerically and experimentally. Key result is the role of fluid inertia that induces partial disintegration of coherent structures of the non-inertial limit into chaotic regions and merger of surviving parts into intricate 3D structures. The response follows a universal scenario and reflects an essentially 3D route to chaotic mixing.



Lagrangian fluid dynamic models for analysis of planktonic predator-prey systems and small-scale fluid motions

John Dabiri

We apply a method of Lagrangian fluid dynamic analysis to a planktonic predator-prey system in which moon jellyfish Aurelia aurita uses its body motion to generate a flow that transports small plankton such as copepods to its vicinity for feeding. With the flow field generated by the jellyfish measured experimentally and the dynamics of plankton described by formulations akin to the Maxey-Riley equation, we use a Lagrangian coherent structure (LCS) method to identify the capture region in which prey can be captured by the jellyfish. The properties of the LCS and the capture region enable analysis of the effect of several physiological and mechanical parameters on the predator-prey interaction, such as prey size and shape, escape force, predator perception, etc. The methods can be readily extended to study the effects of physically or biologically-induced small-scale fluid motions on planktontic systems.



A nonlinear theory of the bimodality of the Kuroshio Extension

Henk Dijkstra


The bimodal behavior of the Kuroshio Extension (KE) in the North Pacific has fascinated physical oceanographers since indications of this phenomenon were found. Why would a western boundary current switch between coherent states of a large-meander and a small- meander in a few years time? Why does this phenomenon not appear in other western boundary currents, such as the Gulf Stream? For the Kuroshio, both large- and small meander states can persist over a period ranging from a few years to a decade. With the analysis of satellite data and those of in situ measurements, a quite detailed description of the different states and their transition behavior is now available.  There is still, however, no consensus on which processes cause the low-frequency variability in the KE.  It appears  that direct interpretation  of the  observations often has been based on  mechanisms  involving external  causes (such as atmospheric forcing) and  linear ocean dynamics while ocean  modelers  have tended to suggest mechanisms  which  involve elements of  nonlinear  ocean dynamics. A theory of the bimodality of the Kuroshio Extension should include a few essential explanations. It should explain (i) why the KE can be in different states and the origin of the spatial patterns of these states, (ii) the interannual time scale of the transition between the two states, and (iii) why there is much irregularity in the large-meander state while spatial and temporal variability are relatively low in the small-meander state.  In this presentation a nonlinear theory is proposed which is able to provide  answers to  the issues  above.

Uncovering fractional monodromy

Konstantinos Efstathiou


Fractional monodromy is a property of the geometry of the fibration of Liouville integrable Hamiltonian systems. Unlike standard monodromy which describes only the regular part of the fibration, fractional monodromy naturally appears when we look at the complete fibration, including singular fibres. In this talk I discuss fractional monodromy for n_1 : (-n_2) resonant Hamiltonian systems with n_1, n_2 coprime natural numbers. The geometry of the fibration is simplified by passing to an appropriate covering space where the same type of standard 1: (-1) monodromy is obtained independently of n_1, n_2. Pushing the results down to the original space gives fractional monodromy.



Coherent structures and particle transport in micro-scale steady streaming flows

Jeff D. Eldredge, Kwitae Chong


It is well known that a body oscillating at high frequency in a viscous fluid will create a steady streaming flow in its vicinity, due to the nonlinear interactions in the thin Stokes layer surrounding the body.  Fluid particles in this streaming flow travel, in a time-averaged sense, along closed orbits about streaming cells adjacent to the body.  Less well understood, however, are the behavior of inertial particles of finite size in such a flow, as well as the interacting streaming flows generated by multiple oscillating bodies in close proximity.  In this talk, we will present high-fidelity numerical simulations of streaming flows, generated by single and multiple oscillating bodies.  The simulations are carried out with the viscous vortex particle method, which has several advantages for such flows, particularly a useful computational efficiency for simulating many bodies separated by arbitrarily large distances.  In post-processing of these flows, we analyze the fluid and inertial particle transport, as well as the finite-time Lyapunov exponent field, in order to reveal the coherent structures..  It will be shown that inertial particles tend to focus at the centers of the streaming cells.  Furthermore, adjacent oscillating bodies can exchange particles under certain conditions.  Finally, future directions will be discussed.



Using LCS to study coherent structures in reacting flows

Melissa Green


Previous research has shown that chemical reactions in a compressible fluid flow interact strongly with the surrounding turbulence both in quantitative measures and qualitative character. In the case of a flame propagating through homogeneous isotropic turbulence, the rapid flow expansion generated in the reaction zone causes a significant attenuation in the vorticity. This suppression of the vorticity magnitude complicates the tracking of individual coherent structures using Eulerian methods, therefore we use Lagrangian coherent structures to study the nature of the vortex dynamics, focusing on structure creation, destruction, and reorientation.



Singularity Theory for KAM tori: General Framework

A. Gonzalez


In this talk we present the geometric and analytic framework underlying the methodology introduced in A. Haro's talk. First we present a parametric KAM result for the existence of the invariant Lagrangian deformations. The required non-degeneracy condition is very weak, indeed this is always satisfied in the in close-to-integrable case. Hereby, the infinite dimensional problem of finding invariant tori is reduced to the finite-dimensional problem of finding zeros of a smooth function. Using symplectic properties of invariant Lagrangian deformations, we show that the latter problem can be transformed to a problem of finding critical points of a real-valued function -- which we call the potential of the Lagrangian deformation. It is also shown that non-twist tori correspond to degenerate > critical points of the potential. This naturally leads to apply Singularity Theory to the critical points of the potential. This work is in collaboration with R. de la Llave and A. Haro.



Variational Theory of Lagrangian Coherent Structures

George Haller


Lagrangian Coherent Structures (LCS) are dynamically evolving surfaces that govern the evolution of complex material patterns in moving fluids and solids. Examples of such patterns include oil spills, ash clouds, plankton populations, schools of fish and moving crowds. Because of their finite lifetime and aperiodic nature, LCS have been challenging to locate, predict or control. At the same time, LCS promise to play a key role in the real-time assessment and short-term forecasting of a number of environmental conditions, including hazardous winds over airports and the spread of contaminants in coastal waters. In this talk, I describe a mathematical theory that enables a rigorous extraction of LCS from observational flow data. In this approach, LCS are defined as invariant surfaces that extremize an appropriate finite-time normal repulsion or attraction measure in the governing dynamical system.

Solving this variational problem leads to computable sufficient and necessary criteria for LCS. I will show recent applications to large scale oceanic and atmospheric flow problems.



On the ubiquity of monodromy in Hamiltonian systems

Heinz Hanßmann


Let us consider a quasi-periodic time-dependent perturbation of an autonomous Hamiltonian system and focus on an elliptic equilibrium point of the unperturbed Hamiltonian. Under suitable hypotheses, this equilibrium point becomes an invariant torus under the effect of the perturbation, with the same internal frequencies as the perturbation. The normal frequencies of this torus are a perturbation of the normal frequencies around the initial equilibrium point.


In this work I consider the interaction between these frequencies. Varying a parameter of the system in such a way that the internal frequencies remain fixed and the normal ones move with the parameter, two normal frequencies may enter into 1:-1 resonance with the internal frequencies, thus violating the second Mel'nikov condition. In fact, this typically occurs for a dense subset of parameters. In the talk I will study the Hamiltonian Hopf bifurcations that take place when the normal and internal frequencies are in resonance. I will also discuss the consequences for the dynamics of the system, in particular concerning the geometry of the bundle of Lagrangean tori.



Singularity Theory for KAM tori: Methodology

A. Haro


We present a novel method to find KAM tori with fixed frequency. The method enables us to prove existence of invariant tori with fixed frequency in degenerate cases, including persistence of invariant tori for perturbations of isochronous systems. Our method also leads to a natural classification of KAM  tori which is based on Singularity Theory. This talk aims to illustrate the main ideas of our approach. Motivating examples are given in the integrable and near-integrable cases --where the interpretation of our method is immediate-- and also in the far from integrable case which highlight the generality and the strength of our approach. This work is in collaboration with R. de la Llave and A. Gonzalez.



Stretching structures in the ocean surface: transport and biological impacts

Emilio Hernandez-Garcia


The Lagrangian description of fluid transport has been largely enriched with the introduction of stretching quantifiers such as the different types of Lyapunov exponents. Lagrangian Coherent Structures can be readily identified from them. Here we will focus on the finite-size Lyapunov exponent case, as applied to horizontal ocean flows estimated from satellite altimetry. The influence on biological organisms of the structures revealed by the Lyapunov analysis will be illustrated with examples from the bottom and from the top of marine ecosystems (phytoplankton and marine birds, respectively). Full understanding of the results requires discussion of the robustness of the approach, as well as the consideration of the vertical dimension. With this aim, preliminary results for three-dimensional flows will be presented.



Elliptic Lagrangian Coherent Structures in geophysical fluid flows

Hüseyin Koçak


This talk is the result of several years of collaboration with oceanographers at the University of Miami's Rosenstiel School of Marine and Atmospheric Science (Francisco Beron-Vera, Michael Brown, and María Olascoaga) and will deal with Lagrangian Coherent Structures (LCS) of elliptic type in incompressible two-dimensional fluid flows with special attention to geophysical fluid flows. Unlike hyperbolic LCS, which are locally the strongest normally attracting or repelling material fluid curves over a finite-time interval, elliptic LCS are material fluid curves which do not experience exponential stretching and folding over a finite-time interval. Thus while hyperbolic LCS can be regarded as finite-time generalizations of invariant manifolds of hyperbolic trajectories, elliptic LCS can be regarded as finite-time generalizations of invariant tori. As such, rather than facilitating mixing as hyperbolic LCS, elliptic LCS inhibit mixing. A vivid example of an elliptic LCS in a geophysical fluid flow system is that one at the perimeter of the austral stratospheric polar vortex. Such an elliptic LCS prevents ozone-depleted air from spreading toward lower latitudes, leading to the formation of the so-called ozone hole. The occurrence of elliptic LCS in various geophysical fluid flow systems will be illustrated. Also, a partial theory that provides support for their occurrence will be discussed. Such a partial theory relies on recent results relating to the Kolmogorov--Arnold--Moser theory for time-quasiperiodic one-degree-of-freedom Hamiltonian systems for which the frequency mapping is degenerate in the Kolmogorov sense.



Transport in aperiodic non-autonomous dynamical systems and applications to geophysical flows

Ana M. Mancho


We introduce and discuss new Lagrangian tools which are successful in achieving a detailed description of transport in aperiodic time dependent dynamical systems. We illustrate applications for purely advective transport events in general aperiodic geophysical flows. First is discussed the concept of Distinguished Trajectory which generalizes the concept of fixed point for aperiodic dynamical systems. It is built on a function that detects simultaneously, invariant manifolds, hyperbolic and non-hyperbolic flow regions, thus insinuating the active transport routes in the flow. Once these are recognized, the transport description is completed by means of the direct computation of the stable and unstable manifolds of the DHTs.



Spatial Structure of Spectral Fluxes in Weak Turbulence

Nicholas Ouellette


The term coherent structure is usually interpreted to mean some kind of dynamical object that is localized and persistent in both space and time. Nonlinear systems, however, also typically have nontrivial spectral properties, since nonlinearities in real space lead to nonlocalities in Fourier space. Here, we extend the idea of a coherent structure to include the spectral properties of the dynamical system. Using a recently developed filtering technique, we study the flux of energy and enstrophy between scales as a function of space and time in an experimental quasi-two-dimensional flow that is weakly turbulent. We show that these fluxes are localized in space, time, and scale, and discuss the implications of this observation.



Coherent structures and biological invasions

Shane Ross


The language of coherent structures provides a new means for discussion of transport and mixing of atmospheric pathogens, paving the way for new modeling and management strategies for the spread of infectious diseases affecting plants, domestic animals, and humans. Atmospheric dynamical structures have an influence on aeroecology, namely the population structure of airborne microbe species. We report on a recent integration of experimental biology and applied mathematics uncovering how Lagrangian coherent structures and their phase space complements, strongly connected (almost-invariant) regions, provide a framework for understanding how airborne microbe populations are dispersed and mixed. Applications to identification of frontiers between qualitatively different kinds of behavior in other areas will also be discussed.



Bifurcations from travelling waves in radially heated rotating spherical shells

Juan Sanchez Umbria, Fernando Garcia Gonzalez and Marta Net Marce


Travelling waves appearing in the thermal convection of a pure fluid contained in a spherical shell with the boundaries at different temperatures are studied. They are computed, by using continuation methods, as steady solutions of a system for the waves, in the frame of reference of the spheres. Navier-Stokes equations are written in terms of two scalar potentials for the velocity, which are expanded, as the temperature, in spherical harmonics, and collocation is employed in the radius. The special block-tridiagonal structure of the linear part of the equations provides a preconditioner, which allows an efficient calculation of the waves. Their stability is also studied, and the secondary bifurcations to subharmonic or modulated waves are detected.



KAM theory for non-autonomous dynamical systems with quasi-periodic time dependence
Mikhail B. Sevryuk

Following mainly our paper [1], we will present a general theory of the birth of quasi-periodic solutions with n+N basic frequencies in perturbations of families of autonomous dynamical systems, under the assumption that the unperturbed systems admit quasi-periodic solutions with n basic frequencies whereas the perturbation is quasi-periodic in time with N Diophantine basic frequencies. The first n frequencies of the perturbed solutions are close to the unperturbed frequencies whereas the last N frequencies of the perturbed solutions coincide with the frequencies of the perturbation itself. In the particular case n=0 one speaks of a quasi-periodic response to the perturbation. The theory is `structured' and treats systems belonging to various symmetry classes (dissipative, volume preserving, Hamiltonian, and reversible) separately but in a unified manner. The nondegeneracy conditions to be imposed on the smooth families of the unperturbed solutions are very weak (of the so-called Ruessmann type). The perturbed quasi-periodic solutions are organized into Whitney-smooth families.

We will also consider some results concerning more conventional nondegeneracy conditions (following, e.g., the fundamental paper [2] devoted to Hamiltonian systems) as well as the situations where the perturbation is quasi-periodic not only in time but also in spatial variables (the recent paper [3] examining planar Hamiltonian and reversible systems provides a key example).

[1] M. B. Sevryuk, Invariant tori in quasi-periodic non-autonomous dynamical systems via Herman's method, Discrete Contin.
Dyn. Syst. 18 (2007) 569-595
[2] A. Jorba and J. Villanueva, On the persistence of lower dimensional invariant tori under quasi-periodic perturbations, J. Nonlinear Sci. 7 (1997) 427-473
[3] H. Hanssmann and J. Si, Quasi-periodic solutions and stability of the equilibrium for quasi-periodically forced planar reversible and Hamiltonian systems under the Bruno condition, Nonlinearity 23 (2010) 555-577



Variability in diffusion-reaction based on Lagrangian Coherent Structures

Wenbo Tang


In this talk we explore the connection between the variability of decaying turbulent diffusion-reaction and Lagrangian Coherent Structures (LCS). In particular the reaction processes work against homogenization by turbulent diffusion. Consequently they are highly dependent on the local flow information, provided by stretching and folding of the chaotic flow, characterized by LCS. The particular example we study addresses nutrient uptake advantage of motile micro-organisms over non-motile species. We find that the uptake advantage, quantified as the difference in total uptake over time and space between the species, can be associated with attracting LCS. We will discuss the implications of our approach to large-scale geophysical fluid dynamics problems, particularly establishment of parameterization on eddy diffusion based on LCS.



Topological detection of Lagrangian coherent structures
Jean-Luc Thiffeault

In many applications, particularly in geophysics, we often have fluid trajectory data from floats, but little or no information about the underlying velocity field. The standard techniques for finding transport barriers, based for example on finite-time Lyapunov exponents, are then inapplicable. However, if there are invariant regions in the flow this will be reflected by a `bunching up' of trajectories. We show that this can be detected by tools from topology. The method relies on examining a large number of topological loops, encoded symbolically. These loops wrap around the trajectories, which are viewed as topological obstacles. As the trajectories move around, they cause most loops to grow. The few loops that do not grow, or grow slowly, can be associated with coherent structures. This is joint work with Michael Allshouse.



Two-dimensional turbulence and coherent vortex structures

GertJan van Heijst


In contrast to its counterpart in the 3D world, turbulence in 2D is characterized by an inverse energy cascade. The presence of this inverse cascade in 2D turbulence is visible in the so-called self-organization of such flows: larger coherent vortex structures are observed to emerge from initially random flow fields.


The lecture will address the evolution of 2D turbulent flows on a finite domain with no-slip walls. The organized state consists of a large, domain-filling cell. Results of both laboratory experiments in rotating / stratified fluids and numerical simulations have revealed the crucial role played by the unsteady boundary layers: the domain boundaries act as important sources of large-amplitude vorticity filaments that may influence the motion in the interior. Attention will be given to global flow quantities like the kinetic energy, the enstrophy, and the total angular momentum. In the case of forced 2D turbulence, the latter quantity may show a remarkable flip-flopping behaviour, associated with a collapse of the organized flow state followed by its re-organization. In the 2D turbulent flow, vortex structures may undergo interactions and deformations due to shear and strain imposed by the ambient flow field. Some theoretical models and laboratory experiments will be discussed in which the effects of merger and time-periodic shear upon individual vortex structures are investigated.



Steady endless vortices with chaotic streamlines

Oscar Velasco


An endless vortex is an eddy whose axis of rotation is a continuous curve with neither beginning nor end. The best-known example is the smoke ring, which is a vortex whose axis of rotation is a circle. Here we study the dynamics and transport properties of more complex examples, namely knotted and linked toroidal vortices. These are thin tubular vortices uniformly coiled on an immaterial torus in an otherwise quiescent fluid. Their evolution depends on the number of vortices in the fluid, the winding number of the vortices (the ratio of the number of coils round the symmetry axis to the number of coils round the centreline), and the aspect ratio of the supporting torus (the ratio of the cross-section radius to the centreline radius). The evolution was computed using the Rosenhead¬Moore approximation to the Biot¬Savart law to evaluate the velocity field and a fourth-order Runge-Kutta scheme to advance in time. It was found that when the supporting torus is thin and the number of vortices is small the system progresses along and rotates around the torus symmetry axis in an almost steady manner, with each vortex approximately preserving its shape. In these cases the velocity field, observed in the comoving frame, has two stagnation points. The stream tube starting at the front stagnation point transversely intersects the stream tube ending at the rear stagnation point along a finite number of streamlines, creating a three-dimensional chaotic tangle. An analysis of the geometry of the tangle showed that more vortices coiled on tori of equal aspect ratio carry more fluid, whereas the same number of vortices coiled on thicker tori carry less fluid.





Coherent Structures in Three-Dimensional Flows

J. H. Bettencourt, C. Lopez and E. Hernandez-Garcia


Coherent Structures are known to drive biological dynamics, from plankton to top predators, thus it is very important to be able to characterize them in realistic three dimensional flows. The Finite-Size Lyapunov Exponent (FSLE) is a measure of particle dispersion in fluid flows and the ridges of this scalar field locate regions of the velocity field where strong exponential separation between particles occur. These regions are referred to as Lagrangian Coherent Structures (LCS). We have identified LCS in two different 3D flows: a canonical turbulent velocity field, that is the turbulent flow between two parallel stationary plates, driven by a pressure gradient in the mean flow direction and a primitive equation model (ROMS) simulation of the oceanic flow in the Benguela region.



Persistence of Cylindrical Manifolds Under Forcing in a Multi-Degree of Freedom Hamiltonian System with Applications to Ship Capsize

M.D. Cooper, L.S. McCue and S.D. Ross


For the abstract, click here.



Persistence of normally hyperbolic invariant manifolds: the noncompact case

Jaap Eldering


Normally hyperbolic invariant manifolds (NHIM's) are a generalization to hyperbolic fixed points. The fixed point is replaced by a whole invariant manifold with corresponding generalized hyperbolicity criteria. Similar to a hyperbolic fixed point, a NHIM has (un)stable manifolds and persists under small perturbations.


The classical theorems [1,2] on persistence assume compactness of the invariant manifold. We formulate a persistence theorem for general noncompact NHIM's. The results include C^{k,\alpha} smoothness with Hölder continuity of the highest derivatives.


The proof is based on the Perron method and uses ideas of Henry [3] and Van Gils and Vanderbauwhede [4]. To properly generalize to arbitrary manifolds, e.g. with non-trivial normal bundle, we require the concept of a Riemannian manifold of bounded geometry.


We illustrate that the smoothness of the perturbed manifold is governed by the spectral gap condition and that it is optimal. Using further examples we show some of the issues specific to the noncompact setting and how bounded geometry comes into play.



[1] Neil Fenichel, Persistence and smoothness of invariant manifolds for flows,

    Indiana Univ. Math. J. 21 (1971/1972), 193–226.

[2] M. W. Hirsch, C. C. Pugh, and M. Shub, Invariant manifolds,

    Lecture Notes in Mathematics, vol. 583, Springer-Verlag, Berlin, 1977.

[3] Daniel Henry, Geometric theory of semilinear parabolic equations,

    Lecture Notes in Mathematics, vol. 840, Springer-Verlag, Berlin, 1981.

[4] A. Vanderbauwhede and S. A. van Gils, Center manifolds and contractions on

    a scale of Banach spaces, J. Funct. Anal. 72 (1987), no. 2, 209–224.



Lagrangian coherent structures in swarming dynamics of bacterial colonies

Mohammad M. Farazmand, George Haller


A recent variational theory of Lagrangian Coherent Structures (LCS) [1] provides a sufficient and necessary criterion for ridges of the Finite-time Lyapunov Exponent (FTLE) field to mark LCS. The criterion requires the computation of the eigenvalues and eigenvectors of the Cauchy–Green strain tensor field over a grid of initial conditions. We propose a numerical method to accurately calculate these eigenvalues and eigenvectors without high computational cost. We then apply this numerical technique to study LCS in experimental data on swarming bacterial colonies [2]. We show that only a subset of the ridges of this FTLE field correspond to hyperbolic LCS. A careful study of the velocity field suggests that the non-hyperbolic ridges result from jet-like structures. Finally, the hyperbolic LCS are carefully advected under the flow map in order to visualize the temporal motion of these bacterial colonies.



[1] G. Haller, Variational theory of hyperbolic Lagrangian Coherent Structures,

Physica D (2010), doi:10.1016/j.physd.2010.11.010

[2] H. P. Zhang et al, EPL, 87(2009), 48011, doi: 10.1209/0295-5075/87/48011



Topological chaos, braiding and breakup of almost-invariant sets

Piyush Grover


In certain two-dimensional time-dependent flows, the braiding of periodic orbits provides a way to analyze chaos in the system through application of the Thurston-Nielsen classification theorem (TNCT). We apply the TNCT to braiding of almost-cyclic sets, which are closely related to almost-invariant sets (AIS). AIS in a fluid flow are regions with high local residence time that can act as stirrers or ghost rods. We also discuss the break up of the AIS as a parameter value is changed, which results in a sequence of topologically distinct braids. We show that, for Stokes flow in a lid-driven cavity, these various braids give good lower bounds on the topological entropy over the respective parameter regimes. Hence we make the case that even in the absence of periodic orbits, almost-cyclic regions identified using a transfer-operator approach can reveal the underlying structure that enables the topological analysis of chaos in the domain.



From the Nearshore and Back Again: Towards a Kinematic Understanding of Larval Transport

Cheryl Harrison, Gary Glatzmaier, Dave Siegel and Satoshi Mitarai


We focus on the transport dynamics of coastal marine larvae in an idealized ROMS Eastern Boundary current model. Coherent structures control two main processes of ecological interest. First, nearshore to far offshore transport is dominated by filamentation correlating with attracting LCS. Second, settlement (transport back to the coast) is moderated by a meandering upwelling jet and an associated upwelling front coincident with a separatrix. We also explore the relationship between mixing efficiency, filamentation and wind forcing.



Horizontal stirring in the global ocean

Ismael Hernandez


Horizontal mixing and the distribution of coherent structures in the global ocean are analyzed using Finite-Size Lyapunov Exponents (FSLE), computed for a surface layer of a velocity field obtained from the Ocean general circulation model For the Earth Simulator (OFES). FSLEs measure horizontal stirring and dispersion, and horizontal structures organizing the oceanic flow can be identified as ridges of the FSLE field, for which we have performed a comparative study between hemispheres and different oceans. The computed Probability Distributions Functions (PDFs) of FSLE, obtained for all the oceans and for the main currents, are broad and symmetric. Horizontal mixing is generally more active in the northern hemisphere than in the southern one. Nevertheless the Southern Ocean is the most active ocean, being the Pacific the less active one. A striking result is that the main currents aggregate into two groups of activity: Western Boundary Currents, which have broad PDFs with large FSLE values, and Eastern Boundary Currents with narrower ranges and lower FSLE values. Both groups of activity are also found when we correlate FSLE fields with Eddy Kinetic Energy (EKE) and vorticity (w), finding dispersion relations.These relations reveal the existence of dynamical and dispersion relations which characterize the dynamics of different ocean regions, which can be used for instance to explain differences in the distribution of quantities of biological interest.



Universality and renormalization of quasiperiodically forced one dimensional maps

A. Jorba, P. Rabassa, J.C. Tatjer


For the abstract, click here.



Surface tracking and efficient computation of Lagrangian coherent structures in 3 dimensions

Doug Lipinski


In this poster, an algorithm for efficiently computing Lagrangian coherent structures (LCS) in three dimensions based on the finite time Lyapunov exponent field (FTLE) is presented. This algorithm relies on a process of surface tracking which computes the FTLE values only near ridges in the FTLE field. This offers a reduction in computational order from about O(dx3) to O(dx2). The ridge points are then used to build a surface triangulation to represent the LCS. This algorithm offers significant speed ups for LCS computations and offers several other benefits including ease of LCS visualization and lobe extraction for further analysis. The properties of the algorithm are discussed in the context of several test cases and corresponding results are presented. Examples include a double gyre flow, Arnold-Beltrami-Childress flow, and a hurricane simulation.



The Hopf-transversal bifurcation in planar Filippov systems

Xia Liu, Konstantinos Efstathiou, Henk W. Broer


Abstract: Discontinuous vector fields find many applications in several fields including mechanical and electrical engineering. In this work we consider a planar Filippov vector field which consists of two parts separated by a discontinuity boundary. On one side a vector field goes through a Hopf bifurcation, and on the other side a constant vector field transversally crossing the boundary. A generic three parameter family of this Hopf-transversal vector field is studied and all the bifurcations and the phase portraits of the system are determined. Furthermore, this bifurcation scenario persists under small perturbations.



Formation of vortex dipole in a numerical model of water flow at the mouth of a river at different Reynolds numbers

Erick Javier López-Sánchez, Gerardo Ruiz-Chavarría


The ocean and atmosphere are full of vortices, i.e., locally recirculating flows with approximately circular streamlines and trajectories. Most often the recirculation is in horizontal planes, perpendicular to the gravitational acceleration and rotation vectors in the vertical direction. Vortices are often referred to as coherent structures, connoting their nearly universal circular flow pattern, no matter what their size or intensity, and their longevity in a Lagrangian coordinate frame that moves with the larger-scale, ambient flow [1]. We model the flow at the mouth of a river, with different flow rates. Coherent vortex dipole is formed at the output of channel. The formation and destruction are not considered by the potential theory. We analyze the vortex formation stage with for different Reynolds numbers using a numerical method. We compare numerical results with potential theory in the case of shallow water, and we analyze the fluid particle trajectories and solid particles trajectories, the later by solving an ordinary differential equation [2]. The vortex causes a suction effect, which can be seen in the trajectories of fluid particles. The simulation is done by solving the system of equations in stream function (y) - vorticity (w) formulation, obtained from the Navier- Stokes and continuity in two dimensions. A pseudo-spectral method for spatial coordinates Chebyshev and difference finite for the time are used [3]. (semi-implicit Adams-Bashfort schema).



[1] McWilliams, J. C. (2006) Fundamentals of Geophysical Fluid Dynamics. Cambridge University Press 105-107

[2] Mordant, N. (2001). Mesure lagrangienne en turbulence: mise en ouvre et analyse. Ph. D. Thesis. ENS de  Lyon, France.

[3] Peyret, R. (2002) Spectral methods for incompressible viscous flow. Applied Mathematical Sciences 148 Springer, N. Y. USA.



E_cient computation of FTLE-LCS in 3D directly from experimental particle tracking data

Samuel G. Raben, Pavlos Vlachos, Shane Ross


For the abstract, click here.



Investigating the connection between complexity of isolated trajectories and Lagrangian coherent structures

Irina I. Rypina, Sherry Scott, Larry J. Pratt and Michael G. Brown


It is argued that the complexity of fluid particle trajectories provides the basis for a new method, referred to as the Complexity Method (CM), for estimation of Lagrangian coherent structures in aperiodic flows that are measured over finite time intervals. The basic principles of the CM are explained and the CM is tested in a variety of examples, both idealized and realistic, and in different reference frames. Two measures of complexity are explored in detail: the correlation dimension of trajectory, and a new measure -- the ergodicity defect. Both measures yield structures that strongly resemble Lagrangian coherent structures in all of the examples considered. Since the CM uses properties of individual trajectories, and not separation rates between closely spaced trajectories, it is well suited for the analysis of ocean float and drifter data sets in which trajectories are typically widely and non-uniformly spaced.



Vortex 'pinch-off' in the Norbury family of vortices

Clara O'Farrell, John O. Dabiri


Jetting swimmers, such as squid or jellyfish, propel themselves by forming axisymmetric vortex rings. It is known that vortex rings cannot grow indefinitely, but rather 'pinch off' once they reach their physical limit, and that a decrease in the efficiency of fluid transport is associated with pinch-off. Therefore, in order to evaluate the performance of jetting swimmers, we must assess the optimality of the vortex wakes they produce, which requires an understanding of their stability. We consider the Norbury family of vortices (Norbury 1973) as a model for the vortex rings produced by jetting swimmers. Pozrikidis (1986) has studied the stability of Hill's spherical vortex under axisymmetric prolate and oblate shape perturbations. However, the stability of other members of the Norbury family to axisymmetric perturbations of the type that might occur during the vortex formation process in jetting swimmers is unknown. In order to assess the stability of different members of the family, we introduce physically pertinent shape perturbations and simulate their development in a manner akin to Pozrikidis' analysis. As we progress through the members of the family, we observe a change in the perturbation response that mirrors the occurrence of pinch-off in jetting flows. We observe the evolution of the Lagrangian coherent structures (LCS) for the different perturbation responses, and compare them to experimental results for pinched-off vortex rings (O'Farrell and Dabiri 2010).



Barriers to Mixing: Lagrangian Coherent Structures In Lake St Clair

Patricia Pernica and M. Wells


Lake St Clair is a large (SA~1100 km2) but shallow lake (maximum depth of 8 m) situated in between Lake Huron and Lake Erie and carries the outflow of the 3 upper Great Lakes.   Empirical evidence including significant differences in residences times and water quality measurements in the east and west basins of Lake St Clair suggests a barrier to mixing between these regions.  Using depth averaged 2D velocity data produced from the NOAA finite volume coastal ocean model (FVCOM) of Lake St Clair we extracted Lagrangian Coherent Structures (LCSs) for July 2009.  The existence of a persistent LCS transport barrier between the east and west basin is observed even with changes in wind speed and direction.  Numerical drifter trajectories are compared with 9 GPS track drifters released during the week of July 17th to July 23rd 2009.  Comparison demonstrates fairly good agreement between the field experiment and the numerical simulation and the actual drifter release results correspond with the presence of the main transport LCS barrier calculated from the model.