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86
Efficient Computation and Visualization of Coherent Structures
 in Fluid Flow Applications.” Transactions on Visualization and Computer Graphics/IEEE Visualization 2007
"... Abstract—The recently introduced notion of FiniteTime Lyapunov Exponent to characterize Coherent Lagrangian Structures provides a powerful framework for the visualization and analysis of complex technical flows. Its definition is simple and intuitive, and it has a deep theoretical foundation. While ..."
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Cited by 50 (6 self)
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Abstract—The recently introduced notion of FiniteTime Lyapunov Exponent to characterize Coherent Lagrangian Structures provides a powerful framework for the visualization and analysis of complex technical flows. Its definition is simple and intuitive, and it has a deep theoretical foundation. While the application of this approach seems straightforward in theory, the associated computational cost is essentially prohibitive. Due to the Lagrangian nature of this technique, a huge number of particle paths must be computed to fill the spacetime flow domain. In this paper, we propose a novel scheme for the adaptive computation of FTLE fields in two and three dimensions that significantly reduces the number of required particle paths. Furthermore, for threedimensional flows, we show on several examples that meaningful results can be obtained by restricting the analysis to a wellchosen plane intersecting the flow domain. Finally, we examine some of the visualization aspects of FTLEbased methods and introduce several new variations that help in the analysis of specific aspects of a flow. Index Terms—Flow Visualization, Feature Detection, 3D Vector Field Visualization. 1
A variational theory of hyperbolic Lagrangian Coherent Structures,
 Physica D,
, 2011
"... a b s t r a c t We develop a mathematical theory that clarifies the relationship between observable Lagrangian Coherent Structures (LCSs) and invariants of the CauchyGreen strain tensor field. Motivated by physical observations of trajectory patterns, we define hyperbolic LCSs as material surfaces ..."
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Cited by 43 (7 self)
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a b s t r a c t We develop a mathematical theory that clarifies the relationship between observable Lagrangian Coherent Structures (LCSs) and invariants of the CauchyGreen strain tensor field. Motivated by physical observations of trajectory patterns, we define hyperbolic LCSs as material surfaces (i.e., codimensionone invariant manifolds in the extended phase space) that extremize an appropriate finitetime normal repulsion or attraction measure over all nearby material surfaces. We also define weak LCSs (WLCSs) as stationary solutions of the above variational problem. Solving these variational problems, we obtain computable sufficient and necessary criteria for WLCSs and LCSs that link them rigorously to the CauchyGreen strain tensor field. We also prove a condition for the robustness of an LCS under perturbations such as numerical errors or data imperfection. On several examples, we show how these results resolve earlier inconsistencies in the theory of LCS. Finally, we introduce the notion of a Constrained LCS (CLCS) that extremizes normal repulsion or attraction under constraints. This construct allows for the extraction of a unique observed LCS from linear systems, and for the identification of the most influential weak unstable manifold of an unstable node.
Relation between Kinematic Boundaries, Stirring, and Barriers for the Antarctic Polar Vortex
 J. Atmos. Sci
"... Maximum stretching lines in the lower stratosphere around the Antarctic polar vortex are diagnosed using a method based on finitesize Lyapunov exponents. By analogy with the mathematical results known for simple dynamical systems, these curves are identified as stable and unstable manifolds of the ..."
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Cited by 32 (0 self)
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Maximum stretching lines in the lower stratosphere around the Antarctic polar vortex are diagnosed using a method based on finitesize Lyapunov exponents. By analogy with the mathematical results known for simple dynamical systems, these curves are identified as stable and unstable manifolds of the underlying hyperbolic structure of the flow. For the first time, the exchange mechanism associated with lobe dynamics is characterized using atmospheric analyzed winds. The tangling manifolds form a stochastic layer around the vortex. It is found that fluid is not only expelled from this layer toward the surf zone but also is injected inward from the surf zone, through a process similar to the turnstile mechanism in lobe dynamics. The vortex edge, defined as the location of the maximum gradient in potential vorticity or tracer, is found to be the southward (poleward) envelope of this stochastic layer. Exchanges with the inside of the vortex are therefore largely decoupled from those, possibly intense, exchanges between the stochastic layer and the surf zone. It is stressed that using the kinematic boundary defined by the hyperbolic points and the manifolds as an operational definition of vortex boundary is not only unpractical but also leads to spurious estimates of exchanges. The authors anticipate that more accurate dynamical systems tools are needed to analyze stratospheric transport in terms of lobe dynamics. 1.
Hyperbolic lines and the stratospheric polar vortex
 Chaos
"... The necessary and sufficient conditions for Lagrangian hyperbolicity recently derived in the literature are reviewed in the light of older concepts of effective local rotation in strain coordinates. In particular, we introduce the simple interpretation of the necessary condition as a constraint on ..."
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Cited by 24 (0 self)
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The necessary and sufficient conditions for Lagrangian hyperbolicity recently derived in the literature are reviewed in the light of older concepts of effective local rotation in strain coordinates. In particular, we introduce the simple interpretation of the necessary condition as a constraint on the local angular displacement in strain coordinates. These mathematically rigorous conditions are applied to the winter stratospheric circulation of the southern hemisphere, using analyzed wind data from the European Center for MediumRange Weather Forecasts. Our results demonstrate that the sufficient condition is too strong and the necessary condition is too weak, so that both conditions fail to identify hyperbolic lines in the stratosphere. However a phenomenological, nonrigorous, criterion based on the necessary condition reveals the hyperbolic structure of the flow. Another ͑still nonrigorous͒ alternative is the finitesize Lyapunov exponent ͑FSLE͒ which is shown to produce good candidates for hyperbolic lines. In addition, we also tested the sufficient condition for Lagrangian ellipticity and found that it is too weak to detect elliptic coherent structures ͑ECS͒ in the stratosphere, of which the polar vortex is an obvious candidate. Yet, the FSLE method reveals a clear ECSlike barrier to mixing along the polar vortex edge. Further theoretical advancement is needed to explain the apparent success of nonrigorous methods, such as the FSLE approach, so as to achieve a sound kinematic understanding of chaotic mixing in the winter stratosphere and other geophysical flows. © 2002 American Institute of Physics. ͓DOI: 10.1063/1.1480442͔ Transport plays an important role in the distribution of chemicals in the stratosphere "the layer of atmosphere between 12 and 55 km in altitude…. This fact is clearly illustrated by, for instance, the formation of the Antarctic ozone hole every austral winter. In the extratropical stratosphere, chemical transport proceeds in quasihorizontal layers, where air parcels practically conserve entropy for up to about 3 weeks. Transport, stirring, and mixing in these isentropic layers is governed by the Lagrangian chaos generated by organized largescale circulations "of several hundred kilometers and larger…. The spatial organization of chaotic stirring is described by the main hyperbolic lines "i.e., the material lines that are locally the most attracting or repelling… forming at any time a skeleton of paths and lobes through the flow. Gradients of longlived tracers tend to orient normal to and intensify along strongly attracting lines, thereby enhancing the mixing process by smallscale vertical circulations. At the same time, a strong vortical circulation exists in the winter polar region. The polar vortex exemplifies an elliptic coherent structure: its edge forms a partial barrier to mixing. Rigorous mathematical criteria were derived recently to characterize hyperbolic lines and elliptic coherent structures. In this paper, we review and test these criteria in a case study using stratospheric winds from the European Center for MediumRange Weather Forecasts. Our work shows that these criteria fail to pick out hyperbolic lines and elliptic coherent structures in the stratosphere, which are, however, readily identified with other less rigorous methods.
Optimal trajectory generation in ocean flows
 In Proc. American Control Conf. 674–679
, 2004
"... Abstract — In this paper it is shown that Lagrangian Coherent Structures (LCS) are useful in determining near optimal trajectories for autonomous underwater gliders in a dynamic ocean environment. This opens the opportunity for optimal path planning of autonomous underwater vehicles by studying the ..."
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Cited by 21 (1 self)
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Abstract — In this paper it is shown that Lagrangian Coherent Structures (LCS) are useful in determining near optimal trajectories for autonomous underwater gliders in a dynamic ocean environment. This opens the opportunity for optimal path planning of autonomous underwater vehicles by studying the global flow geometry via dynamical systems methods. Optimal glider paths were computed for a 2dimensional kinematic model of an endpoint glider problem. Numerical solutions to the optimal control problem were obtained using Nonlinear Trajectory Generation (NTG) software. The resulting solution is compared to corresponding results on LCS obtained using the Direct Lyapunov Exponent method. The velocity data used for these computations was obtained from measurements taken in August, 2000, by HFRadar stations located around Monterey Bay, CA. I.
Coherent structures and isolated spectrum for PerronFrobenius cocycles
, 2008
"... We present an analysis of onedimensional models of dynamical systems that possess “coherent structures”; global structures that disperse more slowly than local trajectory separation. We study cocycles generated by expanding interval maps and the rates of decay for functions of bounded variation u ..."
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Cited by 20 (8 self)
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We present an analysis of onedimensional models of dynamical systems that possess “coherent structures”; global structures that disperse more slowly than local trajectory separation. We study cocycles generated by expanding interval maps and the rates of decay for functions of bounded variation under the action of the associated Perron–Frobenius cocycles. We prove that when the generators are piecewise affine and share a common Markov partition, the Lyapunov spectrum of the Perron–Frobenius cocycle has at most finitely many isolated points. Moreover, we develop a strengthened version of the Multiplicative Ergodic Theorem for noninvertible matrices and construct an invariant splitting into Oseledets subspaces. We detail examples of cocycles of expanding maps with isolated Lyapunov spectrum and calculate the Oseledets subspaces, which lead to an identification of the underlying coherent structures. Our constructions generalise the notions of almostinvariant and almostcyclic sets to nonautonomous dynamical systems and provide a new ensemblebased formalism for coherent structures in onedimensional nonautonomous dynamics.
Scalable Computation of Streamlines on Very Large Datasets
 In Proceedings of the International Conference for High Performance Computing, Networking, Storage and Analysis
, 2009
"... Understanding vector fields resulting from large scientific simulations is an important and often difficult task. Streamlines, curves that are tangential to a vector field at each point, are a powerful visualization method in this context. Application of streamlinebased visualization to very large ..."
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Cited by 20 (7 self)
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Understanding vector fields resulting from large scientific simulations is an important and often difficult task. Streamlines, curves that are tangential to a vector field at each point, are a powerful visualization method in this context. Application of streamlinebased visualization to very large vector field data represents a significant challenge due to the nonlocal and datadependent nature of streamline computation, and requires careful balancing of computational demands placed on I/O, memory, communication, and processors. In this paper we review two parallelization approaches based on established parallelization paradigms (static decomposition and ondemand loading) and present a novel hybrid algorithm for computing streamlines. Our algorithm is aimed at good scalability and performance across the widely varying computational characteristics of streamlinebased problems. We perform performance and scalability studies of all three algorithms on a number of prototypical application problems and demonstrate that our hybrid scheme is able to perform well in different settings.
Visualization of coherent structures in transient 2d flows
 In TopologyBased Methods in Visualization 2007, to appear, Mathematics + Visualization
, 2007
"... The depiction of a timedependent flow in a way that effectively supports the structural analysis of its salient patterns is still a challenging problem for flow visualization research. While a variety of powerful approaches have been investigated for over a decade now, none of them so far has been ..."
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Cited by 15 (3 self)
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The depiction of a timedependent flow in a way that effectively supports the structural analysis of its salient patterns is still a challenging problem for flow visualization research. While a variety of powerful approaches have been investigated for over a decade now, none of them so far has been able
Detection of Lagrangian coherent structures in threedimensional turbulence
 J. Fluid Mech
, 2007
"... We use direct Lyapunov exponents (DLE) to identify Lagrangian coherent structures in two different threedimensional flows, including a single isolated hairpin vortex, and a fully developed turbulent flow. These results are compared with commonly used Eulerian criteria for coherent vortices. We find ..."
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Cited by 14 (1 self)
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We use direct Lyapunov exponents (DLE) to identify Lagrangian coherent structures in two different threedimensional flows, including a single isolated hairpin vortex, and a fully developed turbulent flow. These results are compared with commonly used Eulerian criteria for coherent vortices. We find that, despite additional computational cost, the DLE method has several advantages over Eulerian methods, including greater detail and the ability to define structure boundaries without relying on a preselected threshold. As a further advantage, the DLE method requires no velocity derivatives, which are often too noisy to be useful in the study of a turbulent flow. We study the evolution of a single hairpin vortex into a packet of similar structures, and show that the birth of a secondary vortex corresponds to a loss of hyperbolicity of the Lagrangian coherent structures. 1.
Transport and stirring induced by Vortex Formation
, 2007
"... The purpose of this study is to analyse the transport and stirring of fluid that occurs owing to the formation and growth of a laminar vortex ring. Experimental data was collected upstream and downstream of the exit plane of a piston–cylinder apparatus by particleimage velocimetry. This data was us ..."
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Cited by 14 (1 self)
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The purpose of this study is to analyse the transport and stirring of fluid that occurs owing to the formation and growth of a laminar vortex ring. Experimental data was collected upstream and downstream of the exit plane of a piston–cylinder apparatus by particleimage velocimetry. This data was used to compute Lagrangian coherent structures to demonstrate how fluid is advected during the transient process of vortex ring formation. Similar computations were performed from computational fluid dynamics (CFD) data, which showed qualitative agreement with the experimental results, although the CFD data provides better resolution in the boundary layer of the cylinder. A parametric study is performed to demonstrate how varying the pistonstroke lengthtodiameter ratio affects fluid entrainment during formation. Additionally, we study how regions of fluid are stirred together during vortex formation to help establish a quantitative understanding of the role of vortical flows in mixing. We show that identification of the flow geometry during vortex formation can aid in the determination of efficient stirring. We compare this framework with a traditional stirring metric and show that the framework presented in this paper is better suited for understanding stirring/mixing in transient flow problems. A movie is available with the online version of the paper.