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150
Lagrangian coherent structures and mixing in twodimensional turbulence
 Physica D
, 2000
"... We introduce a Lagrangian definition for the boundaries of coherent structures in twodimensional turbulence. The boundaries are defined as material lines that are linearly stable or unstable for longer times than any of their neighbors. Such material lines are responsible for stretching and foldin ..."
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Cited by 85 (7 self)
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We introduce a Lagrangian definition for the boundaries of coherent structures in twodimensional turbulence. The boundaries are defined as material lines that are linearly stable or unstable for longer times than any of their neighbors. Such material lines are responsible for stretching and folding in the mixing of passive tracers. We derive an analytic criterion that can be used to extract coherent structures with high precision from numerical or experimental data sets. The criterion provides a rigorous link between the Lagrangian concept of hyperbolicity, the Okubo–Weiss criterion, and vortex boundaries. We apply the results to simulations of twodimensional barotropic turbulence. © 2000 Elsevier Science B.V. All rights reserved.
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.
Overview of flow visualization
 The Visualization Handbook
, 2005
"... With increasing computing power, it is possible to process more complex fluid simulations. However, a gap between increasing data size and our ability to visualize them still remains. Despite the great amount of progress that has been made in the field of flow visualization over the last two decades ..."
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Cited by 29 (14 self)
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With increasing computing power, it is possible to process more complex fluid simulations. However, a gap between increasing data size and our ability to visualize them still remains. Despite the great amount of progress that has been made in the field of flow visualization over the last two decades, a number of challenges remain. Whilst the visualization of 2D flow has many good solutions, the visualization of 3D flow still poses many problems. Challenges such as domain coverage, speed of computation, and perception remain key directions for further research. Flow visualization with a focus on surfacebased techniques forms the basis of this literature survey, including surface construction techniques and visualization methods applied to surfaces. We detail our investigation into these algorithms with discussions of their applicability and their relative strengths and drawbacks. We review the most important challenges when considering such visualizations. The result is an uptodate overview of the current stateoftheart that highlights both solved and unsolved problems in this rapidly evolving branch of research.
Topological methods for 2D timedependent vector fields based on stream lines and path lines
 IEEE Transactions on Visualization and Computer Graphics
"... Abstract — This paper describes approaches to topologically segmenting 2D timedependent vector fields. For this class of vector fields, two important classes of lines exist: stream lines and path lines. Because of this, two segmentations are possible: either concerning the behavior of stream lines, ..."
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Cited by 28 (14 self)
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Abstract — This paper describes approaches to topologically segmenting 2D timedependent vector fields. For this class of vector fields, two important classes of lines exist: stream lines and path lines. Because of this, two segmentations are possible: either concerning the behavior of stream lines, or of path lines. While topological features based on stream lines are well established, we introduce path line oriented topology as a new visualization approach in this paper. As a contribution to stream line oriented topology we introduce new methods to detect global bifurcations like saddle connections and cyclic fold bifurcations as well as a method to tracking all isolated closed stream lines. To get the path line oriented topology we segment the vector field into areas of attracting, repelling and saddlelike behavior of the path lines. We compare both kinds of topologies and apply them to a number of test data sets. Index Terms — flow visualization, vector field topology, bifurcations, stream lines, path lines I.
Multifieldgraphs: An approach to visualizing correlations in multifield scalar data
 IEEE Transactions on Visualization and Computer Graphics
"... We present an approach to visualizing correlations in 3D multifield scalar data. The core of our approach is the computation of correlation fields, which are scalar fields containing the local correlations of subsets of the multiple fields. While the visualization of the correlation fields can be do ..."
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Cited by 27 (0 self)
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We present an approach to visualizing correlations in 3D multifield scalar data. The core of our approach is the computation of correlation fields, which are scalar fields containing the local correlations of subsets of the multiple fields. While the visualization of the correlation fields can be done using standard 3D volume visualization techniques, their huge number makes selection and handling a challenge. We introduce the MultifieldGraph to give an overview of which multiple fields correlate and to show the strength of their correlation. This information guides the selection of informative correlation fields for visualization. We use our approach to visually analyze a number of real and synthetic multifield datasets. Index Terms—Visualization, multifield, correlation. 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.
Visualizing lagrangian coherent structured and comparison to vector field topology
 In TopologyBased Methods in Visualization, Proceedings of the 2007 Workshop
, 2007
"... This paper takes a look at the visualization side of vector field analysis based on Lagrangian coherent structures. The Lagrangian coherent structures are extracted as height ridges of finitetime Lyapunov exponent fields. The resulting visualizations are compared to those from traditional instantan ..."
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Cited by 22 (7 self)
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This paper takes a look at the visualization side of vector field analysis based on Lagrangian coherent structures. The Lagrangian coherent structures are extracted as height ridges of finitetime Lyapunov exponent fields. The resulting visualizations are compared to those from traditional instantaneous vector field topology of steady and unsteady vector fields: they often provide more and better interpretable information. The examination is applied to 3D vector fields from a dynamical system and practical CFD simulations. 1
The state of the art in flow visualization: Partitionbased techniques
 In Simulation and Visualization 2008 Proceedings
, 2008
"... Flow visualization has been a very active subfield of scientific visualization in recent years. From the resulting large variety of methods this paper discusses partitionbased techniques. The aim of these approaches is to partition the flow in areas of common structure. Based on this partitioning, ..."
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Cited by 22 (2 self)
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Flow visualization has been a very active subfield of scientific visualization in recent years. From the resulting large variety of methods this paper discusses partitionbased techniques. The aim of these approaches is to partition the flow in areas of common structure. Based on this partitioning, subsequent visualization techniques can be applied. A classification is suggested and advantages/disadvantages of the different techniques are discussed as well. 1