Computing Lagrangian coherent structures from their variational theory

Farazmand, Mohammad; Haller, George

doi:10.1063/1.3690153

Cited by 185 publications

(169 citation statements)

References 25 publications

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“…4a illustrates the coherency of forcing in these experiments, there can be other, perhaps more rigorous measures of the presence of persistent scales in a flow. For example, the persistence can be derived from the analysis of Lagrangian coherent structures using finite-time Lyapunov exponent field (see for example, Farazmand and Haller 25 additional comments on this). The lifetime of a persistent structure t 0 can be determined by varying time interval of the Lyapunov exponent computation and comparing it with the turnover time of the turbulent eddy of size r in the Kolmogorov spectrum, t r ¼ E À 1/3 r 2/3 .…”

Section: Discussionmentioning

confidence: 99%

Lagrangian scale of particle dispersion in turbulence

et al. 2013

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Transport of mass, heat and momentum in turbulent flows by far exceeds that in stable laminar fluid motions. As turbulence is a state of a flow dominated by a hierarchy of scales, it is not clear which of these scales mostly affects particle dispersion. Also, it is not uncommon that turbulence coexists with coherent vortices. Here we report on Lagrangian statistics in laboratory two-dimensional turbulence. Our results provide direct experimental evidence that fluid particle dispersion is determined by a single measurable Lagrangian scale related to the forcing scale. These experiments offer a new way of predicting dispersion in turbulent flows in which one of the low energy scales possesses temporal coherency. The results are applicable to oceanographic and atmospheric data, such as those obtained from trajectories of freedrifting instruments in the ocean.

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Section: Discussionmentioning

confidence: 99%

Lagrangian scale of particle dispersion in turbulence

et al. 2013

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“…Another perspective is the direct computation of manifolds [13,14,15], which has also provided valuable insight into oceanic problems [16,17]. Other approaches in this field have been the geodesic 100 4 and variational theories of LCS [18,19], the trajectory complexity measures [20], mesohyperbolicity measures and ergodic partitions [21,22] and transfer operator methods [23,24]. Lagrangian tools have provided in the past interesting insights into oceanic problems related to oil spills and pollutant release.…”

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confidence: 99%

A dynamical systems perspective for a real-time response to a marine oil spill

García-Garrido

Ramos

Mancho

et al. 2016

Marine Pollution Bulletin

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General rightsThis document is made available in accordance with publisher policies. Please cite only the published version using the reference above. AbstractThis paper discusses the combined use of tools from dynamical systems theoryand remote sensing techniques and shows how they are effective instruments which may greatly contribute to the decision making protocols of the emergency services for the real-time management of oil spills. This work presents the successful interplay of these techniques for a recent situation, the sinking of the Oleg Naydenov fishing ship that took place in Spain, close to the Canary Islands, in April 2015.

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“…The approximation of NHIMs through SLCSs can be further refined by advecting the extracted SLCSs in the time direction in which they are attracting. This zero set necessarily contains all LCSs by the variational theory of Haller 10 and, Farazmand and Haller,11 as well as by the recent geodesic theory of transport barriers developed by Haller and Beron-Vera. 22 Using the classic ABC flow as an example, we have shown how this SLCS detection reveals the stable and unstable manifolds of periodic orbits with high accuracy.…”

Section: Discussionmentioning

confidence: 99%

“…For this reason, Farazmand and Haller 11 give a relaxed formulation for two-dimensional flows. In this formulation, condition IV is replaced with a weaker requirement that the average stretching on the LCS is maximal relative to all nearby material surfaces that satisfy condition III, i.e., are pointwise normal to the dominant eigenvector of the Cauchy-Green strain tensor.…”

Section: IIImentioning

confidence: 99%

Detecting invariant manifolds as stationary Lagrangian coherent structures in autonomous dynamical systems

Teramoto

Haller

Komatsuzaki

2013

Chaos: An Interdisciplinary Journal of Nonlinear Science

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Normally hyperbolic invariant manifolds (NHIMs) are well-known organizing centers of the dynamics in the phase space of a nonlinear system. Locating such manifolds in systems far from symmetric or integrable, however, has been an outstanding challenge. Here, we develop an automated detection method for codimension-one NHIMs in autonomous dynamical systems. Our method utilizes Stationary Lagrangian Coherent Structures (SLCSs), which are hypersurfaces satisfying one of the necessary conditions of a hyperbolic LCS, and are also quasi-invariant in a well-defined sense. Computing SLCSs provides a quick way to uncover NHIMs with high accuracy. As an illustration, we use SLCSs to locate two-dimensional stable and unstable manifolds of hyperbolic periodic orbits in the classic ABC flow, a three-dimensional solution of the steady Euler equations. Invariant manifolds play an important role in transport phenomena that range from chemical reactions through fluid mixing to celestial mechanics. Such manifolds, however, are often challenging to locate in multi-dimensional systems that are far from integrable and possess no special symmetries. Among these invariant manifolds, normally hyperbolic invariant manifolds are particularly important due to their persistence under small perturbations. This renders them robust with respect to modeling errors and numerical inaccuracies. Here, we develop a method to detect codimension-one, normally hyperbolic invariant manifolds in general autonomous systems. We approximate such manifolds as zero level sets of a scalar function derived from the recent variational theory of hyperbolic Lagrangian Coherent Structures (LCSs). This approximation converges exponentially fast to a true invariant manifold as longer and longer flow-map samples are used in its construction. We illustrate this method on the classic steady ABC flow, revealing some of its two-dimensional invariant manifolds at a previously unseen level of detail.

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Computing Lagrangian coherent structures from their variational theory

Cited by 185 publications

References 25 publications

Lagrangian scale of particle dispersion in turbulence

Lagrangian scale of particle dispersion in turbulence

A dynamical systems perspective for a real-time response to a marine oil spill

Detecting invariant manifolds as stationary Lagrangian coherent structures in autonomous dynamical systems

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