Detecting coherent structures using braids

Allshouse, Michael; Thiffeault, Jean-Luc

doi:10.1016/j.physd.2011.10.002

Cited by 86 publications

(129 citation statements)

References 39 publications

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“…Braid theory was initially applied to fluid systems to rapidly study the amount of stirring 12 but was extended to find coherent structures defined as approximate material lines that do not significantly change length during the time interval. 5 Because deformations of the space have been used, this method can only provide approximate positions for material lines that remain geometrically regular. In addition to the coherent structure detection, recent developments allow for the calculation of a finite-time mixing measure referred to as the finite-time braiding exponent (FTBE).…”

Section: B Braid Based Analysismentioning

confidence: 99%

“…The other application was to a closed viscous system periodically stirred by a rod. 5 A velocity field for this system was analytically known and trajectories were calculated for application to the coherent structure approach. While the method successfully identified a number of structures when compared to the system's Poincare map, the application demonstrated the limitations of working with sparse data; namely, the need to have multiple trajectories within a given structure to enable detection.…”

Section: B Braid Based Analysismentioning

confidence: 99%

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Lagrangian based methods for coherent structure detection

Allshouse

Peacock

2015

Chaos: An Interdisciplinary Journal of Nonlinear Science

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There has been a proliferation in the development of Lagrangian analytical methods for detecting coherent structures in fluid flow transport, yielding a variety of qualitatively different approaches. We present a review of four approaches and demonstrate the utility of these methods via their application to the same sample analytic model, the canonical double-gyre flow, highlighting the pros and cons of each approach. Two of the methods, the geometric and probabilistic approaches, are well established and require velocity field data over the time interval of interest to identify particularly important material lines and surfaces, and influential regions, respectively. The other two approaches, implementing tools from cluster and braid theory, seek coherent structures based on limited trajectory data, attempting to partition the flow transport into distinct regions. All four of these approaches share the common trait that they are objective methods, meaning that their results do not depend on the frame of reference used. For each method, we also present a number of example applications ranging from blood flow and chemical reactions to ocean and atmospheric flows. The transport of material by advection in fluid systems is a process of vital and ubiquitous importance, underlying scenarios as diverse as pollutant distribution and searchand-rescue operations in the ocean to blood flow in the human body. The rapidly advancing field of Lagrangian based methods for studying flow transport has demonstrated an effective ability to find robust and significant transport features that underly the organization of flow transport in complex, unsteady flow fields. In this review, we present an overview of four of the leading Lagrangian approaches, each with their own strengths and challenges. Details of each method are presented along with an example application to the same model system, the double-gyre flow. Furthermore, we highlight a number of exciting applications and future directions to bring Lagrangian based analysis closer to implementation in real-time, real-world decision making strategies.

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Section: B Braid Based Analysismentioning

confidence: 99%

Section: B Braid Based Analysismentioning

confidence: 99%

Lagrangian based methods for coherent structure detection

Allshouse

Peacock

2015

Chaos: An Interdisciplinary Journal of Nonlinear Science

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“…This was applied to the ocean (Thiffeault 2010) and might also be useful in the atmospheric context. Furthermore, topological entropy and braid theory have recently been suggested (Allshouse and Thiffeault 2012;Thiffeault 2010) to be efficient tools to determine Lagrangian coherent structures (Shadden et al 2009;Peacock and Haller 2013).…”

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confidence: 99%

Topological Entropy: A Lagrangian Measure of the State of the Free Atmosphere

Haszpra

Tél

2013

Journal of the Atmospheric Sciences

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Topological entropy is shown to be a useful characteristic of the state of the free atmosphere. It can be determined as the stretching rate of a line segment of tracer particles in the atmosphere over a time span of about 10 days. Besides case studies, the seasonal distribution of the average topological entropy is determined in several geographical locations. The largest topological entropies appear in the mid-and high latitudes, especially in winter, owing to the greater temperature gradient between the pole and the equator and the more intense stirring and shearing effects of cyclones. The smallest values can be found in the trade wind belt. The local value of the topological entropy is a measure of the chaoticity of the state of the atmosphere and of how rapidly pollutants and contaminants spread from a given location.

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“…The method starts with computing Lagrangian trajectories for a number of initial positions, i.e., F τ t (x i ), i = 1, 2, · · · , N . The procedure P then consists of computing the topological entropy associated with combinations of these trajectories [92,93,94,95], and then separating trajectories into groups which have nearly identical entropy. Numerical application of this method requires that N be not large (∼ O(50)) to be able to address all possible combinations of trajectories.…”

Section: Lcs Diagnostic Methodsmentioning

confidence: 99%

Generalized Lagrangian coherent structures

Balasuriya

Ouellette

Rypina

2018

Physica D: Nonlinear Phenomena

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The notion of a Lagrangian Coherent Structure (LCS) is by now well established as a way to capture transient coherent transport dynamics in unsteady and aperiodic fluid flows that are known over finite time. We show that the concept of an LCS can be generalized to capture coherence in other quantities of interest that are transported by, but not fully locked to, the fluid. Such quantities include those with dynamic, biological, chemical, or thermodynamic relevance, such as temperature, pollutant concentration, vorticity, kinetic energy, plankton density, and so on. We provide a conceptual framework for identifying the Generalized Lagrangian Coherent Structures (GLCSs) associated with such evolving quantities. We show how LCSs can be seen as a special case within this framework, and provide an overarching discussion of various methods for identifying LCSs. The utility of this more general viewpoint is highlighted through a variety of examples. We also show that although LCSs approximate GLCSs in certain limiting situations under restrictive assumptions on how the velocity field affects the additional quantities of interest, LCSs are not in general sufficient to describe their coherent transport.

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Detecting coherent structures using braids

Cited by 86 publications

References 39 publications

Lagrangian based methods for coherent structure detection

Lagrangian based methods for coherent structure detection

Topological Entropy: A Lagrangian Measure of the State of the Free Atmosphere

Generalized Lagrangian coherent structures

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