Global variational approach to elliptic transport barriers in three dimensions

Oettinger, David; Blazevski, Daniel; Haller, George

doi:10.1063/1.4944732

Cited by 11 publications

(35 citation statements)

References 30 publications

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“…hyperbolic LCSs as generalizations of stable and unstable manifolds [2,8]; elliptic LCSs as generalizations of invariant tori [2,10,20]; and, in two dimensions, parabolic LCSs as generalized jet cores [4].…”

Section: Introductionmentioning

confidence: 99%

“…All the variational LCS theories [2,4,8,10,20] provide particular direction fields to which initial LCS positions must be either tangent (in two dimensions) or normal (in three dimensions). Later LCS positions can then be constructed by forward or backward advection under the flow map.…”

Section: Introductionmentioning

confidence: 99%

“…As a second step, from this large collection of candidate curves, one has to identify smaller families of curves that can be interpolated into surfaces. Moreover, since the normal fields depend on the type of LCS, one has to repeat this complicated analysis for each LCS type [2,20].…”

Section: Introductionmentioning

confidence: 99%

“…The evolution of the ξ 2 -system takes place in the initial configuration of the underlying non-autonomous system, but contains averaged information about the non-autonomous flow. The autonomous Instead of identifying LCSs in three dimensions from various two-dimensional direction fields [2,20], we therefore need to consider only a single three-dimensional direction field. We then locate LCSs by familiar numerical methods developed for autonomous dynamical systems.…”

Section: Introductionmentioning

confidence: 99%

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An autonomous dynamical system captures all LCSs in three-dimensional unsteady flows

Oettinger

Haller

2016

Chaos

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Lagrangian coherent structures (LCSs) are material surfaces that shape the finite-time tracer patterns in flows with arbitrary time dependence. Depending on their deformation properties, elliptic and hyperbolic LCSs have been identified from different variational principles, solving different equations. Here we observe that, in three dimensions, initial positions of all variational LCSs are invariant manifolds of the same autonomous dynamical system, generated by the intermediate eigenvector field, ξ(x), of the Cauchy-Green strain tensor. This ξ-system allows for the detection of LCSs in any unsteady flow by classical methods, such as Poincaré maps, developed for autonomous dynamical systems. As examples, we consider both steady and time-aperiodic flows, and use their dual ξ-system to uncover both hyperbolic and elliptic LCSs from a single computation.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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An autonomous dynamical system captures all LCSs in three-dimensional unsteady flows

Oettinger

Haller

2016

Chaos

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“…The extension[36] of[27] to three dimensions is less aligned with[16], as[36] asks for uniform expansion in all directions in the two-dimensional tangent space to potential LCS surfaces, whereas the approach of[16] in three-dimensions is simply concerned with surface growth without a uniform expansion restriction.…”

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confidence: 99%

A Dynamic Laplacian for Identifying Lagrangian Coherent Structures on Weighted Riemannian Manifolds

Froyland

Kwok

2017

J Nonlinear Sci

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Transport and mixing in dynamical systems are important properties for many physical, chemical, biological, and engineering processes. The detection of transport barriers for dynamics with general time dependence is a difficult, but important problem, because such barriers control how rapidly different parts of phase space (which might correspond to different chemical or biological agents) interact. The key factor is the growth of interfaces that partition phase space into separate regions. The paper [16] introduced the notion of dynamic isoperimetry: the study of sets with persistently small boundary size (the interface) relative to enclosed volume, when evolved by the dynamics. Sets with this minimal boundary size to volume ratio were identified as level sets of dominant eigenfunctions of a dynamic Laplace operator.In this present work we extend the results of [16] to the situation where the dynamics (i) is not necessarily volume-preserving, (ii) acts on initial agent concentrations different from uniform concentrations, and (iii) occurs on a possibly curved phase space. Our main results include generalised versions of the dynamic isoperimetric problem, the dynamic Laplacian, Cheeger's inequality, and the FedererFleming theorem. We illustrate the computational approach with some simple numerical examples.

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A Geometric Heat-Flow Theory of Lagrangian Coherent Structures

Karrasch

Keller

2020

J Nonlinear Sci

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We consider Lagrangian coherent structures (LCSs) as the boundaries of material subsets whose advective evolution is metastable under weak diffusion. For their detection, we first transform the Eulerian advection-diffusion equation to Lagrangian coordinates, in which it takes the form of a time-dependent diffusion or heat equation. By this coordinate transformation, the reversible effects of advection are separated from the irreversible joint effects of advection and diffusion. In this framework, LCSs express themselves as (boundaries of) metastable sets under the Lagrangian diffusion process. In the case of spatially homogeneous isotropic diffusion, averaging the time-dependent family of Lagrangian diffusion operators yields Froyland's dynamic Laplacian. In the associated geometric heat equation, the distribution of heat is governed by the dynamically induced intrinsic geometry on the material manifold, to which we refer as the geometry of mixing. We study and visualize this geometry in detail, and discuss connections between geometric features and LCSs viewed as diffusion barriers in two numerical examples. Our approach facilitates the discovery of connections between some prominent methods for coherent structure detection: the dynamic isoperimetry methodology, the variational geometric approaches to elliptic LCSs, a class of graph Laplacian-based methods and the effective diffusivity framework used in physical oceanography.

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Global variational approach to elliptic transport barriers in three dimensions

Cited by 11 publications

References 30 publications

An autonomous dynamical system captures all LCSs in three-dimensional unsteady flows

An autonomous dynamical system captures all LCSs in three-dimensional unsteady flows

A Dynamic Laplacian for Identifying Lagrangian Coherent Structures on Weighted Riemannian Manifolds

A Geometric Heat-Flow Theory of Lagrangian Coherent Structures

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