Transport Barriers and Coherent Structures in Flow Data: Advective, Diffusive, Stochastic and Active Methods 1009225170, 9781009225175

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T R A N S P O RT BA R R I E R S A N D C O H E R E N T S T RU C T U R E S I N F L OW DATA

Transport barriers are observed inhibitors of the spread of substances in flows. The collection of such barriers offers a powerful geometric template that frames the main pathways, or lack thereof, in any transport process. This book surveys effective and mathematically grounded methods for defining, locating and leveraging transport barriers in numerical simulations, laboratory experiments, technological processes and nature. It provides a unified treatment of material developed over the past two decades, focusing on the methods that have a solid foundation and broad applicability to data sets beyond simple model flows. The intended audience ranges from advanced undergraduates to researchers in the areas of turbulence, geophysical flows, aerodynamics, chemical engineering, environmental engineering, flow visualization, computational mathematics and dynamical systems. Detailed opensource implementations of the numerical methods are provided in an accompanying collection of Jupyter notebooks linked from the electronic version of the book. g e o r g e h a l l e r holds the Chair in Nonlinear Dynamics at the Institute of Mechanical Systems of ETH Z¨urich. Previously, he held tenured faculty positions at Brown University, McGill University and MIT. For his research in nonlinear dynamical systems, he has received a number of recognitions including a Sloan Fellowship, an ASME T. Hughes Young Investigator Award, a Manning Assistant Professorship at Brown and a Faculty of Engineering Distinguished Professorship at McGill. He is an elected Fellow of SIAM, APS, and ASME, and is an external member of the Hungarian Academy of Science. He is the author of more than 150 publications.

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T R A N S P O RT BA R R I E R S A N D C O H E R E N T S T RU C T U R E S IN FLOW DATA Advective, Diffusive, Stochastic and Active Methods GEORGE HALLER ETH Z¨urich

Prepared for publication with the assistance of Alex P. Encinas-Bartos

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Shaftesbury Road, Cambridge CB2 8EA, United Kingdom One Liberty Plaza, 20th Floor, New York, NY 10006, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia 314–321, 3rd Floor, Plot 3, Splendor Forum, Jasola District Centre, New Delhi – 110025, India 103 Penang Road, #05–06/07, Visioncrest Commercial, Singapore 238467 Cambridge University Press is part of Cambridge University Press & Assessment, a department of the University of Cambridge. We share the University’s mission to contribute to society through the pursuit of education, learning and research at the highest international levels of excellence. www.cambridge.org Information on this title: www.cambridge.org/9781009225175 DOI: 10.1017/9781009225199 © George Haller 2023 Prepared for publication with the assistance of Alex P. Encinas-Bartos This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press & Assessment. First published 2023 Printed in the United Kingdom by TJ Books Limited, Padstow Cornwall A catalogue record for this publication is available from the British Library ISBN 978-1-009-22517-5 Hardback Cambridge University Press & Assessment has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

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To the loving memory of my father, János Haller (1934–2018), a man of integrity, kindness and grace.

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Contents

xiii xvi

Preface Acknowledgments 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8

Introduction The Mathematics of Transport Barriers The Physics of Transport Barriers Idealized Transport Barriers vs. Finite-Time Coherent Structures Transport Barriers in Flow Separation and Attachment Transport Barriers in Inertial Particle Motion Barriers to Diffusive and Stochastic Transport Barriers to Dynamically Active Transport Coherent Sets, Coherence Clusters and Coherent States

2 2.1

Eulerian and Lagrangian Fundamentals Eulerian Description of Fluid Motion 2.1.1 2.1.2 2.1.3 2.1.4

2.2

Lagrangian Description of Fluid Motion 2.2.1 2.2.2 2.2.3 2.2.4 2.2.5 2.2.6 2.2.7 2.2.8 2.2.9 2.2.10 2.2.11 2.2.12 2.2.13 2.2.14 2.2.15

2.3

Eulerian Scalars, Vector Fields and Tensors Streamlines and Stagnation Points in 2D Flows Streamsurfaces and Stagnation Points in 3D Flows Irrotational and Inviscid Flows Steady Flows as Autonomous Dynamical Systems The Extended Phase Space The Flow Map and Its Gradient Material Surfaces, Material Lines and Streaklines Invariant Manifolds Evolution of Material Volume and Mass Topological Equivalence and Structural Stability Linearized Flow: The Equation of Variations When Are the Eigenvalues of ∇v Relevant? Lagrangian Aspects of the Vorticity Dynamics Near Fixed Points of Steady Flows Poincaré Maps Revisiting Initial Conditions: Poincaré’s Recurrence Theorem Convergence of Time-Averaged Observables: Ergodic Theorems Lagrangian Scalars, Vector Fields and Tensors

Lagrangian Decompositions of Infinitesimal Material Deformation 2.3.1 2.3.2

Singular Value Decomposition (SVD) Polar Decomposition

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1 2 3 4 4 6 7 8 9 13 14 14 15 17 18 19 20 21 21 23 26 26 28 29 31 32 34 34 40 42 44 48 48 50

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Contents 2.3.3

Dynamic Polar Decomposition (DPD)

2.4 2.5

Are the Eulerian and Lagrangian Approaches Equivalent? Summary and Outlook

3 3.1 3.2 3.3

Objectivity of Transport Barriers Common Misinterpretations of the Principle of Material Frame-Indifference Objectivity Yields the Navier–Stokes Equation in Arbitrary Frames Eulerian Objectivity 3.3.1 3.3.2 3.3.3 3.3.4

3.4

Lagrangian Objectivity 3.4.1 3.4.2 3.4.3

3.5

Objectivity of Lagrangian Scalar Fields Objectivity of Lagrangian Vector Fields Objectivity of Lagrangian Tensor Fields

Eulerian–Lagrangian Objectivity of Two-Point Tensors 3.5.1

3.6 3.7

Objectivity of Eulerian Scalar Fields Objectivity of Eulerian Vector Fields Objectivity of Eulerian Tensor Fields Galilean Invariance

Objectivity of the Decompositions of the Deformation Gradient

Quasi-Objectivity Some Nonobjective Approaches to Transport Barriers 3.7.1 3.7.2 3.7.3

Nonobjective Eulerian Principles Objectivization of Nonobjective Eulerian Coherence Principles Nonobjective Lagrangian Principles

3.8

Summary and Outlook

4 4.1

Barriers to Chaotic Advection 2D Time-Periodic Flows 4.1.1 4.1.2 4.1.3

4.2 4.3

2D Time-Quasiperiodic Flows 2D Recurrent Flows with a First Integral 4.3.1 4.3.2

4.4

Hyperbolic Barriers to Transport Elliptic Barriers to Transport Parabolic Barriers to Transport

Barriers from First Integrals in Time-Periodic Flows Barriers from First Integrals in Time-Quasiperiodic Flows

3D Steady Flows 4.4.1 4.4.2 4.4.3 4.4.4 4.4.5

Definition of Transport Barriers from First-Return Maps Transport Barriers vs. Streamsurfaces and Sectional Streamlines Transport Barriers in 3D Steady Inviscid Flows Transport Barriers in 3D Steady Flows with a Continuous Symmetry Transport Barriers from Ergodic Theory

4.5 4.6 4.7

Barriers in 3D Time-Periodic Flows Burning Invariant Manifolds: Transport Barriers in Reacting Flows Summary and Outlook

5 5.1 5.2

Lagrangian and Objective Eulerian Coherent Structures Tracer-Transport Barriers in Nondiffusive Passive Tracer Fields Advective Transport Barriers as LCSs 5.2.1 5.2.2 5.2.3

Hyperbolic LCS from the Finite-Time Lyapunov Exponent FTLE Ridges Are Necessary (but Not Sufficient) Indicators of Hyperbolic LCS Extraction Interval and Convergence of the FTLE

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53 56 58 60 62 63 64 65 66 69 71 72 72 73 73 74 75 77 77 78 86 90 96 98 99 102 106 110 111 113 115 117 121 121 121 125 130 133 135 136 139 141 144 148 149 150 153

Contents 5.2.4 5.2.5 5.2.6 5.2.7 5.2.8 5.2.9 5.2.10 5.2.11

5.3

Local Variational Theory of LCSs 5.3.1 5.3.2 5.3.3

5.4

Elliptic LCSs in 2D: Black-Hole Vortices Computing Elliptic LCSs as Closed Null-Geodesics Shearless LCSs in 2D: Parabolic and Hyperbolic Barriers Unified Variational Theory of Elliptic and Hyperbolic LCSs in 3D

Adiabatically Quasi-Objective, Single-Trajectory Diagnostics for Transport Barriers 5.5.1 5.5.2 5.5.3 5.5.4

5.6 5.7

Local Variational Theory of Hyperbolic LCSs Local Variational Theory of Elliptic LCSs Local Variational Theory of Parabolic LCSs

Global Variational Theory of LCSs 5.4.1 5.4.2 5.4.3 5.4.4

5.5

Numerical Computation of the FTLE Extraction of FTLE Ridges Hyperbolic LCSs vs. Stable and Unstable Manifolds in Temporally Recurrent Flows Repelling and Attracting LCSs from the Same Calculation FTLE vs. Finite-Size Lyapunov Exponents (FSLE) Parabolic LCSs from FTLE Elliptic LCSs from the Polar Rotation Angle (PRA) Elliptic LCSs from the Lagrangian-Averaged Vorticity Deviation

Instantaneous Limit of the Flow Map Instantaneous FTLE Instantaneous Vorticity Deviation (IVD) Global Variational Theory of OECS in 2D Flows

5.8

Summary and Outlook

6 6.1

Flow Separation and Attachment Surfaces as Transport Barriers Flow Separation in Steady and Recurrent Flows 6.1.1 6.1.2 6.1.3 6.1.4

6.2

Separation in 2D Steady Flows Separation in 2D Time-Periodic Flows Separation in 3D Steady Flows Separation in 3D Recurrent Flows

Unsteady Flow Separation Created by LCSs 6.2.1 6.2.2

2D Unsteady Separation 3D Fixed Unsteady Separation

6.3

Summary and Outlook

7 7.1 7.2 7.3 7.4 7.5 7.6

Inertial LCSs: Transport Barriers in Finite-Size Particle Motion Equation of Motion for Inertial Particles Relationship between Inertial and Fluid Motion The Divergence of the Inertial Equation and the Q Parameter Transport Barriers for Neutrally Buoyant Particles Transport Barriers for Neutrally Buoyant Particles with Propulsion Transport Barriers for Aerosols and Bubbles 7.6.1

154 156 156 158 160 161 164 169 176 177 186 192 193 194 200 203 208 212 213 215

Adiabatically Quasi-Objective Diagnostic for Material Stretching Adiabatically Quasi-Objective Diagnostic for Material Rotation Single-Trajectory, Adiabatically Quasi-Objective LCS Computations for the AVISO Data Set 217 Adiabatically Quasi-Objective Material Eddy Extraction from Actual Ocean Drifters 218

Elliptic-Parabolic-Hyperbolic (EPH) Partition and LCSs Objective Eulerian Coherent Structures (OECSs) 5.7.1 5.7.2 5.7.3 5.7.4

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Attracting iLCS in 2D Steady Flows

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220 226 227 228 229 230 240 242 243 243 248 251 256 259 259 271 273 275 276 277 280 281 283 285 286

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Contents 7.6.2 7.6.3 7.6.4

Attracting iLCS in 3D Steady Flows Attracting iLCS in 2D Time-Periodic Flows Attracting iLCS in General 3D Flows

7.7 7.8 7.9

Inertial Transport Barriers in Rotating Frames Inertial Transport on the Ocean Surface: Modeling and Machine Learning Summary and Outlook

8 8.1

Passive Barriers to Diffusive and Stochastic Transport Unconstrained Diffusion Barriers in Incompressible Flows 8.1.1

8.2

8.2.1

8.3

Unconstrained Diffusion Barriers in 2D Flows

Constrained Diffusion Barriers in Incompressible Flows Constrained Diffusion Extremizers in 2D Flows

Barriers to Diffusive Vorticity Transport in 2D Flows 8.3.1 8.3.2

An Analytic Example: Vorticity Transport Barriers in a Decaying Channel Flow Numerical Examples: Vorticity Transport Barriers as Vortex Boundaries in Turbulence

8.4 8.5 8.6 8.7

Diffusion Barriers in Compressible Flows Transport Barriers in Stochastic Velocity Fields Exploiting Diffusion Barriers for Climate Geoengineering Summary and Outlook

9 9.1 9.2 9.3 9.4 9.5

Dynamically Active Barriers to Transport Setup Active Transport Through Material Surfaces Lagrangian Active Barriers Eulerian Active Barriers Active Barrier Equations for Momentum and Vorticity 9.5.1 9.5.2 9.5.3

9.6

Examples of Active Transport Barriers 9.6.1 9.6.2

9.7

Active Poincaré Maps Active FTLE (aFTLE) and Active TSE (aTSE) Active PRA (aPRA) and Active TRA (aTRA) The Choice of the Maximal Barrier Time, s N , in Active LCS Diagnostics Relationship between Active and Passive LCS Diagnostics

Active Barriers in 2D and 3D Turbulence Simulations 9.8.1 9.8.2 9.8.3 9.8.4 9.8.5

9.9

Active Transport Barriers in General 2D Navier–Stokes Flows Directionally Steady Beltrami Flows

Active LCS Methods for Barrier Detection 9.7.1 9.7.2 9.7.3 9.7.4 9.7.5

9.8

Barriers to Linear Momentum Transport Barriers to Angular Momentum Transport Barriers to Vorticity Transport

2D Homogeneous, Isotropic Turbulence 3D Turbulent Channel Flow Eulerian Active Barriers from the Normalized Barrier Equation Turbulent Momentum Transport Barriers (MTBs) Momentum-Trapping Vortices in Turbulent Channel Flows

Summary and Outlook

Appendix References Index

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Preface

Theme Transport barriers are observed inhibitors of the spread of substances in flows. The collection of such barriers offers a powerful geometric template that frames the main pathways, or lack thereof, in any transport process. The purpose of this book is to survey effective and mathematically grounded methods for defining, locating and leveraging transport barriers in numerical simulations, laboratory experiments, technological processes and nature. A large part of the surveyed material has been developed over the past two decades and has only been covered in scattered research articles, review articles and focus issues. The book takes stock of these methods and focuses on the ones that have a solid foundation and a broad applicability to data sets beyond simple model flows. Scope Chapter 1 gives a brief, nontechnical survey of the material covered in later chapters and reviews some frequently used terminology for transport and coherence. Chapter 2 starts with a review of the relevant mathematical concepts in the language of fluid mechanics. The author believes that this is the broadest review of these concepts that is currently available in the fluid mechanics literature in a language accessible to nonmathematicians. Chapter 3 makes a case for a recently emerging minimal self-consistency requirement for material transport barrier detection: independence of the observer (or objectivity). While objectivity is a fundamental axiom in general continuum mechanics, it is frequently misunderstood or ignored in fluid mechanics. This chapter gives the first available detailed account of objectivity for a fluids audience, pointing out a number of transport barrier definitions and detection methods that fail the litmus test of objectivity. Chapter 4 reviews classic dynamical systems results for mixing and transport. This is a condensed overview as most of these results apply only to idealized flow data. They are surveyed partly for historical reasons and partly as motivation for the more realistic transport barrier problems treated in later chapters. This chapter is the only part of the book whose concepts have been discussed (in smaller breadth but greater detail) in other available books on transport barriers, such as Ottino (1989); Wiggins (1992); Samelson and Wiggins (2006); Bollt and Santitissadeekorn (2013); Balasuriya (2016).

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Preface

Chapter 5 gives a review of available objective barrier detection methods for purely material (advective) transport and shows their applications in various areas of fluid mechanics. Chapter 6 describes the kinematic theory of unsteady flow separation, which is also based on material barriers to transport, with applications to aerodynamic separation and flow control. Chapter 7 deals with material barriers to the transport of finite-size (or inertial) particles. Chapter 8 describes recent results on material barriers to diffusive transport. Chapter 9 describes very recent results on the detection of objective material barriers to the transport of dynamically active fields, such as momentum and vorticity. All methods in these chapters are illustrated on data sets. The Appendix provides further technical results from mathematics, mechanics and fluid mechanics for readers who want a deeper understanding of the methods described in the book.

Audience The intended audience ranges from advanced undergraduates to faculty members doing research in the area of turbulence, geophysical flows, aerodynamics, chemical engineering, environmental engineering, flow visualization, computational mathematics and dynamical systems. This book can be used as a textbook for a course on mixing and transport, with an emphasis on data-driven approaches. It can also serve as a reference book for part of a course on modern methods in applied dynamical systems. Computational resources An open-source Jupyter notebook collection, TBarrier, implementing a number of the transport-barrier-detection algorithms we discuss in this book is available from https://github.com/haller-group/TBarrier Throughout this book, the reader will find direct references to the appropriate Jupyter notebooks after discussions on their underlying algorithms. These links cover all notebooks available at the time of completion of this manuscript. Over time, additional functions and algorithms will be added to the notebook repository. Further numerical implementations of various methods covered in the upcoming chapters include the Matlab-based LCStool, developed by the author’s group for the identification of advective transport barriers in two-dimensional unsteady velocity data. LCStool is available from https://github.com/haller-group/LCStool A more recent Matlab-based resource is BarrierTool. Also developed at ETH Zürich, this graphical user interface (GUI) computes both advective and diffusive Lagrangian coherent structures and their objective Eulerian counterparts for two-dimensional flows. BarrierTool is available from https://github.com/haller-group/BarrierTool

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Preface

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Finally, Julia-based versions of some of the advective and diffusive algorithms discussed here, as well as additional coherent structure detection tools, are implemented for two-dimensional flows in CoherentStructures.jl, available at https://coherentstructures.github.io/CoherentStructures.jl/stable/ This package was developed by Oliver Junge’s research group at the Technical University of Munich.

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Acknowledgments

I am grateful to all my collaborators, postdocs and students from whom I have learned an enormous amount over a period of more than two decades. I am particularly indebted to Chris Jones, a former senior colleague at Brown University, who involved me at the time in a project to develop dynamical systems methods for ocean transport studies. This opportunity shaped my interest for years and led to many of the views and ideas described in this book. I would also like to acknowledge critically important funding to my related research from the US Air Force Office for Scientific Research (AFOSR), the United Technologies Research Center (UTRC), the US National Science Foundation (NSF), the Natural Sciences and Engineering Research Council of Canada (NSERC) and the German Science Foundation (DFG). During the preparation of this book, I greatly benefited from the hospitality of the Departments of Mechanical Engineering at EPFL and MIT. I am particularly thankful to Alex Encinas-Bartos for his outstanding work with the creation of the Jupyter notebook collection, TBarrier, which implements numerically a good portion of the material discussed in this book. His helpful comments on an early draft of the manuscript were also much appreciated. My special thanks go to Bálint Kaszás, who read and commented on the same draft. He also tested TBarrier extensively, made very useful suggestions and implemented some modifications to the code on his own. Additionally, I am pleased to acknowledge the very helpful advice and assistance I have received as author from David Tranah and Anna Scriven of Cambridge University Press. Last but not least, I am immensely grateful to my wife, Krisztina, for her invaluable support and encouragement during the preparation of this book and beyond.

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1 Introduction

Figure 1.1 Oil distribution in the 2010 Deepwater Horizon oil spill in the Gulf of Mexico. Apparent barriers to the transport of oil are readily observable. Image: Daniel Beltrá (used with permission).

Flows in nature tend to generate striking patterns in the tracers they carry. An example is the oil spill distribution in Fig. 1.1, which is framed by apparent barriers to the spread of the oil in certain directions. More often than not, one’s primary interest is to find such barriers and hence understand the overall direction and rate of transport without necessarily identifying pointwise oil concentration values with high accuracy. Figure 1.1 also conveys the strong technological and societal needs for uncovering, forecasting and shaping such barriers. We think of transport barriers as observed inhibitors of the spread of substances in flows. They offer a simplified global template for the redistribution of those substances without the need to simulate or observe numerous different initial distributions in detail. Because of their simplifying role, transport barriers are broadly invoked as explanations for observations in several physical disciplines, including geophysical flows (Weiss and Provenzale, 2008), fluid dynamics (Ottino, 1989), plasma fusion (Dinklage et al., 2005), reactive flows (Rosner, 2000) and molecular dynamics (Toda, 2005). Despite their frequent conceptual use, however, transport barriers are rarely defined precisely or extracted systematically from data. The purpose of this book is to survey effective and mathematically grounded methods for defining, locating and leveraging transport barriers 1

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2

Introduction

in numerical simulations, laboratory experiments, technological processes and nature. In the rest of this Introduction, we briefly survey the main topics that we will be covering in later chapters.

1.1 The Mathematics of Transport Barriers Throughout this book, we will adopt the geometric view of nonlinear dynamical systems theory on transport. That is, rather than focusing on individual fluid particle positions or pointwise concentration values, we seek to identify key invariant surfaces with a major impact on shaping transport patterns. To illuminate the significance of such invariant surfaces, we note that models of transport phenomena are often tested based on their ability to reproduce the evolution of an initial condition or initial distribution. Due to their inherent sensitivity on initial conditions, parameters and uncertainties, however, even predictions from highly accurate models can ultimately display vast discrepancies with individual observations. A more meaningful way to test the validity of models is to assess their ability to reproduce transport barriers of the physical process accurately. Figure 1.2 shows a conceptual example in which a highly accurate dynamical model (black dots) for the true dynamics (solid red) makes a vastly inaccurate prediction for the evolution of a single initial condition (blue), yet reproduces a transport barrier (the unstable manifold of a saddle point) of the true dynamics with high accuracy. Clearly, it is the latter metric in which the model should be assessed and found very accurate. Initial condition

Model trajectory

True trajectory

Model transport barrier

True transport barrier

Figure 1.2 A good model for a transport process should accurately predict transport barriers but not necessarily individual trajectories.

A transport barrier can only block transport locally if its dimension is precisely one less than the dimension of the flow domain, as in Fig. 1.2. Such smooth surfaces are called codimension-one invariant manifolds in dynamical systems theory. They are curves in two-dimensional flows and two-dimensional surfaces in three-dimensional flows. While the impact of transport barriers is apparent in Figs. 1.1–1.9, their ever-shifting location, life span and intrinsic properties are not readily recoverable from instantaneous snapshots of an unsteady flow.

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1.2 The Physics of Transport Barriers

3

Our underlying assumption will be that either the velocity field or a set of its trajectories is available either from observations or numerical simulations. With this assumption, the two fundamental questions we seek to answer are: (1) What defines the fundamental barriers to the transport of active and passive tracers in the flow? (2) How can we locate the most influential transport barriers without solving their underlying transport equation? To review the technical material needed to answer these questions, Chapter 2 recalls relevant concepts from the geometric theory of dynamical systems and continuum mechanics in the context and language of fluid mechanics. In later chapters, we will build on these concepts to answer the two main questions posed above under increasingly fewer assumptions. The order of chapters will roughly mirror actual chronological developments in fluid mechanics and applied mathematics, each describing ideas and results of interest in their own right.

1.2 The Physics of Transport Barriers While the techniques for transport-barrier detection we put forward in this book are quite diverse, we will set one common physical requirement for all of them: their predictions must be experimentally verifiable. Experimental verifiability for predicted transport barriers means that they should be detectable via material tracers such as dye, weakly diffusive smoke or small particles. The requirement of unambiguous experimental reproducibility has an important implication: any self-consistent description of transport barriers must be independent of the frame of reference of the observer. For instance, barriers framing the evolution of an oil spill, such as the one shown in Fig. 1.1, are clearly identified as the same set of material points by an observer on the beach, another one on a ship and a third one in a circling helicopter, as illustrated in Fig. 1.3.

Figure 1.3 Observed transport barriers highlighted by tracers must be objectively defined, i.e., independently of the observer’s frame of reference.

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Introduction

These material barriers move on different orbits in each observer’s frame of reference, yet all three observers agree exactly on where the edges of the oil spill are at any time instance. Accordingly, any self-consistent definition or detection method for transport barriers should be objective, i.e., indifferent to the observer and return the same set of material points forming the barrier in any observer frame. Beyond its theoretical significance, this litmus test for the self-consistency of barrier theories has a very practical motivation. If resource allocation or countermeasures are planned based on the location of transport barriers in a flow, there is no room for disagreement among observers. In Chapter 3, we will discuss the principle of material frame indifference in detail and explore its use as a litmus test for the self-consistency of various views on transport barriers.

1.3 Idealized Transport Barriers vs. Finite-Time Coherent Structures As we will see in Chapter 4, purely advective transport barriers can be defined unambiguously in flows that are known for all times. Such definitions, e.g., those requiring saddle-type behavior for a material surface for all times, exploit properties of the barriers that make them locally unique in the flow. The area of transport studies concerned with such idealized systems is generally referred to as chaotic advection. In its idealized setting, chaotic advection provides very helpful motivation for the main types of material barriers one encounters in nature. It also turns out that some approaches to diffusive and dynamically active transport barriers also yield steady velocity fields (barrier vector fields) that can be analyzed by the methods we review for steady flows in Chapter 4. In practical problems, however, finite-time transport is of interest. Flow behavior over a finite time interval is continuous: all close enough trajectories remain close to each other. Consequently, infinitely many neighboring trajectories will satisfy any barrier definition phrased via inequalities that involve continuous quantities. As a result, the defining features of barriers exploited in chaotic advection no longer yield unique material surfaces. One way to address this nonuniqueness problem is to identify barriers that act as observed centerpieces of material deformation over a finite time interval of interest. We perceive a surface to assume this role if it remains coherent, i.e., keeps its spatial integrity without developing smaller scales (filaments) during its temporal evolution. We will call such uniquely resilient material sets Lagrangian coherent structures (LCSs), as first coined by Haller and Yuan (2000). These LCSs act, for instance, as surfaces bounding the globally complex but temporally coherent cloud patterns in the atmosphere of Jupiter, as shown in Fig. 1.4. Instantaneous limits of these material coherent structures are called objective Eulerian coherent structures (OECSs), which represent short-term, approximately material barriers. Unlike LCSs, OECSs are not material and hence can frame the creation, collision, break-up and disappearance of transport barriers in the flow. We will discuss LCSs and OECSs in detail in Chapter 5.

1.4 Transport Barriers in Flow Separation and Attachment Flow separation involves the detachment of fluid from a rigid boundary, resulting in either the creation of a local recirculation zone (separation bubble) or the global breakaway of fluid particles from the boundary. Highly unsteady flows often display both types of separation, as illustrated by the separation patterns behind the airfoil shown in Fig. 1.5.

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1.4 Transport Barriers in Flow Separation and Attachment

5

Figure 1.4 Enhanced image of cloud patterns in the atmosphere of Jupiter, as seen by the Juno mission in 2020. As we show in later chapters, such patterns are created and shaped by hidden Lagrangian coherent structures (LCSs), such as material jets, eddies and fronts. Image: NASA/JPL-Caltech/SwRI/MSSS/Gerald Eichstädt /Seán Doran.

Figure 1.5 Visualization of aerodynamic separation and reattachment along transport barriers attached to an airfoil in a wind tunnel. Image: German Aerospace Center (DLR), CC-BY 3.0.

Separation depletes the kinetic energy content of the flow near the wall, leading to a degradation in the operational performance of engineering devices. Specifically, separation on a bluff body can increase the pressure drag significantly, whereas separation in a diffuser decreases the pressure recovery. Flow attachment is the opposite phenomenon, involving the sustained convergence of fluid elements toward a well-defined boundary location. Local separation necessarily involves a reattachment of the flow that forms the downstream boundary of the separation zone, as seen in Fig. 1.5. While separation has a clear impact on aerodynamic forces acting on the flow boundary, its experimental detection exploits its impact of fluid particle transport. Indeed, flow visualizations of the type shown in Fig. 1.5 are only possible because particles in the separated and unseparated flow regions are kept apart from each other by material transport barriers. These separation surfaces collect nearby particles along the boundary and eject them into

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Introduction

the mean flow. In contrast, reattachment surfaces guide particles from the mean flow to the wall and then repel them along the boundary. While typical wall-anchored material surfaces will stretch and fold, separation and reattachment surfaces distinguish themselves by remaining coherent. Therefore, using the terminology introduced in §1.3, we can view separation surfaces as attracting LCSs and attachment surfaces as repelling LCSs. Unlike general LCSs, however, separation and reattachment surfaces exert weak attraction and repulsion near flow boundaries characterized by no-slip boundary conditions. In dynamical systems terms, these surfaces are nonhyperbolic invariant manifolds, unless the boundary can be characterized with free-slip conditions. This makes most LCS diagnostic tools inefficient in a small neighborhood of a no-slip boundary. At the same time, the direct connection of these LCSs with the boundary facilitates their more detailed local analysis. This in turn enables the identification of separation and reattachment surfaces near the wall solely based on the derivatives of the velocity field along the boundary, without material advection. We will discuss these results for LCSs and OECSs forming separation surfaces and separation spikes in Chapter 6.

1.5 Transport Barriers in Inertial Particle Motion Patterns formed by finite-size (or inertial) particles carried by fluid flows are often notably different from those seen for fluid particles in dye visualizations. Specifically, while fluid elements in an incompressible fluid cannot exhibit clustering or scattering, both phenomena are well documented in inertial particle motion in incompressible fluids. A spectacular albeit alarming example of inertial clustering is the Great Pacific Garbage Patch, which covers an area of about 1.6 million square kilometers (three times the size of France) (see Fig. 1.6).

Figure 1.6 A transport barrier formed by inertial particles: the Great Pacific Garbage Patch, a floating island of accumulating garbage in the Pacific Ocean. Image: Ocean Cleanup.

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1.6 Barriers to Diffusive and Stochastic Transport

7

Microplastics constitute about 94 % of the roughly 1.8 trillion pieces of plastic in this patch, even though they only make up 8% of the total mass of 80, 000 tons.1 Given the characteristic length scales of the ocean, all objects forming the garbage patch can be considered inertial particles. In Chapter 7, we will discuss the fundamentals of transport barriers for inertial particle motion. It turns out that these barriers can be uncovered by applying the LCS methods of Chapter 5 to a modification of the carrier fluid velocity field that accounts for inertial effects.

1.6 Barriers to Diffusive and Stochastic Transport In Chapter 8, we will discuss barriers to the transport of passive tracers whose diffusion cannot be ignored relative to their advective redistribution over the time scales of interest. An example is the long-term mixing of dye in a gently stirred glass of water, as shown in Fig. 1.7. To model such a mixing process accurately, we can no longer set the diffusivity to zero in the advection–diffusion equation governing the spread of the dye.

Figure 1.7 Diffusive mixing of dye framed by transport barriers in a glass of water. Image: Nathan Dumlao at unsplash.com.

One might wonder why we bother defining and identifying transport barriers in diffusive tracer fields, given that they are readily seen in numerical simulations and experiments such as the one shown in Fig. 1.7. To understand our motivation, one must remember that observed surfaces with large concentration drops across them are not necessarily intrinsic barriers to tracer transport. Rather, many of them are remnants of high gradients in the initial tracer distribution. Indeed, the large concentration drop across the boundary of the initial dye drop in the experiment shown in Fig. 1.7 will persist as a slowly diffusing dye-fluid interface for very long times. Similarly, the lack of large concentration gradients in a given flow domain may simply be due to the concentration being fully mixed or fully absent in that domain. Indeed, there are also transport barriers in the region yet unpenetrated by the dye in the experiment shown in Fig. 1.7. Those barriers will impact other concentration fields (say, the water temperature) with a different initial distribution carried by the same flow. 1

Most of that mass has been found to be made up of abandoned fishing gear (Lebreton et al., 2018).

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Introduction

1224

Finally, one should consider that while large concentration drops seen in finite-time tracer evolution will often reflect the impact of transport barriers, these drops will not coincide with the barriers themselves. The amount of discrepancy between concentration drop locations and actual transport barriers is generally unclear: convergence of the latter to the former can only be expected in flows with well-defined asymptotic behavior. Identifying intrinsic transport barriers that are free from these idiosyncrasies of a specific observed tracer distribution is essential for universal conclusions valid for the transport of all tracer fields. A closely related passive transport problem, stochastic transport, is concerned with the transfer of material fluid elements across spatial domains by a velocity field subject to uncertainties. These uncertainties are most often modeled using Brownian motion.

H. Yang et al. / Continental Shelf Research 19 (1999) 1221}1245

Fig. 2. Observed drifter trajectories from February 1996 to February 1997 with a total of 194 drifters.

Figure 1.8 Drifter trajectories released and tracked in the Gulf of Mexico over a period of one year. The drifters do not cross an apparent transport barrier acting as the boundary of a forbidden zone west off the Florida coastline. (Occasional straight lines indicate captured and transported drifters.) Image: Yang, H. et al. (1999).

For example, drifter trajectories in the Gulf of Mexico, along with an apparent barrier they never cross (see Fig. 1.8), are complex enough to view as trajectories of a random flow. The connection between trajectory distribution in such a random flow and the evolution of a diffusive scalar field under the deterministic component of the velocity field is given by a classic result: the probability density function of the random flow satisfies the Fokker– Planck equation, which can be recast as an advection–diffusion equation driven by the drift component of the velocity field. Using this connection, we will discuss the definition and properties of barriers to stochastic transport in Chapter 8.

1.7 Barriers to Dynamically Active Transport In contrast to the passive transport concepts we have discussed so far, active transport involves the transfer of dynamically active tracers. Such active tracers are scalar or vector fields impacting the flow velocity directly rather than simply being carried by it. Active tracers of practical interest include the kinetic energy, vorticity and linear momentum.

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1.8 Coherent Sets, Coherence Clusters and Coherent States

9

Framing the spatiotemporal evolution of these dynamically active fields informally using the concept of transport barriers is common practice in the literature. As a relevant example, Fig. 1.9 shows apparent barriers to vorticity transport in the turbulent flow near a spinning rotor.

Figure 1.9 Apparent transport barriers in the instantaneous vorticity distribution on a cutting plane for a spinning rotor. Image: Neal M. Chaderjian, NASA Ames Research Center.

Studying dynamically active fields, however, is even more challenging than the transport problems we have already discussed. First, the transport equation for these quantities is a nonlinear partial differential equation (PDE) for the velocity field. For fluid flows, this PDE is the Navier–Stokes equation, whose solution structure in three dimensions is still not fully understood. Second, unlike passive scalar concentration fields, all physically relevant dynamically active fields are nonobjective: they depend on the frame of the observer. This frame dependence is a serious challenge if one wants to conform to the basic requirement of objectivity for transport barriers that we formulated in §1.2. In Chapter 9, we will discuss how barriers to dynamically active transport can also be characterized and located in an observer-independent way. This in turn makes the experimental visualization of these barriers via material tracers feasible in practice.

1.8 Coherent Sets, Coherence Clusters and Coherent States We close this Introduction by mentioning three notions of coherence that we will not be discussing in further detail here. They are beyond the scope of this book because their primary focus is not transport barriers. The first such notion is that of a coherent set: an equivalence class of trajectories that stay closer to each other during their evolution than to other trajectories. Specifically, finite-time coherent sets comprise trajectories that disperse slower than others over a given finite time interval of interest (see, e.g., Froyland et al., 2010; Froyland, 2013; Bollt and Santitissadeekorn, 2013). This view of coherence focuses on sets enclosed by closed transport barriers as opposed to the barriers themselves. A coherent set is defined as a region of initial conditions that continues to have significant overlap with its deterministically advected final position even if a small random perturbation (or diffusion) is added to its originally deterministic advection.

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10

Introduction

Available approaches seek to locate such coherent sets based on properties of the transfer operator (or Frobenius–Perron operator), which maps passive, scalar-valued functions defined over initial positions of fluid particles into the evaluations of the same functions over later positions of those particles.2 More recent reformulations and extensions of the transfer operator approach are given by Froyland (2015); Froyland and Kwok (2017); Froyland et al. (2020) and the references cited therein. The transfer operator can be approximated by its finite-dimensional discretization P, constructed as a matrix of transition probabilities within a finite partition of the flow domain. The second (left) singular vector of P is then expected to characterize a dominant coherent set after appropriate thresholding. Further coherent structures are expected to appear from the appropriate thresholding of higher singular vectors of P. The number of singular vectors to consider and the applied thresholding are typically determined empirically by inspection of the unprocessed singular vectors of P. Transport barriers are then inferred by implication as boundaries of the coherent sets obtained in this fashion. A notion related to coherence sets is the coherence cluster, which comprises a group of trajectories that are more similar to each other than to other groups. Methods for identifying clusters in data sets were originally developed in the computer science literature (see, e.g., Everitt et al., 2011). Using such ideas, Froyland and Padberg-Gehle (2015) use fuzzy Cmeans clustering to identify clusters of initial conditions that remain close to the same virtual cluster center. Hadjighasem et al. (2016) seek clusters as sets of trajectories that maintain small relative distances to each other. They note that clusters can be identified by applying the technique of spectral clustering to the eigenvectors of matrix (the graph Laplacian) that encodes the average distances among trajectories. Schlueter-Kuck and Dabiri (2017) use the same clustering approach but with a different distance function. Banisch and Koltai (2017) apply the spectral clustering approach in the transfer operator framework. These spectral methods targeting coherent sets or coherence clusters are different in spirit from the geometric approaches we will be surveying in this book. First, we will focus on barriers minimizing advective, diffusive, stochastic or active transport of a physical quantity without requiring the barriers to be closed. In contrast, barrier surfaces inferred from spectral methods have to be closed. As a consequence, spectral methods are inapplicable to some of the most frequently observed barriers, such as jets and fronts in the ocean, just because they happen to be open. Second, spectral approaches require empirical user input in the selection of the relevant singular vectors, which is not the case for the geometric methods we will survey. To illustrate the differences between probabilistic and geometric approaches on a specific flow, Fig. 1.10 shows a comparison of coherent sets obtained from transfer operator methods with transport barriers obtained from LCS methods in a two-dimensional, spatially double-periodic turbulence simulation. A third frequently used notion of coherence in contemporary fluid mechanics is that of an exact coherent state3 (or ECS), which refers to an exact, nonlinear solution of the Navier– Stokes equation in canonical shear flows, such as plane Couette, Poiseuille and pipe flows (Waleffe, 1998, Graham and Floryan, 2021). These solutions display simple spatiotemporal 2

3

Technically speaking, the transfer operator is the pushforward operation carried out by the flow on measurable functions. Exact coherent states are sometimes called exact coherent structures, causing occasional confusion with the coherent structures discussed in §1.3 and elsewhere in this book.

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1.8 Coherent Sets, Coherence Clusters and Coherent States

11

tem x_ ¼ vðx; tÞ. We use the auxiliary gird approach with the coherent sets. Most of t distance q ¼ 10 3 to construct the FTLE, FSLE, meso- n of coincidental, rather th chronic, and shape coherence diagnostic fields. The same to the coherence metric auxiliary distance is used to compute the Cauchy–Green tional issue with the hie strain tensor as well as the vorticity for the geodesic and is its convergence on thi Coherent sets: old on the relative im thresholded singular LAVD methods, respectively. Most plots in Figure9 indicate several vortex-type struc- respect to the reference vectors of the transfer tures, except for the shape coherence and transfer operator to be computed and sati operator and its hierarchical version methods. While the boundaries of the large-scale coherent coherent pairs. However sets identified by the latter method indeed do not grow initial numerical diffusio

Transport barriers: hyperbolic (open) and elliptic (closed) Lagrangian coherent structures

Figure 1.10 Coherent sets obtained as thresholded second singular vectors of the transfer operator approach and from its hierarchical version, which applies the original algorithm again within each coherent set recursively n times (Ma and Bollt, 2013). Also shown are transport barriers obtained as Lagrangian coherent structures (LCSs) from the finite-time Lyapunov exponent (FTLE) and from the Lagrangian-averaged vorticity deviation (LAVD). The FTLE highlights hyperbolic LCSs and the LAVD highlights elliptic LCSs, as we will discuss in Chapter 5. Adapted from Hadjighasem et al. (2017).

behavior that is nevertheless reminiscent of coherent features of turbulent solutions that arise recurrently. This is because ECSs are of saddle type and hence other time-evolving Navier–Stokes velocity fields may, from time to time, pass by ECSs. These passing solutions will mimic some of the spatial features of the approached ECSs for a while before departing. Examples of ECSs include steady-state, time-periodic or traveling-wave solutions, such as the one shown in Fig. 1.11. Features of ECSs have traditionally been assessed via heuristic visualization tools, such as instantaneous level surfaces of a velocity component (see, e.g., Fig. 1.11). This visualization approach is useful in a direct comparison between the features of ECS velocity fields and those of turbulent velocity fields. However, the surfaces obtained in this fashion tend to differ from experimentally observable material coherent structures even in two-dimensional steady flows (see §3.7.1 for examples). In summary, while the transport barriers and coherent structures we discuss here are special material surfaces of a given velocity field, the ECSs are special velocity fields themselves. Their particular transport barriers and coherent structures can nevertheless be studied by the methods discussed in the upcoming chapters of this book.

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12

Introduction

98

F. Wale↵e

1

12 10

y 0

8 6

–1 –2

x

4

z

0

2 2

1

Figure 1.11 Exact coherent state (ECS) in a channel flow moving in the positive xdirection of y a0 channel at Reynolds number Re = 376. Level curves of the streamwise velocity in the y = 0 plane are shown together with isosurfaces of the streamwise vorticity. Positive vorticity is shown in blue, negative in red. Adapted from Waleffe –1 2 4 6 8 10 12 (2001). x

–2 –1

z 0 1 2

2

4

6

8

10

12

x Figure 4. Perspective, side and top view of level curves of streamwise velocity u at y = 0 overlayed with isosurfaces of streamwise vorticity (±60% max[!x (x, y, z)], the maximum occurs at the wall). Positive vorticity blue, negative red. Upper branch at Re = 376, R⌧ ⇡ 55. Flow is toward positive x.

solutions are virtually identical to the free-slip solutions except for extra high-vorticity regions at the wall, below the vortices, in the no-slip case and an additional symmetry in plane Couette flow. The lower branch solution (figure 5) develops a distinct and more complex structure. The lower branch streaks are stronger and less wavy than in the upper branch solution. They are flanked by staggered nearly streamwise vortices that split into two pieces, one of which straddles the streak. A second set of similar vortices, slightly weaker and shifted by about half a period, also exists. Both of these branches of solutions are unstable to perturbations with the same wavelengths and symmetry. The unstable eigenvalues consists of a complex conjugate pair and one real eigenvalue on the lower branch and a complex conjugate pair on the upper branch. The real part of the complex pair on the upper branch is almost independent of Re, for the range of Re presented here (252 6 Re 6 423), with a

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2 Eulerian and Lagrangian Fundamentals

Classical continuum mechanics focuses on the deformation field of moving continua. This deformation field is composed of the trajectories of all material elements, labeled by their initial positions. This initial-condition-based, material description is what we mean here by the Lagrangian description of fluid motion (see Fig. 2.1).

∂U

∂U x(t;t0 ,x 0 )

x v(x,t) instantaneus! streamline

trajectory

U

U x0

Lagrangian description

Eulerian description

Figure 2.1 Lagrangian (trajectory-based) and Eulerian (velocity-field-based) descriptions of fluid flow.

In contrast to typical solid-body deformations, however, fluid deformation may be orders of magnitude larger than the net displacement of the total fluid mass. For example, the center of mass of a turbulent, incompressible fluid in a closed tank experiences no displacement in the lab frame, yet individual fluid elements undergo deformations that no material used in structural engineering would withstand. The difficulty of tracking individual fluid elements has traditionally shifted the focus in fluid mechanics from individual trajectories to the instantaneous velocity field and quantities derived from the velocity field, such as the vorticity and momentum. We refer here to the instantaneous-position-based approach to continuum motion as the Eulerian description (see Fig. 2.1). Our discussion of the Eulerian view of fluids will be substantially shorter than usual in fluids textbooks, mainly because the focus here is on flow kinematics rather than the governing equations of fluids. Indeed, all transport barriers and coherent structures to be discussed in this book can be defined and located purely from available velocity data. The only conceptual exceptions to this rule are barriers to the transport of dynamically active quantities (see Chapter 9), whose identification depends on the constitutive equation of the fluid. Once this dependence is clarified, however, the active barriers can still be identified from operations performed solely on the velocity field. 13

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14

Eulerian and Lagrangian Fundamentals

In contrast, our review of Lagrangian aspects of fluid mechanics in this chapter will be more extensive than customary in fluid mechanics textbooks, including those fully dedicated to Lagrangian fluid dynamics (such as Bennett, 2006). This is because our analysis of material barriers to transport will rely extensively on tools from dynamical systems theory and continuum mechanics that are not traditionally part of the toolkit of fluid mechanics.

2.1 Eulerian Description of Fluid Motion We refer to scalar, vector and tensor fields defined over spatial positions x ∈ U ⊂ Rn as Eulerian quantities. The bounded spatial domain U will be either two-dimensional (2D) for n = 2 or three-dimensional (3D) for n = 3. Unless otherwise noted, we will always assume that these fields are known for times t ranging over a finite time interval [t1, t2 ]. The central Eulerian quantity in descriptions of flow evolution is the velocity field v(x, t), which we call steady if it is constant in time (i.e., of the form v(x)). When the velocity field does depend on time explicitly, we call it unsteady.

2.1.1 Eulerian Scalars, Vector Fields and Tensors A scalar field defined at all positions x and times t is an Eulerian scalar field. For instance, any evolving tracer concentration field c(x, t) is an Eulerian scalar field. An Eulerian vector field is a time-dependent vector field u(x, t) defined on the domain U. The velocity field v(x, t) of a moving fluid as well as its vorticity ω (x, t) = ∇ × v(x, t) are examples of Eulerian vector fields. In contrast, an eigenvector field ei (x, t) of an Eulerian tensor field (to be defined below) is, in general, not an Eulerian vector field because it has no well-defined length or orientation. Accordingly, we refer to such eigenvector fields as Eulerian direction fields. An Eulerian tensor field A(x, t) is a linear mapping family that maps each tangent space Tx Rn (see Appendix A.4) into itself at all positions x ∈ U for all times t ∈ [t1, t2 ]. Examples include the velocity gradient tensor ∇v(x, t), the symmetric rate-of-strain tensor S(x, t) and the skew-symmetric spin tensor W (x, t), with the latter two defined as i i 1h 1h (2.1) S = ∇v + (∇v)T , W = ∇v − (∇v)T . 2 2 Here and going forward, the superscript T will refer to transposition. An important relationship between the spin tensor and the vorticity vector is We =

1 ω ×e 2

(2.2)

for all vectors e ∈ R3 (see Appendix A.12). In other words, the vector associated with the spin tensor is half of the vorticity vector. Passive Eulerian scalar, vector and tensor fields are field quantities whose evolution has no impact on the underlying velocity field v. Examples of such fields include a dye concentration field c (x, t), its gradient vector field ∇c(x, t) and its Hessian tensor field ∇2 c(x, t). In contrast, active Eulerian fields, such as the velocity norm |v(x, t)|, the vorticity vector ω (x, t) and the rate-of-strain tensor S(x, t), are fields whose evolution directly impacts the velocity field.

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2.1 Eulerian Description of Fluid Motion

15

2.1.2 Streamlines and Stagnation Points in 2D Flows A parametrized curve that is everywhere tangent to the velocity field v(x, t) at time t is called an instantaneous streamline, as shown in Fig. 2.1. Any streamline x(s; t), parametrized by s ∈ R at time t, is therefore composed of solutions of the autonomous ordinary differential equation (ODE) x0 = v(x, t),

(2.3)

in which the prime denotes differentiation with respect to s and t plays the role of a parameter. We call the flow incompressible if v is divergence-free (or solenoidal), i.e., ∇ · v ≡ 0. For 2D incompressible flows, there exists a stream function ψ(x, t) such that the velocity field can be written as   0 1 v(x, t) = J∇ψ(x, t), J= . (2.4) −1 0 At any time instant t, streamlines are contained in the level curves (or isocontours) of ψ(x, t), given that d ψ(x(s), t) = ∇ψ(x(s), t) · v (x(s), t) = h∇ψ(x(s), t), J∇ψ(x(s), t)i ≡ 0 (2.5) ds holds by the skew-symmetry of the matrix J. Equation (2.4) says that the stream function ∇ψ(x, t) acts as a Hamiltonian function for 2D incompressible fluid particle motion (see Appendix A.5 for more on Hamiltonian systems). We note that in 3D flows, no scalar stream function is guaranteed to exist from which the full velocity field could be derived. An instantaneous stagnation point is a time-dependent point p(t) at which the velocity field v vanishes at time t, i.e., v(p(t), t) = J∇ψ(p(t), t) = 0.

(2.6)

At such points, the linearization

ξ 0 = ∇v(p(t), t)ξ = J∇2 ψ(p(t), t)ξ,

ξ ∈ R2

(2.7)

of the differential equation (2.3) determines the instantaneous local streamline geometry, which depends on the eigenvalues and eigenvectors of ∇v(p(t), t). The eigenvalues of the velocity gradient satisfy the characteristic equation λ2 − (∇ · v) λ + det ∇v = 0.

(2.8)

λ2 + det ∇v = 0

(2.9)

This equation simplifies to for incompressible flows, yielding the two eigenvalues of ∇v(p(t), t) in the form p λ1,2 (t) = ± − det ∇v(p(t), t).

(2.10)

Figure 2.2 shows the possible local streamline geometries for different λ1,2 configurations: hyperbolic (saddle-type) stagnation point; elliptic (center-type) stagnation point; nondegenerate parabolic stagnation point (local shear flow arising near a stagnation point on a free-slip boundary); degenerate parabolic stagnation point (locally quiescent flow arising near a point on a no-slip boundary). At nondegenerate parabolic stagnation points,

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16

Eulerian and Lagrangian Fundamentals

only one independent eigenvector exists. At degenerate parabolic stagnation points, any vector is an eigenvector. Importantly, these instantaneous streamline geometries only give an accurate description of fluid particle motion if the velocity field is steady. In unsteady flows, the fluid motion can differ vastly from the local streamline geometry, as we shall see throughout §2.2.

hyperbolic stagnation point e2

elliptic stagnation point

parabolic stagnation points (nondegenerate)

parabolic stagnation points (degenerate)

e1

e1 = e 2

1

=

2

{0}

1,2

= ±i

1

=

2

= 0, e1 = e2

1

=

2

= 0, e1, e2

2

Figure 2.2 The four possible linearized streamline geometries at an instantaneous stagnation point of a 2D incompressible flow. The eigenvalues of ∇v(p(t), t) are λ1 and λ2 , with corresponding real eigenvectors e1 and e2 , whenever λ1,2 are real numbers.

Instantaneous flow patterns in the velocity field of a 2D flow are separated from each other by special streamlines, or separatrices, across which there is a change in the topology of streamlines. By the implicit function theorem (see Appendix A.1), such a change in the streamline geometry can only occur near points where the gradient ∇ψ(x, t) vanishes. We then conclude by Eq. (2.6) that separatrices between different streamline patterns must necessarily be streamlines that contain at least one stagnation point. With the exception of elliptic stagnation points, all other types of stagnation points shown in Fig. 2.2 can be contained in a streamline. Accordingly, the possible separatrices in 2D incompressible flows are shown in Fig. 2.3. Of these, homoclinic streamlines connecting hyperbolic stagnation points to themselves and heteroclinic streamlines connecting two hyperbolic or nondegenerate parabolic stagnation points on a boundary to each other are structurally stable, i.e., smoothly persist under small perturbations of the stream function (see §2.2.7 for a more formal definition of structural stability). Indeed, the endpoints of these connecting streamlines are forced to be on the same level set of the stream function even after small perturbations, as illustrated in Fig. 2.4 (left). In contrast, heteroclinic streamlines connecting two points away from boundaries or an off-boundary point to an on-boundary point are structurally unstable. Indeed, in the absence of any symmetry that would force the endpoints of such a connection to remain on the same level set of the stream function, small perturbations to the stream function will generally cause the two stagnation points to move to different level curves of the stream function (see Fig. 2.4 (right)).

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p(t)

x2

˜ (t) p sha1_base64="qMwG8nVjgxFdd+LXhT7hLvqu2tc=">AAACBXicbVDLSsNAFJ3UV62vqEtdDBah3ZREKgpuCm5cVrAPaEKZTCft0JlkmJmIJWTjxl9x40IRt/6DO//GaZuFth64cDjnXu69JxCMKu0431ZhZXVtfaO4Wdra3tnds/cP2ipOJCYtHLNYdgOkCKMRaWmqGekKSRAPGOkE4+up37knUtE4utMTQXyOhhENKUbaSH37OPUkh0OJxCireELRSuoFIXzIrnS12rfLTs2ZAS4TNydlkKPZt7+8QYwTTiKNGVKq5zpC+ymSmmJGspKXKCIQHqMh6RkaIU6Un86+yOCpUQYwjKWpSMOZ+nsiRVypCQ9MJ0d6pBa9qfif10t0eOmnNBKJJhGeLwoTBnUMp5HAAZUEazYxBGFJza0Qj5BEWJvgSiYEd/HlZdI+q7nnNee2Xm7U8ziK4AicgApwwQVogBvQBC2AwSN4Bq/gzXqyXqx362PeWrDymUPwB9bnD2Fsl8s=

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graph( ˜(x; t)) AAACDXicbVDLSsNAFJ3UV62vqEs3g1VoNyWRioKbghuXFewDmlAm00k7dPJg5kYsIT/gxl9x40IRt+7d+TdO2yy09cCFwzn3cu89Xiy4Asv6Ngorq2vrG8XN0tb2zu6euX/QVlEiKWvRSESy6xHFBA9ZCzgI1o0lI4EnWMcbX0/9zj2TikfhHUxi5gZkGHKfUwJa6psnqSMDPJQkHmUVB7gYsNSJFc8qqeP5+CG7gmq1b5atmjUDXiZ2TsooR7NvfjmDiCYBC4EKolTPtmJwUyKBU8GykpMoFhM6JkPW0zQkAVNuOvsmw6daGWA/krpCwDP190RKAqUmgac7AwIjtehNxf+8XgL+pZvyME6AhXS+yE8EhghPo8EDLhkFMdGEUMn1rZiOiCQUdIAlHYK9+PIyaZ/V7POadVsvN+p5HEV0hI5RBdnoAjXQDWqiFqLoET2jV/RmPBkvxrvxMW8tGPnMIfoD4/MHGFObgQ==

AAACBXicbVDLSsNAFJ3UV62vqEtdDBah3ZREKgpuCm5cVrAPaEKZTCft0JlkmJmIJWTjxl9x40IRt/6DO//GaZuFth64cDjnXu69JxCMKu0431ZhZXVtfaO4Wdra3tnds/cP2ipOJCYtHLNYdgOkCKMRaWmqGekKSRAPGOkE4+up37knUtE4utMTQXyOhhENKUbaSH37OPUkh0OJxCireELRSuoFIXzIrnS12rfLTs2ZAS4TNydlkKPZt7+8QYwTTiKNGVKq5zpC+ymSmmJGspKXKCIQHqMh6RkaIU6Un86+yOCpUQYwjKWpSMOZ+nsiRVypCQ9MJ0d6pBa9qfif10t0eOmnNBKJJhGeLwoTBnUMp5HAAZUEazYxBGFJza0Qj5BEWJvgSiYEd/HlZdI+q7nnNee2Xm7U8ziK4AicgApwwQVogBvQBC2AwSN4Bq/gzXqyXqx362PeWrDymUPwB9bnD2Fsl8s=

0, Fts Fts0 (x0 ) = Ftt0 (x0 ), (2.24) for any choice of the times t0, s, t ∈ [t1, t2 ]; see, e.g., Arnold (1978) for proofs of these statements. The flow map Ftt0 does not inherit the time-dependence of the velocity field v(x, t). Indeed, a steady velocity field v(x) with no explicit time dependence already generates a flow map Ft with general (aperiodic) time dependence. Similarly, time-periodic and timequasiperiodic velocity fields generate flow maps with general time dependence. Nevertheless, the flow map Ft of a steady velocity field will become a steady map (the identity map) when evaluated on specific initial conditions (fixed points) of the flow (see §2.2.11). Similarly, the time-dependent image, Ftt0 (x0 ), of an initial condition in a time-periodic velocity field is timeperiodic for exceptional x0 initial conditions that lie on periodic orbits of v(x, t) (see §2.2.12). The derivative ∇Ftt0 (x0 ) of the flow map with respect to its argument x0 is the deformation gradient. Technically, ∇Ftt0 (x0 ) is a two-point tensor (see §3.5 for more detail) that maps vectors in the tangent space Tx0 Rn of Rn at x0 to vectors in the tangent space TFtt0 (x0 ) Rn of Rn at the advected position Ftt0 (x0 ).5 More formally, therefore, ∇Ftt0 is defined as

∇Ftt0 : Tx0 Rn → TFtt0 (x0 ) Rn,

ξ0 7→ ∇Ftt0 (x0 )ξ0,

(2.25)

mapping vectors ξ0 based at x0 to vectors based at Ftt0 (x0 ), as shown in Fig. 2.6. Physically, an initial infinitesimal perturbation, ξ0 , to the initial condition x0 at time t0 evolves into a perturbation ∇Ftt0 (x0 )ξ0 to the trajectory location x(t; t0, x0 ) at time t. The deformation gradient also allows us to express the derivative of the
flow map Ftt0 with respect to the initial time t0 . Indeed, differentiating the identity Ftt0 Ftt0 (x) = x with respect to t0 gives the relationship d t F (x0 ) = −∇Ftt0 (x0 )v(x0, t0 ). dt0 t0 For a steady velocity field v(x), this relationship simplifies to

(2.26)

d t F (x0 ) = −v(Ftt0 (x0 )), dt0 t0 5

See Appendix A.4 for a brief summary of concepts from differential geometry, including tangent spaces of R n , that are relevant for derivatives of mappings between manifolds.

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23

2.2 Lagrangian Description of Fluid Motion

TFt

t0

Tx !n

(x 0 )

!n

0

ξ0

∇Ftt (x 0 )ξ0 0

t t0

F (x 0 )

x0

Figure 2.6 The definition of the deformation gradient.

given that v(Ftt0 (x0 )) = ∇Ftt0 (x0 )v(x0, t0 ) holds, as we will deduce later in formula (2.55). As a mapping between two different linear spaces, ∇Ftt0 (x0 ) does not have well-defined eigenvalues and eigenvectors, even though they are sometimes erroneously introduced in the literature for its specific matrix representations. Indeed, it is tempting to introduce coordinates in the domain space Tx0 Rn and the range space TFtt0 (x0 ) Rn , then solve a formal eigenvalue problem for the linear map ∇Ftt0 (x0 ) in those coordinates. These formally computed eigenvalues and eigenvectors, however, will depend on the coordinates introduced and hence will no longer be invariants of ∇Ftt0 (x0 ). The eigenvalues and eigenvectors of ∇Ftt0 (x0 ) are only well defined in a coordinate-free fashion if Tx0 Rn ≡ TFtt0 (x0 ) Rn holds, i.e., the trajectory x(t; t0, x0 ) returns to its initial position x0 at time t. Even then, the eigenvalues are irrelevant for the long-term evolution for the perturbation ξ0 to x0 unless the velocity field v(x, t) is time-periodic with period T = t − t0 , and hence x(t; t0, x0 ) is a periodic trajectory. In that case, Ftt0 (x0 ) is a period-t map or Poincaré map, as discussed in §2.2.12. In contrast, the velocity gradient ∇v(x, t) discussed in §3.3 is a linear operator mapping the tangent space Tx Rn into itself and hence ∇v(x, t) admits well-defined eigenvalues and eigenvectors.

2.2.4 Material Surfaces, Material Lines and Streaklines A material surface M(t) is an evolving surface of fluid particles that move from their initial positions along trajectories of the velocity field v. More precisely, M (t) = Ftt0 (M0 ) ,

M0 = M (t0 ) .

(2.27)

As shown in Fig. 2.7, the surface M (t) is a smooth manifold if M0 is a smooth manifold, because the flow map Ftt0 is a diffeomorphism. In that case, Ftt0 is a smooth mapping between manifolds and hence ∇Ftt0 (x0 ) maps the tangent space Tx0 M0 of M0 at x0 onto the tangent space of M (t) at the point Ftt0 (x0 ), i.e.,

∇Ftt0 (x0 ) Tx0 M0 = TFtt0 (x0 ) M (t) ,

(2.28)

as shown in Fig. 2.7 (see Abraham et al., 1988). Let Nx0 M0 denote the normal space of M0 at x0 , defined as the orthogonal complement of Tx0 M0 in the ambient space Rn , as shown in Fig. 2.7 (see also Appendix A.4). Consequently, if a is a vector in Tx0 M0 and b is a vector in Nx0 M0 , then the inner product of these vectors must vanish by their orthogonality: ha, bi = 0. This identity can be written equivalently as

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24

Eulerian and Lagrangian Fundamentals N x 0

N Ft (x )(t) =

0

t0

Tx 0

Ftt (x 0 ) 0

0

0

T

N x 0 0

TFt (x )(t) = Ftt (x 0 )Tx 0

x0

t0

t t0

F

0

0

0

t t0

F (x 0 ) (t) = Ftt ( 0 )

0

0

Figure 2.7 Geometry near a material surface M (t). Tangent and normal spaces at the same material point of this evolving surface are mapped into each other by the  −T  deformation gradient ∇Ftt0 and by its inverse transpose, ∇Ftt0 , respectively.

E D  E D  −T  −1 hb, ai = b, ∇Ftt0 (x0 ) ∇Ftt0 (x0 )a = ∇Ftt0 (x0 ) b, ∇Ftt0 (x0 )a = 0,

(2.29)

where −T denotes inverse transpose. Note that a ∈ Tx0 M0 and b ∈ Nx0 M0 are arbitrary in   −T the relationship (2.29), and hence Eq. (2.28) implies that ∇Ftt0 (x0 ) b ∈ NFtt0 (x0 ) M (t).  −T  into Therefore, we have obtained that any vector b ∈ Nx0 M0 is mapped by ∇Ftt0 (x0 ) a vector normal to TFtt0 (x0 ) M (t), or, equivalently, 

∇Ftt0 (x0 )

 −T

Nx0 M0 = NFtt0 (x0 ) M (t) .

(2.30)

In other words, a normal of a material surface at the point x0 at time t0 is advected by the   −T inverse transpose ∇Ftt0 (x0 ) of the deformation gradient to a normal to the same material t surface at the point Ft0 (x0 ) at time t. Consequently, as illustrated in Fig. 2.7, we also have 

∇Ftt0 (x0 )

T

NFtt0 (x0 ) M (t) = Nx0 M0 .

(2.31)

Since M0 was arbitrary in this argument, so was the subspace Nx0 M0 ⊂ Tx0 Rn . Therefore,   −T is defined for any vector Tx0 Rn and maps any such vector the linear mapping ∇Ftt0 (x0 ) into TFtt0 (x0 ) Rn : 

∇Ftt0 (x0 )

 −T

: Tx0 Rn → TFtt0 (x0 ) Rn .

(2.32)

Reversing the role of x0 and Ftt0 (x0 ) in formula (2.32) leads to 

∇Ftt0 (x0 )

T

: TFtt0 (x0 ) Rn → Tx0 Rn,

(2.33)

showing that the transpose (or adjoint) of the deformation gradient can be viewed as a mapping of vectors in TFtt0 (x0 ) Rn back to Tx0 Rn .

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2.2 Lagrangian Description of Fluid Motion

A more specific consequence of Eq. (2.30) is the following classic result: if n0 ∈ Nx0 M0 is a unit normal to M0 to at x0 , then the surface element n0 dA0 of M0 at x0 is carried by the flow into the surface element   −T ndA = det ∇Ftt0 (x0 ) ∇Ftt0 (x0 ) n0 dA0 (2.34) of M (t) at the point Ftt0 (x0 ) (see Gurtin, 1981; Truesdell, 1992). A material line (sometimes called a timeline) is an evolving curve composed of the same fluid particles, i.e., the image, L(t) = Ftt0 (L(t0 )) ,

(2.35)

of a curve L(t0 ) of initial particle particle positions under the flow map Ftt0 (see Fig. 2.8). p0 = S(t0 )

x2

sha1_base64="VKbNivUN9yFU8gDJaa4vocYLPLA=">AAAB+nicbVDLSsNAFL3xWesr6tLNYBHqpiSi2GXBjQsXFewD2hAm00k7dDIJM5NCif0TV4KCuPVPXPk3TtostPXAwOGce7lnTpBwprTjfFtr6xubW9ulnfLu3v7BoX103FZxKgltkZjHshtgRTkTtKWZ5rSbSIqjgNNOML7N/c6ESsVi8ainCfUiPBQsZARrI/m23Y+wHhHMs/tZVfvOhW9XnJozB1olbkEqUKDp21/9QUzSiApNOFaq5zqJ9jIsNSOczsr9VNEEkzEe0p6hAkdUedk8+QydG2WAwliaJzSaq783MhwpNY0CM5nnVMteLv7n9VId1r2MiSTVVJDFoTDlSMcorwENmKRE86khmEhmsiIywhITbcoql00L7vKfV0n7suZe15yHq0qjXvRRglM4gyq4cAMNuIMmtIDABJ7hFd6sJ+vFerc+FqNrVrFzAn9gff4AVVOTIw==

L(t0 ) AAAB+nicbVDLSsNAFJ34rPUVdelmsAh1UxJR7LLgxoWLCvYBbQiT6aQdOpmEmZtCif0TV4KCuPVPXPk3TtostPXAwOGce7lnTpAIrsFxvq219Y3Nre3STnl3b//g0D46bus4VZS1aCxi1Q2IZoJL1gIOgnUTxUgUCNYJxre535kwpXksH2GaMC8iQ8lDTgkYybftfkRgRInI7mdV8N0L3644NWcOvErcglRQgaZvf/UHMU0jJoEKonXPdRLwMqKAU8Fm5X6qWULomAxZz1BJIqa9bJ58hs+NMsBhrMyTgOfq742MRFpPo8BM5jn1speL/3m9FMK6l3GZpMAkXRwKU4EhxnkNeMAVoyCmhhCquMmK6YgoQsGUVS6bFtzlP6+S9mXNva45D1eVRr3oo4RO0RmqIhfdoAa6Q03UQhRN0DN6RW/Wk/VivVsfi9E1q9g5QX9gff4AVtmTJA==

L(t1 ) AAAB9nicbVDLSgNBEOz1GeMr6tHLYBDiJeyKYo4BLx48RDAPSNYwO5lNhsw+mOkVw5L/8CQoiFf/xZN/42yyB00sGCiquuma8mIpNNr2t7Wyura+sVnYKm7v7O7tlw4OWzpKFONNFslIdTyquRQhb6JAyTux4jTwJG974+vMbz9ypUUU3uMk5m5Ah6HwBaNopIdeQHHEqExvpxU865fKdtWegSwTJydlyNHol756g4glAQ+RSap117FjdFOqUDDJp8VeonlM2ZgOedfQkAZcu+ks9ZScGmVA/EiZFyKZqb83UhpoPQk8M5ml1IteJv7ndRP0a24qwjhBHrL5IT+RBCOSVUAGQnGGcmIIZUqYrISNqKIMTVHFomnBWfzzMmmdV53Lqn13Ua7X8j4KcAwnUAEHrqAON9CAJjBQ8Ayv8GY9WS/Wu/UxH12x8p0j+APr8weu3ZJP

0, for x ∈ Rn with n = 2 or n = 3. The evolution rule for trajectories, therefore, repeats itself periodically in time, enabling a simplified study of the discretized evolution of trajectories over time intervals that are integer multiples of T. Specifically, the Poincaré map (or period T-map), denoted Pt0 , of the ODE (2.73) is the restriction of its flow map to one time period starting at time t0 : Pt0 := Ftt00 +T .

(2.74)

We illustrate this definition for 2D flows in Fig. 2.12.

x2 Pt (x 0 ) 0

t x0

x(t;t0 , x 0 ) t0

ide

ntif

t0 +T

y

x1 Figure 2.12 The definition of the Poincaré map Pt0 for a 2D velocity field whose time dependence is T-periodic.

By the time-periodicity of the ODE (2.73), the flow map advancing particle positions between times t0 + jT and t0 + ( j + 1) T is the same map for any integer j. Therefore, by the group property of flow maps (see Eq. (2.24)), the repeated iteration of the single mapping Pt0 gives the fluid particle positions at forward and backward times that are separated from t0 by an integer multiple of T: Ftt00 +kT = Ftt00 +kT ◦ · · · ◦ Ftt00 +T = Pt0 ◦ · · · ◦ Pt0 = Ptk0 (x0 ) . +(k−1)T | {z } k

The infinite set of iterations x0, Pt0 (x0 ) , P2t0 (x0 ) , . . . , Ptk0 (x0 ) , . . . is called the orbit of Pt0 starting from the point x0 . We also note the relation
−1 t0 +T Pt1 = Ftt11 +T = Ftt10 +T ◦ Ftt01 = Ftt10 ◦ Pt0 ◦ Ftt01 = Ftt01 ◦ Pt0 ◦ Ftt01 , (2.75) +T ◦ Ft0 

which shows that Poincaré maps based at any two different times, t0 and t1 , are always topologically conjugate. Such maps share the same orbit structure up to the smooth deformation

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36

Eulerian and Lagrangian Fundamentals

represented by Ftt01 (see Guckenheimer and Holmes, 1983). Topological conjugacy is a stronger form of the notion of topological equivalence introduced in §2.2.7: it does not allow for a reparametrization of time (i.e., τ(x0, t) ≡ t must hold). The computation of the Poincaré map for a general 2D, time-periodic velocity field is implemented in Notebook 2.1. Notebook 2.1 (PoincareMap2D) Computes the Poincaré map defined for a 2D, timeperiodic velocity data set. https://github.com/haller-group/TBarrier/tree/main/TBarrier/2D/ demos/AdvectiveBarriers/PoincareMap2D If v(x, t) is divergence-free, then the flow map is volume-preserving (see condition (2.40)) and so is the Poincaré map
det ∇Pt0 = 1, vol Pt0 (V) = vol (V) (2.76) for any set V ⊂ U. For an incompressible time-periodic flow, the volume-preservation of its Poincaré map has important global consequence on the dynamics, which we will discuss in §§2.2.13–2.2.14. While we will discuss the power of Poincaré maps in later chapters, Fig. 2.13 shows a preliminary illustration of their usefulness. The figure shows experimentally observed mixing patterns whose transport barriers (or lack thereof) are revealed by Poincaré maps computed from trajectories of the corresponding model velocity fields.

Quasiperiodic Flows Quasiperiodic time-dependence in a velocity field can be represented by rewriting the general particle equation of motion (2.17) as xÛ = v(x, Ω1 t, . . . , Ωm t),

x ∈ Rn,

t ∈ R,

(2.77)

with a velocity field v whose time-dependence involves several frequencies, Ω j , not just one.11 We assume that the frequencies Ω j are rationally independent, which means that the frequency vector Ω = (Ω1, . . . , Ωm ) satisfies hΩ, ki , 0

(2.78)

for all integer vectors k ∈ Zm . If the frequencies were not rationally independent, then one could select a smaller number of Ω j frequencies to represent the velocity field (2.77). We can turn system (2.77) into a temporally periodic, higher-dimensional dynamical system by introducing the phase vector φ = (φ2, . . . , φm ) with the individual phases φ j = Ω j t for j = 2, . . . , m. We view this phase vector as an element of an (m − 1)-dimensional torus Tm−1 = S 1 × · · · × S 1, | {z } m−1

11

Time periodic flows (m = 1) also fall in this general family.

https://doi.org/10.1017/9781009225199.003 Published online by Cambridge University Press

(2.79)

047104-9

Phys. Fluids 18, 047104 !2006"

(a) (b)

© φÛ = ­­

Ω2 .. ª® , . ® « Ωm ¬

φ0 = (Ω2 t0, . . . , Ωm t0 ) .

https://doi.org/10.1017/9781009225199.003 Published online by Cambridge University Press 10 on’s model gives an excellent result for this case. Figure 10 shows the unstable manifolds of the Poincaré map with period equal to one passage at three phases; the circles indicate the vortical core. The flow visualization photographs are 11 from Yamada and Matsui. The stretching along the manifold is so rapid that even though 4000 particles were placed on the initial segment, the visual appearance of the manifold as a connected curve disappears after the third forward map of the initial segment. For the middle photograph, the manifold winds back and forth between the two vortices, each time passing through the “braid” region. Even the very fine scale features of the manifold agree remarkably well with the experimental photographs. 34 Maxworthy used the photographs together with vorticity diffusion arguments to provide a plausible rendering of the underlying vorticity. The first photograph shows the first passage almost completed, and according to him “thereafter, the latter !the passing vortex" is distorted and wraps around the former, and the two rings become one.” In referring to the third photograph, Maxworthy says that the passing vortex “has become so distorted that it is barely recognizable,” and Yamada and Matsui say that “the core of the first ring was severely deformed and stretched, and it seemed to roll up around that of the second ring….” Maxworthy’s guess of the vorticity field underlying the third photograph is sketched as Stage 3 in Fig. 1 of his paper and shows the two rings diffused together.

On the other hand, the present result suggests that the observed pattern may be due merely to complex motion of tracer in irrotational or weakly vortical fluid, with vortex cores behaving in a simple, nondeforming and almost classical manner. It is only the tracer that appears to deform and roll up around the leading vortex. This suggestion should be tempered with three points. First, our type of interpretation is appropriate only for this experiment in which a smoke wire was stretched across the entire diameter of the orifice, causing smoke to be introduced, not only in the emitted shear layer !which would mark the vorticity aside from Schmidt number effects" but also into nonvortical fluid initially transported with each vortex. If one is careful to ensure that only vortical regions are marked then there is less danger in identifying smoke with vorticity. Second, weak vorticity around the periphery of a core may also become susceptible to passive advection and acquire the structure of the unstable manifold. Third, the question of whether at the Reynolds number of the experiment !1600 based on initial translation speed and orifice diameter" the two rings diffuse together or behave in the simple manner suggested at the Reynolds number of the experiment will have to await viscous numerical computations 35 at higher Reynolds number. Stanaway et al. simulated a passage interaction for the same core size !based on where the maximum velocity occurs" and vortex separation as the present case. The Reynolds number based on initial self-

FIG. 10. Unstable manifold at different phases for two leapfrogging rings. Photographs are from Yamada and Matsui !Ref. 11".

Dynamical systems analysis of fluid transport

2.2 Lagrangian Description of Fluid Motion

37

Figure 2.13 (Bottom) Dye visualization vs. (top) Poincaré maps for two timeperiodic flows. (a) Leapfrogging vortex pair visualized by smoke (reproduced from Shariff et al., 2006, who used the experimental photographs of Yamada and Matsui, 1978). (b) Experiment involving a moving mixer rod in sugar syrup. The figureof-eight path of the mixing rod is shown in the upper Poincaré maps, which were computed numerically from a single trajectory of a model flow. An upper and a lower stagnation point is shown in red. Adapted from Thiffeault et al. (2011).

i.e., the (m − 1)-fold topological product of the unit circle S 1 with itself. We can now rewrite system (2.17) as

xÛ = v(x, Ω1 t, φ),

(2.80)

a time-periodic dynamical system on the (n + m − 1)-dimensional phase space Rn × Tm−1 with period T = 2π/Ω1 . Only some of the trajectories of the extended dynamical system (2.80) will, however, represent trajectories of the ODE (2.17). Indeed, while one can formally select an arbitrary initial phase vector φ(t0 ) as initial condition in the ODE (2.80), the phase variables (Ω2 t, . . . , Ωm t) in the original system (2.77) are constrained to evolve together from the initial vector

(2.81)

38

Eulerian and Lagrangian Fundamentals

Therefore, the elements of this vector cannot be selected arbitrarily relative to each other and relative to the first phase variable Ω1 t0 . For this reason, only an (n + 1)-dimensional family of initial conditions of the (n + m − 1)-dimensional system (2.80) are relevant for the original system (2.17).12 This relevant set of initial conditions is a dense set in the (n + m − 1)-dimensional phase space because the curve of initial conditions satisfying Eq. (2.81) forms a dense curve in the torus Tm−1 due to the assumed rational independence of the frequencies (Ω2, . . . , Ωm ). This dense set, however, has measure zero within the torus: it can be covered by a countable set of open subsets of the torus that have arbitrarily small total volume (total length for m = 2 and total area for m = 3). As a consequence, the trajectories of the extended system (2.80) that are related to the original quasiperiodic velocity field v(x, Ωt) form a measure zero set in the phase space of Eq. (2.80). As a consequence, mathematical statements obtained for almost all (i.e., all but a measure zero set of) trajectories of the extended system (2.80) do not carry over to the original quasiperiodic velocity field (2.77). Examples of such statements are the Poincaré recurrence theorem (§2.2.13) and Birkhoff’s ergodic theorem (§2.2.14). For the extended system (2.80), we can define a Poincaré map in the same way as for the time-periodic system (2.73). Indeed, for any fixed initial time t0 and for the time period T = 2π/Ω1 , an extended Poincaré map Pt0 can be defined on the space Rn × Tm−1 as Pt0 : Rn × Tm−1 → Rn × Tm−1, (x0, φ20, . . . , φm0 ) 7→ (x (t0 + T; t0, x0, Ω1 t0, φ0 ) , φ20 + Ω2T, . . . , φm0 + ΩmT) .

(2.82)

We sketch the geometry of this Poincaré map for 2D flows (n = 2) in Fig. 2.14. Similar Poincaré maps can be defined based on any of the periods Tj = 2π/Ω j associated with the remaining frequencies for j = 2, . . . , m.

x2 Pt (x 0 , 0

t (x 0 , ( 2D , invariant , mtori) 2

m

20

, ,

m0

20

, ,

m0

) t0 + 2π/Ω1

)

1

t0

y ntif ide

x1 Figure 2.14 The definition of the Poincaré map Pt0 for a 2D, temporally quasiperiodic velocity field with m rationally independent frequencies.

By our discussion in §2.2.12, the Poincaré map Pt0 defined for the extended system (2.80) is volume preserving precisely when the velocity field v is incompressible. As we noted for the extended flow (2.80), only a dense but measure zero set of orbits of the Poincaré 12

This restriction leads to conservation laws in the phase space of system (2.80), as we will see in §4.3.2.

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39

2.2 Lagrangian Description of Fluid Motion

Pt0 correspond to sampled fluid trajectories of the original vector quasiperiodic vector field (2.77). Such a dense set of admissible trajectories of Pt0 will give a good description of the overall behavior of the fluid but is still only a measure zero set.

3D Steady Flows As we have already pointed out in the previous section, any steady flow can be considered time-periodic with any period. This enables the sampling of 3D steady flows with arbitrary stroboscopic maps. These sampling maps, however, are still 3D, so there is no immediate practical advantage from such a temporal discretization of the steady flow. However, there is an alternative way to construct Poincaré maps in 3D flows that does bring dimensional reduction, as we discuss next. Consider a 2D surface Σ ⊂ R3 that is everywhere transverse (i.e., nontangent) to the velocity field v(x). For such a Poincaré section Σ, the first-return map, or Poincaré map, PΣ : Σ → Σ can then be defined as the mapping that takes an initial condition x0 ∈ Σ into the first intersection of the trajectory starting from x0 with Σ, if such an intersection exists (see Fig. 2.15).

sha1_base64="S5pVRe5ORFaReFtKb70eeZDwAfM=">AAAB+XicbVDLSsNAFL2pr1pfUZduBovgqiSi6LLoxmUF+4A2hMl00g6dTMLMpFhC/sSNC0Xc+ifu/BsnbRbaemDgcM693DMnSDhT2nG+rcra+sbmVnW7trO7t39gHx51VJxKQtsk5rHsBVhRzgRta6Y57SWS4ijgtBtM7gq/O6VSsVg86llCvQiPBAsZwdpIvm0PIqzHQZg95X7m5LWab9edhjMHWiVuSepQouXbX4NhTNKICk04VqrvOon2Miw1I5zmtUGqaILJBI9o31CBI6q8bJ48R2dGGaIwluYJjebq740MR0rNosBMFjnVsleI/3n9VIc3XsZEkmoqyOJQmHKkY1TUgIZMUqL5zBBMJDNZERljiYk2ZRUluMtfXiWdi4Z71XAeLuvN27KOKpzAKZyDC9fQhHtoQRsITOEZXuHNyqwX6936WIxWrHLnGP7A+vwB4jiTKA==

AAAB7nicbVDLSgNBEOyNrxhfUY9eBoPgKeyKosegF48RjQkkS5idzCZD5rHMzAphyUd48aCIV7/Hm3/jbLIHTSxoKKq66e6KEs6M9f1vr7Syura+Ud6sbG3v7O5V9w8ejUo1oS2iuNKdCBvKmaQtyyynnURTLCJO29H4JvfbT1QbpuSDnSQ0FHgoWcwItk5q9+7ZUOBKv1rz6/4MaJkEBalBgWa/+tUbKJIKKi3h2Jhu4Cc2zLC2jHA6rfRSQxNMxnhIu45KLKgJs9m5U3TilAGKlXYlLZqpvycyLIyZiMh1CmxHZtHLxf+8bmrjqzBjMkktlWS+KE45sgrlv6MB05RYPnEEE83crYiMsMbEuoTyEILFl5fJ41k9uKj7d+e1xnURRxmO4BhOIYBLaMAtNKEFBMbwDK/w5iXei/fufcxbS14xcwh/4H3+AKMUjxs=

0 : x0 ∈ Σ, Ft (x0 ) ∈ Σ .

(2.83)

The computation of the Poincaré map for a general 3D steady velocity field is implemented in Notebook 2.2. Notebook 2.2 (PoincareMap3D) Computes the Poincaré map defined for a 3D steady velocity data set. https://github.com/haller-group/TBarrier/tree/main/TBarrier/3D/ demos/AdvectiveBarriers/PoincareMap3D If Σ and v are smooth and nΣ (x) denotes a unit normal field along Σ with the orientation v · nΣ > 0,13 then the Poincaré map PΣ is a diffeomorphism that preserves the generalized area element 13

It is always possible to choose such an orientation for the unit normal field nΣ globally on Σ. This is because the

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40

Eulerian and Lagrangian Fundamentals

d A˜ = v · nΣ dA,

(2.84)

as we show in Appendix A.14. As a result, PΣ is generally not area preserving but falls in the family of symplectic maps, defined as mappings that preserve a nondegenerate, closed twoform (see, e.g., Abraham et al., 1988). For any symplectic map, appropriate local coordinates can be constructed so that the map preserves the classic area element on Σ in the new coordinates (see Appendix A.14). Therefore, the classification of the linearized Poincaré map ∇PΣ (p) at a fixed point p on Σ is identical to that of the Poincaré map of a 2D, timeperiodic flow (see Fig. 4.3). Global features of 2D symplectic maps coincide with those we already mentioned for area-preserving maps in §4.1 (see Mackay et al., 1984; Meiss, 1992). Similarly, a first-return map PΣ defined on a transverse cross section Σ of a compressible but mass-preserving, steady, 3D flow preserves the generalized area element d A˜ = ρv · nΣ dA,

(2.85)

where ρ(x) > 0 is the mass density field of the fluid (see Appendix A.14). Local and global qualitative properties of Poincaré maps in such flows, therefore, also coincide with those of period-T maps of time-periodic 2D flows.

2.2.13 Revisiting Initial Conditions: Poincaré’s Recurrence Theorem Assume that the Poincaré map Pt0 of a T-periodic velocity field v(x, t) leaves a compact domain D ⊂ U invariant, i.e., Pt0 (D) = D holds. This is the case, for instance, if the velocity field v is defined on a closed and bounded domain D ≡ U surrounded by impenetrable boundaries. In such cases, Poincaré’s recurrence theorem for measure-preserving maps (Arnold, 1989) guarantees that almost all orbits of Pt0 return arbitrarily close to their initial position x0 over long enough time intervals, as illustrated in Fig. 2.16. In other words, trajectories that do not return arbitrarily close to their initial positions over long times form a set of zero volume. Note that Poincaré’s theorem also guarantees repeated returns to D occurring after an arbitrarily large threshold time t ∗ . This can be deduced by applying the theorem to the measure-preserving map P˜ t0 := PtN0 , with the positive integer N selected so that NT > t ∗ holds. Phrased in terms of the orbits of v(x, t) in continuous time, almost all trajectories of a timeperiodic incompressible flow defined on a closed and bounded domain revisit their initial conditions repeatedly with arbitrarily high accuracy over arbitrarily long times. Since steady flows can be considered periodic with any period, Poincaré’s recurrence theorem applies to them as well. Remarkably, therefore, almost all trajectories of a steady incompressible flow confined to a closed and bounded 2D or 3D domain revisit their initial positions arbitrarily closely over arbitrary long times. In its most general form, Poincaré’s recurrence theorem only assumes the conservation of a measure (Arnold, 1989). As we have seen, compressible flows obeying the equation of continuity conserve mass (see formula (2.43)), which makes their Poincaré maps measure preserving. Therefore, almost all trajectories of a compressible but mass-preserving, steady inner product v · nΣ cannot vanish anywhere on Σ by the transversality of v to Σ, and hence this inner product cannot change sign on Σ.

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2.2 Lagrangian Description of Fluid Motion

41

Pt (x 0 ) 0

x0

Ptk (x 0 )

Uδ

0

PtN (x 0 )

D

0

Figure 2.16 The statement of the Poincaré recurrence theorem applied to the Poincaré map Pt0 on a compact invariant domain D. Almost all (i.e., all but a measure zero set of) initial conditions x0 generate trajectories for Pt0 that return for some large enough iteration N of Pt0 to any small neighborhood Uδ of x0 .

or time-periodic flow defined on a closed and bounded domain revisit their initial positions arbitrarily closely over arbitrarily long times. At first sight, it might seem that Poincaré’s recurrence theorem is also applicable to the quasiperiodic vector fields v(x, Ωt) with frequency vector Ω = (Ω1, . . . , Ωm ) discussed in §2.2.12. Indeed, if the original quasiperiodic velocity field (2.77) is defined on a compact invariant domain D, then the extended Poincaré map Pt0 , introduced in formula (2.82), is defined on the compact set D × Tm , given that the m-dimensional torus, Tm , is compact. This enables us to apply the Poincaré recurrence theorem to the extended map Pt0 and conclude that almost all trajectories X(t; X0 ) = (x(t; x0 ), φ0 + Ωt) return arbitrarily close to X0 = (x0, φ0 ) for large enough times t. Note, however, that Poincaré’s recurrence theorem allows for the existence of a measure zero set of nonrecurrent trajectories in the phase space of the extended system (2.80). The trajectories in that extended phase space that correspond to actual trajectories of the original quasiperiodic velocity field (2.77) also form just a measure zero set. Therefore, for all we know from this argument, all trajectories of v(x, Ωt) may be nonrecurrent. Consequently, recurrence in temporally quasiperiodic velocity fields does not follow from Poincaré’s recurrence theorem. The recurrence theorem, however, is applicable to Poincaré maps defined on transverse sections of 3D steady, volume-preserving or mass-preserving flows (see §2.2.12). Namely, on any compact spatial domain invariant under such a flow, typical trajectories will return to any Poincaré section that is transverse to the velocity field. This recurrence guarantees the existence of initial conditions arbitrarily close to Σ that will return to their arbitrarily small neighborhoods. These returning trajectories will, therefore, come back arbitrarily close to Σ over time and hence will have to intersect Σ at some point due to the transversality of the vector field v to Σ. The same argument in backward time guarantees that such trajectories also had an intersection with Σ at some point in the past, and hence these trajectories start from, and return to, Σ. Poincaré’s recurrence theorem, however, does not hold when the domain of definition D of the field (2.77) is not a bounded invariant set, or when the time-dependence of the velocity

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42

Eulerian and Lagrangian Fundamentals

field is not periodic or steady. Indeed, in the latter case one cannot associate an extended dynamical system to v(x, t) with a compact phase space given that t must be taken from all of R.

2.2.14 Convergence of Time-Averaged Observables: Ergodic Theorems Here we discuss how temporal averages of observed scalar fields relate to their spatial averages in perfectly mixing flows. We first state the results for averages taken under stroboscopic samplings (Poincaré maps) of the flow, then directly for the full flow. We first consider a Poincaré map Pt0 : S → S defined on a physical domain S. As a special case of the general flow map Ftt0 , a Poincaré map of a smooth vector field is always a diffeomorphism (see §2.2.3). This will enable us to give a simplified treatment here relative to the more general results of ergodic theory (see, e.g., Walters, 1982). Ergodic theory is concerned with the dynamics of maps on measurable subsets of S. The measures µ relevant for physically observable mixing in fluids are the volume when S is a 3D flow domain, the area (or the scaled areas, given by formulas (2.84) or (2.85)) when S is a 2D surface, and the arc length when S is a one-dimensional curve. Ergodic theory additionally requires S itself to have a finite measure µ(S). For practical applications to fluids, this finiteness assumption on µ amounts to the requirement that the domain S must be bounded. We will assume that Pt0 is measure preserving on S, i.e., the measure of any subset A ⊂ S is preserved under iterations of Pt0 :
(2.86) µ(A) = µ Pit0 (A) for all i ≥ 1. This requirement holds for Poincaré maps of time-periodic incompressible flows and for extended Poincaré maps of time-quasiperiodic incompressible flows (see §2.2.12). As already noted, if the flow is compressible but conserves mass, then its associated stroboscopic maps are still measure preserving with respect to the fluid mass as a measure. Note, however, that these conclusions about measure preservation only follow directly from the volume- or mass-preservation of the flow when S has the same dimension as the underlying velocity field v. For instance, if S ⊂ R2 is an invariant curve in a 2D flow, then the 2D period-T map, Pt0 , restricted to S is generally not a measure-preserving map on S. Examples of such invariant curves are stable or unstable manifolds of saddle-type fixed points, to be discussed later in §4.1.1. Along such invariant curves, Pt0 shrinks or expands the measure (arclength), respectively. Similarly, a measure-preserving Poincaré map, Pt0 , on a 3D domain does not generally preserve any nondegenerate measure on a 2D invariant surface M = Pt0 (M), as examples of 2D stable and unstable manifolds in 3D mappings illustrate. In contrast, Poincaré maps defined as first-return maps to a 2D transverse section S of 3D, steady, volume-preserving or mass-preserving flow are measure preserving on S with respect to the areas computed from the area elements (2.84) or (2.85). We assume that the mapping Pt0 is ergodic on S, i.e., the invariant sets of Pt0 have either full measure or zero measure in S. This means that one cannot identify any experimentally observable part of S that does not mix with the rest of S under iterations of Pt0 . For instance, the Poincaré map shown in Fig. 2.13(a) is not ergodic on the flow domain because the interiors of the two leapfrogging vortices are invariant subsets of nonzero area within the 2D Poincaré section S. The same map, however, is ergodic when restricted to its closed invariant

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2.2 Lagrangian Description of Fluid Motion

43

curves because orbits within such elliptic transport barriers (see §4.1.2) are quasiperiodic and hence densely cover the barrier. As a consequence, there is no invariant set of nonzero arc length within the barrier. As another example, the Poincaré map shown in Fig. 2.13(b) is ergodic on its bounded 2D domain because the only invariant sets smaller than the full flow domain are the one-dimensional stable and unstable manifolds of its fixed points. Consider now an observable, i.e., a scalar field c(x) that is defined and integrable on the domain S of an ergodic, measure-preserving mapping Pt0 .14 Because of the perfect mixing of points in S under iterations of the ergodic map Pt0 , one hopes to uncover dynamical features  of Pt0 by averaging c(x) along the orbits x0, Pt0 (x0 ), P2t0 (x0 ), . . . of Pt0 . Indeed, Birkhoff’s ergodic theorem (Birkhoff, 1931) guarantees that ˆ N
1 1 Õ k c Pt0 (x0 ) = c(x) dµ lim N →∞ N µ(S) S k=1 holds for almost all x0 ∈ S.15 In other words, with the possible exception of an unobservable set of x0 locations, the temporal average of any observable along an orbit starting from x0 converges to the spatial average of that observable over the whole of S. For instance, if c(x) = x is the x-coordinate component of x = (x, y, z), then the averaged x-coordinate´ along 1 a trajectory of a volume-preserving, ergodic Poincaré map, Pt0 (x0 ), is equal to µ(S) x dµ, the x-coordinate of the center of mass of the material set S. In practice, velocity fields will generally not repeat themselves periodically in time and hence concepts from ergodic theory will not apply to them. Nevertheless, ergodicity is a useful conceptual tool for understanding and visualizing certain types of transport barriers (or lack thereof) in idealized model flows. Ergodicity also helps in interpreting mixing phenomena in low Reynolds number experiments whose velocity fields are very close to time-periodic (see Chapter 4). A more general version of the classic ergodic theorem of Birkhoff is the Birkhoff–Khinchin ergodic theorem, which also applies to continuous flows of steady velocity fields (see Cornfeld et al., 1982). To state this theorem, we consider the flow map Ft : S → S of a steady velocity field v(x) and a scalar field c(x) whose integral exists with respect to a measure µ defined on the bounded and invariant 2D or 3D domain S. We assume that Ft preserves the measure µ, i.e., all subsets A ⊂ S satisfy µ(Ft (A)) = µ(A) for all times. For an incompressible flow, the measure µ preserved by Ft on the flow domain S is the volume or the area. For a mass-preserving flow on S, the measure preserved by Ft is the mass. If, however, S is a 2D invariant surface in a 3D flow, then Ft does not necessarily preserve any nondegenerate measure on S, even if it preserves volume or mass on the full 3D flow domain. In analogy with the discrete case already discussed, we also assume that Ft is ergodic on S, i.e., the measure of any invariant sets of Ft in S is either zero or equal to µ(S). Then, according to the Birkhoff–Khinchin ergodic theorem (Cornfeld et al., 1982), for almost all initial conditions x0 ∈ S, we must have 14

15

The observable c can also be a time-periodic scalar field, c(x, t) ≡ c(x, t + T ), where T is the time period of the underlying velocity field v(x, t). The scalar field c does not even have to be smooth, but it has to be integrable with respect to the measure on S. Here “almost all” refers to all points x0 ∈ S with the possible exception of a measure-zero subset of S.

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Eulerian and Lagrangian Fundamentals ˆ ˆ 1 t 1 t 1 t lim c (Fs (x0 )) ds = lim c (F−s (x0 )) ds = lim c (Fs (x0 )) ds t→∞ t 0 t→∞ t 0 t→∞ 2t −t ˆ 1 = c(x) dµ. (2.87) µ(S) S

ˆ

In other words, with the possible exception of an unobservable set of x0 initial conditions, the forward, backward and full temporal averages of any observable along an orbit starting from x0 converge to the spatial average of the observable over the whole of S. As we will see in examples in Chapter 4, typical 3D steady flows are not ergodic on their full domain D but might admit open ergodic subsets. More frequently, they admit 2D invariant surfaces (invariant tori) or 1D closed invariant curves (periodic orbits) restricted to which Ft becomes ergodic. Applying the Birkhoff–Khinchin ergodic theorem on the full domain D does not guarantee anything for averages of scalars taken over these lower-dimensional invariant surfaces given that those have measure zero in D. Thus, for all we know, these surfaces may be part of the set of locations that violate the Birkhoff–Khinchin ergodic theorem. The statement (2.87), therefore, is only meaningful for lower-dimensional invariant sets S ⊂ D if the theorem is applicable directly to S with respect to a measure preserved by Ft on S. This measure is generally not known explicitly, but its existence is guaranteed by the type of dynamics (quasiperiodic or periodic) on these lower-dimensional invariant sets.

2.2.15 Lagrangian Scalars, Vector Fields and Tensors We refer to scalar, vector and tensor fields defined over initial trajectory positions x0 , initial times t0 and current times t as Lagrangian quantities. Specifically, we refer to scalar fields defined at arbitrary initial conditions (x0, t0 ) for all times t as Lagrangian scalar fields. For instance, a concentration field c(x, t) subject to advec
tion and diffusion can be expressed as a Lagrangian scalar field c(x ˆ 0, t0 ; t) := c Ftt0 (x0 ), t , depending on the current time t, initial time t0 and the initial position x0 of the material trajectory that is at the point x = Ftt0 (x0 ) at time t. A Lagrangian vector field is a time-dependent vector field u(x0, t0 ; t) comprising vectors based in the tangent spaces Tx0 Rn of Rn at the initial positions x0 ∈ U. These vectors remain based at the points x0 for all times t ∈ [t1, t2 ] but they generally vary as functions of t. In contrast, an eigenvector field αi (x0 ; t0, t) of a Lagrangian tensor field (to be defined below) is not a Lagrangian vector field because it has no well-defined length or orientation. Accordingly, we refer to such eigenvector fields as Lagrangian direction fields, in line with our terminology in §2.1.1 for the Eulerian case.
Despite their commonly used names, the Lagrangian velocity v Ftt0 (x0 ), t and the La
grangian vorticity ω Ftt0 (x0 ), t are not Lagrangian vector fields, as they comprise vectors based at the time-evolving current location x(t). In other words, these vector fields are pointwise elements of the tangent space TFtt0 (x0 ) Rn , as opposed to Tx0 Rn ,which would be required for a Lagrangian vector field.  ∗  ∗ In contrast, the pullback velocity field Ftt0 v and the pullback vorticity field Ftt0 ω , defined respectively as

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1 AAACI3icbVDJSgNBEO1xjXGLevTSmAjiIcyIouQUEMRjBLNAMhl6Oj1JY89Cd40QmvkXL/6KFw9K8OLBf7GzCJr4oODxXhVV9fxEcAW2/WktLa+srq3nNvKbW9s7u4W9/YaKU0lZncYili2fKCZ4xOrAQbBWIhkJfcGa/sP12G8+Mql4HN3DMGFuSPoRDzglYCSvUCm1dccP8E3W1ZB5Gjw7c7v6NJuoj1mpgnWSCuET+oDjAGe49ON4haJdtifAi8SZkSKaoeYVRp1eTNOQRUAFUart2Am4mkjgVLAs30kVS8we0mdtQyMSMuXqyY8ZPjZKDwexNBUBnqi/JzQJlRqGvukMCQzUvDcW//PaKQRXruZRkgKL6HRRkAoMMR4HhntcMgpiaAihkptbMR0QSSiYWPMmBGf+5UXSOCs7F2X77qxYPZ/FkUOH6AidIAddoiq6RTVURxQ9oRf0ht6tZ+vVGlkf09YlazZzgP7A+voGmFWjig==

0. While the phase-space structure of such ODEs was well understood in the mathematics literature by the 1980s, the fluid mechanics 2

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community first learned about these developments in a seminal work by Hassan Aref (1984); see also Aref (2002) for a historical perspective. Aref’s intent was to demonstrate that even simple (planar and time-periodic) flows can produce complicated conservative tracer patterns in a process he termed chaotic advection. Yet, in studies of such flows, the term transport barrier soon emerged for curves that visibly blocked any exchange of fluid trajectories in stroboscopic images taken at multiples of the period T (see Fig. 4.2).

Figure 4.2 Apparent transport barriers in a stroboscopic, T-periodic sampling of fluid trajectory positions in Aref’s blinking vortex flow. Adapted from Aref (1984).

A more specific definition of these observed barriers to transport was apparently first given by Mackay et al. (1984), who used the term transport barrier for closed curves that remain steady (or invariant) under iterations of the period-T map (or Poincaré map; see §2.2.12) associated with the ODE (4.1). In the same reference, the term partial transport barrier appears in reference to non-closed invariant curves of the time T-map that do not block but slow down the transport of initial conditions between two specific spatial domains. A perfect barrier could indeed be defined as a closed invariant curve of the Poincaré map but many experimentally observed barriers to transport are not closed, as we will see shortly. Indeed, the apparent mixing region in Fig. 4.2 turns out to contain a wealth of non-closed invariant curves of the time-T map that have a profound effect on shaping passive tracer patterns in this region. This prompts us to introduce the following definition: Definition 4.1 An (advective) transport barrier of a temporally T-periodic velocity field v(x, t) = v(x, t + T) is a structurally stable invariant curve for some iterate of the period-T map of v(x, t). By this definition, transport barriers are not necessarily invariant curves under a single iteration of the Poincaré map. Instead, they may be invariant curves only for some higheriterate of the Poincaré map. Examples include the stable or unstable manifolds of period-2 fixed points of saddle type, which are only invariant curves of for the second iterate of the period-T map. For physical relevance, Definition 4.1 also requires a transport barrier to be a structurally stable (see §2.2.7) invariant curve of an iterate of the period-T map. This is to ensure the smooth persistence of the barrier under small perturbations and uncertainties

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in the underlying flow, which are always unavoidable in practice. The physically meaningful perturbations and uncertainties to the flow are generally restricted to volume- or at least masspreserving disturbances, which result in area-preserving perturbations to the period-T map. Such area-preserving perturbations allow for a larger class of structurally stable invariant curves for the period-T map, including the closed invariant curves seen in Fig. 4.2. We recall that the computation of the Poincaré map for a general 2D, time-periodic vector field is implemented in Notebook 2.1. When advected by the full flow map Ftt0 as a material line, the advective transport barriers defined in Definition 4.1 reassume their initial positions periodically in time. This remarkable resilience or material coherence makes these barriers very special among all other material lines, which typically evolve into ever more complicated shapes and continually develop small-scale filaments. For this reason, advective transport barriers in time-periodic flows provide the simplest examples of LCSs, a more general advective barrier concept that we mentioned in §1.3 and will discuss further for nonrecurrent flows in Chapter 5. Next, we look more specifically into what types of invariant curves a Poincaré map may admit. A fixed point p is the simplest element of the dynamics of Pt0 , marking a T-periodic trajectory of the velocity field v. The linear stability of such a periodic trajectory depends on the linearized Poincaré map

ξ 7→ ∇Pt0 (p) ξ

(4.2)

at p, with ξ denoting infinitesimally small displacement vectors emanating from p. By the area-preserving property (2.76) of Pt0 , the product of the two eigenvalues λ1,2 of the Jacobian ∇Pt0 (p) must be equal to one. The eigenvalue configuration, therefore, falls in one of the four categories shown in Fig. 4.3. hyperbolic fixed point e2

elliptic fixed point

parabolic fixed point! (nondegenerate)

parabolic fixed point!! (degenerate)

e1

e1 = e 2 λ1 = 1/ λ2 ∈! − { ±1}

λ1,2 = e ± iα 62 R

λ1 = λ2 = 1 e1 = e 2

λ1 = λ2 = 1 e1 ,e 2 ∈!

2

Figure 4.3 The four possible phase portraits of linearized area-preserving Poincaré maps, along with their corresponding eigenvalue configurations: hyperbolic (saddletype) fixed point; elliptic (center-type) fixed point; nondegenerate parabolic fixed point (shear flow with a line of fixed points); degenerate parabolic fixed point (quiescent flow composed of fixed points). The vector ei denotes the real eigenvector corresponding to a real eigenvalue λi .

These linearized phase portraits already contain the three well-documented barrier types to transport in 2D flows. Specifically, the unstable e2 eigenvector of a hyperbolic fixed point on the left in Fig. 4.3 spans a full line, the unstable subspace, of this saddle-type fixed point, observable as a front that attracts tracers. Approaching tracers are divided and directed

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towards different branches of this front due to the presence of the stable subspace of the saddle point, spanned by the stable eigenvector e1 . Paradoxically, therefore, the stable subspace of the saddle point is an unstable (repelling) material line and the unstable manifold is a stable (attracting) material line. The center-type pattern in the second phase portrait in Fig. 4.3 occurs in vortical flow regions that keep tracers from spreading. All trajectories near this elliptic fixed point are confined to closed transport barriers along which they all travel without shear, i.e., with the same angular velocity α/T. The single real eigenvector of a nondegenerate parabolic fixed point in Fig. 4.3 spans a one-dimensional subspace, which often marks a no-slip wall that fully prevents transport in the normal direction. Finally, any curve is a transport barrier in the linearized map at a degenerate parabolic fixed point, as all points are fixed points in this approximation (see Fig. 4.3). Linearized Poincaré maps are, therefore, perfectly well understood but pose a challenge in defining unambiguously what a transport barrier should be. Indeed, any of the infinitely many other curves approaching the unstable subspace of the saddle points also acts as an attracting material line. Similarly, all horizontal lines in the vicinity of a parabolic fixed point block vertical transport of the fluid, not just the one through the fixed point. Finally, any one of the closed curves near an elliptic fixed point keeps the trajectories in their interior from leaking out. This abundance of invariant curves, however, tends to diminish dramatically once we consider the full, nonlinear Poincaré map Pt0 in the next section. As Fig. 4.2 already illustrates, the full Poincaré map Pt0 is the most broadly applicable and simplest tool to detect barriers to advective transport in time-periodic flows. The visibility of these barriers will, however, depend on the choice of seeding points used in the visualization of Pt0 . If these points are chosen too sparsely, then one may miss some barriers. If the initial positions are chosen too densely, one might just see the whole domain filled with a similarly tightly packed set of current positions by the area preservation of Poincaré maps (see, e.g., Fig. 4.4 for an illustration). This limitation in the visualization of practical examples necessitates a more detailed analytical understanding of the types of transport barriers that may arise in time-periodic planar flows. This will, in turn, help us identify these barriers even from sketchy simulations of a Poincaré map.

4.1.1 Hyperbolic Barriers to Transport We start our discussion of nonlinear Poincaré maps by exploring the fate of the stable and unstable subspaces of a saddle-type fixed point p of Pt0 in the nonlinear Poincaré map   ξ 7→ ∇Pt0 (p) ξ + O | ξ | 2 (4.3) near p. By the classic stable manifold theorem (see Guckenheimer and Holmes, 1983), the stable subspace E s of a saddle point of the linearized Poincaré map ∇Pt0 (p) always has a unique nonlinear continuation, W s (p), that is tangent to E s at p and contains all initial conditions that converge to p under forward iterations of Pt0 . The smooth curve W s (p) is an invariant manifold (see §2.2.5), and hence initial conditions never leave W s (p) under iterations of Pt0 . We call W s (p) the stable manifold of p. Similarly, tangent to the unstable subspace of the linearized mapping ∇Pt0 (p) at a saddle point p, there exists another unique

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4.1 2D Time-Periodic Flows 10 × 10 initial grid

25 × 25 initial grid

50 × 50 initial grid

Figure 4.4 Plot of the Poincaré map for the Navier–Stokes velocity field (2.133) for a = 0.5, b = −0.015 and c = −4 at time t = 0, computed over an increasingly dense grid of initial conditions. Note that a denser grid does not necessarily help in identifying barriers to transport, whereas a sparser grid may also miss important barriers.

invariant manifold, the unstable manifold, W u (p). This invariant curve contains all initial conditions that converge to the saddle point under backward iterations of Pt0 . Both W s (p) and W u (p) are advective transport barriers by Definition 4.1. Because they are attached to hyperbolic fixed points, we refer to them as hyperbolic transport barriers. In physical terms, W s (p) and W u (p) are material lines that deform in time but resume their time-t0 positions exactly at times t0 + kT for any integer k. Just as we have noted for the stable and unstable subspaces of the linearized Poincaré map, the mathematical terms “stable and unstable manifolds” are perhaps misnomers. They refer to the – experimentally unobservable – internal dynamics of W s (p) and W u (p), while the observable external dynamics of these curves is of just the opposite type: stable manifolds repel nearby fluid elements while unstable manifolds attract them. Stable manifolds can neither intersect themselves nor other stable manifolds, as that would imply two different types of forward asymptotic behavior for the initial conditions lying in their intersection. Similarly, unstable manifolds cannot intersect themselves or other unstable manifolds. Stable manifolds may, however, intersect the unstable manifolds of the same or of other fixed points. By the invariance of these manifolds, one intersection between them implies infinitely many other intersections. For weakly unsteady flows, the intersections of stable and unstable manifolds can be analytically predicted using the Melnikov method (see Guckenheimer and Holmes, 1983). The first application of this method to fluid flows was apparently given by Knobloch and Weiss (1987). Further examples and relevant discussions can be found in Wiggins (1992); Samelson and Wiggins (2006); Balasuriya (2016). The infinite sequence of intersection points between W s (p) and W u (p) approaches a hyperbolic fixed point both in forward and backward time, causing an accumulation of stable manifolds on themselves or on other stable manifolds by the classic lambda lemma of Palis (1969). The resulting stretching and folding creates a highly complicated structure, a homoclinic or heteroclinic tangle, as shown in Figs. 4.5(a)–(b). These plots also show the primary intersection points r, r1 and r2 between the invariant manifolds, which then give rise to infinitely many other intersection points in the homoclinic

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heteroclinic tangle

homoclinic tangle e2

lobe dynamics

W s (p)

W u (p)

p

W s (p)

e1

W u (p)

W s (q) W u (p)

p

r1



p

q

r2



W u (q) W s (p)



374

r

Pt (Lin ) 0

L

} turnstile

out W.-L.Chien, 11.Rising and J . M . Ottino

i PI t (Lout )

Lin

I

(a)

(a)

(b)

I i

0

(c)

1

I 374 W.-L.Chien, 11.Rising and J . M . Ottino Figure 4.5 (a) Homoclinic tangle formed by the stable and unstable manifolds of a hyperbolic fixed point p of the Poincaré map P i t0 , with a primary intersection point r. (b) Heteroclinic tangle formed by the stableI and unstable manifolds of two hyperbolic I fixed points, p and q of the map. (c) Forward lobe Lin under (a) i iterations of the external (b) the map Pt0 enter the reference region D, while forward iteration of the internal lobe Lout leave D. These two lobes together form I a turnstile. 1 I

W.-L.Chien, 11.Rising and J . M . Ottino

374

i

I

I I tangles.

and heteroclinic (a) shown in Fig. 4.6. W.-L.Chien, 11.Rising and J . M . Ottino

374

i

An experimental demonstration of such (c)a homoclinic tangle is (b)

1

I

i

I

(a)

I I

1

I (a)

(b)

(c)

(d)

(c)

(b)

i

(b)

(c)

FIGURE 12. Same as figure 10 but changing the location of the blob.

(d)

(d)transverse intersections in a 2D cavity flow Figure 4.6 Experimentally constructed (c) with periodically moving walls. (a) Initial dye blob position. (b) The deformed dye blob after a few forward periods. (c) Same as in (b) but superimposed on the deformations of the same blob under a reversal of the wall motion to mimic backward-time FIGURE 12. Same as figure 10 but changing the location of the blob. evolution. (d) Magnified region showing the transverse intersection of stretched slabs surrounding the stable and unstable manifolds of a saddle-type fixed point of the Poincaré map.(d) Adapted from Chien et al. (1986).

Manifold segments connecting primary intersection points to the underlying hyperbolic FIGURE 12. Same as figure 10 but changing the location of the blob. fixed points enable one to define the reference regions labeled as D in Fig. 4.5. Material transport in and out of these regions has been studied extensively (see, e.g., Mackay et al., (d) 1984; Ottino, 1989; Rom-Kedar and Wiggins, 1990; Rom-Kedar et al., 1990; Wiggins,

FIGURE 12. Same as figure 10 but changing the location of the blob.

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4.1 2D Time-Periodic Flows

105

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The computation of the backward transformations from the forward transformations brings us to an important point which appears in both experimental and computational aspects of the problem: can a picture such as figure 10(b) be unscrambled and restored to its initial configuration, figure 10(a)? Obviously in the FIGURE 9 ( u,b ) . For caption see facing page.

Subplots (a)–(c) of Fig. 4.8 show snapshots of the dye concentration at multiples of the forcing period, illustrating the gradual convergence to a recurrent pattern that slowly fades drastically reduce the striation width that the mixing influences only a small part of the fluid. Also, although the ( 1 , 2 )horseshoe is qualitatively a better mixing device out due to diffusion of the dye. Subplot (d) shows the same pattern for a different choice of than the ( 1 , -2) horseshoe, its development may coincide with loss of good mixing in other parts of the cavity. the phase t0 in the Poincaré map Pt0 . This pattern may look substantially different, but can The computation of the backward transformations from the forward transformato an the important point emerging which appears in in both experimental tions brings us into still be smoothly deformed pattern (a)–(c) due toandthe conjugacy property, computational aspects of the problem: can a picture such as figure 10(b) be its initial configuration, figure 10 (a) ? Obviously in the unscrambled and restored to (2.75), of Poincaré maps. FIGURE 9 ( u,b ) . For caption see facing page. Earlier numerical observations by Pierrehumbert (1994) had already revealed such recurdrastically reduce the striation width that the mixing influences only a small part Also, patterns although the (in 1 , 2 weakly )horseshoe isdiffusive qualitatively atracer better mixing device in a time-periodic velocity field. Pierrehumof the fluid. rent mixing than the ( 1 , -2) horseshoe, its development may coincide with loss of good mixing in other parts of the cavity. bert called these tracer patterns strange eigenmodes, i.e., recurrent, spatially complex tracer DF, /C B C 898 :EC

CB 6E 8 9 CE 7CE9 042 - 6 C 9 7 DF,

The computation of the backward transformations from the forward transformations brings us to an important point which appears in both experimental and computational aspects of the problem: can a picture such as figure 10(b) be unscrambled and restored to its initial configuration, figure 10(a)? Obviously in the IGURE 9 (u,b ) . For caption see facing page. /C B C 898 :EC

FIGURE 9 ( u,b ) . For caption see facing page.

Figure 4.7 Evolution of a dye blob drastically in a time-periodic cavity flow, stroboscopically reduce the striation width that the mixing influences only a small part Also, although the ( 1field. , 2 )horseshoe is qualitatively a better mixing device of the fluid. sampled at multiples of the time period of the velocity Image adapted from than the ( 1 , -2) horseshoe, its development may coincide with loss of good mixing Chien et al. (1986). in other parts of the cavity.

7

196

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1992; Rom-Kedar, 1994; Samelson and Wiggins, 2006; Balasuriya, 2016). The fundamental vehicle for this transport is a turnstile, composed of lobes formed by segments of stable and unstable manifolds between adjacent points of their intersection, as shown in Fig. 4.5(c). One of these lobes, Lin , enters the material reference zone D under iterations of the map, whereas the other lobe, Lout , leaves D. For area-preserving Poincaré maps (i.e., for incompressible flows), the two lobe areas (as well as the areas of their forward and backward iterates) are equal. For homoclinic tangles, Smale (1967) proved the existence of a nearby chaotic invariant set. Restricted to that set, the Poincaré map is chaotic: it behaves like a realization of a Bernoulli shift map, a mathematical model random coin 370 for an infinite, W . - L .Chien, H . Rising and Jtossing . M . Ottino experiment. This remarkable fact has fascinated the applied mathematics and physics community, creating a new paradigm that even simple, time-periodic velocity fields can generate immensely complicated trajectories. The set of chaotic trajectories near a homoclinic tangle, however, is unobservable in incompressible flows: it is of measure zero and unstable, repelling all nearby fluid trajectories.370 Thus, typical Wfluid trajectories vicinity will only experience transient . - L .Chien, H . Rising and in J . Mits . Ottino chaos (Lai and Tél, 2011). If the chaotic set is practically invisible, what governs the formation of tracer patterns in chaotic advection? Shown in Fig. 4.7, a classic experiment by Chien et al. (1986) indicates a complex but asymptotically steady pattern in the stroboscopic images of a dye blob in a time-periodic cavity flow. The dye is visibly attracted to a convoluted material line, forming W . - L .Chien, H . Rising and J . M . Ottino a front that is a de facto barrier to dye transport. The stroboscopic snapshots of the dye distribution gradually align with the material barrier, which is unsteady in time but resumes the same position when viewed at multiples of the flow period. Once the alignment is tight enough, there is virtually no change in the dye configuration, as was also observed in a later experiment by Rothstein et al. (1999), shown in Fig. 4.8.

triation width that the mixing influences only a small part gh the ( 1 , 2 )horseshoe is qualitatively a better mixing device hoe, its development may coincide with loss of good mixing vity. he backward transformations from the forward transformahttps://doi.org/10.1017/9781009225199.005 Published online by Cambridge University Press mportant point which appears in both experimental and

is omitted, on the other hand, two-dimensional weakly turbulent flows can be created11. The fluid is illuminated near 365 nm, and the fluorescent response, imaged by a cooled 12 bit CCD camera operating at 512 ! 512 pixel resolution, is accurately linear in the local concentration under the experimental conditions10. The camera is trigBarriers to Chaotic Advection gered to sample the dye distribution every period or half period. An example of the mixing process is shown in Fig. 1 for the case of time-periodicin forcing 100 mHz after 20 periods, using spatially gradually timeat due to the diffusion of athe tracer. A formal proof for periodic magnet array. The dye solution, initially confined to the existence of strange eigenmodes in tracers mixed by time-periodic velocity fields was ification and magnetic forcing are used to create a the right half of the image, is progressively stretched and folded, al flow with good accuracy. A layer of glycerol– given by Liu, W., and Haller (2004). tion 3 mm deep carries a horizontal electric current of an array of strong permanent magnets just below a b rdered square array with alternating polarity, and a ered array (with magnets having random locations

ye (the passive scalar). We observe the formation of variant but slowly decaying mixing patterns, and the various statistical properties that characterize tration field evolve with time as mixing proceeds cycles. These results show quantitatively how ching of the fluid elements and molecular diffusion 106 to produce mixing of the impurity. We contrast the time-period; c flows and identically forced but nt flows at lower viscosity, wherethat mixing is much patterns fade out

c

of the concentration field for transient magnetically forced chaotic d with fluorescein was initially confined to the right half of the e imaged region is 9:4 ! 9:4 cm. The light intensity under ultraviolet n after 20 periods of forcing by a time periodic electric current at sence of a spatially periodic magnet array. The repetitive stretching haracteristic of chaotic advection is evident.

d

Figure 2 Attainment of a persistent mixing pattern using a disordered array of forcing

magnets. The impurity field develops a complex structure after about 10 periods that Figure 4.8 Strange eigenmode forming in a dye layer over a conducting fluid that is repeats periodically, except for a gradual loss of contrast. a–c, t ¼ 2, 20 and 50 periods, forced time-periodically from below by an irregularly placed array of magnets. The respectively. d, Sampling out of phase at 50.5 periods shows that the tracer distribution changes substantially within same each cycle. The forcing is 70 mHz. period, whereas the plot snapshots (a)–(c) are taken at the phase offrequency the forcing (d) depicts the dye at a different phase in that period. Adapted from Rothstein et al. © 1999 Macmillan Magazines Ltd NATURE | VOL 401 | 21 OCTOBER 1999 | www.nature.com (1999).

As we noted in connection with Fig. 4.4, Poincaré maps may not necessarily provide enough detail to identify the material barriers delineating strange eigenmodes. Indeed, the experimentally constructed map in Fig. 4.9 shows some of the hyperbolic fixed points but indicates little of the global geometry of their stable and unstable manifolds. We will develop methods in Chapter 5 that specifically target generalizations of such invariant manifolds based on their impact on fluid trajectories in velocity fields with general time dependence. Armed with some of those methods, we will revisit the flow shown in Figs. 4.8–4.9 to uncover the transport barriers delineating strange eigenmodes as LCSs directly from the velocity data. As a preview of the results one can obtain from such methods, we show in Fig. 4.10 stable and unstable manifolds extracted as LCSs from the same experimental data.

4.1.2 Elliptic Barriers to Transport Two-dimensional, nonlinear, area-preserving maps have analogues of the elliptic regions we identified for linearized Poincaré maps in Fig. 4.3. While adding nonlinearities to the linearized map tends to destroy a large number of individual closed transport barriers, most of the barriers of the linear map persist in a deformed shape as long as the nonlinear map is a twist map, as we will discuss next.

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concentration field is shown at the same phase we dye discuss the correspondence between the lines Thus, we have the ween and 200. The second Now The mHz,10and typical in Fig. 3(b). In Fig. 3(c), we superimpose the past stretchh length p ! UT!L in one displacements. which therepoints is no net motion over of the stretchingPoints fields at and the fixed of the Poincaré various objects: l periments, half of ing field on the dye image. The dye visualization and parypically in the range 0.5 to 10. a full period are fixed points. Figure 2(b) shows a 43 enmap. In Fig. 4, we show the future stretching field (blue) manifolds; large p hhefluorescein dye, ticle tracking data are measured in separate experiments at lower end of the ranges of largement oftoparameters. part of 2D the Poincaré map. Thesets fixed Time-Periodic Flows 107 This demon in addition the4.1 past stretching field Many of the folds. V the same We observe that the(red). level (con-points xedillumination elliptic regions isare large, lines) of thebeen concentration linecross up with theand lineshyperin thistour map have marked; both elliptic as Re and p are increased. points where the two sets of field lines correspond to of both the stable entration. Experimental Poincaré maplatter rolled experimentally the fixed fixed pointspoints may bethe seen. The have 2b). one axis hyperbolic of Poincaré map (Fig. The time in complicate as been described by bolic ncy, and the fluid viscosity. of approach (stable manifolds) and one axis of departure lines of the future and past stretching fields label the stable of lobe dynamics nt, concentradic the or weakly turbulent in the (unstable manifolds). and unstable manifolds of these fixed points [15]. Qualflows. An animat which nding on stretching p. Figure 3(a) shows the past stretching field for this flow. itatively, this is because particles along these manifolds the superimposed of one component of the velocway that the patand p ! 1. Both components It reveals many sharp maxima which are organized into form homoclinic despite the strong of time, and an animation may manifolds. Care lines, with much smaller stretching values between them. slow overall two fields shown exin Fig. 1 are possible to distin Determination of the stretching field requires the selection gntsprocess may bethe minibefore and after of a time interval Dtfor the map; here it is three periods. fixed points and th he velocity field. Because one e [17]. ther (due to the nonzero Re), Smaller (or larger) values result in broader (or sharper) return to their init arameters. The etrace their paths and chaotic structures. An animation showing how the past stretching We have also c n the mean magperiodicity of the flow and the FIG. 2 (color). Poincaré map of the flow at one phase of the depends on available on-line U, and kinematicFigurefield 4.9 Poincaré map from tracer particles for the time-periodic flowof Re and p, as sh he forcing. forcing "p ! constructed 1,Dtis Re ! 45#. Lines connect the[17]. experimentally measured initial coordinates of each particle with its same coordiexperiment in Fig. 4.5. (a) Line segments connect particle positions with their positionthemselves are ha The dye concentration field is shown at the phase following fluid elements over 200. The second natesBrighter one period later,mark with blue anddisplacements. red designating(b) small and version of one period later. colors larger Zoomed ows the particle displacements stretching is occu in Fig.large 3(b). In Fig. 3(c), we superimpose the past stretch! UT!L in onethe boxed displacements, respectively. Complete of view. marked in (a). Four hyperbolic(a)fixed pointsfield are marked with crosses; caré map. In these maps, lines area (b) Close-up of the region in the box in (a), showing two elliping field on the dye image. The dye visualization and pare range toparticle 10.two posielliptic fixed points are (circles), marked with circles. Adapted from(crosses), Voth et al. (1994). However, the con ed initial to0.5 final tic fixed points four hyperbolic fixed points invariant manifold ticle tracking data arenear measured in separate experiments at dedofto the showranges the largeofand small and the trajectories them.

tion aligning with particularly drama As Re is increa of period doublin where Re is only velocity field repe At Re ! 125, p onent of the velocDespite the bifurca Both components fields measured in an animation may and the past stretc hown in Fig. 1 are pattern at each inst nd after the miniFIG. 4 (color). Lines of the future (blue) and past (red) stretchFigure 4.10 Stable manifolds (red) and unstable manifolds (blue) identified fromat present be perfo eld. Because onethe data ingused field.in The lines markofthe (blue) and unstable (red) the construction thestable experimental Poincaré map of Fig. 4.9. of the hyperbolic fixed points of the Poincaré map. the nonzero Re),Thesemanifolds hyperbolic transport barriers were constructed from the finite-time Lyapunovmanifolds that org Black circles indicate of theinhyperbolic points. Con(FTLE), a method we some will discuss Chapter 5 forfixed locating transport barriersand numerically o paths and chaoticexponent ditions match Figs. 2 and 3(a)–3(c), and the box outlines the in finite-time velocity data. Black circles indicate hyperbolic fixed points recovered flows [14]. This f f the flow and the 2 shown (color). Poincaré map ofofthethe flow atfrom oneVoth phase of(1994). the region in Fig. 2(b). Some fixed points areet associat the FIG. intersection of blue and red curves. Image adapted al. field plays in mixi forcing ! 1,stretching Re ! 45#.lines. LinesViewing connectthethetime experimentally ated with"pweak dependence measured initial coordinates each insight particleinto with coordiThe probability uid elements over of the manifolds [17] allows of greater theitsdynamics nates one period later, with blue and red designating small and Fig. 5 displays th of the mixing process. cle displacements largeisdisplacements, (a) Complete fieldfixed of view. A twist map a nonlinear maprespectively. whose linear part has an elliptic point, but whose these maps, lines (b) Close-up the variation region ininthe in (a), showing ellip- rotation 254501-3 nonlinear part adds shearof(i.e., thebox angular velocity) to two the shearless final particle generated posi- by ticthe fixed points (circles), four hyperbolic fixed points (crosses), linear part. As long as the angular rotation frequency α of the linear part is he large and small the trajectories near sufficientlyand nonresonant (i.e., α/2π is farthem. enough from an integer) and the nonlinear part of the

regions are large, p are increased. mentally by the e fluid viscosity. y turbulent in the

the same parameters. We observe that the level sets (con254501-2 tour lines) of the concentration field line up with the lines

Poincaré map adds nonzero circular shear, most invariant curves of the linearized mapping 254501-2 deform but survive in a neighborhood of the fixed point (see, e.g., Golé, 2001). Those persisting invariant curves of the map are usually referred to as Kolmogorov–Arnold–Moser (KAM) curves. They are transport barriers by Definition 4.1, and they are often referred to as elliptic barriers to advective transport because they generally encircle elliptic fixed points.

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Barriers to Chaotic Advection

Two such elliptic barriers are clearly visible in the blinking vortex flow shown in Fig. 4.2, but they generally occur in concentric families. The outermost member of such a family is the most influential of these barriers: its interior defines a coherent material vortex. While this vortex is steady under iterations of the Poincaré map, it will become a periodically deforming material blob in continuous time, resuming its original shape at multiples of the time period T of the flow. Other examples of elliptic barriers in a twist map can be found in the Poincaré map of the velocity field (2.133) for b , 0. Figure 4.11 shows how a family of closed, nested, elliptic transport barriers gradually breaks up as b grows in norm. Note the creation of elliptic islands from the remnants of invariant curves in annular regions where the rotation frequency is in resonance with the forcing. b=0

b = −0.03

b = −0.05

Figure 4.11 Poincaré map for the Navier–Stokes velocity field (2.133) for a = 0.5 and c = −4, based at time t = 0 for various values of the parameter b. In each case, 150 iterations of the same initial grid of 20×20 particles are shown; the instantaneous streamlines are shown in the background for reference. The linear Poincaré map at b = 0 becomes a twist map for b , 0. The contribution from the nonlinear terms grows with the norm of b, gradually breaking up the closed transport barriers of the linear limit. Simultaneously, the whole elliptic region shrinks in size.

Mackay et al. (1984), Meiss (1992) and Golé (2001) provide more detail on the smallerscale structures that form full or partial barriers to advective transport in area-preserving twist maps. These include resonance islands that are smaller invariant regions created at locations where the primary KAM curves would have a rational rotation frequency α. In these resonance regions, the primary KAM curves break down into discrete elliptic islands of smaller closed invariant curves. Within each resonance island, a microcosm of the full large-scale phase portrait is recreated, including yet smaller-scale KAM curves and their resonant islands, as well as smaller-scale chaotic mixing zones created by intersections of stable and unstable manifolds of periodic orbits. All this implies a beautiful, nested selfsimilarity of transport barriers in typical twist maps and hence in typical time-periodic, incompressible fluid flows, as shown in Fig. 4.12. Figure 4.12 also shows an indications of an Aubrey–Mather set, also called a cantorus (Golé, 2001). First proposed by I. Percival and S. Aubry, then proven mathematically by

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clouds of points (chaos and diffusion due to intersecting stable and unstable manifolds of hyperbolicperiodic orbits). Wealso see some “broken” circlesmade of dashedlines (Cantori or Aubry-Mather sets).

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4.1 2D Time-Periodic Flows

-

K=0.971

K=0.817

Fig. 1.1. The different dynamics in the standard map: the left hand side shows a selection Figure 4.12 Phase portrait of the classic standard map (x , yn+1 ) = The right hand side of orbits for the completely integrable F0 , all on invariant circles. n+1 (xn + yn+1, yn + K sin xn ) for two values of the parameter K. This area-preserving displays orbits for F! with ! = .817.

twist map shows the typical structures that arise in Poincaré maps of time-periodic, planar and incompressible flows. (Left) A few representative trajectories with all points along the same trajectory shown in the same color. Source: Wikipedia. (Right) A smaller set of trajectories illustrates different types of transport barriers and an apparent partial barrier (cantorus or Aubrey–Mather set). Adapted from Golé (2001).

S. Aubry, J. Mather and co-workers, a cantorus is an invariant Cantor set of points, as opposed to a classic continuous invariant curve (see Mackay et al., 1984 for references and further details). This invariant set is reminiscent of a KAM curve with infinitely many small holes. These holes enable some trajectories to pass across the cantori, but this passage is exceedingly slow. Since they are not continuous curves, cantori are technically not transport barriers by Definition 4.1 but nevertheless act as partial barriers to advective transport (see Mackay et al., 1984). Any elliptic transport barrier in a Poincaré map represents the cross-section of an invariant torus for the flow in the extended, 3D phase space of the variables (x, y, t mod T). In Fig. 4.13, we illustrate this geometry for two nested KAM curves and for an invariant curve in a period-six elliptic island. This set of islands lies in an annulus where no KAM curves exist due to a 1 : 6 resonance between the local rotation frequency in the flow and the frequency 2π/T of the T-periodic velocity field. A closed curve in the period-6 resonant island of Fig. 4.13 is an invariant curve of the 6th iterate of the period-T map, giving rise to the green toroidal transport barrier shown in the figure for the full flow. Other barriers and partial barriers (cantori) of the Poincaré map can be similarly extended into barriers and partial barriers in the full, time-periodic fluid flow. Transport barriers in plasma physics are often modeled via 2D Poincaré maps defined by first returns to a selected section of a tokamak. There is ample experimental evidence for the existence of elliptic transport barriers in these flows (see, e.g., Tala and Garbet, 2006), which in turn has inspired a substantial body of related research involving 2D symplectic maps (see, e.g., Viana et al., 2021).

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y t modT

x

Figure 4.13 Closed curves of the period-T Poincaré map of a planar flow give rise to invariant tori filled by fluid trajectories in the extended phase space of (x, y, t mod T). The nested blue tori arise from KAM curves of the Poincaré map. The green torus arises from a closed invariant curve of the map that lies in a period-6 resonant island. Image: courtesy of Brendan Keith.

4.1.3 Parabolic Barriers to Transport The third and fourth possible local phase portraits in Fig. 4.3 for a linearized Poincaré map Pt0 are nondegenerate and degenerate parabolic flows. Nondegenerate parabolic points typically occur near no-slip walls, which then appear as invariant curves for Pt0 . These curves also remain barriers under the addition of nonlinear terms because nonlinearities are also physically constrained to satisfy the same no-slip boundary condition. Therefore, these invariant curves qualify as advective transport barriers by Definition 4.1. In contrast, degenerate parabolic fixed points typically arise in the linearization of the Poincaré map on shearless curves marking jet cores. Indeed, particle motion in the simplest steady, horizontal jet with a shearless core would satisfy xÛ = −y 2, yÛ = 0 in a frame co-moving with the core of the jet along y = 0. For an arbitrary T, the period-T map P0 (x0 ) and its
∗ ∗ ∗ derivative, ∇P0 (x0 ), at a fixed point x0 = x0 , 0 along the jet core are given by P0 (x0 ) =



x0 + y02T y0

 ,

∇P0 (x∗0 )

=



1 0 0 1

 ,

with ∇P0 (x∗0 ) indeed generating the linearized phase portrait shown in the right-most subplot of Fig. 4.3. This phase portrait, however, is structurally unstable: the two-parameter family of fixed points of ∇P0 (x∗0 ) generically breaks up under the inclusion of nonlinear terms in P0 . By a parabolic barrier in nonlinear Poincaré maps away from fixed points, we will mean invariant curves of the Poincaré map that are nonlinear continuations of nondegenerate parabolic barriers of the linearized map ∇P0 at a fixed point. Such barriers frequently arise in planar, area-preserving maps near closed curves where the twist condition (see §4.1.2) fails to hold. While classic KAM curves cannot be guaranteed to exist along non-twist curves, numerical observations of Samelson (1992); del Castillo-Negrete et al. (1996) and others reveal an abundance of closed nearby barriers that appear to be even more robust under perturbations than classic KAM curves (see also Fig. 4.14). The reason for this strong KAM stability, as argued by Rypina et al. (2007), is that resonance regions responsible for the destruction of KAM curves are thinner near shearless curves.

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4.2 2D Time-Quasiperiodic Flows

111

Figure 4.14 Dynamics near a closed parabolic barrier (spatially
periodic jet
core) 2 , yn − b sin xn with in the standard non-twist map (xn+1, yn+1 ) = xn + a 1 − yn+1 a = 0.615 and b = 0.4. This map is a prototype model for planar area-preserving Poincaré maps with a shearless curve at y = 0 (see del Castillo-Negrete et al., 1996). The parabolic barrier arises as a centerpiece of a family of meandering elliptic barriers (KAM curves). Different colors mark different trajectories. Image: courtesy of George Miloshevich.

Delshams and de la Llave (2000) prove the existence of closed parabolic transport barriers near non-twist locations. In a practical situation, one could, in principle, detect these barriers from the period-T Poincaré map of the flow, just as one visualizes hyperbolic and elliptic barriers via the iteration of this map. The plot of the map will, however, generally contain an abundance of elliptic barriers and hence the parabolic barrier serving as the core of a jet is often not immediately visible, as illustrated in Fig. 4.14. For this reason, in Chapter 5 we will discuss methods that identify generalized jet cores in general unsteady flows. Examples of parabolic barriers revealed by these methods include the shearless boundary of the ozone hole and the cores of zonal jets in 2D model flows of planetary atmospheres (Beron-Vera et al., 2008b, 2010).

4.2 2D Time-Quasiperiodic Flows Consider a 2D quasiperiodic velocity field xÛ = v(x, Ω1 t, . . . , Ωm t),

x ∈ R2,

(4.4)

with m rationally independent components in the frequency vector Ω = (Ω1, Ω2, . . . , Ωm ) ∈ Rm . As we discussed in §2.2.12, the ODE (4.4) can be viewed as an extended, 2π/Ω1 -periodic system of the form xÛ = v(x, Ω1 t, φ), Ω2 .. ª® . (4.5) . ® « Ωm ¬ This extended dynamical system is defined on the (2 + m − 1)-dimensional phase space of the (x, φ2, . . . , φm ) variables, where φ j = Ω j t with j = 2, . . . , m. © φÛ = ­­

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As described in §2.2.12, we can define a Poincaré map with period T = 2π/Ω1 for the extended system (4.5) as Pt0 : R2 × Tm−1 → R2 × Tm−1, (x0, φ20, . . . , φm0 ) 7→ (x (t0 + T; t0, x0, φ0 ) , φ20 + Ω2T, . . . , φm0 + ΩmT) ,

(4.6)

where φ0 = (φ20, . . . , φm0 ) is the initial phase vector at time t0 associated with the remaining frequencies. This map is volume preserving precisely when the velocity field v is incompressible. Transport barriers for the map Pt0 in the mapping (4.6) can then be defined as mdimensional invariant surfaces for the map Pt0 within the (m + 1)-dimensional Poincaré section R2 × Tm−1 . The intersections of these barriers with the 2D planes R2 × {φ0 } give physically observable material curves for the original, 2D dynamical system (2.77). These material curves are special in that they deform quasiperiodically as the time t varies in the phase vector φ = (Ω1 t, Ω2 t, . . . , Ωm t). While this is helpful qualitative information about the possible advective transport barriers in quasiperiodic velocity fields, it is usually more expedient to locate these barriers by coherent structure methods that treat system (2.77) as a planar system with general time dependence. We will discuss such methods for temporally aperiodic systems in Chapter 5. An alternative approach to studying advective transport barriers in the quasiperiodic flow (4.4) could be based on time-averaged scalar fields (observables) using Birkhoff’s ergodic theorem (see §2.2.14). As we indicated in our general discussion of quasiperiodic velocity fields in §2.2.12, however, Birkhoff’s ergodic theorem does not apply in this context. To see this in more detail, we reconsider the map Pt0 in Eq. (4.6) on the Poincaré section Σt0 = R2 × Tm−1 . We assume that the velocity field (4.4) is volume or mass preserving and hence Pt0 is measure preserving on Σt0 . If Pt0 is ergodic on a compact subset S ⊂ Σt0 , then by the Birkhoff ergodic theorem (see §2.2.14), for any integrable scalar field c(x, φ) defined on S, we have the equality ˆ N
1 1 Õ k c(x, φ) dµ c Pt0 (x0, φ0 ) = lim N →∞ N µ(S) S k=0

(4.7)

for almost all (x0, φ0 ) ∈ S. Recall, from §2.2.14, that only the initial conditions obeying the phase relationship (2.81) in the extended system represent actual solutions of the original quasiperiodic flow (4.4). This restricted set of relevant initial conditions can, in principle, be in the measure zero set that is not covered by the equality of temporal and spatial averages in Eq. (4.7), and hence the latter formula has no immediate implication for the original quasiperiodic velocity field. Despite this fact, Malhotra et al. (1998) propose to interpret stroboscopic images of scalar fields sampled at the period T = 2π/Ω1 using Birkhoff’s ergodic theorem. Specifically, they assert that for almost all initial conditions X0 = (x0, 0) taken from an ergodic invariant set S of the extended flow map F t of system (4.5), we will find ˆ 1 t c¯t (x0 ) := c (x(s, x0 ), Ωs) ds → const. (4.8) t 0

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4.3 2D Recurrent Flows with a First Integral

113

as t → ∞. Based on this assumption, they suggest that for increasing t, level sets of c¯t defined as   ˆ 1 2 m Ct (S) = X ∈ R × T : c¯t (x0 ) = c(X) dµ µ(S) S will asymptotically highlight ergodic invariant sets S. Strictly speaking, this statement would actually be the converse of Birkhoff’s ergodic theorem which is generally not true. Indeed, asymptotically constant level sets of c¯t (x0 ) do not necessarily delineate ergodic invariant sets of the flow. For instance, c¯t (x0 ) = const. may hold for a set of initial conditions simply because they have not left a region in which the scalar c(x, Ωt) has a constant value. Also note that c¯t (x0 ) → const. will certainly hold on points of all the (non-ergodic) stable manifolds of fixed points, periodic orbits and quasiperiodic invariant tori of the flow. This effect, however, is unrelated to ergodic theory, as the flow map is not ergodic on stable manifolds. Rather, the effect follows from the continuity of the observer field c(x, Ωt), whose average along trajectories in such a stable manifold will converge to its average taken over the target set (a fixed point, a periodic orbit or an invariant torus) that these trajectories approach. Stable manifolds in incompressible flows, however, have measure zero. As a result, numerical computations on a finite grid will not be able capture the time-asymptotic constancy of c¯t (x0 ) on stable manifolds. One may nevertheless look for indications of mixing regions and barriers around them by: (1) releasing trajectories of the quasiperiodic velocity field (4.4) at time t = 0 (or at a fixed initial time t = t0 ) from an initial grid of x0 points; (2) averaging a scalar field c(x, Ωt) along these trajectories; and (3) identifying constant level sets of the resulting function c¯t (x0 ) at times tk = t0 + kT = 2πk/Ω1 for k = 1, 2, . . .. The times required for the convergence of temporal averages to spatial averages can be long, which may lead to a deterioration of the quality of the plot of c¯tk (x0 ) over time. This is especially true along an envisioned ergodic set S that has a dimension less than the dimension of the Poincaré section Σt0 . As a result, points of the initial condition grid are contained in S with probability zero. Figure 4.15 shows the results from this visualization procedure, obtained by Malhotra et al. (1998), for a planar quasiperiodic velocity field with two frequencies. The observable field chosen by these authors was simply c(x, Ωt) = v(x, y, Ω1 t, Ω2 t), the y-component of the velocity field v of a quasiperiodically disturbed steady meandering jet model. As anticipated, the plots suggest both mixing regions and hyperbolic transport barriers formed by stable manifolds of quasiperiodic tori with incommensurate frequencies. Note that unlike all other results discussed in this chapter, selecting c(x, Ωt) to be a velocity component renders this visualization procedure frame dependent even if the Euclidean frame change family (3.5) is restricted to be T-periodic in time.

4.3 2D Recurrent Flows with a First Integral Two-dimensional steady incompressible flows have a stream function ψ(x), whose level curves (stream lines) contain all trajectories, as we have inferred from the conservation law (2.5). This set of level curves gives the complete family of advective transport barriers for such a steady flow by Definition 4.1. Indeed, trajectories of a steady flow are precisely the invariant curves of any time-T map associated with the flow.

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Barriers to Chaotic Advection

1077 Fig. 18. Average y-velocity contours corresponding to initial points having the same time-average along the tracer trajectories at eight different time intervals. Th e jet parameters are fixed at B = 1.2, k = 0.1, c = 0.1, ω = (0.3, π), γ = (0.5, 0.86), δ = (3.5, π), and ε = 0.1.

1078

Int. J. Bifurcation Chaos 1998.08:1053-1093. Downloaded from www.worldscientific.com Int. J. Bifurcation Chaos 1998.08:1053-1093. Downloaded from www.worldscientific.com by SWISS FEDERAL INSTITUTE OF TECHNOLOGY ZURICH (ETH) on 09/02/20. Re-use and distribution is strictly not permitted, except for Openby Access articles. SWISS FEDERAL INSTITUTE OF TECHNOLOGY ZURICH (ETH) on 09/02/20. Re-use and distribution is strictly not permitted, except for Open Access articles.

114

Fig. 18.

(Continued )

Figure 4.15 Contours of the c¯t (x0 ) function computed by averaging the vertical velocity component c(x, Ωt) = v(x, y, Ω1 t, Ω2 t) along trajectories of a planar, meandering jet under quasiperiodic disturbances. The times shown are multiples of T = 2π/Ω1 . Level sets of c¯t (x0 ) suggest ergodic mixing regions and highlight stable manifolds of saddle-type quasiperiodic invariant tori. Adapted from Malhotra et al. (1998).

General unsteady flows also admit conserved quantities (or first integrals), but those integrals generally do not help in identifying transport barriers. For instance, for any unsteady velocity field v(x, t) with a flow map Ftt0 , we can select an arbitrary scalar-valued, smooth function g(x) and obtain a time-dependent first integral of the form

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115

4.3 2D Recurrent Flows with a First Integral
G(x, t) = g Ftt0 (x) ,

(4.9)

given that
d D d G (x(t; t0, x0 ), t) = g Ftt0 (x(t; t0, x0 )) = g (x0 ) ≡ 0. Dt dt dt This means that even chaotic flows have infinitely many time-dependent conserved quantitates and hence their trajectories are typically confined to 2D level sets of these first integrals. Those level sets, however, generally become arbitrarily complicated, temporally aperiodic level surfaces in the extended phase space of the (x, t) variables, as illustrated in Fig. 4.16(a). While different level sets may not intersect, they may still stretch and fold to an arbitrary degree, as well as accumulate on themselves and each other asymptotically in time. Therefore, unlike first integrals of steady flows, first integrals of general unsteady flows do not a priori impose any restriction on the flow and hence cannot be used to identify transport barriers.

(t0 , I0 ) y

y

(t, I0 )

(t0 , I0 )

(t0 + T, I0 )

I(x, t) = I0 I(x, t) = I0

t t0

t

x

t t0

(a)

x

t0 + T (b)

Figure 4.16 (a) A nondegenerate first integral I(x, t) (i.e., one with ∇I(x, t) . 0) of a general unsteady velocity field imposes no a priori topological restriction on the complexity of the trajectories. (b) A nondegenerate first integral I(x, t) of a temporally T-periodic velocity field confines particles to periodically varying surfaces of limited complexity.

4.3.1 Barriers from First Integrals in Time-Periodic Flows In contrast to the general, temporally aperiodic case illustrated in Fig. 4.16(a), if I(x, t) = I(x, t + T) is the first integral of a time-periodic 2D velocity field v(x, Ωt) with frequency Ω = 2π/T, then the level curves  γ(t, I0 ) := x ∈ R2 : I(x, t) = I0 = const. = γ(t + T, I0 ) periodically recur in the x-plane, as shown in Fig. 4.16(b). If, in addition, such a level curve is compact (i.e., closed and bounded) and nonsingular (i.e., ∇ I(x, t) does not vanish on it), then γ(x, t; I0 ) must be a smooth closed curve by the implicit function theorem (see §A.1).

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If γ(t, I0 ) is a closed curve for all t ∈ R, then the family of level curves, I0 = {γ(t, I0 )}t ∈R , is a 2D time-periodic surface in the (x, t)-space. Topologically, I0 is the product of two circles, i.e., a 2D torus that is invariant under the flow generated by v(x, t). Therefore, with the help of the periodic phase variable φ = Ωt, we conclude that closed and nonsingular level curves of a first integral I(x, t) confine trajectories of the extended dynamical system xÛ = v(x, φ), φÛ = Ω

(4.10)

to 2D invariant tori. In other words, a velocity field v(x, Ωt) with a T-periodic, nondegenerate first integral I(x, t) is integrable: its trajectories within nonsingular and bounded level sets of I(x, t) are confined to invariant tori, and hence are either periodic or quasiperiodic in time. This is also the conclusion of the Brown–Samelson theorem (Brown and Samelson, 1994) obtained using a different approach. The intersection of each of these tori with a t = t0 Poincaré section is a closed invariant curve for the Poincaré map Pt0 (see Fig. 4.16(b)), and hence each closed, nondegenerate level set of I(x, t) is an advective transport barrier by Definition 4.1. Nonsingular level sets of time-periodic first integrals in 2D flows, therefore, contain periodic or quasiperiodic motions. Even if the first integral has isolated singularities (i.e., isolated zeros for its gradient), the trajectories are confined to 2D surfaces with singularities. If, however, I(x, t) is constant on a full open region of the flow, then it has no well-defined level surfaces in that region and hence trajectories may display complicated, or even chaotic motion. Example 4.2 As discussed in Chapter 3, the 2D, time-periodic velocity field, v(x, t) =



− sin ct cos ct + a cos ct − a sin ct



x y



−b



y2 − x2 2x y

 ,

(4.11)

is a solution of the 2D Navier–Stokes equation for any value of the viscosity ν (see Pedergnana et al., 2020). This means that this vector field is also a solution for ν = 0, and hence solves the 2D version of the Euler equation (2.16) for inviscid fluids. As we deduced from formula (2.69) in Chapter 2, the scalar vorticity ωz (x, y) = ∂x v(x, y) − ∂y u(x, y) is a first integral for any planar inviscid velocity field, and hence for (4.11) as well. Computing the vorticity for this velocity field, we obtain from (4.11) that ωz (x, y) = −2a = const., and hence the vorticity is a constant (i.e., degenerate) first integral for this time-periodic velocity. Therefore, trajectories are not confined to level sets of ωz and may, in principle, be chaotic. Indeed, Fig. 2.23 shows a parameter configuration in which the Poincaré map of this flow has a homoclinic tangle, which implies chaotic dynamics. At the same time, there can be other nonsingular first integrals for certainparameter values that force the trajectories to

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4.3 2D Recurrent Flows with a First Integral

117

be regular. For instance, for a = 1/2, b = 0 and c = −2, a first integral I(x, t) of the system is given by1      1

cos 2t − sin 2t −5/2 0 T I(x, t) = x, T(t)AT (t)x , T(t) = , A= . sin 2t cos 2t 0 −1/2 2 (4.12) T The gradient ∇ I(x, t) = T(t)AT (t)x only vanishes at x = 0, because both T(t) and A are nonsingular matrices. Furthermore, for any fixed t, the level curves of I(x, t) are closed and hence all level sets of I(x, t) are compact in the extended phase-space of system (4.10). Therefore, by the Brown–Samelson theorem applied to I(x, t), all nonequilibrium trajectories of the velocity field (4.11) are periodic or quasiperiodic orbits on 2D invariant tori, each Physics of Fluids ARTICLE the scitation.org/journal/phf of which forms an elliptic transport barrier. Figure 4.17 confirms existence of elliptic barriers despite the saddle-type dynamics suggested by instantaneous streamlines.

FIG. 1. (a) A typical fluid particle trajectory generated by the linear unsteady velocity field (30) for the time interval [t0 , t1 ] = [0, 200] for the initial condition x 0 = (2, 0). This flow would appear as an unbounded vortex in any flow-visualization experiment involving dye or particles. (b) The instantaneous streamlines of the same velocity field, shown here for t = 0, suggest a saddle point at the origin for all times (streamlines at other times look similar).

Figure 4.17 (a) Orbits of the time-periodic inviscid velocity field (4.11) lie on 2D, quasiperiodic invariant tori. (Parameter values: a = 1/2, b = 0 and c = −2.) (b) In contrast, instantaneous streamlines of the velocity field (4.11) suggest saddle-type from the universal solution family (6). In the notation used for equaExample 3. Building on the discussion of the stability of the x dynamics at all times. Adapted from Pedergnana et al. (2020). tion (25), we now have ω = −1 and C = 4, which give |C − ω| = 0 fixed point of equation (25), we now consider another specific Navier–Stokes velocity field of the form sin 4t u(x, t) = � cos 4t −

1 2

cos 4t + 12 x2 − y2 �x + α(t)� � −2xy − sin 4t

(33)

> 2. Therefore, as discussed in Example 1, the origin of (33) is a center-type fixed point under linearization for the Lagrangian particle motion. At the same time, both the instantaneous streamlines in Fig. 3(a) and the Okubo–Weiss criterion suggest a saddle-type (hyperbolic) behavior for the linearized flow, given that OW > 0

4.3.2 Barriers from First Integrals in Time-Quasiperiodic Flows

While time-periodic velocity fields have a limited ability to approximate real-life fluid flows, time-quasiperiodic velocity fields can approximate general unsteady velocity fields closely over a time interval of interest. We recall from §2.2.12 that temporally quasiperiodic 2D velocity fields are of the form xÛ = v(x, Ωt), 1

t ∈ R,

(4.13)

This I (x, t) can be found by transforming the time-periodic velocity field for x(t) into a steady velocity field in a rotating frame. That can be achieved via the transformation x = T(t)y using the time-dependent rotation tensor T(t) defined in the system (4.12). The stream function for the velocity field in the y-coordinates can be identified
as ψ(y) = 21 hy, Ayi, which is conserved along the y(t) trajectories. This implies that I (x, t) = ψ T−1 (t)x = 
ψ TT (t)x = 21 x, T(t)ATT (t)x is a conserved quantity for system (4.12).

FIG. 2. (a) Instantaneous streamlines and the Okubo–Weiss elliptic region (yellow) for the universal Navier–Stokes solution (32) with α(t) ≡ −0.1 at time t = 0. Other time slices are similar. (b) Stable (blue) and unstable (red) manifolds of the fixed point of the Poincaré map (based at t = 0 with period T = π2 ) for the Lagrangian particle motions under the same velocity field, superimposed on the structures shown in (a). https://doi.org/10.1017/9781009225199.005 Published online by Cambridge University Press

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Barriers to Chaotic Advection

where the frequency vector Ω = (Ω1, . . . , Ωm ) satisfies the non-resonance condition hΩ, ki , 0

(4.14)

for all nonzero integer vectors k. We assume that the flow is incompressible and hence the velocity field admits a stream function ψ(x, Ωt). We now also assume that a nontrivial conserved quantity exists, i.e., there is a temporally quasiperiodic scalar-valued function I(x, Ωt) that satisfies I (x(t; t0, x0 ), Ωt) = const.

(4.15)

on all trajectories x(t; t0, x0 ) of the system (4.13). Differentiation of Eq. (4.15) with respect to t gives ∂x1 I∂x2 ψ − ∂x2 I∂x1 ψ + ∂t I = 0.

(4.16)

In analogy with the periodic case, this system can also be extended into a higherdimensional but autonomous dynamical system by introducing the phase variable vector

φ = (Ω1 t, . . . , Ωm t) ∈ Tm .

(4.17)

Then, as we have already noted in §2.2.12, we obtain the (m + 2)-dimensional differential equation xÛ = v(x, φ), φÛ = Ω,

(4.18)

which is the generalization of system (4.10) from the time-periodic to the time-quasiperiodic case. As we noted in §2.2.12, not all trajectories of the autonomous system (4.18) represent trajectories of the original nonautonomous system (4.13). Indeed, while the phase vector φ can take any initial condition φ0 at time t0 in the extended system (4.10), initial phases in the original flow represented by system (4.13) must fall in the one-parameter family of initial conditions

φ0 (t0 ) = (Ω1 t0, . . . , Ωm t0 )

(4.19)

parameterized by t0 . Another important feature of system (4.13) is the existence of m−1 independent conserved quantities, or first integrals, of the form I j (φ) = Ω j φ1 − Ω1 φ j ,

j = 2, . . . , m,

(4.20)

as noted by Brown (1998). Note that by the constraint (4.19), we have I j (φ) ≡ 0 for all j, and hence only the zero level sets of the first integrals I j (φ) are relevant for the original flow (4.13). Each such zero level set forms an (m − 1)-dimensional torus within Tm . Since the gradients ∂φ I j (φ) are all linearly independent from each other, their joint zero level set is locally a one-dimensional manifold by the implicit function theorem (see Appendix A.1), i.e., a single trajectory on Tm . This trajectory is dense in Tm by the non-resonance assumption (4.14). Consequently, the trajectories of the extended system (4.18) that represent trajectories of the original quasiperiodic system (4.13) form a dense subset in the phase space R2 × Tm . Note, however, that this subset still has measure zero in the phase space. Consequently, measure-theoretic considerations yielding results on sets of full measure in the extended

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119

dynamical system (4.18) generally do not carry over to the original quasiperiodic velocity field (4.13) as we have already indicated in §§2.2.12, 2.2.14 and 4.2.2 One can make use of classic results for integrable Hamiltonian systems by further enlarging the system (4.18) into an (m+1)-degree-of-freedom Hamiltonian system (see Appendix A.5). This can be achieved by introducing a new momentum-type variable p ∈ Rm that is canonically conjugate to the phase variable φ. To this end, we define a Hamiltonian H and the canonical symplectic matrix J as   0 1 H(x, p, φ) = ψ(x, φ) + hΩ, pi , J= , −1 0 which generates the autonomous, canonical Hamiltonian system xÛ = J∂x H(x, φ) = J∂x ψ(x, φ), φÛ = ∂p H(x, p, φ) = Ω,

pÛ = −∂φ H(x, p, φ) = −∂φ ψ(x, p, φ).

(4.21)

As for any autonomous Hamiltonian system, the Hamiltonian I1 (x, p, φ) = H(x, p, φ) = ψ(x, φ) + hΩ, pi

(4.22)

is conserved and hence is automatically a first integral. The conserved quantity assumed in condition (4.15) provides the additional first integral Im+1 (x, φ) := I(x, φ).

(4.23)

Therefore, we have a total of m + 1 first integrals, defined in Eqs. (4.20), (4.22) and (4.23), for the (m + 1)-degree-of-freedom Hamiltonian system (4.21). The gradients of these integrals,
∇ I1 = ∂x H, ∂φ ψ, Ω , !

∇I j =

(j)

0 , (Ω , 0, . . . , 0, −Ω1, 0, . . . ), 0 , |{z} | j {z } |{z} 2

j = 2, . . . m,

m

m

!

∇ Im+1 = ∂x I, ∂φ I,

0 , |{z}

(4.24)

m

are all linearly independent as long as

∂x H , 0, 2

∂x I(x, φ) , 0,

(4.25)

The same issue does not arise in the extension (4.10) of a time-periodic vector field v(x, Ωt) because one can simply restrict attention to the first return map associated with the initial phase φ0 (t0 ) = Ωt0 of this extended system. All orbits of this first-return map will describe sampled trajectories of v(x, Ωt). In contrast, a Poincaré map constructed as a first return map along one of the phases φk of the quasiperiodic system (4.18) will have only a measure zero (albeit dense) set of orbits that describe sampled trajectories of v(x, Ωt), as one concludes from the constraint (4.19) that such trajectories must satisfy.

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Barriers to Chaotic Advection

i.e., as long as we are away from instantaneous stagnation points and I(x, φ) has explicit dependence on the positions x. Furthermore, the pairwise Poisson brackets 





 Ii , I j = ∂x1 Ii ∂x2 I j − ∂x2 Ii ∂x1 I j + ∂φ Ii , ∂p I j − ∂φ I j , ∂p Ii ,

i,j

of the first integrals satisfy the relations 

 Ii , I j = 0,

i, j , 1, *



+

(j)

Ii , I j = − (Ω j , 0, . . . , 0, −Ω1, 0, . . . ), Ω = 0, | {z } 

i = 1, j , m + 1,

m

* 

(j)

+

Ii , I j = (Ω j , 0, . . . , 0, −Ω1, 0, . . . ), Ω = 0, | {z } 

j = 1, i , m + 1,

m

 [I1, Im+1 ] = ∂x1 I1 ∂x2 Im+1 − ∂x2 I1 ∂x1 Im+1 − ∂φ Im+1, ∂p I1 = ∂x1 ψ∂x2 I − ∂x2 ψ∂x1 I − ∂t I = 0,

(4.26)

where we have used the conservation law (4.16). In the language of classical mechanics, therefore, the first integrals I1, . . . , Im+1 are pairwise in involution (see Arnold, 1989). In summary, by the relations (4.24) and (4.26), the (m+1)-degree-of-freedom Hamiltonian system (4.21) has m + 1 independent first integrals in involution on any domain where the two conditions in (4.25) are satisfied. Since I2, . . . , Im are defined on the m-dimensional torus Tm , a joint level set of the m + 1 integrals is closed and connected precisely when the joint level set of I1 and Im+1 , I(a, b) = {(x, t, p) : ψ(x, Ωt) + hΩ, pi = a, I(x, Ωt) = b} , is closed and connected. Then, as long as the conditions (4.25) hold on the joint level set I(a, b), this level set is diffeomorphic to an (m + 1)-dimensional torus in the 2(m + 1)dimensional phase space of (x, φ, p), as proven by Arnold (1989) for general Hamiltonian systems. In that case, the Hamiltonian system is called (completely) integrable. We recall, however, that as we have concluded in our discussion after the conservations laws, only a single trajectory in each joint level set I(a, b) is relevant for the original quasiperiodic velocity field v(x, Ωt). Intersections of such an (m + 1)-dimensional torus with the (m + 2)-dimensional plane {φ = φ0 } in the 2(m+1)-dimensional phase space will typically give one-dimensional closed material lines in the phase space of the original system (4.13). These closed material lines evolve quasiperiodically in time and hence do not satisfy Definition 4.1. Nevertheless, these material lines are exceptional in their quasiperiodic time dependence, which distinguishes them from typical material lines. Indeed, quasiperiodically evolving material lines reapproach their initial shape with arbitrarily high accuracy while blocking advective transport between their interiors and exteriors. Examples of flows admitting such quasiperiodic advective barriers are quasiperiodic potential vorticity conserving flows in geophysics, or quasiperiodic, planar inviscid flows, which conserve their vorticity (see, e.g., Haller and Mezić, 1998).

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4.4 3D Steady Flows

121

4.4 3D Steady Flows Here, we discuss how the ideas developed for transport barriers in 2D steady flows can be applied to define and detect barriers in 3D incompressible steady flows given by xÛ = v(x),

∇ · v(x) = 0,

(4.27)

3

with x ∈ R . For such an autonomous differential equation, trajectories evolve under a flow map that is a one-parameter family of transformations, Ft (x0 ) = x(t; 0, x0 ), as we discussed in relation to system (2.18).

4.4.1 Definition of Transport Barriers from First-Return Maps Using a 2D Poincaré map PΣ : Σ → Σ defined on a 2D transverse section Σ (see §2.2.12), we can proceed as in the case of time-periodic velocity fields to define advective transport barriers for 3D steady flows. Specifically, we can adapt Definition 4.1 to 3D steady flows as follows: Definition 4.3 An (advective) transport barrier of a 3D steady flow (4.27) is a material surface whose intersection with a section Σ transverse to v is a structurally stable invariant curve of the Poincaré map PΣ . Our discussion and classification of hyperbolic, elliptic and parabolic barriers in §§4.1.1– 4.1.3 are then also directly applicable in the present context. This means that all possible barriers and partial barriers to transport shown in Fig. 4.12 will arise in the Poincaré maps of generic 3D steady fluid flows. They can then be extended by the flow map to barriers in the full 3D flow. For instance, a closed invariant curve (elliptic transport barrier) of a first-return map PΣ gives rise to a 2D invariant torus, which in turn acts as an elliptic transport for the full 3D steady flow, as sketched in Fig. 4.18. To show such barriers in a physical flows, Fig. 4.19 depicts a dye visualization of such a torus barrier family, along with numerical simulations of a first-return map capturing the family as closed curves. Also highlighted by the map are four islands born out of a 1:4 resonance. These islands are filled with more localized toroidal barriers similar to those discussed in §4.1.2.

4.4.2 Transport Barriers vs. Streamsurfaces and Sectional Streamlines All 2D transport barriers defined via Poincaré maps in §4.4.1 are composed of fluid trajectories and hence are also streamsurfaces of the steady velocity field v(x). Given that fluid trajectories starting off of a streamsurface cannot cross that streamsurface, it is tempting to think that all streamsurfaces are observed transport barriers in experiments. This is, however, far from being true. Trajectories released from points along any smooth curve of initial conditions form a streamsurface, and hence any point of any steady flow is on infinitely many different streamsurfaces. Using such arbitrary streamsurfaces to explain distinct transport patterns, such as those seen in Fig. 4.19 is, therefore, unrealistic. Indeed, while all transport barriers in steady flows are streamsurfaces, only very special streamsurfaces will act as transport barriers. In particular, trajectories in a typical streamsurface will

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Barriers to Chaotic Advection

closed invariant curve for the Poincar map

transport barrier for the 3D ow

Figure 4.18 Sketch of the reconstruction of a 2D elliptic barrier (invariant torus) to advective transport from a 1D elliptic barrier (invariant closed curve) of a Poincaré map defined on a section Σ transverse to the flow. Adapted from Oettinger (2017). 276

G. O. Fountain, D. V. Khakhar, I. Mezi´c and J. M. Ottino Chaotic mixing in a bounded three-dimensional flow

287

(b)

(b)

(c) (b)

E

D

75E EA E:7 ,

C

7 ,AC7 E7C D A8 D7

7 E :EEBD HHH 5

C

7 AC 5AC7 E7C D :EEBD

A AC

0

(a) (a)

2

(a)

AE:7 A

(b)

Figure 12. Comparison of (a) experimental and (b) numerical Poincar´e sections (↵ = 20 , NII = 30 RPM, NTT = 0 RPM, ReII = 6.5). There is a good match between the position and number of the di↵erent toroidal chains. The red (period-4) and green (period-5) are easy to match. Closer inspection shows matching of the blue (period-6) islands as well. The experimental photo was taken more than seven hours after needle removal.

surface). This problem can be exacerbated by poor grid design or poor convergence. The reader is directed to Souvaliotis, Jana & Ottino (1995) for a discussion of these issues. The calculation of the ratio of frequencies, f(I), for the unperturbed flow (↵ = 0 , figure 13) proceeds in the same manner. As is clear from figure 8, the structure formed

-AH A 7 8CA :EEBD HHH 5

C

7 AC 5AC7 .1/

Figure 4.19 Transport barriers in a 3D steady velocity field created by a rotating, tilted impeller in a cylindrical tank. (a) 3D view of a single 2D invariant torus visualized by red dye. (b) Cross-sectional view of a vortical region filled with invariant tori and resonant islands, obtained by shining a laser sheet across the toroidal region on different types of fluorescent dye injected earlier. (c) Simulation of the Poincaré map based on the same cross section using a numerical model of this flow. Adapted from Fountain et al. (2000).

not keep re-intersecting Poincaré sections along the exact same curve and hence will not Figure 5. Perspective view of tori (↵ = 0 , NI = 30 RPM, NT = 0 RPM, ReI = 6.5). satisfy Definition a transport barrier. (a) Photographed with the room 4.3 lightsfor on. One can see the faint outline of the laser sheet along the surfaces of the red tori. The green dye is not excited to the samevisualization degree and is almostcommunity invisible. An outstanding challenge in the flow is to construct streamsur(b) The same view, but with the room lights extinguished. A weak UV lamp has been placed above the system to add mild supplemental illumination. Here the detail provided by the laser cut is quite faces that give a meaningful skeleton of the transport patterns in the flow (see, e.g., Peikert dramatic.

frames per second. Frame straddling is used to achieve frame pairs with less than 33 ms separation. Data are acquired and stored to RAM during each run and then written to disk at the end. Post processing is then performed on the image data to obtain the velocity information. The software uses two-frame cross-correlation to determine the in-plane velocity components. https://doi.org/10.1017/9781009225199.005 Published online by Cambridge University For each experimental condition, the lower half of the tank Press was imaged in two

123

4.4 3D Steady Flows

and Sadlo, 2009; Born et al., 2010; Sane et al., 2020). From a dynamical systems point of view, this is equivalent to finding 2D invariant manifolds in the flow that are either compact and boundaryless (spheroids and tori) or have a strictly inflowing or a strictly overflowing compact boundary (subsets of 2D stable and unstable manifolds of fixed points or periodic orbits). Finding such invariant manifolds is an unsolved problem at this point, but Mackay (1994) has an insightful characterization of invariant manifolds relevant for transport in 3D steady flows. He examines under what conditions renders a general 2D manifold S the flux functional ˆ Φ(S) = v · dA (4.28) S

of v through S stationary with respect to small deformations of S. In more technical terms, when does the variational derivative δΦ(S) vanish on a surface S? Mackay finds this to be the case if and only if both S and its boundary, ∂S, are invariant under the flow of v. In other words, streamsurfaces that either have empty boundaries or have boundaries that are trajectories of v are the stationary surfaces of the flux Φ(S). Examples of such surfaces are shown in Fig. 4.20. S v

v

v v

v

v

v

v

v

S v

∂S

v

v v

Figure 4.20 Examples of special streamsurfaces S that render the flux Φ(S) of the velocity field v(x) stationary with respect to all small deformations of S. Such streamsurfaces are precisely those with boundaries ∂S composed of streamlines and boundaryless streamsurfaces, such as a torus.

While this result does not provide a recipe for finding transport barriers, it highlights the significance of streamsurfaces with empty or invariant boundaries for material transport. Namely, in a well-defined sense, such surfaces are the most robust transport barriers in steady, volume-preserving flows. Indeed, small O () deformations of these surfaces result in at most
O  2 material transport through the deformed surfaces. In contrast, O () deformations of streamsurfaces with a non-invariant boundary will generally result in O () transport through them due to O (1) flux through their boundaries prior to the deformation. Streamsurfaces with δΦ(S) , 0 do not generally preserve their integrity while one tracks them along evolving streamlines. Indeed, consider a streamsurface formed by streamlines launched along a stable or unstable manifold, such as those in Fig. 4.5, participating in a homoclinic or heteroclinic tangle of a first-return map based on a section Σ. Such a streamsurface, the unstable manifold of a periodic streamline, γ, will undergo drastic stretching and folding, ultimately accumulating on itself by the λ-lemma that we mentioned in §4.1.1 in our discussion of hyperbolic transport barriers in 2D flows. We sketch a part of such a tangling streamsurface in Fig. 4.21; the full streamsurface will accumulate on itself arbitrarily tightly.

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Barriers to Chaotic Advection

Σ W s (p)

S

W u (p) p

γ

Figure 4.21 A subset of a self-accumulating streamsurface, S, in a steady flow. This surface is formed by streamlines launched along the unstable manifold W u (p) of a hyperbolic fixed point p of the first-return map PΣ , defined on a transverse section Σ. The hyperbolic fixed point of the map signals a hyperbolic (saddle-type) closed streamline, γ, for the velocity field v(x). The streamsurface S coincides with the 2D unstable manifold W u (γ) of γ.

Such 2D streamsurfaces are hyperbolic transport barriers by Definition 4.3. They provide the backbones of tracer patterns outside vortical regions, even though they do not fall in the variationally distinguished family of streamsurfaces shown in Fig. 4.20. A more traditional but less justifiable approach to visualizing transport and its barriers in 3D flows is the use of sectional streamlines. Such streamlines are obtained by considering a section (plane or surface) in the flow domain, projecting the velocity vectors at points of this section orthogonally onto the section, and drawing the streamlines of the resulting projected velocity field on the section. This procedure is only justifiable if the selected section is a streamsurface in which case sectional streamlines are actual streamlines. Otherwise, the relationship between sectional and actual streamlines is a priori unclear: the sectional streamlines may or may not give an indication for the one-dimensional intersection of 2D transport barriers with the selected section. An early reference emphasizing the dangers of reliance on sectional streamlines is Perry and Chong (1994). We illustrate the unreliability of sectional streamlines for flow structure detection using the ABC flow (Gromeka, 1881; Arnold, 1965; Dombre et al., 1986), whose velocity field, A sin z + C cos y © ª v(x) = ­ B sin x + A cos z ® , « C sin y + B cos x ¬

(4.29)

is an exact, spatially triple-periodic solution of the steady, incompressible Euler equation (A.39) (see Dombre et al., 1986). This flow is the classic example of a steady Euler flow with chaotic streamlines, an atypical phenomenon in steady inviscid flows, which are generically integrable (see §4.4.3 below).

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4.4 3D Steady Flows

On the y = 2π face of the [0, 2π] × [0, 2π] × [0, 2π] cubic domain shown in Fig. 4.22, the sectional streamlines give a reasonable indication of some (but not all) elliptic barriers revealed by the Poincaré map. These transport barriers in the map give rise to 2D toroidal transport barriers in the full flow. Even on this face of the cube, however, there is no indication of the resonant islands or the parabolic transport barrier separating two mixing regions around the toroidal region. On the other two faces of the cube, none of the further elliptic barriers, resonant islands or mixing regions are indicated by sectional streamlines.

Poincaré maps

sectional streamlines

Figure 4.22 Sectional streamlines vs. trajectories of Poincaré maps computed on three faces of the ABC √ the triply-periodic solution (4.29) of the steady Euler √ flow, equation with A = 3, B = 2 and C = 1. Adapted from Haller et al. (2020b).

4.4.3 Transport Barriers in 3D Steady Inviscid Flows Three-dimensional incompressible, inviscid flows are governed by the Euler equation, the zero-viscosity limit of the Navier–Stokes equation. If such a flow is also steady, then its velocity field obeys the steady Euler equation ρ (∇v) v = −∇ (p + V) ,

(4.30)

B (x(t; x0 )) ≡ B (x0 )

(4.31)

appended with the incompressibility equation ∇ · v = 0. Equation (4.30) involves the steady velocity field v(x), the pressure field p(x), the constant density ρ and the scalar potential function V(x), which generates the vector field g of potential forces via the relationship ρg(x) = −∇V(x). Taking the inner product of both sides of Eq. (4.30) with v(x) and integrating along any trajectory x(t; x0 ) of v(x), we obtain

for any time t, where the scalar function B(x) is the Bernoulli integral B=

1 ρ |v| 2 + p + V, 2

https://doi.org/10.1017/9781009225199.005 Published online by Cambridge University Press

(4.32)

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Barriers to Chaotic Advection

representing the total energy of the fluid. The conservation law (4.31), therefore, reflects the conservation of energy for fluid elements along the streamlines of v. If the Bernoulli integral B(x) is a nonconstant function, then the streamlines (trajectories) of v are confined to lower-dimensional level sets of B(x). In that case, the ODE (4.27) is called integrable. Such integrable systems defined on compact domains cannot have chaotic trajectories because their trajectories are restricted to smooth and bounded 1D or 2D manifolds. Crucial to this argument is that the level sets of B(x) should indeed be compact. Their boundedness is guaranteed if the flow domain is spatially periodic or enclosed by finite boundaries. One must also ensure, however, that the level surfaces of B(x) are manifolds. This is locally the case by the implicit function theorem (see Appendix A.1) at all points where the gradient ∇ B does not vanish. Computing this gradient gives   ∇ B = ρ (∇v)T v + ∇ (p + V) = ρ (∇v)T v − (∇v) v = −2ρWv,

(4.33)

where we have used the Euler equation (4.30) and the definition of the spin tensor W from Eq. (2.1). By the relationship of W to the vorticity vector ω (see Eq. (2.2)), we can rewrite formula (4.33) for the gradient ∇ B as

∇ B = ρv × ω .

(4.34)

Therefore, at any point x in the flow, ∇ B is nonvanishing (and hence the level set of B(x) is locally a smooth, 2D manifold) if the velocity and the vorticity at that point are not collinear, i.e., v(x) × ω (x) , 0.

(4.35)

Arnold (1966) proves (see also Arnold and Keshin, 1998) that if the relation (4.35) holds on an invariant flow domain and v and ω are sufficiently smooth, then all fluid trajectories in that domain are confined to nested families of invariant tori or nested families of boundaryless invariant cylinders (deformations of open annuli), as shown in Fig. 4.23. Figure 4.23 also illustrates how we can select transverse sections, Σ, to both types of streamsurface families and obtain repeated returns of trajectories to the intersection curves of these families with Σ. As these curves are invariant curves for the Poincaré maps based on Σ, the surfaces shown in Fig. 4.23 are barriers to advective transport by the definition we gave in §4.4.1. While each toroidal barrier is disjoint from the singular sets with v × ω = 0, the closures of the boundaryless cylindrical barriers intersect those singular sets. By the continuity of B(x), any bounded singular set is contained in the same level set of B(x) as the boundaryless cylinder whose closure intersects that singular set. Interestingly, B(x) is also a first integral for vortex lines, i.e., trajectories of the vorticity field ω , which are solutions of the ODE x0 = ω (x), with the prime denoting differentiation with respect to a scalar variable s parametrizing the vortex lines x(s). Indeed, by formula (4.34), the derivative of B(x) along vortex lines is

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4.4 3D Steady Flows

Σ

Σ

(a)

(b)

Figure 4.23 The two possible transport barrier configurations in typical velocity fields solving the steady, incompressible Euler equation (4.30): (a) nested family of 2D invariant tori; (b) nested family of open cylindrical surfaces (open annuli). These two types of transport barriers foliate open cells in the flow that are bounded by singular surfaces on which ∇B = ρv × ω vanishes.

dB = ∇ B · ω = −ρ (ω × v) · ω = 0, ds and hence B is conserved along them. Therefore, under condition (4.35), the tori and boundaryless cylinder shown in Fig. 4.23 are also invariant surfaces for the vortex lines of v(x). Example 4.4 A classic explicit solution of the Euler equation (4.30) is Hill’s spherical vortex (Hill, 1894), whose velocity field is xz − x2cy 2 +y 2 © v(x) = ­­ yz + x2cx 2 +y 2
2 « 1 − 2 x + y2 − z2

ª ® (4.36) ®, ® ¬ with x = (x, y, z), with the arbitrary constant c and with the corresponding vorticity field −5y © ª ω = ∇ × v = ­ 5x ® . « 0 ¬

(4.37)

With the nondimensionalized density set to ρ = 1, we obtain the gradient of the Bernoulli integral B as  
5x 2 x 2 + y 2 + z2 − 1 ©   ª
∇ B = v × ω = ­­ 5y 2 x 2 + y 2 + z2 − 1 ®® , (4.38)
2 2 5z x + y « ¬ which gives  5 2 (x + y 2 )2 + (x 2 + y 2 )z2 − (x 2 + y 2 ) . (4.39) 2 As we have seen in this section, for incompressible and steady Euler flows, B(x) is always a first integral for both the velocity and the vorticity fields, and its nonsingular bounded level sets must fall in one of the two categories shown in Fig. 4.23. As ∇ B in Eq. (4.38) only vanishes along the z axis, all level surfaces of B(x) that have a finite distance from the z-axis B(x) =

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are necessarily 2D tori by the integrability results of Arnold (1966). By introducing the polar coordinates (r, ϕ) via x = r cos ϕ, y = sin ϕ, we can rewrite B(x) in Eq. (4.40) as  5  B(r, z) = r 2 r 2 + z2 − 1 , (4.40) 2 whose level curves in the (x, z)-plane are shown in Fig. 4.24(a).

z Σ

x (a)

(b)

Figure 4.24 Transport barriers in Hill’s spherical vortex. (a) Level curves of the Bernoulli integral reveal transport barriers for the first-return map defined on the vertical plane Σ = {(x, y, z) : y = 0}. (b) Dye experiment showing invariant tori in a near-inviscid vortex ring modeled by Hill’s spherical vortex. Adapted from Lim and Nickels (1995).

These level curves are independent of ϕ and depict the intersections of the level surfaces of B with an arbitrary vertical plane ϕ = const. containing the z-axis. Figure 4.24(a) shows that all level surfaces of B within the unit sphere are indeed 2D tori, as each has a finite distance from the z-axis. In contrast, neither the unit sphere without its northern and southern pole nor the level surfaces outside the unit sphere are bounded away uniformly from the z-axis. Accordingly, they all are topologically boundaryless cylinders of the type shown in Fig. 4.23. The cross section of a dye experiment confirming the toroidal transport barriers inside a vortex ring is shown in Fig. 4.24(b). Any steady Euler flow satisfying the Beltrami property v(x) × ω (x) = 0

(4.41)

on some open domain D will violate the integrability condition (4.35). For such flows, the gradient of B(x) is identically zero by formula (4.34), rendering the Bernoulli integral constant on D. One may still obtain a first integral arising from the Beltrami property (4.41) given that this property is equivalent to

ω (x) = α(x)v(x)

(4.42)

for some scalar-valued function α(x). Since the divergence of the vorticity field is always zero, we can write

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4.4 3D Steady Flows

∇ · ω = ∇α · v + α∇ · v = 0.

129

(4.43)

We have assumed the flow to be incompressible, therefore using ∇ · v = 0 in Eq. (4.43) gives

∇α · v = 0,

(4.44)

implying that nondegenerate level sets of α(x) are invariant surfaces for v, even though B(x) is constant. This argument, however, fails when ∇α vanishes on the flow domain, which is the case for strong Beltrami flows defined as

ω (x) = α0 v(x).

(4.45)

For such flows, no general nontrivial first integral arises from Eqs. (4.33) or (4.44). Even in that case, the particle ODE (4.27) might still have another, nontrivial first integral that enables the explicit construction of transport barriers (as level surfaces of that integral) without numerical simulations, as in Example 4.4. Such a first integral is often related to flow-specific symmetries, such as those we will discuss in §4.4.4. In the absence of such a first integral, Poincaré maps remain effective tools for barrier exploration in strong Beltrami flows, even though their construction will require the numerical solution of fluid trajectories starting from arrays of initial conditions. Example 4.5 Arnold (1965) introduced the ABC flow (4.29) as an example of an inviscid, incompressible flow in which the strong Beltrami property (4.45) holds with α0 = 1 on the whole flow domain. As a consequence, the Bernoulli integral B(x) and the Beltrami scalar α(x) are both globally constant and hence cannot be used to construct transport barriers in the way that√we have√done for Hill’s spherical vortex in Example 4.4. For A = 3, B = 2 and C = 1, the Poincaré maps shown in Fig. 4.22 indeed suggest that the ABC flow has chaotic trajectories and hence cannot be integrable. We observe nested but discrete families of tori as transport barriers for this flow. Beyond these nested barriers, there are 2D stable and unstable manifolds (hyperbolic barriers) in the seemingly chaotic mixing zone outside the elliptic barriers. To visualize hyperbolic barriers in mixing zones, one would first have to locate saddletype fixed points of the Poincaré map and then iterate the map on a set of initial conditions close to the fixed points in forward time to see that set shrink onto, and spread out along, one-dimensional unstable manifolds of those fixed points. A similar procedure under backward iteration of the Poincaré map would highlight one-dimensional stable manifolds on Poincaré sections. The full 2D barriers of the flow could then be constructed by releasing trajectories of the velocity field (4.29) from along the stable and unstable manifolds of its Poincaré map. All these steps are flow specific and hence would require substantial effort to automate. In the end, a more efficient approach is to use the transport barrier detection techniques we will discuss in Chapter 5 for general, unsteady velocity field. As a preview of those techniques, we compare in Fig. 4.25 a visualization of the ABC flow (4.29) via Poincaré maps with two simple LCS diagnostics: the Lagrangian-averaged vorticity deviation (LAVD) and the finite-time Lyapunov exponent (FTLE). LCSs are distinguished material surfaces that form

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the backbone curves of tracer patterns over finite times, as will be discussed in Chapter 5. Stable and unstable manifolds, as well as invariant tori, of flows defined for all times qualify as LCSs, and hence LCS methods will approximate them with increasing accuracy over increasing time intervals. Indeed, note in Fig. 4.25 the sharpness of the domains with elliptic barriers (invariant tori) and hyperbolic barriers (stable manifolds) revealed by level surfaces of the LAVD and high values of the FTLE, respectively, without any reliance on the location of fixed points of the Poincaré maps.

6 4 z 2 0

0

6 4 x

2 4

2 0

6

6

4

4

z

z

2

2 0

y

6

0 6

2

4 x

2

4

y

0 6

Elliptic transport barriers from the Lagrangian-averaged vorticity deviation (LAVD)

0

0

6

2

4 x

4

2 0

60

y

Hyperbolic transport barriers from the (FTLE)

√ √ Figure 4.25 Transport barriers in the steady ABC flow (4.29) with A = 3, B = 2 and C = 1, visualized by three different methods: Poincaré maps, the LAVD over the [0, 50] time interval and the FTLE computed over the [0, 15] time interval. The latter two methods will be discussed in detail in Chapter 5 for more general (unsteady and finite-time) velocity fields.

4.4.4 Transport Barriers in 3D Steady Flows with a Continuous Symmetry As we noted in §4.4.3, conserved quantities for 3D flows often arise from continuous symmetries. If the flow is steady and the conserved quantity (first integral) is nondegenerate, then level surfaces of the first integral are streamsurfaces that do not disintegrate due to stretching and folding as trajectories evolve in them. Such streamsurfaces can intersect Poincaré sections in invariant curves and hence satisfy the formal Definition 4.3 we have been using for transport barriers in 3D steady flows.

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131

To find conditions for the existence of a first integral arising from symmetries, Haller and Mezić (1998) consider 3D steady vector fields v(x) on a general 3D manifold that preserve a general 3D volume. For simplicity, here we only consider such a vector field on a 3D spatial domain D ⊂ R3 . We assume that the flow map Ft of v(x) is volume preserving and is equivariant with respect to a continuous group gs : D → D of transformations parametrized by the scalar variable s. This means that
Ft (gs (x)) = gs Ft (x) holds for all s, t ∈ R. We call the vector field d s g (x) , w(x) := ds s=0 the infinitesimal generator of the action of gs . The equivariance of the flow under gs is equivalent to the condition that the vector fields v and w have a vanishing Lie bracket, i.e., [v, w] := (∇w) v − (∇v) w = 0.

(4.46)

We also assume that the symmetry group gs is volume preserving, i.e.,

∇ · w = 0.

(4.47)

Under the conditions (4.46)–(4.47), Haller and Mezić (1998) prove that both v(x) and w(x) admit a common first integral I(x), whose gradient satisfies

∇ I = v × w.

(4.48)

Note the close similarity between this last equation and formula (4.34). This shows that for the purposes of integrability, the infinitesimal generator of a symmetry of a general flow plays the same role as the vorticity of an inviscid flow. Example 4.6 Consider the ABC flow (4.29), which we have seen to admit only a degenerate Bernoulli integral in Example 4.5, as any strong Beltrami flow does. This renders the ABC flow chaotic for most parameter values, Yet, for special parameter values, there may be other, nondegenerate first integrals for this flow that enable a systematic identification of all transport barriers without the numerical construction of Poincaré maps. For instance, selecting C = 0, the corresponding ABC vector field, A sin z © ª v(x) = ­ B sin x + A cos z ® , B cos x « ¬ is equivariant under the volume-preserving group of translations x x © ª © ª g : ­ y ® 7→ ­ y + s ® , « z ¬ « z ¬ s

(4.49)

(4.50)

whose infinitesimal generator is w(x) =

0 d s © ª g (x) = ­ 1 ® . ds s=0 « 0 ¬

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(4.51)

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Barriers to Chaotic Advection

Indeed, we can verify this equivariance by evaluating the equivalent condition (4.46): 0 © [v, w] = − (∇v) w = − ­ B cos x « −B cos x

0 0 A cos z ª© ª 0 −a sin z ® ­ 1 ® = 0. 0 0 ¬« 0 ¬

(4.52)

Then formula (4.48) gives the gradient of the first integral emerging from this symmetry in the form 0 −B cos x A sin z © ª © ª © ª 0 ∇ I = v × w = ­ B sin x + A cos z ® × ­ 1 ® = ­ ®. B cos x « ¬ « 0 ¬ « A sin z ¬

(4.53)

Integrating this last formula, we obtain the first integral I(x) = −(A cos z + B sin x),

(4.54)

noting that the first integral −I(x) was already found heuristically by Dombre et al. (1986). We show the transport barriers obtained as level surfaces of I(x) in Fig. 4.26. Similar integrable cases arise for the parameter choices A = 0 and B = 0.

z

y

x

Figure 4.26 Transport barriers obtained as level surfaces of the first integral (4.54) √ √ arising from a volume-preserving symmetry of the ABC flow with A = 3, B = 2 and C = 0. Image: courtesy of Roshan S. Kaundinya.

Most barrier surfaces in Fig. 4.26 are technically elliptic, as they are 2D invariant tori (doubly periodic surfaces) of various types. Exceptions are the hyperbolic barriers formed by homoclinic manifolds connecting a saddle-type periodic orbit to itself. A more complete description of integrable parameter configurations for the ABC flow is given by Llibre and Valls (2012). Further examples of 3D integrable classes of flows are given in Haller and Mezić (1998) along with 2D time-dependent flows with first integrals.

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4.4.5 Transport Barriers from Ergodic Theory A Poincaré map PΣ : Σ → Σ defined on a transverse section Σ of a volume-preserving or mass-preserving, 3D steady flow preserves an appropriately scaled version d A˜ of the classic area element dA on Σ (see §2.2.12).This renders PΣ measure preserving on Σ, making Birkhoff’s ergodic theorem (§2.2.14) applicable to the study of the dynamics of PΣ via time-averaged observables. Specifically, assume that on a compact subset S ⊂ Σ, the Poincaré map PΣ is ergodic with respect to some measure on S. Then, for any integrable scalar field c(x) defined on Σ, the asymptotic average c¯P (x0 ) := lim

N →∞

N
1 Õ c PΣ k (x0 ) N k=0

of c along trajectories of the map PΣ is well defined for almost all x0 ∈ S, as we have discussed in §2.2.14. For those x0 points, the trajectory average c¯PΣ (x0 ) is equal to the spatial average, ˆ 1 c(S) ¯ = c(x) dµ, (4.55) µ(S) S of c over S. This implies that such x0 initial conditions must be contained in the level set C(S) = {c(x) = c(S)} ¯ of c(x). In an experimental setting, however, first-return maps of individual fluid particles are unrealistic to construct and hence c¯PΣ (x0 ) cannot be evaluated. As an effective alternative, Sotiropoulos et al. (2002) construct what they term an experimental Poincaré map for a 3D steady swirling flow arising from the breakdown of a spherical vortex bubble. In these experiments (see Fig. 4.27(a)), a laser scans a selected cross section Σ of the swirling flow. The resulting laser sheet illuminates within Σ an initially nonuniform concentration of fluorescent, weakly diffusive dye injected into the flow at an earlier initial time. The instantaneous illumination intensities are photographed, recorded and time-averaged. These time-averaged intensities are then plotted over Σ, with one outcome of this procedure shown in the upper plot in Fig. 4.27(b). This plot bears close similarities to a classic Poincaré map constructed earlier by Sotiropoulos et al. (2001) for a numerical model of the same flow. Mezić and Sotiropoulos (2002) invoke Birkhoff’s ergodic theorem to argue that the experimental procedure used in generating Fig. 4.27(a) should indeed highlight the structures captured by a Poincaré map. To see this, we denote the initial dye concentration at time t = t0 in the cross section Σ by the scalar field c0 (x) for x ∈ Σ. Ignoring the diffusivity of the dye, we can view the dye evolution as a purely advective problem governed by the conservation law Dc = ct + ∇c · v = 0, Dt

(4.56)

which is solved by c(x, t) = c0 (F−t (x), t). Sotiropoulos et al. (2002) record and time-average this field at each point x ∈ Σ over a finite time interval [0, t], which means that they construct experimentally the function ˆ 1 t c¯t (x) = c0 (F−s (x)) ds. t 0

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Barriers to Chaotic Advection 2238

I. Mezic´ and F. Sotiropoulos

Phys. Fluids, Vol. 14, No. 7, July 2002

Σ

220

F. Sotiropoulos, D. R. Webster and T. C. Lackey

Nd: YAG laser

Σ

H

Σ

B6 D6B C 9DDAC

: B

0

R

S ∩Σ T ∩Σ p

FIG. 3. "Color# Experimental "Ref. 15# and calculated "Ref. 10# Poincare´ maps for a steady vortex breakdown bubble "H/R!1.75, Re!1850#. The experimental map was constructed by plotting the level sets of the timeaveraged light-intensity field. For the calculated map, same-color markers were released along short straight segments, selectively placed within various regions in the interior of the bubble to elucidate the richness of the dynamics. Note the level of agreement between the two images in terms of both the period and relative location of the various invariant sets.

W u (p)

-

2 6 7B

9DDAC

2

B:

6 B

B6 .1/ : : D96

E

2D

CE 6 D D D96 ,2

B:

6 , B6 D6B C 7 EC6 2 2: 2 6 2D 9DDAC

2

B:

6 B

Variable-speed motor PLIF data acquisition computer

Figure 2. A schematic of the experimental apparatus.

The working solution was a glycerin/water mixture (roughly 3/1 by volume). The viscosity of the mixture was in the neighbourhood of 50 cP at 25 C and was measured using a precision falling-ball viscometer at several temperatures over a range of 10 C. The fluid temperature was measured during the experiment to within 0.1 C in order to determine the viscosity. A precision (a)mercury thermometer, submerged in (b) the rectangular tank immediately adjacent to the test volume, was used intoFig.measure 1. Fix a diametral plane ! that contains the cylinder of the time-averaged light intensity field, let fluorescent dye axis. For a rotating flow, a Poincare´ map P can be defined as be distributed in $ at time t!0 with a nonuniform concenthe temperature. Under normal experimental conditions, the fluid temperature in the the map that sends X0 to the first intersection of the trajectration c 0 (x,y,z)—in LIF experiments the concentration of apparatus was maintained to within 0.1 C for several hours. tory passing through X0 with !. Invariant sets for Poincare´ fluorescent dye is linearly proportional to the intensity of emitted light, which can be readily quantified using digital maps are the intersections of invariant sets for the flow "1# Flow visualization was performed using the planar laser-induced fluorescence photography. Assume the concentration of dye is advected with the plane !. To clarify the relationship between the (PLIF) technique. The dye solution consisted of the glycerin/water mixture with perfectly by the flow "i.e., there is no diffusion# and deso-constructed Poincare´ map of the flow and the iso-contours was suffiRhodamine 6G at a concentration of 500 µg l 1 . The dye concentration Downloaded 09 Oct 2002 to 130.207.135.136. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/phf/phfcr.jsp ciently low to minimize absorption e↵ects. Dye was delivered to the flow through a 1.0 mm hole at the centre of the non-rotating endwall from a dye reservoir elevated 1.5 m above the experiment. The dye flow rate was extremely small, essentially matching the local flow velocity, in order to minimize perturbations to the container flow. Note that if the dye flow rate exceeded this iso-kinetic release, the resulting jet of dye adversely perturbed the bubble and the e↵ects of the perturbation were obvious by visual inspection. For instance, excessive dye flow rates would not only change u the location and overall shape of the bubble but also lead to the onset of visible unsteadiness (with the bubble darting up and down along the axis) even for governing parameters in the steady flow regime. An illumination sheet was generated with a pulsed Nd:YAG laser. The pulse energy was 50 mJ and the sheet was formed by a combination of spherical and cylindrical lenses such that the laser sheet was approximately 1 mm thick in the imaged region. The laser sheet was vertically centred on each bubble before collecting data in order to provide as uniform a distribution of incident light intensity as possible. This was a critical step because the Lagrangian transport is quantified in terms of the intensity of emitted light, which is proportional to both the dye concentration and incident laser light intensity. t

Figure 4.27 (a) The experimental setup for the study of a steady swirling flow with a broken-down spherical vortex (adapted from Sotiropoulos et al., 2002). (b) Timeaveraged dye intensity in the cross section Σ and the classic Poincaré map computed by Sotiropoulos et al. (2001) for a numerical model of this flow (adapted from Mezić and Sotiropoulos, 2002). These plots reveal both elliptic transport barriers and resonance islands of period 2, 3 and 4. The material envelope bounding the broken vortex is formed by a hyperbolic transport barrier, the unstable manifold W (p) of a saddle-type fixed point p of the Poincaré map.

If the flow map is ergodic on a 3D domain S, then the Birkhoff–Khinchin ergodic theorem (2.87) implies lim c¯ (x) = c(S) ¯

t→∞

(4.57)

for almost all x ∈ S, with the spatial average c(S) ¯ defined in Eq. (4.55). This, however, does not necessarily imply yet that Eq. (4.57) holds for any x ∈ Σ, given that any 2D Poincaré section Σ has zero measure (volume) in the 3D domain S. As a consequence, all initial conditions x ∈ Σ may just be in the measure-zero set to which (4.57) does not apply. The arguments given in Mezić and Sotiropoulos (2002) for the applicability of Birkhoff’s ergodic theory to this problem, therefore, need to be augmented with further considerations. If the flow has perfect circular symmetry around the main axis of its cylindrical domain, then if Eq. (4.55) holds at a particular point x ∈ S, then it must also hold along the axially symmetric circle containing x. Therefore, Eq. (4.55) holds along almost all axially symmetric circles, which in turn intersect the Poincaré section subset Σ ∩ S at almost all of its points. Therefore, under perfect cylindrical symmetry for the broken vortex bubble, the experimentally constructed quantity c¯t (x) will converge to c(S) ¯ at almost all points of Σ ∩ S, t as long as the flow map F is ergodic on the 3D domain S. One will, therefore, indeed find

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4.5 Barriers in 3D Time-Periodic Flows

135

(almost all) points of perfectly mixing regions of the Poincaré map PΣ : Σ → Σ to approach the same color in the upper plot of Fig. 4.27(b) for large enough averaging times t. This is indeed confirmed by the numerically computed Poincaré map in the lower plot of Fig. 4.27(b), which shows regions filled with clouds of points where the averaged dye concentration also approaches a constant value. The asymptotic constancy of c¯t (x) on individual closed transport barriers C of the map PΣ requires a slightly different argument. As we have discussed, these closed curves are the intersections of 2D quasiperiodic invariant tori T with Σ, as seen in Fig. 2.15. The flow map is ergodic restricted to these tori with respect to an appropriate 2D area measure defined on the tori. Therefore, Eq. (4.57) will hold at almost all points of such a two-torus T . These trajectories collectively produce intersections with the Poincaré section Σ that cover almost all points of C. Therefore, at almost all points of a closed barrier C of PΣ , one should indeed asymptotically observe the same color in the upper plot of Fig. 4.27(b). This is in agreement with the actual Poincaré map shown in the lower plot of the same figure. Finally, the asymptotic constancy of c¯t (x) over the unstable manifold W u (p) of a hyperbolic fixed point p of the Poincaré map can also be concluded, albeit not from ergodic theory. Indeed, the inverse flow map F−t shrinks area exponentially (and hence cannot be ergodic) within the 2D unstable manifold W u signaled by the invariant curve W u (p). However, all trajectories in W u converge to the fixed point p of the full 3D flow. Therefore, for any continuous scalar field c(x) and for all x ∈ W u (p), we must have lim c¯t (x) = c(p),

t→∞

simply by the continuity of c(x), as indeed suggested by Fig. 4.27(b). The same argument does not hold for the one-dimensional stable manifold W s (p), even though one sees accumulation of red color near W s (p) in Fig. 4.27(b). This accumulation is due to the finite-time effect that dye remains locally captured near a saddle point for long times.

4.5 Barriers in 3D Time-Periodic Flows Temporally periodic velocity fields in 3D also admit Poincaré maps of the type we discussed in §§4.1 and 4.2. For 3D flows, such Poincaré sections are 3D, which makes locating transport barriers through these maps complicated if not impossible. However, the LCS methods we will develop in Chapter 5 for temporally aperiodic, finite-time flows can also be applied to reveal barriers in temporally periodic and quasiperiodic flows over long enough time intervals. Visualizing invariant sets in such flows is also possible by applying Birkhoff’s ergodic theorem to the Poincaré map, as we discussed for Poincaré maps of 3D steady flows in §4.4.5. One has to expect, however, that increasing the computational times will be required for the temporal averages to converge to the spatial averages in 3D flows. Over such increased times, numerical inaccuracies invariably start affecting the quality of the results, as we have already seen on a 2D example in §4.2. Figure 4.28 shows an example of a related 3D computation by Budisić and Mezić (2012) for a periodically forced version of the Hill’s spherical vortex defined in Eq. (4.36). This result is extracted from a clustering analysis of long-term temporal averages of a large number of observables (Fourier harmonics) along trajectories. These observables are velocity-independent

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(passive) scalar fields, and hence, unlike the example shown in Fig. 4.15, the results shown in Fig. 4.28 are objective. As we have noted in §§2.2.14 and 4.2, this approach has no immediate extension supported by Birkhoff’s ergodic theorem to 3D time-quasiperiodic flows.

M. Budi≤i¢, I. Mezi¢ / Physica D 241 (2012) 1255–1269

1264

ϕ

r

z

(a) The Poincaré section at t ⌘ 0 (mod 1). (b) The slice at ✓ = 0 through the Poincar Segments of state space corresponding to clusters panel(2D (a), displaying Figure 4.28 Two vortical domains bounded by elliptic transport barriers invari- all clusters; colors re 0 and 3 shown in blue and red, respectively. The , 1, 2, 3). ant tori) of the Poincaré map in the periodically forced version of labels Hill’s(0spherical remaining clusters are cropped for clarity.

vortex (see Eq. (4.36)). The two sets are obtained from a clustering analysis of trajectory-averaged Fourier harmonics. Adapted from Budisić and Mezić (2012).

4.6 Burning Invariant Manifolds: Transport Barriers in Reacting Flows

We close this chapter by a brief discussion of transport barriers in reacting fluids with recurrent (steady or time-periodic) time dependence, such as the reaction fronts shown in Fig. 4.29. Physically, this phenomenon is different from the purely advective transport phenomena we have surveyed throughout this chapter. Under simplifying assumptions, however, reaction front propagation in temporally recurrent flows can be described by the purely advective techniques of passive mixing and transport that we have been discussing here. Persistent and seemingly coherent reaction front propagation is observed in a variety of settings, including plankton boom, wildfires, spread of disease in moving population or microfluidic chemical reactors and other problems, as reviewed inquotient Goweninto and Solomon As Fig. 4.29 k of the partial q (c) Embedding of the partial first (d) (2015). Diffusion eigenfunctions two diffusion eigenfunctions against , , with color for points in the cluster 2. 1 2 1 illustrates experimentally, reaction front propagation in these phenomena is frame-indifferent. indicating the cluster assigned to the point. Therefore, by our discussion in Chapter 3 on material frame-indifference, transport barriers to reaction front propagation should be described in an objective fashion, just as purely Fig. 8. Structure of the invariant sets of a periodically-forced 3D Hill’s vortex at high perturbation c = " = 0.3495. Diffusion advective algorithm intomaterial 4 clustersmixing. to extract the main features. The Poincaré section used for visualization is at t ⌘ 0 (mod 1). (For inter figureBuilding legend, theon reader is referred theOberlack web version of Cheviakov this article.) (2010) and assuming the reaction related resultstoby and time scales to dominate the advection time scales, Mitchell and Mahoney (2012) put forward Lie group; when time is discrete but more recently, they been analyzed [12]. We useinthem a simplified equation forhave the propagation of ainreaction front a 2D reacting fluid velocity denoted by U. here provide a generalization of the ergodic quotients which can normal fieldtov(x, t) with x ∈ R2 . They assume that propagation takes place to thesimply front at (!) bethe used to analyze periodic and quasi-periodic transport. The harmonic constant, scalar burning speed v0 , independent of the position, the propagation direction averages f˜ For any fixed ! 2 R, and Fourier observables fk , the embedding operator, since it is simple to sh

(. . . , f˜k(!) (x), . . .)k2ZD generates an analog of the ergodic quotient, the harmonic quotient at frequency !. The ergodic

⇡! (x)

=

quotient is then a special case of the harmonic quotient, computed for ! = 0. In understanding the ergodic quotient, the level sets https://doi.org/10.1017/9781009225199.005 Published online by Cambridge Universityrole Press as invariant sets. f˜ played a crucial of averaged observables

Ut f˜ (!) = ei2⇡ !t f˜ (!) .

It follows that, for any observa an eigenfunction of the Koopm i2⇡! for discrete time, ! = e

ant manifolds to find a balance between the outward front propagation and the inward fluid velocity. The BIMs attached to these burning fixed points will also be more separated for larger v0. Experimentally, the increased BIM separation for larger v0 results in a broadening of the steady state, Barrierspinned in Reacting Flows 137 front.

FIG. 5. Motion of passive tracer particles in single vortex flow with an imposed horizontal wind; characteristic vortex velocity U ¼ 0:14 cm=s, dimensionless wind speed w ¼ 0.69 and dimensionless front propagation speed v0 ¼ 0:048. (a) Simulated trajectories, based on measured velocity field of Fig. 4. (b) Experimental trajectories, plotted from a reference frame moving with the vortex.

4.6 Burning Invariant Manifolds: Transport

FIG. 6. Sequence of images for reaction front in time-independent flow as viewed both in the lab frame and a frame moving with the vortex; U ¼ 0.068 cm/s, v0 ¼ 0:095, w ¼ 0.69. Time increases moving downward, with the images spaced by 200 s each.

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded

Figure 4.29 Evolution of a reaction front134.82.105.125 in a time-independent flow near a propaOn: Tue, 09 Jun 2015 15:59:27 gating vortex, shown in the lab frame and in a frame co-moving with the vortex. This experiment illustrates the objectivity of transport barriers even under the simultaneous presence of advection, reaction and diffusion. Adapted from Gowen and Solomon (2015).

and the local curvature of the reaction front.3 If e(t) denotes the unit tangent vector to the propagating front at a position r(t), then the local front element evolves approximately under the equation rÛ = v(r, t) + v0 Je, eÛ = [∇v(r, t) + he, S(r, t)ei I] e,

(4.58)

with the 90◦ rotation matrix J used in Eq. (4.21) and with the rate-of-strain tensor S defined in (2.1). The second equation in (4.58) is the classic equation for the evolution of the unit tangent vector g of a material curve (see, e.g., Haller (2016) for a derivation). The first equation in (4.58) describes a deviation from pure material evolution (v0 = 0) for the front. Therefore, for nonzero v0 , the unit tangent vector e evolves materially under the modified velocity field v(r, t) + v0 Je, which in turn depends on the evolution of e. Enforcing the constraint |e| = 1 for the unit tangent vector of the front, Mitchell and Mahoney (2012) convert Eq. (4.58) into a 3D system of ODEs for the components of v = (u, v). The resulting equation is 3

The latter assumption is somewhat restrictive but holds in the experiments reported in Mahoney et al. (2012).

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xÛ = u + v0 sin θ, yÛ = v − v0 cos θ,

(4.59)

1 θÛ = vy − u x sin 2θ + vx cos2 θ − uy sin2 θ, 2


involving the angle θ between e and the positive x axis. Reaction fronts are represented by one-dimensional, time-dependent invariant manifolds (r(t; λ), θ(t; λ)) of system (4.59) that satisfy the front compatibility condition ∂λ r(t; λ) =



cos θ(t; λ) sin θ(t; λ)

 .

(4.60)

Reaction fronts are, therefore, evolving material lines of the 3D vector field (4.59), determined by their initial position. As such, fronts cannot intersect or self-intersect in the 3D phase space, but their projections on the physical (x, y)-plane can exhibit intersections and self-intersections. In the case of a projected front trailing the other, however, the (x, y)projection of trailing front cannot pass that of the leading front as both travel with the same velocity v0 in the local frame. For a steady 2D velocity field v(x), fixed points of system (4.59) are called burning fixed points. In case such a fixed point has stable and unstable manifolds in the 3D phase space of Eq. (4.59), those manifolds are called burning invariant manifolds (BIMs). Mitchell and Mahoney (2012) argue that the (x, y)-projections of BIMs are one-sided transport barriers to front propagation due to the no-passing property we mentioned above. In other words, no impinging front can burn past any point of the BIM in the same direction that the BIM itself is burning. For a time-periodic 2D velocity field, burning fixed points and burning invariant manifolds can be defined analogously for the Poincaré map associated with the time-periodic 3D dynamical system (4.59). These then mark burning periodic orbits and their 2D invariant manifolds in the full phase space of system (4.59). In both the steady and time periodic cases, the BIMs relevant for observed front propagation are the unstable manifolds of saddletype burning fixed points or of saddle-type burning periodic orbits, since these manifolds behave as attracting material surfaces for system (4.59). Figure 4.30 shows a verification of these predictions in experiments, supported by numerical simulations of Eq. (4.59), for a Belousov–Zhabotinsky chemical reaction in a flow with a vortex (see Gowen and Solomon, 2015 for details). Mahoney and Mitchell (2015) provide a variational formulation for BIMs in 2D steady flows using an adaptation of the theory of shearless advective transport barriers derived by Farazmand et al. (2014) for temporally aperiodic flows (see also §5.4.3). An extension of the front propagation equation (4.58) to 3D flows requires the tracking of a 3D surface normal, resulting in a six-dimensional set of ODEs. Visualizations of the corresponding BIMs in a given steady 3D flow appear in Doan et al. (2018). Similar one-sided barriers arise in the motion of microswimmers in 2D fluid flows, as pointed out by Berman et al. (2021). Finally, more general models and their simulations for chaotic advection in reacting flows are reviewed by Tél et al. (2005).

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4.7 Summary and Outlook Time-periodic reaction front

Steady reaction front

FIG. 7. Persistant reaction patterns for vortex with a steady imposed wind FIG. 7. Persistant reaction (pointing to the right); all patterns images for are vortex with a steady imposed wind viewed in the co-moving reference (pointing to the right); all images frame. (a)–(c) Experimental averages are of viewed in the front co-moving reference the steady-state pattern; (d)–(f) calframe. (a)–(c) Experimental averages of culated BIMs for the same flows. The the steady-state front (d)–(f) calnon-dimensional frontpattern; and wind speeds BIMs for the same flows. The vculated 0 and w are: 0.024 and 0.69 for (a) and non-dimensional front wind (d); 0.095 and 0.069 for and (b) and (e);speeds and v0 and w are: for (a)dots and 0.19 and 0.690.024 for (c)and and0.69 (f). Black (d); 0.095 and 0.069 for (b) and (e); and in (d)–(f) denote location of the burning 0.19 and 0.69and for (c) Black denote dots fixed points, theand red(f).curves in (d)–(f) denote location of the burning the passive invariant manifolds. fixed points, and the red curves denote the passive invariant manifolds.

Experiment

Simulation

BIM bounding the burning region

BIM bounding the burning region

Passive invariant manifold

Second, there is an excluded (darker, unreacted) region in Figure 4.30 Experimental confirmation of the role of burning invariant manifolds (BIMs) as one-sided transport barriers to which reaction fronts are attracted. Also shown (in red) is a nearby homoclinic loop of the passive fluid velocity field in the steady case. Adapted from Gowen and Solomon (2015).

4.7 Summary and Outlook In this chapter, we have surveyed principles for detecting barriers to advective transport in temporarily recurrent (i.e., steady, periodic or quasiperiodic) flows. In contrast to most fluid flows arising in practice, temporarily recurrent flows remain well defined for arbitrarily large forward and backward times. This idealized property enables a unique definition of their transport barriers as material surfaces formed by trajectories starting from codimension one, structurally stable invariant sets of their Poincaré maps. Armed with this definition, we have given a general classification of transport barriers in recurrent flows. Of these barrier types, hyperbolic barriers formed by stable and unstable manifolds can generate chaotic fluid particle motion (chaotic advection) even in simple timeperiodic flows. In contrast, elliptic barriers arising from KAM curves of Poincaré maps surround islands of regular behavior, providing unique definitions of Lagrangian vortex boundaries in this class of flows. Based on the definitions of all these barriers, the fundamental tool for their identification is an appropriate Poincaré map, defined through the temporal or spatial sampling of trajectories. In addition to Poincaré maps, we have also discussed how possible conserved quantities simplify the identification of advective transport barriers. In addition, we have briefly reviewed burning invariant manifolds in steady reactive flows. Conceptually, reactive transport differs from its advective counterpart, but an accepted 3D ODE model of reactive transport can nevertheless be analyzed with the tools of this chapter. Beyond the wealth of technological applications and lab experiments falling in the realm of temporarily recurrent 3D flows (see Speetjens et al., 2021), such flows will resurface in

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Chapter 9 in our discussion of barriers to active transport. Indeed, active barriers turn out to be invariant manifolds of steady, 3D, volume-preserving flows (barrier equations) associated with the underlying fluid flow. Those barrier equations can, therefore, be analyzed using the ideas in this chapter, as well as the LCS and OECS techniques discussed in the next chapter.

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5 Lagrangian and Objective Eulerian Coherent Structures

In Chapter 4, we discussed barriers to nondiffusive tracer transport under velocity fields that were either steady or had recurrent (periodic or quasiperiodic) time dependence. Here, we take our first step to uncover barriers to transport outside this idealized setting. To this end, we still consider nondiffusive passive tracer transport but no longer assume that the velocity field of the fluid is steady or temporally recurrent. The first question is how much we can still utilize from what we learned about transport barriers in the temporally recurrent setting of Chapter 4. Recurrent velocity fields enabled the construction of spatial or temporal stroboscopic maps (Poincaré maps) that are autonomous, i.e., are always the same between two stroboscopic samplings once the underlying sampling phases (Poincaré sections) have been fixed. Iterating an autonomous map instead of tracking the trajectories of a nonautonomous differential equation is a major simplification, which enabled us to define advective transport barriers as invariant curves of Poincaré maps in Definitions 4.1 and 4.3. These invariant curves, in turn, generated recurrent material lines or material surfaces for the full velocity field under advection by the flow map. This strategy fails for more realistic fluid flows that are temporally aperiodic and are known only over finite time intervals. First, Poincaré maps are no longer available. Indeed, while one could still sample fluid trajectories at regular time intervals or at regular spatial locations, the mapping taking subsequent samples to each other along trajectories will now be different at each sampling instant. Such a nonautonomous sequence of mappings is harder to analyze than the full, unsampled flow and it will generally have no invariant curves or surfaces. Second, nonautonomous sampling maps can no longer be iterated asymptotically given the finite-time availability of the flow data. All these preliminary deliberations lead us to the realization that the beautiful intricacies of transport in 2D time-periodic flows and 3D steady flows, as illustrated in Figs. 4.12 and 4.25, will no longer arise in temporally aperiodic flows. In particular, periodic orbits, their stable and unstable manifolds, homoclinic tangles, KAM tori and cantori can no longer be used to explain the tracer transport patterns seen in nature, which nevertheless often bear similarities with those in recurrent flows. Examples of these striking similarities with structures documented in chaotic advection are collected in Fig. 5.1, with the corresponding temporally recurrent transport barriers listed in Fig. 5.2. To understand the source of these similarities, we recall a property that we pointed out for advective transport barriers in recurrent flows in §4.1: their material coherence. While we can no longer hope for even approximately recurring material surfaces in a general unsteady flow, we can certainly look for material surfaces that remain coherent. We regard a material surface as coherent if it preserves its spatial integrity without developing smaller scales. 141

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(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

Figure 5.1 Tracer patterns framed by temporally aperiodic barriers to tracer transport in nature. (a) Spiral eddies in the Mediterranean Sea in 1984. Image: Paul Scully-Power/NASA. (b) Swimmers carried by a rip current at Haeundae Beach, South Korea, in the summer of 2012. Image: Joo Yong Lee/Sungkyunkwan University. (c) A sudden extension of the Deepwater Horizon oil spill in the Gulf of Mexico in June, 2010. Image: NASA. (d) Transport of warm water revealed by the sea surface temperature distribution around the Gulf Stream in 2005. Image: NASA. (e) Jupiter’s Great Red Spot seen from the Voyager 1 mission in February, 1979. Image: NASA/JPL; image processing: Björn Jónsson. (f) Phytoplankton boom east of Tasmania. Image: Jeff Schmaltz/MODIS Rapid Response Team, NASA/GSFC. (g) Water carried by a tornado off the Florida Keys. Image: Joseph Golden/NOAA. (h) Steam rings blown by the volcano Mount Etna in November, 2013. Image: https://www.volcanodiscovery.com/etna/photos/2013/ nov/smoke-rings.html. Adapted from Haller (2015).

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Center fixed point

Homoclinic tangle

Steady shear flow

Invariant cylinder

KAM torus

compare: subplots b,c

compare: subplot d

compare: subplots e,f

compare: subplot g

compare: subplot a

compare: subplot h



KAM curve on a Poincaré section

Figure 5.2 Steady and temporally recurrent transport barriers discussed in Chapter 4 that resemble the perceived temporally aperiodic, finite-time barriers in the referenced subplots of Fig. 5.1. Adapted from Haller (2015).

Those smaller scales would manifest themselves as protrusions from either side of the material surface without a breakup of that surface. In other words, using the terminology of §1.3 of the Introduction, we seek advective transport barriers in nonrecurrent flows as Lagrangian coherent structures (LCSs). As we have already indicated, however, the notion of coherence of an evolving material surface is subject to different interpretations. Even if we fix a general notion of coherence for a material surface, the continuity of the flow map will force very close material surfaces to behave similarly. As a result, we will not be able to find single, isolated material surfaces that show strikingly unique features (such as a stable manifold does asymptotically in time) over finite time intervals. Rather, we will seek LCSs as extremizing surfaces of some physically relevant notion of coherence over a given time interval, with the outcome of this procedure depending on the coherence notion used and the time interval selected. The LCSs obtained in this fashion will only show minute differences from the evolution of their closest neighboring material surfaces. Such an LCS will nevertheless act as theoretical centerpiece of a thin set of material surfaces perceived together as a single barrier to advective transport. The relevant time interval for LCS identification may simply be the maximal temporal length of the data set, may come from physical considerations or may be inferred by extremizing the selected coherence feature over different possible time horizons. Either way, the time scale of the analysis is simply part of the definition of the finite-time dynamical system generated by the nonautonomous, nonrecurrent, finite-time ODE problem xÛ = v(x, t),

x ∈ U ⊂ Rn,

t ∈ [t0, t1 ]

(5.1)

for an at least continuously differentiable velocity field v(x, t). This velocity field will generally be reconstructed from numerical data or experimental measurements. We will focus on the

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dimensions relevant for fluid flows (n = 2, 3) but several of the LCS techniques we will discuss also apply to finite-time dynamical systems of arbitrary (finite) dimensions. As noted in Chapter 3, we will uphold the minimal self-consistency requirement for transport barrier detection by considering only objective approaches to LCSs. In the limit of t1 → t0 ≡ t, any objective Lagrangian coherence principle or identification technique for finite-time dynamical systems turns into an objective Eulerian coherence principle or identification technique. We will refer to this instantaneous time-t limits of LCSs as objective Eulerian coherent structures (OECSs). These Eulerian structures act as LCSs over infinitesimally short time scales and hence their time-evolution is not material. Despite being nonmaterial, OECSs have advantages and important applications in unsteady flow analysis, as we will discuss separately in §5.7.

5.1 Tracer-Transport Barriers in Nondiffusive Passive Tracer Fields As a first approach, we would like to characterize observed barriers to advective transport directly from the tracer concentrations shown in Fig. 5.1. We observe transport barriers as delineators of concentration patterns because the concentration changes abruptly across these surfaces. We then view these surfaces as persistent obstacles to the homogenization of the tracer concentration. More specifically: Definition 5.1 Tracer-transport barriers in a tracer field are material surfaces along which large tracer gradients develop. This definition is illustrated in a magnified version of Fig. 5.1(d) shown in Fig. 5.3.

x = Ftt0 (x0 )

x0

rc (x, t)

observed tracer-transport barrier from the temperature field c(x, t)

Figure 5.3 Locations of large concentration gradients mark observed tracer-transport barriers into the Gulf stream in the magnified version of the image shown in Fig. 5.1(d). We express such a current location as the advected position of a fluid particle starting from an initial position x0 at time t0 .

Definition 5.1 gives us a starting point to look for tracer-transport barriers by analyzing the tracer gradient evolution. In all examples shown in Fig. 5.1, the diffusivity of the tracers is either zero or very small so, for now, we will restrict our investigations to nondiffusive tracers. In contrast, we will consider barriers to diffusive transport in Chapter 8. To proceed with this idea, we consider the conservation law along fluid trajectories for a nondiffusive tracer concentration c(x, t), which can be written as Dc = ∂t c + ∇c · v = 0, Dt

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c(x, t0 ) = c0 (x).

(5.2)

5.1 Tracer-Transport Barriers in Nondiffusive Passive Tracer Fields

145

We take the spatial gradient of both sides of Eq. (5.2) to obtain ∂t (∇c) + ∇2 c v + (∇v)T ∇c = 0,

with ∇2 c denoting the Hessian of c and the superscript T referring, as usual, to transposition. This last equation can be rewritten along any fluid trajectory as


T
D ∇c Ftt0 (x0 ) , t = − ∇v Ftt0 (x0 ) , t ∇c Ftt0 (x0 ) , t , (5.3) Dt which shows that tracer gradients satisfy the adjoint equation of variations we introduced in Eq. (2.48). Therefore, using the solution formula (2.49) for that equation, we obtain that purely advected tracer gradients at points x = Ftt0 (x0 ) evolve under the relation   −T ∇c (x, t) = ∇Ftt0 (x0 ) ∇c0 (x0 ) . (5.4) Specifically, by Eq. (5.4), the squared magnitude of ∇c evolves along trajectories according to the formula    
∇c Ft (x0 ) , t 2 = [∇c0 (x0 )]T ∇Ft (x0 ) −1 ∇Ft (x0 ) −T ∇c0 (x0 ) t0 t0 t0 D E   −1 = ∇c0 (x0 ) , Ctt0 (x0 ) ∇c0 (x0 ) , (5.5)

 T with the right Cauchy–Green strain tensor Ctt0 = ∇Ftt0 ∇Ftt0 that we first introduced in   −1 = 1, and hence at least Eq. (2.90). In an incompressible flow, we have det Ctt0 = det Ctt0  t  −1 will be larger than one of the eigenvalues of the symmetric, positive definite tensor Ct0 one in norm. This eigenvalue is also expected to be growing further exponentially in time   −1 due to the growing deformation exerted by the inverse flow map Ftt0 while it maps highly deformed material elements back to their undeformed initial positions. Equation (5.5) shows that large gradients , observed along the perceived transport barriers of the tracer field in Fig. 5.3, arise at the current time-t positions of trajectories that started   −1 from initial locations x0 where the dominant eigenvalue of the Lagrangian tensor Ctt0 (x0 ) is large. The closer the initial gradient ∇c0 (x0 ) is to the corresponding dominant eigenvector  −1  of Ctt0 (x0 ) at those initial locations, the larger the gradient ∇c will become by time t. These large gradients are generally not seen to be equal to each other along the barriers, but they visibly dominate concentration gradients in directions normal to the barriers. In other words, tracer-transport barriers appear to
form along ridges of the advected concentration t gradient magnitude field ∇c Ft0 (x0 ) , t , defined over the initial positions x0 . There are a number of different mathematical ridge definitions (see e.g., Eberly, 1996), but for our current purposes, we favor the concept of a height ridge (see Appendix A.2 for ridge definitions). We rely on the reader’s intuition for a height ridge, which we also reaffirm with Fig. 5.4. We now set the general time t to be equal to t1 , the end of a finite-time observation window for the flow. Equation (5.5) then shows that as long as the initial tracer gradients have a  t1  −1 general, nondegenerate orientation with respect to the eigenvectors of Ct0 , the ridges of
t1 the advected concentration gradient field ∇c Ft0 (x0 ) , t will coincide with ridges of the   −1 maximum eigenvalue field of Ctt10 (x0 ) when the latter eigenvalues are plotted over the   −1
t1 advected positions x = Ft0 (x0 ) . By the equality of the spectra of Ctt10 (x0 ) and Ctt01 Ftt10 (x0 ) established in formula (2.101), and by the objectivity of these spectra concluded in §3.4.3, we arrive at the following conclusion.

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Figure 5.4 A mountain ridge marked by a walkway is an example of a height ridge. Image: miriadna.com.

Proposition 5.2 Under increasing time t1 , tracer-transport barriers for nondiffusive tracers in the finite-time flow (5.1) generically align with codimension-1, i.e., (n − 1)-dimensional, ridges of the largest eigenvalue λn (x0 ; t1, t0 ) of the backward Cauchy–Green strain tensor Ctt01 (x). By the objectivity of the eigenvalues of Ctt01 (x) (see §3.4.3), Proposition 5.2 provides a frame-indifferent diagnostic principle for typical observed tracer-transport barriers. Its statement is valid at generic locations x0 under generic initial concentrations c0 (x0 ). Genericity here holds under the requirement that ∇c0 (x0 ) is not orthogonal to the dominant eigenvec  −1 tor of Ctt10 (x0 ) (see Eq. 5.5), or, equivalently, to the weakest eigenvector ξ1 (x0 ; t0, t1 ) of Ctt10 (x0 ). This requirement also implies that ∇c0 (x0 ) cannot be zero. Certain initial concentration fields will, therefore, not display visible barriers along certain portions of the ridges of λn (x0 ; t1, t0 ). The simplest example of such an initial concentration is c0 (x0 ) = const., that will develop no nonzero gradients by the conservation law (5.2) even though λn (x0 ; t1, t0 ) may well have ridges. That said, generic tracer concentrations that are free from such degeneracies will display signatures of tracer-transport barriers along the ridges of λn (x0 ; t1, t0 ) for large enough t1 . When t1 is uniformly fixed for the whole flow domain, some barriers may not yet have well-pronounced signatures in a given concentration field, even though a ridge of λn (x0 ; t1, t0 ) signals the presence of the barrier. We will go into more detail regarding the numerical computation of the eigenvalues of the right Cauchy–Green strain tensor Ctt0 in §5.2. For now, we skip those details and revisit the analysis of strange eigenmodes displayed by concentration patterns in time-periodic flows, which we discussed in §4.1.1. As we pointed out in our discussion, experimentally constructed Poincaré maps generally provide insufficient detail to locate the transport barriers delineating the periodically recurrent concentration patterns. Instead, Voth et al. (1994) followed about 800 fluorescent particles of diameter 120 mm and recorded their positions at 10 Hz, providing 40–180 images per forcing period. The resulting 12,000,000 particle positions were determined with a precision of about 40 µm. The periodicity of the flow was then exploited by considering particle positions at times t0 + kT as if they had been released at

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5.1 Tracer-Transport Barriers in Nondiffusive Passive Tracer Fields

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147

time t0 . This process yielded a total of 100,000 precise particle positions at each observational phase of the experiment. One could directly use the displacement of these particle positions over one forcing period as a discrete approximation of the Poincaré map Pt0 . It is more accurate, however, to determine the particle velocities at these positions by differentiating a polynomial fit to particle trajectories, interpolating these velocities onto a rectangular grid G0 , and advecting pseudo-particles from this grid in backward time under the interpolated velocity field. Over the grid G0 , one can then accurately compute the backward-time deformation gradient ∇Ftt01 via finite-differencing, as we will describe in §5.2. With ∇Ftt01 at hand, Voth et al. (1994) computed the backward Cauchy–Green strain tensor over a time interval equal to three times the time period T of the reconstructed velocity field. By Proposition 5.2, ridges of the corresponding eigenvalue field λ2 (x0 ; 3T, 0) should highlight transport barriers framing the strange eigenmodes seen in the tracer concentration patterns of Fig. 4.8. Over this integration time, the strongest ridges are of nearly uniform height and hence they can be extracted by a simple thresholding of the λ2 (x0 ; 3T) field. Figure 5.5(a) shows the dye concentration field at time t = 3T. In Fig. 5.5(b), the concentration field is shown at time t = 30T, with the extracted ridges of the λ2 (x0 ; 3T, 0) field superimposed in red. This confirms the arguments in this section with high accuracy: observed transport barriers extracted at time t0 = 3T as ridges of λ2 (x0 ; t0, t0 − 3T) continue to predict the skeleton of a strange eigenmode 30 periods later! As we will argue in §5.2, these transport barriers are unstable manifolds of hyperbolic fixed points of the corresponding PHYSICAL REVIEW LETTERS 24 JUNE 2002 Poincaré map.

b

c

d

(a) Field showing the stretching experienced during the past three periods at Re ! 45, and p ! 1. This is the same Figure 5.5 Material tracer-transport barriers mark the of strange phase as Fig. 2. (b) Corresponding dye image showing the concentration afterskeleton 30 periods but at eigenmodes the same phase as weakly diffusive dye experiments et al. (1994) at Reynolds number osition of (a) and (b); in thethe contour lines of the concentration fieldof areVoth aligned with the lines of large past stretching. 45. a(Left) Dye distribution at ! time of the past stretchingRe field= with dye image at a higher Re 1003T. and(Middle) p ! 5. Tracer-transport barriers extracted as ridges of the λ2 (x0 ; 3T, 0) field, superimposed at the dye distribution at time 30T. (Right) Same as the middle plot but for the Reynolds number Re = 100.

tretching. Furthermore, we find this to be come very close to a fixed point, and hence experience instant or phase [17]. stretching in the same direction over extended times. While the present, viewwe highlights importance of the singular valuesthe of uss the correspondence between thetracer-based lines Thus, have thethe following correspondence between the deformation gradient in advective transport barrier detection, it does not directly reveal fields and the fixed points of the Poincaré various objects: large future stretching marks the stable the connection these barriers manifolds; with distinguished material linesmarks (i.e., the LCSs) in themaniflow. we show the future stretchingoffield (blue) large past stretching unstable Next, we will take a fluid-trajectory-based alternative view on advective transport barriers he past stretching field (red). Many of the folds. This demonstrated ability to measure the locations clarifies relationship for astable moreand detailed classification he two sets ofthat lines cross this correspond to andofallows both the unstable manifoldsofasLCSs. a function of points of the Poincaré map (Fig. 2b). The time in complicated experimental flows allows the insights re and past stretching fields label the stable of lobe dynamics [3,4] to be applied to practical mixing anifolds of these fixed points [15]. Qualflows. An animation [17] shows the time dependence of because particles along these manifolds the superimposed past and future stretching fields, which form homoclinic and heteroclinic tangles of invariant https://doi.org/10.1017/9781009225199.006 Published online by Cambridge University Press manifolds. Careful study of this animation makes it

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5.2 Advective Transport Barriers as LCSs Some of the subplots in Fig. 5.1 reveal advective transport barriers based on their impact on nearby discrete tracers rather than on concentration fields. The magnified image in Fig. 5.6 illustrates this observation for the rip current from Fig. 5.1(b). x = Ftt0 (x0 ) x0

observed transport barrier as the locally most attracting material surface

Figure 5.6 Locations of locally most attracting material lines mark observed barriers to transport across a rip current in the magnified version of the image shown in Fig. 5.1(b). We express such a current location as the advected position of a fluid particle starting from an initial position x0 at time t0 . The shrinking transverse material lines schematically show attraction to the rip current, as evidenced by shrinking crowd size in the direction normal to the current.

More generally, one may seek advective transport barriers as material surfaces with the locally strongest impact on nearby fluid particles. We start with the following working definition for LCSs, which we will make more specific later. Definition 5.3 Lagrangian coherent structures (LCSs) are codimension-1, structurally stable material surfaces that locally extremize attraction, repulsion or shear among all nearby material surfaces over a finite time interval [t0, t1 ]. Of the LCSs covered by Definition 5.3, an attracting LCS (i.e., a codimension-1 locally most attracting material surface) is illustrated in Fig. 5.6. Unlike attraction, material repulsion remains mostly invisible as the LCS causing it is not highlighted by tracer accumulation. Repelling LCSs (i.e., codimension-1 locally most repelling material surfaces) are nevertheless important as they send tracer particles on their two sides to different parts of the phase space. An example of a repelling LCSs known from chaotic advection is the stable manifold of the saddle point shown in Fig. 5.2, as already noted in §4.1. We will be referring to attracting and repelling LCSs collectively as hyperbolic LCSs, as they are extensions of the 2D hyperbolic barriers to advective transport that we identified in time-periodic flows in §4.1.1 using Poincaré maps. Finally, shear exerted by an LCS manifests itself by predominantly tangential displacement of nearby trajectories along the LCS in comparison to other material surfaces. The shearing action of a shear LCS (i.e., a codimension-1 locally shear-extremizing material surface) can either be maximal or minimal in comparison to neighboring material lines. Maximal shearing turns out to characterize the nested, cylindrical or toroidal material surfaces that make up material vortices of the kind shown in the (e)–(h) subplots of Fig. 5.1. We will refer to codimension-1 locally most shearing material surfaces as elliptic LCSs, as they constitute extensions of the 2D elliptic advective transport barriers we discussed for Poincaré maps in

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§4.1.2. In contrast, we will refer to codimension-1 locally least shearing material surfaces as parabolic LCSs, which typically form centerpieces of jets. An example of a parabolic LCS is the theoretical, minimally shearing backbone curve of the Gulf Stream shown in Fig. 5.1. Parabolic LCSs are, therefore, the extensions of the 2D parabolic advective transport barriers we discussed for Poincaré maps in §4.1.3. This classification of LCSs is based on purely physical arguments and observations but suffices for the development of objective LCS diagnostics, as we will see next. In §5.3–5.4, we will discuss more systematic mathematical descriptions for LCSs based on a precise implementation of Definition 5.3 using methods from the calculus of variations.

5.2.1 Hyperbolic LCS from the Finite-Time Lyapunov Exponent We now seek to find hyperbolic LCSs as the locally most attracting or repelling material surfaces in the flow. Local attraction or repulsion by a material surface M(t) is best assessed by studying the evolution of infinitesimally small material perturbations ξ (t) to M(t). These perturbations evolve as solutions of the equation of variations (2.45) along trajectories x(t; t0, x0 ) ∈ M(t). These solutions are of the form ξ (t) = ∇Ftt0 (x0 )ξ (t0 ), as we discussed in §2.2.8. The magnitude of the evolving perturbation is, therefore, equal to | ξ (t)| =

q

 q

 ∇Ftt0 (x0 )ξ (t0 ), ∇Ftt0 (x0 )ξ (t0 ) = ξ (t0 ), Ctt0 (x0 )ξ (t0 ) ,

(5.6)

and hence is governed by the right Cauchy–Green strain tensor that we have encountered in our concentration-gradient-based preliminary discussion of LCSs in §5.1. By formula (5.6), initial perturbations aligning with the unit dominant eigenvector ξn of √ Ctt0 (as defined in Eq. (2.95)) will be stretched by the largest possible factor, λ2 , as we concluded in formula (2.97). By the same formula, initial perturbations aligning with p the √ weakest eigenvector ξ1 of Ctt0 will be compressed by the factor λ1 . By definition, λ j are the singular values of the deformation gradient ∇Ftt0 (x0 ) (see §2.3.1). The (positive or negative) average growth exponents inferred from the evolution of these singular values are called the finite-time Lyapunov exponents (or FTLEs) associated with the underlying trajectory starting from x0 at time t0 . Often, however, the term FTLE is used simply in reference to the growth exponent FTLEtt10 (x0 ) of the largest singular value λn , defined as FTLEtt10 (x0 ) =

1 log λn (x0 ; t0, t1 ), 2 |t1 − t0 |

(5.7)

with n = 2 or n = 3 for fluid flows. We speak of forward FTLE when t1 > t0 and backward FTLE when t1 < t0 . The latter type of FTLE involves computing the tensor Ctt0 along backward-running trajectories. In either case, FTLEtt10 (x0 ) is an objective Lagrangian scalar field in the sense of §3.4.1. For a more general discussion on Lyapunov exponents, we refer to Appendix A.3.

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Repelling LCSs, defined as locally most repelling codimension-1 material surfaces, are expected to be marked by codimension-1 surfaces of locally largest forward FTLE values. Such codimension-1 surfaces, M(t), are material curves in 2D flows and material surfaces in 3D flows. While FTLE values are not required to be equal along M(t), the FTLE values should reach a maximum along M(t) in directions normal to M(t). Just as in §5.2, these considerations bring us to the following objective diagnostic principles. Proposition 5.4 For t1 − t0 large enough, time t0 positions of repelling LCSs will generically align with codimension-1 ridges of the largest eigenvalue λn (x0 ; t0, t1 ) of the forward Cauchy– Green strain tensor Ctt10 (x0 ), or, equivalently, with the ridges of FTLEtt10 (x0 ). Similarly, for t1 −t0 large enough, time t0 positions of attracting LCSs will generically align with codimension-1 ridges of the largest eigenvalue λn (x0 ; t1, t0 ) of the backward Cauchy–Green strain tensor Ctt01 (x), or, equivalently, with the ridges of FTLEtt01 (x). This proposition provides an objective diagnostic principle because the eigenvalue fields of Ctt10 (x0 ) are objective, as we concluded in §3.4.3. According to this principle, one may reasonably expect forward and backward FTLE ridges to highlight the locally most repelling and attracting material surfaces, respectively. This is indeed the case if the maximal stretching measured locally by the FTLE field occurs in a direction transverse to such a material surface, as opposed to tangent to it, and hence the FTLE indeed characterizes the repulsion from the surface. As this is not necessarily the case, the converse of Proposition 5.4 does not hold: ridges of the FTLE field may not represent hyperbolic LCSs, as we discuss next.

5.2.2 FTLE Ridges Are Necessary (but Not Sufficient) Indicators of Hyperbolic LCS As noted at the end of the previous section, we cannot automatically associate all FTLE ridges with repelling or attracting LCSs, despite suggestions to this end by several authors, starting with the work of Shadden et al. (2005) and Lekien et al. (2007). Indeed, consider, for instance, the 2D incompressible flow xÛ = tanh y + 2, yÛ = 0,

(5.8)

which is a parallel shear flow in the horizontal direction with a line of maximal shear at y = 0. This model flow has explicitly solvable trajectories, which enables an explicit calculation of the flow map and the Cauchy–Green strain tensor. As calculated by Haller (2011), the dominant eigenvalue of Ctt10 (x0 ) with x0 = (x0, y0 ) is r 1 1 2 2 0 (t1 − t0 )4 (tanh0 (y0 ))4 + 1, λ2 (x0 ) = (t1 − t0 ) (tanh (y0 )) + 1 + 2 4 which has an x0 -independent maximum at y0 = 0 both for t1 > t0 and for t1 < t0, given that the derivative tanh0 (y0 ) has a global maximum there. As a consequence, both FTLEtt10 (x0 ) and FTLEtt01 (x0 ) have a ridge of constant height along the x-axis for any choice of the initial and final  time. The associated maximal stretching, however, is tangential to the material line M = (x, y) ∈ R2 : y = 0 and hence M is neither repelling nor attracting, as seen in Fig. 5.7.

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Fig. 5. The backward FTLE field admits a trough along the x axis, which is an observable attracting LCS of example (13). The integration time was selected as t t0 = 0.7450. 5.2 Advective Transport Barriers as LCSs

Fig. 8. FTLE ridge at x = This ridge remains at the s flow. 151

forward and backward FTLE ridge

Note, however, th of (16) crosses it ort flux per unit length t with the flux formu O (1/|t t0 |) flux thr An area-preservin x˙ = 1 + tanh2 x, 2 tanh x y˙ = y. cosh2 x

Figure 5.7 The geometry of the shear flow (5.8) with the deformation of a material blob superimposed. This example shows that FTLE ridges do not necessarily markA numerical computa hyperbolic LCSs. Indeed, both the forward and the backward FTLE have a ridge at thedevelopment of a ridg line of maximal shear at y = 0, which is nevertheless not an attracting or a repelling Fig. 6. A parallel shear flow with an FTLE ridge that is not a repelling LCS, but a times and for any t0 LCS. Adapted from Haller (2011). shear LCS, as shown by the deformation of a set of initial conditions.

exists (cf. Fig. 8). The predicted by formul The simple example (5.8) shows that while hyperbolic LCSs are highlighted by FTLE opposed to the correc

ridges, certain parts of a computed FTLE ridge may just indicate maximal shear along the ridge, as opposed to maximal repulsion from the ridge. This is often the case in high-shear Summary regions near flow boundaries, as illustrated in Fig. 5.8 by a forward FTLE calculation on2.4. a numerical ocean model. In the coastal area, many of the smaller, parallel running, closely aligned ridges tend to indicate curves of maximal shear, or a combination of shear and repulsion.The above exampl

1. Observable LCS a Specifically, Fig. 9 literature for equa applicable, as illus 2. Ridges of the FTL Specifically, FTLE (see Example 3), o any underlying co Fig. 7. A flow with a forward FTLE ridge that has no LCS in its proximity. 3. Ridges of the FTLE structure (see Exam 2.3.4. Example 4: FTLE ridge that is far from any LCS 4. The flux formula p Consider the dynamical system Figure 5.8 Forward FTLE calculation from a numerical ocean model for the Alboran applicable: its high 2 ˙ x = 1 + tanh x , (16) Sea within the 2019 CALYPSO Real-Time Sea Experiment. Image: MIT MSEAS larger than its exp group, http://mseas.mit.edu. y˙ = y, (cf. Examples 4 an whose phase portrait is shown in Fig. 7, along with the deformation

In the following, One needs to keep in mind, therefore, that FTLE ridges are necessary but not sufficient of a set of initial conditions under the flow. As we show in resolves conditions for hyperbolic LCSs. For a definitive conclusion about their meaning, one would these incons Appendix B, the FTLE field computed forofthis flowridges. admitsAsa shown ridge by Haller also have to verify the normal repulsion or attraction those andGreen large enough t > instantaneous t0 . along the y axis for anyetchoice of t0 and (2000) and implemented by Mathur al. (2007) et al. (2007), 3. Hyperbolicity of m Defining an LCS as a time-evolving surface which, at any time t, is ensured normal repulsion of the material line M(t) advected from a ridge under the flow coincides with a ridge of the FTLE field ⇤tt +T , would render the y axis a repelling LCS of the form

Mˆ (t ) ⌘

⇢✓ ◆

0 2 R2 : y 0 2 R . y0

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(17)

In our setting, LC time-interval [↵, ]. properties, therefore hyperbolicity that are

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by the positivity of the inner product hnt , Snt i along M(t). Here, nt is a time-evolving unit normal to M(t) and S is the rate-of-strain tensor defined in Eq. (2.1). We also note that the linear incompressible saddle flow xÛ = x, yÛ = −y

(5.9)

is sometimes regarded as a counterexample to the claim that FTLE ridges are not even necessary conditions for hyperbolic LCSs. This assertion stems from the observation that both the forward and the backward FTLE fields are globally constant (and equal to 1) for any choice of t0 and t1 in the flow (5.9). This is then usually contrasted with the fact that the origin has an unstable manifold along the x-axis and a stable manifold along the y-axis that one expects to be marked by ridges. However, each trajectory (x(t), y(t)) = (x0 et , y0 e−t ) of the ODE (5.9) has co-moving horizontal and vertical unstable and stable manifolds. Indeed, we can pass to a frame comoving with such a trajectory by the Galilean coordinate change ( x, ˜ y˜ ) = (x − x0 et , y − y0 e−t ). The trajectory is now represented by the origin ( x, ˜ y˜ ) = (0, 0) in this moving frame and the transformed equations of motion, xÛ˜ = x, ˜ yÛ˜ = − y˜ are identical to those of the flow (5.9). Consequently, there is an infinite family of horizontal material lines in the phase space of flow (5.9) that attract at the same rate. Likewise, there is an infinite family of vertical material lines that repel at the same rate. Because of equal attraction and repulsion rates all over the phase space, there is no LCS in this flow, which is consistent with the lack of FTLE ridges. We note that the only distinguishing feature of the stable and unstable manifolds of the origin in (5.9) is that these two material lines contain trajectories that are bounded for all times in at least one time direction. Boundedness, however, is a frame-dependent property and hence should not enter our detection of LCSs. Indeed, the stable and unstable manifold of any trajectory becomes bounded in at least one time direction in the frame co-moving with the trajectory, as we have just seen. A repelling LCS may be unsteady even in a steady flow. An example illustrating this is the 2D flow xÛ = 1 + a tanh2 x, (5.10) tanh x y, yÛ = −2a cosh 2 x which is an incompressible version of the example considered by Haller (2011). As Fig. 5.9 illustrates, a repelling LCS coinciding at t = 0 with the y-axis moves to the right and establishes itself as the most repelling material line in the flow by t = 0.6. Accordingly, the field FTLE0.61 0 (x0 ) develops a ridge along the y-axis by that time. The example (5.10) also illustrates that t0 -dependent ridges produced by sliding-window FTLE fields of the form FTLEtt00 +T (x0 ), with t0 varying and T fixed, are generally not material curves or even near-material curves. Indeed, the velocity field (5.10) is steady and hence FTLEtt00 +T (x0 ) will not depend on the initial time and produce the same ridge shown in Fig. 5.9 for all choices of t0 . Therefore, if FTLE ridges in such sliding-window calculations generally indicated tracked material lines, then the y-axis would have to remain fixed under advection. This is clearly not the case, as seen in Fig. 5.9, yet sliding-window FTLE-calculations are commonly used in the literature to infer the time-evolution of LCSs. This practice is only justifiable if the underlying velocity field is time periodic and the sliding window length T is an integer multiple of that period. In that case, the algorithm of Brunton and Rowley (2010)

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5.2 Advective Transport Barriers as LCSs y

trajectory positions at time t=0.6

initial conditons at time t=0

y

x

x

y forward FTLE

x

Figure 5.9 (Top) An unsteady repelling LCS (i.e., a material line M(t) shown in black, advected to the right by the flow) in the steady planar flow (5.10). The positions of trajectories (blue), released from an initially uniform grid in the domain [−2, 2] × [−20, 20] at t0 = 0, reveal that M(t) is the most repelling material line (a unique repelling LCS) in this flow. All horizontal material lines remain horizontal but shrink substantially in the vertical direction near M(t) due to incompressibility. (Bottom) For increasing t1 , the forward FTLE plot indeed shows the development of a ridge that converges quickly to the y axis, correctly indicating with increasing accuracy the initial position of M(t) at t = 0. In the plot, we show FTLE0.61 0 (x0 ) for the parameter value a = 40 selected in Eq. (5.10).

offers substantial speed-up by eliminating the repeated computations of the flow map over overlapping parts of adjacent snapshots of the sliding window. Finally, example (5.10) also disproves the flux formula of Shadden et al. (2005); Lekien et al. (2007) who use their formula to argue that converged FTLE ridges have very small flux through them. As Haller (2011) points out, that flux formula would give a decreasing  material flux of order O |t1 − t0 | −1 through the y axis in example (5.10). In contrast, the Û x=0 = 1 at all times. actual flux through the y-axis can be computed directly as x|

5.2.3 Extraction Interval and Convergence of the FTLE The FTLEtt10 (x0 ) field is associated with the finite-time dynamical system (5.1), whose definition includes the finite time interval [t0, t1 ]. Changes in this time domain of definition change the dynamical system and hence the values and the topology of the FTLE field will change too. Therefore, no convergence can be expected in FTLEtt10 (x0 ) as t1 → ∞ unless the velocity field v(x, t) is temporally recurrent (steady, periodic, quasiperiodic) in time. In that case, the dynamical system (5.1) is either autonomous or can be viewed as autonomous system on an extended phase space P, as seen in Chapter 4. In these autonomous cases, the evolution of the trajectories is governed by an autonomous flow map Ft on P that can be sampled at multiples of a fixed time interval T. If the original

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flow preserves a measure µ (e.g., volume or mass) and FT maps a region D ⊂ P of finite measure into itself, then Oseledec’s multiplicative ergodic theorem (Oseledec, 1968) applies to FT . This theorem will guarantee the existence of the limit Λk (x0 ) = lim

t1 →∞

1 log λk (x0 ; t0, t1 ), 2 |t1 − t0 |

k = 1, . . . , n

for all the finite-time Lyapunov exponents (including the maximal one, FTLEtt10 (x0 )) for almost all x0 ∈ D with respect to the measure µ (see Appendix A.3 for a precise statement of Oseledec’s theorem). For instance, for all points x0 on the stable manifold of a fixed point p of a Poincaré map in a time-periodic flow, FTLEtt10 (x0 ) will converge to the same value as t1 → ∞. If, however, the assumptions made to reach this conclusion (i.e., temporal recurrence, measure preservation and boundedness of the trajectory in P) do not hold, then an asymptotic limit for FTLEtt10 (x0 ) is not guaranteed to exist. Indeed, Ott and Yorke (2008) construct examples of 2D steady compressible flows in which FTLEtt10 (x0 ) has no asymptotic limit on large open sets even though all trajectories in these sets are bounded (see Appendix A.3 for such an example).

5.2.4 Numerical Computation of the FTLE By Proposition 5.4, initial positions of repelling LCSs defined over the interval [t0, t1 ] create ridges in the forward FTLE field. Similarly, time t1 positions of attracting LCSs form ridges in the backward FTLE field FTLEtt01 . In actual flow analysis, however, the converses of these statements become relevant: Do ridges of the FTLEtt10 and FTLEtt01 indicate repelling and attracting LCSs? While we have seen that such converse statements cannot generally be concluded without exceptions, FTLE fields do provide a powerfully simple and effective way to perform a first-order discovery of hyperbolic barriers in a flow. The numerical computation of FTLE hinges on a numerical approximation of the deformation gradient field ∇Ftt0 (x0 ). Solving the equation of variations (2.45) directly along trajectories to obtain ∇Ftt0 (x0 ) tends to be a numerically challenging procedure. Instead, Haller (2001a) proposed to generate an array of trajectories, x1 (t; t0, x0 ) © ª x(t; t0, x0 ) = ­ x2 (t; t0, x0 ) ® , « x3 (t; t0, x0 ) ¬

x10 © ª x0 = ­ x20 ® ∈ G0 ⊂ U, « x30 ¬

by solving the finite-time ODE (5.1) numerically for initial conditions taken from a rectangular grid G0 covering the flow domain U. Assuming a uniform rectangular grid G0 for simplicity, we denote the spacing of this grid by ∆i in the xi0 direction and introduce the grid spacing vector

δi = ∆ i ei ,

i = 1, 2, 3,

with no summation implied over the index i. Then a finite-difference approximation for the deformation gradient ∇Ftt10 (x0 ) can be computed as

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5.2 Advective Transport Barriers as LCSs x1 (t1 ;t0 ,x0 +δ1 )−x1 (t1 ;t0 ,x0 −δ1 )

x1 (t1 ;t0 ,x0 +δ2 )−x1 (t1 ;t0 ,x0 −δ2 )

155

x1 (t1 ;t0 ,x0 +δ3 )−x1 (tt;t0 ,x0 −δ3 )

2∆1 2∆2 2∆3 © ª ­ x2 (t1 ;t0 ,x0 +δ1 )−x2 (t1 ;t0 ,x0 −δ1 ) x2 (t1 ;t0 ,x0 +δ2 )−x2 (t1 ;t0 ,x0 −δ2 ) x2 (t1 ;t0 ,x0 +δ3 )−x2 (t1 ;t0 ,x0 −δ3 ) ® ­ ® . (5.11) 2∆1 2∆2 2∆3 ­ ® x3 (t1 ;t0 ,x0 +δ1 )−x3 (t1 ;t0 ,x0 −δ1 ) x3 (t1 ;t0 ,x0 +δ2 )−x3 (t1 ;t0 ,x0 −δ2 ) x3 (t1 ;t0 ,x0 +δ3 )−x3 (t1 ;t0 ,x0 −δ3 ) 2∆1 2∆2 2∆3 « ¬ t1 For the purposes of revealing the locations of ridges in the eigenvalue fields of Ct0 , this simple calculation returns surprisingly robust results, even if numerical errors in approximating the entries of ∇Ftt10 via approximation (5.11) grow substantially for larger t values. Once the numerical approximation (5.11) for ∇Ftt10 (x0 ) is available, the FTLE defined in Eq. (5.7) can be obtained first by substituting this approximation into the formula (2.90) defining Ctt10 (x0 ). One then solves the characteristic equation of Ctt10 (x0 ) pointwise for its eigenvalues λ j (x0 ; t0, t1 ) and selects the largest, λ3 (x0 ; t0, t1 ), to compute the FTLE field from formula (5.7) with n = 3. For 2D flows (n = 2), the approximation to ∇Ftt10 (x0 ) will comprise only the 2 × 2 main minor matrix of the approximation (5.11), in which case the characteristic equation of Ctt10 (x0 ) is quadratic. This computation in implemented in Notebook 5.1.

Notebook 5.1 (FTLE2D) Computes the finite-time Lyapunov exponent (FTLE) field via singular-value decomposition (SVD) for a 2D unsteady velocity data set. https://github.com/haller-group/TBarrier/tree/main/TBarrier/2D/ demos/AdvectiveBarriers/FTLE2D p A more direct computation of the FTLE targets λn (x0 ; t0, t1 ) as the largest singular value of the numerical approximation (5.7) to ∇Ftt10 (x0 ). Singular-value decomposition (or SVD; see §2.3.1) of the deformation gradient was proposed by Greene and Kim (1987) for computing Lyapunov exponents, but was apparently employed first by Karrasch (2015) to compute LCSs (see also Karrasch et al., 2015). As recalled by Karrasch (2015) from Trefethen and Bau (1997), SVD computations for ∇Ftt10 are generally more stable numerically than eigenvalue computations for Ctt10 and hence are preferable for numerically and experimentally generated vector fields. A further advantage of SVD in the present context is that beyond the singular √ values λi , it will simultaneously render the corresponding left and right singular vectors of the deformation gradient, as we see from the SVD formula (2.108). These singular vectors will be needed in some of the more advanced LCS theories reviewed later in this chapter. The computation of FTLE via SVD for 3D flows is implemented in Notebook 5.2. Notebook 5.2 (FTLE3D) Computes the finite-time Lyapunov exponent (FTLE) field via singular-value decomposition (SVD) for a 3D unsteady velocity data set. https://github.com/haller-group/TBarrier/tree/main/TBarrier/3D/ demos/AdvectiveBarriers/FTLE3D The algorithm of Tang et al. (2010) enables the reliable extraction of FTLE ridges if the velocity is only available on a non-invariant flow domain. The algorithm then smoothly extends the velocity field to a closest-fitting linear velocity field outside the domain. This extension eliminates spurious ridges arising from algorithms that stop trajectory integration at the domain boundary. Finally, we note that in an effort to combine the computation of the velocity field and the FTLE field, Nelson and Jacobs (2015) develop an algorithm that

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computes FTLE fields simultaneously with the time integration of discrete Galerkin-methodbased flow solvers.

5.2.5 Extraction of FTLE Ridges The simplest way to extract ridges of the FTLE field is ridge thresholding: one keeps only those points with FTLE values over a selected high threshold, as in Fig. 5.5. This extraction will undoubtedly lose some of the weaker ridges and will only capture parts of the stronger ridges that reach the required minimal value. For a quick assessment of the most dominant ridges, however, even thresholding is effective enough. The more refined approach of Mathur et al. (2007) is based on the observation that ridges of FTLEtt10 (x0 ) are attractors of the gradient dynamical system1 d x0 (s) = ∇FTLEtt10 (x0 (s)). ds

(5.12)

Indeed, by the growth of FTLEtt10 (x0 ) towards the ridge, the gradient field ∇FTLEtt10 guides all trajectories of the autonomous differential equation (5.12) from a vicinity of any ridge towards the ridge. One can, therefore, launch trajectories of (5.12) in ridge neighborhoods identified from a rough thresholding and simply keep the endpoints of those trajectories as approximation of those ridges after a long enough integration with respect to the evolutionary variable s of system (5.12). Further details of this algorithm are described in Mathur et al. (2007). Examples of repelling and attracting LCSs visualized by numerically extracted FTLE ridges in nonrecurrent 2D and 3D flows are shown in Fig. 5.10. A further application of FTLE analysis to a 3D turbulent channel flow is given by Green et al. (2007), who seek to determine the Lagrangian footprint of hairpin vortices identified previously only from Eulerian considerations in the instantaneous velocity field (see Zhou et al., 1999). Figure 5.11 shows how the threshold-free, frame-indifferent visualization via backward FTLE reveals the bounding material surfaces of a hairpin vortex, in previously unseen detail, as attracting LCSs. Another application of FTLE analysis of passive pollution control in Monterey Bay, California is given by Lekien et al. (2005). They use surface velocity fields reconstructed from high-frequency coastal radar stations to compute the forward FTLE field in the Bay. They then demonstrate by simulations that a timed pollution release scheme based on the location of the most influential FTLE ridge speeds up the clearance of pollutants into the open ocean (see §6.2.1 for details). Shadden (2011) reviews further aspects and applications of FTLE computations in specific flow problems.

5.2.6 Hyperbolic LCSs vs. Stable and Unstable Manifolds in Temporally Recurrent Flows We now examine the relationship between hyperbolic LCSs extracted as FTLE ridges and stable and unstable manifolds of fixed points of Poincaré maps for time-periodic flows. As 1

This observation is used in a more rigorous ridge definition by Karrasch and Haller (2013) that ensures the robustness of the ridge under small perturbations to the data set (see Appendix A.2).

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FL47CH07-Haller

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a

Annu. Rev. Fluid Mech. 2015.47:137-162. Downloaded from www.annualreviews.org Access provided by 80.218.229.121 on 01/08/15. For personal use only.

Repelling LCS Attracting LCS

c

b

Repelling LCS

Attracting LCS Attracting LCS

Figure 5

Repelling LCS

Figure 5.10 Examples of advective transport barriers (hyperbolic LCSs) identified

(a) The Lagrangian skeleton of turbulence in a two-dimensional rotating flow experiment. Repelling Lagrangian coherent structure from forward and backward identified FTLE ridges. (a) The Lagrangian skeleton of (LCS) candidates (red ) are ridges of the forward-time finite-time Lyapunov exponent (FTLE). Attracting LCS candidates (blue) are turbulenceFTLE. extracted Mathur et with al. (2007) fromfrom a 2D rotating flow experiment. Here, ridges of the backward-time Panelby a reproduced permission Mathur et al. (2007). (b) A similar computation for the repelling LCSsa cylinder, (red) arewith marked by ridges thesurface forward FTLE field. Attracting LCSs and street behind the height of the of gray representing the maximum of forwardvon K´arm´an vortex (blue)Panel are marked by ridges of the backward FTLE A similar(red computation backward-time FTLEs. b reproduced with permission from Kasten et al. field. (2010).(b) (c) Repelling ) and attracting (blue) LCSs for a perturbed four-vortex-ring for the 2002 splitting of the Antarctic Panelac cylinder. reproducedThe withheight permission from Lekien by Kasten model et al. (2010) for the von Kármán vortexozone streethole. behind & Ross (2010). of the gray surface represents the maximum of the forward and backward FTLEs. (c)

Repelling (red) and attracting (blue) LCSs computed by Lekien and Ross (2010) for a perturbed four-vortex-ring model of the 2002 splitting of the Antarctic ozone hole. also note that contours of the FTLE align closely with unstable manifolds of classic, infinite-time Adapted from Haller (2015).

dynamical systems. Provenzale (1999) and Bowman (2000) put forward the idea to use relative dispersion, a discretized form of the FTLE, to highlight Lagrangian features. Although relative dispersion as a statistical tool has well-established properties (LaCasce 2008), its spatially discretized nature has prevented attempts to relate its features mathematically to the stability ofis material surfaces. noted in §4.1.1, the observed impact of unstable manifolds on nearby trajectories attraction. The first connecting FTLE ridges repellinglines LCSsare through the above arWhen advected under theworks full flow map, therefore, thesetomaterial expected to act heuristic as gument were by Haller (2001a, 2002). Next, the FTLE was applied to LCS extraction from attracting LCSs. Likewise, stable manifolds in Poincaré maps repel nearby trajectories and laboratory experiments by Voth et al. (2002) and then from turbulent flow experihence should time-periodic be repelling LCSs under advection by the flow map. ments by Mathur et al. (2007), shown in Figure 5a. Further high-end visualizations of LCSs via Without having to construct and iterate Poincaré maps, therefore, we can also directly FTLE ridges by Kasten et al. (2010) and Lekien & Ross (2010) are shown in Figure 5b,c. extract advectiveA number transport barriers as LCSs in time-periodic flows via FTLE ridges, as of applications to numerical and experimental data followed, as surveyed by Peacock indicated in Chapter in connection Fig. Note, however, thatRecent initial studies conditions in in astro& Dabiri4(2010), Shadden with (2012), and4.10. Peacock & Haller (2013). of LCSs our selected numerical grid G fall on stable and unstable manifolds with probability zero, 0 physical plasmas via FTLE ridges appear in Yeates et al. (2012), Rempel et al. (2013), and Chian and hence trajectories starting from G0work willbybeKelley off the stable and unstable manifolds et al. (2014). Experimental et actual al. (2013) suggests that FTLE ridges also tend to with probability one. nearbyenergy initialflux conditions ∈ G0 lying on opposite sides mark theConsequently, zeros of scale-to-scale in weaklyx0turbulent, two-dimensional flows.

of a stable manifold will be repelled from each other in forward time. Similarly, nearby initial 148

Haller

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Lagrangian and Objective Eulerian Coherent Structures Detection of Lagrangian coherent structures in three-dimensional turbulence115 Streamwise cut of backward FTLE ( (a) (a)

(b)

–1.0 –0.5 0

0 –1.0 0.2 –0.5 –1.0 6 y 0.4 0 5 –0.5 4 6 3 y0.60.5 0 2 x 5 1.0 4 1 0.5 0.8 in 4.0three-dimensional 3 turbulence115 3.5 3.0 2.5 2.0 0 structures 2 x 1.0 z 1 1.0 (b)4.0 3.5 0 0 3.0 2.5 2.03.0 2.5 3.5 4.0 ( (b) 0 (c) z z 0.2 0.2 0 (c) Spanwise cut of backward FTLE y(d)0.5 0.44.0 0.4 0 3.5 y1.0 0.6 0.5 1 0.6 2 3 4 6 z x 3.0 0.8 1.0 10.8 2 3 4 6 6 x 1.02.5 2.5 3.0 3.5 4.0 1 2 z 3 plots4 of the6 isolated 1.0 Figure 3.0 Two-dimensional DLE x 2.5 3.0 3.5 2

6 coherent structures in three-dimensional turbulence Detection of Lagrangian y 5

Detection of Lagrangian 4coherent 0.5 (a) 1.0 (a)

–1.0 –1.0 0 –0.5 –0.5 yy 0.5 0 0 y

4.0 3.5 3.0 2.5 2.0 0 z

1

2

3

x

(c)

1.0 0.5 1.0 0.5

4

3 1 2 2 3 x 1 4.0 3.5 3.0 2.5 2.0 0 x 2 1 z 0

5

6

3

44

5

6

x hairpin evolve 4.0 ( Wall-normal cut of backward FTLE of the three planes, with the location maximum λci superimposed 4.0 3.5 3.0 2.5 2.0 3. Two-dimensional DLE plots of z the isolated hairpin evolve (c) (dFigure + (c) constant-spanwise cut, and (d) constant wall-normal cut ( 4.0 2 z Figure 3. Two-dimensional DLE plots of the isolated hairpin evolved to with t =the 63.location ( a) 10 of % the three planes, ( maximum λci superimposed withofthe location of three planes, (b)cut, constant-streamwise cut, maximum λ2ci superimposed 0 3.5 (c) the constant-spanwise and (d) constant wall-normal cut ( signature (c) Eulerian ( d) z y(c)0.5 constant-spanwise cut, and (d) constant wall-normal cut 4.0 ( y + = 98). 3.0 hairpin vortex in the 0 1.0 3.5 2 1 2 ci 3 4 6 same2.5size as z the DNS grid, and linear interpolation is used to y 0.5 x 2 For 3the 4linear 6 1 3.0 between grid points. plots of interpolation the isolated hairpin, thd same0 size as the DNS grid, and is used to x same is used to determine the velocity is between greater by a factor ofFor 6 the in allplots three dimensions. Only a por 1.0 size as the DNS grid, and linear interpolation grid points. of the isolated hairpin, the 2.5 1.0

Figure 5.11 (Left) Eulerian signature of a hairpin vortex in the swirling strength field 2 (see §3.7.1), visualized as a level set corresponding to the 10% of the maximal λci 2 value. (Right) Cuts of the 3D backward FTLE field along the three planes shown λci on the left. Adapted from Green et al. (2007).

conditions lying on opposite sides of the unstable manifold will be attracted to each other in forward time, as we illustrate in Fig. 5.12. Figure 5.12, however, also illustrates an inherent asymmetry in the extraction of attracting and repelling LCSs from flow data. While typical small perturbations to a trajectory in the stable manifold W s (p) will grow exponentially for all times, the forward-time behavior of typical small perturbations to a trajectory in the unstable manifold W u (p) is generally unknown. Indeed, W u (p) will generally lose its exponential attraction away from the fixed points. This is why initial, time t0 positions of stable manifolds are marked by ridges of the forward FTLE, whereas final, time t1 positions of unstable manifolds are marked by ridges of the backward FTLE.2

5.2.7 Repelling and Attracting LCSs from the Same Calculation As we have seen, repelling LCSs at time t0 are highlighted by ridges of FTLEtt10 (x0 ), whereas attracting LCSs at time t1 are highlighted by ridges of FTLEtt01 (x). Therefore, locating both types of LCSs in this fashion requires two separate numerical runs, involving the forward 2

Launching initial conditions very close to p near W u (p) would actually result in an FTLE ridge even along a short segment of W u (p) if t1 − t0 is not too large. This is due to exponential stretching along W u (p) that nearby trajectories experience while they are still close to p (see Karrasch, 2015, for a detailed simulation illustrating this).

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159

5.2 Advective Transport Barriers as LCSs

W u (p)

t +T

Ft 0 0

t −T

p

G0

Ft 0 0

W s (p)

Figure 5.12 Repelling and attracting LCSs associated with a fixed point p of a Poincaré map in time-periodic flows (see §4.1) are expected to be highlighted by forward-time and backward-time FTLE ridges. This is due to the evolution of distances between initial conditions released near W s (p) and W u (p) in forward and backward time, respectively, from a grid G0 .

and the backward advection of two separate sets of initial conditions. This can be a taxing undertaking, especially for 3D unsteady flows. Haller and Sapsis (2011) observe that we can obtain the same information from a single numerical run. To see this, we first define the smallest forward FTLE, Γtt01 (x0 ) =

1 log λ1 (x0 ; t0, t1 ), 2 |t1 − t0 |

(5.13)

over evolving trajectory positions. From the relation (2.101), one obtains that FTLEtt01 (x) = −Γtt01 (Ftt01 (x)).

(5.14)

If one uses SVD to obtain the singular values of the deformation gradient (see §5.2.4), then one also obtains the smallest Cauchy–Green eigenvalue λ1 (x0 ; t0, t1 ) from the same calculation, and hence Γtt01 (x0 ) is readily available from (2.101). Equation (5.14) then shows that from the same numerical run that generates the forward FTLE field, one can also identify attracting LCSs as trenches (or inverted ridges; see §A.2 for definitions) of the smallest FTLE field Γtt01 (x0 ) graphed over current trajectory positions x. Haller and Sapsis (2011) also note that for 2D incompressible flows, formula (5.14) implies FTLEtt01 (x) = FTLEtt10 ((Ftt01 (x)),

(5.15)

given that λ1 λ2 ≡ 1 holds for the Cauchy–Green eigenvalues. In such flows, therefore, repelling LCSs at time t0 are highlighted by the ridges of the maximal FTLE graphed over x0 , whereas attracting LCSs at time t1 are highlighted by the ridges of the maximal FTLE field graphed over the final particle positions x. This one-off computation saves time but also has a shortcoming: formula (5.15) only yields attracting LCSs at time t1 at locations visited by the particles released from the initial grid G0 at t0 . As a consequence, FTLE plots will be limited to a smaller spatial domain and

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orbit(We in the FTLE calculation. showinnothe color where orbit in thex_ 1FTLE show no color where (We orbit FTLE calculation. (W x_ 1 ¼ x2 ; x_ 1 ¼ x2 ; ¼ x2 ; calculation. ! " numerical integration we ! " !numerical " up.) the used integration we used blew up.) By contrast, the blew By contrast, the numerical integration we used 3 2 3 2 _ 3 2 ¼ x " x þ 05x 1 " x þ 01 sin t: x _ ¼ x " x þ 05x 1 " x þ 01 sin t: x _ ¼ x " x þ 05x 1 " x þ 01 sin t: x 2 1 2 2 1 2 2 1 2 1 1 1 1 160 Lagrangian and Objective Coherent 1 1 Fig. 3(b)Structures shows results obtained from the application of Fig. 3(b) Eulerian shows the results obtained from thethe application of Fig. 3(b) shows the results obtained formula (14), with the forward field computed formula (14), with the forward FTLE field computed from FTLEformula (14), withfrom the forward FTL This two-dimensional system admits antorus attracting torus (due This system admits an attracting (due two-dimensional This system admits an attracting two-dimensional torus (due t0 same ¼ 0 to 6 with the time step. example illusstep. This example illust0 ¼ 0 to t ¼ 6 with the samet0time 0 toover t ¼ 6 with the same time s t0 ¼This willderhave lower, nonuniform resolution a full calculation oft ¼FTLE (x j )tangle initialized to encircles the van der Pol term) that encircles a homoclinic tangle to the van Pol term) that a homoclinic tangle tothan the van der Pol term) that encircles homoclinic tgiven 1 strong trates that ina case instabilities given trates that in case of strong instabilities in a of time trates in thata in casetime of strong instab the Duffing-type homoclinic tangle is arising from the Duffing-typearising term. from The homoclinic tangleterm. is The arising from the Duffing-type term. The homoclinic tangle is the lead smallest indirection, the otherusing time the direca uniform gridformed of manifolds points xofjaand ∈ G time t ofwould All toFTLE interpolation direction, using the have. smallestdirection, FTLEthis in using thecan other time direcsmallest FTLE 1 at manifolds by the stable unstable athesmall-amformed by the stable and unstable small-amformed1 by stable and unstable manifolds of a small-amtion view may give more present balanced of all LCSs tion may give a more balanced of allaLCSs inview tion may givepresent a moreinbalanced view plitude periodic orbit near the origin. In this example, the plitude problems periodic orbit in nearthe the origin. In this example, the plitude periodic orbit near the system. origin. In example,intheFig. 5.13(b). FTLE plots computedthe over advected positions, asthisshown the system. the system. smallest FTLE field turns of outtheto yield better FTLE imagesfield of the smallest FTLE field turns out to yield better images smallest turns out to yield better images of the t1 These problems can nevertheless be mitigated by interpolating the values FTLE ((Ftt01 (xi )) t attractor than the largest FTLE. attractor than the largest FTLE. attractor than the largest FTLE. D.around Experimental flow field around0D. a jellyfish D. Experimental flow field a jellyfish Experimental flow field aroun In Fig. 3(a), we show the unstable manifold (attracting In at Fig.the 3(a), current, we show thescattered unstable manifold (attracting InxFig.onto 3(a), we show thex unstable manifold (attracting ˆ1 . One ˆ particle positions points of a regular grid G can i j In this final example, we illustrate the most form we illustra In this final example, we illustrate the most general form LCS),way, identified standard i.e.,t as a ridge of In thisgeneral final example, LCS), identified in the standard i.e., asina the ridge of the way, identified in the standard way, i.e., as a ridge of the t1 LCS), 0 ˆ of our results, Eq. (7), on an experimental velocity our results, Eq. (7), on an experimental velocity then plot interpolated field ((F xback )) toover obtain visualization of Eq.field backward computed 0G t ¼t16(ˆ tfield of our results, (7), on an exp backward FTLE fieldthe computed fromFTLE t ¼ 6 field back to FTLE t0 ¼ 0 from backward computed from t ¼a6 better back to tfield 0¼ jFTLE 1 to 0¼0 t0of obtained from particle image velocimetry (PIV) obtained from velocimetry (PIV) measureobtained frommeasureparticle image velo ¼ 0:01:attraction Observe to that attraction to the image withtheDtstrong thethe strong with Dt ¼ 0:01: Observe that ¼ 0:01:particle Observe that the strong attraction to the with Dt the attracting LCS at time t , as shown in Fig. 5.13(c). 1 ments Details near a swimming jellyfish. ofnear this aexperiment near a swimming jellyfish. of this experiment torus results in a rapid divergence backward-time ments swimming jellyfish. D torus results in a rapid divergence of backward-time trajec- ofments torus results in atrajecrapid divergence of backward-time trajec- Details can be manifold found in Ref. 12. periodic can be found Ref. 12. tories,manifold suppressing the periodic unstable manifold of thein periodic can be found in Ref. 12. tories, suppressing the unstable of the tories, suppressing the unstable of the sha1_base64="Q7SGum/XqONs7PgK+9v5Nfdhneg=">AAACD3icbVC7TsMwFHV4lvAKMLJYVKCyVAkgYKxAIAaGIvUlNSFyXKc1dR6yHUQV5Q9Y+BUWBhBiZWXjb3DaDtByJFtH59yre+/xYkaFNM1vbWZ2bn5hsbCkL6+srq0bG5sNESUckzqOWMRbHhKE0ZDUJZWMtGJOUOAx0vT657nfvCdc0CisyUFMnAB1Q+pTjKSSXGPPDpDs8SC9rF1fZG4qXSu7Vb+ZlVLb8+FD5t7t67prFM2yOQScJtaYFMEYVdf4sjsRTgISSsyQEG3LjKWTIi4pZiTT7USQGOE+6pK2oiEKiHDS4T0Z3FVKB/oRVy+UcKj+7khRIMQg8FRlvr2Y9HLxP6+dSP/USWkYJ5KEeDTITxiUEczDgR3KCZZsoAjCnKpdIe4hjrBUEeYhWJMnT5PGQdk6Lh/eHBUrZ+M4CmAb7IASsMAJqIArUAV1gMEjeAav4E170l60d+1jVDqjjXu2wB9onz+7j5vN

0 and a separation factor r > 1 of interest. The separation time τ(x0 ; t0, δ0, r) can then be defined as the smallest length of time in which the distance between a trajectory starting from x0 at time t0 and some neighboring trajectory starting δ0 -close to x0 at the same time t0 first reaches rδ0 . With these quantities, Artale et al. (1997), Aurell et al. (1997) and Joseph and Legras (2002) define the FSLE associated with the initial location x0 as the Lagrangian scalar field log r FSLEt0 (x0 ; δ0, r) B . (5.16) τ (x0 ; t0, δ0, r) By definition, the FSLE infers a local separation exponent for each initial condition x0 over a different time interval of length τ(x0 ; t0, δ0, r). As a consequence, unlike the FTLEtt10 field, the FSLEt0 field has no direct connection to the flow map Ftt10 and depends on the choice of

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5.2 Advective Transport Barriers as LCSs

161

the initial separation and the separation factor. The geophysics community generally views these differences as advantages of the FSLE over the FTLE, arguing that these enable the statistical detection of material stretching and mixing at different length scales (Cencini and Vulpiani, 2013; Poje et al., 2014). Beyond their use in flow statistics, however, ridges of the FSLE field have also been proposed as delineators of hyperbolic LCSs, based on an analogy with the FTLE field (see, e.g., Joseph and Legras, 2002, d’Ovidio et al., 2004 and Bettencourt et al., 2013). This analogy is heuristic because FSLE ridges are constructed from separation exponents extracted over a range of different time intervals, whereas FTLE ridges represent maximal separation exponents over the single time interval [t0, t1 ]. In addition, FTLE quantifies the separation of infinitesimally close trajectories, wheres FSLE is specifically geared towards assessing separation of trajectories starting at a finite distance δ0 . Seeking to establish a mathematical relationship between FTLE and FSLE ridges, Karrasch and Haller (2013) introduce the infinitesimal-size Lyapunov exponent (ISLE), as the δ0 → 0 limit of the FSLE as follows: ISLEt0 (x0 ; r) B lim FSLEt0 (x0 ; δ0, r) . δ0 →0

(5.17)

They then obtain that robust FSLE ridges (as defined in Appendix A.2) with moderate ISLE variations and high normal steepness at their peaks signal nearby robust FTLE ridges t0 + ISLElog rx ;r t0 ( 0 )

whenever the dominant eigenvalue, λmax , of the tensor field Ct0
∂t λmax Ctt0 (x0 ) t=t0 + log r , 0 ISLE t

(x0 ) is simple and (5.18)

x ;r 0( 0 )

holds in a neighborhood of the robust FSLE ridge. The nondegeneracy condition (5.18), however, will be violated along families of surfaces along which the FSLE field admits jump-discontinuities. Such families of FSLE jumpsurfaces turn out to be generically present in any nonlinear flow, causing sensitivity in FSLE computations, as illustrated for a steady, 2D double gyre flow in Fig. 5.14. This sensitivity may impact the accuracy of FSLE statistics, as already noticed by LaCasce (2008) for observational ocean data. In addition to these jump discontinuities and the temporal sensitivity they cause, Karrasch and Haller (2013) also identify additional challenges in FSLE computations. These include ill-posedness for certain ranges of the separation ratio r and insensitivity to changes in the flow once the separation time τ is reached.

5.2.9 Parabolic LCSs from FTLE Having discussed the diagnostic identification of hyperbolic LCSs in detail, we now turn to parabolic LCSs. We defined these LCSs at the beginning of §5.2 as a subclass of shear LCSs that distinguish themselves as minimally shearing surfaces. The most notable examples of such transport barriers include jet cores, such as the theoretical centerpiece of the Gulf Stream shown in Fig. 5.1(d). The inability of the FTLE to distinguish between normal repulsion or tangential shear for a material surface (see §5.2.2) becomes an advantage in locating parabolic LCS. Since

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higher orde in integratin

162

Lagrangian and Objective Eulerian Coherent Structures

E. Spuriou

Even i and continu repelling L Chaos 23, 043126 after the se FSLE field By contrast, the FTLE field is everywhere continu of the time parameter t and the initialtrenches condition an Furthermore, errors in an FTLE computation areAs typica moving uns higher order with respect to the computational time step The veloc in integrating the velocity field. Hamiltonia

Jump discontinuities in the FSLE field 043126-4

D. Karrasch and G. Haller

FSLE

FSLE

y0

E. Spurious ridges of FSLE

Hð Even in regions where the FSLE field is well-d and continuous, FSLE ridges may signal false wherepositiv L ch repellingx0LCSs. Because of significant changes in theh q2 control after the separation time is reached by keyflow; trajectorie and a Figure 5.14 Jump discontinuities of the FSLE field FSLE computed = 6produce for the such spurious ridges field with may reven tion moves steady limit of the double gyre flow model introduced in Shadden et al. (2005). trenches of the FTLE field.(Left) that moves The FSLE0 (x0 ; 0.1, 6) field as a function of the initial positions = (x0, y0consider ). (Right)a two-dimensional mod As anx0example, We fi The same but restricted to the y0 = 0.48 line. Adapted fromunsteady Karrasch and Haller moving separation along a horizontal free-slip a ¼ 10, and (2013). The velocity field derives from the stream-fun as T ¼ t $ Hamiltonian shows the Hðt; xÞ ¼ $L tanhðq $ 0. atÞÞ; 2 x2 Þ tanhðq a parabolic LCS resides in a domain filled by other (more shearing) material surfaces, the 1 ðx1t ¼ Selecti FTLE will measure predominantly shear-type deformations in such regions. Based on this, where L characterizes the strength of the separation; q Fig. 4(a) an one expects parabolic LCSs to minimize the FTLE field qin2 control directions them, i.e.,of separation hownormal localizedtothe impact is o 0.3. Since form trenches for the FTLE field. This leads to the following diagnostic principle flow;objective and a defines the horizontal speed at which the w se positions moves. The 2010, flow, therefore, first formulated by Beron-Vera and coworkers (Beron-Veration et al., 2008b, 2012). becomes steady arationinata l that moves horizontally with speed a. the FSLE Proposition 5.5 For t1 − t0 large enough, time t positions of parabolic LCSsparameters will generically We fix the L ¼ 4; q ¼ 5; q2 ¼ 1, 0 FIG. 2. (a) Lines of discontinuities in the FSLE field of the autonomous dou- 1 The separa a ¼ 10, length observation align with codimension-1 trenches ble of gyre the flow largest eigenvalue λand ;the t0, tdiscontinuities of maximal the forward (7). (b) A cross section of0choose along the of ther, n (x 1 )the indica t1 2. FTLE For this choice T¼ t $ t0of¼the 0:48 line. (c) with Theasthe root cause jump-discontinuity atof parameters, F Cauchy–Green strain tensor Ctt10 (x0 ),x2;spec or, ¼equivalently, trenches of (x ). bottom-heig 0 t 0 the particle-separation historyvelocity curve withfield the for the model x1;dis % 0:1583: a tangency ofshows the instantaneous At the r ¼ 6 horizontal line. t ¼ 0. The same diagnostic principle is equally valid in both forward and backward time caltion, only a Selecting thetrench separation factor asinr ¼ 2:3, we obse culations launched from t0 , but the two calculations may have differing locations for the FTL Fig. 4(a) anofFSLE ridge along thewith x1-axis, starting at sðx ; d ; rÞ result in a sensitivity FSLE calculations 0 their FTLE fields. Indeed, forward and0 backward computations will technically target two axis se 0.3. Sinceofthe pointdata. moves to This the right, respect to thethe temporal theseparation available flow the most ne different finite-time dynamical systems unless flow isresolution exactly recurrent. Figure 5.15 ilpositions with larger and larger x -values experience Indeed, Fig. 2(c) shows that under a course step size in1 t, the the x1-axis lustrates this difference in an atmospheric wind field simulation et times. al. (2012). aration by at Beron-Vera later and later As a result, heig the first crossing of the dðtÞ=d0 ¼ 6 line by the particlethe FSLE ridge along the x1 robust axis shows large vari The differences between the two computations are visible which suggests separation history curvebut willsmall, be missed altogether for an The separation-time plot in Fig. 4(b) highlights thi FIG. 2. (a) Lines of discontinuities in the FSLE field of the autonomous douzonal jet cores (red and blue curves) open in thissetflow. Figure 5.16 confirms that parabolic LCS ofalong initial Instead ofathe the correct separable gyre flow (7). (b) A cross section of the discontinuities the conditions. ther, indicating separation-time valley with incre barriers advective transport, green and xidentified line.this (c) flow The are root indeed cause ofstrong thetion jump-discontinuity at separation time, atolarger time keeping will be the recorded. The 2;spec ¼ 0:48 in bottom-height. history curve with the x1;dis % 0:1583: a tangency of the particle-separation resulting will typically larger the nature of the se orange tracers released on July 1, 2000 on theerror same side of theAtbe jet cores month later. thesubstantially same even time, aby the than localized r ¼ 6 horizontal line. time-step used in the computation. As Beron-Vera et al. (2012) point out, the numericaltion, extraction of segment FTLE trenches is will generate a only a short of the x1 axis LaCasce (2008) has already observed thatany FSLE statisforbut thetargets FTLE field for choice of the integration conceptually similar to those of FTLE ridges (see §5.2.5), trenches as attractors sðx0 ; d0 ; rÞ result in a sensitivity of FSLE with with ticscalculations show sensitivity respect to the temporal resolution This axis segment is the subset of initial conditions sho of the to negative gradient dynamical system flow data. respect the temporal resolution of the available scales. Such sensitivity beinterval T. The r in the range of smallerthed0most net separation over thecan time Indeed, Fig. 2(c) shows that under stepreduced size in t,for analytic gradually and is numerical model flows by as seen in Fig. d a course in fact an FTLE trench, the x1-axis t1 (s)by =− ∇ FTLEand (s)), time steps. Step-size reduction, (5.19) the particlethe first crossing of the dðtÞ=d0 ¼ 6 xline 0selecting t0 (x0smaller smaller ds separation history curve will be missedhowever, altogether for an an option for in situ observational flow data, is not as opposed to the gradientInstead systemof(5.12). Apart from this sign difference, the details a open set of initial conditions. the correct separawhich comes with a fixed (and typically course) temporalofretion time,trench a larger separation time willagree be recorded. The simple extraction procedure with those given in Mathur et al. (2007) for FTLE solution (LaCasce, 2008). FIG. 3. Inst resulting ridges. error will typically be substantially larger than the time-step used in the computation. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the term LaCasce (2008) has already observed that FSLE statis129.132.169.51 On: Wed, 12 Feb 2014 tics show sensitivity with respect to the temporal resolution in the range of smaller d0 scales. Such sensitivity can be gradually reduced for analytic and numerical model flows by selecting smaller and smaller time steps. Step-size reduction, however, is not an option foronline in situ observational flow https://doi.org/10.1017/9781009225199.006 Published by Cambridge University Pressdata,

x0

5.2 Advective Transport Barriers as LCSs BERON-VERA ET AL.

FEBRUARY 2012

FTLEtt00

762

7days

FTLEtt00 +7days

JOURNAL OF THE ATMOSPHERIC SCIENCES

761

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7days

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+ FTLEtt00 +7days ]

VOLUME 69

t0 = July15, 2000

FIG. 3. (left to right) Zonally averaged zonal wind (thick) and potential vorticity (thin), and backward, forward, and backward-plusforward FTLE fields on 15 Jul 2000 (grayscale). The thin (thick) red and blue curves indicate local minimizing curves or trenches in the backward (forward) FTLE field near 208N and 608S, respectively.

Figure 5.15 Slight differences in parabolic LCSs (extracted as backward and forward FTLE trenches, in the Canadian Middle Atmosphere Model (CMAM). ( respectively) ) similar formula can be derived for an FTLE trench, but dXt (s) (Left Zonally averaged wind (thick) isand potentialmore vorticity (thin); ^[Xzonal !n u[Xtto (s), right) t0 ] 5 u[X (s), t ] 2 its evaluation computationally demanding than t 0 t (s), t0 ], dt0 that of the formula considered here.] presents the backward, forward and the backward-plus-forward FTLE fields onFigure t0 =6July 15, (6) an evaluation of (9) for each of the trenches revealed in 2000 (shown in grayscale). The thin (thick) red and blue curves mark trenches in the the backward and forward FTLE fields on 15 July 2000, backward (forward) FTLE Adapted (2012). the blue thin which vanishes if the FTLE trenchfield. is a material curve. from whichBeron-Vera are depicted in et Fig.al. 3. Specifically, 0

0

762

0

0

of T the FTLE trenches Because at any time J O Ut R5Nt0Aeach L OF HE ATM OSPHER IC SC I E Nin C Fig. E S 6 is the instantaneous VOLUME (thick) curve area 69 flux per of interest here is describable by an invertible function unit length across the backward (forward) FTLE trench July 1, 2000 July 31, 2000 y0 5 h(x0, t0), (6) can be readily computed as follows. associated with the boreal summer subtropical jet, which First, note that wobbles about roughly 208N. The red thin (thick) curve

is the flux across the backward (forward) FTLE trench associated with the austral polar night jet, which un0 dt0 dulates about roughly 608S. The dashed curve is shown y, (7) for reference and corresponds to the flux across the 608S 1 ›t h(x0 , t0 )]^ 0 latitude line. Consistent with the results shown in Fig. 4, the flux across each of the FTLE trenches revealed is where a(x0, t0) is arbitrary. Second, note that very small (in absolute magnitude), particularly compared to that across 608S. The presence of a small flux ^ x y 2 ›x h(x0 , t0 )^ ^[Xt (s), t0 ] 5 qffiffiffiffiffiffiffiffiffiffiffiffiffiffi0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi . (8) across each of the FTLE trenches is explained by the n 0 1 1 [›x h(x0 , t0 )]2 small mismatch in the position of the forward and 0 backward FTLE trenches, which may be attributed to the spatiotemporal coarseness of the velocity field that As a result, supplies advection in the Lagrangian calculations. These results prevent one from rigorously concluding that asu[Xt (s), t0 ] 5 y[x0 , h(x0 , t0 ), t0 ] 2 u[x0 , h(x0 , t0 ), t0 ] 0 sociated withonthe boreal summer FIG. 4. Selected snapshots for July of the evolution of passively advected tracers (dots) superimposed the westward backward FTLE field on the subtropical 2 ›t h(x(or , t ), (9) 3 ›x h(x0 , t0 ) LCSs Figure 5.16corresponding Parabolic jet cores) are indeed barriers to advective transport. jetinand 0 0Black curves are as date (grayscale). Fig.the 3. eastward austral polar jet are invariant tori. 0 0 theylines do provide support to the idea that (left) both Tracers released on July 1, 2000 along fourHowever, different of constant latitude where u (y) is the zonal (meridional) velocity compo- eastward and westward zonal jets behave as barriers for in nent. the [Haller model(2011) already analyzed in Fig. 5.15 do not penetrate the parabolic LCS shown a computable formula the meridional a good approximation. backward FTLE field on 31 Januarypresents 2000 near 208S. rather, it for is a realistic featuretransport has to been observed in in FTLE theinstantaneous back background FTLE image (grayscale) as a that trench (right). Adapted from unit length FTLE Before closing this section we our attention back Such a backward the trench is flux mostperevident fromacross thean transport-barrier diagnostics of Sankey andturn Shepherd ridge that is expressed in terms of the FTLE itself. A to Fig. 5 to note that a trench is seen to develop in the Beron-Vera et al. (2012). dXt (s) 0

5 a(x0 , t0 )^ x 1 [a(x0 , t0 )›x h(x0 , t0 )

February through early March 2000 (not shown). A (2003). trench (not shown) is also seen to be present in the forward FTLE field near 208S in the same period and to 4. Summary and discussion roughly overlap in position with the backward FTLE We close by stressing that the converse of Proposition 5.5 is not true: FTLE trenches are In this paper we have presented support for our hytrench. This backward-plus-forward FTLE trench near pothesized association of transport barriers with both just as FTLE 208S indicates presence of(but an invariant torus, which diagnostic only the necessary not sufficient) indicators of parabolic LCSs, behaves as a meridional transport barrier. This feature is eastward and westward zonal jets. Our association of ridges are only necessary indicators of hyperbolic LCSs (see §5.2.2). A simple example entirely analogous to the transport barrier associated transport barriers with zonal jets independent of the illustrating this is the 2D steady incompressible flow with the boreal summer subtropical jet that we have background potential vorticity (PV) distribution builds FIG. 4. We Selected snapshots for July the evolution of passively tracers (dots) relating superimposed the backward FTLE field on the on recent results to on Kolmogorov–Arnold– described. have not found a of strong indication of advected corresponding date in (grayscale). curvestheory, are as Fig. 3. deals with the stability of 2 inwhich MoserBlack (KAM) the presence of a polar meridional transport barrier Û x = x(1 + 3y ), the boreal winter. This is consistent with the observa- Hamiltonian systems under perturbation. Such KAM (5.20) 2the presence of invariant tori (i.e., predict tion that during year 2000 of the CMAM simulation theory yÛ =results −y(1 ), backward FTLE field January 2000vortex near 208S. it is+ ay realistic been observed in closedrather, material fluid curvesfeature that dothat nothas experience exconsidered, disturbances ononthe31boreal polar Suchstrong, a backward trench is most(e.g., evident the transport-barrier diagnostics of Sankey ponential stretching or folding) in the vicinityand of Shepherd the are quite leadingFTLE to a poorly-defined not from February through early 2000 (not A of(2003). in boreal which thenight y =jet.March 0Itaxis isbeclearly an (exponentially) attracting LCS, for andtheyet both the forward axis a zonal jet, which serve as barriers merpersistent) polar should notedshown). that trench (not shown) is also seen to be present in the idional transport of passive tracers. The basis of such this is not a spurious result of the CMAM simulation; and backward FTLE fields develop trenches along this line, as shown in Fig. 5.17. forward FTLE field near 208S in the same period and to 4. Summary and discussion roughly overlap in position with the backward FTLE In this paper we have presented support for our hytrench. This backward-plus-forward FTLE trench near 208S indicates the presence of an invariant torus, which pothesized association of transport barriers with both behaves as a meridional transport barrier. This feature is eastward and westward zonal jets. Our association of entirely analogous to the transport barrier associated transport barriers with zonal jets independent of the with the boreal summer subtropical jet that we have background potential vorticity (PV) distribution builds described. We have not found a strong indication of on recent results relating to Kolmogorov–Arnold– https://doi.org/10.1017/9781009225199.006 Publishedtransport online by Cambridge Press theory, which deals with the stability of Moser (KAM) the presence of a polar meridional barrier in University

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49

Fig. 6. The tracer evolution for system (22). Left: Initial circular blob of tracers centered at the origin at time t = 0. Right: The advected tracer at time t = 1.5. The forward-time FTLE field with integration time T = 10 is shown in the background.

Figure 5.17 The tracer evolution in the steady flow (5.20). (Left) Initial blob of tracers centered at the origin at time t = 0. (Right) The advected set of tracers at time t = 1.5. The forward-time FTLE field FTLE10 0 is shown in the background, displaying a clear trench along this repelling LCS. Adapted from Farazmand et al. (2014).

5.2.10 Elliptic LCSs from the Polar Rotation Angle (PRA) Fig. 7. The Poincaré map for the standard non-twist map. Left: Integrable parameters: a the red symbols mark the indicator points (24).

0 08, b

0 125 Right: Chaotic parameters: a

0 27, b

0 38. In both panels

= . = . = . As we have seen,⌦ the FTLE is a simple and efficient diagnostic tool for= . transport barriers causing locally the largest growth or decay in Specifically, their normal perturbations (hyperbolic 5. Build smoothly connecting,normal alternating stretchline–strainline points [28]. initial conditions for the shearless barrier heteroclinic chains from the separatrices so obtained. are given by Finally, keepthe only the heteroclinic chains whose individual LCSs) or 6.locally smallest tangential growth a in1 their normal perturbations (parabolic components are weak minimizers of their neutralities. = ± (24) and y = 0, 2 4 distinguish themselves by their global LCSs). In contrast, elliptic LCSs, as defined inx §5.2, 9. Numerical examples and the full barrier can be constructed by iterating these initial conditions underlocally the map (23). These initial conditions are referred (circular, cylindrical or toroidal) geometry beyond their most-shearing character. For 9.1. Standard non-twist map to as indicator points and are shown in Fig. 7 for two choices of parameters ( a , b ) . Therefore, we can compare the parabolic barrier this reason, We FTLE field is not an optimal detection tool for elliptic LCSs. first consider the standard non-twist map (SNTM) computed from finitely many iterations of the SNTM using the stepsvortical in Section 8 with the exact asymptotic shearless barrier of Specifically, valleys tend to signal domains of low stretching in most , = x while + a 1 y FTLE x the map given by indicator points. (23) y =y b sin(2⇡ x ), Fig. 8 shows all heteroclinic tensorlines connecting trisectors to flows of practical interest (see, e.g., Figs. 5.10 and 5.15), these valleys are typically infiltrated wedges (left panel). In the domain [ 0.5, 0.5]⇥[ 2, 2] and for 100 which was first studied in detail in [1], and has since become a iterations of the SNTM, we find 6 singularities: 2 trisectors (green helpful model in understanding shearless KAM curves with ridgesgenerally spiraling into vortex-type regions. Figure 5.18 illustrates this phenomenon on dots) and 4 wedges (black dots). Only 4 alternating sequence of in two-dimensional steady or temporally periodic incompressible tensorlines satisfy conditions P1 and P2 of Definition 1. Fig. 8 flows. the AgulhasForregion of the Southern Ocean using thetheAVISO ocean surface velocity field also shows extracted parabolic barrier, i.e., a heteroclinic b = 0, the map (23) is a discretized version of the canonical chain formed by four tensorlines (note the periodicity in x). This parallel shear flow (1) with vanishing Eulerian shear along y = 0. (see Appendix A.6). The spiraling FTLE ridges seen in this figure simply connect locations parabolic barrier represents the finite-time version of the exactly For steady perturbations of (1), one still has a steady streamfuncknown asymptotic shearless KAM curve. tion whose dynamics is integrable and the shearless barriers can be of high stretching with locations ofb 6=high shear, One yetcanthey arethetypically as indicators also compute parabolic barrier viewed for higher iterations 0, the SNTM understood as the lack of Hamiltonian twist. For of the SNTM map with the same procedure. As the number of corresponds to the evolution of a time-periodic perturbation of (1). iterations increases, the converges to of repellingFormaterial would incomputed turn parabolic implybarrier a complete lack of 0.08, b =hyperbolic 0.125, the SNTM is LCSs the parameter lines. values a =Such the exact asymptotic barrier. In Fig. 9, we show this convergence integrable and well-understood. We choose these parameters to up to 300 iterations. The exact barrier (black curve) in Fig. 9 is illustrate the performance of our theory and extraction methodcoherent material vortices in these domains, ascomputed such from vortices would lose their integrity and 200 iterations of the indicator points (24). ology for parabolic barriers. Fig. 7 (left panel) shows the orbits of The evolution of circular tracersnonobjective off and on the computed SNTM for these integrable parameters. develop filaments under the effect of repelling LCS. In contrast, diagnostics, parabolic barriers is shown in Fig. 10. The purple tracer in the In this integrable case, the location of shearless barriers is left plot of Fig. 10 is located on the computed parabolic barrier no longer trivial, but can be found by the theory of indicator such as streamline plots and Okubo–Weiss Q parameter plots shown in Fig. 5.18, do predict some vortices in these regions, but the validity of their projections for coherent material vortex boundaries is unclear for the reasons discussed in §3.7. For efficient detection of elliptic LCSs, therefore, we need to quantify rotational coherence in Lagrangian terms, as opposed stretching-based coherence. The simplest measure of local material rotation in the flow (5.1) over a finite time interval [t0, t1 ] is the polar rotation tensor Rtt0 , defined via the polar decomposition formula (2.110). Indeed, as we discussed in §2.3.2, the polar decomposition of the deformation gradient ∇Ftt0 (x0 ) yields Rtt0 (x0 ) as the rotation closest to ∇Ftt0 (x0 ) in the Frobenius matrix norm. A scalar diagnostic tool for rotational coherence can then be defined as the polar rotation angle field Θtt0 (x0 ) generated by the tensor field Rtt0 (x0 ) along individual trajectories. One can then approximate time t0 positions of elliptic LCSs as closed level sets of Θtt0 (x0 ). The rationale for this approach stems from the study of temporally T-periodic flows via their Poincaré maps (see §4.1). In such flows, all trajectories starting from an elliptic transport barrier (i.e., closed invariant curve) of the Poincaré map have the same rotation number, which n+1

n

n +1

n

2 n +1 n

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forward FTLE

streamlines

Okubo-Weiss elliptic regions

4 of 7

Figure 5.18 FTLE ridges tend to penetrate the cores of vortical features in the AVISO dataof set, suggesting nonexistence coherent Figure 2. ocean (a) Blow-up the region indicated in the Figuregeneral 1, (b) corresponding sea surfaceof topography map, mesoscale and (c) OW parameter. Indicated in Figure 2a is the region where the FTLEs shown in Figure 3 werecomputed computed. Lagrangian eddies. (Left) Typical mesoscale FTLE features from satellite altimetry data. (Middle) Instantaneous streamlines (constant sea-surface height plots) in the same region. (Right) Vortices suggested by the Okubo–Weiss criterion (i.e., regions of Q > 0; see formula (3.56)). Adapted from Beron-Vera et al. (2008a).

with ξi and ηi corresponding to the left and right singular vectors of ∇Ftt0 (i.e., eigenvectors of Ctt10 and Btt10 ), respectively. If system (5.1) is 2D (n = 2), then we obtain the formula for the PRA from (2.121) as PRAtt10 (x0 ) = cos−1 hξ1 (x0 ; t0, t1 ), η1 (x0 ; t0, t1 )i = cos−1 hξ2 (x0 ; t0, t1 ), η2 (x0 ; t0, t1 )i .

(5.22)

As we have already concluded in the transformation formulas (3.55), the polar rotation tensor transforms under a Euclidean observer change (3.5) as ˜ tt = QT (t)Rtt Q(t0 ), R 0 0

(5.23)

and hence is objective as a two-point tensor. We have also shown, however, that the traces of two-point tensors are not invariant: their value depends on the bases chosen in their domain and range spaces (see §3.5). As a consequence, the PRA, which is purely a function of tr Rtt0 (see Eqs. (2.119) and (2.121)), is not an objective scalar field. For this reason, we will not use the PRA to detect advective transport barriers in 3D fluid flows.3 3

We will, however, use PRA as a diagnostic tool to study vector fields that are objective (unlike fluid velocity fields). Examples of objective vector fields are the active barrier vector fields arising in the study of the transport of dynamically active vector fields, such as the vorticity and the momentum, in Chapter 9. For these vector fields, any time-dependent Euclidean observer change translates to a one-time, initial rotation of the coordinate axes, which does not affect the value of the rotation angle between initial and final configurations.

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L12603

is the infinite-time limit of the total angular displacement along the closed curve divided by the number of iterations (see Golé, 2001). As Rtt00 +T is the best fitting rotation to the linearized Poincaré map, one expects the same number of polar rotations to arise asymptotically along all trajectories starting from the same closed invariant curve. Equivalently, one expects the level curves of the polar rotation angle to approximate closed invariant curves of the Poincaré map more and more accurately for higher and higher iterations of the map. With this motivation at hand, we recall from §2.3.2 that the total polar rotation angle (PRA), can be computed for the finite-time dynamical system (5.1) in the 3D case (n = 3) as " !# 3 Õ t1 −1 1 hξi (x0 ; t0, t1 ), ηi (x0 ; t0, t1 )i − 1 , PRAt0 (x0 ) = cos (5.21) 2 i=1

BERON-VERA ET AL.: OCEANIC MESOSCALE EDDIES AS REVEALED BY LCSS

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In 2D flows, however, the level curves of PRA turn out to be objectively defined, even if the values of PRA on those level curves will be frame dependent. Indeed, as observed by Farazmand and Haller (2016), for the matrix representation     ! t1 t1 cos PRA (x ) − sin PRA (x ) 0 0 t t 0 0     Rtt0 (x0 ) = (5.24) sin PRAtt10 (x0 ) cos PRAtt10 (x0 ) of the polar rotation tensor in a given orthonormal basis, the additivity of rotations in two ˜ tt in (5.23) can be written as dimensions implies that the transformed polar rotation tensor R 0 ˜ tt (y0 )= R 0

  cos PRAtt10 (x0 ) + q(t0 ) − q(t1 )   sin PRAtt10 (x0 ) + q(t0 ) − q(t1 )

  ! − sin PRAtt10 (x0 ) + q(t0 ) − q(t1 )   , cos PRAtt10 (x0 ) + q(t0 ) − q(t1 )

(5.25)

where q(t) is the angle of rotation associated with the tensor Q(t). Computing the traces of the matrices in Eqs. (5.24) and (5.25) then substituting the result into the 2D polar rotation angle formula (2.120), we obtain t

t1 (y0 ) = PRAtt1 (x0 ) + q(t0 ) − q(t1 ), PRA 0 0

(5.26)

with x0 = Q(t0 )y0 + b(t0 ). By formula (5.26), the PRA clearly changes its value in the new frame and hence is not an objective Lagrangian scalar field (see §3.4.1). However, the difference, q(t0 ) − q(t1 ), in PRA values between the two frames is exactly the same at all spatial locations. Therefore, the level curves of PRAtt10 remain unchanged by the frame change, as claimed. These considerations lead to the idea to use closed curves of the PRA as proxies for elliptic LCSs in 2D unsteady flows. A nested family of such level curves surrounding an isolated local maximum of the PRA can in turn be viewed as a coherent Lagrangian vortex (or material vortex) whose vortex center is the location of local maximum, shown in Fig. 5.19. In principle, the vortex boundary is the singular level curve bounding the elliptic LCS family from the outside. Lagrangian vortex center:! local maximum of the PRA

observed vortex boundary:! outermost elliptic LCS ! computable from the available! resolution for the PRA

coherent Lagrangian vortex:! nested family of elliptic LCSs

elliptic LCS:! smooth and closed ! PRA level curve

! ! !

theoretical vortex boundary:! singular PRA level curve

Figure 5.19 Schematic of a coherent Lagrangian vortex identified from the PRA in a 2D flow. The vortex is filled with elliptic LCSs surrounding a PRA maximum at the center of the vortex. The theoretical vortex boundary is the non-smooth outer boundary of the elliptic LCS family.

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We summarize these ideas in a definition that will serve as an objective diagnostic principle for elliptic LCSs, coherent Lagrangian vortices and vortex centers in 2D flows. Definition 5.6 At time t0 , PRA-based elliptic LCSs in the finite-time flow (5.1) are smooth, closed and convex level curves of the polar rotation PRAtt10 (x0 ) surrounding a unique local maximum. PRA-based coherent Lagrangian vortices are formed by nested families of PRAbased elliptic LCSs. PRA-based vortex centers in such Lagrangian vortices are marked by a local maximum location of PRAtt10 (x0 ) surrounded by the elliptic LCSs. This definition is equally applicable to forward and backward calculations launched from t0 , but the two calculations generally return different elliptic LCSs unless the underlying flow is recurrent. Each PRA-based elliptic LCS covered by Definition 5.6 is expected to be observed as a barrier to advective transport inside the Lagrangian vortex. Indeed, material elements along such an elliptic LCS complete exactly the same rotation which does not allow for the development of smaller scales (filaments) along the LCS. The lack of filamentation (coherence) is then observed as no penetration of fluid particles into or out of the interior of the LCS. As we noted at the beginning of this chapter, all material curves block transport across themselves, but only those are observed as barriers to advective transport that keep their coherence and hence disallow protrusions between the spatial regions on their two sides. The theoretical vortex boundary shown in Fig. 5.19 is conceptually justified but unfeasible to compute from a data set. This is because no point of a discrete numerical grid will fall on this theoretical boundary with nonzero probability. For this reason, the practically detected boundary of a PRA-based Lagrangian vortex is the outermost closed PRA level curve computed numerically for the given data resolution. The vortex boundary obtained in this fashion will then necessarily converge to the theoretical boundary upon refinement of the numerical grid. As for the practical numerical implementation of the PRA-based search for elliptic LCSs, we can use the same basic numerical procedure as for hyperbolic and parabolic LCSs to obtain the SVD of the deformation gradient (see §2.3.1). This yields the necessary left and right singular vectors pairs, (ξi , ηi ), either of which can be used in the 2D PRA formula (5.22). The procedure is completed by a level surface plot of the PRA computed in this fashion. If more than a visual identification of Lagrangian vortices is required, then an automated search for nested sets of closed PRA level sets surrounding a PRA minimum is necessary as a last step. The PRA computation in 2D is implemented in Notebook 5.3. Notebook 5.3 (PRA2D) Computes the polar rotation angle (PRA) field for a 2D unsteady velocity data set. https://github.com/haller-group/TBarrier/tree/main/TBarrier/2D/ demos/AdvectiveBarriers/PRA2D An application of PRA-based elliptic LCS detection in a 2D turbulence simulation is shown in Fig. 5.20. For reference, Fig. 5.20 also shows the instantaneous vorticity level sets identified at time t0 . Several of these level sets do give an indication of vortical regions, but the PRA reveals which one of these candidate regions will experience coherent material rotation evidenced

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Lagrangian and Objective Eulerian Coherent Structures M. Farazmand, G. Haller / Physica D 315 (2016) 1–12

7

PRAtt10 (x0 )

!(x0 , t0 )

Fig. 4. Left: Vorticity ! at the initial time t = 50. Right: The PRA ✓tt0 computed from formula (15) over the time interval [50, 100].

Figure 5.20 Instantaneous vorticity (left) at t0 = 50 and the polar rotation angle PRAtt10 (x0 ) computed up to time t1 = 100 (right) in a 2D direct numerical simulation of homogeneous and isotropic turbulence on a doubly periodic domain. Adapted from Farazmand and Haller (2016).

by a family of nested elliptic LCSs. Another strong point of the PRA is the precise locations it provides for coherent vortex boundaries and vortex centers. Finally, in Fig. 5.21 we show more detail for one of the coherent Lagrangian vortices seen in Fig. 5.20, as well as the materially advected positions of two elliptic LCSs inside this vortex. Note the lack of filamentation in these advected curves, which confirms their coherence and hence their role as observed barriers to advective transport. Farazmand, M. Farazmand, G. Haller / Physica D 315M. (2016) 1–12 G. Haller / Physica D 315 (2016) 1–12

7

7

Fig. 5. Left: Contours of the PRA signaling coherent and chaotic regions. Right: Advected image of select contours to the final time t = 100. The local extremum of the PRA (marked by a cross) defining the Lagrangian vortex center by Definition 3. (For interpretation of the references to colour in this figure legend, the reader is referred to the materially LCS! Fig. t4.=Left: Vorticity at the timefrom t = 50. Right:(15) Theover PRAthe ✓ time computed from the time interval [50 , 100]. Fig. 4. Left: Vorticity ! at the initial time 50. Right: The!PRA ✓ initial computed formula interval [50formula , 100]. (15) overadvected web version of this article.) t t0

t t0

observed Lagrangian ! vortex boundary

and vortex boundary

elliptic LCS background turbulence

t0 = 50

t1 = 100

Fig. 5. Left: Contours of the PRA signaling coherentimage and chaotic regions. Right: Advected image select to the final time t = 100. The local extremum of the PRA Fig. 5. Left: Contours of the PRA signaling coherent and chaotic regions. Right: Advected of select contours to the final time t =of100. Thecontours local extremum of the PRA (marked by a center cross) defining the Lagrangian vortex center by Definition interpretation of legend, the references to colour in this (marked by a cross) defining the Lagrangian vortex by Definition 3. (For interpretation of the references3.to(For colour in this figure the reader is referred tofigure the legend, the reader is referred to the web version of this article.) web version of this article.)

Figure 5.21 (Left) Details of an individual coherent Lagrangian vortex from the 2D turbulence simulation shown in Fig. 5.20. (Right) Materially advected positions of one of the elliptic LCSs in the interior of the Lagrangian vortex (blue) and of another one forming the observed boundary of the vortex. Adapted from Farazmand and Haller (2016). Fig. 6. Left panel: Trajectories of the Lagrangian vortex center (red) and nearby passive tracers (blue and black). Middle and Right panels: The coordinates of the vortex center and nearby tracers as a function of time. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) 6. Left vortex panel: center Trajectories of the Lagrangian center and nearby passive tracers (blue The and coordinates black). Middle andvortex Right panels: The coordinates of the vortex Fig. 6. Left panel: Trajectories of the Fig. Lagrangian (red) and nearby passivevortex tracers (blue(red) and black). Middle and Right panels: of the and nearby tracers as of a function of time.to(For interpretation oflegend, the references to colour in this legend, theofreader is referred to the web version of this article.) center and nearby tracers as a functioncenter of time. (For interpretation the references colour in this figure the reader is referred to figure the web version this article.)

6.3. Stratified geophysical fluid flow 6.3. Stratified geophysical fluid flow (1.690 , 5trajectories .380). For reference, two other trajectories aregeophysical also shownfluid flow 6.3. Stratified (1.690at , 5.380 ). For reference, two other are also shown with initial positions 0.05 and 0.1 distance from the vortex initial positions 0.05vortex and 0.1 distance from the vortex with initial positions at 0.05 with and 0.1 distance fromatthe center. Due to the complexity of the flow, the trajectory patterns center. Due to complexity of the flow, the center. Due to the complexity of the flow, thethe trajectory patterns https://doi.org/10.1017/9781009225199.006 Published online by Cambridge University Press We a simplified model for stratified fluid geophysical fluid We trajectory consider apatterns simplified model forconsider stratified geophysical fluid We consider a simplified model forgeophysical stratified

(1.690, 5.380). For reference, two other trajectories are also shown

aretheir not xilluminating. However,astheir x-flow, and the y-coordinates a are not illuminating. However, and y-coordinates a flow, barotropic equation. This equation, in the vorticitybarotropic as equation. Thisthe equation, in the vorticity-

5.2 Advective Transport Barriers as LCSs

169

5.2.11 Elliptic LCSs from the Lagrangian-Averaged Vorticity Deviation To remedy the nonobjectivity of the PRA in 3D flows, we now turn to the dynamic version of the polar decomposition discussed in §2.3.3. Based on the same arguments introduced in the previous section, we continue to seek elliptic LCSs as codimension-1 material surfaces displaying rotational coherence. This time, however, following Haller et al. (2016), we will use a different measure of material rotation instead of the polar rotation angle. To this end, we recall from §2.3.3 that the total accumulated rotation angle (without cancellations) generated by relative rotation tensor Φtt0 around the local relative vorticity
t ¯ (t) can be computed as the intrinsic dynamic rotation angle, ψtt0 . This vector ω Ft0 (x0 ), t − ω prompts us to define the Lagrangian-averaged vorticity deviation (LAVD) as ˆ t 2ψtt0 (x0 ) 1 ω (Fs (x0 ), s) − ω ¯ (s) ds, = (5.27) LAVDtt0 (x0 ) := t0 t − t0 t − t0 t0 ¯ (t) of the vorticity field ω (x, t) (see Fig. 2.22 for with the instantaneous spatial mean ω the underlying geometry). By the objectivity of ψtt0 , LAVDtt0 is an objective Lagrangian scalar field in any dimension. This objective rotation angle also offers an appealingly direct connection with vorticity, the central (nonobjective) quantity in fluid mechanics for the description of flow rotation. Unlike the PRA, the LAVD depends on the reference fluid mass D(t) chosen for the ¯ (t). As noted in §2.3.3, the best choice for D(t) is the full flow domain, computation of ω prompting the LAVD to provide the best available assessment for the deviation of local rotation from the overall rotation of the fluid. The computation of the LAVD for 2D flows in implemented in Notebook 5.4. Notebook 5.4 (LAVD2D) Computes the Lagrangian-averaged vorticity deviation (LAVD) field for a 2D unsteady velocity data set. https://github.com/haller-group/TBarrier/tree/main/TBarrier/2D/ demos/AdvectiveBarriers/LAVD2D The computation of the LAVD for 3D flows in implemented in Notebook 5.5. Notebook 5.5 (LAVD3D) Computes the Lagrangian-averaged vorticity deviation (LAVD) field for a 3D unsteady velocity data set. https://github.com/haller-group/TBarrier/tree/main/TBarrier/3D/ demos/AdvectiveBarriers/LAVD3D

Application of LAVD to 2D Flows In 2D flows, our approximation of elliptic LCSs, coherent Lagrangian vortices and their centers via the LAVD follows the same principles that we have outlined for the PRA. For completeness, we summarize elements of the LAVD-based Lagrangian vortex identification in Fig. 5.22. The corresponding objective diagnostic principle for elliptic LCSs in 2D flows can be stated as the following definition.

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170

Lagrangian and Objective Eulerian Coherent Structures Lagrangian vortex center:! local maximum of the LAVD

observed vortex boundary:! outermost elliptic LCS ! computable from the available! resolution for the LAVD

coherent Lagrangian vortex:! nested family of elliptic LCSs

elliptic LCS:! smooth and closed ! LAVD level curve

! ! !

theoretical vortex boundary:! singular LAVD level curve

Figure 5.22 Schematic of a coherent Lagrangian vortex identified from the LAVD in a 2D flow.

Definition 5.7 At time t0 , LAVD-based elliptic LCSs in the finite-time flow (5.1) with n = 2 are smooth, closed and convex level curves of the Lagrangian-averaged vorticity deviation LAVDtt10 (x0 ) surrounding a unique local maximum. LAVD-based coherent Lagrangian vortices are formed by nested families of LAVD-based elliptic LCSs. LAVD-based vortex centers in such Lagrangian vortices are marked by a local maximum location of LAVDtt10 (x0 ) surrounded by the elliptic LCSs. As in the case of the PRA, the same diagnostic principle applies in both forward and backward time but the two calculations generally return different elliptic LCSs for nonrecurrent flows. Unlike the PRA, however, the LAVD also has a direct connection to observed attractors and repellers of finite-size (or inertial) particles drifting passively in 2D geophysical fluid flows (see Chapter 7 for further discussion on barriers to inertial particle transport). Specifically, building on work on the equations of motion derived by Beron-Vera et al. (2015), Haller et al. (2016) consider a geostrophic flow under the β-plane approximation and assume the presence of an LAVD-based coherent Lagrangian vortex center with initial position x∗0 . They also assume that the relative Eulerian rotation direction, measured at the materially evolving vortex core, is constant in time, i.e.,  
sign ω3 Ftt0 (x∗0 ), t − ω¯ 3 (t) ≡ µ(x∗0 ) = const., t ∈ [t0, t1 ] . (5.28) Furthermore, for a spherical inertial particle of radius r0 > 0 and density ρpart in a fluid of density ρ and viscosity ν > 0, they define the non-dimensional inertial parameter τ=

2r02 ρpart , 9νρ

(5.29)

and also use the Coriolis parameter f , which is twice the local vertical component of the angular velocity of the Earth. In this notation, a particle is considered heavy (relative to the carrier fluid) if ρpart > ρ and light if ρpart < ρ. Using these definitions, Haller et al. (2016) prove the following result: Theorem 5.8 Under assumption (5.28), for τ > 0 small enough, the following hold: (1) In a cyclonic LAVD-based coherent vortex (µ(x∗0 ) f > 0), there exists a finite-time attractor (repeller) for light (heavy) particles that stays O(τ)-close to the evolving vortex center Ftt0 (x∗0 ) over the time interval [t0, t1 ].

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(2) In an anticyclonic LAVD-based coherent vortex (µ(x∗0 ) f < 0), there exists a finite-time attractor (repeller) for heavy (light) particles that stays O(τ)-close to the evolving vortex center Ftt0 (x∗0 ) over the time interval [t0, t1 ].

158

A simulation confirming these statements, as well as the rotational coherence of LAVDbased Lagrangian vortices (mesoscale eddies in the ocean), is shown in Fig. 5.23(a). In these computations of the LAVD, we use the general observation that on large enough ocean domains containing several eddies, the mean vorticity satisfies ω¯ 3 (t) ≈ 0 (see Haller et al., 2016; Abernathey and Haller, 2018; Beron-Vera et al., 2019b). The materially advected eddy boundaries (black) in Fig. 5.23(b) illustrate the difference between stretching-based coherence (which would not allow filamentation in the eddy boundaries) and rotational coherence (which permits some tangential filamentation but no break-away from the rotating fluid mass of the coherent vortex). Also note in Fig. 5.23(b) that heavy (blue) and light (green)

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F IGURE 10. (a) Rotationally coherent Lagrangian vortices at time t0 = 11 November 2006, Figure 5.23 LAVD-based mesoscale eddies and inertial particles in the Agulhas leakt0 +T identified age. from(a)Algorithm 1 using the contours of LAVD (x0 ) extracted with T closed, = 90 days. The Initial positions of the coherent eddy boundaries (outermost t0 t0 +90 days t +T contour plot of LAVD ) is shown in the background for reference. (b) Initial (red) convex contourst00 of (x LAVD ) shown in red at t = November 11, 2006. Back0 0 t0 t0 +90 days and final (black) positions of the Lagrangian vortex boundaries at time t + T, ground: the contour plot of LAVDt0 (x0 ). (b) Simulated inertial 0 particlealong with representative inertialsee particle trajectories. converge to the centres of trajectories; the text for details. Heavy A videoparticles showing (blue) the evolution of material eddy boundaries eddies. and simulated inertial particles available to at the https://www anticyclonic (clockwise) Light particles (green)isconverge centres of cyclonic (clockwise).cambridge.org/core/journals/journal-of-fluid-mechanics/article/ eddies. Here, heavy = 0.99, light = 1.01, r0 = 1 m. (See the online supplementary movie M3abs/defining-coherent-vortices-objectively-from-the-vorticity/ for the complete advection sequence of the vortex boundaries and the inertial particles.) 559F52C404B54265CB7A64D4666EB096\#supplementary-materials.

We show a comparison of the initial positions of rotationally coherent Lagrangian and Eulerian vortices in figure 11(a). Three of the Eulerian eddies (green) are close to Lagrangian eddies (red), and will accordingly show some coherence under advection

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particles converge to the moving anticyclonic (counter-clockwise) and cyclonic (clockwise) vortex centers, respectively, as predicted by Theorem 5.8. Once fluid trajectories have been generated from a velocity field, the computation of the LAVD is more straightforward than that of the PRA because no computation of the deformation gradient is required. At the same time, the LAVD relies on the numerical differentiation of the vorticity. Altogether, the extraction of coherent vortex boundaries from LAVD calculations is more sensitive to numerical errors and discretization effects than PRAbased calculations and the variational approaches discussed in the following sections of this chapter (see, e.g., Andrade-Canto et al., 2020, for an example). A way to mitigate these sensitivities is to restrict the search for outermost closed LAVD level curves in a nested family to curves whose convexity deficiency4 stays below a certain threshold (see Haller et al., 2016, for details of the algorithm). An alternative threshold parameter, the coherency index, has been developed by Zhang, W. et al. (2020). Combined with such numerical control parameters, LAVD is also able to detect coherent Lagrangian swirls, a class of Lagrangian flow structures with relaxed coherence properties in submesoscale-resolving numerical ocean data. The boundaries of these Lagrangian features undergo substantial stretching, yet the fluid mass they enclose preserves its bulk integrity to a high degree, displaying signs of overall material coherence. We show such a feature identified in the Gulf of Mexico in Fig. 5.24, detected as the region enclosed by an outermost closed LAVD curve with convexity deficiency not exceeding 0.25. The figure also demonstrates the inability of the Okubo–Weiss criterion to isolate this coherent features as an elliptic region. More recent applications of LAVD-based coherent vortex detection have appeared in Abernathey and Haller (2018); Liu, T. et al. (2019), yielding systematic mesoscale eddy censuses based on the LAVD. These authors invariably find that only a fraction of eddy locations inferred from the (nonobjective) nonlinear eddy criterion (see §3.7.1) indicate nearby eddies that transport water in a coherent fashion. Liu, T. et al. (2019) specifically find that about half of these nonobjective predictions mark regions that are no more coherent than any other randomly chosen ocean domain of similar size. LAVD-based coherent eddy detection, therefore, suggests an order of magnitude downward correction to earlier estimates for mesoscale eddy transport in the ocean by Zhang, Z. et al. (2014) and Dong et al. (2014). LAVD-based vortex detection has also been used recently by Yang, K. et al. (2021) in medical imaging to identify differences between healthy and unhealthy heart conditions. The authors employ LAVD to determine the size of material vortices in the left ventricular blood flow of healthy subjects and patients. In the pre-ejection periods of their cardiac cycles, healthy patients have consistently displayed a smaller clockwise LAVD vortex than patients with uremic cardiomyopathy, as seen in Fig. 5.25. In the middle of the ejection period, heathy subjects developed a smaller counter-clockwise vortex. No similar systematic differences were observed during the final ejection stage. The authors attribute the formation of larger vortices to cardiac hypertrophy (see Yang, K. et al., 2021, for details and §5.7.3 for further related results involving objective Eulerian coherent vortices). 4

The convexity deficiency of a closed curve on a plane is defined as the ratio of the area between the curve and its convex hull to the area enclosed by the curve.

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5.2 Advective Transport Barriers as LCSs

Figure 5.24 Coherent Lagrangian swirl in the Gulf of Mexico, identified from the LAVD and from the (nonobjective) Okubo–Weiss criterion (see §3.7.1). The first column of plots shows the initial position of the LAVD-based coherent swirl on May 29, 2013 for the extraction times (i.e., t1 − t0 values) of 30, 60 and 90 days, as well as the Okubo–Weiss elliptic region on the same date. The second column shows the materially advected position of the swirls at the corresponding t1 times. The third column shows the corresponding advected positions of the Okubo–Weiss elliptic region. Adapted from Beron-Vera et al. (2019b).

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Med Biol Eng Comput

Figure 5.25 LAVD-based material vortices in the left ventricular (LV) blood flow of a healthy subject and a heart patient. Different rows compare Lagrangian vortices in three different ejection stages of a cardiac cycle, indicated along a red electrocardiogram (ECG) reference line in the upper right corner of each plot. Shown in grayscale in the background are instantaneous ultrasound scans from which a velocity field was inferred using the vector flow mapping (VFM) algorithm. Adapted from Yang, K. et al. (2021).

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T =LAVD-based 120 days. 200Thecoherent yellow surface is extracted Algorithm 2, marking the vortex Figure 5.26 Lagrangian vortexusing (yellow surface) and its core boundary for afrom mesoscale eddy,and extending from 28 m (red curve) identified closedrotationally outermostcoherent LAVDLagrangian level curves the LAVD down to 646 m in depth. The green surface is a nearby level surface of LAVDtt00 +T (x0 ) maximumoutside they surround planes. (Left) Initial position of the vortex, 400 in horizontal the Lagrangian vortex region. The red curve marks the coherent vortex centre, as with nearby, noncylindrical LAVD level surfaces (green) showneddy for reference. (Right) defined in Definition 1. (b) Full view of the Lagrangian boundary and its centre Materiallyatadvected of The the advected vortex 120 days later.andA vortex videocentre showing the later, the initial position time t . (c) eddy boundary 120 days 0 600 extending of from m down tovortex 726 mboundary below theissurface (see atthe online supplementary material evolution the 49 Lagrangian available https://www. movie M4 for the complete advection sequence of the eddy boundary). cambridge.org/core/journals/journal-of-fluid-mechanics/article/ 800 abs/defining-coherent-vortices-objectively-from-the-v orticity/ 559F52C404B54265CB7A64D4666EB096\#supplementary-materials.

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F IGURE 13. (a) Representative level surfaces of LAVDtt00 +T (x0 ) with t0 = 15 May 2006 and Application 3Dvortex Flows T = 120 days. The yellow surface is extracted using Algorithmof2,LAVD markingtothe boundary for a mesoscale rotationally coherent Lagrangian eddy, extending from 28 m +T In depth. 3D flows, advantage of objectivity, down to 646 m in The the greenLAVD surfacehas is the a nearby level surface of LAVDtt00while (x0 ) the PRA is no longer objective, outside the Lagrangian vortex TheIfred the coherent stratified, vortex centre, as noted in region. §5.2.10. thecurve flowmarks is horizontally ouras2D diagnostic principle for the defined in Definition 1. (b) Full view of the Lagrangian eddy boundary and its centre be applied along horizontal at the initial timeLAVD t0 . (c) (Definition The advected 5.7) eddy can boundary and vortex centre 120 daysplanes, later, with the LAVD level curves and they surround in each plane separately and then connected extending from 49 m the downmaximum to 726 m below the surface identified (see the online supplementary movie M4 for thetogether. complete This advection sequenceeffectively of the eddy constructs boundary). elliptic LCSs as tubular level sets around a procedure

one-dimensional height ridge (see Appendix A.2) of the LAVD field. An example of this procedure is shown in Fig. 5.26, with a 3D coherent Lagrangian vortex IVD-based and LAVD-based rotationally coherent vortex boundary surfaces. In this computation, theand initial sliceidentified of the LAVD-based is located itstop center from the vortex LAVDboundary computed along only 2D horizontal slices in a 3D data15 m below the ocean surface, as opposed to the 28 m distance used in figure 13(c). assimilating ocean circulation model (see Haller et al., 2016, for details). The figure also Even this Lagrangian boundary slice remains more coherent than the Eulerian one, theperforms materially position of thisforLagrangian vortex 120 days later, at the endbut the Eulerian shows slice still welladvected under material advection this eddy and for this advectiontime time.t1 of its extraction interval. Note that the vortex boundary indeed remains rotationally coherent, showing only signs of slight tangential filamentation (ribbing). For more general turbulent flows with no preferred orientation for vortices, a fully 3D approach is necessary for LAVD-based vortex detection. Such an approach has been put forward by Neamtu-Halic et al. (2019), who seek vortex center lines as one-dimensional

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ridges of general orientation the fully 3D LAVD field. This diagnostic principle can be formulated as follows: Definition 5.9 At time t0 , LAVD-based elliptic LCSs in the finite-time flow (5.1) are smooth cylindrical level surfaces of the Lagrangian-averaged vorticity deviation LAVDtt10 (x0 ) surrounding a unique, codimension-2 height ridge of LAVDtt10 (x0 ). LAVD-based coherent Lagrangian vortices are formed by nested families of such elliptic LCSs. LAVD-based vortex centers in such Lagrangian vortices are marked by the codimension-2 ridge of LAVDtt10 (x0 ) surrounded by the elliptic LCSs. This principle is general enough to cover 2D flows as well, for which codimension-2 ridges of LAVDtt10 (x0 ) are simply local maximum points. Operationally, one-dimensional height-ridges of the LAVD field can be found in a way similar to the gradient-climbing algorithm we have discussed for FTLE ridges (see formula (5.30)). Specifically, based on the watershed and the structurally stable ridge definitions of Appendix A.2, the ridges we seek are one-dimensional attractors of the gradient dynamical system d x0 (s) = ∇LAVDtt10 (x0 (s)). ds

(5.30)

Following the algorithm we have described for FTLE ridges, we can launch trajectories of (5.30) in ridge neighborhoods identified from a rough thresholding, then seek converged locations spanned by the endpoints of those trajectories. A one-dimensional LAVD ridge located in this fashion marks the center line for a rotationally coherent Lagrangian vortex. Next, the definition of the height-ridge can be invoked from Appendix A.2 to seek outermost closed and convex LAVD level curves in planes normal to the ridge. The collection of these closed curves can then be used to construct the desired rotationally coherent Lagrangian vortex boundary. Details of this algorithm are described in Neamtu-Halic et al. (2019). An example of the application of this algorithm to velocity data obtained from the 3D gravity current experiment analyzed by Neamtu-Halic et al. (2019) is shown in Fig. 5.27.

5.3 Local Variational Theory of LCSs The objective diagnostics we discussed for LCSs in §§5.1–5.2 are very effective for a quick, observer-independent identification of transport barriers in numerical and experimental flow data. They are, however, largely based on approximations to, rather than systematic mathematical implementations of, Definition 5.3 for LCSs. That definition stipulates that LCSs are the locally most influential material surfaces and hence their identification requires the solution of an optimization problem. Specifically, one needs to systematically quantify and compare the influence of a material surface with the influences of all neighboring material surfaces on fluid particles. The added advantage of solving this optimization problem is that the LCSs are no longer simply inferred by visual inspection of a plot, but are rendered as parametrized material curves or material surfaces. Depending on how extensively one wishes to perform this comparison, the resulting optimization problem will vary from a simple, pointwise optimization to a complicated global optimization requiring the solution of nonlinear systems of partial differential equations. In

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F IGURE 2. (Colour online) (a) Three-dimensional fluid trajectories colour-coded with the norm of the vorticity for the flow case Ri20. The time period shown here is equivalent to three turnover times (defined in § 2.4) of the largest eddies. The four alternated red and blue rectangular the Lagrangian four 3-D PTV observation The Figure outlines 5.27 represent Coherent vortices volumes. (blue) (b) identified above the turbucorrespondinglent/nonturbulent 3-D VLCSs, represented by blue tubular surfaces (boundaries) surrounding in interface (TNTI; shown in red) from the LAVD in a gravity one-dimensional (1-D) curves (centres), and the TNTI of the gravity current (red open experiment. curves one-dimensional LAVD ridges. The surface). Thecurrent region above the TNTI isDarker turbulent,blue whereas belowmark the flow is irrotational.

cm) 4 5

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velocity field was obtained from 3D particle tracking velocimetry (PTV). Adapted from Neamtu-Halic et al. (2019). nm). As flow tracers, we used neutrally buoyant polyamide particles with

ple of VLCS extraction. In (a), the final position of the The blue dots aretheselected for the ridge construction, this section, we discuss simplest local optimization setting for locating LCSs, which we refer to as the local variational theory of LCSs. ed. In (b),In this intheory, blue, the connection of the points selected we compare the influence of a material surface in an infinitesimally small of each of The its points corresponding to the influences of all other VLCS material surfaces ve (red) neighborhood are shown. is running shown Duo, 527 a mean diameter of 100 µm (manufactured by Evonik Industries, Germany). Each single 3-D PTV system covered an observation volume of approximately 9 cm ⇥ 9 cm ⇥ 4 cm in the x (streamwise), y (wall-normal) and z (spanwise) directions, respectively. The fields of view of the individual PTV systems overlapped for approximately 2 cm to track particles continuously across the different observation volumes. The start of the measurement volume was located approximately 50 cm away from the inlet and covered roughly 31 cm in the streamwise direction (figure 2). For each observation volume, it was possible to track up to 3000 particles simultaneously. point. Technically, at each point x0 of a codimension-1, initial material This through correspondsthe to asame mean interparticle distance of approximately 3.5 mm, equivalent 1/4 to roughly 10⌘,M(t with0 ), ⌘ =we (⌫ 3 /✏) being the the Kolmogorov microscale, where ⌫ the unit normal n0 into planes surface will perturb tangent space T M(t ) x0 0 iswith kinematic viscosity and ✏ = u03 /L7 is the local dissipation. Here, u0 is the root mean with unitof normals nˆ 0fluctuation . These and perturbed represent thethetangent planes of all nearby square (r.m.s.) the velocity L is the planes integral length scale of turbulence, evaluated as the running integral of through the autocorrelation function of the streamwise material surfaces the point x0 . We then seek conditions under which the velocity along x. The turbulence level was quantified through integral Reynolds (x0, the associated local material stretch factor the ρtt10stable nˆstratification material shear factor σtt01 (x0, nˆ 0 ) 0 ) or localreduces number ReL = u0 L/⌫ (table 1). As can be observed, the turbulence level. Reference evaluated at a height has an extremum at nˆ 0 length = n0 and (seetime Fig.scales 5.28were for an illustration). We will locate hyperbolic and of approximately 7 cm from the top wall, a location that is far from the wall but shear LCS as codimension-1 material surfaces that pointwise extremize (or at least render still sufficiently far from the strongly intermittent region close to the TNTI. The

es. By parametrizing their x, y and z coordinates with stationary) these local stretch and shear factors, which we will compute more precisely in the e fit them with a flows, cubic smoothing spline. next section. For 2D the codimension-1 material surface M(t) will be a material curve.

ple of the application of the last three steps described 5.3.1 Local Variational Theory of Hyperbolic LCSs points in ridge are not along ahave1-D To find a repellingproximity LCS from its local variational theory aligned over the time interval [t , t ], we to find at each point x of the flow domain the tangent space T M(t ) of an initial material select only points withunit sufficiently close neighbouring surface M := M(t ) whose normal n maximizes the stretch factor ρ (x , n ) shown in Fig. 5.28 at x . In contrast, normals to the initial positions of attracting LCSs at t can figure be3b) and apply a smoothing cubic spline (long found by minimizing ρ (x , n ). Once their initial positions are located, arbitrary later 0 1

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f structures, we determined their boundaries using the 7

A full local optimization of the surface M(t0 ), including a perturbation to the point x0 in addition to the normal n0 , was also considered by Haller (2011), but it turns out to lead to an overconstrained problem for LCSs.

ridge, we erect pointwise normal planes to the ridge n-plane outermost almost-convex LAVD contour that curves are 1-D curves in three dimensions. https://doi.org/10.1017/9781009225199.006 Published online by Cambridge University Press

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rFtt10 (x0 )n0

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Figure 5.28 The geometry of local variational LCS theory, which constructs hyperbolic and shear LCSs as pointwise stationary surfaces of the material stretch factor ρtt10 (x0, nˆ 0 ) or local material shear factor σtt01 (x0, nˆ 0 ), with respect to changes in nˆ 0 . These factors are defined as normal and tangential projections, respectively, of the advected unit normal ∇Ftt10 (x0 )nˆ 0 onto the normal space and tangent space of the ˆ 1 ) = Ft1 (M(t ˆ 0 )), at the point xt . Adapted advected perturbed material surface, M(t 1 t0 from Haller (2015).

M(t1 ) = Ftt10 (M0 ). As we concluded from formula (2.30), the normal of a material surface at   −T the point x0 at time t0 is advected by the inverse transpose ∇Ftt10 (x0 ) of the deformation t1 gradient to a normal of M(t1 ) at the point xt1 = Ft0 (x0 ) at time t1 . This advected normal, however, is generally not of unit length, and hence only after normalization does it become the unit normal   −T ∇Ftt10 (x0 ) n0 (5.31) nt1 =   −T ∇Ftt10 (x0 ) n0 to M(t1 ) at the point xt1 . With this expression at hand, we can calculate the stretch factor ρtt10 (x0, n0 ) as the inner product of the advected unit normal, ∇Ftt10 (x0 )n0 , with nt1 : D E  −T ∇Ftt10 (x0 ) n0, ∇Ftt10 (x0 )n0

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 minimal (maximal) precisely when n0, Ctt10 (x0 ) n0 is maximal (minimal), we obtain the following result: Proposition 5.10 The local stretch factor ρtt10 (x0, n0 ) of a codimension-1 material surface M(t) at its initial point x0 ∈ M0 is maximized (or minimized) by the unit normal vector

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n0 ∈ Nx0 M0 aligning with the dominant (or weakest) eigenvector ξn (x0 ; t0, t1 ) (or ξ1 (x0 ; t0, t1 )) of Ctt10 (x0 ) , as defined in Eq. (2.95). We will now explore the implications of this conclusion first for 2D flows and then for 3D flows.

Hyperbolic LCSs in 2D Flows For 2D flows, the locally most repelling and attracting LCSs are, by Proposition 5.10, material curves whose initial positions are everywhere normal to the ξ2 (x0 ; t0, t1 ) and ξ1 (x0 ; t0, t1 ) unit eigenvector fields, respectively. Note that these fields of unit eigenvectors are not uniquely defined: they have two possible orientations at each point x0 . It turns out that even for the simplest flows, one cannot select a smooth global orientation for either of the ξi (x0 ; t0, t1 ) vector fields, and hence it is more appropriate to refer to them as direction fields (see Spivak, 1999; Karrasch et al., 2014). Even though they are globally nonorientable, these direction fields can always be locally oriented except at points where Ctt10 (x0 ) has repeated eigenvalues and hence ξ1 = ξ2 . In the latter case, the distinction between the dominant and weaker eigenvectors of Ctt10 (x0 ) is no longer well defined. For any i , j, being normal to ξi is equivalent to being tangent to ξ j . Therefore, in two dimensions, hyperbolic LCSs can be sought among trajectories of the Lagrangian direction field x00 = ξi (x0 ; t0, t1 ),

i = 1, 2,

(5.33)

where the prime denotes differentiation with respect to a parameter s along the trajectories x0 (s) of (5.33). These trajectories, often called tensorlines, are well defined away from points where λ2 = λ1 . At such points, the ordering of the eigenvalues is no longer well defined, and hence these points are often referred to as tensorline singularities (see Delmarcelle and Hesselink, 1994). For i = 1, we call trajectories of (5.33) shrinklines and for i = 2, we call the trajectories stretchlines. By Proposition 5.10, the most repelling shrinklines mark initial positions of repelling LCSs and the most attracting stretchlines mark initial positions of attracting LCSs. Repelling LCSs can therefore be located as trajectories of the direction field (5.33) with i = 1 that have locally the largest averaged λ2 (x0 (s); t0, t1 ) value among all neighboring shrinklines. Likewise, attracting LCSs can be located as trajectories of the direction field (5.33) with i = 2 that have locally the lowest averaged λ1 (x0 (s); t0, t1 ) value among neighboring stretchlines. To handle orientational discontinuities and tensorline singularities in numerical computations of these LCSs, Farazmand and Haller (2012) use a rescaled version of (5.33) in the form

 x00 (s) = sign ξi (x0 (s); t0, t1 ), x00 (s − ∆) α (x0 (s); t0, t1 ) ξi (x0 (s); t0, t1 ), 2  λ2 (x0 ; t0, t1 ) − λ1 (x0 ; t0, t1 ) , (5.34) α (x0 ; t0, t1 ) = λ2 (x0 ; t0, t1 ) + λ1 (x0 ; t0, t1 ) where ∆ > 0 is the numerical step size used in solving for tensorlines. This rescaling preserves the trajectories of (5.33), but produces a locally oriented vector field on the fly along trajectories during numerical integration and turns tensorline singularities into fixed points. Farazmand and Haller (2012) solve for trajectories of the rescaled direction field

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(5.34) with i = 1, starting from a grid of initial conditions and obtain repelling LCSs at time t0 as shrinklines with locally the largest averaged repulsion rate, λ2 . The computation of hyperbolic LCSs in 2D unsteady flows in implemented in Notebook 5.6. Notebook 5.6 (HyperbolicLCS) Computes hyperbolic LCSs from tensorlines (shrinklines or stretchlines) of the right Cauchy–Green strain tensor in a 2D unsteady velocity data set. https://github.com/haller-group/TBarrier/tree/main/TBarrier/2D/ demos/AdvectiveBarriers/HyperbolicLCS

To obtain later positions of a repelling LCS starting from a shrinkline M(t0 ), Farazmand and Haller (2012) select initial points x0 ∈ M(t0 ) to obtain advected points Ftt0 (x0 ) ∈ M(t) and the tangent vectors ∇Ftt0 (x0 ) ξ1 (x0 ; t0, t1 ) to M(t) at those points.8 They then fit a minimallength Hermite polynomial to the advected points and advected tangents to obtain repelling LCSs as parametrized curves at an arbitrary time t ∈ (t0, t1 ). Farazmand and Haller (2012) discuss a similar computation for an attracting LCSs evolving backwards from its positions M(t1 ). This M(t p1 ) is constructed as a locally most backward repelling shrinkline, using the singular values λi (x1 ; t1, t0 ) and singular vector ξ1 (x1 ; t1, t0 ) of the backward deformation gradient ∇Ftt01 (x1 ) in Eq. (5.34). These computations are costlier than using FTLE ridges to visualize LCSs, but they are also more accurate and render LCSs in the form of parametrized curves, rather than just images. Figure 5.29 shows examples of LCS calculations for 2D flows and maps using shrinklines. Specifically, Fig. 5.29(a) shows the improvement over the FTLE computation (left) of a repelling LCS in the time-periodic double-gyre flow of Shadden et al. (2005) using forward shrinklines (right). Figure 5.29(b) shows trajectories from 100 iterations of a 2D iterated mapping model of Pierrehumbert (1991) (left) and stable (unstable) manifolds computed as locally most repelling forward (backward) shrinklines from only 14 iterations of the same map. Finally, Fig. 5.29(c) shows stable and unstable manifolds computed as locally most repelling forward shrinklines (red) and most attracting backward shrinklines (blue) in the chaotically forced Bickley-jet model developed by del Castillo-Negrete and Morrison (1993). Farazmand and Haller (2013) point out that attracting LCSs in forward time can also be constructed as material curves evolving from stretchlines (i.e., trajectories of the differential equation (5.33) with i = 2) that are locally the most attracting, i.e., have locally the lowest λ1 (x0 ; t0, t1 ) value when averaged along them. This coincidence of attracting LCSs from forward calculations with repelling LCSs from backward calculations follows because the dominant forward-eigenvector field ξ2 (x0 ; t0, t1 ) of Ctt10 (x0 ) (i.e., the tangent vector field of repelling LCSs at time t0 ) will be advected by the flow into the weakest backward-eigenvector field ξ1 (x1 ; t1, t0 ) of Ctt01 (x1 ) by formulas (2.98) and (2.102). An example of this forward-time computation of attracting LCSs is shown in Fig. 5.30. The results of Farazmand and Haller (2013) enable the extraction of both repelling and attracting LCSs at t0 from a single forward computation of the flow map Ftt10 . Similarly, we can obtain both repelling and attracting LCS at time t1 from a single backward computation 8

The vector ∇Ftt0 (x0 ) ξ1 (x0 ; t0 , t1 ) is tangent to M(t) at the point Ftt0 (x0 ) ∈ M(t) by formula (2.28).

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181

5.3 Local Variational Theory of LCSs 013128-9

Computing Lagrangian coherent structures

Chaos 22, 013128 (2012)

(a)

1692

FIG. 10. (Color) (a) The forward time FTLE field for the double-gyre model; integration time is T ¼ 20. (b) The red curve is a strainline segment that qualifies

¼ 241 0:2; ‘min ¼ 1. 1680–1702 as a repelling LCS by satisfying conditions (B), and (D). The filtering are chosen as ‘f D G. Haller, F.J. Beron-Vera / Physica D 241(A), (2012) 1680–1702 1689 G. Haller, F.J. parameters Beron-Vera / Physica (2012)

Among the strainlines initiated from the initial conditions shown in Fig. 9(b), only the ones along which condition (D) is satisfied qualify as repelling LCSs. For instance, for the filtering parameter values ‘f ¼ 0:2 and ‘min ¼ 1, only the single strainline segment shown in Fig. 10(b) qualifies as a repelling LCS over the time interval [0, 20]. Note that the strainline integration stops once it reaches the boundary. In order to obtain longer strainlines (and longer LCSs as a result), a slightly larger domain U ¼ ½#0:02; 2:02$ % ½#0:01; 1:01$ is used in Figure 10. This well-studied example demonstrates the advantages of the variational LCS computation over simply using the FTLE field to infer the location of LCSs:

corresponding LCS is captured without interruption by the strainline shown in Figure 10(b). Similar results and conclusions hold for attracting LCSs, with the computations performed in backward time. B. Randomly forced turbulent flow

In our second example, we test our numerical algorithm for variational LCS detection on a direct numerical simulation of two-dimensional turbulence. The velocity field v is obtained as the numerical solution of the modified Navier–Stokes equation,

@t v þ v ' $v ¼ #$p # !u D2 v þ !i D#1 v þ f; $ ' v ¼ 0; (15) 1. The FTLE field in Figure 10(a) gives an intuitive idea about the location of a LCS. By contrast, Figure 10(b) is based on a mathematical existence result (Theorem 1) by a pseudo-spectral method with 10242 Fourier modes. We that guarantees the existence of a repelling LCS. Note use a Krylov-subspace method8 for temporal integration. We that none of the strainlines (and hence none of the LCSs) set the hyperviscosity as !u ¼ 2 % 10#9 to shift the dissipacoincide precisely with the FTLE ridge. Instead, a part of tion to smaller scales while keeping the resolution relatively a FTLE ridge that remains close to strainlines satisfying low. Energy isFig. removed from largeand scales by the hypoviscos- barriers (red and blue) at time t0 = 0 19. Forwardbackward-hyperbolic Fig. the 16.iteration The three types of(A), zonal perturbation considered, evaluated (B), (D) is 0a.velocities good forall prevent from energy at thefor largest ity manifolds !i ¼ 1 to obtained obtained fromn the Bickley jet with strong chaotic forcing (case Fig. 10. Select transport barriers obtained from ofconditions the advection map Pand with a= 2. Left approximation panel: Stable and unstable =accumulation 9geodesic forward theory LCS. at a point the aaxis the background meandering zonal forcing f isforward mono-scale at [the used the interval t0 , t0 + T ] and the backward-time interval and backward iterations of a small circle of radius 10 6on , centered atofthe known hyperbolic fixed points of P.jet. The KAM tori areavailable obtained scales. from(2)). n The =We 100 iterations of selecttime(active 4 2. The ridge of the FTLE field in Figure 10(a) is easy to recwavenumber k ¼ 2) with random phase. The amplitude ofx /U. We again let ✏⇠1 = 10 6 for the initial conditions. Right panel: Same as on the left, but using n = 14 iterations for the stable and unstable manifolds, and n = 10 iterations for[tthe ] in this analysis with T = 4L 0 , t0KAMTtori. ognize visually. Extracting such ridges for further analythe forcing changes in time, ensuring statistically stationary admissible bound on the pointwise geodesic deviation curvature along hyperbolic sis, however, leads to an image-processing problem that state. Equationbarriers. (15) is solved on the spatial domain ½0 2p$ (For interpretation of the references to colour in this figure legend, the ultimately yields a set of points approximating the LCS periodicis boundary conditions, over theoftime %½0 2p$ with reader referred to the web version this article.) candidate (see Lipinski and Mohseni18 and Senatore and interval [0 100] (see Refs. 5 and 10 for similar simulations of 25 Ross ). By contrast, the strainline segment in Figure the same equation). length data, Fig. that averaged geodesic deviation 10(a) is given in the form of a smoothly parametrized ⌦ ⌘21 ↵ illustrates The hyperbolic LCSofextraction algorithm again follows dg ±fix  are effective in isolating generalvalues condition satisfying curve–a trajectory of the ODE (8)–with precisely defined Table I. To implement (D), we L as0a.0015 set of 20 repulsion properties. This curve can be obtained at arbiequally spacedized vertical and 20 horizontal lines. Local minKAM-type and jet-type structures from open spiraling straintrarily high resolution by selecting a small enough stepsonly kept iflatter the neighboring strainline segima of k!2 ðcÞ are lines. The strainlines also remain close to least-stretching ize D in solving the scaled strainline ODE (9). ments with weaker k!2 values are no more than 0.05 apart geodesics but do not act as closed boundaries of elliptic regions. 3. Even if a FTLE ridge is not well-pronounced, as long as it along the horizontal or vertical lines. Finally, Fig. 22 shows a composite picture of transport barriers correctly signals a nearby LCS, the strainlines will capFigure 11 shows attracting LCSs extracted at time ture that LCS sharply. Indeed, the FTLE ridge shown in fromcalculation our geodesic theory for the Bickley jet with weak backward-time with integration t0 ¼ 70 from aobtained Figure 10(a) fades away in the black rectangle, yet the time T ¼ –30 over a grid of 512 % 512 points. chaotic forcing. Note that the robust extraction of elliptic barriers

(b)

(c)

Figure 5.29 Hyperbolic LCSs in 2D, computed as shrinklines from the local variational theory of hyperbolic LCSs. (a) Improvement over an FTLE calculation. Adapted from Farazmand and Haller, 2012. (b) Refinement of simple iterations of a 2D map. Adapted from Haller and Beron-Vera, 2012. (c) Stable and unstable manifolds of the chaotically forced Bickley jet, computed as forward and backward shrinklines. Adapted from Haller and Beron-Vera, 2012.

required a time interval that is about five times the minimal time scale needed for a robust detection of hyperbolic barriers. This is because shear barriers only prevail as minimally stretching Fig. 11. Forward-hyperbolic barriers (red curves) reconstructed from n = = = . t +T ⇠1 3 Fig. 17. Phase portrait of the the Poincaré Ft00 dpfor Bickley jet strainlines super-minima of relative stretching among all strainline segments satisfying cut-offmap condition the 10 time-periodic pointwise. Some additional (black) are shown for material lines on longer time intervals due to the algebraic growth reference. The unit eigenvector field ⇠1 is depicted select positions, plot shows a blow-up (caseat (1)) with t0 = 0 with and Tdouble-headed = 4Lx /U. arrows emphasizing its lack of a global orientation. The ofthird material length in shear regions vs. exponential growth in highof the barrier obtained for n = 9 near the hyperbolic fixed point at (0, 0). (For interpretation of the references to colour in this figure legend, the reader is referred to the t0 strain regions. ⌦ ⌘ ↵ web version of this article.) barriers with low geodesic deviation ( dg ±  0.0015) do not t1 While the location of the barriers is robust for beyond ⌦ ⌘±times ↵ exist either. This is in Fig. unstable 19, which showsor KAM  0.0015, = 20LAx /blow-up U and for T torus). (stable manifold, manifold, of geodesic deviation values dg the extracted forward-hyperbolic barriers (red curves) in further Fig. 11 illustrated a superposition of forwardand backward-hyperbolic barriers resolving all details of the barriers in the current, chaotically the heteroclinic tangle on the right provides further confirmation reconstructed from fewer iterations (n = 4 and n = 9) using our of the setting proves to be a challenge. For instance, one of the at taccuracy geodesic transport barrier theory. obtained from the geodesic theory 0 = 0. of our barrier reconstruction. 0 forced We conclude that our theory provides highly centered roughly around (x/Lx , y/Ly ) = As for elliptic barriers, Fig. 12 shows By shearlines of the vec-regions are closed barriers contrast, theshear elliptic not destroyed ingeodesic the casetransport of t1elliptic accurate barriers using relatively low numbers of P. appears to turn around and form a lobe. This 15 iterations of the advection tor fields ⌘± (x) obtained from n = weaker .6) in Fig.of22 (4.1, t 0iterations chaotic⌘forcing (case (3)), as 0 seen in Fig. 20. Convergence to 0not provide the Simply iterating P for the same length of time does map P. Those qualifying as ellipticgeneralized barriers withKAM dg ± tori  ✏(closed ⌘± = green is due to numerical errors in the computation of the underlying curves) and generalized shear same quality or detail (compare the left panels of Figs. 10 and 15). 0.25 are highlighted in green. The lower panel of the same figstrainlines over the relatively long time interval [0, 20Lx /U]. jets (open green curves) is apparent for increasing temporal Admittedly, the high accuracylength of the geodesic theory comes at a ure shows the same computation using n = 50 iterations. Note The of artificial lobe disappears once the spatial resolution in our T for the data set. higher computational cost relative to the cost simply iterating that several closed shearlines in these figures still do not yet qual⌦ ⌘± ↵ T = 20L /U emerges as the minimal time scale 1 is doubled in this elliptic region (cf. the right panel computations In general, the map P. ify as near-homogeneous barriers because dg > 0.25 holds xon t 0 in Fig. 22). needed for the robust identification of shear barriers. For this them. Still, they already provide close approximations for KAM Downloaded to 132.216.88.104. Redistribution subject to AIP http://chaos.aip.org/about/rights_and_permissions 4 and 13 n Mar 2012 9 iterations of the advection map P license withoracopyright; 0 2.seeThese barriers were located as

of the backward flow map F . Obtaining other positions of these LCSs by material advection, however, remains problematic in certain time directions. Namely, obtaining an intermediate position M(t) of a repelling LCS known at time t involves the forward-advection of a discrete approximation to M(t ) under the flow map F , which generates exponentially growing errors, as shown in Fig. 5.31. The same numerical instability arises when computing intermediate positions of an attracting LCS known at time t by using backward advection under the inverse flow map Ft9.1 .Example 2: Transport barriers in the chaotically forced

tori; some even highlight secondary KAM tori within the resonant islands. A more detailed study of the convergence of our elliptic barrier reconstruction algorithm is shown in Figs. 13 and 14. Note that 0 used in the reconstruction, the smaller the the more iterations are geodesic deviation becomes, and the closer the closed shearline becomes to ⌦ ⌘an↵invariant curve. Generally, for geodesic deviations of the order dg ±  0.25, KAM tori and reconstructed elliptic barriers appear indistinguishable. Finally, in Fig. 15, we show together the set of reconstructed hyperbolic and elliptic barriers. Gray curves indicate hyperbolic and elliptic barriers from the second, high-resolution phase portrait of P shown in Fig. 10. At this resolution, each reconstructed hyperbolic and elliptic barrier coincides with an observed barrier

Bickley jet

Karrasch et al. (2015) point out that these instabilities can be avoided by advecting time In our second example, we apply our geodesic transport theory toin a spatially more complex, aperiodic dyt positions of attracting LCSs forward timetemporally and time t1planar positions of repelling LCSs namical system. In this setting, no Poincare maps are available, and hence even an indirect visualization barriers from adin backward time. Both of these LCSs behave asof transport attracting material lines in these time vected initial conditions would be problematic. Indeed, there is no known frameis or stroboscopic mapping sequence under which theprocedure, free from the directions and hence their advection a self-stabilizing numerical barriers become steady. the two-dimensional incompressible definedLCSs by instabilities shown in Fig. 5.31. Consider To simplify the selection offlow these from all the possible the stream function tensorlines of Eq. (5.34), Karrasch attracting LCSs at (x, y, t ) = et(xal. , y) +(2015) (x, y, t ) propose constructing (44) time t0 as trajectories of Eq. with (5.34) with i = 2 that cross local minimum locations xmin of 0 λ1 (x0 ; t0, t1 ). These are the locations of generalized saddle points at time t0 with the strongest Fig. 18. Forward-time hyperbolic barriers (red) obtained fromminima the geodesic theory at time = 0 for the Bickley jet withas strong chaotic forcing (case (2))of over the [t , t + T ], for forward-time attraction rate. These local can bet found simply local minima two different choices of the interval length T . The pointwise admissible upper bound on the geodesic deviation is chosen to be ✏ = 10 . Also shown are some additional 0

1

0

⇠1

6

0

0

strainlines (black) and the minimum strain eigenvector field ⇠1 (x, y) (gray) for reference. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

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Lagrangian and Objective Eulerian Coherent Structures

FIG. 9. (a) Forward stretchlines The attracting LCSs (i.e., local stretching stretchlines) are highli red. The green closed curves s boundaries of elliptic regions. (blue circles) are used to visua overall mixing patterns. (b) A image of the attracting LCSs, and elliptic barriers at time t ¼ 50

Figure 5.30constructed Attracting LCSs computed in forward time using the ξimages of and the plane at time t ¼ 2 -tensorlines locally stretch-surfaces do govern the formation advected of the tracer Ctt10 (xof0 )tracer in a patterns 2D turbulence simulation. (Left) General tensorlines (gray) computed in three-dimensional flows as well. shown in Figure 10(b), demonstrating that the stretch-s from Eq.We (5.34) with i =ABC 2 outside outermost elliptic variational LCSs (green; to tracer be blob acts as a de facto use the classic flow (Arnold and Khesin, 1998) through the center of the discussed in §5.3.2); in red: attracting LCSs identified as tensorlines with locally the ble manifold in this three-dimensional example as well lowest average of the attraction λ1 ; in x_ 1 ¼ A sinðx3 Þ rate þ C cosðx 2 Þ;blue: array of tracer initial condition blobs. (Right) Advected images of all these structures at time from Farazmand VI. CONCLUSIONS x_ 2 ¼ B sinðx (14)t1 . Adapted 1 Þ þ A cosðx3 Þ; and Haller (2013). x_ ¼ C sinðx Þ þ B cosðx Þ; We have shown that both repelling and attracting 84

3 2 1 DANIEL KARRASCH, MOHAMMAD FARAZMAND AND GEORGE HALLER (finite-time stable

pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi with A ¼ 1, B ¼ 2=3, and C ¼ 1=3. The Cf strain tensor Exact Numerical is computed over the time interval I ¼ [0, 4] (i.e., a ¼ 0 and b ¼ 4). We release a spherical blob of initial conditions centered at ðp; pÞ with radius 0.1. We approximate the stretchsurface passing through this point by the plane normal to the first eigenvector nf1 of Cf. Figure 10(a) shows this plane together with the sphere of tracers at time t ¼ 0. The

and unstable manifolds) at a time in t ¼ a can be extracted from a single forward-time com tion over a time interval I ¼ [a, b]. This extraction re the computation of the eigenvectors of the fo Cauchy–Green strain tensor Cf . It has been found prev (Haller, 2011; Haller and Beron-Vera, 2012) that a t ¼ a, the position of repelling LCSs are strain-surface are everywhere orthogonal to the dominant eigenvec

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0. By incompressibility, this conservation law then implies a constant tangential stretching factor λ along the curve x0 (s). Therefore, elliptic LCSs satisfying the variational principle (5.50) are special closed curves whose local tangential stretching over the [t0, t1 ] time interval is equal, at each of their points, to the same factor λ > 0. This explains why the γ curves satisfying the variational principle (5.50) create perfect coherence in their vicinity: their pointwise even stretching or compression by the constant factor λ disallows the formation of small-scale material filamentation.
Using the expression for ρtt10 x0, x00 from Eq. (5.48), we rewrite Eq. (5.51) in the form  

0  t1 (5.52) x0, Ct0 (x0 ) − λ2 I x00 = 0, we can seek curves x0 (s) satisfying Eq. (5.52) by expressing their unit tangent vectors as a linear combination x00 = c1 ξ1 (x0 ; t0, t1 ) + c2 ξ2 (x0 ; t0, t1 ),

c12 + c22 = 1

(5.53)

of the eigenvectors of the tensor Ctt10 (x0 ). Substitution of Eq. (5.53) into Eq. (5.52) then leads to the following result (see Haller and Beron-Vera, 2013 for more detail). Theorem 5.12 Initial positions of elliptic LCSs, defined as stationary curves of the average normal repulsion functional (5.49), are closed trajectories of the two one-parameter families of direction fields, s s 2 (x ) λ λ2 − λ1 (x0 ) − λ 2 0 x00 (s) = ηλ± (x0 (s)), ηλ± (x0 ) = ξ1 (x0 ) ± ξ2 (x0 ) , λ2 (x0 ) − λ1 (x0 ) λ2 (x0 ) − λ1 (x0 ) (5.54) defined on the spatial domain  Uλ = x0 ∈ U : λ1 (x0 ; t0, t1 ) < λ2 < λ2 (x0 ; t0, t1 ) . (5.55) Any such elliptic LCS is uniformly λ-stretching: any of its subsets is stretched precisely by the factor λ > 0 over the time interval [t0, t1 ]. Outermost members of nested families of such closed curves serve as initial positions of Lagrangian vortex boundaries. For λ = 1, the λ-lines defined by formula (5.54) recover their initial length at time t1 and hence coincide with the shearlines we have identified from the local variational theory of LCSs in formula (5.39).11 This shows that the present global theory of LCSs is indeed an extension of the local theory, securing a lack of filamentation for the LCS without insisting on a perfect conservation of its length under advection. 11

One p this by setting λ = 1 in formula (5.39), dividing both the numerators and the denominators by p can verify λ2 (x0 ) − λ1 (x0 ) and using the incompressibility condition λ1 (x0 )λ2 (x0 ) = 1.

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197

Formula (5.52) enables the interpretation of λ-lines as null-geodesics (see Beem et al., 1996) of the metric tensor Eλ (x0 ) =

 1  t1 Ct0 (x0 ) − λ2 I , 2

(5.56)

which is a generalization of the Green–Lagrange strain tensor defined in formula (2.105). This metric tensor is always indefinite on the domain of definition Uλ of the ODE (5.54). This fact enables an interesting mathematical analogy between elliptic LCSs (as closed null-geodesics of a Lorentzian metric on a 2D spacetime) and photon spheres surrounding black holes in the relativistic, four-dimensional spacetime (see Claudel et al., 2001). For these reasons, we will refer to coherent Lagrangian vortices identified from formula (5.54) as black-hole vortices, and to the procedure used to extract them as geodesic vortex detection. For details and implications of this analogy between coherent vortices and black holes, we refer to Haller and Beron-Vera (2013, 2014). An alternative, null-geodesic-based computation of black-hole vortex boundaries from their implicit equation (5.52) will be described in §5.4.2. As we will show in §8.1.1, an important theoretical connection can be established between elliptic LCSs obtained from formula (5.54) and closed material barriers to the transport of diffusive tracers under homogeneous and isotropic diffusion. Namely, using the eigenvalues and eigenvectors of the time-averaged Cauchy–Green strain tensor in formula (5.54) instead of those of Ctt10 gives elliptic barriers to diffusive transport with constant diffusive transport density λ2 along them. To extract elliptic LCSs as closed orbits (or tensorlines) of the direction field families (5.54), one can follow the same algorithm that we outlined for the detection of closed shearlines in §5.3.2. The only difference in the present case is that the analysis has to be carried out for a range of λ values. The range to explore is an interval around λ = 1. Indeed, the conservation of the area enclosed by closed λ-lines (for incompressible flows) prevents substantial uniform stretching or shrinking for these curves. This is already evident from formula (5.54), which only returns a real-valued vector for λ values satisfying the bounds min λ1 (x0 ) ≤ λ ≤ max λ2 (x0 )

x0 ∈U

x0 ∈U

on a compact flow domain U ⊂ R2 (see Karrasch et al., 2014, for details). An example of such a geodesic vortex extraction is shown in Fig. 5.41 for the same AVISO ocean surface velocity data set that we analyzed in Fig. 5.36 using shearlines. In comparison to the analysis in Fig. 5.36, the present analysis finds more and larger eddies in the same region, as it allows for deviations from the λ = 1 uniform stretching factor for the elliptic LCSs. All eight detected eddies remain perfectly coherent over their extraction time of 180 days. The boundary of eddy 6 has the largest uniform stretching factor (λ = 1.2) while the boundary of eddy 3 displays slight contraction (λ = 0.9). Some of the eddies remain coherent over periods substantially exceeding their extraction interval. To illustrate how unique it is for initially coherent water masses to remain coherent for months in this ocean region, we show in the bottom plot of Fig. 5.36 the rapid disintegration of a water mass identified using the nonlinear eddy criterion of Chelton et al. (2011a,b) from §3.7.1. Most of such nonlinear eddies show the same lack of coherence in this region, leading to an about ten-fold overestimation of actual coherent transport by black-hole eddies (see Haller and Beron-Vera, 2013, for details).

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Lagrangian and Objective Eulerian Coherent Structures Coherent Lagrangian vortices: the black holes of turbulence (a) Transport of water by black-hole eddies

18.5° S 8 6 25° W BH eddy 8 BH eddy 7 BH eddy 6 BH eddy 5 BH eddy 4 BH eddy 3 BH eddy 2

7

3

10° E

5 2 South Atlantic

35° S Transported water after 0, 45, 90, 135, 180 and 225 days

(b) Transport of water by a sea-surface-height eddy

25° W

10° E

35° S

SSH eddy

South Atlantic

Transported water after 0, 45, 90, 135, 180 and 225 days

F IGURE 4. (a) The evolution of black-hole eddies (extracted from 3 months of data) in the

Figure (Top) of black-hole (BH)move eddies in Agulhas South5.41 Atlantic overEvolution a period of 225 days. The eddies fromdetected east to north-west (fromleakage right to left). material evolution of a nonlinear SSH over the same 225 days. velocity data region of (b) theThe South Atlantic from 180 days ofeddy satellite-based surface (AVISO). (Bottom) Rapid disintegration of the water mass identified from the nonlinearContinually eddy criterion of Chelton et al.data (2011b) (see §3.7.1) over theto same 225 day available observational for the ocean are limited two spatial dimensions. the existence of a coherent period. AdaptedHowever, from Haller and Beron-Vera (2013).material belt outside down- or upwellings should already be well-captured by two-dimensional altimetry data. We therefore also expect black-hole-type vortices to be present in two-dimensional atmospheric wind data, framing the Lagrangian footprints of hurricanes on Earth and of the Great Red Spot on Jupiter.

As a second application of geodesic vortex detection, Fig. 5.42 shows the construction of transport Acknowledgements barriers in Jupiter’s Great Red Spot (GRS), the largest and longest-existing known atmospheric vortex. The altimetry data used here are available from AVISO (http://aviso.oceanobs.com). G.H. acknowledges partial by NSERC This computation is based on support an unsteady windgrant field401839-11. extractedFJBV from acknowledges a video captured by support by NSF grant CMG0825547, NASA grant NNX10AE99G and by a grant from NASA’s Cassini space mission (see Hadjighasem and Haller, 2016a, for details). As seen the British Petroleum/The Gulf of Mexico Research Initiative. in Fig. 5.42(a)–(b), the material footprint of the GRS in the available 2D data falls in the category of a black-hole vortex. This Lagrangian vortex has a perfectly coherent 731 R4-9core with λ = 1 and a boundary with λ = 1.0063. Figures 5.42(c)–(d) show how these LCSs indeed preserve their coherence by the end of the extraction interval (24 Jovian days), acting as material transport barriers inside and around the GRS. In subplot (a) of the figure, elliptic LCSs forming the GRS at the initial (t0 = 0) frame of the Cassini video are shown, extracted from the wind field up to t1 = 24 Jovian days. The colors represent values of the stretching parameter λ on the LCSs. Subplot (b) shows an elliptic LCS with perfect coherence (λ = 1; black), as well as the outermost elliptic LCS

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GEODESIC TRANSPORT BARRIERS IN JUPITER’S ATMOSPHERE

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Zonal velocity profile of Jupiter’s atmosphere. The red line represents the average obtained 1 from the reconstructed velocity field; the green profile .'-/+&'()"0!1,2-,/)*3&"#)4*3,21#())))) is the one reported by Limaye [36]. The Cassini map PIA07782 is used as0.99 background. −11.5

λ

−11.5

Fig. 10

λ

Planetographic Latitude

−5.6

Percent of arc-length change

λ=1 λ=1.0063

Jupiter’s cloud cover into the camera of0.97a spacecraft tends to vary considerably be−17.4 tween two successive images. 0.96 As a consequence, optical flow methods have not gained much popularity in ve0.95 −23.3 locity field reconstruction from planetary observations. A rare exception is [37], which !"#$"%&'()%*+"#",&)-,,"#)%*#"))))) 0.94 works with a single pair of well-lit, high-resolution images from NASA’s Galileo mis−29.2 sion,−58.1 separated −46.1 by one hour. From this data, [37] extracts a single high-quality, steady −34.1 −58.1 −46.1 −34.1 Longitude velocity snapshot. Unfortunately, the data available from the Galileo Longitude mission is insuf(a) ficient for the extraction of a reasonably long unsteady velocity field(b) via the approach of [37]. In comparison to the two high-resolution images used in [37], the more modestly t=24 days t=24 days resolved but temporally extended Cassini data set−5.6used here offers a clear advantage λ=1 1 for unsteady LCS detection. As is now well established for satellite altimetry maps λ=1.0063 0.99 of the ocean, even without capturing smaller (submesocale and lower) features of a −11.5 velocity field, one can accurately capture0.98 its mesoscale LCSs, which in turn agree with in situ float observations [42]. 0.97 −17.4

0.96 4.4. Validation of the Reconstructed Velocity Field. Available observational 0.95 −23.3 indicating that Jupiter’s atmorecords of Jupiter go back to the late 19th century, sphere is highly stable in the latitudinal0.94 direction. The average zonal velocity profile as a function of latitudinal degree is, therefore, an−29.2 important benchmark in assessing −58.1 −46.1 −34.1 −58.1 −46.1 −34.1 Longitude Longitude the quality of the reconstructed velocity field. In Figure 10,(c) we compare the temporally averaged zonal velocity (d)profile obtained from ACCIV with the classic profile reported by Limaye [36]. The Limaye profile has been used and confirmed by several other studies on different data sets from different 2.8 missions (see, e.g., [29, 55, 19]). These studies all support the conclusion that the 2.6 averaged zonal wind field constructed by Limaye [36] has remained fundamentally 2.4 unaltered. Limaye’s velocity profile is based on Voyager I and Voyager II images, covering a 2.2 total of 142 Jovian days in 1979. In contrast, our 2 ACCIV-based velocity profile is based

Figure 5.42 (Top) Image of Jupiter’s atmospheric clouds, with the historically 2 0 averaged zonal velocity profile of Limaye (1986) (green) compared to the av-2 eraged zonal velocities extracted from the Cassini mission video footage. (a)– -4 (d)-6 Evolving Elliptic LCSs forming the GRS (see the text for details). A video 0 2 4 8 10 12 14 16 18 20 22 24 showing the6 material evolution of the elliptic LCS bounding the GRS is available Jovian day (e) (f) at https://epubs.siam.org/doi/suppl/10.1137/140983665/suppl_file/ grs_movie.mov. Adapted from Hadjighasem and Haller (2016a). Aspect Ratio

Downloaded 02/11/16 to 129.132.172.206. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php

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© bydefining SIAM. Unauthorized this article is prohibited. (a) Copyright Elliptic LCSs the Lagrangianreproduction footprint ofofthe GRS at time t = 0. The colorbar refers to values of the stretching parameter λ arising in the construction of the elliptic LCS family. (b) Elliptic LCS with perfect coherence (black), as well as the Lagrangian vortex boundary (blue) of the GRS at time t = 0. The boundary is extracted 0 from velocity data covering 24 Jovian days. The advection sequence is illustrated in http://epubs.siam.org/doi/suppl/10.1137/140983665/suppl file/98366Sup2.mov. 1 (c) Elliptic LCSs defining the Lagrangian footprint of the GRS at time t = 24. (d) Elliptic LCS with perfect coherence (black), as well as the Lagrangian vortex boundary (blue) of the GRS at time t = 24. (e) Relative stretching of the perfectly coherent (λ = 1) inner-core boundary and the slightly expanding outer boundary of the GRS over 24 Jovian days. As predicted by geodesic theory, the arclength of the outer boundary changes about 0.6% in agreement with the theoretical stretching value (λ = 1.0063) of the extracted outer boundary. (f) Plot of aspect ratio (length2 /area) as a function of time.

(blue) forming the black-hole vortex boundary of the GRS at time t = 0. Subplots (c)–(d) show materially advected positions of the LCSs shown in (a)–(b) at time t = 24. Beron-Vera et al. (2018) show observational evidence for the significance of black-hole eddy boundaries as approximate barriers to the transport of diffusive scalars in the ocean (see Fig. 5.43). This observation is consistent with the theory of material barriers to diffusive transport to be discussed in §8.1.1. Indeed, that theory will yield that outermost limit cycles of Advection Corrected Correlation Image Velocimetry (ACCIV) algorithm of Asay– a direction resembling Eq.for (5.54) serve as unsteady the observed vortex barriers for Davis field et al. closely [3]. Next, we identified, the first time, material transport diffusive substances. Andrade-Canto et al. a comparison of black-hole barriers in the wind field that form the(2020) cores ofprovides zonal jets and the boundary of the eddy Great and Red outermost Spot (GRS)diffusion in Jupiter’s atmosphere. boundaries barriers specifically in the context of Kraken shown in The parabolic LCSs (Lagrangian jet cores) we have found show that both easterly Fig. 5.43. Anotherjet application of geodesic vortextransport detection is given by latter Huhnfinding et al. (2015) and westerly cores provide strong material barriers. This confirms the conclusion of Beron-Vera et al. [5] based on a numerical study of a perturbed PV-staircase model relevant for Jupiter. Deforming material blobs placed

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Lagrangian and Objective Eulerian Coherent Structures

Figure 3. Satellite-derived chlorophyll concentration images on selected days with the altimetry-inferred Figure Materially evolving eddy boundaries (solid red: for 1 − long- (solid) and5.43 short-term (dashed) coherent black-hole Lagrangian boundaries of LCR Kraken overlaid. The ttop-right t0 = 90 dashed t0 = area 30 days) inbythe Mexico around an panel shows, as adays; function of timered: sincefor 29 tMay enclosed theGulf long-of (solid red) and short-term 1 − 2013, (dashedexceptionally red) Lagrangian boundaries, the outermost instantaneous SSHwith contour (blue), and the area long-lived Loop Current ring (named closed Kraken), thelevel remote-sensed spanned by the chlorophyll deficient patch (green). Figures constructed MATLAB R2017b (http://www. chlorophyll concentration superimposed. Green marks ausing chlorophyll-deficient patch. mathworks.com/). Adapted from Beron-Vera et al. (2018). and 131 d) within the long-term boundary. The other 6 drifters remained around the ring with a mean minimum

who useshorter this approach distance than 50 km.to identify upstream fluid that will be captured in the material vortices generated by swimming animals their Inertial effects (i.e., of objects’ finiteinsize andwake. buoyancy) might explain the behavior of the relatively slow

outward moving drifters. Indeed, floating objects are predicted to be repelled away from anticyclonic coherent Lagrangian eddies10,15. These effects can be expected to be most effective when the para-drogues attached to the drifters are closed. The outward motion of the drifters may be indicative of this. Windage may also explain deviations of drifter motion from purely Lagrangian motion.as However, effects near mesoscale eddies or 5.4.2 Computing Elliptic LCSs Closedwindage Null-Geodesics rings have not been well constrained for biological tracers51–53, and they may be outbalanced by inertial effects54. additional to be highlighted of the behavior(5.52) of the drifters is the relativeLCSs absencecan of subAs An noted in theimportant previousaspect section, the implicit equation defining elliptic be mesoscaleas wiggles in their hourly-sampled trajectories. The lack of such wiggles drifter trajectories is largely viewed an equation for closed null-geodesics of the tensor Eλin(xthe 0 ) defined in Eq. (5.56). consistent with the convexity of the altimetry-inferred long- and short-term coherent Lagrangian ring boundaThis equation is invariant under re-parametrizations solutions x0 inference. (s). Therefore, we can ries. This observation gives further support to the significance ofof theits altimetry-based It might be argued that the dominance of wiggle-free drifter trajectories is a consequence the drifters choose, without any loss of generality, the tangent vector of x0 (s) to be ofofunit lengthbeing for para-drogued at about 50-m depth, which lies somewhat below the summertime mixed-layer base55. However, 2,38 this should not prevent the drifters from sampling submesoscale motions if they were active . Indeed, loops of about 10-km or smaller diameter can be seen at times mounted on the drifter trajectories. But the frequency of those loops is typically very close to the local Coriolis frequency, as it has been observed earlier in the same region using the same drifters37. So the loops mainly correspond to near-inertial oscillations, which do not impact relative dispersion statistics24 and thus should not be expected to constitute an effective mechanism for the erosion of mesoscale coherent Lagrangian structures such as the resilient material boundary around LCR Kraken.

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5.4 Global Variational Theory of LCSs

each value of the parameter s. Specifically, if e(φ) ⊂ R2 denotes the unit vector enclosing the angle φ ∈ [0, 2π) with the positive horizontal axis of the x0 plane, then we can set   cos φ(s) 0 x0 (s) = e(φ(s)) = (5.57) sin φ(s) for an appropriate angle φ(s) at each point of the null-geodesic curve x0 (s), as shown in the left subplot of Fig. 5.44. x02

x02

gtt01 (x0 0) =

2 (i,j)

null-geodesic of Eλ trajectories returning to the level set of gtt01 (x0 φ)

x0 (s) = e(φ(s)) x0 (s)

φ(s)

x01

(i,j)

(x0

(s) φ(i,j) (s))

x01

Figure 5.44 Computation of an elliptic LCS as a closed null-geodesic. (Left) Parametrization of a null-geodesic of the generalized Green–Lagrange tensor field Eλ (x0 ). (Right) Trajectories of the ODE (5.60) near a closed null-geodesic, released 2 from an initial grid, will re-intersect the level set gtt01 (x0, 0) = λ(i, = const. again j) near the initial condition.

We follow Serra and Haller (2017a) by substituting expression (5.57) into Eq. (5.52). By differentiating the resulting equality with respect to s and using the identity   d − sin φ e(φ) = = −Je(φ) (5.58) cos φ dφ with J defined in Eq. (5.47), we obtain




− 2φ 0 Je(φ), Ctt10 e(φ) + e(φ), ∇Ctt10 e(φ) e(φ) = 0.

(5.59)

Combining Eqs. (5.57) and (5.59) leads to the 3D system of ODEs x00 = e(φ),


t1 e(φ), ∇ C (x )e(φ) e(φ) 0 t 0

 . φ0 = − 2 e(φ), JCtt10 (x0 )e(φ)

(5.60)

This ODE blows up at tensorline singularities of the right Cauchy–Green strain tensor Ctt10 (see §§5.3.2 and 5.3.1). Away from such singularities, however, Eq. (5.60) is a globally oriented vector field (unlike Eq. (5.54)) and hence can be solved directly by any ODE solver. In addition, the ODE (5.60) has no dependence on the parameter family λ. Rather, the

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dependence of any solution of Eq. (5.60) on λ is implicit, imposed by the choice of the initial conditions (x0 (0), φ(0)), which satisfy

 gtt01 (x0, φ) := e(φ), Ctt10 (x0 )e(φ) = λ2 (5.61) by Eq. (5.61). The conservation law (5.61) along trajectories of Eq. (5.60) enables an automated search for closed null-geodesics without setting up Poincaré sections near tensorline singularities, as would be required for the direction field families (5.54). Specifically, for a trajectory launched from a point (x0 (0), φ(0)) of a level curve of gtt01 (x0, φ(0)) to be closed, the trajectory must re-intersect the same level curve at the same point, i.e., at same angle φ(0). As the left plot of Fig. 5.44 shows, all values of φ ∈ [0, 2π) will be taken by points alongs such closed trajectories and hence any of these angles can be selected as initial condition. For simplicity, therefore, we can select φ(0) = 0 for all trajectories of Eq. (5.60) launched j) from an initial grid G0 of the x0 -plane. Each such initial condition x(i, 0 (0) ∈ G0 on this grid then determines a constant   (i, j) t1 2 λ(i, = g x , 0 t j) 0 0     j) (i, j) (i, j) in the conservation law (5.61) for the trajectory x(i, (s), φ (s) starting from x , 0 . 0 0 2 We monitor each trajectory to see if re-intersects the level set gtt01 (x0, 0) = λ(i, j) in a vicinity of   (i, j) x0 , 0 . If yes, then we use the bisection method on the initial conditions to find a nearby trajectory that returns exactly to each initial condition, as sketched in the right plot of Fig. 5.44. Any such closed trajectory, by construction, defines the time t0 position of an elliptic LCS. Outermost members of such orbit families are the boundaries of the black-hole vortices we introduced in §5.4.1. We refer to Serra and Haller (2017a) for more detail on the algorithm and for a comparison with the direction-field-based computation of black-hole vortices described in §5.4.1. An open-source Matlab script BarrierTool for the algorithm null-geodesic-based algorithm discussed in this section is available from Katsanoulis (2020). In addition, the computation is implemented in Notebook 5.7.

Notebook 5.7 (EllipticLCS) Computes elliptic LCSs as closed null-geodesics of the generalized Green–Lagrange strain tensor in a 2D unsteady velocity data set. https://github.com/haller-group/TBarrier/tree/main/TBarrier/2D/ demos/AdvectiveBarriers/EllipticLCS Serra et al. (2017) apply the null-geodesic algorithm for elliptic LCSs to find an objective material barrier to the Arctic polar vortex. Using isentropic surface velocity data from the ECMWF global reanalysis data set (Dee et al., 2011), they piece together this 2D barrier surface from its one-dimensional intersections with various isentropic surfaces that are approximately invariant under the Lagrangian particle dynamics. On each isentropic surface, Serra et al. (2017) employ the identification method sketched in Fig. 5.44 to locate the outermost elliptic LCS surrounding the polar vortex region. The focus of their study is the time period ranging from late December 2013 to early January 2014, in which exceptional cold

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5.4 Global Variational Theory of LCSs

weather was recorded in the Northeastern United States. The null-geodesic-based analysis explains this cold wave by an incursion of the material barrier into the affected regions. Figure 5.45 shows a snapshot of the material boundary of the polar vortex obtained in this fashion. The potential vorticity on the 475-K isentropic surface is plotted over the Earth’s surface in this figure for reference. 3876

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28 December, 2013

VOLUME 74

7 January, 2014

FIG. 4. 5.45 Reconstructed 3D visualization of geodesic polar vortex edges all isentropic surfaces Figure The material boundary of the Arctic polarobtained vortexonduring an extreme under consideration on (a) 28 Dec 2013 and (b) 7 Jan 2014. (Panels not to scale along the radial direction.) PV on cold wave in the Northeastern US, extracted as a set of outermost elliptic LCSs on the 475-K isentropic surface is plotted over Earth’s surface. The full evolution on the vortex boundary over the isentropic surfaces. Adapted from Serra et al. (2017). 10-day time window is available (video 1: https://goo.gl/7CQ9cu).

4. Results

optimality of the geodesic vortex boundary obtained as the outermost elliptic LCS. The optimal boundary of a a. Elliptic LCSs identify the polar vortex edge coherent vortex can be defined as a closed material Shearless LCSsover in 2D: Parabolic andencloses Hyperbolic Barriers surface that the largest possible area around the We perform 5.4.3 a backward time analysis a time interval [t0, t1] of 10 days, with t0 5 7 January 2014 and vortex and undergoes no filamentation over the obseradopt 2013. the global variational in the forweelliptic vational timeprevious period. Tosection this end, considerLCSs, a small 5 28now December From the positions ofapproach the out- used t1 We normal to the geodesic vortex we boundary. ermost elliptic LCSs (i.e., geodesic vortex(i.e., boundaries) but modify it to capture shearless hyperbolic and perturbation parabolic) LCSs. Specifically, will The amount of perturbation is 58, that is, 10% of the onbethe five isentropic surfaces, we construct a threeseeking initial positions of shearless LCSs as material curves along which the averaged dimensional (3D) tvisualization of the geodesic polar equivalent diameter of the outermost elliptic LCSs. We 1 shearedge factor σt0 shown in and Fig.lower 5.28stratosphere is minimal. then advect the geodesic vortex boundary and its pervortex spanning the middle For4).2D flows, the tangentenables spaceusTxtoM(t) only oneover dimensional, which the time window underenables study. us to (cf. Fig. Such a 3D visualization make is turbation an illustration, 5a shows theshear initial factor positionsisof conclusions the overall material compute about a signed version of thedeformation tangentialofshearAsfactor (5.38). Fig. This signed the main vortex over the time interval we analyze. the geodesic vortex t1 edge and its perturbation on the obtained by simply projecting the advected unit normal ∇F (x0 ) n0Figures onto the unit tangent 5b–f, instead, show Specifically, Fig. 4a shows the vortext edge on 28 De- 600-K isentropic t0surface. . The resulting signed tangential factor forand 2Dits vector2013, in the direction of ∇undeformed Ft10 (x0 ) x00 circular the advected images of the shear geodesic vortex cember which has an almost shape, flowswhile is Fig. 4b shows the deformed vortex edge on perturbations on different isentropic surfaces. The geodesic vortex edge 7 January 2014. * + remains coherent under advection t1 0 allF cases, in sharp contrast to the perturbations that Note that the vortex edge deforms toward the north- in ∇ (x ) x 0 t0 0 t1 t1 . filamentation over the 10-day ˆ consistent , substantial eastern coast of the United States,σ with 0 ) n0undergo t0 = ∇F t0 (xthe t ∇window. Ft10 (x0 ) x00 exceptional cold recorded there over the early January time Figure 6 shows a combined visualization of the initial 2014. A video showing the evolution of the 3D geodesic vortex edge over the 10-day time window is available positions of the geodesic vortex (blue) together We again use formula (5.47) to eliminate thetheir dependence of the shear factor σtt01 onpositions n0 with in perturbations (Fig. 6a) and their evolved (video 1: https://goo.gl/7CQ9cu). formula (5.38). We then obtain (Fig. 6b) for all isentropic surfaces. A video showing the OPTIMAL COHERENT TRANSPORT BARRIER complete advection sequence over the 10-day time As noted in the introduction, the vortex edge is window is available (video 2: https://goo.gl/7CQ9cu). thought as the outermost closed material line dividing The main vortex–surf zone distinction (McIntyre and the main vortex from the surf zone. Here, we show the Palmer 1983) divides the polar vortex into a coherent

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t1 t1 0 σ ˆ t0 x0, x0 = ∇Ft0 (x0 ) 0 , Jx ∇Ftt1 (x0 ) x0 0 0 0

0  x0, JCtt10 (x0 ) x00 = − q

. 

x00 , x00 x00 , Ctt10 (x0 ) x00

(5.62)

Following the approach of §5.4.1, we seek shearless LCSs as exceptional γ curves around which an O()-thick material belt will show O( 2 ) variability in material shearing under advection. This is an order of magnitude less than the expected O() variability, implying exceptional material coherence with respect to tangential shearing along γ. In analogy with formula (5.50), we then obtain that shearless LCS must be solutions of the variational principle ˆ
1 s¯ t1 δP(γ) = 0, P(γ) := (5.63) σ ˆ t0 x0 (s), x00 (s) ds s¯ 0 for the averaged shear functional P(γ). In order to solve the variational problem (5.63) using Euler–Lagrange equations, one has to pose appropriate boundary conditions. Farazmand et al. (2014) point out that the variational problem (5.63) has the same symmetry as the one we noted for the variational problem (5.50) for elliptic LCSs (i.e., no explicit dependence on s in the integrand of P(γ)). This symmetry again enables us to invoke Noether’s theorem (see Appendix A.8) for the ˆ tt01 (x0, · ) is a positively homogenous existence of a conserved quantity I(x0, x00 ). Again, σ function of order d =
0 and hence the relevant Noether formula (A.34) gives the first integral ˆ tt01 x0, x00 . This in turn implies that I(x0, x00 ) = σ
(5.64) σ ˆ tt01 x0 (s), x00 (s) = µ = const. along curves x0 (s) solving the variational problem (5.63). Therefore, shearless LCSs satisfying the variational principle (5.50) are special curves that generate the same pointwise Lagrangian shear µ ≥ 0 at each of their points. We pointed out in §5.3.3 that σtt01 reaches its global minimum, σtt01 = 0, whenever the unit normal n0 of the initial material curve γ is aligned with either of the two eigenvectors ξi (x0 ; t0, t1 ) of the Cauchy–Green strain tensor. This
observation prompts us to focus now on perfect shearless LCSs characterized by σ ˆ tt01 x0, x00 ≡ µ = 0. For such LCSs, we therefore have the following result. Theorem 5.13 Perfect shearless LCSs, defined as stationary curves of the average tangential shear functional P(γ) with identically zero shearing rate (µ = 0), are continuous trajectories of the direction field families x00 = ξi (x0 ; t0, t1 ),

i = 1, 2,

(5.65)

with ξi (x0 ; t0, t1 ) denoting the unit eigenvectors of the right Cauchy–Green strain tensor Ctt10 (x0 ). Perfect shearless transport barriers in 2D are, therefore, sets of shrinklines and stretchlines, which agrees with the conclusions of the local variational theory in §5.3.1. Of all the

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trajectories of these direction fields, hyperbolic LCSs will be special ones that maximize local repulsion or attraction. In contrast, parabolic LCSs will be special in that they are the closest to being neutrally stable compared to neighboring sets of shrinklines and stretchlines. We now examine the three possible boundary conditions for the variational problem (5.63): free-endpoint, fixed-endpoint and periodic boundary conditions (see Appendix A.8).

Free-Endpoint Boundary Conditions: Parabolic LCSs
The free-endpoint boundary condition (A.29) takes the particular form ∂x00 σtt01 x0, x00 = 0 for the averaged shear functional P(γ). Farazmand et al. (2014) show that this relationship can only hold at tensorline singularities, i.e., at points satisfying λ1 (x0 ) = λ2 (x0 ) = 1.

(5.66)

Trajectories (or, more generally, heteroclinic chains of trajectories) of the ODE family (5.65) connecting such tensorline singularities are rare but have a profound impact relative to other tensorlines. Indeed, they prevail as stationary curves of the averaged shear functional P(γ) under the broadest possible class of continuous perturbations, including perturbations to their endpoints. To ensure the observability of such heteroclinic chains as parabolic LCSs, Farazmand et al. (2014) further restrict this family of curves to those that are: (P1) robust, i.e., persistent under small changes to the velocity field. (P2) locally the closest to being neutrally stable and hence behaving as the idealized jet core does in Fig. 5.39. To ensure the robustness property (P1), we seek structurally stable chains of alternating stretchlines and shrinklines, i.e., chains that perturb to nearby chains of heteroclinic connections among tensorline singularities under small perturbations of the velocity field. As Farazmand et al. (2014) show, only heteroclinic tensorlines connecting wedge-type singularities to trisector-type singularities can form a structurally stable heteroclinic chain, as illustrated in Fig. 5.46(a). As a preliminary step to ensure the neutrally stable behavior prescribed in (P2), we exclude the possibility of a strictly repelling or attracting LCSs by selecting an alternating chain of shrink and stretchlines, as illustrated in Fig. 5.46(b). This subplot again highlights that each element of the alternating chain must be a trisector-wedge connection. The left plot of Fig. 5.46(c) shows the total set of tensorlines satisfying these requirements for the standard non-twist map. Proposed by del Castillo-Negrete et al. (1996), this can be written as
2 xn+1 = xn + a 1 − yn+1 , (5.67) yn+1 = yn − b sin (2πxn ) , with xk ∈ [−1/2, 1/2] mod 1 and with the parameter values a = 0.08 and b = 0.125.12 12

The strain eigenvector fields ξi (x0 ; t0 , t1 ), with t0 = 0 and t1 = n + 1, of the autonomous discrete mapping T (5.67) can be computed from the Cauchy–Green strain tensor C0n+1 = ∇F0n+1 ∇F0n+1 associated with the
n+1 discrete flow map F n+1 = F10 , where F10 is defined by formula (5.67).

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8

Lagrangian and Objective Eulerian Coherent Structures M. Farazmand et al. / Physica D 278–279 (2014) 44–57

Trisector

Wedge

ig. 3. Schematic representation of the properties of a repelling LCS (red) among earby strainlines (black) and Cauchy–Green singularities (dots). The repelling LCS tays away from singularities of Cauchy–Green singularities. While the length of ny strainline shrinks as advected under the flow map Ftt0 , the length of a repelling

CS shrinks more than any C 1 -close strainline.

50

(a)

M. Farazmand et al. / Physica D 278–279 (2014) 44–57

(b)

Fig. 5. Top: Smooth connection of strainlines (red curve) and stretchlines (blue curve) only occurs at Cauchy–Green singularities. Bottom: An alternating chain of strainlines (red) and stretchlines (blue) connecting trisectors (green) and wedges (black). A schematic phase portrait of strainlines (thin black lines) is shown around one of the trisector singularities. The strainline marked by red color is the unique connection between that trisector and the wedge on its left. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

ig. 4. Topology of tensorlines (black) around a trisector (left) and a wedge (right) ingularity (magenta). The tensorlines shown in red form the separatrices. (For nterpretation of the references to color in this figure legend, the reader is referred o the web version of this article.)

the combined forward and backward FTLE fields was considered in [18]. As advised in [18], the FTLE field should be used with caution for detecting parabolic barriers. While the trench of the FTLE field can indeed be an indicator of a jet core, the following example of a steady two-dimensional incompressible flow shows that this is not necessarily the case. Consider the incompressible flow

variational principle, i.e., their endpoints are Cauchy–Green ingularities. x˙ = x (c) 1 + 3y2 , In addition, in order to provide a generalization of jet cores, we (22) Fig. 8. Left: Heteroclinic tensorlines between the trisector and wedge singularities of the Cauchy–Green strain tensor in the integrable SNTM: strainlines (red) and re interested in non-hyperbolic shearless barriers that have no singularities, y˙ = (respectively. y + y3 ). Right: The extracted parabolic barrier consists of the single alternating stretchlines (blue). The black and green dots mark the wedge and trisector distinct (repelling or attracting) stability type along their interior sequence of tensorlines that satisfy conditions P1–P2 of Definition 1. (For interpretation of the references to color in this figure legend, the reader is referred to the web Figure 5.46 (a) A structurally stable tensorline singularities must The line y between = 0 is antensorline invariant, attracting set, yet numerical points. To this end, we require version of this article.)parabolic barriers to be also weak show that itbe is also a trench forward-time and minimizers of their neutrality sense of Section connection. 3. be ina the trisector-wedge (b)simulations The chain should formed by of anthe alternating backward-time FTLE fields. Fig. 6 shows the forward-time FTLE Finally, for reasons of physical relevance and observability, sequence of shrinklines and our stretchlines. (c) (Left) Structurally stable heteroclinic field and the tracer evolution around the line y = 0. The FTLE definition of a parabolic barrier will further restrict our considertensorlines the integrable non-twist map. (Right) TheLCS, single heteroclinic trench is a hyperbolic (attracting) as opposed to a parabolic tion to strainline–stretchline chains thatin are unique betweenstandard the barrier as a jet core. wo singularities they connect, andof arealternating also structurally stable with (blue) chain stretchlines andacting shrinklines (red) forming the initial posiespect to small perturbations. on our review of tensorline tionBased of a parabolic LCS (jet core) for the same map. Adapted from Farazmand et al. ingularities in Appendix C, strainlines connecting singularities are 8. Automated numerical detection of parabolic barriers (2014). only structurally stable and unique if they connect a trisector singularity to a wedge singularity (see Fig. 4). An identical requirement Definition 1 provides the basis for the identification of parabolic holds for stretchlines. We then have the following definition. barriers in finite-time flow data. Using the numerical details surveyed in Appendix C, we implement conditions P1 and P2 of The second step to ensure that the neutrality condition (P2) holds is to require the neutrality Definition 1 as follows: measure 1. Compute the Cauchy–Green strain tensor C on a twoq dimensional grid 2 in the (x1 , x2 ) variables. 2. Detect the singularities of C by finding the common zeros of λ (x ) − 1 , i,j (5.68) N (x ) = ξi 0 f j= C011 C22 and g = C12 . is an alternating chain of strainlines and stretchlines,

Definition 1 (Parabolic Barriers). Let denote the time t0 position of a compact material line. Then this material line is a parabolic ransport barrier over the time interval [t0 , t ] if the following two onditions are satisfied:

P1

3. For any trisector singularity of the ⇠1 vector field, follow which is a unique connection between a wedge- and a strainlines emanating from the singularity and identify among trisector-type singularity of the tensor field C (x0 ) (see (left), 200 and 300 (right) iterations of the the integrable SNTM to withwedges. parameters them the(middle) separatrices connecting trisectors Fig. 5).Fig. 9. The red curve shows the computed finite-time shearless barrier from 100 a = 0 . 08 and b = 0 . 125. The black curve marks the exact location of the barrier. Repeat the same procedure for the ⇠2 vector field to find is a weak P2 Each strainline and stretchline segment in minimizer of its associated neutrality. trisector–wedge separatrices among stretchlines. 4. Out of the computed separatrices, keep the strainline separatriξi 0 ξi ces satisfying Example 1 (An FTLE Trench is not Necessarily a Parabolic Bar↵ ⌦ 2 ξi 0 ier). Since our notion of a parabolic barrier requires a minimal@r N⇠1 (x0 )⇠2 (x0 ), ⇠2 (x0 ) > 0, ty condition on 2 , one may speculate whether a trench of the and the stretchline separatrices satisfying inite-Time Lyapunov Exponent (FTLE) field will always be a shear↵ ⌦ 2 ess barrier. Such an approach of detecting jet cores by trenches of @r N⇠2 (x0 )⇠1 (x0 ), ⇠1 (x0 ) > 0.

to admit a weak minimum along each shrinkline (i = 1) and each stretchline (i = 2) in the heteroclinic chain. Specifically, we say that a tensorline segment γ is a weak minimizer of its corresponding N (x ) if both γ and the nearest trench of N lie in the same connected component of the set over which the scalar function N (x ) is convex (see Farazmand et al., 2014 for more detail). Applying this weak minimizer condition to all possible alternating shrinkline–stretchline chains on the left of Fig. 5.46(c) selects the initial position of a single parabolic LCS shown on the right of Fig. 5.46(c). In Fig. 5.47, we show the application of the global variational theory of parabolic LCSs to the unsteady wind field data for Jupiter’s atmosphere, already analyzed in Fig. 5.42. The figure also confirms via material advection that the parabolic LCS identified in this

Fig. 10. Parabolic barrier and its impact on tracers in the integrable SNTM. (For interpretation of the references to color in this figure legend, the reader is referred to the https://doi.org/10.1017/9781009225199.006 Published online by Cambridge University Press web version of this article.)

5.4 Global Variational Theory of LCSs

Table 1: ACCIV parameters used to produce the time-resolved velocity field. The box size, search range, and stride are in units of pixels. The box size is the size of the correlation box for the relevant CIV pass. The search range is the range of correlation box displacements used in each dimension. The stride is the number of pixels by which the correlation box is shifted between each measurement. It controls the output resolution of the velocity vector field. The number of image pairs is the total number of pairing of the set of images. The Number of Smooth Fit Control Point Neighbors controls the smoothness of the velocity field on the grid.

207

fashion remains coherent, blocking advective excursion of air between the upper and lower halves of the material jet. This perfect shearless barrier generates the typical chevron-shaped material patterns shown in the bottom plot of Fig. 5.47, providing an explanation for earlier observations !by Simon-Miller et al. (2012) such chevrons. ! of (" (" , ) , ) 20 11

Latitude

2 −8 −17 −26 −35 −100

−83

−67

−50

−33 −17 Longitude

0

17

33

50

Figure 9: A representative snapshot (frame number 105) of the unsteady velocity field using the ACCIV algorithm. Only one tenth of the total number of velocity vectors are shown for the sake of visual clarity.

over the domain ranging from 180 W to 180 E in longitudes and from 35 S to 20 N in latitudes, with a grid resolution of 7202 ⇥ 1102. This supersedes the resolution of earlier velocity fields reconstructed for Jupiter from manual cloud-tracking approaches [39, 12, 49, 33, 53]. In principle, an alternative to the ACCIV method used here would be Digital Particle Image Velocimetry (DPIV), which reproduces fluid velocities from highly resolved tracks of luminescent particles in wellilluminated laboratory flows [26]. Under such conditions, high-speed imaging can reliably detect small particle displacements amidst minimal illumination changes, a basic requirement for the optical flow methods underlying DPIV. In observations of planetary atmospheres, however, these ideal imaging conditions are generally not met. For instance, the sunlight scattered from Jupiter’s cloud cover into the camera of a spacecraft tends to vary considerably between two subsequent images. As a consequence, optical flow methods have not gained popularity in velocity field reconstruction from planetary observations. A rare exception is Ref. [35], which works with a single pair of well-lit, highresolution images from NASA’s Galileo mission, separated by one hour. From this data, [35] extracts a single high-quality, steady velocity snapshot. Unfortunately, the data available from the Galileo mission is insufficient for the extraction of a reasonably long unsteady velocity field via the approach of [35]. In comparison to the two high-resolution images used in [35], the more modestly resolved but temporally extended Cassini data set used here o↵ers a clear advantage for unsteady LCS detection. As is now wellFigure 12: Impact of Jupiter’s southern shearless on smaller tracer disks over 11 jovian established for satellite altimetry mapsequatorial of the ocean, eventransport without barrier capturing (sub-mesocale and days. The deformed tracer disks resemble the can shape of the recently discovered chevrons [49], i.e., in dark clouds in lower) features of a velocity field, one accurately capture its mesoscale LCSs, which turnv-shaped agree with the background. The contrast of images is improved for better visualization. The complete advection sequence over 24 in situ float observations [40]. Jovian days is illustrated in the on-line supplemental movie M1.

Figure 5.47 A perfect shearless zonal transport barrier in Jupiter’s atmosphere, reconstructed as a parabolic LCS from video footage acquired by NASA’s Cassini 4.4 Validation of the reconstructed velocity field LCS (Top) theory described in section 3.2. This necessitates the computation ofwind limit cycles for the family of mission. An instantaneous snapshot of the unsteady field reconstructed Available observational records of Jupiter thediscard late 19th century, that autonomous dynamical systems defined in go eq.back (11).toWe limit cycles indicating obtained for theJupiter’s same atvalue, if from the video. (Middle) heteroclinic tensorlines satisfying exmosphere is highly inA the latitudinal Theof average profile as a(P1)–(P2), function of that they are within twostable velocity grid steps fromdirection. eachchain other. This is tozonal focusvelocity on robust enough limit cycles far enough from undergoing saddle-node bifurcation. latitudinal degree is, therefore, an aimportant benchmark in assessing the quality of the reconstructed velocity tractedare from the reconstructed wind field. (Bottom) Chevron-shaped deformation This computation yields 73 elliptic LCSs, computed as robust limit cycles of the di↵erential equation of (11) (Fig. 13a). set of closed positions curves forms shown a generalized KAM region, filled with sets offamily air pockets, withThis their initial in the middle plot. A material video loops show10 barriers through the entire duration of the underlying that resist filamentation and act as coherent transport video footage. evolution of these air pockets along the parabolic LCS is available ing the material Figure 13b shows separately the elliptic LCS with perfect coherence ( = 1) in black, as well as the at https://epubs.siam.org/doi/suppl/10.1137/140983665/suppl_file/ outermost elliptic LCS in blue that forms the outer boundary of the coherent Lagrangian vortex associated with the GRS. This vortexAdapted boundary isfrom markedHadjighasem by the parameter value = 1.0063,(2016a). which forecasts a roughly chevron_movie.mov. and Haller 0.6% increase in arclength. The two closed curves enclose a highly coherent annular barrier, the Lagrangian counterpart of the outer ring identified within a collar of the GRS described in [35]. This outer ring was constructed as the annulus outside the perceived core of the collar, a closed curve of velocity maxima. Our geometric construction of an annular Lagrangian transport barrier is also motivated by the visual observations in [41]. These often indicate a sharper inner boundary and a more di↵usive outer boundary for coherent jovian rings. The sharp observational boundary suggests an elliptic LCS of the highest possible coherence (zero stretching), while a di↵usive outer boundary is expected near an elliptic LCS of the lowest possible coherence (highest stretching in a family of elliptic LCSs). The exact location of these oval barriers is expected to change under varying data resolution. The structural stability in the construction of elliptic LCSs, however, guarantees that the barriers move only by a small amount under small enough variations in resolution. Advected 0images of the extracted GRS boundary confirm its sustained coherence over the finite time of extraction (see Figures 13c to 13e). Our computation shows that the predicted coherent core of the GRS indeed regains its arc-length after 24 Jovian days. At the same time, the coherent outer boundary of the GRS indeed grows in arclength by about 0.6%, while its longitudinal extent decreases by about 5%. This suggests that the coherent boundary of the GRS is becoming rounder, which is generally consistent with

Fixed-Endpoint Boundary Conditions: Hyperbolic LCSs Hyperbolic LCSs obtained from their 2D global variational theory are non-closed material curves whose time t initial position must stay away from the tensorline singularities (5.66), given that no repulsion or attraction can be exerted on neighboring trajectories at such points. We conclude, therefore, that initial positions of hyperbolic LCSs must be individual trajectories of the direction fields (5.65) that are bounded away from tensorline singularities 13 and their averaged stretching or repulsion is locally maximal among nearby tensorlines. This conclusion is identical to our findings from the local variational theory of LCSs in §5.3.1. The new information obtained from the global theory is that hyperbolic LCSs prevail as stationary curves over a smaller family of material lines and hence their impact on the flow is less pronounced than the impact of parabolic LCSs.

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Periodic boundary conditions: Closed shearless LCSs Periodic boundary conditions for a closed shearless LCS would be satisfied for chains of shrinkline or stretchlines (i.e., trajectories of the direction field (5.65) with i = 1 or i = 2) that form a closed loop. A one-element closed chain, i.e., a closed hyperbolic LCS, cannot arise in these equations for incompressible flows over long enough time intervals. The reason is that such closed loops would have to grow or decrease the length of any of their subsets under advection while still conserving their enclosed area. Although no closed trajectories for the direction field (5.65) with i = 1 or i = 2 have been documented in the literature, it is unclear how one could generally prove their nonexistence. One can only exclude the existence of closed shrinklines and stretchlines that experience substantial change in their lengths in incompressible flows. To see this, we recall that by the isoperimetric inequality in two dimensions, the length p of a closed curve γ and the area of its interior, int(γ), satisfy the inequality length(γ) ≥ 4π area(int(γ)) (see, e.g., Bandle, 1980). This inequality would necessarily be violated by advected positions of a closed shrinkline γ whose length shrinks exponentially yet its area remains constant by material advection. The isoperimetric inequality, however, still does not exclude mild increases in length and hence does not exclude the existence of closed stretchlines under forward advection. A similar argument shows the nonexistence of exponentially shrinking shrinklines when one applies the isoperimetric inequality to their backward advected positions in an incompressible flow. Again, however, mildly deforming shrinklines cannot be excluded by this argument. It is certainly possible to find a closed parabolic LCS formed by alternating stretchlines and shrinklines on a spatially periodic domain. Known examples of such periodic parabolic LCSs include the jet cores of the standard non-twist map (5.67) and the Bickley jet model of del Castillo-Negrete and Morrison (1993) under the addition of various unsteady effects, including chaotic forcing (see Farazmand et al., 2014 for details). Closed zonal jet cores of the type shown in Fig. 5.47 for Jupiter’s atmosphere represent another class of examples.

5.4.4 Unified Variational Theory of Elliptic and Hyperbolic LCSs in 3D Oettinger et al. (2016) develop an extension of the 2D global variational theory of elliptic LCSs, which we discussed in §5.4.1, to 3D flows. They start by examining the existence of material surfaces M(t) ⊂ R3 over a finite time interval [t0, t1 ] that are pointwise uniformly stretching. This means that allh vectors in each iinitial tangent space Tx0 M(t0 ) are stretched p p λ1 (x0 ), λ3 (x0 ) under the linearized flow map ∇Ftt0 (x0 ). by the same factor λ(x0 ) ∈ Here, λi denote, as before, the squared singular values of ∇Ftt0 , i.e., the eigenvalues of the right Cauchy–Green strain tensor Ctt0 (x0 ). Oettinger et al. (2016) show that pointwise uniformly stretching tangent spaces can only exist, in principle, with stretching factor λ(x0 ) = p λ2 (x0 ) (1 + δ) for some constant |δ|  1. For this stretching factor, uniformly stretching tangent spaces must be normal to some member of the two vector field families s n±δ (x0 ) =

λ2 (x0 ) (1 + δ) − λ1 (x0 ) ξ1 (x0 ) ± λ3 (x0 ) − λ1 (x0 )

s

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λ3 (x0 ) − λ2 (x0 ) (1 + δ) ξ3 (x0 ). λ3 (x0 ) − λ1 (x0 )

(5.69)

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5.4 Global Variational Theory of LCSs

As in the case of hyperbolic and elliptic LCSs constructed from their local 3D theory, a smooth surface satisfying (5.69) only exists if

 ∇ × n±δ , n±δ = 0,

 ∇ ∇ × n±δ , n±δ × n±δ = 0

(5.70)

by the results in Appendix A.7. As we concluded in similar settings of the local variational theory, conditions (5.70) will generally not hold along a smooth 2D manifold. Oettinger et al. (2016) circumvent this issue by requiring pointwise nearly-uniformly stretching elliptic LCSs to be only continuous, then proceed to assemble them from their computable intersections with reference planes. This is the same procedure that we discussed for hyperbolic and elliptic LCSs in §§5.3.1 and 5.3.2. An alternative approach to resolving the over-constraining assumptions of the local variational theory and the global variational theory of Oettinger et al. (2016) is based on a relaxed local extremum principle proposed by Oettinger and Haller (2016). This relaxed principle would not require the full tangent space Tx0 M0 of the LCS initial position M0 to be normal to the direction of most repulsion, attraction or shear. Instead, it would require that at least one direction in each tangent space should be normal to these directions. Definition 5.14 A relaxed LCS in a 3D flow over the time interval [t0, t1 ] is a structurally stable material surface M(t) with initial position M0 such at all points x0 ∈ M0 , the tangent space Tx0 M0 contains a vector normal to the direction of highest repulsion, attraction or shear. An inspection of Proposition 5.10, Proposition 5.11 and Eq. (5.69) reveals a single unit vector that is normal to all normal vector fields arising in these formulas: the intermediate Cauchy–Green eigenvector, ξ2 . We, therefore, have the following result. Theorem 5.15 All 2D structurally stable invariant manifolds of the 3D direction field x00 (s) = ξ2 (x0 (s); t0, t1 )

(5.71)

are relaxed LCSs. Remarkably, therefore, all relaxed attracting LCSs, repelling LCSs and elliptic LCSs can be sought as invariant manifolds of the same direction field (5.71), defined by the intermediate eigenvector field of the right Cauchy–Green strain tensor Ctt0 (x0 ). As we noted in §4.4.2, trajectories advected from any smooth curve of initial conditions in a steady 3D flow will form smooth invariant manifolds for finite advection times. Of those manifolds, the ones intersecting a Poincaré section in an invariant curve will keep their integrity for all times and will be observed as transport barriers, as we discussed for steady flows in §4.4.1. Therefore, to extract robust relaxed LCSs, we first locate invariant curves of appropriately defined Poincaré maps for the steady direction field (5.71). We then advect these curves under the flow map of the direction field (5.71) to obtain the full relaxed LCS positions at time t0 , denoted as M0 . Finally, we use the flow map Ftt10 of the velocity field v(x, t) to obtain the evolving relaxed LCS at time t as M(t) = Ftt10 (M0 ).

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For a practical numerical implementation of this procedure, we use a modified version of system (5.71) in the form 

x00 (s) = sign ξ2 (x0 (s); t0, t1 ), x00 (s − ∆) αˆ (x0 (s); t0, t1 ) ξ2 (x0 (s); t0, t1 ),  2  2 λ2 (x0 ; t0, t1 ) − λ1 (x0 ; t0, t1 ) λ3 (x0 ; t0, t1 ) − λ2 (x0 ; t0, t1 ) αˆ (x0 ; t0, t1 ) = , (5.72) λ2 (x0 ; t0, t1 ) + λ1 (x0 ; t0, t1 ) λ3 (x0 ; t0, t1 ) + λ2 (x0 ; t0, t1 ) which locally orients the ξ2 vector field and replaces its tensorline singularities with fixed points (see our discussion surrounding Eq. (5.34)). The computation of LCSs in 3D unsteady flows using the ξ2 eigenvector field is implemented in Notebook 5.8. Notebook 5.8 (UnifiedLCSTheory) Computes LCSs from tensorlines from the intermediate eigenvector field of the right Cauchy–Green strain tensor in a 3D unsteady velocity data set. https://github.com/haller-group/TBarrier/tree/main/TBarrier/3D/ demos/AdvectiveBarriers/UnifiedLCSTheory As a proof of concept, we show in Fig. 5.48 two Poincaré maps, constructed from the same set of points shown in Fig. 5.48(a), for the steady ABC flow already analyzed in Figs. 4.22 and 5.38. The Poincaré map in Fig. 5.48(b) is the classic first-return map for the ABC flow constructed for the z = 0 Poincaré section. In contrast, Fig. 5.48(c) shows the first return map for the same Poincaré section constructed from the trajectories of the ODE (5.72). Remarkably, the ξ2 -Poincaré map shows substantially more clarity from the same set of initial conditions.

Figure 5.48 Classic Poincaré map and ξ2 -Poincaré map computed from Eq. (5.72) for the steady ABC flow, from the same set of initial conditions selected on the z = 0 plane as Poincaré section. Adapted from Oettinger and Haller (2016).

Armed with this proof of concept for steady flows, we now turn to an aperiodically forced version of the ABC flow, given by   B(t) = B0 1 + k0 tanh (k1 t) cos (k2 t)2 , A sin z + C(t) cos y © ª v(x) = ­ B(t) sin x + A cos z ® , (5.73)   2 C(t) sin y + B(t) cos x (k (k C(t) = B 1 + k tanh t) cos t) , 0 0 1 3 « ¬

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with R1 ¼ 2; R2 ¼ 1. In (20), the functions xc ðzÞ; yc ðzÞ are the x, y coordinates of the (approximate) vortex center. (For evaluating xc ðzÞ and yc ðzÞ, we use our numerical values from previous work.20) Mapping the resulting torus under the flow Neither a classical Poincar!e map nor any other method map F50 , we see that it does advect coherently over the inter5.4 due Global Theory of LCSs 211 requiring long trajectories are options here, to theVariational tempoval ½t0 ; t1 ' ¼ ½0; 5' (cf. Fig. 10(b)). Therefore, even though √ k0 ¼ 0:3;√ this surface was just obtained from tangency to n (a necesral aperiodicity of the system. In (19), we choose with parameter values A = 3, B0 = 2, sary C =condition 1, k0 for = Definition 0.3, k 1 4), = it0.5, k = 1.52 and ~ k1 ¼ 0:5; kthe BðtÞ 2 ¼ 1:5 and k3 ¼ 1:8. We show the functions renders2 a full-blown ellip~= ( 1.8. This dependence growing and then saturating oscillatory instability C in Fig.time 8. Elliptic LCSs inmodels similar a time(B;k 3CðtÞ tic LCS. aperiodic ABC-type flows have been obtained in Refs. 2 and We next locallytime whether the complicated green for the steady ABC flow under perturbations. Despite thisexamine aperiodic dependence, Eq. 20; hyperbolic repelling LCSs in Ref. 2, although only to a structure from Fig. 9(b) indeed corresponds to a hyperbolic (5.72) stillz-direction. an autonomous system for whichLCS the(Definition ξ2 -Poincaré map (or first return map) small extent is in the 2): In Fig. 11(a), we take an illustrative part based at z the = n02-system is computable roughly levelandofinterpolate numerical effortfrom as for Considering for the timewith interval ½t0 ; t1 ' the of same the domain a surface the nthe 2-lines e map (cf. Fig. 9(a)). ¼ ½0; 5', we ABC compute the dual Poincar! Centered aroundof a point in the surface, additionsteady flow. Figure 5.49 shows time t0 = (green). 0 initial positions hyperbolic and we elliptic The algorithm and numerical settings are the same as in Sec. ally place a sphere of tracers (purple). Mapping the two LCSs obtained from the ofthat open closed invariant themap ξ2 -Poincaré V B. Compared to Fig. 7(c), there are advection a few structures per- andobjects forward in timecurves under theofflow F10 , we see that the flow perturbation generated(19) by to Eq. sistmap underunder the time-aperiodic the(5.72). velocity the tracers deform into an ellipsoid that is most elongated in ~ ¼ B þ B % k0 tanhðk1 tÞ cos½ðk2 tÞ2 '; B 7! BðtÞ ~ ¼ C þ C % k0 tanhðk1 tÞ sin½ðk3 tÞ2 ': C 7! CðtÞ

(19)

FIG. 9. Time-aperiodic ABC-type flow: Arc segments of n2-lines (corresponding to arclength s 2 ½4 ) 104 ; 5 ) 104 ') reveal locations of elliptic and hyperbolic LCSs. (a) Dual Poincar!e map, showing intersections between the Poincar!e section z ¼ 0 and possible time-t0 locations of elliptic and hyperbolic LCSs. (b) Figure 5.49 (a) Open and closed invariant curves of the ξ2 -Poincaré map based on consists of segments from several n2-lines. The Possible time-t0 locations of elliptic and hyperbolic LCSs: The structure in green (indicating a hyperbolic LCS) the(indicating z = 0elliptic plane the from aperiodically time-dependent ABC flow of(5.73). the periodicity the phase (b) spaceTime to extend the domain tubular surfaces LCSs)for are fitted point data of individual n2-lines. Here we use 3 slightly beyond t ½0;=2p' 0. positions of hyperbolic (green) and elliptic LCSs (red, blue and yellow).

0

Adapted from Oettinger and Haller (2016).

Reuse of AIP Publishing content is subject to the terms at:103111-9 https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 129.132.170.165 Tue, 25 Oct D. Oettinger and G. Haller Chaos 26, 103111On: (2016) 103111-9 G.G.Haller Chaos 103111-9 D.D.Oettinger Oettingerand and Haller D. Oettinger and G. Haller Chaos26, 26,103111 103111(2016) (2016) Chaos 26, 103111 (2016) 103111-9 2016 10:41:00

sha1_base64="yJr+GDWiWj7L9PBNC+Q7LKT21HM=">AAAB8nicbVDLSgNBEOyNrxhfUY9eBoPgKeyKr2NQEI8RzAM2McxOZpMhs7PLTK8QlnyGFw+KePVrvPk3TpI9aLSgoajqprsrSKQw6LpfTmFpeWV1rbhe2tjc2t4p7+41TZxqxhsslrFuB9RwKRRvoEDJ24nmNAokbwWj66nfeuTaiFjd4zjh3YgOlAgFo2glP+sEIbmZPJz13F654lbdGchf4uWkAjnqvfJnpx+zNOIKmaTG+J6bYDejGgWTfFLqpIYnlI3ogPuWKhpx081mJ0/IkVX6JIy1LYVkpv6cyGhkzDgKbGdEcWgWvan4n+enGF52M6GSFLli80VhKgnGZPo/6QvNGcqxJZRpYW8lbEg1ZWhTKtkQvMWX/5LmSdU7r7p3p5XaVR5HEQ7gEI7BgwuowS3UoQEMYniCF3h10Hl23pz3eWvByWf24Recj288E5CR

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1).can, Since stretch (r(r 2r 3 3>>1). 1). Sinceseparation can, stretchininforward-time forward-time 2r https://doi.org/10.1017/9781009225199.006 Published online by Cambridge University Press r > 1). Since separation stretch in forward-time (r 2 3 / expðt1 $ t0 Þ), we can, e.g., grow exponentially in1 $ time (rwe 3 we e.g., inintime 3 //expðt expðt $t0tÞ),Þ), e.g.,grow growexponentially exponentially time(r(r

Figure 5.50 (Top) Behavior of the initial position M of one of the elliptic LCSs shown in Fig. 5.49 under advection by the flow map. The toroidal coordinates used in this visualization are the same as in Eq. (5.45). (Bottom) Impact of one of the hyperbolic LCSs (green) shown in Fig. 5.49 on a nearby material sphere (purple) under advection by the flow map. Adapted from Oettinger and Haller (2016).

Figure 5.50 shows how a specific elliptic (toroidal) LCS indeed retains its coherence under advection throughout its extraction interval. The same figure also illustrates how a repelling

3

1

0

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LCS causes a transversely placed sphere of nearby initial conditions to stretch substantially and uniformly over its extraction interval.

5.5 Adiabatically Quasi-Objective, Single-Trajectory Diagnostics for Transport Barriers If a velocity field v(x, t) is known on a dense enough grid, then the velocity gradient ∇v can be computed with reasonable accuracy. This makes objective Eulerian quantities, such as the rate-of-strain tensor S and its invariants, available for flow analysis. Similarly, a numerical computation and differentiation of the flow map Ftt0 (x0 ) is feasible from a spatiotemporally well-resolved velocity field, and hence the objective Lagrangian strain tensors and their invariants used throughout this chapter can be computed with reasonable accuracy. None of these objective quantities, however, can be computed reliably from sparse velocity data for which spatial differentiation is unfeasible. An important example of an inherently sparse data set is the Global Drifter Program (GDP) of the National Oceanic and Atmospheric Administration (NOAA) of the United States, comprising more than 20,000 ocean drifter trajectories (see Lumpkin and Pazos, 2007). A representative snapshot of the distribution of these drifters is shown in Fig. 5.51.

Figure 5.51 An inherently sparse Lagrangian data set: drifter configuration in the NOAA Global Drifter Program (GDP) on April 5, 2021. Colors indicate the affiliation of the individual drifters. Image: NOAA.

Individual trajectory data  in this data set is available in the form of a time-dependent array of positions xi (ti j ) i, j where the index i labels individual trajectories and ti j labels the jth time instance at which the position of the ith trajectory is known. Originally, the temporal resolution of drifter tracking was at 6 hours, but the majority of the drifters are

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now being tracked by the Argos positioning system. This more advanced tracking provides drifter locations with errors of the order of 100 meters at an average temporal resolution of 1.2 hours since 2005 (see Elipot et al., 2016). While this temporal resolution yields reasonably accurate drifter velocities, the spatial resolution is clearly too coarse for differentiation, as can be deduced from Fig. 5.51. Such sparsity prohibits the computation of the objective scalar, vector and tensor fields we have been using in this chapter. We are, therefore, forced to seek other quantities that could be reliably used in detecting transport barriers and can still be inferred from sparse data. Such an inference is only possible if those quantities are defined on single trajectories, without reference to other trajectories. Yet none of the quantities associated with a single trajectory – such as trajectory length, tangent vectors, curvature or looping number – is objective. This can be seen by noting that in a frame co-moving with the trajectory, the trajectory itself becomes a fixed point and hence all the quantities we have just mentioned vanish identically in that frame. A way to address this issue is to use quasi-objective single-trajectory diagnostics in the extraction of transport barriers from sparse flow data. By the definition of quasi-objectivity given in §3.6, this task requires first the identification of objective single-trajectory diagnostic fields to which their quasi-objective counterparts are expected to be close in all frames satisfying a certain condition. As a second step, we need to find the appropriate quasi-objective diagnostics that do approximate the objective diagnostics under verifiable conditions on their frame of reference. We will address these two tasks in the upcoming sections. First, however, motivated by slowly varying geophysical flow data sets, we restrict the general frame-change family (3.1) to slowly varying (or adiabatic) frame changes of the form x = Q(t)y + b(t),

Q Û , bÛ  1.

(5.74)

Under such observer changes, the velocity transformation formula (3.17) simplifies to the adiabatic velocity transformation formula   Û − bÛ ≈ QT v v˜ = QT v − Qy

(5.75)

over compact spatial domains (in which y is uniformly bounded). Following Haller et al. (2022), we then call an Eulerian or Lagrangian quantity adiabatically quasi-objective under a condition (A) if, in all frames related via adiabatic frame changes (5.74), the quantity approximates the same objective Eulerian or Lagrangian quantity whenever the frame satisfies condition (A).

5.5.1 Adiabatically Quasi-Objective Diagnostic for Material Stretching An objective stretching field candidate is the direction-dependent version of the FTLE field (5.7). This exponent measures the average growth rate of a specific small perturbation ξ0 to an initial condition x0 under the flow map Ftt0 (x0 ) of a velocity field v(x, t) over the time

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interval [t0, t N ]. Using Eq. (A.7) of Appendix A.3, we calculate this averaged stretching exponent, λtt0N (x0, ξ0 ), for an unsteady flow as λtt0N (x0, ξ0 ) =

| ξ (t N )| 1 log , | ξ0 | t N − t0

(5.76)

where ξ (t N ) = ∇Ftt0N (x0 )ξ0 is the vector pointing from the unperturbed trajectory position x(t N ; t0, x0 ) to the perturbed trajectory position x(t N ; t0, x0 + ξ0 ). Indeed, λtt0N (x0, ξ0 ) returns a single positive or negative exponent that characterizes the overall growth ξ (t) in the time interval [t0, t N ]. To obtain more information on how much distortion infinitesimal fluid elements attached to x(t; t0, x0 ) experience, one may use the averaged hyperbolicity strength ˆ tN d | ξ (t)| 1 t N log dt. λ¯ t0 (x0, ξ0 ) := (5.77) | ξ0 | t N − t0 t0 dt The integrand in this expression is always positive, reflecting the fact that even if ξ (t) is shrinking, some other infinitesimal perturbation to the trajectory must be growing in an incompressible flow. Therefore, λ¯tt0N (x0, ξ0 ) is a measure of the total exposure of a trajectory to hyperbolicity in the flow. Under any Euclidean observer change (3.5), formula (3.13) shows the objectivity of | ξ (t)|, which implies that both λtt0N and λ¯tt0N are objective scalar fields.

Steady Flows In a steady velocity field v(x), the Lagrangian velocity vector v(t) = v(x(t; t0, x0 )) evolves as a material element, as we have seen in §2.2.8. Therefore, as long as the assumption |∂t v(x, t)| = 0

(5.78)

holds in the current frame of reference, one can select the specific material element ξ (t) = v(t) in the formulas (5.76)–(5.77) to assess the evolution of the length of v(t) via the diagnostics ˆ tN d |v(t N )| |v(t)| 1 1 tN tN ¯ log , λt0 (x0, v0 ) = log dt. (5.79) λt0 (x0, v0 ) = |v0 | |v0 | t N − t0 t N − t0 t0 dt Adiabatic coordinate changes of the form (5.74) approximately preserve the norms of all velocities by formula (5.75). Therefore, under slowly varying frame changes the fields λtt0N (x0, v0 ) and λ¯ tt0N (x0, v0 ) are nearly constant and approximate true material stretching along trajectories, as long as assumption (5.78) holds. Specifically, as noted by Haller et al. (2021, 2022), if only discretized trajectory data N {x(ti )}i=0 are available for a trajectory x(t; t0, x0 ), then the trajectory stretching exponents (TSE) N −1 tN | xÛ (t N )| | xÛ (ti+1 )| 1 1 Õ TSEtt0N (x0 ) = log , TSEt0 (x0 ) = log , (5.80) | xÛ (t0 )| | xÛ (ti )| t N − t0 t N − t0 i=0 for all xÛ (ti ) , 0, i = 0, . . . , N, are adiabatically quasi-objective measures of trajectory stretching and hyperbolicity strength under assumption (5.78). Under that assumption, the TSE fields do not just approximate an objective scalar field, as required for quasi-objectivity, but in fact coincide with an objective scalar field. There can be multiple frames in which a fluid flow satisfies the steadiness assumption (5.78). For instance, the center-type velocity field v(x) = (x2, −x1 ) is steady in any frame that rotates at a constant speed around x = 0. Similarly, the parallel shear flow v(x) = ( f (x2 ), 0),

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with a arbitrary function f , is steady in any frame that moves at a constant speed in the x1 direction. Under such steady-to-steady frame changes, Lagrangian velocities remain evolving material line elements along a material line in the new frame, given that any trajectory is also a material line in the transformed frame.

Unsteady Flows Differentiating Eq. (2.56) that defines the Lagrangian velocity v(t) with respect to time, we obtain that, in general unsteady flows, v(t) evolves along a trajectory x(t; t0, x0 ) according to the equation vÛ = ∇v (x(t; t0, x0 ), t) v + ∂t v (x(t; t0, x0 ), t) .

(5.81)

Therefore, if the Lagrangian acceleration a(t) := vÛ (t) along x(t; t0, x0 ) satisfies |a|  |∂t v| ,

(5.82)

then v(t) evolves approximately as a material element. Therefore, the TSE diagnostics defined in Eq. (5.80) are adiabatically quasi-objective under assumption (5.82), closely approximating the objective fields λtt0N and λ¯ tt0N in the given frame, as pointed out by Haller et al. (2022).14 We note that (5.82) quantifies the assumption that the Lagrangian time scales dominate Eulerian time scales in the flow (see, e.g., Shepherd et al., 2000). As the evolution of v(t) will not be exactly material in frames satisfying assumption (5.82), the TSE diagnostics will not yield exactly the same values in all those frames. They will nevertheless yield values close to those of their objective counterparts, λtt0N and λ¯tt0N , in those frames. The computation of the TSE diagnostic for 2D flows is implemented in Notebook 5.9. Notebook 5.9 (TSE2D) Computes the trajectory stretching exponent (TSE) field for a 2D unsteady velocity data set. https://github.com/haller-group/TBarrier/tree/main/TBarrier/2D/ demos/AdvectiveBarriers/TSE2D A similar computation of the TSE diagnostic for 3D flows is implemented in Notebook 5.10. Notebook 5.10 (TSE3D) Computes the trajectory stretching exponent (TSE) field for a 3D unsteady velocity data set. https://github.com/haller-group/TBarrier/tree/main/TBarrier/3D/ demos/AdvectiveBarriers/TSE3D

5.5.2 Adiabatically Quasi-Objective Diagnostic for Material Rotation An objective rotation field candidate can be defined as the averaged rotation-rate deviation of ξ (t) from the averaged mean rotation rate of the fluid, based on the considerations used in 14

The original argument of Haller et al. (2021) for the unsteady TSE diagnostics lacked assumption (5.82) and was incorrectly formulated in the extended phase space. This error was demonstrated by Theisel et al. (2022) on a simple counterexample, then corrected by Haller et al. (2022), who also added the requirement of adiabatic time dependence for observer changes.

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the derivation of the LAVD field (see §5.2.11 and Haller, 2016; Haller et al., 2016). Indeed, Haller et al. (2021) show that the averaged rotation deviation ˆ tN d ξ (t) ξ (t) 1 1 tN ¯ (t) × α¯ t0 (x0, ξ0 ) = − ω dt (5.83) | ξ (t)| t N − t0 t0 dt | ξ (t)| 2 is an objective measure of the average relative rotation speed experienced by the tangent ¯ (t) denotes the spatial vector ξ (t) during its evolution along the trajectory x(t; t0, x0 ). Here, ω average of the vorticity over the flow domain, as in §5.2.11.

Steady Flows Haller et al. (2021) also show that under assumption (5.78) and the additional assumption d v(t) 1 v(t) ω (5.84) 2 ¯ (t) × |v(t)|  dt |v(t)| , the trajectory rotation average (TRA) tN TRAt0 (x0 )

N −1 hÛx(ti ), xÛ (ti+1 )i 1 Õ cos−1 = | xÛ (ti )| | xÛ (ti+1 )| t N − t0 i=0

(5.85)

for all xÛ (ti ) , 0, i = 0, . . . , N is an adiabatic quasi-objective measure of total trajectory tN rotation. As Haller et al. (2021) note, TRAt0 (x0 ) becomes fully objective without assumptions (5.78)–(5.84) when applied to objective vector fields. This will be important in Chapter 9 in applications of the TRA field to the detection of active barriers to transport that obey objective differential equations.

Unsteady Flows The extension of the TRA metric to unsteady flows follows the same idea as the unsteady extension of the TSE metric given in §5.5.1. Namely, under assumption (5.82), the evolution of the Lagrangian velocity is nearly material by formula (5.81). Therefore, the trajectory tN rotation average, TRAt0 (x0 ), defined in Eq. (5.85) is an adiabatically quasi-objective measure of total trajectory rotation under assumptions (5.82)–(5.84). The computation of the TRA diagnostic for 2D flows is implemented in Notebook 5.11. Notebook 5.11 (TRA2D) Computes the trajectory rotation average (TRA) field for a 2D unsteady velocity data set. https://github.com/haller-group/TBarrier/tree/main/TBarrier/2D/ demos/AdvectiveBarriers/TRA2D A similar computation of the TRA diagnostic for 3D flows is implemented in Notebook 5.12. Notebook 5.12 (TRA3D) Computes the trajectory rotation average (TRA) field for a 3D unsteady velocity data set. https://github.com/haller-group/TBarrier/tree/main/TBarrier/3D/ demos/AdvectiveBarriers/TRA3D

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5.5 Single-Trajectory Diagnostics for Transport Barriers

5.5.3 Single-Trajectory, Adiabatically Quasi-Objective LCS Computations for the AVISO Data Set We illustrate the adiabatically quasi-objective, single-particle LCS diagnostics TSE, TSE and TRA on the 2D, satellite-altimetry-derived ocean-surface current product (AVISO; see Appendix A.6) that we have already used for Lagrangian analysis in several earlier examples. We focus on the Agulhas leakage region U = {(x, y) ∈ [−2.5◦, 5◦ ] × [−40◦, −30◦ ]}.

(5.86)

Haller et al. (2022) find that assumption (5.82) is satisfied for the majority of initial conditions in this domain, for which they obtain |a| ≈ 10 |∂t v|. Furthermore, Haller et al. (2021) find that assumption (5.84) is satisfied on average for 98.9% of all initial positions, with the left-hand side of the inequality in (5.84) not exceeding 1% of its right-hand side in these cases. Using the AVISO data set, Haller et al. (2021) generate trajectory data over a period of 25 days to compute the TSE25 0 (x0 ) metric over a uniform grid of 250 × 250 25

initial conditions. Subsequently, they repeat these calculations with the metric TSE0 (x0 ) after a gradual, random subsampling of the initial grid to 10%, 1% and 0.1% of the initial conditions. Figure 5.52 shows a comparison of the results, with the squared relative dispersion and FTLE also computed over the same grid and over its randomly subsampled version. Because of the nonuniformity of the subsampled grid, the squared relative dispersion is computed from the formula d 2 (xi0, t0, t) =

|xi (t) − x−i (t)| 2 |xi0 − x−i0 | 2

,

(5.87)

where xi and x−i are trajectory pairs with initial conditions xi0 and x−i0 that are initially close at time t0 (e.g., |xi0 − x−i0 | = r0  1). The FTLE field in these comparisons is computed after the final positions are interpolated using a C 1 -interpolant. A highlighted repelling LCS is clearly visible in the top row of Fig. 5.52 as a ridge in all three diagnostics. The quality of the FTLE25 0 (x0 ) field quickly degrades with decreasing reso2 lution and d loses track of the details of most structures as well. Overall, the TSE diagnostic is on par with multi-trajectory diagnostics at full resolution but has lower computational costs. At decreased resolution, the single-trajectory diagnostic TSE is clearly superior in capturing the most robust fronts even at severe sparsifications of the original numerical grid. Figure 5.53 shows an evaluation of the single-trajectory, adiabatically quasi-objective 25 rotation metric TRA0 (x0 ) over the same full and sparsified grids that were used in Fig. 5.52. In these plots, the TRA is compared with the objective d 2 (xi0, 0, 25) field and with the polar rotation angle field PRA25 0 (x0 ), whose level curves are objective in two dimensions (see §5.2.10). An inset in each plot zooms in on a region with a vortex identified by the geodesic theory of elliptic LCSs (see §5.4.1). All three metrics highlight this vortex in the first row of Fig. 5.53, along with several other elliptic LCSs. Note that PRA25 0 (x0 ) provided the highest level of detail at this full resolution. 25 Subsequent rows of Fig. 5.53 show that TRA0 (x0 ) retains most of its structure under random subsampling, enabling the recognition of three of its maxima, which mark vortices, even at the lowest resolution. For the other two objective, multi-trajectory diagnostics, discerning

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Figure 5.52 Single- and multi-trajectory stretching metric comparisons for the AVISO ocean surface current field in the Agulhas region at full and decreased grid resolutions. Adapted from Haller et al. (2022).

coherent structures becomes unfeasible under progressive subsampling. Haller et al. (2021) and Haller et al. (2022) present further comparisons for steady and unsteady versions of the ABC flow (see Eq. (4.29)), which yield similar conclusions about the TSE and TRA metrics when used for 3D flows.

5.5.4 Adiabatically Quasi-Objective Material Eddy Extraction from Actual Ocean Drifters Encinas-Bartos et al. (2022) use the TRA field defined in Eq. (5.85) to identify Lagrangian (i.e., material) eddies in various drifter data sets. The identified eddies range in size from the submesoscales to the mesoscales. As already noted in §5.5.3, the key assumptions (5.82)– (5.84) securing the adiabatic quasi-objectivity of the TRA field have broadly been found to

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Figure 5.53 Same as Fig. 5.52 but with single- and multi-trajectory rotation metrics. Adapted from Haller et al. (2022).

hold over ocean regions containing multiple eddies (see also Abernathey and Haller, 2018, and Beron-Vera et al., 2019b). tN

With the TRAt0 (x0k ) field computed for individual drifter initial conditions x0k , one can use tN

interpolation to infer the full TRAt0 (x0 ) diagnostic field. Given their focus on locating elliptic LCSs, Encinas-Bartos et al. (2022) propose interpolation of the TRA field via linear radial basis functions. As an example of the results obtained from this approach, the left subplot in Fig. 5.54 shows a material eddy region inferred from the GDP drifter data set already featured in Fig. 5.51. The nominal eddy boundary is identified as the TRA contour corresponding to 70% of the local maximum visible in the plot. The right subplot shows a persistent floating sargassum accumulation inferred from the Maximum Chlorophyl Index (MCI) in the same location, which independently confirms the presence of a coherent material eddy (see also Beron-Vera et al., 2015, for a prior analysis of this feature by other methods).

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manuscript submitted to JGR: Oceans

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Figure 8.

Lagrangian and Objective Eulerian Coherent Structures

(Left) TRA

276

tN at t = 276doy is reconstructed in the GDP data set via linear radial

Figure 5.54 (Left)246 The TRAt0 (x0 ) field reconstructed from the GDP data set with basis function is extracted with algorithm t0 = 246interpolation. days and t N =The 276eddy daysboundary in the year(white 2006.line) A representative levelthe curve of the proposed TRA in Appendix A. Theinwhite indicate driftertrajectory positions pieces at t = of 276doy and the tails field is shown whitetriangles along with the 10-day the six drifters in the selected region at thesuperimposed time. (Right)onSargassum accumulation inferred are 10present days long. (Right) Eddy boundary the floating sargassum concentration from thethe Maximum Chlorophyl suggests a coherent material eddy in inferred from Maximum ChlorophylIndex Index(MCI) (MCI).also Regions covered by clouds are displayed in the same location. Adapted from Encinas-Bartos et al. (2022). white.

5.6 Elliptic-Parabolic-Hyperbolic (EPH) Partition and LCSs fieldclose reveals presenceof of LCSs a Lagrangian eddyan centered at Eulerian around (68 W, 37 N Thisflows We ourthe discussion by deriving objective partition of).fluid 276 eddyregions is visible a blob a local maximum in same the TRA field. The eddy into in as which allaround fluid trajectories share the instantaneous stability type. If a 246 boundary is extracted using the algorithm proposed in Appendix A and coincides with trajectory stays in the same region of the partition for extended times, then its Lagrangian the location the out sargassum patch. The looping exhibited by the two stability typeofturns to coincide with thecyclonic instantaneously inferred Eulerian stability type. trajectories inside the eddy compared to the remaining ones additionally confirms the Therefore, while the upcoming §5.7 examines a connection between objective Eulerian flow presence of an elliptic LCS. We have thus shown that reconstructing TRA fields from features and LCSs the limit of short times, here we of explore suchbarriers. a connection longer sparse drifter data in provides spatially-resolved details transport This over approach time intervals. is free from the limitation of remote sensing data, which require careful preprocessing to We startthe byimpacts recalling setting of §2.2.8, we considered materially evolving eliminate of the external factors, such in as which cloud coverage. infinitesimal perturbations ξ (t) along a fluid trajectory x (t; t0, x0 ) within a flow domain x ∈ U ⊂ Rn with n = 2 or n = 3. Taking the inner product of the equation of variations (2.45) with ξ (t) then leads to the expression

 1 d | ξ | 2 = hξ, ∇v(x, t)ξ i , ξ, ξÛ = (5.88) 2 dt or, equivalently,

1 d | ξ | 2 = C(ξ ; x, t), 2 dt

C(ξ ; x, t) := hξ, S(x, t)ξ i ,

(5.89)

with the rate-of-strain tensor S = ST defined in Eq. (3.34). The symmetric tensor S has zero trace for incompressible flows and hence it is either singular or indefinite. In either case, the local zero strain set, Z(x, t), of instantaneously unstrained perturbations, defined as

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5.6 Elliptic-Parabolic-Hyperbolic (EPH) Partition and LCSs

Z(x, t) = {ξ ∈ Rn : hξ, S(x, t)ξ i = 0} , is nonempty. It is not hard to see that if S(x, t) is nonsingular and hence indefinite, then Z(x, t) is a generalized cone: a set of two orthogonal lines for 2D flows (n = 2) and a 2D elliptic cone for 3D flows (n = 3), as shown in Fig. 5.55. The figure shows the zero set Z(x, t) in the coordinate system defined by the rate-of-strain eigenvectors ei (x, t), defined as Sei = si ei ,

s1 (x, t) ≤ · · · ≤ sn (x, t),

|ei (x, t)| = 1,

i = 1, . . . , n.

(5.90)

By the objectivity of S, the generalized cone-bundle Z(x, t) is an objectively defined feature of a fluid flow.

Figure 5.55 The qualitative geometry of the zero strain set Z(x, t) in 2D and 3D flows at a point x ∈ U satisfying det S(x, t) , 0. For n = 3, we distinguish between two cases depending on the sign of the intermediate rate-of-strain eigenvalue s2 (x, t). Also shown is the sign of the quadratic form C(ξ; x, t) defined in Eq. (5.89) inside and outside the cones formed by Z(x, t).

In the sector with C(ξ ; x, t) > 0, perturbations to the fluid trajectory at the location x grow instantaneously, while perturbations decay instantaneously in the sector with C(ξ ; x, t) < 0. The instantaneous flow geometry near x will crucially depend on how the evolving perturbations cross the moving cone Z (x(t; t0, x0 ), t) along the trajectory x(t; t0, x0 ). This relative motion through Z is determined by the material derivative DC(ξ ; x, t) d = C(ξ (t); x(t; t0, x0 ), t). Dt dt

(5.91)

> 0 is positive at a point along Z, then perturbations move through that point Indeed, if DC Dt from the negative C sector to the positive C sector. Likewise, if DC is negative at a point Dt along Z, then perturbations move through that point from the positive C sector to the negative C sector. A direct calculation utilizing the equation of variations (2.45) gives    

 DC D D DS Û hξ, Sξ i = 2 ξ, Sξ + ξ, Sξ = 2 h∇vξ, Sξ i + ξ, = ξ Dt Dt Dt Dt = hξ, Mξ i,

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(5.92)

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with the strain acceleration tensor, M(x, t), defined as M=

DS + S∇v + (∇v)T S = ∂t S + ∇Sv + S∇v + (∇v)T S. Dt

(5.93)

As noted by Haller (2005), M is an objective material derivative, the Cotter–Rivlin rate, of S (see Cotter and Rivlin, 1955). We denote the pointwise restriction of M, as a linear operator, to the local cone, Z, by MZ (x, t) := M(x, t)|Z(x,t) .

(5.94)

The sign-definiteness of the restricted operator MZ can then be defined via the signdefiniteness of the quadratic form hξ, MZ (x, t)ξ i ξ∈Z(x,t) . It turns out that MZ can only be positive definite (written MZ (x, t) > 0), positive semidefinite (written MZ (x, t) ≥ 0) or indefinite (written MZ (x, t) T 0), as shown by Haller (2001b) and Haller (2005). For incompressible flows, Fig. 5.56 shows the instantaneous local flow geometry at the point x at time t, as a function of the definiteness of MZ , based on the relation (5.92).

Figure 5.56 Possible local instantaneous flow geometries near a fluid trajectory located at a generic point x satisfying det S(x, t) , 0 at time t. For n = 3, the case of s2 > 0 is similar but with all arrows reversed.

Of the instantaneous flow geometries shown in Fig. 5.56, the ones typically arising over open sets of x points in the flow domain are the instantaneous saddle-type (hyperbolic) behavior under a positive definite MZ (x, t) and the instantaneous center-type (elliptic) behavior under an indefinite MZ (x, t). These two types of behaviors are separated from each other

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by surfaces in the flow along which MZ (x, t) is positive semidefinite, marking instantaneous shear-type (parabolic) local behavior. All this implies an objective partition of the nondeˆ generate flow subdomain, U(t) = {x ∈ U : det S(x, t) , 0}, into an elliptic domain E(t), a hyperbolic domain H (t) and a parabolic domain P(t), defined as  ˆ : MZ (x, t) T 0 , E(t) = x ∈ U(t)  ˆ : MZ (x, t) ≥ 0 , P(t) = x ∈ U(t) (5.95)  ˆ : MZ (x, t) > 0 . H (t) = x ∈ U(t) This objective elliptic-parabolic-hyperbolic (EPH) partition applies to any 2D and 3D ˆ in which the rate-of-strain tensor in nondegenerate. The evolving flow on the domain U(t) elements of the EPH partition are sketched qualitatively in the extended phase space in Fig. 5.57. Haller (2001b) shows that for 2D flows, the elliptic and hyperbolic domains of the EPH partition coincide with those obtained form the Okubo–Weiss criterion evaluated in a frame co-rotating with the rate-of-strain eigenvectors {ei (x, t)}3i=1 (see formula (3.59) and our discussion thereafter). For 3D flows, however, no such relation holds between the Q-criterion applied in strain basis and the EPH partition (see Haller, 2005).

Figure 5.57 Schematic evolution of the connected components of the elliptic domain E(t), bounded by connected components of the parabolic domain P(t) and surrounded by the hyperbolic domain H (t).

The EPH partition provides a frame-indifferent mathematical link between Lagrangian behavior and objective Eulerian flow features. Specifically for 2D flows, all fluid trajectories staying in the hyperbolic domain H (t) over a finite time interval I = [t0, t1 ] can be shown to be part of a hyperbolic (attracting or repelling) material line. Furthermore, hyperbolic material lines cannot stay in E(t) for too long: after exceeding a theoretical upper bound on their stay, they must necessarily be contained in an elliptic (vortical) material surface (see Haller, 2001b for details). An objective diagnostic principle suggested by these results is that material surfaces spending the longest times in the hyperbolic region H (t) should be hyperbolic LCSs and those spending the longest time in the elliptic region E(t) should be elliptic LCSs. A computation of times spent in the hyperbolic and elliptic regions by trajectories in a doubly periodic,

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2D turbulence simulation supports this conclusion for elliptic LCS, as seen in Fig. 5.58. In contrast, hyperbolic LCS share long times in H (t) with regions of high shear.

Figure 5.58 Hyperbolicity times and ellipticity times computed for trajectories released from a uniform initial grid in a 2D turbulence simulation. The non-dimensional times are plotted over the initial positions of the trajectories; large values of these times (lighter colors) indicate hyperbolic and elliptic behaviors, respectively. Note that regions of high shear are also highlighted by the hyperbolicity time plot, just as by FTLE plots.

The computation of hyperbolicity and ellipticity times for general 2D unsteady flows is implemented in Notebook 5.13. Notebook 5.13 (MzCriterion2D) Computes times spent in the hyperbolic domain H (t) and elliptic domain E(t) in a 2D unsteady velocity data set. https://github.com/haller-group/TBarrier/tree/main/TBarrier/2D/ demos/AdvectiveBarriers/MzCriterion2D For 3D flows, a further partition of the hyperbolic region exists in the form H (t) = H + (t) ∪ H − (t), where H ± (t) = {x ∈ H (t) : sign [−s2 (x, t)] = ±1}. Using this partition, Haller (2005) proves that fluid trajectories staying in H + (t) over a time interval I are contained in 2D, uniformly repelling material surfaces. Material perturbations to this surface align with a one-dimensional material line traveling with the trajectory. Likewise, fluid trajectories staying in H − (t) over a time interval I are contained in 2D, uniformly attracting material surfaces. Material perturbations to this surface realign with the surface due to its attractivity. No similar mathematical conclusions are available for elliptic material surfaces, but the above results for the 2D case prompt Haller (2005) to give the following objective definition (MZ -criterion) for a material vortex in 3D flows. Definition 5.16 A finite-time material vortex over the time interval I is a connected set of fluid trajectories staying in the elliptic region E(t) for all t ∈ I.

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This criterion identifies a material vortex as a set of fluid trajectories along which material perturbations do not definitely align with a given material line or material surface moving with the fluid trajectory. Indeed, within the elliptic domain E(t), there are always small material perturbations that cross from the positive (C > 0) side of the cone Z to its negative (C < 0) side, defying the alignment suggested by the rate-of-strain eigenvalues si (x, t). As a result, the definitive material alignment observed in the H ± (t) hyperbolic domains is either absent or inconsistent with the eigenvectors of S(x, t) in the elliptic domain. Figure 5.59 shows numerical evaluation of this definition on a time-dependent version of the ABC flow similar to that in Eq. (5.73) (see Haller, 2005, for details). As the figure shows, an even sharper separation of elliptic and hyperbolic structures arises when the total time spent outside the hyperbolic region H (t) is plotted over the initial trajectory positions, as opposed to simply the total time spent in the elliptic region. In applications to more complex flow data, the implementation of the MZ -criterion is challenging due to noisiness introduced by the required spatial differentiation of S and by errors in the advection of trajectories (see Sahner et al., 2007; Urban et al., 2021).

Figure 5.59 Comparison of the results from the M Z -criterion in Definition 5.16 with the FTLE field in a time-dependent ABC flow similar to Eq. (5.73). (Left) Hyperbolic (yellow) and elliptic (red) LCSs indicated by the FTLE field. (Middle) The total time spent by trajectories in the elliptic region E(t), plotted over their initial conditions. Darker colors indicate longer times. (Right) The total time spent by trajectories outside the hyperbolic region H (t), plotted over their initial conditions. Again, darker colors indicate longer times. Adapted from Haller (2005).

The computation of hyperbolicity and ellipticity times for general 3D unsteady flows is implemented in Notebook 5.14. Notebook 5.14 (MzCriterion3D) Computes times spent in the hyperbolic domain H (t) and elliptic domain E(t) in a 3D unsteady velocity data set. https://github.com/haller-group/TBarrier/tree/main/TBarrier/3D/ demos/AdvectiveBarriers/MzCriterion3D

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5.7 Objective Eulerian Coherent Structures (OECSs) A vast number of transport problems are associated with a specific time scale over which one would like to identify LCSs. In the absence of such a time scale, however, it is a priori unclear what finite time interval should be used in these studies. The LCSs one obtains will generally depend on the time interval chosen. Indeed, changes to that time interval alter the finite-time dynamical system (5.1) under consideration. A second limitation of the LCS approach to certain transport problems is precisely its overall robustness: LCSs are, by construction, insensitive to short-lived or transient behavior. In some cases, these short-term developments are also of interest and should be identified in a frame-independent fashion, of which commonly used Eulerian methods are incapable. A third limitation of LCSs in some cases is their strict material nature. While this property is a clear advantage for experimental observability, material invariance limits the ability of LCSs to frame the creation, destruction and merger of flow structures. Indeed, none of these phenomena can be exhibited by material sets in smooth (or at least Lipschitz) velocity field: their birth, annihilation or collision would violate the existence and uniqueness of the trajectories that form these material sets (see §2.2). Figure 5.60 illustrates these considerations on the classic phenomenon of a vortex merger. P. Meunier et al. / C. R. Physique 6 (2005) 431–450

435

Fig. 4. (a)–(c) Cross-cut experimental dye visualizations of two laminar co-rotating vortices, and (d)–(f) vorticity fields obtained by two-dimensional DNS. The snapshots are taken (a), (d) before, (b), (e) during and (c), (f) after merging.

Figure 5.60 Experimental (dye-based) image of a (Top) vortex merger, and (Bottom) its Eulerian footprint in the instantaneous vorticity Adapted from Meunier et al. diffusion of vorticity. The duration of this diffusive stage is proportional to thefield. Reynolds number; it can be calculated using (4) as: (2005). !t1∗ = (a/b)c

Re 8π 2

(7)

The three upper snapshots of the figures from the experiments of Meunier et al. (2005) show that the material interiors of two vortices always remain physically separate open sets, even if they become filamented and wrapped along each other. These two sets lose their individual coherence while their union gains coherence. Therefore, neither the individual vortices nor Once the critical core size is exceeded, the vortex pair becomes unstable and merges, leading to a rapid decrease of the separation distance. This stage appears to be driven by advection of vorticity, and its duration !t2∗ = 0.7 is thus fairly independent of the Reynolds number. The numerical results in Fig. 5 show that, at the end of this stage, the merging of the vortices is never complete: the separation distance b does not vanish but remains at a value close to 0.25. This defines the beginning of a third stage, in which the two vortices are next to each other but still have two separate maxima of vorticity, as can be seen, e.g., in Fig. 4(e). The diffusion of vorticity, coupled to the rotation of the vortex center (close to a solid body rotation), leads to an axisymmetrization of the vortex center in a time scaling as Re1/2 , as has been explained by Bajer et al. [13] for vorticity perturbations at the center of a vortex. Numerically, the duration of this stage has been found to be: !t3∗ = 0.0089 Re1/2

(8)

In the fourth and last stage, the vortex diffuses again due to viscosity and its core size increases with time. This stage will be described in more detail in the next section. The merging phenomenon highly depends on the critical ratio (a/b)c of core size and separation distance at which it begins. The determination of the critical condition has been the subject of numerous works: through numerical simulations on vortex patches, it was found that (a/b)c = 0.3 (see Overman and Zabusky [8], Rossow [11], and Dritschel [14,15]), which was confirmed experimentally by Griffiths andCambridge Hopfinger [9]. Theoretically, Melander et al. [16] used the elliptic moment model (EMM) https://doi.org/10.1017/9781009225199.006 Published online by University Press

5.7 Objective Eulerian Coherent Structures (OECSs)

227

the merged vortex will ever be detected as elliptic LCSs by any self-consistent LCS method over a time interval containing the snapshots (a)–(c). In contrast, the instantaneous vorticity contours in the lower snapshots (obtained from a 2D simulation) lose track of the individual vortices at some point and start displaying a merged vortex without any indication of its still unmixed internal structure. Since instantaneous vorticity contours are objective in two dimensions (see our discussion after formula (3.21)), both the Lagrangian and the Eulerian descriptions of vortex merger in Fig. 5.60 are self-consistent but reveal different aspects of the process. A way to leverage the mathematical foundations and objectivity of LCS methods for coherence analysis in the Eulerian frame is to take their t1 7→ t0 = t limit, as we noted in the introduction of this Chapter. This procedure yields objective Eulerian coherent structures (OECSs) that act as LCSs over infinitely short time intervals. An OECS can appear, persist for a while, then disappear or merge with other OECSs. While active, it still has an impact on material tracers or dye because material notions of attraction, repulsion or shear were used in its construction. In summary, OECSs are ultimately not material objects, but strong OECS can act as organizing centers of temporary coherent patters and able to relinquish this role to other OECSs any time, without any dependence on the time interval of definition for the finite-time dynamical system (5.1). OECSs have their own limitations. The first is precisely their nonmaterial nature, which necessitates their continued re-computation throughout a time interval of interest. Second, as instantaneous limits of LCSs, the OECS are constructed from infinitesimally small trajectory displacements. This generally causes OECS methods to become less sensitive to inhomogeneities in the deformation field in comparison to LCS methods. As an extreme case, the polar rotation angle field discussed in §5.2.10 becomes identically zero in the t1 → t0 limit and hence is unable to detect rotationally coherent structures in the instantaneous sense. Similarly, while the instantaneous limits of the FTLE field are effective in identifying short-term attracting and repelling material lines, the clarity and detail of these instantaneous calculations is inferior to those obtained from the FTLE for t1  t0 . For these reasons, OECSs do not generally represent improvements over LCSs. Rather, they offer complementary information about short-term transport barriers in the flow, while LCSs identify longer-term barriers more efficiently. In the following, we give a survey of OECS methods and discuss examples in which they have shown themselves to be useful alternatives to LCS methods. We will also include pointers to our later discussions of objective Eulerian barriers in flow separation, diffusive transport and dynamically active transport.

5.7.1 Instantaneous Limit of the Flow Map A key element of taking the short-term limit of LCSs is a short-time approximation to the deformation gradient ∇Ft+τ for |τ|  1. This approximation can be computed from the t Taylor expansion of ∇Ft+τ around τ = 0 as t

τ + O τ 2 = I + τ ∇v + O τ 2 , ∇Ft+τ = I + ∂τ ∇Ft+τ (5.96) t t τ=0 where we have used the fact that ∇Ft+τ is the normalized fundamental matrix solution of the t equation of variations (see §2.2.8). With the help of formula (5.96), Serra and Haller (2016)

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also obtain a Taylor series approximation for the short-term right Cauchy–Green strain tensor Ct+τ in the form t  T
Ct+τ = ∇Ft+τ ∇Ft+τ = I + 2τS + O τ 2 , (5.97) t t t which now involves the rate-of-strain tensor S(x, t). The details of the quadratic and cubic terms in the expansion (5.97) are not needed here but can be found in Nolan et al. (2020). Note that for any n × n matrix A with eigenvectors si and corresponding eigenvalues µi , the matrix I + A has the same eigenvectors with corresponding eigenvalues µi + 1. Therefore, using the solutions of the rate-of-strain eigenvalue problem (5.90), we can write the eigenvalues λi and eigenvectors ξi of the symmetric tensor Ct+τ (see Eq. (2.95)) for small t τ values as
λi = 1 + 2τsi + O τ 2 , ξi = ei + O (τ) . (5.98)

5.7.2 Instantaneous FTLE Using formula (5.98), we define the instantaneous FTLE field as the t1 → t0 + limit of formula (5.7), which is equal to the maximal rate-of-strain eigenvalue: FTLEt+ t (x) = lim

τ→0+


 1 log 1 + 2sn (x, t)τ + O τ 2 = sn (x, t). 2 |τ|

(5.99)

Codimension-1 surfaces with instantaneously maximal normal repulsion in forward time are marked by ridges of FTLEt+ t (x) (see §5.2.1). The converse of this statement is not true, given that surfaces of maximal instantaneous tangential shear also generate ridges in the sn (x, t) field (see §5.2.2). Instantaneous limits of forward-time parabolic LCSs (jet cores) are marked by trenches FTLEt+ t (x) (see §5.2.9). The t1 → t0 − limit of the backward-time FTLE can also be computed from formula (5.98) as the minimal rate-of-strain eigenvalue: FTLEt− t (x) = lim

τ→0−


 1 log 1 + 2s1 (x, t)τ + O τ 2 = −s1 (x, t). 2 |τ|

(5.100)

Codimension-1 surfaces with instantaneously maximal normal attraction in forward time are marked by ridges of FTLEt− t (x) (see §5.2.1). Instantaneous limit of backward-time parabolic LCSs (jet cores) are marked by trenches of FTLEt+ t (x) (see §5.2.9). t− For 2D incompressible flows, FTLEt+ (x) = FTLE t t (x) but for compressible 2D flows and for general 3D flows, the two quantities differ from each other. This is illustrated in Fig. 5.61 using the computations of Nolan et al. (2020) on a 2D horizontal slice of the Weather Research and Forecasting Model (WRF) of the National Center for Atmospheric Research (NCAR). Computed from 2D wind data at an altitude of 100, this horizontal wind field t+ represents a compressible flow. Accordingly, FTLEt− t and FTLEt differ noticeably both in magnitude and in parts of their spatial distribution. Note that in this 2D slice of the 3D atmospheric flow, attracting LCS mark upwelling along vertical 2D material surfaces, while repelling LCSs are generally the signatures of downdrafts or shear, as noted by Tang et al. (2011a,b).

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patterns which are shown by the attraction rate field integration times up to 24 h in backward time can be become more sharply defined. This relationship can be found at https://youtu.be/nkqpZ2GcO1E. quantified by the Pearson correlation coefficient, given This data set allows us to test whether iLESs are in Fig. 11a, which shows 5.7 that Objective for short integration effective at predicting the Lagrangian behavior Eulerian Coherent Structures (OECSs) 229 of

rate field. The attraction rate field has been multiplied by "1 to Fig. 14 Comparison of the attraction rate field, s1 , left, and the "1 at T ¼ 0. Structures in thebackward-time, repulsion rate field, 5.61 s2 , right, Figure Differences in the (Left) and (Right) forward-time aid in visualization. The colorbar has units of hrinattraction rate field are noticeably in thefield repulsion stantaneous limitsstronger of thethan FTLE on 2D compressible atmospheric wind field data.

Adapted from Nolan et al. (2020).

123 5.7.3 Instantaneous Vorticity Deviation (IVD) The instantaneous limit of the Lagrangian-averaged vorticity deviation (LAVD) defined in formula (5.27) is the instantaneous vorticity deviation (IVD), defined by Haller et al. (2016) as ¯ (t)| , IVD(x, t) = lim LAVDt+τ t (x) = | ω (x, t) − ω τ→0

(5.101)

¯ (t) denoting the instantaneous spatial mean of the vorticity field ω (x, t) over the flow with ω domain of interest. Based on Definition 5.9, we can now define the instantaneous limits of LAVD-based elliptic LCSs, vortex boundaries and vortex centers in an objective manner using the IVD field. Definition 5.17 At time t, IVD-based elliptic OECSs in the finite-time flow (5.1) are smooth cylindrical level surfaces of the instantaneous vorticity deviation IVD(x, t) surrounding a unique, codimension-2 height ridge of IVD(x, t). IVD-based coherent Eulerian vortices are formed by nested families of such elliptic OECSs. IVD-based vortex centers in such Eulerian vortices are marked by the codimension-2 ridge of IVD(x, t) surrounded by the elliptic OECSs. This definition offers an objective and mathematically well-established alternative to the nonobjective Eulerian vortex criteria surveyed in §3.7.1. We note that Jeong and Hussain (1995) point out the inadequacy of defining vortices as domains where | ω (x, t)| exceeds a selected threshold. Level surfaces of | ω (x, t)| exceeding a threshold may indeed have little to do with vortices; they may simply mark a high-shear region in a parallel channel flow. In contrast to that definition, IVD-based vortices are composed of cylindrical level sets surrounding a codimension-2 ridge of IVD(x, t). As an example, Fig. 5.62 shows IVD-based coherent Eulerian vortices extracted from the same gravity current experiment whose Lagrangian vortices we have already shown in Fig. 5.27. The IVD-based extraction procedure here was identical to the LAVD-based extraction procedure outlined in §5.2.11. A second example of the use of the IVD diagnostic is the work of Yang, K. et al. (2021), which we have already mentioned in the context of the LAVD in §5.2.11. These authors employ the IVD to assess the size of instantaneous vortices in the left ventricular (LV)

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Lagrangian andTNTI Objective Eulerian Coherent Structures Evolution of the in stably stratified turbulent flows

898 A3-9 5

4 5 3

IVD 2

z 4 20

1

18 16

3

x

0

14 10

y

12

9

F IGURE 3. Visualization of the OECSs and the TNTI from Ri11 at t = 100. The OECSs areFigure represented by blue tubular surfaces (boundaries) 1-Dand curves (centres), 5.62 IVD-based coherent objective Euleriansurrounding vortices (blue) vortex center whereas the TNTI is represented by the yellow open surface. The region above the TNTI lines (IVD ridges; red curves) identified below the turbulent/nonturbulent interface is irrotational, whereas below the surface, the flow is turbulent. The OECSs with green (TNTI; transparent yellow surface) from the gravity current experiment shown in Fig. boundaries cross the vertical plane at y = 9.75 almost perpendicularly. On the vertical plane Thethe vortices withisgreen boundaries the plot. vertical plane at y = 9.75 nearly at 5.27. y = 9.75 IVD field shown in the redcross contour

perpendicularly. Adapted from Neamtu-Halic et al. (2020).

(a)

4

2-D 3-D

(b)

(c) 4

15

10 2 as well as the total vorticity carried by these vortices. 2 blood flow, Notable and consistent PDF 5 differences arise among the two subject groups in terms of the size and number of vortices in the contraction and ejection phases of the cardiac cycle, as illustrated on two representative 0 0 -0.5 0 0.5 -0.5 0 0.5 -0.5 0 0.5 subjects in Fig.(∂5.63. Yang, K. et al. (2021) argue that these differences may ultimately be √n◊ · n (∂ij - ninj)S ij - ninj)Sij ij + √n◊ · n used for diagnostic purposes.

F IGURE 4. The PDFs of stretching (a) and curvature/propagation terms (b) and their sum (c) from 2-D (black) and 3-D data (red) for Ri11.

distribution as compared to the PDF of the stretching term. The PDF of the sum of the two terms (figure 4c) shows characteristics of the PDFs of both ( ij ni nj )S ij and 5.7.4 Global Variational Theory of OECS in 2D Flows vn r · n, in that it has a positive peak and a slightly higher negative tail.

Here, we briefly outline the global variational theory of OECS for 2D flows, obtained as the Time evolution of the TNTI area: a 2-D approach instantaneous limit3.2. of the global variational theory of LCSs we discussed in §5.4. For a more In the following, only results from 2-D data are presented. In figure 5, we show the detailed exposition, we refer to averages Serra andofHaller (2016). time evolution of the spatial the terms in (2.4) for each of the flow cases. Note that, (1/ l)d( l)/dt can be computed locally as a sum of ( ij ni nj )S ij and vn r · n only. However, since we consider average values over the entire box, an estimation of the average of (1/ l)d( l)/dt can be made by taking for l the entire

Elliptic OECS

Over a short time interval [t, t + τ], the local repulsion factor (5.48) of a material line γ, parametrized as x(s) with s ∈ [0, s¯], can be approximated by its Taylor expansion as s (x, x0) = ρt+τ t




hx0, x0i hx0, S (x, t) x0i  = 1 − τ + O τ2 , 0 0 t+τ 0 0 hx , x i x , Ct (x) x

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(5.102)

5.7 Objective Eulerian Coherent Structures (OECSs)

Figure 5.63 IVD-based objective Eulerian vortices in the left ventricular (LV) blood flow of a healthy subject and a heart patient. (a)–(b) isovolumic contraction phase. (c)–(d) rapid ejection period. (e)–(f) slow ejection period. These three phases are indicated within the cardiac cycle and superimposed on ECG scans, as in Fig. 5.25. Adapted from Yang, K. et al. (2021).

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Lagrangian and Objective Eulerian Coherent Structures

where we have used the Taylor expansion (5.97) of the right Cauchy–Green strain tensor at the initial time t0 = t. For a 2D incompressible flow, therefore, the local rate of stretching of the material line is obtained from Eq. (5.102) as Û x0, t) = − q(x,

hx0, S (x, t) x0i d t+τ . ρt (x, x0) τ=0 = hx0, x0i dτ

The averaged material stretching rate along γ at time t is, therefore, ˆ ˆ hx0(s), S (x(s), t) x0(s)i 1 1 Û QÛ (γ) = ds. q(x(s), x0(s), t) ds = hx0(s), x0(s)i s¯ γ s¯ γ

(5.103)

(5.104)

In analogy with the global variational theory of elliptic LCSs in §5.4.1, we seek elliptic OECSs as closed stationary curves of the functional QÛ (γ), i.e., as solutions of the calculus of variations problem δQÛ (γ) = 0 under periodic boundary conditions. We note that Û Û q(x(s), x0(s), t), as defined in Eq. (5.103), has no explicit dependence on s, and q(x(s), · , t) is a positively homogenous function of order d = 0 by the definition (A.32). Therefore, as in §5.4.1, Noether’s theorem from Appendix A.8 again applies and guarantees that Û q(x(s), x0(s), t) ≡ µ = const. is a conserved quantity along solutions of this variational problem. Then, proceeding exactly as in §5.4.1, we obtain the instantaneous limit of Theorem 5.12 as follows: Theorem 5.18 Elliptic OECSs, defined as stationary curves of the average tangential stretching rate functional (5.104), are closed trajectories of the two one-parameter families of direction fields x0(s) = χ±µ (x(s); t), s s s2 (x, t) − µ µ − s1 (x, t) ± χµ (x; t) = e1 (x, t) ± e2 (x, t), s2 (x, t) − s1 (x, t) s2 (x, t) − s1 (x, t)

(5.105)

defined on the time-dependent spatial domain Uµ (t) = {x ∈ U : s1 (x, t) < µ < s2 (x, t)} .

(5.106)

Any such elliptic OECS has a uniform stretching rate, i.e., any of its subsets is stretched precisely by a rate µ at time t. Outermost members of nested families of such closed curves serve as instantaneous positions of objective Eulerian vortex boundaries. Therefore, once the rate-of-strain eigenvalue problem (5.90) has been solved, elliptic OECSs can be computed on a numerical grid from the direction field families (5.105) in the same way as elliptic LCSs are computed from the direction field ηλ± (x0 ) in Eq. (5.54). This time, however, the flow map is no longer involved and the full computation can be carried out using the methods of §§5.4.1 and 5.4.2 from a single snapshot of the velocity field. The null-geodesic-based computation using the approach of §5.4.2 is implemented in Notebook 5.15.

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5.7 Objective Eulerian Coherent Structures (OECSs)

Notebook 5.15 (EllipticOECS) Computes elliptic OECSs as closed null-geodesics of the modified rate-of-strain tensor S(x, t) − µI for a 2D unsteady velocity data set. https://github.com/haller-group/TBarrier/tree/main/TBarrier/2D/ demos/AdvectiveBarriers/EllipticOECS

The results of such a calculation are shown for illustration in Fig. 5.64 for a region of the AVISO data set in the Agulhas regions (see Appendix A.6). The figure also illustrates that under short-term material advection, elliptic OECSs keep their material coherence, given that they were derived as short-term limits of elliptic LCSs. In contrast, during the same time interval the strongest vortices in the region inferred from the (nonobjective) Okubo– Weiss principle (see §3.7.1) undergo substantial deformation under material advection in Fig. 5.64. Serra and Haller (2017b) find that elliptic OECS arising as instantaneous limits of elliptic LCSs can be used to forecast the lifetime of those elliptic LCSs. They develop a nondimensional persistence metric that equals the IVD (see §5.7.3) computed over the elliptic OECS, divided by the instantaneous material flux over the boundaries of the IVD. Serra and Haller (2017b) show that high values of this objective vortex-persistence metric correctly forecast long-lived material vortices based on their initial OECS footprint.

Shearless OECSs and Objective Saddle Points in 2D Flows Over a short time interval [t, t + τ], the local shear factor (5.62) of a material line γ, parametrized as x(s) with s ∈ [0, s¯], can be approximated by its Taylor expansion as (x0 ) x x d t+τ 0 σ ˆ t (x, x ) = q 

dτ t+τ 0 0 0 0 hx , x i x , Ct (x) x

0

, JCt+τ t

0



τ + O τ2




τ=0

=


2 hx , JS(x, t)x i τ + O τ2 , 0 0 hx , x i 0

0

(5.107)

where we have again used the Taylor expansion (5.97) of the right Cauchy–Green strain tensor at the initial time t0 = t. For a 2D incompressible flow, therefore, we obtain the local shearing rate of a material line from Eq. (5.107) as Û x0, t) = p(x,

hx0, [JS (x, t) − S (x, t) J] x0i d t+τ 2 hx0, JS(x, t)x0i σ ˆ t (x, x0) τ=0 = = , (5.108) 0 0 hx , x i hx0, x0i dτ

because the symmetric part of the tensor JS is [JS (x, t) − S (x, t) J] /2. Therefore, the averaged material shearing rate along γ at time t is

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Lagrangian and Objective Eulerian Coherent Structures

Figure 5.64 (Top) Material advection of two families of elliptic OECSs (objective coherent Eulerian vortices) over a period of six days in the Agulhas region of the Southern Ocean. (Bottom) Material advection of the three strongest Okubo–Weiss vortices in the region over the same 6-day period. Adapted from Serra and Haller (2016).

1 PÛ (γ) = s¯

ˆ γ

Û p(x(s), x0(s), t) ds =

1 s¯

ˆ γ

hx0(s), [JS (x, t) − S (x, t) J] x0(s)i ds. hx0(s), x0(s)i

(5.109)

As in §5.4.3 for shearless LCSs, we seek shearless OECSs as stationary curves of the functional PÛ (γ), i.e., as solutions of the calculus of variations problem δ PÛ (γ) = 0 under Û free endpoint conditions. We note that p(x(s), x0(s), t), as defined in Eq. (5.108), has no explicit dependence on s and is positively homogenous function of order d = 0. Therefore Noether’s theorem from Appendix A.8 again applies and ultimately yields the following Eulerian version of Theorem 5.13: Theorem 5.19 Shearless OECSs, defined as stationary curves of the average tangential shearing rate functional PÛ (γ) with identically zero shearing rate, are continuous trajectories of the Eulerian direction field families x0 = ei (x, t) ,

i = 1, 2,

(5.110)

with ei (x, t) denoting the unit eigenvectors of the rate-of-strain tensor S (x, t). The Eulerian tensorlines defined by formula (5.110) can be computed as their Lagrangian counterparts discussed in §5.3.1. This computation is implemented in Notebook 5.16.

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5.7 Objective Eulerian Coherent Structures (OECSs)

235

Notebook 5.16 (HyperbolicOECS) Computes hyperbolic OECSs from tensorlines (shrinklines or stretchlines) of the rate-of-strain tensor for a 2D unsteady velocity data set. https://github.com/haller-group/TBarrier/tree/main/TBarrier/2D/ demos/AdvectiveBarriers/HyperbolicOECS In the present case, however, an alternative computation that does not require constant reorientation of the direction field is also feasible.15 Specifically, along a tensorline x(s), let φi (s) ∈ [0, 2π) denote the angle enclosed by ei (x(s), t) and the positive horizontal axis of the x plane, i.e., let   cos φ(s) 0 . (5.111) x (s) = ei (x(s), t) = sin φ(s) Differentiating the eigenvalue problem S(x(s), t)ei (x(s), t) = si (x(s)) ei (x(s), t)

(5.112)

with respect to s and using the notation e⊥i

:= Dφ ei =



− sin φ(s) cos φ(s)

 ,

(5.113)

we obtain (∇Sei ) ei + Se⊥i φ0 = h∇si , ei i ei + si e⊥i φ0,

(5.114)

φ0 [S − si I] ek = [h∇si , ei i I − ∇Sei ] ei ,

(5.115)

or, equivalently, i , k,

where we have used the fact that e⊥i = ek for i , k. Left-multiplying Eq. (5.115) by eTk then gives the equation

φ0 (sk − si ) = −eTk (∇Sei ) ei ,

i , k,

(5.116)

which, together with Eq. (5.111), then yields x0 = ei (φ),

φ0 =

1 eT (φ) (∇S(x, t)ei (φ)) ei (φ), si (x, t) − sk (x, t) k

i , k,

(5.117)

with no summation implied over repeated indices. Note that away from the tensorline singularities marked by s1 (x) = s2 (x), system (5.117) is a well-defined, 3D system of autonomous differential equations for the variables (x, φ) for any given time t. Therefore, in contrast to the 2D direction field (5.110), the numerical solution of the ODE (5.117) does not require constant monitoring and realignment of its 15

This alternative computation is motivated by the geodesic tensorline calculation discussed in §5.4.2. The algorithm is, in principle, also applicable to the Lagrangian tensorlines discussed in §5.3.1 but it involves the partial derivatives of the underlying tensor field. Those derivatives are inherently noisier for Lagrangian tensor fields near shearless LCSs than for the Eulerian tensor field S(x, t) near shearless OECSs.

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Lagrangian and Objective Eulerian Coherent Structures

right-hand side at orientational discontinuities. At the same time, system (5.117) involves the three-tensor ∇S, whose accurate computation requires sufficiently well resolved velocity data. The computation is implemented in Notebook 5.17. Notebook 5.17 (FastTensorlineComputation) Computes hyperbolic OECSs for a 2D unsteady velocity data set using the ODE (5.117). https://github.com/haller-group/TBarrier/tree/main/TBarrier/2D/ demos/AdvectiveBarriers/FastTensorlineComputation Serra and Haller (2016) follow Farazmand et al. (2014) to identify parabolic OECSs as continuous sets of trajectories of the direction field families that are as close to being instantaneously neutrally stable as possible. Repeating the arguments discussed in §5.4.3 for parabolic LCSs, they use the eigenvalues si (x, t) of S (x, t) and the Eulerian version Nei (x, t) =

p

2 si (x, t) − 1 ,

i,j

(5.118)

of the neutrality measure (5.68) to establish the following parabolic OECS detection principle. Definition 5.20 A parabolic OECS at time t is a shearless OECS composed of alternating chains of e1 - and e2 -line segments that connect wedge and trisector singularities of the rate-of-strain tensor field S (x, t). Furthermore, each ei -line segment in the chain is a weak minimizer of the neutrality measure Nei (x, t).16 An example of a mesoscale parabolic OECS (i.e., Eulerian jet core) is shown in Fig. 5.65, extracted from the AVISO data set (see Appendix A.6). This parabolic OECS is a chain formed by two e1 -line segments and one e2 -line segment (red). The material evolution of this objectively identified Eulerian jet core is illustrated via the material advection of small, circular tracer blobs (with their centers initialized on the parabolic OECS). Note the developing boomerang shape, characteristic of a minimally shearing core of a material jet (see §5.4.3). Figure 5.65 shows that the parabolic OECS behaves nearly as a parabolic LCS for at least six days, even though its initial position was identified from a single snapshot of the surface velocity field at time t. In line with the global variational theory of hyperbolic LCSs in §5.4.3, hyperbolic OECSs (i.e., instantaneously attracting or repelling curve) can be defined as stationary curves of the functional PÛ (γ) under fixed endpoint boundary conditions. Specifically, Serra and Haller (2016) arrive at the following identification principle for hyperbolic OECSs. Definition 5.21 A repelling OECS at time t is an open e1 -line segment that contains a local maximum of the dominant strain eigenvalue s2 (x, t) but contains no other local extremum point of s2 (x, t). Similarly, an attracting OECS at time t is an open e2 -line segment that contains a local minimum of the strain eigenvalue s1 (x, t) but contains no other local extremum point of s1 (x, t). Finally, a hyperbolic OECS is either a repelling or an attracting OECS. 16

As in §5.4.3, a tensorline segment γ is a weak minimizer of its corresponding Nei (x, t) if both γ and the nearest trench of Nei (x, t) lie in the same connected component over which Nei (x, t) is convex.

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5.7 Objective Eulerian Coherent Structures (OECSs)

FIG. 15. (a)–(c) Parabolic OECSs as alternating heteroclinic connections of e1-lines (blue) and e2-lines (red) between trisector-type singularities (blue dots) and wedge-type singularities (black dots). Instantaneous streamlines are shown in the background.

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t + 6 days

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t

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s 0.5 AAAB8nicbVDLSsNAFL2pr1pfVZduBovgqiQi6rLoxmUF+4A2lMl00g6dTMLMjVBCP8ONC0Xc+jXu/BsnbRbaemDgcM69zLknSKQw6LrfTmltfWNzq7xd2dnd2z+oHh61TZxqxlsslrHuBtRwKRRvoUDJu4nmNAok7wSTu9zvPHFtRKwecZpwP6IjJULBKFqpl/UjiuMgJMlsUK25dXcOskq8gtSgQHNQ/eoPY5ZGXCGT1Jie5yboZ1SjYJLPKv3U8ISyCR3xnqWKRtz42TzyjJxZZUjCWNunkMzV3xsZjYyZRoGdzBOaZS8X//N6KYY3fiZUkiJXbPFRmEqCMcnvJ0OhOUM5tYQyLWxWwsZUU4a2pYotwVs+eZW0L+reVd19uKw1bos6ynACp3AOHlxDA+6hCS1gEMMzvMKbg86L8+58LEZLTrFzDH/gfP4ATFSRQw==

1.0

AAAB8nicbVDLSsNAFL2pr1pfVZduBovgqiQi6rLoxmUF+4A2lMl00g6dTMLMjVBCP8ONC0Xc+jXu/BsnbRbaemDgcM69zLknSKQw6LrfTmltfWNzq7xd2dnd2z+oHh61TZxqxlsslrHuBtRwKRRvoUDJu4nmNAok7wSTu9zvPHFtRKwecZpwP6IjJULBKFqpl/UjiuMgJMlsUK25dXcOskq8gtSgQHNQ/eoPY5ZGXCGT1Jie5yboZ1SjYJLPKv3U8ISyCR3xnqWKRtz42TzyjJxZZUjCWNunkMzV3xsZjYyZRoGdzBOaZS8X//N6KYY3fiZUkiJXbPFRmEqCMcnvJ0OhOUM5tYQyLWxWwsZUU4a2pYotwVs+eZW0L+reVd19uKw1bos6ynACp3AOHlxDA+6hCS1gEMMzvMKbg86L8+58LEZLTrFzDH/gfP4ATFSRQw==

0.10

AAAB8nicbVDLSsNAFL2pr1pfVZduBovgqiQi6rLoxmUF+4A2lMl00g6dTMLMjVBCP8ONC0Xc+jXu/BsnbRbaemDgcM69zLknSKQw6LrfTmltfWNzq7xd2dnd2z+oHh61TZxqxlsslrHuBtRwKRRvoUDJu4nmNAok7wSTu9zvPHFtRKwecZpwP6IjJULBKFqpl/UjiuMgJMlsUK25dXcOskq8gtSgQHNQ/eoPY5ZGXCGT1Jie5yboZ1SjYJLPKv3U8ISyCR3xnqWKRtz42TzyjJxZZUjCWNunkMzV3xsZjYyZRoGdzBOaZS8X//N6KYY3fiZUkiJXbPFRmEqCMcnvJ0OhOUM5tYQyLWxWwsZUU4a2pYotwVs+eZW0L+reVd19uKw1bos6ynACp3AOHlxDA+6hCS1gEMMzvMKbg86L8+58LEZLTrFzDH/gfP4ATFSRQw==

0.15

AAAB8nicdVDLSgMxFM3UV62vqks3wSK4GjLj0NZd0Y3LCvYB06Fk0kwbmskMSUYoQz/DjQtF3Po17vwbM20FFT0QOJxzLzn3hClnSiP0YZXW1jc2t8rblZ3dvf2D6uFRVyWZJLRDEp7IfogV5UzQjmaa034qKY5DTnvh9Lrwe/dUKpaIOz1LaRDjsWARI1gbyc8HMdaTMIJqPqzWkH3ZrLteHSIboYbjOgVxG96FBx2jFKiBFdrD6vtglJAspkITjpXyHZTqIMdSM8LpvDLIFE0xmeIx9Q0VOKYqyBeR5/DMKCMYJdI8oeFC/b6R41ipWRyaySKh+u0V4l+en+moGeRMpJmmgiw/ijIOdQKL++GISUo0nxmCiWQmKyQTLDHRpqWKKeHrUvg/6bq2U7fRrVdrXa3qKIMTcArOgQMaoAVuQBt0AAEJeABP4NnS1qP1Yr0uR0vWaucY/ID19gmw8pGI

(b) 0.20

> > > vxxxyy (sp , 0, t) dt = 0 vxxxyy (sp , 0, t) dt = 0 > > > > 8 > > t,0 0) = 0, t0(x , 0) > 0 > > v (x v (x , 0) < 0, v (6.13) xxxyy s xxyy s xxxxyy s > > > Z Z (s , 0) = 0 v < < < xxxyy p t0 +Tp

t0 +T

vxxxxyy (sp , 0, t) dt > 0 vxxxxyy (sp , 0, t) dt > 0 vxxxxyy (sp , 0) > 0 > > to be robustly> observable (structurally stable). These t0 t0higher order derivatives are > > :inv order > > Z t0 +Tp Z t0 +T xxyy (sp , 0) < 0 > > > generally cumbersome > to compute, but Eq. (6.13) is nevertheless conceptually important. > > > > : v (s , 0, t) dt < 0 : 0, t) dt AAAB7HicbVBNS8NAEJ34WetX1aOXYBE8haSI9iIUvXisYNpCG8pmu2mXbjZhdyKU0N/gxYMiXv1B3vw3btsctPXBwOO9GWbmhangGl3321pb39jc2i7tlHf39g8OK0fHLZ1kijKfJiJRnZBoJrhkPnIUrJMqRuJQsHY4vpv57SemNE/kI05SFsRkKHnEKUEj+XhTd2r9StV13DnsVeIVpAoFmv3KV2+Q0CxmEqkgWnc9N8UgJwo5FWxa7mWapYSOyZB1DZUkZjrI58dO7XOjDOwoUaYk2nP190ROYq0ncWg6Y4IjvezNxP+8boZRPci5TDNkki4WRZmwMbFnn9sDrhhFMTGEUMXNrTYdEUUomnzKJgRv+eVV0qo53pXjPlxWG7dFHCU4hTO4AA+uoQH30AQfKHB4hld4s6T1Yr1bH4vWNauYOYE/sD5/ALuCjfo=

0.8 0.8

AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE1ItQ9OKxov2ANpTNdtMu3WzC7kQooT/BiwdFvPqLvPlv3LY5aOuDgcd7M8zMCxIpDLrut1NYWV1b3yhulra2d3b3yvsHTROnmvEGi2Ws2wE1XArFGyhQ8naiOY0CyVvB6Hbqt564NiJWjzhOuB/RgRKhYBSt9IDXbq9ccavuDGSZeDmpQI56r/zV7ccsjbhCJqkxHc9N0M+oRsEkn5S6qeEJZSM64B1LFY248bPZqRNyYpU+CWNtSyGZqb8nMhoZM44C2xlRHJpFbyr+53VSDK/8TKgkRa7YfFGYSoIxmf5N+kJzhnJsCWVa2FsJG1JNGdp0SjYEb/HlZdI8q3oXVff+vFK7yeMowhEcwyl4cAk1uIM6NIDBAJ7hFd4c6bw4787HvLXg5DOH8AfO5w/SAY1+

0 –1.6 –1.4 –1.2 –1.0 –0.8 –0.6 –0.4 –0.2 (b) t = 8.2 (d) 1.0 1.0

AAAB8HicbVDLSgNBEJyNrxhfUY9eBoPgKewGXxch6MVjBPOQZAmzk9lkyMzsMtMrhCVf4cWDIl79HG/+jbPJHjSxoKGo6qa7K4gFN+C6305hZXVtfaO4Wdra3tndK+8ftEyUaMqaNBKR7gTEMMEVawIHwTqxZkQGgrWD8W3mt5+YNjxSDzCJmS/JUPGQUwJWegR8jWvnuFTqlytu1Z0BLxMvJxWUo9Evf/UGEU0kU0AFMabruTH4KdHAqWDTUi8xLCZ0TIasa6kikhk/nR08xSdWGeAw0rYU4Jn6eyIl0piJDGynJDAyi14m/ud1Ewiv/JSrOAGm6HxRmAgMEc6+xwOuGQUxsYRQze2tmI6IJhRsRlkI3uLLy6RVq3oXVff+rFK/yeMooiN0jE6Rhy5RHd2hBmoiiiR6Rq/ozdHOi/PufMxbC04+c4j+wPn8AbnzjmU=

0 0–1.6 –1.4 –1.2 –1.0 –0.8 –0.6 –0.4 –0.2 –1.6 –1.4 –1.2 –1.0 –0.8 –0.6 –0.4 –0.2 (c) (e) t = 18.65 1.0 1.0

AAAB8nicbVBNS8NAEN3Ur1q/qh69LBbBU0hEay9C0YvHCvYD0lA22027dLMJuxOhhP4MLx4U8eqv8ea/cdPmoK0PBh7vzTAzL0gE1+A431ZpbX1jc6u8XdnZ3ds/qB4edXScKsraNBax6gVEM8ElawMHwXqJYiQKBOsGk7vc7z4xpXksH2GaMD8iI8lDTgkYyQN8g92GXb/ClUG15tjOHHiVuAWpoQKtQfWrP4xpGjEJVBCtPddJwM+IAk4Fm1X6qWYJoRMyYp6hkkRM+9n85Bk+M8oQh7EyJQHP1d8TGYm0nkaB6YwIjPWyl4v/eV4KYcPPuExSYJIuFoWpwBDj/H885IpREFNDCFXc3IrpmChCwaSUh+Auv7xKOhe2W7edh8ta87aIo4xO0Ck6Ry66Rk10j1qojSiK0TN6RW8WWC/Wu/WxaC1Zxcwx+gPr8wfkA48K

0.4 0.4

AAAB9XicbVDLSsNAFJ3UV42vqks3g0WoUMpERN0IRTcuK9gHtLFMppN26GQSZiZqCP0PNy4Uceu/uPNvnLRZaOuBC4dz7uXee7yIM6UR+rYKS8srq2vFdXtjc2t7p7S711JhLAltkpCHsuNhRTkTtKmZ5rQTSYoDj9O2N77O/PYDlYqF4k4nEXUDPBTMZwRrI93H/aTyVEVVfXyJbLtfKqMamgIuEicnZZCj0S999QYhiQMqNOFYqa6DIu2mWGpGOJ3YvVjRCJMxHtKuoQIHVLnp9OoJPDLKAPqhNCU0nKq/J1IcKJUEnukMsB6peS8T//O6sfYv3JSJKNZUkNkiP+ZQhzCLAA6YpETzxBBMJDO3QjLCEhNtgspCcOZfXiStk5pzVkO3p+X6VR5HERyAQ1ABDjgHdXADGqAJCJDgGbyCN+vRerHerY9Za8HKZ/bBH1ifP5qFkJ4=

0, 0 0 (6.18)   ˆ t ˆ T ´ t vy (x p ,0,s) ds 0 u xy (x p , 0, t) + uy (x p , 0, t) vxy (x p , 0, s) ds dt < 0. e 0

0

For incompressible flows, u xy = −vyy holds everywhere and u x (x, 0, t) = vy (x, 0, t) ≡ 0 holds along the boundary. Substituting these relationships into the general compressible separation formulas (6.18) gives the simple incompressible separation criterion ˆ T ˆ T uy (x p , 0, t) dt = 0, u xy (x p , 0, t) dt < 0, (6.19) 0

0

which reproduces the classic Prandtl separation criterion (6.10) when v is steady. Therefore, one can replace the steady separation criterion with its time-averaged version for timeperiodic incompressible flows with no-slip boundaries. By Eq. (6.18), the same conclusion does not hold for compressible flows. Haller (2004) and Kilic et al. (2005) also derive polynomial approximations for the timedependent unstable manifold M(t) attached to the separation point (see Fig. 6.6). Specifically, for a time-periodic incompressible flow, the leading order approximation to M(t) is given by a straight, time-periodic line whose slope is given by ´t  ´t  t−T uyy (x p , 0, τ) − 3u xy (x p , 0, t) τ uy (x p , 0, s)ds dτ , (6.20) tan α(t) = − ´t 3 t−T u xy (x p , 0, τ) dτ with α(t) denoting the separation angle shown in Fig. 6.6. Consequently, the angle enclosed by the unstable manifold M(t0 ) of the Poincaré map Pt0 and the wall normal is α(t0 ). This represents a generalization of Lighthill’s formula (6.12) for the incompressible case, agreeing with the expression obtained by Shariff et al. (1991). Weldon et al. (2008) provide an experimental verification of the separation criterion (6.19) and the separation angle formula (6.20). Shown in the upper plot of Fig. 6.7, their experiments

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6.1 Flow Separation in Steady and Recurrent Flows

251

sha1_base64="W9LcDqK/fojmruZe4H9XnNuKidI=">AAAB+XicbVDLSsNAFJ3UV42vqEs3g0Wom5KIqMuiGzdCBfuANpTJdNIOnUzCzE2hhP6JGxeKuPVP3Pk3TtostPXAwOGce7lnTpAIrsF1v63S2vrG5lZ5297Z3ds/cA6PWjpOFWVNGotYdQKimeCSNYGDYJ1EMRIFgrWD8V3utydMaR7LJ5gmzI/IUPKQUwJG6jtOLyIwokRkD7MqnNt236m4NXcOvEq8glRQgUbf+eoNYppGTAIVROuu5ybgZ0QBp4LN7F6qWULomAxZ11BJIqb9bJ58hs+MMsBhrMyTgOfq742MRFpPo8BM5jn1speL/3ndFMIbP+MySYFJujgUpgJDjPMa8IArRkFMDSFUcZMV0xFRhIIpKy/BW/7yKmld1Lyrmvt4WanfFnWU0Qk6RVXkoWtUR/eogZqIogl6Rq/ozcqsF+vd+liMlqxi5xj9gfX5AxbAkqQ=

M(t) AAAB8nicbVBNS8NAEN3Urxq/qh69LBahXkoioh6LXjxWsB+QhrLZbtqlm92wOxFK6c/w4kERr/4ab/4bN20O2vpg4PHeDDPzolRwA5737ZTW1jc2t8rb7s7u3v5B5fCobVSmKWtRJZTuRsQwwSVrAQfBuqlmJIkE60Tju9zvPDFtuJKPMElZmJCh5DGnBKwU9IhIR6QG567br1S9ujcHXiV+QaqoQLNf+eoNFM0SJoEKYkzgeymEU6KBU8Fmbi8zLCV0TIYssFSShJlwOj95hs+sMsCx0rYk4Ln6e2JKEmMmSWQ7EwIjs+zl4n9ekEF8E065TDNgki4WxZnAoHD+Px5wzSiIiSWEam5vxXRENKFgU8pD8JdfXiXti7p/VfceLquN2yKOMjpBp6iGfHSNGugeNVELUaTQM3pFbw44L86787FoLTnFzDH6A+fzB50lkCg=

AAAB8nicbVDLSsNAFL2pr1pfVZduBovgqiQi6rLoxmUF+4A2lMl00g6dTMLMjVBCP8ONC0Xc+jXu/BsnbRbaemDgcM69zLknSKQw6LrfTmltfWNzq7xd2dnd2z+oHh61TZxqxlsslrHuBtRwKRRvoUDJu4nmNAok7wSTu9zvPHFtRKwecZpwP6IjJULBKFqpl/UjiuMgJMlsUK25dXcOskq8gtSgQHNQ/eoPY5ZGXCGT1Jie5yboZ1SjYJLPKv3U8ISyCR3xnqWKRtz42TzyjJxZZUjCWNunkMzV3xsZjYyZRoGdzBOaZS8X//N6KYY3fiZUkiJXbPFRmEqCMcnvJ0OhOUM5tYQyLWxWwsZUU4a2pYotwVs+eZW0L+reVd19uKw1bos6ynACp3AOHlxDA+6hCS1gEMMzvMKbg86L8+58LEZLTrFzDH/gfP4ATFSRQw==

0, t0

accurately locates the periodically varying spiking point xs (t) for material lines initialized parallel to the non-slip wall at time t0 . Both the separation criteria (6.19)–(6.20) and the spiking point criterion (6.21) extend to quasiperiodic flows and even to temporally aperiodic flows with a well-defined asymptotic mean. We will review these extensions in §6.2 and refer the reader to Haller (2004) and Kilic et al. (2005) for more detail. An advantage of the latter, more general criteria is that they do not assume periodicity or quasiperiodicity for the underlying velocity field. As a consequence, they can be applied in any observer frame in which the velocity field has an asymptotic temporal mean.

6.1.3 Separation in 3D Steady Flows Simulations and experiments show that 3D steady flows tend to separate along 2D surfaces (see Simpson, 1996, Délery, 2001 and Surana et al., 2006 for reviews). All such experimentally reproducible separation surfaces are attracting material surfaces, given that they

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252

Flow Separation and Attachment Surfaces as Transport Barriers

Front-facing mirror

s

s

s

Figure 6.7 Experimental verification of the separation location and angle formulas (6.19)–(6.20) for a 2D, time-periodic rotor-oscillator flow. The zero wall-shear point is out of range to the left at the times t = 3.7 s and t = 5.0 s, whereas it is at the location marked by a star at the time t = 2.0 s. Image adapted from Weldon et al. (2008).

are highlighted by an accumulation of dye or smoke, just as the separation surface shown in Fig. 6.8. This figure also underlines how separation surfaces act as advective transport barriers, preventing material incursion or fingering between a separation region and the mean flow. By the definition we have used for observed barriers to advective transport in this chapter, separation surfaces must be 2D unstable manifolds emanating from a 2D flow boundary. It is, however, a priori unclear whose unstable manifolds these 2D surfaces are. Indeed, on a steady free-slip boundary, there are infinitely many candidate trajectories along which an invariant material surface may attach to the boundary. Similarly, along a no-slip boundary, all fluid trajectories are fixed points, and hence an invariant material surface may attach to the boundary along any curve of boundary points. As, free-slip boundary conditions are less relevant for 3D flows, we will focus here on no-slip separation. We consider a steady vector field 

xÛ zÛ



= v(x, z) =



u(x, z) w(x, z)



u(x, y, z) © ª = ­ v(x, y, z) ® , « w(x, y, z) ¬

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w(x, y, 0) = 0,

(6.22)

Annu. Rev. Fluid Mech. 2001.33:129-154. Downloaded from www.annualreviews.org Access provided by ETH- Zurich on 02/26/21. For personal use only.

146

´ DELERY

6.1 Flow Separation in Steady and Recurrent Flows

253

Figure 20 Separation over a blunted body at low Reynolds number.

Figure 6.8 Separation surface as a transport barrier, visualized experimentally by smoke released around a 3D bluff body. Image adopted from Délery (2001).

a horseshoe vortex surrounding the obstacle, as shown in Figure 22. Secondary separations may take place in front of the obstacle, giving rise to a succession of such vortices. (x, z) =region with the coordinates (x, y,exists z) chosen soobstacle that the boundary is described A separated behind the with separation surfaces start- by z = 0. The ing from each side of the cylinder, as shown in Figure 23. Because of large2D skin friction field τ (x) on the boundary is defined as scale fluctuations,which blur the photograph, the structure of the mean flow is difficult to visualize so that the interpreter must rely on fragmentary informaτ (x)topology = ρνuofz (x, (6.23) tion. To arrive at the consistent the 0), skin-friction-line pattern represented in Figure 24, one has to introduce two foci F1 and F2 behind the cylinder where ρ is the fluid ν is the andexperiments uz (x, 0) is theanwall-shear (thesedensity, foci are clearly seenviscosity in wind-tunnel using oil-flow tech-vector field. The nique to visualizeonthethe surface flow; Sedney & Kitchens At least three of the projectile smoke streaks accumulating observed separation line1977). on the surface saddle points, S3, S4, and S5, must exist along with the separation lines (S5), (S6), in Fig. 6.8 provide of the friction lines, trajectories of the and (Sa7).visualization Two other separation linesskin (S3) and (S4) start from defined the cylinderasbase: They are the trace on the plate of the separation surfaces visible in Figure 23. vector field τ (x). The flow on the obstacle itself will not be examined, because of the lack of Based on similar observations, Lighthill thatrepresented the convergence of skinclear experimental evidence. One can(1963) propose proposed the organization in friction lines is aFigure necessary for separation. Hebehind also postulated 25 wherecriterion two tornado-like vortices are present the obstacle, inthat addi- separation lines tionsaddle-type to the horseshoeskin-friction vortices forming zeros ahead ofand the obstacle. They result from thespirals or nodes. always start from terminate at stable rolling up of the separation surfaces emanating from the separation lines located Wang (1972), however, provided examples which none ofFthe converging skin friction on the obstacle, their trace on the plateinbeing the foci F1 and 2. If the obstacle has a saddles finite height, thealso two vortices bentPeake, by the general and entrained lines originate from (see Tobakare and 1982 stream and Yates and Chapman, 1992). downstream.

Wang (1974) described this phenomenon as open separation, characterized by a separation line that starts or ends away from skin-friction zeros. In contrast, Lighthill’s closed separation paradigm, with the separation line connecting skin friction zeros, has since become known as closed separation. The first suggestion that separation surfaces are unstable manifolds appears to be by Wu et al. (1987). Van Dommelen and Cowley (1990) also take a material (Lagrangian) view of flow separation, but primarily in the context of the unsteady boundary layer equations. They propose that the deformation gradient ∇Ftt0 should become unbounded at separation points. As we have already noted in connection with Fig. 6.1, however, such singularities cannot arise in bounded physical flows by the basic properties of the flow map (see §2.2.3). Another approach to separation based on material behavior is that of Wu et al. (2000, 2005), who obtain conditions for the simultaneous convergence and upwelling of fluid elements near general boundaries. These conditions yield close approximations to separation lines and separation slopes for steady flows with linear skin-friction fields and hence are accurate indicators of separation lines near zeros of τ (x).

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Flow Separation and Attachment Surfaces as Transport Barriers

A systematic search for separation surfaces should target structurally stable 2D attracting material surfaces (invariant manifolds) emanating from the z = 0 invariant plane of the dynamical system (6.22).3 To carry out this search, we can employ the same blow-up technique that we used in §6.1.1. In our current setting, the trajectory-dependent time-rescaling (6.7) must be carried out with the wall-normal coordinate as ˆ t t˜ = z(s; x0, y0, z0 ) ds, (6.24) 0

leading to a blown-up version of the velocity field (6.22) in the form   0   x u (x, 0) + O(z) z   . = z0 z 12 wzz (x,0) + O(z)

(6.25)

As in the 2D case, all orbits of this ODE coincide with those of system (6.22) away from the z = 0 boundary. For this reason, one can identify separation surfaces as 2D invariant manifolds (streamsurfaces) of the ODE (6.25) that connect to the boundary. As both the boundary and the separation surfaces are invariant, their intersection, the separation line, must also be invariant: it must be a one-dimensional set composed of trajectories of the ODE (6.25) on the boundary. The blown-up flow on the boundary is x0 = uz (x, 0) =

1 τ (x), ρν

(6.26)

generated by the skin friction field τ (x) acting as a velocity field. Therefore, separation lines are always skin-friction lines, in agreement with the classic observations of Lighthill (1963). Another implication of Eq. (6.26) is that the open separation patterns discovered by Wang (1972) must also take place along surfaces emanating from skin friction lines. Surana et al. (2006) obtain a classification of skin friction lines from which experimentally observable separation or attachment surfaces originate. These lines also include pointtrajectories (i.e., skin-friction zeros) of Eq. (6.26) from which observable one-dimensional separation or attachment curves emanate. Robust observability of these separation profiles in experiments requires more than just their structural stability: the separation lines must also attract nearby skin friction lines at each of their points, not just at their endpoints. Using dynamical systems methods, Surana et al. (2006) prove that only four such robustly observable basic separation patterns exist, as shown in Fig. 6.9. The basic separation patterns in Fig. 6.9 can be located from the analysis of the skin friction field τ (x), or its scaled version, the wall shear field uz (x, 0) (see formula (6.26)). This analysis involves finding heteroclinic saddle-node, saddle-focus and saddle-limit-cycle connections, as well as limit cycles in the wall shear field. As already mentioned, the observability of these special skin friction lines as separation profiles requires that they must strictly attract nearby skin friction lines. Surana et al. (2006) show that this pointwise negativity of the normal attraction rate S⊥ (x0 ) of the skin-friction trajectory x(tˆ; x0 ) running through a point x0 at time tˆ = 0 in the ODE (6.26) can be expressed as 3

As we have already remarked for 2D separation, no such material surface would be structurally stable in the general class of 3D velocity fields due to its degenerate stability type, which arises from the continuous family of fixed points filling the z = 0 plane. A separation surface, however, will be structurally stable within the class of physically relevant velocity field perturbations that preserve the no-slip boundary at z = 0.

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Exact theory of three-dimensional flow separation. Part 1 q p

p

(S1)

(S2)

p

(S4)

(S3)

Figure 6.9 The four robust elementary separation patterns in 3D steady flows near a no-slip boundary, classified based on the corresponding separation line γ. (S1) γ is a skin friction line connecting a saddle p to a focus. (S2) γ is a skin friction line connecting a saddle p to a node q along the direction of weaker attraction at the node. (S3) γ is a skin friction line connecting a saddle to p a limit cycle Γ. (S4) γ ≡ Γ is a limit cycle of the skin friction field. Images adapted from Surana et al. (2006).

S⊥ (x) =

hωwall (x), ∇x τ (x)ωwall (x)i ρν | ωwall (x)| 2

< 0,

(6.27)

where ωwall (x) = ω (x, 0) is the wall-parallel component of the vorticity ω = ∇ × v at the wall. Only the part of a skin friction line γ shown in Fig. 6.9 that satisfies the inequality (6.27) will be observed as a separation line. Surana et al. (2006) also calculate the angle θ(x0 ) that the separation surface encloses with the wall-normal direction at each a point x0 along the separation line. Specifically, for incompressible Navier–Stokes flows with a wall-pressure distribution pwall (x), they obtain the expression 1 tan θ(x0 ) = 2ρν

ˆ

tˆ 1 ∇x pwall (x(tˆ; x0 )), ωwall (x(tˆ; x0 )) d tˆ, e− 0 [ 2 ∇x τ (x(s;x0 ))+S⊥ (x(s;x0 ))] ds ωwall (x(tˆ; x0 )) −∞

0

´

(6.28) involving the skin-friction trajectory x(tˆ; x0 ) with x(0; x0 ) = x0 (see also Surana et al., 2007). This angle can be used to construct a leading-order linear approximation (line bundle) along the separation line to which the separation surface will be tangent. Advecting a short segment of this approximate separation surface forward under the flow map of the original velocity field (6.22) then leads to a global approximation of the separation surface. Attachment surfaces can be handled similarly, as explained by Surana et al. (2006, 2007). Applying these results to a steady lid-driven cavity flow, one obtains the global separation surface shown in Fig. 6.10. The separation lines on the walls of the cavity are reconstructed

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N13 N23 N33 Wall-shear zero N113 , S131 or S132 S113 , S231 or S232

256

Wall-shear zero S134 , S143 N136 , S163 Wall-shear zero S14 N14 N24 N34 Wall-shear zero N16

Flow

(0.74, 0.50) 16.49 (0.31, 0.96) %3:24 (0.03, 0.04) %3:45 Wall 3 "z ! 0# corners "x; y# r x $ @y ! (0.34, 0.00) or (0.34, 1.00) 16.63 (0.83, 0.00) or (0.83, 1.00) 27.96 Separation and Attachment Surfaces as Transport Barriers Wall 3 at intersections with wallSURANA, 4 and 6JACOBS, AND HALLER "x; y# r x $ @x ! (1, 0.50) 3.30 (0, 0.50) 131.51 Wall 4 (x ! 1) "y; z# rx $ ! (0.50, 0.24) %3:89 (0.50, 0.15) 8.03 (0.08, 0.41) %7:24 (0.92, 0.41) %7:15 Wall 6 (x ! 0) "y; z# rx $ ! (0.50, 0.15) 8.03

61.60 2.54 2.84 det r x @y ! %3256:11 %81:12

1299

det r x @x ! %9219:04 3180.26 det r x ! %2:10 15.90 9.09 8.36 det r x ! 15.90

(a)

(b)

(c)

Fig. 13 Approximation to the separation surfaces a) local, for walls 1 and 2, obtained from the slope formula in Eq. (28) and b) global, obtained by advecting the local approximate surfaces in time.

Figure 6.10 Identification of a separation surface in a steady lid-driven cavity flow. (a) Skin friction field (red) with its fixed points (black) and extracted separation lines (green). (b) Local approximations to the separation surfaces from the angle formula (6.28), evaluated along the separation lines. (c) Global separation surfaces obtained from the advection of their local approximations. Adapted from Surana et al. (2007). Fig. 12 Wall-shear fields on walls 1, 4, 5, and 6 for the lid-driven-cavity flow. We also indicate special wall-shear lines (stable and unstable manifolds of the saddles) connecting wall-shear zeros. Among these, the solid lines turn out to be actual separation lines (green) or attachment lines (blue) lines.

from the wall-based skin friction field, shown in Fig. 6.10(a) for one of the walls. The By contrast, the application of critical point theory to the (A2) B"p; 0# ! 0 instantaneous velocity field would suggest that the separation identification of the separation line involves the numerical exploration of saddle-node conlocation is time-varying. For details of the time-dependent version of To understand the topology of limiting streamlines near "p; 0#, we present separation theory, we refer the reader to Surana et al. [32], linearize Eq. (A1) at "p;of 0#. The linearization of the scaled velocity nections by starting skinthe friction along theandunstable direction saddle points who extend relatedtrajectories two-dimensional results by Haller [33] Kilic field (A1) at "p; 0# admits the coefficient matrix: et al. [34]. 0 of the skin friction field τ (x). Once these heteroclinic skin friction trajectories x( tˆ; x0 )0 are1 A"p; 0# 0 (A3) M "p# ! @ @ B"p; 0# @ B"p; 0# @ B"p; 0# A Appendix along the wall can be verified from formula found numerically, their pointwise attraction C"p; 0# 0 0 Fig.given 14 inSame as Fig. 13 velocity but for field walland 3. Using the form Eq. (13) for the (6.27) and a linear approximation to the surface introducing the rescaled time separation s through the relation ds= dt! can be computed along x(tˆ; x0 ) Restricted to the z ! 0 plane, the eigenvalues of M"p# are x"t#z"t#, we find that streamlines satisfy from formula (6.28) close to the wall, as shown in Fig. 6.10(b). Advecting trajectories of A"p; 0# ! @ u"p; 0#; @ B"p; 0# ! @ v"p; 0# (A4) d (A1) y; z# ! "xA; B; zC# the full velocity field from this localds "x;approximate separation surface then yields the global thus, p can only be a saddle or a node for the rescaled Eq. (A1) within the z ! 0 plane. A point p ! "0; p# in the z ! 0 plane is a fixed point for Eq. (A1) if approximate separate surface shown in Fig. 6.10(c). x

3 xxz

y

z

y

3 xyz

6.1.4 Separation in 3D Recurrent Flows Unsteady separation in 3D flows continues to receive attention due to its significance in aerodynamics (Smith, 1986; Ruban et al., 2011; Cassel and Conlisk, 2014). Here, we briefly review extensions of the kinematic theory of steady flow separation in §6.1.3 to flows with recurrent (periodic or quasiperiodic time dependence). These results follow from the more general findings of Surana et al. (2008) for flows with a well-defined temporal mean component, to be discussed in §6.2. Following the approach of Kilic et al. (2005), Surana et al. (2008) construct nonlinear, time-dependent coordinates near the no-slip boundary in which the flow becomes locally

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6.1 Flow Separation in Steady and Recurrent Flows

257

slowly varying. Then, by the theory of averaging (see, e.g., Sanders et al., 2007), robust separation lines in the skin-friction field of the leading-order averaged velocity field give rise to similar, fixed separation lines in the full unsteady flow. While the separation locations obtained in this fashion are fixed curves, the separation profiles emanating from these curves will be deforming material surfaces that inherit the recurrent time-dependence of the velocity field. Specifically, consider an incompressible time-periodic velocity field 

xÛ zÛ



= v(x, z, t) =



u(x, z, t) w(x, z, t)



u(x, y, z, t) © ª = ­ v(x, y, z, t) ® = v(x, z, t + T), « w(x, y, z, t) ¬

(6.29)

with period T > 0 and with a no-slip boundary at z = 0. Leading-order approximations to separation lines on the z = 0 boundary can be identified from the averaged skin friction field ˆ 1 1 T 0 uz (x, 0, t) dt = τ¯ (x). (6.30) x = T 0 ρν In this 2D steady velocity field, the four possible robust separation line patterns are just those shown in Fig. 6.9. The question is then: Under what conditions does such a separation line mark the fixed base-curve of a time-periodic separation surface on the wall? To answer this question, we introduce the time-averaged version of the pointwise normal attraction measure (6.27) for trajectories of Eq. (6.30) as ¯ wall (x), ∇x τ¯ (x)ω ¯ wall (x)i hω S¯⊥ (x) = , ¯ wall (x)| 2 ρν | ω

(6.31)

¯ wall denoting the time-average of the wall vorticity field ω (x, 0, t). We also introduce with ω the leading-order averaged wall-normal stretching rate as ˆ 1 T ¯ wzz (x, 0, t) dt. (6.32) C(x) = T 0 With the help of these two quantities, Surana et al. (2008) prove that a heteroclinic trajectory γ of τ¯ (x) that falls in one of the categories (S1)–(S4) of Fig. 6.9 and satisfies the pointwise conditions ¯ S¯⊥ (x) − C(x) < 0,

¯ C(x) > 0,

x∈γ

(6.33)

is a fixed separation line for a time-periodic separation surface M(t) = M(t + T) in the full velocity field v(x, z, t). Surana et al. (2008) also derive a leading-order linear approximation to M(t) by generalizing the angle formula (6.28) to unsteady flows. As an example, Fig. 6.11 shows separation surfaces identified by Surana et al. (2008) for a 3D cavity flow driven by a time-periodically moving lid. The time-averaged wall-shear fields on two different walls of the cavity are shown on the left, with the separation lines (i.e., heteroclinic orbits of τ¯ (x) satisfying the conditions (6.33) pointwise) shown in green and blue, respectively. On the right plots, we show red and purple trajectories initialized on the two sides of the first-order approximations to the predicted time-periodic separation surfaces. These trajectories are indeed guided near the wall by the approximate separation surfaces.

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Flow Separation and Attachment Surfaces as Transport Barriers

y

Figure 6.11 Identification of fixed unsteady separation with time-periodically moving separation surfaces in a cavity flow with a periodically moving lid. Plots adapted from Surana et al. (2008).

For flows with quasiperiodic time dependence, the 3D velocity field v(x, z, t) can be represented, as in §4.2, in the form     xÛ u(x, z, Ωt) = v(x, z, Ωt) = , (6.34) zÛ w(x, z, Ωt) where the frequency vector Ω = (Ω1, Ω2, . . . , Ωm ) ∈ Rm has m rationally independent components. In this case, the general theory of Surana et al. (2008) for 3D unsteady separation again leads to the analysis of the averaged skin friction field τ¯ (x), defined via the formula ˆ 2π ˆ 2π 1 1 x0 = τ¯ (x). (6.35) · · · uz (x, 0, φ) dφ1 · · · dφm = m (2π) 0 ρν 0 In this steady velocity field, the four robust separation line patterns are again those shown in Fig. 6.9. With the time-averaged (i.e., phase-averaged, as in Eq. (6.30)) normal attraction measure ¯ wall (x), ∇x τ¯ (x)ω ¯ wall (x)i hω S¯⊥ (x) = , (6.36) ¯ wall (x)| 2 ρν | ω and with the leading-order averaged wall-normal stretching rate, ˆ 2π ˆ 2π 1 ¯ C(x) = ··· wzz (x, 0, φ) dφ1 · · · dφm, (2π)m 0 0

(6.37)

the results of Surana et al. (2008) again lead to the separation condition (6.33). Specifically, if this condition holds pointwise along a heteroclinic trajectory γ of formula (6.35) that falls in one of the categories (S1)–(S4), then γ is a fixed separation line for a quasiperiodic separation surface M(Ωt) in the full velocity field v(x, z, t). A leading-order linear approximation to this surface is again provided by the more general formulas of Surana et al. (2008) for flows with

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6.2 Unsteady Flow Separation Created by LCSs

259

a well-defined temporal mean component (see §6.2). Criteria for locating material spiking points and spiking profiles for that general class of flows are also available in Serra et al. (2018) and hence can be applied to time-quasiperiodic flows as well.

6.2 Unsteady Flow Separation Created by LCSs A special class of LCSs connects to flow boundaries and creates flow separation or attachment, as discussed in §6.1. We refer the reader to that section for some general discussion on the kinematic theory of flow separation, as well as for separation criteria that apply to steady and temporally recurrent flows. Here, we discuss extensions of those results to flows with more general time dependence. Separation and attachment at free-slip boundaries is due to hyperbolic (attracting and repelling) LCSs, while the same phenomena at no-slip boundaries arise due to nonhyperbolic (yet structurally stable) LCSs. The connection of these LCSs with the boundary enables their more specific local description that only involves the time history of Eulerian quantities along the boundary.

6.2.1 2D Unsteady Separation No Poincaré maps are available for a 2D unsteady flow with general time-dependence. This prevents us from defining and identifying separation points as fixed points of a Poincaré map with a wall-based unstable manifold, as we did in §6.1.2. Instead, we will be seeking material lines anchored at the wall that remain uniformly bounded from the wall in backward time. Allowing infinitely long backward times in this analysis can lead to a uniquely defined separation profile at the present time. Allowing only a finite backward time analysis will result in nonunique separation profiles that highlight a thin set of material lines, as opposed to a single material line as the separation profile.

Unsteady Separation from Free-Slip Walls Our discussion at the beginning of §6.1.1 on the importance of free-slip boundary conditions is most relevant for temporally aperiodic flows arising in geophysics. As we already noted for time-periodic flows, the instantaneous streamline geometry boundaries in such flows cannot generally be taken as a predictor of material flow separation. As an illustration, Fig. 6.12 shows the formation of a separation spike from the free-slip boundary of a 2D rotating flow in a
circular tank. The stream function of this velocity field4 is given by ψ(x, t) = |x| 2 − 1 (x1 sin ωt + x2 cos ωt) − 12 ω |x| 2 , producing the streamlines shown in dashed black lines for ω > 2. All instantaneous streamlines in the flow domain are closed curves for all times, with no connection to the boundary. Yet the flow displays sharp material separation and reattachment, as Fig. 6.12 confirms, at the boundary points marked in red and blue, respectively. To predict such unexpected separation phenomena, Lekien and Haller (2008) consider unsteady velocity fields of the form 4


This flow is obtained by transforming the steady flow with stream function ψ(x) = |x | 2 − 1 x2 into a frame rotating with angular velocity ω. The transformed stream function can then be obtained directly from formula (3.9).

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097101-2

Flow Separation and Attachment Surfaces as Transport Barriers

Phys. Fluids 20, 097101 !2008"

F. Lekien and G. Haller

1.0

1.0

0.5

0.5

0.0

0.0

-0.5

-0.5

-1.0

-1.0 -1.0

-0.5

0.0

0.5

1.0

-1.0

-0.5

0.0

0.5

1.0

FIG. 1. !Color online" Separation in the absence of hyperbolic stagnation points and without any free streamlines detaching from the boundary. The streamfunction is given by !!x , y , t" = !x2 + y 2 − 1"!x cos "t + y sin "t + " / 2". For " # 2, the streamlines are closed curves !dashed lines" and there is only one elliptic stagnation point oscillating near the origin. Numerical pathlines and analytical solutions reveal flow separation from the boundary, which has neither stagnation points nor detaching streamlines. The figure corresponds to " = 4 !enhanced online".

Figure 6.12 Development of free-slip material separation and reattachment in a rotating tank without the presence of any hyperbolic stagnation point in the flow. Green marks fluid particles advected from a small vicinity of the circular flow boundary. amples, the fluid velocity on the boundary is not prescribed time instance and does not have any stagnation points on the boundary. There isfrom not anyLekien free streamline from !free slip". In many other examples, the global separation of Image adapted anddetaching Haller (2008). the boundary. Nevertheless, a thin material spike starting from the boundary reveals the presence of flow separation.13,14 Examples such as this one underline how separation is inherently a Lagrangian concept: The process is defined by the motion of particles detaching from the boundary. In slowly varying flows, separating lines of particles and free streamlines are sufficiently close to approximate the former with the latter. Examples like the one depicted in Fig. 1 show that for quickly varying flows, actual particle separation may be unrelated to streamlines detaching from the boundary. The same conclusion holds for stagnation points. Hence, we seek a separation criterion 0 that does not rely on instantaneous stagnation points or on streamline geometry. Instead, we concentrate on the motion of particles: Separation is the concentration of particles on and near the boundary, followed by ejection of a material spike inside the flow. Lekien et al.15 computed slip separation and reattachment points from a finite-time Lyapunov exponent analysis. This procedure yields correct breakaway locations, but it relies on the global tracking of fluid particles away from the boundary. In some applications, on-wall detection of separation is preferred, if not required. Moreover, Lyapunov exponent studies do not yield explicit separation criteria in terms of familiar boundary quantities. Despite the lack of a complete theory, unsteady separation on slip boundaries occurs in many applications. Ex0 amples include boundary current separation in oceanic flows,16,17 recirculation in coastal areas,18 wake formation behind moving bubbles,19,20 vortex shedding in the wake of a fish fin21 recirculation in microfluidic devices,22,23 and lift generation on free-flying insects.24 In most of these ex-

xÛ = v(x, t),

the boundary layer of a true no-slip flow can be studied efficiently by switching to a free-slip model. In this paper, we present necessary and sufficient criteria for slip2 separation in two-dimensional unsteady flows. Using the theory of normally ∂D hyperbolic invariant manifolds from dynamical system theory, we derive explicit time-dependent formulas for the separation location and angle. At separation, attracting material lines !unstable manifolds" emerge from the boundary, collecting fluid particles from the vicinity of the boundary, then subsequently ejecting them into the main stream. We obtain similar results for flow reattachment by locating repelling material lines !stable manifolds" that0collect and guide fluid from the main stream to stretch it along the boundary. To illustrate our results, we first consider the twodimensional Rayleigh–Bénard convection model of Solomon and Gollub25 with a periodic motion of the roll pattern. In this case, separation is well understood since the separation points are the invariants of the Poincaré map—a stroboscopic view of particle paths in the system.26 We show that, for the periodic case, our separation theory is equivalent to classical results. We then prescribe random motion for the roll pattern in the Solomon–Gollub model and add Gaussian noise to test the robustness of the separation criteria to perturbations. Our aim is to locate separation from, and reattachment to, horizontal convection cell boundaries. We find that separation and reattachment points differ markedly from the randomly appearing instantaneous stagnation points and from the stagnation points of the mean flow. As a first application, we consider separation profiles 0 computed by Coulliette and Wiggins8 for the quasigeo-

x∈D⊂R,

v|

k ∂D,

where D is a bounded, possibly time dependent, 2D domain with a smooth and compact free-slip boundary ∂D. In this setting, the velocity field is assumed to be known at least on a half-bounded time interval (−∞, t ] for separation analysis, or on an interval [t , ∞) for reattachment analysis. In practice, knowledge of the velocity field on long enough time intervals (such that the improper integrals in our upcoming formulas show signs of convergence) is enough. Let e(x, t) and n(x, t) denote a unit tangent vector and unit boundary normal vector, respectively, to ∂D at a point x at time t, oriented in a way that det [e(x, t), n(x, t)] = 1.

We say that flow separation has been taking place along a boundary trajectory {x(t)} ⊂ ∂D up to time t if: (SS1) x(t) attracts all nearby trajectories within ∂D for t ∈ (−∞, t ]; (SS2) x(t) has a unique, time-dependent unstable manifold (separation profile) that stays uniformly bounded away from the boundary ∂D for all t ∈ (−∞, t0 ]; (SS3) small enough perturbations to the velocity field result in a nearby x(t) satisfying (SS1) and (SS2). Using the theory of normally hyperbolic invariant manifolds (Fenichel, 1971; Mañe, 1978), Lekien and Haller (2008) show that (SS1)–(SS3) hold whenever x(t) satisfies the following separation criterion for ˆ 1 t0 he(x(t), t), S(x(t), t)e(x(t), t)i dt < 0, λe (t0 ) = lim sup T →+∞ T t0 −T (6.38) ˆ 1 t0 hn(x(t), t), S(x(t), t)n(x(t), t)i dt > 0, λn (t0 ) = lim inf T →+∞ T t −T 0

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6.2 Unsteady Flow Separation Created by LCSs 261  where S = 2 ∇v + (∇v)T is the rate of strain tensor. Furthermore, the angle α(t0 ) between the separation profile and the boundary at time t0 satisfies ˆ 2 t0 ´tt0 [he,Sei− hn,Sni][x(s), s] ds he(x(t), t), S(x(t), t)n(x(t), t)i dt. (6.39) e cot α(t0 ) = lim T →+∞ T t −T 0

 1

Similar criteria are available in Lekien and Haller (2008) for attachment on flow boundaries. If v is incompressible, then he, Sei = − hn, Sni, and hence the criteria (6.38) simplify to the single separation criterion λe (t0 ) > 0,

(6.40)

or, equivalently, to λn (t0 ) < 0. Note that
for a steady incompressible flow with a free-slip boundary at y = 0, setting x(t) = x p , 0 to be a stagnation point in criterion (6.40) gives ˆ 1 t0 lim sup u x (x p , 0) dt = u x (x p , 0) < 0, T →+∞ T t0 −T which agrees with the condition obtained in Eq. (6.1). Also note that the criterion (6.38) is objective due to the objectivity of the tensor S. Figure 6.13 shows an application of the above criteria to the detection of moving separation and reattachment points in Monterey Bay along the California coastline (see Lekien and Haller, 2008 for details). The 2D velocity field (shown at select locations via black arrows in Fig 6.13) is reconstructed from high-frequency surface current measurements provided by an array of coastal radars (Paduan and Cook (1997)). The grid spacing in these measurements does not allow for a resolution of the boundary layer, which makes free-slip boundary conditions along the coast line a viable option. In the analysis shown in Fig. 6.13, an integration time of T = 48 hours was used to evaluate λe (t0 ), λn (t0 ) and the separation angle formula (6.39). To counteract the effect of noise in the data, a relaxed criterion of the form λn (t0 ) − λe (t0 ) > 0 was used, which was only satisfied at isolated grid points along the boundary at the available resolution. Forward- and backward-evolving streaklines released near the computed separation and reattachment points confirm the predicted locations and angles. Importantly, the separation and reattachment locations cannot be reliably inferred from instantaneous radar data. Specifically, stagnation points along the coast tend to bifurcate on hourly time scales. They are often completely absent near locations of actual fluid breakaway. When they are present, they often coexist with other instantaneous stagnation points that incorrectly indicate local flow attachment. In contrast, identifying material flow separation using the Lagrangian approach described here provides a robust detection of fluid currents from the coast, relevant for pollution control studies (see Lekien et al., 2005; Coulliette et al., 2007). As an example, Fig. 6.14(a) shows the Duke power plant (larger red circle) in Monterey Bay, California, as well as the outlet of its warm-water exhaust pipe and plume (smaller red circle) that extends 200 m off the beach. Figure 6.14(b) shows a snapshot of FTLE ridges extracted from the surface-radar-based velocity field. A black dot marks the time-varying intersection of a strong FTLE ridge with the line of the pipe (dashed). This ridge highlights the main reattachment profile shown in Fig. 6.13, connected to the southern tip of the bay. The instantaneous position of the black dot

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097101-14

Flow and Attachment Surfaces as Transport F. Lekien Separation and G. Haller Phys.Barriers Fluids 20, 097101 !2008"

FIG. 14. !Color online" Separation and reattachment points in Monterey Bay. The upper panels show the separation point near Santa Cruz as well as

fluid flows with free-slip boundary conditions. Based on velocities and velocity derivatives along the boundary, our formulas predict Lagrangian separation location and angle. Only quantities on the boundary must be measured or modeled to use our criteria; they do not use interior information. The Lagrangian view underlying our criteria differs from the traditional Eulerian view that considers instantaneous stagnation points as points of separation or make use of stagnation points to locate separation. Through examples, we have shown that the Lagrangian view is preferable, especially when several transient stagnation points evolve and bifurcate along the boundary; yet, no actual fluid breakaway is generated by most of them. Likewise, our criteria perform well in cases where separation occurs in the absence of persistent stagnation points. Furthermore, the theory and criteria

!""#$%&$

streaklines released near the coast. The panels at the bottom show, in addition, the reattachment point and streamlines obtained from a backward-time Figure Unsteady free-slip and Peninsula. free-slip reattachment in Monterey integration.6.13 In this case, the base of the reattachment profile separation wanders around the Monterey Bay, California. (Top) A moving separation point near Santa Cruz, confirmed by streaklines (green) released nearby. Black arrows mark instantaneous velocity vectors. (Bottom) Same, with a moving reattachment point along the Monterey Peninsula IX. CONCLUSIONS added and with streaklines obtained from backward-time integration. Image adapted We have presented necessary and sufficient criteria for from Lekien andandHaller (2008). unsteady separation reattachment for two-dimensional

relative to the opening of the pipe along the line of the pipe determines whether the material released from the pipe at that time will recirculate in the bay or escape to the open ocean. Figure 6.14(c) shows a white and a black parcel released at times when the orifice is on two different sides of the black ridge-intersections point. The lower right plot then shows how this passive pollution control strategy indeed makes a difference in the simulated evolution of the white and black material parcels. Specifically, the white parcel corresponds to the FIG. 15. !Color online" Sketch illustrating the geometry of transport in Bay based on the separation and reattachment profiles.is undesirably desired clearance of the pollution to the open Monterey ocean, while the black parcel recirculated by surface currents in the bay.

Fixed Unsteady Separation from No-Slip Boundaries For the classic unsteady flow separation problem with no-slip walls, the hyperbolicity-based approach of §6.2.1 is inapplicable. In the absence of relevant invariant manifold results, one is forced to rethink what exactly distinguishes a material separation profile from other material curves attached to the boundary. We will focus here on deconstructing fixed separation created by an attracting material line anchored to a no-slip boundary at a distinguished separation point. In a forward-time movie of an imagined separation experiment, all nearby wall-anchored material lines would converge

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263

6.2 Unsteady Flow Separation Created by LCSs FIGURE 1. Locations of three CODAR SeaSonde HF radar systems around Monterey Bay. Top, bottom, and right photographs show the HF radar antennae at Santa Cruz, Point Pinos, and Moss Landing that were used to measure the current data. Also shown are the footprint at 08:00 GMT on August 8, 2000 (4-6) and bottom topography contours at various depths.

(a)

(b) FTLE max(FTLE)

FIGURE 2. Aerial view of the Elkhorn Slough and the Duke power plant (right circle on the photograph). The plant exhausts warm water through a pipe that extends 200 m off the beach. The circle on the left of the photograph indicates the outlet of the pipe and the plume. release of pollution can be contained until an appropriate release time. Higher frequencies are typically necessary for smaller regions (21).

FIGURE pipeline carries contaminants to be released in the bay from the Moss Landing area. Also shown is the Bay since 1994 (6). In our study, 6. we A usehypothetical data from this intersection point of a peak in the LCS field (i.e., Lt peak) and the axis of the pipeline. installation, acquired by theinstantaneous three HF radar antennas (shown in Figure 1), binned every hour on a horizontal uniform grid with 1 km by 1 km intervals. An example of an HF radar that the reason for their different future behaviors is the footprint of the bay at 08:00 GMT, August 8, 2000, is shown peak difference in their initial position relative to the curve of The Elkhorn Slough and the Duke Energy Moss Landing in Figure 1. t

(c)

(d)

it was already obvious from Figure 7, this means that, Optimal PollutantAs Release Times on one side of during the 22 days observed, the LFigure 7.stays Our analysis makes use of high-frequency (HF) radar In particular, the white parcel exits the bay quickly Power Plant areBay both theabout Moss Landing Harbor,before it crosses The surface current patterns in Monterey are located part of near technology (4-6), which is now able to resolve timethe outlet for 4.3 days because it enters to the the flow other when the point of release lies which is on the eastern a dynamic upwelling system dominated by along-shore wind shore of Monterey Bay. Both dependent Eulerian flow features in coastal surface currents. of Figure the Lt peak (curve side. the Bay. 8.6 day periodbetween was computed from 22 7) and the edge of the forcing. The counterclockwise circulation shown Note in Such an HF radar installation has been operating in Monterey contribute topattern pollutants enteringthat Monterey pipeline (horizontal line on Figure 7). An important days consequence of the analysis is that of data. Itabove is consistent with the major wind reversals VOL. 41, NO. 18, 2007 / ENVIRONMENTAL SCIENCE & TECHNOLOGY 6563 Notice that Figure 7 proves the existence of time intervals it demonstrates the existence of time intervals where released observed during this data collection period, but will most where pollutants are quickly advected outside the bay. The contaminants have either a high or low impact on the change on seasonal inour winds, currents objective of pollution release algorithm is to maximize environment. Ourlikely objective is to showbased that a pollution control changes release these time windows and to store algorithm basedoutside on a nonlinear dynamic the bay, andanalysis other with factorspollutant driving flowduring in the bay. pollutants in a tank outside these intervals. Lyapunov exponents can achieve a significant reduction in To complete our area, prediction the impact of a contaminant in a coastal without procedure, we used all the Real-Time Coastal Pollution Management frequencies of the spectrum of this curve to predict reducing the totalsignificant amount of contaminants released. To facilitate the discussion, we consider an exhaust the location of L peak along pipe the axis of on thethepipeline the section, it is tempting Based analysis in into the previous similar to that mentioned in Figure 2,t which may carry to think thatof thethe intersection curve in Figure 7 predicts times near future. The amplitudes and phases prediction pollution (e.g., chemical, thermal) from Moss Landing and of pollution release that will lead to a quick exit from the bay. the Elkhorn Slough to anare offshore release site shown in curve determined by minimizing the release normpollution of the Why not simply when the curve indicating Figure 6. difference (i.e., the integral of the squared between is well above the horizontal line marking the outlet the Lt peakdifference) Although building a pipeline is not necessary for our of the pipeline? As inFigure the case of the white parcel, such a fitted and real-time L values. The left panel of 11 t method, it is necessary to have some control over the release release would certainly guarantee that the contaminant is peak time and location of the pollutants. Thus to L expedite our shows the predicted together witheast the actual and the t initially of the stable LCS and hence leaves the bay quickly explanation we will imagine a pipeline which carries the peak FIGURE 3. Evolution of two parcels of contaminants released from the same position near Moss Landing at 22:00 August 6, 2000 of the as how it will faithfully approach thethe Monterey Peninsula west real-time locations of Ltoffshore .with Note pre- GMT, FIGURE 7. Oscillations of Lt peak along the axis of the pipeline. The and from contaminants the Moss areaplotted to antogether (black) 09:00 GMT, August Landing 7, 2000 (white), the snapshot of surface currents observed at 08:00 GMT on August separation linethat near Point Pinos. dicted curve reproduces the main ofthethe actual 8,The 2000. motion of thelocation two parcels is shown through daily snapshots over features 8 days. Note black parcel remains in the bay, while release site The at the same that the black and white zero reference time corresponds to 07:00 GMT, August 1, 2000. The above method is flawed for practical applications, thewere whitereleased. parcel departs from the bay. pipeline and release parcels Thisoscillations. hypothetical Lt peak horizontal line marks the location of the outlet of the pipeline. The because any point of the Lt peak curve in Figure 7 is are shown in Figure 6. black and white squares represent the release time andlocation release Figure 1, including strong ajet-like flow from north to without any possible correlation with a filtering method or h. In constructed frompredicts future velocity data over the next 200 particular, the left panel of Figure 11 that For any given time,In wethe consider portion of the previously south across the mouth of the Bay, is representative of the a model. Onceto the existence of and a flow structure has been other words, predict when where to release pollution longitude of the parcels featured in Figure 3. discussed LCS asreleasing it ascends along the coastline of from the bay contaminants pipeline between and currents under strong, upwelling-favorable (from the north- theestablished forwe unfiltered data,3knowledge modal analysis techniques on Monday, would need of the currents in the from west) Mosswind Landing, meandering past Santa Cruz. The conditions. Such winds arepresent common, particularly can be used to increase smoothness measurements 110 hcauses from the time (22:00 6,the2000) willof the bayGMT, up untilAug approximately Tuesday of the following week. meandering of summer the LCS months. itHowever, to intersect theofaxis the peak the periods 3-5ofdays and structures. constant 200 h future window) and the real-timepipeline Lt during Such future data is clearly thet) time when a cause of generally the pollution to Monterey Bay without in several points.most These points can of upwelling favorable winds intersection are followed bybe a exit The HF radar data gives theunavailable velocity fieldatv(x, but we decision haswith to be made. Trying tobased predict theresulting velocity field counted by period following the LCS, starting from as itsrelaxation coastal shorter of weak or reversed winds concerned making deductions on the (computed with a shrinking future window). The actual and recirculation. On known the other hand,are pollution released after in the more than 3 days be unrealistic, periods.point. DuringWe relaxation surface currents ), i.e., the solution of eqmight 1 that satisfies x(t0) ) or at flow x(t;bay t0, x0for attachment refer toperiods, the firstthe intersection pointare as the real-time Lt peak locations, as functions of time, are 110 h is not expected to leave the bay immediately due to peakgenerally least very difficult, because of the spatial and temporal weaker and less organized and they often exhibit x0, where t0 and x0 are the initial time and position of the (t0). Lt peak a narrow of south-to-north across the mouth complexity of the flow. Instead, trajectory. plotted in Figure 8. curve into longitudes on we propose a focused excursion of the as actual Lof The end of band thethe pipe is at the sameflow location the release t Monterey Bay. and black parcels featured in Figure 3. The temporal complexity of the currents becomes evident Lagrangian prediction. site for the white the side of the pipe outlet. from In this case, the algorithm actual The real-time Lt(t0, x0) peak curve approximates the To connect withcoastal the vast literature on dynamical systems, tracking of a fluid parcel (a model As a first different step, weevolutions modify our calculation of Lt(t0, x0). We Figure 6 also shows the instantaneous intersection of the that the availability of measured velocities in the bay prepare for a blob of contaminant) released at the same precise should wait for about 3 h and to release pollutants (200 h) curve with an error less than, approximately, stable 10%notice up fix t ) 22:00 GMT, Aug 6, 2000 as today, or the “present and the axis LCS (revealed the Lon t field) removes the needby foraaridge modelofbased partial differential location, but at slightly different times. We show the results and empty the holding tank for a period ofnumerical about 107 Not time”, when we would likeh. to make our prediction. For any of the hypothetical to 8 h before the “present time.” During the last 8 h, the error equations. If thepipeline. position of a fluid particle in Monterey Bay of two such experiments in Figure 3. peak , we calculate the peak of thethe Lt fluid ridge; this earlier time t The motion of the intersection point along the axis of the referred to only as a vector x, it obeys ordinary differential does the the algorithm predict whether or0 not to release Using available HF velocity data, we advected becomes prohibitive. Note that isthe on the predicted Lt means that the future window our computation is gradually hypothetical equationpipeline is complicated, which is evident from algorithm particles using a fourth orderinRunge-Kutta pollution, but it also provides an estimate of the length of thehistory of the intersection location in Figure 7. combined inserts in Figure 8 show slices of the Lt contours along theand present time t. As shrinkingwith to tricubic zero asinterpolation t0 approaches in space time. These the time hence, also sets the rate which particle trajectories are used approximate flow map, slow) expected, this results inat a to gradual (albeitthe surprisingly onthe this discharge plot are timesand, and release x3 )the v(x,release t)period (1) hours. axis of the pipeline at t - t0 ) 20 hours and t - t0 ) 100Superimposed peak positions, x. which associates positions, x0, toL final (computed with a growth of error initial between the actual location is of the white black parcels be of Figure 3. Recall the and tank should emptied. t We observe from inserts that the position of the barrier These numerical algorithms have been compiled into a where v(x, t) is the velocity at time t and position x. The form software package called ManGen (http://www.lekien.com/ identified by a sharp ridge. It is best to identify the ridgeofnot On the left panel of Figure the next predicted “red” 41, NO. 18, 2007 / ENVIRONMENTAL SCIENCE & TECHNOLOGY 6567 eq 1 is a time-dependent dynamical system (11) and 11,VOL. ∼francois/software/mangen). demonstratesperiod the connection between velocity by its maximum magnitude, but by the gradient in a direction starts 110 ahmeasured from the present time, while thecontaminant actual parcel remains in Figure 3 shows that one field and the vast literature on dynamical systems techniques. whereas other parcel exits the bay and moves that is approximately orthogonal to the Lt ridge. The Lt ridge red period outa to start hbay, after thethe present time. Rather than modeling, we are turned demonstrating method for 118the immediately toward the open ocean. The latter scenario (the analyzing Lagrangian trajectoriesthat computed anyin velocity intersects the axis of our pipeline in a nearly orthogonal This means the from error predicting the end of the release white parcel on Figure 3) is highly desirable, because it field: measured, modeled, or assimilated. In this article, the direction, so we will use the axis of the pipeline to examine minimizes the impactof of the contaminant, causing it to be was by approximately 8 h with a horizon 4 days. velocity field,interval v(x, t), is provided the high-frequency (HF) safely dispersed in the open ocean. This observation inspires of near-surface currents in Monterey the measurements the gradient of the Lt ridge. In Figure 9, we examine radar To illustrate the efficacy of the above pollution us to understand and predictrelease different evolution patterns of Bay. maximum value of the ridge and the gradient of the Lt ridge the same fluid parcel, dependingfor on its Since thescheme, velocity datawe are repeated measured, there some prediction theis same procedure a initial location and time of release. measurement error, as well as vectors that could not be as a function of the time used to compute the Lt ridge. Note different “present time”, t ) 20:00 GMT, August 17, 2000. in some areas or at some times. Various techniques left that the maximum value of the Lt ridge shown in the resolved Lagrangian Coherent Structures The left panel of Figure shows that similar performances such as Open-boundary Modal Analysis (OMA)11 (22) are To understand the evolution of fluid parcels, we use a available filtering, interpolating, and extrapolating this panel of Figure 9 displays no behavior which indicates a for are achieved. In this case, correctly predicts p the algorithm geometric description of mixing from nonlinear dynamical data. Ocean modeling and data assimilation schemes have clear choice for the computational time needed to evolve that LCS is too far east and that should be systemspollutants theory. Autonomous and time-periodic fluid flows also proved to be an the adequate source of dynamical systems the Lt contours. However, the gradient of the Lt is more have long been known to produce chaotic advection (25), (23, 24). redirected to the holding tank. It also predicts that the next i.e., irregular stirring of fluid parcels. Instrumental in this We chose to use HF radar data without any filtering, useful. During the first 8 h, the longitudinal component of interval is 60ofhthisfrom present time. are stable and unstable manifolds of hyperbolic fluid interpolation,“green” or extrapolation. The objective work isthe stirring the Lt gradient increases linearly with the time usedto to trajectories (26). These structures are material curves formed extract and use the coherent structures from the data, It is worth noticing that, in the second case, the period compute the Lt contours. After the first 8 h, the magnitude 6564pENVIRONMENTAL SCIENCE & TECHNOLOGY / VOL. 41, NO. 18, 2007 peak oscillation was 9.26 days. of the dominant mode in the Lt of the longitudinal component of the gradient begins to The difference in oscillation wavelength (8.6 days for t0 ) oscillate due to nonlinear effects. Thus the minimum 22:00 GMT, August 6, 2000 and 9.26 days for t0 ) 20:00 GMT, integration time which provides a well-defined Lt ridge is August 17, 2000) is to be expected since this flow is highly approximately 8 h, which matches the previous qualitative time-dependent. This is evidence that a static analysis of the observation. The magnitude of the longitudinal component flow will never be sufficient to make predictions about of the gradient may still increase after 8 h, but its growth is Lagrangian transport in1Monterey Bay. A nonlinear analysis 1 no longer linear in time, and thus additional computational of real-time current measurements such as that described time is not as beneficial after the first 8 h. Consequently, we in this article is necessary.y y 8 h before the need to stop our real-time Lt calculation about More generally, the “present time” to take advantage of the 0 steep increase in the 0 prediction method described above determines environmentally friendly future time windows. gradient during that time. These windows last for about 100 h, over which most of the As a second step, we identify the main frequency pollution released from the pipeline will advect toward the components of the real-time Lt peak curve over the shortened open ocean. We marked the bottom of Figure 11 with green time interval [t0, t - 8 h]. Shown in Figure 10, the power bars for time periods that result in the pollution exiting the spectrum densityt of the real-time Lt peak curve highlights p p y release p times that cause the pollution bay and with red bars for seven dominant frequency components, with the importance to remain within the bay. of each frequency determined by the area under the corresponding peak in the spectrum. Surprisingly, the most Discussion influential component in this particular time interval is not From the simulations and predictions presented in the the tidal oscillation (with a period of 24 h) or any of its previous section, the following general principles emerge: harmonics, but rather a component with a period of 8.6 days. High-Frequency Radar Measurements

9

Figure 6.14 (a) Aerial view of the Elkhorn Slough and the Duke power plant and the outlet of its warm-water exhaust pipe. (b) Forward FTLE map of Monterey Bay, with ridges highlighting reattachment profiles relevant for pollution control via timed release. (c) Judiciously chosen release times for a white and a black parcel from the pipe. (d) Simulated evolution of the white and black parcels based on the surface current data. Image adapted from Coulliette et al. (2007).

to the separation profile and hence to each other, too. Therefore, the forward-asymptotic behavior of the separation profile is in no way distinguished. In a backward time movie of the same separation event, however, the separation profile would repel all wall-anchored nearby material lines asymptotically towards the wall, as shown in Fig. 6.15. As a result, the separation profile is locally the only wall-anchored material line enclosing an angle with the boundary that remains uniformly bounded away from zero. Any time-evolving material line, attached at a point p = (x , 0) to a no-slip wall along the y = 0 axis, can be written in the form x = x + yF(y, t) 9

(6.41)

for some smooth function F(y, t). Differentiating this expression with respect to time, using the velocity field v(x, t) = (u(x, y, t), v(x, y, t)) and defining the quantities ˆ ˆ A(x, y, t) = u (x, sy, t) ds, B(x, y, t) = v (x, sy, t) ds, (6.42) we obtain a PDE for F in the form

F = A(x + yF, y, t) − B(x + yF, y, t)F − yF B(x + yF, y, t).

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(6.43)

9

x

γ

264

Figure 2. Unsteady separation profile viewed as a time-dependent material line that guides particles away from the wall.

Flow Separation and Attachment Surfaces as Transport Barriers y "(t0)

! (t0)

"(t0 – ∆t) # (t0)

"(t0 – 2∆t)

# (t0 – ∆t)

"(t0 – 3∆t)

pγ

# (t0 – 2∆t) # (t0 – 3∆t) x

Figure 3. Behaviour of typical material lines in backward time near a separation profile M(t).

N(t)Backward-time attaches to the wall away from the separation point. material The line L(t) to The line6.15 Figure behavior of two wall-anchored lines,attaches L(t) and at the separation but does not coincide with M(t) the separation Nthe (t),wall initialized near apoint, material separation profile at timeprofile. t0 . L(t) is also anchored at the separation point p as M(t) is, whereas N (t) is anchored at a nearby boundary Adapted from Haller (2004). 2. Fixed point. unsteady separation

2.1. Set-up Consider a two-dimensional field v(x, = PDE (u(x, y, t),stays v(x, y,uniformly t)), with the We seek the separation profile as avelocity solution F(y, t) y, to t)this that bounded induced motiontime satisfying away from the fluid wall particle in backward for an appropriate choice of the separation location x p

along the wall. This approach can carried x˙ = be u(x, y, t), outy˙by = finding v(x, y, t).an x p for which the coefficients (2.1) ∂i fi (t) = ∂y i F(0, t) in the Taylor expansion

Assume further that a boundary is present in the flow at y = 0 with the no-slip boundary conditions 1 F(y, t) =u(x, f0 (t) + f (t)y + = f2 (t)y 2 + · · · (6.44) 0, t) =1 v(x, 0, t) (2.2) 2 0.

We seek a time-dependent material line M(t) – the separation – thatinto collects remain uniformly bounded in backward time. Substituting this profile expansion the PDE and ejects fluid particles from a vicinity of the boundary (figure 2). As a material line, (6.43), Haller (2004) finds that for a mass-conserving flow with density ρ(x, y, t), the uniform M(t) is anchored to the same boundary point γ for all times owing to the no-slip boundedness of f (t) over a time interval (−∞, t ] can only arise at a location x that satisfies i p boundary conditions. In dynamical systems0 terms, M(t) is an unstable manifold for the conditions a fixed point of the y = 0 boundary. ˆ t weumean theτ)type of separation induced by M(t). By fixed unsteady flow separation y (x p , 0, dτ < ∞, (6.45) limaresup Specifically, fluid particles ejected along, 0,M(t) ρ(x τ) in the form of a thin spike from t→−∞ p t

the vicinity of the boundary into the0 main stream. The spike aligns with M(t), whose ˆ shape ˆ τ remains fixed. As shown in −∞  changes in time, but whose point of attachment u xy (x p , 0, τ) − vyy (x p , 0, τ) uy (x p , 0, s) figure 3, generic material lines emanating theτ)boundary convergedsto the − 2vxyfrom (x p , 0, dτboundary =∞ (6.46) ρ(x punsteady , 0, τ) separation profiles, sucht0as ρ(x s) exceptions to this p , 0,are M(t), t0as t → −∞. Fixed rule, and this property renders them explicitly computable, as we shall see below.

for some choice of the present time t0 . In other words, by condition (6.45), backward-time averages of the density-weighted wall shear need not converge but must remain bounded at the separation location. In addition, by condition (6.46), the backward-time integral of the density-weighted, modified wall-shear gradient must converge to infinity. In 2D flows, these two conditions represent the generalization of Prandtl’s steady, incompressible separation condition for fixed separation in unsteady, mass-conserving flows with arbitrary time dependence. Indeed, Haller (2004) shows that the general conditions (6.45)– (6.46) imply all the more specific 2D separation conditions we have already discussed in §6.1, assuming incompressibility, steadiness, time-periodicity or time-quasiperiodicity. Haller (2004) also obtains formulas for the derivatives fi (t) of the separation profile at the wall. These formulas again follow from the requirement that these derivatives must be bounded. In the incompressible case, the leading-order formula, ´ t0  ´t  uyy (x p , 0, τ) − 3u xy (x p , 0, τ) τ 0 uy (x p , 0, s) ds dτ t f0 (t 0 ) = − lim , (6.47) ´t t→−∞ 3 t 0 u xy (x p , 0, τ) dτ

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6.2 Unsteady Flow Separation Created by LCSs

265

provides a generalization of Lighthill’s classic expression (6.12) for the separation slope relative to the wall-normal direction. While the formulas (6.45)–(6.47) are general enough to capture fixed unsteady separation in 2D flows, they require integration over infinite time intervals and hence are impractical on data. Kilic et al. (2005) show that the formulas become more directly computable for mass-conserving velocity fields with a well-defined steady temporal mean component5   ˆ 1 t0 u(x, ¯ y) v¯ (x) = := lim v(x, t) dt. (6.48) T →∞ T t −T v¯ (x, y) 0 For this class of flows, the mathematical theory of averaging (see, e.g., Sanders et al., 2007) becomes applicable near the y = 0 boundary. Indeed, under the assumption (6.48) of an asymptotic mean velocity field, the velocity field will oscillate slowly around this mean near the wall due to the near-zero velocities arising from the no-slip boundary condition.6 Specifically, by the theorem of averaging, a local nonlinear coordinate change x = (x, y) 7→ ξ = (ξ, η) with   1 exists near the y = 0 no-slip wall such that in these new coordinates, any mass-conserving velocity field becomes

ξÛ = f 0 (ξ ) +  2 f 1 (ξ, t) + O( 3 ). For incompressible flows, the leading-order term in this expansion simplifies to   ηu¯y (ξ, 0) ξÛ = f 0 (ξ ) =  , −η2 u¯ xy (ξ, 0)

(6.49)

(6.50)

but the exact form of f 0 (ξ ) for compressible flows is also computable. Kilic et al. (2005) prove that a separation point identified in the steady leading-order flow (6.50) via the classic Prandtl criterion u¯y (x p , 0) = 0,

u¯ xy (x p , 0) < 0

(6.51)

is also the anchor point of a true time-dependent separation profile in the full unsteady flow generated by v(x, t). Applying higher-order averaging (see Sanders et al., 2007), Kilic et al. (2005) also removes the time-dependence in the higher-order terms in (6.49) by successive local nonlinear coordinate changes near the no-slip wall. The hierarchy of steady flows obtained in this fashion gives increasingly accurate polynomial expansions for the separation profile in terms of the original (x, y)-coordinates. For instance, the general separation slope formula (6.47) simplifies to ´t u¯yy (x p , 0) − 3u xy (x p , 0, t) t 0 uy (x p , 0, s) ds , (6.52) f0 (t0 ) = − 3u¯ xy (x p , 0) with the overbar denoting asymptotic averaging, as in Eq. (6.48). 5

6

´t   A further technical requirement is that the average deviation from this mean, ∆(x, t) = t 0 v(x, s) − v0 (x) ds, as well as the first three spatial derivatives of ∆(x, t), must remain uniformly bounded for t ≤ t0 on bounded domains. For this local slow variation to hold near the no-slip boundary, the velocity field may be compressible but must conserve mass, i.e., satisfy the equation of continuity ρt + ∇ · (ρv) = 0 for a smooth density field ρ(x, t). See Kilic et al. (2005) for details.

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Flow Separation and Attachment Surfaces as Transport Barriers

(mm)

Weldon et al. (2008) verify the separation formulas (6.51)–(6.52) by imposing quasiperiodic motion on the cylinder in the rotor-oscillator flow experiment shown in Fig. 6.7. As in their time-periodic experiment, they evaluate the formulas (6.51)–(6.52) in a direct numerical simulation of the flow and superimpose the predicted separation location at different times on snapshots of an observed dye spike near the wall of the tank. In this experiment, the cylinder rotated at a rate of 20.89 rad/s, while being moved quasiperiodically with 20 incommensurate frequencies with random phases and a common amplitude. A sample of the resulting cylinder motion is shown in the upper plot of Fig. 6.16, designed to mimic chaotic time dependence in the velocity field. The lower plots in the same figure confirm the accuracy of the theory of Kilic et al. (2005) and underline again the irrelevance of instantaneous zeros of the wall shear field for unsteady separation. 35 40 25 20

0

Time (s)

100

γ (mm)

Rotor position (mm)

30

10

experiments numerics

35

0 30 10 0

60 Time (s)

80

100

0 5 Root mean square oscillation amplitude (mm)

20

20

20

y (mm)

40

of unsteady separation 9 (main plot line) and FigureInvestigation 6. (a) 5.97 mm r.m.s. amplitude quasi-periodic rotor location of the fixed separation location, γ (inset); ( b) the fixed separation location, t = 52 3convergence s t = 73 0 s t = 79 0 s ( b ) ( c) a function of the r.m.s. amplitude of oscillation for quasi-periodic forcing.

(a)

the shape of the experimental spike and the linear prediction diverge within a few mm of the wall, though the wall prediction of the separation location remains valid. Finally, an important point to raise is the location of the instantaneous zero-skinfriction point 10 Its location for the 24 mm oscillations is 10 during these experiments. indicated in figure 5 using ( ) for the exact location or ( ) to point to it ifit falls outside the field of view, which is often the case. It is thus clear that the location of the instantaneous zero-skin-friction point does not provide a reliable indicator of the location of separation for these experiments.

10

0

20

25

0

0 Quasi-periodic 304.3. 35 40 x (mm)

and 25 random 30 35forcing 40 x (mm) Quasi-periodic forcing was applied by rotating the cylinder at 20.89 rad s

30 35 x (mm)

40

25

Figure 6.16 Experimental verification of the separation location and angle formulas (6.51)–(6.52) for a 2D temporally quasiperiodic rotor-oscillator flow. A star symbol indicates the location of the instantaneous wall shear zero (whenever in range), which is often believed to be the separation point in unsteady flows as well. Adapted from Weldon et al. (2008).

The general criteria (6.51) can also be used as a basis to control the location of fixed unsteady separation. For use in controlling unsteady attachment, the analogous criteria are u¯y (x p , 0) = 0,

u¯ xy (x p , 0) > 0,

(6.53)

which is used by Alam et al. (2006) to design a closed-loop feedback controller for reattachment behind a 2D backward-facing step. The inflow velocity uin = 0.015 m/s leads to the Reynolds number Re = 200, keeping the 2D approximation to the flow relevant and the model for the two actuators (wall-jets) accurate.The objective is to reduce the recirculation

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267

6.2 Unsteady Flow Separation Created by LCSs

zone by moving the original location of the reattachment point from x ≈ 0.3 m to x = 0.2 m behind the backward-facing step (see Alam et al., 2006 for details of the controller design). Figure 6.17 shows that the controller does indeed reduce the recirculation zone to the required size, as evidenced by the shape of pressure-colored streaklines released from the inlet cross-section of the channel.

043601-9

estimate for convergence sired separation or reattac riously long settling time once the present proporti a proportional-integral-d lizes the integral and

Closed-loop separation control

FIG. 8. Setup for reattachment control behind a backward-facing step im043601-11age Cislonot sedto -looscale p sep. araUncontrolled tion control reattachment

trivial, because the boun two-dimensional forced volved. Still, recent p

Downloaded 28 Apr 2006 to 18.80.4.201. Redistribution subject to AIP license or copyright, see

Controlled reattachment

dimensional separation provides a strong analytical basis for a higher-dimensional extension.25

Finally, we thank the anonymous referees and suggestions. This work was supported No. F49620-03-1-0200 and NSF Grant N

ACKNOWLEDGMENTS

We are grateful to Gustaav Jacobs for advice on numerical simulations and to Tom Peacock for information on wall-

APPENDIX A: DERIVATION OF CONTR

Figure 6.17 Closed-loop reattachment control behind a backward-facing step. Image adapted from Alam et al. (2006).

We note that for free-slip boundaries, the closed-loop control of unsteady separation for mixing enhancement is discussed by Wang et al. (2003). More recent flow control work by Kamphuis et al. (2018) shows that for increased lift and reduced drag on an airfoil, the optimal location for pulse actuation in the boundary layer is near the Lagrangian separation point.

Moving (Off-Wall) Unsteady Separation Near No-Slip Boundaries As in the case of 2D steady and time-periodic separation from no-slip walls (see §§6.1.1 and 6.1.2), the material spikes shown in Fig. 6.16 develop from an initial upwelling whose location is markedly distinct from the location of the asymptotically prevailing Lagrangian separation point. In analogy with the steady case we described in §6.1.1, Serra et al. (2018)

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Flow Separation and Attachment Surfaces as Transport Barriers

seek the wall-based signature of the initial upwelling as a point over which initially wallparallel material lines will reach a curvature maximum. At time t0 , this Lagrangian spiking point, s = (xs (t0 ), 0), satisfies v¯ xxxyy (xs (t0 ), 0) = 0,

v¯ xxyy (xs (t0 ), 0) < 0,

v¯ xxxxyy (xs (t0 ), 0) > 0,

(6.54)

with these derivatives computed from the mean velocity v¯ (x) defined in Eq. (6.48). To avoid the computation of higher-order derivatives appearing in condition (6.54) from data, Serra et al. (2018) focus on the detection of the theoretical centerpiece (or Lagrangian backbone) of the evolving material spike, rather than just its intersection with the wall. They characterize the initial position B(t0 ) of the evolving material backbone of a separation spike as a curve of initial conditions at which initially wall-parallel material lines experience the largest curvature change over a time interval [t0, t). Such a curve is sketched in Fig. 6.18, with r0 (s) denoting a parametrization of an initially wall-parallel material line whose point of locally maximal curvature change falls on B(t0 ).

Figure 6.18 The material backbone B(t) of a separation spike, emanating from a Lagrangian spiking point on a no-slip boundary. The matrix R denotes clockwise rotation by 90◦ , to be used in formula (6.55) for the curvature change experienced by an initially wall-parallel material line parametrized as r0 (s). Image adapted from Serra et al. (2020a).

As a local maximizer of curvature change within each initially wall-parallel material line, B(t0 ) can be formally defined as a wall-transverse height ridge (see §A.2) of the Lagrangian curvature change function κ¯tt0 (r0 ). For incompressible flows, the more general formula of Serra et al. (2018) for this function simplifies to 


κ0 (r0 ) + ∇2 Ftt0 (r0 ) r00 r00 , J ∇Ftt0 (r0 ) r00 t (6.55) − κ0 (r0 ) , κ¯t0 (r0 ) =  3/2

0 t r0, Ct0 (r0 ) r00 involving the right Cauchy–Green strain tensor Ctt0 , the 90◦ rotation tensor J (see Eq. (5.47)), as well as the first and second derivative tensors of the flow map Ftt0 . Approaching separation via the Lagrangian backbone also enables the extension of the notion of wall-based (or fixed) separation to off-wall (or moving) separation. Specifically, a backbone curve (as a height ridge of the κ¯tt0 (r0 )) may or may not reach the no-slip boundary. In case it reaches the boundary, the separation has an on-wall signature in the form of the spiking point originally defined by the formulas (6.54), but detected far more robustly via the ridge B(t0 ) without higher derivatives. In contrast, a ridge κ¯tt0 (r0 ) that terminates before reaching the boundary signals the birth of an off-wall material spike without an on-wall signature. Without an anchor point on the no-slip boundary, the material spike in the latter case will have a moving spiking point which may glide inside, or out of, the boundary layer. Importantly, prior approaches to separation assume either on-wall or off-wall spike formation, whereas the approach of Serra et al. (2018) is a unified treatment of both phenomena.

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269

6.2 Unsteady Flow Separation Created by LCSs

As an example, Fig. 6.19 shows coexisting on-wall and off-wall material spikes near a vertical downward jet impinging on aM.no-slip 883 A30-14 Serrahorizontal and othersplane at y = 0 (see Serra et al., 2020a for details). T=2 T=2 883(a) A30-14 M. Serra (b) and others (a) 400 ˚ tt 200 0

sha1_base64="CnQRFa1ivu/TMp9wOL4cRussm3I=">AAAB6HicbVBNS8NAEJ34WetX1aOXxSJ4KomIeix68dhCv6ANZbOdtGs3m7C7EUroL/DiQRGv/iRv/hu3bQ7a+mDg8d4MM/OCRHBtXPfbWVvf2NzaLuwUd/f2Dw5LR8ctHaeKYZPFIladgGoUXGLTcCOwkyikUSCwHYzvZ377CZXmsWyYSYJ+RIeSh5xRY6V6o18quxV3DrJKvJyUIUetX/rqDWKWRigNE1Trrucmxs+oMpwJnBZ7qcaEsjEdYtdSSSPUfjY/dErOrTIgYaxsSUPm6u+JjEZaT6LAdkbUjPSyNxP/87qpCW/9jMskNSjZYlGYCmJiMvuaDLhCZsTEEsoUt7cSNqKKMmOzKdoQvOWXV0nrsuJdV9z6Vbl6l8dRgFM4gwvw4Aaq8AA1aAIDhGd4hTfn0Xlx3p2PReuak8+cwB84nz+xT4zd

sha1_base64="SJJMYqoR0xns8Q6+/ERbp9h+ybQ=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE1GPRi8cW7Ae0oWy2k3btZhN2N0IJ/QVePCji1Z/kzX/jts1BWx8MPN6bYWZekAiujet+O4W19Y3NreJ2aWd3b/+gfHjU0nGqGDZZLGLVCahGwSU2DTcCO4lCGgUC28H4bua3n1BpHssHM0nQj+hQ8pAzaqzUcPvlilt15yCrxMtJBXLU++Wv3iBmaYTSMEG17npuYvyMKsOZwGmpl2pMKBvTIXYtlTRC7WfzQ6fkzCoDEsbKljRkrv6eyGik9SQKbGdEzUgvezPxP6+bmvDGz7hMUoOSLRaFqSAmJrOvyYArZEZMLKFMcXsrYSOqKDM2m5INwVt+eZW0LqreVdVtXFZqt3kcRTiBUzgHD66hBvdQhyYwQHiGV3hzHp0X5935WLQWnHzmGP7A+fwBer+MuQ==

3) in 6.21 (Left) 3D imagerate of two height ridges of the curvatureseparations rate field κÛt (r) the in turbulent separation bubble. (b) flow. Contour plot the scalar a 2D unsteady separation bubble (Right) Theofridges shown field in redinin (a). the (x, y)-plane, with their connection to the wall marking the Eulerian spiking points of on-wall separation. Image adapted from Serra et al. (2020a).

parallel to the bottom wall. This is accompanied by an unsteady separation that moves with the primary vortices (see e.g. Didden & Ho 1985). It is generally accepted that this separation process, which moving, is not connected 6.2.2is 3D Fixed Unsteady Separationto the wall. In Miron & Vétel (2015), for example, the separation point attached to each vortex was defined The averaging approach used by Kilic et al. (2005) for 2D unsteady flows, described in a priori as a saddle point off the wall. Such a point can be detected by analysing §6.2.1, extends to separation in mass-conserving 3D unsteady flows with a no-slip boundary. an exponent that cumulates the history of the strain rate along a repelling Lagrangian Specifically, consider a mass-conserving, 3D unsteady velocity field coherent structure (LCS), extracted, for example, as in Farazmand & Haller (2012), as  highest   repulsion  rate u(x, y, z, t) a material line with the normal The separation u(x, z, t) xÛ © (Haller 2011). ª v(x, y, z, t) ® , = v(x, z, t) = = (6.57) ­ point can then be subsequently followed in time by material advection. This discussion zÛ w(x, z, t) w(x, y, z, t) ¬ is relevant for the first and second vortex-induced «separation processes indicated in with a no-slip boundary at z = 0, and assume that the velocity field has a well-defined steady figure 11. asymptotic mean v¯ (x, z) =



¯ z) u(x, w(x, ¯ z)



1 T →∞ T

ˆ

t0

:= lim

v(x, z, t) dt

(6.58)

t0 −T

over the time interval (−∞, t0 ]. As in the case of temporally periodic and quasiperiodic velocity fields (§6.1.4), the theorem of averaging can be applied near the wall to transform the ODE (6.57) locally to a slowly

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272

Flow Separation and Attachment Surfaces as Transport Barriers

varying flow that is steady at leading order. In particular, a local nonlinear coordinate change, (x, z) 7→ ζ = (ξ, η) with   1, near the z = 0 no-slip wall transforms the velocity field to the form
ζÛ = f 0 (ζ ) +  2 f 1 (ζ, t) + O  3 . (6.59) For incompressible flows, the leading-order term in this expansion can be written as   ηu¯ z (ξ, 0) 0 Û ζ = f (ζ ) =  , (6.60) ¯ ξ, 0) −η2 ∂z ∇x · u( while the general form of f 0 (ζ ) for compressible flows is given by Surana et al. (2008). Therefore, as in §6.1.4, a leading-order identification of separation lines on the z = 0 boundary can be carried out using the averaged wall shear field x0 = u¯ z (x, 0) =

1 τ¯ (x), ρν

(6.61)

with τ¯ (x) denoting the steady mean component of the skin friction identified from asymptotic averaging, as in Eq. (6.58). Again, the pointwise normal attraction rate of the averaged skin friction lines on the wall can be computed as ¯ wall (x), ∇x τ¯ (x)ω ¯ wall (x)i hω S¯⊥ (x) = , ¯ wall (x)| 2 ρν | ω

(6.62)

¯ wall denoting the asymptotic time-average of the wall vorticity field ω (x, 0, t). with ω Using the leading-order averaged wall-normal stretching rate, ˆ 1 t0 ¯ C(x) = lim ∂z ∇x · u(x, 0, t) dt, (6.63) T →∞ T t −T 0 Surana et al. (2008) obtain a generalized version of the fixed unsteady separation results described in §6.1.4. Specifically, if a heteroclinic trajectory γ of τ¯ (x) falling in one of the categories (S1)–(S4) shown in Fig. 6.9 satisfies the pointwise conditions ¯ S¯⊥ (x) − C(x) < 0,

¯ C(x) > 0,

x ∈ γ,

(6.64)

then γ is a fixed separation line for a time-dependent separation surface M(t) in the full velocity field (6.57). Surana et al. (2008) also obtain a leading-order linear approximation to M(t) in the present context. Higher-order approximations to the separation surface M(t) can be obtained from higher-order asymptotic averaging of the slowly-varying ODE, as noted for the 2D case in §6.2.1. As an example, Fig. 6.22 shows the leading-order unsteady separation surface (green) identified by the criterion (6.64) in a particular realization of a stochastic velocity field. This velocity field represents a mean-zero, random perturbation to a 3D separation bubble model derived by Surana et al. (2006) as an approximate solution of the Navier–Stokes equation. Figure 6.22 shows the time-varying skin friction field, with some general instantaneous skin friction lines (red) and instantaneous heteroclinic skin friction lines (black and blue). The actual separation line is a set of two saddle-focus connections (type (S1) in Fig. 6.9) in the averaged skin friction field τ¯ (x); these connections satisfy the pointwise criteria (6.64). Nearby trajectories of the stochastic velocity field (black) confirm the first-order approximation (green) to the separation surface.

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273

6.3 Summary and Outlook Phys. Fluids 20, 107101 !2008"

Surana et al.

t = 103

t = 215

t = 249

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AAAB7HicbVBNS8NAEJ34WetX1aOXxSJ4Kknx6yIUvXisYNpCG8pmu2mXbjZhdyKU0t/gxYMiXv1B3vw3btsctPXBwOO9GWbmhakUBl3321lZXVvf2CxsFbd3dvf2SweHDZNkmnGfJTLRrZAaLoXiPgqUvJVqTuNQ8mY4vJv6zSeujUjUI45SHsS0r0QkGEUr+XhT9S66pbJbcWcgy8TLSRly1Lulr04vYVnMFTJJjWl7borBmGoUTPJJsZMZnlI2pH3etlTRmJtgPDt2Qk6t0iNRom0pJDP198SYxsaM4tB2xhQHZtGbiv957Qyj62AsVJohV2y+KMokwYRMPyc9oTlDObKEMi3srYQNqKYMbT5FG4K3+PIyaVQr3mXFfTgv127zOApwDCdwBh5cQQ3uoQ4+MBDwDK/w5ijnxXl3PuatK04+cwR/4Hz+ALt5jfo=

0 in the form of a nearby slow manifold, M , in the slow-time version (7.7) of the Maxey–Riley equations (see Fig. 7.2). Fig. 1. (a) The geometry of the domain D . (b) The attracting set of fixed points M ; each point p in M 0

575

0

T , t0 + T ]}

is completely filled with fixed points of (5). Note that M0 is a over the compact domain as can be computed recursively

Thecompute ✏ = 0 limit of system (4), Haller and Sapsis (2008) a Taylor expansion 0  ≥ 0 to obtain the approximation x = 0, 0     @ M
0 = @D ⇥ [t0 T , t = 0,
Dv(x p , t) 3R M = x p , t, v p ∈ Dv0 ×=Ru(x, × Rn): v pv,= v(x p , t) +  −1 − g + O  2 . [ D ⇥ {t0 + T 2 Dt has an n + 1-parameter family of fixed points satisfying v = (7.11) has corners; M0 @ M0 , u(x, ). formally, for any time T > 0, thedynamics compact on mally They also find that, in terms of More the original physical time t, the reduced M are hyperbolic invaria invariant set By the results of F governed by the inertial equation

anytrajectories constant of xMaxey–Riley and . Consequently, M0 is at a compact nearby trajectories a uniform expowevalue find that M 0 attracts Therefore, all of the equation converge to the slow manifold M+ ✏ u (x, ) + O nential rateinvariant of exp( ⌧set ) (i.e., t/✏) in terms of the normally hyperbolic thatexp( has an open domain oforiginal exponentially fast and unscaled synchronize with the M reduced flow on solutions M generated by the the inertial functions uk (x, ) a time). In fact, attracts all the of 0 attraction. Note that M0 is not a manifold because its boundary (5) that (5) equation (7.12). satisfy (x(0), (0)) 2[ D ⇥ [t0 T , t0 + T ]; this can be verified (3). M✏ is a slow manif D ⇥ {t0 T } @ M0 = @D ⇥ [t0 T, t0 + T ] [ D ⇥ {t0 + T }

any compact normally gives rise to a nearby (7.12) (4). (Local invariance m the manifold through its results guarantee the exi all ✏ 2 [0, ✏0 ), system (4 manifold M✏ that is O manifold (7.13) M✏ can be writ n M✏ = (x, , v) : v = u

r r

0 has an n-d

vp

t1 t

xp

s Figure 7.2 (Left) The attracting criticalofmanifold M0 off sM the frozen-time ( = 0) limit geometry the slow manifold ✏ . (b) A trajectory intersecting a stable fiber f ✏ ( p) converges to e attracting set of fixed points M0 ; each point p in Fig. M0 2. has(a) anThe n-dimensional stable manifold 0 ( p) (unperturbed

(7.8) of the Maxey–Riley equation. (Right) The nearby attracting slow manifold M for  > 0 for the full Maxey–Riley equation (7.7). 3. Slow manifold and inertial equation

using the last equation o any constant value of x M in terms of the small parameter normally hyperbolic inv attraction. Note that M0 (5)

280

Inertial LCSs: Transport Barriers in Finite-Size Particle Motion

7.3 The Divergence of the Inertial Equation and the Q Parameter The inertial equation (7.12) enables us to view the velocity field governing the motion of small inertial particles as a small perturbation to the carrier velocity field. Using the identity ∇ · v ≡ 0 as well as the identity ∂vi ∂vi ∂v j © ∂x j + =­ ∂ x j ∂ xi 2 «

∂v j ∂xi

2

∂vi ª © ∂x j − ® −­ 2 ¬ «

∂v j ∂xi

2

ª ® , ¬

(7.14)

with summation implied over repeated indices, we find that the divergence of the inertial velocity field vin (x, t; ) is equal to    

3R Dv 3R 2 ∇ · vin (x, t; ) =  −1 ∇· +O  = − 1 ∇ · [(∇v) v] + O  2 2 Dt 2  
3R = − 1 ∇v · ∇v + O  2 2
=  (2 − 3R) Q(x, t) + O  2 , (7.15)   where Q = 12 kWk 2 − kSk 2 is the second invariant of ∇v that we have already encountered in §3.7.1 (see Maxey, 1987 and Oettinger et al., 2018). By formula (7.15), therefore, the inertial velocity field vin (x, t; ) typically has nonzero divergence and hence can produce the type of inertial clustering and scattering patterns that we mentioned at the beginning of this chapter. An exception arises, however, for neutrally buoyant particles (R = 2/3), also called suspensions. For such particles, not only does the leadingorder divergence of vin (x, t; ) vanish in Eq. (7.15), but we obtain vin ≡ v from the recursive definition in Eq. (7.13). Therefore, the inertial velocity field of a suspension coincides with the incompressible ambient fluid velocity field. This implies that characteristically compressible flow patterns cannot arise for neutrally buoyant inertial particles. In contrast, the inertial velocity fields of aerosols (0 < R < 2/3) and bubbles (2/3 < R < 2) will generally show compressible patterns (clustering or scattering), as we will see in §7.6. Material patterns formed by (material) inertial particles must be indifferent to the observer, which follows from our general discussion in Chapter 3. As a consequence, the main classifier of the nature of these patterns, ∇ · vin , must be objective. This expectation is in apparent contradiction with our finding that the leading-order divergence of the inertial velocity field vin in formula (7.15) is completely determined by the Q(x, t) parameter, a nonobjective quantity when associated with the velocity field v(x, t), as we noted in §3.7.1. The resolution of this paradox is that  (2 − 3R) Q(x, t) only arises as the leading-order term in the divergence ∇ · vin in the inertial frame in which the Maxey–Riley equations (7.5) are valid. In other frames, the Maxey–Riley equations will change and the leading-order term in the otherwise objective scalar field ∇ · v˜ in (see Eq. (3.18)) will no longer be equal to ˜ t) defined via the transformed fluid velocity field v˜ . This leading-order term,  (2 − 3R) Q(x, however, can still be computed to equal  (2 − 3R) Q(x, t), using the representation v of the velocity field in the inertial frame (see Eq. (A.49) of Appendix A.10). We stress that only the observer is transformed in these arguments: the fluid motion and the inertial particle motion are physically unaffected by these observer changes.

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7.4 Transport Barriers for Neutrally Buoyant Particles

281

7.4 Transport Barriers for Neutrally Buoyant Particles For neutrally buoyant particles (R = 2/3), the second equation in system (7.5) can be rewritten as
1 (7.16) vÛ p − ∂t v − (∇v) v = −µ v − v p , µ := .  As first noted by Babiano et al. (2000), this form enables recasting the Maxey–Riley equations (7.5) as  




d v p − v x p , t = − ∇v x p , t + µI v x p , t − v p , dt (7.17) d xp = vp . dt Therefore, the critical manifold M0 defined in Eq. (7.9) remains an invariant manifold M ≡ M0 for neutrally buoyant particle motion for any value of µ = 1/, as we see by the direct substitution of the definition (7.9) into Eq. (7.17). Using the coordinate
z = vp − v xp, t (7.18) transverse to M0 , the system of equations (7.17) can be rewritten as  
zÛ = − ∇v x p , t + µI z,
xÛ p = z + v x p , t .

(7.19)

In this dynamical system, therefore, the invariant manifold M0
appears as the z = 0 invariant subspace. Within this subspace, we obtain xÛ p = z + v x p , t from Eq. (7.19), which means that inertial particle motion coincides exactly with the motion of fluid particles along M0 . Therefore, in spatial regions where the z = 0 plane is is locally attracting, nearby inertial particles with v p , v will synchronize with fluid trajectories. Likewise, in spatial regions where the z = 0 plane is locally repelling, inertial particles with v p , v will further depart from fluid particle motion. Sapsis and Haller (2008) show that the local attraction to, or repulsion from, the z = 0 invariant plane
at a point x p at time t is governed by the smallest eigenvalue of the symmetric tensor S x p , t + µI, with S denoting the rate-of-strain tensor of the carrier velocity field (as defined in Eq. (2.1)). Specifically, Sapsis and Haller (2008) obtain the following partition of the flow domain: λmin [S(x, t) + µI] < 0, =⇒ instability: inertial particles leave fluid trajectories, λmin [S(x, t) + µI] > 0,

=⇒

stability: inertial particles approach fluid trajectories. (7.20) For small inertial particles (µ  1), we have λmin [S(x, t) + µI] ≈ λmin [µI] = µ > 0, and hence small enough inertial particles always synchronize exponentially fast with fluid particle motions (as we have already concluded from formula (7.10)). As the particle size increases, however, the tensors S(x, t) and µI become of comparable magnitudes, and hence λmin [S(x, t) + µI] may take negative values. Sapsis and Haller (2008) propose such negative values as an explanation for the sustained scattering events that Babiano et al. (2000) observe numerically for neutrally buoyant particle motion. For 2D inertial particle motion, the stability criterion in the partition (7.20) can be specifically written as p µ > |det S(x, t)|. (7.21)

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282

Inertial LCSs: Transport Barriers in Finite-Size Particle Motion

Figure 7.3 shows an illustration of this criterion on a kinematic model of Jung et al. (1993) for the 2D unsteady wake flow behind a vertical cylinder.

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