Mathematical and Physical Theory of Turbulence

Mathematical and Physical Theory of Turbulence

Edited by

John Cannon University of Central Florida Orlando, U.S.A.

Bhimsen Shivamoggi University of Central Florida Orlando, U.S.A.

Boca Raton London New York

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Published in 2006 by Chapman & Hall/CRC Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2006 by Taylor & Francis Group, LLC Chapman & Hall/CRC is an imprint of Taylor & Francis Group No claim to original U.S. Government works Printed in the United States of America on acid-free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number-10: 0-8247-2323-6 (Hardcover) International Standard Book Number-13: 978-0-8247-2323-1 (Hardcover) Library of Congress Card Number 2006040268 This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use. No part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC) 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging-in-Publication Data Mathematical and physical theory of turbulence / edited by John Cannon, Bhimsen Shivamoggi. p. cm. -- (Lecture notes in pure and applied mathematics ; 250) Includes bibliographical references. ISBN 0-8247-2323-6 (alk. paper) 1. Turbulence. I. Cannon, John. II. Shivamoggi, Bhimsen K. III. Series. QA913.M38 2006 532’.0527--dc22

2006040268

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Preface Turbulence arises in practically all flow situations that occurs naturally or in modern technological systems. The turbulence problem poses many formidable intellectual challenges and occupies a central place in modern nonlinear mathematics and statistical physics. As the great physicist Richard Feynman mentioned, it is the last great unsolved problem of classical physics. The American Mathematical Society has listed the turbulence problem as one of the four top unsolved problems of mathematics. It is also one of the seven problems for which the Clay Institute has recently announced a $1 million prize. The powerful notions of scaling and universality, which matured when renormalization group theory was applied to critical phenomena, had been manifested in turbulence previously. The recent dynamical system approach has provided several important insights into the turbulence problem. However, deep problems still remain to challenge conventional methodologies and concepts. There is considerable need and opportunity to advance and apply new physical concepts as well as new mathematical modeling and analysis techniques. There is also an ongoing need to bridge the gap between the grand theories of idealized turbulence and the harsh realities of practical applications. The turbulence problem continues to command the attention of physicists, applied mathematicians, and engineers. Several turbulence research groups in Florida collaborated to hold an international turbulence workshop at the University of Central Florida, May 19–23, 2003. The sponsors of this workshop were: University of Central Florida (Department of Mathematics, College of Arts and Sciences, and Office of Research), Florida State University, Florida A&M University, University of Florida, Embry Riddle Aeronautical University, and area industrialist Dr. Nelson Ying. The idea was to bring together experts from physics, applied mathematics, and engineering working on this common problem and promote the influx of expertise into the subject from all these groups to the benefit of all in understanding complex issues of this problem. This workshop aimed at a discussion of recent progress and some major unresolved basic issues in three- and two-dimensional turbulence and scalar compressible turbulence: Three-dimensional turbulence: theory, experiments, computational and mathematical aspects of Navier–Stokes turbulence • Two-dimensional turbulence: geophysical flows and laboratory experiments • Scalar turbulence: theory, modeling, and laboratory experiments • Compressible/magnetohydrodynamics effects •

There was enormous interest in this workshop worldwide. The following leading experts were invited speakers (*overview speakers): Eberhard Bodenschatz (Cornell) George Carnevale (Scripps Institution)* Peter Constantin (University of Chicago)* Gregory Falkovich (Weizmann Institute)* Thomas Gatski (NASA Langley) Sharath Girimaji (Texas A&M) Marvin Goldstein (NASA Glenn) Jerry Gollub (University of Pennsylvania)* Jack Herring (NCAR) Yukio Kaneda (Nagoya) Shigeo Kida (Nagoya)

John Krommes (Princeton) Jacques Lewalle (Syracuse) Tasos Lyrintzis (Purdue) David Montgomery (Dartmouth) Sutanu Sarkar (University of California, San Diego) Eran Sharon (University of Texas, Austin) Siva Thangam (Stevens Institute) Edriss Titi (University of California, Irvine) Zelman Warhaft (Cornell) Victor Yakhot (Boston University)*

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Professor Peter Hilton, Distinguished Professor of Mathematics, University of Central Florida, was the banquet speaker. The workshop program contained introductory overviews, specialist talks, and contributed talks, as well as panel discussions (one every other day). The workshop was a highly stimulating and enjoyable event and a great success. The purpose of these workshop proceedings is to distribute these overviews and contributed talks for the benefit of the turbulence research community at large. The invited talks included in these proceedings are as follows: The invited talk by Gregory Falkovich provides an overview of the Lagrangian description of turbulence in general and the scalar diffusion problem in particular. David Montgomery provides an account of recent results in decaying two-dimensional turbulence via the entropy maximization approach. The discussion by Jack Herring describes the decay mechanism involving phase mixing of gravity waves in stratified turbulence. Jacques Lewalle describes application of wavelet scaling to investigate the regularity of Navier–Stokes equations. The invited talk by John Krommes provides an overview of some aspects of statistical theories of turbulence in strongly magnetized plasmas. Fran¸coise Bataille discusses a general framework to develop eddy viscosity models for turbulent flows which do not exhibit the Kolmogorov energy spectrum. The invited talk by Siva Thangam provides an overview of the development of continuous turbulence models that are suitable for large-eddy simulation and Reynolds averaged Navier–Stokes formulation. The banquet talk by Peter Hilton provides fascinating personal reminiscences and anecdotes about three great mathematicians of the twentieth century — Alan Turing, Henry Whitehead, and Jean-Pierre Serre. The contributed talks included in these proceedings are as follows: The contributed talk by Jacques Lewalle discusses the use of wavelets to describe several aspects of Navier–Stokes turbulence which are not well handled by traditional approaches. Mikhail Shvartsman discusses a connection between the governing equations, the constitutive theory, and the closure problem for the atmospheric boundary layer. The contributed talk by Peter Davidson discusses the significance of the law of conservation of angular momentum for freely evolving, homogeneous turbulence. Ekachai Juntasaro discusses a new concept of turbulence modeling in turbulent channel flow and turbulent boundary-layer flow. John R. Cannon Bhimsen K. Shivamoggi

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The Editors John Cannon completed his B.A. degree in mathematics at Lamar University in 1958. He attended graduate school at Rice University to study mathematics and completed his M.S. degree in 1960 and his Ph.D. degree in 1962. His research expertise is in partial differential equations and numerical analysis with applications to heat flow, fluid dynamics, chemical reactions, change of phase, flow in porous media, and inverse problems, as well as applications to biology and medicine. He has held faculty and research positions at Brookhaven National Laboratory, Universit`a di Genova, Purdue University, University of Minnesota, The University of Texas at Austin, Universit`a di Firenze, Colorado State University, the Hahn Meitner Institute in Berlin, Washington State University, The Office of Naval Research, Lamar University, and the University of Central Florida. He is the author of 2 books, 150+ refereed journal articles, 26 proceedings articles, and 5 technical reports. Bhimsen Shivamoggi received his Ph.D. from the University of Colorado, Boulder. Following postdoctoral research appointments at Princeton University and the Australian National University, Canberra, he joined the mathematics faculty at the University of Central Florida, Orlando in 1984. He has worked on problems of stability and turbulence in fluids and plasmas. He has written several books on these topics and has held visiting appointments in Los Alamos National Laboratory, University of California at Santa Barbara, Technische Hochschule Darmstadt (Germany), Technische Universiteit Eindhoven (The Netherlands), Observatoire de Nice (France), Kyoto University (Japan), International Center for Theoretical Physics at Trieste (Italy), and University of Newcastle (Australia).

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Contributors F. Bataille Centre de Thermique de Lyon Villeurbanne Cedex, France

M. Yousuff Hussaini Department of Mathematics and Computational Science and Engineering Florida State University Tallahassee, Florida

G. Brillant Centre de Thermique de Lyon Villeurbanne Cedex, France

R. James National Center for Atmospheric Research Boulder, Colorado

George F. Carnevale Scripps Institution of Oceanography University of California San Diego La Jolla, California

Ekachai Juntasaro School of Mechanical Engineering Suranaree University of Technology Nakhon Ratchasima, Thailand

J. Clyne National Center for Atmospheric Research Boulder, Colorado P. A. Davidson Department of Engineering University of Cambridge Cambridge, United Kingdom

Varangrat Juntasaro Department of Mechanical Engineering Kasetsart University Bangkok, Thailand

Douglas P. Dokken Department of Mathematics University of St. Thomas St. Paul, Minnesota

Y. Kimura Grade School of Mathematics Nagoya University Nagoya, Japan

Gregory Falkovich Weizmann Institute of Science Rehovot, Israel

J. A. Krommes Plasma Physics Laboratory Princeton University Princeton, New Jersey

J. R. Herring National Center for Atmospheric Research Boulder, Colorado

Jacques Lewalle Department of Mechanical Engineering Syracuse University Syracuse, New York

Peter Hilton Department of Mathematical Sciences State University of New York Binghamton, New York and Department of Mathematics University of Central Florida Orlando, Florida

David C. Montgomery Department of Physics and Astronomy Dartmouth College Hanover, New Hampshire

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Mikhail M. Shvartsman Department of Mathematics University of St. Thomas St. Paul, Minnesota

Siva Thangum Department of Mechanical Engineering Stevens Institute of Technology Hoboken, New Jersey

Stephen L. Woodruff Center for Advanced Power Systems Florida State University Tallahassee, Florida

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Table of Contents Chapter 1 A Mathematician Reflects: Banquet Remarks ................................................................................... 1 Peter Hilton

Chapter 2 Lagrangian Description of Turbulence ..............................................................................................7 Gregory Falkovich

Chapter 3 Two-Dimensional Turbulence: An Overview .................................................................................. 47 George F. Carnevale

Chapter 4 Statistical Plasma Physics in a Strong Magnetic Field: Paradigms and Problems ..........................69 J.A. Krommes

Chapter 5 Some Remarks on Decaying Two-Dimensional Turbulence ........................................................... 91 David C. Montgomery

Chapter 6 Statistical and Dynamical Questions in Stratified Turbulence....................................................... 101 J.R. Herring, Y. Kimura, R. James, J. Clyne, and P.A. Davidson

Chapter 7 Wavelet Scaling and Navier–Stokes Regularity............................................................................. 115 Jacques Lewalle

Chapter 8 Generalization of the Eddy Viscosity Model — Application to a Temperature Spectrum............ 125 F. Bataille, G. Brillant, and M. Yousuff Hussaini

Chapter 9 Continuous Models for the Simulation of Turbulent Flows: An Overview and Analysis ............. 131 M. Yousuff Hussaini, Siva Thangam, and Stephen L. Woodruff

Chapter 10 Analytical Uses of Wavelets for Navier–Stokes Turbulence ......................................................... 145 Jacques Lewalle

Chapter 11 Time Averaging, Hierarchy of the Governing Equations, and the Balance of Turbulent Kinetic Energy ............................................................................................................................................ 155 Douglas P. Dokken and Mikhail M. Shvartsman

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Chapter 12 The Role of Angular Momentum Invariants in Homogeneous Turbulence................................... 165 P. A. Davidson

Chapter 13 On the New Concept of Turbulence Modeling in Fully Developed Turbulent Channel Flow and Boundary Layer ............................................................................................................................. 183 Ekachai Juntasaro and Varangrat Juntasaro

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Mathematician Reflects: 1 ABanquet Remarks Peter Hilton CONTENTS 1.1 1.2 1.3 1.4

Alan Turing............................................................................................................................... 1 Henry Whitehead...................................................................................................................... 2 Jean-Pierre Serre....................................................................................................................... 3 Epilogue.................................................................................................................................... 4

First, let me thank you very much for inviting me to participate in your conference and for giving me this opportunity to say a few informal words to you following an excellent dinner. My area of research intersects yours precisely in our recognition of the importance of a good number system and a good notation, and our use of the same number system. Indeed, as I look back on my own career, my closest contact with turbulence — and this was, as in the case of this conference, a matter of international turbulence — was during World War II, when I was conscripted to work, from January 1942 until the end of the war in Europe in May 1945 at Bletchley Park, breaking the highest-grade German ciphers used for diplomatic and military traffic passing among the German government, the German High Command, and their naval, air force, and army commanders and U-boat captains. Those were indeed turbulent days, and I will say something about them in my remarks this evening. More generally, I will reflect on the wonderful mathematicians I have known during a career spanning more than 60 years, starting in British Military Intelligence, proceeding along a more conventional academic route, and continuing today, though at a gentler pace consistent with my growing maturity.

1.1 ALAN TURING Our work at Bletchley Park, cracking the high-grade German codes, was wonderfully exciting, stimulating us to work tirelessly, over long stretches of time, buoyed by the intellectual challenge and our awareness of the importance of what we were achieving. However, for me, as a very young man — I started work at BP (as we called it) at the age of 18 — there was the thrill of working with some of the greatest British mathematicians of that period, getting to know them well and enjoying their friendship and collegiality. They were absolutely free of any outward awareness of their intellectual superiority and treated their younger colleagues as equals — which, as cryptanalysts but not as mathematicians, of course — we were! They seemed not to be aware that, however adept we might be at applying mathematical thinking and perhaps certain very specialized linguistic skills to the job in hand, we had no mathematical knowledge and experience to match theirs. One lesson I learned from my experience at BP that I would like to share with you is this: to be able to apply mathematical reasoning to problems that are not intrinsically mathematical — in other words, to be able to apply mathematics — the essential prerequisites are a first-class mathematics education and strong interest in and incentive to solve the problem. Experience and familiarity with some scientific discipline, although desirable, are not essential. It is an interesting and, I believe, important fact that, in the group of mathematicians and young would-be mathematicians working on © 2006 by Taylor & Francis Group, LLC

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the most sophisticated ciphers used by the Germans, not one was an “applied” mathematician; all1 were uncompromisingly “pure.” Within this group was one absolutely outstanding thinker: Alan Turing was unmistakably a genius. Alan’s role was different from that of the other members of the team. He played no part in the day-to-day decryption of enemy signals; he was concerned with fundamental questions of cryptanalytical method, especially but not exclusively with the design of high-speed machines to expedite the deciphering process. In 1948, it was my great good fortune to be appointed to a junior position in the mathematics department of the University of Manchester at the same time at which Alan was appointed2 to a readership in mathematics, so I could continue to take advantage of the unique privilege of having Alan as colleague and friend. As I soon realized, Alan Turing had had in mind all along the possibility of building an actual machine to realize the concept of a “Turing machine”; this formed the basis of his seminal prewar paper, “On Computable Numbers.” Of course, throughout the war and in the years immediately following, he was inventing the electronic computer. The realization of this dream in no sense interfered with his enormous success in facilitating the breaking of the German codes. Unfortunately, many people today have the wrong impression of Alan Turing’s personality and character. This misconception arises in part from a popular but quite inaccurate stereotype of genius, especially mathematical genius: such a person is thought to be a very narrow specialist, totally unable to deal with the usual demands of life.3 A further source of confusion with regard to Alan Turing is the fine play by Hugh Whitemore, Breaking the Code, which was, in fact, a work of brilliant imaginative fiction inspired by Turing’s life and the tragedy of his early death. Let me say here that Turing was an inspiring colleague and friend, a wonderful source of ideas, very approachable, and very versatile. In the very early 1950s, his picture of the future social impact of the computer was extraordinarily prophetic. Let me also add, for the benefit of those who do not know the sad story, that Alan Turing committed suicide in 1954, just short of his 42nd birthday. Two years earlier, he had been bound over (i.e., convicted without penalty) on a charge of having a homosexual affair with a consenting adult — at that time a criminal offense — as a result of which he had to undertake not to repeat the “crime” and also lost his security clearance and became ineligible for a U.S. visa. What an appalling way to treat a man who not only was a genius but also had made a contribution of incalculable importance to the winning of World War II. I benefited enormously from having been at BP. I had become friendly at a very early age with some of the greatest mathematicians of the century, and these friendships were to stand me in good stead for many years. Let me tell you about one such friend, Henry Whitehead, my friend and colleague at BP and for many more years afterwards and my research advisor at Oxford after the war.

1.2 HENRY WHITEHEAD Henry, or — to give him his full name — John Henry Constantine Whitehead, was the son of the bishop of Madras and nephew of the famous philosopher Alfred North Whitehead. It was said that, after an academically undistinguished period as an undergraduate at Balliol College, Oxford, Henry took his bachelor’s degree, went into the City (of London), made and lost a fortune in 2 years, and then realized he wanted to be a mathematician. He took a doctorate at Princeton University under the direction of the eminent geometer Oswald Veblen (who coined the term “analysis situs” for what has come to be called topology) and returned to Oxford as a fellow of Balliol College. There it soon 1 One,

Jack Good, did become a statistician, but a theoretical statistician. I would rather describe him as a probabilist. is relevant to mention here that the head of department at Manchester University was Professor Max Newman, who had headed the section at BP concerned with the machine decryption of German ciphers. 3 The recent biography of Paul Erd¨ os, The Man Who Loved Only Numbers, epitomizes this viewpoint. 2 It

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became clear that he was one of the subtlest and most powerful of contemporary mathematicians, combining profound geometric intuition with a masterful capacity to employ and develop deep algebraic methods in the solution of geometric — or, rather, topological — problems. I was fortunate to win an open scholarship in mathematics to Queen’s College, Oxford, and started my undergraduate studies there in the fall of 1940. I met Henry soon after that, taking a course from him in projective geometry. The number of students taking this course rapidly decreased to four (all of whom subsequently became mathematicians); Henry’s lectures were by no means easy to follow. However, we survivors recognized that he was not so much lecturing as thinking out loud, and that we were actually enjoying the privilege of watching how a great mathematician thought out his mathematical ideas. It was stimulating; it was awesome. After four terms at Oxford I had to leave, in January 1942, to begin my service in British Military Intelligence at BP. To my delight, Henry arrived to join our team in the middle of 1943. Thus, the extraordinary situation came about in which I found myself, a mere mathematics student, teaching our cryptanalytic methods to perhaps the greatest British mathematician of his generation. For me, the situation was even more remarkable because Henry Whitehead and I became firm friends! We shared many common interests outside mathematics: cricket, squash, racquets, politics, beer-drinking. Indeed, the irony is that the only significant interest we did not share was algebraic topology, Henry’s areas of research, because I knew nothing about it. In 1945, Henry left BP to take up his new position as Waynflete Professor of Pure Mathematics in the University of Oxford. I, on the other hand, was far younger and had to spend another year in military service. In fact, I spent it very productively at the Post Office Engineering Research Station at Dollis Hill, London, where the Colossi, forerunners of the electronic computer and of crucial importance to us at BP, were built. During that year, Henry invited me to spend a weekend with him and his family at their home in North Oxford. Over that weekend, he invited me to return to Oxford on my release from war service to work on my doctorate under his tutelage and to live in his home as a member of the family. It was a marvelous offer, but I did, briefly, hesitate. “But, Henry,” I said, “I don’t know what algebraic topology is.” “Oh, don’t worry about that,” he responded, “you’ll love it.” Thus should the big decisions in one’s life be made: on the basis of complete trust in the judgment of a very knowledgeable and discerning friend. Henry was right, and I never looked back. I enjoyed the enormous privilege of living with him and his family almost as if I were a member of the family. In fact, they all called me “uncle,” including Henry and his lovely wife, Barbara, and I remained a close family friend until Henry’s sudden and untimely death while on leave of absence in Princeton in 1960.

1.3 JEAN-PIERRE SERRE The third mathematician about whom I would like to speak is the great French mathematician J.-P. Serre, whom I got to know at the start of his distinguished career in 1952 when he was 25 years old. At that time I was a lecturer in mathematics at the University of Manchester, England, in the department headed by my wartime BP boss, Max Newman. Max Newman had built up a remarkably strong department, full of outstanding research mathematicians, but he always insisted on the great importance of good teaching and, to a slightly lesser extent, of fair examining. Max encouraged us to invite outstanding mathematicians who were doing research of great importance to visit our campus so that we could learn from them and be stimulated in our research. Thus it was that I got to know of Serre’s outstanding research in homotopy theory; his thesis, completed and accepted in 1951, was published (in French) in Annals of Mathematics at the end of that year. I am very proud to have been the first mathematician from outside France to have invited Serre to travel abroad to talk about his work. I think it would require at least a semester-long course of lectures, rather than an after-dinner talk, to try to explain Serre’s astonishing contribution in my field of research to a group of experts in a totally different field. Suffice it to say that he introduced into my field totally new methods that © 2006 by Taylor & Francis Group, LLC

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enabled him to answer fundamental questions that, prior to his work, it would have been pointless and frustrating, to ask. Our subject was never the same again. Serre stayed with my wife and me during this visit and was a delightful guest, full of fun. We went for walks in the countryside and he proved to be very athletic, running and leaping over stiles and five-barred gates. Following his visit, we corresponded a lot, almost exclusively about mathematics. Then he invited me to visit Paris and stay with him and his wife. It was an astonishing experience. He was a tremendously hard worker and, after several hours spent discussing mathematics, he would propose we go to the Institut Henri Poincar´e — to play ping-pong! The visit exhausted and stimulated me. It led to what was probably the deepest and most important paper I have ever written.4 I recall that I offered Serre joint authorship, but he declined, saying, very characteristically, that he was not yet old enough for such an honor. Our correspondence continued for some time after my visit; at one point, I felt we had become so friendly that I asked him to call me “Peter.” Such a step was a very significant one for a Frenchman in the 1950s, but Serre readily agreed, adding, “A titre de reciprocit´e, supprimez le Monsieur” (“By way of return, cut out the Mr”). Thus, he wrote “Cher Peter” and I wrote “Dear Serre.” Of course, today, his close colleagues call him “Jean-Pierre.” It was a privilege to know this great mathematician and wonderful to get to understand his marvelous mathematical ideas. I am happy to be able to tell you that, some time after I decided to feature Jean-Pierre Serre as one of the most outstanding mathematicians I have known, he was selected by the Norwegian government as the first recipient of the recently instituted Abel Prize, designed to carry a very substantial financial award as well as a prestige entirely comparable with that of the Nobel prizes.5

1.4 EPILOGUE I could go on to tell you of other great mathematicians I have known over a career spanning more than 60 years and of the fascinating visits I have made to mathematical centers all over the world.6 However, I am well aware that an after-dinner talk should not be too long or too intellectually demanding. Thus, I will close with a few remarks about what I have learned over the years about doing and teaching mathematics. First, I would like to echo Henry Whitehead in declaring my belief that mathematics is a first-class occupation. When trying to persuade a young colleague to remain a mathematician, Henry declared, “It is better to be second rate in a first-rate occupation than first rate in a second-rate occupation.” This, I believe, is profoundly true, though it is also true that Henry was not faced with the agonizing choice confronting the young man seeking his advice. Second, I wish to reject utterly the viewpoint that there is an unavoidable conflict in any university mathematics department between choosing good researchers and choosing good teachers. I argue that one should always try to choose the best mathematicians, for to do otherwise is to miss an opportunity and to risk a gradual descent into mediocrity. Of course, we must always insist on the importance of good teaching; however, the simple fact is that we know how to produce good researchers who love mathematics, but we do not know how to produce good teachers. Moreover, whereas the criteria of good mathematical research are generally agreed and judgments of the quality of an individual’s research are fairly unanimous, the same is by no means the case when it comes to considerations of good teaching. All we can say with virtual certainty is that, if someone is to be a good teacher of mathematics, that person must love mathematics; typically, that person will have demonstrated that love of mathematics

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colleague was recently kind enough to describe it as a “landmark paper.” is no Nobel Prize in mathematics and there are several interesting theories as to why this is so! 6 As I write this, I am looking forward to a visit to Z¨ urich, Switzerland, next week to give a lecture to mark my 80th birthday. 5 There

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by doing some good mathematical research. Thus, although it is certainly appropriate to emphasize to a job candidate that teaching is regarded as very important, the selection criteria should very much concentrate on the quality of the candidate’s research.7 This brings me back to the occasion of this conference and explains why I salute you so wholeheartedly for your work in research in highly significant areas of applied mathematics. I feel confident that you all agree with the dictum I enunciated many years ago that “good mathematics goes from answer to question; it is only elementary arithmetic which goes from question to answer.” I would add that, for good mathematics, it may not be your answer, but it is surely your question; for elementary arithmetic, it is certainly not your question! I salute you and I wish you continued success in your chosen field of research. I thank you again for inviting me to join you in your conference and for giving me not only this glimpse of the feeding habits, gastronomic and intellectual, of a very talented international group of researchers in an active and important field of research in the mathematical sciences, but also an insight into your methods and concerns. Thank you. 7 Of

course, it follows from this argument that senior members of the department must set a good example of conscientious teaching.

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Description of 2 Lagrangian Turbulence Gregory Falkovich CONTENTS 2.1

Introduction .............................................................................................................................. 8 2.1.1 Propagators................................................................................................................... 8 2.1.2 Kraichnan Model.......................................................................................................... 9 2.1.3 Large Deviation Approach ......................................................................................... 10 2.2 Particles in Fluid Turbulence..................................................................................................10 2.2.1 Single-Particle Diffusion............................................................................................ 11 2.2.2 Two-Particle Dispersion in a Spatially Smooth Velocity ........................................... 12 2.2.3 Two-Particle Dispersion in a Nonsmooth Incompressible Flow................................ 15 2.2.4 Two-Particle Dispersion in a Compressible Flow...................................................... 18 2.2.5 Multiparticle Configurations and Zero Modes........................................................... 20 2.3 Unforced Evolution of Passive Fields .................................................................................... 25 2.3.1 Decay of Tracer Fluctuations ..................................................................................... 25 2.3.1.1 Smooth Velocity ......................................................................................... 26 2.3.1.2 Nonsmooth Velocity ................................................................................... 27 2.3.1.3 Scalar Decay with Viscous and Inertial Intervals Present .......................... 28 2.3.1.4 Scalar Decay in a Finite Box ...................................................................... 28 2.3.2 Growth of Density Fluctuations in Compressible Flow............................................. 29 2.3.3 Vector Fields in a Smooth Velocity............................................................................ 30 2.3.3.1 Gradients of the Passive Scalar................................................................... 30 2.3.3.2 Small-Scale Magnetic Dynamo.................................................................. 31 2.4 Cascades of a Passive Tracer .................................................................................................. 31 2.4.1 Direct Cascade ........................................................................................................... 32 2.4.1.1 Direct Cascade in a Smooth Velocity ......................................................... 33 2.4.1.2 Anomalies of Tracer Statistics in a Nonsmooth Velocity........................... 34 2.4.2 Inverse Cascade in a Compressible Flow................................................................... 37 2.5 Active Tracers......................................................................................................................... 37 2.5.1 Activity Changing Cascade Direction........................................................................ 38 2.5.1.1 Burgers Turbulence..................................................................................... 38 2.5.1.2 Two-Dimensional Magnetohydrodynamics ...............................................40 2.5.2 Two-Dimensional Incompressible Turbulence........................................................... 41 2.5.2.1 Direct Vorticity Cascade in Two Dimensions............................................. 41 2.5.2.2 Inverse Energy Cascade in Two Dimensions.............................................. 42 2.6 Conclusion.............................................................................................................................. 42 Acknowledgment ............................................................................................................................. 43 References........................................................................................................................................ 43

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2.1 INTRODUCTION This review is an abridged and updated version of Falkovich et al.[1]. The subject is the combined effect of molecular diffusion and random flow on scalar and vector fields transported by a fluid. We want to understand first when there is mixing and when, on the contrary, inhomogeneities are created and enhanced. We want to distinguish between cases when flow creates small-scale inhomogeneities of the transported fields, which are then killed by molecular diffusion, and cases when large-scale structures of the fields appear. Our goal is to describe temporal and spatial statistical properties of transported fields.

2.1.1 PROPAGATORS If we wish to describe the statistics of different fields transported by the flow, we need formalism to describe the probabilities of different flow trajectories. Consider an evolution of a passive scalar tracer
(r, t) in a random flow. The mean value of the scalar tracer at a given point is an average over values brought by different trajectories:


(r, s) = P(r, s; R, 0)
(R, 0) dR, (2.1) Here, P(r, s; R, t) is the probability density function (PDF) to find the particle at time t at position R, given its position r at time s. That PDF is called the propagator or the Green function. Multipoint correlation functions of the tracer, C N (r, s) ≡

(r1 , s) . . .
(r N , s)
= P N (r, s; R, 0)
(R1 , 0) . . .
(R N , 0) dR,

(2.2)

are expressed via the multiparticle Green functions P N which are the joint PDFs of the equal-time positions R = (R1 , . . . , R N ) of N fluid trajectories. The next chapter is devoted to the analysis of the one-, two-, and multiparticle Green functions. The results of Chapter 2 are used then in the subsequent Chapter 3 and Chapter 4 for the description of the transported passive fields. Chapter 5 describes active tracers, which influence the velocity that transports them. The trajectory of the fluid particle that passes at time s through the point r is described by the vector R(t; r, s), which satisfies R(t; r, t) = r and the stochastic equation [2] ˙ = v(R, t) + u(t). R (2.3) Here, u(t) describes the molecular Brownian motion; it has zero average and covariance function
u i (t) u j (t  ) = 2i j (t −t  ). We shall also consider macroscopic velocity v as random with different statistical properties and different dependencies on space and time in different cases. Molecular diffusivity  is of order 10−1 cm2 s −1 for gases in gases, so it would take many hours for a smell to diffuse across the dinner table. Similarly, to diffuse salt a kilometer depth of the ocean molecular diffusion would take 107 years. It is the motion of fluids that provides large-scale transport and mixing in most cases. There is a clear scale separation between macroscopic velocity v and molecular diffusion u that allows one to treat them separately. Using (2.3), one can write the Green function as an integral over paths that satisfy q(s) = r and q(t) = R (see, e.g., references 1 and 3):  
t ˙ − ıp()·[q()−v(q(),)−u()] d (2.4) P(r, s; R, t) = Dp Dq e s 
=

Dp Dq e 


=

Dqe

1 − 4

−

t s

t s

v,u 2 ˙ [ıp()·(q()−v(q(),))+p ()] d

2 ˙ [q()−v(q(),)] d



(2.5) v

=
P(r, s; R, t|v)v . v

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(2.6)

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The integration over the auxiliary field p in (2.4) enforces the delta function of (2.3). One passes from (2.4) to (2.5) by averaging over the Gaussian Brownian noise, and from (2.5) to (2.6) by calculating Gaussian integral over p. Sometimes it is useful to consider only partial average over the molecular diffusion. Then the tracer satisfies the advection–diffusion equation: ∂
+ (v · ∇)
− ∇ 2
= 0. ∂t

(2.7)

The solution can be expressed via the v-dependent propagator P(r, s; R, t | v) defined by (2.6). It satisfies the initial condition P(r, t; R, t | v) = (R – r) and the equation 

 ∂t − ∇R · v(R, t) − ∇R2 P(r, s; R, t|v) = 0,

(2.8)

when t > s. For a regular velocity with deterministic trajectories, one has at  → 0 P(r, s; R, t|v) = (R − R(t; r, s)).

(2.9)

We shall see later that, even when velocity field is not regular and the notion of a single Lagrangian trajectory does not make sense, the propagators are well defined.

2.1.2 KRAICHNAN MODEL Generally, exact calculations are only possible for Gaussian random processes delta-correlated in time like in (2.5). The simplest case is the Brownian motion when the advection is absent. One then obtains from (2.6) the Gaussian PDF of the displacement, P(R, t) = (4t)−d/2 e−R

2

/(4t)

,

(2.10)

which satisfies the heat equation (∂t − ∇ 2 )P(r, t) = 0. The short-correlated case is far from being an exotic exception but rather presents a long-time limit of an integral of any finite-correlated random function. Indeed, such an integral can be presented as a sum of many independent, equally distributed random numbers; the statistics of such sums is a subject of the central limit theorem. For the long-time description of the advection in finite-correlated flows, it is useful to consider the extreme case of random homogeneous and stationary velocities with a very short correlation time. This case may be regarded as describing the speeded-up-film view of velocity fields with temporal decay of correlations or, more formally, as the scaling limit 1 lim→∞  2 v(r, t). When  → ∞, one gets a Gaussian velocity field with the two-point function
vi (r, t) v j (r , t  ) = 2 (t − t  )Di j (r − r ).

(2.11)

It is common to call the Gaussian ensemble with a white-noise two-point function (2.11) the Kraichnan ensemble [4]. For the Kraichnan velocities v, the Lagrangian velocity v(R, t) has the same white noise temporal statistics as the Eulerian velocity v(r, t) for fixed r and the displacement along a Lagrangian trajectory R(t) − R(0) is a Brownian motion for all times. In exactly the same way as one derives (2.6) and (2.10) from (2.4) one gets P(R, t) = || ˆ 1/2 (4t)−d/2 e−i j Ri R j /4t ,

(2.12)

where (ˆ −1 )i j = Di j (0) + i j . We shall see that the Kraichnan ensemble of velocities constitutes an important theoretical laboratory of the particle behavior in fluid turbulence. © 2006 by Taylor & Francis Group, LLC

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2.1.3 LARGE DEVIATION APPROACH One can move beyond the consideration of the previous section considering the correlation time finite (yet small comparing to the time of evolution). Such generalization is the subject of the large deviation theory. Let us present here the basic idea, which will be used extensively in this course. Consider some quantity X which is an integral of some random function over time t much larger than the correlation time . At t  , X behaves as a sum of many independent identically distributed random numbers yi : X = 1N yi with N ∝ t/. The generating function
e z X  of the moments of X is the product,
e z X  = e N S(z) , where we have denoted
e zy  ≡ e S(z) (assuming that the generating function
e zy  exists for all complex z). The PDF P(X ) is given by the inverse Laplace transform (2i)−1 e− z X +N S(z) dz with the integral over any axis parallel to the imaginary one. For X ∝ N , the integral is dominated by the saddle point z 0 such that S  (z 0 ) = X/N and P(X ) ∝ e−N H (X/N −
y) .

(2.13)

Here, H = −S(z 0 ) + z 0 S  (z 0 ) is the function of the variable X/N −
y. It is called entropy function because it appears also in the thermodynamic limit in statistical physics [5]. A few important properties of H (also called rate or Cram´er function) may be established independently of the distribution P(y). It is a convex function which takes its minimum at zero, i.e., for X taking its mean value
X  = N S  (0). The minimal value of H vanishes because S(0) = 0. The entropy is quadratic around its minimum with H  (0) = −1 , where  = S  (0) is the variance of y. We thus see that the mean value
X  = N
y grows linearly with N . The fluctuations X −
X  on the scale O(N 1/2 ) are governed by the central limit theorem that states that (X −
X )/N 1/2 becomes a Gaussian random variable for large N with variance
y 2 −
y2 ≡  as in (2.10) and 2.12). Finally, its fluctuations on the larger scale O(N ) are governed by the large deviation form (2.13). The possible nonGaussianity of the y’s leads to a nonquadratic behavior of H for (large) deviations from the mean, starting from X −
X /N /S  (0). Note that if y is Gaussian, then X is Gaussian too for any t but the universal formula (2.13) with H = (X − N
y)2 /2N  valid only for t  .

2.2 PARTICLES IN FLUID TURBULENCE As explained in the introduction, understanding the properties of transported fields involves the analysis of the behavior of fluid particles. We present here the results on the time-dependent statistics of the Lagrangian trajectories Rn (t). In this chapter, we sequentially increase the number of particles involved in the problem. We start from a single trajectory whose effective motion is a simple diffusion at times longer than the velocity correlation time in the Lagrangian frame (Section 2.2.1). We then move to two particles. The separation law of two close trajectories depends on the scaling properties of the velocity field v(r, t). If the velocity is smooth, that is, with |v(Rn )−v(Rm )| ∝ |Rn −Rm |, then the initial separation grows exponentially in time (Section 2.2.2). The smooth case can be analyzed in much detail using the large deviation arguments presented in Section 2.2.3. The reader mainly interested in applications to transported fields might wish to take the final results (2.23) and (2.28) for granted, skipping their derivation. If the velocity is nonsmooth, that is, |v(Rn ) − v(Rm )| ∝ |Rn − Rm | with  < 1, then the separation distance between two trajectories grows as a power of time (Section 2.2.3). We discuss important implications of such a behavior for the nature of the Lagrangian dynamics. The difference between the incompressible flows, where the trajectories generally separate, and compressible ones, where they may cluster, is discussed in Section 2.2.4. Finally, in the consideration of three or more trajectories, the new issue of geometry appears. Statistical conservation laws come to light in two-particle problems and then feature prominently in the consideration of multiparticle configurations. © 2006 by Taylor & Francis Group, LLC

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Geometry and statistical conservation laws are the main subject of Section 2.2.5. Although we try to keep the discussion as general as possible, much of the insight into the trajectory dynamics is obtained by studying the Kraichnan model.

2.2.1 SINGLE-PARTICLE DIFFUSION We now consider the single Lagrangian trajectory R(t). For the pure advection without noise, the t displacement R(t) − R(0) = 0 V(s) ds, with V(t) = v(R(t), t) being the Lagrangian velocity. The properties of the displacement depend on the specific trajectory under consideration. We shall always work in the frame of reference with no mean flow:
v = 0. We assume statistical homogeneity of the Eulerian velocities, which implies that the stochastic process V(t) does not depend on the initial position R(0) of the trajectory. If, additionally, the Eulerian velocities are statistically stationary, then so is V(t). This follows by averaging the expectations involving V (t + ) over the initial position R(0) (on which they do not depend) and the change of variables R(0) → R() under the velocity ensemble average. Note that the Jacobian of the change of variables is supposed to be unity, which requires incompressibility. For  = 0, the mean square displacement satisfies the equation: d
[R(t) − R(0)]2  = 2 dt




t


V(0) · V(s)ds.

(2.14)

0

The behavior of the displacement is crucially dependent on the range of temporal correlations of the Lagrangian velocity. Let us define the correlation time  of V(t) by
∞
V(0) · V(s)ds =
v2 . (2.15) 0

The value of  provides a measure of the Lagrangian velocity memory; its divergence is symptomatic of persistent correlations. No general relation between the Eulerian and the Lagrangian correlation times has been established, except for the case of short-correlated velocities. For times t , the two-point function in (2.14) is approximately equal to
V(0)2  =
v2 . The fluid particle transport is then ballistic with
[R(t) − R(0)]2 
v2 t 2 and the PDF P(R, t) is determined by the whole single-time velocity PDF. When the correlation time of V(t) is finite (a generic situation in a turbulent flow where  is of order of a large-scale turnover time), an effective diffusive regime is expected to arise for t   with
(R(t) − R(0))2  2
v2 t [2]. Indeed, the particle displacements over time segments much larger than  are almost independent. At long times, the displacement R(t) behaves then as a sum of many independent variables and falls into the class of stationary processes treated in Section 2.2.2 and Section 2.2.3. In other words, R(t) for t   becomes a Brownian motion in d dimensions, normally distributed with
R i (t)R j (t) Dei j t, where the so-called eddy diffusivity tensor is as follows
∞ 1 ij De =
Vi (0) V j (s) + V j (0) Vi (s) ds. (2.16) 2 0

The symmetric second-order tensor Dei j is the only characteristics of the velocity which matters in this limit of t  . The trace of the tensor is equal to
v2 , i.e., to the large-time value of the integral in (2.14), while its tensorial properties reflect the rotational symmetries of the advecting velocity field. If the latter is isotropic, the tensor reduces to a diagonal form characterized by a single scalar value De . The main problem of turbulent diffusion is to obtain the effective diffusivity tensor given the velocity field v and the value of the molecular diffusivity . A huge amount of work has been devoted © 2006 by Taylor & Francis Group, LLC

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to it from the applied and the mathematical point of view, and exhaustive reviews of the problem are available in the literature [6–9]. The other general issue in turbulent diffusion is the condition on the velocity v(r, t) ensuring that the Lagrangian correlation time  is finite and an effective diffusion regime is taking place for large enough times. A sufficient condition valid for  = 0 and static and time-dependent flow is a finite vector potential variance
A2 , where the three-dimensional incompressible velocity v = ∇ × A [10–12]. The correlation time is experimentally known to be finite in fully developed turbulence, whereas subdiffusion (due to particle trapping) and superdiffusion (due to infinite Lagrangian correlation time) are possible in low Reynolds number flows.

2.2.2 TWO-PARTICLE DISPERSION IN A SPATIALLY SMOOTH VELOCITY Even when velocity v(R, t) is a smooth function of the coordinates, Lagrangian dynamics can be ˙ = v(R, t) generally produce chaotic quite complicated. Indeed, d ordinary differential equations R dynamics (for d ≥ 3 already for steady flows and for d = 2 for time-dependent flows). It is thus natural that the tools for the description of what is called chaotic advection [13] are similar to that of the theory of dynamical chaos. The description in this section consistently exploits two simple ideas: to single out the variables that can be represented by the sum of a large number of independent random quantities and to separate variables that fluctuate on different timescales. We are interested here in the distance R12 = R1 −R2 between two fluid particles with trajectories Ri (t) = R(t; ri ) passing at t = 0 through points ri . In the absence of noise, the distance satisfies the equation ˙ 12 = v(R1 , t) − v(R2 , t). R (2.17) If the distance R12 is smaller than the viscous scale of turbulence, then the velocity field can be considered smooth on such a scale and we may expand: v(R1 , t) − v(R2 , t) = (t, R1 )R12 , introducing the strain matrix which is traceless due to incompressibility. As a function of its spatial argument, changes on a scale that is supposed to be much larger than R12 . Then, can be treated as independent of R12 , which thus satisfies locally a linear ordinary differential equation (we omit subscripts replacing R12 by R) ˙ R(t) = (t)R(t). (2.18) This equation, with the strain treated as given and R(0) = r , may be explicitly solved for arbitrary (t) only in the 1D case
t (s) ds ≡ X, (2.19) ln[R(t)/r ] = ln W (t) = 0

expressing W (t) as the exponential of the time-integrated strain. When t is much larger than the correlation time  of the strain, the variable X is a sum of N independent equally distributed random numbers with N = t/. Using (2.13) we get P(r ; R, t) ∝ exp{−t H [t −1 ln(R/r ) − ]},

(2.20)

Here we denoted =
X /t, which is called the Lyapunov exponent and is the growth (or decay) rate of the interparticle distance R(t). The moments
[R(t)] p  behave exponentially as exp[E( p)t]. The convexity of the entropy function leads to the convexity of E( p). This implies, in particular, that even for = E  (0) < 0, high-order moments of R may grow exponentially in time (see Section 2.2.4). In the multidimensional case, the behavior of the vector R is determined by the product of random matrices rather than just random numbers. Still, the main properties of the propagator (sufficient for most physical applications) can be established for an arbitrary strain. The basic idea is coming back to Lyapunov [14] and it found further development in the multiplicative ergodic theorem of Oseledec [15]. © 2006 by Taylor & Francis Group, LLC

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Introduce the evolution matrix W such that R(t) = W (t)R(0). The modulus R is expressed via the positive symmetric matrix W T W . The main result states that in almost every realization of the strain, the matrix t −1 ln W T W stabilizes at t → ∞, i.e., its eigenvectors tend to d fixed orthonormal eigenvectors fi . To understand that intuitively, consider some fluid volume, say a sphere, which evolves into an elongated ellipsoid at later times. As time increases, the ellipsoid is more and more elongated and it is less and less likely that the hierarchy of the ellipsoid axes will change. The limiting eigenvalues

i = lim t −1 ln |W fi | (2.21) t→∞

are called Lyapunov exponents. The major property of the Lyapunov exponents is that they are realization independent if the flow is ergodic (that is, spatial and temporal averages coincide). We arrange the exponents in nonincreasing order. The relation (2.21) tells that two fluid particles separated initially by r pointing into the direction fi will separate (or converge) asymptotically as exp( i t). The incompressibility constraints det(W ) = 1  and

i = 0 imply that a positive Lyapunov exponent will exist whenever at least one of the exponents is nonzero. Consider indeed E(n) = lim t −1 ln
[R(t)/r ]n , t→∞

(2.22)

whose derivative at the origin gives the largest Lyapunov exponent 1 . The function E(n) obviously vanishes at the origin. Furthermore, E(−d) = 0, i.e., incompressibility and isotropy make
R −d  time independent as t → ∞ [16,17]. Negative moments of orders n < −1 are indeed dominated by the contribution of directions R(0) almost aligned to the eigenvectors f2 , . . . , fd . At n < 1 − d, the main contribution comes from a small subset of directions in a solid angle ∝ exp(d d t) around fd . It follows immediately that
R n  ∝ exp[ d (d + n)t] and that
R −d  is a statistical integral of motion. Apart from n = 0, −d, the convex function E(n) cannot have other zeroes if it does not vanish identically. It follows that d E/dn at n = 0, and thus 1 , is positive. A simple way to appreciate intuitively the existence of a positive Lyapunov exponent is to consider the saddle-point two-dimensional flow vx = x, v y = − y with the axes randomly rotating after time interval T . A vector, initially at the angle  with the x-axis, will be stretched after time T if cos  ≥ [1 + exp(2 T )]−1/2 , i.e., the measure of the stretching directions is larger than 1/2 [17]. A major consequence of the existence of a positive Lyapunov exponent for any random incompressible flow is the exponential growth of the interparticle distance R(t). In a smooth flow, it is also possible to analyze the statistics of the set of vectors R(t) and to establish a multidimensional analog of (2.13) for the general case of a nondegenerate Lyapunov exponent spectrum. The idea is to reduce the d-dimensional problem to a set of d scalar problems for slowly fluctuating stretching variables, excluding the fast fluctuating angular degrees of freedom. Consider the matrix I (t) = W (t)W T (t), representing the tensor of inertia of a fluid element like the previously mentioned ellipsoid. The matrix is obtained by averaging R i (t)R j (t)d/2 over the initial vectors of length  and I (0) = 1. Introducing the variables that describe stretching as the lengths of the ellipsoid axis e2 1 , . . . , e2 d , one can deduce similarly to (2.13) and (2.20) the asymptotic PDF [1,18]: P( 1 , . . . , d ; t) ∝ exp [−t H ( 1 /t − 1 , . . . , d−1 /t − d−1 )] ×
( 1 − 2 ) . . .
( d−1 − d ) ( 1 + · · · + d ).

(2.23)

The entropy function H depends on the details of the statistics of and has the same general properties as the preceding it is non-negative, convex, and vanishes at zero. In the -correlated case, H is everywhere quadratic as in Section 2.2.2: H (x) ∝ d

−1

d  i=1

© 2006 by Taylor & Francis Group, LLC

xi2 , i ∝ d(d − 2i + 1).

(2.24)

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For a generic initial vector r, the long-time asymptotics of ln(R/r ) coincides with that of 1 whose PDF also takes the large-deviation form (2.23) at large times. The quadratic expansion of the entropy near its minimum corresponds to the log-normal distribution for the distance between two particles

P(r ; R, t) ∝ exp − [ln(R/r ) − 1 t]2 /(2t) , (2.25) with r = R(0) and  = C11 . Molecular diffusion is incorporated into the preceding picture by replacing the differential equation (2.18) by its noisy version (both independent noises of two particles contribute, hence the change in the noise coefficient comparing to (2.3)): √ dR(t) = (t)R(t) dt + 2 dq(t). (2.26) This is an inhomogeneous linear stochastic equation whose solution is easy to express via the matrix W (t). The tensor of inertia of a fluid element I i j (t) = R i (t)R j (t)d/2 is now averaged over the initial vectors of length  and the noise, thus obtaining [1,18]: I (t) = W (t)W (t) + T

4 2 d


t

W (t) [W (s)T W (s)]−1 W (t)T ds.

(2.27)

0

The last term in (2.27) is essential for the directions corresponding to negative i . The molecular noise will indeed start to affect the motion of the marked fluid volume when the respective dimension gets sufficiently small. If  is the initial size, the required condition i < − i∗ = − ln(2 | i |/) is typically met for times t i∗ /| i |. For longer times, the Brownian motion does not allow the respective i to decrease much below − i∗ , while the negative i prevents it from increasing. As a result, the corresponding i becomes a stationary random process with a mean of the order − i∗ . The relaxation times to the stationary distribution are determined by , ˜ which is diffusion independent, and they are thus much smaller than t. On the other hand, the components j corresponding to non-negative Lyapunov exponents are the integrals over the whole evolution time t. Their values at time t are thus not sensitive to the latest period of evolution lasting of the order of the relaxation times for the contracting i . Fixing the values of j at times t  i∗ /| i | will not affect the distribution of the contracting i and the whole PDF is thus factorized [1,18–20]. For example, there are two positive and one negative Lyapunov exponents in three-dimensional fully developed Navier–Stokes turbulence [21]. For times t  ∗3 / 3 we have then P( 1 , 2 , 3 , t) ∝ exp [−t H ( 1 /t − 1 , 2 /t − 2 )] Pst ( 3 ),

(2.28)

with the same function H as in (2.23) since 3 is independent of  1 and 2 . The account of the molecular noise violates the condition i = 0 because fluid elements at √ scales smaller than /| 3 | cannot be distinguished. To avoid misunderstanding, note that (2.28) does not mean that the fluid is getting compressible: the simple statement is that if one tries to follow any marked volume, the molecular diffusion makes this volume grow. √ Note that we have implicitly√assumed  to be smaller than the viscous length = /| 3 | but larger than the diffusion scale /| 3 |. Even though  and  are due to molecular motion, their ratio widely varies, depending on the type of material. The theory of this section is applicable for materials with a Schmidt number / large. The universal forms (2.23) and (2.28) for the two-particle dispersion are basically everything we need for physical applications. In Chapter 3 and Chapter 4, we show that the most negative Lyapunov exponent determines the small-scale statistics of a passively advected scalar in a smooth incompressible flow. For other problems, the whole spectrum of exponents and even the form of the entropy functions are relevant. © 2006 by Taylor & Francis Group, LLC

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Generally, the Lyapunov spectrum and the entropy function cannot be derived from a given statistic of except for a few limiting cases. The case of a short-correlated strain allows for a complete solution. As far as a finite-correlated strain is concerned, one can express analytically 1 and  via the correlators of only in two dimensions for a long-correlated strain and at large space dimensionality [1].

2.2.3 TWO-PARTICLE DISPERSION IN A NONSMOOTH INCOMPRESSIBLE FLOW We now assume the Reynolds number sufficiently high and study the separation between two trajectories in the inertial interval of scales r L, where L denotes the integral scale at which the flow is induced and is a viscous scale. Let us describe first the usual phenomenology of two-particle dispersion. In the inertial interval, the velocity differences exhibit an approximate scaling. Let us assume v(r, t)| ∝ r  . Rewriting then ˙ = v(R, t), we infer that d R 2 /dt = Equation (2.17) for the distance between two particles as R 1+ 2R · v(R, t) ∝ R . For  < 1, this is solved (ignoring the proportionality constant) by R(t)1− − R(0)1− ∝ t

(2.29)

For large t, R(t) ∝ t 1/(1−) with the dependence of the initial separation quickly wiped out. Of course, for the random process R(t), relation (2.29) is of the mean field type and should pertain (if true) to the large-time behavior of the averages:
R(t) p  ∝ t p/(1−)

(2.30)

for p > 0, implying their superdiffusive growth, faster than the diffusive one ∝ t p/2 . The power-law scaling (2.30) may be amplified to the scaling behavior of the PDF of the interparticle distance: P(R, t) = P( R, 1− t).

(2.31)

Possible deviations from a linear behavior in the order p of the exponents in (2.30) should be interpreted as a signal of multiscaling of the Lagrangian velocity v(R(t), t) ≡ V(t). The powerlaw growth (2.30) for p = 2 and  = 1/3, i.e.
R(t)2 ∝ t 3 , is the celebrated Richardson dispersion relation stating that d
R(t)2  ∝
R(t)2 2/3 . dt

(2.32)

The Richardson relation was the first quantitative phenomenological prediction in fully developed turbulence. It seems to be confirmed by experimental data [22,23] and by numerical simulations [24,25]. The more general property of self-similarity (2.31) (with  = 1/3) has been observed in the inverse cascade of two-dimensional turbulence [23]. It is likely that (2.32) is exact within the inverse cascade of two-dimensional turbulence while it may be only approximately correct in three-dimensions. It is important to remark that, even assuming the validity of the Richardson relation, it is impossible to establish general large-time properties of the PDF P(R; t) such as those for the single particle PDF in Section 2.2.1 or for the distance between two particles in Section 2.2.2 . The physical reason becomes clear looking at the Lagrangian velocity difference correlation time
t
V(t) · V(s) ds/
(V)2 . (2.33) t = 0

The numerator coincides with d
R 2 /dt and is thus proportional to
R 2 2/3 , while the denominator ∝
R 2 1/3 . It follows that t grows as
R 2 1/3 ∝ t, i.e., the random process V(t) has a correlation © 2006 by Taylor & Francis Group, LLC

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time comparable with its whole span. The absence of decorrelation explains why the central limit theorem and the large deviation theory cannot be applied. There is, in fact, no a priori reason to expect P(R; t) to be Gaussian with respect to a power of R although we shall see that this happens to be the case in the Kraichnan ensemble. It is instructive to contrast the exponential growth (2.22) of the distance between the trajectories within the viscous range with the power-law growth (2.30) in the inertial range. In the viscous regime, the closer that two trajectories are initially, the more time is needed to separate them effectively. As a result, the infinitesimally close trajectories never separate and trajectories in a fixed realization of the velocity field are continuously labelled by the initial conditions. They depend, however, in a sensitive way on the latter due to the exponential magnification of small deviation of the initial point. This sensitive dependence is usually considered as the defining feature of the dynamical chaos. On the other hand, in the inertial range, the trajectories separate in a finite time independent of their initial distance R(0), provided that the latter is also in the inertial range. For very high Reynolds numbers, the viscous scale is negligibly small (a fraction of a millimeter in the turbulent atmosphere) and setting it to zero (or equivalently, setting the Reynolds number to infinity) is an appropriate abstraction if we want to concentrate on the behavior of the fluid trajectories in the inertial range. In such a limit, however, the power law separation extends down to infinitesimal distances between the trajectories: the infinitesimally close trajectories still separate to a finite distance in a finite time. This points to a marked difference in the behavior of trajectories in comparison to that in the chaotic regime: fully developed turbulence and chaos are clearly different phenomena. This explosive separation of trajectories results in a breakdown of the deterministic Lagrangian flow in the limit Re → ∞, a rather dramatic effect [26–28]. Indeed, in this limit the trajectories cannot be labelled by the initial conditions. The sheer existence of the Lagrangian trajectories R(t; r) depending continuously on the initial position r would imply that limr1 →r2
|R(t; r1 ) − R(t; r2 )| p  = 0 and contradict the persistence of a power law separation of the Richardson type for infinitesimally close trajectories. The breakdown of the deterministic Lagrangian flow at Re → ∞ agrees with the fundamental theorem stating that the ordinary differ˙ = v(R, t) has unique solution if v(r, t) is Lipschitz in r, i.e., if |v(r, t)| ≤ O(r ). ential equation R At Re = ∞, however, as first noticed by Onsager [29], the velocities are only H¨older continuous: |v(r, t)| O(r  ) with the exponent  < 1 ( 1/3 in Kolmogorov’s phenomenology. As is shown 1 by the example of the equation x˙ = |x| with two solutions, x = [(1 − )t] 1− and x = 0, starting at zero, one should expect multiple Lagrangian trajectories starting or ending at the same point for velocity fields with  < 1. The Lagrangian description of the fluid then breaks down completely at Re = ∞? Even though the deterministic Lagrangian description breaks down, the statistical description of trajectories is still possible. As we have seen earlier, certain probabilistic questions concerning the flow, like the moments of the distance between initially close trajectories, should still have well defined answers in this limit. We expect that for typical velocity realization at Re = ∞, one can maintain a probabilistic description of Lagrangian trajectories and make sense of such objects as the propagator P(r, s; R, t|v). The mathematical difference between the cases of smooth and rough velocities is that, in the latter case, the propagators are weak solutions of (2.8) rather than strong ones. What happens if we turn off molecular diffusion? If the velocity v(r, t) is Lipschitz in r, then P(r, s; R, t|v) converges to (2.9) (we shall call this collapse “property”). It has been conjectured in Gawedzki [28] that, for a generic Re = ∞ turbulent velocity field, P(r, s; R, t|v) at  → 0 is a weak solution of the pure advection equation, [∂t − ∇R · v(R, t)]P(r, s; R, t|v) = 0, that is a solution not concentrated at a single trajectory R(t; r, s). This way the roughness of turbulent velocities resulting in the explosive separation of the Lagrangian trajectories would ensure the persistence of stochasticity of the noisy trajectories in a fixed generic realization of the velocity field even in the limit  → 0. Let us stress again that, according to this claim, in the limit of large Reynolds numbers, the Lagrangian trajectories behave stochastically already in a given velocity field, for negligible molecular © 2006 by Taylor & Francis Group, LLC

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diffusivity without any random noise or random fluctuations of the velocities. This intrinsic stochasticity of fluid particles seems to constitute an important aspect of fully developed turbulence, an unescapable consequence of the Richardson dispersion law or of the Kolmogorov-like scaling of velocity differences in the limit Re → ∞ and a natural mechanism assuring the energy flux constancy in the inertial range of turbulence. The general conjecture about the existence and diffuse nature of propagators is known to be true for the Kraichnan Gaussian ensemble (2.11) of velocities decorrelated in time. To model a nonsmooth velocity field of turbulence, we choose D i j (r) = D0 i j − (1/2)d i j (r) with D0 = O(L  ) and d i j (r) = D1 [(d − 1 + )i j r  − r i r j r −2 ].

(2.34)

As we discussed in Section 2.2.1, D0 gives the eddy diffusivity of a single fluid particle at long times. Notice that D0 is dominated by the integral scale, indicating that the effective diffusion of a single fluid particle is driven by the velocity fluctuations at the largest scales present. On the other hand, di j (r) describes the statistics of the velocity differences:
v i (r, t)v j (r, t  ) = 2(t −t  )d i j (r). It picks up contributions of all scales. In particular, it has a purely scaling limit when r L. The constant D1 has the dimensionality of length 2− time −1 . For 0 <  < 2, the Kraichnan ensemble is supported on the velocities that are H¨older continuous in space with a fixed exponent  arbitrarily close to /2. It mimics this way the main property of the infinite Reynolds number turbulent velocities characterized by fractional H¨older exponents. The rough (distributional) behavior of Kraichnan velocities in time, although not very physical, is not expected to modify essentially the qualitative picture of the trajectory behavior. (It is the spatial regularity, not the temporal one, of a vector field that is crucial for the uniqueness if its trajectories.) In the Kraichnan ensemble, one can directly calculate the Gaussian integral in (2.5) which gives the Gaussian single-point PDF that satisfies the heat equation [∂t − (D0 + )∇r2 ]P(r, t) = 0. That agrees with the all-time diffusive behavior of a single fluid particle in the Kraichnan ensemble characterized by the enhancement of the molecular diffusivity  by the eddy diffusivity D0 discussed at the end of Section 2.2.1. In much the same way, one can examine the joint PDF of the simultaneous values of the coordinates of two fluid particles averaged over the velocity ensemble: P2 (r1 , r2 , s; R1 , R2 , t) =
P(r1 , s; R1 , t|v)P(r2 , s; R2 , t|v).

(2.35)

For the Kraichnan ensemble, it satisfies the equation (∂t − M2 )P2 (r1 , r2 , s; R1 , R2 , t) = (t − s)(R1 − r1 )(R − r2 ) with an explicit elliptic second-order differential operator M2 = −

2  n,n  =1

D i j (rn − rn  )∇rni ∇r j ,

(2.36)

n

a result which goes back to the original work of Kraichnan [4]. If we are interested only in the separation R = R1 − R2 of two fluid particles at time t, given their separation r at time s, then the relevant PDF P2 (r, s; R, t) is obtained by averaging over the simultaneous translations of the final (or initial) positions of the particle and is governed by the operator M2 restricted to the translationally invariant sector. The latter is equal to −d i j (r)∇r i ∇r j . Note that the eddy diffusivity D0 , dominated by the integral scale, drops out in the action on translation-invariant functions. The preceding result shows that the relative motion of two fluid particles in the Kraichnan ensemble of velocities is an effective diffusion with a distance-dependent diffusivity tensor scaling like r  in the inertial range. This is a precise realization of the scenario for the turbulent diffusion © 2006 by Taylor & Francis Group, LLC

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put up by Richardson as far back as 1926 [30]. Similarly, the PDF P2 (r, s; R, t) of the distance R between two particles satisfies the equation (∂t − M2 )P2 (r, s; R, t) = (t − s)(r − R),

(2.37)

where the restriction of M2 to the homogeneous and isotropic sector is M2 = −D1 (d − 1)r 1−d ∂r r d−1+ ∂r and (2.37) can be readily solved [4,31]. At r R, the PDF has a particularly simple form:  R d−1 R 2− lim P2 (r, s; R, t) ∝ exp −const. . (2.38) r →0 |t − s|d/(2−) |t − s| This confirms the diffusive character of the limiting process describing the Lagrangian trajectories in fixed non-Lipschitz velocities: the endpoints of the process stay at finite distance when the initial points converge. If we set  = 0 but maintain finite integral scale L, then the behavior (2.38) is modified for R  L and crosses over to the simple diffusion with the diffusivity 2D0 . At distances much larger than the integral scale, two fluid particles undergo independent Brownian walks driven by the velocity fluctuations on scale L. The PDF (2.38) changes from Gaussian to log-normal when  changes from zero to two. The PDF has the scaling form (2.31) for  =  − 1 and implies the power law growth (2.30) of the averaged powers of the distance between trajectories. The Richardson dispersion
R 2 (t) ∝ t 3 is reproduced for  = 4/3 rather than for  = 2/3 when the spatial H¨older exponent of the typical Kraichnan ensemble velocities takes the Kolmogorov value 1/3. The reason is that the velocity temporal decorrelation cannot be ignored and we should replace the time t in the right-hand side of (2.29) by the Brownian motion (t). That replacement indeed reproduces for  = /2 the large-time PDF (2.38) up to a geometric power-law prefactor. Note the special case of the average
R 2−−d  in the Kraichnan velocities. Since M r 2−−d is a contact term ∝ (r ) for  = 0, one has ∂t
R 2−−d  ∝ P(r ; 0; t). The latter is zero in the smooth case so that
R −d  is a true integral of motion. In the nonsmooth case,
R 2−−d  ∝ t 1−d/(2−) and is not conserved due to a nonzero probability density to find two particles at the same place even when they started apart.

2.2.4 TWO-PARTICLE DISPERSION IN A COMPRESSIBLE FLOW In discussing the particle dispersion in incompressible fluids and exposing the different mechanisms of particle separation, we paid little attention to the detailed geometry of the flows, severely restricted by the incompressibility. The presence of compressibility allows for more flexible flow geometries with regions of ongoing compression effectively trapping particles for long times and counteracting their tendency to separate. To expose this effect and gauge its relative importance for smooth and nonsmooth flows, we start from the simplest case of a time-independent 1d flow x˙ = v(x). In one-dimensions, any velocity is potential: v(x) = −∂x (x), and the flow is the steepest descent in the landscape defined by the potential . The particles are trapped in the intervals where the velocity has a constant sign and they converge to the fixed points with lower value of  at the ends of those intervals. In the regions where ∂x v is negative, nearby trajectories are compressed together. If the flow is smooth, the trajectories take an infinite time to arrive at the fixed points (the particles might also escape to infinity in a finite time). Let us consider now a nonsmooth version of the velocity, e.g., a Brownian path with H¨older exponent 1/2. At variance with the smooth case, the solutions will take a finite time to reach the fixed points at the ends of the trapping intervals and will stick to them at subsequent times, as in the example of the equation x˙ = |x − x0 |1/2 . The roughness of the velocity clearly amplifies the trapping effects leading to the convergence of the trajectories. © 2006 by Taylor & Francis Group, LLC

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A time dependence of the velocity changes the picture somewhat. The trapping regions, as defined for the static case, start wandering and they do not enslave the solutions which may cross their boundaries. Still, the regions of ongoing compression effectively trap the fluid particles for lengthy intervals. Whether the tendency of the particles to separate or the trapping effects win is a matter of detailed characteristics of the flow. In higher dimensions, the behavior of potential flows is very similar to the one-dimensional case, with trapping totally dominating in the time-independent case; its effects are magnified by the velocity roughness and blurred by the time dependence. The traps might, of course, have a more complicated geometry. Moreover, we might have solenoidal and potential components in the velocity. The dominant tendency for the incompressible component is to separate the trajectories, as we discussed in the previous sections. On the other hand, the potential component enhances trapping in the compressed regions. The net result of the interplay between the two components depends on their relative strength, spatial smoothness, and temporal rate of change. Let us consider first a smooth compressible flow with homogeneous and stationary ergodic statistics. As in the incompressible case discussed in Section 2.2.2, the stretching-contraction variables i , i = 1, . . . , d, behave asymptotically as t i with the PDF of large deviations xi = i /t − i determined by an entropy function H (x1 , . . . , xd ). The dasymptotic growth rate of the fluid volume is given by the sum of the Lyapunov exponents s = i=1 i . Note that density fluctuations do not grow in a statistically steady compressible flow because the pressure provides feedback from the density to the velocity field. This means that s vanishes even though the i variables fluctuate. However, to model the growth of density fluctuations in the intermediate regime, one can consider an idealized model with steady velocity statistics having nonzero s. This quantity has the interpretation of the opposite of the entropy production rate (see Section 2.2.2), and it is necessarily ≤ 0 [32,33]. Indeed, in any statistically homogeneous flow, incompressible or compressible, the distribution of particle displacements is independent of their initial position, as is the distribution of the evolution matrix Wi j (t; r) = ∂ R i (t; r)/∂r j . Since the total volume V (assumed finite in this argument) is conserved, the average
det W  is equal to unity for all times and initialpositions, although the determinant fluctuates in the compressible case. The average of det W = e i is dominated at long times by the saddle-point x∗ giving the maximum  of ( i + xi ) − H (x), which must vanish to conform with the total volume conservation. Since xi − H (x) is concave and vanishes at x = 0, its maximum value must be non-negative. We conclude that the sum of the Lyapunov exponents is nonpositive. The meaning of this result is transparent: there are more Lagrangian particles in the contracting regions, which thus acquire higher weight, leading to negative average gradients in the Lagrangian frame. Let us stress the essential difference between the Eulerian and the Lagrangian averages in the compressible case: a Eulerian average is uniform over space, but in a Lagrangian average every trajectory comes with its weight determined by the local rate of volume change. For quantitative description, we employ again the Kraichnan model. The compressible generalization for smooth velocities has the (nonconstant part of the) pair correlation function defined as follows:   d i j (r) = D1 (d + 1 − 2℘) i j r 2 + 2(℘d − 1) r i r j . (2.39) The degree of compressibility ℘ ≡
(∇i v i )2 /
(∇i v j )2  is between 0 and 1 for the isotropic case at hand, with the two extrema corresponding to the incompressible and the potential cases. The corresponding strain matrix = ∇v has the Eulerian mean equal to zero and two-point function
i j (t) k (t  )  equal to

  2 (t − t  ) D1 (d + 1 − 2℘) ik  j + (℘d − 1) i j k + i  jk . (2.40) t in agreement with the The volume growth rate − 0
ii (t) j j (t  ) dt  is thus strictly negative, ∞ general discussion, and equal to −℘ D1 d(d − 1)(d + 2) if we set 0 (t) dt = 1/2. The PDF © 2006 by Taylor & Francis Group, LLC

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P( 1 , . . . , d ; t) takes again the large deviation form (2.23), with the entropy function and the Lyapunov exponents given by references [1,36,37.] ⎡ 2 ⎤  d d   1 − ℘d 1 ⎣ (2.41) x2 + xi ⎦, H (x) = 4D1 (d + ℘(d − 2)) i=1 i ℘(d − 1)(d + 2) i=1

i = D1 [d(d − 2i + 1) − 2℘ (d + (d − 2)i)].

(2.42)

Compare this expression to (2.24). Note how the form (2.41) of the entropy imposes the condition  xi = 0 in the incompressible limit. The inter-particle distance R(t) has the lognormal distribution (2.25) with ¯ = 1 = D1 (d − 1)(d − 4℘) and  = 2D1 (d − 1)(1 + 2℘). Explicitly, t −1 ln
R n  ∝ n[n + d + 2℘(n − 2)] [37]. The quantity R (4℘−d)/(1+2℘) is thus statistically conserved. The highest Lyapunov exponent ¯

becomes negative when the degree of compressibility is larger than d/4 [36,37]. Low-order moments of R, including its logarithm, would then decrease while high-order moments would grow with time. The decrease of the Lyapunov exponents when ℘ grows clearly signals the increase of trapping. The regime with ℘ > d/4, with all the Lyapunov exponents becoming negative, is the one where trapping effects dominate. The dramatic consequences for the scalar fields advected by such flow will be discussed in Section 2.4.2. Analysis of the Kraichnan model for a nonsmooth case demonstrates even stronger effects of compressibility, with an increased tendency for the fluid particles to aggregate in a finite time [1,38]. When the compressibility degree is large enough, even though the velocity is nonsmooth, the Lagrangian trajectories in a fixed velocity field are determined by their initial positions. Moreover, trajectories starting at a finite distance collapse to zero distance and stay together with a positive probability growing with time.

2.2.5 MULTIPARTICLE CONFIGURATIONS AND ZERO MODES We describe here the time-dependent statistics of multiparticle configurations. Our main interest is in the long-time asymptotics of propagators when final distances far exceed initial ones. A particularly important question is what memory of initial configuration remains in the propagators in that limit. We shall see that, to answer this question, one must analyze the conservation laws of turbulent diffusion. As we have seen in the previous subsections, the two-particle statistics are characterized by the single separation vector. In nonsmooth velocities, the length of the vector grows by a power law, while the initial separation is forgotten. Adding extra particles brings geometry into the game. Many-particle evolution in nonsmooth velocities exhibits nontrivial statistical integrals of motion that are proportional to the positive powers of the distances. The integrals involve geometry in such a way that the distance growth is balanced by the decrease of the shape fluctuations. The existence of multi-particle conservation laws indicates the presence of a long-time memory and is a reflection of the coupling among the particles due to the simple fact that they are all in the same velocity field. The conserved quantities may be easily built for the limiting cases. Already for a smooth velocity, id i1 the d-volume i1 i2 ...id R12 . . . R1d is indeed preserved for (d +1) Lagrangian trajectories. In particular, j for any three trajectories in d = 2, the area i j Ri12 R13 of the triangle defined by the three particles remains constant; the growth of the sides is compensated by the decrease of the angle. In the opposite case of a very irregular velocity, the fluid particles undergo a Brownian motion. 2 2 The distances between the Brownian particles grow according to
Rnm (t) = Rnm (0) + Dt. The 2 2 2 2 4 4 statistical integrals of motion are
Rnm − R pr ,
2(d + 2)Rnm R pr − d(Rnm + R pr ), and an infinity of similarly built polynomials (zero modes of Laplacian) where all powers of t cancel out. Another trivial case is the infinite-dimensional flow where the distances between particles do not fluctuate. The two-particle law Rnm (t)1− − Rnm (0)1− ∝ t implies then that the expectation of any function 1− of Rnm − R 1− pr does not change with time. Away from the degenerate limiting cases, the conserved © 2006 by Taylor & Francis Group, LLC

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quantities continue to exist, but they cannot be generally constructed so easily and they depend substantially on the number of particles. We thus see that the existence of conserved quantities is natural. What is nontrivial in a general case is their precise form and their scaling. The intricate statistical conservation laws of multiparticle dynamics were first discovered for the Kraichnan velocities [39,40]. This discovery has led to a new qualitative and quantitative understanding of intermittency of advected fields, as will be described in Chapter 4. It has also revealed the aspects of the multiparticle evolution that seem present and relevant in generic turbulent flows [1,41]. As for many-body problems in other branches of physics (e.g., in kinetic theory or in quantum mechanics), the multiparticle dynamics may bring about new aspects due to the cooperative behavior of particles. In turbulence, such behavior is mediated by the velocity fluctuations correlated at large scales. If the velocities are statistically homogeneous, it is convenient to separate the absolute motion of particles from the relative one, as in the other many-body problems with spatial homogeneity.  For N particles, we define the absolute motion as the one of the mean position R = Rn /N ; as for any single particle, that motion is also expected to be diffusive on time scales longer than the Lagrangian correlation time (Section 2.2.1). Since for such time scales the particles may be considered as moving independently, the diffusivity of the absolute motion is N times smaller than that of a single particle. The statistics of the relative motion of N particles is described by the joint PDF averaged over rigid translations = ( , . . . , ):
rel P N (s, r; R + , t)d , (2.43) P N (r, s; R, t) = The PDF P Nrel describes the distribution of the separations Rnm = Rn − Rm or the relative positions Rrel = (R1 − R, . . . , R N − R). The PDF P N is again expected to show a different short-distance behavior for smooth and nonsmooth velocities. For smooth velocities, the existence of deterministic trajectories leads  = 0 to the collapse property lim

r N →r N −1

P N (r; R; t) = P N −1 (r ; R ; t) (R N −1 − R N ),

(2.44)

where R = (R1 , . . . , R N −1 ) — and similarly for the relative PDFs. If all the distances between the particles are much less than the viscous length, one may consider velocity smooth and approximate the velocity field differences by linear expressions:
 N  P Nrel (r, 0; R, t) = (Rn + − W (t)rn ) d . (2.45) n=1

Clearly, the preceding PDF depends only on the statistics of the evolution matrix W (t) that has been discussed in Section 2.2.2. Under the evolution governed by W (t), all distances between points grow exponentially for large times, while their ratios Rnm /Rkl tend to a constant. For whatever initial positions, asymptotically in time, the points tend to be situated on the line. This behavior and its dramatic consequences for passive scalar statistics are further discussed in Section 2.4.1.1. The long-time asymptotics of the propagators in the nonsmooth case can be found explicitly for the Kraichnan ensemble of velocities. The great simplification of the Kraichnan model consists in the Markov character of the effective N -trajectory processes, which is due to the time decorrelation of the velocities. In other words, the PDF P N (s, r; R, t) and the relative version (2.43) satisfy the second-order differential equations that can be derived by a straightforward generalization of the arguments employed for two particle (see 2.36): (∂t − M N )P N (r, s; R, t) = (t − s)(R − r), MN = −

N  n,m=1

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D i j (rnm )∇rni ∇rmj ,

(2.46) (2.47)

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where D i j (r) = D0 i j − 12 d i j (r) is the spatial part of the velocity two-point function. For the relative process, (∂t + M N )P Nrel (r, s; R, t) = (t − s)(R − r)  MN = d i j (rnm )∇rni ∇rmj .

(2.48) (2.49)

n<m

Note the multibody structure of M N and M N . Since M N scales as length −2 , time should scale as length 2− and P Nrel (r, 0; R, t) = (N −1)d P Nrel ( r, 0; R, 2− t).

(2.50)

Therefore, the asymptotics of the propagator is the same when initial points get close or final points get far apart and time gets large. We expect the multiparticle PDF to be factorized in that limit: lim P Nrel ( r, 0; R, t) =

→0



 f  (r)g (R, t).

(2.51)



Here, we presume the functions to be scale invariant: f  ( r) =  f (r). To find f , g , consider the composition of two PDFs [26]:

rel rel d(1−N ) P Nrel (x, t; y, 0)P Nrel (z, 0; x, ) dx (2.52) P N ( x, t; y, 0)P N (z, 0; x, ) dx =


= P Nrel (x, t; y, 0)P Nrel ( z, 0; x,  2− ) dx = P Nrel (y, t + 2− ; z, 0).

(2.53)

In deriving  (2.53), we have used the scaling relation (2.50) and the composition property of propagators P(x, t1 ; y, t2 )P(y, t2 ; z, t3 ) dy = P(x, t1 ; z, t3 ). Furthermore, one makes Taylor expansion of (2.53) in  and then applies the expansion (2.51) and compares it order by order with the straightforward expansion (2.51) of (2.52). As a result, one can see that f must be taken as zero modes of M N† and its powers, while ∂t g = −M N g . The first term in the expansion is r-independent with the constant f0 = 1 and g0 (R, t) = P Nrel (0, 0; R, t) being the PDF of N initially overlapping particles. The zero mode of M N† with the lowest positive scaling dimension  gives the first nonvanishing r-dependence in the propagator. The remarkable feature of the zero modes of M N† can be appreciated by considering the Lagrangian average of arbitrary translation-invariant functions F of the simultaneous positions of the particles (we assume that R(0) = r and set R ≡ (R1 , . . . , R N −1 )):   F R(t) =




F(R) P Nrel (r, 0; R, t) dR

(2.54)

When F is taken as a zero mode of M N† , it is conserved in mean by the Lagrangian evolution. Indeed, the time derivative of
f (t) vanishes since it brings down M N† acting on f on the right-hand side of (2.54):
  ∂t f R(t) = f (R)M N P Nrel (r, 0; R, t) dR
= P Nrel (r, 0; R, t)M N† f (R) dR

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The importance of the scale-invariant conserved modes for the transport properties of short correlated velocities has been recognized independently in references 39, 40, and 42. To understand how one can have conserved quantities in turbulent diffusion, think about the evolution of N fluid particles as of that of a discrete cloud of marked points in the physical space. There are two elements in the relative evolution of the cloud: the growth of its size and the change of its shape. We shall define the overall size of the cloud as R = [(2N )−1 R2nm ]1/2 and its “shape”  = Rrel /R. as R For example, three particles form a triangle in the space, with labelled vertices, and the notion of shape that we are using includes the orientation of the triangle. To get convinced that zero modes do exist, let us first consider the limiting case  → 0 of very rough velocity fields. In this limit, the operator M N becomes proportional to ∇ 2 , the (N d)-dimensional Laplacian restricted to the translation-invariant sector (note that, for an incompressible flow, the operator is always self-adjoint: M N† = M N ). The relative motion of particles becomes pure diffusion. With R denoting the size-ofthe-cloud variable,  2, (2.55) ∇ 2 = R −d N +1 ∂ R R d N −1 ∂ R + R −2 ∇  2 is the angular Laplacian on the (d N − 1)-dimensional unit sphere of where d N ≡ (N − 1)d and ∇  The spectrum of the latter may be analyzed using the properties of the rotation group. Its shapes R. eigenfunctions  have eigenvalues −( + d N − 2), where  = 0, 1, . . . is the angular momentum.  ∝ t −/2 . To compensate The averages of the angular eigenfunctions decay as follows:
 (R)   for the decay, we introduce the functions f ,0 = R  (R), which are zero modes of the Laplacian with the scaling dimension  — the contributions coming from the radial and the angular parts in (2.55) indeed cancel out. The averages
f ,0  are thus conserved. All the scale-invariant zero modes of the Laplacian are of that form. The polynomials invariant under d-dimensional translations, rotations and reflections 2 can be reexpressed as polynomials in Rnm . For even N , the irreducible O(d)-invariant zero mode with the lowest scaling dimension has then the form f N (R) = R122 R342 . . . R(N2 −1)N + [. . .]

(2.56)

where [. . .] denotes a combination of terms that depend on positions of (N − 1) or fewer particles. d 2 2 4 4 For example, for four points, the zero mode is R12 R34 − 2(d+2) (R12 + R34 ), the example mentioned before. The terms [. . .] are not uniquely determined since we may add to them degree N zero modes for a smaller number of points. Besides, the functions differing from f N by a permutation of points are also zero modes so that we may symmetrize the preceding expressions and look only at the permutation invariant modes. Clearly, the scaling dimension  N = N . The linear in N growth of the dimension signals the absence of the extra attractive effect between the particles diffusing with a constant diffusivity (no particle binding in the shape evolution for  = 0). As we shall see in Section 2.4.2, this leads to the disappearance of intermittency in the advected scalar, which becomes a Gaussian field in the limit  → 0. For small but positive , the scaling dimension of the irreducible four-point zero mode f 4 was first calculated to the linear order in  by Gaw¸edzki and Kupiainen [40]. Parallelly, a similar calculation in the linear order in 1/d was performed by Chertkov et al. [39]. Those two papers present the first analytic calculations of the anomalous exponents in turbulence. A generalization for larger N has been achieved [43,44]: N =

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N (N − 2) N (2 − ) −  2 2(d + 2)

(2.57)

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giving the leading correction ∝  to the scaling dimension of the lowest irreducible zero mode. Note that, to that order, the scaling dimension  N is a concave function of N . This can be interpreted as a result of particle interaction. From the form (2.49) of the generator of the process Rrel (t), we infer that, in the Kraichnan model, N fluid particles undergo an effective diffusion with the diffusivity depending on the interparticle distances. In the inertial interval of distances r L, where d i j (r) ∝ r  , the effective diffusivity grows as the power  of the distance. It should be intuitively clear that, in comparison to the standard diffusion with constant diffusivity, the particles will tend to spend a longer time together when they are close and to separate faster when they become distant. Let us stress that (2.51) is not a spectral decomposition of the resolvent M N−1 . (Since M N is positive with a continuous spectrum, such decomposition would be a continuous integral involving eigenfunctions.) The scaling zero modes that govern the small-scale asymptotics are rather analogous to resonances in many-body systems with the scaling dimension  playing the role of energy. It is thus instructive to compare the shape vs. size stochastic evolution of the Lagrangian cloud to the imaginary-timeevolution of the  quantum-mechanical many-particle systems governed by the Hamiltonians HN = n pn2 /2m + n
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2.3 UNFORCED EVOLUTION OF PASSIVE FIELDS The qualification “passive” means that we disregard the back reaction of the advected fields on the advecting velocity. The first section of this chapter is devoted to the statistical initial value problem: how does an initially created distribution of a passive scalar evolve in a statistically steady turbulent environment? The simplest question to address is which fields have their amplitudes decaying in time and which growing, assuming the velocity field to be statistically steady. We consider the tracer (scalar density per unit mass) which satisfies the advection–diffusion equation: ∂
(2.58) + (v · ∇)
= ∇ 2
∂t and the scalar density per unit volume n (to be called concentration), whose evolution is governed by the continuity equation ∂t n + ∇ · (n v) = ∇ 2 n. (2.59) For incompressible flows, (2.58) and (2.59) obviously coincide. A tracer field always decays because of dissipative effects, with the law of decay depending on the velocity properties. The fluctuations of a passive density may grow in a compressible flow, with this growth saturated by diffusion after some time. We shall also briefly consider vector fields advected by the flow. A potential vector field can be considered as the gradient of a tracer  = ∇
, obeying ∂t  + ∇(v · ) = ∇ 2 .

(2.60)

A solenoidal vector field (e.g., magnetic field) evolves in an incompressible flow according to ∂t B + v · ∇B − B · ∇v = ∇ 2 B.

(2.61)

The fluctuations of  and B may grow exponentially as long as diffusion is unimportant. After diffusion comes into play, their destinies are different :  decays, while the magnetic field continues to grow. This growth is known as dynamo process and it continues until saturated by the back-reaction of the magnetic field on the velocity. Another important issue here is the presence or absence of a dynamic self-similarity : for example, is it possible to present the time-dependent PDF P(
; t) as a function of a single argument? In other words, does the form of the PDF remain invariant in time apart from a rescaling of the field? We shall show that, for large times, the scalar PDF tends to a self-similar limit when the advecting velocity is nonsmooth, while self-similarity is broken in smooth velocities.

2.3.1 DECAY

OF TRACER

FLUCTUATIONS

For practical applications, e.g., in the diffusion of pollution, the most relevant quantity is the average

(r, t). It follows from (2.2) that the average concentration is related to the single particle propagation discussed in Section 2.2.1. For times longer than the Lagrangian correlation time, the particle diffuses and

 obeys the effective heat equation ∂t

(r, t) = (Dei j + i j ) ∇i ∇ j

(r, t),

(2.62)

with the eddy diffusivity Dei j given by (2.16). The simplest decay problem is that of a uniform scalar spot of size  released in the fluid. Its averaged spatial distribution at later times is given by the solution of (2.16) with the appropriate initial condition. On the other hand, the decay of the scalar in the spot is governed by the multipoint Lagrangian propagators. Another relevant situation is that where a homogeneous statistic with correlations decaying on the scale  is initially prescribed. Taking the point of measurement inside the

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spot or averaging over space for a homogeneous statistics, consider the single-point moment

N (t) described by (2.2):
(2.63)

N (t) = P N (0, t; R, 0)
(R1 , 0) . . .
(R N , 0) dR. If there is no molecular diffusion and the trajectories are unique, particles that end at the same point remain together throughout the evolution and all the moments

N (t) are preserved. From what we have learned in Chapter 2, we expect the preservation at the limit  → 0 when velocity field is smooth or has its nonsmoothness overcame by compressibility. Note that the conservation laws are statistical in the last case: the moments are not dynamically conserved in every realization, but their averages over the velocity ensemble are. On the contrary, when velocity field is nonsmooth and the propagator is diffusive, we expect the decay of the tracer moments even at the limit  → 0. This is an example of the so-called dissipative anomaly, which we shall discuss more later. One calls anomaly a finite effect of symmetry breaking even when the symmetry-breaking factor goes to zero. Here, the symmetry broken by molecular diffusion is time reversibility. 2.3.1.1 Smooth Velocity

Let us start from the simplest problem: consider a small spherical spot of the tracer
released in a spatially smooth incompressible three-dimensional velocity field with 1 > 2 > 0 > 3 . Physically, we imply the Schmidt number Sc = / to be large — that is, the √viscous scale of the flow is much larger than the diffusion scale of the scalar defined as rd = −/ 3 . The initial size of the spot L is assumed to satisfy  L  rd . The spot is stretched and contracted by the velocity field. As we have shown in Section 2.3.1, during the time less that td = | 3 |−1 ln(L/rd ), diffusion is unimportant and
inside the spot does not change. At larger time, the dimensions of the spot with negative Lyapunov exponents are frozen at rd , while the rest keep growing exponentially, resulting in an exponential growth of the total volume exp( 1 + 2 ). This leads to an exponential decay of scalar moments averaged over velocity statistics:
[
(t)]  ∝ exp(− t). The decay rates  can be expressed via the PDF (2.28) of stretching variables i . Since
decays as the inverse volume, then
(2.64)
[
(t)]  ∝ d 1 d 2 exp [−t H ( 1 /t − 1 , 2 /t − 2 ) − ( 1 + 2 )]. At large t, the integral is determined by the saddle point. At small , the saddle point lies within the parabolic domain of H , so  increases with  quadratically. At large , the main contribution is due to the realization with smallest possible spot, which has the volume L 3 , so  is independent of  [18,19,45,46]. Let us consider now an initial random distribution of
(0, r) statistically homogeneous in space. We pass to the reference frame, which moves with the Lagrangian point R(t|T, r0 ), coming, to r0 at T . Such
(t, r) =
˜ (t, r − R(t|T, r0 )) satisfies ∂t
˜ + ˜ r ∇
˜ = ∇ 2
˜ .

(2.65)

Since the correlation functions of
and
˜ coincide at the moment of observation, we omit the tilde sign in what follows. One may treat diffusion in two equivalent ways: by introducing Brownian motion or by making Fourier transform in (2.65). Here, for a change, we choose the second way, defining the time-dependent wavevector k(t  ) = W T (t, t  )k(t) and solving (2.65) as follows

 (2.66)
(t, k) =
0 W T (t)k exp [−Q  k k ],
t  T (2.67) Q(t) =  dt  W (t)W −1 (t  ) W (t)W −1 (t  ) . 0

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 The moments of
(t, 0) = dk(2)−3
(t, k) are to be averaged over velocity statistics and over the initial statistics of the scalar. Because the long-time limit is independent of the statistics of
(0, r) [18], we take it Gaussian with

(0, r)
(0, 0) = (r ) = 0 exp[−r 2 /(8L 2 )]. Then, the moments of
are as follows:

[
(t)]  ∝ d 1 d 2 exp [−t H − ( 1 + 2 )/2] . (2.68) Notice that in (2.68), the scalar amplitude is proportional to the square root of the volume factor as distinct from (2.64). This difference can be intuitively understood by imagining initially different blobs of size L with uncorrelated values of
. At time t, those blobs overlap. The mutual cancellations of
from different blobs lead to the law of large numbers with initial statistics forgotten and the rms value of
proportional to the square root of the number of blobs. The number of blobs is inversely proportional to the volume exp( 1 + 2 ). As in (2.64), the same qualitative conclusions about the decay rates  = limt→∞ t −1 ln{
[
(t)] } can be drawn from (2.68) — in particular, for the Kraichnan model  ∝ (1 − /8) for  < 4 and  = const for  > 4 [18]. Note that in both cases (single spot and random homogeneous distribution),  is not a linear function of , so the scalar decay is not self-similar in a smooth velocity. 2.3.1.2 Nonsmooth Velocity

For the decay in incompressible nonsmooth flow, we shall specifically consider the case of a timereversible Kraichnan velocity field. The comments on the general case are reserved until the end of the subsection. The simplest objects to investigate are the single-point moments

2n (t) and we are interested in their long-time behavior t  2− /D1 . Here,  is the correlation length of the random initial field and D1 enters the velocity two-point function as in (2.34). Using (2.2) and the scaling property (2.50) of the Green function, we obtain
 1  2n

(t) = P2n (0; R; −1) C2n t 2− R, 0 dR. (2.69) There are two universality classes for  this problem, corresponding to nonzero or vanishing value of the so-called Corrsin integral J0 = C2 (r, t) dr. Note that the integral is generally preserved in time by the passive scalar dynamics. We concentrate here on the case J0 = 0 and refer the interested reader to Chaves et al. [47] for d 1 more details. For J0 = 0, the function t 2− C2 (t 2− r, 0) tends to J0 (r) in the long-time limit and (2.69) is reduced to
nd P2n (0; R1 , R1 , . . . Rn , Rn ; −1) dR, (2.70)

2n (t) ≈ (2n − 1)!! J0n t −2 for a Gaussian initial condition. A few remarks are in order. First, the previous formula shows that the behavior in time is selfd d 2(2−) Q(t 2(2−)
). That means that similar. In other √words, the single point PDF P(t,
) takes the form t the PDF of
/ ¯ is asymptotically time independent with ¯ (t) = 
(∇
)2  being time dependent (decreasing) dissipation rate. This should be contrasted with the lack of self-similarity found previously for the smooth case. Second, the result is asymptotically independent of the initial statistics (of course, within the universality class J0 = 0). As in the previous subsection, this follows from the fact that the connected non-Gaussian part of C2n depends on more than n separation vectors. Its nd contribution is therefore decaying faster than t − 2− . Third, it follows from (2.70) that the long-time PDF, although universal, is generally non-Gaussian. Its Gaussianity would indeed imply the factorization of the probability for the 2n particles to collapse in pairs at unit time. Due to the correlations existing among the particle trajectories, this is generally not the case, except for  = 0, where the © 2006 by Taylor & Francis Group, LLC

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particles are independent. The degree of non-Gaussianity is thus expected to increase with  [47]. Finally, notice the dissipative anomaly as the decay laws are independent of . Other statistical quantities of interest are the structure functions S2n (r, t) =
[
(r, t) −
(0, t)]2n  related to the correlation functions by
1
1  S2n (r, t) = ... ∂1 . . . ∂2n C2n (1 r, . . . , 2n r, t) di ≡ (r) C2n (·). 0

0

To analyze their long-time behavior, we proceed similarly as in (2.69) and use the asymptotic expansion (2.51) to obtain
1 1 S2n (r, t) = (t − 2− r) P2n ( · ; R; −1) C2n (t 2− R, 0) dR
 r 2n 2n 1 g2n,0 (R, −1) C2n (t 2− R, 0) dR ∝ ≈ (r) f 2n (·)t −2

2n .  Here, f 2n is the irreducible zero mode in (2.51) with the lowest dimension and the scalar integral 1 scale  ≡ (t) ∝ t 2− . As we shall see in Section 2.4, the scaling dimensions of the zero modes, 2n , give also the scaling exponents of the structure function in the stationary state established in the forced case. Let us briefly discuss the scalar decay for velocity fields having finite correlation times. The key ingredient for the self-similarity of the scalar PDF is the rescaling (2.50) of the propagator. This property is generally expected to hold (at least for large enough times) for self-similar velocity fields regardless of their correlation times. This has been confirmed by the numerical simulations in Chaves et al. [47]. For an intermittent velocity field, the presence of various scaling exponents makes it unlikely that the propagator possesses a rescaling property like (2.50). The self-similarity in time of the scalar distribution might then be broken. 2.3.1.3 Scalar Decay with Viscous and Inertial Intervals Present

Even when the Schmidt/Prandtl number is large and the initial scale of the scalar field l is smaller than the viscous scale , separation of initially close particles brings their distance eventually into the inertial interval. Until the time of order −1 ln( /l), the sizes of scalar blobs are contained within the viscous interval and the decay proceeds as described in Subsection 2.3.1.1. After that time, however, large-scale structures of passive field are created with sizes in the inertial interval. The number of such structures overlapping after time t now grows as power of t as in Subsection 2.3.1.1. As a result, the structure function decays by a power law (that is slower than exponential) even in the viscous interval [48]: S2 (r, t) t −2−d/(2−) ln(r/rd ) at rd r , t  −1 ln( /l).

(2.71)

Logarithmic r -dependence corresponds to a steady cascade of a scalar in a smooth velocity according to (2.84). One can interpret (2.71) as describing a cascade with a time-dependent flux; such interpretation is meaningful since the the flux changes (due to inertial–interval dynamics) much more slowly than the cascade proceeds below the viscous scale. At late times, the inertial interval thus serves as a reservoir of passive scalar. Note that the large-time law of decay of the single-point moments

2n (t) is unknown in this case. 2.3.1.4 Scalar Decay in a Finite Box

Finiteness of the flow restricts the time when the preceding description (based on the separation of fluid particles) is valid. Consider the behavior of the average concentration

(r, t) in a spatially smooth chaotic flow in a finite box of size L. Until time of order −1 ln(L/rd ), scalar decay proceeds exponentially as described in the Section i). After the average size of the scalar blob reaches the box © 2006 by Taylor & Francis Group, LLC

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size, scalar decay in the bulk is getting generally nonuniversal, that is, depending on the large-scale structure of the flow [49,50]. According to Chertkov and Lebedev [48], the main remaining scalar field is concentrated near solid boundaries where stretching is suppressed since flow incompressibility and no-slip conditions require zero velocity gradient perpendicular to the boundary. At the distance q from the boundary, normal velocity is thus proportional to q 2 while tangential velocity is proportional to q. Scalar diffusion is determined by the normal velocity while the correlation time is determined by the (inverse) gradient of the tangential velocity. This makes mixing short-correlated at q L with the eddy diffusivity (2.16) proportional to q 4 . Considering Equation (2.62) with D(q) ∝ q 4 , one can readily establish that the width  of the boundary region where the scalar is preserved shrinks as t −1/2 . During that stage, the leakage from the boundary regions makes the flux into the bulk D(q)∂q

 of decay as 3 ∝ t −3/2 , which makes the scalar concentration in the bulk decrease by the same law

(r, t) ∝ t −3/2 . After the regions shrink to the size of the diffusive boundary layer where D  (the width of the layer is proportional to 1/4 ),

 decays exponentially with the rate proportional to 1/2 (This was also verified experimentally [51]).

2.3.2 GROWTH OF DENSITY FLUCTUATIONS IN COMPRESSIBLE FLOW The evolution of a passive density field n(r, t) is governed by Equation (2.59). Consider smooth velocities and neglect diffusion. The density n changes along the trajectory as the inverse of the ˜ r) = W (t; R(0; r, t)), where W (t; r) volume contraction factor. Let us introduce the matrix W(t; describes the forward evolution of small separations of the Lagrangian trajectories starting at time ˜ r)) and zero near r. The volume contraction factor is det(W(t; ˜ (t; r))]−1 n(R(0; r, t), 0). n(r, t) = [det(W

(2.72)

˜ r) is the inverse of the backward-in-time evolution matrix W  (t; r) with Note that the matrix W(t; i the matrix elements ∂ R (0; r, t)/∂r j . This is indeed implied by the identity R(t; R(0; r, t), 0) = r and the chain rule for differentiation. We shall take the initial field on the right-hand side of (2.72) ˜ (t; r))]−1 . Performing the velocity average and recalling to be uniform. This gives n(r, t) = [det(W ˜ the long-time asymptotics of the W statistics, we obtain  
  
n  ∝ exp (1 − ) i − t H ( 1 /t − 1 , . . . , d /t − d ) d i . (2.73) i

The moments at long times may be calculated by the saddle-point method and they generally behave as ∝ exp( t). The growth rate function  is convex, due to H¨older inequality, and vanishes at the origin and for  = 1 (by the total mass conservation). This leads to the conclusion that  is negative for 0 <  < 1 and is otherwise positive: low-order moments decay, whereas high-order and negative moments grow. For a Kraichnan velocity field, the large deviation function H is given by (2.41) and the density field becomes lognormal with  ∝ ( − 1) [34]. Notethat the asymptotic rate
ln n/t is given by the derivative at the origin of  and it is equal to − ˜ i ≤ 0. The density is thus decaying in almost any realization if the sum of the Lyapunov exponents is nonzero. Since the mean density is conserved, it must grow in some (smaller and smaller) regions, which implies the growth of high moments. The amplification of negative moments is due to the growth of low-density regions. The positive  quantity − i has the interpretation of the mean  (Gibbs) entropy  production rate per unit volume. Indeed, if we define the Gibbs entropy S(n) as − (ln n)n dr = ln det(W (t; r)) dr, then  the entropy transferred to the environment per unit time and unit volume is − ln det(W )/t = − i /t and it is  asymptotically equal to − i > 0 [32]. © 2006 by Taylor & Francis Group, LLC

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The behavior of the density moments discussed previously is the effect of a linear but random hyperbolic stretching and contracting evolution of the trajectory separations. In a finite volume, the linear evolution is eventually superposed with nonlinear bending and folding effects. In order to capture the combined impact of the linear and the nonlinear dynamics at long times, one may observe at fixed time t the density produced from an initially uniform distribution imposed at much earlier times t0 . When t0 → −∞ and if 1 > 0, the density approaches weakly, i.e., in integrals against test functions, a realization-dependent fractal density n ∗ (r, t) in almost all the realizations of the velocity. The resulting density field is the so-called SRB (Sinai–Ruelle–Bowen) measure. The fractal dimension of the SRB measures may be read from the values of the Lyapunov exponents. For the Kraichnan ensemble of smooth velocities, the SRB measures have a fractal dimension equal to 1 + 1−2℘ if 0 < ℘ < 12 in two-dimensions. In three-dimensions, the dimension is 2 + 1−3℘ if 1+2℘ 1+2℘ if 13 ≤ ℘ < 34 , where ℘ is the compressibility degree [36]. 0 < ℘ ≤ 13 and 1 + 3−4℘ 5℘ The preceding considerations show that, as long as one can neglect diffusion, the passive density fluctuations grow in a random compressible flow. One particular case of the prior phenomena is the clustering of inertial particles in an incompressible turbulent flow (see references 33 and 35), where the theory for a general flow and the account of the diffusion effects that eventually stop the density growth were presented.

2.3.3 VECTOR FIELDS IN A SMOOTH VELOCITY 2.3.3.1 Gradients of the Passive Scalar

For the passive scalar gradients  = ∇
in an unforced incompressible situation, we solve (2.60) by simply taking the gradient of the scalar expression (2.66). The initial distribution is assumed statistically homogeneous with a finite correlation length. The long-time limit is independent of the initial scalar statistics [18] and it is convenient to take it Gaussian with the two-point function ∝ exp[− 2d1 (r/)2 ]. The averaging over the initial statistics for the generating function Z(y) =
exp [iy · ] reduces then to Gaussian integrals involving the matrix I (t) determined by (2.27). The inverse Fourier transform is given by another Gaussian integral over y and one finally obtains for the PDF of : √ (2.74) P() ∝
(det I )d/4+1/2 exp[−const. det I (, I )]. As may be seen from (2.27), during the initial period t < td = | −1 d | ln(/rd ), the diffusion is unimportant, the contribution of the matrix Q to I is negligible, the determinant of the latter is unity, and 2 grows as the trace of I −1 . In other words, the statistics of ln  and of − d coincide in the absence of diffusion. The statistics of the gradients can therefore be immediately taken over from Section 2.3.2. The growth rate (2t)−1
ln 2  approaches | d | while the gradient PDF depends on the entropy function. For the Kraichnan model, the PDF is lognormal with the average ∝ d(d − 1)t and the variance ∝ 2(d − 1)t is read directly from (2.24). This result was obtained by Kraichnan [52] using the fact that, without diffusion,  satisfies the same equation as the distance between two particles whose PDF is (2.25). As time increases, the wavenumbers (evolving as k˙ = T k) reach the diffusive scale rd−1 and the diffusive effects start to modify the PDF, propagating to lower and lower values of . First moments first and then lower ones will start to decrease. The law of decay at t  td can be deduced from (2.74). Considering this expression in the eigenbasis of the matrix I , we observe that the dominant component of  coincides with the largest eigendirection of the I −1 matrix, i.e., the one along the d axis. Recalling from the Section 2.3.2 that the distribution of d is stationary, we infer that
|| (t) ∝
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The comparison with (2.68) shows that the decay laws for the scalar and its gradients coincide [18,45]. This is qualitatively understood by estimating  ∼
/min , where min is the smallest size of the spot. Noting that
and min are independent and that min ≈ e d  at large times has a stationary statistic concentrated around rd , it is quite clear that the decrease of  is due to the decrease of
. 2.3.3.2 Small-Scale Magnetic Dynamo

The magnetic fields of stars and galaxies are thought to have their origin in the turbulent dynamo action. In this problem, the magnetic field can be treated as passive. Furthermore, the viscosityto-diffusivity ratio is often large enough for a sizable interval of scales between the viscous and the diffusive cut-offs to be present. That is the region of scales with the fastest growth rates of the magnetic fluctuations. In this section, we consider the generation of inhomogeneous magnetic fluctuations below the viscous scale of incompressible turbulence. The dynamo process is caused by the stretching of fluid elements extensively discussed earlier; the major new point to be noted is the role of diffusion. In an ideal conductor, when the diffusion is absent, the magnetic field satisfies the same equation as the infinitesimal separation between two fluid particles (2.18): dB/dt = B. Any chaotic flow would then produce dynamo, with the growth rate ¯ = lim (2t)−1
ln B 2 , t→∞

(2.75)

equal to the highest Lyapunov exponent 1 . Recall that the gradients of a scalar grow with the growth rate − 3 during the diffusionless stage. If the initial scale of magnetic fluctuations is l, then for time less than td = | 3 | ln(l/rd ), the growth rate is insensitive to diffusion. The long-standing problem solved in Chertkov et al. [53] was whether the presence of a small, yet finite, diffusivity could stop the dynamo growth process at t > td (as is the case for the gradients of a scalar). In a smooth flow, the magnetic field can be expressed in terms of the stretching matrix W and the backward Lagrangian trajectory: B(r, t) = W (t; r) B(R(0; r, t), 0). The realizations contributing to the moments of B are those with the interparticle separations almost orthogonal to the (backward) expanding direction 3 of W −1 . The share of such realizations decreases as the angle ∝ exp( 3 ). As a result, moments of the magnetic field are to be obtained by averaging moments of B 2 ∝ exp(2 1 + 3 ) [1,53]. In particular, the growth rate is now 1 + 3 /2, which is less than in a perfect conductor. Note that the gradients of a scalar field are stretched by the same W −1 matrix that governs the growth of the Lagrangian separations. It is therefore impossible to increase the stretching factor of the gradient and keep the particle separation within the correlation length  at the same time. That is why diffusion eventually kills all the gradients while the component B i , which points into the direction of stretching, survives and grows with ∇ B i perpendicular to it. This simple picture also explains the absence of dynamo in two-dimensional incompressible flow, where the stretching in one direction necessarily means the contraction in the other. To conclude this chapter, note that an important lesson to learn is that the limits  → 0 and t → ∞ do not commute for a smooth flow. Growth/decay rates of scalars and vectors are different before and after time td ∼ −1 ln(l/rd ). This difference is independent of diffusivity.

2.4 CASCADES OF A PASSIVE TRACER This chapter describes forced turbulence of the passive scalar
under the action of pumping which is statistically stationary in time and statistically homogeneous in space. To the advection–diffusion equation ∂t
+ (v · ∇)
= ∇ 2
+  (2.76) we added the pumping , characterized by the variance
(t, r)(0, 0) = (r )(t) with (r ) constant at r < L and decaying fast at r > L. © 2006 by Taylor & Francis Group, LLC

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The following considerations are valid for a finite-correlated pumping too, as long as the pumping correlation time in Lagrangian frame is much smaller than the time of stretching from a given scale to the pumping correlation scale L. In most physical situations, the sources do not move with the fluid, so the Lagrangian correlation time of the pumping is its Eulerian correlation time or L/V , depending on which one is smaller; here, V is the typical fluid velocity. The scalar field along the Lagrangian trajectories R(t) changes as d
(R(t), t) = (R(t), t). dt The N th order scalar correlation function

(r1 , t) . . .
(r N , t) is therefore given by
t
t . . .
(R1 (s1 ), s1 ) . . . (R N (s N ), s N )ds1 . . . ds N , 0

(2.77)

(2.78)

0

with the Lagrangian trajectories satisfying the final conditions Ri (t) = ri . For the sake of simplicity, we have written the expression for the case where the scalar field was absent at t = 0. Averaging (2.78) over the Gaussian pumping, we get for N = 2: 
t 

(r1 , t)
(r2 , t) =  (R12 (s)) ds . (2.79) 0

Higher order correlations are obtained similarly to (2.79) by using the Wick rule to average over the Gaussian forcing statistics. The remaining average is made over the ensemble of Lagrangian trajectories:
t

(t, r1 ) . . .
(t, r2n ) = dt1 . . . dtn 0   × (R(t1 |T, r12 )) . . . (R(tn |T, r2n−1,2n )) + . . . , (2.80) The functions  in (2.79) and (2.80) restrict integration to the time intervals where Ri j < L. If the Lagrangian trajectories separate, the correlation functions reach at long times the stationary form for all ri j . Such steady states correspond to a direct cascade of the tracer (i.e., from large to small scales) and are considered in Section 2.4.1. As we have seen in Section 2.2.4, particles cluster in flows with high enough compressibility. In this case, the correlation functions acquire parts which are independent of r and grow proportional to time: when Lagrangian particles cluster rather than separate, tracer fluctuations grow at larger and larger scales. This phenomenon can be loosely called an inverse cascade of a passive tracer [54,55] and is considered in Section 2.4.2.

2.4.1 DIRECT CASCADE Here we consider incompressible (and weakly compressible) flows where particles separate and the steady state exists. Let us first present the standard flux phenomenology. Assuming stationarity, one derives the flux relation of
2
(v1 · ∇1 + v2 · ∇2 )
1
2  + 2 
∇1
1 · ∇2
2  = (r12 ),

(2.81)

where indices designate spatial points. The relative strength of the two terms on the left-hand side depends on the distance. At some rd (called the diffusion scale), the advection is comparable to diffusion. In the “diffusive interval” r12 rd , the diffusion term dominates in the left-hand side of (2.81). Taking the limit of vanishing separations, we infer that the mean dissipation rate is equal to the mean injection rate © 2006 by Taylor & Francis Group, LLC

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¯ ≡
(∇
)2  = 12 (0). This illustrates the aforementioned phenomenon of the “dissipative anomaly”: the limit  → 0 of the mean dissipation rate is nonzero despite the explicit  factor in its definition. In the “convective interval” rd r12 L, one can drop the diffusive term in (2.81) while still neglecting r -dependence in :
(v1 · ∇1 + v2 · ∇2 )
1
2  ≈ (0).

(2.82)

Relation (2.82) states that the mean flux of
2 stays constant within the convective interval and expresses analytically the downscale scalar cascade. The physical picture is that stretching and contraction by an inhomogeneous velocity provides for a cascade of a scalar from the pumping scale L (where it is generated) to the diffusion scale rd (where it is dissipated). For velocity fields scaling as v ∝ r  , dimensional arguments suggest that [56,57]: 
∝ r (1−)/2 ,

(2.83)

which gives a qualitative understanding that the degrees of roughness of the scalar and the velocity are complementary, yet suggests a wrong scaling for the scalar structure functions of an order higher than two (see Section 2.5.1 below). For smooth velocity, (2.82) correctly suggests

1
2  ∝ ln r12 [58]. 2.4.1.1 Direct Cascade in a Smooth Velocity

In this subsection, all the scales are supposed to be much smaller than the viscous scale of turbulence so that the velocity field can be assumed spatially smooth and we may use the Lagrangian description developed in Section 2.2.2. We restrict ourselves by the incompressible case when 3 < 0 so that particles do separate and the steady state exists. We first treat the interval of scales between the diffusion scale rd and the pumping scale L, which is called convective interval. Formula (2.79) simply tells us that the stationary pair correlation function of a tracer is twice the flux (i.e., (0)) times the average time that two particles spent in the past within the correlation scale of the pumping: −1 2

(0, t)
(r, t) = − −1 3 (0) ln(L/r ),

 = − 3 (0) ln(L/rd ) S2 (r ) = − −1 3 (0) ln(r/rd ).

(2.84)

Deep inside the convective interval when r L, the statistics of passive scalar approaches Gaussian. Indeed, when we average (2.80) over P( ) and perform summation over all sets of the pairs of the points ri , the reducible part in
[r12 e d (t1 ) ] . . . [r2n−1,2n e d (tn ) ] prevails for n less than the ratio between the transfer time | d |−1 ln(L/r ) and the correlation time s of the stretching rate fluctuations. The reason is that the irreducible contributions have less large logarithmic factors than the reducible ones [59]. Therefore, for n n cr ( d s )−1 ln(L/r ), the statistics of the passive tracer is Gaussian. Since L  r , n cr  1. The single-point statistics is Gaussian up to n cr

( d s )−1 ln(L/rd ). Larger n correspond to the exponential tails of tracer’s pdf. The physics behind this is transparent and most likely valid also for a nonsmooth velocity (even though the consistent derivation is absent in the nonsmooth case). Indeed, large values of the scalar can be achieved only if, during a large time, the pumping works are uninterupted by advection (which eventually brings diffusion into play). When the time in question is much larger than the typical stretching time from rd to L, the stretching events can be considered as a Poisson process and the probability is that no stretching occurs during time t is exp(−ct). Integrating that with a pumping-produced distribution, we get: © 2006 by Taylor & Francis Group, LLC

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 √ P(
) ∝ dt exp(−ct −
2 /2t) ∝ exp(−
2c/). The detailed derivation for a smooth case can be found in references 18, 19, 26, and 59. From a general physical viewpoint, it is of interest to understand the properties of turbulence at scales larger than the pumping scale, i.e., at r > L. If only direct cascade exists, one may expect equilibrium equipartition at large scales with the effective temperature determined by small-scale turbulence [60,61]. The peculiarity of our problem is that we consider scalar fluctuations at scales larger than the scale of excitation yet smaller than the correlation scale of the velocity field, which provides for mixing of the scalar. In a smooth flow, the statistics at large scales lack scale invariance and are very far from Gaussian [62]. The probability for two points separated by r12 to belong to the same blob of scalar originated from the pumping scale L is (L/r12 )d . Therefore, the pair correlation function is proportional to −d r12 . Since an advection by a smooth velocity preserves straight lines, the same answer is true for the correlation function of arbitrary order if all the points lie on a line (When the largest distance between points was within L, all other distances were as well.): Cn ∝ r −d . The fact that, for collinear geometry, C2n /C2n ∼ (r/L)(n−1)d  1 is due to strong correlation of the points along the line. When we consider a noncollinear geometry, the opposite takes place, namely, the stretching of different nonparallel vectors is generally anti-correlated because of incompressibility and volume  conservation. Consider the two-dimensional case and the contribution from dt1 dt2
[R12 (t1 )] [R34 (t2 )] into the fourth-order correlation function. Since the area |R12 × R34 | is conserved, the answer is crucially dependent on the relation between |r34 × r12 | and L 2 . When |r34 × r12 | L 2 , we have a collinear answer C4 ∝ r −2 . Let us now consider the case of noncollinear geometry and find the probability of an event that during evolution R12 became of the order L, and then, at some other moment of time, R34 reached L (only such events will contribute into C4 ). There is a reducible part in pumping, which makes C4 nonzero (decaying as power of ri j ) even when |r34 × r12 |  L 2 . 2 . Due to area conservation, there is an anticorrelation The probability that R12 came to L is L 2 /r12 between R12 and R34 : if R12 ∼ L, then R34 ∼ r12r34 /L. So the probability for R34 to come back to L 2 2 is L 2 /(r12r34 /L)2 = L 4 /r12 r34 . Therefore, the total probability can be estimated as L 6 /r 6 , which is much smaller than the naive Gaussian estimation L 4 /r 4 ; the collinear answer L 2 /r 2 is much larger than Gaussian. The breakdown of scale invariance is related to the Lagrangian conservation laws. More details can be found in references 1 and 62. 2.4.1.2 Anomalies of Tracer Statistics in a Nonsmooth Velocity

In this subsection, we shall analyze the steady cascade of a scalar in the inertial interval of scales where the velocities are effectively nonsmooth. Here, the main fundamental issue, as in any turbulence, is the degree of universality of scalar statistics (say, the PDF P(
, r ) of the scalar difference 
measured at two points distance r apart) in the convective interval that is at L  r  rd . One may ask, in particular, what symmetries exist in the convective interval. Finite-scale pumping breaks scale invariance while diffusion breaks time reversibility. Are those symmetries restored when L → ∞ and rd → 0? As discussed in Section 2.2.3, in nonsmooth velocities, an explosive separation of trajectories separates close particles in a finite time. That provides for the dissipation of the single-point moments of the scalar when  → 0. We called this phenomenon dissipative anomaly, which tells that time reversibility remains broken even when the symmetry-breaking factor tends to zero. We shall see in this subsection that the same phenomenon of an explosive separation generally breaks the scale invariance of P(
, r ). Indeed, N th moment of P(
, r ), namely, the structure function SN (r ), is expressed via the N -particle propagator, which generally cannot be reduced to the two-particle propagator even though all particles end up in two points. We shall see that the structure functions are proportional to the respective zero modes which, as we learned, have nontrivial scaling exponents  N . © 2006 by Taylor & Francis Group, LLC

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When  N is not a linear function of N , it is called anomalous scaling since the scale invariance of P(
, r ) is not restored even at the limit L → ∞. Explosive separation of trajectories is necessary but not a sufficient condition of anomalous scaling; as we have seen in Section 2.2.5 the turbulent diffusivity that governs interparticle separation must be scale dependent, which requires velocity field to have power correlation in space. Indeed, anomalous scaling disappears for smooth and for extremely rough white-in-space velocity (respectively, cases  = 2 and  = 0 in the Kraichnan model). In a nonsmooth flow with v ∝ r  , the time to separate is proportional to L 1− −r 1− . Similarly, the pair structure function S2 (r ) =
(
1 −
2 )2  is proportional to the time it takes for two coinciding particles to separate to a distance r . For v ∝ r  , one gets S2 ∝ r 1− in agreement with (2.83). The analytic treatment of the multipoint correlation functions of the tracer is possible for the Kraichnan model (11, 34). Making a straightforward Gaussian averaging of (2.76) over the statistics of pumping and velocity, one gets the following equation for the n-point simultaneous correlation function of the scalar Cn (t, r1 . . . rn ) =

(t, r1 ) . . .
(t, rn ) [19]  ∂t Cn + Mn Cn = (rkl )Cn−2 . (2.85) k,l

Here, the operator Mn is given by (2.49) and, of course, (2.85) can be derived in a Lagrangian way by using the propagator (2.46) [26,63]. The great simplification of scalar description in the Kraichnan model is due to the fact that the set of (2.85) for different n presents a recursive problem since the rhs is expressed in terms of lower order correlation functions. There is no closure problem and any correlation function satisfies closed equation after the lower order functions are found. We consider steady state and drop the time derivative. One starts from the pair correlation function that depends on a single variable and satisfies an ordinary differential equation M2 C2 (r ) = (r ) [4], which is the Yaglom flux relation (2.82) for the Kraichnan model. This equation with two boundary conditions (zero at infinity and finiteness at zero) can be explicitly integrated  r 1−d ∂r (d − 1)D1r d−1+ + 2r d−1 ∂r C2 (r ) = (r ), (2.86)
x
∞ 1−d x dx (y)y d−1 dy, C2 (r ) = x  + rd 0 r where we introduced the diffusion scale rd = 2/D1 (d − 1). Let us remember that we consider the pumping correlated on the scale L assumed to be much larger than rd . There are thus three intervals of distinct behavior. At (D0 /D1 )1/  r  L the ¯ pair correlation function is given by the zero mode of M2 ( = 0): C2 (r ) = r 2−−d /d(d − 1)(d +  ¯   = (k + k )/  − 2)D1 , which may be thought of as Rayleigh–Jeans equipartition


k k k  d−1 2− ¯ with the temperature  = (x)x d x and k = k d(d − 1)(d +  − 2)D1 being an inverse stretching rate. At the convective interval, L  r  rd , C2 is equal to a constant (another zero mode of M2 ) plus the inhomogeneous part (zero mode of M22 ): S2 (r ) =
[
(r) −
(0)]2  = 2C2 (0) − 2C2 (r ) = r 2− (0)/d(d − 1)(d +  − 2)(2 − )D1 . Note that in the convective interval the degrees of roughness of the scalar and velocity are indeed complementary; a smooth velocity corresponds to a roughest scalar and vice versa. Finally, S2 (r ) ≈ r 2 (0)/4d at the diffusive interval. Note though that S2 (r ) is not analytic at zero since its expansion contains noninteger powers r 2n+ , n = 1, 2, . . .. This is an artifact of extending velocity nonsmoothness to the smallest scales i.e., setting the viscous scale to zero (i.e., Schmidt/Prandtl number to zero). Consider now high-order correlation functions in the convective interval. Solving recursively the stationary version of (2.85), one finds that Cn generally contains powers from r 2− to r n(2−) plus a constant and other zero modes of Mn [39,40,43,44]. Note that one cannot satisfy the boundary conditions at large scales without the zero modes. In the structure function, Sn (r ) =
[
(r) −
(0)]n , © 2006 by Taylor & Francis Group, LLC

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all the terms cancel except for the irreducible zero mode. We thus conclude that Sn (r ) = An r n . Note that only S2 is universal (that is, determined by the flux only); all the other An depend on the pumping statistics [1]. As we have seen in Section 2.2.5, the anomalous exponents 2n = n2 − 2n = n(2 − ) − 2n are positive for any d < ∞ and  = 0, 2. That means an anomalous scaling and small-scale intermittency of the scalar field: the ratio S2n /S2n grows as r decreases. In the perturbative domain, n/d(2 − ) 1, the scaling exponents are given by (2.57). At n  d(2 − )/, the dependence (n) saturates, which means that sharp fronts of the scalar determine high moments. The saturation value has been calculated for large d: n → d(2 − )2 /8 [31]. It is instructive to discuss the limits  = 0, 2 and d = ∞ from the viewpoint of the scalar statistics. Since the scalar field at any point is the superposition of fields brought from d directions, it follows from the central limit theorem that scalar’s statistics approache Gaussian when space dimensionality d increases. In the case  = 0, an irregular velocity field acts like Brownian motion so that turbulent diffusion is much like linear diffusion: scalar statistics are Gaussian, provided the input is Gaussian. What is general in limits d = ∞ and  = 0 is that the degree of Gaussianity (say, flatness S4 /S22 ) is independent of the ratio r/L. Quite contrary, we have seen in Section 2.4.1 that ln(L/r ) is the parameter of Gaussianity in the Batchelor limit so that statistics are getting Gaussian at small scales, whatever the input statistics. At  = 2, the mechanism of Gaussianity is temporal rather than spatial: since the stretching is exponential in a smooth velocity field, the cascade time grows logarithmically as the scale decreases. That leads to the essential difference: at small yet nonzero /d, the degree of non-Gaussianity increases downscale, while at small (2 − ) the degree of non-Gaussianity first decreases downscale until ln(L/r ) 1/(2 − ) and then starts to increase. The first region grows with  approaching 2. Already that simple reasoning shows that the perturbation theory is singular at the limit  = 2, which formally is manifested by the many-point correlation functions having singularity (smeared by molecular diffusion only) at the collinear geometry [64]. Note that the dependence n () must be nonmonotonic since n (0) = n (2) = 0. There is a transparent physics behind the nonmonotonic dependence () because the influence of velocity nonsmoothness (measured by ) on scalar intermittency is twofold: if one considers scalar fluctuation of some scale, then velocity harmonics with comparable scales produce intermittency and small-scale harmonics act like diffusivity and smooth it out. At ∗ <  < 2 the first mechanism is stronger and at 0 <  < ∗ the second one takes over. Still, our understanding is only qualitative here; we do not know how the maximum position ∗ depends on n and d. The anomalous exponents determine also the moments of the dissipation field  = |∇
|2 . By a straightforward analysis of (2.85) one can show that
n  = cn
n (L/rd )2n [39,43]. Here, the mean dissipation
 = (0) while the dimensionless constants cn are determined by the fluctuations of dissipation scale. Most likely, they are of the form n qn with yet unknown q. In the perturbative domain, n d(2 − )/, the main factor is (L/rd )2n and the dissipation PDF is close to lognormal since 2n is a quadratic function of n [43]. The forms of the distant PDF tails are unknown. The scalar correlation functions decay by power laws at scales r larger than that of the pumping. Remember that the time two particles spent within L is less than the time they spent within r by the 1−−d small volume factor (L/r )d . Therefore, the pair correlation function is proportional to r12 for  2−−d 2−−d v ∝ r or to r in the Kraichnan model. Note that M2r ∝ (r ). The analysis of higher order correlation functions is simplified in a non-smooth case since straight lines are not preserved and no strong angular dependencies of the type encountered in the smooth case are thus expected. To determine the scaling behavior of the correlation functions,it is therefore enough to focus on a specific geometry. Consider, for instance, the equation M4 C4 = (ri j )C2 (rkl ) for the fourth-order correlation function. A convenient geometry to analyze is that with one distance among the points, say, r12 , much smaller than the other distances, which are of order R. At the dominant order in r12 /R, the solution of the equation is C4 ∝ C2 (r12 )C2 (R) ∼ (r12 R)2−−d . Similar arguments apply to arbitrary orders.

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We conclude that the scalar statistics at r   is scale invariant, i.e., C2n ( r) = n(2−−d) C2n (r) as → ∞. Note that the statistics are generally non-Gaussian when the distances between the points are comparable. As  increases from zero to two, the deviations from the Gaussianity start from zero and reach their maximum for the smooth case described in Section 2.4.1. The results for spatially nonsmooth flows have mostly been derived within the framework of the Kraichnan model with the white forcing. The conditions on the forcing are not crucial and may be easily relaxed since the scaling properties of the scalar correlation functions are universal with respect to the forcing (i.e., independent of its details), while the constant prefactors are not [1]. The situation with the velocity field is more interesting and nontrivial. Even though a short-correlated flow might in principle be produced by an appropriate forcing, all the cases of physical interest have a finite correlation time. The existence of closed equations of motion for the particle propagators, which we heavily relied upon, is then lost. The existing numerical and experimental evidence is that the basic mechanisms for scalar intermittency are quite robust: anomalous scaling is still associated with statistically conserved quantities and the expansion (2.51) for the multiparticle propagator seems to carry over. The specific statistics of the advecting flow affect only quantitative details, such as the numerical values of the exponents [1].

2.4.2 INVERSE CASCADE

IN A

COMPRESSIBLE FLOW

If the trajectories are unique, particles that start from the same point will remain together throughout the evolution. This means that advection preserves all the single-point moments

N (t). Note that the conservation laws are statistical: the moments are not dynamically conserved in every realization, but their average over the velocity ensemble are. In the presence of pumping, the moments are the same as for the equation ∂t
=  in the limit  → 0 (nonsingular now). It follows that the singlepoint statistics are Gaussian, with

2  coinciding with the total injection (0)t by the forcing. That growth is produced by the flux of scalar variance toward the large scales. As explained in Section 2.4.1, correlation functions at very large scales are related to the probability for initially distant particles to come close. In a strongly compressible flow, the trajectories are typically contracting; the particles tend to approach and the distances will reduce to the forcing correlation length L (and smaller) for long enough times. On a particle language, the larger the time is the larger will be the distance starting from which the particle comes within L. The correlations of the field
at larger and larger scales are therefore established as time increases, signaling the inverse cascade process [37,38]. The uniqueness of the trajectories greatly simplifies the analysis of the PDF P(
, r ). Indeed, the structure functions involve initial configurations with just two groups of particles separated by a distance r . The particles explosively separate in the incompressible case and we are immediately back to the full N -particle problem. Conversely, the particles that are initially in the same group remain together if the trajectories are unique. The only relevant degrees of freedom are then given by the intergroup separation and we are reduced to a two-particle dynamics. It is therefore not surprising that the scaling behaviors at the various orders are simply related in the inverse cascade regime [37,38].

2.5 ACTIVE TRACERS As we have have learned in the previous chapters, the most fundamental property of the propagators is whether they describe particles separating or clustering backwards in time. That property alone determines the direction of the cascade for the passive tracer. Another important distinction is whether the propagators possess the collapse property (2.44) in the limit  → 0. The absence of the property makes anomalies possible for the passive tracer. Here, we consider the Lagrangian invariants (conserved along the fluid trajectories without pumping and diffusion) which are active that are related to the velocity that transports them. We shall

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see that the correlation between the Lagrangian tracer and velocity field makes it impossible to derive the direction of the cascade solely from the behavior of trajectories. In some situations, passive and active tracers cascade in opposite directions in the same velocity field. In Section 2.5.1 we first describe Burgers turbulence, which has clustering for the majority of trajectories (going to full measure in the inviscid limit) and collapse property for propagators; a passive tracer then undergoes inverse cascade in such velocity. On the contrary, powers of velocity (active tracers) have their dissipation determined by the minority of trajectories that separate. Velocity statistics thus correspond to the direct cascade with both dissipation anomaly and anomalous scaling. We then consider two-dimensional magnetohydrodynamics where velocity is nonsmooth (separation of trajectories and no collapse) so that passive scalar must have direct cascade and dissipative anomaly. On the contrary, magnetic vector potential (active scalar) influences velocity field via Lorentz force in such a way that only trajectories that carry the same value of the potential can come to the same point. As a result, the potential cascades upscale and there are no anomalies in the magnetic field statistics. In Section 2.5.2, we consider two-dimensional incompressible turbulence and describe the relations between passive tracer and active tracer (vorticity). We argue that in the domain of the direct vorticity cascade, both tracers cascade down to small scales in a very similar way. On the contrary, in the domain of the inverse energy cascade, passive scalar undergoes direct cascade while vorticity has some kind of equipartition and no flux.

2.5.1 ACTIVITY CHANGING CASCADE DIRECTION 2.5.1.1 Burgers Turbulence

We start from the simplest case of Burgers turbulence, whose inviscid version describes a free propagation of fluid particles with velocity being Lagrangian invariant, while viscosity provides for a local interaction: ∂t v + vvx − vx x = f

(2.87)

 Without force, the evolution described by (2.87) conserves total momentum v d x. Burgers equation describes one-dimensional acoustics and many other systems. Under the action of a largescale forcing (or in free decay of large-scale initial data), a cascade of kinetic energy towards the small scales takes place. The nonlinear term provides for steepening of negative gradients and the viscous term causes energy dissipation in the fronts that appear this way. In the limit of vanishing viscosity, the energy dissipation stays finite due to the appearance of velocity discontinuities called shocks. The Lagrangian statistics are peculiar in such an extremely nonsmooth flow and can be closely analyzed even though they do not correspond to a Markov process. Forward and backward Lagrangian statistics are different because they must be in an irreversible flow. Lagrangian trajectories stick to the shocks. This provides for a strong interaction between the particles and results in an extreme anomalous scaling of the velocity field. A tracer field passively advected by such a flow undergoes an inverse cascade. Here we briefly describe the picture of Burgers turbulence at the limit of small viscosity [[1,65] and references therein]. At vanishing viscosity, the Burgers equation may be considered as describing a gas of particles moving in a force field. Indeed, in the Lagrangian frame defined for a regular velocity by X˙ = v(X, t), relation (2.87) becomes the equation of motion of noninteracting unit-mass particles whose acceleration is determined by the force: X¨ = f (X, t).

(2.88)

In order to find the Lagrangian trajectory X (t; x) passing at time zero through x, it is then enough to solve the second-order equation (2.88) with the initial conditions X (0) = x and X˙ (0) = v(x, 0). © 2006 by Taylor & Francis Group, LLC

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For sufficiently short times, such trajectories do not cross and the Lagrangian map x → X (t; x) is ˙ invertible. One may then reconstruct v at time t from the relation v(X (t), t) = X(t). At longer times, however, the particles collide, creating velocity discontinuities, i.e., shocks. Once created, shocks never disappear but they may merge so that they form a tree branching backward in time. The crucial question for the Lagrangian description of the Burgers velocities is what happens with the fluid particles after they reach shocks where their equation of motion x˙ = v(x, t) becomes ambiguous. The question may be easily answered by considering the inviscid case as a limit of the viscous one where shocks become steep fronts with large negative velocity gradients. It is easy to see that the Lagrangian particles are trapped within such fronts and keep moving with them. In other words, the two particles arriving at the shock from the right and the left at a given moment aggregate upon the collision. Momentum is conserved so that their velocity after the collision is the mean of the incoming ones (recall that the particles have unit mass) and is equal to the velocity of the particles moving with the shock that have been absorbed at earlier times. The shock speed is thus the mean of the velocities on both sides of the shock. Note that, in the presence of shocks, the Lagrangian map becomes many-to-one, compressing whole space intervals into the shock locations. The Lagrangian picture of the Burgers velocities allows for a simple analysis of advection of scalar quantities carried by the flow. In the inviscid and diffusionless limit, the advected tracer satisfies the evolution equation ∂t
+ v¯ ∂x
= ,

(2.89)

where  represents an external source. As usual, the solution of the initial value problem is given in terms of the PDF P(x, t; y, 0 | v) to find the backward Lagrangian trajectory at y at time 0, given that at later time t it passed by x. Except for the discrete set of time t shock locations, the backward trajectories are uniquely determined by x. As a result, a smooth initial scalar will develop discontinuities at shock locations but no stronger singularities. Since a given set of points (x1 , . . . , x N ) ≡ x avoids the shocks with probability 1, the joint backward PDF’s of N trajectories P N (x; y; −t) should be regular for distinct xn and should possess the collapse property (2.44). This leads to the conservation of

2  in the absence of scalar sources and to the linear pumping of the scalar variance when a stationary source is present. Such behavior corresponds to an inverse cascade of the passive scalar as in Section 2.4.2. As usual in compressible flows, the advected density n satisfies the continuity equation ∂t n + ∂x (¯v n) = 

(2.90)

different from (2.89) for the tracer. The solution of the initial value problem is given by the forward  Lagrangian PDF: n(x, t) = p(y, 0; x, t|v)n(y, 0)dy. Since the trajectories collapse, a smooth initial density will become singular under the evolution, with -function contributions concentrating all the mass from the regions compressed to shocks by the Lagrangian flow. Since the trajectories are determined by the initial point y, the joint forward PDFs P N (y; x; t) should have the collapse property (2.44), but they will also have contact terms in xn when the initial points yn are distinct. Such terms signal a finite probability for the trajectories to aggregate in the forward evolution — the phenomenon that we have already met in the strongly compressible Kraichnan model discussed in Section 2.2.4. The velocity gradient ∂x v is an example of an (active) density satisfying equation (2.90) with  = ∂x f. The behavior of the Lagrangian PDFs and the advected scalars summarized earlier have been established by a direct calculation in freely decaying Burgers velocities with random Gaussian finitely correlated initial potentials  [66]. The Burgers velocity and all its powers constitute an example of advected scalars. Indeed, the equation of motion (2.87) may be also rewritten as (∂t + v¯ ∂x − f ) e v = 0 from which the relation (2.89) for
= v n and  = n f v n−1 follows. © 2006 by Taylor & Francis Group, LLC

(2.91)

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Of course, v n are active scalars, so in the random case their initial data, the source terms, and the Lagrangian trajectories are not independent, contrary to the case of passive scalars. That correlation makes the unlimited growth of
v 2  impossible: the larger the value of local velocity is, the faster it creates a shock and dissipates the energy. The difference between active and passive tracers is thus substantial enough to switch the direction of the energy cascade from inverse for the passive scalar to direct for the velocity. Indeed, in the presence of the force,
t v(x, t) = v(x(0), 0) +

f (x(s), s) ds ,

(2.92)

0

along the Lagrangian trajectories. The velocity is an active scalar and the Lagrangian trajectories are evidently dependent on the force that drives the velocity. One cannot write a formula like (2.79) obtained by two independent averages over the force and over the trajectories. Nevertheless, the main contribution to the distance-dependent part of the two-point function
v(x, t) v(x  , t) is due, for small distances, to realizations with a shock in between the particles. It is insensitive to a large-scale force and hence approximately proportional to the time that the two Lagrangian trajectories ending at x and x  take to separate backwards to the injection scale L. With a shock in between x and x  at time t, the initial backward separation is linear, so the second-order structure function becomes proportional to x. Other structure functions may be analyzed similarly and give the same linear dependence on the distance (all terms involve at most two trajectories):
|v| p  ∝ x, p ≥ 1.

(2.93)

2n+1 In particular, one can obtain the exact relations
|v|  2n  = −4(2n + 1)n x/(2n − 1) where n are the mean dissipation rates of the inviscid integrals v d x/2, which stay finite in the inviscid limit (consider, for instance, the shock-wave solution v = 2u{1 + exp[u(x − ut)/v]}−1 ). We thus see that the velocity field of forced Burgers equation gives no dissipative anomaly and inverse cascade for a passive tracer while it provides for a dissipative anomaly and direct cascade of the active tracers (powers of velocity itself).

2.5.1.2 Two-Dimensional Magnetohydrodynamics

Another example of an active tracer having its cascade opposite that of a passive one is that of the magnetic vector potential in two-dimensional MHD [67]. Magnetic vector potential a in two dimensions is related to the magnetic field as follows B = (−∂x a, ∂ y a). It satisfies the advection– diffusion equation with forcing: ∂t a + (v · ∇)a = a + f a .

(2.94)

The magnetic field acts on velocity by the Lorentz force: ∂t v + (v · ∇)v = v − ∇ p − a∇a + f v .

(2.95)

Numerics show that the velocity field is nonsmooth so that the passive scalar must undergo a direct cascade and dissipation stays finite when diffusivity tends to zero. On the contrary, vector potential a undergoes inverse cascade [67]. The Lagrangian explanation for that remarkable fact is that even though different trajectories may come to the same point, they must all bring the same value of a (otherwise Lorentz force would be infinite). This type of correlation between the trajectory and the value of active scalar it carries provides for the absence of anomalies in a statistics even for a nonsmooth velocity [68]. The PDF P(a, t) is Gaussian with a variance linearly growing with time; the PDF P(a, r ) of the increments is self-similar [67,68]. © 2006 by Taylor & Francis Group, LLC

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2.5.2 TWO-DIMENSIONAL INCOMPRESSIBLE TURBULENCE Taking curl of two-dimensional Navier–Stokes equation and using incompressibility divv = 0, one obtains the advection–diffusion equation for the vorticity  = curl v: ∂t  + (v · ∇) = a + .

(2.96)

The vorticity and all its powers are thus scalar Lagrangian invariants of the inviscid dynamics in two dimensions. In the presence of an external pumping  injecting energy and enstrophy (squared vorticity), it is clear that both quantities may flow throughout the scales. If both cascades are present, they cannot go in the same direction: the different dependence of the energy and the enstrophy on the scale prevents their fluxes from being both constant in the same interval of scales. Since one cannot provide a turbulent cascade by a potential flow (completely determined by boundaries in two dimensions), energy cannot flow to small scales where a finite energy dissipation would mean an infinite vorticity dissipation at the limit  → 0. The natural conclusion is that, given a single pumping at some intermediate scale, the energy and the vorticity flow towards the large and the small scales, respectively [69–72]. 2.5.2.1 Direct Vorticity Cascade in Two Dimensions

The basic knowledge of the Lagrangian dynamics presented in Section 2.2.2 and 2.4.1.1 is essentially everything one needs for understanding the direct cascade. The vorticity in two dimensions is a scalar and the analogy between the cascades of the vorticity and the passive scalar was noticed by Batchelor and Kraichnan in the 1960s. The vorticity is not passive though and such analogies may be very misleading, as shown in the previous section. (Another misleading analogy is that between the vorticity and the magnetic field in three dimensions which would wrongly suggest that dynamo is absent when viscosity equals magnetic diffusivity.) The basic flux relation for the enstrophy cascade is analogous to (2.82):
(v1 · ∇1 + v2 · ∇2 )1 2  =
1 2 + 2 1  = P2 .

(2.97)

The subscripts indicate the spatial points r1 and r2 and the pumping is assumed to be Gaussian ˜ ˜ with
(r, t)(0, 0) = (t)(r/L) decaying rapidly for r > L. The constant P2 ≡ (0), of dimensionality time−3 , is the input rate of the enstrophy 2 . Equation (2.97) states that the enstrophy flux is constant in the inertial range — that is, for r12 much smaller than L and much larger than the viscous scale. A simple power counting suggests that the velocity difference scales as the first power of r12 . That roughly fits the idea of a scalar cascade in a spatially smooth velocity. As was discussed in Section 2.4, passive scalar correlation functions are logarithmic in that case and we expect the same of vorticity. Of course, logarithm means that velocity is actually (weakly) nonsmooth, which provides for a nonzero vorticity dissipation in the inviscid limit. Hypothetical power-law vorticity spectra [73–75] must be structurally unstable [76]. The physics of the enstrophy cascade is basically the same as that for a passive scalar: a fluid blob embedded into a larger scale velocity shear is stretched along one direction and compressed along another; that provides for the enstrophy flux toward the small scales, with the rate of transfer proportional to the strain. One can show that the vorticity correlation functions at a given scale are indeed solely determined by the influence of larger scales (that give exponential separation of the fluid particles) rather than smaller scales (that would lead to a diffusive growth as the square root of time). The subtle differences from the passive scalar case come from the active nature of the vorticity. The stretching of a blob depends on the vorticity it carries. However, the enstrophy transfer rate is related to the strain (the symmetric part of the tensor of velocity derivatives) rather than to vorticity (the antisymmetric part). The analysis of the relations between the strain and the vorticity correlation functions shows, however, that on average the active nature of the vorticity accelerates the cascade as it goes toward smaller scales [76]. As far as the dominant logarithmic scaling of the correlation © 2006 by Taylor & Francis Group, LLC

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functions is concerned, the active nature of the vorticity simply amounts to the following: the field can be treated as a passive scalar, but the strain and the vorticity acting on it must be renormalized with the scale [69,70,76]. The law of renormalization is then established as follows: from (2.18), one has the dimensional relation that time behaves as −1 ln(L/r ) while the vorticity correlation function is
  ∝ P2 × time according to (2.79). This gives the scaling  ∼ [P2 ln(L/r )]1/3 . The consequences are that 1/2 the distance between two fluid particles satisfies: ln(R/r ) ∼ P2 t 3/2 and that the pair correlation 2/3 function
1 2  ∼ [P2 ln(L/r12 )] . Experiments and numerics are compatible with that conclusion [77–79]. 2.5.2.2 Inverse Energy Cascade in Two Dimensions

The energy is not a Lagrangian tracer and we cannot relate its inverse cascade to the behavior of trajectories. Still, we can get some important Lagrangian insight into the properties of the inverse energy cascade. If one assumes (after Kolmogorov) that ¯ is the only pumping-related quantity that determines the statistics, then the separation between the particles R12 = R(t; r1 ) − R(t; r2 ) must 2  ∝ ¯ t 3 . obey the already-mentioned Richardson law:
R12 The equation for the separation follows from the Euler equation: ∂t2 R12 = f(R(t; r1 ))−f(R(t; r2 ))− ∇(P1 − P2 ). In the inertial range, R12 is much larger than the forcing correlation length. The forcing can therefore be considered short correlated in time and in space. Of the pressure term were absent, 2 one would get the separation growth:
R12 /¯t 3 = 4/3. The experimental data give a smaller numerical factor 0.5 [80], which is quite natural since the incompressibility constrains the motion. 2 What is important to note is that the forcing term already prescribes the law
R12  ∝ t 3 consistent with the scaling of the energy cascade. Another amazing aspect of the two-dimensional inverse energy cascade can be inferred if one considers it from the viewpoint of vorticity. First, there is no dissipative anomaly for enstrophy in the inertial interval of scales of the inverse cascade. Moreover, enstrophy is transferred toward the small scales, and its flux at the large scales (where the inverse energy cascade is taking place) vanishes. By analogy with the passive scalar behavior at the large scales discussed in Section 2.4.1.2, one 1−−d may expect the behavior
1 2  ∝ r12 , where  is the scaling exponent of the velocity. The self-consistency of the argument dictated by the relation  = ∇ × v requires 1 −  − d = 2 − 2, which indeed gives the Kolmogorov scaling  = 1/3 for d = 2. Experiments and numerical simulations indicate that the inverse energy cascade has a normal Kolmogorov scaling for all measured correlation functions [81–83]. No consistent theory is available yet, but the previous arguments based on the enstrophy equipartition might give an interesting clue. Since Kolmogorov scaling corresponds to a nonsmooth velocity (in the limit of pumping scale going to zero), the passive scalar in such velocity field undergoes direct cascade with dissipative anomaly and anomalous scaling while the active scalar (vorticity) has neither dissipative anomaly nor anomalous scaling. From another perspective it is likely that the scale invariance of inverse cascades is physically associated with the growth of the typical times with the scale. As the cascade proceeds, the fluctuations indeed have time to get smoothed out, contrary to direct cascades with typical time decreasing in the direction of the cascade. To conclude this chapter, we note that what matters for the direction of the cascade of active tracers is the correlation between the tracer value and the type of trajectory it traces.

2.6 CONCLUSION We hope that the reader has absorbed by now the two main lessons: the power of the Lagrangian approach to fluid turbulence and the importance of statistical integrals of motion for systems far from equilibrium. © 2006 by Taylor & Francis Group, LLC

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The Lagrangian approach allows the analytical description of most important aspects of the statistics of particles and fields for velocity fields spatially smooth or temporally decorrelated (or both). In a spatially smooth flow, the Lagrangian chaos with the ensuing exponentially separating trajectories is generally present. The respective statistics of passive scalar and vector fields are related to the statistics of the stretching and contraction rates in a way that is well understood. The theory finds a natural physical domain of application in the viscous range of scales. The most important open problem here seems to be the understanding of the back reaction of the advected field on the velocity. This would include an account of the buoyancy force in inhomogeneously heated fluids, the saturation of the small-scale magnetic dynamo, and the polymer drag reduction. In nonsmooth velocities pertaining to the inertial interval of fully developed turbulence, the main Lagrangian phenomenon is the intrinsic stochasticity of the fluid particle trajectories that accounts for the dissipation at short distances. These phenomena are fully captured in the Kraichnan ensemble of nonsmooth time-decorrelated velocities. It is an open problem to exhibit them for more realistic nonsmooth velocities and to relate them to hydrodynamical evolution equations obeyed by the latter. The spontaneous stochasticity of Lagrangian trajectories enhances the interaction between fluid particles, leading to intricate multiparticle stochastic conservation laws. There are already open problems in the framework of the Kraichnan model. First, there is the issue of whether one can build an operator product expansion, classifying the zero modes and revealing their possible underlying algebraic structure, at large and small scales. The second class of problems is related to a consistent description of high-order moments of scalar, vector and tensor fields, especially in the situations where their amplitudes are growing, in a further attempt to describe feedback effects. Another major open problem is to identify the appropriate statistical integrals of motion in the active and the nonlocal cases. One sees there the potential direction of progress: coupling analytical, experimental, and numerical studies to investigate the geometrical statistics of fluid turbulence with the primary aim to identify the underlying conservation laws.

ACKNOWLEDGMENT I am indebted to K. Gawedzki and M. Vergassola for numerous useful discussions. I am grateful to the organizers of the Turbulence Workshop in Orlando, particularly Sen Shivamoggi, for warm hospitality.

REFERENCES 1. Falkovich, G., Gawedzki, K., Vergassola, M. “Particles and fields in fluid turbulence,” Rev. Mod. Phys. 73 (2001) 913–975. 2. Taylor, G.I. Proc. London Math. Soc. 20 (1921) 196. 3. Zinn–Justin, J. Quantum Field Theory and Critical Phenomena (Oxford Science Publ. 1989). 4. Kraichnan, R.H., Phys. Fluids 11 (1968) 945–963. 5. Ellis, R.S., Entropy, Large Deviations, and Statistical Mechanics (Springer–Verlag, Berlin 1995). 6. Bensoussan, A., Lions, J.-L., Papanicolaou, G., Asymptotic Analysis for Periodic Structures (North– Holland, Amsterdam 1978). 7. Pope, S.B. Ann. Rev. Fluid Mech. 26 (1994) 23–63. 8. Fannjang A. and G.C. Papanicolaou, Prob. Theory Rel. Fields 105 (1996) 279. 9. Majda, A.J. and P.R. Kramer, Phys. Rep. 314 (1999) 237. 10. Kraichnan, R.H., in The Pad´e Approximants in Theoretical Physics (Academic Press, New York 1970). 11. Avellaneda, M. and A. Majda, Phys. Rev. Lett. 62 (1989) 753. 12. Avellaneda, M. and M. Vergassola, Phys. Rev. E 52 (1995) 3249. 13. Aref, H. Phys. Fluids 14 (2002) 1315.

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Mathematical and Physical Theory of Turbulence 14. Liapounoff, M.A., “Probl`eme g´en´eral de la stabilit´e du mouvement,” Annales Fac. Sci. Toulouse 9 (1907). 15. Oseledets, V.I. Trans., Moscow Math. Soc. 19 (1968) 197–231. 16. Furstenberg, H. Trans., Am. Math. Soc. 108 (1963) 377–428. 17. Zeldovich, Ya., Ruzmaikin, A., Molchanov, S., Sokolov, V.J. Fluid Mech. 144 (1984) 1–11. 18. Balkovsky, E. and A. Fouxon, Phys. Rev. E. 60 (1999) 4164–4174. 19. Shraiman, B. and E. Siggia, Phys. Rev. E49 (1994) 2912. 20. Chertkov, M., Falkovich G., and Kolokolov, I. Phys. Rev. Lett. 79 (1998) 2121. 21. Girimaji, S. and S.J. Pope, Fluid Mech. 220 (1990) 427. 22. Monin, A. and A. Yaglom, Statistical Fluid Mechanics (MIT Press, Cambridge, MA 1979). 23. Jullien, M.-C., Paret, J., and Tabeling, P. Phys. Rev. Lett. 82 (1999) 2872–2875. 24. Zovari, N. and A. Babiano Physica D 76 (1994) 318. 25. Elliott, F. W., Jr. and A.J. Majda, Phys. Fluids 8 (1996) 1052–1060. 26. Bernard, D., Gaw¸edzki, K., and Kupiainen, A.J. Stat. Phys. 90 (1998) 519–569. 27. Frisch, U., Mazzino, A., and Vergassola, M. Phys. Rev. Lett. 80 (1998) 5532–5535. 28. Gaw¸edzki, K., in Advances in Turbulence VII, ed. Frisch, U. (Kluwer, Dordrecht 1998) 493–502. 29. Onsager, L. Nuovo Cim. Suppl. 6 (1948) 279. 30. Richardson, L.F. Proc. R. Soc. Lond. A 110 (1926) 709–737. 31. Balkovsky. E. and V. Lebedev, Phys. Rev. E 58 (1998) 5776. 32. Ruelle, D. J. Stat. Phys. 86 (1997) 935. 33. Balkovsky, E., Falkovich, G., Fouxon, A. Phys. Rev. Lett. 86 (2001) 2790. 34. Klyatskin, V. and D. Gurarie, Sov. Ph. Usp. 169 (1999) 171. 35. Falkovich, G., Fouxon, A., and Stepanov, M. Nature, 419 (2002) 151. 36. Le Jan, Y. Zeit. f¨ur Warsch. 70 (1985) 609. 37. Chertkov, M., I. Kolokolov, and M. Vergassola, Phys. Rev. Lett. 80 (1998) 512. 38. Gaw¸edzki, K. and M. Vergassola, Physica D 138 (2000) 63. 39. Chertkov, M., Falkovich, G., Kolokolov, I., and Lebedev, V. Phys. Rev. E 52 (1995) 4924–4941. 40. Gaw¸edzki, K. and A. Kupiainen, Phys. Rev. Lett. 75 (1995) 3834–3837. 41. Celani, A. and M. Vergassola, Phys. Rev. Lett., 86 (2001) 424. 42. Shraiman, B. and E. Siggia, C.R. Acad. Sci. 321 (1995) 279–284. 43. Chertkov, M. and G. Falkovich, Phys. Rev Lett. 76 (1996) 2706–2709. 44. Bernard, D., Gaw¸edzki, K., and Kupiainen, A. Phys. Rev. E 54 (1996) 2564–2572. 45. Son, D.T. Phys. Rev. E 59 (1999) 3811–3815. 46. Groisman, A. and V. Steinberg, Nature 405 (2000) 53. 47. Chaves, M., Eyink, G., Frisch, U., and Vergassola, M. Phys. Rev. Lett. 86 (2001) 2305. 48. Chertkov, M. and V. Lebedev, Phys. Rev. Lett. 90 (2003) 034501. 49. Fereday, D., Haynes, P., Wonhas, A., and Vassilicos, J. Phys. Rev. E 65 (2002) 035301. 50. Sukhatme, J. and R. Pierrehumbert, Phys. Rev. E 66 (2002) 056302. 51. Burghelea, T., E. Segre, and V. Steinberg, Phys. Rev. Lett. 92 (2004) 164501. 52. Kraichnan, R.H. J. Fluid Mech. 64 (1974) 737. 53. Chertkov, M., Falkovich, G., Kolokolov, I., and Vergassola, M. Phys. Rev. Lett. 83 (1999) 4065. 54. Chertkov, M., Kolokolov, I., and Vergassola, M. Phys. Rev. Lett. 80 (1998) 512–515. 55. Gaw¸edzki, K. and M. Vergassola, Physica D 138 (2000) 63–90 56. Obukhov, A.M. Izv. Akad. Nauk SSSR, Geogr. Geofiz. 13 (1949) 58–69. 57. Corrsin, S. J. Appl. Phys. 22 (1951) 469–473. 58. Batchelor, G.K. J. Fluid Mech. 5 (1959) 113. 59. Chertkov, M., Falkovich, G., Kolokolov, I., and Lebedev, V. Phys. Rev. E 52 (1995b) 4924. 60. Forster, A., Nelson, D., and Stephen, M. Phys. Rev. A 16 (1977) 732. 61. Balkovsky, E., Falkovich, G., Lebedev, V., and Shapiro, I. Phys. Rev. E 52 (1995) 4537. 62. Balkovsky, E., Falkovich, G., Lebedev, V., and Lysiansky, M. Phys. Fluids. 11 (1999) 2269. 63. Shraiman, B. and E. Siggia, Nature 405 (2000) 639–646. 64. Balkovsky, E., Chertkov, M., Kolokolov, I., and Lebedev, V. JETP Lett. 61 (1995) 1049. 65. Frisch, U. and J. Bec, 2000, in Les Houches 2000: New Trends in Turbulence, ed. Lesieur, M. (Springer EDP-Sciences, 2001). 66. Bernard, D. and M. Bauer, J. Phys. A 32 (1999) 5179. 67. Biskamp, D. and U. Bremer, Phys. Rev. Lett. 72 (1994) 3819.

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Celani, A., Cencini, M., Mazzino, A., and Vergassola, M. arXiv:nlin.CD/0207003. Kraichnan, R.H. Phys. Fluids 10 (1967) 1417. Kraichnan, R.H. J. Fluid Mech. 47 (1971) 525; 67 (1975) 155. Batchelor, G.K. Phys. Fluids Suppl. II 12 (1969) 233–239. Frisch, U. Turbulence. The Legacy of A.N. Kolmogorov, (1995) Cambridge Univ. Press, Cambridge. Saffman, P.G. Stud. Appl. Math. 50 (1971) 277. Moffatt, H.K. in Advances in Turbulence, G. Comte–Bellot and J. Mathieu, eds., p. 284 (Springer– Verlag, Berlin 1986). Polyakov, A., Nucl. Phys. B396 (1993) 367. Falkovich, G. and V. Lebedev, Phys. Rev. E 50 (1994) 3883–3899. Bowman, J.C., Shadwick, B.A., and Morrison, P.J. Phys. Rev. Lett. 83 (1999) 5491. Paret, J., Jullien, M.-C., and Tabeling, P. Phys. Rev. Lett. 83 (1999) 3418. Jullien, M.-C., Castiglione, P., and Tabeling, P. Phys. Rev. Lett 85 (2000) 3636. Jullien, M.-C., Paret, J., and Tabeling, P. Phys. Rev. Lett. 82 (1999) 2872–2875. Paret, J. and P. Tabeling, Phys. Rev. Lett. 79 (1997) 4162–4165; Phys. Fluids 10 (1998) 3126–3136. Boffetta, G., Celani, A., and Vergassola, M. Phys. Rev. E 61 (2000) R29–32. Smith, L. and V. Yakhot, Phys. Rev. Lett. 71 (1993) 352.

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Turbulence: 3 Two-Dimensional An Overview George F. Carnevale CONTENTS 3.1 3.2 3.3

Introduction ............................................................................................................................ 47 Conservation Laws and Cascades........................................................................................... 48 Markovian Closure ................................................................................................................. 49 3.3.1 H-Theorems ............................................................................................................... 50 3.4 Numerical Simulations: The Decay Problem ......................................................................... 51 3.5 A New Scaling Theory for Turbulent Decay.......................................................................... 52 3.6 A New Dynamic Model for Turbulent Decay ........................................................................ 53 3.7 Forced Two-Dimensional Turbulence .................................................................................... 55 3.8 A Question of End States ....................................................................................................... 56 3.8.1 Selective Decay .......................................................................................................... 56 3.8.2 Arnold Stable States................................................................................................... 57 3.8.3 Canonical Equilibrium ............................................................................................... 57 3.8.4 Statistics of Point Vortices and Patches...................................................................... 58 3.9 Flow over Topography............................................................................................................ 59 3.9.1 Subgrid-Scale Modeling ............................................................................................ 59 3.10 Effects of
............................................................................................................................. 61 3.11 Concluding Remarks .............................................................................................................. 64 Acknowledgments............................................................................................................................ 65 References........................................................................................................................................ 66

ABSTRACT A brief overview of the development of the theory of two-dimensional turbulence is presented. The focus is on the transition from a theory based on inertial range cascades and interacting Fourier modes to the study of interacting coherent vortices. Connection to the oceanographic and atmospheric thrust of this session is made via a discussion of some of the important effects of variations of the Coriolis parameter with latitude (in the
-plane model) and topography.

3.1 INTRODUCTION Two-dimensional turbulence is a vast subject with a history of over 50 years of intense investigation. Interest has been stimulated and maintained in great part because of the relevance to atmospheric and oceanographic flows. This is appropriate when considering the evolution of flows with length scales greatly in excess of the thickness of the fluid layer. Additionally, the effect of rotation in planetary applications further supports the two-dimensional flow model since it suppresses vertical variation of the flow field and, in particular, reduces the thickness of the bottom boundary layer. We should also note that a great deal of motivation for two-dimensional turbulence studies has been generated by the consideration of plasmas under magnetic confinement, where a strong magnetic field plays a role similar to that of ambient rotation. © 2006 by Taylor & Francis Group, LLC

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The equation that governs the flow evolution in two-dimensions can be written as ∂ + J (,  + h(x, y)) = F + D ∂t

(3.1)

where the vertical component of the vorticity  is given in terms of the streamfunction by  = ∇ 2 , F represents an externally imposed forcing, and D is a dissipation term whose form varies depending on the application. The term h(x, y) is a time-independent function used to make some connection between the “pure” problem (h = 0) and applications to the atmospheric and oceanographic problems. Two critical features of planetary flows are vortex stretching due to layer depth variations, primarily caused by the presence of topography on the bottom of the layer, and the variation of the importance of rotation with latitude, the so-called
-effect. These effects can be incorporated into h, which then takes the explicit form h(x, y) = f 0


H +
y D

(3.2)

where f 0 is twice the rotation rate times the sine of the mean latitude,
H is the variation of the bottom topography above the mean level, and y is the coordinate along the meridional direction. Another important effect in geophysical applications is that of stratification, but for lack of space this will be ignored here.

3.2 CONSERVATION LAWS AND CASCADES One of the most important features of two-dimensional turbulence is the inviscid conservation of energy
1 (3.3) (∇)2 d xd y, E= 2 and enstrophy
1 ( + h)2 d xd y. (3.4) Q= 2 Paradoxically, the inviscid conservation of enstrophy makes two-dimensional turbulence in many ways more complicated than three-dimensional turbulence, as we will see. The importance of the presence of the enstrophy invariant was first brought to the fore by Fjortoft (1953). Also, from the form of the evolution equations, we see that two-dimensional turbulence possesses an infinity of Lagrangian type invariants. Since the inviscid unforced equation is simply the advection of “potential vorticity” (q =  + h), this quantity q is conserved following fluid particles. There are subtle consequences of this conservation law for two-dimensional turbulence that remained obscure for some time, as we shall see. Perhaps the most important consequence of enstrophy conservation on two-dimensional turbulence can be introduced by considering the evolution in the pure case (h = 0) in a doubly periodic domain. The spectral form of the energy and enstrophy invariants then takes on a particularly revealing form: 1 (3.5) |kk |2 E= 2 and 1 2 (3.6) k |kk |2 Q= 2 where k is the Fourier amplitude of the streamfunction. Imagine a randomly generated initial distribution of energy concentrated at a particular wavenumber, say k1 . The effect of turbulence will be to broaden any such initial distribution of modes by the © 2006 by Taylor & Francis Group, LLC

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nonlinear interaction of its members. The presence of the additional factor of k 2 in the enstrophy compared to the energy implies that a transfer of energy to wavenumbers much higher than k1 must be accompanied by a correspondingly much larger transfer of energy to smaller wavenumber in order to keep the enstrophy invariant (see Rhines, 1975; Salmon, 1982). This kind of reasoning leads to the realization that instead of the single energy cascade familiar from three-dimensional turbulence, there may be two kinds of cascades in two-dimensional turbulence as recognized by Kraichnan (1967) and Batchelor (1969). The transfer of energy to small k, that is, to large scale, suggests an inverse energy cascade. Dimensional scaling arguments, based on the assumption that the only quantity remembered in the cascade process is the energy injection rate , then lead to the prediction of an “inverse” energy cascade range of the form E(k) = C2/3 k −5/3 ,

(3.7)

where E(k) is the isotropic energy spectrum. The form of the spectrum in this case is the same as that in the “forward” inertial range in three-dimensional turbulence. Accompanying the inverse energy transfer to large scale will be a forward cascade of enstrophy to small scale. Again by dimensional analysis, assuming an enstrophy injection rate of magnitude , the spectrum in such a cascade would be (3.8) E(k) = C  2/3 k −3 .

3.3 MARKOVIAN CLOSURE The major part of turbulence research in the 1960s and 1970s was based on closure theory. This came in many varieties depending on the precise method used to “close” the nonlinear statistical problem. We will mention here only the Markovian version of the closures. This gives the rate of change of the energy spectrum entirely in terms of the current spectrum. Early elements of this theory can be found in Edwards (1964), Kraichnan (1964), and Herring (1966). A fuller understanding and explicit definition of the theory were given by Orszag (1970) and Leith (1971). The basic form of the Markovian closure, again limiting the discussion to the h = 0 case, can be expressed as  ∂Uk kpq (akpq Up Uq − bkpq Uq Uk ) + Fk + Dk , = ∂t k+p+q=0

(3.9)

where Uk =< |kk |2 > is the ensemble averaged modal energy and akpq and bkpq are fixed functions of the wavevectors k, p, and q that are readily determined by the kinematics of two-dimensional flow. The important physical approximations and assumptions enter through the factor kpq , which has the dimensions of time. This can be thought of as the “triad” relaxation time, that is, the time during which the three modes k, p, and q remain in phase for effective interaction, or we can think of it as a measure of the efficiency of the interaction. Assuming reasonable forms for the response of a particular mode to infinitesimal perturbation leads to the following expression: kpq =

1 k + p + q

(3.10)

where k is the relaxation rate of mode k. Various assumptions that can be made about the nature of two-dimensional turbulence will lead to diverse prescriptions for the appropriate form of the relaxation time. In one of the most successful formulations, the relaxation rate for mode k is taken to be the cumulative shear of all modes of larger scale. Explicitly, we have
k p 2 E( p)d p, (3.11) k =  0

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where  is taken to be a phenomenological constant whose value can be fixed by comparison with results from simulation or observations (Pouquet et al., 1975). The Markovian closures confirmed that if energy and enstrophy were continuously injected at wavenumber ki , then there would, indeed, be the two cascades predicted by the scaling arguments of Kraichnan (1967) and Batchelor (1969). There were many important contributions to the development of this theory. An overview of Markovian closure can be found in Lesieur (1990).

3.3.1 H-THEOREMS The Markovian closures bear some similarity to the Boltzmann equations of interacting particle systems and the resonant interaction equations for wave–wave interactions in weak-wave interaction theory (Hasselmann, 1966). Both of those theories support H-theorems that show relaxation to statistical equilibrium. However, the structure of the Markovian closures is sufficiently different from those theories that it was not obvious that a corresponding H-theorem could be constructed. In particular, the proof of the H-theorem for the resonant wave theory relied on the presence of a delta function restricting all triad interactions to be resonant (i.e., k + p + q = 0, where k is the dispersion relation for the wave oscillation), while there is no such restriction in turbulent flows. Montgomery (1976) demonstrated that the equation for the marginal probability distribution f k (k ), derived from a BBGKY hierarchy for turbulence, did indeed satisfy an H-theorem, suggesting that such a theorem should also exist directly for the Markovian closures written in terms of the energy spectra. Carnevale et al. (1981) considered the generalized Markovian closure that treats the evolution of two-point correlations < yi y j > for the quadratic nonlinear interaction of an arbitrary set of fields {yi (t)}. They demonstrated that in the unforced nondissipative case, the entropy of the system S = ln det < yi y j >,

(3.12)

increases monotonically toward the stationary state value based on the canonical equilibrium defined by the quadratic invariants of the system. For the pure two-dimensional turbulence problem, this meant that the entropy of the system could be written as  ln U , (3.13) S= k

and the inviscid equilibrium state would be that predicted by the canonical equilibrium distribution P based on simultaneous conservation of energy and potential enstrophy: P ∝ e−a E−bQ ,

(3.14)

where a and b are Lagrange multipliers fixed by equating the average E and Q to the totals for the particular flow considered (Kraichnan, 1967). See Carnevale (1982) for a demonstration of entropy increase in numerical simulations. The canonical equilibrium spectrum for pure two-dimensional turbulence is well defined if we consider the spectrally truncated problem with no wavenumbers greater than kmax . In that case, the equilibrium is given by 1 Uk = . (3.15) a + bk 2 The values of the Lagrange multipliers depend on the initial values of E and Q, as well as the truncation wavenumber kmax . Kraichnan (1967, 1975) considered the limit kmax → ∞ and concluded that, in that limit, all of the energy would collapse into the lowest available mode of the system. He pointed out the analogy with Bose condensation.

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3.4 NUMERICAL SIMULATIONS: THE DECAY PROBLEM The development of the closure theory and its confirmation of the scaling theories was a remarkable achievement. With the advent of increasing computer power in the 1970s came a series of studies which optimistically tried to test the theory through numerical simulation of two-dimensional flow. We can get a clear picture of this quest by first focusing on the decay problem, which consists of starting from an initial randomly generated vorticity field, and watching the subsequent evolution. Typically, these simulations were performed with a Laplacian dissipation D = ∇ 2  for which one could define the usual Reynolds number based on the scale of the most energetic scales of motion and the mean speed. Herring et al. (1974) performed such a decay simulation with a resolution of (128)2 , that is on a grid of 128 by 128 points or the spectral equivalent, which was very high resolution for the time. They found that during the decay, the spectrum at small scales behaved like k −4 , significantly steeper than the expected k −3 spectrum. They also made a comparison with closure theory for the evolving decay problem and pointed out that the closure prediction for the evolution agreed very well with their simulation results. They concluded that their resolution was too low to permit a sufficiently high Reynolds number needed for the development of the k −3 spectrum. The Reynolds number in their flow was 350. They estimated that in order to confidently observe the k −3 spectrum one would need at least a resolution of (256)2 and probably (512)2 . Fornberg (1977) also investigated the decay problem. Even though his resolution (64)2 was less than that in Herring and colleagues (1974) simulations and the spectrum he found was also k −4 , he made an interesting observation that is worth relating. During the decay, he found that some largescale vortices formed and tended to dominate the flow. In words that were prophetic, he stated, “This trend towards well-defined large-scale structures can make it questionable if the 2-D flow should be described as ‘turbulent’ and it casts some doubts on the concept of inertial range and the relevance of energy spectra.” With the availability of the Cray-1 supercomputer, beginning in the late 1970s, access to numerical resolution sufficiently high to resolve the issue of the decay spectrum became possible. McWilliams (1984) clearly demonstrated that the predictions of the spectral scaling theory and the closure theory for the enstrophy cascade were not valid for the high-resolution decay simulations. His demonstration used a spectral model equivalent to resolution (256)2 , and he employed a hyperviscosity dissipation of the form D = −4 ∇ 4 , which allowed for less viscous runs. The resulting spectrum for small scales was approximately k −5 and, more importantly, it became clear that the flow was dominated by the formation and evolution of compact coherent vortices. McWilliams (1984) concluded that the presence of these vortices suppressed the expected turbulence enstrophy cascade. He described the process by which the vortices emerged and how each vortex could be traced back to an identifiable patch in the initial condition. He also described how the number of vortices decayed with time because of mergers of like signed vorticity, a key process subsequently studied in great detail (Melander et al., 1988). Benzi et al. (1986) showed that, in the decay simulations, the spectrum of the flow in between the vortices was given by the k −3 law. The collection of vortices produced a spectrum closer to the total observed spectrum that was steeper than k −3 . Benzi et al. (1987, 1988) described how to relate the observed distribution of vortex properties to the observed steep spectra, which in their particular simulations with resolution (512)2 were k −4.3 . Furthermore, they emphasized that the distribution of vortex properties depends on the initial conditions and that the system of vortices could reasonably well be represented by a Hamiltonian system of point vortices except during periods of vortex merger. In a remarkable demonstration of this, they showed that the trajectories of the 17 remaining vortices in one of their turbulent decay simulations of resolution (512)2 could well be predicted by the evolution of 17 point vortices. Brachet et al. (1986, 1988) investigated the early evolution in the decay problem with resolution going up to (800)2 . They found that, in the period before the formation and dominance of the coherent

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vortices, the k −3 spectrum could be observed as a transient. This reinforced the idea that the vortices suppressed the predicted enstrophy cascade and that the ideas of spectral scaling and closure theory failed because they did not take into account the formation of the vortices. Thus, progress in the 1980s, driven by data from numerical simulations, forced us to consider more and more the role of the vortices in two-dimensional turbulence. In McWilliams (1990), we find the shift in emphasis away from the statistics embodied in the spectra toward the study of the statistics of the vortices. A method of creating a census of vortices and following the statistics of the vortices is described. Of the important results from the approach in that paper, two are particularly relevant to this review. First, the number of vortices is shown to decay as a power law in time with exponent approximately −0.7. Second, the average of the magnitude of the extremum of vorticity found in the centers of the coherent vortices remains approximately constant in time.

3.5 A NEW SCALING THEORY FOR TURBULENT DECAY Based on the results of the simulation studies of the 1980s and the subsequent shift in focus to the statistics of vortices, a new model for two-dimensional turbulent decay was developed. It is useful to first see what Batchelor’s (1969) theory of turbulent decay would predict for the statistics of the vortices. This theory was based on the assumption that, in two-dimensional turbulent decay, all memory of the initial conditions would be lost, except for the total energy of the flow. Thus, if the enstrophy, which has dimensional units of inverse time squared, were to decay as a power law, then it would scale as Q ∼ t −2 . If we apply the same dimensional reasoning to the density of vortices in the flow, then we would predict #vortices 1 ≡ , (3.16) ∼ Area Et 2 where E is the total energy and t is the time. These predictions indicate a decay that is much more rapid than observed. Various simulations had shown the decay of enstrophy to have an exponent between −0.4 and −0.3, and, as mentioned earlier, McWilliams (1990) found the decay exponent for to be about −0.7. Carnevale et al. (1991b) proposed a new scaling theory based on the essential idea that, through the agency of the vortices, two-dimensional turbulence “remembers” more of its initial condition than just the total energy. McWilliams (1984, 1990) and Benzi et al. (1987) had emphasized that properties of the vortices in two-dimensional turbulence could be traced back to the initial conditions. McWilliams (1990) had shown that the mean value of the extremum of the vorticity found in the center of the vortices was approximately invariant with time. The fact that the core of the vortex was somehow preserved from the depredations of turbulence has had an interesting development and was put on a quantitative basis early on in Weiss (1981). Thus, Carnevale et al. (1991b) proposed that scaling theory must take into account this preservation of vortex properties in some way. Since the magnitude of the vorticity extremum values were easily accessible and on average appeared to be only weakly dependent on time, it was natural to consider the possibility of a scaling theory based on the conservation of E and ext , the mean of the magnitude of the peak vorticity within the vortices. On dimensional grounds, then, all one could propose is that the density of vortices should scale like ∼ E −1 2ext f (ext t),

(3.17)

where f could be an arbitrary function. Thus, with the assumption of two-dimensional invariants, the ability to predict the decay exponent on dimensional grounds is lost. However, progress is still possible if it is assumed that decays as a power law with some exponent and one additional assumption is made. Carnevale et al. (1991b) postulated that the total energy should scale like the total energy associated with the vortex cores, an assumption that is difficult to justify a priori since there certainly is energy outside the cores. An argument can be made that the self-energy and the configuration energy © 2006 by Taylor & Francis Group, LLC

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101

101 8/ρ

2/ρ

2Γ 100

100

10 a

0.01 Γ

ζ / 200 2a

10–1 100

101 t (a)

102

10–1 10–1

100 t

101

(b)

FIGURE 3.1 Vortex statistics: = density of vortices,  = mean circulation of the vortices, a = the mean radius, and  = the mean velocity extremum. (a) Data from a high-resolution simulation. (b) Data from the punctuated-Hamiltonian point-vortex model. (Figures from Carnevale et al., 1991)

(interaction energy) should scale this way, and a careful analysis by Weiss and McWilliams (1993) puts this assumption on firmer footing, showing that the inclusion of all energy components only makes corrections to the theory of logarithmic order in time. Modeling the vortices as vortex patches of vorticity ext and radius a then gives
a
1 (r ext )2r dr ∼  2 , (3.18) v 2 d xd y ∼ E= A 0 where A is the total area of the periodic domain and  = ext a 2 is the circulation of the vortex. With this result and the preceding assumptions, the other statistical properties of the flow could be predicted. For example, since E ∼  2 ∼ t 0 , we immediately have  ∼ t /2 , and, hence, a ∼ t /4 . Also, integral quantities could be predicted by assuming that the vorticity is concentrated in the vortices; hence,
−1 Zn ≡ A n d xd y ∼ next a 2 . (3.19) Thus, the enstrophy is predicted to decay as Z 2 ∼ t − /2 and the kurtosis to grow as K ≡ Z 4 /Z 22 ∼ t /2 . These predictions were all verified by high-resolution simulations. In particular, we show the results for comparison between this theory and simulation for the evolution of the mean values of a and  in Figure 3.1a.

3.6 A NEW DYNAMIC MODEL FOR TURBULENT DECAY In addition to the scaling theory discussed in the previous section, Carnevale et al. (1991b) also proposed a new dynamic model of turbulent decay. The results of Benzi et al. (1987) implied that once the coherent vortices formed, their trajectories could be approximated or modeled by the trajectories of point vortices in an inviscid Hamiltonian dynamics for the periodic domain, except when two like-signed vortices approached close enough to merge. This suggested that we could replace the full dynamics by a “punctuated” Hamiltonian dynamics in which the evolution of the system would be that of the usual point vortex dynamics (in a periodic domain) except for instants in which two vortices that have come close together are merged into one according to some rule representative of the continuum problem. © 2006 by Taylor & Francis Group, LLC

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Numerical simulations of merger and theoretical analysis based on elliptical vortex patches (Melander et al., 1988) suggested that two like-signed circular vortex patches of equal relative vorticity would merge if the distance s between their centers was less than 3.3 times the mean of their radii a1 and a2 , that is, if s < 3.3 . (3.20) 1 (a + a2 ) 2 1 To implement such a rule required assigning radii to each of the point vortices. To effect the mergers when this rule was satisfied also required some rule to determine the properties of the product vortex. A hint for how to do this was provided by the scaling relation (3.18). Accordingly, it was assumed that when vortices 1 and 2 merged to form vortex 3, there would be a conservation of core energy, which is equivalent to 21 a14 + 22 a24 = 23 a34 , (3.21) where the ith point vortex is considered to be a patch of vorticity i and radius ai . To apply the simple criterion (3.20) for vortex mergers, we assumed that all i were equal. Based on this assumption, the merger rule becomes one of conservation of radius to the fourth power: a14 + a24 = a34 .

(3.22)

A distribution of i would require a more complicated merger rule. The energy spectrum for the initial conditions for the high-resolution simulations in McWilliams (1990) were highly peaked at one wavenumber. Thus, to reproduce the vortex statistics from that simulation with our model, we assumed an initial condition in which all vortices had the same initial radius. Simulations of the punctuated-Hamiltonian model with this initial condition did remarkably well at reproducing the vortex statistics found in the high-resolution simulations of McWilliams (1990). The results from the point vortex model are shown in Figure 3.1b for comparison to Figure 3.1a. These results are from one particular run and show behavior not very far from the high-resolution simulation or the scaling theory of Section 3.5, with the decay exponent given the value of ≈ 0.75. By averaging over many decay runs with this point vortex model, Weiss and McWilliams (1993) showed the correspondence with the new scaling theory was essentially perfect and they were also able to give an accurate value for the decay exponent for this model. Their result was ≈ 0.72. Benzi et al. (1992) also investigated the point-vortex model with mergers. They used an initial condition with a distribution of vortex radii such that the probability of finding a vortex of radius a scaled as a −3 , which they showed corresponds to an energy spectrum E(k) ∝ k −3 . Starting with such a broad distribution of vortex radii, they found that the punctuated-Hamiltonian dynamics gives a vortex density decay exponent of = 0.6. They found good agreement between the point vortex model evolution and their high-resolution (512)2 spectral simulation that was initialized with a k −3 spectrum (Benzi et al., 1987). Their conclusion was that the value of the exponent depends on the nature of the initial conditions. This lack of universality has been implicit in much of what has been said earlier about the connection of the vortices to the initial conditions of the flow. The details of the evolution of two-dimensional turbulence are actually much more complicated than indicated by the simple scaling and point vortex dynamical model discussed previously. In a spectral simulation with resolution (1024)2 , Santangelo et al. (1989) found two spectral slopes in the enstrophy cascade associated with two separate vorticity populations. The first, the “large” vortices, were those discussed before, which could easily be traced back to the initial data and could be modeled by the punctuated-Hamiltonian dynamics. The second set of “smaller” vortices had a source very different from the first set of vortices. They were created from the debris of the vortex mergers. They were small pieces of the larger vortices that were torn off during the merger, or they resulted from the winding-up of vortex filaments shed during mergers. As for the total number of vortices including these two populations, this need not decay and could even increase, as was verified by Dritschel (1993).

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3.7 FORCED TWO-DIMENSIONAL TURBULENCE We turn now briefly to the forced dissipating problem. The history of the search for the k −5/3 range is more closely associated with the forced turbulence problem than the decay problem, and we will see that vortices play a role in that range as well. Lilly (1969) made an attempt to observe the k −5/3 inverse energy cascade range and the k −3 enstrophy range. In a numerical simulation of resolution (64)2 , he randomly forced modes from wavenumbers 7 to 8 and, indeed, observed the formation of both ranges, with no formation of coherent vortices. The only viscosity used was Laplacian viscosity. The low resolution of these simulations made verification at higher resolution imperative. Basdevant et al. (1981) performed a forced simulation at resolution (128)2 with random forcing also near wavenumber 7. In order to avoid a pile-up of energy at the largest scale of the domain, they added a viscosity that increased toward small wavenumber as k −2 . In their simulation, they observed vortices at approximately the scale of the forcing. The enstrophy cascade took on a power law of k −4 , and the inverse k −5/3 range failed to develop. Frisch and Sulem (1984) performed a forced simulation at resolution (256)2 with forcing in the band 13 < k < 17, with Laplacian dissipation and no additional dissipation at small k. They found a k −5/3 inverse range that was evolving toward smaller scales. Although it is difficult to tell from the streamfunction plots presented, it seems that the vortices were comparable in scale to the forcing scales or perhaps a bit larger. Herring and McWilliams (1985) also found evidence for a k −5/3 inverse range in spite of the formation of vortices. At small scales, the enstrophy cascade produced a spectrum significantly steeper than k −3 unless the vortices were suppressed. They concluded that such suppression could be effected by sufficiently strong random forcing, topography, or
-effect. Maltrud and Vallis (1991) investigated inverse and forward cascade ranges. They demonstrated that, if a Rayleigh friction is applied so as to destroy the vortices in the forward enstrophy range, then a k −3 enstrophy range results; otherwise, the slope is substantially steeper. They also examined the k −5/3 inverse range by forcing at wavenumbers in the range 80 ≤ k ≤ 84. With Rayleigh friction to stabilize the inverse energy flow, they found a stationary k −5/3 regime. Vortices formed with radii only near the forcing scales. Smith and Yakhot (1993) performed an interesting simulation at resolution (512)2 in which the forcing was in the range 100 < k < 105 and the dissipation was taken as eighth power of the Laplacian. No additional viscosity was provided at small k. A k −5/3 inverse range grew from the forcing range toward small k. This continued until energy began to pile up at large scale. A large dipolar pattern formed in the streamfunction field, with two very intense narrow vortices in the centers of the pattern. Apparently the narrow intense vortices resulted from a continuing merger of like-signed vorticity created by the small scale forcing near the centers of the streamfunction field. The high shear that developed in the mean field away from these centers served to prevent vortices from forming elsewhere (Smith and Yakhot, 1994). This was, in a sense, the Bose condensation first discussed by Kraichnan (1967), but different from that prediction, which would have had the streamfunction and the vorticity field on the same scale. Borue (1994) performed simulations of the inverse range using a dissipation given by D = (−−8 k −8 − 8 k 8 ). He found that, given sufficient time and sufficient separation between forcing wavenumber and dissipation wavenumber, the k −5/3 spectrum is unstable to the formation of vortices and becomes nearly as steep as k −3 . He claimed that lack of resolution and/or lack of sufficient integration time had kept this deviation from the predicted −5/3 rule from being observed previously. Dubos et al. (2001) emphasized the importance of the choice of infrared dissipation in determining the nature of the inverse cascade. They compared a forced run with Rayleigh friction (representing bottom drag) to that with no large-scale damping, confirming that, with no friction, vortices may form at all scales and a steep k −3 spectrum results for large scales; however, if Rayleigh friction is

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applied, the scale of the vortices remains confined to the forcing scale and the inverse cascade takes on a k −5/3 spectrum. Tabeling (2002) describes in detail the action of the vortices in the k −5/3 spectrum subject to Rayleigh friction. With this damping, it appears that the cascade to larger scales is a result of clustering of forcing-scale-sized vortices into large-scale structures while the individual vortices remain distinct. These forcing-scale vortices are continually created at random positions and migrate to form larger streamfunction structures (without actually merging). These individual vortices are continually decaying to be replaced by newly formed ones that again migrate into the large-scale formations from their random initiation positions. The need to provide a drain of energy at large scale and the sensitivity of the stationary energy spectrum to the form of infrared viscosity chosen add another element of nonuniversality to the question of the inverse cascade. Even in the evolving case with no-infrared damping, with evolution to ever larger scales, it appears that the formation of vortices on multiple scales may ruin the possibility of an ever elongating k −5/3 range, although it is difficult to prove such an idea with finite resolution simulations. In any case, there has been yet no attempt to apply any kind of model-based vortex statistics or dynamics to the forced problem.

3.8 A QUESTION OF END STATES The question of the final state of a two-dimensional turbulent decay has been approached by some rather different theoretical paths. The prediction for the result of turbulent decay from an initial condition of sufficient randomness is a dipole. Differing theoretical models of the dipole formation are discussed in Matthaeus et al. (1991), Montgomery et al. (1992), and Carnevale et al. (1992). The latter paper argues that the structure of the final dipole will not be universal but, rather, must reflect the properties of the initial conditions captured in the initial set of vortices and then transmitted through the mergers to the final dipole. The question becomes even more interesting when the effect of topography and
are included. We shall consider those effects here.

3.8.1 SELECTIVE DECAY In a study of flow over topography, Bretherton and Haidvogel (1976) argued that a patch of potential vorticity in a turbulent flow would be distorted by turbulence and teased out into finer and finer filaments. These filaments would interleave as they cascaded to ever finer scales. Any coarse grained averaging of this vorticity field would result in a decrease of the enstrophy Q of the flow. The energy, cascading to large scale, would, however, be preserved by this process. Thus, to predict the end result, they minimized Q with the constraint of constant E. The resulting prediction is that the final streamfunction would be given by  = ∇ 2  + h(x, y),

(3.23)

where  is a Lagrange multiplier that depends on the total energy E. This theory of the “selective decay” of the enstrophy with the energy held fixed was generalized and further developed by Montgomery in a series of articles in magnetohydrodynamics and pure two-dimensional turbulence (see Matthaeus and Montgomery, 1980). In an interesting application of the concept, Leith (1984) pointed out that the vortices that evolved in pure decaying two-dimensional turbulence might be the result of such a selective decay process. There is a very interesting appendix in the Bretherton and Haidvogel (1976) article in which they consider the effects of other invariants of the flow. Since potential vorticity q is preserved following particles, all integrals of the form
QF = © 2006 by Taylor & Francis Group, LLC

F(q)d xd y

(3.24)

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are invariant. These may be referred to as generalized enstrophies. As long as F  > 0, the filamentation process and coarse graining should also result in a decay of Q F as with Q. Thus, Bretherton and Haidvogel (1976) suggested that we consider the states obtained by minimizing Q F with E held fixed. The result is the generalized minimum enstrophy state:  = F  (q)

(3.25)

3.8.2 ARNOLD STABLE STATES As pointed out by Bretherton and Haidvogel (1976), subject to the restriction that F  > 0, the generalized enstrophy state (3.25) is nonlinearly stable in the sense of Arnold (1966). Using the existence of the generalized enstrophies, Arnold proved that, in inviscid flow, a norm based on perturbation energy and perturbation relative enstrophy would always be bounded. Specifically, if 0 < c < F  (q) < C, then we have

(3.26)





(∇) + c(∇ ) < 2

2

2

(∇0 )2 + C(∇ 2 0 )2 ,

(3.27)

where the integrals are over the entire spatial domain. Here 0 is an arbitrarily large initial perturbation of the Arnold stable state (3.25). The inequality then shows that the evolving perturbation  is always bounded by the size of the initial perturbation.

3.8.3 CANONICAL EQUILIBRIUM An alternate approach to the question of the final state of decaying two-dimensional turbulence was offered by canonical equilibrium theory. As previously mentioned, this point of view had been used by Kraichnan (1967) to predict the equilibrium spectrum in spectrally truncated two-dimensional flow (3.15) and also to predict the Bose condensation of the flow at infinite resolution. The canonical theory based on the quadratic invariants of the flow was generalized to the case of flow over topography by Salmon et al. (1976). For a spectrally truncated flow, the canonical equilibrium probability distribution, P ∝ exp(−a E − bQ), predicts an ensemble mean field given by a <  >= ∇ 2 <  > + h. b

(3.28)

Although this has the same form as the selective decay state (3.25), quantitatively the results are different. The constants a and b are determined by the values of invariants E and Q, and the ratio a/b is not generally the same as the value of  in (3.25) that is determined by the value of E alone. Additionally, the canonical equilibrium theory differed from the selective decay theory by predicting the existence of persistent fluctuations from the mean state given in spectral notation by < k 2 |k − < k > |2 > =

1 a + bk 2

(3.29)

Note that the spectrum of the fluctuations is the same as that found for the flat bottom case by Kraichnan (1967). The theory of selective decay and the canonical equilibrium theory for flow over topography were published in the same year (1976). For some time afterward, there were numerical simulations verifying both ideas, but little understanding of the connection between them. Carnevale and Frederiksen (1987) analyzed the dependence of the Lagrange multipliers a and b on the truncation wavenumber kmax . It was demonstrated that, in the limit of kmax → ∞, the canonical equilibrium and the selective decay predictions for the mean field became identical. Furthermore, the fluctuations predicted by the © 2006 by Taylor & Francis Group, LLC

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canonical theory at all but the smallest k necessarily vanish in this limit. Some energy — precisely the amount that cannot be accommodated by the mean flow — will still reside at the largest scale of the domain, but this is also an issue for the selective decay theory when there is excess energy in the initial condition. Thus, in the limit of infinite resolution, the mean field becomes statistically sharp, and the canonical theory predicts the same cascade result as that in selective decay theory. Kraichnan’s (1967) Bose condensation to the largest scale is replaced by a collapse to the selective decay state. The fact that the fluctuations all vanish is particularly interesting. The mean state becomes the sole state. Note again that there are other Arnold stable states possible for a given energy of the flow as given by the generalized formulas (3.25) and (3.26). An initial state near one of these states must remain nearby as proven by Arnold (1966). Carnevale and Frederiksen (1987) pointed out that there should be a statistical treatment similar to canonical equilibrium for each stable Arnold state  = F  (q) that could be constructed from the corresponding generalized enstrophy Q F by using the energy– enstrophy norm implicit in the inequality (3.27) derived by Arnold. They speculated that, in the limit of infinite resolution, there would be a collapse to the Arnold stable state just as in the case with the canonical equilibrium based on quadratic enstrophy Q. In this event, the statistical equilibrium states would be identical to those of the generalized minimum enstrophy theory in the appendix of Bretherton and Haidvogel (1976). Thus, we see that the generalizations of the two theories are also equivalent in that all of the predicted states are Arnold stable states. See Carnevale and Vallis (1990) for some examples of these stable states in flow over topography.

3.8.4 STATISTICS OF POINT VORTICES AND PATCHES An alternative to the selective decay theory and the canonical equilibrium theory is the statistical theory of points or patches of vorticity. Onsager (1949) first considered the statistics of point vortices and discussed the interesting possibility of negative temperature states in which vortices of like sign could clump, forming large-scale vortex structures. The theory was later taken up again by Montgomery and Joyce (1974), who showed that assuming an equal distribution of positive and negative point vortices of equal strength in the limit of an infinite number of vortices leads to the sinh–Poisson equation for the equilibrium flow: ∇ 2  = 2 sinh .

(3.30)

A review of the point vortex theory was given by Kraichnan and Montgomery (1980). An alternative approach, which is equivalent to the study of the statistics of nonoverlapping patches of vorticity, reinvigorated this line of enquiry in the early 1990s. Key papers were Robert and Sommeria (1991), Miller et al. (1992), and Whitaker and Turkington (1994). Interestingly, Turkington (1999), among others, points out that the predicted end states of this theory are also Arnold stable states. A recent review can be found in Tabeling (2002). Anticipating that David Montgomery will present a detailed discussion of these matters in his contribution to this volume (see Chapter 5), I will simply refer the reader to that article and to Yin et al. (2003) for a deeper discussion. One further comment, however, may help make some connection with the rest of this section. The generalized selective decay theory of Bretherton and Haidvogel (1976) and the corresponding generalized statistical mechanical arguments of Carnevale and Frederiksen (1987) suffer from the ambiguity inherent in the existence of many possible Arnold stable states. Those theories cannot automatically choose which of the possible Arnold stable states will be selected as an end state. The statistical mechanics of point vortices and patches as has been developed considers an infinity of vortices or patches that make up a microstate of the system. Given a microstate, a maximum entropy principle then determines uniquely the macrostate equilibrium. This seems to avoid the ambiguity inherent in the alternate generalized selective decay and canonical equilibrium approaches. However, it appears that there is an ambiguity in choosing the appropriate initial microstate. For any given © 2006 by Taylor & Francis Group, LLC

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macrostate that one might take as an initial condition, there are many choices of how to represent this in a limiting process of an infinite number of point vortices of arbitrary strength or patches of arbitrary strength and relative sizes. Thus, the ambiguity simply reappears in another guise.

3.9 FLOW OVER TOPOGRAPHY The predictions of the selective decay theory and the canonical equilibrium theory simply based on energy and quadratic enstrophy proved remarkably successful. An example is presented here of a numerical simulation of inviscid flow over topography to better describe the implications of these theories. In spectral notation, the predicted mean streamfunction of the canonical equilibrium theory is given by hk < k >= a . (3.31) + k2 b Thus, we expect the large-scale (small k) streamfunction to be proportional to the topography and, hence, take on the shape of the topography. With positive a/b, as must be the case in a sufficiently large domain, the streamfunction is positively correlated to the topography giving anticyclonic flow around hills and cyclonic flow around valleys. Examples of this are easily provided by numerical simulation of inviscid flow over topography. Consider the topography in Figure 3.2a, which happens to be the topography of the Emperor seamount chain, north of Hawaii, between 40◦ N and 45◦ N. The initial condition for the simulation is taken as a randomly generated vorticity field from which the initial streamfunction, shown in Figure 3.2b, is derived. The energy E and potential enstrophy Q of this initial condition determine the Lagrange multipliers a and b, which define the ensemble mean streamfunction, shown in Figure 3.2c. The thick lines correspond to positive  and the mean flow is such that there is, as expected, anticyclonic flow about the peaks of the topography. As the simulation evolves and there is a transfer of enstrophy to smaller scale, where it simply piles up due to the inviscid and truncated nature of the dynamics, correlation between streamfunction and topography develops. In Figure 3.2d, we see an instantaneous streamfunction plot at a time after many eddy turnover times of evolution. Because there is a maximum wavenumber in such a simulation, the enstrophy cascade cannot be completed and the flow cannot reach the selective decay state or the canonical equilibrium state predicted in the limit of max → ∞. However, the larger kmax is the smaller would be the scale of the fluctuations, and the closer would be the approach of the streamfunction, even instantaneously, to the predicted selective decay state. With finite resolution, however, assuming a fair degree of ergodicity, we should be able to approximate the ensemble average (for finite kmax ) by calculating a long-time average. The resulting time average  is shown in Figure 3.2e, and the correspondence with the prediction, qualitatively and quantitatively, is excellent.

3.9.1 SUBGRID-SCALE MODELING For lack of space, we have not attempted a discussion of the peculiar nature of eddy viscosity in a system for which energy cascades to large scale. Calculations of the negative eddy viscosity results are given in Kraichnan (1976) and Dubrulle and Frisch (1991). It is, however, worth noting here the particular effect that the presence of topography might have on the issue. The selective decay theory and the canonical equilibrium theory imply that the nonlinear interactions cause a cascade of enstrophy to small scales and that there is a resulting inverse energy cascade that, in the presence of topography, requires a buildup of the correlation between streamfunction and topography. The fact of finite resolution in a numerical simulation, which prevents the cascade from going beyond kmax , requires some adjustment to the flow at large scales. To simply force the growth of © 2006 by Taylor & Francis Group, LLC

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(a)

(b)

(d)

(c)

(e)

FIGURE 3.2 Simulation of inviscid flow over topography illustrating the evolution toward canonical equilibrium. (a) Bottom topography corresponding to the Emperor sea mount chain between 40◦ N and 45◦ N. (b) The initial streamfunction. (c) The mean streamfunction predicted by the canonical equilibrium theory based on the energy and enstrophy contained in the initial field shown in panel (b). (d) An instantaneous streamfunction after many eddy-turnover times of evolution. (e) The streamfunction time-averaged over a period of many eddy-turnover times.

large-scale energy with a negative eddy viscosity as in the flat bottom case would miss an effect of the topography. Holloway (1992) has suggested an interesting alternative. He proposed that the actual dynamics (3.1) could be replaced by the truncated dynamics with a tendency toward the mean stationary state predicted by canonical equilibrium s (or s in terms of the relative vorticity field). One variation of this type of model would be ∂ 1 + J (,  + h(x, y)) = − ( − s ) + F + D, ∂t 

(3.32)

for the resolved scales (see Merryfield and Holloway, 1997). Here  is a relaxation time that could be modeled in a variety of ways. Holloway and Sou (1996) performed simulations of such models and showed that they possessed some additional “skill” in comparison to models with no such topographic correction terms. Additional support for such an approach was given by Frederiksen (1999) by calculating the eddy viscosity from closure with topographic effects included. © 2006 by Taylor & Francis Group, LLC

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3.10 EFFECTS OF
The
-effect, which in quasigeostrophic theory is equivalent to the effect of large-scale topography, can drastically alter the nature of two-dimensional turbulence. The conservation of potential vorticity in the form q =  +
y has immediate consequences for the vortices that might form in a turbulent cascade. The circularly symmetric coherent vortex of the type in decaying turbulence is no longer a stationary solution even when isolated. Imagine the motion within and around an initially circularly symmetric cyclonic vortex on a
plane. Particles on the east side of the vortex moving northward must decrease their relative vorticity to conserve q, even developing negative relative vorticity if their initial vorticity was sufficiently low and their northward excursion sufficiently large. Particles on the west side of the vortex will move southward and increase their relative vorticity (becoming more positive). Given the circular nature of the arcs, a net dipolar anomaly to the original circular vortex soon evolves with an orientation toward the northwest. This will drive the cyclone toward the northwest (similarly and anticyclone would be driven to the southwest). The secondary vorticity patches so generated are sometimes called the beta-gyres. Depending on the relative strength of the vortex and the
-effect, the negative, or anticyclonic, vorticity that develops can actually peal off from the main vortex and be left behind in its wake. The resulting vorticity field can become very complicated since there will be repeated generation of anticyclones as the primary cyclone travels further north and each anticyclone left behind will have a similar capacity to generate its beta-gyres and subsequent associated cyclones. An example of the kind of wake that can develop is shown in Figure 3.3, where we see the cyclonic vortex near the northwestern corner of the domain after it has propagated from a point on the eastern border near the southeastern corner (Carnevale, 1995). Note the many anticyclones (light gray) in the wake and the associated small cyclones (dark gray) that they have created. Further discussion of the formation of these vortices in the wake is given in Carnevale and Kloosterziel (1994).

FIGURE 3.3 Shaded contour plot of a cyclonic vortex and its wake on a
-plane. Positive vorticity appears dark gray, while negative vorticity is colored light gray. (Figure taken from color image in Carnevale, 1995) © 2006 by Taylor & Francis Group, LLC

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ky

0

−0.8

−1

0 kx

1

FIGURE 3.4 Plot of the soft barrier at k
as given by (3.34) for
/U = 1. (cf. Vallis and Maltrud, 1993)

If
is sufficiently strong, the entire vortex structure could quickly disperse as Rossby waves. The dispersion relation for these waves is given by
k x
cos  k = − 2 = − . (3.33) k k Note that the frequency of Rossby waves increases with scale. We can imagine a turbulent decay starting from vortices of some small scale generating ever larger vortices through the merger process. Eventually, the vortices will become sufficiently large that the Rossby wave frequencies at the corresponding scale will be comparable to an eddy turnover time associated with the vortex. At that point, the wave motions will dominate and the vortices will disperse ending the two-dimensional turbulence process. Rhines (1975) estimated the scale at which this transition to waves should take place by comparing an eddy turnover time dimensionally given by e = (U k)−1 , where U is taken as the root mean square velocity of the flow, to the inverse Rossby wave frequency. Direct comparison gives the dividing wavenumber as 
| cos | k
= . (3.34) U √ Rhines averaged over angles to obtain an average scale that, up to constant factor, is k
=
/U . With simulations of turbulence on a
-plane, Rhines (1975) demonstrated that there was, in fact, a “soft barrier” to the transfer of energy to wavenumbers smaller than the angularly averaged k
. Furthermore, his simulations showed a tendency toward anisotropy with the formation of zonal jets. Vallis and Maltrud (1993) pointed out that the zonal jet formation could be understood in terms of the anisotropic form of the soft barrier boundary given by (3.34) or one of its analogous forms resulting from various choices of the eddy turnover time. Plotted in the k x − k y plane, the barrier takes on a dumbbell shape (see Figure 3.4) indicating reduced turbulent transfer to points within the dumbbell. The fact that the points k x = 0 lie outside the dumbbell indicates a preference for transfer from smaller scales into the zonal modes. Vallis and Maltrud (1993) studied this effect in great detail, not only considering the effects of varying the size of
, but also showing that zonal jet formation can happen on any broad topographic slope as well. Zonal jet formation was also demonstrated rather dramatically in randomly forced simulations on a sphere by Williams (1978) in an attempt to model the zonal jets on Jupiter. For studies of the evolution of the energy spectrum for two-dimensional turbulence on a sphere, see the recent work by Sukoriansky et al. (2002) and references therein. Alternatively, we can think of the waves as competing with the turbulence at all scales, and a more quantitative and complete theory of the arrest of the inverse cascade in
-plane turbulence and on the formation of zonal currents can be developed using Markovian closure theory. Holloway and Hendershott (1977) and Carnevale and Martin (1982) developed forms for the triad interaction © 2006 by Taylor & Francis Group, LLC

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63

(b)

(c)

FIGURE 3.5 Energy spectra plotted in the k x − k y plane. (a) Initial condition. (b) Actual energy spectrum averaged over six realizations from numerical simulations of inviscid flow starting with the initial energy spectrum shown in panel (a). (c) EDQNM closure prediction for the spectrum after the same time interval represented by the passage from (a) to (b) in the numerical simulation. (Figure taken from Vallis and Maltrud, 1993)

time (3.10) appropriate to
-plane turbulence. Essentially, the result can be written as kpq =

k + p + q . (k + p + q )2 + ( k + p + q )2

(3.35)

Note that under the wave resonance condition k + p + q = 0, the triad interaction time reduces to the original turbulence form of kpq and that the larger the sum k + p + q is the smaller will be kpq . Thus, the energy transfer to the resonant interactions will be just as efficient as in pure two-dimensional turbulence of a given spectrum. However, since the eddy damping rates k remain finite in a turbulent flow, the off-resonant interactions will be somewhat less efficient depending on the ratio of | k + p + q | to (k + p + q ). Thus, the efficiency of the triad interactions decreases significantly only when the sum of the frequencies is much larger than the sum of the eddy damping rates. If we take k << p, q, then kpq

1 ≈ k +  p +  q



2k 1+ (k + p + q )2

−1 .

(3.36)

This indicates a decreased efficiency of transfer from modes with large p and q into modes with small k except for the modes with | k | << (k + p + q ). The resulting flow of energy from small scales to large will preferentially populate modes outside a dumbbell similar to that given by (3.34) except that the size of the dumbbell region will depend on the modeling choice for the eddy decay rates and the ratio of k to the p and q considered. This was verified by Vallis and Maltrud (1993). They made a direct comparison of a numerical simulation of decaying two-dimensional turbulence with closure using (3.35). The result is shown in Figure 3.5. These are plots of the spectral distribution of energy. Panel (a) shows the initial conditions for the simulation and the closure theory. After some period of evolution, the simulation produces zonal jets. At that time, the spectral distribution of energy has evolved to the form shown in panel (b). The evolution of the spectrum in the closure model closely follows that of the simulation, with the spectrum at the time corresponding to that in panel (b) shown in panel (c). Thus, we see how the turbulent energy is channeled into the zonal jets by the turbulent cascade blocked by a soft barrier predicted by k
given in (3.34) inside which kpq becomes small. Interestingly, the zonal jets can be formed only by off-resonant interactions. There is no resonant interaction that can transfer energy into the zonal jets since the resonance condition must be satisfied simultaneously with the conditions k + p + q = 0. This combination of conditions permits resonant interaction with k x = 0 modes only if p = q, but the interaction coefficient for such an interaction © 2006 by Taylor & Francis Group, LLC

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in two-dimensional flow vanishes identically. Thus, it is not that the zonal jets are preferred by resonant interactions, but rather that they are prohibited from forming by that path. It is only that the off-resonant triads interacting with the zonal modes are somewhat more efficient at transferring energy to the modes outside the dumbbell defined by (3.34) than to those within, and the k x = 0 modes are outside the dumbbell. As for the spectrum in
-plane turbulence, the inhibition of vortices by the
-effect might have been expected to bring the spectrum toward the k −3 prediction. Actually, the
-plane spectrum remains steeper than k −3 , but not quite as steep as is found in the pure two-dimensional simulations (Vallis and Maltrud, 1993). It may be of some surprise that the closure equation for inviscid flow predicts a return to isotropy. This follows from the fact that, on the periodic domain, the only quadratic invariants are energy and relative enstrophy — the same invariants as on the flat bottom. Thus, according to the H-theorem given by Carnevale et al. (1981), the closure must force evolution toward the same isotropic spectrum as given in (3.15). This makes fundamental sense from the point of view of canonical theory since all modes are coupled to all other modes, and, to the extent that the quadratic invariants are the only ones preserved by the spectrally truncated dynamics, we must expect a return to isotropy if sufficient time is allowed. The formation of zonal flow in the truncated inviscid system can only be a transitory effect. Finally, we add some further comments on the resonant interaction or weak wave interaction theory. In the limit of infinitesimal amplitude flow, we may imagine the eddy damping rates would tend to vanish. In that case, as noted by Holloway and Hendershott (1977), the limiting value of kpq is given by (3.37) kpq = 2( k + p + q ), and the Markovian closure equation (3.9) reverts to the equation for weak wave interaction theory (see Carnevale and Martin, 1982) developed by various authors (Kenyon, 1964; Hasselmann, 1966; Longet–Higgins and Gill, 1967; Zakharov, 1974; Reznik, 1984; and others). As we have seen, there is some danger in using this approximation for two-dimensional turbulence since the paths to the purely zonal flows are blocked. Additional difficulties are discussed in Carnevale and Martin (1982), where it is pointed out that the set of quadratic invariants of the resonant interactions differs from that for the full set of possible interactions.

3.11 CONCLUDING REMARKS We have presented a view of two-dimensional turbulence that started with the prediction of energy and enstrophy cascades in the mid 1960s. These predictions were also supported by the closure theories of the 1970s. The picture of cascades through wavenumber space seemed well founded based on the closure theory of triadic interactions between Fourier modes. However, as numerical simulations became more powerful, it became ever clearer through the 1980s that the enstrophy cascade was spoiled by the presence of intense vortices that dominated the flow. In the early 1990s a switch in the theoretical approach resulted in the evolution of a statistical theory based on the statistics of vortices, giving a new understanding of decaying turbulence and the enstrophy cascade. One of the casualties of this transition was the idea of universality, which at least for the enstrophy cascade, was lost because so much depended on the initial distribution of vorticity and the ability of the vortices to “remember” their origins. The 1990s saw increasing computer power brought to bear on the question of the inverse cascade. At first, at least, this appeared to obey the classic predictions. However, even this has been challenged and evidence has been presented that this range may also be unstable to vortex formation on all scales and may not be sustainable and universal in any strict sense. This, however, remains somewhat unclear, and even less clear is whether some sort of approach based on statistics of vortices will be of any use in the inverse cascade. The most basic effects of topography and beta were discussed. The existence of many possible Arnold stable flows is a challenge to statistical theory. If these all correspond to states where the © 2006 by Taylor & Francis Group, LLC

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entropy is locally maximum, how can one predict the evolution from any particular initial condition using inviscid equilibrium statistical mechanics? At least, this means that these flows cannot be thought of as ergodic on the energy–enstrophy surface (Shepherd, 1987; Carnevale and Frederiksen, 1987). Although the predictions of selective decay and canonical theory based on energy and enstrophy decay have a great deal of skill in predicting the correlation between streamfunction and topography, simple thought experiments with flows initially near Arnold stable states based on generalized enstrophy show that they cannot succeed in all conditions. Also, the most basic prediction of broad westward flow, or pseudowestward flow, on a beta-plane of large-scale topographic feature is violated routinely by numerical simulations that produce zonal jets instead. Furthermore, laboratory experiments and numerical simulations have demonstrated that cyclones tend to climb hills in a clockwise spiral, seemingly violating the statistical preference for anticyclonic vorticity over hills predicted by canonical equilibrium (Carnevale et al., 1988a, 1991a). Thus, we must conclude that, although the predictions of the energy–enstrophy canonical equilibrium theory may have a wide range of applicability, caution must be exercised in light of these exceptions. We have only had space to develop a few themes in this presentation. A more thorough review of this subject would necessarily examine many other important areas. We have not mentioned the experimental side of the study of two-dimensional turbulence, although such studies have often been extremely important in confirming and developing new ideas (see, for example, Couder and Basdevant, 1986; Kloosterziel and van Heijst, 1989, Carnevale et al., 1991a; Marteau et al., 1995; and the review by Tabeling, 2002). A very important question for geophysical flows, as important as the questions of the effects of topography and beta, is the issue of the effect of vertical density stratification. The first approximation to this three-dimensional effect in the two-dimensional model is by the introduction of a finite deformation radius in the evolution equation, resulting in ∂( − 2 ) + J (,  + h(x, y)) = F + D. (3.38) ∂t Turbulence in this “equivalent barotropic” model has very different statistics, depending on whether the excited scales of motion are much larger or much smaller than the deformation radius 1/. Recent investigations by Iwayama et al. (2001) have suggested that Batchelor scaling may result in the limit of small deformation radius (the opposite limit of the pure two-dimensional turbulence case). The question of negative eddy viscosity was only mentioned briefly, but has been an important topic with many important theoretical and simulation studies (for example, Kraichnan, 1976; Dubrulle and Frisch, 1991; Gamma and Frisch, 1993). Proper formulation of a negative eddy viscosity for use in oceanographic or atmospheric modeling remains little developed. The question of intermittency in the energy and enstrophy cascades as evidenced by the behavior of structure functions has been an important and active topic and may shed some light on what may be universal in two-dimensional turbulence (see Falkovich and Lebedev, 1994). Also, a topic of great fascination and importance in two-dimensional turbulence is that of self-propagating form-preserving coherent structures: solitons, solitary waves, and modons and how they survive in geophysical flow and turbulence (see McWilliams et al., 1981; Carnevale et al., 1988a,b; and more recently, Shivamoggi, 2000). Furthermore, we have not even mentioned the problem of particle dispersion, which has an extensive literature. All these are fascinating topics that deserve further discussion and research. Reviews that were very helpful in preparing this presentation are Kraichnan and Montgomery (1980), Salmon (1982), Holloway (1986), and Tabeling (2002).

ACKNOWLEDGMENTS This work has been supported by National Science Foundation grants OCE 01-29301 and OCE 05-25776, a grant from the Ministero Istruzione Universita e Ricerca (MIUR D.M. 26.01.01 n. 13) and a grant from the San Diego Supercomputer Center. © 2006 by Taylor & Francis Group, LLC

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Merryfield W.J., Holloway, G. 1997 Topographic stress parameterization in a quasi-geostrophic barotropic model. J. Fluid Mech. 341, 1–18. Miller, J., Weichman, P.B., Cross, M.C. 1992 Statistical mechanics, Euler’s equation, and Jupiter’s red spot. Phys. Rev. A 45, 2328–2359. Montgomery, D., Joyce, G. 1974 Statistical mechanics of “negative temperature states.” Phys. Fluids 17, 1139– 1145. Montgomery, D. 1976 A BBGKY framework for fluid turbulence. Phys. Fluids 19, 802–810. Montgomery, D., Matthaeus, W.H., Stribling, W.T., Martinez, D. Oughton, S. 1992 Relaxation in two dimensions and the “sinh-Poisson” equation. Phys. Fluids A 4, 3–6. Onsager, L. 1949 Statistical hydrodynamics. Nuouo Cimento Suppl. 6, 279–287. Orszag S.A. 1970 Analytical theories of turbulence. J. Fluid Mech. 41, 363–386. Pouquet, A., Lesieur, M., Andre, J.C., Basdevant, C. 1975 Evolution of high Reynolds number two-dimensional turbulence. J. Fluid Mech. 72, 305–319. Reznik, G.M. 1984 On the properties of equilibrium spectra of weakly nonlinear Rossby waves. Oceanology 24, 655–657. Rhines, P.B. 1975. Waves and turbulence on a beta-plane. J. Fluid Mech. 69, 417–443. Robert, R., Sommeria, J. 1991 Statistical equilibrium states for two-dimensional flows. J. Fluid Mech. 229, 291–310. Salmon, R. 1982 Geostrophic turbulence. In Topics in Ocean Physics. Proc. Int. Sch. Phys. Enrico Fermi, Varenna, Italy, pp. 30–78. Salmon, R., Holloway, G., Hendershott, M.C. 1976. The equilibrium statistical mechanics of simple quasigeostrophic models. J. Fluid Mech. 75, 691–703. Santangelo, P., Benzi, R., Legras, B. 1989. The generation of vortices in high resolution, two-dimensional decaying turbulence, and the influence of initial conditions on the breaking of self-similarity. Phys. Fluids A 1, 1027–1034. Shepherd, T.G. 1987 Nonergodicity of inviscid two-dimensional flow on a beta-plane and on the surface of a rotating sphere. J. Fluid Mech. 184, 289–302. Shivamoggi, B.K. 2002 A generalized class of nonlinear Rossby localized structures in geophysical flows. Chaos Solitons Fractals, 14, 469–478. Smith, L.M., Yakhot V. 1993 Bose condensation and small-scale structure generation in a random force driven 2D turbulence. Phys. Rev. Lett. 71, 352–355. Smith, L.M., Yakhot V. 1994 Finite size effects in forced two-dimensional turbulence. J. Fluid Mech. 274, 115–138. Sukoriansky, S., Galperin, B., Diovskaya, N. 2002 Universal spectrum of two-dimensional turbulence on a rotating sphere and some basic features of atmospheric circulation on giant planets. Phys. Rev. Lett. 89, 124501 1–4. Tabeling, P. 2002 Two-dimensional turbulence: a physicist approach. Phys. Rep. 362, 1–62. Turkington, B. 1999 Statistical equilibrium measures and coherent states in two-dimensional turbulence. Commun. Pure Appl. Math. LII, 781–809. Vallis, G.K., Maltrud, M.E. 1993 Generation of mean flows and jets on a beta plane and over topography. J. Phys. Ocean. 23, 1346–1362. Weiss, J. 1981 The dynamics of enstrophy transfer in two-dimensional hydrodynamics. La Jolla Inst. Tech. Rep. LJI-TN-81-121. (later published in 1991 in Physica D, 48, 273–294.) Weiss, J.B., McWilliams, J.C. 1993 Temporal scaling behavior of decaying two-dimensional turbulence. Phys. Fluids A 5, 608–621. Whitaker, N., Turkington, B. 1994 Maximum entropy states for rotating patches. Phys. Fluids A 6, 3963–3973. Williams, G.P. 1978 Planetary circulations: 1. Barotropic representation of Jovian and terrestrial turbulence. J. Atmos. Sci. 35, 1399–1426. Yin, Z., Montgomery, D., Clercx, H.J.H. 2003 Alternative statistical–mechanical descriptions of decaying twodimensional turbulence in terms of “patches” and “points.” Phys. Fluids, 15, 1937–1953. Zakharov, V.E. 1974 The Hamiltonian formalism for waves in nonlinear media having dispersion. Izv. Vuzov. SSSR Radiofizica 17, 431–453.

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Plasma Physics in 4 Statistical a Strong Magnetic Field: Paradigms and Problems J.A. Krommes CONTENTS 4.1 4.2 4.3

Introduction ............................................................................................................................ 69 Introductory Plasma-Physics Background, Particularly Gyrokinetics ...................................70 Plasma Applications of Statistical Methods ........................................................................... 74 4.3.1 Gyrokinetic Noise ...................................................................................................... 74 4.3.2 Realizable Statistical Closures ................................................................................... 75 4.4 Statistical Description of Long-Wavelength Flows ................................................................ 76 4.4.1 Asymptotic Long-Wavelength Expansion of the EDQNM Formula for Coherent Damping ................................................................................................ 77 4.4.2 Weakly Inhomogeneous Spectral Kinetics and Convective-Cell Growth Rate ................................................................................................................ 79 4.4.2.1 General Remarks about Weakly Inhomogeneous Statistics ....................... 79 4.4.2.2 Modulated Reynolds Stress, Energy Principles, and Use of the Martin–Siggia–Rose Formalism ...................................................... 80 4.4.3 Hamiltonian Formalism ............................................................................................. 83 4.4.3.1 Hamiltonian Description of Eulerian Partial Differential Equations.......... 83 4.4.3.2 Hamiltonian Description of
q ................................................................... 85 4.4.3.3 The Tensor Triad Interaction Time............................................................. 86 4.5 Discussion............................................................................................................................... 87 Acknowledgments............................................................................................................................ 87 References........................................................................................................................................ 87

ABSTRACT An overview is given of certain aspects of fundamental statistical theories as applied to strongly magnetized plasmas. Emphasis is given to the gyrokinetic formalism, the historical development of realizable Markovian closures, and recent results in the statistical theory of turbulent generation of long-wavelength flows that generalize and provide further physical insight to classic calculations of eddy viscosity. A Hamiltonian formulation of turbulent flow generation is described and argued to be very useful.

4.1 INTRODUCTION The purpose of this chapter is to describe the current state of affairs regarding certain topics in fundamental statistical plasma physics, with emphasis on techniques and recent results especially relevant for strongly magnetized plasmas. Such plasmas are particularly useful in fusion research; however, no detailed knowledge of fusion physics is required or used in the discussion. © 2006 by Taylor & Francis Group, LLC

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In Section 4.2 I provide some introductory plasma-physics background on the gyrokinetic formalism used in modern discussions of low-frequency fluctuations in magnetized plasmas. This is a reduced kinetic (velocity-space) description that efficiently replaces the more comprehensive Vlasov equation when only low-frequency, long-wavelength fluctuations are of concern. Moments of the gyrokinetic equation lead to nonlinear gyrofluid equations that are the plasma analogs of the Navier–Stokes equation. Several important equations are briefly derived or cited, including the Hasegawa–Mima equation (directly analogous to the Charney equation for Rossby waves) and the Hasegawa–Wakatani equations. In Section 4.3 I mention some past and recent plasma applications of statistical methods, including calculations of gyrokinetic noise and the development of realizable Markovian closures appropriate for problems with linear waves (ubiquitous in plasma physics as well as in geophysics and elsewhere). In Section 4.4 I discuss in detail the statistical description of convective cells, including zonal flows (those will be defined later). This problem is closely related to calculations of eddy viscosity for Navier–Stokes fluids; I will note a precise and quantitative connection. The methodology unifies a number of interesting technical specialties, including statistical closure theory, the theory of weakly inhomogeneous spectral evolution equations, field-theoretic techniques, and the Hamiltonian formulation of nonlinear partial differential equations (PDEs). The chapter ends in Section 4.5 with some discussion and suggestions for future work.

4.2 INTRODUCTORY PLASMA-PHYSICS BACKGROUND, PARTICULARLY GYROKINETICS At the level of one-particle probability density functions (PDFs), the most fundamental description of a magnetized, collisional plasma is provided by the kinetic equation ∂t f s (x, v, t) + v · ∇ f s + (q/m)s (E + c−1 v × B) · ∂v f s = −Cs [ f ],

(4.1)

where the collision operator C is the Balescu–Lenard operator or, more practically, the Landau operator. Here s is a species label, q is the charge, m is the mass, E is the electric field, B is the magnetic field, and square brackets denote functional dependence. The fields are determined from Maxwell’s equations. Equation (4.1), which lives in the six-dimensional (6D) phase space of a generic particle (sometimes called the  space), describes all particle motions, including the rapid gyrospiraling around the . magnetic field at frequency c = q B/mc, as well as high- and low-frequency collective fluctuations. Its very completeness, however, poses substantial problems for analysis and numerical simulation. For toroidal plasma systems (Figure 4.1), which are routinely confined long enough to have particle distributions very close to a local Maxwellian, simple arguments [64] show that the most important microinstabilities are of low frequency, characteristically  ∼ ∗ , where ∗ is the so-called diamagnetic drift frequency inversely proportional to the scale length L n of the background density profile. Specifically, if B ∝
z and ∇n ∝ −
x , then ∗ = k y V∗ , where . V∗ = cTe /eB L n = (s /L n )cs , (4.2) . the sound speed is cs = Z Te /m i (Te is the electron temperature and Z is the atomic number of the . ions), and s = cs /ci . One has ∗ /ci = (k y s )(s /L n )  1,

(4.3)

since typically k y s = 0(1) and s  L n . The disparity between ∗ and ci implies that simulations of the complete dynamics are prohibitively lengthy by orders of magnitude, even with modern supercomputers. The advent of the nonlinear gyrokinetic formalism was a major advance. Gyrokinetics is the statistical description of the motion of appropriately defined gyrocenters. Most heuristically, the motion © 2006 by Taylor & Francis Group, LLC

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c eti gn Ma a xi s

Field lines

y b



y

,

a x

o

ra

rb

x

z B (a)

z (b)

FIGURE 4.1 (a) A typical tokamak showing nested flux surfaces. (b) Local coordinate system for studying microturbulence. The turbulence is of much shorter wavelength than the scale length of the background profile.

of a gyrocenter is defined by the well-known particle drift velocities [11] such as the E × B, ∇ B, and curvature drifts. However, there are subtleties, especially surrounding the role of the polarization drift V pol =
−1 c ∂t (cE ⊥ /B) (⊥ means perpendicular to B), that can only be addressed with the aid of a systematic formalism. Following earlier attempts at linear gyrokinetics [1,10], Frieman and Chen [23] published an important paper discussing the nonlinear problem. That work, however, divided the distribution function into separate parts for the background and the fluctuations, so the resulting gyrokinetic equation was not immediately useful for numerical simulation and also disguised certain important conservation properties of the dynamics. Lee [53] attempted to correct those deficiencies; however, his recursive procedure was soon superceded by the noncanonical Hamiltonian formalism of Dubin et al. [18] based on the Darboux techniques of Littlejohn [54]. After the later advent of even more efficient techniques based on differential one-forms [9], the formalism was further refined by Hahm [24]. For the most modern results, see Reference [7] and Reference [63]. The Hamiltonian methods proceed by constructing a perturbative change of variables from an initial set of guiding-center variables appropriate for a static, spatially constant magnetic field. The goal of the transformation is to remove dependence on the gyro-angle  from all components of the fundamental differential one-form [9]; when that is done, the magnetic moment  is (adiabatically [17]) conserved. The procedure thus ensures that the gyrocenter PDF is independent of  (so the phase space is five dimensional rather than six dimensional) and that the left-hand side of the kinetic equation contains no derivative with respect to . These are considerable simplifications. For the simplest case of a constant magnetic field, the gyrokinetic equation is approximately (in the absence of collisions) ∂t Fs (X, , v , t) + v ∇ Fs + V E · ∇Fs + (q/m)s E  ∂v Fs = 0.

(4.4)

Here F is the PDF of gyrocenters and the overlines signify the effective (gyro-averaged) field at the position X of the gyrocenter [in k space, the averaging introduces the factor J0 (k⊥ v⊥ /
ci )]. Notably, only the E × B drift enters in the gyrokinetic equation1 ; the polarization drift does not appear. Instead, the effects of polarization enter through the Poisson equation. That equation holds at the 1 For

nonconstant B, the magnetic drifts appear as well.

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E

E×B

FIGURE 4.2 Let the magnetic field B point out of the page. A net deficiency of ion gyrocenters produces an inward-pointing electric field E; the resulting E × B drift produces counterclockwise motion, i.e., a positive vorticity [in agreement with Eqs. (4.6) and (4.7)].

  position x of the actual particles; it is −∇ 2 (x, t) = 4(x, t), where  = s (nq)s dv f x (x, v, t) is the charge density.2 In this last expression, the coordinates and the distribution function must be transformed in order to calculate  from the gyrocenter distribution F. The polarization drift is contained in that change of coordinates [17,18], the several steps of which are reviewed in Appendix C of Reference [46]. For Ti → 0, the final result for a quasineutral plasma consisting of electrons and a single ion species is −(2De ∇ 2 + 2s ∇⊥2 ) = Ni /n i − n e /n e , (4.5) . . where  De is the electron Debye wavelength [ De = (Te /4n e e2 )1/2 ] and = e/Te . The 2s term pol pol describes the ion polarization density n i (electron polarization is negligible); that is, n i = Ni +n i , . pol pol pol pol . pol where n i obeys ∂t n i + ∇· j i = 0 with j i = (nq)i V i . The quantity ⊥ = 1 + 2s /2De = . 1 + 2pi /2ci [ pi = (4n i qi2 /m i )1/2 ] defines what can be called the dielectric constant of the gyrokinetic vacuum [44]. The gyrokinetic regime is defined [51] by ⊥  1, which is frequently the case.3 Then, since usually k  k⊥ , the 2De term, which describes the deviation from absolute charge neutrality, is negligible. Upon scaling lengths to s , one obtains the gyrokinetic Poisson equation −∇⊥2 = Ni /n i − n e /n e ,

(4.6)

which describes quasineutrality within the gyrokinetic framework. It is quite convenient numerically because one does not need to somehow enforce i = e implicitly but instead can merely calculate the polarization density (or resulting potential) from the given imbalance of gyrocenter densities that exists at time t. A simple physical interpretation can be given of the important 2s ∇⊥2 contribution to the gyrokinetic Poisson equation. Consider the z-directed vorticity  associated with the E × B motion: .  =
z · ∇×VE . One readily finds that /ci = 2s ∇⊥2 ;

(4.7)

that is, for cold ions the ion polarization density is just the vorticity in appropriate units. This important property of strongly magnetized plasma is illustrated in Figure 4.2, where it is shown that a deficiency of ion gyrocenters leads to a positive vorticity. It is thus expected that the dynamics of vorticity will play a crucial role in virtually all gyrokinetic calculations. mean density n enters because of the convenient normalization convention V −1 volume. 3 The guiding-center limit is 2 /2 → 0 ( → 1). ⊥ pi ci 2 The

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dx



dv f = 1, where V is the

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Integration of Equation (4.4) over velocity leads to the continuity equation for gyrocenters. If ion parallel motion is ignored (because the ion inertia is very large) and for simplicity one again considers Ti → 0, one obtains ∂t Ni + ∇·(VE Ni ) = 0. (4.8) Let us divide Ni into a background (mean-profile) part Ni  = n i and a fluctuating part Ni : Ni = Ni  + Ni . The mean profile obeys ∂t Ni  + ∇·Γi = 0, . where Γi =  VE Ni , and the fluctuations obey       ∂ Ni ∂

Ni + V∗ − n i−1 Γ = 0. + ∇· VE ∂t ni ∂y ni

(4.9)

(4.10)

Here the so-called “diamagnetic” term proportional to V∗ is a rewriting of VE · ∇ lnNi ; it thus describes advection of the background density gradient, not literally ion diamagnetism (which vanishes for Ti = 0). For homogeneous statistics, the terms in ∇·Γ vanish. The gyrocenter continuity equation contains both Ni and . The gyrokinetic Poisson equation relates those quantities, but also introduces the electron density as a new unknown. The classic approximation of Hasegawa and Mima (HM) [26] is to assume that the electron response is adiabatic:

n e /n e = (the first-order part of a Maxwell–Boltzmann distribution). Poisson’s equation then becomes

Ni /n i = − ∇⊥2 . (4.11) Finally, substitution of this result into Equation (4.10) leads to the Hasegawa–Mima equation (HME)4 (1 − ∇⊥2 )∂t + V∗ ∂ y + VE · ∇(−∇⊥2 ) = 0.

(4.12)

Note the appearance of the vorticity  = ∇⊥2 in several terms. The original derivation of HM from the Braginskii equations [6] (the fluid equations in particle coordinates) was not as physically transparent. The Hasegawa–Mima equation is the simplest paradigm for the nonlinear dynamics of drift waves. If one omits the two terms related to explicit plasma effects [the term in ∂t (adiabatic electron response)] and the term proportional to V∗ (effect of the background density gradient), one is left with the two-dimensional Navier–Stokes equation (NSE) ∂t  + VE · ∇ = 0.

(4.13)

The implicit assumption of HM was that the parallel wave number k does not vanish. If it does, adiabatic electron response is no longer appropriate because there is no parallel potential modulation for the mobile electrons to neutralize. Because electron polarization is negligible, a frequently used approximation is to ignore electron response altogether for the k = 0 modes (convective cells [CCs]).5 What results is the generalized HME (
− ∇⊥2 )∂t +
V∗ ∂ y + VE · ∇(
− ∇⊥2 ) = 0, where .
=



1 ∇ = 0 0 ∇ = 0

(4.14)

(4.15)

4 Following normal convention, the flux term Γ has been omitted. Rigorously, this is valid only for homogeneous statistics. In the absence of classical dissipation, as in the present derivation, Equation (4.9) shows that it is also a consequence of the assumption of a statistical steady state. 5 A subtle issue regarding the interpretation of this statement in the presence of magnetic shear cannot be discussed here.

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can be interpreted as a projection operator onto the DW subspace. For the CCs, one obtains precisely the two-dimensional NSE, Equation (4.13). Let us call the k = 0 modes the drift waves (DWs). For the DWs, the linear dispersion relation is ∗ = . (4.16) 2 1 + k⊥ In this approximation there is no growth or dissipation. This nonphysical result is a consequence of the neglect of Landau damping (the physically important wave–particle resonance, which is lost in fluid truncations that involve neglect of velocity cumulants beyond some order6 ) and collisions. Both of those effects introduce nonadiabatic electron response. If only collisions are taken into account (through the electron momentum equation), the system of equations due to Hasegawa and Wakatani (HW) [27] results:
( − n) + ∇⊥2 , ∂t  + VE · ∇ =  ∂t n + VE · ∇n = 
( − n) −  ∂ y  + D∇ 2 n, ⊥

(4.17a) (4.17b)

. −1 2  . where n ≡ n e ,  = L −1 n ,
= −D ∇ , and the equations are written for fluctuations only. The dissipative terms involving  and D can be added heuristically or can be derived from more detailed considerations. The virtue of this system is that it is forced (there is a linear instability for some wavevectors [30]) and dissipative, so it can achieve typical states of steady-state turbulence. It provides the simplest paradigm for fluctuations in the cold, collisional edge of toroidal devices7 and has been discussed by a number of workers [30,36,46]. Further nonlinear equations of interest to research on strongly magnetized plasmas are discussed in the review article by Krommes [46].

4.3 PLASMA APPLICATIONS OF STATISTICAL METHODS Much evidence from analysis and simulation shows that steady-state gyrokinetic microturbulence can exist, in agreement with experimental observations. Statistical descriptions are thus appropriate. Before turning in Section 4.4 to the main topic, the statistical description of convective cells, I wish to mention briefly some of the other applications for which statistical formalism has been used in plasma physics. For a much more complete review, see Reference [46].

4.3.1 GYROKINETIC NOISE One method of solution of the gyrokinetic equation is by the so-called particle-in-cell technique (reviewed for unmagnetized plasmas in Reference [2]). A large number of particles is distributed at t = 0 (either randomly or by a quiet start8 ). They are integrated along the characteristic trajectories of the gyrokinetic equation for a small time step, then the charge density is calculated by distributing the particles onto a spatial grid and the potential is updated by solving the gyrokinetic Poisson equation; the process is then repeated.9 The procedure is a Monte Carlo sampling method [28]; as such, one must contend with sampling noise. 6 More

sophisticated fluid closures can model the Landau damping; see Reference [25]. practice, magnetic curvature drive must also be included. 8 The general problem of initializing a particle distribution subject to various constraints is interesting and nontrivial. A Monte Carlo scheme particularly appropriate for generating states of negative temperature was discussed and analyzed in Reference [52]. 9 In practice, one frequently uses a modification of this basic scheme wherein one solves only for perturbations away from a known (e.g., Maxwellian) base state. Some remarks and historical references to this so-called
f method can be found in Reference [28]. 7 In

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In unmagnetized plasmas, long-wavelength Fourier amplitudes are excited almost to thermal level. This would not be a problem if a physically realistic number of particles was used, since the microturbulent fluctuations are far above the thermal level; however, it is an important constraint for simulations, which can use only a restricted number of simulation particles. However, the fluctuation characteristics of gyrokinetic plasmas differ from those of the full many-body problem because high-frequency fluctuations have effectively been excised. The gyrokinetic fluctuation–dissipation theorem was studied in several papers by Krommes [43–45,51]. As one would expect, the large dielectric constant of the gyrokinetic vacuum leads to strongly suppressed gyrokinetic fluctuations. Gyrokinetic particle simulation in fully three-dimensional and global toroidal geometry is now a major industry.

4.3.2 REALIZABLE STATISTICAL CLOSURES In view of the physical complexity of nonlinear plasma dynamics, it is not surprising that the theory and applications of statistical closures for plasmas have lagged far behind their Navier–Stokes counterparts. An attempt at a historical survey of the plasma theories was made in Reference [46]; see in particular the bibliographical timelines presented in Figure 34 through Figure 36 of that work, which clearly show the inflow of information to plasma turbulence theory from the other physics specialties. A very brief historical overview follows. For reference, the direct-interaction approximation (DIA) was proposed by Kraichnan in 1959 [37]. The quasilinear description of plasmas was discussed by Vedenov et al. [69] and Drummond and Pines [16] in the early 1960s. Various calculations on perturbative weak-turbulence theory [14,66] soon followed. The first attempts at renormalization were by Dupree [20,21]. In 1967, Orszag and Kraichnan [60] critically analyzed Dupree’s 1966 resonance-broadening theory (RBT) and gave a thorough discussion of the Vlasov DIA; unfortunately, that important work was ignored for about a decade. A resurgence of interest in renormalized formalisms for plasmas was stimulated by the seminal 1973 paper of Martin, Siggia, and Rose (MSR) [56]. Krommes [41] discussed how RBT and related approximate plasma theories were embedded in the more general formalism. Dubois and Espedal [19] provided important insights about the general form of the nonlinear plasma dielectric function. Many additional references and historical remarks can be found in Reference [46]. By considering a three-wave example, Krommes [42] noted early that the DIA could provide an adequate description of the saturation level of drift waves coupled through a Hasegawa–Mimalike nonlinearity. This was no surprise to the neutral-fluid community in view of the seminal work of Kraichnan [38], but it was the first quantitative calculation of the full DIA in a plasma context. For practical calculations on many-mode problems such as HM or HW, it was readily apparent that Markovian closures were preferred. The eddy-damped quasinormal Markovian (EDQNM) closure was known to be realizable for Navier–Stokes dynamics [59]. It was therefore an unwelcome surprise when numerical integrations of the EDQNM for certain DW problems demonstrated unrealizable behavior. This issue was studied at length by Bowman [3] in collaboration with M. Ottaviani and me. The problem was traced to the triad interaction coefficient k pq , which becomes complex in the presence of linear waves; the triad interaction time Re k pq ≡ rk pq can then easily become negative. This difficulty arises in the evolution of the EDQNM from an initial state. Bowman and collaborators argued pragmatically that one should abandon a faithful description of the transient evolution (in any event, nonrealizable behavior cannot be faithful) while constraining the initial-value problem to evolve to the steady state described by the EDQNM. While there are various ways of doing this for the statistical description of scalar fields, the challenge of implementing this constraint for the evolution of systems of coupled fields proved to be formidable. Bowman et al. [5] discussed a method that appears to work satisfactorily in practice; they called the algorithm the realizable Markovian closure (RMC). Hu and Krommes [29,30] used the RMC to discuss the © 2006 by Taylor & Francis Group, LLC

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statistical dynamics of the HW system; excellent agreement was found between direct numerical simulations and the closure predictions of the particle flux and wave-number spectra. Bowman [4] subsequently made related studies of HM dynamics; he also discussed and studied a realizable test-field model motivated by Kraichnan’s work in Reference [39]. The numerical closure studies that directly stemmed from Bowman’s pioneering work are the most complete and recent in the context of plasma DW models. The successful comparisons between DNS and simple nonlinear DW paradigms such as HM and HW show that the essentials of secondorder closures are reasonably well understood in plasma-physics contexts. At the present time, it appears to be hopeless to perform detailed closure studies on more realistic models involving nontrivial (e.g., toroidal) geometry, three dimensions, and the many practical effects that are inevitably present in real devices; such problems are better addressed by large-scale numerical gyrokinetic simulations, which are ongoing. One fertile area for future research involves the analytical study of intermittent statistics, which are frequently observed in the edges of toroidal devices. This difficult area might be addressed through the mapping-closure techniques originally suggested in Reference [12] and developed by later authors (see Reference [46] and the recent paper by Takaoka [68]). However, application of mapping closure to HW-like systems involves major technical complications, including generalization to coupled fields and the difficulty of adequately handling the nonlocal relation between potential
and vorticity ∇⊥2
. Research in this area [62] is still at a very early stage, so it will not be discussed further in this chapter.

4.4 STATISTICAL DESCRIPTION OF LONG-WAVELENGTH FLOWS I now turn to recent research on the statistical description of the turbulent self-generation of longwavelength flows. This subject is intimately related to the seminal calculations of eddy viscosity by Kraichnan [40]; I will describe some new insights that have emerged recently. The problem is also of considerable practical importance in fusion physics, since it affects the physics of saturation of DW turbulence and thus has implications for the magnitudes of steady-state transport. In the fusion context, it is usual to distinguish various special cases of convective cells (k = 0): zonal flows (ZFs; y-directed E × B velocities arising from potentials with k y = 0); and streamers (x-directed velocities arising from potentials with k x = 0). Streamers can cause direct transport of particles and heat across magnetic flux surfaces, but can be broken up by secondary instabilities [13] and ZFs. Zonal flows do not cause such transport, but they can interact with the DWs and thereby indirectly regulate the level of turbulent transport. Diamond et al. [15] attempted an analytical description of zonal-flow generation. Their basic idea, which was well motivated, was to assume scale separation between short-wavelength DWs and long-wavelength ZFs and to use a weakly inhomogeneous wave kinetic equation (WKE) for the DWs (k = 0) to calculate energy transfer to the ZFs due to the weak modulation of the DWs by the ZFs. A central question is what quantity to use as the “plasmon density” in the WKE. Some insight was provided by Smolyakov and Diamond [67], who derived “action invariants” for various simple paradigms such as the generalized HME. However, that work did not address the question of how to proceed for systems of coupled PDEs such as the HW system. I will discuss an appropriate methodology in Section 4.4.3. Motivated by References [15] and [67], Krommes and Kim (KK) [49] performed a detailed study of the problem of convective-cell generation by generalized HM dynamics in the limit of disparate scales. That work proceeded directly from Markovian statistical closure and thus was closely related to the calculations of Kraichnan [40] on the asymptotic long-wavelength limit of eddy viscosity in two dimensions. As I will describe, the work of KK sheds further light on the calculations of Kraichnan as well as Diamond et al. and Smolyakov and Diamond; it unifies all such calculations for scalar fields into a common framework.

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Subsequent work10 on coupled-field systems [48,50] provided an elegant formalism that subsumes the scalar calculations and essentially completes the formal statistical description of longwavelength CC generation by short-wavelength DWs. That is, however, only one part of the full problem of interacting DWs and CCs. I will not discuss the effect of the CCs on the DWs except to say that preliminary work on systematic bifurcation analysis of the transition to drift-wave turbulence has been accomplished [32]11 ; that work involves the CC backreaction in a fundamental way.

4.4.1 ASYMPTOTIC LONG-WAVELENGTH EXPANSION OF THE EDQNM FORMULA FOR COHERENT DAMPING For the CC generation problem, I first define exactly what statistical quantity is to be computed. Consider a scalar amplitude equation ∂t k − L k k =

1 Mk pq ∗p q∗ . 2 

(4.18)

In a Markovian description of homogeneous statistics, the spectral-balance equation for wave-number . intensity Ck (t) = | k (t)|2  is ∂t Ck − 2(Re L k )Ck + 2(Re k )Ck = 2Fk ,

(4.19)

where k describes coherent nonlinear response and Fk is the variance of the nonlinear noise f k in the Langevin amplitude equation ∂t k − L k k + k k = f k (t).

(4.20)

When I specialize to the DW–CC problem, I shall use k for the DWs and q for the CCs; I always . work in the limit  = q/k  1. Then the CC growth rate
q is defined by .
q ≡
qnl = − Re q .

(4.21)

The linear growth rate is, of course,
qlin = Re L q . Thus, the spectral balance equation for the CCs is  ∂t Cq = 2
qlin +
qnlin Cq + 2Fq . (4.22) Fq is intrinsically positive, so one or both of
qlin or
qnlin must be negative so that a steady state can be achieved. (Typically,
lin is very small [65].) What I shall calculate in this work is the q

portion of
qnlin due to interactions with short-wavelength DWs. Except for a sign, this is exactly the statistical eddy viscosity in the asymptotic limit   1. If
qDW is positive, one is discussing negative eddy viscosity. This does not mean that the total
nlin need be positive. The ultimate value and sign q

of
qnlin involve interactions with neighboring scales as well as distant ones, and the details cannot be available until all of the fully self-consistent modal interactions are analyzed. Such calculations have not been published as yet. 10 At

the May, 2003, conference that these proceedings document, my talk focused on the work of Krommes and Kim [49]. Further progress on coupled systems was made during the subsequent half-year that elapsed before publication of these proceedings was formalized. In the interest of utility and completeness, I will incorporate those latter results into the present discussion. 11 Since this paper was originally submitted in early 2004, work on the transition problem has progressed; its current status is represented by References [35] and [34].

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∆

∆

kmin

p k

D

q

C

kmin ∆

∆

FIGURE 4.3 Integration domains for the evolution of the small q’s. Region C: k ≥ k min . Region D: k < k min , p ≥ k min . Figure reprinted with permission from J. A. Krommes and C.–B. Kim, Phys. Rev. E 62, 8508 (2000). Copyright (2000) by the American Physical Society.

According to the EDQNM, one has
q = −




Mqk p Mk pq rqk p C p .

(4.23)




 Here,
denotes the sum over all triangles such that k + p + q = 0, and qk p is the triad interaction coefficient whose real part r defines the triad interaction time. In the usual EDQNM,  is a symmetric function of its arguments.12 The goal is to expand
q for q  k, p. In detail, this is somewhat tedious because if q is fixed, then p varies with k. The integration domain is thus complicated (Figure 4.3) and one must be very careful. The details were presented in Reference [49], first for isotropic statistics (a test case that is closely related to the calculations of Reference [40]), then for the more general and realistic anisotropic case. For the case of the GHME, the final result is
q = −


  q ·k 2 ∂Zk z · (q×k)2 r q. . k,−k,q 2 ∂k      (1 + k 2 )2   q + q k        (b) (c) (a)

(d)

(4.24)

(e)

. Here, Zk = (1 + k 2 )2 |k |2 . In the next section I will describe a more physical algorithm that recovers this result. However, it is useful now to list the physical interpretation of each term: (a) (b) (c) (d) (e)

Possibly nonadiabatic CC response Nonlinear advection of the DWs by the CCs Interaction time between the DWs and the CCs DW wavevector refraction due to CC modulation Conservation of Z by the DWs due to CC modulation

At this point, the interpretation of at least term (d) is probably unclear. This will be explained by the physical algorithm to be discussed next.

12 Bowman

[5] used an asymmetric function in his RMC.

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4.4.2 WEAKLY INHOMOGENEOUS SPECTRAL KINETICS AND CONVECTIVE-CELL GROWTH RATE Given the rigorous and systematically derived result (4.24), one can discuss various physical algorithms that reproduce it [49]. The basic idea, first proposed in Reference [15], is to use a weakly inhomogeneous spectral balance equation or “wave kinetic equation” to describe the modification of the short-wavelength DW statistics due to the long-wavelength CC modulation. In Diamond’s formalism, it was important that the proper quantity Z was used as the plasmon density evolved by the WKE, and Smolyakov and Diamond [67] discussed the appropriate form of Z for several popular scalar PDEs. Working with a particular Z in the WKE amounts to a particular (k-weighted) choice of dependent field variable in which to write the spectral balance equation. However, general statistical theory does not constrain the choice of dependent variable; moreover, for systems of coupled fields with multiple conserved quantities, it might appear that the choice of dependent variables would be overconstrained. Such paradoxes led KK to reexamine the foundations of the spectral balance equation in the presence of weak inhomogeneity. 4.4.2.1 General Remarks about Weakly Inhomogeneous Statistics

For classical physics, the general problem of weak inhomogeneity (and weak nonstationarity) was addressed in the important paper of Carnevale and Martin [8] by means of multiple-scale expansions in space and time. I will focus on the spatial problem for simplicity. Then, the general two-point . correlation function C(x, x  ) can be rewritten in terms of the sum and difference variables X = . 1 (x + x  ) and ρ = x − x  : C(x, x  ) ≡ C(ρ | X). It is assumed that C varies rapidly with ρ but slowly 2 with X.
represent the abstract operator whose x-space matrix element is C(x, x  ). In general, linear Let C 
C)(x,
operators act nonlocally on C, e.g., ( L x  ) ≡ d x L(x, x)C(x, x  ). According to CM, when this expression is expanded through first order in a weak inhomogeneity (and Fourier transformed with respect to ρ), the result is  
C
→ L k (X)Ck (X) + i L k , Ck , L

(4.25)

where the Poisson bracket for weak spatial inhomogeneity is 

 . ∂A ∂B ∂A ∂B · − · A, B = ∂ X ∂k ∂k  ∂X   ∂ ∂B ∂ ∂B = · A − · A . ∂X ∂k ∂k ∂X

(4.26a) (4.26b)

[I will object to the result (4.25) momentarily.] The form (4.26b) shows that this bracket is conservative in the sense that it is annihilated by integration over X and summation over k. (I will denote this pair of operations by an overline.) That is,   A, B = 0. (4.27) Thus, under the action of a linear operator, which I now write as L = −i, the contribution of weak inhomogeneity to the evolution of the spectral density is apparently   ∂ T C k = k , C k ,

(4.28)

. where the slow time T = 12 (t + t  ) reduces to t for equal-time statistics. I have introduced  because in the application to the CC generation problem the relevant operator stems from the nonlinear advection and is generally purely real in k space. For example, if one uses © 2006 by Taylor & Francis Group, LLC

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the generalized HME to consider DW advection by a long-wavelength CC (denoted by an underline) with velocity V E (X), the relevant operator is
= −i(1 − ∇⊥2 )−1 V E · ∇(1 − ∇⊥2 ), 

(4.29)

whose Fourier transform for constant VE is 2 −1 2 k = (1 + k⊥ ) k · V E (1 + k⊥ ) = k · VE.

(4.30)

For the interaction between DWs and CCs, Smolyakov and Diamond [67] showed that a certain quantity Z was conserved. (In Section 4.4.3, I will show that Z is a Casimir invariant in a noncanonical functional Hamiltonian description of the nonlinear physics.) However, for arbitrary choice of dependent variable, this contradicts the evolution equation (4.28) because barring that equation leads to ∂T C = 0. That is, C is apparently conserved regardless of the choice of variable. This conclusion is obviously incorrect. The resolution of this paradox was given by KK, who took issue with the result (4.25) and its
is the composition of two operators

implication (4.28). Consider the situation where L A and B:







L = A B. Symmetry (or detailed calculation) dictates that the expression for L C = A B C must contain the Poisson bracket of all of the operators taken in pairs:       

C
→ Ak Bk Ck + i Ak , Bk Ck + Ak Bk , Ck + Ak , Ck Bk . AB (4.31) Upon combining the last two terms, this can be written as    
C
→ L k Ck + i Ak , Bk Ck + L k , Ck . L

(4.32)

The underlined, nonconservative term (of first order in the weak gradient) was overlooked by CM.
is composed of noncommuting operators, C is no longer conserved. Thus, to the extent that L Some other quantity, however, may be. Suppose that the quantity Z = (Z) k C k is known to be (Z) conserved, where k is a certain  k-dependent   weight  factor. It is then easy to see that the formalism is consistent provided that (Z) A, B − L , (Z) = 0. [This must be appropriately generalized
is the composition of three or more operators. A good example is generalized HM dynamics, if L which involves the triple operator product given by Equation (4.29).] That is, it must be the case that weak inhomogeneity leads to the spectral evolution equation     −1 −1 ∂T Ck = [(Z) k , Zk = [(Z) k , (Z) (4.33) k ] k ] k Ck , where only the term explicitly related to the inhomogeneity is displayed. In the last form of Equation (4.33), (Z) k may not be passed through the bracket because it is k-dependent. The result (4.33) can be verified explicitly for specific PDEs such as the generalized HME [49]. 4.4.2.2 Modulated Reynolds Stress, Energy Principles, and Use of the Martin–Siggia–Rose Formalism

We wish to calculate
q by somehow invoking the theory of weakly inhomogeneous statistics. However, there is a technical problem:
q is nonzero even when the turbulence is entirely homogeneous. On the one hand, this is clear since, as I have stressed,
q represents only one part of the spectral balance equation. On the other hand, there certainly is a CC–DW interaction, and the intuition that a CC introduces a long-wavelength modulation seems compelling. This paradox is reconciled in a well-known way by paying close attention to the distinction between an ensemble of CCs (distributed homogeneously) and a single test CC that perturbs the background turbulence. There is a precise analogy to the test-particle techniques of classical kinetic theory. Perturbations of background states are best handled by response-function techniques that © 2006 by Taylor & Francis Group, LLC

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generalize thermal-equilibrium fluctuation–dissipation theory [55]. The most elegant formulation was given in Reference [56]; I comment on it briefly here in order to emphasize the variational foundations of
q , which lead rather directly to definitions of
q in terms of either perturbed Reynolds stress or second-order variations of a certain energy functional. Although it is unnecessary, consider statistically homogeneous turbulence for definiteness. In this case, all mean fields can be taken to vanish. A standard way of breaking the symmetry is to add an arbitrary, statistically sharp external source function 
k (t) to the right-hand side of the dynamical equation (4.18). This inevitably induces nonvanishing mean fields 
 that depend functionally on 
. Furthermore, other statistical observables, such as the two-point correlation function C and response function R, also depend functionally on 
. One can use the mean field rather than 
as the control parameter, so one can discuss such functional dependences as C[]. MSR showed that it is also useful to introduce a source  that appears in an equation adjoint to that for . Then, n-point cumulants of  generate (n + 1)-point cumulants by functional differentiation: C(n+1) (1, 2, . . . , n, n + 1) = C(n) (1, 2, . . . , n)/ (n + 1). They also emphasized that proceeding in this way merely leads to an unclosed cumulant hierarchy, the generalization of the BBGKY hierarchy of classical many-body kinetic theory. To effect closure, one must introduce response functions. For the details, see Reference [46] and references therein. The result is that the two-point response  function is


(1)  , (4.34) R(1; 1 ) =


(1 ) 
=0 . where
= (, 
)T . Furthermore, the triplet correlation that appears in the evolution equation for R can be shown by means of the functional chain rule to be equal to  R, where 

N 
(1)   (1; 1 ) = . (4.35)


(1 ) 
=0 Here, N  is the cumulant part of the nonlinear term in the original dynamical equation. This rewriting of the triplet correlation function effects the elimination of the disconnected graphs in the derivation of the famous Dyson equation [22]; the method of sources discussed here is due to Schwinger (for more details and references, see Reference [46]).
q is a Markovian version of −. Now N  is just the generalization of the (divergence of the) Reynolds stress of Navier–Stokes theory to arbitrary nonlinear dynamics. One thus understands that
q is not determined from the Reynolds stress, but rather from a perturbed stress modulated by the CCs. Failure to appreciate this point can lead to paradoxes and incorrect results. Consider, for example, the GHME problem. There, the nonlinear effect on the CCs is the advection of vorticity (not of velocity, as in the usual Navier–Stokes equation). The appropriate nonlinear effect driving the CCs is therefore (with primes denoting the DWs) VE · ∇∇ 2   = (∂x x − ∂ yy )vx v y  + ∂x y (v y 2  − vx 2 ).

(4.36)

Here, the last result, involving the physical Cartesian components of the E × B velocity, follows by integration by parts assuming homogeneous statistics. Now consider isotropic DW statistics for definiteness. In this case, one has vy 2  = vx 2  and   vx v y  = 0. That is, the Reynolds stress vanishes for homogeneous, isotropic turbulence, which is well known. However, the perturbed Reynolds stress on the DWs due to CC modulation does not vanish because the presence of a CC with wavevector q introduces an anisotropy dependent on the direction of q. Now vy 2  = vx 2  and vx vy  = 0. When those modulations are calculated (according to the algorithm to be described shortly), one is led to the proper answer (4.24).13 13 Kim and Diamond [31] suggested that one should ignore the term in v2 −v2  (on the grounds that it vanishes for isotropic y x

statistics) while retaining the term in vx v y . As discussed by Krommes [47], this argument is incorrect; if it is pursued [31], it leads to a formula for
q that is not properly invariant under rotations of q.

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The analysis can be continued to demonstrate that
q can be obtained from the second functional derivative of a certain energy rate-of-change functional. That is, one can derive a generalized Poynting theorem. For HM dynamics, that is ∂t E = E · Γ, (4.37) where E is the DW energy averaged over the long wavelengths and Γ is the flux of DW vorticity. Note the presence of the induced mean field. As in the discussion of the specific example (4.36), the right-hand side of expression (4.37) would vanish for homogeneous turbulence in the absence
or  , and one can of the symmetry-breaking 
. In its presence, E˙ is at least of second order in  show [49] that    

2 E˙ 1  . (4.38)
q = −  2 ∗ q + q

 q   q  
=0

Note that the differentiation is with respect to the induced mean field. A more definitive derivation of Equation (4.38) follows from the Hamiltonian formalism to be discussed in Section 4.4.3. What remains is to calculate E˙ from the weakly inhomogeneous WKE as discussed in Section 4.4.2. Although Z is conserved under the DW–CC interaction, E is not. Upon assuming that the DW energy is related to the spectral density according to Ek = (E) k C k and upon defining . (E) / , one may use Equation (4.33) to find k = (Z) k k       k , k Ek = k , Ek + k , ln k Ek . (4.39) ∂T Ek = −1 k The first term is in conservative form and vanishes under the barring operation, but the last term describes a mean energy transfer from the DWs to the CCs. (The sign of the transfer is not clear at this point.) The last Poisson bracket simplifies to ∇k · ∂k ln k because k depends only on k. Thus [∇k (X) · ∂k k ]Zk (X) (4.40a) ∂T E = k

=

k

(−iq · ∂k k )∗ k;q Zk;q ,

(4.40b)

q

where Parseval’s theorem was used in the last step in order to Fourier-decompose the X dependence of the CCs into the q variable. The functional derivatives required in Equation (4.38) can now be readily performed; only the (1) (1) product (1) k;q Zk;q contributes. For Zk , one may use a variant of Equation (4.33). That equation is not complete, however, since it seems to suggest that the individual spectral components ∂T Zk would never reach a steady state. Such a state is achieved only by the interaction between the CCs and the DWs, and in Markovian closure theory, the duration of that interaction is given by the triad interaction time q,k,−k . Therefore, guided by the rigorous asymptotic closure results, I heuristically generalize the first-order version of Equation (4.33) to  −1 (1)  (1) (0)  [∂T + rq,k,−k ]Zk;q = k;q , Zk . (4.41) (1) (The term {(0) k , Zk;q } vanishes at   = 0.) Now one can consider a true steady state, in which ∂T Zk = 0, and find that  (1) (0)  Zk;q = rq,k,−k (1) . (4.42) k;q , Zk

Substitution of this result into Equation (4.40b) leads to Equation (4.24). The physical effect involved is identified by noting that  (1) (0)  = ∇(1) · ∂k Z (0) . (4.43)  ,Z In the full equation for ∂T Z, the associated characteristic equation is dk/dt = −∇, © 2006 by Taylor & Francis Group, LLC

(4.44)

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which describes a refractive change in wavevector due to the weak inhomogeneity induced by the modulation. This effect (calculated in mean square) gives rise to term (d) in Equation (4.24). This calculation of
q and the result (4.24) are very closely related to Kraichnan’s classic calculation of two-dimensional eddy viscosity [40]. He used the NSE, whereas Equation (4.24) pertains to the HME. However, the plasma problem may be readily reduced to the Navier–Stokes one by removing the term describing adiabatic electron response [the 1 in the factor (1 + k 2 )]. Upon doing that, KK showed that Equation (4.24) reduces precisely to the result of Kraichnan (who considered only homogeneous, isotropic statistics). It is then instructive to compare Kraichnan’s physical explanation with the preceding discussion based on the WKE. Kraichnan considered a special case; however, it can be seen from study of Reference [40] that Kraichnan understood clearly that the underlying mechanism was wavevector refraction. The merits of the present analysis employing the WKE are that the analysis is quite general, holds for arbitrary anisotropic statistics, and has an elegant generalization to the case of multiple coupled fields, as discussed in Section 4.4.3. For isotropic statistics, the eddy viscosity calculated by Kraichnan and also by KK for the HME is negative (
q > 0). In the Appendix of Reference [49], KK commented on some literature in which a positive eddy viscosity was apparently found. The reconciliation was that the present calculations pertain to a self-consistent turbulent steady state in which the total energy of the DWs plus CCs is conserved. The predictions of positive eddy viscosity arise from initial-value calculations in which the DW energy is held fixed; KK identified the specific term in the asymptotic analysis that is omitted in the initial-value calculations. For most purposes, the self-consistent, steady-state version of the formalism is appropriate.

4.4.3 HAMILTONIAN FORMALISM The discussion to this point covers the material presented in the May, 2003, conference represented by these proceedings. Subsequently, new insights into, and generalizations of, the formalism were obtained. Although those have been detailed in other publications [48,50], it is useful for completeness to discuss them briefly here as well. An outstanding question is how one generalizes the heuristic algorithm discussed in Section 4.4.2.2 to the important physical situation of multiple coupled fields, e.g., the HW equations. Systems of PDEs usually possess multiple invariants. Which one, if any, plays the preferred role of the Z in the scalar case? The answer is not immediately clear. In order to address these questions, it is important to realize that the CC generation process is entirely nonlinear. The details of linear physics enter the expression for the CC growth rate only indirectly through the properties of the triad interaction time r . Although the linear evolution matrix may be essentially arbitrary (in particular, its eigenvalues may be complex, representing linear forcing and dissipation), the nonlinear terms are usually conservative. That is certainly true if the nonlinear behavior arises from E × B advection. A powerful way of deriving conservative nonlinear Eulerian PDEs is to use a Hamiltonian formalism. This provides new insights even for a scalar PDE such as the Hasegawa–Mima equation. Moreover, a generalization to multiple coupled fields leads to a particularly efficient representation of the
q for that case. 4.4.3.1 Hamiltonian Description of Eulerian Partial Differential Equations

A Hamiltonian description of certain Eulerian PDEs was first given by Morrison and Greene [58]. An extensive amount of subsequent additional work was reviewed by Morrison [57]. First, consider the scalar case for simplicity. Let (x, t) be the dependent variable.
 The basic idea is to exhibit a Hamiltonian functional H[] and a Poisson-bracket functional ·, · such that the conservative PDE can be represented in the form
 (4.45) ∂t  = , H . © 2006 by Taylor & Francis Group, LLC

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This generalizes to an uncountably infinite number of spatial
points x the familiar equation of finite-dimensional, noncanonical Hamiltonian dynamics z˙ i = z i , H (on which, for example, the modern gyrokinetic formalism requiresthatthe
 [18] is
based).  Consistency of the formalism

bracket be antisymmetric ( A, B = − B, A ) and obey the Jacobi identity ( A, B , C +

 

  B, C , A + C, A , B = 0), where A, B, and C are arbitrary functionals. . As an example, consider the two-dimensional NSE for vorticity  = ∇ 2 : ∂t  + V · ∇ = 0,

(4.46)

. where V = z ×∇. It is well known that V · ∇ = [, ], where . [A, B] = z · ∇A×∇B.

(4.47)

Now, consider the Hamiltonian H[] =

1 (−∇ −2 ) 2

(4.48)

(integration by parts shows that H = 12 |V|2 ) and the bracket


   A B , . A, B =   

(4.49)


 One readily verifies that , H = −[, ], so Equation (4.45) reproduces Equation (4.46). In considering the generalization of this result to multiple-field systems, Krommes and Kolesnikov [50] restricted their attention to PDEs whose nonlinearities can be represented by the square Poisson bracket (4.47). That includes the E × B nonlinearity (as in, for example, the HWEs) as well as the field-line bending terms that arise in weakly electromagnetic generalizations of electrostatic models. Thus, they considered the Hamiltonian functional H[ψ] = and the Lie–Poisson bracket




 A, B = S i j

1 i  gi j  j 2 

 A B . , i  j

(4.50)

(4.51)

Here gi j and S i j are symmetric matrices. Because g is taken to be independent of ψ, H can be considered to be a generalized kinetic energy; g plays the role of a metric tensor that can be used to raise and lower indices; for example, the covariant component of ψ is i = gi j  j . Thus, one has the compact covariant expression 1 H[ψ] = i i . (4.52) 2 S is taken to be linear in ψ, S i j [ψ] = S i j k k .

(4.53)

S is called the structure matrix and the S i j k are called the structure constants. In order that the Jacobi . identity be satisfied, the matrix T i jk m = S i j l S lk m must be fully symmetric in i, j, and k (∀m). It is readily seen that the nonlinear dynamics conserve the Hamiltonian H. Other quantities may be conserved as well. In particular, a quantity that is conserved independently of the form of H is called a Casimir invariant Z. Thus, the conservation of a Casimir depends on only the properties of the Poisson bracket; with the particular choice of bracket given earlier, Casimir conservation is encoded in the properties of the structure tensor S i j k . Consider, for example, how that bracket © 2006 by Taylor & Francis Group, LLC

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C

H&

C

H

FIGURE 4.4 Casimir conservation restricts the dynamics to lie in a subspace of the energy “surface.”

conserves H = 12 i i and the prospective Casimir Z = generalization of Equation (4.45) is

1 i i . 2

In component form, the vector

∂t i = S i j k [k ,  j ].

(4.54)

∂t H = ˙ i i = S i j k [k ,  j ]i = S i j k k [ j , i ] = 0.

(4.55)

One has The last result follows from the contraction of a symmetric form (in contravariant indices) and an antisymmetric form (in covariant indices). Similarly, ∂t Z = ˙ i i = S i j k [k ,  j ]i = S i j k [k , i ] j .

(4.56)

Now the antisymmetric form is in contravariant indices. This expression will therefore vanish if S i j k = S k j i . This symmetry holds sometimes but not always. Thus, the conservation of the particular form Z = 12 i i depends on the up–down symmetry properties of the structure tensor. That this particular form is not always conserved is understandable since i i is not a covariant expression. That is, its form will change under a linear transformation of the field variables. Under that same transformation, the symmetries of the structure tensor will also change. Thus, there may exist a bilinear form in the i ’s that is a Casimir. Generally, there are multiple Casimirs Z (n) . Of course, each such Casimir constrains the dynamics to move in a subspace of the full energy surface; see Figure 4.4. We will see that the Casimirs figure importantly in the CC generation problem. 4.4.3.2 Hamiltonian Description of
q

In order to use this formalism to calculate the generation rate of CCs, one needs to (1) obtain a generalized energy theorem; (2) represent the multivariate spectral balance equation in Hamiltonian form; and (3) use that equation to calculate the requisite functional derivatives. To derive (1), one begins by projecting the dynamical equation (4.54) into the DW subspace (denoted by primes) and the CC subspace (denoted by an underline). (An underline can be interpreted as integration over z, i.e., as extracting the k = 0 component.) For example, one has ∂t i  = S i j k ([k  ,  j ] + [k , j ]) + self-interaction terms.

(4.57)

Next, an equation for the quadratic DW and CC energies can be obtained by contracting Equation (4.57) with i and barring the resulting equation. The details are given in Reference [50]; the result is that the DW power loss is ∂t E = G i P i , (4.58) © 2006 by Taylor & Francis Group, LLC

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where G is the cumulant part of the mean nonlinearity and P is the mean induced field. This covariant result should be compared to formula (4.37). One may now prove an important theorem about the spectral dynamics. In Equation (4.57), the first term on the right-hand side describes advection of the DWs by the CCs; the second term describes the advection of the CCs by the DWs. This latter effect is small and will be neglected. Because the retained interaction involves the covariant component
j (which is H/
j ), the DWs conserve the same Casimirs as the full dynamics. This result is the generalization to multifield systems of the formulas given in Reference [67] for conserved quantities in certain scalar models. That is, in the presence of CC modulation and to lowest order in the scale-separation parameter , the DWs conserve the Casimir invariant(s). In the presence of multiple Casimir invariants, there is no invariant that is a priori preferred (i.e., is the leading candidate for an action density in a scalar WKE). In lieu of other physical information, one should treat all Casimirs on equal footing. The way to accomplish that is simply to work with ij the tensor spectral-balance equation, i.e., the equation for the DW ∂t Ck . The calculation is sketched ij H (i j) ij (i j) in Reference [50]. The final result is ( ) = H ( ), where H denotes the Hermitian part with respect to the indices i and j, and
ks d 2 S ir k [(∂ + 2i)gr s ]  ks H (ks) [S jk l (∂ + 2i)Ckls ], (4.59) qi j = − k

. . d = z · q×k, and ∂ = q · ∂k . This result should be compared to the scalar formula (4.24). The general similarities in form are obvious. The terms proportional to 2i are associated with off-diagonal correlations and vanish in the scalar case. The effects of wave-number refraction shows up here as the derivative of the metric tensor g. The scalar triad interaction time has been heuristically generalized to a fourth-rank tensor  ; further discussion is given in Section 4.4.3.3. Most strikingly, the new result involves ∂C, not the derivative of any particular Casimir. This may be troubling because it may not be apparent how the limit of a scalar system is achieved; this will also be discussed next. However, such a form should really be expected because no Casimir was given preferred treatment; conservation of all Casimirs is automatically built into the result (4.59). This is true even though one may not explicitly know the forms of the Z (n) . Several examples of physically relevant PDEs that can be treated by this Hamiltonian formalism were given in Reference [50]; they include a two-field model describing ion-temperature-gradientdriven fluctuations and a three-field model describing weakly electromagnetic corrections to the electrostatic HW system. One way of using the formula (4.59) is to identify a small parameter  and to calculate the lowest-order corrections to the zeroth-order result. For example, one could calculate q for the twofield electrostatic HW model, then recalculate it for the weakly electromagnetic three-field model (using as small parameter the normalized plasma pressure ). Typically,  corrections show up in the metric tensor g, so the procedure is in principle straightforward for fixed   and C. However, it is inconsistent to hold those quantities fixed because the backreaction of the CCs on the DWs causes -related modifications to the DW spectrum. Those cannot be calculated within the present formalism; a fully self-consistent theory is required. 4.4.3.3 The Tensor Triad Interaction Time

Left undetermined in Equation (4.59) was the form of the triad interaction tensor  . That object is already nontrivial in the scalar case due to issues relating to random Galilean invariance [49]. Its tensor generalization to multiple fields may appear daunting. For some discussions of tensor interaction times, see Reference [5]. One important cross-check is to verify that formula (4.59) reduces properly to a single-field limit. The electrostatic HW model permits such a limit, which may be obtained by letting   approach infinity. In that limit, consistent balance requires that the terms  ( − n) remain finite, i.e., that © 2006 by Taylor & Francis Group, LLC

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n →
(adiabatic electron response). Subtraction of Equation (4.17a) from Equation (4.17b) then leads to the HM equation if the dissipative terms are ignored. However, does Equation (4.59) approach Equation (4.24) in that same limit? An important feature of Equation (4.24) is that it involves ∂Zk .
is taken to be diagonal (an assumption However, formula (4.59) involves ∂C. It is easy to see that if θ that has been frequently made for simplicity [61]), the appropriate limit cannot be obtained. Instead, it can be shown that as
→∞
 approaches a singular matrix whose size is dictated by the scalar q,k,−k and whose structure guarantees the adiabatic relationship between n and
; off-diagonal components are essential. When the details are carried through, it can ultimately be shown [33] that the proper HM limit is, in fact, achieved. This is a necessary cross-check. Unfortunately, it suggests that one must be extremely careful with the use of the tensor
; cavalier assumptions here may seriously vitiate the fidelity of the overall calculation. Further work remains to be done on this topic.

4.5 DISCUSSION I have tried to provide an overview of the general state of affairs of some basic aspects of statistical plasma physics as motivated by confinement in strongly magnetized plasmas. I emphasized (1) the use of the gyrokinetic formalism; (2) the development of realizable Markovian closures and their applications to simple nonlinear plasma paradigms; and (3) the statistical description of convective cells (including zonal flows). Although the latter topic only provides part of the story of self-consistent turbulence, it is quite rich in detail and unifies various technical tools, including asymptotic expansion of Markovian statistical closure, weakly inhomogeneous spectral evolution equations, the use of field-theoretic techniques, the derivation of energy variational principles, and the Hamiltonian formulation of nonlinear PDEs. In general terms, what remains to be done is a detailed investigation of the backreaction of the CCs on the DWs. An analytical formulation of that problem is nontrivial. Energetic consistency in a steady state demands that comparably sized scales be included in the mode coupling; those are not amenable to asymptotic techniques. Drift-wave physics is intrinsically anisotropic, and any sort of analytically tractable model will significantly underrepresent the complexity of real devices. Numerical simulations will be of obvious help, and those are planned by a number of groups. The importance of the drift-wave problem, not only for the understanding of confinement in fusion plasmas but also for its relevance to geophysics, means that the subject will remain of considerable interest for some time.

ACKNOWLEDGMENTS I am grateful for the hospitality of B. Shivamoggi and the University of Central Florida during the period of the conference represented by these proceedings. This work was supported by U.S. Dept. of Energy Contract No. DE-AC02-76-CHO-3073.

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Mathematical and Physical Theory of Turbulence 64. M. N. Rosenbluth. Microinstabilities. In Plasma Physics, pages 485–513. International Atomic Energy Agency, 1965. 65. M. N. Rosenbluth and F. L. Hinton. Poloidal flow driven by ion-temperature-gradient turbulence in tokamaks. Phys. Rev. Lett., 80:724–727, 1998. 66. R. Z. Sagdeev and A. A. Galeev. Nonlinear Plasma Theory. W. A. Benjamin, New York, 1969. 67. A. I. Smolyakov and P. H. Diamond. Generalized action invariants for drift waves-zonal flow systems. Phys. Plasmas, 6:4410–4413, 1999. 68. M. Takaoka. Application of mapping closure to non-Gaussian velocity fields. Phys. Fluids, 11: 2205–2214, 1999. 69. A. Vedenov, E. Velikhov, and R. Sagdeev. The quasi-linear theory of plasma oscillations. In Proceedings of the Conference on Plasma Physics and Controlled Nuclear Fusion Research (Salzburg, 1961) [Nucl. Fusion Suppl. Pt. 2], pages 465–475. International Atomic Energy Agency, Vienna, 1962. Translated in U.S.A.E.C. Division of Technical Information document AEC–tr–5589 (1963), pp. 204–237.

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Remarks on Decaying 5 Some Two-Dimensional Turbulence David C. Montgomery CONTENTS 5.1 Introduction ............................................................................................................................ 91 5.2 The Statistical Mechanics of Vorticity.................................................................................... 92 5.3 Numerical Results: Rectangular Periodic Boundaries ........................................................... 95 5.4 Numerical Results: Material Boundaries................................................................................ 96 5.5 Pressure Determinations and Their Ambiguities.................................................................... 97 5.6 Summary................................................................................................................................. 98 Acknowledgment ............................................................................................................................. 98 References........................................................................................................................................ 99

ABSTRACT Several recent results in decaying two-dimensional turbulence are commented upon. The partial success of two different maximum entropy predictions for the results of computed long-time decays is summarized. The differences between the cases of rectangular periodic boundary conditions and walls with no-slip or stress-free boundaries are emphasized.

5.1 INTRODUCTION The intention of this chapter is to summarize and comment upon several numerical computations of decaying two-dimensional (2D) Navier–Stokes (NS) turbulence that have been reported in the last several years. The work to be described has already been published extensively. Reference will be made to the original articles for the relevant figures, diagrams, tables, and graphs, and they will not be reproduced here. This chapter is intended as a commentary on the original articles and is not written to be accessible in a self-contained way, independently of those articles or of prior knowledge of the subject. The intention is in the nature of providing some elucidation and commentaries that, because of space limitations, were not originally included or have come to seem more interesting since first publication. The computations are of two basic types and are distinguished by the spatial boundary conditions invoked. In the first class, rectangular doubly periodic boundary conditions are imposed. This has been by far the most common framework in which turbulence computations are pursued, for two reasons: (1) the Fourier transformation provides a natural framework into which questions of resolution, partition of excitations among length scales, and the exhibition of spectral properties of the turbulence naturally fit; and (2) the speed and elegance of the “fast Fourier transform” (FFT) has made spectral and pseudospectral methods of computation, developed by Orszag, Patterson, and others the most accurate and efficient method to solve the Navier–Stokes equation with high resolution in two dimensions or three dimensions, in a natural way wherein spatially periodic boundary conditions are automatically satisfied. In the other class of computations to be commented upon, material boundaries, either circular or rectangular, have been presented. The boundary conditions imposed have been “no slip” (zero © 2006 by Taylor & Francis Group, LLC

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fluid velocity at the walls) or “stress free” (vanishing normal velocity at the walls plus vanishing tangential viscous stress there). Two-dimensional turbulence is not easy to approximate in laboratory experiments, and certainly doubly periodic rectangular boundaries cannot ever be achieved in practice. Thus, any hope of close numerical comparisons with laboratory experiments depends upon the ability to enforce boundary conditions that are not spatially periodic ones. Few laboratory two-dimensions turbulent phenomena have exhibited any independence of the size and shape of the container. A computational price is paid for insistence on realistic boundary conditions, in that the computations that can be and have been done so far tend to be for lower Reynolds numbers than in the doubly periodic case and therefore do not involve such a range of spatial scales. This comes about in part because boundaries introduce boundary layers that need to be resolved, and these boundary layers get very thin with increasing Reynolds number. Nevertheless, a nontrivial degree of agreement between the computations and some decaying-turbulence experiments has, perhaps surprisingly, been attained. In both cases, such theory as exists must be statistical–mechanical in nature, but of a rather unorthodox kind. It owes its perspective to that of the early days of the statistical mechanics of classical point particles, in that it starts with an attempt made to define a probability for a given state of the system. One asks whether or not the dynamical evolution is toward states of higher and higher probability, compatible with a few constraints provided by (exact or approximate) conservation laws, and if so, what the character of those more probable states might be. Some convincing evidence has accumulated from numerical solutions that, indeed, there is a strong tendency for the fluid to evolve toward states that increase this postulated probability (or its logarithm, an “entropy”). For this reason, the emphasis in these remarks will be on the degree to which a statistical mechanical perspective has been useful in interpreting the computations. In Section 5.2, we begin by reviewing the 30-year development of the statistical mechanical perspectives that are available for comparison with two-dimensional NS turbulent decays. A mystery exists in the correspondences between these and the computations that to a considerable extent, agree with them, in that the computations are all inherently dissipative, while the statistical–mechanical frameworks in which the analytical predictions are made take no proper account of viscosity. They are for the ideal, Euler dynamics of a perfect fluid. A great deal of obfuscation has gone on in an effort to minimize this discrepancy, and “coarse graining” has been invoked in a variety of ways, usually with some element of mystery, as a satisfactory mimic of viscous decay of vorticity. These efforts are regarded here as misguided and unconvincing; the paradox is seen as worth resolving, but is left ultimately unresolved at present. Having reviewed in Section 5.2 the theoretical framework in which the observed dynamics can be interpreted, we proceed in Section 5.3 to describe the results of some periodic decay computations recently published, due to Yin et al. (2003). This is so far the most extensive set of two-dimensional NS decay calculations known to the writer in which no “hyperviscosity,” or small-scale smoothing, is used. In Section 5.4, the discussion will be on the similarities and differences of turbulent decays in two dimensions, as computed in the presence of material boundaries, both no slip and stress free. Some laboratory experimental comparisons are remarked upon. Section 5.5 is an interlude that backs up and remarks on some as yet unresolved problems with pressure determinations in the presence of material walls; these might be expected to have implications for present and future wall-bounded turbulence computations. Section 5.6 is a brief summary of the results reported and some suggestions for future directions in which the subject might be explored.

5.2 THE STATISTICAL MECHANICS OF VORTICITY In a two-dimensional, incompressible, uniform-density flow, with velocity fields that have only x and y components, say, and are independent of the third coordinate z, the state of the fluid is completely determined by giving the vorticity (curl of the velocity field), a single pseudoscalar quantity, as a

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function of x and y at some time. Its subsequent time evolution occurs according to the Navier–Stokes (NS) equation, and in the absence of external forcing of the system, the velocity and the vorticity will eventually disappear. If the viscosity is set equal to zero, then the system changes fundamentally, becoming conservative and obeying the Euler equations, and there is a sense in which a solution presumably rigorously exists for all time (e.g., Wolibner 1933). This is a rather abstract statement, however, and of limited utility for computations. Because the Euler equation is highly nonlinear and contains no minimum length scale with the viscosity gone, it is in its nature to develop spatial fine structure, locally very large spatial gradients, in very short times. Any Euler code with smooth and continuous initial data quickly overruns its own resolution. In terms of the Fourier spectrum, most computations have shown that, starting from initial conditions that could in any sense be called “turbulent,” there is an order of magnitude or more of increase in the wave numbers present in the spectrum per one eddy-turnover time. Since hundreds of eddy turnover times are required to follow a thorough evolution of a two-dimensional NS turbulent field to any kind of final “asymptotic” state, it is difficult to imagine that numerical continuum Euler codes for continua will, in the foreseeable future, be a useful research tool for settling questions of turbulent evolution. The foregoing statements are a consequence of the vast differences between two dimensions and three dimensions. In the latter, a typical eddy lives about one eddy turnover time. In two-dimensions the energy decay rate is bounded by a product of the viscosity and the initial enstrophy, which can only decay. This means that eddies can live a very long time, and the system has time to visit most of the relevant parts of its phase space. This can happen even in the face of a large fractional decay of enstrophy, which in all computations reported so far is not in even an approximate sense a conserved quantity. Nevertheless, the vorticity distribution seems the natural quantity to which any statistical– mechanical considerations need to be applied, since it determines all other physical quantities and can be regarded as the localized source of the velocity field (through the Biot–Savart law) and the stream function (through Poisson’s equation). One model of the vorticity, due to Helmholtz and developed by Lin (1943) and Onsager (1949), discretizes the vorticity of Euler fluid in terms of conserved, delta-function concentrations of vorticity, which are convected with the local fluid velocity and can be visualized as infinitely long, uniform, parallel “rods” or as “points” in the x y plane, depending upon the taste of the investigator. The dynamics of these discrete “particles” are now Hamiltonian, with the conjugate Hamiltonian coordinates and momenta proportional to the x and y positions of each vortex. After this discretization, there is now only a large but fixed and finite number of particle coordinates to solve for, and the reservations associated with the impossibility of accurately computing the continuum Euler evolution fade away in favor of the more manageable difficulties of following the trajectories of a finite number of particles. There is another advantage to the discrete-vortex model in that the most elementary statistical mechanical considerations can now be brought to bear, following Boltzmann. (The statistical mechanics of continua is a much less well-developed subject, and in practice always seems to require some kind of a discretization.) Namely, a probability of any given vortex configuration can now be assigned by dividing the plane into small cells in the ith one of which an integer “occupation number,” Ni, of vortices may be identified. (If there is more than one strength of vortex present, another index is needed to identify that, but this is no real complication.) Giving a full set of the Ni for all the cells is a complete specification of the state of the system and becomes more and more accurate the more finely the vorticity is divided and the smaller the cell size. Regarding the vortices within any cell as indistinguishable, the Boltzmann combinatorial expression for the probability W of any configuration of the Ni can be adopted: it is simply proportional to N !, where N is the total number of vortices, divided by the product of all the Ni!. The logarithm of W is defined as the entropy S, S = k ln W , and the numerical value of S is taken as the basis of comparison of the relative probability of any two states. The set of Ni that maximizes S gives the “most probable” configuration of the system. The allowed values of the Ni are, however, constrained by a few simple conservation laws (energy and

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vortex number seem to be the only ones for rectangular symmetry), and the variational problem of choosing the most probable set of Ni is handled by including those constraints via the method of undetermined Lagrange multipliers. The Ni are assumed large enough that Stirling’s approximation can be used to eliminate the factorials in favor of logarithms. The result (worked out in detail in Joyce and Montgomery 1973; Montgomery and Joyce 1974; and Pointin and Lundgren 1976; see also particularly Ting et al. 1987) is a set of most probable Ni, which is an exponential of one Lagrange multiplier associated with the conservation of vortex number, plus another times the discretized version of the stream function evaluated at the cell i. Once the combinatorics are accomplished, a mean-field limit can be taken in which the vorticity is infinitely subdivided into infinitesimally weak vortices, of which there are an infinite number. The cell size is also allowed to shrink to zero, with the vortices still identified as points so that an arbitrarily large number of them can be accommodated inside any cell. The limit is then a continuum vorticity that has been selected as “most probable” for given fixed energy and fixed vorticity fluxes for whatever types of point vortices were assumed. It now has the familiar exponential form of an isothermal number density distribution in a particle atmosphere stratified by a gravitational potential. So far, there has been nothing unfamiliar in the development. The novelty comes in when it is required that the (now continuous) vorticity and stream function be self-consistent: that is, the most probable vorticity, which depends exponentially on the stream function, is stuffed back into the Poisson equation, which now necessarily relates the Laplacian of the stream function to one or more exponentials of itself. If there are only two kinds of vorticity, equal and opposite in fluxes, as must be the case for periodic boundary conditions, the two exponentials symmetrically collapse into a hyperbolic sine, and the by now familiar “sinh-Poisson equation” is the result, expressing the most probable vorticity as a hyperbolic sine. The terms contain a constant times an exponential of another constant multiplying the stream function (the constants come from the Lagrange multipliers, in principle to be determined from the given energy and vorticity fluxes). The solutions are numerous. Analytical solutions for free-slip or periodic rectangular boundary conditions can be found, for example, in Ting et al. (1987) and in Kuvshinov and Schep (2000). This is one candidate for predicting what state an ideal Euler fluid might evolve into, in this limit, and we will say more about the extensive comparisons that have been made with weakly viscous (non-Euler) numerical solutions in the following section. More recently, an alternative formulation in which the discretization was in terms of finite-area, mutually exclusive “patches” of vorticity, instead of delta-function points, has been given by Robert and Sommeria (1991, 1992) and by Miller et al. (1992). Again, an entropy for particles is defined and maximized, but the particles are not zero-area “points” and cannot occupy the same areas of the x,y plane. The combinatorics are similar, but the result is that in the denominators of the most probable vorticities, underneath the exponentials are Fermi-Dirac-like expressions that introduce additional complications and nonlinearities, over and above those in the sinh-Poisson equation. (The statistics involved are properly called “Lynden–Bell statistics,” as contrasted with Boltzmann statistics, since they first appeared in an astrophysical paper by Lynden-Bell (1967) dealing with the one-dimensional self-gravitating Vlasov equation where the mass distribution function in the single-particle phase space was given a similar mutually exclusive “patch” representation.) This system, too, is a representation of ideal Euler dynamics and does not involve any viscous dissipation of energy or of the other ideal invariants (moments of the vorticity, such as enstrophy). It has one other troublesome feature, in that there is no physical basis for fixing the “patch” size, no determination of the ratio of the dimension of a vortex to the dimension of the cell; said another way, there is no physical basis for saying what fraction of the space is to be occupied by nonzero vorticity patches in the mean-field limit. “Levels” of vorticity are assigned to the patches, and though the patches may be distorted, neither their areas nor their associated levels are regarded as changing with time. There is a set of fixed measure, or area, associated with each assigned level. There is apparently no unique prescription for choosing levels to approximate a continuous initial vorticity distribution.

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As in the case of the “point” discretization, there is no compulsion to represent the discretization before passage to the mean-field limit. Following the advice of Jaynesian statistical mechanics, the simplest choice, in the absence of reasons to the contrary, has been thought the best. We will discuss the numerical comparisons of these two predictions with the results of Yin et al. (2003) in the following section.

5.3 NUMERICAL RESULTS: RECTANGULAR PERIODIC BOUNDARIES Many groups have found it desirable to employ Orszag–Patterson spectral-method codes (Orszag 1971; Patterson and Orszag 1971) to solve the two-dimensional NS equation in doubly periodic boundary conditions, since these are by far the easiest to implement. Our code is a parallelized MPI Fortran 90 code descended from a Fortran 77 code provided by W.H. Matthaeus. It has been written by Zhaohua Yin. It uses the shifted-grid method of dealiasing and is described in detail elsewhere (Yin and Montgomery 2002; Yin et al. 2004). Many runs were done, at 512 × 512 spatial resolution, starting from a variety of initial conditions (Yin et al. 2003). Two initializations were explored in the most detail: turbulent initial conditions, which were broadband in Fourier space with random phases, of the type employed by McWilliams (1984) and Matthaeus et al. (1991a,b), and vorticity distributions that had large areas of approximately uniform vorticity separated by transition regions as thin as the code’s resolution would allow. The latter set were employed to get intuitively as close as possible to the kinds of vorticity fields that would resemble prominent “levels” in the sense described earlier, and might be expected partially to preserve them. The present brief summary cannot do justice to the variety and complexity of the behavior observed by Yin, but roughly speaking, what happened with the first class of initial conditions consisted of confirmation of previously observed (Montgomery et al. 1992) relaxation to “dipolar” states that were well approximated by a hyperbolic-sinusoidal dependence of vorticity upon stream function — i.e., to solutions of the sinh-Poisson equation. This occurred during a sequence of mergers of like-signed vortices, which continued until all possible like-sign mergers had occurred. In the case of the flat-patch initial conditions, this behavior was sometimes exhibited, but in other cases another previously unobserved behavior occurred: the relaxation to one-dimensional “bar” states, with alternating stripes of positive and negative vorticity. This is an intriguing but ill-understood behavior, one that had not previously emerged in a computation. The differential equations that are supposed to yield most probable stream functions, both in the “point” and “patch” descriptions, have one-dimensional (1D) solutions. Numerical comparisons of their entropies, at fixed vorticity flux and energy, have somewhat different results. For the “point,” or sinh-Poisson, solutions, the dipolar state has higher entropy than the one-dimensional “bar” state. For the “patch” formulation, which topology has the higher entropy depends upon the patch size. For large enough patches (i.e., high enough fraction of the space occupied by vorticity of some sign, before the mean-field limit is taken), the bar is more probable than the dipole. As the patch size decreases, the dipole becomes gradually more probable than the bar. The two plots of entropy vs. energy for fixed fluxes of positive and negative vorticity remain close. As previously remarked, there appears to be no unambiguous basis for the choice of patch size in the theory. It is not known what the essential features of the patch initial conditions are that lead to the bar evolution, but for unknown reasons, those that descend from quadrupolar initial conditions plus noise seem to be favored for evolution toward the bar. Broadband initial conditions that can be unambiguously called “turbulent” have always evolved into the dipolar state. The flat-patch initial conditions, while unstable, never generated broad wavenumber distributions, but rather tended to evolve from one relatively small set of Fourier modes into another, energetically speaking. Whether the evolution to the bar state should properly be called “turbulent” is perhaps debatable. The details may be found in Yin et al. (2003), along with an intriguing variety of related behavior.

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5.4 NUMERICAL RESULTS: MATERIAL BOUNDARIES Two shapes of two-dimensional boundaries have been explored — circular and rectangular — and the results have been different in the two cases. For the circular boundaries, which we describe first, no-slip and stress-free boundary conditions have been imposed, and the results for the two different sets of boundary conditions are different. Li et al. (1996a,b; 1997) used a wholly spectral method to solve two-dimensional NS dynamics in circularly symmetric boundaries. The method was to expand in an orthonormal set of twodimensional vector velocity expansion functions that have both components of velocity zero at the walls. Related to Chandrasekhar–Reid functions (Chandrasekhar 1961), these are obtained by adding to a streamfunction that is an eigenfunction of the Laplace operator, a potential-flow part with the same angular dependence, thereby allowing the freedom to satisfy two boundary conditions. The two-dimensional NS equation is then projected, by taking inner products, onto the various directions in function space that the orthonormal set of functions spans. The orthonormal set appears to be complete for solenoidal velocity fields that vanish at the boundary, but a proof is lacking. The dynamics thereby reduce to a set of coupled nonlinear ordinary differential equations for the expansion coefficients, typically no more than a few thousand, in practice. The computation takes place entirely in the transform space, for there is no equivalent of the FFT for Bessel functions that makes it worthwhile to go back to configuration space for any purpose other than graphing the contours of streamfunction (streamlines) and vorticity contours at subsequent times. It is perhaps worth remarking that there is no troublesome singular behavior associated with the origin r = 0 as there is for finite differences or several other computational methods. The boundary conditions are automatically satisfied for the no-slip and stress-free cases by the proper choice of constants in the expansion functions. Thus, no additional effort is involved in satisfying velocity-field boundary conditions since they are automatically satisfied, term by term, for every term in the expansion. An initially puzzling behavior was displayed in the no-slip decay runs. Starting from a set of randomly chosen expansion coefficients that guaranteed a disordered evolution, the first thing that happened was the appearance, after one eddy turnover time or less, of localized boundary layers at the walls, which pushed the limits of the resolution of the code. Thereafter, one of two scenarios was observed: (1) for some initial conditions, a nearly rotationally symmetric flow pattern emerged after a few tens of eddy turnover times, with a vortex core of one sign surrounded by a ring of opposite vorticity (the total integrated vorticity must be zero) and the pattern thereafter exhibited a slow decay; (2) for other initial conditions, this laminar nearly axisymmetric flow never developed, and the late-time state continued to evolve through dipolar, quadrupolar, etc. patterns irregularly, until by late times the decay had exhausted the nonlinearity. It was eventually determined that what characterized the flows of class (1) was the presence of a finite angular momentum in the initial conditions; in class (2), the initial angular momentum was zero. It was determined that, of the various global ideal invariants, the viscous decay of angular momentum, when nonzero, was the slowest — slower even than the energy. Reasonably good fits of a “most probable” vorticity dependence upon stream function were shown to be possible if the theory was modified to include an angular momentum constraint as well as constraints upon vorticity flux and energy in the statistical–mechanical development. The Reynolds numbers were too low and the runs too few to see what might be called definitively accurate confirmation of these statements, and we must await a more numerically ambitious treatment of the problem with higher resolution. The stress-free case with circular boundaries exhibited a totally different behavior. For the stressfree case, angular momentum is a rigorous constant of the motion, and it is always possible to transform to a rotating set of coordinates in which the angular momentum is zero. In this case, the early stages of decay runs looked similar to those for no-slip boundaries, with some locally intense but transient boundary layers developing at the wall. However then a relaxation always occurred to a dipolar state that persisted and resembled in many ways the periodic boundary condition late-time

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states described previously. No convincing stream-function vs. vorticity scatter plots could be found which correlated well with any most probable state theory, however. The boundary layers, where the viscous terms cease to be negligible, seem often to have no one-to-one correspondence between vorticity and stream function (Jones and Montgomery 1994). These results were confirmed in a set of experiments by Maassen (Maassen et al. 1999), which were performed on a stratified cylindrical tank in which the initial vorticity distribution was excited by dragging a rake horizontally through the fluid, then photographing with a CCD camera the trajectories of many small spheres whose density permitted them to float in a region of approximately uniform density gradient. Extensive computations were done in the presence of rectangular no-slip walls by Clercx et al. (1998; Maassen et al. 2002), again with extraordinarily different results. Here, angular momentum is not an ideal invariant because of the lack of rotational symmetry, and there is no reason to expect it to be even approximately conserved. What was unexpected that characterized many of the runs was the phenomenon of “spontaneous spin-up,” involving the appearance of finite and persistent angular momentum in a significant fraction of the trials. There was no prediction of the sign of the angular momentum; when it developed, it appeared to develop from low-level noise in the initial conditions. It must stand as one of the more spectacular phenomena to have appeared in the subject. The question of how an initially vortex-loaded two-dimensional fluid will behave as a function of the shape of its no-slip walls is, at the time of this writing, still an open one, both numerically and experimentally. What pattern, for instance, would be expected to characterize the late-time decay of turbulence inside elliptical walls?

5.5 PRESSURE DETERMINATIONS AND THEIR AMBIGUITIES The role of the pressure in incompressible flow is apparently straightforward in periodic boundary conditions. One takes the divergence of the equation of motion, invoking the solenoidal nature of the velocity field, and the time derivative term and the viscous term drop out. What remains is a Poisson equation to be solved for the pressure, with a source term proportional to the divergence of the inertial term. Since both are spatially periodic, the solution is only a matter of matching the two Fourier transforms. The resulting pressure is then expressible as a functional of the velocity field and may be formally eliminated from the dynamics. No objections to computing the pressure in this way have ever been raised, to the writer’s knowledge. The presence of material walls with no-slip boundaries makes the problem far less straightforward. Once again, taking the divergence of the Navier–Stokes equation leaves a Poisson equation to be solved for the pressure. However, to solve Poisson’s equation in the presence of boundaries requires boundary conditions, this time on the pressure. How can one infer these? The straightforward way would appear to be to consider the Navier–Stokes equation itself, arbitrarily near the no-slip boundaries. There, the time derivative term and inertial term must go to zero if the velocity does, and this leaves simply that the pressure gradient must approach the viscous term. However, the statement applies to all components of the pressure gradient: the tangential as well as the normal. Specifying the normal component is equivalent to giving Neumann boundary conditions on the pressure, while specifying the tangential components amounts to giving Dirichlet boundary conditions. Thus, the problem is mathematically overdetermined according to the standard theorems of potential theory. This problem was illustrated in detail by Kress and Montgomery (2000) by considering a two-dimensional wall-bounded flow given at the initial instant by a Chandrasekhar–Reid function (Chandrasekhar 1961). This stream function that gives a periodic behavior in the x-direction, and because of an additive potential-flow part of it, it is able to satisfy two boundary conditions at symmetrically placed walls in the y-direction. Thus, both components of velocity may be forced to vanish there, by a proper choice of the constants in the function.

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Moreover, since the only dependences of the velocity on x and y can be expressed as exponentials, the source term for the pressure Poisson equation is sufficiently simple that a particular solution for the Poisson equation can be explicitly found. To this can be added a solution of Laplace’s equation, thereby satisfying the Neumann or Dirichlet boundary condition on the pressure, but not both. The difference in the two pressures so determined is of the order of a few percent at Reynolds numbers in the vicinity of 2000 and is largest near the boundary layers that appear close to the walls. As long as the pressure is considered a function without singularities and no singularity is assumed in the Navier–Stokes equation, this is a puzzle without an answer, to date. The puzzle is not new and has generated a rather large literature that more or less originates in the papers of Heywood (1980). The literature is highly mathematical and has diverged from its physical origins to the point where it is largely inaccessible to physicists. Textbook solutions of the Navier–Stokes equation that are exact and involve no-slip boundaries are mostly one-dimensional ones that make some use of Bernoulli’s theorem to obtain the pressure; explicit and exact two-dimensional solutions with no-slip boundaries are conspicuously absent from the literature. There are none that we know of that are not similarity-variable solutions that reduce to ordinary differential equations. Whether there is an acceptable class of initial solenoidal velocity fields obeying no-slip boundary conditions acceptable for the incompressible Navier–Stokes equation and that will produce nonsingular velocities is simply not known; no such functions have been exhibited to date. The problem has, of course, been confronted by anyone who has ever tried to write a wallbounded, incompressible, shear-flow code. Something must be done about the pressure. A variety of individual, numerical fix-ups has been catalogued (Canuto et al. 1988; Gresho 1991) and seems to give results that do not vary significantly from experimental observations. Kinetic theorists are not unhappy with a wall “slip velocity,” and various numerical experiments with molecular dynamics have been done to shed light on how it might relate to wall stress (e.g., Thompson and Robbins 1990; Thompson and Troian 1997). Unfortunately, the relevant boundary layers tend to be measured in mean-free paths, which are zero in the Navier–Stokes limit. The difficulty is formidable, awkward, and embarrassing.

5.6 SUMMARY Perhaps the biggest gap in classical physics is a satisfactory understanding of the statistical mechanics of dissipative systems, with turbulence probably the number one example. It is respectable to believe that there is none. However, regularities such as the Kolmogorov–Obukhov cascade theory in three dimensions and the relaxation to “maximum entropy” states just noted in several of the previous twodimensional examples are tantalizing suggestions that such an understanding is there, waiting to be achieved. The information we have stems from (1) careful laboratory experiments and/or geophysical measurements, and (2) accurate solution of the dynamical equations by numerical means. What seems likely is that no “extensive” thermodynamic-type theory can be developed that is independent of the boundary conditions. The kinds of statements we can make may well depend on how much more of the data we can acquire that makes explicit how the turbulent evolution relates to what is going on at the boundary of the fluid. A “universal” theory that does not contain boundary conditions as an ingredient may be unachievable.

ACKNOWLEDGMENT The writer’s understanding of decaying two-dimensional turbulence has greatly benefited from conversations and interactions with the following individuals: Wesley B. Jones, Shuojun Li, Herman Clercx, Saskia Maassen, GertJan van Heijst, William Matthaeus, and, especially in recent times, Zhaohua Yin. It has been a pleasure to have worked in the group at the Eindhoven University of Technology in the Netherlands, where many of these results emerged. © 2006 by Taylor & Francis Group, LLC

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REFERENCES 1. C. Canuto, M.Y. Hussaini, A. Quarteroni, and T.A. Zang, 1987, Spectral Methods in Fluid Dynamics, Springer, New York. 2. S. Chandrasekhar, 1961, Hydrodynamic and Hydromagnetic Stability, Dover, New York. 3. H.J.H. Clercx, S.R. Maassen, and G.J.F. van Heijst, 1998, Spontaneous spin-up during the decay of 2D turbulence in a square container with rigid boundaries, Phys. Rev. Lett. 80, 5129–5132. 4. P.M. Gresho, 1991, Incompressible fluid dynamics: some fundamental formulation issues, Annu. Rev. Fluid Mech. 23, 413–453. 5. J.G. Heywood, 1980, The Navier–Stokes equations: on the existence, regularity, and decay of solutions, Indiana Univ. Math. J. 29, 639–681. 6. W.B. Jones and D. Montgomery, 1994, Finite amplitude steady states of high Reynolds number 2-D channel flow, Physica D 73, 227–243. 7. G.R. Joyce and D. Montgomery, 1973, Negative temperature states for a two-dimenisonal guiding-center plasma, J. Plasma Phys. 10, 107–121. 8. B.T. Kress and D.C. Montgomery, 2000, Pressure determinations for incompressible fluids and magnetofluids, J. Plasma Phys. 64, 371–377. 9. B.N. Kuvshinov and T.J. Schep, 2000, Double-periodic arrays of vortices, Phys. Fluids 12, 3282. 10. S. Li and D. Montgomery, 1996a,b, Decaying two-dimensional turbulence with rigid walls, Phys. Lett. A 218, 281–291 and 222, 461 (erratum). 11. S. Li, D. Montgomery, and W.B. Jones, 1997, Two dimensional turbulence with rigid circular walls, Theor. Comp. Fluid Dyn. 9, 167–181. 12. C.C. Lin, 1943, On the Motion of Vortices in Two Dimensions, Univ. Toronto Press, Toronto. 13. D. Lynden–Bell, 1967, Statistical mechanics of violent relaxation in stellar systems, Mon. Not. R. Astron. Soc. 136, 101–121. 14. S.R. Maassen, H.J.H. Clercx, and G.J.F. v. Heijst, 1999, Decaying quasi-2D turbulence in a stratified fluid with circular boundaries, Europhys. Lett. 46, 339–345. 15. W.H. Matthaeus, W.T. Stribling, D. Martinez, S. Oughton, and D. Montgomery, 1991a, Selective decay and coherent vortices in two-dimensional incompressible turbulence, Phys. Rev. Lett. 66, 2731–2734. 16. W.H. Matthaeus, W.T. Stribling, D. Martinez, S. Oughton, and D. Montgomery, 1991b, Decaying two-dimensional turbulence at very long times, Physica D 51, 531–538. 17. J.C. McWilliams, 1984, The emergence of isolated coherent vortices in turbulent flow, J. Fluid Mech. 146, 21–43. 18. J. Miller, P.C. Weichman, and M.C. Cross, 1992, Statistical mechanics, Euler’s equation, and Jupiter’s red spot, Phys. Rev. A 45, 2328–2359. 19. D. Montgomery and G.R. Joyce, 1974, Statistical mechanics of negative temperature states, Phys. Fluids 17, 1139–1145. 20. D. Montgomery, W.H. Matthaeus, W.T. Stribling, D. Martinez, and S. Oughton, 1992, Relaxation in two dimensions and the “sinh-Poisson” equation, Phys. Fluids A 4, 3–6. 21. L. Onsager, 1949, Statistical hydrodynamics, Nuovo Cimento Suppl. 6, 279–287. 22. S.A. Orszag, 1971, Accurate solution of the Orr–Sommerfeld stability equation, J. Fluid Mech. 50, 689–703. 23. G.S. Patterson and S.A. Orszag, 1971 Spectral calculations of isotropic turbulence: efficient removal of aliasing interations, Phys. Fluids 14, 2538–2541. 24. Y.B. Pointin and T.S. Lundgren, 1976, Statistical mechanics of two-dimenisonal vortices in a bounded container, Phys. Fluids 19, 1459–1470. 25. R. Robert and J. Sommeria, 1991, Statistical equilibrium states for two-dimensional flow, J. Fluid Mech. 229, 291–310. 26. R. Robert and J. Sommeria, 1992, Relaxation towards a statistical equilibrium in two-dimensional perfect fluid dynamics, Phys. Rev. Lett. 69, 2776–2780. 27. P.A. Thompson and M.O. Robbins, 1990, Shear flow near solids: epitaxial order and flow boundary conditions, Phys. Rev. A 41, 6830–6837. 28. P.A. Thompson and S.M. Troian, 1997, A general boundary condition for liquid flow at solid surfaces, Nature 389, 360–362. 29. A.C. Ting, H.H. Chen, and Y.C. Lee, 1987, Exact solutions of nonlinear boundary value problem: the vortices of the two-dimensional sinh-Poisson equation, Physica D 26, 37–66. © 2006 by Taylor & Francis Group, LLC

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30. S.R. Maassen, H.J.H. Clercx, and G.J.F. van Heijst, 2002, Self-organization of two-dimensional turbulence in stratified fluids in square and circular containers, Phys. Fluids 14, 2150–2169. 31. W. Wolibner, 1933, Un theorem sur l’existence du mouvement plan d’un fluide parfait, homogene, incompressible, pendent un temps, infinitent long, Math. Z. 37, 698–726. 32. Z. Yin and D.C. Montgomery, 2002, Advances in turbulence IX, ed. by I.P. Castro, P.E. Hancock, and T.G. Thomas, CIMNE, Barcelona, Spain; pp. 669–674. 33. Z. Yin, D.C. Montgomery, and H.J.H. Clercx, 2003, Alternative statistical-mechanical descriptions of decaying two-dimensional turbulence in terms of ”patches” and ”points,” Phys. Fluids 15, 1937–1953. 34. Z. Yin, H.J.H. Clercx, and D. Montgomery, 2004, An easily implemented task-based parallel scheme for the Fourier pseudospectral solver applied to Navier–Stokes turbulence, Computers Fluids 33, 509–520.

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and Dynamical 6 Statistical Questions in Stratified Turbulence J.R. Herring, Y. Kimura, R. James, J. Clyne, and P.A. Davidson CONTENTS 6.1 Isotropic Turbulence and Resolution Issues at Large Scales................................................ 101 6.2 Stably Stratified Turbulence ................................................................................................. 104 6.3 Concluding Comments ......................................................................................................... 111 References...................................................................................................................................... 113

ABSTRACT We examine homogeneous turbulence under stably stratified and neutral conditions, including decaying and randomly forced cases. Our tools include direct numerical simulations (DNSs) and elements of statistical theory. Our DNS at 5123 permit large scales to develop from the dynamics at smaller, energy-containing scales. The preceding resolution permits a Taylor microscale R
∼ 150. The size distribution of such large scales is closely related to conservation principles, such as angular momentum, energy, and scalar variance; we relate these principles to our DNS results. Stratified turbulence decays more slowly than isotropic turbulence with the same initial conditions. We offer a simple explanation in terms of the diminution of energy transfer to small scales resulting from phase mixing of gravity waves. Enstrophy structures in stratified flows (scattered pancakes) are distinctly different from those found from isotropic turbulence (vortex tubes). For the forced case, we examine the modification of the inertial range induced by strong stratification (k −5/3 →∼ k −2 ). We note that the development of the vertically sheared horizontal flow (VSHF) mode of Smith and Waleffe (2002) is closely associated with strong gravity waves at large scales.

6.1 ISOTROPIC TURBULENCE AND RESOLUTION ISSUES AT LARGE SCALES Direct numerical simulations (DNSs) are an important tool in understanding the dynamics of turbulent flows. Often, in an effort to reach as high a Reynolds number as possible, the large-scale dynamics is given short shrift. Thus, the results are box-limited. This defect is easily noticed in the shape of E(k) at small k so that the energy spectra of such simulations, E(k), are at maximum very close to the box size (i.e., k = 1). One of our goals here is to avoid such limits and to display the dynamics that develop via Navier–Stokes in the homogeneous context. Our simulations are of modest resolutions (5123 ), as is the Reynolds number (R
∼ 250). The equations of motion to be investigated are: (∂t − ∇ 2 )u = −∇ p − u · ∇u ∇ ·u = 0 © 2006 by Taylor & Francis Group, LLC

(6.1) (6.2) 101

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We first recall the scaling predicted by Loitsyansky (1939) stemming from an invariance of an integral moment of the two-point velocity correlation. This may be written as:
I=−

d2 u(x) · u(x + )

Here, u = (u, v, w). This translates into spectral language, (u(x) = 1 →0 2




I = lim

∞

d xU (x)G(x),

(6.3) 

G(x) = 64

0

ik·x ˜ dku(k)e ), etc. as

x2 (1 + x 2 )3

(6.4)

with U (k) ≡ E(k)/(2k 2 ). Thus, in order for I to exist, U (k) = C(t)k 2 + Ok 3 .

(6.5)

A consequence of (6.3) would be that C(t) is independent of t. We remark that the validity of (6.3) and (6.4) is independent of viscosity. Much has been written about the existence of I, and we refer to Frisch’s book (Frisch, 1995, pp. 114 and 197) for discussion and references. We only

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101

102 nu Pr Bv2 Om2 1.15E+01 2.06E−02 2.71E+00 3.43E−03 E(k, t), Hyper8

103

FIGURE 6.1 Energy spectrum for decaying isotropic turbulence. Here hyperviscosity damps the high k– scales. Note the k 4 region at small k. Such is converted into a k 2 range if sufficient damping at large k is not used in the DNS. Such damping is realized either via hyperviscosity or an adequate dissipation range. The spectral “bump” at large k is discussed in the text. © 2006 by Taylor & Francis Group, LLC

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103

Kinetic energy spectrum

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10−10 100

101 102 nu Pr Bv2 Om2 1.15E+01 2.06E−02 2.71E+00 3.43E−03 inviscid run

103

FIGURE 6.2 Energy Spectrum, E(k,t), for inviscid run. Notice (1) an approximate k 4 low wave number E(k, t), which has an approximate constant C(t); (2) at high k the development of an approximate inertial range (for short times) with an eventual inviscid k 2 spectrum.

mention here that C(t) is a slowly increasing function of t according to EDQNM closures (Lesieur and Schertzer, 1978). Is this a defect of the closure or a manifestation of a true time-dependence of I? Of importance also is that E(k) ∼ k 2 was also predicted by Birkhoff (1954) and Saffman (1967) and as an inviscid equipartition spectrum (Lee, 1952). Lee’s result is interesting in that if a DNS has too little viscosity, the large scales can develop spuriously as k 2 because of an underresolved vorticity spectrum. Using 5123 –DNS, we explore the question of whether (6.5) is maintained during decay. In general, we find that it is quite easy to observe a transition k 4 → k 2 during the decay. This invariably happens if R
(t = 0) > 40, which is quite surprising. Thus, it is easy to understand why the earlier study of this issue by Chasnov (1993) found it necessary to employ an eddy viscosity to dampen the high wave numbers. Figure 6.1 shows such a result, for which a strong hyperviscosity is applied and for which (6.5) seems valid. Is C(t) constant for this run? Our results are somewhat inconclusive, but not inconsistent with Lesieur and Schertzer (1978). We mentioned inviscid equipartition for which k 2 , k → 0 could be a solution. We explore this case in Figure 6.2, which has the same initial conditions as Figure 6.1. Here, the k 4 spectrum is maintained for a rather long time, with the k 2 inviscid spectrum invading from large k. There remains the question of why, in this inviscid case, the high wave number k 2 does not upset the low wave number k 4 spectrum, forcing it to a k 2 shape. Perhaps the answer is that there is no energy transfer associated with the inviscid k 2 spectrum. This behavior is not physical in homogeneous turbulence because, as Saffman and Birkhoff noted, © 2006 by Taylor & Francis Group, LLC

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the prefactor multiplying k 2 is an invariant, so if it starts out zero it must stay zero. The spurious k 4 to k 2 transition is probably a result of the imposed periodic boundary conditions. Before turning to stratified turbulence, we should note that there are realizable initial conditions for which Loitsyansky’s integral is nonexistent: for example, a spectrum ∼ (k − k0 ), for which the correlation functions u(x)u(x + ) ∼ sin(k0 )/(ko ). (6.6) Because we know that such sharp spectra developed into k 4 , k → 0, we conclude that it may be possible that I diagnoses an attractor for the turbulence.

6.2 STABLY STRATIFIED TURBULENCE The stratified equations of motion generalize (6.1) and (6.2) as: (∂t − ∇ 2 )u = −∇p − u · ∇u − gˆ N  + 2
× u

(6.7)

(∂t − ∇ 2 ) = N w − u · ∇

(6.8)

∇ · u = 0.

(6.9)

Here, u = (u, v, w) and  is the deviation of the temperature field from its mean, whose constant vertical gradient is nondimensionalized to −1. The Loitsyansky invariant for stratified turbulence follows from Davidson’s analysis of the equivalent MHD problem as:
I = − r⊥2 u⊥ · u ⊥ dr (6.10) with r⊥ = r x2 + r y2 , u⊥ = (ˆiu x + ˆju y , 0).

(6.11)

(Davidson, 2001). To represent the anisotropy associated with the buoyancy term, we represent the velocity field u(k) by: ⎫ ⎧ ⎫ ⎧ ⎪ ⎬ ⎪ ⎨ e1 (k)1 (k) + e2 (k)2 (k) ⎪ ⎬ ⎨u ⎪ (k × gˆ )/|k × gˆ | e1 = (6.12) ⎪ ⎪ ⎭ ⎪ ⎩ ⎭ ⎩ ⎪ (k × (k × gˆ )/|k × (k × gˆ ) e2 In what follows, we specify both r and k in polar coordinates so that r⊥ = r 2 sin2 ϑ , dr = r dr sin ϑ dϑ d , and dk = k 2 dk sin ϑdϑd . Where convenient, we call = cos ϑ = cos ϑ. We may now work through the analogous steps leading to (6.4) for isotropic turbulence to get
1
∞ 2
1 x dx d d F( , ) [1 (x, ) + 2 2 (x, )] (6.13) 2 −1 −1 0 2

In this expression,

 F( , ) = (1 − 2 ) G 1 + i

, (1 − 2 )(1 − 2 ) G(a, b) ≡

24a 4 + 9b4 − 72a 2 b2 [a 2 + b2 ]9/2

(6.14)

i = (1, 2).

(6.15)

and i (k) = |i (k)|2 , © 2006 by Taylor & Francis Group, LLC

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Here, we have rescaled r → x. Equation (6.13) and Equation (6.14) are much more complicated than (6.4), but they nonetheless lead to the same conclusion that U (k) ∼ k 2 as k → 0. The

dependence of U (k) is not so restricted, but we should recall the theorem of Cambon et al. (1981): 2 (k, ) − 1 (k, ) ∼ (1 − 2 ), → 1. (6.16) It is useful to record (6.7) through (6.9) in the Fourier representation and use (6.13) for economy ((1 , 2 ) instead of (u, v, w)): ⎛ ⎞ ⎞ ⎛ ⎞ ⎛ 1 1 e1 · f ⎜ ⎟ ⎟ ⎜ ⎟ ⎜ ∂t ⎝2 ⎠ = M ⎝2 ⎠ + ⎝e2 · f ⎠ (6.17) f   Here,

⎛

0 ⎜ M = ⎝−2
cos ϑ 0

2
cos ϑ 0 N sin ϑ

⎞ 0 ⎟ −N sin ϑ ⎠ 0

(6.18)

and {f, f  } ≡ (F1 , F2 , F3 )

(6.19)

are the Fourier amplitudes of the nonlinear terms in (6.7) through (6.9), and
is the nondimensional rotation. Finally, we transform (6.7) through (6.9) with an expansion in terms of the eigenvectors of M, vi (k), and eigenvalues,
i . These are ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 0 0 1 ⎜ ⎟ 1 ⎜ ⎟ 1 ⎜ ⎟ (6.20) (v1 , v2 , v3 ) = ⎝ 0 ⎠ , √ ⎝ i ⎠ , √ ⎝ −i ⎠ 2 2 1 1 0
0,±1 = (0, ±i N sin(ϑ))

(6.21)

and then expand =



i (k, t)vi

(∂t +
i ) i (k, t) = Fi .

(6.22) (6.23)

We record i and
i only for the case
= 0, which is the only case studied here. The relation between and  is explained by ⎛ ⎞ ⎞⎛ ⎞ ⎛ ⎞ ⎛ 1 0 0 1 1 1 √ √ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ (6.24) ⎝ 2 ⎠ = T ⎝ 2 ⎠ ≡ ⎝ 0 i/ 2 −i/ 2 ⎠ ⎝ 2 ⎠ √ √ 3   0 1/ 2 1/ 2 Here, F = T F, with F defined just after (6.16). T has for its inverse its complex  adjoint. Finally, we note via closure for the vi (i.e.,
|

| = 1 and f (M) =
|
 f (
)
|) that ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 0 0 0 0 0 0 1 0 0 1⎜ ⎟ ⎟ ⎜ ⎟ 1⎜ exp(Mt) = ⎝ 0 0 0 ⎠ + ⎝ 0 1 −i ⎠ ei N t sin ϑ + ⎝ 0 1 i ⎠ e−i N t sin ϑ . (6.25) 2 2 0 i 1 0 −i 1 0 0 0 Equation (6.24) describes the evolution of {1 , 2 , } in the absence of the nonlinear terms (the rapid distortion approximation, which has short-term validity if the initial conditions are sufficiently © 2006 by Taylor & Francis Group, LLC

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FIGURE 6.3 Total kinetic energy for decaying strongly stratified turbulence. After Herring and Kimura (2002).

random). We see from (6.23) that for the angular average of {1 , 2 , }, only 1 survives at long times, if the nonlinearities are suppressed and N → ∞. The striking feature of stably stratified turbulence is that it decays slower than isotropic threedimensional turbulence with the same initial conditions. Figure 6.3 illustrates this. It shows the decay of enstrophy for the standard E(k, 0) = k 4 exp(−k 2 ) energy spectrum. Here, R
(0) = 80, and the resolution is 2563 . Stratification (with N = 10) is turned on when the skewness reaches its maximum (about one eddy circulation time). We note that at late times, an approximate E(t) ∼ t −1 obtains. The first-order question is then how to explain the slowdown in the decay, from approximately t −1.5 in isotropic DNS. A significant feature of such flows is its strong anisotropy, with the vorticity organized into scattered pancakes as shown in Figure 6.4. Here, we show the late-time organization of the enstrophy, E(r, t) ≡ |∇ × u(r, t)|2 . The flow is strongly anisotropic, but its degree of anisotropy tends to saturate an N → ∞. This is indicated in Figure 6.5, which shows the angular distribution of the vorticity vector. It is remarkable that although stratified flows contain waves, the evolution of the enstrophy patches shows little wave-like fluctuation. Perhaps this is because pancake organization signifies strong vertical variability, for which the wave frequency, 

=

N 2 sin2 (ϑ) + 4
2 cos2 (ϑ)

(6.26)

is near zero. Here, cos ϑ = k z /k, and we include in a possible rotation rate,
. Perhaps the apparent lack of waves in pancakes may be explained via an analogy to Taylor columns in rotating turbulence. Think of Taylor’s experiment as discussed in the introduction of Greenspan’s book (Greenspan, 1969). A penny is slowly towed across the base of a rapidly rotating tank. A Taylor column is seen to move with it, spanning the fluid from the penny to the top of the tank. How does the fluid lying well above the penny, but within the column know it must move with the penny? © 2006 by Taylor & Francis Group, LLC

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N2 = 0

N2 = 1

N2 = 10

N2 = 100

FIGURE 6.4 Enstrophy profiles for isotropic turbulence, N 2 = 0 (panel 1), and increasingly strong stratification with N 2 = 1 (panel 2), N 2 = 10 (panel 3), and N 2 = 100 (panel 4). After Kimura and Herring (1996). Resolution = 1283 .

Now the Taylor column (which is the analogue of pancakes) exhibits no wave-like features to the casual observer. However, if, as noted by Greenspan, one carefully analyzes the flow on the fast time-scale of inertial wave propagation (a time-scale based on group velocity/tank size, rather than frequency), then one finds that the information telling the column to move is transmitted upward from the penny in the form of fast, low-frequency inertial waves propagating in the vertical direction. (Recall that the fastest waves, in terms of group velocity, have the lowest frequency.) In short, the quasi-steady Taylor column is the manifestation of fast inertial waves. Without the waves there would be no column.

0.15

0.15

N2 = 1000.0 N2 = 100.0 N2 = 10.0 N2 = 1.0

N2 = 1000.0 N2 = 100.0 N2 = 10.0

t = 5.0 P(θ)/ 2π sinθ

April 24, 2006

0.1

t = 10.0

2

N = 1.0 N2 =0.0

0.05

0

0.1

N2 = 0.0

0.05

0

π/2

π

0

0

π/2

π

FIGURE 6.5 Angular distribution of enstrophy for stratified turbulence of various degrees of stratification (N 2 = 0, 1, 10, 100) for two times during the decay of the flow. After Kimura and Herring (1996). © 2006 by Taylor & Francis Group, LLC

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However, if one looks at the flow one sees, no wave-like properties — just a moving column. In a similar way the pancakes could be a manifestation of the fast transmission of information by low-frequency, horizontally propagating gravity waves. As with the Taylor column, there is no obvious wave-like behavior, but without the waves there would be no pancakes. The following is a rough estimate of the decay of kinetic energy for stratified turbulence, based on closure. We recall first a simple wave-number diffusion equation for the evolution of the energy spectrum, E(k, t) as proposed by Leith (1967): EDQNM:   k  2 p d p E( p) E(k) 0 (∂t + k 2 )E(k, t) = ∂k k 4 ∂k . (6.27) (k) k2  k 2 Here, (k) is the eddy relaxation rate, which for isotropic turbulence is 0 p d p E( p). (Actually, √ Leith proposed k 3/2 (k, t).) For strongly stratified turbulence, we expect  ∼ N . Why, though, should waves act to dampen the energy? An essential point here is that gravity waves “phase mix,” thus providing an attenuation of correlations in time. Kaneda (1998) has stressed this point in his application of rapid-distortion theory to stratified turbulence. See also Hunt and Carruthers (1990). Then, by integrating  (6.5) over k 2 2/3 −5/3 , if we take (k) ∼ [0, k], with k in the inertial range, there follows E(k) ∼  k 0 p d p E( p). If, on the other hand, (k) ∼ N , as for strongly stratified flow, there follows, E(k) ∼ (N )1/2 /k 2 .

(6.28)

Note that we have ignored anisotropic effects here, so our argument is rough. However, according to Figure 6.3, anisotropy is not overwhelming, even for strong anisotropy. We may now estimate the decay of E(t) by the following argument. There results, E(t) ∼ t −5/7 .

(6.29)

The exponent is the average of three-dimensional decay (10/7) and two-dimensional (0). The exponent in Figure 6.3 is −1 instead of −5/7. This may be attributable to finite R
∼ 35 in the DNS, just as our estimate of the decay of isotropic turbulence is somewhat faster than indicated by the EDQNM calculations (−1.5 instead of 1.37). Clearly, such estimates should be replaced by more secure EDQNM or DIA calculations, such as those proposed by Godeferd and Cambon (1994). We remark that the spectral form (6.28) has been compared to DNS for rotating turbulence by Yeung and Zhou (1998). A similar slow-down in the cascade to small scales has been noted in MHD turbulence by Galtier et al. (1997). One problem with the brief analysis given here is its near-isotropy assumption. Davidson’s extension of the invariants (Equation (6.19) through Equation (6.13)) suggests that only the horizontal energy u 2⊥ participates in invariants, where our crude analysis involves the total energy. In fact, the decaying (isotopically gathered) energy has k 2 , k → 0. Perhaps a reanalysis of the problem in terms of ( 1 , 2 , 3 ) (see Equation (6.10) through Equation (6.24)) would be more appropriate than using 1 (k), 2 (k), (k). It is nevertheless of interest to examine the k −2 prediction of Equation (6.27). For this purpose, we note that forced turbulence is able to reach further towards the asymptotic regime, where such power laws are expected. Hence, we force the flow with a solenoidal Gaussian random force acting on u(k) at some wave number k0 . It is isotropic. We examine two cases, k0 = 5 and k0 = 10. Of course, the k0 = 5 will reach the higher Reynolds number, and the k0 = 10 will have the better statistics. Figure 6.6 presents the steady-state values of 1 (k⊥ ), 2 (k⊥ ), and (k⊥ ) for k0 = 5. Here, k⊥ denotes the cylindrically gathered spectra, summed over k z . We see some evidence here for a 2 − k −2 . The forcing of 1 is clearly evident, but not for 2 or . Note that  is in near lockstep © 2006 by Taylor & Francis Group, LLC

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100 10−1

Φ1(k⊥, t)

10−2 10−3

Θ(k⊥, t)

10−4 10−5 10−6

Φ2(k⊥, t)

10−7 10−8 10−9 10−10 100

101

102

103

FIGURE 6.6 1 (k⊥ , t), 2 (k⊥ , t), and (k⊥ , t) for horizontally forcing at |k| = 5. Here t = 27.5 by which time the spectra have reached equilibrium.

with 2 , as would be expected from (6.7) through (6.9), for strong stratification. We must note that power-law behavior is observed only for certain ways of forming spectra. For example, Figure 6.7 shows isotropically gathered spectra ((|k|)), etc. Here, there is evidence of power laws for any of the variables. Where is evidence of wave-like behavior? We expect this in 2 and . Clearly, 1 (r, t) displays a spectrum of oscillations, and has been noted by M´etais and Herring (1989), but is there evidence for wave-like behavior in the spectra? Figure 6.8 presents data on this point. Here, we see that after

100

Φ1(|k|, t)

10−1 10−2 10−3

Θ(|k|, t)

10−4 10−5

Φ2(|k|, t)

10−6 10−7 10−8 10−9 10−10 100

101

102

103

FIGURE 6.7 1 (|k|, t), 2 (|k|, t), and (|k|, t) for horizontally forcing at |k| = 5 at t = 27.5. Note the difference between this figure and the previous, indicating highly “layered” structures, as described in the text.

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100 10−1

Θ(k⊥, t)

10−2

Θ(k⊥, t)

Φ1(k⊥, t) Φ1(k⊥, t)

10−3

Φ2(k⊥, t)

10−4

Φ2(k⊥, t)

10−5 10−6 10−7

k⊥ = 1.5

k⊥ = 5.5

10−8 100 Θ(k⊥, t)

10−1

Θ(k⊥, t)

10−2 Φ1(k⊥, t)

10−3 10−4

Φ2(k⊥, t)

Φ2(k⊥, t) Φ1(k⊥, t)

10−5 10−6 10−7 10−8

k⊥ = 10.5 0

5

10

15

k⊥ = 30.5 20

25

30

0

5

10

15

20

25

30

FIGURE 6.8 1 (k⊥ , t), 2 (k⊥ , t), and 2 (k⊥ , t) at k⊥ = 1.5 (upper left), k⊥ = 5.5, (upper right), k⊥ = 10, (lower left), and k⊥ = 30.5 (lower right) as functions of time, t. In each panel,  is the top curve, with 2 (k⊥ , t), the lower, except for k⊥ = 30.5 , for which 1 (k⊥ , t)  2 (k⊥ , t).

the approach to a statically steady state, only the lowest k⊥ modes of 2 and  show wave-like behavior. This mode is the vertically sheared horizontal flow (VSHF) of Smith and Waleffe (2002). Its frequency is about half of N , but notice that it emerges as a vertical response to a horizontal forcing. Finally, we comment on the statistics of stratified flow. We know that isotropic turbulence develops strong non-Gaussianity in its acceleration. For example, Figure 6.9 (top panel) shows the Eulerian acceleration for isotropic turbulence, according to DNS at 1283 resolution. The near exponential distribution is similar to experiments, of which those of Champagne et al. (1977) are the most secure. The lower panel shows the same histogram for strong stratification (N = 10). Note the conversion from exponential to Gaussian, with an accompanying anisotropy (w2 much weaker than u 2 or v 2 , but both near Gaussian). Figures 6.10 and 6.11 illustrate the difference between the enstrophy distribution for isotropic and stratified turbulence. Here, Figure 6.10 shows the vortex-tube like distribution familiar from isotropic turbulence simulations. The initial flow is Gaussian, and the picture shows the enstrophy at the time of maximum total enstrophy, tmax . After tmax , strong stratification is turned on and the enstrophy distribution after several eddy circulation times is as shown in Figure 6.11. We notice the pancake layering of enstrophy in Figure 6.11 and a tendency for the pancakes to exist in pairs. Figure 6.12 is an enlargement of the pancake pair in the upper righthand corner of Figure 6.11. © 2006 by Taylor & Francis Group, LLC

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10 1 0.1 0.01 0.001 0.0001 ? −2 −1 0 1 2 PDF of Eulerian acceleration for unstratified turbulence. Here, Rλ ∼ 30. 10 1 0.1 0.01 0.001 0.0001 ? ? −20

0 10 −10 PDF for stratified turbulence, with N = 10. Again, Rλ ∼ 30.

20

FIGURE 6.9 Distribution function for Eulerian acceleration for unstratified (black curve) and stratified (red horizontal; blue vertical) turbulence. Note that for stratified turbulence, the vertical acceleration is large compared to the horizontal.

6.3 CONCLUDING COMMENTS In this chapter, we have invoked integral invariants of the Loitsyansky type as a basis to discuss the issue of box-limited DNS. An adequate resolution of the large scales of the flow implies a certain behavior of the energy spectrum near k = 0. For isotropic flows, our DNS results at 5123 suggest that if initial conditions have E(k, 0) ∼ k p , p > 4, then p → 4 as t → ∞. However, we fail to confirm that C(t) (as in (6.5)) is constant. Rather, C(t) increases in rough accord with Chasnov’s finding (1993). We do not know whether this slow increase of C(t) holds up as t → ∞. The assumptions needed for the derivation of I include: (1) homogeneity, and (2) spatial reflection covariance, which asserts that (u(−x, t) = −u(x, t), and (−x, t) = (x, t) may be used for quantities such as u(x, t)F and (x, t)F, for any F. For stratified flow, the generalization of the Loitsyansky invariants applies only to the horizontal dynamics. Thus, the strong vertical variability (the VSHF mode of Smith–Waleffe at k⊥ = 0) seems outside its purview. It is this mode that bears the wave-like signature as far as spectra are concerned. It is excited directly through a 1 , 2 coupling, with the scale of 1 near the horizontal forcing. Finally, stratified flows are closer to Gaussian than unstratified flows. This is seen in the distribution © 2006 by Taylor & Francis Group, LLC

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FIGURE 6.10 Enstrophy of a turbulent field at the time of maximum total enstrophy. Initial field is Gaussian. Note the bent vortex tubes typical of such turbulence. Flow is unstratified.

FIGURE 6.11 Enstrophy of a stratified turbulent field at the late time. Initial field is the same as in previous figure. Note the scattered pancake arrangement of the vorticity, with frequently dipole stacking.

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(a)

113

(b)

(c)

FIGURE 6.12 Enstrophy of a stratified turbulent field for the same late time as in Figure 6.11. Here, we show an enlargement of one of the dipoles: a) as viewed from positive and negative y-direction, b) from positive and negative x-direction, c) from positive and negative z-direction.

of Eulerian acceleration (as shown in Figure 6.9), as well as other measures of nonlinear transfer, e.g., skewness factors for velocity and the temperature field. This may suggest that closure approximations will have more success here than for unstratified turbulence. One final comment can be made about the more mathematical aspects of stratified flows setting apart inviscid stratified flows from the unstratified case. For the latter, a finite time singularity in vorticity is thought to develop, but no proof has yet emerged. An examination of this issue via DNS indicates that such is indeed the case, with the singularity developing after a few eddy-circulation times (Kerr, 1993). For stratified flows, similar DNS studies reveal no singularity. Of course, such are always resolution limited, but the fact that most turbulent atmospheric flows are stably stratified suggests that a focus on this issue by mathematicians may be important.

REFERENCES Birkhoff, C. (1954). Fourier synthesis of homogeneous turbulence, Comm. Pure Appl. Math., 7: 1944. Caillol, P. and Zeitlin, V. (1999). Kinetic Equations and Stationary Energy Spectra of Weakly Nonlinear Internal Gravity Waves. Preprint. Cambon, C., Jeandel, D., and Mathieu, J. (1981). Spectral modeling of homogeneous nonisotropic turbulence. J. Fluid Mech, 104: 247–262. Champagne, F.H., Friehe, C.A., Larue, J.C., and Wyngaard, J.C. (1977). Flux measurements, flux estimation techniques and fine-scale turbulence measurements in the unstable surface layer over land. J. Atmos. Sci., 34: 515–530. Chasnov, J.R. (1993). Computation of the Loitsyansky integral in decaying isotropic turbulence. Phys. Fluids A 5 (11): 2579–2581. © 2006 by Taylor & Francis Group, LLC

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Davidson, P.A. (2001). An Introduction to Magnetohydrodynamics. Cambridge, U.K.: Cambridge Univ. Press. Frisch, U. (1995). Turbulence: The Legacy of A.N. Kolmogorov. Cambridge: Cambridge Univ. Press. Galtier, S., Politano, H., and Pouquet, A. (1997). Self-similar energy decay in magnetohydrodynamic turbulence. Phys. Rev. Lett., 79: 2807–2810. Godeferd, F.S. and Cambon, C. (1994). Detailed investigation of energy transfer in homogeneous stratified turbulence. Phys. Fluid, 6 (6): 2084–2100. Greenspan, H.P. (1968). Theory of Rotating Fluids. Cambridge, U.K.: Cambridge Univ. Press. Herring, J.R. and Kimura, Y. (2002). Structural and statistical aspects of stably stratified turbulence. In Y. Kaneda and T. Gotoh, eds. Statistical Theories and Computational Approaches to Turbulence. Springer– Verlag, Berlin, Heidelberg, New York, pp. 15–42. Hunt, J.C.R. and Carruthers, D.J. (1990). Rapid distortion theory and the “problem” of turbulence. J. Fluid Mech., 212: 497–532. Kaneda, Y. and Ishida, T. (2000). Suppression of vertical diffusion in strongly stratified turbulence. J. Fluid Mech., 402: 311–327. Kerr, R.M. (1993). Evidence for a singularity of the three-dimensional incompressible Euler equations. Phys. Fluids A, 5: 1725–1746. Kimura, Y. and Herring, J.R. (1996). Diffusion in stably stratified turbulence. J. Fluid Mech., 328: 253–269. Lee, T.D. (1952). On some statistical properties of hydrodynamical and magneto hydrodynamical fields. Q. Appl. Math., 10: 69–74. Leith, C.E. (1967). Diffusion approximation to the inertial energy transfer in isotropic turbulence. Phys. Fluids., 10: 1409–1416. Lesieur, M. and Schertzer, D. (1978). Amortissement auto similarit´e d´une turbulence a grand nombre de Reynolds. J. de M´ecanique, 17: 609–646. Loitsyansky, L.G. (1939). Some basic laws for isotropic turbulent flow. Trudy Tsentr. Aero–Gidrodin. Inst: 3–32. M´etais, O. and Herring, J.R. (1989). Numerical studies of freely decaying homogeneous stratified turbulence. J. Fluid Mech., 202: 117–148. Saffman, P.G. (1967). The large-scale structure of homogeneous turbulence. J. Fluid Mech., 27: 581–93. Smith, L. and Waleffe, F. (2002). Generation of slow large scales in forced rotating stratified turbulence. J. Fluid Mech., 451: 145–168. Yeung, P.K. and Zhou, Y. (1998). Numerical study of rotating turbulence with external forcing. Phys. Fluids, 10 (11): 2895–3244.

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Scaling and 7 Wavelet Navier–Stokes Regularity Jacques Lewalle CONTENTS 7.1 Background........................................................................................................................... 115 7.2 Navier–Stokes in Wavelet Space .......................................................................................... 117 7.3 Isolated Singularities and Scaling of Wavelet Coefficients .................................................. 118 7.4 Evolution of Singularities..................................................................................................... 119 7.5 Discussion............................................................................................................................. 120 References...................................................................................................................................... 122

ABSTRACT The problem of the regularity of solutions of the Navier–Stokes equations is related to a number of physical and mathematical concepts. A nonexhaustive list includes vorticity, filtering and renormalization, H¨older exponents, Besov spaces, intermittent cascades, Euler dissipation, scaling, ladder inequalities, multifractals, and fractional derivatives. The class of Hermitian continuous wavelet transforms overlaps with many of these topics, and seems well suited to study NS regularity. The asymptotic scaling of the wavelet coefficients is related to the local H¨older exponent h. An anisotropic version of the H¨older exponent is proposed, and the exact evolution equation for the wavelet coefficients is derived. The small-scale asymptotics of the various terms can be estimated, and the dominant exponents evaluated. A runaway singularity corresponds to the smallest exponent associated with the nonlinear terms, which would then dominate the evolution. In the case of isotropic H¨older exponents, a runaway singularity would occur for h < 1/3 for three-dimensional Euler turbulence, but the viscous term would maintain regularity. No firm conclusion is reached yet about anisotropic exponents, but preliminary results do not show evidence of a sharp cross-over value of h.

7.1 BACKGROUND The finite-time regularity of solutions of the Navier–Stokes and Euler equations remains a theoretical as well as a practical problem, with implications for direct numerical simulation (DNS) as well as fractal modeling and physical theory. The pioneering work of Leray [1] showed the relevance of weak solutions, i.e., nondifferentiable fields, but the penetration of these ideas in the broader fluid dynamics community remains limited. A few related ideas and tools have gained wider recognition over the years. The scaling behavior of structure functions was applied to fluid turbulence in Kolmogorov’s theory (K41, e.g., as presented by Frisch [2]). In this approach and its anomalous variants, the moments of velocity differences scale with distance r according to power laws Sn (r ) = <|
u |n > ∼ r n ,

(7.1)

which apply in the inertial range (i.e., the range, if it exists, where neither external forcing nor viscosity affects the dynamics). © 2006 by Taylor & Francis Group, LLC

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For n = 1, the K41 exponent is 1/3, whereas Taylor expansion of continuously differentiable fields would require a value of 1. It is generally accepted that small-scale events are responsible in some way for the specifics of anomalous scaling [3,4]. However, this could hold true even if viscosity smoothes out steep gradients to maintain differentiability. The 1970s saw the introduction of fractal concepts in turbulence. From homogeneous geometrical objects [5] to dynamical intermittent cascades [6,7], scaling laws cease to be valid at some small scale where viscous effects are not negligible. Therefore, fractal models avoid the problem of any singularities that might develop at yet smaller scales. These viscous cut-offs have been, more or less implicitly, related to the Kolmogorov microscale . This is the basis for the often quoted estimate of the number of grid points required for DNS (direct simulation of Navier–Stokes turbulence), since the ratio of the energy-containing eddy size
(proportional to the size of the computational domain) to the Kolmogorov microscale is known to 3/4 9/4 scale as Re
, giving a number of grid points of the order of Re
for three-dimensional turbulence. However, this ratio is only valid as an average concept and should not be misunderstood as relevant to capture the smallest active local event in a flow. This distinction, in conjunction with rapidly growing computational power in the early 1980s, sparked renewed interest in the determination of the smallest scale of any relevant event in fluid turbulence [8–13]. What emerges is a sequence of ever smaller scales associated with rare localized events. It is unclear whether the sequence has a finite lower bound. Three groups of questions can be separated at this point. First, in the case of inviscid flow, one might inquire about the creation of singularities and their spatial and temporal distributions. Second, the more difficult question of the effect of diffusion on such phenomena can be addressed. Third, any macroscopic effect of singularities would make this subfield more relevant to the turbulence practitioners. One of the rigorous results about the appearance of inviscid singularities is related to vorticity [14]. The necessary condition is that the largest value (L ∞ -norm) of vorticity, integrated over a finite time , must become infinite:
 |  |max dt → ∞. (7.2) 0

Such an occurrence requires vortex stretching, absent in two-dimensional flows. Once singularities develop, in isolation or in fractal patterns, classical tools of field theory are no longer applicable. Instead of a first-order truncated Taylor series, the local H¨older estimate is introduced as
u ∼ r h , (7.3) where a singularity corresponds to h < 1. Note that h can vary from point to point. Similarly, the use of Hilbert (finite energy) and Sobolev (finite enstrophy and higher order norms) functional spaces [15] may not be justified. The relevant functional space may be Besov’s [16], which is compatible with the use of H¨older norms. Thus far, it seems that only one nonclassical consequence of singularities has been studied in any depth: Onsager’s conjecture of inviscid dissipation, i.e., a lack of energy conservation in nondifferentiable flow fields. Onsager’s argument has been revisited and increasingly sharp results obtained about Euler dissipation [13,17–20]. It is noteworthy that a filtered (mollified) version of the equations is used, with the filter scale eventually decreased to zero, and that H¨older exponents are most relevant. Although it is now certain that there is no Euler dissipation in two-dimensional flows, the threshold of h = 1/3 seems to hold for three-dimensional flows, and will reappear later. Two more related topics are mentioned for completion. Interesting results have been obtained in the form of ladder inequalities [21], by which rigorous bounds on various norms are established recursively. Another intriguing line of work [22] involves the use of fractional derivatives. When conventional (integer order) derivatives are infinite, fractional order of differentiation may characterize the type of singularity; fractional calculus is endowed with a chain rule and Taylor-type expansions and may be the basis for a theory of singular fields. This will be developed later. © 2006 by Taylor & Francis Group, LLC

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The concepts and techniques listed previously relating to the regularity of Navier–Stokes solutions have little overlap. Each of them appears insulated from some, many, or most of the others. By contrast, the continuous wavelet transform [23,24] happens to connect most of these ideas. Wavelets are bandpass filters, used for the characterization of multifractals sets [25] and of Besov spaces [26]. The scaling of wavelet coefficients is related to H¨older exponents [26]; they can be used in conjunction with exact equations to describe intermittent energy cascades and dissipation [27]. In the case of the Mexican hat and related wavelets, they are closely matched to the concept of vorticity and to the diffusion and Poisson equations. As seen in the companion chapter in this volume (summarized in the following section), they allow for the use of exact equations, including Navier–Stokes.

7.2 NAVIER–STOKES IN WAVELET SPACE Here, we assume incompressible flow, for which ∂i u i = 0

(7.4)

expresses mass balance. The momentum balance is given by the incompressible Navier–Stokes equations p ∂t u i − ∂ 2j j u i = −∂i ( ) − ∂ j (u i u j ). (7.5)  A brief derivation of the wavelet transform version of these equations is provided here so that the chapter is self-contained; additional details are found in the companion chapter and the literature cited there. First, pressure is eliminated by adopting flexion 2 i = ∂kk ui

(7.6)

as the primary variable. Conservation of mass and momentum is then captured by the flexion equation ∂t i − ∂ 2j j i = ∂i3jk (u j u k ) − ∂ 3jkk (u i u j ).

(7.7)

It is easy to see that the familiar Fourier version of the momentum equation [28], multiplied by 2 and transformed back to x-space, is identical to Equation (7.7). The reconstruction of velocity from flexion, required to evaluate the nonlinear terms, is achieved by the Biot–Savart relation
1 i (7.8) d x . ui = 4

| x − x | Next, we turn to filtered flexion and note that it is equivalent to the Mexican hat wavelet transform of velocity. The d-D normalized Gaussian filter at scale s 1 is  2 1 x (7.9) Fs (x) = F(x, s) = √ d/2 exp − (2 s) 4s and the filtered flexion is defined as i>s = Fs ∗ i =




i (x  )F(x − x  , s)d x  .

(7.10)

Then, defining the isotropic Mexican hat wavelet (x, s) as 2 (x, s) = ∂kk F(x, s) = ∂s F(x, s),

1 Note

that s has dimensions L 2 .

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(7.11)

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we find that

i>s = (x, s) ∗ u i = u˜ i (x, s).

(7.12)

Therefore, the filtered flexion field at scale s is identical to the Mexican hat wavelet-transformed velocity field, for which the Navier–Stokes equations can be written as (∂t − ∂s )u˜ i = ∂i jk (u j u k )>s − ∂ jkk (u i u j )>s .

(7.13)

Furthermore, the reconstruction of velocity from flexion can also be expressed as a variant of the inverse wavelet transform, and we have
∞ i>s ds (7.14) ui = 0

as an alternative to Biot–Savart [27]. Wavelet coefficients, with their ability to zoom in on singularities by reducing the scale s, have a history of success with singular fields. In the next sections, we turn to the scaling of wavelet coefficients near singularities and to their evolution under exact dynamics.

7.3 ISOLATED SINGULARITIES AND SCALING OF WAVELET COEFFICIENTS The modulus of continuity
u = u(x) − u(x − y) ∼| y |h

(7.15)

has emerged as a useful diagnostic of regularity2 . For differentiable fields, Taylor series correspond to positive integer hs. Any value h < 1 is an indicator of singularity. The relevance of this index to Navier–Stokes turbulence is apparent from the K41 scaling, in which h = 1/3 on average. The scaling of the wavelet coefficients in the vicinity of a singularity of this type can be estimated easily. Filtering the modulus of continuity,
Fs (y)
u dy = u − u > = u < , (7.16) we see that only the small scales are retained. First we, will consider one-dimensional fields of two types, symmetric and antisymmetric, around the singularity located at the origin:
u s = Cs | x |h ,

(7.17)


u a = Ca sign(x) | x |h ,

(7.18)

with a wake- or jet-like appearance, and

with a shear-layer configuration. Gaussian filtering gives, respectively,      C s 2h 1+h 1 + h 1 x2 h > 2 M , ,
u s = s √ 

2 2 2 4s 

and
u a>

=s

h 2

    Ca 2h+1 h x h 3 x2 √ M 1+ , ,  1+ √

2 2 s 2 2 4s

(7.19)

(7.20)

where M(., ., .) is the Kummer-M (hypergeometric) function. 2 Its

statistical moments, known as structure functions, are the starting point of K41 and anomalous scaling (several chapters in this volume); local behavior is of interest in this chapter.

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Interpreting this result, we see that in the vicinity of an isolated singularity, the filtered √ field combines a self-similar, scale-dependent but singularity-free functional shape with x/2 s as the similarity variable, a numerical constant [in brackets], and a scaling factor proportional to s h/2 . Taking x-derivatives, making use of the properties of Kummer-M functions, and isolating the new self-similar functional shape, it can be seen that a derivative of u > follows the scaling ∂x u > ∼ s

h−1 2

(7.21)

and, in particular, u˜ = ∇ 2 u > ∼ s 2 −1 . h

(7.22)

Making allowance for the arbitrary but consistent scaling factor in the definition of the wavelet transform u˜ , this is in agreement with earlier results [26]. With the intent of examining specific flow models (local shear layers, spiral vortices, etc.), local asymmetry around a singularity needs to be included in the formulation. Indeed, the wellknown distinction between longitudinal and transverse structure functions, discussed elsewhere in this volume, should be reflected in the mathematical description of local singularities. Generalizing the isotropic scaling, we assume that h


j u i = u i (x j ) − u i (x j − y j ) ∼ y j i j is applicable. Then, it follows that

  .. Fs (y j )
j u i dy j ∼ s j

hi j 2

(7.23)

(7.24)

j

and, taking derivatives as before, u˜i ∼ s

 j

hi j 2

−1

.

(7.25)

Similarly, each derivative of the wavelet coefficients will lower the exponent by 1/2. Other rules of manipulation can be verified. Exponents are added for the product of two terms. At small scale, the smallest (algebraic) exponent dominates. Therefore, in a sum over repeated indices such as u˜ i u˜ i , the exponent of s becomes min i j h i j  − 2. Also, a derivative in any direction lowers the exponent by 1/2, so the exponent of s in ∂ j u˜ i is ( m h im − 3)/2 for any j.

7.4 EVOLUTION OF SINGULARITIES With these tools, we can determine the scaling of the various terms in the wavelet-based Navier– Stokes equations (7.13). The key to the evaluation of the nonlinear terms is found in the ConstantinE-Titi (CET) relation [17]:

 Fs (ym )[
m u i
m u j ]dym . (7.26) (u i u j )> = u i> u >j − u i< u <j + .. m

This exact relation was proposed for filters with compact support, but it is easy to show that it holds for Gaussian filters as well. It shows contributions from the coarse-grained (low-pass filtered) field, from the high-pass filtered residue, and from the singularity. For any finite scale s, one can see from Equation (7.19) and Equation (7.20), (7.19–7.20) and the properties of Kummer’s function that the coarse-grained field is regular; the orders of magnitude of the remainder are comparable, and equal to  him +h jm (u i u j )> ∼ s m 2 . (7.27) © 2006 by Taylor & Francis Group, LLC

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TABLE 7.1 Dominant Exponents for Evolution, Pressure, Convective, and Viscous Terms in d-Dimensional Euler and Navier–Stokes Dynamics Anisotropic hi j

Term

Isotropic h

 him −1  m (h2jm +h km ) 3 − min jk  m (him +h2 jm ) 32 min j − 2 m him 2

Evolution Pressure Convective Viscous

m

2

dh 2

−1

dh −

3 2 3 2

dh − dh 2 −2

−2

Consequently, since the smallest exponent of s dominates the small-scale behavior of each term, we have the following estimates for the pressure and convective terms, respectively:  (h jm +hkm ) 3 −2 2 ∂i3jk (u j u k )> ∼ s min jk m (7.28) for each i, and ∂ 3jkk (u i u j )> ∼ s min j

 m

(h im +h jm ) − 32 2

,

(7.29)

also for each i. The dominant exponents for the various terms in the equations are shown in Table 7.1. A separate scaling for time may be introduced, modifying the evolution term to match the smallest exponent elsewhere; this does not affect our conclusions. The central column gives the scaling exponents as derived previously; the right column shows the exponents in the case of isotropic singularities, i.e., for all h im equal to a single value h. Since the exponents are linear in the h im , a dominance of nonlinearities below certain values would only get worse: the generation of ever steeper singularities would be unopposed. Thus, NS regularity has been mapped, for the case of power-law isolated singularities, into a relatively simple criterion. The case of all h equal allows the simplest conclusions. If we first ignore the viscous term (Euler dynamics) and seek the smallest of dh − 1 and dh − 32 , we see that the nonlinearities dominate 2 1 if h < d . Thus, in the case of three-dimensional turbulence, we recover the K41 exponent as the threshold of runaway singularities (and the Onsager threshold of possible Euler dissipation), which is encouraging. However, for d = 2, a threshold of h = 12 has no obvious counterpart in the literature and must be interpreted in the light of Beale et al. [14]. In the absence of vortex stretching, we hypothesize that no incompressible mechanism in two-dimensional turbulence could generate h = 1/2 in the first place, so no singularity arises. Alternative explanations, given the fact that two-dimensional Euler turbulence does not generate singularities, are that all h cannot be equal for some kinematic reason, or again, it is conceivable that some combination of terms disappears under summation so that their exponent would become irrelevant. Furthermore, it is easy to see that the viscous term dominates for any h > − d1 , so we conclude that positive h correspond to Navier–Stokes regularity. A preliminary study of the case of directional h im was carried out by generating random numbers in some interval (H, 1) and tabulating results. A gradual increase of occurence of singularities was observed as H was lowered, but no sharp threshold was noticed among hundreds of samples. No viscous singularities were noted for H > 0. The issues of cancellation under summation and of kinematic constraints on possible distribution of h im remain open.

7.5 DISCUSSION Continuous wavelet transforms are well suited to the study of the regularity of Euler and Navier– Stokes solutions. Continuous wavelets have the ability to characterize Besov spaces and map the H¨older exponent in the modulus of continuity into a small scale exponent [26] of the wavelet © 2006 by Taylor & Francis Group, LLC

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coefficients. In particular, the Mexican hat and other Hermitian wavelets offer a formal representation of the dynamics that combines the advantages of the vorticity and of the velocity-Fourier representations. In addition, they simplify the analytical treatment of the pressure and viscous terms and admit an inverse transform related to the Biot–Savart formula [27], so the exact Navier–Stokes equations can be examined. In this chapter, an anisotropic modulus of continuity is proposed, leading to directional estimates of the small-scale exponents of the wavelet coefficients. Exact estimates for the scaling of wavelet components near isolated singularities and for the filtered stresses have been carried out. All indications are that viscous terms would ensure regularity of Navier–Stokes solutions, unless the dynamics, with kinematic constraints, are capable of generating negative hs; this is an obvious question currently under study. For inviscid flows, the possibility of runaway singularities is much easier to fulfill, and the h = 1/d threshold, when d = 3, corresponds to similar values obtained for Euler dissipation. This provides some vindication for the approach presented in this chapter. Besides regularity of solutions and inviscid dissipation, other topics would be affected by the replacement of truncated series with H¨older-type estimates. They include Kelvin’s theorem, Stokes’ and Gauss’ and Green’s theorems, Cauchy’s relation for internal forces in continua as the divergence of the stress tensor, and the concept of material derivative. The possible breakdown of these familiar relations at singularities generated by three-dimensional turbulence presents many challenges. Untouched here is the case of nonisolated singularities, in particular singularities distributed on a fractal set. Here, the approach of Frisch and Matsumoto [22] may be applicable to identify the local scaling exponents of wavelet coefficients in a probabilistic way. Continuous wavelet transforms seem well suited to study some of these problems.

NOMENCLATURE Ca Cs F(u i ) Fs h hi j M n p Re
s ui uˆ i u˜ i i 
∂ i jk i    i n <>

proportionality constant proportionality constant Fourier transform of u i Gaussian filter of scale s H¨older exponent anisotropic H¨older exponent Kummer-M hypergeometric function order of structure function static pressure large-eddy Reynolds number scale, with dimensions L 2 component of velocity Fourier transform of u i Mexican hat wavelet transform of u i flexion, the Laplacian of u i gamma function spatial difference partial derivative permutation symbol component of the wavenumber vector kinematic viscosity fluid density Mexican hat wavelet component of vorticity anomalous scaling exponent ensemble averaging

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SUBSCRIPTS i, j, k, m

indices for Cartesian directions

SUPERSCRIPTS > <

low-pass filtering (scale implicit or explicit) high-pass filtering (scale implicit or explicit)

REFERENCES 1. J. Leray, Sur le mouvement d’un liquide visqueux emplissant l’espace, Acta. Math. 63, 193–248 (1934). 2. U. Frisch, Turbulence: the legacy of A.M. Kolmogorov, Cambridge University Press (1996). 3. V. Yakhot, Mean-field approximation and a small parameter in turbulence theory, Phys. Rev. E 63, 026307 1–11 (2001). 4. A. Staicu and W. van de Water, Small scale velocity jumps in shear turbulence, Phys. Rev. Lett. 90, 094501 1–4 (2003). 5. B.B. Mandelbrot, Intermittent turbulence in self-similar cascades: divergence of high moments and dimension of the carrier, J. Fluid Mech. 62, 331–358 (1974). 6. I. Procaccia, Fractal structures in turbulence, J. Stat. Phys. 36, 649–663 (1984). 7. K.R. Sreenivasan, Fractals and multifractals in turbulence, Annu. Rev. Fluid Mech. 23, 539–600 (1991). 8. O.P. Manley and Y.M. Treve, Minimum number of modes to approximate solutions to equations of hydrodynamics, Phys. Lett. 82A, 88–90 (1981). 9. W.D. Henshaw, H.O. Kreiss and L.G. Reyna, On the smallest scale for the incompressible Navier–Stokes equations, Theor. Comput. Fluid Dyn. 1, 65–95 (1989). 10. W.D. Henshaw, H.O. Kreiss and L.G. Reyna, Smallest scale estimates for the Navier–Stokes equations for incompressible fluids, Arch. Rat. Mech. Anal. 112, 21–44 (1990). 11. M.V. Bartucelli, C.R. Doering, J.D. Gibbon and S.J.A. Malham, Length scales in solutions of the Navier–Stokes equations, Nonlinearity 6, 549–568 (1993). 12. W.D. Henshaw, H.O. Kreiss and L.G. Reyna, Estimates of the local minimum scale for the incompressible Navier–Stokes equations, Num. Funct. Anal. Opt. 16, 315–344 (1995). 13. P. Constantin, C.R. Doering and E.S. Titi, Rigorous estimates for small scales in turbulent flows, J. Math. Phys. 37 6152–6256 (1996). 14. J.T. Beale, T. Kato and A. Majda, Remarks on the breakdown of smooth solutions for the 3-D Euler equations, Commun. Math. Phys. 94, 61–66 (1984). 15. C. Foias, O. Manley, R. Rosa and R. Temam, Navier–Stokes Equations and Turbulence, Cambridge University Press (2001). 16. G.L. Eyink, Besov spaces and the multifractal hypothesis, J. Stat. Phys. 78, 353–375 (1995). 17. P. Constantin, W.E and E.S. Titi, Onsager’s conjecture on the energy conservation for solutions of Euler’s equation, Commun. Math. Phys. 165, 207–209 (1994). 18. G.L. Eyink, Energy dissipation without viscosity in ideal hydrodynamics, I. Fourier analysis and local energy transfer, Physica D 78, 220–240 (1994). 19. J. Duchon and R. Robert, Inertial energy dissipation for weak solutions of the incompressible Euler and Navier–Stokes equations, Nonlinearity 13, 249–255 (2000). 20. G.L. Eyink, Dissipation in turbulent solutions of 2D Euler equations, Nonlinearity 14, 787–802 (2001). 21. I. Kukavica, A ladder inequality for the Navier–Stokes equation, Nonlinearity 13, 639–652 (2000). 22. U. Frisch and T. Matsumoto, On multifractality and fractional derivatives, J. Stat. Phys. 108, 1181–1202 (2002). 23. I. Daubechies, Ten Lectures on Wavelets, S.I.A.M. (1992).

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24. M. Farge, Wavelet transforms and their applications to turbulence, Annu. Rev. Fluid Mech. 24, 395–457 (1992). 25. E. Bacry, J.F. Muzy and A. Arneodo, Singularity spectrum of fractal signals from wavelet analysis: exact results, J. Stat. Phys. 70, 635–674 (1993). 26. V. Perrier and C. Basdevant, Besov norms in terms of the continuous wavelet transform. Application to structure functions, Math. Models Meth. Appl. Sci. 6, 649–664 (1996). 27. J. Lewalle, A filtering and wavelet formulation for incompressible turbulence, J. Turbulence 1, 004, 1–16 (2000). 28. W.D. McComb, The Physics of Fluid Turbulence, Oxford University Press (1990).

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of the Eddy 8 Generalization Viscosity Model — Application to a Temperature Spectrum F. Bataille, G. Brillant, and M.Yousuff Hussaini CONTENTS 8.1 8.2

Introduction .......................................................................................................................... 125 Eddy Viscosity Model ..........................................................................................................126 8.2.1 Case 1 ....................................................................................................................... 126 8.2.2 Case 2 ....................................................................................................................... 127 8.2.3 Case 3 ....................................................................................................................... 127 8.2.3.1 Case 3.1 .................................................................................................... 128 8.2.3.2 Case 3.2 .................................................................................................... 128 8.3 Application to a Temperature Spectrum............................................................................... 128 8.3.1 Determination of the Eddy Diffusivity..................................................................... 128 8.3.2 Determination of the Eddy Diffusivity for the Smagorinsky Model........................ 129 8.4 Conclusions .......................................................................................................................... 130 References...................................................................................................................................... 130

ABSTRACT Eddy viscosity models for turbulent flows are often based on the assumption of Kolmogorov energy spectrum. The present work provides a general framework to develop models for flow configurations where this hypothesis does not hold or where additional physical parameters such as compressibility and intermittency must be included. The general formalism permits determination of turbulent models not only for turbulent eddy viscosity but also for turbulent thermal diffusivity. In particular instances, it yields models that agree with those existing in the literature.

8.1 INTRODUCTION Large eddy simulation (LES) has become a prevalent tool for computing complex turbulent flows of practical interest [1]. In such a simulation, the large scales of turbulence are computed and the small scales (less than the grid size) that are not resolved are assumed to have a universal character amenable to modeling [2–4]. A key element of LES is therefore the subgrid-scale model (SGS), which parameterizes the effect of the small unresolved scales of turbulence on the resolved large scales. A popular SGS parameterization is the so-called eddy viscosity model and the studies of Kraichnan [5] and Leslie and Quarini [6] are among the early investigations on the subject. There have been attempts to generalize the classical eddy viscosity model to take into account some physical aspects [7]. The purpose of the present work is to generalize the Kolmogorov spectrum to include physical aspects beyond incompressibility and derive an expression for the eddy viscosity for the velocity field and an analogous eddy diffusion for the temperature field.

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In the first instance, we derive eddy viscosity models for different written of energy spectrum. Then, we make an analogy to establish eddy diffusivity models adapted to temperature spectrum and, finally, we determine the eddy diffusivity for the Smagorinsky model.

8.2 EDDY VISCOSITY MODEL The starting point in the derivation of a turbulent eddy viscosity model is to consider the energy spectral density. The simplest and most well-established energy spectral density is due to Kolmogorov [8]: E(K ) = Ck
2/3 K −5/3 ,

(8.1)

where E(K ) is the turbulent energy density, Ck the Kolmogorov constant, K the wave number, and
the energy flux rate. To obtain an expression for the eddy viscosity, Kraichnan [5] proposed an energy balance equation involving an energy flux rate
and the energy contained in the large scales, given by
Kc 2t K 2 E(K )d K . (8.2)
= 0

Combining these two relations yields the well-known result of Leslie and Quarini [6]: t =

2 − 32 1 −1 Ck E(K c ) 2 K c 2 , 3

(8.3)

where K c is the cut-off wave number. To generalize the expression for eddy viscosity, we introduce new functions in the energy spectrum and new exponents for K and
. These functions are designed to represent some physical aspects that we want to take into account, and they depend on the key physical parameters of the problem. The energy spectrum is then written as E(K ) = C
K E1

E2

N 

Sii ,

(8.4)

i=0

where C is a constant (which can be dimensional) and E 1 and E 2 are the exponents of the dissipation rate and of the wave number. The Si are functions representing the key parameters and i are the respective exponents. Three different cases can arise.

8.2.1 CASE 1 At first, we consider the case where the function Si is the identity (or i = 0) and only the coefficients of the energy and of the dissipation rate are modified to account for new physical phenomena. Consequently, the energy spectrum becomes: E(K ) = C
E1 K E2 . Using the energy balance equation (8.2), we obtain the turbulent viscosity,   E2 1 E2 + 3 − 1 −1 − E −3 t = C E1 E(K c ) E1 K c 1 . 2

(8.5)

(8.6)

As E 1 and E 2 tend to the limits 2/3 and −5/3 respectively, we recover the classical equation (8.3) for the eddy viscosity.

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8.2.2 CASE 2 The second case retains the function Si , but it is assumed to be independent of the wave number. The energy spectrum is then given by E(K ) = C
E1 K E2

N 

Sii ,

(8.7)

i=0

where the Si have no dependency with K . For convenience, we assume i = 2 and, as in the first case, we use the energy balance equation to obtain the energy spectrum, (8.8) E(K ) = C
E1 K E2 S11 S22 . After some manipulations, we find the eddy viscosity to be given by  t =

E2 + 3 2

 C

− E1

−1

1

E(K c ) E1

1



E

− E2 −3 − E1 1 S1 1

Kc



− E2

1

S2

.

(8.9)

We remark that the correct Kolmogorov limit is reached as E 1 tends to 2/3 and E 2 to −5/3 and when 1 and 2 tend to zero. This equation can easily be extended to include the variation of i from 0 to N , and it yields  t =

E2 + 3 2

 C

− E1

1

1

E(K c ) E1

−1

E

− E2 −3

Kc

1

N 



− Ei

Si

1

.

(8.10)

i=0

We now apply this general formulation to a compressible case. When the flow is compressible, new parameters such as density, sound speed, and the specific heat ratio of the fluid must be introduced [9]. This results in an energy spectrum of the form, −1

E(K ) = C
3−1 K − 3−1  3−1 c− 3−1 . 2

5−1

2

(8.11)

2 E 1 , E 2 , S1 , S2 , 1 , and 2 are given by E 1 = 3−1 , E 2 = −5+1 , S1 = , and S2 = c, 1 = 3−1 2 2 = − 3−1 . Applying Equation (6.9), we find that the eddy viscosity is

 t =

c3 K c


1   3−1

2 − 1 3 − 1



C − 2 E(K c ) 2 (K c )− 2 . 3

1

1

−1 , 3−1

(8.12)

This equation is consistent with that of Shivamoggi and Hussaini [7].

8.2.3 CASE 3 Here, we consider the most general case where the functions Si are dependent on K . We must assume a functional form for Si . It stands to reason that the most appropriate form is a power expansion. Consequently, we only consider one power expansion function (noting that all the functions Si can be included in it), which we denote by Fi (K): Fi (K ) =

N  i=0

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ai K i .

(8.13)

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8.2.3.1 Case 3.1

Before deriving the global equation for the eddy viscosity, we first consider a simpler form (which is often found in physics), e.g., Fi (K ) = ca K a , (8.14) where a and ca are some constants. The energy spectrum is then given by E(K ) = C
E1 K E2 ca K a , and the eddy viscosity by t =



E2 + 3 + a 2

 C

− E1

1

1

E(K c ) E1

−1

(8.15)

−

Kc

E2 + a E 1 −3

− E1

ca

1

.

(8.16)

If ca is equal to 1, this expression for t reduces to the first studied case. This equation can be used to illustrate the physical case of intermittency. When flows are subject to intermittency, the energy spectrum is (see references 10 and 11 for more details): E(K ) = Ck
3 K − 3 (K lo )− 2

5

3−D 3

,

(8.17)

where D is the fractal dimension. Using Equation (8.16), we obtain the following eddy viscosity:   1+ D 1 −3 1− D 3−D (8.18) Ck 2 E(K c ) 2 K c 2 lo 2 . t = 6 This result agrees with that of Shivamoggi and Hussaini [7]. 8.2.3.2 Case 3.2

We now consider the most general case, where the function Fi is a power expansion. Then, we have for the energy spectrum, N  ai K i . (8.19) E(K ) = C
E1 K E2 i=0

Using the energy balance equation, after some simplifications, we obtain the eddy viscosity, 1 1 −1 −1 t = C E1 E(K c ) E1 2

 N 

− E1

1

ai K ci+E2

+1

 N 

i=0

i=0

ai K i+E2 +3 i + 3 + E2 c

−1 .

(8.20)

This general eddy viscosity model contains as special cases all the models discussed earlier.

8.3 APPLICATION TO A TEMPERATURE SPECTRUM 8.3.1 DETERMINATION OF THE EDDY DIFFUSIVITY Considerations on the eddy viscosity model developed earlier can be adapted to study the eddy diffusivity behavior. We consider a temperature spectrum, which is defined in the convective inertial range [3] by 1 5 E T (K ) = Cco
T
− 3 K − 3 (8.21) where Cco is the Corrsin–Oboukhov’s constant and
T is the scalar flux rate. We suppose again that the energy spectrum follows the classical Kolmogorov spectrum (8.1) and that the energy flux rate
is given by 3

−3

5


= E(K c ) 2 Ck 2 K c2 © 2006 by Taylor & Francis Group, LLC

(8.22)

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129

Consequently, the temperature spectrum can be written as E T (K ) = A
T K − 3 , 5

where

1

(8.23)

−5

A = Cco E(K c )− 2 Ck2 K c 6 . 1

(8.24)

Analogous to modeling the energy spectrum and derivation of eddy viscosity, we model the temperature spectrum and derive the corresponding eddy diffusivity t . The energy spectrum is replaced by the temperature spectrum, the energy flux rate is substituted by the scalar flux rate, and the Kolmogorov constant is replaced by A. Energy balance between the scalar energy flux rate and the thermal energy contained in the large scales of the temperature spectrum is present [3]. Consequently, using Equation (8.6) of case 1 along with E 1 = 1 and E 2 = − 53 , we obtain the eddy diffusivity, 2 −4 (8.25) t = A−1 (E T (K c ))0 K c 3 , 3 which can be reduced to 2 −1 1 −1 −1 t = Cco E(K c ) 2 Ck 2 K c 2 . (8.26) 3 From this expression, we can also deduce the value of the turbulent Prandtl number (for a Kolmogorov spectrum), which is given by the ratio of the eddy viscosity and eddy diffusivity, t Prt = = Cco Ck−1 . (8.27) t If the slopes of the dynamic and thermal spectra are different, then the turbulent Prandtl number may vary with the cut-off wave number or the associated energy.

8.3.2 DETERMINATION OF THE EDDY DIFFUSIVITY

FOR THE

SMAGORINSKY MODEL

Finally, because we would like to derive the eddy diffusivity model along the lines similar to the Smagorinski model, we briefly recall the well-known derivation. The Smagorinski model is based on the assumptions that (1) the subgrid fluctuations are isotropic and homogeneous; (2) there is a Kolmogorov inertial range (Equation 8.1); and (3) the mean dissipation range is t (8.28)
= Si j Si j = t S 2 , 2 where

2 Si j = vi, j + v j,i − vk,k
i, j . 3 Substitution of these assumptions in the expression for the energy flux rate,


= 2t K 2 E(K )d K ,

(8.29)

(8.30)

0

leads to the Smagorinski eddy viscosity, t = Cs2
2 S,

(8.31)

 34 where Cs is the Smagorinsky constant given by Cs = 1 3C2 k . An analogous treatment for temperature using Equations (8.21) and Equation (8.23) yields


T = 2t K 2 E T (K )d K . (8.32) 0

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Using the preceding equations, we obtain 2 −1 1   − 43 . (8.33) C
3 3 co
Consequently, the eddy diffusivity [12] is only a function of the eddy dissipation and of the cut-off wave number. This result is consistent with Galperin and Orszag [12]. The eddy diffusivity can also be written as t =

−1 C k t , t = Cco

(8.34)

which allows one to recover the result for the turbulent Prandtl number previously obtained: t Prt = = Cco Ck−1 . (8.35) t C2

Furthermore, if we use the equation for the eddy viscosity with the assumption C = Prst , we have t = C
2 S, (8.36) which corresponds to the usual form of the eddy diffusivity.

8.4 CONCLUSIONS We have provided a general formalism for determining the eddy viscosity and the eddy diffusivity that includes the current models as particular cases. This formal methodology is particularly useful in cases where a Kolmogorov spectrum does not exist or where the dynamic and thermal spectra behave differently. Furthermore, this methodology is the first step towards deriving new subgrid-scale models for large-eddy simulation of thermal flows. If the slopes of the dynamic and thermal spectra are different, then the turbulent Prandtl number may vary with the cut-off wave number or with the associated energy.

REFERENCES 1. Friedrich, R. and Rodi, W. Advances in LES of Complex Flows, Kluwer Academic Publishers, Dordrecht. 2. Hussaini, M.Y. 1998. On large-eddy simulation of compressible flows. In: AIAA 98-2802, 29th Fluid Dynamics Conference, Albuquerque, NM. 3. Lesieur, M. 1997. Turbulence in Fluids. Kluwer Academic Publisher, Dordrecht. 4. Sagaut, P. 1998. Large Eddy Simulation for Incompressible Flows. Springer–Verlag, Berlin. 5. Kraichnan, R.H. 1959. The structure of isotropic turbulence at very high Reynolds numbers. J. Fluid Mech., 5:497–543. 6. Leslie, Q. 1979. The application of turbulence theory to the formulation of subgrid modelling procedures. J. Fluid Mech., 91:65–91. 7. Shivamoggi, B.K. and Hussaini, M.Y. (2002). Kraichnan’s eddy-viscosity model: further extensions and generalizations, Phys. Letters, 301:315–319. 8. Kolmogorov, A.N. 1941. The local structure of turbulence in an incompressible viscous fluid for very large Reynolds numbers. Dokl. Akad. Nauk. USSR, 30:299–303. 9. Shivamoggi, B.K. 1995. Spatial intermittency in the classical two-dimensional and geostrophic turbulence. Ann. Phys., 243:169–177. 10. Mandelbrot, B. (1976). Lecture Notes in Math., chap. Turbulence and Navier–Stokes Equations, 565. Springer–Verlag, Berlin. 11. Frisch, U., Sulem, P.L. and Nelkin, S. (1978). A simple dynamical model of intermittent fully developed turbulence. J. Fluid Mech., 87:719–736. 12. Galperin, B. and Orszag, A.A. (1993). Large Eddy Simulation of Complex Engineering and Geophysical Flows. Cambridge University Press, Cambridge, U.K.

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Models for the 9 Continuous Simulation of Turbulent Flows: An Overview and Analysis M. Yousuff Hussaini, Siva Thangam, and Stephen L. Woodruff CONTENTS 9.1 Introduction .......................................................................................................................... 131 9.2 Development of Continuous RANS-LES Models — Possible Bases .................................. 134 9.3 DNS of Kolmogorov Flow ................................................................................................... 136 9.4 Continuous RANS-LES Model Development and Application ........................................... 139 9.5 Summary and Conclusions ................................................................................................... 141 Acknowledgments.......................................................................................................................... 141 References...................................................................................................................................... 142

ABSTRACT An overview of the development of continuous turbulence models that are suitable for representing the subgrid scale stresses in large eddy simulation and the Reynolds stresses in the Reynolds averaged Navier–Stokes formulation is described. A brief description of the filters commonly used in conjunction with the LES models and a continuous model capable of bridging the length scale disparity from the cut-off wavenumber to those in the energy-containing range based on recursion approach is provided. The proposed continuous model is analyzed in conjunction with results from direct numerical simulations.

9.1 INTRODUCTION Computational analysis of turbulent flows involves the development of an appropriate mathematical model for the physical problem followed by the discretization of the model and its numerical solution. Such modeling and simulation of turbulent flows play an important role in the design of various engineering devices and systems. Their representation within the framework of direct numerical simulation (DNS) is beyond the reach of current high-performance computing systems since these flow fields are known to exhibit a wide range of length and time scales. Practical applications rely heavily on computations based on Reynolds-averaged Navier–Stokes (RANS) equations as well as on large eddy simulation (LES) for certain practical cases of interest. During the past decade, there has been considerable progress in the development of efficient RANS models capable of capturing the complex features of the flow field, including those associated with the complexities due to surface geometry (Speziale, 1991; Yakhot et al., 1992; Gatski and Speziale, 1993; Zhou et al., 1994; Thangam et al., 1999). However, RANS and LES require turbulence models for closure and the development of the model as well as its numerical solution engender errors, which require careful estimation before embarking on the simulation. The inherent nonlinearities of properly formulated models and the physical and geometrical complexities of the problems of © 2006 by Taylor & Francis Group, LLC

131

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practical interest preclude the use of analytical methodology to fully evaluate the numerical schemes, other than through mesh and time-step refinements. In this context, most applications of the subgrid closures for LES have relied on the isotropic model proposed by Smagorinsky (1963) and its various extensions (see, for example, Bardina et al., 1983; Germano et al., 1991). Such subgrid models do not have the ability to respond to changes in the local state of the flow. In addition, LES schemes also require careful validation of the discrete filters and the commutativity of the derivative and averaging operators. In LES, the Navier–Stokes equations are first operated upon to filter out the chaotic, fluctuating, small-scale high-frequency motions. This is followed by the modeling of the effect of filtered components on the slowly varying large eddies, for which the filtered form of the
equations is applicable. ¯ i , t) = G(xi − xi
, t − t
, xi , t)u(xt
, t)d xi
dt, where G is Specifically, a filtered function, u(x the filter function (or weight with compact support such that it vanishes outside a finite domain and
within which it is finite) that satisfies G(xi − xi
, t − t
, xi , t)d xi
dt = 1. For effective numerical simulation, the spatial and temporal resolution are selected to be much smaller than the characteristic length scales of the large eddies in the flow field. This approach satisfies the first two Reynolds averaging conditions on commutation:
(u + v), =
(u¯ + v¯ ), ∂t u¯ = ∂t u¯ ∂x u = ∂x u¯ (where
is a ¯ = u¯ ¯ v is approximately satisfied. This choice of Reynolds constant), while the third condition, uv averaging ensures that the averaged equations of motion are less complex than the unaveraged set. In LES practice, filter functions without compact support are also utilized, but they are required to be consistent with the intrinsic properties of the Navier–Stokes equations on the following five elements (Speziale, 1991): • • • • •

Time invariance: the Navier–Stokes equations are invariant with respect to a translation of time. Galilean invariance: the Navier–Stokes equations are invariant with respect to a uniform translation in space of the frame of reference. Rotational invariance: the Navier–Stokes equations are invariant with respect to uniform rotation of the frame of reference. Scale invariance: the incompressible Navier–Stokes equations are invariant under the coordinate transformation. Realizability: the modeled stress tensor must be positive definite. A necessary condition is that G be non-negative.

The most common filters used in LES are the Gaussian filter and the top-hat filter defined in physical or Fourier space. In Fourier space, a top hat filter is also called a Fourier cut-off filter. Filters which account for anisotropy (Scotti et al., 1997) and reduce commutation errors (Vasilyev and Lund, 1997) have also been developed and implemented. The top-hat filter has several variants whose form is strongly dependent on the underlying grid structure. Filters which have been used to date in simulations of canonical or prototypical flows are briefly described next.   6 3/2  satisfies time invariance, Isotropic Gaussian filter: G(x,
) = 
exp −6 x•x 2
2 rotation invariance, Galilean invariance, and realizability, but it violates scale invariance. It is primarily used in simulations invariance.  2  of scale 2 sin(y/a) 2 sin(z/a) = 2 sin(x/a) • Fourier cut-off filter: G x, a , which is a top-hat filter x y z in Fourier space, and it satisfies time invariance, and Galilean invariance, but violates rotation and scale invariance and realizability constraint. This filter has been used in LES of incompressible homogeneous turbulent flows. When the Fourier modes that have a wave number greater than the cut-off wave number 2
/a are set to zero in its isotropic form, this filter satisfies the rotational invariance. • Top-hat filter in physical space: the velocity field is directly filtered and represented by
xi +
x 1 u¯ (x) = 2
xi −
x u(x)d x in physical space (through the Heavyside function or nonzero in the region where the quantity is averaged). A discrete representation by quadrature rule is •

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used to implement this filter, and its generalization to multidimension is straightforward. Such top-hat filters have been used in LES and are readily implemented on unstructured meshes. This filter satisfies time invariance and realizability constraint; however, its properties are functions of the mesh geometry and therefore violate Galilean invariance. They also violate rotational invariance except under the special circumstances when the mesh support is spherical.
t • Temporal filters: the Eulerian form of the filter is: u ¯ (x, t) = t− G(x, t − t
)u(x, t
)dt
, where  is the characteristic scale of the filter. In discrete form, the Eulerian filter is often representedusing an impulse-invariant, second-order, nonrecursive form, u¯ i =  2 2 j−1 p j u i− j + k−1 qk u i−k , ( p j , qk , are adjustable parameters that shape the filter).
t The corresponding Lagrangian form along particle paths is u¯ (x, t) = t− G(x, t − t
)u(x(t
), t
)dt
(Meneveau et al., 1996; Pruett, 1997). Since the LES or RANS involve averaging over small scales, a claim can be made that continuous models, which span the entire range from RANS to LES or DNS, are needed for analyzing problems of practical interest (for a review, see, Hussaini, 1998a,b; Speziale, 1998; Hussaini et al., 2002). The overview presented herein focuses on the methodology for developing a suitable continuous RANS–LES model. A renormalization group theory-based method is used to modify an existing RANS model for subgrid scale application so that the correct limiting behavior at the RANS limit and the DNS limit is maintained. To illustrate the model development, the Kolmogorov flow, which is characterized by the presence of sinusoidal shear profile (Figure 9.2), is used to perform benchmark DNS calculations. Such shear flows have been used extensively to analyze the characteristics of turbulent flows and for the testing of LES and RANS models (Borue and Orszag, 1996; Shebalin and Woodruff, 1997). The presence of an unstable mean shear flow, i.e., a nonzero time-averaged velocity gradient, is the essential source of turbulent fluctuations in any given fluid dynamical system. The mean shear is maintained by suitable external forces and, if the system contains a region of fully developed turbulent flow that exists long enough for study, then meaningful statistical statements can be made concerning it. The DNS computations form the basis for the development and analysis of a continuous RANS–LES model. A generalized two-equation RANS model (Speziale, 1987, 1991) that is effective for the representation of the anisotropic part of Reynolds stresses and satisfies the invariance and realizability constraint is utilized in the present study. The explicit k − l representation of this model has been successfully implemented for the prediction of flows in curved noncircular ducts by Hur et al. (1990) and will be used in the present analysis.

FIGURE 9.1 Schematic for the energy spectrum and recursive incorporation of scales in the LES model. © 2006 by Taylor & Francis Group, LLC

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FIGURE 9.2 Physical configuration of the Kolmogorov flow.

9.2 DEVELOPMENT OF CONTINUOUS RANS-LES MODELS — POSSIBLE BASES We write the Navier–Stokes equations in the form, (∂t − v∇ 2 )u + ∇ • N (u) = 0,

(9.1)

where N (u) includes the pressure and body-force terms. A filtering operation Fk (u) in space and time that commutes with differentiation is applied to (9.1). The parameter k is a wavenumber characterizing the magnitude of the filter cut off; k → ∞ corresponds to no filtering, or DNS, and k → 0 corresponds to filtering virtually everything, or RANS. The velocity field is decomposed into u¯ k = Fk (u) and u
k = u
− u¯ k ; then, the equation for the filtered velocity field is

where

(∂t − v∇ 2 )u¯ k + ∇ • N(u¯ k ) = ∇ • Rk ,

(9.2)

Rk = −Fk [N(u¯ k + u
k ) − N(u¯ k )]

(9.3)

represents the subgrid Reynolds stresses, the cross-terms, and the Leonard’s stresses. Through an iterative filtering scheme, which requires filtering (9.2) at k − dk and filtering the filtered equation with Fk−dk , we obtain (∂t − v∇ 2 )u¯ k−dk + ∇ • N(u¯ k−dk ) = −∇ • Fk−dk [N(u¯ k−dk + u˜¯ k−dk ) − N(u¯ k−dk )]

(9.4)

+∇ • Fk−dk [Rk−dk ]. Here, u˜¯ k = obtain

 d u¯ k  dk

∂k + O(∂k)2 ; by comparing the filtered form of (9.2) at k − dk with (9.4), we d Rk = −∇ • Fk−dk [N(u¯ k−dk + u˜¯ k−dk ) − N(u¯ k−dk )],

(9.5)

where, dRk is the gradient of the subgrid Reynolds stress. The model development starts with the following propositions. Proposition I: Rk may be represented for arbitrary k as a function of k times a RANS model M (which is, by definition, independent of k). That is, Rk = f (k)M.

(9.6)

Proposition II: the increment to Rk is a function of k, g(k) × Rk (g(k) is assumed to be positive). That is, dRk = −g(k)dk Rk . (9.7) © 2006 by Taylor & Francis Group, LLC

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From (9.6) and (9.7), we obtain f
+ gf = 0 or

  f = exp −



k

g(s)ds

(9.8)

  = exp −

∞

g(s)




0

 ds , s2

(9.9)

where
= 1/k can be considered to be a filter width and the constant of integration in (9.9) has been chosen so that f → 1 as k → 0, the RANS case. In order to meet the DNS limit, f → 0 as k → ∞ or
→ 0, both integrals in (9.9) must diverge; i.e., 

k

 g(s)ds → ∞

∞

and

g(s)


0

ds → ∞. s2

(9.10)

Consequently, g(k) must decay no faster than k −1 as k → ∞, and g(
) must approach zero no faster than
as
→ 0. It is also anticipated that dRk /dk → 0 in the DNS limit (k → 0, just as Rk does. This condition implies that f → 0 in this limit, and thus from (9.8), we have the additional condition on g(k) that it be less singular than I / f (k). Other than these conditions, the preceding analysis gives no indication of what the function g(k) actually is. However, the following special forms are of relevance. •

Consider g(k) is to be a constant, g(k) = . This corresponds to (based on Proposition II) the assumption that the increment in Reynolds stress as k varies is the same regardless of k. That is, f (k) = exp(−k) or f (
) = exp(−/
). (9.11)

•

In fully turbulent flow, the variation of k is in the inertial range; based on dimensional consistency, dRk  = g(k)dk ⇒ g(k) ∼ 1/k = n/k, (9.12) Rk  where n is a positive dimensionless numerical factor. Since substitution of (9.12) in (9.9) leads to divergence at the low wavenumber limit (a violation of the DNS limit), g(k) has to be regularized (and the following is suggested): g(k) = n/(k +
−1 ) where
−1 is assumed to be small enough that it is negligible in the inertial range. Using this in (9.8), we obtain f (k) = (1 +
k)−n

or

f (
) = (1 + 2
/
)−n

(9.13)

Additional requirements involving dimensional considerations of (9.4) and constraints on M yield n = 2, which is consistent with the form of eddy viscosity proposed by Smagorinsky (1963). • An alternate approach involves the imposition of
2 dependence for small
based on (9.8) and (9.9). That is, g(
) = 2
3 ; as this leads to divergence at the low wavenumber limit, regularization based on (9.9) is used to obtain f (
) = 1 − exp(−
2 ).

(9.14)

This representation of f (
) given by (9.14) is selected for use in the present analysis. In this context, the results for f (
) from (9.13) and (9.14) may be compared with © 2006 by Taylor & Francis Group, LLC

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those proposed by Speziale (1998a,b) on a phenomenological basis, wherein f (
) = [1 − exp(−
/)]n

(9.15)

Here,  and n are constants,
= 2kc−1 is the LES mesh size and  = 2kd−1 is the Kolmogorov length scale. An empirical approach for determining f (
) is described in Woodruff et al. (2000) and Hussaini et al. (2002). In their approach, LES for the benchmark Kolmogorov flow is performed using the Smagorinsky model and the variations in Smagorinsky constant are obtained for a range of
. The resulting empirical expression for f (
) is similar to that shown in (9.13). It should be noted that the subgrid scale model for LES is only needed if the cutoff wavenumber kc < kd . For cutoff wavenumber kc , the subgrid model can be obtained by several methods and the resulting continuous RANS–LES model for eddy viscosity may be written as (with sub representing the subgrid scale eddy viscosity):  = H (kd − k)[sub + (0|k
, kc )],

(9.16)

where H (x) = 1 if x ≥ 0 and H (x) = 0 if x < 0. The RANS eddy-viscosity representation can be obtained from any of the effective anisotropic models (see, for example, Speziale, 1991; Yakhot et al., 1992; Zhou et al., 1994; Thangam et al., 1999). It should also be noted that the procedure outlined previously automatically satisfies the following limits. • • •

DNS: k ≥ kd so that H (x) = 0 (i.e., no model is required) LES: k → kc < kd , and this yields  → sub , since [0|(k, kc )] ∼ B(k −4/3 − k −4/3 ) → 0 RANS: k → k
≈ 0 or k
−4/3 k −4/3 and Bk
−4/3 ysub ; as a result,  → 1/3 k
−4/3

For the purpose of implementation, the expression for eddy viscosity (9.16) could be represented using the form shown in (9.14) and (9.15) f (
) = 1 − exp(−
2 ) Here, the coefficient  must be determined consistent with the choice of representation for turbulent eddy viscosity, vT ,RANS . The corresponding representation of the turbulence stress field is: LES = f (
)RANS . ij ij

(9.17)

In the following, we describe the DNS of a benchmark shear flow followed by comparison to simulations based on the continuous model of the type shown in (9.17). From the point of implementation, a RANS model that can effectively represent the anisotropy of Reynolds stresses and one that meets the invariance and realizability constraints can be utilized (see, for example, Speziale, 1987, 1991).

9.3 DNS OF KOLMOGOROV FLOW The physical configuration for the Kolmogorov flow is shown in Figure 9.2. The governing equations of motion are expressed in the vorticity form, ∂t ¯ i = εi jk εklm ∂ j (u¯ l ¯ m ) + Re−1 ∂ j ∂ j ¯ i + εi jk ∂ j f k .

(9.18)

Here, the total fluid velocity is u¯ l and the vorticity, ¯ m = εmpq ∂ p u¯ q ; Re is the flow Reynolds number, and f t is the external forcing applied to the fluid. For the incompressible fluid motion analyzed here, ∂i u¯ i = 0. Since the Kolmogorov flow is driven by an imposed velocity field, u 0 (with © 2006 by Taylor & Francis Group, LLC

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associated vorticity of 0i = εi jk ∂ j u 0k ), the total velocity and the fluid vorticity may be expressed as u¯¯i = u 0i + u i and ¯ i = 0i + (where, ∂i u i = 0 and i = εi jk (∂ j u k )). The physical space for the flow field is a cube with periodic boundary conditions (Figure 9.2). For the Kolmogorov flow, u 0i = i1 u 0m sin x3 and the corresponding vorticity, 0i = i2 u 0m cos x3 . Using these in (9.18), the resulting equation of motion for Kolmogorov flow has the following form: ∂t i = εi jk εklm ∂ j (u l + u 0l )( m + 0m ) + Re−1 ∂ j ∂ j i .

(9.19)

to be consistent with (9.19), f i = i1 Re u 0m sin x3 . DNS are performed using a spectral scheme that has been effectively applied for DNS and LES of Kolmogorov flows (Shebalin and Woodruff, 1997; Woodruff et al., 2000). The equations of motion (9.19) are transformed to a Fourier space and the resulting equations in Fourier space are solved numerically by using a pseudospectral method with 128 modes. For the purpose of time integration, the dissipative term is treated implicitly using a backward Euler scheme, and a third-order partially corrected Adam–Bashforth method is used for the nonlinear terms (Shebalin and Woodruff, 1997). Computations were performed in a 2.25-tera-flop √ parallel high-performance supercomputer for a turbulence microscale Reynolds number, R = 2E/k avg ≈ 100 (where the rms wavenumber,  √ kavg = /E, with  = <w2 >/2, is the enstrophy and E = <2 > is the energy); this corresponds to R ≈ 50, when the speed is based on the turbulent kinetic energy and the length scale is based on isotropic dissipation, ε). The forcing wavenumber for the flow field, k f = 1 and the initial state of the flow is set to a random vorticity field of low amplitude. The computations are started by advancing with a very small time step (10−4 ) until the transients fade out. Subsequent time steps are slowly increased until a predetermined and viable computational time step value (0.005 to 0.01) is reached, after which computations are performed with the fixed time step. Although computations for Kolmogorov flow at moderate values of R ≈ 30 have been reported by Shebalin and Woodruff (1997) using 643 resolution, the results presented here are the first computations at R ≈ 100 to fully resolve the turbulence scales. In the present study, computations are first performed using a 643 resolution and, after the computations have proceeded to a statistically steady state, the vorticity field was used as the initial condition for the 1283 simulation. Computational results were checked to ensure the viability of the numerical scheme to capture the physical features of the flow field. The time evolution of the kinetic energy and the dissipation for the 1283 DNS computations were monitored to ensure that the computations proceeded sufficiently long to reach a statistically steady state. In addition, the time-averaged energy spectra for the 643 and 1283 simulations were analyzed to ensure that they were essentially exponential and the resolution employed is adequate to capture the spectral variation. The trends observed for the 643 calculation were also consistent with those reported by Shebalin and Woodruff (1997) for R ≈ 30; however, the amplitude of energy is considerably higher due to the higher value of R (≈ 100) used in the present computations. Spatial variation of the velocity field, and the turbulence shear stresses are shown first in Figure 9.3 and Figure 9.4 for DNS computations using 1283 resolution. For the purpose of this analysis, the flow quantities are averaged over the plane and made dimensionless with U0m . Since the spatial variation along the horizontal (x − y) plane is relatively small, only the variations in the vertical (z) direction are shown. The averaged dimensionless velocity profiles are first shown in Figure 9.3. As can be seen, the u-component of the velocity retains essentially the characteristics of the input or the forcing field applied to the domain. In Figure 9.4, the variation of plane-averaged components of the turbulent shear stress along the vertical direction is shown. As anticipated, the zx component of the shear stress is the dominant one and its spatial variation exhibits a trend similar to that of the first derivative of the forcing field. The mean flow profile (Figure 9.2) is inhomogeneous in the vertical direction and thus has the potential to produce significant anisotropy in the turbulence stresses. One of the major weaknesses of standard RANS models and the commonly used LES models is their isotropic characterization of © 2006 by Taylor & Francis Group, LLC

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FIGURE 9.3 Dimensionless plane-averaged velocity profiles in the vertical direction, z/L with 1283 resolution.

the turbulence closure. Thus, the Kolmogorov anisotropy of the spectrum, A
 (k), (
,   {x, y, z}): A
 (k) = where S
 (k) =

 |ki |=k

S
 (k) , Sx x (k) + S yy (k) + Szz (k)

(9.20)

u
(ki )u  (ki ), was also computed and examined.

The Kolmogorov anisotropy (9.20) is a measure of the spectral distribution of turbulent kinetic energy among the different components of the velocity field, since it characterizes the behavior of spectral transfer between the velocity components as a function of increasing wavenumber. The DNS results indicate that this anisotropy is quite significant at low wave numbers (or in the energycontaining range). As the wavenumber increases, the distribution of anisotropy is confined to within

FIGURE 9.4 Plane-averaged profiles of turbulent shear stress components in the vertical direction, z/L with 1283 resolution. © 2006 by Taylor & Francis Group, LLC

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about 10% of the mean value for all modes. This is somewhat different from the trends observed by Shebalin and Woodruff (1997), most likely due√to the significant increase in the turbulence Reynolds number, R for the present computations (i.e., R ≈ 10 to allow for the development of turbulence structure and scales). The spatial distribution of anisotropy is a useful measure for analyzing the physical features of the flow. It also provides a means for comparison of the DNS results with experiments and other simulations. In this context, the anisotropy in Reynolds stress may be expressed in the following form: 2 bi j = i j − K i j . (9.21) 3 The diagonal components of (9.21), which provide a measure of the normal stress anisotropy, are obtained and its plane-averaged representation is expressed in the following dimensionless form: z 2 − ; Kz 3

B
(z) =


 {x, y, z}.

(9.22)

Similarly, the cross-correlation can be used to obtain a measure of the anisotropy in the flow field. We define the cross-correlation, Q
 (z) as z

Q
 (z) = 

z

;
  {x y, yz, zx}.

(9.23)

Here, < . . . >z indicates an average over the xy–plane in physical space corresponding to each value of z; and k z is the kinetic energy in a given z-plane. The components of B
(z) and Q
 (z) were also evaluated as part of this study. There are significant levels of anisotropy of the order of about 10% that are well distributed over the entire height of the fluid layer and were found to compare favorably with those from 643 DNS computations of Shebalin and Woodruff (1997). It should be noted that the microscale Reynolds number R ≈ 100 for the present study (compared to R ≈ 30 used by Shebalin and Woodruff, 1997). In addition, no attempt has been made to force symmetry or smoothing based on long-term averaging, as in the work of Shebalin and Woodruff (1997).

9.4 CONTINUOUS RANS-LES MODEL DEVELOPMENT AND APPLICATION Conventional LES computations rely on subgrid models based on the isotropic model proposed by Smagorinsky (1963) or its extensions (see, for example, Bardina et al., 1983; Germano et al., 1991). While the current methodology provides a means for incorporating efficient RANS models within the framework of the subgrid models for LES, the following points should be noted. Efficient RANS models that characterize the effect of anisotropy of normal stresses, curvature, system rotation, and swirl tend to be more complex in structure and would require the solution of evolution equations for the turbulent kinetic energy and dissipation during each time step of the LES realization. The use of higher order closure schemes will further increase the number of transport equations to be solved. This should be compared with the much simpler algebraic functional calculations needed for isotropic models that are currently in use. In the present study, we consider an efficient anisotropic model that has been shown to be effective for application to complex flows in its explicit form (Speziale, 1991). Computations are performed on a 323 grid using the continuous RANS–LES model outlined in (9.14) with the anisotropic RANS model proposed by Speziale (1987, 1991). The methodology outlined in Hur et al. (1990) is used to represent the RANS model in its explicit form and computations are performed until a statistically steady state is obtained. The results are then compared with those from the DNS calculations to illustrate the efficacy of the model. © 2006 by Taylor & Francis Group, LLC

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FIGURE 9.5 Plane-averaged profiles of turbulent shear stress in the vertical direction, z/L for 323 LES (O) and 1283 DNS (—) calculations.

LES results shown here are performed with the continuous RANS–LES model (9.14) represented as: f (
) = 1 − exp(−
2 ), wherein the coefficient,  is set as C /L 2 to satisfy the isotropic limit and for dimensional consistency. Plane-averaged values of the turbulent shear stress, zx based on 1283 DNS computations are shown in Figure 9.5 along with the LES results based on an anisotropic subgrid model. As can be seen, the 1283 DNS computations are well predicted by the 323 LES calculations. The turbulent kinetic energy K is shown next in Figure 9.6. As can be seen, the overall trend is essentially the same two periods instead of one for the forcing field in the vertical direction. The efficacy of the continuous model can also be clearly seen.

FIGURE 9.6 Plane-averaged profiles of turbulent kinetic energy in the vertical direction, z/L for 323 LES (O) and 1283 DNS (—) calculations. © 2006 by Taylor & Francis Group, LLC

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9.5 SUMMARY AND CONCLUSIONS A methodology for the development and application of continuous RANS–LES models is described. DNS computations based on an efficient spectral algorithm are performed for the benchmark Kolmogorov flow using 1283 resolution and compared with those based on 323 LES resolution. The subgrid model for the LES computations is based on an efficient anisotropic RANS model. The predictions clearly illustrate the potential for efficient subgrid scale representation in LES and its benefits for flows characterized by significant levels of anisotropy. Future work will include applications involving more benchmark test cases; the validation of the methodology with other subgrid models is planned.

ACKNOWLEDGMENTS This research was supported in part by NASA grant HUSS-NAG-1-1889 from NASA Langley Research Center. The supercomputer time for this work was granted by ACNS and CSIT, Florida State University. ST acknowledges the grant of sabbatical leave by Stevens Institute of Technology and the hospitality and support of Prof. M.Y. Hussaini, CSIT/FSU. SLW acknowledges the support from the Office of Naval Research through grant number N00014-02-1-0623.

NOMENCLATURE A bi j B C E H k K l L M N P Qi j R Re R S, Si j t Ui , u i xi
ε  

, i j

Kolmogorov anisotropy parameter Reynolds stress anisotropy tensor plane-averaged normal component of bi j various coefficients used for the models total energy Heavyside function wavenumber turbulent kinetic energy turbulence length scale height of the fluid layer RANS model terms (tensor) nonlinear terms (tensor) pressure cross-correlation of velocity field subgrid stresses, cross terms, and Leonard stresses (tensor) Flow Reynolds number turbulent microstate Reynolds number strain rate and strain rate tensor time men and fluctuating velocity fields coordinate vector computational mesh size turbulent dissipation Kolmogorov microscale fluid kinematic viscosity fluid density mean and fluctuating vorticity Reynolds stress tensor

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SUBSCRIPTS AND SUPERSCRIPTS 0 c d i, j, k K t x, y, z
s,  ε

initial or mean states cut-off dissipation indices kinetic energy time coordinate directions indices dissipation

REFERENCES Bardina, J., Ferziger, J.H., and Reynolds, W.C., 1983, “Improved turbulence models based on large eddy simulations homogeneous, incompressible turbulent flows,” Stanford University Technical Report TF-19. Borue, V. and Orszag, S.A., 1996, “Numerical study of three-dimensional Kolmogorov flow at high Reynolds numbers,” J. Fluid Mech., 306, 293–314. Gatski, T.B. and Speziale, C.G., 1993, “On explicit algebraic stress models for complex turbulent flows,” J. Fluid Mech., 254, 59–78. Germano, M., Piomelli, U., Moin, P., and Cabot, W.H., 1991, “A dynamic subgrid-scale eddy viscosity model,” Phys. Fluids, 3, 1760–1765. Hur, N., Thangam, S., and Speziale, C.G., 1990, “Numerical study of turbulent secondary flows in curved ducts,” J. Fluids Eng., 112, 205–211. Hussaini, M.Y., 1998, “On large eddy simulation of compressible flows,” AIAA Paper 98-2802, 29th Fluid Dynamics Conference, Albuquerque, NM. Hussaini, M.Y., Speziale, C.G., and Woodruff, S.L., 2002, “Continuous models: variants of LES,” 14th U.S. Natl. Congr. Theor. App. Mech., Blacksburg, VA. Kraichnan, R.H., 1976, “Eddy viscosity in two and three dimensions,” J. Atmos. Sci., 33, 1521–1536. Kraichnan, R.H., 1987a, “Kolmogorov’s constant and local interactions,” Phys. Fluids, 30, 1583–1585. Kraichnan, R.H., 1987b, “An interpretation of the Yakhot–Orszag turbulence theory,” Phys. Fluids, 30, 2400– 2405. Meneveau, C., Lund, C.S., and Cabot, W., 1996, “A Lagrangian dynamic sub-grid scale model for turbulence,” J. Fluid Mech., 319, 353–365. Pruett, C.D., 1997, “Time-domain filtering for spatial large eddy simulation,” FEDSM97-3117, ASME. Scotti, A., Meneveau, C., and Fatica, M., 1997, “Dynamic Smagorinsky model on anisotropic grids,” Phys. Fluids, 9, 1856–1858. Shebalin, J.V. and Woodruff, S.L., 1997, “Kolmogorov flow in three dimensions,” Phys. Fluids, 9, 164–170. Smagorinsky, J., 1963, “General circulation experiments with the primitive equations, part I: The basic experiment,” Mon. Weather Rev., 91, 99–164. Smith, L.M. and Woodruff, S.L., 1998, “Renormalization group analysis of turbulence,” Annu. Rev. Fluid Mech., 30, 275–310. Speziale, C.G., 1991, “Analytical methods for the development of Reynolds stress closures in turbulence,” Annu. Rev. Fluid Mech., 23, 107–157. Speziale, C.G. and Durbin, P.A., 1991, “Local anisotropy in strained turbulence at high Reynolds numbers,” J. Fluids Eng., 113, 707–714. Speziale, C.G., 1998a, “Turbulence modeling for time-dependent RANS and VLES: A review,” AIAA J., 36, 173–184. Speziale, C.G., 1998b, “A combined large-eddy simulation and time-dependent RANS capability for high-speed compressible flows,” J. Sci. Comp., 13, 253–274. Thangam, S., Wang, X., and Zhou, Y., 1999, “Development of a turbulence model based on the energy spectrum for flows involving rotation,” Phys. Fluids, 11, 2225–2234. © 2006 by Taylor & Francis Group, LLC

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Vasilyev, O. and Lund, T.S., 1997, “A general theory for discrete filtering for LES on complex geometry,” Center for Turbulence Research Briefs, Stanford University, 67–82. Woodruff, S.L., Seiner, J.M., and Hussaini, M.Y., 2000, “Grid-size dependence considerations for subgrid-scale models for LES of Kolmogorov flows,” AIAA J., 38, 600–604. Yakhot, V., Orszag, S.A., Thangam, S., Gatski, T.B., and Speziale, C.G., 1992, “Development of turbulence models for shear flows by a double-expansion technique,” Phys. Fluids A, 4, 1510–1524. Zhou, Y., Vahala, G., and Thangam, S., 1994, “Development of a turbulence model based on recursion renormalization group theory,” Phys. Rev. E, 49, 5195–5206.

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Uses of Wavelets 10 Analytical for Navier–Stokes Turbulence Jacques Lewalle CONTENTS 10.1 10.2 10.3 10.4

Background........................................................................................................................... 145 Eliminating Pressure............................................................................................................. 147 Filtered Flexion and Wavelet Transforms............................................................................. 148 Applications.......................................................................................................................... 150 10.4.1 Emergence of Structures and Complex Systems Dynamics .................................... 150 10.4.2 Regularity of Euler and NS Solutions...................................................................... 151 10.4.3 Rapid Distortion Theory .......................................................................................... 152 10.4.4 Renormalization Approaches ................................................................................... 152 10.4.5 Structure Functions .................................................................................................. 152 10.5 Conclusion............................................................................................................................ 152 References...................................................................................................................................... 153

ABSTRACT Traditional formulations of Navier–Stokes turbulence do not provide a good match for many physically important concepts: multiscale dynamics, nonlocal effects, spectral transfer and spatial transport, similarity and scaling, intermittency, mixing, regularity, emergence of structures, and others. Defining flexion as the Laplacian of velocity (negative curl of vorticity for incompressible flow), it is shown that filtered flexion is related to most items on this list. The flexion equation is equivalent to the familiar Fourier version of the momentum equation. Gaussian filtering yields the evolution equation for the Mexican hat wavelet transform of velocity at the corresponding scale. The inverse wavelet transform is an alternative form of the Biot–Savart relation, and simple diffusion maps into a uniform translation of the wavelet coefficients toward smaller scale. Work in progress is outlined, including the formulation of nonlinear terms in the framework of complex systems dynamics and emergence of structures, Navier–Stokes regularity (see companion abstract), rapid-distortion theory, and recursive filtering and renormalization.

10.1 BACKGROUND Why is fluid turbulence still an unsolved problem? Commonly mentioned reasons are the nonlinearities, the multiscale dynamics, the dissipative physics, and others. This chapter is based on the premise that the core difficulty is actually in the mismatch between the available mathematical formulations and the physical syndrome: the most successful techniques capture only a few of the important qualitative physical features of turbulence. The ability to calculate the remaining properties, once a solution is known, does not constitute a theory. A better alternative will be proposed later. Conservation of mass and momentum are embedded in the incompressible Navier–Stokes equations (NS) governing the velocity field. In this chapter, we will not consider models loosely related to NS dynamics, such as Burgers’ equation or GOY cascade models, in spite of their interesting © 2006 by Taylor & Francis Group, LLC

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features and extensive literature. Within the NS framework, vorticity is easily displayed, as are energy and dissipation. Pressure is not as simple as an elementary view suggests because of its nonlocal character. The same nonlocality is echoed in the vorticity formulation, which eliminates pressure but includes the same Green’s function in the Biot–Savart kernel. Turbulent transport is readily identified in divergence terms, but spectral transfer and multiscale dynamics are well concealed. Since Taylor’s seminal paper [1], anything spectral has routinely been treated in Fourier space, with considerable success [2]. The corresponding equations have brought us closer to a theory of turbulence than any alternative. The multiscale dynamics are captured in the convolution integrals. Energy is a natural concept, thanks to the Parseval theorem. However the shortcomings are also known: integration over all space amounts to averaging; transport is now concealed, even if no information has been lost in the transform; spectral transfer, buried in convolution integrals, is averaged in space and is not obviously related to vortex stretching or to the dynamics of other flow structures. Regardless of the formulation, a number of important concepts do not even appear in the equations. Even elementary texts mention eddies, but what are they in a mathematical sense? How do they emerge from the dynamics? Intermittence is equally mysterious, a diagnostic on solutions rather than an ingredient in the formulation. The same goes for similarity and scaling, and for regularity. A nonexhaustive but representative list shows the strengths and shortcomings of mainstream options. •

•

• •

•

•

•

Direct numerical simulation (DNS) can be viewed as a computational substitute for experiments. Limited in the accessible range of Reynolds numbers, DNS is extremely valuable in providing information to test theories and models, but it is not a theory. Much closer to a theory is the large body of literature based on the Fourier-transformed NS equations (FNS). In addition to shell models and computational models such as EDQNM, the direct interaction approximation (DIA) and various renormalization approaches have relied on the change from partial derivatives to algebraic operations in wavenumber space. The main shortcoming of FNS is the spatial delocalization (averaging), which obscures the effects of intermittence. The rapid distortion theory (RDT) [3,4] is formulated on the basis of FNS and accounts partially for the presence of some flow structures. Vortices are the “muscles and sinews of fluid flows” [5], and come to the forefront by applying the curl to the NS equations. The vorticity equation (VNS) in conjunction with the Biot–Savart equation has been very successful in two-dimensional problems, but threedimensional turbulence remains a formidable challenge in this representation [6]. Also capable of accounting for some flow structure is the proper orthogonal decomposition (POD) [7]. The decomposition of the flow field with optimal L 2 convergence works best if few modes are active and is essentially statistical rather than local. Scaling theories have their roots in Kolmogorov’s introduction of structure functions (K41) [8]. Also statistical at its core, this approach to modern field theory has recently been enriched by direct links to NS dynamics [9–11]. Large eddy simulation (LES) is a modeling approach with potential for theoretical insights. Based on filtering of the NS equations, the coarse-grained vs. fine-grained description echoes the recursive filtering of renormalization and of mathematical treatment of singularities.

In this chapter, Hermitian1 wavelet transforms [12,13] are shown to offer a viable alternative. Although progress in any one line of study has been modest so far, the relevance of the approach to

1 Hermitian

wavelets are derivatives of Gauss’ function, i.e., products of Hermite polynomials and the bell-shaped kernel. The best known member of the family is the Mexican hat wavelet used in this chapter.

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most concepts in the turbulence syndrome looks very promising. The discussion will expand on this point.

10.2 ELIMINATING PRESSURE In incompressible flow, mass balance takes the form ∂i u i = 0,

(10.1)

and the Navier–Stokes equations are ∂t u i −
∂ 2j j u i = −∂i


p 

− ∂ j (u i u j ).

(10.2)

Vorticity, defined as the curl of velocity i = i jk ∂ j u k ,

(10.3)

is made visible by rewriting the nonlinear and viscous terms, yielding   p u2 + − i jk  j u k ∂t u i +
i jk ∂ j k = −∂i  2

(10.4)

where the Bernoulli term includes pressure and kinetic energy in the absence of body forces. In these forms, it appears that pressure is a local property. However, taking the divergence of Equation (10.2) yields the Poisson equation for pressure
p
p 2 = ∇2 = −∂ 2jk (u k u j ). (10.5) ∂kk   Solving for p, the inverse Laplacian is a nonlocal operator, which can be expressed in terms of Green’s functions: in three dimensions (3D), we have p=

1 4



  ∂ 2 jk (u k u j )

| x − x |

dx

(10.6)

where the integral is over all space. This shows that the pressure term collects information from the entire flow and distributes it among velocity components at any given location. Then, ∂t u i −
∂ 2j j u i = ∂i ∇ −2 ∂ 2jk (u j u k ) − ∂ j (u i u j ).

(10.7)

This picture including nonlocal effects is consistent with the familiar vorticity formulation (VNS). Then, the equations of motion become ∂t i −
∂ 2j j i = −∂ j (u j i ) + ∂ j (u i  j )

(10.8)

Here, the convective and stretching terms on the right hand side imply the reconstruction of velocity from vorticity, using the Biot–Savart equation  (∇  ×  )i  1 dx . (10.9) ui = − 4 | x − x | It can be seen that this is formally equivalent to the inverse Laplacian, since (∇ × )i = −∇ 2 u i in incompressible flows. Now, velocity (hence, kinetic energy) depends explicitly on the spatial distribution of vorticity. © 2006 by Taylor & Francis Group, LLC

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The three-dimensional Fourier transform of the velocity field is defined as  1 u i ei j x j d x F(u i ) = uˆ i = √ ( 2)3

(10.10)

The wavevector i pinpoints the spectral location, but spatial information is now less easily accessible. Applying the transform to Equation (10.7) gives the Fourier–Navier–Stokes equation (FNS)    i k ∂t uˆ i +
2 uˆ i = i j − ik F(u j u k ). (10.11) 2  The inverse Laplacian is recognized as −1/2 , and the transform of the nonlinear terms requires the familiar convolution  F(u j u k ) = uˆ j ( p)uˆ k ( − p) d p (10.12) One of the cornerstones of the renormalization approach is the fact that the diffusion terms on the left-hand side correspond to an exponential attenuation of uˆ i , and spatial derivatives become algebraic operations. As a fourth option following earlier work [14], we introduce flexion2 as the Laplacian of velocity 2 i = ∂kk ui .

(10.13)

For an incompressible velocity field, flexion is the negative curl of vorticity. Therefore, it vanishes identically in potential flows and in uniformly rotating flows. Taking the Laplacian of Equation (10.7) yields the corresponding Navier–Stokes equation for flexion (fNS) ∂t i −
∂ 2j j i = ∂i3jk (u j u k ) − ∂ 3jkk (u i u j ).

(10.14)

Just as for the vorticity equation, the velocity field reconstruction is implied; in terms of flexion, the Biot–Savart law can be rewritten as  1 i ui = (10.15) d x . 4 | x − x | It also can be seen that the Fourier transform of Equation (10.14) is identical (within a factor 2 ) to the preceding Fourier-velocity formulation. Thus, the flexion equation has the potential to combine the insights from the vorticity equation (including spatial localization of vortical structures) and from the Fourier version with its spectral description. With the benefit of several decades of study, the Fourier version appears superior; the algebraic simplicity is partially offset by the complexity of the convolution. However, the flexion equation is much richer than it appears to be at first sight.

10.3 FILTERED FLEXION AND WAVELET TRANSFORMS The interest of the flexion equation arises from its scale-resolved version. Let us introduce the d-D normalized Gaussian filter at scale s (note that s has dimensions L 2 ):  2 x 1 (10.16) F(x, s) = √ d/2 exp − (2 s) 4s 2 This term for the curl of vorticity was used by Truesdell [15], who credits a much earlier source. It was used in Yannacopoulos

et al. [16] in relation to chaos diagnostics in three-dimensional flows. It has also been called di-vorticity [17]. The Laplacian of velocity is needed here; as long as we deal with incompressible flows, flexion appears to be adequate terminology in spite of the sign reversal.

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and the filtered flexion: i>s = F(x, s) ∗ i =



i (x  )F(x − x  , s)d x  .

149

(10.17)

It should be noted that the scale s is not a fixed parameter as in LES, but an independent variable on the positive real line. It is easy to verify that 2 >s ∂kk i = ∂s i>s .

(10.18)

Hence, the general form of the filtered Equation (10.14) is (∂t −
∂s )i>s = ∂i jk (u j u k )>s − ∂ jkk (u i u j )>s = (∂i jk Fs ) ∗ (u j u k ) − (∂ jkk Fs ) ∗ (u j u i ).

(10.19)

This form of NS shows distinct features in the form and meaning of the viscous term, in the need to reconstruct the velocity field to evaluate the stress terms, and in the interplay of derivatives and filtering. These points are briefly discussed in the remainder of this section; for additional details, see Lewalle [14]. The viscous term brings the first surprise. In contrast to Fourier space, where the coefficients decay exponentially, the operator (∂t −
∂s ) reveals an invariant translation of the field. It seems counterintuitive at first that the translation is toward small scales, for which a detailed explanation has been presented [14]. Finally, the translation is at uniform “speed”3
. The wavelet formalism [12,13] provides another physical interpretation of filtered flexion and another formula for the reconstruction of the velocity field. Indeed, within an arbitrary scale factor, the isotropic Mexican hat wavelet (x, s) is given by 2 F(x, s) = ∂s F(x, s). (x, s) = ∂kk

(10.20)

Therefore, the filtered flexion field at scale s is identical to the Mexican hat wavelet-transformed velocity field at the same scale: i>s = (x, s) ∗ u i = u˜ i (x, s).

(10.21)

Hence, Equation (10.19) governs the evolution of the velocity wavelet coefficients. Furthermore, the velocity field can be reconstructed from flexion by using one of several forms [14] of the inverse wavelet transform, in particular  ∞ ds si>s . (10.22) ui = s 0 Reconstructing the velocity from its wavelet coefficients, i.e., ultimately from flexion, Equation (10.22) is another form of the Biot–Savart equation. It gives the velocity at one point in terms of filtered flexion at the same point; nonlocality is introduced with the filtering of flexion, extending farther and farther away as the scale increases. Finally, the spatial-spectral distribution of energy stems from the wavelet version of Parseval’s theorem. For Mexican hat wavelet coefficients, we have  ∞  ∞  ∞ ds 2 d x | ui | = 4 dx | si>s |2 s −∞ −∞ 0  ∞  ∞ ds =2 dx e(x, s). (10.23) s −∞ 0 Here, e is the kinetic energy density (per unit scale and unit volume), and its integral over all space is a smoothed version of the Fourier power spectrum. For a one-dimensional signal, the distribution of energy is shown in Figure 10.1. The lumping of the energy density into quasidiscrete objects in the (x − s)-space suggests that the property s u˜ i is 3 The

author is indebted to Dr. M. Farge for the analogy with a painter’s speed, since s has the dimensions of an area.

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1–D field

0.02 0.00

−0.02

45

50

55

60

50

55

60

1.0 Scale

April 24, 2006

0.1

45

Distance

FIGURE 10.1 Local spectral energy density contours for the signal shown in the upper box; units of space and scale match, but are otherwise arbitrary.

one possible analytical tool to capture eddies. In contrast to the lumpy s u˜ i , the wavelet coefficients u˜ i show a gradual relaxation (see Equation (10.18) from s = 0 toward larger scales and increasing the wavelet scale corresponds to coarse-graining of the flexion field. With this background, the equation for the Mexican hat wavelet coefficients is (∂t −
∂s )u˜ i = (∂i jk Fs ) ∗ (u j u k ) − (∂ jkk Fs ) ∗ (u j u i ).

(10.24)

Having been constructed through Equation (10.22), the stress terms u j u k enter the equations of motion in two equivalent presentations. If filtering is carried out first, the presence of (u j u k )> terms is reminiscent of LES modeling approaches, but one should bear in mind that the scale s is an independent variable. Alternatively, the unfiltered stress term is convolved with derivatives of the Gaussian filter, i.e., a known kernel of zero mean. We shall return to these features in specific contexts in the following section.

10.4 APPLICATIONS 10.4.1 EMERGENCE

OF

STRUCTURES AND COMPLEX SYSTEMS DYNAMICS

One of the contributions of field of complex systems dynamics is the understanding of how simple, small-scale interactions can lead to the emergence of large-scale patterns. Generally speaking, a turbulent flow would qualify as a complex (intermittent, multiscale) system, and the appearance of large eddies from a random background seems like a good example of the general concept of emergence. Badii and Politi [18] point out that “. . . the concept of complexity is closely related to that of understanding. . . .” In the turbulence problem, our lack of understanding is epitomized by the poor match between established descriptive concepts (e.g., eddies) and analytical formulations. In the Nicolis and Prigogine approach [19], chaotic dynamics and generic instabilities provide the background for complexity, but no theory of turbulence seems likely to be based on this approach. A complex system usually consists of many interacting parts, resulting in cooperative (rather than © 2006 by Taylor & Francis Group, LLC

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merely collective) behavior (Bar–Yam [20]). Also significant is the difference between fine- and coarse-grained descriptions, which introduces the concept of scale. Scaling properties can be statistical, although the ergodic assumption is not expected to hold for organized structures. Badii and Politi [18] emphasize a hierarchy of parts, interactions, and scaling as relevant to complex physical systems. For generic parts Y (x), the (arguably) simplest evolution equation associated with complex behavior is dY = A ∗ (Y Y ) (10.25) dt The local interaction kernel A(x) is convolved (asterisk) with nonlinear expressions, and the specific form of A captures the details of the local interactions. Based on such a model, Hansen and Tabeling [21] illustrated the emergence of eddies (see also Bohr et al. [22]). This highlights the need to identify relevant “parts” in the turbulence problem. Among the contenders, Fourier modes are unsatisfactory in spite of significant successes because they conceal the localized nature of eddies. The combination of spatial and spectral resolution makes wavelet analysis a possible alternative, and the analytical framework presented in the previous section confirms the potential for u˜ i or some related quantity to be “Wavelets [that] . . . are, essentially, a mathematical realization of the common physical notion of an eddy in turbulence” [23]. Defining Fi jk = ∂i3jk Fs (10.26) as known functions of space and scale (derivatives of Gaussian, with limited spatial reach), the NS equations take the form (∂t −
∂ 2j j )u˜ i = Fi jk ∗ (u j u k ) − F jkk ∗ (u i u j )

(10.27)

The nonlinear terms (right side) include the convolution with the interaction kernel Fi jk , as in the generic model Equation (10.25). However, the kernel acts not on the wavelet coefficient u˜ i , but, rather through the inverse transform formula, on wavelet coefficients at all scales, each of which is obtained by filtering of the fine-grained coefficient. Preliminary results have been reported previously [24].

10.4.2 REGULARITY

OF

EULER

AND

NS SOLUTIONS

The finite-time regularity of solutions of the Navier–Stokes and Euler equations remains a theoretical as well as practical problem. For nondifferentiable (weak) solutions, the possibility of inviscid dissipation [23,25] could affect our understanding of the dissipation rate  in common models. Also, the relation between regularity and vorticity is well established [26]. Furthermore, the applicability of multifractal models to turbulence implies a lack of differentiability, which is consistent with the scaling of velocity differences in Komogorov’s theory and its modern offsprings. Various mathematical tools appearing in the literature are easily related to wavelets; this topic will be further expanded in the companion chapter in this volume, so only a brief introduction is presented here. The modulus of continuity: u = u(x) − u(x − y) ∼| y |h

(10.28)

has emerged as a useful diagnostic of regularity. Its statistical moments are known as structure functions, for which Kolmogorov scaling predicts an average h = 1/3. For differentiable fields, Taylor series correspond to positive integer hs. Any value h < 1 is an indicator of singularity. Then, it is easy to show [12,27] that  h Fs (y)u dy = u − u > = u < ∼ s 2 (10.29) It can also be proved that derivatives of order n follow the scaling ∂xn u > ∼ s u˜ ∼ s © 2006 by Taylor & Francis Group, LLC

h 2 −1

h−n 2

and, in particular, (10.30)

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Therefore, dominant small-scale terms correspond to the smallest exponents h, and these terms can be identified in the exact equations of motion. In addition to the results described in the companion chapter in this volume, ongoing related work also addresses the possible inviscid dissipation of energy suggested by Onsager.

10.4.3 RAPID DISTORTION THEORY Rapid distortion theory has been successful in accounting to some extent for the emergence of some structures in straining flows. Key ingredients of the theory are the presence of an imposed nonuniform flow, endowed with a time scale (inverse rate of strain) much shorter than the eddy turnover times or the dissipation scale. Under these circumstances, linearization of the equation governing the fluctuations is possible; the evolution of the energy spectra associated with principal directions introduces a strong anisotropy of the field, and structure. The combination of linearization and spectral anisotropy can be applied to variants of the preceding equations. This work is in progress and will be reported at a later time.

10.4.4 RENORMALIZATION APPROACHES Renormalization theory has been applied to the Navier–Stokes equations primarily in Fourier space [2]. The close relation between FNS and the flexion equation and the reduction of nonlinear terms to local convolutions with known coefficients (derivatives of Gaussians) indicate that a renormalization might succeed in wavelet space as well. Differences with the Fourier version are the explicit presence of spatial intermittency of the structures and singularities, the role of filtering, and the nature of the baseline propagator.

10.4.5 STRUCTURE FUNCTIONS Structure functions are the statistical moments of differences in field values at two points. For a given field u, the structure function can be written as Sn (r ) = <| u(x + r ) − u(x) |n > = <| u ∗ ((r ) − (0)) |n > = <| u ∗ (r ) |n >

(10.31)

where is the difference between two Dirac distributions separated by a distance r . Although it has very poor spectral localization, is an admissible wavelet. The structure function can be interpreted as the nth moment of some wavelet coefficient. Also one can conceive of alternatives to Sn , based on better wavelets, more amenable to analytical manipulations and scaling estimates. Continuous wavelets are natural tools to study structure functions (e.g., [27]), i.e., statistical moments of the wavelet coefficients. With Hermitian wavelets, the combination of exact equations and scaling estimates is promising.

10.5 CONCLUSION Wavelet coefficients are a compromise between spatial and spectral descriptions, i.e., between the two sides of Fourier transforms. The concept is appealing whenever we need to describe (or analyze or compute) intermittent fields, since resolution can be focused where it is needed. Within the overlap of wavelets and turbulence, orthogonal wavelets are used exclusively in numerical work, and continuous wavelets take a share of the data analysis. The manipulations of the exact Navier–Stokes equations presented in this chapter is similar in spirit to Kaiser’s treatment of Maxwell’s equations for electromagnetic fields [28]. © 2006 by Taylor & Francis Group, LLC

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Wavelets are the closest thing to a mathematical r epresentation of eddies [23], and the choice of Hermitian wavelets makes the manipulation of exact equations tractable. In fact, as much in spirit (Equation 10.8, Equation 10.11, Equation 10.14, and Equation 10.19) as in the details of treatment of the nonlinear terms, the wavelet formulation is comparable to the best available methods, with the added benefit of local scaling. It is reasonable to expect that everything analytical done in Fourier space (from rapid distortion to renormalization) can be done better in wavelet space where intermittency and individual eddies can be made explicit. Also noteworthy is the fact that diffusion becomes an invariant translation in wavelet space, corresponding to a Hamiltonian formulation [14,29,30]. Its close relation to filtering also makes it a natural fit for renormalization procedures, with fine- and coarse-grained formulations easily implemented. The Navier–Stokes dynamics for the wavelet coefficients are local and nondifferential in space, and the nonlinear interactions are governed by fixed known localized kernels and involve local wavelet coefficients at all scales. Most elements of the turbulence syndrome seem accessible within a single formulation.

NOMENCLATURE A d F(u i ) Fs h hi j p r s ui uˆ i u˜ i Y i  i jk

i
 i

generic local interaction kernel space dimensionality Fourier transform of u i Gaussian filter of scale s H¨older exponent anisotropic H¨older exponent static pressure spatial separation distance scale, with dimensions L 2 ith Cartesian component of velocity Fourier transform of u i Mexican hat wavelet transform of u i generic-state variable flexion, the Laplacian of u i spatial difference or Dirac function permutation symbol difference of Dirac wavelet ith component of the wavenumber vector kinematic viscosity fluid density Mexican hat wavelet ith Cartesian component of vorticity

SUPERSCRIPTS > <

low-pass filtering (scale implicit or explicit) high-pass filtering (scale implicit or explicit)

REFERENCES 1. G.I. Taylor, The spectrum of turbulence, Proc. Roy. Soc. London Series A 164, 476–490 (1938). 2. W.D. McComb, The Physics of Fluid Turbulence, Clarendon Press, U.K. (1990). 3. G.K. Batchelor and I. Proudman, The effect of rapid distortion of a fluid in turbulent motion, Q. J. Mech. Appl. Math. 7, 83–103 (1954). © 2006 by Taylor & Francis Group, LLC

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4. J.C.R. Hunt, D.J. Carruthers and J.C.H. Fung, Rapid distortion theory as a means of exploring the structure of turbulence, in New Perspectives in Turbulence (L. Sirovich, Ed.), Springer–Verlag, Dordrecht, (1991). 5. D. K¨uchemann and J. Weber, Report on the IUTAM symposium on concentrated vortex motions in fluids, J. Fluid Mech. 21, 1–20 (1965). 6. A.J. Chorin, Vorticity and Turbulence, Springer–Verlag, Dordrecht (1994). 7. P. Holmes, J.L. Lumley and G. Berkooz, Turbulence, Coherent Structures, Dynamical Systems and Symmetry, Cambridge University Press (1998). 8. U. Frisch, Turbulence: the Legacy of A.N. Kolmogorv, Cambridge University Press (1995). 9. V.S. L’vov and I. Procaccia, Computing the scaling exponents in fluid turbulence from first principles, Physica A 257, 165–196 (1998). 10. R.J. Hill, Equations relating structure functions of all orders, J. Fluid Mech. 434, 379–388 (2001). 11. M.J. Giles, Anomalous scaling in homogeneous isotropic turbulence, J. Phys. A: Math. Gen. 34, 4389– 4435 (2001). 12. I. Daubechies, Ten Lectures on Wavelets, S.I.A.M. (1992). 13. M. Farge, Wavelet transforms and their applications to turbulence, Ann. Rev. Fluid Mech. 24, 395–457 (1992). 14. J. Lewalle, A filtering and wavelet formulation for incompressible turbulence, J. Turbulence 1, 004, 1–16 (2000). 15. C. Truesdell, The Kinematics of Vorticity, Indiana University Press (1954), p. 84. 16. A.N. Yannacopoulos, I. Mezi´c, G. Rowlands and G.P. King, Eulerian diagnostics of Lagrangian chaos in three-dimensional Navier–Stokes flows, Phys. Rev. E 57, 482–490 (1998). 17. S. Kida, Numerical simulation of two-dimensional turbulence with high-symmetry, J. Phys. Soc. Jpn 54, 2840–2854 (1985). 18. R. Badii and A. Politi, Complexity: Hierarchical Structures and Scaling in Physics, Cambridge University. Press (1997). 19. G. Nicolis and I. Prigogine, Exploring Complexity, W.H. Freeman and Co., New York (1989). 20. Y. Bar–Yam, Dynamics of Complex Systems, Perseus Books (1997). 21. A.E. Hansen and P. Tabeling, Coherent structures in two-dimensional decaying turbulence, Nonlinearity 13, C1–C3 (2000). 22. T. Bohr, M.H. Jensen, G. Paladin and A. Vulpiani, Dynamical Systems Approach to Turbulence, Cambridge University Press (1998). 23. G.L. Eyink, Energy dissipation without viscosity in ideal hydrodynamics, I. Fourier analysis and local energy transfer, Physica D 78, 220–240 (1994). 24. J. Lewalle, Self-organization in Navier–Stokes turbulence, InterJournal, paper #562 (2002). 25. L. Onsager, Statistical hydrodynamics, Nuovo Cimento 6, 279–287 (1949). 26. A.J. Majda, Vorticity, turbulence, and acoustics in fluid flow, SIAM Rev. 33, 349–388 (1991). 27. V. Perrier and C. Basdevant, Besov norms in terms of the continuous wavelet transform. Application to structure functions, Math. Models Methods Appl. Sci. 6, 649–664 (1996). 28. G. Kaiser, Wavelet electrodynamics, Phys. Lett. A 168, 28–34 (1991). 29. J. Lewalle, Hamiltonian formulation for the diffusion equation, Phys. Rev. E. 55, 1590–1599 (1997). 30. J. Lewalle, Formal improvements in the solution of the wavelet-transformed Poisson and diffusion equations, J. Math. Phys. 39, 4119–4128 (1998).

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Time Averaging, Hierarchy of the Governing Equations, and the Balance of Turbulent Kinetic Energy Douglas P. Dokken and Mikhail M. Shvartsman

CONTENTS 11.1 Introduction .......................................................................................................................... 155 11.2 Various Notions of Time Averaging ..................................................................................... 156 11.2.1 Standard (Reynolds) Time Averaging ...................................................................... 156 11.2.2 Running Time Averaging ......................................................................................... 157 11.3 Governing Equations ............................................................................................................ 157 11.4 Constitutive and Closure Theories ....................................................................................... 159 11.5 Turbulent Kinetic Energy .....................................................................................................159 Acknowledgments.......................................................................................................................... 164 References...................................................................................................................................... 164

ABSTRACT We discuss a connection between governing equations, constitutive theory, and closure problems for atmospheric boundary layer. Such a connection is of prime importance in building algorithms for numerical simulations. We consider averaging in time and its relation to the Boussinesq approximation of the governing equations. We introduce a notion of instantaneous turbulent kinetic energy and derive a new balance equation for its material derivative.

11.1 INTRODUCTION Since meteorological data are often regarded as an averaged quantity due to measurement error and other factors, the tools of turbulence become important regardless of a specific event under consideration. The fact that only averaged data are available forces the additional unknowns in the system of governing equations. To close such a system, one has a few choices, and in atmospheric modeling the popular options are the Smagorinsky model, the Germano scheme, and the turbulent kinetic energy equation (see Xue et al. [1]). The kinetic energy is also important since it may serve as a “driving force” for a large variety of atmospheric events. Initiation of some storms is believed to be related to conversion of latent heat energy into the energy of convective motion that in turn may initiate a vertical rotation according to Dauies–Jones et al. [2] and the references therein. This note is organized as follows. In Section 11.2 we discuss some notions of time averaging (spatial and ensemble averaging can be addressed in the same fashion). In Section 11.3 we discuss the basic governing equations that can be applied to atmospheric motion. In Section 11.4 we discuss

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the constitutive and closure theories (we call them the second and the third Boussinesq problems). In Section 11.5 we derive a balance for the material derivative of instantaneous kinetic energy.

11.2 VARIOUS NOTIONS OF TIME AVERAGING A function u(x, t) may be represented as1 u(x, t) =

K


u k (x, t),

(11.1)

k=1

where each u k (x, t) (k = 1, 2, . . . , K ) describes a specific feature of the function, where x is a spatial variable and t is time. In atmospheric science it is customary (see references 3 and 4 to view (11.1) as (11.2) u(x, t) = u(x, t) + u
(x, t), where u(x, t) is a slowly varying average component and u
(x, t) is a rapidly fluctuating or turbulent component.

11.2.1 STANDARD (REYNOLDS) TIME AVERAGING We assume that u(x, t) is a continuously differentiable function of its variables. We assume that there is a typical averaging period T that includes2 a “large” number of oscillations in u
(x, t), and u(x, t) is defined by  1 T u(x, t) ≡ u(x, s) ds, (11.3) T 0 so, in this case, u(x, t) is not slowly varying but a constant (in time) function for t ∈ [0, T ]. It is easy to show that u
(x, t) = 0,

u(x, t)v(x, t) = u(x, t) v(x, t),

u(x, t)v
(x, t) = 0,

(11.4)

and u(x, t) = u(x, t).

(11.5)

Using (11.4), for any functions u(x, t), v(x, t), and w(x, t), satisfying the preceding assumptions, we have u(x, t)v(x, t) = u(x, t) v(x, t) + u
(x, t)v
(x, t), (11.6) and u(x, t)v(x, t)w(x, t) = u(x, t) v(x, t) w(x, t) + u(x, t) v
(x, t)w
(x, t) + v(x, t) u
(x, t)w
(x, t) + w(x, t) v
(x, t)u
(x, t)

(11.7)

+ u
(x, t)v
(x, t)w
(x, t). For the spatial derivatives under our smoothness assumptions, we have ∂ ∂ u(x, t), u(x, t) = ∂x ∂x

∂
∂
u (x, t) = 0. u (x, t) = ∂x ∂x

(11.8)

1 A similar discussion is valid for a vector field u(x, t) where x is a spatial position vector. As a matter of fact, we will use the notions and notations of this section for the Eulerian fluid flow u(x, t) in Section 11.5. 2 Specific T > 0 is determined by the type of the atmospheric event under consideration.

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However, temporal derivatives and time averages do not commute, i.e., ∂ ∂ 1 u(x, t) = u
(x, t) = (u(x, T ) − u(x, 0)) ∂t ∂t T 1 ∂ ∂ = (u
(x, T ) − u
(x, 0)) = 0 = u(x, t) = u
(x, t). T ∂t ∂t

(11.9)

The problem with (11.9) is removed if one assumes that u(x, 0) = u(x, T ),

u
(x, 0) = u
(x, T ) = 0,

(11.10)

but such an assumption will force periodicity of u(x, t) in time and will drastically reduce the class of problems that could be considered. Another way this may be resolved is by considering the case T → ∞. Appropriateness of such an assumption must be determined by scaling considerations.

11.2.2 RUNNING TIME AVERAGING We define

 t+T 1 u(x, t) ≡ u(x, s) ds. 2T t−T Then, assuming (11.2), we see that (11.4) through (11.7) fail, i.e., u
(x, t) = 0,

u(x, t)v(x, t) = u(x, t) v(x, t),

u(x, t)v
(x, t) = 0,

u(x, t)v(x, t) = u(x, t) v(x, t) + u
(x, t)v
(x, t).

(11.11)

(11.12) (11.13)

However, the property similar to (11.8) still holds, i.e., ∂ ∂ u(x, t), u(x, t) = ∂x ∂x

∂
∂
u (x, t) = 0, u (x, t) = ∂x ∂x

(11.14)

and also

∂ 1 ∂ u(x, t) = (u(x, t + T ) − u(x, t − T )) = u(x, t). ∂t 2T ∂t In the remainder of this chapter, we will employ only the Reynolds averaging (11.3).

(11.15)

11.3 GOVERNING EQUATIONS There are two sources for the governing equations of a fluid flow. The first is dynamic (the linear momentum or Navier–Stokes) equations and the second is conservation equations for mass and energy. The set of conservation equations may be augmented since some physical quantities can be assumed constant for a “short enough” period of time3 (see Landau and Livschitz [5]). The question is which of these two sources shall take a priority to represent a phenomenon of interest. We will symbolically call it the first Boussinesq problem. For simplicity, we start with the Euler equations for ideal fluid without gravity: Du 1 = − ∇ p, Dt


(11.16)

supplemented with conservation of energy equation De p = − ∇ · u + J, Dt
3 Usually

such a period is of order T .

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(11.17)

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and with conservation of mass equation D
+
∇ · u = 0, Dt

(11.18)

where u(x, t) = (u(x, t), v(x, t), w(x, t)) is the Eulerian velocity field4 depending on a spatial vector x and time t, p(x, t) is the pressure field,
(x, t) is the density field, e(x, t) is the internal energy density field per unit mass, J (x, t) is the rate of heating per unit mass, and D/Dt stands for the material derivative with respect to time, or, in other words, for the derivative following the trajectory of the air parcel: ∂ ∂ ∂ ∂ ∂ D ≡ +u +v +w = + u · ∇, (11.19) Dt ∂t ∂x ∂y ∂z ∂t where x = (x, y, z), x is longitudinal (arc-length) coordinate, y is latitudinal (arc-length) coordinate, and z is height of the air parcel (see Holton [3]). Note that the system 11.16 through 11.18 includes six unknowns functions: u(x, t), v(x, t), w(x, t), p(x, t),
(x, t), e(x, t) and only five equations. If the total energy includes only kinetic energy and if J = 0, then it forces incompressibility (see Chorin [7]). For incompressible fluids in that case, we have De/Dt = 0, so the Lagrangian internal energy remains constant, while the Eulerian internal energy is allowed to change. If J = 0, where J may include radiation, conduction, and latent heat release (but not convection), then relation between the rate of heating and motion is u·∇e ≈ J (if one may assume ∂e/∂t ≈ 0 for a characteristic time interval T ). A similar situation is repeated if we take into account all the fundamental and apparent forces per unit mass F(x, t) relevant to atmospheric dynamics Du 1 = − ∇ p + F, Dt


(11.20)

supplemented with (11.17) and (11.18), where F = −2 × u + g∗ + 2 R + Fr .

(11.21)

Here,  is the the angular velocity of rotation of the Earth,  = ||, g∗ is the gravitational force per unit mass, R is the position vector from the axis of rotation to the air parcel, and Fr is the viscous force per unit mass, often assumed to be of the form Fr = u, where  is the kinematic viscosity coefficient. The system (11.17) through (11.20) can be written in components: Du uv tan  uw 1 ∂p − + =− + 2v sin  − 2w cos  + Fr x Dt a a
∂x

(11.22)

u 2 tan  vw 1 ∂p Dv + + =− − 2u sin  + Fr y Dt a a
∂y

(11.23)

Dw u 2 + v 2 1 ∂p − =− − g + 2u cos  + Fr z Dt a
∂z p D
De = 2 +J Dt
Dt ∂v ∂w 1 D
∂u + + =− , ∂x ∂y ∂z
Dt

(11.24) (11.25) (11.26)

where  is latitude, g = |g∗ + 2 R|, Fr = (Fr x , Fr y , Fr z ), and a is the radius of the Earth (see Holton [3]). Again, we have six unknowns and only five equations. 4 There

is still a question (see Chorin [6]) whether u(x, t) satisfies the set of governing equations or only its averaged version u(x, t) satisfies such a set.

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11.4 CONSTITUTIVE AND CLOSURE THEORIES From our discussion in Section 11.3 we realize that we need an extra equation to close the system (11.16) through (11.18) or the system (11.17) through (11.20). Again, we have two basic sources of such equations. One class of such equations can be called constitutive equations. Constitutive equations characterize the medium (air in our case). The second class of such equations can be called scaling (or event) equations. Scaling equations characterize events by dictating which terms in the system dominate the phenomenon under consideration. The question of priority here is somewhat more subtle. We could use the theory of ideal gas to close the system (11.16) through (11.18) with p =
RT and e = cV T , where T is the absolute temperature, R is the universal gas constant, and cV is the specific heat measured at the constant volume. However, the resulting set of equations will not be able to distinguish between various scales of atmospheric events. We can symbolically call it the second Boussinesq problem. In many cases the models combine constitutive and scaling equations. Probably the most popular example is the Boussinesq approximation for atmospheric boundary layer: 1 ∂p Du =− + f v + Fr x , Dt
0 ∂ x

(11.27)

1 ∂p Dv =− − f u + Fr y , Dt
0 ∂ y

(11.28)

1 ∂p  Dw =− + g + Fr z , Dt
0 ∂z 0

(11.29)

d0 D = −w , Dt dz ∂v ∂w ∂u + + = 0, ∂x ∂y ∂z

(11.30) (11.31)

where f = 2 sin , where the unknowns are the velocity components u(x, y, z, t), v(x, y, z, t), w(x, y, z, t), the pressure p(x, y, z, t), and the potential temperature (x, y, z, t); and where
0 (z) and 0 (z) are some given functions (see Holton [3] for details). However, since we can use only averaged data, the number of the unknowns doubles and we have ten unknown functions u(x, t), v(x, t), w(x, t), p(x, t), (x, t), and u
(x, t), v
(x, t), w
(x, t), p
(x, t), 
(x, t), that we need to determine. Additional equations in this case are called closure theories and finding such equations we can symbolically call the third Boussinesq problem. In the next section we address one such equation, i.e., the balance for turbulent kinetic energy energy.

11.5 TURBULENT KINETIC ENERGY From (11.19) and (11.31), we have Du ∂u ∂u ∂u ∂u ≡ +u +v +w Dt ∂t ∂x ∂y ∂z

 ∂u ∂v ∂w  ∂ x + ∂ y + ∂z ∂u ∂u ∂u ∂u = +u +v +w + u    ∂t ∂x ∂y ∂z 0 ∂u ∂uu ∂uv ∂uw = + + + . ∂t ∂x ∂y ∂z

Also, by (11.8) and (11.31), we have ∂u ∂v ∂w + + = 0. ∂x ∂y ∂z © 2006 by Taylor & Francis Group, LLC

(11.32)

(11.33)

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We set (see [3]) ∂ D ∂ ∂ ∂ = +u +v +w . Dt ∂t ∂x ∂y ∂z

(11.34)

Definition 11.1

Instantaneous turbulent kinetic energy ITKE (x, y, z, t) is a function defined by ITKE (x, y, z, t) :≡

 1
2 (u ) + (v
)2 + (w
)2 . 2

(11.35)

A standard notion of turbulent kinetic energy is defined by TKE (x, y, z) ≡

 1
2 (u ) + (v
)2 + (w
)2 2

(11.36)

and various balance equations are available for TKE.5 Then, TKE (x, y, z) = ITKE (x, y, z, t).

(11.38)

Theorem 11.1

Let u, v, w, p,  be the functions of (x, y, z, t) satisfying the Boussinesq approximation system of partial differential equations (11.27) through (11.31). Then D (ITKE) = MP + BPL + TR − ε, Dt

(11.39)

where the mechanical production MP ≡ − u
u


∂u ∂u ∂u − u
v
− u
w
∂x ∂y ∂z

− v
u


∂v ∂v ∂v − v
v
− v
w
∂x ∂y ∂z

− w
u


∂w ∂w ∂w − w
v
− w
w
∂x ∂y ∂z

− u
u


∂u
∂u
∂u
− u
v
− u
w
∂x ∂y ∂z

− v
u


∂v
∂v
∂v
− v
v
− v
w
∂x ∂y ∂z

− w
u


∂w
∂w
∂w
− w
v
− w
w
, ∂x ∂y ∂z

the buyoncy production or loss BPL ≡ w



5 The

(11.40)

g , 0

(11.41)

balance equation D (TKE) = MP + BPL + TR − ε Dt

is known but differs from our balance.

© 2006 by Taylor & Francis Group, LLC

(11.37)

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the transport redistribution 1 TR ≡ −
0



∂ p
∂ p
∂ p
u
+ v
+ w
∂x ∂y ∂z

=−

1
u · ∇ p
,
0

161

(11.42)

and the frictional dissipation

  ε ≡ − u
Fr x
+ v
Fr y
+ w
Fr z
= −u
· Fr
.

Proof

(11.43)

We average 11.32 with use of 11.6 and 11.33 to obtain Du ∂u ∂ ∂ ∂ = + (u u + u
u
) + (u v + u
v
) + (u w + u
w
) Dt ∂t ∂x ∂y ∂z =

∂u ∂u ∂u ∂u ∂

∂ ∂

+u +v +w + (u u ) + (u v ) + (u
w
) ∂t ∂x ∂y ∂z ∂x ∂y ∂z

=

∂

∂u
∂u ∂u ∂u ∂

∂ +u +v +w + (u u ) + (u v ) + (u
w
), ∂t ∂x ∂y ∂z ∂x ∂y ∂z

(11.44)

with similar counterparts for the averages Dv/Dt and Dw/Dt. Then the mean equations of (11.27) through (11.31) will be (11.33) and  1 ∂p ∂u ∂u ∂u ∂u ∂

∂

∂

+u +v +w =− + fv− (u u ) + (u v ) + (u w ) + Fr x , ∂t ∂x ∂y ∂z
0 ∂ x ∂x ∂y ∂z (11.45)  ∂v ∂v ∂v ∂v ∂

∂

∂ 1 ∂p +u +v +w =− − fu − (v u ) + (v v ) + (v
w
) + Fr y , ∂t ∂x ∂y ∂z
0 ∂ y ∂x ∂y ∂z (11.46)  ∂w ∂w ∂w ∂ ∂ ∂ ∂w  1 ∂p +u +v +w =− +g − (w
u
) + (w
v
) + (w
w
) + Fr x , ∂t ∂x ∂y ∂z
0 ∂ x 0 ∂x ∂y ∂z (11.47)  ∂

∂ ∂ ∂ ∂ d0 ∂

∂ +u +v +w = −w − ( u ) + ( v ) + (
w
) . (11.48) ∂t ∂x ∂y ∂z dz ∂x ∂y ∂z Now we use ∂u ∂u ∂u u −u = (u − u) +u ∂x ∂x ∂x and rewrite

∂u ∂u − ∂x ∂x



= u


∂u ∂u
+u ∂x ∂x

 Du Du − u Dt Dt

 ∂u ∂u ∂u
∂u
∂u


∂u
∂u
∂u =u − +u +u +v +v +w +w ∂t ∂t ∂x ∂x ∂y ∂y ∂z ∂z


 ∂u ∂u
∂u
∂u
∂u

∂u
∂u
∂u =u − +u +u +v +v +w +w . ∂t ∂t ∂x ∂x ∂y ∂y ∂z ∂z


First, we subtract the left-hand side (11.45) from the left-hand side (11.27) and multiply the result by u
, i.e.,

 ∂u
∂u
∂u
∂u ∂u ∂u ∂u ∂u − + u
+u + v
+v + w
+w u
∂t ∂t ∂x ∂x ∂y ∂y ∂z ∂z


 ∂u
∂u
∂u
∂u ∂u ∂u ∂u ∂u = u
− + u
+u + v
+v + w
+w . (11.49) ∂t ∂t ∂x ∂x ∂y ∂y ∂z ∂z © 2006 by Taylor & Francis Group, LLC

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Similarly, we obtain the terms

 ∂v ∂v ∂v
∂v
∂v


∂v
∂v
∂v − +u +u +v +v +w +w v ∂t ∂t ∂x ∂x ∂y ∂y ∂z ∂z




∂v ∂v ∂v ∂v ∂v ∂v ∂v ∂v
− + u
+u + v
+v + w
+w = v
∂t ∂t ∂x ∂x ∂y ∂y ∂z ∂z

(11.50)

and w


 ∂w ∂w ∂w ∂w ∂w ∂w
∂w
∂w
− + u
+u + v
+v + w
+w ∂t ∂t ∂x ∂x ∂y ∂y ∂z ∂z


 ∂w ∂w ∂w ∂w ∂w ∂w
∂w
∂w
= w
− + u
+u + v
+v + w
+w . ∂t ∂t ∂x ∂x ∂y ∂y ∂z ∂z

(11.51)

Next, we subtract the right-hand side (11.45) from the right-hand side (11.27) and multiply the result by u
, i.e.,

 1 ∂ p
∂

∂

∂




u − (11.52) + fv + (u u ) + (u v ) + (u w ) + Fr x .
0 ∂ x ∂x ∂y ∂z Analogously, we obtain the terms

 1 ∂ p
∂

∂

∂ v
− − f u
+ (v u ) + (v v ) + (v
w
) + Fr y

0 ∂ y ∂x ∂y ∂z and w




 1 ∂ p
∂ ∂ ∂ 







− +g + (w u ) + (w v ) + (w w ) + Fr z .
0 ∂z 0 ∂x ∂y ∂z

(11.53)

The sum of (11.49) through (11.51) equals the sum of (11.52) through (11.54), so 


∂u ∂u ∂u
∂u
∂u


∂u
∂u
∂u u − +u +u +v +v +w +w ∂t ∂t ∂x ∂x ∂y ∂y ∂z ∂z




∂v ∂v ∂v ∂v
∂v

∂v
∂v
∂v +v − +u +u +v +v +w +w ∂t ∂t ∂x ∂x ∂y ∂y ∂z ∂z




∂w ∂w ∂w ∂w ∂w


∂w
∂w
∂w +w − +u +u +v +v +w +w ∂t ∂t ∂x ∂x ∂y ∂y ∂z ∂z


1 ∂p ∂

∂

∂ = u
− + f v
+ (u u ) + (u v ) + (u
w
) + Fr x

0 ∂ x ∂x ∂y ∂z

 1 ∂ p
∂

∂

∂ + v
− − f u
+ (v u ) + (v v ) + (v
w
) + Fr y

0 ∂ y ∂x ∂y ∂z



1 ∂p  ∂ ∂ ∂







+w − +g + (w u ) + (w v ) + (w w ) + Fr z ,
0 ∂z 0 ∂x ∂y ∂z or    ∂ 1
2
2
2
∂u
∂v
∂w − u (u ) + (v ) + (w ) +v +w ∂t 2 ∂t ∂t ∂t + u
u


∂u ∂u ∂u ∂u
∂u
∂u
+ u
u + u
v
+ u
v + u
w
+ u
w ∂x ∂x ∂y ∂y ∂z ∂z

© 2006 by Taylor & Francis Group, LLC

(11.54)

(11.55)

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+ v
u


163

∂v ∂v ∂v ∂v
∂v
∂v
+ v
u + v
v
+ v
v + v
w
+ v
w ∂x ∂x ∂y ∂y ∂z ∂z

∂w ∂w ∂w ∂w
∂w
∂w
+ w
u + w
v
+ w
v + w
w
+ w
w ∂x ∂x ∂y ∂y ∂z ∂z


1 ∂p ∂

∂

∂ = u
− + f v
+ (u u ) + (u v ) + (u
w
) + Fr x

0 ∂ x ∂x ∂y ∂z


1 ∂p ∂

∂

∂




+v − − fu + (v u ) + (v v ) + (v w ) + Fr y
0 ∂ y ∂x ∂y ∂z



1 ∂p ∂ ∂ ∂  + w
− +g + (w
u
) + (w
v
) + (w
w
) + Fr z
.
0 ∂z 0 ∂x ∂y ∂z + w
u


(11.56)

We can rewrite (11.56) in the following way:     ∂ 1
2 ∂ 1
2 (u ) + (v
)2 + (w
)2 + u (u ) + (v
)2 + (w
)2 ∂t 2 ∂x 2     ∂ 1
2 ∂ 1
2 +v (u ) + (v
)2 + (w
)2 + w (u ) + (v
)2 + (w
)2 ∂y 2 ∂z 2  ∂u ∂v ∂w − u
+ v
+ w
∂t ∂t ∂t ∂u ∂u ∂u ∂v ∂v ∂v + u
v
+ u
w
+ v
u
+ v
v
+ v
w
∂x ∂y ∂z ∂x ∂y ∂z

+ u
u


+ w
u


∂w ∂w ∂w + w
v
+ w
w
∂x ∂y ∂z

 1 ∂ p
∂

∂

∂



=u − + fv + (u u ) + (u v ) + (u w ) + Fr x
0 ∂ x ∂x ∂y ∂z

 1 ∂ p
∂

∂

∂ + v
− − f u
+ (v u ) + (v v ) + (v
w
) + Fr y

0 ∂ y ∂x ∂y ∂z



 1 ∂p ∂ ∂ ∂ + w
− +g + (w
u
) + (w
v
) + (w
w
) + Fr z
,
0 ∂z 0 ∂x ∂y ∂z


or D Dt

(11.57)



  ∂u ∂v ∂w 1
2 (u ) + (v
)2 + (w
)2 − u
+ v
+ w
2 ∂t ∂t ∂t

+ u
u


∂u ∂u ∂u ∂v ∂v ∂v ∂w + u
v
+ u
w
+ v
u
+ v
v
+ v
w
+ w
u
∂x ∂y ∂z ∂x ∂y ∂z ∂x

+ w
v


 1 ∂ p
∂

∂

∂



=u − + fv + (u u ) + (u v ) + (u w ) + Fr x
0 ∂ x ∂x ∂y ∂z


∂

∂

∂

1 ∂p


+v − − fu + (v u ) + (v v ) + (v w ) + Fr y
0 ∂ y ∂x ∂y ∂z

 ∂ ∂ ∂ 1 ∂ p

+ w
− +g + (w
u
) + (w
v
) + (w
w
) + Fr z
.
0 ∂z 0 ∂x ∂y ∂z


∂w ∂w + w
w
∂y ∂z

© 2006 by Taylor & Francis Group, LLC

(11.58)

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We average (11.58) and obtain, using (11.4), D Dt



 ∂u ∂u ∂u 1
2
2
2 + u
u
(u ) + (v ) + (w ) + u
v
+ u
w
2 ∂x ∂y ∂z

∂v ∂v ∂v ∂w ∂w ∂w + v
v
+ v
w
+ w
u
+ w
v
+ w
w
∂x ∂y ∂z ∂x ∂y ∂z

 1 ∂ p
∂ p
∂ p
g =− u
+ v
+ w
+ w

+ u
Fr x
+ v
Fr y
+ w
Fr z
,
0 ∂x ∂y ∂z 0 + v
u


(11.59)

or, using also (11.4), (11.7) and (11.8), we have D Dt



 ∂u ∂u ∂u 1
2 (u ) + (v
)2 + (w
)2 + u
u
+ u
v
+ u
w
2 ∂x ∂y ∂z

+ v
u


∂v ∂v ∂v ∂w ∂w ∂w + v
v
+ v
w
+ w
u
+ w
v
+ w
w
∂x ∂y ∂z ∂x ∂y ∂z

+ u
u


∂u
∂u
∂u
∂v
∂v
∂v
+ u
v
+ u
w
+ v
u
+ v
v
+ v
w
∂x ∂y ∂z ∂x ∂y ∂z

(11.60)

∂w
∂w
∂w
+ w
v
+ w
w
∂x ∂y ∂z




 1 ∂p ∂p ∂p g =− u
+ v
+ w
+ w

+ u
Fr x
+ v
Fr y
+ w
Fr z
,
0 ∂x ∂y ∂z 0 + w
u


as claimed in (11.39).

ACKNOWLEDGMENTS The authors express their gratitude to Stuart Antman and Peter Constantin for valuable discussions.

REFERENCES 1. M. Xue, K.K. Droegemeier, V. Wong, The advanced regional prediction system and real-time stormscale weather prediction, International Workshop on Resolution Models, Beijing, China, October, 1995. 2. R. Davies–Jones, R.F. Trapp, H.B. Bluestein, Tornado and tornadic storms, Meteorological Monographs, Severe Convective Storms, Doswell, November, 2001. 3. J.R. Holton, An Introduction to Dynamic Meteorology, 3rd ed., Academic Press, 1992. 4. W.R. Cotton, Averaging and the Parametrization of Physical Processes in Mesoscale Models, Mesoscale Meteorology and Forecasting, Peter S. Ray, Ed., American Meteorological Society, Boston, 1989, pp. 614–635. 5. L.D. Landau, E.M. Livschitz, Statistical Physics. Part 1, 3rd ed., Oxford, Pergamon Press, 1980. 6. A.J. Chorin, Vorticity and Turbulence, Second Printing, Springer–Verlag, New York, 1998. 7. A.J. Chorin, J. Marsden, Mathematical Introduction to Fluid Mechanics, 3rd ed., Springer–Verlag, New York, 1993.

© 2006 by Taylor & Francis Group, LLC

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Role of Angular Momentum 12 The Invariants in Homogeneous Turbulence P. A. Davidson CONTENTS 12.1 12.2 12.3 12.4 12.5

Introduction........................................................................................................................ 165 Loitsyansky’s Integral for Isotropic Turbulence ................................................................ 166 Kolmogorov’s Decay Laws in Isotropic Turbulence ......................................................... 167 Landau’s Angular Momentum in Isotropic Turbulence..................................................... 168 Long-Range Correlations in Homogenous Turbulence ..................................................... 170 12.5.1 The Objections of Birkhoff, Batchelor, and Saffman ........................................... 170 12.5.2 A Reappraisal of the Long-Range Pressure Forces in E ∼ k 4 Turbulence........... 171 12.6 The Growth of Anisotropy in MHD Turbulence ............................................................... 175 12.7 The Landau Invariant for Homogeneous MHD Turbulence .............................................. 177 12.8 Decay Laws at Low Magnetic Reynolds Number ............................................................. 178 12.9 A Loitsyansky-type Invariant for Stratified Turbulence .................................................... 180 12.10 Conclusions........................................................................................................................ 181 References...................................................................................................................................... 181

ABSTRACT We discuss the significance of the law of conservation of angular momentum for freely evolving, homogeneous turbulence. We start by noting that Loitsyansky’s integral can be interpreted as the mean square angular momentum of a large cloud of isotropic turbulence. As noted by Landau, the near invariance of this integral is a direct consequence of the law of conservation of angular momentum. We show that these ideas may be generalized to any homogeneous system which preserves one or more components of angular momentum, such as MHD or stratified turbulence. We focus on the case of MHD turbulence where we derive a Loitsyansky-like invariant and show that a fully nonlinear decay model based on this invariant produces results compatible with the experimental data. The invariant exists for highly conducting and weakly conducting fluids (i.e., high and low Rm ) and for any value of the imposed mean magnetic field, including zero field. The chapter concludes with a brief discussion of how these ideas extend to stratified turbulence.

12.1 INTRODUCTION Our understanding of the large-scale dynamics of homogeneous turbulence has had a checkered career. It started out well with a remarkable result, due to Loitsyansky, which says that, provided certain conditions are satisfied, isotropic turbulence possesses the invariant
(12.1) I = − r2 u · u dr = constant. (Here u and u are the velocities at x and x = x + r, respectively.) © 2006 by Taylor & Francis Group, LLC

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A little later Landau pointed out that the invariance of I is a direct consequence of the general law of conservation of angular momentum. That is to say, I is a measure of the angular momentum possessed by a cloud of turbulence and its invariance is a manifestation of the fact that the cloud maintains its angular momentum as it evolves. Now (12.1) is not just of academic interest. It is important because, as we shall see, it allowed Kolmogorov to predict therate of decay  of energy in a cloud of freely evolving turbulence. Specifically, if we let u 2 = u 2x = u 2y = u 2z , then Kolmogorov predicted u 2 (t) ∼ t −10/7 ,

(12.2)

a law which turns out to be fairly close to some (but by no means all) observations. Thus, for a while, it looked like we had a sound theory of the large scales. However, in 1956, G.K. Batchelor pointed out that, due to rather subtle effects associated with the pressure field, the assumptions underlying Loitsyansky’s proof of (12.1) are suspect. The same objections also invalidate Landau’s later assertion that I is a measure of the angular momentum of the turbulence. Suddenly, (12.1) and by implication (12.2), were in doubt. Around the same time, the quasinormal closure scheme was gaining in popularity and, as we shall see, it too suggested that I is time dependent; thus, a general consensus emerged that (12.1) and (12.2) are wrong. The final nail in the coffin came in 1967 when Saffman showed that, for certain kinds of isotropic turbulence, I does not even exist (it diverges). Loitsyansky’s invariant became Loitsyansky’s integral and Kolmogorov’s decay law looked destined for the scrap heap. Curiously, though, (12.2) provides a reasonably good estimate of the decay of certain types of freely evolving turbulence. In Section 12.2 through Section 12.5, we shall summarize the claims and counterclaims of Landau, Loitsyansky, Batchelor, and Saffman, ending with an overall assessment of the validity, or otherwise, of (12.1) and (12.2). It turns out that the nub of the problem lies in the ability of the pressure field to transmit information (via pressure waves) over long distances. This means that, in principle, remote parts of a turbulent flow can be statistically correlated and those finite, long-distance correlations call into question the validity of (12.1). However, we shall see that, for freely evolving turbulence which has arisen from certain types of initial conditions, the long-range correlations are weak and that, to within a reasonable level of approximation, the classical view of Landau and Loitsyansky prevails. We conclude, in Section 12.6 through Section 12.9, by showing that many of these ideas carry over to MHD turbulence and stratified turbulence.

12.2 LOITSYANSKY’S INTEGRAL FOR ISOTROPIC TURBULENCE The double and triple longitudinal correlations, f (r ) and K (r ), are governed by the Karman–Howarth equation   ∂  2 4 ∂  4 ∂  4   (12.3) u r f (r, t) = u 3 r K (r ) + 2
u 2 r f (r ) . ∂t ∂r ∂r Suppose we integrate this equation from r = 0 to r → ∞. The result is ⎡ ∂ ⎣ 2 u ∂t


∞

⎤

    r 4 f (r ) dr ⎦ = u 3r 4 K ∞ + 2
u 2r 4 f  (r ) ∞ ,

(12.4)

0

Let us now make the plausible (though, as it turns out, questionable) assumption that f (r ) and K (r ) are exponentially small at large r . In effect, we are assuming that remote points in a turbulent flow are statistically independent. If this is true, then we obtain an extraordinary result. The nonlinear © 2006 by Taylor & Francis Group, LLC

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effects, as represented by u 3 K , vanish and we find (Loitsyansky, 1939)
∞ 2 I = 8 u r 4 f dr = constant.

167

(12.5)

0

I was once known as Loitsyansky’s invariant, though it is now more commonly referred to as Loitsyansky’s integral. The factor of 8 is incorporated in (12.5) so that it may be rewritten in the form
I = − r2 u · u dr = constant. (12.6) This is a remarkable equation. If correct, then it is as important for the large scales as Kolmogorov’s theories are for the small. It is important because it can be combined with the energy equation u3 du 2 = − ,  ∼ 1 (12.7) dt  to estimate the rate of decay of energy in isotropic turbulence. (Here l is the integral scale.) That is to say, (12.6) implies that, for freely evolving turbulence, u 2 5 = constant

(12.8) 2

and this may be combined with (12.7) to predict the evolution of u (t). (We are assuming here that the longitudinal correlation function, f , is approximately self-similar, i.e., f (r/l) is of fixed form for all t.) The integral I has physical significance in terms of the energy spectrum E (k). If we expand sin (kr ) in a Taylor series in kr and assume that u · u  falls off sufficiently rapidly with r , then we obtain



E (k) = Lk 2 42 + I k 4 242 + · · · · · ·, (12.9)


where, L=

u · u dr.

(12.10)

The integral L is known as Saffman’s integral and it is thought by many to play a key role in the dynamics of the large scales, possibly as important as I . It is readily confirmed that L = 4u 2 [r 3 f ]∞ .

(12.11)

Thus, following our tentative assumption that f (r ) is exponentially small at large r, we would expect L = 0 and

E (k) = I k 4 242 + · · · . (12.12) We shall refer to spectra of the type E (k) ∼ k 4 + · · · as Batchelor spectra (because their key properties were established by Batchelor and Proudman, 1956), and to spectra like E (k) ∼ k 2 + · · · as Saffman spectra. In the classical theories of Loitsyansky and Landau, [ f (r )]∞ and [K (r )]∞ are assumed to be exponentially small, and thus all spectra are Batchelor spectra and I is an invariant.

12.3 KOLMOGOROV’S DECAY LAWS IN ISOTROPIC TURBULENCE Kolmogorov exploited the alleged invariance of I to estimate the decay of u 2 (t) in freely evolving turbulence. He noted that (12.7) and (12.8) require (12.13) u 2 (t) ∼ t −10/7 ,  ∼ t 2/7 (Kolmogorov, 1941). These are known as Kolmogorov’s decay laws and, by and large, they are a reasonable fit to the experimental data and to certain (but not all) numerical simulations. The growth of the integral scale l is, at first sight, unexpected. This is normally interpreted as follows. Equation 12.6 and Equation 12.12 imply that E (k) is of fixed shape for small k. Thus, as the turbulence decays, E (k) must evolve as shown in Figure 12.1, with the spectrum collapsing from the high-k end. The result is an increase in l. © 2006 by Taylor & Francis Group, LLC

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E(k)

t k

FIGURE 12.1 The decay of E(k) in isotropic turbulence.

12.4 LANDAU’S ANGULAR MOMENTUM IN ISOTROPIC TURBULENCE The simplicity of (12.6) is striking. This suggests that there is some underlying physical principle at work and indeed there is. It was pointed out by Landau, in the first edition of Landau and Lifshitz’s Fluid Dynamics (English edition, 1959), that the invariance of I is a direct consequence of the law of conservation of angular momentum. In fact, Landau showed that
 

I = H2 V, H = (x × u)d V, (12.14) V

where V is the volume of some large cloud of turbulence, V >> 3 ,  being the integral scale. Combining this with (12.6), we obtain the Landau–Loitsyansky equation
 

I = − r2 u · u dr = H2 V = constant. (12.15) Crucially, however, Landau had to make the same assumption as Loitsyansky: that remote points in a turbulent flow are statistically independent in the sense that f ∞ and K ∞ are exponentially small. If this is not true, then (12.15) fails. We now consider Landau’s proof of (12.15). The first question to address is: why should a cloud of turbulence possess any net angular momentum, H? Consider the portion of a wind tunnel grid shown in Figure 12.2. Evidently, angular momentum (vorticity) is generated at the surface of the bars and then swept downstream. In fact, we might suspect that the grid injects angular momentum into the flow because, if loosely suspended, it will shake in response to the hydrodynamic forces. Now it

FIGURE 12.2 Angular momentum is injected into a turbulent flow by a grid. © 2006 by Taylor & Francis Group, LLC

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Turbulence almost isotropic away from surface

FIGURE 12.3 Landau’s thought experiment.

might be thought that, in a large cloud of turbulence, (V >> 3 ), the total angular momentum will be very nearly zero because the vortices have random orientations. In a sense, this is true; however, the central limit theorem tells us that this cancellation will be imperfect. Moreover, the theorem suggests that, provided V >> 3 ,   H = 0, H2 ∼ V.  

In short, we might anticipate that H2 V is finite  and independent of V , as suggested by (12.15). The next question is: in what sense is H2 V an invariant? To answer this, Landau considered the following thought experiment. Consider a large cloud of turbulence freely evolving in a closed sphere of radius R, R >> , as shown in Figure 12.3. That is, we stir up the contents of the sphere and then leave the turbulence to decay. In any one realization of this experiment, we have

2 (x × u )d V  . H = (x × u)d V · V

V

In addition, since u integrates to zero in a closed domain, the flow has zero linear momentum
L = ud V = 0. (Actually, we shall see shortly that the vanishing of L through the use of a closed domain is a crucial ingredient of Landau’s theory. Different results are obtained in an open domain in cases where L is nonzero.) Combining these expressions, it may be shown that

  2 2 x − x u · u d V d V  . (12.16) H =− Now, suppose we repeat the experiment many times and form ensemble averages of (12.16) over many realizations. This yields

 2 (12.17) r2 u · u  dr d V, H =− which is beginning to look remarkably like Loitsyansky’s integral. Finally, Landau, like Loitsyansky before him, assumed that f ∞ is exponentially small. In such a case, only those velocity correlations taken close to the boundary are aware of the presence of this surface and in this sense the © 2006 by Taylor & Francis Group, LLC

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turbulence

is approximately homogeneous and isotropic. Also, the far-field contributions to the integral r2 u · u  dr are small. Integral (12.17) then reduces to
 2   H = −V r2 u · u  dr + O  R V . (12.18)

In the limit R  → ∞ we have 

H

2




V =−

r2 u · u  dr = I.

(12.19)

Thus, provided f ∞ is exponentially small, I is proportional to H2 . However, this still does not explain why I is an invariant, since in each realization of Landau’s thought experiment,

dH dt = T
, where T
is the viscous torque exerted by the boundaries. Luckily, the central limit theorem comes to our rescue again. We can estimate T
on the assumption that the eddies near the boundary are randomly orientated. It turns out that this suggests that T


has negligible influence as R  → ∞. In this sense, then, H (and hence H2 ) is conserved in each realization and it follows that I is an invariant of the flow. In summary, then, in isotropic turbulence we have
 

(12.20) I = − r2 u · u dr = H2 V = constant, provided that the long-range correlations are sufficiently weak. For the case of anisotropic (but homogeneous) turbulence, these arguments may be generalized to yield (Davidson, 1997, 2000):
 2  

Ix = − r y + r z2 u · u  yz dr = Hx2 V = constant (12.21)
Iy = −
Iz = −

 2  

r z + r x2 u · u zx dr = Hy2 V = constant

(12.22)

 2  

r x + r y2 u · u x y dr = Hz2 V = constant,

(12.23)

where u · u x y = u x u x + u y u y  and so on. One implication of (12.21) through (12.23) is that, if at t = 0 the turbulence is anisotropic in the sense that the various components of H are different, then this anisotropy is preserved throughout the subsequent decay. The fact that the invariance of I could be established by two distinct routes and that Kolmogorov’s decay law, which is based on the conservation of I , is reasonably in line with some of the experiments meant that most people were, for some time, happy with (12.15). Everything changed, however, in 1956.

12.5 LONG-RANGE CORRELATIONS IN HOMOGENOUS TURBULENCE 12.5.1 THE OBJECTIONS OF BIRKHOFF, BATCHELOR, AND SAFFMAN Batchelor was struck by the observation of Proudman and Reid (1954) that, according to the quasinormal (QN) closure model, I is time dependent: d2 I d 7 = 8 [u 3r 4 K ]∞ = (4)2 2 dt dt 5


∞ 0

© 2006 by Taylor & Francis Group, LLC

(E 2 k 2 ) dk.

(12.24)

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The implication is that, at least within the framework of the QN closure, long-range correlations are set up, of the form uuu  ∞ ∼ r −4 . Of course, the QN model is now known to be dynamically flawed and thus little weight should be given to (12.24). However, this was not known until the 1960s and thus there was some incentive to try to establish whether these long-range correlations are ageneric feature of homogeneous turbulence or simply an artifact of the QN model. So Batchelor and Proudman (1956) set aside the QN model and asked: under what conditions would we expect long-range correlations to arise? They retained the assumption that f ∞ decays faster than r −3 so that L in (12.11) is zero and E ∼ k 4 . However, they generalized the analysis to include anisotropic, as well as isotropic, homogeneous turbulence. Their key observation was that long-range correlations can be established through the pressure field, which allows remote points in a field of turbulence to interact. In particular, they found that, provided the turbulence is anisotropic, there exist long-range correlations of the form u i u j p   ∼ r −3 and these, in turn, induce triple correlations of the form uuu  ∞ ∼ C3r −4 , for some unknown C3 . The anisotropic version of the Karman–Howarth equation then gives uu  ∞ ∼ r −5 . Under these conditions, the generalized Loitsyansky integral
Ii jmn =

  ri r j u m u n dr

is convergent but time dependant. Curiously, though, Batchelor and Proudman were unable to find any long-range correlations when the symmetries of isotropy are imposed. Moreover, they noted that the wind-tunnel data, though inconclusive, tended to support the classical theory, rather than theirs. Indeed, they commented that “. . . it is disconcerting that the present theory [i.e., theirs] cannot do as well as the old.” A decade later, Saffman (1967), perhaps inspired by Birkhoff (1954), pointed out that f ∞ ∼ r −3 was a dynamical possibility. In such cases, L in (12.11) is nonzero and strictly constant, E ∼ k 2 , and Ii jmn diverges. The physical nature of the invariance of L can be seen from (12.10), rewritten in the form 
2  L= ud V V for some large volume V. Evidently, L is a measure of the linear momentum possessed by a large cloud of turbulence, and its invariance follows from the principle of conservation of linear momentum. Whether we have a Saffman (E ∼ k 2 ) or a Batchelor (E ∼ k 4 ) spectrum depends on the initial conditions. If the linear momentum in some volume V grows as V 1/2 , then L is finite and we have a Saffman spectrum. On the other hand, if the linear momentum grows more slowly with V , then L is zero and E ∼ k 4 . Both types of turbulence can, in principle, exist, and indeed both are seen in computer simulations. However, there are still arguments over which type is observed in, say, grid turbulence. Saffman (1967) suggested that grid turbulence may be of the E ∼ k 4 form, and this is consistent with measurements of the rate of decay of energy in the final period of decay (Monin and Yaglom, 1975). Consequently, in the remainder of this chapter we shall focus on turbulence with an E ∼ k 4 spectrum.

12.5.2 A REAPPRAISAL OF THE LONG-RANGE PRESSURE FORCES IN E ∼ k4 TURBULENCE Let us summarize the position so far. For E ∼ k 4 spectra, Loitsyansky’s integral converges, though it could be time dependent. That is to say, long-range pressure forces induce long-range correlations of the type u i u j p   ∼ r −3 , which in turn induce long-range triple correlations, uuu  ∞ ∼ C3r −4 . The generalized Karman–Howarth equation then tells us that, provided C3 is nonzero, uu  ∞ ∼ r −5 in anisotropic turbulence, but uu  ∞ ∼ r −6 in isotropic turbulence (because symmetry kills the leading order term). More importantly, in anisotropic turbulence, the long-range triple correlations, © 2006 by Taylor & Francis Group, LLC

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uuu  ∞ ∼ C3r −4 , are sufficiently strong to assure the time dependence of the Loitsyansky-like integral Ii jmn (Batchelor and Proudman, 1956). However, Batchelor and Proudman could find no long-range triple correlations when the symmetries of isotropy are enforced; the implication was that C3 might be zero in the isotropic case. Moreover, they found no evidence of long-range correlations in grid turbulence. We are left, therefore, with the problem of knowing the conditions, if any, under which the Landau–Loitsyansky equation is valid in E ∼ k 4 turbulence. We might dispense with Saffman’s objections because these relate to E ∼ k 2 spectra and even point out that Batchelor and Proudman’s paper is inconclusive in the case of isotropic turbulence. However, there is something disconcerting about (12.24). The point is that, although the QN approximation is a flawed dynamical model, it does represent a legitimate kinematic initial condition. Thus, for suitable initial conditions, (12.24) is valid at t = 0, if not for t > 0. The implication is that we cannot throw out Batchelor’s long-range effects in isotropic turbulence on the basis of a purely kinematic argument. The computer simulations of turbulence are interesting in this respect. Sometimes, they have Gaussian initial conditions and often they start with E ∼ k n , n > 4. An example of just such a simulation by Lesieur et al. (2000) is shown in Figure 12.4. One observes the transient growth of a k 4 component in E as the turbulence recovers from its initial condition. However, once the turbulence becomes fully developed, with a full range of length scales and a k 4 spectrum, I seems to be almost constant, as originally claimed by Loitsyansky. Thus, it would seem that, at least for matur e turbulence, Loitsyansky and Landau were, more or less, correct. So what is going on? The first thing to note is the apparent discrepancy between Proudman and Reid (1954) and Batchelor and Proudman (1956). The former show that, for isotropic turbulence, the QN closure model predicts 12.24. Batchelor and Proudman, on the other hand, could find no

10−1 t/te = 0 10−2

10−3

50 100

10−4

k4

E(k, t)

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10−5

850 k−5/3

10−6

10−7

10−8

1

101 k

102

FIGURE 12.4 Decay simulations of Lesieur et al. (2000). This is large eddy simulation using hyperviscosity. © 2006 by Taylor & Francis Group, LLC

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long-range effects when the symmetries of isotropy are imposed, which suggests that I might be constant. Let us go back and repeat the analysis of Batchelor and Proudman. We shall adopt the same approach that they did and take initial conditions in which well separated points are statistically independent. In such a case, f ∞ , K ∞ , and fourth-order cumulants {[u i u j u k u l ]cum }r →∞ are all exponentially small at t = 0. The question, then, is what happens for t > 0. (Note that taking [u i u j u k u  ]cum ∼ 0 for well separated points is not the same as the QN approximation, which requires [u i u j u k u  ]cum = 0 for all combinations of x, x and x . To illustrate the difference, consider the flatness factor, , of the velocity difference. The QN approximation would require  = 3 for all r , which is clearly not true, here we require only that  − 3 is exponentially small for r → ∞, which could well be true.) A re-examination of the isotropic case in the spirit of Batchelor and Proudman reveals that (Davidson, 2000, 2004)
 2  2  3 ∂[u 3 K ]∞ u x 2u x − u 2y − u 2 dr, (12.25) = z 4 ∂t 4r from which

d  3 4  d2 I u r K ∞=6 = 8 2 dt dt

where J and s are give by


J=




ss  dr = 6J,

ss  dr, s = u 2x − u 2y .

(12.26)

(12.27)

Some hint as to where these expressions come from is given by the following argument. Consider a patch of turbulence located near x = 0. It is not difficult to show that the far-field pressure on the x-axis induced by this patch of turbulence is
 2   3 −1 2u x − u 2y − u 2x dx. (12.28) p (r eˆ x ) = (4r ) Thus, we might expect the correlation between u 2x at x = 0 and p  (r eˆ x ) to be
   2  2  2     2  u 2x 2 u x − u y − u z dx . u x p ∞ = (4 r 3 )−1 Now, the governing equation for the triple correlations takes the form     ∂ Si jk ∂p ∂p ∂   + 
(∼) u i u j p  − u k u i + uj + uuuu = − ∂t ∂rk ∂x j ∂ xi

(12.29)

(12.30)

and, in particular, the longitudinal triple correlation obeys 

 ∂ 3 ∂  2  [u K (r )] + uuuu = − u x p (r eˆ x ) − uu  p + 
(∼) , ∂t ∂r

(12.31)

where uu  p is a symbolic representation of terms involving u, p and u  . It turns out that the dominant long-range contribution to u 3 K comes from the term involving u 2x p  and thus we have
    2   2   2  ∂  3  3 2 u u K ∞= 2 ux − u y − uz dr, (12.32) x ∂t 4 r 4 which is (12.25). For isotropic turbulence, this can be written in the more compact form
∂  3  3 ss  dr. u K ∞= ∂t 4 r 4 © 2006 by Taylor & Francis Group, LLC

(12.33)

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Combining this with (12.4) yields d  3 4  d2 I = 8 u r K ∞=6 2 dt dt




ss  dr = 6J

(12.34)

as required. It should be emphasized that (12.34) is true for all t, provided that {[u i u j u k u l ]cum }r →∞ remains exponentially small for t > 0. Now the experimental data suggests that, as far as the fourth-order correlations are concerned, the joint-probability distribution for u (in mature turbulence) is very close to Gaussian for well separated points. (See, for example, Van Atta and Yeh, 1970.) We might hope, therefore, that (12.34) is a good approximation throughout most of the decay; this is the position adopted in Davidson (2000). (It should be noted, however, that it is difficult to tell from the experiments whether {[u i u j u k u l ]cum }r →∞ is exponentially or algebraically small, and thus (12.34) must be regarded with considerable caution.) Actually, it can be shown that the QN result, (12.24), is a special case of (12.34). That is, if we insist that [u i u j u k u  ]cum = 0 for all combinations of x, x , and x , close or distant, (and it certainly is not), then it may be shown that
14 6JQ N = (QN only). (12.35) u · u 2 dr 5 Next, applying Rayleigh’s theorem, we have
∞
 2 2 14 7 2 E k dk u · u  dr = (4)2 6JQ N = 5 5 0

(QN only).

(12.36)

Combining this with (12.34) yields (12.24). It should be emphasized, however, that little credence should be given to the QN estimate of J . Now J is positive since 

J=

ss  dr =

2

sd V

V

(12.37)

and thus we may introduce a dimensionless parameter  and write J =  u 4 3 ,  ≥ 0, for some suitably defined integral scale. Thus, d  3 4  d2 I u r K ∞ = 6 u 4 3 . = 8 dt 2 dt

(12.38)

Therefore, I is indeed time dependent and, as suggested by Batchelor, the flaw in Loitsyansky and Landau’s arguments lies in the existence of long-range pressure forces. However, the results of wind-tunnel experiments tentatively suggest that, if  is based on the estimate  = (I /u 2 )1/5 , then  is relatively small. In fully developed turbulence, for example, it is found that  lies in the range 0 → 0.03 with an average value of, perhaps,  ∼ 0.01 (Davidson, 2000, 2004). This might be compared with the QN estimate of  Q N ∼ 0.6. Thus, it seems likely that, in grid turbulence, the long-range effects are rather weak, and this is why I is approximately conserved in mature turbulence (Figure 12.4). Actually, the fact that I is only a weak function of time has long been acknowledged. (See, for example, Lesieur, 1990.) It seems also that the QN approximation greatly overestimates the strength of the long-range effects, by almost two orders of magnitude. We might note in passing that the EDQNM closure model specifies dI = 8[u 3r 4 K ]∞ ∼  (t) JQ N (EDQNM only), (12.39) dt where  is a somewhat arbitrary model parameter with the dimensions of time. © 2006 by Taylor & Francis Group, LLC

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Evidently this simply amounts to an assertion that [r 4 K ]∞ is nonzero, with its magnitude set by the arbitrary parameter . It is hard to reconcile (12.39) with (12.34). In particular, the arbitrary removal of a time derivative, as a result of Markovianisation, seems difficult to justify. Perhaps we should not be surprised by the weakness of the long-range correlations. This is, after all, in accord with our intuition. If the long-range correlations were strong, then the influence of the boundaries of a closed domain would be felt everywhere in the interior, no matter how large the domain size relative to the integral scale. Thus, for example, the lateral boundaries of a wind tunnel could exert an influence on the decay of grid turbulence, even when the tunnel size greatly exceeds the turbulence integral scale. Yet, there is little evidence that this is the case. In short, our intuition, based on experimental observations, suggests that the long-range correlations are weak in mature turbulence and this is in accordance with the observed smallness of . In summary, then, when we have a E ∼ k 4 spectrum, Loitsyansky’s integral is time dependent and this is a result of Batchelor’s long-range pressure forces. However, we have no reliable means of predicting the strength of these long-range forces and hence no reliable means of predicting the rate of change of I . All we can say with confidence is that the long-range forces appear to be weak in fully developed turbulence. In this respect the classical view of Loitsyansky and Landau is not so far from the truth. It is a beautiful but flawed theory. Let us now redevelop these ideas in the context of MHD turbulence.

12.6 THE GROWTH OF ANISOTROPY IN MHD TURBULENCE Let us start by considering a somewhat contrived thought experiment, first proposed by Davidson (1995,1997), which is designed to bring out the crucial role played by angular momentum conservation in MHD turbulence. For simplicity, we temporarily leave aside viscous forces. Suppose that a nonmagnetic, conducting fluid of conductivity  and permeability is held in a large insulated sphere of radius R (Figure 12.5). The sphere sits in a uniform, imposed field B0 , so the total magnetic field is B = B0 + b, b being associated with the currents induced by the motion u within the sphere. We place no restriction on the size of the magnetic Reynolds number,Rm =  ul, nor on the interaction parameter, which we define to be N = B02  u,  being the initial integral scale of turbulence. When Rm is small, we have |b| << |B0 |, but in general |b|may be as large as |B0 |. At t = 0, the fluid is vigorously stirred and then left to itself. We wish to characterize the anisotropy introduced into the turbulence by B0 . We attack the problem as follows. The global torque exerted on the fluid by the Lorentz force is

T = x × (J × B0 )d V + x × (J × b)d V. (12.40) However, a closed system of currents produces zero net torque when it interacts with its self-field, b, and it follows that the second integral on the right is zero. (This is because an isolated system must B

t=0

t

∞

FIGURE 12.5 A magnetic field organizes turbulence into columnar eddies. © 2006 by Taylor & Francis Group, LLC

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conserve angular momentum so that any isolated system of currents interacting with their associated magnetic field cannot produce a net torque. Alternatively, we can write J × b in terms of Maxwell stresses, so the volume integral becomes a surface integral which tends to zero as the surface recedes to infinity.) The first integral, on the other hand, can be transformed using the identity 2x × [v × B0 ] = [x × v] × B0 + ∇ · [(x × (x × B0 )) v] , where v is any solenoidal field. Setting v = J, we obtain  
1 (x × J)d V × B0 = m × B0 , T= 2

(12.41)

(12.42)

where m is the net dipole moment of the current distribution within the sphere. Evidently, the global angular momentum evolves according to
dH = T = m × B0 ; H = (x × u)d V.  (12.43) dt We see immediately that H// , the component of H parallel to B0 , is conserved. This, in turn, places a lower bound on the total energy of the flow, 
−1 2 2 E = E b + E u ≥ E u ≥ H// 2 x⊥ d V , (12.44)


where Eb =

 2 b 2 d V


,

Eu =



u2 2 d V .

2 (This follows from the Schwarz inequality in the form H2// ≤ u2⊥ d V x⊥ d V .) However, the energy declines due to Joule dissipation and thus we also have


 2 d 1 1 2 d b 2 d V = − J2 d V . (12.45) u d V + dt 2 dt  VR

V∞

VR

Evidently, one component of angular momentum is conserved, requiring that E is nonzero, yet energy is dissipated as long as J is finite. The implication is that the turbulence evolves to a state in which J = 0, yet E u is nonzero, to satisfy (12.44). However, if J = 0, then Ohm’s law reduces to E = −u × B0 , while Faraday’s law requires that ∇ × E = 0. It follows that, at large times, ∇ × (u × B0 ) = (B0 · ∇) u = 0, and thus u becomes independent of z as t → ∞. The ultimate state is therefore two dimensional, of the form u// = 0, u⊥ = u⊥ (x⊥ ). In short, the turbulence eventually approaches a state consisting of one or more columnar eddies, each aligned with B0 (see Figure 12.5). Note that all of the components of H, other than H// , are destroyed during this evolution. When Rm is small, this transition will occur on the timescale of = (B02 /)−1 , the magnetic damping time. The proof of this is straightforward. At low Rm , the current density is governed by J =  (−∇V + u × B0 .) , Thus, the dipole moment becomes

   1 (V x) × dS. x × Jd V =  2 x × (u × B0 ) d V −  2 m= 2 V V S

(12.46)

(12.47)

The surface integral vanishes while the volume integral transforms, with the aid of (12.41), to yield  (12.48) m =  4 H × B0 . © 2006 by Taylor & Francis Group, LLC

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Substituting into (12.43), we obtain H⊥ dH =− , dt 4

−1 = B02 /.

(12.49)

Thus, H// is conserved while H⊥ declines as H⊥ = H⊥0 exp(−t/4 ). In summary, whatever the initial condition and for any Rm or N , our confined flow evolves towards the two-dimensional state u⊥ = u⊥ (x⊥ ) , H// = H// (t = 0) , H⊥ = 0 , u// = 0.

(12.50)

The simplicity of this result is surprising, particularly since we are dealing with the evolution of a fully nonlinear system. This is the first hint that the conservation of angular momentum plays a crucial role in MHD turbulence.

12.7 THE LANDAU INVARIANT FOR HOMOGENEOUS MHD TURBULENCE The preceding arguments involving angular momentum are reminiscent of Landau’s derivation of the Loitsyansky invariant for isotropic turbulence. It is natural to see whether Landau’s arguments can be generalized to MHD turbulence. We shall see that they can and that, in the absence of long-range statistical correlations, MHD turbulence possesses an integral invariant. Let us repeat Landau’s thought experiment, adapted now to MHD turbulence (Figure 12.6). As in Section 12.4, our fluid is held in a large sphere of radius R, R >> , only this time it sits in a uniform, imposed magnetic field, B0 . At t = 0, the fluid is set into turbulent motion and then left to itself. We are interested in the behavior of the turbulence for periods in which the integral scale, , remains much smaller than R. Since R >> , we may follow the strategy of Section 12.4 and ignore the viscous torque exerted by the fluid on the boundary r = R. It follows from (12.43) that H// is conserved during this period and, following Landau’s arguments, we have, for any value of Rm and any N ,

   2 r2⊥ u⊥ · u ⊥ drdx = constant, (12.51) H// = − where r = x − x. If we now ignore Batchelor’s long-range pressure forces, we have, in the spirit of Landau,
 2 

  (12.52) I// = H// V = − r2⊥ u⊥ · u ⊥ dr = constant.

B0

艎

2R

FIGURE 12.6 MHD turbulence evolving in a large sphere and subject to a mean field. © 2006 by Taylor & Francis Group, LLC

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This applies to homogenous turbulence at any value of N . It was first noted in the context of low- Rm turbulence, by Davidson (1997), and then later extended to high-Rm turbulence in Davidson (2000, 2001). As in conventional turbulence, this invariant may also be derived from the Karman–Howarth equation. The argument proceeds as follows. Consider the equation of motion,  

∂u i ∂ p ∂ [u k u i − bk bi / ] − (12.53) =− + (J × B0 )i  +
∇ 2 u i , ∂t ∂ xk ∂ xi  in which b is the induced magnetic field associated with the current J and p includes the magnetic pressure, b2 2 . This yields the generalized Karman–Howarth equation

 

   ∂   ∂  ui u j = u i u k u j − bi bk u j  − u j u k u i − bj bk u i  + 2
∇ 2 u i u j ∂t ∂rk    ∂    1  1 ∂   (J × B0 )i u j + (J × B0 ) j u i . (12.54) pu j − p ui + +  ∂ri ∂r j  Consider first the case where B0 and b are both zero, i.e., conventional hydrodynamic turbulence. Then, following the arguments of Batchelor (1953), it is readily shown that (12.54) yields
  Ii jmn = ri r j u m u n dr = constant, (12.55) provided, of course, that there are no long-range correlations. This is a generalization of Loitsyansky’s integral. When b is finite, but B0 remains zero (no mean field), Batchelor’s arguments may be repeated and again we find that Ii jmn is an invariant. This was first noted by Chandrasekhar (1951) in the context of isotropic turbulence. Let us turn now to the case where B0 is finite. In the absence of long-range correlations, only the final term in (12.54) can contribute to the rate of change of integrals of the type Ii jmn and thus

 2      1 d r2⊥ u⊥ · u ⊥ dr = r⊥ (J × B0 )⊥ · u ⊥ + J × B0 ⊥ · u⊥ dr. (12.56) dt  The integrand on the right consists of terms of the form r⊥2 (Jy ux − Jx uy ) and r⊥2 (Jy u x − Jx u y ). (We take B0 to Such terms can be converted into surface integrals since 3y 2 Jy = point in the z direction.)  2 3 ∇ · y J , 2yu y = ∇ · y u , etc. Moreover, in the absence of long-range correlations, these surface integrals are zero, and thus (12.56) yields
  I// = − r2⊥ u⊥ · u ⊥ dr = constant, (12.57) (any N , any Rm ). We have arrived back at (12.52), but by a different route. However, the Landau-like derivation is to be preferred since it exposes the physical origin of the invariant (12.52). Of course, (12.52) comes with all the usual caveats discussed in Section 12.5. In particular, it applies only if the turbulence has negligible linear momentum, and thus has a low-k spectrum of the form E ∼ k 4 . Moreover, it applies only if the long-range statistical correlations are weak, as seems to be the case for fully developed hydrodynamic turbulence. Despite these limitations, it is an important result because it extends the earlier studies of Landau and Chandrasekhar to MHD turbulence in the presence of a mean magnetic field. We shall now show how (12.57) may be used to predict the rate of decay of energy in low-Rm turbulence.

12.8 DECAY LAWS AT LOW MAGNETIC REYNOLDS NUMBER Let us now repeat Kolmogorov’s arguments of Section 12.3, adapted to homogeneous, low-Rm MHD turbulence. The aim is to determine how a field of initially isotropic turbulence evolves in an imposed magnetic field (Figure 12.7). © 2006 by Taylor & Francis Group, LLC

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B0

∞ t Anisotropic, homogeneous

t=0 Isotropic, homogeneous

FIGURE 12.7 Homogeneous MHD turbulence at low Rm .

We start with the curl of Ohm’s law (12.46), ∇ × J = B0 · ∇u, from which the Joule dissipation can be estimated as  2  

−1  J min 2 u 2 . (12.58) ∼ , = B02   //

(Here min and // are suitably defined integral scales.) Now we know that the effect of B0 is to introduce anisotropy into the turbulence, with // > ⊥ . Thus, we have  2     J  ⊥ 2 u2 = ,  2 //

(12.59)

where  is of order unity. (In fact it can be shown that  = 2 3 when the turbulence is isotropic.) We can use (12.59) to estimate the rate of decay of kinetic energy. That is, the energy equation,    

d 1  2 u = −
2 − J2 , 2 dt can be written as u3 du 2 − = − dt ⊥



⊥ //

2

u2 .

(12.60)

(12.61)

  Here, we define u 2 = 13 u2 and we have made the usual estimate of the viscous dissipation. (In conventional turbulence,  is of the order of unity.) Now our energy equation might be combined with (12.57) in the form u 2 4⊥ // = constant, (12.62) which offers the possibility of predicting u 2 (t) as well as ⊥ and // . In low-Rm turbulence, it is conventional to categorize the flow according to the value of the interaction parameter, N =

B02 ⊥ u. When N is small (negligible magnetic effects), (12.61) and (12.62) reduce to du 2 u3 = − , dt 

u 2 5 = constant,

(12.63)

which yields the familiar Kolmogorov law u 2 ∼ t −10/7 . When N is large, on the other hand, inertia is unimportant and we have du 2 = − dt © 2006 by Taylor & Francis Group, LLC



⊥ //

2

u2 ,

u 2 4⊥ // = constant.

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It may be shown that ⊥ remains constant during the decay of high-N turbulence (Davidson, 1997) and in such a case our equations predict u 2 ∼ u 20 (t / )−1/2 ,

// = 0 [1 + 2t / ]1/2 , ⊥ = 0 ,

(12.65)

which we also know to be correct (Moffatt, 1967). For intermediate values of N , however, we have a problem. Equation (12.61) and Equation (12.62) between them contain three unknowns: u 2 , ⊥ , and // . To close the system, we might tentatively introduce the heuristic equation

d  2 (12.66) // ⊥ = 2 , dt which has the merit of being exact for N → 0 and N → ∞ but cannot be justified for intermediate N . (Essentially the same equation was proposed by Widland et al., 1998, in their one-point closure model of MHD turbulence.) Integrating (12.61), (12.62), and (12.66) yields (Davidson, 1999, 2000):     −10/7 u 2 u 20 = tˆ−1/2 1 + 7 15 tˆ3/4 − 1 N0−1  2 7    ⊥ 0 = 1 + 7 15 tˆ3/4 − 1 N0−1 /

(12.67–12.69)

    2 7 // 0 = tˆ1/2 1 + 7 15 tˆ3/4 − 1 N0−1 / , where N0 is the initial value of N and tˆ = 1 + 2 (t / ). (For simplicity, we have taken  =  = 1.) The high- and low-N results are special cases of (12.67)→(12.69). For the case of N0 = 7 15, we obtain the power laws (12.70) u 2 /u 20 ∼ tˆ−11/7 , // /0 ∼ tˆ5/7 . Indeed, these power laws are reasonable approximations to (12.67) and (12.69) for all values of N0 around unity. Experiments of low-Rm , homogeneous turbulence were carried out by Alemany et al. (1979) and they suggest u 2 ∼ t −1.6 for N0 ∼ 1. This compares favorably with (12.70), which predicts u 2 ∼ t −1.57 .

12.9 A LOITSYANSKY-TYPE INVARIANT FOR STRATIFIED TURBULENCE We conclude with a brief discussion of stratified turbulence. Consider stratified turbulence evolving in a spherical domain, V , whose radius is very much larger than the integral scale. It is readily shown that, when viscous stresses on the boundary are ignored, the vertical component of global angular momentum is conserved. We can therefore use Landau’s argument of Section 12.4, in conjunction with Equation (12.23), to show that, when long-range interactions are weak (Davidson, 2004),
 2      Hz /V = − r x2 + r y2 u x u x + u y u y dr = constant. (12.71) Thus, homogeneous, stratified turbulence has a Loitsyansky-like invariant in cases where the long-range interactions are weak. Of course, it is also possible to show that exactly the same result follows directly from (12.54), generalized to include buoyancy forces. In cases where the large scales evolve in a self-similar fashion, this invariant implies that u 2l⊥4 l// = constant . The dynamical implications of this are still not fully explored. © 2006 by Taylor & Francis Group, LLC

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12.10 CONCLUSIONS There are relatively few situations in which the momentum conservation laws can be used to make specific, nontrivial, testable predictions about the evolution of homogenous turbulence. The Landau– Loitsyansky equation, generalized to incorporate MHD and stratified turbulence, is one of the rare exceptions. Although its justification rests on the tentative assumption that the long-range interactions are weak, its predictions seem to be in line with the little experimental evidence we have.

REFERENCES Alemany, A. et al. (1979). Influence of external magnetic field on MHD turbulence. J. M´ec, 18. Batchelor, G.K. (1953). The Theory of Homogeneous Turbulence. CUP. Batchelor, G.K., Proudman, I. (1956). The large scale structure of turbulence. Phil. Trans. Roy. Soc. A, 248, 369–405. Birkhoff, G. (1954). Fourier synthesis of turbulence. Commun. Pure & Applied Math., 7,19–44. Chandrasekhar, S. (1951). The invariant theory of isotropic turbulence in MHD. Proc. Royal Soc. Lond., A, 204. Davidson P.A. (1995). The role of angular momentum in the magnetic damping of turbulence. J. Fluid Mech., 299. Davidson P.A. (1997). Magnetic damping of jets and vortices. J. Fluid Mech., 336. Davidson, P.A. (1999). In: Magnetohydrodynamics (Proc. of CISM 1999 Summer School), Eds. Davidson, P.A. & Thess A., Springer. Davidson, P.A., (2000). Was Loitsyansky Correct? J. Turbulence. Davidson, P.A. (2001). Introduction to Magnetohydrodynamics. CUP. Davidson, P.A. (2004). Turbulence. Oxford Univ. Press. Kolmogorov, A.N. (1941). On the degeneration of isotropic turbulence. Dokl. Akad. Nauk SSSR, 31(6), 538–541. Landau, L. D., Lifshitz, E.M. (1959). Fluid Mechanics, 1st ed., Pergamon Press. Lesieur, M. (1990). Turbulence in Fluids, 2nd ed., Kluwer Acad. Pub. Lesieur, M. et al. (2000). Eur. Conf. Comp. Methods Sci. Eng., Barcelona, Sept. 2000. Loitsyansky, L.G. (1939). Some basic laws of isotropic turbulence. Trudy Tsentr. Aero-Giedrodin. Inst., 440. Moffatt, H.K. (1967). On the suppression of turbulence by a magnetic field. J. Fluid Mech., 28. Monin, A.S., Yaglom, A.M. (1975). Statistical Fluid Mechanics. MIT Press. Proudman, I., Reid, W.H. (1954). On the decay of normally distributed turbulence. Phil. Trans. Roy. Soc., A, 247, 163–189. Saffman, P.G. (1967). The large-scale structure of homogeneous turbulence. J. Fluid Mech., 27, 581–593. Van Atta, C.W., Yeh, T.T. (1970). Some measurements of multi-point correlations. J. Fluid Mech., 41(1), 169–178. Widland, O. et al. (1998). Developement of a Reynolds stress closure model of MHD turbulence. Phys. Fluids, 10(8).

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the New Concept of 13 On Turbulence Modeling in Fully Developed Turbulent Channel Flow and Boundary Layer Ekachai Juntasaro and Varangrat Juntasaro CONTENTS 13.1 13.2 13.3 13.4

Introduction .......................................................................................................................... 183 Eddy Viscosity Turbulence Modeling .................................................................................. 184 New Concept of Turbulence Modeling ................................................................................ 185 Results and Discussion ......................................................................................................... 185 13.4.1 Fully Developed Turbulent Channel Flow ............................................................... 185 13.4.2 Turbulent Boundary Layer with Constant Pressure ................................................. 187 13.5 Conclusions .......................................................................................................................... 192 Acknowledgments.......................................................................................................................... 193 References...................................................................................................................................... 193

ABSTRACT The new concept of turbulence modeling is proposed as an attempt to simulate the effect of turbulence on the flows as realistically as possible. The turbulence effect is fundamentally not the diffusion phenomena that the concept of eddy viscosity models is based on. It is rather the convection phenomena that are responsible for the turbulence effect on the flows. This can be observed from the time-averaged Navier–Stokes equations that the extra terms, which are called the gradients of the Reynolds stresses, are derived from the convection term of the Navier–Stokes equations. The current work aims to present the new concept on modeling the gradients of the Reynolds shear stresses that does not rely on the eddy viscosity concept. The gradients of the Reynolds shear stresses are modeled in terms of the gradients of the product of the root-mean-square of the velocity fluctuations times a model constant. The direct numerical simulation (DNS) data of the fully developed turbulent channel flow and the turbulent boundary layer with constant pressure are used to evaluate the proposed concept in comparison with the eddy viscosity concept. The proposed concept shows much closer agreement with the DNS data, especially in the regions of viscous sublayer and buffer layer. Furthermore, it is found that the model constant is independent of Reynolds numbers and equal to –1/2 for both flows.

13.1 INTRODUCTION The conventional approach for predicting turbulent flows is to decompose variables in turbulence into a mean (e.g., a time average) plus a fluctuation according to Osborne Reynolds. Taking the mean of the Navier–Stokes equations leaves the mean rates of momentum transfer by the turbulence © 2006 by Taylor & Francis Group, LLC

183

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as extra unknowns in the Reynolds-averaged Navier–Stokes equations. These unknowns are called the gradients of the Reynolds stresses. Modeling of turbulence means replacing the gradients of the Reynolds stresses by the semiempirical equations calibrated by the data from experiments or direct numerical simulation (DNS). Replacing the Reynolds stresses by the product of the eddy viscosity and the mean rate of strain is one of the most popular methods in turbulence modeling. However, the concept of eddy viscosity models is based on the diffusion phenomena, which are fundamentally incorrect since the gradients of the Reynolds stresses are in fact derived from the convection terms of the Navier–Stokes equations. The current work aims to present an alternative approach on modeling the gradients of the Reynolds shear stresses that does not rely on the eddy viscosity concept. Furthermore, the proposed concept can be used to reduce the number of transport equations that must be solved in Reynolds stress models. The gradients of the Reynolds shear stresses can be modeled using the proposed concept instead of being solved from the transport equations in the Reynolds stress models. That means only four transport equations for the Reynolds normal stresses and the dissipation rate of turbulence kinetic energy need to be solved instead of seven transport equations for all Reynolds stresses and the dissipation rate of turbulence kinetic energy. The DNS data of the fully developed turbulent channel flow and the turbulent boundary layer with constant pressure are used to evaluate the proposed concept in comparison with the eddy viscosity concept. The standard high-Reynolds-number k − ε model [1], low-Reynolds-number k − ε model of Launder and Sharma [2], and k − ε − v
2 model of Durbin [3] are chosen to represent the turbulence models based on the eddy viscosity concept. The comparison is made for the gradient of the Reynolds

shear stress, dudyv , between the present concept and the chosen eddy viscosity models with the DNS data of the corresponding test cases.

13.2 EDDY VISCOSITY TURBULENCE MODELING Conventionally, the gradient of the Reynolds shear stress is modeled in terms of the gradient of the product of the eddy viscosity and the mean rate of strain, that is, du
v
d =− dy dy




d u¯
t dy

 ,

(13.1)

where u¯ is the time-averaged streamwise velocity and
t is the eddy viscosity. Hence, the effect of turbulence on the mean flow is taken into account via the eddy viscosity. For the standard high-Reynolds-number k − ε turbulence model [1], the eddy viscosity is modeled as k2
t = C  , (13.2) ε where k is the kinetic energy of turbulence, ε is the dissipation rate of k, and C is equal to 0.09. For the low-Reynolds-number k − ε turbulence model [2], the eddy viscosity is modeled as
t = C  f 

k2 , ε˜

(13.3)

where ε˜ is the modified ε and f  is the damping function. For the k − ε − v
2 turbulence model of Durbin [3], the eddy viscosity is modeled as k
t = C v
2 , ε where C is equal to 0.2, and v
2 is the Reynolds normal stress in the cross-stream direction. © 2006 by Taylor & Francis Group, LLC

(13.4)

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13.3 NEW CONCEPT OF TURBULENCE MODELING



In the proposed approach, the gradient of the Reynolds shear stress, dudyv , is modeled in terms of the gradient of the product of the root-mean-square of the velocity fluctuations times a model constant as follows: d(u r
ms vr
ms ) du
v
=C , (13.5) dy dy where u r
ms and vr
ms are the root-mean-square of the velocity fluctuations in the streamwise and the cross-stream directions, respectively, and C is the model constant. The root-mean-square of the velocity fluctuations in the streamwise and the cross-stream directions, u r
ms and vr
ms , can be found by extracting the values from the DNS data or by solving the transport equations for the Reynolds normal stresses in Reynolds stress models and taking the square root of the Reynolds normal stresses to obtain the root-mean-square of the velocity fluctuations. The former method is used in this chapter to evaluate the proposed concept in modeling the gradients of the Reynolds shear stresses. The model constant, C, in Equation (13.5) can be found by substituting the DNS data of the corresponding test cases for the Reynolds shear stress, u
v
, the root-mean-square of the velocity fluctuation in the streamwise direction, u r
ms , and the root-mean-square of the velocity fluctuation in the cross-stream direction, vr
ms . In the next section, the model constant C will be found for the fully developed turbulent channel flow at two Reynolds numbers and for the turbulent boundary layer with constant pressure at three Reynolds numbers.

13.4 RESULTS AND DISCUSSION 13.4.1 FULLY DEVELOPED TURBULENT CHANNEL FLOW The governing equation for the time-averaged streamwise velocity of steady incompressible fully developed turbulent channel flow is given as follows:


du
v
1 d p¯ d 2 u¯ + , = dy 2  dx dy

(13.6)

where u¯ is the time-averaged streamwise velocity, p¯ is the time-averaged pressure,
is the kinematic

viscosity,  is the fluid density, and dudyv is the gradient of the Reynolds shear stress, which is unknown and needs modeling. The DNS data of the fully developed turbulent channel √ flow of Kim et al. [4] at two Reynolds numbers (Re ≡ u  h/
= 180 and 395, where u  ≡ w / is the friction velocity and h is the channel half-height) are used for comparison.

The comparison is made between the gradient of the Reynolds shear stress, dudyv , from the DNS data and those obtained by using the eddy viscosity concept (Equation 13.1) with the standard highReynolds-number k − ε turbulence model (Equation 13.2) and the low-Reynolds-number k − ε turbulence model (Equation 13.3) in Figure 13.1 and Figure 13.2 at Re = 180 and 395. It is found that both models overpredict the negative peak in the near wall region and predict the positive peak that does not exist in the DNS data. The discrepancy between the gradients of the Reynolds shear stress predicted by the k − ε turbulence models and that obtained from the DNS data is greater at the higher Reynolds number. The gradient of the Reynolds shear stress using the proposed concept (Equation 13.5) in comparison with the k − ε − v
2 model of Durbin [3] (Equation 13.4) and the DNS data is presented in Figure 13.3 and Figure 13.4. The negative peak near the wall can be captured accurately by the proposed concept, although there is some deviation from the DNS data near the centerline region of the channel. It is observed that this deviation is less obvious as the Reynolds number grows higher. © 2006 by Taylor & Francis Group, LLC

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5.0E−04

Gradient of Reynolds shear stress

3.0E−04 1.0E−04 −1.0E−04 −3.0E−04 −5.0E−04 −7.0E−04 −9.0E−04 −1.1E−03

DNS data of Kim, Moin and Moser (4) Standard high-Re k-epsilon model Low-Re k-epsilon model of Launder and Sharma (2)

−1.3E−03 −1.5E−03 −1.7E−03

0.0

0.2

0.4

0.6

0.8

1.0

y

FIGURE 13.1 Gradient of Reynolds shear stress for fully developed turbulent channel flow at Re = 180 using k − ε turbulence models. 5.0E−03 3.0E−03 Gradient of Reynolds shear stress

April 24, 2006

1.0E−03 −1.0E−03 −3.0E−03 −5.0E−03 −7.0E−03 −9.0E−03 −1.1E−02

DNS data of Kim, Moin and Moser (4)

−1.3E−02

Standard high-Re k-epsilon model Low-Re k-epsilon model of Launder and Sharma (2)

−1.5E−02 −1.7E−02

0.0

0.2

0.4

0.6

0.8

1.0

y

FIGURE 13.2 Gradient of Reynolds shear stress for fully developed turbulent channel flow at Re = 395 using k − ε turbulence models.

Figure 13.5 and Figure 13.6 show that the proposed approach in modeling the gradient of the Reynolds shear stress exhibits very close agreement with the DNS data, especially in the regions of viscous sublayer, where 0 ≤ y + ≤ 5, and the buffer layer, where 5 < y + ≤ 30. It is found that the model constant, C, is equal to –1/2 at both Reynolds numbers. The deviation near the centerline of the proposed concept may be further improved by developing the concept as a two-layer turbulence model. This may be done using the current concept in the regions of viscous sublayer and buffer layer with the k − ε turbulence models in the other regions, where the gradient of the Reynolds shear stress predicted by the k − ε models agrees with the DNS data. © 2006 by Taylor & Francis Group, LLC

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1.0E−05 0.0E+00 Gradient of Reynolds shear stress

−1.0E−05 −2.0E−05 −3.0E−05 −4.0E−05 −5.0E−05 −6.0E−05 −7.0E−05 −8.0E−05 −9.0E−05

DNS data of Kim, Moin and Moser (4)

−1.0E−04

k-epsilon-v′v′ model of Durbin (3)

−1.1E−04

Proposed concept

−1.2E−04

0.0

0.2

0.4

0.6

0.8

1.0

y

FIGURE 13.3 Gradient of Reynolds shear stress for fully developed turbulent channel flow at Re = 180 using k − ε − v
2 turbulence model of Durbin [3] and the proposed concept. 1.0E−04 0.0E+00 −1.0E−04 Gradient of Reynolds shear stress

April 24, 2006

−2.0E−04 −3.0E−04 −4.0E-04 −5.0E−04 −6.0E−04 −7.0E−04 −8.0E−04 −9.0E−04

DNS data of Kim, Moin and Moser (4)

−1.0E−03

k-epsilon-v′v′ model of Durbin (3)

−1.1E−03

Proposed concept

−1.2E−03

0.0

0.2

0.4

0.6

0.8

1.0

y

FIGURE 13.4 Gradient of Reynolds shear stress for fully developed turbulent channel flow at Re = 395 using k − ε − v
2 turbulence model of Durbin [3] and the proposed concept.

13.4.2 TURBULENT BOUNDARY LAYER

WITH

CONSTANT PRESSURE

The governing equation for the time-averaged velocity field of steady incompressible turbulent boundary layer with constant pressure is given as follows: ∂ v¯ ∂ u¯ + =0 ∂x ∂y ∂u
v
∂ u¯ ∂ 2 u¯ ∂ u¯ + v¯ =
2 − u¯ ∂x ∂y ∂y ∂y © 2006 by Taylor & Francis Group, LLC

(13.7) (13.8)

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1.0E−05 0.0E+00 Gradient of Reynolds shear stress

−1.0E−05 −2.0E−05 −3.0E−05 −4.0E−05 −5.0E−05 −6.0E−05 −7.0E−05 DNS data of Kim, Moin and Moser (4)

−8.0E−05 −9.0E−05

k-epsilon-v′v′ model of Durbin (3) Proposed concept

−1.0E−04 −1.1E−04 −1.2E−04

0

10

20 y+

30

40

FIGURE 13.5 Gradient of Reynolds shear stress for fully developed turbulent channel flow at Re = 180 in the regions of viscous sublayer and buffer layer.

1.0E−04 0.0E+00 −1.0E−04 Gradient of Reynolds shear stress

April 24, 2006

−2.0E−04 −3.0E−04 −4.0E−04 −5.0E−04 −6.0E−04 −7.0E−04 −8.0E−04 −9.0E−04

DNS data of Kim, Moin and Moser (4)

−1.0E−03

k-epsilon-v′v′ model of Durbin (3)

−1.1E−03

Proposed concept

−1.2E−03

0

10

20 y+

30

40

FIGURE 13.6 Gradient of Reynolds shear stress for fully developed turbulent channel flow at Re = 395 in the regions of viscous sublayer and buffer layer.

where u¯ and v¯ are the time-averaged streamwise and cross-stream velocities,
is the kinematic

viscosity, and ∂u∂ yv is the gradient of the Reynolds shear stress that is unknown and needs modeling. The DNS data of the turbulent boundary layer with constant pressure of Spalart [5] at three Reynolds numbers (Re ≡ u ∞ /
= 300, 670, and 1410, where u ∞ is the freestream velocity and  is the momentum thickness) are used for comparison. The large discrepancy between the gradient © 2006 by Taylor & Francis Group, LLC

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2.0E−04

Gradient of Reynolds shear stress

1.0E−04 0.0E+00 −1.0E−04 −2.0E−04 −3.0E−04 −4.0E−04 −5.0E−04

DNS data of Spalart (5)

−6.0E−04

Standard high-Re k-epsilon model

−7.0E−04 −8.0E−04

Low-Re k-epsilon model of Launder and Sharma (2) 0.0

0.2

0.4

0.6 y

0.8

1.0

1.2

FIGURE 13.7 Gradient of Reynolds shear stress for turbulent boundary layer with constant pressure at Re = 300 using k − ε turbulence models.

2.0E−03 1.0E−03 Gradient of Reynolds shear stress

April 24, 2006

0.0E+00 −1.0E−03 −2.0E−03 −3.0E−03 −4.0E−03 −5.0E−03 DNS data of Spalart (5)

−6.0E−03

Standard high-Re k-epsilon model

−7.0E−03 −8.0E−03

Low-Re k-epsilon model of Launder and Sharma (2) 0.0

0.2

0.4

0.6 y

0.8

1.0

1.2

FIGURE 13.8 Gradient of Reynolds shear stress for turbulent boundary layer with constant pressure at Re = 670 using k − ε turbulence models.

of the Reynolds shear stress predicted by the k – ε turbulence models and that from the DNS data are clearly observed near the wall in Figure 13.7 to Figure 13.9, as in the case of the fully developed turbulent channel flow. Figure 13.10 to Figure 13.12 show that the proposed concept compared with the DNS data can capture the gradient of the Reynolds shear stress across the flow domain at all three Reynolds numbers. © 2006 by Taylor & Francis Group, LLC

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2.0E−02

Gradient of Reynolds shear stress

1.0E−02 0.0E+00 −1.0E−02 −2.0E−02 −3.0E−02 −4.0E−02 −5.0E−02 −6.0E−02

DNS data of Spalart (5)

−7.0E−02

Low-Re k-epsilon model of Launder and Sharma (2)

−8.0E−02

Standard high-Re k-epsilon model

0.0

0.2

0.4

0.6 y

0.8

1.0

1.2

FIGURE 13.9 Gradient of Reynolds shear stress for turbulent boundary layer with constant pressure at Re = 1410 using k − ε turbulence models.

1.0E−05 0.0E+00 Gradient of Reynolds shear stress

April 24, 2006

−1.0E−05 −2.0E−05 −3.0E−05 −4.0E−05 −5.0E−05

DNS data of Spalart (5)

−6.0E−05

k-epsilon-v′v′ model of Durbin (3) Proposed concept

−7.0E−05

0.0

0.2

0.4

0.6 y

0.8

1.0

1.2

FIGURE 13.10 Gradient of Reynolds shear stress for turbulent boundary layer with constant pressure at Re = 300 using the k − ε − v
2 turbulence model of Durbin [3] and the proposed concept.

The gradient of the Reynolds shear stress in the viscous sublayer and buffer layer regions is presented in Figure 13.13 to Figure 13.15. It is found that the prediction using the proposed concept shows closer agreement with the DNS data compared to the k −ε−v
2 turbulence model of Durbin [3]. The model constant is found to be universally equal to –1/2 and is also independent of Reynolds number, as in the case of the fully developed turbulent channel flow.

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1.0E−04

Gradient of Reynolds shear stress

0.0E+00 −1.0E−04 −2.0E−04 −3.0E−04 −4.0E−04 −5.0E−04

DNS data of Spalart (5) k-epsilon-v′v′ model of Durbin (3)

−6.0E−04

Proposed concept −7.0E−04

0.0

0.2

0.4

0.6 y

0.8

1.0

1.2

FIGURE 13.11 Gradient of Reynolds shear stress for turbulent boundary layer with constant pressure at Re = 670 using the k − ε − v
2 turbulence model of Durbin [3] and the proposed concept.

1.0E−03 0.0E+00 Gradient of Reynolds shear stress

April 24, 2006

−1.0E−03 −2.0E−03 −3.0E−03 −4.0E−03 −5.0E−03

DNS data of Spalart (5) k-epsilon-v′v′ model of Durbin (3)

−6.0E−03 −7.0E−03

Proposed concept 0.0

0.2

0.4

0.6 y

0.8

1.0

1.2

FIGURE 13.12 Gradient of Reynolds shear stress for turbulent boundary layer with constant pressure at Re = 1410 using the k − ε − v
2 turbulence model of Durbin [3] and the proposed concept.

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1.0E−05

Gradient of Reynolds shear stress

0.0E+00 −1.0E−05 −2.0E−05 −3.0E−05 −4.0E−05 −5.0E−05

DNS data of Spalart (5) k-epsilon-v′v′ model of Durbin (3)

−6.0E−05

Proposed concept −7.0E−05

0

10

20 y+

30

40

FIGURE 13.13 Gradient of Reynolds shear stress for turbulent boundary layer with constant pressure at Re = 300 in the regions of viscous sublayer and buffer layer.

1.0E−04 0.0E+00 Gradient of Reynolds shear stress

April 24, 2006

−1.0E−04 −2.0E−04 −3.0E−04 −4.0E−04 −5.0E−04

DNS data of Spalart (5)

−6.0E−04 −7.0E−04

k-epsilon-v′v′ model of Durbin (3) Proposed concept 0

10

20 y+

30

40

FIGURE 13.14 Gradient of Reynolds shear stress for turbulent boundary layer with constant pressure at Re = 670 in the regions of viscous sublayer and buffer layer.

13.5 CONCLUSIONS The new concept on modeling the gradients of the Reynolds shear stresses is proposed in the current work and evaluated for the fully developed turbulent channel flow and the turbulent boundary layer with constant pressure where the DNS data are available. The comparison of the gradient of the

Reynolds shear stress, dudyv , between the proposed concept and the chosen turbulence models using © 2006 by Taylor & Francis Group, LLC

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DK3004˙C013

On the New Concept of Turbulence Modeling

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1.0E−03 0.0E+00 Gradient of Reynolds shear stress

April 24, 2006

−1.0E−03 −2.0E−03 −3.0E−03 −4.0E−03 −5.0E−03

DNS data of Spalart (5) k-epsilon-v′v′ model of Durbin (3)

−6.0E−03 −7.0E−03

Proposed concept 0

10

20 y+

30

40

FIGURE 13.15 Gradient of Reynolds shear stress for turbulent boundary layer with constant pressure at Re = 1410 in the regions of viscous sublayer and buffer layer.

the eddy viscosity concept shows that the proposed concept provides the best agreement with the DNS data. The model constant, C, is found to be equal to –1/2 for both flows and is independent of Reynolds numbers. The regions of viscous sublayer and buffer layer that most turbulence models find it difficult to simulate are accurately predicted by the proposed concept.

ACKNOWLEDGMENTS This research is supported by the National Electronics and Computer Technology Center and the Thailand Research Fund for the Senior Scholar Professor Pramote Dechaumphai. The authors also would like to thank Professor Mike M. Gibson for helpful discussion.

REFERENCES 1. Launder, B.E., Spalding, D.B. (1974). The numerical computation of turbulent flows. Computer Methods Appl. Mech. Eng. 3:269–289. 2. Launder, B.E., Sharma, B.I. (1974). Application of the energy-dissipation model of turbulence to the calculation of flow near a spinning disk. Lett. Heat Mass Transfer. 1:131–138. 3. Durbin, P.A. (1991). Near-wall turbulence closure modeling without ”Damping Functions.” Theor. Computational Fluid Dyn. 3:1–13. 4. Kim, J., Moin, P., Moser, R. (1987). Turbulent statistics in fully developed channel flow at low Reynolds number. J. Fluid Mech. 177:133–166. 5. Spalart, P.R. (1988). Direct simulation of a turbulent boundary layer up to R = 1410. J. Fluid Mech. 187:61–98.

© 2006 by Taylor & Francis Group, LLC