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Frontiers in Turbulence and Coherent Structures Proceedings of the COSNet/CSIRO Workshop on Turbulence and Coherent St...

Author: Jim Denier | Jorgen S. Frederiksen

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Frontiers in Turbulence and Coherent Structures Proceedings of the COSNet/CSIRO Workshop on Turbulence

and Coherent Structures inFluids, Plasmas and Nonlinear Media

WORLD SCIENTIFIC LECTURE NOTES IN COMPLEX SYSTEMS Editor-in-Chief: A.S. Mikhailov, Fritz Haber Institute, Berlin, Germany

H. Cerdeira, ICTP, Triest, Italy B. Huberman, Hewlett-Packard, Palo Alto, USA K. Kaneko, University of Tokyo, Japan Ph. Maini, Oxford University, UK

AIMS AND SCOPE The aim of this new interdisciplinary series is to promote the exchange of information between scientists working in different fields, who are involved in the study of complex systems, and to foster education and training of young scientists entering this rapidly developing research area. The scope of the series is broad and will include: Statistical physics of large nonequilibrium systems; problems of nonlinear pattern formation in chemistry; complex organization of intracellular processes and biochemical networks of a living cell; various aspects of cell-to-cell communication; behaviour of bacterial colonies; neural networks; functioning and organization of animal populations and large ecological systems; modeling complex social phenomena; applications of statistical mechanics to studies of economics and financial markets; multi-agent robotics and collective intelligence; the emergence and evolution of large-scale communication networks; general mathematical studies of complex cooperative behaviour in large systems.

Published Vol. 1 Nonlinear Dynamics: From Lasers to Butterflies Vol. 2 Emergence of Dynamical Order: Synchronization Phenomena in Complex Systems Vol. 3 Networks of Interacting Machines Vol. 4 Lecture Notes on Turbulence and Coherent Structures in Fluids, Plasmas and Nonlinear Media Vol. 5 Analysis and Control of Complex Nonlinear Processes in Physics, Chemistry and Biology

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World Scientific Lecture Notes in Complex Systems - Vol. 6

editors

Jim Denier The University of Adelaide, Australia

Jorgen S. Frederiksen CSIRO Marine and Atmospheric Research, Australia

Frontiers in Turbulence and Coherent Structures Proceedings of the COSNet/CSIRO Workshop on Turbulence

and Coherent Structures in Fluids, Plasmas and Nonlinear Media The Australian National University, Canberra,, Australia

10 - 13 January 2006

World Scientific N E W J E R S E Y • L O N D O N • S I N G A P O R E • B E I J I N G • S H A N G H A I • H O N G K O N G • TA I P E I • C H E N N A I

Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

World Scientific Lecture Notes in Complex Systems — Vol. 6 FRONTIERS IN TURBULENCE AND COHERENT STRUCTURES Proceedings of the COSNet/CSIRO Workshop on Turbulence and Coherent Structures in Fluids, Plasmas and Nonlinear Media Copyright © 2007 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN-13 978-981-270-393-4 ISBN-10 981-270-393-4

Printed in Singapore.

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PREFACE Over the past decade there have been signiﬁcant advances in our understanding of turbulence and the emergence of coherent structures in ﬂuids, plasmas and nonlinear media. New theoretical, modeling and experimental and observational techniques have been developed for tackling the complex interactions of turbulence with coherent structures, mean ﬂows and waves. A workshop on Turbulence and Coherent Structures in Fluids, Plasmas and Nonlinear Media was held at the Australian National University, Canberra, 10 to 13 January 2006 and was sponsored by the Australian Research Council’s Complex Open Systems Network and CSIRO Complex Systems Science. The aim of the workshop was to bring together researchers from the often-disparate disciplines of ﬂuid mechanics, plasma physics, atmosphere and ocean dynamics and dynamical systems theory to codify recent developments in our understanding of the dynamics and statistical dynamics of turbulence and coherent structures. The workshop covered topics ranging from instability theory, to bifurcation and singularity theory, and stochastic modeling, through chaos and predictability theory, turbulence, coherent structures, multiple equilibria and hysteresis, to subgrid-scale processes and statistical dynamics and renormalization. It is expected that many of the methodologies presented may also be applicable to other Complex Systems. Presentations were given on the theoretical, numerical modeling, observational and experimental studies of turbulence and coherent structures in quasi-two-dimensional geophysical ﬂows, such as oceans and atmospheres, and in plasmas and in three-dimensional ﬂows, such as the turbulent boundary layer. The works presented form the basis of this volume in the World Scientiﬁc Lecture Notes in Complex Systems entitled Frontiers in Turbulence and Coherent Structures. The ﬁrst two chapters consider the topics of dynamical systems and instability theory. The chapter by Ball and Holmes details the historical development of dynamical systems theory, stability and chaos as well as interesting applications to a wide range of phenomena. The following chapter by Frederiksen focuses on atmospheric applications of instability theory, predictability and chaos. These papers were presented at the Summer School

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on Turbulence and Coherent Structures in Fluids, Plasmas and Nonlinear Media held at the Australian National University, Canberra immediately following the workshop. They are included here to provide a pedagogical basis for many of the subsequent papers that rely heavily upon an understanding of dynamical systems and instability theory. For the remainder of the volume the papers are grouped into four general themes that emerged during the workshop. Following on from the introductory chapter by Frederiksen on atmospheric disturbances, Chapters 3 to 7 collect papers on the interaction of turbulence and coherent structures in the atmosphere and ocean. Works are presented employing bifurcation and instability theory, examining regime transitions and multiple equilibria, detailing the properties of coherent structures and teleconnection patterns, and presenting modeling and observational studies of atmospheric and oceanic ﬂows. These large-scale ﬂows in the atmosphere and oceans are quasi-geostrophic, with an approximate balance between Coriolis and pressure forces, and share many properties with two-dimensional turbulence. Chapters 8 to 12 review recent progress in three-dimensional turbulence including boundary layer turbulence and the formation of coherent structures. The studies include dynamical systems theory approaches and Lagrangian dispersion in three-dimensional turbulence as well recent developments in experiments and modeling of high Reynolds number turbulent ﬂows. Chapters 13, 14 and 15 appraise the current state of turbulence closure models, based on renormalized perturbation theory, for both twodimensional and three-dimensional homogeneous turbulence. Recent developments in the generalization of closures to inhomogeneous turbulence interacting with mean ﬂows, coherent structures and topography, and to Rossby wave turbulence, are outlined. Applications of closure theory to subgrid-scale parameterizations, and ensemble prediction and data assimilation in the presence of developing coherent blocking structures, are presented. The ﬁnal theme, in chapters 16 through 21, brings together a collection of studies on regime transitions in magnetized fusion plasmas. A major focus in fusion plasma research has been understanding the low- to highconﬁnement transitions that can occur due to the formation of zonal shear ﬂows that break up coherent eddies responsible for turbulent transport. Theoretical, modeling and experimental studies of the dynamics of plasmas are presented. The close connections between the equations for the ﬂuid

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description of magnetized plasmas and those for quasi-geostrophic geophysical ﬂuids are brought out. We thank the ARC Complex Open Systems Research Network convened by Prof. Robert Dewar, and the Commonwealth Scientiﬁc and Industrial Research Organization, through CSIRO Complex Systems Science directed by Dr. John Finnigan, for sponsoring this workshop. We thank Dr. Michael Shats for help in organizing the workshop. The Workshop in Canberra was immediately followed by the 19th Canberra International Physics Summer School and the Lecture Notes (editors M. Shats and H. Punzmann) have been published by World Scientiﬁc, 2006 under the title “Turbulence and Coherent Structures in Fluids, Plasmas and Nonlinear Media”. Jim Denier (Adelaide) Jorgen Frederiksen (Melbourne) Co-Conveners of Workshop on Turbulence and Coherent Structures in Fluids, Plasmas and Nonlinear Media January 2007

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CONTENTS

Preface

v

Dynamical systems, stability, and chaos R. Ball and P. Holmes

1

Instability theory and predictability of atmospheric disturbances J. S. Frederiksen

29

Multiple equilibria and atmospheric blocking M. J. Zidikheri, J. S. Frederiksen and T. J. O’Kane

59

Coherent patterns of interannual variability of the atmospheric circulation: the role of intraseasonal variability C. S. Frederiksen and X. Zheng

87

Regimes of the wind-driven ocean ﬂows H. A. Dijkstra

121

Nonlinear resonance and chaos in an ocean model A. Kiss

149

Low frequency ocean variability: feedbacks between eddies and the mean ﬂow A. McC. Hogg, W. K. Dewar, P. D. Killworth and J. R. Blundell Periodic motion versus turbulent motion: scaling laws, bursting and Lyapunov spectra L. van Veen, S. Kida and G. Kawahara

171

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Maximum entropy N-particle Lagrangian clusters in turbulence and application to scalar ﬁelds M. S. Borgas

203

A review of recent investigations into high Reynolds number wall-turbulence J. P. Monty and M. S. Chong

227

What are we learning from simulating wall turbulence? J. Jim´enez

247

Coherent structures generated by a synthetic jet J. H. Watmuﬀ

261

Two-point turbulence closures revisited W. D. McComb

281

Turbulence closures and subgrid-scale parameterizations J. S. Frederiksen and T. J. O’Kane

315

Statistical dynamical methods of ensemble prediction and data assimilation during blocking T. J. O’Kane and J. S. Frederiksen

355

Distilled turbulence. A reduced model for conﬁnement transitions in magnetic fusion plasmas R. Ball

395

Zonal ﬂow generation by modulational instability R. L. Dewar and R. F. Abdullatif

415

Nonlinear simulation of drift wave turbulence R. Numata, R. Ball and R. L. Dewar

431

The transition to ion-temperature-gradient-driven plasma turbulence J. A. Krommes

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Spectral transfer analysis in plasma turbulence studies H. Xia and M. G. Shats Coherent structures in toroidal electron plasmas: simulation and experiments R. Ganesh and S. Pahari

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DYNAMICAL SYSTEMS, STABILITY, AND CHAOS ROWENA BALL Mathematical Sciences Institute and Department of Theoretical Physics, The Australian National University, Canberra, Australia [email protected]

PHILIP HOLMES Department of Mechanical and Aerospace Engineering and Program in Applied and Computational Mathematics, Princeton University, NJ 08544, USA

In this expository and resources chapter we review selected aspects of the mathematics of dynamical systems, stability, and chaos, within a historical framework that draws together two threads of its early development: celestial mechanics and control theory, and focussing on qualitative theory. From this perspective we show how concepts of stability enable us to classify dynamical equations and their solutions and connect the key issues of nonlinearity, bifurcation, control, and uncertainty that are common to time-dependent problems in natural and engineered systems. We discuss stability and bifurcations in three simple model problems, and conclude with a survey of recent extensions of stability theory to complex networks.

1. Introduction Deep in the heart of northern England, on the banks of a river near a village at the edge of the Lancashire Pennines, there is a ﬁne brick building dating from the late nineteenth century. Here dwell two stout, well-preserved old ladies named Victoria and Alexandra. They will never invite you in for tea though, for the building is the Ellenroad Mill Engine House and the two Ladies are a giant, twin compound steam engine operating in tandem, originally built in 1892. On weekends willing teams of overalled maids and butlers oil and polish the Ladies and ﬁre up the old Lancashire boiler that delivers the steam to their cylinders to move the pistons that drive the giant, 80-ton ﬂywheel.

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The speed of the engines is controlled by a centrifugal governor∗, and the motions of this device, occurring on time and spatial scales that can be appreciated by the human visual cortex, are fascinating to watch. Originally patented by James Watt in 1789, the centrifugal steam engine governor is the most celebrated prototype example of a self-regulating feedback mechanism. The device consists of two steel balls hinged on a rotating shaft which is spun from a belt or gears connected to the ﬂywheel, Figure 1. In stable operation, as the speed of the engine increases the inertia of the ﬂyballs swings the arms outwards, contracting the aperture of a valve which controls the speed of rotation by restricting the steam supply. If the engine lags due to an additional, imprecisely known, load (in the mill this might have been another loom connected up to the engine by a belt drive) the ﬂyballs are lowered and the valve opens, increasing the steam supply to compensate. Thus the design of the governor cleverly uses the disturbance itself, or deviation from set-point or desired performance, to actuate the restoring force.

Fig. 1. The centrifugal ﬂyball governor (after Pontryagin20 ). See Equations 3 and accompanying text in section 3 for deﬁnitions of the labels.

∗ The Greek word for governor is kubernetes, from which the mathematician Norbert Wiener (1894–1964) coined the term cybernetics as a name for the collective ﬁeld of automated control and information theory.

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In certain operating regimes the motions of the governor may lose stability, becoming oscillatory and spasmodic, amplifying the eﬀect of the disturbance and thwarting control of the engine. Nineteenth century engineers called this unstable behaviour hunting and devoted much eﬀort to improving the design of centrifugal governors. James Clarke Maxwell was the ﬁrst to formulate and analyse the stability of the equations of motion of the governor, explaining the onset of hunting behaviour in mathematical terms,1,2 followed (independently) by Vyshnegradskii.3 We analyse Vyshnegradskii’s equations for the governor’s motion in section 3, as an exemplary three dimensional stability problem. The self-correcting centrifugal governor is a simple feedback control system because the changes in velocity are fed back to the steam valve. Its widespread adoption during the 18th and 19th centuries dramatically transformed the steam-driven textile mills, the mining industry, and locomotion. (In 1868, the year Maxwell published “On Governors”1 there were an estimated 75,000 Watt governors in England alone.4 ) Without this device the incipient industrial revolution could not have progressed, because steam engines lacking self-control would have remained hopelessly ineﬃcient, monstrous, contraptions, requiring more than the labour that they replaced to control them. Watt’s iconic governor also embodies a radical change in the philosophy of science. For several hundred years the mechanical clock, with its precise gears and necessity for human intervention to rewind it or correct error and its complete absence of closed-loop feedback, had been the dominant motif in scientiﬁc culture. In a common metaphor, the universe was created and ordered by God the Clockmaker. Isaac Newton had no doubt that God had initiated the celestial mechanics of the motions of the planets and intervened when necessary to keep His creation perfectly adjusted and on track.5,6 The clockwork view was also deeply satisfying to Laplace, one of the most inﬂuential mathematicians of the eighteenth and early nineteenth centuries. Stability theory was developed some two centuries and more after Newton published his Principia (1687), so he could not have known that the planetary orbits may be what Poincar´e called Poisson stable7,8 (small perturbations are self-correcting) — or they may be chaotic† . As concepts of feedback and stability were developed rigorously and applied in the late nineteenth and early twentieth centuries, Divine open-loop † A fact which might cause you some queasiness to learn. Fear not — it is believed that chaotic motions were important in the early evolution of the solar system,9 and a slow chaotic drift may be noticeable a few billion years hence.6,10 What luck for us!

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control began to wane and there came a growing awareness of systems as dynamical entities that can regulate their own destiny and internally convert uncertain inputs into stable outputs. The technological advances in transport, power, and communications made possible by feedback control and applied stability theory are agents of change, the vectors of liberty, liberalism, and literacy in societies, themselves enabling the blossoming and seeding of more sophisticated ideas of feedback and stability in complex environmental, socio-economic, and biological systems. Now, due to stability theory and feedback control, we may contemplate “the fundamental interconnectedness of all things”,11 but back then, in the clockwork days, people could not. It is surely no coincidence that totalitarian governments favour clockwork metaphors. Today, we are so comfortable with the concept of feedback control inducing stable dynamics that we barely notice how it permeates most aspects of our lives. Control theory, then, is a major strand in the development of modern nonlinear dynamics, but it is not the ﬁrst. The centrifugal governor also transformed the practice of astronomy, in that it enabled ﬁne control of telescope drives and vastly improved quantitative observations, and it is this earlier force (already alluded to above in mention of Newton’s and Laplace’s work) in the development of dynamical systems and stability theory — celestial mechanics — on which we now focus attention. The next stage of our nonlinear dynamics odyssey takes us from the post-industrial north of England to the miraculously intact (given the destructions of WWII) medieval city of Regensburg in Germany, to an older, humbler but no less important building than that which houses the Ladies, the Kepler museum. In addition to celebrating the life and work of Johannes Kepler (1571– 1630) the museum houses priceless manuscripts, letters, publications, and astronomical instruments and interpretive exhibits that tell a lively and inspiring story, that of the development of celestial mechanics from Galileo to its culmination, in analytic terms, in the work of Poincar´e. An exhibit from the 18th century, an exquisitely engineered brass orrery, or clockwork model of the solar system, in its detail and precision expresses the satisfaction and conﬁdence of the clockwork aﬁcionados of the Age of Enlightenment. But a nearby exhibit expresses, rather presciently, the need for a new metaphor for scientiﬁc endeavour and achievement. It is an early 19th century relief in which Kepler unveils the face of Urania, the Muse of astronomy, whereupon she insouciantly hands him a telescope and a scroll inscribed with his own laws, as if to say: “Hmm. . . not a bad job; now take

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Fig. 2. Johannes Kepler is given cheek by his Muse after two long centuries of ellipses and clockworks.

these back and do some more work then tell me why your elliptical orbits are non-generic”. (See Figure 2.) And in fact the one-dimensional Kepler ellipse can be transformed into a harmonic oscillator with Hamiltonian12 1 H(Q, P ) = P 2 − EQ2 . (1) 8 Despite Laplace’s conﬁdence the problem of the stability of the solar system refused to go away, but instead took on a central role in the preoccupations of mathematicians, physicists, astronomers, and navigators postNewton. It was by no means clear, even to Newton, that Newton’s law was suﬃcient to describe the motions of three or more celestial bodies under mutual gravitational attraction. The problem also refused to be solved, in the sense of what was accepted as a “solution” during the latter 18th century and ﬁrst half of the 19th century, i.e., analytically in terms of elementary or previously-known special functions. Progress was made in the mid-1800s in improving series approximations but, not surprisingly, the hydra of nonconvergence soon raised one after the other of its ugly (of course!) heads. By 1885, when it was chosen by Weierstrass as one of four problems in the mathematics competition sponsored by King Oscar II of Sweden, the n-body problem had achieved notoriety for

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its recalcitrance — but in doing so it had also driven many of the seminal advances in mathematics and produced many of the greatest mathematicians of the 19th century. The ﬁrst problem in King Oscar’s competition was to show that the solar system as modeled by Newton’s equations is stable. In his (corrected) entry7 Poincar´e invented or substantially extended integral invariants, characteristic exponents, and Poincar´ e maps (obviously), invented and proved the recurrence theorem, proved the nonexistence of uniform ﬁrst integrals of the three body problem, other than the known ones, discovered asymptotic solutions and homoclinic points, and wrote the ﬁrst ever description of chaotic motion — in short, founded and developed the entire subject of geometric and qualitative analysis. Then he concluded by saying he regarded his work as only a preliminary survey from which he hoped future progress would result. Poincar´e’s “preliminary survey” is still inspiring new mathematics and applications, but during the 20th century the collective dynamic of dynamical systems development was highly nonlinear. Homoclinic points and homoclinic chaos were partially treated by the American mathematician George Birkhoﬀ (1884–1944) — he obtained rigorous results on the existence of periodic orbits near a homoclinic orbit — and by Cartwright and Littlewood in their study of Van der Pol’s (non-Hamiltonian) equation,13 y¨ − k(1 − y 2 )y˙ + y = bλk cos(λtα ). Cartwright and Littlewood stated numerous “bizarre” properties of solutions of this diﬀerential equation, implying the existence of an invariant Cantor set, but their very concise paper was not easy to penetrate, and their results remained largely unknown until Levinson14 pointed them out to Stephen Smale. During the 1960s and 1970s Smale’s representation of homoclinic chaos in terms of symbolic dynamics and the horseshoe map15 stimulated renewed interest in dynamical systems (although we have skipped a lot of mathematical history here, most notably KAM theory). Happily, this coincided with the advent of desktop digital computers subject to Moore’s law. Since the 1980s improvements in processor speed have both driven and been driven by the use of computational simulations of dynamical systems as virtual experiments, and inspired advances in ﬁelds such as network stability, numerical instabilities, and turbulence. Essentially these advances are sophisticated and technologically facilitated applications of Poincar´e’s and Lyapunov’s stability theory, and in the next section we present the basics and some working deﬁnitions.

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It is somewhat ironic that improvements in processor speed have also led to renewed interest in low dimensional dynamical systems, which usually only require small-time computing and are at least partially amenable to rigorous stability analysis. For large dynamical systems usually mean turbulent ones, and computation is, in essence, the notorious “problem of turbulence”. In a turbulent ﬂow energy is distributed among wavenumbers that range over perhaps seven orders of magnitude (for, say, a tokamak) to twelve orders of magnitude (for a really huge system, say a supernova). To simulate a turbulent ﬂow in the computer it is necessary to resolve all relevant scales of motion in three dimensions. It is a fair estimate16 that such calculations would take 400 years at today’s processor speeds, therefore a faster way to do them would be to rely on Moore’s law and wait only 20 years until computers are speedy enough. Many of us in the turbulence business have realized that while we are waiting we can, more expediently, apply reduced dynamical systems methods to the problem, such as Karhunen-Lo´eve (KL) decomposition‡ , to distill out a much-reduced, but nevertheless sophisticated, approximation to the dynamics and spatial structure of a turbulent ﬂow.17 To introduce KL decomposition, we imagine a fractional distillation tower for which the feedstock is not crude oil but a high Reynolds number ﬂow. Then instead of a natural distribution over hydrocarbon molecular weights we have an energy distribution over scales of motion. We know, in principle, how hydrocarbons are separated in the still according to their boiling points (even if we do not work at an oil reﬁnery), but what properties may we exploit to separate and re-form the energy components of a turbulent ﬂow? Our turbulence reﬁnery does not deﬁne the skyline of a seamy port city in complicated chiaroscuro, but exists more conveniently in constrained ﬂuid ﬂow experiments or as direct numerical simulations of the Navier-Stokes equations in silico. The KL transform operates on data to yield eigenfunctions that capture in decreasing order most of the kinetic energy of the system, so it is especially useful for highly self-structured ﬂows. 2. To understand stability is to understand dynamics Very few dynamical systems have known, exact solutions. For the vast majority it cannot even be proved that general solutions exist. Stability theory is quite indiﬀerent to such issues; instead it tells us how families of solutions ‡ Also known by the aliases proper orthogonal or singular value decomposition, principal component analysis, and empirical eigenfunction analysis.

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would behave, assuming they do exist. Loosely we understand stability to mean that a solution does not run away, or to refer to the resilience of a solution to changes in initial conditions or to changes to the equation that generates it. Stability is a qualitative property of dynamical equations and their solutions. For practical applications stability analysis allows us to say whether a given system conﬁguration will exhibit runaway dynamics (catastrophic failure) or return to a stable quasi-equilibrium, limit cycle, or other attractor, in response to perturbation. We have indicated in section 1 above how the issue of stability of the planetary orbits drove the development of celestial mechanics, but stability is equally important in control theory — from a design and operational point of view it could be said that control is applied stability. It is a grave issue because, as we show in section 3, feedback can result in systems that fail due to instabilities, as well as create ones which maintain homeostasis. Thermal explosions, ecological “arms races”, and economic depressions are all more-or-less disastrous consequences of unstable feedback dynamics. A big stability question that occupies many scientists today concerns the long-term stability of the world’s climate in response to the enhanced greenhouse eﬀect; questions related to stability of other complex systems will be explored in section 6. In this section we give precise mathematical expression to these concepts of stability, for later reference. For more detail and discussion the reader is referred to the article in Scholarpedia curated by Holmes and Shea-Brown.18 Consider the general dynamical system in vector form x˙ = f (t, x),

(2)

where f i (t, x) and the derivatives ∂f i (t, x)/∂xj are deﬁned and continuous on a domain Γ of the space of t, x. Let γt (x) = x(t) with the initial value x(0) = x. Then, the (forward) orbit is the set of all values that this trajectory obtains: γ(x) = {γt (x)|t ≥ 0}. Deﬁnition 1. Two orbits γ(x) and γ(ˆ x) are -close if there is a reparameterization of time (a smooth, monotonic function) tˆ(t) such that x)| < for all t ≥ 0. |γt (x) − γtˆ(t) (ˆ Deﬁnition 2. Orbital or generalized Lyapunov stability. γ(x) is orbitally stable if, for any > 0, there is a neighbourhood V of x so that, for ˆ in V , γ(x) and γ(ˆ all x x) are -close.

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Deﬁnition 3. Generalised asymptotic stability. If additionally V may ˆ ∈ V , there exists a constant τ (ˆ be chosen so that, for all x x) so that x)| → 0 as t → ∞ then γt (x) is asymptotically stable. |γt (x) − γt−τ (ˆx) (ˆ These general deﬁnitions of Lyapunov stability and asymptotic stability are indiﬀerent to the choice of initial values t0 , x(0). Lyapunov stability is intimated in Figure 3, which sketches a segment of an orbit γ(x) and a segment of a neighbouring orbit γ(ˆ x), in periodic and non-periodic cases.

Fig. 3. The orbit γ(x) is orbitally stable. The black lines indicate the boundary of an -neighborhood of γ(x).

In the particular case where the system (2) is autonomous and the solution is an equilibrium xe we have the following speciﬁcations: Deﬁnition 4. Lyapunov stability of equilibria. xe is a stable equilibrium if for every neighborhood U of xe there is a neighborhood V ⊆ U of xe such that every solution x(t) starting in V (x(0) ∈ V ) remains in U for all t ≥ 0. Notice that x(t) need not approach xe . Lyapunov stability means that when all orbits starting from a small neighbourhood of a solution remain forever in a small neighborhood of that solution the motion is stable, otherwise it is unstable. If xe is not stable, it is unstable. Deﬁnition 5. Asymptotic stability of equilibria. An equilibrium xe is asymptotically stable if it is Lyapunov stable and additionally V can be chosen so that |x(t)−xe | → 0 as t → ∞ for all x(0) ∈ V . An asymptotically

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stable equilibrium (stationary state) and its local environment is sketched in Figure 4.

Fig. 4.

An asymptotically stable equilibrium is also called a sink.

It is all very well to settle the stability properties of a solution, but what then? If, as is usually the case, we are studying Eq. 2 as a model for coupled physical motions or a system of rate processes, and therefore necessarily imperfect, we also need information about how those properties fare under perturbations to the model, or structural stability. The question usually goes something like this: When are suﬃciently small perturbations of a dynamical system equivalent to the original unperturbed dynamical system? And if a system is not structurally stable, how may one unfold it until it is? And what (new mathematics, physics) do the unfoldings reveal? The concept of structural stability has yielded a rich taxonomy of bifurcations and of diﬀerent classes of vector ﬁelds. Structural stability is thus fundamentally a classiﬁcation science, a binomial key of the type that has been used in biology since the method was devised by the Swedish botanist Linnaeus (1707-1778). It is more distracting than useful to deﬁne structural stability rigorously at this stage (although authoritative deﬁnitions can be found in the literature, e.g., Hirsch and Smale19 ); instead, we shall illustrate some of the concepts in section 4 in relation to a perturbed simple pendulum as a simpliﬁed surrogate for the restricted three body problem. 3. Governor equations of motion: a simple case study Now that we have some background and theory resources to draw on, let us carry out a stability analysis of the centrifugal governor. This analysis is all the more important for being elementary because it introduces many of the

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key concepts of dynamical systems theory in a setting that is understandable to non-mathematician physical scientists and engineers and also sets the scene for the more complicated motions we describe in sections 4 and 5. Vyshnegradskii’s equations of motion for the ﬂyball governor sketched in Figure 1 were given as a 3-dimensional, autonomous, ﬁrst-order dynamical system by Pontryagin:20 dϕ =ψ dt dψ b = n2 ω 2 sin ϕ cos ϕ − g sin ϕ − ψ dt m

(3)

k F dω = cos φ − , dt J J where ϕ is the angle between the spindle S and the ﬂyball arms L, ω is the rotational velocity of the ﬂywheel, the transmission ratio n = θ/ω, θ is the angular velocity of S, g is the gravitational acceleration, m is the ﬂyball mass, J is the moment of inertia of the ﬂywheel, F represents the net load on the engine, k > 0 is a constant, and b is a frictional coeﬃcient. The length of the arms L is taken as unity. For a given load F the engine speed and ﬂy-ball angle are required to remain constant, and the unique steady state or equilibrium coordinates are easily found as ψ0 = 0, cos ϕ0 = F/k, n2 ω02 = g/ cos ϕ0 . So far, so dull. Dull, too, are the designers of engines, according to Maxwell. In his treatment of the governor problem, which was more general than that of Vyshnegradskii, he wrote: “The actual motions corresponding to these impossible roots are not generally taken notice of by the inventors of such machines, who naturally conﬁne their attention to the way in which it is designed to act; and this is generally expressed by the real root of the equation.” The impossible roots he referred to are the complex roots of the characteristic equation obtained from the linearized equations of motion. Maxwell and Vyshnegradskii both used this method to investigate the mathematical stability of the engine-governor dynamical system and relate the results closely to observed misbehaviours of the physical system. Their linear stability analyses provide criteria for which the system returns to its equilibrium engine speed ω0 and ﬂyball angle ϕ0 when subjected to a small perturbation. Let us represent the perturbed system by setting ϕ = ϕ0 + δϕ,

ψ = ψ0 + δψ,

ω = ω0 + δω,

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with |δϕ|, |δψ|, |δω| 1, and recasting equations (3) as d δϕ = δψ dt g sin2 ϕ0 2g sin ϕ0 d b δψ = − δϕ − δψ + δω dt cos ϕ0 m ω0

(4)

d k δω = − sin ϕ0 δϕ, dt J where we have neglected terms that are quadratic in the small perturbations δϕ, δψ, and δω. Equations (4) are a linear system with constant coeﬃcients that may be written succintly in matrix form x˙ = Ax, where x˙ =

d



dt δϕ d   dt δψ  , d dt δω



0

1

 sin2 ϕ0 b A =  − g cos ϕ0 − m

(5) 0



2g sin ϕ0  , ω0

− Jk sin ϕ0 0



δϕ



  x = δψ  .

0

δω

Equation (5) has nontrivial, linearly independent solutions of the form x = ueλt

(6)

where the constant components of u and the constant λ may be complex. Diﬀerentiating (6) with respect to t and substituting in (5) gives the eigenvalue problem (A − λI) u = 0

(7)

where I is the identity matrix. The requirement that u = 0, needed to obtain nontrivial solutions, satisﬁes (7) if and only if the factor det (A − λI) = 0, or −λ 1 0 g sin2 ϕ0 ϕ0 = 0. b (8) − cos ϕ0 − m − λ 2g sin ω0 − k sinϕ 0 −λ 0 J The determinant may be evaluated and equation (8) expressed in terms of the characteristic polynomial: λ3 +

b 2 g sin2 ϕ0 2gk sin2 ϕ0 λ + λ+ = 0. m cos ϕ0 Jω0

(9)

The roots λ1 , λ2 , λ3 of (9) are the eigenvalues of A and the solutions u1 , u2 , u3 of (7) are the corresponding eigenvectors. By inspection of equation (6) stability can ensue only if the real eigenvalues, or real parts

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of complex eigenvalues are negative. From analysis of the characteristic equation (9) this condition can be written as bJ ω0 > 1. m 2F

(10)

Now let us consider the dynamical behaviour of the engine-governor system in the light of (10) and with the aid of Figure 5. In (a) and (b) the equilibria and linear stability of equations 3 have been computed numerically and plotted as a function of the friction coeﬃcient b. This is a bifurcation diagram, where the bifurcation or control parameter b is assumed to be quasistatically variable, rendered in the variables ϕ (a) and ω (b). We see immediately that stable, steady state operation of the enginegovernor system requires frictional dissipation above a critical value. As b is decreased through the Hopf bifurcation point HB the real parts of a pair of conjugate eigenvalues become positive, the equilibrium becomes unstable, and the motion becomes oscillatory. The envelope of the periodic solutions grows as b is decreased further, which is also deduced in the inequality (10): a decrease in the coeﬃcient of friction can destabilize the system. As the bifurcation parameter b is decreased through the marked value with the label NS the stable periodic solution, for which the Floquet multipliers have modulus < 1, undergoes a Niemark-Sacker bifurcation. A conjugate pair of multipliers leaves the unit circle, and a two-dimensional asymptotically stable invariant torus bifurcates from the limit cycle21,22 § . For b < bN S the periodic solutions are unstable but the torus is stable. The behaviour of the system has become essentially 3-dimensional. In the governor problem we have studied the stability of solutions. In the next section we consider structural stability, in relation to the the restricted three body problem from celestial mechanics. 4. The restricted three body problem, homoclinic chaos, and structural stability This section assumes a working knowledge of Hamiltonian mechanics from a text book such as Goldstein23 or from undergraduate lecture notes such as Dewar.24 Rather than presume to capture the entire content and context § The discovery of torus bifurcations ﬁrst by Niemark in the USSR and ﬁve years later independently by Sacker in the USA seems to be a classic case of unnecessarily duplicated development of mathematics during the cold war.

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Fig. 5. Bifurcation diagrams rendered for the variables ϕ (a) and ω (b), stable equilibria are marked by a solid line, unstable equilibria are marked by a dashed line, HB stands for Hopf bifurcation, NS stands for Neimark-Sacker bifurcation, black dots mark the amplitude envelope of the oscillations. (c) The period τ of the oscillations decreases with b. (d) Continuations at the Hopf bifurcation in the parameters J and F .

of the restricted three body problem within the space of one chapter section we again summarize a small vignette from the panorama, a surrogate for the restricted three body problem. Homoclinic chaos and the associated topics of Poincar´e maps, symbolic dynamics, and the Smale horseshoe construction, are ﬂeshed out in Guckenheimer and Holmes25 and Holmes.26 First let us return to Kepler’s ellipse, or the two-body problem of Newton, which at the end of section 1 we gave in terms of the Hamiltonian for the transformed harmonic oscillator, Eq. 1. The well-known simple pendulum is also a harmonic oscillator, with Hamiltonian H = p2 /2 + (1 − cos q)

(11)

and equations of motion q˙ = p,

p˙ = − sin q.

(12)

The phase portrait of the ﬂow, Figure 6, shows the three families of periodic solutions bounded by the separatrices H = 2, which are emphasized in Figure 6. The ﬁxed point (or equilibrium) at (q, p) = (0, 0) represents the pendulum at rest and that at (q, p) = (±π, 0) represents the upside-down

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Fig. 6.

The phase space of the simple pendulum

position of the pendulum, keeping in mind that the ﬂat phase portrait should be wrapped around a cylinder of circumference 2π. Elementary linear analysis tells us that the the ﬁxed point at (q, p) = (0, 0) is a centre, with the solution matrix of the linearization having a pair of pure imaginary eigenvalues, and that at (q, p) = (±π, 0) is a hyperbolic (or non-degenerate) saddle point, with the solution matrix of the linearization having having one positive and one negative eigenvalue. Each point of the H = 2 separatrices is homoclinic, or asymptotic to to the ﬁxed point (q, p) = (±π, 0) as t → ±∞. In fact the separatrices are simultaneously the stable and unstable manifolds for the saddle point. Thus the phase portrait of the pendulum contains qualitative information about the global dynamics of the system. Now consider the restricted three body problem that featured in Poincar´e’s memoir, in which two massive bodies move in circular orbits on a plane with a third body of negligible mass moving under the resulting gravitational potential. In a rotating frame the system is described by the position coordinates (q1 , q2 ) of the third body and the conjugate momenta (p1 , p2 ). Poincar´e studied the following two degree of freedom Hamiltonian as a proxy for this system: H(q1 , q2 , p1 , p2 ) = −p2 − p21 + 2µ sin2 (q1 /2) + µε sin q1 cos q2 ,

(13)

with corresponding equations of motion q˙1 = −2p1 ,

q˙2 = −1;

p˙ 1 = −µ sin q1 − µε cos q1 cos q2 ,

p˙2 = µε sin q1 sin q2 .

(14)

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By inversion of Eq. 13 we have p2 = Ph (q1 , p1 ; q2 ) = h − p21 + 2µ sin2 (q1 /2) + µε sin q1 cos q2 ,

(15)

from which we can obtain the reduced equations of motion q1 = −∂Ph /∂p1 = 2p1 ,

p1 = ∂Ph /∂q1 = µ sin q1 + µε cos q1 cos q2 , (16)

where (·) denotes d/dq2 ). We see that Eqs 16 have the form of a periodically forced one degree of freedom system in which the angle variable q2 plays the role of time. For ε = 0 Eqs 16 are isomorphic to those for the simple pendulum, Eqs 12, and the phase portrait is that of Figure 6 (to make the origin (q1 , p1 ) = (0, 0) a center we set µ < 0). When a time-periodic perturbation is applied to the pendulum the stable and unstable manifolds that form the separatrix level set typically break up, but some homoclinic points may persist and with them small neighbourhoods of initial conditions, which are repeatedly mapped around in the region formerly occupied by the separatrixes. Such regions can now fall on both sides of the saddle point so that of two solutions starting near each other, one may ﬁnd itself on the rotation side and the other on the oscillation side. At each juncture near the saddle point such solutions must decide which route to take. The global structure of the stable and unstable manifolds rapidly becomes very complicated. Poincar´e prudently decided that, in this case, a thousand words are worth more than a picture: “When we try to represent the ﬁgure formed by [the stable and unstable manifolds] and their inﬁnitely many intersections, each corresponding to a doubly asymptotic solution, these intersections form a type of trellis, tissue or grid with inﬁnitely ﬁne mesh. Neither of the two curves must ever cross itself again, but it must bend back upon itself in a very complex manner in order to cut across all of the meshes in the grid an inﬁnite number of times.”(Poincar´e,27 quoted in Diacu and Holmes28 ). We have computed some orbits and rendered the data in Figure 7, which may or may not help to clarify the issue. Thus did Poincar´e describe homoclinic chaos, after years of careful and productive analysis of the phenomenon. In particular, Poincar´e obtained the following results: • Transverse homoclinic points exist for ε = 0. A transverse homoclinic orbit occurs when the stable and unstable manifolds intersect transversally, i.e., the unstable manifold intersects and crosses the stable manifold. In two dimensions, continuous dynamical systems do not have transverse homoclinic orbits, but

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Fig. 7. Segments of orbits belonging to the stable (blue) and unstable (yellow) manifolds of the saddle type periodic orbit of the periodically perturbed pendulum, Eqs 16 with µ = −1 and ε = 0.1.

a two-dimensional Poincar´e map deﬁned near a periodic orbit of a continuous dynamical system may have them. • Transverse homoclinic points obstruct the existence of second integrals of the motion. • Transverse homoclinic points imply that chaotic motions exist nearby. The model problem, Eq. 13, is essentially a simple pendulum coupled weakly to a linear oscillator. For the restricted three body problem itself, Poincar´e showed that after applying perturbation methods and truncating certain higher order terms in the expansion the Hamiltonian becomes completely integrable. He also showed that the reduced system, and therefore its Poincar´e map, possesses hyperbolic saddle points whose stable and unstable manifolds, being level sets of the second integral, coincide, as they do for the pendulum illustrated in Figure 6. He then asked the key question in the qualitative approach to dynamical systems: Should I expect this picture to persist if I restore the higher order terms? In other words, is the reduced system structurally stable? It is now known that integrable Hamiltonian systems of two or more degrees of freedom are not structurally stable. It is for this reason, even if no other, that they are exciting and productive to study.

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In this section we have described how the structural stability of a Poincar´e map of a continuous dynamical system can be evaluated, even though in general such a map cannot be computed explicitly. In the next section we look at stability and chaos in an explicit discrete dynamical system.

5. Discrete dynamics, blowﬂies, feedback, and stability In a series of population dynamics experiments, May and Oster and coworkers29 chose to rear blowﬂies in boxes (for reasons we cannot entirely fathom — surely there are more alluring model species), and count their numbers at every generation. The blowﬂies in their boxes are a simple ecological system consisting of a single species limited by crowding and food supply, but with no predation. The system was analysed as a model of discrete chaos, and, in a diﬀerent paradigm, as a control system by Mees.30 Assuming discrete generations, the data for the population dynamics of the blowﬂies can be ﬁtted by a ﬁrst-order diﬀerence equation Nt+1 = f (Nt ),

(17)

where N is the number of blowﬂies in the time period t. The function f is chosen so that f (Nt ) increases when the population is small, because there is plenty of food and living space in the box, but decreases when the population is large, because of competition for food and living space. The simplest single-humped function for f that one can think of is a parabola: f (N ) = rN (1 − N ),

(18)

for which Equation 17 is known as the logistic map. The parameter r is then the reproduction rate constant. Equation 17 then says that due to reproduction the population will increase at a rate proportional to the current population, and due to starvation the population will decrease at a rate proportional to the square of the current population. For example, if there is a large number of ﬂies in a box in one time period, they will eat most of the food, and the next generation of ﬂies will be few in number. The weird properties of this simple model never fail to delight people. Their implications for ecologies were explored in May;31 a good modern mathematical treatment, accompanied by downloadable software to play with, is given in Chapter 1 of Ball.32

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The evolution of the population N starting initially at N0 may be found graphically as indicated in the cobweb diagram of Figure 8(a), where the f (N ) of Eq. 18 is plotted against N for a given value of r (dashed curve). A vertical line takes the eye from Nt , the population in time-window t, to the corresponding f (Nt ) and an adjoining horizontal line takes you from f (Nt ) to Nt+1 , the population in the time-window t+ 1. The solution converges to a point of zero population growth where the graphs of f (N ) = rN (1 − N ) and f (N ) = N intersect. This period-1 ﬁxed point (or equilibrium) is a stable attractor: all nearby orbits converge to it as t → ∞.

Fig. 8. (left) The logistic function f (N ), Eq. 18, is plotted against the population N for r = 2.9, (right) the second composition f (f (N )) is plotted against N for r = 3.4.

Increasing the height of the hump, r, means increasing the reproduction rate in the blowﬂy model. For example, at r = 3.4 the equilibrium has become unstable and two new stable equilibria have appeared. These new equilibria are not ﬁxed points of f . They are ﬁxed points of the second composition map, f2 (N ) ≡ f (f (N )), as shown in Figure 8(b). Here, the initial condition N0 is the same as in (a), and the iterates at ﬁrst take the population toward the old ﬁxed point. But then they are repelled from it, because it is unstable, and converge instead to the two intersections of f2 (N ) = f (f (N )) and f2 (N ) = N , between which they oscillate in a period 2 orbit. This situation corresponds to the population N switching between two states: a highly populated generation results in the next generation being poorly populated, but then resources are plentiful enough to induce a populous generation again, and so on.

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One cannot help but be curious as to what happens when the parameter r is increased again, and again . . . We could compute many more of these cobweb diagrams, each at a diﬀerent value of r, but both the diagrams and this chapter would become very crowded. Our curiosity can be assuaged (or whetted!) more succinctly by inspecting the bifurcation diagram of stable solutions in Figure 9. One can easily make out the branch points at r ≈ 3,

Fig. 9. Bifurcation diagram over r for the logistic map, where a point is plotted for each solution at every increment in r.

3.449, and 3.544 corresponding to bifurcations to period 2, 4 and 8 orbits. Beyond that, the period-doubling repeats until the periodic behaviour of the population becomes chaotic. The population never settles to discernibly regular n-periodic oscillations, although the window at r ∼ 3.8 suggests the resumption of some sort of regularity. 5.1. Blowﬂy dynamics as a feedback system So far we have viewed the blowﬂy system as a diﬀerence equation, to model the generational delay, and as a bifurcation problem, to study the stability of the dynamics. Picking up the theme of section 3, it is also instructive to view the blowﬂy system as a simple feedback system. The output of the system (number of adult blowﬂies) is sensed by a controller which implements a mechanism, approximated here by the model function f (N ) = rN (1 − N ) to control the level of input, or number of larvae. The actuating mechanism which transforms the larvae into adult

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Fig. 10. Schematizing the blowﬂy system as a block diagram brings out the feedback nature of the dynamics.

ﬂies is simply the delay time of one generation. Figure 10 represents the feedback system as a block diagram. This diagram may seem rather facile, and nowhere near as interesting as the cobweb or bifurcation diagrams, but it does highlight a diﬀerent side to the problem. For instance we see that the block components are independent. We could change the function f (N ) without changing the simple delay model. Inspecting this diagram also makes it easy to build in perturbations such as predation or injecting more ﬂies from outside. The conceptual diﬀerence between modelling the blowﬂy population as a diﬀerence equation and as a feedback system is how information is treated. In the block diagram representation the information ﬂow is explicit and the feedback is obvious, and we can immediately think up ways of adding additional regulations to it. In this sense feedback is an information science. This information about connectivity is subsumed in the discrete dynamical model, which allows us to analyse the stability of the population but glosses over the fact that the instabilities are caused by feedback. 6. Stability of complex networks The third (and ﬁnal, for this chapter) destination in our world tour of nonlinear dynamics is the 41st ﬂoor of an oﬃce tower in the district of Wan Chai, Hong Kong. It is here that the transport operations and infrastructure of Hong Kong, Kowloon, and the New Territories (which together constitute a Special Administrative Region of the People’s Republic of China, or HKSAR) are controlled and coordinated day-to-day, and planning and policy development for future transport needs are carried out.

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The job of the HKSAR Department of Transport is formidable. Consider the problem: The public transport network carries over 11 million passenger trips each day and this number will increase. It consists of railways, franchised buses, public light buses, private buses, ferries, trams, and taxis. Each of these components is a complex sub-network in its own right. The area is geographically diverse, with islands, harbour, waterways, steep hills, airport, and old built-up districts with limited road space to be traversed or accessed. Environmental imperatives require the use of or conversion to low or zero emissions locomotive units. Eﬃcient integration with transport in the densely populated economic-tiger zones of the Pearl River Delta is becoming necessary. The network as a whole must be safe, aﬀordable, reliable, and robust. It must minimize redundancy and duplication of services, yet be ﬂexible enough to match new demand without undue time-lags and provide services to new and changing population and employment centres. This means it must be capable of response and adaptatation on two time scales, daily and long-term (approximately yearly). What a tall order! Can one tackle this complex network problem using the tools of dynamical systems theory? In dynamical systems language we ask: Is the HKSAR public transport network stable? Intuitively (or through direct experience) we expect such a complex network to exhibit sensitive dependence on initial conditions. One blinking red LED on a signal-room console leads to a log-jam of peak hour trains. Even with no perturbations on the network itself we know (with depressing certitude) that leaving for work ﬁve minutes later than usual is likely to result in arriving at work an hour late. These sorts of cascade eﬀects in networks seem to occur when a small disturbance in one element of a network is transmitted through it leading to instability as it spreads, but what lies behind these phenomena? Studying networks such as the HKSAR public transport network is about building models of how they function, and then analysing those models to understand how changes in the structure of the network will result in changes in behaviour. Na¨ively, one expects that increasing the fraction of interacting elements or increasing the strength of interaction will enhance the stability of a complex network, but as we will show in the next example, that is not necessarily so. In a paper in Nature in 1972 Robert May33 used random matrix theory to show that in a large, linear, randomly coupled network the system dimension and the coupling strength must together satisfy a simple inequality.

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Let us revisit the matrix equation (5): x˙ = Ax.

(5)

May considered this as the linearization of a (large) set of nonlinear ﬁrstorder diﬀerential equations that describe an ecology, or populations of n interacting species, but it could equally well describe rates of passenger turnover at each of n nodes in a public transport network. The elements of the n × 1 column vector x are the disturbed populations xj and the elements ajk of the n × n interaction matrix A describe the eﬀect of species k on species j near equilibrium. Each ajk is assigned from a distribution of random numbers that has a mean of zero, so that any element is equally likely to be positive or negative, and a mean square value α, which expresses the average interaction strength. Then A = B − I, where B is a random matrix and I is the unit matrix. The probability that any pair of species will interact is expressed by the connectance C, measured as the fraction of non-zero elements in A. The elements in the random matrix B are drawn from the random number distribution with probability C or are zero with probability 1 − C. For any given system of size n, average interaction strength α, and connectance C we ask what is the probability P (n, α, C) that any particular matrix drawn from the ensemble gives a stable system? May found that for large n the system (5) is almost certainly stable (P (n, α, C → 1)) if α < (nC)−1/2 , and almost certainly unstable (P (n, α, C → 0)) if α > (nC)−1/2 . This result suggests that an ecology that is too richly connected (large C) or too strongly connected (large α) is likely to be unstable and that the eﬀect is more dramatic the larger the number of species n. May’s result is based ﬁrmly on stability theory as it was developed by Poincar´e and Lyapunov over a hundred years ago, as are more recent results on stability and control of dynamical network systems. For example, Yao et al 34 in proposing a control method for chaotic systems with disturbances and unknown parameters (imprecisely modelled or unmodelled dynamics) rely on Lyapunov stability theory, as do almost all of the applications mentioned by Boccaletti and Pecora35 in the preface to a special issue of the journal Chaos devoted to stability of complex networks.

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7. Conclusions and inconclusions Although dynamical systems and stability theory was born and bred in celestial mechanics and control engineering, we now see that the concepts and methods have much wider application in the biological and environmental sciences and in socio-economic modelling and forecasting. A goal that is shared by many researchers in both hard and soft science is the improved management, and ultimately a priori design, of complex dynamical networks that are intrinsically imprecise or error-prone. To this end there is a need to disseminate the principles of stability and chaos outside mathematics, so that non-mathematical scientists are better-equipped to understand and manage the dynamics of complex natural and anthropogenic systems, and channel uncertainty into stable output. How will these problems, fundamental and applied, be tackled? How will the science of dynamical systems, stability and chaos advance? We suggest that the three main approaches will be used in synergy: qualitative and asymptotic analysis, interdisciplinary collaboration, and computation. The rapid growth of interest in dynamical systems and chaos over the past 30 years is, in a sense, quite diﬀerent from the way that areas of mathematics and physics developed in earlier times. It is not driven by industrialization, as for example was thermodynamics in the 19th century and classical control in the early 20th century, or by defence and cold war imperatives, as was nuclear physics from the 1940s to the 1960s. What we are seeing now is the reverse: theory and mathematics of dynamical systems and chaos together with faster computers are actually driving developments in a wide range of very diverse ﬁelds, from medical imaging to art restoration, traﬃc control to ecosystems, neuroscience to climatology. References 1. J. C. Maxwell, On Governors. Proc. Royal Soc. London, 16, 270–283 (1868). Reprinted in: R. Bellman and R. Kalaba. Selected Papers on Mathematical Trends in Control Theory. Dover Publications Inc. New York, 1964. 2. A. T. Fuller, The early development of control theory. II. Transactions of the AMSE Journal of Dynamic Systems, Measurement, and Control, pages 224–235, September 1976. 3. J. Vyshnegradskii, On the general theory of governors. (Sur la th´eorie g´enerale des r´egulateurs) . Comptes Rendus de l’Acad´ emie des Sciences de Paris, 83, 318 (1876). (Translation in C.C. Bissell: Stodola, Hurwitz and the genesis of the stability criterion. Int. J. Control 50 (6), 2313–2332, 1989). 4. M. Denny, Watt steam governor stability. European Journal of Physics, 23, 339–351 (2002).

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5. G.E. Christianson, In the Presence of the Creator: Isaac Newton and his Times. Free Press, New York; Collier Macmillan, London, 1984. 6. I. Peterson, Newton’s Clock: chaos in the solar system. Freeman, 1993. 7. H. Poincar´e, Sur le probl´eme des trois corps et les ´equations de la dynamique. Acta, 13, 1–270 (1890). 8. J. Barrow-Green, Poincar´e and the Three-Body Problem. The American Mathematical Society, 1997. 9. A. Morbidelli, H. F. Levison, K. Tsiganis, and R. Gomes, Chaotic capture of Jupiter’s Trojan asteroids in the early Solar System. Nature, 435, 462–465 (2005). 10. G. J. Sussman and J. Wisdom, Chaotic evolution of the solar system. Science, 257, 56–62 (1992). 11. D. Adams, Dirk Gently’s Holistic Detective Agency. William Heinemann Ltd UK; Pocket Books USA (1998 reissue edition), 1987. 12. P. Cvitanovi´c, R. Artuso, R. Mainieri, G. Tanner, and G. Vattay, Chaos: Classical and Quantum. Niels Bohr Institute, Copenhagen, 2005. ChaosBook.org/version11. 13. M. L. Cartwright and J. E. Littlewood, On nonlinear diﬀerential equations of the second order, I: the equation y¨ − k(1 − y 2 )y˙ + y = bλk cos(λtα ), k large. J. London Math. Soc., 20, 180–189 (1945). 14. N. Levinson, A second-order diﬀerential equation with singular solutions. Ann. Math., 50, 127–153 (1949). 15. S. Smale, How I got into dynamical systems. Springer, New York, 1980. 16. E. S. Oran and V. N. Gamezo, Origins of DDT in gas-phase combustion. Preprint, NRL Laboratory for Computational Physics and Fluid Dynamics, Washington DC, 2006. 17. P. Holmes, J. L. Lumley, and G. Berkooz, Turbulence, Coherent Structures, Dynamical Systems and Symmetry. Cambridge University Press, Cambridge, 1996. 18. P. Holmes and E. T. Shea-Brown, Stability. Scholarpedia, www.scholarpedia.org, page 4208, 2006. 19. M. W. Hirsch and S. Smale, Diﬀerential equations, dynamical systems, and linear algebra. Academic Press, New York and London, 1974. 20. L. S. Pontryagin, Ordinary Diﬀerential Equations. Addison-Wesley, 1962. 21. J. Niemark, On some cases of periodic motions depending on parameters. Dokl. Acad. Nauk SSSR, 129, 736–739 (1959). 22. R. J. Sacker, On invariant surfaces and bifurcation of periodic solutions of ordinary diﬀerential equations. PhD thesis, Technical Report, IMM-NYU, October 1964. A corrected version is published at www-rcf.usc.edu/∼rsacker/. 23. H. Goldstein, Classical Mechanics. Addison-Wesley, 2 edition, 1980. 24. R. L. Dewar, Classical Mechanics lecture notes. wwwrsphysse.anu.edu.au/∼rld105/C01 ClassMech/index.html, 2001. 25. J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer–Verlag, New York, 1983. 26. P. Holmes, Poincar´e, celestial mechanics, dynamical systems theory and “chaos”. Physics Reports (Review Section of Physics Letters, 193 (3), 137– 163 (1990).

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27. H. Poincar´e, Les M´ethodes Nouvelles de la M´ecanique c´eleste. Vols 1–3. Gauthier-Villars, Paris, 1892, 1893, 1899. 28. F. Diacu and P. Holmes, Celestial Encounters. The Origin of Chaos and Stability. Princeton University Press, 1996. 29. R. M. May and G. Oster, Bifurcations and dynamic complexity in simple ecological models. The American Naturalist, 110, 573–599 (1976). 30. A. I. Mees, Dynamics of Feedback Systems. Wiley, New York, 1981. 31. R. M. May, Stability and Complexity in Model Ecosystems. Princeton University Press, Princeton, NJ, 1974. 32. R. Ball, editor. Nonlinear Dynamics: From Lasers to Butterﬂies. World Scientiﬁc, Singapore, 2003. Chapter 1, B. Davies. 33. R. M. May, Will a large complex system be stable? Nature, 238, 413 (1972). 34. J. Yao, Z-H. Guan, and D. J. Hill, Adaptive switching control and synchronization of chaotic systems with uncertainties. International Journal of Bifurcation and Chaos, 15(10), 3381–3390 (2005). 35. S. Boccaletti and L. M. Pecora, Introduction: Stability and pattern formation in networks of dynamical systems. Chaos: An Interdisciplinary Journal of Nonlinear Science, 16:015101, 2006.

Glossary The terms highlighted in bold-faced type in their ﬁrst appearance in the text are deﬁned or described in this glossary. More comprehensive glossaries of dynamical systems terminology may be found easily on the web; for example, mrb.niddk.nih.gov/glossary/glossary.html, www.dynamicalsystems.org/gl/gl/. Asymptotic solutions: Solutions which asymptotically approach an unstable periodic solution. Homoclinic points or doubly asymptotic solutions: Points at which stable and unstable manifolds intersect transversally. In a Hamiltonian ﬂow the stable and unstable manifolds must intersect transversally inﬁnitely often (or coincide, as in the harmonic oscillator, Equation 11) because otherwise one of them would shrink and volume conservation would be violated. This remains true for dissipative systems.25 Homoclinic chaos or homoclinic tangle or sensitive dependence on initial conditions: A region densely packed with homoclinic points, where the dynamics is equivalent to and described by the Smale horseshoe map. Arbitrarily close initial conditions must actually belong to totally diﬀerent parts of the homoclinic tangle, therefore they evolve quite diﬀerently in time.

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Poincar´ e map and cross section: A sort of stroboscopic map; an extremely useful way of representing the dynamics of a two degree of freedom system on a plane. Consider the set of trajectories of a two degree of freedom Hamiltonian system that satisfy H(p1 , p2 , q1 , q2 ) = C, where C is a constant and p1 , q1 and p2 , q2 are canonical action-angle variables. Each energy level H = C is therefore three-dimensional. To construct a Poincar´e map we take a two-dimensional transverse surface or cross section Σ such as that deﬁned by q2 =0. Then, for given C the value of p2 can be computed by solving the implicit equation H(p1 , p2 , q1 , 0) = C, so that we may locally describe Σ by the two variables (q1 , p1 ). Successive punctures of the surface Σ in one direction by each trajectory form a stroboscopic map of the time evolution of the trajectory in phase space. Recurrence theorem: A volume-preserving system has an inﬁnite number of solutions which return inﬁnitely often to their initial positions, or an inﬁnite number of Poisson stable solutions. Hopf bifurcation: The real parts of a pair of conjugate eigenvalues become positive and a family of periodic orbits bifurcates from a “spiral” ﬁxed point (a focus). Neimark-Sacker bifurcation or secondary Hopf bifurcation: Consider a periodic orbit with period T = 2π/ω1 and suppose that a pair of Floquet multipliers crosses the unit circle at ±eiω2 at an isolated bifurcation point. An invariant torus is born. Solutions on the torus are quasi-periodic, and if qω1 = pω2 for integers p and q the motion is said to be phase-locked. The Floquet multipliers are related to the eigenvalues of the Poincar´e map linearised at the ﬁxed point corresponding to the original T -periodic orbit.

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INSTABILITY THEORY AND PREDICTABILITY OF ATMOSPHERIC DISTURBANCES JORGEN S. FREDERIKSEN CSIRO Marine and Atmospheric Research, Private Bag No. 1 Aspendale, Australia [email protected] Instability theory and chaos theory are applied to study the generation mechanisms of large-scale atmospheric disturbances and for determining their predictability in weather and seasonal climate forecasts. Regime transitions associated with weather prediction, climate prediction and climate change are examined. The reasons for the dramatic reduction in observed winter rainfall in the South West of Western Australia since the mid-1970s are described. The ensemble predictability of strong zonal ﬂow to blocking regime transitions is analysed. The seasonal variability of large-scale instabilities and teleconnection patterns is examined. The causes of the boreal spring predictability barrier associated with seasonal climate prediction of coupled ocean-atmosphere models are examined.

1. Introduction In this chapter we review the application of instability theory and chaos theory for the understanding of the formation of atmospheric disturbances and for improving the predictability of weather and climate regime transitions. Instability and bifurcation theory and stochastic methods can be employed to study the formation of atmospheric disturbances. Ensemble prediction methods based on chaos theory are used to improve forecasts of weather and climate and to study atmospheric and coupled ocean-atmosphere regime transitions. Stochastic modelling approaches together with Floquet methods for analyzing the instability properties of time-dependent ﬂows over the whole annual cycle are applied to examine the causes of the boreal spring predictability barrier in coupled ocean-atmosphere models of seasonal climate predictions. It is shown that with three-dimensional steady observed climatological basic states the properties of most of the important large-scale atmospheric ﬂuctuations can be explained using instability and bifurcation theory.

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Explanations can be provided for the causes of localised cyclogenesis and the structures of the major storm tracks in both hemispheres. The theory predicts the life-cycles and preferred geographical locations of blocks, which prevents the regular eastward progression of storms. It can also be used to elucidate the origins of the major large-scale low-frequency atmospheric circulation anomalies, which characterise climate variability over extensive parts of the globe, and to explain the genesis mechanisms of tropical disturbances such as intraseasonal oscillations and the classes of convectively coupled equatorial waves. Normal mode instability theory has a long history with the foundations laid during the 19th century. In the area of dynamic meteorology, Charney,6 Eady11 and Phillips48 pioneered the generally accepted theory of cyclogenesis, in which synoptic scale storms result from the baroclinic instability of the large-scale atmospheric ﬂow ﬁeld, which they represented by simple zonal mean ﬂows on beta- and f-planes. Frederiksen15 examined the instability properties of three-dimensional forced basic state ﬂows consisting of baroclinic zonal mean ﬂows and planetary waves in two-level models on a sphere. In this study and that of Frederiksen,16 it was found that the theoretical model could capture the locations of regional cyclogenesis poleward and slightly downstream of the jetstream maxima. The causes of localized cyclogenesis and the structures of the major storm tracks in both the Northern (Frederiksen;18,19 Frederiksen and Frederiksen;13 Whitaker and Barcilon;60 Lee;41 Whitaker and Dole61 ) and Southern (Frederiksen;22 Frederiksen and Frederiksen33 ) Hemispheres have been explained on the basis of instability theory. It has also been possible to explain the dynamical causes of many other atmospheric phenomena including blocking and teleconnection patterns (Frederiksen;18,19 Simmons et al.;53 Branstator;3 Frederiksen and Bell;28 Anderson;1 Frederiksen and Frederiksen;34 Branstator and Held;4 de Pondeca et al.;8,9 Li et al.;43 Frederiksen and Branstator29), intraseasonal oscillations and convectively coupled equatorial waves (Frederiksen and Frederiksen;33,35 Frederiksen26 ) and Australian north-west cloud band disturbances (Frederiksen and Frederiksen14 ). Most recently, instability theory has been applied to explain the climate regime transitions and changes in storm track activity associated with the dramatic reduction in winter rainfall in the south-west of Western Australia (SWWA) that occurred in the early to mid 1970s (Frederiksen and Frederiksen36,37 ).

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In recent years, methods of chaos theory (Lorenz;44 Farrell;12 Frederiksen and Bell;28 de Pondeca et al.;8,9 Frederiksen;25 Wei and Frederiksen58 ) have been applied to develop ensemble weather prediction schemes. These methodologies have been implemented at major weather forecasting centres and have been shown to improve medium range weather forecasts and to provide estimates of the reliability of the forecasts (Toth and Kalnay;54,55 Molteni et al.;47 Frederiksen et al.31 and references therein). Instability theory has also been applied to examine the normal mode instability of ﬂows varying in both space and time (Frederiksen;24,25 de Pondeca et al.;8,9 Li et al.;43 Frederiksen and Branstator;29,30 Wei and Frederiksen58,59 ). In particular, both Floquet instability methods (Frederiksen and Branstator29) and stochastic modelling approaches (Frederiksen and Branstator30) have been used to analyse modes of variability of timedependent ﬂows over the whole annual cycle. These methods have been applied to examine the causes of the seasonal variability of predictability and in particular the boreal spring predictability barrier in coupled oceanatmosphere models of seasonal climate prediction (Latif and Graham;40 Webster and Yang62 ). In section 2, we brieﬂy summarize the methodology for studying the instability properties of steady ﬂows that may be three dimensional. We also present the details of a simple separable model of baroclinic instability that is analytically solvable. In this section we review the application of instability theory for explaining the causes of regional cyclogenesis and the structure of the storm tracks in both hemispheres. We also discuss the applications to blocking, large-scale teleconnection patterns, intraseasonal oscillations, equatorial waves and Australian north-west cloud bands. In section 3, we discuss the application of instability theory for understanding the causes of the dramatic reduction in winter rainfall in the south-west of Western Australia in the early to mid-1970s. The application of chaos theory to ensemble weather prediction is discussed in section 4. In section 5, the generalization of instability theory and stochastic modelling approaches to study time-dependent ﬂows is considered. Applications of these methods for examining the variability of instabilities and teleconnection patterns over the whole annual cycle are explored. The conclusions are summarised in section 6.

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2. Instability Theory 2.1. Theoretical approach The prognostic equations for ﬂuid motion including that describing atmospheric dynamics may be written formally in the form dX(t) = N(X(t)) dt where N denotes a nonlinear matrix operator and X(t) is a vector specifying the state of the ﬂuid. Suppose now that we are interested in the development of small perturbations x(t) on a basic state x ¯(t) where X(t) = x ¯(t) + εx(t). Then for small ε we obtain the tangent linear equation for the development of the perturbations: dx(t) = M(t)(x(t)) dt where M(t) = M[¯ x(t)] and M is a linear matrix that depends on the basic state. For stationary basic states such that M is constant the solution to the linear perturbation equation is x(t) =

N

κν φν exp(−iω ν t)

ν=1

where N is the length of the vectors. Here ω ν are the eigenvalues and φν are the eigenvectors of the eigenvalue problem (ω ν I − iM)φν = 0,

ν = 1, . . . , N.

Also, ω ν = ωrν + iωiν and ωiν is the growth rate and ωrν is the frequency. The coeﬃcients κν depend on the initial conditions (Frederiksen and Bell28 ) and the eigenvalues and eigenvectors are determined above.

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2.1.1. Separable Example To illustrate the above instability theory we consider a separable example for which a simple analytical solution is derivable. The example is taken from Frederiksen15,17 and is a generalization to spherical geometry of Phillips’48 model of baroclinic instability. The two-level nondimensional equations for quasigeostrophic ﬂow on a sphere are given in Eq. (2.1) of Frederiksen15 and when linearized about a basic state of solid body rotational ﬂow take the form: ∂(∇2 ψ) ¯ ∇2 ψ) − J(ψ, ∇2 ψ¯ + f ) − J(¯ = − J(ψ, τ , ∇2 τ ) − J(τ, ∇2 τ¯) ∂t ∂(∇2 τ − Γτ ) ¯ (∇2 τ − Γτ )) − J(ψ, (∇2 τ¯ − Γ¯ = − J(ψ, τ )) − J(¯ τ , ∇2 ψ) ∂t − J(τ, ∇2 ψ¯ + fo ) where J(A, B) =

∂A ∂B ∂A ∂B − ∂λ ∂µ ∂µ ∂λ

Here ψ is the average of the upper (250hP a) and lower (750hP a) level perturbation streamfunctions and τ is the perturbation shear streamfunction, being one half the upper minus lower level ﬁelds. Also f is the nondimensional Coriolis parameter, fo is the Coriolis parameter at 45◦ latitude, λ is longitude and µ is sin φ where φ is latitude. We also have Γ=

fo2 ; σ ¯

fd = 2µ = 2 sin φ; Ω ψ¯ = −αµ = −α sin φ;

f=

bcp ∆θ , 2a2 Ω2 fd fo = o = 2µo = 2 sin φo , Ω τ¯ = −βµ = −β sin φ, σ ¯=

u ¯1 = (α + β)(1 − µ2 ) 1/2 = (α + β) cos φ, u ¯3 = (α − β)(1 − µ2 ) 1/2 = (α − β) cos φ. In these expressions Ω is the earth’s angular frequency, a is the earth’s radius, b = 0.124, cp is the gas constant at constant pressure, ∆θ is the 250-750 hPa potential temperature diﬀerence and σ ¯ is the nondimensional static stability. Also the basic state streamfunctions, denoted by a bar, are u3 ) level speciﬁed by parameters α and β as are the upper (¯ u1 ) and lower (¯

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zonal velocities. Then it may be shown that for disturbances of the form ψ = Re

∞

ψmn Pnm (µ)ei(mλ−ωt) ;

τ = Re

n=m

∞

τmn Pnm (µ)ei(mλ−ωt)

n=m

the eigenvalues are given by

2 2 ω 1 =− −α 1− + m n(n + 1) n(n + 1) n(n + 1)(n(n + 1) + Γ)

1 q 3 Γ/4] . × Γ(1 + α) ± [β 2 y 2 (n) + qoµ oµ Here, Pnm (µ) are orthonormalized Legendre functions. Also y(n) = n(n + 1){n(n + 1) − 2} − Γ(Γ + 2)/2 and the potential vorticity gradients are given by: u1 − u¯3 ) d2 {(1 − µ2 )1/2 u¯1 } 2µ20 (¯ + dµ2 σ ¯ (1 − µ2 )1/2 = 2 + 2α + (Γ + 2)β,

1 q0µ ≡ 2−

u3 − u¯1 ) d2 {(1 − µ2 )1/2 u¯3 } 2µ20 (¯ + 2 dµ σ ¯ (1 − µ2 )1/2 = 2 + 2α − (Γ + 2)β.

3 q0µ ≡ 2−

Thus a necessary condition for instability is that 1 3 q¯0µ q¯0µ ≤ 0

and this ﬁrst occurs at a wavenumber n for which y(n) = 0. 2.2. Storm tracks and regional cyclogenesis 2.2.1. Northern Hemisphere Frederiksen18 examined the instability properties of observed threedimensionally varying climatological basic states; the Northern hemisphere winter ﬂow averaged over the eight winters 1963-64 to 1970-71 was used as the basic state in a two-level quasigeostrophic model. Fig. 1 shows the 250hP a disturbance streamfunction of the fastest growing mode (eigenvector) with a growth rate of 0.43 day −1 and a period of 3.3 days for the case when the potential temperature diﬀerence between the 250 hPa and 750 hPa levels is 23K, typical of the atmosphere. The mode consists of a train of eastward propagating highs and lows with a monopole

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(a)

(b)

(c) Fig. 1. N.H. winter storm track disturbances showing 250hP a (a) disturbance streamfunction (b) amplitude of disturbance streamfunction of fastest growing cyclogenesis mode in the theory of Frederiksen18 and (c) standard deviation of band pass ﬁltered 500hP a geopotential height ﬂuctuations from the observations of Blackmon.2

structure in the latitudinal direction and a dominant zonal wave number of about 10. The mode has similar structure but slightly smaller amplitude at the lower level and has a westward tilt with height (not shown) typical of baroclinic disturbances. The amplitude envelope of the 250hP a disturbance is shown in Fig. 1b. The disturbance has largest amplitude over the North Atlantic ocean and a secondary maximum over the North Paciﬁc ocean. The regions of preferential development are slightly downstream and poleward of the jetstream maxima of the basic state (not shown). The locations of these

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regions are in quite good agreement with the standard deviations of band pass ﬁltered geopotential height ﬂuctuations that characterize the Northern Hemisphere winter storm tracks shown in Fig. 1c (reproduced from Fig. 5 of Blackmon2). The geographical locations of the storm tracks in Fig. 1 are also captured by higher vertical resolution calculations for northern winter as shown in Fig. 9 of Frederiksen19 for a ﬁve-level quasigeostrophic model. Robertson and Metz50 subsequently investigated the feedbacks of transient eddies generated by the instability of three-dimensional Northern Hemisphere ﬂows. They compared storm tracks based on the instability of general circulation model basic states with the general circulation model storm tracks and found general good agreement as far as geographical locations are concerned. Frederiksen and Frederiksen33 found that convective processes increase storm track mode growth rates but have little eﬀect on structures. Instability theory with three-dimensional basic states has also been used to explain the deﬂection and splitting of storm tracks from their usual climatological positions by blocking highs (Frederiksen;23 Frederiksen and Bell28 ) and other large-scale persistent anomalies (Robertson and Metz49 ). Frederiksen and Frederiksen13 examined the roles of variable static stability and non-geostrophic eﬀects, as well as planetary wave structure, in the location of storm tracks by comparing results from two-level primitive equation and quasigeostrophic models. They used the monthly averaged basic states for January 1979 and found noticeable diﬀerences for the North American storm track due to a region of very low static stability just oﬀ the coast of the United States. Subsequent instability studies of localized N.H. cyclogenesis include those of Whitaker and Barcilon,60 Lee,41 Whitaker and Dole.61 2.2.2. Southern Hemisphere In the Southern Hemisphere, the dynamics of the highs and lows that make up the storm track in the diﬀerent seasons can again be understood using instability theory with three-dimensionally varying basic states. Frederiksen18 and Frederiksen and Frederiksen34 examined the Southern Hemisphere storm track modes in 5-level quasigeostrophic and 2-level primitive equation models respectively. The fastest growing modes again are monopole cyclogenesis modes with dominant zonal wave number of about 10 (not shown). Fig. 2a shows the amplitude of the 700 hPa disturbance streamfunction for the fastest growing mode in the 5-level calculation of Frederiksen22 with a January basic state obtained from monthly averages between 1972 and 1976. We note the maxima in the eastern Hemisphere

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over the Southern ocean. In Fig. 2b we reproduce Fig. 11c of Trenberth56 which shows the geopotential height variance at 500 hPa in the 2-8 day band in summer (4 November to 11 March) and based on analyses for the period 1972 to 1980. Again we note that the leading mode of instability successfully captures the regions of largest geopotential height variance associated with developing storms.

(a)

(b)

Fig. 2. S.H. summer storm track disturbances showing (a) amplitude of 700hP a disturbance streamfunction of fastest growing cyclogenesis mode in theory of Frederiksen22 and (b) variance of 2-8 day band 500hP a geopotential height fluctuations from the observations of Trenberth.56

Three-dimensional instability theory also been applied to study the structure of storm tracks in other seasons. Frederiksen and Frederiksen34 used a two-level primitive equation model with January and July basic states. For January the storm track structure of the leading instability mode was found to be very similar to that shown in Fig. 2a for the ﬁvelevel quasigeostropic model. In July, the leading instability mode (Figs. 5a,b of Frederiksen and Frederiksen34 ) in contrast has elongated eddies and two branches of the storm track downstream of Australia where the mode grows on both atmospheric jetstreams. More recent applications of instability theory to storm track formation are discussed in section 3 below.

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2.3. Blocking The study of Frederiksen18 indicated that instability theory with threedimensional basic state ﬂows could produce a variety of disturbances in addition to the monopole cyclogenesis modes, shown in Fig. 1, associated with the storm tracks. It was suggested that the theory could provide a similar basis for understanding both blocking and localized cyclogenesis as zonally averaged instability theory has traditionally provided for cyclone scale disturbances. It was proposed that large-scale anomalies such as mature blocks would be initiated by the generation of so-called onsetof-blocking dipole modes upstream of the regions of the large amplitude blocks. The onset modes have westward tilt with height indicating the importance of baroclinic processes in their formation. As they propagate eastward they increase in amplitude, become quasi-stationary and become essentially equivalent barotropic. Figs. 3a and b show the 250hPa disturbance

(a)

(b)

Fig. 3. Paciﬁc-North American winter blocks showing 250hP a streamfunction for (a) onset-of-blocking mode and (b) mature blocking mode in theory of Frederiksen.18,19

streamfunction for the onset-of-blocking (from Frederiksen,18 Fig. 6a) and mature blocking modes (from Frederiksen,20 Fig. 3a) respectively, relevant to blocking in the Paciﬁc-North American region. The basic state is again the Northern hemisphere winter ﬂow averaged over the eight winters 196364 to 1970-71 and the calculation used the same two-level quasigeostrophic model as for the storm track results in Fig. 1. The onset-of-blocking mode

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is the third fastest growing mode while the mature Paciﬁc-North American mode is the twenty-third fastest. Modes very similar to the mature PaciﬁcNorth American mode, which is equivalent barotropic, are also found to be the leading modes in barotropic calculations with similar upper level zonally varying basic states (Simmons et al.;53 Frederiksen20 ). The predicted sequence of development of the Paciﬁc-North American mature anomaly based on instability theory is very similar to that subsequently obtained from analysis of observations by Dole.10 Dole examined the time sequence of composite analyses of 15 positive anomaly cases leading to the establishment of a mature Paciﬁc anomaly pattern. In Fig. 4a (taken from Fig. 4a of Dole10 ) is shown unﬁltered 500hPa geopotential data for the perturbation on day -3 before the appearance on day 0 of a quasi-stationary large-scale positive anomaly in the key region in the north central Paciﬁc; Fig. 4b (taken from Fig. 1f of Dole10 ) shows the corresponding low-pass ﬁltered data for the mature anomaly on day 6. We note the

(a)

(b)

Fig. 4. Paciﬁc-North American winter blocks showing 500hP a geopotential height anomalies for (a) unﬁltered data on day -3 and (b) low-pass ﬁltered data on day 6 from the observations of Dole.10

broad similarities particularly in the Paciﬁc-American region between the onset-of-blocking mode in Fig. 3a and the day -3 anomaly in Fig. 4a; as well, there is a close correspondence between the mature instability mode in Fig. 3b and the Paciﬁc-North American anomaly in Fig. 4b. As proposed in the instability theory, the development of the blocking anomalies consists of an initial phase in which rapidly growing and relatively rapidly eastward

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propagating dipole wave-trains form in the east Asia-Paciﬁc Ocean region. These disturbances are westward tilting with height and develop through largely baroclinic processes. As the disturbance grows it propagates into the central Paciﬁc, increases its scale, becomes quasi-stationary and essentially equivalent barotropic. Thereafter the anomaly further ampliﬁes through essentially barotropic eﬀects and obtains the structure of the Paciﬁc-North American pattern. Subsequent applications of instability theory have focused on block development on instantaneous ﬂows (Frederiksen;23 Frederiksen and Bell28 ) and time-dependent ﬂows (Frederiksen;24,25 de Pondeca et al.,;8,9 Li et al.,;43 Wei and Frederiksen58,59 ). Instability theory has also been employed to understand block development in the Southern hemisphere (Frederiksen;21 Frederiksen and Frederiksen;34 Wei and Frederiksen58,59 ). 2.4. Other large-scale atmospheric disturbances Normal mode instability theory has also been applied to explain the generation mechanisms of other large-scale teleconnection patterns in both the Northern (Branstator;3 Frederiksen and Bell;27 Anderson;1 Frederiksen and Frederiksen;13 Branstator and Held;4 Frederiksen and Branstator29) and Southern (Frederiksen and Frederiksen33 ) hemispheres. It has been employed to explain the genesis of intraseasonal oscillations and convectively coupled equatorial waves (Frederiksen and Frederiksen;33,35 Frederiksen26 ) and Australian north-west cloud band disturbances (Frederiksen and Frederiksen14 ). 3. Climate Regime Transitions The early to mid-1970s was a time of major shift in the structure of the large-scale circulation of both the Northern and Southern Hemispheres (Trenberth 1990). In the Southern Hemisphere there was a dramatic reduction of 20% in winter rainfall in the south-west of Western Australia (SWWA) associated with an increase in Perth mean sea-level pressure (MSLP) (Sadler et al.;51 Smith et al.;52 IOCI39 ). Recently, Frederiksen and Frederiksen36,37 studied the causes of the SWWA July rainfall reduction and the associated changes in the large-scale circulation and in transient disturbances in the Southern Hemisphere. Most noticeably they ﬁnd a reduction of 20% in the peak strength of the SH subtropical jet stream together with a southward shift. This reduction in turn

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is related to in the SH Hadley circulation. As well, the thermal structure of the SH atmosphere has altered with a signiﬁcant warming south of 30S and a reduction in the equator-pole temperature gradient. They also ﬁnd that this transition in the structure of the Southern Hemisphere circulation has a dramatic eﬀect on the nature of the SH storms, which have a major impact on southwest Western Australia (SWWA), and on other modes of weather variability. In fact, the SH atmosphere has generally become less unstable in those regions associated with the generation of mid-latitude storms. Frederiksen and Frederiksen36,37 used a primitive equation insta-

(b)

(a)

(c) Fig. 5. Leading storm tracks modes inﬂuencing Western Australian winter rainfall showing 250hP a streamfunction for (a) mode 1 for 1949–1968 basic state, (b) mode 8 for 1975–1994 basic state and (c) mode 9 for 1975–1994 basic state.

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bility model to identify the dominant unstable weather modes in July for the two 20 year periods 1949-68 and 1975-94 before and after the transition. In the earlier period, the dominant weather mode (mode 1) is a SH cyclogenesis mode which aﬀects southern Australia, and has largest impact over SWWA; Figure 5a shows the amplitude of this leading storm track mode. The associated wind divergence also has largest amplitude over SWWA and is indicative of enhanced rainfall. Of the ﬁrst 10 dominant weather modes there are about half a dozen cyclogenesis modes that have similar structure. By contrast, in the latter period, the dominant SH cyclogenesis mode (mode 8) has a diﬀerent horizontal structure. In particular, this weather mode eﬀectively bypasses SWWA and impacts more on the eastern seaboard (Figure 5b) and has a growth rate which is around 30% less than for the leading mode for 1949-68. There are, however, other subdominant weather modes (Figure 5c which shows mode 9), with a similar structure and frequency to the dominant mode from the earlier period, but their growth rates have been reduced by more than 30%. These results are consistent with the observed reduction in rainfall over southern Australia, and in particular, SWWA. Also, their largest impact has shifted to be over eastern Australia (Figure 5b). Overall, there has been more than a 30% reduction in the intensity of storm development associated with changes in the winter climate. There may be other contributing causes to the observed rainfall reduction over SWWA since the mid 1970s such as changes to the land surface and associated ﬂuxes due to land clearing (IOCI39 ). However, the reduction in the intensity of cyclogenesis and the related changes in the instability properties of the large scale Southern Hemisphere circulation are so dramatic that they are expected to be the primary cause of the rainfall reduction. 4. Ensemble Weather Prediction during Regime Transitions During the last decade or so methods of chaos theory have been applied to develop ensemble prediction schemes based on fast growing perturbations (Toth and Kalnay,54,55 Molteni et al.,47 Frederiksen et al.31 ). Here we discuss the study of Frederiksen et al.31 in which the skill of ensemble prediction during blocking regime transitions was examined for Northern Hemisphere ﬂows within two atmospheric models. An ensemble prediction scheme based on fast growing perturbation was implemented for the Commonwealth Scientiﬁc and Industrial Research Organisation (CSIRO) conformal-cubic model (McGregor and Dix46 ) and the Bureau of Meteorology Research Centre (BMRC) spectral model (Hart & et al.,38 Frederiksen

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& et al.32 ). The methodology uses a breeding method, based on an implicit linearization of the models, in which perturbations likened to leading Lyapunov vectors are obtained and used to perturb the initial conditions. Detailed comparisons of the skill of ensemble mean forecasts with control forecasts were carried out for Northern Hemisphere initial conditions in October and November 1979. A particular focus was the variability of forecast skill during regime transitions associated with the development, maturation and decay of the large-scale blocking dipoles that occurred in the major blocking regions over Europe, over the Gulf of Alaska, over the North Atlantic and as well over North America. The aim was to show that despite using quite diﬀerent models veriﬁed against diﬀerent analysis, the forecasts have similar variability in forecast skill that is determined by the instability regimes for the diﬀerent synoptic situations. 4.1. Methodology The ensemble prediction scheme used is an iterative breeding method31,54,55 which is relatively easy to implement and computationally relatively cheap. In the method of Frederiksen et al.31 it consists of a period of 10 days of breeding of perturbations starting at the beginning of October 1979 followed by a period of self-breeding and 10 day ensemble forecasts starting at 0000 UTC on 11 October and ﬁnishing at 1200 UTC on 21 November 1979. During the breeding period a speciﬁed small perturbation is added to the analysis and 12-hour forecasts are performed from both the perturbed and unperturbed (control) initial conditions. The diﬀerence between the 12hour perturbed and control forecasts is then scaled to have the same rootmean-square (RMS) magnitude as the initial perturbation and is added to the subsequent 12-hour analysis. Again 12-hour forecasts are performed from both the new perturbed and unperturbed analysis and the process is repeated until the end of the 10-day breeding period. The breeding methodology corresponds essentially to an implicit linearization of the nonlinear dynamics about the time-dependent analyses and would with time result in the bred perturbation converging to the leading Lyapunov vector were it not for stochastic eﬀects associated with the convective parameterizations in the numerical weather prediction models. In these studies31 eight separate breeding cycles starting from 8 diﬀerent perturbations over the ﬁrst 10 days were performed. The self-breeding was also performed for 8 diﬀerent perturbations with the ensemble forecast consisting of 16 forecasts employing the 8 initial bred perturbations and 8 initial identical perturbations with opposite signs. This ensures that

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the ensemble mean forecast starts from the same initial condition as the control. 4.2. Synoptic situations The performance of the ensemble prediction method was examined for initial conditions taken from the period between 0000 UTC on 11 October 1979 and 1200 UTC on 21 November 1979. This interval was a time of large-scale Northern Hemisphere blocking in the major blocking regions over the North Atlantic Ocean and over Europe and over the Gulf of Alaska and as well over North America. 4.3. Error growth and predictability In Fig. 6a is shown for the CSIRO and BMRC models the 500 hP a Northern Hemisphere RMS zonal wind errors (between 20◦ and 90◦ N) for the ensemble mean and control forecasts averaged between 0000 UTC on 11 October and 1200 UTC on 21 November 1979. We note that for forecasts longer than about 3 days the average errors of the ensemble mean forecasts are lower than for the control forecasts. Quite similar results are obtained in terms of the 500 hP a geopotential height as shown for the two models in Fig. 6b. For both BMRC and CSIRO models 500 hP a geopotential height errors in m are typically ten times 500 hP a zonal velocity errors in m/s for forecast times longer than 4 days. This relationship which is evident from Figs. 6a and b for average forecast errors also applies in good measure for errors on a particular day. More detailed information on the variability of error growth in diﬀerent synoptic situations may be seen from the 12 hourly forecast results. This may best be seen by focusing on 60◦ longitude sectors of the Northern Hemisphere extratropics (20◦ − 90◦ N) where the diﬀerent blocks form. Although blocking appears to be primarily a local phenomenon42,45 the development of a block or the presence of a mature block in a given sector aﬀects not only the error growth in the sector but tends to have a similar eﬀect on sectors both upstream and downstream. For this reason, and because the Atlantic sector is a region for a major Northern Hemisphere storm track, we focus here on the sector 20◦ − 90◦ N, 0◦ − 60◦ W. Fig. 7 shows the RMS zonal wind errors of CSIRO ensemble, control and control-ensemble 10 day forecasts started every 12 hours for the Atlantic sector; results for the BMRC model are very similar as shown in Fig. 10 of Frederiksen et al.31 The signatures of the life cycle of the four blocks mentioned in Section 4b, and detailed in Section 4 of Frederiksen et al.,31

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Fig. 6. Northern Hemisphere RMS errors of (a) 500 hP a zonal wind (m s−1 ) and (b) 500 hP a geopotential height (m), averaged between 0000 UTC on 11 October and 1200 UTC on 21 November 1979, for ensemble mean and control forecasts with the CSIRO and BMRC models.

are particularly clear in the control and especially ensemble errors shown in Fig. 7. Forecasts on diagonals ending on 25 October, 5 November, 13 November and 19 November, when the blocks in the respective European,

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Gulf of Alaska, Atlantic and North American regions are rapidly developing, tend to have relatively large errors for a given forecast time. In contrast, when the blocks have reached their mature phase, on 30 October, 8 November and 23 November, forecast errors are somewhat smaller than average for a given forecast time. This is consistent with the notion that errors grow rapidly when dynamical development is rapid (Frederiksen and Bell28 ) and are suppressed in the presence of large-scale equivalent barotropic waves such as mature blocks (Frederiksen15 and Colucci and Baumhefner7 ). We may quantify the similarities of the error growth shown in the Hovmoeller diagrams for the Atlantic sector in the CSIRO and BMRC models. The pattern correlations are 0.983 for ensemble errors, 0.980 for control errors and 0.412 for control-ensemble. The pattern correlations between anomalies from the time mean are 0.611 for ensemble errors and 0.540 for control errors. Quite comparable ﬁndings for the whole Northern Hemisphere extratropics are also found as discussed by Frederiksen et al.31 4.4. Discussion Studies of the variability of predictability depending on the particular synoptic situations, contrasting the growth, maturation and decay of blocks and alternating times of strong zonal ﬂows have been reviewed. Primary ﬁndings are as follows: • For both the CSIRO and BMRC models, on average the ensemble mean forecast is better than the control forecast for forecast times longer than 3 or 4 days. • 500 hP a Northern Hemisphere RMS errors of zonal wind and geopotential height (and meridional velocity and temperature), averaged between 0000 UTC on 11 October and 1200 UTC on 21 November 1979, are lower for the ensemble forecasts than for the control forecasts. • Despite the diﬀerent model formulations, the average error growth curves in the two models are quite similar. • There is considerably variability in the skill of both ensemble and control forecasts related to particular synoptic events: errors tend to be larger for forecasts validating when blocks are developing or decaying and smaller for mature blocks. • The use of an ensemble forecasting approach generally improves forecast skill including during blocking events; at timescales beyond 4 days, it is possible to extend the skillful range of model forecasts by an extra half or one day by using ensemble mean prediction.

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Fig. 7. RMS errors in the North Atlantic sector of 500 hP a zonal wind (m s−1 ) in 10 day forecasts initiated every 12 hours between 0000 UTC on 11 November and 1200 UTC on 21 November 1979 and for ensemble, control and control-ensemble in the CSIRO model. The tick marks on both axes denote a day.

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5. Teleconnection Pattern Variability and the Predictability Barrier As reviewed in section 2, during the last two decades, normal mode instability theory with time-invariant basic states has provided explanations for the generation mechanisms and dynamical properties of the major classes of synoptic- and large-scale atmospheric disturbances. However, it is only recently that Frederiksen and Branstator;29 hereafter FB1) studied how the inclusion of the annual cycle of basic states aﬀects the properties of leading eigenmodes. They found dramatic seasonal ﬂuctuations in the growth rates and amplitudes of ﬁnite-time normal modes (FTNMs) of the barotropic vorticity equation. Branstator and Frederiksen5 studied the ability of stochastic models to capture low-frequency variability. Frederiksen and Branstator;30 hereafter FB2) used stochastic modelling techniques to examine the seasonal variability of corresponding observed low-frequency atmospheric anomalies. They employed an analogous methodology to that of FB1 to study the seasonal variability of teleconnection patterns determined as ﬁnite-time principal oscillation patterns (FTPOPs) from reanalysed observations. Here we review the extent to which ﬁnite-time normal mode instability theory is able to provide insights into the structural and amplitude variability of teleconnection patterns as they ﬂuctuate during the annual cycle. We also discuss the relationships between the seasonal variability of teleconnection patterns and large-scale instabilities and the boreal spring predictability barrier. Models of climate prediction over the tropical Paciﬁc commonly encounter a predictability barrier in boreal spring when correlations between observations and predictions rapidly decline (Latif and Graham40 ). Lagged correlations between the mean monthly Southern Oscillation Index are also found to decrease rapidly in boreal spring (Webster and Yang62 ). 5.1. Theory The FTPOPs are the eigenvectors of the propagator obtained by ﬁtting a linear stochastic model to a statistically cyclostationary data set with ¯ (t) that is time-dependent and ﬂuctuations about the mean x(t). mean x Our linear stochastic model has the form dx = M(t)x(t) + f (t) dt

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where M(t) is a time-dependent matrix determined from the data and f (t) represents noise. The solution to this equation is t dsG(t, s)f (s) x(t) = G(t, 0)x(0) + 0

Here G(t, s) is the propagator which has the integral representation t G(t, s) = T exp dσM(σ) s

where T is the chronological time-ordering operator (FB1, Eq.(2.4)). Note that G(t, s) may be constructed as the product of propagators over suﬃciently short time intervals (tk , tk−1 ), taken as half-hour time steps (FB2, Eq.(2.8)): G(t, s) = G(t, tl−1 )G(tl−1 , tl−2 ) . . . G(t1 , s). Since our data are only sampled every ∆t hrs we estimate the stability matrix M(t) through the associated ﬁnite-diﬀerence equation. The estimate of M that minimizes the noise is then given through Gauss’ theorem of least squares as −1 − I (∆t)−1 M = x(t + ∆t)x+ (t) x(t)x+ (t) where I is the unit matrix, + denotes Hermitian conjugate and angular brackets denote ensemble (or time) means. The FTPOPs between an initial time t = 0 and a ﬁnal time T, taken here to be 1 year, are the eigenvectors of the eigenvalue-eigenvector problem (λν [T, 0]I − G(T, 0))φν [T, 0] = 0 ν = 1, . . . , N where λν = λνr + iλνi are eigenvalues and φν are eigenvectors. The global growth rate and frequency may be deﬁned by: λν = λνr + iλνi = exp[−i(ωrν + iωiν )T ] Here, ωiν is the global growth rate and ωrν is the global phase frequency. We also deﬁne the relative ampliﬁcation factor, the ratio of the evolved to initial amplitudes scaled by the global growth factor, by Rν (t) =

xν (t) exp(−ωiν t)

xν (0)

and the local total growth rate by ω ˜ iν (t) = ωiν +

d ln Rν (t), dt

ν = 1, . . . , N.

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We note that the FTPOPs satisfy exactly the same eigenvalueeigenvector problem as do FTNMs except that for the FTNMs of FB1 the matrix M(t) is obtained from the linearized barotropic vorticity equation while for the FTPOPs it is obtained as described above from the observational data. 5.2. FTPOPs and FTNMs The column vector x(t) consists of spherical harmonic spectral components truncated rhomboidally at wave number 15 and based on twice daily National Centers for Environmental Prediction-National Center for Atmospheric Research reanalysis 300-hPa streamfunction ﬁelds. The data is taken from the 40 year period starting on 1 January 1958.

(a)

(b)

Fig. 8. (a) Local total growth rate (dashed) and relative ampliﬁcation factor (thick solid) of FTPOP1 as functions of time starting and ﬁnishing on 15 January, (b) corresponding average local total growth rate (dashed) of ﬁve leading FTPOPs and relative ampliﬁcation factor (thick solid)

Fig. 8a shows two measures of the change with the annual cycle of the root mean square streamfunction amplitude of FTPOP1, the leading (least damped) empirical mode. The ﬁrst is the local total growth rate, ω ˜ i1 (t) (dashed), the tendency of the logarithm of the amplitude, and the second is the relative ampliﬁcation factor R1 (t) (thick solid), the ratio of the evolved to initial amplitudes (scaled by expωi1 t where ωi1 is the global or annual average growth rate). Comparing this diagram with the corresponding results in Fig. 9a for FTNM1 we note a number of general similarities. Firstly the maximum relative ampliﬁcation factor occurs in early boreal spring and

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has a magnitude of around twice the value in January. It then plummets in late boreal spring attaining low values in boreal summer and autumn and with generally increasing values in late autumn and early winter. We also note that the minimum in R1 (t) for both FTPOP1 and FTNM1 occurs between October and November. These similarities are also reﬂected in the local growth rates. In Fig. 8b we show the average local total growth rate of the ﬁve leading FTPOPs (in dashed) and the relative ampliﬁcation factor R(t) (in thick solid). The ampliﬁcation factor again has largest values in the ﬁrst half of the year, as for the ﬁve leading FTNMs in Fig. 9b, decreases rapidly in late boreal spring and summer and then increases gradually in boreal autumn and winter. The growth rates tend to be smallest on average in northern summer, and largest in northern autumn and winter.

(a) Fig. 9.

(b)

(a) As in Fig. 8a for FTNM1, (b) as in Fig. 8b for average of ﬁve leading FTNMs.

Fig. 10 shows the 300-hPa disturbance streamfunction for FTPOP1 on 15 January on Northern Hemisphere stereographic projection. FTPOP1 displays the distinct Paciﬁc-North American pattern. In each month the leading FTPOPs have similar structures to some of the leading empirical orthogonal functions (EOFs), the eigenvectors of the monthly averaged streamfunction covariance matrix, and to leading principal oscillation patterns (POPs), the eigenvectors of the propagator with the monthly averaged empirical stability matrix M (Frederiksen and Branstator30). The structural and amplitude changes of the leading FTPOPs as they evolve (not shown) have a similar complexity to that shown in Fig. 5 of FB1 for their FTNM1. The FTPOPs however tend to have larger relative amplitudes in the subtropical regions and in the Southern Hemisphere than do the FTNMs.

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Fig. 10.

Streamfunction at 300hP a (arbitrary units) for FTPOP1 in January.

5.3. Discussion In many respects the properties of theoretical FTNMs established in FB1 do carry over to their observational counterparts, the FTPOPs. The most striking similarity between FTNMs and FTPOPs is in the seasonality of the growth rates of the leading modes. In both theoretical and empirical settings there is a distinct annual cycle of these rates with the maximum occurring during the middle of the boreal cold season and a broad minimum being present during the boreal warm season. The similarity in the seasonality of growth characteristics is even more evident if one considers the time-integrated eﬀects of growth, as given by our relative ampliﬁcation rate. In this case one ﬁnds that for both theoretical and empirical modes maximum amplitudes are reached near the end of March and minimum amplitudes occur in early November. A further similarity that we have found between growth properties of leading FTNMs and FTPOPs is that both attain growth rates during each season that are similar to the growth one would expect from normal modes and POPs calculated for that season (not shown). This means that in both cases perturbations are reacting to the seasonally changing basic state faster than the state is changing. Both leading FTPOP teleconnection patterns and the leading FTNM instabilities have peak amplitudes in boreal spring. These results suggest a close relationship between the boreal spring predictability barrier of some

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models of climate prediction over the tropical Paciﬁc Ocean (Latif and Graham;40 Webster and Yang62 ) and the amplitudes of the large-scale instabilities and teleconnection patterns of the atmospheric circulation. 6. Conclusions In this chapter we have reviewed the application of instability theory and chaos theory for understanding the dynamical origins of large-scale atmospheric disturbances and for determining their predictability in weather and seasonal climate forecasts. We have discussed the application of instability theory with steady three-dimensional basic states for understanding localized cyclogenesis and the structure of the storm tracks in both hemispheres, for understanding the life-cycles of blocks and other large-scale low frequency anomalies and for elucidating the genesis mechanisms of tropical disturbances such as intraseasonal oscillations, the classes of equatorial waves and the formation of Australian north-west cloud band disturbances. We have discussed the application of methods from instability and chaos theory for predicting regime transitions associated with weather prediction, with climate prediction and with climate change. The reasons for the dramatic reduction in observed winter rainfall in the South West of Western Australia since the mid-1970s have been discussed and related to the reduction and southward shift in the peak strength of the SH subtropical jet stream and, in turn, in a 30% reduction in the intensity of storm development. We have reviewed recent developments in ensemble prediction and how ensemble methods can improve predictability including during strong zonal ﬂow to blocking regime transitions. Methods for analyzing the instability of time-dependent ﬂows have been discussed and applied for understanding the causes of the seasonal variability of large-scale instabilities. Stochastic methods have been examined and their applications for studying the seasonality of teleconnection patterns discussed. The causes of the boreal spring predictability barrier associated with seasonal climate prediction of coupled ocean-atmosphere models have been examined. Acknowledgements It is a pleasure to thank Stacey Osbrough for assistance with the preparation of this chapter. This work was partly supported by the Indian Ocean Climate Initiative of the W.A. Department of Environment, Water and Catchment Protection and by the Australian Greenhouse Oﬃce.

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MULTIPLE EQUILIBRIA AND ATMOSPHERIC BLOCKING MEELIS J. ZIDIKHERI Department of Theoretical Physics, Research School of Physical Sciences and Engineering, The Australian National University, Canberra, ACT 0200, Australia [email protected] JORGEN S. FREDERIKSEN AND TERENCE J. O’KANE CSIRO Marine and Atmospheric Research, Aspendale, Victoria 3195, Australia It has been proposed that large-scale atmospheric mid-latitude ﬂows may possess multiple equilibrium states and that this may be related to observed lowfrequency variability phenomena such as blocking. Since the pioneering work of Charney and DeVore, which was based on a highly simpliﬁed system, many studies have been conducted with varying degrees of complexity. Nevertheless, the issue of the relevance of multiple equilibrium states to atmospheric regime transitions, such as blocking, is still controversial. In this article, we present a systematic account of the theory starting with a highly simpliﬁed system, physically identical to that considered by Charney and DeVore. We then successively increase the complexity by the addition of extra modes, more realistic topographic distribution, and zonal jet structure. We ﬁnd multiple equilibria in all of these systems, suggesting that this mechanism is likely to play a role in large-scale atmospheric dynamics.

1. Introduction Blocking refers to the formation of a quasi-stationary high-pressure system in the atmospheric mid-latitudes. This is associated with a reduction in the strength of the zonal circulation and a corresponding enhancement of the meridional motion, a situation which may persist on a time-scale of the order of a week or longer. In a pioneering study, Charney and DeVore1 —hereafter CdV—proposed a possible mechanism for blocking events in the atmosphere with their Multiple Equilibria hypothesis. They proposed that the atmosphere possesses a variety of steady states corresponding to the observed multiple weather regimes, the blocked and unblocked weather patterns being examples of such regimes. They used a severely truncated

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barotropic beta-plane model to make their case, ﬁnding two stable equilibrium states, which they identiﬁed with the above weather patterns, and one unstable state. One of the stable equilibria consists of strong zonal ﬂow and weak wave activity, corresponding to the unblocked regime, while the other has weaker zonal ﬂow and strong wave activity, corresponding to the blocked regime. A similar study was independently conducted by Wiin-Nielsen2 using spherical geometry; he also reported the existence of multiple states. The physical mechanism which generates multiple equilibria, as proposed by CdV, can be described as follows. Meridional temperature gradients and the Coriolis eﬀect create strong zonal (eastward) jets in midlatitudes; this eﬀect can be parameterized by an appropriate zonal forcing in a barotropic model. The underlying topography, on the other hand, generates Rossby waves, which create drag on the ﬂow, pushing it westward. At some value of the zonal wind, usually lower in value than the zonal wind forcing, and under suitable conditions, the waves might exhibit a sharp increase in amplitude, usually referred to as a resonant response, in which case the large-scale ﬂow becomes locked near the resonant wind value. Thus, for some zonal forcing, dissipation, or topographic height parameter values, the ﬂow will settle into either the state with winds near the zonal forcing value or to one with winds near the resonant wind value, depending on the initial conditions. The advantage of using severe truncation, where only a few dominant modes are retained, is that the simpliﬁed dynamics makes the problem analytically tractable, general results can be deduced, and the physics is transparent. With this procedure, the hope is that some qualitative features of the full (high resolution) model can be captured by retaining only the ‘essential’ modes. On the other hand, as the problem is non-linear, one has to parameterize the eﬀects of the discarded modes in some way. Egger3 attempted to do this by introducing a stochastic forcing in his severely truncated model. He obtained a probability distribution function with maxima corresponding to the stable equilibria found by CdV. Similar studies were conducted, for example, by Benzi et al.,4 Speranza,5 and Sura.6 Another eﬀect of the discarded modes is to facilitate the drain of energy from the retained modes. O’Brien and Branscome,7 for example, used an artiﬁcial damping term to parameterize this eﬀect in their severely truncated twolevel baroclinic model. A less ad-hoc approach was taken by Rambaldi and Mo8 who constructed a low-order model which took into account the eﬀects of non-linear interactions excluded by severe truncation.

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A less insightful but more reliable approach is to perform high resolution numerical experiments. Such experiments have been run, for example, by Tung and Rosenthal9 who showed, with a channel model, that the range of parameters for which multiple equilibria are found is reduced when enough modes are retained. Holloway and Eert10 —hereafter HE, on the other hand, found that multiple equilibria appear over a wide and realistic range of parameters in a barotropic beta-plane model run at high resolution. The same conclusion was reached by Yoden,11 using a similar model but with a different method for obtaining the equilibrium points. More recently, Tian et al.12 performed experiments on a rotating annulus and found two states resembling blocked and unblocked patterns in the atmosphere; numerical simulation, with a barotropic beta-plane model, of the experiment also yielded two equilibria. Moreover, they found that these states undergo two-way spontaneous transitions over time. Another issue that arose with this work is the eﬀect of baroclinic instability, a synoptic-scale atmospheric instability not captured by barotropic models. CdV proposed that the instability caused the atmosphere to intermittently switch from one state to the other. Baroclinic models have been investigated, for example, by Charney and Strauss16 and by Rheinhold and Pierrehumbert.17 Both used severely truncated models and found multiple equilibria. However, Cehelsky and Tung18 argued that when enough modes are taken into account in these models, the multiple equilibria do not appear. On the other hand, HE attempted to trigger transitions by introducing random torques in their barotropic model but were unsuccessful. The same was done by Tian et al.,12 who observed two-way spontaneous transitions in their rotating annulus experiment but did not ﬁnd any in their numerical simulation, suggesting that simple barotropic models are unable to capture this process. Of course, as has been pointed out by Tung and Rosenthal,9 for example, the transitions observed in the atmosphere might be simply due to the diﬀerent parameters changing; for instance, the zonal driving, which is due to diﬀerential heating, changes seasonally. A complementary point of view regarding the role of instabilities in the formation (as well maintenance and decay) of blocks is provided by Three Dimensional Instability Theory, as developed by Frederiksen.19–21 He ﬁnds that baroclinic instability plays an important role in the development of patterns resembling those observed during blocking, particularly in the early stages of their development, with barotropic processes more important in the mature stage. Frederiksen’s results are supported by the observational studies of Dole,22 for example.

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Concerns have also been raised whether resonance, which is a crucial element for the existence of multiple equilibria, is possible in spherical models of the atmosphere. High resolution experiments of this type, using barotropic models, have been carried out, for example, by Kallen,13 Legras and Ghil,14 and by Gravel and Derome.15 In some of these experiments, the zonal wind forcing used was of suﬃciently high value, e.g. 60 m s−1 , as to raise doubts about their realism. However, Yang et al.23 — hereafter YRK—showed, with a baroclinic model on a sphere, that multiple weather regimes exist for realistic values of parameters. Furthermore, they hypothesized that the zonal jet structure found in the atmosphere, which is simulated quite well in baroclinic models, plays a part in developing the resonant behaviour needed for the appearance of multiple equilibria in the CdV scenario by conﬁning the Rossby waves in latitudinal bands. As well as re-examining the theory of CdV, this study seeks to address a number of issues raised by previous investigations. These are as follows: (a) the eﬀect of transient eddies on the ﬂow; (b) the eﬀect of a more complex—and more realistic—topography consisting of more than one mode on the system dynamics; (c) whether resonance is possible on a spherical (global) domain; and (d) whether the zonal jet structure found in the atmosphere can act as a waveguide as proposed by YRK. The article is arranged as follows. In Section 2 we describe the governing equations used in this study, namely, the barotropic vorticity equation and the form-drag equation. In Section 3 we perform a spectral decomposition and construct a low-order version of these equations. In Section 4 we perform a direct numerical simulation of the model equations and compare the results with those obtained from the low-order system. In Section 5 we look at the eﬀect of a bimodal topography on the dynamics of the system. In Section 6 we discuss the problem associated with working on a global domain. In Section 7 we investigate the idea of YRK that a realistic zonal jet structure tends to conﬁne the Rossby waves in latitudinal circles by constructing a global beta-plane model with zonal jets resembling those found in the atmosphere. We then search for multiple equilibria using a realistic distribution of global topography. Section 8 comprises the summary and conclusion.

2. Governing Equations We use the generalized beta-plane barotropic vorticity equation: ∂ζ = −J(ψ − U y, ζ + h + βy + k02 U y) − αζ − ν∇4 ζ, ∂t

(1)

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as discussed by Frederiksen and O’Kane.25 The large-scale zonal ﬂow, U , which is of particular interest in this study, has been separated from the rest of the ﬂow, with the latter being rather loosely referred to as the ‘small-scale’ ﬂow. The small-scale ﬂow, minus small-scale zonal components, will at times also be referred to as the ‘wavy’ ﬂow. Here ζ = ∇2 ψ, where ψ is the streamfunction and ζ is the vorticity, which correspond to the ∂B small-scale ﬂow. J is the Jacobian operator, deﬁned as J(A, B) = ∂A ∂x ∂y − ∂B ∂A ∂x ∂y . U is the large scale ﬂow, deﬁned as the average zonal ﬂow over the domain. h = f0 HHs , where H is the topography; Hs is the scale height of atmosphere taken to be approximately 10 km (see for example Frederiksen24 on how to calculate the scale height); and f0 is the Coriolis parameter at the location of the beta plane (which is assumed to be constant over the domain), deﬁned as f0 = 2Ω sin φ0 , where Ω is the angular velocity of the Earth, and φ0 is the latitude at which the beta plane is located. α is the Ekman dissipation constant. ν is the parameter controlling the strength of the scale-selective dissipation, which is needed to limit the growth of the smallest scales retained in our model. All the variables in this equation have been made dimensionless by scall and the times by Ts = Ω1 , where l is the size ing the lengths by Ls = 2π of the square domain that we are considering. The term containing k0 is a quantitatively insigniﬁcant addition to the β-eﬀect due to solid-body rotation, as outlined by Frederiksen and O’Kane.25 It has little impact on our study of multiple equilibria but is included here for completeness and to enable a one-to-one correspondence with the spherical case considered in Section 6. The corresponding equation for the large scale ﬂow is 1 ∂U ∂ψ = α(U∗ − U ) + dS. (2) h ∂t S ∂x Here S is the area of the domain, and U∗ is the forcing on the large scale ﬂow. The latter is usually identiﬁed with the transfer of momentum from the tropics to the mid-latitudes. We work in the spectral domain, which means that the ﬁelds (ψ, ζ, and h) are expanded in terms of complex Fourier series. For example, ζk (t) exp(ik · x), (3) ζ(x, t) = k

where x = (x, y) are the coordinates in physical space, and k = (kx , ky ) are the coordinates in wavenumber space. This choice of basis functions implies that, in physical space, our domain is doubly periodic. The choice of boundary conditions is not crucial for the arguments presented in Sections

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3, 4, and 5. We shall address the relevance of the chosen boundary conditions in Section 7, where we attempt to make connections to realistic atmospheric ﬂows. In the sum, for each value of k there is a corresponding −k. Note that the Fourier (wave) amplitudes ζk , which together with U are the dynamical variables in our model, are in general complex; to ensure that the physical ﬁelds are real, we need to impose the condition ζ−k = ζk∗ (the star, here, implies complex conjugation). We similarly expand the other ﬁelds ψ and h. 3. Three-Component System Before attempting a direct numerical simulation (DNS) of the above equations at high resolution, i.e., with numerous modes, it is worthwhile to consider a simpliﬁed system consisting of only the large scale ﬂow and the topographic Rossby waves. This is a useful approach as multiple equilibria, in the CdV scenario, arise from the existence of two competing forces: the zonal driving, due to a meridional temperature gradient, which pushes the ﬂow eastward, and the topography, which generates Rossby waves that become stationary at the resonant wind value, giving the ﬂow a westward push due to form drag. As mentioned in the introduction, it is useful to consider the reduced system in order to gain a deeper insight into the behaviour of the high resolution model. This is done as follows. When (3) and the corresponding ψ and h expansions are substituted in (1), we get the following spectral evolution equation: ∂ζk = δ(k + p + q)[K(k, p, q)ζ−p ζ−q + A(k, p, q)ζ−p h−q ] ∂t p q +i

kx (β + k02 U )ζk − ikx U ζk − ikx hk U − (α + νk 4 )ζk , k2

(4)

where K and A are the wave-wave and wave-topography interaction coefﬁcients, respectively. They are given by K(k, p, q) =

(p2 − q 2 ) 1 (px qy − py qx) 2 p2 q 2

(5)

−(px qy − py qx ) . p2

(6)

and A(k, p, q) =

Here δ is the Kronecker delta function; it is equal to 1 if the argument is 0, otherwise it is 0. When (3) as well as the corresponding h expansion are

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substituted in (2), we get ikx ζk h∗ ∂U k = α(U∗ − U ) − . ∂t k2

(7)

k

We can then choose which modes we wish to retain in our model by specifying ζk and hk . In order to simplify the system as much as possible, we initially use a topography consisting of only a single mode, the zonal wavenumber 3 mode. This is not entirely unreasonable, as far as the atmosphere is concerned, as the observed energy spectrum of stationary waves has a peak at that wavenumber. Here we note that the topography entering our model should not be considered as a parametrization for the eﬀect of mountains only but also of other sources of zonal asymmetry such as meridional temperature gradients. To carry out the severe truncation, we retain only the large-scale ﬂow and the (wavy ﬂow) mode corresponding to the retained topographic mode, whose wavenumber we denote by k; to ensure that the physical space ﬁelds are real, we must also retain its complex conjugate. With this choice, all the non-linear wave-wave and wave-topography interaction terms disappear, and we obtain the following two coupled equations for ζk and U: ∂ζk = −iωk (U )ζk − ikx hk U − αζ ζk ∂t

(8)

∂U 2kx ¯ − U ), = 2 (ζk h∗k ) + αU (U ∂t k

(9)

and

where

β + k02 U ωk (U ) = kx U − k2

(10)

is the Doppler-shifted Rossby-wave frequency, and k = (kx , ky ) = (±3, 0). We have also deﬁned αζ = α and αU = α. The scale-selective dissipation term has been ignored at this step. To see that the eddy-eddy interaction terms disappear, observe that there is no way to satisfy the condition k + p + q = 0 in (4) if p, q = ±k. We should also note that the resulting system, represented by (8) and (9), is still non-linear as is evident from the U ζk term on the right hand side of (8). It is illuminating to write equations (8) and (9) in the following form: 2kx ∂U ¯ − U) = 2 (zk ) + αU (U ∂t k

(11)

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∂ (zk ) = ωk (U )(zk ) − αζ (zk ) ∂t

(12)

∂ (zk ) = −ωk (U )(zk ) − kx |hk |2 U − αζ (zk ), ∂t

(13)

zk = ζk h∗k .

(14)

where

Here (zk ) and (zk ) are the real and imaginary parts of zk , respectively. This is a three component system consisting of two waves in a background ﬂow. It is straightforward to show that (zk ) is proportional to the amplitude of the wave in phase with the topography and that (zk ) is proportional to the amplitude of the wave out of phase with the topography. We can then see that it is the out-of-phase wave that is responsible for the drag on the large scale ﬂow: if (zk ) is small, then the ﬂow will simply relax towards U∗ , while if it is large, and negative, then the ﬂow will experience a strong drag and settle to a lower value of U . The equilibrium points can be obtained by plotting (zk ) as functions of U , which are obtained from the steady-state versions of (11), (12), and (13). Equation (11) yields a straight line while equations (12) and (13) yield a resonance curve. The intersections of the two curves yield the equilibrium points. The curves whose intersections yield the equilibria are given by −k 2 αU ¯ (U − U ) 2kx

(15)

−kx |hk |2 αζ U . (αζ )2 + (ωk (U ))2

(16)

(zk ) = and (zk ) =

We also obtain an equation for (zk ): (zk ) =

−kx |hk |2 ωk (U )U , (αζ )2 + (ωk (U ))2

(17)

which does not help us in locating the positions of all the equilibria but is useful as it tells us how the component of the wave motion in phase with the topography varies with U . Equation (16) yields the resonance speed, Ures , at which the topographic drag becomes a maximum:

2 2 β αζ 1 + . (18) Ures = k02 kx k2 (1 − 2 ) k

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As an example, we use H(x, y) = Hm cos(3x), where Hm = 1000 m (from which we can easily work out the wave amplitude hn ), α−1 = 8.7 days, ¯ = 13.3 m s−1 , β = 2.0 × 10−11 m−1 s−1 (corresponding to 30o latitude), U and l (size of our square domain) = 6000 km. We have also set k0 = 0 in (10), which corresponds to the standard beta-plane (see Frederiksen and O’Kane25 ). The resulting plots, for (zk ), are shown as solid lines in Fig. 1 for Hm = 1000 m. We can see from these plots that multiple equilibria in the CdV model are associated with the resonance peak, at around 2 m s−1 , as predicted by (18), with the blocked state being close to this peak, and a zonal forcing which pushes the ﬂow towards a state where the drag (outof-phase component) is weak, near 13 m s−1 . The intermediate equilibrium is an unstable one. The dotted lines represent the amplitude of the wave in phase with the topography ((zk )). We can see that in the unblocked state this is the only wave with signiﬁcant amplitude; the drag is consequently negligible. We can also obtain the equilibrium points algebraically by solving the cubic equation for U resulting from the elimination of (zk ) between (15) and (16). We ﬁnd that the three-component model predicts multiple equilibria for the following range of values of Hm : 600 m< Hm <3000 m. In the next section, we discuss higher resolution DNS results depicted by asterisks in Fig. 1. 4. Direct Numerical Simulation We now carry out higher resolution DNS of the spectral equations with the maximum circular truncation wavenumber of 16. We shall refer to this as the high-order system to distinguish it from the system considered in Section 3. We have found, through experimentation, that this truncation wavenumber is suﬃcient for exploring high resolution behaviour for our chosen parameters. Obviously, for diﬀerent sets of parameters, this resolution may not be suﬃcient; for example, if the dissipation is much weaker. Unless otherwise stated, we keep the same set-up and parameters as stated in the previous section. In addition, we set the parameter ν = 6.06 × 1013 m4 s−1 in (1). These values are plausible for the atmosphere and were speciﬁed by HE, whose results we seek to conﬁrm (although they did not specify the value of the parameter ν and scale height Hs used). In our experiments, unlike HE, we have not added modes of random amplitude to the sinusoidal topography; this enables us to have the best possible comparison with the severely truncated results. The full spectral equations, (4) and (7), are stepped forward in time

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Fig. 1. Resonance curves at equilibrium for three-component model (solid line) and for higher resolution DNS model (asterisks), with Hm = 1000 m. The straight line representing the form drag equation is common to both models. The dotted line represents the amplitude of the wave in phase with the topography.

using a predictor-corrector algorithm. We start oﬀ the ﬂow simulations with both a low initial U = 2.0 m s−1 and a high U = 10.0 m s−1 for a range of values of Hm : 0 < Hm ≤ 2500 m. The initial small-scale ﬁeld is set to zero. The timestep is 1/60 day. We shall, at times, use a scaled time, αt, for convenience (when αt = 1, t = 8.7 days). A bifurcation results as the parameter Hm is varied, as shown in Fig. 2. As Hm is increased from zero, the ﬂows exhibit one (unblocked) equilibrium. Then as Hm reaches a critical value (around 1000 m), the ﬂows suddenly exhibit two equilibria (blocked and unblocked). When Hm reaches another critical value (around 2300 m), the ﬂows suddenly revert to one equilibrium (blocked). All the evolved ﬁnal ﬂow velocities are for t = 167 days. This curve conﬁrms the result obtained by HE although the values for the topography are larger by roughly a factor of two. This might be due to a diﬀerent scale height (not speciﬁed) used in that study. We can see that, at high resolution, the range of values of Hm for which multiple equilibria exist is reduced (now 1000 m< Hm <2300 m) although clearly this range remains signiﬁcant.

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H Fig. 2. Equilibrium values of U for two ﬂows with diﬀerent initial conditions as a function of maximum topographic height Hm .

The ﬁrst point to branch in Fig. 2, at Hm = 1000 m for initial U = 2.0 m s−1 , is interesting because it enables us to see how adding extra modes aﬀects the low-order system described in Section 3. This is because the ﬂows have been initialized with no energy in the small-scales; thus, in the initial stages the high-order system behaves exactly like the loworder system (which has multiple states at this topographic height). When the other modes have picked up suﬃcient energy, as a result of non-linear interactions, the behaviour of the two systems starts to diverge. As can be seen in Fig. 3, the ﬂow seems to settle in the blocked state until αt ≈ 45 and then suddenly makes a transition to the unblocked state. This brings up the question of whether all the ﬂows in the blocked state eventually end up in the unblocked state. For this system at least, the answer is no: in the long run, the dual states seem to persist once the topography is high enough. We can see this in Fig. 4: when Hm = 1100 m, after αt ≈ 60, the ﬂow develops an instability but remains in the blocked state. We have evolved the ﬂow for as long as αt ≈ 320 but no transition was observed; it appears to have settled permanently in this unsteady equilibrium state.

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α

α

α

α

Fig. 3. Kinetic energy time series of (a) large scale ﬂow, (b) kx = ±3, (c) kx = 0, and (d) kx = 0, ±3 modes for Hm = 1000m

It is worthwhile to take a closer look at what happens to the diﬀerent components of the ﬂow for both Hm = 1000 m and Hm = 1100 m. The former being the case where the transient eddies actually ‘destroy’ the multiple states while the latter being a case where they are preserved. For Hm = 1000 m (Fig. 3), the small-scale ﬂow is dominated by the kx = ±3, ky = 0 modes, as a result of interaction with the topography, until αt ≈ 40. When 40 < αt < 50, the ﬂow suddenly jumps to the unblocked state; during this intermediate time, there is a dramatic drop in wavenumber 3 energy and a subsequent rise in the energies of the other modes while the large scale ﬂow rapidly relaxes towards U∗ . For αt > 50, the wavenumber 3 energy has settled to a lower—but not insigniﬁcant—value, and the other modes have vanished. It is important to note that the energy in wavenumber 3 that remains in the unblocked state is not due to topographic drag in the ﬂow, of which there is virtually none. We can see this in Fig. 5, which shows (zk ) and (zk ) (zk has been deﬁned in (14)). It is clear from equation (11) that

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Fig. 4. Kinetic energy time series of (a) large scale ﬂow, (b) kx = ±3, (c) kx = 0, and (d) kx = 0, ±3 modes for Hm = 1100 m

(zk ) represents the topographic drag on the ﬂow; this drag disappears once the transient eddies have perturbed the system. The remaining energy in wavenumber 3 is due to the real part of zk , which represents the amplitude of the wave in phase with the topography. We have also shown, in Fig. 1, the resonance curve (asterisks) for the high-order system. When compared to the resonance curve for the low-order system (solid curve), it is clear that the transient eddies have a damping eﬀect on the resonant wave; hence, this topographic height represents the boundary at which multiple equilibria start to appear in the high-order system whereas for the low-order system the boundary is further down. The resonance curve for the high-order system has been calculated by evolving the small-scale ﬂow, equation (4), with U kept constant, for a range of values of U (0 ≤ U ≤ 15 m s−1 ). (zk ), for k = (3, 0), was then averaged over a series of time-steps once the system had reached a steady state, for each value of U . It is also worthwhile noting that the resonant wind has shifted

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α

α

Fig. 5. Time-evolution of (a) real part, and (b) imaginary part of z for Hm = 1000 m, obtained by DNS.

α

Fig. 6.

α

Same as in ﬁgure 5, but for Hm = 1100 m.

to a somewhat lower value as compared to the low-order system. A similar eﬀect is seen in the article by Speranza,5 for example, which discusses the eﬀects of wave-wave interactions on the low-order system. For the case when Hm = 1100 m (Fig. 4), the ﬂow is dominated by wavenumber 3 until αt ≈ 60 after which there is an analogous drop in that wavenumber’s energy. The diﬀerence is that the extraction of energy from wavenumber 3 is not suﬃcient to allow the large scale ﬂow to relax ¯ . The ﬂow instead becomes dominated by transient eddies of towards U various scales. In Fig. 6, the diagnostics (zk ) and (zk ) show that the mean drag remains more or less the same, even after instability sets in. The drop in wavenumber 3 energy is clearly due to the drop in the real part of zk , the in-phase component.

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So it seems that with this simple model at least, the destabilizing eﬀect of the transient eddies is not enough to limit the ﬂow to just one equilibrium state (the unblocked state). Multiple equilibria as envisioned by CdV can appear in high-order systems. In the following sections we explore further a number of issues related to the behaviour of models exhibiting multiple equilibria. 5. Multi-mode Topographies In this section, we take one step closer towards more realistic physical systems by considering topographies with more than one mode. How does the simple picture of dual equilibrium states, considered previously, change in this case? By extending the reasoning of the previous sections, it might be possible to obtain more than one wavy equilibrium state. The reasoning is as follows: if there are several topographic modes of the right magnitudes, then each one could start to resonate at a particular (unique) value of U and therefore lock the ﬂow near that state, leading to many wavy equilibria. To investigate this possibility, we perform a high resolution experiment with topography of the form H(x, y) = Hm1 cos(x) + Hm2 cos(2x), where Hm1 = 400 m and Hm2 = 1200 m. This topography consists of the zonal wavenumbers 1 and 2. The reason for this choice is the values of the resonance speeds as determined by equation (18): wavenumber 1 has a resonance speed of 18 m s−1 while for wavenumber 2 it is 4 m s−1 . Thus, for this case, the resonance speeds are well separated, and we have a better chance of observing distinct wavy equilibrium states. Since the resonance speeds fall as n2 , the equilibria tend to be ‘bunched’ closer together for higher wavenumbers and hence harder to distinguish in terms of the value ¯ = 30 of the large-scale ﬂow, U ; this is what we wish to avoid. We also use U −1 −1 14 4 −1 m s , α = 18 days, and ν = 3.03 × 10 m s in this section. The results are shown as time-series plots, after the ﬂow has settled into a steady state, in Fig. 7. The large-scale ﬂow has been initialized with three diﬀerent values of U : 2, 18, and 30 m s−1 . The small-scales have been initialized with the same, non-zero, value in all three cases. As can be seen in Fig. 7, there are three distinct equilibrium states: the familiar unblocked state with strong zonal ﬂow and relatively weak waves (dashed lines); the wavy state with a strong wavenumber 1 component and lower value of U (dotted lines); and the wavy state with a strong wavenumber 2 component with lowest value of U (solid lines). It is also clear that the latter has much stronger transient activity than the other states. This could be related to the higher values of the zonal wind possessed by the ﬁrst two states, for,

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as we saw in the previous section, a high value of U seems to inhibit the growth of the other modes in the system. In particular, this might account for the lack of transient activity in the ‘wavenumber 1’ state.

Fig. 7. Time-evolution of (a) the large-scale ﬂow, (b) wavenumber 1 kinetic energy, and (c) and (d) wavenumber 2 kinetic energy for ﬂow with zonal wavenumbers 1 and 2 topographic components. The ﬂows have three diﬀerent initial conditions: the solid lines with U = 2 m s−1 , the short dashed lines with U = 18 m s−1 , and the long dashed lines with U = 30 m s−1 . Three distinct equilibrium states emerge as shown.

6. Global Models So far we have been considering blocking as a localized phenomenon. It is however useful to consider whether we can use a global model to produce blocks in mid-latitudes. To accomplish this as realistically as possible, we need to use a spherical model. As mentioned in the introduction, most of the investigations using spherical models quoted in the literature, which

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have found multiple equilibria, required excessive zonal wind forcing not found in nature (greater than 50 m s−1 ). In this section, we investigate why this is the case. The barotropic vorticity equation in spherical coordinates is given by ∂ψ ∂ζ = −J(ψ, ζ + h) − 2 + α(ζ¯ − ζ) − ν∇4 ζ. ∂t ∂λ

(19)

Here ζ = ∇2 ψ, where ψ is the streamfunction, and ζ is the vorticity in spherical coordinates, λ (longitude) and µ (sine of latitude). J is the Ja∂B ∂A ∂B cobian operator, deﬁned as J(A, B) = ∂A ∂λ ∂µ − ∂λ ∂µ . The topographic contribution to the (potential) vorticity is given by h = f HHs , where H is the topography; Hs is the scale height of atmosphere as in section 2; and f is the variable Coriolis parameter, deﬁned as f = 2Ω sin φ, where Ω is the angular velocity of the Earth, and φ is the latitude. α and ν are the Ekman and scale-selective dissipation constants as in previous sections. ζ¯ is some prescribed value towards which ζ(t) is being relaxed. All the variables in this equation have been made dimensionless by scaling the lengths by Ls = a and the times by Ts = Ω1 , where a is the radius of the Earth. We should note that the ζ and ψ deﬁned here contain both the large and small scales in contrast to their deﬁnition in Section 2, and hence we do not explicitly display an evolution equation for the large scale ﬂow as in (2). If we expand the ﬁelds in terms of spherical harmonics, for example ζmn (t)Pnm (µ) exp(imλ), (20) ζ(λ, µ, t) = mn

where Pnm (µ) are (normalized) associated Legendre polynomials of order m and degree n, with m and n being deﬁned as the zonal and total wavenumbers, respectively, we get the following evolution equation for the wave amplitudes ζmn : ∂ζmn mpr =i δ(m + p + r)[Knqs ζ−pq ζ−rs + Ampr nqs ζ−pq h−rs ] ∂t pq rs +

2im ζmn + α(ζ¯mn − ζmn ) − ν(n(n + 1))2 ζmn . n(n + 1)

(21)

Here K and A are, again, the wave-wave and wave-topography interaction coeﬃcients, respectively. In spherical geometry, they are given by 1 1 1 1 mpr − dµPnm (µ)P (22) Knqs = 2 s(s + 1) q(q + 1) −1

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and Ampr nqs = −

1 q(q + 1)

1

−1

dµPnm (µ)P,

(23)

which are analogous to the planar coeﬃcients deﬁned in (5) and (6). Here we have deﬁned d d P = pPqp (µ) Psr (µ) − rPsr (µ) Pqp (µ). dµ dµ We can then follow the procedure of section 3 to truncate (21). We choose to retain the (0, 1) mode, which represents (in physical space) solid body rotation—the analogue of U on the beta-plane, and a topographicallyexcited mode, which we denote in general by (m, n). We then obtain the two-component system consisting of ∂ζk = −iωk(U )ζk − imhk U − αζ ζk ∂t

(24)

4 ∂U 2m ¯ − U ), = (ζk h∗k ) + αU (U 3 ∂t n(n + 1)

(25)

and

where

2 + 2U ωk (U ) = m U − n(n + 1)

(26)

is, again, the Doppler-shifted Rossby-wave frequency, and U = 12 32 ζ01 is ¯ is deﬁned similarly. Equations (24), (25), the zonal wind at the equator. U and (26) are analogous to (8), (9), and (10). Here we have chosen ζ¯mn =

0 if (m, n) = (0, 1), which means we are only forcing the (0,1) mode. To obtain (24) and (25), we have explicitly calculated thefollowing interac-

m,0,−m 1 = − 12 32 [ 12 − n(n+1) ]m, tion coeﬃcients in terms of m and n: Kn,1,n m,0,−m 0,m,−m 1 3 3 m An,1,n = − 2 2 m, and A1,n,n = 2 n(n+1) . We have also deﬁned

αζ = α and αU = 43 α. The vector k has been deﬁned as k = (m, n), with −k = (−m, n). Equations (25) and (9) diﬀer by the constant factor multiplying the derivative ∂U ∂t in the former, but the steady state versions of these equations have exactly the same form. Hence, the steady-state expressions (15) to (18) are valid for the spherical case with the substitutions kx → m, k 2 → n(n + 1), β → 2, and k02 → 2. Now the analog of the planar (3,0) mode on the sphere is the (m, n) = (3, 3) mode, but in the spherical model the topography is multiplied by the variable Coriolis parameter. This introduces a factor of µ, which means

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that we need to use (m, n) = (3, 4) for h and ζ in our equations if the topography, H, consists of the wavenumber (3, 3). Using this mode, we ﬁnd that the resonant speed (at φ = 30o ), as given by (18), with the appropriate substitutions, is Ures ≈ 44 m s−1 . This might explain why spherical models need such unrealistically strong wind forcing for multiple equilibria to appear, for, as we saw in the planar case, the low-index state occurs near resonance while the high-index state occurs far away from resonance. This means that we need a forcing substantially greater than 44 m s−1 to observe the dual states. The diﬀerence in resonant speeds between the planar, regional model and the spherical, global model is partly due to geometry and partly due to domain size. To see the eﬀect of domain size, consider what happens if we ‘scale-up’ our localized beta-plane so that it ‘covers’ the whole globe but keep the original dimensional value of β. If we denote by U old , the typical speed in the former (smaller) domain; U new , the corresponding speed in the new new , the (larger) domain; Lold s , the length scaling in the old domain; and Ls scaling in the new domain, then it is easy to show that

new 2 Ls new = U old . (27) U Lold s Let us choose the new (global) scaling such that the (non-dimensional) domain is still [0, 2π] × [0, 2π] but the (dimensional) surface area now being 3000 = √aπ while Lold equal to 4πa2 . This gives Lnew s s = π , in km, as in section old ≈ 2 m s−1 is ‘equivalent’ 2. Using equation (27), this implies that the Ures new −1 to Ures ≈ 28 m s in the new, ‘global’, domain. Thus, the low-index state would have zonal winds near this value; the high-index state would have values (talking the old 13.3 m s−1 as typical) near 188 m s−1 , which is unrealistic. On the face of it, this suggests that it might not be realistic to consider multiple equilibria as arising from global resonance of Rossby waves. 7. Zonal Jets and Conﬁnement of Rossby Waves We have seen in the previous section that if we simply scale up our regional model onto the globe, either in planar or spherical geometry, the parameters required to sustain multiple equilibria fall into an unrealistic range. On the other hand, if we view the multiple equilibria as a regional phenomenon, what is the mechanism responsible for conﬁning the waves in a given region? YRK have suggested that the zonal jet structure found in the atmosphere, with strong westerlies at mid-latitudes and easterlies both equator-ward

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and pole-ward, results in conﬁnement of the topographic waves generated in northern hemisphere mid-latitudes, and hence the possibility for regional resonance. To test this idea further, we have scaled up our beta-plane model onto a global domain. This may result in some distortion of the ﬂow outside the mid-latitude regions; however, the waveguide eﬀect of the zonal jets ensures that the mid-latitude ﬂow, which is primarily due to the interaction between the large scale zonal ﬂow and topographic Rossby waves in our model, is largely unaﬀected by this. To keep the non-dimensional form of the equations the same, we have placed the beta-plane at 60 degrees North. This ensures that the model domain remains a square. To obtain a jet structure reminiscent of that found in the real atmosphere or that obtained from more complex models, we have added a term of the form αζ∗ (t) to the vorticity equation (1), where ζ∗ is given by ζ∗ (t) = 2A cos(k · x), and we choose k = (0, ±2). A is an amplitude chosen so that in this case the maximum strength of the jet is 17.5 m s−1 (at 45 degrees). Together with a uniform relaxation, U∗ = 12.5 m s−1 , over the whole domain, this gives a maximum wind of 30 m s−1 at 45 degrees and a minimum of −5 m s−1 at the equator and poles. We choose a relaxation time α−1 = 10 days. Fig. 8 demonstrates that the jet structure conﬁnes the topographic wave within latitudinal circles (in mid-latitudes). In this experiment, we have placed a conical mountain with a radius of 22.5o and height hC = 3275 m at 45o North, 180o East. The ﬁgure shows the eddy-streamfunction, which is the streamfunction with zonal components removed. Clearly the topographic wave is mainly conﬁned in the region of the Westerly winds, which is between 22.5o and 67.5o North. The waves do not propagate outside of this region as the jet structure acts as a waveguide, conﬁning them within this latitudinal band. To further demonstrate that the waveguide eﬀect of the jet structure is responsible for this, we carry out experiments without such jets but with U∗ simply relaxed to 30 m s−1 over the whole domain. The result is shown in Fig. 9. In this case, the topographic wave propagates over the whole domain as is evident in that ﬁgure. Having conﬁrmed that the jet structure can eﬀectively conﬁne the topographic waves to localized regions and hence potentially lead to (local) resonance needed for multiple equilibria in the CdV scenario, we now investigate whether such multiple states do indeed occur. To achieve this with as much realism as possible, we use a representation of the Earth’s topography as shown in Fig. 10. We have discovered robust multiple equilibria in the vicinity of the following values of parameters: A corresponding to

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Fig. 8. Eddy streamfunction contour plot showing topographic waves in ‘blocked’ state in the presence of latitudinally dependent jets for Northern Hemisphere conical mountain.

Fig. 9. Eddy streamfunction contour plot showing topographic waves propagating throughout the domain when no latitudinally-dependent jet structure is imposed.

a maximum of 50 m s−1 and U∗ = 40 m s−1 , giving a maximum wind of 90 m s−1 at 45o and a minimum of −10 m s−1 at the equator and poles; α−1 = 20 days; and ν = 6.06 × 1014 m4 s−1 . Fig. 11 shows the zonal winds averaged over a latitudinal band between 22.5o and 67.5o . The ﬂows have been initialized with two diﬀerent zonal winds: high (90 m s−1 ) and low (2 m s−1 ). We indicate with solid lines the ﬂows in the (more interesting) Northern Hemisphere and with dotted lines the corresponding ﬂows in the

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Fig. 10. Global distribution of topography used in the experiment of Section 7. Only the highest elevations are shown.

Southern Hemisphere. The Northern Hemisphere ﬂow is clearly bimodal with a high wind state consisting of weak transients and a low wind state with stronger transients. The Southern Hemisphere ﬂow also seems to have two very closely spaced equilibrium states. This is probably due to resonance of Rossby waves excited by the relatively weaker topography that is present in the mid-latitudes of that hemisphere. Fig. 12 shows the timeseries plot of wavenumber 2 kinetic energy for the two states. Clearly it is much stronger in the blocked state than in the unblocked one. It is also clear that in the blocked state (upper curve), there are more interactions between diﬀerent wavenumbers as is evident from the more complicated patterns of ﬂuctuations than those in the unblocked state. This is consistent with the studies of the simpler models of the previous sections. Zonal wavenumber 2 is the dominant resonating mode in this experiment even though the topographic distribution is dominated by zonal wavenumber 1 rather than 2. The reason for this is that the resonant speed of the former is a formidable 116 m s−1 , as calculated from equation (18), which is probably not observable in the Earth’s atmosphere. Figures 13 and 15 show the instantaneous zonal wind contours for the two states. Clearly the former shows the blocked state, with reduced winds and pronounced meanders in the Northern Hemisphere, while the latter shows the unblocked state with its stronger, more zonal winds in both hemispheres. Figures 14 and 16 show the corresponding contours of the eddy-streamfunction. Clearly the blocked state has ‘stronger’ highs and lows than the unblocked state, i.e., the waves

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Fig. 11. Zonal winds averaged over a mid-latitude band. The solid lines are for the Northern Hemisphere while the dotted lines are for the Southern Hemisphere. Two ﬂows are shown, with diﬀerent initial conditions. The Southern hemisphere ﬂows converge to two very ﬁnely spaced equilibria while the Northern Hemisphere ones have two distinct equilibria.

in the blocked state have a stronger amplitude. Another thing that is apparent from these contours is that there is a phase diﬀerence between the waves in the two states. The streamfunction contours in the unblocked state are roughly correlated with the topography while in the blocked state they are not; in fact, in the Northern Hemisphere mid-latitudes, in particular, they seem to be anti-correlated. This is quite consistent with the discussion of Section 3, where we saw that in the unblocked state the waves consist of just the in-phase (with topography) component while in the blocked state there is a strong out-of-phase component. Although our global model is more complex than that described in Section 3, we can still see this behaviour. We can also deduce the positions of the centres of the ‘blocks’ (two of them) from these ﬁgures by locating the positions of the highs in the eddy-streamfunction contours or, alternatively, from the positions of maximum poleward deﬂection of the zonal wind contours. It is interesting that these positions, near 0o and 180o longitude, are at, or slightly to the west of the longitudinal positions of the observed North Atlantic and North Paciﬁc blocks respectively.

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Fig. 12. Time series of wavenumber 2 kinetic energy in a (statistically) steady state. The top curve is the ‘blocked’ ﬂow while the bottom one is ‘unblocked’.

Fig. 13. Zonal wind contour plot in ‘blocked’ state in the presence of latitudinally dependent jets with underlying topography of ﬁgure 10.

The values of the winds used in this experiment are probably twice as large as would be expected in the atmosphere; nonetheless, the results are still of interest as the aim here is mainly qualitative understanding

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Fig. 14. Eddy Streamfunction contour plot in ‘blocked’ state in the presence of latitudinally dependent jets with underlying topography of ﬁgure 10.

Fig. 15. Zonal wind contour plot in ‘unblocked’ state in the presence of latitudinally dependent jets with underlying topography of ﬁgure 10.

of the processes involved in generating these phenomena. We should note that there are many unknowns and features that have only been crudely represented in our model such as the eﬀect of zonally asymmetric heating, the precise latitudinal proﬁle the zonal jets, and the appropriate relaxation time for the jet structure as compared to the Ekman dissipation time-scale, to mention just a few.

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Fig. 16. Eddy Streamfunction contour plot in ‘unblocked’ state in the presence of latitudinally dependent jets with underlying topography of ﬁgure 10.

8. Summary and Conclusion We have found that a severely truncated, regional barotropic beta-plane model of the atmosphere consisting of just the topographically-excited waves in a background ﬂow possesses multiple equilibria for a wide range of physically-plausible parameters. Furthermore, when numerous other modes are introduced in the ﬂow, multiple equilibria still exist albeit with a reduced range of parameters. We have also investigated how some of the ﬂows at the parameter ‘boundaries’ spontaneously ﬂip from the blocked state to the unblocked one as instabilities develop in the ﬂow. We have discussed why some global models require excessive winds to maintain multiple equilibria and have investigated further the ideas of YRK that models with a realistic zonal jet structure can ameliorate this problem. By introducing mid-latitudinal jets in our (barotropic) model, we have indeed found that the zonal jet structure in global baroclinic models helps to conﬁne the topographic Rossby waves within a given latitudinal band and that this can lead to resonance without the excessive winds needed in a simple global barotropic scenario, where the waves are allowed to travel over the whole domain. Although we have found multiple equilibrium states resembling the blocked and unblocked patterns in the atmosphere in this case, the winds needed in our experiments are still probably twice as large as the observed ones. Nevertheless, contrary to objections raised by some previous studies, we have been able to show that multiple equilibria can appear in the

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presence of transient activity and realistic topographic distribution and may be conﬁned in mid-latitude regions in global models. Acknowledgements Meelis Zidikheri was a recipient of an ANU postgraduate scholarship funded by RSPhysSE as well as a CSIRO postgraduate scholarship funded by CSIRO Complex Systems Science during the course of this work. He also wishes to thank Dr. Rowena Ball and Prof. Robert (Bob) Dewar, his supervisors at ANU, for their encouragement. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25.

J. Charney and J. DeVore, J. Atmos. Sci. 36, 1205 (1979). A. Wiin-Nielsen, Tellus 31, 375 (1979). J. Egger, J. Atmos. Sci. 38, 2606 (1981). R. Benzi, A. Hansen, and A. Sutera, Quart. J. Roy. Meteor. Soc. 110, 393 (1984). A. Speranza, Adv. Geophys. 29, 199 (1986). P. Sura, J. Atmos. Sci. 59, 97 (2002). E. O’Brien and L. Branscome, Tellus 40A, 358 (1988). S. Rambaldi, and K. Mo, J. Atmos. Sci. 41, 3135 (1984). K. Tung and A. Rosenthal, J. Atmos. Sci. 42, 2804 (1985). G. Holloway and J. Eert, J. Atmos. Sci. 44, 2001 (1987). S. Yoden, J. Meteor. Soc. Japan 63, 1031 (1985). Y. Tian, E. Weeks, K. Ide, J. Urbach, C. Baroud, M. Ghil, and H. Swinney, J. Fluid. Mech. 438, 129 (2001). E. Kallen, Tellus 37A, 249 (1985). B. Legras and M. Ghil, J. Atmos. Sci. 42, 433 (1985). S. Gravel and J. Derome, Tellus 45A, 81 (1993). J. Charney and D. Strauss, J. Atmos. Sci. 37, 1157 (1980). B. Reinhold and R. Pierrehumbert, Mon. Weath. Rev. 110, 1105 (1982). P. Cehelsky and K. Tung, J. Atmos. Sci. 44, 3282 (1987). J. Frederiksen, J. Atmos. Sci. 39, 969 (1982). J. Frederiksen, J. Atmos. Sci. 40, 2593 (1983). J. Frederiksen, Trends in Atmospheric Science 1, 239 (1992). R. Dole, Adv. Geophys. 29, 31 (1986). S. Yang, B. Reinhold, and E. Kallen, J. Atmos. Sci. 54, 1397 (1997). J. Frederiksen, J. Atmos. Sci. 39, 2477 (1982). J. Frederiksen and T. O’Kane, J. Fluid. Mech. 539, 137 (2005).

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COHERENT PATTERNS OF INTERANNUAL VARIABILITY OF THE ATMOSPHERIC CIRCULATION: THE ROLE OF INTRASEASONAL VARIABILITY CARSTEN S. FREDERIKSEN Bureau of Meteorology Research Centre P.O. Box 1289, Melbourne, Victoria, Australia [email protected] X. ZHENG National Institute of Water and Atmospheric Research Wellington, New Zealand. In this chapter, methods, using both daily and monthly data, are proposed for studying coherent patterns of interannual variability in seasonal means of the atmospheric circulation that arise from sub-seasonal, or intraseasonal, variability. Meteorological phenomena that vary signiﬁcantly within a season include atmospheric blocking and intraseasonal oscillations such as, for example, the Madden-Julian oscillation. Such phenomena may be regarded as essentially comprising a random process. By removing the inﬂuence of this intraseasonal variability, a slow and more potentially predictable component can be derived that is more associated with very slowly varying external forcing and internal dynamics. The eﬃcacy of the methodology is shown by testing it on synthetic data where the intraseasonal and slow components are known a priori. For both the Northern Hemisphere and Southern Hemisphere observed wintertime atmospheric circulation, coherent patterns of variability in the intraseasonal component are shown to be related to atmospheric blocking and intraseasonal dynamics. Similarly, coherent patterns associated with the slow component are more related to external forcings, such as sea surface temperature variability, El Nino/Southern Oscillation and very slow internal dynamics.

1. Introduction A substantial proportion of the interannual variability of seasonal mean atmospheric ﬁelds, especially in the extra-tropics, is known to arise from variability within the season.4,22,23,25 This component of interannual variability is essentially unpredictable on interannual, or longer, timescales. Until recently,3,5,6,24 it has not been possible to determine how this sub-seasonal, or

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intraseasonal, variability inﬂuences coherent patterns of interannual variability in the seasonal mean. In the past, such patterns could not be identiﬁed because patterns of interannual variability were typically derived from the sampling covariance matrix10 using yearly time series of the seasonal mean ﬁeld. As a result, these patterns tend to be the result of a number of factors including slowly varying external forcings, such as sea surface temperatures (SSTs), very slowly varying (interannual to supra-annual) internal dynamics and intraseasonal variability. In this chapter, we shall review some recently developed techniques,3,5,6,24 using daily or monthly mean climate data, to extract coherent spatial patterns in seasonal mean atmospheric circulations arising from the eﬀects of intraseasonal dynamics. Central to the derivation of these methods is the commonly used assumption that monthly and seasonal climatic means may be regarded as random variables. Leith13 was one of the ﬁrst to discuss the errors of ﬁnitetime averaged estimates of climatic means and to estimate their properties using a simple stochastic model for atmospheric variables. Within such a framework14,17 each season (or seasonal mean of an atmospheric variable) represents a single realisation of an ensemble of possible realisations, all subject to the same external factors such as, for example, sea surface temperature, radiative forcing, ice cover etc. The stochastic ﬂuctuations in the climate means represent the natural internal variability of the climate system and is also referred to as the “climatic noise”.17 The magnitude of this natural variability is important when considering climate predictability, and, in particular, the signal-to-noise ratio in long range prediction. For seasonal means, the majority of this climatic noise can be attributed to variability within the season,22,24 that is, intraseasonal variability. Implicit in the work of Leith13,14 and Madden17 is the assumption that under the same external conditions, although individual realisations exhibit a natural variability due to the nature of statistical sampling, each is associated with a process of constant statistical properties. Leith13 also argued that the magnitude of the natural variability is related to the auto-correlation of daily weather ﬂuctuations. The daily time series of many meteorological variables can be approximated by a ﬁrst-order autoregressive (AR(1)) Markovian process, or rednoise time series13 xt+1 − µ = φ(xt − µ) + δt+1 ,

(1)

with lagged time (auto-)correlation function R(τ ) = e(−ν|τ |). Here, x represents the climate variable, t is time in days, τ is the lag time in days,

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µ is the mean of the time series, φ is the autoregressive parameter, and δt+1 is a random quantity. Lorenz16 showed, for example, that the Northern Hemisphere geopotential height ﬁeld approximately satisﬁes a red-noise model with ν = 0.3 day−1 . The geopotential height ﬁeld is commonly used to describe coherent structures of the atmospheric circulation and is the ﬁeld we will concentrate on in this chapter. In addition, the corresponding power spectrum, of many meteorological ﬁelds, tends to a non-zero value as the frequency approaches zero. This implies that there is always some contribution to the natural variability from these daily ﬂuctuations. Recognising that day-to-day weather events are unpredictable beyond a week or two, it would be expected that this low frequency natural variance is also unpredictable. Conversely, low frequency internal variability over and above this low-frequency extension of the weather contribution might be potentially predictable.17 With this in mind, the rationale behind the techniques described herein is as follows. Within a particular season, daily or monthly climate time series can be conceptualised as comprising a component of the seasonal “population” mean (note, not the seasonal sample mean) and a component of the departure from the seasonal population mean. The former is unchanged within the season and is, therefore, more likely to be related to physical processes that do not vary signiﬁcantly within the season. Hence, it is more likely to arise from the slowly varying boundary, or external, forcings on the climate system (e.g. SST, sea ice coverage and greenhouse gas concentration) and from slowly varying (interannual to supra-annual) internal atmospheric variability (e.g. the equatorial stratospheric quasi-biennial oscillation). The latter component changes daily or monthly and is more likely to be related to meteorological phenomena that vary signiﬁcantly within the season (e.g. storms and atmospheric blocking in the extratropics, and the Madden-Julian oscillation in the tropics). A seasonal mean of a climate variable, calculated from either daily or monthly mean data, can then be thought of as consisting of two components. The ﬁrst is just the seasonal population mean. The second is the seasonal mean of the departures of the daily or monthly means from the seasonal population mean. The latter component should average to zero if sampled over a very large number of realisations, but not necessarily if averaged over a single season. In mid-latitudes, the second component can explain about 50% of the variance of seasonal means.22,25 Since the variability, at frequencies less than two weeks (e.g. midlatitude storms), is largely smoothed out by the seasonal-mean operator, this component is related

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more closely to meteorological phenomena at frequencies longer than two weeks (i.e. to intraseasonal variability associated with, for example, atmospheric blocking in the extratropics and the Madden-Julian oscillation in the tropics). Two techniques will be presented using, respectively, daily and monthly means of atmospheric data, and based on extensions of the techniques of Madden17 and Zheng et al.22 but adapted for estimating the covariance (or cross-covariance) matrix of the intraseasonal components of a seasonal mean ﬁeld (or pair of ﬁelds). The empirical orthogonal functions (EOFs) (see Appendix A), calculated from the resulting covariance matrix, or the singular vectors (see Appendix B) of the cross covariance matrix, represent the coherent patterns related to intraseasonal variability. The removal of the contribution of this component from the sample covariance matrix, or sample cross covariance matrix, allows one to derive a “residual” covariance matrix, or cross covariance matrix. Patterns derived from these residual matrices, would be expected to be more closely related to very slowly varying (interannual/supra-annual) external forcings and internal dynamics, and in this sense can be regarded as more “potentially” predictable.17 We shall refer to these patterns as the “slow” patterns of interannual variability.6,24 2. Methodology In this section, we review two techniques for estimating the contribution of atmospheric processes, associated with intraseasonal variability, to the interannual variability of seasonal means of the atmospheric circulation. Of particular interest is the contribution of such sub-seasonal variability to the generation of coherent spatial patterns. The ﬁrst method,3 based on ideas ﬁrst proposed by Madden,17 uses the frequency domain and daily data to estimate the interannual covariance matrix associated with the intraseasonal component. The second method is an estimation of the covariance matrix in the time domain using moment estimation and monthly mean data22,24 . Both methods are shown in section 4 to give equivalent results. In section 3, we test the methods by applying them to synthetic data and Monte Carlo simulations24 where the slow and intraseasonal components are known a priori. This allows us to compare the normal sampling errors inherent in calculating the EOF patterns, given the corresponding covariance matrices, and the additional estimation errors that occur in using our estimations for separating the variability into the two components. Here, we also describe how the methods can be modiﬁed to handle coupled patterns

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of variability between two climate ﬁelds such as surface temperature and the atmospheric circulation. 2.1. Estimates using daily data The methodology described herein is based on the assumptions described above in the introduction, and relies on the availability of a daily time series of the meteorological variable (say x) over a number of decades. The mean annual cycle is also assumed to have been removed from the daily time series. Conceptually, we consider a daily anomaly in a particular year, day and geographical location, as consisting of a slow, or potentially predictable, component and an unpredictable stochastic component. The former is related to the interannual and supra-annual variability of external forcings and internal dynamics and may therefore be regarded as a constant over the season. The latter arises from day-to-day weather variability. The slow component represents the seasonal population mean of the data. As in Zheng and Frederiksen,24 we shall refer to the stochastic component as the intraseasonal component, because the seasonal mean operator eﬀectively ﬁlters out variability less than about 2 weeks. Thus, a daily anomaly xyt (r) is represented by a simple stochastic linear model as3 xyt (r) = µy (r) + εyt (r).

(2)

Here, y = 1, . . . , Y is the year; t = 1, . . . , T is the day in a season of length T days; r = 1, . . . , R denotes a location in a ﬁeld with R spatial locations; µy (r) represents the slow component; εyt (r) represents daily weather noise, modelled here as the residual daily departure of xyt (r) from the seasonal value µy (r). The set {εyt , t = 1, . . . , T } is assumed to represent a stationary normal stochastic process in time with mean zero and to be statistically independent and identically distributed with respect to year y. Note, that in this model externally forced intraseasonal variability is included in the intraseasonal component. It is convenient to introduce the convention of using a “circle” as a subscript whenever an average is done over that index. Thus, for example, from (2), a seasonal mean anomaly derived from daily values is written as (3) xyo (r) = µy (r) + εyo (r), T where, for example, xyo (r) = T1 t=1 xyt . Also, the symbol Vˆ will be used to denote the estimated variance of a single variable or the covariance of two variables. The covariance of the seasonal mean anomaly, at two diﬀerent

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locations r1 and r2 , can be estimated by the sample covariance, Vˆ (xyo (r1 ), xyo (r2 )) =

Y 1 [xyo (r1 ) − xoo (r1 )] [xyo (r2 ) − xoo (r2 )] . Y − 1 y=1

(4) Given daily data, an estimate of the covariance of the weather noise component can be derived from the sample covariance, and using the deﬁnition of the seasonal mean, as,

=

1 Y T2

Y 1 yo (r1 )yo (r2 ) Vˆ (εyo (r1 ), εyo (r2 )) Y y=1 T T Y 1 = yt (r1 ) yt (r2 ) Y T 2 y=1 t=1 t=1 T T Y iωo t −iωo t Re yt (r1 )e yt (r2 )e , y=1

t=1

(5)

t=1

where, ωo ≡ 0 denotes the zero frequency and therefore eiωo t = 1. The right hand side of (5) represents the cross periodogram between yt (r1 ) and yt (r2 ) at frequency zero. If we assume that the cross spectrum at zero frequency is smooth and T is suﬃciently large,17 then the intraseasonal covariance can be approximated2 at the nearest frequency 2π/T . Thus, 1 Vˆ (εyo (r1 ), εyo (r2 )) × Y T2 T Y T it2π/T −it2π/T Re yt (r1 )e yt (r2 )e . (6) y=1

t=1

t=1

Because yt (r) is not observable, (6) can not be used directly to estimate Vˆ (εyo (r1 ), εyo (r2 )). However, because the Fourier transform of any constant with respect to t at frequency 2π/T is identically zero, it follows, using (2), that, 1 Vˆ (εyo (r1 ), εyo (r2 )) × Y T2 T Y T it2π/T −it2π/T Re xyt (r1 )e xyt (r2 )e , (7) y=1

t=1

t=1

since µy (r1 ) and µy (r2 ) are constant with respect to t. Madden17 derived an expression for the variance of the climate noise at individual geographical points that is equivalent to that derived from (7)

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in the limit as point r1 approaches the point r2 . That is, T 2 Y 1 it2π/T Vˆ (εyo (r1 ), εyo (r1 )) x (r )e , yt 1 Y T 2 y=1 t=1 From (4) and (7), it is possible to deﬁne a residual covariance component Vˆ (xyo (r1 ), xyo (r2 )) − Vˆ (εyo (r1 ), εyo (r2 )) Vˆ (µy (r1 ), µy (r2 )) + Vˆ (µy (r1 ), εyo (r2 )) + Vˆ (µy (r2 ), εyo (r1 )).

(8)

For the case when the sets {µy } and {εyo } are assumed to be statistically independent, (8) reduces to an estimate for V (µy (r1 ), µy (r2 )). Nevertheless, even if this is not the case, the left hand side of (8) will still be a better estimate of the variability in the slow component than is Vˆ (εyo (r1 ), εyo (r2 )), because the intraseasonal component is largely removed, and in the extratropics this component can be as much as 50% of the total variance.6,22 An EOF analysis10 (see Appendix A), or eigen-analysis, can then be applied to each of the spatial covariance matrices given by (4), (7) and (8) to determine dominant patterns of interannual variability (the eigenvectors of the covariance matrix), which we shall refer to as the total-EOFs, the intraseasonal-EOFs (I-EOFs) and the slow-EOFs (S-EOFs), respectively. The corresponding eigenvalues then give the amount of variance, in each component, explained by each pattern. The corresponding principal component (PC), or daily time series, {pyt , y = 1, . . . , Y, t = 1, . . . , T }, associated with an EOF {e(r), r = 1, . . . , R} is then deﬁned as the projection of the climate data onto the EOF, that is, pyt =

R r=1

e(r)xyt (r) =

R r=1

e(r)µy (r) +

R

e(r)εyt (r) ≡ µ y + εyt ,

r=1

where, µ y and εyt represent the slow and intraseasonal components, respectively, of the PC time series. The potential predictability of the PC is then deﬁned as 1 − Vˆ ( εyo , εyo )/Vˆ (pyo , pyo ), that is, as the fraction of the interannual variance in the PC series that remains after the intraseasonal component has been removed. This ratio can be determined from (4), (7) and (8), with xyt replaced by pyt and r1 = r2 = r. Here, pyo represents the interannual PC time series of the corresponding EOF pattern. For convenience, we shall use the notation total-PC, intraseasonal-PC (I-PC) and slow-PC (S-PC) to diﬀerentiate between the three types of possible PCs.

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2.2. Estimates using monthly data In the derivation above, the crucial step is the estimation of the covariance V (εyo (r1 ), εyo (r2 )) for the seasonal mean of the intraseasonal component. Here, we propose a simpler method using only monthly data. This is possible because, as discussed above, the power spectrum of large-scale atmospheric motions is fundamentally red.16,17,22 This implies that a non-zero seasonal mean of the weather noise is largely contributed to by intraseasonal weather events with monthly or longer time scales, including such phenomena as persistent blocking events in the extratropics and the Madden-Julian Oscillation in the tropics. That is, the seasonal mean operator is a very eﬀective ﬁlter of the higher frequency daily weather events such as midlatitude storms. A method using monthly means is desirable because the availability of daily time series is limited in comparison with that of monthly time series for some meteorological variables. It is also computationally more eﬃcient, using only 3 monthly means for a typical season, compared with 90 daily means. In addition, the estimation in (7) requires an assumption of normality for the daily time series which may not be true for many meteorological variables. With monthly data, (2) can be replaced by6,22,24 xym (r) = µy (r) + εym (r),

(9)

where, m(=1, 2, 3) denotes a month within a given 3-month season, xym (r) represents a monthly anomaly, and εym (r) the monthly mean intraseasonal component. The other indices and variables have the same meaning as for (2). Equations (3), (4) and (8) also hold with the understanding that the average over the day index is replaced by an average over the 3 months. The vector eT (r) = (εy1 (r), εy2 (r), εy3 (r)) is assumed to comprise a stationary and independent annual random vector. Equation (9) implies that monthto-month ﬂuctuations, that is intraseasonal variability, arise entirely from e(r) (for example, xy1 (r) − xy2 (r) = εy1 (r) − εy2 (r)). It is possible to estimate V (εyo (r1 ), εyo (r2 )), using monthly means, by using the following assumptions. Since the daily time series of a climate variable, within a season, is in general assumed to be stationary, so are the monthly statistics. Thus, the covariance between two locations can be assumed to be independent of months. That is, V (εy1 (r1 ), εy1 (r2 )) = V (εy2 (r1 ), εy2 (r2 )) = V (εy3 (r1 ), εy3 (r2 )).

(10)

The same can be assumed to be true for the inter-monthly covariance, such

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that, V (εy1 (r1 ), εy2 (r2 )) = V (εy2 (r1 ), εy3 (r2 )).

(11)

In adddition, we assume that the intraseasonal components are uncorrelated if they are a month or more apart. This is a reasonable assumption because daily weather events are unpredictable beyond a week or two. Hence, V (εy1 (r1 ), εy3 (r2 )) = 0.

(12)

Under the assumptions contained in (10)-(12),   1β 0 E e(r1 )eT (r2 ) + E e(r2 )eT (r1 ) = 2α  β 1 β  , 0β 1

(13)

where E denotes the expectation value21 based on all years and α = V (εym (r1 ), εym (r2 )), m = 1, 2, 3

(14)

and 1 [V (εy1 (r1 ), εy2 (r2 )) + V (εy1 (r2 ), εy2 (r1 ))] 2α 1 [V (εy2 (r1 ), εy3 (r2 )) + V (εy2 (r2 ), εy3 (r1 ))]. = 2α

β=

Using (14)-(15) it follows also that, T εy1 (r1 ) − εy2 (r1 ) εy1 (r2 ) − εy2 (r2 ) E εy2 (r1 ) − εy3 (r1 ) εy2 (r2 ) − εy3 (r2 ) T εy1 (r2 ) − εy2 (r2 ) εy1 (r1 ) − εy2 (r1 ) +E εy2 (r2 ) − εy3 (r2 ) εy2 (r1 ) − εy3 (r1 )

2 − 2β 2β − 1 = 2α . 2β − 1 2 − 2β

(15)

(16)

From (9), xy1 (r) − xy2 (r) = εy1 (r) − εy2 (r) and xy2 (r) − xy3 (r) = εy2 (r) − εy3 (r)). Therefore, the left hand side of (16) can be evaluated using the given data xym (r). Thus, solving for α and β α=a+b and β=

a + 2b 2 (a + b)

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where, a=

+

b=

+

Y 1 1 { [xy1 (r1 ) − xy2 (r1 )] [xy1 (r2 ) − xy2 (r2 )] 2 Y y=1 Y 1 [xy2 (r1 ) − xy3 (r1 )] [xy2 (r2 ) − xy3 (r2 )]} Y y=1

(17)

Y 1 1 { [xy1 (r1 ) − xy2 (r1 )] [xy2 (r2 ) − xy3 (r2 )] 2 Y y=1 Y 1 [xy2 (r1 ) − xy3 (r1 )] [xy1 (r2 ) − xy2 (r2 )]} Y y=1

(18)

Using these estimates for α and β in (13), 1 ˆ [V (εy1 (r1 ), εy2 (r2 )) + Vˆ (εy1 (r2 ), εy2 (r1 ))] 2 1 = [Vˆ (εy2 (r1 ), εy3 (r2 )) + Vˆ (εy2 (r2 ), εy3 (r1 ))] 2 = αβ (19) Thus, taking into account assumption (12), an estimate for the covariance of the intraseasonal weather noise at locations r1 and r2 , using monthly data, is 1 Vˆ (εyo (r1 ), εyo (r2 )) = [Vˆ (εyo (r1 ), εyo (r2 )) + Vˆ (εyo (r2 ), εyo (r1 ))] 2 3 1 ˆ = [V (εym (r1 ), εyn (r2 )) + Vˆ (εym (r2 ), εyn (r1 ))] 18 m,n=1 α(3 + 4β) (20) 9 Zheng and Frederiksen24 show that the estimation error can be reduced if we constrain β to lie within the interval [0, 0.1]. Brieﬂy, the lower limit can be justiﬁed as follows. Owing to persistence, daily meteorological variables are positively auto-correlated, and so are their monthly means. Thus, V (εy1 (r1 ), εy1 (r2 )), V (εy2 (r1 ), εy2 (r2 )) and V (εy2 (r1 ), εy1 (r2 )) are likely to have the same sign. This leads to the lower bound (see (14) and (15)). The upper bound is based on the assumption that the daily meteorological ﬁeld is a multivariate ﬁrst-order autoregressive process with ν ≤ 0.9 day−1 (see (1) and deﬁnition of auto-correlation function above), which is generally true for meteorological variables. =

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In section 4, we show that using either (7) or (20) gives almost identical results. 2.3. Coupled patterns of variability Many important surface climate variables, such as seasonal mean temperatures and seasonal mean rainfall, are related to local or hemispheric seasonal mean pressure ﬁelds (that is, the atmospheric circulation). From the point of view of seasonal prediction, a knowledge of the spatial patterns relating the slow (potentially predictable) component of the seasonal mean pressure ﬁelds to the slow component of the surface climate variable should help us to understand the meteorological phenomena associated with forecast skill. Conversely, the identiﬁcation of the spatial patterns that relate the intraseasonal components of the pressure ﬁeld and the surface climate variable should help us to understand the meteorological phenomena mainly responsible for the uncertainty in forecast skill, at the long range. Here, we present a generalisation of the method in section 2.2 allowing for an estimate of the interannual cross-covariance matrices associated with the long range (in advance of a season) intraseasonal and slow components of a pair of climate variables. From the intraseasonal and slow cross-covariance matrices it is possible to construct coupled patterns of the intraseasonal and slow components of covariability of the pair of climate variables. In particular, it is possible to identify coherent patterns in the atmospheric circulation relating the intraseasonal and slow components of the circulation to patterns of the corresponding intraseasonal and slow spatial patterns of the surface climate variable. Let tym (r) and hym (s) represent monthly anomalies of two climate variables (e.g. surface air temperature and global 500hPa geopotential height), in year y (=1, . . . , Y ), month m (=1,2,3) and at some location r = 1, . . . , R and s = 1, . . . , S, not necessarily the same. Then, as in section 2.2, we assume that tym (r), tym (r) = ty (r) +

(21)

hym (s), hym (s) = hy (s) + where ty (r) and hy (s) represent slow components and tym (r) and hym (s) ty1 (r), the intraseasonal component of each variable. The vectors tT (r)= ( ty2 (r), ty3 (r)) and hT (s)= ( hy1 (s), hy2 (s), hy3 (s)) are assumed to comprise stationary and independent annual random vectors.

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Hence, the seasonal means of variables t and h can be expressed as tyo (r), tyo (r) = ty (r) + hyo (s), hyo (s) = hy (s) + hyo (s) are associated with intraseasonal variability, and where tyo (r) and ty (r) and hy (s) with the interannual variability of external forcings and slowly varying (interannual/supra-annual) internal dynamics. hyo (s)) of the intraseaAn estimate of the (cross-)covariance V ( tyo (r), sonal components, using monthly means, can be made with similar assumptions to the case with one variable (see section 2.2). In particular, hy1 (s)) = V ( ty2 (r), hy2 (s)) = V ( ty3 (r), hy3 (s)), V ( ty1 (r),

(22)

hy2 (s)) = V ( ty2 (r), hy3 (s)), V ( ty1 (r),

(23)

hy3 (s)) = 0. V ( ty1 (r),

(24)

Assumptions (22)-(24) imply that,

  1δ0 E t(r)hT (s) + E h(s)tT (r) = 2γ  δ 1 δ  , 0δ1

(25)

where E denotes the expectation value based on all years and hym (s)), m = 1, 2, 3 γ = V ( tym (r),

(26)

and 1 hy2 (s)) + V ( hy1 (s), ty2 (r))] [V ( ty1 (r), 2γ 1 [V ( ty2 (r), hy3 (s)) + V ( hy2 (s), ty3 (r))]. = 2γ

δ=

In addition, using (26)-(27),  T    hy1 (s) − hy2 (s) ty1 (r) − ty2 (r) E   ty3 (r) ty2 (r) − hy2 (s) − hy3 (s) T hy1 (s) − hy2 (s) ty2 (r) ty1 (r) − +E ty2 (r) − ty3 (r) hy2 (s) − hy3 (s)

2 − 2δ 2δ − 1 = 2γ . 2δ − 1 2 − 2δ

(27)

(28)

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From (21), ty1 (r) − ty2 (r) = ty1 (r) − ty2 (r) and ty2 (r) − ty3 (r) = ty2 (r) − ty3 (r)), with a similar relationship for h. Thus, the left hand side of (28) can be evaluated using the given data tym (r) and hym (s). It follows then that γ =c+d and δ=

c + 2d 2 (c + d)

where, c=

+

d=

Y 1 1 { [ty1 (r) − ty2 (r)] [hy1 (s) − hy2 (s)] 2 Y y=1 Y 1 [ty2 (r) − ty3 (r)] [hy2 (s) − hy3 (s)]} Y y=1

(29)

Y 1 1 { [ty1 (r) − ty2 (r)] [hy2 (s) − hy3 (s)] 2 Y y=1

Y 1 + [ty2 (r) − ty3 (r)] [hy1 (s) − hy2 (s)]} Y y=1

(30)

Using these estimates for γ and δ in (25), 1 ˆ [V (ty1 (r), hy2 (s)) + Vˆ ( hy1 (s), ty2 (r))] 2 1 ty2 (r), = [Vˆ ( hy3 (s) + Vˆ ( hy2 (s), ty3 (r))] 2 = γδ

(31)

From (24), it follows further that 1 Vˆ ( tyo (r), tyo (r), hyo (s)) = [Vˆ ( hyo (s)) + Vˆ ( hyo (s), tyo (r))] 2 3 1 ˆ = [V (tym (r), hyn (s)) + Vˆ ( hym (s), tyn (r))] 18 m,n=1 γ(3 + 4δ) (32) 9 As in section 2.2, we constrain δ to lie within the interval [0, 0.1] in order to reduce the estimation error. The covariance V (tyo (r), hyo (s)), which we =

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shall refer to as the total (cross-)covariance matrix, can be estimated by the sample (cross-)covariance, Vˆ (tyo (r), hyo (s)) =

Y 1 [tyo (r) − too (r)] [hyo (s) − hoo (s)] . Y − 1 y=1

(33)

Thus, using (32) and (33), a residual (cross-)covariance can be deﬁned as Vˆ (tyo (r), hyo (s)) − Vˆ ( tyo (r), hyo (s) = Vˆ (ty (r), hy (s)) + Vˆ (ty (r), hyo (s)) + Vˆ (hy (s), tyo (r)).

(34)

When the intraseasonal and slow components of the two variables are independent, the residual (cross-)covariance reduces to the (cross-)covariance of the slow components only. Even when this is not the case, the coupled patterns associated with the residual (cross-)covariance matrix can be shown to be more related to slow covariability than those from the total (cross-)covariance matrix, because the intraseasonal component has been largely removed. Equations (32)-(34) can be used to construct the corresponding cross-covariance matrices from which the coupled spatial patterns can be derived using a standard singular value decomposition (SVD) analysis (see, Appendix B). 3. Monte Carlo Simulation

Synthetic Data

To show the eﬃcacy of our methodology, we shall test it on synthetic data, using Monte Carlo simulations of the daily 500-hPa geopotential heights for boreal winter (December-January-February, DJF) over the North Paciﬁc/North American (NP/NA) region (150◦ E-75◦W, 20◦ N-70◦ N). To this end, we have used the National Centers for Environmental Prediction (NCEP) and National Center for Atmospheric Research (NCAR) re-analysis 500hPa geopotential height dataset11 for the period 1958-1996. The synthetic data has been constructed in such a way that the intraseasonal and slow components of interannual variability are known a priori and explicitly satisfy the assumptions inherent in (1). Here, we will test our methodology using monthly data constructed from synthetic daily time series within the season. The ﬁrst step is to construct the daily time series of weather events at R spatial points. Suppose we let Eyt = [ε1yt , . . . , εRyt ]T , t = 1, . . . , 90 be an R-dimensional column vector of daily weather events during a 90 day season for year y and day t, and which is assumed to be a multivariate ﬁrst order autoregressive process2 generated by the following iterative procedure, Eyt = ΦEy(t−1) + Zyt , t = 1, . . . , 90.

(35)

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Here, Φ, the innovation matrix, is an R × R matrix with eigenvalues within T the unit circle. Zyt = [ζ1yt , . . . , ζRyt ] is a R-dimensional multivariate normal random vector with mean zero, covariance matrix Σ, and is statistically independent with respect to day t and year y. To maintain stationarity with respect to t, Ey0 is selected to be a normal vector with mean zero and covariance matrix Vo and to be statistically independent of Zy1 . The innovation error covariance matrix is selected as Σ ≡ Vo − ΦVo ΦT . In this case, the covariance matrix of Eyt for all t is Vo and the covariance matrix of the intraseasonal component is E 1 902

1 T 1 Eyt E 90 t=1 90 t=1 yt 90

90

=

t t t T (1 − ) Vo Φ + Vo Φ 90Vo + . 90 t=1 90

(36)

The corresponding vector of the seasonal population mean (denoted by My ) in year y is generated as T

T

My ≡ [µ1y , . . . , µRy ] = Vµ1/2 [ξ1y , . . . , ξRy ]

(37)

T

where [ξ1y , . . . , ξRy ] is a R-dimensional standard normal vector which is statistically independent with respect to y and with Zyt , and Vµ is the covariance matrix associated with the seasonal population means My .Thus, the synthetic daily time series Xyt generated is just the linear combination Xyt = My + Eyt . Here, without loss of generality, My and Eyt have been constructed to be statistically independent of each other and hence the covariance matrix of My equals the diﬀerence of the covariance matrix of Xyo and the covariance matrix of Eyo . The NCEP-NCAR data has been sub-sampled onto a 5o x5o latitude/longitude grid resulting in R =308 geographical points. To generate the synthetic data, we have used our method to estimate the intraseasonal and slow covariance matrices of the NCEP-NCAR data using the methodology of section 2.2 and have calculated the corresponding signiﬁcant EOFs (see Appendix A). This analysis shows that there are, by Kaiser’s rule of thumb,18 6 signiﬁcant I-EOFs and 4 signiﬁcant S-EOFs (see Fig. 1). The Y =40 or intraseasonal & component Eyt is generated using (35) by setting& 80, Φ = 1/2ηI (where I is an R × R identity matrix), Vo = 1/2ηVε .

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The columns of Vε consists of the six most dominant I-EOFs of the intraseasonal covariance matrix and weighted with the square root of their respective eigenvalues. Also, to ensure that the covariance matrix of the intraseasonal component is Vε , the normalisation constant η has to be set to, ' ( 90 ( ) t/2 η ≡ 45/ 1 + 2 (1 − t/90) 0.5 . (38) t=1

The vector of slow components My is generated using (37) by setting 1/2 the columns of Vµ equal to the 4 dominant S-EOFs of the slow covariance matrix weighted with the square root of their eigenvalues. Constructed in this way, the “true” EOF patterns of each component are known a priori (i.e. they are just the I-EOFs and S-EOFs of Fig. 1). In total, one thousand samples of the daily time series Xyt were generated and monthly means constructed. As determined here, the two components of the simulated seasonal mean ﬁelds are known. This means that the sampling covariance matrices of both components can be estimated (see above), and EOF patterns of both components can be derived. The EOFs derived by this approach will be estimates of the actual EOFs used to generate the data time-series and will be subject to sampling errors. For convenience, this procedure will be referred to as the “ideal” approach. In actuality, the two components for real climate data are, of course, not known, and, as a consequence, the covariance matrix of each component needs to be estimated, as outlined in the previous sub-sections. In order to evaluate the skill of our method, we shall compare our results with those derived from the ideal approach. This is achieved by comparing the varimax rotated10 intraseasonal and slow patterns derived by our methodology, and those derived by the ideal approach, with the true rotated patterns (rotated versions of EOFs in Fig. 1). Varimax rotation involves ﬁnding a linear combination of the original patterns such that the resulting patterns have a “simple structure” with largest loadings in a small number of geographical regions such that the variance of the loadings is maximized. For each of the 1000 daily data sets, the correlation coeﬃcients between the true patterns and the estimated patterns can be estimated. Table 1 shows the root mean square of these correlation coeﬃcients averaged over all samples. It is obvious from Table 1 that the skill increases for a longer time series (40 years compared with 80 years). Also, the root mean squares of the

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Fig. 1. Spatial patterns of interannual variability of DJF 500hPa geopotential height over the North Paciﬁc/North America region for the intraseasonal (left column) and slow (right column) components for the period 1958-1996. Contour interval is 0.04 and positive contours are shaded. Percentage variance explained is in brackets.

pattern correlations for the intraseasonal component, are all above 0.90. Thus all patterns estimated by both methods are close to the true patterns. The skill of our method is even slightly better than that of the ideal

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method, because our method accounts for the monthly autocorrelation in more detail than the ideal method does. The root mean squares of the pattern correlations for the slow component are not as good as those of the intraseasonal component, but are still statistically signiﬁcant (the 1% signiﬁcant level is about 0.35). Our method has skill comparable but not as good as the ideal method. This is because uncertainty for our method comes from the estimation of both the sample covariance matrix and the intraseasonal covariance matrix. However, the skill of our method is still at an acceptable level. Also, it is worth recalling that the data for the ideal method is not available in practice, and therefore our proposed method is clearly a reasonable approach for estimating the EOF patterns of both components. Table 1. Root mean square of the correlation coeﬃcients between estimated and true patterns for 1000 samples. Ideal Method

Estimation Method

40 years

80 years

40 years

80 years

I-REOF1

0.95

0.97

0.97

0.98

I-REOF2 I-REOF3

0.97 0.93

0.98 0.96

0.98 0.96

0.99 0.98

I-REOF4

0.94

0.98

0.97

0.99

I-REOF5 I-REOF6

0.92 0.93

0.97 0.95

0.96 0.95

0.98 0.96

S-REOF1 S-REOF2

0.97 0.84

0.98 0.89

0.92 0.72

0.94 0.79

S-REOF3

0.87

0.91

0.73

0.77

S-REOF4

0.59

0.55

0.42

0.47

4. Coherent Patterns of the Atmospheric Circulation In this section, we apply our methodology to identify coherent intraseasonal and slow patterns of interannual variability in the Southern Hemisphere (SH) wintertime (June-July-August, JJA) and Northern Hemisphere (NH) wintertime (December-January-February, DJF) atmospheric circulation. It is worthwhile reiterating that our principal interest here is to determine how intraseasonal variability inﬂuences, or can generate, coherent patterns of interannual variability in the seasonal mean extratropical SH and NH atmospheric circulation. We also illustrate our coupled methodology by identifying the coupled intraseasonal and slow patterns between Australian

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surface temperature and the global atmospheric circulation during DJF. The data we use is the NCEP-NCAR reanalysis11 500hPa geopotential height ﬁeld. This ﬁeld eﬀectively gives the height of the 500hPa pressure level in the atmosphere and is often used to study coherent spatial patterns, or teleconnections, of interannual variability in the atmospheric circulation. For the hemispheric studies, the data runs from latitude 90◦ S-20◦ S (SH) or 20◦ N-90◦ N (NH) on a 5◦ x5◦ latitude/longitude grid and is for the period 1956-1998. Also, in order not to bias the analysis to high latitudes, a variable longitude spacing has been used. In particular, in the NH (SH) the longitude spacing is taken as 5◦ between 20◦ N and 45◦ N (20◦ S and 45◦ S), 10◦ between 50◦ N and 65◦ N (50◦ S and 65◦ S), 15◦ at 70◦ N (70◦ S), 20◦ at 75◦ N (75◦ S), 30◦ at 80◦ N (80◦ S), 60◦ at 85◦ N (85◦ S) and a single point at 90◦ N (90◦ S). 4.1. Southern Hemisphere winter circulation We have applied both methods ((7) and (20)) to our study of the Southern Hemisphere wintertime atmospheric circulation to show that each one gives essentially the same results. Figure 2 shows the ﬁrst four dominant intraseasonal weather patterns using both daily (left column) and monthly (right column) data, paired appropriately. The two sets of patterns are remarkably similiar with only subtle diﬀerences and this is reﬂected in high pattern correlations of 0.99, 0.98, 0.93 and 0.93 for patterns 1-4, respectively. The associated slow patterns, derived from the residual matrix (8) are shown in Fig. 3. Again, the two sets of patterns are remarkably similar with high pattern correlations of 0.99, 0.99, 0.94 and 0.92 for patterns 1-4, respectively. The minor diﬀerences are due to sampling errors (daily versus monthly) and estimation errors ((7) versus (20)). Clearly, these are not large and conﬁrm that our assumptions contained in (10)-(12) are reasonable. For the rest of this chapter, we will only present results using monthly data in the analysis. I-EOF1, the dominant intraseasonal mode of variability, has largest (negative) loadings in the high latitudes south of about 60◦ S. A wavenumber 3 pattern of positive loadings is evident between about 55◦ S and 45◦ S. It is very reminiscent of the Southern Annular Mode12 (SAM) which has been identiﬁed in variability studies using the sampling covariance matrix ((4)). However, dynamical studies9 have shown that there are high latitude dynamical modes with similar structures that can occur with periods from about 18 days (i.e. intraseasonal fequencies) to essentially inﬁnity (stationary). I-EOF1 appears to capture the inﬂuence of the high latitude modes

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with intraseasonal frequencies, whereas S-EOF1 is related to the stationary and longer period modes. The remaining intraseasonal-EOFs can best be categorized as midlatitude to high-latitude (40◦ S-70◦ S) wavenumber 3 patterns with highest loadings generally over the south Paciﬁc. These modes of interannual variability have similar horizontal structures to the bandpass ﬁltered modes of intraseasonal variability of Kiladis and Mo,12 although in their case they were looking at variability within the season and not its eﬀects on the interannual variability of the seasonal mean circulation. But it is expected6,24 that our intraseasonal-EOFs would reﬂect the structure of these modes. Kiladis and Mo12 show that the bulk of the intraseasonal variability in winter is due to propagating wavenumber 3 or 4 modes. This has also been conﬁrmed in dynamical studies.9 These intraseasonal-EOFs also appear to have a high correlation with SH atmospheric blocking. To show this we have used a blocking index deﬁned as 0.5(u(20◦S) + u(30◦ S) + u(60◦ S) + u(70◦ S) − u(40◦ S) − u(50◦ S) − 2u(45◦S)) for each longitude. Here, u is the NCEP-NCAR reanalysis 500hPa zonal velocity. The correlations between this index and the intraseasonal-PCs are 0.71 (I-PC2), 0.63 (I-PC3) and 0.52 (I-PC4) at longitudes 122.5◦W, 160◦E and 170◦ W, respectively. In these regions the I-EOFs tend to have heavy (positive or negative) weighting. These regions are also associated with persistent (5-10 days or more) quasi-stationary anomalies associated with blocking.12,19,20 Frederiksen and Frederiksen9 also found dynamical blocking modes with largest amplitudes in these regions. As mentioned above, S-EOF1 is associated with low frequency high latitude variability. The second and fourth dominant slow patterns appear to be related to El Nino/Southern Oscillation (ENSO) variability associated with sea surface temperature (SST) variability in the eastern and western Paciﬁc Ocean. In particular, S-EOF2, at the opposite phase to that shown in Fig. 3, is remarkably similar to the pattern associated with the composite of warm ENSO events (1956-1998),12 with a pattern correlation of 0.63. Warm ENSO events are associated with warm eastern Paciﬁc SST anomalies and cooler western Paciﬁc anomalies. Similarly, S-EOF4 is remarkably similar to the composite of cold ENSO events12 with a pattern correlation of 0.78. Both of the patterns have signiﬁcant correlations with one-season (MAM) lead SSTs (not shown) suggesting that they are forced by Paciﬁc SST variability.

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Fig. 2. Southern Hemisphere patterns of interannual variability in the JJA 500hPa geopotential height ﬁeld associated with intraseasonal variability using daily reanalysis data (left column) and monthly mean data (right column).Contour interval is 0.2 with positive contours shaded. Percentage variance explained is in brackets.

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Fig. 3. Southern Hemisphere patterns of interannual variability in the JJA 500hPa geopotential height ﬁeld associated with slow variability using daily reanalysis data (left column) and monthly mean data (right column).Contour interval is 0.2 with positive contours shaded. Percentage variance explained is in brackets.

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Fig. 4. Northern Hemisphere patterns of interannual variability in the DJF 500hPa geopotential height ﬁeld associated with intraseasonal variability. Contour interval is 5 and 10 with positive contours shaded. Percentage variance explained is in brackets.

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Fig. 5. Northern Hemisphere patterns of interannual variability in the DJF 500hPa geopotential height ﬁeld associated with slow variability. Contour interval is 5 and 10 with positive contours shaded. Percentage variance explained is in brackets.

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Finally, S-EOF3 is a wavenumber 3 pattern with heaviest loadings in the South Paciﬁc sector. It shows a characteristic wavetrain with source over northern Australia and extending poleward into the South Paciﬁc returning into the extratropical Atlantic and Indian Oceans. Correlations with SSTs (not shown) suggest the possibility of some contemporaneous Indian Ocean SST forcing of this pattern. 4.2. Northern Hemisphere winter circulation Here, we describe the coherent patterns for NH winter. In line with other NH studies,1 we have chosen to use varimax rotation10 of our EOFs to describe the patterns. This tends to localise the maximum weighting of the patterns into the North Atlantic, North Paciﬁc and Eurasian sectors of the hemisphere. Shown in Fig. 4 are the top six NH rotated I-EOFs (IREOFs) in order of the interannual variance explained. Figure 5 shows the top six NH S-REOFs. I-REOF1 has a localised dipole (positive/negative weighting) structure over Greenland and the North Atlantic Ocean in a region of preferred atmospheric blocking. Frederiksen and Zheng6 discuss this pattern in some detail and show that it appears to be related to blocking in this region, with a horizontal structure reﬂecting that seen in dynamical blocking modes with intraseasonal periods. In addition, they show that rotated I-PC1 (I-RPC1) has a correlation of 0.77, at longitude 45◦ W, with a NH blocking index.15 S-REOF1, has a similar structure but with the negative centre shifted further downstream and reminiscent of the North Atlantic Oscillation (NAO).1,6 Interestingly, the dominant REOF derived from the sampling covariance matrix (4) has a similar structure but with the negative centre further westward in the Atlantic Ocean about half way between the negative centres of I-REOF1 and S-REOF1. It appears to be almost a linear combination of I-REOF1 and S-REOF1. Both I-REOF2 and S-REOF2 display a coherent wavetrain structure connecting the Paciﬁc Ocean to the Atlantic Ocean via the North American continent. This pattern is commonly referred to as the Paciﬁc North American (PNA) pattern.1 Frederiksen and Frederiksen8 have found this NH extratropical structure in dynamical modes with periods between 20 to 60 days, in association with intraseasonal oscillations, as well as longer period quasi-stationary modes. I-REOF2 and S-REOF2 appear to be associated with intraseasonal and quasi-stationary modes, respectively. Another mode that shows large amplitude over this region is I-REOF3. It displays a meridional dipole structure in the North Paciﬁc region reminiscent of blocking over the Alaska/northwest Canada and the Beaufort Sea, an area

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where blocking is prevalent. It has a correlation of -0.71 with a NH blocking index at 180◦ E.6 S-REOF3 and I-REOF6 both have heavy loading in the Western Paciﬁc and similar structure. They have much in common with the Western Paciﬁc (WP) pattern1 with a distinct meridional dipole in the western Paciﬁc, and appear to be the slow (quasi-stationary) and intraseasonal components, respectively, of this mode of interannual variability. In addition, I-REOF6 also appears to be associated with blocking in this region (correlation with NH blocking index of -0.676 ). The essential features of the Eastern Atlantic (EA) pattern1 are captured in S-REOF4 and I-REOF5. Again, they appear to capture the slow and intraseasonal components, respectively, of this mode of variability. SREOF6 and I-REOF4 also have large weighting in the Atlantic region, and both display a characteristic Eurasian wavetrain seen in the Eurasian pattern (EU)1 and dynamical studies.8 Frederiksen and Frederiksen,8 in particular, found intraseasonal oscillation modes with similar features to I-REOF4. Finally, S-REOF5, while somewhat similar to S-REOF2, is the NH pattern that is most closely associated with ENSO and Paciﬁc Ocean SST variability. The pattern resembles the Tropical Northern Hemisphere pattern (TNH),1 but when compared with the pattern derived from the sampling covariance matrix, usually used to deﬁne this pattern, it has much higher (20% larger) correlation (-0.62) with lagged (SON) Paciﬁc Ocean SST variability. Of the other slow patterns only S-REOF4, the EA slow pattern, has any appreciable correlation with lagged tropical SST forcing. In this case also, the highest correlation is with Paciﬁc Ocean SSTs (-0.46). S-REOF3, the slow WP pattern, has some contemporaneous (DJF) correlation with tropical Indian Ocean SST variability (0.50), but relatively weak SST-forced predictability at one season lag. 4.3. Coherent atmospheric patterns and Australian surface temperature The development of seasonal climate forecast schemes, whether statistical or dynamical, is predicated on an understanding of the sources of predictive skill as well as the sources of uncertainty in the variability of relevant climate variables. Surface variables, such as rainfall and surface temperature, are closely related to local and/or hemispheric (global) pressure ﬁelds. Hence, the identiﬁcation of the relationships between coherent patterns of variability in the atmospheric circulation and similar patterns in the surface

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climate can give an indication of the sources of uncertainty and predictability. For example, relationships between the intraseasonal components of the atmosphere and surface climate give an indication of the types of weather phenomena, or atmospheric teleconnections, that are responsible for the unpredictable variability in the surface climate. Conversely, relationships between the slow components indicate atmospheric circulations associated with predictable sources of variability. Here, we apply the method described in section 2.4 to a study of Australian summer (DJF) surface air temperature variability and its relationship to the global atmospheric circulation. The temperature data is taken from the Australian Bureau of Meteorology high quality surface temperature dataset interpolated onto a 2.5◦ x2.5◦ latitude/longitude grid. Again, we have used the NCAR-NCEP reanalysis 500hPa geopotential height data to identify coherent patterns in the atmospheric circulation that relate directly to variability in the Australian surface temperature. This data is global and on a 5◦ x5◦ latitude/longitude grid, except for latitudes south (north) of 45◦ S (45◦ N) where the longitude grid is variable as described above in the introduction to this section. We consider the period 1951-1999. Figure 6 shows the ﬁrst three dominant intraseasonal coupled patterns. They explain 24%, 16% and 10%, respectively, of the covariability in these ﬁelds. For surface temperature, they represent patterns of variability in the seasonal mean that are essentially unpredictable at the long range (a season or more ahead). The atmospheric patterns are the associated seasonal mean circulations (teleconnections). Thus, for example,the ﬁrst pattern (Figure 6(a)) depicts surface temperature variability over southern Australia associated with, at the particular phase shown, a dipole structure in the geopotential height ﬁeld with positive (shaded) and negative (unshaded) height anomalies. The associated atmospheric winds go approximately anticlockwise (clockwise) around the positive (negative) centre. Thus, at this phase, positive temperature anomalies are related to anticyclonic ﬂow over southern Australia and cyclonic ﬂow in the Australian Bight. This is a typical synoptic situation related to atmospheric blocking associated with intraseasonal internal dynamics. The second pattern (Figure 6(b)), shows opposite temperature anomalies over northern and southern Australia associated with a meridional wavetrain of height anomalies extending from over northern Australia southward and then eastward into the Paciﬁc Ocean. Again, at the phase shown, the negative (positive) temperature anomalies are consistent with the presence of cyclonic (anticyclonic) ﬂow. Finally, pattern three

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Fig. 6. The ﬁrst three dominant DJF coupled patterns of the intraseasonal components of Australian surface air temperature and global 500hPa geopotential height. Positive contours are shaded.

(Figure 6(c)) is associated with a geopotential height wavetrain emanating from the northwest of Australia. These patterns represent the essentially unpredictable components of surface temperature and atmospheric height variability.

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Fig. 7. The ﬁrst two dominant DJF coupled patterns of the slow components of Australian surface air temperature and global 500hPa geopotential height. Positive contours are shaded.

Fig. 8. Correlation between observed SON sea surface temperature and the slow components of Australian surface air temperature variability.Positive contours are shaded.

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The two most important slow coupled patterns are shown in Figure 7. They explain 41% and 12%, respectively, of the covariability in the slow components. The variability is dominated by the ﬁrst pattern (Figure 7(a)). In contrast to the intraseasonal patterns, where the geopotential height patterns show anomalies that are more localised in the Australian region, the slow geopotential height patterns have more global characteristics, usually indicative of some large scale forcing or slow internal dynamics. The most important forcing is generally sea surface temperature. For the ﬁrst pattern, positive temperature anomalies over central eastern Australia are related to positive geopotential height anomalies in a global band between 30◦ S to 30◦ N. There is also evidence of a Northern Hemisphere wavetrain (or teleconnection). This atmospheric pattern is known to be associated with ENSO and sea surface temperature forcing in the tropical Paciﬁc Ocean. To illustrate this, we have also calculated correlations between the time series of our temperature patterns and corresponding time series of one season lead (i.e. SON) SSTs. We use one season lead SSTs to indicate a possible source of forcing, or predictor, of these patterns. Here, we have used the UK Meteorological Oﬃce Hadley Centre SST dataset. Figure 8(a) shows the correlation with this dominant temperature pattern. There are clearly large correlations between eastern Australian temperature and SST anomalies in the tropical eastern Paciﬁc. In particular, positive (negative) Australian temperature anomalies are associated with positive (negative) SST anomalies in the eastern Paciﬁc. The second coupled pattern (Figure 7(b)) shows positive temperature anomalies over the northwest and negative anomalies over the southeast of Australia, associated predominantly with geopotential height anomalies, of opposite sign, in the middle and high latitudes of the Southern Hemisphere. This height pattern has features of the Southern Annular Mode, an important teleconnection generally associated with slow internal dynamics, as shown above. Figure 8(b) shows that there is little large scale SST forcing of this coupled pattern. 5. Conclusions Two methods, using daily and monthly means of climate data, have been proposed for studying the inﬂuence of intraseasonal variability on the interannual variability of the seasonal mean atmospheric circulation. Both methods were shown to give essentially the same results with minor diﬀerences due to sampling and estimation errors. The methods can be adapted to provide estimates of the interannual variance, covariance and cross-covariance

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in the seasonal mean ﬁeld(s), which arise from internal dynamical processes associated with intraseasonal variability. Removing this intraseasonal component from the sample variance, covariance or cross-covariance, derived from the seasonal mean ﬁeld(s), allows one to identify patterns which should be closely related to external forcings and very slowly varying (interannual/supra-annual) internal dynamics. Such a decomposition of the interannual variability of climate variables allows us to identify sources of uncertainty in climate variability as well as sources of predictive skill. The methodology was tested using synthetic data, in a Monte Carlo simulation study, of the circulation in the NP/NA region using NCEP 500hPa geopotential height data. In this case, the slow and intraseasonal components of the datasets were known a priori as were the slow and intraseasonal patterns of variability. Comparing the skill of our methodology, in retrieving the signiﬁcant rotated EOF patterns, with that of an ideal method, it was found that the skill in identifying correctly the intraseasonal component was comparable and slightly better in our case. For our method and the ideal method, the skill in identifying the slow component decreases with decreasing dominance (i.e. variance explained) of the pattern. In this case, the skill of our method was less than the ideal method but still quite acceptable. Considering both components are not observable in practice, and therefore it is not possible to use the ideal method, our method is clearly useful. In the case of the wintertime SH circulation, both daily and monthly mean 500hPa geopotential height data were used to derive the intraseasonal and slow patterns, and to show that both estimation methods gave essentially the same results. For both the SH and NH winter circulation, the intraseasonal patterns of interannual variability were shown to reﬂect the horizontal structures of well known intraseasonal dynamical modes of variability. Thus, for example, in the SH case, the patterns were shown to be related to the intraseasonal component of high latitude variability, associated with the SAM, and persistent blocking in the South Paciﬁc. Similarly, for the NH, they reﬂected the structures of the intraseasonal components of teleconnection patterns such as the PNA, the EA, and EU teleconnection patterns, and blocking activity over Greenland, the West Paciﬁc and Alaska. The patterns associated with the slow component were shown to have horizontal structures similar to many well known quasi-stationary teleconnections, such as the SAM, South Paciﬁc Wave, PNA, EA, EU, WP and ENSO teleconnections. Finally, we showed how the methodology could be adapted to identify the intraseasonal and slow patterns of the atmospheric circulation that

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inﬂuence the intraseasonal and slow components of variability in Australian surface temperature. In this case, the dominant intraseasonal patterns showed consistent relationships between the temperature anomalies and geopotential height anomalies with positive (negative) temperature anomalies associated with anti-cyclonic (cyclonic) circulation. The intraseasonal height patterns were fairly localised, with largest weighting in the Australia region, and displayed features of Southern Hemisphere blocking and meridional wavetrains, typically associated with internal dynamics at the intraseasonal time scale. The two most important slow patterns were related to ENSO variability and high latitude variability through an association with the SAM, which are global scale atmospheric teleconnections. SST forcing plays an important role in the former, and slow internal dynamics in the latter. The methodology presented here has provided some valuable new insights into the interannual variability of the SH and NH wintertime atmospheric circulation. The patterns associated with intraseasonal variability can be regarded as patterns that are, at the very long range (a season or more ahead), essentially unpredictable. The slow patterns, because they are related to the combined eﬀects of external forcings and internal dynamics on a longer time scale, are potentially predictable. Thus the technique has the potential to identify sources of uncertainty and predictability, in the variability of seasonal means, and thereby possibly improve statistical forecast schemes (through a better identiﬁcation of predictors) or give a better understanding of the limitations of dynamical forecast schemes. Appendix A. Principal Component Analysis Here we provide a brief summary of the PC analysis technique used to produce spatial patterns of variation from a given covariance matrix. The interested reader is referred to Joliﬀe10 for more details. Let uy = {ury |r = 1, . . . , R} (y = 1, . . . , Y ) be a multivariate column vector of data at R locations, in a given year y of Y years and with sample covariance matrix V. That is, it represents R time series of length Y years. A PC analysis seeks to deﬁne another vector wy = {wpy |p = 1, . . . , R} (y = 1, . . . , Y ) with wpy =

R

cpr ury = cTp uy

(A.1)

r=1

deﬁned as the pth PC whose sample variance can be shown10 to be cTp Vcp . The PCs are chosen in such a way as to maximize their sample variance with

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the largest variance assigned to the ﬁrst PC, the second largest variance to the second PC and so forth. If the coeﬃcients cp are constrained so that cTp cq = 1, p = q = 0, p = q, 10

then it can be shown problem

that this can be achieved by solving the eigen Vc = λc

with the eigenvalues (λp ) giving the sample variances of the PCs and the eigenvectors (cp ) providing the coeﬃcients in (A.1). The eigenvectors also provide the spatial patterns of variability and are sometimes referred to as “empirical orthogonal functions” (EOFs). Appendix B. Singular Value Decomposition Here we provide a brief summary of the SVD analysis technique used to produce coupled spatial patterns of covariation from a given cross-covariance matrix. The interested reader is referred to von Storch21 for more details. Let Xm×t and Yn×t be data matrices representing data from two climate variables at m and n geographical locations, respectively, and over t years. Assume further that the columns of the data matrices consists of deviations from the corresponding vector of sample means over all years. Without loss of generality, assume also that m ≤ n. Then the cross-covariance matrix can be written as 1 T Am×n = Xm×t Yn×t t T where Yn×t represents the transpose of Yn×t . Any rectangular matrix can be given a singular value decomposition21 T Am×n = Um×n Dn×n Vn×n

where the columns of U and V are orthonormal vectors of dimension m and n and are called left and right singular vectors, respectively. Matrix D is a diagonal matrix with non-negative elements dii = di for i = 1, . . . , n, called singular values. Furthermore, the column vectors of U and V are the eigenvectors of the AAT and AT A, respectively, with eigenvalues equal to the square of the singular values. For example, AAT U = UD(VT V)DT (UT U) = UD2 . The pairs of eigenvectors from U and V, corresponding to the same singular value, are the coupled patterns we consider above.

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References 1. A. G. Barnston and R. E. Livezey, Mon. Wea. Rev. 115, 1083 (1987). 2. P. J. Brockwell and R. A. Davis, Time Series Theory and Methods (SpringerVerlag, 1987). 3. C. S. Frederiksen, A. P. Kariko and X. Zheng, ANZIAM J. 44(E), C160 (2003). 4. C. S. Frederiksen and X. Zheng, ANZIAM J. 42(E), C608 (2000). 5. C. S. Frederiksen and X. Zheng, ANZIAM J. 45(E), C378 (2004). 6. C. S. Frederiksen and X. Zheng, Clim. Dyn. 23 193 (2004). 7. C. S. Frederiksen and X. Zheng, ANZIAM J. 46(E), C101 (2005). 8. J. S. Frederiksen and C. S. Frederiksen, J. Atmos. Sci. 50, 1349 (1993). 9. J. S. Frederiksen and C. S. Frederiksen, J. Atmos. Sci. 50, 3148 (1993). 10. I. T. Joliﬀe, Principal Component Analysis, (Springer Verlag, 1986). 11. E. M. Kalnay, et al. Bull. Am. Meteorological Soc. 77, 437 (1996). 12. G. N. Kiladis and K. C. Mo, AMS Meteor. Monogr. 49, 307 (1998). 13. C. E. Leith, J. Appl. Meteor. 12, 1066 (1973). 14. C. E. Leith, The Physical Basis of Climate and Climate Modelling, (World Meteorological Organisation) 16, Appendix 2.2 (1975). 15. H. Lejen¨ as and H. Økland, Tellus 35A, 350 (1983). 16. E. N. Lorenz, J. Appl. Meteor. 12, 1543 (1973). 17. R. A. Madden, Mon. Wea. Rev. 104, 942 (1976). 18. K. V. Mardia, J. T. Kent and J. M. Bibby Multivariate analysis (Academic Press, 1979). 19. J. A. Renwick, Mon. Wea. Rev. 133, 977 (2005). 20. M. R. Sinclair, Mon. Wea. Rev. 124, 245 (1996). 21. H. von Storch and F. W. Zwiers, Statistical Analysis in Climate Research (Cambridge University Press, 1999) 22. X. Zheng, H. Nakamura and J. A. Renwick, J. Clim. 13 2591 (2000). 23. X. Zheng and C. S. Frederiksen, J. Clim. 12 2386 (1999). 24. X. Zheng and C. S. Frederiksen, Clim. Dyn. 23 177 (2004). 25. X. Zheng, M. Sugi and C. S. Frederiksen, J. Meteorol. Soc. Japan 82 1 (2004).

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REGIMES OF THE WIND-DRIVEN OCEAN FLOWS HENK A. DIJKSTRA Institute for Marine and Atmospheric Research, Utrecht, Department of Physics and Astronomy, Utrecht University, Utrecht (IMAU), the Netherlands, [email protected] An overview is provided of recent results on patterns of internal variability of the midlatitude wind-driven ocean circulation. This variability arises through successive instabilities of these ﬂows. Bifurcation diagrams are presented for single-layer ﬂows and physical mechanisms associated with some of the instabilities are brieﬂy described. A homoclinic bifurcation is central for the lowfrequency, aperiodic variability in these ﬂows. The diﬀerent ﬂow regimes, as found in transient ﬂow computations in both single- and multi-layer ﬂows, can be interpreted with the help of the bifurcation diagrams.

1. Introduction The large-scale ocean circulation is driven by momentum ﬂuxes, the tides and aﬀected by ﬂuxes of heat and freshwater at the ocean–atmosphere interface. The near-surface circulation is dominated by horizontal currents that are mainly driven by the wind-stress forcing, while buoyancy diﬀerences are involved in the much slower motions of the deep ocean. Figure 1 gives an impression of the near-surface ﬂow in the northwestern part of the North Atlantic basin, based on a multi-pass satellite image of the sea-surface temperature (SST) ﬁeld. The wind-stress curl induced by the easterly winds in very low and very high latitudes, on the one hand, and the midlatitude westerlies, on the other, induces midlatitude cellular ﬂows, called gyres. The North Atlantic is typical of several other ocean basins in exhibiting a dominant anticyclonic cell, called the subtropical gyre, and a smaller cyclonic cell, called the subpolar gyre (Fig. 1). Each of these gyres has a narrow, fast-ﬂowing western boundary current and a slower, more diﬀuse eastern boundary current. The major surface current in the North Atlantic is the Gulf Stream, an eastward jet that arises through the merging of two western boundary

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currents, the northward-ﬂowing Florida Current and the southward ﬂowing Labrador Current. The variability of the Gulf Stream has been studied for decades through time-continuous in situ measurements, at a few locations, as well as by more detailed one-time-only hydrographic surveys. As the Gulf Stream penetrates further east into the open ocean, it spreads out due to meandering. In this region, cut-oﬀ eddies are formed and move away from the main jet, generally in a westward or southwestward direction. Their average wavelength is about 100 km and their propagation speed is of the order of 10 km/day. The scale of the eddies is related to an internal length scale of the ocean, the internal Rossby deformation radius; both stratiﬁcation and rotation effects contribute to this length scale.23 In the oceans, this scale of motion is commonly referred to as the mesoscale. Generally, the presence of mesoscale eddies causes variability on a subannual, 2–3-month time scale. The last decade has seen a huge increase in the observational information available on the ocean circulation on the basin and global scales.39 As a result, attention has focused more and more on the temporal variability of the wind-driven circulation that is associated with larger spatial scales and involves lower frequencies. Various observations — though limited in spatial and temporal coverage — suggest the existence of distinct scales of temporal variability from subannual15,28 to interannual1 scales. The sources of this low-frequency variability and of the associated spatio-temporal patterns have become an object of intense scrutiny. The classical view is that the overall red spectrum of the oceans’ variability in time is due to its “ﬂywheel” integration of atmospheric white noise12 and that any peaks that rise above this broad-band spectrum also result primarily from changes in the external forcing, especially in wind stress or buoyancy ﬂuxes. The forced variability does not always account, however, for all or even most of the observed variability. Internal ocean dynamics — i.e., intrinsic variability due to nonlinear interactions between two or more physical processes that aﬀect the wind-driven ocean circulation — may therefore play an important role on these time scales. In this paper, an overview is given of results of internal variability of the wind-driven ocean circulation. Using dynamical systems theory, we will ﬁrst analyse successive bifurcations in single-layer (constant density) models and next use these results to interpret ﬂow regimes as found in multi-layer (stratiﬁed) models.

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Fig. 1. Composite, multi-pass satellite image of the average SST ﬁeld in May 1996 within the Gulf Stream region. The infrared data used to obtain this picture were obtained from high-resolution (0.5◦ horizontally) observations from the Advanced Very High Resolution Radiometer (AVHRR, see http://fermi.jhuapl.edu/avhrr/index.html). The dark shading indicates a warm sea surface, with SSTs of typically 25◦ C in the Gulf of Mexico and the Florida Straits; lighter shading indicates a colder sea surface.

2. Single-layer ﬂows The theory of the homogeneous wind-driven ocean circulation20,35,36 is one of the cornerstones in physical oceanography. This theory describes the mid-latitude wind-driven ocean ﬂow in an active layer of ocean water with constant density ρ in an idealized L × B rectangular basin. Below this layer, with equilibrium thickness H, there is a very deep motionless layer of density ρ + ∆ρ (Fig. 2). The basin is located on a midlatitude β-plane with Coriolis parameter f = f0 + β0 y. Let the ﬂow be characterized by a horizontal length scale L and a horizontal velocity scale U . When the Rossby number = U/(f0 L) is small, quasi-geostrophic theory is an adequate description of the large-scale ﬂow.23 Let ψ indicate the geostrophic streamfunction in the horizontal plane, then

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Fig. 2. plane.

Sketch of the ocean model set-up in a rectangular basin on a midlatitude β-

the zonal velocity u, the meridional velocity v and the vorticity ζ are given by u = −∂ψ/∂y, v = ∂ψ/∂x and ζ = ∂v/∂x − ∂u/∂y = ∇2 ψ, respectively. When the ﬂow is driven by a zonal wind stress τ , the governing equation in this theory is the (equivalent) barotropic vorticity equation, given by ∇×τ ∂q + J(ψ, q) = AH ∇4 ψ + , ∂t ρ0 H f2 q = ∇2 ψ − 0 ψ + β0 y . Hg

(1a) (1b)

Here, q is the potential vorticity, g = g∆ρ/ρ is the reduced gravity and the Jacobian operator J is deﬁned as J(F, G) = Fx Gy − Fy Gx where the subscripts indicate diﬀerentiation. The quantity AH represents the turbulent lateral friction coeﬃcient. The strict homogeneous case is obtained when the second layer is a solid, for which g → ∞. No-slip boundary conditions are usually prescribed at the east-west boundaries and slip conditions at the north-south boundaries, i.e. x = 0, L : ψ = 0, y = 0, B : ψ = 0,

∂ψ =0 ∂x ∂2ψ = 0. ∂y 2

The wind-stress proﬁle often considered with (1a) is τ x (x, y) = −τ0 (σ cos π

y y + (1 − σ) cos 2π ) ; B B

τ y (x, y) = 0 ,

(2)

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where the dimensionless parameter σ controls the shape and τ0 is a typical amplitude. Following Veronis,37 much attention14 has focussed on the subtropical (single) gyre system as obtained above with the choice σ = 1. The single-gyre wind-stress forcing consists of easterlies (westerlies) at the south (north) part of the basin. The so-called double-gyre case has more recently received much attention and is obtained with σ = 0 in (2). In this case, both the subtropical and subpolar gyres are forced and the wind stress is symmetric with respect to the mid-axis of the basin (Fig. 3).

Fig. 3.

Plots of the zonal wind stress (2) for three diﬀerent values of σ.

Under a given steady wind-stress forcing, the linear steady quasigeostrophic theory predicts a Sverdrup interior ﬂow and a frictional western boundary layer. The linear theory provides a ﬁrst order explanation of the existence of western boundary currents, such as the Gulf Stream. The nonlinear theory is, however, far from complete. Although the strong eﬀect of inertia on the ﬂows was already shown by Veronis,37 the work to determine systematically the solution structure of (1a) versus the lateral friction parameter AH did not start until the mid 1990s.7

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Value

Parameter

Value

L H f0 ρ0

1.0 × m 6.0 × 102 m 1.0 × 10−4 s−1 103 kgm−3

τ0 β0 U B

1.5 × 10−1 Pa 1.6 × 10−11 (ms)−1 1.6 × 10−2 ms−1 1.0 × 106 m

106

3. Primary bifurcations For large values of AH , a unique and globally stable ﬂow state for both single- and double-gyre cases is found.10 To investigate the solution structure of the equations when AH is decreased, continuation methods9 have been used on discretized versions of (1a). In the results below, a 128 × 128 equidistant grid is used and the steady states are computed versus Re = U L/AH . In other studies, also the ratio of boundary layer thicknesses δI /δM , where δI = (U/β0 )1/2 and δM = (AH /β0 )1/3 is used. Other parameters are ﬁxed at values shown in Table 1 and g → ∞. 3.1. Single-gyre ﬂows In the bifurcation diagram (Fig. 4a) for the single-gyre ﬂows (σ = 1), a value of the streamfunction at a certain gridpoint (ψR ) is plotted versus Re. Each point on the curve represents a steady state and its stability is indicated by the linestyle, with solid (dashed) curves indicating stable (unstable) solutions. At small and large values of Re, there is a unique steady solution, while between the two-saddle node bifurcations L1 and L2 there is a regime of multiple equilibria. Plots of the streamfunction ψ at labelled locations in Fig. 4a are shown in the Figs. 4b-d. The pattern in Fig. 4b near Re = 10 deviates already from the symmetric linear MunkSverdrup solution. The eﬀects of strong nonlinearities on the ﬂow can be seen in the streamfunction for both solutions at Re = 60. A strong northsouth asymmetry appears in the solution in Fig. 4c and a gyre ﬁlling up the basin can be seen in Fig. 4d). 3.2. Double-gyre ﬂows For the case σ = 0, the structure of the steady solutions is shown through the bifurcation diagram in Fig. 5a, where the value of the streamfunction at a point in the southwest part of the domain (ψR ) is plotted versus Re = U L/AH . At large values of AH (small Re), the anti-symmetric double-

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(a)

(b)

(c)

(d)

Fig. 4. (a) Bifurcation diagram for the single-gyre (σ = 1) barotropic quasi geostrophic model for a square basin with Re = U L/AH as the control parameter. (b) Pattern of ψ near Re = 10 on the lower stable branch in (a). (c) Same for Re = 60 along lower branch and (d) for Re = 60 along the upper stable branch.

gyre ﬂow (Fig. 5b) is a unique state. When lateral friction is decreased, this ﬂow becomes unstable at the pitchfork bifurcation P1 and two branches of stable asymmetric states appear for smaller values of AH (larger Re). The solutions on these branches have the jet displaced either southward or northward (Fig. 5c) and are exactly symmetrically related for the same value of Re. For even smaller friction, the anti-symmetric ﬂow becomes

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(a)

(b)

(c)

(d)

Fig. 5. (a) Bifurcation diagram for the double-gyre (σ = 0) barotropic quasi geostrophic model for a square basin with Re = U L/AH as the control parameter. (b) Pattern of ψ near Re = 10 on the lower stable branch in (a). (c) Same for Re = 60 along the branch A1u ; the pattern on the branch A1d at Re = 60 is the mirror image of (c) with respect to reﬂection through the midaxis of the basin. (d) The pattern at Re = 60 on the branch A2d .

inertially dominated and ψR increases rapidly. A pitchfork bifurcation P2 occurs on the anti-symmetric branch where an additional pair of asymmetric solution branches appear (Fig. 5d); all these solutions are unstable.

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The existence of the bifurcation P1 (Fig. 5a) captures the heart of the physics of symmetry breaking in these ﬂows. The physical mechanism of the instability can be analyzed with help of the patterns of the steady state and the eigenvector of the linear stability analysis which has a zero growth rate just at P1 . The streamfunction and vorticity ﬁeld of the steady state are presented in Fig. 6a, while those of the streamfunction and vorticity perturbation (determined from the eigenvector) are shown in Fig. 6b. The streamfunction perturbation has a tripole-like structure with a negative vorticity centre along the jet-axis and two positive vorticity centres at either side. The special property of these patterns is that the centre negative vorticity lobe is exactly localized within the vorticity extrema of the antisymmetric basic state. If we consider the region just above the symmetry line of the eastward jet (y = 0.5), the perturbation zonal ﬂow is eastward, and therefore in the same direction as that of the basic state. More northward (above y = 0.7), the perturbation ﬂow is westward and therefore also in the same direction as the steady ﬂow. If we consider the ﬂow just below the symmetry line of the steady jet, it is observed that the ﬂow perturbations are in the opposite direction to that of the basic state. Hence, the ﬂow perturbation weakens the subtropical gyre and strengthens the subpolar gyre. The asymmetric change in the strength of the basic ﬂow due to the perturbations leads to increased horizontal shear in the eastward jet, which leads to an additional negative vorticity. This extra vorticity just ampliﬁes the original perturbation ﬂow in this region leading to instability. 3.3. Connection between single-gyre and double-gyre ﬂows We can follow the branches of the double-gyre case in the parameter σ (controlling the asymmetry of the wind stress) to connect to the singlegyre case. The bifurcation diagram for σ = 0.1 (Fig. 7a) shows the basic imperfections of the double-gyre ﬂow. The branch S1 in Fig. 7a connects to the branch A1u ; this is expected because the σ = 0.1 wind stress induces a preference for the jet-up solution. Both the branches containing jet-down solutions (the branches A1d and A2d ) connect to the branch S2 to give the branch labeled with A1d −S2 −A2d in Fig. 7a. The branch of jet-up solutions A2u connects to the branch S3 and forms the S3 − A2u branch. When the wind stress is made more asymmetric through an increase in σ, both the A1d − S2 − A2d and S3 − A2u branches move quickly to higher values of Re and eventually move out of the computational domain. What remains is the S1 − A1u branch. Up to σ = 0.75, there are no saddle-node bifurcations on this branch (Fig. 7b), but these appear for values of σ just

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(a)

(b) Fig. 6. (a) Contour plots of the steady state at the pitchfork bifurcation P1 in Fig. 5a ¯ in the left panel and the vorticity (ζ) ¯ in the right panel. with the streamfunction (ψ) (b) Contour plots of the perturbation destabilizing the steady state of (a) with the ˜ in the left panel and the vorticity (ζ) ˜ in the right panel (from Ref. 11). streamfunction (ψ)

before σ = 0.9. It is clear that the S1 − A1u branch eventually deforms into the single branch of the single-gyre wind stress case (σ = 1). 4. Secondary bifurcations: double-gyre case The secondary bifurcations of the single-gyre ﬂows have been studied in detail by Sheremet et. al 29 and time-dependent behaviour of the singlegyre ﬂows has also been studied extensively.4,5,18,19 We will here only focus on the double-gyre case since it appears to be a better prototype situation for the actual North Atlantic Ocean circulation than the single-gyre case.9

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(a) Fig. 7. of σ.

(b)

Shown are (a) bifurcation diagram for σ = 0.1 and (b) for several larger values

4.1. Hopf bifurcations The symmetry-breaking associated with the pitchfork bifurcation P1 leads to two branches of stable asymmetric steady states. However, these states also become unstable at larger values of Re due to the occurrence of Hopf bifurcations. The pattern of the oscillatory mode which destabilizes the asymmetric double-gyre ﬂow at each Hopf bifurcation can be determined from the solution of the linear stability problem. At the Hopf bifurcation, a complex conjugate pair of eigenvalues σ = σr ± iσi crosses the imaginary ˆ=x ˆ R + iˆ xI provides the axis. The corresponding complex eigenfunction x disturbance structure Φ(t) with angular frequency σi and growth rate σr to which the steady state becomes unstable, i.e., ˆ I sin(σi t)] . Φ(t) = eσr t [ˆ xR cos(σi t) − x Propagation features of a neutral eigenmode (σr = 0.0) can be determined ˆ I and then at Φ(0) = x ˆ R . The period by ﬁrst looking at Φ(−π/(2σi )) = x P of the oscillation is given by P = 2π/σi . The ﬁrst Hopf bifurcation is associated with the destabilization due to a so-called Rossby-basin mode. These modes can be described by a sum of free Rossby waves where the coeﬃcients are chosen such that the boundary conditions are satisﬁed. The simplest patterns of these modes can be determined by solving the normal mode problem for the motionless, unforced, non-viscous ﬂow in (1a). For the gravest Rossby basin mode, the

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(a)

(b)

(c)

Fig. 8. Sketch of the streamfunction pattern of the gravest Rossby-basin mode at three diﬀerent instances during its propagation. (a) t/P = 0; (b) t/P = 1/4; (c) t/P = 3/8.

period P is about 20 days. The pattern of this mode is shown in Fig. 8 at three instances during its propagation. The transition patterns at the Hopf bifurcations of the asymmetric double-gyre ﬂows (near Re = 52 in Fig. 5a) are deformations of the pattern in Fig. 8. The growth rate of these modes are determined by the horizontal shear within the asymmetric double-gyre ﬂow. At the second Hopf bifurcation, the asymmetric state destabilizes to a mode which has an interannual period and the perturbations strengthen and weaken the eastward jet during both phases of the oscillation. These interannual, so-called gyre modes do not have their origin in the spectrum of the linear operator related to free Rossby-wave propagation. Simonnet and Dijkstra33 clariﬁed the spectral origin of the gyre mode and presented a physical mechanism of its propagation. The gyre mode destabilizes the asymmetric solutions at a Hopf bifurcation located near Re = 83. The gyre mode, therefore has a negative growth factor σr for Re < 83 (Fig. 9). In the rightmost panels of Fig. 9, the patterns of the real and imaginary parts ˆ I ) are shown near Re = 40. The path of the of this eigenmode (ˆ xR and x gyre mode (the dash-dotted curve in Fig. 9) ends at the point M , where it splits into two stationary eigenmodes. These stationary modes exist up to the point P1 where the asymmetric solutions cease to exist. Also shown in Fig. 9 are the growth factors of the leading eigenmodes on the symmetric solution branch. The non-oscillatory mode responsible for the ﬁrst pitchfork bifurcation (P1 ) has a symmetric tripolar structure (Fig. 9, upper-left panel), similar to the streamfunction pattern in Fig. 6b and is called the P-mode. At P1 , the growth rate σr of this mode becomes positive which means that the symmetric steady ﬂow becomes unstable to this perturbation pattern. The non-oscillatory mode responsible for the

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Fig. 9. Real part σr of the eigenvalues closest to the imaginary axis of the linear stability problem of the symmetric double-gyre ﬂow (thin lines) and along one of the asymmetric steady states (thick lines). The P-mode (streamfunction pattern in the upper left panel) destabilizes the symmetric state at the pitchfork P (from [33, ]). Along the asymmetric states, however, it deforms and merges with the L-mode (streamfunction pattern in the lower left panel) at the point M. This gives rise to the gyre mode (streamfunction patterns in the right panel).

saddle-node bifurcation at L in Fig. 5a has a dipolar anti-symmetric structure (Fig. 9, lower-left panel). It thus acts on both gyres simultaneously so that they either increase or decrease in intensity. Simonnet and Dijkstra33 called this non-oscillatory mode the L-mode. At P1 , the L-mode is damped but σr becomes positive at the saddle-node bifurcation L. Relevant for the spectral origin of the gyre mode is the path of both the P -mode and L-mode on the asymmetric branches for Re > 29.4. For Re slightly above P1 , both modes are still non-oscillatory and have negative growth factor since the asymmetric branch is stable. The paths of the eigenvalues of both modes are indicated by the thick lines in Fig. 9 (starting at Re = 29.4). The growth factor of the P-mode decreases with Re, whereas that of the L-mode increases. Both modes meet at the point M (Fig. 9), which Ref. 33 called the merging point, and give birth to the gyre mode.

4.2. Homoclinic bifurcations We now consider the transient behaviour of the double-gyre ﬂows for values of Re beyond the ﬁrst Hopf bifurcation. Trajectories computed for the 1000 × 1000 km basin show that indeed intermonthly variability ﬁrst occurs

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with increasing Re in the form of periodic oscillations. Subsequently, when Re is increased, a quasi-periodic orbit is obtained with both interannual and intermonthly frequencies. Soon after Re = 85, the ﬂow becomes irregular. In Ref. 17, transient ﬂows in a basin of 1024 × 2048 km with no-slip boundary conditions on the lateral walls are considered. Transient solutions are computed versus lateral friction and either steady, periodic and aperiodic solutions are found. The structure of the steady states and periodic orbits can be understood with help of the bifurcation diagram Fig. 4a, where the periodic orbits are coming from the Hopf bifurcations. In some aperiodic solutions, large excursions are made and ultra low-frequency variability arises; Ref. 17 suggests that it arises through a homoclinic orbit. In Ref. 21, the location of the homoclinic orbit in the double-gyre ﬂows is precisely located for ﬂows in a basin of size 1000 × 2000 km. Many transient computations are performed for diﬀerent parameters and spectra are versus parameters. In this way, they ﬁnd evidence for the occurrence of a homoclinic orbit of Ref. 30 type. This behaviour is characterized by speciﬁc periodic and aperiodic orbits that can be observed in the spectrum of the time series. Ref. 21 also demonstrated the importance of this dynamical phenomenon in explaining low-frequency variability in these ﬂows. For a 2560 × 2560 km basin, Ref. 8 shows that the anti-symmetric ﬂow also destabilizes through a pitchfork bifurcation and that the asymmetric double-gyre ﬂows subsequently destabilize through Hopf bifurcations. The ﬁrst periodic orbit that appears has a subannual time scale and interannual variability occurs at slightly larger values of δI /δM . They monitor the transition to aperiodicity in detail by plotting the transport diﬀerence ∆Φ between the subtropical and subpolar gyre versus the basin kinetic energy E of the ﬂow for diﬀerent ratios δI /δM (Fig. 10). The quantity ∆Φ is deﬁned as ∆Φ =

−ψpo − ψtr , max | ψ |

(3)

where ψpo < 0 is the maximum transport of the subpolar gyre and ψtr > 0 the maximum transport of the subtropical gyre. Note that ∆Φ = 0 for an anti-symmetric ﬂow, with ψpo = −ψtr . In Fig. 10a, the projection of a periodic orbit around an asymmetric steady state can be seen and it has a period of about 148 days. As δI /δM increases, the periodic orbit at some instant of time reaches the anti-symmetric double-gyre solution, for which ∆Φ = 0 (Fig. 10b-d). For slightly larger values the ﬂow becomes aperiodic while the trajectory now attains both positive and negative values of ∆Φ (Fig. 10e-f). It appears as

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Fig. 10. Phase space projections of trajectories computed by Chang et. al.8 On the horizontal axis, the basin averaged kinetic energy (BKE) of the ﬂow and on the vertical axis the asymmetry of the ﬂow measured through T D = ∆Φ is plotted. The diﬀerent panels are for several values of the ratio BLR = δI /δM . (a) 0.880, (b) 0.884, (c) 0.888, (d) 0.892, (e) 0.897 and (f) 0.900.

though the periodic orbit makes a connection with the branch of steady anti-symmetric solutions and then connects to the periodic orbit which is present around the symmetry-related asymmetric state: this is characteristic of the presence of a homoclinic bifurcation. The connection between the pitchfork bifurcation, the gyre modes and the occurrence of the homoclinic bifurcation was clariﬁed in Ref. 34 and an overview of the bifurcation behaviour leading to the homoclinic orbit is plotted in Fig. 11. The symmetry-breaking pitchfork bifurcation P is responsible for the asymmetric states; the P-mode is involved in this instability. The merging of the P-mode and the L-mode on the branches of the asymmetric states (at the points M) is responsible for the Hopf bifurcations H associated with the gyre modes. Finally, the periodic orbits arising from these Hopf bifurcation points on both asymmetric branches connect with the unstable anti-symmetric steady state at the point A; this gives rise to the homoclinic orbit. The type of homoclinic orbit depends on the eigenvalues associated with the linear stability of the anti-symmetric state at the connection point A.38 In case there are only real eigenvalues, there

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Fig. 11. Schematic bifurcation diagram of the solutions of the barotropic vorticity equation, plotted in terms of a measure of the asymmetry of the solution (for example, ∆Φ) versus either wind-stress intensity, the ratio δI /δM or simply the Reynolds number Re (from Ref. 34).

is a homoclinic connection of Lorenz-type and when the second and third eigenvalue form a complex-conjugate pair, there is a homoclinic bifurcation of Shilnikov type. Ref. 34 shows that both types can occur and that a Shilnikov type is more likely to occur at small lateral friction, in accordance with the results in Ref. 21. 5. Low-frequency variability McCalpin and Haidvogel16 investigated the time-dependent solutions of (1a) for a basin of realistic size (3600 × 2800 km), as well as the sensitivity of solutions to the magnitude of the wind stress and its meridional proﬁle. They classiﬁed solutions according to their basin-averaged kinetic energy, and found three persistent states in their simulations (Fig. 12a). High-energy states are characterized by near-symmetry with respect to the mid-axis, weak meandering, and large jet penetration into the basin interior (Fig. 12b, left panel). Low-energy states have a strongly meandering jet that extends but a short way into the basin (Fig. 12b, right panel), while intermediate-energy states resemble the time-averaged ﬂow and have a spatial pattern somewhere between high- and low-energy states (not shown). The persistence of the solutions near either state is irregular but can last for more than a decade (Fig. 12a). Primeau25 reproduces the time-dependent behaviour of the ﬂows found

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(a)

(b) Fig. 12. (a) Typical variation of the basin averaged kinetic energy of the double-gyre ﬂow in a large basin for the high-forcing or low-dissipation regime (from McCalpin and Haidvogel16 ). (b) Typical patterns of the streamfunction for the high-energy state (left panel) and the low-energy state (right panel).

by McCalpin and Haidvogel.16 By projecting the instantaneous ﬂow ﬁelds onto four of the steady solutions, he found that a signiﬁcant amount of the low-frequency variability of the trajectories are associated with transitions between these steady solutions. Furthermore, he explained that the reduction of the low-frequency variability associated with an increased asymmetry of the wind forcing is a result of the fact that some of the steady states cease to exist. However, knowing that there are many branches of steady solutions, many gyre modes32 and possibly several homoclinic connections, much more work is needed to ﬁgure out the precise dynamics causing the diﬀerent energy states found in Ref. 16. At very high values of Re, Greatbatch and Nadiga13 analyze the ﬂows

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for a 1000 × 2000 km basin and ﬁnd that typically a four-gyre pattern arises in the mean ﬂow. The upper and lower gyres (which circulate against the wind direction) are driven by mesoscale variability and are associated with a homogenization of potential vorticity.27 6. Multi-layer ﬂows We next consider a model with a more detailed representation of the stratiﬁcation through the increase in the number of layers. In this three-layer model, the stratiﬁcation is idealized by three stacked layers of water with constant densities ρi , ρ1 < ρ2 < ρ3 , and mean layer thicknesses Hi , with H = H1 + H2 + H3 . The governing equations of the model are ∂q1 ∇×τ , + J(ψ1 , q1 ) = AH ∇4 ψ1 + ∂t ρ 0 H1 f0 q1 = ∇2 ψ1 − h 1 + β0 y H1 ∂q2 + J(ψ2 , q2 ) = AH ∇4 ψ2 , ∂t f0 (h2 − h1 ) + β0 y q2 = ∇2 ψ2 − H2 ∂q3 + J(ψ3 , q3 ) = AH ∇4 ψ3 , ∂t f0 h 2 + β0 y , q3 = ∇2 ψ3 + H3 where the qi represent the potential vorticity and the ψi the streamfunction in layer i. The quantities hi represent interface perturbations related to the streamfunctions, ψi , through hi = f0 (ψi −ψi+1 )/gi for i = 1, 2, in which the gi = g(ρi+1 − ρi )/ρ0 , i = 1, 2, represent the reduced gravity parameters. Conditions of no normal ﬂow and no-slip, ψi = 0 and ∂ψi /∂n = 0, are applied to the lateral boundaries. Because of the representation of vertical shear in this model, the ﬂows become susceptible to baroclinic instabilities23 which show up in the bifurcation diagrams as additional Hopf bifurcations.11 Nauw et. al 22 used the three-layer model to investigate the diﬀerent ﬂow regimes (in a 2000 × 2000 km basin) which appear when the lateral friction coeﬃcient AH is decreased from AH = 2400 m2 s−1 to AH = 300 m2 s−1 . Parameter values are as in Table 1 with additional parameter values H1 = 600 m, H2 = 1400 m, H3 = 2000 m, g1 = 0.02 ms−2 and g2 = 0.03 ms−2 . Four ﬂow regimes are identiﬁed by the analysis of a combination of the maximum northward transport of the time-mean ﬂow and the normalized

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transport diﬀerence between the subtropical and subpolar gyre (∆Φ). In this case, ∆Φ is deﬁned as in (3) in which the streamfunction ψ is that of the depth averaged ﬂow. With decreasing AH , the regimes found are the viscous anti-symmetric regime (for AH ≥ 2100 m2 s−1 ), the asymmetric regime (for 1400 ≤ AH ≤ 2100 m2 s−1 ), the quasi-homoclinic regime (for 700 ≤ AH ≤ 1400 m2 s−1 ) and the inertial anti-symmetric regime (for AH ≤ 700 m2 s−1 ). For four diﬀerent values of AH , the value of ∆Φ is plotted versus time in Fig. 13 and time-mean plots of the barotropic transport streamfunction Ψ of the vertically averaged ﬂow are shown in Fig. 14. The time-series of ∆Φ in the viscous anti-symmetric regime (Fig. 13a) displays a low-frequency modulation of a high-frequency signal, while the time-mean state (Fig. 14a) is anti-symmetric. A transition to an asymmetric regime occurs at smaller AH and a typical time-series of ∆Φ in that regime is shown in Fig. 13b. The value of ∆Φ remains positive after a spin-up of slightly more than 25 years and the amplitude of the high-frequency oscillation changes on a decadal time-scale. The time-mean barotropic transport streamfunction is asymmetric and displays a jet-down solution (Fig. 14b) in correspondence with the positive value of ∆Φ. For the ﬂow in the quasi-homoclinic regime, several intervals can be distinguished in which there is a preference for either positive or negative values (Fig. 13c). The time-mean ﬂow in this regime is slightly asymmetric (Fig. 14c). The time-series of the case in the inertial anti-symmetric regime (Fig. 13d) consists of a mainly high-frequency signal. The time-mean ﬂow in this regime (Fig. 14b) is also anti-symmetric, but the midlatitude jet is much stronger than in the anti-symmetric viscous regime. Moreover, the large-scale gyres are accompanied by small-scale subgyres near the northern and southern boundary (Fig. 14d), similar to those in Ref. 13. The four regimes are also characterized by diﬀerent types of variability. In Ref. 22, the spatio-temporal variability of the ﬂows (of which time series were shown for diﬀerent values of AH in Fig. 13) was analyzed with the MSSA technique.24 In Fig. 15, a histogram is shown of the variance explained by each of the statistical modes, classiﬁed into groups that can be related to an internal mode. Case (a) is for a symmetric wind-stress forcing (σ = 0.0), while case (b) is for a slightly asymmetric wind stress (σ = 0.1). Most of the variance in the viscous anti-symmetric regime can be explained by two classical baroclinic modes (CB1 and CB2), both with a period of about 3 months. In the inertial anti-symmetric regime, the variability is controlled by Rossby basin modes (RB) with intermonthly periods. Part

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(a)

(b)

(c)

(d)

Fig. 13. Time series (including spin-up) of the transport diﬀerence, ∆Φ, for diﬀerent values of AH in the large basin case. (a) AH = 2400 m2 s−1 , viscous anti-symmetric regime; (b) AH = 1600 m2 s−1 , asymmetric regime; (c) AH = 900 m2 s−1 , quasi-homoclinic regime; (d) AH = 600 m2 s−1 , inertial anti-symmetric regime.

of the variance in the asymmetric and quasi-homoclinic regimes can be explained by a gyre mode (G). It causes low-frequency variability with a period of about 3 years. The case with asymmetric wind-stress forcing demonstrates that the presence of the gyre mode is linked to the asymmetry of the time-mean state (Fig. 15b). Nauw et. al 22 also explain the transitions between the diﬀerent regimes. The transition from the viscous anti-symmetric regime to the asymmetric regime is associated with a symmetry-breaking pitchfork bifurcation. A homoclinic bifurcation, caused by the merging of two mirror-symmetric lowfrequency relaxation oscillations and the unstable symmetric steady state, marks the transition from the asymmetric regime to the quasi-homoclinic regime. The transition from the quasi-homoclinic regime to the inertial anti-symmetric regime occurs through symmetrization of the zonal velocity ﬁeld of the time-mean state. The interactions of high-frequency modes (such as CB1 and CB2) introduces forcing terms that oppose the wind-

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(a)

(b)

(c)

(d)

Fig. 14. The patterns of the barotropic transport function, Ψ, averaged over the ﬁnal 75 years of integration for selected values of AH . (a) AH = 2400 m2 s−1 , viscous antisymmetric regime; (b) AH = 1600 m2 s−1 , asymmetric regime; (c) AH = 900 m2 s−1 , quasi-homoclinic regime; (d) AH = 600 m2 s−1 , inertial anti-symmetric regime.

stress forcing, thereby moving the system towards a regime where both multiple equilibria and the gyre mode cease to exist. The results in Ref. 22 indicate that the study of the steady states, the bifurcations and the internal modes of variability provide an interpretation framework for complex time-dependent multi-layer ﬂows. But, as the bifurcation diagrams become more complicated for ‘realistic’ size basins, much work is needed to obtain a more detailed dynamical interpretation of time-dependent ﬂows in these basins.

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(a)

(b)

Fig. 15. Histograms of explained variances (%) for each of the diﬀerent statistical oscillations, as determined from the M-SSA technique. (a) symmetric wind-stress forcing and (b) asymmetric wind-stress forcing. CB1 = classical baroclinic mode, causing meandering of the midlatitude jet; CB2 = classical baroclinic mode causing strengthening and weakening of the midlatitude jet; D = dipole oscillation; G = gyre mode; RB = Rossby basin mode; WT = wall-trapped mode. The D and WT modes are not discussed here.

Berloﬀ and McWilliams2 computed numerical solutions of the doublegyre ﬂows in a two-layer model for ﬁve values of the lateral friction coefﬁcient AH in a basin of realistic size (3200 × 2800 km). For AH = 1200 m2 s−1 , an asymmetric steady state is found. At AH = 1000 m2 s−1 , quasiperiodic variability is found containing two dominant frequencies in the subannual and interannual range. At even smaller friction, a broadband spectrum appears, with the spectral power of the total energy increasing towards lower frequencies. At AH = 800 m2 s−1 , the behaviour of the solutions is called ‘chaotic’, while at AH = 600 m2 s−1 the ﬂow patterns hover near three states with distinct total energy. These states are characterized by a diﬀerent penetration length of the eastward jet and the presence or absence of dipole-pattern oscillations in the recirculation region. Characteristics of the asymmetric and quasi-homoclinic regimes are found in Ref. 2. The spatial pattern of the interannual mode in the symmetrically forced case at AH = 1000 m2 s−1 (their Fig. 15) is similar to that of the gyre mode. The meridional position of the separation point of their time-dependent solution at AH = 600 m2 s−1 alternates between locations to the north and to the south of the mid-axis of the basin on a decadal timescale (their Fig. 18). This indicates an alternation between a jet-up and a jet-down solution and provides support for the nearby presence of a homoclinic orbit. Hence, this solution is likely to reside in a quasi-homoclinic regime. Berloﬀ and McWilliams3 investigate the double-gyre ﬂows at even

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Fig. 16. Upper layer streamfunction snapshots of the ocean in a 3200 km square basin for varying Reynolds numbers, Re, with Re = 0.375, 1.5, 6.0, 24 for the panels (A)(D), respectively. Here, Re = U L/AH , with U = 10−3 ms−1 and L = 3200 km. The time-mean ﬂow consists of an anticyclonic midlatitude subtropical gyre and a cyclonic subpolar gyre. The resolution in the computations increases from 25 km in (A) to 1.56 km in (D). Note the appearance of coherent vortices throughout the circulation in the highest value of Re results (from Ref. 31).

smaller values of AH in a three-layer model and ﬁnd a destabilization of the western boundary current at very small values of AH . In Ref. 31, ﬂows for very small values of AH are computed in a high resolution 6-layer model for a 3200 km square basin. In Fig. 16, upper layer streamfunction plots are shown from several numerical experiments diﬀering in the values of AH . The displayed sequence goes from relatively low Reynolds numbers Re (see deﬁnition in caption of Fig. 16) in panel A to very high values in panel D. In panel D, numerous small-scale coherent vortices are displayed. Comparable features occasionally appear in the ﬂow in panel C, but are essentially absent in the panels A and B. In C, the

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vortices are far sparser and do not survive for ‘long’ times (relative to vortex turnover time scales). The highest Re computations possess eddy kinetic energies approaching values like those observed in the open ocean. The dynamical origin of this so-called coherent vortex regime is still unknown. 7. Summary and outlook With the dynamical systems approach as presented here, the idea is that an understanding of the physics of the observed complex ocean ﬂows can be obtained by approaching the ‘real’ situation from particular limiting ﬂows. One path proceeds from simple to complex situations through a hierarchy of models. We considered only single- and multi-layer quasi-geostrophic models here, but the hierarchy is much larger including shallow-water models and primitive equation models. A second path was taken within one particular member of the model hierarchy where we proceeded from steady, highly-dissipative or weakly-forced ﬂows to irregular, weakly dissipative or strongly forced ﬂows by varying parameters, here only AH . By proceeding along both paths, two important issues have become apparent. The ﬁrst issue is the existence of multiple steady ﬂow patterns in wind-driven midlatitude ocean ﬂows. These multiple states have been robust in the model hierarchy and their origin is a symmetry breaking shear instability, most apparent in the single-layer model. The second issue is that a classiﬁcation of internal modes of variability is appearing. From a mathematical point of view, there are two types of modes. One type of modes, the Rossby-basin modes (RB), comes from the basic linear operator arising from the linear stability analysis of the no-ﬂow state. The other types of modes (CB and G) do not have an origin in this basic linear operator. The oscillatory classical baroclinic modes (CB) arise when vertical shear is present in the background state. The low-frequency gyre modes (G) arise through a merger process of stationary (P and L) modes. The gyre modes play an important role in the generation of aperiodic ﬂows through the occurrence of homoclinic bifurcations. It is less clear at the moment, why the transition behaviour to aperiodic behaviour diﬀers in both single- and double-gyre ﬂow. In Ref. 6, it is suggested that the route to chaos in the baroclinic single-gyre case is the classical three-frequency route.26 which appears diﬀerent from the Lorenz34 and Shilnikov21 route (through a homoclinic connection) as found in double-gyre ﬂows. The study of the route to complex ﬂows over the model hierarchy, however, is in its infancy and many exciting new results can be expected in the near future.

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Acknowledgements I would like to thank Eric Simonnet, Michael Ghil, Janine Nauw and Maurice Schmeits for the pleasant and productive collaboration over the years.

References 1. S. J. Auer, Five-year climatological survey of the Gulf Stream system and its associated rings. J. Geophys. Res., 92, 11,709 – 11,726 (1987). 2. P. S. Berloﬀ and J. C. McWilliams, Large-scale, low-frequency variability in wind-driven ocean gyres. J. Phys. Oceanogr., 29, 1925–1949 (1999). 3. P. S. Berloﬀ and J. C. McWilliams, Quasi-geostrophic dynamics of the western boundary current. J. Phys. Oceanogr., 29, 2607–2634 (1999). 4. P. .S. Berloﬀ and S. P. Meacham, The dynamics of an equivalent barotropic model of the wind-driven circulation. J. Mar. Res., 55, 407–451 (1997). 5. P. S. Berloﬀ and S. P. Meacham, The dynamics of a simple baroclinic model of the wind-driven circulation. J. Phys. Oceanogr., 28, 361–388 (1998). 6. P. S. Berloﬀ and S. P. Meacham On the stability of the wind-driven circulation. J. Mar. Res., 56, 937–993 (1998). 7. P. Cessi and G. R. Ierley, Symmetry-breaking multiple equilibria in quasigeostrophic, wind-driven ﬂows. J. Phys. Oceanogr., 25, 1196–1205 (1995). 8. K. -I. Chang, M. Ghil, K. Ide, and C. -C. A. Lai, Transition to aperiodic variability in a wind-driven double-gyre circulation model. J. Phys. Oceanogr., 31, 1260–1286 (2001). 9. H. A. Dijkstra, Nonlinear Physical Oceanography: A Dynamical Systems Approach to the Large Scale Ocean Circulation and El Ni˜ no, second edition. (Springer, Dordrecht, the Netherlands, 2005) 10. H. A. Dijkstra and W. P. M. De Ruijter, Finite amplitude stability of the wind-driven ocean circulation. Geophys. Astrophys. Fluid Dyn., 83, 1–31 (1996). 11. H. A. Dijkstra and C. A. Katsman, Temporal variability of the wind-driven quasi-geostrophic double gyre ocean circulation: Basic bifurcation diagrams. Geophys. Astrophys. Fluid Dyn., 85, 195–232 (1997). 12. C. Frankignoul and K. Hasselmann, Stochastic climate models. II: Application to sea-surface temperature anomalies and thermocline variability. Tellus, 29, 289–305 (1997). 13. R. J. Greatbatch and B. Nadiga, Four-gyre circulation in a barotropic model with double-gyre wind forcing. J. Phys. Oceanogr., 30, 1461–1471 (2000). 14. G. R. Ierley and V. A. Sheremet, Multiple solutions and advection-dominated ﬂows in the wind-driven circulation. I: Slip. J. Mar. Res., 53, 703–737 (1995). 15. D. Lee and P. Cornillon, Temporal variation of meandering intensity and domain-wide lateral oscillations of the Gulf Stream. J. Geophys. Res., 100, 13,603–13,613 (1995). 16. J. D. McCalpin and D. B. Haidvogel, Phenomenology of the low-frequency variabiliity in a reduced gravity quasi-geostrophic double-gyre model. J. Phys. Oceanogr., 26, 739–752 (1996).

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17. S. P. Meacham, Low frequency variability of the wind-driven circulation. J. Phys. Oceanogr., 30, 269–293 (2000). 18. S. P. Meacham and P. S. Berloﬀ, Instabilities of a steady, barotropic, winddriven circulation. J. Mar. Res., 55, 885–913 (1997). 19. S. P. Meacham and P. S. Berloﬀ, Barotropic, wind-driven circulation in a small basin. J. Mar. Res., 55, 523–563 (1998). 20. W. Munk, On the wind-driven ocean circulation. J. Meteorol., 7, 79–93 (1950). 21. B. T. Nadiga and B. Luce, Global bifurcation of Shil˜ nikov type in a doublegyre model. J. Phys. Oceanogr., 31, 2669–2690 (2001). 22. J. Nauw, H. A. Dijkstra and E. Simonnet, Regimes of low-frequency variability in a three-layer quasi-geostrophic model. J. Mar. Res., 62, 684–719 (2004). 23. J. Pedlosky, Geophysical Fluid Dynamics. 2nd Edn. (Springer-Verlag, New York, 1987). 24. G. Plaut, M. Ghil and R. Vautard, Interannual and interdecadal variability in 335 years of Central England Temperature. Science, 268, 710–713 (1995). 25. F. W. Primeau, Multiple Equilibria and Low-Frequency Variability of WindDriven Ocean Models. Ph.D. thesis, M.I.T. and Woods Hole, Boston, MA, U.S.A. 26. D. Ruelle and F. Takens, On the nature o turbulence. Comm. Math. Phys., 20, 167–192 (1970). 27. R. Salmon, Lectures on Geophysical Fluid Dynamics. (Oxford Univ. Press, 1998). 28. F. Schott and R. L. Molinari, The western boundary circulation of the subtropical warm watersphere. In W. Krauss, editor, The Warmwatersphere of the North Atlantic Ocean, pages 229–252. Borntraeger, Berlin-Stuttgart, Germany. 29. V. A. Sheremet, G. R. Ierley and V. M. Kamenkovich, Eigenanalysis of the two-dimensional wind-driven ocean circulation problem. J. Mar. Res., 55, 57–92 (1997). 30. L. P. Shilnikov, A case of the existence of a denumerable set of periodic motions. Sov. Math. Dokl., 6, 163–166 (1965). 31. A. Siegel, J. B. Weiss, J. Toomre, J. C. McWilliams, P. Berloﬀ and I. Yavneh, Eddies and coherent vortices in ocean basin dynamics. Geophys. Res. Letters, 28, 3183–3186 (2001). 32. E. Simonnet, Quantization of the low-frequency variability of the double-gyre circulation. J. Phys. Oceanogr., 35, 2268–2290 (2005). 33. E. Simonnet and H. A. Dijkstra, Spontaneous generation of low-frequency modes of variability in the wind-driven ocean circulation. J. Phys. Oceanogr., 32, 1747–1762 (2002). 34. E. Simonnet, M. Ghil and H. A. Dijkstra, Homoclinic bifurcations of barotropic qg double-gyre ﬂows. J. Mar. Res., 63, 931–956 (2005). 35. H. Stommel, The westward intensiﬁcation of wind-driven ocean currents. Trans. Amer. Geophysical Union, 29, 202–206 (1948). 36. H. U. Sverdrup, Wind-driven currents in a baroclinic ocean with application

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to the equatorial current in the eastern Paciﬁc. Proc. Natl. Acad. Sci. Wash., 33, 318–326 (1947). 37. G. Veronis, An analysis of the wind-driven ocean circulation with a limited number of Fourier components. J. Atmos. Sci., 20, 577–593 (1963). 38. S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos. (Springer-Verlag, Heidelberg-Berlin, Germany, 1990). 39. WOCE (2001). Ocean Circulation and Climate: Observing and Modeling the Global Ocean Ocean [Siedler, G. and Church, J. and Gould, J. (eds)]. (Academic Press, San Diego, USA, 2001).

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NONLINEAR RESONANCE AND CHAOS IN AN OCEAN MODEL ANDREW KISS School of Physical, Environmental and Mathematical Sciences, University of New South Wales at ADFA, Australia [email protected] The response of an idealised barotropic midlatitude ocean gyre to periodic wind forcing is investigated. The model displays an intrinsic periodic oscillation (periodic eddy-shedding from the western boundary current jet) under steady wind forcing, and the interaction of this oscillation with variable forcing can lead to eddy-shedding which is quasiperiodic or locked onto a rational multiple of the forcing period (nonlinear resonance). It can also lead to chaotic partially locked states with variability on timescales exceeding either the natural or forcing periods. The resonant states are arranged in interleaved regimes reminiscent of the “Arnol’d tongues” found in simple forced nonlinear oscillator models.

1. Introduction Subtropical western boundary currents (WBCs), such as the Gulf Stream and East Australian Current, are major ocean currents which carry warm water poleward along the western side of each ocean basin to compensate for a wind-driven equatorward drift in the rest of the basin. Their heat transport is an important part of the global climate system, and ﬂuctuations in these currents have been implicated in climate variability. Previous studies1–6 have investigated the dynamical origins of this variability in idealised numerical models under steady wind forcing. These have revealed the bifurcation structures by which the ﬂow evolves from a steady state to periodic and then quasiperiodic and chaotic states as the WBC becomes more nonlinear (i.e. as the wind forcing is increased and/or the friction reduced). Depending on the model, this intrinsic ocean variability can occur on seasonal to decadal timescales. In reality, mid-latitude gyres are driven by a surface wind stress curl with variability on timescales ranging from days to decades which includes pronounced annual and

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interannual cycles.7 The overlap between these timescales and those intrinsic to the ocean raises the question of how such intrinsic variability might be aﬀected by variations in the wind forcing. This paper reports a ﬁrst step towards addressing this question, using an idealised gyre model in which the intrinsic variability is periodic with period Tn , and studying how this time-dependence is altered by sinusoidal temporal variability (with period Tw ) in the spatially uniform wind forcing. We can regard this system as a periodically-forced nonlinear oscillator. There is an extensive literature on the behaviour of such systems,8,9 much of it focusing on the the dynamics of the “circle map” model. In the circle map, the long-term behaviour depends in a complicated way on the nonlinearity of the oscillator and the ratio of the intrinsic and forcing periods. When the oscillator is linear the response will be periodic only if the frequency ratio is rational; otherwise it will be quasiperiodic. Since the rationals form a set of measure zero, the probability of a periodic response is nil. When the oscillator is nonlinear the probability of a periodic response becomes nonzero, because the oscillator can change its period (by changing its amplitude) in response to the periodic forcing. This allows “nonlinear resonance”, by which the oscillator shifts its period slightly and locks onto a rational multiple of the forcing period. Thus with small nonlinearity there is a narrow range about each rational frequency ratio for which the response is periodic. As nonlinearity increases, these bands of locked periodic behaviour (called “Arnol’d tongues”) occupy increasingly wide frequency ranges. Under sufﬁciently nonlinear conditions the tongues overlap, allowing hysteresis and chaos in addition to locking. We anticipate that a periodically-forced ocean model would exhibit qualitatively similar behaviour as a function of Tf /Tn and the amplitude A of the forcing variation, as suggested by Dijkstra.10 However this situation is more complex than the circle map, since the ocean model is actually very high-dimensional and its low-order behaviour is due to strong damping of most degrees of freedom. The circle map is limited to changing the amplitude and period of its single oscillator, whereas periodic forcing could conceivably excite otherwise decaying modes in the ocean model, in addition to modifying the natural oscillation. Nevertheless, circle-map dynamics have been observed experimentally in Rayleigh-B´enard convection11 and have been implicated in the irregularity of ENSO.12,13 This paper investigates the degree to which simple circle map ideas provide understanding of the response of an unstable western boundary current jet to periodic forcing. To the author’s knowledge this is the ﬁrst attempt to

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identify circle-map dynamics in this system. Earlier studies14,15 addressed the gyre response to periodic forcing with zero mean, and therefore no interaction with a western boundary current. Veronis16 investigated the gyre response to wind with both steady and periodic components, but with a stable western boundary current. The model and its numerical implementation are described in the next section, followed by results and discussion of the ﬂow response under steady and periodic forcing, and conclusions.

2. The Model 2.1. Model geometry We use a numerical model of the “sliced cylinder” laboratory apparatus used by Griﬃths and Kiss1 (see ﬁgure 1), which simulates the basic dynamics governing wind-driven ocean gyres. The apparatus consists of a cylindrical tank of radius a = 0.49m and central depth H = 0.125m, with a ﬂat bottom of slope s = 0.1 relative to the horizontal. The tank rotates about a vertical axis at the rate Ω > 0 (anticlockwise). A ﬂat, horizontal lid is in contact with the water of uniform density ρ which ﬁlls the tank. The lid rotates at a slightly slower rate (1 + (t))Ω to drive the ﬂow with a timedependent but spatially uniform imposed surface vorticity 2Ω(t) < 0 relative to the rotating frame, producing variable downward Ekman pumping which simulates the ﬂuctuating anticyclonic wind stress curl which drives subtropical gyres. We decompose into two parts: (t) = ¯ + (t), where ¯ is the steady component of the diﬀerential lid rotation, and (t) is the time-dependent part (with zero mean). The potential vorticity is higher at the shallow end than the deep end, so these can be regarded as the poleward and equatorward ends of the basin. The tank rotates in the northern hemisphere sense, and we therefore use the compass points shown in ﬁgure 1 to label regions of the boundary. This model is essentially the same as those used by previous authors,17–19 except that it is much wider relative to its depth and therefore produces a narrower and more nonlinear western boundary current (WBC). We focus on the ﬂow in the bulk of the ﬂuid, outside the thin Ekman layers which form against the top and bottom boundaries. Horizontal ﬂow in this region is depth-independent (columnar) and almost nondivergent in the horizontal,20 in accordance with the “strong” Taylor-Proudman theorem. The horizontal ﬂow is driven by vortex compression due to downward Ekman pumping from the top Ekman layer, and retarded by viscosity

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Fig. 1.

The “sliced cylinder” model of a mid-latitude gyre.

acting on lateral shear and by Ekman pumping via both top and bottom Ekman layers in regions of vortical ﬂow. The ﬂow in most of the basin is essentially irrotational, in a topographic Sverdrup balance21 whereby ﬂuid columns move to deeper water (southward) in order to accommodate the addition of water from the top Ekman layer without horizontal divergence. This balance breaks down near the western boundary, where strong vorticity allows a northward return ﬂow via lateral viscous torque (a Munk balance22 ), suction of water into both top and bottom Ekman layers (a topographic Stommel balance23 ), and advection of potential vorticity (an inertial boundary layer balance24 ). 2.2. Governing equations We calculate the horizontal ﬂow between the Ekman layers using the modiﬁed quasigeostrophic vorticity equation derived by Kiss.20 If we scale all lengths by the central depth H, and scale velocity by |¯ Ω|H and time by |¯ Ω|−1 , this can be written 1 1 E2 ∂Q E2 + J(ψ, Q) = ζ + E∇2 ζ, (1) − ∂t D |¯ | D where ψ is the streamfunction for the horizontal velocity, ζ = ∇2 ψ is the relative vorticity, Q = Ro ζ − 2 ln D is the potential vorticity,∗ D = 1 − sy is the scaled ﬂuid depth, y is the scaled northward displacement from the ∂a ∂b ∂a ∂b tank’s centre, and J(a, b) ≡ ∂x ∂y − ∂y ∂x is the Jacobian. The Rossby ∗ This

form of the potential vorticity includes a partial correction for the ﬁnite depth variation, and reduces to the quasigeostrophic form Ro ζ + 2sy in the limit |sy| 1.

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153 ν and Ekman numbers are Ro = |¯ | and E = ΩH 2 , respectively, where ν is the kinematic viscosity. |¯ | = −1 + |¯ | is half the dimensionless relative vorticity of the lid, with a steady component and a periodic perturbation |¯ | = −A sin(fw t). Equation (1) states that material changes in potential vorticity are due to imbalance between Ekman pumping (due to the lid vorticity and relative vorticity) and lateral viscous torque (an O(s2 ) correction to the bottom Ekman layer due to the slope25 has been neglected). With this scaling, the duration of one “day” (i.e. one tank rotation period) is (2πRo)−1 . The boundary conditions at dimensionless radius r = a/H are ψ = 0 (no through-ﬂow) and ∂ψ ∂r = 0 (no slip).

2.3. Numerical scheme The numerical model is based on a code developed by Page,26,27 which was extensively modiﬁed to improve its conservation properties,28 incorporate Sheremet’s method29 for obtaining unstable steady states, and solve for the evolution of linear and nonlinear perturbations to steady states. The code is based on the algorithm of Beardsley30 and uses conservative second-order ﬁnite diﬀerences on a polar grid, with a ﬂux-conserving integral treatment at the origin. The potential vorticity equation (1) was time-stepped using the alternating-direction implicit method, and the Poisson equation ζ = ∇2 ψ was solved for ψ in each timestep via a fast Fourier transform in θ. An in-timestep iteration of these two operations ensures convergence of the Jacobian linking ψ and ζ. There were 512 azimuthal and 161 radial nodes (equally-spaced), suﬃcient to resolve the western boundary current scales δS , δM and δI given in section 3. Runs were conducted with three diﬀerent time steps, either ∆t = 5.918e − 3 = 1.570e − 2 “days”†, or twice or four times this long. The forcing period Tw = 2π/fw was chosen to be an integer number of time steps, to allow the model to lock exactly onto this period when possible. Under steady forcing the ﬂow spins up to a steady state at low Ro,1 but with the parameters used here this steady state is unstable and the ﬂow displays periodic eddy shedding from the separated WBC (see section 3.1). The unstable steady state was calculated by applying Sheremet’s method,29 in which an artiﬁcial term is added to equation (1) which damps Q towards its exponentially-weighted running time-average. If the damping and averaging timescales are correctly chosen, perturbations to the steady state † At

Ro = 0.06, E = 6.273e − 5.

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will decay and under forward timestepping the model will (unphysically) converge to the unstable steady state of the unmodiﬁed equations (since, by construction, the artiﬁcial damping term vanishes in the steady state). The method is simple to implement and has far smaller storage demands than solving for the steady state directly via Newton’s method. ¯ ζ, ¯ with = 0) we can ¯ ψ, Having obtained a steady state solution (Q, ¯ ¯ ¯ substitute Q = Q + Q , ψ = ψ + ψ , ζ = ζ + ζ into equation (1) to obtain the perturbation evolution equation 1 2 ∂Q ¯ Q + J ψ , Q ¯ + J(ψ , Q ) = E + J ψ, − ζ + E∇2 ζ , (2) ∂t D |¯ | where ζ = ∇2 ψ . The numerical model has a perturbation mode which calculates the evolution of Q via equation (2), or by its linearisation about the steady state (obtained by neglecting the J(ψ , Q ) term on the left-hand side). 3. Results and Discussion We ﬁrst a survey of the ﬂow behaviour as a function of Ro and E under steady forcing, then focus on a particular periodic eddy-shedding state and investigate its dynamics under steady forcing before exploring how its timedependence can be altered by forcing ﬂuctuations of diﬀerent amplitudes and frequencies. 3.1. The basic state under steady forcing (A = 0) Kiss28,31 and Griﬃths and Kiss1 describe the ﬂow regimes in this model under steady forcing (see ﬁgure 2). At very low Ro the western boundary current has a north-south symmetry, but as Ro increases the WBC becomes asymmetric, with a faster and narrower outﬂow which separates from the coast to form a jet (via the mechanism described by Kiss31 ) above the lowest dashed line in the ﬁgure. If Ro is further increased beyond a threshold Roc (the lower solid line in ﬁgure 2) the jet becomes unstable and the asymptotic time-dependence is periodic, with cyclonic eddies shed from the end of the WBC jet. At yet larger Ro there is a transition to aperiodic eddy shedding under steady forcing. The experiments with variable forcing used ﬁxed parameters Ro = 0.06, E = 6.273e − 5, s = 0.1 and H/a = 0.2551, putting the ﬂow in the middle of the periodic regime. With these parameters the western boundary current occupies a small fraction of the basin width, as in the oceans: character E 1/3 H = istic WBC scales31 relative to the basin diameter are δM = 2a 2s

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Fig. 2. Flow regimes identiﬁed in the sliced cylinder numerical model as a function of Rossby number Ro and Ekman number E. All runs have s = 0.1 and H/a = 0.2551. Transitions from stable (•) to unstable ﬂow with periodic eddy-shedding (◦) and then to aperiodic () eddy shedding can be seen; the colours indicate ﬁner distinctions within these regimes. The grey lines show for comparison the transitions found in the laboratory by Griﬃths1 for both s = 0.1 and s = 0.15. Figure from Kiss.28

√

8.67e − 3 for a Munk22 balance, δS = H2asE = 1.01e − 2 for a topographic H E 1/4 Ro1/2 = 2.78e − 2 for an inertial Stommel23 balance, and δI = 2as 24 boundary layer. Unfortunately, this choice of s and H/a also produces a large relative depth variation, yielding an ambient Q gradient β = 2s/D which varies between 72% and 164% of its central value from the southern to the northern end of the basin (this variation is opposite to the variation of β on a sphere). Figure 3 shows a sequence ψ and ψ contour plots over one period for this basic state under steady forcing. The clockwise circulation is very slow in the interior, but much faster in the narrow western boundary current which has separated to form a jet which sheds cyclonic eddies at the “natural” period Tn . The eddy shedding cycle involves a pulsation in the strength location of the jet. Each eddy originates as a growing negative perturbation on the eastern side of the jet (frames (a) and (b)), i.e. the jet is becoming longer and forming a larger loop at its end. This loop pinches oﬀ as a cyclonic eddy in frame (c), which propagates southwestward and rapidly decays in

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frames (d) and (a). Weak Rossby waves are also generated, as can be seen by the westward propagation of roughly north-south aligned wavefronts (the ψ = 0 contour) in the rest of the basin. These waves are refracted in the WBC inﬂow region due advection and the altered potential vorticity gradient. There was a very weak dependence of the eddy-shedding period on the time step: Tn was 6.3919 (16.954 “days”; 1080 time steps) with time step ∆t = 5.918e − 3 = 1.570e − 2 “days”, 6.4037 (16.986 “days”; 541 time steps) with a time step of 2∆t, and 6.4866 (17.206 “days”; 274 time steps) with a time step of 4∆t. Tn therefore appears to have converged to within 0.2% for the runs with the shortest time step. The oscillation period is signiﬁcantly shorter than the shortest Rossby basin mode period17 of 37.8 “days”, so the basin mode resonance described by Meacham and Berloﬀ 32 does not occur. The eddy shedding periods at various E collapse onto almost the same power-law curve when plotted against Ro E −1/3 (ﬁgure 4), indicating that the period is controlled by the dimensional velocity of the western boundary current divided by the width of the cyclonic inner part of the jet (which scales as δM ). Thus the period is controlled by the dimensional vorticity of the cyclonic part of the jet, i.e. the time taken for a ﬂuid parcel to travel around the cyclonic loop at the end of the WBC jet. Although the ﬂow spins up to a steady state for Ro < Roc , it does so via decaying periodic oscillations of the WBC jet when close to this threshold. The period of these oscillations decreases continuously as Ro increases across the stability threshold, whilst the asymptotic amplitude increases smoothly from zero past the threshold (see ﬁgure 5). This indicates that the stability boundary in ﬁgure 2 is a supercritical Hopf bifurcation, i.e. that the steady solution exists beyond the boundary but has become linearly unstable to an oscillatory mode. Figure 6 shows the linear growth of this unstable oscillatory mode, whose ﬁnite-amplitude form is the eddy-shedding shown in ﬁgure 3. The stable steady state ψ¯ (red streamlines) was calculated via Sheremet’s method,29 which was time-stepped until the time-dependence of the stabilised ﬂow had been reduced close to the double-precision numerical noise level. The most unstable eigenmode was then found by the “power method”: the linearised version of equation (2) was evolved from an arbitrary initial condition until the perturbation kinetic energy had grown exponentially over about 28 orders of magnitude, by which time it had become completely dominated by the most unstable eigenmode (shown by the black contours). Comparing ﬁgures 6 and 3, it is clear that the spatial structure and frequency of the eddy-shedding instability originate from

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(a)

(b)

(c)

(d)

Fig. 3. Periodic eddy-shedding under steady forcing, with Ro = 0.06, E = 6.273e − 5, s = 0.1 and H/a = 0.2551. The frames show (a) t = Tn /4, (b) t = Tn /2, (c) t = 3Tn /4 and (d) t = Tn where Tn = 6.3919 = 16.954 “days” represents one period. Red streamlines indicate the total streamfunction ψ = ψ¯ + ψ , and black streamlines indicate ¯ negative values are dashed. the perturbation ψ relative to the unstable steady state ψ;

those of the eigenmode, although the frequency of the saturated mode is slightly lower and the symmetry between positive and negative perturbation structures has been lost. 3.2. Response to variable forcing A striking feature of the laboratory results was locking of the eddy-shedding to the lid rotation period (see ﬁgure 4) despite every eﬀort being made to ensure that the lid was ﬂat and rotating at a constant speed.1 This

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Fig. 4. Oscillation periods in the sliced cylinder in the periodic, unstable regime, in “days” (tank rotation periods), plotted on log-log axes as a function of Ro E −1/3 . Except where indicated, s = 0.1 for all the numerical results, and s = 0.15 for all the laboratory results; all data shown are for anticyclonic forcing and H/a = 0.2551. The laboratory results are for E = 3.15e − 5, E = 6.29e − 5, and E = 12.6e − 5. The lines through the numerical data points are power-law regressions. The lines through the laboratory data points are not regressions, but Ro−1 , the lid period in “days”, for the three Ekman numbers investigated. Figure from Kiss.28

tendency of the ﬂow to lock onto small forcing ﬂuctuations was investigated in more detail by a suite of numerical experiments which surveyed the two-dimensional parameter space deﬁned by A and Tw /Tn , with all other parameters ﬁxed at Ro = 0.06, E = 6.273e − 5, s = 0.1 and H/a = 0.2551. An overview of these results is shown in ﬁgure 7. The main diagnostic used to determine the time-dependence of each ﬂow ** ψζdydx. Flows locked was the basin-integrated kinetic energy K = − 21 to a rational multiple of the forcing period (Nw time steps) were identiﬁed by calculating 1 Nw

(n+1)Nw 2

(Kj+mNw − Kj )

(3)

j=nNw +1

as a function of the forcing cycle n and lag mNw , where Kj denotes K at time step j . The ﬂow was considered to be phase-locked to the forcing if there was an integer m for which this mean squared diﬀerence was exponentially decreasing as a function of n (for large n). In this case the

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Fig. 5. Change in the oscillation amplitude and period as a function of Ro across the instability threshold in the sliced cylinder, with anticyclonic forcing, E = 3.14e − 5, s = 0.1 and H/a = 0.2551. The amplitude shown is the peak-to-peak amplitude of the ** ψζdydx, and the period is oscillation in the basin-integrated kinetic energy K = − 12 in “days” (tank rotation periods). For the stable runs the period shown is that of the exponentially decaying oscillation. The lines are interpolations to assist the eye. Figure from Kiss.28

smallest m for which this was true indicated the number of forcing periods required to see a repeat of the ﬂow (these m values are shown on the data points in ﬁgure 7). The m values found by this method can be regarded as the denominators of rational winding numbers. In cases where no such m could be found the ﬂow was either quasiperiodic or chaotic. The distinction between these two cases was made by inspection of Fourier spectra, three-dimensional delay-space trajectories and Poincar´e sections,33 and recurrence plots,34 all based on K(t). Most aperiodic ﬂows could be unambiguously identiﬁed as either quasiperiodic or chaotic (indicated by Q or C, respectively, in ﬁgure 7), but there were a few cases in which the time series were insuﬃciently long to decide (indicated by a question mark in the ﬁgure). Calculation of Lyapunov exponents was not attempted, as most time series were insuﬃciently long. The ﬂow adjusts to wind variations via Rossby waves which propagate away from the eastern boundary and force ﬂuctuations in the strength of the boundary current when they arrive at the west. This in turn drives

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(a)

(b)

(c)

(d)

Fig. 6. Unstable steady state ψ¯ (red contours) and linearised growth of its most unstable eigenmode ψ (black contours; negative values are dashed) under steady forcing, with the same parameters as in ﬁgure 3. The frames show perturbation growth over one eigenperiod: (a) t = T /4, (b) t = T /2, (c) t = 3T /4 and (d) t = T where T = 6.2972 = 16.704 “days”. Nonlinear saturation of this eigenmode produces the periodic ﬂow shown in ﬁgure 3.

periodic variations in the strength of the jet, which modiﬁes the eddyshedding process. It is clear from ﬁgure 7 that there are extensive regions in this parameter space in which the eddy-shedding has locked onto a rational multiple of the forcing period (locked jet instabilities were also observed in laboratory experiments with periodic wind forcing). The largest region has a 1:1 frequency ratio, but there also appear to be connected regions with m values of 2, 3 and 4. At small A the locked regions are aligned with rational Tw /Tn values with denominator m. The locked regions tend to

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Fig. 7. Time-dependence of the ﬂow as a function of the amplitude A and period Tw = 2π/fw of the perturbation in the wind forcing (relative to the natural period Tn ), with all other parameters the same as in ﬁgure 3. The top plot shows the whole parameter space investigated; the coloured rectangles are shown in more detail in the lower plots. Circles indicate runs with time step ∆t = 5.918e − 3 = 1.570e − 2 “days”, and squares indicate runs with twice this timestep. Periodic ﬂows are labelled by numbers indicating how many forcing periods were needed for the ﬂow to repeat. Aperiodic ﬂows are labelled Q for quasiperiodic, and C for chaotic. Aperiodic runs whose character could not be determined due to insuﬃcient time series length are indicated with a question mark.

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Fig. 8. Time series for quasiperiodic ﬂow with Tw /Tn = 0.30, A and all other **= 0.0333 ψ ζ dydx (black) and parameters the same as in ﬁgure 3. (a): time series of K = − 12

− 1 (blue); (b): projection of a delay-space reconstruction of the the wind forcing || attractor (K (t) vs. K (t − tdelay )); (c): power spectrum of K , based on a much longer time series than that shown in panel (a) (the dashed line indicates the natural frequency fn = 2π/Tn ). From Kiss.35

widen with increasing A, as increasingly nonlinear response allows locking at forcing periods which are less closely matched to rational multiples of Tn . At very small A the eddy shedding period typically does not lock onto the forcing and quasiperiodic responses prevail (there may still be signiﬁcant modiﬁcations to the time-dependence in these cases, however). A few regions displaying chaotic behaviour are also evident. The overall structure of these regimes closely resembles the interleaved “Arnol’d tongues” found in the circle map model of a forced nonlinear oscillator. The threshold A at which the tongues overlap appears to decrease with increasing Tw /Tn . Although the possibility was not investigated, it is likely that hysteresis is present where the tongues appear to abut one another (e.g. the m = 1 and m = 2 regimes near Tw /Tn = 1.7 and A = 0.05), so that the boundary between them could be diﬀerent if other initial conditions had been used. A typical quasiperiodic time series is shown in ﬁgure 8 (a), together with the forcing. In this case the spectrum (panel (c)) consists of sharp peaks at the forcing frequency (solid line) and natural frequency fn = 2π/Tn (dashed line) together with their cross-harmonics. The two incommensurate frequencies result in a delay-space trajectory which winds densely over the surface of a torus (panel (b)). The Hovmoller plots in ﬁgure 9 show the Rossby waves associated with this state. The middle panel displays ψ (x, t),

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** Fig. 9. Quasiperiodic ﬂow with parameters as in ﬁgure 8. Top: K = − 12 ψ ζ dydx (blue) and the wind forcing W = (green) versus time step (∆t = 5.918e − 3 = 1.570e − 2 “days”); middle: perturbation streamfunction ψ across the centre of the basin versus east-west position x and time (yellow/red: positive, cyan/blue: negative); bottom: as for the middle ﬁgure, but with the average cycle of ψ at the forcing period Tw subtracted out to show the component of the response which is not at the forcing frequency.

which is dominated by waves at the forcing frequency propagating from the eastern boundary. The last panel in ﬁgure 9 shows the diﬀerence between the perturbation ψ and its average cycle over a forcing period, which reveals the shorter Rossby waves associated with the jet instability at the lower frequency fn (these are strongest in the western half of the basin, since they propagate westward from the jet). In this case the jet behaviour is essentially unaﬀected by the forcing ﬂuctuations. Figure 10 shows a ﬂow locked to twice the forcing period, despite its mismatch from half the natural period (Tw = 0.519Tn). The eddy-shedding period 2Tw is signiﬁcant only near the jet; in the rest of the basin the perturbations are dominated by the forcing period Tw . The latter perturbations are strongest in the western half of the basin, and are phase-locked to weak Rossby waves arriving from the eastern boundary. The western half can be

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(a) t = Tw /4

(b) t = Tw /2

(c) t = 3Tw /4

(d) t = Tw

(e) t = 5Tw /4

(f) t = 3Tw /2

(g) t = 7Tw /4

(h) t = 2Tw

Fig. 10. Periodic eddy-shedding locked to double the forcing period Tw /Tn = 0.519, with A = 0.1 and other parameters the same as in ﬁgure 3. The frames show one ﬂow period (two forcing periods). Red streamlines indicate the total streamfunction ψ = ψ¯ + ψ , and black streamlines indicate the perturbation ψ relative to the unstable ¯ negative values are dashed. steady state ψ;

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interpreted as a resonator in which perturbations from the jet feed back onto the WBC via Rossby waves. The phase relationship between the “western resonator” and the waves arriving from the east is shown by the Hovmoller plot of ψ (x, t) in ﬁgure 11 (middle panel). The last panel in ﬁgure 11 shows the shorter Rossby waves associated with the eddy-shedding period 2Tw .

Fig. 11. As for ﬁgure 9 but for ﬂow locked to twice the forcing period, with the same parameters as in ﬁgure 10.

Figure 12 shows the time-dependence of a ﬂow in the chaotic region shown in the lower-left panel of ﬁgure 7. In this case Tw /Tn is close to 4/5, and the “western resonator” couples strongly to the wind perturbation but is unable to lock (i.e. the ﬂow is inﬂuenced by an unstable periodic orbit with Tw /Tn = 4/5). This eﬀect produces broad spectral peaks at multiples of fw /5, with signiﬁcant power at timescales far longer than either Tw or Tn . The delay-space trajectory appears to lie on a strange attractor in this case.

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Fig. 12. Time series for chaotic ﬂow with Tw /Tn = 0.777, A = 0.0167 and all other parameters the same as in ﬁgure 3. Panels as in ﬁgure 8. From Kiss.35

4. Conclusions The behaviour of a simple gyre model under steady and time-dependent wind forcing was investigated. Under steady forcing the model exhibits an intrinsic periodicity in the western boundary current jet which was shown to originate from a supercritical Hopf bifurcation. The oscillatory jet instability gives rise to periodic eddy-shedding from the jet at ﬁnite amplitude. Under periodic forcing, Rossby waves carry the Sverdrup adjustment westward across the basin, producing periodic variations in the strength of the western boundary current and the jet, and thereby aﬀecting the eddy-shedding process. A survey of the ﬂow response to wind perturbations of diﬀerent frequencies and amplitudes shows that if the variation is suﬃciently large and has a period close to a rational multiple of the natural eddy-shedding period, it can drive the jet instability, resulting in eddy shedding which is locked onto a multiple of the forcing period. Under other conditions the eddy-shedding period may remain independent of the forcing variation, producing a quasiperiodic time-dependence. Chaotic states were also observed; these may be of particular geophysical interest, as they provide a mechanism for variability at timescales much longer than those of the forcing or intrinsic ocean variability. The locked regimes are interleaved in a similar way to the “Arnol’d tongues” we would predict if we regard this system as a forced nonlinear oscillator, with an intrinsic period due to the jet instability (driven by the time-mean wind), and an independent forcing period driven by the wind ﬂuctuations.

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Acknowledgements I thank Claire M´enesguen (ENS, Paris) for assisting with the calculations. This research was partly supported by the Australian Research Council (PDF F00104281). References 1. R. W. Griﬃths and A. E. Kiss, Flow regimes in a wide ‘sliced-cylinder’ model of homogeneous β-plane circulation, J. Fluid Mech. 399, 205–236, (1999). 2. S. Speich, H. A. Dijkstra, and M. Ghil, Successive bifurcations in a shallowwater model applied to the wind-driven ocean circulation, Nonlin. Process. Geophys. 2, 241–268, (1995). 3. S. Jiang, F.-F. Jin, and M. Ghil, Multiple equilibria, periodic, and aperiodic solutions in a wind-driven, double-gyre, shallow-water model, J. Phys. Oceanogr. 25, 764–786, (1995). 4. H. A. Dijkstra and C. A. Katsman, Temporal variability of the wind-driven quasi-geostrophic double gyre ocean circulation: Basic bifurcation diagrams, Geophys. Astrophys. Fluid Dyn. 85, 195–232, (1997). 5. F. W. Primeau, Multiple equilibria of a double-gyre ocean model with superslip boundary conditions, J. Phys. Oceanogr. 28, 2130–2147, (1998). 6. E. Simonnet, M. Ghil, and H. Dijkstra, Homoclinic bifurcations in the quasigeostrophic double-gyre circulation, J. Marine Res. 63, 931–956, (2005). 7. K. E. Trenberth, W. G. Large, and J. G. Olson, The mean annual cycle in global ocean wind stress, J. Phys. Oceanogr. 20, 1742–1760, (1990). 8. J. A. Glazier and A. Libchaber, Quasi-periodicity and dynamical systems: an experimentalist’s view, IEEE Trans. Circuits and Systems. 35, 790–809, (1988). 9. P. Bak, T. Bohr, and M. H. Jensen, Mode-locking and the transition to chaos in dissipative systems, Physica Scripta. T9, 50–58, (1985). 10. H. A. Dijkstra, Nonlinear physical oceanography: A dynamical systems approach to the large scale ocean circulation and El Ni˜ no. Atmospheric and oceanographic sciences library ; v. 22., (Kluwer Academic Publishers, Dordrecht, 2000). 11. M. H. Jensen, L. P. Kadanoﬀ, A. Libchaber, I. Procaccia, and J. Stavans, Global universality at the onset of chaos: Results of a forced Rayleigh-B´enard experiment, Phys. Rev. Lett. 55, 2798–2801, (1985). 12. E. Tziperman, L. Stone, M. A. Cane, and H. Jarosh, El Ni˜ no chaos: Overlapping of resonances between the seasonal cycle and the Paciﬁc oceanatmosphere oscillator, Science. 264, 72–74, (1994). 13. E. Tziperman, M. A. Cane, and S. E. Zebiak, Irregularity and locking to the seasonal cycle in an ENSO prediction model as explained by the quasiperiodicity route to chaos, J. Atmos. Sci. 52, 293–306, (1995). 14. J. Pedlosky, A study of the time dependent ocean circulation, J. Atmos. Sci. 22, 267–272, (1965). 15. R. C. Beardsley, The ‘sliced-cylinder’ laboratory model of the wind-driven

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16. 17. 18. 19. 20.

21.

22. 23. 24. 25. 26. 27. 28.

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ocean circulation. Part 2: Oscillatory forcing and Rossby wave resonance, J. Fluid. Mech. 69, 41–64, (1975). G. Veronis, Eﬀect of ﬂuctuating winds on ocean circulation, Deep-Sea Res. 17, 421–434, (1970). J. Pedlosky and H. P. Greenspan, A simple laboratory model for the oceanic circulation, J. Fluid Mech. 27, 291–304, (1967). R. C. Beardsley, A laboratory model of the wind-driven ocean circulation, J. Fluid Mech. 38, 255–271, (1969). A. Becker and M. A. Page, Flow separation and unsteadiness in a rotating sliced cylinder, Geophys. Astrophys. Fluid Dyn. 55, 89–115, (1990). A. E. Kiss, A modiﬁed quasigeostrophic formulation for weakly nonlinear barotropic ﬂow with large-amplitude depth variations, Ocean Modelling. 5, 171–191, (2003). H. Sverdrup, Wind-driven currents in a baroclinic ocean: with application to the equatorial currents of the eastern Paciﬁc, Proc. Natl. Acad. Sci. USA. 33, 318–326, (1947). W. H. Munk, On the wind-driven ocean circulation, J. Meteorology. 7, 79–93, (1950). H. Stommel, The westward intensiﬁcation of wind-driven ocean currents, Trans. Am. Geophys. Union. 29, 202–206, (1948). N. P. Fofonoﬀ, Steady ﬂow in a frictionless homogeneous ocean, J. Mar. Res. 13, 254–262, (1954). J. Pedlosky, Geophysical Fluid Dynamics. (Springer, New York, 1987), 2 edition. M. A. Page, Rotating Fluids at Low Rossby Number. PhD thesis, University College, London, (1981). M. A. Page, A numerical study of detached shear layers in a rotating sliced cylinder, Geophys. Astrophys. Fluid Dyn. 22, 51–69, (1982). A. E. Kiss, Dynamics of laboratory models of the wind-driven ocean circulation. PhD thesis, Australian National University, (2000). Available from http://thesis.anu.edu.au/public/adt-ANU20011018.115707/index.html. V. A. Sheremet, A method of ﬁnding unstable steady solutions by forward time integration: relaxation to a running mean, Ocean Modelling. 5, 77–89, (2002). R. C. Beardsley, A numerical investigation of a laboratory analogy of the wind-driven ocean circulation. In NAS Symposium on Numerical Models of Ocean Circulation, p. 311, Durham, New Hampshire, (1972). National Academy of Sciences. A. E. Kiss, Potential vorticity “crises”, adverse pressure gradients, and western boundary current separation, J. Mar. Res. 60, 779–803, (2002). S. P. Meacham and P. S. Berloﬀ, Instabilities of a steady, barotropic, winddriven circulation, J. Mar. Res. 55, 885–913, (1997). F. Takens. Detecting strange attractors in turbulence. In eds. D. A. Rand and L.-S. Young, Dynamical Systems and Turbulence, vol. 898, Lecture notes in mathematics, pp. 366–381. Springer-Verlag, Berlin, (1981).

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34. J.-P. Eckmann, S. Oliﬀson Kamphorst, and D. Ruelle, Recurrence plots of dynamical systems, Europhysics Letters. 4(9), 973–977, (1987). 35. A. E. Kiss and C. M´enesguen, Response of ocean circulation to variable wind forcing. In 15th Australasian Fluid Mechanics Conference, Sydney, Australia, (2004). University of Sydney.

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LOW FREQUENCY OCEAN VARIABILITY: FEEDBACKS BETWEEN EDDIES AND THE MEAN FLOW ANDREW McC. HOGG Research School of Earth Sciences, The Australian National University, Canberra, ACT, 0200, Australia [email protected] WILLIAM K. DEWAR Florida State University PETER D. KILLWORTH AND JEFFREY R. BLUNDELL National Oceanography Centre, Southampton Many ocean and climate models are incapable of resolving ocean eddies, which are generated on the Rossby radius scale (∼ 50 km) in all ocean basins. However, there is an emerging body of evidence supporting the idea that eddies may have several important eﬀects which cannot be simply parameterised. In particular, eddies may play a role in generating low frequency (decadal) variability in the dynamics of mid-latitude ocean circulation. In this manuscript two speciﬁc examples – double gyre circulation and wind-driven channel ﬂow – are used to analyse the dynamical mechanisms of low frequency variability in the ocean.

1. Introduction Most climate modellers recognise the importance of the ocean in modulating, increasing or damping climate variability and include a dynamical ocean in their climate simulations. However the ocean components of most climate models are necessarily coarse resolution and invariably have, as a consequence, high viscosity, thereby omitting some potentially important aspects of the ocean circulation. Resolution of eddies can be achieved with idealised ocean models that use only a few layers in the vertical. Such models have shown that low frequency variability can spontaneously arise in these systems, although the strong eddy ﬁeld may obscure many features of the circulation, making it

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diﬃcult to agree upon the mechanisms underpinning this variability.1–3 A number of possible mechanisms of low frequency variability have been proposed. In this chapter we investigate the possibility of feedback between eddies and the large scale mean ﬂow, originally proposed by Spall.4 Spall used an eddy resolving simulation to investigate an oscillation in the position of the Gulf Stream and the deep western boundary current (DWBC) with a period of about 10 years. The proposed mechanism involved a feedback between baroclinic instability, the position of the DWBC and the strength of barotropic, eddy-driven inertial recirculating gyres near the Gulf Stream separation point. A similar mechanism was proposed to explain variability ◦ in a 16 resolution model of the North Paciﬁc.5 In this chapter we show results from time-dependent simulations of a large, high-dimensional, low-viscosity ocean. We investigate two diﬀerent cases: a midlatitude double gyre case (resembling the North Paciﬁc or North Atlantic Oceans) and a reentrant channel ocean which mimics the Southern Ocean. In each case we examine the low frequency variability of the system in the context of eddy–mean ﬂow feedback. The model is brieﬂy described in Sec. 2. The double gyre results are shown in Sec. 3, and channel ocean case in Sec. 4. 2. The quasi-geostrophic coupled model (Q-GCM) The model used for this study is the ocean component of the QuasiGeostrophic Coupled Model (Q-GCM). The model is described more fully in a technical note6 and a second paper which provides a justiﬁcation for the model.7 We use the model in its ocean-only mode, with three (constant potential temperature) quasigeostrophic (QG) layers governed by the equation qt =

1 A2 4 A4 6 J(q, p) + Be + ∇ p− ∇ p. f0 f0 H f0 H

(1)

In this equation J(q, p) = py qx − px qy is the Jacobian, and we use vectors of length three to represent ﬁelds in the three ocean layers such as pressure (p) and potential vorticity (q), where ˜ f0 (q − βy) = ∇2H p − f02 Ap + f0 D, ˜ represents the topographic height. Other parameters include in which D the midlatitude Coriolis parameter f0 , the (assumed constant) gradient of the Coriolis parameter with distance northward β, a Laplacian diﬀusion coeﬃcient A2 and a biharmonic diﬀusion coeﬃcient A4 .

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Q-GCM diﬀers from many QG models in that we include an explicit diabatic term due to the Ekman pumping of surface temperature through the entrainment vector e in (1) which includes the eﬀect of a linear bottom drag on layer 3. The matrices A and B include coeﬃcients describing the vortex stretching and forcing respectively. In all experiments shown here, the wind forcing is steady. The wind stress ﬁeld is speciﬁed to represent a time mean realistic forcing for the midlatitudes. For further details see Hogg at al.7 We use a geometrically simple rectangular domain for each of the two cases discussed below, but the large domain size, high horizontal resolution (10 km) and low viscosity (A2 = 0 m2 /s and A4 = 1×1010 m4 /s) conspire to yield a dynamically rich circulation. Baroclinic instability, which is much stronger when three or more layers are used than with only two layers, produces ocean eddies which form on length scales comparable with the larger of the Rossby radii (∼ 50 km in this model), and the low viscosity increases the longevity of the eddies. The simulations therefore have a higher dimensionality than most ocean models.

3. Double gyre circulation 3.1. Mean circulation The double gyre simulations are performed in a rectangular ocean which is 3840 × 4800 km, and a wind stress which is small in the north and south of the domain and maximum (0.1 N/m2 ) near the centre. We use f0 = 1 × 10−4 s−1 , β = 2×10−11 (ms)−1 , partial slip lateral boundary conditions and a value for bottom drag which produces a spindown timescale of 350 days. The wind stress drives a mean circulation which is represented by streamfunctions ψi for each layer in Fig. 1(a–c). The upper layer (or layer 1) transport shows a broad double gyre pattern, which is well represented by a linear solution over large parts of the domain. But the gyres are intensiﬁed to the west, creating a narrow, fast western boundary current which feeds a nonlinear jet dividing the two gyres. Near the separation of the jet from the coast there are intense, nonlinear recirculating gyres. These gyres are driven by excess transport of potential vorticity by the western boundary currents8 and extend to full depth (as shown in panels b and c). The eddy ﬁeld is shown in Fig. 1(d ). Here the western boundary currents and jets are identiﬁable, but the strong eddies mask many characteristics of the circulation. It can be shown9 that these eddies can be damped by

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high viscosity or greater friction on the lateral boundaries – this weakens the nonlinear components of the circulation. 3.2. Variability Simulation of a dynamically rich ocean circulation, containing eddies which exist on a range of spatial scales (see Fig. 1d ), results in high temporal

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variability which occurs on short (monthly) timescales due to the direct eﬀect of eddies and a meandering jet. Of greater interest to climate modelling is the variability of the circulation on timescales longer than a year. To investigate this low frequency variability we record the pressure at 15 day intervals in each layer over 160 model years; these pressure ﬁelds necessarily include the high temporal variability of the eddy ﬁeld which is ﬁltered out using a 2-year low-pass Fourier ﬁlter, applied at every point in space. The data are further reduced by calculating the Hilbert (or complexiﬁed) Empirical Orthogonal Functions (EOFs). Hilbert EOFs are calculated using a complexiﬁed dataset; the imaginary part of this dataset is constructed by taking the Hilbert transform of the real data.10 One then calculates the EOFs in the standard way (eigenvector analysis of the covariance matrix of the data) which decomposes the data set into complex eigenvectors (spatial patterns of the data which maximise variance) and their principal components (PCs). This allows extraction of statistical modes of variability from the ﬁltered data set, and in many cases the bulk of the variance can be attributed to only a few spatial modes. Fig. 2 shows the spatial pattern of the ﬁrst two Hilbert EOFs for the ﬁltered uppermost layer transport streamfunction in the default case. For each mode we obtain two spatial ﬁelds; one of these is the real part of the (a) Hilbert EOF 1 ( 47 %)

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Hilbert EOF, the other is the imaginary part. These two spatial ﬁelds describe the shape of the statistical mode at diﬀerent phases of the oscillation. In Fig. 2(a) the ﬁrst spatial mode (which represents 47% of the total variance in the ﬁltered dataset) is shown. The statistical oscillatory mode consists of two EOF patterns: a real and an imaginary component. The real part of the spatial mode forms a dipole focussed on the inertial recirculations; the imaginary component comprises a tripole pattern in the same region. The dipole phase of this oscillation describes a strengthening or weakening of the inertial recirculations (and hence the ocean jet which forms in the western boundary current extension), while the tripole pattern represents a shifting of the ocean jet, which is out of phase with the amplitude modulation. The second mode shows diﬀerent patterns from the ﬁrst mode, but represent only 9% of the variance respectively and therefore this mode and higher modes are of secondary importance. At any point in time, the data can be reconstructed by multiplying each spatial mode by its own time-dependent complex coeﬃcient (called a principal component) and summing over all modes. The spectra of the principal components (Fig. 2c) therefore give information about the frequency of any periodic behaviour in the modes. At higher frequencies, both modes have similar power, but the ﬁrst mode has a broad, low frequency peak centred on periods of 16 years (indicated by the arrow). Therefore, the ﬁrst EOF mode for simulations with the default parameter set describes a strong interdecadal oscillation which centres on the inertial recirculation region. 3.3. Mechanisms of variability The data presented above are not the ﬁrst simulations to suggest that lowfrequency variability may arise spontaneously in wind-driven ocean circulation. Previous studies have identiﬁed similar trends in a range of diﬀerent models; however, the cause of this variability is unclear. To help in this regard, we ﬁrst refer to Fig. 3, in which two timeseries are plotted (this data is from a simulation with double the bottom drag to that shown above). The jet position and jet velocity are calculated from the position and magnitude (respectively) of the maximum eastward velocity, and ﬁltered with a 6 month low-pass ﬁlter. These timeseries show three distinct phases which form the oscillation, and which are labelled A, B and C (the latter of these shown by the grey bands) in the diagram. These phases are determined from the jet position. In phase A the jet is in its northernmost position (north of 2000 km, and close to the zero wind stress curl line). Phase B describes a gradual

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southward shift, while in phase C the jet jumps northwards once again. Jet velocity is greatest in phase A and smallest in phase C. The phases marked are approximate, but can be used to identify the primary ﬁndings of the Hilbert EOF analysis: that the oscillation involves shifting and modulation of the strength of the jet which are out of phase. The phases shown in Fig. 3 show how the jet builds in strength during regime A, and weakens as it shifts south in regime B (away from the maximum wind stress position). The dynamics of regime C act to push the jet suddenly back to the north. This ﬂow has been analysed for the existence of eddy–mean ﬂow feedbacks, such as changes in mixing of PV by eddies, or variability of the eddy-driven momentum ﬂux into the lowest layers. However, a range of diagnostics applied to this case do not distinguish

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whether or not the mechanism operating in phase C is due to eddy–mean ﬂow feedback.9 There is an alternative explanation for low frequency variability in midlatitude ocean circulation, derived from nonlinear dynamical systems theory. The general approach here is to solve a simple, idealised lowdimensional model at high viscosity for a steady solution. Then, by gradually decreasing viscosity (or increasing forcing strength) one sees the development of other stable states,11 bifurcations leading to periodic behaviour12 quasi-periodic attractors and secondary bifurcations13 and ultimately chaotic behaviour.14,15 Such approaches have the advantage that one is able to pinpoint the processes which lead to, for example, a particular Hopf bifurcation which is responsible for periodic behaviour in the high viscosity model. For example, Simonnet and Dijkstra16 examine the evolution of a particular low frequency mode of instability which they call the gyre mode. The gyre mode is the product of two stationary modes which merge to produce a single oscillatory mode which ultimately has positive growth rate (see their Fig. 3). It is possible that the manifestation of the gyre mode at high Reynolds number is responsible for the behaviour observed in Fig. 3, but this cannot be proved with existing methods. Therefore, there are two plausible mechanisms for the low frequency variability observed in these models. The ﬁrst is a feedback between eddies and the mean ﬂow, while the second can be characterised by lowdimensional mean ﬂow dynamics. Resolution of this question remains an active area of research.

4. Channel ocean circulation 4.1. Mean ﬂow The channel ocean case diﬀers from the previous case in that the ocean is now very long (11520 × 3000 km – approximately half the length of the Southern Ocean at its central latitude) and has periodic boundary conditions allowing ﬂow through the channel. We include a realistic topography (truncated at ±900 m to ensure the validity of the QG assumption) which strongly controls the ﬂow. Other parameters are similar to the double gyre case, except for f0 = −1.2 × 10−4 s−1 and β = 1.3 × 10−11 (ms)−1 . This case is analysed in detail by Hogg & Blundell.17 The mean ﬂow is shown in Fig. 4 for each layer. The upper layer shows a strong circulation through the channel, with a series of strong, nonlinear

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jets (numbering between 1 and 3 jets at any lateral position). These jets are a realistic feature of the Southern Ocean circulation. The lower layer is constrained by topography, with large areas of recirculation conﬁned to topographic basins. The recirculations are also nonlinear, and in some cases extend to the surface and help to steer the upper layer ﬂow. 4.2. Variability The variability of this circulation can be demonstrated by simple energetic diagnostics as shown in Fig. 5(a). Here the potential (solid line) and kinetic (dashed line) energy show clear and regular cycles with a period of approximately 13 years. Peaks in potential energy lead peaks in kinetic energy by

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several years. It can be shown that the bulk of this excess kinetic energy resides in the transient eddies.17 The timeseries has been divided into four regimes, A–D. Regime A is characterised by low energy in both ﬁelds. In regime B, potential energy builds, but kinetic energy is still low. The transition from regime B to C occurs at the peak of potential energy, and produces a rapid increase in kinetic energy. Regime D is deﬁned by falling potential and kinetic energy.

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Diagnosis of the energetics of this system is achieved by determining the transfer of energy between diﬀerent parts of this system. The total kinetic

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energy equation (after McWilliams18 ) can be simpliﬁed to 2 ρg k 2 (ηk )t dA + ρ KE t = − u1 τ x dA 2 k=1 2 ρδek |f0 | u3 + v3 2 dA − 2 3 − A4 ρHk uk ∇4H uk + vk ∇4H vk dA, k=1

where ρ is the mean density, g k the reduced gravity between layers k and k + 1, and (uk , vk ) the velocity components of layer k. The double integrals are taken over the entire domain. The terms on the RHS of this equation are: (1) Transfer between the total potential and kinetic energy ﬁelds, or equivalently, the negative of the time rate of change of potential energy (ηk is the perturbation to the height of the interface dividing layers k and k + 1). This contribution may be either positive or negative, and is plotted with the solid line in Fig. 5(b). (2) Energy input from wind stress forcing, τ x , which is always positive (dashed line). (3) Linear drag in the lowest layer (δek is the thickness of the bottom Ekman layer) which is always a sink of energy (dotted line). (4) Viscous dissipation, the dot-dash line in Fig. 5(b), which is the loss of energy due to the biharmonic viscosity. These energy diagnostics are shown as timeseries in Fig. 5(b). The energy input in this system comes from the wind stress term, which depends upon the local ﬂuid velocity in the upper layer as well as the wind stress. Thus, despite the constant wind stress, the wind energy input varies by about 40% over the cycle. In regimes A and B, when energy input from wind stress is rising (due to an increasing correlation between u1 and τ x as shown below) much of the excess energy is absorbed by transfer into the potential energy ﬁeld (that is, the solid line is negative). These regimes are the low kinetic energy states, so both dissipation and drag are small. In regime C the system switches to a new phase, through massive and sudden transfer of potential to kinetic energy. This surge of energy feeds into the turbulent kinetic energy ﬁeld, presumably caused by ﬂuid instability, so that there is a loss of correlation between u1 and τ x and energy input from the wind decreases. Furthermore, the sink of energy through drag and

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dissipation is increased. In regime D, the store of available potential energy is exhausted, so that conversion of potential to turbulent kinetic energy begins to abate. When this is combined with the reduction in wind stress forcing, the kinetic energy ﬁeld dies away over a period of several years, and the system returns to its low energy state. 4.3. Mechanism of variability The mechanism governing the observed variability can be diagnosed by examining the spatial variability of the ﬂow in diﬀerent regimes. The ﬂow in the regimes A and B is more zonal than in the regimes C and D.17 There is a substantial body of work19 which predicts that parallel nonzonal shear ﬂow is less stable to baroclinic processes than an equivalent zonal ﬂow. We calculate the nonzonal nature of the ﬂow in diﬀerent regimes as a proxy for enhanced baroclinic instability. Nonzonality is deﬁned through a new parameter, |v1 |

, (2) |u1 | where values of ζ close to 1 indicate a nonzonal ﬂow and small values are more zonal. This parameter is plotted in Fig. 5(c) as a function of time, and demonstrates a coherent relationship between the nonzonality and the energetics used to deﬁne the four regimes. The lowest energy state, regime B, shows a minimum in ζ, while regimes C and D, where the production of turbulent kinetic energy is high, have increased nonzonality. The evidence here is consistent with the notion that the extent of nonzonality in the mean ﬂow controls the production of baroclinic eddies. While all regimes are unstable, in regimes C and D the nonzonal nature of the ﬂow is likely to cause enhanced baroclinic instability. This result leads to an hypothesis which explains the low frequency periodic oscillations in the circumpolar ﬂow. We outline this hypothesis by summarising the dynamics of each regime, and the mechanisms associated with the transition between each regime: ζ=

Regime A This is a low energy state. While this ﬂow is unstable, the store of available potential energy is low compared to states C and D. For this reason, production of eddies is relatively weak. Thus, ﬂow is accelerated zonally by the wind stress, and this energy is stored in the mean potential energy ﬁeld. Regime B This regime is characterised by a minimum in the nonzonality parameter, and hence the most stable ﬂows. As a result, turbulent

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kinetic energy is minimised. The wind stress input continues to build a zonal ﬂow and a store of potential energy. Energy input from the wind increases signiﬁcantly during this regime. Regime C In regime C the ﬂow appears to exceed a critical threshold for instability. As a result, the mean ﬂow generates baroclinic eddies which drive the deep circulation. The deep ﬂow is steered by topography, and as it strengthens it deforms the interfaces above, thereby permitting some steering of layers 1 and 2. This increases the nonzonality of the current in all layers, thereby further destabilising the ﬂow. This is a positive feedback on the instability. Thus, the large stores of potential energy which were built during phase B are transferred to turbulent kinetic energy in a sudden burst lasting for several years. Regime D The positive feedback between eddies and the mean ﬂow is limited by the stores of potential energy. During phase D the stores of potential energy decline, and turbulent kinetic energy decays over a period of several years. Note that ﬂow is still nonzonal, and therefore unstable, during this time, and production of turbulent kinetic energy continues – but at a lower rate than previously. Phase D ends when stores of potential energy reach a minimum level, and the system returns to the lower energy state, regime A. This description of the process is based on the feedback between eddies and the mean ﬂow. The eddies drive a deep ﬂow which is steered by topography and further destabilises the system, enhancing the eddy ﬁeld in a short burst at the end of phase C. The process is limited by the store of potential energy.

5. Conclusions Eddy resolving ocean models are used to demonstrate that low frequency variability, with a period of order one hundred eddy timescales, occurs in two diﬀerent conﬁgurations. In the ﬁrst, the double gyre case, the variability takes the form of a shifting and modulation in strength of the jet dividing the two gyres. This variability is robust across a wide parameter regime, and has been found by a number of authors. However, the mechanisms driving this variability are not clear. In this chapter we demonstrate that it is possible that feedbacks between eddies and the large scale ﬂow play a role in the variability. However, the available data does not distinguish whether eddy processes or mean ﬂow self interaction is the dominant mechanism.

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The second case is a channel ocean which mimics ﬂow in the Southern Ocean. Here the variability is stronger than in the double gyre case, allowing more accurate determination of the mechanisms. The results are consistent with the hypothesis of eddy–mean ﬂow feedback: the eddies drive a deep ﬂow which is steered by topography and which further destabilises the ﬂow. It is currently not clear (due to the sparsity of ocean data) whether or not this mechanism is operating in the ocean. Acknowledgments AMH was supported by an Australian Research Council Postdoctoral Fellowship (DP0449851) during this work. Numerical computations were supported by an award under the Merit Allocation Scheme on the National Facility of the Australian Partnership for Advanced Computing. References 1. P. S. Berloﬀ and J. C. McWilliams, Large-scale, low-frequency variability in wind-driven ocean gyres. J. Phys. Oceanogr., 29, 1925–1949 (1999). 2. F. W. Primeau, Multiple equilibria and low-frequency variability of the winddriven ocean circulation. J. Phys. Oceanogr., 32, 2236–2252 (2002). 3. J. J. Nauw, H. A. Dijkstra, and E. Simonnet, Regimes of low-frequency variability in a three-layer quasi-geostrophic ocean model. J. Mar. Res., 62, 684– 719 (2004). 4. M. A. Spall, Dynamics of the Gulf Stream/Deep Western Boundary Current crossover. Part II: Low-frequency internal oscillations. J. Phys. Oceanogr., 26, 2169–2182 (1996). 5. B. Qiu and W. Miao, Kuroshio path variations south of Japan: Bimodality as a self-sustained internal oscillation. J. Phys. Oceanogr., 30, 2124–2137 (2000). 6. A. McC. Hogg, J. R. Blundell, W. K. Dewar, and P. D. Killworth, Formulation and users’ guide for Q-GCM (version 1.0). Southampton Oceanography Centre, 2003. Internal Document No. 88. Most recent version available from http://www.noc.soton.ac.uk/JRD/PROC/Q-GCM/. 7. A. McC. Hogg, W. K. Dewar, P. D. Killworth, and J. R. Blundell, A quasigeostrophic coupled model: Q-GCM. Mon. Wea. Rev., 131 (10), 2261–2278 (2003). 8. P. Cessi, A stratiﬁed model of the inertial recirculation. J. Phys. Oceanogr., 18, 662–682 (1988). 9. A. McC. Hogg, P. D. Killworth, J. R. Blundell, and W. K. Dewar, Mechanisms of decadal variability of the wind-driven ocean circulation. J. Phys. Oceanogr., 35, 512–531 (2005). 10. H. von Storch and F. W. Zwiers, Statistical analysis in climate research. (Cambridge University Press, 1999).

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11. F. W. Primeau, Multiple equilibria of a double-gyre ocean model with superslip boundary conditions. J. Phys. Oceanogr., 28, 2130–2147 (1998). 12. J. J. Nauw and H. A. Dijkstra, The origin of low-frequency variability of double-gyre wind-driven ﬂows. J. Mar. Res., 59, 567–597 (2001). 13. P. Berloﬀ and S. P. Meacham, The dynamics of a simple baroclinic model of the wind-driven circulation. J. Phys. Oceanogr., 28, 361–388 (1998). 14. S. Jiang, F. Jin, and M. Ghil, Multiple equilibria, periodic,and aperiodic solutions in a wind-driven, double-gyre, shallow-water model. J. Phys. Oceanogr., 25, 764–786 (1995). 15. B. T. Nadiga and B. P. Luce, Global bifurcation of Shilnikov type in a doublegyre ocean model. J. Phys. Oceanogr., 31, 2669–2690 (2001). 16. E. Simonnet and H. A. Dijkstra, Spontaneous generation of low-frequency modes of variability in the wind-driven ocean circulation. J. Phys. Oceanogr., 32, 1747–1762 (2002). 17. A. McC. Hogg and J. R. Blundell, Interdecadal variability of the southern ocean. J. Phys. Oceanogr. (2006). In Press. 18. J. C. McWilliams, W. R. Holland, and J. H.S. Chow, A description of numerical Antarctic Circumpolar Currents. Dyn. Atmos. Oceans, 2, 213–291 (1978). 19. J. Pedlosky, Geophysical Fluid Dynamics. (Springer-Verlag, 1987).

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PERIODIC MOTION VERSUS TURBULENT MOTION: SCALING LAWS, BURSTING AND LYAPUNOV SPECTRA LENNAERT VAN VEEN Department of Mathematical and Statistical Science, La Trobe University Victoria 3086 Australia [email protected] SHIGEO KIDA Department of Mechanical Engineering and Science, Kyoto University Yoshida–Honmachi, Sakyo–ku, Kyoto 606-8501, Japan [email protected] GENTA KAWAHARA Department of Mechanical Science and Bioengineering, Osaka University 1–3 Machikaneyama-cho, Toyonaka-shi, Osaka 560-8531, Japan [email protected] Developed turbulence is traditionally deﬁned in terms of, and described by, mean quantities, extracted from statistical analysis of measurements and data from simulations. The merit of the statistical approach can hardly be overestimated as it unveils universal laws such as the scaling of the energy spectrum in the universal range and the velocity proﬁle in near-wall ﬂows. Averaging over an ensemble of ﬂow ﬁelds or a long time series does, however, obscure the instantaneous properties of the ﬂow. Therefore a study of the dynamical processes that collectively produce the universal laws requires a diﬀerent approach. We propose to study these processes by means of time-periodic solutions of the Navier–Stokes equation. From the point of view of chaos theory such periodic solutions are expected to be ubiquitous in turbulence. In order to ﬁnd and analyse them, however, we need to overcome some fundamental diﬃculties related to the complexity of their bifurcation diagrams and the large number of degrees of freedom. As an example we present several periodic solutions of the Navier–Stokes equations on a triply periodic domain at moderate Reynolds number.

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188 “The ideas of low-dimensional dynamics and, in particular, chaos has not shed much light on what has often been called the last unresolved problem of classical physics: turbulence. (. . . ) This is more a measure of the depth of the problem which will probably require a combination of innovative ideas for a significant breakthrough to be made. I remain convinced that an essential ingredient of this will be based in nonlinear dynamical systems.” Tom Mullin

1. Introduction Although a straightforward, generally accepted deﬁnition of developed turbulence does not seem to exist, two key elements are certainly part of it: motion on a wide range of spatial scales and coherent structures. The existence of coherent structures, conﬁrmed both by experiment and by simulations, gives us some hope that we can understand turbulence in terms of relatively simple, computationally tractable objects. Rather than to process information about the entire ﬂow ﬁeld we can concentrate on a number of coherent structures and see if and how they interact and break down into smaller scale structures, as the conventional picture of the energy cascade suggests. For this purpose we need an objective procedure to identify coherent structures and track their behaviour in time. This proves to be a hard problem indeed, and a substantial amount of work in turbulence research is going into this issue. The common approach to this problem is based on analysis of data in physical space, such as isosurfaces of enstrophy or velocity correlations. A diﬀerent approach, more common in mathematical literature, is to think of a space and time dependent ﬂow ﬁeld as an orbit in a high-dimensional phase space. The study of such orbits lies in the domain of dynamical systems theory. In particular, dynamical systems theory gives us tools to study special solutions, such as equilibrium points, corresponding to stationary ﬂows, and periodic orbits, corresponding to time-periodic ﬂows. In a dissipative system, such as viscous ﬂuid ﬂow, the orbits in phase space settle on an attractor, and for high enough Reynolds number this attractor will be chaotic. Chaos theory tells us that this “turbulent attractor” contains inﬁnitely many unstable periodic orbits. If we study a certain orbit segment, i.e. a ﬁnite time series, which corresponds to the formation and breakdown of a coherent structure, we will ﬁnd that it is not periodic. However, there is always a periodic orbit close to it. In the words of Henri Poincar´e: “Given the equations (. . . ) and a particular solution one can always ﬁnd a periodic solution (of which the period may indeed be very long) such that the diﬀerence between the two remains as small as one likes for as long as one likes.”

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Thus, if we can ﬁnd one or more periodic orbits which lie in the turbulent attractor we can use them to study turbulence and its fundamental dynamical processes. In this chapter we show how this idea can be applied to the study of isotropic turbulence.1 This work was preceded by a study of Kawahara and Kida into plane Couette ﬂow.2 In a separate development, Hof et al. found evidence for the relevance of periodic solutions to pipe ﬂow.3 Although this is all work of the last few years, the application of ideas from dynamical systems theory to ﬂuid dynamics is certainly no novelty. In the study of laminar or weakly turbulent ﬂows this approach has proven rather successful. Helped by a highly symmetric geometry, ﬂows such as Rayleigh-B´enard and Taylor-Couette have been analysed in terms of equilibria and periodic orbits and their bifurcations. Examples can be found, e.g., in Ref. 4. Typically, the ﬂow is ﬁrst analysed at low Reynolds number, where it is stationary and laminar. At higher Reynolds number the ﬂow bifurcates as it turns unstable to, e.g., travelling waves, represented by periodic orbits or equilibria, depending on the frame of reference. If we further increase the Reynolds number solutions of increasing complexity are created, representing more complex physics. We might ﬁnd invariant tori representing modulated waves or homoclinic orbits representing solitary waves. At some point chaos sets in, and by studying the critical bifurcation we can determine some properties of the resulting, intrinsically unpredictable, ﬂuid motion. When we try to extend this work to high Reynolds number ﬂows we encounter several theoretical and practical problems. In the ﬁrst place, bifurcation analysis does not always turn out as nicely as it does in the celebrated examples mentioned above. A notoriously hard problem occurs in certain shear ﬂows, such as pipe ﬂow and plane Couette. In these cases, the laminar equilibrium solution is stable for all Reynolds number. Thus there is no straightforward way to ﬁnd any of the more complex, space and time dependent solutions that play a role in the onset of turbulence. Such solutions may be unstable and bifurcate from inﬁnite Reynolds number, which makes them hard to detect. We face another problem if many sub critical bifurcations occur, in which unstable solutions are created. One soon ends up with large numbers of coexisting, unstable equilibria and periodic solutions. There is no easy way to decide which are the most important to track and the system is prone to sudden transitions caused by collisions of global manifolds that cannot be predicted on the basis of local bifurcation analysis. Thirdly, even if we can track all solutions and study their bifurcations it is

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not guaranteed that this gives us the necessary information to understand the physical transition to developed turbulence. In isotropic turbulence, for instance, the transition from weak to developed turbulence appears to be gradual rather than related to a particular bifurcation. The derivative of the energy dissipation rate with respect to the viscosity changes when developed turbulence sets in. From the point of view of dynamical systems theory the behaviour is chaotic all along and it seems unlikely that a particular bifurcation scenario could explain the transition. Bearing in mind these diﬃculties, we do not attempt to compute the bifurcation diagram at low Reynolds numbers. This diagram is known to involve bifurcating tori5,6 which spawn inﬁnitely many periodic orbits in a myriad of scenarios. Instead, we ﬁlter periodic orbits directly from a time series obtained in the weakly turbulent regime. Subsequently we continue the periodic orbits to higher Reynolds number. Along the continuation curve we compute the time-mean energy dissipation rate along the periodic orbits and compare it to the values found in turbulence. Thus, we can see if the periodic orbits reproduce the onset of developed turbulence. One of the orbits we study here reproduces the turbulent time-mean values very well and is analysed in some detail by means of its Lyapunov spectrum. A practical problem with this work is the large number of degrees of freedom we need to take into account. Generally speaking, we need to study systems of millions of diﬀerential equations in order to apply tools of dynamical systems theory to turbulence. We used symmetry considerations to bring this number down to manageable proportions. All computations were done on parallel processors in a time frame of months. As faster and bigger computers are developed and smarter algorithms form numerical linear algebra are becoming available, we expect this to be a transient problem. Within years, we will be able to study periodic solutions in ﬂows with fewer (symmetry) constraints and at higher Reynolds number. This might be the key to signiﬁcant progress in turbulence research. 2. Turbulence, chaos and the cycle expansion The mathematical motivation of this work is based on the special role of periodic orbits in chaos theory. For now, it is impossible to apply this theory to turbulence with mathematical rigour. There is no proof of existence of periodic solutions to the three-dimensional Navier–Stokes equations, nor of a ﬁnite-dimensional attractor for high-Reynolds number ﬂows. Nevertheless, the concept of the cycle expansion is intuitively clear and inspiring.

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In the lingo of chaos theory the observation by Henri Poincar´e, cited in the introduction, might be rephrased as follows: “Unstable periodic orbits lie dense in a chaotic attractor.” Thus, a chaotic orbit can be thought of as composed of inﬁnitely many segments, each of which closely follows a periodic orbit. This observation led Cvitanovi´c and coworkers to the idea that we might replace the time mean of any quantity along a chaotic orbit by a weighted average over the timemean values of this quantity along periodic orbits. This idea is not unlike the Bolztmann ergodic hypothesis in statistical physics, in which the time mean is replaced by an ensemble average. The resulting theory is known as cycle expansion theory because it provides us with an approximation of the time-mean value by a sum of terms generated by periodic orbits (cycles). A full explanation of this approach can be found in Ref 7. Here, we will only provide a sketch to illuminate the similarity to our work.

A

Poincar´e section

B nearly one-dimensional map 0

1 symbolic dynamics

2
Time averaged quantities computed as a¯ = limn→∞ i a¯γin µ(γin)

Ordering of all periodic points x: P n(x) = x of period n and the corresponding orbits γin ∈ Γn.

D Fig. 1.

C

A schematic representation of the cycle expansion.

Consider a chaotic dynamical system with an attractor of dimension D < 3. In order to write down a cycle expansion for a time-mean quantity a ¯ we proceed in four steps, illustrated in Fig. 1:

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A In a Poincar´e intersection plane we see thin, folded ﬁlaments, reminiscent of the H´enon attractor. Periodic orbits correspond to ﬁxed points of the iterated Poincar´e map, i.e. P n (x) = x, with discrete period n. By a smart parameterisation of the ﬁlaments we can reduce the problem to the study of a one-dimensional map. B Now we deﬁne a partition on the one-dimensional domain, and translate the dynamics of the continuous-time system into symbolic dynamics. C If we know the symbolic dynamics we can list all periodic points up to a given symbolic length, i.e. period. Denote the set of k(n) points of k(n) period n by Γn = {γin }i=1 . D For each periodic point we compute a weight, µ, that depends on its Lyapunov spectrum. The cycle expansion is now given by a ¯γin µ(γin ) a ¯ = lim n→∞

i

where a ¯γin is the time-mean value of a along the periodic orbit γin ∈ Γn and µ(γin ) is the weight of that orbit. These steps are worked out in detail in Ref. 8, in which the cycle expansion is applied to the Kuramoto-Shivashinski equation in a mildly chaotic regime. There are some technical conditions that the chaotic attractor has to satisfy in order for the prove of validity of this expansion to hold. For our present purposes, the most restrictive condition is that on the attractor dimension. In turbulent ﬂows we do not ﬁnd such low-dimensional dynamics. The dimension of the turbulent attractor has been estimated to be of order 100 for shear ﬂow9 and this estimate is probably on the low side. Consequently, an eﬀort has been made to formulate a cycle expansion for high-dimensional chaos. Basically, we can use the expansion as given above, except that there is no way to ensure we know all orbits up to a certain period and the weight of each orbit does not follow from the theory. An ad hoc expansion was applied successfully to the Kuramoto-Shivasinsky equations in a regime of full-ﬂedged spatio-temporal chaos.10 The method by which periodic orbits were localised is based on near recurrences, ﬁltered from a long time series, and has an element of chance to it that might seem unsettling to the exact scientist. Statistically speaking, however, this approach gives us those orbits that are likely to have a large impact on the chaotic dynamics. This idea was formulated in the context of a coupled map lattice.11 An interesting prediction in that work is that the larger the number of degrees of freedom, the fewer periodic orbits we need to obtain a good estimate of any time-mean quantities.

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The interpretation of this work in terms of ﬂuid mechanics is not straightforward, however, and hinges on some crude assumptions on the decay of correlations in space and time. The cycle expansion for high-dimensional chaos and the statistical approach to this problem are by no means a complete mathematical theory. There are far too many open technical questions and ambiguities. However, the ideas are consistent with all results on periodic orbits in turbulence that we know of, and it would not be the ﬁrst time that physicists apply successfully a series expansion without a proof of convergence. . . 3. Isotropic turbulence Consider an incompressible, viscous ﬂuid on a triply periodic domain. Its motion is governed by the Navier–Stokes equation and the divergence-free condition, which are conveniently solved in terms of the Fourier compo(k, t), and vorticity, ω (k, t). Energy is dissipated at the nents of velocity, v ω (k, t)|2 rate = 2νQ, where ν is the kinematic viscosity and Q = 12 k | is the enstrophy. In order to input energy we ﬁx the Fourier components time. Under the symmetry of vorticity at the smallest wave number kf in √ constraints explained in Sec. 3.2 we have kf = 11. Like the energy dissipation rate , the energy input rate e is a function of time. The complexity of the ﬂow is measured by Taylor’s microscale Reynolds number, deﬁned as + 10 1 E(t) & (1) Rλ (t) = 3 ν Q(t) where E(t) = 12 k | u(k, t)|2 is the energy. The microscale Reynolds number scales as the square root of the geometric Reynolds number commonly employed in shear ﬂows. We integrate the Navier–Stokes equation numerically, truncating the Fourier series at −N/2 ≤ k1 , k2 , k3 ≤ N/2. The nonlinear terms are computed by the spectral method in which the aliasing interaction is suppressed by eliminating all the Fourier components beyond the cut-oﬀ wavenumber kmax = N/3. The fourth-order Runge-Kutta-Gill scheme with step size ∆t = 0.005 is employed for time stepping. 3.1. The number of degrees of freedom The maximal microscale Reynolds number that can be attained in simulations at a given truncation level is determined by the ratio of the typical size of the largest eddies, L, to that of the the smallest eddies, l. The former is associated with the length scale of the forcing, i.e. L = 2π/kf , and

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the latter with the Kolmogorov length scale, i.e. l = ηk = (ν 3 /¯ )1/4 . In developed turbulence E¯ and ¯ are independent of ν so that L 3/2 ≈ γRλ l

(2)

where γ ≈ 0.13 does not depend on ν.12 As a rule of thumb, we resolve small enough scales if kmax ≈ 1/ηk , i.e. L/l ≈ 2πkmax /kf . The lowest resolution that allows us to simulate turbulence is N = 128, corresponding to a maximal microscale Reynolds number of Rλ = 67. At this Rλ , E¯ and ¯ are approximately constant as a function of ν. However, we capture the inertial range only marginally. At N = 128 we have about 3×(2128/3+1)3 ≈ 2·105 independent Fourier modes. A simulation of this size can be run comfortably on a present day personal computer. As will be explained below, however, for the purposes of tracking periodic orbits this number of degrees of freedom is too high. In the next section, we will introduce a reduction by symmetry that makes the problem tractable. In order to observe the inertial range spectrum over one decade we need to have Rλ 100 and N = 256 is the minimal resolution. This truncation level has about 1.5 · 107 independent Fourier modes. For time being, this number is too large to handle even after reduction by symmetry. 3.2. Reduction by symmetry In order to reduce the number of degrees of freedom in our simulations we impose spatial symmetries on the solutions. These symmetries form a subgroup of the full symmetry group of the boundary-free Navier–Stokes equation, which consists of translations, rotations and reﬂections. The maximal set of symmetries that allows for turbulent solutions was determined by Kida in Ref. 13 and reduces the number of degrees of freedom by a factor of nearly 200. The resulting, so called high-symmetric, ﬂow can be studied at high microscale Reynolds number with a relatively small computational effort. In the eighties and early nineties, this enabled Kida et al.12 to observe the Kolmogorov scaling laws in numerical experiment for the ﬁrst time. With present-day computers, these scaling laws can be reproduced without any symmetry constraints. The equations for high-symmetric ﬂow found a new use in the search for ﬁnite-time blowup solutions to the Navier–Stokes equations, and are in that context referred to as Kida-Pelz ﬂow. As will be explained below, the method we use for ﬁnding and continuing periodic orbits requires the computation and decomposition of matrices of the size of the number of degrees of freedom. Therefore, the reduction by

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symmetry from 2 · 105 down to 1 · 104 degrees of freedom is essential. Thus, we recycled an old idea for reduction of the size of the problem in order to test and explore a new idea for the analysis of the problem. 3.3. How to ﬁnd unstable solutions? After reduction by symmetry we have a system of n ≈ 10, 000 coupled, nonlinear ordinary diﬀerential equations with one parameter, ν. Symbolically, we can write the dynamical equations as d x = f (x, ν) (3) dt where x is the n-dimensional vector that holds the linearly independent Fourier components of vorticity. In this system of equations we want to ﬁnd periodic solutions. It is convenient to regard them as ﬁxed points of an iterated Poincar´e map in phase space. We ﬁx a plane of intersection S by setting one of the small wave number components of vorticity to a constant, e.g. x1 = c. The coordinates y in the intersection plane are the remaining (n − 1) components. Fixed points of the Poincar´e map P on S satisfy P

m

(y) − y = 0

(m = 1, 2, 3, · · · ),

(4)

for some “discrete period” m. The Poincar´e map is found by integrating (3) over a ﬁnite time interval and consequently (4) is highly nonlinear. In order to ﬁnd solutions numerically we use Newton–Raphson iteration. Essential for the convergence of this method is an accurate initial guess. As explained in the introduction, we do not extract initial data from the bifurcation diagram at low microscale Reynolds number. Instead, we ﬁlter initial guesses for the Newton-Raphson iterations directly from a long, weakly turbulent time series. We run a simulation at Rλ ≈ 55 and keep track of the iterates of the Poincar´e map. If a point is mapped close to itself, i.e.

P

m

(y) − y Q < δ,

(5)

we use it as an initial guess. The distance is measured as the enstrophy of the diﬀerence ﬁeld and δ is a threshold that we found by trial and error to be around 10% of the standard deviation of enstrophy at this Rλ . Less accurate initial guesses usually diverge under Newton–Raphson iteration. This is essentially the same procedure as the one used in Ref. 10. In order to perform Newton-Raphson iteration on the initial guesses we need to compute the matrix of derivatives of the Poincar´e map. For this end we used ﬁnite diﬀerencing, which means that we have to integrate (3) over

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a ﬁnite time interval once for each degree of freedom, which can be done eﬃciently on parallel processors. This is computationally the hardest step, and the time and money spent on this part of the process is certainly the largest obstacle we have to overcome to study periodic orbits in turbulence. In the case at hand, we ﬁltered about 50 initial guesses from the time series, from which we distilled a dozen or so periodic orbits. The discrete period of these orbits is closely related to their period in the continuoustime system. An orbit with a discrete period m has a period roughly equal to mTR , where TR is the most probable return time of the Poincar´e map. Thus we can refer to the periodic orbits we found as period-m orbits. Note, that the symmetries we impose on the velocity ﬁeld do not allow for travelling wave solutions. The periodic orbits we compute are solutions of the Navier-Stokes equation in a Eulerian frame of reference.

3.4. Continuation in the Reynolds number In order to see if periodic motion represents the onset of developed turbulence we continued orbits with period 1 up to 5 in ν and compared the time-mean energy dissipation rate to that of turbulence. The result is shown in Fig. 2. Interestingly, the time-mean values produced by the periodic orbits are all close to that of turbulence at ν = 0.0045 (Rλ = 55), where they have been ﬁltered from a time series. At ν = 0.0035 (Rλ = 67), only the orbit of longest period reproduces the turbulent values accurately. The period of this orbit is about 2.5TT , where TT is the large-eddy turnover time, i.e. the longest intrinsic time scale of turbulence. We can compare the periodic motion to an ensemble of non-periodic time series of the same length, randomly selected from a simulation of turbulence. Surprisingly, the diﬀerence between the time-mean energy dissipation rate of the period-5 orbit and that of turbulence is signiﬁcantly smaller then the standard deviation of this quantity in the random ensemble. In other words, the time-mean value found along the period-5 orbit is closer to that of turbulence then we would expect if we treat the orbit as random sampling of the attractor over a period of 2.5TT. This observation leads us to the conjecture that the period-5 orbit represents the onset of developed turbulence, in other words it is embedded in the turbulent attractor in the whole range of Rλ . In Ref. 1 it is shown that also the energy spectrum agrees extremely well with that of turbulence both simulations, experiments and theory.

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5p 0.1

¯ 0.096

3p

4p

0.092

2p 0.088

1p 0.084 0.0035

0.00375

0.004

ν

0.00425

0.0045

Fig. 2. Continuation of the periodic orbits in the viscosity. Shown is the time-mean energy dissipation rate for turbulence (solid), period-1 to period-4 (dashes) and period-5 (dots). The period-5 orbit reproduces the values measured in turbulence well, especially in the regime of developed turbulence, i.e. ν < 0.004 (Rλ > 60).

3.5. Analysis of embedded periodic motion By comparing time-mean quantities we have established that the period-5 orbits represents the turbulent motion. Now we want to learn about the turbulent dynamics by studying the time-periodic solution. One interesting quantity that is hard, if not impossible, to compute for turbulence is the Lyapunov spectrum. The largest exponent can readily be computed, but exponents deeper in the spectrum are swamped by numerical error in practical computations. Consequently, little is known about the properties of the Lyapunov spectrum of turbulence and its relation to physical processes. Here, we use the period-5 orbit as a reference solution to obtain a number of time-mean and local exponents. These data indicate that the Lyapunov spectrum is related to the energy spectrum through preferred spatial scales of growing and decaying perturbations. The computation of the Lyapunov spectrum requires integration of the linearised equations d v = Jv, dt

(6)

where J is the matrix of derivatives of the vector ﬁeld f in (3) and v is a perturbation vorticity ﬁeld. Through J, the time evolution of the perturbation ﬁeld depends on the reference orbit x(t). The Lyapunov exponents

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are deﬁned by Λ = lim

t→∞

v(t) Q 1 ln 2t v(0) Q

(7)

The Oseledec theorem, also known as the multiplicative ergodic theorem, ensures that for each initial point x(0) there exist n independent initial perturbations vi (0) that give rise to exponents Λ1 ≥ Λ2 ≥ . . . ≥ Λn . This spectrum does not depend on the initial point, as long as it lies on an orbit which samples the whole attractor.a The computation of more than one exponent is necessary for estimating such quantities central to chaos theory as the attractor dimension and the Kolmogorov-Sinai entropy.14 In addition to the time-mean exponents we can compute the local Lyapunov exponents, corresponding to the instantaneous growth rate of the perturbation ﬁelds. The local exponents are deﬁned as 1 d ln vi (t) Q (8) 2 dt so that their time-mean value coincides with the Lyapunov spectrum, i.e. ¯i = Λi . A very long time integration is necessary for the computation of λ the spectrum as the local exponents typically have a standard deviation an order or magnitude larger than their mean value. In order to compute the largest k exponents we need to integrate (3) and (6) for k independent initial perturbations. At regular time intervals during the integration an orthogonalisation procedure is executed to separate the distinct growth rates. As some of the perturbations are exponentially growing while others are exponentially decaying, the orthogonalisation introduces a signiﬁcant error which accumulates. Consequently, accurate estimates are hard to obtain, and in Ref. 9 the authors take care to state their results as approximate lower bounds on the attractor dimension. If we compute the Lyapunov exponents relative to a periodic orbit we can apply Floquet theory to (6). It then follows that the exponents Λi are directly related to the eigenvalues of the monodromy matrix, which, after a projection onto the plane of intersection, is equal to the matrix of derivatives of the Poincar´e map. The eigenvectors of the monodromy matrix correspond to the initial perturbations vi (0), also called Lyapunov vectors. Therefore, once we have computed the eigenspectrum of the monodromy matrix we can compute any number of time-mean Lyapunov exponents and a large number of local Lyapunov exponents. The number of local exponents λi (t) =

a Technically, the spectrum is equal for almost all initial points relative to the natural invariant measure on the attractor. It diﬀers if the initial point lies on a periodic orbit.

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we can compute is only restricted by the accumulation of numerical error in the integration of (3) and (6) over one period. We computed the leading 50 local Lyapunov exponents along the period5 orbit at Rλ = 67. The largest exponent is close to that of turbulent motion. In units TT−1 , we ﬁnd that Λ1 = 0.88 and Λ5p 1 = 1.05 for the turbulent and the period-5 motion, respectively, compared to a standard deviation of the corresponding local exponent in turbulence of 1.74. In Fig. 3 we visualise the behaviour of the local exponents over one period, and compare it to the instantaneous energy spectrum. In each subplot, we include the time series of the energy input rate on the left and the energy dissipation rate on the right. The way to read this ﬁgure is as follows: energy is input at small wave numbers, and therefore the energy content at small wave numbers is strongly correlated to the energy input rate. In agreement with the conventional picture of the energy cascade process, we see the local maxima progress to larger wave numbers in the course of time. The energy content in the dissipation range is strongly correlated to the energy dissipation rate. In the second subplot we can see a similar correlation: the local exponents with small indices are correlated to the energy input rate, whereas those with a large index are correlated to the energy dissipation rate. This is intuitively correct, as we tend to associate instabilities with external forcing and damped perturbations with dissipation. However, apart from some speculations arising from shell model turbulence,15 to our knowledge no relation had ever been established between the Lyapunov spectrum of turbulence on one hand and spatial scales and physical processes on the other. We expect to see an even clearer correlation between the Lyapunov spectrum and the energy cascade process in turbulence with a developed inertial range. Ultimately, we should be able to establish a direct relation between the processes in physical space which contribute to the energy cascade, and the mathematical properties of the attractor in phase space. 4. Conclusions and outlook The work on isotropic turbulence is ongoing. We are studying the spatial patterns represented by, in particular, the period-5 solution. Also, we are exploring ways to increase the truncation level and study periodic orbits in the presence of an inertial range. Thirdly, an interesting issue is the role played by the period of the orbits. We are gathering orbits of longer period for intercomparison to shine a light on this issue.

(b)

40

50

2

1

30

3

3

20

((t) − ¯)/σ

4

10

(λi(t) − Λi)/σλi

4

(e(t) − e¯)/σe

0

0

i

1

1

Fig. 3. (a) The instantaneous (3D) energy spectrum along the period-5 orbit, with on the left the energy input rate and on the right the energy dissipation rate. Local maxima of e(t) give rise to local maxima in the energy content at small wave number, which cascade to larger wave numbers and show correlation to (t) with a small time delay in the dissipation range. (b) Time series of the local Lyapunov exponents visualised with contours just like for the energy spectrum. There is a positive correlation between the local Lyapunov exponents with a small index (Λ1 to Λ19 ) and e(t) and between those with large index (Λ20 and higher) and (t). Incidentally, the Kaplan-Yorke dimension computed from the Λi is 19.7. Contours denote deviation from time mean, normalised by standard deviation. White denotes positive and black negative deviations.

(a)

0

0

0

kη

1

1

1

0.6 0.8 1

2

2

1

0.4

3

3

3

0.2

4

t

((t) − ¯)/σ

4

¯ (E(k, t) − E(k))/σ E

4

t

12:18

t

(e(t) − e¯)/σe

t

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Obviously the work presented here poses more questions than it answers. It is only a ﬁrst glance at the results that can be obtained by applying dynamical systems theory to turbulent ﬂows. Moreover, this ﬁrst glance is blurred by the practical restrictions we face when performing the numerical computations. We had to impose spatial symmetries, which renders the resulting ﬂow non-homogeneous. We had to choose a fairly low resolution. We had to restrict ourselves to the analysis of ﬁve orbits of fairly short period. Yet we obtained promising results, and similarly rapid progress is made in the study of shear ﬂows and the development of eﬃcient numerical methods.2,3,16,17 This fresh attack on one of the oldest standing problems of physics will, at the very least, bring the theory of dynamical systems and the theory of turbulence closer together and be an inspiration to both. Acknowledgements LvV was supported by the Japan Society for Promotion of Science and the ARC COE for Mathematics and Statistics of Complex Systems. References 1. L. van Veen, S. Kida and G. Kawahara, Fluid Dyn. Res. 38 19–46 (2005). 2. G. Kawahara and S. Kida, J. Fluid Mech. 449 291–300 (2001). 3. B. Hof, C. W. H. van Doorne, J. Westerweel, F. T. M. Nieuwstadt, H. Faisst, B. Eckhardt, H. Wedin, R. R. Kerswell, F. Waleﬀe, Science 305 1594–1598 (2004). 4. K. A. Cliﬀe, A. Spence and S. J. Tavener, Acta Numer. 9 39–131 (2000). 5. S. Kida, M. Yamada and K. Ohkitani, Physica D 37 116–125 (1989). 6. L. van Veen, Phys.D 210 249–261 (2005). 7. R. Artuso, E. Aurell and P. Cvitanovi´c, Nonlinearity 3 325–359 (1990). 8. F. Christiansen, P. Cvitanovi´c and V. Putkaradze, Nonlinearity 10 55–70 (1997). 9. L. Keefe, P. Moin and J. Kim, J.Fluid Mech. 242, 1–29 (1992). 10. S. M. Zoldi and H. S. Greenside, Phys. Rev. E 57 R2511 (1998). 11. M. Kawasaki and S.-i. Sasa, Phys. Rev. E 72, 037202 (2005). 12. S. Kida and Y. Murakami, Phys. Fluids 30, 2030–2039 (1987). 13. S. Kida, J. Phys. Soc. Japan 54, 2132–2136 (1985). 14. J-P. Eckmann and D. Ruelle, Rev. Mod. Phys. 57, 617–656 (1985). 15. M. Yamada and K. Ohkitani, Phys. Rev. E 57, R6257–R6260 (1998). 16. G. Kawahara, Phys. Fluids 17, 041702 (2005). 17. J. S´ anchez, M. Net, B. Garc´ıa-Archilla and C. Sim´ o, J.Comput.Phys. 201, 13–33 (2004).

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MAXIMUM ENTROPY N-PARTICLE LAGRANGIAN CLUSTERS IN TURBULENCE AND APPLICATION TO SCALAR FIELDS MICHAEL S. BORGAS CSIRO Marine and Atmospheric Research P.O. Bag #1, Aspendale, VIC 3195 Australia [email protected] A solution is given for n particle (arbitrary n) probability density functions (PDFs) for Lagrangian displacements in turbulent ﬂows. This is based on maximum entropy conditioned on scale of separation, and ultimately accounts for the non-Gaussian properties of separation statistics. The model encodes in an optimal way existing knowledge about separation processes in turbulence, solves many practical problems for prediction of concentration properties in scalar mixing, and can be generalised.

1. Introduction The process of mixing in turbulent ﬂows may be deﬁned by the Lagrangian representation of material particle displacements.1 A general result for the n-th moment of the concentration of a passive scalar ﬁeld, θ(x, t) , for an instantaneous source release is2 θn (x, t) = φ(x(1) ) . . . φ(x(n) )Pn (x(1) , . . . , x(n) , t0 |x, . . . , x, t)d3 x(1) . . . d3 x(n) ,

(1)

for a prescribed instantaneous source distribution φ(x) at initial time t0 < t. The joint transition probability density function (PDF), Pn , for n marked (Lagrangian) particles contains the key dynamics. This expression ﬁnds applications mostly for mean concentrations (n = 1) in atmospheric dispersion problems, or in generalised forms for scalar structure functions, and the prediction of probability density functions for concentration.3–5 Understanding the properties of higher-order concentration moments, including the probability density function, is useful and is in widespread practice,

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from environmental pollution studies to engineering estimates of combustion processes.6 In this paper we brieﬂy recap in §2 the statistical representation of Lagrangian particle dynamics in turbulence. In particular, the relatively well known single particle dispersion and pair separation behaviour is highlighted. Analysis of higher-order cluster behaviour is facilitated by a transformation to centre-of-mass and uncorrelated separation vectors. In §3 the known mean-square single and relative dispersion are used as constraints to derive the n-particle cluster PDF by maximising the entropy, introduced as the concept of plume information ‘written’ by a scalar source. The jointGaussian outcome of §3 is generalised in §4 to allow just the length scale of the cluster to carry information, say from the dynamics of the energy cascades in turbulence. A generalised maximum-entropy approach to minimise the information content in cluster shapes, for ﬁxed scale, is developed. The simple mathematical outcome permits the non-Gaussian single-pair relative dispersion PDF to parameterise the full non-Gaussian cluster statistics for arbitrary numbers of particle pairs. The pair-separation PDF plays the role of the spectrum of scales and the knowledge of the PDF is assumed to encode the turbulent dynamics. This input information links speciﬁc physical properties to cluster statistics. The optimal n-particle solution is summarised in §7. It yields the correct one-particle and relative dispersion statistics, and all symmetries, while imposing no arbitrary internal structure on the shapes within the cluster. Applications of the solution are made in §8 to study the mixing of a unit mass point source in turbulence. For a robust parameterisation of the relative dispersion PDF, the cluster solution essentially gives the mass-fraction distribution analytically as a function of time. In the ideal inertial-range limit, this solution remarkably gives power-law behaviour for the high-concentration limit of the mass fraction. Interestingly, this ‘heavy’ tail corresponds to inﬁnite concentration skewness (unbounded third moment), but which is bounded by molecular eﬀects whenever the inertial ranges are truncated. 2. The Lagrangian Representation The probability density function for Lagrangian transitions, Pn (x(1) , . . . , x(n) , t0 |x, . . . , x, t), represents the ‘backwards’ displacements given the ﬁxed receptor point x, at a ﬁxed later time t. In practice we think of the displacements of, say, molecules backwards in time from some inﬁnitesimal volume surrounding

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the receptor point, to their release points at source. We also more commonly consider the equivalent ‘forward’ transitions for stationary ﬂows,2,7 Pn (x(1) , . . . , x(n) , t|x, . . . , x, t0 ) = Pn (x(1) , . . . , x(n) , t)

(2)

where for simplicity we set the receptor point at the origin (x = 0) and set the initial time point to zero (t0 = 0). This interpretation of Lagrangian displacements relies on a familiar limiting argument in high Reynolds number turbulence that formally permits ‘inﬁnitesimal’ initial separations provided that they are greater than the viscous microscale.8,9 The microscale vanishes in the limit of inﬁnite Reynolds number. In practice, initial separations have some ﬁnite scale, but in the ﬁrst instance we only model the process after any initial structure apart from smallness of scale is forgotten. We should also note that the role of molecular diﬀusion is important in the initial phase of loss-of-memory of the mixing process. The dynamics of turbulence are represented solely by the function Pn (x(1) , . . . , x(n) , t) and considerable eﬀort has gone into understanding these Lagrangian properties: ﬁrst by G.I. Taylor10 essentially for mean ﬁeld properties P1 (x(1) ) , then by L.F. Richardson11 essentially for P2 (x(1) , x(2) , t), in the form of the nearest neighbour formulation P2 (r, t) for separations r = x(1) − x(2) . Subsequently there have been many developments based on Lagrangian modelling, direct numerical simulations, and laboratory experiments, primarily focused on more accurate speciﬁcation of both P1 (x(1) ) and P2 (x(1) , x(2) , t).12–16 To a large degree, the process of single particle dispersion and relative dispersion is well understood.17 The extension to general cluster modelling for n-particles, for arbitrary n, is less well advanced. However, simple models for this general problem emerge from an argument based on maximum entropy for conditioned clusters of particles and are the basis of this paper. To do the analysis we use a general ortho-normal transformation of the cluster coordinates, x(1) , . . . x(n) , to an equivalent coordinate representation using the centre-of-mass, Σ , and n − 1 separations, r(1) , . . . r(n−1) .18 In this case we have 1 , Σ = √ x(1) + · · · + x(n) , n and the explicit forms for the statistically identical uncorrelated separation vectors are given in the Appendix. The choice of transformation also leads to the PDF identity (interpreted as PDFs of the arguments, not a single function of the arguments) Pn (x(1) , . . . , x(n) , t) = Pn (Σ, r (1) , . . . , r(n−1) , t).

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There is some knowledge of the cluster statistics, at least at the level of mean-square displacements: for the standard Cartesian representation of vectors , (k) (k) (k) , Σ = (Σ1 , Σ2 , Σ3 ) and r (k) = r1 , r2 , r3

2 σΣij = Σi Σj

/ . (k) (k) . and σr2 ij = ri rj

One particularly important ‘known’ property is Richardson’s famous law,11 . / (k) (k) ri ri = ∆2 ∼ t3 , where is the rate of dissipation of kinetic energy in the turbulence. More generally the energy cascade in turbulence and its spectrum of spatial scales19 is responsible for the non-Gaussian character of relative dispersion.11,13 In addition, Taylor’s single-particle dispersion10 also gives another known property / . (k) (k) = 2σu2 t2L (t/tL − 1 + exp (−t/tL)) δij = σ12 δij , xi xj where tL is the Lagrangian integral time scale and σu is the turbulent velocity ﬂuctuation. In contrast to the separation statistics which are sensitive to the small-scale distribution of turbulent energy ﬂuxes, the single particle dispersion depends mainly on the energy of the large scales and is close to Gaussian. The one-particle dispersion length scale, σ1 , is used as the unit of measure (σ1 = 1) in most of this paper. We can also write these mean-square constraints as / . / . (1) (1) (1) (2) 2 + (n − 1) xi xj = σ12 ij + (n − 1)ρij σ12 = xi xj σΣij and

/ . / . (1) (1) (1) (2) − xi xj = σ12 ij − ρij σ12 σr2 ij = xi xj

in terms of one-particle variances and two-particle correlations. For the purposes of this note we will in the main deal with homogeneous isotropic turbulence in which case the mean-square displacements are 2 = (1 + (n − 1)ρ)σ12 δij σΣij

and σr2 ij = (1 − ρ)σ12 δij .

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3. Maximum Entropy Methods Subject to just these mean-square displacement constraints, we seek the n-particle cluster PDF that maximises the entropy S (as deﬁned in information theory where the shape of the cluster at any point in time represents ‘information’),20,21 S=− Pn log Pn d3 Σd3 r (1) . . . d3 r (n−1) . (3) Ω

For the full ensemble of cluster conﬁgurations, the solution is

3

1

˜2Γ (2π)− 2 n Γ

n−1 2

exp −

1 2

Pn (Σ, r(1) , . . . , r(n−1) , t) = (k) (k) ˜ ij Σj + Σi Γ ri Γij rj ,

(4)

k

˜ −1 = σ 2 , Γ−1 = σ 2 simply a jointly Gaussian form. Here we have Γ r ij ij ij Σij 0 0 0˜ 0 ˜ with Γ = 0Γij 0 and Γ = Γij . This solution follows from the standard calculus-of-variations maximisation of (3) using the functional n−1 , (k) , (k) (k) 2 − log Pn λij ri rj − σr2 ij + µij Σi Σj − σΣij Ω

k=1

×Pn (Σ, r (1) , . . . , r(n−1) , t)d3 Σd3 r . . . d3 r (n−1) , (k)

where λij and µij are Lagrange multipliers used to impose the mean-square constraints. The functional variation Pn + δPn is stationary when log Pn = −

n−1

(k) (k) (k)

λij ri rj

− µij Σi Σj + C

k=1

for some constant C. The normalisations and constraints then ﬁx the Lagrange multipliers and the constant and the solution is obtained, in this case the jointly Gaussian form as given in (4). However, from the earliest considerations of relative dispersion5 it has been known that the separation statistics are non-Gaussian and therefore that (4) is not a good model of the cluster statistics, although mean-ﬁeld (n = 1) Gaussian plumes are the fundamental bedrock of pollution dispersion models in the turbulent atmosphere.22 The non-Gaussian character of the separation process is essentially one of scale; for example in isotropic turbulence the separation PDF has the form P2 (r, t) = P2 (r, t) which is independent of the orientation of the separation vector and depends on just the magnitude of the separation. The non-Gaussian properties of the scale of the separation thus determine the

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non-Gaussian properties of the full separation. Similarly, scaled joint separations describing the shape of three particles have simple distributions: , , P3 ˆr (1) , ˆr (2) , t = P3 ˆr (1) .ˆr(2) , t = (4π)−2 . In direct numerical simulations of turbulence where normalised separations, ˆr (k) = r (k) /r(k) , are tracked, it is found that the properties of random triangles embodied by these vectors are reasonably well described by Gaussian clusters independent of scale.14,23 The conclusion of this section is that, overall, the turbulent mixing behaviour represented by the dynamics of clusters of particles initially released at a point cannot act to simply just maximise the entropy of the cluster conﬁguration (with a ﬁnite variance). Thus some information is preserved in the cluster. For example, the ﬂux of dispersing particles through the length scales characterising the energy cascade should have some signature of the cascade in the cluster statistics, at least for scale. We next consider a simple way that scale-information alone is carried by turbulent mixing and seek to maximise the entropy of the cluster for ﬁxed measures of scale. This is eﬀectively a matter of rescaling the cluster at each instant in time to maintain a ﬁxed ﬁdelity of ‘resolution’ of the cluster, like zooming out with a microscope. The question is whether the rescaling creates shape information in the cluster, or whether a self preserving property maximises the entropy in the cluster, i.e. the minimal amount of information survives the turbulent mixing process at any ﬁxed scale. 4. Conditioned Ensembles: Cluster Separation Conditioning A powerful method for isolating the eﬀect of scale on the cluster properties is to condition the statistics, that is, select particular sub ensembles with a ﬁxed-scale property. For example, we can take an ensemble of clusters with a ﬁxed net magnitude of separations for M pairs of particles: M

2

r(k) = M R2

, 2 (k) (k) ri ri = r(k) .

k=1

This particular choice is the most useful mathematically for our purposes and leads to a fully self-consistent prescription of n-particle statistics. Essentially it is a cluster with a prescribed mean-square separation, R2 , between pairs of particles, without any prescribed directional or shape information for the cluster. It is ultimately necessary to take the limit M → ∞ and obtain a ‘ﬁxedpoint’ solution independent of this parameter. This ﬁxed-point solution

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(which we have obtained) is the unique maximum entropy model we seek. There are other pathways to this solution, with other more abstract rescaling methods, but the end result given below is useful in its own right, independently of precisely how it is derived. The beneﬁt of the derivation, however, is that it can be understood in the context of (minimised) information content of the shape distributions in Lagrangian clusters. We now seek the conditioned probability density function for separations , Pn|R r (1) , . . . , r(n−1) , t that maximises the conditioned entropy, Pn|R log Pn|R d3 r (1) . . . d3 r (n−1) , SR = − ΩR

for the conditioned sub-ensemble denoted ΩR . This is again subject to the minimalist set of constraints . / (k) (k) σr2(k) |R = ri rj |R = σr2(k) |R δij , ij

where the critical dependence on the net separation scale, R, is made explicit. We also note that the conditioned mean separations vanish in homogeneous isotropic turbulence by symmetry: / . r(k) |R = 0. As before the maximum-entropy solution is jointly ‘Gaussian’ in the form

, n−1 1 (k) (k) (1) (n−1) 2 , t = Cn (R)Γ exp − Pn|R r , . . . , r ri Γij (R)rj 2 normalised such that , Pn|R r (1) , . . . , r(n−1) , t|R d3 r(1) . . . d3 r (n−1) = 1, 2 ≤MR2 r12 +···+rn

which determines the function 3

Cn (R) → (2π)− 2 (n−1) as R → ∞ (or M → ∞) and is given in detail in the Appendix. However, the critical diﬀerence now is that the unconditioned statistics, averaged over the scale of the net cluster separations, are non-Gaussian. To do this average we use the PDF for scale, pM (R), which is calculated

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in the Appendix, but may simply be treated as an unknown function. The conditional probability identity we use is of course , , - ∞ (1) (n−1) ,t = Pn|R r (1) , . . . , r(n−1) , t pM (R)dR Pn r , . . . , r 0

which for the unconditional cluster separation PDF is written as , Pn r (1) , . . . , r(n−1) , t =

∞ n−1 1 (k) (k) 2 ri Γij rj Cn (R)Γ exp − pM (R)dR. 2 0 In fact, for two-particle separation statistics, with n = 2, the result simpliﬁes to

∞ 1 −2 −3 2 P2 (r, t) = C2 (R)σr(1) |R exp − σr(1) |R r pM (R)dR, 2 0 which by rearranging the dummy integration, by using a new variable s, and deﬁning a weighting function w(s), can be written as

∞ 3 1 2 − 32 2 w(s)s exp − sr ds. P2 (r, t) = (2π) (5) 2 1 Explicitly we have used the transformation deﬁned by s=

σr−2 (1) |R

and w(s) =

C2 σr−3 (1) |R pM

dσr(1) |R −1 1 dR . = C2 pM ds 2 dR

In this transformation the limits of the integration are crucial and are determined in (5) for the large M limit using the conditioned cluster properties that √ 1 σr2(1) |R = R2 and pM ∼ 0 for R > 3σ1 asM → ∞, 3 thus σr2(1) |R → 0 as R → 0 and σr2(1) |R → σ12 = 1 as R → ∞. Note that this last limit is based on the large-M limit which we must formally take at some point in our analysis.√As a consequence of the large M limit, pM has a sharp peak around R = 3σ1 (pM details are given in the Appendix). It corresponds to the cluster separating as if all particles were uncorrelated, which is the maximal possible spreading rate. We have also absorbed the unknown properties into a single weighting function, which by construction is independent of M (implying some relationship between the conditioned variance and the conditioned-variable PDF and also interpreting our results as a ﬁxed point limit as M tends to inﬁnity).

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For any prescribed single-pair separation PDF, P2 (r, t), we can determine the weighting function (see below for details and an example) and, more generally for n − 1 pairs, we then have n−1 , - ∞ 2 3 1 wn (s)s 2 (n−1) exp − s r(k) ds (6) Pn r (1) , . . . , r(n−1) , t = 2 1 k=1

as the generic form of the maximum entropy PDF for cluster separations. 3 However, it is simple to show that wn (s) = (2π)− 2 (n−1) w(s), thus (6) is entirely parameterised by the single-pair relationship (5). This is essentially the fundamental result of this paper, although a slight generalisation is given below. Note that this result is formally only possible for n-clusters with fewer particles than in the conditioning cluster overall, that is, n < M . If we do condition on clusters which are a smaller subset of the cluster whose entropy we seek to maximise, i.e. with n > M , then the results cannot be fully self consistent (with (5)) and can only be made consistent with a ﬁltered two-particle separation PDF, which is more nearly Gaussian than the speciﬁc form in naturally occurring turbulence. Only when n < M are we free to prescribe speciﬁc forms for P2 (r, t) (as say occurs in naturally occurring turbulence). In any case, we are formally taking the limit that M → ∞ with n ﬁxed, so there is not an issue with n < M . The fundamental ﬁxed point solution in this limit is (6) and it is independent of M . The speciﬁc choice to condition the cluster on the separations is to focus on internal structure within the cluster. It also exploits symmetry, but it is nevertheless restricted and provides only the marginal PDF for separations. It is possible, however, to generalise this approach to also account for bulk displacements, sometimes called the meandering, of the cluster. This generalisation is considered before applying the maximum entropy solution to scalar ﬁeld prediction. 5. Conditioned Ensembles: Joint Centre-of-Mass/ Separation PDF Within the conditioned M -cluster ensemble, we again consider properties of sub-clusters. The statistics for this ﬁxed net-separation cluster have properties for a centre-of-mass variable, and n − 1 other separation variables, which have the second-order moments . / (1) (1) σΣ|r ij = Σi Σj |R and σr(1) |R ij = ri rj |R , the latter essentially identical to the cluster form considered above.

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The maximum-entropy joint PDF for these constraints therefore has the form , Pn|r Σ, r(1) , . . . , r(n−1) , t =

n−1 1 ˜ 1 (k) 1 (k) ˜ ˜ 2 2 Cn Γ Γ exp − Σi Γij (R)Σj − , ri Γij (R)rj 2 2 with a straightforward reinterpretation of the variables. Next we have the unconditional non-Gaussian probability density function , Pn Σ, r(1) , . . . , r(n−1) , t =

∞ 1 ˜ 1 (k) (k) ˜ 12 Γ n−1 2 Cn Γ exp − Σi Γ Σ − Γ r pM dR. r ij j ij j i 2 2 0 For isotropic homogeneous turbulence this reduces to , Pn Σ, r(1) , . . . , r(n−1) , t =

∞ 1 −2 2 1 −2 −3 −3(n−1) (k) 2 ˜ Cn σ σr(k) |R r ˜Σ|R σr(k) |R exp − σΣ|R Σ − pM dR, 2 2 0 where we write 2 σΣ|R = (1 + (n − 1)ρ(R))σ12

and σr2(k) |R = (1 − ρ(R))σ12 ,

with the important limits being ρ(R) → 1 as R → 0 and ρ(R) → 0 as 2 R → ∞. In writing this we also assume that σ1|R = σ12 , which is reasonable and can be checked for self consistency with the ﬁnal solution (see below). By transforming the dummy integration variable (as above) we may therefore write , Pn Σ, r (1) , . . . , r(n−1) , t =

n

3 ∞ s (2π)−n 2 sΣ2 1 (k) 2 1 − w(s) exp − ds, (7) sr sn − n + 1 2 sn − n + 1 2 1 which is essentially equivalent to the previous form derived. Moreover we ﬁnd that, consistent with maximum entropy for clusters conditioned on scale of separation, the centre-of-mass is also non-Gaussian (although it is expected to be only weakly non-Gaussian):

∞ 3 sΣ2 3 w(s)s 2 1 exp − ds. (8) Pn (Σ, t) = (2π)− 2 3 2 sn − n + 1 1 (sn − n + 1) 2 For example, in the limit of large n, the centre-of-mass PDF explicitly becomes Gaussian (with a broad peak centred about zero displacement).

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Similarly, if the function w(s) is only signiﬁcantly diﬀerent from zero for large s, then the PDF in (12) also explicitly becomes Gaussian (plus small corrections). Some calculations for this PDF in the latter case (which is typical) are given below. We also note that representation (7) always gives the one-particle result

, 3 1 P1 x(1) , t = (2π)− 2 σ1−3 exp − σ1−2 |x(1) |2 , 2 which is fully anticipated and represents the probability density that maximises the entropy in the full ensemble subject just to the mean-square dispersion. Finally, we have from the Appendix the result for large M that the 2 = 1, is focused on a narrow band distribution of pM deﬁned with σ1|R 2 2 around R ∼ 3σ1 ∼ 3. Thus in the large M limit, the cluster of particles conditioned on average particle pair separations, tends on average towards weakly correlated pair motions, where each individual trajectory has single statistics close to unconditioned behaviour. For example, using √ particle (1) (2) 2r = x − x(1) from the Appendix, we have 2 2 3σ1|R − 3ρ(R)σ12 = 3σ1|R − 3ρ(R) = R2 ∼ 3. 2 Thus in this limit, ρ ∼ 0 and σ1|R ∼ σ12 = 1 and it is possible to demonstrate self-consistency for the appropriate limits of integration in (7). Despite the limiting behaviour of weak correlations in the overall M cluster, the behaviour of sub-clusters can be prescribed with arbitrarily strong pairseparation coherence. The single-pair PDF, for example, can be prescribed arbitrarily.

6. Scaled Separations The conditioned maximal entropy PDFs considered above dealt with subensembles. It is not possible to easily deal with the maximal entropy PDF for the full ensemble with all the known constraints consistent with nonGaussian separation behaviour. However, the PDF for scaled separations , Pn ˆr(1) , . . . , ˆr (n−1) , t that maximises the entropy for the full ensemble is simply , Pn ˆr (1) , . . . , ˆr (n−1) , t = (4π)1−n , which is embodied in the full Gaussian cluster behaviour. Importantly, this prediction is also a property of the PDFs (6) and (7), when scale is removed and the joint PDF for ˆr(k) vectors is obtained. In this restricted

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sense these PDFs also represent maximum entropy for the full ensemble, and emphasise the role of scale in controlling the non-Gaussian aspects of relative dispersion. 7. Summary of the n Particle Lagrangian PDF We have derived a general solution for the n particle Lagrangian transition PDF with the following properties: correct one-particle displacement statistics; correct two-particle separation statistics; correct symmetries and transformation properties; and maximal entropy for clusters conditioned on scale, where the entropy is related to the ‘information’ content of the cluster shape. Initially molecular diﬀusion will destroy any shape information in the cluster and subsequently the cluster characteristics are only described by scale (provided the Schmidt number is not too large).24 This model essentially represents the ‘best’ that can be embodied with the information available, and formally imposes the ‘least’ arbitrary structure according to these constraints. It imagines that the turbulent mixing process is one that has non-Gaussian scale properties but seeks to maximally randomise the internal cluster orientations for ﬁxed scale. This may or may not be strictly true, although some evidence suggests that this is precisely the case beyond the dissipation range of viscous eﬀects.14,23 However, if speciﬁc internal cluster orientation structure emerges from further investigations, say properties . for three-particle/ Lagrangian clusters, embodied (m) (m) (n) (n) say in moments like ri rj rk rl |R , then it is possible to generalise the maximum entropy approach used here to incorporate such new information. Note that this framework anticipates that no new informa25,26 because all tion can be encoded from four-particle ,. / - (tetrad) dynamics (l) (m) (n)

triple products ri rj rk |R = 0 must vanish. This suggests that simple cluster topologies for two, three and four particles (deﬁning a length, area and volume respectively) are equivalent, but more complex structure is potentially possible with ﬁve or more particles. However, preliminary applications of the model to scalar mixing problems, both for laboratory experiments and for atmospheric dispersion, suggest that the information encoded in these higher-order conditional moments is not practically important. 8. Applications to Scalar Statistics We brieﬂy return now to the scalar statistics to illustrate some of the properties. For deﬁniteness, consider an instantaneous unit mass point source

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at the origin. The concentration moments according to (1) are θn (x, t) = Pn (0, 0, . . . , 0, 0|x, x, . . . , x, t) =

32 ∞ 3 sn 1 snx2 (2π)− 2 n w(s) exp − ds, sn − n + 1 2 sn − n + 1 1 where x2 = x.x and the weight, w(s), satisﬁes

∞ 1 3 3 w(s)s 2 exp − sr2 ds. P2 (r, t) = (2π)− 2 2 1

(9)

(10)

We are regarding the left-hand-side of (10) as known, usually by some other modelling process.11,15,16 In (9) the reverse displacements expressed as centre-of-mass and separation vectors are √ Σ = nx, r (1) = · · · = r (n−1) = 0. We can write (10) as 1

P2 (r, t) = e− 2 r

2

∞ 0

1 w(s) ˜ exp − sr2 ds, 2

(11)

where 3

3

w(s) ˜ = (2π)− 2 w(s + 1)(s + 1) 2 which uses the Laplace-transform framework for more convenient manipulations. In fact, the two-particle separation PDF is directly related to the ‘plume integral’ concentration covariance . / √ θ(x, t)θ(x + 2r, t) d3 x = 23/2 P2 (r, t) where the net scalar dissipation rate is d d χ=− θ(x, t)2 d3 x = −23/2 P2 (0, t). dt dt Thus the Corrsin-Obukhov inertial-sub-range property for scalar ﬁelds implies that 2

1

2

P2 (r, t) = P2 (0, t) − 2− 3 Cθ χ− 3 r 3 + . . . where is the energy dissipation rate in the turbulent ﬂow and Cθ is a constant with a value near three.27 This implies that the weighting function 4 is a power-law for large s with w(s) ˜ ∼ s− 3 as s → ∞ . To see this we take the derivative with respect to r in (11) ∞ 1 1 2 ∂P2 (r, t) (− 12 sr2 ) ds = − 2 3 C χ− 13 r− 13 + . . . = −re− 2 r (s + 1)w(s)e ˜ θ ∂r 3 0

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where the singular structure function behaviour corresponds to power law decay of the weighting function w ˜ ∼ s−λ for large s, where as r → 0 ∞ 2 1 4 s1−λ e(− 2 sr ) ds ∼ r− 3 0

which then determines the value of the exponent λ = 43 . A simple approach is to use a parameterised form for the pair separation PDF: Richardson’s nearest neighbour function,11

1 r2 P2 (r, t) = α exp −βr2/3 − , (12) 2 σ12 where α and β are functions of time. In fact, a direct functional dependence, * α = α(β), simply follows by normalising the PDF ( P2 d3 r = 1). The elapsed dispersion time can also be related to the pair correlation ρ(t) = ρ(β) = 1 − The relation

1 r2 . 3 σ12

r = 2

r2 P2 d3 r,

which is also a function of β, then ﬁxes β = β(t) as a function of time, with say the Richardson’s law r2 ∼ t3 and Taylor’s dispersion for σ12 . More general models for pair dispersion can extend this selection of β to arbitrary times.13 The addition of the r2 term in the argument of the exponential in (12) permits sensible large-scale separation behaviour. Very large separations necessarily follow Gaussian statistics and are related to maximal separation rates. These are, in general, uncorrelated single-particle motions for each of the pair of particles. Accounting for this behaviour also simpliﬁes the application of the Laplace transform model (11). Richardson’s form (12) has had reasonable veriﬁcation from experiments15,16 and the maximumentropy weight function √

3 1/3 1 α ∞ βz sin (13) exp − βz 1/3 − sz dz, w(s) ˜ = π 0 2 2 is obtained from (11) using the inverse Laplace transform. Figure 1 shows the behaviour of Richardson’s pair separation PDF (plotted as the structure function C2 (r, t) = P2 (0, t) − P2 (r, t)). Curves at three times are plotted, and with increasing time the correlation ρ = ρ(t) falls

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and concentration ﬂuctuations also decay. The corresponding centre-ofmass PDFs, calculated from (8), are shown in ﬁgure 2. The time choice corresponds to ρ = 0.65, with the particles in the cluster still relatively well correlated. Figure 1 emphasises the asymptotic power-law Corrsin-Obukhov behaviour,1 while ﬁgure 2 demonstrates the near Gaussianity of the centreof-mass PDFs (compared with the Gaussian forms which are plotted as the dashed lines). 102 ~ r2/3 10

C2(r,t)

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100

10

U=0.75

-1

U=0.65 U=0.50

10

-2

-3

10

10

-2

10

-1

10

0

1

10

r/V1 Fig. 1. The scalar structure function for an instantaneous point source for decreasing pair correlation: ρ = 0.75, ρ = 0.65 and ρ = 0.5, corresponding to increasing time. Corrsin-Obukhov power law scaling applies for small scales and r 2/3 is shown as dashed line.

For the concentrations in this paper we use the time dependent concentration scale deﬁned by m θ∗ = 3 , σ1 where m is the mass input (assumed to be unity) and the Taylor-dispersion length scale σ1 grows with time. We now have θn (x, t) − 32 (n−1)

∞

w(s) ˜

= (2π)

0

(s + 1)n−1 sn + 1

32

(s + 1)nx2 exp − ds, 2(sn + 1) (14)

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P26(6V)

Maximum Entropy Gaussian

n=2

0.03

U=0.65

n=3

0.02

n=4 0.01

n=5

0 0

1

2

3

4

5

6V

Fig. 2. A selection of centre-of-mass PDFs for clusters of n = 2, 3, 4 and 5 particles, when the pair correlation is high at ρ = 0.65, corresponding to the separation behaviour in ﬁgure1. The dashed curves show Gaussian PDFs for the respective clusters.

102

Mass fraction, Pmass(T)

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1

10

0

~ T

10-1 10

-2

10

-3

U=0.75 U=0.65

10-4 10

U=0.50

-5

10-2

10-1

100

101

102

103

Concentration,T

Fig. 3. Mass fraction levels for an instantaneous puﬀ corresponding to Richardson’s pair separation function for small scales as shown in Figure 1 (correlation times ρ = 0.75, ρ = 0.65 and ρ = 0.5). The power law tail for large concentrations means that the concentration variances must diverge.

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which is a fairly complex and unwieldy expression. However, we can simplify the interpretation by integrating over volume (with respect to x). These plume-integral moments, formally deﬁned for n ≥ 1 ∞ 3 3 3 w(s)(s ˜ + 1) 2 (n−2) ds, θn = (2π)− 2 (n−2) n− 2 0

will then formally correspond (see below) to the scalar plume integral distribution (the PDF integrated over volume) P˜ (θ) = P (θ, x, t)d3 x . For the unit mass source, the plume integral leads to the mass-fraction distribution, Pmass (θ) = θP˜ (θ), which can be manipulated to give ∞ 3 −3 Pmass (θ) = (2π) 2 (1 + s) w(s)ds ˜ (θ > 1) , 2/3 θ −1 ∞ 3 (1 + s)−3 w(s)ds ˜ (0 < θ < 1). (15) = (2π) 2 0

The mass-fraction simply gives the proportion of mass in all space between concentration levels θ and θ + dθ. For purely Gaussian n-particle turbulent mixing (with (4) in §3), these results are all exactly equivalent to Giﬀord’s meandering Gaussian plume (or puﬀ in this case).28 For our case, the non-Gaussian behaviour, corresponding to internal ﬂuctuations - which is equivalent to information content in the plume - is imposed by the weighting function w(s). ˜ The purely Gaussian case has w(s) ˜ as a function concentrated at a single value of s. This corresponds precisely to no internal ﬂuctuations or structure in the scalar plume and clearly maximises the loss of information as a consequence of turbulent mixing within the plume. The existence of internal ﬂuctuations requires non-Gaussian eﬀects with w(s) ˜ generally distributed over a semi-inﬁnite band of s values. The Richardson parameterisation, which corresponds to w ˜ in (13), is used to determine the mass fraction as a function of scalar concentrations and is shown in ﬁgure 3. This ﬁgure emphasises the power-law tail for the mass fraction (frequency) of high concentrations in the turbulent mixing process. The ﬁgure also shows that as time increases (decreasing ρ), the frequency of larger concentrations decreases. This reﬂects the inexorable march of mixing, diluting the high concentrations as the puﬀ mixes into the ‘clean’ environment. However, for low concentrations (whenever θ < 1) it is remarkable to note the similarity between purely Gaussian mixing and Richardson mixing. The purely Gaussian case has a uniform mass fraction

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distribution, and a sharp cut-oﬀ tail at θ = 1. The results in ﬁgure 3 show that the main eﬀect of non-Gaussian mixing is to add a power-law tail to the mass fraction distribution. The detailed calculations transferring between (14) and (15) are quite technical and rely on being able to mathematically solve for the scalar PDFs exactly in the Gaussian mixing case. This procedure is outlined elsewhere (along with veriﬁcation of the model with wind tunnel data for truncated inertial ranges).4 As a consequence of the power-law weighting function, w ˜ ∼ s−4/3 for large s, the tail of the mass-fraction distribution is Pmass (θ) ∼ θ−20/9

as

θ→∞

and the plume integral tail behaves like P˜ (θ) ∼ θ−29/9

as θ → ∞ ,

implying that concentration skewness in the plume is inﬁnite. It is remarkable that we have used a Lagrangian moment generating technique to derive our results but where, formally, only the ﬁrst two moments are ﬁnite. For other initial source distributions the power-law tail is less ‘heavy’ than for the instantaneous point source. The large concentration limiting case has application to atmospheric dispersion predictions of inertial subrange properties, with power-law tails in concentration PDFs for line and area sources imposing power-law peak-to-mean predictions in remarkable agreement with power-laws in practical use.5,29 On the other hand, when the scalar ﬁeld small-scale properties are analytic (say in the dissipation subranges) with the Taylor series expansion P2 (r, t) = P2 (0, t) −

1χ 2 r + ,... 2κ

where, for scalar molecular diﬀusivity κ, 2 1 ∂θ ∂θ d3 x , χ = 2κ ∂xi ∂xi the weight function falls oﬀ faster than any power of s and therefore w ˜ is exponentially small for large s. This means that all moments of concentration are deﬁned and moreover it means that the ‘inﬁnite’ inertial-range moments are in fact bounded by the small-scale molecular properties of the scalar mixing. These dissipation range (moderate Reynolds/Pecl´et number) results have found application for the analysis of wind-tunnel plumes, implying exponentially small PDF tails, and are found to predict overall concentration PDFs very well.4

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The truncation of the power-laws by molecular eﬀects is equivalent to an ultra-violet renormalisation of the system, permitting the divergent moments in (14) to formally contribute to the construction of the PDF.30 These details about puﬀ statistics are important, particularly for experiments undertaken to measure puﬀ behaviour. Moments that depend on molecular properties require high-resolution measurements, to resolve the small scales, and intensive sampling to obtain stable statistics when the tails are heavy. At present there are no reliable detailed experiments for puﬀs. The instantaneous puﬀ in turbulence with inertial ranges (say in the atmosphere or ocean) is quite expensive and diﬃcult to measure and our knowledge remains incomplete.31 To connect to real world experiments it may also be necessary to account for ﬁnite-size source characteristics, for example, by using equation (1) with a unit-mass ﬁnite source

− 3 1 x2 φ(x) = 2πσ02 2 exp − 2 2 σ0 centred on the origin with size σ0 . This selection will also truncate the tails − 3 of the scalar concentration distribution (θ ≤ 2πσ02 2 ) and ‘renormalise’ the system, but the details are more diﬃcult to obtain analytically. 9. Conclusions A general solution is provided for a practically important property in the Lagrangian theory of turbulent mixing. The prediction gives the joint PDF for displacements of particles in clusters which are released from a point (assuming uncorrelated molecular diﬀusion processes initially spread the cluster). The complex function embodied in this n-body solution seems an unlikely tool for simple practical application, but the results obtained here are ideally suited for application in scalar ﬁeld prediction in turbulent environments when the sources are localised. The application of these Lagrangian cluster statistics allows predictions of scalar properties in ‘puﬀs’ of material giving remarkable detail which cannot currently be computed or estimated by other methods at geophysical-ﬂow scales. These include: predictions of the probability density functions of scalar concentrations, high-order concentration moments, high-order scalar structure functions, and shape statistics for Lagrangian clusters. The solution relies on a minimum of input information. The main external input, the knowledge of the pair separation PDF, is increasingly well known in practical situations, from either models or simple theoretical parameterisations. The solution given here emerges from an application of a maximum entropy argument which

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states that turbulent mixing, on average, does not generate shape information in clusters of particles as the cluster disperses. Generalisation of this powerful principle allows clear pathways for incorporating greater explicit Lagrangian structure in due course, and also for extension to anisotropic and inhomogeneous ﬂow, but it is already developed enough for practical applications for small-scale properties in turbulence. Finally, the key practical result from the analysis is that as a consequence of the inertial-range scalar structure function, which is a power law in separation distance, r, for small r, the tail of the scalar concentration distribution is also a power law function, but of scalar concentration for large values. There is some practical evidence that this is in fact the case. Remarkably the result implies that the concentration skewness (the third moment) in a puﬀ is inﬁnite, although in practice molecular material properties (truncating the inertial ranges) bound the value. Acknowledgement The CSIRO Complex System Science Program helped support this work with funding for the project Insect Tracking in Chemical Plumes. Appendix A. The ortho-normal transformation The transformation of the n particle coordinates, from the absolute positions x(1) , x(2) , . . . , x(n) to an equivalent centre-of-mass, Σ , and n − 1 separations, r (1) , . . . , r(n−1) , is uniquely given by 1 , Σ = √ x(1) + · · · + x(n) , n , 1 r (1) = √ x(2) − x(1) , 2 1 , (3) (2) r = √ 2x − x(2) − x(1) , 6 .. . , 1 r (n−1) = & (n − 1)x(n) − x(n−1) − · · · − x(1) . n(n − 1) The properties of this transformed system are . / . / . / . / (k) (k) (l) (k) (k) (l) (l) Σi rj = 0, ri rj = 0, ri rj = ri rj

∀k, l

(k = l)

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and the Jacobian of the transformation is ∂ x(1) , . . . , x(n) = 1 . ∂ Σ, r(1) , . . . , r(n−1) Thus the variables are all uncorrelated and the separations are all statistically identical. Appendix B. Normalisations The normalisation of the PDF for conditioned clusters is given by integrating over an M dimensional hypersphere (scaled for unit radius): −1 √ 3(M−1)

Cn (M ) = (4π)1−n

Sn

1

2

r3n−4 e− 2 r dr

0

where, in terms of Gamma functions32 n−1 1 Γ 12 . Sn = 2 Γ 32 (n − 1) This permits the normalisation of the maximum entropy PDF for the conditioned clusters. Appendix C. Scale PDF The fundamental PDF for net separation (scaled by σ1 ) can be derived from 2 equation (6) with n = M that under change of variables (R2 = r(k) plus 3(M − 1) − 1 hypersphere angle variables which do not explicitly appear in the PDF for isotropy) simply transforms into

∞ 1 2 3 − 32 (M−1) 3(M−1)−1 ˜ (M−1) 2 SM R w(s)s exp − sR ds pM (R, t) = (2π) 2 1 where the normalisation factor S˜M is given by

∞ 1 2 −1 − 32 (M−1) 3(M−1)−1 ˜ SM = (2π) r exp − r dr 2 0

3 3 1 (M − 1) . = (π)− 2 (M−1) Γ 2 2 The limit of large M is of most interest, and the steepest-descent calculation & for ﬁxed R gives (with some care for R close to 3(M − 1))

& 3(M − 1) −3 pM (R, t) = 3(M − 1)R w , 0 ≤ R 3(M − 1). 2 R

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Moreover, for ﬁnite R, and in the limit M → ∞, the large s behaviour of 17 w(s) is signiﬁcant, namely, w(s) = w∞ s− 6 as s → ∞, so that & 8 pM (R, t) = w∞ (3(M − 1))−11/6 R 3 , 0 ≤ R 3(M − 1). In this approximation we also have √3(M−1) pM (R, t)dR ≈ 1. 0

& Beyond the pM peak at R = 3(M − 1), pM vanishes exponentially fast as M increases. ˜ 2 = (M − 1)−1 R2 , When written as the average separation variable, R the M independent limit solution is easily obtained. References 1. A.S. Monin and A.M. Yaglom, Statistical Fluid Mechanics, vol. II, (MIT Press, Cambridge, Mass., 1975). 2. G.K. Batchelor, Diﬀusion in a ﬁeld of homogeneous turbulence. II. The relative motion of particles. Proc. Camb. Phil. Soc., 48. 345-362 (1952). 3. B.L. Sawford, Recent developments in the Lagrangian stochastic theory of turbulent dispersion. Bound. Layer Meteorol., 62: 197-215 (1993). 4. M.S. Borgas, Lagrangian-Particle Models of PDF Integrals and Small Scale Mixing In Turbulent Plumes. Boundary Layer Meteorology, (Submitted) 5. M.S. Borgas, The Mathematics of Whiﬀs and Pongs, Proc. Enviro2000 Odour Conf., Clean Air Society of Australia & New Zealand. Sydney (2000). 6. S.B. Pope, Turbulent Flows, (Cambridge University Press, 2000). 7. B.L. Sawford, P.K. Yeung, M.S. and Borgas, Comparison of backwards and forwards relative dispersion in turbulence. Physics of Fluids, 17 095109 (2005). 8. P.A. Durbin, A stochastic model of two-particle dispersion in isotropic homogeneous turbulence. J. Fluid Mech., 100, 279-302 (1980). 9. D.J. Thomson, A stochastic model for the motion of particle pairs in isotropic high-Reynolds-number turbulence, and its application to the problem of concentration variance. J. Fluid Mech., 210, 113-153 (1990). 10. G.I. Taylor, Diﬀusion by continuous movements. Proc. Lond. Math. Soc. (2), 20. 196-211 (1921). 11. L.F. Richardson, Atmospheric diﬀusion shown on a distance-neighbour graph. Proc. Roy. Soc. London A, 110, 709-737 (1926). 12. A. La Porta, G.A. Voth, A.M. Crawford, J. Alexander and E. Bodenschatz, Fluid particle accelerations in fully developed turbulence. Nature, 409, pp. 1017-1019 (2001). 13. M.S. Borgas and P.K. Yeung, Relative Dispersion in Isotropic Turbulence: Part 2 Stochastic model simulations with Reynolds number dependence. J. Fluid Mech., 503: 125-160 (2004).

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14. P.K. Yeung, S. Xu, M.S. Borgas and B.L. Sawford, Scaling of multi-particle Lagrangian statistics in direct numerical simulations. Proceedings of In IUTAM Symposium on Reynolds Number Scaling in Turbulent Flow, edited by A.J. Smits, Kluwer Academic Publishers, Dordrecht; pp. 163-168 (2004). 15. M. Bourgoin, N.T. Ouellette, H. Xu, J. Berg and E. Bodenschatz, The Role of Pair Dispersion in Turbulent Flow. Science, 10 February 2006: 835-838 (2006). 16. S. Ott and J. Mann, An experimental investigation of the relative diﬀusion of particle pairs in three-dimensional turbulent ﬂow. J. Fluid Mech., 422, 207-223 (2000). 17. B.L. Sawford, Turbulent relative dispersion. Ann. Rev. Fluid Mech., 33, 289-317 (2001). 18. B.L. Sawford, Spatial Structure of Concentration Moments in Homogeneous Turbulence. Proc. 10th Symposium on Turbulence & Diﬀusion, American Meteorological Society. 180-183 (1992). 19. A.N. Kolmogorov, The local structure of isotopic turbulence in an incompressible viscous ﬂuid. Dokl. Akad. Nauk. SSSR 30, 301-305 (1941). 20. C.E. Shannon, A mathematical theory of communication, Bell System Tech. Journal, 27, 379-423; 623-656. Reprinted in Shannon, C. E. and Weaver, W., The Mathematical Theory of Communication, University of Illinois Press, Urbana, 1949. 21. E.T. Jaynes, Information Theory and Statistical Mechanics Phys. Rev., 106, 620-630 (1957). 22. F. Pasquill, Atmospheric Diﬀusion, 2nd Edition, (Ellis Horwood, Ltd., Chichester, England, 1974). 23. M.S. Borgas, Meandering plume models in turbulent ﬂows. Proc. 13th Australasian Fluid Mech. Conf. (Monash University, Melbourne, Australia, Dec. 1998), 139-142. 24. M.S. Borgas, B.L. Sawford, S. Xu, D.A. Donzis and P. Yeung, High Schmidt number scalars in turbulence: structure functions and Lagrangian theory. Physics of Fluids, 16 (11): 3888-3899 (2004). 25. A. Pumir, B. I. Shraiman and M. Chertkov, Statistical geometry of Lagrangian dispersion in turbulence, Phys. Rev. Lett., 85 (25) 5324-5327 (2000). 26. B.I. Shraiman and E.D. Siggia, Scalar Turbulence. Nature, 405, 639-646 (2000). 27. K.R. Sreenivasan, The passive scalar spectrum and the Obukhov-Corrsin constant. Phys. Fluids, 8(1), 189-196 (1996). 28. F.A. Giﬀord, Statistical properties of a ﬂuctuating plume dispersion model. Adv. Geophys., 6:117-137 (1959). 29. M.S. Borgas, The Use of Peak to Mean Ratios. Proceeding of CASANZ (Clean Air Society of Australia & New Zealand) Odour SIG, November, 2000. 30. V. Yakhot, Ultraviolet dynamic renormalization group: Small-scale properties of a randomly stirred ﬂuid. Phys. Rev. A 23, 1486-1497 (1981).

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31. R.J. Munro, P.C. Chatwin and N. Mole, The high concentration tails of the probability density function of a dispersing scalar in the atmosphere. Boundary-Layer Meteorology 98: 315–339 (2001). 32. M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions, (Dover, New York, 1975).

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A REVIEW OF RECENT INVESTIGATIONS INTO HIGH REYNOLDS NUMBER WALL-TURBULENCE JASON P. MONTY AND MIN S. CHONG Department of Mechanical and Manufacturing Engineering, The University of Melbourne Parkville, VIC 3010, Australia [email protected] Recent progress in the study of turbulent boundary layers is reviewed, beginning with the current understanding of mean velocity scaling. A comparison between Nickels’ mean velocity curve-ﬁt and high Reynolds number pipe ﬂow data is provided, along with further evidence of a logarithmic mean velocity proﬁle from a new, high Reynolds number boundary layer facility at Melbourne. Discoveries of physical mechanisms of wall-turbulence are then reviewed, in particular the evidence and signiﬁcance of hairpin vortices. The attached eddy model of Perry and Chong is revisited and implications of its predictions for high Reynolds number ﬂows are discussed. Finally, results from a recent ﬁeld trip to the SLTEST atmospheric surface layer facility are shown to agree well with wind-tunnel measurements and also provide evidence of hairpin vortex packets at extremely high Reynolds numbers.

1. Introduction Fluid dynamics has now entered into a second century of boundary layer research following the recent 100 year anniversary of the discovery of the ﬂuid ﬂow phenomenon. It was in 1904 that Ludwig Prandtl introduced the concept of Grenzschicht (boundary layer) to the world (see Tani1 for an excellent review on the history of boundary layers). Considering the amount of work devoted to boundary layers, one might have expected a somewhat better understanding of turbulent shear ﬂows in general than has since been attained. Even for the most tractable of all such ﬂows, the ﬂow in a circular pipe ﬂow, there is still some uncertainty in our understanding, especially for rough pipes. Although progress has been made and should not be understated, the last century of turbulent boundary layer research has left many questions unanswered, and arguments over the form of even the simplest of attributes — the mean streamwise velocity distribution —

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continue to pervade the literature.2,3 Indeed, researchers are yet to agree on the scaling of velocity statistics of any order. The frequent revival of such arguments could possibly be attributed to a focus on statistical rather than physical aspects of turbulence in wall bounded ﬂows. The task of measuring turbulence statistics is generally easier than visualising the ﬂow; the latter being practically impossible at high Reynolds numbers. Improvements in measurement technology, including hot-wire anemometry (mid-century) and laser Doppler anemometry (more recently), combined with high-speed computing, have resulted in an increasing number of statistical studies of turbulence. While less frequent, there have been important contributions made toward a physical understanding of largescale structures in wall-bounded turbulence, beginning with the work of Theodorsen in 1952.4 Theodorsen was able to show that the boundary layer should be comprised of ‘horseshoe’-shaped vortex loops. Townsend (1956)5 presented a more general approach to the physical problem; namely, that a ﬁeld of organised eddies could describe the turbulent ﬂow, provided that: “The velocity ﬁelds of the main eddies, regarded as persistent, organised ﬂow patterns, extend to the wall and, in a sense, they are attached to the wall.” p152, Townsend (1976).6 Townsend’s idea is now generally known as the Attached Eddy Hypothesis (AEH). Taking this hypothesis, Theodorsen’s ‘horseshoe’ or ‘hairpin’∗ vortex model, and the ﬂow visualisations of Head and Bandyopadhyay7 and Perry, Lim and Teh,8 Perry and Chong9 proposed a physical, kinematic model of wall-turbulence. In this model, the boundary layer is considered to be ﬁlled with a forest of geometrically similar hairpin vortices. In the past decade this concept has gained wider acceptance, due mainly to the increasing change in focus of turbulence research toward further understanding the physical mechanisms using both ﬂow visualisation and statistical data. This has been aided by rapid improvements in high-speed computing which has allowed the Direct Numerical Simulation (DNS) of laboratory-scale wall-turbulence. Also, Particle Image Velocimetry (PIV) techniques are now extensively employed and readily available. Examples of recent research conﬁrming the existence of hairpin structures includes the PIV measurements of both Adrian et al.10 and Ganapathisubramani et al.,11 and the channel ﬂow DNS analyses of Liu and Adrian12 and Jimenez et al.13 ∗ ‘Hairpin’

is the preferred description of characteristic structure geometry in recent years and will, therefore, be used hereafter.

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The development of physical mechanisms in conjunction with statistical methods is undoubtedly a positive step. Yet, regardless of how the data are are to be analysed, there remains a fundamental deﬁciency: there is a lack of high Reynolds number turbulent shear ﬂow measurements. While it has always been thought that the concepts developed from low to moderate Reynolds number ﬂow studies could be extended to high Reynolds numbers, it has now become clear that wall-turbulence arguments will not be settled until attempts are made to take accurate measurements at high Reynolds numbers. However, high Reynolds number data often create more uncertainty, especially if they produce results contradicting the outcomes of lower Reynolds number investigations. The interest in high Reynolds number turbulence is not purely academic. The turbulent shear ﬂows occurring on the wings of aeroplanes, on the hulls of ships and submarines and in the gas and oil pipelines are generally classiﬁed as ‘high’ Reynolds number ﬂows — that is, Reθ = θUτ /ν > O(105 ), where θ is the momentum thickness, Uτ is the friction velocity and ν the kinematic viscosity. Enormous energy loss is attributed to turbulence in shear layers, so that the lack of understanding of high Reynolds number boundary layers and how to control them is likely inhibiting the progress of many of our technologically advanced vehicles. Although there is limited high Reynolds number wall-turbulence data, there are ongoing investigations that aim to remedy this problem. For example, the pressurised Princeton ‘Superpipe’ continues to provide major contributions to our understanding of pipe ﬂow; the German-Dutch windtunnel (DNW-LLF) generates a zero-pressure-gradient boundary layer having Reθ = 1.15 × 105; and at the University of Melbourne, a large boundary layer wind-tunnel capable of reaching Reθ = 8 × 104 has been constructed (see Nickels et al.14 ). A novel high Reynolds number facility is the very large boundary layer occurring naturally over the salt ﬂats of Utah, USA. This was the site to which the authors conducted a ﬁeld trip in the Northern summer of 2005. Having an estimated Reynolds number of Reθ = O(106 ), the Surface Layer Turbulence and Environmental Science Test (SLTEST) facility at Utah has been studied regularly for over a decade (see Metzger and Klewicki15 ). Unfortunately, acceptance of data from this facility has been diﬃcult due to the perceived uncertainties concerning the relevance of the results to wind-tunnel boundary layer measurements. Some evidence of similarities between SLTEST results and typical laboratory results will be presented later in this article.

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In summary, the future of turbulence research is far from bleak. Rapidly advancing technology is improving measurement techniques, providing more accurate statistics, better visualisations and higher Reynolds number numerical simulations. A renewed focus on the physical nature of turbulence, combined with an increasing number of high Reynolds number studies, encourages one to look forward with anticipation to a second century of wall-turbulence research. 2. Mean streamwise velocity 2.1. A Brief History of Mean Flow Scaling In 1911, Heinrich Blasius16 (a student of Prandtl’s) gathered all available data and found that non-dimensional pipe ﬂow resistance was a function of Reynolds number only; that is, the friction factor, λ=

1 8Uτ2 = 0.3164Re− 4 , Ub2

(1)

where Re = Ub D/ν, D is the pipe diameter and Ub is the bulk velocity. Note that Ub /Uτ is simply the inner scaled† local mean velocity (U/Uτ ) integrated over the pipe cross-section, so that a frictional resistance power law would be expected to admit an inner scaled mean velocity power law. If one assumes such a power law for U + = U/Uτ , it can be shown (see Appendix B of Monty17 ) that

1 yUτ 7 U = 8.562 , (2) Uτ ν if Blasius’ frictional resistance law (1) is to be satisﬁed; y is the wall-normal coordinate. A more rigorous derivation of this velocity power law from the resistance law (1) can be made if one assumes the following functional forms for the mean velocity:

yUτ U =f (y → 0), (3) Uτ ν and U Umax † Throughout

length, ν/Uτ .

=G

,yR

(y 0),

(4)

this article, ‘inner’ ﬂow scales are the friction velocity, Uτ and the viscous

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where Umax is the maximum mean velocity and R is the pipe radius. While equations (1) and (2) agreed with the low Reynolds number pipe ﬂow data considered by Blasius,16 it soon became apparent that these laws would not apply to ﬂow at higher Reynolds numbers. This led to further research into mean velocity scaling, particularly by Prandtl’s group. In 1925 Prandtl18 presented a logarithmic mean velocity scaling law. This law was valid for ﬂow very near a wall and was derived using a mixing length argument. The now familiar log law is written as

yUτ 1 U + A, (5) = ln Uτ κ ν where κ and A are universal constants. In 1930, another of Prandtl’s students, Theodore von K´ arm´ an,19 suggested an alternative outer ﬂow scaling to (4): ,yUmax − U . (6) =g Uτ R From its inception, Prandtl’s log law “found practically universal acceptance among workers in ﬂuid mechanics,” (Clark Millikan20 ). However, over a decade would pass before a more elegant derivation of this scaling was proposed by Millikan.20 In his paper, equations (3) and (6) were assumed to simultaneously apply in a region of the ﬂow termed the ‘overlap’ region, which is the region of overlap between inner (3) and outer (6) ﬂow regimes. The resulting equations describing mean velocity in the overlap region were the log law (5) and a velocity defect law, 1 ,yUmax − U + B, (7) = − ln Uτ κ R where B is a Reynolds-number-independent, non-universal constant. It should be noted that Millikan’s argument leads to the log law only when equation (6) is accepted for the outer ﬂow. Alternatively, if one begins with (4), the same argument produces a power law inner scaling of the mean velocity. In other words, if the outer ﬂow velocity scale is taken as Uτ , a log law results, but if this scale is Umax , a power law results. The combination of a new derivation and superior ﬁt to the data furthered the acceptance of the log laws, which are now universally employed by the industrial engineering community. 2.2. The Renaissance of the Power Law Recently, there has been a renewed interest in power laws21,22 which, according to the discussion above, implies a rejection of Uτ as the outer

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35 30 25 U Uτ

20 15

U+ =

1 0.406

ln(y + ) + 4.547

U = 9.95 (y + ) +

10

1

10

2

10

3

10

0.112

4

yUτ ν

Fig. 1. University of Melbourne high Reynolds number boundary layer mean velocity data for Reθ = 52103 (Reτ = 20000). The broken vertical lines indicate y + = 50 and y/δ = 0.2, where superscript ‘+’ denotes inner ﬂow scaling and δ is the boundary layer thickness. The power law was ﬁtted to data in the region 400 < y + < 4000.

velocity scale for ﬁnite Reynolds number. A recent paper by George et al.21 claims that the apparent collapse of most experimental data on to a log law “has never been entirely true, and as better data have been acquired it has become even more evident to be false.” Recent hot-wire measured results from a high Reynolds number boundary layer facility at the University of Melbourne (see Nickels et al.14 for details of this facility) are shown in ﬁgure 1. Visually, the most striking feature of ﬁgure 1 is the extraordinary extent of the log law. Note that the log law constants, κ = 0.406 and A = 4.547, were found from a least squares error ﬁt to data in the overlap region deﬁned by 100 < y + < 3000; the outer limit corresponding to y/δ = 0.15, where δ is the boundary layer thickness. The limits shown in the ﬁgure (i.e. y + = 50 and y/δ = 0.2) simply indicate the range where the data lie very close to the log law. A variety of diagnostics could be plotted to further assert the validity of the log law (see, for example, Osterlund et al.23 ), but are not considered in this paper. For comparison, this ﬁgure also includes a power law ﬁtted to the data. As there appears to be no guidance for choice of power law region limits in the literature (it was obviously not possible to ﬁt a power law to the overlap region deﬁned above), the method of Barenblatt et al.24 was employed. This essentially involved plotting the data in log-log coordinates

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and curve-ﬁtting the most linear region of that data. It should be noted that the linear region is always near the outer edge of the overlap region. The result is shown in ﬁgure 1 and it can be seen that: i. The wall-normal extent of the power law has decreased relative to the size of the inner ﬂow region as Reynolds number has increased (c.f. Barenblatt et al.24 ). Conversely, the extent of the log law increases with Reynolds number. In fact, the log region increases linearly with Karman number, Reτ = Uτ δ/ν. ii. Much of the power law region overlaps the log region. Where this occurs, it appears that the power law is tangent to the log law and so both laws are appropriate. Above this region, the power law describes the data well, however, it could be argued that this is really a description of the wake ﬂow (outside the inner ﬂow region), rather than the overlap region. This is an important point because the wake behaviour depends on large-scale geometry; therefore, if it does describe the wake region, the power law should not be universal across diﬀerent shear ﬂows. It is not the intention of the authors to enter into yet another log law versus power law argument. It was, however, considered relevant in the context of high Reynolds number boundary layer discussion, to simply present the new data which appears to lend support to the universal log law. This new boundary layer data is not only unique in Reynolds number magnitude, but in providing data close to the wall at high Reynolds numbers. It was only possible to realise such measurements because of the thickness of the boundary layer, δ ≈ 320mm at Reτ = 20, 000. Thus, ﬁgure 1 testiﬁes to the importance, indeed the requirement, of high Reynolds number studies in large facilities. 2.3. Curve-ﬁtting the mean velocity proﬁle Since 1950, there have been several attempts to provide formulations with minimal free parameters that eﬀectively describe the entire mean velocity proﬁle.25–27 Most recently, Nickels28 presented a formulation for the entire mean velocity proﬁle of a turbulent boundary layer. A revised version was published in 2004.29 The formulation addressed a number of major deﬁciencies in existing curve-ﬁts, most notably an inapplicability to pressure gradient ﬂows. By introducing the term p+ x =

ν dp , ρUτ3 dx

(8)

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Nickels determined a three-part form for the mean velocity distribution which should be applicable to any turbulent boundary layer. The threepart structure combines descriptions of the viscous sublayer, the overlap region and the wake region. Furthermore, it can be conveniently expressed in a law of the wall/law of the wake formulation (after Coles30 ), i.e. U + = f1 (y + ) + g1 (η),

(9)

where η = y/δ. The ﬁnal formulation has three free parameters, yc+ , b and δ, and is given by:

+ 2

3 + + y y+ 1 3 + + y+ + p y ) − y yc+ e−3y /yc U + = yc+ − 1 + 2 + + (3 − p+ x c x c + + 2 2 yc yc yc

1 + (0.6y + /yc+ )6 1 5(η 4 + η 8 ) ln + + b 1 − exp − . (10) 6κ 1 + η6 1 + 5η 3 The free parameter yc+ is physically related to the sublayer thickness (having a value of approximately 12 for zero-pressure-gradient boundary layers), while b is analogous to Coles wake factor, Π. b is therefore found by curveﬁtting and is obviously non-universal. Nickels29 provides examples of his curve-ﬁt to experimental and numerical boundary layer data and clearly establishes the accuracy of (10) for a wide range of Reynolds number and pressure gradient ﬂows. In order to extend this formulation to duct ﬂows where the velocity gradient is always zero at the centreline, Nickels pointed out that the second and third terms should be modiﬁed to

1 + (0.6y + /yc+ )6 5(η 4 + η 8 ) 1 ln and b 1 − exp − , (11) 6κ 1 + η 6 + η 12 1 + 5η 3 + 10.5η 8 respectively. To test this pipe ﬂow curve-ﬁt, the original Princeton Superpipe data‡ and a low Reynolds number DNS pipe ﬂow data set from Loulou et al.,32 are plotted in ﬁgure 2 along with the corresponding Nickels curve-ﬁt. The quality of the ﬁts to the entire Reynolds number range are immediately obvious. It is also clear that the DNS data, which has a Karman number, Reτ = Uτ R/ν of only 190, rises above the log law ﬁtted to the majority of superpipe data sets. Such behaviour is also evident in

‡ The

superpipe data were taken from Perry et al.;31 the data have the MacMillan correction and a turbulence intensity correction applied. See this reference for further details.

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30

U Uτ

20

10

Superpipe Loulou et al.(1997) Nickels (2004) 0 0 10

10

1

10

2

10 yUτ ν

3

10

4

10

5

10

6

Fig. 2. Nickels (2004) curve-ﬁts to Princeton superpipe experimental data and the DNS data of Loulou et al. (1997). The broken line represents a log law with constants κ = 0.39 and A = 4.2 found by Perry et al. from a ﬁt to the majority of the superpipe data.

low Reynolds number channel ﬂow DNS data (e.g. Kim et al.33 ). This eﬀect can be accounted for by the duct pressure gradient, i.e. the p+ x term, in Nickels’ curve-ﬁt. At what Reynolds number then, does the data collapse onto the log law? Taking the deﬁnition of p+ x given by equation (8), it + −1 is easily shown that px = −2Reτ for fully developed pipe ﬂow. If it is conservatively assumed that pressure gradient eﬀects are negligible when |p+ x | < 0.002, then a Karman number of Reτ ≈ 1000 is required for collapse onto higher Reynolds number data (this, of course, applies to duct ﬂow data only). For the Princeton Superpipe data considered, there is a downward shift in mean velocity for the highest Re case. Perry et al.31 suggest this could be due to roughness eﬀects. This is a contentious issue which needs no further discussion. However, it is likely that such issues could be resolved with high Reynolds number measurements in geometrically large facilities. 3. Attached eddy model Figure 3 shows the downstream ﬂow structures in a smoke-ﬁlled laminar boundary layer which has been disturbed with a trip-wire. The clarity of the induced hairpin shaped eddies led Perry and Chong9 to propose a physical model of boundary layer turbulence.9 This model essentially consisted of a random array of such vortices which grow out of the viscous sublayer

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Fig. 3. Top and front views of hairpin vortices created by a tripwire in a laminar boundary layer. Taken from Perry, Lim and Teh (1981).

‘material’. These characteristic vortices were considered to be inclined at 45o to the wall. For simplicity, Perry and Chong (PC) considered only the vortex shape shown in ﬁgure 4(a), rather than a more complex geometry (e.g., horseshoe or true hairpin shapes). In order to achieve an inﬁnite logarithmic region of the velocity proﬁle at inﬁnite Reynolds number, it was found that a system of hierarchies of geometrically similar eddies was

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required. As illustrated in ﬁgure 4(b), the characteristic height of these eddies doubles from hierarchy to hierarchy. The ﬁrst hierarchy is assumed to contain eddies with inner scaled characteristic height, he Uτ /ν = O(100); that is, the ﬁrst hierarchy follows the Kline34 scaling. It was shown by PC 5th hierarchy

4th

3rd 2nd 1st

(a)

(b)

Fig. 4. Illustrations of attached eddy concepts from Perry and Chong (1982). (a) Details of individual attached eddies. (b) The system of discrete hierarchies of eddies.

that their attached eddy model predicts a k1−1 region for the streamwise velocity spectra, Φ11 , i.e. C Φ11 (k1 y) , = Uτ2 k1 y

(12)

where C is a universal constant and k1 is streamwise wavenumber. Note that the same scaling can be shown to exist from an overlap argument.14 However, PC also indicated that at least 10 hierarchies of eddies (using the discrete system of hierarchies) are required before a noticeable ‘-1’ slope appears in the predicted spectra. If the ﬁrst hierarchy has height, he = O(100ν/Uτ ), then the Karman number, Reτ = δUτ /ν, required for the streamwise spectra to exhibit the k1−1 law is Reτ = O(105 ). This order of Karman number is the same as that of boundary layers on aerodynamic vehicles or industrial pipe/channel ﬂows; most wind-tunnel turbulence studies have Reτ of O(103 ) which means they have less than 5 hierarchies of eddies present. Table 1 provides various properties of ﬁve high Reynolds number ﬂows, listed in order of decreasing number of hierarchies, Nh , present. Note that all values in this table are only approximate. The surface layer at SLTEST

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238 Table 1.

Properties of selected high Reynolds number test facilities. δ or R(m)

+ ymin

Nh

10 × 105

100

< 100

14

2

4 × 105

0.06

2000

13

640

4 × 104 2.6 × 104

0.26 0.30

40 50

9 9

30 17

4 × 104

0.70

10

9

11

Facility

Reτ (max)

SLTEST Superpipe35 Knobloch and Fernholz36 Melbourne wind-tunnel Proposed large D pipe

l+

has the most hierarchies, so it is not surprising that the ‘-1’ law has been observed by atmospheric boundary layer researchers.37,38 Unfortunately there are diﬃculties inherent in atmospheric measurements, such as wind speed and directional variations, which make it diﬃcult to obtain adequately resolved energy spectra. The Princeton Superpipe has around 13 hierarchies at the highest Reynolds number. However, a limiting factor is the mini+ at Reτ (max), which is higher than mum wall-distance measureable, ymin + y = 100. This is a limitation because eddies in a given hierarchy only make a signiﬁcant contribution to the ﬂow beneath them; that is, if a measuring probe is placed above a hierarchy of eddies, that probe will not “see” those eddies. Another obstacle which has to be taken into account while analysing turbulence measurements is the spatial resolution problem. This problem arises because of the non-dimensional probe size near the wall. For hot-wire measurements, if a ﬂow has high freestream velocity and/or low viscosity, then the non-dimensionalised hot-wire ﬁlament length, l+ , becomes very large. There is some evidence39 that l+ 20 is required to suﬃciently resolve the wavenumber region of interest. Table 1 provides l+ estimates at Reτ (max) for all facilities. Note that in all cases the hot-wire ﬁlament is conservatively assumed to be 1µm in diameter§ with a length-to-diameter ratio of 200. Although l+ is acceptably low at SLTEST, this is not true of all the experiments. Spatial resolution problems could theoretically be resolved by employing smaller hot-wires and preliminary work in this area is underway, although this is not a simple task (e.g. Kunkel et al.40 ). An alternative approach is to work with larger facilities which will have lower l+ (e.g. SLTEST) in addition to various other beneﬁts described earlier.

§ 1µm

this is the smallest diameter hot-wire used in any of the studies of table 1.

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4. A very large boundary layer The SLTEST facility on the great salt lakes of Utah, USA has a number of unique physical properties that make it favourable for turbulence research. The terrain in the vicinity of the measurement site is extremely ﬂat and smooth¶ with an elevation change of less than 1m for at least 13km upstream of the measurement location.15 The thickness of the surface layer is at least 100m as measured by Metzger and Klewicki15 with long periods of consistent wind speeds in excess of 5m/s. In June, 2005, the authors were involved in a collaborative investigation into large-scale motions of the surface layer turbulence at SLTEST. Part of the motivation of this study was to determine the similarities between controlled wall-turbulence and atmospheric surface turbulence. Two arrays of sonic anemometers were erected; the ﬁrst was a 30m vertical tower mounted with nine sonic anemometers, having logarithmic spacing; the second was a linearly spaced horizontal array of 10 anemometers. Figure 5 displays the initial setup of the experiment. A shear stress sensor was also inserted directly beneath the vertical tower, ﬂush with the desert ﬂoor. Details of this sensor can be found in Heuer and Marusic.41 Data was sampled from all anemometers and the shear stress sensor at 20Hz continuously. Since all parameters are uncontrolled, the ﬁrst step in analysing the data is to determine periods of consistent wind speed and direction. These variables tended to be more consistent in the evening and through the night. This is fortuitous, since it is also the period of neutral stability of the surface layer. Neutral stability occurs whenever the modulus of the Monin-Obukhov length, ΘUτ3 , (13) L= 9.81κv θ becomes very large. In the above deﬁnition Θ and θ are the mean and ﬂuctuating temperatures respectively and v is the ﬂuctuating component of wall-normal velocity. Physically, a large value of L means buoyancy eﬀects are insigniﬁcant in comparison to turbulent shear, which is, of course, the case in typical laboratory wall-turbulence without heat addition. Given that the mean temperature and friction velocity are relatively weak functions of time over the diurnal cycle (provided wind speed does not vanish), the stability of the surface layer is primarily determined by the heat ﬂux, v θ . ¶ During the author’s visit in the summer of 2005, a recent rainstorm had rendered the surface even smoother than usual.

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A photograph showing the sonic anemometer arrays at SLTEST. Fig. 5.

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35

U Uτ

25

Melbourne wind-tunnel

15

SLTEST U+ = 10

1

10

2

10

3

10

4

1 0.406

10

5

ln(y + ) + 4.547 10

6

yUτ ν

Fig. 6.

SLTEST mean velocity distribution.

That is, if the heat ﬂux approaches zero, L → ∞. It was found, somewhat surprisingly, that the measured heat ﬂux remained very near zero throughout the night. This meant almost 10 hours of neutrally stable surface layer ﬂow was available for analysis over a 24 hour period. The mean velocity proﬁle is given in ﬁgure 6 together with the Melbourne wind-tunnel data presented earlier. The closeness of the data to the log law occurs because the Clauser42 method was used to determine Uτ . The purpose of this plot is to display the logarithmic nature of the proﬁle, that is, the data points appear highly linear on the log-linear plot of ﬁgure 6. Such a velocity distribution was observed almost throughout the night, except during periods of inconsistent wind speed and/or wind direction. The two-point correlation of wall shear stress and freestream velocity ﬂuctuations, Rτ u , at a given wall-normal location was determined; if this is done for each sonic anemometer in the vertical tower, a total of 9 correlation plots are available. These plots were combined as a contour plot which is shown in ﬁgure 7. Each of the correlations were normalised by their peak values for this plot, therefore, the ridge of dark red indicates a line of peak correlation. From the solid lines plotted over the ﬁgure, it appears that a peak correlation angle of 12 – 15◦ is a feature of this ﬂow. This is consistent with a similar experiment conducted in a low Reynolds number wind-tunnel The

shear stress sensor was employed to measure ﬂuctuating wall shear stress, τ only — not mean shear stress as required to calculate Uτ .

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242 25 0.9

0.8 20 Surface Normal Distance, z (m)

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0.7

0.6

15

0.5

0.4

10

0.3 5

0.2

0.1 −100

−80

−60

−40

−20 0 20 Streamwise Distance, x (m)

40

60

80

100

Fig. 7. Correlation of wall shear stress with streamwise velocity contour plot. Solid lines indicate angles of 12, 15 and 18◦ to the surface.

by Brown and Thomas.43 In that case an angle of 18◦ was calculated from correlations between hot-wire measured velocity and hot-ﬁlm measured wall shear stress. At this point, an interpretation of this angle is oﬀered, which ﬁrst requires further discussion of attached eddies. It has recently been found that a random array of single hairpin eddies as proposed by Perry and Chong9 does not correctly model the large-scale motions of wall shear ﬂow.44 A solution to this problem is a random array of coherent packets of hairpin eddies, all at diﬀerent stages of growth. This idea was originally proposed by Adrian et al.10 Thus, the angle determined above could be thought of as a structure angle, γ; that is, the angle between the wall and a line connecting the heads of characteristic eddies in a packet. This is illustrated in ﬁgure 8. If this interpretation is correct, the correlations of ﬁgure 7 lend support to the vortex packet model of a boundary layer even up to Reτ = O(106 ). One ﬁnal example of similarities between the atmospheric surface layer and lower Reynolds number wind-tunnel tests is provided in ﬁgure 9. In both parts of this ﬁgure, the colours represent the momentum of the ﬂuid relative to the mean ﬂow. Thus, low speed ﬂuid is represented by the colour

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γ

Fig. 8.

A sketch of a coherent packet of attached eddies moving from left to right.

blue, while red indicates high momentum ﬂuid. Hutchins et al.45 have already shown that wind-tunnel boundary layers exhibit elongated regions of low momentum ﬂuid in the log region. These low momentum regions are commonly ﬂanked by high momentum regions. Such behaviour would be expected if coherent packets of eddies are signiﬁcant features of wallturbulence. Figure 9b shows a similar long region of low momentum ﬂuid wandering past the horizontal sonic anemometer array at SLTEST. The wall-normal distance, y, of the array was 3m which is expected to be inside the log region (y/δ ≈ 0.03, y + ≈ 40000). Figure 9 presents not only another example of similarity between the atmospheric surface layer and laboratory boundary layers, but also a second indication that packets of hairpin vortices persist at very high Reynolds numbers.

5. Conclusions and outlook Over the past decade there have been encouraging steps made toward a clearer understanding of wall-turbulence, especially as Reynolds number becomes very high. Particularly signiﬁcant is the progress made in the study of physical mechanisms of wall-turbulence. The existence of hairpin-type vortices has been conﬁrmed in all wall bounded shear ﬂows and research continues into the coherency of individual vortices in the ﬂow. Recent investigations carried out in the SLTEST facility has revealed that there are indeed close similarities with controlled, wind-tunnel wallturbulence. Strong evidence of packets of attached eddies and elongated low and high momentum ﬂuid regions were found. The combination of improved measurement techniques, investigations employing large-scale facilities and a focus on the physical understanding of wall-turbulence suggests a very bright outlook for this ﬁeld of research.

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244 z = 0.3δ x = 8δ

7δ 6δ

5δ

4δ

3δ

2δ z/δ

δ

0

x/δ

a) Wind-tunnel, Reτ = 103

b) SLTEST, Reτ = 106

Fig. 9. Evidence of elongated low momentum regions in the logarithmic region. The coloured contours represent the instantaneous streamwise velocity relative to the mean. Figure a) was extracted from the low Reynolds number PIV data of Hutchins, Ganapathisubramani and Marusic (2004). The second ﬁgure was created from the horizontal sonic anemometer array at SLTEST by N. Hutchins (used here with permission).

Acknowledgments The authors wish to acknowledge the ﬁnancial assistance of the Australian Research Council. Also, the assistance of Dr N. Hutchins and Prof. I. Marusic is greatly appreciated. References I. Tani, Annu. Rev. Fluid Mech. 9, 87 (1977). E. S. Zanoun, F. Durst and H. M. Nagib, Phys. Fluids 16(9), 3509 (2004). M. H. Buschmann and M. Gad-el-Hak, Phys. Fluids 16(9), 3507 (2004). T. Theodorsen, in Proceedings of the Second Midwestern Conference on Fluid Mechanics (Ohio State University, 1952). 5. A. A. Townsend, The structure of turbulent shear ﬂow (Cambridge University Press., 1956), 1st edn. 6. A. A. Townsend, The structure of turbulent shear ﬂow (Cambridge University Press., 1976), 2nd edn. 7. M. R. Head and P. Bandyopadhyay, J. Fluid Mech. 107, 297 (1981).

1. 2. 3. 4.

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8. A. E. Perry, T. T. Lim and E. W. Teh, J. Fluid Mech. 104, 387 (1981). 9. A. E. Perry and M. S. Chong, J. Fluid Mech. 119, 173 (1982). 10. R. J. Adrian, C. D. Meinhart and C. D. Tomkins, J. Fluid Mech. 422, 1 (2000). 11. B. Ganapathisubramani, E. K. Longmire and I. Marusic, J. Fluid Mech. 478, 35 (2003). 12. Z. C. Liu and R. Adrian, in Proc. Turb. Shear Flow Phen. 1 (Santa Barbara, U.S.A., 1999). 13. J. Jimenez, J. C. Del Alamo and O. Flores, J. Fluid Mech. 505, 179 (2004). 14. T. B. Nickels, I. Marusic, S. Hafez and M. S. Chong, Phys. Rev. Letters 95(074501) (2005). 15. M. M. Metzger and J. C. Klewicki, Phys. Fluids 13, 692 (2001). 16. H. Blasius, Physikalische Zeitschrift 12, 1175 (1911), or refer to H. Schlicting, Boundary layer theory, 560–566, McGraw-Hill, 1968. 17. J. P. Monty, Developments in smooth wall turbulent duct ﬂows, Ph.D. thesis, University of Melbourne (2005). 18. L. Prandtl, ZAMM 5, 136 (1925). 19. T. von K´ arm´ an, Mechanische ahnlichkeit und turbulenz, Tech. rep., Nachr. Ges. Wiss. Gottingen (1930). 20. C. B. Millikan, in Proc. 5th Int. Congress of Appl. Mech. (Cambridge, Mass, 1938), 386–392. 21. W. K. George, L. Castillo and P. Knecht, in Reynolds Symposium on Turbulence (Asilomar, CA, 1993). 22. G. I. Barenblatt, J. Fluid Mech. 248, 513 (1993). ¨ 23. J. M. Osterlund, A. V. Johansson, H. M. Nagib and M. H. Hites, Phys. Fluids 12(1), 1 (2000). 24. G. I. Barenblatt, A. J. Chorin and V. M. Protoskishin, Phys. Fluids 12, 2159 (2000). 25. D. B. Spalding, J. Applied Mechanics 455–457 (1961). 26. H. Reichardt, Z. angew. Math. Mech. 31, 208 (1951). 27. E. R. Van Driest, J. Aerosp. Sci. 23, 1007 (1956). 28. T. B. Nickels, in 14th Australasian Fluid Mechanics Conference (Adelaide University, Adelaide, Aust., 2001). 29. T. B. Nickels, J. Fluid Mech. 521, 217 (2004). 30. D. E. Coles, J. Fluid Mech. 1, 191 (1956). 31. A. E. Perry, S. Hafez and M. S. Chong, J. Fluid Mech. 439, 395 (2001). 32. P. Loulou, R. Moser, . M. N and B. Cantwell, Direct simulation of incompressible pipe ﬂow using a b-spline spectral method, Tech. Rep. TM 110436 , NASA (1997). 33. J. Kim, P. Moin and R. Moser, J. Fluid Mech. 177, 133 (1987). 34. S. Kline, W. Reynolds, F. Shrub and P. Rundstadler, J. Fluid Mech. 30, 741 (1967). 35. M. V. Zagarola and A. J. Smits, J. Fluid Mech. 373, 33 (1998). 36. K. Knobloch and H.-H. Fernholz, in A. J. Smits, ed., Proceedings of the IUTAM symposium (Princeton, NJ, 2002), 11–16. 37. J. C. R. Hunt and P. Carlotti, Flow, Turb. and Comb. 66, 453 (2001).

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38. P. L. Fuehrer and C. A. Friehe, Boundary Layer Met. 90, 241 (1999). 39. P. M. Ligrani and S. Bradshaw, Experiments in Fluids 5, 407 (1987). 40. G. J. Kunkel, G. B. Arnold and A. J. Smits, in 36th AIAA Fluid Dynamics Conference (San Francisco, CA, 2006). 41. W. D. C. Heuer and I. Marusic, Meas. Sci. Tech. 16, 1644 (2005). 42. F. H. Clauser, J. Aero. Sci. 21, 91 (1954). 43. G. L. Brown and A. S. W. Thomas, Phys. Fluids 20(10), S243 (1977). 44. I. Marusic, Phys. Fluids 13, 735 (2001). 45. N. Hutchins, B. Ganapathisubramani and I. Marusic, in Proc. 15th Australasian Fluid Mech. Conf. (The University of Sydney, Sydney, Aust., 2004).

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WHAT ARE WE LEARNING FROM SIMULATING WALL TURBULENCE? ´ JAVIER JIMENEZ School of Aeronautics, Universidad Polit´ ecnica, 28040 Madrid SPAIN, [email protected] The study of turbulence near walls has experienced a renaissance in the last decade, largely because of the availability of high-quality numerical simulations. The viscous and buﬀer layers over smooth walls are essentially independent of the outer ﬂow, and there is a family of numerically-exact nonlinear structures that account for about half of the energy production and dissipation. The other half can be modelled in terms of their unsteady bursting. Many of the bestknown characteristics of the wall layer, such as the dimensions of the dominant structures, are well predicted by those models, which were essentially completed at the end of the last century. It is argued that this was made possible by the increase in computer power that made the kinematic simulations of the late 1980s cheap enough to undertake dynamic experiments. We are today at the early stages of simulating the logarithmic layer. A kinematic picture of the various cascades present in that part of the ﬂow is beginning to emerge. Dynamical understanding can be expected in the next decade.

1. Introduction Some of the ﬁrst systems in which turbulence was identiﬁed were wallbounded ﬂows,9,12 but they remain to this day worse understood than homogeneous or free-shear turbulence. Part of the reason is that we are interested in diﬀerent things for the diﬀerent types of ﬂows. While in homogeneous turbulence the emphasis is on the self-similar energy cascade,29 and in free shear ﬂows the main interest is in the energy-containing range of structures controlled by large-scale instabilities,7 wall-bounded turbulence is essentially inhomogeneous and anisotropic. The eddy sizes containing most of the energy at one wall distance are in the midst of the inertial cascade when they are observed farther away from the wall. The local Reynolds number, deﬁned as the scale disparity between energy and dissipation, also changes continuously with wall distance. The main emphasis in studying

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wall turbulence is not so much in understanding the inertial energy cascade between eddies at the same geometric location. That process is probably roughly the same as in other turbulent ﬂows. The challenge is to understand the interplay between diﬀerent scales at diﬀerent distances from the wall, particularly for the larger eddies carrying the energy and the Reynolds stresses. A particularly simple part of wall-bounded turbulent ﬂows is the thin region formed by the viscous and buﬀer layers in the immediate vicinity of smooth walls, where viscosity is important and the energetic and dissipative scales overlap. This layer, although geometrically negligible when compared with the bulk of the ﬂow, is both technologically and scientiﬁcally important, because a relatively large fraction of the velocity diﬀerence across boundary layers resides in it. Its modern study began experimentally26,31 in the 1970’s, and got a strong impulse with the advent of high-quality direct numerical simulations27 in the late 1980’s and in the 1990’s. The numerical emphasis of the present review is partly a personal bias of the author, but it is not altogether arbitrary. The near-wall region is relatively well suited to numerical simulation, and diﬃcult to explore experimentally. Much of the available information is numerical. The logarithmic law is located just above the near-wall region, and it is also unique to wall turbulence. It has been studied experimentally for a long time, but its numerical simulation is only now beginning to be possible. It is still much worse understood than the viscous layers. This paper is organized as follows. In the next section we deﬁne the near-wall layer and outline the classical models for it. In section 3 we review the recent work on equilibrium solutions for wall-bounded shear ﬂows, and how they are related to turbulence, and in section 4 we discuss brieﬂy the present status of our understanding of the logarithmic layer. Finally some conclusions are oﬀered. 2. The structure of near-wall turbulence It is well known40 that wall-bounded turbulence over smooth walls can be described to a good approximation in terms of two sets of scaling parameters. Viscosity is important near the wall, and the units for length and velocity in that region are constructed with the kinematic viscosity ν and with the friction velocity uτ = (τw /ρ)1/2 , which is based on the shear stress at the wall τw , and on the ﬂuid density ρ. Magnitudes expressed in these ‘wall units’ are denoted by + superscripts. If y is the distance to the wall, y + is a Reynolds number for the size of the structures. The near-wall layer

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extends at most33 to y + = 150, and it is because of those relatively low values that turbulence near smooth walls has always been a good candidate for simple modelling. Far from the wall the velocity also scales with uτ , but the length scale is the ﬂow thickness h. Between the inner and the outer regions there is an intermediate layer where the only length scale is the wall distance y. The mean velocity in that ‘logarithmic’ layer is given approximately by U + = κ−1 log y + + A.

(1)

The K´ arm´ an constant κ ≈ 0.4 is approximately universal. The intercept constant is A ≈ 5 for smooth walls, but depends on the details of the near-wall region. The viscous inner layer is extremely important for the ﬂow as a whole. The ratio between the inner and the outer length scales is the friction Reynolds number, h+ , which ranges from 200 for barely turbulent ﬂows to h+ = 5 × 105 for large water pipes. In the latter, the near-wall layer is only about 3 × 10−4 times the pipe radius, but it follows from (1) that, even in that case, 40% of the velocity drop takes place below y + = 50. Turbulence is characterized by the expulsion towards the small scales of the energy dissipation, away from the large energy-containing eddies. In wall-bounded ﬂows that separation occurs not only in the scale space for the velocity ﬂuctuations, but also in the shape of the mean velocity proﬁle. The singularities are expelled both from the large scales, and from the centre of the ﬂow towards the walls. Because of this singular nature, the near-wall layer is not only important for the rest of the ﬂow, but it is also essentially independent from it. That was for example shown by numerical experiments with ‘autonomous’ simulations22 in which the outer ﬂow was artiﬁcially removed above a certain wall distance δ. The near-wall dynamics was unaﬀected as long as δ + 60, at least when compared with low-Reynolds number ﬂows. We will however see below that there is some inﬂuence of the outer ﬂow even in the viscous sublayer, but that it happens at scales that are diﬀerent from those of the typical inner-layer dynamics. Most of the velocity diﬀerence that does not reside in the near-wall region is concentrated in the logarithmic layer, which extends experimentally up to y = 0.2h (ﬁgure 1). It follows from (1) that the velocity diﬀerence above the logarithmic layer is only 20% of the total when h+ = 200, and decreases logarithmically for higher Reynolds numbers. In the limit of inﬁnite Reynolds numbers, all the velocity drop is in the logarithmic layer.

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Fig. 1. Premultiplied spectra, kE(k), of the kinetic energy, |u |2 (shaded), and of the enstrophy, |ω |2 (lines), as functions of the streamwise wavelength λx = 2π/kx , and of the wall distance y. At each y the lowest contour is 0.86 times the local maximum. The two horizontal lines are the y + = 80 and y/h = 0.2, and represent the limits of the logarithmic layer. The diagonal line is λx = 2y. From a numerical channel14 at h+ = 2000.

The buﬀer, viscous and logarithmic layers are the most characteristic features of wall-bounded ﬂows, and the main diﬀerence between those ﬂows and other types of turbulence. The ﬁrst two layers are today relatively easy to compute, because the local Reynolds numbers are low, and we will see below that they can be described in terms of relatively simple eddies. Because of the velocity estimates in the previous paragraphs, understanding those ﬁrst two layers has practical implications. They are for example responsible for most of the friction drag of turbulent boundary layers, and any attempt to control wall friction has to centre on them. The logarithmic layer, on the other hand, is an intrinsically highReynolds number region. Its existence requires at least that its upper limit should be above the lower one, so that 0.2h+ 100, and h+ 500. The local Reynolds numbers y + of the eddies are also never too low. Numerical simulations with an appreciable logarithmic region have only recently become available.3,14 We have seen on the other hand that, as the Reynolds number increases, this layer becomes the dominant ﬂow feature.

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2.1. The classical model The region below y + ≈ 100 has been intensively studied. It is dominated by coherent streaks of the streamwise velocity and by quasi-streamwise vortices. The former are an irregular array of long (x+ ≈ 1000) sinuous alternating streamwise jets superimposed on the mean shear, with an average spanwise separation39 of the order of z + ≈ 100. The quasi-streamwise vortices are slightly tilted away from the wall,15 and stay in the near-wall region only for x+ ≈ 200. Several vortices are associated with each streak,20 with a longitudinal spacing of the order of x+ ≈ 400. Most of them merge into disorganized vorticity outside the immediate neighbourhood of the wall.35 It was proposed very soon that streaks and vortices were involved in a mutual regeneration cycle in which the vortices were the results of an instability of the streaks,36 while the streaks were caused by the advection of the mean velocity gradient by the vortices.6,26 While there is still some discussion on how the vortices are generated, it is known that they derive from the streaks, because disturbing the latter inhibits their formation.22 That manipulation is only eﬀective if the ﬂow is perturbed below y + ≈ 60, and fails if it is applied only below y + ≈ 10, suggesting that it is predominantly between those two levels that the streaks are involved in the vortex-generation process. There is a substantial body of numerical13,37,44 and analytic24,34 work on the linear instability of model streaks. It shows that streaks are unstable to a variety of sinuous perturbations associated with inﬂection points of the perturbed velocity proﬁle, whose eigenfunctions correspond well with the shape and location of the observed vortices. The model implied by these instabilities is a time-dependent cycle in which streaks and vortices are created, grow, generate each other, and eventually decay. Reference 22 discusses other unsteady models of this type, and gives additional references.

3. Exact solutions for the sublayer A slightly diﬀerent point of view is that the regeneration cycle is organized around a nonlinear travelling wave, a ﬁxed point in some phase space, which represents a nonuniform streak. This is actually not too diﬀerent from the previous model, which essentially assumes that the undisturbed streak is a ﬁxed point in phase space, and that the cycle is an approximation to an orbit along its unstable manifold. The new model however considers ﬁxed points which are non-trivially perturbed streaks, and therefore separates the dynamics of turbulence from that of transition.

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Fig. 2. Exact permanent-wave solution for the Navier-Stokes equations in an ‘autonomous’ domain below δ+ = 40. The ﬂow is from top-right to bottom-left. The central object is an isosurface of the streamwise perturbation velocity, u + = −3.5, and deﬁnes the streak. It is ﬂanked by two staggered streamwise vortices of opposite signs, ωx+ = ±0.18, whose eﬀect is to create an upwash that maintains the streak.23

The organization of the buﬀer layer does not require the chaos observed in fully-turbulent ﬂows. Simulations in which the ﬂow is substituted by an ordered array of identical single structures20 reproduce the correct statistics. In a further simpliﬁcation, that occurred at roughly the same time as the previous one, nonlinear equilibrium solutions of the three-dimensional Navier–Stokes equations were obtained numerically,32 with characteristics that suggested that they could be useful in a dynamical description of the near-wall region. Other such solutions were soon found for plane Couette ﬂow,32,47 plane Poiseuille ﬂow,41,46,47 and for an autonomous wall ﬂow.23 All those solutions look qualitatively similar,24,45 and take the form of a wavy low-velocity streak ﬂanked by a pair of staggered quasi-streamwise vortices of alternating signs, closely resembling the spatially-coherent objects educed from the near-wall region of true turbulence. An example is shown in ﬁgure 2. Their mean and ﬂuctuation intensity proﬁles are reminiscent of experimental values,23,47 as shown in ﬁgure 3(a). Other properties are also suggestive of real turbulence. For example, the range of spanwise wavelengths in which the nonlinear solutions exist is always in the neighbourhood of the observed spacing of the streaks in the sublayer.19 In those cases in which the stability of the equilibrium solutions has been investigated, they have been found to be unstable saddles in phase space at the Reynolds numbers at which turbulence is observed. They are not therefore expected to exist as such in real turbulence, but any turbulent

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Fig. 3. (a) Proﬁles of the root-mean-square velocity ﬂuctuations in a full channel27 with h+ = 180 (lines without symbols), and in the permanent-wave autonomous solution in , streamwise velocity; , wall-normal velocity. (b) ﬁgure 2 (with symbols). Comparison of some exact solutions with the near-wall turbulent structures, in terms of + + the maxima of the u and v proﬁles taken over boxes of size b+ x ×bz ×y = 380×110×50. , Nagata’s solutions for Couette ﬂow, at diﬀerent Reynolds numbers.19 Solid symbols are ‘upper branch’ solutions, and open ones are ‘lower branch’. • , autonomous permanent waves.23 The solid loop is an exact limit cycle in plane Couette ﬂow.25 Other open symbols are probability isocontours from large-box Poiseuille ﬂows:1,3 , h+ = 1880; ♦ , 950; , 550; ◦ , 180. They contain 90% of the boxes in the p.d.f.

ﬂow could spend a substantial fraction of its lifetime in their neighbourhood. Exact limit cycles and heteroclinic orbits based on these ﬁxed points have been found numerically,25,42 and several reduced dynamical models of the near-wall region have been formulated in terms of low-dimensional projections of such solutions.5,38,44 Two questions have to be addressed. The ﬁrst one is whether all the exact solutions that have been published for wall-bounded ﬂows are related to each other and to near-wall turbulence. The second question is whether real turbulence can be described in terms of those steady structures or whether it requires the inclusion of unsteadiness. That latter question cannot be addressed here because of lack of space, but it turns out that a well-deﬁned unsteady bursting cycle can be identiﬁed both in simpliﬁed solutions and in fully turbulent ﬂows, and that the unsteadiness accounts for roughly 50% of the production and of the dissipation of turbulent energy in the buﬀer region.19 The adequacy of the models based on equilibrium structures is addressed in ﬁgure 3(b). The earliest and best-understood nontrivial steady solutions of a wall-bounded Navier-Stokes shear ﬂow are those by Nagata32 for a plane Couette ﬂow, which were recently extended by Kawahara19 to a wider

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range of parameters. They can be classiﬁed into ‘upper’ and ‘lower’ branches in terms of their mean wall shear, and both branches have very diﬀerent proﬁles of their ﬂuctuation intensities. It can be shown19 that most of the known wall-bounded solutions by other authors can also be classiﬁed into one or another of Nagata’s branches. The ‘upper’ solutions have relatively weak sinuous streaks ﬂanked by strong vortices. They consequently have relatively weak root-mean-square streamwise-velocity ﬂuctuations u , and strong wall-normal ones v , at least when compared to those in the lower branch. The solution in ﬁgure 2 belongs to the upper branch, and we already saw in ﬁgure 3(a) that its r.m.s. velocity-ﬂuctuations proﬁles agree well with those of a full channel. ‘Lower’ solutions have stronger and essentially straight streaks, and much weaker vortices. The relative strength of both types of ﬂuctuations for a particular solution can be characterized by the maximum values of its u and v proﬁles, both of which are usually attained within the near-wall layer. Diﬀerent solutions can then be compared among themselves, and with fully-turbulent ﬂows, by comparing those two numbers. This is done in ﬁgure 3(b), but the comparison with full turbulence is not straightforward. Statistics compiled over small boxes of diﬀerent sizes are not comparable, even within the same ﬂow, because they are not ‘converged’. In particular the r.m.s. proﬁles of the exact solutions, which are computed over periodic domains + of size L+ x × Lz ≈ 400 × 100 parallel to the wall, cannot be compared directly with the standard intensity proﬁles compiled from full experiments or from computations, which typically have domains of the order + of L+ x × Lz ≈ 10, 000 × 5, 000. To allow the comparison in ﬁgure 3(b), each wall of the large computational boxes is divided into ‘minimal’ sub-boxes with the same wallparallel dimensions as the smaller computational boxes of the exact solu+ tions, b+ x × bz ≈ 380 × 110, and the statistics are compiled over them. In addition, each wall of the full channels is treated independently, and the maxima of the intensity proﬁles are deﬁned only from the wall to y + = 50. Each sub-box is characterized by its maximum r.m.s. intensities, and the values for diﬀerent sub-boxes are summarized as a joint probability density function of the two quantities, compiled over the diﬀerent sub-boxes and over time. Each ﬂow is not therefore characterized by a single point, but by the probability distribution of the possible states of the sub-boxes. The results of the ﬁgure suggest that only the ‘upper-branch’ exact solutions are representative of real turbulence, at least at the scales corresponding to a single streak and to a single vortex pair. They also show

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Fig. 4. Two-dimensional spectral energy density of the streamwise velocity in the nearwall region (y + = 15), in terms of the streamwise and spanwise wavelengths. Numerical , h+ = 547; , 934; , 2003. Spectra are normalized channels.1,3,14 in wall units, and the two contours for each spectrum are 0.125 and 0.625 times the maximum of the spectrum for the highest Reynolds number. The heavy straight line is λz = 0.15λx , and the heavy dots are λx = 10h for the three cases.

that the correspondence is reasonably good, but only for the weaker turbulent ﬂuctuations. Specially for the wall-normal velocities, there are turbulent ﬂuctuations in the near-wall region which are substantially stronger than those of the exact solutions. Note that the probability densities of the turbulent ﬂows depend on the Reynolds number, but that they saturate beyond approximately h+ = 1000. This is not the case for the velocity ﬂuctuation proﬁles compiled over full ﬂows, which keep increasing for much higher Reynolds numbers.10 That eﬀect is due to large outer-ﬂow velocity ﬂuctuations reaching the wall,1,18 and is unrelated to the structures being considered here. This is shown in ﬁgure 4, which contains two-dimensional premultiplied energy spectra of the streamwise velocity, kx kz Euu (kx , kz ), displayed as functions of the streamwise and spanwise wavelengths. The three spectra in the ﬁgure correspond to large numerical channels at diﬀerent Reynolds numbers. They diﬀer from each other almost exclusively in the long and wide structures represented in the upper-right corner of the spectrum, whose sizes are of the order of λx × λz = 10h × h. Those spectra are fairly well understood.1,14,18 The lower-left corner contains the structures discussed in this section, which are very approximately universal and local to the

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near-wall layer. The larger structures in the upper edge of the spectra, and specially those in the top-right corner, extend into the logarithmic layer, scale in outer units, and correspond approximately to the ‘attached eddies’ proposed by Townsend.43 Because they are too large to be contained within the averaging boxes used in this section, they do not inﬂuence the statistics in ﬁgure 3(b). 4. The logarithmic layer We noted in the introduction that the logarithmic layer is expensive to compute. The ﬁrst simulations with an appreciable logarithmic range have only appeared in the last few years, and even in them the range of wall distances is short. In Ref. 14, for example, h+ = 2000 and the upper and lower logarithmic limits are approximately y + = 400 and y + = 100. Some of the results of those simulations have been used in this article. The linear range in the lengths of the energy-containing eddies in ﬁgure 1, and the linear behaviour in the large-scale tails of the two-dimensional spectra in ﬁgure 4, are logarithmic-layer results. These simulations, as well as corresponding advances in experimental methods, have greatly improved our kinematic understanding of the structures in this region. It is for example known that there is a self-similar hierarchy of compact ejections extending from the buﬀer layer into the outer ﬂow, within which the vorticity is more intense than elsewhere. They are associated with extremely long, conical, low velocity regions in the logarithmic layer4 – “wakes” – which agree well with the energy-containing structures of the streamwise velocity spectrum. The overall arrangement is reminiscent of the association of vortices and streaks in the buﬀer layer, but at a much larger scale. It is also known that the low-velocity regions are almost identical to the transient-growth structures forced on the mean velocity proﬁle by a concentrated ejection,2 but that whatever is causing them does not grow directly from the buﬀer layer. The lifetimes of the observed ejections are too short for that,4 and the spectra of ﬂows in which the buﬀer region has been purposefully destroyed are essentially identical to those over smooth walls.11 We know much less about how the ejections are created, although there are indications that their association with the low-velocity regions goes both ways. Not only are the wakes seen when the statistics are conditioned to the ejections, but viceversa. The knowledge that we are gaining from the simulations is however essentially kinematic.

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The problem is one of cost, and was shared by the original low-Reynolds number simulations that eventually led to the understanding of the buﬀer layer. The simulation in Ref. 14 took six months on 2000 supercomputer processors. It took a similar time to run the simulation in Ref. 27 at h+ = 180. As long as each numerical experiment takes such long times, it is only possible to observe the results, and simulations are little more than betterinstrumented laboratory experiments. As computers improve, however, other things become possible. When the low-Reynolds number simulations of the 1980s became roughly 100 times cheaper in the 1990s, it became possible to experiment with them in ways that were not possible in the laboratory. The series of ‘conceptual’ simulations that led to the results in section 3 were of this kind. The perturbed-wall simulations cited above11 are one of the ﬁrst examples of this type of simulations for the logarithmic layer, but their Reynolds numbers are still only marginal, and they are in any case conceptually similar to ﬂows over rough walls. There is however no reason to believe that computer improvements have stopped, and the next decade will bring the cost of the simulations of the logarithmic layer to the level at which dynamical experiments should become commonplace. It is only then that we can expect a dynamical theory for this part of the ﬂow to emerge from simulations. 5. Conclusions We have brieﬂy reviewed the present state of the understanding of the dynamics of turbulent ﬂows near smooth walls. This is a subject that, like most others in turbulence, is not completely closed, but which has evolved in the last two decades from empirical observations to relatively coherent theoretical models. It is also one of the ﬁrst cases in turbulence, perhaps together with the structure of small-scale vorticity in isotropic turbulence, in which the key technique responsible for cracking the problem has been the numerical simulation of the ﬂow. The reason is that the Reynolds numbers of the important structures are low, and therefore accessible to computation, while experiments are diﬃcult. For example the spanwise Reynolds number of the streaks is only of the order of z + = 100, which is less than a millimetre in most experiments, but we have seen that it is well predicted by the range of parameters in which the associated equilibrium solutions exist. We have seen that the larger structures coming from the outside ﬂow interfere only weakly with the near-wall region, because the local dynamics are intense enough to be always dominant. The spacing of the streaks just

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mentioned has been observed up to the highest Reynolds numbers of the atmospheric boundary layer.28 On the other hand the thinness of the layer in which the dynamics takes place makes the ﬂow very sensitive to small perturbations at the wall. Roughness elements with heights of the order of a few wall units, microns in a large pipe, completely destroy the delicate cycle that has been described here, and can increase the friction coeﬃcient by a factor of two or more.17 Conversely it only takes a concentration of polymers of a few parts per million in the near-wall region30 to decrease the drag coeﬃcient by 40%. The same can be said of the control strategies based on the manipulation of the near-wall structures.8,16 The next few years will probably be dominated by modelling eﬀorts for the logarithmic layer similar to those described here for the viscous ones. The cost of simulations is higher in that case, but it is beginning to be within the reach of modern computers. The motivation is both theoretical and technological. The cascade of momentum across the range of scales in the logarithmic layer is probably the ﬁrst three-dimensional self-similar cascade that will be accessible to computational experiments. Its simplifying feature is the alignment of most of the net transfer along the direction normal to the wall. The main practical drive is probably large-eddy simulation, in which the momentum transfer across scales in the inertial range has to be modelled for the method to be practical.21 Only by understanding the structures involved would we be sure of how to accomplish that. Acknowledgments The preparation of this paper was supported in part by the CICYT grant ´ DPI2003–03434. I am deeply indebted to J.C. del Alamo, O. Flores, S. Hoyas, G. Kawahara and M.P. Simens for providing most of the data used in the ﬁgures. References ´ 1. J.C. del Alamo and J. Jim´enez, Spectra of very large anisotropic scales in turbulent channels. Phys. Fluids, 15, L41–L44 (2003). ´ 2. J.C. del Alamo and J. Jim´enez, Linear energy ampliﬁcation in turbulent channels. J. Fluid Mech., 2006. In press. ´ 3. J.C. del Alamo, J. Jim´enez, P. Zandonade and R.D. Moser, Scaling of the energy spectra of turbulent channels. J. Fluid Mech., 500, 135–144 (2004). ´ 4. J.C. del Alamo, J. Jim´enez, P. Zandonade and R.D. Moser, Self-similar vortex clusters in the logarithmic region. J. Fluid Mech., 2006. In press.

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5. N. Aubry, P. Holmes, J.L. Lumley and E. Stone, The dynamics of coherent structures in the wall region of a turbulent boundary layer. J. Fluid Mech., 192, 115–173 (1988). 6. H.P. Bakewell and J.L. Lumley, Viscous sublayer and adjacent wall region in turbulent pipe ﬂow. Phys. Fluids, 10, 1880–1889 (1967). 7. G.L. Brown and A. Roshko, On the density eﬀects and large structure in turbulent mixing layers. J. Fluid Mech., 64, 775–816 (1974). 8. H. Choi, P. Moin and J. Kim, Active turbulence control and drag reduction in wall-bounded ﬂows. J. Fluid Mech., 262, 75–110 (1994). 9. H. Darcy, Recherches exp´erimentales r´elatives au mouvement de l’eau dans les tuyeaux. M´em. Savants Etrang. Acad. Sci. Paris, 17, 1–268 (1854). 10. D.B. DeGraaf and J.K. Eaton, Reynolds number scaling of the ﬂat-plate turbulent boundary layer. J. Fluid Mech., 422, 319–346 (2000). 11. O. Flores and Jim´enez, Eﬀect of wall-boundary disturbances on turbulent channel ﬂows. J. Fluid Mech., 2006. In press. ¨ 12. G.H.L. Hagen, Uber den Bewegung des Wassers in engen cylindrischen R¨ ohren. Poggendorfs Ann. Physik Chemie, 16 (1839). 13. J.M. Hamilton, J. Kim and F. Waleﬀe, Regeneration mechanisms of near-wall turbulence structures. J. Fluid Mech., 287, 317–248 (1995). 14. S. Hoyas and J. Jim´enez, Scaling of the velocity ﬂuctuations in turbulent channels up to reτ = 2003. Phys. Fluids, 18, 011702 (2006). 15. J. Jeong, F. Hussain, W. Schoppa and J. Kim, Coherent structures near the wall in a turbulent channel ﬂow. J. Fluid Mech., 332, 185–214 (1997). 16. J. Jim´enez, On the structure and control of near wall turbulence. Phys. Fluids, 6, 944–953 (1994). 17. J. Jim´enez, Turbulent ﬂows over rough walls. Ann. Rev. Fluid Mech., 36, 173–196 (2004). ´ 18. J. Jim´enez, J.C. del Alamo and O. Flores, The large-scale dynamics of nearwall turbulence. J. Fluid Mech., 505, 179–199 (2004). 19. J. Jim´enez, G. Kawahara, M.P. Simens, M. Nagata and M. Shiba, Characterization of near-wall turbulence in terms of equilibrium and ‘bursting’ solutions. Phys. Fluids, 17, 015105 (2005). 20. J. Jim´enez and P. Moin, The minimal ﬂow unit in near-wall turbulence. J. Fluid Mech., 225, 221–240 (1991). 21. J. Jim´enez and R.D. Moser, LES: where are we and what can we expect? AIAA J., 38, 605–612 (2000). 22. J. Jim´enez and A. Pinelli, The autonomous cycle of near wall turbulence. J. Fluid Mech., 389, 335–359 (1999). 23. J. Jim´enez and M.P. Simens, Low-dimensional dynamics in a turbulent wall ﬂow. J. Fluid Mech., 435, 81–91 (2001). 24. G. Kawahara, J. Jim´enez, M. Uhlmann and A. Pinelli, Linear instability of a corrugated vortex sheet – a model for streak instability. J. Fluid Mech., 483, 315–342 (2003). 25. G. Kawahara and S. Kida, Periodic motion embedded in plane Couette turbulence: regeneration cycle and burst. J. Fluid Mech., 449, 291–300 (2001). 26. H.T. Kim, S.J. Kline and W.C. Reynolds, The production of turbulence near

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40. 41.

42. 43. 44. 45. 46. 47.

a smooth wall in a turbulent boundary layer. J. Fluid Mech., 50, 133–160 (1971). J. Kim, P. Moin and R.D. Moser, Turbulence statistics in fully developed channel ﬂow at low Reynolds number. J. Fluid Mech., 177, 133–166 (1987). J.C. Klewicki, M.M. Metzger, E. Kelner and E.M. Thurlow, Viscous sublayer ﬂow visualizations at Rθ ≈ 1, 500, 000. Phys. Fluids, 7, 857–863 (1995). A.N. Kolmogorov, The local structure of turbulence in incompressible viscous ﬂuids a very large Reynolds numbers. Dokl. Akad. Nauk. SSSR, 30, 301–305 (1941). Reprinted in Proc. R. Soc. London. A, 434, 9–13 (1991). W.D. McComb, The physics of ﬂuid turbulence. Oxford U. Press, 1990. W.R.B. Morrison, K.J. Bullock and R.E. Kronauer, Experimental evidence of waves in the sublayer. J. Fluid Mech., 47, 639–656 (1971). M. Nagata, Three-dimensional ﬁnite-amplitude solutions in plane Couette ﬂow: bifurcation from inﬁnity. J. Fluid Mech., 217, 519–527 (1990). ¨ J.M. Osterlund, A.V. Johansson, H.M. Nagib and M. Hites, A note on the overlap region in turbulent boundary layers. Phys. Fluids, 12, 1–4 (2000). S.C. Reddy, P.J. Schmid, J.S. Baggett and D.S. Henningson, On stability of streamwise streaks and transition thresholds in plane channel ﬂows. J. Fluid Mech., 365, 269–303 (1998). S.K. Robinson, Coherent motions in the turbulent boundary layer. Ann. Rev. Fluid Mech., 23, 601–639 (1991). J.D. Swearingen and R.F. Blackwelder, The growth and breakdown of streamwise vortices in the presence of a wall. J. Fluid Mech., 182, 255–290 (1987). W. Schoppa and F. Hussain, Coherent structure generation in near-wall turbulence. J. Fluid Mech., 453, 57–108 (2002). L. Sirovich and X. Zhou, Dynamical model of wall-bounded turbulence. Phys. Rev. Lett., 72, 340–343 (1994). C.R. Smith and S.P. Metzler, The characteristics of low speed streaks in the near wall region of a turbulent boundary layer. J. Fluid Mech., 129, 27–54 (1983). J.L. Tennekes, H. & Lumley, A ﬁrst course in turbulence. MIT Press, 1972. S. Toh and T. Itano, On the regeneration mechanism of turbulence in the channel ﬂow. In T. Kambe, Nakano T. and T. Muiyauchi, editors, Proc. Iutam Symp. on Geometry and Statistics of Turbulence, pages 305–310. Kluwer, 2001. S. Toh and T. Itano, A periodic-like solution in channel ﬂow. J. Fluid Mech., 481, 67–76 (2003). A.A. Townsend, The structure of turbulent shear ﬂow. Cambridge U. Press, second edition, 1976. F. Waleﬀe, On a self-sustaining process in shear ﬂows. Phys. Fluids, 9, 883– 900 (1997). F. Waleﬀe, Three-dimensional coherent states in plane shear ﬂows. Phys. Rev. Lett., 81, 4140–4143 (1998). F. Waleﬀe, Exact coherent structures in channel ﬂow. J. Fluid Mech., 435, 93–102 (2001). F. Waleﬀe, Homotopy of exact coherent structures in plane shear ﬂows. Phys. Fluids, 15, 1517–1534 (2003).

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COHERENT STRUCTURES GENERATED BY A SYNTHETIC JET JONATHAN H. WATMUFF School of Aerospace, Mechanical and Manufacturing Engineering, RMIT University, PO Box 71, Bundoora, 3083 Australia jon.watmuﬀ@rmit.edu.au Most synthetic–jet–actuator ﬂow-control applications utilize relatively large amplitude forcing. Under these conditions the asymmetrical inﬂow and outﬂow characteristics lead to a net momentum transfer while the net mass ﬂux is zero. Hot-wire measurements are presented which demonstrate that the response is linear and symmetrical for suﬃciently small actuator amplitudes. The base ﬂow is a Blasius boundary layer with an exceptionally small background disturbance level. For small actuator amplitudes the disturbance has the form of three-dimensional TS (Tollmien–Schlichting) waves which conform with the results of computations using the Parabolized Stability Equations (PSE). For larger actuator amplitudes, other short-wavelength instabilities develop and grow with streamwise development and they ultimately breakdown to form a turbulent wedge. The evidence suggests that the instabilities have the form of an absolute Rayleigh-type instability associated with locally inﬂectional velocity proﬁles. There is an actuator amplitude threshold below which these short-wavelength instabilities do not form, and a larger threshold below which the instabilities do not grow with streamwise development. Characteristics of the wedge are considered in some detail.

1. Introduction The synthetic–jet–actuator is a device in which periodic forcing of a sealed cavity by a membrane leads to alternating inﬂow and outﬂow through a small oriﬁce. At moderate membrane amplitudes the ﬂuid separates from the lip of the oriﬁce during the outﬂow to create a coherent jet. However during the inﬂow, the ﬂow will always resemble a point sink, no matter how large the inﬂow. The asymmetrical response leads to a net momentum transfer to the external ﬂow but the net mass ﬂux always remains zero. The ability of the synthetic–jet–actuator to act as a point source of momentum without the need for external plumbing has made it an ideal candidate for ﬂow control applications. The device was ﬁrst studied by Ingard1 but it was

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rediscovered and further developed for ﬂow control applications by Glezer and co-workers.2–4 Subsequent ﬂow control application studies include control of boundary layer separation,5–7 control of jets8 and small-scale control of turbulence.8–10 These studies have focused on applications which use relatively large amplitude disturbances. In contrast, the present work is concerned with much smaller actuator amplitudes. A potential application area for the use of small actuator amplitudes is Laminar Flow Control (LFC), e.g. cancellation of background disturbances by superposition with counter disturbances generated by the actuator.20 2. Apparatus and techniques The experiments were conducted in a small stand-alone open-return wind tunnel that was speciﬁcally modiﬁed for boundary layer transition studies. Painstaking ﬂow quality improvements were made over a period of several years by improving the quality of the wind tunnel screens. For example, small span-wise variations in the porosity of the screens were discovered by Watmuﬀ 12 by traversing each screen between a laser and a photo detector. The background unsteadiness, u /U1 in the Blasius boundary layer was ultimately reduced by a factor of 30 compared to the original conﬁguration. Flow quality is an important consideration for the type of ﬂows considered here since extraneous background disturbances could interact with disturbances introduced by the actuator leading to facility-dependent behaviour. The base-ﬂow consists of a highly span-wise uniform Blasius boundary layer. The mean ﬂow development closely follows the Blasius similarity solution. The background unsteadiness levels are extremely small: in the freestream, u /U1 < 0.05%; and within the layer, u /U1 < 0.08%. Vibrating ribbon experiments by Watmuﬀ 13 demonstrate the growth of distortionfree two-dimensional Tollmien–Schlichting (TS) waves that closely follow predictions from the linearized Parabolized Stability Equations (PSE) provided by Bertolotti [private communication]. The synthetic–jet–actuator consists of a small speaker located within a hermetically-sealed container which is connected via a ﬂexible tube to a 1 mm diameter oriﬁce in the test surface as shown schematically in Fig. 1. Disturbances are introduced by applying a harmonic drive signal to the actuator which is generated by a custom hybrid circuit in precise synchronization with the signals that control data acquisition. The signal from a miniature single hot-wire probe is ensemble averaged on the basis of the phase of the actuator drive signal. A total of 64 phase intervals are used for

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Blasius boundary layer: Test Section Flow Orifice: d = 1 mm

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3 mm thick stainless steel test plate Flexible Tubing

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Unit Re = 6.68 ? 105 m-1 Conditions at orifice xs = 0.352 m, R = 485 d*=1.25 mm, d99 = 3.63 mm

Sample

Computer

Control Signals for Hot-Wire DACQ

Fig. 1. Schematic of synthetic–jet–actuator and hybrid circuit for generating harmonic drive signal for actuator in precise synchronization with control signals for 64-interval phase-averaged data acquisition.

ensemble averages using 600 phase cycles at each data point (38,400 samples over an approximately 10 s long sampling period). A key feature of the apparatus is that all experimental functions are totally automated, including frequent hot-wire calibration. Total automation of the facility allows unattended computer controlled experiments to be programmed ahead of time and performed continuously 24 h/day for up to several weeks without the need for scheduled manual intervention. Phase-averaged data can be acquired on large spatially dense measurement grids to provide a level of detail that is usually associated with computations, see Watmuﬀ.13,14 The measurements are limited to streamwise velocities using a normal hot-wire probe owing to the small thickness and high shear of the laminar boundary layer. Custom hot-wire probes were specially developed to

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minimize intrusive ﬂow disturbances. The prongs consist of a pair of 20 mm long jeweller’s broaches which are inclined at a minimal angle to freestream while still providing close access to the test surface. The probes were ﬁtted with 2.5 µm diameter Platinum-coated Tungsten ﬁlaments. A 40 µm thick copper coating was applied to the wire using an electrolytic technique to facilitate soldering onto the prongs. The copper was then etched away with an acid solution to expose the 0.5 mm long active portion of the ﬁlament. The phase-averaging technique forms the basis for all measurements of the phase and amplitude of the disturbance motions. The phase-averaging ¯ a periprocess treats any ﬂow variable, S, as a global mean component, S, odic ﬂuctuating component, s˜, and a random component, s . This notation was used by Watmuﬀ 14 and conforms with that used by Reynolds and Hussain15 and Cantwell and Coles16 with the exception of the nomenclature for the random component, s . The reason for the minor departure in notation is to retain the conventional notation for Reynolds decomposition into a temporal mean value, S, and a random ﬂuctuating component, s . In order to establish a notation that is more in conformance with that in the transition literature, the tilde will be replaced with the subscript φ (to indicate phase). By deﬁnition, the total variable is given by the sum, S = S + sφ + s = S + s Interpretation of the results in this paper is dependent on implications derived from a full consideration of the results of the phase-averaging process. In transition studies, the quantity of most interest is usually just the amplitude of the instability, which is most often expressed as the rms (rootmean-square) amplitude. To conform with prior usage in the literature, the overscore notation will not be used and the term, u, will signify the rms value, obtained by using all phase intervals. Of particular interest in the present work is the random component of the phase-averaged velocity ﬂuctuations. It should be emphasized that this is also a phase-averaged quantity and it can provide valuable additional information that has proved useful in the study of turbulent ﬂows. However this quantity has been generally ignored in transition studies. In turbulent ﬂows this quantity arises from fully random motions. However in transitional ﬂows this quantity can also be used to identify the onset of randomness, such as cycle-to-cycle variations at a particular phase interval. It is also particularly useful for identifying regions of large velocity gradient. For consistency, the overscore will also be

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dropped and the term uφ will be used for the rms velocity ﬂuctuation at constant phase, and the term, u , will be used for the rms amplitude over all phases.

(a)

(b)

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50

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Fig. 2. Contours of rms streamwise wave amplitude in span-wise plane at x∗ = 765 for Mack–Kendall parameters: F = 2πf ν/U1 2 = 60 × 10−6 , R = 485. (a) PSE results of Mack and Herbert.11 (b) Experimental results using synthetic–jet–actuator in Fig. 1 with erms = 360 mV.

3. Three-dimensional Tollmien–Schlichting waves Gaster17 developed Linear Stability Theory (LST) to predict the characteristics of the wave packet generated by an impulsive point source observed by Gaster and Grant.18 He assumed that an impulsive point source excites wave motions with a broad wavenumber spectra and that a wave-packet forms and develops through selective ampliﬁcation and interference of linear instability waves. Gaster calculated the overall characteristics of the wave-packet by performing a summation over all span-wise wavenumbers and frequencies of least-damped linear modes. However the agreement between observation and predictions was only qualitative and corrections were required to account for the boundary-layer growth. The corrections required by Gaster were attributed to nonparallel effects. However, other anomalies between predictions from LST and observations were discovered subsequently, which led Mack19 to propose a combined study of the simplest possible form of three-dimensional disturbance, i.e. the Harmonic Point Source (HPS). Only a single frequency is involved but the HPS still generates a broad spectrum of span-wise wave numbers. However even for this simple form of disturbance Mack and Kendall19 found predictions of the centreline growth to be much greater than observations. Mack and Herbert11 considered the HPS with Mack–Kendall parameters; F = 2πf ν/U1 2 = 60 × 10−6 , R = Rx 0.5 = 485 (f is disturbance

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Fig. 3. Proﬁles of the total phase-averaged velocity immediately above the oriﬁce for range of rms actuator drive signal amplitudes: (a) erms = 120 mV, (b) erms = 360 mV, (c) erms = 720 mV, and (d) erms = 1152 mV. (31 y-points, 64 proﬁles each case).

frequency, x is distance of source from layer origin). They used three numerical techniques: Linear Stability Theory; Parabolized Stability Equations and Direct Numerical Simulation (DNS). Each technique gave diﬀerent results. Resolution of the disparities between the results is diﬃcult because the near ﬁeld cannot be captured by LST or PSE, while DNS has a strong dependence on the source geometry. Experimental observations are required to resolve the anomalies, particularly near the source of the disturbance. During the period of wind-tunnel ﬂow-quality improvements the synthetic–jet–actuator shown in Fig. 1 was used to generate disturbances which provided a sensitive test case. TS waves generated for small actuator amplitudes were found to conform with the PSE results of Mack and 1/2 Herbert11 as shown in Fig. 2. The quantity x∗ = Rx /Rs , is the nondimensional streamwise coordinate used by Mack and Herbert, where Rs and Rx are the Reynolds numbers based on the streamwise distance from the origin to the source and measurement plane respectively. The HPS is treated as an idealized linear disturbance in the PSE so the agreement between the observations and the computations suggests that the asymmetry between the amplitude of the positive and negative disturbance levels is insigniﬁcant

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y = 0.18 mm y = 0.23 mm

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Fig. 4. Phase-averaged velocity perturbations for positions close to test surface. Same source data as proﬁles in Fig. 3 (a) erms = 120 mV, (b) erms = 360 mV, (c) erms = 720 mV, and (d) erms = 1152 mV.

at small actuator amplitudes. The results presented in this paper are the outcome of more direct investigations into the asymmetry and the linearity of the response. 4. Asymmetry and linearity with actuator amplitude Proﬁles of the total phase-averaged velocity obtained directly over the centre of the oriﬁce for small to moderately large drive signal amplitudes are shown in Fig. 3. The position of the hot-wire ﬁlament was veriﬁed directly in situ with a high-powered microscope. Proﬁles located 10mm downstream

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of the oriﬁce for larger actuator amplitudes are shown in Fig. 6. The proﬁles demonstrate that the disturbance is essentially conﬁned to the inner third of the layer thickness. Phase-averaged velocity perturbations for the proﬁles over the oriﬁce at the six positions closest to the test surface are shown in Fig. 4. The positive and negative perturbations are almost perfectly symmetrical for the smallest drive signal, erms = 120 mV, shown in Fig. 4(a). The results for the case erms = 360 mV shown in Fig. 4(b) correspond to the contours in the span-wise plane further downstream shown in Fig. 2. A small nonlinear departure from symmetric behaviour is apparent for the point closest to the test surface (y=0.18 mm) for this case between phase 45 and phase 55. The disturbance amplitude decays rapidly with streamwise distance and the small nonlinearity is not considered to have signiﬁcantly distorted the results shown in Fig. 2. Highly nonlinear asymmetrical behaviour is observed for moderately large actuator drive signals. For small actuator amplitudes the outﬂow corresponds with a negative phase-averaged velocity perturbation since the air issuing from the oriﬁce has zero streamwise velocity. However for larger actuator amplitudes the velocity component normal to the streamwise direction is likely to make a signiﬁcant contribution to the eﬀective cooling velocity experienced by the wire. Evidence of this eﬀect can be seen for the larger actuator amplitudes where the indicated velocity perturbations have an appearance resembling hot-wire rectiﬁcation under conditions of ﬂow reversal. The case erms = 1152 mV shown in Fig. 4(d) corresponds to an actuator amplitude just below that required for breakdown of the disturbance. This case will be examined in more detail in Sec. 5. 1.0

5

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Fig. 5. Linearity of disturbance amplitude with actuator drive signal amplitude. (a) Above oriﬁce: x = 0.4094 m, y = 1.0 mm, R = 485; (b) Centreline: x = 0.45 m, y = 1.25 mm, R = 508; (c) Oﬀ-centreline: x = 0.675 m, y = 1.25 mm, z = −30 mm, R = 623. Spanwise position for maximum amplitude at each streamwise position.

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The variation of the rms disturbance amplitude with the rms drive signal amplitude for three ﬁxed probe positions is shown in Figs. 5(a–c). Linear response of the disturbance amplitude is limited to small drive signal amplitudes i.e. in the immediate vicinity of the oriﬁce, erms < 180 mV. A linear response is maintained for larger actuator amplitudes further downstream and away from the centreline, e.g. erms < 260 mV, as shown in Fig. 5(c). The “smart-wall” concept20 proposes the use of Neural Networks and Micro-Electro-Mechanical-Systems (MEMS) based sensors and actuators for the detection and cancellation of TS waves at the linear stage of their development. The basic idea is that optimum counter disturbances can be generated to cancel small amplitude incoming disturbances based on the wave superposition principle. The MEMS-based actuator considered for the smart-wall consists of a ﬂush mounted impermeable membrane. The results in Fig. 5(a–c) demonstrate that the synthetic–jet–actuator operating at small amplitudes could act as an alternative MEMS-based device for Laminar Flow Control (LFC) applications. 5. Formation of instabilities and breakdown of disturbance Proﬁles of the total phase-averaged velocity located 10 mm (≈ 2.7 δ99 ) downstream of the oriﬁce and corresponding to a range from moderate to large actuator amplitudes are shown in Figs. 6(a–f). It is clear that locally inﬂectional proﬁles occur for all these actuator amplitudes. However, the strength of the shear associated with the locally inﬂectional proﬁles decays with streamwise distance for erms ≤ 1152 mV . Breakdown of the disturbance further downstream only occurs for erms ≥ 1440 mV, i.e. for the proﬁles shown in Figs. 6(d–f). It is diﬃcult to determine the physical characteristics of any instabilities associated with the disturbance from velocity proﬁles. A better depiction of the instabilities can be obtained by using pseudo-ﬂow visualization, i.e. the use of phase as the streamwise coordinate. The pseudo-ﬂow ﬁeld will only be equivalent to the true (spatial) ﬂow ﬁeld if the disturbance remains frozen as it propagates downstream, i.e. Taylor’s Hypothesis in turbulent ﬂow. While this is strictly not the case, the spatial development of the instability with streamwise distance is suﬃciently small so that the pseudo-ﬂow ﬁeld provides a reasonably accurate representation of the disturbance. Pseudoﬂow visualization oﬀers the advantage that a contour plot on a streamwise section through the disturbance can be constructed from a single proﬁle, and that a compete three-dimensional representation of the disturbance can be constructed from measurements on a spanwise plane.

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Fig. 6. Proﬁles of the total phase-averaged velocity 10 mm downstream of oriﬁce for range of rms actuator drive signal amplitudes: (a) erms = 720 mV, (b) erms = 960 mV, (c) erms = 1152 mV, (d) erms = 1440 mV, (e) erms = 2000 mV, and (f) erms = 2400 mV. (31 y-points, 64 proﬁles each case).

Pseudo-ﬂow contour lines constructed from the same proﬁle data as Fig. 6 are shown for the larger actuator-amplitudes in Fig. 7. Two consecutive phase cycles are shown for each quantity to aid interpretation, since the regions of interest lie towards the extremities of the phase cycle. The results for erms = 1152 mV in Fig. 7(a) reveal four small amplitude corrugations in the total velocity contours in the region of high shear. The locally-concentrated elevated levels in the random ﬂuctuation contours coincide with the region of high shear and are most likely the result of small cycle-to-cycle variations. For the next largest actuator amplitude,

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Fig. 7. Formation of short wavelength instabilities associated with disturbance 10 mm downstream of oriﬁce. Pseudo-ﬂow contour lines from centreline proﬁles. Actuator drive signal amplitude: (a) erms = 1152 mV; (b) erms = 1440 mV; (c) erms = 2000 mV; and (d) erms = 2400 mV. For each case: contours of total phase-averaged velocity, (U + uφ )/U1 ; phase-averaged ﬂuctuations, uφ /U1 ; and rms random ﬂuctuations at constant phase, u φ /U1 . CR is Contour Range. CI is Contour Increment. For uφ /U1 : dashed lines are positive and black lines negative-levels; zero levels not shown. Two phase cycles shown successively for each contour plot to aid interpretation.

erms = 1440 mV shown in Fig. 7(b), there are also four corrugations in the total velocity contours, but they have a greater amplitude and they have a considerably larger spatial extent. The appearance of corrugations in the region of high shear in the total velocity contours suggests the formation of an inviscid Rayleigh-type instability. It is curious that only two corrugations appear in the total phaseaveraged velocity contours for the two largest actuator amplitudes in Figs. 7(c) and (d), since the shear associated with the disturbance is likely to be larger for these cases. Examination of individual time records can provide an explanation for this apparent anomaly. Representative time record sequences are shown in Figs. 8(a–c) at a wall distance y=1.6 mm. The

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Fig. 8. Perturbation time records: 64 sequences sampled at precisely 256 intervals per phase cycle. Hot-wire located on centreline 10 mm downstream of the oriﬁce at a wall distance, y = 1.6 mm. Resulting phase-averaged waveforms using 64 intervals shown at top of each sequence, with curves uφ , (uφ + u φ ) and (uφ − uφ ). Actuator drive signal amplitude: (a) erms = 1440 mV; (a) erms = 2000 mV; (a) erms = 2400 mV. (Arbitrary velocity scale chosen for ﬁt on each plot.)

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(c) erms = 2400 mV. Oriﬁce diameter, d, and length of hot-wire ﬁlament, hw indicated on (U + uφ )/U1 contours for erms = 2400 mV. Source data is full y–z plane, (Ny , Nz )=(31,41)=1271 data points, ∆z = ±5 mm, δz=0.25 mm. CI is Contour Increment as labelled on each contour. For uφ /U1 : dashed lines are positive and black lines negative-levels; zero levels not shown.

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Fig. 10. Development of short wavelength instabilities associated with disturbance 30 mm downstream of oriﬁce. Pseudo-ﬂow contour lines from centreline proﬁles. Actuator drive signal rms amplitude: (a) erms = 1152 mV; (b) erms = 1440 mV; (c) erms = 2000 mV; and (d) erms = 2400 mV. Refer to caption of Fig. 7 for deﬁnition of quantities, contour levels and other terms.

sequences were recorded at precisely four times the phase-average sampling frequency for improved resolution (i.e. 256 samples per phase interval). Three curves resulting from the phase-averaging process using 64 samples per cycle are located above the time record sequences, i.e. curves of uφ , (uφ + uφ ) and (uφ − uφ ). These curves provide a compact method for revealing the strength of both uφ and uφ . The phase-averaged perturbations for erms = 1152 mV shown in Fig. 8(a) have an appearance consistent with each time record sequence, e.g. there are four minima, and the unsteadiness at constant phase, uφ , is small. However, the time record sequences for the larger actuator amplitudes in Figs. 8(b–c) clearly demonstrate a larger number of velocity oscillations than realized in the phase-averaged waveform. This is particularly evident for the actuator drive level, erms = 2400 mV, where between eight and nine oscillations are evident in the time

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record sequences but only the ﬁrst oscillation appears in the phase-averaged waveform and the phase jitter associated with subsequent oscillations leads to elevated values of uφ . Therefore one explanation for the appearance of fewer corrugations in the total velocity contours at larger actuator amplitudes, is that a larger number of instability waves do in fact form further upstream on the disturbance, but that the phase jitter arising from randomness in their relative position on the disturbance from cycle to cycle results in washout of the phase-averaged contours. Pseudo-ﬂow gray-scale contours and contour lines in a plane parallel to the test surface st y = 1 mm are shown in Figs. 9(a–c). These contours have been generated from measurements on a full spanwise plane. The oriﬁce diameter is shown in the total velocity contours in Fig. 9(c) and the narrow spanwise extent of the primary disturbance is clearly evident in every case. Note that the spacing of the spanwise grid for these measurements is half the length of the hot-wire ﬁlament, which is also indicated in the ﬁgure. Spatial ﬁltering by the ﬁlament implies that the spanwise velocity gradients will be even sharper than those indicated in the contours. An interesting feature that is apparent in both the total and perturbation velocity contours is that the short-wavelength motions associated with the instabilities exist over a much wider spanwise extent than the primary larger-scale disturbance introduced by the actuator. Oﬀ-centreline (z =1.5 mm) time record sequences clearly show large amplitude ﬂuctuations that are more coherent than the corresponding ﬂuctuations on the centreline that are shown in Fig. 8(c). Pseudo-ﬂow contour lines corresponding to a position 30 mm (≈ 8 δ99 ) downstream of the oriﬁce are shown in Figs. 10(a–d). The reduced concentration of shear in the total velocity contours with streamwise development is evident in comparison with the corresponding results in Figs. 7(a–d). Most noticeable is the further development of corrugations in the total velocity contours for erms = 1152 mV as shown in Fig. 10(a). However, the corrugations have not reached a wall distance corresponding with the undisturbed layer thickness and further downstream, there are fewer corrugations in the contours for this case and they occur at much reduced strength. However, for the larger actuator drive signals, the short-wavelength instabilities maintain much the same shape and form with further streamwise development. A feature common to all cases further downstream is a marked increase in the amplitude of the random component, e.g. uφ /U1 > 10%. By the streamwise position x = 600 mm downstream, all disturbances corresponding to the larger actuator amplitudes have undergone breakdown to

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form a turbulent wedge, while the disturbance for the case erms = 1152 mV has continued to decay. Characteristics of the turbulent wedge are considered in Sec. 6 below. 6. Characteristics of the turbulent wedge Pseudo-ﬂow contours in a plane parallel to the test surface have been generated from a spanwise proﬁle at the streamwise position x = 600 mm for the cases, erms = 1440 mV, 2000 mV and 2400 mV. These contours are not shown in this paper but they reveal an almost identical ﬂow structure for each case. On this basis, representative detailed characteristics of the turbulent wedge have been measured for the case erms = 2400 mV. True spatial contours of the rms phase-averaged ﬂuctuation, u/U1 , are shown in a surface nearly parallel to test surface in Fig. 11(a). The wedge is fully turbulent, but remnants of the disturbance remain following breakdown, although the relative contribution of the phase-averaged motions to the total rms broadband unsteadiness decreases with streamwise development owing to the presence of ﬁne-scale turbulent motions and phase jitter. The relative contribution of u/U1 and u /U1 to the total broadband rms level, u /U1 , can be seen more clearly in the cross-stream planes shown in Figs. 11(b–c). The maxima in the broadband turbulence intensity occur at the spanwise extremities of the wedge which coincide with the location of the maxima in the phase-averaged contribution. Pseudo-ﬂow contour surfaces of total phase-averaged velocity corresponding to the measurements in each spanwise plane are shown in Figs. 12(a–c). The contour level, (U + uφ )/U1 =0.9, of these surfaces means that they are located further from the test surface where the washout introduced by ﬁne-scale turbulent motions is minimal. The contours corresponding to the position x = 600 mm in Fig. 12(a) have an appearance that is reminiscent of the Λ-shaped vortex loops that appear in fully turbulent boundary layers. A more complete manifestation of the Λ-shaped structure is apparent in the contour surface of the random ﬂuctuations, uφ , shown in Fig. 12(d). It is not possible to diﬀerentiate whether this surface is the result of (i) phase jitter associated with the appearance of a single large-scale structure each cycle, or (ii) a collection of ﬁne-scale motions organized into a preferred shape. In any case, this surface has a clearly deﬁned Λ-shape. The -shaped patterns present in the total phase-averaged velocity contours further downstream in Figs. 12(b–c) suggest that the vortex loops may have become larger with the increased width of the wedge. However, the contour surfaces of the random component further downstream in

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Fig. 11. Development of turbulent wedge structure. Gray-scale contours and contour lines of rms quantities for actuator drive signal amplitude, erms = 2400 mV, showing: (a) rms phase-averaged ﬂuctuations, u/U1 , in surface nearly parallel to wall. Gray-scale legend maximum, Max = 4.0 %. Cross-stream planes showing development at x = 0.6 m, 0.8 m and 1.0 m, with legend maxima, Max, labelled in each contour: (b) rms phase-averaged ﬂuctuations, u/U1 ; (c) rms random ﬂuctuations at constant phase, u /U1 ; and (d) rms broadband ﬂuctuations, u /U1 . All contour lines use same increment of 1.0% and zero levels are not shown.

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Fig. 12. Contour surfaces for actuator drive signal rms amplitude, erms = 2400 mV, showing early development of turbulent wedge structure. Pseudo-ﬂow contour surfaces of total phase-averaged velocity, (U + uφ )/U1 =0.9, derived from data in spanwise plane located at (a) x = 0.6 m, (b) x = 0.8 m and (c) x = 1.0 m. Pseudo-ﬂow contour surfaces of u φ /U1 and contours parallel to test surface at (d) x = 0.6 m, also with contour lines at 1% increment; zero levels not shown. (e) x = 0.8 m and (f) x = 1.0 m. Three consecutive phase-cycles shown to aid interpretation.

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Figs. 12(e–f) suggest that structures responsible for elevated level do not have a double-sided Λ-shape. It is almost certain the ﬂow structure that forms during breakdown of the disturbance introduced by the actuator is responsible for the spanwise growth of the wedge with streamwise distance. One plausible mechanism for the spanwise growth is cross-ﬂow instability. 7. Conclusions The case of a synthetic–jet–actuator in a Blasius boundary layer is a multiple parameter problem. Only a single oriﬁce-diameter to layer-thickness ratio and a single operating frequency have been considered. Nevertheless, detailed measurements have shown that it is short-wavelength instabilities that form on the primary large-scale disturbance which are responsible for breakdown to form a turbulent wedge. There is an actuator amplitude threshold below which these instabilities do not form, and another larger threshold below which the instabilities do not grow with streamwise development. A more complete understanding will guide more sophisticated implementations of the synthetic–jet–actuator for ﬂow control applications. Acknowledgements The measurements were obtained in the Fluid Mechanics Laboratory, at NASA Ames Research Center, in California, USA. Most of the subsequent analysis has been performed at the School of Aerospace, Mechanical Manufacturing Engineering at RMIT University in Australia. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.

U. Ingard, J. Acoust. Soc. of America, 25 No. 6, 1037–1060 (1953) B. Smith and A. Glezer, Phys. Fluids, 10 No. 9, 2281–2297 (1998) B. Smith and A. Glezer, J. Fluid Mech., 458 1–34 (2002) A. Glezer and M. Amitay, Ann. Rev. of Fluid Mech., 34 503–529 (2002) D. C. McCormick, AIAA paper 2000–0519 (2000). M. Honohan, M. Amitay and A. Glezer, AIAA paper 2000–2401 (2000). A. Seifert and L. Pack, AIAA J., 37 No. 9 (1999) R. Rathnasingham and K. S. Breuer, Phys. Fluids, /bf 9 No. 7, 1867–1869 (1997) R. Rathnasingham and K. S. Breuer, J. Fluid Mech., /bf 495 209–233 (2003) C. Lee and D. Goldstein, AIAA paper 2001–1013 (2001). L. M. Mack and T. Herbert, AIAA paper 95–0774 (1995). J. H. Watmuﬀ, AIAA J. 36, No. 3 (1998) J. H. Watmuﬀ, J. Fluids Eng. 128, No. 2, 247–257, (2006)

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14. J. H. Watmuﬀ, J. Fluid Mech. 397, 119–170 (1999) 15. W. C. Reynolds and A. K. M. F. Hussain, J. Fluid Mech., 54 263–288 (1972) 16. B. J. Cantwell and D. Coles, J. Fluid Mech., 136 321–374. (1983) 17. M. Gaster, Proc. Roy. Soc., A347 271–289, (1975) 18. M. Gaster and I. Grant, Proc. Roy. Soc., A347 253–269, (1975) 19. L. M. Mack and J. M. Kendall, AIAA paper 83–0046 (1983). 20. X. Fan, T. Herbert and J. Haritonidis, AIAA-1995–674 (1995).

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TWO-POINT TURBULENCE CLOSURES REVISITED DAVID McCOMB School of Physics, University of Edinburgh, James Clerk Maxwell Building, The King’s Buildings, Mayﬁeld Road, Edinburgh EH9 3JZ, Scotland, UK. [email protected]

1. Introduction 1.1. Renormalized perturbation theories and two-point turbulence closures In this chapter we begin with a short history of the turbulence closure problem, and then state the basic equations in wavenumber space (or kspace). This is followed by an outline of the quasi-normality theory, as the best known ad hoc method. We turn then to perturbation theory, as the only general method, and state the second-order covariance equations as an approximation to the required statistical closure. A comparison between these and the quasi-normality equations is used to motivate the introduction of renormalized perturbation theory or RPT; and at that point we list the main pioneering RPTs, outline the method (Kraichnan-Wyld perturbation theory) and state the resulting second-order covariance equations in terms of an unknown renormalized response function. At this lowest non-trivial order, the turbulence closure problem can be seen as one of ﬁnding the renormalized response function. We then outline the derivation of the Direct-Interaction Approximation (DIA) equation for the response function, showing that it is a natural outcome of the Kraichnan-Wyld formulation. In Section 4.7 we state the resulting DIA response equation in detail, and anticipate later discussions by also stating the Local Energy Transfer (LET) response equation in order to facilitate comparisons. In Section 5 we discuss the failure of ﬁrst-generation RPTs, as indicated by their incompatibility with the Kolmogorov ‘-5/3’ spectrum (K41),1,2 along with the re-interpretation of this failure which led to the LET theory,

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which is compatible with K41. In Section 5.2 some numerical results are presented for the free decay of isotropic turbulence and also for stationary forced isotropic turbulence, as predicted by both DIA and LET theories. The thorny problem of unresolved and contentious issues is discussed in Section 6. As will be seen, some of these aﬀect the way in which the turbulence community perceives RPTs, and others are wider, aﬀecting the whole of turbulence theory. The chapter closes with a derivation of the LET response equation, making use of the latest developments in this subject, and also gives the extension to new single-time and Markovianized forms of the theory. 1.2. A brief history of closures The turbulence closure problem was formulated for turbulent shear ﬂows by Reynolds in the 1890s (see McComb3 and references therein): the equation ¯ contains the unknown covariance of two ﬂuctuating for the mean velocity U velocities uu: the Reynolds stress. It was later formulated for isotropic turbulence by Taylor in the 1930s (see Batchelor4 and references therein): the equation for the correlation of two velocities uu contains the unknown correlation of three velocities uuu. The equation for uuu contains the unknown uuuu, and so on . . . The problem is seen as an open hierarchy of statistical equations (onepoint for Reynolds, two-point for Taylor) which requires some procedure for closure. This is the fundamental problem of turbulence theory and is consistent with an interpetation of turbulence as a problem in many-body physics, such as occurs in the theory of dense gases, structure of liquids, magnetism, nuclear binding, and so on. Eddy-viscosity, mixing-length and n-equation models are all eﬀectively ‘statistical closure approximations’ for the single-point problem. The Heisenberg eddy viscosity and the quasi-normality theory are eﬀectively closure approximations for the two-point problem. Quasi-normality in 1954 (see Leslie5 or McComb3,6 ) was the ﬁrst formal treatment of the closure problem: in this approach we solve the next equation in the hierarchy for uuu by factorizing uuuu in terms of uu × uu. Quasi-normality failed when computed numerically in the 1960s. As it predicts negative spectra, it is not physically realizable. Eulerian renormalized perturbation theories, viz., the DIA7 of Kraichnan in 1959; the Edwards-Fokker-Planck (EFP)8 of Edwards in 1964; the Self-Consistent Field (SCF)9 of Herring in 1965; were physically realizable: but were found not to be compatible with K41.

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Lagrangian-history theories due to Kraichnan,10 Kraichnan and Herring,11 Kaneda,12 and Kida and Goto13 are claimed to be random Galilean invariant: the implication is that they are compatible with K41. However, the LET theory of McComb14 is compatible with K41, despite being purely Eulerian. 2. Basic equations in K-space The velocity ﬁeld in a ﬂuid u(x, t) can be expressed in terms of its Fourier transform u(k, t), thus: u(x, t) ≡ uα (x, t) = d3 k uα (k, t) exp(ik · x). The Fourier transform pair is completed by

3 1 uα (k, t) = d3 x uα (x, t) exp(−ik · x). 2π The covariance of velocities for homogeneous and isotropic turbulence takes the form: uα (k, t)uβ (k , t ) = δ(k + k )Cαβ (k; t, t ) = δ(k + k )Pαβ (k)C(k; t, t ), where the projector Pαβ (k) is expressed in terms of the Kronecker delta as Pαβ (k) = δαβ −

kα kβ . |k|2

(1)

The continuity equation is kα uα (k, t) = 0, and the solenoidal Navier-Stokes equation (NSE) takes the form

∂ + ν0 k 2 uα (k, t) = Mαβγ (k) d3 j uβ (j, t)uγ (k − j, t). ∂t

(2)

The inertial transfer operator Mαβγ (k) is symmetric under the interchange of the indices β and γ and is given by Mαβγ (k) = (2i)−1 [kβ Pαγ (k) + kγ Pαβ (k)]. It is sometimes convenient to deﬁne the linear operator L0 (k) as:

∂ 2 L0 (k) = + ν0 k . ∂t

(3)

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If we wish to study the initial value problem posed by the free decay of turbulence then the turbulence ensemble is speciﬁed by the initial conditions. It is usual to do this by choosing the arbitrary velocity ﬁeld at the initial time to be random with a Gaussian distribution. If we wish instead to study stationary turbulence, this requires the addition of a stirring force fα (k, t) to the Navier-Stokes equation. In this case we choose the stirring forces to be random with a Gaussian distribution and complete their speciﬁcation by taking their autocorrelation to be: fα (k, t)fβ (k , t ) = 2(2π)3 Pαβ (k)D(k)δ(k + k )δ(t − t ),

(4)

where D(k) is the arbitrarily chosen force spectrum. 2.1. Equation for the velocity covariance The velocity-ﬁeld covariance Cαβ (k; t, t ) is deﬁned by equation (1). To obtain a governing equation for this, we multiply each term in (2) by uσ (−k, t ) and then average each term. The resulting equation is L0 (k)Pασ (k)C(k; t, t ) = Mαβγ (k) d3 jCβγσ (j, k − j, −k; t, t ). Here Cαβγ (k, j, −k − j; t, t ) stands for the three-velocity correlation. The aim of a closure approximation is to express the three-velocity correlation in terms of the pair correlation or covariance C(k; t, t ). 2.2. Equation for the energy spectrum The energy spectrum E(k, t) is related to the spectral density C(k, t) by E(k, t) = 4πk 2 C(k, t).

(5)

To obtain an equation for this, we ﬁrst multiply each term in (2) by uσ (−k, t). Then we form a second equation from (2) for uσ (−k, t), multiply this by uα (k, t), add the two resulting equations together and average the ﬁnal expression. The resulting equation is

∂ + 2ν0 k 2 Pασ (k)C(k, t) = Mαβγ (k) d3 jCβγσ (j, k − j, −k; t) ∂t −Mσβγ (k d3 jCβγα (j, −k − j, k; t). (6) Here Cαβγ (k, j) stands for the three-velocity correlation.

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We then set σ = α, sum over α (noting that T r Pαβ = 2) and multiply each term in (6) by 2πk 2 , to obtain:

∂ 2 (7) + 2ν0 k E(k, t) = T (k, t). ∂t The energy transfer spectrum T (k, t) is given by 2 T (k, t) = 2πk Mαβγ (k) d3 j {Cβγα (k − j, −k, t) − Cβγα (−k − j, k, t)} . (8) This is an example of the statistical closure problem: an equation for a second-order moment contains an unknown third-order moment. And so on . . . 2.3. Dissipation rate ε in k-space By deﬁnition, ε = −dE/dt for freely decaying turbulence. Integrating over wavenumber, the energy balance becomes: ∞ dE = −ε = − 2ν0 k 2 E(k, t) dk. dt 0 This is because the inertial transfer term vanishes when integrated over all k. The region in k-space where the dissipation mainly occurs is characterised by the Kolmogorov dissipation wavenumber: kd = (ε/ν0 )1/4 . 2.4. Inertial transfer in k-space Write T (k, t) as

∞

S(k, j) dj,

T (k, t) = 0

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∞

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∞

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∞

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(9)

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where D(k) is deﬁned by equation (4). Then dE(k, t)/dt = 0, and the energy balance becomes: T (k) + W (k) − 2ν0 k 2 E(k) = 0. At suﬃciently high Reynolds numbers, assume there is a wavenumber κ such that input eﬀects are below it and dissipation eﬀects above it. That is, for a well-posed problem: ∞ κ W (k)dk ε − 2ν0 k 2 E(k) dk. 0

κ

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0

κ

i.e. energy supplied directly by the input term to modes with k ≤ κ is transferred by the nonlinearity to modes with j ≥ κ. Thus T (k) behaves like a dissipation and absorbs energy. Second, κ ∞ ∞ dk dj S(k, j) − 2ν0 k 2 E(k) dk = 0. κ

0

κ

i.e. nonlinearity transfers energy from modes with j ≤ κ to modes with k ≥ κ, where it is dissipated into heat. In this range of wavenumbers T (k) behaves like a source and emits energy which is then dissipated by viscosity. 3. Theoretical approaches to the closure problem We begin with a statement of the equations of the quasi-normality theory and then go on to introduce perturbation theory and consider the relationship between the two approaches. We identify the need for renormalization and brieﬂy introduce the application of renormalization methods to turbulence. 3.1. Covariance equation from the quasi-normality (QN ) hypothesis With an assumption of quasi-normality, it can be shown3,5 that the covariance equation on the time diagonal takes the form:

∂ 2 + 2ν0 k C (k, t) = d3 jL (k, j) ∂t t × ds R0 (k; t, s) R0 (j; t, s) R0 (|k − j| ; t, s) 0

×2 [C (j, s) C (|k − j| , s) − C (k, s) C (|k − j| , s)] . (10)

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The viscous response function is given by: R0 (k; t, s) = exp[−ν0 k 2 (t − s)], and the coeﬃcient L(k, j) is deﬁned as: L(k, j) = −2Mαβγ (k)Mβαδ (j)Pγδ (k − j). The coeﬃcient L(k, j) is discussed later in Section 4. 3.2. Perturbation theory Add a stirring force fα (k, t) and a book-keeping parameter λ to the NSE: L0 (k)uα (k, t) = fα (k, t) + λMαβγ (k) d3 j uβ (j, t)uγ (k − j, t). (11) We may choose either λ = 0 (linear system) or λ = 1 (nonlinear system), thus λ is also a control parameter. We make the perturbation expansion (1) 2 (2) uα (k, t) = u(0) α (k, t) + λuα (k, t) + λ uα (k, t) + . . . .

(12)

where we take u(0) to be Gaussian, through our choice of the random stirring force f , and calculate the higher-order coeﬃcients u(1) , u(2) , . . . in terms of products of the u0 . Then we can obtain an equation for the exact covariance C in terms of an expansion in the bare covariance C0 and the viscous response function R0 , as an expansion where all the coeﬃcients are given by Gaussian averages. 3.3. Second-order covariance equations Evaluating the various coeﬃcients and truncating the resulting primitive perturbation series at lowest non-trivial order gives us unrenormalized second-order equations for the covariance.3,6 The second-order equation for the velocity covariance is: ∂ + ν0 k 2 C(k; t, t ) ∂t t

d3 j L(k, j)

=

− 0

dsR0 (k; t , s)C0 (j; t, s)C0 (|k − j|; t, s)

0

t

ds R0 (j; t, s)C0 (k; s, t )C0 (|k − j|; t, s) + O λ3 .

(13)

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And on the time diagonal:

∂ + 2ν0 k 2 C (k, t) ∂t t ds R0 (k; t, s) R0 (j; t, s) R0 (|k − j| ; t, s) = 2 d3 j L (k, j) 0 × [C0 (j, s) C0 (|k − j| , s) − C0 (k, s) C0 (|k − j| , s)] + O λ3 . (14) Note the ‘zero’ subscripts on the right hand side. It should be emphasised that at this stage we are considering the lowest-order nontrivial truncation of a perturbation expansion which is not in terms of a small parameter. Accordingly, we do not expect the above two equations to provide a satisfactory approximation: in fact, quite the reverse is the case! 3.4. Quasi-normality versus perturbation theory Compare equation (10) from QN (right hand side in terms of R0 and C) with (14) from perturbation theory (right hand side in terms of R0 and C0 ): i.e. QN looks already partially renormalized. In fact the QN equation is an approximation (i.e. quasi-Gaussian) whereas equation (14) is an exact second-order truncation of the perturbation series and is fully Gaussian in nature. QN can be fully renormalized by the ad hoc replacement of the coeﬃcient ν0 in the viscous response by an eﬀective turbulence viscosity (introducing an adjustable constant in the process): this leads on to the EDQNM family of closures. In perturbation theory a renormalization programme consists of making the consistent replacements R0 → R and C0 → C on the right hand side of equations (29) and (30). Additionally we require some principle to determine R. 3.5. Application of renormalization methods to turbulence It is well known that in wavenumber (k) space, the Navier-Stokes equations are equivalent to a quantum ﬁeld theory with the Reynolds number as the coupling constant. The molecular viscosity ν0 may be renormalized by the collective eﬀects of turbulent eddies to an eﬀective form νT (k). Correspondingly the viscous response function R0 (k; t − t ) = exp[−ν0 k 2 (t − t )] may be renormalized to an eﬀective form R(k; t − t ). Note that the latter is not necessarily of exponential form. Renormalization may lead to other eﬀects such as eddy noise or stochastic backscatter: see, for example, McComb, Hunter and Johnston15 and references therein.

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Perturbation theory requires a soluble zero-order model. This is normally obtained by setting the nonlinear term equal to zero and introducing random stirring forces to generate random ﬂuid motion. Gaussian distribution, and, as in equation (4), their covariance is chosen to be of the form: f (k, t) · f (k , t ) = 2(2π)3 D(k)δ(k + k )δ(t − t ). The stirring forces do work on the ﬂuid at a rate given by 3 εW = d k D(k) ≡ dk W (k), where the spectrum W (k) is deﬁned by equation (9) In general, the coupling constant (the Reynolds number) is large. Hence a successful renormalized perturbation theory (RPT) relies on the partial summation of certain classes of terms. However, if we apply Renormalization Group to turbulence, then the coupling is determined by a local (in wavenumber) Reynolds number R0 (k). The local coupling is related to the energy spectrum E(k) by R0 (k) = [E(k0 )]1/2 /ν0 k 1/2 . As Fig. 1 shows, this allows one to do perturbation theory as k → 0 and k → ∞. The former case has been much studied as it allows one to take over the results of the theory of critical phenomena. But it is remote from real turbulence. In the present case, we are considering the full closure problem and accordingly we need some method of summing the primitive perturbation series. 4. Renormalised perturbation theory (RPT): the general idea Unlike problems in microscopic physics, there is no additional control parameter (e.g. density or temperature) in the turbulence problem, so the perturbation series has to be renormalised by partial summation. The primitive perturbation series for the actual convariance C and response R can be written as coupled equations of the form: (1) L0k Ck = inﬁnite series involving C 0 and R0 . (2) L0k Rk = inﬁnite series involving C 0 and R0 . Renormalisation means making the replacements Ck0 → Ck and Rk0 → Rk , on the right hand side. In practice some other step is also needed.

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E(k)

Line of fixed points

inertial range dissipation range UV asymptotic

IR asymptotic freedom

freedom

k

1

k

k1

0

Gaussian fixed point k=0

k0

k

Non−Gaussian fixed point k=k*

Direction of mode elimination Fig. 1.

The turbulence energy spectrum indicating the application of RG methods.

The replacement of Ck0 and Rk0 by Ck and Rk can be justiﬁed (to some extent) on topological grounds using diagrams. It can also be partially justiﬁed by reversion of power series. It means replacing the mythical Ck0 by the observable Ck . It also means replacing the observable Rk0 by the mythical Rk . In this respect it diﬀers from quantum ﬁeld theory where bare Green functions are replaced by renormalised observable Green functions. 4.1. Pioneering RPTs We now list the pioneering renormalized perturbation theories, as follows: (1) Kraichnan (1959): direct interaction approximation (DIA). Successful in both qualitative and quantitative terms but does not give the K41 ‘-5/3’ spectrum at large Reynolds numbers.7 (2) Wyld (1961): formal analysis of the Navier-Stokes perturbation expansion in terms of diagrams with renormalisation arising from purely topological considerations. Showed DIA response equation emerged naturally at lowest nontrivial order.16 (3) Edwards (1964): derived Liouville equation for the probability distribution function of turbulent velocities. Approximation by Fokker-Planck equation determined renormalised response. Strong point is that is an expansion about Gaussian.8 Weak point is its restriction to single-time forms. Can be related to DIA by assumed exponential time dependences.

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(4) Herring (1965): self-consistent ﬁeld theory. A more abstract version of the Edwards theory. A two-time generalisation (1966) is very similar to DIA.9,17 (5) Lee (1965): Extended Wyld’s analysis to magneto-hydrodynamics including diagrammatic expansions up to sixth order, and in the process corrected a double-counting problem in the Wyld analysis.18 This particular problem need not have arisen in the ﬁrst place McComb.3 4.2. Kraichnan-Wyld perturbation theory Again we add a stirring force fα (k, t) and a book-keeping parameter to the NSE, in order to obtain the form (11), repeated here for convenience: L0 (k)uα (k, t) = fα (k, t) + λMαβγ (k) d3 j uβ (j, t)uγ (k − j, t). As before, we put λ = 1 at the end of the calculation. Note that if we scaled variables in a suitable way, we could replace λ by a Reynolds number. Also note that in the absence of a small parameter the expansion is eﬀectively in terms of the complexity of the interactions. Hence the perturbation expansion takes the form (1) 2 (2) uα (k, t) = u(0) α (k, t) + λuα (k, t) + λ uα (k, t) + . . . .

Now let us introduce a simpliﬁed notation, such that the NSE becomes: L0k uk = fk + λMkjl uj ul . Writing the inverse of the operator on the left hand side as: L−1 0k ≡ Rk ≡ the viscous response function, (0)

the zero-order solution is: (0)

(0)

uk = Rk fk . For convenience, we re-write the NSE as: (0)

(0)

uk = uk + λRk Mkjl uj ul . Substitute the perturbation series into this, multiply out, and equate terms at each order of λ. The coeﬃcients in the perturbation series are given by: (0)

(0)

order λ0 : uk = Rk fk ;

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(0)

(0) (0)

order λ1 : uk = Rk Mkjl uj ul ; (2)

(0)

(0)

(0)

(0) order λ2 : uk = 2Rk Mkjl uj Rl Mlpq u(0) p uq ,

and so on. The exact covariance is then given by: (0) (0)

(0) (2)

Ck = uk u−k = uk u−k + uk u−k (1) (1) (2) (0) +uk u−k + uk u−k + O λ4 . (1)

(2)

Substituting for the coeﬃcients uk , uk , . . . (0)

(0)

(0)

(0) (0)

(0) Ck = Ck + 2Rk Mkjl Mlpq Rl uk uj u(0) p uq (0)

(0) (0)

(0) +Rk Mkjl M−kpq R(0) uj ul u(0) p uq

4 (0) (0) (0) (0) (0) +2Rk Mkjl Mlpq Rl uk uj u(0) . p uq + O λ Factorise the moments, using Gaussian statistics: (0) (0)

(0)

(0) (0) uk uj u(0) p uq = δkj δpq Pk Pp Ck Cp (0)

(0)

+δkp δqj Pk Pj Ck Cj

(0)

+ δkq δjp Pk Pq Ck Cq(0) .

We can combine all the M ’s and P ’s into a simple coeﬃcient L(k, j): details are given in the next section. 4.3. The L coeﬃcients in turbulence theory The coeﬃcient L(k, j) is deﬁned as: L(k, j) = −2Mαβγ (k)Mβαδ (j)Pγδ (k − j). The coeﬃcient L(k, j) can be evaluated as: 4 3 2 1 − µ2 kj µ k + j 2 − kj 1 + 2µ2 , L(k, j) = − k 2 + j 2 − 2kjµ where µ is the cosine of the angle between the vectors k and j. Alternatively, the coeﬃcient L(k, k − j) is deﬁned as: L(k, k − j) = −2Mαβγ (k)Mβαδ (k − j)Pδγ (j). The coeﬃcient L(k, k − j) can be evaluated as: L(k, k − j) =

(k 4 − 2k 3 jµ + kj 3 µ)(1 − µ2 ) . k 2 + j 2 − 2kjµ

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4.4. The Kraichnan-Wyld covariance equations The second-order equation for the velocity covariance is: ∂ 2 + ν0 k C(k; t, t ) ∂t t 3 = d j L(k, j) ds R(k; t , s)C(j; t, s)C(|k − j|; t, s) −

0

t

ds R(j; t, s)C(k; s, t )C(|k − j|; t, s) ;

0

and on the time diagonal:

∂ 2 + 2ν0 k C (k, t) ∂t t ds R (k; t, s) R (j; t, s) R (|k − j| ; t, s) × = 2 d3 j L (k, j) 0

[C (j, s) C (|k − j| , s) − C (k, s) C (|k − j| , s)] . These equations are an exact second-order truncation of renormalized perturbation theory. Their derivation can rely on either the topology of Feynmann-type diagrams or reversion of power series. They satisfy all the required symmetries and in particular the closure conserves energy, displaying the correct antisymmetric behaviour of the transfer spectrum. With an appropriate choice of renormalized response function R they can predict the free decay of isotropic turbulence, without invoking arbitrary constants, in agreement with experiment and numerical simulation. The turbulence problem in this context becomes one of ﬁnding a principle to determine the renormalised response function. Such a principle leads to a renormalised perturbation theory or RPT. 4.5. Renormalised response functions and RPTs All RPTs rely on some use of linear response theory which in turn is in some sense renormalized. All RPTs can be interpreted as mean-ﬁeld theories. The DIA response function corresponded to the summation of certain classes of terms to all orders in perturbation theory.16 The EFP8 and SCF9 theories are single-time self-consistent theories which are cognate to DIA. A later two-time version version of SCF17 is very similar to DIA. All these theories have an incorrect interpretation of energy conservation in terms of their response function. The LET theory14 used a local (in wavenumber) energy balance to determine the response function.

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4.6. Derivation of DIA We now have our equation for Ck as: (0)

(0)

(0)

(0)

(0)

Ck = Ck + Rk L(k, j)Rl Cj Ck

(0) (0) (0) (0) (0) (0) (0) +Rk L(k, j) Rk Cj Cl − Rj Cl Ck + O λ4 . Let us remind ourselves what we are trying to do. Multiply the NSE through by u−k and average: L0k Ck = fk u−k + Mkjl uj ul u−k . The ﬁrst term on the rhs is the input term due to stirring forces. It can be worked out exactly and this process yields a renormalised Rk . The second term on the rhs is the energy transfer due to the nonlinearity. It can only be treated approximately. We deal with the input term ﬁrst: (0) (0) (0) (0) (0) recall uk = Rk fk and Rk Rk = Rk . Re-write the expansion for Ck as: (0) (0) (0) (0) (0) (0) Ck = Rk fk Rk f−k + Rk L(k, j)Rl Cj Ck + O λ4 (0) (0) (0) (0) (0) (0) (0) +Rk L(k, j) Rk Cj Cl − Rj Cl Ck + O λ4 . (0)

Express Ck the rhs.

in terms of stirring forces in the ﬁrst nonlinear term on

(0)

Ck = Rk

.

/ (0) (0) (0) (0) fk Rk + Rk L(k, j)Rl Cj + O λ4 f−k

(0) (0) (0) (0) (0) (0) (0) +Rk L(k, j) Rk Cj Cl − Rj Cl Ck + O λ4 . Identify the quantity in curly brackets as the exact response function (to order λ4 ). Hence, recalling that fk f−k = Wk : (0) (0) (0) (0) (0) (0) (0) (0) Ck = Rk Rk Wk + Rk L(k, j) Rk Cj Cl − Rj Cl Ck + O λ4 . This equivalence leads to (0)

(0)

(0)

(0)

Rk = Rk + Rk L(k, j)Rl Cj

+ O λ4 .

Now take three steps with the Ck and Rk equations: (1) Operate from the left with L0k . (0) (0) (2) Replace Ck → Ck and Rk → Rk on the rhs. 4 (3) Drop terms of order λ and higher.

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For the covariance we get L0k Ck = Wk + L(k, j) [Rk Cj Cl − Rj Cl Ck ] . For the response function we get L0k Rk = δ(t − t ) + L(k, j)Rl Cj . In the next section we write this equation with full notation restored. 4.7. The DIA and LET response equations The DIA response equation is: ∂ + ν0 k 2 R(k; t, t ) ∂t t 3 + d j L(k, j) dt R(k; t , t )R(j; t, t )C(|k − j|; t, t ) t

= δ(t − t ).

(15)

For convenience in making comparisons, we state the LET response equation at this stage, thus:

∂ 2 + ν0 k θ (t − t ) R (k; t, t ) − θ(t − t )R(k; t, t )δ (t − t ) θ (t − t ) ∂t t 3 + d j L (k, j) θ (t − t ) ds R (j; t, s) R (k; s, t ) θ (t − s) C (|k − j| ; t, s)

t

t

θ (t − s) C (|k − j| ; t, s) × C (k; t , t ) 0 × {R (k; t , s) θ (t − s) C (j; t, s) − R (j; t, s) θ (t − s) C (k; t , s)} .

=

d3 jL (k, j) θ (t − t )

ds

(16)

Apart from the addition of the second term on the left hand side, this is the LET response equation as it appears in the literature. The addition of this extra term as a consequence of time-ordering, ﬁxes the problem of the singularity in the time-derivative of the response equation (16) which occurs when one takes t = t . More importantly, if we compare (16) with the DIA response equation (15), the additional terms on the right hand side of (16) cancel the infra-red divergence and ensure compatibility with the Kolmogorov K41 spectrum. The ﬂuctuation-dissipation relation can be used directly to calculate the LET theory instead of the above response equation. With this simpliﬁcation it is much easier to calculate than DIA. We shall return to this later.

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5. Assessment of the pioneering RPTs DIA is both quantitatively and qualitatively very good at predicting the free decay of isotropic turbulence. However, it is not consistent with K41. We turn to the simpler Edwards theory to understand the diﬃculty. If we take the two-time dependences as Rk ∼ e−ωk (t−t ) and Ck ∼ qk e−ωk (t−t ) then for stationary turbulence the Edwards equation for the spectral density qk is: L(k, j)qk−j (qk − qj ) d3 j = W (k) − 2ν0 k 2 qk , ωk + ωj + ωk−j while the Edwards equation for the response ωk is: 1 L(k, j)qk−j ωk = . d3 j 2 ωk + ωj + ωk−j Re-arrange the energy (density) balance equation as: L(k, j)qk−j qj W (k) + d3 j − 2ν0 k 2 + 2ωk qk = 0. ωk + ωj + ωk−j With the obvious interpretation: ωk = νT (k)k 2 , where νT (k) is an ‘eﬀective viscosity’ due to nonlinear interactions. Its form can be deduced from the Edwards equation for the response. This leads on to the idea of the inﬁnite Reynolds number limit and the infra-red divergence. Edwards took the limit ν0 → 0 such that ε = constant. Then K41 applies for all k such that 0 ≤ k ≤ ∞. That is: qk ∼ ε2/3 k −11/3 and ωk ∼ ε1/3 k 2/3 for all k. Under these circumstances the energy balance becomes: L(k, j)qk−j (qk − qj ) = εδ(k) − εδ(k − ∞). d3 j ωk + ωj + ωk−j The integral in this energy balance is well-behaved due to cancellations. In contrast, the corresponding integral for the response function is divergent due to qk−j → ∞ as k − j → 0. 5.1. The local energy transfer (LET) theory It was pointed out (McComb 1974) that the above interpretation of the energy transfer spectrum in terms of ωk is wrong. The entire transfer spectrum, not just some part of it, behaves like a sink for small k and like a

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source for large k. A re-interpretation led to 1 L(k, j)qk−j (qk − qj ) d3 j ωk = . 2 (ωk + ωj + ωk−j )qk This is well-behaved and leads on to the two-time LET theory. 5.2. Numerical computation of RPTs The ﬁrst numerical investigation of a renormalized perturbation theory was in 1964 by Kraichnan,19 who calculated the free decay of DIA from several arbitrary initial spectra, and for initial values of the (microscale-based) Taylor-Reynolds number up to Rλ ∼ 35. Later, Herring and Kraichnan investigated the DIA, EFP and SCF closures, along with the Test-ﬁeld Model (TFM);20 and later still the Lagrangian-history closures, ALHDI and SBALHDI.21 The free decay of the LET theory has been investigated22 -,24 using the same numerical methods and initial spectra as those of Kraichnan. Comparisons were made with results from DIA, TFM and the Lagrangian-history theories, along with representative experimental results. More recently McComb and Quinn25 have obtained results for both DIA and LET with forcing, in order to study stationary turbulence. All of this work shows that RPT closures perform quite well, in that they give good qualitative and quantitative agreement with the results of both experiments and direct numerical simulations. In particular, the DIA and LET closures are very similar under most circumstances. These investigations have all been for three-dimensional turbulence but broadly similar conclusions have been drawn for two-dimensions by Frederiksen26 -29 and co-workers, in an extensive programme of calculations with application to atmospheric turbulence. In order to illustrate the behaviour of RPT closures, we now present some of the calculations by my student Anthony Quinn.30 As well as considering forced turbulence, we also repeated calculations of decaying turbulence where the new feature is a comparison with a direct numerical simulation which had the same initial spectrum. Noting that the ‘initial spectrum’ for the DNS is the result of an ensemble average, its equivalence to the LET/DIA initial spectrum can only be an approximation and in Fig. 2 this is represented by error bars. We note that as time evolves, the spectrum decreases in amplitude, but also spreads out as energy is transferred to higher wavenumbers by the nonlinear term; as indicated in Fig. 3 by the development of the transfer spectrum. Note that at t = 0 the trans-

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fer spectrum is zero at all wavenumbers, corresponding to the Gaussian initial conditions. In Fig. 4 we see that the energy decreases monotonically whereas the dissipation increases initially and then declines. Both forms of behaviour are as one would expect and one notes reasonable agreement between the closures and the DNS. Fig. 5 shows similar behaviour for the microscale but the results for skewness reinforce the view that it is the most sensitive index of diﬀerences between theories. For sake of completeness, we note that the ‘skewness’ here is the skewness of the one-dimensional derivative of the velocity ﬁeld as given by S(t) = −

(∂u1 /∂x1 )3 . (∂u1 /∂x1 )2 3/2

For a more complete discussion of the performance of LET and DIA as assessed by the skewness, reference should be made to the relevant papers by McComb and co-workers.22,23 However, it is well known that these closures systematically underestimate the skewness and that this corresponds to an underestimate of energy transfer to the small scales. This eﬀect can be attributed to lack of vertex renormalization and recently successful results have been obtained with DIA in two-dimensional ﬂows by introducing an emperical vertex renormalization.31,32 It would be interesting to see how this approach would work with LET in three dimensions. Turning now to the stationary case, as the investigation of forced turbulence is rather preliminary in nature, we just show two reasonably encouraging results. In Fig. 6, we plot the compensated energy spectrum at Rλ = 232 for LET and DIA, and compare with both the DNS result and the ad hoc result due to Qian33 which is adjusted to give good agreement with experiment. We see that the closures agree with each other, but that neither they nor the Qian spectrum agree with the DNS at the lower numbers. To some extent, this reﬂects the fact that the role of the forcing is crucial at low wavenumbers. Evidently many more studies are needed in this area (as in most areas of turbulence!) and also at somewhat higher Reynolds numbers. Lastly, the LET theory has also been applied to passive scalar transport in freely decaying isotropic turbulence by McComb, Filipiak and Shanmugasundaram,24 with generally favourable results. In particular, it was found that while LET overestimates both the Kolmogorov prefactor with α = 2.3 (experimental value ∼ 1.61) and the Corrsin-Oboukhov constant with β = 1.13 (experimental value 0.7 − 0.8), the ratio β/α = 0.45, which is the eddy Prandtl number at high Reynolds number, is close to the experimental value ∼ 0.44.

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0.8 DNS Ensemble Average tu(0)/L(0)=0.00 LET/DIA tu(0)/L(0)=0.00 DNS Ensemble Average tu(0)/L(0)=0.98 LET tu(0)/L(0)=0.98 DIA tu(0)/L(0)=0.98 DNS Ensemble Average tu(0)/L(0)=1.94 LET tu(0)/L(0)=1.94 DIA tu(0)/L(0)=1.94

0.6 2

<E(k)/[L(0)u(0) ]>

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0.2

0

0

5

10 k

15

20

Fig. 2. Free decay of the energy spectrum: DIA and LET compared with DNS at initial Taylor-Reynolds number Rλ =95.

6. Issues which aﬀect perceptions of RPTs These theories come from theoretical physics and are seen by the turbulence community (who are mostly ﬂuid dynamicists) as being somewhat ‘alien’. Such perceptions are reinforced by articles in journals which are supposed to be on turbulence, yet contain numerous Feynman-type diagrams and phrases like ‘one-loop renormalisation’ and ‘Ward-Takahashi identities’ which merely bemuse most turbulence researchers. The community notes that such theories are never applied to ‘real’ problems in real space. In the absence of understanding, ﬂuid dynamicists look to the theoretical physicists for some consensus. Instead they see disagreement and wild claims that they suspect cannot be true. Accordingly they tend to write oﬀ the whole ﬁeld as being ‘mired in controversy’. 6.1. What are the issues? The main unresolved issues can be summarized as follows: (1) Disagreement on the causes of the failure of DIA and EFP to give k −5/3 . ‘Lack of convective invariance’ versus ‘lack of scale invariance’.

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0

−10

2

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−20

DNS Ensemble Average tu(0)/L(0)=0.98 LET tu(0)/L(0)=0.98 DIA tu(0)/L(0)=0.98 DNS Ensemble Average tu(0)/L(0)=1.44 LET tu(0)/L(0)=1.44 DIA tu(0)/L(0)=1.44 DNS Ensemble Average tu(0)/L(0)=1.94 LET tu(0)/L(0)=1.94 DIA tu(0)/L(0)=1.94

−30

−40

−50

0

5

10

15

k

Fig. 3. The transfer spectrum for DIA and LET compared with DNS at initial TaylorReynolds number Rλ =95.

(2) The need for ad hoc corrections to the Wyld formalism. (3) Conﬂict between Wyld (diagram) and MSR (functional operator)34 formalisms on vertex renormalisation. (4) MSR extends ideas from canonical Hamiltonian systems in thermal equilibrium to macroscopic ﬂuid motion. Is this valid? (5) Lagrangian versus Eulerian formulations. (6) Galilean invariance (GI): does it suppress vertex renormalisation? (7) IR and U V divergences: do they exist? 6.2. Wider issues General disagreement on Galilean invariance and K41 make the turbulence picture even more confused. Thus: A Galilean invariance (GI) is widely invoked, often as a ‘low-speed’ version of Lorentz invariance, and used to justify many things. We shall argue that it is a trivial symmetry in most cases, being ‘satisﬁed’ by the constant mean velocity. B Correctness of Kolmogorov k −5/3 spectrum is challenged by two groups: (1) Those who believe that intermittency of the local dissipation rate invalidates the argument.

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1.5

1

0.5

0

0

0.5

1 t<[u(0)/L(0)]>

1.5

2

Fig. 4. Variation of energy and dissipation rate during free decay at initial TaylorReynolds number Rλ =95.

(2) Those who draw analogies with critical phenomena and see ‘−5/3’ as the ‘canonical dimension’ corresponding to mean-ﬁeld theory and wish to use RG to establish the ‘anomalous dimension’. 6.3. Incompatibility with K41: ‘Convective invariance’ versus ‘scale invariance’ To a considerable extent, the ‘disagreement’ between Kraichnan and Edwards is due to the fact that they work with two-time and single-time theories respectively. Kraichnan argued that the non-simultaneous covariance C(x, x ; t, t ) = u(x, t)u(x t ) cannot be Galilean invariant, whereas the single-time form C(x, x ; t, t) is Galilean invariant. Accordingly, a closure in terms of an expansion in time-convolutions of two-time covariances (see equation (13)) cannot be correct on the time diagonal (see equation (14)). The left hand side is invariant, the right hand side may not be. This can be interpeted as a lack of convective invariance of the theory. Edwards considered a time-independent situation and introduced the inﬁnite-Reynolds number limit at constant dissipation rate ε. Under these

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0.4 DNS S(t) LET S(t) DIA S(t)

0.2 0

0

0.5

1 t<[u(0)/L(0)]>

1.5

2

Fig. 5. Variation of skewness and microscale during free decay at initial Taylor-Reynolds number Rλ =95.

circumstances the k −5/3 spectrum applies for all k such that 0 ≤ k ≤ ∞. He found that this leads to an inﬁnite value of the response integral as k → 0. This is interpreted as lack of scale invariance. A discussion and comparison of the two explanations may be found on pages 267-274 of the book by McComb.3 Further study is needed to understand the relationship between these two approaches and recent work by McComb and Kiyani35 on the connection between DIA and EFP may assist in this. However, Kraichnan’s work led on to the concept of Random Galilean Invariance (RGI) and we discuss this in the following subsection. 6.3.1. Random Galilean Invariance We know that classical mechanical systems must satisfy the principle of Galilean invariance and this includes the NSE and its solutions (in which we include closure approximations). Kraichnan extended this requirement to invariance (in some statistical sense) under an ensemble of Galilean transformations. That is, if the Galilean shift velocity is denoted by c (constant in both space and time) then we introduce an ensemble of Galilean transformations in which c varies randomly from one realization to another.

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0

E(k,t)/ε k

2/3 −5/3

10

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−2

10

−3

10

−2

−1

10 k/kd

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0

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1/4 5/4

Fig. 6. Compensated energy spectra for forced turbulence at an evolved TaylorReynolds number of Rλ = 232.

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Grant, Stewart & Moilliet Comte−Bellot & Corrsin Champagne, Harris & Corrsin Van Atta & Chen LET Rλ=232 LET Rλ=88 −4

10

−3

10

−2

10

−1

10

0

10

1

k/kd

Fig. 7. One-dimensional spectra for forced LET compared with experiment at an evolved Taylor-Reynolds number of Rλ = 232.

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Given the complicated nature of the arguments leading to its formulation it is perhaps not surprising that there is some degree of uncertainty about the nature of this transformation. For instance in the book by Leslie5 (see pages 186-188), it is interpreted as consisting of an ensemble of random Galilean transformations for each turbulence realization. A similar interpretation was given by McComb, Shanmugasundaram and Hutchinson,23 who argued that the random Galilean transformations rendered the turbulence averaging procedure non-ergodic, and further argued that there was no essential diﬀerence between the deterministic and random Galilean transformations. However, it later appeared that these interpretations were not quite correct. My student Mark Filipiak studied the topic further and in his thesis36 quoted a later paper by Kraichnan10 to the eﬀect that the ensemble is a joint one, in which the set of turbulence velocities {u(x, t)} is combined with the set of random shift velocities {c}, to add one individual member of {c} to each individual member of {u} such that we generate a new ensemble corresponding to the set {u + c}. After a careful analysis, Filipiak concluded that the random Galilean transformation amounted to a change of ensemble, rather than a symmetry transformation. He also concluded that ‘This change of ensemble makes the derivation of an Eulerian renormalized perturbation theory impossible as the zero-order propagator/response function becomes a random variable.’. 6.4. Wyld’s formalism versus MSR Lee18 made ad hoc corrections to eliminate a double-counting problem with the Wyld formalism.16 However, a more correct initial procedure would have made this correction unnecessary.3 It is not a real issue. Using a formalism based on functional integrals, Martin, Siggia and Rose34 (MSR) predicted some additional vertex corrections which Wyld does not mention. Apparently Kraichnan (in unpublished work) agrees with MSR. The MSR formalism is a synthetic formalism, which seeks to apply methods which have been derived for Hamiltonian systems to the nonHamiltonian case of macroscopic ﬂuid turbulence. In eﬀect it has been set up to reproduce DIA and this it duly does. As a formalism, it stimulated considerable interest37,38 and has been further developed as a path-integral formalism.39,40 The general view is that this puts the formalism on a better mathematical footing (although other such formalisms exist: see the paper by Thacker41 and references therein). However some elements in it (e.g.

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Legendre transformation) need more critical examination, from a ‘physics’ point of view. While it is important that this disagreement should be resolved, it is worth noting that the immediate practical consequences are not likely to be signiﬁcant. Essentially all two-point closures are at lowest non-trivial order and hence are unaﬀected by vertex corrections.

6.5. Lagrangian versus Eulerian formulations Kraichnan introduced Lagrangian-history coordinates,10 along with a generalized velocity: u(x, t|s) ≡ velocity at time s of a particle which was at x at time t. t = labelling time (Eulerian). s = measuring time (Lagrangian). On this basis DIA was reworked as Lagrangian History Direct Interaction or LHDI, where now one works with a two-point, four-time covariance C(k; t, t |s, s ), with a similar generalization of other statistical quantities. This formulation achieved random Galilean invariance (for a discussion, see page 210 et seq of the book by Leslie5 ) but the resulting equations were too complicated for computation to be possible so it was abridged to ALHDI, but did not perform as well as hoped. Later Kraichnan and Herring11 introduced strain-based version of ALHDI. This gave much better numerical predictions and is known as SBALHDI. In the abridged theory, Kraichnan ends up working with C(k; t|s) rather than C(k; t, t ) and claims42 that this is the form which is compatible with random Galilean invariance. Later, Kaneda12 produced a version of Kraichnan’s LHDI formalism by working with measuring-time derivatives rather than labelling-time derivatives. Kaneda’s theory is in terms of C(k; t, t ) and appears to be very similar (possibly indentical) to the purely Eulerian SCF theory. This raises questions about the random Galilean invariance of the formulation, when the whole point of the exercise is that a purely Eulerian theory like SCF cannot be random Galilean invariant. (And of course there is the fact that the purely Eulerian LET theory is compatible with K41!) For completeness, we should mention that Kida13 has, in eﬀect, rederived Kaneda’s Lagrangian equations, and also that a more recent (and very interesting) discussion of quasi-Lagrangian theories may be found in the paper by Frederiksen and Davies.31

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6.6. Galilean invariance and vertex renormalisation In recent years Galilean invariance has been invoked to constrain the nature of perturbation expansions of the NSE, and related equations in soft condensed matter. Arguing by analogy with Lorentz invariance in quantum ﬁeld theory, Ward identities have been derived using Galilean invariance. These imply that there is no vertex renormalisation which in turn leads to nontrivial relationships between critical exponents. This view has been challenged by McComb43 who asserts that Galilean invariance is satisﬁed trivially by the mean velocity, even (or, especially) when this is constant. Recently Berera and Hochberg44 have argued that there is an exception to this for the particular case k = 0. 6.7. IR and U V divergences: do they exist? In quantum ﬁeld theory there are well known divergences in the primitive (unrenormalised) perturbation theory. In recent decades, many theorists have claimed that such divergences exist in the perturbation expansion and then they claim to ﬁnd ways of dealing with them. It is curious that none of the pioneers (Kraichnan, Wyld, Edwards, Herring, Lee, Balescu and Senatorski ...) noticed or commented on them! The perturbation expansion is in terms of R0 and C0 , and as C0 is not an observable we are unable to say whether or not there are divergences. The exception is where we calculate C0 = R0 D(k), where D(k) is the covariance of the stirring forces, and choose a power law for D(k). In other words, we only get divergences if we ourselves put them into the problem. 6.8. The wider issue of the K41 ‘-5/3’ power law The moments/spectra of the velocity ﬁeld are not solutions of the NSE. Either they are obtained by averaging operations on the actual solution of the NSE (the velocity ﬁeld); or they are connected together by the open moment hierarchy of the NSE and are indeterminate. The one exception to this general rule is that the second- and thirdorder moments are rigorously connected by conservation of energy. Scaleinvariance leads to a de facto closure of the moment hierarchy. This enforces K41 and is in accord with experiment and direct numerical simulation, although higher-order moments probably depart from K41 behaviour. References to ‘intermittency corrections’ beg the question! If corrections to higher order moments exist, they may not be due to intermittency.

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References to ‘intermittent dissipation rate’ ignore the irrelevance of the dissipation rate to the K41 arguments. The Kolmogorov energy spectrum is determined by the inertial transfer rate. For stationary ﬂows the two quantities are numerically equal. Fine-scale (or, better) internal intermittency is the fact that the turbulence cascade in any one realization is not space ﬁlling. This behaviour is part of the dynamics of turbulence and is true for virtually all length scales. Questions one might ask are: does it have consequences for energy conservation? For scale invariance? Bear in mind that both these properties are tested using averaged quantities. Analogies with the theory of critical phenomena should be drawn with care. Dimensional analysis in equilibrium problems is a relatively weak tool which relies on the introduction of densities, and relates only to length. In turbulence, energy conservation associated with a ﬂux through the modes is a controlling symmetry which has no analogue in equilibrium critical phenomena. There is no justiﬁcation for calling K41 a mean-ﬁeld theory. The ‘-5/3’ law was derived by two diﬀerent methods and neither is a meanﬁeld theory. On the contrary, EFP, DIA and SCF are mean-ﬁeld theories, yet are not compatible with K41. In K41 the exponent is determined by dimensional analysis, conﬁrmed by a de facto closure of the Karman-Howarth equation. It turns out that the prefactor can be determined by renormalization group methods.45 7. New developments in LET This work was done in conjunction with my student Khurom Kiyani.46 A new symmetrized time-ordered covariance is introduced. This eliminates problems encountered in using exponential time dependences. It also reconciles conﬂicting requirements of time-reversal symmetry (of the covariance) and causality (of the response function) in the ﬂuctuation dissipation relation. An improved derivation of the LET response equation has been given in terms of closing the Kraichnan-Wyld perturbation series at second-order by means of a local (in wavenumber) energy balance. By specialising to a particular initial condition, the response equation is reduced to a ﬂuctuation-dissipation relation. The instantaneous propagator (velocity-ﬁeld response function) is trivially shown to be transitive with respect to intermediate time. The mean-ﬁeld propagator (‘covariance’ response function) is non-trivially shown to be transitive with respect to intermediate times. This is a new result. In addition, new relationships have been obtained linking single-time covariances. Modiﬁed two-time LET equations have been obtained. These eliminate

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minor problems which arose on the time diagonal when t = t . A partial propagator representation has been introduced allowing the two-time LET equations to be represented solely in terms of single-time covariances. Single-time and Markovianized versions of LET have been derived. A timeindependent form of the LET equations has been obtained and shown to be well-behaved in the limit of inﬁnite Reynolds numbers.35,47 7.1. Two problems with time-ordering The ﬁrst problem arises as follows. Isotropy also implies time-reversal symmetry, which requires that C (k; t, t ) = C (k; t , t) .

(17)

The renormalized response is not an observable but must nevertheless satisfy the causality condition R(k; t, t ) = 0

for

t > t.

However, the standard form of ﬂuctuation-dissipation relationship cannot satisfy both these symmetries simultaneously. The second problem arises because in practice it would be helpful if we could assume exponential forms for the covariance and renormalized response function, thus:

C(k; t − t ) = C(k)e−ω(k)|t−t | ;

R(k; t − t ) = e−ω(k)(t−t ) .

(18)

In this case, there is a basic problem with these forms in that the timereversal symmetry of (17) is in practice not satisﬁed, and that diﬀerentiating the steady-state covariance with respect to diﬀerence time leads to a nonzero result at the origin, where t = t 7.2. Derivation of the mean propagator equation By using an integrating factor and integrating the covariance equation over time we can write it as (0) (k; t, s) C σ (k; s, t ) + Cασ (k; t, t ) = Rαε t (0) dt Rα (k; t, t ) M βγ (k) × + λ s 3 × d j uβ (j, t ) uγ (k − j, t ) uσ (−k, t ) ,

(19)

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where s is some initial time and the integrating factor is 2 Pα (k) e−ν0 k (t−t ) t ≥ t , (0) Rα (k; t, t ) = 0 t < t . From the primitive perturbation series (12), we have (0) (2) (k; t, t ) + λ2 Cασ (k; t, t ) + · · · . Cασ (k; t, t ) = Cασ (0) Rα (k; t, s)

When (20) is substituted in (19) we can see that zero-order response for the zero-order covariance, thus:

(20) acts as a

(0) (0) (0) (k; t, t ) = θ (t − s) Rα (k; t, s) C σ (k; s, t ) . Cασ

(21)

This is an exact result. We call this the zero-order or bare result. Rearranging (19) to prompt the next step, (0) (k; t, s) Cασ (k; t, t ) = Rαε t 1 (0) λ + dt Rα (k; t, t ) M βγ (k) × C σ (k; s, t ) s 3 × d j uβ (j, t ) uγ (k − j, t ) uσ (−k, t ) × ×C σ (k; s, t ) .

(22)

We postulate that we may write this in its renormalized form as Cασ (k; t, t ) = θ (t − s) Rα (k; t, s) C σ (k; s, t ) .

(23)

7.3. The ﬂuctuation-dissipation relationship We begin by putting equation (23) in terms of isotropic forms, thus: C (k; t, t ) = θ (t − s) R (k; t, s) C (k; s, t ) , where the θ(t − s) incorporates the causality condition. We have eﬀectively replaced the zero-order equation (21) by its renormalized version using the replacements C (0) → C and R(0) → R. Choosing the time-ordering t > t say, is merely a matter of applying the Heaviside unit function to both sides: θ(t − t )C (k; t, t ) = θ(t − t )θ (t − s) R (k; t, s) C (k; s, t ) .

(24)

If we now set s = t in (24), which amounts to a choice of the initial condition, we get θ (t − t ) C (k; t, t ) = θ (t − t ) R (k; t, t ) C (k; t , t ) .

(25)

This result takes the form of a ﬂuctuation-dissipation relationship (or FDR).

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7.4. Symmetrized time-ordered covariance We now introduce a representation of the covariance which preserves the symmetry under interchange of time arguments as C (k; t, t ) = θ (t − t ) C (k; t, t ) + θ (t − t) C (k; t, t ) −δt,t C (k; t, t ) .

(26)

Equation (26) may be written in the form of a time-ordered ﬂuctuationdissipation relation by using (25) to construct it: C (k; t, t ) = θ (t − t ) R (k; t, t ) C (k; t , t ) +θ (t − t) R (k; t , t) C (k; t, t) −δt,t C (k; t, t ) . Turning now to the problem of the exponential forms as given by (18) it is readily veriﬁed that this time-ordered representation (26) has the required property that lim

t→t

∂ C(k; t, t ) = 0. ∂t

7.5. Group-closure properties of the LET ˆ ασ (k; t, s) For the velocity ﬁeld, we have the instantaneous propagator R deﬁned by ˆ ασ (k; t, s)uσ (k, s). uα (k, t) = R It is easily shown by successive applications that this is transitive with respect to intermediate times: ˆ ασ (k; t, s)R ˆσρ (k; s, t ). ˆ αρ (k; t, t ) = R R ˆ ασ (k; t, s), we can show35 For the mean-ﬁeld propagator, Rασ (k; t, s) = R that the renormalized response is also transitive with respect to intermediate times. R (k; t, t ) = R (k; t, s) R (k; s, t ) , where we have specialized to the isotropic case. We can also write linked single-time covariances as C(k; t, t) = θ(t − s)R(k; t, s)R(k; t, s)C(k; s, s).

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8. Single-time LET equations With the ansatz of local energy transfer to determine the response, along with an assumption of an exponential relationship between the response function and the eddy damping, as given by equation (40), we ﬁnd

∂ 2 + 2ν0 k C(k; t) = 2 d3 j L(k, j)D(k, j, |k − j|; t) ∂t × C(|k − j|; t)[C(j; t) − C(k; t)] = −2ω(k; t)C(k; t). The eddy damping is given by C(|k − j|; t) [C(j; t) − Ck; t]. ω(k; t) = − d3 j L(k, j)D(k, j, |k − j|; t) × C(k; t) The triple-mode damping function satisﬁes: 3 ∂D(k, j, |k − j|; t) = 1 − ν0 k 2 + ν0 j 2 + ν0 |k − j|2 + ∂t + ω(k; t) + ω(j; t) + ω(|k − j|; t)] × D(k, j, |k − j|; t).

(27)

The initial conditions can be taken as: C(k; t = 0) =

E(k; t = 0) , 4πk 2

D(k, j, |k − j|; t = 0) = 0,

where E(k; t = 0) is some arbitrarily chosen initial energy spectrum. This set of equations is similar to the test-ﬁeld model, but has an extra term in the equation for the eddy damping. The extra term cancels infra-red divergences and this means that (unlike the test-ﬁeld model) it does not require an additional hypothesis and adjustable constant to be compatible with the Kolmogorov distribution. 9. Conclusion This new single-time LET theory still has to be tried out on the standard test problems of isotropic turbulence. However, the two-time LET theory performs well on such problems and we may hope that the single-time and Markovianized forms will perform adequately while oﬀering computational advantages. The theory has potential for generalization to form realizable Markovian closures to include the eﬀect of waves.48 In particular, Bowman and Krommes49 found that it was necessary to modify the ﬂuctuationdissapation relationship in order to ensure realizability in the precence of

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waves, although other non-Markovian wave-turbulence studies encountered no problems with realizability.29 The theory also has potential for application to inhomogeneous and shear ﬂow turbulence. References 1. A. N. Kolmogorov, The local structure of turbulence in incompressible viscous ﬂuid for very large Reynolds numbers. C. R. Acad. Sci. URSS, 30, 301 (1941). 2. A. N. Kolmogorov, Dissipation of energy in locally isotropic turbulence. C. R. Acad. Sci. URSS, 32, 16 (1941). 3. W. D. McComb, The Physics of Fluid Turbulence. (Oxford University Press, 1990). 4. G.K. Batchelor, The theory of homogeneous turbulence. (Cambridge University Press, Cambridge, 2nd edn edition, 1971). 5. D.C. Leslie, Developments in the theory of modern turbulence. (Clarendon Press, Oxford, 1973). 6. W. D. McComb, Theory of turbulence. Rep. Prog. Phys., 58, 1117–1206 (1995). 7. R. H. Kraichnan, The structure of isotropic turbulence at very high Reynolds numbers. J. Fluid Mech., 5, 497–543 (1959). 8. S.F. Edwards, The statistical dynamics of homogeneous turbulence. J. Fluid Mech., 18, 239 (1964). 9. J.R. Herring, Self-consistent ﬁeld approach to turbulence theory. Phys. Fluids, 8, 2219 (1965). 10. R. H. Kraichnan, Lagrangian-history closure approximation for turbulence. Phys. Fluids, 8 (4), 575–598 (1965). 11. R. H. Kraichnan and J.R. Herring, A strain-based Lagrangian-history turbulence theory. J. Fluid Mech., 88, 355 (1978). 12. Y. Kaneda, Renormalized expansions in the theory of turbulence with the use of the Lagrangian position function. J. Fluid Mech., 107, 131–145 (1981). 13. S. Kida and S.Goto, A Lagrangian direct-interaction approximation for homogeneous isotropic turbulence. J. Fluid Mech., 345, 307–345 (1997). 14. W. D. McComb, A local energy transfer theory of isotropic turbulence. J.Phys.A, 7 (5), 632 (1974). 15. W. D. McComb, A. Hunter, and C. Johnston, Conditional mode-elimination and the subgrid-modelling problem for isotropic turbulence. Physics of Fluids, 13, 2030 (2001). 16. H.W Wyld, Jr, Formulation of the theory of turbulence in an incompressible ﬂuid. Ann.Phys, 14, 143 (1961). 17. J.R. Herring, Self-consistent ﬁeld approach to nonstationary turbulence. Phys. Fluids, 9, 2106 (1966). 18. L. L. Lee, A formulation of the theory of isotropic hydromagnetic turbulence in an incompressible ﬂuid. Ann.Phys, 32, 292 (1965). 19. R. H. Kraichnan, Decay of isotropic turbulence in the Direct-Interaction Approximation. Phys. Fluids, 7 (7), 1030–1048 (1964).

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20. J.R. Herring and R.H. Kraichnan, Comparison of some approximations for isotropic turbulence Lecture Notes in Physics, volume 12, chapter Statistical Models and Turbulence, page 148. (Springer, Berlin, 1972). 21. J.R. Herring and R.H. Kraichnan, A numerical comparison of velocity-based and strain-based Lagrangian-history turbulence approximations. J. Fluid Mech., 91, 581 (1979). 22. W. D. McComb and V. Shanmugasundaram, Numerical calculations of decaying isotropic turbulence using the LET theory. J. Fluid Mech., 143, 95–123 (1984). 23. W. D. McComb, V. Shanmugasundaram, and P. Hutchinson, Velocity derivative skewness and two-time velocity correlations of isotropic turbulence as predicted by the LET theory. J. Fluid Mech., 208, 91 (1989). 24. W. D. McComb, M. J. Filipiak, and V. Shanmugasundaram, Rederivation and further assessment of the LET theory of isotropic turbulence, as applied to passive scalar convection. J. Fluid Mech., 245, 279–300 (1992). 25. W. D. McComb and A.P. Quinn, Two-point, two-time closures applied to forced isotropic turbulence. Physica A, 317, 487–508 (2003). 26. J.S. Frederiksen, A. G. Davies, and R.C. Bell, Closure theories with nongaussian restarts for truncated two dimensional turbulence. Phys. Fluids, 6 (9), 3153 (1994). 27. J.S. Frederiksen and A. G. Davies, Eddy viscosity and stochastic backscatter parameterizations on the sphere for atmospheric circulation models. J. Atmos. Sci., 54, 2475–2492 (1997). 28. Jorgen S. Frederiksen and Antony G. Davies, Dynamics and spectra of cumulant update closures for two-dimensional turbulence. Geophys. Astrophys. Fluid Dynamics, 92, 197 (2000). 29. Jorgen S. Frederiksen and Terence J. O’Kane, Inhomogeneous closure and statistical mechanics for Rossby wave turbulence over topography. J. Fluid. Mech., 539, 137–165 (2005). 30. A. P. Quinn, Local Energy Transfer theory in forced and decaying isotropic turbulence. (PhD thesis, University of Edinburgh, 2000). 31. Jorgen S. Frederiksen and Antony G. Davies, The Regularized DIA Closure For Two-Dimensional Turbulence. Geophys. Astrophys. Fluid Dynamics, 98, 203 (2004). 32. Terence J. O’Kane and Jorgen S. Frederiksen, The QDIA and regularized QDIA closures for inhomogeneous turbulence over topography. J. Fluid Mech., 504, 133 (2004). 33. J. Qian, Universal equilibrium range of turbulence. Phys. Fluids, 27 (9), 2229–2233 (1984). 34. P.C. Martin, E.D. Siggia, and H.A. Rose, Statistical Dynamics of Classical Systems. Phys. Rev. A, 8 (1), 423–437 (1973). 35. W. D. McComb and K. Kiyani, Eulerian spectral closures for isotropic turbulence using a time-ordered ﬂuctuation-dissipation relation. Phys. Rev. E, 72, 016309–1–12 (2005). 36. M. J. Filipiak, Application of LET theory to passive scalar transport and turbulent shear ﬂows. (PhD thesis, University of Edinburgh, 1992).

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37. R. Phythian, The operator formalism of classical statistical dynamics. J.Phys.A, 8, 1423 (1975). 38. R. Phythian, Further application of the Martin, Siggia, Rose formalism. J.Phys.A, 9, 269 (1976). 39. R. Phythian, The functional formalism of classical statistical dynamics. J.Phys.A, 10, 777 (1977). 40. R. V. Jensen, Functional Integral Approach to Classical Statistical Dynamics. J. Stat. Phys., 25, 183 (1981). 41. W. D. Thacker, A path integral for turbulence in incompressible ﬂuids. J. Math. Phys., 38, 300 (1997). 42. R. H. Kraichnan, Eddy Viscosity and Diﬀusivity: Exact Formulas and Approximations. Complex Systems, 1, 805–820 (1987). 43. W. D. McComb, Galilean invariance and vertex renormalization. Phys. Rev. E, 71, 037301 (2005). 44. A. Berera and D. Hochberg, Galilean invariance and homogeneous anisotropic randomly stirred ﬂows. Phys. Rev. E, 72, 057301 (2005). 45. W. D. McComb, Asymptotic freedom, non-Gaussian perturbation theory, and the application of renormalization group theory to isotropic turbulence. Phys. Rev. E, 73, 026303–1–7 (2006). 46. K. Kiyani, An assessment of renormalization methods in the statistical theory of isotropic turbulence. (PhD thesis, University of Edinburgh, 2005). 47. K. Kiyani and W. D. McComb, Time-ordered ﬂuctuation-dissipation relation for incompressible isotropic turbulence. Physical Review E, 70, 066303–1–4 (2004). 48. J. C. Bowman, J. A. Krommes, and M. Ottaviani, The realizable Markovian closure. I. general theory, with application to three-wave dynamics. Phys. Fluids B, 5, 3558–3589 (1993). 49. J. C. Bowman and J. A. Krommes, The realizable markovian closure and realizable test-ﬁeld model 11. application to anisotropic drift-wave dynamics. Phys. Plasmas, 4, 3895 (1997).

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TURBULENCE CLOSURES AND SUBGRID-SCALE PARAMETERIZATIONS JORGEN S. FREDERIKSEN AND TERENCE J. O’KANE CSIRO Marine and Atmospheric Research, 107-121 Station St, Aspendale, Australia [email protected] Recent developments in renormalized closure theory, including the generalization to inhomogeneous turbulence and its application for developing subgridscale parameterizations, are reviewed, focusing on two-dimensional and Rossby wave turbulence. Developments in the formulation and application of the quasi-diagonal direct interaction approximation (QDIA) closure for the interaction of general mean ﬂows and topography with inhomogeneous two-dimensional and Rossby wave turbulence are discussed. It is also noted that for both homogeneous turbulence and inhomogeneous turbulence a regularization process in which transfers are localized in wavenumber space can overcome the small scale deﬁciencies of direct interaction approximation (DIA) type closures and that the localization parameter appears to be almost universal. For a wide range of large-scale Reynolds numbers and ﬂow conﬁgurations there is in general very good agreement between the regularized DIA and QDIA closures and the statistics of DNS, except where sampling problems aﬀect the DNS results (even with very large ensembles). The application of both DIA type closures and corresponding Markovianized versions for developing subgrid-scale parameterizations of renormalized viscosity and stochastic backscatter is discussed. Such parameterizations are needed for large-eddy simulations (LES) in which simulations are carried out at relatively coarse resolution to reduce computational costs and focus on the large-scale features of the ﬂows. Kinetic energy spectra of LES with dynamical subgrid-scale parameterizations have been found to compare closely with those of DNS at the scales of the LES.

1. Introduction The theory of two-dimensional turbulence has direct application to geophysical ﬂows and to plasmas. In particular, large-scale motions in the atmosphere are quasi-geostrophic, with an approximate balance between Coriolis and pressure forces, and in many respects share properties with two-dimensional turbulence. Indeed, the ﬁrst numerical weather predic-

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tions more than 50 years ago were carried out with a generalization of the barotropic vorticity equation that describes two-dimensional turbulence. The development of a theory of turbulence has been one of the major challenges in classical physics due to the problem of “closure”. The diﬃculty in developing prognostic equations describing the statistics of turbulence, represented for example by the Navier Stokes equations with quadratic nonlinearity, is that the resulting equation for the moment of a given order involves the next higher order moment. The problem is how to close the inﬁnite hierarchy of moment or cumulant equations in terms of the lower order terms. Kraichnan27 made a major contribution to the development of renormalized closure theory for homogeneous turbulence with his Eulerian direct interaction approximation closure (DIA) which he derived using heuristic methods, as reviewed by Frederiksen.10 The DIA equations were subsequently derived by more formal mathematical approaches that had previously been developed for quantum ﬁeld theories. These methods include the Feynman diagram approach (Wyld51 ), the Schwinger-Dyson functional formalism (Martin et al.39 ) and elegant and powerful path integral formalisms (Phythian;46 Jensen25 ). In Kraichnan’s27 Eulerian DIA closure the propagators, the secondorder cumulant and response functions, are renormalized but the vertex function is bare and equal to the interaction coeﬃcient. Closely related closures such as Herring’s22 self-consistent ﬁeld theory (SCFT) and McComb’s local energy-transfer theory (LET, McComb;40 McComb et al.41 ) were developed independently. In retrospect, the SCFT and LET closures may be shown to be derivable from the DIA equations by replacing either the twotime cumulant or response function equation by the ﬂuctuation-dissipation theorem (FDT; Frederiksen and Davies12 and references therein). Markovianized versions of the closures such as the eddy-damped quasi-normal Markovian model (EDQNM, Orszag;45 Leith;36 Bowman, Krommes & Ottaviani;2 Frederiksen & Davies11 ), test ﬁeld model (TFM, Kraichnan;31,32 Leith & Kraichnan38) and realizable TFM (Bowman & Krommes3 ) have also been developed and successfully applied to a variety of important problems. These have been further discussed and reviewed in the previous chapter by McComb and by Krommes35 and McComb.42 The Eulerian DIA contains no arbitrary parameters and is in satisfactory agreement with simulations in the energy containing range of the large scales. However, at high Reynolds numbers, it leads to power laws which 5 diﬀer slightly from the k − 3 energy and k −3 enstrophy cascading inertial ranges (Kraichnan;29 Herring et al.24 ). In some problems such as plasma

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transport calculations (Bowman et al.2 ) or large-scale atmospheric ﬂows (Frederiksen and Davies11 ) the interest is primarily with the energy containing eddies and inertial range deﬁciencies may not be terribly important. In this chapter, we shall also consider two other closure models for homogeneous turbulence, the regularized DIA (RDIA) (Kraichnan;30 Frederiksen and Davies13 ) and eddy-damped quasi-normal Markovian model (EDQNM) (Orszag45) which do yield the correct inertial ranges. Both of these closures involve one speciﬁed parameter and may be established as modiﬁcations of the Eulerian DIA. Until recently studies with statistical closure theories have been largely restricted to homogeneous or isotropic turbulence because of the diﬃculty of computing inhomogeneous turbulence with the statistical DIA closure equations (Kraichnan;33 McComb41 ). However, Frederiksen9 developed a computationally tractable quasi-diagonal DIA (QDIA) closure for ﬂows, with general mean and ﬂuctuating components, over single realization mean topography on an f -plane and O’Kane and Frederiksen44 examined the performance of the closure compared with the statistics of direct numerical simulations (DNS). They found in their experiments that the QDIA for inhomogeneous f -plane two-dimensional turbulence has similar performance to the DIA for homogeneous turbulence (Frederiksen and Davies12 ), that it is only a few times more computationally intensive than the DIA and that a one-parameter regularized version of the QDIA (RQDIA), in which transfers are localized, is in excellent agreement with DNS at all scales. Frederiksen and O’Kane17 generalized the QDIA closure theory to the interaction of Rossby wave turbulence with mean ﬁelds and topography on a β-plane. In fact, on examining this problem they noted that the standard β-plane approximation neglects a small term that corresponds to the solid body rotation vorticity on the sphere and therefore formulated their theory for a generalized β-plane including a corresponding term. In their moderate resolution studies of Rossby wave dispersion over topography and ensemble prediction they found the QDIA closure was generally in very good agreement with the statistics of direct numerical simulations (DNS). Exceptions to the agreement were found in situations of strong turbulence and weak mean ﬁelds where ensemble averaged DNS fails to correctly predict mean ﬁeld amplitudes due to sampling problems even with as many as 1800 ensemble members. Closure models may be used for determining self-consistent subgridscale parameterizations needed in LES at relatively coarse resolution to successfully simulate the same large-scale ﬂow structures as in higher reso-

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lution DNS (Frederiksen and Davies;11 Frederiksen9 and references therein). Here we also review recent developments in the application of closure theory for deriving subgrid scale parameterizations of renormalized viscosity and stochastic backscatter. The plan of this chapter is as follows. In section 2, we present the barotropic vorticity equation for ﬂow over topography and Rossby wave turbulence on a generalized β-plane and in the presence of a large scale ﬂow U . We present the form drag equation for U . Here we also discuss how to transform the generalized β-plane equations for the ‘small-scales’ and the large scale ﬂow U into the standard f -plane form by generalizing the expressions for the interaction coeﬃcients and considering the large-scale ﬂow as a zero wavenumber ﬁeld. This allows us in section 3 to write down the QDIA closure equations on the generalized β-plane in the same form as the f -plane QDIA (Frederiksen9 ) with the sums over wave number just extended to include the zero wavenumber component. In section 4 we deﬁne a number of diagnostics that we subsequently use to analyze the results of the prognostic closure and DNS equations. In section 5, we consider the generation of Rossby waves when large scale ﬂows interact with isolated topography in the form of a conical mountain located at mid-latitudes. Two diﬀerent cases of Rossby wave development in the presence of moderate to strong turbulence are considered and the results of large ensembles of DNS are compared with those of the QDIA closure. In section 6, we discuss how the barotropic vorticity equation and DIA, SCFT and LET closures for homogeneous turbulence may be obtained from the more general DNS and QDIA closure equations for inhomogeneous turbulence. The performance of these closures and the RDIA in comparison with the statistics of DNS are then examined in section 7. High resolution inhomogeneous turbulence over topography on an f -plane is discussed in section 8 where the performance of the QDIA and RQDIA closures are compared with DNS. In Section 9, we consider the barotropic vorticity equation in spherical geometry and note the relationship between the DIA closure for isotropic turbulence on the sphere and on the doubly periodic domain. Section 10 describes how the integro-diﬀerential equations for the DIA closure can be reduced to the simpler diﬀerential equations for the EDQNM closure by using the FDT and at the cost of introducing a parameter specifying the strength of the eddy damping in the response function. In Section 11, we consider the problem of establishing self-consistent subgrid-scale parameterizations based on DIA and EDQNM closures. Section 12 describes

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comparisons of LES, employing renormalized drain viscosity together with stochastic backscatter parameterizations or renormalized net viscosity, with the results of higher resolution DNS. The conclusions are summarized in Section 13. Appendix A gives expressions for the oﬀ-diagonal covariance and response functions and for the three point cumulant. 2. Barotropic ﬂow on a β−plane Both the simulations and statistical closure calculations considered in this paper are based on the barotropic vorticity equation for two-dimensional turbulence. In the case of ﬂow over topography the results are based on the generalized β-plane model described by Frederiksen and O’Kane.17 As noted there the full streamfunction is written in the form Ψ = ψ−U y, where U is a large-scale east-west ﬂow and ψ represents the ‘small scales’. The evolution equation for the two-dimensional motion of the small scales over a mean topography is then described by the barotropic vorticity equation ∂ζ = −J(ψ − U y, ζ + h + βy + k02 U y) + νˆ2 ζ + f 0 . ∂t Here f 0 is the forcing, νˆ the viscosity and J(ψ, ζ) =

∂ψ ∂ζ ∂ψ ∂ζ − ∂x ∂y ∂y ∂x

(1a)

(1b)

is the Jacobian. The vorticity is the Laplacian of the streamfunction ζ = 2 ψ. The scale height for the topography h is given by h = 2µgAH where RT0 −1 −1 H is the height of the topography, R = 287Jkg K is the gas constant for air, T0 = 273K is the horizontally averaged global surface temp, g is the acceleration due to gravity, µ = sin(φ), φ is the latitude and A = 0.8 is the value of the vertical proﬁle factor. The term k02 U y generalizes the standard β-plane by the inclusion of an eﬀect corresponding to the solid-body rotation vorticity in spherical geometry where k0 is a wave number that speciﬁes the strength of this large-scale vorticity. Frederiksen and O’Kane17 noted that this additional small term results in a one-to-one correspondence between the dynamical equations, Rossby wave dispersion relations, nonlinear stability criteria and canonical equilibrium theory on the generalized β-plane and on the sphere. The barotropic vorticity equation and the form-drag equation for U can be made nondimensional by introducing suitable length and time scales, which we choose to be a/2, where a is the earth’s radius, and Ω−1 , the inverse of the earth’s angular velocity. With this scaling we consider ﬂow on the domain 0 ≤ x ≤ 2π, 0 ≤ y ≤ 2π.

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The form-drag equation for the large-scale ﬂow U is the same as on the standard β−plane. With the inclusion of relaxation towards the state U it takes the form 1 ∂U ∂ψ = dS + α(U − U ). h (2) ∂t S S ∂x Here, α is a drag coeﬃcient and S is the area of the surface 0 ≤ x ≤ 2π, 0 ≤ y ≤ 2π. In the absence of forcing and dissipation, (1) and (2) together conserve kinetic energy and potential enstrophy. We derive the corresponding spectral space equations by representing each of ‘small-scale’ terms by a Fourier series; for example ζk (t) exp (ik · x) ζ(x, t) = k

where ζk (t) =

1 (2π)2

2π

d2 x ζ(x, t) exp (−ik · x)

0

and x = (x, y), k = (kx , ky ), k = (kx2 + ky2 )1/2 and ζ−k is conjugate to ζk . As noted in (4.1) of Frederiksen and O’Kane17 the sums in the consequent spectral equations run over the set R consisting of all points in discrete wavenumber space except the point (0,0). However, it was also observed that the form-drag equation for U can be written in the same form as for the small scales by deﬁning suitable interaction coeﬃcients, representing the large-scale ﬂow as a component with zero wavenumber and extending the sums over wavenumbers. The spectral form of the barotropic vorticity equation with diﬀerential rotation, describing the evolution of the ‘small scales’, and the form-drag equation, may then be written in the same compact form as for the f -plane:

∂ 2 + ν0 (k)k ζk (t) = δ(k + p + q) [K(k, p, q)ζ−p ζ−q ∂t p∈T q∈T

+A(k, p, q)ζ−p h−q ] + fk0 where T = R ∪ 0,

δ(k + p + q) =

1 if k+p+q=0, 0 otherwise.

The interaction coeﬃcients are deﬁned by A(k, p, q) = −γ(px qˆy − pˆy qx )/p2 ,

(3)

(4)

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K(k, p, q) =

1 [A(k, p, q) + A(k, q, p)] 2

γ [px qˆy − pˆy qx ](p2 − q 2 )/p2 q 2 2 where our deﬁnitions of the interaction coeﬃcients are generalized to include the zero wave vector as any of the three arguments by specifying γ, qˆy and pˆy as follows:  k0   − 2 if k = 0 γ = k0 (5) if q = 0 or p = 0    1 otherwise 1 if k = 0 or p = 0 or q = 0 qˆy = (6) qy otherwise 1 if k = 0 or p = 0 or q = 0 (7) pˆy = py otherwise. =

Here the complex ν0 (k) is related to the bare viscosity νˆ and the intrinsic Rossby wave frequency ωk by the expression: ν0 (k)k 2 = νˆk 2 + iωk

(8a)

where βkx . (8b) k2 ∗ and introduced a term h−0 that we We have deﬁned ζ−0 = ik0 U , ζ0 = ζ−0 take to be zero but which could more generally be related to a large-scale topography. We note that U is real and we have deﬁned ζ0 to be imaginary. This is done to ensure that all the interaction coeﬃcients that we use are deﬁned to be purely real. Also with ζ 0 = −ik0 U , f00 and ν0 (k0 ) are deﬁned by ωk = −

f00 = αζ 0 , ν0 (k0 )k02

= α.

(9) (10)

These spectral equations are then the basis for our subsequent studies and theoretical developments. 3. The QDIA closure equations The QDIA closure equations were derived by Frederiksen9 for general barotropic mean ﬂows interacting with inhomogeneous turbulence over topography on an f -plane. O’Kane and Frederiksen44 tested the performance

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of the closure including cumulant update and regularized variants while the generalization to Rossby wave turbulence on a β-plane was formulated and tested by Frederiksen and O’Kane.17 Here we very brieﬂy summarize the β-plane QDIA equations. We consider an ensemble of ﬂows satisfying the generalized spectral barotropic vorticity equation (3) that describes the evolution of both the ‘small scales’ and the large scales through the form-drag equation. Throughout this section the wave vectors range over the set T = R ∪ 0. We can express the vorticity ζk and forcing fk0 in terms of their ensemble average means, denoted by , and the deviations from the means, denoted byˆ: ζk = ζk + ζˆk , f 0 = f 0 + fˆ0 . k

k

k

(11a) (11b)

The equations for the ensemble mean and the deviation can then be written in the form:

∂ 2 + ν0 (k)k ζk = δ(k + p + q)K(k, p, q) ∂t p q

× [ζ−p ζ−q + C−p,−q (t, t)] δ(k + p + q)A(k, p, q)ζ−p h−q + fk0 ,

(12a)

∂ + ν0 (k)k 2 ζˆk = δ(k + p + q)K(k, p, q) ∂t p q × ζ−p ζˆ−q + ζˆ−p ζ−q + ζˆ−p ζˆ−q − C−p,−q (t, t)] δ(k + p + q)A(k, p, q)ζˆ−p h−q + fˆk0 . +

(12b)

+

p

q

p

q

Here the two-point cumulant is deﬁned by C−p,−q (t, s) = ζˆ−p (t)ζˆ−q (s). Thus, we see from (12) that to determine the mean ﬁeld we need an equation for the two-point cumulant C−p,−q (t, t). Similarly the second order cumulant requires knowledge of the third order cumulant which in turn depends on the fourth and so on. We are consequently faced with two problems, namely, the cost of computing the full covariance matrix, which is prohibitive at any reasonable resolution (see Kraichnan33), and secondly the closure problem. The quasi-diagonal DIA closure equations overcome the problem of computational cost by expressing the oﬀ-diagonal two-point cumulant and response functions and higher order cumulants in terms of

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the diagonal elements. The resulting equations for the mean ﬁeld, two-point cumulant and response functions are then expressed entirely as functionals of the diagonal elements of the two-point cumulant and response functions. These QDIA equations are computationally much more eﬃcient than the general inhomogeneous closure equations.33 Speciﬁcally, we have the following functional forms: QDIA (t, t )[Ck , Rk , ζk , hk ] Ck,−l (t, t ) ≈ Ck,−l QDIA Rk,−l (t, t ) ≈ Rk,−l (t, t )[Ck , Rk , ζk , hk ]

where the oﬀ-diagonal elements of the response function, which measures the change in the vorticity perturbation due to an inﬁnitesimal change in the forcing, is given by ˆ ˆ k,l (t, t ) = δ ζk (t) R δ fˆ0 (t ) l

ˆk,l (t, t ). Rk,l (t, t ) = R We also use the abbreviations for the diagonal elements Ck (t, t ) = Ck,−k (t, t );

Rk (t, t ) = Rk,k (t, t ).

As well the following terms are represented as functionals of the diagonal elements: ζˆk (t)ζˆ−l (t)ζˆl−k (t ) ≈ ζˆk (t)ζˆ−l (t)ζˆl−k (t )QDIA [Ck , Rk , ζk , hk ] ˆ l−k (t, t )ζˆ−l (t)QDIA [Ck , Rk , ζk , hk ]. ˆ l−k (t, t )ζˆ−l (t) ≈ R R QDIA QDIA The explicit expressions for Ck,−l (t, t ), Rk,−l (t, t ) and ˆ l−k (t, t )ζˆ−l (t)QDIA are given in Apζˆk (t)ζˆ−l (t)ζˆl−k (t )QDIA and R pendix A. The mean-ﬁeld equation can be written in the form

∂ 2 + ν0 (k)k ζk = δ(k + p + q)[K(k, p, q)ζ−p (t)ζ−q (t) ∂t p q t t ds ηk (t, s)ζk (s) + hk ds χk (t, s) +A(k, p, q)ζ−p (t)h−q ] − t0

t0

+fk0 (t)

(13)

Here, the nonlinear damping ηk (t, s) = −4

p

δ(k + p + q)

q

×K(k, p, q)K(−p, −q, −k)R−p (t, s)C−q (t, s),

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measures the interaction of transient eddies with the mean ﬁeld while χk (t, s) = 2

p

δ(k + p + q)

q

×K(k, p, q)A(−p, −q, −k)R−p (t, s)C−q (t, s), measures the strength of the interaction of transient eddies with the topography. The second-order expression for the diagonal two-time cumulant can then be written in terms of the QDIA two- and three-point cumulants:

∂ + ν0 (k)k 2 Ck (t, t ) = δ(k + p + q)A(k, p, q) × ∂t p q QDIA C−p,−k (t, t )h−q + δ(k + p + q)K(k, p, q)[ζ−p (t) × p

QDIA (t, t ) C−q,−k

+

q

QDIA C−p,−k (t, t )ζ−q (t)

+ ζˆ−p (t)ζˆ−q (t)ζˆ−k (t )QDIA ]

+fˆk0 (t)ζˆ−k (t )

(14)

where fˆk0 (t)ζˆ−k (t ) =

t

t0

ds Fk0 (t, s)R−k (t , s)

(15)

and Fk0 (t, s) = fˆk0 (t)fˆk0∗ (s) is the variance of the random forcing. Thus, from the expressions in Appendix A we ﬁnd

t

t0

∂ 2 + ν0 (k)k Ck (t, t ) = ∂t

3 4 ds Sk (t, s) + Pk (t, s) + Fk0 (t, s) R−k (t , s) −

t

ds [ηk (t, s) + πk (t, s)] C−k (t , s)

(16)

t0

where Sk (t, s) = 2

p

δ(k + p + q)

q

K(k, p, q)K(−k, −p, −q)C−p (t, s)C−q (t, s),

(17a)

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is the nonlinear noise and δ(k + p + q)C−p (t, s) × Pk (t, s) = p

q

[2K(k, p, q)ζ−q (t) + A(k, p, q)h−q ] × [2K(−k, −p, −q)ζq (s) + A(−k, −p, −q)hq ] , πk (t, s) = − δ(k + p + q)R−p (t, s) × p

(17b)

q

[2K(k, p, q)ζ−q (t) + A(k, p, q)h−q ] × [2K(−p, −k, −q)ζq (s) + A(−p, −k, −q)hq ] ,

(17c)

are noise and dissipation terms associated with eddy-mean ﬁeld and eddytopographic interactions. The equation for the diagonal response function is as described in Frederiksen:9

t ∂ 2 + ν0 (k)k Rk (t, t ) = − ds [ηk (t, s) + πk (t, s)] Rk (s, t ) (18) ∂t t with Rk (t, t) = 1 and for t < t we have Rk (t, t ) = 0. The equation for the diagonal single-time two-point cumulant is

∂ + 2Reν0 (k)k 2 Ck (t, t) ∂t t 3 4 = 2Re ds Sk (t, s) + Pk (t, s) + Fk0 (t, s) R−k (t, s) t0

−2Re

t

ds [ηk (t, s) + πk (t, s)] C−k (t, s)

(19a)

t0

since ∂Ck (t, t) = lim t →t ∂t

∂Ck (t, t ) ∂Ck (t, t ) + ∂t ∂t

(19b)

and Ck (t , t) = C−k (t, t ) = Ck∗ (t, t ). 4. Diagnostics In the sections that follow prognostic DNS and closure equations are analyzed using a number of diagnostics. Firstly we deﬁne the kinetic energy by 1 E(t) = [Ck (t, t) + ζk (t)ζ−k (t)] /k 2 . (20) 2 k

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The components appearing in (20) are used to calculate band-averaged energy spectra deﬁned as E(ki , t) = E T (ki , t) + E M (ki , t) where the transient E T and mean E M energies are deﬁned by 1 E T (ki , t) = [Ck (t, t)] /k 2 , 2 k∈S 1 [ζk (t)ζ−k (t)] /k 2 . E M (ki , t) = 2 k∈S

The set S is deﬁned as

1 S = k|ki = Int.[k + ] 2

(21)

where the subscript i indicates that the integer part is taken in (21) so that all k that lie within a given radius band of unit width are summed over. We deﬁne the palinstrophy production and enstrophy dissipation as K(t) = k 2 Nk (t, t) k

η(t) =

νˆk 2 Ck (t, t).

k

Following Frederiksen and Davies12 and Herring et al.24 we also deﬁne the large-scale Reynolds number RL (t) and the skewness SK (t) by ˆ ν η 1/3 ) RL (t) = E/(ˆ SK (t) = 2K/(Pˆ Fˆ 1/2 ) where, the transient energy, enstrophy, and palinstrophy are given by ˆ = 1 E(t) Ck (t, t)/k 2 , 2 k 1 ˆ F (t) = Ck (t, t), 2 k ˆ =1 P(t) Ck (t, t)k 2 . 2 k

The skewness SK is a sensitive measure of the small scale diﬀerences between the DNS and QDIA closure while RL provides a measure of the strength of the turbulence.

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Fig. 1. Circular conical mountain of height 2500m centred at 30◦ N , 180◦ W with a diameter of 45◦ latitude.

5. Topographic Rossby waves in a turbulent environment Kasahara26 examined the generation of Rossby waves in numerical simulations of two-dimensional eastward and westward zonal ﬂows impinging on isolated topography. He was able to explain much of the behaviour in his simulations and in earlier laboratory experiments by Fultz and Long,19 and Fultz, Long and Frenzen.20 Further contributions to the basic understanding of topographic Rossby wave generation were made in the numerical simulation studies of Egger,7 Vergeiner and Ogura,49 Edelmann6 and Grose and Hoskins.21 Frederiksen8 compared linear steady state theory with canonical equilibrium solutions while Verron and Le Provost50 studied quasi-geostrophic homogeneous ﬂow impinging on isolated topography for both f - and β-planes. In this section we examine the generation of Rossby waves when eastward zonal currents impinge on a conical mountain in the presence of turbulence. We compare QDIA closure calculations with averages based on large ensembles of DNS for ﬂow on the generalized β-plane. Here we discuss the accuracy of the QDIA closure in describing the evolution of topographic Rossby waves in two cases of medium to strong turbulence. Our examples test the validity of the QDIA in describing Rossby wave turbulence (Frederiksen and O’Kane17 ) for strong transients and relatively weak mean ﬁelds. In this section we use a length scale of a/2 and a time scale of Ω−1 ; the ﬁelds are mapped onto the doubly periodic domain, evolved and displayed on spherical projections. We examine the dynamics of Rossby waves for a β−eﬀect of 1.15 × 10−11 m−1 s−1 or a non-dimensional value of β = 1/2

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and k02 = 1/2, (typical of the β−eﬀect at 60◦ latitude). Our experiments are carried out at circular C16 truncation for which k ≤ 16 and this is adequate for our purposes since we focus on the dynamics of relatively large-scale Rossby waves. As in Fig. 4 of Frederiksen,8 the conical mountain shown in our Fig. 1 is 2500m high and is centred at 300 N , and 1800W with a diameter of 45◦ latitude at the base. 5.1. Case 1 For case 1 the initial large-scale ﬂow U is eastward at 7.5ms−1 (a nondimensional value of 0.0325) and β = 1/2 and k02 = 1/2. In this case the term k02 U only makes just over 3% contribution to the β−eﬀect and we expect little quantitative diﬀerence between our results for the generalized β−plane and corresponding standard β−plane results. The results presented in this subsection are for dissipative ﬂows with a viscosity of 2.5 × 104 m2 s−1 or a nondimensional value of νˆ = 3.378 × 10−5 . 102

100

2 -2

10-2 E(k) in m s

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10-4

10-6

10-8

10-10 0

5

10

15

k

Fig. 2. The evolved kinetic energy spectra in m2 s−2 for case 1. Shown are: initial mean energy (thin dotted), initial transient energy (thin solid), evolved DNS transient energy (thick solid), evolved CUQDIA transient energy (thin dashed), evolved DNS mean energy (thick dotted) and evolved CUQDIA energy (thick dashed). Parameters used are given in table 1.

The kinetic energy spectra of the initial transient and mean ﬁeld contributions of the ‘small scales’ are as shown in Fig. 2. The initial turbulent transients are Gaussian and isotropic and are several orders of magnitude

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larger than the mean ﬁeld at all wave numbers except for the large-scale mean ﬁeld. The kinetic energy of the large-scale ﬂow is plotted at zero wave number. The initial DNS ﬁelds have been constructed by ﬁrst taking a Gaussian sample with zero mean and unit variance. For a given realization further members of the ensemble are then obtained by moving its origin by an increment in the x -direction and then in the y-direction. The initial realization is moved successively by 2π/n in the x -direction to form n realizations. Each of these n realizations is then shifted by 2π/n in the y-direction to form a total of n2 realizations. The negative value of each of the n2 elements is then taken so that we now have an ensemble of 2n2 realizations. The same process is then repeated with further initial Gaussian samples until an ensemble of the required number is obtained. This method ensures that not only are the initial ﬁelds accurate but that the initial DNS covariance matrix is eﬀectively isotropic for large enough n. The closure was integrated forward for 10 days . A time step of 1/30 day, or a non-dimensional value of t = 0.21, was used and a restart carried out every day for the QDIA44 closure. For the DNS 1800 realizations were integrated forward from the same initial mean and transient spectra as for the closure and again using a time step of 1/30 day. Fig. 2 also shows the evolved QDIA and ensemble averaged DNS kinetic energy spectra on day 10. The transient and mean components for the evolved results for the DNS and closure are virtually indistinguishable. We see large increases in the mean ﬁeld energies with peaks at wave number 4 and a drop in the tail of the transient components. Table 1. Case 1

2

Ck (0, 0) 0.01k 2 a + bk 2 k2 a + bk 2

Parameters for Figs. 2-4

ζk (0)

a

b

−10bhk Ck (0, 0)

4.824E + 04

2.511E + 03

−bhk Ck (0, 0)

4.824E + 04

2.511E + 03

The mean ﬁeld peaks are a consequence of the topographic Rossby waves generated in the presence of diﬀerential rotation (Kasahara;26 Frederiksen;8 Verron and Le Provost50). Fig. 3 shows the corresponding eddy, or zonally asymmetric, contribution to the streamfunction of the mean ﬁelds for case 1. These ﬁelds show the characteristic Rossby wave trains downstream of

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the conical mountain also seen in linear steady state solutions (eg Fig. 6 of Frederiksen8 ). The pattern correlation between the closure and DNS mean ﬁelds for the zonally asymmetric component of the streamfunction is 0.9999 in both cases indicating the excellent agreement between closure and DNS results. 5.2. Case 2 For case 2 (corresponding to case 3 of Frederiksen and O’Kane17 ) the same closure and DNS calculations as for case 1 above were repeated but with the initial transient kinetic energy spectrum increased by a factor of 100. The ensemble averaged results based on both 800 and 1800 realizations of direct numerical simulations have been calculated for case 2. The aim was to examine the sampling problem that may arise in calculating smallamplitude mean ﬁelds and spectra from ensemble averaged DNS in the presence of strong turbulence (O’Kane and Frederiksen44 ). For case 2 the initial and days 2 and 10 evolved energy spectra for the CUQDIA17 closure and DNS are shown in Figs. 4 a), b) and c). We note from Fig. 4 b) that on day 2 there is close agreement between the closure and DNS mean energy spectra for k < 12 but at higher wave numbers the DNS results overestimate the mean spectra, increasingly so for fewer realizations. Fig. 4 c), which shows the evolved spectra on day 10, indicates that the DNS error in calculating the mean ﬁeld increases with time and then saturates at a level depending on the number of realizations in the DNS ensemble. 6. Homogeneous and isotropic closure theories Homogeneous turbulence may be simulated with the barotropic vorticity (1a) but with h = 0, U = 0 and β = 0. Then with no forcing or with a random forcing function and with zero mean vorticity ζ = 0 we expect to be able to simulate homogeneous turbulence. The spectral barotropic equation is again given by (3) but with p ∈ R, q ∈ R, the topographic term missing and ωk = 0 in (8b). The QDIA closure also reduces to the DIA closure for homogeneous turbulence. The mean ﬁeld equation (13) no longer applies. However, the equations for the two-time cumulant and response function are as in (16) and 18 and the single-time cumulant equation is again given by (19), but with Pk (t, s) = 0, πk (t, s) = 0 in (17). As noted in the introduction, the SCFT closure of Herring22 and the LET closure of McComb40 (see also McComb et al.43 ) are non-Markovian

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a)

b)

Fig. 3. The evolved day 10 DNS (a) and CUQDIA (b) zonally asymmetric streamfunction in units of 105 m2 s−1 for case 1.

closures that are closely related to Kraichnan’s DIA.27 In retrospect the SCFT and LET closure equations may be formally obtained from the DIA by invoking the ﬂuctuation-dissipation theorem (FDT, Kraichnan;28 Carnevale and Frederiksen5 ) out of strict statistical mechanical equilibrium. The FDT states that Ck (t, t )θ(t − t ) = Rk (t, t )Ck (t , t )

(22)

where θ(t − t ) is the Heaviside step function that vanishes for t < t and is otherwise unity. The FDT approximation (22) gives us an additional relationship between the two-time cumulant and response function and the single-time cumulant. For the SCFT, the equations are identical to the homogeneous DIA equations but the FDT equation replaces the prognostic equation for the two-time cumulant. For the LET the FDT relation instead replaces the response function equation.

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a) 10

b)

2

102

100

10-2

10-2

2 -2

100

E(k) in m s

2 -2

E(k) in m s

10-4

10-4

10-6

10-6

10-8

10-8

10-10 0

5

10

10-10 0

15

5

c)

10

15

k

k

102

100

2 -2

10-2

E(k) in m s

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10-4

10-6

10-8

10-10 0

5

10

15

k

Fig. 4. The evolved kinetic energy spectra (m2 s−2 ) at 0 (a), 2 (b) and 10 (c) days for case 2. The line types are as described for Fig. 2. Parameters used are given in table 1.

7. Performance of DIA, SCFT and LET closures for moderate Reynolds number turbulence Frederiksen and Davies12 compared the performance of discrete wavenumber non-Markovian closures with DNS and with the continuous wavenumber DIA closure results of Herring et al.24 for very similar spectra and large-scale Reynolds numbers (RL ) between ≈ 50 and ≈ 300. Most of these runs used a C48 or C63 circular truncation with maximum wavenumber kmax = 48 or 63 and 1 ≤ k ≤ kmax . They found that the discrete wavenum-

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ber closures perform considerably better than the continuous wavenumber closures of Herring et al.24 in comparison with DNS as far as the evolved kinetic energy and transfer spectra and skewness are concerned. As well, they compared closure and DNS runs for spectra at the higher resolution of C96, with kmax = 96, and high evolved large-scale Reynolds number of RL = 4000. For the range of Reynolds numbers considered it was found that the discrete wavenumber DIA, SCFT and LET closures are in reasonable agreement with DNS in the energy containing range of the large scales but that these closures, without vertex renormalization, underestimate the enstrophy ﬂux to high wavenumbers, particularly at high Reynolds number, resulting in an underestimation of small-scale kinetic energy. The performance of the DIA, SCFT and LET closures compared with the statistics of 40 DNS starting in each case from Gaussian initial conditions is shown in Fig. 5. The results presented there are based on integration of the DNS and closure from t = 0 to t = 0.8 non-dimensional units with an initial spectrum given by

2 −3 2 (23) Ck (0, 0) = 1.8 × 10 k exp − k . 3 We use a non-dimensional viscosity of νˆ = 0.0025 giving an initial Reynolds number of RL (0) = 307 for the C48 circular truncation used in these studies. We see from Fig. 5 that the kinetic energy spectra of the closures agree well with the evolved DNS for k ≤ 20. However, at the smaller scales the closures under-represent the kinetic energy spectra. The systematic small scale errors of the closures at moderate Reynolds number appears to be associated with spurious convection eﬀects of the of the small-scale eddies by the large eddies in the closure formalism (Kraichnan30 ). As discussed in detail by Frederiksen and Davies,13 this may be overcome by a regularized version of the DIA closure where the interaction coeﬃcient or vertex function is renormalized such that K(k, p, q) → θ(p − k/α)θ(q − k/α)K(k, p, q) and this modiﬁcations applied to (16) and (18) but not to (19). Here θ is the Heaviside step function which vanishes for negative argument and is otherwise unity. It restricts the interactions between large and small wavenumbers, depending on the cut-oﬀ ratio α, making the interaction more local and in the process can be shown to improve small-scale aspects of the DIA closure (Frederiksen and Davies13 and references therein). Frederiksen and Davies13 ﬁnd that on the whole the RDIA closure with the above empirical vertex renormalization and regularization cut-oﬀ ratio

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Fig. 5. Initial (t = 0) and evolved (t = 0.8) kinetic energy spectra E(k) for runs starting from initial spectrum B of Frederiksen and Davies (2000) at C48 for DNS and closures. Some results have been scaled by a factor of 10 as shown.

α = 6 performs remarkably well compared with DNS. In particular, for their spectrum B speciﬁed by (23) the evolved spectra are in close agreement. The RDIA and DNS runs have resolution C63 and start from Gaussian initial conditions for both the closure and DNS. The integrations have been performed between nondimensional times t = 0 and t = 0.8 with a timestep of t = 0.016 for the closure and t = 0.004 for DNS to ensure computational stability. The non-dimensional viscosity is speciﬁed as νˆ = 0.0025 and the Reynolds number starts at RL = 307 and the ﬁnal evolved Reynolds number RL (0.8) = 280 for both DNS and RDIA and RL (0.8) = 291 for the DIA closure. Fig. 6 shows the initial and evolved (t = 0.8) kinetic energy spectra E(k) for the RDIA and DIA closures and compares them with DNS results, where the DNS results are the average over 40 realizations. The RDIA closure compares closely with DNS and represents a considerable improvement over the DIA closure for k > 20. We note the close agreement between the RDIA closure and DNS except at the very smallest scales where k > 58. We also note that the RDIA closure results appear to be generally superior to the

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Fig. 6. Initial (t = 0) and evolved (t = 0.8) kinetic energy spectra E(k) for runs starting from initial spectrum B of Frederiksen and Davies (2004) at C63 for DNS and RDIA and DIA closures.

abridged Lagrangian-history direct interaction (ALHDI) and strain-based abridged Lagrangian-history direct interaction (SBALHDI) closure results shown in Fig. 8b of Herring and Kraichnan23 for a closely similar spectrum. Frederiksen and Davies13 show that the results reproduced here for spectrum B with moderate Reynolds number are in fact representative for evolved Reynolds numbers between ≈ 50 and ≈ 4000 in terms of comparison of RDIA and DIA closures with DNS. 8. Inhomogeneous turbulence on an f -plane Next we consider inhomogeneous turbulence interacting with mean ﬂows and topography on an f -plane. Again the barotropic vorticity equation 1a is employed but with β = 0 and U = 0. Unlike the studies in section 6, the topography h and mean vorticity ζ are both nonzero and inhomogeneous. The spectral barotropic vorticity equation is again given by (3) but with p ∈ R, q ∈ R and ωk = 0 in (8b). The QDIA closure equations for the mean ﬁeld, single-time and two-time cumulants and response function are again as discussed in section 3.

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2

|h | =1.8x10 k exp(-2k/3) k 10

0

(a)

E(k)

10

10

10

-2

-4

-6

t=0 10

-8

1

10 k

α1 = 6, α2 = 3 10

10

0

α1 = 4, α2 = 4

0

(f)

(b)

10

10

10

-2

E(k)

E(k)

10

-4

10

10

-6

-2

-4

-6

t=0.4 10

10

-8

1

t=0.4

-8

10

1

10

k 10

k

3

10

3

(g)

(c)

10

10 10

2

10

1

P(k)

P(k)

10

0

10

10

-1

10

2

1

0

-1

t=0.4

t=0.4 10

-2

1

10

10

-2

1

10 k

k 1.5

1.5

(h)

(d) 1.0

S (t) K

S (t) K

1.0

0.5

0 0

0.5

0.2 t

0

0.4

1.6

0

0.2 t

0.4

1.6

(e) 1.4

(i) 1.4

R (t)/R (0) L L

R (t)/R (0) L L

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1.2

1.2

1.0

1.0

0.8 0

0.2 t

0.4

0.8

0

0.2 t

0.4

Fig. 7. Case BII. Comparison of DNS with CUQDIA and RCUQDIA closures at C48 resolution. Parameters can be found in table 2. (a) Initial mean (thin dashed) and transient (thin solid) kinetic energy. (b) and (f) Mean and transient kinetic energy spectra at tf = 0.4. (c) and (g) Mean and transient palinstrophy spectra at tf = 0.4; Component ﬁeld diagrams: mean ﬁeld; DNS (100 member ensemble, thick dashed: 1000 member ensemble, thick long dashed), CUQDIA (thick dotted), RCUQDIA (thin dashed): transient ﬁeld; DNS (100 member ensemble, thick solid: 1000 member, thin long dashed), CUQDIA (thin dot dashed), RCUQDIA (thin solid). (d) and (h) Skewness and (e) and (i) RL (t)/RL (0) evaluated at time t. DNS (thick solid), CUQDIA (thick dotted) and RCUQDIA (thin solid) with restarts calculated at every 20 timesteps.

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O’Kane and Frederiksen44 examined the performance of the QDIA closure equations compared with DNS for a wide variety of ﬂow regimes, resolutions, topographies and Reynolds numbers. Here we discuss a particular case that they denoted case BII which provides a very stringent test of the performance of the QDIA closure with and without regularization. For case BII of O’Kane and Frederiksen,44 the conditions from which the simulations and closure calculations were started have initial mean and transient kinetic spectra as shown in Fig. 7 (a). Case BII has mean and transient kinetic energy spectra which have broad similarities with typical atmospheric spectra (Boer and Shepherd1 ). In particular, at large scales the mean ﬂow dominates while at intermediate and small scales the transients dominate the kinetic energy spectrum. This situation continues as the ﬂow evolves. These properties are in part related to the choice of topography (table 2) that ensures strong mean-ﬁeld topographic interactions at large scales but negligible coupling at the smallest scales allowing the eddy-eddy interactions to dominate there. For this initial state the large-scale Reynolds number is RL 202. We discuss the performance of cumulant update versions of the QDIA and a regularized QDIA closure with empirical vertex renormalizations. The cumulant update methodology (Rose;48 Frederiksen et al.14 ) is a restart scheme in which the time history integrals are periodically truncated. It is considerably more eﬃcient for long time integrations and the details of the methodology for the QDIA are described by O’Kane and Frederiksen.44 For the QDIA, the regularized version replaces the interaction coeﬃcients

k k θ q− K(k, p, q) K(k, p, q) → θ p − α1 α1

k k θ q− A(k, p, q) A(k, p, q) → θ p − α2 α2 in (16) and (18) but not in (19) as described in more detail by O’Kane and Frederiksen.44 Fig. 7 compares the performance of the cumulant update version of the RQDIA closure (RCUQDIA) for α1 = 6, α2 = 3 and α1 = α2 = 4 with the results of DNS and the cumulant update version of the QDIA (CUQDIA) closure calculations. The statistics are evolved to a ﬁnal non-dimensional time tf = 0.4 and the timesteps used for the closures and DNS, as well as the viscosity in these viscous decay calculations, are given in table 2. For the DNS results for 100 and 1000 member ensembles are compared in order to examine the sampling problem of determining the mean ﬁeld and variance.

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Figs. 7 (b) and (f) compare the evolved kinetic energy spectra and Figs. 7 (c) and (g) the evolved palinstrophy spectra for the two DNS calculations, the two RCUQDIA calculations and the CUQDIA results. We note that the CUQDIA mean and transient spectra have too small amplitudes compared with DNS at the smaller scales. In contrast, the RCUQDIA transient spectra, for both choices of cut-oﬀ parameters, compare closely with the DNS ensemble averages for both the 100 and 1000 member ensembles. In particular, the transient spectra for the RCUQDIA for both pairs of cutoﬀ ratios are nearly indistinguishable from the DNS results. The RCUQDIA mean ﬁeld also compares most closely to the 1000 member ensemble DNS results at small scales, although there does appear to be a slight over estimation of the RCUQDIA mean ﬁeld for the small scales. The diﬀerences in the two DNS are due to the sampling problem with the DNS spectra ﬂattening out at small scales (Figs. 7f and g) where the mean ﬁeld in the 100 member DNS ensemble approaches the 1 percent sampling error barrier. In contrast, the 1000 member ensemble is better able to resolve the mean ﬁeld at small scales. Table 2. Fig.

tCU QDIA

7

0.004

Parameters for Fig. 7

tDNS 0.002

|hk |2

νˆ 0.002

1.8 ×

10−3 k 2

exp (− 23 k)

Figs. 7 (d) and (h) show the evolution of the skewness for the closures and DNS. We note that the RCUQDIA skewness, for both pairs of cut-oﬀ parameters, is in excellent agreement with the DNS skewness. In contrast the CUQDIA skewness is signiﬁcantly less than for the DNS; at tf = 0.4 RCUQDIA RCUQDIA DN S = 0.86, SK = 0.83, SK = 0.95 while we ﬁnd that SK α =6,α =3 α =4,α =4 1

2

1

2

1

2

1

2

CUQDIA = 0.32. Again, Figs. 7 (e) and (f) show that the regularized SK closures accurately predict the evolution of the DNS Reynolds number while the CUQDIA closure slightly overestimates the evolved Reynolds number. RCUQDIA RCUQDIA DN S ≈ 223, RL ≈ 233, RL ≈ 229 while At tf = 0.4 RL α =6,α =3 α =4,α =4 CUQDIA ≈ 257. RL

9. Vorticity equation and DIA closure on the sphere Next, we consider the problem of determining self-consistent subgrid-scale parameterizations based on closures. Since the aim is to apply these to

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atmospheric circulation models the formulation is developed in spherical geometry. In spherical geometry the barotropic vorticity equation takes the same form as in (1a) and the relation between the vorticity and streamfunction is as for planar geometry, ζ = ∇2 ψ, but the Laplacian is now for spherical geometry. If we non-dimensionalize the equations by taking a (the earth’s radius) and Ω−1 (the inverse of the earth’s angular velocity) as length and time scales then the Jacobian is again given by (1b) but with x replaced by λl , the longitude, and y replaced by µ, the sine of the latitude. We shall use a slightly more general formulation of the dissipation than that in previous sections. The corresponding spectral equation can be obtained by expanding each of the functions in spherical harmonics; for example ζ(λl , µ, t) =

T

T

ζmn (t)Pnm (µ) exp(imλl ),

m=−T n=|m|

where m is zonal wavenumber, n is total wavenumber, T is the triangular truncation wavenumber, and Pnm are orthonormalized Legrendre functions. Now let Pk ≡ Pnm , k = (m, n) = (mk , nk ), −k = (−m, n), k = n. Then the spectral equation for homogeneous turbulence (see section 6) is

∂ + ν0 (k)k(k + 1) + iωk ζk (t) ∂t =i δ(mk + mp + mq ) K(k, p, q) ζ−p (t) ζ−q (t) + fk0 (t), p

q

where ωk = − and

Bm , n(n + 1)

1 p(p + 1) − q(q + 1) × 2 p(p + 1)q(q + 1)

1 dPq dPp − mq Pq dµ Pk mp Pp , dµ dµ −1

K(k, p, q) =

(24a)

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δ(mk + mp + mq ) =

1 if mk + mp + mq = 0, 0 otherwise .

(24b)

Here B represents the beta eﬀect on the sphere, which with the current scalings, would take value 2 for the atmosphere; we shall also be interested in the case B =0 applicable to isotropic turbulence. In spherical geometry, the DIA closure for homogeneous turbulence is again given by the equations described in sections 6 and 7 but with the interaction coeﬃcient given by (24a) and the Kronecker delta function (24b) replacing that of (4) in (3) and k 2 → k(k + 1) in the viscous damping term. The selection rules on the total wavenumbers for non-zero interaction coeﬃcients are given in (2.3) of Frederiksen and Sawford.18 In the following sections we shall also specialise to isotropic turbulence for which the statistics of the cumulants and response functions only depend on k = n rather than on k = (m, n). 10. EDQNM closure on the sphere The time-history integrals associated with the non-Markovian DIA closure reﬂect memory eﬀects associated with turbulent eddies. Other nonMarkovian closure theories such as the SCFT and LET closures have very similar performance to the DIA closure. The SCFT and LET closures replace either the second-order two-time cumulant or the response function with the ﬂuctuation dissipation theorem (FDT). Unlike quantum ﬁeld theories which deal with Hermitian operators, classical systems tend not to be self-adjoint and the FDT in general only applies at canonical equilibrium (Kraichnan,28 Carnevale and Frederiksen,5 and references therein). Nevertheless, as noted above it appears to be a reasonable approximation more generally. We specialise to isotropic turbulence with B = 0. Then the nonMarkovian DIA closure can be simpliﬁed to a Markovian closure called the eddy-damped quasi-normal Markovian (EDQNM) closure by replacing the equation for the two-time cumulant (16) by the stationary form of the FDT: Ck (t, t )θ(t − t ) = Rk (t, t )Ck (t, t), and assuming an exponential decay form for the response functions Rk (t, t ) = exp[−µk (t − t )].

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Here, θ is the Heaviside step function which is zero for negative argument and otherwise is unity and the eddy damping is given by the empirical form 1

µk = γ [k(k + 1)Ck (t, t)] 2 + ν0 (k)k(k + 1) (Orszag;45 Leith;36 Frederiksen and Davies11 ). This form of the eddy damping is consistent with the −3 enstrophy cascading inertial range power law of two-dimensional turbulence. We ﬁnd that taking γ = 0.6, gives good comparison with DNS. The single-time cumulant equation (19) then reduces to the ordinary diﬀerential equation ∂Ck (t, t) + 2 ν0 (k)k(k + 1)Ck (t, t) − F0 (k; t) ∂t = 2Sk (t) − 2ηk (t)Ck (t, t) .

(25)

We have also specialised to white noise forcing for which 0 (t ) = F0 (k; t)δ(t − t ), fk0 (t) f−k

so that, from (19), t ds Fk0 (t, s) Rk (t, s) 2

−→

F0 (k; t).

t0

The nonlinear damping and nonlinear noise then reduce to δ(k + p + q)K(k, p, q)K(−p, −q, −k) ηk (t) = −4 p

q

×θkpq Cq (t, t), Sk (t) = 2

p

δ(k + p + q)[K(k, p, q)]2

q

×θkpq Cp (t, t)Cq (t, t) (> 0) δ(k + p + q)K(k, p, q)K(−p, −q, −k) = −4 p

q

×θkpq Cp (t, t)Cq (t, t).

(26)

We note that the nonlinear noise is positive. The time-history integrals can be calculated analytically to determine the triad relaxation time as t ds Rk (t, s)Rp (t, s)Rq (t, s) θkpq (t) = t0

1 − exp[−(µk + µp + µq )(t − t0 )] = , µk + µp + µq

(27a)

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with θkpq (∞) = (µk + µp + µq )−1 .

(27b)

Here the summations over p and q are determined by T (T ) = {p, q −T ≤ mp ≤ T , |mp | ≤ p ≤ T , −T ≤ mp ≤ T , |mp | ≤ p ≤ T }. 11. Subgrid-scale parameterizations Next, we examine how to establish self-consistent subgrid-scale parameterizations when the resolution is reduced from triangular truncation T to TR < T where TR is the triangular truncation wavenumber of the resolved scales. We deﬁne the set of resolved scales by R = T (TR ), and the set of subgrid-scales by S = T (T ) − R, Then, the nonlinear damping and nonlinear noise due to the resolved scales (respectively subgrid-scales) are given by (25) and (26), with subscript R (respectively S) for p, q ∈ R (respectively S). 11.1. The EDQNM Closure For the EDQNM closure, the equation for (twice) the enstrophy components of the resolved scales can then be written in the form ∂Ck (t, t) + 2 [ν0 (k)k(k + 1) + ηkS (t)] Ck (t, t) ∂t − [F0 (k; t) + 2SkS (t)] = 2SkR (t) − 2ηkR (t)Ck (t, t). It is then clear that ηkS modiﬁes the bare viscous damping and SkS modiﬁes the random forcing variance. As noted in (26), SkS is positive and corresponds to an injection of enstrophy from the subgrid scale eddies to the resolved scales; that is, it represents stochastic backscatter. We therefore deﬁne the eddy drain viscosity νd (k) = [k(k + 1)]−1 ηkS ,

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the renormalized viscosity νr (k) = ν0 (k) + νd (k), the stochastic backscatter Fb (k) = 2SkS , and the renormalized noise variance Fr (k) = F0 (k) + Fb (k). The injection of enstrophy due to the stochastic backscatter term may contribute to the growth of instabilities, which may be suppressed in lower resolution LES unless the ﬂow is randomly forced or suitably chaotic (Leith;37 Piomelli et al.47 ). Most atmospheric circulation models have however not accounted for the stochastic backscatter term, but have tried to account for the diﬀerences between the drain and injection terms by an effective or net viscosity. This may be achieved as in Frederiksen and Davies11 by deﬁning a (negative) eddy backscatter viscosity νb through νb (k) = −[k(k + 1)Ck (t, t)]−1 SkS , so that, Fb (k) = −2νb (k)k(k + 1)Ck . The net eddy viscosity is deﬁned by νn (k) = νd (k) + νb (k), and the renormalized net viscosity by νrn (k) = νn (k) + ν0 (k). 11.2. The DIA closure For the DIA closure the viscosities again have the relationships deﬁned for the EDQNM closure but with the eddy drain viscosity deﬁned by t νd (k) = [k(k + 1)Ck (t, t)]−1 dsηkS (t, s)Ck (t, s), t0

the stochastic backscatter by

t

Fb (k) = 2 t0

ds SkS (t, s)Rk (t, s),

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and the eddy backscatter viscosity by νb (k) = −[k(k + 1)Ck (t, t)]−1

t

t0

ds SkS (t, s)Rk (t, s)

11

(Frederiksen and Davies ). Here the integral terms of course come from the integrals in the DIA closure equation 19. 12. Comparisons of DNS and LES with subgrid-scale parameterizations Frederiksen and Davies11 compared DNS with LES at various resolutions and including dynamic subgrid-scale parameterizations as deﬁned in section 11. They focused on DNS simulations which reproduced the observed January 1979 isotropized kinetic energy spectrum as a statistically stationary spectrum in the presence of random forcing and dissipation. The DNS runs have been performed at triangular T63 resolution with the dissipation taken as a linear combination of surface drag and a Laplacian dissipation for which the nondimensional bare viscosity takes the form  1.014 × 10−2   for 2 ≤ k ≤ 15,   k(k + 1) (28) ν0 (k) =   1.014 × 10−2  −5  + 4.223 × 10 for 16 ≤ k ≤ 63. k(k + 1) The drag corresponds to 7.4 × 10−7 s−1 , or an e-folding decay time of 15.6 days, and the Laplacian contribution corresponds to 1.25 × 105 m2 s−1 in dimensional units. To determine the random forcing variance spectrum needed to balance the dissipation, the steady state EDQNM equation is used as follows. With the enstrophy components speciﬁed by the T63 January 1979 isotropic spectrum, and the triad relaxation time given by the stationary form in (27b), F0 (k) must be speciﬁed by F0 (k) = (2ν0 (k)k(k + 1) + 2ηk ) Ck − 2Sk

(29)

for a steady state solution to (25). The bare viscosity in (28) and bare forcing speciﬁed by (29) (with γ = 0.6) when used in DNS of the barotropic vorticity equation also produce a statistically steady state which is very close to the January 1979 spectrum. This is shown in Figs. 8 and 9 (redrawn from Frederiksen and Davies11 ) which, at T63, compare the results of a DNS run (including rotation B=2) averaged over the last 100 days of a 150-day simulation with the January 1979 spectrum. Fig. 8 shows the dimensional

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Fig. 8. Kinetic energy spectra E(m) in m2 s−2 for (A) isotropized January 1979, (B) DNS at T63, (C) LES at T31 with B=2, and (D) E(m) ± Σ(m) for DNS or LES. LES results with EDQNM-based renormalized stochastic backscatter and renormalized viscosity (×10−1 ) and LES with EDQNM-based renormalized net viscosity (×10−2 ). Note that results for m = 0 are plotted at 10−1 . Integrations use B = 2.

zonal wavenumber kinetic energy spectrum E(m) while Fig. 9 shows the dimensional total wavenumber kinetic energy spectrum e(n). Also shown are these spectra ± the standard deviations Σ(m) and σ(n). The zonal wavenumber spectra are in close agreement at most scales while for the total wavenumber spectra the agreement at the largest scales, where there are few m components to average over and where the eddy turn-over time is long, is not as good as at small scales. The DNS spectra in Figs. 8 and 9, or, almost equivalently, the T63 January 1979 spectra, are regarded as the benchmark or truth against which LES at lower resolution are to be compared. The various EDQNM subgrid-scale terms deﬁned in section 11, are shown in Figs. 10 and 11 (redrawn from Figs. 3b and d of Frederiksen and Davies11 ) for the case when the initial T63 January 1979 spectrum is truncated back to T31. Fig. 10 depicts the eddy viscosities νd (n), νb (n) and νn (n) while Fig. 11 shows the net dissipation function νn (n)n(n + 1). We note that these subgrid-scale parameterizations all have a cusp behaviour at

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Fig. 9.

As in Fig. 8 for e(n) shown as (A), (B), (C) and e(n) ± σ(n) shown as (D).

the smallest scales as is also characteristic of eddy viscosity parameterizations for three-dimensional turbulence in Cartesian geometry (Kraichnan;34 Chasnov4 ). Very similar subgrid-scale parameterizations are obtained when the January 1979 spectrum is truncated back to other lower resolutions as shown at T15 in Fig. 4 of Frederiksen and Davies.11 Figs. 8 and 9 also compare the spectra of LES using the renormalized viscosity νr (n) = νd (n) + ν0 (n) and renormalized random forcing Fr (n) = Fb (n)+F0 (n) at resolution T31. Again the kinetic energy spectra shown are averaged over the last 100 days of 150-day simulations. The diagrams also show the spectra ± the standard deviation Σ(m) and σ(n) and the energy spectra of the T63 DNS, and January 1979 observations, truncated back to T31. For the zonal wavenumber spectra, there is good agreement between the LES kinetic energies at T31 and both the DNS and January 1979 results truncated to the resolution of the LES. For the total wavenumber spectra, the agreement between LES and DNS or January 1979 kinetic energies is good at the smaller scales; at the larger scales there are some deviations, presumably because of the few m components to average over and the long eddy turn-over times.

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Fig. 10. EDQNM nondimensional subgrid-scale parametrizations for T31. Shown are viscosities corresponding to (A) νd (n), (B) νb (n), and (C) νn (n).

Frederiksen and Davies11 show that the generally good agreement between DNS and LES using the EDQNM based renormalized viscosity and renormalized noise forcing is also found if the EDQNM based renormalized net viscosity is used instead or if the corresponding very similar DIA based parameterizations are employed. This is also the situation in the presence of a diﬀerential rotation speed characteristic of the earth. They also ﬁnd that the comparison between DNS and LES is much better than when a number of ad hoc viscosity parameterizations are used in the LES. 13. Discussion and conclusions We have reviewed recent developments in renormalized closure theory and its application for developing subgrid-scale parameterizations for two dimensional turbulence. It has recently become possible to generalise the statistical closure theories for homogeneous turbulence to inhomogeneous turbulence and thus to tackle directly the statistical properties of more realistic ﬂows. The quasi-

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Fig. 11.

As in Fig. 10 for νn (n)n(n + 1).

diagonal direct interaction approximation (QDIA) closure has been formulated and applied to the interaction of general mean ﬂows and topography with inhomogeneous two-dimensional and Rossby wave turbulence (Frederiksen,9 O’Kane and Frederiksen,44 Frederiksen and O’Kane17 ). For both homogeneous turbulence and inhomogeneous turbulence it has been found that a regularization process in which transfers are localized in wavenumber space can overcome the small scale deﬁciencies of direct interaction approximation (DIA) type closures and that the localization or cut-oﬀ parameter appears to be almost universal. For a wide range of large-scale Reynolds numbers and ﬂow conﬁgurations it is found that there is in general very good agreement between the regularized DIA and QDIA closures and the statistics of DNS, where sampling problems do not aﬀect the DNS results. Dynamical subgrid-scale parameterizations based on either DIA type closures or corresponding Markovianized versions have been successful when applied to large eddy simulations (LES) in that the statistics of the LES agree closely with the statistics of higher resolution direct numerical simulations (DNS) at the scales of the LES. These closure based subgrid-scale

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parameterizations have also been used to improve the mean climate and transient atmospheric circulations in more complex global climate models (Frederiksen, Dix, and Davies15 ). Recently, Frederiksen and Kepert16 have developed similarly successful subgrid-scale parameterizations by using stochastic modelling methods and transforming the statistics of DNS into a form similar to the non-Markovian closures. Further applications of the regularized QDIA closure to ensemble prediction studies and to data assimilation are presented in the following chapter by O’Kane and Frederiksen. Appendix A. Two- and three point cumulants and response function equations The QDIA approach uses renormalized perturbation theory to derive the following expression for the oﬀ-diagonal elements of the covariance matrix purely in terms of diagonal cumulant and response functions:

QDIA (t, t ) = Ck,−l

t

ds Rk (t, s)Cl (s, t ) ×

t0

[A(k, −l, l − k)h(k−l) + 2K(k, −l, l − k)ζ(k−l) (s)]

t

+

ds R−l (t , s)Ck (t, s) ×

t0

[A(−l, k, l − k)h(k−l) + 2K(−l, k, l − k)ζ(k−l) (s)] ˜ (2) (t0 , t0 ) + Rk (t, t0 )R−l (t , t0 )K k,−l

(A.1)

(2)

˜ where K k,−l (t0 , t0 ) is the contribution to the oﬀ-diagonal covariance matrix at initial time t0 .44 The response function may also be written in purely diagonal terms where the oﬀ-diagonal elements of the response function take the form

QDIA (t, t ) = Rk,l

t

t

ds Rk (t, s)Rl (s, t ) ×

[A(k, −l, l − k)h(k−l) + 2K(k, −l, l − k)ζ(k−l) (s)]. (A.2) To close (14) we also need an expression for the three-point cumulant and this is derived in the same way as for the DIA closure for homogeneous

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turbulence: ζˆ−l (t)ζˆ(l−k) (t)ζˆk (t )QDIA =

t

2

ds K(k, −l, l − k)C−l (t, s)C(l−k) (t, s)Rk (t , s)

t0

t

+2 t0 t

+2

ds K(−l, l − k, k)R−l (t, s)C(l−k) (t, s)Ck (t , s) ds K(l − k, −l, k)R(l−k) (t, s)C−l (t, s)Ck (t , s)

t0

˜ (3) +R−l (t, t0 )R(l−k) (t, t0 )Rk (t, t0 )K −l,(l−k),k (t0 , t0 , t0 )

(A.3)

(3)

˜ where K −l,(l−k),k (t0 , t0 , t0 ) allows for non-Gaussian initial conditions. The evolution of ζk (t) and Ck (t, t ) can now be expressed by substitution of (A.1), (A.2) and (A.3) into (14) and (12). The prognostic equation (18) for the response function Rk (t, t ) is obtained by using (A.2) and the fact that ˆ l−k (t, t )ζˆ−l (t)QDIA R t =2 ds K(l − k, −l, k)Rl−k (t, s)C−l (t, s)Rk (s, t ). t

References 1. G.J. Boer and T.G. Shepherd, Large-scale two-dimensional turbulence in the atmosphere. J. Atmos. Sci. 40, 164–184 (1983). 2. J.C. Bowman, J.A. Krommes and M. Ottaviani, The realizable Markovian closure I. General theory, with applications to 3-wave dynamics. Phys. Fluids 5, 3558–3589 (1993). 3. J.C Bowman and J.A. Krommes, The realizable Markovian closure and realizable test-ﬁeld model II. Application to anisotropic drift-wave dynamics. Phys. Plasmas 4, 3895–3909 (1997). 4. J.R. Chasnov, Simulation of the Kolmogorov inertial subrange using an improved subgrid model. Phys. Fluids A 3, 188–200 (1991). 5. G.F. Carnevale and J.S. Frederiksen, Viscosity renormalization based on direct-interaction closure. J. Fluid Mech. 131(289), 289–303 (1983). 6. W. Edelmann, Numerical experiments with a barotropic current across mountains. Beitr. Phys. Atmos. 45, 196–229 (1972). 7. J. Egger, On the simulation of subgrid orographic eﬀects in numerical forecasting. J. Atmos. Sci. 27, 896–902 (1970). 8. J.S. Frederiksen, Eastward and westward ﬂows over topography in nonlinear and linear barotropic models. J. Atmos. Sci. 39, 2477–2489 (1982). 9. J.S. Frederiksen, Subgrid-scale parameterizations of eddy-topographic force, eddy viscosity, and stochastic backscatter for ﬂow over topography. J. Atmos. Sci. 56, 1481–1494 (1999).

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10. J.S. Frederiksen, Renormalized closure theory and subgrid-scale parameterizations for two-dimensional turbulence. Nonlinear Dynamics: From Lasers to Butterﬂies World Scientiﬁc Lecture Notes in Comples Systems, Vol. 1, p452, Editors Ball, R. and Akhmediev, N., 6, 225–256 (2003). 11. J.S. Frederiksen and A.G. Davies, Eddy viscosity and stochastic backscatter parameterizations on the sphere for atmospheric circulation models. J. Atmos. Sci. 54, 2475–2492 (1997). 12. J.S. Frederiksen and A.G. Davies, Dynamics and spectra of cumulant update closures for two-dimensional turbulence. Geophys. Astrophys. Fluid Dyn. 92, 197–231 (2000). 13. J.S. Frederiksen and A.G. Davies, The regularized DIA closure for twodimensional turbulence. Geophys. Astrophys. Fluid Dyn. 98, 203–223 (2004). 14. J.S. Frederiksen, A.G. Davies and R. Bell, Closure equations with nonGaussian restarts for truncated two-dimensional turbulence. Phys. Fluids 6, 3153–3163 (1994). 15. J.S. Frederiksen, M. Dix and A.G. Davies, The eﬀects of closure based eddy diﬀusion on the climate and spectra of a GCM. Tellus 55A, 31–44 (2003). 16. J.S. Frederiksen and S.M. Kepert, Dynamical subgrid-scale parameterizations from direct numerical simulations. J. Atmos. Sci. 16, 3006–3019 (2006). 17. J.S. Frederiksen and T.J. O’Kane, Inhomogeneous closure and statistical mechanics for Rossby wave turbulence over topography. J. Fluid Mech. 539, 137–165 (2005). 18. J.S. Frederiksen and B.L. Sawford, Statistical dynamics of two-dimensional inviscid ﬂow on a sphere. J. Atmos. Sci. 37, 717–732 (1980). 19. D. Fultz and R.R. Long, Two-dimensional ﬂow around a circular barrier in a rotating spherical shell. Tellus 3, 61–68 (1951). 20. D. Fultz, R.R. Long and P. Frenzen, A note on certain interesting ageostrophic motions in a rotating hemispherical shell. J. Meteor. 12, 323–338 (1955). 21. W.L. Grose and B.J. Hoskins, On the inﬂuence of orography on large-scale atmospheric ﬂow. J. Atmos. Sci. 36, 223–234 (1979). 22. J.R. Herring, Self-consistent-ﬁeld approach to turbulence theory. Phys. Fluids 8, 2219–2225 (1965). 23. J.R. Herring and R.H. Kraichnan, A numerical comparison of velocity-based and strain based Lagrangian-history turbulence approximations. J. Fluid Mech. 91, 581–597 (1979). 24. J.R. Herring, S.A. Orszag, R.H. Kraichnan and D.G. Fox, Decay of twodimensional homogeneous turbulence. J. Fluid Mech. 66, 417–444 (1974). 25. R.V. Jensen, Functional integral approach to classical statistical dynamics. J. Stat. Phys. 25, 183-210 (1981). 26. A. Kasahara, The dynamical inﬂuence of orography on the large scale motion of the atmosphere. J. Atmos. Sci. 23, 259–270 (1966). 27. R.H. Kraichnan, The structure of isotropic turbulence at very high Reynolds numbers. J. Fluid Mech. 5, 497–543 (1959). 28. R.H. Kraichnan, Classical ﬂuctuation-relaxation theorem. Phys. Rev. 113, 1181–1182 (1959).

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29. R.H. Kraichnan, Decay of isotropic turbulence in the direct interaction approximation. Phys. Fluids 7, 1030–1048 (1964). 30. R.H. Kraichnan, Diagonalizing approximation for inhomogeneous turbulence. Phys. Fluids 7, 1169–1177 (1964). 31. R.H. Kraichnan, An almost-Markovian Galilean-invariant turbulence model. J. Fluid Mech. 47, 513–524 (1971). 32. R.H. Kraichnan, Inertial-range transfer in two and three dimensional turbulence. J. Fluid Mech. 47, 525–535 (1971). 33. R.H. Kraichnan, Test-ﬁeld model for inhomogeneous turbulence. J. Fluid Mech. 56, 287–304 (1972). 34. R.H. Kraichnan, Eddy viscosity in two and three dimensions. J. Atmos. Sci. 33, 1521–1536 (1976). 35. J.A. Krommes Analytical Descriptions of Plasma Turbulence, in Lecture notes on turbulence and coherent structures in ﬂuids, plasmas and nonlinear media (Eds. M. Shats and H. Punzmann), World Scientiﬁc 2006. 36. C.E. Leith, Atmospheric predictability and two-dimensional turbulence. J. Atmos. Sci. 28, 145–161 (1971). 37. C.E. Leith, Stochastic backscatter in a subgrid-scale model: Plane shear mixing layer. Phys. Fluids A 2, 297–299 (1990). 38. C.E. Leith and R.H. Kraichnan, Predictability of turbulent ﬂows. J. Atmos. Sci. 29, 1041–1058 (1972). 39. P.C. Martin, E.D. Siggia and H.A. Rose, Statistical dynamics of classical systems. Phys. Rev. A 8, 423–437 (1973). 40. W.D. McComb, A local energy-transfer theory of isotropic turbulence. J. Phys. A 7, 632–649 (1974). 41. W.D. McComb, The Physics of Fluid Turbulence. Clarendon Press, Oxford (1990). 42. W.D. McComb Renormalization and Statistical Methods, in Lecture notes on turbulence and coherent structures in ﬂuids, plasmas and nonlinear media (Eds. M. Shats and H. Punzmann), World Scientiﬁc 2006. 43. W.D. McComb, M.J. Filipiak and V. Shanmugasundaram, Rederivation and further assesment of the LET theory of isotropic turbulence, as applied to passive scalar convection. J. Fluid Mech. 245, 279–300 (1992). 44. T.J. O’Kane, and J.S. Frederiksen, The QDIA and regularized QDIA closures for inhomogeneous turbulence over topography. J. Fluid Mech. 504, 133–165 (2004). 45. S.A. Orszag, Analytical theories of turbulence. J. Fluid Mech. 41, 363–386 (1970). 46. R. Phythian, The functional formalism of classical statistical dynamics J. Phys. A 10, 777-789 (1977). 47. U. Piomelli, W.H. Cabot, P. Mion and S. Lee, Subgrid-scale backscatter in turbulent and transitional ﬂows. Phys. Fluids A. 3, 1766–1771 (1991). 48. H.A. Rose, An eﬃcient non-Markovian theory of non-equilibrium dynamics. Physica D 14, 216–226 (1995). 49. I. Vergeiner and Y. Ogura, A numerical shallow ﬂuid model including orography with a variable grid. J. Atmos. Sci. 29, 270–284 (1972).

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50. J. Verron, and C. Le Provost, A numerical study of quasi-geostrophic ﬂow over isolated topography. J. Fluid Mech. 154, 231–252 (1985). 51. W.H. Wyld, Formulation of the theory of turbulence in an incompressible ﬂuid. Ann. Phys. 14, 143–165 (1961).

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STATISTICAL DYNAMICAL METHODS OF ENSEMBLE PREDICTION AND DATA ASSIMILATION DURING BLOCKING TERENCE J. O’KANE AND JORGEN S. FREDERIKSEN CSIRO Marine and Atmospheric Research 107-121 Station St, Aspendale, Australia Terence.O’[email protected] In this chapter we discuss the application of inhomogeneous statistical closure models to ensemble prediction and data assimilation. The quasi-diagonal direct interaction approximation (QDIA) represents a tractable closure for investigating general turbulent geophysical ﬂuids. The QDIA may also be used to examine ensemble prediction methods and to develop novel statistical dynamical data assimilation methods. Using the QDIA and the method of bred perturbations we ﬁrst examine the role of non-Gaussian terms in ensemble prediction. We demonstrate both the importance of the cumulative contribution of non-Gaussian correlations to the evolved error tendency as well as the role of the covariances in the small scale instantaneous errors, thereby extending the work of Epstein13 and Fleming19,20 to models without any explicit cumulant discard hypothesis. Further, the QDIA closure is shown to be valid for both strongly non-Gaussian and strongly inhomogeneous ﬂows. The formal similarities between statistical turbulence theory and ensemble averaged data assimilation techniques leads naturally to the development of closure based spectral data assimilation techniques. In the second part of this chapter we compare ensemble averaged Kalman (stochastic), square root (deterministic) and statistical dynamical data assimilation methodologies, examining the use of random and ﬂow correlated observational error perturbations and issues related to sampling error. Forecast error covariances, typically under-predicted in the ensemble Kalman ﬁlter (EnKF), are shown to occur largely due to sampling error resulting from the traditional use of random observational error perturbations. Flow correlated observational errors are shown to be demonstrably preferable for generating forecast error perturbations in order to correctly capture error evolution. Atmospheric regime transitions are often associated with the formation of large scale coherent structures, commonly known as blocks. These blocking events are typically associated with rapidly growing ﬂow instabilities that lead to a loss of predictability and enhanced error growth. Results from the QDIA statistical closure and ensemble averaged direct numerical simulations are presented speciﬁcally for Northern Hemisphere ﬂow during the formation of a series of large and persistent blocking events between mid-October and mid-November 1979.

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1. Introduction Flow instabilities, analysis errors and model deﬁciencies determine the predictability of atmospheric ﬂows. Early work aimed at establishing the theoretical limits to predicting the state of the atmosphere considered the divergence of pairs of initially close states for deterministic forecasts.9,39,63 More recently weather forecasting has become to be regarded as a statistical problem where one aims to forecast the atmospheric state probability density function or alternately calculate the moments of typical meteorological variables. Early predictability studies focused on two-dimensional turbulent ﬂows. The studies of Kraichnan and Leith42,44,45 pioneered the use of renormalized self-consistent second-order closure models in an examination of evolving error spectra and their convergence towards typical climatological atmospheric variance spectra. Low resolution stochastic dynamical models were also developed by Epstein13 and Fleming19,20 to directly calculate the ﬁrst (mean) and second (variances) moments, although computationally more eﬃcient Monte Carlo methods were subsequently used to approximate the stochastic dynamic model in order to reach higher resolution. In these stochastic methods moments higher than second order were typically discarded. In the ﬁrst part of this chapter (2) we show how the advent of tractable inhomogeneous statistical dynamical closure models,22,27,53 discussed in chapter 14, enable us to quantify the eﬀects of non-Gaussian correlations and ﬂow instabilities on the predictability of the atmosphere. Sampling is most often cited as the limiting factor in determining the statistics of error prediction. Sampling error is typically associated with an insuﬃcient number of realizations i.e. when the sample size is less than the number of degrees of freedom. It may also be caused by poor or inadequate observational data. In order to gather speciﬁc information about the probability distributions of atmospheric forecast models, ensemble data assimilation techniques such as the ensemble Kalman ﬁlter17 (EnKF) were developed. As with ensemble prediction methods the EnKF is inherently susceptible to sampling error. The use of randomly perturbed observations introduces uncorrelated noise and sampling error into the assimilated ensemble destroying prior covariances between model state variables thus removing information about the higher order moments. The EnKF propagates an ensemble of model states with a fully nonlinear model allowing the error covariance matrix to be calculated with no moment closure required. This allows the construction of the forecast error variance at any given time by averaging over the ensemble. The focus of the second part of this chapter

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(section 10) is the development and performance of a novel statistical dynamical deterministic ﬁlter with no sampling error to rigorously test the eﬀects of uncorrelated and ﬂow correlated observational noise on data assimilation and the generation of initial forecast perturbations. The closure based ﬁlter is compared with the EnKF and deterministic square root ﬁlters (EnSF). As with the statistical closure method we apply the quasi-diagonal assumption to both the EnKF and EnSF so that the oﬀ-diagonal and higher order elements are represented in terms of renormalized diagonal elements and develop as the ensemble evolves. In section 3 we brieﬂy describe the spectral barotropic or vorticity form of the 2-D Navier-Stokes equation for ﬂow over topography, including Rossby wave turbulence, on a generalized β-plane and in the presence of a large scale ﬂow U . A more detailed explanation of the generalized βplane, conserved quantities, the statistical mechanical equilibrium theory and nonlinear stability theory for ﬂow on the generalized β-plane can be found in Frederiksen and O’Kane.27 The moment closure problem and form of the QDIA closure equations and with additional contributions from the initial oﬀ-diagonal covariance matrix and from initial non-Gaussian terms have been discussed in detail in chapter 14 and in detail by O’Kane and Frederiksen27,53 and we refer the interested reader to these references for the details. Section 5 describes the diagnostic quantities used throughout the chapter. In sections 6 we outline our approach to ensemble prediction using the RQDIA closure and variants. In 7 the method of bred perturbations is outlined then in 8 ensemble prediction studies are performed in which comparisons are made between closure and DNS for error growth studies with a 5 day cycle during which bred perturbations are generated followed by a 5 day forecast period. The numerical experiments are relaxed toward interpolated daily 500mb observed streamfunction ﬁelds starting on the 26th October 1979 for the Northern Hemisphere. In this period a very strong high/low blocking dipole formed over the Gulf of Alaska on the 5th November 1979 and so our ensemble prediction study covers the period of strong instability associated with the development and maturation of this large scale coherent structure. In addition to small scale inhomogeneity the eﬀect of non-Gaussian correlations and variances on error growth is quantiﬁed by comparing four variants of the QDIA closure; the ﬁrst includes contributions from the variances, non-Gaussian correlations and higher order moments (QDIA). The second is a regularized variant of the QDIA (RQDIA). The third removes information due to the two- and three-point time history

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integrals and the fourth is based on the cumulant discard hypothesis. The discussion and conclusions are contained in section 9. In section 11 statistical dynamical Kalman ﬁlter methodologies are described. In section 12 we state the functional statistical dynamical Kalman ﬁlter (QSDKF) in terms of the QDIA closure equations. In section 13 and 14 the ensemble Kalman and ensemble square root ﬁlters are described. In the section 15.1 we compare the ensemble Kalman ﬁlter (EnKF) method to the QSDKF model in an examination of data assimilation for the same blocking event over the Gulf of Alaska considered in the ensemble prediction studies. In section 15.2 we extend the comparisons to include ensemble square root ﬁlter results. We also describe a 30 day assimilation using the QEnSF during a period in which 3 large scale regime transitions occurred. The discussion and conclusions are presented in section 16. 2. Ensemble prediction Epstein’s12,13 stochastic dynamic prediction model uses probabilistic representations of the initial conditions to solve the deterministic equations that govern atmospheric ﬂows as a means to gather information on the forecast uncertainty. Due to nonlinearity and the presence of instabilities initial forecast errors grow causing deterministic forecasts to fail over reasonable prediction periods. Epstein’s13 stochastic dynamic model solves an inhomogeneous anisotropic model of the statistical hydrodynamics directly forecasting the mean and variance but discarding the third and higher order moments in order to incorporate probabilistic information into a deterministic forecast. The low resolution performance of the stochastic dynamic model compared with the non-realizable quasi-normal (QN: Millionshtchikov51 and Proudman and Reid59 ) and eddy-damped quasinormal (EDQN: Orszag56) closure schemes as well as Monte Carlo (MC) based ensemble prediction and deterministic methods was examined by Fleming.19 The stochastic dynamic model was found to only be valid for very short range predictions of the order of a few days deviating signiﬁcantly from the Monte-Carlo (MC) solution after ≈ 13 days. In contrast the EDQN with empirical damping of the three-point non-Gaussian correlations was in very close agreement with the MC solution out to ≈ 18 days. Subsequent stochastic prediction studies of error growth in a spectral barotropic model were carried out by Pitcher58 and Epstein14 for atmospheric data (500mb ﬂow pattern) with random forcing. Leith46 also compared the predictability of MC approximations to the stochastic prediction technique of Epstein against Kraichnan’s test ﬁeld model (TFM) (Leith and

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Kraichnan45) for atmospheric ﬂows. The principal contribution of the stochastic dynamic methods to the problem of atmospheric forecasting was to enable the prediction of the uncertainty in the ensemble mean. Kraichnan’s direct interaction approximation (DIA)40,41 marks the beginning of modern statistical turbulence closure theory. Herring’s self consistent ﬁeld theory (SCFT32 ) and McComb’s local energy transfer theory (LET49 ) were later found to be related to Kraichnan’s DIA through diﬀering applications of the ﬂuctuating dissipation theorem. Herring33 and Holloway35 developed statistical closures for homogeneous turbulence interacting with ensembles of zero mean random topography. The ﬁrst closure theory for inhomogeneous turbulent ﬂow over non-zero mean topography was developed by Frederiksen.22 Frederiksen’s quasi-diagonal direct interaction approximation (QDIA) was formulated for barotropic ﬂows on an f -plane using a quasi-diagonal approximation for the covariances and Kraichnan’s DIA closure to treat the higher order non-Gaussian correlations. Computationally tractable variants of the QDIA were developed and their performance examined in comparison with ensemble averaged DNS by O’Kane and Frederiksen.53 Frederiksen and O’Kane27 extended the methods to include Rossby waves eﬀects at the large scales as well as strong turbulence eﬀects at the small scales. Because the QDIA has analytic expressions for the two- and three-point correlations it enables an improvement not just on the third moment discard hypothesis but also on empirically damped approximations to the higher order moments. The QDIA is realizable unlike the EDQN closures (see Fleming19 ) and naturally incorporates time history information into the growing error ﬁelds. Importantly this statistical dynamical model has no sampling error. In comparison to a single realization control forecast, ensemble averaged forecasts provide estimates of the forecast error variance and possibly the higher order moments. In ensemble prediction methods one generates independently perturbed initial conditions such that the covariance of the ensemble perturbations is approximately equal to that of the initial analysis error covariance matrix and similarly at the time of the forecast. The method of bred perturbations developed by Toth and Kalnay67 enables forecast perturbations to be transformed into analysis perturbations through the use of a globally or regionally constant scaling factor with magnitude less than 1. Thus, in the case of the atmosphere, the error growth associated with the evolving state develops within the breeding cycle and subsequently dominates the forecast error growth. That is, breeding is a method for incorporating information about the fastest growing errors into the initial

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conditions for the forecast. Bred vectors have been shown to be superpositions of the leading local time dependent Lyapunov vectors (LLV’s) of the ﬂow.67 In the atmosphere random perturbations will, after an initial transient period (≈ 7 days), converge on the structure of the LLV’s and for low order span the system. In dynamic ﬂows errors typically arise due to fast growing large scale ﬂow instabilities. In turbulent ﬂows such instabilities are often associated with the emergence of coherent structures and in the atmosphere these coherent structures are called blocks. In addition to the stochastic dynamic and statistical closure models discussed above, low order models (Lorenz,48 Anderson1 ), barotropic models (Frederiksen and Bell;23 Molteni and Palmer52 ) and simpliﬁed and low vertical resolution baroclinic models (Farrell;18 Frederiksen and Bell;23 Molteni and Palmer;52 Houtekamer and Derome36 ) have all been used to study atmospheric predictability. In this section we examine perturbation or ‘error’ growth during a period of block formation, maturation and decay in the Northern Hemisphere in November 1979. Error structures in the early stages of block development may have a baroclinic component however, a large component of large scale error growth can be described by barotropic dynamics particularly during the more mature phase of blocking (Veyre;68 Frederiksen21 ). In the later sections we examine these processes by separating the two- and three-point contributions to the diagonal cumulant tendency equation. We also test the validity of the cumulant discard hypothesis over short forecast periods by examination of the relative contributions of both instantaneous and cumulative contributions from the non-Gaussian correlations to error growth. 3. Barotropic ﬂow on a β−plane We base our studies of both ensemble prediction and the formulation of statistical dynamical prediction on the generalized β-plane barotropic model for ﬂow over topography described in detail by Frederiksen and O’Kane.27 The full streamfunction is written in the form Ψ = ψ − U y, where U is a large-scale east-west ﬂow and ψ represents the small scales. The evolution equation for the two-dimensional motion of the small scales over a mean topography is then described by the barotropic vorticity equation ∂ζ = −J(ψ − U y, ζ + h + βy + k02 U y) + νˆ2 ζ + f 0 . (1a) ∂t Here f 0 is the forcing, νˆ the viscosity and ∂ψ ∂ζ ∂ψ ∂ζ − (1b) J(ψ, ζ) = ∂x ∂y ∂y ∂x

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is the Jacobian. The vorticity is the Laplacian of the streamfunction where H is the height of the topography, R = ζ = 2 ψ, h = 2µgAH RT0 287Jkg −1K −1 is the gas constant for air, T0 = 273K is the horizontally averaged global surface temp, g is the acceleration due to gravity, µ = sin(λ), λ is the latitude and A = 0.8 is the value of the vertical proﬁle factor. The term k02 U y generalises the standard β-plane by the inclusion of an eﬀect corresponding to the solid-body rotation vorticity in spherical geometry where k0 is a wave number that speciﬁes the strength of this large-scale vorticity. Frederiksen and O’Kane27 noted that this additional small term results in a one-to-one correspondence between the dynamical equations, Rossby wave dispersion relations, nonlinear stability criteria and canonical equilibrium theory on the generalized β-plane and on the sphere. The barotropic vorticity equation and the form-drag equation for U can be made nondimensional by introducing suitable length and time scales, which we choose to be a/2, where a is the earth’s radius, and Ω−1 , the inverse of the earth’s angular velocity. With this scaling we consider ﬂow on the domain 0 ≤ x ≤ 2π, 0 ≤ y ≤ 2π. The form-drag equation for the large-scale ﬂow U is the same as on the standard β−plane. With the inclusion of relaxation towards the state U it takes the form 1 ∂ψ ∂U = dS + α(U − U ). h (2) ∂t S S ∂x Here, α is a drag coeﬃcient and S is the area of the surface 0 ≤ x ≤ 2π, 0 ≤ y ≤ 2π. In the absence of forcing and dissipation, Eqs. (1) and (2) together conserve kinetic energy and potential enstrophy. We derive the corresponding spectral space equations by representing each of small-scale terms by a Fourier series; for example ζk (t) exp (ik · x) ζ(x, t) = k

where 1 ζk (t) = (2π)2

2π

d2 x ζ(x, t) exp (−ik · x)

0

and x = (x, y), k = (kx , ky ), k = (kx2 + ky2 )1/2 and ζ−k is conjugate to ζk . As noted in Eq. (4.1) of Frederiksen and O’Kane27 the sums in the consequent spectral equations run over the set R consisting of all points in discrete wavenumber space except the point (0,0). However, it was also observed that the form-drag equation for U can be written in the same form as for the small scales by deﬁning suitable interaction coeﬃcients,

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representing the large-scale ﬂow as a component with zero wavenumber and extending the sums over wavenumbers. The spectral form of the barotropic vorticity equation with diﬀerential rotation, describing the evolution of the ‘small scales’, and the form-drag equation, may then be written in the same compact form as for the f −plane:

∂ + ν0 (k)k 2 ζk (t) = δ(k + p + q) [K(k, p, q)ζ−p ζ−q ∂t p∈T q∈T

+A(k, p, q)ζ−p h−q ] + fk0

(3)

where T = R ∪ 0 and the interaction coeﬃcients are deﬁned in the chapter 14. Here the complex ν0 (k) is related to the non-dimensional viscosity νˆ and the intrinsic Rossby wave frequency ωk by the expression: ν0 (k)k 2 = νˆk 2 + iωk where βkx . k2 ∗ We have deﬁned ζ−0 = ik0 U , ζ0 = ζ−0 and introduced a term h−0 that we take to be zero but which could more generally be related to a large-scale topography. We note that U is real and we have deﬁned ζ0 to be imaginary. This is done to ensure that all the interaction coeﬃcients that we use are deﬁned to be purely real. Also with ζ 0 = −ik0 U , f00 and ν0 (k0 ) are deﬁned by ωk = −

f00 = αζ 0 ,

ν0 (k0 )k02 = α.

These spectral equations are then the basis for our subsequent studies and theoretical developments. We can also consider the case where the bare forcing is replaced by a relaxation term of the form γ(ζobs − ζ) where γ is the strength of the relaxation and ζobs are linearly interpolated daily observed ﬁelds. 4. The QDIA closure equations with memory eﬀects The QDIA closure equations were derived by Frederiksen22 for general barotropic mean ﬂows interacting with inhomogeneous turbulence over topography on an f -plane. O’Kane and Frederiksen53 tested the performance of the closure including cumulant update and regularized variants while the generalisation to Rossby wave turbulence on a β-plane was formulated

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and tested by Frederiksen and O’Kane.27 The β-plane QDIA model and variants are discussed in some detail in chapter 14 and we simply assume the forms in this chapter. 5. Diagnostics Next, we deﬁne a number of diagnostics that we employ throughout this chapter for analyzing both the predictability and data assimilation studies. We deﬁne the zonally averaged transient (error) eT (kx , t) and mean eM (kx , t) kinetic energy spectra by eT (kx , t) =

1 [Ck (t, t)] /k 2 , 2 ky

1 eM (kx , t) = [ζk (t)ζ−k (t)] /k 2 2 ky

where Ck (t, t) = ζˆk (t)ζˆ−k (t) and denotes ensemble average. The kinetic energy of the large-scale ﬂow is plotted at kx = 0. We shall also use a measure based on palinstrophy production to characterise the error growth during forecasts. For this it is useful to separate the enstrophy production terms into the contribution from the two-point inhomogeneous N I (t, t), and three-point non-Gaussian N S (t, t), terms. The palinstrophy production measure based on the three-point non-Gaussian terms is just the skewness S K that has been commonly used to examine the small scale behaviour of homogeneous25,34 and inhomogeneous turbulence.53 The current regime of small amplitude transient errors growing on larger amplitude mean ﬂows however diﬀers from these previous studies in that most of the transfer results from the two-point inhomogeneous production. One of our aims will be to quantify the relative contribution of the two-point inhomogeneous and three-point and higher order non-Gaussian terms to both the instantaneous and cumulative error growth. Examination of both the palinstrophy production and kinetic energy spectra allows this quantiﬁcation. ˆ transient palinstroWe begin by specifying the transient enstrophy F, phy Pˆ and palinstrophy production K by 1 ˆ = 1 Ck (t, t), P(t) Ck (t, t)k 2 , Fˆ (t) = 2 2 k k I S I K(t) = K (t) + K (t), K (t) = k 2 NkI (t, t) KS (t) = k 2 NkS (t, t). k

k

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Then the palinstrophy production measure P M can be deﬁned in terms of these quantities by P M (t) = 2K/(Pˆ Fˆ 1/2 ) = I K (t) + S K (t)

(4)

I K (t) = 2KI /(Pˆ Fˆ 1/2 ), S K (t) = 2KS /(Pˆ Fˆ 1/2 ).

(5a)

where

(5b)

6. Error growth during blocking using initial bred perturbations The ensemble predictability over the period towards the end of October up to the 12th of November 1979 for 500-hPa ﬂow in the Northern Hemisphere is of particular interest due to the formation (5th November), maturation and decay (12th November) of a large-scale blocking high-low dipole over the Gulf of Alaska. This blocking event has been chosen because it was particularly large and persistent and its dynamics are well understood having been examined previously by Frederiksen, Collier and Watkins28 using two diﬀerent general circulation models (GCMs) with a breeding scheme to generate initial perturbations. Our method54 is to examine the performance of the QDIA closure and variants thereof with diﬀerent approximations made in the treatment of the higher order cumulants as compared to DNS in order to determine the processes that contribute to error growth. It will be shown that error variance growth is driven by contributions from 1) the oﬀdiagonal variances as they develop from an initially isotropic error ﬁeld and 2) the role of non-Gaussian correlations associated with the higher order moments. Initially we compare 5 day forecast results whose initial conditions are bred over a 5 day period. Using the QDIA closure representations of the diagnostics Eqs. 5a and 5b allow the separation and quantiﬁcation of the various contributions to error growth while the comparison to DNS of the total palinstrophy production measure Eq. 4 is used as a test of the accuracy of the closure. An examination of the evolution of transient error ﬁelds over the period between the 26th October and the 8th November 1979 on trajectories similar to those taken by the observed atmospheric 500-hPa ﬁeld requires that the mean ﬁelds within a barotropic model be forced by suitable time-evolving source terms. Thus we specify a relaxation term of the form Sk (t) = γ(ζk∗ − ζk )

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b)

Fig. 1. 500 hPa streamfunction ﬁeld at 1200 UTC on the 6th November 1979 in km2 s−1 (a) and Northern hemisphere topography in m (b).

which is added to the right hand side of the spectral barotropic vorticity equation where ζk∗ are the linearly interpolated daily observed ﬁelds. The simulation uses an initial observed 500-hPa streamfunction for 1200 UTC days taken between the 26th − 29th of October with an e-folding relaxation time of 2 days over the 10-day period (5 day breeding, 5 day forecast). The source term, which is calculated at each time step of the unperturbed simulation, is then stored and applied to both perturbed ensemble DNS runs and to the mean ﬁeld equation of the closure. We present results for a nondimensional β = 1/2 which has a dimensional value of 1.15 × 10−11m−1 s−1 and a corresponding nondimensional k02 = 1/2 typical of the β-eﬀect at 600 latitude. The nondimensional ∇2 viscosity used throughout is νˆ = 3.378×10−4 which has a corresponding dimensional value of 2.5 × 105 m2 s−1 typical of atmospheric ﬂows at the resolutions considered here. Both closure and DNS are run with a 1 hour timestep. The initial error ﬁelds have Gaussianly distributed isotropic spectra that are taken to be approximately constant with wave number and are several orders of magnitude less than the mean ﬁeld at all wave numbers. In order to guarantee isotropy the initial DNS ﬁelds are constructed by ﬁrst taking a Gaussian sample with zero mean and unit variance and for a given realization obtaining further members of the ensemble by moving its origin by an increment in the x-direction and then in the y-direction. The initial realization is moved successively by 2π/n in the x-direction to form n realizations. Each of these n realizations is then shifted by 2π/n in the y-direction to form a total of n2 realizations. Initial sampling errors are minimized by taking the negative value of each of the n2 elements giving an ensemble of 2n2 realizations to which we give the required weights

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for the random and mean ﬁelds. The process is then repeated with further initial Gaussian samples until an ensemble of the required number is obtained. In the DNS calculations that follow ensemble average results from 1800 simulations are shown although comparable results for obtained for 32 realizations. This is in agreement with the ﬁndings of Leith46 and Houtekamer and Derome.36 All calculations are carried out in wavenumber space circularly truncated at a maximum resolution of k = 16 (C16). 7. Bred initial forecast errors The method of bred perturbations developed by Toth and Kalnay67 was formulated as a means of generating fast growing perturbations that would typically be bred in the data assimilation scheme. Our approach to breeding starts with an initially isotropic error ﬁeld that is subsequently scaled by a global scaling factor after every evolved timestep. Atmospheric ﬂows are typically dominated by the large scales and so we implement our global scaling parameter in terms of streamfunction. The global scaling factor for adjusting the perturbation ﬁeld is deﬁned as 1/2

Ck (t0 , t0 )/k 4 k g(t) = 4 k Ck (t, t)/k then ζˆk (t+ ) = g(t)ζˆk (t− )

(6a)

Ck (t+ , t+ ) = g(t)g(t)Ck (t− , t− )

(6b)

Ck (t+ , t+ )

(6c)

= g(t)g(t

)Ck (t− , t− )

where − and + indicate prior and posterior ﬁelds. In both the DNS and closure the oﬀ-diagonal elements of the covariances are adjusted. In the DNS this occurs through direct application of the global scaling factor to the vorticity ﬁeld (Eq. 6a) whereas in the closure it occurs because the oﬀdiagonal covariances are expressed in terms of the renormalized diagonal covariances through the diagonal cumulant, consistent with the assumptions of the QDIA. The breeding period prepares the initial forecast error (perturbation) ﬁelds to contain information about the fastest growing modes arising due to dynamical ﬂow instabilities. In the closure studies of ensemble prediction that follow large inhomogeneities and very strong ﬂow instabilities required a 5 day restart period in order to ensure numerical stability and an accurate representation of the two- and three-point terms. All regularized calculations presented in this

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chapter are in the manner of O’Kane and Frederiksen53 with α1 = α2 = 4 (see also chapter 14). 8. Ensemble prediction results In an eﬀort to examine the eﬀect of initial error ﬁelds on forecast accuracy we examine the growth of errors as the ﬂow regime moves from one dominated by the mean to one where the error ﬁeld saturates the mean. In the strong turbulence regime O’Kane and Frederiksen27,53 found that very large numbers of ensemble members are required in order to resolve the small scale mean ﬂow. In the opposite regime where the ﬂow is dominated by inhomogeneities, we expect that smaller ensemble sizes may be suﬃcient.

10-2

10-2

a)

b) 10-3

10-4

10-4

e(kx)

e(kx)

10-3

10-5

10-5

10-6

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10-7

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10-8 1

10-8 1

10

10-2

10

kx+1

kx+1 4

c)

d)

10-3 3

P (t)

10-4

M

e(kx)

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10-6

10-7

<e> ^ e

RQDIA . . . . . . ...... QDIA DNS

2

1

RQDIA QDIA DNS

RQDIA . . . . . . QDIA . . . . . . DNS

10-8 1

0 10

kx+1

26th

31st day

5th

Fig. 2. Bred perturbations are generated over a 5 day period starting on the 26th October 1979, followed by a 5 day forecast period. The initial, day 5 and day 10 mean and perturbation kinetic energy as a function of zonal wavenumber are shown for DNS, QDIA and RQDIA calculations in ﬁgures a), b) and c) respectively. Figure d) shows the DNS, QDIA, and RQDIA palinstrophy production measure over the entire 10 day period.

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Fig. 2 shows a comparison of DNS, QDIA and RQDIA calculations in which the 5 day breeding cycle, beginning on the 26th October 1979, is initialized with isotropic initial perturbations (Fig. 2a). At the end of a 5 day breeding period (Fig. 2b) the mean zonally averaged kinetic energy of all three calculations are in good agreement with the QDIA transient energy slightly underestimated at the small scales in comparison to the regularized variant. At the conclusion of the breeding cycle the fastest growing mode for the initial forecast is clearly at kx = 3. At the end of the 5 day forecast period (Fig. 2c) the error ﬁeld is saturating the small scales with the RQDIA showing generally close agreement with the DNS while the QDIA clearly under-predicts the evolved small scale error energy. This result suggests that higher order moments are small at any given instant but are cumulatively important. The palinstrophy production measure (Fig. 2d) displays four distinct regimes in the structural organization and growth of errors over the depicted 10 day experiment. The ﬁrst is over day 1 of the breeding period and corresponds to a dramatic and rapid growth as the initial error ﬁeld evolves from Gaussian isotropic initial conditions towards the leading instability vectors ie growth of the oﬀ-diagonal covariances. The second is from approximately day 2 through to day 5 where the P M continues to grow but now at a signiﬁcantly reduced and nearly constant rate as shown by the overall slope in this period. On day 5 the restart procedure is implemented and the bred perturbations are used to generate initial forecast perturbations. The third period of error growth occurs between days 5 and 6 and is associated with accelerated growth as the perturbation ﬁeld amplitude rapidly increases due to the use of ﬂow dependent perturbations. This is evident by the rapid drop in the palinstrophy measure. From day 7 onwards the growth of the error ﬁeld amplitude is much reduced as the error starts to saturate at about day 10. In Fig. 3 the evolved kinetic energy for the regularized closure with and without cumulant update terms is contrasted with the unregularized closure and DNS for a similar 5 day breeding, 5 day forecast period beginning on the 29th of October. The eﬀect of the two- and three-point time history integrals in determining the error growth rate can be ascertained in the closure model (ZRQDIA) by simply zeroing them at the restart. The ZRQDIA removes all information contained in the time history integrals and restarts the closure calculation at day 5 with only the initial amplitudes of the homogeneous component of the bred error ﬁeld. In this way all information about the oﬀ-diagonal inhomogeneities and non-Gaussian correlations

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is removed allowing the calculation to be restarted with Gaussian initial conditions. Fig. 3 shows the ZRQDIA systematically under-predicting the evolved energy at all scales. In comparison the QDIA is in good agreement with the RQDIA and DNS at the large scales (k < 7) falling away sharply to more severely underpredict the small scale kinetic energy than even the ZRQDIA. The evolution of the forecast P M for the ZRQDIA (Fig. 4a) run demonstrates the rapid transient growth resulting from using ﬂow dependent perturbations where strong ﬂow instabilities associated with the onset of the blocking event (7 − 8th November 1979) are present during the breeding cycle. Using the QDIA closure integral representations we are able to quantify the relative contributions from S K and I K to P M at each time step (Fig. 4b). We immediately see (Fig. 4b) that at any given time S K , the three-point contribution to P M , is less than 1 percent and that the growth of the error ﬁeld is completely dominated by inhomogeneities through the twopoint covariances. Thus the ﬂow regime is dominated by eddy-topographic and eddy-mean ﬁeld interactions and it is not until the error ﬁeld saturates that we begin to see some contribution from the three-point (eddy-eddy) correlations. For cases where the forecast is run with isotropic initial forecast perturbations the saturation times are strongly dependent on e-folding times with smaller e-folding times corresponding to larger transient energy amplitudes after 10 day evolution periods. In cases where the perturbation 10-2

10-3

10-4

e(k x )

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10-6

10-7

10-8 1

<e>

RQDIA . . . . . . ZQDIA . . . . . . QDIA DNS

^ e

ZQDIA RQDIA QDIA . . . . . . DNS

kx +1

10

Fig. 3. Kinetic energy as a function of zonal wavenumber after 5 days breeding and 5 days forecast on the 8th November for RQDIA, QDIA, ZRQDIA and DNS calculations.

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amplitudes are able to dominate the mean over most scales, such as after saturation, this results in much larger values for S K (up to ≈ 20%) on day 10. In contrast to the results in Fig. 4b) for cases of strong perturbations and weak mean ﬁelds such as those considered by O’Kane and Frederiksen,53 the non-Gaussian correlations dominate the small scales and consequently form the major contribution to the palinstrophy production measure. Further in homogeneous isotropic turbulence our palinstrophy production reduces to the standard deﬁnition of skewness which is purely a measure of small scale non-Gaussian correlations (see the study of Frederiksen and Davies26 ). Comparing the regularized QDIA closure with and without time-history integral contributions illustrates the point that to neglect the two- and

4

a)

M

P (t)

3

2

1 ZRQDIA . . . . . . RQDIA . . . . . . DNS

0 29th

3rd

8th

day 4

b) 3

K

K

P (t),S (t),I (t)

2

M

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K

K

. . . . . . S (t)

QDIA

M

M

K

K

I (t)

I (t) 0 29th

. . . . . . S (t) P (t)

P (t)

3rd

8th

day Fig. 4. a) P M (t) for the period 29th October - 8th November 1979 are shown for both the RQDIA and ZRQDIA. b) P M (t), S K (t) and I K (t) for the same period for both the RQDIA and QDIA.

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three-point integral terms is to reduce the rate of growth of the perturbation or error ﬁeld. This was demonstrated in the evolved error ﬁelds in Fig. 3a) on the 8th November 1979 where the eﬀect on the small scales of zeroing the time history integral contributions (only) on day 5 of the breeding period (ZRQDIA), is contrasted with inclusion of those terms (RQDIA) in P M . Further illustration of this point is provided in Fig. 5 where the evolution of the ratio of the evolved perturbation to mean ﬁeld over the 10 day period again shows the ZRQDIA lagging the RQDIA and DNS. In this approach we take saturation to have occurred when the perturbation kinetic energy eˆ(kx ) is approximately equal to the mean kinetic energy e(kx ) over most scales. Using bred perturbations this was found to typically occur at approximately day 5 of the forecast period (Fig. 5a) whereas for isotropic initial conditions 10 days was more typical for 1-2 day e-folding times (not shown). Fig. 6 compares physical space DNS and RQDIA results for the zonally asymmetric or eddy streamfunction for the same runs depicted in Figs. 3, 4 and 5.

10 6

^ e(k ) / <e(k x) >=1 X 10 x

X 10

^ e(k ) / <e(k x) > x

X 10

X 10

X 10

X 10

4

10

10 4

4

7.5 10 2

5

3

2

2.5

10 0

1

0

day

0 10 -2 ZRQDIA . . . . . . RQDIA DNS

10 -4 1

kx+1

10

Fig. 5. The ratio of perturbation to mean kinetic energy as a function of zonal wavenumber at every 2.5 days. Each measurement has been scaled by factors of 10 for ease of viewing.

The use of a global scaling factor has the potential for the bred perturbations to converge on to Lyapunov vector 1. Toth and Kalnay67 have shown that bred vectors are superpositions of the leading local time dependent Lyapunov vectors (LLV’s) of the ﬂow and that after an initial transient period (≈ 7 days for atmospheric ﬂows) all random perturbations assume the structure of the LLV’s. In order to examine the issue of spread in the

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29th October 1979

DNS

RQDIA

3rd Nov

8th Nov

Fig. 6. The physical space plots of the eddy streamfunction for both DNS and RQDIA models at the initial day (29th October), day 5 (3rd November)of the breeding cycle and after 5 forecast days (8th November)in m2 s−1 .

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perturbation ﬁeld we have re-examined the run starting on the 29th October with an additional stochastic backscatter forcing of the small scales. Stochastic backscatter forcing corresponds to an injection of energy from the sub-grid scale eddies to the retained scale eddies thereby randomly forcing the small scale perturbations. We next apply a stochastic backscatter forcing Fb (k) = fb (k)fb∗ (k) (Fig. 7d) throughout the 5 day breeding period. The stochastic backscatter forcing Fb (k) is as discussed by Frederiksen and Davies24 for subgrid-scale parameterizations based on eddy damped quasi-normal Markovian methods. Little qualitative diﬀerence is found between model runs with and without stochastic backscatter. However the quantitative diﬀerence is not negligible with a clearly appreciable eﬀect to P M (Fig. 7c) via injection of small scale kinetic energy and increased small scale isotropy∗. As another approach to quantifying the cumulative contribution of the non-Gaussian correlations to the evolved kinetic energy we examine a variant in which the cumulant discard assumption is applied ie ζˆk (t)ζˆ−l (t)ζˆl−k (t ) = 0.

(7)

In Fig. 8a) the ratio of the mean and transient kinetic energy amplitudes e(kx )RQDIA and e(kx )CD /e(kx )RQDIA for both cumulant ie eˆ(kx )CD /ˆ discard (CD) and RQDIA closure calculations is compared on days 5 and 10 for the same parameters as the results shown in Fig. 7. When compared with the cumulant discard results the RQDIA demonstrate the considerable cumulative contribution that the three-point terms make to the evolved transient kinetic energy Fig. 8. Figure 8a also demonstrates an over prediction in day 10 mean kinetic energy for the CD model as a result of the systematic under prediction of the transients. In Fig. 8b the forecast kinetic energy amplitudes at day 10 (8th November) are seen to under predict the evolved transient energy over a much larger range of scales than occurs for an unregularized QDIA calculation (not shown) further demonstrating that the cumulative non-Gaussian correlations are also an important inﬂuence on the larger scales. ∗ Wang

and Bishop69 made direct comparison between both globally constant and regionally rescaled or masked breeding methods and a form of the ensemble Kalman ﬁlter (EnKF) in which the forecast error covariance is used to predict the analysis error covariance matrix but not for updating the mean state. The advantage of masked breeding is that the eﬀective regional rescaling ensures that the background error covariances have as many directions as regional rescaling factors, whereas for a globally constant scaling factor the background error covariances quickly converge to the dominant Lyapunov vector.

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......

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Fig. 7. 10 day forecast starting with isotropic initial error ﬁeld with nudging ﬁeld speciﬁed by linearly interpolated daily 500mb streamfunction ﬁelds starting from 29th October 1979 and ending on the 8th November 1979. a) and b) Kinetic energy as a function of zonal wavenumber on days 5 and 10 respectively. c) Palinstrophy measure (Stochastic backscatter: DNS thick solid line, RQDIA thick dotted line) (No stochastic backscatter: DNS thin solid line, RQDIA thin dotted line). d) The stochastic backscatter.

9. Discussion Our examination of the performance of the closure and DNS during the period when a large scale high-low blocking dipole was forming over the Gulf of Alaska was conducted using sequential 5 day breeding / 5 day forecast experiments over the 26th -29th of October 1979. The block that developed on the 5th and persisted until the 12th November 1979 is typical of a large scale coherent structure in the atmosphere associated with markedly increased ﬂow instability and a corresponding loss of predictability. The closure model has been able to accurately capture the evolution of our main diagnostic namely the palinstrophy production measure P M throughout the forecast periods for the cases depicted. The three variants of the closure implemented were the regularized QDIA (RQDIA), QDIA and ZRQDIA allowing quantiﬁcation of the relative importance of the time-history information built up about the non-Gaussian correlations and oﬀ-diagonal covariances and higher order moments. The ratio of error to mean kinetic energy as examined in Fig. 5 demonstrated a large lag at day 7.5 for the ZRQDIA (dashed)

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Fig. 8. Figure a) displays the ratio of the mean and error kinetic energy as a function of zonal wavenumber from cumulant discard and RQDIA calculations after 5 days (3rd November 1979) and 10 days (8th November) evolution. Figure b) compares day 10 (8th November) kinetic energy for DNS and the cumulant discard approximation.

which extends out to day 10 even though the DNS and RQDIA error kinetic energies have clearly saturated. Comparison of the DNS, QDIA, RQDIA and ZRQDIA calculations further allowed us to quantify the respective contributions of oﬀ-diagonal and non-Gaussian correlations to error variance growth. A cumulant discard (CD) variant of the closure model further demonstrated the cumulative eﬀect of the non-Gaussian correlations on the evolved kinetic energy in experiments including stochastic backscatter forcing. These CD results illustrate the consequences of higher order moment discard methods such as the quasi-normal results of Fleming19 and the stochastic dynamic moment discard results of Epstein and Pitcher.14 The results of this section have demonstrated that instantaneous error growth, resulting from the weak transient strong mean ﬁeld regime, is largely due to growth in the inhomogeneity and not the non-Gaussian correlations which at any given time are small. However the cumulative eﬀect of the non-Gaussian correlations determines the correct amplitude of the evolved kinetic energy variances. It is only after the error ﬁeld has saturated the mean that instantaneous non-Gaussian eﬀects start to become of increasing importance and eventually dominate in the strong transient, weak mean ﬁeld regime typical of strongly non-Gaussian weakly inhomogeneous ﬂows as discussed by O’Kane and Frederiksen.53 The potential for reduced spread associated with the collapse of bred perturbations onto the leading Lyapunov vector was countered by perturbing the small-scale transients through stochastic backscatter forcing. Qualitatively similar behaviour was found between forced and unforced cases; however, decreased

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amplitudes were observed in P M which are indicative of energy being injected at the small scales by forcing with Fb (k) thereby increasing the small scale isotropy. 10. Data assimilation The mean ﬁeld and error covariance of an evolving nonlinear system involves an inﬁnite number of equations for the moments. The problem of closing this moment hierarchy is formally identical to the closure problem of statistical turbulence theory.50 The Kalman ﬁlter methodology applies either a linear (Kalman6 ) or tangent linear (extended Kalman15 ) approximation to the equation for the second moments while discarding moments of third order and higher. These moments are of critical importance in nonlinear models in that they are required to enable tracking of regime transitions.50 As a consequence of an inherent lack of information about the covariances we are generally unable to accurately specify observational and model error covariance matrices in atmospheric and oceanic ﬂows. In data assimilation methods for large systems such as numerical weather prediction models one of the major issues is the inability to specify the complete forecast error covariance matrix. Thus observational and model error covariances are often speciﬁed as diagonal or quasi-diagonal matrices11 for which the information can be accurately speciﬁed e.g. the 3-D Variational operational method developed at ECMWF.10 Modern statistical turbulence methods22,26,27,53 successfully describe the dynamics and statistics of strongly non-Gaussian ﬂows for which a range of spatial scales are simultaneously excited. Due to secular divergences standard perturbative and expansion methods fail in strongly nonlinear, non-Gaussian ﬂows requiring renormalized closure treatments of inhomogeneities and higher order moments. The formal similarities between data assimilation and the moment closure problem suggest statistical dynamical methods would provide a natural tool for the related problem of incorporating higher order moments and memory eﬀects into general ensemble Kalman ﬁlter based data assimilation schemes. The ensemble Kalman ﬁlter (EnKF)37 propagates an ensemble of model states with a fully nonlinear model allowing the error covariance matrix to be calculated with no moment closure required. This allows the construction of the forecast error variance at any given time by averaging over the ensemble. Due to computational restraints current operational numerical weather prediction systems are only able to perform of the order of a hundred forecasts requiring additional assumptions to be made about the

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nature of the background correlations so that methods may be employed to reduce the sampling error. Given low rank representations of the forecast and analysis error covariances the task of transforming the forecast into an analysis ensemble has been examined using general ensemble ﬁlter methods that allow ﬂow dependent perturbations. This approach falls within the family of square root ﬁlters (see Tippett et al64 ), examples of which are the ensemble adjustment Kalman ﬁlter (EnAF) of Anderson2 and the ensemble transform Kalman ﬁlter of Wang and Bishop.69 Where sample sizes are far too small to give reliable statistics about the model state, conditional on the availability of data describing the observed state, the ﬁltering solution may diverge from the observations and covariances may be under represented. In light of these problems Hamill et al31 (EnKF) and Anderson2 (EnAF) further developed the use of an empirically selected linear factor or covariance inﬂation factor. Whitaker and Hamill70 examined the role of sampling error by comparing the EnKF, which is predicated on random perturbations and white noise, to a ﬁlter method based on a square root approximation of Andrews4 (EnSF) in which the perturbations are correlated with the evolving ﬂow. Square root methods such as the EnAF and EnSF share a common approach based on the use of unperturbed observations and form a class of deterministic ensemble square root ﬁlters. The square root ﬁlters are also complemented by the stochastic approaches to analysis ensemble formation of Houtekamer and Mitchell37 and Burgers et al.8 In this section we compare EnKF and ensemble square root ﬁlter methods (EnSF) to a statistical dynamical Kalman ﬁlter (QSDKF: O’Kane and Frederiksen55 ) model under a general assumption of quasi-diagonality. The deterministic (EnSRF), stochastic (EnKF) and statistical dynamical (QSDKF) methods we develop are derived from the DNS and quasi-diagonal QDIA spectral barotropic models on a generalized β-plane described in the previous chapter and sections 3 and 4. The ensemble Kalman ﬁlter (EnKF) propagates an ensemble of model states with a fully nonlinear model. This allows the construction of the forecast error variance at any given time by averaging over the ensemble without the need to close a moment hierarchy. In the QSDKF we apply the quasi-diagonal assumption i.e. that the initial forecast error covariances are assumed diagonal to zeroth order, then renormalize by replacing all zeroth and ﬁrst order diagonal covariances by their exact representations. Renormalization incorporates higher order terms while removing secular terms that may cause divergences. Next the interaction coeﬃcients are regularized in order that all

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higher order moments be accurately represented.26,53 Finally we employ a cumulant update or restart procedure29,53,61 which, in addition to enabling non-Gaussian initial conditions, allows eﬃcient, tractable computations of the non-Markovian time history integrals. Thus the application of the QDIA methods of Frederiksen,22 O’Kane and Frederiksen53 and Frederiksen and O’Kane27 to data assimilation within the ensemble Kalman ﬁlter formalism accurately incorporates all higher order moments. The assumption of quasi-diagonality is not only consistent with the QDIA closure methodology but with many operational data assimilation schemes such as the spectral statistical interpolation (SSI) schemes used at NCEP57 and the ECMWF.3,10,60 Our approach updates the forecast error covariance matrix and non-Gaussian terms based on the functional dependence of the diagonal elements. These elements are then updated using the diagonal Kalman ﬁlter methodology resulting in a quasi-diagonal statistical 4-D ﬂow dependent methodology. Many operational approaches use climatological variances of which the SSI method is one example. We assume that inhomogeneities can be combined with a diagonal Kalman ﬁlter.7 As a consequence of using the quasi-diagonal statistical closure methodology the QSDKF oﬀ-diagonal and higher order elements are functionals of the diagonal elements and similarly the resulting QSDKF Kalman gain is deﬁned in terms of diagonal forecast and observational error covariances. We focus on 500-hPa Northern Hemisphere atmospheric ﬂows during a 30 day period starting on the 16th of October 1979. During this period three major Northern Hemisphere atmospheric blocking high low dipoles formed. The ﬁrst large-scale block began developing over the United Kingdom around the 21st of October then matured to form a high (over Scandinavia) - low (over Spain) dipole around the 27th of October. The second occurred in the middle of this period over the Gulf of Alaska on the 5th of November, ampliﬁed and persisted until the 12th of November and then weakened and moved downstream with the resulting instability initiating a third blocking high that formed over the Atlantic Ocean around the 13th maturing into a high-low dipole on the 16th and ﬁnally decaying during the period between 17th − 18th of November. By conducting our data assimilation experiments over this period we are rigorously testing the ability of the various assimilation methods to track the truth trajectory through a number of atmospheric regime transitions. As such our results are analogous to the study of Miller et al50 examining the ability of the general extended Kalman ﬁlter (GEKF) methodology to track regime transitions associated with the chaotic trajectory of the Lorenz model. Our studies include 5 day

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assimilation experiments followed by 5 day forecast studies and as well 30 day assimilations experiments. The aim of this section is to examine the role of sampling and the eﬀects of correlated versus uncorrelated observational error covariances within the quasi-diagonal framework using statistical dynamical Kalman, ensemble square root and ensemble Kalman ﬁlter methods for atmospheric data assimilation. We will also discuss the role of inhomogeneity and higher order moments in the data assimilation cycle as well as the spectrum of the Kalman gain and the generation of initial perturbations during the analysis cycle. 11. Statistical dynamical ﬁlters In this section we brieﬂy describe a hierarchy of Kalman ﬁlter schemes in order of increasing complexity. Useful derivations of the matrix Kalman ﬁlter equations can be found in section 5.4 of Brown6 or alternately Lorenc47 using Bayes’ theorem. The Kalman ﬁlter theory implicitly assumes that both the observations and priori (forecast) distributions, based on the background state, are Gaussianly distributed enabling the posteriori (analysis) distribution to be derived based on their products.2,47 For Kalman ﬁlter equations that are diagonal in spectral space, observational data dk = dk + dˆk with error variance Dk = dˆk (t)dˆ∗k (t ) is assimilated using linear interpolation with the forecast weighted by the Kalman gain Kk ie ζka = ζkf + Kk (dk − ζkf ),

(8a)

Cka (t, t) = (1 − Kk )Ckf (t, t)

(8b)

where such that Ckf (t, t) = ζˆkf (t)ζˆkf ∗ (t) is the forecast (background prior) error variance. Here a denotes analysis (posteriori), f forecast (priori), is the mean and d are the observations. The diagonal Kalman gain is given by Kk (t, t) =

Ckf (t, t) (Ckf (t, t) + Dk (t, t))

.

(8c)

As we are dealing with scalars and a diagonal gain the linear observation operator that maps the model state vector into the space of observations H is simply the identity matrix I. The extended Kalman ﬁlter (EKF)15,30 is formulated entirely in terms of covariances employing a tangent linear approximation for calculating

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Ck (t, t). Speciﬁcally the EKF uses a tangent linear model to transform an ˆ i ) to a ﬁnal perturbation at time initial perturbation at some initial time ζ(t 38 ˆ ζ(ti+1 ). Linearizing about the nonlinear model trajectory will however be inaccurate for unstable systems, where there are poor or infrequent observations or more generally whenever perturbations are large, resulting in the EKF failing to track the true solution. This motivated the development of a generalized extended Kalman ﬁlter (EKF) in which a moment expansion method in terms of Taylor series was used to estimate the contributions of the third and fourth order moments to the Kalman gain.50

12. Quasi-diagonal statistical dynamical Kalman ﬁlter The QDIA theory allows us to further extend the Kalman ﬁlter methodology beyond the generalized EKF by incorporating accurate tractable time history integral representations of all higher order moments using functionals of purely diagonal functions and regularized interaction coeﬃcients as applied to a high dimensional barotropic model. Suppose we have an ensemble of ﬂows satisfying the spectral barotropic vorticity equation and we express the vorticity and forcing ζk = ζk + ζˆk f 0 = f 0 + fˆ0 k

k

k

(9) (10)

in terms of their ensemble means and perturbations, then the equation for the ensemble mean can be written in the form:

∂ 2 + ν0 (k)k ζk = δ(k + p + q) × ∂t p q

QDIA K(k, p, q){ζ−p ζ−q + C−p,−q (t, t)} + A(k, p, q)ζ−p h−q + fk0 whereas the deviation forms the two-point cumulant as deﬁned by QDIA (t, s) C−p,−q (t, s) = ζˆ−p (t)ζˆ−q (s) = C−p,−q

(11)

where p and q both range over the set T = R ∪ 0. The second order expressions for the diagonal two-time and single time

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cumulants in terms of two- and three-point terms are respectively

∂ QDIA 2 δ(k + p + q)A(k, p, q)C−p,−k (t, t )h−q + ν0 (k)k Ck (t, t ) = ∂t p q + δ(k + p + q)K(k, p, q) p

q

QDIA QDIA (t, t ) + C−p,−k (t, t )ζ−q (t) + ζˆ−p (t)ζˆ−q (t)ζˆ−k (t )QDIA ] ×[ζ−p (t)C−q,−k

+fˆk0 (t)ζˆ−k (t) = Nk (t, t )

and

∂ + 2ν0 (k)k 2 Ck (t, t) = 2Nk (t, t). ∂t

The method of deriving the QDIA closure equations22 and its variants the cumulant update QDIA and regularized CUQDIA53 with k in the set T = R ∪ 0, has been described in detail in chapter 14 and elsewhere27 as have the general expressions for the interaction coeﬃcients, governing QDIA equations and the update terms. In the present study the QDIA closure equations have been implemented in regularized form in order that the amplitudes of the small scale transient perturbations arising from higher order moments be accurately calculated. 13. Ensemble Kalman ﬁlter Our approach to ensemble ﬁltering diﬀers from the EnKF formulation of Evensen16 through the fact that the Kalman gain is implemented in diagonal form only and the oﬀ-diagonal covariances are again generated from the resulting posteriori analysis error ﬁelds. This approach generalizes the SSI method of Parrish and Derber57 to ﬂow dependent perturbations allowing, within the framework of the quasi-diagonal assumption, the calculation of the full analysis covariance matrix. It also enables direct comparison to the statistical closure based assimilation method (QSDKF) developed in section 12 which assumes quasi-diagonality in order to derive tractable representations of inhomogeneous terms which speciﬁcally are the two-time covariances that couple to the mean ﬁeld and topography.22 The QEnKF method has a prescribed observational error whose perturbations have Gaussian statistics ie ζkobs = dk and dik = dk + dˆik where i = 1, . . . , N runs over the ensemble and the observational variance is deﬁned as Dk = dˆk dˆ∗k . Throughout the following discussion we will simply assume that the perturbation ﬁeld denotedˆruns over the entire ensemble ie

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ζˆi will be assumed. The forecast f and analysis a ﬁelds can now be deﬁned as ζkf = ζk f + ζˆkf ζ a = ζk a + ζˆa . k

k

Given the initial scalar vorticity ﬁeld the basic equation for the analysis at time t is again in terms of a linear interpolation between observations and predictions weighted by the Kalman gain K thus the perturbation ﬁeld assimilates data according to ζˆkai = ζˆkfi + Kk (dˆik − ζˆkfi )

(12a)

dˆk (t)dˆ∗k (t ) = δ(t − t )Dk (t, t).

(12b)

where

The blending factor or Kalman gain is again given by Eq. 8c. The prior ﬁeld ζkf satisﬁes Eq. 3 in which we may include a model error through Eq. 10 where fˆk0 is generally taken to be white. Eqs. 12 correspond to the quasidiagonal ensemble Kalman ﬁlter (QEnKF) when applied to the ensemble averaged direct numerical simulation methodology. At any point in time we may directly calculate the full covariance matrix by simply averaging over i . the perturbation ﬁeld ie ζˆki ζˆ−l 14. Quasi-diagonal ensemble square root ﬁlter We next consider a quasi-diagonal formulation of the square root ﬁlter. Our approach follows that belonging to the class of ﬁlters developed by Bishop et al,5 Anderson,2 Whitaker and Hamill70 and Tippet et al64 but has its genesis in the work of Andrews.4 The EnKF methodology requires that observations be perturbed in order that the Kalman ﬁlter update equation be satisﬁed ie Eq. 8a. If one naively omits the perturbations on the observations then it has been shown70 that a term Kk Dk Kk is omitted and we ﬁnd Cka = (1 − Kk )Ckf (1 − Kk ) resulting in Cka being systematically under-estimated. The square root ﬁlter methods were developed to address the associated problem of under-estimated analysis error covariances due to insuﬃcient sampling of the ensemble Kalman ﬁlter. This approach requires an ensemble ﬁlter that does not use perturbed observations yet satisﬁes the Kalman ﬁlter equation for the analysis error. This requires a square root formulation of Eq. 8b. The solution developed by Andrews4 and employed in various guises

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by Bishop et al,5 Anderson,2 Whitakerand Hamill70 and Tippet et al64 is as follows ˜ k )ζˆf , ζˆka = (1 − K k ζka = ζkf + Kk (dk − ζkf ) Whitaker and Hamill70 implement an ensemble square root ﬁlter based on ˜ whose a form ﬁrst derived by Andrews.4 This approach seeks to deﬁne K analysis error covariance satisﬁes ˜ k )C f (1 − K ˜ k ) = (1 − Kk )C f Cka = (1 − K k k for which the appropriate scalar solutions are4,70

−1 & −1 f f f ˜ Kk = Ck Ck + Dk Ck + Dk ± Dk =

1±

= 1∓

Dk Ckf + Dk

−1 Kk

& 1 − Kk

which is equivalent to

& ζˆka = ± 1 − Kk ζˆkf

(13)

˜ ≤1 in the scalar case and also satisﬁes Eqs. 8c and 8b. In order that 0 ≤ K we choose the positive in Eq. 13. We refer to this solution as the quasidiagonal square root ﬁlter (QEnSF). 15. Results In the following sections we compare the use of correlated observational (QSDKF and QEnSF) and uncorrelated (QEnKF) error perturbations in data assimilation experiments over periods during which large scale atmospheric regime transitions occurred. In all experiments to follow the QSDKF has been used in regularized form in order that the amplitudes of the small scale transient perturbations arising from higher order moments be accurately calculated. 15.1. A comparison of quasi-diagonal ensemble Kalman and statistical dynamical ﬁlters We now compare QEnKF and QSDKF models in a range of experiments from 5 day assimilation followed by 5 day forecast runs over the same period

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as the ensemble prediction experiments of section 2. Throughout, the truth trajectory is calculated by running the barotropic vorticity equation over the desired period with a relaxation γ(ζk∗ − ζk ) where γ = 1/24hrs. The source term is calculated at each timestep of the unperturbed simulation, stored and then applied to both the perturbed ensemble DNS and mean ﬁeld equation of the closure for data assimilation and prediction. In all the studies that follow we use the general dissipation ν0 (k)k 2 = νˆk 2 + iωk where the dimensional ν = 2.5 × 105 m2 s−1 . We choose the observational error to have an rms of 1×106m2 s−1 and deﬁne the nondimensional variance Dk (t, t) = 1.826 k, which results in an almost ﬂat kinetic energy spectrum. Throughout we assume the model error variance Qk (t, t) to be zero. Our choice of observational error variance is similar to that employed by Anderson2 in studies of data assimilation using the ensemble adjustment Kalman ﬁlter, which is of the same family of square root ﬁlters that are being considered here. Anderson2 assimilated T21 daily spherical streamfunction data for the winter of 1991/1992 using a spherical barotropic model at T42 truncation. In common with our studies, Anderson2 regards interpolated data as the truth with the observational noise added to obtain the forecast while the zonal ﬂow is relaxed toward the observed time mean ﬂow according to a speciﬁed e-folding time for perfect model runs. The Ensemble adjustment ﬁlter (EnAF) constrains the updated ensemble covariance to satisfy the standard Kalman ﬁlter analysis error-covariance relation. This is achieved by applying a linear operator to the prior ensemble70 which is functionally equivalent to the square root ﬁlter. In EnAF studies of models whose state spaces are much larger than the ensemble size it was shown that the deterministic ﬁlter out-performed the EnKF.2 Anderson hypothesized that the poor performance of the EnKF as compared to the EnAF was a result of the use of randomly perturbed observations (see also Sakov et al62 ) and small ensembles and that with increasing ensemble size the performance of traditional EnKF would improve. Figure 9a displays the day 5 (3rd November 1979) zonally averaged kinetic energy for QSDKF and QEnKF 6 hourly data assimilation models run with 3600 and 20 member ensembles respectively. Only small diﬀerences were found between the 20 and 3600 QEnKF results. However it is clear that both QEnKF calculations underestimate the evolved error variances at day 5 as compared to QSDKF. Additionally in Fig. 9b we see that the initial forecast perturbations generated by the QEnKF assimilation method fail to grow signiﬁcantly as evident in the 5 day (8rd November 1979) evolved forecast error variances (Fig. 9b). Remember that the QEnKF assimilates

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data according to ζˆka = (1 − Kk )ζˆkf + Kk dˆk ζka = ζkf + Kk (dk − ζkf ) where dˆk has newly sampled random phases at every assimilation point. The eﬀect of the use of white noise generating the phases of dˆk is to randomize the evolved analysis vorticity ﬁeld perturbations thus destroying phase correlations. Our calculations were carried out at circularly truncated k = 16 (C16) resolution which has 797 degrees of freedom and that at 3600 realizations the QEnKF calculation has a much larger ensemble size than degrees of freedom. The reason for this lack of growth is the inability of the perturbed ﬁeld to organize due to the decorrelating eﬀect of sampling error introduced through the use of randomly perturbation observations at each assimilation point. The reason that the QEnKF data assimilation kinetic energy results

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depicted in Fig. 9 do not show strong dependence on the number of forecast perturbations in the vorticity ﬁeld are twofold. Firstly data assimilation studies are typically in a regime where the perturbation kinetic energy or error ﬁeld is orders of magnitude weaker than the mean and thus very few realizations are required to resolve the mean at reasonable resolutions.27 The problem of accurately resolving the second moment is signiﬁcantly more complex than simply resolving the mean. Second order error arises through the Jacobian terms as evident in the evolution equation for the

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diagonal cumulant ie

∂ ˆ + 2ν0 (k)k 2 Ckf (t, t) = −2ζˆk (t){Jk [ψ(t), ζ(t)] ∂t ˆ ˆ ˆ +Jk [ψ(t), ζ(t) + h] + Jk [ψ(t), ζ(t)]} where Jk [A, B] if the Fourier transform of J(A, B) deﬁned in Eq. 1b. Frederiksen and O’Kane27,55 have discussed the mechanisms by which errors in representing the Jacobian arise starting from diagonal (homogeneous or isotropic) initial conditions in the context of turbulent barotropic ﬂows. In the present study this initial sampling problem arises for each ζˆkfi at each assimilation point. The sampling error in estimating ζˆkfi may in reality be even greater were it not for the ameliorating eﬀect due to convergence towards the leading ﬂow instabilities. 15.2. The performance of the ensemble square root ﬁlter In Fig. 10 we contrast the QSDKF and QEnKF with the ensemble square root ﬁlter (QEnSF) on the 3rd after a 5 day period of 6 hourly data assimilation. The QEnSF shows much closer agreement to the QSDKF with very close agreement for k ≥ 3. Although the results presented in Fig. 10 are for 3600 realizations similar results were found for calculations down to 20 member ensembles. The mean streamfunction ﬁelds are shown in Fig. 11. For small sample sizes in the QEnKF there is always the possibility that spurious cross correlations of the form dˆk ζˆk∗ may be aﬀecting the results so we have also carried out calculations in which the vorticity ﬁeld amplitude was rescaled to try to minimize the eﬀects of such terms. The importance of the Kalman gain in determining the analysis error covariance cannot be underestimated. The gain adjusts itself in order to assimilate observations as required therefore it is of interest to consider the spectra of the gain. In Fig. 12 we plot the band averaged Kalman gain Ck (t, t)/ Ck (t, t) + Dk (t, t) K(ki ) = k

k

where the set S is deﬁned as

1 S = k|ki = Int.[k + ] 2

k

(14)

and the subscript i indicates that the integer part is taken in Eq. (14) so that all k that lie within a given radius band of unit width are summed over.

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10-3

10-4

e(k x )

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10-5

10-6

10-7

QEnKF (3600) QEnSF (3600) QSDKF

10-8 1

^ e

<e>

10

k x +1

Fig. 10. Evolved mean and error kinetic energy as a function of zonal wavenumber after 5 day data assimilation on the 3rd November 1979 for the QSDKF, QEnKF and the square root ﬁlter QEnSF.

We see that the summed gain is initially constant over all modes and that as the ﬂow evolves the gain spectra takes up a proﬁle that peaks at the fastest growing mode. That is, more information is required to be assimilated about the most unstable mode (k = 4) in the evolving ﬂow which in this particular case corresponds to the developing large scale coherent structure (the block). We also notice that the reason the QEnKF under-predicts the forecast error covariance arises due to sampling error manifesting as a systematically weak Kalman. The square root ﬁlters performance is much improved because they are better able to capture the eﬀects of the developing large scale ﬂow instability. We next examine the performance of the ensemble square root ﬁlter QEnSF in a 30 day 12 hourly data assimilation calculation beginning on the 16th October 1979. We estimate the systematic diﬀerences between the QEnSF calculation and the control by plotting

ψ

S (t) = abs

k

f ψkf (t)ψ−k (t)

−

t ψkt (t)ψ−k (t)

,

(15)

k

where f indicates background forecast and t indicates truth, in comparison

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QEnKF

QEnSF

Fig. 11. Evolved streamfunction in m2 s−1 after 5 day of 6 hourly data assimilation on the 3rd November 1979 for the QSDKF, QEnKF and the QEnSF.

to Dψ (k, t) =

Dk (t, t)/k 4 .

k

In Fig. 13 we ﬁnd that the QEnSF methodology performs very well over the 30 day assimilation period. It maintains a very close trajectory to the truth with little evidence of a systematic drift over the period. We note that our test (Eq. 15) is similar to that used by Anderson2 except that whereas Anderson considered a single grid point we are summing over the entire spectral space and so incorporate all regions of instability. In Fig. 14 we compare the evolved zonally asymmetric or eddy streamfunction and variances in physical space on the 6th November 1979 when the block was forming over the Gulf of Alaska. 16. Discussion We have considered comparisons of the ensemble Kalman ﬁlter to the ensemble square root ﬁlter and the statistical dynamical Kalman ﬁlter within

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K(k)

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0.10

Initial QSDKF QEnSF QEnKF

0.01

1

10

k+1

Fig. 12. Band averaged Kalman gain at the initial day and after the 5th day of data assimilation.

10-4

10-5

10-6

10-7

10-8 Dψ Sψ

10-9 0

......

5

10

15 day

20

25

30

Fig. 13. S ψ (t) versus D ψ (t) during the 30 day assimilation period starting on the 16th October 1979.

the framework of the quasi-diagonal approximation more typical of operational data assimilation systems such as at NCEP and ECMWF. We again consider a period in which a strong, rapidly growing large scale Northern Hemisphere high low blocking dipole formed and matured over the Gulf of

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b)

Fig. 14. The 500hPa QEnSF a) physical space eddy streamfunction (km2 s−1 ) and b) variance (km4 s−2 ) calculations respectively for the 6th October 1979 showing the forming block over the Gulf of Alaska.

Alaska. Many of the problems associated with the EnKF are ascribed to arise because of sampling error in the resolved Jacobian terms. In addition to the statistical closure based ﬁlter we have been able to run ensemble averages over very large numbers of realizations (up to 3600) to further elucidate the role of sampling. It was shown that even when the sample size is signiﬁcantly larger than the state space the QEnKF systematically under-predicts not only the evolved covariances but the zonally averaged kinetic energy. It was noted that unlike the generalized extended Kalman ﬁlter of Miller et al50 in which only estimates of the third and fourth order moments are incorporated the QDIA based QSDKF ﬁlter incorporates information on all higher order moments via a renormalized covariance and regularized interaction coeﬃcients (see chapter 14). A 5 day 6 hourly assimilation followed by a 5 day forecast experiment starting on the 29th October 1979, further demonstrated that the initial forecast perturbations generated by the QEnKF method were not able to match the growth demonstrated by the QSDKF error variances. The QEnSF deterministic model was much better able to match the QSDKF results due to an improved estimate of not only the amplitude but just as importantly the shape of the Kalman gain. The failure of the QEnKF was then shown to result because of sampling through the use of perturbed observations whose phases are random. The deterministic square root ﬁlters were able to demonstrate improved growth of the error covariances because the forecast perturbations were rescaled while preserving the phase directions. Thus information about the fastest growing modes and ﬂow instabili-

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ties accumulates throughout the assimilation cycle in an analogous manner to the method of bred perturbations developed by Toth and Kalnay.65–67 More stringent comparisons using 30 day 12 hourly data assimilation experiments starting from the 16th October 1979 were then made between the truth and the ensemble square root ﬁlter variant QEnSF. The square root ﬁlter was demonstrated to remain close to the truth streamfunction even through a period in which 3 large-scale atmospheric regime transitions occurred.

References 1. J.L. Anderson, A method for producing and evaluating probabalistic forecasts from ensemble model integrations J. Climate 9, 1518–1530 (1996). 2. J.L. Anderson, An ensemble adjustment kalman ﬁlter for data assimilation Mon. Wea. Rev. 129, 2884–2903 (2001). 3. E. Andersson, J. Haseler, P. Und´en, C. Courtier, G. Kelly, D. Vasiljevi´c, C. Brankovi´c, C. Cardinali, C. Gaﬀard, A. Hollingsworth, C. Jakob, P. Janssen, E. Klinker, A. Lanzinger, M. Miller, F. Rabier, A. Simmons, B. Strauss, J. Th´epaut, P. and Viterbo, The ECMWF implementation of three-dimensional variational assimilation (3D-Var). III: Experimental results Q. J. R. Meteorol. Soc. 124, 1831–1860 (1998). 4. A. Andrews, An square root formulation of the Kalman covariance equations AIAA J 6, 1165–1168 (1968). 5. C.H. Bishop, B.J. Etherton and S.J. Majumdar Adaptive sampling with the ensemble Kalman ﬁlter. PartI: Theoretical aspects Mon. Wea. Rev. 129, 420–436 (2001). 6. R.G. Brown, Introduction to Random Signal Analysis and Kalman Filtering. (J. Wiley and Sons, New York, 1983). 7. M. Buehner, Ensemble-derived stationary and ﬂow-dependent backgrounderror covariances: Evaluation in a quasi-operational NWP setting Q. J. R. Meteorol. Soc. 131, 1013–1043 (2005). 8. G. Burgers, P.J. van Leeuwen and G. Evensen, On the analysis scheme in the ensemble Kalman ﬁlter. Mon. Wea. Rev. 126, 1719–1724 (1998). 9. J.G. Charney, The feasibility of a global observation and analysis experiment Bull. Amer. Meteor. Soc. 47, 200–220 (1966). 10. P. Courtier, E. Andersson, W. Heckley, J. Pallieux, D. Vasiljevi´c, M. Hamrud, A. Hollingsworth, F. Rabier and M. Fisher, The ECMWF implementation of three-dimensional variational assimilation (3D-Var). I: Formulation Q. J. R. Meteorol. Soc. 124, 1783–1807 (1998). 11. D.P. Dee, On-line estimation of error covariance parameters for atmospheric data assimilation Mon. Wea. Rev. 123, 1128–1144 (1995). 12. E.S. Epstein, The role of initial uncertainties in prediction J. Atmos. Sci. 8, 190–198 (1968). 13. E.S. Epstein, Stochastic dynamic prediction Tellus 21, 739–759 (1969).

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14. E.S. Epstein and E.J. Pitcher, Stochastic analysis of meteorological ﬁelds J. Atmos. Sci. 29, 244–257 (1972). 15. G. Evensen, Using the extended Kalman ﬁlter with a multi-layer quasigeostrophic ocean model. J. Geophys. Res. 97 C11, 17905–17924 (1992). 16. G. Evensen, Sequential data assimilation with a nonlinear quasi-geostrophic model using Monte-Carlo methods to forecast error statistics J. Geophys. Res. 99 C5, 10143–10162 (1994). 17. G. Evensen, The Ensemble Kalman Filter: theoretical formulation and practical implementation (2003). 18. B.F. Farrell, Optimal excitation of baroclinic waves J. Atmos. Sci. 46, 1193– 1206 (1989). 19. R.J. Fleming, On stochastic dynamic prediction: I. The energetics of uncertainty and the question of closure Mon. Wea. Rev. 99, 851–872 (1971). 20. R.J. Fleming, On stochastic dynamic prediction: II. Predictability and utility Mon. Wea. Rev. 99, 927–938 (1971). 21. J.S. Frederiksen, Precursors to blocking anomalies: The tangent linear and inverse problems J. Atmos. Sci. 55, 2419–2436 (1998). 22. J.S. Frederiksen, Subgrid-scale parameterizations of eddy-topographic force, eddy viscosity, and stochastic backscatter for ﬂow over topography, J. Atmos. Sci. 56, 1481–1494 (1999). 23. J.S. Frederiksen and R.C. Bell, North Atlantic blocking during January 1979: Linear theory Q. J. R. Met. Soc. 116, 1289-1313 (1990). 24. J.S. Frederiksen and A.G. Davies, Eddy viscosity and stochastic backscatter parameterizations on the sphere for atmospheric circulation models J. Atmos. Sci. 54, 2475–2492 (1997). 25. J.S. Frederiksen and A.G. Davies, Dynamics and spectra of cumulant update closures for two-dimensional turbulence Geophys. Astrophys. Fluid Dyn. 92, 197–231 (2000). 26. J.S. Frederiksen and A.G. Davies, The regularized DIA closure for twodimensional turbulence Geophys. Astrophys. Fluid Dyn. 98, 203–223 (2004). 27. J.S. Frederiksen and T.J. O’Kane, Inhomogeneous closure and statistical mechanics for Rossby wave turbulence over topography J. Fluid Mech. 539, 137–165 (2005). 28. J.S. Frederiksen, M.A. Collier and A.B. Watkins, Ensemble prediction of blocking regime transitions Tellus 56 A, 485–500 (2004). 29. J.S. Frederiksen, A.G. Davies and R.C. Bell, Closure equations with nonGaussian restarts for truncated two-dimensional turbulence. Phys. Fluids 6, 3153–3163 (1994). 30. M. Ghil, S.E. Cohn, J. Tavantzis, K. Bube and E. Isaacson, Applications of estimation theory to numerical weather prediction. Dynamic Meteorology: Data Assimilation Methods L. Bengtsson, M. Ghil, and E. K¨ allen, Eds., Springer Verlag, 139–224, 1981. 31. T.M. Hamill, J.S. Whitaker and C. Snyder, Distance-dependent ﬁltering of background error covariance estimates in an ensemble Kalman ﬁlter Mon. Wea. Rev. 129, 2776–2790 (2001).

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32. J.R. Herring, Self-consistent-ﬁeld approach to turbulence theory Phys. Fluids 8, 2219–2225 (1965). 33. J.R. Herring, On the statistical theory of two-dimensional topographic turbulence J. Atmos. Sci. 34, 1731–1750 (1977). 34. J.R. Herring, S.A. Orszag, R.H. Kraichnan and D.G. Fox, Decay of twodimensional homogeneous turbulence J. Fluid Mech. 66, 417–444 (1974). 35. G. Holloway, A spectral theory of nonlinear barotropic motion above irregular topography J. Phys. Oceanogr. 8, 414–427 (1978). 36. P.L. Houtekamer and J. Derome, Methods for ensemble prediction Mon. Wea. Rev. 123, 2181–2196 (1995). 37. P.L. Houtekamer and P.L. Mitchell, Data assimilation using an ensemble Kalman ﬁlter technique Mon. Wea. Rev. 126, 796–811 (1998). 38. E. Kalnay, Atmospheric Modelling, Data Assimilation and Predictability. (Cambridge University Press, Cambridge, 2003). 39. A. Kasahara, Simulation experiments for meteorological observing systems for GARP Bull. Amer. Meteor. Soc. 53, 252–264 (1972). 40. R.H. Kraichnan, Irreversible statistical mechanics of incompressible hydrodynamic turbulence Phys. Rev. 109, 1407–1422 (1958). 41. R.H. Kraichnan, The structure of isotropic turbulence at very high Reynolds numbers J. Fluid Mech. 5, 497–543 (1959). 42. R.H. Kraichnan, Instability in fully developed turbulence Phys. Fluids 13, 569–575 (1970). 43. R.H. Kraichnan, Test-ﬁeld model for inhomogeneous turbulence J. Fluid Mech. 56, 287–304 (1972). 44. C.E. Leith, Atmospheric predictability and two-dimensional turbulence J. Atmos. Sci. 28, 145–161 (1971). 45. C.E. Leith and R.H. Kraichnan, Predictability of turbulent ﬂows J. Atmos. Sci. 29, 1041–1058 (1972). 46. C.E. Leith, Theoretical skill of Monte Carlo forecasts Mon. Wea. Rev. 6, 409–418 (1974). 47. A.C. Lorenc, Analysis methods for numerical weather prediction. Quart. J. Roy. Met. Soc. 112, 1177–1194 (1986). 48. E.N. Lorenz, A study of the predictability of a 28-variable atmospheric model Tellus 17, 321–333 (1965). 49. W.D. McComb, A local energy-transfer theory of isotropic turbulence J. Phys. A 7, 632–649 (1974). 50. R.N. Miller, M. Ghil and F. Gauthiez, Advanced data assimilation in strongly nonlinear dynamical systems J. Atmos. Sci. 8, 1037–1056 (1994). 51. M. Millionshtchikov, On the theory of homogeneous isotropic turbulence Compte Rendus (Doklady) de l’Academie Des Sciences de l’U.S.S.R. 32, 615– 618 (1941). 52. F. Molteni and T. Palmer, Predictability and ﬁnite-time instability of the northern winter circulation Quart. J. Meteor. Soc. 119, 268–298 (1993). 53. T.J. O’Kane and J.S. Frederiksen, The QDIA and regularized QDIA closures for inhomogeneous turbulence over topography J. Fluid Mech. 504, 133–165 (2004).

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54. T.J. O’Kane and J.S. Frederiksen, A comparison of statistical dynamical and ensemble prediction during blocking, submitted. 55. T.J. O’Kane and J.S. Frederiksen, Comparison of statistical dynamical, square root and ensemble Kalman ﬁlters, submitted. 56. S.A. Orszag, Analytical theories of turbulence J. Fluid Mech. 41, 363–386 (1970). 57. D.F. Parish and J.C. Derber, The national meteorological center’s spectral statistical-interpolation analysis system Mon. Wea. Rev. 120, 1747–1763 (1992). 58. E.J. Pitcher, Application of stochastic dynamic prediction to real data J. Atmos. Sci. 34, 3–21 (1977). 59. I. Proudman and W.H. Reid, On the decay of a normally distributed and homogeneous turbulent vector ﬁeld Philosophical transactions of the Royal Society of London. Ser. A 247, 163–189 (1954). 60. F. Rabier, A. McNally, E. Andersson, P. Courtier, P. Und´en, J. Eyre, A. Hollingsworth and F. Bouttier, The ECMWF implementation of three-dimensional variational assimilation (3D-Var). II: Structure functions Q. J. R. Meteorol. Soc. 124, 1809–1829 (1998). 61. H.A. Rose, An eﬃcient non-Markovian theory of non-equilibrium dynamics Physica D 14, 216–226 (1985). 62. P. Sakov, P.R. Oke and S.P. Corney, On the form of the ensemble transformation matrix in ensemble square root ﬁlters submitted Mon. Wea. Rev. (2006). 63. J. Smagorinsky, Problems and promises of deterministic extended range forecasting Bull. Amer. Meteor. Soc. 50, 286–311 (1969). 64. M.K. Tippett, J.L. Anderson, C.H. Bishop, T.M. Hamill and J.S. Whitaker, Ensemble square root ﬁlters Mon. Wea. Rev. 131, 1485–1490 (2003) 65. Z. Toth and E. Kalnay, Ensemble forecasting at NMC: The generation of perturbations Bull. Amer. Meteor. Soc. 74, 2317–2330 (1993). 66. Z. Toth and E. Kalnay, Ensemble forecasting at NMC: The use of the breeding method for generating perturbations Proc. Tenth Conference on Numerical Weather Prediction, 1994. Portland. 67. Z. Toth and E. Kalnay, Ensemble forecasting at NCEP and the breeding method Mon. Wea. Rev. 125, 3297–3319 (1997). 68. P. Veyre, Direct prediction of error variances by the tangent linear model: A way to forecast uncertainty in the short range? New developments in predictability; Proceedings of a workshop held at ECMWF, Reading U.K., 65–86 (1991). 69. X. Wang and C.H. Bishop, A comparison of breeding and ensemble transform Kalman ﬁlter ensemble forecast schemes J. Atmos. Sci. 60, 1140–1158 (2003). 70. J.S. Whitaker and T.M. Hamill, Ensemble data assimilation without perturbed observations Mon. Wea. Rev. 130, 1913–1924 (2002).

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DISTILLED TURBULENCE. A REDUCED MODEL FOR CONFINEMENT TRANSITIONS IN MAGNETIC FUSION PLASMAS ROWENA BALL Mathematical Sciences Institute and Department of Theoretical Physics The Australian National University, Canberra ACT 0200 Australia, [email protected] Two diﬀerent views of the physics behind plasma conﬁnement transitions are uniﬁed in a reduced, or distilled, model for the coupled dynamics of potential, turbulence, and shear ﬂow energy subsystems. A regime of super-suppressed turbulence is shown to occur at low power input.

1. Introduction Imagine a fractional distillation column for which the feedstock is not crude oil but a ﬂow of high purity and Reynolds number. Then the feedstock is not some natural distribution over hydrocarbon molecular weights but an energy distribution over scales of motion. We know, in principle, how hydrocarbons are separated in the still according to their boiling points (even if we do not work at an oil reﬁnery), but how do we separate and reform the energy components of the pure turbulent ﬂuid? And why should we do this anyway? First, our turbulence reﬁnery does not deﬁne the skyline of a seamy port city in complicated chiaroscuro, but exists only inside the computer. And computation is, in essence, the notorious “problem of turbulence”. In a turbulent ﬂow energy is distributed over wavenumbers that range over perhaps seven orders of magnitude (for, say, a tokamak) to twelve orders of magnitude (for a really huge system, say a supernova). To simulate a turbulent ﬂow in the computer it is necessary to resolve all relevant scales of motion in three dimensions. As pointed out recently1 such calculations would take 400 years at today’s processor speeds, therefore a faster way to do them would be to rely on Moore’s law and wait only 20 years until computers are speedy enough. While we are waiting, we can study various

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classes of reduced, or distilled, dynamical models for turbulent systems: (1) (2) (3) (4)

Metaphors, e.g., predator-prey models. Fourier mode truncations, e.g., the Lorenz equations. Spatial averages with semi-empirical closures. Mixed metaphors, or elements of all of the above.

For this workshop I describe an application of class 3, in which the components of the global dynamics are distilled rather than the motions on individual spatial scales. 2. Two-dimensional turbulence in plasmas Magnetized fusion plasmas are strongly driven dissipative ﬂows in which the kinetic energy of small-scale turbulence can drive the formation of large-scale coherent structures such as sheared mass ﬂows. This tendency to self-organise is typical of ﬂows where Lagrangian ﬂuid elements see globally prevalent or soliton-like two-dimensional velocity ﬁelds, and is a consequence of non-negligible inverse energy cascades.2 The distinctive properties of quasi two-dimensional ﬂuid motion are the basis of natural phenomena such as zonal structuring of planetary ﬂows and the segregation of pebbles washed by the sea on a beach (Fig. 1), but are under-exploited in technologies. The propensity of two-dimensional ﬂows to structure themselves — and hence organise any advected or diﬀusively transported particles or heat — is a fair prospect for achieving directed management of turbulent transport in industrial ﬂows, but it is only in fusion plasmas that this potential has been realized to some extent. In fusion devices the most useful consequence of two-dimensional ﬂuid motion is a dramatic enhancement of sheared mass ﬂows concomitant with suppression of the high wavenumber turbulence at the edge that degrades conﬁnement.3 These low- to high-conﬁnement (L–H) transitions have been investigated intensively since the 1980s. Two major strands in the literature emerged early and are supported by experiments: (1) They are an internal phenomenon that occurs spontaneously when upscale energy transfer from turbulence to shear ﬂows exceeds nonlinear dissipation;4,5 (2) They are due to edge ion orbit losses or induced biasing, the resulting electric ﬁeld providing a torque which drives shear ﬂows nonlinearly.6–8 In this contribution I give a condensed account of how these two diﬀerent strands may be uniﬁed in a reduced dynamical model, the essence of which is distilled from MHD equations using technique (3). The model is developed and strengthened by interrogating degenerate singularities and matching

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(a)

(b)

(c)

(d)

(e)

(f)

Fig. 1. (a) – (d) On Pebbly Beach, NSW, the sea has sorted and segregated the stones according to size. (e) A rare cloud over Canberra. Secondary instabilities grow from the edge of this streaming cloud formation, a fast-moving zonal ﬂow. (f) Sullivans Creek, Canberra. Kinetic energy seems to cascade from both turbulent and laminar ﬂow regions into coherent structures. (Photos by the author.)

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their unfoldings to appropriate physics, using singularity theory methods9 that were ﬁrst applied to dynamical modelling of magnetic fusion plasmas by Ball and Dewar.10 3. Recipe for the reduced model Take the following MHD momentum and pressure convection equations: dv = −∇p + J × B + µ∇2⊥ v + Ω p˜x ˆ − ρν (v − V (x) yˆ) (1) dt dp = χ∇2⊥ p, (2) dt where d/dt = ∂/∂t + v · ∇, together with an incompressibility condition ∇ · v = 0 and a resistive Ohm’s law E + v × B = ηJ. A guide to the symbols and notation used in this section is provided in the table at the end. Equations 1 and 2 are the electrostatic limit of the reduced MHD equations that were originally derived by Strauss.11 They have been shown to describe well the nonlinear dynamics of a high beta, large aspect ratio, tokamak plasma.12–15 Here an additional term ρν (v − V (x) yˆ) has been included in the momentum balance, which breaks shear ﬂow reversal symmetry. ρ

Put them in slab geometry:

Fig. 2. The plasma edge region is −δ < x < δ, with x = δ at the plasma surface. ∇p0 < 0 is the y, z-averaged pressure gradient. · denotes the average over the (y, z) plane.

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Extract the dynamics of the mean ﬂow: From Eq. 1 averaged over the (y, z) plane we obtain ∂t v0 − µ∂x2 v0 + ∂x ˜ vx v˜y = −ν(v0 − V ), ˜ . We multiply by v0 and integrate over x to where v0 = vy , v == v0 + v obtain the evolution of the background shear ﬂow kinetic energy F : 2 dv0 1 δ d 1 δ dx 2 2 v =− dx µ + ν v0 dt δ −δ 2 0 δ −δ dx 1 δ 1 δ dv0 + + dx ˜ vx v˜y dx νV v0 , δ −δ dx δ −δ which we re-write as d F = −F + EF + Eϕ , dt 2 *δ * dv0 1 δ 2 2 where F ≡ 1δ −δ dx v , ≡ dx µ + ν v F 0 0 , EF 2 δ −δ dx * * dv0 1 δ 1 δ vx v˜y dx , and Eϕ ≡ δ −δ dx νV v0 . δ −δ dx ˜

≡

Total kinetic energy: The energy moment of Eq. 1 gives the dynamics of the background shear ﬂow plus ﬂuctuation kinetic energy. d 1 δ dx 2 ˜ p v˜x 1 δ 2 v0 + v˜ = dx Ω dt δ −δ 2 δ −δ ρ 6 2 7 ∂˜ vi 1 δ η . ˜2 / J + µ − dx δ −δ ρm ∂xj 2 δ dv0 1 2 − dx µ + ν v0 δ −δ dx 1 δ + dx νV v0 , δ −δ which we re-write as d [F + N ] = EN − N − F + Eϕ , dt *δ *δ where N ≡ 1δ −δ dx ˜2 , EN ≡ 1δ −δ dx Ω ˜p ρv˜x 2 v 1 . / 2 , -2 * η ∂˜ vi 1 δ ˜2 + µ J dx . δ −δ ρm ∂xj

and N

≡

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Potential energy dynamics: We assume the cross-ﬁeld thermal diﬀusivity can be neglected and obtain the evolution of potential energy in the pressure gradient as: ˜ pv˜x |δ−δ 1 δ d 1 δ ˜ pv˜x p0 = Ω − Ω, dx (−x) Ω dx dt δ −δ ρ ρ δ −δ ρ which we re-write as

d 1 δ P = EP − EN , P ≡ dx (−x) Ω p0 /ρ, dt δ −δ *δ where εP ≡ 1δ −δ dx (−x) Ω p0 /ρ, ε is the speciﬁc heat capacity, and EP ≡ ε

p˜ ˜vx |δ−δ Ω. ρ

Closure and approximations: We need explicit expressions for the averaged rates of energy input, transfer, and loss on the right hand sides. In general, such averaged rate laws for mass or energy conversions cannot be derived from theory but must be postulated using physical arguments and tested by experiments. We make use of the approximations and deﬁnitions given in the table at the end to obtain the closed dynamical system, ε

dP = Q − γN P dt

dN = γN P − αv 2 N − βN 2 dt dv = αv N − µ(P, N )v + ϕ. 2 dt

(10) (11) (12)

The easy way: A schematic showing the energy ﬂuxes through the P , N , and v subsystems is shown in Fig. 3(a). The skeleton dynamical system can be written down directly from the schematic by inspection, and ﬂeshed out using the approximations and deﬁnitions given in the table. 4. Analysis of and modiﬁcations to the distilled model In this work the terms bifurcation structure or bifurcation analysis refer collectively to the stability and singularity analysis of equilibria and periodic

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power source

(a) Q

γ

P

ϕ

α

N

shear flow drive

(b) Q

v’

γ

P

β

µ (N,P)

β

γ

P

χ

v’

κ

ϕ µ (N,P)

r(P)

(c) Q

α

N

α

N

β

κ

ϕ v’

µ (N,P)

(d)

Q

γ

P

χ

α

N

β

κ

ϕ v’

µ (N,P)

Fig. 3. Energy ﬂux schematics through the potential energy P , turbulent kinetic energy N , and shear ﬂow v subsystems. See text for explanations of each subﬁgure. Curly arrows indicate dissipative channels, straight arrows indicate inputs and transfers between energetic subsystems.

solutions and assessment of the structural stability of a dynamical system. The principles of stability theory are well-known; singularity theory less so. A brief exposition is given in the Appendix. Singularity theory is a systematic methodology for characterizing the equilibria of dynamical systems, that involves perturbing around high-order singularities to map the bifurcation landscape. In this paper singularity theory is used as a diagnostic tool to probe the relationship between the bifurcation structure of dynamical models for and the physics of conﬁnement transitions. We study the stability of solutions, interrogate degenerate or pathological singularities, and compute and present key bifurcation diagrams. The bifurcation structure of Eqs 10–12 predicts behaviours observed repeatably in magnetic fusion experiments, such as shear ﬂow suppresion of turbulence, and hysteretic, non-hysteretic, and oscillatory transitions.16,17 However, there are several outstanding issues that suggest the model is still incomplete. This section can be read therefore as a sequel to refs. 16 and 17 in which (unlike most sequels) the most interesting parts of the story are told. The ﬁrst issue is a trapped degenerate singularity, revealed by inquisitive review of the bifurcation structure of Eqs 10–12. It is responsible for an unphysical prediction of the model: that the shear ﬂow increases without bound as the power input is withdrawn. The second issue is the thermal diﬀusivity term that was neglected in the derivation in section 3. It turns out to have profound eﬀects on the bifurcation structure. The third concerns

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the strand (2) view of conﬁnement transitions. I propose a uniﬁed model in which this physics is included as a direct conversion channel between gradient potential energy and shear ﬂow kinetic energy. 4.1. The case of the trapped singularity A bifurcation diagram of the equilibria (or steady states) of Eqs 10–12 is displayed in Fig. 4(a), rendered for all three variables v , P , and N , with the power input Q as the control parameter. Stable and unstable equilibria are marked with solid and dashed lines respectively, and amplitude envelopes of enclaves of periodic solutions, which begin to occur at Hopf bifurcations on the equilibrium curves, are marked by large black dots.∗ Note that P and N , being quadratic energy variables, have a positive domain only, while the shear ﬂow v , as a velocity variable, may be positive or negative. In the P and N renderings, therefore, solution curves are annotated to distinguish between energies that correspond to the +v and −v domains. The distinction is important: if we are interested, for example, in knowing the turbulent kinetic energy N that results from evolving this system to the −v branch in (a), the na¨ive use of the +v value of N gives an underestimate. I highlight two other interesting features that have particular relevance to this discussion. (1) On the −v branch if the power input ebbs below the turning point labeled s0 the shear ﬂow must spontaneously reverse direction to the +v domain. The zoom-in shows a small range of Q over which there are ﬁve steady state solutions, or ﬁvefold multiplicity, comprising three stable and two unstable branch segments. Other kinds of ﬁvefold multiplicity also occur in this system which are qualitatively diﬀerent from this and from each other. They will be shown later. In the remainder of this discussion I ignore the −v domain. (2) Considering the branch of unstable solutions occurring in the top lefthand corner of the v and P renderings, and in the lower left-hand corner of the N rendering, leads us to the crucial issue of unbounded shear ﬂow growth. For ϕ = 0 this branch is nonexistent, but because we are ∗ The

bifurcation diagrams in this work were computed numerically using the continuation code AUTO, http://indy.cs.concordia.ca/auto/. Given a starting equilibrium, AUTO continues the solutions as a function of the selected bifurcation parameter, computes the eigenvalues at each step (and therefore the stability), ﬂags bifurcation points, where it can continue solution branches in a second parameter, and computes the amplitude, period, and stability (using Floquet exponents) of periodic solutions.

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2

(a)

v’

2

0

s0

s0

-2

-2

0.44

0.1 12

(a)

0

0.48

10

+v’

(a)

(a)

P

+v’

+v’ -v’

4

1

N .01

+v’ 0.1

10

+v’

-v’ +v’

+v’

+v’ 0.1

10

3

(b)

2

(c)

v’

1 0

s1 0.1

1 Q 10

s1 100

0.1

1 Q 10

100

Fig. 4. (a) ϕ = 0.05. For clarity in the N and P renderings the limit cycle envelopes are not shown and the solution curves are annotated to indicate whether they correspond to the +v or −v domain. (b) ϕ ≈ 0.08059 = ϕT m . (c) ϕ = 0.11. Other parameters: β = 1, γ = 1, α = 2.4, b = 1, a = 0.3, ε = 1.5.

smoothly increasing ϕ from zero it is more illuminating to consider the branch as representing the interaction of ϕ with the nonlinear viscosity term in Eq. 12. From this point of view the branch is trapped as a singularity — a turning point — at (v , Q) = (∞, 0) for ϕ = 0. As ϕ is increased smoothly from zero the “new” branch, retaining the turning point, is released from its trap and begins to interact with the “old” branches and at ϕT m the new and old branches exchange arms, (b). Singularity theory describes ϕT m as an organizing centre, a loose but useful and evocative name for the highest order singularity in some

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locality of the state variable and parameter space. We can imagine standing on a local organizing centre, like standing on top of a hill, and looking around. In our ﬁeld of view is all of the possible qualitative structure of the system, local to that organizing centre. The relationship between the bifurcation structure around the organizing centre and known strand (2) physics was explored in Ball et al .16 In (c) a transition must still occur at the turning point labeled s1, but classical S-shaped hysteresis is locally forbidden, because ϕT m is classiﬁed as a transcritical bifurcation (see Appendix), the deﬁning conditions for which are given in Eq. A.5. The non-degeneracy condition Gxx = 0 must be violated for classical hystersis to occur. However there is still a degenerate singularity in the bifurcation structure of the system. What is not evident in Fig. 4 (because a log scale is used) is a highly degenerate branch of equilibria that exists at Q = 0 where N = 0 and v = (P 3/2 ϕ)/b; it is shown in Fig. 5(a). (Hopf bifurcations are annotated with asterisks, where limit cycle envelopes are not plotted for clarity.) For ϕ > 0 there is a trapped degenerate turning point, labeled s4, where the new branch crosses the Q = 0 branch. The singularity s4 is nonlocal to the organizing centre because it is not seen from ϕT m , nor do conditions that unfold (or dissolve) ϕT m , i.e., variation of existing parameters of Eqs 10–12, aﬀect s4. This latter attribute of s4 also ensures that it is eﬀectively trapped. To see how the presence of the trapped s4 infers an unphysical extreme it is helpful to track the surviving Hopf bifurcation in Fig. 4(c) as ε is decreased. (Note that ε is a dynamical parameter; variation may change the stability but not the position of the steady states, hence its utility in diagnosis.) Numerical experiments show that the Hopf bifurcation slides along the curve to high v and low Q, the branch of limit cycles shrinks, and by conjecture, is eliminated through a double zero eigenvalue (DZE) pair at (Q, v ) = (0, ∞). Thus the entire solution branch can become stable. The anomaly in this low-capacitance picture is that as Q ebbs the shear ﬂow can grow without bound due to the piece of the viscosity function µ(P, N ) with negative power-law dependence on the pressure gradient. A formal two-timing analysis using ε to examine the dynamics on the the stretched (or shrunken) timescale τ = t/ε leads to the same inference. Thus we have two linked pieces of evidence that some essential physics is still missing from the model: a trapped degenerate singularity, which is unphysical because it persists under perturbation of existing parameters in the model, and the prediction of unbounded shear ﬂow growth.

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(a)

(b)

*

v’

0 -0.1

s4

*

s4 0

0.1

*

0

-0.1

0.1

10

(c)

*

v’ s4

* s2

s3 0

0.01

0.1

(d)

* s4

*

*

Q

1

*

s1

s1 10

0.01

0.1

Q 1

10

Fig. 5. (a) κ = 0, ϕ = 0.08; (b) κ = 0.001, ϕ = 0.08; (c) κ = 0.001, ϕ = 0.083; (d) κ = 0.001, ϕ = 0.084. Other parameters: β = 0.3, α = 2.4, γ = 1, b = 1, a = 0.3, ε = 1.

4.2. Energy must ﬂow in both directions The key to the dissolution (or unfolding) of s4 lies in recognising that in ﬂows where Lagrangian ﬂuid elements (i.e., labeled ﬂuid particles that are tracked by the observer) see local two-dimensional velocity ﬁelds there will be a strong tendency to upscale energy transfer, but the net rate of energy transfer from large-scale structures to high wavenumbers is not negligible. What amounts to an infrared catastrophe in the physics when downscale energy ﬂux is neglected maps to a trapped degenerate singularity in the mathematical structure of the model. A simple, conservative, downscale transfer rate between the shear ﬂow and turbulence subsystems unfolds s4 smoothly: dN = γN P − αv 2 N − βN 2 + κv 2 dt dv = αv N − µ(P, N )v + ϕ − κv . 2 dt

(13) (14)

The enhanced model consists of Eqs 10, 13, and 14, and the corresponding energy ﬂux diagram is Fig. 3(b). The downscale transfer rate coeﬃcient κ may be treated, at this level, simply as a lumped dimensionless parameter. The consequences of unfolding s4 can be appreciated from Fig. 5(b). A

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maximum in v is created because s4 formally is a turning point. At the given values of the other parameters an islet of steady-state solutions is formed, but the bifurcation diagram should be visualized as a slice of a surface where the third coordinate is a second parameter of the system. Two slices of this surface are shown in Fig. 5(c) and (d) where the other turning points are labeled s1, s2, and s3. From the low shear ﬂow solution branch we make the forward transition at s1 and progress through the onset of an oscillatory regime, as in Fig. 4. This segment is henceforth designated as the intermediate shear ﬂow branch, and the islet or peninsula as the high shear ﬂow branch. In (c) a back-transition to the low shear ﬂow state (it is actually not zero) occurs at s2. The islet can only be reached via a transient. In (d) we see a radical diﬀerence in the global dynamics: with diminishing Q the shear ﬂow grows, passes through a second oscillatory regime, reaches the maximum then falls steeply, and the back-transition at s4 occurs at relatively low power input. 4.3. Thermal diﬀusivity is not negligible The next step is to modify Eq. 10 to include a linear thermal energy sink: ε

dP = Q − γN P − χP, dt

(15)

where χ is a lumped dimensionless parameter that represents cross-ﬁeld thermal diﬀusivity18 and other non-turbulent or residual losses.19 The model now consists of Eqs 13, 14, and 15, and the corresponding energy ﬂux schematic is Fig. 3(c). Figure 6 shows a series of bifurcation diagrams computed for increasing values of χ. A qualitative change from Fig. 5(d) is immediately apparent — two new turning points s5 and s6 have appeared in (a). They were born from a local cusp singularity but the global eﬀect is to stoop and shift the entire pensinsular towards higher Q. As in Fig. 5, the system transits to an intermediate shear ﬂow state at s1, but from this branch the eﬀect of decreasing Q is radically diﬀerent: at s6 another discontinuous transition occurs to a high shear ﬂow state on the peninsula. In (b) between s5 and s6 there is a range of ﬁvefold multiplicity, which in (c) has disappeared in a surprisingly mundane way: not through a singularity but merely by the shift of the peninsula toward higher Q. But the shift creates a new and qualitatively diﬀerent range of ﬁvefold multiplicity through the creation of s7 and s8 at another cusp singularity. In (d) s4 and s7 have been annihilated at yet another cusp. Thus there are four cusps, the origins of which can be seen

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(a)

s5

v’ s4

s6

* *

*

s1

0

(d)

s5

s8

s6 s1

*

*

10

s5

(e)

v’

0

10

(e) s1

N

s8

0.1

s8 s1 0.1

s6

Q

s6

.001 100

s5 0.1

Q

100

Fig. 6. (a) χ = 0.005, (b) χ = 0.01, (c) χ = 0.05, (d) χ = 0.1, (e) χ = 0.2. Other parameters: κ = 0.001, ϕ = 0.088, β = 0.3, α = 2.4, b = 1, a = 0.3, γ = 1, ε = 1.

in the 2-parameter diagram of Fig. 7, onto which the lines of s1, s4, s5, s6, s7, and s8 over χ are projected. From the high shear ﬂow branch in Fig. 6(c), (d), and (e), increasing Q induces the system to transit at s5 to a limit cycle rather than to a steady state on the intermediate branch. In (e) we see super-suppression of the turbulence N corresponding to this enormous uptake of energy by the shear ﬂow, but the hard onset of oscillations with the transition at s5 causes the turbulence to rise again dramatically, with other parameters as given. Note that the Hopf bifurcation that is starred near s6 in (a) has vanished in (b)–(e). It coalesces with s6 at a degenerate point characterized by one real zero and a pair of purely imaginary complex conjugate eigenval-

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Fig. 7. In the two black areas there are ﬁve steady states and in the dusted areas there are three steady states.

ues, known as a saddle-node Hopf (SNH) bifurcation. Evidently the SNH bifurcation occurs between 0.005 < χ < 0.01. 4.4. Strands (1) and (2) are uniﬁed Finally I address the strand (2) view of conﬁnement transition physics. The published models for this process have no coupling to the potential energy and turbulence dynamics. Here, in contrast, the ion orbit loss physics associated with edge electric ﬁelds is included as a piece of a holistic picture, so that the model now consists of Eq. 13 and ε

dP = Q − γN P − v 2 r(P ) − χP dt

(16)

dv = αv N − µ(P, N )v + v r(P ) − κv + ϕ. (17) dt 2 simply says that the rate at which The new term r(P ) = ν exp − w2 /P ions are lost, and ﬂow generated, is proportional to a collision frequency ν times the fraction of collisions that give ions suﬃcient energy to escape. The energy factor assumes a Maxwellian ion distribution and w2 is proportional to the square of the critical escape velocity. This is the simpliﬁed essence of expressions used by earlier authors,6,7 except that I have included explicitly the temperature dependence of r through P . The corresponding energy schematic is Fig. 3(d) where it is seen that r(P ) is a competing potential energy conversion channel that can dominate the dynamics when the critical escape velocity w is low or the pressure is high. 2

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(a)

v’

2

* 0

*

0

1

0

0

1

(d)

10

(c)

10

(b)

4

N

v’

0.1

0.1 .001

0.1

1

Q

10

100

0.1

1

Q

10

100

Fig. 8. (a) ν = 0.015, (b), (c), and (d) ν = 0.05. Other parameters: χ = 0.01, κ = 0.001, ϕ = 0.088, β = 0.3, γ = 1, α = 2.4, b = 1, a = 0.3, w = 1, ε = 1.

This is exactly what we see in the bifurcation diagrams, Fig. 8. The high shear ﬂow peninsula is elongated and ﬂattened. The Hopf bifurcations in (a), where r(P ) is small, have disappeared in (b) through a DZE, leaving the intermediate branch unstable until the remaining Hopf bifurcation (c) is encountered. In the transition region the bifurcation diagram begins to look more like the simple S-shaped, cubic normal form schematics featured in numerous papers by earlier authors.6,7 However, this uniﬁed model accounts for shear ﬂow suppression of the turbulence (d), whereas theirs could not. 5. Discussion In summary, the strands (1) and (2) physics of plasma conﬁnement transitions are smoothly uniﬁed for the ﬁrst time in a physics-based, predictive dynamical model, analysis of which distils the global dynamics of the system. With incorporation of a simple term for downscale energy ﬂux, allowing eﬀective small scale dissipation independent of the temperature, when the direct shear ﬂow drive ϕ is high enough to join the islet to the peninsula, the shear ﬂow can rise as the power input ebbs. There is concomitant supersuppression of turbulence and a back-transition at very low power input.

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A report in,20 where a parameter equivalent to ϕ was used as the control parameter, supports the super-suppression of turbulence found here. Nonturbulent dissipation introduces the capacity for discontinuous transitions to a super-suppressed turbulence regime, but oscillations may onset. These results suggest new and interesting—and potentially cost-saving in terms of heating power—strategies for controling conﬁnement in fusion experiments. Appendix A. Singularity theory A non-technical reference on singularity theory that is accessible to the physics reader is Ball,9 which also gives other key references on the mathematics and physical applications. Systematic bifurcation, singularity, and stability analysis provide qualitative information on the global behaviour of a dynamical system† , in the following sense: In attempting to understand the behavior of a complex system with many degrees of freedom a standard approach is to begin by making a simpliﬁed model of the process — a dynamical system — and within the conﬁnes of that model analyse what it has to tell us about the process. Yet very few dynamical systems have known, exact solutions. For the vast majority it cannot even be proved that general solutions exist. Qualitative methods, pioneered by Poincar´e in his work on the restricted three-body problem, can tell us how solutions would behave, assuming they do exist, as parameters are varied. For an experimental or real-world macroscopic dynamical system we are often interested in questions that qualitative analysis can answer, rather than numerical solutions. Is it capable of discontinuous, periodic, or chaotic dynamics? Can we draw the bounds of this behaviour in parameter space, so that we may design or manage experiments to include, forbid, or modify such action? We can also use qualitative analysis to improve the model itself. In the following section these capabilities of singularity theory are illustrated using the model described by Eqs 10–12. Persistent, degenerate singularities inform us about physics: For a dynamical system dx = F (x, λ1 , · · · λn ) dt † This

means the behaviour over the m + n-dimensional space of m dynamical state variables and n independent parameters.

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where the components of x are dynamical state variables and the λi are parameters, the equilibria are found by setting the right hand sides to zero and solving to obtain a single algebraic equation in terms of one of the state variables x: G(x, λ1 , · · · λn ) = 0.

(A.2)

The function G(x, λ1 , · · · λn ) is called the bifurcation problem. Solutions of (A.2) that also satisfy the additional condition Gx = 0,

(A.3)

where the subscript notation denotes partial diﬀerentiation with respect to x, are called singular points or singularities. Points where conditions (A.2) and (A.3) hold and, in addition, one or more higher order partial derivatives is zero are known as degenerate or higher order singularities. An example of a degenerate singularity that has physical importance in the current context is the pitchfork, for which the deﬁning and nondegeneracy conditions are G = Gx = Gxx = Gλ1 = 0, Gxxx = 0, Gxλ = 0.

(A.4)

Applying these conditions to Eqs 10–12 we ﬁnd, with the aid of some computer algebra, the unique pitchfork P at & α2 γ 2 2α3 γ α/a √ (v , Q, β, ϕ) = 0, 2 , ,0 . (P) 9a b 27 3 a2 b At P the two non-degeneracy conditions in Eq. A.4 evaluate as gP Q = 8a/α, gP P P = −18aγ 2/(αβ). We see that the pitchfork is a twice-degenerate, or codimension 2, singularity because two parameters in addition to the principal bifurcation parameter Q are required to deﬁne it. The bifurcation diagram for the equilibria of Eqs 10–12 at the critical values of the dissipative parameter β and and the symmetry-breaking parameter ϕ is shown in in Fig. A1(a). In (b) and (c) β is relaxed either side of the critical value, but ϕ is held at zero. The other singularity T on v = 0 in Fig. A1 satisﬁes the deﬁning and non-degeneracy conditions for a transcritical bifurcation, G = Gx = Gλ1 = 0, Gxx = 0,

det

Gxx Gλ1 x Gxλ1 Gλ1 λ1

≡ det d2 G < 0. (A.5)

It is once-degenerate but also requires the symmetry-breaking parameter for full determination.

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Fig. A1. Bifurcation diagrams showing the fully degenerate (a) and partially unfolded (b) and (c) pitchfork P. (a) β = βcrit ≈ 18.58, (b) β = 50, (c) β = 1. Other parameters: ϕ = 0, γ = 1, b = 1, a = 0.3, α = 2.4, ε = 1.5.

We see from Fig. A1 that the pitchfork is persistent to variations in β. (In fact it is persistent to variations in all parameters of Eqs 10–12 other than ϕ. Persistence of a degenerate singularity when there are not enough independent parameters to unfold it was not recognized in some previous models for conﬁnement transitions, where such points were wrongly claimed to represent second-order phase transitions.) Furthermore, as long as the degenerate singularity P persists the model cannot be predictive near it. This may be viewed as an overdetermination problem: with ϕ ﬁxed at zero we see from (A.4) there are four deﬁning conditions but we have only three variable quantities, v , Q, and β. Typically the pitchfork is associated with a fragile symmetry in the physics of the modelled system. In this case the symmetry is obvious from Fig. A1: in principle the shear ﬂow can be in either direction equally. In real life (or in numero), experiments are always subject to perturbations that determine a preferred direction for the shear ﬂow (such as friction with neutrals, or any other asymmetric shear-inducing mechanism), and the pitchfork is inevitably dissolved, or unfolded. In the bifurcation diagrams of Figs 4, 5, 6, and 8 the pitchfork in Fig. A1 is fully unfolded, giving us a more realistic picture of conﬁnement transition dynamics. More generally, in analysing a dynamical model we are interested in the mapping between the bifurcation and stability structure and the physics of the process the model is supposed to represent. If we probe this relationship we ﬁnd that degenerate singularities correspond to some essential physics (such as fulﬁlling a symmetry-breaking imperative, or the onset of hysteresis, or resolving an “infrared catastrophe” type of anomaly), or they are pathological. In the ﬁrst case we can usually unfold the singularity in a physically meaningful way; in the other case we know that something is amiss and we should revise our assumptions.

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Deﬁnitions of symbols, notation, and approximations. ˜ v = B10 ˆ z × ∇φ = v0 + v v0 = v ˜ v p = p0 + p˜ p0 = p p˜ B0 ρ µ η ν Ω ≡ dΩ/dx > 0 ∇2⊥ χ V p0 (x) p0(x=δ) + xdp0 /dx ≡ xdp0 /dx v0 (x) v0(x=δ) + xdv0 /dx ≡ xdv0 /dx √ v = ± F Ep ≡ Q = constant EN γ/P N EF αF N

N βN 2

F µ(P, N )F µ(P, N ) = bP −3/2 + aP N Eϕ ϕF 1/2 ϕ δνV

E × B ﬂow velocity average background component ﬂuctuating or turbulent component plasma pressure average background component ﬂuctuating or turbulent component magnetic ﬁeld along the z axis average mass density of ions, constant ion viscosity coeﬃcient resistivity frictional damping coeﬃcient average ﬁeld line curvature, constant ∂x2 + ∂y2 cross-ﬁeld thermal diﬀusivity coeﬃcient external ﬂow † †

poloidal shear ﬂow power input to the pressure gradient turbulence growth rate transfer rate due to the Reynolds stress turbulent energy dissipation rate damping rate due to viscosity neoclassical and turbulent viscosities shear ﬂow driving rate

†

The constant terms p0(x=δ) and v0(x=δ) are dropped in the deﬁnitions of p0 (x) and v0 (x).

Acknowledgments This work is supported by the Australian Research Council. References 1. E. S. Oran and V. N. Gamezo, Origins of DDT in gas-phase combustion. Preprint, NRL Laboratory for Computational Physics and Fluid Dynamics, Washington DC, 2006. 2. R. H. Kraichnan and D. Montgomery, Two-dimensional turbulence. Reports on Progress in Physics, 43, 547–619 (1980). 3. P. W. Terry, Suppression of turbulence and transport by sheared ﬂow. Reviews of Modern Physics, 72(1), 109–165 (2000).

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4. P. H. Diamond and Y.-B. Kim, Theory of mean poloidal ﬂow generation by turbulence. Physics of Fluids B, 3, 1626–1633 (1991). 5. E-J. Kim, P. H. Diamond, and T. S. Hahm, Transport reduction by shear ﬂows in dynamical models. Physics of Plasmas, 11(10), 4554–4558 (2004). 6. S.-I. Itoh and K. Itoh, Model of L- to H-mode transition in tokamak. Physical Review Letters, 60(22), 2276–2279 (1988). 7. K.C. Shaing and E.C. Jr Crume, Bifurcation theory of poloidal rotation in tokamaks: a model for the L–H transition. Physical Review Letters, 63(21), 2369–2372 (1989). 8. S Magni, C. Riccardi, and H. E. Roman, Statistical investigation of transport barrier eﬀects produced by biasing in a nonfusion magnetoplasma. Physics of Plasmas, 11(10), 4564–4572 (2004). 9. R. Ball, Singularity theory. In Alwyn Scott, editor, Encyclopedia of Nonlinear Science, New York, 2005. Routledge. 10. R. Ball and R. L. Dewar, Singularity theory study of overdetermination in models for L–H transitions. Physical Review Letters, 84(14), 3077–3080 (2000). 11. H. R. Strauss, Dynamics of high β tokamaks. The Physics of Fluids, 20(8), 1354–1360 (1977). 12. B. A. Carreras, L. Garcia, and P. H. Diamond, Theory of resistive pressuregradient-driven turbulence. Physics of Fluids, 30(5), 1388–1400 (1987). 13. H. Sugama and M. Wakatani, A transport study for resistive interchange mode turbulence based on a renormalized theory. J. Phys. Soc. Japan, 57(6), 2010–2025 (1988). 14. H. Sugama and W. Horton, Shear ﬂow generation by Reynolds stress and suppression of resistive g modes. Physics of Plasmas, 1(2), 345–355 (1994). 15. H. Sugama and W. Horton, Transport suppression by shear ﬂow generation in multihelicity resistive-g turbulence. Physics of Plasmas, 1(7), 2220–2228 (1994). 16. R. Ball, R. L. Dewar, and H. Sugama, Metamorphosis of plasma shear ﬂow– turbulence dynamics through a transcritical bifurcation. Physical Review E, 66, 066408–1–066408–9 (2002). 17. R. Ball, Suppression of turbulence at low power input in a model for plasma conﬁnement transitions. Physics of Plasmas, 12, 090904–1–8 (2005). 18. S. I. Braginskii, Transport processes in a plasma. In M. A. Leontovich, editor, Reviews of Plasma Physics, volume 1. Consultants Bureau, New York, 1965. 19. A. Thyagaraja, F. A. Haas, and D. J. Harvey, A nonlinear dynamical model of relaxation oscillations in tokamaks. Physics of Plasmas, 6(6), 2380–2392 (1999). 20. Y. and Kishimoto, Plasma Physics and Controlled Nuclear Fusion Research (IAEA-CN-60/D-10, Vienna, 3, 299 (1994).

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ZONAL FLOW GENERATION BY MODULATIONAL INSTABILITY ROBERT L. DEWAR AND R. F. ABDULLATIF Department of Theoretical Physics, Research School of Physical Sciences and Engineering, The Australian National University, Canberra ACT 0200, Australia [email protected] This paper gives a pedagogic review of the envelope formalism for excitation of zonal ﬂows by nonlinear interactions of plasma drift waves or Rossby waves, described equivalently by the Hasegawa–Mima (HM) equation or the quasigeostrophic barotropic potential vorticity equation, respectively. In the plasma case a modiﬁed form of the HM equation, which takes into account suppression of the magnetic-surface-averaged electron density response by a small amount of rotational transform, is also analyzed. Excitation of zonal mean ﬂow by a modulated wave train is particularly strong in the modiﬁed HM case. A local dispersion relation for a coherent wave train is calculated by linearizing about a background mean ﬂow and used to ﬁnd the nonlinear frequency shift by inserting the nonlinearly excited mean ﬂow. Using the generic nonlinear Schr¨ odinger equation about a uniform carrier wave, the criterion for instability of small modulations of the wave train is found, as is the maximum growth rate and phase velocity of the modulations and zonal ﬂows, in both the modiﬁed and unmodiﬁed cases.

1. Introduction As of January 1, 2006, Wikipedia1 introduces the term “zonal ﬂow” thus: Fluid ﬂow is often decomposed into mean and deviation from the mean, where the averaging can be done in either space or time, thus the mean ﬂow is the ﬁeld of means for all individual grid points. In the atmospheric sciences, the mean ﬂow is taken to be the purely zonal ﬂow of the atmosphere which is driven by the temperature contrast between equator and the poles. In geography, geophysics, and meteorology, zonal usually means ‘along a latitude circle’, i.e. ‘in the east-west direction’. In atmospheric sciences the zonal coordinate is denoted by x, and the zonal wind speed by u.

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(a)

(b)

Fig. 1. (a) NASA image PIA04866: Cassini Jupiter Portrait, a mosaic of 27 images taken in December 2000 by the Cassini spacecraft. (b) A simulation of plasma potential ﬂuctuations in a tokamak cut at a ﬁxed toroidal angle as produced by the GYRO code (courtesy Jeﬀ Candy http://fusion.gat.com/theory/pmp/). Note that, in the plasma case, zonal ﬂows are in the y-direction when slab geometry is used.

The above deﬁnition omits the further qualiﬁcation that zonal ﬂows are restricted to bands (zones) of latitude. This is most clearly seen in the banded cloud patterns on Jupiter (see left panel of Fig. 1) where the magnitude, and even sign, of the zonal ﬂows varies with latitude in a quasiperiodic fashion. (On Earth, topographic variations like mountain ranges disrupt the zonal symmetry.) The term “zonal ﬂow” has also recently come to be much used in toroidal magnetic conﬁnement plasma physics (see e.g. the review of Diamond et al.2 ) to refer to a mean poloidal ﬂow with strong variation in minor radius. The sheared nature of this ﬂow is thought to have the strongly beneﬁcial effect of reducing radial transport by suppressing turbulence, thus improving the conﬁnement of heat required to achieve fusion conditions. The use of the same phrase “zonal ﬂow” in the context of both geophysics and magnetic plasma conﬁnement is no coincidence, as the existence of strong analogies between these ﬁelds has become well recognized.3 In this paper we work in the plasma context, but point out the relation to the geophysical context when appropriate.

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The right panel of Fig. 1 depicts a section of a simulated tokamak, showing turbulence excited by gradients in temperature and density (the plasma being hotter and denser in the middle section, which has been cut out to aid in simulation and visualization). The radial coherence length of the driftwave eddies has been reduced by spontaneously excited zonal ﬂows. The magnetic ﬁeld is predominantly in the toroidal direction, but there is some component in the poloidal direction so that the magnetic ﬁeld lines wind helically around the torus, mapping out nested toroidal magnetic surfaces that conﬁne the magnetic ﬁeld in topologically toroidal magnetic ﬂux tubes. The helical nature of the magnetic ﬁeld lines can be seen in the ﬁgure from the fact that the turbulent eddies have their cores essentially aligned with the magnetic ﬁeld, making the turbulence quasi-two-dimensional despite the manifestly three-dimensional nature of the tokamak. A generalized polar representation would clearly be most appropriate for representing the cross-sectional plane of the torus, but, for the purpose of gaining physical insight with a minimum of formalism in this paper we use slab geometry. That is, the toroidal magnetic surfaces are imagined as ﬂattened into planes, so that Cartesian coordinates, x, y, z, can be used, with y and z replacing the poloidal and toroidal angles, respectively, and x the minor radius. The x and y directions are indicated in the right panel of Fig. 1. The slab approximation is the analogue of the β-plane approximation in geophysics, but note the axis convention is opposite to that used in geophysics, with y now the zonal direction. Modulational instability of drift waves (the analogue of planetary Rossby waves) is a strong candidate4–9 for generating these zonal ﬂows through a feedback mechanism, in which modulations of the wave envelope excite zonal ﬂows through a nonlinear mechanism (Reynolds stress) and the zonal ﬂows enhance the modulation through a self-focusing mechanism. It is the aim of this paper to elucidate this theory in a pedagogic way using as simple a plasma description as possible, namely the one-ﬁeld Hasegawa–Mima equation.10,11 This provides a simple theoretical starting point for describing the nonlinear interaction of drift waves and zonal ﬂows.5,11 The same equation also describes Rossby wave turbulence in planetary ﬂows in the quasigeostrophic and barotropic approximations.11–13 To emphasize its geophysical connections, we shall follow a common practice in the plasma physics literature and call the original form of the Hasegawa– Mima equation the Charney–Hasegawa–Mima (CHM) equation (although in the geophysical literature the equation is called the “quasigeostrophic barotropic potential vorticity equation”14 ).

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Unfortunately for the interdisciplinarity aﬀorded by the use of the CHM equation, it became recognized in the early ’90s15,16 that a corrected form for toroidal plasma applications should be used, which we shall call the Modiﬁed Hasegawa–Mima (MHM) equation. Although the modiﬁcation seems at ﬁrst glance to be minor, we shall show that it makes a profound diﬀerence to the modulational stability analysis because it enhances the generation of zonal ﬂows. Some early works on modulational instability of drift waves can be found in Refs. 17–19 but these predate the recognition of the need to use the MHM equation to enhance the nonlinear eﬀect of zonal ﬂows in a toroidal plasma. Both Majumdar18 and Shivamoggi19 add a scalar nonlinearity, arising from polarization drift and/or temperature gradient, to the CHM equation in order to ﬁnd a nonlinear frequency shift. In the Mima and Lee17 paper, the nonlinear frequency shift comes from time-averaged ﬂow and density proﬁle ﬂattening. In Sec. (2) we introduce the CHM and MHM equations and in Sec. (4) we introduce the generic form of the nonlinear Schr¨ odinger equation, which describes the time evolution of modulations on a carrier wave, and use it to derive a criterion for modulational instability. In Sec. (5) we use the MHM and CHM equations to derive the nonlinear frequency shift of a ﬁnite-amplitude drift/Rossby wave and use it to determine the criteria for modulational instability of drift and Rossby waves, respectively. Section (6) contains conclusions and directions for further work. 2. The CHM and MHM equations The Charney–Hasegawa–Mima equation (CHM)11,13 is an equation for the evolution in time, t, of the electrostatic potential ϕ(x, y, t) (or, in the Rossby wave application, the deviation of the atmospheric depth from the mean11 ). Here x and y are Cartesian coordinates describing position in a two-dimensional domain D, representing a cross section of a toroidal plasma with a strong magnetic ﬁeld, B, predominantly in the z-direction (unit vector zˆ). In the slab model we take D to be a rectangle with sides of length Lx and Ly . A circular domain would clearly be more realistic geometrically because it has a unique central point, representing the magnetic axis, but it is unlikely to add any qualitatively new physics. In strongly shaped tokamaks, like the one depicted in Fig. 1, one might be tempted to give D the noncircular shape of the plasma edge to add yet more realism. However, we caution against this line of thinking because each point in D represents an

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extended section of ﬁeld line, over which the drift-wave amplitude is signiﬁcant. That is, D does not represent any given cross section of the tokamak, but rather a two-dimensional surface in a ﬁeld-line coordinate space (see e.g. Ref. 20), onto which behaviour in the third dimension is projected. We assume the ion temperature to be negligible with respect to the electron temperature Te , assumed constant throughout the plasma. The strong magnetic ﬁeld allows the plasma to support a cross-ﬁeld gradient in the time-averaged electron number density, n ¯ , and the wave dynamics is taken to be suﬃciently slow that, along the ﬁeld lines, the electrons respond adiabatically to ﬂuctuations in ϕ. That is, on a given ﬁeld line they remain in local thermodynamic equilibrium, with distribution function f (r, v, t) = const exp(−E/Te ), where Te is the electron temperature in energy units (eV) and E is the total electron energy 12 mv 2 − eϕ, with m the electron mass and e the electronic charge. Following Hasegawa and Mima, the shear in the magnetic ﬁeld is assumed very weak, so that z-derivatives and the parallel component, k , of the wave vector k can be neglected, thus reducing the problem to a two-dimensional one. However, the existence of magnetic shear is crucial in one qualitative respect—the foliation of the magnetic ﬁeld lines into nested toroidal magnetic surfaces (x = const in slab geometry). Field lines cover almost all magnetic surfaces ergodically, so the constant in the expression for the distribution function is a surface quantity. Integrating over velocity we ﬁnd

eϕ˜ eϕ˜ + O(ϕ˜2 ) , (1) = n0 1 + n = n0 (x, t) exp Te Te where we have decomposed ϕ into a surface-averaged part, ϕ(x, t) ≡ P ϕ(x, y, t) (absorbed into n0 ), and the surface-varying part, ϕ˜ ≡ P ϕ ≡ ϕ−ϕ. Here we have used the magnetic-surface-averaging operator P deﬁned in slab geometry by Ly 1 P·≡ dy · , Ly 0 and its complementary projector P ≡ 1 − P . (Note that P and P commute with ∂t and ∇.) Equation (1) can also be derived purely from ﬂuid equations, without introducing the distribution function explicitly. One can show, by surface-averaging the continuity equation for the electron ﬂuid in the absence of sources or sinks, that the surface-averaged electron density is independent of t. Thus, to O(ϕ), ˜ n0 is independent of t and equals the prescribed average density n ¯ . This would not be the case if we

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had not subtracted oﬀ ϕ in Eq. (1), and in this we diﬀer from Hasegawa and Mima but follow most modern practice since Dorland et al.15,16 pointed out the importance of modifying the electron response in this way. Use of Eq. (1) leads to what we shall call the Modiﬁed Hasegawa–Mima equation (MHM equation). Deﬁning a switch parameter s such that s = 0 selects the original CHM equation and s = 1 the MHM equation, and a stream function ψ ≡ ϕ/B0 , we write Eq. (10) of Hasegawa et al.11 as

d ζ eB0 ˜ ωci + − (ψ + δs,0 ψ) = 0 , (2) ln dt n0 ωci Te where d/dt ≡ ∂t + vE ·∇, with vE ≡ −

∇ϕ×ˆ z = ˆz×∇ψ , B0

(3)

ˆ·∇×vE = ∇2 ψ the vorticity, being the E×B velocity (SI units), ζ ≡ z ˆ ∂x + y ˆ ∂y the perpenωci ≡ eB0 /mi the ion cyclotron frequency, and ∇ ≡ x dicular gradient. As shown in the Appendix of Meiss and Horton,13 this is an approximate form of Ertel’s theorem for the conservation of potential vorticity under Lagrangian advection at the E×B velocity. Note that the MHM equation satisﬁes the expected∗ Galilean invariance under boosts ˆ (so ψ = ψ − V x), in the poloidal direction, y = y − V t, E = E + V B0 x whereas the original CHM equation does not and is therefore unsatisfactory for plasma physics purposes. We now rewrite Eq. (2) in a more explicit way [cf. Eq. (1) of Smolyakov et al.5 ] (∂t + vE ·∇ + v∗ ·∇)(ψ˜ + δs,0 ψ) − (∂t + vE ·∇)ρ2s ∇2 ψ = 0 ,

(4)

−1 where the characteristic drift-wave scale length ρs ≡ ωci (Te /mi )1/2 is the sound speed divided by ωci , and the electron diamagnetic drift † is deﬁned by

v∗ ≡ − ∗ Even

Te zˆ×∇n0 . eB0 n0

in the absence of topography, we do not expect Galilean invariance in geophysical application of the CHM equation, as the β-plane is not an inertial frame. In the plasma conﬁnement application, a poloidal boost in polar coordinates would also be to a rotating frame, but the slab approximation implies we should ignore any Coriolis eﬀects and Galilean invariance in the poloidal direction should apply. † v ≡ |v | is the analogue of β in the geophysical application of the CHM equation. ∗ ∗

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The ordering in Ref. 11 makes all terms in Eq. (4) of the same order. Thus, balancing ∂t ϕ and ∂t ρ2s ∇2 ϕ2 we see that ρs is indeed the characteristic scale length for spatial ﬂuctations. Balancing ∂t ϕ and v∗ ·∇ϕ we see that the characteristic time scale is ρs /v∗ , and balancing vE and v∗ we see that the characteristic amplitude of potential ﬂuctuations is (Te /e)ρs /Ln , where Ln is the scale length for radial variation of n0 . We assume ρs /Ln 1, so the waves have small amplitudes compared with the thermal potential. However, kξ, with k a typical ﬂuctuation wavelength, and ξ a typical displacement of a ﬂuid element by the waves, can be order unity, and thus the equation can describe strong turbulence. Projecting Eq. (4) with P and P we can split it into two equations, one for the surface-varying part and one for the zonal-ﬂow part (∂t + ˆ z×∇ψ·∇)(1 − ρ2s ∇2 )ψ˜ + [v∗ − zˆ×∇(δs,0 − ρ2s ∇2 )ψ]·∇ψ˜ 2˜ ˜ = ρ2 P ˆz×∇ψ·∇∇ ψ

(5)

2˜ ˜ ∂t (δs,0 − ρ2s ∇2 )ψ = ρ2s P ˆz×∇ψ·∇∇ ψ.

(6)

s

In the MHM case, s = 1, Eq. (6) reduces to Eq. (2) of Ref. 5, ∂t ∇2 ψ = 2˜ ˜ ψ. −P zˆ×∇ψ·∇∇ Although physically an approximation, we shall in this paper regard the CHM/MHM equations as given and treat them as exact equations even for the mean ﬂow component of ψ, which we assume to vary on longer and slower length and time scales than assumed in the maximal balance ordering discussed above. 3. Waves and mean ﬂow Assuming there is a scale separation between ﬂuctuations and mean ﬂow, we introduce an averaging operation · which ﬁlters out the ﬂuctuating, wavelike component of whatever it acts on, leaving only a slowly varying component related to the mean ﬂow. This operation can be realized explicitly by convolution with a smooth, bell-shaped kernel of width (in time and space) long compared with the ﬂuctuation scale but short compared with the mean ﬂow scale. Alternatively we can deﬁne it implicitly via the test-function formalism introduced in Appendix A of Ref. 22. Either way, averaging can be shown to commute with ∂t and ∇ to all orders in , the ratio of ﬂuctuation scales to mean-ﬂow scales. We then split ψ into a slowly varying mean ﬂow part, ψ0 ≡ ψ, and a ﬂuctuating part, ψ1 ≡ ψ − ψ.

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Fig. 2. Visualization showing disruption (lower right) of a coherent wave train (lower left) through generation of zonal ﬂows by modulational instability in a tokamak simulation. (Courtesy of Z. Lin, http://w3.pppl.gov/~zlin/visualization/. See also Refs.21 and.6 )

Note that, except when the mean ﬂows are purely zonal, · is distinct from the surface averaging operation P · we used to set up the MHM equation. In this we diﬀer from Champeaux and Diamond,9 who, in eﬀect, take P to be the same as · irrespective of the direction of the mean ﬂows. We consider the case of modulations carried on a coherent wave (see e.g. Fig. 2), rather than a broad turbulent spectrum.23 (As the CHM and MHM equations include no drift-wave instability mechanism, the origin of this wave is outside the theory—it is an initial condition.) Taking, for simplicity, v∗ to be a global constant we assume the carrier wave (also called the pump wave in some analyses) to be a plane wave and write ψ1 = A(r, t) exp(ik·r − ωk t) + c.c. ,

(7)

where A is a slowly varying complex amplitude and c.c. denotes complex conjugate. To begin, we take A and the mean ﬂow, vE , to be constant and treat the carrier wave using linear theory. (Nonlinear eﬀects will be discussed in Sec. 4.) Linearizing Eq. (4) we ﬁnd the dispersion relation in the CHM case

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to be ωk =

ky v∗ ρ2s k 2 + k·vE , 1 + ρ2s k 2 1 + ρ2s k 2

(8)

whereas, in the MHM case, s = 1, and assuming purely zonal mean ﬂow ˆ ), it is (vE = v¯E y ωk =

ky v∗ + k·vE . 1 + ρ2s k 2

(9)

In the latter case, the mean ﬂow causes a simple Doppler shift of frequency, but for the unmodiﬁed CHM equation the Doppler shift is reduced by a factor ρ2s k 2 /(1 + ρ2s k 2 ). We shall use the frequency shift due to mean ﬂow to calculate the nonlinear frequency shift. Otherwise we can ignore it. The group velocity, vg ≡ ∂ωk /∂k, in the absence of a mean ﬂow, is the same in both cases ˆ y 1 ∂ωk 2ρ2s kky = − . 2 2 v∗ ∂k 1 + ρs k (1 + ρ2s k 2 )2

(10)

We shall also need the dispersion dyadic ∇k ∇k ωk ˆ k + ky I 1 ∂ 2 ωk ky kk kˆ y+y = 8ρ4s − 2ρ2s , 2 2 3 v∗ ∂k∂k (1 + ρs k ) (1 + ρ2s k 2 )2

(11)

where I is the unit dyadic. 4. Nonlinear Schr¨ odinger equation and modulational instability We largely follow the simple introduction to modulational instability theory odinger equation given in Dewar et al.,24 starting with the nonlinear Schr¨

∂ ∂ωk 1 ∂ 2 ωk + ·∇ A = ∆ω[|A|]A − :∇∇A , (12) i ∂t ∂k 2 ∂k∂k where ∂/∂k denotes the gradient in k-space and ∆ω the nonlinear frequency shift, a nonlinear functional of the amplitude |A| (cf. e.g. Ref. 25). If the scale length of the modulations is O(−1 ) compared with the wavelength of the carrier, then the vg ·∇ term on the LHS of Eq. (12) is O() whereas the ∇k vg :∇∇ term on the RHS is smaller, O(2 ). Assuming the nonlinear frequency shift to be of the same order, we see that Eq. (12) expresses the fact that, on a short time scale, modulations simply advect with the group velocity, while on a longer timescale the nonlinear frequency shift causes a slow drift in the phase while the dispersion dyadic ∂ 2 ωk /∂k∂k causes spreading of the modulations.

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An amplitude-modulated wave can be represented as the sum of the unmodulated carrier wave and upper and lower sidebands A = A0 exp(−i∆ω0 t) 9 8 × 1 + a+ exp(iK·r − iΩt) + a∗− exp(−iK·r + iΩ∗ t) , where ∆ω0 ≡ ∆ω[|A0 |]. Linearizing in |a± |, |A| = |A0 |[1 + 12 (a+ + a− ) exp i(K·r − Ωt) + c.c.], and using this in Eq. (12) we ﬁnd   KK ∂ 2 ωk 1 1 k − : − δω − δω Ω − K· ∂ω K K ∂k 2 ∂k∂k 2 2  a+  1 KK ∂ 2 ωk 1 a− k Ω − K· ∂ω 2 δωK ∂k + 2 : ∂k∂k + 2 δωK = 0,

(13)

where δωK (denoted α∆ω0 in Ref. 24) is deﬁned by δ∆ω exp(iK·r) . δωK ≡ |A0 | exp(−iK·r) d2 x δ|A|

(14)

Setting the determinant of the matrix in Eq. (13) to zero gives the dispersion relation for plane-wave modulations

2

∂ 2 ωk ∂ 2 ωk 1 1 ∂ωk = KK: Ω − K· δωK + KK: . (15) ∂k 2 ∂k∂k 2 ∂k∂k The criterion for modulational instability is that Ω be complex, Ω = Ωr +iΓ, Γ > 0, and from Eq. (15) we immediately see that this occurs, for suﬃciently small K, if and only if there exist directions for K in which δωK KK:

∂ 2 ωk < 0. ∂k∂k

(16)

5. Nonlinear frequency shift The nonlinear frequency shift ∆ω in a general ﬂuid or plasma is composed of two parts. The ﬁrst is that due to the intrinsic nonlinearity of the medium and the second is that due to Doppler-like shifts [see Eqs. (8) and (9)] associated with nonlinearly induced mean ﬂows. However, in the case of drift or Rossby waves described by the CHM or MHM equations, the intrinsic nonlinear frequency shift is zero (or, at most, of higher order than quadratic). To see this, consider the terms in Eq. (4) describing nonlinear wave-wave (including self) interactions: {ψ1 , ψ˜1 + δs,0 ψ 1 } and {ψ1 , ∇2 ψ1 }, where the Poisson bracket of two functions f and g is deﬁned by {f, g} ≡ ˆ z×∇f ·∇g = ∂x f.∂y g − ∂x g.∂y f .

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Since we assume ky = 0, ψ 1 is zero (to leading order, at least) and ψ˜1 = ψ1 . As the Poisson bracket {f, f } ≡ 0 for any f , the ﬁrst nonlinear self-interaction term vanishes. Similarly, because we are considering a monochromatic carrier wave, the second self-interaction term also vanishes to leading order: {ψ1 , ∇2 ψ1 } ≈ −k 2 {ψ1 , ψ1 } ≡ 0. For the calculation of the nonlinearly excited mean ﬂows, we will need to evaluate the above term more accurately, which is best done via the useful identity 3 4 (17) {f, ∇2 f } = ∂x ∂y (∂x f )2 − (∂y f )2 − ∂x2 − ∂y2 (∂x f.∂y f ) . (Some earlier discussion of this identity can be found in Ref. 26.) Averaging Eqs. (5) and (6) over the ﬂuctuation scale, ˜ = ρ2 P{ψ˜1 , ∇2 ψ˜1 } , (18) (∂t + v∗ ·∇)(1 − ρ2 ∇2 )ψ s

s

∂t (δs,0 − ρ2s ∇2 )ψ = ρ2s P {ψ˜1 , ∇2 ψ˜1 } .

(19)

For both the CHM and MHM cases, the small ρ2s ∇2 = O(ρ2s K 2 ) term on the LHS in Eq. (18) is negligible compared with 1. 5.1. Modulational instability for MHM equation However, on the LHS of Eq. (19) the leading term 1 does not occur in the MHM case, s = 1, so the ρ2s ∇2 term must be retained. Consequently, in this ˜ is smaller than ψ by a factor O(ρ2 K 2 ). That is, the mean ﬂow, case ψ s ∝ˆ z×K, is predominantly zonal so we lose no real generality in assuming ˜ = 0. Then, using the identity Eq. (17), dividing ˆ and setting ψ K = Kx 2 Eq. (19) by ρs and integrating twice with respect to x we ﬁnd5 ∂t ψ0 (x, t) = P ∂x ψ1 .∂y ψ1 = 2kx ky |A|2 ,

(20)

with the second form following from Eq. (7). We can convert the time derivative to a spatial derivative by noting that the RHS of Eq. (12) is small (assuming |A| is small) so, to leading order, ∂t A = −vg ·∇A; that is, the modulations move at the group velocity. This also applies to quantities like ψ0 driven by |A|, so, to leading order, Eq. (20) becomes 2kx ky |A|2 . ∂x ψ0 = − vgx ˆ , so Eqs. (9) and (10) give the nonlinear frequency By Eq. (3), vE = ∂x ψ0 y shift for case s = 1 as a simple function of |A| ∆ω =

(1 + ρ2s k 2 )2 ky 2 |A| . ρ2s v∗

(21)

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As the nonlinear frequency shift in this case is a simple quadratic function (rather than functional) of |A|, the modulated frequency shift parameodinger equation is now given simter δωK , Eq. (14), in the nonlinear Schr¨ ply by δωK = d∆ω/d ln |A| = 2∆ω0 . The modulational instability criterion, Eq. (16), now becomes −ky ∂ 2 ωk /∂kx2 < 0. That is, from Eq. (11), the modulational instability criterion for the modiﬁed Hasegawa–Mima equation case, s = 1, is 1 − 3ρ2s kx2 + ρ2s ky2 > 0 ,

(22)

which agrees with Ref. 5 and Ref. 27 but not with Ref. 9 who, due to a misprint28 reproduced in Ref. 2, omit the factor 3 multiplying ρ2s kx2 . (An apparently similar inequality to that in Ref. 9 appears in Ref. 7, but this is not really relevant as they consider only a drift wave propagating in the poloidal direction—their kx is our Kx .) If criterion Eq. (22) is fulﬁlled, the growth rate curve, Γ2 vs. K 2 , is an inverted parabola with maximum at Γmax = ∆ω0 ,

Kmax =

|A| (1 + ρ2s k 2 )5/2 . (1 − 3ρ2 kx2 + ρ2s ky2 )1/2 ρ2s v∗

This extends the small-K result in Eq. (15) of Ref. 5 (who use the notation q for our K) to get a turnover in Γ at large K, as was also found using a mode-coupling approach by Chen et al.6 via the gyrokinetic equation in toroidal geometry within the ballooning approximation, and by LashmoreDavies et al.8 using the modiﬁed Hasegawa–Mima equation. As the latter authors base their analysis on the same model as used in the present paper, we can make a precise comparison between our Eq. (15) and their modulational dispersion relation, Eq. (43) in the small q ≡ Kx limit implied by our envelope approach. Expanding their quantities δ± in q, it is easily seen that, to leading order, their δ+ + δ− = q 2 ∂ 2 ωk /∂kx2 and (δ+ − δ− )/2 = q∂ωk /∂kx , while their expression 2Ω20 |A0 |2 F0 (k0 , q) is δωK = 2∆ω0 , with ∆ω0 given by Eq. (21) above. Completing the square in their Eq. (43), we see that correspondence between the two modulational dispersion relations is exact in the small q = Kx limit. Note also that the modulations and zonal ﬂows have ﬁnite frequency, even without geodesic eﬀects,29 as they propagate radially with phase velocity equal to the carrier group velocity, ∂ωk /∂kx = −2ρ2s kx ky v∗ /(1 + ρ2s k 2 )2 from Eq. (10).

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5.2. Modulational instability for CHM equation In the unmodiﬁed case, s = 0, there is actually no compelling reason to make the split into zonal and nonzonal components. Averaging Eq. (4) over the ﬂuctuation scale (noting that vE ·∇ψ ≡ 0 and neglecting small terms) we ﬁnd (∂t + v∗ ·∇)ψ0 = ρ2s vE ·∇∇2 ψ1 = ρ2s {ψ1 , ∇2 ψ1 } . Integrating from t = −∞, where ψ0 is assumed to vanish, along the trajectory of a ﬂuid element moving at the drift speed and assuming the modulations in the forcing term on the RHS to be advecting at the group velocity we ﬁnd

0 ∂ωk ∂ωk dτ {ψ1 , ∇2 ψ1 } x − τ, y + (v∗ − )τ, t . ψ0 (x, y, t) = ρ2s ∂kx ∂ky −∞ Using the ansatz Eq. (7), the dispersion relation Eq. (8), and the identity Eq. (17) we ﬁnd the nonlinear frequency shift to be the functional ∆ω =

2ρ4s k 2 (ky ∂x − kx ∂y )[(kx2 − ky2 )∂x ∂y − kx ky (∂x2 − ∂y2 )] 1 + ρ2s k 2 0 × dτ |A|2 (x − vgx τ, y + (v∗ − vgy )τ, t) . −∞

Perturbing ∆ω with a small modulation δ|A|, replacing δ|A| by exp iK·r, and substituting in Eq. (14) we get the modulated frequency shift 4ρ4s k 2 |A0 |2 (ky Kx − kx Ky )[(kx2 − ky2 )Kx Ky − kx ky (Kx2 − Ky2 )] . 1 + ρ2s k 2 Kx vgx + Ky (vgy − v∗ ) (23) Equation (15) then gives the dispersion relation for small modulations. Clearly, this is considerably more complicated than found in the MHM case and will not be analyzed further here except to make comparison with the results of Ref. 5, who take Ky = 0. In this case δωK =

2|A0 |2 Kx2 2 2 ρs k (1 + ρ2s k 2 ) , v∗ which gives a modulational dispersion relation in the small Kx limit in essential agreement with Eq. (19) of Ref. 5, who note that the modulational instability criterion is the same as that for the MHM, Eq. (22). However, the resonance at K·(vg −v∗ ) = 0 arising from the vanishing of the denominator in Eq. (23) will give higher growth rates for oblique modulations, so it is not clear that this special case is of great signiﬁcance for Rossby waves in the absence of boundaries. δωK =

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6. Conclusions We have derived a nonlinear Schr¨ odinger equation for modulations on a train of drift or Rossby waves in a very universal, if heuristic, fashion. The nonlinear Schr¨ odinger equation has been widely studied in other applications and is known to have soliton solutions. However, we have analyzed it only for stability to small modulations and have found criteria in agreement with those found by Smolyakov et al.5 for modulation waves with zonal phase fronts. Our results are encouraging as a step towards explaining the experimental discovery by Shats and Solomon30 of modulational instability associated with low-frequency zonal ﬂows, but the Hasegawa–Mima equation is rather too simpliﬁed for direct comparison with experiment and further work remains to be done in this regard. Acknowledgments This work was supported by the Australian Research Council and AusAID. We thank Dr F.L. Waelbroeck for explaining the importance of the modiﬁcation of the electron adiabatic response leading to the Modiﬁed Hasegawa– Mima Equation and Dr G.W. Hammett for discussions on the history of this modiﬁcation. Also we thank the referee for constructive suggestions and Dr J.S. Frederiksen for commenting on the nomenclature diﬀerence between the plasma and geophysical communities regarding the CHM equation, and Dr R. Ball for bringing Ref. 14 to our attention. References 1. Wikipedia http://en.wikipedia.org/ is a free online encyclopedia that anyone can edit. It is thus, in principle, self-correcting and inﬁnitely expandable. The quotes in the text are from the articles http://en.wikipedia.org/wiki/ Mean_flow and http://en.wikipedia.org/wiki/Zonal. 2. P. H. Diamond, S.-I. Itoh, K. Itoh, and T. S. Hahm, Zonal ﬂows in plasma—a review. Plasma Phys. Control. Fusion, 47, R35 (2005). 3. W. Horton, Nctp research workshop. In R. L. Dewar and R. W. Griﬃths, editors, Two-Dimensional Turbulence in Plasmas and Fluids, pages 3–36, Woodbury, New York, USA, 1997. National Centre for Theoretical Physics (now Centre for Complex Systems, http://wwwrsphysse.anu.edu.au/ccs), The Australian National University, American Institute of Physics. 4. P. H. Diamond, M. N. Rosenbluth, F. L. Hinton, M. Malkov, J. Fleischer, and A. Smolyakov, Dynamics of zonal ﬂows and self-regulating drift-wave turbulence. In Plasma Physics and Controlled Fusion Research, pages IAEA– CN–69/TH3/1, 8 pages, Vienna, 1998. 17th IAEA Fusion Energy Conference, Yokohama, Japan, 1998, International Atomic Energy Agency.

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5. A. I. Smolyakov, P. H. Diamond, and V. I. Shevchenko, Zonal ﬂow generation by parametric instability in magnetized plasmas and geostrophic ﬂuids. Phys. Plasmas, 7, 1349 (2000). 6. L. Chen, Z. Lin, and R. White, Excitation of zonal ﬂow by drift waves in toroidal plasmas. Phys. Plasmas, 7, 3129 (2000). 7. P. N. Guzdar, R. G. Kleva, and L. Chen, Shear ﬂow generation by drift wave revisited. Phys. Plasmas, 8, 459 (2001). 8. C. N. Lashmore-Davies, D. R. McCarthy, and A. Thyagaraja, The nonlinear dynamics of the modulational instability of drift waves and the associated zonal ﬂows. Phys. Plasmas, 8, 5121 (2001). 9. S. Champeaux and P. H. Diamond, Streamer and zonal ﬂow generation from envelope modulations in drift wave turbulence. Phys. Letters A, 288, 214 (2001). 10. A. Hasegawa and K. Mima, Pseudo-three-dimensional turbulence in magnetized nonuniform plasma. Phys. Fluids, 21, 87 (1978). 11. A. Hasegawa, C. G. Maclennan, and Y. Kodama, Nonlinear behavior and turbulence spectra of drift waves and Rosbby waves. Phys. Fluids, 22, 2122 (1979). 12. J. G. Charney, On the scale of atmospheric turbulence. Geofys, Publikasjoner, Norske Videnskaps-Akad. Oslo, 17, 3 (1948). 13. J. D. Meiss and W. Horton, Solitary drift waves in the presence of magnetic shear. Phys. Fluids, 26, 990 (1983). 14. P. Lynch, Resonant Rossby wave triads and the swinging spring. Bull. Am. Meteor. Soc., 84, 605 (2003). 15. W. Dorland, G. W. Hammett, L. Chen, W. Park, S. C. Cowley, Hamaguchi S., and W. Horton, Numerical simulations of nonlinear 3-D ITG ﬂuid turbulence with an improved Landau damping model. Bull. Am. Phys. Soc., 35, 2005 (1990). 16. W. Dorland and G. Hammett, Gyroﬂuid turbulence models with kinetic effects. Phys. Fluids B, 5, 812 (1993). 17. K. Mima and Y. C. Lee, Modulational instability of strongly dispersive drift waves and formation of convective cells. Phys. Fluids, 23, 105 (1980). 18. D. Majumdar, Eﬀect of temperature gradient on the modulational instability of drift waves and anomalous transport due to drift-wave turbulence. J. Plasma Phys., 40, 253 (1988). 19. B. K. Shivamoggi, Modulational instability of electrostatic drift waves in an inhomogenous plasma. Phys. Rev. A, 40, 471 (1989). 20. R. L. Dewar and A. H. Glasser, Ballooning mode spectrum in general toroidal systems. Phys. Fluids, 26, 3038 (1983). 21. Z. Lin, T. S. Hahm, W. W. Lee, W. M. Tang, and R. B. White, Turbulent transport reduction by zonal ﬂows: Massively parallel simulations. Science, 281, 1835 (1998). 22. R. L. Dewar, Interaction between hydromagnetic waves and a timedependent, inhomogeneous medium. Phys. Fluids, 13, 2710, (1970). 23. J. A. Krommes and C. B. Kim, Interactions of disparate scales in drift-wave turbulence. Phys. Rev. E, 62, 8508 (2000).

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24. R. L. Dewar, W. L. Kruer, and W. M. Manheimer, Modulational instabilities due to trapped electrons. Phys. Rev. Letters, 28, 215 (1972). 25. R. L. Dewar, A lagrangian theory for nonlinear wavepackets in a collisionless plasma. J. Plasma Phys., 7, 267 (1972). 26. J. A. Krommes, Comment on ”dynamics of zonal ﬂow saturation in strong collisionless drift wave turbulence” [phys. plasmas 9, 4530 (2002)]. Phys. Plasmas, 11, 1744 (2004). 27. J. A. Krommes, Analytical description of plasma turbulence. In M. Shats and H. Punzmann, editors, Turbulence and Coherent Structures in Fluids, Plasma and Nonlinear Medium: Selected Lectures from the 19th Canberra International Physics Summer School, Singapore, In press. World Scientiﬁc. 28. P. H. Diamond, private communication (2006). 29. N. Winsor, J. L. Johnson, and J. M. Dawson, Phys Fluids, 11, 2448 (1968). 30. M. G. Shats and W. M. Solomon, Zonal ﬂow generation in the improved conﬁnement mode plasma and its role in conﬁnement bifurcations. New J. Phys, 4, 30.1 (2002).

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NONLINEAR SIMULATION OF DRIFT WAVE TURBULENCE RYUSUKE NUMATA, ROWENA BALL AND ROBERT L. DEWAR Department of Theoretical Physics, Research School of Physical Sciences and Engineering The Australian National University, Canberra ACT 0200, Australia [email protected] In a two-dimensional version of the modiﬁed Hasegawa-Wakatani (HW) model, which describes electrostatic resistive drift wave turbulence, the resistive coupling between vorticity and density does not act on the zonal components (ky = 0). It is therefore necessary to modify the HW model to treat the zonal components properly. The modiﬁed equations are solved numerically, and visualization and analysis of the solutions show generation of stable zonal ﬂows, through conversion of turbulent kinetic energy, and the consequent turbulence and transport suppression. It is demonstrated by comparison that the modiﬁcation is essential for generation of zonal ﬂows.

1. Introduction In quasi two-dimensional (2D) plasma and ﬂuid ﬂows the energy ﬂux from small scale turbulent modes toward lower wavenumber modes can dominate the classical Kolmogorov cascade to dissipative scales, with the result that energy can accumulate in large scale coherent structures. Zonal ﬂows in planetary atmospheres and in magnetically conﬁned fusion plasmas are well-known examples of such coherent structures. Quasi two-dimensional ﬂuid systems in which turbulent activities and coherent structures interact can undergo a spontaneous transition to a turbulence-suppressed regime. In plasmas such transitions dramatically enhance the conﬁnement and are known as L–H or conﬁnement transitions. From theoretical and experimental works the importance of shear or zonal ﬂows for suppression of cross-ﬁeld transport and conﬁnement improvement is now widely appreciated. Several low-dimensional dynamical models, comprised of a small number of coupled ordinary diﬀerential equations, have been proposed to describe and predict the L–H transition.1–3 Ball et al. have analyzed a

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three-variable model using bifurcation and singularity theories.3 The model is based on the reduced resistive magnetohydrodynamic equations with the electrostatic approximation, and describes the pressure-gradient-driven turbulence–shear ﬂow energetics. This approach using low-dimensional modelling greatly simpliﬁes the problem, and when validated against simulated or real experimental data, will provide an economical tool to predict transitions over the parameter space. In this work we report the results of numerical simulations that both complement the low-dimensional modelling results and raise some interesting issues in their own right. We focus on a model for electrostatic resistive drift wave turbulence, the Hasegawa-Wakatani (HW) model,4 and solve the equations by direct numerical simulation in 2D slab geometry. The HW model has been widely used to investigate anomalous edge transport due to collisional drift waves.5 Moreover, self-organization of a shear ﬂow has been shown by numerical simulation of the HW model in cylindrical geometry.6 Thus we consider the HW model is a good starting point for studying self-consistent turbulence–shear ﬂow interactions, even though it does not describe physics that can be important in speciﬁc situations, such as magnetic curvature, magnetic shear, and electromagnetic eﬀect.

2. Modiﬁed Hasegawa-Wakatani Model The physical setting of the HW model may be considered as the edge region of a tokamak plasma of nonuniform density n0 = n0 (x) and in a constant equilibrium magnetic ﬁeld B = B0 ∇z. Following the drift wave ordering,7 the density n = n0 + n1 and the electrostatic potential ϕ perpendicular to the magnetic ﬁeld are governed by the continuity equation for ions or electrons and the ion vorticity equation, 1 ∂ d n= jz , dt e ∂z ∂ mn d 2 ∇ ϕ = B0 jz , B0 dt ⊥ ∂z T

(1a) (1b)

where ∇⊥ = (∂/∂x, ∂/∂y) , d/dt = ∂/∂t+V E ·∇⊥ is the E ×B convective derivative (V E ≡ −∇⊥ ϕ × ∇z/B0 , E = −∇⊥ ϕ), m is the ion mass, jz is the current density in the direction of the magnetic ﬁeld. The continuity equation (1a) can refer to ions and electrons because ∇ · j = 0 under the quasi-neutral condition, and (1b) holds because the current density is divergence-free. Since the ion inertia is negligible in the parallel direction

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(z), the parallel current is determined by the Ohm’s law, 1 ∇pe = ηj. en If the parallel heat conductivity is suﬃciently large, the electrons may be treated as isothermal: pe = nTe (p is the pressure, T is the temperature, and subscript e refers to electrons.) This gives the parallel current as

1 ∂ Te ln n . jz = − ϕ− η ∂z e E+

If we eliminate jz from (1a), (1b) and normalize variables as x/ρs → x,

ωci t → t,

eϕ/Te → ϕ,

n1 /n0 → n,

& −1 where ωci ≡ eB0 /m is the ion cyclotron frequency, and ρs ≡ Te /mωci is the ion sound Larmor radius, we ﬁnally obtain the resistive drift wave equations known as the Hasegawa-Wakatani (HW) model,4 ∂ ζ + {ϕ, ζ} = α(ϕ − n) − Dζ ∇4 ζ, ∂t ∂ϕ ∂ n + {ϕ, n} = α(ϕ − n) − κ − Dn ∇4 n, ∂t ∂y

(2a) (2b)

where {a, b} ≡ (∂a/∂x)(∂b/∂y) − (∂a/∂y)(∂b/∂x) is the Poisson bracket, ∇2 = ∂ 2 /∂x2 + ∂ 2 /∂y 2 is the 2D Laplacian, ζ ≡ ∇2 ϕ is the vorticity. We omit ⊥, and use ∇ for the 2D derivative. The dissipative terms with constant coeﬃcients Dζ and Dn have been included as adjuncts without derivation, for numerical stability. The background density is assumed to have an unchanging exponential proﬁle: κ ≡ −(∂/∂x) ln n0 . α ≡ −Te /(ηn0 ωci e2 )∂ 2 /∂z 2 is the adiabaticity operator describing the parallel electron response. In a 2D setting the coupling term operator α becomes a constant coeﬃcient, or parameter, by the replacement ∂/∂z → ikz . This resistive coupling term must be treated carefully in a 2D model because zonal components of ﬂuctuations (the ky = kz = 0 modes) do not contribute to the parallel current.8 Recalling that the tokamak edge turbulence is considered here, ky = 0 should always coincide with kz = 0 because any potential ﬂuctuation on the ﬂux surface is neutralized by parallel electron motion. Let us deﬁne zonal and non-zonal components of a variable f as 1 f dy, non-zonal: f˜ = f − f , zonal: f = Ly where Ly is the periodic length in y, and remove the contribution by the zonal components in the resistive coupling term in (2a) and (2b).

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By subtracting the zonal components from the resistive coupling term α(ϕ − n) → α(ϕ˜ − n ˜ ), we end up with the modiﬁed HW (MHW) equations, ∂ ζ + {ϕ, ζ} = α(ϕ˜ − n ˜ ) − Dζ ∇4 ζ, ∂t ∂ϕ ∂ n + {ϕ, n} = α(ϕ˜ − n ˜) − κ − Dn ∇4 n. ∂t ∂y

(3a) (3b)

The evolution of the zonal components can be extracted from (3a) and (3b) by averaging in the y direction: ∂ ∂ ∂ϕ ˜ f + f vx = D∇2 f , vx ≡ − , ∂t ∂x ∂y where f stands for ζ and n, and D stands for the corresponding dissipation coeﬃcients. The HW model spans two limits with respect to the adiabaticity parameter. In the adiabatic limit α → ∞ (collisionless plasma), the nonzonal component of electron density obeys the Boltzmann relation n ˜ = ˜ and the equations are reduced to the Hasegawa-Mima equan0 (x) exp(ϕ), tion.7 In the hydrodynamic limit α → 0 and the equations are decoupled. Vorticity is determined by the 2D Navier-Stokes (NS) equation, and the density ﬂuctuation is passively advected by the ﬂow obtained from the NS equation. In the ideal limit (α = ∞, Dζ = Dn = 0) the modiﬁed HW system has two dynamical invariants, the energy E and the potential enstrophy W , 1 E= 2

(n + |∇ϕ| ) dx, 2

2

1 W = 2

(n − ζ)2 dx,

where dx = dxdy, which constrain the ﬂuid motion. According to Kraichnan’s theory of 2D turbulence,9 the net ﬂux of enstrophy is downscale while that of energy is upscale. This inverse energy cascade is behind the development of large scale, stable coherent structures in a HW ﬂow. Conservation laws are given by dE = Γn − Γα − D E , dt

dW = Γn − D W . dt

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Fluxes and dissipations are given by ∂ ϕ˜ Γn = −κ n dx, ˜ ∂y Γα = α (˜ n − ϕ) ˜ 2 dx, DE = [Dn (∇2 n)2 + Dζ |∇ζ|2 ] dx, DW = [Dn (∇2 n)2 + Dζ (∇2 ζ)2 − (Dn + Dζ )∇2 n∇2 ζ] dx. These quantities constitute sources and sinks. As will be seen in the simulation results, they are mostly positive (Γα and DE are positive deﬁnite), thus only Γn can act as a source. The energy absorbed from the background supplies the turbulent ﬂuctuations through the drift wave instability. Note that the same conservation laws hold for the unmodiﬁed original HW (OHW) model except * that Γ2α is deﬁned by both zonal and non-zonal OHW ≡ α (n − ϕ) dx. In the OHW model, the zonal modes components; Γα as well as the non-zonal modes suﬀer the resistive dissipation. 2.1. Linear Stability Analysis Since the zonal modes have linearly decaying solutions, we only consider the form ei(kx x+ky y−ωt) (ky = 0). Linearization of the equations (3a) and (3b) yields the dispersion relation, ω 2 + iω(b + (1 + Pr−1)k 4 Dζ ) − ibω∗ − αk 2 (k 2 + Pr−1 )Dζ − k 8 Pr−1 Dζ2 = 0, (4) where we deﬁned k 2 = kx2 + ky2 , b ≡ α(1 + k 2 )/k 2 , the drift frequency ω∗ ≡ ky κ/(1 + k 2 ), and the Prandtl number Pr ≡ Dζ /Dn . Solutions to the dispersion relation (4) are given by 1 1 θ (ω) = ± (σ 2 + 16b2ω∗2 ) 4 cos , 2 2 1 θ −1 4 2 2 2 14 (ω) = − b + (1 + Pr )k Dζ ∓ (σ + 16b ω∗ ) sin , 2 2

σ = 4αk 2 (k 2 +Pr−1 )Dζ +4k 8 Pr−1 Dζ2 −(b+(1+Pr−1 )k 4 Dζ )2 , tan θ = 4bω∗ /σ. In the limit where Dζ = Dn = 0, it is readily proved that one of the growth rate γ ≡ (ω) is positive if bω∗ is ﬁnite, thus unstable. However, there exists a range of Dζ where the drift wave instability is suppressed. The stability threshold is given by 4 θ b + (1 + Pr−1 )k 4 Dζ ≥ (σ 2 + 16b2 ω∗2 ) sin4 , 2

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and is depicted in Fig. 1. The left panel shows the stability boundary in Dζ − κ plane. If we enhance the drive by increasing κ, the system becomes unstable. However, the instability is stabilized by increasing the dissipation. The stability threshold in kx − ky plane is shown in the right panel. We see that in a highly driven-dissipative system only low wavenumber modes are unstable. The stability boundary in parameter space is a region where interesting dynamics are expected to occur, such as bifurcations or sudden changes to a suppressed (or enhanced) turbulence regime. Figure 2 shows the dispersion relation for cases where Dζ = Dn = 0. To provide a test of the simulation code, we plot growth rates obtained from numerical simulations together with the analytic curves. We can see that the growth rates obtained numerically agree very well with that calculated analytically. We also note that, in the parameter range plotted in Fig. 2 (α = 1, κ = 1), the most unstable mode is kx ∼ 0, ky ∼ 1. 3. Simulation Results The HW equations are solved in a double periodic slab domain with box size (2L)2 = (2π/∆k)2 where the lowest wavenumber ∆k = 0.15. The equations are discretized on 256 × 256 grid points by the ﬁnite diﬀerence method. Arakawa’s method is used for evaluation of the Poisson bracket.10 Time stepping algorithm is the third order explicit linear multistep method.11 Since we are focusing in this work on how the modiﬁcation (3a), (3b) inﬂuences nonlinearly saturated states, we ﬁx the parameters to κ = 1, Dζ = 10−6 , α = 1, and Pr = 1, and compare the results obtained using the MHW model with those computed from the OHW model. For these parameters the system is unstable for most wavenumbers. During a typical evolution, initial small amplitude perturbations grow linearly until the nonlinear terms begin to dominate. Then the system arrives at a nonlinearly saturated state where the energy input Γn and output due to the resistivity Γα and the dissipations DE,W balance. In Fig. 3, we contrast the zonally elongated structure of the saturated electrostatic potential computed from the MHW model with the strong isotropic vortices in that* from the OHW model. Time evolution of the kinetic energy E K = 1/2 |∇ϕ|2 dx, and its partition to the zonal and the non-zonal components are shown in Fig. 4. The saturated kinetic energy is not aﬀected by the modiﬁcation (E K ∼ 1 for both cases). In the OHW model, the zonal ﬂow grows in the linear phase, as well as the other modes, up to a few percent of the kinetic energy, and saturates. On the other hand, in the MHW model the zonal kinetic energy continues to grow after the

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102

kx=0, ky=1 kx=1, ky=1 kx=5, ky=1 kx=1, ky=5 kx=5, ky=5

Unstable 0

κ

10

10-2 α=1, Pr=1

Stable 10-4 10-8 10-6 10-4 10-2 100 Dζ

κ=1, Dζ=10-5 κ=1, Dζ=10-4 κ=1, Dζ=10-3

20 Stable

15 ky ρs

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10 5 α=1, Pr=1

0 0

5

10 15 kx ρs

20

Fig. 1. Stability diagram of the MHW model. Top panel shows the stability thresholds in Dζ − κ plane. The drift wave instability can be stabilized by strong dissipation. In the bottom panel, stability thresholds are plotted in kx − ky plane. For certain parameters, only some low wavenumber modes are unstable.

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0.1

0.1

kx=0.5 kx=1.0 kx=1.5 kx=2.0

0.08 0.06

Growth Rate (γ/ωci)

Growth Rate (γ/ωci)

0.04 0.02 0

Fig. 2.

0

1

2

3 kyρs

4

5

ky=1 ky=2 ky=3 ky=4 ky=5

0.08 0.06 0.04 0.02 0

6

0.5

1

1.5 kxρs

2

2.5

Dispersion relation of the dissipationless MHW model. α = 1, κ = 1.

y/ρs

linear phase, and dominates the kinetic energy. The kinetic energy contained in other modes decreases to a few percent of the total kinetic energy. In the original 2D HW model, the resistive coupling term is retained for the zonal modes, the eﬀect of which is to prevent development of zonal ﬂows. But since the zonal modes do not carry parallel currents it is clearly unphysical to retain resistive action on them. Subtraction of the zonal components from the resistive coupling term is necessary to permit the generation of zonal ﬂows.

y/ρs

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x/ρs

x/ρs

Fig. 3. Contour plots of saturated electrostatic potentials for the modiﬁed and the original HW models. Zonally elongated structure is clearly visible for MHW case.

The density ﬂux in x direction Γn (transport across the magnetic ﬁeld), together with the energy partition to the kinetic energy E K and the po* 2 P tential energy E = 1/2 n dx, is plotted in Fig. 5. We observe that once the zonal ﬂow is generated in the MHW model, the transport level is

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10 MHW, Total and Zonal (left axis) 1 MHW, Non-Zonal (left axis)

0.1

10

0.01

1 OHW, Total and Non-Zonal (right axis)

0.001 1e-04

Total OHW, Zonal (right axis) Zonal Non-zonal

1e-05 0

500

1000

1500

2000

0.1 0.01

0.001 2500

Kinetic Energy of Original HW [n0Te]

439

Kinetic Energy of Modified HW [n0Te]

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Time [ωci-1] Fig. 4. Time evolution of the kinetic energy, and its partition to the zonal and the nonzonal components. In the modiﬁed HW model, the zonal mode contains most of kinetic energy, while non-zonal turbulence contains most of the kinetic energy in the original HW model.

signiﬁcantly suppressed. The transport suppression is mostly because the saturated potential energy (or amplitude of saturated density ﬂuctuation) is reduced. The potential energy and the turbulence kinetic energy are converted into the zonal kinetic energy. By contrast the energy of the OHW model is almost equi-partitioned between the kinetic and potential energy. The kinetic energy spectra averaged over the x or y direction for the MHW and the OHW models are shown in Fig. 6. The x (y) averaged kinetic K ) are deﬁned from the Fourier amplitude of the kinetic energy spectra (Ex(y) K energy E by Ky 1 EyK (kx ) = E K (kx , ky ) dky , Ky 0 Kx 1 K E K (kx , ky ) dkx , Ex (ky ) = Kx 0 where Kx , Ky are the highest wavenumbers. The spectra of the modiﬁed model again show strong anisotropic structure whereas there is no marked diﬀerence in the original HW model. In the modiﬁed model, potential energy stored in the background density is converted into turbulent kinetic energy through the drift wave instability at ky ∼ 1, kx = 0 and then is distributed

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MHW, Kinetic Energy (left axis)

1 MHW, Potential Energy (left axis) 0.1

MHW, Flux (left axis)

0.01

OHW, Potential Energy (right axis)

0.001

10 1

OHW, Kinetic Energy (right axis) Flux OHW, Flux Kinetic Energy (right axis) Potential Energy

1e-04 1e-05 0

500

1000

1500

Time [ωci-1]

2000

0.1 0.01

2500

Flux [n0Te/ωci] and Energy [n0Te] of Original HW

440

Flux [n0Te/ωci] and Energy [n0Te] of Modified HW

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Fig. 5. Time evolutions of the radial density transport and the kinetic and the potential energies for the modiﬁed and the original HW models. Once zonal ﬂow is generated in MHW model, the turbulent ﬂuctuation level and transport are signiﬁcantly reduced.

to smaller wavenumbers. The drift wave structure, which is elongated in the x direction, is break up into rather isotropic vortices after the nonlinear eﬀect sets in, and those isotropic vortices merge in the y direction to produce the zonal ﬂow. We can recognize this non-negligible inverse energy cascade in the y direction from a slight negative slope of Ex (ky ) spectrum in ky 1 region. The y averaged spectrum Ex (ky ) shows the strong peak at the zonal wave number kx ∼ 0.45. 4. Conclusion We have performed nonlinear simulations of the 2D HW model. As suggested recently,8 the electron response parallel to the background magnetic ﬁeld must be treated carefully in the 2D model. The model should be modiﬁed to exclude the zonal (ky = 0) contribution from the resistive coupling term. By comparing the numerical results of the modiﬁed and the unmodiﬁed original HW models, we have revealed that a remarkable zonal ﬂow structure in the nonlinearly saturated state is only observed in the modiﬁed model. Thus, the modiﬁcation is crucial to the generation of the zonal ﬂow in this model. Time evolutions of the macroscopic quantities, such as the energies and ﬂuxes show that, after the zonal ﬂow is built up by turbulent

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0.001 1e-04 1e-05 1e-06 1e-07

1

Linear Drive

0.1

1e-08 1e-09

0.01

1e-10

0.001

1e-11

0.1

1 kx ρs, ky ρs

10

0.0001

Linear Growth Rate (γ/ωci)

MHW: εy (kx) MHW: εx (ky) OHW: εy (k x) OHW: εx (k y) γ (ky; kx=0) γ (kx; ky=1.05)

0.01 Kinetic Energy Spectra (A.U.)

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Fig. 6. The x and y averaged kinetic energy spectra for the MHW and the OHW models. The top two lines (solid line for Ey (kx ) and broken line for Ex (ky )) for the OHW model are almost overlapped indicating isotropy. The middle two lines (dot-dashed line for Ey (kx ) and dotted line for Ex (ky )) for MHW show highly anisotropic structure in low k region. The energy injected at (kx , ky ) = (0, 1) cascades inversely to the zonal mode of the wave number (0.45, 0). The bottom two series of symbols show the linear growth rates of modes for reference.

interaction, the generated zonal ﬂow signiﬁcantly suppresses the turbulent ﬂuctuation level and the cross-ﬁeld density transport. The build up of the zonal ﬂow and resulting transport suppression indicate bifurcation structure of the system. If we increase a parameter (say, strength of the linear drive term κ), the system may undergo sudden transition from a high transport to a low transport regime. The state shown in this paper can be a bifurcated state. A systematic parameter study and comparison with the low-dimensional dynamical model are possible next steps. Acknowledgements The simulation code used in this paper is provided by B.D. Scott. The authors would like to thank J.A. Krommes, F. Jenko and H.A. Dijkstra for fruitful discussions and comments during the Workshop on Turbulence and

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Coherent Structures. This work is supported by the Australian Research Council. References 1. P.H. Diamond et al., Phys. Rev. Lett. 72, 2565, (1994). 2. H. Sugama and W. Horton, Plasma Phys. Control. Fusion 37, 345 (1995). 3. R. Ball, R.L. Dewar, and H. Sugama, Phys. Rev. E 66, 066408 (2002); R. Ball, Phys. Plasmas 12, 090904 (2005). 4. A. Hasegawa and M. Wakatani, Phys. Rev. Lett., 50, 682 (1983). 5. H. Sugama, M. Wakatani, and A. Hasegawa, Phys. Fluids 31, 1601 (1988); A.E. Koniges, J.A. Crotinger, and P.H. Diamond, Phys. Fluids B 4, 2785 (1992); S.J. Camargo, D. Biskamp, and B.D. Scott, Phys. Plasmas 2, 48 (1995); G. Hu, J.A. Krommes, and J.C. Bowman, Phys. Lett. A 202, 117 (1995). 6. A. Hasegawa and M. Wakatani, Phys. Rev. Lett. 59, 1581 (1987). 7. A. Hasegawa and K. Mima, Phys. Rev. Lett. 39, 205 (1977). 8. W. Dorland and G.W. Hammett, Phys. Fluids B 5, 812 (1993); G.W. Hammett et al., Plasma Phys. Control. Fusion 35, 973 (1993). 9. R.H. Kraichnan and D. Montgomery, Rep. Prog. Phys. 43, 547 (1980). 10. A. Arakawa, J. Comput. Phys. 1, 119 (1966). 11. G.E. Karniadakis, M. Israeli, and S.A. Orszag, J. Comput. Phys. 97, 414 (1991).

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THE TRANSITION TO ION-TEMPERATURE-GRADIENT-DRIVEN PLASMA TURBULENCE

JOHN A. KROMMES Plasma Physics Laboratory, Princeton University, P.O. Box 451, MS 28, Princeton, NJ 08543–0451 USA, [email protected]

Recent advances in the understanding of the transition to collisionless iontemperature-gradient-driven (ITG) plasma turbulence are described. A brief introduction to the physics of ITG modes is provided, then a model nonlinear system is motivated. The results of systematic bifurcation analysis of the model are reported, with emphasis on the unusual properties of the center manifold stemming from the existence of undamped zonal ﬂows. The model is shown to exhibit a “Dimits shift” of the critical temperature gradient for the onset of ITG activity similar to that observed in large computer simulations and understood to be due to the excitation of zonal ﬂows by a transient burst of ITG activity. For the model, the shift can be calculated from ﬁrst principles.

1. Introduction In this paper I discuss some aspects of the transition to plasma turbulence. Many details have already been published,16,17 and the dissertation of Kolesnikov18 can be consulted for further information. The purpose of the present article is primarily to provide some context for students and researchers not working directly in this specialized ﬁeld.

1.1. Drift waves and ion-temperature-gradient-driven modes For more than forty years, it has been recognized that in plasma containment devices such as tokamaks ﬂuctuations driven by gradients in the background proﬁles of density and temperature can have important adverse

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consequences for conﬁnement.a The most basic (“universal”) mode, the “drift wave” (DW) (see Krommes,21 Sec. 1.4), arises from a competition between the cross-ﬁeld E × B advection of ion gyrocenter density and the rapid parallel (to the conﬁning magnetic ﬁeld B) response of the electrons (which neutralizes charge ﬂuctuations). In slab geometry and in the limit of vanishing ion temperature Ti , the drift-wave frequency is ΩDW = k

ω∗ (k) 2 ρ2 , 1 + k⊥ s

(1)

. . where the so-called diamagnetic frequency is ω∗ (k) = ky ρs cs /Ln (= is . used for deﬁnitions); the “sound radius” is ρs = cs /ωci ; the sound speed is . 1/2 cs = (ZTe /mi ) , where Z is the atomic number and mi is the ion mass; . . the ion gyrofrequency is ωci = qi B/mi c, where qi = Ze is the ion charge; and the density scale length Ln (of the background or mean density proﬁle . n) is given by L−1 n = −d lnn/d ln x (and is assumed to be constant). 2 2 The k⊥ ρs term in Eq. (1) describes shielding due to the ion polarization drift. In the absence of that eﬀect, the DW propagates in the y direction (orthogonal to the directions of both B and the density gradient −> x). Its frequency is usually much smaller than ωci because ω∗ /ωci = (ky ρs )(ρs /Ln ) and, while typically ky ρs = O(1), one generally has ρs /Ln 1. The fact that ω∗ /ωci 1 permits a reduced, gyrokinetic description of the ﬂuctuations in which the rapid particle gyration around the magnetic ﬁeld is removed analytically. Some remarks on gyrokinetics are given by Krommes21 (Sec. 1.3). Although the linear theory of drift waves has been analyzed extensively, many other modes of oscillation are possible in magnetized plasmas. In particular, when nonzero ion temperature is considered, a new mode emerges called the ion-temperature-gradient-driven (ITG) mode. In slab geometry, the limiting form of the dispersion relation for large ion . = e2πi/3 ΩT , where ΩT = (Ω2s |ω∗Ti |)1/3 and temperature gradient isb ΩITG k a Quoting

from the important early monograph of Kadomtsev,14 “It has now become evident, however, that the coeﬃcient of turbulent diﬀusion cannot be obtained without a detailed investigation of the instability of an inhomogeneous plasma and in particular of its drift instability. . . .” b The complete slab dispersion relation is 0 = ω 3 −Ω ∗n ω 2 −Ω2s ω−Ω3T , where Ω∗n ≡ ΩDW k . For ΩT → 0, one root is ω = 0; this is the seed of the ITG mode. The other two roots describe the co- and counter-propagating sound waves modiﬁed by the drift frequency; the drift wave is the k → 0 limit of the forward-going sound wave. The three roots at large ΩT are ω = 11/3 ΩT , where 11/3 = {1, e2πi/3 , e−2πi/3 }; the real root at ω = ΩT has evolved from the drift wave. Thus the ITG mode and the drift wave lie on diﬀerent branches of the dispersion relation. (Nevertheless, ITG modes are sometimes referred to generically as “drift waves.”)

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. 2 2 Ω2s = k2 c2s /(1 + k⊥ ρs ). (Ωs is the eigenfrequency of the ion acoustic mode.) The ITG mode is of ﬂuid nature (it involves only the nonresonant ions) and is unstable due to an eﬀective negative compressibility of the plasma; a detailed physical picture was given in Ref. 3. In a torus, it is also strongly driven by magnetic curvature, and a limiting form of the toroidal dispersion relation shows a purely growing instability, = i(ωd |ω∗Ti |)1/2 , ΩITG k

(2) . where ωd = 2ky ρs /R, R being the major radius of the torus (a measure of the curvature). The physics behind ωd involves the (sign-dependent) ∇B and curvature drifts of the charged particles moving along the curved magnetic lines, as well as the space charge and subsequent E × B motion resulting from those drifts. Many other diﬃculties of magnetic conﬁnement stem from those drifts as well,c but those are well beyond the scope of this article. A discussion that presents a unifying picture of the various limiting ITG cases and describes interesting physics issues is given in Ref. 26. 1.2. The Dimits shift It is believed that ITG modes are responsible for the anomalous ion heat losses routinely observed in toroidal devices. Dimits et al.6 reported large gyrokinetic particle simulations that computed the ion heat ﬂux Γ as a . function of the normalized temperature gradient κ = R/LT , where LT is the temperature gradient scale length. Let κc be the threshold for linear instability. Naive reasoning would suggest that Γ, or at least ITG activity, would turn on as κ is raised past κc .d (One might, for example, expect a Hopf bifurcation at κ = κc .) Instead, the simulations reported by Dimits et al.6 showed that in purely collisionless situations there is an order-unity shift in the turn-on value, here called κ∗ . (For a certain set of reference . parameters, κc ≈ 4 whereas κ∗ ≈ 6.) The diﬀerence ∆κ = κ∗ − κc has come to be known as the Dimits shift. Subsequently, the Dimits shift has been observed in various other simulations as well. 1.3. Zonal ﬂows Both Dimits et al.6 and Rogers et al.27 recognized that the Dimits shift was associated with the excitation of zonal ﬂows. In slab geometry, a zonal c Some

examples are (i) there can be no conﬁnement in a purely toroidal magnetic ﬁeld because the particle drifts create a vertical electric ﬁeld that expels the plasmas by the E × B drift; (ii) the drifts drive interchange modes and resistive ballooning modes. d This belief ignores the possibility of submarginal turbulence.

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ﬂow (ZF) arises from an electrostatic potential φ with no y or z variation, i.e., φ = φ(x). Such potentials lead to y-directed shear ﬂows. (In a torus, they are poloidally directed and radially sheared.) Both general intuition and numerical experiments suggest that such ﬂows can suppress drift-wave ﬂuctuations and transport by shearing apart drift-wave eddies. (Further discussion of zonal ﬂows is given by Krommes,21 Sec. 3.) Note that turbulent heat ﬂux across the background temperature gradient is expressed by

cTe Γ = δVE,x δT = ky Imδϕk δTk∗ . (3) eB k

Here a δ preﬁx denotes the ﬂuctuating part, the asterisk denotes complex conjugation, VE,x is the radial or x component of the E × B velocity > with E = −∇φ, and I have introduced the normalized VE = cE × b/B . potential ϕ = eφ/Te . The presence of ky in this formula shows that zonal ﬂuctuations do not contribute directly to transport; however, they can help to regulate the size of the drift-wave ﬂuctuations and, therefore, indirectly moderate the transport. To the extent that shearing is a relevant concept, one expects that larger zonal-ﬂow activity leads to reduced transport. The Dimits shift arises from an extreme limit in which zonal ﬂows suppress the ITG ﬂuctuations and transport altogether. However, the phenomenon does not seem to be associated with a conventional Hopf bifurcation. In the work reported here, I discuss a simple model of interacting ITG modes and zonal ﬂows (a minor variant of models well established in the literature, combined with a low-order Galerkin truncation) and describe the nature of the Dimits shift from the point of view of bifurcation theory. The Dimits shift is shown to be associated with an atypically large center manifold that arises from the assumption of vanishing linear ZF damping. Because many approximations have been made, the calculation is not expected to have quantitative predictive power; certainly no attempt is made to calculate the Dimits shift in realistic toroidal geometry. Rather, the focus is on conceptual issues related to the unusual nature of the bifurcations in the presence of undamped zonal ﬂows. 2. Model building Most fundamentally, a description of interacting zonal ﬂows and drift waves should begin with the nonlinear gyrokinetic equation (GKE). That description is essential if resonant wave–particle interactions (Landau damping) are to be considered. Indeed, Ref. 28 employed the linear GKE to show that in toroidal geometry a residual, kinetically-undamped component of

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poloidal ﬂow survives at long times; such ﬂow is ultimately dissipated only by classical collisions, which are quite weak. There are beneﬁts to a gyrokinetic description even for nonresonant particles; for hot ions,e the eﬀective potentials felt by ion gyrocenters are reduced by the factor J0 (k⊥ v⊥ /ωci ), which signiﬁcantly complicates attempts to construct closed ﬂuid moment models by integrating the GKE over velocity. However, detailed work by Hammett and collaborators1,2,7–9,11–13 show that, with care, reasonable gyroﬂuid models can be constructed that well reproduce linear dispersion relations and also capture the dominant nonlinear eﬀects associated with E × B advection. Related work on reactive ﬂuid models has been done by Weiland et al.29 (and references therein). Dastgeer et al.4 demonstrated a Dimits shift in computer simulations of a reactive ﬂuid model. The model of interacting ITG modes and zonal ﬂows adopted by Kolesnikov and Krommes17 is most directly rooted in the work of Beer cited above. Speciﬁcally, one begins with coupled equations for vorticity and pressure P in which the basic nonlinearity is due to E × B advection. An immediate question concerns the boundary conditions. If such equations were studied globally (between two walls in slab geometry, or between r = 0 and r = a in cylindrical or toroidal geometry), one would impose nontrivial boundary conditions on the pressure in order to ensure that some sort of pressure gradient exists over the entire domain. That gradient would be self-consistently determined by the nonlinear ITG activity. However, one is not attempting here to model an actual discharge in detail, and one wants to employ a constant background gradient as a bifurcation parameter. Therefore, one can follow a standard procedure in which ﬁrst the dependent variables are decomposed into the sum of mean and ﬂuctuating parts, then the mean-ﬁeld evolution is ignored and the assumption of constant background gradient is inserted by hand into the equation for ﬂuctuations. As an example of the procedure, consider the model advection– dissipation equation >P = 0 ∂t P + VE · ∇P + χ

(4)

in slab geometry. (I postpone a statement of the boundary conditions.) Here χ > is a positive-semideﬁnite operator that represents wave-numberdependent dissipation. Speciﬁcally, motivated by the work of Ref. 28, one ek

⊥ ρi

. . = O(1), where ρi = vti /ωc is the ion gyroradius; vti = (Ti /mi )1/2 .

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assumes that χ > vanishes when acting on zonal amplitudes (weak collisional dissipation is omitted), and is a positive constant χ otherwise (representing Landau damping). In terms of an as-yet-unspeciﬁed averaging operator . . . that is assumed to commute with spatial gradients, and upon assuming that there is no mean ﬂow, one has χP = 0. ∂t P + ∇ · δVE δP + >

(5)

All ﬁelds have been decomposed into the sum of mean and ﬂuctuating parts, e.g., P = P + δP (δP ≡ 0). One possible choice for . . . is the . * zonal projection operator . . . Z = dy dz . . . , which extracts the Fourier > has been assumed to vanish amplitude having ky = 0 and kz = 0. Since χ for zonal amplitudes, one ﬁnds ∂t P Z + ∂x ΓZ (x) = 0,

(6)

. where ΓZ (x) = δVE · δP Z . However, this choice of averaging operator has two undesirable consequences. First, it requires that zonal ﬂows have no ﬂuctuating parts; the mean ﬁeld is the zonal ﬂow. Second, if ΓZ is assumed to be positive (as would follow from a turbulent Fick’s law in the presence of a leftward-directed gradient of P ), then the zonal amplitudes cannot achieve a steady state unless ΓZ is independent of x. Whether or not ΓZ (x) can be made independent of x depends on the boundary conditions. For situations in which a constant background gradient is assumed, it is usual to impose periodic boundary conditions in x on the ﬂuctuations. However, one is interested in maintaining some meager connection with the physics of toroidal conﬁnement, which includes the existence of magnetic shear (in the rotational transform of the magnetic lines). In the presence of magnetic shear, radial eigenfunctions are no longer plane waves but rather become localized around rational surfaces. To crudely model that localization, one can require that the ﬂuctuations vanish at both sides of a domain (microscopic in size with respect to the macroscopic container). But then ΓZ (x) also vanishes at the edges of the domain. That implies that statistical homogeneity is precluded, which makes the . . . Z average unsuitable for present purposes. Instead, let us deﬁne the averaging operation by the complete volume integral over the domain, which I denote by an overline. This operation commutes with the divergence operator applied to a ﬂuctuating quantity δA, * since by Gauss’s law ∇ · δA = dS·δA = 0. (In the periodic directions, the surface integral cancels between the opposing faces, and it vanishes in the x direction by virtue of the zero boundary conditions.) Now if nontrivial

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boundary conditions were to be applied to the background ﬁelds, this averaging operator would not necessarily commute with the gradients of those ﬁelds. Instead, one ignores the mean-ﬁeld equation altogether and, in the equation for ﬂuctuations, > δP = 0, ∂t δP + δVE · ∇P + ∇ · (δVE δP − Γ) + χ

(7)

−L−1 T

so that δVE · ∇P = V∗T ∂y ϕ. In fact, asserts that d ln P /dx = ∇ · Γ = 0, but it is useful to retain it in Eq. (7) in order to remind ourselves that the average of that equation vanishes term by term. In particular, the ﬂuctuating nonlinear term has indeﬁnite sign and does not drive unidirectional growth. Instead, it may be proven that Γ > 0 by multiplying Eq. (7) by δP and averaging: > δP . ∂t ( 12 δP 2 ) + V∗T δP (−∂y ϕ) + ∇ · [δVE ( 12 δP 2 )] = −δP χ

(8)

The second term is proportional to Γ, the third (triplet-correlation) term vanishes under the average, and the last term is negative-deﬁnite: > δP . ∂t ( 12 δP 2 ) = κT Γ − δP χ

(9)

One thus ﬁnds that steady states are possible with mean ﬂux in balance with the (positive-deﬁnite) nonzonal dissipation.f Having explained how one can extract equations for ﬂuctuating quantities from the original PDEs, I now turn to the speciﬁc model. Because one wishes to apply dynamical-systems techniques, one seeks a Galerkin truncation with relatively few degrees of freedom. In the absence of zonal dissipation, there seems to be no entirely systematic way of accomplishing this; modes that are neglected may be physically or quantitatively important. The work must therefore be considered to be preliminary and merely suggestive. In terms of fundamental wave numbers kx and ky characteristic of the microturbulence, modes at (lkx , mky ) are labeled by (l, m), and one retains the Fourier amplitudes (1, 1) — bifurcating, primary ITG mode, (3, 1) — stable ITG sideband (SB), (2, 0) — zonal mode. These amplitudes are uniquely identiﬁed by their l index, so one can write, for example, 1 instead of 1,1 . The boundary conditions can be satisﬁed f The manipulations leading to Eq. (9) are at the heart of the bounding method for turbulent ﬂuxes see Ref. 20 (and references therein). The interpretation of such “entropy”balance equations was discussed by Ref. 19.

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by using a Fourier sine series in x and a traveling plane wave in y. One is ultimately ledg to the model system ˙ 1 = −i(Ω1 − iη1 )1 − ib1 P1

1 1 1 1 1 + − − 1 2 − 3 2 , 2i D1 D2 D3 D2 ˙ P1 = iκ1 − d1 P1 (1)

(0)

(2)

(3)

? @A B ? @A B - , ? @A B ? @A B - 2 3 2 1 , 1 + P2 − P1 − P2 − P3 , 2i D1 D2 D3 D2 ˙ 3 = −i(Ω3 − iη3 )3 − ib3 P3

1 1 1 − − 1 2 , 2i D1 D2 (4)

(2)

(10b)

(10c)

(3)

? @A B ? @A B 2 1 , 1 P˙3 = iκ3 3 − d3 P3 − P2 − P1 , 2i D1 D2

1 1 ˙2 = − Im(1 3∗ ), D1 D3 (4)

(10a)

(10d) (10e)

(1)

, ?@A B , ?@A B ?@A B 1 ∗ 3 ∗ 1 ∗ P˙2 = Im P3 + P1 − Im P . D1 D3 D1 1

(10f)

Here the bi describe the magnetic curvature and are responsible for the instability drive; the di are measures of ITG dissipation, including both Landau damping and collisional eﬀects; the Di ’s relate the ion gyrocenter response δN to a potential ﬂuctuation δϕ according toh δN = Di−1 δϕ; and κ is the bifurcation parameter. The ITG and sideband amplitudes are complex, while the zonal amplitudes are real. The system thus has ten real degrees of freedom. Its nonlinear terms conserve the quantities . . W = |1 |2 + |3 |2 + 22 and P = |P1 |2 + |P3 |2 + P22 . The numbered overbraces indicate the cancellations of terms contributing to P˙ (term 0 vanishes separately). On the right-hand side of Eq. (10f), note the presence of two kinds of interactions: DW–SB–ZF (1–3–2), and DW–DW–ZF (1–1–2). The ﬁrst one is easily understandable and is the standard interaction considered in modulational instability calculations of zonal-ﬂow generation.5 The second g Some

details are provided in Ref. 16 (Appendix A). . >= in real space D α > − ∇2⊥ , where α > vanishes for zonal modes and is equal to one otherwise.

h Thus

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one may be confusing, because in detail the involved Fourier amplitudes are apparently (1, 1), (1, 1)∗ = (−1, −1), and (2, 0), so the momentum conservation law k + p + q = 0 does not seem to hold. The resolution is the assumption of a standing wave in the x direction. Then, for example, the presence of a (−1, −1) amplitude also implies a (1, −1) excitation, and one has (1, 1) + (1, −1) + (−2, 0) = (0, 0). The presence of both kinds of interaction are crucial in determining the Dimits shift; the second one has been frequently omitted in similar analyses, White et al.30 (and references therein). 3. Bifurcation theory In bifurcation theory,10,22 the concept of the center manifold (CM) plays a crucial role. Let the phase-space axes be the eigenvectors associated with the linear matrix. At the point of bifurcation (κ = κc ), some of the eigenvalues λ (with linear perturbations assumed to vary as eλt ) will have zero real part. The space spanned by the associated eigenvectors is called the center eigenspace. In essence, the CM is the nonlinearly curved generalization of the ﬂat center eigenspace and is tangent to that space at the point of bifurcation. Precise statements of the center manifold theorem can be found in the references cited above. The important practical result is that if that there are no unstable eigenvalues at the point of bifurcation, then at long times all solutions are attracted to the CM. The dimensionality of the CM is crucial. In a standard Hopf bifurcation,i in which a complex conjugate pair of eigenvalues crosses the imaginary λ axis, the CM is 2D. In the present model, there is such a pair of eigenvalues associated with the primary bifurcating ITG mode. However, in the absence of zonal dissipation, each zonal mode contributes a null eigenvalue as well. The present two-ﬁeld model thus possesses a 4D center eigenspace and a corresponding 4D CM. In local analysis, construction of the CM can be accomplished by a standard projection technique.22 First the dynamical system is extended by adjoining the equation κ˙ = 0. Next, to describe the technique for the simplest case of a single bifurcating eigenvalue, the dynamical variables x are decomposed into a part parallel to the right eigenvector q of the bifurcating mode and a part y orthogonal to the associated left orthonormal eigenvector p: x = Dq + y. Thus the modal amplitude is D = p† · x; the iA

laboratory experiment on drift waves in which a Hopf bifurcation was observed was described by Ref. 15.

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procedure can be viewed as a way of determining contravariant coordinates on the CM.j The center manifold theorem guarantees that y is at least quadratic in D and/or κ. Equating the time derivative of the assumed Taylor expansion for y to the independent equation for y˙ that follows from the system of ODEs provides order-by-order constraints that can be solved to determine the Taylor coeﬃcients. Then substitution into the equation for D˙ leads to a nonlinear equation for the local dynamics on the CM. For the present model, the procedure becomes technically more involved because the zonal variables are also coordinates on the CM; one must derive coupled equations for the set {D, z}, where D (the amplitude of the bifurcating ITG mode) is complex and z ≡ (2 , P2 ). The detailed construction of the 4D CM is described by Ref. 16. The most important qualitative result is the existence of a special ﬁxed point F corresponding to zero ITG . intensity I = |D|2 but nonzero zonal amplitudes z. The position and indeed the very existence of F stem from the balance between the two types of nonlinear interactions present in Eq. (10f). While F is attractingk in all directions for suﬃciently small κ, it becomes destabilizedl in the I direction for suﬃciently large κ > κ∗ . Kolesnikov and Krommes identiﬁed κ∗ as the upper boundary of the Dimits-shift regime. The value of κ∗ can be calculated analytically for the model, both perturbatively from construction of the local CM and exactly from the full 10D set of ODEs. The result agrees precisely with numerical experiments performed on Eqs. (10). Unfortunately, the value of κ∗ depends on the order of truncation. At best, therefore, the 10D model is only suggestive of the kind of interactions that determine the Dimits shift. For a reﬁned calculation of κ∗ , alternate PDE methods are highly desirable. When more zonal modes are retained in the Galerkin truncation, the dimensionality of the CM rises accordingly. j For

an illustrative diagram, see Fig. 12 of Ref. 16 (Appendix C). are subtleties associated with the notion of attraction. It is not the case that for κ < κ∗ all dynamical trajectories asymptote to F as t → ∞. By eliminating I and t ˙ one can perform a phase-plane analysis of the z plane. From from the equations for z, that point of view, F is always attracting in the z plane. However, because time has been eliminated, one can make no deﬁnitive statement about temporal asymptotics. In the time domain, some trajectories may as t → ∞ end up on the z plane (I = 0) at positions other than F . An exactly solvable example illustrating this behavior is given by Ref. 16. l The very brief discussion here is oversimpliﬁed. In more detail,16 one must determine the so-called marginal curve, which divides the z plane into stable and unstable regions. (Linear ITG stability must be recalculated in the presence of a zeroth-order zonal ﬂow.) The boundary κ∗ is identiﬁed with the κ value at which the nontrivial ﬁxed point crosses from the stable to the unstable side of the marginal curve. k There

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It may not be technically feasible to treat a very high-dimensional CM by the ODE methods used here. However, one class of interactions can be treated by generalization of the present techniques. The possibility of modulational instability of zonal ﬂows of long-wavelength m relative to the basic wave can be treated by multiple-scale techniques following the work of Newell and Whitehead.25 [For authoritative and pedagogical discussion, see Ref. 24 (Chap. 8).] This is the appropriate method for spatially extended systems such as a tokamak in the presence of microturbulence.n Some initial steps in this direction have been described in Refs.18 and 16, who derived an appropriate Ginzburg– Landau equation. The tentative conclusion is that above κ∗ the transition to turbulence may be initiated by closely spaced bifurcations related to the Benjamin–Feir instability. However, considerable further work is required to fully characterize the scenario. 4. Discussion In summary, an attempt has been made to calculate the Dimits shift of the temperature gradient for the onset of collisionless ion-temperature-gradientdriven turbulence for a simple yet nontrivial nonlinear model. The bifurcation scenario is unusual because of the higher-than-usual dimensionality of the center manifold in the presence of undamped zonal ﬂows. An important question is how the analysis is altered by the addition of a small amount of collisional dissipation µZF on the zonal ﬂows. Large simulations23 reveal intermittent behavior, and that is reproduced by Galerkin truncations of higher order than the 10D one studied here.18 It was argued in Refs.16,17 that this behavior can be understood qualitatively by adding a small amount of dissipation to the 4D CM equations (not by beginning with a dissipative description of the zonal ﬂows, which would have 2D CM dynamics). Note that the distinction between collisional and purely collisionless dynamics can be described in terms of the interchange of the limits t → ∞ and µZF → 0. (One always retains at least Landau damping on the ITG modes.) To analyze the collisionless case, one must take µZF → 0 ﬁrst.o Apparently that limit is also relevant to the case of very weak damping. m Note

that the (2, 0) zonal ﬂow considered in the basic model (10) is of shorter wavelength than the fundamental ITG mode. n A somewhat more detailed discussion of this point is given by Krommes21 (Sec. 1.7). o For Navier–Stokes physics, one speaks of the interchange of limits t → ∞ and µ → 0, where µ is the viscosity. Taking t → ∞ ﬁrst yields forced, dissipative Kolmogorov spectra, while taking µ → 0 ﬁrst yields Gibbsian equilibrium spectra. In the present problem, taking µZF → 0 is not fully analogous to the nondissipative Gibbsian limit because dissipation is always retained on the non-zonal ﬂuctuations.

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Much work remains to be done. While the weakly dissipative intermittency is easy to understand qualitatively, no attempt has been made to probe the depths of the associated nonlinear dynamics or to calculate higher-order statistics. Retention of more modes introduces both the possibility of qualitatively new physical eﬀects as well as severe technical diﬃculties. As suggested above, alternate PDE approaches may be required to calculate the Dimits shift precisely. Such methods will no doubt be essential to treat realistic (sheared) magnetic geometry. Further exploration of the role of modulational instabilities on the transition to ITG turbulence would also be very desirable. In conclusion, the problem of the transition to ITG turbulence involves both interesting physical eﬀects as well as subtle and challenging analytical techniques. It is an interesting area for future work on the foundations of nonlinear plasma physics. Acknowledgements The research reported here was performed in collaboration with R. Kolesnikov. This work was supported by U. S. Dept. of Energy Contract No. DE-AC02-76-CHO-3073. References 1. M. A. Beer and G. W. Hammett, Toroidal gyroﬂuid equations for simulations of tokamak turbulence, Phys. Plasmas, 3, 4046 (1996). 2. M. A. Beer, Gyroﬂuid Models of Turbulent Transport in Tokamaks, PhD thesis, Princeton University, 1995. 3. S. C. Cowley, R. M. Kulsrud, and R. Sudan, Considerations of iontemperature-gradient-driven turbulence, Phys. Fluids B, 3, 2767 (1991). 4. S. Dastgeer, S. Mahajan, and J. Weiland, Zonal ﬂows and transport in ion temperature gradient turbulence, Phys. Plasmas, 9, 4911 (2002). 5. R. L. Dewar and R. F. Abdullatif, Zonal ﬂow generation by modulational instability, in Proceedings of the COSNet/CSIRO Workshop on “Turbulence and Coherent Structures in Fluids, Plasmas and Granular Flows”, Canberra, Australia, Jan. 10–13, 2006, edited by J. Denier and J. S. Frederikson, World Scientiﬁc, Singapore, 2006. 6. A. M. Dimits, G. Bateman, M. A. Beer, B. I. Cohen, W. Dorland, G. W. Hammett, C. Kim, J. E. Kinsey, M. Kotschenreuther, A. H. Kritz, L. L. Lao, J. Mandrekas, W. M. Nevins, S. E. Parker, A. J. Redd, D. E. Shumaker, R. Sydora, and J. Weiland, Comparisons and physics basis of tokamak transport models and turbulence simulations, Phys. Plasmas, 7, 969 (2000). 7. W. Dorland and G. W. Hammett, Gyroﬂuid turbulence models with kinetic eﬀects, Phys. Fluids B, 5, 812 (1993).

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8. W. Dorland, G. W. Hammett, T. S. Hahm, and M. A. Beer, Nonlinear gyroﬂuid model of ITG turbulence, in U.S.–Japan Workshop on Ion Temperature Gradient-Driven Turbulent Transport, edited by W. Horton, M. Wakatani, and A. Wootton, p. 344, New York, 1994, AIP Press. 9. W. D. Dorland, Gyroﬂuid Models of Plasma Turbulence, PhD thesis, Princeton University, 1993. Available as GAX94–07080 from University Microﬁlm Int., 300 N. Zeeb Road, Ann Arbor, MI 48106–1346 (phone 800–521–3042). 10. J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer, New York, 1983. 11. G. W. Hammett and F. W. Perkins, Fluid moment models for Landau damping with application to the ion-temperature-gradient instability, Phys. Rev. Lett., 64, 3019 (1990). 12. G. W. Hammett, W. Dorland, and F. W. Perkins, Fluid models of phase mixing, Landau damping, and nonlinear gyrokinetic dynamics, Phys. Fluids B, 4, 2052 (1992). 13. G. W. Hammett, M. A. Beer, W. Dorland, S. C. Cowley, and S. A. Smith, Developments in the gyroﬂuid approach to tokamak turbulence simulations, Plasma Phys. Control. Fusion, 35, 973 (1993). 14. B. B. Kadomtsev, Plasma Turbulence, Academic Press, New York, 1965, Translated by L. C. Ronson from the 1964 Russian edition Problems in Plasma Theory, edited by M. A. Leontovich. Translation edited by M. C. Rusbridge. 15. T. Klinger, A. Latten, A. Piel, G. Bonhomme, T. Pierre, and T. D. de Wit, Route to drift wave chaos and turbulence in a bounded low-β plasma experiment, Phys. Rev. Lett., 79, 3913 (1997). 16. R. A. Kolesnikov and J. A. Krommes, Bifurcation theory of the transition to collisionless ion-temperature-gradient-driven plasma turbulence, Phys. Plasmas, 12, 122302 (2005). 17. R. A. Kolesnikov and J. A. Krommes, The transition to collisionless iontemperature-gradient-driven plasma turbulence: A dynamical systems approach, Phys. Rev. Lett., 94, 235002 (2005). 18. R. A. Kolesnikov, Bifurcation theory of the transition to collisionless iontemperature-gradient-driven plasma turbulence, PhD thesis, Princeton U., 2005. 19. J. A. Krommes and G. Hu, The role of dissipation in simulations of homogeneous plasma turbulence, and resolution of the entropy paradox, Phys. Plasmas, 1, 3211 (1994). 20. J. A. Krommes and R. A. Smith, Rigorous upper bounds for transport due to passive advection by inhomogeneous turbulence, Ann. Phys. (N.Y.), 177, 246 (1987). 21. J. A. Krommes, Analytical descriptions of plasma turbulence, in Turbulence and Coherent Structures in Fluids, Plasmas and Nonlinear Medium: Selected Lectures from the 19th Canberra International Physics Summer School, edited by M. Shats and H. Punzmann, World Scientiﬁc, Singapore, 2006. 22. Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, Springer, New York, 1998.

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23. Z. Lin, T. S. Hahm, W. W. Lee, W. M. Tang, and P. H. Diamond, Eﬀects of collisional zonal ﬂow damping on turbulent transport, Phys. Rev. Lett., 83, 3645 (1999). 24. P. Manneville, Dissipative Structures and Weak Turbulence, Academic Press, Boston, 1990. 25. A. C. Newell and J. A. Whitehead, Finite bandwidth, ﬁnite amplitude convection, J. Fluid Mech., 38, 279 (1969). 26. M. Ottaviani, M. Beer, S. Cowley, W. Horton, and J. Krommes, Unanswered questions in ion-temperature gradient driven turbulence, Phys. Rep., 283, 121 (1997). 27. B. N. Rogers, W. Dorland, and M. Kotschenreuther, Generation and stability of zonal ﬂows in ion-temperature-gradient mode turbulence, Phys. Rev. Lett., 85, 5336 (2000). 28. M. N. Rosenbluth and F. L. Hinton, Poloidal ﬂow driven by ion-temperaturegradient turbulence in tokamaks, Phys. Rev. Lett., 80, 724 (1998). 29. J. Weiland, S. Dastgeer, R. Moestam, I. Holod, and S. Gupta, Excitation of zonal ﬂows and ﬂuid closure, J. Plasma Fusion Res. SERIES, 6, 74 (2004). 30. R. White, L. Chen, and F. Zonca, Zonal-ﬂow dynamics and size scaling of anomalous transport, Phys. Rev. Lett., 92, 75004 (2004).

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SPECTRAL TRANSFER ANALYSIS IN PLASMA TURBULENCE STUDIES HUA XIA AND MICHAEL G. SHATS Research School of Physical Sciences and Engineering, Australian National University, Canberra ACT 0200, Australia In this paper, we describe the application of the spectral transfer analysis to plasma turbulence studies. The analysis is based on the wave coupling equation in which three wave interactions play a dominant role. The applicability of the spectral transfer technique to our plasma conditions is justiﬁed experimentally. The analysis results are consistent with the inverse energy transfer, as expected for two-dimensional ﬂuid turbulence.

1. Introduction Turbulence is one of the most important topics in physics of magnetically conﬁned plasma due to its role in the anomalously large transport of energy and particles across magnetic ﬁeld. Turbulence in magnetised plasma, in some cases, shows properties similar to those in two-dimensional (2D) ﬂuids. One of the properties of 2D ﬂuid turbulence is the existence of dual cascades of energy and enstrophy, which has been predicted theoretically and observed in both numerical simulations and experiments. In the 2D ﬂuid turbulence, energy is transferred from small to larger scales while enstrophy is transferred from large to smaller scales. The shape of the resulting energy spectrum is characterised by diﬀerent power-law scalings in the energy and in the enstrophy inertial ranges. Dual cascades of energy and enstrophy have also been predicted theoretically for turbulence in magnetised plasma. However, no conclusive experimental evidence of the dual cascades in toroidal plasma has been presented. The problem arises from the diﬃculty in describing turbulent plasma as a single ﬂuid. Fluctuations in the plasma density and in the electrostatic potential often do not coincide and need to be described separately as two coupled ﬁelds. To overcome this diﬃculty, we produce plasma in the H-1 heliac in which ﬂuctuations in the density and potential can, to some extent,

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be described as a single ﬁeld. Fluctuations in H-1 have been characterized using Langmuir probes providing data for the spectral transfer analysis. The wave coupling equation describes the evolution of a spectral component: 1 ∂φ(k, t) = (γk + i¯ ωk )φ(k, t) + ΛQ (1) k (k1 , k2 )φ(k1 , t)φ(k2 , t). ∂t 2 k1 ,k2 , k=k1 +k2

Here, φ(k, t) is the spatial Fourier spectrum of the ﬂuctuation ﬁeld ψ(x, t) = φ(k, t)eikx . The wave coupling equation gives the rate of change of the k

spectrum due to linear and nonlinear eﬀects, namely, due to the mode ¯ k , and the three-wave coupling ΛQ growth rate γk , its dispersion ω k (k1 , k2 ). The power transfer function (PTF) technique was developed and applied to hydrodynamic turbulence described by Eq. (1).1,2 This technique allows quantitative estimation of the coupling coeﬃcients (γk , ΛQ k (k1 , k2 )) from the experimental measurements. These coeﬃcients can give an estimation of the energy transfer in the spectrum due to the nonlinear coupling. Details of the PTF technique are reviewed in.3 In this paper, the applicability of the PTF technique to the H-1 plasma turbulence is discussed, then the PTF analysis results are presented. 1.1. Applicability of PTF technique to plasma turbulence in the H-1 heliac 1.2. Fluctuation measurement in H-1 The H-1 heliac is a Major National Research Facility at the Australian National University. Research on H-1 focuses on detailed understanding of the behavior of magnetically conﬁned plasma in the heliac conﬁguration and fundamental studies of turbulence and transport in plasma. Plasma is generated using about 100 kW of radio-frequency waves at 7 MHz. Magnetic ﬁeld of B = (0.05 − 0.15) T is generated for the plasma conﬁnement. Details of design and of the structure of magnetic ﬁeld are given in.4,5 Detail review of the ﬂuctuation measurement techniques can be found in.3 Fluctuations in the plasma electrostatic potential are measured using a quadruple probe whose schematic is shown in Fig. 1. Quadruple probe is a variation of the triple probe6 with additional pin for measuring ion saturation current. For each quadruple probe, a ﬂoating potential Vf , a potential of the positively biased tip V+ and the ion saturation current Is are measured.

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Fig. 1. Schematic of the quadruple probe used for measurements of the ﬂoating potential Vf , the positive potential, V+ , and the ion saturation current, Is .

The plasma potential and electron density can be derived from the measurements as: V+ − Vf kTe = , (2) e ln2 kTe , (3) Vp = Vf + α e 2Is ne = , (4) ecs Ap where α is a constant deﬁned by the plasma and probe parameters, cs = k(ZTe +Ti ) and Ap is the probe surface area. mi The ﬂoating potential Vf , rather than the plasma potential Vp , is sometimes used to derive poloidal electric ﬁeld to achieve the best signal to noise (S/N) ratio since Vp is a derived quantity while Vf is measured directly. This simpliﬁcation is also used in the H-1 spectral energy transfer analysis, because the S/N ratio is important in the calculation of spectral quantities, especially higher order spectra. In Fig. 2 (a) and (b), the power spectra of the plasma potential, V˜p , and ﬂoating potential, V˜f , ﬂuctuations are shown for a typical discharge. The phase shift between V˜p and V˜f ﬂuctuations is shown in Fig. 2 (c). This phase

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~ (a) P(Vp) (a.u.)

10 10-1 10-3 -5

10 10

~ (b) P(Vf) (a.u.)

-1

10 10

-3

10-5 1

~ ~ (c) Dj/p (Vf^Vp) (rad)

0

f ( kHz)

-1 0

40

20

60

Fig. 2. (a) Power spectrum of the ﬂuctuations in the plasma potential V˜p .(b) Power spectrum of the ﬂuctuations in the plasma ﬂoating potential V˜f . (c) Spectra of the phase shift between ﬂuctuations in the plasma potential, V˜p , and the ﬂoating potential, V˜f .

shift is close to 0 in a wide spectral range, from 0 to 60 kHz. For spectral energy transfer analysis, the phase information is more important than the amplitude of the ﬂuctuations because most spectral quantities used in the analysis are normalized, (e.g., auto and cross-bicoherence). From the result of Fig. 2, we conclude that by replacing Vp with Vf we will not aﬀect phases of the ﬂuctuations and thus, the spectral transfer analysis.

10

10

~ P(Vf) (a.u.)

-1

-3

10

10-5

0

20

40

f (kHz) 60 80

Fig. 3. Power spectrum of the ﬂuctuations in the ﬂoating potentials, V˜f , at r/a = 0.5, Bφ = 0.052 T, Prf = 55 kW.

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A typical frequency power spectrum of the V˜f ﬂuctuations is presented in Fig. 3. The spectrum decays in the frequency range of f ≤ 80 kHz, with several coherent features in the frequency range of f < 20 kHz.

1.3. Justiﬁcation of the single ﬁeld model To justify application of the single ﬁeld model, Eq. 1, one needs to investigate the phase relationship between ﬂuctuations in the electron density and in the electrostatic potential. Figure 4 (a) shows the power spectra of the ﬂuctuations in the ion saturation current (I˜s ) and in the ﬂoating potential (V˜f ). Is and Vf are measured using a quadruple probe. The electron density √ ﬂuctuations n ˜ e can be related to I˜s since Is ∼ ne Te , while the ﬂoating potential Vf can represent the plasma electrostatic potential φ as discussed above. Frequency spectra of the phase shift and of the coherence between I˜s and V˜f are shown in Fig. 4 (b) and (c), respectively. They are representative of the phase shift and coherence between ﬂuctuations in the electron ˜ Figure 4 demonstrates density n ˜ e and the plasma electrostatic potential φ. that the density and potential ﬂuctuations are well correlated (coherence between n ˜ e and φ˜ is larger than 0.6) in the spectral range of interest. The phase diﬀerence between them is almost constant at around 0.3 π for the frequency range of 0 to 80 kHz. To understand the phase shift between n ˜e ˜ one needs to consider the probe geometry used in the measurements. and φ, Figure 5 (a) shows geometry of the quadruple probe with respect to the magnetic ﬂux surface. Figure 5 (b) shows the probe head used for the measurements. Floating potentials are measured using the two poloidally separated tips of the probe in the center (3.5 mm apart) while the ion saturation current is measured using a tip on the right hand side. Since both of the Vf tips are shifted poloidally with respect to the Is tip, V˜f and I˜s ﬂuctuations have a phase shift of about 0.3π seen in Fig. 5(c). Poloidal electric ﬁeld is derived from the two ﬂoating potentials as Eθ = (Vf1 − Vf2 )/∆y. This poloidal electric ﬁeld is measured at the poloidal position, θEθ , which is very close to that of the Is , θIs . Thus the phase ˜θ and I˜s should not be aﬀected by the plasma drift relationship between E in the poloidal direction. The electron density ﬂuctuations are a sum of adiabatic (Boltzmann) ˜ ) + δne . and non-adiabatic parts: n ˜ e = n0 exp(eφ/T For a single ﬁeld description of plasma turbulence to be valid, the electron response should be adiabatic, δne = 0. In other words, when δne = 0, ˜ and electron density, n ﬂuctuations in the plasma potential, φ, ˜ e , should be

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P(~ j, ~ne)

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(a)

(c)

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g(j~ ^ n~e)

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~ Is

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~ ~ Dj/p (n^Eq) (rad)

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5

0

10 f (kHz) 15

Fig. 4. (a) Power spectra of the ﬂuctuations in the measured ion saturation current I˜sat (dashed line) and the ﬂoating potential V˜f (solid line). Spectra of (b): the phase shift, and (c): the coherence between ﬂuctuations in the electron density, n ˜ e , and the ˜ electrostatic potential, φ.

poloidal

poloidal

Eq=Vf1-Vf2

Bf

Vf1

*

Is

Is

(a)

(b)

qVf

Bf

qIs qEq

Vf2

Fig. 5. (a) Orientation of the probe array with respect to the magnetic ﬂux surface in the H-1 heliac, (b): a photo of the quadruple probe. Arrows show the toroidal (Bφ ) and poloidal directions.

in phase. This would also mean no turbulent transport, . / ˜θ /B = 0, Γf l = n ˜eE

(5)

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˜θ , would have a π/2 because n ˜ e and the poloidal electric ﬁeld ﬂuctuations, E ˜ phase shift (E˜θ = −∇θ φ).

1.0 0.5 0

~ ~ Dj/p (n^Eq) (rad)

-0.5 -1.0

f (kHz) 0

20

40

60

80

Fig. 6. Spectrum of the phase shift between ﬂuctuations in the electron density, n ˜e, ˜θ . and poloidal electric ﬁeld, E

˜θ . This Figure 6 shows the spectrum of the phase shift between n ˜ e and E phase shift is close to π/2 over the entire spectrum. Thus the adiabatic electron response is approximately valid, and application of the single ﬁeld model to the plasma turbulence in H-1 is justiﬁed. To summarise, the above experimental results justify application of the single-ﬁeld model, described by Eq. 1, to the analysis of spectral power transfer in the H-1 turbulence. Low frequency coherent ﬂuctuations are observed in H-1 as shown in Fig. 3. In Fig. 7 (a), spectra of ﬂuctuations in the electron density and potential are shown for low frequencies (0 - 15 kHz). The coherence between them is shown in Fig. 7 (b). The ﬂuctuations are strongly correlated with coherence of about 0.9. The phase diﬀerences between the density and poloidal electric ﬁeld for those coherent structures are around 0.7 π. This phase shift leads to the particle transport as can be seen from Eq. 5. Actually, most of the transport driven by the ﬂuctuations are driven by those coherent structures and there is practically no particle transport driven by the broadband turbulence. The broadband part of the ﬂuctuation spectra in H-1 probably has different origin from that of the coherent structures. It is necessary to separate them in the spectral analysis. Numerical simulations7 suggest that the coherent structures may be an energy sink of the f ≈ 0 zonal ﬂows due to toroidal eﬀects. Experimental measurements in H-1 also conﬁrm this,8 but this will not be discussed in this paper.

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10 f (kHz) 15

Fig. 7. Spectra of (a): the electron density, n ˜ e (dashed line) and the potential, φ˜ (solid line), (b):the coherence between ﬂuctuations in n ˜ e and V˜f , (c): the phase shift between ˜θ . ﬂuctuations in n ˜ e , and poloidal electric ﬁeld, E

2. Experimental results of plasma turbulence studies in the H-1 heliac Two ﬂoating potentials measured at two poloidally separated positions are digitized at 0.3 MHz. Typical plasma pulse length is ∼(60 - 80) ms. The signals are divided into 460 overlapping segments, so that each segment contains 80 data points. Such a severe statistical averaging results in the reduced frequency resolution of the spectra of the NETF, ∆f ≈ 4 kHz. The averaging is needed for the numerical convergence of the higher order spectra. Fluctuations are measured in laboratory frame of reference, such that frequency spectra are Doppler shifted due to the large E×B drift in poloidal direction. Wave numbers of ﬂuctuations have been measured using two poloidally separated probes, as kθ = ∆φ/∆y. This has been analyzed in the frequency domain and the kθ (f ) dependence is shown in Fig. 8 (b) as the grey line. In Fig. 8(a), the frequency power spectrum of the ﬂuctuation is shown. This dependence has a linear trend, however a large ripple is usually observed. This ripple can be reduced by statistical averaging over a large number of realizations, except for strong coherent structures. The relation between the ripple and the power spectrum of the ﬂuctuation has been

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(a)

150

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(b)

kq (m )

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50 -3

10

0 f (kHz)

-5

10

0

20

40

60

-50 80

f (kHz) 0

20

40

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Fig. 8. (a): Power spectrum of the ﬂuctuation in the ﬂoating potentials, V˜f , at r/a = 0.5, Bφ = 0.052 T, Prf = 55 kW; (b): the measured poloidal wave number spectrum kθ (f ) (grey line) with the linear ﬁt (black line).

investigated. The amplitude of the kθ (f ) ripple correlates with the relative strength of the coherent features to the level of broadband ﬂuctuations. It has been shown9 that in the presence of zonal ﬂow, phases of potential ﬂuctuations become more random, which could therefore aﬀect the kθ (f ) measurements. The linear approximation of the kθ (f ) dependence is shown in Fig. 8 (b) as the black line. The ﬂuctuation phase velocity derived from this linear ﬁt is within 10% of the measured E × B drift velocity in this radial region. As discussed in,3 in laboratory frame of reference, frequencies of the ﬂuctuations are Doppler shifted due to the presence of the E × B drift: Vlab = Vplasma + kθ VE×B . In many cases, E × B drift dominates over the phase velocity in the plasma frame. Thus, the ﬂuctuation frequencies in the lab frame are proportional to the poloidal wave numbers of the ﬂuctuations. Since in the broadband turbulence the √ wave number spectra are isotropic, kθ ∼ kr , one can assume that k = 2kθ ∝ ω. The E × B Doppler shift plays in such cases a role of the wave number spectrograph. Note that the isotropy breaks for the coherent structures. However, as mentioned in the last section, coherent structures may have diﬀerent origin to that of the broadband turbulence. Thus, as a ﬁrst approximation, the temporal change of the wave structure in the wave number space can be studied by measuring the change of the wave structure in the frequency space between two poloidally separated positions. The wave number domain is transformed into the frequency domain, with the three-wave interactions satisfying matching conditions, k = k1 + k2 , obeying the frequency selection rules, f = f1 + f2 .

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Two plasma conﬁnement modes, namely the low and high conﬁnement, have been identiﬁed in H-1.10 Plasma transits spontaneously from low to high conﬁnement mode within ∼ 1 ms, such that the mean electron density doubles. The transition is accompanied by the ﬂuctuation suppression and the onset of the strongly sheared radial electric ﬁeld, Er . The turbulence spectrum is strongly altered across the transition, as will be shown later. First, we present the spectral analysis results in the low mode plasma. The wave-kinetic equation for the spectral power Pf = φf φ∗f can be written as:

∂Pf ≈ 2γf Pf + ∂t

Tf (f1 , f2 ),

(6)

f1 ,f2 , f =f1 +f2

where the linear growth rate γf and the energy transfer coeﬃcient Tf (f1 , f2 ) can be derived from the coupling coeﬃcients in Eq. 1. The nonlinear energy transfer (NETF) is deﬁned as: WNf L = (1 + f 2 )

Tf (f1 , f2 ).

(7)

f1 ,f2 , f =f1 +f2

The NETF, linear growth rate γf and the power spectrum are shown in Fig. 9 for a typical low mode discharge. The frequency resolution here is ∆f ≈ 4 kHz. As a result of this low frequency resolution, coherent spectral features seen in Fig. 8 (a) do not show up in Fig. 9 (a). The power spectrum in Fig. 9 (a) represents mostly the broadband part of the spectrum as shown in Fig. 8 (a) in dashed line. The NETF WNf L is presented in Fig. 9 (b). Negative WNf L in the broadband spectral region of f = (20 − 50) kHz suggests that waves in this range lose energy, whereas the lower frequency spectral components (f < 20 kHz) gain energy. Two spectral ranges of interest, shaded regions (I and II) in Fig. 9, can be identiﬁed from the power transfer analysis. One is the region with maximum linear growth rate (region I) and the other has zero energy transfer (region II). The linear growth rate shown in Fig. 9(c) has a maximum at f ≈ 25 kHz. This region of maximum linear growth rate is indicative of the underlying linear instability. This instability range identiﬁed from the energy transfer analysis coincides with the spectral range of residual ﬂuctuations in the high mode.

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Fig. 9. (a): The power spectrum of the ﬂoating potential ﬂuctuations; (b): the nonlinear f ; (c): the linear growth rate γf derived from Eq. (6). The energy transfer function WNL frequency resolution is ∆f ≈ 4 kHz.

Figure 10 shows two power spectra of the ﬂoating potential ﬂuctuations for low and high conﬁnement mode, respectively. The wave number (k) spectra are shown based on the measured kθ (f ) for low and high modes. It can be seen that turbulence spectra are strongly altered across the transition. The ﬂuctuation level is reduced in high mode over the entire spectrum except for the region around k ≈ 200 m−1 , where the ﬂuctuation level even increases in high mode, shown as region I in Fig. 10. This is also the region with maximum linear growth rate as shown in Fig. 9(c). Region II in Fig. 9(b) shows zero nonlinear energy transfer, which is indicative of the enstrophy cascade range similar to two-dimensional ﬂuid turbulence. This range obeys the power law ﬁt of P ∼ k −α , where α ≈ −5.8 as shown in Fig. 10. This power law holds over 3 decades and is observed in all the low mode spectra. Summarising the nonlinear energy transfer in the H-1 plasma turbulence, when nonlinear spectral transfer is high in the low mode, energy is transferred from the initially unstable region of k ≈ 200 m−1 up the spectrum through the inverse cascade. An enstrophy cascade range is formed at the same time with a spectral scaling of close to k −6 . In the high mode, the level of WN L around k ≈ 200 m−1 is reduced. The decrease of WNk L at

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~ P(Vf) (a.u.)

1

I

II

L mode

0.1 0.01

H mode

-5.8

k

0.001

ki 0.0001 10

100

-1

k(m )

1000

Fig. 10. Wave number spectra of the ﬂoating potential ﬂuctuation in the low (dashed line) and high (solid line) mode. The shaded regions I and II are the same range as shown in Fig. 9.

200 m−1 means that there is less energy transferred out of this frequency band. As a result, the spectral energy becomes peaked in this spectral region. The free energy for this instability (pressure gradient) is increased across the transition from low to high mode due to the more peaked density proﬁle in the high mode.11 Consequently, one would expect a linear growth rate to increase. This expectation agrees with the observed increase in the ﬂuctuation level at k ≈ 200 m−1 seen in Fig. 10. 3. Summary To summarize, the applicability of the governing equation of the spectral transfer analysis, the wave kinetic equation, to the H-1 plasma turbulence has been justiﬁed experimentally. The spectral analysis results show the inverse energy cascade and the enstrophy inertial range scaling. These results are consistent with the theoretical predictions for quasi-2D turbulence. References 1. C. P. Ritz and E. J. Powers, Estimation of nonlinear transfer functions for fully developed turbulence. Physica D, 20 D(2-3), 320–34 (1986). 2. C. P. Ritz, E. J. Powers, and R. D. Bengtson, Experimental measurement of three-wave coupling and energy cascading. Physics of Fluids B-Plasma Physics, 1 (1), 153–63 (1989). 3. M. Shats and H. Xia, Experimental studies of plasma turbulence. In M. Shats and H. Punzmann, editors, Turbulence and Coherent Structures in Fluids,

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4. 5.

6.

7. 8.

9.

10.

11.

Plasmas and Nonlinear Media. World Scientiﬁc Lecture Notes in Complex Systems, ISBN 981-256-698-8, 2006. S. M. Hamberger, B. D. Blackwell, L.E. Sharp, and D.B. Shenton, H-1 design and construction. Fusion Technology, 17 (1), 123–130 (1990). M. G. Shats, D. L. Rudakov, B. D. Blackwell, L. E. Sharp, R. Tumlos, S. M. Hamberger, and O. I. Fedyanin, Experimental investigation of the magnetic structure in the h-1 heliac. Nuclear Fusion, 34 (12), 1653–61 (1994). S.L. Chen and T. Sekiguchi, Instantaneous direct-display system of plasma parameters by means of triple probe. Journal of Applied Physics, 36, 2363–75 (1965). B.D. Scott, Energetics of the interaction between electromagnetic exb turbulence and zonal ﬂows. New Journal of Physics, 7, 92/1–45 (2005). M. G. Shats, H. Xia, and M. Yokoyama, Mean e x b ﬂows and gam-like oscillations in the h-1 heliac. Plasma Physics Controlled Fusion, 48 (4), S17– S29 (2006). M. G. Shats, W. M. Solomon, and H. Xia, Turbulent transport reduction and randomization of coherent ﬂuctuations by zonal ﬂows in toroidal plasma. Physical Review Letters, 90 (12), 125002/1–4 (2003). M. G. Shats, D. L. Rudakov, B. D. Biackwell, G. G. Borg, R. L. Dewar, S. M. Hamberger, J. Howard, and L. E. Sharp, Improved particle conﬁnement mode in the h-1 heliac plasma. Physical Review Letters, 77 (20), 4190–3 (1996). M. G. Shats, Eﬀect of the radial electric ﬁeld on the ﬂuctuation-produced transport in the h-1 heliac. Plasma Physics Controlled Fusion, 41 (11), 1357– 70 (1999).

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COHERENT STRUCTURES IN TOROIDAL ELECTRON PLASMAS: SIMULATION AND EXPERIMENTS R. GANESH AND S. PAHARI Institute for Plasma Research, Bhat Village Gandhinagar, 382 428, Gujarat, India [email protected] Incompressible two dimensional (2D) Euler ﬂows are hard to realize in experimental ﬂuid dynamic laboratories. For example, ﬁnite dissipation at the boundaries however small makes the ﬂow nonideal. Recently, pure electron plasmas in homogeneous guiding magnetic ﬁeld as working ﬂuid have proven to be ideal experimental test-beds for studying ideal, 2D, incompressible turbulence, formation of coherent structures and vortex dynamics. Here we focus on electron plasmas trapped in an inhomogeneous guiding magnetic ﬁeld in a small aspect ratio torus which may well become a similar test-bed for compressible ﬂuid dynamical experiments. Using a simple ﬂuid model with cold, massless electrons, the dynamical evolution for various initial states are studied. Final states are shown to be turbulent or coherent or could possess both the features, depending on the initial state. Though comparison with experiments show remarkable qualitative agreement, many questions remain unanswered pointing out the need for more sophisticated models and experiments.

1. Introduction Pure electron plasmas (PEP) conﬁned in an uniform external magnetic ﬁeld have become an important part of ﬂuid dynamic research. A particular reason is the morphological similarity between two dimensional (2D) drift-Poisson equations governing a PEP and that of a 2D Euler ﬂow. For example, ﬂuid vorticity Ω(r) ⇔ (n(r)e)/(0 B0 ), ﬂuid stream function Ψ(r) ⇔ φ/B0 , free-slip boundary conditions pertain to φ|wall = 0 and regions of zero density in PEP is equivalent to regions of irrotational ﬂow in the Euler context. This similarity has lead to a plethora of experimental veriﬁcation of ideas and theories from 2D vortex dynamics with PEP as the “working” medium. Such experiments have led to the study of a host of linear and/or nonlinear modes, notably, the surface waves (Kelvin waves in ﬂuid dynamics)

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and the Diocotron modes. For example, a uniform density PEP column embedded in a uniform magnetic ﬁeld (a vortex patch in ﬂuid dynamics) with circular bounding wall, exhibits surface waves with poloidal mode number m > 1. For small amplitudes, the mode numbers m are not coupled whereas ﬁnite amplitude eﬀects could lead to nonlinear mode coupling. Oscillations for m = 1 are called Diocotron or centre-of-charge oscillations. On the other hand, for a plasma conﬁned by highly inhomogeneous toroidal magnetic ﬁeld (whose magnitude B ∝ 1/R, R− is the toroidal radial variable), the mode numbers m are strongly coupled even in the linear limit. In the following, for ease of discussion, we continue using m as if it is an “independent poloidal mode number”, but it would be helpful to remind ourselves that the various modes m are indeed strongly coupled for the toroidal system studied here. Historically, experiments in conﬁning PEP in an inhomogeneous external B ﬁeld date back to late the 1960s starting from the pioneering work of J. D. Daugherty et al.6 In these experiments toroidal axisymmetric magnetic ﬁeld with 1/R dependence was used with thermionic electron fuelling. Vertical grad-B drift inherent in inhomogeneous B ﬁelds led to poor conﬁnement times. In the late 1990s, SMall Aspect Ratio Toroidal EXperiments in a Torus (SMARTEX-T) at the Institute for Plasma Research, India reported improved conﬁnement times up to 100 µs with cross-ﬁeld thermionic fuelling of electrons. This was one of the ﬁrst experiments where an “equilibrium” electron cloud was observed.7 However, attempts to further improve conﬁnement time scales were marred by competition between losses due to grad-B drift overtaking the self-consistent E × B rotational transform. Recently several large aspect ratio experiments world-wide have been reported. These experiments range from a simple Toroidal C-trap8 to complex levitated, super-conducting magnetic coil-based experiment9 to nonneutral stellarator10 commissioned recently. The ﬁrst two experiments have reported improved conﬁnement times either in the presence of an external electric ﬁeld Eext ne 0 and/or in the presence of a poloidal B ﬁeld both of which impart an external rotational transform. It may be of interest to note that in the absence of Eext , no plasma formation is seen.8 In this paper, we report ﬁndings from a series of simulations and of experiments conducted in SMARTEX-C which is a small aspect ratio C-trap (i.e, an incomplete torus). Experimentally, we regard these measurements as the longest conﬁnement time in a small aspect ratio toroidal

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geometry without any external Eﬁeld or poloidal B ﬁeld.2 These results are corroborated with qualitative results from numerical simulations. To this end, a new 2D ﬂuid code TEPFCT4 has been developed for arbitrary vessel aspect ratio A = R0 /a where R0 − is the vessel major radius and 2a = R2 − R1 , R1,2 − is inner (outer) wall respectively. Since the zeroth order electric ﬁeld dominates the rest of the physics, as a ﬁrst nontrivial step we treat the electrons as a cold, compressible ﬂuid undergoing a simple E × B dynamics in an inhomogeneous B ﬁeld. The code is tested using an equilibrium model obtained via a mean ﬁeld theory.11 Under E × B approximation, several experimentally relevant initial conditions are studied. The numerical results thus obtained show remarkable qualitative agreement with experimental results. It is demonstrated numerically that almost always the unstable states evolve via a turbulent path toward a coherent structure. A possible explanation in terms of a dual cascade phenomenology is posed. A numerical study of various initial conditions suggests that it is possible to control the oscillation amplitude of the coherent structure. Also a study of cross-sectional aspect ratio is reported. However, several issues remain unanswered. For example, it is experimentally seen that the oscillation amplitude of the coherent structure may grow, damp or stay put depending on initial conditions. Our present simple model is unable to explain this observed feature. Role of neutrals, residual ions, machine asymmetries and compressibility have not been fully understood yet. An improved model using the full set of ﬂuid equations will be reported elsewhere.4 Our present ﬁndings lead us to believe that toroidal pure electron plasmas trapped in small aspect ratio system such as ours would be an ideal “test bed” for studying compressible ﬂuid-dynamics related issues, similar to the pure electron plasmas in uniform magnetic ﬁeld for incompressible ﬂuids. However, more work needs to be done both experimentally and via simulations to further elaborate the aforementioned similarity. In Sec. II, we discuss the experimental setup and present our major results. Numerical model is presented in Sec. III with crucial diﬀerences between the actual experiments and the simulation. Sec. IV contains discussions, conclusion and future direction of experiments and simulation. 2. Experimental Setup and Some Major Results The experimental setup is shown in Fig.1. Various device-related parameters are deﬁned below.

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2.1. Device Parameters • • • • • • • • •

B0 100 − 200 gauss, tﬂat-top 1.2 ms, tB = 4.5 ms R0 13.5 cm, Rinner = 5.0 cm, Router = 22.0 cm a 8.5 cm, A = R0 /a 1.6, Lz 32 cm Acs = Lz/(2a) 2 Circ. Injector : Rinj 5 & 3 cm Vacuum 5 × 10−8 − 3 × 10−7 torr Trap-Voltage Vtrap −410 volts Injector-Voltage Vinj −330 volts C-trap angle 330 o

Relevant physical parameters such as time and length scales for the above said device parameters is given below. 2.2. Physical Parameters • ωce = eB0 /m 300 − 600 × 106 Hz • Bounce frequency ωB,e = v||,e /(1.83π ∗ R) 42 MHz • ωpe = ne2 /0 m 9 MHz 2 2 /ωce 0.02 (low density limit) ωpe 2 /(2ωce) 100 KHz • Diocotron freq. ωD (measured) = ωpe • Estimated density (not-measured) n = 106 cm−3 • Estimated Temp. Te (not-measured) = 1 − 300 eV Larmor radius λD (Te 300 eV ) = 0.2 cm − 300 eV) = 0.7 − 13.0 cm • Debye Length λD (Te 1& 9 • e-n collision (Hz) P [torr] & 3 (T [eV])7.4 × 10 740 Hz • e-e collision (Hz) 120 (T [eV]) 120 Hz Note that the bounce frequency (ωB,e ) is much larger than the any other frequency in the system. Our interest is in cross-ﬁeld Diocotron physics 2 /2ωe ) 10 µs) whose time scales are much slower than the (tD = 2π/(ωpe bounce time (2π/ωB,e 150 ns) rendering the system essentially 2D in (R, Z) for our time scales of interest. 2.3. Toroidal Diocotron Oscillations With Eext = 0 In SMARTEX-C, perhaps for the ﬁrst time, the longest ever conﬁnement of electron plasma in a grounded torus, without the aid of any Eext , has been achieved. The experimental setup has 5 wall probes mounted ﬂush with the chamber wall on the “inboard” side on one toroidal location. Another wall probe is placed on the outer wall at the same toroidal location.

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Fig. 1. A schematic illustration of the the experimental setup SMARTEX-C. Device parameters are listed in the text.

The inner wall probe signal is presented in ﬁg.2. Except during the ﬁrst diocotron period, the rest of the 400 diocotron times shows coherent wall probe oscillations. In the later sections, it will be numerically demonstrated that the initial “noisy” period is due to the Kelvin instability followed by formation of large number of small vortices and inverse cascade leading to coherent structure and its toroidal dioctron oscillation. In ﬁg.3 the amplitude of the oscillation envelope and frequency (as a function of time) has been shown.2 Note the unmistakable growth and the unusual frequency variation in time. In cylindrical or large aspect ratio toroidal devices8 the frequency dependence on time is quite monotonic. However, in small aspect ratio device the dependency is seen to be nonmonotonic. As mentioned in the Introduction, the amplitude growth and strong time variation of frequency seen in SMARTEX-C has not yet been explained. As will be discussed in the next section, reason(s) could be one or

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Fig. 2. Wall probe signal obtained in SMARTEX-C: Plasma lifetime is seen only to be limited by the magnetic ﬁeld lifetime tB . 400 Diocotron oscillations seen here is the longest conﬁnement time seen in a small aspect ratio device in the absence of Eext .

more dissipation mechanism(s) from the following : 1. electron-neutral collisional dissipation. 2. residual ion resonant damping 3. magnetic pumping in the presence of electron-neutral collisions 4. physics beyond two dimensions (end-plug eﬀects). Some of these will be discussed in the section for Simulation. In ﬁg.4, we have shown a “blow-up” of the oscillations near the saturation ie., at about 100 TD (or 100 toroidal diocotron times). This picture clearly reveals the “double-peaked” nature of the oscillations. Finally, in ﬁg.5 the power spectrum or mode structure is shown. In a frequency sheared system such as the present one, instead of standard Fourier transform, a model-based estimation is used to obtain the local frequency which is then used to generate a phase-function φ(t); the original signal sampled uniformly in φ(t) yields a set of samples without any shear.2,3 The power spectrum obtained after this “timewarping” (of the entire signal) is shown in ﬁg.2. Note that the power peaks at m = 2 and 16 harmonics are seen showing the coherent nature of the oscillations. Though the dynamics is not yet understood, we attribute this mode structure to the small aspect ratio of the device where “elliptical” component is the most dominant one as compared to others and to the cross-sectional aspect ratio. This feature is perhaps seen for the ﬁrst time in toroidal devices.

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Fig. 3. Amplitude and frequency evolution of the signal shown in ﬁg.2. While the amplitude growth waxes and wanes, the non-monotonic time-variation of frequency is a new feature. Experiments in cylindrical devices or large aspect ratio toroidal devices exhibit monotonic behaviour.

3. Numerical Model The code TEPFCT is setup in cylindrical co-ordinates (R, Φ, Z) with axisymmetry ∂/∂Φ ≡ 0. The schematic of the system is shown in Fig.6. Thus all quantities are two dimensional (2D) in (R, Z). The governing equations are the following:

Momentum: mn

D v = −en(E + v × B) − ∇p − mn(v − v 0 )νc Dt

where

∂ D = +v·∇ Dt ∂t

is material derivative

(1)

The other quantities are m− electron mass, n− density, v− electron ﬂuid ˆ external velocity, E = −∇φ the self-consistent electric ﬁeld, B = B(R)Φ ﬁeld with B = B0 R0 /R, R0 is the major radius of the vessel, p is the particle pressure, νc is collision frequency with ions/neutrals moving with velocity v0 .

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Fig. 4. Close-up of the oscillations: 10 Diocotron periods are shown from 400 periods starting from 100 to 110 oscillations. During this period the growth is nearly saturated.

Poisson: en with E = −∇φ 0 en becomes ∇2 φ = 0

∇·E = −

(2)

Appropriate boundary conditions for φ(R, Z) are to be used. In the simulation, φ(R = R1 , Z) = φ(R = R2 , Z) = φ(R, Z1 ) = φ(R, Z2 ) ≡ 0.

Continuity: ∂n + ∇ · (nv) = 0 (3) ∂t Here no source or sinks are to considered, i.e., rhs is zero. Importantly, compressibility eﬀects such as ∇ · vne 0 are included fully.

Equation of State: D p =0 Dt (n)γ

(4)

where the ratio of speciﬁc heat γ = 1 + (2/Ndof ) where Ndof is number of degrees of freedom and p is the pressure. Note that the adiabatic pressure

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Fig. 5. Mode structure obtained using a “timewarping” technique reveals that the predominant amplitude is m = 2.

closure renders the system of equations self consistent with total number of unknowns [vR , vz , φ, n, T ] and the number of equations equal. For the purposes of the present work, we have used the following simpliﬁcations. 3.1. Model and Assumptions: (1) Physics Issues: 2 2 /ωce = (nm)/(0 B02 ) 1. This approximation alWe assume that ωpe lows us to neglect the following eﬀects: (i). any self consistent magnetic ﬁeld produced by the plasma is small and negligible, i.e., electrostatic limit (ii) Magnetic pressure (B02 ) is much stronger than the repulsive electrostatic pressure i.e., the plasma is well below the Brillouin limit deﬁned as ωp2 /ωc2 = 1. A simple E × B dynamics for electron-ﬂuid element (m → 0 limit) has been used. Residual ions are considered absent (v0 → 0). The plasma is considered as a cold ﬂuid (i.e., T → 0 limit) and no dissipation is included(νc → 0 limit). Convective nonlinearity is absent (m → 0 limit) and streaming along Φ−direction is considered as absent.

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Fig. 6. A schematic illustration of the cylindrical geometry used in the simulation. Note that experimental device breaks the toroidal symmetry ∂/∂Φne 0, but, simulations use axisymmetry geometry ∂/∂Φ = 0. The bounding walls are grounded. Electron plasma trapped in this device is shown schematically.

(2) Normalization: Length scales are normalized to a deﬁned as 2a = R2 − R1 where R1,2 are the radial locations of the inner and outer walls of the device. Time scales are normalized to n0 e/0 B0 where B0 is the magnetic ﬁeld at the major radius of the vessel e is the electronic charge and n0 is the typical density length scale. (3) Computational Issues: In SMARTEX-C experimental setup, the axisymmetry has been violated. In the simulation however, we employ cylindrical co-ordinates with axisymmetry That is, we believe that the physics is entirely two dimensional (R, Z) and would be an adequate physical model at least qualitatively. This assumption will thus allow us to estimate how crucial is axisymmetry for the problem. Uniform gridding is employed in R and Z. Time step Dt is computed using Courant number (C) as vp ×Dt < 1. Here vp is a typical follows: Courant number C = min (DR,DZ) physical velocity. For example, in our model it is the E×B velocity estimate from the previous time step. A Flux Corrected Transport package LCPFCT developed by Oran et al5 is used to solve the electron continuity in each direction [4]. The 2D continuity equation is solved ﬁrst for

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each Z long R and then for each R along Z using LCPFCT and averaged. This method is called multidimensional time splitting. Presently, the code works on a single processor. It has been bench marked in a newly obtained supercomputer CRAY X1E at IPR. 3.2. Equilibrium or Steady State One can obtain an equilibrium or steady state using the “momentum” equation E×B (5) v= B2 where B = B(R)φˆ and B(R) = B0 R0 , the continuity equation R

∂n + ∇ · (nv) = 0 ∂t and by setting ∂/∂t → 0. We get n ∇ · ( 2 E × B) = 0 → n/B 2 = f (Φ) B where f (Φ) is any arbitrary function of Φ. Therefore we get ∇2 Φ =

ef (Φ)B 2 (R) 0

(6)

(8)

This is a general nonlinear equilibrium or steady state problem. The question now is how to decide which f (Φ) to choose ? A relationship between n and Φ could be obtained using a variational principle. For example, by extremizing enstrophy subject to the constraints of total circulation (or particles) yields nB02 = n0 (1 + Φ) (9) B2 One could also extremize entropy which yields,11 for pure electron plasmas, a Fermi-like distribution q nB02 = 2 B 1 + exp q(βΦ − µ)

(10)

Here q is the normalization on density, β and µ are the Lagrange multipliers whose value should be obtained from the conservation of electrostatic energy and total circulation or particles. A typical equilibrium or steady state solution using Fermi-like distribution is shown in ﬁg.7a. The coherent structure obtained above is a equilibrium solution of the underlying equations. The code TEPFCT was initialized with the above shown equilibrium. A test-particle initialized [Rp(t = 0), Zp(t = 0)] = [1.68, 2.0] is shown in

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Fig. 7. (a) An equilibrium or steady state constructed using a Fermi-like distribution (see text) is shown. The magnetic ﬁeld B points “into” the plane of the paper. The arrow shown indicates the sense of vortex rotation. A test particle initialized at (1.68, 2.0) [(b),(c)] would move downwards, closer to the inner wall and complete a single rotation in a vortex turn-over time. Equilibrium solution has been obtained numerically by solving the nonlinear diﬀerential equations and the constraints of total particle or circulation and energy.

ﬁg.7b,c to complete a single rotation which is one vortex-turn-over period. The time evolution of the conserved quantities is shown in ﬁg.8. These studies show that our simulations give reasonably accurate results. Note that the conservation is demonstrated to be less than 1% for over 50 tD or vortex turn-over periods and does not show signs of diverging. Also, the code has been tested for higher grid resolutions. 3.3. Experimentally Relevant Initial Conditions and Their Evolution As described in the experimental section, SMARTEX-C has injector rings of two diﬀerent radii at the mid-plane and mounted coaxially. In ﬁg.9, the three initial conditions studied here are shown. “Strip-1” and “Strip2” are identical in size to those used in the experiments “Strip-3” is an initial condition for which the simulations results show lower amplitude oscillations. This will be discussed later. In the following section, the results of numerical simulation for strip-1 will be discussed.

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t/t

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D

Fig. 8. Conserved quantities such as total particles and energy are shown as a function of time for 50 vortex-turnover periods. While total particles are conserved quite accurately, energy is conserved to with ±0.7%. ∆N, W are deﬁned as ∆Q(t) = (Q(t) − Q(0)) ∗ 100/Q(0), Q = N or W .

3.3.1. Numerical Evolution of Strip-1 For about 10 oscillation periods or toroidal diocotron period, the global conserved quantities such as the total energy W , total number of particles (or total circulation) N for for strip-1 are shown in ﬁg.10. These quantities are deﬁned as : N (t) = drn(r, t) (11) 1 drn(r, t)Φ(r, t) (12) W (t) = 2 Note that our boundary conditions are such that all the four walls are grounded. Hence angular momentum P deﬁned as (13) P (t) = − dr ln(R(t)/R0 )n(r, t) is not a conserved quantity. However, it serves as a useful numerical diagnostic to track oscillations. In the simulation, to emulate the experiments, the potential close to four walls (“wall probe”) are recorded as function of time. The time derivative of

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Fig. 9. Experimentally relevant density initial conditions are shown viz., strip-1, strip-2. In normalized units the radii are 0.38 and 0.58 respectively. Strip-3 is a initial condition for which the simulations predicts lower amplitude oscillations. Experiments for strip-3 are being planned in SMARTEX-C.

this potential is shown in ﬁg.10. After an initial noisy phase, we see a clear coherent double peak oscillation setting in. As will be discussed, strip-1 is an unstable initial condition which undergoes a velocity shear instability and ﬁnally evolves into a coherent nonlinear rotating vortex structure via inverse cascade. To attempt a qualitative comparison with the experiments, in ﬁg.11 three rows are shown. Top row shows the full length of experimentally obtained wall probe signals (normalized to the toroidal diocotron time tD1 = 10 µs). There are over 400 oscillation periods. Note that the amplitude of the oscillations grow, saturate and damp oﬀ. For comparison purposes, only 10 (ten) oscillations are shown in the Middle row where the amplitude saturates around 100 tD1 . In the last row, the simulational results of the wall probe data are presented. A comparison shown a clear qualitative reproduction of the “double peaked oscillation” demonstrating that the basic phenomenon is E × B dynamics. We attribute the double peak to two physical reasons : (i) low aspect ratio eﬀects couple the poloidal modes resulting in a “D” shaped orbit of the centre-of-charge. (ii) the vessel cross-sectional aspect ratio Acs = b/a where 2a = R2 −R1 and 2b = Z2 −Z1 for both experiments and simulations is Acs = 2, which adds to the ellipticity of the centre-of-charge trajectories. These two eﬀects together result in “double-peaked” oscillations. Note that the simulations are shown only

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for 10 (ten) oscillations. There is no growth of the amplitude is seen in the simulational results. This failure of numerical simulations to reproduce the amplitude growth implies that though E × B adequately reproduces the nature of the oscillation cycle, the amplitude growth is more than just E × B dynamics. It is well known in cylindrical pure electron plasmas with uniform B ﬁeld that the m = 1 Diocotron oscillations (or centre-of-charge oscillations) is a negative energy mode. Meaning, if energy is extracted from the mode, the mode has a tendency to grow and vice versa. This is exactly the opposite to what happens with ordinary positive energy modes. Assuming a similar idea to be applicable to the toroidal problem, we attribute the amplitude growth to one of the following energy damping mechanisms (1) Neutral-electron collisions (2) Residual ion resonant damping (3) Compressible E × B rotation-driven magnetic pumping (4) A combination of magnetic pumping and eﬀects due to incomplete symmetry in the toroidal direction. TEPFCT code has been generalized to solve the full set of ﬂuid

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Fig. 11. Wall Probe Signal: A qualitative comparison of experimentally observed oscillations on the wall probe and those found in the TEPFCT simulations: (Top) experimentally found wall probe signal with oscillation period tD1 = 10µs. Note that over 400 oscillation periods exist. Oscillation amplitude grows slowly, saturates and damps oﬀ. (Middle) Experimental oscillations between 100 and 110 where amplitude nearly saturates. (Bottom) Oscillations obtained in the simulations. Signal shows a clear qualitative agreement between experiments and simulation. Compressible E × B dynamics adequately explains the “double peak” oscillations but fails to explain the amplitude growth.

equations for electrons and drift-kinetic equations for ions and is being benchmarked presently. The results will be reported elsewhere. 3.3.2. First Diocotron Period: Turbulence and Coherent Structure It is seen in the experiments that in the very ﬁrst Diocotron period, the initial conditions undergoes massive re-organization via a noisy phase and followed by the birth of a coherent toroidal vortex. Experimentally, the noisy phase and the birth of coherent vortex is indirectly inferred from wall probe signals. From numerical simulation of the ﬁrst diocotron period, this inference is indeed seem to be justiﬁed. In ﬁg.12, we show the density contours of the

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Fig. 12. First Toroidal Diocotron Period : Starting from Strip-1, density contours in (R,Z) plane are shown for six successive times within one toroidal diocotron period. The strip undergoes velocity-shear instability or Kelvin instability, followed by ﬁlamentation, formation of many small vortices, re-merger and emergence of coherent structure. Simultaneously, the centre-of-charge of the system undergoes a E × B motion in counterclockwise direction.

ﬁrst oscillation which clearly shows (i) Kelvin instability (ii) ﬁlamentation into large number of small vortices (iii) coalescence of these vortices into a coherent structure, probably via inverse cascade and ﬁnally in the end of one diocotron period clear coherent oscillations set in. 3.3.3. Nonlinearity and Matched Injection : Strip-1 and Strip-3 For a small aspect ratio torus, the plasma radial force balance or “equilibrium” point is close to the inner wall (see Equilibrium section). Simulation of wall probe signals in the two cases viz., Strip-1 and Strip-3 (ﬁg.13) suggests that the nonlinearity (amplitude of the oscillation) can be con-

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trolled by “matched injection”. The red curve is the Toroidal Diocotron Oscillations due to Strip-1. While the curve in magenta is the oscillation amplitude due to Strip-3 showing substantial reduction in the amplitude and near-complete suppression of centre-of-charge Diocotron (quasi-linear) oscillation. Since the centre-of-charge excursion is reduced, so is the nature of the “double peaked” oscillations demonstrating that the double peaked signal is both due to toroidicity and due also to cross sectional aspect ratio Acs = b/a (where 2b is the height of the vessel). Eﬀect of Acs on the “ellipticity” of the system etc will be discussed elsewhere4 4. Conclusion and Future work In conclusion, (1) Experimentally, SMARTEX-C results are the longest conﬁnement times reported so far. (2) Using a simple compressible E × B model many qualitative features

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observed in experiments are shown to be reproducible using arbitrary aspect ratio simulation. Incomplete toroidal symmetry has no eﬀect on the toroidal diocotron oscillations : C-Traps retain eﬀective two-dimensionality. Qualitative reproduction of experimentally obtained “double peaked oscillations” with grounded wall boundaries: External electric ﬁeld Eext = 0 Double peaked nature of oscillation is due to a combination of strong toroidicity due to inhomogeneous B ﬁeld and the cross-sectional aspect ratio4 of the conﬁning vessel Within the ﬁrst one or two toroidal diocotron oscillation time scale, compressible, 2D, E×B dynamics leads via inverse cascade to formation of coherent vortex. Our results indicate that in the future, toroidal experiments such as SMARTEX-C would become ideal “test-beds” for performing “compressible ﬂuid-dynamic” experiments.

In the future work, (1) An explanation for growth of mode amplitude is being sought. Few eﬀects that we are looking at are (i) residual Ions/Magnetic Pumping (ii) electron-neutral dissipation (iii) 3D eﬀects (end plug eﬀects) One (or many) of these eﬀects could provide with an explanation for the slow amplitude growth observed in the experiments. (2) Frequency variation and Q/B scaling (where Q is the total vortex charge and B is the magnetic ﬁeld strength) observed in the experiments (the later one is not discussed here, see Ref.[2]). (3) Current version of TEPFCT includes the complete dynamical equations. Simulations with complete dynamics will be reported elsewhere.

Acknowledgements The numerical part of this work was done in MODG Linux cluster and CRAY X1E at the Institute for Plasma Research (IPR). Authors are indebted to the Computer Centre, IPR for their constant support. One of the authors (RG) acknowledges support for data analysis of the numerical results by D. Raju, the details of this analysis will be published elsewhere. Authors thank S. Sengupta and R. Srinivasan for sharing their experiences with FCT algorithms.

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References 1. S. Pahari, Electron plasmas in a toroidal penning trap, PhD Thesis, Institute for Plasma Research, India (2004). 2. S. Pahari et al, Phys. Plasmas, 13, 092111 (2006) 3. S. Pahari et al, Submitted to Physics of Plasmas (2005) 4. R. Ganesh et al, To be submitted for publication (2006). 5. E. S. Oran and J. P. Boris, Numerical Simulation of Reactive Flow, 2nd Edn, Cambridge University Press, Cambridge, UK (2000), Chapter 8. 6. J. D. Daugherty, J. E. Eninger and G. S. Janes, Phys. Fluids 12, 2677 (1969) 7. P. Zaveri at al, Phys. Rev. Lett., 68, 3295 (1992), S. S. Khirwadkar et al, Phys. Rev. Lett., 71, 4334 (1993) 8. M. R. Stoneking et al, Phys. Rev. Lett., 92, 095003 (2004) 9. H. Saitoh et al, Phys. Rev. Lett., 92, 255005 (2004) 10. T. S. Pedersen (Private Communication by S. Pahari) 11. R. Ganesh, Studies in statistical mechanics of magnetized plasmas, PhD Thesis, Institute for Plasma Research, India (1998).