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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

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

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

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

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

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

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Nonlinear resonance and chaos in an ocean model A. Kiss

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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

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

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

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What are we learning from simulating wall turbulence? J. Jiménez

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

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Two-point turbulence closures revisited W. D. McComb

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

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

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

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

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

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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 Poincaré 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 Poincaré. 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 Poincaré 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. Poincaré’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 Poincaré’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-Loéve (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 Poincaré 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 Poincaré’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 ). Poincaré 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. Poincaré 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.”(Poincaré,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 Poincaré describe homoclinic chaos, after years of careful and productive analysis of the phenomenon. In particular, Poincaré 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 Poincaré 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, Poincaré 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 Poincaré 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 Poincaré 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. Naïvely, 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 Poincaré 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 théorie génerale des régulateurs) . 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. Poincaré, Sur le probléme des trois corps et les équations de la dynamique. Acta, 13, 1–270 (1890). 8. J. Barrow-Green, Poincaré 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. Cvitanović, 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, Poincaré, 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. Poincaré, Les Méthodes Nouvelles de la Mécanique céleste. 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 Poincaré 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 Poincaré 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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38. T.L. Hart, W. Bourke, B.J. McAvaney, W. Forgan and J.L. McGregor, Atmospheric general circulation simulations with the BMRC global spectral model: The impact of revised physical parameterizations. J. Climate, 3, 436– 459 (1990). 39. IOCI, Climate variability and change in south west Western Australia. Indian Ocean Climate Initiative Panel, Department of Environment, Water and Catchment Protection, Perth, W.A., September, 34 pp. (2002). 40. M. Latif and N.E. Graham, How much predictive skill is contained in the thermal structure of an OGCM? TOGA Notes, 2, 6–8 (1991). 41. S. Lee, Linear modes and storm tracks in a 2-level primitive equation model. J. Atmos. Sci., 52, 1841-1862 (1995). 42. H. Lejenas and H. Okland, Characteristics of Northern Hemisphere blocking as determined from a long time series of observational data. Tellus, 35A, 350–362 (1983). 43. Z. Li, A. Barcilon and I.M. Navon, Study of block onset using sensitivity perturbations in climatological flows. Mon. Wea. Rev., 127, 879–900 (1999). 44. E.N. Lorenz, A study of the predictability of a 28-variable atmospheric model. Tellus, 17, 321–333 (1965). 45. A.R. Lupo, A diagnosis of two blocking events that occurred simultaneously over the midlatitude Northern Hemisphere. Mon. Wea. Rev., 125, 1801–1823 (1997). 46. J.L. McGregor and M.R. Dix, The CSIRO conformal-cubic atmospheric GCM. In: IUTAM symposium on Advances in Mathematical Modelling of Atmosphere and Ocean Dynamics, 298 pp. Ed. P.F. Hodnett. Kluwer Academic Publishers, Dordrecht, 197–202 (2001). 47. F. Molteni, R. Buizza, T. Palmer and T. Petroliagis, The ECMWF ensemble prediction system: Methodology and validation. Quart. J. Roy. Meteor. Soc., 122, 73–119 (1996). 48. N.A. Phillips, A simple three-dimensional model for the study of large-scale extratropical flow patterns. J. Meteor.,8, 381-394 (1951). 49. A.W. Robertson and W. Metz, Three-dimensional linear instability of persistent anomalous large-scale flows. J. Atmos. Sci., 46, 2783–2801 (1989). 50. A.W. Robertson and W. Metz, Transient-eddy feedbacks derived from linear theory and observations. J. Atmos. Sci., 47, 2743–2764 (1990). 51. B.S. Sadler, G.W. Mauger and R.A. Stokes, The water resources implications of a drying climate in south-west Western Australia. In Greenhouse: Planning for Climate Change, 296- 311, ed. G.I. Pearman, Commonwealth Scientific and Industrial Research Organisation, Australia, 752 pp. (1988). 52. I.N. Smith, P. McIntosh, T.J. Ansell, C.J.C. Reason and K. McInnes, Southwest Western Australian winter rainfall and its association with Indian Ocean climate variability. Int. J. Clim., 20, 1913–1930 (2000). 53. A.J. Simmons, J.M. Wallace and G.W. Branstator, Barotropic wave propagations and instability, and atmospheric teleconnection patterns. J. Atmos. Sci., 40, 1363–1392 (1983). 54. Z. Toth and E. Kalnay, Ensemble forecasting at NMC: the generation of perturbations. Bull. Amer. Meteor. Soc., 174, 2317–2330 (1993).

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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

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