CSIRO Workshop on Turbulence and Coherent Structures in Fluids, Plasmas and ... Scientific Lecture Notes in Complex Systems)

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. Jim´enez

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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 Poincar´e called Poisson stable7,8 (small perturbations are self-correcting) — or they may be chaotic† . As concepts of feedback and stability were developed rigorously and applied in the late nineteenth and early twentieth centuries, Divine open-loop † A fact which might cause you some queasiness to learn. Fear not — it is believed that chaotic motions were important in the early evolution of the solar system,9 and a slow chaotic drift may be noticeable a few billion years hence.6,10 What luck for us!

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

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

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

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

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

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

(2)

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

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

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

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

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

Fig. 4.

An asymptotically stable equilibrium is also called a sink.

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

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

(3)

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

ψ = ψ0 + δψ,

ω = ω0 + δω,

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

(4)

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

d



dt δϕ d   dt δψ  , d dt δω



0

1

 sin2 ϕ0 b A =  − g cos ϕ0 − m

(5) 0



2g sin ϕ0  , ω0

− Jk sin ϕ0 0



δϕ



  x = δψ  .

0

δω

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

(6)

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

(7)

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

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

(9)

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

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

(10)

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

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

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

(11)

and equations of motion q˙ = p,

p˙ = − sin q.

(12)

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

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Fig. 6.

The phase space of the simple pendulum

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

(13)

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

q˙2 = −1;

p˙ 1 = −µ sin q1 − µε cos q1 cos q2 ,

p˙2 = µε sin q1 sin q2 .

(14)

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

(15)

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

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

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

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

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

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

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

(17)

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

(18)

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

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

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

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

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

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

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

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

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

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

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

(5)

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

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

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

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

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

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

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

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

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

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

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

N 

κν φν exp(−iω ν t)

ν=1

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

ν = 1, . . . , N.

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

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

∂A ∂B ∂A ∂B − ∂λ ∂µ ∂µ ∂λ

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

fo2 ; σ ¯

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

f=

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

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

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

∞ 

ψmn Pnm (µ)ei(mλ−ωt) ;

τ = Re

n=m

∞ 

τmn Pnm (µ)ei(mλ−ωt)

n=m

the eigenvalues are given by

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

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

1 q0µ ≡ 2−

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

3 q0µ ≡ 2−

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

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

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(a)

(b)

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

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

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

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

(a)

(b)

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

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

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

(a)

(b)

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

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

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

(a)

(b)

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

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

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

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

(b)

(a)

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

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

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

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

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

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

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

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

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

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

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

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

xν (t) exp(−ωiν t)

xν (0)

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

d ln Rν (t), dt

ν = 1, . . . , N.

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

(a)

(b)

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

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

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

(a) Fig. 9.

(b)

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

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

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Fig. 10.

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

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

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

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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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55. Z. Toth and E. Kalnay, Ensemble forecasting at NCEP and the breeding method. Mon. Wea. Rev., 125, 3297–3319 (1997). 56. K.E. Trenberth, Seasonality in Southern Hemisphere eddy statistics at 500 mb. J. Atmos. Sci., 39, 2507–2520 (1982). 57. K.E. Trenberth, Recent observed interdecadal climate changes in the Northern Hemisphere. Bull. Amer. Meteor. Soc., 71, 988–993 (1990). 58. M. Wei and J.S. Frederiksen, Error growth and dynamical vectors during Southern Hemisphere blocking. Nonlin. Process. Geophys., 11, 99–118 (2004). 59. M. Wei and J.S. Frederiksen, Finite-time normal mode disturbances and error growth during Southern Hemisphere blocking. Adv. Atmos. Sci., 22, 69–89 (2005). 60. J.S. Whitaker and A. Barcilon, Type B cyclogenesis in a zonally varying flow. J. Atmos. Sci., 49, 1877–1862 (1992). 61. J.S. Whitaker and R.M. Dole, Organization of storm tracks in a zonally varying flow. J. Atmos. Sci., 52, 1178–1191 (1995). 62. P.J. Webster and S. Yang, Monsoon and ENSO: Selectively interactive systems. Quart. J. Roy. Meteor. Soc., 118, 877–926 (1992).

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

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

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

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

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

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

(1)

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

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

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

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

(4)

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

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

(5)

−(px qy − py qx ) . p2

(6)

and A(k, p, q) =

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

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

(7)

k

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

(8)

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

(9)

and

where

 β + k02 U ωk (U ) = kx U − k2

(10)

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

(11)

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∂ (zk ) = ωk (U )(zk ) − αζ (zk ) ∂t

(12)

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

(13)

zk = ζk h∗k .

(14)

where

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

(15)

−kx |hk |2 αζ U . (αζ )2 + (ωk (U ))2

(16)

(zk ) = and (zk ) =

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

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

(17)

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

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

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

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

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

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

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

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α

α

α

α

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

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

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

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

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α

α

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

α

Fig. 6.

α

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

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

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

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

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

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

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

(19)

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

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

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

(21)

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

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and Ampr nqs = −

1 q(q + 1)



1

−1

dµPnm (µ)P,

(23)

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

(24)

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

(25)

and

where

 2 + 2U ωk (U ) = m U − n(n + 1)

(26)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

(1)

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

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

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

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

(2)

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

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

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

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

=

1 Y T2

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

t=1

(5)

t=1

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

t=1

t=1

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

t=1

t=1

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

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

(8)

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

R  r=1

e(r)xyt (r) =

R  r=1

e(r)µy (r) +

R 

e(r)εyt (r) ≡ µ y + εyt ,

r=1

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

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

(9)

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

(10)

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

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

(11)

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

(12)

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

(13)

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

(14)

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

β=

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

 2 − 2β 2β − 1 = 2α . 2β − 1 2 − 2β

(15)

(16)

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

a + 2b 2 (a + b)

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where, a=

+

b=

+

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

(17)

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

(18)

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

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

(21)

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

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

(22)

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

(23)

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

(24)

Assumptions (22)-(24) imply that,

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

(25)

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

(26)

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

δ=

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

 2 − 2δ 2δ − 1 = 2γ . 2δ − 1 2 − 2δ

(27)

(28)

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

c + 2d 2 (c + d)

where, c=

+

d=

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

(29)

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

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

(30)

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

(31)

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

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

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

(33)

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

(34)

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

Synthetic Data

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

(35)

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

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

90

 =

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

(36)

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

T

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

(37)

T

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

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

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

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

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

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

Estimation Method

40 years

80 years

40 years

80 years

I-REOF1

0.95

0.97

0.97

0.98

I-REOF2 I-REOF3

0.97 0.93

0.98 0.96

0.98 0.96

0.99 0.98

I-REOF4

0.94

0.98

0.97

0.99

I-REOF5 I-REOF6

0.92 0.93

0.97 0.95

0.96 0.95

0.98 0.96

S-REOF1 S-REOF2

0.97 0.84

0.98 0.89

0.92 0.72

0.94 0.79

S-REOF3

0.87

0.91

0.73

0.77

S-REOF4

0.59

0.55

0.42

0.47

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

R 

cpr ury = cTp uy

(A.1)

r=1

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

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

then it can be shown problem

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

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

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

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

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

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

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

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

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

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Fig. 2. plane.

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

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

(1a) (1b)

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

∂ψ =0 ∂x ∂2ψ = 0. ∂y 2

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

y y + (1 − σ) cos 2π ) ; B B

τ y (x, y) = 0 ,

(2)

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

Fig. 3.

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

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

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Value

Parameter

Value

L H f0 ρ0

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

τ0 β0 U B

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

106

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

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(a)

(b)

(c)

(d)

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

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

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(a)

(b)

(c)

(d)

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

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

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

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(a)

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

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

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(a) Fig. 7. of σ.

(b)

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

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

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(a)

(b)

(c)

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

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

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

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

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

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

−ψpo − ψtr , max | ψ |

(3)

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

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

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

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

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

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(a)

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

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

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

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

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(a)

(b)

(c)

(d)

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

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

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(a)

(b)

(c)

(d)

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

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

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(a)

(b)

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

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

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

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

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

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

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