Relaxation Dynamics in Laboratory and Astrophysical Plasmas

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Author: Diamond P.H. | Garbet X. | Ghendrith P. | Sarazin Y. (eds.)

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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.

Biennial Reviews of Fusion Physics — Vol. 1 RELAXATION DYNAMICS IN LABORATORY AND ASTROPHYSICAL PLASMAS Copyright © 2010 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-13 978-981-4291-54-5 ISBN-10 981-4291-54-4

Printed in Singapore.

Lakshmi - Relaxation Dynamics.pmd

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In 2003, three of the most distinguished and brightest plasma physicists passed away: Professor Marshall Rosenbluth, Professor Masihiro Wakatani and Doctor Ren´e Pellat. This book is dedicated to them.

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Foreword

The material for this book was written shortly after the second “Festival de Th´eorie,” held in Aix-en-Provence (France) in July 2003. This Festival de Th´eorie was initiated in 2000 by Jean Jacquinot, and Ren´e Pellat chaired the Director Committee. From the very beginning, the Festival de Th´eorie was also supported by Professor Masihiro Wakatani and Professor Marshall Rosenbluth. But Professor Masihiro Wakatani, Professor Marshall Rosenbluth and Doctor Ren´e Pellat passed away in 2003. This book is dedicated to them. A speciﬁc article addresses the crucial achievements of Professor M. Wakatani. Tributes to the outstanding contributions of M. Rosenbluth and R. Pellat will be paid in the following volumes of the series. The Festival de Th´eorie has taken place every two years in Aix-enProvence, since 2001. Its scientiﬁc committee is chaired by Professor Patrick Diamond and co-chaired by Doctor Xavier Garbet. This international meeting gathers during three weeks about 30 senior experts on well-focused subjects in magnetised plasma physics, both in controlled fusion and in astrophysics. The objective is three-fold. First, the participants present the most recent results in the ﬁeld and promote new ideas. Secondly, subgroups are formed and start new common work on the subject. The third objective is to provide educational training for PhD and post-doc students. This latter is being given a growing attention, and dedicated lectures are being organised for the younger physicists. This book explores in a pedagogical way the present understanding of “Relaxation Dynamics in Magnetised Plasmas,” which was the focus of the Festival in 2003. It is suited for physicists and advanced students in the ﬁeld, as well as researchers in other ﬁelds who would like to acquire an exhaustive and up-to-date view on these problems. vii

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Contents

Foreword

vii

In memory of ...

1

1. Prof. Masahiro Wakatani and Fusion Research in His Days

3

K. Itoh 1.1 1.2 1.3 1.4

Research period of Prof. Wakatani . Research on nonequilibrium systems Wakatani-sensei’s gifts to scientists . Closing words . . . . . . . . . . . . .

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Magnetic relaxation and self-organization in astrophysical and laboratory plasmas

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2. An Introduction to Mean Field Dynamo Theory

15

D.W. Hughes and S.M. Tobias 2.1 2.2

2.3

Introduction . . . . . . . . . . . . . . . . . . . . . Kinematic Mean Field Theory . . . . . . . . . . . 2.2.1 Formulation . . . . . . . . . . . . . . . . 2.2.2 Calculation of the transport coeﬃcients . 2.2.3 Linear mean ﬁeld dynamo models . . . . Incorporating Nonlinear Eﬀects . . . . . . . . . . 2.3.1 The nonlinear behaviour of the transport coeﬃcients . . . . . . . . . . . . . . . . . ix

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2.4

2.3.2 Other nonlinear eﬀects . . . . . . . . . . . . . . . 2.3.3 Nonlinear mean Field Models . . . . . . . . . . . Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . .

3. Origin, Structure and Stability of Astrophysical MHD Jets

40 41 45 49

P.-Y. Longaretti 3.1 3.2

3.3

3.4

3.5

3.6 3.A

3.B

3.C 3.D 3.E

Introduction . . . . . . . . . . . . . . . . . . . . . . . Generalities on astrophysical jets . . . . . . . . . . . 3.2.1 Accretion . . . . . . . . . . . . . . . . . . . . 3.2.2 Magnetic ﬁelds in jets . . . . . . . . . . . . . 3.2.3 Relativistic vs non-relativistic jets in AGNs . Origin of jets . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Central object . . . . . . . . . . . . . . . . . 3.3.2 Disk/central-object interaction region . . . . 3.3.3 Disk . . . . . . . . . . . . . . . . . . . . . . . MHD jet structure . . . . . . . . . . . . . . . . . . . 3.4.1 Jet launching and magnetic surface opening 3.4.2 Current structure, and jet acceleration and collimation . . . . . . . . . . . . . . . . . . . Stability of MHD jets . . . . . . . . . . . . . . . . . 3.5.1 Global disk-jet stability and stationarity . . 3.5.2 Motion driving: the Kelvin-Helmholtz instability . . . . . . . . . . . . . . . . . . . 3.5.3 Magnetic driving: the current and pressure instabilities . . . . . . . . . . . . . . . . . . . Summary and open issues . . . . . . . . . . . . . . . A brief compendium of relevant astrophysical facts . 3.A.1 Mostly AGNs . . . . . . . . . . . . . . . . . 3.A.2 Origin of magnetic ﬁelds . . . . . . . . . . . 3.A.3 On the MHD approximation in astrophysics MHD axisymmetric steady-state equations . . . . . . 3.B.1 General form . . . . . . . . . . . . . . . . . . 3.B.2 Ideal MHD form . . . . . . . . . . . . . . . . The Kelvin-Helmholtz instability of a single vortex sheet layer . . . . . . . . . . . . . . . . . . . . . . . . MHD stability and the energy principle . . . . . . . Pressure-driven instabilities asymptotic limit . . . .

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Turbulence and turbulent transport - the agents of relaxation and structure formation

117

4. A Tutorial on Basic Concepts in MHD Turbulence and Turbulent Transport

119

P.H. Diamond, S.-I. Itoh and K. Itoh 4.1 4.2 4.3 4.4 4.5 4.6

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . K41 beyond dimensional analysis - revisiting the theory of hydrodynamic turbulence . . . . . . . . . . . . . . . . . . Kraichnan-Iroshnikov, Goldreich-Sridhar and all that: a scaling theory of MHD turbulence . . . . . . . . . . . . . Steepening of nonlinear Alfven waves - a little compressibility goes a long way... . . . . . . . . . . . . . . . . . . . Turbulent ﬂux diﬀusion in 2D MHD - a ‘minimal’ problem which is not so simple... . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . .

5. Intermittency Like Phenomena in Plasma Turbulence

119 120 127 137 142 149 151

A. Das, P. Kaw and R. Jha 5.1 5.2 5.3

5.4 5.5 5.6

Introduction . . . . . . . . . . . . . . . . . . . . . Concept of intermittency in hydrodynamic ﬂuids Evidence for intermittency in plasmas . . . . . . 5.3.1 Laboratory fusion plasma . . . . . . . . . 5.3.2 Space plasmas . . . . . . . . . . . . . . . 5.3.3 Remarks . . . . . . . . . . . . . . . . . . Plasma intermittency: ﬂuid approach . . . . . . Alternative approach for plasma turbulence . . . Summary and conclusion . . . . . . . . . . . . .

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6. Nonlinear Cascades and Spatial Structure of Magnetohydrodynamic Turbulence

151 153 158 159 165 166 168 174 178

187

W.-C. M¨ uller and R. Grappin 6.1 6.2 6.3 6.4

Introduction . . . . . . . . . . . . . . . Magnetohydrodynamics . . . . . . . . Basic concepts . . . . . . . . . . . . . Phenomenologies of turbulent cascades 6.4.1 Kolmogorov phenomenology .

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6.5 6.6

6.7 6.8

6.4.2 Iroshnikov-Kraichnan phenomenology 6.4.3 Goldreich-Sridhar phenomenology . . Numerical simulation . . . . . . . . . . . . . . 6.5.1 Pseudospectral method . . . . . . . . Nonlinear energy dynamics . . . . . . . . . . 6.6.1 Isotropic energy spectra . . . . . . . . 6.6.2 Anisotropic energy spectra . . . . . . 6.6.3 Residual energy spectra . . . . . . . . Spatial structure . . . . . . . . . . . . . . . . 6.7.1 Intermittency modelling . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . .

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7. Scale Covariance and Scale-ratio Covariance in Turbulent Front Propagation

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A. Pocheau 7.1 7.2 7.3

7.4

7.5 7.6

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Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . Turbulent front and scale symmetries . . . . . . . . . . . . From scale invariance to scale covariance . . . . . . . . . . 7.3.1 Single variable relationship . . . . . . . . . . . . . 7.3.2 Geometry . . . . . . . . . . . . . . . . . . . . . . . 7.3.3 Equilibrium systems . . . . . . . . . . . . . . . . . 7.3.4 Out-of-equilibrium . . . . . . . . . . . . . . . . . . On the scale-ratio invariance . . . . . . . . . . . . . . . . 7.4.1 The meaning of scale-ratio invariance . . . . . . . 7.4.2 Single variable function . . . . . . . . . . . . . . . 7.4.3 Multivariable function . . . . . . . . . . . . . . . . 7.4.4 Front propagation . . . . . . . . . . . . . . . . . . Front propagation laws both scale covariant and scale-ratio covariant . . . . . . . . . . . . . . . . . . . . . . . . . . . Experimental evidence of scale covariance in turbulent front propagation . . . . . . . . . . . . . . . . . . . . . . . 7.6.1 Experimental set-up and data processing . . . . . 7.6.2 Integral analysis . . . . . . . . . . . . . . . . . . . 7.6.3 Local analysis . . . . . . . . . . . . . . . . . . . . 7.6.4 Sensitivity of scale covariance . . . . . . . . . . . Scale construction of turbulent fronts . . . . . . . . . . . . 7.7.1 Deterministic construction of front geometry in scale space . . . . . . . . . . . . . . . . . . . . . .

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7.7.2 7.7.3 7.8

“Apparent” fractal dimensions . . . . . . An artifact of ﬁnite size: the variation of dimensions” with turbulence intensity . . Conclusion . . . . . . . . . . . . . . . . . . . . .

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. . . . . 246 “fractal . . . . . 248 . . . . . 249

Transport bifurcations and relaxation

253

8. Transport Barrier Relaxations in Tokamak Edge Plasmas

255

P. Beyer 8.1 8.2 8.3 8.4 8.5 8.6

Introduction . . . . . . . . . . . . . . . . . . . . . . Model for resistive ballooning turbulence . . . . . . Formation of a transport barrier . . . . . . . . . . Appearance of relaxation oscillations . . . . . . . . Low dimensional model and non-linear short-term dynamics of shear ﬂow stabilization . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . .

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9. Dynamics of Edge Localized Modes

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X. Garbet, P. Ghendrih, Y. Sarazin, P. Beyer, G. Fuhr-Chaudier and S. Benkadda 9.1 9.2

9.3

9.4

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Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . An introduction to edge localized modes . . . . . . . . . . 9.2.1 Tokamak geometry . . . . . . . . . . . . . . . . . 9.2.2 A brief description of edge localized modes . . . . 9.2.3 MHD stability . . . . . . . . . . . . . . . . . . . . Dynamical models of edge localized modes . . . . . . . . . 9.3.1 Minimal model . . . . . . . . . . . . . . . . . . . . 9.3.2 Quasi-linear MHD model . . . . . . . . . . . . . . 9.3.3 Status of the modelling of ELM’s based on the MHD quasi-linear model . . . . . . . . . . . . . . L-H transition and edge localized modes . . . . . . . . . . 9.4.1 Models based on a subcritical bifurcation . . . . . 9.4.2 Models based on shear ﬂow stabilization . . . . . 9.4.3 Models based on shear ﬂow generation . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . .

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10. On the Onset of Collapse Events in Toroidal Plasmas Turbulence Trigger

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K. Itoh, S.-I. Itoh, M. Yagi, S. Toda and A. Fukuyama 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Phenomenological observations and problem deﬁnition . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.1 Onset of collapse and topological change of magnetic surfaces . . . . . . . . . . . . . . . . . . . . . 10.2.2 Approaches for bifurcation phenomena . . . . . . 10.3 Theoretical Framework . . . . . . . . . . . . . . . . . . . . 10.3.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.2 Stochastic equation . . . . . . . . . . . . . . . . . 10.3.3 Statistical properties . . . . . . . . . . . . . . . . 10.4 Example of Neoclassical Tearing Mode . . . . . . . . . . 10.4.1 Description of nonlinear instability and stochastic equation . . . . . . . . . . . . . . . . . . . . . . . 10.4.2 Nonlinear instability, subcritical excitation and cusp catastrophe . . . . . . . . . . . . . . . . . . 10.4.3 Statistical property . . . . . . . . . . . . . . . . . 10.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 1

Prof. Masahiro Wakatani and Fusion Research in His Days

K. Itoh National Institute for Fusion Science, Toki 509-5292, Japan Prof. Wakatani has passed away on 9th January 2003 at the age of 57. It is a big tragedy that we have lost Prof. Wakatani. This article is to remember his achievements in fusion and plasma science.

1.1

Research period of Prof. Wakatani

Prof. Masahiro Wakatani (Fig. 3.7) was born on 15 May 1945, and was given the PhD degree from Kyoto University in 1973. He has worked at JAERI, IPP Nagoya University and PPL Kyoto University. He has been a full professor at Kyoto since 1985, (and on that occasion I have joined his chair as an associate professor to him). It was the thirty years, from 1973 to 2003, when he has played a leading role in fusion and plasma research. Progress in the fusion and plasma research during these thirty years, from 1973 to 2003, can be seen in Fig. 1.1, which shows the starts of experiments of tokamaks and helical devices (stellarators). Each symbol corresponds to one experimental device. Devices are clariﬁed into four: circular cross section (conventional) tokamaks (open circle), noncircular tokamaks (closed circle), helical devices (triangle), and spherical tokamaks (square). This ﬁgure clearly illustrates that the steady progress was ﬁrst made on the circular cross-section tokamaks. Then the noncircular tokamaks are developed. In parallel, helical devices have been explored, slightly slower but steadily. In the fourth category, we observe that the spherical tokamaks have been attracting focused interest recently. 3

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1000 3

ITER ITER

V (m )

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TFTR

JET JT-60U LHD

10 1

NSTX T-3

Stellarator-C 0.1 1960

Fig. 1.1

START 1970

1980

1990

2000

2010

Year

2020

Plasmas in the research period of Prof. Wakatani.

The research in the last three decades may be characterized by the following: (1) (2) (3) (4)

the exploration of conﬁgurations was pursued, the tokamak plasma has been evolving to fusion regime, the optimization of toroidal systems was investigated, the plasma physics has been recognized as a frontier of general modern physics.

In particular, deeper and wider recognition of the fact that “the shape and size change the property of the matter in the conﬁned plasma”. Prof. Wakatani has provided many keys for the development in the above four aspects of the recognition in the fusion research. Initial phase An explicit goal of the fusion research has been to establish the scientiﬁc feasibility for the realization of controlled thermonuclear fusion. I would like to stress that, at the same time, our goal has been to systematize the research as a discipline. This latter goal has been recognized as important as the former, from the time of early 70s when Prof. Wakatani has started his academic carrier. For instance, we could quote from, e.g., Feynman Lectures on Physics, “The next great era of awakening of human intellect

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may well produce a method of understanding the qualitative content of equations. Today we cannot. Today we cannot see that the water ﬂow equations contain such things as the barber pole structure of turbulence · · · ” [1]. It was 1974, when Prof. S. Yoshikawa, being collaborated by Prof. K. Nishikawa, has initiated “Workshop on Theoretical Research of Fusion” at Hachiouji near Tokyo. In the ﬁrst and consecutive workshops, Prof. Yoshikawa has tried to establish a methodology for the research on conﬁned plasmas in Japanese academic society. Those who have joined the series of this workshop, as a newcomer to this ﬁeld then, have later grown up as leading researchers in Japan. Prof. Wakatani has been one of the leading ﬁgures in the series of the workshop. Four examples from Prof. Wakatani’s outstanding achievements Prof. Wakatani’s work has been always in the frontier of fusion research for the last three decades, which is illustrated in Fig. 1.1, and may be organized into four categories corresponding to the four characterization listed above. The ﬁrst is the problem of stability and plasma shape. The initial works are typical examples including the toroidal eﬀect on vertical displacement of elliptic tokamak [2] and nonlinear calculation of the m = 1 internal kink instability in current carrying stellarators [3]. The second is the nonlinear instability problem, in which he proposed the so-called ‘Hasegawa-Wakatani equations’ [4]. This set of equations has been an elegant and elementary tool for theory and simulational studies of plasma turbulence. Thus, the inﬂuences of the plasma size and parameters on the turbulent transport have been clariﬁed. The third, he has devoted continuously for understanding of helical systems [5] including the design study. In this area, he has developed a systematic theory of the stability of Heliotron systems, played a central role in the theoretical design of the large helical device (LHD) and also developed a helical axis system (heliac or helias). The fourth is the study of the structure formation in turbulent plasmas. By use of his ‘HasegawaWakatani equations’ he has demonstrated that the global structures are generated by plasma turbulence [6]. It is evident that these four categories are the central issues in the fusion research of the last three decades as is illustrated in Fig. 1.1. In the next section, his originality is highlighted by selecting some of his leading achievements [7].

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1.2

Research on nonequilibrium systems

In the research of complex phenomena in nonequilibrium systems, such as conﬁned plasmas, it is essential to identify the appropriate equations. In particular, reduction of variables illustrates the physical processes and stimulates the ﬁnding of physics law. Prof. Wakatani has recognized this importance of selecting proper equations for proper problems, and has given illustrative results of key issues in conﬁnement physics. One example is his research on the relaxation phenomena in helical systems. Pressure driven relaxation instability in a current-free high-shear helical system has been studied by him and his students (e.g., [8]), and repetitive sequence of onsets of crash and following gradual recovery was shown by numerical simulation (Fig. 1.2). Thus he and his colleagues have opened the way that the nonlinear MHD dynamics of pressure-driven collapse events, which may limit the performance of helical devices, is treated quantitatively.

Fig. 1.2

Evolution of pressure proﬁle in relaxation phenomena of Heliotron [8].

More famous example of his achievements is the investigation of the ‘Hasegawa-Wakatani’ equations. ∂ 2 (1.1) ∇ φ + φ, ∇2⊥ φ = d (φ − n) + µc ∇4⊥ φ ∂t ⊥ ∂ ∂φ n + [φ, n] = d (φ − n) − + Dc ∇2⊥ n (1.2) ∂t ∂y −1 −1 2 νei ωci Ln ρ−1 where d ≡ k2 vth,e s . (Plasma parameters such as the density, temperature, characteristic scale of gradient, magnetic ﬁeld are included

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into this one parameter d .) This set of equations is simple, and includes the drive by instabilities and the process of generation of structures (such as zonal ﬂow or convective cells, etc.). This is a prototypical set of equations which clariﬁes many of essential elements in the plasma turbulence, and has given tremendous impacts for the progress of physics for the conﬁned plasmas. This has opened the area of research where mutual interactions between turbulence and meso-scale structure are studied. Two most important results are illustrated here. The ﬁrst is the spectrum of plasma turbulence which is driven by inhomogeneity. When one talks about a generality of turbulent spectrum, many eﬀorts have been devoted to the universality of the inertial regime [9]. It is true that, after the evolution of nonlinear turbulent interactions, external energy which is injected to a relevant scale is transferred to smaller scales and is ﬁnally dissipated by molecular collisions. In such a study to pursuit ‘universality’, the longerwavelength components are often treated as external parameter. However, the longer-wavelength component contains dominant part of turbulent energies. Thus impact of turbulence to observable phenomena (or that to human recognition of nature) is mainly carried by the dynamics of the energy containing region. The determination of the energy spectrum in a region where free energy is injected by various instability mechanisms has vital importance. Such a problem needs a formulation a set of equations which include both the nonlinear dynamics of vortex together with instability mechanism. Thus, the Hasegawa-Wakatani equations have served the simplest set of equations, which provides insights for the evolution of turbulence. By use of this set of equation, Prof. Wakatani has obtained the spectrum −3 , E(k) ∝ k⊥

(1.3)

where E(k) is the energy spectrum [10]. This spectral form tells that the dominant part of turbulent energy is contained in the region of longest wavelength. The rate of the nonlinear transfer is summarized in the index, −3. Another important ﬁnding is the ﬂow generation by turbulence. Figure 1.3 is one early example [4], in which the formation of potential contour, loosely resembling that of the magnetic surface, is demonstrated. These contour structures indicate the presence of banded poloidal E × B ﬂow, called zonal ﬂow. It has been pointed out that in the two-dimensional turbulence, the turbulence energy can cascade, not only into the ﬁner scales, but also in the direction of longer scale lengths. As the injection of energy to

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a mode continues, the wave energy starts to pile up in the longer-wavelength region if the molecular dissipation is weak. Turbulence in toroidal plasmas is quasi-two-dimensional owing to the strong magnetic ﬁeld. The HasegawaWakatani equations are formulated in a form of two dimensional turbulence, where the wavenumber in the direction of magnetic ﬁeld is treated as a constant parameter. The accumulation of wave energy in the longer wavelength region is demonstrated by the simulation of the Hasegawa-Wakatani equations. As is seen from Fig. 1.3, the potential contour is akin to the magnetic surface. This means that the ﬂuctuation electric ﬁeld in the poloidal direction is weak, i.e., the ﬂuctuating velocity in the radial direction is small. Thus, the reduction of radial transport is expected to occur when such a global electrostatic potential is established. It is well known now that the studies of the improved conﬁnement in 90’s are developed based upon several key ideas, such as the suppression of turbulence by radial electric ﬁeld, bifurcation of the radial electric ﬁeld, and the formation of zonal ﬂow by turbulence [11]. All of us are aware of the leading role of Prof. Wakatani’s work in the progress of the theory of plasma conﬁnement.

Fig. 1.3

Contour of electrostatic potential from the simulation of [4].

We could now enlighten his work as a historical achievement. We recall Helmholtz’s theorem, i.e., vortex lines move with ﬂuid (in invicid ﬂuid). In a terminology of plasma physics [12] one may express the evolution of the vorticity as ∂ 2 ∇ φ + φ, ∇2⊥ φ = · · · (1.4) ∂t ⊥ The Helmholtz theorem tells that the RHS vanishes, · · · = 0. In the nature, however, there are abundant of examples that the vortical ﬂow structure is ‘generated’. More generally, the global axial vector ﬁelds are generated

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by the turbulent convection of thermal energy. Such as, geodynamo, solar dynamo, Jet formation in accretion disk, Jovian zonal ﬂow, Jet stream, Venus super rotation, etc. In the problems of plasma structure formation, one can point out the relaxation to Taylor state, sawtooth oscillation, Hmode, ITB, zonal ﬂow, dynamo, etc. These phenomena will be explained if “· · · ” in the right hand side of this equation is properly modelled. In this aspect, Prof. Wakatani has given a critical step for the development of general physics. Prof. Wakatani has thus clariﬁed an essence of the problem. This attitude of him was described as “· · · Perhaps because Wakatani-sensei lived in the ancient cultural region of Kyoto and Nara, he liked theories that were simple, orderly, and yet elegant, like a Zen garden. · · · ” by Prof. J. W. Van Dam and Prof. C. W. Horton, Jr. in the tribute to Prof. Wakatani. Looking into waves of pebbles in Zen garden, one may see waves, tide and vortices in nature. 1.3

Wakatani-sensei’s gifts to scientists

There are a lot of key elements in order to promote the future progress. Among them, Prof. Wakatani has contributed much in the following aspects: (1) Systematization of achievements as a branch of academic research, (2) Improve creativity, (3) Improve environment of research. What he has done in relation to these issues is now considered as a gift from Prof. Wakatani to the future researchers. Systematization of research results as an academic discipline What should be emphasized ﬁrst is that, Prof. Wakatani has been an excellent teacher for the students. Education of students was a joy for him, and his devotion to the education has been awarded by the energetic scientiﬁc production by those in Wakatani school. They are now central members in the theory and simulation researchers in Japan. In relation with the education, he has also made eﬀorts in publication of books. The achievement includes: “The Beta Equilibrium Stability and Transport Codes: Applications to Design of Stellarators” by F. Baure,

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O. Betancourt, R. Garabedian, M. Wakatani (Academic Press, Inc., 1987); “Plasma Physics Basic Theory with Fusion Applications” by K. Nishikawa, M. Wakatani (Springer-Verlag, 1990 (Second ed. 1993, Third ed. 1999)); and “Stellarator and Heliotron Devices” by M. Wakatani (Oxford University Press, 1998). The fact that the third edition of his “Plasma Physics Basic Theory with Fusion Applications” was published shows that this book is widely accepted by the international academic community. Improve creativity Next, he played a role in improving the creativity of researchers. I would point out three keys. “Increasing creativity” might be heard as a miracle, but what Prof. Wakatani has done has been full of the essence for it. The ﬁrst key for the creativity is ‘Self-belief’. Without self-belief, one cannot achieve a really new scientiﬁc achievement on which nobody has thought or analyzed. Only by the self-belief, we can continue the work until the ﬁnal breakthrough is realized. Prof. Wakatani has been always encouraging his colleague and students. By warm encouragement by him, many researchers have been injected self-belief. The next key is ‘chemistry’ among researchers. From the time of ancient peripatetic school at Greek, to the modern Copenhagen spirit, and to recent group researches, the communication between many researchers has been the central scheme for driving creative ideas that had shed lights for human understanding. In the area of fusion and plasma research, Prof. Wakatani has been always leading friendship among researchers. He has been always stressed ‘loose-coupling’ among scientists: this was the key of him so that researchers keep good chemistry among them while each of the members can grow their own originality. Third, his eﬀorts to grow the chemistry has been most impressive in his contribution in the international collaborations. Interactions between diﬀerent ways of thinking, no doubt, is the key for the emergence of the new idea. For this, the interactions between Oriental-Occidental ways of thinking, as well as International and Interdisciplinary way of thinking, are very eﬀective way of progress. Wakatani-sensei’s eﬀorts have covered many endevours, e.g., IUPAP (plasma physics), US-Japan collaboration on fusion research, in particular Joint Institute for Fusion Theory, International Stellarator Workshop, International Toki-conference, Festival de Theorie, and others. Every participant in the ﬁrst Festival de Theory remembers Wakatani-sensei’s warm and leading work in the festival (see Fig. 3.8).

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Improve environment of research Prof. Wakatani was the leader in Japanese education for plasma and fusion science. He was elected as councilor of Kyoto University, and everyone has believed that he would soon be a dean. He was the key promoter for the 21st Century COE (center of excellence) Program of MEXT (ministry of education, culture, sports, science and technology) on “Establishment of COE on Sustainable-Energy System”. For this, he was the task leader for “Plasma Group”, and provided a splendid conditions for researcher who belonged to this programme. He has also made a lot of eﬀorts for academic agreement between Kyoto U. and Universite de Province and others. His devotion beneﬁted the communication between the scientists, government oﬃcials and citizens. He was a member of Fusion Council of Japan, which was the highest subcommittee specializing at the national policy of the fusion research. He was also a subcommittee member of Science Council of Japan, in which academic society has shown opinions to the government. Prof. Wakatani has played a role of spokesman for the researchers in the ﬁeld of plasma and fusion study to the government and to wider academic society. Among these activities, his role for the big project ITER was outstanding. He was a member of Technical Advisory Committee, and was the chairman for scientiﬁc and technical evaluation of ITER for Fusion Council of Japan. Through these activities, he guided Fusion Council of Japan to have scientiﬁc judgement for the decision of participating the ITER project. In the end of this article, let me be a bit sentimental. We had started an activity of “Group of Seven” since 1985. Members were (in alphabetical order) M. Fujiwara, K. Itoh, S.-I. Itoh, S. Matsuda, O. Motojima, Y. Shimomura and M. Wakatani. At that time, foreseeing the start of the LHD project, an endeavour to make an inter-institutional research collaboration project in Japan has been studied. One of the ﬁrst outcome of this activity was ‘Method for design of Large Helical Device’. It strengthened the spirit of collaboration, domestic as well as international, which has been the essence of Wakatani sensei’s research moral.

1.4

Closing words

All of us have been impressed by his dedication to the science, students, colleague, academic society and, last but not least, his family. The enlightening ideas and selﬂess devotion for research have sprung from his heart.

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Prof. Wakatani’s spirit lives with us so long as the plasma physics and fusion research continue to evolve. Acknowledgements The author would like to thank the “Director Committee”, the chairman and local organizers for giving this opportunity for a memorial talk at Festival Th´eorie 7 July 2003 (Aix-en-Provence, France). He acknowledges discussions and help in materials by colleague in preparing the talk; in particular, Drs. S.-I. Itoh, M. Yagi, H. Sugama, N. Kasuya, A. Fukuyama, S. Hamaguchi, R. M. More, H. Sanuki, S. Toda, A. Fujisawa, Y. Miura, A. Yoshizawa, P. H. Diamond, T. S. Hahm, S. Benkadda, X. Garbet, V. Naulin, C. W. Horton, and J. van Dam. I would like to thank a partial support in completing this manuscript from Grant-in-Aid for Specially-Promoted Research (16002005) and Grant-in-aid for Scientiﬁc Research (15360495) from MEXT Japan. References [1] Feynman R P, Leighton R B and Sands M L 1965 The Feynman Lectures on Physics (Addison-Wesley, Reading) §41-6. [2] Wakatani M 1975 J. Phys. Soc. Jpn. 38 1555. [3] Wakatani M 1978 Nucl. Fusion 18 1499. [4] Hasegawa A and Wakatani M 1983 Phys. Rev. Lett. 50 682. [5] Wakatani M 1998 Stellarator and Heliotron Devices (Oxford University Press). [6] Hasegawa A and Wakatani M 1987 Phys. Rev. Lett. 59 1581. [7] A thorough list of Prof. Wakatani’s work is now under preparation. [8] Wakatani M, Shirai H and Yamagiwa M 1984 Nucl. Fusion 24 1407. [9] Kolmogorov A N 1941 Dokl. Akad. Nauk SSSR [Sov. Phys. Dokl.] 30 299. [10] Wakatani M and Hasegawa A 1984 Phys Fluids 27 611. [11] See, e.g., Yoshizawa A, Itoh S -I, Itoh K 2002 Plasma and Fluid Turbulence (IOP, Bristol). [12] Taylor J B and McNamara B 1971 Phys. Fluids 14 1492.

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PART 2

Magnetic relaxation and self-organization in astrophysical and laboratory plasmas

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Chapter 2

An Introduction to Mean Field Dynamo Theory

D.W. Hughes and S.M. Tobias Department of Applied Mathematics, University of Leeds

2.1

Introduction

Magnetic ﬁelds are essentially ubiquitous, being detected over a tremendous range of scales in planets, stars, accretion discs and in the interstellar medium. The dynamic behaviour of such ﬁelds is then responsible for a vast range of astrophysical phenomena (see, for example, Parker 1979). For instance, the solar magnetic ﬁeld gives rise to sunspots, solar ﬂares and coronal mass ejections; it also plays a major role in shaping the solar wind which, on interacting with the Earth’s magnetic ﬁeld, causes aurorae. Starspots, analogous to sunspots but covering a much greater surface area, have been detected on a number of cool stars. The pulsed emission of pulsars is a consequence of an extremely strong magnetic ﬁeld. On the largest scales, the interstellar magnetic ﬁeld plays a role in star formation, mediating angular momentum transport as the star collapses. The ultimate question in the study of astrophysical magnetic ﬁelds must then be that of the origin of the magnetic ﬁeld in cosmical objects. In particular, one might ask whether the observed magnetic ﬁelds are simply ‘fossil ﬁelds’, or whether, alternatively, they are being continually regenerated — i.e. whether some sort of dynamo process is taking place. For collision-dominated plasmas with short mean free paths, such as those found in stellar interiors, the evolution of magnetic ﬁelds is very well described by the equations of single-ﬂuid magnetohydrodynamics (MHD). 15

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(Although, for example, in studies of stellar atmospheres — such as the solar corona — the use of MHD is not based on such solid foundations.) In this review we shall be concerned only with physical systems for which the MHD description is appropriate. The key equation of MHD is the magnetic induction equation, derived from the pre-Maxwell equations (in which the displacement current is neglected) and the simplest form of Ohm’s law in a moving conductor, namely J = σ(E + U × B),

(2.1)

where J is the electric current, E the electric ﬁeld, U the velocity ﬁeld, B the magnetic ﬁeld, and σ the electrical conductivity. The induction equation takes the form: ∂B (2.2) = ∇ × (U × B) + η∇2 B, ∂t where η (= 1/µ0 σ, where µ0 is the magnetic permeability) is the magnetic diﬀusivity (assumed constant in this derivation). The ﬁrst term on the right hand side of Eq. (2.2) describes the advection of the magnetic ﬁeld by the velocity, the second term describes collisional dissipation of the ﬁeld. An order of magnitude estimate of the relative size of these two terms is given by the magnetic Reynolds number Rm = U L/η, where U and L are a typical velocity and length. Signiﬁcantly, in most astrophysical contexts Rm is huge. Clearly, in the absence of any ﬂuid motions the induction equation simply reduces to a vector diﬀusion equation, with a timescale for decay of the magnetic ﬁeld (the Ohmic timescale) given by τη ∼ L2 /η. For the Earth, taking L as the radius of the metallic core and η as the appropriate molecular value, the Ohmic decay time τη is of the order of 104 years, several orders of magnitude shorter than the timescale over which the Earth’s magnetic ﬁeld has existed, namely O(109 ) years (see, for example, Backus et al. 1996). Thus it follows immediately that the advective term in Eq. (2.2) is of paramount importance — any fossil ﬁeld would by now have decayed to a negligible strength. In other words, the ﬂuid motions are critical to maintenance of the magnetic ﬁeld. This is what is meant by a (hydromagnetic) dynamo. For the case of the Sun, whose magnetic ﬁeld can be studied in far greater detail than that of the Earth, the Ohmic time is very long (roughly speaking, L is large and η small), and is comparable with the lifetime of the Sun. Solely on these grounds it is not therefore possible to assert that the Sun’s ﬁeld cannot be primordial in origin. However, conversely, it is the

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fact that the observed solar ﬁeld varies on a short timescale, with reversals roughly every eleven years, that strongly suggests that the magnetic ﬁeld is a product of dynamo action. The alternative — an observed rapidly reversing ﬁeld fed somehow by a (hidden) slowly decaying and non-reversing ﬁeld (a hydromagnetic oscillator ) — is much harder to envisage. Magnetic ﬁelds with short-term variability have also been observed on a number of other solar-type stars, and it is natural to assume that they are also the result of some kind of dynamo action (see, for example, the review by Rosner 2000). Within accretion discs, although there is no compelling observational evidence of dynamo action, numerical simulations of the nonlinear development of the magneto-rotational instability, which is believed to play a crucial role in angular momentum transport, suggest a novel ‘bootstrapping’ dynamo mechanism in which the instability of a weak ﬁeld leads to turbulent motions, which can then act to amplify the magnetic ﬁeld (see the review by Balbus & Hawley 1998). On the largest, galactic, scale the question of whether the magnetic ﬁeld is a product of dynamo action is rather more open (see Zweibel 2005). The Ohmic decay time of the ﬁeld is about ten orders of magnitude larger than the age of the Universe, so on these grounds the ﬁeld could obviously be primordial. Although there are possible diﬃculties in reconciling a wound-up primordial ﬁeld with the observed spatial scale between galactic ﬁeld reversals, there is currently no overwhelming observational evidence to decide the issue either way. A complete mathematical description of MHD dynamo action requires the solution of the induction equation (2.2) and also of the momentum equation, which accounts for the back-reaction of the magnetic ﬁeld on the ﬂuid ﬂow through the Lorentz force J × B. For compressible ﬂuids an energy equation and an equation of state must also be solved. Thus the full problem requires the solution of a set of coupled nonlinear partial diﬀerential equations, which represents a formidable mathematical and computational challenge. A simpler, but still non-trivial problem, is to consider the evolution of the magnetic ﬁeld under a prescribed velocity ﬁeld, which requires solution only of Eq. (2.2). Investigating the growth of the magnetic ﬁeld governed by this linear equation is known as the kinematic dynamo problem; a ﬂow for which the magnetic ﬁeld grows exponentially (governed solely by Eq. (2.2)) is said to act as a kinematic dynamo. Determination of kinematic dynamo action, even though it requires solution of only one linear partial diﬀerential equation, is nonetheless not straightforward. Indeed, from the initial formulation of the problem in a

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short paper by Larmor (1919) (reproduced in Ruzmaikin et al. 1988), it took a period of nearly forty years before the ﬁrst rigorous demonstrations of kinematic dynamo action (Herzenberg 1958; Backus 1958). The main obstacle to success came from the fact that simple ﬁelds and ﬂows of the kind that allow analytical progress with Eq. (2.2) typically do not act as dynamos; indeed, there are a number of anti-dynamo theorems that expressly rule out ﬂows and ﬁelds with certain symmetries. The most famous are due to Cowling (1934), who showed that a purely axisymmetric magnetic ﬁeld cannot be maintained by dynamo action, and to Zeldovich (1957), who showed that dynamo action cannot result from plane two-dimensional motion. Subsequently there have been many extensions to these fundamental results (see, for example, Hide & Palmer 1982; Ivers & James 1984). Consequently, the dynamo problem is inherently three-dimensional and, as such, decidedly non-trivial. A distinction of a diﬀerent kind may be drawn between small-scale and large-scale dynamos. A small-scale dynamo is one in which a growing magnetic ﬁeld occurs on scales comparable with or smaller than those of the driving ﬂow. A large-scale dynamo, on the other hand, has signiﬁcant energy on scales very large compared with those of the velocity ﬁeld. (The magnetic ﬁeld of the Sun, which gives rise to sunspots and is global in scale, is envisaged to be the product of a large-scale dynamo.) However, this distinction, although sometimes useful, is not clear-cut. A large-scale dynamo will, typically, also have strong small-scale ﬁelds; conversely a small-scale dynamo ﬂow, given the opportunity, will typically generate a certain amount of large-scale ﬁeld — i.e. the spectrum of the ﬁeld will not have an abrupt cut-oﬀ at the driving scale, but will decrease gradually to small wavenumbers (large scales). Thus a certain amount of caution is needed in categorising any dynamo as being of small or large scale. Moreover, in a turbulent ﬂow there may be a wide range of energetic scales of motion; in such cases even the idea of a separation of scales becomes unclear. A major breakthrough in the study of dynamo action occurred with the formulation of mean ﬁeld electrodynamics by Steenbeck, Krause and R¨ adler (1966 and subsequent papers; see the monographs by Krause & R¨ adler 1980 and Moﬀatt 1978). This theory considers the generation by dynamo action of magnetic ﬁelds on scales large compared with those of the driving ﬂow, and is formulated in terms of the large-scale (or mean) component of the ﬁeld, with the inﬂuence of small-scale motions and ﬁeld encapsulated in a few fundamental coeﬃcients of the problem (such as the famous ‘α-eﬀect’). In a few special cases these can be calculated exactly; more generally though

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they are parameterised in a physically plausible manner. For the past forty years astrophysical dynamo theory has been dominated by mean ﬁeld electrodynamics. In this review we aim to provide a brief introduction to mean ﬁeld electrodynamics. We shall concentrate on the generic aspects of the theory, rather than on object-speciﬁc models aimed at describing the magnetic ﬁelds of particular astrophysical bodies. The literature relating to mean ﬁeld electrodynamics is vast, and so we make no claim here to be exhaustive in our choice of references; a fuller reference list may be obtained from the reviews of Roberts & Soward (1992), Ossendrijver (2003) and Brandenburg & Subramanian (2005). In Sec. 2.2 we discuss kinematic considerations — the formulation of mean ﬁeld electrodynamics, the calculation of the governing coeﬃcients in the theory, and simple linear mean ﬁeld dynamo models. The incorporation of nonlinear eﬀects is described in Sec. 2.3. Research into mean ﬁeld electrodynamics remains an active, and indeed controversial, topic in astrophysical MHD; some of the diﬃculties of the theory and the issues that arise are discussed in Sec. 2.4.

2.2 2.2.1

Kinematic Mean Field Theory Formulation

Mean ﬁeld electrodynamics is, at heart, a linear (or kinematic) theory, being based on the induction equation. In this section we shall concentrate solely on the linear formulation; nonlinear eﬀects, which may be incorporated through a variety of approaches, are described in Sec. 2.3. The fundamental assumption underlying mean ﬁeld electrodynamics is that there exists a strict separation of length scales for both the magnetic and velocity ﬁelds. The velocity and magnetic ﬁelds have a small-scale component, with characteristic length scale , and a large-scale — or mean — component with scale L . For the case of a turbulent velocity ﬁeld,

may be regarded as a typical size of the energetic eddies. Averages, which we denote by · , may then be taken over an intermediate length scale a satisfying a L. The velocity and magnetic ﬁelds may be decomposed into their mean and ﬂuctuating components: B = B0 + b,

U = U0 + u,

(2.3)

where, by deﬁnition, b = u = 0. Averaging the induction equation

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(2.2) yields an equation for the evolution of the mean magnetic ﬁeld: ∂B0 = ∇ × (U0 × B0 ) + ∇ × E + η∇2 B0 , (2.4) ∂t where E = u × b is the mean electromotive force (emf), resulting from interactions between the small-scale velocity ﬁeld and the small-scale magnetic ﬁeld. It is this term that provides the crucial distinction between the (un-averaged) induction equation and the mean induction equation. Clearly, in order to make progress with Eq. (2.4) it is necessary to close the system by expressing E in terms of the mean ﬁeld B0 and its derivatives. This is done by considering the equation governing the ﬂuctuating magnetic ﬁeld, obtained by subtracting Eq. (2.4) from Eq. (2.2): ∂b (2.5) = ∇ × (U0 × b) + ∇ × (u × B0 ) + ∇ × G + η∇2 b, ∂t where G = u × b − u × b. Symbolically, Eq. (2.5) may be written as L(b) = ∇ × (u × B0 ),

(2.6)

where L is a linear operator. There are two distinct possible modes of behaviour for the small-scale magnetic ﬁeld b. One is if b is ‘slaved’ to the mean ﬁeld B0 ; here b is driven solely by the inhomogeneous term ∇ × (u × B0 ) — in the absence of B0 the small-scale ﬁeld simply decays to zero. The second is when the operator L can support non-trivial solutions even in the absence of a large-scale ﬁeld B0 — in other words, the velocity ﬁeld supports small-scale dynamo action. As we shall see, the implications for mean ﬁeld theory for the two cases can be rather diﬀerent. Let us consider ﬁrst the case when b exists only if the right hand side of Eq. (2.6) is non-zero; it is then linearly and homogeneously related to B0 . Hence, for a given velocity ﬁeld u, the mean emf E is also linearly and homogeneously related to B0 . Since, by construction, the mean magnetic ﬁeld varies on a long length scale, one may therefore postulate an expansion for E of the form ∂B0j + ··· . (2.7) Ei = αij B0j + βijk ∂xk For consistency with Eq. (2.4), which is a second order partial diﬀerential equation, it is important that the ﬁrst two terms (and only the ﬁrst two terms) of this expansion are retained. Since the vector E is a polar vector, whereas B0 is an axial vector, it follows therefore that αij and βijk are pseudo-tensors (i.e. they change their sign, compared with tensors, under transformation). Their physical interpretation is brought out most clearly

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by considering the case of isotropic turbulence. In the kinematic regime the pseudo-tensors αij and βijk can depend only on the statistical properties of the ﬂow ﬁeld and on the diﬀusivity η; therefore, for isotropic turbulence, they must take the form αij = αδij ,

βijk = β ijk ,

(2.8)

where α is a pseudo-scalar and β a true scalar. Substitution from Eq. (2.8) into Eq. (2.4) gives ∂B0 = ∇ × (U0 × B0 ) + ∇ × αB0 + (η + β)∇2 B0 , ∂t

(2.9)

where we have made the further simplifying assumption that β is constant. Clearly β is an additional, turbulent, contribution to the magnetic diﬀusivity. On physical grounds we expect this to be positive for a genuinely turbulent ﬂow; however it is of interest to note that there are certain types of ﬂow (e.g. random potential ﬂows) for which β is negative (Krause & R¨adler 1980). The most important term though is ∇ × αB0 , the ‘α-eﬀect’ of mean ﬁeld dynamo theory, which may lead to growth of the mean magnetic ﬁeld. Whereas in the full induction equation (2.2) the advective term has the form of the curl of a vector perpendicular to the magnetic ﬁeld, in the mean induction equation (2.9) the new (α) term is the curl of a vector parallel to the mean ﬁeld. This turns out to have profound consequences. One is that Cowling’s theorem no longer holds (because it is possible to have dynamos that are axisymmetric on average); the problem then becomes much simpler, analytically and computationally, and allows ready progress to be made in dynamo modelling. We explore some such models in Sec. 2.2.3. However, even without any calculation, it is possible to gain considerable insight into the nature of any turbulence that can lead to a non-zero value of α. Consider the case when the turbulence is reﬂectionally symmetric; i.e. its statistics have no sense of ‘handedness’. Then, on changing from a right-handed to a left-handed frame of reference, the quantity α, which depends only on the statistics of the turbulence, is invariant. Conversely though, α, being a pseudo-scalar, must change sign under this reﬂection of coordinates. Thus it follows that α must vanish for reﬂectionally symmetric turbulence. Equivalently, α can be non-zero only for turbulence possessing some sense of handedness. In nature, reﬂectional symmetry is broken in any rotating system. The simplest measure of the lack of reﬂectional symmetry in a ﬂow is the helicity, H = u · ∇ × u.

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For non-isotropic turbulence it is instructive to split the αij tensor into (s) its symmetric and antisymmetric parts. The symmetric pseudo-tensor αij may be referred to its principal axes:  (1)  α (s) . αij =  (2.10) α(2) (3) α Arguing as above, α(1) , α(2) and α(3) must vanish for turbulence that lacks reﬂectional symmetry. (a) The antisymmetric part of αij may be written as αij = − ijk γk , leading to a contribution γ ×B0 to the emf. From comparison with Eq. (2.4), it can be seen that γ is to be regarded as a velocity acting on the mean magnetic ﬁeld; it can be non-zero provided the statistics of u are either anisotropic or inhomogeneous, but is not dependent on the symmetry properties of the turbulence (Cattaneo et al. 1988). A detailed review of the transport eﬀects of turbulence on both scalar and vector ﬁelds is given by Moﬀatt (1983). For any given ﬂow, one can calculate αij (at least in principle, and often in practice) by imposing a uniform ﬁeld and measuring the resulting emf; here the scale separation is between the scale of variation of the ﬂow ﬁeld and that of the uniform magnetic ﬁeld (which is formally inﬁnite). This is not a dynamo calculation as such, since the uniform ﬁeld remains constant in time. However, the resulting value of αij can then be used (once βijk is known) to calculate the growth of magnetic ﬁelds of large but ﬁnite scales. Conceptually (though with great diﬃculty in practice) βijk may, provided dynamo action is not taking place, be calculated by considering the diﬀusion of a magnetic ﬁeld that is non-uniform but has a scale of variation far in excess of the characteristic scale of velocity. If, however, the ﬂow acts as a large-scale dynamo then such a ﬁeld will of course not decay but will grow. The discussion above has focused on the case where the ﬂuctuating ﬁeld b is driven by the ∇ × (u × B0 ) term in Eq. (2.5). However, turbulent ﬂows, regardless of their symmetry properties, typically act as small-scale dynamos at suﬃciently high values of Rm; i.e. b can grow quite happily of its own accord with B0 = 0. The growth rate of such a dynamo will then be determined not by α and β but by completely diﬀerent properties of the ﬂow, such as its Lyapunov exponents (see, for example, the monograph by Childress & Gilbert 1995). The diﬃculty comes in determining the nature of the dynamo action that may result from a helical, turbulent ﬂow at high Rm — the typical astrophysical situation. If large-scale growth is

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forbidden, through restricting the size of the domain and through the choice of boundary conditions, any dynamo action that ensues will be unambiguously small-scale. However, if the same ﬂow is embedded in an extended domain there are two possibilities: a modiﬁed version of the small-scale dynamo with some magnetic energy on the large scales, or alternatively a genuine large-scale — but not small-scale — dynamo, though this too will have associated with it strong small-scale ﬁelds. Thus, given one particular realisation of kinematic dynamo action in an extended domain, drawing a meaningful distinction between these two types of dynamo may not be possible. 2.2.2

Calculation of the transport coeﬃcients

Calculation of the mean emf E, and hence of the tensors αij and βijk , requires solution of Eq. (2.5) for the ﬂuctuating magnetic ﬁeld b. Unfortunately, for a general turbulent ﬂow this is impossible. There are, however, certain limiting cases in which E can be calculated — or at least wellapproximated. These results can be complemented by numerical solutions — away from the limiting regimes — for particular ﬂows, although it is not possible to probe the astrophysical regime of extremely high Rm. The term that renders Eq. (2.5) so troublesome is ∇ × G. It is therefore obviously of interest to consider situations in which this term can safely be ignored. (The neglect of this term is sometimes referred to as the ﬁrst order smoothing approximation or the second order correlation approximation.) Suppose we denote the time and length scales of the velocity ﬁeld by τ and

respectively, and the rms velocity by u. Then if either Rm = u /η 1 or S = uτ / 1 (where S is the Strouhal number), the ∇ × G term can be neglected. If Rm 1 Eq. (2.5) for b (assuming, for simplicity, U0 = 0) simpliﬁes to 0 = ∇ × (u × B0 ) + η∇2 b, (2.11) assuming that dissipative eﬀects are so strong that the time-dependence of b can be neglected. As described by Moﬀatt (1978), Eq. (2.11) can be solved via Fourier transforms to give (for isotropic turbulence) F (k) 1 dk, (2.12) α=− 3η k2 where F (k) is the helicity spectrum function. The corresponding expression for β is E(k) 2 β= dk, (2.13) 3η k2

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where E(k) is the energy spectrum function. If, on the other hand, S 1 then b is governed by ∂b = ∇ × (u × B0 ), ∂t

(2.14)

leading to the expressions τ τ and β = u2 α = − u · ∇ × u (2.15) 3 3 (see Krause & R¨adler 1980). The signiﬁcant feature of expressions (2.15) is that they are independent of the magnetic diﬀusivity η. Analogous, but more complicated expressions can be obtained for αij and βijk for non-isotropic turbulence; these are discussed at length in Krause & R¨ adler (1980). New physical eﬀects can result, but they fall outside the scope of this introductory review. The crucial points to note from the above expressions are that α is directly related to the ﬂow helicity (it is proportional to the helicity in Eq. (2.15) and to a weighted average of the helicity in Eq. (2.12)), and that β is determined by the energy of the ﬂow. The clear link between α and the ﬂow helicity can be understood in terms of Parker’s picture of helical motions that raise and twist ﬁeld lines to provide a net current anti-parallel to the initial ﬁeld (Parker 1955), as sketched in Fig. 2.1. It is clear though that for this picture to be valid the loops must undergo only small twists — which will be true provided that either diﬀusion dominates (Rm 1) or the motions act only for a very short time (S 1). Astrophysical turbulence, however, is characterised by Rm 1 and S = O(1); it is therefore of fundamental importance to see whether the close link between the helicity of the ﬂow and the α-eﬀect carries through into this regime. A very diﬀerent approach can be taken by the formal neglect of the diﬀusive terms in the induction equation. The magnetic ﬁeld is then frozen into the (perfectly conducting) ﬂuid and can be expressed in terms of its initial conﬁguration via the Cauchy solution. This leads to the following expression for α in terms of the Lagrangian displacement ξ (Moﬀatt 1974): d ξ · ∇ × ξ. (2.16) dt It is of interest to note that the helicity again appears, albeit in a Lagrangian sense. However, it is by no means straightforward to calculate α from Eq. (2.16), and indeed it is not clear that the expression converges as t → ∞. One might hope that expression Eq. (2.16), derived under the assumption of inﬁnite Rm, sheds some light on the behaviour for large but α=−

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Fig. 2.1 Distortion of the magnetic ﬁeld by a rising, twisting motion (after Moﬀatt 1978). For this small value of the twist the associated current is anti-parallel to the initial magnetic ﬁeld.

ﬁnite Rm. If so then this would provide a possible avenue into exploring the astrophysically important case of Rm 1. However, the link between the cases of Rm inﬁnite and Rm large but ﬁnite is a subtle one that is not fully understood. Expression Eq. (2.16) thus currently remains a relatively unexplored approach to understanding the α-eﬀect. In order to investigate the relationship between the α-eﬀect and the properties of the velocity ﬁeld, many calculations of αij have considered spatially periodic, helical motions. This was an idea pioneered by G.O. Roberts (1972), who considered two-dimensional, incompressible, steady ﬂows, i.e. u(x, y) = ∇ × (ψˆ z) + wˆ z.

(2.17)

Such ﬂows have the simplifying feature that the induction equation (2.2) supports monochromatic solutions of the form B(x, y, z, t) = B(x, y)ept+ikz ,

(2.18)

has the same spatial periodicity as the ﬂow. However they also where B pose a diﬃculty in deﬁning precisely what is meant by a large-scale magnetic ﬁeld. For any given value of z a ﬁeld of the form Eq. (2.18) has a component that is independent of x and y. Clearly, in some sense, this is a large-scale component — it has no variation in x or y and so has an inﬁnite length

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scale in these directions. Its evolution can be described through an α-eﬀect in which the averaging procedure involves averaging over planes of constant z. Conversely though, this purely z-dependent component of the magnetic ﬁeld may be thought of as simply part of the ‘small-scale’ eigenfunction B(x, y). This indeterminacy seems unavoidable for these special — but computationally tractable — ﬂows. Roberts (1972) calculated the α-eﬀect for four diﬀerent ﬂows in the limit as k → 0, exploiting symmetry arguments to determine the form of the anisotropic tensor αij . He focused particularly on the case of ψ = sin x sin y,

w = Kψ,

(2.19)

using analytical and numerical solutions to obtain α, and hence also the dynamo growth rate, for a range of Rm. The nature of the α-eﬀect for the Roberts ﬂow in the limit of Rm → ∞ was considered by Childress (1979) and later by Soward (1987), who showed that the growth rate scales as ln (ln Rm) . (2.20) ln Rm Childress & Soward (1989) extended their analysis to calculate the α-eﬀect for large Rm for the ‘cat’s eye’ ﬂow with ψ = sin x sin y + δ cos x cos y. Plunian & R¨ adler (2002) considered the k-dependence of the α-eﬀect for the Roberts ﬂow, and the possible relationship between this dynamo and that of the Karlsruhe dynamo experiment. In order for a dynamo to be ‘fast’ — i.e. to have a positive growth rate in the limit as Rm → ∞ — it is necessary that it has chaotic particle paths (Vishik 1989; Klapper & Young 1995). Steady two-dimensional ﬂows are integrable and hence, as such, cannot act as fast dynamos — although, as for the Roberts ﬂow analysed by Soward (1987), the growth rate may be an extremely slowly decaying function of Rm. It is of interest therefore to consider the α-eﬀect produced by non-integrable ﬂows as Rm → ∞. A much-studied ﬂow in the context of possible fast dynamo action is the timedependent version of the Roberts ﬂow introduced by Galloway & Proctor (1992), with p∼

ψ = sin(x + cos ωt) + cos(y + sin ωt),

w = Kψ.

(2.21)

(When ω = 0 ﬂows Eq. (2.19) and Eq. (2.21) are equivalent after a simple transformation.) Courvoisier et al. (2006) have investigated the nature of the α-eﬀect for the ﬂow Eq. (2.21), for both ﬁnite and zero k, and demonstrated that, for this time-dependent ﬂow, α is a sensitive function of Rm which, furthermore, does not tend to zero for large Rm.

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Another widely used approach to investigating the kinematic α-eﬀect, which is one step closer to astrophysical reality, is to consider ﬂows that are driven by rotating thermal convection. Here the ﬂow is not speciﬁcally prescribed, but results from the nonlinear evolution of a convective instability; the spatial and temporal dependence of the ﬂow depends on the parameters of the problem (such as the Rayleigh number) and on the boundary conditions. The simplest conﬁguration is the plane layer model, with the rotation axis parallel to gravity, which was investigated in the pioneering studies of Childress & Soward (1972) and Soward (1974), who explored the nature of dynamo action for mildly supercritical Boussinesq convection under the assumptions of ﬁrst order smoothing. The antisymmetry of Boussinesq convection about the mid-plane leads to an antisymmetric helicity distribution and hence an antisymmetric α— where the large-scale ﬁeld is horizontal and dependent solely on the vertical coordinate. Weakly supercritical convection is characterised by a relatively simple planform. Soward (1974) was able to demonstrate large-scale dynamo action for some simple steady ﬂows — though the very simplest planform consisting of a single roll with a single horizontal wavenumber does not exhibit dynamo action. The plane layer dynamo has been investigated further numerically (kinematically and dynamically), particularly in the regime of rapid rotation, by Jones & Roberts (2000), Rotvig & Jones (2002) and Stellmach & Hansen (2004). The convection in all of these studies, although outside the weakly nonlinear regime considered by Soward, is nonetheless fairly wellordered; it gives rise to a signiﬁcant α-eﬀect and, hence, to dynamo action with a pronounced large-scale component. Cattaneo & Hughes (2006) took a somewhat diﬀerent approach to the plane-layer convective dynamo, concentrating on more turbulent ﬂows which, although rotationally inﬂuenced (O(1) Rossby number) were not rotationally dominated; furthermore they considered horizontally extended domains, thereby allowing the convection to evolve without inﬂuence from sidewall boundaries. Their results diﬀer considerably from those that emerge from considerations of ‘well-ordered’ convection. In particular they found that, despite the helical nature of the ﬂow, two signiﬁcant problems arose in determining α — even in the parameter regime where there is no self-sustaining dynamo action and hence E is due entirely to the imposed mean ﬁeld. One problem is that α, even after spatial averaging over many convective cells, is a strongly ﬂuctuating quantity in time (see Fig. 2.2); the second is that, even if one is bold enough to assign a mean value to α then this is extremely small — only O(u/Rm) (where u is the rms velocity) rather than O(u) as one might expect in a tur-

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bulent process. The underlying problem — and one that seems inescapable — is that at large Rm the ﬂuctuations completely dominate the mean. This result is potentially extremely bad news for mean ﬁeld dynamo modelling, and will be discussed further in Sec. 2.4.

Fig. 2.2 Time history of the spatially-averaged x-component of the emf for rotating Boussinesq convection, with the rotation axis vertical and with an imposed magnetic ﬁeld of unit strength (measured in terms of the equivalent Alfv´en velocity) in the xdirection. Thus this component of the emf corresponds to α11 . The computational domain has size 5 × 5 × 1; the rms velocity u ≈ 50 and Rm ≈ 250. The spatial average at any time involves many convective cells. Despite this, it can be seen that α is a wildly ﬂuctuating quantity in time. The thick line in the upper panel corresponds to the time average up to time t of the signal. The same curve is plotted again in the lower panel with a more appropriate vertical axis, showing a painfully slow convergence to a value much smaller than that of u.

2.2.3

Linear mean ﬁeld dynamo models

In the modelling of astrophysical magnetic ﬁelds it is natural to think of the averaging procedure as an azimuthal average, and the mean ﬁelds that

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result as therefore axisymmetric. In terms of cylindrical polar coordinates (s, φ, z) we may then write U = sω(s, z)eφ + Up ,

B = B(s, z)eφ + Bp ,

E = Eφ eφ + E p ,

(2.22)

where the subscript p denotes the poloidal component. The quantities ω, Up and Bp therefore represent the diﬀerential rotation, large-scale meridional ﬂow and large-scale meridional ﬁeld. For simplicity, but without losing the essence of mean ﬁeld dynamo theory, we may consider the case of isotropic turbulence. The mean emf then takes the form E = αB − β∇ × B.

(2.23)

On writing Bp = ∇ × A(s, z)eφ (by axisymmetry), the mean induction equation (2.4) can be expressed as the two scalar equations: ∂A 2 A, + s−1 (Up · ∇) (sA) = αB + ηT ∇ (2.24) ∂t

∂B 2 B, (2.25) + s (Up · ∇) s−1 B = s (Bp · ∇) ω + (∇ × αBp )φ + ηT ∇ ∂t 2 = ∇2 − s−2 , and where ηT = η + β is the eﬀective magnetic where ∇ diﬀusivity (assumed constant in the above equations). It can be seen from Eq. (2.24) that the α-eﬀect can generate poloidal ﬁeld from the toroidal component, and from Eq. (2.25) that it can also generate toroidal from poloidal ﬁeld; the diﬀerential rotation ω can generate toroidal ﬁeld by stretching out the poloidal ﬁeld. A mean ﬁeld dynamo must continually regenerate magnetic ﬁeld through some combination of the αand ω-eﬀects; the relative magnitudes of these generation terms allows us to classify the nature of the dynamo action. If, on the right hand side of Eq. (2.25), the term involving α dominates that involving ω then we may neglect the s (Bp · ∇) ω term and speak of an α2 -dynamo; if the ω term dominates we may neglect the (∇ × αBp )φ term and speak of an αωdynamo; if they are comparable then we speak of an α2 ω-dynamo. Note that the α-term in Eq. (2.24) is indispensable for mean ﬁeld dynamo action. In its absence A simply decays; the removal of the source terms for B then leads to the eventual decay of the toroidal ﬁeld also. The key characteristics of these dynamos can be brought out in a Cartesian model, for which Eq. (2.24) and Eq. (2.25) simplify to ∂A + (Up · ∇) A = αB + ηT ∇2 A, ∂t ∂B + (Up · ∇) B = (Bp · ∇) U − α∇2 A + ηT ∇2 B, ∂t

(2.26) (2.27)

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where U = Up + U ey , B = Bp + Bey . Under the assumption that Up , α and ∇U may be treated as uniform, we may seek local solutions of these equations, of the form A = A˜ exp (pt + ik · x) ,

(2.28)

leading to the following dispersion relation governing the growth rate p:

2 p + i(Up · k) + ηk 2 = α2 k 2 − iα(k × ∇U )y (2.29) (Parker 1955). Exponential growth of the ﬁeld — a kinematic mean ﬁeld dynamo — occurs if Re(p) > 0. For an α2 -dynamo with no meridional circulation (Up = 0), Eq. (2.29) simpliﬁes to p = ±αk + ηk2 ,

(2.30)

showing that steady dynamo action (Re(p) > 0, Im(p) = 0) will occur provided |α| > ηk; i.e. growth is guaranteed for suﬃciently small k. For an αω-dynamo with no meridional circulation, Eq. (2.29) reduces to 2

(2.31) p + ηk 2 = −iα(k × ∇U )y = 2iΓ, say. Dynamo action occurs if Γ > η 2 k 4 ; i.e. if the dynamo number D = Γ/η2 k 4 is greater than unity. The growing solution takes the form of a travelling wave, propagating in the +k (−k) direction if Γ < 0 (Γ > 0). (More generally it can be shown that for spatially varying underlying quantities, dynamo waves tend to propagate along lines of constant rotation (Yoshimura 1978).) If the equatorial migration of sunspots is viewed as the surface manifestation of an underlying, equatorially propagating, dynamo wave, then this suggests that in the northern (southern) hemisphere of the Sun the product α∂ω/∂r should be negative (positive). From Eq. (2.29) it can be seen that, at least within this simple model, a meridional ﬂow aﬀects the frequency, but not the growth rate, of any dynamo, with the α2 -dynamo now also assuming an oscillatory character, and with the direction of propagation for the αωdynamo dependent (for Γ > 0) on the sign of Γ1/2 − Up · k. Although all the fundamental ingredients of mean ﬁeld dynamos are contained in the extremely simple plane layer model described above, clearly many important eﬀects are omitted, even when attention is restricted to kinematic dynamos. Linear mean ﬁeld dynamos in a sphere were considered by Steenbeck & Krause (1969a, b), Roberts & Stix (1972) and P.H. Roberts (1972). The evolution of dynamo waves in a spherical αω-dynamo for one

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Fig. 2.3 Evolution of the magnetic ﬁeld through one cycle of a kinematic αω-dynamo (from P.H. Roberts 1972). Meridian sections are shown, with poloidal ﬁeld lines on the right and lines of constant toroidal ﬁeld strength on the left. For this choice of parameters it can be seen that the pattern of the ﬁeld progresses from the equator to the poles.

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speciﬁc choice of α and ω is shown in Fig. 2.3. In certain astrophysical circumstances there are good reasons for believing that the α-eﬀect and/or the ω eﬀect are concentrated into regions that are thin in comparison with the body as a whole (for example, the solar tachocline, a thin region of radial velocity shear at the base of the convection zone, is widely conjectured to be the location of the ω-eﬀect of the solar dynamo). Mathematically, such concentrated generation and shear can be modelled by δ-function representations of the α- and ω-eﬀects. Away from the generation regions, A and B satisfy diﬀusion equations, with the generation terms brought in through the interface conditions; this approach has been investigated by Steenbeck & Krause (1966), Moﬀatt (1978) and Kleeorin & Ruzmaikin (1981). The role of a weakly modulated background state on the propagation of short wavelength dynamo modes was ﬁrst considered by Kuzanyan & Sokoloﬀ (1995), who showed that the eigenfunction of the magnetic ﬁeld is localised away from the maximum of the dynamo number. The related problem of the inﬂuence of lateral boundaries (i.e. boundaries in the direction of propagation of the dynamo waves) has been considered in the linear regime by Worledge et al. (1997). 2.3 2.3.1

Incorporating Nonlinear Eﬀects The nonlinear behaviour of the transport coeﬃcients

The discussion above indicates that the validity of mean ﬁeld electrodynamics is unclear even in the kinematic regime, where the back-reaction of the Lorentz force on the plasma motions is ignored. In this section, we discuss the role of the nonlinearity in modifying the transport coeﬃcients. We proceed on the (far-from-certain) assumption that mean ﬁeld electrodynamics yields sensible results in the kinematic regime, with transport coeﬃcients (the α-eﬀect and turbulent diﬀusivity) of the right order of magnitude, and ask how the generation of a mean magnetic ﬁeld aﬀects the transport in the ﬂow. The question above may be restated as, ‘What is the strength of the mean magnetic ﬁeld that can be generated by the α-eﬀect before the αeﬀect is itself suppressed via nonlinear eﬀects in the momentum equation?’ The momentum equation for an incompressible ﬂow (with unit density) is given by ∂u 1 2 (2.32) + (u · ∇) u = −∇p + j × b + ∇ u + F, ∂t Re where F represents any external forces. The Lorentz force in a dynamo is

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believed to be signiﬁcant once the magnetic energy reaches the same order of magnitude as the kinetic energy of the ﬂow that is generating it. Within the mean ﬁeld framework this has traditionally been assumed to occur when the magnetic energy of the mean ﬁeld reaches equipartition with the kinetic energy of the ﬂow, i.e. when ρu2 ∼ B02 /µ0 .

(2.33)

Hence it is convenient to model the mean ﬁeld transport coeﬃcients as having the dependence α0 β0 α= , β= , (2.34) 2 2 1 + B0 /B 1 + B02 /B 2 where α0 and β0 are the kinematic values of the transport coeﬃcients and B is the strength of the equipartition ﬁeld. These, and other similar expressions, have been extensively used in models of astrophysical dynamos (Jepps 1975; see the review by Ossendrijver 2003 for further references). They are particularly appealing for a number of reasons. Clearly their inclusion will lead to a model that is capable of generating a substantial mean ﬁeld — they are designed precisely to allow a mean ﬁeld that has an energy comparable with that of the turbulence. In addition, these formulae are simple to implement. In the mean ﬁeld ansatz only the mean ﬁeld is calculated, and so a nonlinearity that relies solely on the amplitude of the mean ﬁeld sits self-consistently within the framework of the theory. There is, however, signiﬁcant doubt over the validity of these expressions at high Rm. Moreover, the uncertainties exist at such a fundamental level that even the order of magnitude of the mean ﬁeld generated in the nonlinear regime is open to question. The issue of the level of saturation of the mean magnetic ﬁeld is contentious and subtle, and a comprehensive discussion is well beyond the scope of this paper (see, for example, the recent review by Diamond et al. 2005a). However, as the issue is of such importance for astrophysical dynamo theory, a brief introduction will be included here. Perhaps the clearest (and most convincing) argument for the level of suppression of the transport coeﬃcients is on physical grounds (see, for example, Vainshtein & Cattaneo 1992). It is generally accepted (as argued above) that the kinematic phase in a mean ﬁeld dynamo will continue until the magnetic energy becomes comparable with that of the turbulence. Moreover at high Rm, both theory and numerical experiments indicate that the magnetic energy of the small-scale ﬁeld will be much larger than that of the mean ﬁeld. Turbulence is known to amplify magnetic ﬁelds locally (i.e. on the scale of the small-scale eddies with short turnover times)

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so that the small-scale ﬁelds are more energetic than those on large scales — indeed most numerical experiments indicate that the ratio of the energy of the small-scale ﬁeld to that of the large-scale ﬁeld is given by b2 ∼ Rmq B02 ,

(2.35)

where 0 ≤ q ≤ 2 is a ﬂow- and geometry-dependent coeﬃcient. This, of course, poses a major problem for mean ﬁeld theory. In particular, from this simple order of magnitude argument it appears as though the transport coeﬃcients will be altered signiﬁcantly once the strong smallscale ﬁeld reaches equipartition with the kinetic energy of the turbulence, i.e. when ρu2 ∼ Rmq B02 /µ0 .

(2.36)

(Though the dynamic eﬀect of the small-scale ﬁeld will depend also on its ﬁlling factor — i.e. on how much of the volume is occupied by strong smallscale ﬁelds.) If Eq. (2.36) rather than Eq. (2.33) is the correct estimate of the strength at which the large-scale ﬁeld becomes dynamically signiﬁcant, then the expressions in Eq. (2.34) should be replaced by α=

α0 , 1 + Rmq B02 /B 2

β=

β0 . 1 + Rmq B02 /B 2

(2.37)

This leads to what is sometimes referred to as ‘catastrophic’ quenching of the transport coeﬃcients; i.e. a dramatic reduction in their eﬃciency even when the mean ﬁeld is many orders of magnitude smaller than equipartition. The above argument relies on simple physical balances, but seems plausible nonetheless. However, calculation of the transport coeﬃcients relies on correlations between the turbulence and the small-scale ﬁeld, and these may have unexpected properties. There are currently two possible ways of discriminating between the expressions in Eq. (2.34) and Eq. (2.37), neither of which is entirely satisfactory. The ﬁrst is to derive analytical expressions for α and β by applying conservation laws and closure approximations. The second is to compute straightforwardly the correlations between the small-scale ﬁelds at moderate Rm — and then to extrapolate the results to higher Rm. Analytical progress may be made in certain circumstances by ﬁrst recognising that the induction equation (and indeed Ohm’s Law, which leads to its derivation) results in a number of relations which, in the absence of dissipation, reduce to conservation laws. These take the form (see Diamond

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et al. 2005a; Brandenburg & Dobler 2001; Blackman & Field 2000) 1 E · B0 = − j · b + e · b = αB02 , (2.38) σ da · b = −2e · b + ∇ · (bφ) − ∇ × (a × e), (2.39) dt dHM (2.40) = −2ηQH − FH , dt where φ is an arbitrary scalar (gauge) that is used to deﬁne the vector potential A, HM is the total magnetic helicity A · B, QH is the total current helicity J · B, and FH is the surface-integrated magnetic helicity ﬂux through the boundaries of the domain. The ﬁrst of these equations follows simply from Ohm’s Law, and relates α and the mean ﬁeld to correlations between the small-scale magnetic ﬁelds, currents and emfs. The second is a conservation law for the averaged small-scale magnetic helicity — for statistically steady ﬁelds and suitable boundary conditions this may be simpliﬁed substantially to yield an equation for e · b (see Diamond et al. 2005a). The ﬁnal equation expresses conservation of magnetic helicity. In the absence of magnetic diﬀusion and for boundary conditions that do not allow magnetic helicity to leave the domain the total magnetic helicity is conserved. Hence, under these constraints, if the large-scale magnetic helicity is to grow it must be at the expense of the small-scale magnetic helicity. These relations all allow the achievement of a similar goal — they permit the expression of the average emf E, which relies on correlations between u and b, in terms of correlations between j and b. This substitution is not much more than a matter of taste, since a priori we know none of u, j or b in the dynamic regime. However, the above expressions do become useful if they can be combined with a further expression relating α to b · j. As described below, this has been done, in an approximate sense, by Pouquet et al. (1976). As stated above, in order to calculate α it is necessary to solve for the ﬂuctuating magnetic ﬁeld and velocity. The dynamics is all contained in the role of the Lorentz force in modifying the small-scale velocity, which at high Rm may be subtle and unexpected. A widely used, though often misunderstood, result utilises the eddy-damped quasi-normal Markovian (EDQNM) framework to approximate this back-reaction for turbulence possessing a short correlation time. This approach has been used extensively for purely hydrodynamic turbulence (see, for example, Frisch 1995; Davidson 2004), although the range of its applicability is still an open question. For magnetised ﬂows, in which the turbulence at high Rm appears to deviate

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signiﬁcantly from being normally distributed, there is no convincing evidence that the theory is applicable in any regime. Nonetheless, in the limit of short correlation time, and assuming that the magnetic ﬁeld has no eﬀect on the correlation time of the turbulence, the EDQNM formalism yields a relationship between α and b · j, namely τc α = − u · ω − b · j, (2.41) 3 where ω = ∇ × u (Pouquet et al. 1976). Note that in the kinematic limit this expression reduces to that given in Eq. (2.15), and that the eﬀect of the Lorentz force is thus to add to the kinematic expression a magneticallydriven α-eﬀect αM = τc /3 b · j. There are two other ways to derive this approximate result; either by using closure mode such as the ‘minimal τ approximation’ (Kleeorin et al. 2002), or by linearising about a pre-existing turbulent MHD state with low Rm and short τc (Proctor 2003). (In the second approach, great care must be taken in the interpretation of the meaning of the variables u, ω, b and j, as noted by Proctor.) It is worth stressing again that this approach has circumvented the need to calculate correlations between u and b in favour of calculating those between b and j — clearly some dynamical relationship between j and u has been assumed (or approximated). Ignoring for the minute any reservations about the validity of expression (2.41), it is now clear how to derive expressions for the dynamic eﬀects of the magnetic ﬁeld on the α-eﬀect (and similarly for β). Combining one of the conservation laws with Eq. (2.41) yields either a dynamic result for the evolution of the magnetically-driven α-eﬀect (αM ) (e.g. Blackman & Brandenburg 2002) or, under the assumption of stationarity, a simpliﬁed expression for α in the nonlinear regime — see, for example, the expressions in Gruzinov & Diamond (1994, 1995, 1996) and Kleeorin et al. (1995). In general these expressions support the notion that the α-eﬀect is catastrophically quenched at high Rm (with the nature of the β-eﬀect much less clear), though they indicate that the presence of a large-scale current (J) or of boundary conditions that allow magnetic helicity to escape from the domain may lead to alleviation of the catastrophic quench. The other approach to the question of the nonlinear suppression is to perform numerical simulations to calculate explicitly the emf as a function of Rm and B0 . This has the advantage that no approximations are necessary, but the disadvantage that results can only be obtained for fairly moderate values of Rm. Despite claims to the contrary in the literature, this is actually a reasonably severe restriction (as discussed below), and

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considerable care should be taken in assessing the claimed values of Rm in diﬀerent calculations. As an order of magnitude estimate, a calculation performed with 10003 spectral modes (or equivalently 16003 ﬁnite diﬀerence points), with a separation of scales of about a decade between domain size and the size of the most energetic turbulent eddy, is able accurately to reach magnetic Reynolds numbers of O(103 ). Two classes of numerical calculations have been performed in order to calculate the nonlinear dependence of the transport coeﬃcients. The ﬁrst employs a prescribed body force F(x, t) in Eq. (2.32) in order to drive the ﬂow u. This can be carefully selected so as to drive a ﬂow with particularly advantageous properties for large-scale dynamo action (in a similar manner to the way kinematic ﬂows are selected to be maximally helical so that they yield strong α-eﬀects). The forcing is typically chosen in order to drive a ﬂow with large pointwise and net helicity and also strong time-dependence, so that particle trajectories within the ﬂow are chaotic. Separation of scales can be achieved in two ways. One is to force the ﬂow on a scale much smaller than that of the computational domain; e.g. in a spectral calculation applying the forcing at a wavenumber kf = 8, which determines the small-scale, and ascribing ﬂows and ﬁelds with k = 1 to be large-scale (e.g. Brandenburg 2001). Here, however, there is no natural separation of scales and a continuous spectrum of scales emerges. The other (Cattaneo & Hughes 1996) is to drive the small-scale ﬂow on the scale of the computational domain (kf = 1) and to achieve the separation of scales by imposing an inﬁnite-scale (k = 0) mean magnetic ﬁeld (B0 ), as in the kinematic case described above. The second approach to calculating the α-eﬀect numerically in the nonlinear regime is to drive the ﬂow via the action of buoyancy in a rotating convective ﬂow (either compressible or incompressible), as discussed for the kinematic case in Sec. 2.2.2. A mean magnetic ﬁeld is inserted and the emf E (and hence α) calculated self-consistently, taking account of the Lorentz force. There are now many examples of calculations employing both these approaches (e.g. Cattaneo & Hughes 1996; Cattaneo et al. 2002). Here we proceed by giving details of one such calculation (Cattaneo & Hughes 1996) and then discuss its relation to other such calculations later in the section. Here, the forcing F(x, t) is chosen so that in the absence of a magnetic ﬁeld the Galloway-Proctor ﬂow Eq. (2.21) is driven. A mean magnetic ﬁeld is then imposed in the z-direction and the full magnetohydrodynamic system (Eq. (2.2) and Eq. (2.32)) evolved to a statistically steady state. The components of the mean emf E = (Ex , Ey , Ez ) are then calculated as a

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Fig. 2.4 Time histories of components of the emf for the 3d turbulent ﬂow driven by the Galloway-Proctor forcing with a magnetic ﬁeld imposed in the z-direction (from Cattaneo & Hughes 1996). The emf has strong temporal ﬂuctuations, but a meaningful time average (thick line) can clearly be deﬁned. The top two panels show Ez and Ex for a weak magnetic ﬁeld; the time average of Ex converges to zero, as expected from symmetry considerations. The lowest panel shows Ez for a ﬁeld of almost equipartition strength.

function of Rm and B0 , as shown in Fig. 2.4. These time series show clearly that the (space-averaged) emf is a wildly ﬂuctuating quantity and that, in order to achieve average values for α that have converged to a meaningful result, temporal averages over signiﬁcant time periods (very many turnover times of the ﬂow) are necessary. However it is possible to integrate the equations for long enough to obtain a meaningful average value for α. This is plotted as a function of B0 (at ﬁxed Rm) and Rm (at three representative values of B0 ) in ﬁgures 2.5a and 2.5b respectively. Taken together, these results strongly suggest that the α-eﬀect is suppressed with a very strong dependence on ﬁeld strength at high Rm. In particular, it appears possible to discriminate between the formulae in Eq. (2.34) and Eq. (2.37), with the numerical results indicating that Eq. (2.37) is a considerably more accurate description of the behaviour of the α-eﬀect in the nonlinear regime.

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Fig. 2.5 α versus B02 (upper panel, from Cattaneo & Hughes 1996) and versus Rm (lower panel, from Cattaneo et al. 2002), for the same ﬂow as in Fig. 2.4. The upper panel shows the predictions of the two quenching formulae Eq. (2.34) and Eq. (2.37) (with q = 1), and shows that the numerical simulations strongly support the latter.

Many other numerical calculations with various assumptions and interpretations (e.g. Brandenburg 2001; Ossendrijver et al. 2003) also indicate that the transport coeﬃcients may be catastrophically quenched in the nonlinear regime. These results have, however, been criticised on the grounds that they are all undertaken in closed domains with boundary conditions that do not allow magnetic helicity to leave the computational domain (Blackman & Field 2000). The argument put forward is that large-scale helicity can only grow at the expense of small-scale helicity, as discussed above, and that the system is thus overly constrained. This is an interesting point and one that merits further investigation. It is diﬃcult however to envisage a situation where the ﬂux of magnetic helicity out of the domain can alleviate the constraint on the transport coeﬃcients without removing ﬂux from the domain and limiting the eﬃciency of the dynamo. Moreover,

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the physical argument for strong suppression outlined above relies solely on the dynamics of the small scales and it is not clear how the modiﬁcation of boundary conditions (possibly far away from the turbulent small-scale eddies) can lead to a signiﬁcant enhancement of the local transport properties of the turbulence. 2.3.2

Other nonlinear eﬀects

The suppression of the transport coeﬃcients in the nonlinear regime is contentious and is likely to remain so for the foreseeable future. This issue lies at the very heart of mean ﬁeld electrodynamics as it determines the level of saturation of nonlinear mean ﬁeld models. However, there are other nonlinear processes that may be of importance for saturating the exponential growth of ﬁeld in a kinematic mean ﬁeld model. In Sec. 2.2.3 we described how diﬀerential rotation is a key feature of the dynamo process, stretching out a poloidal ﬁeld and converting it into toroidal ﬁeld (the ω-eﬀect). Therefore a possible saturation mechanism involves the modiﬁcation of the diﬀerential rotation by the Lorentz force. In order to model this, a detailed understanding of the mechanism for the generation and maintenance of diﬀerential rotation by rotating turbulent motions is required. In general, mean ﬂows are set up in turbulent rotating systems via small-scale correlations, leading to Reynolds stresses that naturally transport angular momentum, allowing the formation of a pattern of diﬀerential rotation. As in mean ﬁeld electrodynamics, the role of correlations in this picture is crucial — and poorly understood (see Diamond et al. 2005b for a comprehensive review of the generation of zonal ﬂows in plasmas, and Brummell et al. 1998 or Brun et al. 2004 for a discussion of the generation of mean ﬂows and diﬀerential rotation by convection). In the mean ﬁeld formalism, the generation of mean ﬂows by small-scale correlations is often parameterised using the Λ-eﬀect, a closure scheme that relates the average Reynolds stresses to the local angular velocity and its derivatives (see R¨ udiger 1989 for a complete discussion). It is suﬃcient here to state that the theory underpinning the Λ-eﬀect is a two-scale theory in a similar vein to mean ﬁeld electrodynamics — and so has the same strengths and weaknesses. Just as for the transport coeﬃcients α and β, the role of magnetic ﬁelds in modifying angular momentum transport, and hence diﬀerential rotation, is also poorly understood. However what is known is that the magnetic ﬁeld can modify the diﬀerential rotation in a straightforward manner, with

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a large-scale magnetic ﬁeld B0 driving a large-scale ﬂow via the action of the large-scale Lorentz force J0 × B0 in the momentum equation. This mechanism is often termed the ‘Malkus-Proctor eﬀect’ (Malkus & Proctor 1975) in astrophysical dynamo theory (especially in solar dynamo theory, where this saturation mechanism has been extensively invoked). In addition, in a high Rm environment the presence of a large-scale ﬁeld and a small-scale ﬂow inevitably implies the presence of a strong small-scale ﬁeld, as argued above. This small-scale ﬁeld can itself inﬂuence the angular momentum transport in two ways; either by modifying the dynamics of the small-scale velocity (in particular the correlations that lead to the driving of mean ﬂows in the ﬁrst place) via the small-scale Lorentz force, or by itself driving mean ﬂows via average Maxwell stresses (which arise through small-scale/small-scale interactions of magnetic ﬁelds). This process in mean ﬁeld models has usually been either crudely parameterised in the form of ω-quenching (e.g. Roald & Thomas 1997), or Λ-quenching (e.g. Kitchatinov et al. 1994) in which the back-reaction of the magnetic ﬁeld on the angular momentum transport is calculated within the same framework as the mean ﬁeld equations. It is though important to note that all the diﬃculties and uncertainties surrounding the transport coeﬃcients in mean ﬁeld electrodynamics arise a fortiori in any theory of the modiﬁcation of angular momentum transport.

2.3.3

Nonlinear mean Field Models

We discussed brieﬂy in Sec. 2.2.3 how the framework of mean ﬁeld electrodynamics could be used to construct kinematic dynamo models. As for the kinematic theory, most of the computational eﬀort in the nonlinear regime has been ‘object speciﬁc’, with computations being undertaken with the speciﬁc aim of explaining the generation of magnetic ﬁelds in either planets, stars or galaxies. In these cases, plausible proﬁles for the diﬀerential rotation, α-eﬀect and turbulent diﬀusivity are selected for the object to be modelled and, likewise, a nonlinear saturation mechanism is selected that may be of importance for the dynamo operating in that object. In this paper we do not attempt to discuss these astrophysically-motivated mean ﬁeld models, as it is extremely diﬃcult to draw robust conclusions from studies that aim speciﬁcally to model a particular astrophysical object in detail. Instead we focus on aspects of nonlinear behaviour that are inherent to all nonlinear mean ﬁeld models, and refer the interested reader to reviews by Ossendrijver (2003) and Tobias & Weiss (2007), who discuss solar and

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stellar models, by Jones (2003), who discusses planetary dynamos, and by Beck et al. (1996), who discuss the galactic dynamo. 2.3.3.1

Nonlinear Travelling Waves

Here we give a brief discussion of the role of nonlinearities in modifying the behaviour of the kinematic mean ﬁeld dynamos discussed in Sec. 2.2.3 — i.e. plane-wave dynamos, bounded dynamos and spherical dynamos. The simplest example of a mean ﬁeld dynamo (as discussed earlier) is the local travelling-wave solution of Parker (1955). There the solution is assumed to be independent of the z-coordinate and wave-like solutions depending on the x-coordinate are constructed. In order to extend this model to the nonlinear regime, where computational techniques are required, a ﬁnite domain must be considered. In two dimensions, boundary conditions must be imposed in both directions. Here we choose appropriate matching conditions in the z-direction and periodic boundary conditions in the x-direction (see Tobias 1997 for a detailed discussion). Within this framework it is now permissible to introduce the possibility that the transport coeﬃcients (α and β) and the shear have an underlying dependence on the z-coordinate (though they remain independent of x). Such a model is local in the x-direction, but global in the z-direction. The nonlinear dynamics of such a model was investigated in detail in Tobias (1997) and the results summarised here. The linear theory (as for the completely localised Parker model described earlier) indicates that as the dynamo number is increased past a threshold value, instability sets in to travelling wave solutions, with a welldeﬁned preferred wavelength and frequency (which depend on the critical dynamo number). The waves now possess non-trivial spatial structure in the z-direction, determined by the underlying proﬁles of the transport coeﬃcients. What happens to the nonlinear solutions as the dynamo number D is increased further depends on the form of the nonlinearity adopted. The ﬁrst class of nonlinearities considered are ‘static nonlinearities’ (such as α, β or ω-quenching), in which the mean magnetic ﬁeld acts back instantaneously on the transport coeﬃcient or diﬀerential rotation through a parameterised formula such as Eq. (2.34). For these instantaneous nonlinear eﬀects the travelling wave solutions that are destabilised in the initial bifurcation remain stable as the dynamo number is increased and no further temporal structure is added. It is also found that α- and ω-quenching have similar properties, acting purely as equilibration mechanisms that have

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little eﬀect on the spatial dependence of the waves. Furthermore, for these cases, the wave-speed remains largely independent of the amplitude of the magnetic ﬁeld as the dynamo number is increased. For β-quenching both the spatial structure and the wave speed of the travelling waves are modiﬁed in the nonlinear regime. Waves tend to travel more slowly as the magnetic diﬀusivity is quenched, and the dynamo becomes more ineﬃcient. This change in the frequency of the waves due to diﬀusivity quenching may provide a constraint for oscillatory dynamo models. In the second class of nonlinearities the magnetic ﬁeld acts back via a separate dynamical equation. This could be due to the mean ﬁeld driving large-scale ﬂows directly (the Malkus-Proctor eﬀect), or to the mean magnetic ﬁeld suppressing the small-scale turbulence dynamically (which may involve the addition of a dynamic equation for the evolution of the transport coeﬃcients, α, β or Λ). If the dynamic Malkus-Proctor eﬀect is included as a nonlinearity, then the travelling wave solutions created in the initial bifurcation rapidly lose stability to states with more spatio-temporal variability. In particular, a complicated series of bifurcations, involving a secondary and tertiary Hopf bifurcation to modulated travelling waves and quasi-periodic waves, leads to a transition to chaotic oscillations. The basic magnetic cycle becomes modulated on a time-scale that is determined by the ratio of the turbulent diﬀusivities for magnetic ﬁeld (β) and angular velocity (νT ). This natural modulation of the basic magnetic cycle is often used as an explanation of intermittent behaviour in nonlinear dynamos — for example, the solar cycle is modulated and undergoes recurrent periods of reduced activity known as Grand Minima. 2.3.3.2

Nonlinear Dynamos in Finite Cartesian and Spherical Domains

The behaviour of nonlinear travelling waves described above is instructive, and yields some important results for the dependence of the amplitude and period of the mean ﬁeld as a function of dynamo number. It also gives an indication of the role of dynamical nonlinearities in producing complicated spatio-temporal behaviour. However, we noted at the end of Sec. 2.2.3 that, even in the kinematic regime, dynamo solutions may have very diﬀerent properties in ﬁnite domains to those found for periodic boundary conditions. Of particular importance is the interaction of the dynamo solutions with inhomogeneities in the underlying proﬁles of either the transport coeﬃcients or the diﬀerential rotation, or the interaction with

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boundaries. In the linear regime, both the frequency and the spatial form of the dynamo solutions were found to be sensitive to the imposition of boundary conditions and the inhomogeneities. It is therefore of no surprise that the behaviour of nonlinear mean ﬁeld dynamo models in ﬁnite domains such as spheres, spherical shells, discs and Cartesian slabs may be very diﬀerent to that of nonlinear travelling wave solutions. As for the kinematic case, most of the calculations of nonlinear mean ﬁeld models in ﬁnite domains are targeted at modelling dynamos in speciﬁc astrophysical objects, with the structure of the transport coeﬃcients and rotation proﬁles usually chosen using some plausible physical assumptions for the object to be modelled. In a similar manner the nonlinearity is chosen in a plausible, but ad hoc, manner. There are therefore a vast number of nonlinear mean ﬁeld dynamo calculations, whose conclusions diﬀer owing to the diﬀerent assumptions behind the model in question (see, for example, the reviews cited above by Beck et al. 1996; Jones 2003; Ossendrijver 2003; Tobias & Weiss 2007). Discriminating between such models is extremely diﬃcult and largely subjective. We prefer here to focus on generic properties of nonlinear models in ﬁnite domains — i.e. properties that appear robust whichever assumptions are included in the model. For nonlinear dynamo models in ﬁnite domains, the following properties tend to be observed. In general, the amplitude of the energy in the mean magnetic ﬁeld is an increasing function of dynamo number. More specifically, the magnetic energy is often observed initially to increase linearly with supercriticality (i.e. B2 ∝ (D − Dc )) for D suﬃciently close to its critical value Dc . As D − Dc is increased further there is a saturation of the amplitude of the mean ﬁeld as a function of dynamo number. Moreover, as D − Dc increases, the dynamo solution becomes more irregular for all types of nonlinearities used. A sequence of bifurcations leads to the spatio-temporal fragmentation of solutions and eventually to chaotic time dependence of the dynamo solutions. This pattern of spatio-temporal chaos may in general be the result of either nonlinear interactions between the magnetic ﬁeld and the mean ﬂow (as discussed above for nonlinear dynamo waves), or the interaction between mean magnetic modes with diﬀerent symmetry properties. These interactions can be understood with reference to the theory of nonlinear dynamical systems (see, for example, Knobloch et al. 1998, Tobias 2002 for an in-depth discussion of the mechanisms that can lead to the formation of chaotic solutions). For all nonlinearities, the primary frequency of oscillation of the dynamo appears to be an increasing function of dynamo number. This is due to the interaction of the nonlinear

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wavetrain with the boundaries of the domain, and the subsequent change in lengthscale for the dynamo wavetrain (see, for example, Tobias 1998). Despite some common themes running through the nonlinear development of mean ﬁeld dynamo models, there is still great uncertainty as to the range of applicability of such models. We stress again here that these models are based upon a theory for which even the kinematic behaviour is uncertain. It is clear that one must proceed with great care in evaluating the results of such nonlinear mean ﬁeld models, though they may yet provide some insight into the mechanisms for the generation of magnetic ﬁeld in astrophysical bodies.

2.4

Discussion

Since the formulation of mean ﬁeld electrodynamics by Steenbeck, Krause and R¨ adler in the 1960’s — foreshadowed by the work of Parker (1955) — it has been used extensively in the modelling of the generation and maintenance of magnetic ﬁelds in planets, stars, galaxies and accretion discs. It is, in some sense, a remarkably successful theory, being able to reproduce temporal and spatial features of a tremendous range of observed astrophysical magnetic ﬁelds. This suggests strongly therefore that the ‘true’ equation describing the evolution of the mean ﬁeld does indeed take the form of Eq. (2.9) — although this is possibly not all that surprising, since the αeﬀect closes the dynamo loop and β is (in its simplest form) a turbulent diﬀusivity. However, that is not to say that the all is well with mean ﬁeld electrodynamics. Even at the level of reproducing observed cosmical ﬁelds, the freedom in the choice of parameters means that models inspired by different physical assumptions can lead to the same global features. Thus it is not possible to assert that speciﬁc magnetic behaviour (such as the cyclic evolution of a star’s magnetic ﬁeld, for example) results from speciﬁc forms of αij and βijk , and hence from speciﬁc physical considerations. Conversely it is possible to have two rather similar mean ﬁeld models that produce very diﬀerent behaviour. Thus great care needs to be taken in the description — and, a fortiori, in the prediction — of astrophysical magnetic ﬁelds via parameterised mean ﬁeld models. Detailed investigation of the micro-physics of the mean ﬁeld transport coeﬃcients leads to further diﬃculties. Although for turbulent ﬂows with low magnetic Reynolds numbers or very short correlation times the mean ﬁeld description seems to hold, for the astrophysically relevant case of

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turbulence with Rm 1 and O(1) correlation times the very foundations of mean ﬁeld electrodynamics seem unsure. Here it seems unavoidable that the ﬂuctuations in the magnetic ﬁeld overwhelm the mean. This leads, as discussed earlier, to problems both in the kinematic regime, in which it can be hard even to determine α, and in the dynamic regime, where any α-eﬀect is ‘catastrophically’ quenched by an extremely weak mean magnetic ﬁeld. It appears that maybe what is required are less turbulent, more ordered ﬂows. In the Sun these may arise from the presence of the tachocline, a thin shear region at the base of the convection zone. However, what happens in fully convective stars, for example, or with the evolution of galactic magnetic ﬁelds, remains very much an open question. References [1] Backus G E 1958 Ann. Phys. 4 372–447. [2] Backus G, Parker R and Constable C 1996 Foundations of Geomagnetism (Cambridge University Press, Cambridge). [3] Balbus S A and Hawley J F 1998 Rev. Mod. Phys. 70 1–53. [4] Beck R, Brandenburg A, Moss D, Shukurov A and Sokoloﬀ D 1996 Ann. Rev. Astron. Astrophys. 34 155–6. [5] Blackman E G and Brandenburg A 2002 Astrophys. J. 579 359–73. [6] Blackman E G and Field G B 2000 Astrophys. J. 534 984–8. [7] Brandenburg A 2001 Astrophys. J. 550 824–40. [8] Brandenburg A and Dobler W 2001 Astrophys. J. 369 329–38. [9] Brandenburg A and Subramanian K 2005 Phys. Rep. 417, 1–209. [10] Brummell N H, Hurlburt N E and Toomre J 1998 Astrophys. J 493 955–69. [11] Brun A S, Miesch M S and Toomre J 2004 Astrophys. J 614 1073–1098. [12] Cattaneo F and Hughes D W 1996 Phys. Rev. E 54 R4532–5. [13] —- 2006 J. Fluid Mech. 553, 401–18. [14] Cattaneo F, Hughes D W and Proctor 1988 Geophys. Astrophys. Fluid Dyn. 41 335–42. [15] Cattaneo F, Hughes D W and Thelen J-C 2002 J. Fluid Mech. 456 219–37. [16] Childress S 1979 Phys. Earth Planet. Inter. 20 172–80. [17] Childress S and Gilbert A D 1995 Stretch, Twist, Fold: The Fast Dynamo (Springer-Verlag, Berlin). [18] Childress S and Soward A M 1972 Phys. Rev. Lett. 29 837–9. [19] Childress S and Soward A M 1989 J. Fluid Mech. 205 99–133. [20] Courvoisier A, Hughes D W and Tobias S M 2006 Phys. Rev. Lett. 034503. [21] Cowling T G 1934 Mon. Not. R. Astr. Soc. 94 39–48. [22] Davidson P A 2004 Turbulence: an Introduction for Scientists and Engineers Oxford University Press: Oxford. [23] Diamond P H, Hughes D W and Kim E 2005a In Fluid Dynamics and Dynamos in Astrophysics and Geophysics (eds A M Soward, C A Jones,

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D W Hughes and N O Weiss) (CRC press, Boca Raton) p 145–92. [24] Diamond P H, Itoh S-I, Itoh K and Hahm T S 2005b Plasma Phys. Control. Fusion 47 R35–R161. [25] Frisch U 1995 Turbulence: the Legacy of A N Kolmogorov (Cambridge University Press, Cambridge). [26] Galloway D J and Proctor M R E 1992 Nature 356 691–3. [27] Gruzinov A V and Diamond P H 1994 Phys. Rev. Lett. 72 1651–4. [28] Gruzinov A V and Diamond P H 1995 Phys. Plasmas 2 1941–6. [29] Gruzinov A V and Diamond P H 1996 Phys. Plasmas 3 1853–7. [30] Herzenberg A 1958 Phil. Trans. R. Soc. Lond. A250 543–83. [31] Hide R and Palmer T N 1982 Geophys. Astrophys. Fluid Dyn. 19 301–9. [32] Ivers D J and James R W 1984 Phil. Trans. R. Soc. Lond. A312 179–218. [33] Jepps S A 1975 J. Fluid Mech. 67 625–45. [34] Jones C A 2003 In Stellar Astrophysical Fluid Dynamics (eds. M J Thompson and J Christensen-Dalsgaard) (Cambridge University Press, Cambridge) p 159–176. [35] Jones C A and Roberts P H 2000 J. Fluid Mech. 404 311–43. [36] Kitchatinov L L, R¨ udiger G and K¨ uker M 1994 Astron. Astrophys. 292 125–32. [37] Klapper I and Young L S 1995 Commun. Math. Phys. 173 623–46. [38] Kleeorin N and Ruzmaikin A A 1981 Geophys. Astrophys. Fluid Dyn. 17 281–96. [39] Kleeorin N, Rogachevskii I and Ruzmaikin A 1995 Astron. Astrophys. 297 159–167. [40] Kleeorin N, Rogachevskii I and Sokoloﬀ D 2002 Phys. Rev. E 65 36303-7. [41] Knobloch E, Tobias S M and Weiss N O 1998 Mon. Not. R. Astr. Soc. 297 1123–38. [42] Krause F and R¨ adler K-H 1980 Mean Field Electrodynamics and Dynamo Theory (Pergamon Press, Oxford). [43] Kuzanyan K M and Sokoloﬀ D D 1995 Geophys. Astrophys. Fluid Dyn. 81 113–29. [44] Larmor J 1919 Rep. Brit. Assoc. Adv. Sci. 1919 159–60. [45] Malkus W V R and Proctor M R E 1975 J. Fluid Mech. 67 417–44. [46] Moﬀatt H K 1974 J. Fluid Mech 65 1–10. [47] Moﬀatt H K 1978 Magnetic ﬁeld generation in electrically conducting ﬂuids (Cambridge University Press, Cambridge). [48] Moﬀatt H K 1983 Rep. Prog. Phys. 46 621–64. [49] Ossendrijver M 2003 Astron. Astrophys. Rev. 11 287–367. [50] Parker E N 1955 Astrophys. J. 122 293–314. [51] Parker E N 1979 Cosmical Magnetic Fields: Their Origin and Their Activity (Clarendon Press, Oxford). [52] Plunian F and R¨ adler K-H 2002 Geophys. Astrophys. Fluid Dyn. 96 115–33. [53] Pouquet A, Frisch U and L´eorat J 1976 J. Fluid Mech. 77 321–54. [54] Proctor M R E 2003 In Stellar Astrophysical Fluid Dynamics (eds. M J Thompson and J Christensen-Dalsgaard) (Cambridge University Press, Cambridge) p 143–158.

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[55] [56] [57] [58] [59] [60] [61] [62] [63] [64] [65] [66] [67] [68] [69] [70] [71] [72] [73] [74] [75] [76] [77] [78] [79] [80]

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Roald C B and Thomas J H 1997 Mon. Not. R. Astr. Soc. 288 551–64. Roberts G O 1972 Phil. Trans. R. Soc. Lond. A271 411–54. Roberts P H 1972 Phil. Trans. R. Soc. Lond. A272 663–98. Roberts P H and Soward A M 1992 Ann. Rev. Fluid Mech. 24 459–512. Roberts P H and Stix M 1972 Astron. Astrophys. 18 453–66. Rosner R 2000 Phil. Trans. R. Soc. Lond. A358 689–709. Rotvig J and Jones C A 2002 Phys. Rev. E 66 056308-1–15. R¨ udiger G 1989 Diﬀerential Rotation and Stellar Convection: Sun and SolarType Stars (Gordon and Breach, New York). Ruzmaikin A A, Shukurov A M and Sokoloﬀ D D 1988 Magnetic Fields of Galaxies (Kluwer, Dordrecht). Soward A M 1974 Phil. Trans. R. Soc. Lond. A275 611–46. Soward A M 1987 J. Fluid Mech. 180 267–95. Steenbeck M and Krause F 1966 Z. Naturforsch. 21a 1285–96. Steenbeck M and Krause F 1969a Astron. Nachr. 291 49–84. Steenbeck M and Krause F 1969b Astron. Nachr. 291 271–86. Steenbeck M, Krause F and R¨ adler K-H 1966 Z. Naturforsch. 21a 369–76. Stellmach S and Hansen U 2004 Phys. Rev. E 70 056312-1–16. Tobias S M 1997 Geophys. Astrophys. Fluid Dyn. 86 287–343. Tobias S M 1998 Mon. Not. R. Astr. Soc. 296 653–61. Tobias S M 2002 Astron. Nachr. 323 417–423. Tobias S M & Weiss N O 2007 In The Solar Tachocline (eds D W Hughes, R Rosner and N O Weiss) (Cambridge University Press, Cambridge) p319–50. Vainshtein S I and Cattaneo F 1992 Astrophys. J. 393 165–71. Vishik M M 1989 Geophys. Astrophys. Fluid Dyn. 48 151–67. Worledge D, Knobloch E, Tobias S and Proctor M 1997 Proc. R. Soc. Lond. A453 119–43. Yoshimura H 1978 Astrophys. J. 226 706–19. Zeldovich Ya B 1957 Sov. Phys. JETP 4 460–2. Zweibel E G 2005 In Fluid Dynamics and Dynamos in Astrophysics and Geophysics (eds A M Soward, C A Jones, D W Hughes and N O Weiss) (CRC press, Boca Raton) p115-43.

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Chapter 3

Origin, Structure and Stability of Astrophysical MHD Jets

P.-Y. Longaretti LAOG, Observatoire de Grenoble, BP53X, 38041 Grenoble Cedex 9, France The dynamics of the powerful jets observed in a variety of astrophysical objects is discussed, with special emphasis on the question of jet stability. This introduction is primarily aimed at researchers and graduate students of the fusion and plasma physics community, but is expected to provide a useful review to astrophysicits as well, especially the section concerning jet stability. The exposition begins with well-established background notions and more or less direct observational constraints on jet physics. Then the question of MHD jet origin and structure is discussed. Finally, jet stability is addressed in particular with respect to Kelvin-Helmholtz, current-driven and pressure-driven instabilities. The ﬁrst half of the paper is rather descriptive, and the second half somewhat more technical. Although obviously useful, no prior exposure to astrophysics is required.

3.1

Introduction

Astrophysical jets span a surprisingly wide range of physical conditions. The largest ones are produced in some categories of Active Galactic Nuclei1 (AGNs in short). They can propagate over2 ∼ 106 pc, at nearly 1 A short description of AGNs, along with a brief survey of the relevant background astrophysical notions is provided in Appendix 3.A. 2 1 pc ∼ 3 × 1018 cm; 1 M 33 g; 1 L = solar mass ∼ 2 × 10 = solar luminosity ∼ 4 × 1033 erg/s.

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the velocity of light. They originate from the environment of giant black holes of ∼ 106−9 M (see Fig. 3.11). At the opposite end of the mass and size spectrum, jets originating from Young Stellar Objects (YSOs), in their various early stages of formation, have typical extents of ∼ 1 pc, and propagation velocities of the order of a few 100 km/s; their parent protostars have masses ∼ 1 M (see Fig. 3.12). In between, one ﬁnds jets associated with neutron stars, X-ray binary systems and stellar mass black holes. For deﬁniteness, only AGN and YSO jets are considered as speciﬁc examples of the physical processes discussed in this review. The ejection appears to be bipolar, with one jet on two opposite sides of the source. One of the jets may be weaker or even absent on images, due to observational selection eﬀects; for example, relativistic emission in AGNs is often detected close to near alignment with the observer, so that one jet is strongly ampliﬁed, while the other one is strongly de-ampliﬁed. Somewhat paradoxically, these energetic ejection phenomena occur in conjunction with a vigorous accretion of gaseous matter onto the associated central object. The details of the involved processes are not and by far all understood, nor are they all agreed upon. However, there is a large consensus over the fact that large scale infalling gaseous matter fuels the small scale engine. A disk most likely forms in the process because energy is dissipated into heat and radiation during the collapse, while angular momentum is conserved in mechanical form. In the course of time, the matter of the disk is partially accreted onto the central object and partially expelled by the jet. For such a large scale collapse to proceed, the concentration of the ambient magnetic ﬁeld ﬂux must be largely limited through some sort of ambipolar, or turbulent diﬀusion process. The present observational limits do not allow us to pinpoint the source of large-scale jets, as the central region is too small to be directly observed: the jet driving source can either be the central object, its accretion disk, or the interaction zone between the two. As the central engine (central object and inner disk) is not directly accessible to observation, even in YSOs, its dynamics is only indirectly constrained. Nevertheless, the coexistence of accretion and ejection phenomena is well supported by the available body of observations. All the most powerful jets observed in all classes of astrophysical objects — the so-called FR2 jets in AGNs, and most YSO jets — appear as very narrow, highly collimated outﬂows over most of their propagation length, with opening angles not exceeding a few degrees. They end up in gigantic, much less well-organized, lobes of hot matter, which form through the

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interaction of the jet with the ambient medium (see Fig. 3.11 and 3.12); this interaction takes on the form of a (usually double) bow shock in YSOs. These lobes often seem much more prominent than the well-organized jets themselves3 . An important feature which is often observed in both AGN and YSO jets is the presence of bright knots travelling along the jet, known as Herbig-Haro objects in the case of YSOs. The origin of these structures is still highly debated, and is possibly system dependent. Also, their relation to the global dynamics of the jets is unclear, as radiation is not always a reliable tracer of mass in astrophysical objects. Finally, it is worth noting that the jet radiation is mostly produced through thermal emission lines in the case of YSOs, while it is mostly a non-thermal continuum in the case of AGNs. A striking feature in both cases is the propagation distance compared to the size of their region of birth. Indeed, both for AGNs and YSOs, the characteristic scale of the central region is ∼ 1 AU (Astronomical Unit4 ). Therefore jets can propagate over about 105 times their initial radial extent in YSOs, and 1011 times in AGNs. Even if one restricts attention to the quasi-cylindrical propagation section, jets propagate over at least tens to hundreds of times (for YSOs) to 106 times (for AGNs) their radius. The question of jet stability has at least two diﬀerent aspects in this context. On the one hand, one would like to understand how these jets can propagate over such enormous distances without losing coherence. On the other, one would like to understand the origin of the jet apparently inhomogeneous spatial and temporal structure (this last question is barely addressed in the present review). Our present lack of observational constraints makes these questions ill-posed to some extent. As a consequence, our present understanding is only partial, and partially model-dependent, and in any case, less well-developed than (although on some issues related to) the theory of fusion devices. Our present core knowledge of the systems in which jets are observed is presented in Sec. 3.2. What is known (or not known) on jet launching mechanisms is discussed in Sec. 3.3. Models of jet structure are exposed in Sec. 3.4, with special emphasis on disk-driven jets, the most studied ones; our understanding of jet acceleration and collimation is also discussed in this context. Jet stability, the main object of these notes, is reviewed in Sec. 3.5; this topic is the one most directly connected 3 The 41

dynamics of the formation of this lobe is not considered in the present review. AU ≡ Sun-Earth distance 150 × 106 km∼ 10−5 pc.

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to the physics of fusion devices. Finally, a summary is proposed and some open issues are gathered in Sec. 3.6. The material presented here is rather descriptive at ﬁrst, before moving to a more theoretically oriented exposition. No attempt is made at a complete or even fair bibliographical coverage, as this would easily constitute an overwhelming fraction of this chapter. Instead, well-established facts and ideas are exposed with reference to more complete reviews or books, but limited in number, to provide an easier entry point to the subject. References to original articles are reserved for more specialized or less wellknown aspects, for which no recent review is available. They are therefore more numerous in the later sections. Peripheral or more technical matters are relegated to various appendices. The exposition is intended to be self-contained, and to provide a guideline to the more specialized literature.

3.2

Generalities on astrophysical jets

A discussion of the stability of jets supposes some knowledge of their structure and fundamental physical properties. Unfortunately, we have no detailed or direct observational constraints on some of the most important physical parameters. Furthermore, the type of information we have depends on the class of objects considered. The purpose of this section is to brieﬂy survey what we know, in order to focus as much as possible on the relevant physics. An underlying theme is that one can deﬁne a “core” knowledge, common to all object classes, while taking into account particular features when needed. This idea is reasonably consistent with our current body of knowledge. Accretion and ejection seem to be highly interdependent phenomena. For example, accretion and ejection eﬃciencies appear to be closely related in YSOs through their various evolutionary stages [1] and an understanding of jet stability partially relies on the identiﬁcation of the constraints put by this interdependence on jet essential characteristics. Arguments for the presence of accretion-ejection structures, as well as some of their physical characteristics, are brieﬂy surveyed here in both AGNs and YSOs. The aim of this section is to make the whole picture plausible, rather than to establish its various aspects in full detail. For more in-depth discussions of our core body of knowledge on AGNs, see, e.g., [2, 3] and references therein. Concerning AGN jets more speciﬁcally, see [4, 5] and references therein. For YSOs, see, e.g., [6–8, 1] and references therein.

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Accretion Active Galactic Nuclei

The existence of a massive central object, most probably a Black Hole, was recognized relatively early on after the now more than forty years old discovery of AGNs (see [2] for a review; for an introduction to the relevant Black Hole physics, see [3] pp. 192-208). Of course the Black Hole itself is not visible; its existence is inferred from the radiation emitted by the material in its neighborhood, or from the motion of this material, if accessible to observation. The spectra of AGNs suggest that the innermost medium producing the high energy emission is fully ionized. In such conditions, the material, mostly composed of electrons and protons, must be bound against the radiation pressure by the gravitational attraction of the central body. The radiation pressure on electrons considerably exceeds that on protons5 , while the gravitational attraction is larger on protons. Assuming that both species are bound by electrostatic forces leads to the deﬁnition of the Eddington luminosity LE : LE =

4πcGM mp ∼ 1046 M8 erg/s, σT

(3.1)

where mp is the proton mass σT is the Thomson cross-section, and M8 ≡ M/(108 M ). This argument, combined with the observed luminosity of AGNs, explains the M = 106−9 M mass range quoted in the introduction for the nucleus. That this nucleus must also be a Black Hole follows from various lines of arguments (for a more complete critical discussion, see [3]): • AGNs exhibit rapid variability, with time-scales tv as short as a few minutes (they are longer for the more luminous objects). The coherence of this variability implies that the size of the emission region is l < ctv , i.e., of the order of the Schwarzschild radius (RS = 2GM/c2 ) of Black Holes in the above quoted mass range. • The eﬃciency of conversion into radiation (i.e., the ratio of the emitted radiation to the nuclear mass, E/M c2 ) is of the order of a few percents, making nuclear reactions an unlikely source of energy. Actually, gravitation is the most eﬃcient known reservoir, as the energy 5 The resulting force is F = Lσ /4πcr 2 where L is the luminosity of the central object T and σT the Thomson cross-section. Spherical symmetry is assumed in this deﬁnition, but an asymmetric matter or radiation distribution would only change the result by a factor of order unity.

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that can be released by a test particle falling from inﬁnity onto a Black Hole is comprised between 6% and 42% of the particle rest mass energy. • The analysis of stellar motions in the vicinity of a Black Hole galactic nucleus implies an increase in the stars’ velocity dispersion. This has been reported in a few galaxies (in particular, our own, although it does not host an AGN). • The density of a large mass Black Hole is fairly small6 . Consequently, most scenarios of nucleus formation end up in a Black Hole being formed, by lack of ways to halt gravitational collapse. An AGN must steadily accrete gaseous matter to maintain its radiation output7 . For example, assuming an eﬃciency = 0.1 for deﬁniteness, the mass accretion rate of an AGN at the Eddington limit of Eq. (3.1) is dME /dt ∼ 2M8 M /yr. There is enough time since the formation of galaxies and appearance of AGNs in the past to build up their inferred mass, and enough gas in the host galaxy to do this. Furthermore, as pointed out in the introduction, the speciﬁc angular momentum of this gaseous supply implies that it must form a disk, suﬃciently close to the Black Hole8 . An accretion disk has at least two main virtues for the theorist: it allows the angular momentum to be evacuated outwards and/or upwards if the disk launches a jet while mass is accreted9 , and in the process explains how the energy gained by infall in the gravitational potential well is liberated as radiation. The weakest point of this picture is that the transfer requires anomalous (wave or turbulent) transport in the disk, as the eﬃciency of the process exceeds by many orders of magnitude the expected microscopic transport. The physical origin and properties of this transport are not yet fully understood, although the last 15 years have witnessed an important breakthrough on this issue (see [9, 10] for an introductory review). The disk physics in AGNs is badly constrained: it is still unclear whether disks are thin or thick in the direction perpendicular to the disk plane (“vertical”), whether angular momentum transport is primarily driven by 6 As m n ∼ M/R3 ∝ M −2 , only stellar mass Black Holes are very dense objects; note p S that space curvature changes the usual estimates of volumes by factors of order unity only. 7 Only gas, as opposed e.g. to stars, can collisionnally dissipate the kinetic energy of infall into radiation. 8 Although spherical or quasi-spherical accretion has been largely discussed in the literature, and is probably relevant far enough from the nucleus, the mere presence of jets indicates the breaking of spherical symmetry, anyway. 9 The maximum angular momentum density theoretically allowed in the central Black Hole is much smaller than that of the accreted gas.

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turbulent stresses or Maxwell stresses associated to a magnetic ﬁeld threading the disk, and so on. Thick disks radiate less eﬃciently than thin disks, in part because they must be optically thick to geometrically thicken in the ﬁrst place. As the physics is simpler to formulate in the latter case than in the former, a thin disk is often assumed, resulting in a keplerian rotation law10 [v(r) ∝ r1/2 ].

3.2.1.2

Young stellar objects

YSOs are observed in our Galaxy, i.e., at distances which are considerably smaller than AGNs (hundreds of parsecs instead of mega- or gigaparsecs). They are rather numerous in giant interstellar molecular complexes (tens of parsecs in size), such as the Taurus ones. The size of the “central engine” is roughly the same in AGNs and YSOs; consequently, because of their relative proximity, the latter are much better known from an observational point of view, and the various stages of star formation, from the initial collapse of a cloud core, to a young individual or multiple star, are rather well constrained. After the initial stage of cloud core collapse, taking at most 104 years, a protostar forms and evolves to a main sequence star in about 106 to 107 years. This factor of 10 of uncertainty is the object of a lively debate in the community, as it may rule out ambipolar diﬀusion as a mechanism of magnetic ﬂux expulsion in the collapse stage of cloud cores (see, e.g., [11, 12]). The accretion disk forms and dissipates on the same time-scale, as well as the jet. The overall accreted/ejected mass amounts to ∼ 1 M /10−1 M , for a solar mass star. Accretion disks around protostars or young stars, such as T Tauri stars, have initially been postulated to explain the infrared continuum excess observed in these objects (which in this picture would arise from the viscous heating and the subsequent black-body radiation, of a disk in nearly keplerian rotation). With the advent of high resolution imaging techniques, such as adaptative optics and infrared interferometry, outer disk regions are now directly seen in a number of instances, out to a few hundreds of astronomical units (see Fig. 3.12 for example). In a few instances, the velocity proﬁle has been determined from a Doppler eﬀect measurement at millimeter radio wavelengths, and was found to be compatible with a keplerian law ([13] p. 192). 10 For the disk to be thin, the internal pressure must be small compared to the gravitational attraction of the central object, which then dominates the motion.

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Magnetic ﬁelds in jets

The present evidence and constraints on the magnetic ﬁeld in astrophysical jets is rather indirect. In fact, one of the best arguments in favor of dynamically signiﬁcant magnetic ﬁelds in astrophysical jets is theoretical: a magnetic ﬁeld oﬀers the best way to create, accelerate and collimate the jet; if furthermore the jet originates in the accretion disk, then the magnetic stress can also contribute to the accretion ﬂow by extracting angular momentum from the disk and transferring it to the jet (see Sec. 3.3.3). Furthermore, purely hydrodynamic jets are prone to destructive KelvinHelmholtz instabilities (see Sec. 3.5). The idea of a dynamically signiﬁcant magnetic ﬁeld is supported by the available evidence: • In AGNs, the jet emission is linearly polarized. This indicates both that the jet is magnetized, and that the jet material is ionized. The inferred direction of the magnetic ﬁeld is either longitudinal (in compact sources) or transverse (mostly in extended radio sources) (see, e.g., [14]). The signiﬁcance of these directions measurements is diﬃcult to assess, as it is biased by the dominant emitting region. For example, the ﬁeld will become predominantly transverse in MHD shocks due to plasma compression, even if it is mostly longitudinal elsewhere. The evidence from radio sources is usually compatible either with a predominantly longitudinal ﬁeld in the jet core, and predominantly toroidal further out, or with a helical ﬁeld (e.g., [15, 16]). • The continuum of AGN jets is best interpreted in terms of synchrotron and inverse Compton emission. For a synchrotron source, at ﬁxed spectral emissivity, the total pressure of the plasma is minimized at equipartition between the gaseous and magnetic energy density (see, e.g., [3] p. 170-172). This minimal pressure exceeds the intergalactic pressure, in particular in FRII sources, as constrained by X ray observations of the hot and tenuous intergalactic gas. This implies in turn that these jets must be magnetically conﬁned, as free expansion is observationally excluded (ibid, p. 230-232). Magnetic conﬁnement implies that the gaseous pressure cannot exceed equipartition on average. • Polarization measurements in YSO jets prove the existence of a ∼ 1 G magnetic ﬁeld at distances of a few tens of astronomical units from the jet source in at least one case [17]. The temperature, density and pressure are also constrained from the analysis of line emission [1, 18]. Depending on the assumed magnetic ﬁeld conﬁguration, this results in

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Table 3.1 Estimates of the physical parameters relevant for AGN and YSO jets. Values are typical in YSOs, but refer to strong AGN jets. Quantities in parentheses belong to the ambient medium, when applicable. For the jet radius and jet speed, the range of values is indicative, not only of intrinsic uncertainty, but of the trend along the jet. An Alfv´ en speed of the order of the sound speed is usually assumed. The magnetic ﬁelds estimates are obtained with the same type of equipartition argument.

Jet radius Jet length Jet speed Vj Magnetic ﬁeld Sound speed

AGNs 10−2 – 103 pc 1 Mpc 10−3 c – c 1 – 10−4 (< 10−5 ) G 10−2 – 1 Vj (?) (10−3 c)

YSOs 30 – 300 AU 1 pc ∼ 300 km/s 10−4 (< 10−5 ) G 10 (0.2) km/s

badly constrained values of β = 2µo P/B 2 , which can be as low as 10−3 or as high as 103 . However, recent observations such as the one on Fig. 3.12 imply that jets are collimated very close to the source (down to 0.1 pc), while the external pressure is unlikely to play any role. As in the case of AGNs, this suggests that jets are self-collimated, which excludes values of β much in excess of unity. • Conversely, in YSOs, the ratio of the jet velocity to the jet sound speed is consistent with freely expanding jets. This however raises the question of the mechanism responsible for their initial acceleration, as purely hydrodynamical mechanisms lead to destructive Kelvin-Helmholtz type of instabilities ([19, 20]; see also Sec. 3.5). Furthermore, these jets are signiﬁcantly more extended in the transverse direction than the region which gives them birth. The size of these regions is comparable to the central object’s, as jets asymptotic velocities are comparable to the central object escape velocity. The most likely collimation mechanism consistent with this feature is magnetic in origin (see Sec. 3.4). Order of magnitude estimates of the most important physical parameters are summarized in Table 3.1. The quoted values are highly uncertain concerning pressure and magnetic ﬁelds in AGNs, as they result from the very rough equipartition argument exposed above. Note however that this argument is self-consistently checked in some synchrotron self-Compton (SSC) sources where the ratio of the synchrotron to the Compton emissions can be measured and constrains the magnitude of the magnetic ﬁeld, so that the gas pressure can then be derived from the synchrotron emissivity itself.

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In any case, the overall jet β parameter is probably not smaller than, say, 10−2 or 10−3 , and not larger than ∼ 1. Magnetic ﬁeld values in YSOs are also highly uncertain, as the eﬀect of the magnetic ﬁeld on line emission is more diﬃcult to detect, but temperatures and pressures are more stringently constrained from the analysis of the line thermal emission. A similar crude equipartition is usually assumed in YSO jets as well. The jet velocities are measured from the outﬂow speed in YSOs. It is more uncertain in AGNs, because the measurement is less direct: the absence of emission lines makes a Doppler measurement impossible; also the relation that the speed of the material producing the jet high emission possibly bears to the speed of the bulk of the jet is highly uncertain (see next section). The origin of the magnetic ﬁelds observed in jets and postulated in disks is unclear11 . They can either have been advected along in the initial collapse of the central object, or generated in the disk through some sort of dynamo process. Actually, as pointed out in the introduction, there is more than ample ambient ﬁeld to do that, and the ﬁeld ﬂux must be evacuated somehow during the infall and disk formation. If too much ﬂux is expelled out, a dynamo action must be invoked, and the problem relies not so much in explaining the magnitude of the ﬁeld than in understanding its conﬁguration. In models where the jet originates from the disk, a dipolar-like ﬁeld is usually assumed (see Fig. 3.14), threading the disk vertically at the disk mean-plane. Quadrupolar conﬁgurations, with the disk midplane as a plane of symmetry, have also been considered. Analytical investigations indicate that when the coupling of the jet to the disk is taken into account only dipolar conﬁgurations are capable of producing stationary jets, at least for keplerian accretion disks [21]. Either dipolar or quadrupolar geometries can be produced by a disk dynamo, depending on the sign of the α eﬀect. Both signs can apparently emerge from disk dynamo simulations, but simulations also seem to conﬁrm the analytic argument on the preferred dipolar conﬁguration for jet production (see e.g. [22] and references therein).

3.2.3

Relativistic vs non-relativistic jets in AGNs

Bulk Lorentz factors of 10 or larger can be inferred from the high energy emission of AGNs, and it has long been debated in the community whether this implied that the entire jet is relativistic, or only a more energetic but 11 The question of the origin of astrophysical magnetic ﬁelds is brieﬂy overviewed in Appendix A.

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highly sub-dominant population12 . The ﬁrst type of models is conceptually simpler, but the second one is energetically more favorable. Over the years, various lines of arguments have been put forward, favoring the second option over the more widely accepted ﬁrst one (note however the possible role of the light cylinder13 [23]): • Recent numerical simulations have shown a lack of cylindrical collimation in relativistic jets [24, 25]. This problem is well-known in the pulsar wind community (where Lorentz factors are usually rather large) and arises because the energy in the electric ﬁeld is always comparable to or even larger than the energy in the azimuthal magnetic ﬁeld (for some details, see e.g. [5]), counteracting the conﬁning action of the azimuthal magnetic ﬁeld tension14 . Parabolic conﬁnement is possible once relativistic jets have become large enough in radius [26, 27], but this type of conﬁnement may be diﬃcult to reconcile with the jet-like appearance of the emission regions, due to the relativistic focusing of the radiation in the forward direction. • The total emission observed in radio lobes is compatible with comparatively low (∼ 0.1) bulk Lorentz factors. • High bulk Lorentz factors can only be achieved if the ejection eﬃciency (the mass-loss rate in the outﬂow compared to the mass accretion rate in the disk) is extremely low, 10−5 , typically; this follows because the energy reservoir is the electromagnetic ﬁeld, and is ﬁnite, so that the asymptotic speed achieved decreases with increasing mass ejection, a qualitative consequence of Bernoulli’s invariant (see e.g. [26] for a more quantitative assessment). However, such low mass ejection eﬃciencies cannot be obtained for stationary jets if the jet originates in the disk [21, 28]. Of these arguments, probably the most critical, although not yet widely acknowledged, is the ﬁrst one. Taken together, they suggest that relativistic 12 It is also not clear that large bulk Lorentz factors are intrinsic, as they can be biased by simplifying assumptions made in the analysis of the data. 13 The issue is the following. Because jets widen substantially before collimation takes place, relativistic velocities can be achieved if the jet radius r reaches r = Ω/c. Then, the local neutrality assumption of MHD must fail, and a fully relativistic treatment is called for. On the other hand, the other terms in the generalized Ohm’s law must also become non-negligible, and the question has apparently not yet been convincingly addressed in the literature. 14 The electric ﬁeld contribution in the nonrelativistic regime is smaller by a factor (v/c)2 , as seen from the ideal MHD Ohm’s law E + v × B = 0.

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ejections are less important than previously anticipated for the dynamics of most of the ejected material, a point which is however still debated in the community. In such a context, the relativistic motions and emissions could be explained by the existence of a subdominant electron-positron developing and accelerating to relativistic velocities in the core of a more massive nonrelativistic, or mildly relativistic MHD jet, which would act as an energy reservoir for the whole structure, as sketched on Fig. 3.13. The MHD jet would provide the turbulence which is responsible for the continuous reheating of the pair plasma, while the radiation force from the disk can accelerate it to relativistic velocities (see [29, 30] and references therein). Alternatively, a Blandford-Znajek type of mechanism (described in Sec. 3.3.1) can produce the inner jet. In any case, the idea of a twocomponent model, of this type of another, is slowly attracting attention, due to the problems that one component models are facing (see, e.g. [31]).

3.3

Origin of jets

As mentioned in the introduction, jets can be produced either by the central object, the disk/central-object interaction region, or the disk itself. These three possibilities are brieﬂy discussed here.

3.3.1

Central object

YSO jets are usually not believed to originate from the protostar, as they carry too much momentum for this to be a viable option [32]; in particular, the large mass ejection rates observed in the very ﬁrst stages of protostellar formation (∼ 10−6 M /yr) exceed by several orders of magnitude the largest rates that can be produced by a stellar wind. This option is therefore not further discussed here. Conversely, the central object of an AGN is often believed to be a viable source of AGN jets. This may be somewhat surprising, because, as discussed in Sec. 3.2.1, this central object is believed to be a Black Hole, a fact requiring some explanation. Black Holes are described by only three quantities, as all other characteristics of the material which ended up in the Black Hole are lost in the Hole formation process: their mass M (or gravitational radius rg = GM/c2 ), their speciﬁc angular momentum a (< rg c), and their charge, which is

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usually assumed to be zero (presumably, Black Holes form from globally neutral material). The gravitational radius characterizes the Black Hole event horizon, i.e., the limit between the inner and outer space of the Black Hole (the inner space is curved and closed on itself). The speciﬁc angular momentum deﬁnes a region outside the horizon, called ergosphere, of maximum radius r+ = rg + (rg2 − a2 /c2 )1/2 . In the ergosphere, no object can maintain itself static with respect to an observer at inﬁnity. RL

ΩL

Magnetic flux surface (Φ)

Load electric resistance

Ω I (current) I (current) RH Hole electric resistance Black Hole

ΩH

( ΩH − Ω ) ∆Φ = I RH ( Ω − ΩL ) ∆Φ = I RL

Fig. 3.1 Sketch of a rotating Black Hole equivalent circuit in the Blandford-Znajek mechanism. The Hole rotation generates by induction an electromotive force between the pole and the equator. If the Hole and “astrophysical load” rotation velocities diﬀer, a current ﬂows, generating energy dissipation. This system is powered by the Hole rotation (after [3]). The energy sink may be a jet.

Although no energy and momentum can be extracted from a nonrotating Black Hole, this is not true for a rotating one, as ﬁrst demonstrated by Penrose. Indeed, inside the ergosphere, some particle orbits have a total negative energy so that the energy of the Hole decreases when such particles cross its horizon. This can lead to energy extraction, if e.g., a pair of particles is created out of vacuum ﬂuctuations in the Hole potential well, one with total negative energy and an orbit entering the Hole, and one with positive energy and an orbit escaping the Hole. Because the area of Black Holes cannot decrease (a theorem due to Hawking), there is a limit to energy extraction of a rotating Black Hole, which cannot exceed 29% of its total mass and binding energy (see [3] p. 194-200, and references therein for more details).

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In a famous paper [33], Blandford and Znajek have devised a more powerful way to extract energy and momentum from the rotation of a Black Hole, by transfer from the Hole to an electromagnetic ﬁeld anchored on the Hole horizon. The involved processes are rather complex, and are best grasped within the framework of the “membrane paradigm” [34], which establishes the idea that the Black Hole horizon acts as a conducting membrane of ﬁnite resistivity equal to the vacuum resistivity, and endowed with ﬁctitious charges and currents to account for the behavior of electric and magnetic ﬁelds in the vicinity of the horizon. Based on this idea, MacDonald and Thorne [35] have reformulated the Blandford and Znajek mechanism as a form of general relativistic direct current electric circuit problem. In this picture, a Hole rotating in a magnetic ﬁeld induces an electromotive force between the pole and the equator V ∼ ΩH rg2 B ∼ ΩH Φ (Φ is the magnetic ﬂux at the membrane). Energy extraction comes about due to the diﬀerence in angular velocity between the hole itself and some far away “astrophysical load”, as sketched on Fig. 3.1 (for a more complete heuristic description of this picture, see [3] p. 205-207). A pair plasma can be produced and accelerated into a jet in this context [36], but most of the energy remains in the electromagnetic ﬁeld. To date, the exact eﬃciency of the mechanism remains to be precisely quantiﬁed, due to the high level of complexity of the involved general relativistic apparatus, and the poor knowledge of the characteristics of the astrophysical load. A potential problem of principle has been raised by Punsly and Coroniti [37], who argued that the outgoing wind that might be driven in this picture cannot be in causal contact with the event horizon. This point has recently been shown to be incorrect [38, 39]. Also, it seems to appear that the process is eﬃcient only for AGNs radiating well below the Eddington limit of Eq. (1) unless the magnetic ﬁeld on the Hole horizon largely exceeds the maximum ﬁeld allowed in the accretion disk by stability considerations [5], and that in any case the power source from the gravitational binding energy exceeds that from the Black Hole rotation energy, at least for thin accretion disks [40]. 3.3.2

Disk/central-object interaction region

In the context of AGNs, the coupling between the disk and the central object (Black Hole) is formulated as a modiﬁcation of the Blandford-Znajek mechanism, in which the astrophysical load is partly or wholly constituted by the disk itself (e.g., [40, 41] and references therein). This line of work

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mostly emphasizes the eﬀect of this coupling on the radiation emitted by the AGN central engine, and virtually nothing is known on the eﬀect on a potential jet, beyond what has been said above in the previous subsection. The situation is somewhat clearer in the YSO context, where the physics does not reach such a high level of intricacy, and investigations have been started by various groups both semi-analytically (see [42–44] and references therein) and numerically (see [45] and references therein). These investigations are rather preliminary, and some important aspects of the involved physics are not yet understood. However, a few elements seem unavoidable: • The protostar magnetosphere plays an essential role in its interaction with its accretion disk. Protostars rotate at about 10% of their breakup velocity [6], and seem to maintain a constant period during their accretion phase. If the disk extended down to the protostar surface, it would rotate much faster than the star in the accretion layer, and would spin up the protostar. • Conversely, in presence of a magnetosphere, the disk may be truncated near the corotation radius, i.e., the radius at which the star rotation velocity matches the disk’s. Magnetic coupling with the material beyond corotation will slow down the star, and accretion could occur through magnetospheric columns, as suggested by recent observations [46]. In fact, this type of model was initially put forward to explain the low rotation of newly formed stars. • The interaction region must be narrow otherwise the magnetospheric ﬁeld lines inﬂate and diﬀuse out, because of the turbulent diﬀusivity whose existence is inferred from the accretion itself [47]. Two basic conﬁgurations have been explored in the literature, to match the disk ﬁeld to the magnetospheric one. They are sketched on Fig. 3.2. In one of them [42, 43], two azimuthally extended “Y-points” are present in the ﬁeld above and below the disk mid-plane, noted as n on the sketch. In this picture, the angular momentum of the star is deposited in the disk via the magnetic coupling of the magnetospheric ﬁeld. It is then presumably carried outwards by the disk turbulent viscosity, and then evacuated from the system by a conventional disk jet (see next section). In the second picture [44], the magnetospheric ﬁeld and the disk one reconnect through an azimuthally extended “X-point”, a jet forms from mass transferred from the disk to the newly opened ﬁeld lines, and is expelled by the centrifugal force of the rotating protostar. The spin down of the star is a priori more

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outflow accretion

X type magnetic configuration

Y type magnetic configuration

Fig. 3.2 Sketches of the two proposed conﬁgurations of protostar-disk magnetic interaction, presented as cuts in the (r, z) plane (axisymmetry is assumed). The relative sizes of the stars and the various distances are somewhat arbitrary. The conﬁguration of the right involves two “Y-type” neutral points, denoted by the letter n, while the conﬁguration on the left relies on the existence of an X reconnection point (adapted from [44] and [43]).

eﬃcient in this model but it requires reconnection to go on, possibly in a time-dependent fashion, throughout the life-span of the structure, thus implying that the star continuously replenishes its magnetic ﬂux, or that more and more stellar magnetospheric lines are open. 3.3.3

Disk

Although criticized in the literature (a point addressed right below), diskdriving is by far the most studied case. On the analytical front, original papers and some noteworthy developments include references [48–56, 21]. The constant progress of computers has also prompted numerical investigations of the problem in the last decade (see e.g., [57–61] and references therein). Disk-driving has been criticized on two diﬀerent fronts. First, it has been pointed out that in a standard disk model supplied with turbulent resistivity and viscosity, the ﬁeld lines are either not bent enough due to ﬂux diﬀusion [62], or the ﬂux is totally expelled [63]. However, the existing global disk-wind solutions involve an inﬂow velocity which is substantially larger than the standard model prediction [56, 21], with suﬃcient resultant ﬁeld line bending, thereby escaping escaping the self-consistency problem, but possibly not the formation one. Secondly, disk-jet models have been argued to be unstable [64, 65]; however, the global disk-wind solutions

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just mentioned possibly escape this problem as well, as brieﬂy discussed in Sec. 3.5.1. In any case, one reason behind the favor enjoyed by the disk-driving scenario is undoubtedly that, as complex as the physics of jet launching from accretion disks may be, it is signiﬁcantly simpler to analyze than the other two options previously discussed. Another reason (circumstantial evidence, rather), is that an accretion disk is the only physical ingredient common to all the systems where jets are observed ([66] and references therein). In fact, some care should be in order in the use of this argument, as what is shown by the observations of a wide variety of physical systems, besides YSOs and AGNs, is that: • All systems in which jets are observed have also direct or indirect signs of the presence of an accretion disk. • However, the reverse is not true. In particular, some cataclysmic variable (CV) systems (a particular kind of stellar binaries) have an accretion disk, but no jet. • All observed jets have velocities comparable to the central object escape velocity. The last point implies that if jets are produced by the disk and not by the central object, they originate from the disk inner region. As the central object does not clearly play a role in YSOs and in the other systems except those containing a Black Hole, such as AGNs or microquasars, the prevailing opinion (or bias) is that jets are produced by accretion disks. In any case, the absence of jet in CVs is an important open issue. The basic MHD processes which allow a disk to drive a jet are sketched15 on Figs. 3.14 and 3.15. An accretion disk is threaded by a magnetic ﬁeld, producing axisymmetric magnetic ﬂux surfaces. The rotation velocity provides for the vertical winding of the ﬁeld lines, and the self-consistently produced accretion ﬂow for the inward (at the disk plane) motion of the magnetic ﬂux. Turbulence in the disk is assumed to provide a turbulent diﬀusivity which allows the structure to reach steady state, and the matter in the disk ﬂows through the magnetic surfaces, both azimuthally and radially. In the process, the magnetic ﬁeld is made to rotate at a velocity slightly smaller than the rotation velocity of the matter where the magnetic surface crosses the disk midplane. 15 Beware that in more realistic conﬁgurations, the radius at which jet magnetic surfaces recollimate towards the axis is considerably larger than shown on Fig. 3.14.

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From an electrodynamic point of view, this diﬀerence of rotation velocity between the ﬁeld and the disk material induces an electromotive ﬁeld E ∼ uφ × Bz , and, correlatively, an outward radial current in the disk. This current induces in turn a torque on the disk, which acts against the disk rotation (Lenz’s law). This torque dominate any possible turbulent torque in the disk (see Appendix 3.B.2), and causes matter to fall onto the central object. Matter lying close enough to the disk midplane eventually accretes onto the central object. Matter lying somewhat farther away from the mid-plane gradually undergoes an uplifting force due to the disk vertical pressure gradient, until it ends up in an ideal MHD region. It is then entrained by the magnetic ﬁeld rotation, which provides an outward centrifugal force and feeds the ejection. This follows because the corresponding ﬁeld lines are anchored in the disk closer to the central object, and therefore rotate faster than the matter at the location of “loading” of the matter onto the ﬁeld line, i.e., at the transition between resistive and ideal MHD. Note that the eﬃciency of turbulent diﬀusion in the jet is not constrained. It is a necessary assumption of this whole picture that it can be neglected, at least in ﬁrst approximation. In the initial stage of the ejection process (close to the disk), the ﬁeld energy density dominates over the matter one. However, in the centrifugal acceleration process just described, more and more of the energy stored in the ﬁeld is transferred to the matter, until the inertia of the matter dominates. Magnetic ﬂux surfaces are then forced by the matter inertia to widen further; both the azimuthal and poloidal components of the ﬁeld decrease due to matter acceleration for the one, and magnetic ﬂux conservation for the other, but the ratio of Bθ /Bz increases. The process halts when the rotational inertia becomes negligible, and magnetic tension due to the azimuthal ﬁeld is dynamically signiﬁcant, and the jet recollimates (see next section for more details). Let us specify a bit more some of the aspects of the picture described above. Theorists who wish to construct steady-state solutions meet with three diﬃculties: i/ one needs some speciﬁcation for the turbulent transport coeﬃcients; ii/ the full disk/jet resultant problem is non-ideal in the disk, and ideal in the jet; iii/ the jet equations possess three critical surfaces associated to the Alfvenic, the slow and the fast magnetosonic waves, where regularity conditions need to be imposed for the looked-for steady-state solution to cross smoothly through these critical surfaces (see Fig. 3.16). Because of this, a vast majority of papers does not consider the coupled

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problem: either the disk is treated, or the jet. Notable exceptions are provided in the work of Sauty and coworkers, Ferreira and co-workers. Analyses concentrating on the jet dynamics directly assume ideal MHD, and treat the disk as a boundary condition. Even with these simplifying assumptions, only self-similar solutions have been amenable to a semi-analytic treatment (see next section for a more detailed discussion of jet models). Here, only the principal results directly connected to the disk-jet interaction are reported ([21, 28, 60] and references therein). For simplicity, the disk is assumed to be cold, i.e. radiatively eﬃcient; most of the literature is related to thin and cold disks in any case. In such disks, because all forces are small compared to the central body gravitational attraction (a point still debated out of the disk midplane), the rotation velocity Ω(r, z) is close to keplerian (Ω2K ∝ (r2 + z 2 )−3/2 ). It is useful to deﬁne a local vertical scale height h such that PT = ρΩ2K h2 at the disk midplane, where PT is the total (gaseous and magnetic) pressure. Cold disks are thin, as measured by ≡ h(r)/r, believed to be of the order of 10−2 to 10−1, from the available evidence in YSOs. The steady state equations used in disk-jet modelling are presented in Appendix 3.B. Writing u = Ωreφ + up , the toroidal component of the momentum equation yields an equation for angular momentum conservation

rBθ dΩ Bp − 2πρνv r2 ∇. ρΩr2 up − (3.2) er = 0, µo dr where νv is the turbulent viscosity used in the modelling of the turbulent Reynolds stress tensor. In the plane of the disk, the ﬁrst term is the advection of angular momentum due to the radial motion, the second is the Maxwell stress term, which describes the extraction of angular momentum from the disk by the magnetic structure of the jet, and the last one is the turbulent transport of angular momentum (directed outwards for most known processes [9]). Let us deﬁne Λ = |jet torque|/|turbulent torque|, which measures the ratio of the last two terms. Deﬁning a turbulent magp netic Reynolds number16 Rm = rup /νm , Eq. (3.2) implies p νm . (3.3) 1 + Λ ∼ Rm νv p /νv ∼ 1 is expected, it turns out that disks dominated by the As νm turbulent torque (Λ 1) require Rm ∼ 1 to launch a jet, with a large 16 The

p

t , α quantities νm , νm m and αSS are introduced in Appendix 3.B.

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t p turbulent resistivity anisotropy (νm /νm ∼ 1/α2SS −1 ). Such an anisotropy seems unrealistic to achieve; furthermore, the resulting jets are rather weak. As a consequence, this option is not favored. On the contrary, in disks dominated by the Maxwell stress, Rm ∼ Λ ∼ −1 is required for jet launching, t p with a reduced implied anisotropy, νm /νm ∼ 1/α2SS , which nevertheless requires αSS 1 to be realistic, a rather stringent constraint. Note that a large eﬃciency of the magnetic torque is in any case expected on general grounds (see Sec. 3.B.2), although “disk people” often assume the contrary. The turbulent resistivity in this case implies a signiﬁcant bending of the magnetic surfaces towards the disk plane, at the disk surface. Correlatively, the magnetic ﬁeld tends to compress the disk. Turning to the vertical component of the momentum equation, one ﬁnds, in the thin disk approximation, to leading order in

up .∇uz −Ω2K z +

1 ∂P 1 ∂Br2 + Bθ2 − . 2µo ρ ∂z ρ ∂z

(3.4)

This shows that the gas pressure gradient is the only force responsible for deﬂecting matter from the accretion ﬂow into an upward motion. Consequently, the heating of the disk surface controls in an large way the eﬃciency of the ejection. For cold disk surfaces, the ratio of the accreted to ejected mass is limited to 1%, while it can represent a signiﬁcant fraction of unity otherwise. It must be noted that ejection is possible from a disk only when a rough equipartition between the gaseous and magnetic energy holds at the disk midplane (β ∼ 1 at z = 0), and requires large values of the “magnetic Shakura-Sunyaev parameter” (αm 0.3). Although generally speaking αSS < αm , this last contraint is rather stringent. All this also shows that the vertical structure of the disk needs to be realistically represented, to obtain relevant constraints on the ejection; this is true even if one relaxes the stationarity assumption made here. This point is largely underestimated in the literature. Overall, the constraints to produce ejection from a disk are rather tight, a point which is sometimes taken against a disk origin for astrophysical jets. Finally, the disk-jet interaction also constrains the way the global poloidal current is closed. It has already been pointed out that the disk carries an outward radial current (it also carries an azimuthal current, obviously, but this is of no concern here). There are two possible ways to close the electric circuit. Either the vertical current ﬂows into the disk, and the return current ﬂows outside the disk-jet structure (in this case, the radial current increases outwards in the disk), or

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the vertical current ﬂows out from the disk, and return current ﬂows close to the jet axis (in this case, the radial current decreases outwards in the disk; this conﬁguration is represented on Fig. 3.3). Ejection is possible in both cases, but in the ﬁrst case, the jets are not able to cross the Alfv´enic critical surface, so that the second conﬁguration is favored. It is also more favorable from the point of view of jet collimation (see next section).

3.4

MHD jet structure

Having discussed jet origin, let us now turn to jet structure. For simplicity, only non-relativistic MHD will be used in this section and in the next one. Concerning YSOs, jet poloidal velocities are well below the velocity of light. The situation is less clear for toroidal velocities, if the jet widens a lot. It is usually assumed that relevant rotation velocities (i.e., for magnetic surfaces carrying most of the jet ﬂow) are also subrelativistic (but see [23] for a diﬀerent opinion). The situation is a more complex in AGNs. However, it has been argued in Sec. 3.2.3 that nonrelativistic jet models may exhibit a signiﬁcant part of the relevant physics. In any case, understanding their properties is a necessary ﬁrst step towards an understanding of relativistic MHD jet models. An extension of the description made here to a relativistic context can be found, e.g., in Ref. [5]. The equations of jet dynamics are reduced in Appendix 3.B.2 to a set of invariants on magnetic surfaces, and to the Bernoulli and generalized Grad-Shafranov equation. The invariants are χ(a), which follows from the continuity equation and measures the mass ﬂux (as it is issued from the continuity equation), Ω∗ (a), which can be interpreted as the magnetic ﬁeld line angular velocity, l(a), the total speciﬁc angular momentum, and K(a), the coeﬃcient in the polytropic equation of state. In these equations, only the Alfv´enic point explicitly appears as a critical point (through the poloidal Alfv´enic Mach number m) in the generalized Grad-Shafranov equation, but the other two critical points are hidden in the fact that E(a) (Bernoulli’s integral) must be positive for ejection to occur. The three critical points appear explicitly in the important case of self-similar solutions. As a matter of fact, the only known solutions to these equations are self-similar. Nearly all such solutions assume that z/r is the self-similar variable, so that a ﬂow quantity X ∝ rαX fX (z/r). A notable exception is the work of Sauty, Tsinganos and co-workers who assume a separation of variables of the form X = fX (r)gX (θ) in spherical coordinates r, θ, φ (see

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[67] and references therein). An important limitation of the self-similar approach is that neither the jet axis, nor the radial outer jet boundary, are accounted for. As such, self-similar jet solutions solely rely on the speciﬁcation of boundary conditions at the jet base. Only in the case of disk-driven jets are some of these boundary conditions constrained, but with still a certain degree of arbitrariness. This explains both why these jets are so much studied, and why there exists such a wide range of solutions in the literature. The most constrained ones are those in which the disk-jet coupling is explicitly taken into account in a full solution, as discussed in the previous section. Note also that most solutions cross only one critical surface (the Alfv´enic one), as they are started beyond the slow magnetosonic surface, and no regularity condition is imposed on the fast one. Here again, notable exceptions are the work of Sauty, Tsinganos and co-workers on the one hand, and Ferreira and co-workers on the other, which cross all three critical surfaces.

3.4.1

Jet launching and magnetic surface opening

An often quoted constraint on the magnetic surface opening needed to launch a jet has been derived by Blandford and Payne for cold jets [52]. It follows from the following consideration. With the help of Eq. (3.26), Eq. (3.27) can be rewritten E(a) − Ω∗ (a)l(a) = u2p /2−Ω2 (r2 +z 2 )/2−ΩΩ∗ (r2 +z 2 )+ΩΩ∗ z 2 . Note that the ideal MHD situation applies very close to the disk, and that for z 0, Ω∗ Ω and up 0. Assuming that matter is ejected along a magnetic surface from (r = r0 , z = 0) to (ro + δro , z), a Taylor expansion of the reexpression of Bernoulli’s integral given above yields u2p /2 Ho −H +Ω2o (3δro2 −z 2)/2 > 0, which translates into the condition

1/2 √ 2 Ho − H z < 3 1+ . tan θ = δro 3 Ω2o δro2

(3.5)

Thus, cold jets (H = Ho = 0) require that magnetic surfaces are bent by more than 30◦ with respect to the vertical axis at the disc surface. This limit keeps some relevance when the jet enthalpy is not neglected. Heavy mass injection in jet is hardly possible below it; reversely, eﬃcient ejection requires a bending suﬃciently in excess of 30◦ , but not arbitrarily large, other wise the magnetic pressure vertically compresses the disk, which limits the ejection or even totally suppresses it.

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Current structure, and jet acceleration and collimation

A complete understanding of the dynamics of steady-state jet structures can only be gained from an exact resolutions of the equations presented in Appendix 3.B.2. However, some features can be understood by direct inspection. MHD jets are dominated by the magnetic force. As such, it is useful to look at the expressions of the toroidal and poloidal components of this force. They read Bp ∇ I 2πr Bφ F = − ∇ I 2πr Bφ F⊥ = Bp Jφ − ∇⊥ I 2πr Fφ =

(3.6)

where ∇ = ∇·Bp /Bp , ∇⊥ ≡ (∇a·∇)/|∇a|, and I = 2πrBφ /µo is the total poloidal current ﬂowing inside the magnetic surface. Note that Bφ < 0, as a consequence of the disk rotation and of the outwards radial current in the disk. This shows that when the current results in acceleration of the ﬂuid along the magnetic surface (F > 0), it also results in azimuthal acceleration, i.e., in speciﬁc angular momentum increase. The total speciﬁc angular momentum invariant, Eq. (3.26) implies in turn that Bφ decreases along a magnetic surface. It turns out, as mentioned earlier that, in selfsimilar solutions at least, the magnetic ﬂux conservation implies that Bp decreases faster, so that Bφ eventually dominates over Bp . Furthermore, ρ decreases very fast along a magnetic surface, because the ﬂow velocity increases along the magnetic surface, and because the surfaces expand. Consequently, the variation of the density in fact dominates over that of Bφ , and the tension of bφ eventually dominates over the rotational inertia in Eq. (3.29). Similarly, past the Alfv´enic critical point, the gravitational potential term becomes negligible, and the transﬁeld equation reduces to

Bφ2 Bp2 B (3.7) −m2 − − ∇⊥ P + ∇⊥ r 0. µo R 2µo µo r The jet recollimates if the curvature of the magnetic surface R < 0. As the gas pressure decreases outwards, this requires that the total (pressure and tension) magnetic force is directed inwards. Equivalently, as F⊥ −(Bφ /2πr)∇⊥ I, |I| must increase outwards (as Bφ < 0, I < 0 by Ampere’s theorem). A general theorem about collimation in the asymp-

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Fig. 3.3 View of the poloidal current (dashed lines) and magnetic surfaces (solid lines) in a self-similar solution of the coupled disk-jet problem (adapted from [21]).

totic regime (inﬁnite distance from the source) has been demonstrated by Heyvaerts and Norman [68] from the set of steady state equations assembled in Appendix 3.B.2. This theorem states that the collimation of a given surface is asymptotically cylindrical if I = 0, and parabolic if I = 0. In both cases, the curvature term is asymptotically negligible in Eq. (3.7). Note however that as the electric circuit is closed, the parallel current density must change sign in the radial direction. This implies that at a given vertical distance z along the jet, not all magnetic surfaces will be suﬃciently collimated, a point (over)emphasized by Okamoto [69]. Note however that one must have j > 0 right above the accretion disk for super-Alfv´enic ejection to occur. Fig 3.3 shows how a disk-driven jet manages all these contraints. The solid lines represent magnetic surfaces, and the dashed line the poloidal current. The return current is carried by the jet core. For a given magnetic surface, a high degree of collimation is eventually reached, far enough from the disk, where the enclosed current I is suﬃciently negative. However, at a given altitude z, the inner magnetic surfaces are collimated, while the outer ones are still fairly open, as a consequence of the return current ﬂowing inside the structure, and not outside. If, in a real disk-driven jet, most of the ejection comes from the inner disk regions, as constrained by the observations, this implies that the jet would appear collimated, even though not all magnetic surfaces would be. Two connected issues concerning the jet asymptotic structure are usually not addressed in the type of model discussed here, mostly by lack of

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relevant constraint on the physical conditions prevailing in jets. They are both connected to the existence of turbulence and/or dissipation heating inside the jet, compared to radiation losses. For one thing, the simple polytropic gas law assumed in most jet models implies that the gas pressure becomes asymptotically negligible, as the density drops considerably during the expansion. However, jets are believed to be in rough equipartition (see Sec. 3.2), i.e. β ∼ 1. In fact, if one applies polytropic models to YSOs, where the physics is best constrained, even the minimum heating source known to be present (the ambipolar diffusion) easily leads to equipartition [70], implying that a more realistic energy equation needs to be included in the model for realistic predictions on this issue17 . However, the jet β could remain small if radiation losses are underestimated. One may also ask if jets can relax to a force-free conﬁguration while they propagate. K¨ onigl and Choudhuri [71] have transposed Taylor’s relaxation theory [72, 73] to MHD jets. In fusion devices known as Reversed Field Pinches (RFP), such a relaxation occurs because an m = 1 instability transforms the azimuthal ﬁeld Bφ in axial one Bz , and turbulent resistive diﬀusion and reconnection allows for the ﬁeld reversal to take place and be maintained through a form of non-kinematic turbulent dynamo (see, e.g. [74] chapters 4-5 and references therein). With our present state of knowledge, it is diﬃcult to assess whether any such relaxation mechanism can develop over the time-scale associated to the propagation of the jet, and the subject is not mature enough from an astrophysical point of view for such an attempt to be undertaken. It seems likely that the large velocity gradient due to the propagation in the ambient medium may alter such a mechanism in essential ways. Nevertheless, a relaxed force-free equilibrium is sometimes considered in studies of cold jet stability (see next section). More generally, the jet models adopted for the study of jet stability usually ignore most of the complexities discussed here. For simplicity, a purely cylindrical conﬁguration is generally assumed, with a total current which may or may not be vanishing. However, it is yet unclear how the stability properties of such structures relate to those of the jet models discussed in this section. The question of the origin of the knotty structure observed in AGN and YSO jets is not addressed here. Birkinshaw [75] reviews the possible role of the Kelvin-Helmholtz instability as a cause of these structures in AGNs. 17 The eﬀect could be mimicked within the present framework by an isothermal (Γ = 1) model, with substantial heating at the jet base, though.

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Similarly, the eﬀect of magnetically-driven instabilities in YSOs has been illustrated in the simulation of Todo [76].

3.5

Stability of MHD jets

The question of jet stability was raised as soon as jets were invoked to explain the power supply of extended radio galaxies, and before they were directly observed. Indeed, in their original paper, Blandford and Rees [77] have pointed out that in laboratory experiments, jets do not maintain their coherence over propagation lengths exceeding a few times their diameter, in clear distinction to observed astrophysical jets. As such jets are dominated by their bulk kinetic energy — a feature shared by both HD and MHD jet models — a major source of concern for theorists was to understand how large scale jets could survive the Kelvin-Helmholtz instability. It has become clear over the years that purely hydrodynamic jets would be too quickly destroyed by the instability by comparison to observed jets; see, e.g. the simulations of Bodo and co-workers [19, 20]. Conversely, other simulations have shown that a jet can propagate over distances much larger than its radius if it contains a signiﬁcant azimuthal ﬁeld, a somewhat unexpected result [78, 79]. As a matter of fact, the catastrophic disruption of purely hydrodynamic jet has been one of the incentives behind the development of MHD jet models, long before the three-dimensional simulations just mentioned could have been performed. In any case, the question of the stability of jets with respect to the Kelvin-Helmholtz instability largely dominates the astrophysical literature on jet stability; it is discussed in Sec. 3.5.2. The progressive construction of MHD jet models has raised other sources of concern relating to the stability of the required magnetic structure when the toroidal ﬁeld is largely dominant. For example, in the asymptotic conﬁnement and propagation domain of disk-driven jets, the toroidal ﬁeld (Bφ ) is expected to dominate by a factor of 10 to 100 over the axial (Bz ) one. In the fusion context, such conﬁgurations are well-known to be prone to disruptive MHD instabilities. This is often invoked in the astrophysical community as an argument against the viability of magnetically self-conﬁned jet models. Several important points are usually ignored in such statements. For one thing, the current structure is sensibly diﬀerent in jets and toroidal fusion devices (the current provides its own return current by construction in the latter). Furthermore, the velocity gradient (or more generally, the

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existence of large gradients) between the jet and the ambient medium, which is the cause of the Kelvin-Helmholtz instability, may also be a source of stabilization of the MHD instabilities, a feature which has attracted considerable attention in the fusion community in the last decade as a means to generate “transport barriers” and to increase fusion devices stability (see, e.g. [80] and references therein). Also, the assumed role of the kink instability is often oversimpliﬁed. Finally, recent simulations indicate that a helical magnetic ﬁeld helps to stabilize the Kelvin-Helmholtz modes in the nonlinear regime. The question of ideal MHD pressure- and current-driven instabilities in jets has unfortunately not yet attracted a lot of attention in the astrophysical community, because of the rather late developement of MHD jet models, and because of the long focus of the community on the question of the Kelvin-Helmholtz instability. What is known on this question is discussed in Sec. 3.5.3. Note that in a realistic jet equilibrium conﬁgurations, all sources of instability just mentioned may act at once. However, in many of the studies performed so far, they can still be individually identiﬁed, and the distinction is kept, as it provides a useful way to organize the literature. Finally, is is legitimate to ask whether the picture of stationary MHD background structure described in the previous section is meaningful. Indeed, the mere presence of knots along AGN and YSO jets as well as the existence of the high energy emission in AGN jets suggest that the structure is at least turbulent, and may be non-stationary in an essential way. Some aspects of this question have partially been addressed in the literature, in the context of disk-driven jets. This is discussed right below. For completeness, it is worth mentioning that the possible role of the magneto-rotational instability, now regarded a major source of turbulence in accretion disks, has recently been investigated in jets [81]. However, it is unlikely that this instability can lead to the jet disruption. The Kelvin-Helmholtz instability of jets has been extensively reviewed by Birkinshaw [75]. Consequently, only the major points will be collected here, as well as the advances performed since this earlier review. On the contrary, the sections on current and pressure-driven instabilities will be a bit more detailed, as this material has not yet been signiﬁcantly reviewed, and as some aspects of it are not known in the astrophysics community. Note that instabilities could be absolute (i.e., grow at ﬁxed space points) or convective (grow in the jet rest frame), a question which may obviously depend on the nature of the considered instability. Although the latter

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option is a priori favored, a temporal approach has generally been adopted in the literature as it is simpler to implement in numerical simulations; the present discussion of linear stability follows this choice for consistency with most of the literature, unless otherwise speciﬁed. 3.5.1

Global disk-jet stability and stationarity

The question partially addressed here is twofold: do disk instabilities induce instability of the whole accretion/ejection structure? Are jet structures essentially stationary or intrinsically time-varying? Concerning the ﬁrst point, it is unlikely that local disk instabilities can endanger the entire structure. For example, the most studied local disk instability (the magneto-rotational instability) leads to local turbulence, and may be the source of the required turbulent transport (see, e.g. [82] and references therein), although, as β ∼ 1 is required in disks for jet launching, the instability would operate close to its threshold. Also, global instabilities do not necessarily lead to disk-jet disruption, either. For example, the well-known Papaloizou and Pringle [83] instability, as all global instabilities, is rather sensitive to boundary conditions [84], and furthermore, its nonlinear development appears to produce large scale spiral waves instead [85]. Wardle and K¨ onigl [86] have argued that jet-driving disks should not be affected by the radial interchange instability of Spruit et al. [87]. It has been argued that wind-driving disks in which the magnetic torque is the dominant source of angular momentum transport may be inherently unstable [62, 64, 65], but this may either result from oversimplifying assumptions concerning the disk vertical structure or apply to structures which are not in the regimes identiﬁed in Sec. 3.3 for jet stationary launching to occur [88, 89, 21]. In particular, it must be noted that opening the magnetic surfaces at the disk surface does not necessarily produce an increase in the ejection eﬃciency, as often assumed in these stability studies. On the contrary, the increased vertical magnetic pressure gradient compresses the disk and results in a reduced ejection eﬃciency, and prevents runaway. Recently, numerical simulations which incorporate a realistic disk physics have been performed [90, 60], and show that accretion/ejection structures are stable over at least tens of dynamical times, although they may not be strictly stationary. These structures are found for disk eﬀective transport parameters which are less constrained than their stationary self-similar counterpart discussed in Sec. 3.3.3: the time-varying accretion-ejection structures are found for αSS 10−1 and isotropic turbulent resistivity.

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These simulations were performed in 2.5D cylndrical geometry, thus excluding the possibility of purely 3D modes of instability. Extension to 3D is therefore required before ﬁrm conclusions can be reached on this issue. Other indications that jets may not be stationary also come from simulations in which the disk is not resolved, but treated as a boundary condition, in much the same way as in most jet models. For example, Ouyed and Pudritz [91–93] ﬁnd a transition from steady ejection to episodic one when the mass loading of the jet, which is imposed as a boundary condition at the jet base, is decreased. However, Anderson et al. [94] reach an opposite conclusion. The diﬀerence between the two works is attributed by Anderson et al. to a diﬀerence in the treatment of the boundary conditions: the magnetic ﬁeld is imposed to be vertical at the jet base in the Ouyed and Pudritz simulations, while this constraint is released in Anderson et al. . In any case, the self-similar disk-jet solutions exist only in somewhat restricted mass-loading window [21, 28]. It is likely that non-stationary but long-lived jets exist outside it, but a deﬁnite answer on this issue must probably await the implementation of realistic back-reactions of the jet on the disk in numerical simulations, as well as full 3D calculations. 3.5.2

Motion driving: the Kelvin-Helmholtz instability

Because the bulk kinetic energy of a jet in the superfast asymptotic regime (discussed in Sec. 3.4.2) dominates the energy budget of the jet, the KelvinHelmholtz instability is certainly the most obvious source of potential disruption of a jet. The instability occurs when a free velocity shear layer is present in a ﬂuid, stratiﬁed or not. An extreme case is represented by the free interface between two ﬂuids moving at diﬀerent velocities (the two ﬂuids may be identical). The instability is illustrated on Fig. 3.4. A ripple in the velocity shear layer (inﬁnitely thin in this case) creates variations of velocity between peaks an troughs; correlatively (e.g., because of Bernoulli’s integral), such variations translate into pressure variations, and the resulting pressure gradient ampliﬁes the motion. In the most simple situation, all wavelengths are unstable (see Appendix 3.C). Another useful view on this instability stems from Rayleigh’s inﬂexion point theorem, which states that in a pure shear ﬂow (i.e., without stratiﬁcation and such), a necessary condition of instability is that the velocity proﬁle contains an inﬂexion point (see, e.g., [95] p. 131). Such an inﬂexion point is unavoidable when a free velocity shear layer is formed to match diﬀerent ﬂuid velocities. Fig. 3.4 provides an extreme example, where the shear layer has zero width (the so-called vortex sheet limit).

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U

+

z

Fig. 3.4

+

yv ar

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tio

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cit

x

-

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pr es su re va ria tio n

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Sketch of the mechanism of the Kelvin-Helmholtz instability (see text).

The Kelvin-Helmholtz linear stability has been studied for cylindrical jet conﬁgurations, by a number of authors (most notably for our purposes here: [96–103]; for a more extensive list of references, see [75]; see also [4] for a more recent but briefer survey of the subject). In this geometry, perturbations of the form exp(iωt−ikz−imφ) are looked for. The geometry as well as the inclusion of a ﬁnite thickness of the velocity shear layer, of relativistic motions, of longitudinal, azimuthal or helical ﬁeld, of density contrast between the jet and the ambient medium, all lead to considerable complexities in the form of the dispersion relation D(ω, k, m) = 0 with respect to the simplest case derived in Appendix 3.C. In all but the simplest cases, this dispersion relation is not analytic, but a systematic method of exploration of the stability domain is presented in Bodo et al. [102]. Because of this complexity, realistic cylindrical equilibrium conﬁgurations have rarely been studied, either in the linear or the nonlinear regime. The major features of all these investigations are summarized here. • Because of the geometry, two types of mode exist in MHD jets: ordinary surface modes, with amplitude which decrease steeply with the distance from the jet interface; these are the generalization to cylindrical geometry of the modes found in Appendix 3.C, and they are nodeless in the radial direction. And reﬂected body modes, which become resonantly unstable in cylindrical, or planar slab geometry, where the mode can reﬂect oﬀ the interface of the jet with the ambient medium. • In unmagnetized jets, the most unstable wavelength is λ ∼ 2πRM , with growth times τ πM × 2R/cs (M = Vj /cs is the sonic Mach number based on the jet velocity Vj ). Ordinary surface modes dominate for M < 2 — 3, and reﬂected body modes dominate otherwise; reﬂected modes are completely suppressed in low Mach number jets. When the velocity shear layer has a ﬁnite extent δ, wavelengths smaller than ∼ δ are stabilized. Higher density in the jet, as well as larger Mach numbers, result in enhanced stability (as inertia is increased).

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• An azimuthal magnetic ﬁeld is destabilizing while a longitudinal one is stabilizing. In particular, all modes are stabilized by a longitudinal ﬁeld in subalfvenic jets (Vj < vA ). • Jet rotation produces a stabilization of ordinary surface modes at all wavelengths at small Mach number, and at small wavelengths at large Mach numbers. • The eﬀects of jet heating and radiative cooling on the development of the Kelvin-Helmholtz instability have been studied by various authors [104–106]. Not surprisingly, the radiative losses can either enhance or reduce the instability, depending on the (not well-constrained) cooling functions. The interested reader is referred to these publications for details. The nonlinear evolution of the Kelvin-Helmholtz instability has been addressed through 3D numerical simulations in the last ten years, and some signiﬁcant ﬁndings have been made. It has already been noted earlier that purely hydrodynamic jets loose coherence due to this instability once they propagate over a few times their radial extent. The linear Kelvin-Helmholtz unstable modes tend to develop the so-called “cat’s eye” vortices, i.e., the interface ondulations roll over themselves a number of times, to form a vortex aligned with the ﬂow vorticity, before disruption takes place. The linear stability analysis suggests that the inclusion of a longitudinal magnetic ﬁeld component might signiﬁcantly change this picture, and this expectation has been conﬁrmed by the numerical simulations of Jones et al. [107] and Jeong et al. [108] at least for small enough fast-Mach numbers. These authors have studied a plane slab geometry (i.e., a conﬁguration analogous to Fig. 3.4, except that the ﬂuid is sheared at two diﬀerent interfaces), in which a longitudinal magnetic ﬁeld (along the ﬂuid motion) is added. They have maintained a jet sonic Mach number smaller than unity, and varied the jet Alfvenic Mach number. They found basically four regimes of nonlinear evolution: • When MA 2, the ﬂuid is essentially linearly stable, due to the magnetic tension. • When 2 MA 4, the perturbation produces a nonlinear increase in the magnetic tension which relaminarizes the ﬂow before the cat’s eye Kelvin-Helmholtz rolls brieﬂy described above have time to form. In the process, the velocity shear interface is widened (strong ﬁeld regime). • For 4 MA 20, a Kelvin-Helmholtz vortex forms. However, at

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the same time, the magnetic ﬁeld is sheared, the magnetic ﬁeld energy correlatively increased, and magnetic ﬁeld reversal takes place in the vortex. The subsequent evolution is dominated by the reconnection dynamics of the magnetic ﬁeld, which disrupts the vortex, but also leads to a nearly complete relaminarization of the ﬂow, with a signiﬁcantly widened velocity interface, and nearly aligned velocity streamlines and magnetic ﬁeld lines. This evolution avoids the disruption of the ﬂuid slab (weak ﬁeld regime). • Finally for large Mach numbers (MA 20), the ﬁeld plays very little role, except to enhance the dissipation of the Kelvin-Helmholtz vortices. These ﬁndings oﬀer the ﬁrst real window into an understanding of processes through which jets manage to avoid a catastrophic outcome because of the Kelvin-Helmholtz instability. However, they suﬀer from a number of drawbacks which need to be addressed in the future before a more complete picture of the problem can be obtained. First, the exact relevance of the temporal approach to the instability adopted in these simulations seems questionable at the light of the fact that jets are supersonic and probably superfast: new material continuously enters and leaves the jet, at speeds which greatly exceed the advection speed of the Kelvin-Helmholtz vortices at the jet interface. Secondly, the structure studied is far from being realistic: in the jet asymptotic regime, the azimuthal ﬁeld dominates largely over the longitudinal one, and it has been noted that in the linear regime, an azimuthal ﬁeld has a destabilizing inﬂuence. Although the jets outer region may not have reached this asymptotic regime, leading to a possibly more dynamically signiﬁcant longitudinal ﬁeld, such regions are not in asymptotic cyclindrical collimation either (see Sec. 3.4). Recent simulations implementing a more realistic structure have been performed [109]. As these simulations involve some interplay between current-driven and Kelvin-Helmholtz instabilities, their discussion is deferred to the end of Sec. 3.5.3.1. Finally, the fast Mach number plays an important role in the problem. Cat’s eye structures do not develop for fast Mach numbers 2. On the other hand, superfast jets are liable to global and eventually destructive instabilities when the velocity interface has a width comparable to the jet radius (Hubert Baty, private communication). Simulations with non-periodic boundary conditions along the jet are required to draw ﬁrmer conclusions on this issue, as periodicity implements a temporal approach

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to the instability, whereas a spatial one would be required, especially for superfast modes of instability. At present, these results suggest that jets reaching the fast magnetosonic critical point become unstable; this may be related, e.g., to the observed large scale braking of AGN jets and the formation of large radio lobes. 3.5.3

Magnetic driving: the current and pressure instabilities

In the asymptotic regime, the transﬁeld equilibrium equation for a given magnetic surface [Eq. (3.7)] reduces to

Bφ2 d P + B 2 /2µo − − 0, (3.8) dr µo r i.e., the jet is approximated as the familiar screw pinch18 of the fusion literature, with two important diﬀerences: at a given altitude z, not all magnetic surfaces may be in this asymptotic regime (at least for disk-driven jets) due to the fact that the jet carries its own return current, and that jets are not static columns. If furthermore, one assumes that |Bφ | |Bz |, which is supposed to hold over most of the jet radial extent, the resulting conﬁguration is close to a Z-pinch19 . The stability of these columns has been largely studied in the fusion literature. The relevant background concepts directly useful to this review are recalled in Appendix 3.D. Because of the adopted cylindrical geometry, the displacement with respect to the chosen static equilibrium can be chosen as a Fourier mode as in Sec. 3.5.2: ξ = ξo exp [i(ωt − mφ − kz z)] ,

(3.9)

where ξo is the Fourier transformed displacement vector, whose components ξr (r), ξφ (r), ξz (r) depend on radius only (the displacement and its Fourier transform are no longer distinguished in the rest of the text). Magnetic resonances play an important role in the stability of cylindrical columns. Magnetic resonances are deﬁned by k · B ≡ Bk = 0, and are radially located on surfaces where the wave number k = (m/r)eφ + kez is locally perpendicular to the unperturbed magnetic ﬁeld. In a sheared 18 A screw pinch is a cylindrical plasma column, with arbitrary magnetic shear s = d ln(rBz /Bφ )/d ln r. 19 A Z-pinch is a cylindrical column where the magnetic ﬁeld is only azimuthal. Note that with respect to the fusion labelling, the toroidal and poloidal components are exchanged between a screw pinch and a jet, as the symmetry axis is the z-axis in a jet, while it is the axis of the torus in fusion devices.

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magnetic ﬁeld, where s = d ln(rBz /Bφ )/d ln r = 0, such resonances occur on a given cylindrical magnetic surface, for any given wave-number20. High wave-number pressure-driven unstable modes are localized near the resonance. Usually current-driven unstable modes have a non-zero displacement inside the resonance radius, and a vanishing displacement outside it, but the mode may be broadened if inertial terms are important. Current-driven and pressure-driven instabilities are examined in turn in the next two subsections. 3.5.3.1

Current-driven instabilities

Generally speaking, the fastest growing current-driven instability is the kink (m = 1) mode. This is a global instability, and no general criterion is available. Quite often, a Kruskal-Shafranov type of criterion is used as a simple semi-quantitative guideline (for a derivation of the KruskalShafranov criterion, see ref.[110] pp. 342-343). The basic quantity entering such criteria is the safety factor21 q(r) = rBz (r)/Lbφ (r); L is the length of the column. Equivalently, two quantities relating to the safety factor are used in the literature: the pitch Pi = rBz /Bφ (2πP is the length of a ﬁeld line circling once around the axis at radius r), and the twist Φ = LBφ /(rBz ), which measures the amount be which a line is twisted over the length L of the column. The Kruskal-Shafranov criterion states that Φ > 2π at the column surface r = a is a condition of instability with respect to the kink (m = 1). This is in fact one example of such limits. For example, in an equilibrium with constant pitch, the limit is close to 4π (see, e.g., [111] pp. 259-264); also, higher order current-driven modes (m > 1) usually have a more stringent safety factor, but smaller growth rates. The Kruskal-Shafranov limit is not dependent on the vertical current density proﬁle jz (this is usually not true for higher order modes), and therefore indicates the existence of a generic and potentially very disruptive instability. Generally speaking, the twist limit varies by a factor of order unity, when varying the equilibrium conﬁguration. This type of criterion can also be understood as giving a minimum wavelength in the z direction, for instability to occur. In a suﬃciently long column, such as a jet, this type instability is bound to occur. The mechanism underlying the instability is often discussed in the lit20 Not all wavenumbers may be resonant in a given equilibrium; they can also be multiply resonant. 21 Note also that the magnetic shear is the logarithmic derivative of the safety factor.

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Fig. 3.5 Growth-rate of the most unstable current-driven mode, for various equilibrium conﬁgurations, as a function of the inverse relative pitch R/Po . In the “astrophysical regime” (R/Po 1), the growth-rate scales like the inverse relative pitch, independently of the details of the current structure. See text for detail. Adapted from [114].

erature (see, e.g., [110] p. 323): as the plasma column is kinked sideways, the azimuthal ﬁeld Bφ at the column surface is compressed on the concave side of the kinked column, while the contrary is true at the radially opposite convex side, inducing a magnetic pressure gradient which ampliﬁes the kinking motion. Another widely used picture is the analogy with a rubber band that is twisted upon itself; at some point, the increase in elastic tension energy will cause the band to kink and fold on itself. Based on this picture, it is often assumed that an MHD jet will kink and fold on itself, leading to disruption and/or decollimation (e.g. [112, 113]). This possibility has prompted a number of linear numerical analyzes of current-driven instabilities for a variety of simple cylindrical equilibria [115–118, 114]. The most signiﬁcant ﬁndings are those of Appl et al. [114], and are brieﬂy summarized and discussed here. These authors have discussed the stability of a variety of cold (P = 0) equilibria with respect to both current-driven and Kelvin-Helmholtz instabilities, for static columns (where the KH instability disappears) or moving jets, in the vortex sheet limit. For cold jets, the equilibrium relation Eq. (3.7) implies that the equilibrium is completely deﬁned once a relation between Bz and Bφ is speciﬁed. To this eﬀect, they choose to impose the radial variation of the pitch Pi . They have considered diﬀerent families of equilibria, with constant or quadratically varying pitch proﬁle. They have also considered a Bessel function model chosen to represent an energetically relaxed, force-free equilibrium. In what follows, Po refers to the value of the pitch on the axis.

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At a radius R (identiﬁed with the jet radius), the jet is interfaced with an unmagnetized outer medium of constant density and appropriately deﬁned gas pressure to maintain equilibrium. If Po /R is small enough, this external conﬁning pressure is small, and the jet model is essentially magnetically self-conﬁned. Furthermore, for r Po , Bφ Bz . Therefore, the ﬁrst two models share some important features with the self-similar jet structures discussed in Sec. 3.4 when Po /R 1. The last one is studied in case the jet relaxes as it propagates.

Fig. 3.6 Radial displacement of m = −1 mode, for constant (left) and increasing quadratic (right) pitch radial variation, and a relative pitch Po /R = 1/3. Various boundary conditions are used (rigid, or coupling with the appropriate radiating outgoing decaying solution). The mode is already essentially internal in spite of the still substantial relative pitch, and largely independent of the chosen boundary condition. The stabilization of the boundary increases with increasing Alfv´enic Mach number, deﬁned as the jet velocity over the Alfv´en speed evaluated on the jet axis. The radial ﬁeld perturbation is also shown; it vanishes at the mode resonance, if any. Adapted from [114].

For illustration purposes, the growth-rate of the most unstable currentdriven mode (usually an m = −1 mode; k is assumed positive) is shown on Fig. 3.5 as a function of R/Po , while a typical mode displacement is shown on Fig. 3.6, for the constant and quadratic pitch proﬁles. The most important results are the following: • For R/Po 1 the growth rate is dependent on the equilibrium and related current structure, while for R/Po 1 — 10, the growth-rate becomes largely independent of the equilibrium (except for the relaxed Bessel function model, which corresponds to an energy minimum at given magnetic helicity, and therefore has particular favorable stability properties).

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• The growth-rate is ∝ vA /Po for large enough R/Po (vA is the Alfv´en speed on the axis). It is substantially larger in these conﬁgurations compared to the equilibria which are usually considered in fusion, where instead R/Po 1 is rather chosen, precisely for this reason. It is comparable to the Kelvin-Helholtz growth-rate for these conﬁgurations (except for the relaxed model). This question has also been reinvestigated by Baty [119], who ﬁnds that the current-driven instability growth-rate dominates over the Kelvin-Helmholtz one for R/Po 10, for a variety of equilibria. • The mode is internal, independently of the assumed boundary condition. This is related to the fact that the parallel current peaks around r ∼ Po when Po /R is small. • The velocity gradient present at the jet surface due to the jet global motion with respect to the ambient medium has a stabilizing eﬀect on the jet boundary. This feature reinforces the internal character of the modes with increasing Alf´enic Mach number MA = V /vA , where V and vA are evaluated on the jet axis. For MA 3 – 4, the boundary eﬀectively becomes rigid. The internal character of the modes of instability is the major reason why jets, although not stable with respect to current instabilities, are not expected to disrupt. This expectation is borne out in the nonlinear evolution of these instabilities, as computed by Lery et al. [120]. These authors have followed some of the unstable equilibria discussed above in the nonlinear stabilization phase. They conﬁned themselves to static equilibria, to focus on pure current-driven instabilities, and with the idea that the jet superalfv´enic motion is stabilizing, as discussed above. They did indeed ﬁnd that the jet does not disrupt, but reorganizes itself internally. This reorganization leads to the development of some internal helical structure, with a prominent, conﬁned helical current sheet for equilibria having a radially increasing pitch. The time-scale of the process is consistent with the results of the linear analysis. Internal reorganization is also observed for relaxing unstable equilibria; magnetic reconnection plays an important role in the process, as it allows a kink unstable magnetic ﬂux tube buckling on itself to change its structure to a more stable state. However, the process of relaxation, if present, could not be followed in these simulations. The obtained results are consistent with earlier simulations of the kink instability which were performed to understand if the kink instability could account for the knot structure observed in YSO jets [76].

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Baty and Keppens [109] have considered the stability of cylindrical moving jets, with an increasing pitch proﬁle and a constant axial ﬁeld (the velocity shear is conﬁned to a small layer at the jet interface), which are liable to both Kelvin-Helmholtz and current-driven instabilities. They found that in the “weak” longitudinal ﬁeld regime discussed in Sec. 3.5.2, the development of a current-driven instability in the jet core, as discussed above, leads to the interaction of the resulting azimuthal ﬁeld distribution and the K-H nonlinear vortices, which helps to saturate the latter, leading to reduced disruption eﬀects. Increased Mach number, as well as allowing for vortex coalescence, further contributes to this mechanism. These results are still preliminary, and only a few equilibrium structures have been tested, but this certainly constitutes an important step towards an understanding of the processes which allow jets to avoid disruption for astrophysically relevant magnetic structures. 3.5.3.2

Pressure-driven instabilities

Because jets are believed to be close to equipartition between the gaseous and magnetic energy, and because the azimuthal ﬁeld is believed to dominate over the axial one, it would appear that, in fact, pressure-driven instabilities can be potentially much more critical from the point of view of jet disruption than the previously discussed current-driven ones. This point has unfortunately largely gone unnoticed in the astrophysical community, where only few papers have been devoted to this question, most notably Begelmann [121]. Here again, the analysis of the stability properties of cylindrical columns (or pinches) performed in the fusion context provides useful guidelines. From the Energy Principle and Eq. (3.44), it appears that the pressure term is destabilizing when the pressure gradient and the ﬁeld line curvature are directed in the same half-space. More precisely, for cylindrical static equilibria, they are aligned, and a necessary condition of instability is Kc Kρ > 0,

(3.10)

where Kc ≡ |e · ∇e | = (Bφ /B) /r is the ﬁeld line curvature, and Kρ ≡ −d ln ρ/dr is the density length scale, directly related to the pressure lengthscale by the assumed polytropic equation of state. Eq. (3.44) also implies that, in order of magnitude, the contribution of the pressure term to ω 2 is ∼ −c2s Kc Kρ (cs is the sound speed). In the absence of magnetic shear, and in an inﬁnite cylinder, one can always ﬁnd modes that will be destabilized. One can partially understand 2

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this result by considering the dispersion relation of Alfv´en and slow magnetosonic waves in an homogeneous medium in the quasi-perpendicular 2 2 2 propagation limit, ω 2 = vA k and ω 2 = c2s k2 /(c2s + vA ), where k is the component of the total wavenumber along the unperturbed magnetic ﬁeld. This suggests that instability will set in in inhomogeneous media, for small enough parallel wavenumbers, when this stabilizing contribution to the dispersion relation becomes smaller than the destabilizing eﬀect due to the plasma pressure and ﬁeld line curvature ∼ −c2s Kc Kρ . The magnetic shear s = −d ln Pi /d ln r (Pi = rBz /Bφ is the pitch) has an important eﬀect on this simple picture. When the magnetic ﬁeld is sheared, unstable modes are conﬁned to the vicinity of magnetic resonances, where k = 0, as the shear makes k2 increase with radial distance to the resonance. Furthermore, a large enough magnetic shear can stabilize radially localized modes. Indeed, Suydam’s criterion states that the magnetic shear stabilizes the plasma if Bz2 2 dP s + > 0. 8µo r dr

(3.11)

This criterion is a necessary condition of instability for all modes (see e.g. [110] section 9.4 for a derivation). It is a necessary and suﬃcient condition of stability for high wavenumber modes [122, 123]. Two other criteria have been derived by Kadomtsev for the stability of the Z-pinch (see [110] section 9.3 for a demonstration). They read 1 d ln |Bφ | < m2 − 1, d ln r 2

(3.12)

γβ/2 − 1 d ln |Bφ | < , d ln r γβ/2 + 1

(3.13)

for m = 0 modes, and

for the m = 0 mode; β = 2µo P/B 2 as usual. Both criteria are diﬃcult to meet in practice. The physics of this type of instability is rather involved, and its physical origin diﬃcult to grasp. For a Z-pinch in the large wavelength, quasiperpendicular limit, perturbations are pressureless (δP∗ = 0), and the origin of both the destabilizing term and stabilizing one is to be found in the variation of the magnetic tension. For example, in a ﬂux tube exchange, as one would operate in a typical interchange reasoning, a direct computation of the magnetic tension variation for the m = 0 mode shows that the destabilizing term must not exceed the contribution of the plasma compression

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for the plasma to be stable with respect to this type of disturbance. Once the equilibrium relation is used, Eq. (3.13) is recovered. One would think at ﬁrst glance that Kadomtsev criteria should provide guidelines to understand the stability behavior of cylindrical equilibria in which the toroidal ﬁeld dominates over the axial one, as in jets. In fact, this is not quite true, as pointed out by Begelman [121]. As this result is directly relevant to the present discussion, but has only been derived in the relativistic framework in the published literature, an alternative derivation based on material more familiar in the fusion context is outlined in Appendix 3.E. Eqs. (3.55) and (3.56) imply that low m modes can always be made unstable once Eq. (3.10) is satisﬁed, by choosing the resonant k for m = 0, and for an appropriate range of k for m = 0. This is at variance with Kadomtsev criteria: even a negligibly small Bz lifts the degeneracy which imposed a special treatment of the m = 0 mode; as a consequence, this mode cannot be made stable by compressibility eﬀects as in Z-pinches, even if |Bφ | |Bz |. The most unstable wavenumbers are always close to k2 γβKc Kρ /4, in which case the growth rates are always ∼ γβKc Kρ /4 ∼ 2 /R2 for β ∼ 1. These are very large growth-rates, pertaining to large vA scale modes, thus making the question of stabilization of the pressure-driven modes in MHD jets rather acute. The most obvious question to ask in this context is the role of the axial ﬂow of the jet on the stability properties of pressure-driven modes. This problem has not yet been investigated in the astrophysics literature, but some of its important aspects have been addressed in the fusion literature, and are brieﬂy discussed here, as they are directly relevant to the question of jet stability. In all the investigations cited below, the adopted velocity proﬁle contains no inﬂexion point, in order to avoid triggering the KelvinHelmhotz instability (see Sec. 3.5.2). Bondeson et al. [124] have generalized Suydam’s analysis to a sheared ﬂow. They provide the appropriate generalization of the linearized momentum equation Eq. (3.37), which is not reproduced here, as it is rather complex and not needed for the purpose of this simpliﬁed discussion. Focusing attention on axial ﬂows (v = vz (r)er ), their analysis shows that the behavior of localized modes depends on the magnitude of M ≡ ρ1/2

vz , Pi Bz /Pi

(3.14)

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where the prime denotes radial derivative, and Pi is the pitch introduced for the discussion of current-driven instabilities. This quantity is a form of Alfv´enic Mach number based on the velocity and magnetic shear. When M 2 < β, the ﬂow shear destabilizes Suydam modes (i.e., modes such that k · B = 0). Above this limit, Suydam modes are stable, but in this case, unstable modes exist at the edge of the slow continuum, and may be global. The authors found however that the growth rates are small (comparable to resistive instabilities growth rates). Note also that, as Pi /Pi ∼ 1/r, M ∼ (Bφ /Bz )(r/d)(vz /vA ) 1 in MHD jets (d is the width of the velocity layer). These results seem to suggest that the velocity shear layer at the jet boundary is substantially linearly stabilized in MHD jets. This seems to be conﬁrmed by global linear stability analyzes. Shumlak and Hartman [125] and Arber and Howell [126] have examined the stability analysis of a variety of equilibria in a pure Z-pinch (Bz = 0), with a smooth axial velocity proﬁle (similar analyzes of the ballooning mode stabilization have also been performed; see, e.g. [127–129] and references therein). The m = 1 mode is stabilized for modest velocity shear in equilibria for which the m = 0 mode is marginally stable in the static limit. This is not the case in more general equilibria (even though the m = 0 mode is apparently easily stabilized by the velocity shear), but in all cases, increasing the ﬂow Mach number produces a vanishing displacement of the unstable modes at the plasma boundary. An sheared azimuthal velocity also tends to stabilize the kink mode [130]. Nonlinear stabilization of pressure-driven modes has been studied by Hassam [131] and DeSouza-Machado et al. [132]. Hassam exploits an analogy between the m = 0 pressure-driven interchange mode and the Rayleigh-Taylor instability in an appropriately chosen magnetized plasma conﬁguration: the magnetic ﬁeld is perpendicular both to the velocity and the direction of the density gradient responsible for the instability. From this analysis, he concludes that the m = 0 pressure-driven mode is nonlinearly stabilized by a smooth velocity shear (dVz /dr ∼ V /R) if Ms = Vz /cs [ln(τd /τg )]1/2 , where τg is the instability growth time-scale (τg ∼ cs (Kρ Kc )1/2 ) and τd the diﬀusion time-scale (τd ∼ νKρ Kc where ν is the viscosity, assumed comparable to the resistivity). The nonlinear evolution of an unstable, slightly viscous and resistive Z-pinch, was simulated by DeSouza-Machado et al. [132]. They found that the plasma relaminarizes over almost all its volume for applied velocity shears that are in good agreement with this analytic estimate. The core of the plasma

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still has some residual unstable “wobble”, which can apparently be stabilized if by the magnetic shear if some longitudinal ﬁeld Bz is added to the conﬁguration. This nonlinear stabilization mechanism of Hassam [131] apparently hinges on the existence of a small, but nonvanishing dissipation. Even assuming that the eﬀective Reynolds number22 R = τd /τg ∼ 1030 , one ﬁnds [ln R]1/2 ∼ 8. The resulting Mach number is within what can be reasonably expected for astrophysical jets. On the other hand, the extent to which the result of Hassam [131] depends on the form of the nonlinearity involved in the solution, and is applicable to pressure-driven instabilities, is unclear. Consequently, the relevance of this result to the simulation of DeSouza-Machado [132] remains to be ascertained. 3.6

Summary and open issues

In guise of conclusions, I would like to stress here a few points which I believe to be now well-established, although some of them may not yet be widely recognized, and to point out some of their consequences in terms of open problems. This list may reﬂect my own interests in the problem, though. It appears that, with respect to all the issues addressed here (jet production, collimation, acceleration and stability), MHD as opposed to HD is an unavoidable framework. It seems that all the elements are now gathered to begin understanding the ability of jets to avoid disruption, and the presence of a magnetic ﬁeld in the jet seems an essential factor in this problem. On the other hand, it may also turn out that some of the stability problems discussed in these notes may eventually become irrelevant, because of the very crude assumptions made on the badly known jet structure, and on its interface with the ambient medium. Of particular importance is the question of the true current distribution, and its connection to the jet structure, a problem somewhat ignored in the literature. In any case, although the necessary ingredients allowing the jet to avoid disruption seem to have been identiﬁed, it is fair to say that the global physics of the problem is not yet understood. Developing such an understanding is certainly an important issue. 22 This estimate comes from the assumption that in AGN jets, plasma microturbulence provides a minimum diﬀusion, which involves velocities of the order of a fraction of the Alfv´ en velocity, and typical sizes comparable to the ions Larmor radii which are of the order of 10−15 times the jet radius. In YSO jets, the eﬀective diﬀusion, collisional or other, is expected to be much larger and the constraint on the Mach number weaker.

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In what concerns AGN jets, the issue of the conﬁnement of relativistic jets (discussed in Sec. 3.2.3) has apparently not yet reached a wide audience in the astrophysics community. Nevertheless, this problem seems to make two-components models, such as the one brieﬂy alluded to in Sec. 3.4.2, an almost unavoidable construct to explain both jet collimation and the high energy emission of AGNs. On a related front, the question of the light cylinder needs to be further clariﬁed. The various options through which a jet may be launched seem to be more or less known by now. However, none of the scenarios discussed in Sec. 3.3 is understood in full detail yet. Also, the Blandford-Znajek mechanism may require some modiﬁcation at the light of the conﬁnement problem just recalled. The disk-central object scenarios are still in their infancy. Even the best understood launching mechanism (disk-driving) suﬀers from a number of weak points. Indeed, although a signiﬁcant number of studies of disk-driven jet structure have been performed, all of them are formulated in a self-similar framework, and most of them treat the disk as a boundary condition. However, it appears that a careful treatment of the disk-jet coupling is required to get realistic jet ejection eﬃciencies, and realistic stability properties of the disk-jet structure. Furthermore, it appears that stationary accretion-ejection structures require rather contrived parameters to exist: extremely eﬃcient turbulent transport in the disk, anisotropy in this transport, and near equipartition between gas and magnetic pressure. The evidence from numerical simulations is that non-stationary accretionejection structures do most probably exist, and appear to be long-lived at least on the time-scales accessible to numerical computation, i.e., up to several tens of dynamical time-scales. This certainly questions the true relevance of stationary structures in spite of their obvious usefulness to deﬁne parameter regimes in which accretion-ejection structures can be found in numerical simulations, especially that observed jets do not appear to be stationary. This last point is also related to the question of the observed jet variability. Quite obviously, such a variability can either be the result of an instability in the jet23 , or of a variability at the jet source. Both options have been explored in the literature. However, the recent developments reported here on the nonlinear behavior of the most important dynamical instabilities probably calls for a reexamination of this issue. 23 In

this respect, there are instabilities which have not been discussed in these notes, in particular in relation to radiation processes rather than dynamical ones, such as the well-known ﬁlamentary instability; see, e.g. [4] pp. 568-569, and references therein.

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Appendices 3.A

A brief compendium of relevant astrophysical facts

A few properties of stars, galaxies, AGNs, and astrophysical magnetic ﬁelds, are brieﬂy summarized here. More details can be found in any lecture book on general astrophysics.

3.A.1

Mostly AGNs

Galaxies are ensembles of ∼ 1011−12 stars, with a varying amount of mass in gaseous form (from basically none for ellipticals to about 10% for spirals), and of sizes ∼ 10 kpc. Typical intergalactic distances are the order of a few megaparsecs (Mpc) in clusters of galaxies. Most galaxies are grouped in such clusters, of a few tens to a few hundreds of galaxies. The gas observed in spirals is clumpy, with a hierarchy of structures at almost every possible scale from the galactic one down to a parsec or smaller, the smallest structures usually being the densest and coldest ones (except star forming cores). Present day star formation proceeds from the gravitational collapse of a parsec scale (or less), cold, magnetized, cloud core of interstellar matter of a few tens of solar masses at most, a solar mass or so being the typical value. Newly forming or formed stars are grouped under the general denomination of Young Stellar Objects (YSOs). It is now known that accretion disks and outﬂows are always present in the early stages of star formation. Only a small fraction of galaxies, of the order of a few percents, contain extremely active central regions, baptized “Active Galactic Nuclei” (AGNs in short). These regions are extremely compact, as indicated by the very short time-scales of variability of the radiation output observed in a number of cases, but nevertheless produce a power exceeding by a factor of ∼ 103 the output of a whole normal galaxy. The most active have a power output ∼ 1046 erg/s, i.e., several solar masses per year, assuming all the mass is converted into energy. The activity of these galaxies signals itself in two diﬀerent forms: radiation from the region surrounding the nucleus, and, when present, large scale jets. The radiation output is reprocessed many times, to larger and larger scales, and correlatively larger and larger wavelengths. This emission comes out in wide variety of spectral form, intensity, variability etc, which has led to a fairly complex classiﬁcation of these object. Understanding the relation between these various objects,

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and possibly uniting them in a single framework, is a major open problem in this ﬁeld. Jets were initially mostly observed in the radio wavelengths; their progenitors were divided into extended or compact radio sources, depending on whether their radio emission was produced within one kiloparsec from the nucleus. It is now apparent that both types are diﬀerent aspects of the same underlying phenomenon. Extended radio sources are also subdivided into FR1 and FR2 objects (after Faranoﬀ and Riley), with the ﬁrst ones being the most energetic. Apparent super-luminal motions are often observed in compact sources. This proper motion is interpreted as a special relativistic eﬀect: i/ relativistically moving sources emit in a small cone in the forward direction, and can consequently only be observed if the jet is nearly aligned with the observer; ii/ for such small subtended angles, the relativistic Doppler eﬀect results in an apparent super-luminal motion on the velocity component projected on the sky plane. Most compact radio sources are associated with quasars, and the remainder with a nearby class of objects, known as BL Lac. A new category of objects, named blazars, has been more recently deﬁned, based on their high variability. They include BL Lac and the most variable quasars. Their properties are thought to emerge when the emission of the jet dominates the emission from the central object. Most AGNs have a rather extended spectrum. For example, the blazar spectrum can extend up to TeV energies in some cases. The list of observational characteristics of these objects can become fairly involved, although they are largely correlated in a given class. It is believed that at least some, perhaps most, of this classiﬁcation arises from orientation eﬀect of the source with respect to the observer. A simpliﬁed diagram is presented in Fig. 3.17. For more details see, e.g., references [133, 134] and references therein. 3.A.2

Origin of magnetic ﬁelds

The key role played by magnetic ﬁelds in various aspects of astrophysical phenomena has only relatively recently been recognized by the astrophysical community, through a slow evolution which took place over the last two or three decades. For the problem at hand, MHD phenomena are unescapable in the formation, acceleration, and conﬁnement of jets, as well as on some essential aspects of disk physics. As magnetic ﬁelds are not believed to be primordially present in the Universe, one is faced with the task of explaining their origin.

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Fields are most probably not generated in the early Universe with their presently observed magnitude, as they would strongly aﬀect the now well established mechanisms of structure formation24 . Therefore, they are thought to arise as rather weak seed ﬁelds, which are subsequently ampliﬁed after structure formation has started, either through elementary mechanisms of ﬂux concentration and/or streamline shearing, or through dynamo processes (see e.g. [135] for a recent review). At present two categories of mechanisms have been proposed: either quantum ﬁeld processes taking place during inﬂation or phase transitions in the primordial Universe; and charge separation eﬀects (battery eﬀects) in the “late” Universe. The ﬁrst ones have extremely varying and usually extremely weak eﬃciencies, from 10−65 to 10−9 G, and they are all compatible with the probably very conservative upper bound of ∼ 10−8 G recently deduced from the analysis of the CMB ﬂuctuations [136]. In any case, a nanogauss upper limit is required at the recombination epoch if one does not want to aﬀect our present understanding of large scale structure formation. The late Universe mechanisms usually produce amplitudes ∼ 10−20 G, to the notable exception of the most recently proposed one [137], which reaches magnitudes of ∼ 10−5 G from anisotropic and inhomogeneous radiation pressure eﬀects at the epoch of reionization of the Universe25. This process seems to provide magnetic ﬁelds with the right coherence scale and at the right magnitude for astrophysical purposes. 3.A.3

On the MHD approximation in astrophysics

Discussing in detail the validity of the MHD approximation in astrophysics in general, or even only in disks and jets, would take us too far aﬁeld. I will only brieﬂy discuss the various MHD approximations, i.e., electro-neutrality, the one ﬂuid approximation, and Ohm’s law, in 24 For a cosmologist, structures are everything from the largest structures observed in the Universe, such as low density contrast “walls” made of galaxies, down to clusters of galaxies and galaxies themselves. They are explained through the gravitational collapse of small primordial inhomogeneities in the matter and radiation of the Universe. These inhomogeneities are themselves directly observed in the temperature ﬂuctuations of the cosmic microwave background (CMB), and thought to originate from quantum ﬂuctuations in the primordial inﬂationary Universe. 25 It is now well established that the Universe, which was fully ionized in its ﬁrst few 105 yrs of existence after the quark-baryon phase transition and until the recombination which produced the CMB, has been reionized by the radiation emitted by the ﬁrst generations of objects formed, without recombination since then due to the decreasing density implied by the cosmic expansion.

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the context of astrophysical media. For more extensive discussions of the MHD approximation, see, e.g., [138] pp. 290-294, or [139] pp. 92-95. For an introduction to MHD with astrophysical applications in mind, see Ref. [138, 111]. Generally speaking, any relevant astrophysical scale L is always gigantic compared to the Debye length λD of the ﬂuid, which is typically (way) smaller than centimeter scales, so that the local quasi-neutrality assumption is always satisﬁed to a high degree of precision in nonrelativistic plasmas. Relatedly, the plasma parameter (1/ne λ3D ) is small (although orders of magnitude larger than λD /L). Furthermore, astrophysical plasmas are often characterized by values of β ∼ 1 within an order of magnitude or so, so that all relevant velocities, except drift velocities, are of the order of the Alfv´en velocity vA = Bo2 /(µo mi no ). Requiring that dynamical time-scales exceed −1 = mi /eBo of cyclotron motions of the ions (usually the the time-scale ωci most directly relevant one capable of distinguishing ion and electron motions) is equivalent to requiring that astrophysical scales L > ro = vA /ωci , where ro is of the order of the Larmor radius of the ions. Correlatively, the drift velocity |ue − ui | ∼ Jo /(no e) ∼ Bo /(µo Lno e) ∼ vA ro /L. In general, again because astrophysical scales are so huge, L ro , and the one ﬂuid approximation is well-satisﬁed. Similarly, for typical dynamical time-scales T ∼ L/vA , ωci T ∼ L/ro (vA L/T )2 ∼ 1, and ωci T ∼ L/ro (cs L/T )2 ∼ 1, where cs is the sound speed; this implies that the Hall term and the electronic pressure term are negligible in the generalized Ohm’s law, which reduces to E + v × B = ηJ. Mean free paths are usually signiﬁcantly larger than the ion Larmor radius, but nevertheless typically still many orders of magnitude smaller than relevant dynamical scales, so that magnetic Reynolds numbers are enormous, and that perfect MHD holds to a very good approximation as well. Nevertheless, the effect of small scale turbulence on larger scales is often modelled in terms of turbulent viscosities and resistivities in astrophysics, so that an eﬀective, possibly anisotropic resistivity is retained, e.g., in the description of disk dynamics, with all the caveats involved with this type of approximations. Jet instabilities usually have growth rates which can be signiﬁcantly shorter than the collision time-scale, making them eﬀectively collisionless. No step has been taken yet, though, to take this point into account. Note that some non-thermal high energy populations are known to exist in a number of instances, in particular to explain the AGN high energy emission. This population is thought to be accelerated by shock waves or turbulent motions through Fermi-type mechanisms. In this case, the

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Larmor radii of these particles become comparable to the size of the accelerating region. Also, not all astrophysical plasmas are ionized enough to allow good coupling between charged particles and neutrals, leading to the breakdown of the one-ﬂuid approximation, or implying that, e.g., the ambipolar resistivity and/or Hall term can no longer be neglected in the analysis of some types of ﬂuid instabilities. This is in particular the case in some regions of YSO disks, the so-called dead-zone. 3.B

MHD axisymmetric steady-state equations

For deﬁniteness, and reference in the main text, the basic form of the equations most commonly used in the analysis of steady-state accretion-ejection structures are compiled here. Axisymmetry is assumed in the (r, φ, z) coordinate system. Because of this symmetry, poloidal ﬁeld components (up , Bp , Jp ) belong to the (r, z) plane, while the toroidal (uφ = rΩ(r, z), Bφ , Jφ ) component is identical to the azimuthal one26 . Instead of the poloidal magnetic ﬁeld component, it is convenient to make use of the ﬂux function a(r, z), deﬁned such that 1 ∇a × eφ , r which characterizes a surface of constant magnetic ﬂux: Φ= B · dS = 2πa(r, z). Bp =

(3.15)

(3.16)

S

The variables of the problem are the density ρ, the pressure P , the velocity u, the magnetic ﬁeld B (the temperature may be included too, but quite often, a polytropic equation of state is assumed, due to our poor knowledge of the heating and cooling properties of disks and jets). A turbulent viscosity and resistivity modelling is assumed, and a separation of mean and ﬂuctuating quantities is implicitly performed in all published studies of disks and disk/jet global dynamics. 3.B.1

General form

The continuity equation reduces to, without surprise ∇ · ρu = 0.

(3.17)

26 If one views a jet as a form of straight fusion device, note that the convention on poloidal and toroidal components just described diﬀers from the one used in fusion.

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The momentum equation writes ρu · ∇u = −∇P − ρ∇ΦG + J × B + ∇ · T

(3.18)

where Tij = ρδui δuj is the turbulent Reynolds stress tensor (the turbulent contribution to the Maxwell stress is usually included as well in the deﬁnition of T). In general, only the Trφ component of this tensor (responsible for the radial turbulent transport) is kept. The turbulent viscosity prescription amounts to assume that Trφ νv ρrdΩ/dr with νv = αss cs h, where cs is the sound speed and h the local disk scale-height; αss is a parameter introduced by Shakura and Sunyaev [140] to account for the eﬀect of the unspeciﬁed mechanisms producing the transport27 . Depending on the considered systems, the value of αSS varies between 10−3 and 1. The next equation is the induction equation. Because of the assumed axisymmetry, the toroidal component of the electric ﬁeld vanishes (Eφ = 0). The poloidal component of the induction equation (or equivalently, the toroidal component of Ohm’s law) reduces to p νm Jφ = up × Bp ,

(3.19)

p is the poloidal turbulent resistivity. Similarly, the toroidal comwhere νm ponent of the induction equation reads t

νm 1 ∇· = ∇ · (Bφ up − Bp Ωr), ∇rB (3.20) φ 2 r r t is the toroidal turbulent resistivity. where νm The turbulent resistivity has never been speciﬁcally studied nor modelled, although it is needed in the analysis of the coupling of the magnetic ﬁeld to the disk in disk-driven jet models. A Shakura-Sunyaev type of prep scription is used: νm = αm vA h, where vA is the local Alfv´en velocity, and t p /νm and αm are h the local disk scale-height. In this picture, the ratio νm free parameters. The system is closed with the help of a polytropic equation of state:

P = KρΓ ,

(3.21)

where Γ is the polytropic index (Γ = γ = 5/3 for nonrelativistic adiabatic jets, and Γ = 1 for isothermal ones). 27 The transport may be due to 2D-waves as well as to 3D turbulence, although it is always referred to as a turbulent transport coeﬃcient; for an overview of our present understanding of mechanisms contributing to this coeﬃcient, and of their eﬃciency, see Ref. [9, 10].

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Ideal MHD form

In the ideal MHD context that is relevant in jets, it is possible to reduce the problem to only two equations, namely, Bernoulli’s energy integral, which describes the ﬂow evolution along magnetic surfaces, and a generalized Grad-Shafranov (transﬁeld) equilibrium equation, which deﬁnes the shapes of the magnetic surfaces. First, note that in ideal MHD, Eq. (3.19) implies that the poloidal velocity and magnetic ﬁeld are parallel, so that one can deﬁne without loss of generality, χ such that χ up = Bp . (3.22) µo ρ The continuity equation (3.17) then implies that Bp · ∇χ = 0,

(3.23)

i.e., χ = χ(a) is constant on magnetic surfaces. All such constant quantities are most conveniently evaluated on the Alfv´en critical surface (designated by a subscript “A”), where the poloidal velocity is equal to the poloidal √ Alfv´en velocity, so that χ(a) = µo ρA . Similarly, the toroidal component of the induction equation , Eq. (3.20), yields Ω∗ (a) = Ω − χ

Bφ . µo ρr

(3.24)

With the help of Eq. (3.22) and (3.24), one can compute the electric ﬁeld, which reads E = −u × B = −Ω∗ reφ × Bp ,

(3.25)

which shows that Ω∗ can be interpreted as the rotation velocity of the magnetic ﬁeld. The toroidal component of the momentum equation, Eq. (3.2) yields an equation of conservation of the total angular momentum on magnetic surfaces: rBφ 2 , (3.26) l(a) ≡ Ω∗ rA = Ωr2 − χ which implicitly deﬁnes the magnetic “lever arm” rA . This denomination derives from the fact that, for disk-driven jets, the torque ratio (introduced in Sec. 3.3.3) Λ ∼ βrA /h, where h is the disk scale height (see, e.g., [52, 53]; as β ∼ 1, and as most jet models have rA h, this implies that the magnetic torque always dominates over the turbulent one in these jets.

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An assumption commonly made about the polytropic equation of state is that K = K(a). The only remaining dynamical information is contained in the poloidal part of the momentum equation. It is equivalent to Bernoulli’s equation and the generalized Grad-Shafranov equation; they can be expressed as E(a) =

u2 Bφ , + H + ΦG − Ω∗ r √ 2 µo ρA

where H = ΓP/(Γ − 1) is the gas enthalpy, and 2 dΩ∗ rA ∇a dE 2 2 dΩ∗ ∇ · (m − 1) − Ω + (Ωr2 − Ω∗ rA = ρ ) 2 µo r da da da 2 2 2 2 B + m B d ln K Cs p d ln χ φ + , − γ(γ − 1) da µo da

(3.27)

(3.28)

2 where m2 ≡ u2p /VAp is the poloidal Alfv´enic Mach number and Cs2 = γkB T /µmp is the jet sound speed. The derivation of these last two equations is rather involved and is not performed here (see, e.g., [53] for intermediate steps). This form of the generalized Grad-Shafranov equation stresses the fact that once the functional dependence on a of the various invariants just derived is given (i.e., χ, Ω∗ , l, E and K), the shape of the magnetic surfaces is determined. Note that the Poynting ﬂux in the jet is E × Bφ = −Ω∗ rBφ Bp ; it appears as the magnetic term in Bernoulli’s equation. It is the conversion of this magnetic energy into mechanical energy which accelerates the jet. Jets in which the Poynting ﬂux everywhere dominates over the matter kinetic energy are called Poynting, or electromagnetic jets. An equivalent form of the generalized Grad-Shafranov equation can be derived, which has a somewhat more transparent physical content. It reads

2 Bφ2 B2 2 Bp 2 − ρ∇⊥ ΦG + ρΩ r − (1 − m ) − ∇⊥ P + ∇⊥ r = 0 µo R 2µo µo r (3.29) where ∇⊥ X ≡ ∇a·(∇X)/|∇a| is the gradient of a quantity X perpendicular to a magnetic surface; it is negative if X decreases with increasing magnetic ﬂux. One also deﬁnes R, the curvature radius of the magnetic surfaces, by

∇a (Bp · ∇)Bp 1 , ≡ · R |∇a| Bp2

(3.30)

When R > 0, the surface is bent outwards while for R < 0, it bends inwards.

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3.C

The Kelvin-Helmholtz instability of a single vortex sheet layer

One wishes to study the stability of the ﬂuid time-independent equilibrium sketched on the left-hand side of Fig. 3.4, which is made of two regions (hereafter 1 and 2), endowed with uniform velocities U and −U respectively, uniform pressure, and constant density. A cartesian reference system is erected, such that region 1 corresponds to x > 0 and region 2 to x < 0. The unperturbed interface is at x = 0. The direction z is the direction of the velocity ﬁeld, the direction y is transverse both to x and z. The ﬂuid is assumed incompressible. The only force acting on the ﬂuid is the pressure. The ﬂuid is perturbed, and the interface between the two regions is designated by ζ = ζ(y, z, t). Region 1 now corresponds to x > ζ, and region 2 to x < ζ. The linearized equations of motion in regions i = 1, 2 read ∂δvi ∇δP i ∂δvi + U =− , (3.31) ∂t ∂z ρ where = +1 in region 1, and = −1 in region 2, and ∇ · δvi = 0.

(3.32)

The boundary conditions are that the perturbation vanishes when x → ±∞, and that the perturbed pressure and velocity appropriately match at the boundary x = ζ. Perturbations of the form exp(iωt − ikz) are assumed (a wavenumber could be added in the direction y). The two preceding equations imply that ∆δP i = 0 so that δP i = pi exp[iωt − ikz − k(x − ζ)] pi exp[iωt − ikz − kx].

(3.33)

The last approximate equality follows from the linearization. The velocity perturbation is connected to the pressure one through ∇δP i δvi = − , (3.34) iρω i where ω i = ω − kU is the Doppler shifted frequency in regions 1 and 2. The boundary condition at the interface on the pressure reduces to δP 1 = δP 2 . To express the boundary condition on the velocity a small detour through lagrangian dynamics is necessary. Assume a ﬂuid particle on the boundary is at position (ζo , yo , zo ) at time to , and deﬁne the displacement from this position as ξ = ξ(ζo , yo , zo , t); note that ζ(yo , zo , t) = ζo +ξx . One has δvxi (ζo , yo , zo , to ) = (Dξx /Dt)t=to (∂to + U ∂zo )ζo = iω i ζo ,

(3.35)

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where the approximate equality results from the linearization. Collecting all these results together yields the dispersion relation ω = ±ikU.

(3.36)

One of these two modes is unstable.

3.D

MHD stability and the energy principle

The stability of MHD astrophysical ﬂuids is rarely considered around static equilibria, so that the variational formulation of perturbation MHD around such equilibria is not a well-known topic among astrophysicists. The notions needed in this review are brieﬂy recalled here. Only ideal MHD stability is discussed in this review (for a discussion of non-ideal MHD stability, see [141]). For static equilibria, the MHD equation of motion can be written as (to ﬁrst order): −ρo ω 2 ξ = −∇δP∗ + δT ≡ F(ξ),

(3.37)

where δT is the perturbation of the magnetic tension, δP∗ = δP + B · δB the total pressure perturbation, and ξ the ﬂuid element displacement from equilibrium (a Fourier transform in time has been performed). The density, pressure and magnetic ﬁeld perturbations follow from the time integration of the respective equations, which read δρ = −∇·(ρξ),

(3.38)

δB = ∇×(ξ×B),

(3.39)

while the expression of the total pressure perturbation follows from the assumed adiabatic equation of state: B·δB , (3.40) δP∗ = −ξ·∇P − γP ∇·ξ + µ δP δPm The linear operator F of Eq. (3.9) is self-adjoint (see [110] p. 242, and his Appendix A, and references therein). Consequently, an Energy Principle can be formulated. Deﬁning 1 ∗ ξ ∗ · F(ξ)d3 r, (3.41) δW (ξ , ξ) = − 2 and 1 K(ξ ∗ , ξ) = (3.42) ρ|ξ|2 d3 r, 2

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and taking the scalar product of Eq. (3.37) with ξ ∗ leads to ω2 =

δW . K

(3.43)

The self-adjointness of F implies that Eq. (3.37) is the extremum of Eq. (3.43) with respect to a variation of ξ. A useful form of δW has been derived by Bernstein et al. [142], which reads (see [110] p. 259) 1 d3 r |Q⊥ |2 + Bo2 |∇ · ξ⊥ + 2κc · ξ⊥ |2 + γPo |∇ · ξ|2 δW = − 2 ∗ ∗ − 2(ξ⊥ · ∇Po )(κc · ξ⊥ ) − J (ξ⊥ × e ) · Q⊥ , (3.44) where ξ⊥ is the component of the displacement perpendicular to the unperturbed ﬁeld B, Q⊥ = ∇ × (ξ⊥ × Bo ) is the perturbation in the magnetic ﬁeld, and κc ≡ (e · ∇)e ] = −Bθ2 /(B 2 r)er is the curvature vector of the magnetic ﬁeld (e is the unit vector in the direction of the unperturbed magnetic ﬁeld. The ﬁrst term describes the ﬁeld line bending energy. The second term is the energy in the ﬁeld compression, while the third is the energy in the plasma compression. The fourth term arises from the perpendicular current (as ∇P = J⊥ × B in a static equilibrium), and the last one arises from the parallel current J . Only these two terms can be negative, and give rise to an instability if they are large enough to make ω 2 < 0. These instabilities are referred to as pressure-driven and current-driven, respectively. They include the well-known pressure-driven “sausage” (m = 0) and current-driven “kink” (m = 1) modes; these denominations follow from the generic shape assumed by the deformation, for any given, ﬁnite vertical wave-number k. Pressure-driven instabilities are further subdivided into interchange and ballooning modes (see [110] for details). Also, it is important to know whether unstable modes are internal or external, i.e., have vanishing or non-vanishing displacement on the plasma surface. The distinction is important, as unstable external modes are prone to disrupt the plasma. 3.E

Pressure-driven instabilities asymptotic limit

In this Appendix, successive analytic reductions of the momentum equation, Eq. (3.37), are outlined, leading to a simple dispersion relation for pressuredriven modes in the |Bφ | |Bz | limit, which is of interest for MHD jet models.

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After some algebra (detailed in Appendix C of [110]), the perturbation equations can be combined in a single, exact but somewhat complex equation for the radial displacement ξ ≡ ξor [143, 144]: d d A (rξ) + C(rξ) = 0 (3.45) dr dr with 2 (ω 2 − v 2 k 2 )(ω 2 − v 2 c2 k 2 /(v 2 + c2 )) ρ(c2s + vA ) s A A s A , r D 2 2 2 4k2 Bφ vA Bφ d 2 2 2 2 2 k ) − µo Dr C = ρr (ω 2 − vA 3 (ω − cs k ) − dr r2 µo 2B B 2 2 2 2 2 2 ⊥ + µφo kk v k − (c + v )ω c , 2 s s A A r D

A=

D = (ω 2 − ωf2 )(ω 2 − ωs2 ),

(3.46)

(3.47) (3.48)

where ωf and ωs are the frequencies of the usual fast and slow magnetosonic modes, vA and cs the local Alfv´en and sound speed, and ﬁnally k = (mBφ /r + kBz )/B

(3.49)

k⊥ = (mBz /r − kBφ )/B

(3.50)

are the components of the wavenumber k parallel and perpendicular to the unperturbed magnetic ﬁeld. A ﬁrst simpliﬁcation is obtained by noting that unstable modes have |ω 2 | ∼ c2s Kc Kρ , so that, in the large wavelength limit (|k⊥ r| 1), ω 2 ωf2 . If one furthermore restricts oneself to quasi-perpendicular wavenumbers and keeps only the leading order terms in the |k |/|k⊥ | expansion, A and C reduce to ρ 1

2 A = − 2 ω 2 − ωA , (3.51) r k⊥ 2Bφ2 C =− 3 r

rP B2

−

4β ∗ Bφ4 ω2 , (1 + β ∗ )r3 B 2 ω 2 − ωs2

(3.52)

2 2 ) in this limit, and where β ∗ = c2s /vA . with ωs2 = c2s k2 /(c2s + vA The remainder of this Appendix is devoted to the large toroidal ﬁeld limit. Note that, in the vicinity of a resonance, the parallel wavenumber can be expanded to ﬁrst order in the distance δr to the resonance rc to read

|k | = |k⊥ |

|Bφ Bz | |δr| |s|. B2 rc

(3.53)

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In the large toroidal ﬁeld limit, the magnetic shear term in the Suydam criterion Eq. (3.11) is small, and, correlatively, for |k⊥ | |Bφ /Bz | and β ∼ 1, the mode can extend over distances ∼ r without imperiling the 2 2 k < c2s Kc Kρ (to have unstable modes). Consequently, one requirement vA can choose ξr (r) = ξo exp(ikr r), with 1 |kr r| |k⊥ r|, and the term involving A in the momentum equation Eq. (3.45) becomes negligible. The remaining terms lead to the looked for dispersion relation: 2 2 2 2 + vSM ) + ωc2 ω 2 + k2 k2 − 2β ∗ Kc Kρ vSM vA = 0, (3.54) ω 4 − k2 (vA 2 2 β ∗ vA /(1 + β ∗ ) is the slow magnetosonic square speed (β ∗ = where vSM 2 2 cs /vA = γβ/2), and

4β ∗ Kc2 2 2 . (3.55) ωc = vA −2βKρ Kc + 1 + β∗

As seen by direct inspection, only one root of Eq. (3.54) is negative at a time, and only when k2 < 2β ∗ Kc Kρ

(3.56)

Acknowledgments I thank Jonathan Ferreira and Guy Pelletier for the many exchanges we have had over the years, and in the course of the preparation of this work, on various aspects of the physics discussed here. I also thank Hubert Baty for discussions on some aspects of current-driven instabilities in jets. References [1] Cabrit S 2002 Constraints on accretion-ejection structures in young stars. In Star formation and the physics of Young stars, J Bouvier and J-P Zahn, eds (EDP Sciences). [2] Rees M J 1984 Ann. Rev. Astron. Astrophys. 22 471. [3] Blandford R D 1990 Physical Processes in Active Galactic Nuclei. In Active Galactic Nuclei, 20th Saas-Fee Advanced Course in Astronomy and Astrophysics (Springer-Verlag). [4] Ferrari A 1998 Ann. Rev. Astron. Astrophys. 36 359. [5] Pelletier G 2004 Black Hole induced ejection. In the Cargese lectures on Black Holes in the Universe (to be published; available as astroph/0405113). [6] Bertout C 1989 Ann. Rev. Astron. Astrophys. 27 351. [7] Andr´e P 2002 The initial conditions for protostellar collapse: observational constraints. In Star formation and the physics of Young stars, J Bouvier and J-P Zahn, eds (EDP Sciences).

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[8] Ferreira J 2002 Theory of magnetized accretion disks driving jets. In Star formation and the physics of Young stars, J Bouvier and J-P Zahn, eds (EDP Sciences). Available as astro-ph/0211621. [9] Papaloizou J C B and Lin D N C 1995 Annu. Rev. Astron. Astrophys. 33 505. [10] Lin D N C and Papaloizou J C B 1996 Annu. Rev. Astron. Astrophys. 34 703. [11] Hartmann L 2001 Astrophys. J. 121 1030. [12] Tassis K and Mouschovias T C 2004 Astrophys. J. 616 283. [13] M´enard F and Bertout C 2002 Accretion disks around young stars: an observational perspective. In Star formation and the physics of Young stars, J Bouvier and J-P Zahn, eds (EDP Sciences). [14] Gabuzda D C 1997 in Relativistic Jets in AGNs, M Ostrowski, M Sikora, G Madejski, M Begelman eds, Cracow. [15] Asada K, Inoue M, Ushida Y, Kameno S, Fujisawa K, Igushi S and Mutoh M 2002 Pub. Astron. Soc. Pac. 54 L39. [16] Krause M and L¨ or A 2004 Astron. Astrophys. . 420 115. [17] Ray T P, Muxlow T W B, Axon D J, Brown A, Corcoran D, Dyson J and Mundt R 1997 Nature 385 415. [18] Reipurth B, and Bally J 2001 Ann. Rev. Astron. Astrophys. 39 403. [19] Bodo G., Massaglia S., Rossi P., Rosner P., Malagoli A. and Ferrari A. 1995 Astron. Astrophys. 303 281. [20] Bodo G., Rossi P., Massaglia S., Ferrari A., Malagoli A. and Rosner P. 1998 Astron. Astrophys. 333 1117. [21] Ferreira J 1997 Astron. Astrophys. 319 340. [22] Rekowski M v, R¨ udiger G and Elstner D 2000 Astron. Astrophys. 353 813. [23] Camenzind M 1986 Astron. Astrophys. 156 137. [24] Bogovalov S 2001 Astron. Astrophys. 371 1155. [25] Bogovalov S and Tsinganos K 2001 Monthly Not. Roy. Astron. Soc. 325 249. [26] Vlahakis N and K¨ onigl A 2004 Astrophys. J. 605 656. [27] Vlahakis N 2004 Astrophys. J. 600 324. [28] Casse F and Ferreira J 2000 Astron. Astrophys. 361 1178. [29] Phinney E S 1982 Monthly Not. Roy. Astron. Soc. 198 1109. [30] Renaud N and Henri G 1998 Monthly Not. Roy. Astron. Soc. 300 1047. [31] Gracia J, Tsinganos K and Bogovalov S V. 2005 Astron. Astrophys. 442 L7-L10. [32] K¨ onigl A 1986 Can. J. Phys. 64 362. [33] Blandford R D and Znajek R L 1977 Monthly Not. Roy. Astron. Soc. 179 433. [34] Thorne K S, Price R H and MacDonald D A 1986 Black Hole: the Membrane Paradigm (Yale University Press). [35] MacDonald D A and Thorne K S 1982 Monthly Not. Roy. Astron. Soc. 198 345. [36] Phinney E S 1982 In Astrophysical jets, 1st Torino workshop p. 201. Ferrari and Pacholczyk eds (Reidel). [37] Punsly B and Coroniti F V 1990 Astrophys. J. 350 518.

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Komissarov S S 2004 Monthly Not. Roy. Astron. Soc. 350 427. Komissarov S S 2004 Monthly Not. Roy. Astron. Soc. 350 1431. Li L X 2000 Phys. Rev. D 61 084016. Wang D.-X., Ma, R.-Y. and Lei W.-H. 2003 Astrophys. J. 595 109. Shu F H, Najita J, Ostriker E and Wilkin F 1994 Astrophys. J. 429 781. Lovelace R V E, Romanova M M and Bisnovatyi-Kogan G S 1999 Astrophys. J. 513 368. Ferreira J, Pelletier G and Appl S 2000 Monthly Not. Roy. Astron. Soc. 312 387. Romanova M M, Ustyugova G V, Koldoba A V and Lovelace R V E 2004 Astrophys. J. 610 920. Bouvier J, Chelli A, Allain S, Carrasco L, Costero R, Cruz-Gonzalez I, Dougados C, Fern´ andez M, Martn E L, M´enard F, Mennessier C, Mujica R, Recillas E, Salas L, Schmidt G and Wichmann R 1999 Astron. Astrophys. 349 619. Bardou A and Heyvaerts J 1996 Astron. Astrophys. 307 1009. Blandford R D 1976 Monthly Not. Roy. Astron. Soc. 176 465. Lovelace R V E 1976 Nature 242 649. Benford G 1978 Monthly Not. Roy. Astron. Soc. 183 29. Chan K L and Henriksen R N 1980 Astrophys. J. 241 534. Blandford R D and Payne D G 1982 Monthly Not. Roy. Astron. Soc. 199 883. Pelletier G and Pudritz R E 1992 Astrophys. J. 394 117. Contopoulos J and Lovelace R V E 1994 Astrophys. J. 429 139. Sauty C and Tsinganos K 1994 Astron. Astrophys. 287 893. Li Z-H 1996 Astrophys. J. 465 855. Ustyugova G V, Koldoba A V, Romanova M M, Chetchetkin V M and Lovelace R V E 1999 Astrophys. J. 516 221. Krasnopolski R, Li Z Y and Blandford R D 1999 Astrophys. J. 526 631. Ouyed R, Clarke D A and Pudritz R E 2003 Astrophys. J. 582 292. Casse F and Keppens R 2004 Astrophys. J. 601 90. Kato S, Kudoh T and Shibata K 2002 Astrophys. J. 565 1035. Lubow S H, Papaloizou J C B, Pringle J E 1994 Monthly Not. Roy. Astron. Soc. 267 235. Heyvaers J, Priest E R and Bardou A (1996) Astrophys. J. 473 403. Lubow S H, Papaloizou J C B, Pringle J E 1994 Monthly Not. Roy. Astron. Soc. 268 1010. Cao X and Spruit H C 2002 Astron. Astrophys. 385 289. Livio M. 1997, in IAU 163, Accretion Phenomena and Related Outﬂows, ASP Conf Series 121, p. 845. Sauty C, Trussoni A, and Tsinganos K 2002 Astron. Astrophys. 389 1068. Heyvaers J and Norman C (1989) Astrophys. J. 347 1055. Okamoto I 2001 Monthly Not. Roy. Astron. Soc. bf 327 55. Garcia P J V, Ferreira J, Cabrit S and Binette L (2001) Astron. Astrophys. 377 589. K¨ onigl A and Choudhuri A R 1985 Astrophys. J. 289 173.

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[72] Taylor J B 1974 Phys. Rev. Lett. 33 139. [73] Taylor J B 1986 Rev. Mod. Phys. 58 741. [74] Ortolani S and Schnack D D 1993 Magnetohydrodynamics of Plasma Relaxation (World Scientiﬁc). [75] Birkinshaw M 1991 The stability of jets. In Beams and Jets in Astrophysics, P A Hugues, ed (Cambridge University Press). [76] Todo Y, Uchida Y, Tetzuya S and Rosner R 1993 Astrophys. J. 403 164. [77] Blandford R D and Rees M J 1974 Monthly Not. Roy. Astron. Soc. 169 395. [78] Hardee P E, Clarke D A and Rosen A 1997 Astrophys. J. 485 533. [79] Rosen A, Hardy P E, Clarke D A and Johnson A 1999 Astrophys. J. 510 136. [80] Terry P W 2000 Rev. Mod. Phys. 72 109. [81] Hanasz M, Sol, H and Sauty C (2000) Monthly Not. Roy. Astron. Soc. 316 494. [82] Balbus S A 2003 Annu. Rev. Astron. Astrophys. 41 555. [83] Papaloizou J C B and Pringle J E 1984 Monthly Not. Roy. Astron. Soc. 208 721. [84] Blaes O M 1987 Monthly Not. Roy. Astron. Soc. 227 975. [85] Hawley J F 1991 Astrophys. J. 496 507. [86] Wardle M and K¨ onigl A 1993 Astrophys. J. 410 218. [87] Spruit H C, Stehle R and Papaloizou J C B 1995 Monthly Not. Roy. Astron. Soc. 275 1223. [88] K¨ onigl A and Wardle M 1996 Monthly Not. Roy. Astron. Soc. 279 L61. [89] K¨ onigl A 2004 Astrophys. J. 617 1267. [90] Casse F and Keppens R 2002 Astrophys. J. 581 988. [91] Ouyed R and Pudritz R E 1997 Astrophys. J. 482 712. [92] Ouyed R and Pudritz R E 1997 Astrophys. J. 484 794. [93] Ouyed R and Pudritz R E 1999 Monthly Not. Roy. Astron. Soc. 309 233. [94] Anderson J M, Li Z-H, Krasnopolsky R and Blandford R D 2005 Astrophys. J. 630 945. [95] Drazin P G and Reid W H 1981 Hydrodynamic Stability (Cambridge University Press). [96] Ferrari A, Trussoni E and Zaninetti L 1981 Monthly Not. Roy. Astron. Soc. 196 1051. [97] Ray T P 1981 Monthly Not. Roy. Astron. Soc. 196 195. [98] Ray T P 1982 Monthly Not. Roy. Astron. Soc. 198 617. [99] Cohn H 1983 Astrophys. J. 269 500. [100] Fiedler R L and Jones T 1984 Astrophys. J. 283 532. [101] Roy Choudhury S and Lovelace R V E 1986 Astrophys. J. 302 188. [102] Bodo G, Rosner R, Ferrari A and Knobloch E 1989 Astrophys. J. 341 631. [103] Bodo G, Rosner R, Ferrari A and Knobloch E 1996 Astrophys. J. 470 797. [104] Hardee P E and Stone J M 1997 Astrophys. J. 483 121. [105] Stone J M, Xu J and Hardee P E 1997 Astrophys. J. 483 136. [106] Xu J, Hardee P E and Stone J M 2000 Astrophys. J. 543 161. [107] Jones T W, Gaalaas J B, Ruy D and Frank A 1997 Astrophys. J. 482 230.

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Jeong H, Ryu D, Jones T W and Frank A 2000 Astrophys. J. 529 536. Baty H and Keppens R 2002 Astrophys. J. 580 800. Freidberg J P 1987 Ideal Magnetohydrodynamics. Plenum Press. Priest E R 1982 Solar Magnetohydrodynamics. Reidel. Eichler D 1993 Astrophys. J. 419 111. Spruit H C, Foglizzo T and Stehle R 1997 Monthly Not. Roy. Astron. Soc. 288 333. Appl S, Lery T and Baty H 2000 Astron. Astrophys. 355 818. Istomin Y N and Pariev V I 1994 Monthly Not. Roy. Astron. Soc. 267 629. Istomin Y N and Pariev V I 1996 Monthly Not. Roy. Astron. Soc. 281 1. Appl S 1996 Astron. Astrophys. 314 995. Lyubarskii Y E 1999 Monthly Not. Roy. Astron. Soc. 308 1006. Baty H 2005, Astron. Astrophys. 430 9. Lery T, Baty H and Appl S 2000 Astron. Astrophys. 355 1201. Begelman M C 1998 Astrophys. J. 493 291. Dewar R L, Tatsuno T, Yoshida Z, N¨ urenberg C and McMillan B F 2004 Phys. Rev. E 70 066409. Cheremhykh O K and Revenchuk S M 1992 Plasma Phys. Contr. Fusion 34 55. Bondeson A, Iacono R and Bhattacharjee A 1987 Phys. Fluids 30 2167. Shumlak U and Hartman C W 1995 Phys. Rev. Lett. 75 3285. Arber T D and Howell D F 1996 Phys. Plasmas 3 554. Waelbroeck F L and Chen L 1991 Phys. Fluids B 3 601. Miller R L and Waltz R E 1994 Phys. Plasmas 1 2835. Chiueh T 1996 Phys. Rev. E 54 5632. Wanex L F, Sotnikov V I and Leboeuf J N 2005 Phys. Plamas 12 042101. Hassam A B 1992 Phys. Fluids B 4 485. DeSouza-Machado S, Hassam A B and Sina R 2000 Phys. Plasmas 7 4632. Woltjer L 1990 Phenomenology of Active Galactic Nuclei. In Active Galactic Nuclei, 20th Saas-Fee Advanced Course (Springer-Verlag). Netzer H 1990 AGN emission line. In Active Galactic Nuclei, 20th Saas-Fee Advanced Course (Springer-Verlag). Vall´ee J P 2004 New Astron. Rev. 48 763. Yamazaki D G, Ichiki K and Kajino T 2005 Astrophys. J. Lett. 625 L1. Langer M, Puget J-L and Aghanim N 2003 Phys. Rev. D 67 043505. Shu F H 1992 The Physics of Astrophysics. II: Gas Dynamics. University Science Books. Krall N A and Trievelpiece A W 1986 Principles of Plasma Physics. San Francisco Press. Shakura N I and Sunyaev R A 1973 Astron. Astrophys. 24 337. Bateman G 1978 MHD Instabilities. MIT Press. Bernstein I B, Frieman E A, Kruskal M D and Kulsrud R M 1958 Proc. Royal Soc. London A 244 17. Hain K and L¨ ust R 1958 Z. Naturforsch. Teil A 13 936. Goedbloed J P 1974 Physica 53 501.

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Fig. 3.7

Fig. 3.8

Professor Masahiro Wakatani.

Prof. Wakatani and friends at the ﬁrst ‘Festival de Th´eorie’.

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Fig. 3.9 Magnetic ﬁeld lines in globally isotropic turbulence as in the simulation underlying Fig. 6.2 (left) and in a system with strong mean ﬁeld component, b0 = 5 (right).

Fig. 3.10 Dissipative micro current sheets in isotropic (left) and anisotropic (right) turbulence. The anisotropic system is permeated by a mean magnetic ﬁeld of strength b0 = 5 pointing upwards (cf. Fig. 3.9). Every current sheet is accompanied by vorticity sheets (not shown).

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Fig. 3.11 Views of the jets of the galaxies M87 (left) and 3C175 (right). The blow-up insert on the left image shows a gas disk, whose size is considerably larger than that of the central engine. The disk fast rotation strongly suggests the existence of a massive black hole at the galaxy center. The image of 3C175 shows the large scale collimated jet, and the end lobes (of rather modest size in this case). There are in fact two jets, one on each side of the galaxy, but one of them is not visible because of Doppler shift eﬀects. Credits: Hubble Space Telescope Science Institute (left); National Radio Astronomy Observatory (right).

Fig. 3.12 Views of jets from three diﬀerent forming stars. The sketch at the bottom shows them in relative scale, and gives some indication of the typical global appearance of YSO jets. The HH-30 image shows not only a jet, but an edge-on disk. The disk corresponds to the dark region of the “diabolo”-shaped feature of the upper-left image; it is darkened by the aborption due to the presence of dust in the disk plane. Light from the protostar is processed and reemitted towards the observer by the top and bottom of the disk (green regions). The protostar itself is hidden by the disk. As in AGNs, jets are two-sided. Colors are artiﬁcial. Credits: Space Telescope Science Institute.

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MHD Jet

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Gb Plasma

non thermal radiative emission synchrotron+ICS

plasma - MHD Waves interaction

radiative acceleration

accretion disk

central engine

(supermassive black hole)

Fig. 3.13 Poloidal cut through a two-component model of AGN jet. A conventional, non-relativistic, disk-driven MHD jet provides the energy to conﬁne and heat a pair plasma in its core, which is responsible for AGN high energy radiation (ﬁgure by Ludovic Saug´ e).

Fig. 3.14 A diﬀerentially rotating disk threaded by a dipolar ﬁeld can give rise to a jet, which is centrifugally accelerated by the ﬁeld. Collimation eventually follows once the jet has widened enough (magnetic surfaces opening and recollimation are not shown to scale).

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Bp Ideal MHD

- dP/dz

up

resistive MHD

Central Object

Matter streamline

Fig. 3.15 Vertical cut through an accretion-disk. The disk is assumed to be eﬀectively resistive and viscous through turbulent processes, while the jet is assumed ideal. The polo¨idal velocity and magnetic ﬁeld are shown (ﬁgure by Fabien Casse).

Fig. 3.16 Poloidal cut through an accretion-disk and jet. The sketch illustrates the existence of three critical surfaces in steady-state solutions to the ideal MHD jet problem, with a change of type of the PDEs at the surfaces (ﬁgure by Jonathan Ferreira).

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Radio power Radio galaxies

Distance (redshift)

Seyfert galaxies Sf 1 :broad and narrow lines Sf 2 : narrow lines QSO radio-quiet

FR I

FR II

Quasars Radio quasars BL Lacs (very weak lines)

Fig. 3.17 Qualitative classiﬁcation of AGNs based on their radio power, and their distances to the observer (measured by their cosmological redshift). Figure by Gilles Henri.

Fig. 3.18 Snapshots of electric potential ﬂuctuations in a poloidal plane in a quiescent phase (left) and during a relaxation (right). The radial co-ordinate is stretched by a factor 4 (comp. Fig. 8.1). The relaxation is dominated by a (m, n) = (5, 2) mode which is the lowest order (m, n) mode localized at the barrier position.

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Fig. 3.19 Radial proﬁles of density (top) and electron and ion temperature (bottom) in the tokamak JET (from [21]).

Fig. 3.20

Radial shape of a global mode calculated with the MISHKA code [24].

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Fig. 3.21

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Domain of existence for type I and type III edge localized modes [12].

Fig. 3.22 Map of the pressure gradient versus radial coordinate and time for the model Eq. (9.16) with RS = 0.

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PART 3

Turbulence and turbulent transport the agents of relaxation and structure formation

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Chapter 4

A Tutorial on Basic Concepts in MHD Turbulence and Turbulent Transport P.H. Diamond∗† , S.-I. Itoh† and K. Itoh‡ ∗

University of California, San Diego, La Jolla, CA 92093-0319 USA † Research Institute for Applied Mechanics, Kyushu University, Kasuga City 816-8580 Japan ‡ National Institute for Fusion Science, Toki-Shi 509-5292 Japan

4.1

Introduction

In this paper, we present an eclectic tutorial on some of the basic ideas in MHD turbulence and turbulent transport, with special attention to incompressible and weakly compressible dynamics with a mean magnetic ﬁeld. The approach throughout is conceptual - we emphasize intuition, ideas and basic notions rather than detailed results. The latter are already well documented in the research literature. Also, we place primary emphasis on understanding the case of magnetized MHD turbulence, in which an externally ﬁxed, large scale magnetic ﬁeld breaks symmetry, produces anisotropy and restricts nonlinear interactions. This case should be contrasted with that considered in many (but not all) discussions of MHD turbulence, which focus on weakly magnetized or unmagnetized systems. This paper is organized into four sections, each of which discusses an essential paradigm in MHD turbulence theory. These four sections should be thought of as four related but distinct vignettes, rather than one continuous narrative. Each attempts to give the reader the chance to peep at some essential ideas in the theory. The four sections are the following. Section 4.2: K41 beyond dimensional analysis - revisiting the theory of hydrodynamic 119

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turbulence. Section 4.3: Kraichnan-Iroshnikov, Goldreich-Sridhar and all that: a scaling theory of MHD turbulence. Section 4.4: Steepening of nonlinear Alfven waves - a little compressibility goes a long way... Section 4.5: Turbulent ﬂux diﬀusion in 2D MHD - a ‘minimal’ problem which is not so simple... While these four topics are distinct, they do have a common theme, namely the eﬀect of Alfvenically induced ‘memory’ on turbulence and transport. These four sections are followed by a brief discussion and conclusion, Sec. 4.6. We indicate some possible future research directions throughout the paper, where appropriate.

4.2

K41 beyond dimensional analysis - revisiting the theory of hydrodynamic turbulence

Surely everyone has encountered the basic ideas of Kolmogorov’s theory of high Reynolds number turbulence [1]! Loosely put, empirically motivated assumptions of (I) spatial homogeneity - i.e. the turbulence is uniformly distributed in space, (II) isotropy - i.e. the turbulence exhibits no preferred spatial orientation, (III) self-similarity - i.e. all inertial range scales exhibit the same physics and are equivalent. Here ”inertial range” refers to the range of scales

smaller than the stirring scale 0 but larger than the dissipation scale ( d < < 0 ), (IV) locality of interaction - i.e. the (dominant) nonlinear interactions in the inertial range are local in scale, i.e. while large scales advect small scales, they cannot distort or destroy small scales, only sweep them around. Inertial range transfer occurs via like-scale straining, only. Assumptions (I)-(IV) and the basic idea of an inertial range cascade are summarized in Fig. 4.1 and Fig. 4.2. Using assumptions (I) - (IV), we can state that energy through-put must be constant for all inertial range scales, so

∼ v03 / 0 ∼ v( )3 / ,

(4.1)

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and v( ) ∼ ( )1/3 ,

(4.2)

2/3 −5/3

E(k) ∼

k

,

(4.3)

which are the familiar K41 results. The dissipation scale d is obtained by balancing the eddy straining rate 1/3 / 2/3 with the viscous dissipation rate ν/ 2 to ﬁnd the Kolmogorov microscale.

d ∼ ν 3/4 / 1/4

ut np i y erg ing En stirr by

(4.4)

Inertial range Local energy transfer in the inertial range

En vis ergy co us outp he ut b a ti ng y

Fig. 4.1 Basic cartoon explanation of the Richardson-Kolmogorov cascade. Energy transfer in Fourier-space.

l0

.. .. .

.. .. .

.. .. .

.. .. .

.. .. .

.. .. .

.. .. .

.. .. .

..................

l1 = α l0 .. .. . l n = αn l 0 .. .. . ld

Fig. 4.2 Basic cartoon explanation of the Richardson-Kolmogorov cascade. That in real space.

A related and important phenomenon, which also may be illuminated by scaling arguments, is how the distance between two test particles grows in time in a turbulent ﬂow. This problem was ﬁrst considered by Louis Fry Richardson, who was stimulated by observations of the rate at which pairs of weather balloons drifted apart from one another in the (turbulent) atmosphere [2]. Consistent with the assumption of locality of interaction

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in scale, Richardson made the ansatz that the distance between two points in a turbulent ﬂow increases at the speed set by the eddy velocity on scales corresponding (and comparable) to the distance of separation. Thus, for distance , d

= v( ) dt Figures 4.3, 4.4, and 4.5 so using the K41 results gives

(t) ∼ 1/3 t3/2 ,

(4.5)

(4.6)

l

t'

v (l) t

Fig. 4.3 Basic idea of the Richardson dispersion problem. The evolution of the separation of the two points (black and white dots) l follows the relation.

t' l

t

v

Fig. 4.4 If the advection ﬁeld scale exceeds l, particle pair swept together, so l unchanged.

l

t' t

Fig. 4.5

v

If the advection ﬁeld scale is less than l, there is no eﬀect on pair dispersion.

a result which Richardson found to be in good agreement with observations. Notice that the distance of separation grows super-diﬀusively, i.e.

(t) ∼ t3/2 , and not ∼ t1/2 , as in the textbook case of Brownian motion.

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The super-diﬀusive character of (t) is due to the fact that larger eddies support larger speeds, so the separation process is self-accelerating. Note too, that the separation grows as a power of time, and not exponentially, as in the case of a dynamical system with positive Lyapunov exponent. This is because for each separation scale , there is a unique corresponding separation velocity v( ), so in fact there is a continuum of Lyapunov exponents (one for each scale) in the case of a turbulent ﬂow. Thus, (t) is algebraic, not exponential! By way of contrast, the exponential rate of particle pair separation in a smooth chaotic ﬂow is set by the largest positive Lyapunov exponent. We also remark here that while intermittency corrections to the K41 theory based upon the notion of a dissipative attractor with a fractal dimension less than three have been extensively discussed in the literature [3], the eﬀects of intermittency in the corresponding Richardson problem have received relatively little attention. This is unfortunate, since, though it may seem heretical to say so, the Richardson problem is, in many ways, more fundamental than the Kolmogorov problem, since unphysical eﬀects due to sweeping by large scales are eliminated by deﬁnition in the Richardson problem. An exception to the lack of advanced discussion of the Richardson problem is the excellent review article by Falkovich, Gawedski and Vergassola, 2001 [4]. Of course, ”truth in advertising” compels us to emphasize that the scaling arguments presented here contain no more physics than that which was inserted ab initio. To understand the physical mechanism underpinning the Kolmogorov energy cascade, one must consider structures in the ﬂow. As is well known, the key mechanism in 3D Navier-Stokes turbulence is vortex tube stretching, schematically shown in Fig. 4.6. There, we see that alignment of strain ∇v with vorticity ω (i.e. ω·∇v = 0) generates small scale vorticity, as dictated by angular momentum conservation in incompressible ﬂows. The enstrophy (mean squared vorticity) thus diverges as ω 2 ∼ /ν,

(4.7)

for v → 0. This indicates that enstrophy is produced in 3D turbulence, and suggests that there may be a ﬁnite time singularity in the system, an issue to which we shall return later. By ﬁnite time singularity of enstrophy, we mean that the enstrophy diverges within a ﬁnite time (i.e. with a growth rate which is faster than exponential). In a related vein, we note that ﬁniteness of as ν → 0 constitutes what is called an anomaly in quantum ﬁeld theory. An anomaly occurs when symmetry breaking (in this case, breaking of time reversal symmetry by viscous dissipation) persists as the

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v

r1

Ω1

L1 = Ω1 r12 A fterstretching

r2

Ω2

L2 = Ω2 r22 Fig. 4.6 The mechanism of enstrophy generation by vortex tube stretching. The vortex tube stretching vigorously produces small scale vorticity.

symmetry breaking term in the ﬁeld equation asymptotes to zero. The scaling ω 2 ∼ 1/ν is suggestive of this. So is the familiar simple argument using the Euler vorticity equation (for ν → 0): dω = ω · ∇v, dt

(4.8)

d 2 (4.9) ω ∼ ω3 . dt Of course, this ”simple argument” is grossly over-simpliﬁed, and incorrect. In fact, a mathematical proof of ﬁnite time singularity of enstrophy remains an elusive goal, with an as-yet-unclaimed Clay prize of $1,000,000. In two dimensions ω · ∇v = 0, so enstrophy is conserved. As ﬁrst shown by Kraichnan, this necessitates a dual cascade, in which enstrophy forward cascades to small scales, while energy inverse cascades to large scales [5]. The mechanism by which the dual conservation of energy and enstrophy force a dual cascade in 2D turbulence is shown via the cartoon in Fig. 4.7. As elegantly and concisely discussed by U. Frisch in his superb monograph ”Turbulence - The Legacy of A.N. Kolmogorov”, the K41 theory can be systematically developed from a few fundamental hypotheses or postulates. Upon proceeding, the cynical reader will no doubt conclude that the hypotheses (H1)-(H4) stated below are simply restatements of assumptions (I)-(IV). While it is diﬃcult to refute such a statement, we remark here that (H1)-(H4), are indeed of value, both for their precise presentation of

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E(k) E1 E2 E1'

E2' k1

k

k2

Z(k) Z2

Z1 ∆k1

Z1

∆k2 Z2

k1

k

k2

Fig. 4.7 A conceptual explanation of the inverse is cascade of energy in two-dimensional turbulence. The energy spectrum E(k) and the enstrophy spectrum is Z(k). In a short time period, cascade events for enstrophy Z1 and Z2 occur. Because the enstrophy is conserved, the associated variations in energy satisﬁes the relation E1 −E1 = −2∆k1 K1−3 Z1 and E2 − E2 = 2∆k2 k2−3 Z2 . As a whole, the energy is transported to lower-k.

Kolmogorov’s deep understanding and for the insights into his thinking which they provide. As these postulates involve concepts of great relevance to MHD turbulence, we revisit them here in preparation for our subsequent discussion of MHD turbulence. The ﬁrst fundamental hypotheses of the K41 theory is: (H1) As Reynolds number Re → ∞, all possible symmetries of the NavierStokes equation, usually broken by the means of turbulence initiation or production, are restored in a statistical sense at small scales, and away from boundaries.

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The reader should note that H1.) is a deceptively simple, and fundamentally quite profound hypothesis! The onset or production of turbulence nearly always involves symmetry breaking. Some examples are: (i) shear ﬂow turbulence: the initial Kelvin-Helmholtz instability results from breaking of translation and rotation symmetry. (ii) turbulence in a pipe with a rough boundary: the wall and roughenings break symmetry. (iii) turbulence in a ﬂushing toilet: the ﬂow has ﬁnite chirality and is nonstationary. Naively, one might expect the turbulent state to have some memory of this broken symmetry. Indeed, the essence of β-model and multi-fractal theories of intermittency is the persistence of some memory of the large, stirring scales into the smallest inertial range scales. Yet, the universal character of K41 turbulence follows directly from, and implies a restoration of, initially broken symmetry at small scales. Assumptions (i) and (ii) really follow from hypothesis (H1). The second K41 hypothesis is: (H2) Under the assumptions of H1.), the ﬂow is self-similar at small scales and has a unique scaling exponent h, such that v(r, λ ) = λh v(r, ).

(4.10)

Here, v(r, ) refers to the velocity wavelet ﬁeld at position r and scale

. Clearly, (H2) implies assumptions (III) and (IV), concerning selfsimilarity and locality of interaction. Hypotheses (H1) and (H2) pertain to ﬂow structure and scaling properties. Two additional postulates pertain to dynamics. These are: (H3) Given the assumptions of (H1) and (H2), turbulent ﬂow has a ﬁnite, non-vanishing mean rate of dissipation per unit mass , as ν → 0 (H4) In the limit of high but ﬁnite Re , all small-scale statistical properties are uniquely and universally determined by and . Hypothesis (H3) is tacitly equivalent to stating that an anomaly exists in K41 turbulence. Note that is independent of ν. However, notice also that

, the “mean rate of dissipation per unit mass” is not related to physical, calculable quantities, and is left as a more-than-slightly ambiguous concept.

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Introduction of ﬂuctuations (which relax the statement ‘uniquely’ in (H4) in the local dissipation rate (which in reality are usually associated with localized dissipative structures such as strong vortex tubes) and of a statistical distribution of dissipation, leads down the path to intermittency modelling, a topic which is beyond the scope of this paper. The reader is referred to Frisch ’95 for an overview, and to seminal references such as Frisch, Sulem, Nelkin ’78, She and Leveque ’94 [6], Falkovich, Gawedski and Vergassola, 2001, and others for an in depth discussion of intermittency modiﬁcations to the K41 theory. Finally, hypothesis (H4) relates all statistics to and , the only two possible relevant parameters, given (H1), (H4).

4.3

Kraichnan-Iroshnikov, Goldreich-Sridhar and all that: a scaling theory of MHD turbulence

We ﬁnally have arrived at the main topic of this paper, namely MHD turbulence in strongly magnetized systems. In this section, the focus will be exclusively on incompressible MHD, which for uniform B 0 = B0 z , is described by the well known equations for the coupled ﬂuid v and magnetic namely: ﬁeld B, · ∇B −∇P B0 ∂ B ∂ v + + ν∇2 v + f˜v , (4.11) + v · ∇ v = B+ ∂t ρ0 4πρ0 ∂z 4πρ0 ∂B · ∇ = B0 ∂ v + B + f˜ . + v · ∇B v + η∇2 B (4.12) m ∂t ∂z Here ρ0 is constant, and magnetic pressure has been absorbed into p. Equations 4.11-4.12 describe the evolution of two inter-penetrating ﬂuids, which are strongly coupled for large magnetic Reynolds number Rm ∼ v0 0 /η. is ‘frozen into’ the ﬂuid, up to the resistive dissipation. Equivalently put, B The system can have two external stochastic forcings f˜v and f˜m , though we take f˜m → 0 here. There are two control parameters, Re and Rm, or equivalently Rm and magnetic Prandtl number Pm = ν/η. For a strongly magnetized system, we are concerned with small scale turbulence consisting of amplitude ﬂuctuations with (|δB| < B0 ) and which are isotropic in the plane perpendicular to B 0 . Forcing is taken to be localized to large scales, and assumed to result in a mean dissipation rate

. Note that in contrast to the corresponding hydrodynamic system, MHD turbulence has two components or constituents, namely

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2 (i) shear Alfven waves, with frequency ωk = k|| vA , where vA = B02 /4πρ0 . Note that a shear Alfven wave is an exact solution of the incompressible MHD equations. In the absence of dissipation or non-Alfvenic perturbations, then, an Alfven wave will simply persist ad-inﬁnity.

(ii) ‘eddys’, namely zero frequency hyrdodynamic and magnetic cells, which do not bend magnetic ﬁeld lines (i.e. have k · B 0 = 0). Eddies are characterized by a ﬁnite self-correlation time or lifetime τk . For strong B0 , k|| vA > 1/τk , which is equivalent to |δB| B0 . Note that in MHD, the waves are high frequency with respect to ﬂuid eddies. Thus, as ﬁrst recognized by Kraichnan and Iroshnikov, two Alfven waves must beat together and produce a low frequency virtual mode, in order to interact with ﬂuid eddy turbulence [7, 8]. Such interaction is necessary for any cascade to small scale dissipation. Indeed, the generation of such non-Alfvenic perturbations is a key to the dynamics of MHD turbulence! At this point, it is instructive to discuss an analogy between magnetized MHD turbulence and Vlasov turbulence, the latter system a paradigm universally familiar to plasma physicists. Like MHD, Vlasov turbulence also consists of two constituents, namely collective modes or ‘waves’, and ‘particles’. For example, ion acoustic turbulence consists of ion-acoustic waves and ions. The analogue in MHD of the ‘collective mode’ is the Alfven wave, while the analogue of the ‘particle’ is the eddy. In both cases, the dispersive character of the collective modes (N.B.: Alfven waves are dispersive via anisotropy, since k|| = k · B0 / |B0 |. Most plasma waves of interest are also dispersive) implies that strong nonlinear interaction occurs when two waves interact to generate a low frequency ‘beat’ or virtual mode. In the case of Vlasov turbulence, such a low frequency beat wave may resonate and exchange energy with the particles, even if the primary waves are nonresonant (i.e. have ω >> kv). This occurs via the familiar process of nonlinear Landau damping, which happens when: ωk − ωk = (k − k )v.

(4.13)

In the case of MHD, the frequency and wave number matching conditions for Alfven wave interaction require that: k1 + k2 = k 3 ,

(4.14)

k||1 vA + k||2 vA = k||3 vA .

(4.15)

Thus, the only way to generate higher |k⊥ |, and thus smaller scales, through the coupling with vortical motion at ω ∼ 0 as in a cascade, is to have

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k||1 k||2 < 0, which means that the two primary waves must be counterpropagating! Note that counter-propagating waves necessarily generate low frequency modes, which resemble the quasi-2D eddies or cells referred to earlier. Indeed, for k||3 vA < ∼ 1/τk3 , the distinction between these two classes of ﬂuctuations is lost. Hence, in strongly magnetized MHD turbulence, interaction between counter-propagating populations generates smaller perpendicular scales, thus triggering a cascade. Note that parallel propagating packets cannot interact, as each Alfven wave moves at the same speed and is, in fact, an exact solution of the incompressible MHD equations. Instead, Alfven populations must pass through one another for cascading to occur (see Fig. 4.8 and Fig. 4.9). This seminal insight is due to Kraichnan and Iroshnikov.

t=0

t=t

Fig. 4.8

Counter-propagating Alfven wave streams interact.

t=0

t=t

Fig. 4.9

Parallel propagating wave streams do not interact.

We note here that the requirement of counter-propagating populations constrains the cross-helicity of the system. The Elsasser variables Z+ , where Z+ = v + B,

(4.16)

each correspond to one of the two Elsasser populations. The net imbalance

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in the two population densities is thus N+ − N− = Z + · Z + − Z − · Z − = 4v · B, where the total cross helicity is just Hc = d3 xv · B.

(4.17)

(4.18)

Thus, for a system with counter propagating populations of equal intensity, Hc necessarily must vanish. Similarly, maximal cross helicity (|v · B| = (|v|2 |B|2 )1/2 ) implies that either N+ = 0 or N− = 0, meaning that no Alfven wave cascade can occur. Hereafter in this section, we take Hc = 0. We now present a heuristic derivation of the MHD turbulence spectrum produced by the Alfven wave cascade [9, 10]. As in the K41 theory, the critical element is the lifetime or self-correlation time of a particular mode k. Alternatively put, we seek a time scale τk such that (v · ∇v)k ∼ vk /τk .

(4.19)

This is most straightforwardly addressed by extracting the portion of the nonlinear mixing term which is phase coherent with the ‘test mode’ of interest. Thus, we wish to determine (2) v −k v k+k , (4.20) vk /τk = k · k

(2)

where vk+k is determined via perturbation theory by solving: (2) ∆ωk v k

−

ik|| 4πρ0

(1) (2) = v (1) B0 B vk , k+k k · k

(2) (2) k . ∆ωk B k = B0 ik|| v

(4.21) (4.22)

Here k = k + k , ∆ωk is the self-correlation rate of the best mode, and nonlinearities other than v · ∇v are ignored. This results in no loss of generality, as all nonlinear couplings are of comparable strength in the case of nonlinear Alfven interaction. Most important of all, we take the k virtual mode to be low frequency since, as discussed above, such interactions maximize the power transfer to small scales. Equations 4.20, 4.21 and 4.22 then yield: 1/∆ωk 2 |k · v k | , (4.23) 1/τk = 1 + (k|| vA /∆ωk )2 k

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which, for kz vA > ∆ωk , reduces to: 1/τk = |k · vk |2 πδ(k|| vA ).

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(4.24)

k

Note that Eq. 4.24 is equivalent to the estimate 1/τk ∼ k |k· vk |2 τac|| , where τac|| ∼ 1/|∆k|| vA is the auto-correlation time of the Alfven spectrum. Here, ∆k|| refers to the bandwidth of the k|| spectrum. Of course, the need for counter-propagating populations emerges naturally from the resonance condition. Similarly, anisotropy is clearly evident, in that the coupling coeﬃcients, (i.e. k⊥ · k ⊥ × z ), depend on k⊥ while the selection rules depend on k|| . Finally, the correspondence with nonlinear Landau damping in Vlasov turbulence is also clear. For that process,   ∂f  |Ek |2 /τk ∼  |Ek |2 F (k, k )πδ(ωk + ωk − (k + k )v)vT2 |Ek |2 ∂v vb k

(4.25) where f (k, k ) refers to a coupling function and interaction occurs at the beat phase velocity vb = (ω + ω )/(k + k ) [11]. Having derived the correlation time, we now can proceed to determine the spectrum. In the interests of clarity and simplicity, we derive a scaling relation, using the expression for τk given in Eq. 4.23. Despite the facts that: (i) there are no a priori theoretical reasons or well documented experimental evidence that energy transfer in MHD turbulence is local in k, (ii) there is no reason whatsoever to expect that the (as yet unproven!) anomaly or ﬁnite time singularity which underlies the independence of from dissipation in hydrodynamic turbulence should necessarily persist in MHD, we plunge ahead and write a cascade energy transfer balance relation. The old proverb, “Fools rush in, where angels fear to tread” comes vividly to mind at this point. However, so does another ancient aphorism, “Nothing ventured, nothing gained”. Anticipating the role of anisotropy, the transfer balance relation is:

= v( ⊥ )2 /τ ( ⊥ ), where 1/τ ( ⊥ ) = 1/τk =

k

|k · vk |2 πδ(k|| vA ) ∼ =

(4.26) 1 v( ⊥ )2 ,

2⊥ k|| vA

(4.27)

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so

=

v( ⊥ )4 .

2⊥ k|| vA

(4.28)

Those readers who are skeptical of the simple arguments presented in the past few paragraphs can arrive at Eq. 4.28 by the even simpler reasoning that, as is generic in weak turbulence theory, the energy transfer will have the form

∼ (coupling coeﬃcient)2 ∗ (interaction time) ∗(scatter − er energy) ∗ (scatter − ee energy).

(4.29)

Taking the coupling ∼ 1/ ⊥, interaction time ∼ 1/k|| vA , and scatterer and scatteree energy ∼ v( ⊥ )2 then yields Eq. 4.28. In comparison to the familiar (and deceptive) relation = v( )3 / for K41 turbulence, Eq. 4.28 contains two new elements, namely: (a) anisotropy - the distinction between perpendicular and parallel remains, (b) reduction in transfer note - notice that in comparison to its hydrodynamic counterpart, energy transfer in MHD turbulence is reduced by a factor of v⊥ / ⊥ k|| vA , the ratio of a parallel Alfven transit time to a perpendicular eddy shearing rate, which is typically much less than unity. The reduction in transfer rate in comparison to hydrodynamic turbulence is commonly referred to as the Alfven eﬀect. The Alfven eﬀect is a consequence of the enhanced memory of MHD turbulence, as compared to that of hydrodynamic turbulence. The memory enhancement is due to the reversibility intrinsic to Alfven waves. It is now possible to consider several related cases and incarnations of the MHD cascade. First, we revisit the original paradigm of Iroshnikov and Kraichnan. Here, we consider a weakly magnetized system, where Brms >> B0 . Note that in contrast to hydrodynamics, Alfvenic interaction in MHD is not constrained by Galilean invariance. Thus, Equation ˜ 2 1/2 . Furthermore, as there is no (14c) applies, with B0 → Brms = B large scale anisotropy (B0 is negligible!), we can dare to take k|| ⊥ ∼ 1, so that the energy transfer balance Eq. (4.28) becomes:

∼ v( )4 / ˜ vA ,

(4.30)

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˜ where v˜A = vA computed with Brms. The value of Brms is dominated by the large eddies, and is sensitive to the forcing distribution and the geometry. In this system, the rms ﬁeld is not straight, but does possess some large scale order. Thus, ‘here the Alfven waves’ should be thought of as propagating along a large scale ﬁeld with some macroscopic correlation length and a stochastic component. This in turn immediately gives: v( ) ∼ 1/4 ( ˜ vA )1/4 ,

(4.31)

and (4.32) E(k) ∼ ( ˜ vA )1/2 k −3/2 , ! where we use the normalization dkE(k) = Energy. Equation (16c) gives the famous Iroshnikov-Kraichnan (I.-K.) spectrum for weakly magnetized incompressible MHD turbulence. Concomitant with the departure from k −5/3 , reconsidering the onset of dissipation when (for Pm = 1) ν/ 2d = vA / )1/3 . Interestingly, v( ⊥ )/ ⊥ gives the I.-K. dissipation scale d = ν 2/3 (˜ there is nothing in this argument which is speciﬁc to three dimensions! Indeed, since the J × B force breaks enstrophy conservation for inviscid 2D MHD, a forward cascade of energy is to be expected there, ab initio. Thus, it is not completely surprising that the results of detailed, high resolution numerical simulations of 2D MHD turbulence are in excellent agreement with both the I.-K. spectrum and dissipation scale [12]. The success of the I.-K. theory in predicting the properties of weakly magnetized 3D MHD will be discussed later in this article. Finally, we note that two rather subtle issues have been ‘swept under the rug’ in this discussion. First, the ˜ large scale ﬁeld Brms is tangled, with zero mean direction but with a local coherence length set by the turbulence integral scale. Thus, while there is no system averaged anisotropy, it seems likely that strong local anisotropy will occur in the turbulence. The theory does not account for this local anisotropy. Second, it is reasonable to expect that some minimum value ˜ of Brms is necessary to arrest the inverse energy cascade, characteristic of 2D hydrodynamics, and to generate a forward cascade. The scaling of this ˜ Brms and possible dependence on forcing scaling and statistics are as yet unknown. Both of the subtle issues mentioned here are topics of active, ongoing research. We now turn to the case of strongly magnetized, anisotropic turbulence. In that case, Eq. 4.28 states the energy ﬂux balance condition, which is

∼

1 v( ⊥ )4 .

2⊥ k|| vA

(4.33)

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Here again v( ⊥ )/( ⊥ k|| vA ) < 1. Now !using !the normalization for an anisotropic spectrum where (Energy E = dk|| dk⊥ E(k|| , k⊥ )), Eq. 4.33 directly suggests that 2 , E(k⊥ ) ∼ ( k|| vA )1/2 /k⊥

(4.34)

a steeper inertial range spectrum than that predicted by I.-K. for the weakly magnetized case. Note that consistency with the ordering |δB| < B0 , or equivalently v( ⊥ )/ ⊥ < k|| vA , requires that 1/3

⊥ 1/3 /vA ≤ k|| ⊥ 1,

(4.35)

symptomatic of the anisotropic cascade of Goldreich and Sridhar (G.-S.) [13, 14]. It is interesting to note that Eq. 4.35 says that the anisotropy increases as the cascade progresses toward smaller scales, so that initially spheroidal eddies on integral scales produce progressively more prolate and extended (along B0 ) eddies on smaller (cross-ﬁeld) scales, which ultimately fragment into long, thin cylindrical ‘rods’ on the smallest inertial range scales. This anisotropic cascade process is compared to the isotropic eddy fragmentation picture of Kolmgorov in Fig. 4.10. Recognition of the intrinsically anisotropic character of the strongly magnetized MHD cascade was the important contribution of the series of papers by Goldreich and Sridhar.

B0

Fig. 4.10 Comparison of the isotropic Kolmogorov cascade with the anisotropic Alfven turbulence cascade. In the latter case, anisotropy increases as the cascade progresses.

A particularly interesting limit of the anisotropic MHD cascade is the “critically balanced” or “marginally Alfvenic” cascade, which occurs in the

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limiting case where v⊥ ( ⊥ )/ ⊥ ∼ k|| vA , i.e. when the parallel Alfven wave transit time through an (anisotropic) eddy is equal to the perpendicular straining or turn-over time of that eddy. In this limit, Eq. 4.33 reduces to ∼ v( ⊥ )3 / ⊥ , (i.e. back to K41!) albeit with rather diﬀer−5/3 ent physics. Thus in the critically balanced cascade, E(k⊥ ) ∼ = 2/3 k⊥ 1/3 2/3 and k|| ⊥ ∼ = ⊥ 1/3 /vA , so that k|| ∼ k⊥ 1/3 /vA , which deﬁnes a trajectory or ‘cone’ in k space along which the cascade progresses. On this cone −5/3 (taken dominant here), one has the spectrum E(k⊥ ) ∼ k⊥ . Note that for v( ⊥ )/ ⊥ > k|| vA , the turbulence shearing rate exceeds the Alfven transit rate, so the dynamics are eﬀectively ‘unmagnetized’ and so the spectrum will approach that of I.-K. in that limit. We can summarize this zoology of MHD turbulence spectra by considering a magnetized system with ﬁxed ν = η and variable forcing. As the forcing strength increases, so that increases at ﬁxed B0 , ν, η, the turbulence spectra should transition through three diﬀerent stages. These three stages correspond, respectively, to: 2 and (i) ﬁrst, the anisotropic cascade, with E(k⊥ ) ∼ ( k|| vA )1/2 /k⊥ 1/3

k|| ⊥ > ⊥ 1/3 /vA throughout the inertial range, (ii) then, the critically balanced anisotropic cascade, with E(k⊥ ) ∼ −5/3 2/3 and k|| ∼ k⊥ 1/3 /vA throughout the inertial range

2/3 k⊥ (iii) and ﬁnally, the weakly magnetized cascade for Brms > B0 , with E(k⊥ ) ∼ ( v˜A )1/2 k −3/2 and k isotropic, on average. Note that the spectral power law index decreases with increasing stirring strength, at ﬁxed B0 . After reading through all this theory, the patient reader surely is entitled to a discussion of just how well the theory performs when compared to numerical calculations. As discussed before, the weakly magnetized I.-K. cascade theory is quite successful in explaining 2D MHD turbulence at moderate Re with P m = 1. Three numerical calculations for strong B0 in 3D have recovered results which agree with the predictions of Goldreich and Sridhar, albeit only over intervals of scale of a decade, or less [15–17]. Interestingly, the numerical study with the best resolution to date yields a spectrum which appears closer (for strongly magnetized 3D!) to the I.−3/2 K.-like prediction of E(k⊥ ) ∼ k⊥ than the G.-S. predictions [18]. The

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deviation from G.-S. scaling may be due to intermittency corrections or to a more fundamental departure from the physical picture of G.-S. In −3/2 particular, it is tantalizing to speculate that the E(k⊥ ) ∼ k⊥ scaling at strong B0 result suggests that the turbulence assumes a quasi-2D structure consisting of extended columns along B0 . The viability of this speculation is strengthened by the observation of a clear departure from the accompanying 2/3 k|| ∼ k⊥ scaling also predicted by G.-S., though perpendicular vs. parallel anisotropy clearly remains. In physical terms it seems plausible that the turbulence might form such a quasi-2D state, since:

(i) a state of extended columns aligned with the strong B 0 is the ‘TaylorProudman state’ for the system. Such a state naturally minimizes the energy spent on magnetic ﬁeld line bending, which is necessary for Alfven wave generation. (ii) a state of extended, ﬁeld-aligned columns which are re-arranged by approximately horizontal eddy motions is also the state in which the translational symmetry along B0 , which is broken by the excitation mechanism, is restored to the maximal extent.

Thus, formation of such a quasi-2D state seems consistent with considerations of both energetics and of probability. Further detailed study of the k|| and k⊥ spectra is required to clarify the extent and causes of the apparent two dimensionalization. This issue is one of the most fundamental ones confronting researchers in MHD turbulence today. Of course, diﬃcult to believe as it may be, there is a lot more to understanding MHD turbulence than simply computing spectral indexes. The nature of the dissipative structures in 3D MHD turbulence remains a mystery, and the dynamical foundations of intermittency eﬀects are not understood. In 2D, numerical studies suggest that inertial range energy may be dissipated in current sheets, but much further study of this phenomenon is needed. In both 2D and 3D, the structure of the probability distribution function of hydrodynamic and magnetic strain (i.e. ∇ v and ∇ B) at high Rm and Re remains terra incognita. Finally, the dependence of the large scale structure of Brms upon stirring properties, geometry, etc. has not been addressed. Note that this structure ultimately is responsible for the breaking of local rotational symmetry and the origin and extent of domains of local anisotropy in 2D MHD turbulence.

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Steepening of nonlinear Alfven waves - a little compressibility goes a long way...

At this point, the alert reader may be wondering how the nonlinear evolution of Alfven waves proceeds in the absence of counter-propagating wave streams. This is an important question, since many physical situations and systems do involve nonlinear Alfven dynamics but do not have counterpropagating wave streams of comparable intensity. Indeed, any situation involving emission of Alfven waves from an astrophysical body (i.e. star) falls into this category. The answer, of course, is that introduction of even modest compressibility (i.e. parallel compressibility, associated with acoustic perturbations) is suﬃcient to permit the steepening of uni-directional shear Alfven wave packets [19]! Wave steepening then generates small scales by the familiar process of shock formation. Steepening terminates in either dissipation at small scales, as in a dissipative or collisional shock, or the arrest of steepening by dispersion, as in the formation of a collisionless shock or solitary wave. Alfven wave steepening is thus the ‘mechanism of (nature’s) choice’ for generating small scales in uni-directional wave spectra, and naturally complements the mechanism of low frequency beat generation, which is the key to the Alfvenic wave cascade in counter-propagating wave streams. Quasi-parallel Alfven wave steepening is especially important to the dynamics of the solar wind, since high intensity streams of outgoing Alfven waves are emitted from solar coronal holes. These high intensity wave streams play a central role in generating and heating the ‘fast solar wind’. We now present a simple, physical derivation of the theory of Alfven wave steepening due to parallel compressibility. Just as in the case of shock formation in gas dynamics, Alfven wave trains steepen in response to modulations in density. As in gas dynamics, the density dependence of the wave speed (here the Alfven speed) is the focus of the modulational coupling. So, starting from the Alfven wave dispersion relation " ω = k|| vA = k|| B0 / 4π(ρ0 + ρ˜), (4.36) where a localized density perturbation ρ˜ enters the wave speed. Straightforward expansion gives an ‘envelope’ equation for the slow space and time variation of the wave function of the perturbation

δB, i.e. vA ∂ ρ˜ ∂δB =− δB . (4.37) ∂t 2 ∂z ρ0 We understand that, in the spirit of reductive perturbation theory, ρ˜(2) /ρ0 is second order in perturbation amplitude. Here, “perturbation” refers to a

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modulation of the uni-directional Alfven wave train. We assume that this modulation has parallel scale L|| > 2π/k|| . ρ˜(2) /ρ0 is easily determined by considering of the parallel ﬂow dynamics. In addition to the linear acoustic force, parallel forces are also induced by the gradient of the carrier Alfven wave energy ﬁeld, i.e. since |v|2 − v × ω, 2 |B|2 + B · ∇ B, J × B = −∇ 2 and since z · (v × ω) = z · (B · ∇ B) = 0, to second order, we have

∂ |δB|2 |δv|2 ∂ ρ ∂ − + v|| = −c2s . ∂t ∂z ρ0 ∂z 8πρ0 2 v·∇v = ∇

(4.38) (4.39)

(4.40)

Note that the parallel gradient of the ponder motive pressure of the Alfven wave train drives the parallel ﬂow perturbation, which then couples to the density perturbation. The loop of couplings is closed by the linearized continuity equation relating v || to ρ /ρ0 , i.e. ∂ ∂ ρ = − v || . ∂t ρ0 ∂z

(4.41)

Equations (21c.) and (21d.) may then be combined to obtain 2

2 ∂ 2 |δB|2 ∂ ρ 2 ∂ − c = , (4.42) s ∂t2 ∂z 2 ρ0 ∂z 2 4πρ0 √ where we have used the fact that v⊥ ∼ δB/ 4πρ0 for Alfven waves. At this point, it is convenient to transform to a frame of reference comoving with the Alfven carrier wave, so that ρ = ρ (z − vA t), etc. In this frame, we can simplify Eq. 4.42 to:

|δB|2 1 ρ /ρ0 = , (4.43) (1 − β) B02 where β = 8πPth /B02 . Substituting ρ /ρ0 into the wave function equation for δB gives 2 δB ∂ vA ∂ δB + δB = 0. (4.44) ∂t ∂z 2(1 − β) B0 As mentioned above, the fast Alfvenic dependence of δB has already cancelled, so this equation almost fully describes the slow dependence of the perturbation envelope. Equation 4.44 describes the steepening of an Alfven wave train. One more ingredient is necessary, however - namely

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a term which represents possible limitation and saturation of the steepening, once it generates suﬃciently small scale. This is accomplished by adding a diﬀusion and/or dispersion term to Eq. 4.44, such as η∂ 2 δB/∂z 2 or id2i Ωi ∂ 2 δB/∂z 2 , respectively. In that case, the envelope equation for δB becomes the well known Derivative Nonlinear Schr¨ odinger (DNLS) equation 2 ∂ δB ∂ vA δB δB + ∂t ∂z 2(1 − β) B0 ∂2 ∂2 δB + id2i Ωi 2 δB. (4.45) 2 ∂z ∂z Here di = c/ωpi , the ion inertial scale, and Ωi is the ion cyclotron frequency [20]. In most expositions and discussions, the resistive dissipation term is dropped, and ion inertial scale physics (associated with Hall currents, etc.) is invoked to saturate Alfvenic steepening by dispersion. Thus, the stationary width of a modulated Alfven wave train is set by the balance of steepening with dispersion, and so the steepened Alfven wave packet is often referred to as a quasi-parallel Alfvenic collisionless shock. In contrast to systems with counter-propagating Alfven streams, in a uni-directional wave train modulations can generate small scale via a coherent process of wave train steepening, which is ultimately terminated via balance with small scale dispersion. The physics of the steepening process encapsulated by the back-of-anenvelope (albeit a large one!) calculation presented here can also be described graphically, by a series of cartoons, as shown in Fig. 4.11. The unperturbed Alfven wave train is shown in Fig. 4.11, and its modulation (a parallel rarefaction) is shown in Fig. 4.12. The modulation induces a perturbation in the pondermotive energy ﬁeld of the wave train, which in turn produces a pondermotive force couple (i.e. dyad) along B 0 , as shown in Fig. 4.13. Note that the resulting parallel ﬂow is yet another example of a Reynolds stress driven ﬂow, though in this case, the ﬂow is along B 0 and a diagonal component of the Reynolds stress tensor is at work, symptomatic of the fact that the ﬂow is compressible. The resulting parallel ﬂow re-enforces δB via ∇ × v × B, as depicted in Fig. 4.14, thus enhancing the initial modulation. At this point, the alert reader is no doubt wondering about what happens to Eq. 4.45 when β → 1?! This natural question touches on two interesting issues in the theory of Alfvenic steepening. First, it should be readily apparent that the crucial nonlinear eﬀect in this story is the second order parallel ﬂow, driven by the parallel pondermotive force. Thus, any =η

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B

0

Fig. 4.11

Unperturbed wave train and its envelope.

Fig. 4.12

Localized modulational perturbation.

Fig. 4.13

Force couple along B 0 .

dissipation, dephasing, etc. such as parallel viscosity, Landau damping, etc., (which are surely present but not explicitly accounted for) immediately resolves the β → 1 singularity and also can be expected to have an impact on the steepening process for a range of β values. An extensive literature on the important topic of dissipative and kinetic modiﬁcations to the DNLS theory exists. One particularly interesting generalization of the DNLS is the KNLS or KDNLS, i.e. the kinetic nonlinear Schr¨ odinger equation or the k-derivative - NLS [21–23]. A second point is that for β = 1, the sound and Alfven speeds are equal, so it no longer makes sense to ‘slave’ the

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Fig. 4.14

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Growth of modulation and steepening of initial perturbation.

density perturbation to the Alfven wave. Rather, the acoustic and Alfven dynamics must be treated on an equal footing, as in the analysis by Hada [24]. The DNLS is integrable, via the inverse scattering method. The KNLS, an integro-diﬀerential equation, is not so easily tractable, but its numerical solutions seemingly can ‘explain’ MHD shock phenomena observed in the solar wind, such as rotational discontinuities. The moral of this little story is, then, that one should take care to avoid a tunnel vision focus on only the incompressible theory. Indeed, in this section, we saw that introducing weak compressibility completely changed the nonlinear Alfven wave problem, by: (i) allowing strong nonlinear interaction and wave steepening, leading to the formation of shocks, solitons and other structures. (ii) allowing a mechanism for the nonlinear evolution of a uni-directional wave train. Thus, the alert reader should be wary of exclusive reliance upon the I.-K., G.-S. theory and its perturbative ﬁx-ups as a framework for understanding nonlinear Alfven phenomena. Rather, one might more proﬁtably expect that most natural Alfvenic turbulence phenomena will involve some synergism between the incompressible dynamics ala’ I.-K., G.-S. and the compressible, DNLS-like steepening dynamics. Indeed, recent numerical studies of weakly compressible MHD turbulence have shown both a cascade to small scales in the perpendicular direction and the formation of residual DNLS-like structures along the ﬁeld to be at work in the nonlinear dynamics! A theoretical understanding of such weakly compressible MHD turbulence remains elusive.

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Turbulent ﬂux diﬀusion in 2D MHD - a ‘minimal’ problem which is not so simple...

Up until now, our discussion has focused primarily on the structure and dynamics of MHD turbulence. In this section, we shift gears somewhat, to discuss the mean ﬁeld theory of magnetic ﬂux diﬀusion in two dimensions [25]. This is, no doubt, the simplest, “minimal” problem in the theory of mean ﬁeld electrodynamics of a turbulent magnetic ﬂuid. However, as we shall see, even the ‘simple’ problem is not so simple. Indeed, the problem of ﬂux diﬀusion is a splendid example of the impact of ‘dynamical memory’ or ‘elasticity’, both of which are intrinsic to Alfvenic turbulence, upon transport. The upshot of this elasticity in turbulence is the prediction that turbulent diﬀusion is severely quenched, in comparison to its expected kinematic value. A similar ﬁnding is relevant to the alpha eﬀect in three dimensions. The equations of 2D MHD are ∂A + (∇φ × z ) · ∇A = η∇2 A, ∂t

(4.46)

∂ 2 ∇ φ + (∇φ × z ) · ∇∇2 φ = (∇A × z ) · ∇∇2 A + ν∇2 ∇2 φ, (4.47) ∂t where A is the magnetic potential (B = ∇ × A z ), φ is the velocity stream function (v = ∇ × φ z ), η is the resistivity, ν is the viscosity and z is the unit vector orthogonal to the plane of motion. We shall consider the case where the mean magnetic ﬁeld is in the y-direction, and is a slowly varying function of x. Equations! 4.46 and 4.47 have non-dissipative quadratic 2 2 2 magnetic invariants, the energy ! 2 2 E = [(∇A) + (∇φ) ]d!x, mean-square potential HA = A d x and cross helicity Hc = ∇A·∇φd2 x. Throughout this section, we take Hc = 0 ab initio, so there is no net Alfvenic alignment in the MHD turbulence considered here. The basic dynamics of 2D MHD turbulence are well understood [26]. For large-scale stirring, energy is self-similarly transferred to small scales and eventual dissipation via an Alfvenized cascade, as originally suggested by Kraichnan and Iroshnikov, and clearly demonstrated in simulations. Mean square magnetic potential HA , on the other hand, tends to accumulate at (or cascade toward) large scales, as is easily demonstrated by equilibrium statistical mechanics for non-dissipative 2D MHD. Here, Hc is the second conserved quadratic quantity (in addition to energy), which thus suggests a dual cascade. In 2D, the mean ﬁeld quantity of interest is the spatial ﬂux of magnetic potential ΓA = vx A. An essential element of the physics

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of ΓA is the competition between advection of scalar potential by the ﬂuid, and the tendency of the ﬂux A to coalesce at large scales. The former is, in the absence of back-reaction, simply a manifestation of the fact that turbulence tends to strain, mix, and otherwise “chop up” a passive scalar ﬁeld, thus generating small-scale structure. The latter manifests the fact that A is not a passive scalar, and that it resists mixing by the tendency to coagulate on large scales. The inverse cascade of A2 , like the phenomenon of magnetic island coalescence, is ultimately rooted in the fact that like-signed current ﬁlaments attract. Not surprisingly then, the velocity ﬁeld drives a positive potential diﬀusivity (turbulent resistivity), while the magnetic ﬁeld perturbations drive a negative potential diﬀusivity. Thus, we may anticipate a relation for the turbulent resistivity of the form ηT ∼ v 2 − B 2 , a considerable departure from expectations based upon kinematic models. A similar competition between mixing and coalescence appears in the spectral dynamics. Note also that ηT vanishes for turbulence at exact Alfvenic equipartition (i.e., v 2 = B 2 ). Since the presence of even a weak mean magnetic ﬁeld will naturally convert some of the ﬂuid eddies to Alfven waves, it is thus not entirely surprising that questions arise as to the possible reduction or “quenching” of the magnetic diﬀusivity relative to expectations based upon kinematics. Also, note that any such quenching is intrinsically a synergistic consequence of both: (i) the competition between ﬂux advection and ﬂux coalescence intrinsic to 2D MHD, (ii) the tendency of a mean magnetic ﬁeld to “Alfvenize” the turbulence. The close correspondence between the problems of 2D ﬂux diﬀusion and that of the 3D mean ﬁeld electromotive force is remarkable. The 3D EMF is central to the theory of the turbulent dynamo. Both seek a representation of a mean product of ﬂuid and magnetic ﬂuctuations in terms of local transport coeﬃcients. In each case, the magnetic dynamics are critically constrained by the conservation, up to resistive dissipation, of magnetic helicity in 3D and of HA in 2D. Both magnetic helicity and HA inverse cascade to large scales, and thus produce an interesting dual cascade, since energy ﬂows to small scales in each case. The inverse cascade of magnetic helicity and mean-square potential underpin the appearance of magnetic “back-reaction” contributions to α and ηT . Thus, both tend to vanish for fully Alfvenized turbulence. This trend, then, naturally suggests the possibility of both α-quenching in 3D, and magnetic diﬀusivity quenching in 2D. Of course, there are crucial diﬀerences between the two problems.

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Obviously, in 2D only decay of the magnetic ﬁeld is possible, whereas 3D admits the possibility of dynamo growth. Furthermore, magnetic helicity and α (the pertinent quantities in 3D) are pseudo-scalars while HA and ηT are scalars; thus, the eﬀect of helicity conservation on β, the magnetic diﬀusivity in three dimensions, remains far from clear. An important element of the basic physics, common to both problems, is the process of “Alfvenization”, whereby ﬂuid eddy energy is converted to Alfven wave energy. This may be thought of as a physical perspective on the natural trend of MHD turbulence toward an approximate balance between ﬂuid and magnetic energies, for Pm ∼ 1. Note also that Alfvenization may be thought of as the development of a dynamical memory, which constrains and limits the cross-phase between vx and A. This is readily apparent from the fact that vx A vanishes for Alfven waves in the absence of resistive dissipation. For Alfven waves then, ﬂux diﬀusion is directly proportional to resistive dissipation, an unsurprising conclusion for cross-ﬁeld transport of ﬂux which is, in turn, frozen into the ﬂuid up to η. As we shall soon see, the ﬁnal outcome of the quenching calculation also reveals the explicit proportionality of ηT to η. For small η, then, ΓA will be quenched. Another perspective on Alfvenization comes from the studies of Lyapunov exponents of ﬂuid elements in MHD turbulence. These showed that as small-scale magnetic ﬁelds are ampliﬁed and react back on the ﬂow, Lyapunov exponents drop precipitously, so that chaos is suppressed [27]. This observation is consistent with the notion of the development of a dynamical memory. A key element in our discussion of ﬂux diﬀusion in 2D MHD is the Zeldovich theorem, which is an expression of the balance between turbulent transport and resistive dissipation for a stationary, 2D MHD system [28]. The Zeldovich theorem is derived by multiplying the magnetic potential equation by A and integrating over space, yielding

∂A 1 ∂ 2 (4.48) A + ∇ · (vA2 ) = −vx A − ηB 2 . 2 ∂t ∂x We assume a clear-cut separation of scales between mean quantities and ﬂuctuations. For a periodic domain and a stationary state, the relation above reduces to ηT vx A ∂A = B2 , (4.49) B 2 = − η ∂x η where we have used Fick’s law to represent ΓA . Equation (27) states the Zeldovich theorem.

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The Zeldovich theorem, as expressed in Eq. 4.49, has several interpretations and implications. We list these below. (i) It indicates that the eﬀective turbulent resistivity ηT must scale directly with the collisional resistivity η, in proportion to B 2 /B2 . Note that B 2 = (∇A)2 itself is ﬁnite as η → 0 (consider the I.-K. spectrum, for example), so there is no singularity. This is in distinct contrast to the case of a passive scalar concentration ﬁeld c(x, t), where (∇˜ c)2 diverges in the absence of scalar diﬀusivity. (ii) It states that the mean square ﬂuctuation level B 2 can be large, even if the mean ﬁeld B is weak, i.e. B 2 /B2 ∼ Rm >> 1. (iii) It may be taken as a statement of Prandtl mixing-length theory for the magnetic potential. This is because Eq. 4.49 states an equality between the decay rate of the mean magnetic potential (∼ ηT (∂A/∂x)2 - i.e. the rate at which large scales are dissipated) and ηB 2 , the dissipation rate on small scales. Such a balance constitutes an important constraint on the mean magnetic ﬂux transport, ΓA . Now we discuss the mean ﬁeld theory of ﬂux diﬀusion in 2D. In the discussion of ΓA , we do not address the relationship between the turbulent velocity ﬁeld and the mechanisms by which the turbulence is excited or stirred. However, a weak large-scale ﬁeld (the transport of which is the process to be studied) will be violently stretched and distorted, resulting in the rapid generation of a spectrum of magnetic turbulence. As discussed above, magnetic turbulence will likely tend to retard and impede the diﬀusion of large-scale magnetic ﬁelds. This, of course, is the crux of the matter, as ΓA depends on the full spectrum arising from the external excitation and the back-reaction of the magnetic ﬁeld, so, as suggested above, the net imbalance of v 2 and B 2 determines the degree of ηT quenching. Leverage on B 2 is obtained by considering the evolution of mean-square magnetic ! potential density HA . In particular, the conservation of HA = HA d2 x straightforwardly yields the identity ∂A 1 ∂HA = −ΓA − ηB 2 , 2 ∂t ∂x

(4.50)

where the surface terms vanish for periodic boundaries. For stationary turbulence, then, this give

2 ΓA ∂A ηT ∂A 2 , (4.51) B = − = η ∂x η ∂x

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which is the well-known Zeldovich theorem discussed earlier. The key message here is that when a weak mean magnetic ﬁeld is coupled to a turbulent 2D ﬂow, a large mean-square ﬂuctuation level can result, on account of stretching iso-A or ﬂux contours by the ﬂow. To calculate ΓA , standard closure methods yield ΓA = [vx−k δAk − Bx−k δφk ] = ΓA (k ), (4.52) k

k

where δA(k) and δφ(k) are, in turn, driven by the beat terms (in 4.46 and 4.47) that contain the mean ﬁeld A. The calculational approach here treats ﬂuid and magnetic ﬂuctuations on an equal footing, and seeks to determine ΓA by probing an evolved state of MHD turbulence, rather than a kinematically prescribed state of velocity ﬂuctuations alone. The calculation follows those of Pouquet et al. and yields the result ∂A ΓA = − τcφ (k )v2 k − τcA (k )B 2 k ∂x k ∂ τcA (k )A2 k J. (4.53) − ∂x k √ The magnetic ﬁeld is expressed in units of velocity (i.e. 4πρ0 ≡ 1). Here, consistent with the restriction to a weak mean ﬁeld, isotropic turbulence is assumed. The quantities τcφ (k) and τcA (k) are the self-correlation times (lifetimes), at k, of the ﬂuid and ﬁeld perturbations, respectively. These are not at all necessarily equivalent to the coherence time of vx (−k ) with A(k ), which determines ΓA . For a weak mean ﬁeld, both τcφ (k) and τcA (k) are determined by nonlinear interaction processes, so that 1/τcφ,A (k ) ≥ k B, i.e., ﬂuctuation correlation times are short in comparison to the Alfven time of the mean ﬁeld. In this case, the decorrelation process is controlled by the Alfven time of the r.m.s. ﬁeld (i.e., [kB 2 1/2 ]−1 ) and the ﬂuid eddy turnover time. Consistent with the assumption of unity magnetic Prandtl number, we take τcφ (k) = τcA (k) = τc (k), hereafter. The three terms on the right-hand-side Eq. 4.53 correspond respectively to (a) a positive turbulent resistivity (i.e., ΓA proportional to ﬂux gradient) due to random ﬂuid advection and straining of ﬂux, (b) a negative turbulent resistivity symptomatic of the tendency of magnetic ﬂux to accumulate on large scales, (c) a positive turbulent hyper-resistive diﬀusion, which gives ΓA proportional to current gradient. Such diﬀusion of current has been proposed

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as the mechanism whereby a magnetoﬂuid undergoes Taylor relaxation [29, 30]. Note that terms (b) and (c) both arise from Bx (k)δφ(k ), and show the trend in 2D MHD turbulence to pump large-scale HA while damping smallscale HA . For smooth, slowly varying mean potential proﬁles, the hyperresistive term is negligible in comparison with the turbulent resistivity, (i.e., k2 > (1/A)(∂ 2 A/∂x2 )), so that the mean magnetic potential ﬂux reduces to ∂A , (4.54) ΓA = −ηT ∂x where

(4.55) ηT = τc (k ) v 2 k − B 2 k . k

As stated above, the critical element in determining ΓA is to calculate B 2 k in terms of v2 k , ΓA itself, etc. For this, mean-square magnetic potential balance is crucial! To see this, note that the Zeldovich theorem states that ˜ 2 = −ΓA ∂A , B (4.56) η ∂x assuming incompressibility of the ﬂow. An equivalent, k-space version of Eq. 4.56 is ∂A 1 ∂ 2 A k + T (k) = −ΓA (k) − ηB 2 k , (4.57) 2 ∂t ∂x where T (k) is the triple correlation T (k) = ∇ · (vA2 )k ,

(4.58)

which controls the nonlinear transfer of mean-square potential, and ΓA (k) = vx Ak is the k-component of the ﬂux. Equations (35) and (36) thus allow the determination of B 2 and B 2 k in terms of ΓA , ΓA (k), T (k), etc. At simplest, crudest level (the so-called) τ -approximation), a single τc is assumed to characterize the response or correlation time in Eq. 4.55. In that case, we have ∂A . (4.59) τc (v2 k − B 2 k ) ΓA = − ∂x k

For this, admittedly over-simpliﬁed case, Eq. 4.59 then allows the determination of B 2 in terms of ΓA , the triplet and ∂t A2 . With the additional restrictions of stationary turbulence and periodic boundary conditions (so that ∂A2 /∂t = 0 and ∇ · (vAA) = 0), it follows that

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B 2 = −(ΓA /η)∂A/∂x, so that magnetic ﬂuctuation energy is directly proportional to magnetic potential ﬂux, via HA balance. This corresponds to a balance between local dissipation and spatial ﬂux in the mean-square potential budget. Inserting this into Eq. 4.59 then yields the following expression for the turbulent diﬀusivity: 2 ηk k τc v k ηT = = , (4.60) 2 2 1 + τc vA0 /η 1 + Rm vA0 /v 2 where η k refers to the kinematic turbulent resistivity τc v 2 , vA0 is the Alfven speed of the mean B, and Rm = v 2 τc /η. It is instructive to note that Eq. 4.60 can be rewritten as ηT =

ηηk 2 . η + τc vA0

(4.61)

Thus, as indicated by mean-square potential balance, ΓA ultimately scales directly with the collisional resistivity, a not unexpected result for Alfvenized turbulence with dynamically interesting magnetic ﬂuctuation intensities. This result supports the intuition discussed earlier. It is also inter2 /v2 > 1 and v2 ∼ B 2 , ηT ∼ esting to note that for Rm vA0 = ηB 2 /B2 , consistent with the Zeldovich theorem prediction. Equation 4.60 gives the well-known result for the quenched ﬂux diﬀusivity. There, the kinematic diﬀusivity ηTk is modiﬁed by the quenching or 2 suppression factor [1 + Rm vA0 /v2 ]−1 , the salient dependencies of which 2 are on Rm and B . Equation 4.60 predicts a strong quenching of ηT with increasing Rm B2 . Despite the crude approximations made in the derivation, numerical calculations indicate remarkably good agreement between the measured cross-ﬁeld ﬂux diﬀusivity (as determined by following marker particles tied to a ﬂux element) and the predictions of Eq. 4.60 [31, 32]. In particular, the scalings with both Rm and B2 have been veriﬁed, up to Rm values of a few hundred. The quench may be viewed as one consequence of the Alfvenization of turbulence by the stretching of a weak mean magnetic ﬁeld by the ﬂow. Limitations of space and time availability force us to leave the fascinating subject of turbulent diﬀusion of magnetic ﬁelds at this point. Truth be told, we have only scratched the surface of the 2D problem, and have not dared to even touch the 3D diﬀusion or alpha eﬀect issues. In 2D, several aspects of the problem merit further discussion. Perhaps the most important of these is concerned with the eﬀects of a ﬂux or inhomogeneity-driven transport of A2 upon the Zeldovich theorem balance [33]. If such a process were at work, it would alter the balance between resistive dissipation and

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turbulent diﬀusion, and thus change the quench of ηT . This issue is an area of ongoing research. 4.6

Conclusion

This brief pedagogical article has surveyed some of the interesting problems in MHD turbulence theory and has only explored the ‘tips’ of a few ‘icebergs’ ﬂoating in the ‘sea’ of that large topical area. The reader is referred to the research literature for further discussion, and for treatments of other related topics. The authors hope that the discussion of key concepts presented in this article will stimulate and facilitate the reader’s future explorations. Acknowledgments The authors would like to express their gratitude to (listed alphabetically): D.W. Hughes, M. Malkov, W.-C. M¨ uller, S.M. Tobias, E.T. Vishniac and A. Yoshizawa for stimulating conversations on the material of this article. P.D. and K.I. wish to acknowledge the hospitality of Kyushu University, where part of this article was written. This work was supported by U.S. Department of Energy Grant No. DE-FG02-04ER54738 and by the Grantin-Aid for Specially-Promoted Research of MEXT (16002005). References [1] Frisch U 1995 “Turbulence”, Cambridge University Press; Yoshizawa A, Itoh S-I and Itoh K 2003 Plasma and Fluid Turbulence, I.O.P. [2] Richardson LF 1926, Proc. Roy. Soc. London, Ser. A 110, 709. [3] Frisch U, Sulem P-L and Nelkin M 1978 J. Fluid Mech. 87 719. [4] Falkovich G Gawedski K and Vergassola M 2001 Rev. Mod. Phys. 73 913. [5] Kraichnan RH 1967 Phys. Fluids 10 1417. [6] She ZS and Leveque E 1994 Phys. Rev. Lett. 72 336. [7] Kraichnan RH 1965 Phys. Fluids 8 1385. [8] Iroshnikov TS 1964 Sov. Astron. 7 566. [9] Craddock G and Diamond PH 1990 “On the Alfven Eﬀect in MHD Turbulence” Comments in Plasma Phys. Control. Fusion 13(6) pp. 287-297. [10] Lazarian A and Vishniac E 1999 Ap. J. 517. [11] Manheimer WM and Dupree TH 1968 Phys. Fluids 11 2709. [12] Biskamp D and Welter H 1989 Phys. Fluids B1 1964. [13] Goldreich P and Sridhar S 1995 Ap. J. 458 763.

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[14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33]

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Goldreich P and Sridhar S 1997 Ap. J. 485 680. Cho J Lazarian A and Vishniac ET 2002 Ap. J. 564 291. Maron J and Goldreich P 2001 Ap. J. 554 1175. Maron J et al. 2004 Ap. J. 603 569. Muller WG Biskamp D and Grappin R 2003 Phys. Rev. E 67 066302. Cohen, RH and Kulsrud RH 1974 Phys. Fluids 17 2215. Rogister A 1971 Phys. Fluids 14 2733. Medvedev MV Diamond PH et al. 1997 Phys. Rev. Lett. 78 4934. Medvedev MV et al. 1997 Phys. Plasmas 4 1257. Passot T and Sulem PL 2003 Phys. Plasmas 10 3906. Hada T 1994 Geophys. Res. Lett. 21 2275. Diamond PH, Hughes DW and Kim E in “Fluid Dynamics and Dynamos in Astrophysics and Geophysics”, 2005 A. Soward, et al. ed., CRC Press 145. Pouquet A 1978 J. Fluid Mech. 88 1. Cattaneo F Hughes DW and Kim E 1996 Phys. Rev. Lett. 76 2057. Zeldovich Ya B 1957 Sov. Phys. JETP 4 460. Taylor JB 1986 Rev. Mod. Phys. 58 741. Strauss HR 1986 Phys. Fluids 29 3668. Cattaneo F and Vainshtein SI 1991 Ap. J. 376 L21. Silvers Lara 2004 Ph.D. Thesis University of Leeds. Kleeorin N and Rogachevskii I 1999 Phys. Rev. E 59 6724.

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Chapter 5

Intermittency Like Phenomena in Plasma Turbulence

A. Das, P. Kaw and R. Jha Institute for Plasma Research, Bhat Gandhinagar - 382428, India The article contains an overview of the intermittency like phenomena in plasma turbulence.

5.1

Introduction

The understanding of turbulent phenomena in ﬂuids has remained an outstanding, fascinating and important problem of physics for more than a century. The major diﬃculty encountered in providing a suitable description of turbulence is the existence of a multiplicity of scales (degrees of freedom). Other problems of physics where a multiplicity of scales was involved have made progress by devising rules which simplify the problem. For example, in the mathematical description of phase transition and critical phenomena, the self similarity of scales at the critical transition provides the requisite simpliﬁcation. Similarly, in the study of ideal gases, the principle of complete randomness helps in providing an exact statistical description. In the description of turbulence it has not been possible to devise simplifying rules which would assist one in dealing with the problem of inﬁnite degrees of freedom. The most successful work on turbulence is that of Kolmogorov [1]. His ansatz of local interaction in wavenumber space was an attempt to provide a simplifying rule to describe the properties of the ‘inertial range’ of turbulent scales. He used this ansatz, along with dimensional arguments and constancy of energy ﬂux in the inertial regime, to obtain power laws for spectral dependence of velocity moments of various orders. Thus he basi151

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cally postulated the existence of scale similarity over a limited wavenumber regime viz. the so called ‘inertial’ range which was well removed from the energy containing and dissipative regimes. However, unlike the previous two examples of physics (viz., phase transition and ideal gases) where the descriptions are exact, in this case there are clear indications of deviations from Kolmogorov’s picture of turbulence. The deviations have largely been attributed to the phenomena of intermittency (presence of random bursty events in space and time) exhibited by the turbulent ﬂuids. The presence of these events violates Kolmogorov’s scale similarity ansatz in the inertial range. These events are also responsible for introducing some amount of coherence in the turbulent state. This intermixing of coherence and randomness in the turbulent state is a major cause of failure of the statistical descriptions developed so far. The study of intermittency thus remains a crucial issue on which the turbulence community is focussing much of its current interests. The study of turbulence in plasmas has started in ernest only fairly recently (i.e., only in the last couple of decades). Since the plasma state of matter exhibits ﬂuid like behaviour in a variety of physical contexts, the approach towards studying turbulence in this phase has closely followed some of the well known ﬂuid approaches. However, the additional complexity of this state due to the response of its constituent species to external and self generated electromagnetic ﬁelds and the associated wave aspect in plasmas has also led to an alternative approach to this problem. Both these approaches, however, ultimately encounter similar problems, viz., that of dealing with a large number of scales with no apparent simpliﬁcations. In Sec. 5.2 we dwell on the notion of intermittency in neutral ﬂuid turbulence. We present in short the various measures employed for the quantiﬁcation of intermittency, and the variety of theories that have been put forth for providing an explanation for the origin of this particular phenomenon. In Sec. 5.3 we present evidences from experimental data for intermittency in plasmas. We then summarize in Sec. 5.4 research activities carried out by adopting ﬂuid concepts for quantiﬁcation and understanding of the notion of intermittency in plasmas. We also outline open areas where interesting study yet needs to be carried out in plasmas. In Sec. 5.5 we present an alternative approach to the study of plasma turbulence that has been adopted by plasma physicists. Such an approach has emerged in view of certain simpliﬁcations which arise in plasma turbulence due to distinctive features of this particular state. Basically such theories are best at describing plasma state in which the wave aspects (instead of eddy like features)

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have dominance. Typically a plasma turbulent state can have a whole set of possibilities wherein wave and eddy aspects can have diﬀerent levels of importance. The intermixing of randomness and order due to waves and eddies can also lead to bursty transient events in plasmas, thereby producing some characteristic features of intermittency. Therefore we have pointed out in Sec. 5.6 that it is important that the notion of intermittency be broadened in the context of plasma. Intermittency like phenomena in plasmas should encompass all phenomena which exhibit transient bursty events showing an intermixing of randomness and coherence, leading to non - gaussian distributions, deviations from scale similarity etc., whether they arise due to ﬂuid like behaviour or whether they arise due to characteristic plasma like features of mixed eddy and wave like states. 5.2

Concept of intermittency in hydrodynamic ﬂuids

As mentioned in the introduction the plasma state of matter exhibits ﬂuid like features in a wide variety of circumstances. In fact a number of ﬂuid models are adopted for describing distinct plasma phenomena viz., electrostatic and electromagnetic phenomena, fast and slow time scale phenomena (fast and slow are deﬁned corresponding to the response times of the electron and ion species respectively). It is then reasonable to presume that turbulent state in a plasma would have characteristic features which are somewhat similar to those observed for neutral ﬂuids. Thus the guidance from the neutral ﬂuid studies of the turbulent phase are highly desirable and are being pursued. This section summarizes the notion of intermittency in the context of neutral hydrodynamic ﬂuids. The evolution equation for velocity ﬁeld in the incompressible neutral ﬂuid is given by the Navier-Stokes equation which is ∂ v + v · ∇v = ν∇2v − ∇p + f ; ∇ · v = 0 ∂t

(5.1)

Here ν is the coeﬃcient of viscosity, p denotes pressure and f is the external forcing in the system. The nonlinear term v · ∇v is responsible for exciting a multitude of scales, which poses major diﬃculty in providing a suitable description of turbulent phase. It should be noted here that even though there is no characteristic scale in the system arising from physical considerations (apart from the forcing and the dissipation scales), still one observes deviations from the concept of strict scale similarity even in regimes where dissipation and/or forcing do not have any direct role to play.

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A characteristic result of such a multiscaling is the observation of random bursty events in space and time which is termed as intermittency. The above is, however, only a qualitative description of intermittency. A quantitative deﬁnition of this phenomenon is desirable and is sought by seeking the statistics of the ﬂuctuating ﬁelds in the turbulent ﬂow. The presence of intermittent events are responsible for giving rise to high amplitude events with a greater probability than what a completely random ensemble or a Gaussian distribution might yield. Thus deviations from gaussianity is a likely quantitative measure for intermittency. The single point statistics of any variable, however, carries no scale information of the system, whereas the observation of spatial and temporal bursty events are pointers at strong interplay and dependence on scales for this phenomenon. Thus it is desirable to study at least two point joint probability distribution function of the ﬁelds (at various separation scales) and its deviation from the multi (bi) variate Gaussian distribution. A convenient measure indicating deviations from bivariate Gaussianity and yielding scale information is the structure function Sp (l). The structure function is deﬁned as Sp (l) = |δv(x, t, l)|p , where δv(x, t, l) = [v (x+l, t)−v (x, t)]·l/l is the diﬀerence between longitudinal velocity components separated by a spatial distance of l. One could use in the deﬁnition of structure functions, instead of velocity v , any other ﬁeld φ on whose turbulent properties one is interested in. For a homogeneous turbulence, Sp is independent of x and depends on the separation distance of the two ﬁeld points l. The deviations from the bivariate gaussian distribution can get reﬂected by the properties of Sp as follows. Let φ1 = φ(x) and φ2 = φ(x + l) be the two ﬁelds, with (φ1 )2 = (φ2 )2 = a from homogeneity and φ1 φ2 = b. Thus (δφ)2 = (φ1 − φ2 )2 = 2(a − b) = f (l) a function of l. The bivariate Gaussian joint probability distribution function of φ1 and φ2 is given by # $ 2 1 2 + φ ) − 2bφ φ P (φ1 , φ2 ) ∼ exp −ΦT Σ−1 Φ = exp − 2 a(φ 1 2 1 2 (a − b2 ) (5.2) where Φ is the column vector having two components φ1 and φ2 , and Σ−1 is a 2 × 2 inverse correlation matrix. One can obtain the probability distribution function for δφ from Eq. (5.2) as follows. (δφ)2 (5.3) P (δφ) = P (φ1 , φ1 − δφ)dφ1 = exp − 2(a − b) This is a Gaussian distribution having a width of 2(a − b), which is a function of l, viz., f (l). Clearly for such a distribution Sp = (δφ)p ∼

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(δφ)2 p/2 ∼ [f (l)]p/2 , i.e. the scaling exponent depends linearly on p. Moreover, it has been shown exactly from the evolution equation that the third order structure functions S3 ∼ l [2]. Thus f (l) ∼ l2/3; and Sp ∼ lp/3 if the ﬂuctuating ﬁeld has a Gaussian distribution.Any deviation from linearity of the exponent is a clear indication of non-Gaussianity of the joint distribution function. Kolmogorov [1] had conjectured that in the strongly turbulent regime the energy cascade is local in wavenumber space, that there is a constant (independent of scales) ﬂux of energy through various scales, and that this energy ﬂux eventually gets dissipated at short scales where dissipative mechanisms are present. On using dimensional consistency arguments he was then able to obtain a spectral scaling law for energy as Ek ∼ 2/3 k −5/3 . This scaling also implies that that [δv(x, t, l)]2 ∼ l2/3 . A Gaussian ansatz for the probability distribution function of δv(x, t, l) then gives the scaling for all powers as Sp (l) = [δv(x, t, l)]p ∼ lp/3 , which is consistent with the derivation in the last paragraph. Both non-Gaussianity as well as the ﬂuctuating character of lead to deviations from the Kolmogorov scaling for Sp . Thus a quantitative characterization for intermittency is provided by the deviation of the structure function scaling index (Sp ∼ l ζp ) ζp from a linear value of p/3. Experimental and numerical observations indicate that in three dimensional (3-d) neutral ﬂuid turbulence ζp > p/3 for p ≤ 2, ζ3 = 1 and for p > 3, ζp is smaller than p/3. These results are very well summarized by Frisch [3] and presented in Fig. 5.1. The nonlinear dependence of ζp on p signiﬁes deviations from spatial scale similarity. There is no evidence of any such multiscaling in two dimensional (2-d) hydrodynamic turbulence in both cascade regimes. Several theories and models have been put forth to provide an understanding of the physical mechanism responsible for intermittency. Landau [4] was the ﬁrst to point out the ﬂuctuating character of energy dissipation rate (contrary to the assumption used by Kolmogorov) which results in deviations from the spectral scaling law proposed by Kolmogorov [1]. Attempts have been made at providing explanation of the observed scaling of the structure functions at various levels of sophistication. A variety of phenomenological theories were put forth assuming suitable distributions for . The energy dissipation rate being positive deﬁnite, Kolmogorov and Obukhov [5] assumed that a lognormal distribution would be appropriate for it. This led to ζp =

p(p − 3) p −µ 3 18

(5.4)

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Fig. 5.1 Exponent ζp of the pth order structure function vs p adapted from Frisch [3]. The symbols represent experimental data from various sources cited in the book Ref. [3] and smooth lines represent various models. It is observed that the log-Poisson model of She and Leveque [10] ﬁts the data points up to the highest p−value.

where the index µ is given by (x) (x + l) ∼ l−µ . A good ﬁt to observed numerical and experimental data was obtained upto p = 10 for a value of µ ≈ 0.2. The β and multifractal models are based on the geometrical representation of the non-space ﬁlling nature of turbulent activity. The β model was proposed by Novikov and Stewart [6] and Frisch et al. [7] and assumed that at each level of cascade any eddy of scale length lp splits into 2D β eddies of scale lp+1 = lp /2. Here D is the dimensionality of the space. Clearly the factor β which is chosen to lie between 0 < β ≤ 1, deﬁnes the fraction of space which is ﬁlled at each subsequent scale by the turbulent activity. The expression for structure function exponent in this case is ζp =

p δ − (n − 3) 3 3

(5.5)

Here β = 2−δ . The linearity of ζp with p in above equation fails to provide a good ﬁt to the observed data for any value of δ. As a modiﬁcation to the β model a multifractal model was developed by Parisi and Frisch [8] and Maneveau and Sreenivasan [9]. To obtain the nonlinear scaling it was

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assumed that at each level of cascade energy was distributed unequally. The simplest possibility was when at each level a certain fraction say n of energy got distributed equally amidst half of the new eddies and the remaining (1−n) fraction to the other half of the eddies. Such a distribution continues upto dissipation scale length. This yields following expression for the structure function index. p ζp = (5.6) − 1 Dp + 1 3 Here Dp = log2 [np + (1 − n)p ]1/(1−p) . The agreement of this model is fairly good for n ∼ 0.7. All the above phenomenological models require an adjustable parameter to achieve a good ﬁt with observations. Recently, however, She and Leveque [10] have put forth a model involving a hierarchy of ﬂuctuating structures satisfying log-Poisson statistics which gives ζpSL = p/9 + 2[1 − (2/3)p/3 ]

(5.7)

an expression which ﬁts the experimental data very well and uses no ad hoc adjustable parameter. This model has thus gained a wide acceptance. Apart from phenomenological attempts listed above there have also been eﬀorts to explain the phenomena of intermittency on the basis of governing set of evolution equations. The fractal properties of ﬂuctuating structures as evidenced from the phenomenological picture have been sought on the basis of the singularity structure of governing set of evolution equations e.g. Navier-Stokes etc. A direct derivation of the scaling of various structure functions from the dynamical equations have also been sought. There has been partial success in this direction. As pointed out earlier in this section an exact derivation of the scaling index of the third order structure function has been possible which shows that S3 ∼ l [2]. The scaling of the third order structure function is now being successfully utilized for the purpose of benchmarking. The scaling index for the passive scalar advection up to all orders has been determined exactly for space dimension 2 and above and for a speciﬁed delta correlated velocity in time whose equal time spatial correlation function scales as rξ , where 0 < ξ < 2 [11]. The analytically determined index shows a clear deviation from linearity and hence provides for the ﬁrst time analytical evidence for intermittency. There have also been attempts at capturing the phenomena of intermittency by statistical theories. The statistical descriptions based on the ansatz of near gaussianity (DIA and its variants) [12] are typically counter

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to the notion of intermittency. However, a new statistical scheme introduced by Kraichnan [13] in the last decade has the capability of capturing wildly non-Gaussian features and hence provides a suitable paradigm for these phenomena. The method is termed as the mapping closure technique and relies on the nonlinear mapping relationship of the ﬁeld with a surrogate Gaussian ﬁeld [14, 15]. The method is attractive as it captures nonlinearity exactly, the approximations being involved only for representing the gradient and other higher spatial derivative terms that arise in the dynamics. It is thus clear that a lot of eﬀort towards proper characterization and development of appropriate physical understanding of the phenomenon of intermittency has taken place for neutral ﬂuids.

5.3

Evidence for intermittency in plasmas

In this section we review some of the experimental (laboratory plasma), observational (space plasma) and simulation studies which provide signiﬁcant evidence for the presence of intermittency like phenomena in plasma turbulence. In the last section we saw that the structure functions and the non-Gaussian statistics of various ﬁelds were two major ways to characterize the phenomena of intermittency. We will see in this section that the scaling studies with the help of structure functions have been widely adopted in the context of space plasma. For laboratory plasma the adoption of such a characterization scheme is as yet at a very preliminary stage due to the in-adequate range of available spatial scales. The spatial scales in the laboratory plasma get severely restricted due to the ﬁniteness of the containing device, rendering such a characterization diﬃcult. Statistical studies on moments and determination of probability distribution function (PDF), for variety of ﬂuctuating ﬁelds (e.g. plasma potential, plasma density, electron temperature, particle ﬂux etc.) have, however, been carried out for laboratory plasmas. Certain other additional methods such as diagnosing the presence of coherent structures etc. also provide indirect evidences for violation of scale similarity aspect in plasma turbulence. We summarize below such studies carried out for laboratory as well as space plasmas, which provide evidence for multiscaling in plasma turbulence. We end this section with some remarks highlighting subtle aspects of the analysis carried out in experimental and observational data for plasma turbulence.

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159

Laboratory fusion plasma

The subject of intermittency has attracted the attention of the fusion community with the report of non-Gaussian PDFs of plasma ﬂuctuations and ﬂuctuation induced ﬂux in the edge plasma of ADITYA tokamak [16, 17]. These results have now been conﬁrmed in a large number of fusion devices [18–25]. While most of these measurements are obtained with Langmuir probes, results from CT-6B [20] and Alcator C-Mod [23] use Hα and Dα light intensity for the measurements of plasma density (or plasma pressure). Figure 5.2 shows an example of time series of plasma density, radial ﬂow velocity and ﬂuctuation induced particle ﬂux in the turbulent edge plasma of ADITYA tokamak. The PDFs of ﬂuctuation data are non-Gaussian and leptokurtic (peaked central region and fatter wings [16]. The nonGaussianity is quantiﬁed in terms of skewness (S, the third order moment) and kurtosis (K, the fourth order moment) of the PDF. Typical results for density and potential ﬂuctuations show the values of S and K in the range of 0.1-1 and 1-3 respectively above their Gaussian levels (S=0, K=3). On the other hand the PDF of particle ﬂux is strongly non-Gaussian (see Fig. 5.3). Typical values of skewness and kurtosis are 1-2 and 5-30 respectively. 20

15 (c)

10

(b)

5

(a) 0 0

100

200 300 Time (µs)

400

500

Fig. 5.2 Experimental time series of (a) plasma density, (b) radial velocity and (c) particle ﬂux in the scrape-oﬀ boundary layer of ADITYA tokamak measured using Langmuir probes. The measured quantities are normalized to the respective standard deviations. The bursts in the measurements are clearly visible.

The shear layer, near the last closed ﬂux surface can inﬂuence the statistical properties of ﬂuctuations. Results from TJ-I tokamak indicate that PDFs of ﬂoating potential and density ﬂuctuations in the main plasma near the shear layer (rsh ) are Gaussian, but in the SOL region (r/rsh > 1),

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Probability Distribution Function

0

10

<Γ> =0.26 σ =0.93 Γ −2

10

−4

10

−5

0

5

10

Particle Flux, Γ

15

Fig. 5.3 The probability distribution function of particle ﬂux. The bullets show the experimental data joined by a solid line. The dashed curve represent a Gaussian distribution using the values of mean (Γ) and standard deviation σΓ as shown in the ﬁgure. The calibration of the ﬂux is 3.6×1016 cm−2 s−1 .

the PDFs become clearly non-Gaussian [19]. However, particle ﬂux is nonGaussian throughout the edge region (r/rsh =0.9- 1.1). Similar observations have been reported from Tore Supra [21, 22], Alcator C-Mod [23] and D-IIID [26] tokamaks. Sanchez et al. [27] have presented results for boundary regions of Advanced Toroidal Facility (ATF) and Wendelstein 7-Advanced Stellarator (W7-AS) and Joint European Torus (JET). In all the three machines, ﬂuctuations in ion saturation current and ﬂoating potential have a near-Gaussian character in the proximity of the shear layer. However, ﬂuctuations deviate from a Gaussian distribution when moving inside of the plasma edge (r < rsh ) and into the SOL region (r > rsh ). It is now understood that velocity shear can decorrelate ﬂuctuations and destroy localized coherent structures which are primarily responsible for the non-Gaussian PDFs. The statistics of ﬂuctuations of diﬀerent scales have been determined on ADITYA tokamak using band pass wavelet ﬁlter and it is concluded that low frequency components are nearly Gaussian [28]. The non-Gaussianity appears at frequencies above 20 kHz. Since ﬂuctuations propagate with phase velocity of ∼ 2×105 cm/s), the time scales correspond to structures of size < 10 cm. The ﬂoating potential ﬂuctuation in the Reverse Field Pinch device RFX [29] and the stellarator L-2M [24] have also been analysed using similar technique and the same conclusion has been arrived at. Magnetic ﬂuctuation, δb, has been analysed in the RFX device using digital ﬁltering and it has been concluded that non- Gaussianity increases at small scales

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and on moving towards the wall [30]. Measurement of scaling exponent of the structure function, Sp (τ ) = |δb(τ )|p ∼ τ ζp , has also been attempted and it is found that deviation of ζp from p/3 increases on moving towards the wall. However, the available scaling regime data being limited, the thoroughness and general applicability of such results remains questionable. The non-Gaussian PDF is related to coherent features in the time series of plasma ﬂuctuations. Measurements of two-dimensional structures of plasma ﬂuctuations in the radial-poloidal (r − θ) plane have been reported from the Caltech tokamak using Langmuir probes [31]. The structures in the edge plasma (r/a = 0.85-0.95) appear to be localized and have a short life time (a few tens of µs). A detailed measurement of two-dimensional structures have been reported from ADITYA tokamak using conditional averaging technique [32]. It is observed that coherent structures are found on both sides of the LCFS (i.e., r/a = 1) and they are generally isolated (see Fig. 5.4]. The poloidal span of the structure is generally larger than the radial span except when the isolation is broken. This happens at large amplitude. When the isolation breaks, the potential contour gets aligned in the radial direction indicating that the coherent structure has crossed the LCFS. The structure crosses the LCFS with an estimated radial velocity (Eθ /Bφ ) in the range (0.5-1) km/s. This convection velocity is about an order of magnitude larger than the diﬀusion velocity (vdif f = D⊥ /Ln , where D⊥ is the perpendicular diﬀusivity). Such fast transport events can be recognized as streamers in the framework of low frequency magnetized plasma turbulence such as turbulence due to drift waves, resistive ballooning modes etc. [33]. Normally the coherent structures are poloidally elongated and radially localized which can be recognized as the well known low frequency modes saturated by the self-consistently generated velocity shear due to associated zonal ﬂows. The streamer events can break and propagate through a transport barrier that is generated by this velocity shear. Sometimes, the streamer events can be seen in the form of dense, radially convecting blobs of plasma moving across the magnetic ﬁeld in the low density background plasma. The convection of density blobs in the edge plasma has directly been observed in D-IIID tokamak using beam emission spectroscopy [34]. The density blobs move in poloidal as well as radial direction. The poloidal velocity is typically twice the radial velocity (0.5-2 km/s). Similar observation has also been reported using gas puﬀ imaging (GPI) diagnostics on NSTX and Alcator C-Mod tokamaks [35, 36]. The GPI diagnostics allow observation in the r − θ plane when a gas puﬀ is released in the limiter shadow. The observation plane is deﬁned by the

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Fig. 5.4 The snapshots of ﬂoating potential structures in the edge region of ADITYA tokamak obtained using conditional averaging technique [32]. The contours represent equal amplitudes of potential and the observed region is 4.5×3 cm2 in the radial (Xaxis) and poloidal (Y-axis) plane. The X=25 cm represents the limiter location. The transient structures observed in the edge (X < 25 cm) and the scrape-oﬀ boundary layer (X > 25 cm) are typically isolated but the isolation is broken at large amplitude.

leading ionization front and the rear emission front. The observation shows blobs of Hα (or Dα ) light moving in radial and poloidal directions. The typical size of the blob is 1-2 cm and the life time is few tens of microseconds. The blobs actually represent localized regions of plasma pressure. In most C-Mod and NSTX discharges, the blobs are seen outside the LCFS

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while DIII-D measurements show that they are formed near the LCFS. It appears that separatrix and/or open ﬁeld lines are involved in their generation and/or transport. Detailed experiment has further shown that as the mean density (ne ) is raised towards the Greenwald limit (ne /nGW ∼ 0.7), the blobs can also be seen inside the LCFS. Thus, GPI observations complement the measurements with Langmuir probes and demonstrate that intermittently occuring blobs are responsible for the large amplitude nonGaussian distributed events on the probes. In addition to the direct observation in the r − θ plane, the coherent structures being the cause of intermittency has also been inferred from advanced analysis of time series data. Langmuir probe data from ADITYA and ASDEX have been analysed using biorthogonal decomposition technique to characterize coherent structures [37]. They have shown that such structures eﬀectively contribute to bursty radial transport. Similarly, coherent structures of density ﬂuctuations have been observed in CT-6B tokamak plasma by analysing the measured Hα intensity using wavelet correlation technique [20]. The structures exhibit strong temporal and spatial localization with a life time of 20-100 µs and radial size of 1-4 cm. One also employs certain experimental techniques such as bicoherence analysis which look into the weak turbulence (which is discussed in detail later) aspect by providing evidences for three wave nonlinear couplings occuring in the observed spectrum. Such three wave nonlinear coupling in the edge plasma of fusion devices has been reported [17, 38–40]. The nonlinear coupling is measured in terms of bicoherence between three frequency components, ω1 , ω2 and ω3 such that ω3 = ω1 ± ω2 . A ﬁnite bicoherence in the ﬂuctuation time series indicates that ω3 is produced by frequency mixing of ω1 and ω2 . The bispectral analysis of ﬂuctuation time series at two probe points can give bicoherence among wave numbers k3 = k1 ± k2 as well as power transfer due to linear and nonlinear wave coupling. Such studies have shown that both forward (i.e., ω3 = ω1 + ω2 ) and inverse (i.e., ω3 = ω1 − ω2 ) cascade of energy do take place in the turbulent edge plasma [39]. A new bispectral analysis technique has been introduced by Milligen et al. [41] in which wavelet transform is used in place of Fourier transform. Since a wavelet is a localized basis function (as against nonlocal sinusoid), this technique allows studies of intermittency phenomena. The presence of nonlinear coupling among wavelet components is measured in terms of wavelet bicoherence. The analyses of probe and reﬂectometry data in ATF stellarator [42] and probe data in ADITYA tokamak [28] show that ﬂuctuation time series have episodes of strong nonlinear coupling among wavelet

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components punctuated by quiescent periods when the coupling is absent. Analysis of ADITYA discharges has further shown that wavelet bicoherence is strong in presence of well separated (sparse) structures [43]. When the structures are closely packed, the wavelet bicoherence is negligible. The wavelet bicoherence measurements of plasma ﬂuctuations in D-IIID tokamak have shown that total bicoherence temporally varies in both L- and Hmode discharges but L-mode displays much weaker episodic behaviour than H-mode discharges [44]. Thus, though these diagnostics are based on weak turbulence wave matching criterion, they give ample indications for certain strong turbulence aspects, viz. the temporally episodic coupling amongst the waves. There is also a signiﬁcant amount of work in the context of plasma turbulence which either invokes or is suggestive of absence of multiscaling. For example, it is believed that some experimental data which displays the characteristic 1/f scaling (even in a limited frequency domain), either in the turbulent ﬂuctuation time series itself or in derived quantities such as turbulent ﬂux, may be explained by self organized criticality models of the underlying physical phenomena. Such models would lead to self-similarity in the data. It is for this reason that several attempts have been made to analyse the ﬂuctuation time series for self-similarity using a variety of techniques. The measure of self-similarity is the Hurst parameter (H) which has been determined using the rescaled range R/S [45], structure function [46, 47] and L´evy analysis [48, 25] techniques. Using the R/S approach, ﬂuctuation data from several tokamaks shows H ∼ 0.7 in regions away from the velocity shear layer and H ∼ 0.5 close to the shear layer. Although these studies have not speciﬁcally looked at the probability distribution function (PDF), but based on what is known from the literature, the PDF is expected to be Gaussian. The structure function approach has been followed in some recent studies on DIII D and TEXTOR tokamaks. The analysis shows again that in the mesoscale range, the exponent of the pth order structure function of the integrated time series, ζp = pH . The Hurst parameter obtained from this relation is ∼ 0.7. However, for several reasons, including short data length and measurement noise, it has not been possible to measure structure functions of order higher than 5 where deviation due to intermittency is expected to be signiﬁcant. Self-similarity studies have also been carried out following the PDF approach. Studies on RFX reversed ﬁeld pinch [29] and ADITYA tokamak [48] show that mesoscale ﬂuctuations are Gaussian whereas small time scale ﬂuctuations are non-Gaussian. The PDF of small scale ﬂuctuations can be ﬁtted to a truncated Levy distribution

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which exhibits self-similarity (with H ∼ 0.8 − 0.9). Similar results have been obtained from the Uragan 3-M stellarator [25]. It is to be emphasized however, that the above data on self similarity and hence the basis for SOC or levy ﬂight models, is still neither universal nor suﬃciently convincing (because of short data lengths, measurement noise etc); considerable more work is necessary to ﬁx the range of validity of such models and/or to make a convincing case for their applicability. 5.3.2

Space plasmas

Observations on solar wind plasma have been carried out using satellites during the last couple of decades. Measurements of bulk plasma ﬂow velocity and local magetic ﬁeld show ﬂuctuations of time period ranging from 1 minute to 15 hours. It is observed that energy spectrum follows k−5/3 dependence and the scaling range of wave number k increases with increasing distance from the sun [49, 50]. The hallmark of intermittency is the deviation of exponent of the structure function from the Kolmogorov prediction of ζp = p/3. This is clearly demonstrated in the data of solar wind speed and magnetic ﬁeld [51–53] and proton temperature and density [54]. The deviation from Kolmogorov prediction is then explained in terms of multifractality of turbulent ﬂuctuations. With the exception of slow wind [55], the ﬂuctuations in solar wind plasma is almost always anisotropic and have non-Gaussian leptokurtic PDF [56]. The non-Gaussian characteristics increase at small scales whereas large scales are nearly Gaussian [57, 58]. The PDF ﬁts a model proposed by Castiang (1990), which basically represents a PDF obtained from a superposition of several gaussian distributions which have variances that are themselves random and are distributed according to the lognormal distribution. Sorriso-Valvo et al. [53] have ﬁtted solar wind data with the model expression which indicates the importance of multiplicative process in generating intermittency. A new approach based on wavelets, has also been applied which identiﬁes the presence of coherent structures within the time series of experimental solar wind data [56] and MHD simulation data [59]. The earth’s magnetosphere responds to the incoming solar wind by storing its kinetic energy in the form of magnetic energy and releasing it sporadically in the form of geomagnetic substorm. The magnetosphere exhibits substantial complexity in many of its physical properties. Particle populations wax and wane and magnetic ﬁelds ﬂuctuate on virtually all observed time scales from less than a minute to many days. The eﬀects of particle

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release on the ionosphere is measured by such global indices as Auroral Electrojet or AE index and Polar Cap or PC index. These indices contain information about the spatiotemporal scales of magnetic ﬁeld reconnection which causes the release of particles and energy. Extensive study of these indices as well as in-situ satellite measurements show enormous complexity of substorm dynamics including signatures of low phase space dimension, hysteresis, power-law spectra of ﬂuctuations on diﬀerent scales and nonGaussian PDFs of ﬂuctuations. Various notions which can explain the magnetospheric complexity are still being evaluated and debated [60]. One of the notions emerging out of these debates is that magnetospheric activity might be explained as a manifestation of multifractal behaviour. The multifractal model gained support from the observations of Consolini and De Mechelis [61] on the non-Gaussianity of AE index ﬂuctuations. They observed that PDFs of AE index ﬂuctuations are always non-Gaussian for the time scales in a range of 1-120 minutes for both quiet and disturbed magnetosphere. Stepanova et al. [62] have examined the ﬂuctuations in PC index using similar technique. The analysis reveals that the ﬂuctuation PDFs are non-Gaussian and ﬁt the lognormal distribution of Castaing et al. [63]. The ﬁt parameters are close to those reported for solar wind magnetic ﬂuctuations indicating that PC index adequately represents the solar wind eﬀects on the polar region of the ionosphere. The direct in-situ satellite measurements of turbulence in the magnetospheric plasma sheet also show that bulk ﬂows and magnetic ﬂuctuations exhibit non-Gaussianity [64, 65]. It should be noted that unlike solar wind, the magnetospheric plasma sheet represents high beta (β = kinetic pressure/ magnetic pressure) plasma where magnetic reconnection dominates the dynamics. Borovsky et al. [64] showed that frequency spectra of ﬂuctuations in bulk ﬂow and magnetic ﬁeld are power-laws. The PDF of ﬂow velocity is non-Gaussian (stretched exponential) and has positive skewness towards the earthward ﬂow. Angelopolous et al. [65] have demonstrated that bulk ﬂows are bursty and PDFs can be ﬁtted to lognormal distribution indicating multifractal behaviour of ﬂuctuations in plasma sheet. 5.3.3

Remarks

It is thus clear from the summary of activities presented above that a lot of emphasis is presently being put on the study of intermittency/ multiscaling aspects in plasma turbulence. These studies have primarily focussed on issues very similar to the ones studied in the ﬂuid context. They constitute

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in particular statistical studies, spatial and temporal scaling features of ﬂuctuating ﬁelds and sporadic behaviour of ﬂuctuations. There are a few underlying aspects which need to be emphasized in the context of the above studies. While the presence of non-Gaussian features in the primary ﬁelds such as density and potential ﬂuctuations do point at the presence of multiscaling/intermittent aspects of plasma turbulence, any non-Gaussian measurements on ﬂux need to be closely looked at. The ﬂux is a product of two variables, density (or any other ﬁeld whose ﬂux is being studied) and the advecting velocity. Even if both these ﬁelds themselves follow Gaussian distribution (and hence are not intermittent), any correlation amongst these ﬁelds would automatically render their product (the ﬂux) with non-Gaussian traits in its distribution. Thus, in such cases nonGaussianity in the turbulent ﬂux may arise purely as an artifact of using a derived variables (ﬂux) instead of the primary ﬁelds. Hence one needs to be wary of this eﬀect while carrying out any interpretation of intermittency phenomena on the basis of non-Gaussian distribution of any such derived ﬁelds. This was ﬁrst pointed out by Carreras et al. [19]. For laboratory plasmas the assumption of statistical homogeneity of the turbulence becomes an issue. This is because the size of the conﬁning device and/or the equilibirum scale length of various ﬁelds which drive instabilities, are comparable to the scales at which one is making the observations. Furthermore, the permitted scales in the laboratory are also very limited. We thus feel that before interpreting the statistical and the structure function studies made on laboratory plasmas to be indicative of intermittency, these issues need to be thoroughly addressed. There is another point in the context of plasma turbulence which one needs to consider seriously. Plasma, unlike incompressible ﬂuid supports a variety of characteristic scales which appear inherently in the dynamical evolution equations themselves. The various instability mechanisms also drive certain speciﬁc scales preferentially. Unlike neutral ﬂuids the energy driving mechanism does not occur in plasma at some isolated region of a very long scale, it lies well within the ﬂuctuation spectrum. Thus in these cases the violation of scale similarity is an inherent characteristics of the system. The variety of localized structures which are observed in various conditional and wavelet analysis measurement, can have scale lengths associated with certain physical and/or dynamical characteristics. This is a novel aspect of plasma turbulence and does not have any counterpart in Navier-Stokes hydrodynamic ﬂuids.

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Plasma intermittency: ﬂuid approach

As the behaviour of plasma medium resembles ﬂuid features in a number of ways, it is only but natural that the ﬂuid approach towards turbulence state be pursued with appropriate modiﬁcations pertinent to the plasma medium. Such an approach has indeed been adopted in the context of several ﬂuid models of plasmas. The most extensive study in this regard has been carried out on the incompressible magnetohydrodynamics (MHD) [66] model for plasmas. This model is a single ﬂuid depiction of the plasma medium restricted to the depiction of electromagnetic phenomena in a plasma occuring at very slow ion dynamics time scales. The model resembles the force Navier-Stokes equation in some way, however it has additional J × B acting on the evolution equation of velocity of the conducting plasma ﬂuid. given There is also a second evolution equation for the magnetic ﬁeld B by the Maxwell’s equation in which the electric ﬁeld E is replaced by the Ohm’s law for the moving plasma ﬂuid. The model is thus described by the following coupled set: ∂v + v · ∇v = µ∇2v − ∇p + J × B ∂t

(5.8)

∂B =B · ∇v + η∇2 B (5.9) + v · ∇B ∂t along with the incompressibility condition on ﬂuid velocity and magnetic = 0 respectively, and the deﬁnition ﬁeld vector, viz., ∇ · v = 0 and ∇ · B J = ∇× B. Here µ represents the viscous damping coeﬃcient of the plasma ﬂow and η denotes the resistivity of the plasma. Following ﬂuid analogy it can be shown that in 2-d! the equations support two square ! integral invariants viz., energy E = (v 2 + B 2 )dr and helicity, H = ψ 2 dr, where ψ is the component of the magnetic vector potential along the symmetry direction. In 3-d energy is the only square integral invariant. In 2-d as well as in 3-d the energy cascade is direct towards short scales. In 2-d there is an additional inverse cascade regime for H. A Kolmogorov like analysis then leads to the energy spectrum of Ek ∼ 2/3 k −5/3 in the energy cascade regime. However, an important aspect that is overlooked in the derivation of the above scaling is the existence of the natural Alfven wave excitations supported by the system. Unlike Navier-Stokes equation, basically the model itself contains a characteristic oscillation time scale which can be obtained upon the linearization of Eq. (5.8), Eq. (5.9) in the presence of a mean magnetic ﬁeld. This leads to the Alfven wave dispersion

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relation of ω 2 = kz2 Va2 , here Va is the Alfven velocity. A correction to the energy spectral scaling due to the presence of this wave was envisaged by Kraichnan and Iroshnikov (KI) [67]. They assumed that the energy cascade rate in the MHD ﬂuid gets altered by a factor of τa /τl , the ratio of the Alfven time with the eddy turn over time. This was to account for the fact that only oppositely travelling Alfven wave packets can interact nonlinearly and they can be within the interaction distance only for a fraction of time viz. (τa /τl ). This led to Ek ∼ 1/2 k −3/2 as the new scaling for the energy spectrum. This modiﬁcation is termed as the Alfven eﬀect on energy scaling. A large amount of simulation work has been devoted to ascertaining whether Alfven eﬀect really exists or not. The diﬀerence between 3/2 and 5/3 being very small, it is very diﬃcult to resolve between the two scalings due to computational limitations. On the basis of isotropic MHD simulation studies with a resolution of 5123 Biskamp and his colleagues [68] seem to have arrived at the conclusion that the Alfven eﬀect is not present in MHD. They have also investigated the scaling behaviour of very high order structure functions (upto 8) and have shown that the scaling of ζp with p is nonlinear and can be explained by a generalization of the She-Leveque [10] theory for MHD. The generalized She-Leveque expression can be written as p (5.10) ζp = (1 − x) + C[1 − (1 − x/C)p/g ] g Here g is related to the basic scaling of the relevant ﬁeld δzl ∼ l1/g . The scaling of the energy transfer time at smallest scale deﬁnes x by the scaling tl ∼ lx and C = 3 − D where D is the dimension of dissipative structures. For neutral ﬂuids, z = v and g = 3, x = 2/3 and the dissipative structures being vortex ﬁlaments D = 1 i.e. C = 2; this reproduces the conventional She Leveque scalings. For MHD, z = z ± = v ± B are the Elsasser variables. Here one can utilize either the Kolmogorov’s scalings [1] or the KI scalings (which incorporate Alfven eﬀect) [67] to determine the values of g and x. In the former case g = 3 and x = 2/3, whereas for KI theory the values are g = 4 and x = 1/2. Numerical simulations by Biskamp et al. [68, 69] show that the numerical results on structure function scalings ﬁt the analytical curve only if one assumes g = 3 and x = 2/3 (the Kolmogorov’s scalings).Furthermore, they show that the dissipative structures in MHD are two dimensional sheets [69] i.e. D = 2 and so C = 1. Thus the scaling exponents from MHD simulations agree well with the expression ζp = p/9 + 1 − (1/3)p/3 inducing Biskamp et al. [68, 69] to make the claim that the KI conjecture on modiﬁcations of the spectral scalings due to the

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Alfven wave eﬀect is absent in isotropic MHD turbulence. The deviation of ζp from linearity is a clear evidence for the presence of multiscaling and intermittency in MHD ﬂuid. The distinction in multiscaling characteristics between MHD and the neutral ﬂuid dynamics arises due to the dimensionality of the dissipative structures. In MHD dissipation occurs on two dimensional current sheets whereas in neutral ﬂuids, dissipative structures are one dimensional vortex ﬁlaments. A recent work by Biskamp et al. [70] with and without the mean magnetic ﬁeld, however, seems to support the k −3/2 scaling for the perpendicular (to local magnetic ﬁeld) energy spectrum. Furthermore, the structure function analysis seems to indicate that the ﬁeld-perpendicular scalings display increased intermittency compared to the isotropic structure functions. The parallel scalings are observed to be less intermittent. For B0 = 0 too there is a diﬀerence between the ﬁtted value of g for the parallel and perpendicular structure functions. Basically, g = 2.7 and g⊥ = 3.16 (Note that the K41 and KI scalings correspond to g = 3 and 4 respectively). As the value of B0 is increased Biskamp et al. [70] observe that the diﬀerence between g and g⊥ increases. In fact for a value of B0 = 10 in their simulations they observe that g = 2.45 and g⊥ = 4.4. This study seems to indicate that there is a change in turbulent characteristics with increasing B0 . The linear modes of the system start playing a role in terms of not only anisotropizing the spectrum but also inﬂuencing the multiscaling aspects. In the light of these simulations the phenomenological approach of Goldreich and Sridhar [71] in terms of weak, intermediate and strong turbulence seems promising. Another model of magnetohydrodynamic activity in plasmas, namely the electron Magnetohydrodynamic (EMHD) model [72] is relevant for the study of fast time scale phenomena and has attracted a lot of attention recently. This is primarily due to the present research trend which has been promoted by the technological tools that have been made available to investigate shorter and faster phenomena in general. Also the fast ignition concept of laser fusion, plasma acceleration concepts and certain plasma devices operating at fast time scales (e.g. fast plasma opening switches etc.) has given a trememdous boost to the study of this particular regime. The turbulence research on this particular model is at a very nascent stage. There are no three dimensional studies on turbulence for this model conducted so far. In two dimensions also only limited aspects of turbulence studies have been carried out. The model in two dimensional case can be

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described by the following coupled set of equations: ∂ (b − d2e ∇2 b) − zˆ × ∇b · ∇∇2 b = −ˆ z × ∇ψ · ∇∇2 ψ ∂t

(5.11)

∂ (ψ − d2e ∇2 ψ) + zˆ × ∇b · ∇(ψ − d2e ∇2 ψ) = 0 (5.12) ∂t Here de = c/ωpe is the electron skin depth. Note that this is a natural spatial scale length present in the model. It is interesting to observe that Eq. (5.11) of the model reduces to two dimensional vorticity equation for the hydrodynamic ﬂuid when the excitation scales are shorter than de provided ψ = 0. Equation (5.12) gives the convection of the scalar ﬁeld ψ − d2e ∇2 ψ in the ﬂow for which the velocity potential is given by b. The ﬁeld ψ − d2e ∇2 ψ, however, actively inﬂuences the ﬂow potential through a driving term −ˆ z × ∇ψ · ∇∇2 ψ appearing in Eq. (5.11). The skin depth de is an important characteristic scale of the model. As in MHD, the EMHD model too supports characteristic waves as normal modes in the presence of ambient magnetic ﬁeld. The waves are dispersive in this case and are known as whistler waves. The evolution equations !support two non-dissipative square invariants viz.,! the total energy E = [(∇ψ)2 + b2 + d2e ((∇2 ψ)2 + ! (∇2 b)2 )]dr and H = (ψ − ∇2 ψ)2 dr. The energy in this case cascades directly towards short scales as in MHD. The energy spectra for this model have been investigated recently by high resolution numerical simulations [73]. The numerically obtained energy spectra scales as k −5/3 conﬁrming its similarity to hydrodynamic ﬂuid ﬂow in the regime of kde 1. In the kde 1 regime where whistler waves are expected to play some role, the observed scaling spectra are Ek ∼ 2/3 k −7/3 . Biskamp et al. [73] have ruled out any eﬀect of whistler interactions even in this wave length regime. This is because their analysis shows that on the basis of Kolmogorov’s scaling hypothesis energy should scale as Ek ∼ k −7/3 , whereas if IK like analysis is used and eﬀects due to whistler wave interactions are incorporated the energy ought to scale as ∼ k−2 . However, it would be inappropriate to rule out whistler eﬀect, on the basis of energy scaling alone as the presence of intermittency correction on the exponent can produce signiﬁcant diﬀerence in the scaling exponent from KI and/or Kolmogorov’s phenomenology. The scaling study of a hierarchy of structure functions are desirable (as done in the context of MHD) to arrive at more conclusive evidence of the absence of whistler eﬀect. It is worthwhile to point out here that a later numerical work by Dastgeer et al. [74, 75] demonstrates that even if the whistler waves do not inﬂuence the energy scaling exponents, they inﬂuence the cascade

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mechanism. They show that in the presence of a uniform magnetic ﬁeld the spectrum anisotropizes revealing a cascade mechanism where whistler wave interactions are inﬂuential. The multiscaling in 2-d EMHD was demonstrated recently with the help of high resolution numerical studies which were carried out by Germaschewski et al. [76]. The structure functions (upto p = 14) for both the ﬁelds ψ and b show deviation from linearity for the scaling exponent ζp . A theoretical single parameter ﬁt to the numerical curves for the two scaling exponents (with diﬀerent values of the parametrs) corresponding to b and ψ was provided by the authors using a derivation based on logPoisson statistics of generalized She Leveque [10] expression. Note that unlike MHD, no physical justiﬁcation for the ﬁtting parameters could be given by them. The other evidence for intermittency in EMHD has been provided by Boﬀetta et al. [77] who have measured the scaling exponent of various powers of energy dissipation function numerically, viz., they study p 1 (5.13) d2 x (x) ∼ lτp (l)p = V (l) and show a nontrivial scaling of τp with p signifying the presence of intermittency in 2-d electron magnetohydrodynamics. We ﬁnally discuss the work carried out in the context of ﬂuid models describing electrostatic phenomena in plasmas. These models have been adopted extensively in the description of turbulent transport property of plasma conﬁnement devices viz., tokamaks, stellarators, reverse ﬁeld pinches etc. We select one of the most generic models, namely the 2-d Hasegawa Mima (HM) [78] model on which a lot of fundamental studies on turbulent phenomena have been carried out. In fact this is the model which has been subjected to a variety of (almost all kinds of) ﬂuid turbulence techniques for the study of strong turbulence regime. This model has also been utilized by plasma physicists to approach the turbulence description in the case when wave features dominates. The model equations for the scalar potential φ resemble the evolution equation for b of the EMHD equation (Eq. (5.11)) when ψ is taken = 0. It also has an additional linear term associated with weak plasma inhomogeneity which gives rise to drift waves: ∂φ ∂ (φ − ρ2s ∇2 φ) + zˆ × ∇φ · ∇∇2 φ + vd = µ∇2 ∇2 φ (5.14) ∂t ∂y The support two square invariants !viz., the energy E = ! 2 above equations [φ + (∇φ)2 ]dr and the generalized vorticity Ω = [(∇φ)2 + (∇2 φ)2 ]dr.

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Thus similar to 2-d ﬂuids here too apparantly there are two cascade regimes. Note that in the kρs 1 regime the equation is identical to the evolution equation for vorticity in the 2-d hydrodynamic ﬂuid with the identiﬁcation of v = ∇φ. Fyfe and Montgomery [79] use Kolmogorov’s ansatz for this particular range kρs 1, which gives the hydrodynamic ﬂuid scaling for Ek , viz., k −5/3 in inverse energy cascade regime and k −3 in the vorticity cascade regime. These scalings for energy along with the identiﬁcation that ∇φ ∼ v leads to |δφl |p similar to l4p/3 and l2p in the energy and vorticity cascade regimes respectively. The scaling relationship in the other range of wavenumbers k 1 (lengths normalized by ρs ) has been obtained by various authors [80–82] as Ek similar to k−11/3 and k −5 in the energy and vorticity cascade regime. Such a scaling was veriﬁed numerically by Bofetta et al. [83] recently. Note that the energy scaling of HasegawaMima equation diﬀers from that of EMHD in this regime, in the latter as mentioned earlier in this section energy Ek scales as k −7/3 . The diﬀerence essentially arises due to coupling with the ﬁeld ψ in the case of EMHD. For HM when kρs 1 the equation reduces to ∂φ − zˆ × ∇φ · ∇∇2 φ = 0 (5.15) ∂t which implies that the eddy turn over time scales for this equation as l4 /φ. This leads to k−11/3 . On the other hand the EMHD equations which reduce in the k 1 (length normalized by skin depth de ) limit to ∂b z × ∇ψ · ∇∇2 ψ − zˆ × ∇b · ∇∇2 b = −ˆ ∂t

(5.16)

∂ψ + zˆ × ∇b · ∇ψ = 0 (5.17) ∂t Here while Eq. (5.16) implies an eddy turn over time to be l 4 /b, Eq. (5.17) suggests a faster turn over time of l2 /b (note we are in k 1 regime hence l 1). Clearly, the latter is more eﬀective in governing the turbulent dynamics. It also aﬀects the evolution of b ﬁeld through the coupling term and hence deﬁnes a distinct scaling law for EMHD in contrast to HM, which is k −7/3 . An interesting consequence of the reduced eddy turn over time in k 1 regime for the HM equation is that of a slower transfer of power toward long scales in comparison to the 2-d Navier-Stokes ﬂuid. This results in accumulation of power and formation of quasistationary vortices [84] in the driven dissipative simulations of the above equation. In this quasistationary stage the various structure functions do not show a monotonic scaling

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with separation distance, instead there are distinct crystal like oscillatory behaviour [84] displayed by them. Higher the order of structure function the oscillation amplitude increases (as the higher powers emphasize the high intensity regions). The probability distribution function for the velocity ﬁeld shows highly non-Gaussian features due to these quasi-stationary structures as has been shown by the simulations of Bofetta et al. [83]. For the decaying HM turbulence numerical studies on the properties (skewness and kurtosis parameters) of single point probability distribution function of variables φ and ∇2 φ have also been carried out [85]. The results suggest that while these parameters for the single point PDF of φ remains close to the Gaussian value, for the variable ∇2 φ the deviation from Gaussian values is evident. This indicates that the short scale part of the turbulent ﬂuctuations can have intermittency. An explanation on the basis of Kraichnan’s mapping closure technique has also been provided [86] to show that the single point probability distribution function for φ can be gaussian. The reason was attributed to the absence of nonlinear straining terms in the evolution equation Eq. (5.14). It was also shown that the statistics of ∇2 φ will be non-Gaussian in general.

5.5

Alternative approach for plasma turbulence

The conventional description of plasma turbulence adopts an approach which utilizes the intrinsic property of the plasma medium viz. that of supporting a wide variety of characteristic normal oscillatory modes. These modes are the solutions of the linearized dynamical equations obtained in the small amplitude limit and are characterized by a dispersion relation which relates frequency ω to the wavenumber k. Thus for a given wavenumber k, the frequency ω has a unique (or a few in case of degeneracy) discrete value. As the amplitude of the excitations becomes large the linear dispersion relation is no longer valid as the nonlinear interactions become important. The nonlinear interaction in the medium produces a width ∆ωN L proportional to the interaction coeﬃcient around the natural mode frequency ω. Thus, in the presence of nonlinearity the individual waves start losing their signiﬁcance. However, when the nonlinearity is weak such that ∆ωN L /ω 1, the wave aspects dominate the dynamics. In this regime (usually termed as the weak turbulence regime) the dynamics is adequately described by the well known Wave Kinetic Equation (WKE) [87]. The wave kinetic equation describes the evolution of action density

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of waves as individual wave packets propagate along the ray trajectories in the k − x phase space and weakly interact due to nonlinear eﬀects. When the amplitude of the waves becomes very high so that ∆ωN L /ω ∼ 1, the notion of individual wave packets loses signiﬁcance and one approaches the strongly turbulent regime. One of the spectacular successes of the weak turbulence theory has been in the context of gravity wave problem. Zakharov and Filonenko [88, 89] obtained Kolmogorov like spectral scaling law for the gravity wave problem by seeking exact solution of the WKE. The simulations and experimental observations [90] were found to agree very well with the theoretical value of k −2.5 predicted by this theory for the spectrum. Since then the weak turbulence theory has found applications in a variety of other ﬁelds in physics. In plasma physics it has been applied for the Langmuir waves, drift waves and also Alfven waves. In the context of Alfven as well as acoustic waves the derivation from weak turbulence wave kinetic treatment predicts a k −3/2 [91] scaling for energy. It is interesting to note that for these dispersionless waves the scaling index for energy from weak turbulence considerations not only gives a power law prediction but also matches with the KI [67] derivations which employ Kolmogorov’s adhoc dimensional approach along with correction to the energy transfer rate due to limited interactions amidst the Alfven wave packets as shown in the previous section. The agreement of predictions of the weak turbulence theory with a Kolmogorov like power spectrum lends credence to the idea that the observation of power laws can in general be explained on the basis of weakly interacting nonlinear waves. It is important to emphasize that Kolmogorov’s conjecture in the context of hydrodynamic ﬂuids is still only a highly successful and intuitive hypothesis. In contrast, the weak turbulence theory oﬀers an exact derivation of the powerlaw spectra observed in the turbulent regime where wave features dominate. One must, however, not forget that the weak turbulence theory itself is based on assumptions e.g., phase stochasticity and absence of coherent structures, and hence a derivation based on it is also to that extent non-rigourous. The inherent assumption of phase stochastization amidst the collection of waves in the weak turbulence approximation precludes the possibility of intermittency phenomena. However, as one enters the strongly turbulent regime a variety of physical eﬀect takes place which can lead to intermittency in plasma turbulence. The work on Langmuir wave turbulence illustrates the approach towards the strong turbulence limit very succinctly. As the amplitudes of the individual plasma modes become high and the mode

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widths start overlapping, phase correlations start developing. In one dimension it leads to the development of modulational instability which causes the formation of coherent soliton structures. Kingsep et al. [92] have attempted to explain the numerically observed energy spectral scalings (for which the weak turbulence predictions were found to be inadequate) by assuming a suitable speciﬁcally chosen distribution of such solitonic structures. In higher than one dimension the modulational instability of the Langmuir waves packets leads to collapse and formation of cavitons, as the dispersion of waves is unable to balance the nonlinear terms [93]. Clearly, the possibility of formation of such nonlinear localized structures in physical space is indicative of the presence of nonlocal interactions in wave number space. This is in clear contrast to the basic premise of Kolmogorov’s formalism, namely that of local cascade in wave number space and hence provides for a possible novel physical reason behind the origin of intermittency like phenomena in plasma. In general as the nonlinear width of any natural mode becomes of the order of the linear frequency (or the width of linear frequency space in the case of weakly dispersive waves), i.e., ∆ωN L /ω 1 one enters the strongly turbulent regime which causes the transition from phase incoherent to phase coherent nonlinear interactions. As a result of such interactions a collection of uniform wave packets can undergo modulational instabilities which can further lead to the formation of a variety of nonlinear localized structures such as solitons, cavitons etc. While in the case of Langmuir waves the modulational instability arises from the coupling of the high frequency plasma waves with the low frequency ion modes, there are other low frequency modes whose modulational instability in the strongly turbulent regime has been widely studied. An interesting and practically useful example is the case of the drift waves (natural modes of an inhomogeneous plasma) which constitute an important part of ﬂuctuation spectrum in any plasma conﬁnement device. The modulation of a collection of drift-wave wave packets leads to the self consistent generation of linearly stable poloidally symmetric shear ﬂow patterns known as zonal ﬂows [94]. In such ﬂow patterns the velocity ﬂow is orthogonal to the shear which is directed along the plasma inhomogeneity direction. In the case of drift waves, this self consistent generation of zonal shear ﬂow patterns produces an interesting feature of self regulation of the turbulent spectrum, which has important implications in terms of reducing the radial transport losses of the plasma. The coupled physics of drift wave spectrum and zonal ﬂow patterns and the physics of self regulation

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has been discussed theoretically by a number of authors [92–98]. The underlying physics of such a self regulation process has been identiﬁed with the radial decorrelation of eddies causing transport. The diﬀerential advection due to the shear ﬂow can stretch the eddies to a distance longer than the eddy coherence length in a time which is shorter than the eddy turn over time (the typical life time of the eddy in the absence of shear), resulting in reducing the lifetime of eddies. This reduction in the eddy decorrelation time causes faster dissipation of the turbulent energy, resulting in suppression of the turbulent ﬂuctuations and the transport. There is ample experimental evidence of the drop in turbulent transport of density and energy in magnetic conﬁnement devices in the presence of shear ﬂow to values as low as due to classical collisional eﬀects. A neccesary requirement for such a stabilization is that the shear ﬂow itself should be stable and extended enough so that the turbulent ﬂuid spends time longer than the decorrelation time in the shear ﬂow environment. This does not happen in the case of neutral hydrodynamic ﬂuids, where the shear ﬂows are inherently unstable to the Kelvin - Helmholtz instability. In fact in the neutral ﬂuids the ﬂuctuations generated from the Kelvin - Helmholtz instability acts as an additional source for turbulence. In plasma ﬂuids the stability of the shear ﬂow is to an extent ensured by the existence of magnetic shear. This is thus a novel aspect appearing in the context of plasmas and can be emulated in neutral ﬂuids only under special conﬁgurations, viz., constraining the system in two dimensions by strong rotations and/or by the presence of gravity. A detailed discussion of shear ﬂow in plasmas has been provided in a review by Terry [99]. In addition to suppression of the turbulent intensity it is also observed that the presence of strong shear prevents the mixing of the ﬂuid constrained in it with the turbulent background and thereby inhibits the normal cascade mechanism that is typically operative in turbulence. This results in spatially patchy regions of enhanced vorticity, which can be a source for intermittency. In a recent work Kaw et al. [100] have studied the strong turbulence aspect of the combined system of drift and zonal ﬂows. They show the possibility of the formation of nonlinear stationary structures in such a coupled system. These nonlinear stationary structures were obtained in a moving frame by retaining novel eﬀects associated with trapped and untrapped drift wave trajectories. They showed that the drift wave turbulence can self consistently sustain coherent, radially propagating modulation envelope structures such as solitons, shocks, nonlinear wave

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trains etc. The possibility of such solutions indicate that the combined drift and zonal ﬂow system can be a source for intermittency. The experimental evidence of the formation of zonal shear ﬂow layers, along with the observations of sporadic bursts of transport activities and the theoretical possibility of the existence of a variety of nonlinear structures shows that plasma turbulence can not be structureless as envisaged by Kolmogorov’s ansatz. Thus as one approaches towards the strong turbulence limit within the weak turbulence formalism in plasmas the intermittency aspect seems to appear inherently. It is necessary that an appropriate identiﬁcation, quantiﬁcation and catagorization of these novel intermittency like phenomena in plasmas be also made. This, however, is a diﬃcult task experimentally. This is basically because the typical plasma experiments are pulsed, thereby restricting the data severely. The plasma boundaries and inhomogeneity of the system cause further restrictions in obtaining a large amount of data. Due to the lack of suﬃcient data an appropriate identiﬁcation of inertial regime (well separated from driving and dissipation regimes) and the evaluation of higher order structure functions in this regime becomes a diﬃcult task. 5.6

Summary and conclusion

It is clear that the plasma turbulence oﬀers exciting and challenging issues for investigation. We summarize below the salient features of the present understanding of the phenomena of intermittency in the context of plasma turbulence. We start by summarizing the basic notions of intermittency in NavierStokes ﬂuids. The Navier-Stokes equations have no spatial or temporal scales in them. There are also no linear propagating modes in the equation in the incompressible limit. Thus scale dependence may get introduced only by stirring forces, dissipation and/or nonlinearity. The K41 theory applies for the inertial range, which is well separated from the former two scales. In the lowest order a near-Gaussian ansatz for the turbulence and self similarity of scales leads to the K41 spectral law based on simple dimensional arguments. Experiments show deviations from self similarity and the K41 ansatz. They show that the turbulence exhibits scale dependent non-Gaussianity in the probability distribution of velocity gradients etc. This non-Gaussianity exists because nonlinearity creates long lived vortex structures within the turbulence making it patchy in space and time. Quan-

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titatively, the structure function Sp (l) ∼ lζp where ζp is a nonlinear function of p provides for the measure of the deviation from monoscaling. Thus turbulent ﬂuctuations show multiscaling although there are no natural scales in the equations. This kind of scale dependence is entirely because of nonlinearity. This succinctly is the fundamental problem of intermittency in ﬂuid turbulence. Plasma intermittency studies closest to the above are those dealing with three dimensional MHD turbulence. These equations too have no characteristic spatial scale in them. However, there are linear non-dispersive modes of propagation called Alfven waves. KI argued that the presence of these wave like characteristics leads to the Alfven eﬀect causing modifed interaction involving the ratio of Alfven time to eddy turnover time. In the inertial range this led to a modiﬁed Kolmogorov spectral law with 3/2 power. Computer simulations with 3-d MHD equations by Biskamp and his colleagues have argued against the existence of the Alfven eﬀect for isotropic MHD turbulence studies. Thus the 3-d MHD turbulence is very much like hydrodynamic turbulence in this view. The intermittency features are slightly diﬀerent and may be understood by assuming that the characteristic dissipative structures are 2-d current sheets rather than 1-d vortex ropes as in hydrodynamic turbulence. However, a recent work by them in which they distinguish between the parallel and perpendicular directions (with respect to local mean magnetic ﬁeld), the spectral index for perpendicular energy and perpendicular structure functions exhibit scaling which point towards the role of Alfven eﬀect. Thus, whether the Alfven eﬀect exists and/or plays a role in intermittency remains a question for future studies. There is magnetic ﬁeld ﬂuctuation data from solar wind, interplanetary medium, solar atmosphere, reversed ﬁeld pinches etc which can be compared with these theories. The next interesting model is the EMHD model. Here the linear waves are dispersive and there is a characteristic length scale, the skin depth, in the equations. The nature of turbulence is diﬀerent on two sides of the skin depth, viz. on the short scale side it is like hydrodynamic equations and on the long scale side it is dominated by whistler wave like perturbations. Numerical simulations show distinct spectral scaling indices in the two sides of the electron skin depth. Basically, this suggests that the casacde rates are dissimilar in the two regions. Thus if turbulent ﬂuctuations are forced at one region and the ﬂuctuation energy moves through the skin depth into the other scale the diﬀerence in cascade rates may lead to the formation

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of interesting structures near the characteristic skin depth regime. Such a phenomenon has indeed been demonstrated in the context of another ﬂuid model (viz., the Hasegawa-Mima model) about which we discuss later. This would be an intermittency like phenomenon which has no ﬂuid analogue. The features of scale similarity and deviations from it on the two sides are perhaps similar to those observed in MHD and Navier-Stokes turbulence; they would thus correspond to intermittency of the normal variety with dissipative structures of diﬀerent dimensionality arising in diﬀerent regimes and eﬀects like whistler eﬀect (analogue of Alfven eﬀect) arising in long scale EMHD turbulence. Another ﬂuid model which has been extensively used in plasma turbulence studies is the Hasegawa Mima model for drift waves. This model also involves a natural spatial scale, namely the ion Larmor radius and a natural temporal scale namely, the drift frequency. In the long scale and the short scale regimes, notions of self similarity again work. In the context of this model the formation of quasicrystalline structures around the larmor radii scale length has indeed been demonstrated by the simulation work of Kukharkin [84] due to dissimilar casacde rates in the two regime. Thus the boundary region breaks self similarity of scales and introduces intermittency like behaviour just as discussed in the EMHD model above. Furthermore, in each wavelength regime there may be standard multiscaling eﬀects due to nonlinearity unless the eﬀective two dimensionality of the model precludes this. Another interesting feature exhibited by the Hasegawa-Mima like model and its variants is the phenomena of self regulation of turbulence. A totally novel possibility of an intermittency like phenomenon in plasma turbulence arises in the process of self regulation. The background turbulence due to drift waves generate shear ﬂow patterns which are known as zonal ﬂows. Zonal ﬂows are relatively long scale anisotropic ﬂows which extract energy from the turbulence by a modulational instability mechanism and then try to limit the growing turbulent ﬂuctuations by scattering them to shorter scales where dissipation is important. This leads to a stochastic self regulating system where interactions between very disparate scales are involved unlike the cascade mechanisms of standard ﬂuid turbulence. The study of such self regulated turbulent states in plasmas has played a very prominent role in ﬁnding low transport conﬁgurations in magnetically conﬁned fusion plasmas. Considerable attention is also being paid to intermittency like phenomenon which may arise in such situations. It is of interest to point out that the various ﬂuid like models discussed

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above have their own ranges of validity and there may arise physical situations where the turbulence involves a range of space and time scales which may span more than one model. A case in point is turbulence which may span all scales from longer than ion larmor to ion skin depth down to electron skin depth; this could be a problem which can be discussed by two ﬂuid plasma equations or by a combination of simpliﬁed models like standard MHD, Hall MHD and EMHD. In this case the turbulence may again acquire interesting features as excitation scale lengths hits and crosses the boundaries associated with certain characteristic scales, say by nonlinear cascade phenomena, and moves from description by one type of a model to a diﬀerent type. This is another example of how novel intermittency like phenomena may arise in plasmas when characteristic scale boundaries are crossed. Plasmas are also known to support a wide variety of nonlinear structures unlike incompressible ﬂuids which can only support vortices and eddies. The solitons, shocks, tangential discontinuities, vortices, blobs, magnetic island chains, current sheets, current ﬁlaments, magnetic ropes and so on are some examples of nonlinear structures which are known to form in compressible plasmas. Given this great variety of possible structures, the sources of intermittency in plasmas may also exhibit greater variety. It seems that the origin of intermittency like phenomena in plasmas (bursty patchy spatial and temporal behaviour) has a variety of manifestations. It can be classiﬁed broadly in the following catagories (i) normal hydrodynamic ﬂuid like intermittency arising through nonlinearity, (ii) the characteristics scale induced structure formations and (iii) the presence of inherent linear and nonlinear instabilities providing direct connections between disparate scales. The existence of linear waves, a variety of nonlinear structures and dissipative patterns may add further ﬂavour to intermittency in plasmas. It would be interesting if the experimental study of plasma turbulence identiﬁes intermittency and is able to distinguish between the three varieties. It should be recognized that most laboratory experiments are pulsed in nature and deal with bounded plasmas with signiﬁcant inhomogenities in the turbulence. One may also not have a clearly deﬁned inertial regime of turbulence, which is well separated from the driven and dissipative regimes. The data sets used for analysis must take account of these limitations and natural features of laboratory plasma turbulence. Moreover, at the moment the typical tools employed for the study of this phenomena are in general the same as the ones used in the study of intermittency in ﬂuid turbulence,

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namely, a study of scale dependent non-Gaussianity in the PDF of some ﬂuctuation variable, the study of burstiness in time or patchiness in space of the time series, the study of coherent long lived structures arising primarily through nonlinearity as seen by quantitative study of structure factors etc. However, new diagnostics need to be designed to distinguish and classify this phenomena as per the discussions above. Finally, given the richness of interactions (electromagnetic ﬁelds) in plasma it is conceivable that a detailed study on turbulence might lead to an altogether new, hitherto unknown aspect of turbulence. It is also possible that the diversity and in some cases simplicity of the turbulent state (e.g. natural two dimensionality, predominance of wave eﬀects) may provide us with essential and signiﬁcant clues to the phenomenon of turbulence itself.

References [1] Kolmogorov A N 1941 Dokl. Akad. Nauk SSSR 30 9-13 (reprinted in 1991 Proc. R. Soc. London A 434 9-13). [2] Kolmogorov A N 1941 Dokl. Akad. Nauk SSSR 32 16-18 (reprinted in 1991 Proc. R. Soc. London A 434 15-17). [3] Frisch U 1999 Turbulence (Cambridge University Press, New Delhi) 127-133. [4] Landau L D and Lifsitz E M 1987 Fluid Mechanics Pergamon Press, Oxford. [5] Kolmogorov A N 1962 J. Fluid Mech. 13 82-85; Obukhov A M J. Fluid Mech. 13 77-81. [6] Novikov E A and Stewart R W 1964 Izv. Acad. Nauk SSSR, Ser. Geoﬃz 3 408–412. [7] Frisch U, Sulem P-L and Nelkin M 1978 J. Fluid Mech. 87 719–736. [8] Parisi G and Frisch U 1985 Turbulence and Predictability in Geophysical Fluid dynamics. Ed: Ghil M, Benzi R and Parisi G (Amsterdam North Holland) 744–745. [9] Maneveau C and Sreenivasan K R 1987 Phys. Rev. Letts. 59 1424–1427. [10] She Z S and Leveque E 1994 Phys. Rev. Letts. 72 336–339. [11] Frisch U, Mazzino A and Vergassola M 1998 Phys. Rev. Letts. 80 5532–5539. [12] Krommes J A and Kleva R G 1979 Phys. Fluids 22 2168-2177. [13] Kraichnan R H 1990 Phys. Rev. Lett. 65 575–578. [14] She Z S 1991 Phys. Rev. Lett. 66 600–603. [15] Gotoh T and Kraichnan R H 1993 Phys. Fluids A 5 445. [16] Jha R, Kaw P K, Mattoo S K, Rao C V S, Saxena Y C and the ADITYA Team 1992 Phys. Rev. Lett. 69 1375–1378. [17] Jha R, Joseph B K, Kalra R, Kaw P K, Mattoo S K, Raju D, Rao C V S,

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[18]

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Saxena Y C, Sen A and the ADITYA team 1995 Proc. 15th International Conf. Plasma Phys. and Control. Nucl. Fusion, Seville, 26 September-1 October, 1994, Vol. 1, 583–591. Endler M, Niedermeyer H, Giannone L, Holzhauer E, Rudyj A, Theimer G, Tsois N, Zoletnik S, the ASDEX team and the W7-AS team 1995 Nucl. Fusion 35 1307–1339. Carreras B A, Hidalgo C, Sanchez E, Pedrosa M, Balbin R, Garcia-Cortes I, van Milligen B, Newman D and Lynch V E 1996 Phys. Plasmas 3 2664–2672. Dong L, Wang L, Feng C, Li Z, Zhao Q and Wang G 1998 Phys. Rev. E 57 5929–5936. Antar G Y, Krasheninnikov S I, Devynck P, Doerner R P, Hallmann E M, Boedo J A, Luckhardt S C and Conn R W 2001 Phys. Rev. Lett. 87 65001–4. Antar G Y, Devynck P, Garbet X and Luckhardt S C 2001 Phys. Plasmas 8 1612–1624. LaBombard B et al. 2002 Proc. 19th International Conf. Plasma Phys. and Control. Nucl. Fusion, Lyon, October 14-19, 2002, IAEA-CN-94/ EX/D2-1. Kharchev W K, Skvortsova N N and Sarksyan K A 2001 J. Math. Sci. 106 2691–2703. Gonchar V Yu, Chechkin A V, Sorokovoi E L, Chechkin V V, Grigoreva I and Volkov E D 2003 Plasma Physics Reports 29 380–390. Boedo J A et al. 2001 Phys. Plasmas 8 4826–4833. S´ anchez E et al. 2000 Phys. Plasmas 7 1408–1416. Jha R and Saxena Y C 1996 Phys. Plasmas 3 2979–2989. Carbone V, Regnoli G, Martines E and Antoni V 2000 Phys. Plasmas 7 445–447. Carbone V, Sorriso-Valvo L, Martines E, Antoni V and Veltri P 2000 Phys. Rev. E 62 R49–R52. Zweben S J and Gould R W 1985 Nucl. Fusion 25 171–183. Joseph B K, Jha R, Kaw P K, Mattoo S K, Rao C V S, Saxena Y C and the ADITYA team 1997 Phys. Plasmas 4 4292–4300. Beyer P, Benkadda S, Garbet X and Diamond P H 2000 Phys. Rev. Lett. 85 4892–4895. Boedo J A et al. 2003 Phys. Plasmas 10 1670–1677. Zweben S J et al. 2002 Phys. Plasmas 9 1981–1989; 2004 Nucl. Fusion 44, 134–153. Terry J L et al. 2003 Phys. Plasmas 10 1739–1747. Benkadda S, de Wit T D, Verga A, Sen A, the ASDEX team and Garbet X 1994 Phys. Rev. Lett. 73 3403–3406. Ritz Ch P and Powers E J 1986 Physica 20 D 320–334. Ritz Ch P, Powers E J and Bengtson R D 1989 Phys. Fluids B 1 153–163. Tsui H Y, Rypdal K, Ritz Ch P and Wootton A W 1993 Phys. Rev. Lett. 70 2565–2568.

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[41] Milligen B Ph van, Hidalgo C and S´ anchez E 1995 Phys. Rev. Lett. 74 395–398. [42] Milligen B Ph van, Sanchez E, Estrada T, Hidalgo C, Branas B, Carreras B A, and Garcia L 1995 Phys. Plasmas 2 3017–3032. [43] Jha R, Mattoo S K and Saxena Y C 1997 Phys. Plasmas 4 2982–2988. [44] Moyer R A, Lehmer R D, Evans T E, Conn R W, Schmitz L 1996 Plasma Phys. Control. Fusion 38 1273–1278. [45] Carreras B A et al. 1998 Phys. Rev. Lett. 80 4438–4441; Phys. Plasmas 5, 3632–3643. [46] Xu Y H, Jachmich S, Weynants R R, Huber A, Uterberg B and Samm U, 2004 Phys. Plasmas 11 5413–5422. [47] Yu C X, Gilmore M, Peebles W A and Rhodes T L 2003 Phys. Plasmas 10 2772–2779. [48] Jha R, Kaw P K, Kulkarni D R, Parikh J C and the ADITYA team 2003 Phys. Plasmas 10 699–704. [49] Burlaga L F and Goldstein M L 1984 J. Geophys. Res. 89 6813–6817. [50] Roberts D A, Goldstein M L and Klein L W 1990 J. Geophys. Res. 95, 4203–4215. [51] Burlaga L F 1991 J. Geophys. Res. 96 5847–5851. [52] Marsch E and Liu S 1993 Ann. Geophys. 11 227–238. [53] Sorriso-Valvo L, Carbone V, Giuliani P, Veltri P, Bruno R, Antoni V and Martines E 2001 Planet. Space Sci. 49 1193–1200. [54] Burlaga L F 1992 J. Geophys. Res. 97 4283–4293. [55] Tu C-Y and Marsch E 1990 J. Geophys. Res. 95 4337–4341; 1993 J. Geophys. Res. 98 1257–1276; 1995 Space Sci. Rev. 73 1–210. [56] Bruno R, Carbone V, Veltri P, Pietropaolo E and Bavassano B 2001 Planet. Space Sci. 49 1201–1210. [57] Marsch E and Tu C-Y 1997 Nonlinear Proc. Geophys. 4 101–124. [58] Sorriso-Valvo L, Carbone V, Consolini G, Bruno R and Veltri P 1999 Geophys. Res. Lett. 26 1801–1804. [59] Chang T and Sunny W Y 2004 Phys. Plasmas 11 1287–1299. [60] Sintov M I, Sharma A S and Papadopoulos K 2001 Phys. Rev. E 65 161161–10. [61] Consolini G and De Michelis P 1998 Geophys. Res. Lett. 25 4087–4090. [62] Stepanova M V, Antonova E E and Troshichev O 2003 Geophys. Res. Lett. 30 271–274. [63] Castaing B, Gagne Y and Hopﬁnger E J 1990 Physica D 46 177–200. [64] Borovsky J E, Elphic R C, Funsten H O and Thomsen M F 1997 J. Plasma Phys. 57 1–34. [65] Angelopoulos V, Mukai T and Kokubun S 1999 Phys. Plasmas 6 4161–4168. [66] Biskamp D 1993 Nonlinear Magnetohydrodynamics (Cambridge University Press, Cambridge, Great Britain) 217–228. [67] Iroshnikov P S 1964 Sov. Astron. 7 566-567; Kraichnan R H 1965 Phys. Fluids 8 1385–1387. [68] Muller W C and Biskamp D 2000 Phys. Rev. Letts 84 475–478. [69] Biskamp D, Muller W C 2000 Phys. Plasmas 7 4889–4900.

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[70] Muller W C, Biskamp D and Grappin R 2003 Phys. Rev. E 67 066302–4. [71] Goldreich P and Sridhar S 1995 Astrophys. J. 438 763. [72] Kingsep A S, Chukbar K V and Yankov V V 1990 Reviews of Plasma Physics Consultant Bureau, New York 16. [73] Biskamp D, Schwarz and Drake J F 1996 Phys. Rev. Letts. 76 1264–1267. [74] Dastgeer S, Das A, Kaw P and Diamond P H 2000 Phys. Plasmas 7 571–579. [75] Dastgeer S, Das A and Kaw P 2000 Phys. Plasmas 7 1366–1373. [76] Germaschewski K and Grauer R 1999 Phys. Plasmas 6 3788–3793. [77] Boﬀetta G, Celani A, Cresanti A and Prandi R 1999 Phys. Rev. E 59 3724–3726. [78] Hasegawa A and Mima K 1978 Phys. Fluids 21 87. [79] Fyfe D and Montgomery D 1979 Phys. Fluids 22 246–248. [80] Ottaviani M and Krommes J A 1992 Phys. Rev. Lett. 69 2923–2926. [81] Watanabe T and Fujisaka H 1997 Phys. Rev E 55 5575–5580. [82] Larichev V D and McWilliams J C 1991 Phys. Fluid A 3 938–950. [83] Boﬀetta G, Lillo F De, Musacchio S 2002 Europhys. Letts. 59 687–693. [84] Kukharkin N, Orszag S A and Yakhot V 1995 Phys. Rev. Letts 75 2486–2489. [85] Crotinger J A and Dupree T H 1992 Phys. Fluids B 4 2854. [86] Das A and Kaw P 1995 Phys. Plasmas 2 1497–1505. [87] Sagdeev R Z and Galeev A A 1969 Nonlinear Plasma Theory (W A Benjamin Inc., New York, USA) 5-35. [88] Zakharov V E and Filonenko N N 1967 Sov. Phys. Dokl. 11 881. [89] Zakharov V E 1968 J Appl. Mech. Tech Phys. 9 190. [90] Toba Y 1973 J. Ocean Soc. Japan 29 209. [91] Zakharov V E and Sagdeev R Z 1970 Sov. Phys. Dokl. 15 1970. [92] Kingsep A S, Rudakov L I and Sudan R N 1973 Phys. Rev. Letts. 31 1482–1484. [93] Zakharov V E 1972 Sov. Phys. JETP 35 908. [94] Busse F H 1994 Chaos 4 123-134. [95] Hasegawa A and Wakatani M 1987 Phys. Rev. Lett. 59 1581–1584. [96] Lin Z, Hahm T S, Lee W W Tang W M and White R B 1998 Science 281 1835. [97] Sydora R D, Decyk V K and Dawson J M 1996 Plasma Phys. Controlled Fusion 38 A281. [98] Diamond P H, Rosenbluth M N, Sanchez E, Hidalgo C, Milligan B. Van, Estrada T, Branas B Hirsch M, Hartfuss H J and Carreras B A 2000 Phys. Rev. Lett. 84 4842–4845. [99] Terry P W 2000 Rev. Mod. Phys. 72 109–165. [100] Kaw P K, Singh R and Diamond P H 2002 Plasma Phys. and Cont. Nuc. Fusion 44 51–59.

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Chapter 6

Nonlinear Cascades and Spatial Structure of Magnetohydrodynamic Turbulence W.-C. M¨ uller and R. Grappin∗ Max-Planck-Institut f¨ ur Plasmaphysik, 85748 Garching, Germany ∗ Observatoire de Paris-Meudon, 92195 Meudon, France

6.1

Introduction

Plasma turbulence constitutes a complex dynamical state of ﬂow which is encountered in a multitude of diﬀerent physical settings. In experiments for magnetically conﬁned thermonuclear fusion, turbulence leads to anomalous particle transport that reduces the eﬃciency of the magnetic cage conﬁning the fusion plasma [1]. In astrophysical systems, turbulence governs, e.g., the generation of large-scale magnetic ﬁelds of planets and stars by the turbulent dynamo eﬀect [2, 3], the evolution of stellar winds [4], the dynamic state of the interstellar medium especially with regard to star formation [5] and the transport of angular momentum in accretion disks prone to magneto-rotational instability [6]. Although the detailed physical processes as well as parameters such as length and time scales can diﬀer signiﬁcantly between these conﬁgurations, certain inherent features of plasma turbulence remain unchanged. The supposedly universal characteristics include approximate self-similarity of turbulent ﬁelds, cascades of ideal invariants, or intermittency of small-scale structure. They are fundamental and underlie the dynamics of each of the above-mentioned physical realizations. The fundamental properties even of homogeneous incompressible hydrodynamic turbulence are only partially understood (for comprehensive accounts of the developments in hydrodynamic turbulence theory, see [7, 8]). 187

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Additionally, while the nature of the hydrodynamic energy cascade can meanwhile be regarded as clariﬁed, this is not true for its magnetohydrodynamic counterpart. The knowledge of processes which lead to small-scale intermittency, i.e. deviations from spatial self-similarity of the turbulence, is also incomplete for both systems. This chapter will cover theoretical and phenomenological approaches towards turbulent energy dynamics and intermittency in magnetohydrodynamic (MHD) turbulence as well as their experimental and numerical veriﬁcation. Section 6.2 introduces the necessary formulations of the MHD equations, Sec. 6.3 outlines basic concepts of turbulence theory, Sec. 6.4 presents basic cascade phenomenologies, Sec. 6.5 covers technical aspects of numerical turbulence simulation, Sec. 6.6 highlights recent results on energy spectra in isotropic and anisotropic systems, Sec. 6.7 deals with spatial two-point statistics and intermittency followed by conclusions in Sec. 6.8.

6.2

Magnetohydrodynamics

As the main focus lies on the inherent properties of turbulence, the adopted physical model is strongly simpliﬁed. The plasma is assumed to be a quasineutral, electrically conducting single ﬂuid with ﬁnite viscosity and electrical resistivity. This implies that the characteristic spatial scales of the system are much larger than the mean free path of its microscopic constituents and that the shortest resolved time scale equals the time-scale of the fast magnetosonic wave. As for inner fricton the MHD ﬂuid is assumed to be Newtonian. The MHD equations can be derived in a formally consistent way from the conservation laws of mass, momentum and electric charge in combination with Maxwell’s equations [9]. A more heuristic approach can be found in [10]. As an additional simpliﬁcation the turbulence is considered to be incompressible. This restricts the validity of the approximation to ﬂows where the ﬂuid velocity is signiﬁcantly smaller than the speed of the slow magnetosonic wave and eﬀectively eliminates compressible modes. The mass density ρ0 is assumed to be spatially uniform. Under these assumptions the equations of incompressible MHD read ∂t − Re−1 ∆ v = −v · ∇v − ∇p − b × (∇ × b) ,

∂t − Rm−1 ∆ b = ∇ × (v × b) ,

(6.1) (6.2)

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and ∇·v =∇·b =0 ,

(6.3)

with the dimensionless ﬂuid velocity v = V/v0 , magnetic ﬁeld b = B/B0 , and thermodynamic pressure p = P/(ρ0 V 2 ). All quantities can be dressed with units using a characteristic length L0 , velocity v0 and mass density ρ0 as reference quantities. The interaction parameter, B02 (µ0 ρ0 )−1 /v02 , in front of the Lorentz force term is set to unity by proper choice of B0 . Incompressibility turns the pressure into a passive variable determined by a Poisson equation which is the divergence of the momentum balance (6.1). It can be eliminated from the equations by considering the rotational part of (6.1), only. This gives an evolution equation for the vorticity ω = ∇ × v, (∂t − Re−1 ∆)ω = ∇ × (v × ω − b × j) ,

(6.4)

with the electric current density j = ∇ × b. The dimensionless parameters characterizing the dynamical state of the system are Re =

L0 v0 , µ

Rm =

L0 v0 , η

where Re is the kinetic Reynolds number deﬁned with the kinematic viscosity µ and Rm the magnetic Reynolds number involving the magnetic diffusivity η of the magnetoﬂuid. The Reynolds numbers are rough estimates of the strength of the nonlinearities in the respective equation compared to dissipative terms on the left-hand sides of (6.1) and (6.2). If both Re and Rm are signiﬁcantly larger than unity, the ﬂow undergoes a transition from the laminar state characterized by a stationary stream-line topology to turbulence where the ﬂuid motion seems to be erratic and unpredictable. An equivalent formulation of (6.1)–(6.3), introducing the Els¨ asser ﬁelds z± = v ± b [11], reads b2 ) 2 −1 −1 Re + Rm Re−1 − Rm−1 ∆z± + ∆z∓ , + 2 2 ∇ · z± = 0 . ∂t z± = −z∓ · ∇z± − ∇(p +

(6.5) (6.6)

In incompressible MHD the Els¨asser variables are more fundamental quantities than v and b alone as the symmetry of (6.6) and (6.5) suggests. Additionally, in the presence of a background magnetic ﬁeld, |b0 | |b|,

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the z± -ﬁelds may be interpreted as shear Alfv`en-wave packets travelling along b0 where z+ and z− propagate in opposite directions. In the following {v, b} or Els¨ asser variables are used, whichever makes the presentation clearer. on the ratio of the kinetic energy of the ﬂow ﬁeld, E K = ! Depending ! 1 1 2 M dV v , to the magnetic energy, E = 2 V dV b2 with V denoting the 2 V spatial volume of the system, three regimes can be distinguished. The case E K E M , which will not be treated here, represents the dynamo problem, i.e. the ampliﬁcation of a magnetic ﬁeld by plasma turbulence. To study the nonlinear interaction between v and b, we will further on concentrate on turbulence with E K ∼ E M and E K E M . The latter situation can be found in the solar corona or terrestrial laboratory experiments with magnetic plasma conﬁnement. The magnetic Prandtl number, Prm = Rm/Re, is assumed to be unity throughout this chapter..

6.3

Basic concepts

The apparent randomness of turbulent ﬁelds naturally leads to a statistical approach involving smoothly evolving ensemble averages, •. Under the assumption of quasi-ergodicity the ensemble average!can be replaced by more practical deﬁnitions. Time averaging, f t = T1 T dtf (t),!is applicable under stationary conditions. The spatial average, f x = V1 V dV f (x), is meaningful when statistical homogeneity, i.e. translation invariance of statistical properties, holds. This assumption will be adopted in the following. Though only approximately valid in small regions of real-world turbulence, it strongly reduces the mathematical complexity of the problem. For some derivations the more restrictive symmetry of statistical isotropy, i.e. invariance under rotations about arbitrary axes, is imposed locally. The turbulence is assumed to be ‘fully developed’. That is to say the turbulent system has reached dynamical equilibrium between the energy inﬂow supplied by driving mechanisms such as a large-scale shear instability and the energy outﬂow caused by small-scale dissipation. The equilibrium does not have to be stationary in time. Neglecting the inﬂuence of (faraway) boundaries, periodic boundary conditions are applied at the outer surfaces of the volume V containing the turbulent ﬂow. Besides its chaotic nature, turbulence has remarkable similarity properties. Certain functions, including the moments of the two-point probability distribution functions of the turbulent ﬁelds, exhibit invariance under scal-

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ing transformations if the inﬂuence of initial and boundary conditions is negligible, i.e. x → λx ⇒ f (x) → λα f (x). Such a self-similar function asymptotically follows a power-law, f (x) = Cxα , where the scaling exponent α can often be derived phenomenologically without exact knowledge of either the functional form or the numerical value of the coeﬃcient C.

6.4

Phenomenologies of turbulent cascades

The fundament of virtually all phenomenologies of turbulence is the K41 picture put forward by Kolmogorov in the 1940s [12, 13]. Since the model is intended to describe hydrodynamic ﬂows, it does not include magnetic ﬁelds. The turbulent velocity is viewed as a superposition of structures or ‘eddies’ characterized by a spatial scale, , and the associated velocity ﬂuctuation δv = [v(r + ) − v(r)] · / . The ﬁeld is assumed to be statistically isotropic, therefore the ﬂuctuation amplitude depends only on . This allows to deﬁne the characteristic eddy velocity, v = δv 2 1/2 . As illustrated in Fig. 6.1, the K41 picture distinguishes diﬀerent scales of motion: the energy-containing scales driving the ﬂow, the dissipation range at smallest scales where energy is removed from the system and the ‘inertial’ range where nonlinear interactions govern the dynamics and the inﬂuence of driving and dissipation is negligible. It is within the latter region that spatial self-similarity is observed experimentally in the structure functions of order p, Spv ( ) = δv p ∼ ζp , the ζp being constant, p-dependent scaling exponents. In the limiting case of inﬁnite Reynolds number, the K41 theory predicts values for these scaling exponents (cf. Sec. 6.7). The structure functions are statistical moments of the two-point probability distribution of the respective turbulent ﬁeld. Their scaling exponents provide fundamental statistical information, e.g., S2v ( ) is linked to the onedimensional energy spectrum while the p-dependence of the ζp characterizes the intermittency of ﬂow structures. The relation of the K41 picture to the asymptotic statistical invariance properties of fully developed turbulence is treated in detail in [8]. The inertial-range dynamics of the spectral energy ﬂow is pictured as a Richardson cascade [14]. The turbulent eddies are forming a spatial hierarchy where energy is transferred to smaller scales by eddies breaking up into smaller ﬂuctuations. The overall process is termed ‘direct’ because the resulting energy ﬂux is headed towards smaller spatial scales. The name ‘cascade’ reﬂects the small diﬀerence in size between an eddie and its direct

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descendants which means that the transfer proceeds in small local steps in spatial Fourier-space. Turbulent cascades also work in the opposite direction. In this case they are called ‘inverse’, a case in point being the inverse cascade of magnetic helicity in three-dimensional MHD turbulence. It leads to the build-up of large-scale magnetic structures by the turbulent dynamo [15]. The magnetic ! helicity, H M = 12 V dV a · b, b = ∇ × a, is a macroscopic measure for the twist and linkage of the magnetic ﬁeld lines [16].

Energy (arb. units)

0

10

Large eddies

Small-scale structures

Direct cascade

-1

10

-2

10

Inverse cascade

Drive range -2

10

Dissipation range

Inertial range -1

10

0

10

k (arb. units)

Fig. 6.1 A schematic view of the Kolmogorov (K41) picture of turbulence using the energy spectrum as an example.

The cascades are caused by the quadratic nonlinearities in the NavierStokes and MHD equations which redistribute the cascading quantities dissipationless among diﬀerent scales of motion. In particular, the spectral energy ﬂux can be estimated by order of magnitude as v 2 /τ with the nonlinear eddy turnover time τ = /v .

6.4.1

Kolmogorov phenomenology

In fully developed isotropic turbulence the energy ﬂux generated by the direct cascade in the inertial range equals! the dissipation of energy per unit mass at the smallest scales, ε = − 12 ∂t V dV v 2 . This yields by order of

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magnitude v 2 v3 =

τ

1/3 ⇒ v ∼ (ε ) !

ε∼

⇒ Spv ( ) ∼ (ε )p/3 .

(6.7)

For p = 3 an exact relation exists, the four-ﬁfth law (see, e.g., [8]): 4 S3v ( ) = − ε . 5 The one-dimensional energy spectrum deﬁned as 1 dk2 dk3 (v(k) · v∗ (k) + b(k) · b∗ (k)) E(k1 ) = 2

(6.8)

with ki = (2π)/ i scales like the Fourier transform of S2v (). For future reference magnetic energy has already been included in the expression even though it does not play a role in the K41 picture. In isotropic turbulence one usually considers the angle-integrated energy spectrum 4π E(k) = dΩ E(k)||k|=k , (6.9)

1

0

deﬁning E(k) = 2 |v(k)|2 + |b(k)|2 . The real-space exponent ζ2 of S2v () corresponds to an inertial range Fourier-space scaling ∼ k −(1+ζ2 ) which for hydrodynamic turbulence yields the experimentally well-supported Kolmogorov spectrum E(k) = CK ε2/3 k −5/3 ,

(6.10)

with the Kolmogorov constant CK ≈ 1.6 . The Kolmogorov dissipation length, K41 , gives an order of magnitude estimate of the spatial scales where energy dissipation ∼ v 2 µ/ 2 begins to dominate the nonlinear energy dynamics ∼ v 2 /τ , marking the beginning of the dissipation range. With the approximations introduced above, µ ! −1 ∼ τ = v / ∼ (ε )1/3 /

2 3 1/4 ⇒ K41 = µε .

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6.4.2

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Iroshnikov-Kraichnan phenomenology

The Iroshnikov-Kraichnan (IK) phenomenology [17, 18] for MHD turbulence modiﬁes the K41 approach by introducing a diﬀerent model of the nonlinear energy transfer. The mutual eddy scrambling which is underlying the K41 cascade is replaced by an energy transfer by Alfv´en waves propagating along the magnetic ﬁeld. The energy is spectrally redistributed by nonlinear scattering of colliding Alfv´en-wave packets. In the following, the Els¨ asser quantity z is used analogously to v in the hydrodynamic case. By restricting consideration to MHD turbulence with C K M 1/2 as small mean ! correlation, ρ = |H |/(E E ) , deﬁning+the cross-helicity − 1 C H = 2 V dV v · b, it is not necessary to distinguish z and z . For ﬁnite ρ, the respective energy spectra can diﬀer considerably [19], demanding a more complex theoretical approach [20]. The coupling of magnetic and velocity ﬁelds through the generation and attenuation of Alfv´en waves, termed Alfv´en eﬀect, leads to local equipartition of magnetic and kinetic energy. The process is based on the interaction of eddies of size with the magnetic ﬁeld b0 generated by the largest energycontaining swirls or imposed externally. As a consequence, the associated velocity perturbations v are triggering Alfv´en waves by locally deforming the guide ﬁeld b0 . The incompressible transverse waves then travel along b0 . If the involved perturbations δv, δB are small compared to b0 , one has approximately δv = ±δB. For two colliding Alfv´en waves of extent the interaction time to exchange energy nonlinearly is given by τA = /b0 (In the chosen nondimensional representation b0 is measured in Alfv´en speed multiples of the reference velocity v0 ). Due to b0 b this time is much shorter than the corresponding K41 transfer time τ = /z . In fact, the expression τ∗ entering the IK energy ﬂux z 2 /τ∗ is enlarged by the factor τ /τA compared to the K41 case [21]. Accordingly, the IK model can be obtained from the K41 phenomenology by replacing v with z and by rewriting the characteristic energy cascade time as τ∗ ∼ ττA τ . This gives the following inertial range scaling of the Els¨ asser ﬁelds: z ∼ (εb0 )1/4 ⇒ Spz ( ) ∼ (εb0 )p/4 . Analogously the energy spectrum is obtained as E(k) = CIK (εb0 )1/2 k −3/2

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and the IK dissipation length reads 2 1/3

IK = b0εη . There also exists an exact relation corresponding to the four-ﬁfth law for incompressible three-dimensional MHD ([22], [23]), 3

!

i=1

4 δz ∓ (δi z ± )2 = − ε± , 3

dV [µω + ηj 2 ± (µ + η)ω · j], δz ∓ denoting the longitudiwith ε = nal ﬁeld increments introduced above and δi z ± = (z ± (r + ) − z ± (r)) · ei introducing the unit vector ei from the orthogonal base of an arbitrary co-ordinate system. ±

6.4.3

1 2 V

2

Goldreich-Sridhar phenomenology

In the K41 and the IK phenomenology turbulence ﬂuctuations are characterized by a single length scale, , that makes these models spatially isotropic. However, the presence of a magnetic ﬁeld renders the turbulence locally anisotropic. Alfv´en waves of extent λ interact along the magnetic ﬁeld b0 within the time scale τλ ∼ λ/b0 . Under typical conditions, this is much shorter than the nonlinear eddy-scrambling rate perpendicular to the ﬁeld, τl ∼ l/zl , where l is the ﬁeld-perpendicular extent and zl the amplitude of the ﬂuctuation. In addition, the nonlinear energy ﬂux is much weaker along the direction of the magnetic ﬁeld [24–26]. Goldreich and Sridhar put forward a phenomenology which takes into account the spatial anisotropy caused by the magnetic ﬁeld [27, 28]. Originally, the phenomenology is based on nonlinear 4-wave interactions claiming that three-wave interactions, which underlie the IK model, are absent [29]. The latter conjecture is wrong [30, 31, 26, 32] and to what extent 4-wave interactions play a role in MHD turbulence remains unclear. There are two versions of the GS phenomenology: one for intermediate and one for strong turbulence which give predictions for the ﬁeldperpendicular energy spectrum. Intermediate Turbulence The intermediate MHD turbulence model [28] is an anisotropic generalization of the IK phenomenology which is basically a weak turbulence theory. By diﬀerentiating ﬁeld perpendicular eddy scrambling and ﬁeld parallel

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wave interaction in the IK expression τ∗ ∼ written as τ∗ ∼ ττλl τl leading to zl ∼ (εb0 /λ)1/4 l1/2

⇒

τ τ, τA

the cascade rate can be

Spz (l) ∼ (εb0 /λ)p/4 l p/2 .

The space increment l is perpendicular to the local magnetic ﬁeld, contrary to the quantity in the K41 and IK phenomenologies which denotes an increment in an arbitrary direction. By zl2 ∼ kl E(kl ) with kl ∼ l−1 and kλ ∼ λ−1 the perpendicular energy spectrum is obtained as E(kl ) ∼ (εb0 kλ )1/2 kl−2 . This is the scaling derived in the context of weak turbulence [26]. Since the ﬁeld-parallel spectral energy transfer is assumed to be weak, kλ is constant by order of magnitude under variation of kl . Strong Turbulence If the nonlinear interactions are more eﬃcient than in the intermediate case, the picture of strong turbulence [27] might apply. The phenomenology can be derived from the intermediate model by assuming equality by order of magnitude of the characteristic time scales parallel and perpendicular to the local magnetic ﬁeld. This amounts to setting the prefactor in τ∗ , namely τl /τλ , equal to unity, i.e. τl ∼ l/zl ∼ λ/b0 , yielding zl ∼ (εl)1/3

⇒

Spz (l) ∼ (εl)p/3 ,

(6.11)

which results in a Kolmogorov-like perpendicular energy spectrum, −5/3

E(kl ) ∼ ε2/3 kl

.

The strong turbulence model, which is an anisotropic K41 phenomenology, yields a relation between the corresponding spatial scales of the turbulent eddies. By λ/b0 ∼ l/zl in combination with (6.11) one obtains λ ∼ ε−1/3 b0 l2/3 ,

(6.12)

which implies that eddies become elongated along b0 with decreasing spatial scale. 6.5

Numerical simulation

All phenomenologies of turbulence have in common that they need to be veriﬁed by experimental or numerical data. For MHD turbulence, this is a

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particularly diﬃcult undertaking since experiments that allow testing theoretical claims are scarce, for example regarding the inertial-range scaling exponent of the energy spectrum. Though MHD turbulence seems to be omnipresent in the universe, most of the turbulent ﬂows are simply too far away to allow measurements with the required precision. One of the rare exceptions is the solar wind which is accessible to in-situ measurements by space probes. It has meanwhile been explored within a radial distance between 0.3 and 80 astronomical units from the sun and shows ﬂuctuations of velocity and magnetic ﬁeld over a wide range of spatial scales. Especially the slow component of the stream (see, e.g., [33]) displays turbulent energy spectra with an exponent close to -1.7 [34, 35] in agreement with the Kolmogorov and Goldreich-Sridhar phenomenologies. In the laboratory, conditions for high-Reynolds-number fully developed MHD turbulence are achieved in experimental setups for magneticallyconﬁned thermonuclear fusion. However, the onset of turbulent MHD activity generally leads to a catastrophic disruption of the discharge in Tokamaks [10] or, in the case of the reversed-ﬁeld pinch [36], cannot be examined with suﬃcient spatial resolution (for an exception, see [37]). Consequently, direct numerical simulations (DNS) in the framework of resistive MHD as outlined in Sec. 6.2 are the main source of experimental data. The simulations allow for ‘perfect’ diagnostics under well controlled conditions but limitations in computational resources restrict the attainable Reynolds numbers to the order of 103 [38, 39]. This has to be compared to realistic values of at least 108 in astrophysical turbulence [40]. Nevertheless, high-resolution DNS are a valuable tool for scrutinizing the presented phenomenologies when being aware of ﬁnite-Reynolds-number eﬀects. 6.5.1

Pseudospectral method

Pseudospectral methods, which discretize the turbulent ﬁelds in Fourierspace, have become the standard numerical technique for DNS of homogeneous MHD turbulence (see, e.g., [41]). The Fourier-space representation corresponds to the view of Kolmogorov which regards the quasi-random ﬁelds as an ensemble of ﬂuctuations at diﬀerent spatial scales. The spatial homogeneity of the system naturally leads to periodic boundary conditions. Furthermore, spatial diﬀerentiation is reduced to wave-vector multiplication, a valuable advantage over ﬁnite-diﬀerence schemes. In contrast, the quadratic nonlinearities in the MHD equations become numerically expensive convolution sums. The numerical eﬀort can be re-

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duced to the level of ﬁnite-diﬀerence schemes by computing the underlying quadratic products in real-space and applying fast Fourier transforms (FFTs) to shuttle between real- and Fourier-space. However, mapping the result of a realspace product (the Fourier-space convolution) back on a ﬁnite Fourier-space grid leads to an unwanted eﬀect known as ‘aliasing’ [42]. This error can be eliminated by surrounding the Fourier-space grid of linear extension N with a buﬀer zone of size N/2 taking up the aliasing modes. In practice, the aliasing error can be reduced to the order of the discretization error by restricting the Fourier-space volume of interest to a sphere of radius N [43]. In general, pseudospectral Fourier codes display higher precision than ﬁnite-diﬀerence algorithms at the same spatial resolution . Moreover, with pseudospectral codes insuﬃcient spatial resolution for a given Reynolds number leads to an energy pile-up at the largest wavenumbers which is not the case for ﬁnite-diﬀerence methods. For a comparative study of ﬁnitediﬀerence and pseudospectral schemes in turbulence, see [44].

6.6

Nonlinear energy dynamics

The angle-integrated energy spectrum (6.9) is the trace of the correlation tensors of velocity and magnetic ﬁeld, v(k)v∗ (k) and b(k)b∗ (k). Within the inertial range, it follows a power law, E(k) ∼ k−α . The value of the exponent α allows to draw conclusions about the validity of the phenomenologies presented in Sec. 6.4, e.g. α = 5/3 for the Kolmogorov picture and α = 3/2 for the Iroshnikov-Kraichnan phenomenology. For the Goldreich-Sridhar model the angle-integrated spectrum is not appropriate, since the considered system is spatially anisotropic. In this case deﬁnition (6.8) has to be used instead, with α = 5/3 for the perpendicular spectrum in the ‘strong’ version. However, the diﬀerence in the exponents is small and only high-resolution simulations yield an inertial range broad enough to distinguish the diﬀerent numerical values with suﬃcient accuracy.

6.6.1

Isotropic energy spectra

Figure 6.2 shows a spectrum from a pseudospectral simulation of incompressible and isotropic decaying MHD turbulence with vanishing magnetic and cross helicity. The system is represented by a periodic box in Fourier-space of edge

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length 2π. The initial ﬁelds are smooth with random phases while the ﬂuctuation amplitudes ∼ exp(−k2 /(2k02 )) with k0 = 4. For the global Alfv´en ratio of kinetic and magnetic energy E K /E M = 1. A trapezoidal leapfrog scheme is applied to evolve the MHD equations (6.4) and (6.2) in time. In this highly idealized setup the inverse Reynolds numbers occurring in the equations are just dimensionless dissipation parameters which are set to Re−1 = Rm−1 = 1 × 10−4 . The run was performed over 9 eddy turnover times deﬁned as the time required to reach the maximum of dissipation when starting from smooth initial ﬁelds. The spectrum is normalized in wavenumber using the Kolmogorov dissipation length (6.11) and in amplitude assuming a Kolmogorov spectrum (6.10). It was time-averaged over the period of self-similar decay, t = 6 − 8.9. The simulation involves 10243 collocation points and is one of the largest runs attempted to far. The spectrum shows a well developed inertial range with an associated scaling exponent α = 5/3 in agreement with the solar wind measurements. The same observation has been made in related work [45, 39]. This invalidates the Iroshnikov-Kraichnan model for isotropic turbulence without mean magnetic ﬁeld although Alfv´en waves are present in the system (see below). In forced three-dimensional compressible supersonic and super-Alfv´enic MHD turbulence the picture is not so clear with the kinetic energy spectrum observed to be steeper, E K (k) ∼ k−1.74 [46] for M ∼ 10 and MA ∼ 3, where M = v/c is the sonic Mach number deﬁned with the sound speed a and MA = v/b is the Alfv´enic Mach number. The scaling exponent also shows a strong dependence on the sonic Mach number [47]. 6.6.2

Anisotropic energy spectra

The numerical data on isotropic turbulence is in agreement with the Kolmogorov phenomenology and the strong version of Goldreich and Sridhar’s picture. To scrutinize the latter model the system has to be made globally anisotropic by imposing a strong mean magnetic ﬁeld b0 (for an illustration, see ﬁgure 3.9). In this case, a 5123 forced turbulence simulation is carried out. The forcing is realized by freezing all modes with k ≤ kf = 2. Its purpose is to keep the value of ﬂuctuating ﬁeld to mean ﬁeld approximately constant. The simulation with Re−1 = Rm−1 = 3 × 10−4 and |b0 | = 5 covers about 25 eddy turnover times. Kinetic and magnetic energy as well as the Alf´ ven

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Fig. 6.2 Compensated total energy spectrum in direct numerical simulation of decaying isotropic MHD turbulence averaged from t = 6 − 8.9 which shows a Kolmogorov ∼ k−5/3 inertial range. Amplitude and abscissa are normalized to prevent secular changes caused by the attenuation of the turbulence.

ratio E K /E M are of order one. The system has relaxed to a state with cross helicity around 15% and the magnetic helicity around 0.2 of the theoretical maximum ∼ E M 2π/kf . Figure 6.3 depicts the normalized parallel and perpendicular energy spectra averaged over t = 20 − 25 when the system is quasi-stationary. The parallel spectrum shows a signiﬁcant reduction of turbulence compared to the perpendicular spectrum since the nonlinear energy transfer is depleted in the parallel direction as theoretically expected (see above). The drop in amplitude of the ﬁeld-parallel ﬂuctuations compared to the ﬁeld-perpendicular ones has also been observed in shell-model calculations of MHD turbulence [48]. An inertial range is not discernible. For the perpendicular energy spectrum a short scaling range is observed in which the spectrum exhibits Iroshnikov-Kraichnan scaling E(k⊥ ) ∼ −3/2 k⊥ . This is caused by strong Alfv´enic ﬂuctuations propagating along b0 . Their existence can be inferred from quasi-periodic oscillations of the Alfv´en ratio (not shown) and the approximate equipartition of kinetic and magnetic energy on all scales of motion (cf. ﬁgure 6.4). In [49], α = 3/2 was also observed for a much stronger mean ﬁeld (the ratio of ﬂuctuations to mean component was about 300). However, the scaling was explained with the bottleneck eﬀect [50, 51] caused by the use of higher-order dissipation terms, e.g. µν (−1)ν−1 ∆ν ω for dissipativities

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Fig. 6.3 Normalized and time-averaged one-dimensional parallel (dashed) and perpendicular (solid) energy spectra. The dotted line shows the diﬀerence between IroshnikovKraichnan scaling ∼ k−3/2 (in this compensation horizontal) and Goldreich-Sridhar behavior ∼ k −5/3 .

Fig. 6.4 Kinetic (dashed) and magnetic (dotted) perpendicular energy spectra (normalized and averaged) for the anisotropic run depicted in ﬁgure 6.3.

ν > 1. Hyperviscosities of this kind are used to enlarge the inertial scaling range but result in a non-physical steepening of the spectrum close to the dissipative fall-oﬀ (see, e.g., [38]). The simulations presented here use normal viscosities, i.e. ν = 1, and do not exhibit a signiﬁcant bottleneck eﬀect.

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The numerical results are at variance with recent simulations which claim to support the Goldreich-Sridhar model by observing α = 5/3 in the perpendicular energy spectrum [52, 49, 53]. However, these studies were performed at signiﬁcantly lower resolution (2563 ) narrowing the inertial range. Additionally, fourth- and eighth-order hyperviscosities were used. Both factors make a precise estimate of the scaling exponent rather diﬃcult. The presented ﬁndings put the Goldreich-Sridhar picture in question. While it is plausible that the IK phenomenology holds for this conﬁguration, the appearance of Kolmogorov spectra in isotropic MHD turbulence remains an open problem. We also note that in simulations of twodimensional MHD turbulence IK scaling seems to hold even without mean magnetic ﬁeld [54].

6.6.3

Residual energy spectra

The residual energy spectrum, E R (k) = E M (k) − E K (k) , is of interest because it links kinetic and magnetic energy spectra and exhibits self-similar scaling. For isotropic decaying turbulence and anisotropic forced turbulence with a mean magnetic ﬁeld E R (k) = EkR displays fundamentally diﬀerent behaviour which becomes evident when comparing kinetic and magnetic energy spectra for the two cases (Fig. 6.4 and Fig. 6.5). In both systems the Alfv´en eﬀect is present. However, while it dominates the anisotropic system and leads to energetic equipartition at all scales of the ﬂow, in the isotropic simulation this is only true for the dissipation range. The excess of magnetic energy with increasing spatial scale, visible in Fig. 6.5, is due to the turbulent small-scale dynamo. This mechanism ampliﬁes the magnetic ﬁeld locally through the stretching of magnetic ﬁeld lines by turbulent ﬂuid motions. A generalization of previous theoretical work [20] allows to correctly predict the resulting scaling exponent of the residual energy spectrum in both cases and gives some insight into the turbulent interplay between kinetic and magnetic energy. 6.6.3.1

Closure theory

Among the diﬀerent approaches towards a statistical theory of turbulence (see, e.g., [55–61]), the eddy-damped quasi-normal Markovian approximation (EDQNMA), originally derived for incompressible hydrodynamic turbulence [59], has proven to be a useful compromise between mathematical rigour and phenomenological ﬂexibility. In its magnetohydrodynamic form

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Fig. 6.5 Kinetic (dashed) and magnetic (dotted) energy spectra (normalized and averaged) for the isotropic run shown in Fig. 6.2.

[15], neglecting helicity eﬀects, the equations governing the spectral dynamics of kinetic and magnetic energy read

K

K K , (6.13) dpdqΘkpq Tkin + Tmag + Tcrs ∂t + 2Re−1 k 2 EkK =

M M ∂t + 2Rm−1 k 2 EkM = , (6.14) dpdqΘkpq Tmag + Tcrs

with the ﬂux-density contributions K k 2 K K k3 K Tkin k Ep Eq − p2 EqK EkK , Tmag = bkpq = ckpq EpM EqM , pq pq 3 kp k K M = ckpq EqM EkK , Tmag = −ckpq EqM EkM , Tcrs q pq

k k5 M Tcrs k 2 EpM EqK − p2 EqK EkM + ckpq 3 EpK EqM , = hkpq pq p q and geometric coeﬃcients bkpq , ckpq , hkpq deﬁned, e.g., in [15]. The triangle symbol, ‘’, denotes integration over mode numbers which fulﬁll k = p + q. The time Θkpq is characteristic of the relaxation of the nonlinear energy ﬂux involving the modes k, p, and q and can be approximated by Θkpq = t . Here, µkpq is a phenomenological expression for the damping rate 1+µkpq t of the ﬂux by higher order moments with µkpq = µk + µp + µq ensuring energy conservation. −1 + τA−1 + A straightforward choice for the damping rates is µk = τNL −1 τD which combines the three physical processes that underlie turbulent

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energy dynamics in MHD: ﬁeld-line deformation by turbulent motions on

−1/2 the time-scale τNL ∼ /v ∼ k3 Ek , energy equipartition in interacting shear Alfven waves characterized by τA ∼ /b0 ∼ (kb0 )−1 and molecular dissipation, τD ∼ (µ + η)−1 k −2 . Under realistic conditions, diﬀusion is associated with the longest timescale of the turbulent system. Thus, for t τD one has Θkpq µ−1 kpq min(τNL , τA ). The evolution equation for the residual energy spectrum, EkR , can be derived by linear combination of equations (6.13) and (6.14) [62] and reads in the case of negligible v–b alignment:

R

! R R + Tcrs + Tloc (6.15) ∂t + Re−1 + Rm−1 k 2 EkR = dpdqΘkpq Tres with 2

2

R = −mkpq kp EpR EqR + rkpq pq EqR EkR , Tres R Tcrs R Tloc

= mkpq pEq EkR + tkpq pEqR Ek ,

2 s = − kpq k Ep Eq − p2 Eq Ek . k

The geometric coeﬃcients mkpq , rkpq , skpq , tkpq are deﬁned in [62]. Spectral interactions are called non-local if the modulus of one wavenumber, say k, in the interacting triad diﬀers signiﬁcantly from the wavenumbers of the other two, p ∼ q. Non-local interactions are associated with mutual Alfv´en-wave scattering and for this special case a simpliﬁed version of equation (6.15) can be derived:

(6.16) ∂t EkR = −Γk k EkM − EkK , where Γk = 43 k 6.6.3.2

! ak 0

dqΘkpq EqM [15].

A phenomenology for the residual energy

It is now assumed that the right hand side of (6.15) can be written as R R Tnonloc +Tloc [20]. This states that the ratio EkK /EkM is a result of a dynamic equilibrium between the (spectrally local) dynamo eﬀect which ampliﬁes the magnetic ﬁeld and the (spectrally nonlocal) Alfv´en eﬀect tending towards energetic equipartition. For stationary conditions and negligible molecular dissipation (k kdiss ) dimensional analysis yields k 3 Ek2 ∼ k 3 EkM EkR . (6.15)

(6.16)

(6.17)

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With the deﬁnitions of τNL and τA given above and assuming that the largescale magnetic ﬁeld sets the time scale for Alfv´enic interactions, k3 EkM 2 (kb0 ) , this expression can be re-written as

2 τA R Ek ∼ Ek . (6.18) τNL For Iroshnikov-Kraichnan scaling, Ek ∼ k −3/2 , as seen in the simulation with mean magnetic ﬁeld (see Fig. 6.3), the known result EkR ∼ k −2 [20] is obtained. This is in good agreement with the ﬁeld-perpendicular residual energy spectrum of the same run shown in 6.6.

Fig. 6.6

Averaged and normalized perpendicular residual energy spectrum, EkR

EkM − EkK , for the simulation with mean magnetic ﬁeld (cf. Fig. 6.3). ⊥

⊥

=

⊥

The simulation with vanishing mean magnetic ﬁeld displays a Kolmogorov inertial range scaling (cf. Fig. 6.2) for which relation (6.18) predicts EkR ∼ k −7/3 . As in this simulation the mean magnetic ﬁeld vanishes, the b0 term in the expression above denotes the mean magnetic ﬁeld carried by large-scale ﬂuctuations. Figure 6.7, depicting EkR for this case, conﬁrms that the theoretical prediction is well fulﬁlled. Apart from its meaning for the fundamental mechanism converting kinetic into magnetic energy (and vice versa), relation (6.18) also serves a more practical task. It ampliﬁes the inertial range scaling exponent of the residual energy spectrum by a factor of two with regard to the corresponding total energy spectrum. Thus, the diﬀerence between IK and K41 scaling which amounts to only 1/6 (represented by dotted line in Fig. 6.3)

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Fig. 6.7 Averaged and normalized residual energy spectrum, EkR = EkM − EkK , for the isotropic simulation (cf. Fig. 6.2). The dotted line shows the scaling associated with the Iroshnikov-Kraichnan model.

is enlarged to a much more obvious diﬀerence of 1/3 in the residual energy spectrum (see dotted line in Fig. 6.7). Consequently, the ﬁnding of −3/2 −2 can be regarded as an independent indication for Ek⊥ ∼ k⊥ EkR⊥ ∼ k⊥ in Fig. 6.3.

6.7

Spatial structure

All phenomenologies presented in Sec. 6.4 assume that the turbulence is spatially self-similar, i.e. that it has no characteristic internal length scale. This implies that the spatial distribution of turbulent structures is spaceﬁlling and uniform. Experimental data, however, shows signiﬁcant deviations from this behaviour in hydrodynamic turbulence (cf. [63] for a review), in the turbulent solar wind [64, 65] as well as in DNS of incompressible [66, 45, 38, 53] and compressible [47] MHD turbulence. The departure from self-similarity can be linked to the spatial distribution of dissipative turbulent structures by Kolmogorov’s reﬁned similarity hypothesis [67]. In contrast to relation (6.7), it postulates that p/3 Spv ∼ ε p/3 . The quantity ε is the local energy dissipation in a sphere of radius . This approach is motivated by the observation that turbulent energy dissipation is not homogeneously distributed in space. Instead, small regions of intense dissipation are embedded into a weakly dissipative environment making the associated spatial distribution intermittent.

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The Kolmogorov and Iroshnikov-Kraichnan models, which do not take this eﬀect into account, predict the isotropic structure-function exponents as ζpK41 = p/3 and ζpIK = p/4, respectively. The intermittency corrections to the linear predictions are found by examination of structure function scaling exponents of various orders. Figure 6.8 shows, as an example, a third order structure function from a 5123 simulation of decaying isotropic MHD turbulence. The dissipation terms in equations (6.4) and (6.2) are hyperdiﬀusive (ν = 2) to bring out the scaling range more clearly. The exponent of the approximate power law observed in the increment interval

∈ [20, 200] is estimated via the logarithmic derivative depicted in the inset.

S +3

1.00

Slope

0.10 1.6 1.4 1.2 1.0 0.8 0.6 0.4

0.01

100

10

100

1000

L Fig. 6.8 Normalized and averaged longitudinal |z+ |-structure function in a 5123 simulation of decaying isotropic MHD turbulence. The inset displays the scaling exponent via the logarithmic derivative.

The ﬁnite domains in space and time available to experiment and numerical simulation limit the statistical convergence of the associated averages necessary for determining structure functions. This behaviour is particularly pronounced for functions of a higher order since they react more sensitively to extreme ﬂuctuations of the turbulent ﬁelds. The statistical noise can be reduced by exploiting the fact that structure functions of a different order deviate qualitatively in the same way from their ‘ideal’ shape, a property termed extended self-similarity (ESS) [68]. Hence, the scaling range can substantially be enlarged by considering relative scaling exponents ξp,q = ζp /ζq . The ξp,q are obtained by supposing

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the structure functions of order p to depend on a reference structure function of ﬁxed order q whose scaling exponent can be measured with suﬃcient accuracy. Figure 6.9 displays examples of ESS curves together with their estimated slopes in units of the S3+ scaling exponent.

Slope: 0.39

S+ 2

1.0

S+ 1

1.0

Slope: 0.72

0.1

0.1 0.01

0.10 S+ 3

1.00

0.01 10.000

Slope: 1.23

1.00

0.10 S+ 3

1.00

Slope: 1.42

1.000

0.10

S+ 5

S+ 4

1.00

0.10 S+ 3

0.100 0.010

0.01

0.001 0.01

0.10 S+ 3

1.00

0.01

Fig. 6.9 The |z+ |-structure functions (solid) of a 5123 -simulation of decaying MHD turbulence. They display extended self-similarity when drawn against the corresponding + are determined by taking the third-order function. The relative scaling exponents ξp,3 logarithmic slope of linear least-square ﬁts (dashed).

In a pseudospectral simulation with 5123 collocation points the use of ESS allows to compute scaling exponents up to order eight with suﬃcient precision. The second-order value, ζ2 , is related to the inertial-range behaviour of the energy spectrum, E(k) ∼ k −(1+ζ2 ) . The whole family of exponents gives more general information about the small-scale structure of the corresponding turbulent ﬁelds and represents a framework for the veriﬁcation of intermittency phenomenologies. 6.7.1

Intermittency modelling

There are a number of diﬀerent phenomenologies [8] which predict the characteristic change of two-point scaling exponents with increasing order. The Log-Poisson model, which was ﬁrst proposed for hydrodynamic turbulence by She and L´evˆeque [69], takes a unique position among them as it achieves very good agreement with experiments and simulations and only contains parameters which can be estimated by physical reasoning.

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The Log-Poisson phenomenology assumes a hierarchical relation be(p) tween the functions ε = εp+1 /εp which reads

(p) β (p+1) ε

ε

∼ (∞) , β ∈ [0, 1] . (6.19) (∞) ε

ε

(∞)

The quantity ε

stands for the dissipation due to the topologically most singular structures while β parameterizes the degree of intermittency: β → 1 corresponds to spatially homogeneous dissipation, β → 0 stands for the most intermittent conﬁguration where ε is concentrated in the most singular structure. Relation (6.19) expresses a generalized scale-covariance of dissipation [70, 71] which is equivalent to a logarithmic Poisson distribution of the ε [70], In the inertial range, where z p ∼ ζp and εp ∼ τp , the reﬁned similarity hypothesis leads to !

p/g

z p ∼ ε p/g ∼ τp/g p/g ∼ ζp

⇒ ζp = p/g + τp/g .

(6.20)

Equation (6.20) links the scaling exponents of turbulent ﬁelds, ζp , and dissipation, τp . Assuming that the energy to be dissipated in the most singular structures, E ∞ , is scale-independent, (∞)

ε

−x ∼ E ∞ /t∞

∼

x with t∞

∼ ,

gives lim (τp+1 − τp ) = −x

p→∞

⇒ τp = −xp + C0 + f (p) .

(6.21)

With (6.19) and τ0 = 0 one obtains f (p) = −C0 β p which yields τp = −xp + C0 (1 − β p ) . Furthermore, τ1 = 0 results in β = 1 − x/C0 and consequently τp (6.20)

= −xp + C0 (1 − (1 − x/C0 )p )

⇒ ζp = (1 − x)p/g + C0 (1 − (1 − x/C0 )p/g )

(6.23)

This is the general Log-Poisson model (see, e.g. [72]). It depends on the parameters x, g, and C0 which have to be determined on physical grounds. The non-intermittent scaling, z ∼ 1/g , ﬁxes g (Kolmogorov g = 3, Iroshnikov-Kraichnan g = 4). Equation (6.21) is analogous to a Legendre transformation. Therefore, C0 can be interpreted as the co-dimension of a set of singularities of strength τ∞ which is equivalent to the most singular dissipative structures. The parameter x is related to the dissipation rate in x these structures t∞

∼ .

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Hydrodynamics

The hydrodynamic She-L´evˆeque model is obtained for Kolmogorov scaling, g = 3, and quasi one-dimensional dissipative structures, i.e., vorticity ﬁlaments with C0 = 2. The most singular dissipation rate is assumed to be set by the inertial range energy cascade rate, (∞) !

E (∞) /t

∼ ε ∼ z 2 /t

(∞)

⇒ t

∼ 2/g (x = 2/g)

⇒ ζp = p/9 + 2 1 − (2/3)p/3 . 6.7.1.2

Isotropic MHD

In the isotropic MHD case Kolmogorov scaling also holds (g = 3). Maintaining the dissipation rate assumption (x = 2/g) and observing that the most singular dissipative structures are current and vorticity sheets (C = 1, cf. Fig. 3.10) one gets the isotropic MHD model, ⇒ ζp = p/9 + 1 − (1/3)p/3 .

(6.24)

The formula predicts structure function scaling in isotropic threedimensional incompressible MHD turbulence with good precision and is moreover consistent with the ﬁnding of aK41-like energy spectrum [45, 38]. It also agrees well with solar wind data [65, 73] and simulations of compressible [46] MHD turbulence. In contrast, the relation based on IroshnikovKraichnan scaling, g = 4 [74, 72], is not in accordance with experimental and numerical data. The MHD model is indicated in Fig. 6.10 by the solid line. The dotted line shows the non-intermittent Kolmogorov prediction, ζp = p/3, for comparison. 6.7.1.3

Anisotropic MHD

Though the picture of Goldreich and Sridhar does not seem to hold for the energy cascade, spatial anisotropy is nevertheless displayed by higherorder statistics shown in Fig. 6.10 [75] (see also [24, 76, 52, 53] and [77] for the compressible case). The data stems from a globally isotropic 5123 -simulation (b0 = 0, circles) of decaying turbulence and from forced anisotropic 5122 × 256-simulations with mean magnetic ﬁeld (b0 = 5, 10, squares/triangles). For the anisotropic runs, the resolution along b0 can be reduced to 256 collocation points as the turbulence is depleted in this direction (cf. Fig. 6.3) due to the stiﬀness of the magnetic ﬁeld lines.

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Fig. 6.10 Scaling exponents ζp of the |z + |-structure functions against the corresponding order. The MHD Log-Poisson model (6.24) is depicted by the solid line. Non-intermittent K41 scaling is represented by the dotted line. Anisotropic exponents are shown for b0 = 0, 5, 10 (circles, squares, triangles). Structure functions parallel to the mean magnetic ﬁeld are given by open symbols, ﬁeld-perpendicular data by ﬁlled symbols.

The underlying structure functions are calculated with the space increment, , taken either parallel or perpendicular to mean ﬁeld direction. This direction is found locally by applying a top-hat ﬁlter of width to the magnetic ﬁeld (for details, cf. [75]). In the ﬁeld-parallel direction the structure function exponents show an asymptotic approach towards a straight line with increasing b0 . This indicates a decrease of intermittency in this direction, i.e. more homogeneous dissipation. The ﬁeld-perpendicular exponents display a gradual transition towards values known from two-dimensional turbulence simulations; in fact the perpendicular exponents for b0 = 10 are identical within the error margin with two-dimensional results [19, 54]. These ﬁndings become plausible when regarding Fig. 3.10. While the dissipative current sheets (and the associated vorticity sheets) show no preferred orientation in the isotropic case, they tend to align with an applied mean magnetic ﬁeld. The aligned conﬁguration results in increased homogeneity of dissipation in the b0 -direction and makes the system look two-dimensional in ﬁeld-perpendicular planes. A generalized version of the p/g isotropic MHD intermittency model (6.24), ζp = p/g 2 + 1 − (1/g) , where g is a free parameter setting the energy cascade rate in the respective parallel/perpendicular direction is able to reproduce the observed scaling exponents [75].

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Conclusion

The understanding of the nonlinear energy cascade and the spatial smallscale structure of incompressible MHD turbulence is largely based on phenomenology and direct numerical simulation. This chapter brieﬂy described the level of knowledge at present, outlined numerical methods commonly used for turbulence simulations, and also highlighted recent developments in this ﬁeld of research. Particular stress has been put on the diﬀerence between isotropic turbulence and conﬁgurations permeated by a mean magnetic ﬁeld. Properly taking into account the spatial anisotropy of the turbulence induced by the magnetic ﬁeld is probably one of the most important challenges to current MHD turbulence theory. The physical picture of the turbulent energy cascade is still under discussion. While there is ample evidence that in isotropic MHD the Kolmogorov picture holds, recent high-resolution simulations with a strong mean magnetic ﬁeld suggest that the Iroshnikov-Kraichnan phenomenology is correct for ﬁeld-perpendicular ﬂuctuations in such systems. This rules out the Goldreich-Sridhar model which is the ﬁrst anisotropic phenomenology of MHD turbulence and predicts ﬁeld-perpendicular Kolmogorov spectra. These ﬁndings have been corroborated by results of EDQNM closure theory calculations which give a simple relation between the residual and the total energy spectrum veriﬁed by numerical simulation. The intermittent small-scale structure, which is probed by higher-order two-point statistics, is visible in the structure function scaling exponents. In isotropic MHD turbulence, their characteristic behaviour can be matched well by a Log-Poisson model which takes into account that the energy cascade is Kolmogorov-like and that the most singular dissipative structures are quasi-two-dimensional current and vorticity sheets. The model can be generalized to reproduce the structure function scalings parallel and perpendicular to an applied mean magnetic ﬁeld. Numerical simulations with increasing ﬁeld strength show, furthermore, that the system in the ﬁeld-perpendicular direction becomes gradually two-dimensional while the parallel dissipation turns out to be more homogeneous and less intermittent. This behaviour is the consequence of the alignment of dissipative current and vorticity sheets with the mean magnetic ﬁeld. While this seems to be a consistent view of incompressible MHD turbulence, the transition of the energy cascade from Kolmogorov to IroshnikovKraichnan behaviour and generally the Kolmogorov scaling in isotropic turbulent systems remains an open question. The approach towards inter-

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mittency has to be veriﬁed by means of higher-resolution numerical simulations and to be ﬁtted into a general framework explaining some of the ad-hoc assumptions that enter the phenomenology. Finally, the validity of the results should be tested in physically more complex setups including, for instance, buoyancy, compressibility, or rotation. Some of this work is already under way.

Acknowledgements The authors would like to thank Jacques L´eorat for fruitful discussions. WCM thanks Dieter Biskamp for his supportive interest and gratefully acknowledges ﬁnancial support by the CNRS, Meudon.

References [1] M. Wakatani, V. S. Mukhovatov, K.H. Burrell, et al. Chapter 2: Plasma conﬁnement and transport. Nuclear Fusion, 39(12):2175–2249, 1999. [2] D. Biskamp. Magnetic Reconnection in Plasmas. Cambridge University Press, Cambridge, 2000. [3] A. Yoshizawa, S.-I Itoh, K. Itoh, and N. Yokoi. Dynamos and MHD theory of turbulence suppression. Plasma Physics and Controlled Fusion, 46:R25–R94, 2004. [4] C.-Y. Tu and E. Marsch. MHD structures, waves and turbulence in the solar wind: observations and theories. Space Science Reviews, 73:1–210, 1995. [5] M.-M. Mac Low. MHD turbulence in star-forming regions and the interstellar medium. In E. Falgarone and T. Passot, editors, Turbulence and Magnetic Fields in Astrophysics, volume 614 of Lecture Notes in Physics, pages 182– 212. Springer Berlin, 2002. [6] S. A. Balbus. Enhanced angular momentum transport in accretion disks. Annual Review of Astronomy and Astrophysics, 41:555–597, 2003. [7] M. Lesieur. Turbulence in Fluids. Kluwer Academic Publishers, Dordrecht, 1997. [8] U. Frisch. Turbulence. Cambridge University Press, Cambridge, 1996. [9] D. I. Braginskii. Transport processes in a plasma. Reviews of Plasma Physics, 1:205–311, 1965. [10] D. Biskamp. Nonlinear Magnetohydrodynamics. Cambridge University Press, Cambridge, 1993. [11] W. M. Els¨ asser. The hydromagnetic equations. Physical Review, 79:183, 1950. [12] A. N. Kolmogorov. The local structure of turbulence in incompressible viscous ﬂuid for very large Reynolds numbers. Proceedings of the Royal Society A, 434:9–13, 1991. [Dokl. Akad. Nauk SSSR, 30(4), 1941].

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[13] A. N. Kolmogorov. Dissipation of energy in the locally isotropic turbulence. Proceedings of the Royal Society A, 434:15–17, 1991. [Dokl. Akad. Nauk SSSR, 32(1), 1941]. [14] L. F. Richardson. Weather Prediction by Numerical Process. Cambridge University Press, Cambridge, 1922. [15] A. Pouquet, U. Frisch, and J. L´eorat. Strong MHD helical turbulence and the nonlinear dynamo eﬀect. Journal of Fluid Mechanics, 77(2):321–354, 1976. [16] H. K. Moﬀatt. The degree of knottedness of tangled vortex lines. Journal of Fluid Mechanics, 35(1):117–129, 1969. [17] P. S. Iroshnikov. Turbulence of a conducting ﬂuid in a strong magnetic ﬁeld. Soviet Astronomy, 7:566–571, 1964. [Astron. Zh., 40:742, 1963]. [18] R. H. Kraichnan. Inertial-range spectrum of hydromagnetic turbulence. Physics of Fluids, 8(7):1385–1387, 1965. [19] H. Politano, A. Pouquet, and P. L. Sulem. Inertial ranges and resistive instabilities in two-dimensional magnetohydrodynamic turbulence. Physics of Fluids B, 1(12):2330–2339, 1989. [20] R. Grappin, A. Pouquet, and J. L´eorat. Dependence of MHD turbulence spectra on the velocity ﬁeld-magnetic ﬁeld correlation. Astronomy and Astrophysics, 126:51–58, 1983. [21] M. Dobrowolny, A. Mangeney, and P. Veltri. Fully developed anisotropic hydromagnetic turbulence in interplanetary space. Physical Review Letters, 45(2):144–147, 1980. [22] H. Politano and A. Pouquet. Von K´ arm´ an-Howarth equation for magnetohydrodynamics and its consequences on third-order longitudinal structure and correlation functions. Physical Review E, 57(1):R21–R24, 1998. [23] H. Politano and A. Pouquet. Dynamical length scales for turbulent magnetized ﬂows. Geophysical Research Letters, 25(3):273–276, 1998. [24] J. V. Shebalin, W. H. Matthaeus, and D. Montgomery. Anisotropy in MHD turbulence due to a mean magnetic ﬁeld. Journal of Plasma Physics, 29(3):525–547, 1983. [25] R. Grappin. Onset and decay of two-dimensional magnetohydrodynamic turbulence with velocity-magnetic ﬁeld correlation. Physics of Fluids, 29(8):2433–2443, 1986. [26] S. Galtier, S. V. Nazarenko, A. C. Newell, and A. Pouquet. A weak turbulence theory for incompressible magnetohydrodynamics. Journal of Plasma Physics, 63(5):447–488, 2000. [27] P. Goldreich and S. Sridhar. Toward a theory of interstellar turbulence. II. Strong Alfv´enic turbulence. Astrophysical Journal, 438:763–775, 1995. [28] P. Goldreich and S. Sridhar. Magnetohydrodynamic turbulence revisited. Astrophysical Journal, 485:680–688, 1997. [29] S. Sridhar and P. Goldreich. Toward a theory of interstellar turbulence. I. Weak Alfv´enic turbulence. Astrophysical Journal, 432:612–621, 1994. [30] D. Montgomery and W. H. Matthaeus. Anisotropic model energy transfer in interstellar turbulence. Astrophysical Journal, 447:706–707, 1995. [31] C. S. Ng and A. Bhattacharjee. Interaction of shear-alfv´en wave packets: implication for weak magnetohydrodynamic turbulence in astrophysical plasmas. Astrophysical Journal, 465:845–854, 1996.

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[32] S. V. Nazarenko, A. C. Newell, and S. Galtier. Non-local MHD turbulence. Physica D, 152–153:646–652, 2001. [33] D. Biskamp. Magnetohydrodynamic Turbulence. Cambridge University Press, Cambridge, 2003. [34] M. L. Goldstein, D. A. Roberts, and W. H. Matthaeus. Magnetohydrodynamic turbulence in the solar wind. Annual Review of Astronomy and Astrophysics, 33:283–325, 1995. [35] R. J. Leamon, C. W. Smith, N. F. Ness, W. H. Matthaeus, and H. K. Wong. Observational constraints on the dynamics of the interplanetary magnetic ﬁeld dissipation range. Journal of Geophysical Research, 103(A3):4775–4787, 1998. [36] S. Ortolani and D. D. Schnack. Magnetohydrodynamics of Plasma Relaxation. World Scientiﬁc, Singapore, 1993. [37] V. Carbone, L. Sorriso-Valvo, E. Martines, V. Antoni, and P.Veltri. Intermittency and turbulence in a magnetically conﬁned fusion plasma. Physical Review E, 62(1):R49–R52, 2000. [38] D. Biskamp and W.-C. M¨ uller. Scaling properties of three-dimensional isotropic magnetohydrodynamic turbulence. Physics of Plasmas, 7(12):4889– 4900, 2000. [39] N. E. L. Haugen, A. Brandenburg, and W. Dobler. Is nonhelical hydromagnetic turbulence peaked at small scales ? Astrophysical Journal, 597:L141– L144, 2003. [40] Ya. B. Zeldovich, A. A. Ruzmaikin, and D. D. Sokoloﬀ. Magnetic Fields In Astrophysics. Gordon and Breach Science Publishers, New York, 1983. [41] C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang. Spectral Methods in Fluid Dynamics. Springer-Verlag, New York, 1988. [42] S. A. Orszag. Comparison of pseudospectral and spectral approximation. Studies in Applied Mathematics, 51(3):253–259, 1972. [43] A. Vincent and M. Meneguzzi. The spatial structure and statistical properties of homogeneous turbulence. Journal of Fluid Mechanics, 225:1–20, 1991. [44] J. R. Herring, S. A. Orszag, R. H. Kraichnan, and D. G. Fox. Decay of two-dimensional homogeneous turbulence. Journal of Fluid Mechanics, 66(3):417–444, 1974. [45] W.-C. M¨ uller and D. Biskamp. Scaling properties of three-dimensional magnetohydrodynamic turbulence. Physical Review Letters, 84(3):475–478, 2000. [46] S. Boldyrev, ˚ A. Nordlund, and P. Padoan. Scaling relations of supersonic turbulence in star-forming molecular clouds. Astrophysical Journal, 573:678– 684, 2002. [47] P. Padoan, R. Jimenez, ˚ A. Nordlund, and S. Boldyrev. Structure function scaling in compressible super-Alfv´enic MHD turbulence. Physical Review Letters, 92(19):191102–1–191102–4, 2004. [48] V. Carbone and P. Veltri. A shell model for anisotropic magnetohydrodynamic turbulence. Geophysical and Astrophysical Fluid Dynamics, 52:153– 181, 1990. [49] J. Maron and P. Goldreich. Simulations of incompressible magnetohydrodynamic turbulence. Astrophysical Journal, 554:1175–1196, 2001.

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[50] D. Lohse und A. M¨ uller-Groeling. Bottleneck eﬀects in turbulence: Scaling phenomena in r versus p space. Physical Review Letters, 74(10):1747–1750, 1995. [51] G. Falkovich. Bottleneck phenomenon in developed turbulence. Physics of Fluids, 6(4):1411–1414, 1994. [52] J. Cho and E. T. Vishniac. The anisotropy of magnetohydrodynamic Alfv´enic turbulence. Astrophysical Journal, 539:273–282, 2000. [53] J. Cho, A. Lazarian, and E. T. Vishniac. Simulations of magnetohydrodynamic turbulence in a strongly magnetized medium. Astrophysical Journal, 564:291–301, 2002. [54] D. Biskamp and E. Schwarz. On two-dimensional magnetohydrodynamic turbulence. Physics of Plasmas, 8(7):3282–3292, 2001. [55] R. H. Kraichnan. The structure of isotropic turbulence at very high Reynolds numbers. Journal of Fluid Mechanics, 5:497–543, 1959. [56] R. H. Kraichnan. Lagrangian-history closure approximation for turbulence. Physics of Fluids, 8(4):575–598, 1965. [57] S. F. Edwards. The statistical dynamics of homogeneous turbulence. Journal of Fluid Mechanics, 18:239–273, 1964. [58] J. R. Herring. Self-consistent-ﬁeld approach to turbulence theory. Physics of Fluids, 8(12):2219–2225, 1965. [59] S. A. Orszag. Analytical theories of turbulence. Journal of Fluid Mechanics, 41(2):363–386, 1970. [60] R. H. Kraichnan. An almost Galilean-invariant turbulence model. Journal of Fluid Mechanics, 47(3):513–524, 1971. [61] U. Frisch, M. Lesieur, and A. Brissaud. A Markovian random coupling model for turbulence. Journal of Fluid Mechanics, 65(1):145–152, 1974. [62] R. Grappin, U. Frisch, J. L´eorat und A. Pouquet. Alfv´enic ﬂuctuations as asymptotic states of MHD turbulence. Astronomy and Astrophysics, 105:6– 14, 1982. [63] K. R. Sreenivasan and R. A. Antonia. The phenomenology of small-scale turbulence. Annual Review of Fluid Mechanics, 29:435–472, 1997. [64] L. F. Burlaga. Intermittent turbulence in the solar wind. Journal of Geophysical Research, 96(A4):5847–5851, 1991. [65] T. S. Horbury and A. Balogh. Structure function measurements of the intermittent MHD turbulent cascade. Nonlinear Processes in Geophysics, 4:185– 199, 1997. [66] H. Politano, A. Pouquet, and V. Carbone. Determination of anomalous exponents of structure functions in two-dimensional magnetohydrodynamic turbulence. Europhysics Letters, 43(5):516–521, 1998. [67] A. N. Kolmogorov. A reﬁnement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible ﬂuid at high Reynolds number. Journal of Fluid Mechanics, 13:82–85, 1962. [68] R. Benzi, S. Ciliberto, R. Tripiccione, C. Baudet, F. Massaioli, and S. Succi. Extended self-similarity in turbulent ﬂows. Physical Review E, 48(1):R29– R32, 1993. [69] Z.-S. She and E. L´evˆeque. Universal scaling laws in fully developed turbulence. Physical Review Letters, 72(3):336–339, 1994.

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[70] B. Dubrulle. Intermittency in fully developed turbulence: Log-Poisson statistics and generalized scale covariance. Physical Review Letters, 73(7):959–962, 1994. [71] Z.-S. She and E. C. Waymire. Quantized energy cascade and log-Poisson statistics in fully developed turbulence. Physical Review Letters, 74(2):262– 265, 1995. [72] H. Politano and A. Pouquet. Model of intermittency in magnetohydrodynamic turbulence. Physical Review E, 52(1):636–641, 1995. [73] A. Bershadskii. Three-dimensional isotropic magnetohydrodynamic turbulence and thermal velocity of the solar wind ions. Physics of Plasmas, 10(12):4613–4615, 2003. [74] R. Grauer, J. Krug, and C. Marliani. Scaling of high-order structure functions in magnetohydrodynamic turbulence. Physics Letters A, 195:335–338, 1994. [75] W.-C. M¨ uller, D. Biskamp, and R. Grappin. Statistical anisotropy of magnetohydrodynamic turbulence. Physical Review E, 67:066302–1–066302–4, 2003. [76] L. J. Milano, W. H. Matthaeus, P. Dmitruk, and D. C. Montgomery. Local anisotropy in incompressible magnetohydrodynamic turbulence. Physics of Plasmas, 8(6):2673–2681, 2001. [77] W. H. Matthaeus, S. Ghosh, S. Oughton, and D. Roberts. Anisotropic three-dimensional MHD turbulence. Journal of Geophysical Research, 101(A4):7619–7629, 1996.

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Chapter 7

Scale Covariance and Scale-ratio Covariance in Turbulent Front Propagation A. Pocheau IRPHE, UMR 6594 CNRS, Universit´es Aix-Marseille I & II, 49 rue Joliot-Curie, B.P. 146, Technopˆ ole de Chˆ ateau-Gombert, F-13384 Marseille, Cedex 13, France

We address the implications of scale symmetries in out-of-equilibrium systems within the paradigm of turbulent front propagation. Usually, in this kind of issues, the attention is laid on the scale properties of objects, i.e. of their geometry, as for instance in fractal analysis. In contrast, the main objective here will be to infer the scale properties of laws, i.e. of the relationships between physical variables. This leads us to introduce the concept of scale covariance of laws which may include, but goes beyond, the scale similarity of objects. In particular, emphasizing the fact that turbulent front propagation refers not to isolated scales but to scale ranges, we put forward a scale symmetry diﬀerent than the usual scale invariance: the scale-ratio invariance, based on the similarity with respect to changes of scale-ratios. Implementing the resulting constraints on turbulent front propagation provides the selection of a relationship that gives the eﬀective front velocity. Both the form of this relation and it scale properties are validated from experiment. Interestingly, this scale symmetric law enables us to recover the construction of fronts in scale space from the euclidean regime to the fractal regime. This provides an explanation of the variation of the fractal dimensions of fronts with turbulence intensity and uniﬁes in a single framework the euclidean and the fractal regimes which were previously thought to be intrinsically diﬀerent.

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Introduction

The concept of symmetry stands as a powerful tool for characterizing physical objects or for determining the relationships which govern them. For instance, translational and rotational symmetries enable the classiﬁcation of crystals into crystallographic groups or yield the derivation of momentum conservation from the Noether’s theorem. The ﬁrst example refers to a characterization which provides no physical explanation by itself: it thus stands on a descriptive ground. The second example brings about a physical law that system dynamics have to obey: it thus stands on a causal ground. Regarding scale symmetries, the descriptive approach is worked out in fractal analysis since its objective consists in using scale invariance to classify geometry by a number: the fractal dimension. This has provided useful determination of the kind of objects that can be produced in various out-of-equilibrium systems ranging from clouds in environmental science to fractures in material science [1]. However, taken solely, fractal analysis cannot grasp the causal relationships that might be implied by scale invariance. To obtain them, it thus appears relevant to look for applying the causal approach to scale symmetries, especially in out-of-equilibrium systems where deﬁnite laws are usually hard to infer from ﬁrst principles. The purpose of this work is to develop, on the deﬁnite example of turbulent fronts, the implications of scale symmetries on relevant relationships of far-from-equilibrium systems. Meanwhile, the procedures and the alternatives brought about by this kind of approach will be evidenced, so as to make them applicable to other systems. We shall thus consider a reactive front propagating in a turbulent medium and focus attention on its eﬀective properties in a scale invariant regime of the front-turbulence interaction. Seeking for the selection of relationships satisfying this symmetry will lead us to the concept of scale covariance. Further analyzing the concept of scale symmetries will make us face two kinds of symmetries: the usual invariance with respect to a change of scale (the scale invariance) and a more original invariance, the invariance with respect to a change of scale-ratio (the scale-ratio invariance). Solving for the relationships which satisfy both symmetries will provide us with the law which governs the eﬀective propagation of a front in a multi-scale stirred medium. Looking for its implications on front geometry will yield us to a statistical picture of front shape in scale space which, although linked to a scale symmetry, will be found to diﬀer from the scale similarity inherent to fractals.

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Fig. 7.1 Tomographic cut of a turbulent ﬂame front. A plane laser sheet is diﬀused transversely by fresh gases (white domain) but not by burnt gases (dark domain). This enables the front to be evidence as the frontier between them. The ratio of the turbulence intensity u to the normal velocity UN is u /UN = 1.30 here.

The paper is organized as follows. We ﬁrst address in Sec. 7.2 the relevance of scale symmetries to the analysis of turbulent propagative fronts. We then clarify in Sec. 7.3 the link between the scale invariance of a system and the scale covariance of its laws. For this, various issues are successively considered: geometry, equilibrium systems and out-of-equilibrium systems. The analysis of scale symmetries is deepened in Sec. 7.4 by drawing attention to the scale-ratio invariance, by pointing out its diﬀerence with the usual scale invariance, and by stressing its relevance to turbulent front propagation. The turbulent propagation law which satisﬁes both the scale covariance and the scale-ratio covariance is then determined in Sec. 7.5. The experimental evidence of this symmetry and of the corresponding law is reported in Sec. 7.6 on turbulent fronts. Its implications on the statistical construction of front geometry in scale space are addressed in Sec. 7.7 with emphasis on the diﬀerences with fractal analysis. A conclusion about this work and its meaning for the role of scale symmetries in out-of-equilibrium systems is ﬁnally drawn. 7.2

Turbulent front and scale symmetries

We consider a reactive front which propagates in a medium that is randomly stirred on a large scale range Fig. 7.1. This issue may be interesting both on a practical point of view and on a fundamental ground. On the one

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hand, it addresses a number of practical processes, ranging from engine industry to the propagation of epidemics or of forest ﬁres ; on the other hand, it provides geometric objects, the turbulent fronts, which may help investigating the implications of turbulent mixing with a geometric support. We experimentally achieved this issue by making premixed ﬂames propagate in turbulent gases (see Refs. [2, 3] and [4]). Visualization of front cuts was obtained by laser tomography. For this, gases were previously inseminated with an oil spray that was designed to be vaporized by the front. Then, diﬀusion by a laser sheet could only occur in the still unburnt medium, so that the frontier of the diﬀused sheet simply provided the intersection of the front with the laser plane. A typical tomographic cut, reported in Fig. 7.1, shows front wrinkles at many diﬀerent scales. These may be attributed to turbulence since they do not appear in a laminar ﬂow. They increase the front surface and thus its eﬃciency in burning the medium, insofar as the normal velocity of the front is unchanged by front curvature (i.e. for wrinkle scales large compared to the front thickness [5, 6]). This results in an ampliﬁcation of the mean velocity of the front in the medium. In order to understand this enhancement of the mean transport property of the medium by turbulence, one would be tempted to directly address the interplay between all the various phenomena that are essential to front propagation: hydrodynamic or thermo-diﬀusive instabilities, modiﬁcation of normal front velocity by ﬂow stretch or front curvature, cusp generation in ﬁnite-time, sensitivity to noise ...[5, 6]. Needless to say, this corresponds to a formidable task which lies beyond the ability of present analytical technics. Hopefully, the use of statistical symmetries enables this rich phenomenology to be by-passed to select the few possible relationships compatible with it. This will provide less information about the inner mechanisms of the system but will in turn enable widespread conclusions to be made somewhat elegantly on a general ground. Owing to the multi-scale nature of turbulent front propagation, the relevant symmetries to be used in a convenient regime to clarify will be the scale symmetries. In applying scale symmetry arguments to front propagation, two diﬀerent approaches may be put forward. • The ﬁrst approach, largely implemented in litterature, consists in describing the statistical features of front geometry in scale space [1] by identifying fronts with fractals. It then leads to characterize them by a single number, the fractal dimension, and is thus mostly descriptive.

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• The second approach that we propose here consists to change the spirit of the scale analysis by addressing not the objects themselves (i.e. the fronts) but the physical process that is responsible for their formation (i.e. the wrinkling process) [2, 7]. This entails considering neither the eﬀects of the process alone (here the front geometry or the mean front velocity), nor its cause alone (here the turbulent ﬂow) but the relationship between them.

This second approach, which focusses attention on the link between a cause and an eﬀect may thus be termed “causal” as opposed to the more descriptive fractal analysis. It turns out studying the structure of laws instead of the structure of objects, i.e. the relationships between measurements rather than the measurements themselves. In the present case, it will lead us to go beyond fractal analysis by considering, not only the large scales or the large mixing regimes to which fractal analysis is restricted, but also the small scales and the weakly turbulent regimes where information relevant to the physical process may also be gained. Applying scale invariance to turbulent front propagation will turn out stating that the corresponding laws rely on no characteristic scale and, as we shall see, on no characteristic scale ratio. The following section addresses the procedure for explicitly formalizing the resulting scale properties of laws in term of so-called scale covariances.

7.3

From scale invariance to scale covariance

The central issue to consider later on consists in expressing the implications of a scale symmetry of the system on the laws which monitor its behaviour. As we shall see, this is an issue more subtle than might be thought and which may have several kinds of answers depending on the system and on the kind of law that is considered. To better illustrate the transposition between a symmetry property of the system and a symmetry property of its laws, we restrict attention here to the well-known scale invariance and we review the way it has been addressed in three familiar topics: single variable relationship, geometry, and phase transitions. We then report how to apply it to turbulent front propagation.

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7.3.1

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Single variable relationship

On a general ground, the symmetry of scale invariance means that all magnitudes are equivalent in the sense that no observation of any kind could particularize any of them. In particular, changing a standard ls for another ls should yield no objective diﬀerence for an observer studying the system. How can this be possible insofar as the change of standard ls modiﬁes for sure the measurement nl = l/ls of a scale l ? Simply because the objective diﬀerence that is sought refers to the relationship between data instead of the data themselves. To be more precise, consider an observer which faces an object made of points of coordinates (x, y) and which questions the scale invariance of its shape. He will ﬁrst determine the form of this object by performing measurements of the coordinates nx = x/xs and ny = y/ys of each of its points within a given choice of standards (xs , ys ). This will provide a relationship f (.) between the coordinates of the points on this shape: ny = f (nx ). Notice that, without loss of generality, one can reduce to 1 = f (1), simply by choosing ys so that the point S of coordinates (xs , ys ) belongs to the object. We shall adopt this convention from now on for simplicity. The test for scale invariance will now consist in changing the standard xs , and thus the standard point S, and investigate whether this modiﬁes the shape function f (.), Fig. 7.2. If f (.) involves a characteristic scale lc , the answer will be positive since the corresponding speciﬁc point of f (.), nc = lc /xs , will change accordingly: xs → xs ⇒ nc → nc = lc /xs . For instance, if f (.) is a sinusoide (Fig. 7.2-a), its period T will be dilated (Fig. 7.2-b), this being an objective change implied by the standard change: T → T = T xs /xs . The surprising case occurs when there is no such characteristic scale so that function f (.) remains the same despite the change of standard S → S (Fig. 7.2-c and d). Then, the shape of the object is such that its observation cannot give any information on the scale used by the observer. The concepts of “small” or “large” are then destroyed since all scale magnitudes are found to be equivalent: one faces scale invariance. The criterion for this to happen is the invariance of the shape function f (.) with the standard change S → S . Taking, as above, the new standards so that (xs , ys ) still belongs to the object, the criterion for scale invariance thus corresponds to: f (.) = f (.)

(7.1)

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where the new shape function f (.) satisﬁes ny = f (nx ) with nx = x/xs , ny = y/ys . Then, writing nx = x/xs xs /xs = nx nxs and ny = y/ys ys /ys = ny nys , we obtain: ny = f (nx nxs ) = f (nx ) f (nxs )

(7.2)

∀(a, b), f (ab) = f (a)f (b)

(7.3)

or :

so that f (.) corresponds to an aﬃne function in log-log coordinates. As is well known [8, 9], the functional relationship Eq. (7.3) then selects power laws as the only functions not everywhere discontinuous satisfying its constraint: ∃(α, β); ∀z ; f (z) = α z β

(7.4)

Accordingly, there thus exists a way of simultaneously varying the standards (xs , ys ), i.e. the standard points S, so that the shape function f (.) keeps the same (Fig. 7.2-c and d). The existence of such a combined way of dilating the data nx and ny without changing their relationship expresses the property of covariance Eq. (7.1) of function f (.). Although stated on a simple case, the above procedure exempliﬁes the way to go, in more complex situations, from a property of scale invariance to a property of scale covariance (see [10] for an application to the so-called signiﬁcant-digit law). In particular, covariance with respect to scale or, as will be considered later on, with respect to scale-ratio, will always expresses itself as a functional relationship to be satisﬁed by the sought law. 7.3.2

Geometry

Use of similarity arguments in geometry dates back at least to the Greeks. This was mainly done in a descriptive way however, i.e. without evidencing deﬁnite geometrical relationships. In particular, the main assumption was that, owing to the scale invariance of geometry, the internal relationships of geometrical ﬁgures could not involve explicitly the ﬁgure scale. Then, applying this similarity argument to circles yields the ratio of their perimeter to their diameter to be a universal constant π, independent of the actual size of circles ; however, this leaves its value undetermined. Also, applying similarity arguments to triangles that only diﬀer by a dilation, i.e. to similar triangles, reveals that the ratios of the diﬀerent sides of a triangle are the same for all of them. This statement then shows that these ratios only depend on the angle between the corresponding triangle sides but leaves

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n

n ’ y

y S S’

1

1

S’ n ’ x

n

x

1 (a) n

1 (b) n ’ y

y S’ S

1 1

(c)

1 n

x

S’ 1

n ’ x

(d)

Fig. 7.2 Sketch of scale invariance. Two functions f (.) are considered, one in (a,b) and the other in (c,d). On each of their graph, two points S and S serve as standards. Changing standard S = (xs , ys ) for standard S = (xs , ys ) corresponds to a dilation of standard on the x-axis completed with a standard change on the y-axis so that f (1) remains unity. This is achieved in the change (a) → (b) for the upper curve and in the change (c) → (d) for the lower curve. The change of standard modiﬁes the upper curve, a sinusoide, which is thus not scale invariant. The standard change does not modify the lower curve, a power law, which is thus scale invariant.

the resulting trigonometric value undetermined. Finally, applying similarity on a couple of similar triangles yields the Thales relationship according to which all ratios of similar sides are equal to the triangle dilation factor; however, this does not tell further on the proper geometry of triangles. Expressing scale invariance in geometry in a more predictive way can be obtained with the concept of scale covariance. This may be exempliﬁed on right triangles so as to recover the Pythagoras relationship. The idea, which is reported in the famous book by G.I.Barenblatt [8], consists in splitting a right triangle T into two others t1 , t2 , by drawing the altitude perpendicular to its hypotenuse, Fig. 7.3. Here, the area S of each of these right triangles is a universal function of the length of their hypotenuse h and of their smallest angle φ: S = S(h, φ). However, due to scale invariance, S expresses as a power law of h Eq. (7.4) with an exponent of 2 for dimensional reasons and a prefactor therefore only function of φ: S = g(φ) h2 . The important thing

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for the sequel is that changing right triangles from T to t1 or t2 does not change neither their smallest angle φ nor the function g(.): gT (.) = gt1 (.) = gt2 (.) = g(.)

(7.5)

Relation Eq. (7.5) actually expresses scale covariance. Now, summing the areas of triangles t1 , t2 to recover that of triangle T yields: gT (φ) c2 = gt1 (φ) a2 + gt2 (φ) b2

(7.6)

where a, b, c denote their respective hypotenuses. Thus, applying scale covariance Eq. (7.5) yields the Pythagoras theorem, i.e. a predictive geometrical relationship on right triangles.

b

T a

φ t1

φ

t2 c

Fig. 7.3 Pythagoras theorem recovered from scale covariance. All triangles T , t1 or t2 are right triangles. Additivity of areas Eq. (7.6) using the scale covariance Eq. (7.5) of the relationship between areas to angle in right triangles yields the theorem.

7.3.3

Equilibrium systems

In equilibrium systems, the approach to critical points of phase transitions is accompanied by a divergence to inﬁnity or a convergence to zero of some relevant variables. To address this phenomenon, one may notice that, at the critical point, two condensed phases merge. Then, the resulting medium does not correspond to a mix between distinct phases but, instead, to a peculiar state from which no distinction between phases can be made [11]. This diﬀers from a usual homogeneous state however, owing to the large implication of ﬂuctuations. Instead, the medium appears as a statistically homogeneous mix of states displaying all ﬂuctuation scales. In particular, there is no way for identifying any speciﬁc condensed phase from scales: the critical point satisﬁes scale invariance [11]. Usually, thermodynamic descriptions yield a relevant variable V to be considered as a function of the distance to the critical point in the parameter space: V ≡ V ( ). Then, as discussed in Sec. 7.3.1, scale invariance applied to this single-scale function selects a power law relationship

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Eq. (7.4): V ( ) = V ( )( / )e

(7.7)

However, the exponent e is free so that deeper investigations are necessary to elucidate the nature of the critical point. They have been performed in the 70’s still by using scale invariance but in a more reﬁned way [12]. In particular, the approach to the critical point has been formalized by the renormalization theory so as to ensure an actual convergence to a deﬁnite state at the transition. This provided additional covariant relationships among variables that gave correlations between their critical exponents. Compared to the somewhat descriptive power law-type relationship Eq. (7.7), renormalization theory thus succeeded in extracting more predictive informations from scale invariance. 7.3.4

Out-of-equilibrium

We now consider scale invariance in turbulent front propagation and wish to express it, in a way similar as above, as a scale covariance. The ﬁrst step consists in identifying the regime in which scale invariance can apply. For this, we consider turbulent combustion in scale space and we view it as an interaction between the scales of the front and the scales of the ﬂow (Fig. 7.4). The front scales involve the front thickness δ and the transit time τc of the fresh medium through the front. The ﬂow scales involve the eddy sizes Li ranging from the Kolmogorov scale Lη to the integral scale LI and their corresponding turn-over times τi . We then consider a window W of size L larger than the integral scale of turbulence LI , L > LI , comoving with the front and within which front propagation is observed at a ﬁne resolution scale Lr below the Kolmogorov scale Lη : Lr < Lη . We denote UN the front normal velocity, UT the front turbulent velocity deﬁned as the mean velocity of the window W and u the turbulent intensity of the ﬂow in W (Fig. 7.4a). The increase of velocity UT /UN is a priori a function of the ratio u /UN , of the scales of the ﬂow non-dimensionalized by those of the front and of other numbers specifying the physical regime such as the Reynolds number Re of the ﬂow at the integral scale LI , the Lewis number Le of the reacting species, etc...: UT u δ τc =L , , , Re, Le (7.8) UN UN Li i=0,...,I τi i=0,...,I A geometrical interpretation of the velocity enhancement UT /UN may be obtained by considering the mean consumption rate of the fresh medium

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by the propagating front (Fig. 7.4a). On one hand, it writes UN ST since the actual front surface ST moves with the normal velocity UN . On the other hand, it writes UT SL since the projected front surface SL on the mean propagation direction moves with a mean velocity UT . We thus obtain: ST UT = =R UN SL

(7.9)

where R denotes the front surface enhancement, i.e. the front roughness. Hereafter, we shall restrict ourselves to the regime where the scales of the front are much smaller than those of the ﬂow [13]: δ Lη < L i

(7.10)

τc τη < τi

(7.11)

Then, any vortex of the ﬂow sees the front as an inﬁnitely small interface with an inﬁnitely fast kinetics: ∀i, δ/Li 1 ; τc /τi 1

(7.12)

This suggests that the conditions of interaction between the vortices and the front are the same for all vortices, whatever their size. In particular, assuming regular limits of function L(.) in this regime, this turns out assimilating the present state to that obtained for δ = 0 and τc = 0. Then, most variables vanish in relation Eq. (7.8), whatever the vortex scales. Disregarding a dependence of this relationship with the Reynolds number in a turbulent regime and omitting in a given gaseous mixture its explicit dependence with the Lewis number for simplicity, we thus obtain: UT u (7.13) ≡L δ → 0, τc → 0, UN UN We may notice that this regime, called the ﬂamelet regime, has practical implications since it is largely encountered in internal combustion engines for instance [13]. As, in this regime, vortices see the front the same way, one can expect the wrinkling process to behave statistically the same way for all of them, irrespective of their scales. This turns out stating the statistical scale invariance of this process. Our goal will now consist in extracting as much information as possible from this property. To express the resulting scale covariance, we have to consider the front velocity relationship Eq. (7.13) in any scale range and state that no scale can be identiﬁed from it, by any means. On a formal viewpoint, this property simply follows from the fact that the law L(.) involves no explicit

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scale in the ﬂamelet regime so that the same relationship would have been obtained if the analysis had been performed in another scale range. In order to formalize it, let us observe the front within another window Wj of size Lj and at another resolution scale Li (Fig. 7.4b). The mean velocity of the window Wj co-moving with the front then corresponds to the turbulent velocity Uj observed at the scale Lj . Similarly, the mean velocity Ui of windows Wi of size Li co-moving with the front plays the role of a mean normal velocity at scale Li . Both are respectively the analogues of the turbulent velocity UT and of the normal velocity UN that have been considered previously. It then remains to express the analogue of the turbulence intensity u . Assuming that the interaction between front and turbulence is local in scale space, the turbulence intensity to consider in the window Wj and at the resolution scale Li is simply that found in the scale range [Li , Lj ]: ui,j . The analogue of the relationship Eq. (7.13) in this scale range then writes: ui,j Uj (7.14) = Li,j Ui Ui Here, the indexation of functions Li,j by i and j recalls that they a priori change with the scales Li , Lj , except if scale covariance is satisﬁed. Scale covariance however requires and implies that all the functions Li,j (.) are the same: ∀i, ∀j, Li,j (.) = L(.)

(7.15)

Its physical interpretation is the following: as soon as one rejects the intrinsic scales of the front (δ, τc ) as relevant variables of the physical process, all the resolution scales Li and all the observation scales Lj become physically equivalent. Then the wrinkling laws Li,j (.) should not be able to make a distinction between them. In other words, they must all be scale covariant.

7.4

On the scale-ratio invariance

As may be noticed on relation Eq. (7.15), scale covariance of front propagation involves a couple of scales (Li , Lj ) which bound the relevant scale range of interaction [Li , Lj ]. The fact that the relevant law relies on two scales instead of one will make us face another scale symmetry: the scaleratio invariance. Prior to addressing it on front propagation, we discuss below its meaning and its potential implications.

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U

231

U j

T UN

u'

U δ S

S

L

W L (a)

T

i W i

W j L

r

L

L

j

i

(b)

Fig. 7.4 Sketch of a turbulent front: a) usual representation in a window larger than the integral scale of turbulence and at a resolution ﬁner than the dissipation scale of turbulence; b) generalization of the usual representation to various resolution scales Li and various observation scales Lj .

7.4.1

The meaning of scale-ratio invariance

The well-known symmetry of scale invariance is based on the concept of magnitude. However, it is usually referred to the concept of scale since “magnitude” reduces to “scale” in a geometrical context. Scale invariance actually reﬂects the property following which changing all the magnitudes of a variable by the same factor would have no implication on the system behaviour. Notice that, in this change, the ratio of diﬀerent magnitudes, i.e. of scales, remains unchanged : Li → Li = λLi ; Lj → Lj = λLj ⇒

Li L Li → i = Lj Lj Lj

(7.16)

This, in particular, means that the relative extents Lj /Li of the scale ranges [Li , Lj ] remain constant when one applies the homogeneous dilation that is used to probe scale invariance. However, scale invariance might not be the only scale symmetry satisﬁed by the system under study. In particular, in systems for which the concept of scale range is relevant, one way wonder whether an additional symmetry changing the extent of scale ranges could apply. In practice, this could mean increasing the resolution Li of observation while keeping the same ﬁeld scale Lj in which the system is considered, or changing the integral scale of a turbulent ﬂow Lj while keeping the same dissipation scale Li for it. Should there be fundamental reasons for the system to keep the same behaviour in these transformations, then another symmetry would be in order.

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We thus explore the implications of the existence of an additional scale symmetry based on the change of the “magnitude” of scale ranges, a socalled scale-ratio invariance. However, if scale invariance is already known to be satisﬁed, a natural constraint on the transformations linked to scaleratio invariance is to generate no characteristic scale. Otherwise, applying them could maintain the same laws while singling out some scales, in contradiction with the property of scale invariance. We thus consider those scale transformations S(.) which particularize no scale by their own. They thus correspond to scale invariant single-variable functions and can therefore only be power laws Eq. (7.4): x x → S(x) = xf ( )ν ≡ Sxf ,ν (x) (7.17) xf As they depend on two parameters (xf , ν), we shall label them Sxf ,ν (.). For ν = 1, these transformations actually change the extent of scale ranges and thus the scale ratios: Sxf ,ν (Lj ) Li Li = ( )ν → Lj Sxf ,ν (Li ) Lj

(7.18)

When scale-ratio covariance is satisﬁed, the relevant laws of the system must remain the same, despite the change of scale-ratios. However, as the corresponding scale transformations are not single parameter transformations as for scale invariance but instead two-parameters transformations, the above statement needs to be clariﬁed. In particular, we notice that the parameter xf , i.e. the ﬁxed point of the transformations Sxf ,ν (.), is of no importance for the change of scale ratios Eq. (7.18) so that it stands as a free parameter for this process. Accordingly, scale-ratio invariance will be satisﬁed provided that, for each change of scale-ratios, i.e. each ν, there exists at least a scale transformation Sxf ,ν (.), i.e. at least a xf , keeping the system laws the same. In this case, the resulting laws could thus equally correspond to one or another extent of scale ranges, so that the concept of large or small scale ratio will be destroyed. We now look for the implications of this additional scale symmetry on single-scale functions, multi-scale functions, and front propagation, in the case where scale invariance is yet satisﬁed. 7.4.2

Single variable function

Single-scale functions f (.) satisfying scale covariance are power laws Eq. (7.4): f (z) = α z β . On the other hand, the criterion for scale-ratio

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covariance calls that, for each change of scale ratio of variable z, i.e. each ν, there exists at least a xf and a scale transformation Sxf ,ν (.) of variable f (z), so that the relation between variables remains the same: ∀ν, ∃(xf , ν , xf ) ; Sxf ,ν [f (z)] = f [Sxf ,ν (z)]

(7.19)

It takes here the particular form: ∀ν, ∃(xf , ν , xf ) ; xf (

αz β ν z νβ ) ] ) = α[xf ( xf xf

(7.20)

which is actually satisﬁed for ν = ν and xf = f (xf ), whatever xf . Accordingly, scale covariant single-scale functions are also scale-ratio covariant. They thus change scales without singling out neither particular scales nor particular scale-ratios. Presumably, this degeneracy of symmetry properties is responsible for the fact that scale-ratio covariance has been mostly overlooked in multi-scale systems so far. 7.4.3

Multivariable function

The above degeneracy is actually speciﬁc of single-variable functions and ceases to be the rule as soon as functions depend on several variables. An example of this is given by the statistics of fully-developped turbulent ﬂow. Let us consider this statistics by the series of momentum < (δv)i > (l), i ∈ N , where δv denotes the velocity increments on distance l and < . > the time-average. As moments are single-scale functions, invoking scale invariance in the inertial range of turbulence yields all of them to be power laws of l: l ∀i, < δv)i > (l) =< δv)i > (l0 )( )ζi (7.21) l0 Then, the nature of the statistics is given by the family of scale exponents (ζi )i∈N . In particular, similarity arguments by Kolmogorov have predicted linear scaling exponents, ζi = i/3 [14], but detailed measurements have revealed deviations from linearity, a feature which is referred to as intermittency [15]. However, beyond scale invariance, one may wonder whether turbulence statistics rely on particular scale-ratios or on none [16]. In the former case, the nature of turbulence would either exhibit a particular number or involve no universal structure. The latter case would be encountered for instance if the Reynolds number, no matter its magnitude, was explicitly governing turbulence, even in the fully developped limit Re → ∞: turbulence would

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then be a singular problem. However, the conclusions of experimental and numerical investigations point towards a regular problem exhibiting no speciﬁc number. Accordingly, turbulence should rather single out no speciﬁc scale-ratio in the large Reynolds number limit: it should be scale-ratio invariant. Expressing this requirement would give no additional information on each moment taken separately, since all are single-scale functions. However, taken together, scale covariance calls for speciﬁc correlations between their scaling exponents, i.e. for speciﬁc kinds of relationships ζi = ζ(i). These expressions, derived in [16] and [17], provide the possible intermittent statistics of fully developed turbulence satisfying both scale covariance and scale-ratio covariance. 7.4.4

Front propagation

The criterion for scale covariance of front propagation Eq. (7.15) involves the two scales Li , Lj which bound the scale range of observation. It expresses that whatever these scales, i.e. whatever the scale range that they deﬁne, the same relationship between velocities must be found. Accordingly, any changes of scales Li → Li , Lj → Lj , must maintain the common velocity law L(.). These general changes include ﬁrst those changes that preserve scale ratios: Lj /Li = Lj /Li . These simply correspond to usual changes of units, i.e. Lk → γLk with constant γ, and thus, in logarithmic coordinates, to a uniform translation in scale space (Fig. 7.5a). Covariance with respect to these changes means that the value of scale is physically irrelevant so that the system does not single out - nor rely on - any speciﬁc one. We call the covariance with respect to these restricted scale changes “global covariance” [2]. It corresponds here to the usual scale covariance. However, other changes of scales do modify scale ratios: Lj /Li = Lj /Li . In scale space and in logarithmic coordinates, they thus imply not only a translation but also a stretch of scale ranges (Fig. 7.5b). Covariance with respect to them means that the size of scale ranges is physically irrelevant so that the system does not single out - nor rely on - any speciﬁc scale ratio. We call this symmetry “local covariance” [2]. For functions of a single scale, global covariance is suﬃcient for selecting power laws, Sec. 7.3.1. Then local covariance is redundant (Sec. 7.4.2) and thus usually overlooked. The present case is however diﬀerent since the function Li,j (.) involves an argument ui,j /Ui which depends not only on the

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Ln(L )

235

Ln(L )

k

k

Lj' L i'

Lj' Lj Li

Lj Li a)

L i' b)

Fig. 7.5 Sketch of scale covariance: the wrinkling law keeps the same expression in the scale ranges [Li , Lj ] or [Li , Lj ]: a) global covariance: scale ranges have the same size: Lj /Li = Lj /Li ; b) local covariance: scale ranges have diﬀerent sizes: Lj /Li = Lj /Li .

scale range [Li , Lj ] by the turbulence intensity ui,j but also on the sub-scales L0 < Lk < Li by the normal velocity Ui : Ui /UN = L0,i (u0,i /UN ) (here U0 = UN since the scale L0 is taken small enough for avoiding any velocity enhancement). Accordingly, functions Li,j (.) stand as implicit functions of many scales for which the simple selection of power laws does not apply. In particular, here, global covariance simply states that functions Li,j (.) do not rely on explicit scales: Li,j (.) = L(.). This is naturally the case for all algebraic functions that might be proposed for L(.). Among them, some will be scale-ratio covariant but most will not. Accordingly, global covariance stands here as a weaker constraint than local covariance which therefore needs to be explicitly implemented [2].

7.5

Front propagation laws both scale covariant and scaleratio covariant

Global covariance of L(.) means that this law involves no explicit length scale. As this is validated, within relation Eq. (7.13), by any algebraic form of L(.), this requirement is actually fulﬁlled by a large number of laws proposed in turbulent combustion [18]. However, those laws which do not satisfy local scale covariance make a distinction between various sizes of scale ranges and thus, for instance, between diﬀerent deﬁnitions of the integral scale of turbulence LI : L0,I (.) = L0,I (.). This diﬀerence would

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provide an objective way for diﬀerentiating scales that we reject in this system. A way for solving for scale-ratio covariance consists, as in special relativity, in expressing the compatibility between diﬀerent observers, i.e. here, between diﬀerent scale ranges. We thus consider consecutive scales ranges [Lk , Lk+1 ], k = i, . . . , j − 1, starting from scale Li and ending at scale Lj . Each of them relates the eﬀective normal velocity Uk at scale Lk to the eﬀective turbulent velocity Uk+1 at scale Lk+1 Eq. (7.14). In particular, Uk+1 diﬀers from Uk by the enhancement implied by the turbulence intensity uk,k+1 in this scale range. Combining step by step these scale ranges, one can then relate this way Ui to Uj , given the turbulence intensities uk,k+1 , k = i, . . . , j −1. However, this must be equivalent to relating these velocities directly by the law Eq. (7.14) using the turbulence intensity ui,j in the scale range [Li , Lj ]. This call for self-consistence may be formally written using the laws Ll,m (.): Li,j (.) = Lj−1,j (.) ◦ . . . ◦ Li,i+1 (.)

(7.22)

where ◦ denotes the composition of laws. It then expresses transitivity and must be satisﬁed in any regime and for any law, scale covariant or not, scale-ratio covariant or not. However, when scale invariance and scale-ratio invariance are satisﬁed, all laws Ll,m (.) must be the same. Then, transitivity implies a speciﬁc relationship: L(.) = L(.) ◦ . . . ◦ L(.)

(7.23)

This peculiar relation means that a ﬁxed point of the procedure Eq. (7.22) has been reached. As this procedure deduces a law on a given scale range from the laws valid on its sub-scale ranges, it corresponds to renormalization. Accordingly, relation Eq. (7.23) expresses the condition for the existence of a ﬁxed point of renormalization in the law space. The important thing for the sequel is that this requirement Eq. (7.23) provides a sharp selection of the possible scale covariant and scale-ratio covariant laws [2]. To show how selection can operate, let us consider the following law which has been proposed at ﬁrst order in turbulence intensity in the low turbulence regime u /UN << 1: 1 u 2 UT =1+ ( ) (7.24) UN 2 UN The origin of this relationship lays on an approximate evaluation of front roughness at low turbulence intensity u /UN << 1. In this regime,

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the front can be assumed to be planar. Then, the ﬂow component on its tangent direction can be overlooked since it provides no deformation of the front. However, the ﬂow component on the normal direction actually distorts the front in a way which, at the dominant order, can be assumed to be a simple rotation of the planar front. The surface increase, i.e. the front roughness, can thus be evaluated from simple geometry. On a vortex with scale L, the front resident time is close to L/UN since the front velocity is about UN in this regime. After crossing the vortex, the most advected points of the front get separated by a distance d = Lu /UN on the mean propagation direction. As their distance on the front is L, this corresponds to a rotation angle θ such that tan(θ) = d/L = u /UN and to a front roughness R = 1/ cos(θ) ≈ 1 + 1/2 (u /UN )2 . This, together with relation Eq. (7.9), gives relation Eq. (7.24). As the resulting law involves no characteristic scale, it can be applied in any scale range. However, as it relies on a low turbulence intensity, we shall consider it on elementary scale ranges [Lk , Lk+1 ], k = i, . . . , j − 1, and evaluate the corresponding law in a larger scale range [Li , Lj ] by transitivity Eq. (7.22). Writing the velocity enhancement in any elementary scale range in a diﬀerential form, relation Eq. (7.24) writes: Uk+1 1 uk,k+1 2 = 1+ ( ) (7.25) Uk 2 Uk dUk 1 2 = u k,k+1 (7.26) Uk (Uk+1 − Uk ) = Uk dk 2 (7.27) Integration from scale Li to scale Lk gives: Uj2 = Ui2 +

k=j−1 1 2 u k,k+1 2

(7.28)

k=i

1 (7.29) Uj2 = Ui2 + u2 2 i,j following the additivity of turbulence intensities with respect to scale ranges implied either by the Parseval relation or by energetic considerations. Interpreted in the full turbulence scale range, the resulting law then writes: 1 u 2 UT 2 ) =1+ ( ) (7.30) ( UN 2 UN It thus diﬀers from the initial law Eq. (7.24) so that the criterion Eq. (7.15) for a ﬁxed point of renormalization is not fullﬁlled: the initial law Eq. (7.24) is thus not scale-ratio covariant.

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However, let us now consider the resulting law Eq. (7.30) and apply the same procedure. We obtain the following laws in elementary scale ranges [Lk , Lk+1 ]: 1 2 2 Uk+1 = Uk2 + u k,k+1 2

(7.31)

Integration of Eq. (7.31) then gives straightforwardly relation Eq. (7.29) and, ﬁnally, the same relationship Eq. (7.30) from which we started. This shows that the law Eq. (7.30) is a ﬁxed point of the renormalization procedure. It thus satisﬁes scale-ratio covariance. 7.6

Experimental evidence of scale covariance in turbulent front propagation

This section is devoted to experimentally evidence the relevance of the scale symmetric law Eq. (7.30) and of its intrinsic properties [3, 4, 7]. 7.6.1

Experimental set-up and data processing

The experimental set-up is made of a closed combustion chamber involving a piston at one end, a grid in the middle, a matrix of sparks at the other end and three optical windows on the sides, Fig. 7.6. The chamber is initially ﬁlled with premixed propane-air gases seeded with oil particles. Then the piston is pushed suddenly to compress gases through the grid. This produces turbulent motions within which a ﬂame front generated by spark ignition propagates. Vizualization of the ﬂame front is provided by tomography [19]. A plane laser sheet illuminates the chamber in its middle and along the mean direction of propagation. It is diﬀused by oil particles towards a camera. As these particles are burnt by the ﬂame, observation of the laser sheet in a transverse direction reveals a bright zone not yet burnt, a dark zone already burnt and the front cut in between, Fig. 7.1. These images are registered on a CCD camera and then processed on a computer. Their resolution, 120µm, is typically ﬁve times smaller than the lowest noticeable wrinkling scale. The normal velocity UN is varied by changing the mixture composition. As the piston is pushed at the highest available rate, the turbulence intensity u is changed by monitoring its decay by an ignition delay. The measurements of u by anemometry laser doppler [20] ﬁt with an exponen-

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Photodiode

Piston

239

Turbulence Laser sheet generator

Spark Matrix

Electrogate Camera

Trigger Monitoring Fig. 7.6

Sketch of the experimental set-up.

tial decay involving a characteristic time scale τ = 136ms. Compared to τ , the propagation time (around 10ms) is then short enough for preventing noticeable further decay of u until front observation. Combustion of the medium induces pressure and temperature rises and adiabatic compression of the fresh gases. These eﬀects are included in the calculation of UN according to the empirical laws determined in laminar experiments [21]. In contrast, these eﬀects are not used to renormalize u so that one might question the relevance of its value at the observation time. However, since the thermodynamic conditions were the same for each front at the time they were observed, one may guess that the modiﬁcations of u from ignition to observation were the same for all. For this reason, the value of u presumably remains relevant to the turbulent ﬂow at the observation time, up to an assumed constant prefactor. The ranges of parameters scanned in our study are: combustion volume 10 × 6 × 6cm3 , normal velocity at observation time (T 520K, p 7.2atm) 0.40ms−1 ≤ UN ≤ 0.86ms−1 , integral scale of turbulence LI = 5.2mm, 0.30 ≤ u ≤ 0.93ms−1 and 0.7 ≤ u /UN ≤ 2.2. Image processing involves the measurement of the roughness Λi,S of the front cut at increasing resolutions Li , the scale LS denoting the size of the chamber. The surface roughness Ri,S is deduced by the relation 2 Ri,S = 2Λ2i,S − 1 which is valid both at ﬁrst order in Λi,S − 1 and at large values of Λi,S following diﬀerential geometry and the proportionality of surface roughnesses to linear roughnesses in quasi-fractal regimes.

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Surface roughnesses Ri,j at various resolutions Li and within diﬀerent windows Wj of size Lj , are then determined from the series {Ri,S }i=0,...,I by a property inherent to the deﬁnition of roughness, the multiplicativity: Ri,j = Ri,S /Rj,S . Surface roughnesses drive an enhancement of front velocity which may be derived by considering the volume burnt by a ﬂame per unit time. At a given resolution scale Li , it corresponds to the product of the normal velocity of the front Ui by the front surface Si seen at this resolution. Equating this determination for two diﬀerent scales Li and Lj gives Ui Si = Uj Sj . Since the ratio of surfaces Si /Sj is simply the surface roughness Ri,j , one obtains: (7.32) Ri,j = Uj /Ui which generalizes the relation Eq. (7.9) in scale space. According to Eq. (7.14), they are thus related by the law Li,j (.) to the ratio ui,j /Ui that we call the mixing variable mi,j : mi,j = ui,j /Ui (7.33) (7.34) Ri,j = Li,j (mi,j ) Turbulence intensities ui,j are calculated according to the power law: ui,i+1 = u0,1 (Li /L0 )K (7.35) for a given value of K, the value of the prefactor u0,1 being ﬁxed so that the net turbulent intensity in the scale range [Lη , LI ] is u . Being a power law, relation Eq. (7.35) implies the absence of any characteristic scale in the ﬂow. Although this property is invalid close to the dissipative ﬂow scale, we shall nevertheless invoke it as a useful way of ﬁtting turbulence intensities in our scale range. However, as the range of wrinkling scales is more dissipative than inertial and because of the feedback of combustion on turbulence, the classical value K = 1/3 of the Kolmogorov theory is not expected. Instead, ﬁtting the intensities given by viscous damping in non-reactive ﬂows yields K = 1/2 [3]. Normal velocities Ui are deduced from roughness measurements by the relationship Ui = R0,i U0 , using the fact that, for a scale L0 smaller than the smallest wrinkling scale, the velocity U0 is simply the normal velocity UN . Since both roughnesses Ri,j and mixing variables mi,j may thus be determined at all scales, a comparison between the various laws Li,j (.) can be made. It is performed below in two ways: ﬁrst by ﬁxing the lowest scale Li at the value L0 and, second, by taking for the scale Lj the scale immediately larger than Li .

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1.2 Ri,i+12

2.0

1.1

Ro,j2

241

b)

a) 1.0 0.0

mo,j2 1.0

2.0

1.0 0.0

mi,i+12 0.1

0.2

Fig. 7.7 Plots of roughnesses Ri,j versus mixing variables mi,j . a) integral analysis: scale ranges [L0 , Lj ]; b) local analysis: scale ranges [Li , Li+1 ]. Symbols refer to diﬀerent fronts. The invariance of curves in one (a) or the other (b) representation evidences scale-covariance.

3.0

1.2

Ro,j2

Ri,i+12

2.0

1.1

a) 1.0 0.0

b)

m0,j2 1.0

2.0

1.0 0.0

mi,i+12 0.1

0.2

Fig. 7.8 Theoretical wrinkling laws Eq. (7.36) (crosses), Eq. (7.38) (circles), Eq. (7.39) (squares) within the experimental scale range [L0 , LI ] for a = 1.2 and u /UN ≤ 1.3. Plots (a) and (b) as in Fig. 7.7. In (b), the data relevant to relations Eq. (7.38) and Eq. (7.39) even extend beyond the plotted range. The invariance of the law Eq. (7.36) and the evolution of the laws Eq. (7.38), Eq. (7.39) evidence both the scale-covariance of the law Eq. (7.36) and the sensitivity of this property.

7.6.2

Integral analysis

We consider the scale ranges [L0 , Lj ] at a ﬁxed L0 but a variable Lj . This corresponds to including in our description the eﬀects of more and more vortices or, equivalently, to taking the integral scale at larger and larger values. The measurements (m0,j , R0,j ) are made, for each mixture composition, on a dozen fronts and, for each front, on 20 scales Lj in between L0 and LI (LI /L0 = 44 with an elementary scale-ratio a = 1.2). They are plotted in Fig. 7.7a for a given mixture composition in quadratic coordinates 2 ) computed with K = 1/2. (m20,j , R0,j Figure 7.7a shows almost straight lines. This suggests that, to our

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experimental uncertainty in u and UN , the plots are linear for each front. This is conﬁrmed by the fact that the slope dispersion is much smaller on each front (4%) than on the whole data base (18%). Since, at a given Lj , the values of both m0,j and R0,j vary with the turbulence intensity undergone by the front, another piece of information can be extracted from Fig. 7.7a: all the laws L0,j (.) are actually linear in quadratic coordinates and involve the same slope β, β = 1.14 ± 0.21: ∀j,

2 R0,j = 1 + βm20,j

(7.36)

This means that the laws L0,j (.) are the same: ∀j, L0,j (.) = L(.). This property corresponds to a local scale covariance since the change of Lj modiﬁes the size of the scale ranges studied. On the other hand, the quadratic expression of the laws Eq. (7.36) agrees with the expected scale covariant forms Eq. (7.29). This suggests that scale covariance could extend to any other scale change. 7.6.3

Local analysis

We now consider consecutive scales [Li , Li+1 ] for various scales Li . This corresponds to isolating the eﬀect of each vortex family of scale Li and thus to scanning the wrinkling process from small scales to large scales. The results are plotted in Fig. 7.7b for the fronts analyzed in Fig. 7.7a, 2 still with K = 1/2 and in quadratic coordinates (m2i,i+1 , Ri,i+1 ). Here again, a line with a small scatter is observed. As in the previous analysis, this implies that the laws Li,i+1 (.) are linear in quadratic coordinates and involve the same slope β , β = 1.18 ± 0.17. Interestingly, to our level of accuracy, β and β appear to be equal, so that: 2 = 1 + β m2i,i+1 ∀i, Ri,i+1

(7.37)

Here too, the fact that the laws Li,i+1 (.) are the same means that there is scale covariance. More precisely, this covariance corresponds to a global scale covariance since the size of the scale ranges studied are the same: Li+1 /Li = a , ∀i. In addition, the equality between these laws Eq. (7.37) and the former laws Eq. (7.36) shows that scale covariance extends to any kind of scale changes, as expected from the analytic property of the quadratic laws investigated in Sec. 7.5. This equality, although apparent over the front set by comparison between the plots of Fig. 7.7a and Fig. 7.7b, may even be better evidenced by comparing the data from front to front. ∗ Then, the variation of their slope in one (β ∗ ) or the other (β ) analysis are ∗ ∗ ∗ much smaller than that displayed in average: |(β − β )/β | < 5%.

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Although the scale covariant laws in both kinds of analysis are linear in a quadratic representation, we stress that the change of representation from integral analysis to local analysis is actually non-linear and non-local, owing to the use of the normal velocity Ui in the mixing variable mi,i+1 Eq. (7.33). For this reason, the plot of Fig. 7.7b is deﬁnitely not an enlargement of that of Fig. 7.7a but the result of a non-linear and non-local change of variables. Obtaining another line, moreover the same as in Fig. 7.7a, is therefore actually meaningful. 7.6.4

Sensitivity of scale covariance

To better judge about the above evidence of scale covariance, it is instructive to apply the same procedure to two analytical laws proposed in the literature: the Clavin-Williams law Eq. (7.24) Eq. (7.38) [22], and the Yakhot law Eq. (7.39) [23], in addition to the scale covariant law Eq. (7.36). β 2 m 2 0,j 2 = exp(βm20,j /R0,j )

∀j,

R0,j = 1 +

(7.38)

∀j,

2 R0,j

(7.39)

As may be seen in Fig. 7.8a, these three laws are actually tangent at the origin (0,1) and, within the present experimental range, close to one another in integral analysis. However, turning to the local analysis produces dramatic changes of both the Clavin-Williams law and of the Yakhot law but, on the opposite, leaves the scale covariant law unchanged (Fig. 7.8b). This behaviour shows that the non-linear change of representation applied when going from the integral analysis to the local analysis enhances the diﬀerences between non-covariant laws and covariant ones. In particular, whereas our data agree equally well with any of the three laws in integral analysis Fig. 7.8a, they deﬁnitely lie much farther from the non-covariant laws Eq. (7.38) and Eq. (7.39) in local analysis Fig. 7.8b than allowed by our experimental uncertainties Fig. 7.7b. The sensitivity of scale covariance thus accurately selects a covariant law in our experiment, without need of larger turbulence intensities. 7.7

Scale construction of turbulent fronts

Up to now, we have focussed attention to the relationship which governs front propagation. In particular, the object displayed in this issue, the turbulent front, was only considered as a mean for identifying the wrinkling

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law L(.). Now that we have determined this law, we propose to change goal by addressing the way the geometry of the present object, the front, is statistically constructed in scale space. We thus consider the front roughness Ri,i+1 brought about by each scale range [Li , Li+1 ] following the additional wrinklings produced by its vortices. Its evolution from scale to scale then describes statistically the evolution of front geometry in scale space. In particular, elementary roughnesses Ri,i+1 all equal to unity in some scale range correspond to an euclidean geometry in this range. In contrast, elementary roughnesses Ri,i+1 all equal to the same value diﬀerent than unity in a given scale range correspond to a fractal geometry in this range. In between, roughnesses evolve with scale so that the front no longer appears either euclidean nor fractal. In particular, it is thus not scale invariant. However, as it nevertheless has to satisfy the scale covariant wrinkling law L(.), we propose below to determine its geometry from it, i.e. from relations Eq. (7.30) and Eq. (7.29). At this point, it is instructive to stress the relevance of studying a nonscale invariant object with a scale covariant law. This follows from the fact that scale covariance results from the scale invariance of a process, here the interaction between front and ﬂow, and not from the scale invariance of an object. Then, the scale invariant process may or may not give rise to a scale invariant object ; however, all the objects produced by it will nevertheless follow the scale covariant law. This situation is analogous to that found in mechanics where special relativity, which refers to covariant laws, may apply to a uniform motion or not, depending on initial conditions. Here, somewhat similarly, a scale covariant wrinkling law may give rise to a fractal geometry or not depending on initial conditions in the scale space. This, in particular, illustrates the fact that the scale symmetry of a process goes beyond the scale symmetry of the objects that it produces. 7.7.1

Deterministic construction of front geometry in scale space

2 and we take a geometFor simplifying the notations, we denote ρi = Ri,i+1 rical series of scales: ∀i, Li+1 /Li = a. Using relations Eq. (7.35) Eq. (7.37), the link between roughness and front velocities Eq. (7.32) and the deﬁnition of mixing variables Eq. (7.33), we straightforwardly obtain with ρ∞ = a2K the following series:

∀i,

ρ∞ ρi+1 − 1 = ρi − 1 ρi

(7.40)

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provided that all ρi diﬀer from unity. The solution of this series is: 1 + σ0 ρi+1 ∞ ρi = (7.41) 1 + σ0 ρi∞ where σ0 depends on the initial condition ρ0 : ρ0 − 1 σ0 = (7.42) ρ∞ − 1 These relations show that, in the limit of an inﬁnite number of scales, the iteration of roughnesses converges towards an attracting value ρ∞ , provided that all ρi diﬀer from unity, i.e. that all scale ranges involve vortices. In this limit regime, each scale then brings about the same additional front roughness: the front has reached a fractal regime. Its fractal dimension D corresponds to the dimension of the euclidean surface plus the ratio of the increase of roughness to the increase of length scale, in logarithmic coordinates: D = 2 + lim ln(Ri,i+1 )/ ln(Li+1 /Li) i→∞

As, in this limit regime, Ri,i+1 =

1/2 ρ∞

(7.43)

= aK , we obtain:

D =2+K ; i→ ∞

(7.44)

Of course, if there is a hole in the turbulence range, i.e. a sub-range in which turbulence intensity vanishes, ui,j = 0, the front roughness will stagnate: ρj = ρi . In particular, if turbulence stops before the fractal regime is reached, uk,I = 0, elementary roughnesses will not have reached the limit value ρ∞ : ρI = ρk < ρ∞ . The determination Eq. (7.44) of the fractal dimension agrees with those found by other arguments [6, 24, 25]. However, the way the front reaches the fractal regime, i.e. the series Eq. (7.40), is original since it fully relies on scale covariance. Its prediction is compared in Fig. 7.9a to the behaviour of one of the most turbulent fronts generated in our experiment (u /UN = 1.85). Crosses correspond to the iteration Eq. (7.41) for an inﬁnite number of scales, F to its ﬁxed point, i.e. to the fractal regime, and circles to the experimental points. In agreement with Eq. (7.41), the iteration of experimental points with scales shows a decrease of their distance to F and a near stagnation in the vicinity of both the euclidean (ρi = 1) and the fractal (ρi = ρ∞ ) regimes. On the other hand, it reveals that the roughness brought about by each scale of the turbulent ﬂow is actually not a constant, even at large scales. This means that this front, although largely turbulent, is, strictly speaking, not a fractal. It actually displays “fractal dimensions” varying with scales and could thus be better termed “scale dependent fractal” [26].

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Ri,i+12 1.20 F

1.15 1.10 1.05 1.00 0.0

0.1

0.2

mi,i+12

(a)

Y = Ln(Ri,I) 0.8

Slopes exp(2KX)

0.6 0.4 0.2 0.0 0.0

exp(- 2KX)

X = Ln(Li/L0) 1.0

2.0

3.0

4.0

(b) Fig. 7.9 a) Geometry formation in scale space, a = 1.2, K = 1/2. Circles: experimental front u /UN = 1.85. Crosses: simulation of fractal construction for β = 1.15 Eq. (7.37) and for an infinite number of scales. F : fractal regime. b) Classical scale analysis of the front studied in (a). The evolution of curvature according to Eq. (7.48) and Eq. (7.51) shows the diﬃculty of appreciating the distance to a pure fractal regime F in the representation (b).

7.7.2

“Apparent” fractal dimensions

To further clarify the diﬀerence between the turbulent front and a fractal, we now confront our analysis to the conclusions that might be drawn from a classical scale analysis. In particular, we seek to accurately determine the way the apparent “fractal dimensions” drift with scales. Classical scale analysis relates the front roughness Ri,I measured at increasing resolution scales Li to the scale ratio Li /L0 , in logarithmic coordinates. The corresponding curve Y (i) = ln(Ri,I ) versus X(i) = ln(Li /L0 ) is plotted in Fig. 7.9b for the front studied in Fig. 7.9a. It shows a zero slope domain (the euclidean regime) followed by an increasing slope do-

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main (the transition from the euclidean to the fractal regime) and ends up with a zone with a seemingly constant slope which corresponds to a fractal-like behaviour. The latter evidence is at variance with the conclusion drawn in Sec. 7.7.1 and in Fig. 7.9a where the front was recognized not having reached the fractal regime. This disagrement traces back to the accuracy of the conclusions drawn above: is the seemingly constant slope of Fig. 7.9b an actual constant slope? The fact that it is not shows that the classical scale analysis might be misleading as to the determination of the actual scale similarity of forms, as conﬁrmed by the alternate analysis of Fig. 7.9a. To better clarify this disagreement, we consider below local boxcounting dimensions D(i) at scale Li : ln(Ri,i+1 ) ln(Li+1 /Li) ln(ρi ) = 2+ 2 ln(a)

D(i) = 2 +

(7.45) (7.46)

and determine their scale evolution according to the scale covariant iteration Eq. (7.40). We notice that this will provide the local slope of the curve (X(i), Y (i)) since dY /dX(i) = 2 − D(i). (1) Quasi-euclidean regime This regime corresponds to roughnesses close to unity, ρi ≈ 1, or following Eq. (7.41), σ0 ρi∞ 1. It is encountered in practice in the vicinity of the resolution scale L0 . Relation Eq. (7.41) then yields: ρi ≈ 1 + (ρ∞ − 1)σ0 ρi∞

(7.47)

As ρ∞ − 1 ≈ 2K ln(a) is much smaller than unity here, this implies from Eq. (7.45): D(i) ≈ 2 + Kσ0 ρi∞

(7.48)

and, since ln(ρi∞ ) = 2KX(i), dY /dX ≈ −Kσ0 exp(2KX)

(7.49)

The curve (X(i), Y (i)) thus shows a negative slope whose magnitude increases exponentially from zero at Li = L0 . This provides an exponential growth of its curvature which corresponds to the bent part of the transition from the euclidean regime to the fractal regime.

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(2) Quasi-fractal regime This regime corresponds to roughnesses close to that found in the fractal regime, i.e. following Eq. (7.41), to σ0 ρi∞ 1. It is encountered in practice in the vicinity of the largest observed scale LI and at the largest turbulence intensities. Relation Eq. (7.41) then yields: ] ρi ≈ ρ∞ [1 − (ρ∞ − 1)σ0−1 ρ−(i+1) ∞

(7.50)

As ρ∞ − 1 ≈ 2K ln(a) 1, this implies: D(i) ≈ D − Kσ0−1 ρ−(i+1) ∞

(7.51)

or: dY /dX ≈ 2 − D +

K exp(−2KX) σ0 ρ ∞

(7.52)

In this large scale regime, the slope of the curve (X(i), Y (i)) thus reaches its asymptotic value 2 − D in an exponentially slow manner. This implies, for this curve, an exponentially small curvature which makes the distance to a constant slope regime, i.e. a fractal regime, especially hard to evidence. This is responsible for the diﬃculty in distinguishing pure scale similarity from an approximate scale similarity in the classical representation (X(i), Y (i)) of Fig. 7.9b. In particular, this would have led us to abusively consider the most turbulent fronts as fractal here and with a fractal dimension diﬀerent than the predicted value Eq. (7.44). On a more general ground, relation Eq. (7.51) shows that, according to the scale covariance of the wrinkling law, the local box-counting dimension D(i) varies with scale. This corroborates the observation of “scale dependent fractal” in this system and simply links it here to the construction of a multi-scale front by a scale covariant process, starting from an euclidean geometry at small scale. 7.7.3

An artifact of ﬁnite size: the variation of “fractal dimensions” with turbulence intensity

Classical scale analysis leads one to attribute a fractal dimension to the experimental fronts insofar as their curve (X(i), Y (i)) displays an apparently constant slope. Figure 7.10, shows the measurements of these numbers for the experimental fronts observed at various mixing variables m = u /UN . They are not constant and show an increase with the turbulence intensity, as already found in other studies [27, 28].

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These features are somewhat surprising since the fractal regime is usually expected to be reached in the limit of large turbulence intensity. In particular, if the vicinity of the fractal regime is thought to be approached enough at ﬁnite m = u /UN to allow fronts to be considered as fractal, is it meaningful that the resulting fractal dimensions be so diﬀerent? In other words, how can we conciliate the existence of a single asymptotic regime, the fractal regime, with so diﬀerent asymptots, i.e. so diﬀerent fractal dimensions? To elucidate this paradox, we calculate the evolution with m = u /UN of the apparent fractal dimension at the integral scale LI , D(I, m), according 2 is the product of ρi to the scale covariant behaviour Eq. (7.41). Since R0,I 2 from i = 0 to I − 1, iteration of Eq. (7.41) yields R0,I = (1 + σ0 ρI∞ )/(1 + σ0 ) and, by elimination of σ0 : 2 − 1) 1 + f (ρ∞ )(R0,I (7.53) ρI = 2 R0,I where f (ρ∞ ) = ρ∞ + (ρ∞ − 1)/(ρI∞ − 1). As ρ∞ − 1 1, ρI∞ 1 and 2 R0,I = 1 + β(u /UN )2 , one obtains: 1 + ρ∞ βm2 K ln[ D(I, m) = 2 + ] (7.54) ln(ρ∞ ) 1 + βm2 Relation Eq. (7.54) evidences a variation of D(I, m) with m = u /UN . This may be interpreted, at ﬁxed scale LI and ﬁxed normal velocity UN , as a variation of D(I, m) with the turbulence intensity u : ∂D(I, m)/∂u = 0. Within our framework, this simply appears as a corollary of the dependence of D(i) with scale Li at ﬁxed turbulence intensity: ∂D(i, m)/∂i = 0 Eq. (7.51). As shown in Fig. 7.10, taking for β and K the values selected by scale covariance (β = 1.15 and K = 1/2), yields a good agreement between relation Eq. (7.54) and the curve obtained experimentally. This corroborates our analysis of “apparent” fractal dimensions in this system. According to it, the evolution of these dimensions with m = u /UN can be traced back to the fact that they are calculated at a ﬁnite scale, the integral scale LI of turbulence, instead of at an inﬁnite scale. This therefore corresponds to an artifact due to the ﬁnite size of the system.

7.8

Conclusion

The implications of scale-invariance in far-from-equilibrium systems have been addressed on the paradigm of turbulent front propagation. Whereas

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D(I,m) 2.5 2.4 2.3 2.2 2.1 m=u'/U

2.0 0

1

2

3

4

N

Fig. 7.10 Fractal dimension D measured at the integral scale of turbulence LI for various turbulence intensities or mixing variables m. Circles: experimental points; solid line: evolution of D(I, m) with m = u /UN according to scale covariance Eq. (7.54) with β = 1.15, ρ∞ = a2K , a = 1.2, and K = 1/2. The scale-covariant law reproduces well the evolution of the apparent fractal dimension D.

usual scale analysis are devoted to the characterization of the geometry of objects, for instance by a fractal dimension, our main goal has been to determine the statistical laws that relate the relevant variables of the system. This led us to deal with a property of law directly transposed from the scale symmetry of the system: the scale covariance. Interestingly, this scale symmetry of laws, the scale covariance, has proved to be more extended than the scale symmetry of forms, i.e. the scale similarity. On the other hand, the dependence of the propagation law on a scale range has led us to consider an additional scale symmetry diﬀerent than the pure scaleinvariance: the scale-ratio invariance and its related scale-ratio covariance. Applying these covariances provided a sharp selection of the relationship which sets the eﬀective front velocity. In addition, the equivalence between velocity enhancement and surface increase in this problem enabled us to transpose this selection in scale space to determine the evolution of the front roughness from scale to scale. The main advantages of scale-covariance upon scale-similarity are the following: Scale-covariance reveals an unexpected universality in this stochastic system: the physical equivalence of the euclidean regime, the fractal regime and the transition regime between both. This corresponds to a uniﬁed description of the regimes of weak or strong turbulence intensities or of weak or large wrinkling. This diﬀers from the picture derived from scale

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similarity following which the weak turbulence regime is usually overlooked since not displaying scale similar forms. By contrast, scale covariance shows that it contains as much information regarding statistical laws as the strong turbulence regime. On another level, the scale representation inherent to scale covariance evidences the distance to the fractal regime much better than that used in classical scale analysis. In particular, it reveals that the most turbulent fronts are abusively qualiﬁed as fractal since they are actually not scale similar. This means that their apparent fractal dimensions do not refer to an asymptotic fractal regime and should thus be considered with care. In particular, they show a paradoxical evolution with turbulence intensity, which simply traces back to an artifact due to the ﬁnite size of the system. Since scale covariance relies on the scale invariance of physical processes, it should be valid in a number of systems presently studied under the heading of “scale similarity”. Given its larger advantages, it might be worth focussing attention on it to determine not only the characteristics of the forms generated in scale invariant systems, but also the shape of the laws that actually govern them in a large range of regimes. References [1] Dynamics of Fractal Surfaces (1991) Family F. and Vicsek T. Eds. (World Scientiﬁc, Singapore). [2] Pocheau A. (1994) Phys. Rev. E 49, pp. 1109. Pocheau A. (1992) Europhys. Lett. 20, pp. 401. Pocheau A. (1992) C. R. Acad. Sci. Ser.II 315, pp. 21. [3] Pocheau A. and Queiros-Cond´e D. (1996) Phys. Rev. Lett. 76, pp. 3352. [4] Pocheau A. and Queiros-Cond´e D. (1996) Europhys. Lett. 35, pp. 439. [5] Williams F. A. (1985) Combustion theory 2nd Edition (The Benjamin Cummins Publishing Company, Menlo Park). [6] Clavin P. (1987) in Theory of Flames in Disorder and Mixing Guyon E. Eds, NATO Advanced Study Institute Series E: Applied Science Vol.152 (Plenum, New-York), pp. 293. [7] Pocheau A. (1996) in “Mixing, Chaos, and Turbulence” Carg`ese France Nato ASI Series B (Physics) 373 Plenum Press, pp. 187-204. Pocheau A. (1997) in “Scale Invariance and Beyond” Les Houches France Dubrulle B., Graner F., Sornette D. Eds, EDP Sciences Springer, pp. 209-223. [8] Barenblatt G. I. (1979) Similarity, Self-similarity and Intermediate Asymptotics (Consultants Bureau, New York). [9] Sedov L. I. (1993) Similarity and dimensional methods in mechanics (CRC Press).

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[10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]

[21] [22] [23] [24] [25] [26] [27] [28]

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Pocheau A. (2006) Eur. Phys. J. B 49, pp. 491-511. Domb C. (1996) The critical point (Taylor & Francis). Wilson K. G. (1983) Rev.Mod.Phys. 55, pp. 583. Borghi R. (1988) Prog.Energy Combust. Sci. 14, pp. 245. Kolmogorov A. N. (1941) C.R.Acad.Sci. URSS 30, pp. 301. Frisch U. Turbulence (1995) Cambridge University Press. Pocheau A. (1996) Europhys. Lett. 35, pp. 183. Dubrulle B. and Graner F. (1996) J. Phys.(France) 6, pp. 7976. G¨ ulder O. L. (1990) XXIII Symposium (International) on Combustion, The Combustion Institute, pp. 743. Boyer L. (1980) Comb. Flame 39, pp. 321. Floch A., Trinit´e M., Fisson M., Kageyama T., Kwon C. and Pocheau A. (1989) in Dynamics of Deﬂagrations and Reactive Systems Kuhl A.L. Eds, Progress in Astronautics and Aeronautics 131, pp. 378. Metgalchi M. and Keck J. C. (1982) Combust. Flame 48, pp. 191. Clavin P. and Williams F. A. (1979) J. Fluid Mech. 90, pp. 589. Yakhot V. (1988) Combust. Sci. Tech. 60, pp. 191. Peters N. (1986) XXI Combustion Symposium, The Combustion Institute, pp. 1231. Kerstein A. (1988) Combust. Sci. Tech. 60, pp. 441. Catrakis H. J. and Dimotakis P.E. (1996) J. Fluid. Mech. 317, pp. 369. Gouldin F. C. (1987) Combust. Flame 68, pp. 249. Mantzaras J., Felton P. G. and Bracco F. V. (1989) Combust. Flame 77, pp. 295.

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Transport bifurcations and relaxation

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Chapter 8

Transport Barrier Relaxations in Tokamak Edge Plasmas

P. Beyer LPIIM, Universit´e de Provence, 13397 Marseille cedex 20, France

8.1

Introduction

Transport barriers at the plasma edge are key elements of high conﬁnement modes (H–modes) in fusion devices. These barriers, characterized by a local steepening of density and temperature gradients, are strongly linked to shear ﬂows that reduce signiﬁcantly turbulent heat and particle transport. During a transition from low to high conﬁnement (L–H transition), an edge transport barrier builds up spontaneously [1–3]. A barrier can also be produced by externally driving an ExB shear ﬂow via edge biasing techniques [4–6]. In the most promising operational regime of future reactors, the edge transport barrier is not stable but relaxes quasi-periodically. During such fast relaxation events, turbulent transport through the barrier increases strongly and the pressure inside the barrier drops. Thereafter, the barrier builds up again on a slow, collisional time scale. The basic physical mechanism underlying these relaxation oscillations is not fully understood. In particular, there is no universal explanation why the plasma, instead of remaining in a state of marginal stability, oscillates close to stability limits. Currently, transport barrier relaxations are modelled by phenomenologically constructed dynamical equations for the amplitudes of relevant modes [7–10]. Here, we propose three dimensional (3D) ﬂuid turbulence simulations 255

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and subsequent one dimensional (1D) modelling to investigate non linear barrier dynamics [11]. In the simulations, a barrier is produced by an externally imposed ExB shear ﬂow and it is found to relax quasi-periodically in a range of ExB shear rates. This behavior persists even if the ExB ﬂow is frozen, i.e. turbulent ﬂow generation is suppressed. Hence, the mechanism at work departs from previously reported explanations based on turbulent shear ﬂow generation [12, 13]. The relaxation dynamics is found to be governed by the intermittent growth of a mode localized at the barrier center, characterized by low poloidal and toroidal wavenumbers. A one dimensional (1D) model for the dynamics of this mode is derived. An analytical study reveals that the eﬀect of the ExB shear ﬂow is diﬀerent from a shift of the linear instability threshold. In fact, the dynamics is found to be governed by a time delay for eﬀective velocity shear stabilization.

8.2

Model for resistive ballooning turbulence

Resistive ballooning mode (RBM) turbulence at the edge of a tokamak plasma is modelled by reduced resistive magneto–hydrodynamical (MHD) equations for the electrostatic potential φ and pressure p [14], n 0 mi µi⊥0 4 (∂t + uE · ∇) ∇2⊥ φ = ∇ j − Gp + ∇ φ, (8.1) 2 B0 B02 ⊥ (∂t + uE · ∇) p = Γp0 Gφ + χ0 ∇2 p + χ⊥0 ∇2⊥ p + S(r) ,

(8.2)

where the coeﬃcients are evaluated with reference values of the density n0 , the pressure p0 , the magnetic ﬁeld B0 , the perpendicular collisional ion viscosity µi⊥0 and the eﬀective parallel and perpendicular collisional heat diﬀusivities χ0 , χ⊥0 . The ion mass is designated by mi and the adiabatic index is Γ = 53 . Equation (9.14) corresponds to the charge balance in the drift approximation involving the divergences of the polarization current, the parallel current, and the diamagnetic current, and viscous eﬀects, respectively. Equation (9.15) corresponds to the energy balance where S(r) represents an energy source located at the plasma core. The compressibility 2 ×∇p and electric drift uE = B/B 2 ×∇φ of diamagnetic current jdia = B/B gives rise to curvature terms, B × ∇f ≡ Gf , with f = p, φ . (8.3) ∇· B2 In this MHD model, the diamagnetic velocity is neglected with respect to the ExB velocity, and the parallel current is evaluated using a simpliﬁed

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Fig. 8.1 Electrostatic potential corresponding to a linear ballooning mode in the poloidal plane ϕ = 0.

electrostatic Ohm’s law, η0 j = −∇ φ, where η0 is a reference value of the parallel resistivity. The magnetic ﬁeld is described by = Bϕ eˆϕ + r eˆθ B Rq(r) in toroidal co-ordinates (r, θ, ϕ), where R is the major radius and q(r) is the safety factor. Assuming a monotonically increasing safety factor q(r), the domain chosen here covers a region between q = 2 and q = 3 at the plasma edge. At the vicinity of a reference surface r0 corresponding to q = q0 = 2.5, a linear approximation of the inverse safety factor is used, 1 1 R0 = − (r − r0 ) . q q0 Ls r0 Here Ls is the magnetic shear length at the reference surface. Resistive ballooning modes are the eigenmodes of the system Eq. (9.14), Eq. (9.15), linearized with respect to the equilibrium state r 1 1 ¯ φ(r) = 0 , ∂r p¯(r) = − r S(r )dr = − Γtot (r) χ⊥ r rmin χ⊥ (8.4) p0 (r − a) for r ≥ rq=2 , → p¯(r) = − Lp where the source S(r) is radially located between rmin and rq=2 such that !r the total energy ﬂux Γtot = (1/r) rmin r S(r )dr is constant in the main computational domain between rq=2 and rq=3 . Here, Lp = χ⊥ p0 /Γtot is

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a characteristic value of the pressure gradient length and a is the outer radial boundary of the plasma. The equilibrium Eq. (8.4) is unstable if the pressure gradient −∂r p¯ is larger than a critical value κ0 . In this case, resistive ballooning modes φ˜ φˆn (r, θ) exp(inϕ + γt) , (r, θ, ϕ, t) = p˜ pˆn characterized by the toroidal wavenumber n, have a positive growth rate γ. Figure 8.1 shows the structure in the (r, θ) plane of such a mode for n = 18. The mode balloons at the low ﬁeld side θ ≈ 0 (to the right in Fig. 8.1), where the magnetic curvature B 1 B a sin θˆ eθ − cos θˆ ·∇ ≈ er + O B B R0 R0 is in the same direction as pressure gradient (∂r p¯) eˆr . The mode can be decomposed into a series of Fourier modes φˆmn φˆn (r, θ) = (r) exp imθ , pˆn pˆmn m where m is the poloidal mode number. Each Fourier component (φˆ mn , pˆmn) 2 (r) has a characteristic radial width ξbal with ξbal = mi n0 η0 L2s / τint B02 . In a strongly ballooned case, i.e. if (φˆn , pˆn )(r, θ) is strongly localized at the low ﬁeld side θ ≈ 0, the growth rate γ is close to the interchange growth 2 = R0 Lp / 2c2S0 , where cS0 is the reference sound rate 1/τint with τint speed with c2S0 = p0 / (n0 mi ). Note however, that in typical cases as the one shown in Fig. 8.1, the growth rate γ is considerably lower compared to 1/τint , as the mode is stabilized by its components on the high ﬁeld side θ ≈ π where the magnetic curvature is opposed to the pressure gradient. The system Eq. (9.14), Eq. (9.15) can be normalized using the characteristic time is τint and perpendicular length ξbal , t (r−r0 , r0 θ) R0 ϕ τint φ Lp p →t, → (x, y) , →z, →φ, →p. 2 τint ξbal Ls B0 ξbal ξbal p0 The normalized system takes the form # $ ∂t ∇2⊥ φ + φ, ∇2⊥ φ = −∇2 φ − Gp + ν∇4⊥ φ ,

(8.5)

∂t p + {φ, p} = δc Gφ + χ ∇2 p + χ⊥ ∇2⊥ p + S(r) . with the coeﬃcients Lp τint µi⊥0 δc = 2Γ , ν= 2 , R0 ξbal mi n0

χ =

τint χ0 , L2s

χ⊥ =

(8.6)

τint χ⊥0 . 2 ξbal

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U(r)

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0 rmin

rq=2

Fig. 8.2

rq=2.5

rq=3

rmax

0r

min

rq=2

rq=2.5

rq=3

rmax

Radial proﬁles of the poloidal ﬂow U (left) and the source S (right).

As the width of the radial domain considered here is small compared to the minor radius of the reference surface, i.e. rq=3 − rq=2 r0 , a slab geometry can be used in the numerical code leading to a simpliﬁcation of the poloidal and toroidal derivatives via r−1 ∂θ → r0−1 ∂θ and R−1 ∂ϕ → R0−1 ∂ϕ . The normalized operators then take the form {φ, · } = ∂x φ ∂y − ∂y φ ∂x , G = sin θ ∂x + cos θ ∂y , ζ Ls r0 , ∇2⊥ = ∂x2 + ∂y2 . ∇ = ∂z + ∂y with ζ = q R0 ξbal 8.3

Formation of a transport barrier

In the present model, a transport barrier is generated by externally imposing a locally sheared poloidal ExB ﬂow. A corresponding drive is added to the equation for the poloidal ﬂow, i.e. the magnetic ﬂux surface average . . .θϕ of Eq. (8.5), 1 1 ∂r r2 ˜ uθ u ˜r + ν∂r ∂r r¯ uθ − µ (¯ uθ − U ) , (8.7) 2 r r where u ¯θ = uθ θϕ is the ﬂow proﬁle and u ˜r,θ = ur,θ − u ¯r,θ are the ﬂuctuations of radial and poloidal velocity. The ﬁrst two terms on the right hand side of Eq. (8.7) correspond to the divergences of the Reynolds stress and the viscosity stress, respectively, and the last term has been added artiﬁcially to account for the friction with an external ﬂow U . The latter is chosen to be strongly sheared at the position r0 , U (r) = ωE ext d tanh [(r − r0 ) /d] + U0 , where ωE ext is the maximal shear, d is width of the shear layer (here d/ξbal = 13), and! the constant U0 is adapted such that the correspondr U dr vanishes at the boundaries of the radial domain. ing potential Φ = The radial proﬁle of the ﬂow U is plotted in Fig. 8.2 (left). In the absence of external drive (i.e. µ = 0), a poloidal ﬂow is generated by turbulent ﬂuctuations via Reynolds stress. This mechanism generates both, a mean component (ﬁnite time average) and a ﬂuctuating (in time) ∂t u ¯θ = −

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q=2.5 radius

q=3

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q=2.5 radius

q=3

Fig. 8.3 Time average proﬁles of the velocity shear (left) and root mean square ﬂuctuations of these proﬁles (right) for diﬀerent values of the friction coupling constant µ, and ωE ext = 2, Γtot = 8.

component of the poloidal ﬂow. The latter corresponds to the so called zonal ﬂows [15, 16]. With increasing µ, the friction with the imposed ﬂow gets important, the time averaged proﬁle becomes dominated by the imposed one, and the deviations from that proﬁle get smaller (Fig. 8.3). In the limit µ → ∞, the poloidal ﬂow u ¯θ becomes identical to the external ﬂow U (frozen ﬂow case). This limit is simulated in the numerical code by using 2 a ﬁnite value µ much larger then ν (ξbal /d) (here µ = 2) and suppressing the Reynolds stress term in Eq. (8.7). The localized ﬂow shear leads to a local reduction of turbulent transport [17–19]. A steepening of the pressure gradient then follows from the energy ﬂux conservation. The latter is a consequence of the magnetic ﬂux surface average of Eq. (8.6), 1 ∂t p¯ = − ∂r r (Γturb + Γcoll ) + S , (8.8) r where p¯ = pθϕ corresponds to the pressure proﬁle, Γturb = ˜ pu ˜r θϕ and Γcoll = −χ⊥ ∂r p¯ correspond to the turbulent and collisional radial energy ﬂuxes, respectively. Here, p˜ = p − p¯ are the pressure ﬂuctuations. The source S is located in an artiﬁcial (“buﬀer”) zone outside the main computational domain between rq=2 and rq=3 (see Fig. 8.2, right). It determines the total energy ﬂux Γtot across a magnetic surface in this radial domain. In time average, according to Eq. (8.8), a local reduction of turbulent ﬂux Γturb by an ExB shear ﬂow leads to an increase of collisional ﬂux Γcoll , i.e. a steepening of the pressure gradient |∂r p¯|. Figure 8.4 shows radial proﬁles of the time averaged turbulent ﬂux and pressure for diﬀerent values of the maximal shear ωE ext .

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turbulent flux

30 400

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200

q=2.5

0 q=2

q=3

radius

q=2.5

q=3

Fig. 8.4 Time averaged proﬁles of turbulent ﬂux and pressure for diﬀerent values of the maximal shear and Γtot = 36, µ → ∞.

8.4

Appearance of relaxation oscillations

In typical RBM turbulence simulations with a transport barrier generated as described above, the barrier is not stationary but relaxes quasiperiodically. In Fig. 8.5 (left), the dynamics of the pressure gradient, the turbulent ﬂux, and the poloidal ﬂow shear are shown, all values are taken at the barrier center (r = r0 ). The evolution of the pressure gradient is characterized by phases of a slow increase quasi periodically interrupted by rapid crashes. The latter correspond to relaxations of the barrier and are associated with large peaks of the turbulent ﬂux at the barrier center as well as ﬂuctuations of the velocity shear at the barrier position. These relaxation oscillations are found to persist even if the poloidal 1 0.5

normalized pressure gradient

1

normalized pressure gradient

0.5

0 0 normalized turbulent flux normalized turbulent flux 15 15 10 10 5 5 0 0 normalized velocity shear fluctuations normalized velocity shear fluctuations 0 0 −0.2 −0.2 −0.4 −0.4 −0.6 −0.6 3000 5000 7000 9000 3000 5000 7000 9000 time time

Fig. 8.5 Time evolution of pressure gradient normalized to the diﬀusive value, ∂r p¯/ (−Γtot /χ⊥ ), turbulent ﬂux normalized to the total incoming ﬂux ˜ pu ˜ r θϕ /Γtot , and relative deviations of the poloidal ﬂow shear from the imposed value ¯θ − ωE ext ) /ωE ext , at the center of the barrier. Here, ωE ext = 8, Γtot = 36, and (∂r u µ = 0.02 (left) respectively µ → ∞ (right).

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3 2 1 0

(a)

normalized turbulent flux

20 (b) 10 0 3

6

9 time / 103

12

15

Fig. 8.6 Dynamics of the turbulent ﬂux at the barrier center for (a) Γtot = 9, ωE ext = 2 and (b) Γtot = 36, ωE ext = 12.

ﬂow proﬁle is frozen. Figure 8.5 (right) shows the corresponding results from a simulation with the same parameters as in Fig. 8.5a except that the friction coeﬃcient is set to µ → ∞. In this case, the velocity shear proﬁle is constant in time but intermittent ﬂux peaks with relaxation of the barrier do appear. When varying the maximal ExB shear ωE ext and the total energy ﬂux Γtot , two opposite trends concerning the behavior of the oscillation frequency are observed. The frequency increases with Γtot (for ﬁxed ωE ext ) and decreases with ωE ext (for ﬁxed Γtot ). However, in tokamaks, the ExB ﬂow shear increases with the heating power [20]. It is found here that if this increase is faster than linear, the actual relaxation frequency decreases with power. This is illustrated in Fig. 8.6 where the dynamics of the turbulent ﬂux is shown for two cases. With respect to case (a), the heating power is four times larger and the ExB shear is six times higher in case (b). Obviously, in the latter case, the relaxation frequency is lower. For the relaxation oscillations of transport barriers observed here, several possible mechanisms can be excluded. (1) As mentioned above, the relaxation mechanism at work here departs from explanations based on turbulent shear ﬂow generation. In fact, the behavior persists even if the ExB ﬂow is frozen, i.e. turbulent ﬂow generation is suppressed. (2) Resistive ballooning modes are global modes with a large radial extend. In general, in the simulations presented here, these modes are linearly unstable due to the components localized in the regions far from the

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mode amplitude 5−2 40 theory

mode amplitude

40 30 20 10 0

5−2 4−2 6−2

40 30 20 10 0 480

10−4 9−4 11−4

263

30 20

500

10

520

time

540

560

0 0

10

time

20

Fig. 8.7 (left) Time evolution of amplitudes (of the potential) of diﬀerent (m, n) modes during a ﬂux peak. (right) Time evolution of a perturbation of the (m, n) = (5, 2) mode in a restarted simulation (solid line) compared to the curve obtained from Eq. (8.12) (dashed line), as explained in the text.

barrier. As will be shown in the following, the components of the global modes localized at the barrier center are almost vanishing during quiescent phases but get rapidly excited during a relaxation event. This could be an eﬀect of the toroidal coupling of the Fourier components of a global mode, leading to a pumping of the part localized at the barrier center by the unstable components localized at the shoulders. However, as can shown in Fig. 8.7 (left), no precursor on the directly coupled neighbors at the barrier shoulder, (m, n) = (4, 2) and (6, 2), is observed prior to the growth of the central (m, n) = (5, 2) mode. This is also true for the neighbors (m, n) = (8, 4) and (12, 4) of the next order central mode (m, n) = (10, 4). (3) The growth of a perturbation at the barrier center could in principle also be triggered by large scale transport events [21, 22, 18], as these high ﬂux perturbations propagate radially over large distances. However, the amplitude of such bursts decreases strongly as they approach the barrier [18]. On the contrary, one observes that a relaxation event leads to a ﬂattening of the pressure gradient at the barrier center and an isolation of two regions of steep gradient on both sides. These steep gradients are then propagating radially away from the barrier center (Fig. 8.8). (4) The strong velocity shear at the barrier center can in principle generate

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a Kelvin–Helmholtz instability. In contrast to the resistive ballooning instability, this type of instability is independent of the magnetic curvature. However, if the magnetic curvature is suppressed artiﬁcially in our simulation, all turbulent ﬂuctuations rapidly die out, indicating that the system is Kelvin–Helmholtz stable. The barrier relaxation oscillations observed here are governed by the intermittent growth of a low poloidal (m) and toroidal (n) wavenumber mode localized at the barrier center. As illustrated in Fig. 3.18, a relaxation event is dominated by a (m, n) = (5, 2) mode which is the lowest order (m, n) mode localized at the barrier position. This is a rather surprising result because one expects ﬂuctuations localized at this position to be strongly stabilized by the velocity shear. However, as will be shown in the following by means of a reduced model, a transitory growth of a perturbation is possible due to the existence of a time delay for velocity shear stabilization which is an intrinsically nonlinear eﬀect. 8.5

Low dimensional model and non-linear short-term dynamics of shear ﬂow stabilization

A 1D model reproducing the dynamics of barrier relaxations is constructed by decomposing the pressure into the proﬁle p¯(r, t) and a perturbation

Fig. 8.8 Mean pressure gradient versus radius and time during a relaxation (top) and in a quiescent phase (middle), and its time average during both phases (bottom).

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p˜(r, t)eimθ−inϕ localized at the barrier center, i.e. m/n = 2.5. In order to obtain a simpliﬁed model involving only two ﬁelds, potential ﬂuctuations

are 2 ˜ supposed to follow pressure ﬂuctuations via the relation φ = ikθ / γ0 k⊥ p˜. The latter is obtained from balancing the two terms governing the ballooning instability in Eq. (9.14), i.e. the time derivative and the curvature term. Here, γ0 is the linear growth rate in the case of a constant pressure gradient, ∂r p¯ = −1, and in the absence of mean velocity, magnetic shear and dissipation. kθ = m/r0 and k⊥ are respectively, the poloidal and perpendicular wavevectors of the corresponding mode, and a cylindrical curvature G → r−1 ∂θ has been assumed. Supposing further a uniformly sheared poloidal ﬂow, u ¯θ = ωE (r − r0 ), projections of the pressure equation Eq. (9.15) on the proﬁle and the perturbation, respectively, lead to 2

p| + χ⊥ ∂x2 p¯ + S , ∂t p¯ = −2γ0 ∂x |˜ ∂t p˜ = γ0 (−∂x p¯ − κ0 ) p˜ −

iωE x˜ p−

(8.9) χ x2 p˜ + χ⊥ ∂x2 p˜ . and χ = kθ2 χ . The

(8.10)

with x = r − r0 , κ0 = kθ2 χ⊥ /γ0 , ωE = kθ ωE , system Eq. (8.9), Eq. (8.10) reproduces barrier relaxation oscillations for ﬁnite (Fig. 8.9). Note that for a ﬁxed pressure gradient ∂x p¯ = −κ, values of ωE Eq. (8.10) is linear and the growth rate for the most unstable radial mode is given by % ω 2 (8.11) γ = γ0 (κ − κ0 ) − χ⊥ χ − E . 4χ

This implies that for large enough ﬂow shear ωE , linear modes are completely stabilized. However, the dynamics of these modes describes only the long-term behavior of the system, i.e. an asymptotic decay of the ﬂuctuations for suﬃciently large ﬂow shear. For the short-term dynamics, a description in terms of linear modes is not appropriate. Indeed, if the advection with the sheared ﬂow in Eq. (8.10) is replaced by a shift of the linear instability threshold, the modiﬁed system does always evolve to a stable ﬁxed point. In order to obtain relaxation oscillations with such a model, one has to introduce phenomenologically further elements such as a Heaviside function multiplying the instability term [10]. In fact, the short-term dynamics in the model Eq. (8.9), Eq. (8.10) is better described by the evolution of an initial pulse p˜(x, t = 0) = pˆδ(x), inﬁnitely localized at x = 0, that can be calculated analytically from Eq. (8.10) for a given pressure gradient −∂x p¯ = κ and when neglecting the χ term. The solution takes the form

pˆ t3 x x2 iωE p˜ = √ exp − , (8.12) − t + γ0 t − 3 4χ⊥ t 2 3τD 4πχ⊥ t

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normalized pressure gradient 1 0.5 0 normalized turbulent flux 20 10 0 3

6

9 3 time / 10

12

15

Fig. 8.9 Time evolution of pressure gradient [∂r p¯/ (−Γtot /χ⊥ )] and turbulent ﬂux p|2 /Γtot ) at the barrier center obtained from the 1D model Eq. (8.9), Eq. (8.10). (2γ0 |˜ Parameters are the same as in the 3D model and kθ = 0.05, γ0 = 0.8, ωE = 1.1, Γtot = 1.4.

2 −1/3 where γ0 = γ0 (κ − κ0 ) and τD = 14 χ⊥ ωE . Note that for ωE = γ0 = 0, the usual solution of the diﬀusion equation is recovered. The expression Eq. (8.12) describes an initial transient growth of the perturbation for

3 1/2 γ0−1 < t < τD γ0 before the cubic term in the exponential dominates the linear term, leading to a stabilization. The characteristic time τD for the transient growth is large for small values of the perpendicular diﬀusivity χ⊥ (close to the collisional value at the barrier center) and low poloidal wave numbers m. From this analysis, the mechanism for relaxation oscillations is as follows. During a quiescent phase, the pressure gradient increases slowly on a collisional time scale. When it crosses the linear instability threshold, ﬂuctuations set in and the associated anomalous ﬂux keeps the pressure gradient close to the threshold, which tends to saturate the ﬂuctuations. However, the latter trigger the nonlinear mechanism described above, leading to a growth of the mode during a characteristic time of the order of τD , even though the pressure gradient drops well below the linear stability threshold due to the large anomalous ﬂux. For the mode shown in Fig. 3.18 (right) we have τD = 8.7 in normalized units (with ωE = 5.5) which agrees well with the temporal width of a ﬂux peak. The transient growth of the (m, n) = (5, 2) mode is reproduced (Fig. 8.7 (right)) when restarting the simulation, initializing the (5, 2) mode with a perturbation and all other (m, n) modes with noise, and keeping the pressure and velocity proﬁles obtained from the simulation shown in Fig. 8.7 (left) just before a relaxation.

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Moreover, this short term dynamics of the central mode is well described by Eq. (8.12), as can be seen from the dashed curve in Fig. 8.7 (right) that 3 corresponds to p˜ = p1 exp γ0 t − t3 /(3τD ) with p1 = 1.2, γ0 = 0.42, and τD = 10. Note that well pronounced relaxation oscillations are found for values such that the linear growth rate Eq. (8.11) for the diﬀusive pressure of ωE gradient κdiff = Γtot /χ⊥ is only slightly above the instability threshold. In this case, the recovery phase of the proﬁle between to relaxations is long compared to τD . In agreement with this picture, we ﬁnd that a slight increase of the value of ωE leads to a completely stable situation without any ﬂuctuations. This is also consistent with the fact that the frequency of relaxation oscillations decreases with the velocity shear ωE , as observed in the 3D simulations.

8.6

Conclusions

In conclusion, 3D turbulence simulations with imposed ExB shear ﬂow show the formation of a transport barrier and the appearance of relaxation oscillations. The analysis of these simulations reveals that this dynamics is governed by an eﬀective time delay in the stabilization by the shear ﬂow. This is conﬁrmed by a reduced 1D model. The ExB ﬂow shear in tokamaks increases with heating power. It is found here that if this increase is faster than linear, the relaxation frequency decreases with power. These properties, onset of a transport barrier, relaxation oscillations associated to resistive ballooning modes, and the oscillation frequency that decreases with power, are reminiscent of so-called type III edge localized mode (ELM) dynamics in tokamak edge transport barriers [23].

References [1] [2] [3] [4]

F. Wagner, G. Becker, K. Behringer, et al., Phys. Rev. Lett. 49, 1408 (1982). R. J. Groebner, Phys. Fluids B 5, 2343 (1993). K. H. Burell, Phys. Plasmas 4, 1499 (1997). R. J. Taylor, M. L. Brown, B. D. Fried, et al., Phys. Rev. Lett. 63, 2365 (1989). [5] R.R. Weynants, G. van Oost, G. Bertschinger, et al., Nucl. Fusion 32, 837 (1992). [6] J. Cornelis, R. Sporken, G. van Oost, et al., Nucl. Fusion 34, 171 (1994). [7] S.-I. Itoh, K. Itoh, A. Fukuyama et al., Phys. Rev. Lett. 67, 2485 (1991).

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[8] V. B. Lebedev, P. H. Diamond, I. Gruzinova et al., Phys. Plasmas 2, 3345 (1995). [9] S. C. Cowley, H. Wilson, O. Hurricane et al., Plasma Phys. Controll. Fusion 45, A31 (2003). [10] J.-S. L¨ onnroth, V. Parail, C. Figarella et al., Plasma Phys. Controll. Fusion 46, A249 (2004). [11] P. Beyer, S. Benkadda, G. Fuhr-Chaudier et al., Phys. Rev. Lett. 94, 105001 (2005). [12] H. Sugama, W. Horton, Plasma Phys. Controll. Fusion 37, 345 (1995). [13] R. J. Colchin, M. J. Schaﬀer, B. A. Carreras et al., Phys. Rev. Lett. 88, 255002 (2002). [14] P. Beyer, X. Garbet, S. Benkadda et al., Plasma Phys. Controll. Fusion 44, 2167 (2002). [15] Z. Lin, T. S. Hahm, W. W. Lee et al., Phys. Rev. Lett. 83, 3645 (1999). [16] P. H. Diamond et al., Plasma Phys. Control. Nucl. Fusion Res. (IAEA, Vienna, 1998). [17] T. S. Hahm, K. H. Burell, Phys. Plasmas 2, 1648 (1995). [18] P. Beyer, S. Benkadda, X. Garbet, et al., Phys. Rev. Lett. 85 4892 (2000). [19] C. F. Figarella, S. Benkadda, P. Beyer et al., Phys. Rev. Lett. 90, 015002 (2003). [20] B. A. Carreras, D. Newman, P. H. Diamond et al., Phys. Plasmas 1, 4014. [21] Y. Sarazin, Ph. Ghendrih, Phys. Plasmas 5, 4214 (1998). [22] X. Garbet, Y. Sarazin, P. Beyer, et al., Nucl. Fusion 39, 2063 (1999). [23] J. Connor, Plasma Phys. Controll. Fusion 40, 531 (1998).

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Chapter 9

Dynamics of Edge Localized Modes

X. Garbet, P. Ghendrih, Y. Sarazin, P. Beyer∗ , G. Fuhr-Chaudier∗ and S. Benkadda∗ Association Euratom-CEA, CEA/DSM/DRFC Cadarache 13108 St Paul-lez-Durance, France ∗ LPIIM, Universit´e de Provence, 13397 Marseille cedex 20, France. This overview focusses on dynamical models that describe the time evolution of Edge Localized Modes (ELM’s). It is proposed here to describe the dynamics of Edge Localized Modes by using a set of reduced models coupling transports equation to evolution equations for the instability amplitude. It appears that the simplest model does not produce relaxation oscillations. Two large classes of models have been proposed previously to overcome this diﬃculty. The ﬁrst one is based on a quasi-linear approach, which retains the main features of MHD modes. The second line of research addresses models that describe both the transition from low conﬁnement to high conﬁnement plasmas (L-H transition) and the onset of Edge Localized Modes. This category covers models that involve a subcritical bifurcation in the evolution equation of one of the ﬁelds, theories based on mean shear ﬂow stabilization, and mechanisms that invoke shear ﬂow turbulent generation.

9.1

Introduction

Edge Localized Modes (ELM’s) are relaxation oscillations that take place in tokamak plasmas [1]. They usually appear when the plasma is in the “H-mode”, where “H” stands for “high conﬁnement” [2, 3]. In H-mode plasmas, turbulence is quenched within a layer at the plasma edge. As a 269

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result, a transport barrier is formed where turbulent transport is reduced and gradients become larger. Hence edge localized modes and transport barriers are intimately related. The energy loss induced by these relaxation oscillations is moderate and acceptable for fusion. However the relaxation events also lead to large heat excursions on plasma facing components, which may damage these elements. Hence it is quite important to predict the characteristics of edge localized modes in a fusion reactor, in particular their frequency, relaxation time and energy release. The underlying instabilities are now well identiﬁed for the most common type of ELM’s and belong to the class of MHD modes driven by pressure and/or current gradients [4–6] (see Ref. [7] for an overview). This identiﬁcation has been done by comparing experimental gradients before the instability onset to the values predicted by ideal MHD. This comparison has become a routine exercise and can be considered as reliable, in spite of large uncertainties on measured proﬁles in this region of the plasma [8–10]. Several overviews are available, which describe the principal experimental observations and the comparison to MHD stability diagrams [9, 11, 12]. While linear MHD stability is rather well mastered, the mechanisms which dictate the dynamics of edge localized modes are not well understood. This is unfortunate since an accurate description of the dynamics is needed to predict the associated energy loss and characterize the heat transient on plasma facing components. No reliable quantitative prediction is available yet for the frequency and amplitude of these relaxation oscillations, in spite of encouraging results [13]. Also the dependence of the frequency on heating power is not well understood. Indeed experimentalists classify edge localized modes as “type I” or “type III” depending whether their frequency increase or decrease with heating power [14]. It is rather easy to understand why the frequency increases with the heat source (type I ELM’s), whereas the behavior of type III ELM’s is a challenge for theory. The aim of this overview is to summarize the status of various existing models to describe the dynamics of edge localized modes. Simple analytical tools will be used to introduce the various underlying mechanisms. The main drawback of this approach is to oversimplify a phenomenon whose nature is intrinsically non linear and three dimensional. On the other hand it provides a useful tool to assess this fast evolving ﬁeld of research. In particular it will be seen that the simplest model coupling the evolution of an instability to a transport equation does not work. In the most com-

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mon approach, based on quasi-linear MHD, this diﬃculty is overcome by using the properties of MHD stability. Indeed the frequency of an ideal MHD mode moves from the real to the imaginary axis when the instability threshold is crossed, and the eigenmodes are spatially extended in a tokamak. These two features lead to relaxation oscillations [9, 10]. In this picture edge localized modes appear in H-mode plasmas due to large gradients in the transport barrier. This steepening is usually obtained using heuristic transport models. However one would ideally look for a uniﬁed picture describing both the transition from “Low” conﬁnement (L-mode) to “High” conﬁnement (H-mode) plasmas (called L-H transition), and the onset of ELM’s. This is why a considerable attention has been given to a large class of models, which involve a subcritical bifurcation (“S-curve”) in the evolution equation of one of the ﬁelds [15–17]. This type of models is able to cover both the physics of the L-H transition and the dynamics of edge localized modes. In fact subcriticality is not mandatory. Transport barriers produced with a frozen shear ﬂow do exhibit relaxation oscillations without invoking a subcritical bifurcation. This result has been obtained with 3D numerical simulations of resistive ballooning modes and also with reduced models. Also the class of “S-curve models” contains models based on the concept of “explosive” growth of instability [19], which is a possible route for the non linear evolution of pressure driven MHD modes [20]. Finally there exists also models which exhibit oscillations driven by turbulent ﬂow generation. Whether these are edge localized modes is still subject to discussion since oscillations are not the same as cyclic relaxation events. The remainder of this article is organized as follows. An introduction to edge localized modes is given in Section 2. Section 3 presents general considerations on dynamical models and the status of models based on quasi-linear MHD. Section 4 deals with models that describe both the L-H transition and oscillation relaxations. A conclusion follows.

9.2 9.2.1

An introduction to edge localized modes Tokamak geometry

In a tokamak, the magnetic ﬁeld has toroidal and poloidal components. Field lines are helical and are winded on tori, called magnetic surfaces (see Fig. 9.1. Magnetic surfaces are nested around a magnetic axis. Each torus is labelled by an eﬀective radius r, deﬁned for instance as the square root

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of the ﬂux of toroidal magnetic ﬁeld normalized to a reference magnetic ﬁeld. The set of coordinates is completed by toroidal and poloidal angles ϕ and θ. The poloidal angle can be chosen such that the winding number q of the ﬁeld lines, called safety factor, is constant on each magnetic surface, i.e. depends on r only. At the very edge of the plasma, ﬁeld lines are intercepted by plasma facing components, i.e. are open. The magnetic surface separating the region where ﬁeld lines are closed from the region where they are open is called separatrix. The description of transport in tokamaks usually makes use of time and spatial scale separation. Mean ﬁelds are density, velocity and temperature averaged over fast time and spatial scales (in practice over poloidal and toroidal angles and fast radial scales). This averaging procedure allows for writing transport equations in the radial direction (1D mean ﬁeld theory). Relaxation events due to edge localized modes are fast events, so that the hypothesis of scale separation may be questionable. Examples will be shown where this assumption is not done.

Fig. 9.1

9.2.2

Set of nested magnetic surfaces in the tokamak JET.

A brief description of edge localized modes

Edge localized modes appear in H-mode plasmas. In these plasmas, turbulence is quenched in a layer at the edge, in the vicinity of the separatrix.

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Transport is lower in this region and therefore gradients of density and temperature become larger at given particle and heat ﬂuxes. The region with steep proﬁles is called pedestal (see Fig. 3.19). This situation is obviously favorable for fusion. The formation of a pedestal usually leads to a doubling of the energy content. This is why the achievement of H-mode plasmas is the standard scenario for future fusion reactors. However, the onset of edge localized modes brings limitations that remain to be fully assessed. A relaxation event starts with a rapid growth, of the order of 100µs, followed by a crash, whose duration is also of the order or 100µs. This crash leads to a decrease of the gradients in the pedestal, and a concomitant release of energy. The energy release corresponds to a small fraction of the total energy content. After the crash, a slow recovery phase takes place. Its duration is a diﬀusion time (pressure and/or current). Once the conditions for the onset of an instability are met, the process starts again. The typical frequency of these relaxation oscillations is a few 10Hz. Edge localized modes are usually classiﬁed depending on the behavior of the frequency fELM with heating power P . Type III ELM’s correspond to dfELM /dP < 0, whereas Type I ELM’s are characterized by dfELM /dP > 0. The power loss, equal to the frequency times the energy release, is moderate (typically it results in a less than 10 per cent reduction of the total energy conﬁnement time). However, the energy is expelled beyond the separatrix, then propagates along open ﬁeld lines, and induce a power excursion on plasma facing components. This transient may exceed the technology limit of these components for the most violent events. This is why the control of edge localized modes is a key issue for fusion. 9.2.3

MHD stability

An important question is the nature of the instability that leads to the relaxation event. When a pedestal develops in the H-mode, the local increase of pressure gradient pushes the plasma closer to the MHD stability limit. Moreover, the pressure gradient also changes the local density of current. This is due to an oﬀ-diagonal term in the generalized Ohm’s law that is proportional to the pressure gradient (similar to a thermo-electric eﬀect). This current is called “bootstrap” current. A local alignment of the pressure gradient with the magnetic ﬁeld curvature drives interchange modes if the pressure gradient is large enough. In a tokamak, the magnetic ﬁeld decreases with the major radius. Hence the plasma is prone to interchange instabilities on the low ﬁeld side of the torus. On the other hand, the high

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ﬁeld side is locally stable, while both regions are connected by helical ﬁeld lines. The complete MHD stability calculation is quite complex and will not be reproduced here. It is found that modes are unstable above a critical value of the parameter α = −q 2 R dβ dr , where β is the ratio of the plasma pressure and magnetic pressure and R is the major radius of the magnetic axis [22]. and corresponds The threshold depends on the magnetic shear s = rdq qdr to high toroidal wave numbers, which are the most unstable. A stability diagram is usually shown for inﬁnite toroidal wave numbers in the space (α, s). A simple criterion is α = 0.7s in a simpliﬁed large aspect ratio geometry [22]. However, the parameter α also controls the displacement of magnetic surfaces. This shift, called Shafranov shift, is stabilizing. As a consequence, ballooning modes become stable again if α is high enough. This is called second stability (see for instance [23]). First and second stable domains communicate in the (α, s) space due to geometry eﬀects (elongation, triangularity,...) so that a continuous transition from 1st to 2nd regions of stability is in principle possible. Eigenmodes are localized on the low ﬁeld side of the tokamaks, and are therefore called ballooning modes. Each unstable mode is a global mode whose radial extent is intermediate between macroscopic and microscopic scales (see Fig. 3.20). Hence a global mode aﬀects a signiﬁcant portion of the plasma once it is excited. This property will play an important role in the following. A large current density close to the separatrix leads to another instability called external kink mode, or peeling mode depending on the wave number. The most unstable external kink modes are characterized by low wave numbers, whereas peeling mode are characterized by larger wave numbers. This branch is characterized by a threshold which is expressed in terms of a critical current density at the separatrix ja . However both pressure and current driven modes couple at intermediate wave numbers to give a spectrum of unstable modes, called peeling/ballooning modes. The resulting stability diagram is expressed in the set of coordinates (α, s) or equivalently (α, ja ). An example is shown on Fig. 9.2. Comparing the measured value of α in the plasma edge to the MHD threshold indicates that the plasma is close to the stability limit just before the onset of type I ELM’s. This was found to be the case on DIII-D, ASDEX-Upgrade and JET [6, 8, 29]. The nature of type III ELM’s is still subject to discussion. On ASDEX-Upgrade, it is found that type III ELM’s appear at edge temperatures which are lower than for type I ELM’s [12]

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Fig. 9.2 Top panel: stability of peeling/balloning modes. The vertical coordinate is the current density in the pedestal and the horizontal coordinate is the pressure gradient in the pedestal. Low panel: schematic cycles associated to ELM’s for peeling-ballooning modes [6].

3

(cf. Figure 3.21). Since the resistivity behaves with temperature as T − 2 , this observation suggests that resistive MHD is at play [7]. In particular, resistive ballooning modes appear as a natural candidate for triggering the event [5]. On the other hand, plasma parameters are found to be close to the ﬁrst stability limit of ideal ballooning modes on JET, although resistive ballooning modes cannot be fully excluded [8]. On DIII-D, type III ELM’s are believed to be peeling modes at low density whereas at high density their behavior is better described in terms of resistive ballooning modes [6]. This short summary of experimental results explains why most models based on stability analysis relate type I ELM’s to ideal MHD instabilities, and type III ELM’s to resistive ballooning modes, although other instabilities cannot be excluded.

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9.3

Dynamical models of edge localized modes

9.3.1

Minimal model

A dynamical model for edge localized modes usually contains two ingredients: • a transport equation characterized by a diﬀusion coeﬃcient that increases with the amplitude of the mode • an evolution equation of the ELM amplitude, which grows above an instability threshold. A minimal set of equations that satisfy these constraints is: ∂t peq = ∂x D0 + Dm |ξ|2 ∂x peq + S ∂t ξ = γ0 (∂x peq − κc ) ξ

(9.1) (9.2)

where x = 1 − r/a is the distance to the separatrix r = a, and the ﬂux averaged pressure peq is normalized to some reference value p0 . The ﬁrst equation Eq. (9.1) is a heat equation where S is the heat source. The diﬀusion coeﬃcient D0 corresponds to collisions and Dm |ξ|2 is the contribution of the underlying instability. An equation over the pressure gradient ∂x peq is easily obtained after a derivation with respect to x. It is assumed that the transport associated to the edge localized mode is diﬀusive. This assumption is justiﬁed if the relaxation time is larger than a correlation time. An obvious limitation of this approach is the description of an electromagnetic and 3D MHD mode by a single scalar amplitude ξ. It must be noted at this stage that transport may be produced by two diﬀerent processes, namely E × B drift convection, or magnetic ﬂuttering. The equation Eq. (9.2) describes the evolution of the mode amplitude. The threshold is a critical value of the pressure gradient κc and γ0 is a growth rate. Obviously the equation on the amplitude ξ can be modiﬁed to account for various damping terms. For instance a diﬀusion term D∂xx ξ can be added, or a non linear saturation term −µ |ξ|2 ξ , or a damping term of the form −ν (ξ − ξbg ) , where ξbg represents some level of excitation due to a background turbulence. The latter eﬀect can also be reproduced with a noisy source. At this stage, the mode amplitude is real, but it is a complex quantity when shear ﬂow is added (see Sec. 9.4). A 0D version of this set of equations can be obtained by using the following recipe. Assuming a constant heat ﬂux and taking the radial derivative

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of the heat equation (Eq. (9.1)) yields the equation ∂t ∂x peq = ∂xx D0 + Dm |ξ|2 ∂x peq − ΦT where ΦT is the heat ﬂux. If the radial shape of the pressure gradient ∂x peq is prescribed, the heat equation can be written as an equation over a scalar Πeq that characterizes the pressure gradient. After projecting the heat equation onto the radial shape of the pressure gradient, the system (9.1),(9.2) becomes ∂t Πeq = S0 − ν0 + νm |ξ|2 Πeq (9.3) ∂t ξ = γ0 (Πeq − κc ) ξ The set of equations (9.3) has a ﬁxed point with non zero mode amplitude Π0 = κc and |ξ0 |2 = [S0 /κc − ν0 ] /νm when the heat source satisﬁes the condition S0 > ν0 κc . It turns out that this ﬁxed point is stable. This property can be veriﬁed by perturbing Eq. (9.3) around the ﬁxed point. This equation is of the form dt X = M · X where the vector X is deﬁned as = Π − Π0 Π X= ξ = ξ − ξ0 and the matrix M is :

M=

−S0 /κc −2νm κc ξ0 γ0 ξ0 0

The two eigenfrequencies are 1/2 2 −iω± = −S0 /2κc ± (S0 /2κc ) − 2νm κc γ0 ξ02 It is found that all solutions are decaying in time. These decaying solutions exhibit oscillations in a range of heat source S0 . For large values of the growth rate γ0 , i.e. γ0 >> ν0 /κc , this range is given by ν0 κc < S0 < 8κ2c γ0 . However these are not steady relaxation oscillations, and cannot be considered as edge localized modes. This result can be extended to the 1D version of this model Eqs.(9.1) and (9.2) for a heat source that is spatially localized (delta function). Numerical calculation show that this is still true for more general shapes of the heat source and a large choice of boundary conditions. Thus a transport equation coupled to an evolution equation of the mode amplitude does not generically exhibit relaxation oscillations.

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Quasi-linear MHD model

In fact the system Eqs.(9.1),(9.2) does not represent well the properties of an ideal MHD mode. The frequency of an ideal MHD mode satisﬁes the relation δW ω2 = 1 ! (9.4) 2 d3 xρ |ξ| 2 where δW is the MHD energy functional, ρ is the mass density and ξ is the MHD displacement. Hence the frequency is a pure real or a pure imaginary number. This disagrees with the ELM evolution equation Eq. (9.2), which corresponds to a growth rate that changes sign when the stability threshold is crossed. A ﬁrst way to cure this problem is to use a set of equations with more ﬁelds representing the instability so that the eigenfrequency is of the form Eq. (9.4). An well-known example is based on an interchange-like instability, i.e. ∂t peq = ∂x [D0 ∂x peq + 2pE] + S ∂t p = (∂x peq − κc ) E + D∇2⊥ p ∂t ∇2⊥ E = −gky2 p + D∇4⊥ E

(9.5)

where p and E are perturbed pressure and electric ﬁeld with an appropriate normalization, ky is the poloidal wave number of the most unstable mode, and ∇2⊥ = ∂xx − ky2 is the perpendicular Laplacian. This system was studied by Herring to study thermoconvection with gravity (Rayleigh-B´enard convection) [25]. In the original version of the Herring model, there is no explicit threshold κc . However the inclusion of a ﬁnite threshold is equivalent to shift the heat ﬂux. When the diﬀusion D is zero, the dispersion 2 g (∂x peq − κc ), i.e. the frequency moves relation is of the form ω 2 = −ky2 /k⊥ from the real to the imaginary axis at the threshold. The 0D version of the Herring model is ∂t Πeq = S0 − D0 Πeq + 2pE ∂t p = (Πeq − κc ) E − νp ∂t E = gp − νE

(9.6)

where p and E are now scalars with appropriate normalization. The system Eq. (9.6) is nothing but the Lorenz system [26]. It is well known that it exhibits an oscillatory behavior above a threshold (see Figure 9.3), and transition to chaos further on. On the other hand, the Herring model does not show oscillating solutions since its single ﬁxed point is stable. This calls for several remarks. First, a relation dispersion of the form Eq. (9.4) does not guarantee the onset of oscillations. Second, even if oscillations appear

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Fig. 9.3 Pressure gradient, perturbed pressure and electric ﬁeld versus time for the model described by Eq. (9.6).

in the 0D version of a model, they may not show up in the 1D formulation of the same model. The system Eq. (9.6) can be improved to include more physics [28]. However it remains hard to recover the right dynamics for type I and type III ELM’s. Turning back to the question of edge localized modes, and to a reduced 2 ﬁeld formulation, the simplest model that produce oscillations is of the form ∂t peq = ∂x

D0 + Dm (x) |ξ|2 ∂x peq + S

∂t ξ = γ0 (∂x peq − κc ) H (∂x peq − κc ) ξ

(9.7)

The Heaviside function H introduces the jump of growth rate at the threshold. Moreover the diﬀusion coeﬃcient Dm (x) associated to an ELM has a ﬁnite radial extent. In the frame of the quasi-linear theory, this radial extent is the width of a global mode. If an MHD stability analysis is performed, the eigenmodes that are calculated can be used to prescribe the shape of Dm (x).

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Status of the modelling of ELM’s based on the MHD quasi-linear model

The model Eq. (9.7) has been used under various forms to model edge localized modes in a predictive way [9, 13]. In reference [13], the evolution equation is of the form ∂t ξ = γ0 (1 − αc /α) H (1 − αc /α) ξ − ν1 (ξ − ξbg )

(9.8)

1/2

where γ0 = cs / (RLp ) is the growth rate of an ideal ballooning mode far from threshold (Lp is the pressure gradient length and cs the sound speed). This model exhibits the properties of type I ELM’s, in particular the frequency increases with the additional power (cf. Figure 9.4).

Fig. 9.4

Flux versus time for increasing power using a quasi-linear MHD model.

This model has been recently updated to cover the physics of peeling modes. This was done by adding an equation, similar to Eq. (9.8), which describes the amplitude of a peeling mode. In this case the threshold is given by a critical value of the edge current density. This critical value depends on the pressure gradient since the latter quantity is stabilizing for peeling modes. In addition to the heat equation, an anomalous resistivity proportional to the peeling amplitude is added in the current diﬀusion equation. A limit cycle is found, where the ballooning limit is reached in a ﬁrst stage. The enhanced heat diﬀusivity induces a decrease of the pressure gradient, and therefore a lowering of the peeling threshold. Thus the peeling mode becomes unstable in a second stage, with a subsequent relaxation of

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the current proﬁle. Both pressure and current proﬁles are re-established in the recovery phase (see Figure 9.5). The frequency is found to increase with power and the energy release has the right order of magnitude. Thus this model provides a reasonable description of type I ELM’s.

Fig. 9.5 ELM cycle in the diagram (edge current density, pressure gradient) using a quasi-linear MHD model (from [13]).

9.4

L-H transition and edge localized modes

As mentioned before, oscillations can be obtained by introducing a bifurcation of the growth rate at the threshold. However there exists other ways to get oscillations. A useful tool to assess the existence of relaxation oscillations is the Poincar´e-Bendixson theorem (see Ref. [27] for an overview). It applies to dynamical systems characterized by a non trivial ﬁxed point (second ﬁxed point) that becomes unstable above a critical value of the control parameter. If all trajectories are bounded, i.e. if the system is repelling on a closed boundary encircling the ﬁxed point (conﬁning boundary), then there exists limit cycle solutions. Two generic situations lead to this type of behavior for dynamical systems when the control parameter is changed. In the ﬁrst case, one of the evolving ﬁelds experiences a subcritical bifurcation. The second situation corresponds to a sequence of pitchfork bifurcations leading to an unstable nontrivial ﬁxed point. In fact a 3rd route exists where a constant shear ﬂow is suﬃcient to produce an H-mode and to trigger oscillation relaxations. This section describes the physics associated to these three routes.

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Models based on a subcritical bifurcation

The onset of relaxation oscillations is generic when the dynamics of one of the ﬁeld is controlled by an S-curve. The relaxation aspect requires the existence of two time scales. A typical example is the Van der Pol equation. This type of explanation is attractive for ELM’s because S-curves appear naturally in many ways. Also there exists two diﬀerent time scales related to the time scale of the instability and the diﬀusive time. Starting from the system Eq. (9.7), a more general form is 2 ∂t peq = ∂x F (∂x peq ) ∂x peq + Dm |ξ| ∂x peq + S (9.9)

∂t ξ = γ0 (∂x peq − κc ) ξ + G ξ 2 ξ This system can be transformed into a 0D model using the recipe described in Sec. 9.3.1: 2 ∂t Πeq = S0 − F (Πeq ) Πeq − νm |ξ| Πeq (9.10) 2 ∂t ξ = γ0 (Πeq − κc ) ξ + G ξ ξ Two cases are interesting. The ﬁrst case, which is called case ’LH’, is such that G = 0. There exists a non trivial ﬁxed point Πeq = κc and |ξ|2 = [S0 /κc − F (κc )] /νm when the heat source is large enough, S0 > κc F (κc ). This ﬁxed point is unstable if κ2c ∂Πeq F |Πeq =κc + S0 < 0, which implies that ∂Πeq F must be negative at the ﬁxed point. This is an attractive condition in view of the physics of the L-H transition [15–17]. The second case, called ‘explosive’, corresponds to F = ν0 . A non trivial ﬁxed point exists, which is unstable if ∂ξ G > 0 (the derivative is calculated at the ﬁxed point). This second option corresponds to an “explosive” growth of the mode amplitude. It is interesting regarding recent results related to explosive MHD instabilities [20]. The system Eq. (9.10) satisﬁes the conditions for applying the Poincar´e-Bendixon theorem, provided that suitable damping terms are added to make the system repelling at inﬁnity. Not surprisingly the condition for getting an unstable non trivial ﬁxed point corresponds to an “S-curve” shape in the heat equation (‘LH’ case) or in the evolution equation of the mode amplitude (‘explosive’ case). This result can be extended to systems with more than 2 ﬁelds, adding for instance the mean velocity shear. The various options are detailed in the following sections. Bifurcation of the mean shear ﬂow Itoh et al. [15] and Lebedev and Diamond [16] pointed out that the physics of edge localized modes and L-H transition should be connected in some

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way since edge relaxation oscillations are observed in H-mode plasmas only. Although the physics of the L-H transition is not completely understood, it is widely admitted that the quench of turbulence is due to the onset of a sheared poloidal velocity [31]. A shear ﬂow is known to stabilize a turbulence if the shear ﬂow rate is large enough when compared to a typical turbulence time (e.g. correlation time) [32]. In a tokamak, the poloidal velocity is determined by the radial electric ﬁeld, which increases with the gradients transverse to the magnetic ﬁeld. This property opens the way to a positive feedback loop that leads to a bifurcation toward a regime with improved conﬁnement (H-mode) [36]. Hence the idea naturally comes that the dynamics of the radial electric ﬁeld may also control the dynamics of edge localized modes. A possible formulation is ∂t peq = ∂x F ∂x peq + Dm |ξ|2 ∂x peq + S (9.11) ∂t ξ = γ0 (∂x peq − κc ) ξ + Gξ ∂t V = −νneo (V − Vneo ) + L

where V is the poloidal velocity and Vneo is the neoclassical velocity due to collisional eﬀects. Shear ﬂow is stabilizing and a reasonable parameterization of the function F is [33, 34] Dan F = 2 + D0 1 + C (∂x V ) 2 or F = Dan 1 − C (∂x V ) + D0 [35, 16]. The shape of the ﬂux versus the pressure gradient is shown in Figure 9.6. If collisions control the dynamics of the radial electric ﬁeld, then L = 0. In absence of toroidal velocity, the neoclassical velocity is given by the relation 1 ∂r neq ∂r Teq Teq + (1 − kneo ) neq (9.12) Vneo = eB neq Teq where neq and Teq are the equilibrium density and temperatures. The parameter kneo depends on the collisionality and is close to 1 in the regimes of interest. At this stage it is assumed that the pedestal is much higher in density than in temperature, as observed in experiments. The radial proﬁle of the pressure is then essentially given by the shape of density. Also the collisional shear ﬂow rate is reasonably approximated by 2 2 Teq ∂r neq Teq ∂r peq ∂r Vneo ≈ − ≈− (9.13) eB neq eB peq The dynamics of the radial electric ﬁeld is in fact more complicated than Eq. (9.11). Turbulence ﬂow generation and non ambipolar particle losses

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Fig. 9.6 Flux versus pressure gradient in a model that accounts for shear ﬂow stabilization (from [16]).

in the edge may produce a non trivial behavior, resulting in a ‘S-curve’ shape. The interested reader is encouraged to read the overview [36] on this question. This compex dynamics is actually covered by the original work of Lebedev and Diamond [16] and will be discussed in Sec. 9.4.2. Returning to the case where the mean ﬂow is controlled by collisional eﬀects and assuming furthermore that the collisional time 1/νneo is smaller than the other time scales (ELM growth time and recovery diﬀusive time), it appears that the set Eq. (9.11) can be replaced by an equivalent two ﬁeld model of the form Eq. (9.9) with G = 0 and F (Πeq ) =

Dan + D0 1 + CΠ4eq

Analysing the ﬁxed point, it is found that a subcritical bifurcation occurs above a critical value of the heat ﬂux. This can be interpreted as a transition to an H-mode plasma. This state of improved conﬁnement experiences relaxation oscillations if the pressure gradient is larger than the threshold for the instability that underlie edge localized modes. The oscillation frequency increases with power, suggesting type I ELM’s. Explosive growth Artun and Cowley [19] showed that a pressure driven MHD instability may exhibit an explosive growth. This work was recently extended to ideal ballooning modes in a tokamak [20]. The physics of this behavior is quite complex and will not be detailed here. The amplitude equation is actually

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diﬀerent from the one mentioned in section 4, Eq. (9.9) (with ∂ξ G > 0 ), but nevertheless belongs to the same family. It predicts the formation of ﬁngers that grow explosively (faster then exponential), typically as a ﬁnite −λ time singularity (t0 − t) , with λ > 0. The growth time is found to be of the order of 50µs in JET, i.e. in rough agreement with observations. Thus it provides a possible description for the non linear evolution of ELM’s. One attempt has been done to study the non linear evolution of a ballooning mode with the BOUT code. This code contains all the relevant physics for electromagnetic instabilities in the H-mode, covering domains of closed and open ﬁeld lines. It was found that a ballooning mode is excited when the threshold is exceeded and grows very fast. However the whole dynamics remain to be simulated [37]. The question of type III ELM’s The models that have been presented till now have advantages and drawbacks. As already mentioned, models that involve shear ﬂow physics oﬀer a uniﬁed picture for the L-H transition and ELM’s. On the other hand, is is known that shear ﬂow has little eﬀect on ideal ballooning modes. To be eﬀective, the shear ﬂow rate has to be typically larger than the growth rate of a micro-instability, and this criterion is marginally reached for an ideal MHD mode, in particular for large wave numbers. Models that involve shear ﬂow stabilization seem to be more relevant for type III ELM’s then for type I ELM’s. The reason is that type III ELM’s are likely driven by less violent instabilities than ideal ballooning or kink modes, such as resistive ballooning modes. Still the frequency is often found to increase with additional power, whereas the frequency of type III ELM’s decreases with power. The key parameter here is the threshold of the instability that drives a relaxation event. The higher the threshold, the lower the frequency. Hence an attractive solution is to introduce some dependence of the threshold on a parameter that is sensitive to the additional power. Lebedev and Diamond have proposed a model where the threshold depends on the square of the poloidal velocity V , which itself depends on the gradient of temperature (see Eq. (9.12)) and therefore on the power [16] (see Figure 9.7). The physics behind this choice is a destabilizing eﬀect of the centrifugal force. Snyder suggested that type III ELM’s are likely peeling modes in low density H-mode plasmas. Since the threshold of peeling modes increases with the pressure gradient, this should lead to a frequency that increases with power. Indeed the larger the power, the longer the time

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to reach a threshold that increases with the pressure gradient [6]. Finally, it is stressed here that conventional ideal MHD may not be appropriate to describe the physics of these modes, due to geometry or to non ideal eﬀects. The physics of resistive interchange mode in a geometry of open ﬁeld lines (beyond the separatrix) has been studied by Pogutse et al. [38]. In this model, it is predicted that the ELM frequency decreases with power, and is thus related to type III ELM’s.

Fig. 9.7 ELM frequency versus normalized heat ﬂux for the Lebedev-Diamond model when including the eﬀect of poloidal velocity.

9.4.2

Models based on shear ﬂow stabilization

It has been shown recently that relaxation oscillation can be obtained in a transport barrier due to shear ﬂow without invoking a subcritical bifurcation [18]. This was done with the RBM3D code, which computes resistive ballooning modes in the electrostatic limit. The results were also recovered with a reduced 1D model. Transport barriers and resistive ballooning modes Resistive ballooning mode (RBM) turbulence appears in the edge of a tokamak plasma if it is cold enough so that resistive eﬀects are dominant. In the limit of high resistivity, an electrostatic description is appropriate, i.e. a set of two equations for the normalized electrostatic potential φ and pressure p, dt ∇2⊥ φ = −∇2 φ − Gp ,

(9.14)

dt p = δc Gφ + χ ∇2 p + Sp .

(9.15)

Eq. (9.14) corresponds to the charge balance in the drift approximation involving the divergences of the polarization current, the parallel current, and the diamagnetic current, respectively. Eq. (9.15) represents the energy

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balance, where χ is an eﬀective parallel collisional heat diﬀusivity and Sp (r) is an energy source. The curvature terms Gp and Gφ arise from the compressibility of diamagnetic current and E × B drift, respectively. The curvature operator is G is proportional to cos θr−1 ∂θ + sin θ∂r . The time derivative associated to the E × B drift operator dt is the Lagrangian

velocity and ∇ = 1/R0 ∂ϕ + q −1 ∂θ is the gradient along the ﬁeld lines. A transport barrier is produced by enforcing a shear ﬂow at mid-radius. As expected, the pressure gradient increases at this position. Relaxation oscillations appear with the following properties: • • • •

the frequency increases with additional power, the frequency decreases with the shear ﬂow rate, above a critical value of the shear ﬂow, the oscillations disappear, if the shear ﬂow increases faster than linear with power, then dfelm /dP < 0.

The later property is reminiscent of type III ELM’s. These results can be found in [18] and are presented in detail in this book [39]. It appears that relaxation oscillations are essentially due to a time delay, the Dupree time, in the stabilizing process due to the shear ﬂow. Reduced 1D model The results obtained for a Resistive Ballooning Mode turbulence can be recovered with the following simpliﬁed 1D model [47], which is an extension of a previous reduced model for interchange turbulence [48] ∂t peq = ∂x [pE ∗ + E ∗ p + D0 ∂x peq ] + S ∂t V = iεRS (E∂xx E ∗ − E ∗ ∂xx E) − µ (V − V0 ) + D∂xx p ∂t p = (∂x peq − κc ) E − iV p + D∇2⊥ p ∂t ∇2⊥ E = −gky2 p − iV ∇2⊥ E + D∇4⊥ E

(9.16)

where ∇2⊥ = ∂xx − ky2 . The ﬁrst equation is a heat equation, the second describes the evolution of the mean ﬂow, whereas the two last equations give the evolution of the perturbed pressure and poloidal electric ﬁeld. This model is obtained by keeping only one mode describing an ELM in the original system (9.14, 9.15). The system Eq. (9.16) reproduces most of the features of the full 3D simulations. It is found that a transport barrier appears when the shear rate of the mean ﬂow is large enough. For a range of shear ﬂow rates, relaxation oscillations appear (cf. Figure 9.8). The physics of the time delay due to shear ﬂow can be recovered from the

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Fig. 9.8 Top 4 panels: Proﬁles of mean pressure, perturbed ﬂux, poloidal velocity, shear ﬂow rate. Lower panel: Time evolution of perturbed ﬂux.

structure of the evolution equations of the ELM amplitude. This equation is typically of the form (9.17) ∂t ξ = γ0 ξ − iV ξ + D∇2⊥ ξ

3 where τD = The solution evolves in time as exp γ0 t − t3 /3τD

−1/3 2 is the Dupree time. Hence the stabilization due to shear DV /4

3 1/2 γ0 . As in 3D simulations, the ﬂow occurs with a time delay τ ∗ = τD relaxation usually starts from the barrier and propagates outward (see Figure 3.22). It is also found that the ELM frequency increases with additional power, and decreases with the shear ﬂow rate. On the other hand, the shear ﬂow rate is an increasing function of gradients (see Eq. (9.13)), and thus of the power. Hence these two eﬀects compete. Including all ingredients show that the frequency increases ﬁrst, then decreases with power, until oscillations disappear [49] (cf. Figure 9.9). This is reminiscent of type III ELM’s. It is stressed here that this eﬀect exists even if the shear ﬂow is frozen, i.e.

RS = 0 in Eq. (9.16). It is quite remarkable that this mechanism does not require a subcritical bifurcation. Work is currently under way to assess the

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case where the radial electric ﬁeld is linked to the pressure gradient via the relation Eq. (9.13). 60

100

Turbulent Flux

10

40 30

1

20

0.1

10 0

0

1000

Fig. 9.9

9.4.3

2000

3000 Time

4000

5000

Shearing rate ωE'

50

0.01

Flux and shearing rate versus time.

Models based on shear ﬂow generation

Turbulence in a magnetized plasma is known to generate a mean shear ﬂow due to a non zero divergence of the Reynolds stress. The shear ﬂow back reacts on turbulence and reduces the level of ﬂuctuations. This mechanism was proposed as a plausible mechanism for the L-H transition [40]. A typical system that represents this process can be produced starting from Eq. (9.11), adding contributions that represent the Reynolds stress and the action of shear ﬂow on turbulence, i.e. ∂t peq = ∂x [(D0 + Dm ξ) ∂x peq ] + S ∂t ξ = γ0 (∂x peq − κc ) ξ − U ξ − µξ 2 ∂t U = −νneo U + ξU

(9.18)

where ξ represents the turbulence energy and U is the shear ﬂow rate. Assuming a frozen pressure gradient, the system Eq. (9.18) is nothing else than the prey-predator model proposed by P.H. Diamond et al [40]. It possesses two ﬁxed points: one with large level of turbulence and no ﬂow, and the other one with no turbulence and large shear ﬂow. The transition occurs above a critical value of the pressure gradient, i.e. of heat source. Thus it represents a minimal model for an L-H transition due to turbulent ﬂow generation. At this stage, it is interesting to note that with µ = 0 (no non linear saturation), this system would exhibit oscillations since this is nothing else than the Lodka-Volterra system [27]. However this result is not

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robust since any dissipation (here µ = 0) leads to stable ﬁxed points (this is a generic property of oscillation models based on Hamiltonian dynamics). A complete analysis including the dynamics of the pressure gradient and neoclassical eﬀects can be found in [41]. Although it gives a better description of the L-H transition, it does not exhibit ELM’s. Oscillations appear when the generation of ﬂuctuating shear ﬂow (zonal ﬂows) by turbulence is included in the model, on top of the mean ﬂow. This is now recognized as a generic behavior of turbulence in magnetized plasmas. Fluctuations of the poloidal ﬂow (zonal ﬂows) also regulate the turbulence (for an overview, see [45]). The interplay between shear ﬂow and turbulence was demonstrated by Howard and Krishnamurti [42] in the context of Rayleigh-Benard convection. It was studied by several authors for interchange-type turbulence [43, 44]. The detailed presentation of these models is beyond the scope of this paper. However it can be illustrated with a simpliﬁed set of equations obtained by P. Beyer et al. [44] ∂t a0 = −νa0 + νa1 a1 ∂t a1 = γ1 a1 − a0 a1 − ν1 a21 a1 − ν2 a31 (9.19) ∂t a1 = γ1 a1 + a0 a1 − ν1 a21 a1 where a0 is the amplitude of the shear ﬂow and are a1 and a1 are the amplitudes of the most unstable modes. This system was obtained by projecting the initial equations on the most correlated modes. These modes were determined using a Singular Value Decomposition technique. A similar exercise can be done by projecting the system onto linear eigenmodes. A sequence of bifurcations is found where a transition occurs ﬁrst from large to low mode amplitude (L-H transition). In a second stage a (pitchfork) bifurcation takes place and leads to an oscillatory regime. This sequence of bifurcations has been found in several simulations dealing with interchange or resistive ballooning mode turbulence [30, 44, 50]. These works show that oscillations may appear after an L-H transition driven by ﬂow turbulent generation. The question whether these are type III ELM’s or dithering H-mode plasmas is controversial. First ELM’s exhibit a magnetic signature that is not predicted by these models. However this criticism is quite unfair since these models are based on the assumption of electrostatic modes. Future simulations including electromagnetic eﬀects should be able to determine whether these oscillations persist and exhibit a magnetic signature in agreement with observations. A more serious diﬃculty is the tendency of these oscillations to be sinusoidal, i.e. they do not exhibit clear relaxation events. Finally the behavior of the frequency with power remains unclear. Thus more work is needed to assess this type of models.

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Conclusion

Many models have been developed up to now to describe the dynamics of ELM’s, covering a large variety of mechanisms. Thus it has become quite diﬃcult to get a clear picture. It is proposed in this overview to classify these mechanisms using a set of reduced models coupling evolution equations for transport and the instability amplitude. This approach provides a useful guide to read the abundant literature on this topic. It appears that the simplest model coupling the evolution of an instability to a transport equation does not produce relaxation oscillations. This property illustrates the main diﬃculty met when describing the evolution of an instability with a threshold: there exists often a non trivial ﬁxed point that is stable. This ﬁxed point corresponds to a physical situation where gradients are tied to the threshold and the mode amplitude is constant. In the most common approach, called here “quasi-linear MHD model”, this diﬃculty is overcome by noticing that the frequency of an ideal MHD mode moves from the real to the imaginary axis when the instability threshold is crossed. Also MHD eigenmodes are spatially extended in a tokamak. These two features, bifurcation from real to imaginary axis and spatial extent of eigenmodes, lead to a dynamics of relaxation oscillations. In this picture ELM’s appear in H-mode plasmas because gradients are large in a transport barrier. This steepening is usually obtained using heuristic transport models. Another line of research consists of a model that describes both the transition from “normal” (L-mode) to H-mode plasmas (L-H transition) and the onset of ELM’s. Hence considerable attention has been given to a large class of models that involve a subcritical bifurcation in the evolution equation of one of the ﬁelds. This type of models is indeed able to cover both the physics of the L-H transition and the ELM’s dynamics. However it can be shown also that the physics of transport barriers produced with a constant shear ﬂow does lead to relaxation oscillations without invoking a subcritical bifurcation. This was found in 3D numerical simulations of resistive ballooning modes and also with reduced models. A third class of model covers oscillation relaxations associated to turbulent ﬂow generation. This class of models traces back to studies done in the frame of Rayleigh-B´enard convection. All these models suﬀer from several limitations. First it is diﬃcult to explain why the frequency sometimes decrease with the heat source (type III ELM’s). It seems that the best explanation invokes a dependence of the threshold on the pressure gradient. A few models only are able to reproduce this feature. Second several models are built in the frame of an

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electrostatic assumption, whereas it is known from experiment that ELM have a magnetic signature. Also plasma parameters are deep in a domain where modes are expected to be electromagnetic. Thus an extension to include electromagnetic eﬀects seem mandatory. Finally two points remain unclear: the link with the L-H transition and the role of the interface between regions of closed and open ﬁeld lines. At the moment there exists no simulations able to produce all features of ELM oscillations: transport barrier, electromagnetic relaxation oscillations and right behavior of the frequency with power. It is nevertheless foreseen that the rapid progress in theory and simulations will lead to a clearer picture in a near future.

Acknowledgments The authors wish to thank M. B´ecoulet, P.H. Diamond, G. Huysmans, J. L¨ onnroth, P. Monier-Garbet, and V. Parail for fruitful discussions.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]

M. Keilhacker et al., Plasma Physics and Controlled Fusion 26, 49 (1984). F. Wagner et al., Phys. Rev. Letters 49, 1408 (1982). The ASDEX Team, Nucl. Fusion 29, 1959 (1989). J. Connor et al., Physics of Plasmas 5, 2687 (1998). G.T.A. Huysmans, Physics of Plasmas 8, 4292 (2001). P.B. Snyder, Physics of Plasmas 9, 2037 (2002). J. Connor, Plasma Physics ans Control. Fusion 40, 531 (1998). V. Parail et al., in Proceedings of the Fusion Energy Conference, Lyon, 2002 (Editor IAEA). M. Becoulet et al., Plasma Physics ans Control. Fusion, 45, A93 (2003). J.-S L¨ onnroth et al., Plasma Physics ans Control. Fusion, 46, 767 (2004). H. Zohm, Plasma Physics ans Control. Fusion, 38, 105 (1996). W. Suttrop, Plasma Physics ans Control. Fusion, 42, A1 (2000). J.-S L¨ onnroth et al., Plasma Physics ans Control. Fusion, 46, 1197 (2004). E.J. Doyle, Phys. Fluids B 3, 2300 (1991). S-I Itoh et al., Phys. Rev. Letters, 67, 2485 (1991). V.B. Lebedev and P.H. Diamond, Phys. Plasmas, 2, 3345 (1995). S-I Itoh et al., Phys. Rev. Letters, 76, 920 (1996). P. Beyer et al., Phys. Rev. Letters 94, 105001 (2005). S. Cowley et al., Phys. Plasmas, 3, 1848 (1996). H. Wilson and S. Cowley, Phys. Rev. Letters, 92, 175006-1 (2004). G. Matthews, private communication. J. W. Connor, R.J. Hastie, and J.B. Taylor, Phys. Rev. Lett. 40, 396 (1978).

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[23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50]

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B. Coppi, A. Ferreira, J-W-K Mark, J.J. Ramos, Nucl. Fusion 19, 715 (1979). MISHKA code. J. Herring J. Atmospheric Science, 20, 326 (1963). E. Lorenz J. Atmospheric Science, 20, 130 (1963). J.D. Murray “Mathematical Biology”, Springer, Berlin, 1993. A. Thyagaraja et al., Phys. Plasmas, 6, 2380 (1993). T. Onjun et al., Phys. Plasmas, 11, 3006 (2004). A. Takayama and M. Wakatani, Plasma Phys. Cont. Fusion 38, 1411 (1996). K.H. Burrell, Phys. Plasmas 6, 4418 (1999). H. Biglari, P.H. Diamond, and P.W. and Terry, Phys. Fluids B 2, 1 (1990). G.M. Staebler, et al., Phys.Plasmas 1, 909 (1994). C. Figarella et al., Phys. Rev. Lett. 90, 015002 (2003). R.E. Waltz, G.D. Kerbel, J. Milovitch, Phys. Plasmas 1, 2229 (1994). P.W. Terry, Rev. Mod. Phys. 72, 109 (2000). P. B. Snyder, H. R. Wilson, and X. Q. Xu, Phys. Plasmas 12, 056115 (2005). O. Pogutse et al., Plasma Phys. Cont. Fusion 36, 1963 (1994). P. Beyer et al., this book. P.H. Diamond, Y.M. Liang, B.A. Carreras, P.W. Terry, Phys. Rev. Lett. 72, 2565 (1994). B.A. Carreras, D. Newman, P.H. Diamond, Y.M. Liang, Phys. Plasmas 1, 4014 (1994). L. Howard and R. Krishnamurti, J. Fluid Mech 170, 385 (1986). H. Sugama and W. Horton, Plasma Physics and Cont. Fusion, 37, 345 (1995). P. Beyer, S. Benkadda, X. Garbet, Phys. Rev. E 61, 813 (2000). P.H. Diamond, et al., submitted to Plasma Physics and Controlled Fusion. X.Q. Xu et al., Phys. Plasmas 7, 1951 (2000). N. Bian et al., Phys. Plasmas 10, 1382 (2003). Y. Sarazin, X. Garbet, P. Ghendrih, S. Benkadda, Phys. Plasmas 7, 1085 (2000). Y. Sarazin, private communication. V. Naulin et al., Phys. Plasmas 10, 1075 (2003).

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Chapter 10

On the Onset of Collapse Events in Toroidal Plasmas Turbulence Trigger K. Itoh, S.-I. Itoh∗ , M. Yagi∗ , S. Toda and A. Fukuyama† National Institute for Fusion Science, Oroshi-cho 322-6, Toki 509-5292, Japan ∗ Research Institute for Applied Mechanics, Kyushu University, Kasuga 816-8580, Japan † Department of Nuclear Engineering, Kyoto University, Kyoto, 606-8501, Japan

A theoretical framework is discussed to investigate collapse events in toroidal conﬁnement plasmas. The key issues in the observations of collapse events in toroidal plasmas are (i) the rapid increase of growth rate of a symmetry-breaking mode (trigger-problem), (ii) the unpredictability of the time of onset, and (iii) the persistence of initial state (life time problem). The theoretical methodology here is composed of two approaches. The ﬁrst step is the study of the nonlinear instability mechanisms that allows a subcritical excitation of the perturbation. This provides a ‘deterministic view’ (or ‘mean ﬁeld approach’) for the problem of the collapse. The second step is an application of the statistical theory of turbulence to establish a stochastic equation for the onset of perturbation. By these theoretical methodologies, the sudden onset and following collapse, the probabilistic occurrence of trigger, the average of large number of observations, the phase boundary and the life time of the state are explained. An example for the onset of neoclassical tearing mode in high beta plasmas is given. It is also stressed that the theoretical results are extensions of the principles in thermodynamical equilibrium to those in turbulent plasmas which are far from thermal equilibrium. 295

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Introduction

The evolution of plasma proﬁle and structure in magnetic conﬁnement devices has been subject to intensive study throughout the history of thermonuclear fusion research. This is because establishing plasma parameter in the regime of thermonuclear fusion reaction, without damaging the device of conﬁnement (at room temperature or cooler), requires to maintain a steep pressure gradient of plasmas. The path of this quest turned out to be not easy nor well-paved by established scientiﬁc methodology. As a result, the new area of modern physics has been progressed, i.e., the study of high temperature plasmas in a condition far from thermal equilibrium. One of the main issues is the strong turbulence and related turbulent transport, which are induced by strong inhomogeneity. The relation between the gradient and ﬂow in the direction of the gradient (‘gradient-ﬂux relation’) has been studied, and it is now widely understood that the nonlinearlity in the gradient-ﬂux relation allows a variety of the structures in conﬁned plasmas. The understanding of turbulence and structure formation has been the primary element that gave rise to the progress of modern plasma physics. The other view point to understand the plasmas, which are realized in laboratory, space and astrophysical objects, and are clearly discriminated from thermal death, to investigate the dynamical change of structures of plasmas. Experimental observations have clariﬁed that in many circumstances plasma proﬁles evolve slowly (being characterized by ‘transport time scale’) and a sudden onset takes place so that a new quasi-stationary state is realized. One has a picture that the plasmas are in one metastable state, and jumps to the other metastable state via transition (see ref [1] for a survey of this issue). Such transitions are the key to establish a law for inhomogeneous and far-nonequilibrium plasmas. Among many types of transition phenomena, there is a class of phenomena that is often called ‘collapse event’ (see, e.g., a review [2]). That is, the initial state and the ﬁnal state have certain (quasi) symmetry, and an abrupt symmetry breaking occurs. The onset of symmetry breaking has been interpreted as an onset of global instability. It is no doubt that a growth of symmetry-breaking perturbation needs an instability. Researches of plasma instabilities have been developed, and a variety of linear instabilities has been tabulated taking into account of detailed plasma proﬁles [3–6]. Nevertheless, mysteries remains, i.e., (1) why the growth rate is

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switched-on so rapidly (trigger-problem), (2) why the time of onset is so unpredictable, (3) how long can the initial state persist (life time problem) and others. In this article, we explain one way to understand the physics of sudden collapse. Two key physics issues are necessary. One is the presence of nonlinear instability, i. e., the growth rate of perturbation can be larger as the perturbation amplitude increases. This allows a subcritical excitation [7]. The other is the statistical approach [8, 9]. In turbulent plasmas, the perturbation mode of interest is subject to the ﬂuctuating force from back ground turbulence. By taking this process into account, we have a stochastic evolution equation for the perturbation. By combining the subcritical excitation and stochastic equation approach, we present a picture for the onset of collapse events. Example is taken from plasma conﬁnement in toroidal plasmas as an explicit application of theoretical methodology. In section 10.2, we brieﬂy explain the situation and experimental observations, providing a problem deﬁnition. In section 10.3, a brief description is made for the stochastic equation approach. In section 10.4, we take an example of nonlinear instability of tokamak plasmas. Short summary is given in a ﬁnal section.

10.2

Phenomenological observations and problem deﬁnition

High temperature plasmas in laboratory experiments, in space and astrophysical circumstances as well, are often in such a state that they are conﬁned by strong magnetic ﬁeld. Plasma conﬁnement is established by forming toroidal magnetic surfaces. We are interested in a situation ρi /L << 1, where ρi is the gyroradius of ions at the representative energy of plasma particles and L is a characteristic scale of plasma inhomogeneity across the magnetic ﬁeld. If the magnetic ﬁeld is constructed such that the magnetic ﬁeld lines form nested toroidal surfaces (See ﬁgure 10.1), charged particles do not escape from this surface by the free motion in the direction of the magnetic ﬁeld line. On a magnetic surface, plasma parameters are nearly constant owing to a rapid motion along the ﬁeld line. The isolation of a magnetic surface from others allows to sustain steep gradient of plasma parameters across the magnetic surface. We call the direction across the magnetic surfaces ‘radial direction’ in this article.

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Magnetic island formation in toroidal plasma.

Onset of collapse and topological change of magnetic surfaces

Topology of the magnetic surfaces is determined by the self-organized dynamics of plasmas. In the presence of perturbation current, the magnetic surfaces often lose the topology of nested toroidal surfaces, but are composed of large number of small tori. The small tori are called magnetic islands (See ﬁgure 10.1). The change of topology often occurs very abruptly, and is followed by a global change of plasma proﬁles. Figure 10.2 illustrates an example from experimental observation in toroidal plasmas [10]. Figure 10.2a shows the spatial proﬁle of the soft Xray emission that reﬂects electron temperature. A proﬁle of central peaking is realized as an initial state, and a broad one is realized then, both of which are considered as quasi-stationary states. Figure 10.2b shows that the temporal evolution at the center and half radius, illustrating that the change from the peaked proﬁle to the broad one occurs very suddenly. A detailed two-dimensional proﬁle of the X-ray emission was obtained. A Fourierdecomposition has revealed that a symmetry breaking perturbation which has a dependence ∝ exp (imθ + inζ) starts to grow abruptly, so as to induce a collapse (θ: poloidal angle, ζ: toroidal angle, (m, n): poloidal and toroidal mode numbers). Figure 10.2c explicitly shows the temporal change of the Fourier-decomposed proﬁles at the collapse event.

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Fig. 10.2 Observation of a collapse event on Heliotron-E. (a): Proﬁles of the brightness of soft X-ray on the midplane before, during and after a collapse. (b): Time evolution of the SX data at the center and at the mid-radius. (c): Contour of the S-X brightness on the poloidal cross-section at diﬀerent times. The proﬁle of the total emission (top) and the m = 2 Fourier component (bottom).

Standard method of studying the onset of such collapse events is the stability analysis of large scale MHD perturbations. Linear stability theory captures some important aspect, nevertheless, it is insuﬃcient for understanding collapse events [11]. Linear theory has given a linear growth rate γL as a function of the global plasma parameters γL [β, q, η, R/a, · · ·]

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(β-value: plasma pressure normalized to the magnetic pressure, R/a: the aspect ratio of torus, η: electric resistivity, q = Bt r/(Bp R) : the safety factor). γL [β, q, η, R/a, · · ·] is a regular function and the controlling parameters (β, q, η, R/a, · · ·) change in a time scale of global transport time τE . Therefore, γL does change in a time scale of τE . In the vicinity of linear stability boundary, one may have t − t0 γL [β, q, η, R/a, · · ·] ∼γL0 + · · ·, (10.1) τglobal where γL0 is a characteristic value of strong linear instability, t0 is the time at which the stability boundary is reached, and τglobal is the global time scale that characterizes the change of global plasma parameters as well as γL . The perturbation starts to grow after t = t0 , as

γL0 ∝ exp (10.2) (t − t0 )2 2τglobal " such that the characteristic time for the onset of grow is τglobal /γL0 (The linear expansion in Eq. 10.1 is based on the consideration of global instabilities in the presence of dissipations such as resistivity. If one assumes an " ideal MHD limit, one may employ a temporal dependence like γL ∝ (t − t0 ) /τglobal . In this case, one has another hybrid time of 1/3 −2/3 τglobal γL0 [12]). The appearance of the hybrid time for the increase of growth rate is in contrast to the observation that the onset of growth of symmetry-breaking perturbation is very abrupt. A detailed analysis of the experimental data requires the change of the conventional picture. Figure 10.3a shows how the instantaneous growth rate γ of the perturbation in ﬁgure 10.2c depends on the pressure gradient |∇p0 |, which is considered to be a primary origin of the instability. It is clear that the increase of the growth rate occurs associated with the reduction of pressure gradient, and the increment of growth rate continues until the pressure gradient becomes substantially small as compared to the initial values. The relation γ [|∇p0 |] turns to be a multiple-valued function, revealing a hysteresis relation, in contrast to a naive expectation like Eq. 10.1. A schematic diagram for the instability domain is shown in Fig. 10.3b, on the plane of the global control parameter and perturbation amplitude, for the case of subcritical excitation. The other observation is the dependence of the instantaneous growth rate γ as One sees that γ increases as a function of the perturbation amplitude A. A grows while the driving source of the linear instability decreases. We call such a case nonlinear insatiability. A list of nonlinear instabilities are

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Growth rate of the m = 2 perturbation.

given in ref. [13]. (It should be noticed that the increase of γ with respect to the%amplitude can be seen in a case of Eq. 10.2. In this case, one has A &0 , where A &0 is the initial amplitude at t = t0 . − A &0 at A γ ∝ A is a necessary and That is, the fact that γ is an increasing function of A not suﬃcient condition for nonlinear instability. In the linear instability, the increment of linear drive is necessary. In addition, for the cases like &0 can be a random initial value and no threshold of amplitude Eq. 10.1, A has an explicit dependence on the global control parameters in the case of nonlinear instabilities as has been explained in the literature [13].) We also notice that the trigger happens after a long hesitation in the state with small but ﬁnite amplitude perturbation. The unpredictability on the onset of the trigger is the other issue to be understood.

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Approaches for bifurcation phenomena

These facts indicate that a nonlinear picture of bifurcation/transition at critical point is more suitable to explain the collapse phenomena than a linear instability picture. In the following, some characteristic theory and modelling are surveyed. Essential for the study is an appropriate choice of variables. Among a very large number of degree of freedoms of conﬁned plasmas, one must extract a set of reduced variables that are essential for the realizability of bifurcated states and the transition between them. The ﬁrst step is the identiﬁcation of the nonlinear instability mechanism. By this, a deterministic picture of the bifurcation in terms of the averaged quantities is established. The second step is the analysis of the excitation of transition by noises. This allows to calculate the transition probability. Both the deterministic modelling of bifurcations (the 1st step) and the statistical picture of bifurcation (the 2nd step) are necessary for understanding the collapse events in toroidal plasmas. 10.3 10.3.1

Theoretical Framework Model

We study a magnetized plasma, and employ a reduced set of equations of ﬂuctuation ﬁelds as ∂ f + L (0) f = N (f ), ∂t

(10.3)

where L (0) denotes the linear operator f denotes the ﬂuctuating ﬁeld, e.g., f = (φ, J, p) where φ is an electrostatic potential perturbation, J is the perturbation of the current in the direction of the main magnetic ﬁeld, and p is the pressure perturbation [8]. The term N (f ) stands for the nonlinear terms. For the study of turbulence driven by plasma pressure, one may use   −2 ∇⊥ φ, ∇2⊥ φ . N (f ) = −  (10.4) [φ, J ] [φ, p] where the bracket [f, g] denotes the Poisson bracket, showing the Lagrangian nonlinearity as [f, g] = (∇f × ∇g) · b, (b = B/B0 ), ∆⊥ = ∇2⊥ . For a case of electromagnetic turbulence, explicit form of N (f ) is explained in [14]. Cases of larger number of ﬁeld variables are explained in ref. [8].

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Length, time, static potential and pressure are normalized to the global plasma size a, the Alfven transit time τAp = a/vAp , avAp B0 and B02 q 2 R2 /a2 µ0 , respectively (a and R are minor and major radii of torus, −1/2 aR−1 q −1 , mi is the ion mass, and ni is the ion vAp = B0 (2µ0 mi ni ) density). One can also introduce a thermal ﬂuctuation in the right hand side of Eq. 10.3, in order to study the transition from thermal ﬂuctuation to turbulent ﬂuctuation. 10.3.2

Stochastic equation

The key issue is the modelling of the Lagrangian nonlinearity term, Eq. 10.4. First, we choose a variable of interests, e.g., a Fourier component, level of micro ﬂuctuation, etc. (These variables are usually belonging to ‘experimentally-observable variables’.) We divide the ﬂuctuations into 1) the component of the interest and 2) the ones with a shorter scale length than the former. The component of interest is often chosen from the collective modes. The nonlinear interaction along with it is called selfnonlinearlity in this article and the notation Nself f is used. The others are nonlinear interactions related to ﬂuctuations with much ﬁner scales. The latter, the turbulent nonlinearity, is the subject of the statistical theory. A rigorous argument for the separation of the coherent nonlinear part from the ﬂuctuating term in N (f ) has been discussed by Mori [15]. A short description is made on the turbulent nonlinearity following [16]. The Lagrangian nonlinearity term, Eq. 10.4, that comes from ﬁner scales is modelled in two terms. One eﬀect of Eq. 10.4 is the turbulent drag to the collective mode and this part is renormalized to the eddy-viscosity type nonlinear transfer rate γj . The other part is regarded for simplicity as a random noise, which has a faster decorrelation time than γj according to RCM [17]. The nonlinear drag term is written in an linear term from the projection operator P as

apparent 2 PN (f ) = µN ∇⊥ f1 , µN e ∇2⊥ f2 , χN ∇2⊥ f3 = − (γ1 f1 , γ2 f2 , γ3 f3 ). The operator to the k-th component, Lk , Lk fk = L0,k fk − Pk Nk (f ), is the renormalized operator, which includes the renormalized transfer rates of ∗ ∗ 2 γi,k = − Mi,kpq Mi,qkp θqkp f1,p . (10.5) ∆

= N (f ) − PN (f ), has a much shorter correlation The self-noise, S time, and is approximated to be given by the Gaussian white noise term T = S1 , S2 , S3 is w (t). The self-noise term for the k-th component S

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expressed as

gi,

k

≡

Si,

= w (t) gi, k , " Mi, kpq θkpq ζ1, p ζi, q . k

(10.6a) (10.6b)

∆

In these expressions, summation ∆ indicates the constraint k + p + q = 0. The explicit form of the nonlinear interaction matrix is given as, e.g.,

2 2 p⊥ − q⊥ , or M(2,3),kpq = (p × q) · b (10.7) M1,kpq = (p × q) · b 2 k⊥ and the propagator satisﬁes the relation (∂/∂t + L (k) + c.p.) θkpq = 1, where c.p. indicates the counter part, i.e., L (p) + L (q). The term ζj,p in a random noise represents the j-th ﬁeld of p-component in the nonlinear term N , and their correlation functions satisfy the average relations of the mode, which we call an Ansatz of equivalence in correlation in the following, as ζi ζj = fi fj with ζi,p ζj,q ∝ δpq where the bracket indicates the statistical average. Then a stochastic equation is derived as ∂ f + L f + Nself f = S ∂t

(10.8)

(0)

with Lij = Lij + γi δij δij is the Kronecker’s delta). This equation has two key features. One is the ‘deterministic part’ L f + Nself f , which determines (nonlinear) instability of the perturbation of interests. The which represents the stochastic impact. other is the ﬂuctuating force S We perform the spectral decomposition (for given amplitude of f for the operator L + Nself , and obtain the least stable nonlinear eigenmode and the associated eigenvalue Λ [18–20]. In many circumstances, one variable has a primary importance. The amplitude of the variable of interests (often the least stable eigenmode) is in the direction of written as X, and the projection of the ﬂuctuation force S the least stable eigenmode is written as S = gw (t). By this simpliﬁcation, one has a stochastic equation ∂ X + ΛX = gw (t) (10.9) ∂t In this equation −Λ [X] is the nonlinear growth rate (−Λ > 0 if unstable) and gw (t) is the random kick where g is the magnitude and w (t) indicates white-noise. Extensions to the cases of multiple variables in plasma turbulence have been discussed in [21–23].

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Statistical properties

Equation 10.9 serves a basic equation to study the onset of collapse events. The quasi-stationary states are given by the condition ΛX = 0.

(10.10)

When the nonlinear instability exists, the equation Λ=0

(10.11)

can have a nontrivial solution of X = 0, i.e., a ﬁnite-amplitude and selfsustaining state is allowed. When multiple solutions of X are allowed, a rule of selecting the state is deduced from Eq. 10.9. The rate of transition from one state to the other is calculated, giving the life time of state. The statistical theory is next explained. 10.3.3.1

Nonlinear dissipation function

In a stationary state, probability density function (PDF) of amplitude X is given as Peq (X) ∝ g −1 exp (−S (X)) Here, S (X) is a nonlinear dissipation function A 2Λ (X ) g −2 X dX S (X) =

(10.12)

(10.13)

0

It is shown that this quantity is proportional to the entropy production rate in the limit of the thermal equilibrium [19] (Note that both of Λ and g in the integrand of Eq. 10.13 can be dependent on X ). Let us consider the case that multiple states are allowed (e.g., the nonlinear marginal condition Λ = 0 has triple solutions.) Figure 10.4a illustrates the schematic proﬁle of S (X) as a function of the helical deformation. In this case, S (X) has two minima at X = XA and X = XB , being separated by a local maximum at X = Xm . Statistical transitions take place between these solutions. Local minima represents metastable states, and the random noise gw (t) induces the transition between metastable states. 10.3.3.2

Rate of transition

Transitions between the state ‘A’ (X = XA ) and state ‘B’ (X = XB ) occur owing to the random kick by noise as is schematically shown in Fig. 10.4b.

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Fig. 10.4 Schematic drawing of the transition by noise (a). Schematic drawing that shows the excitation and decay of NTM perturbations owing to the turbulent noise (b).

The frequency of excitation (from (X = XA to X = XB ) and that for decay (from X = XB to X = XA ) are expressed as [19, 20]

rdec

√ ΛA Λm exp (− S (Xm )) , rex = 2π √ ΛB Λm exp (S (XB ) − S (Xm )) , = 2π

(10.14a) (10.14b)

respectively, where the time rates ΛA,m,B are given as ΛA,m,B = 2X |∂Λ/∂X| at X = XA , X = Xm and X = XB . (If one of X = XA,m,B vanishes,"say XA = 0, then ΛA is evaluated as ΛA = Λ0 ≡ Λ (X0 ), where X0 = X 2 for the state ‘A’.)

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Phase boundary and the rule for selection of the state

The long time average of multiple state is given as X = (XB rex + XA rdec ) (rex + rdec )− 1 .

(10.15)

X approaches to XB if rex > rdec holds. It reduces to XA , if rex < rdec holds. The phase boundary for the statistical average is determined by the condition rex = rdec . From Eq. 10.14b, one obtains

ΛA 1 S (XB ) = . (10.16) ln 2 ΛB Equation 10.16 determines the critical value of plasma parameters (say, βp = βp ∗ ). Figure 10.5 illustrates an example of PDF for 2S (XB ) > ln (ΛA /ΛB ) (Fig. 10.5a), for 2S (XB ) ∼ ln (ΛA /ΛB ) (Fig. 10.5b) and 2S (XB ) < ln (ΛA /ΛB ) (Fig. 10.5c). In the case of Fig. 10.5a, the solution X = XA is statistically-selected, and X = XB is the most probable state for 2S (XB ) < ln (ΛA /ΛB ). This is a selection rule of turbulent plasmas, corresponding to the generalized form of the Maxwell’s construction rule. In other words, Eq. 10.16 expresses the selection rule in a form of the minimum principle of S (X). This is a generalization of the Prigogine’s ‘minimum entropy production rate’ [24]. As is explained at Eq. 10.13, S (X) is in proportion to the entropy production rate in the limit of thermal equilibrium. 10.3.3.4

Life time near phase boundary

The transition rate rex means that the metastable state X = XA will be lost owing to the trigger by background turbulence, and the average − 1 . When the state X = XA is chosen as an time for the onset is rex − 1 , owing to the statistical initial condition, this state is lost in a time of rex excitation. Thus we deﬁne the life time of the quiescent state as −1 tlif e = rex , i.e., 2π exp (S (Xm )) . tlif e = √ ΛA Λm

10.3.3.5

(10.17a) (10.17b)

Short summary

Above results are compared to the cases of thermal equilibrium in the Table 10.1. The roles of nonlinear structure formation mechanisms in Λ and ﬂuctuating force from background turbulence are integrated in this table.

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P (A) eq

0

0

5

10

15

A/W

20 1

(a)

P (A) eq

0

0

10

20

30

A/W

40

1

(b)

P (A) eq

0

0

10

20

30

A/W

40

1

(c) Fig. 10.5 Typical examples of the PDF in the stationary state. The cases of (a) βp /βpc = 1.35 (NTM is not excited as an average), (b) βp /βpc = 1.4 (near marginal condition), and (c) βp /βpc = 1.51 (most unstable, and NTM is excited), are shown. (Other parameters are: W1 = W2 , C1 /2C2 W1 = 1, hW1 = 0 and ΓC2 W1 = 5.)

10.4 10.4.1

Example of Neoclassical Tearing Mode Description of nonlinear instability and stochastic equation

The tearing mode is a perturbation that induces a magnetic island (an example is illustrated in Fig. 10.1b) [24, 25]. The primary origin of the drive

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Table 10.1 Comparison of rules of nonequilibrium plasmas and those near thermal equilibrium. Near thermal equilibrium

Far non-equilibrium plasma

Mini/max principle

Minimum entropy production rate

S(X) minimum

Phase boundary

Maxwell’s construction

Rate of transition

ln K ∝ −∆Q/T Arrhenius law

S (XB ) =

1 2

ln

ΛA ΛB

K ∝ exp (−S (Xm ))

is the current density gradient and the pressure gradient. A resonant perturbation with the (m, n)-Fourier component at the mode rational surface q = m/n causes the break of magnetic surface and topology change. Under the transport process, this mode can be sub-critically unstable [26–29], being named ‘neoclassical tearing mode’ (NTM). The mode has been studied intensively, in conjunction with an achievable beta value and the steadystate tokamak plasmas [30–34]. ∗ q 2 R/Brs3 q which As a relevant dynamical parameter, we choose A ≡ A is the normalized amplitude of the (m, n)-Fourier component of helical vec∗ at the mode rational surface, r = rs where tor potential perturbation A q = m/n holds. A detailed argument of the nonlinear dependence of Λ on A has been given in, e.g., [35–42]. We choose a prototypical form of Λ [A] and study the problem of the probabilistic onset. An explicit form of the growth rate is given by

C1 C2 −1/2 − 2 + , −Λ = η 2∆ A W1 + A2 W2 + A

(10.18)

within the neoclassical transport theory, where the ﬁrst, second and third terms of RHS stand for the eﬀects of current density gradient, polarization drift and bootstrap current, respectively. The coeﬃcient η is the inverse of −1 −2 resistive diﬀusion time η = η// µ−1 0 rs τAp = RM , where η// stands for a parallel resistivity, and RM is the Lundquist number (magnetic Reynolds number). The ﬁrst term in the RHS of Eq. 10.18 stands for the eﬀect of the current proﬁle. A simple formula √ ∆ = ∆ 0 − h A (10.19)

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has been found useful, and the coeﬃcients ∆ 0 and h are determined by the current proﬁle. ∆ 0 is the parameter that speciﬁes the linear stability. The second and third terms in the RHS of Eq. 10.18 represent the stabilization eﬀect by polarization drift and the destabilization eﬀect by bootstrap current, respectively. Coeﬃcients C1 and C2 are given as 1 − 1 − 1 β with the normalizing β and C2 = βp βpn C1 = ρ2b rs−2 Lq L− p p pn poloidal beta value of βpn = Lp /2abs ε1/2 Lq ,

(10.20)

where ρb is the banana width, Lq and Lp are the gradient scale lengths of safety factor and pressure, respectively, and abs is a numerical constant. In Eq. 10.18, W1 represents the cut-oﬀ due to the banana orbit eﬀect, and W2 represents the cut-oﬀ determined by the cross-ﬁeld energy transport. We 1/2

− 1 choose a simple model, W1 = ρ2b rs− 2 , and W2 = χ⊥ /χ|| (ε s n) , where χ⊥ and χ|| are thermal diﬀusivities perpendicular to and parallel to the magnetic ﬁeld line, ε is the inverse aspect ratio, n is the toroidal mode number and s = rs q / q. (When the eﬀect of the perpendicular

1/2 diﬀusion is not eﬀective, χ⊥ /χ|| /εsn < ρ2b rs− 2 , W2 is evaluated as 2 − 2 W2 = ρb rs .) The magnitude g is evaluated based on the reduced set of equations as [43, 44] g 2 = k kh− 2 −

1

S

2

τac = − 1 k 3 kh4 C 2 A4h τac ,

(10.21)

where Ah is the vector potential of microscopic ﬂuctuations, is the radial scale length of the NTM, and k is the poloidal wave number of NTM, kh is the typical " number of the micro ﬂuctuations, kh C = mode − s f δ 2 rs− 2 + kh− 2 + s βmi /me δ 2 rs− 2 with f ≡ φh /Ah , and τac is the autocorrelation time of microturbulence. 10.4.2

Nonlinear instability, subcritical excitation and cusp catastrophe

Equation 10.18 shows the nonlinear instability in the deterministic model. Figure 10.6a illustrates the nonlinear growth rate −Λ as a function of the perturbation amplitude A. When A becomes larger, the growth rate −Λ becomes positive, owing to the third term in the RHS of Eq. 10.18. The nonlinear marginal condition ΛA = 0 has triple solutions. Two metastable states are written as A = 0 and A = As , being separated by an unstable state A = Am .

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1

A 0

A

-1

0

A m

10

s

20

30

A/W

40

1

(a) 40

A/W

' W

1

30

0

1/2

1

=0.1 ' W 0

20

A

10

0

A 0

1

2

pc

1/2

=0

1

s

m

3

4/ p

5

pn

(b) Fig. 10.6 (a) Normalized growth rate multiplied by amplitude, γA ≡ − ΛA/C2 , is shown by dashed line. Zeros indicate the nonlinear marginal stability conditions for the deterministic model. (Parameters are: W1 = W2 , C1 /2C2 W1 = 1, h = 0 and 1/2 ∆ 0 W1 /C2 = − 0.0922.) (b) Marginal stability solutions A = As and A = Am as a function of βp for ﬁxed value of ∆ 0 . The solid line shows the case of ∆ 0 = 0, and 1/2 the dashed line indicates the linearly unstable case ∆ 0 W1 = 0.1. (Other parameters are: W1 = W2 , C1 /2C2 W1 = 1, and hW1 = 1/20.)

The cusp catastrophe of the perturbation amplitude in the stationary state is expected, because the marginal stability condition can have multiple solutions. The structure of the cusp is studied in this subsection. Figure 10.6b illustrates the amplitude Am and As as a function of βp for the ﬁxed value of ∆ 0 . The solid line shows the case of ∆ 0 = 0, and the dashed line indicates the linearly unstable case ∆ 0 W1 = 0.1. Subcritical excitation appears in the linearly stable case. Figure 10.6a shows that the equation Λ = 0 has multiple solutions, when the plasma pressure becomes high enough. The relation for the cusp ridge is rewritten as that of βp for the ﬁxed value of ∆ 0 . From Eq. 10.18, the

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ridge of cusp can be written as βp = βp

c

" √ ≡ 3 3 b βpn W1 (− ∆ 0 ) .

(10.22)

for ﬁxed negative value of ∆ 0 . The ridge at βp = βp c is denoted in Fig. 10.6b. Figure 10.7 illustrates the ridges of cusp on the plane of (∆ 0 , βp ). The case of hW1− 1 = 1/40 is drawn by the solid line. A critical point of the cusp appears in the low pressure gradient region.

Fig. 10.7 Ridge of cusp on the plane of (∆ 0 , βp ). (Parameters are hW1 = 1/40, W1 = W2 and C1 /2C2 W1 = 1.)

10.4.3

Statistical property

10.4.3.1

Nonlinear dissipation function

Substituting Eqs. 10.18 and 10.21 into Eq. 10.13, we have

4 ∆0 3/2 h 2 C1 A2 A + A + ln 1 + 2 S (A) = Γ0 × − 3 W0 W1 2W1 W1

A C2 − A − W2 ln 1 + (10.23) W1 W2 with Γ0 = (2 ρ2b )/(RM k 3 kh4 C 2 A4h τac rs2 ). The coeﬃcient Γ0 shows a characteristic value of the ratio between the dissipation for crossing over the barrier and excitation by turbulence noise. Substituting Eq. 10.23 into Eq. 10.14b, the transition rates of NTM are calculated.

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Explicit value of the transition rate is examined by specifying a micro mode for typical experimental parameters. The parameter Γ0 is the key for the transition frequency. For the L-mode plasmas, when one employs the current-diﬀusive ballooning mode as the micro mode [45, 46], one has Γ0 =

2 × 10−4

rs6 ρ2b −11/2 α , (10.24) 2 " δ8 k 3 −α−1/2 (1 + α) + s βmi / me s4 RM

where δ is the collisionless skin depth and α = εrs βp /Lp is the normalized pressure gradient. 10.4.3.2

Statistical average and phase boundary

Figure 10.7 shows the statistical average A as a long time average, together with threshold and saturation amplitudes (Am and As ), as a function of βp . Solid line shows the statistical average A. A thin dotted line indicates the threshold Am and saturation amplitude As from the deterministic model. A drastically changes across the condition βp = βp∗ .

Fig. 10.8 Amplitude of NTM as a function of the plasma pressure for ﬁxed value of ∆ 0 . Solid line shows the statistical average A. A thin dotted line indicates the threshold Am and saturation amplitude As of the deterministic model. Normalized βp is given by Eq. (10b). (Parameters are: ∆ 0 = 0, hW1 = 1/40, W1 = W2 , C1 /2C2 W1 = 1, ΓW1 = 5.)

It must also be noted that the long time average in Fig. 10.8 is a monotonous curve while the deterministic model (dotted line) predicts a multiple-valued curve. Whether a hysteresis is observed or not depends on

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−1 −1 the competition between the life time of transitions, rex and rdec , and the characteristic time for variation of the global parameters τglobal . When the −1 −1 change of the global parameters is slow and the relation τglobal >> rex , rdec holds, the value of long time average is realized. And the monotonous dependence of A is observed. In contrast, if the global change is fast, −1 −1 , rdec , the hysteresis characteristics are observed. τglobal ∼ rex Equations 10.16 and 10.23 are numerically solved, and the boundary of phase that satisﬁes Eq. 10.16 is obtained. Figure 10.9 shows the phase boundary βp∗ on the (∆ 0 , βp ) plane by the solid line. Thin dashed lines in Fig. 10.9 show the ridge points of the cusp, βpc , in the deterministic model. Away from the critical point of the cusp,which is denoted by “C” in Fig. 10.9, the boundary of phase is approximated as a straight line.

Fig. 10.9 Phase diagram of the statistical average of NTM amplitude. In the region of “excited”, the NTM is found excited after statistical average. In the region “quenched”, NTM is not excited as an average. Thin dashed line is the boundary of the cusp in the deterministic model, which is shown in Fig. 10.3a. Symbol “C” denotes the critical point of the cusp. (Other parameters are: hW1 = 1/40, W1 = W2 and C1 /2C2 W1 = 1.)

10.4.3.3

Life time

The transition rate rex means that the metastable state A = 0 will be lost owing to the trigger by background turbulence, and the average occurrence −1 time for the onset is rex . The life time can be long at the phase boundary (Eq. 10.17b), but can become short as the plasma parameter exceeds the phase boundary. The life time depends on two factors. One is the functional

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dependence of the barrier height S (Am ) /Γ0 and the other is Γ0 , which mainly comes from the magnitude of noise. The evaluation of Γ0 truly depends on the theoretical modelling of the back ground turbulence. Owing to immaturity of the theory of strong plasma turbulence, the estimate of Γ0 still includes ambiguity. An order of magnitude estimate is possible, as is given in [45, 46]. For a set of typical parameters, rs /ρb ∼10, rs /δ∼102 , RM ∼108 , βmi / me ∼ 10, krs = 3, s = 1, α ∼ βp /βpn , one has Γ0 ∼ 3 (βp /βpn )

− 11/2

.

(10.25)

By use of Eqs. 10.17 and 10.25, the life time of the state A = 0 is obtained. In Fig. 10.10, the life time of the state A = 0 is illustrated. The √ unit used in this ﬁgure is 2π/η Λ0 Λm , which is of the order of Rutherford growth time. As is shown by Fig. 10, the life time becomes shorter as the plasma beta exceeds the phase boundary βp∗ . The life time is a decreasing function of ∆0 , and the sustaining time is ﬁnite even if ∆0 is positive (i.e., the classical tearing mode is unstable). This is the consequence of the stabilization eﬀect by the polarization drift.

Fig. 10.10 Mean life time before the onset of NTM as a function of the plasma beta value. ∆0 = 0 (Other parameters are hW1 = 1/40, W1 = W2 and C1 /2C2 W1 = 1.)

Here, other mechanisms that may limit the duration of the state being free from NTM (e.g., onset by global events like sawtooth and ﬁshbone, lack of control, limit of power supply, etc.) are not taken into account. In this sense, Eq. 10.16 provides an upper bound of the life time of the state.

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The result provides the following perspective in conjunction with sustainment of the plasma state free from NTM. The sustainable time (life time) is a function of plasma pressure. As the plasma pressure becomes high, the statistically-expected value of the sustainable time becomes shorter. This gives a basis for the understanding of the sustainable time of improved state at high beta. In another words, the upper bound of the beta value due to NTM is a function of the sustainable time. The plasma can be free from NTM at higher beta value in a shorter time. An explicit example is given in Fig. 4. In this sense, the beta limit by NTM is a diﬀused boundary depending on the sustaining time.

10.5

Conclusion

In this article, we discussed a theoretical framework to investigate the collapse events in toroidal conﬁnement plasmas. First, the problem deﬁnition is made. The key issues in the collapse events in toroidal plasmas were postulated as (1) why the growth rate is switched-on so rapidly (trigger-problem), (2) why the time of onset is so unpredictable, (3) how long can the initial state persist (life time problem). Then the theoretical methodology is explained. This is composed of two nonlinear theoretical approaches. The ﬁrst one is the study of the nonlinear instability that allows a subcritical excitation of symmetry-breaking perturbation. This provides a ‘deterministic view’ for tunderstanding the collapse. The second one is an application of the statistical theory to turbulence, establishing a stochastic equation for temporal evolution of perturbation. By these theoretical methodologies, we explained the sudden onset and following collapse, the probabilistic occurrence of trigger, the average of large number of observations, the phase boundary and the life time of the state. The analysis was given, taking an example of the onset of neoclassical tearing mode in high beta plasmas. Other application has been developed for the problem of the L-H transitions. The statistical description has been developed [47, 48], and the statistical excitation of the L/H transition has been examined [49]. In addition, the interpretation for the experimental database has been explained in refs. [50, 51]. It is also stressed here that the results are extensions of the principles in thermodynamical equilibrium to turbulent plasmas which are far from thermal equilibrium as is being highlighted by the table. The research

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in this direction clearly illustrates that the modern plasma physics has extended the frontier of modern physics. Acknowledgements Discussions with colleagues, in particular, Prof. Y. Miura, Prof. H. Zohm, Prof. H. Zushi, Dr. A. Isayama, and Prof. A. Yoshizawa are acknowledged. Authors would like to thank a partial support in completing this manuscript by Grant-in-Aid for Specially-Promoted Research (16002005) and Grant-in-Aid for Scientiﬁc Research (15360495) from Ministry of Education, Culture, Sports Science, and Technology Japan. This work is also partly supported by the collaboration programme of National Institute for Fusion Science and by the collaboration programme of the Research Institute for Applied Mechanics of Kyushu University. References [1] S.-I. Itoh and N. Kawai (ed.), Bifurcation Phenomena in Plasmas (Kyushu Univ., 2002). [2] S.-I. Itoh, K. Itoh, H. Zushi, A. Fukuyama, Plasma Phys. Contr. Fusion 40 (1998) 879. [3] A. B. Mikhailovskii, Theory of Plasma Instabilities (transl. Barbour J B, Consultants Bureau, 1974, New York). [4] B. B. Kadomtsev, Plasma Turbulence (Academic Press, 1965, New York). [5] K. Miyamoto, 1976 Plasma Physics for Nuclear Fusion (The MIT, Cambridge). [6] B. B. Kadomtsev, 1992 Tokamak Plasma: A Complex Physical System (IOP Publishing, Bristol). [7] K. Itoh, S.-I. Itoh and A. Fukuyama: Transport and Structural Formation in Plasmas (IOP, England, 1999). [8] A. Yoshizawa, S.-I. Itoh, K. Itoh, Plasma and Fluid Turbulence (IOP, England, 2002). [9] J. A. Krommes: Phys. Reports 360 (2002) 1. [10] H. Zushi, et al., in Proc. 16th International Conference on Fusion Energy (IAEA, Montreal, 1996) paper IAEA-CN-64/CP-4. [11] J. A. Wesson, in Theory of Fusion Plasmas, (International School of Plasma Physics Piero Caldirola, Varenna) (ed. A. Bondeson, E. Sindoni, F. Troyon, 1987, Edtrice Compositori) p.253. [12] S. Cowley, presented at 13th Toki Conference (2003, Toki). [13] Examples are found in a following list: P. H. Rebut and M. Hugon, in Plasma Physics and Controlled Nuclear Fusion Research 1984 (IAEA, 1985, Vienna) Vol.2, p197;

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[14] [15]

[16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28]

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P. H. Diamond, et al., Phys. Fluids 27 (1984) 1449; D. Biskamp and M. Walter, Phys. Lett. 109A (1985) 34; R. D. Sydora, et al., Phys. Fluids 28 (1985) 528; R. E. Waltz, Phys. Rev. Lett. 55 (1985) 1098; B. D. Scott, Phys. Fluids B4 (1992) 2468; K. Itoh, S.-I. Itoh, A. Fukuyama, Phys. Rev. Lett. 69 (1992) 1050; B. A. Carreras, et al., Phys. Fluids B4 (1992) 3115; A. J. Lichtenberg, et al., Nucl. Fusion 32 (1992) 495; A. Fukuyama, et al., in Plasma Physics and Controlled Nuclear Fusion Research 1992 (IAEA, 1993, Vienna) Vol.2, p363; M. Ottaviani and F. Porcelli, Phys. Rev. Lett. 71 (1993) 3802; H. Nordman, V. P. Pavlenko, J. Weiland, Phys. Plasmas B5 (1993) 402; D. Biskamp and A. Zeiler, Phys. Rev. Lett. 74 (1995) 706; M. Yagi, S.-I. Itoh, K. Itoh, A. Fukuyama, M. Azumi, Phys. Plasmas 2 (1995) 4140; K. Itoh et al., J. Phys. Soc. Jpn. 65 (1996) 2749; S. Cowley, M. Artun, B. Albright, Phys. Plasmas 3 (1996) 1848; E. Knobloch, N. O. J. Weiss, Physica D 9 (1983) 379; N. Bekki, T. Karakisawa, Phys. Plasmas 2 (1995) 2945; J. F. Drake, A. Zeiler, D. Biskamp, Phys. Rev. Lett. 75 (1995) 4222; M. A. Malkov, P. H. Diamond, Phys. Plasmas 8 (2001) 3996; Y. Ishii, M. Azumi, Y. Kishimoto, Phys. Rev. Lett. 89 (2002) 205002; See also related work in ﬂuid dynamics: T. Herbert, AIAA J 18 (1980) 243; F. Walleﬀe, Phys. Fluids 9 (1997) 883. M. Yagi, et al. : Plasma Phys. Contr. Fusion 39 (1997) 1887. H. Mori and H. Fujisaka, Phys. Rev. E 63 (2001) 026302; H. Mori, S. Kurosaki, H. Tominaga, R. Ishizaki, N. Mori, Prog. Theor. Phys. 109 (2003) 333. J. A. Krommes, Plasma Phys. Contr. Fusion 41 (1999) A641. R. H. Kraichnan, J. Fluid Mech. 41 (1970) 189. S.-I. Itoh and K. Itoh, J. Phys. Soc. Jpn. 68 (1999) 1891, 2611. S.-I. Itoh and K. Itoh, J. Phys. Soc. Jpn. 69 (2000) 408, 3253. S.-I. Itoh and K. Itoh: J. Phys. Soc. Jpn. 69 (2000) 427. S.-I. Itoh and K. Itoh: Plasma Phys. Contr. Fusion 43 (2001) 1055. S.-I. Itoh, K. Itoh, M. Yagi, M. Kawasaki, A. Kitazawa, Phys. Plasmas 9 (2002) 1947. K. Itoh, S.-I. Itoh, F. Spineanu, M. O. Vlad and M. Kawasaki, Plasma Phys. Control. Fusion 45 (2003) 911. I. Prigogine, Introduction to Thermodynamics of Irreversible Processes, 2nd ed. (Interscience Publishers, New York, 1961). H. P. Furth, J. Killeen, M. N. Rosenbluth, Phys. Fluids 6 (1963) 459. P. H. Rutherford, Phys. Fluids 6 (1973) 1903. P. H. Rebut and M. Hugon, in Plasma Physics and Controlled Nuclear Fusion Research 1984 (IAEA, Vienna, 1985) Vol.2, p197. J. D. Callen, et al., in Plasma Physics and Controlled Nuclear Fusion Research (IAEA, Vienna, 1986) Vol.2, p157.

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[29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50]

[51]

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A. I. Smolyakov, Plasma Phys. Contr. Fusion 35 (1993) 657. Z. Chang, et al., Phys. Rev. Lett. 74 (1995) 4663. O. Sauter, et al. Phys. Plasmas 4 (1997) 1654. R. J. Buttery, et al., Plasma Phys. Contr. Fusion 42 (2000) B61. A. Gude, et al., Nucl. Fusion 39 (2001) 127. A. Isayama, et al. : Nucl. Fusion 41 (2001) 761. A. Bergmann, E. Poli, A. G. Peeters, presented at 19th IAEA Conference on Fusion Energy (2002, Lyon) paper TH/P1-01. F. Waelbroeck and R. Fitzpatrick, Phys. Rev. Lett. 78 (1993) 1703; R. Fitzpatrick, Phys. Plasmas 2 (1994) 825. E. Poli, et al., Phys. Rev. Lett. 88 (2002) 075001. H. R. Wilson, et al., Plasma Phys. Contr. Fusion 38 (1996) A149. A. B. Mikhailovskii, et al., Phys. Plasmas 7 (2000) 3474. M. Yagi, S.-I. Itoh, A. Fukuyama and K. Itoh, 7th IAEA Technical Committee Meet. H-mode Physics and Transport Barriers, Oxford, Sept 1999. A. Furuya, S.-I. Itoh, M. Yagi, J. Phys. Soc. Jpn. 70 (2001) 407. A. Furuya, S.-I. Itoh, M. Yagi, J. Phys. Soc. Jpn. 71 (2002) 1261. S.-I. Itoh, K. Itoh, M. Yagi, Phys. Rev. Lett. 91 (2003) 045003. S.-I. Itoh, K. Itoh, M. Yagi, Plasma Phys. Control. Fusion 46 (2004) 123. K. Itoh, M. Yagi, S.-I. Itoh, A. Fukuyama and M. Azumi, Plasma Phys. Contr. Fusion 35 (1993) 543. A. Fukuyama, K. Itoh, S.-I. Itoh, M. Yagi and M. Azumi, Plasma Phys. Control. Fusion 37 (1995) 611. S.-I. Itoh, K. Itoh, S. Toda, Phys. Rev. Lett. 89 (2002) 215001. S.-I. Itoh, K. Itoh and S. Toda, Plasma Phys. Control. Fusion 45 (2003) 823. S. Toda, S. -I. Itoh, M. Yagi, K. Itoh and A. Fukuyama, J. Phys. Soc. Jpn 68 (1999) 3520. S.-I. Itoh, et al., “Statistical theory of L-H transition and its implication to threshold database”, presented at 19th IAEA Fusion Energy Conference (Lyon, October 2002) paper PDP/04. S.-I. Itoh, K. Itoh, M. Yagi, S. Toda, Plasma Phys. Contr. Fusion. 46 (2004) A341.