H.K. Moffatt Magnetic field generation in electrically conducting fluids: Preface
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C A M B R I D G E M O N O G R A P H SON M E C H A N I C S A N D A P P L I E D M A T H E M A T I C S
GENERAL EDITORSG. K . BATCHELOR, PH.D., F.R.S.
J . W. M I L E S , PH.D.Professor of Applied Mathematics in the Universityof Cambridge
Professor of Applied Mathematics, Universityof California,La JollaMAGNETIC FIELD GENERATION IN
ELECTRICALLY CONDUCTING FLUIDS
www.moffatt.tc

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MAGNETICFIELD GENERATION I N
ELECTRICALLYCONDUCTING FLUIDS
H. K. MOFFATTP R O F E S S O R OF A P P L I E D M A T H E M A T I C S , U N I V E R S IT Y OF B R I S T O L
C A M B R I D G E U N I V E R S I T Y P R E S SC A M B R I D G E
L O N D O N . N E W Y O R K . M E L B O U R N E

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Published by the Syndicsof the Cambridge University PressThe Pitt Building, Trumpington Street, Cambridge CB 2 I R PBentleyHouse, 200 Euston Road, London NW 12 D B32 East 57th S treet, New York, NY 10022, U SA296 Beaconsfield Parade, Middle Pa rk, Melbourne 3206 , Australia@ Cambridge University Press 1978
First published 19 78Printed in Gr ea t Britain at the University P ress, Cambridge
Library ofCongress Cataloguing in Publication DataMoffatt, Henry Keith, 1935
Magnetic field generation in electrically conducting fluids.(Cambridge monographs o n mechanics and applied mathematics)
Bibliography: p. 3251.Dy nam o theory (Cosmic physics)I.Title.
QC809.M25M63 538 774398ISBN 0 521 21640 0

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CONTENTS
page ix122.12.22.32.42.52.62.733.13.23.33.43.53.63.73.83.9
PrefaceIntroduction and historical backgroundMagnetokinematic preliminariesStructural properties of the BfieldMagnetic field representationsRelations between electric curren t and m agnetic fieldForcefree fieldsLagrangian variables and magnetic field evolutionKinematically possible velocity fieldsFre e decay modesConvection, distortion and diffusion of magnetic fieldAlfvBns theo rem and related resultsThe analogy with vorticityT he analogy with scalar transpo rtMaintenance of a flux rope by uniform rate of strainAn example of accelerated ohmic diffusionEquation for vector potential and fluxfunction underparticular symmetriesField distortion by differential rotationEffect of plane differential rotation on an initially uniformfieldFlux expulsion for general flows with closed streamlines
3.10 Expulsion of poloidal fields by m eridional circulation3.11 Generation of toroidal field by differential rotation3.12 Topological pum ping of magnetic flux4 The magnetic field of the Earth4 .1 Planetary magnetic fields in general4.2 Spherical harmonic analysis of the Earths field
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vi CONTENTS4.34.44.54.6 The coremantle interface4.7
Variation of the dipole field over long timescalesParam eters and physical state of the lower mantle a nd coreThe need for a dynamo theory for the Earth
Precession of the Earths angular velocity vector5 The solarmagnetic field5.1 Introduction5.2 Observed velocity fields5.35.45.5 Magnetic stars
Sunspots and the solar cycleThe general poloidal magnetic field of the Sun
6 Laminar dynamo theory6.1 Formal statement of the kinematic dynamo problem6.2 Rate of strain criterion6.3 Rate of change of dipole moment6.4 The impossibility of axisymmetric dyaam o action6.5 Cowlings neutral point argum ent6.6 Some comm ents on the situation B. A B=06.7 The impossibility of dynamo action with purely toroidal
motion6.8 The impossibility of dynamo action with plane two
dimensional motion6.9 Ro tor dynamos6.10 Dynamo action associated with a pair of ring vortices6.11 The BullardGellman formalism6.12 Th e stasis dynamo of Backus (1958)77.17.27.37.47.57.67.77.8
The mean electromotive force generated by random motionsTurbulence and random wavesThe linear relation between 8 and B,,The a effectEffects associated with the coefficients pijkFirstorder smoothingSpectrum tensor of a stationary ran dom vector fieldDetermination of aijfor a helical wave motionDetermination of ajj or a random ufield under firstordersmoothing
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101105108108109111113115117118121122131137142145145149150154156157162165

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C O N T E N T S7.9 Determination of pilkunder firstorder smoothing7.10 Lagrangian ap pro ach to the weak diffusion limit7.1188.18.28.38.48.58.68.78.899.19.29.39.49.59.69.79.89.99.109.119.1210
10.110.210.310.410.510.6
Effect of helicity fluctuations on effective turbuleflt diffusivityBraginskiis theory for weakly asymm etric systemsIntroductionLagrangian transformation of the induction equation whenh=OEffective variables in a Cartesian geometryLagrang ian transformation including weak diffusion effectsDyn am o equatio ns for nearly rectilinear flowCorresponding results for nearly axisymmetric flowsA limitation of the pseudoLagrangian approachMatching conditions and the external fieldStructure and solution of the dynamo equationsDynam o models of a and awtypeFree m odes of the adynamoFre e modes when a,, is anisotropicT he a dynamo in a spherical geom etryT he a dynamo with antisymmetric aFr ee modes of the a w dynamoCo nce ntrated generation and shearSymmetric U ( z ) nd antisymm etric a (2)A model of the galactic dynamoGeneration of poloidal fields by the ae ffectThe am dynam o with periods of stasisNumerical investigations of aw dynamosWaves of helical structure influenced by Coriolis, Lorentzand buoyancy forcesThe momentum equation and some elementary consequencesWaves influenced by Coriolis and L ore ntz forcesModification of a effect by L ore ntz forcesDynam ic equilibration du e to reduction of a effectHelicity generation due to interaction of buoyancy andCoriolis forcesExcitation of magnetostrophic waves by unstable stratification
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viii C O N T E N T S10.7 Instability due to magnetic buoyancy10.8 Helicity generation due to flow over a bumpy surface11 Turbulence with helicity and associated dynamo action11.1 Effects of helicity on homogeneous turbulence11.2 Influence of magnetic helicity conservation in energy transfe r
processes11.3 Modification of inertial range due to largescale magnetic
field11 .4 Nonhglical turbulent dynamo action12 Dynamically consistent dynamos12.1 The Taylor constraint and torsional oscillations12.2 Dynam o action incorporating mean flow effects12 .3 Dynamos driven by buoyancy forces12.4 Reversals of the Earths field, as modelled by coupled disc
dynamos
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References 325Index 337

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PREFACE
Understanding of the process of magnetic field generation byselfinductive action in electrically conducting fluids (or dynamotheory as the subject is commonly called) has advanced dramatically over the last decade. The subject divides naturally into itskinematic and dynamic aspects, neither of which were at all wellunderstood prior to about 1960. The situation has been transformed by the development of the twoscale approach advocated byM. Steenbeck, F. Krause and K.H. Radler in 1966, an approachthat provides essential insights into the effects of fluid motionhaving either a random ingredient, or a spaceperiodic ingredient,over which spatial averages may usefully be defined. Largely as aresult of this development, the kinematic aspect of dynamo theory isnow broadly understood, and recent inroads have been made on themuch more difficult dynamic aspects also.
Although a number of specialised reviews have appeared treatingdynamo theory in both solar and terrestrial contexts, this monograph provides, I believe, the first selfcontained treatment of thesubject in book form. I have tried to focus attention on the morefundamental aspects of the subject, and to this end have included inthe early chapters a treatment of those basic results of magnetohydrodynamics that underly the theory. I have also howeverincluded two brief chapters concerning the magnetic fields of theEarth and the Sun, and the relevant physical properties of thesebodies, and I have made frequent reference in later chapters tospecific applications of the theory in terrestrial and solar contexts.Thus, although written from the point of view of a theoreticallyoriented fluid dynamicist, I hope that the book will be found usefulby graduate students and researchers in geophysics andastrophysics, particularly those whose main concern is geomagnetism or solar magnetism.
My treatment of the subject is based upon a course of lecturesgiven in various forms over a number of years to graduate students

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X P R E F A C Ereading Part I11of the Mathematical Tripos at Cambridge University. I was also privileged to present the course to students of the3me Cycle in Theoretical Mechanics at the UniversitC Pierre etMarie Curie, Paris, during the academic year 19756. The materialof all the chapters, except the difficult chapter 8on the theory of S . I.Braginskii, has been subjected in this way to student criticism, andhas greatly benefited in the process.
The single idea which recurs throughout and which I hope givessome unity to the treatment is the idea of lack of reflexionalsymmetry of a fluid flow, the simplest measure of which is itshelicity. In a sense, this is a book about helicity; the invariance andtopological interpretation of this pseudoscalar quantity are discussed at an early stage (chapter 2) and the central importance ofhelicity in the dynamo context is emphasised in chapters 7 and 8.Helicity is also the theme of chapter 10 (on helical wave motions)and of chapter 11,in which its influence in turbulent flows with andwithout magnetic fields is discussed. A preliminary and muchabbreviated account of some of these topics has already appeared inmy review article (Moffatt, 1976) in volume 16 of Advances inApplied Mechanics.
It is a pleasure to record my gratitude to many colleagues withwhom I have enjoyed discussions and correspondence on dynamotheory; in particular to Willi Deinzer, David Gubbins, Uriel Frisch,Robert Kraichnan, Fritz Krause, Willem Malkus, KarlHeinzRadler, Paul Roberts, Michael Stix and Nigel Weiss; also to GlynRoberts, Andrew Soward and Michael Proctor whose initialresearch it was my privilege to supervise, and who have since madestriking contributions to the subject; and finally to George Batchelor who as Editor of this series, has given constant encouragementand advice. To those who have criticised the manuscript and helpedeliminate errors in it, I offer warm thanks, while retaining fullresponsibility for any errors, omissions and obscurities that mayremain.
I completed the writing during the year 19756 spent at theUniversitC Pierre et Marie Curie, and am grateful to M. PaulGermain and M. Henri Cabannes and their colleagues of theLaboratoire de MCcanique ThCorique for inviting me to work insuch a stimulating and agreeable environment.