H.K.MOFFATT I’ 1 L L ‘G ;1II r r411 1 1 CAMBRIDGE MONOGRAPHS ON MECHANICS AND APPLIED MATHEMATICS GENERAL EDITORS G. K. BATCHELOR, PH.D., F.R.S. Professor of Applied Mathematics in the University of Cambridge J. W. MILES, PH.D. Professor of Applied Mathematics, Universityof California,La Jolla MAGNETIC FIELD GENERATION IN ELECTRICALLY CONDUCTING FLUIDS MAGNETIC FIELD GENERATION IN ELECTRICALLY CONDUCTING FLUIDS H. K. MOFFATT PROFESSOR OF APPLIED MATHEMATICS, UNIVERSITY OF BRISTOL CAMBRIDGE UNIVERSITY PRESS CAMBRIDGE LONDON. NEW YORK. MELBOURNE Published by the Syndicsof the Cambridge University Press The Pitt Building, Trumpington Street, Cambridge CB2 IRP BentleyHouse, 200 Euston Road, London NW12DB 32 East 57th Street, New York, NY 10022, USA 296 Beaconsfield Parade, Middle Park, Melbourne 3206, Australia @ Cambridge University Press 1978 First published 1978 Printed in Great Britain at the University Press, Cambridge Library of Congress Cataloguing in Publication Data Moffatt, Henry Keith, 1935- Magnetic field generation in electrically conducting fluids. (Cambridge monographs on mechanics and applied mathematics) Bibliography: p. 325 1. Dynamo theory (Cosmic physics) I. Title. QC809.M25M63 538 77-4398 ISBN 0 521 21640 0 CONTENTS Preface page ix 1 Introduction and historical background 1 2 Magnetokinematic preliminaries 13 2.1 Structural properties of the B-field 13 2.2 Magnetic field representations 17 2.3 Relations between electric current and magnetic field 23 2.4 Force-free fields 26 2.5 Lagrangian variables and magnetic field evolution 31 2.6 Kinematically possible velocity fields 35 2.7 Free decay modes 36 3 Convection, distortion and diffusion of magnetic field 43 3.1 AlfvBn’s theorem and related results 43 3.2 The analogy with vorticity 46 3.3 The analogy with scalar transport 48 3.4 Maintenance of a flux rope by uniform rate of strain 49 3.5 An example of accelerated ohmic diffusion 50 3.6 Equation for vector potential and flux-function under particular symmetries 51 3.7 Field distortion by differential rotation 53 3.8 Effect of plane differential rotation on an initially uniform field 54 3.9 Flux expulsion for general flows with closed streamlines 62 3.10 Expulsion of poloidal fields by meridional circulation 64 3.11 Generation of toroidal field by differential rotation 65 3.12 Topological pumping of magnetic flux 70 4 The magnetic field of the Earth 76 4.1 Planetary magnetic fields in general 76 4.2 Spherical harmonic analysis of the Earth’s field 79 vi CONTENTS 4.3 Variation of the dipole field over long time-scales 83 4.4 Parameters and physical state of the lower mantle and core 85 4.5 The need for a dynamo theory for the Earth 89 4.6 The core-mantle interface 89 4.7 Precession of the Earth’s angular velocity vector 91 5 The solar magnetic field 94 5.1 Introduction 94 5.2 Observed velocity fields 95 5.3 Sunspots and the solar cycle 96 5.4 The general poloidal magnetic field of the Sun 101 5.5 Magnetic stars 105 6 Laminar dynamo theory 108 6.1 Formal statement of the kinematic dynamo problem 108 6.2 Rate of strain criterion 109 6.3 Rate of change of dipole moment 111 6.4 The impossibility of axisymmetric dyaamo action 113 6.5 Cowling’s neutral point argument 115 6.6 Some comments on the situation B.V A B = 0 117 6.7 The impossibility of dynamo action with purely toroidal motion 118 6.8 The impossibility of dynamo action with plane two- dimensional motion 121 6.9 Rotor dynamos 122 6.10 Dynamo action associated with a pair of ring vortices 131 6.1 1 The Bullard-Gellman formalism 137 6.12 The stasis dynamo of Backus (1958) 142 7 The mean electromotive force generated by random motions 145 7.1 Turbulence and random waves 145 7.2 The linear relation between 8 and B,, 149 7.3 The a -eff ect 150 7.4 Effects associated with the coefficients pijk 154 7.5 First-order smoothing 156 7.6 Spectrum tensor of a stationary random vector field 157 7.7 Determination of aij for a helical wave motion 162 7.8 Determination of ajjfor a random u-field under first-order smoothing 165 CONTENTS vii 7.9 Determination of pilk under first-order smoothing 169 7.10 Lagrangian approach to the weak diffusion limit 170 7.11 Effect of helicity fluctuations on effective turbuleflt diffusivity 175 8 Braginskii’s theory for weakly asymmetric systems 179 8.1 Introduction 179 8.2 Lagrangian transformation of the induction equation when h=O 182 8.3 Effective variables in a Cartesian geometry 185 8.4 Lagrangian transformation including weak diffusion effects 187 8.5 Dynamo equations for nearly rectilinear flow 188 8.6 Corresponding results for nearly axisymmetric flows 190 8.7 A limitation of the pseudo-Lagrangian approach 192 8.8 Matching conditions and the external field 194 9 Structure and solution of the dynamo equations 197 9.1 Dynamo models of a’ and aw-type 197 9.2 Free modes of the a’-dynamo 199 9.3 Free modes when a,, is anisotropic 202 9.4 The a ’-dynamo in a spherical geometry 205 9.5 The a ’-dynamo with antisymmetric a 209 9.6 Free modes of the aw-dynamo 212 9.7 Concentrated generation and shear 216 9.8 Symmetric U(z)and antisymmetric a (2) 219 9.9 A model of the galactic dynamo 221 * 9.10 Generation of poloidal fields by the a-effect 230 9.11 The am-dynamo with periods of stasis 233 9.12 Numerical investigations of aw -dynamos 234 10 Waves of helical structure influenced by Coriolis, Lorentz and buoyancy forces 244 10.1 The momentum equation and some elementary conse- quences 244 10.2 Waves influenced by Coriolis and Lorentz forces 248 10.3 Modification of a -effect by Lorentz forces 252 10.4 Dynamic equilibration due to reduction of a -effect 257 10.5 Helicity generation due to interaction of buoyancy and Coriolis forces 262 10.6 Excitation of magnetostrophic waves by unstable strati- fication 264 viii CONTENTS 10.7 Instability due to magnetic buoyancy 270 10.8 Helicity generation due to flow over a bumpy surface 276 11 Turbulence with helicity and associated dynamo action 280 11.1 Effects of helicity on homogeneous turbulence 280 11.2 Influence of magnetic helicity conservation in energy transfer processes 288 11.3 Modification of inertial range due to large-scale magnetic field 294 11.4 Non-hglical turbulent dynamo action 295 12 Dynamically consistent dynamos 298 12.1 The Taylor constraint and torsional oscillations 298 12.2 Dynamo action incorporating mean flow effects 303 12.3 Dynamos driven by buoyancy forces 307 12.4 Reversals of the Earth’s field, as modelled by coupled disc dynamos 318 References 325 Index 337 PREFACE Understanding of the process of magnetic field generation by self-inductive action in electrically conducting fluids (or ‘dynamo theory’ as the subject is commonly called) has advanced dramati- cally over the last decade. The subject divides naturally into its kinematic and dynamic aspects, neither of which were at all well understood prior to about 1960. The situation has been trans- formed by the development of the two-scale approach advocated by M. Steenbeck, F. Krause and K.-H. Radler in 1966, an approach that provides essential insights into the effects of fluid motion having either a random ingredient, or a space-periodic ingredient, over which spatial averages may usefully be defined. Largely as a result of this development, the kinematic aspect of dynamo theory is now broadly understood, and recent inroads have been made on the much more difficult dynamic aspects also. Although a number of specialised reviews have appeared treating dynamo theory in both solar and terrestrial contexts, this mono- graph provides, I believe, the first self-contained treatment of the subject in book form. I have tried to focus attention on the more fundamental aspects of the subject, and to this end have included in the early chapters a treatment of those basic results of magneto- hydrodynamics that underly the theory. I have also however included two brief chapters concerning the magnetic fields of the Earth and the Sun, and the relevant physical properties of these bodies, and I have made frequent reference in later chapters to specific applications of the theory in terrestrial and solar contexts. Thus, although written from the point of view of a theoretically oriented fluid dynamicist, I hope that the book will be found useful by graduate students and researchers in geophysics and astrophysics, particularly those whose main concern is geomagnet- ism or solar magnetism. My treatment of the subject is based upon a course of lectures given in various forms over a number of years to graduate students X PREFACE reading Part I11 of the Mathematical Tripos at Cambridge Univer- sity. I was also privileged to present the course to students of the 3me Cycle in Theoretical Mechanics at the UniversitC Pierre et Marie Curie, Paris, during the academic year 1975-6. The material of all the chapters, except the difficult chapter 8 on the theory of S. I. Braginskii, has been subjected in this way to student criticism, and has greatly benefited in the process. The single idea which recurs throughout and which I hope gives some unity to the treatment is the idea of ‘lack of reflexional symmetry’ of a fluid flow, the simplest measure of which is its ‘helicity’. In a sense, this is a book about helicity; the invariance and topological interpretation of this pseudo-scalar quantity are discus- sed at an early stage (chapter 2) and the central importance of helicity in the dynamo context is emphasised in chapters 7 and 8.