MEAN-FIELD MAGNETOHYDRODYNAMICS AND DYNAMO THEORY F.KRAUSE and K.-H.RADLER Zentralinstitut für Astrophysik der Akademie der Wissenschaften der DDR H Tedmisdie Hochschule Dormstodt Fachbereich Mechanik PERGAMON PRESS Oxford • New York • Toronto • Sydney • Paris • Frankfurt CONTENTS Chapter.1 Introduction 1.1. 1.2. 1.3.' 11 Turbulence and large-scale structures , On the gener,al ooncept of mean-field rnagnetohydrodynainics Technical remarks v Chapter 2 Basic ideas of mean-field electrodynamics . .- 2.1. 2.2. 2.3. 2.4. Chapter 3 11 13 14 - .. . • i , 15 . . . .. Basic equations ,..Averaging operations \ The equations for the mean f ields General properties, of the turbulent electromotive force , ; Elementary treatment of a simple example . . . : 19 3.1. 3.2. 3.3. 3.4. Assunxptions Homogeneity, isotropy and mirrorsymmetry of turbulent fields, Symmetry laws .' '. The structure of the turbulent eleotromotive force 3.5. Ohm'slaw .....; ••:••••. : 3.6. . Preliminary steps for &• determination of a and ß on Special assumptions . . . 3.7. The high-conductivity limit 3.8. Applications to the solar' convection zone . 3.9. The low-conductivity limit • 3.10. Illustration of the a-effect and the a-experiment 3.11. The mean Square of the f luctuations Chapter 4 4.1. 4.2. 4.3. 4.4. • 4.5. 4.6. 4.7. 15 15 16 17 General methods for a calculation of the turbulent electromotive force ' 19 19 20 22 22 23 25 ,27 28 33 42 45 Intröductory remarks. Definitions •,•••• 45 The hierarchy of equations for the correlation tensors 47, Second order correlation approximation 48 Higher order correlation approximation 50 Green's function tensor of the jnduction equation 51 Application of the Green's function tensor to the equations of mean-field electrodynamics • 53 On the convergence of the correlation approximation 56 8 Chapter 5 5.1. 5.2. 5.3. 5.4". 5.5. Contents Two-scalc turbulence Introductory remarks Isotropie tensors : Structures of the tensors ^-...B Examples for the turbulent electromotive force Representation of the tensors ga„M 60 60 61 64 65 68 Chapter 6 ' Homogeneous turbulence .• 71 6.1. , 6.2. 6.3. 71 72 .• 6.4. 6.5. 6.6. 6.7. Introductory remarks Fourier transformation of homogeneous steady random fields A basic relation connecting the means of the Fourier transforms with the Fourier transform of the correlation tensor Bochner's theorem Isotropie turbulence : Two Special cases: Incompressible turbulence and random sound waves .. Fourier transform of the Green's function tensor. Evaluation of integrals in the limiting cases 73 74 75 76 78 Chapter 7 Mean-field electrodynamics for homogeneous turbulence in the .7.1. 7.2. 7.37.4. ' 7.5. 7.6. • 7.7. 718. 7.9. << Chapter 8 The turbulent electromotive force in the case of non-vanishing mean flow 107 8.1. " Introductory remarks • 107 8.2. The Green's tensor for velocity fields with oonstant rates of strahl 108 8.3. Representation of the turbulent electromotive force 112 8.4. On the influence of a mean motion on the correlation tensor 115 8.5. On the influence of a rotational motion on the correlation tensor 117 Chapter 9 9.1. 9.2. 9.3. 9.4. | ' case ol vanishing mean flow 81 Determination of the tensor a^j 81 The pumping effect 83 Dynamo action of homogeneous turbulence 84 Determination öf the tensor 6;^.: The turbulent magnetic diffusivity 89 Turbulence undergoing the influence of Coriolis forces 92 Two-dimensional turbulence 94 Higher order correlation approximation: Vainshtein's recurrence formula . 97 The dispersion relation -. 101 The mean Square of the fluetuating magnetic field : 105 9.5. 9.6. The turbulent electromotive force in the case of rotational mean motion 119 IUustrating examples 119 The correlation tensor of an inhomogeneous turbulence s 120 Determination of the tensor b^ for an inhomogeneous turbulence influenced by Coriolis forces • 123 Determination of the tensor «jj, for an inhomogeneous turbulence influenced • by Coriolis forces 124 Discussion of the tensor a^ 128 Further results concerning the tensor a^p 130 Contents , Chapter 10 10.1. 10.2. 10.3. 10.4. 10.5. The dynamo problem of magnetohydrodynamics . .-. Fundamentals of the theory of the turbulent dynamo v 144 147 148 152 153 156 • . Toroidal and poloidal vector fields Preliminary remarks \ Toroidal and poloidal vector fields in the axisymmetric case A Special representation of a vector field Toroidal and poloidal vector fields in the general case Expansions in spherical harmonics Chapter 14 14.1. 14.2. 14.3. 14.4. 14.5. 132 133 135 137 139 Basic concept . 156 Remarks concerning averaging procedures and the scales of mean and f luctuating quantities .159 Chapter 13 13.1. 13.2. 13.3. . 13.4. 13.5. 132 144 The question of the origin of cosmical magnetic fields General view of the dynamo problem Mathematical formulation of the dynamo problem and simple consequences Some necessary conditions for dynamos Successful attempts to construct kinematic dynamo models Chapter 12 12.1. 12.2. On the back-reaction of the magnetic field on the motions Introductory remarks The influence of a uniform magnetic field-on the correlation tensor Discussion of the result Two-dimensional turbulence ' Applications to the decay of sunspots . .' Chapter 11 11.1. 11.2. 11.3. 11.4. 11.5. 9 A simple model of an a-effect dynamo Description of the model Basic equations and their reduotion to equations for.scalar functions . . . . The steady case The non-steady case . Considerations involving the back-reaction of the magnetic field on the motions .'. Chapter 15 Spherical models of turbulent dynamos as suggested ,by cosmical bodies. General aspects 15.1. General,description of the models 15.2. Basic equations and some of their symmetry properties • 15.3. Special magnetic field modes * 15.4. Speoification of the mean velocity field and the turbulent electromotive force 15.5. A further symmetry property of the basic equations 15.6. Reduction of the basic equations 15.7. , Possibilities of dynamo mechanisms . .' 16.8. Further reduction of the basic equations •. » 162 162 163 164 167 169 171 171 172 175 178 183 187 187 188 189 191 195 196 203 212 10 , Contents Chapter 16 Spherical models, of turbulent dynamos as suggested by cosmical bodies. Besnlts of computations 215 16.1. General definitions 215 16.2. Definitions for Special types of models 216 16.3. • Models with <x2-mechanism 219 16.4. Models with aco-mechanism ,•.•••. 228 16.5. Models with (5cu-mechanism 237 Chapter 17 Applications to cosmical objects : 240 17.1. Observational facts on the magnetic fields of the Earth, the Moon and the planets 240 17.2. Dynamo theory of the Earth's magnetic field ......: 241 17.3. Observational facts on magnetic fields at the Sun 245 17.4. Dynamo theory of the solar cycle .' 247 17.5. Observational facts on magnetic fields öf stellar objects 251 17.6. Remarks on dynamo mechanisms on magnetic stars : ,252 Bibliography Index 254 •. : '. 270