Mean-Field Magnetohydrodynamics and Dynamo Theory (eBook)

Front Cover 1
Mean-Field Magnetohydrodynamics and Dynamo Theory 
4 
Copyright Page 5
PREFACE 6
Table of Contents 8
CHAPTER 1. INTRODUCTION 12
1.1. Turbulence and large-scale structures 12
1.2. On the general concept of mean-field magnetohydrodynamics 14
1.3. Technical remarks 15
CHAPTER 2. BASIC IDEAS OF MEAN-FIELD ELECTRODYNAMICS 16
2.1. Basic equations 16
2.2. Averaging operations 16
2.3. The equations for the mean fields 17
2.4. General properties of the turbulent electromotive force 18
CHAPTER 3. ELEMENTARY TREATMENT OF A SIMPLE EXAMPLE 20
3.1. Assumptions 20
3.2. Homogeneity, isotropy and mirrorsymmetry of turbulent fields 20
3.3. Symmetry laws 21
3.4. The structure of the turbulent electromotive force 23
3.5. Ohm's law 23
3.6. Preliminary steps for a determination of a and ß on special assumptions 24
3.7. The high-conductivity limit 26
3.8. Applications to the solar convection zone 28
3.9. The low-conductivity limit 29
3.10. Illustration of the a-effect and the a-experiment 34
3.11. The mean square of the fluctuations 43
CHAPTER 4. GENERAL METHODS FOR A CALCULATION OF THE TURBULENT ELECTROMOTIVE FORCE 46
4.1. Introductory remarks. Definitions 46
4.2. The hierarchy of equations for the correlation tensors 48
4.3. Second order correlation approximation 49
4.4. Higher order correlation approximation 51
4.5. Green's function tensor of the induction equation 52
4.6. Application of the Green's function tensor to the equations of mean-field electrodynamics 54
4.7. On the convergence of the correlation approximation 57
CHAPTER 5. TWO-SCALE TURBULENCE 61
5.1. Introductory remarks 61
5.2. Isotropic tensors 62
5.3. Structures of the tensors 65
5.4. Examples for the turbulent electromotive force 66
5.5. Representation of the tensors 69
CHAPTER 6. HOMOGENEOUS TURBULENCE 72
6.1. Introductory remarks 72
6.2. Fourier transformation of homogeneous steady random fields 73
6.3. A basic relation connecting the means of the Fourier transforms with the Fourier transform of the correlation tensor 74
6.4. Bochner's theorem 75
6.5. Isotropic turbulence 76
6.6. Two special cases: Incompressible turbulence and random sound waves 77
6.7. Fourier transform of the Green's function tensor.Evaluation of integrals in the limiting cases 79
CHAPTER 7. MEAN-FIELD ELECTRODYNAMICSFOR HOMOGENEOUS TURBULENCE IN THE CASE OF VANISHING MEAN FLOW 82
7.1. Determination of the tensor 82
7.2. The pumping effect 84
7.3. Dynamo action of homogeneous turbulence 85
7.4. Determination of the tensor bijk: The turbulent magnetic diffusivity 90
7.5. Turbulence undergoing the influence of Coriolis forces 93
7.6. Two-dimensional turbulence 95
7.7. Higher order correlation approximation:Vainshtein's recurrence formula 98
7.8. The dispersion relation 102
7.9. The mean square of the fluctuating magnetic field 106
CHAPTER 8. THE TURBULENT ELECTROMOTIVE FORCE IN THE CASE OF NON-VANISHING MEAN FLOW 108
8.1. Introductory remarks 108
8.2. The Green's tensor for velocity fields with constant rates of strain 109
8.3. Representation of the turbulent electromotive force 113
8.4. On the influence of a mean motion on the correlation tensor 116
8.5. On the influence of a rotational motion on the correlation tensor 118
CHAPTER 9. THE TURBULENT ELECTROMOTIVE FORCE IN THE CASE OF ROTATIONAL MEAN MOTION 120
9.1. Illustrating examples 120
9.2. The correlation tensor of an inhomogeneous turbulence 121
9.3. Determination of the tensor bipq for an inhomogeneous turbulence influenced by Coriolis forces 124
9.4. Determination of the tensor aip for an inhomogeneous turbulence influenced by Coriolis forces 125
9.5. Discussion of the tensor 129
9.6. Further results concerning the tensor 131
CHAPTER 10. ON THE BACK-REACTION OF THE MAGNETIC FIELD ON THE MOTIONS 133
10.1. Introductory remarks 133
10.2. The influence of a uniform magnetic field on the correlation tensor 134
10.3. Discussion of the result 136
10.4. Two-dimensional turbulence 138
10.5. Applications to the decay of sunspots 140
CHAPTER 11. THE DYNAMO PROBLEM OF MAGNETOHYDRODYNAMICS 145
11.1. The question of the origin of cosmical magnetic fields 145
11.2. General view of the dynamo problem 148
11.3. Mathematical formulation of the dynamo problem and simple consequences 149
11.4. Some necessary conditions for dynamos 153
11.5. Successful attempts to construct kinematic dynamo models 154
CHAPTER 12. FUNDAMENTALS OF THE THEORYOF THE TURBULENT DYNAMO 157
12.1. Basic concept 157
12.2. Remarks concerning averaging procedures and the scales of mean and fluctuating quantities 160
CHAPTER 13. TOROIDAL AND POLOIDAL VECTOR FIELDS 163
13.1. Preliminary remarks 163
13.2. Toroidal and poloidal vector fields in the axisymmetric case 164
13.3. A special representation of a vector field 165
13.4. Toroidal and poloidal vector fields in the general case 168
13.5. Expansions in spherical harmonics 170
CHAPTER 14. A SIMPLE MODEL OF AN a-EFFECT DYNAMO 172
14.1. Description of the model 172
14.2. Basic equations and their reduction to equations for scalar functions 173
14.3. The steady case 176
14.4. The non-steady case 179
14.5. Considerations involving the back-reaction of the magnetic field on the motions 184
CHAPTER 15. SPHERICAL MODELS OF TURBULENT DYNAMOS AS SUGGESTED BY COSMICAL BODIES. GENERAL ASPECTS 188
15.1. General description of the models 188
15.2. Basic equations and some of their symmetry properties 189
15.3. Special magnetic field modes 190
15.4. Specification of the mean velocity field and the turbulent electromotive force 192
15.5. A further symmetry property of the basic equations 196
15.6. Reduction of the basic equations 197
15.7. Possibilities of dynamo mechanisms 204
15.8. Further reductions of the basic equations 213
CHAPTER 16. SPHERICAL MODELS OF TURBULENT DYNAMOS AS SUGGESTED BY COSMICAL BODIES.RESULTS OF COMPUTATIONS 216
16.1. General definitions 216
16.2. Definitions for special types of models 217
16.3. Models with a2-mechanism 220
16.4. Models with a. -mechanism 229
16.5. Models with ..-mechanism 238
CHAPTER 17. APPLICATIONS TO COSMICAL OBJECTS 241
17.1. Observational facts on the magnetic fields of the Earth, the Moonand the planets 241
17.2. Dynamo theory of the Earth's magnetic field 242
17.3. Observational facts on magnetic fields at the Sun 246
17.4. Dynamo theory of the solar cycle 248
17.5. Observational facts on magnetic fields of stellar objects 252
17.6. Remarks on dynamo mechanisms on magnetic stars 253
BIBLIOGRAPHY 255
INDEX 271