Krause / Rädler Mean-Field Magnetohydrodynamics and Dynamo Theory

1. Auflage 2016
ISBN: 978-1-4831-5806-8
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 272 Seiten, Web PDF

Mean-Field Magnetohydrodynamics and Dynamo Theory provides a systematic introduction to mean-field magnetohydrodynamics and the dynamo theory, along with the results achieved. Topics covered include turbulence and large-scale structures; general properties of the turbulent electromotive force; homogeneity, isotropy, and mirror symmetry of turbulent fields; and turbulent electromotive force in the case of non-vanishing mean flow. The turbulent electromotive force in the case of rotational mean motion is also considered. This book is comprised of 17 chapters and opens with an overview of the general concept of mean-field magnetohydrodynamics, followed by a discussion on the back-reaction of the magnetic field on motion; the structure of the turbulent electromotive force; homogeneous and two-scale turbulence; turbulent electromotive force in the case of rotational mean motion; and the dynamo problem of magnetohydrodynamics. The dynamo theory, which is based on mean-field magnetohydrodynamics, is explained and its applications to cosmical objects are described. The remaining chapters explore toroidal and poloidal vector fields; a simple model of an a-effect dynamo; and spherical models of turbulent dynamos as suggested by cosmical bodies. This monograph will be of interest to physicists.

Autoren/Hrsg.

Krause, F.

Rädler, K. -H.

Inhaltsverzeichnis

1;Front Cover;1
2;Mean-Field Magnetohydrodynamics and Dynamo Theory
;4
3;Copyright Page;5
4;PREFACE;6
5;Table of Contents;8
6;CHAPTER 1. INTRODUCTION;12
6.1;1.1. Turbulence and large-scale structures;12
6.2;1.2. On the general concept of mean-field magnetohydrodynamics;14
6.3;1.3. Technical remarks;15
7;CHAPTER 2. BASIC IDEAS OF MEAN-FIELD ELECTRODYNAMICS;16
7.1;2.1. Basic equations;16
7.2;2.2. Averaging operations;16
7.3;2.3. The equations for the mean fields;17
7.4;2.4. General properties of the turbulent electromotive force;18
8;CHAPTER 3. ELEMENTARY TREATMENT OF A SIMPLE EXAMPLE;20
8.1;3.1. Assumptions;20
8.2;3.2. Homogeneity, isotropy and mirrorsymmetry of turbulent fields;20
8.3;3.3. Symmetry laws;21
8.4;3.4. The structure of the turbulent electromotive force;23
8.5;3.5. Ohm's law;23
8.6;3.6. Preliminary steps for a determination of a and ß on special assumptions;24
8.7;3.7. The high-conductivity limit;26
8.8;3.8. Applications to the solar convection zone;28
8.9;3.9. The low-conductivity limit;29
8.10;3.10. Illustration of the a-effect and the a-experiment;34
8.11;3.11. The mean square of the fluctuations;43
9;CHAPTER 4. GENERAL METHODS FOR A CALCULATION OF THE TURBULENT ELECTROMOTIVE FORCE;46
9.1;4.1. Introductory remarks. Definitions;46
9.2;4.2. The hierarchy of equations for the correlation tensors;48
9.3;4.3. Second order correlation approximation;49
9.4;4.4. Higher order correlation approximation;51
9.5;4.5. Green's function tensor of the induction equation;52
9.6;4.6. Application of the Green's function tensor to the equations of mean-field electrodynamics;54
9.7;4.7. On the convergence of the correlation approximation;57
10;CHAPTER 5. TWO-SCALE TURBULENCE;61
10.1;5.1. Introductory remarks;61
10.2;5.2. Isotropic tensors;62
10.3;5.3. Structures of the tensors;65
10.4;5.4. Examples for the turbulent electromotive force;66
10.5;5.5. Representation of the tensors;69
11;CHAPTER 6. HOMOGENEOUS TURBULENCE;72
11.1;6.1. Introductory remarks;72
11.2;6.2. Fourier transformation of homogeneous steady random fields;73
11.3;6.3. A basic relation connecting the means of the Fourier transforms with the Fourier transform of the correlation tensor;74
11.4;6.4. Bochner's theorem;75
11.5;6.5. Isotropic turbulence;76
11.6;6.6. Two special cases: Incompressible turbulence and random sound waves;77
11.7;6.7. Fourier transform of the Green's function tensor.Evaluation of integrals in the limiting cases;79
12;CHAPTER 7. MEAN-FIELD ELECTRODYNAMICSFOR HOMOGENEOUS TURBULENCE IN THE CASE OF VANISHING MEAN FLOW;82
12.1;7.1. Determination of the tensor;82
12.2;7.2. The pumping effect;84
12.3;7.3. Dynamo action of homogeneous turbulence;85
12.4;7.4. Determination of the tensor bijk: The turbulent magnetic diffusivity;90
12.5;7.5. Turbulence undergoing the influence of Coriolis forces;93
12.6;7.6. Two-dimensional turbulence;95
12.7;7.7. Higher order correlation approximation:Vainshtein's recurrence formula;98
12.8;7.8. The dispersion relation;102
12.9;7.9. The mean square of the fluctuating magnetic field;106
13;CHAPTER 8. THE TURBULENT ELECTROMOTIVE FORCE IN THE CASE OF NON-VANISHING MEAN FLOW;108
13.1;8.1. Introductory remarks;108
13.2;8.2. The Green's tensor for velocity fields with constant rates of strain;109
13.3;8.3. Representation of the turbulent electromotive force;113
13.4;8.4. On the influence of a mean motion on the correlation tensor;116
13.5;8.5. On the influence of a rotational motion on the correlation tensor;118
14;CHAPTER 9. THE TURBULENT ELECTROMOTIVE FORCE IN THE CASE OF ROTATIONAL MEAN MOTION;120
14.1;9.1. Illustrating examples;120
14.2;9.2. The correlation tensor of an inhomogeneous turbulence;121
14.3;9.3. Determination of the tensor bipq for an inhomogeneous turbulence influenced by Coriolis forces;124
14.4;9.4. Determination of the tensor aip for an inhomogeneous turbulence influenced by Coriolis forces;125
14.5;9.5. Discussion of the tensor;129
14.6;9.6. Further results concerning the tensor;131
15;CHAPTER 10. ON THE BACK-REACTION OF THE MAGNETIC FIELD ON THE MOTIONS;133
15.1;10.1. Introductory remarks;133
15.2;10.2. The influence of a uniform magnetic field on the correlation tensor;134
15.3;10.3. Discussion of the result;136
15.4;10.4. Two-dimensional turbulence;138
15.5;10.5. Applications to the decay of sunspots;140
16;CHAPTER 11. THE DYNAMO PROBLEM OF MAGNETOHYDRODYNAMICS;145
16.1;11.1. The question of the origin of cosmical magnetic fields;145
16.2;11.2. General view of the dynamo problem;148
16.3;11.3. Mathematical formulation of the dynamo problem and simple consequences;149
16.4;11.4. Some necessary conditions for dynamos;153
16.5;11.5. Successful attempts to construct kinematic dynamo models;154
17;CHAPTER 12. FUNDAMENTALS OF THE THEORYOF THE TURBULENT DYNAMO;157
17.1;12.1. Basic concept;157
17.2;12.2. Remarks concerning averaging procedures and the scales of mean and fluctuating quantities;160
18;CHAPTER 13. TOROIDAL AND POLOIDAL VECTOR FIELDS;163
18.1;13.1. Preliminary remarks;163
18.2;13.2. Toroidal and poloidal vector fields in the axisymmetric case;164
18.3;13.3. A special representation of a vector field;165
18.4;13.4. Toroidal and poloidal vector fields in the general case;168
18.5;13.5. Expansions in spherical harmonics;170
19;CHAPTER 14. A SIMPLE MODEL OF AN a-EFFECT DYNAMO;172
19.1;14.1. Description of the model;172
19.2;14.2. Basic equations and their reduction to equations for scalar functions;173
19.3;14.3. The steady case;176
19.4;14.4. The non-steady case;179
19.5;14.5. Considerations involving the back-reaction of the magnetic field on the motions;184
20;CHAPTER 15. SPHERICAL MODELS OF TURBULENT DYNAMOS AS SUGGESTED BY COSMICAL BODIES. GENERAL ASPECTS;188
20.1;15.1. General description of the models;188
20.2;15.2. Basic equations and some of their symmetry properties;189
20.3;15.3. Special magnetic field modes;190
20.4;15.4. Specification of the mean velocity field and the turbulent electromotive force;192
20.5;15.5. A further symmetry property of the basic equations;196
20.6;15.6. Reduction of the basic equations;197
20.7;15.7. Possibilities of dynamo mechanisms;204
20.8;15.8. Further reductions of the basic equations;213
21;CHAPTER 16. SPHERICAL MODELS OF TURBULENT DYNAMOS AS SUGGESTED BY COSMICAL BODIES.RESULTS OF COMPUTATIONS;216
21.1;16.1. General definitions;216
21.2;16.2. Definitions for special types of models;217
21.3;16.3. Models with a2-mechanism;220
21.4;16.4. Models with a. -mechanism;229
21.5;16.5. Models with ..-mechanism;238
22;CHAPTER 17. APPLICATIONS TO COSMICAL OBJECTS;241
22.1;17.1. Observational facts on the magnetic fields of the Earth, the Moonand the planets;241
22.2;17.2. Dynamo theory of the Earth's magnetic field;242
22.3;17.3. Observational facts on magnetic fields at the Sun;246
22.4;17.4. Dynamo theory of the solar cycle;248
22.5;17.5. Observational facts on magnetic fields of stellar objects;252
22.6;17.6. Remarks on dynamo mechanisms on magnetic stars;253
23;BIBLIOGRAPHY;255
24;INDEX;271