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.
Inhalt
Chapter 1 Introduction
1.1. Turbulence and Large-Scale Structures
1.2. On the General Concept of Mean-Field Magnetohydrodynamics
1.3. Technical Remarks
Chapter 2 Basic Ideas of Mean-Field Electrodynamics
2.1. Basic Equations
2.2. Averaging Operations
2.3. The Equations for the Mean Fields
2.4. General Properties of the Turbulent Electromotive Force
Chapter 3 Elementary Treatment of a Simple Example
3.1. Assumptions
3.2. Homogeneity, Isotropy and Mirrorsymmetry of Turbulent Fields
3.3. Symmetry Laws
3.4. The Structure of the Turbulent Electromotive Force
3.5. Ohm's Law
3.6. Preliminary Steps for a 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 Fluctuations
Chapter 4 General Methods for a Calculation of the Turbulent Electromotive Force
4.1. Introductory Remarks. Definitions
4.2. The Hierarchy of Equations for the Correlation Tensors
4.3. Second Order Correlation Approximation
4.4. Higher Order Correlation Approximation
4.5. Green's Function Tensor of the Induction Equation
4.6. Application of the Green's Function Tensor to the Equations of Mean-Field Electrodynamics
4.7. On the Convergence of the Correlation Approximation
Chapter 5 Two-Scale Turbulence
5.1. Introductory Remarks
5.2. Isotropic Tensors
5.3. Structures of the Tensors gij...n
5.4. Examples for the Turbulent Electromotive Force
5.5. Representation of the Tensors gij...n
Chapter 6 Homogeneous Turbulence
6.1. Introductory Remarks
6.2. Fourier Transformation of Homogeneous Steady Random Fields
6.3. A Basic Relation Connecting the Means of the Fourier Transforms with the Fourier Transform of the Correlation Tensor
6.4. Bochner's Theorem
6.5. Isotropic Turbulence
6.6. Two Special Cases: Incompressible Turbulence and Random Sound Waves
6.7. Fourier Transform of the Green's Function Tensor. Evaluation of Integrals in the Limiting Cases
Chapter 7 Mean-Field Electrodynamics for Homogeneous Turbulence in the Case of Vanishing Mean Flow
7.1. Determination of the Tensor aij
7.2. The Pumping Effect
7.3. Dynamo Action of Homogeneous Turbulence
7.4. Determination of the Tensor bijk: The Turbulent Magnetic Diffusivity
7.5. Turbulence Undergoing the Influence of Coriolis Forces
7.G. Two-Dimensional Turbulence
7.7. Higher Order Correlation Approximation: Vainshtein's Recurrence Formula
7.8. The Dispersion Relation
7.9. The Mean Square of the Fluctuating Magnetic Field
Chapter 8 the Turbulent Electromotive Force in the Case of Non-Vanishing Mean Flow
8.1. Introductory Remarks
8.2. The Green's Tensor for Velocity Fields with Constant Rates of Strain
8.3. Representation of the Turbulent Electromotive Force
8.4. On the Influence of a Mean Motion on the Correlation Tensor
8.5. On the Influence of a Rotational Motion on the Correlation Tensor
Chapter 9 the Turbulent Electromotive Force in the Case of Rotational Mean Motion
9.1. Illustrating Examples
9.2. The Correlation Tensor of an Inhomogeneous Turbulence
9.3. Determination of the Tensor bipq for an Inhomogeneous Turbulence Influenced by Coriolis Forces
9.4. Determination of the Tensor aip for an Inhomogeneous Turbulence Influenced by Coriolis Forces
9.5. Discussion of the Tensor aip
9.6. Further Results Concerning the Tensor aip
Chapter 10 on the Back-Reaction of the Magnetic Field on the Motions
10.1. Introductory Remarks
10.2. The Influence of a Uniform Magnetic Field on the Correlation Tensor
10.3. Discussion of the Result
10.4. Two-Dimensional Turbulence
10.5. Applications to the Decay of Sunspots
Chapter 11 the Dynamo Problem of Magnetohydrodynamics
11.1. The Question of the Origin of Cosmical Magnetic Fields
11.2. General View of the Dynamo Problem
11.3. Mathematical Formulation of the Dynamo Problem and Simple Consequences
11.4. Some Necessary Conditions for Dynamos
11.5. Successful Attempts to Construct Kinematic Dynamo Models
Chapter 12 Fundamentals of the Theory of the Turbulent Dynamo
12.1. Basic Concept
12.2. Remarks Concerning Averaging Procedures and the Scales of Mean and Fluctuating Quantities
Chapter 13 Toroidal and Poloidal Vector Fields
13.1. Preliminary Remarks
13.2. Toroidal and Poloidal Vector Fields in the Axisymmetric Case
13.3. A Special Representation of a Vector Field
13.4. Toroidal and Poloidal Vector Fields in the General Case
13.5. Expansions in Spherical Harmonics
Chapter 14 a Simple Model of an a-Effect Dynamo
14.1. Description of the Model
14.2. Basic Equations and Their Reduction to Equations for Scalar Functions
14.3. The Steady Case
14.4. The Non-Steady Case
14.5. 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. Specification 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
15.8. Further Reduction of the Basic Equations
Chapter 16 Spherical Models of Turbulent Dynamos as Suggested By Cosmical Bodies Results …