# Mean-Field Magnetohydrodynamics and Dynamo Theory

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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 α-effect dynamo; and spherical models of turbulent dynamos as suggested by cosmical bodies. This monograph will be of interest to physicists.

## Table of Contents

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 α 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 α-Effect and the α-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 α-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 of Computations

16.1. General Definitions

16.2. Definitions for Special Types of Models

16.3. Models with α2-Mechanism

16.4. Models with αω-Mechanism

16.5. Models with δω-Mechanism

Chapter 17 Applications to Cosmical Objects

17.1. Observational Facts on the Magnetic Fields of the Earth, the Moon and the Planets

17.2. Dynamo Theory of the Earth's Magnetic Field

17.3. Observational Facts on Magnetic Fields at the Sun

17.4. Dynamo Theory of the Solar Cycle

17.5. Observational Facts on Magnetic Fields of Stellar Objects

17.6. Remarks on Dynamo Mechanisms on Magnetic Stars

Bibliography

Index

## Product details

- No. of pages: 272
- Language: English
- Copyright: © Pergamon 1980
- Published: January 1, 1980
- Imprint: Pergamon
- eBook ISBN: 9781483158068

## About the Authors

### F. Krause

### K.-H. Rädler

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