# Methods in Nonlinear Plasma Theory

## 1st Edition

**Authors:**Ronald Davidson

**eBook ISBN:**9780323153386

**Imprint:**Academic Press

**Published Date:**28th February 1972

**Page Count:**376

## Description

Methods in Nonlinear Plasma Theory is from lectures given in graduate classes in both University of Maryland and University of California at Berkeley. To be able to understand fully the contents in this book, the reader is assumed to be a graduate student with background of classical physics and linear plasma waves and instabilities.

This text is divided into two major parts. Part I deals with the coherent nonlinear phenomena, while Part II discusses the turbulent nonlinear phenomena. Six chapters comprise Part I, where basic equations and methods are described and discussed. Some of these methods are Vlasov-Maxwell equations and Korteweg-de Vries equation. Part II meanwhile has eight chapters that discuss frameworks and theories for weak plasma turbulence. Specifically, the weak turbulence theory is presented as it is applied to electromagnetic wave-particle interactions, nonlinear wave-wave interactions, and nonlinear wave-particle interactions.

This book is a useful reference for students and researchers in the study of classical physics and plasma theory.

## Table of Contents

Preface

Acknowledgments

General References

Part I Coherent Nonlinear Phenomena

Chapter 1. Introduction

1.1 Basic Equations and Methods

1.2 Terminology and Classification of Interactions

1.3 Multiple-Time-Scale Perturbation Analysis of the Van der Pol Equation

References

Chapter 2. The Korteweg-de Vries Equation-A Weakly Nonlinear Theory of Ion Sound Waves

2.1 Introduction and Basic Equation

2.2 Derivation of the Korteweg-de Vries Equation for Nonlinear Ion Sound Waves

2.3 Solution of the Korteweg-de Vries Equation

2.4 Examples and General Properties of the Solution

2.5 Nonlinear Theory of Ion Sound Waves with Collisionless Dissipation

References

Supplementary References

Chapter 3. Large-Amplitude Electron Plasma Oscillations

3.1 One-Dimensional Cold-Plasma Model

3.2 Exact Solution in Lagrangian Variables

3.3 Nonlinear Example with Inversion to Eulerian Variables

3.4 Extensions of the Model

3.5 Modifications Due to Finite Electron Temperature

References

Supplementary References

Chapter 4. Strong Wave-Particle Interactions-Particle Trapping

4.1 Introduction

4.2 Collisionless Damping of Electron Plasma Oscillations

4.3 Large-Amplitude Bernstein-Greene-Kruskal Waves

4.4 Stability of Bernstein-Greene-Kruskal Waves

References

Supplementary References

Chapter 5. Plasma Echoes

5.1 Introduction and Basic Echo Mechanism

5.2 Theory of Second-Order Temporal Echoes

References

Supplementary References

Chapter 6. Coherent Nonlinear Wave-Wave Interactions

6.1 Introduction and Basic Equation

6.2 Wave-Wave Coupling in Macroscopic Plasma Models

6.3 Wave-Wave Coupling in Vlasov Plasma Models

6.4 Nonlinear Theory of Resonant Three-Wave Coupling

6.5 Examples of Resonant Three-Wave Coupling

References

Supplementary References

Part II Turbulent Nonlinear Phenomena

Chapter 7. Statistical Frameworks for Weak Plasma Turbulence

7.1 Basic Definitions and Terminology

7.2 The Random Phase Approximation

7.3 Evolution of Collective Correlations in a Vlasov Ensemble

7.4 Discrete Particle Interactions

7.5 Summary

References

Supplementary References

Chapter 8. Weak Turbulence Theory of Electrostatic Wave-Particle Interactions

8.1 Introduction and Basic Assumptions

8.2 The Weak Turbulence Kinetic Theory

8.3 Weak Turbulence Kinetic Equations for a Multicomponent Plasma

8.4 General Properties of the Kinetic Equations

8.5 Electrostatic Turbulence in a Magnetoplasma

8.6 Phase-Mixing of Free-Streaming Correlations

References

Supplementary References

Chapter 9. Stabilization of the Weak Electron Beam (Bump-in-Tail) Instability

9.1 Introduction and Dispersion Relation

9.2 Stabilization in One Dimension

9.3 Stabilization in Higher Dimensions

9.4 Range of Validity of the Theory

9.5 Computer Simulation of the Weak Bump-in-Tail Instability

9.6 Resonance Broadening

References

Supplementary References

Chapter 10. Stabilization of the Ion Loss-Cone Instability

10.1 Introduction and Basic Assumptions

10.2 Dispersion Relation

10.3 Stabilization Process

References

Chapter 11. Electron Heating by Electron-Ion Beam Instabilities

11.1 Introduction

11.2 Dispersion Relation

11.3 Time Evolution of the Electron-Ion Beam System

11.4 Computer Simulation of the Electron-Ion Beam Instability

References

Supplementary References

Chapter 12. Weak Turbulence Theory of Electromagnetic Wave-Particle Interactions

12.1 The Vlasov-Maxwell Description

12.2 Weak Turbulence Kinetic Theory for Parallel Propagation

12.3 Stabilization of the Electron Whistler Instability

12.4 Stabilization of the Garden-Hose Instability

References

Supplementary References

Chapter 13. Weak Turbulence Theory of Nonlinear Wave-Wave Interactions

13.1 Introduction

13.2 The Hierarchy for Wave Correlations

13.3 The Weak Turbulence Kinetic Theory

13.4 General Properties of the Wave Kinetic Equation

13.5 Exact Solution of the Wave Kinetic Equation in One Dimension

13.6 Kinetics of Resonant Three-Wave Interactions in Three Dimensions

13.7 Kinetic Equation for Resonant Four-Wave Processes

References

Supplementary References

Chapter 14. General Weak Turbulence Theory of Nonlinear Electrostatic Interactions

14.1 Introduction and Basic Assumptions

14.2 Derivation of the Kinetic Equations

14.3 Extension of the Kinetic Equations to the Multicomponent Many-Mode Case

14.4 Properties of the Kinetic Equations

14.5 Nonlinear Landau Damping of Electron Plasma Oscillations

14.6 Extensions of the Weak Turbulence Theory

References

Supplementary References

Appendix A. The Korteweg-de Vries Equation and the Inverse Scattering Problem

Appendix B. Matrix Symmetries

Appendix C. Phase-Mixing of Free-Streaming Correlations

Appendix D. Second-Order Two-Body Correlations in a Weakly Turbulent Plasma

D.1 Evaluation of the Two-Body Correlation Function

D.2 Kinetic Equation for the Waves

D.3 Kinetic Equation for the Particles

Reference

Appendix E. Nonlinear Wave-Particle Coupling Coefficients

E.1 Coupling Coefficient for the Wave Kinetic Equation

E.2 Coupling Coefficient for the Particle Kinetic Equation

Reference

Author Index

Subject Index

## Details

- No. of pages:
- 376

- Language:
- English

- Copyright:
- © Academic Press 1972

- Published:
- 28th February 1972

- Imprint:
- Academic Press

- eBook ISBN:
- 9780323153386