Table Of ContentPhysics and Chemistry in Space
Volume 8
Edited by
J. G. Roederer, Denver
J. T. Wasson, Los Angeles
Editorial Board:
H. Elsasser, Heidelberg' G. Elwert, Tiibingen
L. G. Jacchia, Cambridge, Mass.
J. A. Jacobs, Cambridge, England
N. F. Ness, Greenbelt, Md .. W. Riedler, Graz
Akira Hasegawa
Plasma Instabilities
and Nonlinear Effects
With 48 Figures
Springer-Verlag
Berlin Heidelberg New York 1975
L--"V",-!
A.KlRA HASEGAWA
Bell Telephone Laboratories, Murray Hill, NJ 07974/USA
The illustration on the cover is adapted from Fig. 30, which
shows various shapes of instabilities of cylindrical current
pinches.
ISBN -13: 978-3-642-65982-9 e-ISBN -13: 978-3-642-65980-5
DOl: 10.1007/978-3-642-65980-5
This work is subject to copyright. All rights are reserved, whether the whole or part of
the material is concerned, specifically those of translation, reprinting, re~use of ilIus~
tratians, broadcasting, reproduction by photocopying machine or similar means, and
storage in data banks.
Under § 54 of the German Copyright Law, where copies are made for other than
private use, a fee is payable to the publisher, the amount of the fee to be determined
by agreement with the publisher.
© by Springer-Verlag Berlin' Heidelberg 1975.
Softcover reprint of the hardcover 1st edition 1975
Library of Congress Cataloging in Publication Data. Hasegawa, Akira, 1934-. Plasma
instabilities and nonlinear effects. (physics and chemistry in space, 8). Bibliography: p.
1. Plasma instabilities. I. Title. II. Series. QC801.P46 vol 8 [QC718.5.S7]523.01'08s
[530.4'4].74-18072
The use of registered names, tradem~rks, etc. in this publication does not imply, even
in the absence of a specific statement, that such names are exempt from the relevant
protective laws and regulations and therefore free for general use.
To my mother
Kaoru Takata
Preface
In recent years the significant progress in satellite-based observations
of plasma states and associated electromagnetic phenomena in space
has resulted in the accumulation of much evidence of various plasma
instabilities. Today plasma instabilities are believed to be responsible
for electromagnetic radiation as well as for many of the macroscopic
dynamics of plasmas in space.
Most students who begin to study plasma physics are intrigued by
the unstable nature of plasmas compared with other states of matter;
however, they often become frustrated because there are so many in
stabilities. Such frustration explains in part why there is no textbook
which treats this subject exclusively. A description of plasma instabilities
in a systematic way is nontrivial and takes a pertinacious effort. This
book is an attempt to provide a basic introduction on the subject and
covers most of the important instabilities. However, the author must
apologize for any omission of references to contributions of individuals
who deserve more credit.
The reader is assumed to have a general knowledge of plasma physics
obtainable in an undergraduate course. The book is intended to be used
as a reference text on the subject of plasma instabilities at the under
graduate level as well as for a text in a special course in graduate school.
Because the book is part of a series on physics and chemistry in space,
emphasis is placed on plasma instabilities relevant in space plasmas.
However, most of the instabilities discussed here are also applicable to
laboratory plasmas; hence the book should serve also as an introductory
text for scientists interested in controlled thermonuclear fusion. In fact,
most of the instabilities introduced in the text have been discovered in
the course of research in laboratory plasmas.
The book is divided into four chapters. The first is an introduction
to plasma instabilities, where their general concept and physical origin
are presented. They are classified in standard form as microscopic in
stabilities (those associated with a velocity space nonequilibrium) and
macroscopic instabilities (those associated with a coordinate space non
equilibrium). These are described in the second and the third chapters.
Following the theoretical derivations, the implications of the instabilities
VIII Preface
and possible satellite observations are presented for as many cases as
possible. To consider some possible consequences of the instabilities,
associated nonlinear processes are discussed in chapter 4.
MKS units are used throughout the book.
The author wishes to thank Mr. T -Y Huang for his careful checking
of algebra, Ms. C. G. Maclennan for her editing of English and Mr.
B. A. Stevens for collection of references, as also to express his gratitude
to Dr. H. L. Berk, Dr. L. Chen, Prof. H. P. Furth, Dr. L. J. Lanzerotti,
Dr. L. M. Linson, Prof. T. M. O'Neil, Prof. T. Taniuti and Dr. F. D.
Tappert for their valuable discussions and many important suggestions.
The book has grown in part from the lecture notes for graduate
courses in Columbia University Plasma Physics Group by the author
in spring terms of 1972 and 1974, and in part from the author's review
article "Plasma Instabilities in the Magnetosphere", Review of Geo
physics and Space Physics 9, 703 (1971). Discussions with the members
of the Columbia Plasma Physics Group, in particular Profs. C. K. Chu,
T. C. Marshall and A. K. Sen have been very valuable.
The author wishes to extend his appreciation for the encouragement
given by Dr. W. L. Brown and Dr. J. A. Giordmaine of Bell Labora
tories, Prof. C. K. Birdsall of the University of California, Prof. A. J.
Dessler of Rice University and Prof. J. G. Roederer of the University
of Denver.
The author is indebted to Mrs. E. H. Nitchie, Miss D. G. Bracht
and many typists in Bell Laboratories typing pool for their careful
typing of the manuscript.
Finally the author is grateful to his wife Miyoko for her encourage
ment to the project and her gracious hospitality to my companions,
Johann Sebastian Bach, Miles Davis, A. Stockly, and Sachiko Nishida
(and some Scotch) during the preparation of the manuscript.
New Jersey, 1975 AKIRA HASEGAWA
Contents
1. Introduction to Plasma Instabilities 1
1.1 Introduction . . . . 1
1.2 Basic Equations. . . 3
1.3 Dispersion Relations. 12
1.4 Analysis of Instability 17
References for Chapter 1 26
2. Microinstabilities-Instabilities Due to Velocity Space Nonequi-
librium 0 0 ..... 0 ......... 0 ..... 0 . 27
2.1 Introduction 27
2.2 Instabilities Due to Two Humped Velocity Distributions 27
2.2a Electrostatic Instabilities. . 28
2.2b Electromagnetic Instabilities 0 . . 0 . . . 0 . . 44
2.2c Effect of Magnetic Mirror. . . . . . . . . . . 52
2.2d Application to the Field Aligned Current - Anomalous
Resistivity . . . . . 0 . . . . . 0 . . . . . . . 62
2.3 Instabilities Due to Anisotropic Velocity Distributions 68
2.3a Electrostatic Instabilities. . . . . . . . . . . 68
2.3b Electromagnetic Instabilities . . . . . . . . . 79
2.3c Effect of Nonuniformity on Anisotropic Distributions . 83
2.3d Quasi-Linear Diffusion in Velocity Space 88
2.4 Hydromagnetic Instabilities. 92
2.4a Introduction 0 0 . 92
2.4b Mirror Instability. . . 94
2.4c .......... . 98
2.5 Instabilities in Partially Ionized Plasmas 99
205a Introduction. . 0 . . . 0 0 0 . 0 99
2.5b General Properties of Instabilities. . 99
205c Application to Ionospheric Plasma Instabilities . 103
References for Chapter 2. 0 . . 0 0 . . 0 0 . 0 . 106
x Contents
3. Macroinstabilities-Instabilities Due to Coordinate Space Non-
equilibrium . 110
3.1 Introduction 110
3.2 Drift Wave Instabilities 110
3.2a Low f3 Case . . 110
3.2b Medium f3 Case . 113
3.2c High f3 Case 118
3.2d Notes on the Application of Drift Wave Instabilities 120
3.3 Rayleigh-Taylor Instability . . . 122
3.4 Kelvin-Helmholtz Instability . . 125
3.4a Hydromagnetic Instability. 125
3.4 b Electrostatic Instability . . 129
3.5 Current Pinch Instabilities . . . 132
3.5a Instability of Cylindrical Current with Infinite Conduc-
tivity . . . . . . . . . . . . . . . . . . . . . . . . . 132
3.5 b Instability of a Current with Finite Resistivity: Tearing
Mode Instability . . . . 138
References for Chapter 3 ........... . 143
4. Nonlinear Effects Associated with Plasma Instabilities 145
4.1 a Introduction ..... 145
4.1 b Methods of Approach 145
4.2 Nonlinear Effects on Particles. 147
4.2a Introduction 147
4.2b Quasilinear Theory. . . 148
4.2c Resonance Broadening . 154
4.2d Particle Trapping and Wave Overtaking 160
4.3 Nonlinear Effects on Waves . . . . . 165
4.3 a Introduction 165
4.3b Coherent Three-Wave Interactions ... 166
4.3c Three Wave Interactions with Random Phase 174
4.3d Nonlinear Two Wave Interactions Due to Scattering
by Particles-Nonlinear Landau Damping 183
4.4 Nonlinear Waves and Envelopes . . . . 190
4.4 a Introduction .......... 190
4.4b Nonlinear Waves-Solitons and Shocks 191
4.4c Nonlinear Envelopes - Envelope Solitons, Envelope
Shocks, and Envelope Holes . 194
4.4d Modulational Instability 201
References for Chapter 4 204
Contents XI
Appendix 207
General References 210
List of Symbols . 212
Subject Index. . 215
1. Introduction to Plasma Instabilities
1.1 Introduction
Particles in space plasmas generally have a large mean free distance
between interparticle collisions. For example, the mean free distance of
solar wind electrons is known to be approximately 1 A. U. near the earth.
In terms of the mean free time, this means that such a particle will
collide only once every few days. Furthermore, if we note that the energy
equipartition time is longer than the interparticle collision time by the
mass ratio of proton and electron, we can expect that the plasmas in
space, either interplanetary or magnetospheric, are not in thermodyna
mical equilibrium. However, these plasmas can in most cases be assumed
to be under dynamical equilibrium: forces acting on the plasma body
are balanced to zero. This is because the time scale of a dynamic response
of a plasma will be ~W;l or we- 1 where wp and We are plasma and
cyclotron (angular) frequencies respectively. For the example of the
solar wind these numbers are approximately 10-5 and 10-3 sec for
electrons and 10-2 and 1 sec for protons.
The fact that a plasma is not in thermodynamic equilibrium implies
that a certain amount of free energy is stored in the plasma; this energy
may be converted into a violent motion of the plasma or into radiation
of electromagnetic waves. Plasma instability is a process where such
a conversion takes place in a collective way: this leads to the fact that
a small deviation from the dynamic equilibrium becomes the cause of
a further deviation. Mathematically, if we write such a deviation x, the
above statement means that the time rate change of x, dx/dt, is pro
portional to x itself, say, dx/dt =y x. Solving this equation, we obtain
x = el"t, showing that the deviation grows exponentially. The constant y
is called the growth rate of the instability.
There are basically two ways by which a plasma can deviate from
thermodynamic equilibrium. One way is by localizing in space with a
locally higher (or lower) density, temperature, pressure or some other
thermodynamic quantity; the other is by having a velocity distribution
other than the Maxwell-Boltzmann distribution. When an instability
occurs because of the former reason, the plasma as a whole will change
Description:In recent years the significant progress in satellite-based observations of plasma states and associated electromagnetic phenomena in space has resulted in the accumulation of much evidence of various plasma instabilities. Today plasma instabilities are believed to be responsible for electromagnetic
