Non-Linear Theory and Fluctuations. Plasma Electrodynamics

312 Pages·1975·8.76 MB·English

by  A. I. Akhiezer, I. A. Akhiezer and R. V. Polovin (Auth.)

INTERNATIONAL SERIES OF MONOGRAPHS IN
NATURAL PHILOSOPHY
GENERAL EDITOR: D. TER HAAR
VOLUME 80
PLASMA ELECTRODYNAMICS 2
PLASMA
ELECTRODYNAMICS
VOLUME 2. NON-LINEAR THEORY AND FLUCTUATIONS
A. I. Akhiezer, I. A. Akhiezer,
R. V. Polovin, A. G. Sitenko and K. N. Stepanov
Translated by D. ter Haar
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First edition 1975
Library of Congress Catalog Card No. 74-3323
Printed in Hungary
ISBN 0 08 018016 7
Contents of Volume 1. Linear Theory
Chapter 1. Kinetic and Hydrodynamic Methods of Describing a Plasma
1.1. Kinetic equations hierarchy
1.2. The Vlasov equation
1.3. The pair correlation function of an equilibrium plasma and the Landau collision integral
1.4. Relaxation of a plasma
1.5. The hydrodynamical description of a plasma
Chapter 2. Small Amplitude Magneto-hydrodynamic Waves
2.1. Magneto-sound and Alfven waves
2.2. Characteristics of the magneto-hydrodynamical equations
Chapter 3. Simple Waves and Shock Waves in Magneto-hydrodynamics
3.1. Simple waves
3.2. Discontinuities |
3.3. Stability and structure of shock waves
3.4. Study of discontinuities
Chapter 4. High-frequency Oscillations in an Unmagnetized Plasma
4.1. Hydrodynamical theory of high-frequency oscillations of an unmagnetized plasma
4.2. Kinetic theory of longitudinal plasma oscillations
4.3. Kinetic theory of electromagnetic waves in a plasma
Chapter 5. Oscillations of a Plasma in a Magnetic Field
5.1. Hydrodynamical theory of oscillations of a plasma in a magnetic field
5.2. Kinetic theory of plasma oscillations in a magnetic field
5.3. Damping of high-frequency electromagnetic waves in a magneto-active plasma
5.4. Absorption of Alfven and fast magneto-sound waves
5.5. Low-frequency oscillations of a hot plasma in a magnetic field
5.6. Cyclotron waves in a plasma for the case of quasi-transverse propagation
5.7. Cyclotron waves in the case of transverse propagation
Chapter 6. Interaction Between Charged Particle Beams and a Plasma. Stable and Unstable Particle
Distributions in a Plasma
6.1. Interaction of charged particle beams with the oscillations of an unmagnetized plasma
6.2. Interaction of a charged particle beam with plasma oscillations in a magnetic field
6.3. Excitation of electromagnetic waves in a plasma by oscillator beams
6.4. Excitation of electromagnetic waves in a plasma by relativistic charged particle beams
6.5. General criteria for the stability of particle distributions in a plasma
6.6. Absolute and convective instabilities
Chapter 7. Oscillations of a Partially Ionized Plasma
7.1. Electron distribution function and high-frequency electron oscillations in an external electrical
field
7.2. Ion-sound oscillations in a strong electrical field
7.3. Low-frequency oscillations of a partially ionized plasma
V l l l
Preface
THE properties of plasmas as a specific state of matter are to an important extent determined
by the fact that there are between the particles which constitute the plasma electromagnetic
forces which act over macroscopic distances. Processes occurring in a plasma are therefore
as a rule accompanied by the excitation of electromagnetic fields which play a fundamental
role in the way these processes develop.
The electromagnetic interactions which extend over macroscopic distances show up first
of all in the occurrence in the plasma of collective oscillations in which a large number of
particles takes part simultaneously. The existence of these specific collective electromagnetic
oscillations is just as much a characteristic of a plasma as a specific state of matter as, for
instance, the crystalline ordering is for the solid state of matter.
This explains the place occupied in plasma physics by plasma electrodynamics, that is,
the theory of electromagnetic fields in a plasma—and, in the first instance, the theory of
electromagnetic oscillations of a plasma—and the theory of macroscopic electrical and
magnetic properties of a plasma.
Such problems as the theory of magnetic traps, the problem of plasma heating by external
fields or currents, and the theory of instabilities in a non-uniform plasma belong also to the
field of plasma electrodynamics in its widest sense. We shall not consider these problems in
the present book—not because they are not important; to the contrary, they are of great
importance. We restrict ourselves to an exposition of the theory of the electromagnetic
properties of a uniform plasma as this theory is the basis of the whole of plasma electro­
dynamics.
Although there are several monographs (see, for example, Alfven, 1950; Artsimovich,
1963; Akhiezer, Akhiezer, Polovin, Sitenko, and Stepanov, 1967; Vedenov, 1965; Ginz-
burg, 1970; Cowling, 1957; Kulikovskii and Lyubimov, 1962; Leontovich, 1965, 1966,
1967, 1968, 1970; Silin and Rukhadze, 1961; Spitzer, 1956; Stix, 1962; Tsytovich, 1970)
devoted to the problems of plasma electrodynamics, we decided all the same to write
yet another book on this topic having in mind to give the theory of both low- and
high-frequency oscillations—without restricting ourselves to small amplitude oscillations
only—from a unified point of view and to give the basic fundamental applications of
this theory.
The book starts with an exposition of the general methods of describing a plasma. Chapter
1 is devoted to this problem; in this chapter we construct the BBGKY-hierarchy of kinetic
equations, introduce the self-consistent field, and introduce the Vlasov kinetic equation
and the Landau collision integral. We give an account of Boltzmann's //-theorem as applied
to a plasma and study the problem of the relaxation of a plasma. Finally, in that chapter
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PREFACE
we elucidate the transition from a kinetic to a hydrodynamic description of a plasma and
derive the equations of magneto-hydrodynamics.
The methods for describing a plasma which we have discussed allow us then to start a
detailed study of both low- and high-frequency plasma oscillations.
We start with the theory of low-frequency oscillations in the case of frequent collisions
when the concise hydrodynamic description of a plasma suffices. Chapters 2 and 3 are
devoted to the low-frequency oscillations.
We give in Chapter 2 the linear theory of magneto-hydrodynamic waves. We define
there phase velocities, damping, and polarization of different waves and study the conic
refraction of magneto-hydrodynamic waves and the excitation of these waves, as well as
the problem of the formation of lacunae when two-dimensional excitations propagate from
a point source. Finally, we study the characteristics of magneto-hydrodynamic flow.
After that we turn to non-linear magneto-hydrodynamic waves, both simple waves and
shock waves (Chapter 3). Here we study the distortion of the profile of a simple wave leading
to the formation of discontinuities. We integrate the equations for simple waves and, in
particular, we evaluate the Riemann invariants.
Then follows an exposition of the theory of shock waves. We prove the Zemplen theorem
and study simple and shock waves in relativistic magneto-hydrodynamics. We put the pro­
blem of the evolutionarity and structure of shock waves. Finally, we solve the problem of the
formation and splitting-up of an arbitrary discontinuity in magneto-hydrodynamics.
Having studied magneto-hydrodynamic waves for the case of frequent collisions we turn
to a consideration of another limiting case—oscillations in a collisionless plasma. Chapters
4 and 5 deal with this problem. In the first of these chapters we give the theory of oscillations
in an unmagnetized plasma, and in the second one the theory of oscillations in a collisionless
plasma in an external magnetic field.
Chapter 4 starts with an exposition of the theory of oscillations in a collisionless plasma
in the hydrodynamic approximation and then these oscillations are studied using a kinetic
equation. The spectra of both the high- and the low-frequency oscillations (Langmuir
waves and ion-sound waves) are studied in detail. We consider the collisionless (Landau)
damping of the oscillations and we solve the problem of the anomalous skin effect.
In Chapter 5 we study in detail the spectra and damping of oscillations in a collisionless
magneto-active plasma. At the start of the chapter we consider oscillations in a "cold"
magneto-active plasma. Then we determine the dielectric tensor of a magneto-active plasma,
using a kinetic equation, and we introduce a dispersion relation for electromagnetic waves,
taking spatial dispersion, caused by the thermal motion of the electrons and ions in the
plasma, into account. We find the frequencies and damping rates (Cherenkov and cyclotron
damping) of practically all branches of the oscillations which can propagate in a magneto-
active plasma with a Maxwell particle velocity distribution—the ordinary, fast and slow
extra-ordinary, fast magneto-sound, and Alfven waves, fast and slow ion-sound oscillations,
electron-sound oscillations in a non-isothermal plasma, and different branches of electron
and ion cyclotron waves.
Having studied the oscillation spectra in an equilibrium plasma, we turn to the study of
oscillations in a non-equilibrium, uniform plasma (Chapter 6).
First of all we study the interaction of a beam of charged particles with the oscillations
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of an unmagnetized plasma and show that the plasma-beam system is unstable, that is, that
the interaction between the beam particles and the plasma oscillations leads to an exponen­
tial growth in time of a small initial perturbation. We then find the growth rates for different
kinds of oscillations, consider the problem of the stability of a plasma in an electric field,
and study the excitation of non-potential (electromagnetic) waves in a plasma with aniso-
tropic particle velocity distributions. We study the interaction of charged particle currents
with slow waves in a magneto-active plasma (the particles in the currents are characterized
either by isotropic or by anisotropic distribution functions). Finally, we consider the ex­
citation of electromagnetic waves in a plasma by currents of relativistic particles.
Having studied the interaction of charged particle currents with the plasma, we elucidate
the general criteria of the stability of different particle distributions in a plasma. We con­
sider separately an unmagnetized plasma and a plasma in an external magnetic field. We
solve the problem of the two-beam instability.
Concluding Chapter 6 we study the general problem of the nature of the instability, give
a definition of absolute and convective instabilities, and establish criteria for those two
kinds of instability. We also establish criteria for the amplification and blocking of waves
and,f inally,c onsider the global instability caused by the reflection of waves from the system
boundaries.
The problem of the interaction between charged particle currents and the plasma is
related to the problem of the behaviour of a partially ionized plasma in an external electric
field. As the stationary states of such a plasma are characterized by a directed motion of
the electrons relative to the ions there can arise in such a plasma an instability analogous
to the beam instability of a collisionless plasma.
Having studied the interaction of charged particle currents with the plasma we consider
the oscillations of a partially ionized plasma in an external electric field (both with and
without an external magnetic field). This problem is treated in Chapter 7. We derive there
the kinetic equation describing the electron component of a partially ionized plasma in
external electric and magnetic fields and we determine the stationary electron distribution
function in such a plasma (Druyvesteyn-Davydov distribution). We then study high-
frequency (transverse electromagnetic and Langmuir), ion-sound, and magneto-sound
oscillations and show that ion-sound and magneto-sound oscillations in an external electric
field turn out to be growing oscillations. Finally, we study in Chapter 7 the peculiar oscilla­
tions of a partially ionized plasma—the ionization-recombination oscillations in which not
only the charged particle density, but also their total number changes.
In Chapter 8 we turn again to the study of a completely ionized plasma. In this chapter
we study the non-linear oscillations in such a plasma (in contrast to Chapters 4 to 6 in
which we restricted our considerations to oscillations with a small amplitude). We discuss
here non-linear high-frequency waves in a cold plasma, Langmuir waves in a non-relativistic
plasma, and longitudinal, transverse, and coupled longitudinal-transverse waves in a
relativistic plasma. We study non-linear waves in a plasma in which the average electron
energy appreciably exceeds the average ion energy (ion-sound and magneto-sound waves
off initea mplitude) and consider both simple (Riemann) waves and waves with a stationary
profile (periodic, isolated, and quasi-shock waves with an oscillatory structure). We show
that the nature of simple and stationary waves depends greatly on the electron velocity
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distribution. Finally, we study in Chapter 8 non-linear low-frequency waves in a cold
magneto-active plasma.
Chapter 9 is devoted to a study of oscillations in the quasi-linear approximation in which
the simplest non-linear effect is taken into account—the influence of oscillations on reson­
ance particles. We first consider the interaction between resonance particles with longitudinal
oscillations of an unmagnetized plasma and we give the derivation of the basic equation
of a quasi-linear theory—the particle diffusion equation in velocity space. We then consider
the quasi-linear relaxation process which leads to the formation of a plateau on the dis­
tribution function of resonance particles and study the quasi-linear wave transformation.
In the same chapter we consider the quasi-linear theory of the interaction between resonance
particles and the oscillations of a magneto-active plasma and study the problem of the
quasi-linear relaxation of wave packets for the cases of cyclotron and Cherenkov resonance.
Finally, we consider the influence of collisions on the quasi-linear relaxation process and
on the damping of oscillations.
Having expounded the quasi-linear theory, which describes the effects of the first approx­
imation in terms of the plasma wave energy, we turn to a study of the processes of higher
order in the energy of the oscillations: the interaction between waves and waves and the
non-linear interaction of waves and particles. This is the subject of Chapter 10 in which
we obtain a kinetic equation for waves which takes into account three-wave processes and
the non-linear interaction between waves and particles (sometimes called the non-linear
Landau damping). We then study turbulent processes in which Langmuir waves take part:
their interaction with ion sound and the decay instability and non-linear damping of
Langmuir waves, and we study in detail ion-sound turbulence which occurs in a plasma
with a directed motion of electrons relative to ions. Finally, we consider the interaction
between Alfven and magneto-sound waves in a magneto-active plasma.
The last three chapters of the book deal with the theory of fluctuations and of the wave
and particle-scattering processes in a plasma caused by the fluctuations.
We give in Chapter 11 the theory of electromagnetic fluctuations in a plasma. We start
with the derivation of the general fluctuation-dissipation relation which establishes a connec­
tion between the spectral distribution of the fluctuations and the energy dissipation in the
medium; we use this relation to determine the fluctuations first in an equilibrium and then
in a two-temperature plasma, both for an unmagnetized plasma and for a plasma in a mag­
netic field.
We then develop the theory of fluctuations in a non-equilibrium plasma and a kinetic
theory of fluctuations; we find the fluctuations in the particle-distrubution function and
consider the critical fluctuations near the instability limits of the plasma and study the
fluctuations in the plasma-beam system. We elucidate how one can proceed to a hydrody-
namic theory of fluctuations and, finally, we study fluctuations in a partially ionized plasma
in an electric field.
Chapter 12 is devoted to the theory of scattering processes and the transformation of
waves in a plasma. We study here the scattering of electromagnetic waves in an unmagne­
tized plasma and determine the spectral distribution of the scattered radiation. We consider
critical opalescence connected with the scattering of waves in a plasma near the limits of
instability, and we study the transformation of transverse and longitudinal waves in an
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unmagnetized plasma and also the spontaneous emission in an non-equilibrium plasma.
We give the theory of incoherent reflection of electromagnetic waves from a plasma. We
study scattering and transformation of waves in a magnetoactive plasma, in a partially
ionized plasma in an external electric field, and in a turbulent plasma. Finally, we discuss
echo effects in a plasma; these are connected with undamped oscillations of particle distri­
bution functions in a plasma.
In Chapter 13 we study the scattering of charged particles in a plasma. We determine
here the polarization energy losses when charged particles move in a plasma; we find the
energy losses caused by the fluctuations of the field in the plasma, and we determine the
coefficients of dynamic friction in diffusion. We also study the propagation of charged par­
ticles through a magneto-active plasma and the interaction of charged particles with a non-
equilibrium plasma, as well as the scattering of particles by critical fluctuations and the
interaction between particles and a turbulent plasma.
We are well aware that the problems considered by us do not cover the complete theory,
even of a uniform plasma, and that we have not given equal weight to the different pro­
blems. However, this is apparently unavoidable when writing a relatively large book. A very
apt quotation comes from one of the best books on elementary particle theory (Bernstein,
1968): "No doubt another physicist writing the same book would have emphasized different
aspects of the subject or would have treated the same aspects differently. One of the few
pleasures in writing such a book is that the author can present the subject as he would like
to see it presented . . . and if this encourages someone else to write a better book, then the
present author will be among its most enthusiastic readers."
The authors express their gratitude for assitance and useful remarks to V. F. Aleksin,
V. V. Angeleiko, A. S. Bakai, A. B. Mikhailovskii, S. S. Moiseev, V. A. Oraevskii, J. R. Ross
and V. P. Silin.
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Preface to the English Edition
"PLASMA ELECTRODYNAMICS", which is here brought before the English-speaking public,
is devoted to the theory of collective oscillations in a plasma—strictly speaking, of a uniform
plasma.
We have endeavoured to collect here the most important aspects of the theory of plasma
oscillations and decided therefore to present not only the theory of oscillations in a colli­
sionless plasma, but also the theory of magneto-hydrodynamic waves. Of course, our consi­
derations include both linear oscillations and large amplitude oscillations.
The book is an expanded and extended version of our booklet Collective Oscillations
in a Plasma, the English edition of which appeared in 1967.
Although we restricted our considerations solely to a uniform plasma, nevertheless the
material referring to the electromagnetic properties of such a plasma is so extensive that
we considered it appropriate to split the English edition into two volumes. The first volume
contains the theory of magneto-hydrodynamical waves and the theory of linear oscillations
of a collisionless plasma. The second volume contains the theory of non-linear waves in a
collisionless plasma, including the quasi-linear theory, the theory of plasma turbulence,
and the theory of electromagnetic fluctuations in a plasma.
The publication of an English edition of our book would have been impossible without
the active participation of D. ter Haar: not only was it his initiative which led to the publi­
cation in England, but he also undertook the translation of the book, which—as far as
we can judge with our knowledge of the English language—is excellent. Both for this and
for his many useful comments we want to thank Professor ter Haar most sincerely.
A. I. AKHIEZER
I. A. AKHIEZER
R. V. POLOVIN
A. G. SITENKO
K. N. STEPANOV
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