Classical Kinetic Theory of Weakly Turbulent Nonlinear Plasma Processes 9781107172005

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Classical Kinetic Theory of Weakly Turbulent Nonlinear Plasma Processes
 9781107172005

Table of contents :
Cover......Page 1
Front Matter
......Page 3
CLASSICAL KINETIC THEORY OF WEAKLY TURBULENT NONLINEAR
PLASMA PROCESSES......Page 5
Copyright
......Page 6
Contents
......Page 7
Preface......Page 11
Part I - Vlasov Weak Turbulence Theory:
Electrostatic Approximation
......Page 15
1 Nonlinear Electrostatic Equations for
Collisionless Plasmas......Page 17
2 Electrostatic Vlasov Weak Turbulence Theory
Wave Kinetic Equation......Page 60
3 Electrostatic Vlasov Weak Turbulence Theory
Particle Kinetic Equation......Page 74
Part II - Klimontovich Weak Turbulence Theory:
Electrostatic Approximation
......Page 87
4 Electrostatic Klimontovich Weak Turbulence Theory......Page 89
5 Spontaneous Emission and Collisional Kinetic Equation......Page 119
6 Langmuir Turbulence and Electron Kappa Distribution......Page 136
Part III - Vlasov Weak Turbulence Theory:
Electromagnetic Formalism
......Page 167
7 Nonlinear Electromagnetic Equations in
Vlasov Plasmas......Page 169
8 Electromagnetic Vlasov Weak Turbulence Theory......Page 212
Part IV - Klimontovich Weak Turbulence Theory:
Electromagnetic Formalism
......Page 237
9 Electromagnetic Klimontovich Weak
Turbulence Theory......Page 239
10 Applications of Electromagnetic Klimontovich
Weak Turbulence Theory......Page 282
Epilogue......Page 303
Appendix A.

Time Irreversible Small Amplitude Perturbations......Page 305
Appendix B.

Resonant Velocity Integral......Page 310
Appendix C.

Nonlinear Dispersion Relations......Page 313
Appendix D.

Plasma Dispersion Function......Page 323
Appendix E.

Weak Turbulence Theory for Reactive Instabilities......Page 327
Appendix F.

On Higher-Order Perturbative Expansion......Page 332
Appendix G.

On Renormalized Kinetic Turbulence Theory......Page 339
Appendix H.

One-Dimensional Normalized Equations......Page 349
References......Page 354
Index......Page 365

Citation preview

C L A S S I C A L K I N E T I C T H E O RY O F W E A K LY T U R BU L E N T N O N L I N E A R P L A S M A P RO C E S S E S

Kinetic theory of weakly turbulent nonlinear processes in plasma helped form the foundation of modern plasma physics. This book provides a systematic overview of the kinetic theory of weak plasma turbulence from a modern perspective. It covers the fundamentals of weak turbulence theory, including the foundational concepts and the mathematical and technical details. Applications to some key space plasma problems are also covered, including the origin of nonthermal charged particle population and radio burst phenomena from the sun. Treating both collective and discrete particle effects, it provides a valuable reference for researchers looking to familiarize themselves with plasma weak turbulence theory. p e t e r h . yo o n is a senior research scientist at the Institute of Physical Science and Technology, University of Maryland, international scholar professor at Kyung Hee University, Korea and senior research leader at Korea Astronomy and Space Science Institute. He is a leading expert in theoretical space plasma physics and a fellow of the American Physical Society.

C L A S S I C A L K I N E T I C T H E O RY O F W E A K LY T U R BU L E N T N O N L I N E A R P L A S M A P RO C E S S E S P E T E R H . YO O N University of Maryland

University Printing House, Cambridge CB2 8BS, United Kingdom One Liberty Plaza, 20th Floor, New York, NY 10006, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia 314–321, 3rd Floor, Plot 3, Splendor Forum, Jasola District Centre, New Delhi – 110025, India 79 Anson Road, #06-04/06, Singapore 079906 Cambridge University Press is part of the University of Cambridge. It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning, and research at the highest international levels of excellence. www.cambridge.org Information on this title: www.cambridge.org/9781107172005 DOI: 10.1017/9781316771259 © Peter H. Yoon 2019 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2019 Printed in the United Kingdom by TJ International Ltd, Padstow Cornwall A catalogue record for this publication is available from the British Library. Library of Congress Cataloging-in-Publication Data Names: Yoon, Peter H., 1958– author. Title: Classical kinetic theory of weakly turbulent nonlinear plasma processes / Peter H. Yoon (University of Maryland, College Park). Description: Cambridge ; New York, NY : Cambridge University Press, 2019. | Includes bibliographical references and index. Identifiers: LCCN 2019010668 | ISBN 9781107172005 (hardback : alk. paper) Subjects: LCSH: Plasma turbulence. | Kinetic theory of matter. Classification: LCC QC718 .Y66 2019 | DDC 530.4/46–dc23 LC record available at https://lccn.loc.gov/2019010668 ISBN 978-1-107-17200-5 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

Contents

Preface

page ix

Part I Vlasov Weak Turbulence Theory: Electrostatic Approximation

1

1

Nonlinear Electrostatic Equations for Collisionless Plasmas 1.1 Preamble: Fundamental Concepts 1.2 Electrostatic Vlasov Equation 1.3 Fast-Time Scale Solution 1.4 Perturbed Wave Equation 1.5 Formal Wave Kinetic Equation for Eigenmodes 1.6 Formal Particle Kinetic Equation 1.7 Linear and Nonlinear Susceptibilities 1.8 Linear Waves and Weak Instabilities

3 4 9 13 16 26 29 31 39

2

Electrostatic Vlasov Weak Turbulence Theory: Wave Kinetic Equation 2.1 Induced Emission 2.2 Spontaneous and Induced Decay/Coalescence 2.3 Induced Scattering

46 48 50 54

3

Electrostatic Vlasov Weak Turbulence Theory: Particle Kinetic Equation 3.1 Quasilinear Particle Kinetic Equation 3.2 Null Effects of Decay Processes on the Particles 3.3 Effects of Nonlinear Induced Scattering on the Particles 3.4 Summary of Electrostatic Vlasov Weak Turbulence Theory

60 60 61 63 68

v

vi

Contents

Part II Klimontovich Weak Turbulence Theory: Electrostatic Approximation

73

4

Electrostatic Klimontovich Weak Turbulence Theory 4.1 Plasma Equations Based upon Klimontovich Formalism 4.2 Formal Klimontovich Weak Turbulence Theory 4.3 Weak Turbulence Theory with Discrete Particle Effects 4.4 Summary of Klimontovich Weak Turbulence Theory

75 75 87 97 103

5

Spontaneous Emission and Collisional Kinetic Equation 5.1 Spontaneous Emission of Electrostatic Fluctuations 5.2 Collisional Kinetic Equation 5.3 Properties of Collisional Kinetic Equation 5.4 Collisional Kinetic Equation with Maxwellian Background

105 106 112 115 118

6

Langmuir Turbulence and Electron Kappa Distribution 6.1 Normalized Equations 6.2 Numerical Analysis of Weak Langmuir Turbulence 6.3 Langmuir Turbulence and Electron Kappa Distribution

122 124 130 135

Part III Vlasov Weak Turbulence Theory: Electromagnetic Formalism

153

7

Nonlinear Electromagnetic Equations in Vlasov Plasmas 7.1 Nonlinear Electrodynamic Equations in Plasmas 7.2 Formal Electromagnetic Kinetic Equations for Eigenmodes 7.3 Nonlinear Susceptibilities

155 155 170 183

8

Electromagnetic Vlasov Weak Turbulence Theory 8.1 Linear Theory of Electromagnetic Plasma Eigenmodes 8.2 Electromagnetic Vlasov Weak Turbulence Theory for Eigenmodes 8.3 Summary of Equations

198 198 204 218

Part IV Klimontovich Weak Turbulence Theory: Electromagnetic Formalism 9

Electromagnetic Klimontovich Weak Turbulence Theory 9.1 Nonlinear Electrodynamic Equations in Plasmas 9.2 Formal Equations of Electromagnetic Klimontovich Weak Turbulence Theory 9.3 Electromagnetic Klimontovich Weak Turbulence Theory: Summary

223 225 225 239 260

Contents

10 Applications of Electromagnetic Klimontovich Weak Turbulence Theory 10.1 Spontaneous Emission of Magnetic Field Fluctuations 10.2 Electromagnetic Radiation in Thermal Plasmas 10.3 Electromagnetic Collisional Kinetic Equation 10.4 Plasma Emission by EM Weak Turbulence Process Epilogue Appendix A Appendix B Appendix C Appendix D Appendix E Appendix F Appendix G Appendix H References Index

Time Irreversible Small Amplitude Perturbations Resonant Velocity Integral Nonlinear Dispersion Relations Plasma Dispersion Function Weak Turbulence Theory for Reactive Instabilities On Higher-Order Perturbative Expansion On Renormalized Kinetic Turbulence Theory One-Dimensional Normalized Equations

vii

268 269 272 276 278 289 291 296 299 309 313 318 325 335 340 351

Preface

Plasma is an ionized gas in which dynamical processes are governed predominantly by a collective behavior rather than that of individual particle interactions. For this reason, the plasma is often regarded as free of inter-particle collisions. However, the discrete nature of the plasma particles or, equivalently, the single-particle behavior of plasma can be important, and often the description of plasma is incomplete if the discrete particle effects are not properly taken into account. This monograph is concerned with the kinetic theory of weakly turbulent plasma processes in which collective and discrete particle effects are systematically treated. The methodology is classical since dilute high-temperature plasmas in laboratory, near-Earth space, and most astrophysical environments do not require quantum mechanical approaches. The book is exclusively concerned with incoherent nonlinear plasma phenomena. Such processes are known as microscopic or kinetic plasma turbulence, as opposed to macroscopic or fluid turbulence. The purpose of this book is to present essential ideas for treating the weak plasma turbulence on the basis of kinetic theory. In order to simplify the discussion, the plasma of interest is considered to be free of influence from externally applied electric or magnetic fields. The plasma is also considered as uniform on average, without spatial inhomogeneity associated with it. Despite these simplifying assumptions, basic theoretical tools and concepts developed in the present monograph can, in principle, be applied to situations in which the plasma is immersed in external fields, especially, the ambient magnetic field. Realistic laboratory or space/astrophysical plasmas are magnetized. Consequently, applications of the theory developed in this book are limited to high-frequency phenomena for which the approximation of field-free plasma may be valid. The weak plasma turbulence theory is a perturbative nonlinear theory, and the essential formalism was developed by the pioneers of modern plasma physics beginning in the late 1950s and 1960s, and continued on through the early 1980s.

ix

x

Preface

It is the purpose of the present author to systematically overview various aspects of this venerable theory, but also certain new developments are included in the discussion. This monograph comprises four parts. In Part I, the simplest kinetic theory is discussed, where it is assumed that the particles interact primarily through electrostatic force and without collisions. Chapter 1 formulates the perturbative kinetic theory based upon the relatively simple Vlasov–Poisson equation. Chapter 1 establishes conceptual foundations for a statistical description of plasmas, and also includes a detailed discussion of various linear and nonlinear response functions. Chapter 2 takes the formal results developed in Chapter 1 in a further direction, and derives the wave kinetic equation that describes weakly turbulent plasma processes. Chapter 3 derives the corresponding particle kinetic equations for electrons and ions interacting with the waves. In Part II, the discrete particle effects are discussed within the framework of electrostatic approximation. The conceptual approach is based upon the Klimontovich equation, hence Part II is entitled “Klimontovich Weak Turbulence Theory.” Chapter 4 revisits the perturbative nonlinear theory already discussed in earlier chapters, but additional effects that arise from particle discreteness are incorporated. The result is a weak turbulence theory in which various processes are expressed in the balanced form between the so-called spontaneous and induced processes. Chapter 5 considers applications. The first example is on the spontaneous emission of electrostatic fluctuations in thermal equilibrium plasma. The second application relates to plasmas slightly out of equilibrium, where collisional relaxation processes bring such plasmas back to thermal equilibrium. In Chapter 6, further application is made to a situation where collisional processes are not sufficient to relax the system to equilibrium, but rather the relaxation is achieved via excitation of instability. The sample problem considered involves an electron beam interacting with the background plasma, leading to an excitation of the socalled Langmuir turbulence. Numerical solutions of the basic equations show that the long-time evolution of electron beam-plasma instability involves the formation of a nonthermal population in the electron velocity distribution function. Chapter 6 closes with the discussion of the time asymptotic state of the plasma turbulence. Relevance to the near-Earth space environment is also discussed. Part III returns to the collision-free problem, but it generalizes Part I to fully electromagnetic situation. Chapter 7 formulates the general framework of Vlasov– Maxwell weak turbulence theory, and also discusses the electromagnetic linear and nonlinear response functions. Chapter 8 derives the specific equations of electromagnetic Vlasov weak turbulence theory for plasma normal modes. In Parts III and IV where theories concerning the electromagnetic weak turbulence formalism

Preface

xi

are discussed, various intermediate steps are deliberately included so that the readers may be able to retrace the derivations of various mathematical formulae. Part IV generalizes Part III by adding the particle discreteness effects, and it also generalizes Part II by including electromagnetic effects in the Klimontovich weak turbulence formalism, thus bringing all the necessary ingredients together, and completing the generalized formalism of weak turbulence theory. Chapter 9 presents the general formalism, followed by Chapter 10, where we make specific applications, first to the issue of spontaneous emission of magnetic field fluctuations in thermal plasmas, and the radiation emission in thermal plasma. We also discuss the problem of relativistic collisional kinetic equation. Finally, we apply the electromagnetic “Klimontovich” weak turbulence formalism to the problem of plasma emission, which is a physical process that is intimately related to the solar radio bursts phenomena. The backdrop on this problem is also discussed. Various topics and supplemental discussions are included in the appendices.

Part I Vlasov Weak Turbulence Theory: Electrostatic Approximation

1 Nonlinear Electrostatic Equations for Collisionless Plasmas

This monograph presents a perturbative nonlinear kinetic theory of plasma turbulence, known as the weak turbulence theory. At the outset, it should be pointed out that this book does not include the effects of ambient magnetic field. Plasmas in real situations are usually magnetized, so that applications of the method discussed in this book will be somewhat limited, but the purpose is to lay out the fundamental methodology and conceptual foundations so that more general applications for magnetized plasmas may be developed on the basis of this book. This book also limits the discussions to spatially homogeneous plasma. Plasma kinetic theory has a long history, and many early papers can be found in the literature that discuss the perturbative nonlinear kinetic theory of plasma turbulence – see, for example, papers by Vedenov and Velikhov (1962); Kovrizhnykh and Tsytovich (1964, 1965); Kovrizhnykh (1965); Gorbunov and Silin (1965); Gorbunov et al. (1965); Tsytovich (1967); Rogister and Oberman (1968, 1969), to name just several. These are merely sample papers, among those that personally influenced the author of this book. If one is interested in the general background on plasma kinetic theory, there are some excellent early monographs, among which may be, for instance, those by Montgomery and Tidman (1964); Kadomtsev (1965); Klimontovich (1967, 1982); Pitaevskii and Lifshitz (1981); Sagdeev and Galeev (1969); Tsytovich (1970, 1977a,b); Davidson (1972); Ichimaru (1973); Krall and Trivelpiece (1973); Akhiezer et al. (1975); Hasegawa (1975); Kaplan and Tsytovich (1973); Sitenko (1967, 1982); Melrose (1980a, 1986); Nicholson (1983); Alexandrov et al. (1984); Chen (1987), etc. This list is incomplete, but they represent some standard works that treat the foundations of plasma kinetic theory and/or weak plasma turbulence theory. More recent books are also available. See, for example, those by Musher et al. (1995); Sitenko and Malnev (1995); Treumann and Baumjohann (1997); ˇ Tsytovich (1995); Kono and Skori´ c (2010); Diamond et al. (2010), etc., which deal with the subject of plasma kinetic theory and nonlinear phenomena. 3

4

Nonlinear Electrostatic Equations for Collisionless Plasmas

So, as the readers may appreciate, there is an abundance of resources on the topic of plasma kinetic theory, and one may ask why another book? The rationale for this book is as follows: Discussions of nonlinear plasma theories, particularly those concerning the weak turbulence theory found in many of the above-cited works, are sometimes not so easy to follow, especially for young researchers. Moreover, many of the monographs cover wide-ranging topics with generally brief descriptions for each subject area without going much into in-depth discussions. It is the purpose of this book to focus only on the kinetic theory of weak plasma turbulence, but to present the detailed fundamental discussions and derivations as clearly as possible, without sacrificing the intermediate mathematical steps. Talking of the latter, many authors omit too many intermediate steps, which can be a source of much frustrations for young scientists. This book does not spare the readers the mathematical details. This strategy means that some materials in the book can be a bit lengthy, and casual readers may get lost in the maths. However, if one approaches the material with enough patience, he or she will be rewarded with the intimate knowledge on how the weak turbulence theory actually works, what are the essential assumptions behind the theory, and so forth. Owing to the space devoted to mathematical details, some standard topics often included in the textbooks and monographs on nonlinear plasma theory are left out. For instance, parametric instabilities, solitary wave theory, coherent nonlinear structure formation in plasma, etc., are not covered in this book. This book is intended for advanced undergraduate, graduate students, or young researchers who are already familiar with the introductory level of plasma kinetic theory, but wishing to familiarize themselves with a more in-depth understanding on nonlinear theory of weak plasma turbulence. In spite of this, this book expounds on foundational principles at the conceptual level as much as possible without assuming too much prior knowledge on the part of the readers.

1.1 Preamble: Fundamental Concepts We are interested in physical phenomena that are described as turbulent, which loosely means physical quantities that are fluctuating in space and time. In order to characterize such fluctuations, we employ statistical methods and concepts. That is, we deal with averages in time, space, or over hypothetical collection of different possible states called the ensemble. One is particularly interested in how fluctuating quantities measured in two or more different times or in two or more different spatial locations are correlated. We begin by considering many-body correlations associated with fluctuating physical quantities, and the spectral transformation of such quantities in space and time.

1.1 Preamble: Fundamental Concepts

5

The statistical correlation is an important concept that characterizes the nature of turbulence. Suppose that one measures a particular physical quantity, say velocity or electromagnetic field, in a turbulent medium at a given time. Suppose also that one measures the same quantity at another time separated by an interval. If one repeats such series of measurements over and over again, then if the physical quantities are uncorrelated, that is, if there is no cause and effect relationship between the two measurements, then on average, the product of two measurements made at two different time intervals may be zero, since by the very nature of turbulence, velocity or field may have random directions. On the other hand, if the first measurement affects the second measurement because there exists an underlying cause-and-effect relationship, then the average of the products may be finite. A systematic way to characterize how the statistical average of the products of physical quantities, or equivalently, their correlation function, behaves in space and time can thus be useful for understanding and characterizing the turbulence. Consequently, in this book we will be concerned with the description of how  statistical average of the  the correlation of fluctuating (i.e., turbulent) quantity, δa 2 , dynamically evolves. Here, δa represents any dynamical quantity, and the symbol · · ·  denotes the statistical average. The convention adopted in this book for the definition of spatial Fourier transformation and its inverse is   −3 −ik·r dr f (r) e , f (r) = dk fk eik·r . (1.1) fk = (2π) Here f (r) is any physical quantity, which is a function of spatial coordinate r, and which is bounded in space. The Fourier transformation of a product of two functions is represented by the convolution    (1.2) (2π)−3 dr f (r) g(r) e−ik·r = dk fk gk−k = dk fk−k gk . The proof of this “convolution theorem” is straightforward. All one has to do is to insert for f (r) and g(r), their respective Fourier transformations, and make use of the well-known delta function identity  (1.3) dr eik·r = δ(k). Fourier transformation of a function f (r,t) in both space and time can be defined by   −4 dr dt f (r,t) e−ik·r+iωt , fk,ω = (2π)   f (r,t) = dk dω fk,ω eik·r−iωt . (1.4)

6

Nonlinear Electrostatic Equations for Collisionless Plasmas

Convolution theorem for the spatio-temporal Fourier transformation is   −4 (2π) dr dt f (r,t) g(r,t) e−ik·r+iωt    dω fk,ω gk−k,ω−ω = dk    dω fk−k,ω−ω gk,ω . = dk

(1.5)

When the angular frequency ω satisfies the dispersion relation ω = ωk +iγk , that is, when (generally complex) ω is a function of k, then the Fourier representation of function f (r,t) can be re-expressed by virtue of the fact that we may write the spectral amplitude as fk,ω = fk δ(ω − ωk − iγk ), or

(1.6)

 f (r,t) =

dk fk exp(ik · r − iωk t + γk t).

(1.7)

If f (r,t) is real then obviously f ∗ (r,t) = f (r,t), where the asterisk ∗ represents the complex conjugate. From this it follows that   ∗ dk fk exp(−ik · r + iωk t) = dk fk exp(ik · r − iωk t), (1.8) which leads to the following symmetry relations: fk∗ = f−k,

ω−k = −ωk,

γ−k = γk .

(1.9)

Let δf (r,t) represent a fluctuating quantity whose ensemble average is zero: δf (r,t) = 0.

(1.10)

In our notation, any quantity preceded by δ indicates that this quantity is fluctuating in space and time, that is, turbulent. By “ensemble average” we may mean an average over phase, space, or time. Or it could mean an average over all possible configurations. Turbulence is called “homogeneous” if the spatial dependence of the two-body correlation is only upon the relative distance, δf (r,t) δf (r,t) = δf 2 r−r,t,t  = δf 2 r −r,t,t  ,

(1.11)

and “stationary” if the temporal two-body correlation is a function of relative time difference, δf (r,t) δf (r,t  ) = δf 2 r,r,t−t  = δf 2 r,r,t  −t .

(1.12)

1.1 Preamble: Fundamental Concepts

7

Thus, for homogeneous and stationary turbulence the two-body correlation function is given by δf (r,t) δf (r,t  ) = δf 2 r−r,t−t  .

(1.13)

It should be noted that not all fluctuating quantities in nature satisfy the zero ensemble average property (1.10). Physical processes whose fluctuations satisfy (1.10) are called “incoherent” phenomena, while “coherent” processes may be associated with a nonvanishing ensemble average. For incoherent processes different phases are uncorrelated such that when averaged over them, the result vanishes; hence, such processes are characterized by the zero ensemble average property specified by (1.10). In a similar way, the three-body correlation function for homogeneous and stationary turbulence is a function of distances between any two points, say (r,t) and (r,t  ), among three points (r,t), (r,t  ), (r,t  ), in coordinate-time space: δf (r,t) δf (r,t  ) δf (r,t  ) = δf 3 r−r,r−r ;t−t ,t−t  .

(1.14)

The four-body correlation function for homogeneous and stationary turbulence can be defined likewise: δf (r,t) δf (r,t  ) δf (r,t  ) δf (r,t  ) = δf 2 r−r ;t−t  δf 2 r −r ;t  −t  + δf 2 r−r ;t−t  δf 2 r −r ;t  −t  + δf 2 r−r ;t−t  δf 2 r −r ;t  −t  + δf 4 r−r,r −r,r −r ;t−t ,t  −t ,t  −t  .

(1.15)

Let us represent the two-body correlation function in spectral form:       δf (r,t) δf (r ,t ) = dk dω δf 2 k,ω eik·(r−r )−iω(t−t )          dω δfk,ω δfk,ω  eik·r+ik ·r −iωt−iω t , = dk dω dk (1.16) where in the second line we have made use of the spectral representations for individual functions δf (r,t) and δf (r,t  ). From this, it is seen that the equality can be obtained if the following condition is satisfied: δfk,ω δfk,ω  = δ(k + k ) δ(ω + ω ) δf 2 k,ω .

(1.17)

If we write the spectral component δfk,ω with an explicit phase factor, δfk,ω = fˆk,ω eiφk,ω ,

(1.18)

8

Nonlinear Electrostatic Equations for Collisionless Plasmas

where φk,ω represents the phase, then we have δfk,ω δfk,ω  = fˆk,ω fˆk,ω eiφk,ω +iφk,ω .

(1.19)

For homogeneous and stationary turbulence the phase is assumed to be random (or uncorrelated). As such, the ensemble average over random phases becomes nonzero only if φk,ω + φk,ω = 0,

(1.20)

which can be satisfied under the assumption that, if for k = −k and ω = −ω , the following is also satisfied: φ−k,−ω = −φk,ω .

(1.21)

This is but the rephrasing of condition (1.17). The assumption of homogeneous and stationary turbulence is thus equivalent to the “random phase approximation.” In short, the property δf 2 k,ω = δfk,ω δf−k,−ω 

(1.22)

is a useful spectral characteristic for homogeneous and stationary turbulence, or equivalently, fluctuations with random phases. Next, consider the three-body correlation, which we may write as          dω δf 3 k,ω;k,ω δf (r,t) δf (r ,t ) δf (r ,t ) = dk dω dk 















ik·(r−r )+ik ·(r −r )−iω(t−t )−iω (t −t ) ×     e      dω dk dω = dk dω dk

× δfk,ω δfk,ω δfk,ω   

× eik·r+ik ·r +ik

 ·r −iωt−iω t  −iω t 

.

(1.23)

From this we obtain the identity δfk,ω δfk,ω δfk,ω  = δ(k + k + k ) δ(ω + ω + ω ) δf 3 k,ω;k+k,ω+ω . (1.24) A similar analysis can be carried out for the four-body correlation. The derivation is tedious but straightforward, and is thus omitted. We summarize the general properties of the many-body correlations, or manybody cumulants for homogeneous and stationary turbulence: δfk,ω δfk,ω  = δ(k + k ) δ(ω + ω )δf 2 k,ω, δfk,ω δfk,ω δfk,ω  = δ(k + k + k )δ(ω + ω + ω )δf 3 k,ω;k+k,ω+ω ,

1.2 Electrostatic Vlasov Equation

9

δfk,ω δfk,ω δfk,ω δfk ω  = δ(k + k + k + k ) δ(ω + ω + ω + ω ) × [δ(k + k ) δ(ω + ω ) δf 2 k,ω δf 2 k,ω + δ(k + k ) δ(ω + ω ) δf 2 k,ω δf 2 k,ω + δ(k + k ) δ(ω + ω ) δf 2 k,ω δf 2 k,ω + δf 4 k,ω;k+k,ω+ω ;k+k +k,ω+ω +ω ].

(1.25)

An important consequence of this result is that an ensemble average of two fluctuating quantities δf and δg, where they are related to each other, can be expressed in terms of their spectral counterparts as follows:   (1.26) δf (r,t) δg(r,t) = dk dω δfk,ω δg−k,−ω .

1.2 Electrostatic Vlasov Equation A simple and intuitive definition of plasma is that it is an ionized gas. Individual electrons and ions that make up the plasma interact through collective electromagnetic force. Collective behavior of a plasma is described by a statistical means. In this book we are concerned with a fully ionized plasma. For partially ionized plasma, atomic processes such as the recombination and collisions between charged particles and neutrals cannot be ignored, which complicate the matter. Vlasov equation (Vlasov, 1938) describes the statistical property of a plasma governed by collective processes. The system under consideration is a spatially uniform plasma made of single-species ions (protons) and electrons, and there is no net electric or magnetic field. We also assume zero average charge or current in the system. If we make the simplifying approximation that the plasma particles interact primarily through electrostatic field, then the dynamics can be described by the Vlasov– Poisson system of equations   ∂ ea ∂ fa = 0, E· +v·∇ + ∂t ma ∂v   ∇ · E = 4π ea dv fa, (1.27) a

where ea and ma are charge and mass of species a (= e,i) for electrons and ions (ea = e for protons and ea = −e for electrons). The one-particle distribution function fa (r,v,t) is the probability density of finding a collection of plasma particles of species a, at a particular state in phase space (r,v) at a given time t. Consequently, if we integrate fa (r,v,t) over v, or equivalently, if we collect all possible configuration in velocity space, then the result becomes the density of charged particle species labeled a,

10

Nonlinear Electrostatic Equations for Collisionless Plasmas



ρa (r,t) =

dv fa (r,v,t).

(1.28)

Multiplying the charge ea and summing over all charged particle species leads to the total charge density  ρ(r,t) = ea ρa (r,t). (1.29) a

Since fa (r,v,t) is the probability density, it is normalized to the ambient charged particle number density na ,   1 dr dv fa (r,v,t) = na, (1.30) V where V is the volume of the system. That is, if we collect all possible configurations in velocity space at a given time, and integrate over the entire volume under consideration and divide by V , that is, take the spatial average, then the result should be the total number of particles per volume, na = Na /V , or equivalently, the ambient density. Since in the absence of source or sink, plasma particles cannot be created or annihilated (that is, no recombination into neutrals or reionization), the one-particle distribution function must be conserved. Hence,   dfa ∂ ∂ fa = 0. (1.31) = + r˙ · ∇ + v˙ · dt ∂t ∂v By virtue of the equation of motion, r˙ = v and v˙ =

ea E, ma

(1.32)

we obtain the Vlasov equation in (1.27). Because of the charge neutrality condition, the ambient density is the same for both ions and electrons, ne = ni = n.

(1.33)

Let us separate the physical quantities into average and fluctuating parts. The average particle distribution function is independent of the spatial coordinate r since we assume uniform plasma, and there is no average electric field, so that we may write fa (r,v,t) = na Fa (v,t) + δfa (r,v,t), E(r,t) = δE(r,t),

(1.34)

where δ represents fluctuating quantities whose phases are supposed to be random. When averaged over their phases, these quantities vanish. In (1.34)  we have introduced a normalized one-particle distribution function Fa (v,t) [ dv Fa (v,t) = 1]. Inserting (1.34) back into the coupled Vlasov–Poisson equation, we obtain





1.2 Electrostatic Vlasov Equation

11





∂ ∂ ∂ ∂ ea ea na Fa + δfa = 0, δE · δE · + +v·∇ + ∂t ma ∂v ∂t ma ∂v ∇ · δE = 4π



 ea

dv δfa .

a

(1.35) Upon averaging (1.35) over random phases of the fluctuations, we obtain the formal particle kinetic equation: ea ∂ ∂na Fa =− · δfa δE. ∂t ma ∂v

(1.36)

Let us subtract the formal particle kinetic equation from the original equation in order to obtain the equation for perturbed distribution function:   ∂ ea ∂na Fa δE · + v · ∇ δfa = − ∂t ma ∂v ea ∂ (1.37) − · (δfa δE − δfa δE) . ma ∂v Note that (1.37) is nonlinear since it contains terms of order O(δ 2 ). We assume that the fluctuations can be decomposed in the sense of Fourier– Laplace transformation over the fast-time scales of fluctuations while the spectral amplitudes may vary slowly in time:   a (v,t) eik·r−iωt , δfa (r,v,t) = dk dω δfk,ω L  ∞ 1 a dr dt δfa (r,v,t) e−ik·r+iωt , δfk,ω (v,t) = (2π)4 0   δφ(r,t) = dk dω δφk,ω (t) eik·r−iωt , L  ∞ 1 dr dt δφ(r,t) e−ik·r+iωt , (1.38) δφk,ω (t) = (2π)4 0 where we have assumed δE(r,t) = −∇δφ(r,t),

or

δEk,ω = −ik δφk,ω,

(1.39)

since we work under the electrostatic approximation. In (1.38), the integration  dω is taken along the path L that stretches from ω = −∞ + iσ to ω = ∞ + iσ L (σ > 0 and σ → 0). The infinitesimal positive imaginary part σ signifies that we are only interested in causal solutions. The causality requirement is related to the Laplace transformation being defined only over 0 < t < ∞ rather than the entire integral domain, −∞ < t < ∞. The reason for the positive infinitesimal imaginary part, σ > 0, in the ω integration along the path L, is to ensure the

12

Nonlinear Electrostatic Equations for Collisionless Plasmas

temporal convergence for t → ∞, hence the causality. By the choice of posi∞ tive integral 0 dt, one is effectively breaking the time reversal symmetry, and this forces physical processes to proceed in forward time, t > 0. That is, the Laplace transformation in place of the symmetric temporal Fourier transformation is equivalent to imposing the causal relationship to the otherwise time-reversible Vlasov equation. In Appendix A we review the treatment of time-irreversible small amplitude plasma perturbation, as discussed originally by Landau (1946). We also discuss the notion of Landau damping in Appendix A. In the Fourier–Laplace transformation defined in (1.38), we have made an a (v,t) and δφk,ω (t) have slow and assumption that the spectral amplitudes δfk,ω adiabatic time dependence. These quantities are assumed to vary slowly in time, while the temporal dependence dictated by exp(−iωt) is assumed to be fast. That is, the time dependence of amplitudes is much weaker than that associated with the wave time scale, δf , δφ ∼ O(tslow ), where O(tslow ) O(ω−1 ). These amplitudes are calculated as if they are independent of time on the fast wave time scale (t ∼ ω−1 ). Employing the transformation (1.38), the equation for fluctuating field, formal particle kinetic equation, and the equation for perturbed distribution function are expressed, respectively, as follows:  4πea  a dv δfk,ω (v,t), δφk,ω (t) = 2 k a     ∂na Fa (v,t) iea  dk dω dk dω = ∂t ma      ∂ a δφk,ω (t)δfk,ω × k· (v,t) ei(k+k )·r−i(ω+ω )t , ∂v   ea ∂na Fa (v,t) ∂ a δfk,ω (v,t) = − δφk,ω (t) k · ω−k·v+i ∂t ma ∂v   ea ∂ dk dω k · − ma ∂v a × δφk,ω (t) δfk−k ,ω−ω (v,t)

a −δφk,ω (t) δfk−k (1.40) ,ω−ω (v,t) . The present treatment of slow and adiabatic time dependence of spectral amplitudes is not rigorous. Mathematically consistent treatment involves the multiple timescale analysis as employed by Davidson (1972). The present discussion resorts to the shortcut approach, following the method employed in the standard literature, for example, Sitenko (1982), Akhiezer et al. (1975), etc.

1.3 Fast-Time Scale Solution

13

1.3 Fast-Time Scale Solution Formal solutions for the fast-time scale quantities necessitate the inversion of differential operator, ω − k · v + i∂/∂t. To simplify the matter, we temporarily ignore the adiabatic time derivative by absorbing the time derivative as part of the “new” definition for ω, ω + i ∂/∂t → ω,

(1.41)

within the definition for ω − k · v + i0 + i∂/∂t. We will reinstitute the explicit slow-time derivative when we discuss the wave kinetic equation later. Meanwhile, let us define ∂ 1 ea gk,ω ≡ − . (1.42) ma ω − k · v + i0 ∂v Note that +i0 is meant to indicate that the angular frequency ω is always to be interpreted as having an infinitesimally small but positive imaginary part, which is intimately related to the causality condition associated with the Laplace transformation, and discussed in Appendix A. The notion of ω having positive imaginary part arises from the asymptotic treatment associated with the analytic continuation, and it is further discussed in detail in Appendix B. Then the equation for perturbed particle distribution function in (1.40) is re-expressed as follows: a = k · gk,ω na Fa δφk,ω (1.43) δfk,ω  

a a + dk dω k · gk,ω δφk,ω δfk−k ,ω−ω − δφk,ω δfk−k,ω−ω  . a a , we expand δfk,ω in a formal perturbation series with To solve the equation for δfk,ω successive terms proportional to increasing powers of δφk,ω : a(1) a(2) a(3) a = δfk,ω + δfk,ω + δfk,ω + ··· , δfk,ω

(1.44)

a(n) a(n) n where δfk,ω scales as O(δφk,ω ). It is not so difficult to see why δfk,ω should n ). If we ignore the nonlinear coupling term in (1.43), then it is scale as O(δφk,ω a is proportional to the readily obvious that the perturbed distribution function δfk,ω perturbed wave amplitude δφk,ω . From this it logically follows that the expansion parameter should scale as δφk,ω . This consideration also leads to the conceptually “necessary” condition for the validity of so-called weak turbulence expansion (1.44). If we consider that the characteristic energy associated with the average particle distribution na Fa is the thermal energy,    na ma v 2  ma v 2 Ta na dv na Ta , (1.45) Eparticle = Fa (v) = = 2 2 a a a

14

Nonlinear Electrostatic Equations for Collisionless Plasmas

and that the perturbed distribution function, which is defined by δfa = fa − na Fa , scales as the wave amplitude, δfa ∝ δφ, then in order for perturbative expansion (1.44) to converge, the characteristic energy associated with the perturbed distribution function, which is proportional to the wave energy density,  2  δE ma v 2 dv δfa (v) ∝ Ewave = , (1.46) 2 8π a where δE = −∇δφ, must be sufficiently lower than the thermal energy. In short, the following condition must be satisfied for the weak turbulence expansion to be valid: Eparticle Ewave .

(1.47)

This is the general requirement for the validity of weak turbulence theory. a Inserting the series expansion (1.44) to the original equation for δfk,ω (v), we readily obtain, order by order, a(1) δfk,ω = k · gk,ω na Fa δφk,ω,  

a(2) a(1) a(1)  dω k · gk,ω δφk,ω δfk−k δfk,ω = dk ,ω−ω − δφk,ω δfk−k,ω−ω  ,  

a(3) a(2) a(2)  dω k · gk,ω δφk,ω δfk−k δfk,ω = dk ,ω−ω − δφk,ω δfk−k,ω−ω  ,

···

(1.48)

a(2) a(3) and δfk,ω are given below: etc. Iterative solutions for δfk,ω  

a(2) = dk dω (k · gk,ω ) (k − k ) · gk−k,ω−ω δfk,ω × na Fa δφk,ω δφk−k,ω−ω − δφk,ω δφk−k,ω−ω  ,     a(3) δfk,ω = dk dω dk dω (k · gk,ω ) (k · gk−k,ω−ω )

× (k − k − k ) · gk−k −k,ω−ω −ω na Fa × δφk,ω δφk,ω δφk−k −k,ω−ω −ω

− δφk,ω δφk,ω δφk−k −k,ω−ω −ω 

− δφk,ω δφk,ω δφk−k −k,ω−ω −ω  .

Let us introduce the following simplified notations:    = dk1 dk2 δ(k1 + k2 − k), k1 +k2 =k



ω1 +ω2 =ω



 =

dω1

dω2 δ(ω1 + ω2 − ω),

(1.49)





=

k1 +k2 +k3 =k



1.3 Fast-Time Scale Solution



 dω1

15

dk3 δ(k1 + k2 + k3 − k),

dk2

dk1 

=



 dω2

dω3 δ(ω1 + ω2 + ω3 − ω),

(1.50)

ω1 +ω2 +ω3 =ω a is given by etc. Then, the series solution for δfk,ω a = k · gk,ω na Fa δφk,ω δfk,ω   + (k1 · gk,ω ) (k2 · gk2,ω2 ) na Fa k1 +k2 =k ω1 +ω2 =ω

× (δφk1,ω1 δφk2,ω2 − δφk1,ω1 δφk2,ω2 )   + (k1 · gk,ω ) (k2 · gk2 +k3,ω2 +ω3 ) k1 +k2 +k3 =k ω1 +ω2 +ω3 =ω

× (k3 · gk3,ω3 ) na Fa (δφk1,ω1 δφk2,ω2 δφk3,ω3 − δφk1,ω1 δφk2,ω2 δφk3,ω3  − δφk1,ω1 δφk2,ω2 δφk3,ω3 ). .. ..

(1.51)

The right-hand side of (1.51) is not symmetric with respect to the interchange of dummy variables, (k1,ω1 ) and (k2,ω2 ). The second-order nonlinear term should be symmetrized with respect to variables (k1,ω1 ) and (k2,ω2 ), while for the thirdorder term, the symmetrization should be implemented with respect to (k2,ω2 ) and (k3,ω3 ). The interchange of these dummy variables leaves (1.51) intact. When the above quantities are symmetrized with respect to these variables, then the correct symmetrized expression emerges, a = α(k,ω) na Fa δφk,ω δfk,ω   + α (2) (k1,ω1 |k2,ω2 ) na Fa k1 +k2 =k ω1 +ω2 =ω

× (δφk1,ω1 δφk2,ω2 − δφk1,ω1 δφk2,ω2 )   + α (3) (k1,ω1 |k2,ω2 |k3,ω3 ) na Fa k1 +k2 +k3 =k ω1 +ω2 +ω3 =ω

× δφk1,ω1 δφk2,ω2 δφk3,ω3 − δφk1,ω1 δφk2,ω2 δφk3,ω3   −δφk1,ω1 δφk2,ω2 δφk3,ω3  , where α(k,ω) = k · gk,ω, 1 α (2) (k1,ω1 |k2,ω2 ) = [(k1 · gk1 +k2,ω1 +ω2 ) (k2 · gk2,ω2 ) 2 + (k2 · gk1 +k2,ω1 +ω2 ) (k1 · gk1,ω1 )],

(1.52)

16

Nonlinear Electrostatic Equations for Collisionless Plasmas

α (3) (k1,ω1 |k2,ω2 |k3,ω3 ) =

1 (k1 · gk1 +k2 +k3,ω1 +ω2 +ω3 ) 2 × [ (k2 · gk2 +k3,ω2 +ω3 ) (k3 · gk3,ω3 ) + (k3 · gk2 +k3,ω2 +ω3 ) (k2 · gk2,ω2 ) ],

(1.53)

1.4 Perturbed Wave Equation a We insert the iterative solution for δfk,ω (1.52) to the right-hand side of perturbed Poisson equation in (1.40),

 0= 1−

 4πea na  a





k2 

k1 +k2 =k ω1 +ω2 =ω



 dv α(k,ω) Fa δφk,ω

 4πea na  a

k2

dv α (2) (k1,ω1 |k2,ω2 ) Fa

× δφk1,ω1 δφk2,ω2 − δφk1,ω1 δφk2,ω2     4πea na  − dv Fa α (3) (k1,ω1 |k2,ω2 |k3,ω3 ) 2 k k1 +k2 +k3 =k ω1 +ω2 +ω3 =ω a × δφk1,ω1 δφk2,ω2 δφk3,ω3 − δφk1,ω1 δφk2,ω2 δφk3,ω3  (1.54) − δφk1,ω1 δφk2,ω2 δφk3,ω3  . Let us define the linear dielectric response function, (k,ω) = 1 + χ (k,ω) = 1 +



χa (k,ω)

a

4π 4π  σa (k,ω), σ (k,ω) = 1 + ω ω a  4π 4πea na χa (k,ω) = dv α(k,ω) Fa , σa (k,ω) = − ω k2 =1+

(1.55)

where χ (k,ω) is the linear dielectric susceptibility and σ (k,ω) is the linear dielectric conductivity. The second-order nonlinear response function, or equivalently, the second-order nonlinear susceptibility, is likewise defined χ (2) (k1,ω1 |k2,ω2 ) =



χa(2) (k1,ω1 |k2,ω2 ),

a

χa(2) (k1,ω1 |k2,ω2 )

4πiea na =− k1 k2 |k1 + k2 |

 dv α (2) (k1,ω1 |k2,ω2 ) Fa,

(1.56)

1.4 Perturbed Wave Equation

17

and the third-order nonlinear response function, or third-order nonlinear susceptibility, may also be defined  χ¯ a(3) (k1,ω1 |k2,ω2 |k3,ω3 ), χ¯ (3) (k1,ω1 |k2,ω2 |k3,ω3 ) = a

χ¯ a(3) (k1,ω1 |k2,ω2 |k3,ω3 ) =

4πea na k1 k2 k3 |k1 + k2 + k3 |  × dv α (3) (k1,ω1 |k2,ω2 |k3,ω3 ) Fa,

χ (3) (k1,ω1 |k2,ω2 |k3,ω3 ) =

1 (3) [ χ¯ (k1,ω1 |k2,ω2 |k3,ω3 ) 3 + χ¯ (3) (k2,ω2 |k1,ω1 |k3,ω3 ) + χ¯ (3) (k3,ω3 |k2,ω2 |k1,ω1 ) ].

(1.57)

To sum up, the linear and nonlinear susceptibilities can be expressed as  4πea na dv k · gk,ω Fa, (1.58) χa (k,ω) = − k2  4πiea na 1 dv gk1 +k2,ω1 +ω2 χa(2) (k1,ω1 |k2,ω2 ) = − 2 k1 k2 |k1 + k2 | · [ k1 (k2 · gk2,ω2 ) + k2 (k1 · gk1,ω1 ) ] Fa, (1.59) 4πea na 1 χ¯ a(3) (k1,ω1 |k2,ω2 |k3,ω3 ) = 2 k1 k2 k3 |k1 + k2 + k3 |  × dv (gk1 +k2 +k3,ω1 +ω2 +ω3 · k1 )gk2 +k3,ω2 +ω3

· k2 (k3 · gk3,ω3 ) + k3 (k2 · gk2,ω2 ) Fa . (1.60) After explicitly writing out the various objects, making use of their respective definitions, we may also rewrite the susceptibilities in concrete forms as follows: χa (k,ω) = χa(2) (k1,ω1 |k2,ω2 ) =

2 ωpa

k2

 dv

k · ∂Fa /∂v , ω − k · v + i0 2 ωpa

−i ea 2 ma k1 k2 |k1 + k2 |  1 × dv ω1 + ω2 − (k1 + k2 ) · v + i0    k2 · ∂Fa /∂v ∂ × k1 · ∂v ω2 − k2 · v + i0   k1 · ∂Fa /∂v ∂ , + k2 · ∂v ω1 − k1 · v + i0

(1.61)

(1.62)

18

Nonlinear Electrostatic Equations for Collisionless Plasmas

χ¯ a(3) (k1,ω1 |k2,ω2 |k3,ω3 ) =

2 ωpa (−i)2 ea2 2 m2a k1 k2 k3 |k1 + k2 + k3 |  1 × dv ω1 + ω2 + ω3 − (k1 + k2 + k3 ) · v + i0  1 ∂ × k1 · (1.63) ∂v ω2 + ω3 − (k2 + k3 ) · v + i0    k3 · ∂Fa /∂v ∂ × k2 · ∂v ω3 − k3 · v + i0   k2 · ∂Fa /∂v ∂ , + k3 · ∂v ω2 − k2 · v + i0

where

 ωpa =

4πna ea2 ma

1/2 (1.64)

is the plasma frequency for species a. In (1.64), we again iterate that the positive infinitesimal imaginary part associated with the angular frequency within the resonant denominators is the result of imposing the causality condition in the Laplace transformation (1.38) – see also Appendix B. Various linear and nonlinear susceptibilities in (1.55)–(1.63) are known by their respective names: • • • • • •

(k,ω); linear dielectric constant χ (k,ω); linear dielectric susceptibility σ (k,ω); linear dielectric conductivity χ (2) (k1,ω1 |k2,ω2 ); second-order nonlinear susceptibility χ¯ (3) (k1,ω1 |k2,ω2 |k3,ω3 ); partial third-order nonlinear susceptibility χ (3) (k1,ω1 |k2,ω2 |k3,ω3 ); fully symmetrized third-order nonlinear susceptibility.

These definitions and designation of conventions (and even the notations) follow Sitenko (1982). In terms of these susceptibilities, (1.54) can be written compactly as   k1 k2 χ (2) (k1,ω1 |k2,ω2 ) 0 = k (k,ω) δφk,ω − i k1 +k2 =k ω1 +ω2 =ω

× δφk1,ω1 δφk2,ω2 − δφk1,ω1 δφk2,ω2    k1 k2 k3 χ¯ (3) (k1,ω1 |k2,ω2 |k3,ω3 ) − k1 +k2 +k3 =k ω1 +ω2 +ω3 =ω

× δφk1,ω1 δφk2,ω2 δφk3,ω3 − δφk1,ω1 δφk2,ω2 δφk3,ω3  − δφk1,ω1 δφk2,ω2 δφk3,ω3  .

(1.65)

Recall that in the statistical theory of turbulence we are interested in the correlation function, δφ 2 . We may construct the equation for correlation from (1.65).

1.4 Perturbed Wave Equation

19

Let us therefore multiply δφk,ω to (1.65), and take the ensemble average of the resulting equation: 0 = k (k,ω) δφk,ω δφk,ω    −i k1 k2 χ (2) (k1,ω1 |k2,ω2 ) δφk1,ω1 δφk2,ω2 δφk,ω  k1 +k2 =k ω1 +ω2 =ω







k1 k2 k3 χ¯ (3) (k1,ω1 |k2,ω2 |k3,ω3 )

k1 +k2 +k3 =k ω1 +ω2 +ω3 =ω

× δφk1,ω1 δφk2,ω2 δφk3,ω3 δφk,ω 

− δφk1,ω1 δφk,ω  δφk2,ω2 δφk3,ω3  .

(1.66)

This equation relates the two-body electric field amplitude correlation δφ 2  to three- and four-body correlations δφ 3  and δφ 4 . As a consequence, (1.66) is not closed, but instead forms an infinite chain, or hierarchy, of correlations. To break the hierarchy one must introduce certain assumptions. This is the “closure problem.” To break the hierarchy, we write the four-body cumulant δφ 4  as the sum of products of two body cumulants while ignoring the irreducible four-body correlation. This is equivalent to ignoring the last term on the right-hand side of (1.15). Let us thus consider the third-order nonlinear term associated with χ¯ (3) . Making use of the shorthand notations, K = (k,ω),

1 ≡ K1 = (k1,ω1 ),

2 ≡ K2 = (k2,ω2 ),

3 ≡ K3 = (k3,ω3 ),

(1.67)

we may write δφ1 δφ2 δφ3 δφK   − δφ1 δφK   δφ2 δφ3  = δ(1 + 2 + 3 + K  ) δ(1 + 2) δφ 2 1 δφ 2 3

+ δ(1 + 3) δφ 2 1 δφ 2 2 ,

(1.68)

where we have ignored the irreducible four-body correlation. This leads to  k1 k2 k3 χ¯ (3) (1|2|3) (δφ1 δφ2 δφ3 δφK   − δφ1 δφK   δφ2 δφ3 ) 1+2+3=K

=



k1 k2 k3 χ¯ (3) (1|2|3) δ(1 + 2 + 3 + K  )

1+2+3=K

× δ(1 + 2) δφ 2 1 δφ 2 3 + δ(1 + 3) δφ 2 1 δφ 2 2  k k12 χ¯ (3) (1| − 1|K) δφ 2 1 δφ 2 K . = 2δ(K + K  ) 1

(1.69)

20

Nonlinear Electrostatic Equations for Collisionless Plasmas

Thus, (1.66) simplifies 0 = k (k,ω) δφ 2 k,ω δ(k + k ) δ(ω + ω )   −i k1 k2 χ (2) (k1,ω1 |k2,ω2 ) δφk1,ω1 δφk2,ω2 δφk,ω  k1 +k2 =k ω1 +ω2 =ω

−2 δ(k + k ) δ(ω + ω )



k k

2

k,ω

× χ¯ (3) (k,ω | − k, − ω |k,ω) δφ 2 k,ω δφ 2 k,ω .

(1.70)

In deriving this result, we have made use of the properties of homogeneous and stationary turbulence (1.25) to write δφk,ω δφk,ω  = δ(k + k ) δ(ω + ω ) δφ 2 k,ω,δφk,ω δφk,ω δφk,ω δφk,ω  = δ(k + k + k + k ) δ(ω + ω + ω + ω ) × [ δ(k + k ) δ(ω + ω ) δφ 2 k,ω δφ 2 k,ω + δ(k + k ) δ(ω + ω ) δφ 2 k,ω δφ 2 k,ω + δ(k + k ) δ(ω + ω ) δφ 2 k,ω δφ 2 k,ω + δφ 4 kω;k+k,ω+ω ;k+k +k,ω+ω +ω ],

(1.71)

and after having done so, we ignored the irreducible four-body correlation δφ 4 kω;k+k,ω+ω ;k+k +k,ω+ω +ω in order to truncate the hierarchy of correlations. The resultant wave equation (1.70) still contains the three-body correlation δφk1,ω1 δφk2,ω2 δφ−k,−ω , hence, not completely closed yet. To compute this threebody correlation, we return to (1.65) and consider up to second-order nonlinearity,   k1 k2 χ (2) (k1,ω1 |k2,ω2 ) 0 = k (k,ω) δφk,ω − i k1 +k2 =k ω1 +ω2 =ω

× δφk1,ω1 δφk2,ω2 − δφk1,ω1 δφk2,ω2  .

(1.72)

The third-order nonlinear term makes contributions to higher-order corrections only, thus can be ignored at the outset. We impose the iterative solution of (1.72), (0) (1) + δφk,ω + ··· δφk,ω = δφk,ω

(1.73)

If we ignore the nonlinear correction in (1.72), and truncate the solution by δφk,ω = (0) , then we have δφk,ω (0) 0 = (k,ω) δφk,ω .

(1.74)

(0) The solution δφk,ω represents a sinusoidal (or plane wave) solution, which is uncorrelated with other spectral components. For such a plane-wave solution, ensemble (0) are zero: averages of all odd moments of δφk,ω

1.4 Perturbed Wave Equation (0) δφk,ω  = 0,

(0) (0) δφk,ω δφk(0) ,ω δφk,ω  = 0, . . .

21

(1.75)

This is because the plane-wave solution has a phase dependence of the form (0) ∼ φˆ k,ω eiϕk,ω , so that odd moments are associated with odd products of δφk,ω sinusoidal functions each with phase ϕk,ω . When integrated over ϕk,ω , odd moments thus disappear. When the iterative solution (1.73) is inserted to (1.72), the next order solution emerges:   i (1) = k1 k2 χ (2) (k1,ω1 |k2,ω2 ) δφk,ω k (k,ω) k +k =k ω +ω =ω 1 2 1 2 

(0) (0) (0) δφ  . (1.76) × δφk1,ω1 δφk2,ω2 − δφk(0) k2,ω2 1,ω1 The three-body correlation of interest can be approximately expressed as follows: δφk1,ω1 δφk2,ω2 δφk,ω  = δφk(1) δφk(0) δφk(0) ,ω  1,ω1 2,ω2 (0) (0) (1) + δφk(0) δφk(1) δφk(0) ,ω  + δφk ,ω δφk ,ω δφk,ω  1,ω1 2,ω2 1 1 2 2

+ ··· .

(1.77)

, δφk(1) , and δφk(1) Upon making use of solution (1.76) and substituting for δφk(1) ,ω , 1,ω1 2,ω2 it can be shown that the result is (the intermediate steps are somewhat tedious but straightforward) δφk1,ω1 δφk2,ω2 δφk,ω  2i k |k − k1 | χ (2) (−k + k1, − ω + ω1 |k,ω) = δφ 2 k−k1,ω−ω1 δφ 2 k,ω k1 (k1,ω1 ) 2i k k1 χ (2) (−k1, − ω1 |k,ω) + (1.78) δφ 2 k1,ω1 δφ 2 k,ω |k − k1 | (k − k1,ω − ω1 ) 2i k1 |k − k1 | χ (2)∗ (k1,ω1 |k − k1,ω − ω1 ) − δφ 2 k1,ω1 δφ 2 k−k1,ω−ω1 , k ∗ (k,ω) where we have deleted the superscripts “(0)” after everything is said and done, and have made use of the symmetry properties, (−k, − ω) = ∗ (k,ω), χ (2) (−k1, − ω1 | − k2, − ω2 ) = −χ (2)∗ (k1,ω1 |k2,ω2 ), χ (2) (k1,ω1 |k2,ω2 ) = χ (2) (k2,ω2 |k1,ω1 ).

(1.79)

We have not yet discussed these symmetry properties associated with various susceptibilities, although the complex conjugate properties can be easily deduced from definitions (1.61)–(1.64). In the subsequent section we will discuss the symmetries as well as other properties associated with the susceptibilities in detail.

22

Nonlinear Electrostatic Equations for Collisionless Plasmas

With (1.78) the wave equation (1.70) is now written as 0 = (k,ω)  k δφ k,ω + 2 2



2



χ (2) (k,ω |k − k,ω − ω )

k,ω

(−k + k , − ω + ω |k,ω)  |k − k |2 δφ 2 k−k,ω−ω (k,ω )  χ (2) (−k, − ω |k,ω) 2 2   + (1.80)  k δφ k ,ω  k 2 δφ 2 k,ω   (k − k ,ω − ω )  |χ (2) (k,ω |k − k,ω − ω )|2 2 2  2 2     − δφ   |k − k | δφ   k k ,ω k−k ,ω−ω ∗ (k,ω)  −2 χ¯ (3) (k,ω | − k, − ω |k,ω)  k 2 δφ 2 k,ω  k 2 δφ 2 k,ω . ×

χ



(2)

k,ω

This equation is known as the nonlinear spectral balance equation since the linear term, which appears as the first term on the right-hand side of the equality, is balanced by the rest that represents nonlinear response. We may rewrite the (1.80) by noting that the electrostatic potential correlation can be rewritten as the electric field correlation  k 2 δφ 2 k,ω = δE 2 k,ω,

(1.81)

where δE 2  is related to the spectral electric field energy density, Ewave = δE 2 / (8π). As a consequence, (1.80) is equivalently written as 0 = (k,ω) δE k,ω + 2 2





χ (2) (k,ω |k − k,ω − ω )

k,ω

χ (2) (−k + k, − ω + ω |k,ω) δE 2 k−k,ω−ω (k,ω )  χ (2) (−k, − ω |k,ω) 2   δE 2 k,ω +  δE k ,ω (k − k,ω − ω )  |χ (2) (k,ω |k − k,ω − ω )|2 2 2 − δE k,ω δE k−k,ω−ω ∗ (k,ω)  −2 χ¯ (3) (k,ω | − k, − ω |k,ω) δE 2 k,ω δE 2 k,ω . ×

(1.82)

k,ω

To determine the adiabatic time evolution of the spectral wave energy density, we now reinstitute the slow time dependence implicit in the wave-particle resonance denominator: (ω − k · v + i0 + i∂/∂t)−1 .

1.4 Perturbed Wave Equation

23

Recall that in deriving the final spectral balance equation (1.82), we had “absorbed” the derivative i∂/∂t in the “new definition” of ω – see (1.41). This resulted in turning the differential equation (1.40) into an algebraic equation. After the desired equation (1.82) has now been obtained, we reintroduce the factor i∂/∂t in the arguments of the response functions. As a consequence, various dielectric susceptibility response functions become operators in the slow time t of the amplitude evolution. However, i∂/∂t was present in the original equation (1.40) only on the left-hand side, while the angular frequency ω appeared on both sides. Consequently, when we reintroduce the slow-time derivative i∂/∂t to the angular frequency, we do so by treating this object as a small correction, and we introduce it only to the leading term, which is the linear response function. To reiterate, this whole procedure is heuristic, and as we noted already, the proper way to treat this type of problem is via multiple time scale analysis (Davidson, 1972). In this book, we take the present shortcut method nonetheless. When the slow-time derivative is thus reintroduced, the linear term is modified as   ∂ 2 δE 2 k,ω (k,ω) δE k,ω → k,ω + i ∂t   i ∂ (k,ω) ∂ δE 2 k,ω, → (k,ω) + (1.83) 2 ∂ω ∂t where 1/2 in front of the time derivative in the last expression stems from the fact that ∂ δE−k,−ω (t) ∂ δEk,ω (t) = . (1.84) ∂t ∂t a , which automatiRecall that the slow-time derivative originally affects only δfk,ω cally implies that i∂/∂t is meant to operate on δEk,ω or δE−k,−ω separately, but not on their products. Consequently, when the slow-time derivative is reintroduced, the proper procedure is as follows:   (k,ω) δE 2 k,ω = δE−k,−ω (k,ω) δEk,ω     ∂ δEk,ω → δE−k,−ω k,ω + i ∂t     ∂ (k,ω) ∂ δEk,ω = δE−k,−ω (k,ω) + i ∂ω ∂t   = (k,ω) δE 2 k,ω   ∂δEk,ω ∂δE−k,−ω i ∂ (k,ω) δE−k,−ω . + + δEk,ω 2 ∂ω ∂t ∂t After making use of (1.84), then we readily arrive at (1.83).

24

Nonlinear Electrostatic Equations for Collisionless Plasmas

This procedure results in the formal wave kinetic equation i ∂ (k,ω) ∂ δE 2 k,ω 0 = (k,ω) δE 2 k,ω + 2 ∂ω ∂t  +2 χ (2) (k,ω |k − k,ω − ω ) k,ω



χ (2) (−k + k, − ω + ω |k,ω) δE 2 k−k,ω−ω (k,ω )  χ (2) (−k, − ω |k,ω) 2   + δE k ,ω δE 2 k,ω   (k − k ,ω − ω )  |χ (2) (k,ω |k − k,ω − ω )|2 2 2     −  δE  δE k ,ω k−k ,ω−ω ∗ (k,ω)  −2 χ¯ (3) (k,ω | − k, − ω |k,ω) δE 2 k,ω δE 2 k,ω . ×

(1.85)

k,ω

This equation is further manipulated by separating the linear dielectric function (k,ω) into real and imaginary parts, (k,ω) = Re (k,ω) + i Im (k,ω),

(1.86)

where it is assumed that |Im (k,ω)| |Re (k,ω)|.

(1.87)

This assumption is equivalent to the weak growth/damping approximation. The imaginary part of the derivative, ∂ Im (k,ω)/∂ω, which couples with the slowtime derivative ∂/∂t, is also ignored. Before we present the final result, let us invoke another useful symmetry property associated with the second-order nonlinear susceptibility, χ (2) (k1,ω1 |k2,ω2 ) = χ (2) (k1 + k2,ω1 + ω2 | − k2, − ω2 ),

(1.88)

which will be derived in the next section. Then (1.85) can be expressed as 0=

i ∂ Re (k,ω) ∂ δE 2 k,ω + Re (k,ω) δE 2 k,ω + i Im (k,ω) δE 2 k,ω 2 ∂ω ∂t      dω { χ (2) (k,ω |k − k,ω − ω ) }2 + 2 dk   δE 2 k−k,ω−ω δE 2 k,ω δE 2 k,ω × (1.89) + (k,ω ) (k − k,ω − ω )  |χ (2) (k,ω |k − k,ω − ω )|2 2 2 − δE k,ω δE k−k,ω−ω ∗ (k,ω)    − 2 dk dω χ¯ (3) (k,ω | − k, − ω |k,ω) δE 2 k,ω δE 2 k,ω .

1.4 Perturbed Wave Equation

25

The real part of this equation,       0 = Re (k,ω) + 2 Re dk dω { χ (2) (k,ω |k − k,ω − ω ) }2  ×

δE 2 k−k,ω−ω δE 2 k,ω + (k,ω ) (k − k,ω − ω )

−χ¯

(3)









(k ,ω | − k , − ω |k,ω) δE  

− 2 Re

dk





dω

2

  δE 2 k,ω

k,ω

(1.90)

|χ (2) (k,ω |k − k,ω − ω )|2 δE 2 k,ω δE 2 k−k,ω−ω , ∗ (k,ω)

determines the wave dispersion relation, ω = ωk , while the imaginary part, 0=

1 ∂ Re (k,ω) ∂ δE 2 k,ω + Im (k,ω) δE 2 k,ω 2 ∂ω ∂t      dω { χ (2) (k,ω |k − k,ω − ω ) }2 + 2 Im dk  ×

δE 2 k−k,ω−ω δE 2 k,ω + (k,ω ) (k − k,ω − ω )

 δE 2 k,ω

(1.91)

 |χ (2) (k,ω |k − k,ω − ω )|2 2 2 − δE k,ω δE k−k,ω−ω ∗ (k,ω)    − 2 Im dk dω χ¯ (3) (k,ω | − k, − ω |k,ω) δE 2 k,ω δE 2 k,ω, corresponds to the wave kinetic equation. In general, (1.90) is not quite a “dispersion equation” yet, since in the last term different spectral components are inexorably coupled. However, if we ignore the non-diagonal term, δE 2 k,ω δE 2 k−k,ω−ω , that is, if we approximate (1.90) by       dω { χ (2) (k,ω |k − k,ω − ω ) }2 0 = Re (k,ω) + 2 Re dk 

δE 2 k−k,ω−ω δE 2 k,ω × + (k,ω ) (k − k,ω − ω ) −χ¯

(3)









(k ,ω | − k , − ω |k,ω) δE k,ω

≡ Re ε˜ (k,ω) δE 2 k,ω,

2

  δE 2 k,ω (1.92)

then (1.92) represents nonlinear dispersion equation in the proper sense. If we are concerned with a situation where the waves excited in the plasma can be characterized by linear dispersion relation, but where we are interested in waves interacting

26

Nonlinear Electrostatic Equations for Collisionless Plasmas

with the particles and among themselves via linear and nonlinear wave-particle and wave-wave interactions, then one may simplify the real part of the spectral balance equation, namely, (1.92), by simply ignoring nonlinear mode coupling terms altogether, Re (k,ω) δE 2 k,ω = 0.

(1.93)

In Appendix C, we discuss the result of retaining the nonlinear correction terms in (1.92), but in the main body of this book we focus on linear eigenmodes and nonlinear interactions among the linear modes and the particles. 1.5 Formal Wave Kinetic Equation for Eigenmodes Linear wave equation (1.93) implies that the angular frequency is a function of wave vector, that is, ω and k satisfy a “dispersion relation.” ω = ωkα,

(1.94)

where α designates the eigenmode. There could be more than a single solution, hence the superscript α. Equation (1.93) implies that the spectral electric field wave energy density can be represented in terms of the wave intensity  δE 2 k,ω = [ Ik+α δ(ω − ωkα ) + Ik−α δ(ω + ωkα ) ]. (1.95) α=L,S

In this equation, Ik±α represents the intensity of electrostatic waves associated with eigenmode α, propagating in forward/backward (±) direction. We will discuss the linear wave properties later, but it is well known that linear electrostatic eigenmodes of a uniform, unmagnetized plasma are high-frequency Langmuir wave (α = L) and low-frequency ion-sound (or ion acoustic) wave (α = S). It is important to distinguish the forward versus backward propagation, as nonlinear interactions of these modes depend on the wave propagation direction. Inserting (1.95) to the wave kinetic equation (1.91) we have     1 ∂ Re (k,σ ωα ) ∂I σ α k k α σα δ(ω − σ ωkα ) + Im (k,σ ωk ) Ik 0= α 2 ∂σ ω ∂t k α σ =±1       + 2 Im dk ×

γ σ  =±1 γ γ − σ  ωk−k |k − k,σ  ωk−k ) }2 γ (k,σ ωkα − σ  ωk−k )

α σ =±1 (2)  { χ (k ,σ ωkα

σ  γ

Ik−k

1.5 Formal Wave Kinetic Equation for Eigenmodes

+

  { χ (2) (k,σ  ωβ |k − k,σ ωα − σ  ωβ ) }2 k k k β

(k −

σ  =±1

− 2 Im







dk

σ  γ



β σ  ωk )



σ β Ik ⎠ Ikσ α δ(ω − σ ωkα )

|χ (2) (k,σ  ωk |k − k,σ  ωk−k )|2 γ

β



β,γ σ ,σ  =±1 σ β

k,σ ωkα

27



β

γ

∗ (k,σ  ωk + σ  ωk−k )

× Ik Ik−k δ(ω − σ  ωk − σ  ωk−k )    β β dk χ¯ (3) (k,σ  ωk | − k, − σ  ωk |k,σ ωkα ) − 2 Im β

γ

α,β σ,σ  =±1 σ β

× Ik Ikσ α δ(ω − σ ωkα ),

(σ = ±1).

(1.96)

Let us focus on the quantity [ (k,ω)]−1 . If ω lies in the vicinity of linear eigenmode, ω ∼ σ ωkα , where Re (k,σ ωkα ) = 0, since we had assumed that |Im (k,ω)| |Re (k,ω)|, we may approximately express 1 1 ≈ (k,ω) Re (k,ω)   1 ≈ α (ω − σ ωk + i0) [∂Re (k,ω)/∂ω]ω=σ ωkα α σ =±1 =−

  α

iπ δ(ω − σ ωkα ) , ∂Re (k,σ ωkα )/∂σ ωkα σ =±1

(1.97)

where we have taken the series expansion,   ∂Re (k,ω)  α Re (k,ω) ≈ (ω − σ ωk + i0) .  ∂ω ω=σ ωα σ α

(1.98)

k

In the second line of (1.97) we have summed over all possible poles. If we include contributions from those ω’s that are sufficiently far away from linear eigenmodes in the complex frequency space, then we must add the principal part contribution to the right-hand side of (1.97) as well:   iπ δ(ω − σ ωkα ) 1 1 . =P − (k,ω) (k,ω) ∂Re (k,σ ωkα )/∂σ ωkα α σ =±1   1 iπ δ(ω − σ ωkα ) 1 . = P + ∗ (k,ω) ∗ (k,ω) ∂Re (k,σ ωkα )/∂σ ωkα α σ =±1

(1.99)

28

Nonlinear Electrostatic Equations for Collisionless Plasmas

Principal value P is meant to exclude those ω’s in the vicinity of linear eigenmodes, ω = ±ωkα . Making use of (1.99), the wave kinetic equation (1.96) is now expressed as follows:     1 ∂ Re (k,σ ωα ) ∂I σ α k k α σα 0= + Im (k,σ ωk ) Ik 2 ∂σ ωkα ∂t α σ =±1

   β β dk 2 { χ (2) (k,σ  ωk |k − k,σ ωkα − σ  ωk ) }2 + 2 Im α,β σ,σ  =±1

×P − χ¯

1 (k −

(3)



(k ,σ

− 2 Im





β

k,σ ωkα

− σ  ωk )

β ωk |



−k, −σ 







β ωk |k,σ ωkα )

σ β

Ik Ikσ α

dk |χ (2) (k,σ  ωk |k − k,σ  ωk−k )|2 γ

β

β,γ σ ,σ  =±1

×P + 2π 

1 β ∗ (k,σ  ωk



+

σ  β σ  γ Ik Ik−k γ  σ ωk−k )





δ(σ ωkα − σ  ωk − σ  ωk−k ) γ

β

dk |χ (2) (k,σ  ωk |k − k,σ ωkα − σ  ωk )|2 β

β

α,β,γ σ,σ ,σ  =±1 σ  γ

Ik−k Ikσ α

σ β

Ik Ikσ α × + γ γ β β ∂Re (k − k,σ  ωk−k )/∂σ  ωk−k ∂Re (k,σ  ωk )/∂σ  ωk  σ  β σ  γ Ik Ik−k γ β δ(σ ωkα − σ  ωk − σ  ωk−k ). − (1.100) ∂Re (k,σ ωkα )/∂σ ωkα In deriving this result, we have invoked the fact that, to the leading order, χ (2) is purely imaginary so that {χ (2) (k1,ω1 |k2,ω2 )}2 ≈ −|χ (2) (k1,ω1 |k2,ω2 )|2,

(1.101)

which we imposed for the last term that contains the three-wave resonance delta γ β function condition δ(σ ωkα − σ  ωk − σ  ωk−k ). This property will be discussed in γ β the subsequent section. In (1.100), the principal part, P[1/ ∗ (k,σ  ωk + σ  ωk−k )], γ β and the condition, σ ωkα = σ  ωk + σ  ωk−k , where σ ωkα satisfies ∗ (k,σ ωkα ) = 0,

1.6 Formal Particle Kinetic Equation

29

are mutually exclusive. Thus, by definition, this term is zero. This leaves us with   ∂ Re (k,σ ωα ) ∂I σ α k

k

∂σ ωkα ∂t α σ =±1   =− 2 Im (k,σ ωkα ) Ikσ α α σ =±1

−4

  k

×P − χ¯



Im 2 {χ (2) (k,σ  ωk |k − k,σ ωkα − σ  ωk )}2 β

β

α,β σ,σ  =±1

1

(1.102)

β

(k − k,σ ωkα − σ  ωk )

(3)

− 4π







β ωk |

(k ,σ 







β ωk |k,σ ωkα )

σ β

−k, −σ Ik Ikσ α  β β |χ (2) (k,σ  ωk |k − k,σ ωkα − σ  ωk )|2

k α,β,γ σ,σ ,σ  =±1 σ  γ

σ β

Ik−k Ikσ α

Ik Ikσ α × + γ γ β β ∂Re (k − k,σ  ωk−k )/∂σ  ωk−k ∂Re (k,σ  ωk )/∂σ  ωk  σ  β σ  γ Ik Ik−k γ β δ(σ ωkα − σ  ωk − σ  ωk−k ). − ∂Re (k,σ ωkα )/∂σ ωkα This equation is the formal wave kinetic equation governing the dynamics of linear eigenmodes as they undergo nonlinear interactions among themselves as well as with the plasma particles. At this stage in the development of formalism, however, the result is not practically useful since the various susceptibilities are expressed at a formal level. These quantities are yet to be explicitly calculated in forms that readily lend themselves to further analysis.

1.6 Formal Particle Kinetic Equation Formal particle kinetic equation (1.40) can be further manipulated by considering the quantity given in (1.103), which follows from (1.52), a δφk,ω  = δ(k + k ) δ(ω + ω ) α(k,ω) δφ 2 k,ω na Fa δfk,ω   + α (2) (k1,ω1 |k2,ω2 ) k1 +k2 =k ω1 +ω2 =ω

× δφk1,ω1 δφk2,ω2 δφk,ω  na Fa  + 2 δ(k + k ) δ(ω + ω ) α (3) (k1,ω1 | − k1, − ω1 |k,ω) k1

× δφ k1,ω1 δφ k,ω na Fa . 2

2

ω1

(1.103)

30

Nonlinear Electrostatic Equations for Collisionless Plasmas

Upon making use of (1.78), we have

 a δφk,ω  = δ(k + k ) δ(ω + ω ) α(k,ω) δφ 2 k,ω δfk,ω 

+

k1 ω1

2i α (2) (k1,ω1 |k − k1,ω − ω1 ) k k1 |k − k1 |



χ (2) (k1,ω1 |k − k1,ω − ω1 ) |k − k1 |2 k 2 (k1,ω1 ) × δφ 2 k−k1,ω−ω1 δφ 2 k,ω

×

+

χ (2) (k1,ω1 |k − k1,ω − ω1 ) 2 2 k1 k δφ 2 k1,ω1 δφ 2 k,ω (k − k1,ω − ω1 )

χ (2)∗ (k1,ω1 |k − k1,ω − ω1 ) 2 k1 |k − k1 |2 ∗ (k,ω)  2 2 × δφ k1,ω1 δφ k−k1,ω−ω1



+2



α (3) (k1,ω1 | − k1, − ω1 |k,ω) δφ 2 k1,ω1 δφ 2 k,ω .

k1,ω1

(1.104) Substituting (1.104) to the right-hand side of formal particle kinetic equation in (1.40), we obtain    δE 2 k,ω ∂ ∂ Fa iea α(k,ω) dk dω k · =− ∂t ma ∂v k2   M(k,ω |k − k,ω − ω ) + 2i dk dω α (2) (k,ω |k − k,ω − ω ) k k  |k − k |    δE 2 k,ω δE 2 k,ω   (3)     dω α (k ,ω | − k , − ω |k,ω) + 2 dk Fa , k 2 k 2 (1.105) where

  δE 2 k2,ω2 δE 2 k1,ω1 δE 2 k1 +k2,ω1 +ω2 M(k1,ω1 |k2,ω2 ) = χ (k1,ω1 |k2,ω2 ) + (k1,ω1 ) (k2,ω2 ) χ (2)∗ (k1,ω1 |k2,ω2 ) − ∗ (1.106) δE 2 k1,ω1 δE 2 k2,ω2 . (k1 + k2,ω1 + ω2 ) (2)

It is instructive to rewrite this equation explicitly by making use of (1.53). Taking the real part only we obtain

1.7 Linear and Nonlinear Susceptibilities



31



k ∂ δE 2 k,ω k ∂Fa · · k ∂v ω − k · v k ∂v     k ∂ M(k,ω |k − k,ω − ω ) e3 dk dω dk dω · + Re a3 ma k ∂v ω−k·v   1 (k − k ) ∂ k ∂ ×  · · k ∂v ω − ω − (k − k ) · v |k − k | ∂v  1 k ∂ (k − k ) ∂ Fa · · + |k − k | ∂v ω − k · v k  ∂v     k ∂ δE 2 k,ω δE 2 k,ω k ∂ e4 dk dω dk dω · + Im a4 · ma k ∂v ω−k·v k  ∂v   k ∂ 1 k ∂ 1 × · ·    ω − ω − (k − k ) · v k ∂v ω − k · v k ∂v  1 k ∂ k ∂ Fa . · (1.107) − · k ∂v ω − k · v k  ∂v

∂Fa e2 = −Im a2 ∂t ma

dk



We will make explicit use of the spectral wave energy density (1.95) in (1.107), thereby eliminating ω and ω integrals by virtue of the delta functions. This will be done later in Chapter 4. For now, we treat (1.107) as constituting the formal particle kinetic equation, and as an interlude, we next discuss the properties of various dielectric susceptibility response functions. 1.7 Linear and Nonlinear Susceptibilities Linear and nonlinear susceptibilities have already been introduced in (1.55)–(1.64). It is instructive to rewrite these response functions by means of partial integrations,  Fa 2 , (1.108) χa (k,ω) = −ωpa dv (ω − k · v + i0)2  1 χa(2) (k1,ω1 |k2,ω2 ) = − dv Fa (ω1 − k1 · v + i0)(ω2 − k2 · v + i0) 1 ω1 + ω2 − (k1 + k2 ) · v + i0  2 k k2 · (k1 + k2 ) k22 k1 · (k1 + k2 ) + × 1 ω1 − k1 · v + i0 ω2 − k2 · v + i0  (k1 + k2 )2 (k1 · k2 ) , + ω1 + ω2 − (k1 + k2 ) · v + i0

×

(1.109)

32

Nonlinear Electrostatic Equations for Collisionless Plasmas

χ¯ a(3) (k1,ω1 |k2,ω2 |k3,ω3 ) =

2 [k1 · (k1 + k2 + k3 )] 1 ea2 ωpa 2 2 ma k1 k2 k3 |k1 + k2 + k3 |  1 × dv Fa (ω2 − k2 · v + i0) (ω3 − k3 · v + i0) 1 × [ω1 + ω2 + ω3 − (k1 + k2 + k3 ) · v + i0]2  2 × ω1 + ω2 + ω3 − (k1 + k2 + k3 ) · v + i0  3 [k2 · (k1 + k2 + k3 )] [k3 · (k1 + k2 + k3 )] × ω1 + ω2 + ω3 − (k1 + k2 + k3 ) · v + i0

+

k22 [k3 · (k1 + k2 + k3 )] ω2 − k2 · v + i0

 k32 [k2 · (k1 + k2 + k3 )] + ω3 − k3 · v + i0 1 + ω + ω3 − (k2 + k3 ) · v + i0 2 2 (k2 · k3 ) [(k2 + k3 ) · (k1 + k2 + k3 )] × ω1 + ω2 + ω3 − (k1 + k2 + k3 ) · v + i0 k22 [k3 · (k2 + k3 )] k32 [k2 · (k2 + k3 )] + ω2 − k2 · v + i0 ω3 − k3 · v + i0  2 (k2 + k3 ) (k2 · k3 ) . (1.110) + ω2 + ω3 − (k2 + k3 ) · v + i0

+

1.7.1 Symmetry Relations The first useful symmetry property involves the permutation of arguments. The second- and third-order susceptibilities, χ (2) (k1,ω1 |k2,ω2 ) and χ (3) (k1,ω1 |k2, ω2 |k3,ω3 ), are fully symmetric with respect to permutations of arguments. However, the partial third-order susceptibility χ¯ (3) (k1,ω1 |k2,ω2 |k3,ω3 ) is symmetric only with respect to the permutation of the last two sets of arguments: χ (2) (k2,ω2 |k1,ω1 ) = χ (2) (k1,ω1 |k2,ω2 ), χ¯ (3) (k1,ω1 |k3,ω3 |k2,ω2 ) = χ¯ (3) (k1,ω1 |k2,ω2 |k3,ω3 ), χ

(3)

(k1,ω1 |k2,ω2 |k3,ω3 ) = χ

(3)

(1.111)

(k1,ω1 |k3,ω3 |k2,ω2 )

= χ (3) (k2,ω2 |k1,ω1 |k3,ω3 ) = χ (3) (k2,ω2 |k3,ω3 |k1,ω1 ) = χ (3) (k3,ω3 |k2,ω2 |k1,ω1 ) = χ (3) (k3,ω3 |k1,ω1 |k2,ω2 ).

1.7 Linear and Nonlinear Susceptibilities

33

The next useful symmetry property involves arguments of opposite signs: (−k, − ω) = ∗ (k,ω), χ (2) (−k1, − ω1 | − k2, − ω2 ) = −χ (2)∗ (k1,ω1 |k2,ω2 ), χ¯

(3)

χ

(3)

(−k1, − ω1 | − k2, − ω2 | − k3, − ω3 ) = χ¯

(3)∗

(k1,ω1 |k2,ω2 |k3,ω3 ),

(−k1, − ω1 | − k2, − ω2 | − k3, − ω3 ) = χ

(3)∗

(k1,ω1 |k2,ω2 |k3,ω3 ).

(1.112)

These symmetry relations can easily be checked from definitions (1.62)–(1.64) or (1.108)–(1.110). If the small positive imaginary part in the resonant denominator is ignored, then we obtain a useful approximate symmetry relation from (1.109), χ (2) (k1,ω1 |k2,ω2 ) = χ (2) (k2,ω2 |k1,ω1 ) = χ (2) (k1 + k2,ω1 + ω2 | − k2, − ω2 ),

(1.113)

which we have already invoked in (1.88). 1.7.2 Linear Dielectric Susceptibility In the long-wavelength limit (k 2 → 0), the linear dielectric function χ a (k,ω) takes on the limiting form, χ a (0,ω) = −

2 ωpa

. (1.114) ω2 To include thermal corrections, we expand the resonant denominator for small argument, k · v/ω 1, to obtain  2  ωpa 3k 2 Ta a χ (k,ω) ≈ − 2 1 + ω ma ω 2  2 ωpa ∂Fa − iπ 2 dv k · δ(ω − k · v), (1.115) k ∂v where we have defined the kinetic temperature,  ma vT2 a . (1.116) Ta = ma dv v 2 Fa = 2 √ Here, vT a = 2Ta /ma represents the thermal speed for species a. In deriving (1.115) we assumed that the plasma species a has zero net drift, dv v Fa ≈ 0. The expression (1.115) assumes the so-called fast-wave condition, ω kvT a . For the opposite case when ω kvT a is satisfied (the slow-wave condition), we may approximate the principal part of the resonant denominator by P

1 1 ≈− . ω − k · v + i0 k·v

(1.117)

34

Nonlinear Electrostatic Equations for Collisionless Plasmas

If we assume that the distribution Fa is given by a quasi-Maxwellian form so that we may write ∂Fa 2(k · v) Fa , ≈− ∂v vT2 a

(1.118)

then this leads to the top-and-bottom cancelation of the factor k · v within the velocity integral. The resulting linear dielectric susceptibility is χ (k,ω) = a

2 2ωpa

− iπ

k 2 vT2 a

2 ωpa

 dv k ·

k2

∂Fa δ(ω − k · v). ∂v

(1.119)

For isotropic thermal equilibrium distribution, the linear dielectric function is given in terms of the plasma dispersion function Z(z), as (k,ω) = 1 −

2  ωpa

ω2

a

a2 a ζk,ω Z  (ζk,ω ),

(1.120)

2Ta , ma

(1.121)

where a = ζk,ω

ω , kvT a

vT2 a =

and 



dx e−x 2 = 2i e−z √ π x−z −∞  Z (z) = −2[1 + z Z(z)]. Z(z) =

2



iz

2

dt et , −∞

(1.122)

The plasma dispersion function and its properties are well known, for example, see Huba (2009). Series and asymptotic expansions of Z(z) are given by √ 4z3 8z5 16z7 2 − + + · · · , (z2 < 1), Z(z) = i π e−z − 2z + 3 15 105 √ 1 3 15 1 2 Z(z) = i π σ e−z − − 3 − 5 − 7 + · · · , (z2 > 1), z 2z 4z 8z

(1.123)

where σ = 0 if Imz > 1/Rez, σ = 1 if |Imz| < 1/Rez, and σ = 2 if −Imz > 1/Rez. The plasma dispersion or Fried–Conte function (Fried and Conte, 1961) and its properties are further discussed in Appendix D. Making use of the asymptotic expansion we obtain an approximate expression for the linear dielectric response function for thermal equilibrium,

1.7 Linear and Nonlinear Susceptibilities



2  ωpa

3k 2 Ta ω2 ma ω 2 a   2  ωpa ω ω2 1/2 + 2iπ exp − 2 2 . k vT a k 3 vT3 a a

(k,ω) = 1 −

35



1+

(1.124)

As it turns out, this expression, derived under the assumption of fast wave condition, is applicable to high-frequency Langmuir waves where both conditions, ω kvT e and ω kvT i , are valid. For ion-sound waves, on the other hand, ω kvT e (slowwave condition for electrons), while ω kvT i (fast wave condition for protons). In this case, we have  2  2 ωpi 2ωpe 3k 2 Ti (k,ω) = 1 + 2 2 − 2 1 + ω mi ω 2 k vT e   2  ωpa ω ω2 1/2 (1.125) + 2iπ exp − 2 2 . k vT a k 3 vT3 a a These properties will be used later when we review the linear wave theory. 1.7.3 Second-Order Nonlinear Susceptibility The following limiting forms for the second-order susceptibility can easily be derived: χa(2) (0,ω1 |0,ω2 ) = 0, χa(2) (k1,0|k2,0) = −

2 ωpa 1 iea . Ta k1 k2 |k1 + k2 | vT2 a

(1.126)

If the fast-wave condition is satisfied for all frequencies and wave numbers, ω1 k1 vT a ,

ω2 k2 vT a ,

ω1 + ω2 |k1 + k2 |vT a,

(1.127)

then the approximate second-order nonlinear susceptibility can be obtained by assuming that the temperature is zero (Ta → 0), or equivalently by choosing Fa (v) = δ(v): χa(2) (k1,ω1 |k2,ω2 ) =

2 ωpa 1 −i ea 2 ma ω1 ω2 (ω1 + ω2 ) k1 k2 |k1 + k2 |  2 k2 k1 k2 · (k1 + k2 ) + 2 k1 · (k1 + k2 ) × ω1 ω2  2 (k1 + k2 ) (1.128) k1 · k2 . + ω1 + ω 2

36

Nonlinear Electrostatic Equations for Collisionless Plasmas

If one of the frequencies, say ω1 , does not satisfy the fast-wave condition while ω2 and ω1 + ω2 are characterized by ω2 k2 vT a and ω1 + ω2 |k1 + k2 |vT a , then we may obtain the following expression: χa(2) (k1,ω1 |k2,ω2 ) =

2 ωpa 1 i ea 2 ma ω2 (ω1 + ω2 ) k1 k2 |k1 + k2 |  k1 · ∂Fa /∂v dv × k2 · (k1 + k2 ) , ω1 − k1 · v + i0

where we have made use of the identity   k1 · ∂Fa /∂v Fa 1 dv = − 2 dv , 2 (ω1 − k1 · v + i0) ω1 − k1 · v + i0 k1

(1.129)

(1.130)

and have invoked the fast wave conditions to ignore terms k22 /ω and (k1 + k2 )2 /(ω1 + ω2 ). The velocity integral is related to the linear response function so that we have χa(2) (k1,ω1 |k2,ω2 ) =

k2 · (k1 + k2 ) k1 i ea χa (k1,ω1 ). 2 ma ω2 (ω1 + ω2 ) k2 |k1 + k2 |

(1.131)

If we further assume that ω1 k1 vT a , that is, the slow-wave condition for ω1 , then we obtain χa(2) (k1,ω1 |k2,ω2 ) =

2 ωpa k2 · (k1 + k2 ) i ea 2 ma ω2 (ω1 + ω2 ) k1 k2 |k1 + k2 |    ∂Fa 2 − iπ dv k1 · × δ(ω1 − k1 · v) . ∂v vT2 a

(1.132)

If ω2 represents an arbitrary wave, then by following the same steps as in the previous case, we may derive the following result: χa(2) (k1,ω1 |k2,ω2 ) = ≈

k1 · (k1 + k2 ) k2 i ea χa (k2,ω2 ) 2 ma ω1 (ω1 + ω2 ) k1 |k1 + k2 | 2 ωpa k1 · (k1 + k2 ) i ea 2 ma ω1 (ω1 + ω2 ) k1 k2 |k1 + k2 |    ∂Fa 2 − iπ dv k · − k · v) . × δ(ω 2 2 2 ∂v vT2 a

(1.133)

Finally, if ω1 + ω2 represents the low frequency while ω1 and ω2 satisfy the fastwave condition (which becomes possible if ω1 and ω2 have opposite signs), then we have

1.7 Linear and Nonlinear Susceptibilities

37

i ea |k1 + k2 | k1 · k2 χa (k1 + k2,ω1 + ω2 ) 2 ma ω1 ω2 k1 k2  2 2 k1 · k2 i ea ωpa (1.134) = 2 ma ω1 ω2 k1 k2 |k1 + k2 | vT2 a   ∂Fa δ[ω1 + ω2 − (k1 + k2 ) · v] . − iπ dv (k1 + k2 ) · ∂v

χa(2) (k1,ω1 |k2,ω2 ) =

If we ignore the resonant contribution to the second-order nonlinear susceptibility, then it becomes evident that the leading order expression satisfies {χ (2) (k1,ω1 |k2,ω2 )}2 ≈ −|χ (2) (k1,ω1 |k2,ω2 )|2,

(1.135)

which we have invoked in (1.101). 1.7.4 Third-Order Nonlinear Susceptibility To simplify the discussion, herewith we only consider the third-order susceptibility of the form, χ¯ a(3) (k,ω | − k, − ω |k,ω): χ¯ a(3) (k,ω |

 2 1 1 ea2 ωpa dv − k , − ω |k,ω) = − 2 2 2 2 ma k k ω − k · v + i0  1 ∂ × k ·  ∂v ω − ω − (k − k ) · v + i0     k · ∂Fa /∂v ∂ × k· ∂v ω − k · v + i0   k · ∂Fa /∂v ∂  , (1.136) −k · ∂v ω − k · v + i0 



or equivalently (after partial integrations) χ¯ a(3) (k,ω | − k, − ω |k,ω) =

2 (k · k ) 1 ea2 ωpa 2 m2a k 2 k 2  Fa × dv (ω − k · v + i0)3 (ω − k · v + i0)   4 (k · k ) 2k 2 × ω − k · v + i0 ω − k · v + i0  1 k 2 + +    ω − k · v + i0 ω − ω − (k − k ) · v + i0

38

Nonlinear Electrostatic Equations for Collisionless Plasmas



2 (k × k )2 + 3 k 2 [k · (k − k )] ω − k · v + i0  2 k [k · (k − k )] |k − k |2 (k · k ) . +  + ω − k · v + i0 ω − ω − (k − k ) · v + i0 (1.137)

×

It is straightforward to obtain the following limiting expressions: χ¯ a(3) (k,0| − k,0|k,0) = −

2 k · k 1 ea2 ωpa , Ta2 k 2 k 2 k 2 vT2 a

χ¯ a(3) (0,ω |0, − ω |0,ω) = 0.

(1.138)

The first expression is obtained from (1.136), while the second result follows from (1.137). When all three frequencies, ω, ω , and ω − ω , satisfy the fast-wave conditions ω kvT a , ω k  vT a , ω − ω |k − k |vT a , then we have χ¯ a(3) (k,ω | − k, − ω |k,ω) =

 2 k · k 2 k 2 k 2 k 2 [k · (k − k )] 1 ea2 ωpa + 2 m2a ω3 ω k 2 k 2 ω ω ω (ω − ω ) 2   2 8 k (k · k ) 2 (k × k ) + 3k 2 [k · (k − k )] + + ω2 ω (ω − ω )  (k − k )2 (k · k ) . (1.139) + (ω − ω )2

If ω represents an arbitrary wave frequency, but the fast-wave condition is applicable for other two frequencies, ω kvT a and ω − ω |k − k | vT a , then we obtain χ¯ a(3) (k,ω | − k, − ω |k,ω)   2 k · k 2 k 2 k · (k − k ) k · ∂Fa /∂v 1 ea2 ωpa dv + =− 2 m2a ω3 k 2 k 2 ω ω − ω ω − k · v + i0   1 e2 k · k 2 k 2 k · (k − k ) χa (k,ω ). (1.140) = − 3 a2 2 + 2ω ma k ω ω − ω Note that this result can be alternatively expressed as χ¯ a(3) (k,ω | − k, − ω |k,ω) i ea (k · k ) |k − k | = 3 { 2k 2 (ω − ω ) + [k · (k − k )] ω } ω ma k k  [k · (k − k )] × χa(2) (k,ω |k − k,ω − ω ).

(1.141)

1.8 Linear Waves and Weak Instabilities

39

Next, let us consider the case when ω − ω represents an arbitrary frequency, while the other two frequencies satisfy the fast-wave condition. In this case, we have χ¯ a(3) (k,ω | − k, − ω |k,ω)  2 (k · k )2 (k − k ) · ∂Fa /∂v 1 ea2 ωpa dv =− 2 m2a ω3 ω k 2 k 2 ω − ω − (k − k ) · v + i0 1 e2 |k − k |2 (k · k )2 χa (k − k,ω − ω ), = − a2 2 ma ω3 ω k 2 k 2

(1.142)

which can be alternatively expressed as χ¯ a(3) (k,ω | − k, − ω |k,ω) =

iea k · k |k − k | (2)   χa (k ,ω |k − k,ω − ω ). ma k k  ω2 (1.143)

1.8 Linear Waves and Weak Instabilities In this section we review the textbook theory of small amplitude electrostatic (linear) waves and weak instabilities operative in unmagnetized plasmas, which includes the electron beam-plasma (or bump-on-tail) instability. Linear dispersion relation is determined from the solvability condition of (1.93), 2   ωpa k · ∂Fa /∂v dv = 0. (1.144) Re (k,ω) = 1 + Re 2 k ω − k · v + i0 a Let us assume that isotropic thermal Maxwellian forms represent the bulk electron and ion distributions but a tenuous electron beam may also exist. For Langmuir waves satisfying the fast wave condition for both electrons and ions, |ωkL |/k vT e

1 and |ωkL |/k vT i 1, where vT a is the thermal speed defined via (1.116), upon making use of (1.115) Re (k, ± ωkL ) is approximately given by  2  ωpe 3k 2 Te , (1.145) Re (k, ± ωkL ) = 1 − L2 1 + ωk me ωkL2 where ω = ωkL denotes the Langmuir wave dispersion relation. By setting Re (k, ± ωkL ) equal to zero one readily obtains the following:   3 L ω−k = −ωkL, (1.146) ωkL = ωpe 1 + k 2 λ2De , 2 where Te is the electron bulk temperature and λ2De =

vT2 e Te = , 2 4πne2 2ωpe

(1.147)

40

Nonlinear Electrostatic Equations for Collisionless Plasmas

is the square of the electron Debye length. In (1.146) the symmetry property (1.9) L = −ωkL . is invoked in order to have the relation ω−k For ion-sound mode characterized by |ωkS |/k vT e 1 and |ωkS |/k vT i ≥ 1, upon combining (1.115) and (1.119) we have  2  ωpi 3k 2 Ti 1 S , (1.148) Re (k, ± ωk ) = 1 + 2 2 − S2 1 + k λDe ωk mi ωkS2 where Ti is the ion (proton) temperature and ωkS denotes the ion-sound wave dispersion relation. Setting Re (k, ± ωkS ) equal to zero leads to ωkS =

kcS (1 + 3Ti /Te )1/2 , (1 + k 2 λ2De )1/2

where

S ω−k = −ωkS ,

(1.149)

 cS =

Te mi

(1.150)

is the ion sound (or ion acoustic) speed. It is useful to evaluate the derivatives of real parts of the dielectric constants, ∂ Re (k, ± ωkL ) 2 = , L ∂(±ωk ) (±ωkL ) ∂ Re (k, ± ωkS ) 2 1 , =  (k, ± ωkS ) ≡ S L μ (±ωk ) k ∂(±ωk )  1/2   me 3Ti 1/2 3 3 1+ . μk = k λDe mi Te

 (k, ± ωkL ) ≡

(1.151)

Figure 1.1 displays the three basic plasma eigenmodes  (normal modes) of T 2 + c2 k 2 , which unmagnetized plasma. These are the transverse modes ωk = ωpe we did not discuss under the present electrostatic approximation (we will deal with the transverse mode in later chapters), and Langmuir and ion acoustic (or ion sound) modes, which we have already covered. The instability (or damping) of plasma eigenmodes can be discussed on the basis of the imaginary part of dielectric response function 2   πωpa ∂Fa dv k · Im (k,ω) = − δ(ω − k · v). (1.152) 2 k ∂v a Consider the linear term in the formal wave kinetic equation (1.102), ∂Ikσ α Im (k,σ ωkα ) I σ α ≡ 2γkα Ikσ α . = −2 ∂t ∂Re (k,σ ωkα )/∂σ ωkα k

(1.153)

1.8 Linear Waves and Weak Instabilities w

41

Transverse EM w kT = (w pe2 + c2k2)1/2

Langmuir w pe

w kL = (w pe2 + 3k2lD2)1/2 Ion acoustic w kS = kcs(1 + 3Ti /Te)1/2 /(1 + k2lD2)1/2 k

Figure 1.1 Three basic eigenmodes of unmagnetized plasma; transverse mode ωkT , which we could not discuss under the present electrostatic treatment, and Langmuir and ion acoustic (or ion sound) modes, ωkL and ωkS , respectively.

In this equation we have introduced the linear “growth rate” or equivalently “Landau damping rate,” – see Appendix A; γkα = −

Im (k,σ ωkα ) . ∂Re (k,σ ωkα )/∂σ ωkα

(1.154)

Depending on whether the sign of γkα is positive or negative, the quantity γkα denotes either the growth rate for instabilities or the damping rate associated with waves. The gentle electron beam-plasma or bump-on-tail instability will play a crucial role as a test bed for weak turbulence theory to be developed in the present monograph. The weak (or gentle) electron beam-plasma instability has been studied since the beginning of modern plasma physics, and there is a substantive body of literature on the topic, so that it is practically impossible to cite them all, but some selective references are those by Vedenov and Velikhov (1962); Drummond and Pines (1962); Frieman and Rutherford (1964); Bernstein and Engelmann (1966); Dawson and Shanny (1968); Morse and Nielson (1969); Vahala and Montgomery (1970); Roberson et al. (1971); Joyce et al. (1971) among earlier works, and Appert et al. (1976); Ivanov et al. (1976); Grognard (1982); Dum (1990); Muschietti and Dum (1991); Tsunoda et al. (1987); Nishikawa and Cairns (1991); Dum and Nishikawa (1994), to cite some representative papers up to the mid-1990s. Since the decade of 1990s, researches on beam-plasma interaction have moved on to more application-oriented topics, so that problems on pure or fundamental aspects of the beam-plasma instability, especially in relations to linear or quasilinear aspects, seem to have been exhausted. However, as we shall see later in this book, certain fundamental aspects associated with the electron beam-plasma interaction in the

42

Nonlinear Electrostatic Equations for Collisionless Plasmas

weak turbulence regime are still being investigated. These relate to the electron acceleration by Langmuir turbulence and the radiation generation. Let us focus on the Langmuir wave, α = L, in (1.154), and restrict ourselves to the forward propagation, σ = +1. Here, the directionality of forward versus backward is with respect to the beam propagation direction. Making use of (1.151) and (1.152), the growth rate (1.154) for forward-propagating Langmuir wave is given by 2  ∂Fe πωkL ωpe dv k · (1.155) δ(ωkL − k · v), γkL = 2 k2 ∂v where we have omitted the plus sign and have retained the electron terms only in Im (k,ω). For isotropic Maxwellian, Fe = (πvT e )−3 exp(−v 2 /vT2 e ), it is obvious that γkL < 0. That is, Langmuir waves for thermal plasma is subject to Landau damping. If, on the other hand, we consider that the electron distribution is composed of a core thermal population plus an energetic but tenuous beam, take for instance, the drifting Gaussian beam distribution,   2 2 2 2 nb e−v /vT e nb e−(v−Vb ) /vT b + , (1.156) Fe (v) = 1 − n0 π 3/2 vT3 e n0 π 3/2 vT3 b √ where vT b = 2Tb /me is the thermal spread (or beam temperature), and where we assume nb n0,

(1.157)

then by assuming that the background population largely determines the real frequency, we may determine the damping (or growth) rate:    ω2 v nb −v2 /vT2 e L 1/2 L pe 1− e γk = − π ωk 2 k n0 vT3 e  (v −Vb ) nb −(v −Vb )2 /v2 T b + e . (1.158) vT3 b n0 v =ωL /k k

In (1.158) we have decomposed the velocity vector into components perpendicular and parallel to the beam vector, and without loss of generality, we have taken the k vector to be directed along the beam. In the absence of electron beam (1.158) reduces to   ω2 (ωkL )2 (ωkL )2 L 1/2 pe γk = −π exp − 2 2 k v k 3 vT3 e   T e  1/2 4 π ωpe 3k 2 vT2 e vT2 e 3 1+ exp − − 2 2 . (1.159) =− 3 3 2 2ωpe 2 k ωpe k vT e

1.8 Linear Waves and Weak Instabilities

43

0

Γ

–0.1 –0.2 –0.3 –0.4

0

0.5

1

1.5

2

q Figure 1.2 Normalized Landau damping rate  versus normalized wave number q.

This is the Landau damping rate for thermal plasma. Plotted in Figure 1.2 is the normalized Landau damping rate  = γkL /ωpe versus normalized wave number q = kvT e /ωpe . Notice that the damping rate exponentially decreases in magnitude for k → 0. Suppose that the background electrons are cold, vT e vT b . In such a case, the damping rate associated with the background component can be ignored, and the beam electrons determine the net growth/damping rate only,   3 ωpe (kVb − ωpe )2 L 1/2 nb . (1.160) (kVb − ωpe ) exp − γk ≈ π n0 k 3 vT3 b k 2 vT2 b This shows that γkL > 0 over the unstable range of k corresponding to k>

ωpe . Vb

(1.161)

The instability of Langmuir wave driven by a gentle (or weak) electron beam is called the bump-on-tail instability, and (1.160) represents the approximate growth rate for the said instability. In the low-frequency regime similar processes involving ion-acoustic wave with its associated damping phenomena as well as instabilities may be operative. It is a straightforward exercise to obtain the Landau damping rate for ion acoustic wave,    π 1/2 kλD (1 + 3Ti /Te ) me me 1 + 3Ti /Te S exp − γk = − ωpi 3/2 2 mi 2mi 1 + k 2 λ2D (1 + k 2 λ2D )2     3/2 Te 1 + 3Ti /Te Te . (1.162) exp − + Ti 2Ti 1 + k 2 λ2D

44

Nonlinear Electrostatic Equations for Collisionless Plasmas

If we approximate that me /mi ≈ 0, and introduce the dimensionless variables, = then we have

23/2 γk , π 1/2 ωpi

κ = kλD,

τ=

Ti , Te

  1 1 + 3τ κ (1 + 3τ ) 1 . exp − =− (1 + κ 2 )2 τ 3/2 2τ 1 + κ 2

(1.163)

(1.164)

Damping generally increases for increasing κ, so let us focus on small κ behavior. If we consider /κ for κ 1, we have   1 + 3τ  1 + 3τ . (1.165) ≈ − 3/2 exp − κ τ 2τ Below in Figure 1.3 is the plot of −/κ versus τ = Ti /Te in horizontal logarithmic scale. As one can see, if Ti Te then the damping rate becomes exponentially small, and ion sound wave may propagate in a plasma undamped. For Ti Te , the ion-sound speed cS2 = Te /mi is much higher than ion thermal speed vT2 i = 2Ti /mi , but lower than electron thermal speed vT2 e = 2Te /me , so that   2Ti 2Te ωkS < . (1.166) < mi k me In this case the ion-sound mode does not suffer Landau damping by ions as the wave phase speed is sufficiently higher than ion thermal speed. On the other hand,

Figure 1.3 Normalized ion acoustic wave damping rate −/κ versus Ti /Te .

1.8 Linear Waves and Weak Instabilities

45

the wave also does not suffer from electron Landau damping since the electron distribution is practically constant over the velocity range corresponding to the ion acoustic speed. This shows that the excitation and persistence of ion sound wave requires hot electrons. Note that the ion sound damping rate is maximum for T i ∼ Te . Ion acoustic mode can become unstable when there is a net mild drift between the ions and electrons. Let us consider the stationary ions and drifting Gaussian electrons (without loss of generality, we assume the electron drift direction to be along z axis), where the electron drift speed is significantly lower than electron thermal speed,   1 v2 Fi (v) = 3/2 3 exp − 2 , vT i π vT i   1 (v − Ve zˆ )2 . (1.167) Fe (v) = 3/2 3 exp − vT2 e π vT e Then, assuming k = kˆz, and for Te /Ti 1 and me /mi 1, we obtain the growth rate for ion acoustic instability, ⎛ ⎞ 1/2 1/2  k 2 cS2 me π ⎝ Ve −  1 ⎠ . (1.168) γkS = 3/2 2 mi cS (1 + k 2 λ2De )2 1 + k 2 λ2 De

If electron drift speed Ve is higher than the ion acoustic speed cS , the ion sound mode becomes unstable.

2 Electrostatic Vlasov Weak Turbulence Theory Wave Kinetic Equation

In this chapter and in Chapter 3, we will take the formal wave and particle kinetic equations, (1.102) and (1.107), respectively, and make specific applications to the problem of electrostatic Langmuir and ion-sound modes interacting with the particles via linear wave-particle resonance, interacting among themselves via threewave resonance, as well as two Langmuir or ion-sound waves resonantly interacting with the particles via nonlinear wave-particle resonance. In (1.102) let (α,β,γ ) denote either L (Langmuir) or S (ion-sound). The delta γ β function condition δ(σ ωkα − σ  ωk − σ  ωk−k ) describes resonant three-wave decay/coalescence processes among L and S modes. A careful consideration leads γ β to the conclusion that the resonance condition, σ ωkα − σ  ωk − σ  ωk−k = 0, cannot be satisfied if all three waves are of the same kind. Langmuir waves possess frequency close to ωpe . If all three waves are Langmuir waves, then one of them must possess wave frequency either close to 2ωpe or to zero, neither of which is possible. For ion-sound waves, the situation is a bit subtle, and we will in fact discuss the three-wave coupling of ion acoustic waves, but a careful analysis later reveals that a similar conclusion can be drawn for ion-sound waves. That is, threewave decay interaction involving only ion-sound waves can be ignored. Likewise, it is not possible to satisfy the three-wave resonance condition by considering two ion-sound waves and a Langmuir wave, since ion-sound waves are low-frequency waves. Thus, the only possible three-wave process is when two Langmuir waves and an ion-sound wave are involved. It is instructive to write out (1.102) explicitly by taking all permissible interactions into account: ∂Ikσ L ∂I σ S +  (k,σ ωkS ) k ∂t ∂t L σL = −2 Im (k,σ ωk ) Ik − 2 Im (k,σ ωkS ) Ikσ S ! "    σ,σ   σ L  σ S dk Aσ,σ − 4 Im Ikσ L L,L (k,k ) Ik + AL,S (k,k ) Ik

 (k,σ ωkL )

σ  =±1

46

Electrostatic Vlasov Weak Turbulence Theory

!

σ,σ 

σ,σ 





"

47

#

+ AS,L (k,k ) Ikσ L + AS,S (k,k ) Ikσ S Ikσ S   dk |χ (2) (k,σ  ωkL |k − k,σ ωkL − σ  ωkL )|2 + 4π  ×

σ ,σ  =±1 





σ S Ikσ L Ik−k 



σ S σL Ik−k  Ik

Ikσ L Ikσ L − − S  (k,σ ωkL )  (k,σ  ωkL )  (k − k,σ  ωk−k )

S × δ(σ ωkL − σ  ωkL − σ  ωk−k )

+ |χ

(2)



(k ,σ



ωkS |k





k ,σ ωkL



−σ



ωkS )|2







σ L Ikσ S Ik−k 

 (k,σ ωkL )

  Ikσ S Ikσ L L −    S −  δ(σ ωkL − σ  ωkS − σ  ωk−k ) L ) (k ,σ ωk ) (k − k,σ  ωk−k    σ  L Ikσ L Ik−k  (2)   L  S  L 2 + |χ (k ,σ ωk |k − k ,σ ωk − σ ωk )| (2.1) S  (k,σ ωk )    σ  L σ S Ik−k Ikσ L Ikσ S  Ik L δ(σ ωkS − σ  ωkL − σ  ωk−k −    L −  ) , L (k ,σ ωk ) (k − k,σ  ωk−k ) 

σ L σL Ik−k  Ik

where σ = ±1 signify forward/backward propagation, and   Aσ,σ α,β (k,k )

2 {χ (2) (k,σ  ωk |k − k,σ ωkα − σ  ωk )}2 β

=P

β

β

(k − k,σ ωkα − σ  ωk )

−χ¯ (3) (k,σ  ωk | − k, − σ  ωk |k,σ ωkα ). β

β

(2.2)

We may balance the terms relevant to Langmuir waves and sound waves separately in (2.1). Making use of the fact that k and k − k are interchangeable, we obtain !    ∂Ikσ L 4 2 Im (k,σ ωkL ) σ L   σ L dk − I Im Aσ,σ =−  k L,L (k,k ) Ik L L  ∂t (k,σ ωk ) (k,σ ωk ) σ  =±1 "   8π σ,σ   σ S σL dk + AL,S (k,k ) Ik Ik +  (k,σ ωkL ) σ ,σ  =±1   σ  S σ  S σ L Ikσ L Ik−k Ik−k   I (2)   L  L  L 2 × |χ (k ,σ ωk |k − k ,σ ωk − σ ωk )| −   k L L  (k,σ ωk ) (k ,σ ωk )   Ikσ L Ikσ L S δ(σ ωkL − σ  ωkL − σ  ωk−k −  (2.3)  ), S (k − k,σ  ωk−k )

48

Electrostatic Vlasov Weak Turbulence Theory

∂Ikσ S 2 Im (k,σ ωkS ) σ S Ik =−  ∂t (k,σ ωkS ) ! "   4 σ,σ  σ,σ    σ L  σ S dk −  (k,k ) I + A (k,k ) I Im A Ikσ S   k k S,L S,S (k,σ ωkS ) σ  =±1   4π dk |χ (2) (k,σ  ωkL |k − k,σ ωkS − σ  ωkL )|2 +  (k,σ ωkS ) σ ,σ  =±1     σ  L σ  L σ S Ikσ L Ik−k Ik−k Ikσ L Ikσ S   Ik ×  − − L (k,σ ωkS )  (k,σ  ωkL )  (k − k,σ  ωk−k ) L × δ(σ ωkS − σ  ωkL − σ  ωk−k  ).

(2.4)

We make use of (1.146), (1.149), and (1.151), namely, 1 (σ ωkL ) ≈ , 2  (k,σ ωkL )

(σ ωkL ) 1 ≈ μ , k 2  (k,σ ωkS )

ωkL = ωpe (1 + 3k 2 λ2De /2),

L ω−k = ωkL,

ωkS = kcS (1 + 3Ti /Te )1/2 /(1 + k 2 λ2De )1/2, μk = k 3 λ3De (me /mi )1/2 (1 + 3Ti /Te )1/2 .

S ω−k = ωkS ,

(2.5)

In what follows, we consider various interaction terms on the right-hand sides of (2.3) and (2.4). 2.1 Induced Emission We begin with the first terms representing linear wave-particle interactions, which depict induced emissions of plasma eigenmodes. The notion of induced emission stems from the following concept: If the particles travel in a wave field and their velocities are in resonance with the wave phase velocities, then the particles may gain or lose energy as they interact with the waves. Waves in turn may also lose or gain energy, in reversal to the particle behavior. This represents a linear waveparticle resonant interaction, but in this picture, a preexisting wave field is assumed. Consequently, the change in the wave intensity by induced emission process is proportional to the wave intensity, with the multiplicative coefficient that dictates the wave-particle resonance affecting the dynamical change also appearing in the formula. The induced emissions are described by the first terms on the right-hand sides of (2.3) and (2.4). The changes in wave intensities due to the influence of linear wave-particle resonance are given in a straightforward manner. For Langmuir waves, we have

2.1 Induced Emission

 ∂Ikσ L  2 Im (k,σ ωkL ) σ L =−  Ik  ∂t ind.em. (k,σ ωkL ) 2  ωpe ∂Fe σ L L dv δ(σ ωkL − k · v) k · = π (σ ωk ) 2 I , k ∂v k while for ion-sound waves, the same is given by  ∂Ikσ S  2 Im (k,σ ωkS ) σ S = − Ik ∂t ind.em.  (k,σ ωkS ) 2  ωpe dv δ(σ ωkS − k · v) = π μk (σ ωkL ) 2 k   ∂ me Fe + ×k · Fi Ikσ S . ∂v mi

49

(2.6)

(2.7)

For L mode only electrons participate in the linear wave-particle resonance, while for S mode both charged particle species, namely, electrons and ions (protons), may resonantly interact with the waves. If we rewrite (2.6) and (2.7) as  ∂Ikσ α  = 2γkσ α Ikσ α, (α = L,S), ∂t ind.em. 2  ωpe π ∂Fe dv δ(σ ωkL − k · v) k · γkσ L = (σ ωkL ) 2 , 2 k ∂v 2  ωpe σS L dv δ(σ ωkS − k · v) γk = π μk (σ ωk ) 2 k   ∂ me Fe + ×k · (2.8) Fi , ∂v mi then one may realize that the basic form of evolution equation is none other than that given by (1.153) and that γkσ L and γkσ S are none other than the linear growth/damping rate introduced in (1.154). The delta-function resonance conditions ωkα − k · v = 0 and ωkα + k · v = 0 signify that (2.6) and (2.7) depict linear wave-particle resonant interactions, that is, these depict the particles moving in the wave field with their velocities in resonance with the wave phase velocities. In this process, whether waves gain or lose energy depends on whether more particles move faster or slower than the wave phase velocities. That is, the growth or damping rate depends on the slope of the particle distribution function in velocity space, hence the derivative ∂Fa /∂v. This form of wave kinetic equation that only includes linear wave-particle resonance is called the “quasilinear” wave kinetic equation. Consequently, the induced emission and quasilinear wave processes are equivalent.

50

Electrostatic Vlasov Weak Turbulence Theory

2.2 Spontaneous and Induced Decay/Coalescence Spontaneous and induced decay/coalescence processes are those terms dictated by the three-wave resonant interaction delta functions.    8π ∂Ikσ L  dk |χ (2) (k,σ  ωkL |k − k,σ ωkL − σ  ωkL )|2 =   L ∂t decay (k,σ ωk )   σ ,σ =±1     σ  S σ L σ L σ  S Ik Ik−k Ik−k Ikσ L Ikσ L  Ik ×  − − S (k,σ ωkL )  (k,σ  ωkL )  (k − k,σ  ωk−k ) S × δ(σ ωkL − σ  ωkL − σ  ωk−k  ),

(2.9)

   4π ∂Ikσ S  dk |χ (2) (k,σ  ωkL |k − k,σ ωkS − σ  ωkL )|2 = ∂t decay  (k,σ ωkS )   σ ,σ =±1     σ L σ  L σ  L σ S Ik Ik−k Ik−k Ikσ L Ikσ S  Ik ×  − − L (k,σ ωkS )  (k,σ  ωkL )  (k − k,σ  ωk−k ) L × δ(σ ωkS − σ  ωkL − σ  ωk−k  ),

(2.10)

To compute the various three-wave decay coefficients |χ (2) |2 , we note that ωkL k vT e,

ωkL k  vT e,

L  ωk−k  |k − k | vT e,

ωkL k vT i ,

ωkL k  vT i ,

L  ωk−k  |k − k | vT i ,

ωkS k vT e,

ωkS k  vT e,

S  ωk−k  |k − k | vT e,

ωkS > k vT i ,

ωkS > k  vT i ,

S  ωk−k  > |k − k | vT i ,

(2.11)

which we have already discussed in Section 1.7.3. Note that within the context of three-wave resonance, S  |ωkL − ωkL | = |ωk−k  | |k − k | vT e .

(2.12)

S  |ωkL − ωkL | = |ωk−k  | > |k − k | vT i .

(2.13)

For ions, we have

On the basis of these considerations, the following approximate expressions for electron and ion partial second-order nonlinear susceptibilities can be obtained: S χe(2) (k,σ  ωkL |k − k,σ ωkL − σ  ωkL ) = χe(2) (k,σ  ωkL |k − k,σ  ωk−k )   |k − k | k · k i e ≈− 2 me (σ ωkL )(σ  ωkL ) k k 

× χe (k − k,σ ωkL − σ  ωkL ) ≈−

2 ωpe ie k · k , 2Te (σ ωkL )(σ  ωkL ) k k  |k − k |

2.2 Spontaneous and Induced Decay/Coalescence

51

S χi(2) (k,σ  ωkL |k − k,σ ωkL − σ  ωkL ) = χi(2) (k,σ  ωkL |k − k,σ  ωk−k ) 2 ωpi 1 ie ≈− S  2mi (σ ωkL )(σ  ωkL )(σ  ωk−k k k |k − k | )  |k − k |2 k 2  k · (k − k ) + k · k × S σ  ωkL σ  ωk−k   k2  + k · (k − k ) L σ ωk 2 ωpi (k·k ) |k − k | ie . ≈− S 2 2mi (σ ωkL )(σ  ωkL )(ωk−k k k ) (2.14)

From this, it is seen that |χe(2) (k,σ  ωkL |k − k,σ ωkL − σ  ωkL )|2

|χi(2) (k,σ  ωkL |k − k,σ ωkL − σ  ωkL )|2,

(2.15)

which shows that the ion response can be ignored. L For the case where the decay is dictated by σ ωkS − σ  ωkL − σ  ωk−k  = 0, we obtain e2 [k · (k − k )]2 2 4 k λDe |χe (k,σ ωkS )|2 4Te2 k 2 |k − k |2 e2 [k · (k − k )]2 ≈ . (2.16) 4Te2 k 2 k 2 |k − k |2

|χ (2) (k,σ  ωkL |k − k,σ ωkS − σ  ωkL )|2 =

Again, in this result, the ion response has been ignored. Thus, for both Langmuir and ion-sound waves, the three-wave decay processes are dominated by electron nonlinear response. The changes in the wave energy densities owing to the three-wave decay processes are therefore given by    ∂Ikσ L  L L L  L  S =2 σ ωk dk Vk,k  δ(σ ωk − σ ωk − σ ωk−k ) ∂t decay σ ,σ  =±1

  σ  S  L σ  S σ L L σ L σ L × σ ωkL Ikσ L Ik−k − σ  μk−k ωk−k Ik ,  Ik  − σ ωk Ik−k Ik    ∂Ikσ S  L S S  L  L dk Vk,k = σ μk ωk  δ(σ ωk − σ ωk − σ ωk−k ) ∂t decay σ ,σ  =±1

  σ  L  L σ  L σ S L σ L σ S × σ μk ωkL Ikσ L Ik−k − σ  ωk−k Ik ,  − σ ωk Ik−k Ik  Ik  (2.17)

52

Electrostatic Vlasov Weak Turbulence Theory

where π 4 π ≈ 4 π = 4 π ≈ 4

L Vk,k  =

S Vk,k 

e2 (k · k )2 S 2 |k − k |2 λ4De |χe (k − k,σ  ωk−k  )| Te2 k 2 k 2 e2 (k · k )2 , Te2 k 2 k 2 |k − k |2 e2 [k · (k − k )]2 2 4 k λDe |χe (k,σ ωkS )|2 Te2 k 2 k 2 e2 [k · (k − k )]2 . Te2 k 2 k 2 |k − k |2

(2.18)

These two equations describe a Langmuir wave decaying into another Langmuir wave and an ion-sound wave, or a Langmuir wave and an ion-sound wave coalescing (merging) in order to generate another Langmuir wave. The wave frequency and wave number conservation conditions (energy and momentum of the wave mode) are dictated by the delta function. The first terms within the parenthesis on the right-hand sides of (2.17), where the wave intensities have integral variables inexorably coupled via Ik and Ik−k , depict the “spontaneous” decay processes, while the second and third terms with the intensity Ik , which is independent of the integral variable k , represent the “induced” decay processes. The spontaneous decay process is conceptually understood as a given wave mode with wave vector (or wave momentum) k spontaneously decaying into two other waves, one with k and the other with k − k , while conserving total wave energy and momentum. The induced decay takes place when a wave with k travels in the background of two other waves with k and k − k interacting with each other. The original wave with k may resonantly interact with the other two waves, during the course of which the original wave with k may gain or lose energy. It is interesting to observe that the following conservation relation exists within the context of three-wave processes:   L    ∂ Ik 2IkS  dk = 0. (2.19) + ∂t decay ωkL μk ωkL σ =±1 This can be directly proven by adding the two quantities ∂(IkL /ωkL )/∂t and ∂(2IkS /μk ωkL )/∂t and integrating over k. The quantities NkL =

IkL IkS S and N = k h¯ ωkL h¯ μk ωkL

(2.20)

can be considered as the “plasmon number densities” for eigenmodes, where h¯ = 1.0545718 × 10−34 m2 kg/s is the Planck constant. This is a quantum mechanical concept. In the literature, sometimes the quantities NkL and NkS are

2.2 Spontaneous and Induced Decay/Coalescence

53

used as the fundamental quantities instead of IkL and IkS (Tsytovich, 1970, 1977a,b; Melrose, 1980a, 1986). Equation (2.19) shows that the decay process does not change the total plasmon number densities involved in the three-wave process. The decay process involves two Langmuir waves and an ion-sound wave for a given k value. As a consequence, the total number of conserved plasmon number densities equals the total number density of Langmuir modes and twice the total number density of ion-sound modes. Let us consider the contribution to ion-sound wave kinetic equation that comes from three-wave decay processes involving only ion-sound waves,    4π ∂Ikσ S SS dk |χ (2) (k,σ  ωkS |k − k,σ ωkS − σ  ωkS )|2 = ∂t decay  (k,σ ωkS )   σ ,σ =±1     σ S σ  S σ  S σ S Ik Ik−k Ik−k Ikσ S Ikσ S  Ik ×  − − S (k,σ ωkS )  (k,σ  ωkS )  (k − k,σ  ωk−k ) S × δ(σ ωkS − σ  ωkS − σ  ωk−k  ).

(2.21)

Making use of the wave properties ωkS kvT e,

ωkS k  vT e,

ωkS > kvT i ,

ωkS > k  vT i ,

|ωkS − ωkS | |k − k |vT e, |ωkS − ωkS | > |k − k |vT i ,

(2.22)

we may proceed to evaluate the second-order susceptibility 2 ωpe ieme − −σ ≈ , 2Te2 kk  |k − k |  2   ωpe ieme (2)   S  S  S   k · (k − k ) σ σ χi (k ,σ ωk |k − k ,σ ωk − σ ωk ) ≈ − 2Te2 kk  |k − k | k  |k − k |  k · (k − k )  k·k . + σ σ + σ σ  k |k − k | kk  (2.23)

χe(2) (k,σ  ωkS |k



k ,σ ωkS



ωkS )

Making use of these we obtain  4   μ μ  μk−k ωpe ∂ SS Ikσ S π e2 m2e L  k k dk = σ ω k ∂t decay μk 4 Te4 k 2 k  2 |k − k |2 σ ,σ  =±1    2 k · (k − k )  k · (k − k )  k·k × 1 − σ  σ   − σ σ − σ σ k |k − k | k |k − k | kk    σ  S σ  S I σ  S σ S σ S σS I I I I I     k−k k−k k k L k × σ ωkL k − σ  ωkL − σ  ωk−k  μk μk−k μk−k μk μ k μ k S × δ(σ ωkS − σ  ωkS − σ  ωk−k  ).

(2.24)

54

Electrostatic Vlasov Weak Turbulence Theory

The decay process involving only S waves is described by (2.24). In reality, however, contribution to the net decay process from this mechanism is absent. To see this, consider the three-wave resonance condition, which can be expressed explicitly as    3Ti 1/2 k S S S ωk − ωk − ωk−k = cS 1 + 2 Te (1 + k λ2De )1/2  k k = 0. (2.25) − − (1 + k  2 λ2De )1/2 (1 + k  2 λ2De )1/2 This resonance condition is satisfied for k = k  , but for such a situation the coupling coefficient in (2.24) identically vanishes by virtue of the factor μk−k . Consequently, decay processes involving only S modes can be ignored. 2.3 Induced Scattering We next discuss nonlinear scattering processes, which are described by  "   ! ∂Ikσ L   σ L  σ S = − (k,k ) I + a (k,k ) I a Ikσ L,   L,L L,S k k ∂t ind.sc.   k σ =±1  " σS   !  ∂Ik   σ L  σ S = − (k,k ) I + a (k,k ) I a Ikσ S ,   S,L S,S k k ∂t ind.sc.   k σ =±1  β β 2 {χ (2) (k,σ  ωk |k − k,σ ωkα − σ  ωk )}2 4  Im P aα,β (k,k ) =  β (k,σ ωkα ) (k − k,σ ωkα − σ  ωk )  − χ¯ (3) (k,σ  ωk | − k, − σ  ωk |k,σ ωkα ) . β

β

(2.26)

2.3.1 Induced Scattering Involving Two Langmuir Waves The first process of interest involves a Langmuir wave scattering off particles into another Langmuir wave. During the course of such a scattering, the original Langmuir wave may gain or lose energy. This is similar to the induced emission where the particles moving in the background wave field resonantly interacting with the waves give off energy or gain energy. Only this time, the particles resonantly interact with two waves. The scattering coefficient is dictated by aL,L (k,k ). The situation of interest is when the frequency difference |ωkL − ωkL | satisfies the slow wave S condition. Note that |ωkL − ωkL | = |ωk−k  |, since the nonlinear scattering process does not involve three-wave decay/coalescence resonance condition. The scattering process is similar to the decay interaction also, except that instead of three waves

2.3 Induced Scattering

55

resonantly interacting, it is the two waves resonantly interacting with a virtual third wave, which is the particles. For electrons, we have ωkL k vT e,

ωkL k  vT e,

ωkL − ωkL |k − k | vT e .

(2.27)

ωkL k  vT i ,

|ωkL − ωkL | |k − k | vT i .

(2.28)

For ions, we expect that ωkL k vT i ,

Nonlinear wave-particle interaction is important for the range of resonant velocity and wave numbers that is inaccessible from the standpoint of linear wave-particle interaction. For instance, linear wave-particle resonance between the electrons and Langmuir waves dictates that the resonant velocity is vres ∼ ±ωpe /k. Nonlinear wave-particle interaction that involves two Langmuir waves leads to vres ∼ ±(3ωpe /2)(k 2 − k 2 )λ2De /|k − k |, which is much lower than the resonant velocity for linear interaction. Thus, in evaluating aLL (k,k ), we are mainly interested in the case where the difference between σ ωkL and σ  ωkL is small. If σ and σ  have opposite signs, or even for the same signs of σ and σ  , if k and k have opposite signs, then the two quantities can be added up to produce σ ωkL − σ  ωkL ∼ 2ωpe . Such cases are not of interest, since it implies that all three arguments, σ ωkL , σ  ωkL , and σ ωkL − σ  ωkL , satisfy the fast-wave condition. When such a situation occurs, then the second- and third-order susceptibilities become exceedingly small, and aL,L (k,k ) will become insignificant. In what follows, although we will consider all signs of σ and σ  in the formal sense, and do not distinguish a priori the possibility of two Langmuir waves adding up to twice the plasma frequency, or different signs of k and k , which are not of interest, in the end one must discard the resonant velocities such as vres ≈ 2ωpe /|k ± k  |. What is presented subsequently should be interpreted with this caveat. Imaginary parts of the third-order nonlinear susceptibilities relevant for aLL (k,k ) are given by Im χ¯ e(3) (k,σ  ωkL | − k, − σ  ωkL |k,σ ωkL ) |k − k |2 1 e2 (k · k )2 Im χe (k − k,σ ωkL − σ  ωkL ), ≈− 2 m2e k 2 k 2 (σ ωkL )3 (σ  ωkL ) Im χ¯ i(3) (k,σ  ωkL | − k, − σ  ωkL |k,σ ωkL ) (2.29) 2  2  2 |k − k | 1 e (k · k ) ≈− Im χi (k − k,σ ωkL − σ  ωkL ). 2 2 2 2 mi k k (σ ωkL )3 (σ  ωkL ) Again, the electron nonlinear response is the dominant term. To reiterate, this 2 approximation assumes that σ ωkL − σ  ωkL |k − k |2 vTj for both j = e,i.

56

Electrostatic Vlasov Weak Turbulence Theory

Next, we consider the second-order nonlinear susceptibility under the same 2 . Since the electron response is implicit assumption, σ ωkL − σ  ωkL |k − k |2 vTj dominant we have { χ (2) (k,σ  ωkL |k − k,σ ωkL − σ  ωkL ) }2 1 e2 (k · k )2 |k − k |2 ≈− { χe (k − k,σ ωkL − σ  ωkL ) }2 . 4 m2e k 2 k 2 ωkL2 ωkL2

(2.30)

The linear dielectric constant (k−k,σ ωkL −σ  ωkL ) that appears within the denominator of the principal value P [1/ (k − k,σ ωkL − σ  ωkL )] can be approximated by (k − k,σ ωkL − σ  ωkL ) ≈ χe (k − k,σ ωkL − σ  ωkL ) + i Im χi (k − k,σ ωkL − σ  ωkL ),

(2.31)

where χe (k − k,σ ωkL − σ  ωkL ) =

2 2ωpe

|k − k |2 vT2 e + i Im χe (k − k,σ ωkL − σ  ωkL ).

(2.32)

As a result, we have all the ingredients to evaluate aL,L (k,k ). Putting everything together we arrive at    ∂Ikσ L LL  = − aL,L (k,k ) Ikσ L Ikσ L  ∂t ind.sc k σ  =±1   (k · k )2 π e2   dk dv 2 2 =− 2 2 (2.33) ωpe me  k k σ =±1

∂ × δ[σ ωkL − σ  ωkL − (k − k ) · v] (k − k ) · ∂v   m  e (σ ωkL ) Fi Ikσ L Ikσ L . × (σ ωkL − σ  ωkL ) Fe − mi This expression was derived under the specific assumption that the difference in frequencies, σ ωkL − σ  ωkL , is small when compared with |k − k | vT2 e . Consequently, the leading term is the induced scattering of Langmuir waves mediated by ions, or scattering off thermal ions,    (k · k )2 ∂Ikσ L LL π e2   dk dv = 2 m2 ∂t ind.sc ωpe k 2 k 2 e  σ =±1 L δ[σ ωk − σ  ωkL

− (k − k ) · v] ∂Fi σ  L σ L me (σ ωkL ) (k − k ) · I I . × mi ∂v k k

×

(2.34)

2.3 Induced Scattering

57

The induced scattering involving thermal ions can be considered as the nonlinear Landau damping, but unlike the linear Landau damping, it is the ions that mediate the scattering of two Langmuir waves rather than the electrons. 2.3.2 Induced Scattering Involving Langmuir and Ion-Sound Waves Induced scattering processes that involve Langmuir and ion-sound waves are dictated by coefficients aL,S (k,k ) and aS,L (k,k ). These coefficients include inverse linear dielectric constants with arguments involving shifted frequencies and wave vectors, 1 , (k − k,σ ωkL − σ  ωkS )

and

1 . (k − k,σ ωkS − σ  ωkL )

(2.35)

As we have seen in the discussion of the previous subsection that led to (2.34), inverse dielectric constants are associated with resonance delta function conditions δ[σ ωkL −σ  ωkS −(k−k )·v] and δ[σ ωkS −σ  ωkL −(k−k )·v], respectively. However, as we have noted, nonlinear wave-particle resonant interaction and induced scattering processes are supposed to become important in the range of velocity space for which the linear wave-particle resonance δ(σ ωkL − k · v) or δ(σ ωkS − k · v) becomes ineffective. In the case of induced scattering involving Langmuir and ion-sound waves because of the fact that |ωkL | |ωkS |, nonlinear wave-particle resonance conditions practically overlap with that of linear wave-particle resonance, σ ωkL − σ  ωkS − (k − k ) · v ∼ σ ωkL − (k − k ) · v, σ ωkS − σ  ωkL − (k − k ) · v ∼ σ  ωkL − (k − k ) · v.

(2.36)

For this reason, the induced scattering involving a high-frequency Langmuir wave and a low-frequency ion-sound wave can at best be treated as a small correction to the essentially linear wave-particle interaction process, and thus can be ignored. Besides this physical ground for ignoring the coupled L and S induced scattering, such a case has some inherent mathematical problems too. The inverse dielectric constants (2.35) can become extremely large because of the fact that the arguments with shifted frequencies and wave vectors practically satisfy the linear wave-particle resonance conditions, as discussed in (2.36). This indicates the failure of perturbative solution to nonlinear plasma equation. Recall that in the formal derivation of the nonlinear spectral balance equation one had to deal with the three-body cumulant, δφk1,ω1 δφk2,ω2 δφ−k,−ω . To obtain this quantity, we solved the nonlinear Poisson equation (1.72) iteratively. In doing so the nonlinear correction term to the perturbed electrostatic potential was obtained via (1.76), which contained the inverse dielectric constant on the right-hand side. However, a fundamental assumption there is that the quantity (k,ω) is not small, so that

58

Electrostatic Vlasov Weak Turbulence Theory

its inverse is well defined and mathematically meaningful. For the case of induced scattering involving two Langmuir waves, P

1 (k −

k,σ ωkL

− σ  ωkL )

1,

(2.37)

so that the fundamental assumption in the iterative solution is not violated. However, in the present case of combined L and S modes, the denominator, (k − k,σ ωkL − σ  ωkS ), can become very small. As a consequence, the iterative solution fails. The principal part must avoid the zeros, but since in this case, the denominator contains possible zeros, the leading contribution must be coming from the residue (i.e., the imaginary part, which is already accounted for in the decay process), and since the precise method of computing the principal part is not available, we must ignore the principal part altogether. This is another reason for ignoring induced scatterings involving L and S modes. 2.3.3 Induced Scattering Involving Two Ion-Sound Waves For the case of ion-sound waves scattering off particles, the scattering coefficient is governed by aS,S (k,k ). For this case, we have ωkS k vT e,

ωkS k  vT e,

|ωkS − ωkS | |k − k | vT e,

(2.38)

ωkS > k  vT i ,

|ωkS − ωkS | > |k − k | vT i ,

(2.39)

for electrons, and ωkS > k vT i ,

for ions. Note that the ion-sound wave can exist in a plasma where Te Ti . As a result, ωkS − ωkS ∼ (k − k  ) cS , and cS /vT i ∼ (Te /Ti )1/2 1. This leads to 2(k · k − σ σ  k k  ) |k − k |2 (σ k − σ  k  )2 λ2De      me (σ k− σ  k  )2 π me 1/2 σ k − σ  k  exp − +i 2 mi 2mi |k − k |2 |k − k |3 λ2De    1/2  3/2 Te Te (σ k − σ  k  )2 mi . exp − + me Ti 2Ti |k − k |2

(k − k,σ ωkS − σ  ωkS ) ≈ 1 +

(2.40) As for the second-order nonlinear susceptibility, unlike the situation involving high-frequency Langmuir waves, both electron and ion nonlinear responses are important,

2.3 Induced Scattering

59

2 ωpe 1 ie ,   Te k k |k − k | vT2 e 2 ωpi i e χi(2) (k,σ  ωkS |k − k,σ ωkS − σ  ωkS ) ≈ − 2 mi σ ωkS σ  ωkS (σ ωkS − σ  ωkS )  2 k 1 × k · (k − k ) k k |k − k | σ ωkS  |k − k |2 k 2   k·k , +  S k · (k − k ) + σ ωk σ ωkS − σ  ωkS (2.41)

χe(2) (k,σ  ωkS |k − k,σ ωkS − σ  ωkS ) ≈

from which, we obtain { χ (2) (k,σ  ωkS |k − k,σ ωkS − σ  ωkS ) }2  2 4 (k · k )2 |k − k |2 e2 m2e ωpe 1+ = − 2 2 4 4 . Te 4Te k k |k − k |2 (σ k − σ  k  )2

(2.42)

The third-order nonlinear response in the present case is dominated by the ions, Im χ¯ (3) (k,σ  ωkS | − k, − σ  ωkS |k,σ ωkS ) ≈ Im χ¯ i(3) (k,σ  ωkS | − k, − σ  ωkS |k,σ ωkS ) σ σ  e2 |k − k |2 (k · k )2 ≈− Im χi (k − k,σ ωkS − σ  ωkS ). 2 Te2 k 5 k 3

(2.43)

Putting all the results together, we now have    ∂Ikσ S   = aS,S (k,k ) Ikσ S Ikσ S ,  ∂t ind.sc.   σ =±1

 e2 (k · k )2 π dv Wk,k 2 m m k 4 k 4 λ4 ωpe e i De ∂Fi × (k − k ) · δ[σ ωkS − σ  ωkS − (k − k ) · v], ∂v k



aS,S (k,k ) = μk (σ ωkL )

where

 = 1+

2

 1  k + σ σ , k |k − k |4 λ4De | (k − k,σ ωkS − σ  ωkS )|2 2  2(k · k − σ σ  k k  )  S  S 2 | (k − k ,σ ωk − σ ωk )| ≈ 1 + |k − k |2 (k − σ σ  k  )2 λ2De    me (k − σ σ  k  )2 π me (k − σ σ  k  )2 exp − + 2 mi |k − k |6 λ4De 2mi |k − k |2 2   1/2  3/2 Te Te (k − σ σ  k  )2 mi exp − . + me Ti 2Ti |k − k |2

Wk,k

|k − k |2 (σ k − σ  k  )2

(2.44)

(2.45)

3 Electrostatic Vlasov Weak Turbulence Theory Particle Kinetic Equation

In this chapter we pick up where we left off in (1.107) and complete the discussion of particle kinetic equation. The simplest form of particle kinetic equation is the quasilinear velocity space diffusion equation, which can be discussed by ignoring nonlinear effects. 3.1 Quasilinear Particle Kinetic Equation If we ignore nonlinear wave-wave and wave-particle interaction terms in (1.107), we obtain   ∂Fa ∂ a ∂Fa Dij , = ∂t ∂vi ∂vj  ki kj ea2   a dk 2 Ikσ α δ(σ ωkα − k · v), Dij = π 2 (3.1) ma α σ =±1 k $ +α δ(ω − ωkα ) + where we have made use of the relationship δE 2 kω = α [ Ik −α α Ik δ(ω + ωk ) ] and have kept only the imaginary contribution in the resonant denominator,

ωkα

1 1 =P α − i π δ(ωkα − k · v). − k · v + i0 ωk − k · v

(3.2)

Energetic electrons near the tail of velocity distribution interact with Langmuir waves (ωkL ≈ k · v), while the low-energy bulk electrons can, in principle, absorb the ion-sound waves (ωkS ≈ k · v). However, the range of resonant velocities for ion-sound waves is extremely narrow. As such, we can ignore the interaction of

60

3.2 Null Effects of Decay Processes on the Particles

61

electrons with ion-sound waves. Consequently, the electron quasilinear diffusion equation can be discussed by considering only L mode:   ki kj +L ∂Fe ∂Fe πe2 ∂ −L L L dk 2 [ Ik δ(ωk − k · v) + Ik δ(ωk + k · v) ] . = 2 ∂t me ∂vi k ∂vj (3.3) In contrast, the Langmuir wave frequency is too high for ions to respond directly. Thus, the ion kinetic equation is affected only by the presence of ion-sound waves:   ∂Fi ki kj +S ∂Fi πe2 ∂ −S S S dk 2 [ Ik δ(ωk − k · v) + Ik δ(ωk + k · v) ] . = 2 ∂t k ∂vj mi ∂vi (3.4) 3.2 Null Effects of Decay Processes on the Particles We next consider the corrections to particle kinetic equations that arise from various nonlinear processes. Let us first consider the influence of three-wave decay/coalescence processes. This effect comes from the second term on the righthand side of (1.107). In particular, the decay process only involves the residue contributions from inverse linear dielectric functions. Principal part contribution primarily gives rise to nonlinear wave-particle interactions. Therefore, in the present section, we only consider the residue terms within the second term on the right-hand side of (1.107):      k k  (k − k )k 1 ea3 ∂ ∂Fa    i j dk dω dk dω = Re  3   ∂t decay ma ∂vi k k |k − k | ω − k · v   2 δE k−k ω−ω δE 2 kω × χ (2) (k,ω |k − k,ω − ω ) (k,ω )  δE 2 k ω δE 2 kω (3.5) + (k − k,ω − ω )  2 2 (2)∗     δE k ω δE k−k ω−ω − χ (k ,ω |k − k ,ω − ω ) ∗ (k,ω)   ∂ ∂ 1 1 ∂ ∂ Fa . + × ∂vj ω − ω − (k − k ) · v ∂vk ∂vk ω − k · v ∂vj Making use of (1.99), explicitly writing the spectral electric field fluctuations in terms of eigenmode intensities (1.95), and expressing the second-order susceptibility as χ (2) (k,σ  ωk |k − k,σ  ωk−k ) ≈ i Im χ (2) (k,σ  ωk |k − k,σ  ωk−k ), β

γ

γ

β

χ (2)∗ (k,σ  ωk |k − k,σ  ωk−k ) ≈ −i Im χ (2) (k,σ  ωk |k − k,σ  ωk−k ), β

γ

β

γ

(3.6)

62

Electrostatic Vlasov Weak Turbulence Theory

we have  ∂Fa  πe3 ∂ = −Re 3a  ∂t decay ma ∂vi





 dk

σ,σ ,σ  =±1 α,β,γ

dk

ki kj (k − k )k k k  |k − k |

Im χ (2) (k,σ  ωk |k − k,σ  ωk−k ) σ ωα − k · v   σ  β σ  γ k σ  γ σ β Ik−k Ikσ α Ik Ik−k Ik Ikσ α − − ×  γ β (k,σ ωkα )  (k − k,σ  ωk−k )  (k,σ  ωk ) γ

β

×

× δ(σ ωkα − σ  ωk − σ  ωk−k ) (3.7)   ∂ ∂ ∂ 1 1 ∂ + Fa . × γ β     ∂vj σ ωk−k − (k − k ) · v ∂vk ∂vk σ ωk − k · v ∂vj β

γ

Recalling that the only permissible three-wave interactions are those that correspond to (α,β,γ ) = (L,L,S), (L,S,L), and (S,L,L), and making use of the fact that (L,S,L) can be re-expressed as (L,L,S), thus leading to an overall factor 2, we may explicitly write (3.7) as   2   k k  (k − k )k πωpe e ea3 ∂ ∂Fa   i j dk dk = Re ∂t decay 4 Te m3a ∂vi k 2 k 2 |k − k |2 σ,σ ,σ  =±1 

2(k · k ) 1  σ  S × P σ ωkL Ikσ L Ik−k  L L L  (σ ωk )(σ ωk ) σ ωk − k · v  σ  S σ L L σ L σ L − σ  ωkL Ik−k − σ  μk−k ωk−k Ik  Ik   Ik S × δ(σ ωkL − σ  ωkL − σ  ωk−k )  ∂ 1 ∂ P × L L   ∂vj σ ωk − σ ωk − (k − k ) · v ∂vk ∂ ∂ − iπ δ[σ ωkL − σ  ωkL − (k − k ) · v] ∂vj ∂vk  ∂ ∂ 1 + P  L ∂vk σ ωk − k · v ∂vj

k · (k − k ) 1  σ  L σ μk ωkL Ikσ L Ik−k −  L P  L S  (σ ωk ) (σ ωk−k ) σ ωk − k · v  L σ L σ S σ  L σ S L − σ  ωk−k Ik − σ  ωkL Ik−k δ(σ ωkS − σ  ωkL − σ  ωk−k  Ik   Ik )  ∂ ∂ 1 × P  L  ∂vj σ ωk−k − (k − k ) · v ∂vk  ∂ 1 ∂ Fa, + P  L (3.8) ∂vk σ ωk − k · v ∂vj

3.3 Effects of Nonlinear Induced Scattering on the Particles

63

where we have made use of approximate forms for the second-order susceptibilities. We are concerned with a situation where linear wave-particle resonant interaction is ineffective, i.e., ωkL − k · v = 0 and ωkS − k · v = 0, while nonlinear wave-particle resonant interactions are effective. As a result, we have retained only the principal parts of the factors containing the wave-particle resonant denominators in (3.8). If we ignore the imaginary contribution, then other terms associated with the principal part contributions are supposed to be small, and thus negligible. From this, it can be seen that the three-wave resonant interaction has little or no contribution to the particle evolution,  ∂Fa  ≈ 0. (3.9) ∂t decay That the three-wave decay processes have no effect on the particles is not difficult to understand on physical grounds. Since no energy or momentum is exchanged between the particles and waves, it is only expected that the decay process should not influence the particle evolution. 3.3 Effects of Nonlinear Induced Scattering on the Particles We next consider the effects that the induced scattering has on the particle dynamics, which come from the principal parts associated with the second term and the last term on the right-hand side of generalized particle kinetic equation (1.107). We write the contribution from the induced scattering in three separate terms for the sake of convenience:     ∂Fa  ∂Fa  ∂Fa  ∂Fa  = + + , ∂t ind.sc. ∂t 1st ∂t 2nd ∂t 3rd

(3.10)

where    k k  (k − k )k ∂Fa  2ea3 ∂  i j dk dk = Re ∂t 1st m3a ∂vi k k  |k − k | ×

  σ,σ  =±1 α,β



χ (2) (k,σ  ωk |k − k,σ ωkα − σ  ωk ) β

P

β

(σ ωkα − k · v) (k − k,σ ωkα − σ  ωk )

∂ ∂ 1 β ∂vj σ ωkα − σ  ωk − (k − k ) · v ∂vk  ∂ 1 ∂ Fa, + ∂vk σ  ωkβ − k · v ∂vj ×

β

σ β Ik Ikσ α

(3.11)

64

 ∂Fa  ∂t 

2nd

Electrostatic Vlasov Weak Turbulence Theory



e3 ∂ = −Re a3 ma ∂vi ×



 dk



σ ,σ  =±1 β,γ





dk

ki kj (k − k )k k k  |k − k | χ (2)∗ (k,σ  ωk |k − k,σ  ωk−k ) γ

β

P

γ

β

β

∂ 1 ∂ γ   ∂vj σ ωk−k − (k − k ) · v ∂vk  ∂ 1 ∂ Fa, + ∂vk σ  ωkβ − k · v ∂vj

×

γ

(σ  ωk − σ  ωk−k − k · v) ∗ (k,σ  ωk − σ  ωk−k )

σ  β σ  γ Ik Ik−k

(3.12)

and  ∂Fa  ea4 ∂ = Im ∂t 3rd m4a ∂vi ×

 

 dk

σ,σ  =±1 α,β



dk

ki kj kk kl k 2 k 2

σ β

Ik Ikσ α ∂ α σ ωk − k · v ∂vj

1 σ ωkα





β σ  ωk

− (k − k ) · v

∂ ∂ 1 1 ∂ ∂ − × α β ∂vk σ ωk − k · v ∂vl ∂vl σ  ωk − k · v ∂vk

 Fa .

(3.13)

We evaluate each term. First, (∂Fa /∂t)1st . Taking only the linear eigenmodes, L and S, into consideration, we note that the second-order susceptibility has an overall i factor. Consequently, the real-part contributions come from the residue terms associated with various wave-particle resonance denominators. However, when the range of wave-particle interaction overlaps that of linear wave-particle response, then we ignore such contributions. Then it becomes evident that the only surviving terms are those dictated by nonlinear wave-particle interactions involving either two Langmuir waves, σ ωkL − σ  ωkL − (k − k ) · v = 0, or two ion-sound waves, σ ωkS − σ  ωkS − (k − k ) · v = 0. Noting that the range of velocity space for which linear wave-particle resonance is effective is characterized by the conditions ωkL − k · v ≈ 0 and/or ωkS − k · v ≈ 0, we associate the resonant denominators such as 1/(σ ωkL − k · v) and 1/(σ ωkS − k · v) with the principal part integrations. Of the various nonlinear wave-particle resonance denominators, the interactions characterized by 1/[σ ωkL − σ  ωkS − (k − k ) · v] or 1/[σ ωkS − σ  ωkL − (k − k ) · v] effectively reduce to the linear wave-particle resonance, 1/[σ ωkL − (k − k ) · v] or 1/[−σ  ωkL − (k − k ) · v], since ωkL ωkS . Consequently, these are not interesting because the resonant velocity regions dictated by these conditions overlap with that dictated by the linear wave-particle resonance conditions.

3.3 Effects of Nonlinear Induced Scattering on the Particles

65

Making use of approximate properties of various linear and nonlinear response functions, it is possible to simplify (∂Fa /∂t)1st as follows:     πe ea3 ∂ ∂Fa   k·k dk dk = − ki kj (k − k )k ∂t 1st me m3a ∂vi k 2 k 2 %   Ikσ L Ikσ L 1 ∂ × P δ[σ ωkL − σ  ωkL − (k − k ) · v] L  L L ∂v σ ωk σ ωk σ ωk − k · v j σ,σ  =±1 

2me Ikσ S Ikσ S 1 − P S  Te kk σ ωk − k · v     2 k k − σ σ (k + k 2 − k · k ) Re (k − k,σ ωkS − σ  ωkS ) × |k − k |2 (k − σ σ  k  )2 λ2De | (k − k,σ ωkS − σ  ωkS )|2  ∂Fa ∂ δ[σ ωkS − σ  ωkS − (k − k ) · v] . × ∂vj ∂vk

(3.14)

For electrons we may ignore the contribution from ion sound turbulence:     πe4 ∂ ∂Fe   k·k dk dk = ki kj (k − k )k ∂t 1st m4e ∂vi k 2 k 2  I σ  L I σ L ∂ 1 k k × P L  L L σ ωk σ ωk σ ωk − k · v ∂vj  σ,σ =±1

× δ[σ ωkL − σ  ωkL − (k − k ) · v]

∂Fe . ∂vk

(3.15)

Interchanging integral variables k and k , adding the result to the right-hand side, dividing the result by two, and making use of the delta function condition we may rewrite (3.15) as     2  ∂ πe4 ∂Fe    (k · k ) dk dk = − (σ σ  ) Ikσ L Ikσ L  ∂t 2m4 ω2 ∂v k 2 k 2 e

1st

i

pe

σ,σ  =±1

1 (k − k )i (k − k )j − k · v)2 ∂Fe × δ[σ ωkL − σ  ωkL − (k − k ) · v] . ∂vj ×P

(σ ωkL

(3.16)

For ions, the contribution from Langmuir modes can be ignored:      ∂Fi  2π e4 ∂  k·k   dk dk = − k k (k − k ) i k j ∂t 1st k 3 k 3 cS2 m4i ∂vi 

σ,σ =±1



× (σ σ  ) Ikσ S Ikσ S P

σ ωkS

 ∂ 1 Aσ,σ k,k ∂vj −k·v

66

Electrostatic Vlasov Weak Turbulence Theory

× δ[σ ωkS − σ  ωkS − (k − k ) · v] 

Aσ,σ k,k = 

∂Fi , ∂vk

(3.17)

k 2 + k 2 − k · k − σ σ  k k  Re (k − k,σ ωkS − σ  ωkS ) . |k − k |2 (k − σ σ  k  )2 λ2De | (k − k,σ ωkS − σ  ωkS )|2 

σ ,σ Noting that Aσ,σ k,k = Ak,k , and following the similar steps as in the derivation for (3.16), we obtain the ion equation:

    2 π e4 ∂ ∂Fi   (k · k ) dk dk = ∂t 1st cS2 m4i ∂vi k 3 k 3 ×P

(σ ωkS





(σ σ  ) Ikσ S Ikσ S

σ,σ  =±1

 1 (k − k )i (k − k )j Aσ,σ k,k 2 − k · v)

× δ[σ ωkS − σ  ωkS − (k − k ) · v]

∂Fi . ∂vj

(3.18)

We next evaluate the second term (∂Fa /∂t)2nd mentioned in (3.12). By noting that the second-order susceptibility has an overall factor i, we can discard those terms that contain linear wave-particle resonance conditions, or those that effectively reduce to linear wave-particle resonance. For (∂Fa /∂t)2nd , a careful examination reveals that the nonlinear wave-particle interaction of the form ω ± ω − (k − k ) · v = 0 does not exist. Only interactions of the type ω ± ω − k · v = 0 or ω − (k − k ) · v = 0 exist, which practically overlap with linear wave-particle resonance conditions. Consequently, we conclude that  ∂Fa  = 0, ∂t 2nd

(3.19)

to the lowest order in the description. Finally, consider the third term, (∂Fa /∂t)3rd , mentioned in (3.13). First, the electrons. Retaining only the nonlinear wave-particle interaction terms, and ignoring ion-sound waves, we have    1 πe4 ∂Fe  dk dk 2 2 = −  4 ∂t 3rd me k k

 σ,σ  =±1

∂ k· ∂v

%



Ikσ L Ikσ L ∂ k · L ∂v σ ωk − k · v

(3.20) × δ[σ ωkL − σ  ωkL − (k − k ) · v]   1 1 ∂ ∂ ∂ ∂ Fe . k· k · × k · −k· L L   ∂v σ ωk − k · v ∂v ∂v σ ωk − k · v ∂v

3.3 Effects of Nonlinear Induced Scattering on the Particles

67

Making use of the resonance conditions we may manipulate (3.20) and obtain     2 πe4 ∂Fe   (k · k ) dk dk = ∂t 3rd 2m4e k 2 k 2 ×

(σ ωkL





Ikσ L Ikσ L

σ,σ  =±1

1 1 ∂ (k − k ) · L 2 ∂v (σ ωk − k · v)2 − k · v)

× δ[σ ωkL − σ  ωkL − (k − k ) · v] (k − k ) ·

∂Fe . ∂v

(3.21)

Noting that ∂G ∂ 1 (k − k ) · = (k − k ) · L 2 ∂v ∂v (σ ωk − k · v) −



G L (σ ωk − k · v)2

k 2 − k 2 G, (σ ωkL − k · v)3



(3.22)

and upon symmetrizing with respect to k and k , we have     2 πe4 ∂ ∂Fe   (k · k ) dk dk = ∂t 3rd 2m4e ∂vi k 2 k 2 ×P

(σ ωkL





Ikσ L Ikσ L

σ,σ  =±1

1 (k − k )i (k − k )j − k · v)4

× δ[σ ωkL − σ  ωkL − (k − k ) · v]

∂Fe . ∂vj

(3.23)

Next, consider the ions. For energetic ions, we may ignore linear wave-particle resonances and contributions from Langmuir waves. The structure of the ion equation is the same as that of the electrons, except that Langmuir mode intensities are replaced by ion-sound mode intensities. Thus, without going through the detailed manipulations, the result can be written down readily:     2 πe4 ∂ ∂Fi   (k · k ) dk dk =  ∂t 3rd 2m4i ∂vi k 2 k 2 ×P

(σ ωkS





Ikσ S Ikσ S

σ,σ  =±1

1 (k − k )i (k − k )j − k · v)4

× δ[σ ωkS − σ  ωkS − (k − k ) · v]

∂Fi . ∂vj

(3.24)

68

Electrostatic Vlasov Weak Turbulence Theory

3.4 Summary of Electrostatic Vlasov Weak Turbulence Theory In this section, we summarize the basic set of equations relevant for describing weakly turbulent nonlinear processes dictated by electrostatic interactions in unmagnetized plasmas. At this level of description, the plasma is considered as purely collisionless. The effects of discreteness associated with the plasma particles will be discussed in Part II, but the discussion thereof is built upon the findings in Part I. The following are the basic physical quantities and equations that summarize Part I. • Linear dispersion relations for Langmuir (L) and ion-sound or ion-acoustic (S) modes     3 2 2 3 k 2 vT2 e L , ωk = ωpe 1 + k λDe = ωpe 1 + 2 2 4 ωpe √ √  me 1 + 3Ti /Te kcS 1 + 3Ti /Te S  = kvT e . (3.25) ωk =  2mi 1 + k 2 v 2 /2ω2 1 + k 2 λ2De pe Te • Fundamental plasma quantities  4πne2 ωpe = , plasma frequency; me  Te λDe = , Debye length; 4πne2  2Te , electron thermal speed; vT e = me  2Ti , ion (proton) thermal speed; vT i = mi  Te cS = , ion sound (ion acoustic) speed. mi • Wave kinetic equations 2 ωpe ∂Ikσ L = π(σ ωkL ) 2 ∂t k &



∂Fe σ L dv δ(σ ωkL − k · v) k · I ∂v k '( )

induced emission (linear growth/Landau damping)     Iσ S L  LS L σ  L k−k + σ ωk dk Vk,k σ ωk Ik μ  σ ,σ  =±1 & '( k−k) spontaneous decay

(3.26)

3.4 Summary of Electrostatic Vlasov Weak Turbulence Theory





σ  S

I  L σ L Ikσ L − σ  ωkL k−k + σ  ωk−k  Ik μk−k & '( )

69



induced decay (3.27)    me ∂Fi σ  L σ L LL  + dk dv Uk,k (σ ωkL ) I I ,  (k − k ) · mi ∂v k k σ  =±1 & '( ) induced scattering (of Langmuir wave off thermal ions)   σS  2  ωpe ∂ I ∂ me L S Fe + dv δ(σ ωk − k · v) k · = π (σ μk ωk ) 2 Fi k ∂t μk k ∂v mi μk & '( ) Ikσ S

induced emission for ion-sound mode     L  SL σ  L σ ωk dk Vk,k σ ωkL Ikσ L Ik−k +  & '( )   σ ,σ =±1



σ L − (σ  ωkL Ik−k  &

spontaneous decay σS  L σ  L Ik + σ  ωk−k I )  k μk '( )

(3.28)

induced decay    ∂Fi σ  S σ S L  SS  + (σ ωk ) dk dv Uk,k I I ,  (k − k ) · ∂v k k σ  =±1 & '( ) induced scattering (of ion-sound wave) Various processes are indicated by their respective names, and various coupling coefficients are defined by  μk =

k 3 λ3De

   k 3 vT3 e me me 3Ti 3Ti 1+ = 3 1+ , mi Te ωpe 8mi Te

LS Vk,k  =

π e2 μk−k (k · k )2 S δ(σ ωkL − σ  ωkL − σ  ωk−k  ), 2 2 2  2 2 Te k k |k − k |

SL Vk,k  =

π e2 μk [k · (k − k )]2 L δ(σ ωkS − σ  ωkL − σ  ωk−k  ), 4 Te2 k 2 k 2 |k − k |2

LL Uk,k  =

π e2 (k · k )2 δ[σ ωkL − σ  ωkL − (k − k ) · v], 2 m2 k 2 k 2 ωpe e

SS Uk,k  =

π e2 (k·k )2 Wk,k δ[σ ωkS −σ  ωkS −(k−k )·v], 2 m m k 4 k 4 λ4 ωpe e i De

70

Electrostatic Vlasov Weak Turbulence Theory



Wk,k

|k − k |2 = 1+ (σ k − σ  k  )2

2

 1  k + σ σ , k |k − k |4 λ4De | (k − k,σ ωkS − σ  ωkS )|2 2  2(k · k − σ σ  k k  )  S  S 2 | (k − k ,σ ωk − σ ωk )| = 1 + |k − k |2 (k − σ σ  k  )2 λ2De    me (k − σ σ  k  )2 π me (k − σ σ  k  )2 exp − + 2 mi |k − k |6 λ4De 2mi |k − k |2  1/2  3/2   mi Te Te (k − σ σ  k  )2 2 + exp − . me Ti 2Ti |k − k |2 (3.29)

×

• Electron particle kinetic equation   ∂Fe ∂ el ∂Fe Dij , = ∂t ∂vi ∂vj  ki kj  πe2 dk 2 Dijel = 2 δ(σ ωkL − k · v) Ikσ L me k σ =±1     2  Ikσ L Ikσ L πe4  (k · k ) dk dk + (k − k )i (k − k )j L 2 2m4e k 2 k 2 (σ ω − k · v)  k σ,σ =±1    1 σσ × − (3.30) δ[σ ωkL − σ  ωkL − (k − k ) · v]. L 2 ωpe (σ ωk − k · v)2 • Ion particle kinetic equation   ∂Fi ∂ ion ∂Fi Dij , = ∂t ∂vi ∂vj  ki kj  πe2 ion dk 2 Dij = 2 δ(σ ωkS − k · v) Ikσ S k σ =±1 mi     2  Ikσ S Ikσ S πe4  (k · k ) dk dk + k 2 k 2 2m4i (σ ωkS − k · v)2 σ,σ  =±1   2(σ σ  ) Aσ,σ  1 k,k × + k k  cS2 (σ ωkS − k · v)2 × (k − k )i (k − k )j δ[σ ωkS − σ  ωkS − (k − k ) · v], 

Aσ,σ k,k =

1 k 2 + k 2 − k · k − σ σ  k k  . 2  2   2  |k − k | (k − σ σ k ) λDe (k − k ,σ ωkS − σ  ωkS )

(3.31)

3.4 Summary of Electrostatic Vlasov Weak Turbulence Theory

71

In (3.30) and (3.31), we have presented formal electron and ion particle kinetic equations in which the weak turbulence ordering is carried out up to the third order in perturbative expansion (Tsytovich, 1977b; Yoon et al., 2003b). This was done for the sake of being consistent with the same procedure in the wave kinetic equation. However, after the formal theory is being developed and carried out, it becomes clear that the nonlinear correction simply modifies the velocity space diffusion coefficient in the velocity range for which linear wave-particle interaction is not forthcoming. The nonlinear diffusion coefficient is, however, quite insignificant, since the overall magnitude is second order in wave intensity. For a wide range of applications, the simple quasilinear diffusion equation is sufficient. Nevertheless, the present chapter formally discusses the wave and particle kinetic equations with consistent weak turbulence ordering scheme fully implemented, which is done for the sake of consistency. At the closing of this chapter, we note that the final set of wave kinetic equations contain various terms with designated names indicating different wave-particle and wave-wave interaction processes. Some processes are designated with the term “spontaneous” while others are denoted by “induced.” These terms originate from quantum mechanical concepts, and we have also discussed the meaning of these processes on the basis of intuitive concepts. However, the important point is that under the present Vlasov weak turbulence theory, not all processes appear in balanced form where every induced process has a spontaneous counterpart. In order to provide a complete description where spontaneous and induced processes appear in balanced forms, one must turn to the Klimontovich formalism, which is the subject of Part II of this book. Before we move on to Part II, we discuss a number of caveats inherent in the present weak turbulence formalism. One is that throughout the main part of this monograph we work under the inherent assumption of weak wave growth or damping, which is exemplified by the treatment of resonant denominator, 1/(ω − k · v + i0). Such a treatment is applicable when the plasma is unstable to weakly growing instability, such as the bump-on-tail instability, or when one is dealing with a weakly damped mode. However, when the free energy source of the instability is sufficiently high, then it is no longer valid to treat the problem by assuming that ω is essentially real and that the velocity space resonant denominator can be treated by adding an infinitesimally small but positive imaginary part to real ω. For a more general situation, the wave dispersion relation ω = ωk may no longer be determined by considering only the real part of linear dielectric function, Re (k,ωk ) = 0, but instead, for such cases, one must treat ω as complex, and the dispersion relation ω = ωk + iγk must be solved from the complete plasma dielectric function, Re (k,ωk + iγk ) + iIm (k,ωk + iγk ) = 0. The plasma instability that necessitates this kind of treatment is known as the “reactive instability.” The linear theory of waves and instabilities of this type of problem is well established – see, e.g.,

72

Electrostatic Vlasov Weak Turbulence Theory

excellent monographs on linear plasma instability by Mikhailovskii (1974), Cap (1976), Stix (1992), Swanson (1989), Gary (1993), Melrose (1986), or Baumjohann and Treumann (1997) and even quasilinear description of such types of situations are also well developed, e.g., Sagdeev and Galeev (1969) and Davidson (1972). However, the question is whether one may also formulate the perturbative nonlinear theory, i.e., whether the weak turbulence theory may be extended to the reactively unstable problem. This question is not completely addressed in the literature, but Appendix E describes a possible direction – see also the works by Ishihara et al. (1980, 1981) and Yoon (2010). Another limitation of the fundamental formalism adopted in this book is the assumption that the iterative solution of nonlinear equation is valid and that the truncation of series expansion can effectively be done at the third order. However, it is natural to ask whether the infinite series δfa = δfa(1) + δfa(2) + δfa(3) + · · · converges or not, and, if so, under what conditions? Or whether one may perform at least a partial summation of infinite series? We have shown that the “necessary” condition for the series convergence is that the wave energy be sufficiently lower than the particle energy, Ewave Eparticle – see (1.47). However, this is not the “sufficient” condition, and, under some conditions, the series may appear to diverge. Nevertheless, the prima facie divergence may be avoided if one can sum up the most divergent terms in the infinite series. Appendix F overviews this problem, where the iterative solution up to fifth order is explicitly computed. There, it is shown on the basis of the pattern that emerges that the partial summation of leading order terms points to the need for the so-called renormalized resonance denominator. The partial summation idea of the type discussed in Appendix F leads to what is known as the “renormalized” kinetic theory. There are a number of different versions of renormalized kinetic theory, starting from the works by Dupree (1966), Weinstock (1969), etc. – see also works by Kono and Ichikawa (1973); Horton and Choi (1979); Krommes (2002); Itoh and Itoh (2009); Diamond et al. (2010). The main attention of this book is on the standard weak turbulence theory, so that such methods are not systematically reviewed, but Appendix G surveys one such approach. The basic outline of the method overviewed in Appendix G is that of Rudakov and Tsytovich (1971).

Part II Klimontovich Weak Turbulence Theory: Electrostatic Approximation

4 Electrostatic Klimontovich Weak Turbulence Theory

In Part II of this book we discuss the effects of particle discreteness. In order to include such effects we resort to the statistical mechanical framework pioneered by Klimontovich (1967, 1982). Vlasov equation and Klimontovich equation are mathematically identical, but in the Vlasov approach, the particle discreteness is smoothed out, whereas the Klimontovich formalism retains the individual nature of the plasma particles. As such, the two approaches differ essentially on how the initial state is described. While in the Vlasov approach the initial state is prescribed by a smooth distribution function, in the Klimontovich approach, the initial state is a summation of infinitely sharp delta functions. 4.1 Plasma Equations Based upon Klimontovich Formalism Let us begin the discussion by considering a collection of N ions (protons) and N electrons, labeled respectively by a = i and e. If the charged particles interact according to the classical nonrelativistic Newtonian dynamics, then their motions, or particle orbits rai (t) and vai (t), obey drai (t) = vai (t), dt dva (t) ea ma i = ea E[rai (t),t] + vai (t) × B[rai (t),t], dt c

(4.1)

where i = 1, . . . ,N. The self-consistent fields E(r,t) and B(r,t) must be determined from Maxwell’s equation: ∂ 1 ∂ × E(r,t) + B(r,t) = 0, ∂r c ∂t ∂ · B(r,t) = 0, ∂r 75

76

Electrostatic Klimontovich Weak Turbulence Theory

  ∂ ea δ[r − rai (t)], · E(r,t) = 4π ρ = 4π ∂r a i=1 N

∂ 1 ∂ 4π × B(r,t) − E(r,t) = j ∂r c ∂t c N 4π   a ea vi (t) δ[r − rai (t)]. = c a i=1

(4.2)

The net charge and current densities in (4.2) are computed by collecting contributions from all charged particles. We may rewrite ρ and j by   ρa , j= ja , ρ = a

ρa =

N 

a

δ[r −

rai (t)]

=

 dv

i=1

ja =

N 

N 

δ[r − rai (t)] δ[v − vai (t)],

(4.3)

i=1

 vai (t) δ[r − rai (t)] =

i=1

dv v

N 

δ[r − rai (t)] δ[v − vai (t)].

i=1

From this, the notion of N body phase space density, namely, the “Klimontovich function,” emerges: Na (r,v,t) =

N 

δ[r − rai (t)] δ[v − vai (t)].

(4.4)

i=1

Physically, the function Na (r,v,t) represents the density in six-dimensional phase space (r,v) at time t. This is because when integrated over v and r, and averaged over the volume, we obtain the total number of particles. It also represents the phase space probability density of finding a collection of N particles. As is obvious from the definition Na (r,v,t) is composed of infinitely sharp delta functions. In terms of Na (r,v,t), Maxwell’s equation is now written as ∂ 1 ∂ × E(r,t) + B(r,t) = 0, ∂r c ∂t ∂ · B(r,t) = 0, ∂r   ∂ ea dv Na (r,v,t), · E(r,t) = 4π ∂r a  ∂ 1 ∂ 4π  ea dv v Na (r,v,t). × B(r,t) − E(r,t) = ∂r c ∂t c a

(4.5)

4.1 Plasma Equations Based upon Klimontovich Formalism

77

The Klimontovich function Na (r,v,t) is a six-dimensional phase space density. Since particles cannot be created or destroyed, Na (r,v,t) must be conserved. Consequently, the equation that Na (r,v,t) obeys is given by   dN ∂ dr ∂ dv ∂ N(r,v,t), (4.6) =0= + · + · dt ∂t dt ∂r dt ∂v which can be rewritten as  N   drai (t) ∂ dvai (t) ∂ ∂ δ[rai (t) − r] δ[vai (t) − v] = 0. + · + · ∂t dt ∂r dt ∂v i=1 (4.7) By virtue of the exact microscopic equation of motion (4.1), the dynamical equation (4.6) or (4.7) can be expressed by   ∂  ∂ v ∂ ea Na (r,v,t) = 0. E(r,t) + × B(r,t) · +v· + ∂t ∂r ma c ∂v (4.8) Note that this is formally identical to the Vlasov equation, but the difference is that Na (r,v,t) is not a smooth distribution, but rather it is given by a collection of point-like delta functions (4.4). Both Vlasov and Klimontovich equations advance the phase space distribution, fa (r,v,t) in the case of Vlasov equation and Na (r,v,t) for Klimontovich equation, in time, while preserving the total phase space volume. Consequently, both equations represent exact phase space mapping of the initial functions, fa (r,v,0) and Na (r,v,0), along their trajectories, fa [r(t),v(t),t] and Na [r(t),v(t),t], but the difference is that for Vlasov system fa (r,v,0) is a smooth function, while for Klimontovich system, Na (r,v,0) is made of the summation of infinitely singular delta functions. The singularities associated with the delta functions are what lead to the discreteness effects, as we will see. The microscopic Klimontovich function (4.4) and the dynamical equation it satisfies, namely, (4.7), contain all the necessary dynamical information to describe the collection of N charged particle, and it is equivalent to solving the coupled Newtonian equations of motion for N particles. However, the Klimontovich function, as with the Newtonian equations of motion, contains too much information. In reality, we cannot deal with all the phase space information, but we can only deal with some gross average quantities. Therefore, we resort to the statistical method. We write the physical quantities into averages and perturbations – see (1.34):   Na (r,v,t) = Na (r,v,t) + δNa (r,v,t),   E(r,t) = E(r,t) + δ E(r,t),   B(r,t) = B(r,t) + δ B(r,t), (4.9)

78

Electrostatic Klimontovich Weak Turbulence Theory

where < · · · > represents the (ensemble) average, and δ · · · represents the remainder (i.e., fluctuations). In this book, we are interested in a situation where average quantities are spatially uniform, and where there are no average fields. Thus, we write       (4.10) Na (r,v,t) = fa (v,t), E(r,t) = 0, B(r,t) = 0.   Physical interpretation of Na (r,v,t) = fa (v,t) is that the quantity * N +    a a Na (r,v,t) = δ[r − ri (t)] δ[v − vi (t)] = fa (v,t) (4.11) i=1

represents the average probability distribution of all the particles at a phase space location (r,v) at time t. The quantity fa (v,t) is the smoothed Klimontovich function, and is the same quantity that one encounters in the Vlasov theory. The idea is that when the ensemble average is taken over the Klimontovich function, which is made of infinitely sharp delta functions, the singularities average out to generate a smooth one-particle distribution function. Since the system is assumed to be spatially uniform, the smoothed probability distribution fa (v,t) is everywhere equal in r. Making use of the short-hand notation, x = (r,v),

(4.12)

we may also consider the ensemble averages of the products of Klimontovich functions. Take, for instance, 

N  N     δ(x − xia ) δ(x  − xjb ) Na (x,t) Nb (x ,t) = i=1 j =1

=

N  

N  N     δ(x − xia ) δ(x  − xib ) + δ(x − xia ) δ(x  − xjb ) . i=1 j =1 (i=j )

i=1

(4.13) In the first term, if a = b, then we are dealing with a product of two delta functions that belong to two different particle species (say, one electron and one ion). For such a combination, we generally assume that there is no cross correlation. Consequently, we demand that such quantities are zero unless a = b: * N +    δ(x  − xia ) Na (x,t) Nb (x ,t) = δab δ(x − x  ) i=1

+

N  N 



i=1 j =1 (i=j )

 δ(x − xia ) δ(x  − xja ) .

(4.14)

4.1 Plasma Equations Based upon Klimontovich Formalism

 $N

79



a  Note that i=1 δ(x − xi ) = fa (x,t), that is, (4.11). We define the two-body phase space distribution function

f2ab (x,x ,t)

=

* N N 

+ δ(x −

xia ) δ(x 



xja )

.

(4.15)

i=1 j =1 (i=j )

Then we have 

 Na (x,t) Nb (x ,t) = δab δ(x − x  ) fa (x,t) + f2ab (x,x ,t).

(4.16)

Likewise, we may conceive of the following three-body correlation: 

     δ(x − xia ) δ(x  − xjb ) δ(x  − xkc ) Na (x,t) Nb (x ,t) Nc (x ,t) = i

=

j



k



δ(x − xia ) δ(x  − xib ) δ(x  − xic )

i

+

 



δ(x − xia ) δ(x  − xjb ) δ(x  − xic )

i j (j =i)

+

 



δ(x − xia ) δ(x  − xib ) δ(x  − xkc )

i k (k=i)

+

 



δ(x − xja ) δ(x  − xjb ) δ(x  − xkc )

j k (k=j )

+

  



δ(x − xia ) δ(x  − xjb ) δ(x  − xkc )

i

j k (i=j =k)

*

= δab δbc δ(x − x  ) δ(x  − x) 

δ(x  − xia ) +

δ(x − xia ) δ(x  − xjb )

i j (j =i)

* + δab δ(x − x  )

+

i

* + δac δ(x − x  )



 i k (k=i)

+ δ(x  − xib ) δ(x  − xkc )

80

Electrostatic Klimontovich Weak Turbulence Theory

*

+ δbc δ(x  − x  )

+

*  i



+

δ(x − xja ) δ(x  − xkc )

j k (k=j )

δ(x −

xia ) δ(x 

+ −

xjb ) δ(x 



xkc )

.

j k (i=j =k)

(4.17) Defining the three-body phase space distribution function, * +  abc   a  b  c δ(x − xi ) δ(x − xj ) δ(x − xk ) , f3 (x,x ,x ,t) = i

(4.18)

j k (i=j =k)

we have   Na (x,t) Nb (x ,t) Nc (x ,t) = δab δbc δ(x − x  ) δ(x  − x) fa (x ,t) + δac δ(x − x  ) f2ab (x,x ,t) + δab δ(x − x  ) f2bc (x ,x ,t) + δbc δ(x  − x  ) f2ac (x,x ,t) + f3abc (x,x ,x ,t). (4.19) The two- and three-body distributions, f2ab (x,x ,t) and f3abc (x,x ,x ,t), respectively, can be further expressed in an alternative way, in terms of the twoand three-body correlations, g2ab (x,x ,t) and g3abc (x,x ,x ,t), defined respectively as follows: f2ab (x,x ,t) = fa (x,t) fb (x ,t) + g2ab (x,x ,t), f3abc (x,x ,x ,t) = fa (x,t) fb (x ,t) fc (x ,t) + fa (x,t) g2bc (x ,x ,t) + fb (x ,t) g2ac (x,x ,t) + fc (x ,t) g2ab (x,x, ,t) + g3abc (x,x ,x ,t).

(4.20)

In terms of these expressions, (4.11), (4.16), and (4.19) can be alternatively written as follows:   Na (x,t) = fa (x,t),   Na (x,t) Nb (x ,t) = fa (x,t) fb (x ,t) + δab δ(x − x  ) fa (x,t)   +g2ab (x,x ,t), Na (x,t) Nb (x ,t) Nc (x ,t) = fa (x,t) fb (x ,t) fc (x ,t) + fa (x,t) g2bc (x ,x ,t) +fb (x ,t) g2ac (x,x ,t)+fc (x ,t) g2ab (x,x ,t)+g3abc (x,x ,x ,t).

(4.21) + δac δ(x − x  ) fa (x,t) fb (x ,t) + g2ab (x,x ,t)

4.1 Plasma Equations Based upon Klimontovich Formalism

81



+ δab δ(x − x  ) fb (x ,t) fc (x ,t) + g2bc (x ,x ,t)

+ δbc δ(x  − x  ) fa (x,t) fc (x ,t) + g2ac (x,x ,t)

+ δab δbc δ(x − x  ) δ(x  − x) fa (x ,t) + · · · From (4.9) and (4.10), we have   δNa (x,t) = Na (x,t) − Na (x,t) = Na (x,t) − fa (x,t). Making use of the short-hand notation     A(x,t) B(x ,t) = A B x,x ,t ,

(4.22)

(4.23)

let us consider the ensemble averages of correlations of fluctuations. Some straightforward manipulations lead to the following:   δNa (x,t) = 0,   δNa (x,t) δNb (x ,t) = δab δ(x − x  ) fa (x,t) + g2ab (x,x ,t),   δNa (x,t) δNb (x ,t) δNc (x ,t) = δab δbc δ(x − x  ) δ(x  − x) fa (x ,t) + δab δ(x − x  ) g2bc (x ,x ,t) + δbc δ(x  − x  )g2ac (x,x ,t) + δac δ(x − x  ) g2ab (x,x ,t) + g3abc (x,x ,x ,t). (4.24) The Vlasov approximation is equivalent to ignoring all discrete particle effects. In the Klimontovich formalism, discrete particle effects arise from singular functions associated with delta function terms such as δ(x − x  ) δab fa (x,t), etc. Even though the ensemble average led to the smoothed Vlasovian one-particle distribution out of infinitely sharp Klimontovich function, the delta function terms mentioned earlier represent what’s left over from the smoothing procedure. These delta function terms are intimately related to the discrete particle effects, which we will eventually see. This proves that the singular nature of the infinitely sharp delta functions that make up the Klimontovich function is what eventually leads to the discrete particle effects. If we ignore all such terms, Na (x,t) = fa (x,t), Na (x,t) Nb (x ,t) = fa (x,t) fb (x ,t) + g2ab (x,x ,t), Na (x,t) Nb (x ,t) Nc (x ,t) = fa (x,t) fb (x ,t) fc (x ,t) + fa (x,t) g2bc (x ,x ,t) + fb (x ,t) g2ac (x,x ,t) + fc (x ,t) g2bc (x ,x ,t) + g3abc (x,x ,x ,t),

(4.25)

82

Electrostatic Klimontovich Weak Turbulence Theory

etc., and



 δNa (x,t) = 0,

δNa δNb x,x  ;t = g2ab (x,x ,t), δNa δNb δNc x,x ,x  ;t = g3abc (x,x ,x ,t),

(4.26)

etc., then we obtain the Vlasov turbulence theory. Retaining the discrete particle terms will result in spontaneous emission/scattering and collisions, which will be discussed later. We can clearly see that the Vlasov theory is an approximation since there is no a priori justification for ignoring terms such as δ(x − x  ) δab fa (x), etc., although, later, we will see that all the new terms that arise from retaining the discrete particle effects will have an overall proportionality to a quantity that represents the inverse of total number of plasma particles in a sphere with the radius equal to the Debye length. If the number of particles within the “Debye” sphere is large, then the discrete particle terms will be less important. From this, we will see that the Vlasov approximation is valid if the number of particles in the Debye sphere is infinite. At the moment, however, this is not evident. Returning to the Klimontovich formalism, Let us now introduce another quantity, that is, the Klimontovich function for free or noninteracting particles: Na0 (x,t) =

N 

a0 δ[r − ra0 i (t)] δ[v − vi (t)],

(4.27)

i=1 a0 where the free-streaming particle orbits ra0 i (t) and vi (t) satisfy

dra0 dva0 i (t) i (t) (t), = va0 = 0. (4.28) i dt dt Then the Klimontovich function for noninteracting particles obeys a trivial equation,   ∂ ∂ Na0 (x,t) = 0. (4.29) +v· ∂t ∂r Upon taking the ensemble average, we have    ∂ ∂  0 +v· Na (x,t) = 0. ∂t ∂r

(4.30)

The quantity Na (x,t) is the exact phase space distribution that describes charged particles interacting with each other via collective electromagnetic force. In contrast, the object Na0 (x,t) is the phase space distribution that describes noninteracting free particles. The difference between the two, Na (x,t) − Na0 (x,t), represents, conceptually, the phase space information in which the collective effects are emphasized while the pure single particle effects are subtracted out. Consequently, it is

4.1 Plasma Equations Based upon Klimontovich Formalism

83

useful to subtract (4.29) from (4.8), in order to obtain the dynamical equation for the quantity Na (x,t) − Na0 (x,t):   ∂ ∂ [Na (x,t) − Na0 (x,t)] +v· ∂t ∂r   ∂  ea v Na (x,t) = 0. + (4.31) δE + × δB · ma c ∂v Let us define the fluctuations of unperturbed Klimontovich function for free particles:   δNa0 (x,t) = Na0 (x,t) − Na0 (x,t) . (4.32) Then (4.31) can be rewritten as   ∂ ∂ [fa (x,t) + δNa (x,t) − δNa0 (x,t)] +v· ∂t ∂r     v ∂ ea [fa (x,t) + δNa (x,t)] = 0, δE + × δB · + ma c ∂v

(4.33)

where we have made use of (4.30). Maxwell’s equation in (4.6) coupled with (4.9) and (4.10) is written as  

ea dv fa (x,t) + δNa (x,t) , ∇ · δE = 4π a

1 ∂ ∇ × δE + δB = 0, c ∂t 

1 ∂ 4π  ea dv v fa (x,t) + δNa (x,t) , ∇ × δB − δE = c ∂t c a ∇ · δB = 0.

(4.34)

Taking the ensemble average of (4.33), we obtain the equation for average particle distribution:      ∂ v ea ∂ δE + × δB · + v · ∇ fa (x,t) = − δNa (x,t) . (4.35) ∂t ma c ∂v Averaging Maxwell’s equation leads to the average charge and current neutrality conditions   0= ea dv fa (x,t), a

0=

 a

 ea

dv v fa (x,t).

(4.36)

84

Electrostatic Klimontovich Weak Turbulence Theory

Subtracting (4.35) from (4.33), we obtain the equation for perturbed Klimontovich function. Together with the wave equation for perturbed field, we have    ∂f (x,t)

∂ ea v a δE + × δB · + v · ∇ δNa (x,t) −δNa0 (x,t) = − ∂t ma c ∂v    ∂ ∂ − δa · δNa (x,t)− δa· δNa (x,t) , ∂v ∂v   ea v δa = δE + × δB , ma c  4π  1 ∂ ea dv v δNa (x,t), δE = ∇ × δB − c ∂t c a   ∇ · δE = 4π ea dv δNa (x,t), a

∇ × δE +

1 ∂ δB = 0, c ∂t

∇ · δB = 0.

(4.37)

This is a closed set of equations involving δNa , δE, and δB, except that an additional quantity δNa0 is also present. One must be able to compute this quantity, which is discussed next. It is useful to discuss the two-time correlation function of the free particle fluctuations. Subtracting (4.30) from (4.29) and making use of definition (4.32), we have   ∂ (4.38) + v · ∇ δNa0 (x,t) = 0. ∂t Multiplying δNb0 (x ,t  ) and taking the ensemble average, we obtain     ∂ + v · ∇ δNa0 δNb0 x,t;x ,t  = 0. ∂t The following equation is equally valid:     0 ∂   +v ·∇ δNa δNb0 x,t;x ,t  = 0.  ∂t The initial condition of either (4.39) or (4.40) must be   0 δNa δNb0 x,t;x ,t  =t = δab δ(x − x  ) fa (x,t).

(4.39)

(4.40)

(4.41)

This is because according to (4.24), δNa δNb x,x  ;t = g2ab (x,x ,t) + δab δ(x − x  ) fa (x,t). Since δNa0 and δNb0 are fluctuations for noninteracting particles, their correlation must not possess the correlation function g2ab (x,x ,t). Hence, (4.41).

4.1 Plasma Equations Based upon Klimontovich Formalism

85

 Another  more straightforward way to see this is to directly compute δNa0 δNb0 x,t;x ,t  :  0       δNa (x,t) δNb0 (x ,t  ) = Na0 (x,t) Nb0 (x ,t  ) − Na0 (x,t) Nb0 (x ,t  ) * N  a0 δ[r − ra0 = i (t)] δ[v − vi (t)] i=1

×

N 

+ 

δ[r −

 vb0 j (t )]



δ[v −

 vb0 j (t )]

j =1



  − Na0 (x,t) Nb0 (x ,t  ) .

(4.42)

a0 a0 a0 a0 Since va0 i = const, we have ri (t) = vi t + ri , with ri = const. Consequently, (4.42) reduces to  0  δNa (x,t) δNb0 (x ,t  ) x,t;x ,t  = δab δ[r − r − v (t − t  )] δ(v − v ) fa (x,t), (4.43)

which is the desired result (4.41). Here, an assumption is made that the ensemble average of the Klimontovich function for noninteracting, or free, particles is equal to that of the complete Klimontovich function,   0 (4.44) Na (x,t) = Na (x,t) = fa (x,t).   0 The Fourier transformation of δNa δNb0 x,t;x ,t  with respect to the fast timeand spatial-scales, t − t  and r − r , is of interest. In the Laplace transformation (1.38) we restricted ourselves to positive t integral because we are interested in the causal problem. This prescription is equivalent to adding an infinitesimal positive imaginary part to ω. However, in the present case, the spectral representation is to be defined with respect to the time difference, t − t  , which may be defined over an entire range −∞ < t − t  < ∞. This contrasts to individual time variables, which are defined over 0 < t < ∞ and 0 < t  < ∞ (that is, causal). Consequently, in the present case, following Klimontovich (1982), we define   ∞ +  0 1  d(r − r ) d(t − t  ) δNa (v) δNb0 (v ) k,ω = (2π)4 0     +  × δNa0 δNb0 x,t;x ,t  e−ik·(r−r )+iω(t−t )−0 (t−t )  ∞ 1 +  = δab δ(v − v ) dτ e−ik·v τ +iωτ −0 τ fa (x,t). 4 (2π) 0 (4.45) Then we construct the total spectral representation, !   + + " ∗  0 δNa (v) δNb0 (v ) k,ω = δNa0 (v) δNb0 (v ) k,ω + δNa0 (v) δNb0 (v ) k,ω .

(4.46)

86

Electrostatic Klimontovich Weak Turbulence Theory

This quantity reduces to the customary Fourier transformation for 0+ = 0 exactly. From this we now have a proper definition, which leads to  0  δNa (v) δNb0 (v ) k,ω = (2π)−3 δab δ(v − v ) δ(ω − k · v) fa (x,t). (4.47) The set of equations, (4.35)–(4.37), with the source fluctuation (4.43) and its spectral representation (4.47), can now be reformulated in terms of Fourier–Laplace transformation (1.38). With the redefinition of angular frequency that includes the slow-time derivative, (1.41), the set of equations that constitute the basis of subsequent analysis readily emerges:      ki vj k·v ∂fa (v,t) ea ∂ δij + dk dω 1 − =− ∂t ma ∂vi ω ω , j a × δE−k,−ω δNk,ω (v) ,     ki kj c2 k 2 4πi  j a δij − 2 δij − 2 δEk,ω = − ea dv vi δNk,ω (v), ω k ω a a

a0 × i(ω − k · v) δNk,ω (v) − δNk,ω (v)    k·v vi kj ∂fa (v,t) ea i δij + δEk,ω 1 − = ma ω ω ∂vj   ∂ ea dk dω + ma a ∂vj    vi kj k · v δij +  (4.48) × 1− ω ω  i 

a a × δEki ,ω δNk−k ,ω−ω (v)− δEk,ω δNk−k,ω−ω (v) . In the rest of Part II, we are interested in the electrostatic approximation. Consequently, the system of equations to be solved can be simplified by taking the electrostatic approximation, δB = 0, k · v/ω 1, and δE = ik δφ. This simplifies (4.48):    ∂  ∂fa (v,t) −iea a dk dω k · (v) , = δφ−k,−ω δNk,ω ∂t ma ∂v

a (v,t) e ∂f a a a0 (v) = − δφk,ω k · (ω− k·v) δNk,ω (v)−δNk,ω ma ∂v   ∂ ea a δφk,ω δNk−k dk dω k · − ,ω−ω (v) ma ∂v 

 a − δφk,ω δNk−k ,ω−ω (v,t) ,  4πea  a dv δNk,ω (v). (4.49) δφk,ω (t) = 2 k a

4.2 Formal Klimontovich Weak Turbulence Theory

87

These equations directly generalize (1.40), which formed the basis for Vlasov weak a0 a a (v) and replace δNk,ω (v) by δfk,ω (v), then we turbulence theory. If we ignore δNk,ω recover (1.40). Note that the resonance factor ω − k · v + i0 in (4.49) contains implicit slow-time derivative, ω − k · v + i0 + i ∂/∂t. 4.2 Formal Klimontovich Weak Turbulence Theory We now reformulate the plasma weak turbulence theory on the basis of (4.49). We follow the same steps as taken in the Vlasov weak turbulence analysis of Chapter 1. We remind the readers of the shorthand notation (1.42), gak,ω = −

∂ 1 ea , ma ω − k · v + i0 ∂v

with which the equation for perturbed phase space density distribution can be expressed compactly as a a0 = δNk,ω + k · gak,ω fa δφk,ω (4.50) δNk,ω    

a a + dk dω k · gak,ω δφk,ω δNk−k . ,ω−ω − δφk,ω δNk−k,ω−ω

For the sake of notational simplicity, let us introduce a convention, q ≡ (k,ω).

(4.51)

We also drop “δ” in front of the field perturbation, δφ, and perturbed Klimontovich function, δN. Following the same steps as in arriving at (1.49), the iterative solution can be obtained:   

(k1 · gaq ) φq1 Nqa02 − φq1 Nqa02 Nqa = Nqa0 + q1 +q2 =q

+



q1 +q2 +q3 =q

(k1 · gaq ) (k2 · gaq−q1 )

   

× φq1 φq2 Nqa03 − φq1 φq2 Nqa03 − φq1 φq2 Nqa03   

(k1 · gaq ) (k2 · gaq2 ) fa φq1 φq2 − φq1 φq2 + (k · gaq ) fa φq + +

 q1 +q2 +q3 =q

q1 +q2 =q

(k1 ·

gaq ) (k2

· gaq−q1 ) (k3 · gaq3 ) fa

   

× φq1 φq2 φq3 − φq1 φq2 φq3 − φq1 φq2 φq3 .

(4.52)

In the present scheme, we take the view that the effects arising from the particle discreteness are supposed to provide the next-order correction to what is essentially Vlasovian nonlinear theory. This is equivalent to the assumption that the plasma

88

Electrostatic Klimontovich Weak Turbulence Theory

parameter g ∝ 1/(nλ3De ) is small. This allows us to ignore nonlinear responses that arise from Nqa0 , and leads to the simplification, Nqa = Nqa0 + (k · gaq ) fa φq  

 + (k1 · gaq ) (k2 · gaq2 ) fa φq1 φq2 − φq1 φq2 q1 +q2 =q



+

q1 +q2 +q3 =q

(k1 · gaq ) (k2 · gaq−q1 ) (k3 · gaq3 ) fa

   

× φq1 φq2 φq3 − φq1 φq2 φq3 − φq1 φq2 φq3 .

(4.53)

Fully symmetric version of this iterative solution is  

 αa(2) (q1 |q2 ) fa φq1 φq2 − φq1 φq2 Nqa = Nqa0 + αa (q) fa φq + q1 +q2 =q



+

αa(3) (q1 |q2 |q3 ) fa

(4.54)

q1+q2 +q3 =q

   

× φq1 φq2 φq3 − φq1 φq2 φq3 − φq1 φq2 φq3 , where αa (q), αa(2) (q1 |q2 ), and αa(3) (q1 |q2 |q3 ) are quantities that we have already encountered in (1.53). Combining the solution (4.54) and perturbed Poisson’s equation in (4.49), we obtain   

i k k  |k − k | χ (2) (q  |q − q  ) φq  φq−q  − φq  φq−q  k 2 (q) φq − q



 q



k k  k  |k − k − k | χ¯ (3) (q  |q  |q − q  − q  ) φq  φq  φq−q  −q 

q 

  

− φq  φq  φq−q  −q  − φq  φq  φq−q  −q    4πea dv Nqa0 (v), =

(4.55)

a

where linear and nonlinear response functions are defined exactly as in (1.55)– (1.63). Let us multiply φ−q to (4.55) and take the ensemble average. Then, we have      k 2 (q) φ 2 q − i k k  |k − k | χ (2) (q  |q − q  ) φq  φq−q  φ−q −2 =

 a

 q

q

    k 2 k 2 χ¯ (3) (q  | − q  |q) φ 2 q  φ 2 q

4πea



  dv φ−q Nqa0 (v) .

(4.56)

4.2 Formal Klimontovich Weak Turbulence Theory

89

a0 Next, we multiply N−q (v) to (4.55) and take the average. We subsequently replace q by −q:      k k ,|k − k | χ (2)∗ (q  |q − q  ) φ−q  φ−q+q  Nqa0 (v) k 2 (−q) φ−q Nqa0 (v) = − i q

+2

 q

+



    k 2 k 2 χ¯ (3)∗ (q  | − q  |q) φ 2 q  φ−q Nqa0 (v) 

4πeb

 b0  a0  dv N−q (v ) Nq (v) .

(4.57)

b

Making use of (4.47), we have 

 φ−q Nqa0 (v) =

4πea δ(ω − k · v) fa (v) (2π)3 k 2 ∗ (q)     2  2 (3)∗  k χ¯ (q | − q  |q) φ 2 q  φ−q Nqa0 (v) + ∗ (q) 

(4.58)

q





i k ∗ (q)

  k  |k − k | χ (2)∗ (q  |q − q  ) φ−q  φ−q+q  Nqa0 (v) .

q

  Since the quantity φ−q Nqa0 (v) appears on both sides of (4.58) we iteratively solve for this quantity, making use of the lowest-order solution first:  φ−q Nqa0 (v) ≈



4πea δ(ω − k · v) fa (v). (2π)3 k 2 ∗ (q)

(4.59)

This procedure leads to   φ−q Nqa0 (v) =

4πea (4.60) (2π)3 k 2 ∗ (q) ⎞ ⎛  2 χ¯ (3)∗ (q  | − q  |q)   2 2 ⎠ φ δ(ω − k · v) fa (v) × ⎝1 + k q ∗ (q)  q



i k ∗ (q)



  k  |k − k | χ (2)∗ (q  |q − q  ) φ−q  φ−q+q  Nqa0 (v) .

q

Inserting (4.60) to the right-hand side of (4.56), we arrive at      (q) E 2 q − i dq  χ (2) (q  |q − q  ) k k  |k − k | φq  φq−q  φ−q   (4πea )2  (3)   2 2 dv − 2 dq χ¯ (q | − q |q) E q  E q = (2π)3 k 2 ∗ (q) a

90

Electrostatic Klimontovich Weak Turbulence Theory

 2 χ¯ (3)∗ (q  | − q  |q) 2  × 1 + dq (4.61) E q δ(ω − k · v) fa (v) ∗ (q)     (4πea ) χ (2)∗ (q  |q − q  )  dq  dv −i k |k − k | φ−q  φ−q+q  Nqa0 (v) , ∗ k (q) a 





where  2  we have  resorted back to the electric field representation by making use of 2 2 E q = k φ q . In (4.61), we note that the first term on the right-hand side of the equality has an overall factor δ(ω − k · v). This is the linear wave-particle resonance delta function condition. This means that the nonlinear correction associated with this factor can be ignored, as it represents a small correction.    To evaluate the third-order cumulants, φq  φq−q  φ−q and φ−q  φ−q+q  Nqa0 (v) , we make use of (4.55) without the third-order nonlinearity: φq 1 =

  

1 i k1 k  |k1 − k | χ (2) (q  |q1 − q  ) φq  φq1 −q  − φq  φq1 −q  (q1 ) q    1 4πea dv Nqa01 (v). (4.62) + 2 k1 (q1 ) a k12

Following the same steps as in arriving at (1.78), the quantity φq  φq−q  φ−q  can be constructed by successively making use of the expression (4.62) iteratively for each of φq  , φq−q  and φ−q :  (2)  χ (q| − q + q  ) 2 φq  φq−q  φ−q = 2i k k  |k − k | φ q−q  φ 2 q k 2 (q  )



+

χ (2) (q| − q  ) φ 2 q  φ 2 q |k − k |2 (q − q  )

 χ (2)∗ (q  |q − q  ) 2 2 − φ q  φ q−q  k 2 ∗ (q)     φq−q  φ−q Nqa0 (v) + 4πea dv k 2 (q  ) a +

a0 φq  φ−q Nq−q  (v)

|k − k |2 (q − q  )

+

a0 (v) φq  φq−q  N−q

k 2 ∗ (q)

 ,

(4.63)

where we made use of the property χ (2) (−q1 | − q2 ) = −χ (2)∗ (q1 |q2 ) and decomposed the four-body cumulants as products of two-body cumulants, ignoring irreducible components.

4.2 Formal Klimontovich Weak Turbulence Theory

91

Inserting (4.63) to (4.61), we have   (2)  χ (q| − q + q  ) 2 2  (2)   E q−q  (q) E q + 2 dq χ (q |q − q ) (q  )   χ (2) (q| − q  ) 2 (3)   2 + E q  − χ¯ (q | − q |q) E q  E 2 q (q − q  )  |χ (2) (q  |q − q  )|2 2 − 2 dq  E q  E 2 q−q  ∗ (q)   (4πea )2 dv δ(ω − k · v) fa (v) = 3 k 2 ∗ (q) (2π) a %      φq−q  φ−q Nqa0 (v)    (2)   dv k k |k − k | χ (q |q − q ) 4πea dq +i k 2 (q  ) a  a0 a0 φq  φ−q Nq−q (v) φq  φq−q  N−q  (v) + +  2  2 ∗ |k − k | (q − q ) k (q)  a0  φ−q+q  N (v) φ −q q . (4.64) − χ (2)∗ (q  |q − q  ) k 2 ∗ (q) This shows that we need to evaluate the remaining third-order cumulants,  φq−q  φ−q a0 a0 a0 Nqa0 (v), φq  φ−q Nq−q  (v), φq  φq−q  N−q (v), and φ−q  φ−q+q  Nq (v). These a0 quantities are special cases of a generic form  φq1 φ−q1 +q2 N−q (v). Making use of 2 (4.62) in order to evaluate φq1 and φ−q1 +q2 successively, we have a0 φq1 φ−q1 +q2 N−q (v) = 2

 1 i k1 k  |k1 − k | χ (2) (q  |q1 − q  ) k12 (q1 ) q  a0 (v) × φq  φq1 −q  φ−q1 +q2 N−q 2

a0 − φq  φq1 −q   φ−q1 +q2 N−q (v) 2  i |k1 − k2 | k  |k1 − k2 + k |χ (2) (q  |− q1 + q2 − q  ) + |k1 − k2 |2 (−q1 + q2 ) q a0 × φq  φ−q1 +q2 −q  φq1 N−q (v) 2

a0 − φq  φ−q1 +q2 −q   φq1 N−q2 (v)   1 a0 4πeb dv Nqb01 (v ) φ−q1 +q2 N−q (v) + 2 2 k1 (q1 ) b   1 + dv 4πe (4.65) b |k1 − k2 |2 (−q1 + q2 ) b b0 a0 × N−q (v ) φq1 N−q (v). 1 +q2 2

92

Electrostatic Klimontovich Weak Turbulence Theory

Of the various terms in (4.65), those that contain three-body cumulants of the form φ N N are all of purely particle effects, with no wave intensity appearing in the expressions. These terms can be ignored since the discrete particle effects are assumed to be a correction to what is essentially a Vlasovian weak turbulence formalism. As a result, we now have a simplification: 2i k2 χ (2) (q2 |q1 − q2 ) 2 a0 (v) E q1 −q2 φq2 N−q 2 k1 |k1 − k2 | (q1 ) 2i k2 χ (2) (−q1 |q2 ) a0 + (v) E 2 q1 φq2 N−q 2 k1 |k1 − k2 | (−q1 + q2 )  (2) χ (q2 |q1 − q2 ) 2 8πea i = E q1 −q2 3 (2π) k1 k2 |k1 − k2 | (q2 ) (q1 )  χ (2) (−q1 |q2 ) 2 + (4.66) E q1 δ(ω2 − k2 · v) fa (v). (−q1 + q2 )

a0 (v) =  φq1 φ−q1 +q2 N−q 2

The remaining task is to identify (q1,q2 ) = (q − q , − q  ), (q1,q2 ) = (q , − q + q  ), (q1,q2 ) = (q ,q), and (q1,q2 ) = (−q , − q) successively. Actual evaluations of the a0 (v) with the corresponding identifications of (q1,q2 ) as quantity  φq1 φ−q1 +q2 N−q 2 specified in (4.66) are omitted, and the intermediate steps can easily be filled by the readers. Inserting the final results to (4.64), we have the desired nonlinear spectral balance equation including the effects of spontaneous fluctuations:  | χ (2) (q  |q − q  ) |2 2 2 E q  E 2 q−q  (q) E q − 2 dq  ∗ (q)    2  E 2 q   (2)   2 E q−q  + 2 dq { χ (q |q − q ) } + (q  ) (q − q  )  − χ¯ (3) (q  | − q  |q) E 2 q  E 2 q  (4πea )2 dv δ(ω − k · v) fa (v) = (2π)3 k 2 ∗ (q) a  (2)   2 (4πea )2 { χ (q |q − q  ) }2 2  dq (4.67) − E q (2π)3 k 2 | (q  )|2 (q − q  ) a  | χ (2) (q  |q − q  ) |2 2  dv δ(ω − k · v) fa (v)  − E q−q ∗ (q)  (2)   2 (4πea )2 { χ (q |q − q  ) }2 2  dq − E q (2π)3 |k − k |2 | (q − q  )|2 (q  ) a  | χ (2) (q  |q − q  ) |2 2 dv δ[ω − ω − (k − k ) · v] fa (v). − E q  ∗ (q) 

4.2 Formal Klimontovich Weak Turbulence Theory

93

In (4.67) we have ignored the nonlinear correction terms on the right-hand side of the  equality, which are associated with the linear wave-particle resonance factor dv δ(ω − k · v)fa (v), since they are but small corrections. At this point, we reintroduce the slow-time derivative, as in (1.83). This leads to the formal nonlinear spectral balance equation that generalizes (1.89): i ∂ Re (k,ω) ∂δE 2 k,ω + Re (k,ω) δE 2 k,ω + i Im (k,ω) δE 2 k,ω 2 ∂ω ∂t     2   (2)     2 δE k−k,ω−ω dω { χ (k ,ω |k − k ,ω − ω ) } + 2 dk (k,ω )   δE 2 k,ω (3)     2 − χ¯ (k ,ω | − k , − ω |k,ω) δE k,ω δE 2 k,ω + (k − k,ω − ω )   | χ (2) (k,ω |k − k,ω − ω ) |2  dω − 2 dk δE 2 k,ω δE 2 k−k,ω−ω ∗ (k,ω)   1 2 = e2 dv δ(ω − k · v) fa (v) π k 2 ∗ (k,ω) a a  (2)     1 { χ (k ,ω |k − k,ω − ω ) }2 4   dk dω 2 − δE 2 k,ω π k | (k,ω )|2 (k − k,ω − ω )  | χ (2) (k,ω |k − k,ω − ω ) |2 2 − δE k−k,ω−ω ∗ (k,ω)   ea2 dv δ(ω − k · v) fa (v) × a

  1 4 dk dω −  2 π |k − k | | (k − k,ω − ω )|2  (2)   { χ (k ,ω |k − k,ω − ω ) }2 × δE 2 k,ω (k,ω )  | χ (2) (k,ω |k − k,ω − ω ) |2 2 − δE k,ω ∗ (k,ω)   × ea2 dv δ[ω − ω − (k − k ) · v] fa (v),

(4.68)

a

where we have resorted back to the longhand notation. If we ignore everything to the right of equality then we recover (1.89). We next discuss the particle kinetic equation. From (4.49), the particle kinetic equation is given by   ∂ iea ∂fa dk dω k · φ−q Nqa . (4.69) =− ∂t ma ∂v

94

Electrostatic Klimontovich Weak Turbulence Theory

To this, we insert the expression for Nqa as given in (4.54). The quantity of interest is φ−q Nqa  = φ−q Nqa0  + αa (q) fa φ 2 q + +2





αa(2) (q  |q − q  ) fa φ−q φq  φq−q  

q

αa(3) (q  | − q  |q) fa φ 2 q φ 2 q  .

(4.70)

q

Making use of (4.60), (4.63), and (4.66), we obtain 4πea φ−q Nqa  = αa (q) φ 2 q fa + (2π)3 k 2 ∗ (q) ⎞ ⎛  2 χ¯ (3)∗ (q  | − q  |q) k 2 φ 2 q  ⎠ δ(ω − k · v) fa × ⎝1 + ∗ (q)  q  (2)  χ (q| − q + q  ) 2 (2)     + 2i αa (q |q − q ) k k |k− k | φ q−q  φ 2 q 2 (q  ) k q  χ (2) (q|− q  ) χ (2)∗ (q  |q − q  ) 2 2 2 2    + φ q φ q − φ q φ q−q fa |k− k |2 (q − q  ) k 2 ∗ (q)      φq−q  φ−q Nqb0 (v) (2)   + αa (q |q − q ) 4πeb dv k 2 (q  ) b q  b0 b0 φq  φ−q Nq−q (v) φq  φq−q  N−q  (v) + fa + |k − k |2 (q − q  ) k 2 ∗ (q)  +2 αa(3) (q  | − q  |q) fa φ 2 q φ 2 q  . (4.71) q

Among the terms in (4.71), those terms that contain the factors of the form φ φ N will be ignored, since these factors are proportional to fa , leading to an overall contribution of the form fa2 . These are higher-order terms associated with the single-particle effects, which we ignore. Thus, we simplify (4.71) as φ−q Nqa  =

1 4πea αa (q) E 2 q fa + 2 k (2π)3 k 2 ∗ (q) ⎞ ⎛  2 χ¯ (3)∗ (q  | − q  |q) E 2 q  ⎠ δ(ω − k · v) fa × ⎝1 + ∗ (q)  q (2)   α (q |q − q  )  χ (2) (q  |q − q  ) a + 2i E 2 q−q  E 2 q  |k − k | ) k k (q  q

4.2 Formal Klimontovich Weak Turbulence Theory



95

χ (2) (q  |q − q  ) 2 χ (2)∗ (q  |q − q  ) 2 2  E q −  E E q  E 2 q−q  fa q (q − q  ) ∗ (q)  α (3) (q  | − q  |q) a +2 E 2 q E 2 q  fa . (4.72) 2 k 2 k  +

q

Making explicit definitions for αa(2) (q  |q − q  ) and αa(3) (q  | − q  |q) as defined in (1.53), we have ea E 2 q k ∂fa · ma ω − k · v k 2 ∂v ⎞ ⎛  4πea ⎝1+ 2 χ¯ (3)∗ (q  |−q  |q) E 2 q ⎠δ(ω− k·v)fa + 3 2 ∗ (2π) k (q) ∗ (q)  q  (2)  2  χ (q |q − q  ) 2 1 ie + 2a E q−q  E 2 q ma  k k  |k − k | (q  ) q  (2)  χ (q |q − q  ) 2 χ (2)∗ (q  |q − q  ) 2 2 2    +  E  −  E  E E q q q q−q (q − q  ) ∗ (q)  1 ∂ ∂ 1 k · × (k − k ) ·   ω−k·v ∂v ω − ω − (k − k ) · v ∂v  1 ∂ ∂ fa + (k − k ) · k · ∂v ω − k · v ∂v 1 e3  1 E 2 q E 2 q   ∂ + a3 k · 2 2  ma  k k ω−k·v ∂v ω − ω − (k − k ) · v q   1 1 ∂ ∂ ∂ ∂   fa . (4.73) × k · k· −k· k · ∂v ω − k · v ∂v ∂v ω − k · v ∂v

φ−q Nqa  = −

Inserting this result to (4.69), we obtain   ki kj δE 2 k,ω ∂fa ∂fa ie2 ∂ dk dω 2 = 2a ∂t ma ∂vi k ω − k · v ∂vj   2 2 ki i (4πea ) ∂ 1+ ∗ dk dω 2 ∗ − 3 (2π) ma ∂vi k (k,ω) (k,ω)     (3)∗     2 × dk dω χ¯ (k ,ω | − k , − ω |k,ω) δE k,ω δ(ω − k · v) fa    k k  (k − k )k (2)   ea3 ∂   i j χ (k ,ω |k − k,ω − ω ) dk dω dk dω + 3 ma ∂vi k k  |k − k |

96

Electrostatic Klimontovich Weak Turbulence Theory

 δE 2 k−k,ω−ω δE 2 k,ω δE 2 k,ω + (k,ω ) (k − k,ω − ω )

 ×

(4.74)

 δE 2 k,ω δE 2 k−k,ω−ω − χ (k ,ω |k − k ,ω − ω ) ∗ (k,ω)   ∂ ∂ ∂ 1 1 1 ∂ × fa + ω − k · v ∂vj ω − ω − (k − k ) · v ∂vk ∂vk ω − k · v ∂vj     ki kj kk kl δE 2 k,ω δE 2 k,ω ∂ ie4 ∂ dk dω dk dω − 4a ma ∂vi k 2 k 2 ω−k·v ∂vj   ∂ ∂ ∂ 1 1 1 ∂ fa . × −     ω − ω − (k − k ) · v ∂vk ω − k · v ∂vl ∂vl ω − k · v ∂vk (2)∗









Taking only the real part of the right-hand side, we obtain the following:     k ∂ ∂fa πe2 δ(ω − k · v) dk dω = 2a · ∂t ma k ∂v %   k ma (k,ω) ∂f a × Im 3 fa + δE 2 k,ω · 2π k | (k,ω)|2 k ∂v e2 + a2 ma



 dk

 dω





dk







 1 k ∂ Im 2 · ∗ k ∂v π k [ (k,ω)]2

× χ¯ (3)∗ (k,ω | − k, − ω |k,ω) δE 2 k,ω δ(ω − k · v) fa

+ Re

ea3 ∂ m3a ∂vi



 dk

 dω

dk



dω

ki kj (k − k )k M(k,ω|k,ω ) k k  |k − k |

ω−k·v   ∂ ∂ ∂ 1 1 ∂ × fa + ∂vj ω − ω − (k − k ) · v ∂vk ∂vk ω − k · v ∂vj     ki kj kk kl δE 2 k,ω δE 2 k,ω ∂ ea4 ∂  dk dω dk dω + Im 4 ma ∂vi k 2 k 2 ω−k·v ∂vj   ∂ ∂ ∂ 1 1 1 ∂ fa , × −     ω − ω − (k − k ) · v ∂vk ω − k · v ∂vl ∂vl ω − k · v ∂vk (4.75) where we have indicated the new terms that arise from adding the discrete particle effects by enclosing them within boxes, and M(k,ω|k,ω ) is defined in (1.106).

4.3 Weak Turbulence Theory with Discrete Particle Effects

97

4.3 Weak Turbulence Theory with Discrete Particle Effects Let us now make use of the formal results (4.68) and (4.75) and derive specific forms of equations of weak turbulence theory in which linear eigenmodes interact among themselves and with the particles, and where discrete particle effects are incorporated. Let us first dispense with the particle kinetic equation (4.75), since it is relatively straightforward. 4.3.1 Particle Kinetic Equation with Discrete Particle Effects Consider the electron equation first. Recall that Chapter 3 was devoted to the analysis of particle kinetic equation based upon (1.107). In the end, it was shown that the modified kinetic equation for electrons is given by (3.30). In the present Klimontovich formalism the formal particle kinetic equation is generalized to (4.75). It suffices to consider the new, boxed terms in order to address the additional effects that arise from the particle discreteness. Thus, the new terms in the electron kinetic equation are first for the linear wave-particle resonance,      k ∂ e2 ∂fe lin. dk 2 · = σ ωL δ(σ ωkL − k · v)fe, (4.76) ∂t new 4πme k ∂v σ =±1 k while for the nonlinear term,        ∂fe nonlin. 1 e2 ∂   k Im dk dω dk dω = ·  ∂t new m2e k ∂v π 2 k[ ∗ (k,ω)]2   × χ¯ (3)∗ (k,ω | − k, − ω |k,ω) δE 2 k,ω δ(ω − k · v)fe      1 e2  ∂  k Im 2 ∗ dk dω dk = 2 · me  k ∂v π k[ (k,ω)]2 σ =±1



× χ¯ (3)∗ (k,σ  ωkL | − k, − σ  ωkL |k,ω)Ikσ L δ(ω − k · v)fe .

(4.77)

If we make use of χ¯ (3)∗ (k,σ  ωkL | − k, − σ  ωkL |k,ω) 1 e2 |k − k |2 (k · k )2 =− χe (k − k,ω − σ  ωkL ), 2 m2e ω3 (σ  ωkL ) k 2 k  2

(4.78)

and assume that ω is in the vicinity of Langmuir wave dispersion relation, then we may approximate 

χe (k − k ,ω − σ



ωkL )



2 2ωpe

|k − k |2 vT2 e

.

(4.79)

98

Electrostatic Klimontovich Weak Turbulence Theory

This leads to χ¯ (3)∗ (k,σ  ωkL | − k, − σ  ωkL |k,ω) = −

2 (k · k )2 1 1 e2 ωpe , 2 me Te ω3 (σ  ωkL ) k 2 k  2

(4.80)

resulting in      2   ∂fe nonlin. 1 e4 ωpe ∂  k Im dk dω dk = − ·  ∂t new m3e 2π 2 Te  k ∂v k[ ∗ (k,ω)]2 σ =±1

1 (k · k )2 σ  L × 3  L I  δ(ω − k · v)fe . ω (σ ωk ) k 2 k  2 k

(4.81)

Writing  iπ δ(ω − σ ωL ) 1 1 k = P , [ ∗ (k,ω)]2 ∗ (k,ω) σ =±1  (k,σ ωkL )

(4.82)

we obtain the nonlinear correction term to the electron kinetic equation that stems from the discrete particle effects,     2   1 ∂fe nonlin. e4 ωpe ∂  k P ∗ dk dk =− 3 ·  ∂t new me 4πTe  k ∂v (k,σ ωkL ) σ,σ =±1 ×

1 (k · k )2 σ  L I  δ(σ ωkL − k · v)Fe . (σ ωkL )2 (σ  ωkL ) k 3 k  2 k

(4.83)

Turning to the ion equation, the new terms are      ∂fi lin. k ∂ e2 dk 2 · = σ μk ωkL δ(σ ωkS − k · v)fi ,  ∂t new 4πmi k ∂v σ =±1 nonlin.     1 ∂fi  e2 1  σ μk ωkL ∂  k P dk dk = · ∂t new k ∂v m2i 2π σ,σ  =±1 ∗ (k,σ ωkS ) k 

× χ¯ (3)∗ (k,σ  ωkS | − k, − σ  ωkS |k,σ ωkS ) Ikσ S δ(σ ωkS − k · v)fi . (4.84) Since ωkS > kvT i , ωkS kvT e , ωkS > k  vT i , ωkS k  vT e , and ωkS − ωkS |k − k |vTj , j = i,e, we may approximate χ¯ (3)∗ (k,σ  ωkS | − k, − σ  ωkS |k,σ ωkS ) ≈ χ¯ i(3)∗ (k,σ  ωkS | − k, − σ  ωkS |k,σ ωkS ) 2 1 e2 ωpi 1 (k · k )2 =− . (4.85) 2 mi Ti (σ ωkS )3 σ  ωkS k 2 k  2 Making use of the this result, we may simplify the nonlinear term in (4.84).

4.3 Weak Turbulence Theory with Discrete Particle Effects

99

In summary, modified particle kinetic equations, which include the particle discreteness effects, are given by   ∂fa ∂fa ∂ , (a = e,i), = · Aa fa + Da · ∂t ∂v ∂v  k  e2 dk 2 Ae = σ ωkL δ(σ ωkL − k · v) 4πme k σ =±1    2 σ L  2  I e2 ωpe (k · k ) 1  k × 1− 2 P ∗ dk 2  2 , L 3  L me T e (k,σ ωkL ) (σ ω ) k k k (σ ωk ) σ  =±1  k  e2 dk 2 Ai = σ μk ωkL δ(σ ωkS − k · v) 4πmi k σ =±1     2  2  Ikσ S e2 ωpi 1  (k · k ) , × 1− 2 P ∗ dk mi T i (k,σ ωkS ) k 2 k  2 σ  =±1 (σ ωkS )3 (σ  ωkS )  kk  πe2 De = 2 dk 2 δ(σ ωkL − k · v) Ikσ L me k σ =±1    2  (k − k ) (k − k ) I σ  L I σ L πe4 k  (k · k ) k + 4 dk dk L 2 2 2  2me (σ ω − k · v) k k  k σ,σ =±1    1 σσ δ[σ ωkL − σ  ωkL − (k − k ) · v], × − L 2 ωpe (σ ωk − k · v)2  kk  πe2 dk 2 δ(σ ωkS − k · v) Ikσ S Di = 2 k σ =±1 mi     2  (k − k ) (k − k ) Ikσ S Ikσ S πe4  (k · k ) dk dk + k 2 k 2 2m4i (σ ωkS − k · v)2 σ,σ  =±1   2(σ σ  ) Aσ,σ  1 k,k × + δ[σ ωkS − σ  ωkS − (k − k ) · v], k k  cS2 (σ ωkS − k · v)2 

Aσ,σ k,k =

1 k 2 + k 2 − k · k − σ σ  k k  . 2  2   2  |k − k | (k − σ σ k ) λDe (k − k ,σ ωkS − σ  ωkS )

(4.86)

This form of particle kinetic equation is given in a balanced form where the velocity space diffusion term, dictated by D, appears with the velocity space friction, or drag, term, governed by A. In steady state these two terms balance each other out. Such a form is reminiscent of the Fokker–Planck equation. The velocity friction term is the result of single (or discrete) particle effects. As it will be shown eventually, this term has an overall inverse proportionality to the number of plasma particles

100

Electrostatic Klimontovich Weak Turbulence Theory

within a Debye sphere. If such a number is infinite, then this term disappears, and we recover the particle kinetic equation of the Vlasov weak turbulence theory. 4.3.2 Wave Kinetic Equation with Discrete Particle Effects As we have already discussed, taking the real part of nonlinear spectral balance equation (4.68) leads to the dispersion relation, while taking the imaginary part leads to the wave kinetic equation for specific eigenmodes. Let us approximate the dispersion relation by restricting the solution to linear eigenmodes, Re (k,ω) = 0 (see [1.93]), so that the dispersion relation for each mode is given by ω = ωkα $ $ (α = L,S). Then we express the wave intensity by δE 2 k,ω = σ =±1 α Ikσ α δ(ω − σ ωkα ), as in (1.95). Inserting this to the imaginary part of (4.68), we have ∂Ikσ α 2 Im (k,σ ωkα ) σ α =−  Ik ∂t (k,σ ωkα ) +

 a

4ea2 k 2 [  (k,σ ωkα )]2

 dv δ(σ ωkα − k · v) fa (v)

   4 dk −  (k,σ ωkα )  σ =±1 β=L,S  β β 2 { χ (2) (k,σ  ωk |k − k,σ ωkα − σ  ωk ) }2 × Im P β (k − k,σ ωkα − σ  ωk )  β β σ β − χ¯ (3) (k,σ  ωk | − k, − σ  ωk |k,σ ωkα ) Ik Ikσ α α (2)   β   β 2    16ea2  |χ (k ,σ ωk |k − k ,σ ωk − σ ωk ) | dk β  (k,σ ωkα )  |k − k |2 | (k − k,σ ωkα − σ  ωk )|2 a σ =±1 β=L,S   σ β I k Ikσ α β dv δ[σ ωkα − σ  ωk − (k − k ) · v] fa (v) × −  α β   (k,σ ω ) (k ,σ ωk ) k





   4π γ β dk | χ (2) (k,σ  ωk |k − k,σ  ωk−k ) |2  (k,σ ωkα )   σ ,σ =±1 β,γ =L,S   σ  γ σ  β σ  γ σ β σα Ik Ik−k Ik−k Ik Ik Ikσ α − × + γ α β  (k,σ  ωk )  (k − k,σ  ωk−k )  (k,σ ωk ) −

× δ(σ ωkα − σ  ωk − σ  ωk−k ), β

γ

(4.87)

4.3 Weak Turbulence Theory with Discrete Particle Effects

101

which is a direct generalization of (1.102). The first term on the right-hand side, the induced emission, is present in (1.102) but the new term is the second term, which represents the spontaneous emission process (encased in a box). The third term, which represents the induced scattering, is also present in (1.102), but the fourth term, also encased in a box, which is new, represents the spontaneous scattering process. The final term, which depicts the decay processes, is already present in (1.102). Induced emission terms were already discussed in Section 2.1. We write the velocity space distribution function with the ambient density explicitly taken out of the definition fa (v) = na Fa (v) = nFa (v) (see [1.34]). Induced emissions of L and S modes are described, respectively, with (2.6) and (2.7). Spontaneous emission terms contributing to L and S modes are given by    4ea2 ∂Ikσ L  dv δ(σ ωkL − k · v) fa (v) = 2 [  (k,σ ωL )]2 ∂t spont.emiss. k k a 2 2  ne ωpe dv δ(σ ωkL − k · v) Fe, = k2    ∂Ikσ S  4ea2 dv δ(σ ωkS − k · v) fa (v) = 2 [  (k,σ ωS )]2 ∂t spont.emiss. k k a 2 2 2  μk ne ωpe dv δ(σ ωkS − k · v) (Fe + Fi ). (4.88) = k2 The spontaneous emission takes place when randomly moving charged particles emit plasma waves. Such an emission process does not depend on the pre-existing level of waves, which contrasts to the induced emission. Recall that the induced emission is a process that involves charged particles moving in the wave field with their velocities in resonance with the wave phase velocities, and exchanging energy. For such a case, the particles may give off energy to the waves or gain energy from the waves depending on whether more particles move slightly faster or slower than the wave phase velocities; hence, the process is dependent on the derivative of the distribution function. In contrast, the spontaneous emission only requires that the charged particles have finite random velocities, or that the particles have a finite temperature. This means that the spontaneous emission is proportional to the particle distribution function rather than its derivative. And this process does not require pre-existing level of waves, so that the spontaneous emission formula is not proportional to the wave intensity. The spontaneously emitted waves will have phase velocities that match the particle velocities; hence, the process is governed by the linear wave-particle resonance condition. The spontaneous emission is also known as the Cerenkov emission.

102

Electrostatic Klimontovich Weak Turbulence Theory

Decay processes have already been discussed in Section 2.2, and induced scattering processes have been dealt with in Section 2.3. Spontaneous scattering terms are given by   16e2    ∂Ikσ α  a dk =−  (k,σ ωα ) ∂t spont.scatt. k  a β=L,S σ =±1

| χ (2) (k,σ  ωk |k − k,σ ωkα − σ  ωk ) |2 β

×

β

β

|k − k |2 | (k − k,σ ωkα − σ  ωk )|2   σ β I k Ikσ α × − α β (k,σ  ωk )  (k,σ ωk )  β × dv δ[σ ωkα − σ  ωk − (k − k ) · v] fa (v).

(4.89)

The spontaneous scattering is similar to the spontaneous emission in the sense that the random particle motion leads to the emission of plasma waves, but, in this case, instead of giving off a single wave with the wave phase velocity matching the particle velocity, the scattering process involves two waves with the difference in the wave phase velocities equaling the particle random velocity. Conceptually, the process of spontaneous scattering can be viewed as being triggered by an incoming wave carrying wave momentum k (or k ), which impinges on thermal distribution of particles. The particles spontaneously give off another wave with k (or k), but the particle velocity equals the net difference in the incoming wave phase velocity and the spontaneously emitted wave phase velocity. Consequently, this process is proportional to the incoming wave intensity and the particle distribution function, and the interaction is dictated by nonlinear wave-particle resonance condition. For Langmuir waves, we retain the interaction involving only another Langmuir mode (β = L). This is consistent with the discussion of induced scattering. Following the same steps as before we obtain   4   (k · k )2 ∂Ikσ L  L ne 4  dk dv 2 2 = −(σ ωk ) 2 λDe  ∂t spont.scatt. Te k k 

σ =±1



× σ  ωkL Ikσ L − σ ωkL Ikσ L



× δ[σ ωkL − σ  ωkL − (k − k ) · v] (Fe + Fi ).

(4.90)

For ion-sound modes, we also follow the earlier discussion and retain the interaction with another ion-sound mode only. Again, such a treatment retains the consistency with the induced scattering process. Following the same steps as taken before we obtain

4.4 Summary of Klimontovich Weak Turbulence Theory

103

  4 ∂Ikσ S  1  L ne dk = −μk (σ ωk ) 2 4 ∂t spont.scatt. Te λDe  

σ =±1

|k − k |2 (k · k )2 1 + × 4 4 k k |k − k |4 (σ k − σ  k  )2

2



μk σ  ωkL Ikσ S − μk σ ωkL Ikσ S × (4.91) | (k − k,σ ωkS − σ  ωkS )|2  × dv δ[σ ωkS − σ  ωkS − (k − k ) · v] (Fe + Fi ).

4.4 Summary of Klimontovich Weak Turbulence Theory • Langmuir and Ion-Sound Wave Kinetic Equations In the summary of equations we express the ion-sound wave intensity with the factor μk – see (3.29) for definition – explicitly absorbed into the definition: Ikσ S →

Ikσ S . μk

(4.92)

For Langmuir (L) mode, the wave kinetic equation is given by  ne2 ∂Fe L σL dv − k · v) F e + σ ωk Ik k · ∂v) '( &π'( ) & spontaneous & induced emissions 

 LS L σ  L σ  S +2 σ ωkL dk Vk,k I   σ ω k Ik  & '( k−k)  

2 πωpe ∂Ikσ L = ∂t k2





δ(σ ωkL

σ ,σ =±1

spontaneous decay  L σ L σ L Ik − σ  ωkL Ik−k Ikσ L − σ  ωk−k  Ik & '( ) induced decay (4.93)     ∂Fi LL me σ  L σ L dk dv Uk,k + σ ωkL Ik Ik (k − k ) ·  m ∂v σ  =±1 & i '( ) induced scattering (off thermal ions)  ne2 L σ  L  L σL σ ωk Ik − σ ωk Ik (Fe + Fi ) . + 2 πωpe '( ) & spontaneous scattering σ  S

104

Electrostatic Klimontovich Weak Turbulence Theory

For ion-sound (S) mode, the same is given by  2 2  πμk ωpe ∂Ikσ S ne S dv δ(σ ωk − k · v) = (Fe + Fi ) 2 ∂t k ) &π '( spontaneous emission    ∂ m e L σS Fe + Fi + σ ωk Ik k · ∂v mi '( ) & induced emission 

  L SL σ  L dk Vk,k + σ ωk σ ωkL Ikσ L Ik−k   & '( )   σ ,σ =±1

−σ &



σ  L σ S ωkL Ik−k  Ik



spontaneous decay 

L σ L σ S ωk−k Ik  Ik 

(4.94) −σ '( ) induced decay     ∂Fi L  SS σ S σ S  dk dv Uk,k Ik (k − k ) · + μk μk σ ωk  Ik  ∂v) & '( σ  =±1 induced scattering 

 2 ne mi SS L σ S  L σS + Yk,k σ ωk Ik − σ ωk Ik (Fe + Fi ) , 2 m πωpe e '( ) & spontaneous scattering where the various coefficients are defined in (3.29), except that an additional object is defined by   (k · k )2 πe2 SS  k Wk,k − σ σ Yk,k = 2 k 4 k 4 λ4 me mi ωpe k De × δ[σ ωkS − σ  ωkS − (k − k ) · v].

(4.95)

The wave equations (4.93) and (4.94) now feature various spontaneous and induced processes appearing in balanced forms. • Electron and Ion Particle Kinetic Equations For the summary of particle kinetic equation see (4.86).

5 Spontaneous Emission and Collisional Kinetic Equation

One of the most important effects that arise from the particle discreteness is the spontaneous emission of electromagnetic fluctuations in thermal plasmas. We have already discussed the process of spontaneous emission of plasma waves within the context of wave kinetic equations. However, spontaneous emission of fluctuations, which we will discuss in depth, is a more general concept. Fluctuations are not necessarily the same as waves. Waves are defined by their dispersion relations, which relate the wave frequency ω with the wave vector k. Fluctuations, on the other hand, do not necessarily satisfy the dispersion relation. It is still possible to define the frequency and wave vector associated with the fluctuations, but they do not have to be related to each other, but, instead, they can remain independent of each other. Those ω’s and k’s that are mathematically related, i.e., the dispersion relation, can be termed the “eigenmodes” or, equivalently, “waves.” In contrast, when ω’s and k’s are independent of each other, they represent simply “fluctuations.” These are obviously, “non-eigenmodes.” When the plasma contains free energy, which excites eigenmodes, then the non-eigenmodes become ignorable. However, for thermal equilibrium plasma where there is no free energy source, both types of modes become important. Another important effect that arises from the particle discreteness relates to the collisional kinetic equation. In plasmas, collisions are generally rare, but when the plasma is slightly out of equilibrium, yet the system does not have sufficient free energy to excite instabilities, then the only recourse to bring the system back to equilibrium is through collisions. The collisional relaxation process cannot be discussed if one relies solely on the Vlasovian description of plasma. In contrast, with the discrete particle effects inherent in the Klimontovich formalism, the collisional relaxation process becomes a natural part of the theory. An interesting point to note is that both Vlasov and Klimontovich equations are time reversible at the microscopic level, before averages are taken over them.

105

106

Spontaneous Emission and Collisional Kinetic Equation

Nevertheless, the Klimontovich formalism, which treats the particle discreteness properly, leads to the collisional kinetic equation, while the Vlasovian approach does not. The collisional processes represent dissipation and time irreversibility, so a conceptual question that may arise is how did the time irreversibility creep in from what is essentially a time-reversible equation. But the answer to this hypothetical question is well known in the non-equilibrium statistical mechanics. That is, at the microscopic level, classical Newtonian equation of motion, or quantum mechanical Schr¨odinger equation, is time reversible, but at the macroscopic level, physical theories are time irreversible. It is well known that the irreversibility of macroscopic system of equations arises out of statistical averages and subsequent loss of information. In this regard, it is no surprise that the collisional kinetic equation, which results from the statistically averaged Klimontovich theory, i.e., the weak turbulence theory, represents the dissipation. As a matter of fact, the particle and wave kinetic equations derived for the plasma eigenmodes in both Vlasov and Klimontovich weak turbulence theory are also time irreversible, so that the statistical averaging procedure already had produced macroscopically time irreversible theory in both approaches. It is just that, in the Vlasovian theory, the dissipation is not of the collisional variety, but rather it is of the collective dissipation type. For example, the quasilinear diffusion process in velocity space by resonant waveparticle interaction is an irreversible collective dissipation process. 5.1 Spontaneous Emission of Electrostatic Fluctuations For plasmas in thermal equilibrium, the Maxwell–Boltzmann–Gaussian model defines the velocity distribution function,   3/2 nma ma v 2 . (5.1) exp − fa (v) = (2πT )3/2 2T This means that we do not have to be concerned with the particle kinetic equation, since the distribution remains constant in time. As for the waves, the spontaneous emission of fluctuations takes place with all possible ω’s and k’s, including those ω’s that satisfy the dispersion relation ω = ωk , but also those so-called noneigenmodes that do not satisfy the condition (k,ω) = 0. These non-eigenmodes characterized by ω = ωk are naturally emitted by particles with thermal distribution. When there exists an instability, however, the eigenmodes grow exponentially from the noise level and completely dominate the dynamics. In such cases, the excitation of plasma eigenmodes far outweighs the non-eigenmodes in intensity, so that we may ignore the non-eigenmodes altogether. However, for thermal equilibrium plasmas, the naturally occurring fluctuations must include all possible ω and k values.

5.1 Spontaneous Emission of Electrostatic Fluctuations

107

To discuss the spontaneous emission of fluctuations, we return to the formal nonlinear spectral balance equation (4.68). Since for thermal plasmas the electric field spectral energy density does not evolve in time, we may set ∂  2 δE k,ω = 0. ∂t

(5.2)

  We expect the level of δE 2 k,ω to be sufficiently low when compared with the particle thermal energy so that we may ignore all nonlinear terms. Consequently, the non-vanishing terms in (4.68) can be collected and written down as  2 nea2 2 dv δ(ω − k · v) Fa (v), P 2 δE k,ω = π a k | (k,ω)|2 2   ωpa 1 ∂Fa (v) dv (k,ω) = 1 + k· . (5.3) 2 k ω − k · v + i0 ∂v a For isothermal plasmas, whose velocity distribution function is given by (5.1), we have ∂Fa (v) T . (5.4) δ(ω − k · v) Fa (v) = −δ(ω − k · v) k · ∂v ma ω This leads to T Im (k,ω) , P 2π 3 ω | (k,ω)|2   2   2ωpa ω ω 1 + , Z (k,ω) = 1 + 2v2 kv kv k T a T a T a a  2T vT a = . ma

δE 2 k,ω =

(5.5)

Figure 5.1 plots the high-frequency portion of normalized spontaneous thermal fluctuation spectrum that shows enhanced emission near Langmuir wave dispersion relation, ω = ωkL , but the spectrum also shows a broad region of non-vanishing level of thermal fluctuations in the non-eigenmode spectral range (ω = ωkL ). Figure 5.1 also indicates the enhanced emission in the low-frequency part surrounding the ion acoustic mode, but because of the plotting scales it is not so easy to visually identify the structure, so, in the next figure, we display the blown-up version emphasizing the low-frequency portion of the spectrum. Thus Figure 5.2 displays the spontaneous emission of electrostatic fluctuations in the low-frequency portion of the angular frequency that covers the range of ionacoustic wave dispersion relation. In this case, the clear enhancement along the ion-acoustic dispersion curve is not apparent, but instead the fluctuation spectrum

108

Spontaneous Emission and Collisional Kinetic Equation

(2π 3ω pe / T) δ E 2Ò k,ω Ò

1.4 1.2 L

z = ω/ωpe

1 0.8 0.6 0.4 0.2 S 0

0.2

0.4

0.6

0.8

1

κ = kλDe Figure 5.1 Spontaneously emitted electrostatic fluctuations including the singular eigenmode range satisfying the Langmuir wave dispersion relation, ω = ωkL , as well as non-eigenmode spectral range (ω = ωkL ). In the low-frequency domain, ion acoustic mode range S is also specified.

(2π 3ω pe / T ) δ E 2 Ò

0.1

Ò

k,ω

z = ω/ω

pe

0.08

0.06

0.04

S

0.02

0

0

0.5

1

1.5

2

κ = ck/ωpe Figure 5.2 Extension of the spontaneously emitted electrostatic fluctuations in the low-frequency portion that shows the enhancement in the vicinity of ion-sound wave dispersion relation, ω = ωkS .

5.1 Spontaneous Emission of Electrostatic Fluctuations

109

is generally enhanced in the broad vicinity of ω = ωkS . This is because the ionsound wave for thermal plasmas (Te = Ti = T ) is heavily damped, which means that a substantial imaginary part of (k,ω) is associated with the low-frequency range in the vicinity of ion-sound wave. Consequently, the denominator in (5.5) does not remain extremely small even for the eigenmode ω = ωkS . This explains why the spectrum is not highly enhanced for ω = ωkS , which contrast to the spectral behavior near ω = ωkL , especially for low k region. It is well known that Langmuir wave damping rate becomes exponentially small for k → 0 (see Figure 1.2). As a result, the spontaneous emission spectrum strictly diverges for Langmuir waves in the limit of k → 0. However, this does not happen for ion sound mode. Lastly, we show several cuts along angular frequency for fixed values of wave numbers. The result is shown in Figure 5.3. Note that as κ = kλDe approaches small value, the peak intensity becomes higher and higher, until for k = 0, the peak value becomes infinite. Again, this behavior is intimately related to the disappearance of Landau damping rate for Langmuir waves in the infinite wavelength regime. The fact that the spontaneous emission formula diverges for Langmuir waves in the infinite wavelength regime may indicate that the theory fails as one approaches k = 0. However, if we are only interested in the plasma eigenmodes with weak damping rates, then we know how to deal with such a situation. We simply expand the dielectric constant around the pole, ω − ωk , in Taylor series, and implement

102

κ = kλ

De

= 0.2

κ = 0.3

(2π3/ T) áδ E 2 ñ k,ω

κ = 0.4 0

10

10–2

10–4 0.8

0.9

1

1.1

z = ω/ωpe

1.2

1.3

1.4

Figure 5.3 Plot of spontaneous emission spectrum along angular frequency for fixed values of wave numbers.

110

Spontaneous Emission and Collisional Kinetic Equation

the residue theorem in order to take care of the singularity, which is what we have done in the derivation of particle and wave kinetic equations. Nevertheless, when one applies the spontaneous emission formula to the “fluctuations,” then we have a potential problem when the denominator, which contains (k,ω), becomes exceedingly small. This implies that the spontaneous emission theory requires some sort of “renormalization.” In regard to this issue, we note that Sitenko (1967), Akhiezer et al. (1975), and Sitenko (1982) discuss the so-called nonlinear theory of spontaneous emission for fluctuations. In the present book, however, we omit such a discussion. It is interesting to discuss the fluctuation as a function of k only:  (2π) δE k = 3

2

dω (2π)4 δE 2 k,ω = 4T 2π



dω Im (k,ω) . P ω | (k,ω)|2

(5.6)

If we note that for any complex number z = x + iy, we may write   1 y i 1 Im z 1− , = 2 = −1+ |z|2 x + y2 2 x − iy x + iy

(5.7)

  Im (k,ω) 1 i 1 1− ∗ . = −1+ | (k,ω)|2 2 (k,ω) (k,ω)

(5.8)

then we have

Let us define f + (ω) = 1 −

1 , (k,ω)

f − (ω) = 1 −

1 . ∗ (k,ω)

(5.9)

Then we have  (2π) δE k = 2iT 3

2

dω − f (ω) − f + (ω) . ω

(5.10)

We may further write 1 1 = ω 2



 1 1 . + ω − i0 ω + i0

(5.11)

Consequently, we have  (2π) δE k = iT 3

2

 −

1 1 f (ω) − f + (ω) . dω + ω − i0 ω + i0 

(5.12)

5.1 Spontaneous Emission of Electrostatic Fluctuations

111

The function f + is analytic in the upper half ω plane, while f − is analytic in the lower half complex ω plane. It is also to be noted that lim f ± (ω) = 0.

(5.13)

ω→∞

Making use of the Kramers–Kr¨onig relations,  f ± (ω ) 1 dω = ±f ± (ω), − 2πi ω − ω ± i0  f ∓ (ω ) 1 dω = 0, − 2πi ω − ω ± i0 we find that

(2π)3 δE 2 k = 2πT f − (0) + f + (0) = 4πT

  Re (k,0) . 1− | (k,0)|2

(5.14)

(5.15)

For thermal equilibrium plasma we have (k,0) = 1 +

 a

1 2 ≈ 1 + , 2 k 2 λDa k 2 λ2De

(5.16)

where λ2Da = Ta /(4πnea2 ). As a consequence, we obtain δE 2 k ≈

1 T . 2 2π 2 + k 2 λ2De

(5.17)

So, as one can see, when the spontaneous emission formula, which contains potential singularity associated with the plasma eigenmodes, is integrated over ω, one obtains a well-behaved finite valued function. For more discussions on the spontaneous emission theory including the nonlinear correction, see Sitenko, 1967, 1982; Akhiezer et al., 1975, for example. Applications of the spontaneous emission theory for space plasmas were pioneered by Meyer-Vernet (1979). Her method, known as the quasi-thermal noise spectroscopy, takes into account the antenna response to the spontaneously emitted electrostatic fluctuations naturally occurring in space plasma, and it is being used for passively diagnosing the underlying space plasma properties (Meyer-Vernet et al., 2017). The basic theory of spontaneous emission of fluctuations in plasmas has been quite extensively discussed in the literature, which include the extension to magnetized plasmas (Meyer-Vernet et al., 1993; Sentman, 1982), generalization for suprathermal plasma distribution (Mace et al., 1998; Le Chat et al., 2009), full extension to electromagnetic formalism in magnetized plasma (Kim et al., 2017; L´opez and Yoon, 2017), etc., although these generalizations and extensions are beyond the scope of the present book.

112

Spontaneous Emission and Collisional Kinetic Equation

5.2 Collisional Kinetic Equation We now consider the situation where the system is slightly out of thermal equilibrium, but not to the extent that any instabilities can be excited in the plasma. For such a case, while the electrostatic field fluctuations can be described by the spontaneous emission theory, δE k,ω 2

  1 2 = e2 dv δ(ω − k · v) fa (v), π k 2 | (k,ω)|2 a a

(5.18)

the dynamical evolution of particle distribution function toward a genuine thermal equilibrium must be discussed on the basis of particle kinetic equation (4.75), but without the nonlinear correction term,     k ∂ πea2 ∂fa δ(ω − k · v) dk dω = 2 · ∂t ma k ∂v     2 ma (k,ω) k ∂fa . × Im 3 f + δE · a k,ω k 2π k | (k,ω)|2 ∂v

(5.19)

When (5.18) is inserted to (5.19), we have 





ma Im (k,k · v) fa (v) 2π 3 k | (k,k · v)|2     1 2 (v) k ∂f a , + e2 dv δ(k · v − k · v ) fb (v ) · π k 2 | (k,k · v)|2 b b k ∂v

∂fa πe2 = 2a ∂t ma

k ∂ dk · k ∂v

(5.20) where we have carried out the ω integration. Making use of  4πe2  ∂fa a dv k · Im (k,ω) = −π δ(ω − k · v), 2 ma k ∂v a

(5.21)

we readily obtain the well-known Balescu–Lenard–Guernsey collisional kinetic equation (Balescu, 1960; Guernsey, 1960; Lenard, 1960),     ∂fa 2ea2  2 ∂ δ(k · v − k · v )  k dk dv e = 2 · ∂t ma b b k ∂v k 2 | (k,k · v)|2      k ∂fa (v) ma k ∂fb (v )  fb (v ) − fa (v) . × · · k ∂v mb k ∂v

(5.22)

5.2 Collisional Kinetic Equation

113

When we take the ambient density factor n out of the velocity distribution function, fa = nFa , then we have    ∂Fa (v)  2nea2 eb2 ∂  ki kj δ(k · v − k · v ) dk dv = ∂t m2a ∂vi k 4 | (k,k · v)|2 b   ∂ ma ∂ × − Fa (v) Fb (v ). ∂vj mb ∂vj

(5.23)

Equation (5.22) or equivalently (5.23) contains the quantity (k,k · v), which, in turn, contains the particle distribution function, hence, nonlinearly coupled to the distribution function. A simplifying approximation is to replace (k,k · v) by 1. The resulting equation is   ki kj ∂Fa (v)  2nˆ ea2 eb2 ∂ dk dv 4 δ(k · v − k · v ) = 2 ∂t ma ∂vi k b   ∂ ma ∂ − Fa (v) Fb (v ). × ∂vj mb ∂vj The following identity is useful:   ∞ kk dk dk δ(k · u) 4 = U π , k k 0 1 ui uj  Uij = δij − 2 , u u

(5.24)

(5.25)

where u = v − v . To prove this, let u = u zˆ and k = kx xˆ + ky yˆ + kz zˆ , without loss of generality. Then  ∞  ∞ kk kk dkz δ(k · u) 4 = dkz δ(kz u) 4 k k −∞ −∞ 1 (kx xˆ + ky yˆ )(kx xˆ + ky yˆ ) = , (5.26) u (kx2 + ky2 )2 which leads to 

kk 1 dk δ(k · u) 4 = k u



 dkx

dky

kx2 xˆ xˆ + ky2 yˆ yˆ (kx2 + ky2 )2

,

(5.27)

where we have made use of the fact that the cross term ky kx does not contribute to the integral. When (5.27) is expressed in polar coordinate integrals, then the rest easily leads to the proof

114



Spontaneous Emission and Collisional Kinetic Equation

  kk 1 ∞ dk 2π dk δ(k · u) 4 = dφ cos2 φ xˆ xˆ + sin2 φ yˆ yˆ k u 0 k 0   π ∞ dk  u u  π ∞ dk (ˆx xˆ + yˆ yˆ ) = I− 2 . = u 0 k u 0 k u

(5.28)

Making use of identity (5.25), we have   1 ∂Fa (v)  2π nˆ ea2 eb2 ∞ dk ∂ dv = 2 ∂t ma k ∂vi |v − v | 0 b    ∂ (v − v )i (v − v )j ma ∂ × δij − − Fa (v) Fb (v ). |v − v |2 ∂vj mb ∂vj (5.29) Since the k integral diverges at both limits, we must introduce cutoffs,   kmax dk dk kmax = ln . = = ln k k kmin kmin

(5.30)

The resulting equation is called the Landau collisional equation (Landau, 1937):    ∂Fa (v)  2π nˆ ea2 eb2 ln  ∂ 1 (v − v )i (v − v )j  δij − dv = ∂t m2a ∂vi |v − v | |v − v |2 b   ∂ ma ∂ − (5.31) × Fa (v) Fb (v ). ∂vj mb ∂vj To determine the limits, kmin and kmax , we make note of the fact that in Figure 5.1, while the spontaneously emitted fluctuations attain high intensity for ω in the vicinity of ωkL , the area (or measure) in (ω,k) space over which this occurs is quite limited. The broad swath of regions where the spontaneous emission is finite actually takes place in the non-eigenmode region, excluding the narrow domain satisfying ω = ωkL . The approximation of replacing (k,k · v) by 1 is equivalent to ignoring the eigenmode contribution to the spontaneously emitted fluctuations. Mathematically, this approximation is essentially equivalent to restricting the integral range of k to kλDe > 1. That is, we may choose kmin =

1 . λDe

(5.32)

For large k, we make note of the fact that we cannot resolve physical processes whose spatial scale is shorter than the atomic scales. Considering that the average particle kinetic energy is the thermal energy, E ∼ kB Ta or simply E ∼ Ta , we

5.3 Properties of Collisional Kinetic Equation

115

estimate that the smallest physical scale can be determined by balancing the kinetic energy with electrostatic potential energy, |ea eb | ∼ Ta . rmin

(5.33)

It is the convention of this book to adopt the energy units in eV, thus absorbing the Boltzmann constant kB = 1.38064852 × 10−23 m2 kg s−2 K−1 in the definition for temperature. From this, we have rmin ∼ |ea eb |/Ta , which, for a = b = e, is the classical radius of electron, or kmax =

Ta . |ea eb |

From this we obtain the Coulomb logarithm,   λDe Ta . ln  = ln |ea eb | For a = b = e and for equilibrium, we have ln  = ln 4πnλ3De .

(5.34)

(5.35)

(5.36)

The quantity g≡

1 nλ3De

(5.37)

is known as the “plasma parameter,” and it is inversely proportional to the total number of plasma particles in the so-called Debye sphere, that is, a sphere with radius equal to the Debye length, and whose volume is thus, (4π/3)λ3De . We already encountered this quantity when we mentioned the difference between the Klimontovich versus Vlasov paradigm. Recall that when the number of particles per Debye sphere is infinite, or, equivalently, when g = 0, we have the strict Vlasov limit. Obviously, the larger the number of particles in the Debye sphere, the slower the collisional relaxation toward establishing a new thermal equilibrium will proceed, starting from an initial system, which is slightly out of equilibrium. Since the righthand side of the collisional integral (5.31) is proportional to g, it is evident that the plasma parameter determines the relative importance of inter-particle collisions versus collective processes. 5.3 Properties of Collisional Kinetic Equation The collisional integral possesses certain invariants. Starting from (5.22), we show that thermal equilibrium Maxwellian–Boltzmann (MB) distribution function is an

116

Spontaneous Emission and Collisional Kinetic Equation

exact solution for steady state. It is sufficient to consider the following quantity within the velocity integral on the right-hand side of (5.22) for MB distribution:    ∂ mb ∂   fa (v) fb (v ) dv δ(k · v − k · v ) k · − ∂v ma ∂v    k · v k · v   fa (v) fb (v ), (5.38) − = −mb dv δ(k · v − k · v ) Tb Ta which is zero, if Ta = Tb = T . This shows that thermal equilibrium distribution satisfies the steady-state kinetic equation. Alternatively, this shows that the collisional kinetic equation should lead any particle distribution slightly out of equilibrium, to evolve toward thermal equilibrium distribution. Consider now an integral  dv (v) Ca (r,v,t), (5.39) J (r,t) = a

where (v) is an arbitrary function and Ca (r,v,t) is the collision integral,    2 e2 e2 ∂  a b  k k δ(k · v − k · v ) Ca (r,v,t) = · dk dv m2a ∂v k 4 | (k,k · v)|2 b   ∂ ma ∂ fa (v) fb (v ). · − ∂v mb ∂v

(5.40)

It can be shown that J (r,t) = 0 for ma v 2 . (5.41) 2 Obviously these three quantities are related to conservations of total number density, momentum, and kinetic energy. To prove this, we proceed as follows:     e2 e2  ∂ k k δ(k · v − k · v ) a b  dk dv dv (v) · 4 J (r,t) = 2 2 m ∂v k | (k,k · v)|2 a a b   ∂ ma ∂ fa (v) fb (v ) · − ∂v mb ∂v       e2 e2  ma ∂(v ) a b  ∂(v) dk dv dv =− − m2a ∂v mb ∂v a b   k k δ(k · v − k · v ) ∂ ma ∂ · 4 fa (v) fb (v ), · (5.42) − 2 k | (k,k · v)| ∂v mb ∂v (v) = 1,

(v) = p = ma v,

and

(v) =

where we have taken successive partial integrations. It readily follows that (v) = 1 and (v ) = ma v; (v ) = mb v renders the right-hand side to vanish. For (v) = ma v 2 /2, (v ) = mb v  2 /2, the right-hand side becomes

5.3 Properties of Collisional Kinetic Equation

rhs = −

  e2 e2 a b

a

b

ma



 dk

δ(k · v − k · v ) × 4 k· k | (k,k · v)|2

 dv



117

dv (k · v − k · v )

 ∂ ma ∂ fa (v) fb (v ), − ∂v mb ∂v

(5.43)

which again vanishes by virtue of the delta function δ(k · v − k · v ). If we choose (v) = − ln fa (v,t),

(5.44)

then we may prove that the total entropy of the plasma, where the entropy is defined by  S(t) = − dv fa (v,t) ln fa (v,t), (5.45) a

always increases and tends toward a constant value in the time-asymptotic limit. To see this, we write ∂(v) ma ∂(v ) + ∂v mb ∂v ∂ ma ∂ = ln fa (v,t) ln fa (v,t) − ∂v mb ∂v   ∂ 1 ma ∂ = fa (v,t) fa (v,t). − fa (v,t) fa (v,t) ∂v mb ∂v

A(v,v,t) = −

(5.46)

Then we have     e2 e2  k k δ(k · v − k · v ) a b   J (t) = dk dv dv A(v,v ,t) · m2a k 4 | (k,k · v)|2 a b   ∂ ma ∂ fa (v) fb (v ) · − ∂v mb ∂v    2   e2 e2  a b  [k · A(v,v ,t)] dk dv dv = m2a k 4 | (k,k · v)|2 a b × δ(k · v − k · v ) fa (v) fb (v ) > 0, since the integral is positive definite. Upon taking the time derivative of S(t), we have  ∂fa dS dv (1 + ln fa ) =− dt ∂t a     2ea2 eb2 ∂ dv (1 + ln fa ) dk dv =− 2 ma ∂v a b

(5.47)

118

Spontaneous Emission and Collisional Kinetic Equation

  kk δ(k · v − k · v ) ∂ ma ∂ fa (v) fb (v ) ×· 4 · − k | (k,k · v)|2 ∂v mb ∂v       e2 e2  ∂ 1 ma ∂ a b  dk dv dv = − m2a fa (v) fb (v ) ∂v mb ∂v a b   kk δ(k · v − k · v ) ∂ ma ∂  fa (v) fb (v ) · × fa (v) fb (v ) · 4 − 2  k | (k,k · v)| ∂v mb ∂v    2 2  2  e e a b  [k · A(v,v )] dk dv dv = m2a k 4 | (k,k · v)|2 a b × δ(k · v − k · v ) fa (v) fb (v ) = J (t) > 0.

(5.48)

This shows that dS > 0, (5.49) dt meaning that the entropy always increases. This is the plasma equivalent of Boltzmann’s H-theorem. 5.4 Collisional Kinetic Equation with Maxwellian Background The collision integrals, either Balescu–Lenard form (5.22) or Landau form (5.31), are nonlinear in their mathematical structures, since the right-hand side involves the product fa (v)fb (v ) such that one must solve the nonlinear integro-differential equation (Pezzi et al., 2016). However, the problem can be substantially simplified if we treat fb (v ) as a Maxwellian distribution. That is, the problem can be approximately treated as collisions between an arbitrary species (a) and a Maxwellian species (b), which can be taken to be at rest. Let us express the collisional operator as   ∂ ∂fa (v) Ai (v)fa (v) + Dij (v) , Cab (fa,fb ) = ∂vi ∂vi   4πea eb 2 ma ∂ϕb (v) Ai (v) = ln  , ma mb ∂vi   4πea eb 2 ∂ 2 ψb (v) , Dij (v) = − ln  ma ∂vi ∂vj  fb (v ) 1 dv , ϕb (v) = − 4π v − v  1 dv (v − v )fb (v ), (5.50) ψb (v) = − 8π

5.4 Collisional Kinetic Equation with Maxwellian Background

119

√ where fb given by fb (v) = fb0 (v) = nb (π 1/2 vT b )−3 e , vT2 b = 2Tb /mb . The quantities ϕb (v) and ψb (v) are known as the Rosenbluth potentials (Rosenbluth et al., 1957). Since fb is isotropic, the Rosenbluth potentials are also isotropic,  1 1 dv fb (v  ), ϕb (v) = − 4π u  1 dv u fb (v  ), ψb (v) = − (5.51) 8π −v 2 /vT2 b

where u = |v − v |. Since ϕb and ψb are isotropic, we have ∂ϕb vk  = ϕ, ∂vk v b   ∂ 2 ψb ∂ vl  vk vl = ψb = Wkl ψb + 2 ψb, ∂vk ∂vl ∂vk v v Wkl = This leads to

v 2 δkl − vk vl . v3

(5.52)



 ∂f  ∂ ma vk  vk vl a ϕb fa − Wkl ψb + 2 ψb ∂vk mb v v ∂vl      ∂ 1 ∂fa  4πea eb 2 ma ∂ v   · ϕb fa − · ψb = ln  ma mb ∂v v ∂v v ∂v        ∂fa ∂ ∂fa  v v ∂  ψb − v· v· . (5.53) · · ψ + ∂v v 3 ∂v ∂v v 2 ∂v b 

Cab (fa,fb ) = ln 

4πea eb ma

2

We may rewrite (5.53) in terms of the velocity space spherical coordinate system: ∂ ∂ ∂ 1 ∂ 1 = vˆ + θˆ + ϕˆ , ∂v ∂v v ∂θ v sin θ ∂ϕ       ∂fa  ∂fa 1 ∂ ∂ v ψb = 2 v · 3 v· ψ , ∂v v ∂v v ∂v ∂v b       ∂ ∂fa 1 ∂ v  2 ∂fa  ψb = 2 v v· · ψ . ∂v v 2 ∂v v ∂v ∂v b This leads to

(5.54)

  2 4πea eb 2  ln  ψb L (fa ) v3 ma     1 4πea eb 2 ∂ 2 ma  2  ∂fa v , + 2 ln  ϕ fa − v ψb v ma ∂v mb b ∂v     1 ∂ ∂ 1 ∂2 1 sin θ + 2 fa . L (fa ) = (5.55) 2 sin θ ∂θ ∂θ sin θ ∂ϕ 2

Cab (fa,fb ) = −

120

Spontaneous Emission and Collisional Kinetic Equation

Let us define three basic collision frequencies,    ma ϕb (v) 4πea eb 2 ab νF (v) = ln  1+ , ma mb v   2 4πea eb 2  ab νD (v) = − 3 ln  ψb (v), v ma   2 4πea eb 2  ab νE (v) = − 2 ln  ψb (v), v ma

(5.56)

where νFab (v) represents the collisional velocity friction frequency, νDab (v) designates the collisional frequency associated with diffusion in velocity pitch angle and azimuthal angle space, and νEab (v) corresponds to the collisional frequency associated with velocity space diffusion, or, equivalently, diffusion in energy space. In terms of these frequencies, we have Cab (fa,fb ) = νDab (v) L (fa )   ma vνEab (v) ∂fa 1 ∂ 3 ab . ν (v) fa + + 2 v v ∂v ma + m b F 2 ∂v The next task is to calculate the Rosenbluth potentials:  fb0 (v  ) 1 dv ϕb (v) = − 4π |v − v |  1  ∞ 2 2 v  2 e−v /vT b nb  = − 3/2 3 d(cos θ) dv . 2π vT b −1 0 v 2 + v  2 − 2vv  cos θ  v/vT b nb nb 2 2 dx e−x = − =− erf(xb ), √ 4πv π 0 4πv v xb = , vT b 1 ψb (v) = − 8π



(5.57)

(5.58)

dv |v − v | fb0 (v  )  1  ∞ . nb 2 2 2 = − 3/2 3 d(cos θ) dv  v  v 2 + v  2 − 2vv  cos θ e−v /vT b 4π vT b −1 0    xb  nb vT b 1 1 −x 2 −x 2 xb + = − 3/2 dx e + e b 4π 2xb 2 0    1 1 −x 2 nb vT b b xb + erf(xb ) + √ e , (5.59) =− 8π 2xb π

where erf(x) is the error function.

5.4 Collisional Kinetic Equation with Maxwellian Background

Quantities of interest are the derivatives of the Rosenbluth potentials: nb mb G(xb ), ϕb (v) = 4πTb nb [erf(xb ) − G(xb )], ψb (v) = − 8π nb G(xb ) ψb (v) = − , 4πvT b xb √ 2 erf(xb ) − (2/ π) xb e−xb G(xb ) = . 2xb2 The three collision frequencies defined in (5.56) are thus expressed as   2Ta mb G(xb ) ab 1+ , νF (v) = νˆ ab Tb ma xa erf(xb ) − G(xb ) , νDab (v) = νˆ ab xa3 G(xb ) νEab (v) = 2ˆνab , xa3 4πnb ea2 eb2 ln  v , xb = . νˆ ab = 3 2 vT b ma vT a

121

(5.60)

(5.61)

The reduced collisional kinetic equation expressed in terms of the Rosenbluth potentials are linear, which greatly facilitates analytical and/or numerical investigations (Tigik et al., 2016). For systematic discussions of collisional relaxation processes in space or laboratory plasmas, the readers are referred to excellent monographs by Helander and Sigmar (2002) and Zank (2014).

6 Langmuir Turbulence and Electron Kappa Distribution

Having formulated the electrostatic weak turbulence theory that includes the effects of discrete nature of plasma particles, which extends the Vlasov weak turbulence theory that assumes collision-free plasma, we now consider an application that involves a plasma instability. Chapter 5 already pertained to applications, but it was for plasmas where collective modes are of no particular significance. In this chapter, in contrast, we consider a situation where the plasma carries significant free energy source so as to excite collective oscillations. Specifically, we consider the problem of Langmuir wave instability excited by gentle (or weak) bump-on-tail instability, and the Langmuir/ion-acoustic turbulence generated during the beamplasma interaction process. The long-time, quasi-asymptotic state of the Langmuir turbulence state also involves the acceleration/heating of electrons such that the end state corresponds to the celebrated kappa distribution (Olbert, 1968; Vasyliunas, 1968; Tsallis, 2009; Livadiotis, 2017). Since the 1960s, when the near-Earth space environment was explored with artificial satellites, it became quite evident that the measured electron distribution functions in the solar wind deviate substantially from the Maxwellian thermal distribution. Specifically, while the core part of the electron distribution can be modeled with the Maxwellian distribution function, the high-velocity tail population features a quasi-inverse power law distribution (Vasyliunas, 1968; Feldman et al., 1975; Gosling et al., 1981; Armstrong et al., 1983; Wang et al., 2012a). Olbert (1968) and Vasyliunas (1968) famously introduced the kappa distribution function, −κ−1  v2 , (6.1) fe ∝ 1 + κ2 to empirically fit the measurement. Note that the kappa distribution approaches the Maxwellian model for κ → ∞,  −κ−1   v2 v2 (6.2) = exp − 2 . lim 1 + κ→∞ κ2  122

Langmuir Turbulence and Electron Kappa Distribution

123

0 –1 FKappa

log10[F(v)]

–2 –3 F

–4

κ=2 Max

κ=5

–5 –6 –10

–5

0

5

10

v/Θ Figure 6.1 Maxwellian versus kappa distributions.

To see this, we display in Figure 6.1 the Maxwellian distribution FMax = exp (−v 2 /2 ) versus kappa distribution FKappa = (1 + v 2 /κ2 )−κ−1 for two different values of κ, which equals to 2 and 5, respectively. As one can see, the case of κ = 5 is more close to the Maxwellian distribution than that of κ = 2. It is also obvious that the kappa distribution effectively models the low-energy population characterized by quasi-Maxwellian form, while for high-energy regime, the model describes a supra-thermal tail population. This is why kappa model is widely adopted for space plasmas, where charged particle distributions are often measured to have quasi-inverse power law tail populations. The empirical fitting of in situ spacecraft measurements by kappa distribution, however, did not enjoy any first-principle justifications when it was first introduced. However, in the 1980s, Tsallis proposed the non-extensive thermo-statistical theory, for which the microscopic state of the system that maximizes the non-extensive (or Tsallis) entropy is none other than the kappa distribution (Tsallis, 1988, 2009). This has led to a renewed interest in the observed kappa-like space plasma distribution (Leubner, 2002; Livadiotis and McComas, 2009). Tsallis’ original theory is quite general, and not specific to plasmas, but when applied to plasmas, the implication was quite obvious. In a parallel effort, Treumann (1999a,b) attempted to formulate an alternative statistical mechanics based upon Lorentzian distribution function, which is related to the kappa distribution. Independent of these developments, Yoon et al. (2005) investigated the Langmuir turbulence problem from the standpoint of weak turbulence theory. Numerical solutions of the equations of weak turbulence led to the discovery that the end state of Langmuir turbulence also coincides with the generation of non-thermal tail population of the electron velocity distribution function, which superficially resembles

124

Langmuir Turbulence and Electron Kappa Distribution

the kappa distribution. This has led the present author to seek the genuine time asymptotic solution of the weakly turbulent state. It was found that the kappa distribution is a unique solution that characterizes the steady-state Langmuir turbulence (Yoon, 2014). The fact that the end state of Langmuir turbulence corresponds to the kappa distribution function is a major theoretical breakthrough in understanding the physical origin of measured electron distribution in the solar wind, and this problem represents a prime example of the success of weak turbulence theory. Conceptually, the quasi-equilibrium state of Langmuir turbulence represents a collection of electrons interacting among themselves through long-range collective electrostatic fluctuations. Tsallis entropy also describes a system interacting with long-range force such that the entropy of the system exhibits non-extensive thermostatistical characteristics. In this regard, the quasi-equilibrium Langmuir turbulence state and the non-extensive statistical state might be equivalent. The profound interrelationship that may exist between the two approaches has yet to be worked out in a rigorous mathematical sense, but this is an area that is being investigated at a fundamental level – see the recent monograph by Livadiotis (2017), where various aspects related to the kappa distribution and non-extensive statistical mechanics is thoroughly surveyed. This chapter discusses the application of weak turbulence theory to the beamplasma instability, dynamical development of the Langmuir turbulence, the steadystate Langmuir turbulence, and the associated electron kappa distribution function. 6.1 Normalized Equations The electron beam-plasma instability is a classic problem in any standard plasma physics literature. The linear stage of the instability is well understood, and is widely covered in many textbooks, and quasilinear phase of the instability is also well known (Drummond and Pines, 1962; Vedenov and Velikhov, 1962; Frieman and Rutherford, 1964; Bernstein and Engelmann, 1966; Dawson and Shanny, 1968; Morse and Nielson, 1969; Vahala and Montgomery, 1970; Joyce et al., 1971; Roberson et al., 1971; Appert et al., 1976; Ivanov et al., 1976; Grognard, 1982; Tsunoda et al., 1987; Dum, 1990; Muschietti and Dum, 1991; Nishikawa and Cairns, 1991; Dum and Nishikawa, 1994). Many advanced textbooks discuss the saturation of gentle beam-plasma instability via velocity space plateau formation. The nonlinear stage of the instability that takes place after the quasilinear plateau formation, in contrast, is not widely discussed in the elementary textbooks. The weak turbulence theory may be employed in order to discuss various stages of the beam-plasma instability development including the quasilinear saturation by velocity space plateau formation, but also the same theory naturally and quite rigorously applies to processes that take place much beyond the quasilinear

6.1 Normalized Equations

125

saturation stage. The best way to illustrate the nonlinear development is via numerical solution. We begin the discussion with the particle kinetic equation (4.86), but for the present purpose, let us retain the linear wave-particle interaction terms and focus only on the electrons:   ∂Fe ∂ ∂Fe Ai Fe + Dij , = ∂t ∂vi ∂vj  ki  e2 dk 2 Ai = σ ωL δ(σ ωkL − k · v), 4πme k σ =±1 k  ki kj  πe2 Dij = 2 dk 2 δ(σ ωkL − k · v) Ikσ L . (6.3) me k σ =±1 We employ the wave kinetic equation summarized by (4.93) and (4.94), but in doing so, we neglect the scattering terms involving ion-sound waves, as such processes are extremely slow ones. Thus, the wave kinetic equations for Langmuir (L) and ion-sound (S) modes adopted in the present discussion are   2 2  πωpe ne ∂Fe ∂Ikσ L L L σL dv δ(σ ωk − k · v) = F e + σ ωk I k k · ∂t k2 π ∂v 

 LS L σ  L σ  S +2 σ ωkL dk Vk,k Ik−k  σ ω k Ik  σ ,σ  =±1

 σ  S σ L L σ L σ L − σ  ωkL Ik−k − σ  ωk−k Ik  Ik   Ik     ∂Fi L  LL me σ  L σ L dk dv Uk,k Ik Ik (k − k ) · + σ ωk mi ∂v σ  =±1   ne2 L σ  L σ ωk Ik − σ  ωkL Ikσ L (Fe + Fi ) , + 2 πωpe 2 πμk ωpe ∂Ikσ S = ∂t k2

(6.4)





ne2 (Fe + Fi ) π    ∂ me L σS Fe + Fi + σ ωk Ik k · ∂v mi 

 L SL L σ  L σ  L dk Vk,k σ ωk Ik−k +  σ ω k Ik  dv

δ(σ ωkS

− k · v)

σ ,σ  =±1

 σ  L σ S L σ L σ S − σ  ωkL Ik−k − σ  ωk−k Ik ,  Ik  Ik 

(6.5)

126

where

Langmuir Turbulence and Electron Kappa Distribution

   2 2 v k 3 3 Te ωkL = ωpe 1 + k 2 λ2De = ωpe 1 + , 2 2 4 ωpe √ √  me 1 + 3Ti /Te kcS 1 + 3Ti /Te S  = kvT e . ωk =  2mi 1 + k 2 v 2 /2ω2 1 + k 2 λ2De pe Te     k3v3 me me 3Ti 3Ti 1+ = 3T e 1+ , μk = k 3 λ3De mi Te ωpe 8mi Te 

π e2 μk−k (k · k )2 S δ(σ ωkL − σ  ωkL − σ  ωk−k  ), 2 Te2 k 2 k 2 |k − k |2 π e2 μk [k · (k − k )]2 L = δ(σ ωkS − σ  ωkL − σ  ωk−k  ), 4 Te2 k 2 k 2 |k − k |2 π e2 (k · k )2 = 2 2 2 2 δ[σ ωkL − σ  ωkL − (k − k ) · v]. ωpe me k k

LS Vk,k  = SL Vk,k  LL Uk,k 

(6.6)

The initial exponential growth (or inverse Landau damping rate) was already discussed in Section 1.8. After the onset of bump-on-tail instability, the saturation is achieved via velocity space plateau formation. The velocity space diffusion is effective only in the range where the electron distribution has a positive slope. This range corresponds to the positive growth of the Langmuir waves. It is convenient to consider one-dimensional situation. If we approximate ωk ≈ ωpe , then upon ignoring the velocity friction term for now, the particle velocity space diffusion equation may be approximately written as   ∂ ∂Fe (v,t) πe2 ∂ 1 (6.7) = 2 Iωpe /v (t) Fe (v,t) . ∂t me ∂v v ∂v In the time asymptotic limit, ∂/∂t → 0, we must have  ∂Fe (v,t)  → 0, Iωpe /v (t) ∂v t→∞

(6.8)

over the resonant velocity space. Outside the resonant velocity space, the diffusion coefficient is zero, so that Fe does not change in time. The asymptotic condition (6.8) can be satisfied if   → 0, (6.9) Iωpe /v (t) t→∞

or

 ∂Fe (v,t)  → 0. ∂v t→∞

(6.10)

6.1 Normalized Equations

127

Condition (6.9) is generally not satisfied, since at saturation the wave level will be finite. Consequently, the second condition (6.10) must be true. This means that, in the resonant velocity space, the electron distribution must form a plateau. Since the growth rate is proportional to ∂Fe /∂v, but, in the time asymptotic regime, ∂Fe /∂v → 0 in the resonant velocity range, the asymptotic growth rate must be zero. The wave intensity, which grows from some initially low noise level, must saturate to some finite value: γk → 0,

t → ∞,

Ik (t) → finite value,

t → ∞.

(6.11)

This overall picture of the quasilinear development of the gentle beam-plasma, or bump-in-tail instability, is a well-known velocity plateau formation by quasilinear process. Physical processes that take place beyond the quasilinear saturation stage involve wave-wave interaction dictated by nonlinear decay interaction terms in L and S mode equations, and nonlinear wave-particle interaction dictated by induced and spontaneous scattering processes in L mode equation. Toward the end of the mode coupling stage, the electron velocity distribution function develops a quasi-inverse power law energetic tail population. This involves energy redistribution among the electrons, which, in the end, resembles the so-called kappa distribution function. Before we discuss the various stages of bump-on-tail instability in detail, it is useful to discuss the dimensionless form of weak turbulence equations (6.3)–(6.6). Let us introduce the following normalization scheme: T = ωpe t,

xqα =

ωkα , ωpe

a (u) = vT3 e Fa (v), M=

mi , me

τ=

Ti , Te

q=

k vT e , ωpe

Eqσ α = (2π)2 g g=

u=

v , vT e

Ikσ α , me vT2 e

1 . 23/2 (4π)2 n λ3De

(6.12)

√ In (6.12), vT e = 2Te /me represents the thermal speed for bulk quasi-Maxwellian electrons. The quantity 3/(4πnλ3De ) represents the inverse of the total number of plasma particles in a spherical volume with the radius equal to the Debye length, the so-called Debye sphere. This quantity is the plasma parameter already introduced in (5.37). As we already discussed in Section 5.2, the plasma parameter is associated with the collisional process. As we will shortly see, the plasma parameter is also related to the discrete particle effects. This will become obvious once we rewrite the basic equations (6.3)–(6.5) in normalized form. All the terms that are associated with the particle discreteness, namely, velocity friction term in the electron

128

Langmuir Turbulence and Electron Kappa Distribution

kinetic equation and spontaneous emission and spontaneous scattering terms in the wave kinetic equation, will turn out to have an overall proportionality to the factor 3/(4πnλ3De ) or, equivalently, g defined in (6.12). Note that the definition of g in (6.12) is the same as that of (5.37), except for a trivial numerical factor. If g → 0, then all the terms associated with the particle discreteness will disappear, and the system reduces to that of Vlasov weak turbulence theory. Recall that in the same limit of g → 0, collisional processes also disappear, so that both binary collisions and discrete particle effects are controlled by the same g parameter. Employing the normalization convention (6.12), the relevant equations (6.5)– (6.6) can be shown to reduce to the following. First, the electron particle kinetic equation is given by   ∂e (u) ∂ ∂e (u) el el Ai (u) e (u) + Dij (u) , = ∂T ∂ui ∂uj  qi  Ael (u) = g dq 2 σ x L δ(σ xqL − q · u), i q σ =±1 q  qi qj  el Dij (u) = dq 2 δ(σ xqL − q · u) Eqσ L . (6.13) q σ =±1 Note that the velocity space friction (or drag) term has an overall proportionality to g, which proves our previous point that the finiteness of the g parameter is intimately related to the particle discreteness effect. Second, the wave kinetic equations for Langmuir and ion-sound waves are given in dimensionless form by    ∂Eqσ L ∂e (u) π L L σL = 2 du δ(σ xq − q · u) g e (u) + σ xq Eq q · ∂T q ∂u   μq−q (q · q )2 S +2 σ xqL dq 2 2 δ(σ xqL − σ  xqL − σ  xq−q )  2 q q |q − q | σ ,σ  =±1 

 σ  S  L σ  S σL L σ L σ L − σ  xq−q Eq × σ xqL Eqσ L Eq−q  Eq  − σ xq Eq−q Eq −

  σ  =±1



dq

 du

(q · q )2 δ[σ xqL − σ  xqL − (q − q ) · u] q 2 q 2

!  × g σ xqL σ  xqL Eqσ L − σ xqL Eqσ L [e (u) + i (u)] (6.14)   L σ x ∂  q (σ xqL − σ  xqL ) e (u) − + Eqσ L Eqσ L (q − q ) · i (u) , ∂u M

6.1 Normalized Equations

∂Eqσ S ∂T

=

129



du δ(σ xqS − q · u) g [e (u) + i (u)]    ∂ i (u) L σS e (u) + (6.15) q· + σ xq Eq ∂u M   μq [q · (q − q )]2 L L + dq σ xq δ(σ xqS − σ  xqL − σ  xq−q ) 2 q 2 |q − q |2 q σ ,σ  =±1

  σ  L  L σ  L σ S L σ L σ S × σ xqL Eqσ L Eq−q − σ  xq−q Eq .  − σ xq Eq−q Eq  Eq πμq q2

The dispersion relations and the auxiliary quantity μk defined in (6.6) are given in dimensionless form by   2 1 + 3τ 3q L S 3 1 + 3τ xq = 1 + , xq = q , μq = q . (6.16) 4 M(2 + q 2 ) 8M Again, note that in both wave kinetic equations in (6.14) and (6.15), spontaneous emission and scattering terms have an overall proportionality ∝ g, which demonstrates that the particle discreteness determines the relative importance of these terms in relations to their induced counterparts. In the textbooks, it is stated that one of the conditions for ionized gas to qualify as the plasma is that g should be sufficiently small. The limit of g → 0 represents the Vlasov limit, and, in such a case, the number of plasma particles within a Debye sphere is infinite. In general, even for a finite g, its value must be sufficiently low in order for the tools of plasma physics to apply in a rigorous sense. If g is not so low, or, equivalently, the number of charged particles in the Debye sphere is not high enough, then the problem reduces to that of a small collection of charged particle interacting with electromagnetic fields and among themselves in a vacuum. The dimensionless equations (6.13)–(6.15) have been numerically solved by many researchers (Grognard, 1982; Muschietti and Dum, 1991; Kontar and P´ecseli, 2002), but also by the present author and his colleagues in one dimension and for two dimensions (and with cylindrical symmetry, three dimensions) (Ziebell et al., 2001, 2008b). The relatively simple one-dimensional limit may be of interest to the readers. In Appendix H, we further discuss the one-dimensional version of the normalized equations, and present the derivations of concrete forms of equations that readily lend themselves to the numerical finite difference analysis. Here, we mention a salient point regarding the limit of one dimension. In order to reduce (6.13)–(6.15) to one-dimensional form, one may simply consider one-dimensional velocity and wave vector, u → u, q → q, but, in doing so, one should bear in mind that the parameter g is a result of normalizing the three-dimensional equation. For a one-dimensional world, the plasma parameter, or g, should no longer be interpreted

130

Langmuir Turbulence and Electron Kappa Distribution

as the inverse of number of particles in a Debye sphere, but rather it should be the inverse of number of particles in a line segment equal to the Debye length, g1D ∝

1 . nλDe

(6.17)

Since the number of particles in a sphere is obviously much higher than the same in a line segment, the effective plasma parameter in one dimension must be significantly higher than the three-dimensional plasma parameter, g1D g3D .

(6.18)

3 , such that if g1D , which is much less than unity, Roughly, we expect that g3D ∝ g1D is given by a value, then g3D is smaller still by many orders of magnitude.

6.2 Numerical Analysis of Weak Langmuir Turbulence Dynamical evolution of weak electron beam and Langmuir/ion-sound turbulence as described by the equations of weak turbulence theory is best discussed by numerical solutions. Figure 6.2 plots the result of numerical analysis based upon (6.13)– (6.15), except that the equations are reduced to one dimension – see Appendix H. The initial electron distribution and the initial Langmuir wave intensity profile are specified by 10 10

0

–2

ω pe t = 200

ω t pe ω pe t ω pe t ω t

10–3

pe

2

3

v/vth

ln WL ( k)

ln f( v)

–2

= 200 = 100 = 50 =0 4

10

ω t = 100 pe

ω t

10–4

pe

= 50

5

10–6 0

ω pe t = 0 0.2

0.4

kvth/ωpe

0.6

Figure 6.2 Excitation of bump-on-tail Langmuir wave instability followed by subsequent quasilinear saturation during the relatively early time, 0 < t < −1 . 200ωpe

0.8

6.2 Numerical Analysis of Weak Langmuir Turbulence

 Fe (v) = 1 − IkL =

nb n0



−v 2 /vT2 e

e nb e + 1/2 π vT e n0 π 1/2 vT b

1 Te , 2 4π 1 + 3k 2 λ2De

131

−(v−Vb )2 /vT2 b

, (6.19)

where we assumed nb /n0 = 10−2 , Tb = Te , Vb /vT e = 4, and the dimensionless plasma parameter of g = (nλDe )−1 = 10−3 is adopted. Note that this is a onedimensional plasma parameter. The three-dimensional value should be roughly g3D ∝ 10−9 . The normalization of Langmuir wave spectral energy density is gIkσ L /(8me vT2 e ). For t = 0, there exists a positive velocity gradient associated with the beam electrons – see (6.18). The gentle, or weak, beam with positive gradient in velocity space, ∂Fe /∂v > 0, over the range ∼ 2.5 < v/vT e < ∼4 (see the left-hand panel, for t = 0), corresponds to the unstable wave number range ∼0.2 < kvT e /ωpe < ∼0.55. Over this range of normalized k, the bump-on-tail instability operates so that the initially low Langmuir wave intensity (shown on the −1 , it is seen right-hand panel) exponentially increases. Between t = 0 and 50ωpe that the wave intensity is greatly amplified (ωpe t = 50), but by the time the system evolves to ωpe t = 100, the instability is practically saturated, and the Langmuir energy density does not show much increase between ωpe t = 100 and 200. During the same time period it is seen that the positive velocity gradient associated with the resonant portion of electron distribution function gradually disappears until when ωpe t = 200, the velocity space plateau formation is almost complete. This is the well-known textbook example of linear instability excitation and subsequent quasilinear saturation. Physical processes that take place after the quasilinear saturation stage is dictated by the nonlinear wave dynamics. Beyond the velocity space plateau formation stage, the electron velocity distribution function does not undergo much change in the bulk velocity and the flattened beam range. However, near the energetic suprathermal region with v much higher than vT e , quasi-inverse power law tail population is gradually and spontaneously generated. We will discuss such an acceleration later, but for now let us focus on the nonlinear wave dynamics. Nonlinear processes are of two types. One is the resonant three-wave interaction, or decay process, and the other is the nonlinear wave-particle interaction, or scattering. In Figures 6.3 and 6.4, we show the time development of wave intensities. First, Figure 6.3 shows the dynamic spectrum of Langmuir wave intensity. In the plot, positive k region corresponds to the forward-propagating Langmuir wave (signified by σ = 1), while the negative k region designates the backward-propagating L mode (corresponding to σ = −1). We combined the two modes into a single figure, plotting Ik+L over positive k range, and Ik−L in k < 0 space. Actual numerical computation was done over k > 0 space for both Ik+L and Ik−L . For early time

132

Langmuir Turbulence and Electron Kappa Distribution log( ILk ) –2 –4 –6 –8 4000 3500 3000

ω pe t

2500 2000 1500 1000 500 0 –1

–0.5

0

0.5

1

kv th /ω pe

Figure 6.3 Nonlinear progression of Langmuir wave turbulence in the mode coupling stage.

periods between t = 0 and ωpe t = 200, or so, the Langmuir wave dynamics is simply dictated by the exponential growth and subsequent quasilinear saturation, which is already shown in Figure 6.2. During this stage we begin to see the growth associated with the backward Langmuir waves (k < 0 region). This is the result of combined three-wave decay process (that is, forward or primary L mode decaying into backward L mode and an ion-sound mode S) and scattering of forward L mode off thermal ions. It is also seen that Langmuir waves near k ∼ 0 slowly but steadily grow in intensity. This is again the result of combined decay and scattering, and such a long wavelength Langmuir mode generation is known as the Langmuir condensation effect. Nonlinear mode coupling processes involve multiple back-and-forth wave-wave interaction processes and scattering off thermal ions, and the entire mode coupling processes continue on well beyond the quasilinear saturation phase. Figure 6.4 is the corresponding dynamic spectrum associated with the ion-sound mode turbulence. In the early stage, between t = 0 and ωpe t = 200 or so, no ion-sound mode is apparent above initial noise level in both positive and negative k regions. However, around the time when the backward Langmuir wave begins to appear, it can be see that, first, the forward-propagating S mode wave becomes slightly enhanced, followed by the backward (k < 0) ion-sound mode waves.

6.2 Numerical Analysis of Weak Langmuir Turbulence

133

log( ISk ) –4 –6 –8 4000 3500 3000

ω pe t

2500 2000 1500 1000 500 0 –1

–0.5

0

0.5

1

kv th /ω pe

Figure 6.4 Nonlinear progression of ion-sound wave turbulence in the mode coupling stage.

The production of S mode turbulence is entirely owing to the decay process since the scattering process does not involve ion-sound waves. In the solar wind, the ratio Ti /Te varies (Gurnett et al., 1979; Newbury et al., 1998), but in general, unless Ti /Te is small, ion-sound waves may heavily damp thus prohibiting the three-wave resonant interaction. However, numerical solutions of the equations of electrostatic weak turbulence theory as well as electrostatic particle-in-cell simulation show that the three-wave interaction may still be operative even when Ti /Te is as high as an order unity (Yoon et al., 2012a; Rha et al., 2013). It is interesting to note that the ion-sound turbulence is a transient phenomenon. As the system evolves toward the end of the present numerical computation, that is, ωpe t = 4000, it is seen that the S mode wave intensity gradually settles down back toward the initial noise level. This is important for later discussion of asymptotically steady-state solution, or, equivalently, the so-called turbulent quasi-equilibrium (Treumann, 1999a,b). In the asymptotically steady state, t → ∞, we will solve for the steady-state particle distribution function as well as for the steady-state Langmuir wave spectral intensity, which form a coupled set of solutions. The notion of turbulent quasiequilibrium is the state of plasma in which the electrons and Langmuir fluctuations continuously exchange momentum and energy but, on average, the whole system is in dynamical equilibrium. Since the wave-wave or decay processes exchange

134

Langmuir Turbulence and Electron Kappa Distribution

10

0 4

fe ( v)

10–2

ω pe t = 2×10 4 1.8×10 1.4×104 1×104

10–4

4×103 3

1×10 10–6

6×102

10–8 t=0 –10

10

–15

–10

–5

0 v/ vTe

5

10

15

Figure 6.5 Development of energetic tail during the nonlinear mode-coupling stage.

momentum and energy only among the waves, asymptotically they are assumed to be insignificant. The present numerical calculation bears out such a feature. Returning to the Langmuir mode dynamic spectrum, the Langmuir condensation is intimately related to the acceleration of small amount of electrons to form an energetic tail population. This is because the long wavelength, or small k mode, leads to the resonant electron speed given by vres ≈ ωpe /k, such that, for small k, suprathermal electrons can resonate with the long wavelength Langmuir mode to form the high-energy tail distribution. In Figure 6.5, we display the long-time evolution of electron distribution function much beyond the mode coupling stage. Recall that Figures 6.3 and 6.4 show turbulent wave dynamics only up to ωpe t = 4000. In contrast, the electron velocity distribution function shown in Figure 6.5 is calculated up to ωpe t = 2 × 104 . Observe the formation of energetic tail population in the suprathermal energy range, which superficially resembles the kappa distribution. Ryu et al. (2007) confirmed the present findings based upon the weak turbulence formalism by means of particle-in-cell (PIC) simulation. That is, they demonstrated by means of PIC code simulation that the generation of kappa-like tail population in the late stages of beam-plasma instability process is consistent with the theoretical calculation. As we noted already, the kappa-like electron distribution function is pervasively observed in the near-Earth space environment. The question that

6.3 Langmuir Turbulence and Electron Kappa Distribution

135

naturally arises is whether the genuine electron kappa distribution function corresponds to the time asymptotic state of weak Langmuir turbulence or not. Or could it be that the numerical solution continually flattens the suprathermal portion of the electron velocity distribution with asymptotically steady-state solution? These questions cannot be addressed by the brute-force numerical solution. Instead, one must attack this issue with some form of analytical method. To address this issue, we next turn to an analytical approach and consider the steady-state solution for both electrons and Langmuir turbulence. As already noted, in the time asymptotic state, the ion-sound turbulence may be ignored since the S mode generation is only part of a transient phenomenon. The analytical approach to understand the formation of electron kappa distribution function in the asymptotically steady stage of Langmuir turbulence is useful since the numerical demonstration presented herewith is based upon one-dimensional system. The actual solar wind electron distribution is observed to be highly isotropic in three velocity dimensions. It is desirable to carry out the numerical analysis for fully three-dimensional system and demonstrate the formation of genuine three-dimensional electron kappa distribution function, but, thus far, such a demonstration has not been done. Instead, we bypass the issue of numerical demonstration, and directly attack the problem of three-dimensional kappa distribution by means of analytical approach with the assumption of threedimensional isotropy at the outset. The next section discusses such a solution. 6.3 Langmuir Turbulence and Electron Kappa Distribution 6.3.1 Steady-State Electron Distribution Consider the particle kinetic equation for electrons (6.3). In what follows, we assume that the wave dispersion relation depends only on the magnitude of k, and that the forward and backward wave intensities are identical and isotropic, ωkL = ωkL and Ikσ L = IkL , which are valid assumptions, provided the electron distribution function is isotropic, Fe (v) = Fe (v). We assume the steady state, ∂Fe /∂t = 0, by virtue of the velocity friction and diffusion terms balancing each other out, ∂Fe ∂ ∂Fe ∂ Ai Fe + Dij , =0= ∂t ∂vi ∂vi ∂vj  ki  e2 dk 2 σ ωL δ(σ ωkL − k · v), Ai = 4πme k σ =±1 k  ki kj  πe2 dk 2 δ(σ ωkL − k · v) IkL . Dij = 2 me k σ =±1

(6.20)

136

Langmuir Turbulence and Electron Kappa Distribution

We may decompose the wave vector into components perpendicular and parallel to the velocity vector: (v × k) × v (v · k) v + = k⊥ + k v2 v2 v = k⊥ eˆ ⊥ + k eˆ  = k⊥ eˆ ⊥ + k . |v|

k=

(6.21)

With this decomposition, we may write the drag (or friction) coefficient as follows:   ∞ k⊥ eˆ ⊥ + k eˆ  e2  ∞ A= dk⊥ k⊥ dk k v δ(σ ωkL − k v) 2 2 2me σ =±1 0 k + k −∞ ⊥    ∞ k2 v e2  ∞ ≈ eˆ  dk⊥ k⊥ dk 2 δ(σ ωpe − k v) 2me σ =±1 0 k⊥ + k2 −∞   e2 (k · v)2 dk δ(σ ωpe − k · v) = eˆ  4π me v σ =±1 k2

2   dk e2 ωpe δ(σ ωpe − k · v), = eˆ  4π me v σ =±1 k2

(6.22)

where the perpendicular component of A vanishes upon summing over both signs of σ = ±1. Upon making use of the notation eˆ  = v/v, we thus obtain A = v G(v),

  dk δ(σ ωpe − k · v). G(v) = 4π me v 2 σ =±1 k2 2 e2 ωpe

(6.23)

Consider next the diffusion coefficient,  2 2 eˆ ⊥ eˆ ⊥ k⊥ v + eˆ  eˆ  k2 v 2 πe2  dk δ(σ ωpe − k · v) IL (k) Dij = 2 2 me v σ =±1 k2  eˆ ⊥ eˆ ⊥ [1 − (k · v)2 ] + eˆ  eˆ  (k · v)2 πe2  dk = 2 2 me v σ =±1 k2 × δ(σ ωpe − k · v) IL (k) (6.24)  2 2 eˆ ⊥ eˆ ⊥ (1 − ωpe ) + eˆ  eˆ  ωpe πe2  dk δ(σ ωpe − k · v) IL (k), = 2 2 me v σ =±1 k2 where, again, the mixed terms disappear when summed over both signs of σ = ±1. We may rewrite this result as (Hasegawa et al., 1985)

6.3 Langmuir Turbulence and Electron Kappa Distribution

vi vj vi vj  Dij = D⊥ δij − 2 + D 2 , v v  2 2  πe (1 − ωpe ) dk D⊥ = δ(σ ωpe − k · v) IL (k), 2 2 me v k2 σ =±1 2   πe2 ωpe dk δ(σ ωpe − k · v) IL (k). D = 2 2 me v σ =±1 k2

137

(6.25)

It is instructive to rewrite the electron kinetic equation (6.20) in spherical velocity space coordinate system by making use of the following transformation properties between cylindrical and spherical coordinate representations: Av = A⊥ sin θ + A cos θ, Aθ = A⊥ cos θ − A sin θ, Dvv = D⊥⊥ sin2 θ + 2D⊥ sin θ cos θ + D cos2 θ, Dθθ = D⊥⊥ cos2 θ − 2D⊥ sin θ cos θ + D sin2 θ,

(6.26)

Dvθ = Dθv = (D⊥⊥ − D ) sin θ cos θ − D⊥ (sin θ − cos θ). 2

2

This leads to ∂ ∂Fe 1 1 ∂ =0= 2 (v 2 Av Fe ) + (Aθ Fe ) ∂t v ∂v  v sin  θ ∂θ   1 ∂ ∂Fe ∂Fe 1 ∂ 2 v Dvv + 2 vDvθ (6.27) + 2 v ∂v ∂v v ∂v ∂θ     1 ∂ 1 ∂ ∂Fe ∂Fe + sin θDθv + 2 sin θDθθ , v sin θ ∂θ ∂v v sin θ ∂θ ∂θ  2 where v = v⊥ + v2 and cos θ = v /v. If we denote μ = cos θ,

(6.28)

then we may alternatively write ∂Fe 1 ∂ 1 ∂ =0= 2 (v 2 Av Fe ) − (Aμ Fe ) ∂t v ∂v v ∂μ    ∂Fe ∂Fe 1 ∂ (1 − μ2 )1/2 2 v Dvv (6.29) + 2 − Dvμ v ∂v ∂v v ∂μ    ∂Fe ∂Fe 1 ∂ (1 − μ2 )1/2 2 1/2 (1 − μ ) . Dμv − − Dμμ v ∂μ ∂v v ∂μ From (6.23) and (6.24), we have A = eˆ v v G = eˆ v Av, Dvv = D,

Dvμ = Dμv = 0,

Dμμ = D⊥,

(6.30)

138

Langmuir Turbulence and Electron Kappa Distribution

which reduces (6.29) to   ∂Fe ∂Fe 1 ∂ 1 ∂ 2 2 v Dvv =0= 2 (v Av Fe ) + 2 ∂t v ∂v v ∂v ∂v   ∂F 1 ∂ e (1 − μ2 ) Dμμ . + 2 v ∂μ ∂μ For isotropic Fe , we further reduce (6.31) as    ∂Fe ∂Fe 1 ∂ 2 v Av Fe + Dvv . = 2 ∂t v ∂v ∂v

(6.31)

(6.32)

The steady-state solution is thus given by       vG Av Fe = C exp − dv = C exp − dv (6.33) Dvv D ⎞ ⎛ $ dk  δ(σ ω − k · v) pe σ =±1 me v ⎟ ⎜ k2 = C exp ⎝− dv ⎠. dk 4π 2 $ δ(σ ωpe − k · v) IL (k) σ =±1 k2





To compute the quantity vG/D , we define the angle between k and v by γ , or α = cos γ , dk = 4πk 2 dk d cos γ = 4πk 2 dk dα.

(6.34)

Then we have 1 ∞ $ vG me v σ =±1 −1 dα 0 dk δ(σ ωpe − kvα) =   $ D 4π 2 σ =±1 1 dα ∞ dk δ(σ ωpe − kvα) IL (k)

∞

me v = 4π 2



ωpe /v ∞ dk

ωpe /v

k

−1

dk k

0

.

(6.35)

IL (k)

∞ In (6.35) the integral ωpe /v dk/k formally diverges for k → ∞. In order to avoid such a problem, we invoke the following approach. We formally define  ∞  ∞ dk dk H (v) = (6.36) , H (v) I (v) = IL (k). ωpe /v k ωpe /v k Then we may formally remove the divergence, so that we have vG me v 1 = . D 4π 2 I (v)

(6.37)

6.3 Langmuir Turbulence and Electron Kappa Distribution

139

At this point, we anticipate the final result in that the electron distribution function in the asymptotic state of weak Langmuir turbulence is the kappa distribution. We thus force our steady-state solution to be the kappa-like solution, Fe =

C (1 + me

v 2 /2κ 

θe )κ+1

.

The normalization constant C can be obtained by requiring the condition   ∞ 4πC (2κ  θe )3/2 ∞ x2 1 = 4π dv v 2 Fe = dx . 3/2 (1 + x 2 )κ+1 me 0 0 Making use of the integral identity  ∞ μ−1 x dx (μ/2) (κ − μ/2) = , 2 κ (1 + x ) 2 (κ) 0

(6.38)

(6.39)

(6.40)

we readily obtain the normalized Fe , 3/2

(κ + 1) me 1 Fe = . 3/2 3/2 2  (2π θe ) κ (κ − 1/2) (1 + me v /2κ  θe )κ+1

(6.41)

The effective temperature for this kappa-like model can be computed on the basis  of definition Ta = dv(me v 2 /3)Fe . The result is Te = θe

κ . κ − 3/2

(6.42)

The proof is straightforward, which is omitted. The relevant question is what kind of wave spectrum IL (k) will be consistent with the kappa-like distribution (6.41). To answer this question, we consider the relationship    me v 1 1 . (6.43) = exp − dv (1 + me v 2 /2κ  θe )κ+1 4π 2 I (v) Taking the logarithm of both sides, making use of the integral identity    2v v2 dv 2 , = b ln 1 + v /b + 1 b and identifying b = 2κ  θe /me , we obtain   me v/θe κ +1 me v 1 dv = dv ,  2  κ me v /(2κ θe ) + 1 4π 2 I (v) or, equivalently,

  me v 2 θe κ  1+  . I (v) = 4π 2 κ + 1 2κ θe

(6.44)

(6.45)

(6.46)

140

Langmuir Turbulence and Electron Kappa Distribution

From (6.36), which can be rewritten upon replacing ωpe /v by k,  ∞   ∞  dk dk , H (k) I (k) = IL (k  ), H (k) =   k k k k we obtain 1 I (k) =  ∞   k dk /k

 k



dk  IL (k  ). k

(6.47)

(6.48)

The quantity I (k) on the left-hand side is already given by (6.46). As a consequence, we have    ∞  2 ∞ me ωpe θe κ  dk dk  IL (k  ). (6.49) 1 + = 4π 2 κ + 1 k k  2κ  k 2 θe k k Taking the derivative d/dk to both sides, %    2 2  ∞ me ωpe me ωpe 1 1 θe κ  dk  − 1 + − = − IL (k), 4π 2 κ + 1 k 2κ  k 2 θe κ  k 3 θe k k  k

(6.50)

we readily obtain

%  ∞   2  me ωpe dk θe κ  1+2 . 1+  2 IL (k) = 2 4π κ + 1 2κ k θe k k

(6.51)

Thus, we obtain the turbulence spectrum that is consistent with the kappa-like electron distribution function (6.41):   2 me ωpe θe κ  1 +  2 [1 + 2H (k)] , IL (k) = 4π 2 κ + 1 2κ k θe    ∞ 2 me ωpe dk 1 θe κ  1+  2 , I (k) = IL (k) = H (k) k k 4π 2 κ + 1 2κ k θe  ∞ dk H (k) = . (6.52) k k It should be noted that the forced solution (6.41) and the corresponding Langmuir turbulence intensity (6.52) are by no means unique solutions. In fact, at this stage, the coupled solutions (6.41) and (6.52) are but one in a class of infinitely many solutions. Even if we accept that these are the only acceptable coupled solutions, the indices κ and κ  are free parameters at this point. In order to determine the specific values for κ and κ  as well as to prove the uniqueness theorem for the kappa-like distribution function (6.41), we need to turn to the steady-state wave equations. As it turns out, when we consider the balance of spontaneous and induced processes in the steady-state wave kinetic equation, the

6.3 Langmuir Turbulence and Electron Kappa Distribution

141

resulting constraints on possible solutions drastically gets reduced, and the coupled solutions (6.41) and (6.52) become the only acceptable solution, provided κ and κ  are given by specific values. To show this, we next consider the steady-state wave kinetic equation. Before we do that, however, let us revisit the problem of wave dispersion relations when the kappa-like model defines the electron distribution function. 6.3.2 Electrostatic Waves with Kappa Distribution Consider the electrostatic dispersion relation 2   ωpa k · ∂Fa /∂v dv (k,ω) = 0 = 1 + . 2 k ω − k · v + i0 a

(6.53)

Decomposing v into component perpendicular and parallel to k, v = v⊥ + v = we have (k,ω) = 0 = 1 +

(k × v) × k (k · v) k + , k2 k2 2   ωpa a

k

dv

∂fa /∂v . ω − kv + i0

(6.54)

(6.55)

For model distribution (6.41) we obtain the reduced distribution and its derivative,  −κ  ∞ 1/2 me v2 (κ) me dv⊥ v⊥ Fa = , 1+  2π (2πθe )1/2 κ  1/2 (κ − 1/2) 2κ θe 0  −κ−1  ∞ 3/2 me v2 ∂Fa κ(κ) me 2π dv⊥ v⊥ =− . v 1 +  3/2 3/2 ∂v 2κ θe (2π)1/2 θe κ  (κ − 1/2) 0 (6.56) For high-frequency Langmuir waves, one may expand the resonant denominator by   1 1 kv k 2 v2 k 3 v3 ≈ (6.57) 1+ + 2 + 3 + ··· . ω − kv ω ω ω ω Under this assumption and making use of (6.56), one may obtain  2  ωpe 3k 2 θe κ 0=1− 2 1+ , ω κ − 3/2 me ω2

(6.58)

which leads to the desired Langmuir wave dispersion relation for kappa-like distribution (6.41),

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Langmuir Turbulence and Electron Kappa Distribution



ωkL = ωpe

k 2 θe 3 κ 1+ 2 2 κ − 3/2 me ωpe



.

(6.59)

The ion-sound mode dispersion relation can be obtained by approximating the resonant denominator by 1 1 ≈− , ω − kv kv

(6.60)

for electrons, but the ion response can be treated in the same way as we did for the Langmuir wave dispersion relation. For ions, we assume Mawellian distribution. Then it is straightforward to show that (6.55) reduces to  2  2 ωpi me ωpe κ − 1/2 3k 2 Ti + , (6.61) 0=1− 2 1+ ω mi ω 2 k 2 θe κ from which we obtain



1 + (3Ti /θe )[(κ − 1/2)/κ  ] , (κ − 1/2)/κ  + k 2 λ2De θe θe , λDe = . cS2 = mi 4πne2

ωkS = kcS

(6.62)

It is useful for later purposes to consider specific forms of dielectric constants corresponding to each wave mode, and the derivatives of dielectric constants for Langmuir and ion-acoustic waves:   2  2 ω κ m k θ 3 pe e e (k,σ ωkL ) = 1 − 1+ , 2 2 κ − 3/2 me ωpe (σ ωkL )2   2 ωpi 3Ti κ − 1/2 1 κ − 1/2 S 1+ + 2 2 , (k,σ ωk ) = 1 − S 2  θe κ κ k λDe (σ ωk ) σ ωkL 1 = , 2  (k,σ ωkL ) μk = k 3 λ3De

σ ωkL 1 = μ , k 2  (k,σ ωkS )  1/2   me κ 3Ti 1/2 . + mi κ − 1/2 θe

(6.63)

6.3.3 Steady-State Wave Kinetic Equation The starting point is the nonlinear spectral balance equation (4.87). The only difference in the customary weak turbulence theory that assumes a Maxwellian background versus the present formalism that assumes a kappa-like state is that the wave dispersive properties are modified via dispersion relations (6.59) and (6.62),

6.3 Langmuir Turbulence and Electron Kappa Distribution

143

as well as the properties (6.63). We find that the induced and spontaneous emission terms are formally identical to the previous results, except for definitions (6.59), (6.62), and (6.63). To compute the various three-wave decay coefficients, we must evaluate the quantity |χ (2) |2 , which is given by |χe(2) (k,σ  ωkL |k |χ

(2)



(k ,σ



ωkL |k

− −



k ,σ ωkL 

k ,σ ωkS



−σ



ωkL )|2

−σ



ωkL )|2

≈  ≈

κ − 1/2 κ κ − 1/2 κ

2 2

e2 (k · k )2 , 4θe2 k 2 k 2 |k − k |2 e2 [k · (k − k )]2 . 4θe2 k 2 k 2 |k − k |2 (6.64)

With these modified modified results, the equations that describe the decay processes are identical as before, except that the coupling coefficients are given by  L Vk,k 

= 

S Vk,k 

=

κ − 1/2 κ κ − 1/2 κ

2 2

π e2 (k · k )2 , 4 θe2 k 2 k 2 |k − k |2 π e2 [k · (k − k )]2 . 4 θe2 k 2 k 2 |k − k |2

(6.65)

Similarly, equations describing the induced and spontaneous scattering processes are formally identical except that the coupling coefficient is modified to  LL Uk,k 

=

κ − 1/2 κ

2

π e2 (k · k )2 2 m2 k 2 k 2 ωpe e

× δ[σ ωkL − σ  ωkL − (k − k ) · v].

(6.66)

Putting all the results together, we obtain the steady-state L mode weak turbulence kinetic equation for the kappa-like model: 2 πωpe





ne2 ∂Fe dv − k · v) Fe + σ ωkL Ikσ L k · 0= 2 k π ∂v  2   2 κ − 1/2 π e2  L  μk−k (k · k ) − dk σ ω k κ 2 θe2   k 2 k 2 |k − k |2 δ(σ ωkL

σ ,σ =±1



S  L σ L σL × δ(σ ωkL − σ  ωkL − σ  ωk−k Ik  ) σ ωk−k Ik     (k · k )2 κ − 1/2 2 πe2   dk dv − 2 κ m2e ωpe k 2 k 2  σ =±1



144

Langmuir Turbulence and Electron Kappa Distribution

%

 ne2 L  L σL L σ L σ ω ω I − σ ω I σ Fi   k k k k k 2 πωpe  me ∂Fi  . (6.67) − σ ωkL Ikσ L Ikσ L (k − k ) · mi ∂v × δ[σ ωkL − σ  ωkL − (k − k ) · v]

Let us assume that the forward- and backward-propagating Langmuir waves have the same isotropic spectral forms, Ik+L = Ik−L = Ik , as before, thus, omitting the subscript L from the wave intensity and Langmuir wave dispersion relation for the sake of simplicity. In deriving the result in (6.67), we have ignored the S mode contribution in the decay process. This is because, as we have seen in Figure 6.4, the ion-sound turbulence generation process by decay instability is a transient phenomenon, so that in the asymptotic stage, ion-sound turbulence is at the level of initial noise, hence S mode intensity can be ignored. Since the wave intensity is symmetric in k, it is sufficient to consider only σ = 1 in the wave kinetic equation. Thus, the relevant equation further reduces to   2 2  πωpe ne ∂Fe dv δ(ω − k · v) + ω I k · F 0= k e k h k2 π ∂v 2    2 κ − 1/2 π e2   μk−k (k · k ) dk − ω k κ 2 θe2   k 2 k 2 |k − k |2 

σ ,σ =±1  S σ ω|k−k | ) σ  ω|k−k | Ik Ik

× δ(ωk − σ ωk −     (k · k )2 κ − 1/2 2 ωk   dk dv − δ[ωk ∓ ωk − (k − k ) · v] κ 4πnTi +,− k 2 k 2   Ti (6.68) × (±ωk Ik − ωk Ik ) + Ik Ik (ωk ∓ ωk ) Fi , 4π 2 where we have taken advantage of the fact that the ion distribution is assumed to remain stationary and given by the Maxwellian form. The steady-state equation (6.68) must be solved for Langmuir wave intensity, Ik , but, at first sight, the general solution appears to be extremely difficult to obtain. However, the three basic processes, namely, spontaneous versus induced emission, decay, and scattering, can be separately considered. This is physically meaningful, since spontaneous and induced processes are supposed to balance each other out, according to the principle of detailed balance in the steady state. We thus consider the balance of each process. Balance of Induced and Spontaneous Emission Making use of 1 me (κ + 1)v ∂Fe Fe, =−  ∂v κ θe 1 + me v 2 /2κ  θe

(6.69)

6.3 Langmuir Turbulence and Electron Kappa Distribution

145

we may express the balanced equation of spontaneous and induced emission terms within (6.68),  2 πωpe ne2 dv δ(ωpe − kvμ) 0= k2 π   IL (k) κ + 1 4π 2 Fe . (6.70) × 1− κ  θe 1 + me v 2 /2κ  θe Let us integrate this equation in positive k space,    ∞  IL (k) κ + 1 4π 2 Fe dk dv δ(ωpe − kvμ) 1 − κ  θe 1 + me v 2 /2κ  θe 0  ∞  1  ∞ dk ωpe  = 4π dv v dμ δ μ− k kv 0 −1 0   IL (k) κ + 1 4π 2 Fe × 1−  κ θe 1 + me v 2 /2κθe    ∞ κ + 1 4π 2 H (v) I (v) Fe , = 4π dv v H (v) − (6.71) κ  θe 1 + me v 2 /2κθe 0 from which we obtain

  me v 2 κ  θe 1+ , I (v) = κ + 1 4π 2 2κθe

(6.72)

which is none other than (6.52), upon replacing v by ωpe /k. We thus see that the balance of induced and spontaneous emissions leads to the same solution as before. Let us next consider the balance of nonlinear terms. Absence of Contributions from Decay Process Consider the decay term,   κ − 1/2 2 π e2  κ 2 θe2  

σ ,σ =±1

 ωk

dk

μk−k (k · k )2 k 2 k 2 |k − k |2

S  × δ(ωk − σ  ωk − σ  ω|k−k  | ) σ ω|k−k | Ik  Ik     2 κ − 1/2 2 π e2  μk−k (k · k ) S δ(ωk − ωk − ω|k−k dk = ω |) k κ 2 θe2 k 2 k 2 |k − k |2 S S − δ(ωk − ωk + ω|k−k  | ) + δ(ωk + ωk  − ω|k−k | )

S − δ(ωk + ωk + ω|k−k  | ) ω|k−k | Ik  Ik .

(6.73)

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Langmuir Turbulence and Electron Kappa Distribution

Of the four delta function terms in (6.73), those involving ωk +ωk obviously, cannot be satisfied. Ignoring these terms, we have     2 κ − 1/2 2 π e2  μk−k (k · k ) dk ω ω|k−k | Ik Ik k κ 2 θe2 k 2 k 2 |k − k |2

S S × δ(ωk − ωk − ω|k−k (6.74)  | ) − δ(ωk − ωk  + ω|k−k | ) . The remaining two delta function terms cancel each other. To see this, let us consider the following: S δ(ωk − ωk ± ω|k−k |)      3 κ 3 κ 2 = δ ωpe 1 + k 2 λ2De − ωpe 1 + k  λ2De 2 κ − 3/2 2 κ − 3/2     1/2 me /θ )[(κ − 1/2)/κ ] 1 + (3T i e ± ωpe |k − k | λDe mi (κ − 1/2)/κ  + |k − k |2 λ2De    1 me |k − k | λDe 3 κ 2 2 2 = δ (k − k )λDe ± ωpe 2 κ − 3/2 mi [(κ − 1/2)/κ  ]1/2 1 2 κ − 3/2 2 ≈ δ[(k 2 − k  )λ2De ]. (6.75)  ωpe 3 κ

This shows that the two decay terms cancel each other upon ignoring terms of order √ me /mi , and hence proves that the three-wave term can be ignored. Balance of Induced and Spontaneous Scattering We next consider the balance of induced and spontaneous scattering terms (off thermal ions):     (k · k )2 κ − 1/2 2 ωk   dk dv 0=− δ[ωk ∓ ωk − (k − k ) · v] κ 4πnTi +,− k 2 k 2   Ti     × + I I − ω I I (ω ∓ ω ) Fi . (6.76) (±ω ) k k k k k k k k 4π 2 Explicitly writing down the summation over two signs and ignoring resonance conditions with the factor ωk + ωk , we obtain   (k · k )2  dv 2 2 δ[ωk − ωk − (k − k ) · v] 0 = dk k k   Ti × (6.77) (ωk Ik − ωk Ik ) + Ik Ik (ωk − ωk ) Fi . 4π 2 Let us write k = k + δk.

(6.78)

6.3 Langmuir Turbulence and Electron Kappa Distribution

147

The nonlinear wave-particle resonance condition ωk − ωk − (k − k ) · v = 0 can be satisfied only if the magnitude of resonant velocity is on the order of ion thermal speed, vres ∼ (ωk − ωk )/|k − k | ∼ vT i vT e . Making use of the Langmuir wave dispersion relation, it can be shown that the resonant velocity is roughly given by  vres ∼ 3 k − k θe /(2me ωpe ), when k and k are in opposite directions. This shows that in order for vres ∼ vT i vT e to be satisfied, k and k must be sufficiently close to each other, or |k − k | ∼ |δk| 1. Making use of this consideration, we may employ the Taylor series expansion to obtain dω(k) dI (k) I (k) − ω(k) δk · , dk dk dω(k) Ik Ik (ωk − ωk ) = −[I (k)]2 δk · , (6.79) dk from which we have   (k · k )2 (6.80) 0 = d(δk) dv 2 2 δ[ωk − ωk − (k − k ) · v] k k   dI (k) 4π 2 dω(k) dω(k) × (δk) ω(k) + [I (k)]2 − I (k) Fi . dk Ti dk dk ωk Ik − ωk Ik = δk ·

Making use of dispersion relation (6.59) and its derivative with respect to k, we obtain the necessary condition for equality: dI (k) 4π 2 dω(k) dω(k) + [I (k)]2 − I (k) dk Ti dk dk   κ 3kθe dI (x) 1+ = x me ωpe κ − 3/2 dx  2  κ κ 4π 2 [I (x)] − I (x) , + Ti κ − 3/2 κ − 3/2

0 = ω(k)

(6.81)

where we have defined a dimensionless variable x≡

3k 2 θe . 2 2me ωpe

Equation (6.81) is exactly satisfied for   2 κ − 3/2 Ti 1+ . I (k) = 2 4π 2 3 κ  k 2 θe /me ωpe

(6.82)

(6.83)

Upon comparing (6.83) with the earlier solution (6.52), it immediately becomes obvious that the two expressions are identical, provided we identify   3 κ 4 κ− = 1 + 2H , (6.84) , Ti = θe κ +1 3 2

148

Langmuir Turbulence and Electron Kappa Distribution

or, equivalently, κ=

9 3 + H = 2.25 + 1.5H , 4 2

κ  θe = (κ + 1) Ti ,

(6.85)

where we have treated the correction factor H as constant. This reconciliation between (6.52) and (6.83) would not have been possible had we assumed any other form of electron distribution than the kappa-like model (6.41). Note that the notion of electron distribution function did not enter in the derivation of (6.83). Rather, (6.83) was derived on the basis of balancing the nonlinear scattering terms, which involved ions only. Nevertheless, the end result is mathematically identical to (6.52). Had we assumed some other form of electron velocity distribution function, then one would have obtained a totally different form of wave intensity when compared with (6.52), which would have made the reconciliation impossible. This shows that the kappa distribution is the only solution that is allowed in the asymptotically steady-state Langmuir turbulence system, thus proving the uniqueness theorem for kappa distribution. Here, we should note that if the system is initially placed at an isotropic thermal equilibrium state (that is, Maxwellian distribution), then the system will never spontaneously evolve toward the turbulent equilibrium state characterized by kappa distribution function. The system must be first perturbed, say, by an instability (in this case, bump-on-tail instability), or subject the system to some other forms of large amplitude disturbance. Then one must “wait and watch,” as it were, before the system resettles to a new quasi-equilibrium. In the case of Langmuir turbulence, the new quasi-equilibrium is characterized by the electron kappa velocity distribution function, but it is important to note that an initially Maxwellian electron distribution can never spontaneously make the transition to kappa distribution without some disturbances. Of course, one such disturbance is the gentle electron beam, and the mechanism that allows the transition is the bump-on-tail instability. The discussion presented in this section did not address how the initial system is perturbed, but the numerical analysis of bump-on-tail instability in the previous section already provided some clues as to how the initial system can be perturbed, and how the perturbed system gradually tends toward the turbulent quasi-equilibrium. In the present section we did not address the issue of how long it takes for the system to make the transition to quasi-equilibrium either, but again, the numerical demonstration of the previous section provides some clues as to the time scales involved. Instead, in the present section, we have addressed the issue of how to describe the final state when the dust is settled, and the system has finally managed to make the transition to the new turbulent quasi-equilibrium. Even though, strictly speaking, we do not quite have any exact way to estimate the turbulent quasi-equilibrium time scale, we may surmise on the basis of

6.3 Langmuir Turbulence and Electron Kappa Distribution

149

numerical solution of the previous section that the nonlinear development of bumpon-tail instability and Langmuir turbulence proceeds over distinct time scales, that is, quasilinear saturation time scale, tQL ; nonlinear mode coupling time scale, tNL ; turbulent quasi-equilibrium time scale, tturb ; and finally collisional relaxation time scale, tcoll . On the basis of numerical demonstration of the previous section, for the given set of parameters adopted in the numerical example, we estimate the following: ωpe tQL ∼ O(102 ), ωpe tNL ∼ O(103 ) to O(104 ), ωpe tturb ∼ O(104 ) to O(105 ), ωpe tcoll O(105 ). Another point we should make is that the kappa distribution function is a quasiequilibrium state, rather than a true thermodynamic equilibrium. In the derivation of time asymptotic solution, we have relied on the particle kinetic equation in which the velocity space diffusion process driven by the wave intensity is balanced by the velocity space drag effect. The kappa distribution exactly satisfies the steady-state condition for the particle kinetic equation. However, over time scales much longer than even the turbulent quasi-equilibrium time scale, eventually binary collisional processes will kick in. Processes that are operative over collisional time scales have not been included in the present discussion – but the recent work by Tigik et al. (2016) addresses some aspects of this problem. In any case, once we subject the kappa distribution function to collisional kinetic equation, then, over much longer collisional time scale, the system will ultimately evolve toward a genuine thermodynamic equilibrium state. Before we close, let us address the issue of H : ∞  ∞   dk  = ln k (6.86) H =  . k k k =k The quantity H formally diverges for k  = ∞ as well as for k  = 0. To remedy the divergent situation, we introduce the lower and upper cutoffs, kmin and kmax : H = ln

kmax . kmin

(6.87)

If we choose these quantities as kmin =

1 , λDe

kmax =

θe , e2

(6.88)

150

Langmuir Turbulence and Electron Kappa Distribution

in analogy with (5.34) and (5.35), then we obtain the customary Coulomb logarithm H = ln

λDe θe ≡ ln , e2

3/2

=

θe 1 = , (4πn)1/2 e3 g

g=

1 . 4πnλ3De

(6.89)

Alternatively, we may simply consider kmin and kmax as undetermined and treat H as a free adjustable parameter. 6.3.4 Electron Kappa Distribution and Steady-State Langmuir Turbulence Let us summarize our findings concerning the quasi-equilibrium state of weak Langmuir turbulence and electron kappa distribution function. Since according to (6.42), θe = Te

κ − 3/2 , κ

we may express our final solution as  −κ−1 3/2 1 me v 2 (κ + 1) me 1+ , Fe (v) = (2πTe )3/2 (κ − 3/2)3/2 (κ − 1/2) κ − 3/2 2Te   1 + 2H 2πne2 Te κ − 3/2 1+ , (6.90) I (k) = 4π 2 κ + 1 κ − 3/2 k 2 Te κ=

9 3H + = 2.25 + 1.5H , 4 2

κ − 3/2 Ti 3 + 6H = = . Te κ +1 13 + 6H

This solution, which was obtained by Yoon (2014), represents the steady-state Langmuir turbulence and electron kappa distribution that together form a quasiequilibrium physical state in which the electrons and Langmuir waves exchange momenta and energies in a dynamically steady state fashion. The steady-state Langmuir turbulence characterized by kappa electron velocity distribution function may be highly relevant to the solar wind plasma. The kappa electron distribution (6.90) predicts that for suprathermal energy range, the velocity distribution should behave as an inverse power law, Fe ∼ v −6.5,

for v vT e,

(6.91)

since κ ≈ 9/4 = 2.25, assuming H can be ignored. Such a velocity power-law spectral index compares favorably against observations made in the solar wind. Typically, the solar wind electrons can be modeled as being comprised of three components, the Maxwellian “core,” a suprathermal “halo,” and a highly fieldaligned component called the “strahl” that is observed to be streaming away from the Sun. The strahl is typically associated with the high-speed or fast solar wind,

6.3 Langmuir Turbulence and Electron Kappa Distribution

151

while the core-halo two-component feature is more common in the slow-speed or slow solar wind. While customary discussions of the solar wind electrons often do not include the fourth component known as the “superhalo,” these highly energetic electrons are observed to possess nearly isotropic angular distribution and an inverse power law velocity spectrum (Lin, 1998). The characteristic energy for the Maxwellian core is tens of eV, while the halo component is defined up to ∼102 –103 eV or so. The highly energetic superhalo is characterized by 103 –105 eV. The superhalo electrons are observed in all solar wind conditions, whether active or quiet, and their velocity power law indices do not show much dependence on solar conditions (Wang et al., 2012a). The strahl component occupies a similar energy range as the halo (∼102 –103 eV). These electron components are distinguishable from each other on the basis of their morphology in velocity (or energy) space distribution functions. It is generally known from observations that at 1 AU (= 149.6 × 106 km, the astronomical unit, which is the distance from the Sun to Earth), the Maxwellian component makes up about ∼95%, and, in terms of the number density, the halo is roughly estimated to possess ∼4% density, and the strahl component takes up, approximately, the remaining 1%. The superhalo component is very low in terms of number density (almost negligible, on the order of 10−9 –10−6 when compared with the core density), yet owing to their high energy, their presence can be clearly identified in the velocity distribution function. Instead of fitting the solar wind electron velocity distribution function by various components, it is possible to model the entire solar wind electrons by a single kappa model (Olbert, 1968; Vasyliunas, 1968; Maksimovic et al., 1997, 2005). Measured electron velocity distribution functions for suprathermal energy range (the superhalo component of the solar wind) near Earth orbit behave as Fe ∼ v −5.0 to v −8.7 with average behavior (Wang et al., 2012a) Feobs ∼ v −6.69,

v vT e .

(6.92)

Note that (6.92) agrees quite well with (6.91). Moreover, taking into account only ˇ the so-called halo electron component, Stver´ ak et al. (2009) analyzed Helios, Cluster, and Ulysses spacecraft data in order to show that the value of κ decreases from ∼9 near 0.3 AU to ∼4 near 1 AU, to ∼2.25 near ∼5 AU. This shows that as the solar wind evolves radially, the solar wind electrons seem to increasingly evolve toward the turbulent quasi-equilibrium state predicted by Yoon (2014). As for the steady-state Langmuir turbulence, the solar wind is also replete with pervasive electrostatic fluctuations with peak frequency near plasma frequency. Such an electrostatic fluctuation is known in the literature as quasi-thermal noise fluctuations (Meyer-Vernet, 1979). The quasi-thermal noise represents the spontaneously emitted electromagnetic fluctuations that take place in a plasma, which may be close to but not necessarily at thermal equilibrium state (Sitenko, 1967).

152

Langmuir Turbulence and Electron Kappa Distribution

A theoretical tool to analyze quasi-thermal noise is known as the quasi-thermal noise “spectroscopy,” which was developed by Meyer-Vernet (1979) in the context of solar wind. The quasi-thermal noise spectroscopy is a versatile research tool that can be employed to indirectly determine not only the total electron density and temperature (Meyer-Vernet et al., 1986; Maksimovic et al., 1995) but also properties of energetic and, often, nonthermal electrons (Zouganelis, 2008). In this regard, the thermal noise spectrum ubiquitously observed in the solar wind may be none other than the steady-state Langmuir turbulence intensity as specified by (6.90).

Part III Vlasov Weak Turbulence Theory: Electromagnetic Formalism

7 Nonlinear Electromagnetic Equations in Vlasov Plasmas

In Part III, we extend the formalism outlined in Part I to the fully electromagnetic case. One of the most prominent examples demonstrating the usefulness of electromagnetic weak turbulence theory is on the radiation emission with frequency close to the plasma frequency and/or harmonics. Such a process is known as plasma emission, and it is the basic mechanism for type II and type III solar radio bursts. It is the process that is intimately related to the beam-plasma instability. We have seen in Chapter 6 that the bump-on-tail instability leads to the excitation of Langmuir waves, which undergo nonlinear mode coupling process involving ion-sound waves. If one includes electromagnetic effects, then mode coupling also involves the transverse radiation. The emission at the plasma frequency and/or its second harmonic is the consequence of the mode coupling process involving transverse mode. The electromagnetic formalism to be developed in Parts III and IV of this book thus can be applied to the problem of plasma emission. Before we do that, however, let us proceed with the fundamental development of formalism. 7.1 Nonlinear Electrodynamic Equations in Plasmas The starting point is the Vlasov–Maxwell system of equations for unmagnetized plasmas:   ∂  v ∂ ea ∂ fa (r,v,t) = 0, E(r,t) + × B(r,t) · +v· + ∂t ∂r ma c ∂v ∂ 1 ∂B(r,t) × E(r,t) + = 0, ∂r c ∂t   ∂ ∂ ea dv fa (r,v,t), · E(r,t) = 4π · B(r,t) = 0, ∂r ∂r a  ∂ 1 ∂E(r,t) 4π  ea dv v fa (r,v,t). (7.1) × B(r,t) − = ∂r c ∂t c a 155

156

Nonlinear Electromagnetic Equations in Vlasov Plasmas

We again separate physical quantities into the customary average and fluctuating parts: fa (r,v,t) = nFa (v,t) + δfa (r,v,t), E(r,t) = δE(r,t),

B(r,t) = δB(r,t).

(7.2)

This leads to the following:

 ∂nF (v,t) v ea ∂nFa (v,t) a δE(r,t) + × δB(r,t) · + ∂t ma c ∂v   ∂ ∂ δfa (r,v,t) + +v· ∂t ∂r  ∂ ea v + δE(r,t) + × δB(r,t) · δfa (r,v,t) = 0, ma c ∂v ∂ 1 ∂ δB(r,t) × δE(r,t) + = 0, ∂r c ∂t   ∂ ea dv δfa (r,v,t), · δE(r,t) = 4π ∂r a ∂ · δB(r,t) = 0, ∂r  ∂ 1 ∂ δE(r,t) 4π  ea dv v δfa (r,v,t). × δB(r,t) − = ∂r c ∂t c a

(7.3)

Upon averaging (7.3) over the random phases of the fluctuations, we obtain the formal particle kinetic equation:  ∂nFa (v,t) v ea ∂ =− · δE(r,t) δfa (r,v,t) + × δB(r,t) δfa (r,v,t) . ∂t ma ∂v c (7.4) Subtracting (7.4) from (7.3), we obtain the equation for perturbed distribution function:    ∂nF (v,t) ∂ ea v ∂ a δfa (r,v,t) = − δE(r,t) + × δB(r,t) · +v· ∂t ∂r ma c ∂v v ea ∂ − · δE(r,t) δfa (r,v,t) + × δB(r,t) δfa (r,v,t) ma ∂v c  v − δE(r,t) δfa (r,v,t) − × δB(r,t) δfa (r,v,t) . c (7.5) Taking spectral transformation for the fluctuations following the definitions in (1.38), but including now the magnetic field fluctuation, and including the slow a (v,t), δEk,ω (t), and and adiabatic time dependence for spectral amplitudes, δfk,ω δBk,ω (t), we now have the formal particle kinetic equation

7.1 Nonlinear Electrodynamic Equations in Plasmas

157

   ∂nFa (v,t) k·v vj ki ea ∂ δij + dk dω 1 − =− ∂t ma ∂vi ω ω , j a × δE−k,−ω (t) δfk,ω (v,t) , 

the wave equation



∂ ω+i ∂t



(7.6)



δBk,ω (t) = ck × δEk,ω (t),   a k · δEk,ω (t) = −4πi ea dv δfk,ω (v,t),



∂ ck × δBk,ω (t) + ω + i ∂t

a

 δEk,ω (t) = −4πi



 ea

a dv v δfk,ω (v,t),

a

k · δBk,ω (t) = 0, and the equation for perturbed distribution function   ∂ a δfk,ω ω−k·v+i (v,t) ∂t    vi kj ∂nFa (v,t) k·v ea i δij + δEk,ω (t) 1 − = −i ma ω ω ∂vj       vi kj k ·v ea ∂   dk dω 1− δij +  −i ma ∂vj ω ω i  i 

a a × δEk,ω (t) δfk−k,ω−ω (v,t) − δEk,ω (t) δfk−k,ω−ω (v,t) .

(7.7)

(7.8)

On the right-hand side of (7.8), we have ignored the adiabatic time-dependence in converting the magnetic field fluctuation to electric field fluctuation. That is, we have made use of the relation v v δEk,ω + × δBk,ω = δEk,ω + × (k × δEk,ω ) c ω  k k·v δEk,ω + (v · δEk,ω ). (7.9) = 1− ω ω Note that, in (7.8), the arguments (k,ω ) and (k − k,ω − ω ) are not symmetrized yet. Eventually, these arguments should be fully symmetrized with respect to dummy integral variables. Let us consider the fast time-scale solution by absorbing the slow time derivative as part of the new definition for ω, as in (1.41). Let us also define the shorthand notation as in (1.42), except that we now generalize to electromagnetic formalism: 1 , ω − k · v + i0    vi kj ∂ k·v iea δij + 1− =− . ma ω ω ∂vj

gk,ω = Lik,ω

(7.10)

158

Nonlinear Electromagnetic Equations in Vlasov Plasmas

Then, omitting δ in front of perturbed quantities for the sake of simplicity, (7.8) is re-expressed as follows:   a i i  dω Lik,ω fk,ω = gk,ω Lk,ω Fa Ek,ω + gk,ω dk  a  a i i . (7.11) × fk−k ,ω−ω Ek ω − fk−k,ω−ω Ek ω a We again solve (7.11) for fk,ω in an iterative fashion, as in (1.44): a(1) i = gk,ω Lik,ω Fa Ek,ω , (7.12) fk,ω   ,

- a(2) a(1) a(1) i i fk,ω = gk,ω dk dω Lik,ω fk−k , ,ω−ω Ek,ω − fk−k,ω−ω Ek,ω   ,

- a(3) a(2) a(2) i i fk,ω = gk,ω dk dω Lik,ω fk−k , ,ω−ω Ek,ω − fk−k,ω−ω Ek,ω

etc., or, inserting lower-order solution to successively higher-order equation, we have a(1) i = gk,ω Lik,ω Fa Ek,ω , fk,ω   j a(2) = gk,ω dk dω Lik,ω gk−k,ω−ω Lk−k,ω−ω nFa fk,ω

, - j j × Eki ,ω Ek−k,ω−ω − Eki ,ω Ek−k,ω−ω ,     j a(3)    dω dk dω Lik,ω gk−k,ω−ω Lk,ω fk,ω = gk,ω dk

j l × gk−k −k,ω−ω −ω Llk−k −k,ω−ω −ω nFa Eki ,ω Ek,ω Ek−k  −k,ω−ω −ω , - , - j j l i l −Eki ,ω Ek,ω Ek−k ,  −k,ω−ω −ω − Ek,ω Ek,ω Ek−k −k,ω−ω −ω

(7.13) a(1) a a is given by adding each component, fk,ω = fk,ω + etc. The series solution for δfk,ω a(2) a(3) fk,ω + fk,ω . We reiterate that the iterative solution (7.13) is not yet symmetrized. The symmetrized expression will be given below in (7.16), but before we do that, let us introduce some further simplified notations:



K = (k,ω), K1 = (k1,ω1 ), K2 = (k2,ω2 ), K3 = (k3,ω3 ),     = dk1 dk2 δ(k1 + k2 − k) dω1 dω2 δ(ω1 + ω2 − ω),

K1 +K2 =K





 =

K1 +K2 +K3 =K

×



dk1

dk2

dk3 δ(k1 + k2 + k3 − k)







dω1

dω2

dω3 δ(ω1 + ω2 + ω3 − ω),

7.1 Nonlinear Electrodynamic Equations in Plasmas

g(K) = gk,ω,

159

g(K1 ) = gk1,ω1 ,

g(K2 ) = gk2,ω2 ,

g(K1 + K2 ) = gk1 +k2,ω1 +ω2 , i Li (K) = Lk,ω, Li (K1 ) = Lik1,ω1 , Li (K2 ) = Lik2,ω2 , Li (K1 + K2 ) = Lik1 +k2,ω1 +ω2 , Ei (K1 ) = Eki 1,ω1 , Ei (K2 ) = Eki 2,ω2 , Ei (K1 + K2 ) = Eki 1 +k2,ω1 +ω2 .

(7.14)

Sometimes we will use even more abbreviated notations, such as g(1) = g(K1 ),

L(1 + 2) = L(K1 + K2 ),

etc.

(7.15)

In terms of these notations, the fully symmetrized expression for iterative solution to (7.8) finally emerges, fa (K) = g(K) Li (K) nFa Ei (K)

1  g(1 + 2) Li (1) g(2) Lj (2) + Lj (2) g(1) Li (1) + 2 1+2=K  

× nFa Ei (1) Ej (2) − Ei (1) Ej (2) 1  g(1 + 2 + 3)Li (1) g(2 + 3) (7.16) + 2 1+2+3=K

× Lj (2) g(3) Lk (3) + Lk (3) g(2) Lj (2) nFa    

× Ei (1) Ej (2) Ek (3) − Ei (1) Ej (2) Ek (3) − Ei (1) Ej (2) Ek (3) . We next insert (7.16) to the right-hand side of wave equation (7.7), which, upon absorbing the slow-time derivative to ω, is expressed as 

   c2 k 2 ki kj 4πi  j a δij − 2 δij − 2 δEk,ω = − ea dv vi δfk,ω . ω k ω a

(7.17)

The result, expressed in terms of shorthand notations, is the following:   c2 k 2 ki kj Ej (K) 0 = δij − 2 δij − 2 ω k  4πea n  dv vi g(K) Lj (K) Fa Ej (K) +i ω a  i   4πea n dv vi g(1 + 2) + 2 1+2=K a ω 

(7.18)

160

Nonlinear Electromagnetic Equations in Vlasov Plasmas



 

× Lj (1) g(2) Lk (2) + Lk (2) g(1) Lj (1) Fa Ej (1) Ek (2) − Ej (1) Ek (2)  i   4πea n dv vi g(1 + 2 + 3) Lj (1) g(2 + 3) + 2 1+2+3=K a ω

× Lk (2) g(3) Ll (3) + Ll (3) g(2) Lk (2) Fa  

× Ej (1) Ek (2) El (3) − Ej (1) Ek (2) El (3) − Ej (1) Ek (2) El (3) .

This result can be compactly rewritten as follows:  (2)  

χij k (1|2) Ej (1) Ek (2) − Ej (1) Ek (2) 0 = ij (K) Ej (K) + 1+2=K



+

χij(3)kl (1|2|3) Ej (1) Ek (2) El (3)

(7.19)

1+2+3=K

 

− Ej (1) Ek (2) El (3) − Ej (1) Ek (2) El (3) , where

  c2 k 2 ki kj ij (K) = ij (K) − 2 δij − 2 , ω k  ij (K) = δij + χij (K) = δij + χija (K), a

 4πiea n a dv vi g(K) Lj (K) Fa, χij (K) = ω  χij(2)k (1|2) = χija(2) k (1|2), a

 i 4πea n dv vi g(1 + 2) χija(2) (1|2) = k 2 ω1 + ω 2

× Lj (1) g(2) Lk (2) + Lk (2) g(1) Lj (1) Fa,  a(3) χij kl (1|2|3), χij(3)kl (1|2|3) =

(7.20)

a

χija(3) kl (1|2|3)

 4πea n i dv vi g(1 + 2 + 3) = 2 ω1 + ω 2 + ω 3

× Lj (1) g(2 + 3) Lk (2) g(3) Ll (3) + Ll (3) g(2) Lk (2) Fa .

Equation (7.19) directly generalizes (1.65) to fully electromagnetic circumstances. The electromagnetic response functions defined in (7.20) also directly generalize the electrostatic counterparts (1.55)–(1.57). It is instructive to write the various response functions in explicit forms:    2  ωpa vi k·v ∂ vj ∂ a dv 1− Fa, + k· χij (k,ω) = ω ω − k · v + i0 ω ∂vj ω ∂v

7.1 Nonlinear Electrodynamic Equations in Plasmas

161



2 vi i ea ωpa dv 2 m a ω1 + ω 2 ω1 + ω2 − (k1 + k2 ) · v + i0    1 vj ∂ k1 · v ∂ + k1 · × 1− ω1 ∂vj ω1 ∂v ω2 − k2 · v + i0    k2 · v ∂ vk ∂ × 1− + k2 · ω2 ∂vk ω2 ∂v    1 ∂ k2 · v vk ∂ + 1− + k2 · ω2 ∂vk ω2 ∂v ω1 − k1 · v + i0    ∂ k1 · v vj ∂ Fa, × 1− + k1 · ω1 ∂vj ω1 ∂v 2 ωpa (−i)2 ea2

χija(2) k (k1,ω1 |k2,ω2 ) = −

χija(3) kl (k1,ω1 |k2,ω2 |k3,ω3 ) =

2 

m2a ω1 + ω2 + ω3

vi dv ω1 + ω2 + ω3 − (k1 + k2 + k3 ) · v    ∂ vj ∂ k1 · v + k1 · × 1− ω1 ∂vj ω1 ∂v 1 × ω2 + ω3 − (k2 + k3 ) · v + i0    1 ∂ k2 · v vk ∂ × 1− + k2 · ω2 ∂vk ω2 ∂v ω3 − k3 · v + i0    k3 · v ∂ vl ∂ × 1− + k3 · ω3 ∂vl ω3 ∂v    1 k3 · v ∂ vl ∂ + 1− + k3 · ω3 ∂vl ω3 ∂v ω2 − k2 · v + i0    ∂ vk ∂ k2 · v Fa . + k2 · (7.21) × 1− ω2 ∂vk ω2 ∂v ×

These are now linear dielectric susceptibility tensor, second-order nonlinear susceptibility tensor, and (the partial) third-order nonlinear susceptibility tensor, rather than constants. As with (1.66), we take the dot product of (7.19) with Ei (K  ), and take the ensemble average:  (2)     χij k (1|2) Ei (K  ) Ej (1) Ek (2) 0 = ij (K) Ei (K  ) Ej (K) + + −



1+2=K

  χij(3)kl (1|2|3) Ei (K  ) Ej (1) Ek (2) El (3)

1+2+3=K  

Ei (K  ) Ej (1) Ek (2) El (3)

162

Nonlinear Electromagnetic Equations in Vlasov Plasmas

 (2)     = ij (K) Ei Ej K δ(K + K  ) + χij k (1|2) Ei (K  ) Ej (1) Ek (2) 1+2=K



+ δ(K + K  )

  χij(3)kl (1|2|3) δ(K  + 2) Ei Ek K  Ej El 1

1+2+3=K

 

+ δ(K1 + 2) Ei El K  Ej Ek 1   = ij (K) Ei Ej K δ(K + K  )  (2)   χij k (1|K − 1) Ei (K  ) Ej (1) Ek (K − 1) + 1

+ 2 δ(K + K  )



  χij(3)kl (1| − 1|K) Ei El K Ej Ek 1 ,

(7.22)

1

where in the last equality, 1, 2, 3, etc. stand for K1 , K2 , K3 , etc. To determine the third-body correlation, we approach the problem in the same way as in (1.73), and take the steps, which we have taken thereafter, until we arrive at (1.78). In this regard, it is useful to consider the solution to (7.18) without the third-order nonlinear terms, and express it in direct analogy with and generalization of (1.76), Ei(1) (K  ) = −−1 il (K)



(2) χlmn (2|K  − 2)

2

 

× Em(0) (2) En(0) (K  − 2) − Em(0) (2) En(0) (K  − 2) ,  (2) Ej(1) (1) = −−1 χlmn (2|1 − 2) j l (1) 2



 

− 2) − Em(0) (2) En(0) (1 − 2) ,  (2) Ek(1) (K − 1) = −−1 χlmn (2|K − 1 − 2) kl (K − 1) ×

Em(0) (2) En(0) (1

(7.23)

2

 

× Em(0) (2) En(0) (K − 1 − 2) − Em(0) (2) En(0) (K − 1 − 2) , and upon writing 

 , Ei (K  ) Ej (1) Ek (K − 1) ≈ Ei(1) (K  ) Ej(0) (1) Ek(0) (K − 1) , + Ei(0) (K  ) Ej(1) (1) Ek(0) (K − 1) , + Ei(0) (K  ) Ej(0) (1) Ek(1) (K − 1) ,

(7.24)

7.1 Nonlinear Electrodynamic Equations in Plasmas

163

we may obtain the third-body cumulant by virtue of (7.23). After everything is said and done, as it were, we then drop the superscript (0). This results in the following generalization of (1.78):   Ei (K  ) Ej (1) Ek (K − 1)  (2)    χlmn (2|K  − 2) Em (2) En (K  − 2) Ej (1) Ek (K − 1) = −−1 il (K ) 2

 

− Em (2) En (K  − 2) Ej (1) Ek (K − 1)  (2)   − −1 (1) χlmn (2|1 − 2) Ei (K  ) Ek (K − 1) Em (2) En (1 − 2) jl 

2



− Ei (K ) Ek (K − 1) Em (2) En (1 − 2)  (2) − −1 χlmn (2|K − 1 − 2) kl (K − 1) 



2

 × Ei (K ) Ej (1) Em (2) En (K − 1 − 2) 

 − Ei (K  ) Ej (1) Em (2) En (K − 1 − 2) !   (2) = −2 δ(K + K  ) −1 il (−K) χlmn (−1| − K + 1) Ej Em 1 Ek En K−1 



(2) + −1 j l (1) χlmn (K|1 − K) Ei En K Ek Em K−1  " (2) E  E + −1 (K − 1) χ (K| − 1) E E . i n j m K lmn kl 1

(7.25)

Inserting (7.25) to (7.22), we obtain the formal nonlinear spectral balance equation,   0 = ij (K) Ei Ej K !  (2) (2)   −2 χij k (K  |K − K  ) −1 j l (K ) χlmn (−K + K |K) Ek Em K−K  K

 "  (2)   Ei En K + −1 (K − K ) χ (−K |K) E E j m lmn kl K

#   (2)   E  + −1 (−K) χ (−K | − K + K ) E E E  j m K k n K−K lmn il  (3)   χij kl (K  | − K  |K) Ei El K Ej Ek K  . +2

(7.26)

K

7.1.1 Formal Nonlinear Wave Kinetic Equation To determine the adiabatic time evolution of wave energy, we reintroduce the slow time dependence to ij (K), as in (1.83), except that we now deal with the electromagnetic linear response function

164

Nonlinear Electromagnetic Equations in Vlasov Plasmas



ij (K) → ij

∂ K +i ∂t



≈ ij (K) +

i ∂ij (K) ∂ . 2 ∂ω ∂t

(7.27)

If we further assume that |Im ij (K)| |Re ij (K)|, as before, which implies weakly growing or damped mode, then we rewrite (7.26) in long-hand notation by 0=

i ∂ Re ij (k,ω) ∂ Ei Ej k,ω 2 ∂ω ∂t + Re ij (k,ω) Ei Ej k,ω + i Im ij (k,ω) Ei Ej k,ω      dω χij(2)k (k,ω |k − k,ω − ω ) −2 dk ! (2)     × −1 j l (k ,ω ) χlmn (−k + k , − ω + ω |k,ω) Ek Em k−k,ω−ω " (2)     ,ω Ei En k,ω + −1 (k − k ,ω − ω ) χ (−k , − ω |k,ω) E E  j m k lmn kl + χij(2)k (k,ω |k − k,ω − ω ) −1 il (−k, − ω) (2) (−k, − ω | − k + k, − ω + ω ) Ej Em k,ω Ek En k−k,ω−ω × χlmn    + 2 dk dω χij(3)kl (k,ω | − k, − ω |k,ω) Ei El k,ω Ej Ek k,ω .



(7.28) Note that the following symmetry relations can be immediately deduced from definitions (7.21): χija (−k, − ω) = χija∗ (k,ω), a(2)∗ χija(2) k (−k1, − ω1 | − k2, − ω2 ) = χij k (k1,ω1 |k2,ω2 ),

χija(3) kl (−k1,

− ω1 | − k2, − ω2 | − k3, − ω3 ) =

(7.29)

χija(3)∗ kl (k1,ω1 |k2,ω2 |k3,ω3 ).

Moreover, the following symmetries hold for nonlinear susceptibilities upon interchange of indices and arguments (more on the symmetries will be discussed subsequently): a(2) χija(2) k (k1,ω1 |k2,ω2 ) = χikj (k2,ω2 |k1,ω1 ), a(3) χija(3) kl (k1,ω1 |k2,ω2 |k3,ω3 ) = χij lk (k1,ω1 |k3,ω3 |k2,ω2 ).

(7.30)

The second-order nonlinear susceptibility enjoys an additional (approximate) symmetry property, which is a direct generalization of (1.113) (we will discuss this property in more detail later also), a(2) χija(2) k (k1 + k2,ω1 + ω2 | − k2, − ω2 ) = −χj ik (k1,ω1 |k2,ω2 ).

(7.31)

7.1 Nonlinear Electrodynamic Equations in Plasmas

165

It also follows from the definition that the linear susceptibility tensor and its inverse are expressed as   ki kj ki kj ij (k,ω) = 2  (k,ω) + δij − 2 ⊥ (k,ω), k k   ki kj ki kj 1 1 −1 ij (k,ω) = 2 + δij − 2 , k  (k,ω) k ⊥ (k,ω) ⊥ (k,ω) = ⊥ (k,ω) −

c2 k 2 . ω2

(7.32)

From this it follows that the spectral wave energy density tensor can be decomposed into longitudinal and transverse parts in diagonal form,   ki kj 1 ki kj 2 δij − 2 E⊥2 k,ω . Ei Ej k,ω = 2 E k,ω + (7.33) k 2 k The properties (7.32) and (7.33) can be easily proven if we take, without loss of generality, k = kˆz, for instance, and proceed to evaluate the tensor quantity ij (k,ω). The factor 12 associated with E⊥2 k,ω is related to the fact that the transverse mode has two independent polarizations. That is, the transverse mode in unmagnetized plasmas is degenerate. This is intuitively obvious. Since there are two orthogonal axes perpendicular to the wave vector, the transverse mode electric field can be linearly polarized in either directions, yet they both satisfy the same dispersion relation. The factor 12 ensures that the total electric field energy, which is the determinant, Ei Ej k,ω δij , is properly partitioned between the longitudinal and transverse components, E 2 k,ω = E2 k,ω + E⊥2 k,ω . These properties lead to     ki kj ∂ Re ⊥ (k,ω) i ki kj ∂ Re  (k,ω) − 0= + δ ij 2 k2 ∂ω k2 ∂ω     ki kj ∂ 1 ki kj ∂ 2 2 × δij − 2 E k,ω + E k,ω k 2 ∂t  2 k ∂t ⊥     ki kj ki kj + Re ⊥ (k,ω) Re  (k,ω) + δij − 2 k2 k     ki kj 1 ki kj 2 2 δij − 2 × E⊥ k,ω E k,ω + k2 2 k     ki kj ki kj +i Im  (k,ω) + δij − 2 Im ⊥ (k,ω) k2 k     1 ki kj ki kj 2 2 δij − 2 E⊥ k,ω E k,ω + × k2 2 k

166

Nonlinear Electromagnetic Equations in Vlasov Plasmas

 −2

 dω

dk

 + δj l − 



kj kl k 2







χij(2)k (k,ω |k

    kj kl 1 − k ,ω − ω ) − 2 k  (k,ω ) 



 1 (2) χnlm (k,ω |k − k,ω − ω )   ⊥ (k ,ω )

(k − k )k (k − k )m 2 E k−k,ω−ω |k − k |2    1 (k − k )k (k − k )m 2 δkm − E⊥ k−k,ω−ω + 2 |k − k |2  1 (k − k )k (k − k )l −  2 |k − k |  (k − k,ω − ω )    1 (k − k )k (k − k )l + δkl − |k − k |2 ⊥ (k − k,ω − ω ) ×

(2) × χnml (k,ω |k − k,ω − ω )        kj km 2 kj km 1 2     δj m − 2 × E⊥ k ,ω E k ,ω + k 2 2 k     ki kn 2 1 ki kn 2 δin − 2 E⊥ k,ω × E k,ω + k2 2 k  %   1 1 ki kl ki kl (2)     + χij k (k ,ω |k − k ,ω − ω ) + δil − 2 k 2 ∗ (k,ω) k ∗⊥ (k,ω)        kj km 2 kj km 1 (2)∗     2 δj m − 2 E⊥ k,ω × χlmn (k ,ω |k − k ,ω − ω ) E k,ω + k 2 2 k  (k − k )k (k − k )n 2 × E k−k,ω−ω |k − k |2    1 (k − k )k (k − k )n 2   δkn − E +  ⊥ k−k ,ω−ω 2 |k − k |2    dω χij(3)kl (k,ω | − k, − ω |k,ω) + 2 dk

   1 ki kl ki kl 2 2 δil − 2 E⊥ k,ω E k,ω + × k2 2 k       kj kk kj kk 2 1 2 δj k − 2 E⊥ k,ω . × E k,ω + k 2 2 k 

(7.34)

7.1 Nonlinear Electrodynamic Equations in Plasmas

167

Carrying out the various matrix multiplications, one may obtain 0=

i ∂ Re  (k,ω) ∂ i ∂ Re ⊥ (k,ω) ∂ E2 k,ω + E 2 k,ω 2 ∂ω ∂t 2 ∂ω ∂t ⊥ + E2 k,ω Re  (k,ω) + E⊥2 k,ω Re ⊥ (k,ω)   + i E2 k,ω Im  (k,ω) + i E⊥2 k,ω Im ⊥ (k,ω) + 2 dk dω 1% (2) kj kl χij(2)k (k,ω |k − k,ω − ω ) χnlm (k,ω |k − k,ω − ω ) × k 2  (k,ω )    (2) (k,ω |k − k,ω − ω ) kj kl χij(2)k (k,ω |k − k,ω − ω ) χnlm + δj l − 2 k ⊥ (k,ω )  ki kn (k − k )k (k − k )m 2 × E k−k,ω−ω E2 k,ω k2 |k − k |2   1 ki kn (k − k )k (k − k )m δkm − E⊥2 k−k,ω−ω E2 k,ω + 2 k2 |k − k |2   1 ki kn (k − k )k (k − k )m 2 δin − 2 + E k−k,ω−ω E⊥2 k,ω 2 k |k − k |2    1 ki kn (k − k )k (k − k )m δin − 2 δkm − + 4 k |k − k |2   (k − k )k (k − k )l × E⊥2 k−k,ω−ω E⊥2 k,ω + |k − k |2 ×

(2) (k,ω |k − k,ω − ω ) χij(2)k (k,ω |k − k,ω − ω ) χnml

 (k − k,ω − ω )  (k − k )k (k − k )l + δkl − |k − k |2 

×

(2) (k,ω |k − k,ω − ω ) χij(2)k (k,ω |k − k,ω − ω ) χnml

 × + + +

⊥ (k − k,ω − ω )

 ki kn kj km E2 k,ω E2 k,ω k 2 k 2    kj km 1 ki kn δj m − 2 E⊥2 k,ω E2 k,ω 2 2 k k     1 ki kn kj km 2 δin − 2 E k,ω E⊥2 k,ω 2 k k 2      kj km 1 ki kn δin − 2 δj m − 2 E⊥2 k,ω E⊥2 k,ω 4 k k



168

%

Nonlinear Electromagnetic Equations in Vlasov Plasmas

ki kl 1 (2)∗  (k ,ω |k − k,ω − ω ) χ (2) (k,ω |k − k,ω − ω ) χlmn ∗ 2 k  (k,ω) ij k   1 ki kl + δil − 2 ∗ k ⊥ (k,ω)  (2)  (2)∗        × χij k (k ,ω |k − k ,ω − ω ) χlmn (k ,ω |k − k ,ω − ω ) −

 ×

 kj km (k − k )k (k − k )n

k 2

|k − k |2

E2 k−k,ω−ω E2 k,ω

   1 kj km (k − k )k (k − k )n δkn − E2 k,ω E⊥2 k−k,ω−ω + 2 k 2 |k − k |2    kj km (k − k )k (k − k )n 2 1 δj m − 2 + E⊥ k,ω E2 k−k,ω−ω 2 k |k − k |2     kj km 1 (k − k )k (k − k )n δj m − 2 δkn − + 4 k |k − k |2  × E⊥2 k,ω E⊥2 k−k,ω−ω   + 2 dk dω χij(3)kl (k,ω | − k, − ω |k,ω) 

ki kl kj kk E2 k,ω E2 k,ω × k 2 k 2   kj kk 1 ki kl δj k − 2 E⊥2 k,ω E2 k,ω + 2 k2 k     1 ki kl kj kk 2 δil − 2 + E k,ω E⊥2 k,ω 2 k k 2     kj kk 1 ki kl 2 2   δil − 2 δj k − 2 E⊥ k ,ω E⊥ k,ω . + 4 k k In deriving this result, we have made use of the relations   ki kj ki kj ki kj ki kj δij − 2 = 0, = 1, k2 k2 k2 k    ki kj ki kj δij − 2 = δii − 1 = 2. δij − 2 k k

(7.35)

(7.36)

Equation (7.35) represents the formal nonlinear wave kinetic equation, which directly generalizes (1.85).

7.1 Nonlinear Electrodynamic Equations in Plasmas

169

7.1.2 Formal Particle Kinetic Equation In Section 1.6, we discussed the formal particle kinetic equation under electrostatic approximation in which the effects of nonlinear mode coupling were included. This was done for the sake of being consistent with the weak turbulence ordering in the wave kinetic equation. It turns out, however, that the nonlinear mode coupling term could be ignored for reasons, which we have already discussed at the end of Chapter 3. On such a basis, in the present electromagnetic formalism, we take the simple view that quasilinear approximation is valid as far as the particle kinetic equation is concerned. Thus, we simply write      k·v vj ki ea2 ∂ ∂Fa δij + Ej Ek k,ω dk dω 1 − = −Im 2 ∂t ma ∂vi ω ω    k·v 1 vk kl ∂Fa 1− δkl + × . (7.37) ω − k · v + i0 ω ω ∂vl Decomposition of Ei Ej k,ω into longitudinal and transverse components as in (7.33) leads to      vj ki ∂Fa k·v ea2 ∂ δij + dk dω 1 − = −Im 2 ∂t ma ∂vi ω ω     kj kk 2 1 kj kk × δj k − 2 E⊥2 k,ω E k,ω + k2 2 k    k·v vk kl ∂Fa 1 1− δkl + . (7.38) × ω − k · v + i0 ω ω ∂vl Note that       ki kl k·v vi kj kj kk vk kl k·v δij + δ = 2 . 1 − + 1− kl ω ω k2 ω ω k

(7.39)

Note also the following:        k·v vj ki kj kk vk kl k·v δij + δj k − 2 δkl + 1− 1− ω ω k ω ω     ki [(k × v) × k]k k·v ki kk δik − 2 + = 1− ω k ω k2     [(k × v) × k]k kl kk kl k·v δkl − 2 + , (7.40) × 1− ω k k2 ω where we have made use of (7.36), and have also made use of the following: kk [(k × v) × k]k = k · v⊥ = 0.

(7.41)

In (7.41), we have decomposed the velocity vector into components parallel and perpendicular to the wave vector – see (6.54),

170

Nonlinear Electromagnetic Equations in Vlasov Plasmas

k (k · v) (k × v) × k , v⊥ = . 2 k k2 Making use of (7.39) and (7.40), the particle kinetic equation (7.37) reduces to    ki kj 1 ∂Fa ea2 ∂ dk dω E2 k,ω = −Im 2 ∂t ma ∂vi ω − k · v + i0 k 2     k·v ki [(k × v) × k]k 1 2 ki kk δik − 2 + + E⊥ k,ω 1 − 2 ω k ω k2     [(k × v) × k]k kj ∂Fa kk kj k·v δkj − 2 + , × 1− 2 ω k ωk ∂vj (7.42) v = v + v⊥,

v =

or, equivalently, ∂Fa ∂Fa ∂ Dij , = ∂t ∂vi ∂vj   1 ea2 dk dω Dij = −Im 2 m ω − k · v + i0   a 1 ki kj 2 2 E k,ω + aki akj E⊥ k,ω , × k2 2    [(k × v) × k]i kj k·v ki kj aij (k,ω;v) = 1 − δij − 2 + . ω k k2 ω

(7.43)

7.2 Formal Electromagnetic Kinetic Equations for Eigenmodes In this section, we discuss further manipulations of formal theory developed in the previous section, namely, formal wave kinetic equation (7.35) and formal particle kinetic equation (7.42), and render them applicable for linear eigenmodes. Since the particle equation (7.42) is relatively simple, major portion of the discussions will be devoted to the wave kinetic equation. The difference between electrostatic and electromagnetic theory is the addition of a new eigenmode, namely, transverse polarized mode. Electrostatic linear eigenmodes have already been discussed in Part I, of this book, and they satisfy the formal longitudinal wave dispersion relation, Re  (k,ωα ) = 0.

(7.44)

We already discussed that α = L and S correspond to Langmuir and ion-sound modes, and these are the two longitudinal modes satisfying (7.44). The electromagnetic mode satisfies the transverse wave dispersion relation, which is formally expressed by Re ⊥ (k,ωT ) = 0.

(7.45)

7.2 Formal Electromagnetic Kinetic Equations for Eigenmodes

171

As we already discussed in (1.95), the longitudinal mode can be represented by   Ikσ α δ(ω − σ ωkα ). (7.46) E2 k,ω = α=L,S σ =±1

Likewise, the transverse mode can be represented by  E⊥2 k,ω = Ikσ T δ(ω − σ ωkT ).

(7.47)

σ =±1

The dispersion relations (7.44) and (7.45) effectively replace the real part of formal wave kinetic equation (7.35). We thus need only to consider the imaginary part. In explicitly writing down the imaginary part of (7.35), we make use of the shorthand notations – see (1.151), ∂Re  (k,ω) , ∂ω ∂Re⊥ (k,ω) ⊥ (k,ω) = . ∂ω  (k,ω) =

The result is   ∂I σ α  ∂I σ T k k δ(ω − σ ωkα ) + ⊥ (k,ω) δ(ω − σ ωkT ) ∂t ∂t α=L,S σ =±1 σ =±1    σα + 2 Im  (k,ω) Ik + 2 Im ⊥ (k,ω) Ikσ T δ(ω − σ ωkT )

0 =  (k,ω)

 + 4 Im

dk



α σ =±1

dω

σ =±1 (2) kj kl χij k (k,ω |k − k,ω − ω ) k 2  (k,ω )   kj kl χij(2)k (k,ω |k − k,ω − ω ) − ω ) + δj l − 2 k ⊥ (k,ω )     ki kn (k − k )k (k − k )m  − ω ) k2 |k − k |2 α,γ =L,S σ,σ  =±1



(2) × χnlm (k,ω |k − k,ω (2) (k,ω |k − k,ω × χnlm σ  γ

× Ik−k Ikσ α δ(ω − ω − σ  ωk−k ) δ(ω − σ ωkα )    (2)   kj kl χij k (k ,ω |k − k,ω − ω ) (2)   + χnlm (k ,ω |k − k,ω − ω ) k 2  (k,ω )    kj kl χij(2)k (k,ω |k − k,ω − ω ) (2)     + δj l − 2 χnlm (k ,ω |k − k ,ω − ω ) k ⊥ (k,ω )   (k − k )k (k − k )m   1 ki kn δkm − × 2 k2 |k − k |2  α γ

σ,σ

×

σ  T σ α Ik−k  Ik



δ(ω − ω − σ



T ωk−k )

δ(ω − σ ωkα )

172

Nonlinear Electromagnetic Equations in Vlasov Plasmas

 +

kj kl χij(2)k (k,ω |k − k,ω − ω )



k 2

+ δj l −

kj kl



 (k,ω )

(2) (k,ω |k − k,ω − ω ) χnlm

χij(2)k (k,ω |k − k,ω − ω )

⊥ (k,ω )    1 ki kn (k − k )k (k − k )m (2) δin − 2 (k,ω |k − k,ω − ω ) × χnlm 2 k |k − k |2   σ  γ γ × Ik−k Ikσ T δ(ω − ω − σ  ωk−k ) δ(ω − σ ωkT ) γ

σ,σ 

kj

kl χij(2)k (k,ω |k − k,ω − ω )

 +

× × ×

k 2

(k,ω )

(2) (k,ω |k − k,ω − ω ) χnlm

   (2)  kl χij k (k ,ω |k − k,ω − ω ) (2)   δj l − 2 χnlm (k ,ω |k − k,ω k ⊥ (k,ω )    ki kn (k − k )k (k − k )m  σ  T σ T 1 δin − 2 δkm − Ik−k Ik 4 k |k − k |2 σ,σ   (k − k )k (k − k )l   T T δ(ω − ω − σ ωk−k ) δ(ω − σ ωk ) + |k − k |2 

+

k 2

kj

χij(2)k (k,ω |k − k,ω − ω ) 

 (k − k,ω − ω )

(k − k )k (k − k )l + δkl − |k − k |2



−ω)

(2) (k,ω |k − k,ω − ω ) χnml



χij(2)k (k,ω |k − k,ω − ω )

⊥ (k − k,ω − ω )   ki kn kj km σ β (2)     × χnml (k ,ω |k − k ,ω − ω ) Ik Ikσ α 2 2 k k α,β σ,σ   (k − k )k (k − k )l β × δ(ω − σ  ωk ) δ(ω − σ ωkα ) + |k − k |2 ×

χij(2)k (k,ω |k − k,ω − ω )  (k − k,ω − ω )



(2) (k,ω |k − k,ω − ω ) χnml

 (2) (k − k )k (k − k )l χij k (k,ω |k − k,ω − ω ) + δkl − |k − k |2 ⊥ (k − k,ω − ω )     kj km 1 ki kn  (2) × χnml δ (k,ω |k − k,ω − ω ) − Ikσ T Ikσ α jm 2 k2 k 2 α σ,σ     (k − k )k (k − k )l × δ(ω − σ  ωkT ) δ(ω − σ ωkα ) + |k − k |2 

×

χij(2)k (k,ω |k − k,ω − ω )  (k −

k,ω



ω )

(2) (k,ω |k − k,ω − ω ) χnml



7.2 Formal Electromagnetic Kinetic Equations for Eigenmodes

173





(2) (k − k )k (k − k )l χij k (k,ω |k − k,ω − ω ) |k − k |2 ⊥ (k − k,ω − ω )      ki kn kj km 1 σ β (2) δin − 2 × χnml (k,ω |k − k,ω − ω ) Ik Ikσ T 2 2 k k β σ,σ   (k − k )k (k − k )l β × δ(ω − σ  ωk ) δ(ω − σ ωkT ) + |k − k |2

+ δkl −

×

χij(2)k (k,ω |k − k,ω − ω ) 

 (k − k,ω − ω )

(2) (k,ω |k − k,ω − ω ) χnml

 (2) (k − k )k (k − k )l χij k (k,ω |k − k,ω − ω ) + δkl − |k − k |2 ⊥ (k − k,ω − ω )      kj km 1 ki kn (2)     δin − 2 δj m − 2 × χnml (k ,ω |k − k ,ω − ω ) 4 k k    ki kl × Ikσ T Ikσ T δ(ω − σ  ωkT ) δ(ω − σ ωkT ) − k2  σ,σ

χij(2)k (k,ω |k − k,ω − ω )

(2)∗  (k ,ω |k − k,ω − ω ) χlmn ∗ (k,ω)  (2)   ki kl χij k (k,ω |k − k,ω − ω ) (2)∗     + δil − 2 χlmn (k ,ω |k − k ,ω − ω ) k ∗⊥ (k,ω)  kj km (k − k )k (k − k )n   σ  γ σ  β × Ik−k Ik k 2 |k − k |2   β,γ σ ,σ  ki kl   γ   β × δ(ω − ω − σ ωk−k ) δ(ω − σ ωk ) − k2

×

χij(2)k (k,ω |k − k,ω − ω )

(2)∗  (k ,ω |k − k,ω − ω ) χlmn ∗ (k,ω)  (2)   ki kl χij k (k,ω |k − k,ω − ω ) (2)∗     + δil − 2 χlmn (k ,ω |k − k ,ω − ω ) k ∗⊥ (k,ω)    1 kj km (k − k )k (k − k )n   σ  β σ  T × δkn − Ik Ik−k 2 k 2 |k − k |2   β

×

σ ,σ



ki kl χij k (k,ω |k − k,ω − ω ) × δ(ω − σ δ(ω − ω − σ − k2 ∗ (k,ω)  (2)  ki kl χij k (k,ω |k − k,ω − ω ) (2)∗     × χlmn (k ,ω |k − k ,ω − ω ) + δil − 2 k ∗⊥ (k,ω) 



β ωk )





T ωk−k )

(2)

174

Nonlinear Electromagnetic Equations in Vlasov Plasmas

× ×

(2)∗  χlmn (k ,ω |k

 γ

 − + × ×

    kj km (k − k )k (k − k )n 1 δj m − 2 − k ,ω − ω ) 2 k |k − k |2 



σ  γ



Ikσ T Ik−k δ(ω − σ  ωkT ) δ(ω − ω − σ  ωk−k ) γ

σ ,σ 

ki kl χij k (k,ω |k − k,ω − ω ) χlmn (k,ω |k − k,ω − ω ) k2 ∗ (k,ω)  (2)   ki kl χij k (k,ω |k − k,ω − ω ) (2)∗     δil − 2 χlmn (k ,ω |k − k ,ω − ω ) k ∗⊥ (k,ω)     kj km (k − k )k (k − k )n 1 δj m − 2 δkn − 4 k |k − k |2    σ T σ  T   T   T Ik Ik−k δ(ω − σ ωk ) δ(ω − ω − σ ωk−k )

σ ,σ 

(2)



(2)∗





dω χij(3)kl (k,ω | − k, − ω |k,ω) + 4 Im dk  ki kl kj kk   σ α σ  β β Ik Ik δ(ω − σ ωkα ) δ(ω − σ  ωk ) × 2 2 k k α,β σ,σ    kj kk   σ  T σ α 1 ki kl δj k − 2 Ik Ik δ(ω − σ  ωkT ) δ(ω − σ ωkα ) + 2 k2 k α σ,σ       ki kl kj kk 1 σ β β δil − 2 Ik Ikσ T δ(ω − σ  ωk ) δ(ω − σ ωkT ) + 2 2 k k β σ,σ     kj kk ki kl 1 δil − 2 δj k − 2 + 4 k k    σ T σT   T T Ik Ik δ(ω − σ ωk ) δ(ω − σ ωk ) . (7.48) × σ,σ 

We make use of the following – see (1.97) and (1.99):   δ(ω − σ ωα ) 1 1 k =P − iπ α ,   (k,ω)  (k,ω) (k,σ ω  k) α σ   δ(ω − σ ωα ) 1 1 k = P + iπ α ,  ∗ (k,ω)  (k,ω) (k,σ ω  k) α σ

 δ(ω − σ ωT ) 1 1 k , =P − iπ  T ⊥ (k,ω) ⊥ (k,ω)  (k,σ ω ⊥ k) σ 1 ∗⊥ (k,ω)

=P

 δ(ω − σ ωT ) 1 k . + iπ  T ⊥ (k,ω)  (k,σ ω ) ⊥ k σ

(7.49)

7.2 Formal Electromagnetic Kinetic Equations for Eigenmodes

175

Then (7.48) can be further manipulated by separating the principal parts and the residues. In writing the final result, it turns out that the final form can be separated into terms that are associated with an overall delta function δ(ω − σ ωkα ), and terms that have an overall factor δ(ω − σ ωkT ). (Actual demonstration is left as an exercise for the readers.) This shows that those terms with the factor δ(ω − σ ωkα ) constitute the formal wave kinetic equation for longitudinal modes, while those terms with the factor δ(ω − σ ωkT ) collectively constitute the formal wave kinetic equation for transverse mode. Thus, we may separate the equations for longitudinal mode ω = σ ωkα and transverse mode ω = σ ωkT . 7.2.1 Formal Wave Kinetic Equation for Longitudinal Modes Collecting all the terms associated with δ(ω − σ ωkα ), we have ∂Ikσ α + 2 Im  (k,σ ωkα ) Ikσ α ∂t      kj kl (2)     dω + 4 Im dk χ (k ,ω |k − k,σ ωkα − ω ) k 2 ij k  1 (2)    α  × χnlm (k ,ω |k − k ,σ ωk − ω ) P  (k,ω )    δ(ω − σ  ωβ )   kj kl k + δj l − 2 χij(2)k (k,ω |k − k,σ ωkα − ω ) − iπ β    k β σ   (k ,σ ωk )   δ(ω − σ  ωT )  1 (2)    α  k × χnlm (k ,ω |k − k ,σ ωk − ω ) P − iπ  ,σ  ωT ) ⊥ (k,ω )  (k ⊥ k σ

0 =  (k,σ ωkα )

ki kn (k − k )k (k − k )m   σ  γ σ α γ Ik−k Ik δ(σ ωkα − ω − σ  ωk−k ) k2 |k − k |2 γ σ     kj kl (2)   (2) + χ (k ,ω |k − k,σ ωkα − ω ) χnlm (k,ω |k − k,σ ωkα − ω ) k 2 ij k     δ(ω − σ  ωβ )   kj kl 1 k + δj l − 2 − iπ × P  ,σ  ωβ )  (k,ω ) k (k    k β σ

×

(2) (k,ω |k − k,σ ωkα − ω ) × χij(2)k (k,ω |k − k,σ ωkα − ω ) χnlm   δ(ω − σ  ωT )  1 ki kn 1 k × P − iπ  ,σ  ωT ) ⊥ (k,ω ) 2 k2  (k ⊥ k σ   (k − k )k (k − k )m  σ  T σ α T × δkm − Ik−k Ik δ(σ ωkα − ω − σ  ωk−k ) |k − k |2  σ

176

Nonlinear Electromagnetic Equations in Vlasov Plasmas



(k − k )k (k − k )l (2)   χij k (k ,ω |k − k,σ ωkα − ω ) |k − k |2  1 (2)    α  × χnml (k ,ω |k − k ,σ ωk − ω ) P   (k − k ,σ ωkα − ω )    γ   δ(σ ωkα − ω − σ  ωk−k (k − k )k (k − k )l ) + δ − − iπ kl γ |k − k |2  (k − k,σ  ωk−k )  γ +

σ

×

χij(2)k (k,ω |k

×

χij(2)k (k,ω |k



(2) − k,σ ωkα − ω ) χnml (k,ω |k − k,σ ωkα − ω )

 T  δ(σ ωkα − ω − σ  ωk−k 1 ) × P − iπ T ⊥ (k − k,σ ωkα − ω ) ⊥ (k − k,σ  ωk−k ) σ      (k − k )k (k − k )l ki kn kj km σ β σ α   β I I δ(ω − σ ω ) + × 2   k k k k k 2 β |k − k |2  σ



(2) − k,σ ωkα − ω ) χnml (k,ω |k − k,σ ωkα − ω )

  δ(σ ωkα − ω − σ  ωk−k ) 1 × P − iπ γ  (k − k,σ ωkα − ω )  (k − k,σ  ωk−k ) γ σ    (k − k )k (k − k )l χij(2)k (k,ω |k − k,σ ωkα − ω ) + δkl −  2 |k − k |  1 (2) × χnml (k,ω |k − k,σ ωkα − ω ) P ⊥ (k − k,σ ωkα − ω )     T  δ(σ ωkα − ω − σ  ωk−k kj km 1 ki kn ) − iπ δj m − 2 T 2 k2 k ⊥ (k − k,σ  ωk−k )  γ



σ

×





Ikσ T Ikσ α δ(ω − σ  ωkT ) − iπ

σ

× ×

(2)∗  χlmn (k ,ω |k



σ  γ





k ,σ ωkα

σ β

Ik−k Ik



−ω)

(2) α ki kl χij k (k,ω |k − k,σ ωk − ω ) k2  (k,σ ωkα )

 kj km (k − k )k (k − k )n

|k − k |2

k 2

δ(σ ωkα − ω − σ  ωk−k ) δ(ω − σ  ωk ) γ

β

β,γ σ ,σ  (2) α ki kl χij k (k,ω |k − k,σ ωk − ω ) (2)∗   χlmn (k ,ω |k − k,σ ωkα − ω ) k2  (k,σ ωkα )    1 kj km (k − k )k (k − k )n   σ  β σ  T × δkn − Ik Ik−k 2 k 2 |k − k |2   β

− iπ

σ ,σ

T × δ(ω − σ  ωk ) δ(σ ωkα − ω − σ  ωk−k  ) − iπ β

ki kl k2

7.2 Formal Electromagnetic Kinetic Equations for Eigenmodes

177

χij(2)k (k,ω |k − k,σ ωkα − ω )

(2)∗  (k ,ω |k − k,σ ωkα − ω ) χlmn  (k,σ ωkα )    kj km (k − k )k (k − k )n   σ  T σ  γ 1 δj m − 2 Ik Ik−k × 2 k |k − k |2   γ

×

× − × ×

σ ,σ T α   γ δ(ω − σ ωk ) δ(σ ωk − ω − σ ωk−k ) (2) α ki kl χij k (k,ω |k − k,σ ωk − ω ) (2)∗   iπ 2 χlmn (k ,ω |k k  (k,σ ωkα )     kj km (k − k )k (k − k )n 1 δj m − 2 δkn − 4 k |k − k |2 





 Ikσ T

σ  T Ik−k 

σ ,σ 





δ(ω − σ



ωkT )

δ(σ ωkα



−ω −σ



− k,σ ωkα − ω ) (7.50) 

T ωk−k )



 σ β ki kl kj kk β β Im χij(3)kl (k,σ  ωk | − k, − σ  ωk |k,σ ωkα ) Ik  2 2 k k β σ    kj kk 1 ki kl (3)   T   T α σ T δ Im χ Ikσ α . + − (k ,σ ω | − k , − σ ω |k,σ ω )I jk k k k k ij kl 2 k2 k 2 +4



dk

Proceeding from (7.50), one may make use of various delta functions in order to carry out the ω integrations, thus leaving the results with only k integrals. The result is that we obtain the following formal wave kinetic equation for longitudinal modes (α = L,S). The result is only formal at this point since the various nonlinear coupling coefficients that are expressed in terms of the susceptibility tensors are yet to be evaluated. In what follows, we present the result together with brief descriptive texts encased in boxes: ∂I σ α Induced emission 0 =  (k,σ ωkα ) k + 2 Im  (k,σ ωkα ) Ikσ α ∂t     ki kn kj km (k − k )k (k − k )l + 4π Im dk k 2 k 2 |k − k |2 β,γ   σ ,σ

×

β χij(2)k (k,σ  ωk |k



σ  γ

Ik−k Ikσ α

(2)∗  − k,σ  ωk−k ) χnml (k ,σ  ωk |k − k,σ  ωk−k ) γ

σ β

β

γ

Ik Ikσ α × + γ  β  (k,σ  ωk )  (k − k,σ  ωk−k ) σ  γ σ  β  Ik−k Ik γ β δ(σ ωkα − σ  ωk − σ  ωk−k ) −  Decay (α,β,γ )  (k,σ ωkα )      ki kn kj km (k − k )k (k − k )l δ + − kl k 2 k 2 |k − k |2   β σ ,σ

178

Nonlinear Electromagnetic Equations in Vlasov Plasmas (2)∗  T    T × χij(2)k (k,σ  ωk |k − k,σ  ωk−k  ) χnml (k ,σ ωk |k − k ,σ ωk−k )  σ  T σ α σ β Ik−k Ik 2 Ik Ikσ α × + β T )  (k,σ  ωk ) ⊥ (k − k,σ  ωk−k σ  β σ  T  Ik Ik−k  β T δ(σ ωkα − σ  ωk − σ  ωk−k −  Decay (α,β,T ) ) α  (k,σ ωk )     1 ki kn  kj km (k − k )k (k − k )l δ δ + − − jm kl 4 k2 k 2 |k − k |2   β

β

σ ,σ

(2)∗  T  T   T × χij(2)k (k,σ  ωkT |k − k,σ  ωk−k  ) χnml (k ,σ ωk |k − k ,σ ωk−k )   σ  T σ α 2 Ik−k 2 Ikσ T Ikσ α  Ik × + T ⊥ (k,σ  ωkT ) ⊥ (k − k,σ  ωk−k )     σ T Ikσ T Ik−k  α  T  T δ(σ ωk − σ ωk − σ ωk−k ) Decay (α,T ,T ) −   (k,σ ωkα )      ki kn kj km (k − k )k (k − k )l  + 4 Im dk k 2 k 2 |k − k |2  β σ

(2) − k,σ ωkα − σ  ωk ) χnml (k,σ  ωk |k − k,σ ωkα − σ  ωk )   2 (k − k )k (k − k )l ×P + δkl − β |k − k |2  (k − k,σ ωkα − σ  ωk )

×

β χij(2)k (k,σ  ωk |k

β

β

β

(2) × χij(2)k (k,σ  ωk |k − k,σ ωkα − σ  ωk ) χnml (k,σ  ωk |k − k,σ ωkα − σ  ωk )  2 (3)   β   β α ×P (k ,σ ω | − k , − σ ω |k,σ ω ) + χ k ij mn k k β ⊥ (k − k,σ ωkα − σ  ωk ) β

β

σ β

× Ik Ikσ α +



1  ki kn δj m − 2  k2

σ (2)  χij k (k ,σ  ωkT |k

  km (k 2 k

kj

β

β

Induced scattering α,β − k )k (k − k )l |k − k |2

(2) − k,σ ωkα − σ  ωkT ) χnml (k,σ  ωkT |k − k,σ ωkα − σ  ωkT )   (k − k )k (k − k )l 2 + δkl − ×P |k − k |2  (k − k,σ ωkα − σ  ωkT )

×

(2) (k,σ  ωkT |k − k,σ ωkα − σ  ωkT ) × χij(2)k (k,σ  ωkT |k − k,σ ωkα − σ  ωkT ) χnml  2 (3)   T   T α + χij mn (k ,σ ωk | − k , − σ ωk |k,σ ωk ) ×P ⊥ (k − k,σ ωkα − σ  ωkT )  σ T σ α × I k I k . Induced scattering α,T . (7.51)

7.2 Formal Electromagnetic Kinetic Equations for Eigenmodes

179

When compared with the electrostatic Vlasov weak turbulence formalism, the present electromagnetic generalization shows that the time evolution of longitudinal modes is modified by the presence of additional nonlinear processes. These are the decay processes involving two longitudinal modes and a transverse mode, as well as the three-wave decay involving two transverse modes and a single longitudinal mode. They also include an additional induced scattering process involving a longitudinal mode and a transverse mode. In (7.51), we have made use of the fact that the leading term associated with the second-order susceptibility in the context of decay processes is the imaginary part, χij(2)k (k1,ω1 |k2,ω2 ) ≈ i Im χij(2)k (k1,ω1 |k2,ω2 ) = −χij(2)∗ k (k1,ω1 |k2,ω2 ).

(7.52)

7.2.2 Formal Wave Kinetic Equation for Transverse Mode Collecting all the terms associated with δ(ω − σ ωkT ) in (7.48), one may show that the resulting equation is ∂I σ T 0 = ⊥ (k,σ ωkT ) k + 2 Im ⊥ (k,σ ωkT ) Ikσ T ∂t      kj kl (2)     dω + 4 Im dk χij k (k ,ω |k − k,σ ωkT − ω ) 2 k    δ(ω − σ  ωβ )  1 (2)    T  k × χnlm (k ,ω |k − k ,σ ωk − ω ) P − iπ  ,σ  ωβ )  (k,ω ) (k  k β σ    kj kl (2) + δj l − 2 χij(2)k (k,ω |k − k,σ ωkT − ω ) χnlm (k,ω |k − k,σ ωkT − ω ) k    δ(ω − σ  ωT )  1  ki kn 1 k δin − 2 × P − iπ ⊥ (k,ω ) k ⊥ (k,σ  ωkT ) 2 σ (k − k )k (k − k )m   σ  γ σ T γ Ik−k Ik δ(σ ωkT − ω − σ  ωk−k ) × |k − k |2 γ σ     kj kl (2)   (2) + χij k (k ,ω |k − k,σ ωkT − ω ) χnlm (k,ω |k − k,σ ωkT − ω ) 2 k     δ(ω − σ  ωβ )   kj kl 1 k + δj l − 2 × P − iπ β  (k,ω ) k  (k,σ  ωk )  β σ

×

χij(2)k (k,ω |k 

× P





(2) − ω ) χnlm (k,ω |k − k,σ ωkT − ω )  δ(ω − σ  ωT ) 

k ,σ ωkT

1 − iπ ⊥ (k,ω )

k

σ

⊥ (k,σ  ωkT )

180



Nonlinear Electromagnetic Equations in Vlasov Plasmas

  1 ki kn (k − k )k (k − k )m δin − 2 δkm − × 4 k |k − k |2    (k − k )k (k − k )l σ T σT T   T Ik−k Ik δ(σ ωk − ω − σ ωk−k ) + × |k − k |2  σ

(2) × χij(2)k (k,ω |k − k,σ ωkT − ω ) χnml (k,ω |k − k,σ ωkT − ω )   γ   δ(σ ωkT − ω − σ  ωk−k 1 ) − iπ × P γ  (k − k,σ  ωk−k )  (k − k,σ ωkT − ω ) γ σ    (k − k )k (k − k )l + δkl − χij(2)k (k,ω |k − k,σ ωkT − ω ) |k − k |2  1 (2)    T  × χnml (k ,ω |k − k ,σ ωk − ω ) P  ⊥ (k − k ,σ ωkT − ω )    T  δ(σ ωkT − ω − σ  ωk−k 1 ki kn ) δin − 2 − iπ T 2 k ⊥ (k − k,σ  ωk−k ) σ     kj km (k − k )k (k − k )l σ β β Ik Ikσ T δ(ω − σ  ωk ) + × 2 k |k − k |2  β σ

(2) − k,σ ωkT − ω ) χnml (k,ω |k − k,σ ωkT − ω )   γ   δ(σ ωkT − ω − σ  ωk−k 1 ) − iπ × P γ  (k − k,σ  ωk−k )  (k − k,σ ωkT − ω ) γ σ    (k − k )k (k − k )l + δkl − χij(2)k (k,ω |k − k,σ ωkT − ω ) |k − k |2  1 (2) × χnml (k,ω |k − k,σ ωkT − ω ) P ⊥ (k − k,σ ωkT − ω )    T  δ(σ ωkT − ω − σ  ωk−k 1 ki kn ) δin − 2 − iπ T 4 k ⊥ (k − k,σ  ωk−k ) σ      kj km  Ikσ T Ikσ T δ(ω − σ  ωkT ) × δj m − 2 k σ  (2)    ki kl χij k (k ,ω |k − k,σ ωkT − ω ) (2)∗   χlmn (k ,ω |k − k,σ ωkT − ω ) − iπ δil − 2 k ⊥ (k,σ ωkT )  kj km (k − k )k (k − k )n   σ  γ σ  β × 2 Ik−k Ik k |k − k |2 β,γ  

×

× −

χij(2)k (k,ω |k

σ ,σ γ β δ(σ ωkT − ω − σ  ωk−k ) δ(ω − σ  ωk )  (2)  ki kl χij k (k,ω |k − k,σ ωkT − iπ δil − 2 k ⊥ (k,σ ωkT )

ω )

(2)∗  (k ,ω |k − k,σ ωkT − ω ) χlmn

7.2 Formal Electromagnetic Kinetic Equations for Eigenmodes

× ×

  km δkn k 2

kj

1 2 



σ β

(k − k )k (k − k )n |k − k |2 

181



σ T   T   T Ik Ik−k  δ(ω − σ ωk ) δ(σ ωk − ω − σ ωk−k ) β

σ ,σ 

β

 (2)  ki kl χij k (k,ω |k − k,σ ωkT − ω ) (2)∗   χlmn (k ,ω |k − k,σ ωkT − ω ) − iπ δil − 2 k ⊥ (k,σ ωkT )    kj km (k − k )k (k − k )n   σ  T σ  γ 1 δj m − 2 Ik Ik−k × 2 k |k − k |2   γ × − × ×

σ ,σ T   γ δ(ω − σ δ(σ ωk − ω − σ ωk−k )  (2)    ki kl χij k (k ,ω |k − k,σ ωkT − ω ) (2)∗   χlmn (k ,ω |k iπ δil − 2 k ⊥ (k,σ ωkT )     kj km (k − k )k (k − k )n 1 δj m − 2 δkn − 4 k |k − k |2 





 Ikσ T

ωkT )

σ  T Ik−k 



δ(ω − σ



ωkT )

δ(σ ωkT



−ω −σ



− k,σ ωkT − ω )



T ωk−k )

σ ,σ 

   1 ki kl kj kk β β δil − 2 dk Im χij(3)kl (k,σ  ωk | − k, − σ  ωk |k,σ ωkT ) +4 2 2 k k σ      σ β kj kk 1 ki kl σT δil − 2 δj k − 2 Ik Ik + × 4 k k β σ  (3)   T   T T σ T σ T (7.53) × Im χij kl (k ,σ ωk | − k , − σ ωk |k,σ ωk ) Ik Ik . 



Proceeding from (7.53) by focusing on eigenmodes, and carrying out the ω integrations by virtue of the delta functions, we obtain the formal wave kinetic equation for transverse wave, which is to be considered in conjunction with (7.51). Brief descriptive texts, which appear in boxes, are also added to each term: ∂I σ T Induced emission 0 = ⊥ (k,σ ωkT ) k + 2 Im ⊥ (k,σ ωkT ) Ikσ T ∂t       1 (k − k )k (k − k )l ki kn kj km  δin − 2 + 4π Im dk 2 k k 2 |k − k |2 β,γ   σ ,σ

×

β χij(2)k (k,σ  ωk |k



σ  γ

Ik−k Ikσ T

(2)∗  − k,σ  ωk−k ) χnml (k ,σ  ωk |k − k,σ  ωk−k ) γ

σ β

Ik Ikσ T × + γ  β  (k,σ  ωk )  (k − k,σ  ωk−k )

β

γ

182

Nonlinear Electromagnetic Equations in Vlasov Plasmas σ  γ

σ β

2 Ik−k Ik



δ(σ ωkT − σ  ωk − σ  ωk−k ) Decay (T ,β,γ ) ⊥ (k,σ ωkT )      1 kj km (k − k )k (k − k )l ki kn δin − 2 δj m − 2 + 2 k k |k − k |2   γ −

γ

β

σ ,σ

(2)∗  × χij(2)k (k,σ  ωkT |k − k,σ  ωk−k ) χnml (k ,σ  ωkT |k − k,σ  ωk−k )   σ  γ 2 Ik−k Ikσ T Ikσ T Ikσ T + × γ ⊥ (k,σ  ωkT )  (k − k,σ  ωk−k )    σ  γ 2 Ik−k Ikσ T T  T  γ Decay (T ,T ,γ ) δ(σ ωk − σ ωk − σ ωk−k ) −  ⊥ (k,σ ωkT )        1 (k − k )k (k − k )l ki kn kj km  δin − 2 + 4 Im dk 2 k k 2 |k − k |2  β γ

γ

σ

(2) − k,σ ωkT − σ  ωk ) χnml (k,σ  ωk |k − k,σ ωkT − σ  ωk )   (k − k )k (k − k )l 2 − ×P + δ kl β |k − k |2  (k − k,σ ωkT − σ  ωk )

×

β χij(2)k (k,σ  ωk |k

β

β

β

(2) (k,σ  ωk |k − k,σ ωkT − σ  ωk ) × χij(2)k (k,σ  ωk |k − k,σ ωkT − σ  ωk ) χnml  2 (3)   β   β T + χ (k ,σ ω | − k , − σ ω |k,σ ω ) ×P k ij mn k k β ⊥ (k − k,σ ωkT − σ  ωk ) β

β

β

β

σ β

Induced scattering β,T × Ik Ikσ T     1 kj km (k − k )k (k − k )l ki kn δin − 2 δj m − 2 + 4 k k |k − k |2  σ

(2) × χij(2)k (k,σ  ωkT |k − k,σ ωkT − σ  ωkT ) χnml (k,σ  ωkT |k − k,σ ωkT − σ  ωkT )   (k − k )k (k − k )l 2 ×P + δkl − |k − k |2  (k − k,σ ωkT − σ  ωkT ) (2) (k,σ  ωkT |k − k,σ ωkT − σ  ωkT ) × χij(2)k (k,σ  ωkT |k − k,σ ωkT − σ  ωkT ) χnml  2 (3)   T   T T ×P + χij mn (k ,σ ωk | − k , − σ ωk |k,σ ωk ) ⊥ (k − k,σ ωkT − σ  ωkT ) 

× Ikσ T Ikσ T .

Induced scattering T ,T

(7.54)

The wave kinetic equation for T mode is an entirely new equation that could not be discussed within the context of electrostatic approximation.

7.3 Nonlinear Susceptibilities

183

7.3 Nonlinear Susceptibilities In this section, we revisit the formal definitions of various linear and nonlinear susceptibility tensors, defined in (7.21), and evaluate specific forms in approximate manners, in order to facilitate further analysis. 7.3.1 Linear Susceptibility Tensor The linear susceptibility tensor is given by    2  ωpa vi k·v ∂ vj ∂ a dv 1− Fa + χij (k,ω) = k· ω ω − k · v + i0 ω ∂vj ω ∂v    2  ωpa vi ∂Fa k ∂Fa dv = . (7.55) + × v× ω ω − k · v + i0 ∂v ω ∂v j It is instructive to rewrite this expression by means of partial integration,   2  ωpa ki vj + kj vi k 2 vi vj a χij (k,ω) = − 2 dv Fa δij + . + ω ω−k·v (ω − k · v)2

(7.56)

The linear susceptibility satisfies the following property: χija (−k, − ω) = χija∗ (k,ω).

(7.57)

From the first expression (7.55), the imaginary (or anti-Hermitian) part of linear susceptibility tensor is given by 2  ωpa ∂Fa a dv vi vj δ(ω − k · v) k · . (7.58) Im χij (k,ω) = −iπ 2 ω ∂v For fast waves (ω k · v), assuming a quasi-Gaussian form of Fa , we obtain   2  ωpa 3k 2 vT2 a ki kj a 1+ Re χij (k,ω) = − 2 ω k2 2ω2    k 2 vT2 a ki kj + δij − 2 1+ . (7.59) k 2ω2 In (7.59), we remind the readers that vT2 a = 2Ta /ma is the square of thermal speed, Ta being the temperature defined by Ta = ma dv v 2 Fa . For slow waves (ω k · v), upon making use of the property of quasi-Gaussian distribution, ∂Fa 2v ≈ − 2 Fa, ∂v vT a

184

Nonlinear Electromagnetic Equations in Vlasov Plasmas

we obtain Re χija (k,ω) = −

2 2ωpa



vT2 a ω

dv

vi vj Fa . ω−k·v

(7.60)

We may decompose v into components parallel and transverse to k, as is customary, v=

k k (k · v) (k × v) × k + = v + v⊥, k2 k2 k

where k·v , k As a consequence, we may write

(k × v) × k . k2

(7.61)

    ki kj 2 1 ki kj 2 v⊥ Fa, v + δ − ij ω − k v k 2  k2

(7.62)

v =



vi vj dv Fa = ω−k·v

 dv

v⊥ =

where we have made use of the fact that non-diagonal elements vanish since the velocity integrals associated with those terms are odd. For the parallel component, we may expand the denominator,   ω 1 1 1+ . ≈− ω − k v k v k v

(7.63)

The first term cannot contribute to the integral since it is odd. The second term leads to the cancellation of v2 from top and bottom. This leads to Re χija (k,ω)

 2 2  2 2ωpa 2ωpa ki kj v⊥ ki kj dv = 2 2 − − Fa . δ ij k2 ω − k v k vT a k 2 vT2 a ω

(7.64)

For Gaussian distribution, we have  dv

  2 v⊥ vT a ω . Fa = − Z ω − k v 2k k vT a

(7.65)

For slow waves, we may approximate the real part of the Z function by Z(ω/k vT a ) ≈ −2(ω/k vT a ). Thus, we have Re χija (k,ω)

=

2 2ωpa

k 2 vT2 a



  ki kj ki kj . − δij − 2 k2 k

(7.66)

7.3 Nonlinear Susceptibilities

185

Note that the imaginary part of linear susceptibility tensor (7.58) can be more specifically written as   ki kj ki kj a a Im χij (k,ω) = 2 Im χ (k,ω) + δij − 2 Im χ⊥a (k,ω), k k 2  2 ωpa (k · v) ∂Fa dv Im χa (k,ω) = −π 2 δ(ω − k · v) k · , 2 ω k ∂v   2  (k · v) k 2 π ωpa ∂Fa a dv v − δ(ω − k · v) k · Im χ⊥ (k,ω) = − . (7.67) 2 2 2 ω k ∂v We have already discussed the general diagonal property in (7.32), but for Gaussian thermal distribution, one can show that   ki kj a ki kj a χij (k,ω) = 2 χ (k,ω) + δij − 2 χ⊥a (k,ω), k k 2 2 ωpa ωpa χa (k,ω) = − 2 ξa2 Z  (ξa ), χ⊥a (k,ω) = 2 ξa Z(ξa ), ω ω ω . (7.68) ξa = k vT a Let us discuss some further alternative expressions. From (7.55), we have   2  ωpa 1 ki (k · v) [(k × v) × k]i a dv + ij (k,ω) = δij + ω ω − k · v + i0 k2 k2    k ∂Fa ∂Fa . (7.69) + × v× × ∂v ω ∂v j From the property (7.32), the tensor ij is given by the diagonal form   ki kj ki kj ij = 2  + δij − 2 ⊥ . k k From this, we see that ki kj  = 2 ij , k

  1 ki kj δij − 2 ij . ⊥ = 2 k

(7.70)

(7.71)

Now,  =

2 ωpa ki kj = 1 + ij k2 ω k2

 dv

k·v ∂Fa k· . ω − k · v + i0 ∂v

This allows us to obtain the familiar expression for longitudinal dielectric constant 2  ωpa k · ∂Fa /∂v dv . (7.72)  (k,ω) = 1 + χ (k,ω) = 1 + 2 k ω − k · v + i0

186

Nonlinear Electromagnetic Equations in Vlasov Plasmas

The transverse dielectric tensor can be obtained in a similar way:   1 ki kj δij − 2 ij ⊥ = 2 k 2  ωpa 1 (k × v) × k dv =1+ 2ω ω − k · v + i0 k2    ∂Fa k ∂Fa . · + × v× ∂v ω ∂v

(7.73)

We may rewrite this result as ⊥ (k,ω) = 1 + χ⊥ (k,ω) 2  ωpa [(k × v) × k]i dv =1+ 2ω k2    ∂Fa /∂vj k·v vi kj δij + × 1− . ω ω ω − k · v + i0

(7.74)

7.3.2 Second-Order Nonlinear Susceptibility Tensor The second-order susceptibility is given by (7.21), which is repeated here for the sake of convenience: χija(2) k (k1,ω1 |k2,ω2 )

 2 vi i ea ωpa dv =− 2 m a ω1 + ω 2 ω1 + ω2 − (k1 + k2 ) · v + i0    k1 · v ∂ 1 vj ∂ × 1− + k1 · ω1 ∂vj ω1 ∂v ω2 − k2 · v + i0    vk ∂ k2 · v ∂ + k2 · × 1− ω2 ∂vk ω2 ∂v    k2 · v ∂ vk ∂ + 1− + k2 · ω2 ∂vk ω2 ∂v    vj k1 · v ∂ 1 ∂ 1− Fa . × + k1 · ω1 − k1 · v + i0 ω1 ∂vj ω1 ∂v

Let us take partial integrations in order to obtain an alternative expression. Let us adopt the shorthand notation, ω = ω1 + ω2,

k = k1 + k2 .

(7.75)

7.3 Nonlinear Susceptibilities

187

Then we have χija(2) k (k1,ω1 |k2,ω2 ) 2 i ea ωpa =− 2 ma ωω1 ω2



vi ω − k · v + i0  ∂ ∂ ∂ vk ∂ Fa + v j k1 · + × (ω1 − k1 · v) k2 · ∂v ∂v ∂vk ω2 − k2 · v + i0 ∂v  j + (1 ↔ 2,j ↔ k)    2 ∂Fa i ea ωpa vk ∂Fa dv = + k2 · 2m ∂vk ω2 − k 2 · v ∂v a ωω1 ω2  (ω1 − k1 · v) δij + k1i vj (ω1 − k1 · v) kj vi + k1 · k vi vj × + ω − k· v (ω − k · v)2 dv

+ (1 ↔ 2,j ↔ k)    2 vk i ea ωpa ∂ ∂ dv Fa =− + k2 · 2ma ωω1 ω2 ∂vk ∂v ω2 − k2 · v ω2 − k 2 · v k1i vj + vi kj × δij − δij + ω−k·v ω−k·v   k1 · k ω2 − k2 · v vi kj + vi vj + (1 ↔ 2,j ↔ k) − (ω − k · v)2 (ω − k · v)2   2 i ea ωpa δik k1j δij k2k δj k ki dv Fa =− + + 2 m a ω ω1 ω2 ω1 − k1 · v ω2 − k2 · v ω − k · v 2 δij k22 vk δj k k 2 vi δik k1 vj + + + (ω1 − k1 · v)2 (ω2 − k2 · v)2 (ω − k · v)2 k2i vk k1i + vi kk k1j + k2i k1k vj + k1i kk vj + (ω1 − k1 · v)(ω − k · v) k1i vj k2k + vi kj k2k + k1i k2j vk + k2i kj vk + (ω2 − k2 · v)(ω − k · v) 2 k (k2i vj vk + kk vi vj ) k22 (k1i vj vk + kj vi vk ) + 1 + (ω1 − k1 · v)2 (ω − k · v) (ω2 − k2 · v)2 (ω − k · v) (k1 · k) (k2i vj vk + kk vi vj ) + (k2 · k) (k1i vj vk + k1j vi vk + k1k vi vj ) + (ω1 − k1 · v)(ω − k · v)2 (k2 · k) (k1i vj vk + kj vi vk ) + (k1 · k) (k2i vj vk + k2j vi vk + k2k vi vj ) + (ω2 − k2 · v)(ω − k · v)2 k12 (k2 · k) vi vj vk k22 (k1 · k) vi vj vk + + (ω1 − k1 · v)2 (ω − k · v)2 (ω2 − k2 · v)2 (ω − k · v)2  2(k1 · k)(k2 · k) vi vj vk 2(k1 · k)(k2 · k) vi vj vk , + + (ω1 − k1 · v)(ω − k · v)3 (ω2 − k2 · v)(ω − k · v)3 ω = ω1 + ω2,

k = k 1 + k2 .

(7.76)

188

Nonlinear Electromagnetic Equations in Vlasov Plasmas

It is possible to further rearrange (7.76) into a slightly more compact form,  2 i ea ωpa a(2) dv Fa χij k (k1,ω1 |k2,ω2 ) = − 2 m a ω ω 1 ω2  δik k1j δij k2k δj k ki × + + ω1 − k 1 · v ω2 − k 2 · v ω − k · v δik k12 vj δij k22 vk δj k k 2 vi + + (ω1 − k1 · v)2 (ω2 − k2 · v)2 (ω − k · v)2 k1i vj k2k + k2i k1j vk vi kj k2k + ki k2j vk + + (ω1 − k1 · v)(ω2 − k2 · v) (ω2 − k2 · v)(ω − k · v) vi k1j kk + ki vj k1k + (ω1 − k1 · v)(ω − k · v) 1 + (ω1 − k1 · v)(ω2 − k2 · v)(ω − k · v)  ω2 − k 2 · v ω1 − k 1 · v × k12 kk vi vj + k22 kj vi vk ω1 − k 1 · v ω2 − k 2 · v ω2 − k2 · v ω1 − k1 · v + k 2 k2j vi vk + k 2 k1k vi vj ω−k·v ω−k·v ω − k · v ω−k·v + k12 k2i vj vk + k22 k1i vj vk ω1 − k1 · v ω2 − k2 · v +



+ (k1 · k2 ) ki vj vk + (k1 · k) k2k vi vj + (k2 · k) k1j vi vk vi vj vk (ω1 − k1 · v)(ω2 − k2 · v)(ω − k · v)   2 k1 (k2 · k) k22 (k1 · k) k 2 (k1 · k2 ) . × + + ω1 − k1 · v ω2 − k2 · v ω−k·v +

(7.77)

In (7.77), one may replace the first argument by (k1,ω1 ) → (k1 + k2,ω1 + ω2 ) and the second argument by (k2,ω2 ) → (−k2, − ω2 ). Then one can easily show the following symmetry property, a(2) χija(2) k (k1 + k2,ω1 + ω2 | − k2, − ω2 ) = −χj ik (k1,ω1 |k2,ω2 ),

(7.78)

which we have already invoked in (7.31). Infinite Wavelength Limit In this limiting case, all the k vectors may be set equal to zero. From the alternative expression (7.77), we can immediately deduce that χija k (0,ω1 |0,ω2 ) = 0.

(7.79)

7.3 Nonlinear Susceptibilities

189

This can also be seen directly from definition (7.21). If we take all k vectors to zero, then we are left with  2 ωpa ∂ 2 Fa i ea a(2) dv vi χij k (0,ω1 |0,ω2 ) = − 2 ma ω1 ω2 (ω1 + ω2 ) ∂vj ∂vk  2 ωpa ∂Fa i ea dv = δij = 0. 2 ma ω1 ω2 (ω1 + ω2 ) ∂vk Fast Wave Limit Suppose that ω1 k1 vT a ,

ω2 k2 vT a ,

and

(ω1 + ω2 ) |k1 + k2 | vT a .

(7.80)

Then we have from (7.77) χija k (k1,ω1 |k2,ω2 )

 2 ωpa δik k1j i ea =− 2 ma ω1 ω2 (ω1 + ω2 ) ω1  δij k2k δj k (k1 + k2 )i . + + ω2 ω1 + ω2

(7.81)

One could also obtain (7.81) directly from the original definition (7.21). One Slow Wave and Two Fast Waves If ω1 remains arbitrary but the combined frequency ω = ω1 + ω2 as well as ω2 satisfy the fast-wave condition, then retaining the leading terms only, we obtain  2 i ea ωpa dv Fa χija k (k1,ω1 |k2,ω2 ) = − 2 m a ω ω1 ω2   δik k1j δj k ki δij k2k δik k12 vj . × + + + ω1 − k 1 · v ω2 ω (ω1 − k1 · v)2 (7.82) Consider the quantity

 dv Fa

k1 vj . (ω1 − k1 · v)2

This is zero unless vj is parallel to k1 . Consequently, we may write   k1 vj k1 v k1j dv Fa = dv Fa 2 (ω1 − k1 · v) k1 (ω1 − k1 v )2    1 k1j ω1 dv Fa − . = + k1 ω1 − k1 v (ω1 − k1 v )2

190

Nonlinear Electromagnetic Equations in Vlasov Plasmas

This leads to χija k (k1,ω1 |k2,ω2 )

If we make use of  dv Fa

 2 i ea ωpa dv Fa =− 2 m a ω ω1 ω2   δij k2k δj k ki δik k1j ω1 . × + + ω2 ω (ω1 − k1 · v)2

1 1 =− 2 2 (ω1 − k · v) k1

 dv

(7.83)

k1 · ∂Fa /∂v , ω1 − k 1 · v

and retain the most important term, then we have χija k (k1,ω1 |k2,ω2 )

2 ωpa k1j i ea = δik 2 2 ma ω2 (ω1 + ω2 ) k1

 dv

k1 · ∂Fa /∂v . ω1 − k1 · v

(7.84)

If ω2 is a slow wave, then we may simply interchange 1 and 2 as well as j and k in (7.84) in order to arrive at χija k (k1,ω1 |k2,ω2 )

2 ωpa k2k i ea = δij 2 2 ma ω1 (ω1 + ω2 ) k2

 dv

k2 · ∂Fa /∂v . ω2 − k2 · v

(7.85)

If ω1 and ω2 are fast waves but ω = ω1 + ω2 is a slow mode, then starting from the expression    2 δj k ki i ea ωpa δj k k 2 vi dv Fa + χija k (k1,ω1 |k2,ω2 ) = − 2 m a ω ω1 ω2 ω − k · v (ω − k · v)2 and following the same steps as before, we obtain χija k (k1,ω1 |k2,ω2 )

2 (k1 + k2 )i i ea ωpa = δj k 2 m a ω1 ω2 |k1 + k2 |2

 dv

(k1 + k2 ) · ∂Fa /∂v . ω1 + ω2 − (k1 + k2 ) · v (7.86)

7.3.3 Third-Order Nonlinear Susceptibility Tensor Third-order nonlinear susceptibility tensor is defined by (7.21), which is again repeated here for the sake of convenience: χija(3) kl (k1,ω1 |k2,ω2 |k3,ω3 )

 2 ωpa (−i)2 ea2 dv = 2 m2a ω1 + ω2 + ω3 vi × ω1 + ω2 + ω3 − (k1 + k2 + k3 ) · v

7.3 Nonlinear Susceptibilities







191

k1 · v ∂ vj ∂ + k1 · ω1 ∂vj ω1 ∂v 1 × ω2 + ω3 − (k2 + k3 ) · v + i0    k2 · v ∂ vk ∂ × 1− + k2 · ω2 ∂vk ω2 ∂v    k3 · v ∂ 1 vl ∂ 1− × + k3 · ω3 − k3 · v + i0 ω3 ∂vl ω3 ∂v    1 k3 · v ∂ vl ∂ + 1− + k3 · ω3 ∂vl ω3 ∂v ω2 − k2 · v + i0    k2 · v ∂ vk ∂ Fa . × 1− + k2 · ω2 ∂vk ω2 ∂v ×

1−

Since we are only interested in the situation where (k1,ω1 ) = (k,ω ), (k2,ω2 ) = (−k, − ω ), and (k3,ω3 ) = (k,ω), we henceforth consider  2 (−i)2 ea2 ωpa     dv χija(3) (k ,ω | − k ,ω |k,ω) = kl 2 m2a ω2 ω2   vi ∂ ∂    (ω − k · v) × + vj k · ω−k·v ∂vj ∂v 1 × ω − ω − (k − k ) · v   ∂ ∂    × (ω − k · v) + vk k · ∂vk ∂v   ∂ vl ∂ × + k· ∂vl ω−k·v ∂v   ∂ ∂ − (ω − k · v) + vl k · ∂vl ∂v   ∂ vk ∂  Fa . × +  · (7.87) k ∂vk ω − k · v ∂v The third-order susceptibilities of actual relevance, that is, combinations that appear in the wave kinetic equation, are always given in the following forms: ki kl kj kk a(3)   χ (k ,ω | − k,ω |k,ω), T1 ≡ 2 k k 2 ij kl   kj kk ki kl     T2 ≡ 2 δj k − 2 χija(3) kl (k ,ω | − k ,ω |k,ω), k k   ki kl kj kk a(3)   χ (k ,ω | − k,ω |k,ω). T3 ≡ δil − 2 k k 2 ij kl

(7.88)

192

Nonlinear Electromagnetic Equations in Vlasov Plasmas

Moreover, the actual situation corresponds to the case when both ω and ω satisfy the fast wave condition, but ω − ω remains arbitrary. Writing out the quantities in (7.88) explicitly, we obtain, first for T1 ,  2 vi ki kl kj kk (−i)2 ea2 ωpa dv 2 T1 = 2 2 2 2 2 ma ω ω k k ω−k·v   1 ∂ ∂ + v j k · × (ω − k · v) ∂vj ∂v ω − ω − (k − k ) · v    ∂ ∂ ∂ vl ∂    × (ω − k · v) + vk k · + k· ∂vk ∂v ∂vl ω−k·v ∂v    ∂ ∂ ∂ vk ∂  Fa − (ω − k · v) + vl k · +  k · ∂vl ∂v ∂vk ω − k · v ∂v    2 1 1 ω 1 ea2 ωpa ∂ k · dv 1 − = 2 2 2  2 ma ω k k ω−k·v ∂v ω − ω − (k − k ) · v       ∂ k · ∂Fa /∂v ∂ k · ∂Fa /∂v −k· × k · ∂v ω − k · v ∂v ω − k · v   2 1 e2 ωpa 1 1 ∂ = − a2 2 2 dv k · 2 ma k k ω−k·v ∂v ω − ω − (k − k ) · v      ∂ k · ∂Fa /∂v ∂ k · ∂Fa /∂v  −k· . (7.89) × k · ∂v ω − k · v ∂v ω − k · v In the last inequality, we have made use of the fact that  ∂ dv k · G(v) = 0, ∂v for bounded function G(v). The result (7.89) shows that T1 = −χ¯ a(3) (k,ω | − k, − ω |k,ω),

(7.90)

where χ¯ a(3) (k,ω | − k, − ω |k,ω) is given by (1.136), and is the partial third-order nonlinear susceptibility associated with the purely electrostatic case. Consequently, the approximate expression already derived in (1.142) for ω kvT e,

ω k  vT e,

(ω − ω ) < |k − k | vT e,

is immediately applicable for the electrons: ki kl kj kk e(3)   χ (k ,ω | − k,ω |k,ω) k 2 k 2 ij kl e2 (k · k )2 (k − k )2 e ≈ χ (k − k,ω − ω ). 2m2e k 2 k 2 ω3 ω

T1 =

(7.91)

7.3 Nonlinear Susceptibilities

193

Next, we move on to evaluating two other related quantities. First, the quantity T2 . Upon a series of partial integrations, we have    2 kj kk (−i)2 ea2 ωpa ki kl T2 = 2 δj k − 2 dv k k 2 m2a ω2 ω2   1 vi ∂ ∂    (ω − k · v) × + vj k ·  ω−k·v ∂vj ∂v ω − ω − (k − k ) · v    ∂ ∂ ∂ vl ∂    × (ω − k · v) + vk k · + k· ∂vk ∂v ∂vl ω−k·v ∂v    ∂ ∂ vk ∂ ∂  Fa + vl k · +  − (ω − k · v) k · ∂vl ∂v ∂vk ω − k · v ∂v    2 kj kk 1 1 e2 ωpa = − a2 2 2 dv δj k − 2 2m ω k k ω−k·v   a 1 ∂ ∂ + v j k · × (ω − k · v)  ∂vj ∂v ω − ω − (k − k ) · v   1 ∂ ∂ ∂    × (ω − k · v) + vk k · k· ∂vk ∂v ω − k · v ∂v   ∂ vk ∂ ∂ Fa . +  (7.92) −k · k · ∂v ∂vk ω − k · v ∂v This result can be re-expressed in terms of T1 , which we have already discussed in (7.90) and (7.91), and a new quantity P , as follows: T2 = P − T1,    2 1 ∂ ∂ 1 e2 ωpa (ω − k · v) + v j k · P = − a2 2 2 dv 2 ma ω k ω−k·v ∂vj ∂v    1 ∂ ∂ (ω − k · v) × + v j k ·   ω − ω − (k − k ) · v ∂vj ∂v    ∂ 1 vj ∂ ∂ ∂  Fa . × +  k· −k· k · ω−k·v ∂v ∂v ∂vj ω − k · v ∂v

(7.93)

The third quantity T2 can similarly be manipulated as follows in terms of T1 and a new quantity, Q: T3 = Q − T1   2 1 vi 1 ea2 ωpa ∂  dv Q=− k · 2 2 2  2 ma ω k ω−k·v ∂v ω − ω − (k − k ) · v    ∂ ∂ vi ∂  + k· × k · ∂v ∂vi ω−k·v ∂v     1 ∂ ∂ ∂  Fa . (7.94) − (ω − k · v) + vi k · k · ∂vi ∂v ω − k · v ∂v

194

Nonlinear Electromagnetic Equations in Vlasov Plasmas

To compute the quantities P and Q, we introduce shorthand notations r = ω − k · v, r  = ω − k · v, ∂ , R = ω − ω − (k − k ) · v. ∂i = ∂vi

(7.95)

Then, we have  2 1 ea2 ωpa 1 P =− dv (r  ∂j + vj k · ∂) 2 2 2 2 ma ω k r     1 1   (r ∂j + vj k · ∂) × (k · ∂) Fa R r   vj  , −(k · ∂) ∂j Fa +  (k · ∂) Fa r

(7.96)

where we may manipulate the various terms in (7.96) as shown below in (7.97),   r 1 r  r ∂j (k · ∂) Fa = ∂j (k · ∂) Fa + kj 2 (k · ∂) Fa, r r r   v (k · k ) vj 1 j (k · ∂) Fa, (k · ∂) Fa = (k · ∂)(k · ∂) Fa + vj (k · ∂) r r r2   vj  vj (k · ∂)  (k · ∂) Fa =  (k · ∂)(k · ∂) Fa r r (k · k ) vj  kj + (k · ∂) Fa +  (k · ∂) Fa . (7.97) 2 r r Making use of these results, we obtain  2 1 ea2 ωpa 1 P =− dv (r  ∂j + vj k · ∂) 2 2 2 2 ma ω k r       1 1  1  r ∂j × (k · ∂) Fa + vj (k · ∂) (k · ∂) Fa R r r   vj  − (k · ∂) ∂j Fa − (k · ∂)  (k · ∂) Fa r    2 ω2  1 ea r 1 pa  = dv ∂j + (k · ∂) vj 2 m2a ω2 k 2 r r  1 1 1 × − ∂j (k · ∂) Fa − kj 2 (k · ∂) Fa − vj (k · ∂)(k · ∂) Fa  r r rr 1 1 + kj (k · ∂) Fa − kj  (k · ∂) Fa rR r R  (k · k ) (k · k )  (7.98) + vj 2 (k · ∂) Fa − vj 2 (k · ∂) Fa . r R r R

7.3 Nonlinear Susceptibilities

195

Making use of ∂j

r 1 r (k · k ) , + (k · ∂) vj = kj 2 + vj r r r r2

(7.99)

we have

  2 kj r  + vj (k · k ) 1 ea2 ωpa dv − (k · ∂) ∂j Fa P = 2 m2a ω2 k 2 r3 k 2 r  + (k · k )(k · v) (k · v) r  + (k · k ) v 2 − (k · ∂) F − (k · ∂)(k · ∂) Fa a r4 r3 r k2 k2 (k · k ) (k · v) − 3 (k · ∂) Fa + 2  (k · ∂) Fa − (k · ∂) Fa r r r r4 (k · k ) (k · v)  k2 k2 + (k · ∂) F + − (k · ∂) F (k · ∂) Fa a a r 2 r 2 r2 R r r R (k · k ) (k · v) (k · k ) (k · v)  + − (k · ∂) F (k · ∂) Fa a r3 R r r 2 R (k · k ) (k · v) (k · k ) (k · v)  + − (k · ∂) F (k · ∂) Fa a r3 R r2 r R  (k · k )2 v 2 (k · k )2 v 2  (7.100) + (k · ∂) Fa − 2 2 (k · ∂) Fa . r4 R r r R

If we retain only the leading terms, that is, those terms that contain R in the denominator and without the velocity-dependent terms in the numerator, then we have  2   2 k k2 1 ea2 ωpa  dv (7.101) − · ∂) F (k · ∂) F (k P = a a . 2 m2a ω2 k 2 r2 R r r R Upon approximating r ≈ ω and r  ≈ ω , we further simplify (7.101) as  2 1 1 ea2 ωpa dv P = 2 2  2 ma ω ω ω − ω − (k − k ) · v   ∂Fa ∂Fa 1 1 . k· −  k · × ω ∂v ω ∂v

(7.102)

For actual applications of this formula to the induced scattering problem, we may further approximate ω ∼ ω ∼ ωpe . In what follows, we are only interested in the electron response. Then, we finally have  (k − k ) · ∂Fe /∂v 1 e2 1 dv P = 2 2 m2e ωpe ω − ω − (k − k ) · v =

1 e2 (k − k )2 e χ (k − k,ω − ω ). 4 2 m2e ωpe

(7.103)

196

Nonlinear Electromagnetic Equations in Vlasov Plasmas

Inserting (7.103) to (7.93), we arrive at   kj kk ki kl     T2 = 2 δj k − 2 χije(3) kl (k ,ω | − k ,ω |k,ω) k k e2 (k × k )2 (k − k )2 e χ (k − k,ω − ω ). = 4 2m2e k 2 k 2 ωpe

(7.104)

We proceed in a similar manner for Q:      2 vi  vi 1 1 ea2 ωpa  (k · ∂) ∂i Fa + (k · ∂) Fa dv (k · ∂) Q=− 2 m2a ω2 k 2 r R r   1 −(r ∂i + vi k · ∂)  (k · ∂) Fa . (7.105) r Making use of





1 r  1 vi (k · ∂)  r r ∂i



vi (k · ∂)(k · ∂) Fa r ki (k · k ) vi + (k · ∂) F + (k · ∂) Fa, a r2 r  r r (k · ∂) Fa =  ∂i (k · ∂) Fa + ki 2 (k · ∂) Fa, (7.106) r r  vi (k · k ) vi   (k · ∂) Fa =  (k · ∂)(k · ∂) Fa + (k · ∂) Fa, r r 2

vi (k · ∂) (k · ∂) Fa r 

=

it is possible to show that  2 vi 1 ea2 ωpa dv (k · ∂) Q=− 2 2 2 2 ma ω k r     1 vi   (k · ∂) ∂i Fa + (k · ∂) × (k · ∂) Fa R r     1  1  (k · ∂) Fa − vi (k · ∂)  (k · ∂) Fa − r∂i r r     2 ω2  ki 1 ea (k · k ) vi pa = dv + 2 m2a ω2 k 2 r r2  1 1 1 × −  (k · ∂) ∂i Fa − ki 2 (k · ∂) Fa − vi  (k · ∂)(k · ∂) Fa r r rr    k 1 ki + (k · ∂) Fa − i (k · ∂) Fa R r r   (k · k ) vi (k · k ) vi  + (k · ∂) Fa − (k · ∂) Fa r2 r 2   2  1 ea2 ωpa 1 1 1  ≈ dv (7.107) (k · ∂) Fa −  (k · ∂) Fa . 2 m2a ω2 rR r r

7.3 Nonlinear Susceptibilities

197

For electrons in the context of induced scattering, we have  (k − k ) · ∂Fe /∂v e2 dv Q= 2 2m2e ωpe ω − ω − (k − k ) · v = which leads to

e2 (k − k )2 e χ (k − k,ω − ω ) = P , 2 2m2e ωpe

 ki kl kj kk e(3)   T3 = δil − 2 χ (k ,ω | − k,ω |k,ω) k k 2 ij kl e2 (k × k )2 (k − k )2 e = χ (k − k,ω − ω ). 4 2m2e k 2 k 2 ωpe

(7.108)



(7.109)

In summary, the third-order nonlinear susceptibility of interest is given by ki kl kj kk a(3)   χ (k ,ω | − k,ω |k,ω) T1 ≡ 2 k k 2 ij kl e2 (k · k )2 (k − k )2 e = χ (k − k,ω − ω ), 2m2e k 2 k 2 ω3 ω   kj kk ki kl     T2 ≡ 2 δj k − 2 χija(3) kl (k ,ω | − k ,ω |k,ω) k k e2 (k × k )2 (k − k )2 e χ (k − k,ω − ω ), = 2 2 2 4 2me k k ωpe     ki kl kj kk a(3)   χ (k ,ω | − k,ω |k,ω) T3 ≡ δil − 2 k k 2 ij kl e2 (k × k )2 (k − k )2 e = χ (k − k,ω − ω ). 4 2m2e k 2 k 2 ωpe

(7.110)

8 Electromagnetic Vlasov Weak Turbulence Theory

In this chapter, we formulate the detailed weak turbulence theory for unmagnetized plasmas interacting through fully electromagnetic force. Unlike Chapter 7, where longitudinal and transverse mode equations are given in formal representations, the results to be derived in this chapter relate directly to the plasma eigenmodes, and the mathematical expressions of the governing equations are explicit in that they can be solved by direct numerical means, or may be analyzed in detail from a theoretical standpoint. The development of the theory is based upon formal equations (7.51) and (7.54) for wave kinetic equations, as well as the simplified quasilinear particle kinetic equation (7.42) or, equivalently, (7.43). The discussion also makes use of specific forms of linear and nonlinear susceptibilities, which were extensively analyzed in Section 7.3. We begin with the discussion of linear theory of plasma eigenmodes.

8.1 Linear Theory of Electromagnetic Plasma Eigenmodes 8.1.1 Linear Dispersion Relation Linear dispersion relations for longitudinal and transverse waves determine their respective wave properties, Re  (k,ωkα ) = 0,

(α = L,S),

Re ⊥ (k,ωkT ) = 0.

(8.1)

For α = L,S, we have already discussed the linear dispersive properties in Section 1.8. Specifically, the Langmuir and ion-acoustic mode dispersion relations are given, respectively, by   3 2 2 L L ωk = ωpe 1 + k λDe , ω−k = −ωkL, 2 kcS (1 + 3Ti /Te )1/2 S , ω−k = −ωkS . (8.2) ωkS = 2 1/2 2 (1 + k λDe ) 198

8.1 Linear Theory of Electromagnetic Plasma Eigenmodes

199

In (8.2), Debye length and ion sound speed are defined by λ2De = Te /(4π ne ˆ 2) = 2 2 2 vT e /(2ωpe ) and cS = Te /mi , as already discussed previously – see, e.g., (3.26). We have also derived in (1.151) the following useful properties associated with longitudinal modes: ∂ Re  (k,σ ωkS ) 2 , ≈ S σ μk ωkL ∂(σ ωk )

∂ Re  (k,σ ωkL ) 2 , ≈ L ∂(σ ωk ) σ ωkL where  μk =

k 3 λ3De

(8.3)

     me 3Ti kvT e 3 me 3Ti 1+ = 1+ mi Te ωpe 8mi Te

is defined in (3.29). In the present electromagnetic theory of plasma waves, we now have an additional transverse wave dispersion relation, which, under the cold-plasma approximation, is given by 2 ωpe

c2 k 2 . (8.4) ω2 ω2 Solving for the transverse mode dispersion relation, we obtain the well-known result  T 2 + c2 k 2, ω−k = −ωkT . (8.5) ωkT = ωpe Re ⊥ (k,ωkT ) ≈ 1 −



It is also useful to compute the following (σ = ±1): ∂ Re ⊥ (k,σ ωkT ) 2 . ≈ T ∂(σ ωk ) σ ωkT

(8.6)

8.1.2 Linear Instabilities for Unmagnetized Plasmas We next discuss some well-known electrostatic and electromagnetic linear instabilities for unmagnetized plasmas. Weak electrostatic electron beam-plasma, or bump-on-tail instability, was already discussed in Section 1.8. The present section further extends the discussion to include other instabilities. It is convenient to start from linear dielectric constants for longitudinal and transverse modes,  (k,ω) and ⊥ (k,ω). The waves and instabilities correspond to the solutions of complex algebraic (generally transcendental) equations, 0 =  (k,ω) = 1 +

2   ωpa a

k2

dv

k · ∂Fa /∂v , ω−k·v

200

Electromagnetic Vlasov Weak Turbulence Theory

c2 k 2 c2 k 2 0 = ⊥ (k,ω) = ⊥ (k,ω) − 2 = 1 − 2 ω ω   2   ωpa [(k × v) × k]i ∂Fa vi ∂Fa . dv + + k· 2ω2 k2 ∂vi ω−k·v ∂v a

(8.7)

Without loss of generality, we may take k = zˆ k. Then k × v = xˆ k 2 vx + yˆ k 2 vy . If 2 = vx2 + vy2 , then we have we assume that Fa depends only on vz and v⊥ 0=1+

2   ωpa a

k

dv

∂Fa /∂vz , ω − k vz

2 2  ωpa c2 k 2  ωpa + 0=1− 2 − ω ω2 2ω2 a a

where we have made use of



 dv

2 ∂Fa kv⊥ , ω − k vz ∂vz

(8.8)

2 2 dv v⊥ ∂Fa /∂v⊥ = −1.

Electrostatic Instabilities 2 is irrelevant. We may For electrostatic instabilities, the dependence of Fa on v⊥ simply define the reduced distribution,  ∞ 2 ˆ dv⊥ v⊥ Fa (v⊥ ,vz ), (8.9) fa (vz ) = 2π 0

and rewrite the dispersion equation as 0 =  (ω,k) = 1 +

2   ωpa a

k



−∞

dvz

1 ∂ fˆa (vz ) . ω − k vz ∂vz

(8.10)

In the bulk part of this book, we take the approach of treating ω as real, and discuss the growth of unstable mode by wave kinetic equation. However, in the customary approach of linear theory of plasma waves, the angular frequency is generally treated as complex, ω = Re(ω) + iIm(ω) = ωr + iγ . If the growth rate or imaginary part, Im(ω), is much lower than the real frequency, Re(ω), then the growth rate γ is computed on the basis of the weak growth rate formula (1.154). Such a situation is known as the “dissipative” instabilities in the literature. The weak turbulence theory of this book is thus applicable to dissipative instabilities. The growth rate (1.154) is equivalent to the induced emission terms in the wave kinetic equation, except for a factor 2. When the condition Im(ω) Re(ω) is not satisfied, the instability is called “reactive,” and such instabilities generally do not lend themselves to the present weak turbulence theoretical treatment. Rather, more precisely speaking, the weak turbulence theory of this book must be generalized to treat the reactive instability, a short discussion of which is given in Appendix E.

8.1 Linear Theory of Electromagnetic Plasma Eigenmodes

201

An example of reactive instabilities may be the electromagnetic Weibel instability, to be discussed later. In this section, we will take the approach of treating the growth rate via (1.154). The first term on the right-hand side of wave kinetic equation (3.27), (3.28), (4.93), or (4.94) is equivalent to twice the growth rate (1.154): 2ωi ≡ − =

2 Im  (ω,k) ∂Re  (ω,k)/∂ω 2   ωpa 2π

∂Re  (ω,k)/∂ω

k

a

(8.11) ∞

−∞

dvz δ(ω − k vz )

∂ fˆa (vz ) . ∂vz

An example of weakly growing, or dissipative instability, is the bump-on-tail instability, already discussed Section 1.8. For Langmuir waves, the bump-on-tail instability growth rate (1.155) is given by  2 ∂ fˆa (vz )  πω  ωpa . (8.12) ωi = 2 a k 2 ∂vz vz =ω/k For a combination of thermal plus a tenuous drifting Gaussian model (1.156), e−vz /vT e δe−(vz −Ub ) /vT b fˆa (vz ) = 1/2 + , π vT e π 1/2 vT b 2

2

2

2

(8.13)

where δ 1 represents the density ratio between the beam and background electrons, and Ub designates the drift associated with tenuous energetic beam, the weak growth rate for Langmuir wave is given by    1 + 3k 2 λ2De 1 ωi π 1/2 =− 2 2 √ exp − ωpe 2k λDe 2k 2 λ2De 2 k λDe   (ωpe − k Ub )2 Te Ub − ωpe /k −δ . (8.14) exp − Tb vT b k 2 vT2 b This result was already discussed in (1.158), and the first term on the right-hand side depicts Landau damping, while the second term indicates bump-on-tail instability. If the beam speed Ub is higher than the Langmuir wave phase speed, vϕ = ωpe /k, then the instability (ωi > 0) is possible, provided the growth rate exceeds the Landau damping rate. Chapter 6 was devoted to the analysis of bump-on-tail instability and its nonlinear consequence. Another example of electrostatic dissipative instability is the ion-acoustic instability. For slow ion-sound waves, whose dispersion relation is given by (8.2), or   3Ti 1 2 2 Te 1+ , (8.15) ω =k 2 mi Te 1 + k 2 vT2 e /2 ωpe

202

Electromagnetic Vlasov Weak Turbulence Theory

it follows from (1.155) that the growth rate is given by  2 ∂ fˆa (vz )  πω4  ωpa ωi = . 2 k 2 ∂vz vz =ω/k 2ωpi a

(8.16)

In writing this equation, we have already assumed that Ti /Te is small. It is well known that the ion-acoustic mode is heavily damped if Ti /Te is large – see Figure 1.3. For stationary ions and drifting Gaussian electrons with the drift speed significantly lower than electron thermal speed, we already discussed the ion-acoustic instability, whose growth rate in the limiting case of Te /Ti 1 is given by (1.168), which is reproduced here, for the sake of convenience,     k 2 cS2 π 1/2 me 1/2 Ve 1 − . (8.17) ωi = 3/2 2 2 2 2 mi cS 1 + k 2 λ2De (1 + k λDe ) It is possible to apply the weak turbulence formalism for studying the nonlinear phase of ion-acoustic instability (Ishihara and Hirose, 1981), as we have done for the bump-in-tail instability, but we do not discuss such a problem in this book. Electromagnetic Instability Electromagnetic branch of the dispersion relation in (8.8) does not support any dissipative instabilities. Instead, it enjoys a purely growing unstable solution, that is, the Weibel instability (Weibel, 1959), which is a prime example of reactive instability, since the purely growing (or aperiod) mode satisfies ωi = 0 and ωr = 0. As we already noted, Weibel instability does not lend itself to the weak turbulence formalism developed in this book. However, we may discuss the Weibel instability for the sake of completeness. Consider the electromagnetic dispersion equation given by the second line of (8.8), 2 2  2  ωpa ∂Fa kv⊥ c2 k 2  ωpa dv + . (8.18) 0=1− 2 − 2 2 ω ω 2ω ω − kvz ∂vz a a The Weibel instability is characterized by zero real frequency, and it is characterized by c2 k 2 /|ωi |2 1. This is a generalized condition for slow mode, and thus (8.18) can be simplified by ignoring the unity term (the displacement current term),  2 2   ωpa k v⊥ ∂Fa 2 2 2 dv ωpa = . (8.19) c k + 2 ω − k vz ∂vz a a If we define the perpendicular temperature  ∞ 2T⊥a ˆ 3 fa (vz ), dv⊥ v⊥ Fa (v) = 2π ma 0

(8.20)

8.1 Linear Theory of Electromagnetic Plasma Eigenmodes

203

then (8.19) can be expressed as c k + 2 2



2 ωpa

=−



a

k 2 T⊥a ma

2 ωpa

a





−∞

dvz

fˆa (vz ) . (ω − k vz )2

(8.21)

This equation supports a purely growing or damped solution, ω = iωi , with zero real frequency (that is, an aperiod mode). If we assume that fˆa (vz ) is given by the Maxwellian form, e−vz /vT a fˆa (vz ) = 1/2 π vT a 2

2

vT2 a =

2Ta , ma

then upon inserting the model distribution, we obtain      ω ω 2 2 2 2 T⊥a 1+ , ωpa + ωpa Z 0 = −c k − Ta kvT a kvT a a a

(8.22)

(8.23)

where Z(ζ ) is the plasma dispersion function (1.122). The purely growing unstable mode is called the Weibel instability (Weibel, 1959). If we assume that      ω   ωi      (8.24)  kv  =  kv  1, z z then we may approximately write (8.23) as 

0 ≈ −c k − 2 2

 2 ωpa

a

and solving for ωi2 , we obtain ωi2

=

k2

$

a c2 k 2

 k 2 T⊥a 1− , ma ωi2

2 ωpa T⊥a /ma $ 2 . + a ωpa

The maximum growth occurs for k → ∞,   2 T⊥a 2 1/2 max ω . ωi = ma c2 pa a

(8.25)

(8.26)

(8.27)

To add finite k effects, we include thermal correction term associated with the plasma dispersion function – see the series expansion (1.123), √     π 2 2 2 2 T⊥a 1− (8.28) ωpa + ωpa γk . 0 ≈ −c k − Ta kvT a a a

204

Electromagnetic Vlasov Weak Turbulence Theory

This leads to

  k2 k 1− 2 , ωi = γ k0 k0 c2 k02 k0 vT a γ = √ $ 2 , π a ωpa (T⊥a /Ta )  2   ωpa T⊥a − 1 . k02 = c2 Ta a

(8.29)

According to this approximate analytical formula, the maximum growth should take place at k 2 /k02 = 1/3, and the maximum growth rate is given by ωimax =

2γ . 33/2

(8.30)

The instability also has a finite range of wave numbers over which the mode grows, 0 < k < k0 . An interesting extension of the aperiodic mode solution is when the plasma is isotropic, T⊥a = Ta = Ta . Then the dispersion relation (8.28) leads to the solution c2 k 3 , γk = − √ $ 2 π a (ωpa /vT a )

(8.31)

which is a purely damped mode with zero real frequency. 8.2 Electromagnetic Vlasov Weak Turbulence Theory for Eigenmodes We next move on to the discussion of electromagnetic weak turbulence theory that involves linear eigenmodes, that is, two longitudinal modes, L and S, and the transverse mode, T . Linear wave properties determine the characteristics of the eigenmodes. We start from the formal wave kinetic equations (7.51) and (7.54), and quasilinear particle kinetic equation (7.42) or (7.43), which we derived in Chapter 7. For Langmuir and ion-sound waves, the nonlinear wave kinetic equation at the formal level is given by (7.51). If we take α = L explicitly, then we obtain the formal wave kinetic equation for L mode. For ion-sound wave, we take α = S explicitly. Similarly, the formal wave kinetic equation for transverse wave is already given by (7.53). Formal wave kinetic equations already obtained can be further manipulated by making use of various approximate but explicit forms of linear and nonlinear susceptibility response tensors discussed in Section 7.3.

8.2 Electromagnetic Vlasov Weak Turbulence Theory for Eigenmodes

205

8.2.1 Induced Emission Induced emission terms are given by  2 Im  (k,σ ωkL ) σ L ∂Ikσ L  = − Ik , ∂t ind.em.  (k,σ ωkL )  ∂Ikσ S  2 Im  (k,σ ωkS ) σ S Ik , = − ∂t ind.em.  (k,σ ωkS )  2 Im ⊥ (k,σ ωkT ) σ T ∂Ikσ T  = − Ik . ∂t ind.em. ⊥ (k,σ ωkT )

(8.32)

Making use of Im  (k,σ ωkL )

= −π

2   ωpa

k2

a

Im  (k,σ ωkS ) = −π

2   ωpa

k2

a

Im ⊥ (k,σ ωkT )

= −π

dv δ(σ ωkL − k · v) k ·

∂Fa , ∂v

dv δ(σ ωkS − k · v) k ·

∂Fa , ∂v

 2  ωpa a

(ωkT )2

dv

(8.33)

(k × v)2 ∂Fa δ(σ ωkT − k · v) k · , 2 k ∂v

and 2 , σ μk ωkL  1/2   me 3Ti 1/2 3 3 μk = k λDe 1+ , mi Te 2 , ⊥ (k,σ ωkT ) = σ ωkT  (k,σ ωkL ) =

2 , σ ωkL

 (k,σ ωkS ) =

(8.34)

we obtain the desired expressions describing induced emissions of plasma eigenmodes,  2   ωpa ∂Fa σ L ∂Ikσ L  L dv δ(σ ωkL − k · v) k · = π (σ ωk ) I ,  2 ∂t ind.em. k ∂v k a  2   ωpa ∂Ikσ S  ∂Fa σ S L dv δ(σ ωkS − k · v) k · = π (σ μk ωk ) I ,  2 ∂t ind.em. k ∂v k a  2   ωpa (k × v)2 ∂Fa σ T ∂Ikσ T  dv = π δ(σ ωkT − k · v) k · I .  T 2 ∂t ind.em. k ∂v k σ ωk a

(8.35)

206

Electromagnetic Vlasov Weak Turbulence Theory

In (8.35), we note that the induced emission for transverse electromagnetic wave is practically zero since the resonance condition cannot be satisfied for fast T mode. Nevertheless, we include the expression for the sake of completeness. 8.2.2 Decay/Coalescence L Mode Decay processes for L mode involve the following terms:    ∂Ikσ L  L  σ  L σ L Q1 μk σ  ωkL Ik−k = − π (σ ωk ) Im dk  Ik  ∂t decay σ ,σ  L σ S σ L σ  L σ  S L δ(σ ωkL − σ  ωkS − σ  ωk−k + σ  ωk−k Ik − σ ωkL Ik−k  Ik   Ik  )  L σ  S σ L  L σ L σ L + Q2 σ ωk Ik−k Ik + μk−k σ ωk−k Ik Ik σ  S σ  L S − σ ωkL Ik−k δ(σ ωkL − σ  ωkL − σ  ωk−k  Ik  )  L σ  T σ L   T σ L σL + Q3 σ ωk Ik−k Ik + σ ωk−k 2 Ik Ik  σ  T L  L  T − σ ωkL Ikσ L Ik−k  δ(σ ωk − σ ωk − σ ωk−k ) σ  T σ L T σ S σ L + Q4 μk σ  ωkL Ik−k + σ  ωk−k Ik  Ik  2 Ik   σ T L  S  T − σ ωkL Ikσ S Ik−k  δ(σ ωk − σ ωk − σ ωk−k )  T σ  T σ L T σ T σ L + Q5 σ ωk 2 Ik−k + σ  ωk−k Ik  Ik  2 Ik   σ  T L  T  T − σ ωkL Ikσ T Ik−k − σ ω − σ ω ) (8.36) δ(σ ω  k k k−k , where the coupling coefficients are given by Q1 =

 (k − k )k (k − k )l ki kn kj km k 2 k 2 |k − k |2 (2)∗  L  S   L × χij(2)k (k,σ  ωkS |k − k,σ  ωk−k  ) χnml (k ,σ ωk |k − k ,σ ωk−k ),

Q2 =

 (k − k )k (k − k )l ki kn kj km k 2 k 2 |k − k |2

(2)∗  S  L   S × χij(2)k (k,σ  ωkL |k − k,σ  ωk−k  ) χnml (k ,σ ωk |k − k ,σ ωk−k ),    ki kn kj km (k − k )k (k − k )l δ − Q3 = 2 kl k k 2 |k − k |2 (2)∗  T  L   T × χij(2)k (k,σ  ωkL |k − k,σ  ωk−k  ) χnml (k ,σ ωk |k − k ,σ ωk−k ),    ki kn kj km (k − k )k (k − k )l δkl − Q4 = 2 k k 2 |k − k |2 (2)∗  T  S   T × χij(2)k (k,σ  ωkS |k − k,σ  ωk−k  ) χnml (k ,σ ωk |k − k ,σ ωk−k ),

8.2 Electromagnetic Vlasov Weak Turbulence Theory for Eigenmodes



Q5 =

1 ki kn δj m − 4 k2

  km δkl k 2

kj



(k − k )k (k − k )l |k − k |2

207

 (8.37)

(2)∗  T  T   T × χij(2)k (k,σ  ωkT |k − k,σ  ωk−k  ) χnml (k ,σ ωk |k − k ,σ ωk−k ).

In nonlinear response functions, only the electron contribution is of importance, since lighter electrons rapidly respond to finite amplitude perturbations, while the heavier ions lag behind. The approximate expressions for various second-order nonlinear susceptibility tensors in the context of decay processes in (8.36) are given by L χij(2)k (k,σ  ωkS |k − k,σ  ωk−k ) = −

(2)∗  L (k ,σ  ωkS |k − k,σ  ωk−k χnml ) =

 χe (k,σ  ωkS ) i e δnl km , L L 2 me (σ  ωk−k  )(σ ωk )

S χij(2)k (k,σ  ωkL |k − k,σ  ωk−k ) = −

(2)∗  S (k ,σ  ωkL |k − k,σ  ωk−k χnml ) =

T χij(2)k (k,σ  ωkL |k − k,σ  ωk−k ) =

e S i e δik kj χ (k,σ  ωk ) , L L 2 me (σ  ωk−k  )(σ ωk )

e S i e δij (k − k )k χ (k − k,σ  ωk−k ) , 2 me (σ  ωkL )(σ ωkL )

S ) i e δnm (k − k )l χe (k − k,σ  ωk−k , L L  2 me (σ ωk )(σ ωk ) 2 ωpe i e T L 2 me (σ  ωkL )(σ  ωk−k  )(σ ωk )   kj (k − k )k ki , + δj k × δik  L + δij  T σ ωk σ ωk−k σ ωkL 2 ωpe i e T L 2 me (σ  ωkL )(σ  ωk−k  )(σ ωk )   k (k − k )l kn , + δml × δnl  mL + δnm  T σ ωk σ ωk−k σ ωkL

(2)∗  T χnml (k ,σ  ωkL |k − k,σ  ωk−k ) = −

T χij(2)k (k,σ  ωkS |k − k,σ  ωk−k ) = −

(2)∗  (k ,σ  ωkS |k χnml

χij(2)k (k,σ  ωkT |k





T ωk−k )





T ωk−k )

− k ,σ − k ,σ

e S i e δik kj χ (k,σ  ωk ) , T L 2 me (σ  ωk−k  )(σ ωk )

 χe (k,σ  ωkS ) i e δnl km = , T L 2 me (σ  ωk−k  )(σ ωk ) 2 ωpe i e = T L 2 me (σ  ωkT )(σ  ωk−k  )(σ ωk )   kj (k − k )k ki , + δj k × δik  T + δij  T σ ωk σ ωk−k σ ωkL

208

Electromagnetic Vlasov Weak Turbulence Theory 2 ωpe i e T L 2 me (σ  ωkT )(σ  ωk−k  )(σ ωk )    km (k − k )l kn . + δml × δnl  T + δnm  T σ ωk σ ωk−k σ ωkL (8.38)

(2)∗  T χnml (k ,σ  ωkT |k − k,σ  ωk−k ) = −

Substituting (8.38) to (8.37), we obtain Q1 =

e S e2 k 2 [k · (k − k )]2 [χ (k,σ  ωk )]2 , L 2 4m2e k 2 |k − k |2 (ωkL )2 (ωk−k )

S 2  )] e2 |k − k |2 (k · k )2 [χe (k − k,σ  ωk−k , Q2 = 4m2e k 2 k 2 (ωkL )2 (ωkL )2  2  4 ωpe (k × k )2 k e2 k 2 2 Q3 = + , T L 2 2 4m2e k 2 k 2 |k − k |2 (ωkL )2 (ωk−k σ ωkL σ  ωkL  ) (ωk )

Q4 =

e2 k 2 (k × k )2 [χe (k,σ  ωkS )]2 , T L 2 2 4m2e k 2 |k − k |2 (ωk−k  ) (ωk )

Q5 =

4 ωpe e2 k 2 {k 2 |k − k |2 + [k · (k − k )]2 } . T L 4 2 16m2e k 2 |k − k |2 (ωkT )2 (ωk−k  ) (ωk )

(8.39)

Inserting the coupling coefficients (8.39) to (8.36), we obtain the expressions for decay processes in the wave kinetic equation,   2  e μk [k · (k − k )]2 ∂Ikσ L  L  dk = −π (σ ω ) k ∂t decay 4Te2 k 2 k 2 |k − k |2 σ ,σ    σ S  Ikσ S σ L  L σ  L σ L  L L σ  L Ik × σ ωk Ik−k Ik + σ ωk−k I − σ ωk Ik−k μ k k μ k L × δ(σ ωkL − σ  ωkS − σ  ωk−k )   Iσ S e2 μk−k (k · k )2  L k−k σ L + σ ωk I 4Te2 k 2 k 2 | k − k |2 μk−k k   Iσ S  L σ L σ L L k−k σ  L S δ(σ ωkL − σ  ωkL − σ  ωk−k + σ ωk−k Ik Ik − σ ωk I ) μk−k k   Iσ T e2 μk (k × k )2  L k−k σ L + σ ωk I 2Te2 k 2 k 2 |k − k |2 2 k  σ  S I σ  T  Ikσ S σ L k−k  T L Ik T δ(σ ωkL − σ  ωkS − σ  ωk−k + σ ωk−k I k − σ ωk ) μ k μ k 2

8.2 Electromagnetic Vlasov Weak Turbulence Theory for Eigenmodes



2 

209

σ  T

I  (k × k )2 k2 e2 k 2 σ  ωkL k−k Ikσ L + L L 2 2 2 2  2  8 me ωpe k k |k − k | σ ωk 2 σ ωk   I σ T  T σ L σ L T + σ  ωk−k Ik − σ ωkL Ikσ L k−k δ(σ ωkL − σ  ωkL − σ  ωk−k  Ik  ) 2 +

1 e2 k 2 {k 2 |k − k |2 + [k · (k − k )]2 } T T 2 2  2 2 4me k |k − k | (ωk )2 (ωk−k )    σ  T I σ  T  I σ T Ikσ T σ L k−k T L Ik  × σ  ωkT k−k Ikσ L + σ  ωk−k − σ ω I  k 2 2 k 2 2  T × δ(σ ωkL − σ  ωkT − σ  ωk−k (8.40) ) . +

The first two terms on the right-hand side dictated by three-wave resonance conL L  L  S ditions, δ(σ ωkL − σ  ωkS − σ  ωk−k  ) and δ(σ ωk − σ ωk − σ ωk−k ), are the same processes already present in the electrostatic theory, and they describe a Langmuir wave decaying into another Langmuir wave and an ion-sound wave (L → L + S). T The third decay term dictated by δ(σ ωkL − σ  ωkS − σ  ωk−k  ) describes a Langmuir wave decaying into an ion-sound wave and a transverse wave (L → S + T ), and this process cannot be described by the electrostatic theory. The term dictated by T δ(σ ωkL − σ  ωkL − σ  ωk−k  ) describes the decay of a Langmuir wave into another Langmuir wave and a transverse mode (L → L + T ). The final term associated T with the resonance delta function δ(σ ωkL − σ  ωkT − σ  ωk−k  ) dictates the decay of an L mode into two T modes, or L → T + T . All the processes pertaining to T modes are new, which could not be discussed with the electrostatic theory. S Mode Contribution to the S mode wave kinetic equation from decay processes is given by 

 ∂Ikσ S  L  σ  L σ S dk = − π μ σ ω Im Q σ  ωkL Ik−k  Ik k 1 k  ∂t decay σ ,σ   L σ L σ S σ  L σ  L L + σ  ωk−k Ik − μk σ ωkL Ik−k δ(σ ωkS − σ  ωkL − σ  ωk−k  Ik   Ik  )

σ  T σ S T σ L σ S + 2 σ  ωk−k Ik + Q2 σ  ωkL Ik−k  Ik  Ik    σ  T T − μk σ ωkL Ikσ L Ik−k δ(σ ωkS − σ  ωkL − σ  ωk−k  )

σ  T σ S T σ T σ S + 2 σ  ωk−k Ik + Q3 2 σ  ωkT Ik−k  Ik  Ik  #   σ  T T − μk σ ωkL Ikσ T Ik−k (8.41) δ(σ ωkS − σ  ωkT − σ  ωk−k  ) ,

210

Electromagnetic Vlasov Weak Turbulence Theory

where   (k − k )k (k − k )l ki kn kj km Q1 = 2 k k 2 |k − k |2 (2)∗  L  L   L × χij(2)k (k,σ  ωkL |k − k,σ  ωk−k  ) χnml (k ,σ ωk |k − k ,σ ωk−k ),    ki kn kj km (k − k )k (k − k )l δkl − Q2 = 2 k k 2 |k − k |2 (2)∗  T  L   T × χij(2)k (k,σ  ωkL |k − k,σ  ωk−k  ) χnml (k ,σ ωk |k − k ,σ ωk−k ),     kj km 1 ki kn (k − k )k (k − k )l δj m − 2 δkl − (8.42) Q3 = 4 k2 k |k − k |2 (2)∗  T  T   T × χij(2)k (k,σ  ωkT |k − k,σ  ωk−k  ) χnml (k ,σ ωk |k − k ,σ ωk−k ).

The relevant susceptibilities are L χij(2)k (k,σ  ωkL |k − k,σ  ωk−k ) = −

(2)∗  L (k ,σ  ωkL |k − k,σ  ωk−k χnml ) =

i e δml kn χe (k,σ ωkS ) , L 2 me (σ  ωkL )(σ  ωk−k )

T χij(2)k (k,σ  ωkL |k − k,σ  ωk−k ) = −

(2)∗  T (k ,σ  ωkL |k − k,σ  ωk−k χnml ) =



− k ,σ



T ωk−k )

i e δj k ki χe (k,σ ωkS ) , T 2 me (σ  ωkL )(σ  ωk−k )

i e δml kn χe (k,σ ωkS ) , T 2 me (σ  ωkL )(σ  ωk−k )

T χij(2)k (k,σ  ωkT |k − k,σ  ωk−k ) = −

(2)∗  (k ,σ  ωkT |k χnml

i e δj k ki χe (k,σ ωkS ) , L 2 me (σ  ωkL )(σ  ωk−k )

i e δj k ki χe (k,σ ωkS ) , T 2 me (σ  ωkT )(σ  ωk−k )

i e δml kn χe (k,σ ωkS ) = . T 2 me (σ  ωkT )(σ  ωk−k )

(8.43)

Making use of (8.43), the quantities Q1 , Q2 , and Q3 defined in (8.42) become Q1 =

e2 [k · (k − k )]2 , 4Te2 k 2 k 2 |k − k |2

(k × k )2 e2 , 4Te2 k 2 k 2 |k − k |2   [k · (k − k )]2 e2 1 1 + 2 , Q3 = 16 Te2 k 2 k |k − k |2

Q2 =

(8.44)

8.2 Electromagnetic Vlasov Weak Turbulence Theory for Eigenmodes

211

which leads to   2  σS  ∂  Ikσ S e μk [k · (k − k )]2 L   L σ  L Ik dk σ = − π σ ω ω I   k k k−k ∂t decay μk 4Te2 k 2 k 2 |k − k |2 μk σ ,σ   σS L σ  L Ik σ  L σ  L L δ(σ ωkS − σ  ωkL − σ  ωk−k + σ  ωk−k − σ ωkL Ik−k  Ik   Ik  ) μk   I σ T σS e2 μk (k × k )2  L k−k Ik σ ω (8.45) + k 2Te2 k 2 k 2 |k − k |2 2 μk    σS Iσ T  T σ  L Ik L σ  L k−k S  L  T δ(σ ωk − σ ωk − σ ωk−k ) . + σ ωk−k Ik − σ ω k Ik  μk 2 In deriving this result, we have ignored T + T → S term since such a process cannot satisfy the wave energy and momentum conservation conditions. Note that the first term on the right-hand side is the same as the electrostatic counterpart, but the second term that involves a T mode is new. T Mode Next, we consider decay processes contributing to the wave kinetic equation for T mode. In doing so, let us ignore T + T → S process at the outset:     σT  Ikσ T ∂  T   L σ  L Ik Q σ Im dk ω I = −πσ ω   1 k k k−k ∂t decay 2 2 σ ,σ   σT L σ  L Ik σ  L σ  L L + σ  ωk−k δ(σ ωkT − σ  ωkL − σ  ωk−k − σ ωkT Ik−k  Ik   Ik  ) 2  σT σT σ  L Ik L σ  S Ik + Q2 σ  μk ωkL Ik−k + σ  ωk−k   Ik  2 2  σ  L σ  S L δ(σ ωkT − σ  ωkS − σ  ωk−k − σ ωkT Ik−k  Ik  )  σT σT σ  S Ik L σ  L Ik + Q3 σ  ωkL Ik−k + σ  μk−k ωk−k  Ik   2 2  σ  S σ  L S δ(σ ωkT − σ  ωkL − σ  ωk−k − σ ωkT Ik−k  Ik  )  + Q4 σ





σ L − σ ωkT Ik−k 



Ikσ T Ikσ T Ikσ T L + σ  ωk−k  2 2 2  σ T  Ik  T  T  L δ(σ ωk − σ ωk − σ ωk−k ) , 2

σ  L ωkT Ik−k 

(8.46)

212

Electromagnetic Vlasov Weak Turbulence Theory

where    (k − k )k (k − k )l 1 ki kn kj km δin − 2 Q1 = 2 k k 2 |k − k |2 (2)∗  L  L   L × χij(2)k (k,σ  ωkL |k − k,σ  ωk−k  ) χnml (k ,σ ωk |k − k ,σ ωk−k )  2  e2 (k × k )2 k |k − k |2 2 ≈ −  L , 2 k 2 k 2 |k − k |2 σ  ωL 32 m2e ωpe σ ωk−k k    (k − k )k (k − k )l 1 ki kn kj km δin − 2 Q2 = 2 k k 2 |k − k |2 (2)∗  L  S   L × χij(2)k (k,σ  ωkS |k − k,σ  ωk−k  ) χnml (k ,σ ωk |k − k ,σ ωk−k )

(k × k )2 e2 , 8Te2 k 2 k 2 |k − k |2    (k − k )k (k − k )l 1 ki kn kj km δin − 2 Q3 = 2 k k 2 |k − k |2 ≈

(2)∗  S  L   S × χij(2)k (k,σ  ωkL |k − k,σ  ωk−k  ) χnml (k ,σ ωk |k − k ,σ ωk−k )

(k × k )2 e2 , 8Te2 k 2 k 2 |k − k |2     kj km (k − k )k (k − k )l ki kn δj m − 2 Q4 = δin − 2 k k |k − k |2 ≈

(2)∗  L  T   L × χij(2)k (k,σ  ωkT |k − k,σ  ωk−k  ) χnml (k ,σ ωk |k − k ,σ ωk−k )   |k − k |2 (k · k )2 1 e2 1+ . (8.47) ≈ 4 m2e (σ ωkT )2 (σ  ωkT )2 k 4

Inserting the Q coefficients to the basic equation, we arrive at    Ikσ T (k × k )2 e2 ∂  T  dk = −π σ ω k 2 k 2 k 2 |k − k |2 ∂t decay 2 32 m2e ωpe σ ,σ    2 σT k |k − k |2 2  L σ  L Ikσ T  L σ  L Ik σ × − ω ω + σ  Ik−k  Ik  k k−k L 2 2 σ  ωkL σ  ωk−k   σ  L σ  L L δ(σ ωkT − σ  ωkL − σ  ωk−k − σ ωkT Ik−k  Ik  )   σT Ikσ S Ikσ T e2 μk (k × k )2  L σ  L Ik  L + σ ωk Ik−k + σ ωk−k 8Te2 k 2 k 2 |k − k |2 2 μ k 2   σ S σ  L Ik L δ(σ ωkT − σ  ωkS − σ  ωk−k − σ ωkT Ik−k  ) μ k

8.2 Electromagnetic Vlasov Weak Turbulence Theory for Eigenmodes



213

σ  S

σT σT I e2 μk−k (k × k )2  L k−k Ik  L σ  L Ik σ ω ω I + σ    k k−k k 8Te2 k 2 k 2 |k − k |2 μk−k 2 2   σ S I   S − σ ωkT k−k Ikσ L δ(σ ωkT − σ  ωkL − σ  ωk−k ) μk−k   σT 1 e2 |k − k |2 (k · k )2  T σ  L Ik + 1 + σ ω I (8.48)   k k−k 4 m2e (σ ωkT )2 (σ  ωkT )2 k 4 2   σ T  Ikσ T Ikσ T L T σ  L Ik T  T  L δ(σ ω I − σ ω − σ ω ) − σ ω + σ  ωk−k  k k−k k k k−k . 2 2 2

+

8.2.3 Induced Scattering L Mode As for Langmuir waves, we will ignore induced scattering involving S mode. The rationale for such a treatment is already discussed in Section 2.3.2. Consequently, contribution to the L mode wave kinetic equation from induced scattering processes includes the following terms: 1     ki kn kj km 4 ∂Ikσ L   = − Im dk ∂t ind.scatt. k 2 k 2  (k,σ ωkL ) σ  (k − k )k (k − k )l (2)   L × χij k (k ,σ ωk |k − k,σ ωkL − σ  ωkL ) |k − k |2 2 (2) (k,σ  ωkL |k − k,σ ωkL − σ  ωkL ) P × χnml   (k − k ,σ ωkL − σ  ωkL )   (k − k )k (k − k )l + δkl − χij(2)k (k,σ  ωkL |k − k,σ ωkL − σ  ωkL )  2 |k − k | 2 (2) (k,σ  ωkL |k − k,σ ωkL − σ  ωkL ) P × χnml  ⊥ (k − k ,σ ωkL − σ  ωkL )   + χij(3)mn (k,σ  ωkL | − k, − σ  ωkL |k,σ ωkL ) Ikσ L Ikσ L    kj km (k − k )k (k − k )l 1  ki kn δ + − j m 2  k2 k 2 |k − k |2 × ×

σ (2)  χij k (k ,σ  ωkT |k − k,σ ωkL − σ  ωkT ) (2) χnml (k,σ  ωkT |k − k,σ ωkL − σ  ωkT )

  (k − k )k (k − k )l 2 + δkl − ×P |k − k |2  (k − k,σ ωkL − σ  ωkT )

× χij(2)k (k,σ  ωkT |k − k,σ ωkL − σ  ωkT )

214

Electromagnetic Vlasov Weak Turbulence Theory (2) × χnml (k,σ  ωkT |k − k,σ ωkL − σ  ωkT ) 2 ×P  ⊥ (k − k ,σ ωkL − σ  ωkT )

+

χij(3)mn (k,σ  ωkT |



−k, −σ



(8.49) 

ωkT |k,σ ωkL )

 Ikσ T

 Ikσ L

.

Induced scattering involving transverse wave is relevant only for fundamental T mode. We thus have the following inequalities: ωkL kvT e,

ωkL k  vT e,

ωkL − ωkL |k − k | vT e,

ωkL kvT i ,

ωkL k  vT i ,

ωkL − ωkL |k − k | vT i ,

ωkL kvT e,

ωkT k  vT e,

ωkL − ωkT |k − k | vT e,

ωkL kvT i ,

ωkT k  vT i ,

ωkL − ωkT |k − k | vT i ,

 2

c |k − k |

2

(σ ωkL

−σ



ωkL )2,

 2

c |k − k |

2

(σ ωkL

−σ

(8.50) 

ωkT )2 .

Because of the inequality in the last line of (8.50), the induced scattering term that involves ⊥ is unimportant. This means that we can simplify (8.49) by  4 ∂Ikσ L  =−  ∂t ind.scatt.  (k,σ ωkL ) 1    2  σ L σ L I1 P × Im dk + J 1 I k I k ,σ ωL − σ  ωL ) (k − k  k k σ 2   1 2  I2 P + + J2 Ikσ T Ikσ L , (8.51) ,σ ωL − σ  ωT ) 2  (k − k   k k σ where the quantities of relevance are defined by  (k − k )k (k − k )l ki kn kj km I1 = 2 k k 2 |k − k |2

× χij(2)k (k,σ  ωkL |k − k,σ ωkL − σ  ωkL ) (2) × χnml (k,σ  ωkL |k − k,σ ωkL − σ  ωkL ),    kj km (k − k )k (k − k )l ki kn I2 = 2 δj m − 2 k k |k − k |2

× χij(2)k (k,σ  ωkT |k − k,σ ωkL − σ  ωkT ) (2) × χnml (k,σ  ωkT |k − k,σ ωkL − σ  ωkT ), (8.52)   ki kn kj km (3)   L χ (k ,σ ωk | − k, − σ  ωkL |k,σ ωkL ), J1 = 2 k k 2 ij mn    kj km ki kn χij(3)mn (k,σ  ωkT | − k, − σ  ωkT |k,σ ωkL ). J2 = 2 δj m − 2 k k

8.2 Electromagnetic Vlasov Weak Turbulence Theory for Eigenmodes

215

Making use of approximate forms of second-order nonlinear susceptibilities, χij(2)k (k,σ  ωkL |k − k,σ ωkL − σ  ωkL ) e i ≈− (k − k )k δij χe (k − k,σ ωkL − σ  ωkL ), 2 2 me ωpe (2) χnml (k,σ  ωkL |k − k,σ ωkL − σ  ωkL ) e i ≈− (k − k )l δnm χe (k − k,σ ωkL − σ  ωkL ), 2 2 me ωpe

χij(2)k (k,σ  ωkT |k − k,σ ωkL − σ  ωkT ) e i ≈− (k − k )k δij χe (k − k,σ ωkL − σ  ωkT ), 2 2 me ωpe (2) χnml (k,σ  ωkT |k − k,σ ωkL − σ  ωkT ) e i (k − k )l δnm χe (k − k,σ ωkL − σ  ωkT ), ≈− 2 2 me ωpe

(8.53)

we obtain I1 = −

e2 (k · k )2 (k − k )2 e [χ (k − k,σ ωkL − σ  ωkL )]2, 4 4m2e k 2 k 2 ωpe

I2 = −

e2 (k × k )2 (k − k )2 e [χ (k − k,σ ωkL − σ  ωkT )]2 . 4 4m2e k 2 k 2 ωpe

(8.54)

Similarly, by making use of approximate third-order nonlinear susceptibilities discussed in Section 7.3.3, we obtain J1 ≈

e2 (k · k )2 (k − k )2 e χ (k − k,σ ωkL − σ  ωkL ), 4 2m2e k 2 k 2 ωpe

J2 ≈

e2 (k × k )2 (k − k )2 e χ (k − k,σ ωkL − σ  ωkT ). 4 2m2e k 2 k 2 ωpe

This leads to 1    2  2 e2 ∂Ikσ L  L  (k · k ) (k − k ) = (σ ω ) Im dk k 4 ∂t ind.scatt. m2e k 2 k 2 ωpe %  [χe (k − k,σ ωkL − σ  ωkL )]2 × P  (k − k,σ ωkL − σ  ωkL ) σ  

− χe (k − k,σ ωkL − σ  ωkL ) Ikσ L Ikσ L

(8.55)

216

Electromagnetic Vlasov Weak Turbulence Theory

% [χe (k − k,σ ωkL − σ  ωkT )]2 1 (k × k )2 (k − k )2  + P 4 2 k 2 k 2 ωpe  (k − k,σ ωkL − σ  ωkT ) σ   e  L  T σ T σ L (8.56) − χ (k − k ,σ ωk − σ ωk ) Ik Ik .

The remaining terms to evaluate are related to the principal parts of inverse dielectric constants, which are treated as shown in (8.57):   Im χi [χe ]2 [χe ]2 e ≈ Im P e ≈ Im P χ 1 − i Im P  χe χ + i Im χi = Im (χe − i Im χi ) = Im χe − Im χi  2  π ωpa dv (k − k ) =−  )2 (k − k a=e,i ·

(8.57)

∂Fa δ[σ ωk − σ  ωk − (k − k ) · v]. ∂v

Of the two terms associated with the summation over particle species, the ion term is the dominant term since it is related to the scattering off thermal ions – see the discussion in Section 2.3.1. Substituting (8.57) to (8.56), we finally obtain     π e2 ∂Fi ∂Ikσ L  L  dk dv (k − k ) · = 2 σ ωk  ∂t ind.scatt. ωpe me mi ∂v σ  =±1  (k · k )2  δ[σ ωkL − σ  ωkL − (k − k ) · v] Ikσ L × 2 2 k k   Ikσ T (k × k )2 L  T  Ikσ L . (8.58) + δ[σ ωk − σ ωk − (k − k ) · v] k 2 k 2 2 T Mode The induced scattering process involving ion-sound modes, which is discussed in Section 2.3.3, will not be considered here. It is a slow process, and, as we have seen in Chapter 6, especially Figure 6.4, the ion-sound mode excitation by decay processes during the beam-plasma instability development is generally a transient process with low intensity. Consequently, the scattering of protons mediated by interaction with ion-sound wave can be ignored, which is a process that is expected to take place, if at all, over a time scale even longer than the ion-sound decay processes. This is because the ion-sound decay involves the Langmuir wave intensity such that the time scale of relevance is governed by the products of the wave intensities, tS decay ∝ O(IS IL ). In contrast, the scattering process involving two ionsound modes is expected to be characterized by tS scatt. ∝ O(IS2 ) O(IS IL ).

8.2 Electromagnetic Vlasov Weak Turbulence Theory for Eigenmodes

217

We thus consider the induced scattering process involving transverse waves. For this process, we only consider L–T mode coupling.      1 4 ki kn kj km ∂Ikσ T   δ = − − Im dk in ∂t ind.scatt. 2 k2 k 2 ⊥ (k,σ ωkT ) σ  (k − k )k (k − k )l (2)   L × χij k (k ,σ ωk |k − k,σ ωkT − σ  ωkL ) |k − k |2 2 (2) (k,σ  ωkL |k − k,σ ωkT − σ  ωkL ) P × χnml   (k − k ,σ ωkT − σ  ωkL )   (k − k)k (k − k)l + δkl − χij(2)k (k,σ  ωkL |k − k,σ ωkT − σ  ωkL ) 2 |k − k| 2 (2) (k,σ  ωkL |k − k,σ ωkT − σ  ωkL )P × χnml  ⊥ (k − k ,σ ωkT −σ  ωkL )   + χij(3)mn (k,σ  ωkL | − k, − σ  ωkL |k,σ ωkT ) Ikσ L Ikσ T . (8.59) Of these, contribution from the second term that contains the inverse of ⊥ in the denominator can again be ignored, since the frequency- and wave number–shifted arguments satisfy inequalities (8.50). The remaining terms for induced scattering of transverse waves involving the second-order nonlinear susceptibility are relevant only if the shifted frequency, σ ωkT − σ  ωkL remains small, which implies that this term is important only if the transverse mode has angular frequency close to the plasma frequency. We make use of approximate expressions for the second-order susceptibilities to simplify the following quantity:   (k − k )k (k − k )l ki kn kj km δin − 2 k k 2 |k − k |2



(2) × χij(2)k (k,σ  ωkL |k − k,σ ωkT − σ  ωkL ) χnml (k,σ  ωkL |k − k,σ ωkT − σ  ωkL )

≈ −

e2 (k − k )2 (k × k )2 e [χ (k − k,σ ωkT − σ  ωkL )]2 . 4 4m2e ωpe k 2 k 2

(8.60)

Likewise, making use of approximate third-order nonlinear susceptibility, we have 

  ki kn kj km δin − 2 χij(3)mn (k,σ  ωkL | − k, − σ  ωkL |k,σ ωkT ) 2 k k e2 (k × k )2 (k − k )2 e χ (k − k,σ ωkT − σ  ωkL ). ≈ 4 2m2e k 2 k 2 ωpe

(8.61)

218

Electromagnetic Vlasov Weak Turbulence Theory

From this, we arrive at    Ikσ T ∂  ∂Fi 1 π e2   dk dv (k − k ) · =  2 ∂t ind.scatt. 2 2 ωpe me mi  ∂v σ =±1

×

 2

σT (k × k ) T  L  σ  L Ik δ[σ ω − σ ω .  − (k − k ) · v] Ik k k k 2 k 2 2

(8.62)

8.3 Summary of Equations Let us summarize the equations of electromagnetic Vlasov weak turbulence theory within the framework of Vlasovian (or collisionless) approximation. In this summary, the final set of equations directly generalize the equations of electrostatic Vlasov weak turbulence theory, but, as in that case, the present electromagnetic Vlasov theory is incomplete in that various terms representing induced processes do not all have their spontaneous counterparts. To complete the theory requires the consideration of Klimontovich formalism, which is done in Part IV, but the discussion thereof builds upon the findings in Part III, which is summarized next. 8.3.1 Particle Kinetic Equation From (7.43), upon taking the eigenmodes to express the spectral wave electric field wave energy densities,    2 Ikσ α δ(ω − σ ωkα ), E k,ω = 



E⊥2 k,ω

α=L,S σ =±1

=



Ikσ T δ(ω − σ ωkT ),

σ =±1

we have the particle kinetic equation ∂Fa ∂Fa ∂ Dij , = ∂t ∂vi ∂vj     ki kj Ikσ α ea2 dk Dij = −Im 2 ma k 2 σ ωkα − k · v + i0 α σ  Ikσ T 1 aki akj + , 2 σ σ ωkT − k · v + i0    [(k × v) × k]i kj k·v ki kj δij − 2 + aij = 1 − . ω k k2 ω

(8.63)

Note that the contribution from transverse mode to the diffusion coefficient tensor is included only for the sake of completeness. In reality, because of the fact that

8.3 Summary of Equations

219

the transverse mode is a fast mode (superluminal mode with the phase speed higher than the speed of light in vacuo, or ω/k > c), the resonance condition cannot be satisfied, so that for all practical purposes, the T mode has no contribution to the velocity space diffusion. 8.3.2 L Mode Wave kinetic equation for Langmuir wave is given in (8.64), together with brief descriptions for each process, displayed as boxed texts, and indicated with underbraces: 2 ωpe ∂Ikσ L = π (σ ωkL ) 2 ∂t k

−π

(σ ωkL )



∂Fe σ L dv δ(σ ωkL − k · v) k · I & '( ) ∂v k

 σ ,σ 

ind. emiss. L

 e2 μk [k · (k − k )]2 σ  L σ L dk σ  ωkL Ik−k  Ik 2 2 2  2 4Te k k |k − k | 







L ωk−k 



σ S Ikσ S σ L σ  L Ik Ik − σ ωkL Ik−k  μ k μ k



L δ(σ ωkL − σ  ωkS − σ  ωk−k ) '( ) &

decay L ↔ L + S   2  Iσ S μk−k (k · k )2 L  e  L k−k σ L dk σ ωk − π (σ ωk ) I 4Te2 k 2 k 2 | k − k |2 μk−k k   σ ,σ







L σ L σ L ωk−k Ik  Ik 



σ ωkL

σ S Ik−k   Ikσ L μk−k



S δ(σ ωkL − σ  ωkL − σ  ωk−k ) '( ) &

decay L ↔ L + S   2  Iσ T μk (k × k )2 L  e  L k−k σ L dk σ ωk − π (σ ωk ) I 2Te2 k 2 k 2 |k − k |2 2 k   σ ,σ 





T ωk−k 





σ T Ikσ S σ L I σ S Ik−k  Ik − σ ωkL k μ k μ k 2



T δ(σ ωkL − σ  ωkS − σ  ωk−k ) '( ) &

decay L ↔ T + S  2   e2 (k × k )2 k k 2 2 L  dk − π (σ ωk ) +  L 2 k 2 k 2 |k − k |2 σ ωL 8 m2e ωpe σ ωk k σ ,σ    Iσ T  L k−k σ L T σ L σ L × σ ωk Ik Ik + σ  ωk−k  Ik  2

220

Electromagnetic Vlasov Weak Turbulence Theory







 σ ωkL Ikσ L

σ T Ik−k  2

T δ(σ ωkL − σ  ωkL − σ  ωk−k ) '( ) &

decay L ↔ T + L



e2 k 2 {k 2 |k − k |2 + [k · (k − k )]2 } T 2 4m2e k 2 |k − k |2 (ωkT )2 (ωk−k ) σ ,σ     σ  T I σ  T  I σ T Ikσ T σ L k−k T L Ik  × σ  ωkT k−k Ikσ L + σ  ωk−k − σ ω I  k 2 2 k 2 2 L  T  T × δ(σ ωk − σ ωk − σ ωk−k ) '( ) & − π (σ ωkL )

dk

decay L ↔ T + T    π e2 ∂Fi L  dk dv (k − k ) · + 2 σ ωk ωpe me mi ∂v σ  =±1  (k · k )2  × δ[σ ωkL − σ  ωkL − (k − k ) · v] Ikσ L '( ) k 2 k 2 & ind. scatt. L ↔ L + i 

Ikσ T (k × k )2 L  T  + δ[σ ω − σ ω  − (k − k ) · v] k k '( ) 2 k 2 k 2 &

 Ikσ L .

(8.64)

ind. scatt. L ↔ T + i The L mode equation mentioned in (8.64) is to be considered in conjunction with wave kinetic equations for S and T modes as well as with the particle kinetic equations. 8.3.3 S Mode The equation that dictates the time evolution of S mode intensity is given by 2 ωpe ∂ Ikσ S = π (σ μk ωkL ) 2 ∂t μk k

− π σ ωkL

 σ ,σ  

L σ L + σ  ωk−k  Ik

 dv δ(σ ωkS − k · v) k · & '( )

∂(Fe + Fi ) Ikσ S ∂v μk

ind. emiss. S

 σS e2 μk [k · (k − k )]2  L σ  L Ik σ ω  Ik−k k 4Te2 k 2 k 2 |k − k |2 μk  Ikσ S L σ  L σ  L L δ(σ ωkS − σ  ωkL − σ  ωk−k − σ ωk Ik−k Ik ) & '( ) μk dk

decay S ↔ L + L

8.3 Summary of Equations

− π σ ωkL

 σ ,σ  

T σ L + σ  ωk−k  Ik 

221



σ  T

σS I e2 μk (k × k )2  L k−k Ik σ ω (8.65)  k 2Te2 k 2 k 2 |k − k |2 2 μk   I σ T Ikσ S  T − σ ωkL Ikσ L k−k δ(σ ωkS − σ  ωkL − σ  ωk−k ) . '( ) & μk 2

dk

decay S ↔ L + T 8.3.4 T Mode The transverse mode wave intensity dynamically evolves according to 2   ωpa (k × v)2 ∂ Ikσ T ∂Fa Ikσ T T dv δ(σ ω − k · v) k · =π k T & '( ) ∂t 2 k2 ∂v 2 σ ωk a ind. emiss. T − π σ ωkT



dk

σ ,σ 

e2 (k × k )2 2 k 2 k 2 |k − k |2 32 m2e ωpe

 σT k 2 |k − k |2 2  L σ  L Ikσ T  L σ  L Ik σ × − ω I ω I + σ     k k−k k−k k L 2 2 σ  ωkL σ  ωk−k   σ  L σ  L L δ(σ ωkT − σ  ωkL − σ  ωk−k − σ ωkT Ik−k  Ik  ) & '( ) 

decay T ↔ L + L  σT 2  μk (k × k )2 T  e  L σ  L Ik dk σ − π σ ωk ω I   k k−k 8Te2 k 2 k 2 |k − k |2 2   σ ,σ



L + σ  ωk−k 

−π

σ ωkT

 σ ,σ  

L σ L + σ  ωk−k  Ik 

− π σ ωkT



σ S Ikσ S Ikσ T σ  L Ik − σ ωkT Ik−k  μ k 2 μ k

 σ ,σ 



L δ(σ ωkT − σ  ωkS − σ  ωk−k ) & '( )



decay T ↔ L + S 

I σ S σT e2 μk−k (k × k )2  L k−k Ik dk σ ω  k 8Te2 k 2 k 2 |k − k |2 μk−k 2   I σ S  Ikσ T S − σ ωkT k−k Ikσ L δ(σ ωkT − σ  ωkL − σ  ωk−k ) '( ) & 2 μk−k 

decay T ↔ L + S   2 |k − k |2 (k · k )2  1 e dk 1 + 2 2 4 m2e (σ ωkT )2 (σ  ωkT )2 k k

222

Electromagnetic Vlasov Weak Turbulence Theory

 × σ





σ L − σ ωkT Ik−k 

+



Ikσ T Ikσ T Ikσ T L + σ  ωk−k  2 2 2 σ T  Ik L δ(σ ωkT − σ  ωkT − σ  ωk−k ) & '( ) 2

σ  L ωkT Ik−k 

1 π e2 2 m m 2 ωpe e i

  σ  =±1

decay T ↔ L + T  ∂Fi (k × k )2  dk dv (k − k ) · ∂v k 2 k 2

I σT  × δ[σ ωkT − σ  ωkL − (k − k ) · v] Ikσ L k . & '( ) 2

(8.66)

ind. scatt. T ↔ L + i The particle and wave kinetic equations summarized in (8.63)–(8.66) constitute the self-consistent system of equations for electromagnetic Vlasov weak turbulence theory. These equations are not complete since not all terms appear in balanced form between the spontaneous and induced processes (for the waves), and the particle kinetic equations do not contain the velocity space friction (or drag) term. In order to supplement the set of equations in (8.63)–(8.66), we need to include the effects owing to particle discreteness, which we will turn to in Part IV next.

Part IV Klimontovich Weak Turbulence Theory: Electromagnetic Formalism

9 Electromagnetic Klimontovich Weak Turbulence Theory

In Part IV, we finalize the discourse by formulating the electromagnetic weak turbulence theory that includes the discrete particle effects, thereby completing the formalism. 9.1 Nonlinear Electrodynamic Equations in Plasmas The convenient starting point of the discussion is the set of equations already derived in Chapter 4, namely, (4.48), which was derived on the basis of Klimontovich equation combined with Maxwell’s equation. In what follows, it is convenient to re-express the result by making use of the momentum space representation, as such a representation naturally lends itself to relativistic formalism:  

∂ dω ∂fa (ω − k · v) δij + vj ki dk = −ea ∂t ∂pi ω , j a (p) , × δE−k,−ω δNk,ω    c2 2 4πi  j a δij − 2 k δij − ki kj δEk,ω = − ea dp vi δNk,ω (p), ω ω a

∂fa

a iea a0 i (ω − k · v) δij + vi kj (p)− δNk,ω (p) = − (ω −k · v) δNk,ω δEk,ω ω ∂pj   

∂ dω (ω − k · v) δij +vi kj dk − i ea  ∂pj ω a × [δEki ,ω δNk−k ,ω−ω (p)   i a − δEk,ω δNk−k,ω−ω (p) ],

(9.1)

where 

 δNa0 (p) δNb0 (p ) k,ω = (2π)−3 δab δ(p − p ) δ(ω − k · v) fa (p).

(9.2) 225

226

Electromagnetic Klimontovich Weak Turbulence Theory

Making use of definition (7.10), which is repeated here, 1 , ω − k · v + i0   ∂ k·v vi kj δij + = −iea 1 − , ω ω ∂pj

gk,ω = Lik,ω

(9.3)

and omitting δ in front of perturbed quantities, the equation for perturbed distribution δNa in (9.1) is re-expressed as follows: a a0 i = Nk,ω + gk,ω Lik,ω fa Ek,ω Nk,ω    a a   i i dω Lik,ω Nk−k + gk,ω dk . (9.4) ,ω−ω Ek,ω − Nk−k,ω−ω Ek,ω

This equation directly generalizes (7.11). The iterative solution that generalizes (7.12) is a(1) a0 i = Nk,ω + gk,ω Lik,ω fa Ek,ω , Nk,ω   ,

- a(2) a(1) a(1) i i Nk,ω = gk,ω dk dω Lik,ω Nk−k , ,ω−ω Ek,ω − Nk−k,ω−ω Ek,ω   ,

- a(3) a(2) a(2) i i Nk,ω = gk,ω dk dω Lik,ω Nk−k E − N E ,     ,ω−ω k ,ω k−k,ω−ω k ,ω

(9.5) a0 . Succesetc. Note that (9.5) is similar to (7.12), except for the “source” term Nk,ω sively inserting the lower-order solution to the next-order solution, and expressing the results in terms of the shorthand notations introduced in (7.14), we have

NKa(1) = NKa0 + gK LiK fa EKi ,  a0   a0 a(2) i i NK = dK  gK LiK  NK−K  EK  − NK−K  EK   ,

- j j j + dK  gK LiK  gK−K  LK−K  fa EKi  EK−K  − EKi  EK−K  ,  

j j a(3)  a0 i dK  gK LiK  gK−K  LK  NK−K NK = dK  −K  EK  EK  , - , - j j a0 i a0 − EKi  NK−K  −K  EK  − EK  NK−K  −K  EK    j  dK  gK LiK  gK−K  LK  gK−K  −K  LlK−K  −K  fa + dK

, - , - j j j l i l i l × EKi  EK  EK−K ,  −K  − EK  EK  EK−K  −K  − EK  EK  EK−K  −K  (9.6)

9.1 Nonlinear Electrodynamic Equations in Plasmas

227

etc. The symmetrized iterative solution thus emerges, NKa = NKa0 + gK LiK fa EKi ,

 - 1  j j j j + g1+2 Li1 g2 L2 + L2 g1 Li1 fa E1i E2 − E1i E2 2 1+2=K

 1  j j g1+2+3 Li1 g2+3 L2 g3 Ll3 + Ll3 g2 L2 fa + 2 1+2+3=K

, - , - j j j × E1i E2 E3l − E1i E2 E3l − E1i E2 E3l . (9.7) This generalizes (7.16), and constitutes the perturbative solution of nonlinear Klimontovich equation for perturbed distribution function. Substituting (9.7) to the right-hand side of the wave equation in (9.1), we arrive at 

ij (K) Ej (K) + 

+

 

χij(2)k (1|2) Ej (1) Ek (2) − Ej (1) Ek (2)

1+2=K

χij(3)kl (1|2|3) Ej (1) Ek (2) El (3)

1+2+3=K





− Ej (1) Ek (2) El (3) = −i

 4πea  a

ω

− Ej (1) Ek (2) El (3) dp vi NKa0,

(9.8)

where ij (K), χij(2)k (1|2), and χij(3)kl (1|2|3) are linear dielectric response tensor and second- and third-order nonlinear susceptibility tensors already defined in (7.20). Upon taking the dot product of (9.8) with Ei (K  ), we have  (2)     ij (K) Ei Ej K δ(K + K  ) + χij k (1|K − 1) Ei (K  ) Ej (1) Ek (K − 1) 

+ 2 δ(K + K ) = −i

 4πea  a

ω



1

χij(3)kl (1|

  − 1|K) Ei El K Ej Ek 1

1

  dp vi NKa0 Ei (K  ) .

(9.9)

This generalizes the Vlasov weak turbulence formalism given in (7.22) by including the source term on the right-hand side. Equation (9.9) also generalizes the electrostatic Klimontovich formalism described by (4.56).  To determine the third-body cumulant, Ei (K  ) Ej (1) Ek (K − 1) , and thereby close the infinite hierarchy of correlations, we write Ei (K) = Ei(0) (K) + Ei(1) (K) and express

228

Electromagnetic Klimontovich Weak Turbulence Theory

 , Ei (K  ) Ej (1) Ek (K − 1) ≈ Ei(1) (K  ) Ej(0) (1) Ek(0) (K − 1) , + Ei(0) (K  ) Ej(1) (1) Ek(0) (K − 1) , + Ei(0) (K  ) Ej(0) (1) Ek(1) (K − 1) ,



as before. We then drop the superscript (0) later. Quantities with superscript (1) can be obtained from (9.8) by ignoring the third-order nonlinearity,  4πea  (1) −1 dp vj NKa0 Ei (K) = − i ij (K) (9.10) ω a ! , -"  (2) (0) (0) (0) (0) (K) χ (1|2) E (1) E (2) − E (1) E (2) , − −1 j kl k l k l ij 1+2=K

which generalizes (7.23). Recall that (7.23) pertains to collisionless electromagnetic formalism. Equation (9.10) also generalizes (4.62), which relates to electrostatic formalism with discrete particle effects. From (9.10), it is rather straightforward to obtain other related quantities, Ei(1) (K  ), Ej(1) (1), and Ek(1) (K − 1), by permutation of the arguments. The result is given in (9.11), where we have dropped the superscript (0): Ei (K  ) Ej (1) Ek (K − 1)  4πea   dp vl NKa0 Ej (1) Ek (K − 1) (K ) = − i −1 il  ω a  4πea  dp vl Ei (K  ) NKa01 Ek (K − 1) − i −1 j l (1) ω 1 a  4πea  a0 − i −1 dp vl Ei (K  ) Ej (1) NK−1  kl (K − 1) ω − ω 1 a

(9.11)

(2)   − 2 δ(K + K  ) −1 il (K ) χlmn (1|K − 1) Em Ej 1 En Ek K  −1 (2)   − 2 δ(K + K  ) −1 j l (1) χlmn (−K |K + 1) Ei Em K  Ek En K−1 (2)   − 2 δ(K + K  ) −1 kl (K − 1) χlmn (−K |K + K − 1) Ei Em K  Ej En 1 .

Inserting (9.11) to (9.9), we obtain ij (K) Ei Ej K !  (2) (2)   −2 χij k (K  |K − K  ) −1 j l (K ) χlmn (−K + K |K) Ek Em K−K  K

 (2)    + −1 (K − K ) χ (−K |K) E E  j m K Ei En K lmn kl

9.1 Nonlinear Electrodynamic Equations in Plasmas (2)   + −1 il (−K) χlmn (−K | − K + K ) Ej Em K  Ek En K−K   (3) χij kl (K  | − K  |K) Ei El K Ej Ek K  +2

"

229

K

= −i

 4πea  ω

a

+i



4πea

dp vi NKa0 (p) Ei (−K)



dp χij(2)k (K  |K − K  )

K

a

 vl  a0 × −1 j l (K )  Ei (−K) Ek (K − K ) NK  (p) ω vl  a0 Ei (−K) Ej (K  ) NK−K + −1  (p) kl (K − K ) ω − ω  vl   a0 − −1 (−K) (K ) E (K − K ) N (p) . E j k −K il ω

(9.12)

To compute the terms on the right-hand side of (9.12), we return to (9.8), multiply and take the average,

a0 N−K (p),

ij (−K) Ej (−K) NKa0 (p)  4πeb  b0 dp vi N−K =i (p ) NKa0 (p) ω b  (2) χij k (−K  | − K + K  ) Ej (−K  ) Ek (−K + K  ) NKa0 (p) − K

−2



χij(3)kl (−K  |K  | − K) Ej Ek K  NKa0 (p) El (−K),

(9.13)

K

where we have replaced K by −K. Making use of (4.47), or to reiterate, b0 (p ) NKa0 (p) = Na0 (p) Nb0 (p )K N−K

= (2π)−3 δab δ(p − p ) δ(ω − k · v) fa (p), we have 4πea −1 (−K) vj δ(ω − k · v) fa (p) (2π)3 ω ij  (2) − −1 (−K) χj kl (−K  | − K + K  ) ij

NKa0 (p) Ei (−K) = i

K 

× Ek (−K ) El (−K + K  ) NKa0 (p)

(9.14)

230

Electromagnetic Klimontovich Weak Turbulence Theory

− 2−1 ij (−K) ×



  χj(3) klm (−K |K | − K)

K a0 Ek El K  NK (p) Em (−K).

(9.15)

In the last term of (9.15), we may use the iterative solution, NKa0 (p) Em (−K) = i

4πea −1 (−K) vn δ(ω − k · v) fa (p), (2π)3 ω mn

(9.16)

to obtain NKa0 (p) Ei (−K)

  (3) 4πea −1 =i (−K) v + 2 χj klm (−K  |K  | − K)  j (2π)3 ω ij K  × Ek El K  −1 mn (−K) vn δ(ω − k · v) fa (p) − −1 ij (−K)



  χj(2) kl (−K | − K + K )

K

× Ek (−K  ) El (−K + K  ) NKa0 (p).

(9.17)

The remaining quantities to evaluate are Ek (−K  ) El (−K + K  ) NKa0 (p), Ei (−K) Ek (K − K  ) NKa0 (p), a0 Ei (−K) Ej (K  ) NK−K  (p), a0 (p), Ej (K  ) Ek (K − K  ) N−K

all of which are of a generic form, a0 (p). Ea (K1 ) Eb (−K1 + K2 ) N−K 2

To evaluate this generic quantity, we return to (9.8) without the third-order term. We again resort to the iterative method and make use of (9.10). We then write a0 a0 (p) = Ea(1) (K1 ) Eb(0) (−K1 + K2 ) N−K (p) Ea (K1 ) Eb (−K1 + K2 ) N−K 2 2 a0 (p). + Ea(0) (K1 ) Eb(1) (−K1 + K2 ) N−K 2 (9.18)

Making use of (9.10) and the associated quantities with alternative arguments, we may evaluate the right-hand side of (9.18). In doing so, we ignore terms

9.1 Nonlinear Electrodynamic Equations in Plasmas

231

such as E N 0 N 0 . After some manipulations, we obtain, upon dropping the superscript (0), a0 Ea (K1 ) Eb (−K1 + K2 ) N−K (p) 2  (2) = −−1 χlmn (K  |K1 − K  ) al (K1 ) K a0 × [ Em (K  ) En (K1 − K  ) Eb (−K1 + K2 )N−K (p) 2 a0 − Em (K  ) En (K1 − K  ) Eb (−K1 + K2 )N−K (p) ] 2  (2) − −1 χlmn (K  | − K1 + K2 − K  ) bl (−K1 + K2 ) K a0 × [ Em (K  ) En (−K1 + K2 − K  ) Ea (K1 ) N−K (p) 2 a0 − Em (K  ) En (−K1 + K2 − K  ) Ea (K1 ) N−K (p) ] 2

(9.19)

(2) a0 = −2−1 al (K1 ) χlmn (K1 − K2 |K2 ) Em Eb K1 −K2 En (K2 )N−K2 (p) (2) a0 − 2−1 bl (−K1 + K2 ) χlmn (−K1 |K2 ) Em Ea K1 En (K2 )N−K2 (p).

Making use of the lowest-order solution – see (9.16), a0 En (K2 ) N−K (p) = −i 2

4πea −1 (K2 ) vp δ(ω2 − k2 · v) fa (p), (2π)3 ω2 np

(9.20)

the right-hand side of (9.19) can be evaluated: a0 Ea (K1 ) Eb (−K1 + K2 ) N−K (p) 2  8πea i (2) −1 = al (K1 ) χlmn (K1 − K2 |K2 ) Em Eb K1 −K2 (2π)3 ω2  (2) −1 + bl (−K1 + K2 ) χlmn (−K1 |K2 ) Em Ea K1

× −1 np (K2 ) vp δ(ω2 − k2 · v) fa (p).

(9.21)

Identifying a = k, b = l, K1 = −K  , and K2 = −K, we have Ek (−K  ) El (−K + K  ) NKa0 (p)  8πea i  (2)  −1 =− km (−K ) χmnp (−K + K| − K) En El K−K  (2π)3 ω  −1  (2)   + lm (K − K) χmnp (K | − K) En Ek K × −1 pq (−K) vq δ(ω − k · v) fa (p).

(9.22)

232

Electromagnetic Klimontovich Weak Turbulence Theory

With the choice of a = i, b = k, K1 = −K, and K2 = −K  , we have Ei (−K) Ek (K − K  ) NKa0 (p)  8πea i (2)   −1 =− im (−K) χmnp (−K + K | − K ) En Ek K−K  (2π)3 ω  −1  (2)  + km (K − K ) χmnp (K| − K ) En Ei K    × −1 pq (−K ) vq δ(ω − k · v) fa (p).

(9.23)

Next, upon identifying a = i, b = j , K1 = −K, and K2 = −K + K  , we obtain a0 Ei (−K) Ej (K  ) NK−K  (p)  8πea i (2)   −1 =− im (−K) χmnp (−K | − K + K ) En Ej K  (2π)3 (ω − ω )   (2)  + −1 (K ) χ (K| − K + K ) E E  n i K mnp jm    × −1 pq (−K + K ) vq δ[ω − ω − (k − k ) · v] fa (p).

(9.24)

Finally, for a = j , b = k, K1 = K  , and K2 = K, we arrive at a0 Ej (K  ) Ek (K − K  ) N−K (p)  8πea i  (2)  −1 = j m (K ) χmnp (K − K|K) En Ek K−K  (2π)3 ω  −1  (2)  + km (−K + K) χmnp (−K |K) En Ej K 

× −1 pq (K) vq δ(ω − k · v) fa (p).

(9.25)

Inserting (9.22) to (9.17), we obtain the closed-form expression for the source term, 4πea i NKa0 (p) Ei (−K) = −1 (−K) (2π)3 ω ij   (2) × vj + 2 χj kl (−K  | − K + K  ) K



 (2)  × −1 (9.26) km (−K ) χmnp (−K + K| − K) En El K−K   −1  (2)  + lm (K − K) χmnp (K | − K) En Ek K  −1 pq (−K) vq   (3)   −1 +2 χj klm (−K |K | − K) Ek El K  mn (−K) vn K

× δ(ω − k · v) fa (p).

9.1 Nonlinear Electrodynamic Equations in Plasmas

233

Substituting (9.23)–(9.26) to the right-hand side of (9.12), we obtain the intermediate form of nonlinear spectral balance equation including the effects of particle discreteness:  (2) χij k (K  |K − K  ) ij (K) Ei Ej K − 2 K

 (2)   × −1 j l (K ) χlmn (−K + K |K) Ek Em K−K   (2) −1    + kl (K − K ) χlmn (−K |K) Ej Em K Ei En K (2)  + −1 il (−K) χlmn (−K |

+2







− K + K ) Ej Em K  Ek En K−K 

χij(3)kl (K  | − K  |K) Ei El K Ej Ek K 

K

=

 (4πea )2  dp vi −1 ij (−K) vj δ(ω − k · v) fa (p) 3 ω2 (2π) a   (4πea )2   dp χij(2)k (K  |K − K  ) −1 +2 j l (K ) vl 3 ω2 (2π) a K  (2)   × −1 im (−K) χmnp (−K + K | − K ) En Ek K−K   −1  (2)     + km (K − K ) χmnp (K|− K ) En Ei K −1 pq (−K ) vq δ(ω − k · v) fa (p)   (4πea )2  dp χij(2)k (K  |K − K  ) −1 +2 kl (K − K ) vl 3 (ω − ω )2 (2π) a K  (2)   × −1 im (−K) χmnp (−K | − K + K ) En Ej K   −1  (2)  + j m (K ) χmnp (K| − K + K ) En Ei K    × −1 pq (−K + K ) vq δ[ω − ω − (k − k ) · v] fa (p).

(9.27)

This can be simplified by making use of the symmetry properties – see Section 7.3, ij (−K) = ∗ij (K), χij(2)k (−K1 | − K2 ) = χij(2)∗ k (K1 |K2 ), χij(3)kl (−K1 | − K2 | − K3 ) = χij(3)∗ kl (K1 |K2 |K3 ), (2) χij(2)k (K1 |K2 ) = χikj (K2 |K1 ),

234

Electromagnetic Klimontovich Weak Turbulence Theory

χij(3)kl (K1 |K2 |K3 ) = χij(3)lk (K1 |K3 |K2 ), χij(2)k (K1 + K2 | − K2, − K2 ) = −χja(2) ik (K1 |K2 ).

(9.28)

Then the nonlinear spectral balance equation (9.27) reduces to    ij (k,ω) δEi δEj k,ω − 2 dk dω χij(2)k (k,ω |k − k,ω − ω )  (2)       × −−1 j l (k ,ω ) χnlm (k ,ω |k − k ,ω − ω ) δEk δEm k−k,ω−ω  (2) −1         − kl (k − k ,ω − ω ) χnml (k ,ω |k − k ,ω − ω ) δEj δEm k ,ω (2)∗  (k ,ω |k − k,ω − ω ) × δEi δEn k,ω + ∗il −1 (k,ω) χlmn  × δEj δEm k,ω δEk δEn k−k,ω−ω    + 2 dk dω χij(3)kl (k,ω | − k, − ω |k,ω)

× δEi δEl k,ω δEj δEk k,ω  (4πea )2  dp vi ∗ij −1 (k,ω) vj δ(ω − k · v) fa (p) = 3 ω2 (2π) a    (4πea )2    + 2 dk dω dp χij(2)k (k,ω |k − k,ω − ω ) 3 ω2 (2π) a    ∗ −1 (2)∗     × −1 j l (k ,ω ) vl im (k,ω) χmpn (k ,ω |k − k ,ω − ω )   × δEn δEk k−k,ω−ω − −1 km (k − k ,ω − ω )  (2) × χnpm (k,ω |k − k,ω − ω ) δEn δEi k,ω

× ∗pq −1 (k,ω ) vq δ(ω − k · v) fa (p)    (4πea )2  dω + 2 dk (2π)3 (ω − ω )2 a    × dp χij(2)k (k,ω |k − k,ω − ω ) −1 kl (k − k ,ω − ω ) vl  (2)∗  × ∗im −1 (k,ω) χmnp (k ,ω |k − k,ω − ω ) δEn δEj k,ω  −1   (2)     − j m (k ,ω ) χnmp (k ,ω |k − k ,ω − ω ) δEn δEi k,ω

(9.29)

× ∗pq −1 (k − k,ω − ω ) vq δ[ω − ω − (k − k ) · v] fa (p), where we have written down the final result in terms of the longhand notation.

9.1 Nonlinear Electrodynamic Equations in Plasmas

235

Let us further manipulate (9.29) by making use of (7.32), (7.33), and (7.36). By taking the series of steps similar to that which led to (7.34), it is possible to show, albeit the intermediate steps are quite lengthy and tedious, that (9.29) reduces to  (k,ω) E2 k,ω + ⊥ (k,ω) E⊥2 k,ω    (2)     (2) (k,ω |k − k,ω − ω ) kj kl χij k (k ,ω |k − k,ω − ω ) χnlm   dω + 2 dk k 2  (k,ω )    (2) (k,ω |k − k,ω − ω ) kj kl χij(2)k (k,ω |k − k,ω − ω ) χnlm + δj l − 2 k ⊥ (k,ω )  ki kn (k − k )k (k − k )m 2 × E k−k,ω−ω E2 k,ω k2 (k − k )2   1 ki kn (k − k )k (k − k )m δ E⊥2 k−k,ω−ω E2 k,ω + − km 2 k2 (k − k )2   1 ki kn (k − k )k (k − k )m 2 δin − 2 + E k−k,ω−ω E⊥2 k,ω 2 k (k − k )2     1 ki kn (k − k )k (k − k )m 2 2   δin − 2 δkm − E +  E  ⊥ k−k ,ω−ω ⊥ k,ω 4 k (k − k )2        (k − k )k (k − k )l dω + 2 dk (k − k )2 ×

(2) (k,ω |k − k,ω − ω ) χij(2)k (k,ω |k − k,ω − ω ) χnml

 (k − k,ω − ω )  (k − k )k (k − k )l + δkl − (k − k )2 

×

(2) (k,ω |k − k,ω − ω ) χij(2)k (k,ω |k − k,ω − ω ) χnml



⊥ (k − k,ω − ω )



    kj km ki kn kj km 1 ki kn 2 2 × δj m − 2 E⊥2 k,ω E2 k,ω E k,ω E k,ω + k 2 k 2 2 k2 k    1 ki kn kj km δin − 2 + E2 k,ω E⊥2 k,ω 2 k k 2      kj km 1 ki kn 2 2 δin − 2 δj m − 2 E⊥ k,ω E⊥ k,ω + 4 k k    (2)∗  χ (2) (k,ω |k − k,ω − ω ) χlmn (k ,ω |k − k,ω − ω )   ki kl ij k dω − 2 dk k2 ∗ (k,ω)   (2)  (2)∗ ki kl χij k (k,ω |k − k,ω − ω ) χlmn (k,ω |k − k,ω − ω ) + δil − 2 k ∗⊥ (k,ω)

236

Electromagnetic Klimontovich Weak Turbulence Theory



kj

 km k 2

(k − k )k (k − k )n 2 E k,ω E2 k−k,ω−ω  2 (k − k )     k k 1 j m (k − k )k (k − k )n δ E2 k,ω E⊥2 k−k,ω−ω + − kn 2 k 2 (k − k )2    kj km (k − k )k (k − k )n 2 1 δj m − 2 + E⊥ k,ω E2 k−k,ω−ω 2 k (k − k )2     kj km 1 (k − k )k (k − k )n δj m − 2 δkn − + 4 k (k − k )2    2 2  dω χij(3)kl (k,ω | − k, − ω |k,ω) × E⊥ k,ω E⊥ k−k,ω−ω + 2 dk    kj kk 1 ki kl ki kl kj kk 2 2   δj k − 2 E⊥2 k,ω E2 k,ω E k ,ω E k,ω + × 2 2 2 k k 2 k k     k k 1 ki kl j k 2 δil − 2 + E k,ω E⊥2 k,ω 2 k k 2     kj kk 1 ki kl δil − 2 δj k − 2 E⊥2 k,ω E⊥2 k,ω + 4 k k    2  (4πea ) 1 1 (k · v)2 (k × v)2 dp = + (2π)3 ω2 k 2 ∗ (k,ω) k2 ∗⊥ (k,ω) a    (4πea )2    × δ(ω − k · v) fa (p) + 2 dk dω dp (2π)3 ω2 a       kj kl kj kl 1 1 × + δj l − 2 k 2  (k,ω ) k ⊥ (k,ω )     1 1 ki km ki km × + δim − 2 k 2 ∗ (k,ω) k ∗⊥ (k,ω) ×

(2)∗  × χij(2)k (k,ω |k − k,ω − ω ) χmpn (k ,ω |k − k,ω − ω )       kp kq kp kq 1 1 + δpq − 2 × k 2 ∗ (k,ω ) k ∗⊥ (k,ω )  (k − k )n (k − k )k 2 × E k−k,ω−ω (k − k )2    1 (k − k )n (k − k )k 2 δnk − E⊥ k−k,ω−ω vl vq δ(ω − k · v) fa (p) + 2 (k − k )2    (4πea )2    dω dp − 2 dk (2π)3 ω2 a       kj kl kj kl 1 1 × + δj l − 2 k 2  (k,ω ) k ⊥ (k,ω )



9.1 Nonlinear Electrodynamic Equations in Plasmas 



1 (k − k )k (k − k )m  2 (k − k )  (k − k,ω − ω )    1 (k − k )k (k − k )m + δkm − (k − k )2 ⊥ (k − k,ω − ω ) ×

(2) (k,ω |k − k,ω − ω ) × χij(2)k (k,ω |k − k,ω − ω ) χnpm       kp kq kp kq 1 1 + δpq − 2 × k 2 ∗ (k,ω ) k ∗⊥ (k,ω )     kn ki 1 kn ki 2 2 × δni − 2 E⊥ k,ω vl vq δ(ω − k · v) fa (p) E k,ω + k2 2 k     (4πea )2 dp + 2 dk dω (2π)3 (ω − ω )2 a  (k − k )k (k − k )l 1 ×  2 (k − k )  (k − k,ω − ω )     1 (k − k )k (k − k )l + δkl − (k − k )2 ⊥ (k − k,ω − ω )     1 ki km 1 ki km × − + δ im k 2 ∗ (k,ω) k 2 ∗⊥ (k,ω) (2)∗  × χij(2)k (k,ω |k − k,ω − ω ) χmnp (k ,ω |k − k,ω − ω )  1 (k − k )p (k − k )q × ∗  2 (k − k )  (k − k,ω − ω )    1 (k − k )p (k − k )q + δpq − (k − k )2 ∗⊥ (k − k,ω − ω )       kn kj kn kj 1 2 2 δnj − 2 E⊥ k,ω × E k,ω + k 2 2 k

× vl vq δ[ω − ω − (k − k ) · v] fa (p)     (4πea )2 dp − 2 dk dω 3 (ω − ω )2 (2π) a  (k − k )k (k − k )l 1 ×  2 (k − k )  (k − k,ω − ω )    1 (k − k )k (k − k )l + δkl − (k − k )2 ⊥ (k − k,ω − ω )        kj km kj km 1 1 × − + δ jm k 2  (k,ω ) k 2 ⊥ (k,ω ) (2) (k,ω |k − k,ω − ω ) × χij(2)k (k,ω |k − k,ω − ω ) χnmp

237

238



Electromagnetic Klimontovich Weak Turbulence Theory

1 (k − k )p (k − k )q ∗  2 (k − k )  (k − k,ω − ω )    1 (k − k )p (k − k )q + δpq − (k − k )2 ∗⊥ (k − k,ω − ω )     kn ki 1 kn ki 2 2 × δni − 2 E⊥ k,ω E k,ω + k2 2 k ×

× vl vq δ[ω − ω − (k − k ) · v] fa (p).

(9.30)

This generalizes (7.35), and if we ignore everything to the right of equality, then we recover (7.35). Equation (9.30) can further be separated into longitudinal and transverse parts once we write the electric field into eigenmode forms,  2spectra  2  $ $ $ σα α E k,ω = α=L,S σ =±1 Ik δ(ω − σ ωk ) and E⊥ k,ω = σ =±1 Ikσ T δ(ω − σ ωkT ) – see (7.48), and thus derive separate nonlinear spectral balance equations for longitudinal and transverse modes, as it was done in Sections 7.2.1 and 7.2.2. For the moment, we discuss the particle kinetic equation. We again take the simple view that the quasilinear approximation is valid. As a consequence, we adopt the following formal particle kinetic equation:      vj ki , a ∂ k·v ∂fa j δij + Nk,ω (p) E−k,−ω , dk dω 1− = −ea ∂t ∂pi ω ω    k Ek,ω k·v vk kl ∂fa a a0 1− δkl + . (9.31) Nk,ω (p) = Nk,ω (p) − iea ω−k·v ω ω ∂pl We may construct the relevant quantity by ignoring the nonlinear term, , - , j j a a0 Nk,ω (p) E−k,−ω = Nk,ω (p) E−k,−ω      Ej Ek k,ω k·v vk kl ∂fa 1− δkl + − iea ω−k·v ω ω ∂pl 4πea =i (−k, − ω) vk δ(ω − k · v) fa −1 3 (2π) ω j k      Ej Ek k,ω k·v vk kl ∂fa 1− δkl + − iea . ω−k·v ω ω ∂pl Inserting (9.32) to (9.31), we obtain      ∂fa k·v vj ki 2 ∂ δij + dk dω 1− = πea ∂t ∂pi ω ω   kj (k · v) Im  (k,ω) 1 × 3 2π ω k2 |  (k,ω)|2

(9.32)

9.2 Formal Equations of Electromagnetic Klimontovich Weak Turbulence Theory 239

  kj (k · v) Im ⊥ (k,ω) δ(ω − k · v) fa + vj − k2 |⊥ (k,ω)|2     kj kk 2 1 kj kk 2 δj k − 2 E⊥ k,ω + δ(ω − k · v) E k,ω + k2 2 k     vk kl ∂fa k·v δkl + , (9.33) × 1− ω ω ∂pl 

where we have made use of (7.32) and (7.33). If we make explicit use of the resonance condition, ω = k · v, to write (k · v)2 ω2 = , k2 k2

v2 −

(k · v)2 (k × v)2 = , k2 k2

then we obtain the formal particle kinetic equation,    1 k ∂ Im  (k,ω) δ(ω − k · v) · 3 k ∂p 2π k |  (k,ω)|2  (k × v)2 Im ⊥ (k,ω) fa + (k · v)2 k |⊥ (k,ω)|2    k ∂fa 1 (k × v)2 2 2 . (9.34) + E k,ω + E⊥ k,ω · 2 (k · v)2 k ∂p

∂fa = πea2 ∂t







dk



Nonlinear spectral balance equation (9.30) and formal particle kinetic equation (9.34) constitute the foundational equations subject to further analysis, which is done next. 9.2 Formal Equations of Electromagnetic Klimontovich Weak Turbulence Theory 9.2.1 Electron and Ion Particle Kinetic Equations For further reduction of particle kinetic equation, we start from (9.34) and take into account the contributions from linear eigenmodes. Electrons can resonate only with high-frequency modes, while ions resonate with low-frequency fluctuations and waves. Transverse electromagnetic modes cannot resonate with the particles since the wave speed is higher than the speed of light. As a consequence, we only need to retain the effects of L mode for electrons, while for ions only S mode needs to be taken into consideration. Making use of various properties of linear dielectric susceptibilities discussed in Section 7.3.1, it is possible to derive the particle kinetic equations for electrons and ions,

240

Electromagnetic Klimontovich Weak Turbulence Theory

  ∂fe (p) k ∂ 2 dk · = πe δ(σ ωkL − k · v) ∂t k ∂p σ =±1   σ ωkL σ L k ∂fe (p) , × fe (p) + Ik · 4π 2 k k ∂p   ∂fi (p) k ∂ 2 dk · = πe δ(σ ωkS − k · v) ∂t k ∂p σ =±1   σ μk ωkL σ S k ∂fi (p) . × fi (p) + Ik · 4π 2 k k ∂p

(9.35)

When compared with the corresponding particle kinetic equation (8.63) derived under the Vlasov kinetic approach, it becomes evident that the forms in (9.35) are more general in that the right-hand side is characterized by the Fokker–Planck form in which the velocity friction (or drag) term appears in balanced form together with the velocity space diffusion term,   ∂fa ∂fa ∂ , (a = e,i), = · Aa fa + Da · ∂t ∂p ∂p  k  e2 Ae = dk 2 σ ωL δ(σ ωkL − k · v), 4π k σ =±1 k  k  e2 dk 2 σ μk ωkL δ(σ ωkS − k · v), Ai = 4π k σ =±1  kk  2 dk 2 De = πe δ(σ ωkL − k · v)Ikσ L, k σ =±1  kk  2 dk 2 Di = πe δ(σ ωkS − k · v)Ikσ S . (9.36) k σ =±1 The basic equation here is the same as that discussed in Chapter 4 without the nonlinear term – see (4.86). This shows that the electromagnetic generalization does not modify the particle kinetic equations. In Chapter 6, we made use of the form of electron particle kinetic equation in (9.36) – see (6.3) – together with electrostatic wave kinetic equations, in order to analyze the weak Langmuir turbulence problem and the formation of asymptotic electron kappa velocity distribution function. 9.2.2 Formal Wave Kinetic Equations We now separate the nonlinear spectral balance equation (9.30) into longitudinal and transverse parts and derive the wave kinetic equations of electromagnetic weak

9.2 Formal Equations of Electromagnetic Klimontovich Weak Turbulence Theory 241

turbulence theory at the formal level. We first reintroduce the slow time derivative as is customary,  (k,ω) E2 k,ω + ⊥ (k,ω) E⊥2 k,ω → Re  (k,ω) E2 k,ω + Re ⊥ (k,ω) E⊥2 k,ω + iIm  (k,ω) E2 k,ω + iIm ⊥ (k,ω) E⊥2 k,ω i  ∂ i ∂ (9.37)  (k,ω) E2 k,ω + ⊥ (k,ω) E⊥2 k,ω, 2 ∂t 2 ∂t where  (k,ω) = ∂[Re  (k,ω)]/∂ω and ⊥ (k,ω) = ∂[Re ⊥ (k,ω)]/∂ω. With these, upon taking the linear eigenmode representation, the imaginary part of (9.30) can further be manipulated. Here, we write down the full expressions for the sake of completeness. In what follows, we designate various terms with boxed numerical identification. This is done in order to aid readers in tracing the steps in subsequent derivations. i   ∂I σ α 1 0=  (k,σ ωkα ) k δ(ω − σ ωkα ) '( ) 2 ∂t & +

σ

α

i   ∂I σ T + ⊥ (k,σ ωkT ) k δ(ω − σ ωkT ) '( ) 2 σ ∂t &  2 +i Im  (k,σ ωkα ) Ikσ α δ(ω − σ ωkα ) & '( ) σ α  Im ⊥ (k,σ ωkT ) Ikσ T δ(ω − σ ωkT ) +i & '( ) σ    ki kj (k − k )k  3 + 2i Im dk k k  |k − k |  α,γ σ,σ

× χij(2)k (k,σ ωkα

− σ  ωk−k |k − k, − σ  ωk−k ) γ

γ

kn kl (k − k )m (2)  γ γ χ (k ,σ ωkα − σ  ωk−k |k − k, − σ  ωk−k ) k k  |k − k | nlm   kj kl ki (k − k )k 1 + δj l − 2 ×P γ k k |k − k |  (k,σ ωkα − σ  ωk−k ) ×

× χij(2)k (k,σ ωkα − σ  ωk−k |k − k, − σ  ωk−k ) γ

γ

kn (k − k )m (2)  γ γ χ (k ,σ ωkα − σ  ωk−k |k − k, − σ  ωk−k ) k |k − k | nlm   ki kj (k − k )k 1 − iπ P γ k k  |k − k | ⊥ (k,σ ωkα − σ  ωk−k )  β

×

σ

242

Electromagnetic Klimontovich Weak Turbulence Theory

× χij(2)k (k,σ  ωk |k − k,σ  ωk−k ) γ

β

kn kl (k − k )m k k  |k − k | δ(σ ωkα − σ  ωk − σ  ωk−k ) γ

β

(2) (k,σ  ωk |k − k,σ  ωk−k ) × χnlm γ

β

− iπ



δj l −

σ

kj kl k 2



 (k,σ  ωk ) β

ki (k − k )k (2)   T γ χ (k ,σ ωk |k − k,σ  ωk−k ) k |k − k | ij k

1 kn (k − k )m (2)   T γ χnlm (k ,σ ωk |k − k,σ  ωk−k )    T  k |k − k | ⊥ (k ,σ ωk )  γ σ  γ × δ(σ ωkα − σ  ωkT − σ  ωk−k ) Ik−k Ikσ α δ(ω − σ ωkα ) & '( )      (k − k )k (k − k )m ki kj δkm − 4 + 2i Im dk (k − k )2 k k  α ×

σ,σ

T   T × χij(2)k (k,σ ωkα − σ  ωk−k  |k − k ,σ ωk−k )

kn kl k k

(2) T   T × χnlm (k,σ ωkα − σ  ωk−k  |k − k ,σ ωk−k ) P

 + δj l −

kj kl



k 2

(k − k )k (k − k )m δkm − (k − k )2



 (k,σ ωkα

1 T − σ  ωk−k )

ki (2)  T   T χ (k ,σ ωkα − σ  ωk−k  |k − k ,σ ωk−k ) k ij k 1 kn (2)  T   T (k ,σ ωkα − σ  ωk−k × χnlm  |k − k ,σ ωk−k ) P α T k ⊥ (k,σ ωk − σ  ωk−k )    (k − k )k (k − k )m ki kj δkm − − iπ (k − k )2 k k  β ×

σ

kn kl (2)   β T χ (k ,σ ωk |k − k,σ  ωk−k ) k k  nlm  β T  kj kl δ(σ ωkα − σ  ωk − σ  ωk−k ) δj l − 2 − iπ × β k  (k,σ  ωk )  σ   (k − k )k (k − k )m ki (2)   T T × δkm − χ (k ,σ ωk |k − k,σ  ωk−k ) (k − k )2 k ij k  T δ(σ ωkα − σ  ωkT − σ  ωk−k kn (2)   T )   T × χnlm (k ,σ ωk |k − k ,σ ωk−k ) k ⊥ (k,σ  ωkT ) 1 σ  T σ α δ(ω − σ ωkα ) × Ik−k  Ik & '( ) 2

T × χij(2)k (k,σ  ωk |k − k,σ  ωk−k ) β

9.2 Formal Equations of Electromagnetic Klimontovich Weak Turbulence Theory 243

5 + 2i



 Im





dk

kj (k − k )k k  |k − k |

σ,σ  γ



ki kn δin − 2 k

× χij(2)k (k,σ ωkT − σ  ωk−k |k − k,σ  ωk−k ) γ

γ

kl (k − k )m k  |k − k |

(2) (k,σ ωkT − σ  ωk−k |k − k,σ  ωk−k ) P × χnlm γ

 + δj l −

kj kl k 2



γ

 ki kn (k − k )k δin − 2 k |k − k |

× χij(2)k (k,σ ωkT − σ  ωk−k |k − k,σ  ωk−k ) γ

γ

γ

  kj (k − k )k 

γ

σ

β

k  |k − k |

 (k,σ ωkT

1 γ − σ  ωk−k )

(k − k )m |k − k |

(2) (k,σ ωkT − σ  ωk−k |k − k,σ  ωk−k ) P × χnlm

− iπ



1 γ − σ  ωk−k )

⊥ (k,σ ωkT

 ki kn γ β δin − 2 χij(2)k (k,σ  ωk |k − k,σ  ωk−k ) k

δ(σ ωkT − σ  ωk − σ  ωk−k ) kl (k − k )m (2)   β   γ χ (k ,σ ωk |k − k ,σ ωk−k ) ×  β k |k − k | nlm  (k,σ  ωk )    kj kl ki kn (k − k )k (2)   T γ δj l − 2 δin − 2 − iπ χij k (k ,σ ωk |k − k,σ  ωk−k ) | k k |k − k σ  γ δ(σ ωkT − σ  ωkT − σ  ωk−k ) (k − k )m (2)   T   γ × (k ,σ ω |k − k ,σ ω ) χ k k−k |k − k | nlm ⊥ (k,σ  ωkT ) 1 σ  γ × Ik−k Ikσ T δ(ω − σ ωkT ) & '( ) 2      kj ki kn  T   T 6 + 2i Im dk  δin − 2 χij(2)k (k,σ ωkT − σ  ωk−k  |k − k ,σ ωk−k ) k k σ,σ    kl (k − k )k (k − k )m (2) T   T ×  δkm − χnlm (k,σ ωkT − σ  ωk−k  |k − k ,σ ωk−k )  2 k (k − k )    kj kl ki kn 1 δj l − 2 + δin − 2 ×P T k k  (k,σ ωkT − σ  ωk−k )   (k − k )k (k − k )m (2)  T  T   T × χij k (k ,σ ωk − σ ωk−k |k − k ,σ ωk−k ) δkm − (k − k )2 1 (2) T   T (k,σ ωkT − σ  ωk−k × χnlm  |k − k ,σ ωk−k ) P T T  ⊥ (k ,σ ωk − σ  ωk−k )      kj ki kn β T − iπ δin − 2 χij(2)k (k,σ  ωk |k − k,σ  ωk−k )  k k  β β

σ

γ

244

 kl δkm k

Electromagnetic Klimontovich Weak Turbulence Theory

 (k − k )k (k − k )m β (2) T χnlm × − (k,σ  ωk |k − k,σ  ωk−k ) (k − k )2   β T  kj kl δ(σ ωkT − σ  ωk − σ  ωk−k ki kn ) − iπ δin − 2 δj l − 2 × β k k  (k,σ  ωk ) σ   (k − k )k (k − k )m T δkm − × χij(2)k (k,σ  ωkT |k − k,σ  ωk−k ) (k − k )2  T δ(σ ωkT − σ  ωkT − σ  ωk−k ) (2)   T   T × χnlm (k ,σ ωk |k − k ,σ ωk−k ) ⊥ (k,σ  ωkT )

1 σ  T σ T δ(ω − σ ωkT ) I I '( ) 4 k−k k &    ki kj (k − k )k (2)   β β 7 + 2i Im dk χij k (k ,σ ωk |k − k,σ ωkα − σ  ωk )  |k − k | k k  α,β ×

σ,σ

 (k − k )l (2)   β kn km β χ (k ,σ ωk |k − k,σ ωkα − σ  ωk ) k k  |k − k | nml   ki kj 1 (k − k )k (k − k )l + ×P δkl − β k k (k − k )2  (k − k,σ ωkα − σ  ωk )  β β kn km × χij(2)k (k,σ  ωk |k − k,σ ωkα − σ  ωk ) k k 1 β β (2) × χnml (k,σ  ωk |k − k,σ ωkα − σ  ωk ) P β ⊥ (k − k,σ ωkα − σ  ωk )   ki kj (k − k )k (2) γ β − iπ χij k (k,σ  ωk |k − k,σ  ωk−k )  |k − k | k k  γ

×

σ

 kn km (k − k )l (2)   β γ × χ (k ,σ ωk |k − k,σ  ωk−k ) k k  |k − k | nml  γ β  ki kj  δ(σ ωkα − σ  ωk − σ  ωk−k ) (k − k )k (k − k )l − iπ δ × − kl γ k k (k − k )2  (k − k,σ  ωk−k )  σ

 kn km β β T T χ (2) (k,σ  ωk |k − k,σ  ωk−k × χij(2)k (k,σ  ωk |k − k,σ  ωk−k ) ) k k  nml  β T δ(σ ωkα − σ  ωk − σ  ωk−k ) σ β × Ik Ikσ α δ(ω − σ ωkα )  T   & '( ) ⊥ (k − k ,σ ωk−k )      kj km ki (k − k )k δj m − 2 8 + 2i Im dk k k |k − k |  α σ,σ

× χij(2)k (k,σ  ωkT |k − k,σ ωkα − σ  ωkT )

kn (k − k )l k |k − k |

9.2 Formal Equations of Electromagnetic Klimontovich Weak Turbulence Theory 245

1

(2) × χnml (k,σ  ωkT |k − k,σ ωkα − σ  ωkT ) P

 (k −      kj km (k − k )k (k − k )l ki δj m − 2 δkl − + k k (k − k )2 kn × χij(2)k (k,σ  ωkT |k − k,σ ωkα − σ  ωkT ) k 

(2) (k,σ  ωkT |k − k,σ ωkα − σ  ωkT ) P × χnml

− iπ

  σ 

γ

k,σ ωkα

− σ  ωkT )

1 ⊥ (k − k,σ ωkα − σ  ωkT )

  kj km ki (k − k )k (2)   T γ δj m − 2 χ (k ,σ ωk |k − k,σ  ωk−k ) k k |k − k | ij k

δ(σ ωkα − σ  ωkT − σ  ωk−k ) kn (k − k )l (2)   T   γ × (k ,σ ω χ  |k − k ,σ ωk−k ) γ k nml k |k − k |  (k − k,σ  ωk−k )     ki  kj km (k − k )k (k − k )l δj m − 2 δkl − − iπ k k (k − k )2  γ

σ

kn (2)   T T T χ (k ,σ ωk |k − k,σ  ωk−k × χij(2)k (k,σ  ωkT |k − k,σ  ωk−k ) ) k nml  T δ(σ ωkα − σ  ωkT − σ  ωk−k 1 σ T σ α ) × I  Ik δ(ω − σ ωkα )  T & '( ) 2 k ⊥ (k − k,σ  ωk−k )     ki kn kj (k − k )k δin − 2 9 + 2i Im dk k k  |k − k |  β σ,σ

× χij(2)k (k,σ  ωk |k − k,σ ωkT − σ  ωk ) β

β

 (k − k )l km k  |k − k |

1

(2) (k,σ  ωk |k − k,σ ωkT − σ  ωk ) P × χnml β

β

β

 (k − k,σ ωkT − σ  ωk )      ki kn kj km (k − k )k (k − k )l δkl − + δin − 2 k k 2 (k − k )2 (2) (k,σ  ωk |k − k,σ ωkT − σ  ωk ) × χij(2)k (k,σ  ωk |k − k,σ ωkT − σ  ωk ) χnml    1 ki kn kj (k − k )k ×P δin − 2 − iπ β k k  |k − k | ⊥ (k − k,σ ωkT − σ  ωk )  γ β

β

× χij(2)k (k,σ  ωk |k − k,σ  ωk−k ) γ

β

β

β

σ  km (k − k )l k  |k − k |

δ(σ ωkT − σ  ωk − σ  ωk−k ) γ

β

β (2) (k,σ  ωk |k × χnml



− k ,σ



γ ωk−k )

 (k − k,σ  ωk−k ) γ

246

Electromagnetic Klimontovich Weak Turbulence Theory

− iπ

10



    ki kn kj km (k − k )k (k − k )l δin − 2 δkl − k k 2 (k − k )2

σ  β (2) T   β   T × χij(2)k (k,σ  ωk |k − k,σ  ωk−k  ) χnml (k ,σ ωk |k − k ,σ ωk−k )  β T δ(σ ωkT − σ  ωk − σ  ωk−k 1 σ β σ T ) δ(ω − σ ωkT ) I I ×  T   '( ) 2 k k & ⊥ (k − k ,σ ωk−k )        kj km (k − k )k ki kn δin − 2 δj m − 2 + 2i Im dk k k |k − k | σ,σ  (k − k )l × χij(2)k (k,σ  ωkT |k − k,σ ωkT − σ  ωkT ) |k − k | (2) (k,σ  ωkT |k − k,σ ωkT − σ  ωkT ) P × χnml

 + δin −

ki kn k2

 δj m −

 kj km

 δkl −

k 2

1  (k −

− σ  ωkT ) 

k,σ ωkT 

(k − k )k (k − k )l (k − k )2

× χij(2)k (k,σ  ωkT |k − k,σ ωkT − σ  ωkT ) 1 ⊥ (k − k,σ ωkT − σ  ωkT )      kj km (k − k )k ki kn δin − 2 δj m − 2 − iπ k k |k − k |  γ

(2) (k,σ  ωkT |k − k,σ ωkT − σ  ωkT ) P × χnml

σ

× χij(2)k (k,σ  ωkT |k − k,σ  ωk−k ) γ

(k − k )l |k − k | δ(σ ωkT − σ  ωkT − σ  ωk−k ) γ

(2) (k,σ  ωkT |k × χnml



− k ,σ



γ ωk−k )

 (k − k,σ  ωk−k )      kj km ki kn (k − k )k (k − k )l − iπ δin − 2 δj m − 2 δkl − k k (k − k )2  γ

σ

× χij(2)k (k,σ  ωkT |k

(2) T   T   T − k,σ  ωk−k  ) χnml (k ,σ ωk |k − k ,σ ωk−k )  T δ(σ ωkT − σ  ωkT − σ  ωk−k 1 σ T σ T ) × I  Ik δ(ω − σ ωkT )  T & '( ) 4 k ⊥ (k − k,σ  ωk−k )     ki kj (k − k )k  11 − 2i Im dk iπ k k  |k − k |   β,γ σ α σ ,σ

× χij(2)k (k,σ  ωk |k − k,σ  ωk−k ) β

γ

 (k − k )n kl km k k  |k − k |

(2)∗  (k ,σ  ωk |k − k,σ  ωk−k ) × χlmn β

γ

9.2 Formal Equations of Electromagnetic Klimontovich Weak Turbulence Theory 247

δ(σ ωkα − σ  ωk − σ  ωk−k ) δ(ω − σ ωkα ) & '( )  (k,σ ωkα )      kj (k − k )k ki kl γ β δ χij(2)k (k,σ  ωk |k − k,σ  ωk−k ) − + iπ il  |k − k | 2 k k σ β

γ

×

 (k − k )n (2)∗   β km γ χlmn (k ,σ ωk |k − k,σ  ωk−k )   k |k − k |  γ β δ(σ ωkT − σ  ωk − σ  ωk−k ) σ  β σ  γ T × δ(ω − σ ωk ) Ik Ik−k  T & '( ) ⊥ (k,σ ωk )      ki kj 12 − 2i Im dk iπ k k σ α σ ,σ  β   (k − k )k (k − k )n β T χij(2)k (k,σ  ωk |k − k,σ  ωk−k × δkn − ) (k − k )2  kl km β T χ (2)∗ (k,σ  ωk |k − k,σ  ωk−k × ) k k  lmn β T δ(σ ωkα − σ  ωk − σ  ωk−k ) × δ(ω − σ ωkα ) α  & '( )  (k,σ ωk )      kj (k − k )k (k − k )n ki kl + iπ δ δ − − kn il k (k − k )2 k2 σ

×

 km β (2)∗  T χlmn (k ,σ  ωk |k − k,σ  ωk−k )  k  β T δ(σ ωkT − σ  ωk − σ  ωk−k 1 σ  β σ  T ) T × ) δ(ω − σ ω I I  k & '( ) 2 k k−k ⊥ (k,σ ωkT )        kj km ki (k − k )k  δj m − 2 13 − 2i Im dk iπ k k |k − k |   γ σ α T × χij(2)k (k,σ  ωk |k − k,σ  ωk−k ) β

σ ,σ

kl (k − k )n k |k − k | γ δ(σ ωkα − σ  ωkT − σ  ωk−k ) γ (2)∗  (k ,σ  ωkT |k − k,σ  ωk−k ) δ(ω − σ ωkα ) × χlmn & '( )  (k,σ ωkα )        kj km (k − k )k ki kl δil − 2 + iπ δj m − 2  k |k − k | k σ × χij(2)k (k,σ  ωkT |k − k,σ  ωk−k ) γ

(k − k )n (2)∗   T γ χ (k ,σ ωk |k − k,σ  ωk−k ) |k − k | lmn  γ δ(σ ωkT − σ  ωkT − σ  ωk−k ) 1 σ  T σ  γ T × δ(ω − σ ωk ) I  Ik−k  T & '( ) 2 k ⊥ (k,σ ωk ) × χij(2)k (k,σ  ωkT |k − k,σ  ωk−k ) γ

248

Electromagnetic Klimontovich Weak Turbulence Theory

14 − 2i



 Im



dk

σ ,σ 

 iπ

  ki  σ

α

k

 kj km

δj m −



k 2

  (k − k )k (k − k )n kl T χij(2)k (k,σ  ωkT |k − k,σ  ωk−k × δkn − ) (k − k )2 k α  T  T δ(σ ωk − σ ωk − σ ωk−k ) (2)∗  T (k ,σ  ωkT |k − k,σ  ωk−k × χlmn δ(ω − σ ωkα ) ) & '( )  (k,σ ωkα )       kj km (k − k )k (k − k )n ki kl δj m − 2 δkn − δil − 2 + iπ  2 k (k − k ) k σ (2)∗  T  T   T × χij(2)k (k,σ  ωkT |k − k,σ  ωk−k  ) χlmn (k ,σ ωk |k − k ,σ ωk−k )  T δ(σ ωkT − σ  ωkT − σ  ωk−k 1 σ  T σ  T ) T δ(ω − σ ωk ) × I  Ik−k  T & '( ) 4 k ⊥ (k,σ ωk )   ki kl kj kk (3)   β β 15 + 2i Im dk 2 χij kl (k ,σ ωk | − k, − σ  ωk |k,σ ωkα ) 2 k k σ,σ  α,β     kj kk σ β σ α α  1 ki kl δj k − 2 × Ik Ik δ(ω − σ ωk ) +2i Im dk & '( ) 2 k2 k  α σ,σ



× χij(3)kl (k,σ  ωkT | − k, − σ  ωkT |k,σ ωkα ) Ikσ T Ikσ α δ(ω − σ ωkα ) & '( )       1 ki kl kj kk δil − 2 Im dk +2i 2 k k 2  β σ,σ

β × χij(3)kl (k,σ  ωk |

σ β

− k, − σ  ωk |k,σ ωkT ) Ik Ikσ T δ(ω − σ ωkT ) & '( )        kj kk 1 ki kl δil − 2 + 2i δj k − 2 Im dk 4 k k  β

σ,σ

× χij(3)kl (k,σ  ωkT |



− k, − σ  ωkT |k,σ ωkT ) Ikσ T Ikσ T δ(ω − σ ωkT ) & '( )    (4πea )2  1 dp iπ 16 = i Im  3 2 (2π) k  (k,σ ωkα ) a σ α  × δ(σ ωkα − k · v) fa (p) δ(ω − σ ωkα ) & '( )    (4πea )2  1 dp  17 + 2i Im dk 3 (2π) (k · v)2 a σ,σ  α,γ    kj (k · v) 1 × γ α 2  k  (k ,σ ωk − σ  ωk−k )

9.2 Formal Equations of Electromagnetic Klimontovich Weak Turbulence Theory 249

 + vj −  ×

kj (k · v)

k 2 kp (k · v) k 2



1 γ α  ⊥ (k ,σ ωk − σ  ωk−k )

∗ (k,σ ωkα  

1 γ − σ  ωk−k )



  kp (k · v) 1 + vp − γ k 2 ∗⊥ (k,σ ωkα − σ  ωk−k ) ki (k − k )k (2)  γ γ × χ (k ,σ ωkα − σ  ωk−k |k − k,σ  ωk−k ) k |k − k | ij k km (k − k )n (2)∗  γ γ × χ (k ,σ ωkα − σ  ωk−k |k − k,σ  ωk−k ) k |k − k | mpn   α  γ  σ  γ δ(σ ωk − σ ωk−k − k · v) fa (p) δ(ω − σ ωkα ) × iπ Ik−k & '( )  (k,σ ωkα )    (4πea )2  1 dp  18 + 2i Im dk 3 (2π) (k · v)2 a σ,σ  γ    kj (k · v) 1 × γ T 2  k  (k ,σ ωk − σ  ωk−k )    kj (k · v) 1 + vj − γ k 2 ⊥ (k,σ ωkT − σ  ωk−k )    kp (k · v) 1 × γ ∗ T 2  k  (k ,σ ωk − σ  ωk−k )    kp (k · v) 1 + vp − γ k 2 ∗⊥ (k,σ ωkT − σ  ωk−k )   ki km (k − k )k (2)  γ γ × δim − 2 χ (k ,σ ωkT − σ  ωk−k |k − k,σ  ωk−k ) k |k − k | ij k (k − k )n (2)∗  γ γ × χ (k ,σ ωkT − σ  ωk−k |k − k,σ  ωk−k ) |k − k | mpn   T  γ  σ  γ δ(σ ωk − σ ωk−k − k · v) × iπ Ik−k fa (p) δ(ω − σ ωkT ) & '( ) ⊥ (k,σ ωkT )   2  (4πea )  1 dp 19 + 2i Im dk (2π)3 (k · v)2 a σ,σ  α    kj (k · v) 1 × α T 2  k  (k ,σ ωk − σ  ωk−k )      kj (k · v) 1 + vj − T k 2 ⊥ (k,σ ωkα − σ  ωk−k )

250

Electromagnetic Klimontovich Weak Turbulence Theory

 ×

kp



(k · v) k 2

∗ (k,σ ωkα  

1 T − σ  ωk−k )

  kp (k · v) 1 ki + vp − α ∗ T 2   k ⊥ (k ,σ ωk − σ ωk−k ) k    (k − k )n (k − k )k T   T χij(2)k (k,σ ωkα − σ  ωk−k × δnk −  |k − k ,σ ωk−k ) (k − k )2 km (2)∗  T   T × χmpn (k ,σ ωkα − σ  ωk−k  |k − k ,σ ωk−k ) k   1 σ  T 1 α  T  × iπ  I  δ(σ ωk − σ ωk−k − k · v) fa (p) δ(ω − σ ωkα ) & '( )  (k,σ ωkα ) 2 k−k   2   (4πea ) 1 dp 20 + 2i Im dk (2π)3 (k · v)2 a σ,σ     kj (k · v) 1 × T T 2  k  (k ,σ ωk − σ  ωk−k )      kj (k · v) 1 + vj − T k 2 ⊥ (k,σ ωkT − σ  ωk−k )    kp (k · v) 1 × ∗ T T 2  k  (k ,σ ωk − σ  ωk−k )      kp (k · v) 1 + vp − T k 2 ∗⊥ (k,σ ωkT − σ  ωk−k )   ki km T   T χij(2)k (k,σ ωkT − σ  ωk−k × δim − 2  |k − k ,σ ωk−k ) k   (k − k )n (k − k )k (2)∗  T   T × δnk − χmpn (k ,σ ωkT − σ  ωk−k  |k − k ,σ ωk−k ) (k − k )2   1 σ  T 1 T  T  × iπ  I  δ(σ ωk − σ ωk−k − k · v) fa (p) δ(ω − σ ωkT ) & '( ) ⊥ (k,σ ωkT ) 2 k−k    (4πea )2  1 dp  21 − 2i Im dk 3 (2π) (k · v)2 a σ,σ  α,γ    kj (k · v) 1 × γ α 2  k  (k ,σ ωk − σ  ωk−k )    kj (k · v) 1 + vj − γ k 2 ⊥ (k,σ ωkα − σ  ωk−k )    kp (k · v) 1 × γ α ∗  2 k  (k ,σ ωk − σ  ωk−k )

9.2 Formal Equations of Electromagnetic Klimontovich Weak Turbulence Theory 251

 + vp −

kp (k · v)



1 γ α ∗  ⊥ (k ,σ ωk − σ  ωk−k )



k 2 ki (k − k )k (2)  γ γ × χ (k ,σ ωkα − σ  ωk−k |k − k,σ  ωk−k ) k |k − k | ij k kn (k − k )m (2)  γ γ × χ (k ,σ ωkα − σ  ωk−k |k − k,σ  ωk−k ) k |k − k | npm  1 × (−iπ)  I σα γ   (k − k ,σ  ωk−k ) k  α  γ  × δ(σ ωk − σ ωk−k − k · v) fa (p) δ(ω − σ ωkα ) & '( )  (4πea )2    1 dp  22 − 2i Im dk 3 (2π) (k · v)2 a σ,σ  γ    kj (k · v) 1 × γ T 2  k  (k ,σ ωk − σ  ωk−k )    kj (k · v) 1 + vj − γ k 2 ⊥ (k,σ ωkT − σ  ωk−k )    kp (k · v) 1 × γ ∗ T 2  k  (k ,σ ωk − σ  ωk−k )    kp (k · v) 1 + vp − γ k 2 ∗⊥ (k,σ ωkT − σ  ωk−k )   (k − k )k kn ki γ γ δni − 2 χij(2)k (k,σ ωkT − σ  ωk−k |k − k,σ  ωk−k ) ×  |k − k | k (k − k )m (2)  γ γ χ (k ,σ ωkT − σ  ωk−k |k − k,σ  ωk−k ) |k − k | npm  1 1 σT × (−iπ)  I γ    (k − k ,σ ωk−k ) 2 k  T  γ  × δ(σ ωk − σ ωk−k − k · v) fa (p) δ(ω − σ ωkT ) & '( )    (4πea )2  1 dp  23 − 2i Im dk 3 (2π) (k · v)2 a σ,σ  α    kj (k · v) 1 × k 2  (k,σ ωkα − σ  ωk−k )    kj (k · v) 1 + vj − k 2 ⊥ (k,σ ωkα − σ  ωk−k ) ×

252

Electromagnetic Klimontovich Weak Turbulence Theory



kp



(k · v)

1 ∗ (k,σ ωkα − σ  ωk−k )    kp (k · v) 1 + vp − k 2 ∗⊥ (k,σ ωkα − σ  ωk−k )   ki (k − k )k (k − k )m T δkm − χij(2)k (k,σ ωkα − σ  ωk−k |k − k,σ  ωk−k × ) k (k − k )2 kn (2)  T (k ,σ ωkα − σ  ωk−k |k − k,σ  ωk−k × χnpm ) k  1 × (−iπ)  Ikσ α T ⊥ (k − k,σ  ωk−k )  α  T  × δ(σ ωk − σ ωk−k − k · v) fa (p) δ(ω − σ ωkα ) & '( )     (4πea )2  kj (k · v) 1  dp  24 − 2i Im dk (2π)3 (k · v)2 k 2 a σ,σ     kj (k · v) 1 1 × + vj − T T k 2  (k,σ ωkT − σ  ωk−k ⊥ (k,σ ωkT − σ  ωk−k ) )    kp (k · v) 1 × ∗ T T 2  k  (k ,σ ωk − σ  ωk−k )    kp (k · v) 1 + vp − T k 2 ∗⊥ (k,σ ωkT − σ  ωk−k )   kn ki T   T × δni − 2 χij(2)k (k,σ ωkT − σ  ωk−k  |k − k ,σ ωk−k ) k   (k − k )k (k − k )m × δkm − (k − k )2 ×

k 2

(2) T   T (k,σ ωkT − σ  ωk−k × χnpm  |k − k ,σ ωk−k )  1 1 × (−iπ) Ikσ T  T 2 ⊥ (k − k,σ  ωk−k )  T  × δ(σ ωkT − σ  ωk−k − k · v) f (p) δ(ω − σ ωkT )  a & '( )    (4πea )2  1 dp 25 + 2i Im dk 3 (2π) [(k − k ) · v]2 a σ,σ  α,β  1 (k − k )k [(k − k ) · v] × β  2 (k − k )  (k − k,σ ωkα − σ  ωk )

9.2 Formal Equations of Electromagnetic Klimontovich Weak Turbulence Theory 253

  1 (k − k )k [(k − k ) · v] + vk − β (k − k )2 ⊥ (k − k,σ ωkα − σ  ωk )  (k − k )p [(k − k ) · v] 1 × β ∗  (k − k )2  (k − k ,σ ωkα − σ  ωk )    1 (k − k )p [(k − k ) · v] + vp − β (k − k )2 ∗⊥ (k − k,σ ωkα − σ  ωk ) ki kj (2)   β β × χ (k ,σ ωk |k − k,σ ωkα − σ  ωk ) k k  ij k km kn (2)∗   β β × χmnp (k ,σ ωk |k − k,σ ωkα − σ  ωk )  k k   1 σ β α  β  × (iπ)Ik  δ[σ ωk − σ ωk − (k − k ) · v] fa (p) δ(ω − σ ωkα ) & '( )  (k,σ ωkα )    (4πea )2  1 dp 26 + 2i Im dk 3 (2π) [(k − k ) · v]2 a σ,σ  β  1 (k − k )k [(k − k ) · v] × β  2  (k − k )  (k − k ,σ ωkT − σ  ωk )    1 (k − k )k [(k − k ) · v] + vk − β (k − k )2 ⊥ (k − k,σ ωkT − σ  ωk )  (k − k )p [(k − k ) · v] 1 × β  2 (k − k ) ∗ (k − k,σ ωkT − σ  ωk )    1 (k − k )p [(k − k ) · v] + vp − β (k − k )2 ∗⊥ (k − k,σ ωkT − σ  ωk )   kj ki km β β ×  δim − 2 χij(2)k (k,σ  ωk |k − k,σ ωkT − σ  ωk ) k k 

kn (2)∗   β β χmnp (k ,σ ωk |k − k,σ ωkT − σ  ωk )  k   1 σ β T  β  × (iπ)Ik δ[σ ωk − σ ωk − (k − k ) · v] fa (p) δ(ω − σ ωkT ) & '( ) ⊥ (k,σ ωkT )     (4πea )2 1 dp 27 + 2i Im dk 3  ) · v]2 (2π) [(k − k a σ,σ  α  1 (k − k )k [(k − k ) · v] × (k − k )2  (k − k,σ ωkα − σ  ωkT )    1 (k − k )k [(k − k ) · v] + vk − (k − k )2 ⊥ (k − k,σ ωkα − σ  ωkT ) ×

254



Electromagnetic Klimontovich Weak Turbulence Theory

1 (k − k )p [(k − k ) · v] ∗  2 (k − k )  (k − k,σ ωkα − σ  ωkT )    1 (k − k )p [(k − k ) · v] + vp − (k − k )2 ∗⊥ (k − k,σ ωkα − σ  ωkT )   kn kj ki δnj − 2 χij(2)k (k,σ  ωkT |k − k,σ ωkα − σ  ωkT ) × k k km (2)∗   T × χmnp (k ,σ ωk |k − k,σ ωkα − σ  ωkT ) k   1 1 σ T α  T  × (iπ) Ik δ[σ ωk − σ ωk − (k − k ) · v] fa (p) δ(ω − σ ωkα ) & '( ) 2  (k,σ ωkα )    (4πea )2  1 dp 28 + 2i Im dk 3  ) · v]2 (2π) [(k − k a σ,σ   1 (k − k )k [(k − k ) · v] × (k − k )2  (k − k,σ ωkT − σ  ωkT )    1 (k − k )k [(k − k ) · v] + vk − (k − k )2 ⊥ (k − k,σ ωkT − σ  ωkT )  (k − k )p [(k − k ) · v] 1 × ∗  2  (k − k )  (k − k ,σ ωkT − σ  ωkT )    (k − k )p [(k − k ) · v] 1 + vp − (k − k )2 ∗⊥ (k − k,σ ωkT − σ  ωkT )   ki km χij(2)k (k,σ  ωkT |k − k,σ ωkT − σ  ωkT ) × δim − 2 k   kn kj (2)∗  (k ,σ  ωkT |k − k,σ ωkT − σ  ωkT ) × δnj − 2 χmnp k   1  T  T  × (iπ)Ikσ T  − σ ω − (k − k ) · v] f (p) δ(ω − σ ωkT ) δ[σ ω a k k & '( ) ⊥ (k,σ ωkT )   2   (4πea ) 1 dp 29 − 2i Im dk 3 (2π) [(k − k ) · v]2 a σ,σ  α,β  1 (k − k )k [(k − k ) · v] × β  2  (k − k )  (k − k ,σ ωkα − σ  ωk )    1 (k − k )k [(k − k ) · v] + vk − β (k − k )2 ⊥ (k − k,σ ωkα − σ  ωk )  1 (k − k )p [(k − k ) · v] × β  2 ∗ (k − k )  (k − k,σ ωkα − σ  ωk ) ×

9.2 Formal Equations of Electromagnetic Klimontovich Weak Turbulence Theory 255

  1 (k − k )p [(k − k ) · v] + vp − β (k − k )2 ∗⊥ (k − k,σ ωkα − σ  ωk ) ki kj (2)   β β × χij k (k ,σ ωk |k − k,σ ωkα − σ  ωk )  kk  kn km β β (2) × χnmp (k,σ  ωk |k − k,σ ωkα − σ  ωk )  k k  1 σα α  β  × (−iπ) Ik δ[σ ωk − σ ωk − (k − k ) · v] fa (p) δ(ω − σ ωkα ) β    & '( )  (k ,σ ωk )  (4πea )2    1 dp 30 − 2i Im dk 3 (2π) [(k − k ) · v]2 a σ,σ  α  1 (k − k )k [(k − k ) · v] ×  2  (k − k )  (k − k ,σ ωkα − σ  ωkT )    1 (k − k )k [(k − k ) · v] + vk − (k − k )2 ⊥ (k − k,σ ωkα − σ  ωkT )  1 (k − k )p [(k − k ) · v] × ∗  2  (k − k )  (k − k ,σ ωkα − σ  ωkT )    1 (k − k )p [(k − k ) · v] + vp − (k − k )2 ∗⊥ (k − k,σ ωkα − σ  ωkT )    kj km ki δj m − 2 χij(2)k (k,σ  ωkT |k − k,σ ωkα − σ  ωkT ) × k k  kn (2)   T  α  T × χnmp (k ,σ ωk |k − k ,σ ωk − σ ωk ) (−iπ ) k  1 σα α  T  ×    T Ik δ[σ ωk − σ ωk − (k − k ) · v] fa (p) δ(ω − σ ωkα ) & '( ) ⊥ (k ,σ ωk )     (4πea )2 1 dp 31 − 2i Im dk 3 (2π) [(k − k ) · v]2 a σ,σ  β  1 (k − k )k [(k − k ) · v] × β  2 (k − k )  (k − k,σ ωkT − σ  ωk )    1 (k − k )k [(k − k ) · v] + vk − β (k − k )2 ⊥ (k − k,σ ωkT − σ  ωk )  (k − k )p [(k − k ) · v] 1 × β (k − k )2 ∗ (k − k,σ ωkT − σ  ωk )    (k − k )p [(k − k ) · v] 1 + vp − β (k − k )2 ∗⊥ (k − k,σ ωkT − σ  ωk ) 

256

Electromagnetic Klimontovich Weak Turbulence Theory

×

kj k



kn ki δni − 2 k



χij(2)k (k,σ  ωk |k − k,σ ωkT − σ  ωk ) β

β

  km (2)   β  T  β 1 ×  χnmp (k ,σ ωk |k − k ,σ ωk − σ ωk ) (−iπ ) k 2 ×

1



β ωk

 − (k − k ) · v] fa (p) δ(ω − σ ωkT ) & '( ) 

−σ β  (k,σ  ωk )    (4πea )2  1 dp 32 − 2i Im dk 3 (2π) [(k − k ) · v]2 a σ,σ   1 (k − k )k [(k − k ) · v] ×  2  (k − k )  (k − k ,σ ωkT − σ  ωkT )    1 (k − k )k [(k − k ) · v] + vk − (k − k )2 ⊥ (k − k,σ ωkT − σ  ωkT )  1 (k − k )p [(k − k ) · v] × ∗  2  (k − k )  (k − k ,σ ωkT − σ  ωkT )    1 (k − k )p [(k − k ) · v] + vp − (k − k )2 ∗⊥ (k − k,σ ωkT − σ  ωkT )   kn ki × δni − 2 χij(2)k (k,σ  ωkT |k − k,σ ωkT − σ  ωkT ) k     kj km (2)   T  T  T 1 (9.38) × δj m − 2 χnmp (k ,σ ωk |k − k ,σ ωk − σ ωk ) (−iπ ) k 2  1 σT T  T  ×    T Ik δ[σ ωk − σ ωk − (k − k ) · v] fa (p) δ(ω − σ ωkT ) . & '( ) ⊥ (k ,σ ωk ) Ikσ T

δ[σ ωkT

In (9.38), the various terms identified by boxed numerical scheme contain individual terms identified by either δ(ω − σ ωkα ) or δ(ω − σ ωkT ). We have indicated these by making use of under-braces, in order to highlight these delta functions. This means that one may collect all the terms containing either of the two delta functions and separate the formal equations into formal longitudinal versus transverse wave kinetic equations. We discuss these two equations separately next. Collecting various terms containing the factor δ(ω − σ ωkα ) and removing the $ $ common factor (i/2) σ α , we obtain the following general form of wave kinetic equation, arranged according to the types of interaction. In deriving the final form, we have ignored nonlinear scattering terms associated with the factor of the type 1/⊥ . The result is

9.2 Formal Equations of Electromagnetic Klimontovich Weak Turbulence Theory 257

∂Ikσ α ∂t

=− +

2 Im  (k,σ ωkα ) σ α Ik  (k,σ ωkα )   4e2 a

a

k 2 |  (k,σ ωkα )|2

induced emission dp δ(σ ωkα − k·v)fa (p)

spontaneous emission

 k k  (k − k )k (2)   β 4π γ  i j dk −  χ (k ,σ ωk |k − k,σ  ωk−k )  (k,σ ωkα )   β,γ k k  |k − k | ij k ×

σ ,σ (k − k )m k k  |k − k |

kn kl 

σ  γ

Ik−k Ikσ α

(2)∗  (k ,σ  ωk |k − k,σ  ωk−k ) χnlm β

γ

σ β

Ik Ikσ α × + γ  β  (k,σ  ωk )  (k − k,σ  ωk−k ) σ  β σ  γ  Ik Ik−k γ β δ(σ ωkα − σ  ωk − σ  ωk−k ) decay (α,β,γ ) −  α  (k,σ ωk )   ki kj (2)   β 8π  T dk −  χ (k ,σ ωk |k − k,σ  ωk−k )  (k,σ ωkα )   β k k  ij k

×

 kn kl δkm k k





σ ,σ

 (k − k )k (k − k )m β (2)∗  T χnlm − (k ,σ  ωk |k − k,σ  ωk−k ) (k − k )2 

σ β σ T σα Ik Ikσ α 1 Ik−k  Ik + × T 2  (k,σ  ωkβ ) ⊥ (k − k,σ  ωk−k ) β   σ 1 I σ T I  β T decay (α,β,T ) − k−k k α δ(σ ωkα − σ  ωk − σ  ωk−k ) 2  (k,σ ωk )    kj kl 4π  ki T δj l − 2 χij(2)k (k,σ  ωkT |k − k,σ  ωk−k dk −  )  (k,σ ωkα )   k k σ ,σ   (k − k )k (k − k )m kn (2)∗  T δkm − χnlm (k ,σ  ωkT |k − k,σ  ωk−k × ) k (k − k )2   σ  T σ α Ikσ T Ikσ α 1 1 Ik−k  Ik + × T 2 ⊥ (k,σ  ωkT ) 2 ⊥ (k − k,σ  ωk−k )    σ T 1 I σ T Ik−k  T δ(σ ωkα − σ  ωkT − σ  ωk−k − k decay (α,T ,T ) ) α 4  (k,σ ωk )   4 −  Im dk  (k,σ ωkα )  β σ   ki kj (k − k )k (2)   β β × χ (k ,σ ωk |k − k,σ ωkα − σ  ωk ) k k  |k − k | ij k

258

Electromagnetic Klimontovich Weak Turbulence Theory

×

 (k − k )l kn km k k  |k − k |

(2) (k,σ  ωk |k − k,σ ωkα − σ  ωk ) χnml β

β

ki kl kj kk β k 2 k 2  (k − k,σ ωkα − σ  ωk )  σ β (3)   β   β α × χij kl (k ,σ ωk | − k , − σ ωk |k,σ ωk ) Ik Ikσ α ind. scatt. (α,β)    4ea2 4  dk dp +  β  (k,σ ωkα )  β |k − k |2 |  (k − k,σ ωkα − σ  ωk )|2 a σ 2

×P

+

ki kj (k − k )k

χ (2) (k,σ  ωk |k − k,σ ωkα − σ  ωk ) k k  |k − k | ij k km kn (k − k )p (2)∗   β β × χmnp (k ,σ ωk |k − k,σ ωkα − σ  ωk ) k k  |k − k |   σ β I k Ikσ α β δ[σ ωkα − σ  ωk − (k − k ) · v] fa (p) ×  α −  β    (k,σ ωk )  (k ,σ ωk ) ×

β

β

spont. scatt. (α,β) −

 4 Im α   (k,σ ωk ) 



dk

σ

   kj km ki (k − k )k δj m − 2 k k |k − k |

× χij(2)k (k,σ  ωkT |k − k,σ ωkα − σ  ωkT )

kn (k − k )l k |k − k |

(2) (k,σ  ωkT |k − k,σ ωkα − σ  ωkT ) P × χnml



+

+

 

kj kk ki kl δ − jk k2 k 2 4   (k,σ ωkα ) 

 σ



2

− σ  ωkT )  σ T I (3)   T   T α χij kl (k ,σ ωk | − k , − σ ωk |k,σ ωk ) k Ikσ α 2

dk



 (k −

k,σ ωkα

ind. scatt. (α,T ) dp

4ea2 (k − k )2 |  (k − k,σ ωkα − σ  ωkT )|

a    kn kj χij(2)k (k,σ  ωkT |k k 2

ki (k − k )k δnj − − k,σ ωkα − σ  ωkT ) k |k − k | km (k − k )p (2)∗   T (9.39) × χmnp (k ,σ ωk |k − k,σ ωkα − σ  ωkT )  k |k − k |    1 Ikσ T Ikσ α δ[σ ωkα − σ  ωkT − (k − k ) · v] fa (p). × − 2  (k,σ ωkα ) ⊥ (k,σ  ωkT )

×

spont. scatt. (α,T )

9.2 Formal Equations of Electromagnetic Klimontovich Weak Turbulence Theory 259

Taking α = L or S leads to the wave kinetic equation for L or S mode, which will be discussed later. At this point, the formalism is still given in terms of various susceptibilities. Next we consider the transverse mode associated with the overall factor δ(ω − σ ωkT ): ∂Ikσ T 2 Im ⊥ (k,σ ωkT ) σ T Ik =− ∂t ⊥ (k,σ ωkT ) +

 a

4ea2 |ωkT ⊥ (k,σ ωkT )|2

induced emission  dp

(k × v)2 δ(σ ωkT − k · v) fa (p) k2

spontaneous emission     k  (k − k )k 4π ki kn  j δin − 2 dk  −  k |k − k | k ⊥ (k,σ ωkT ) σ ,σ  β,γ kl (k − k )m k  |k − k |  σ  γ σ β Ik Ikσ T 1 Ik−k Ikσ T 1 (2)∗   β   γ × χnlm (k ,σ ωk |k − k ,σ ωk−k ) + γ 2  (k,σ  ωkβ ) 2  (k− k,σ  ωk−k )   σ β σ γ  I I  γ β − k k−kT decay (T ,β,γ ) δ(σ ωkT − σ  ωk − σ  ωk−k ) ⊥ (k,σ ωk )      kj kl 8π ki kn  dk δj l −  2 δin − 2 −  k ⊥ (k,σ ωkT )   β k × χij(2)k (k,σ  ωk |k − k,σ  ωk−k ) γ

β

σ ,σ

(k − k )k (k − k )m (2)   T β χij k (k ,σ ωk |k − k,σ  ωk−k ) × |k − k |2  σ  T σ  β σ  β Ik Ik−k Ik−k Ikσ T (2)∗   T   β − × χnlm (k ,σ ωk |k − k ,σ ωk−k ) ⊥ (k,σ ωkT ) ⊥ (k,σ  ωkT )   Ikσ T Ikσ T 1 β δ(σ ωkT − σ  ωkT − σ  ωk−k ) decay (T ,T ,β) − β    2  (k − k ,σ ωk−k )     4 ki kn kj (k − k )k  δ −  Im dk − in k2 k  |k − k | ⊥ (k,σ ωkT )  β σ

× χij(2)k (k,σ  ωk |k − k,σ ωkT − σ  ωk ) β

β

 (k − k )l km k  |k − k |

(2) (k,σ  ωk |k − k,σ ωkT − σ  ωk ) P × χnml β

β

2 β

 (k − k,σ ωkT − σ  ωk )

260

Electromagnetic Klimontovich Weak Turbulence Theory

  σT ki kl kj kk (3)   β σ  β Ik   β T + δil − 2 χ (k ,σ ω  | − k , − σ ωk |k,σ ωk ) Ik ij kl k k k 2 2 





ind. scatt. (T ,β)

4ea2 4  dk dp β ⊥ (k,σ ωkT ) σ  β (k− k )2 |  (k −k,σ ωkT −σ  ωk )|2 a   kj (k − k )k ki km β β δim − 2 χij(2)k (k,σ  ωk |k − k,σ ωkT − σ  ωk ) ×   k |k − k | k kn (k − k )p (2)∗   β β (9.40) ×  χ (k ,σ ωk |k − k,σ ωkT − σ  ωk ) k |k − k | mnp   σ β I k Ikσ T 1 β δ[σ ωkT − σ  ωk − (k − k ) · v] fa (p). × − β  T    2 ⊥ (k,σ ωk )  (k ,σ ωk ) +

spont. scatt. (T ,β) (9.40) describes nonlinear interaction and time evolution of transverse wave. Together with the equation for longitudinal modes (9.39), the wave kinetic equations of electromagnetic Klimontovich weak turbulence theory are formally expressed in terms of linear and nonlinear susceptibilities. Next we evaluate various response functions by approximate means, as in Section 8.2. The resulting set of equations constitutes the fully general electromagnetic weak turbulence formalism that can be readily solved by numerical means or that readily lend them to analytical treatment. Such a formalism that incorporates discrete particle effects under fully electromagnetic framework is the culmination of all the discussions presented in this book. 9.3 Electromagnetic Klimontovich Weak Turbulence Theory: Summary To derive the desired final and most general equations for electromagnetic Klimontovich weak turbulence theory involving plasma eigenmodes, that is, longitudinal eigenmodes, L and S, and transverse mode, T , all we need to do is evaluate extra terms that arise from the particle discreteness effects that were missing in the equations derived in Chapter 8. For Langmuir wave, we already discussed the induced emission terms in Section 8.2.1. The first extra term relates to the spontaneous emission,   (ωkL )2  2 ∂Ikσ L  = ea dp δ(σ ωkL − k · v) fa (p). (9.41)  2 ∂t spont.em. k a This term is the counterpart to (8.35). The decay terms for L mode are already derived, and the result is (8.40). The discrete particle effects do not influence the

9.3 Electromagnetic Klimontovich Weak Turbulence Theory: Summary

261

decay terms. Induced scattering term for L mode is given by (8.58). The spontaneous scattering term is new, and is given formally by    4 ∂Ikσ L   dk dp = ∂t spont.scatt.  (k,σ ωkL )  a σ 4ea2 |k − k |2 |  (k − k,σ ωkα − σ  ωkL )|2 ki kj (k − k )k (2)   L × χij k (k ,σ ωk |k − k,σ ωkL − σ  ωkL )   k k |k − k |  (k − k )l (2)∗   L kn km × χ (k ,σ ωk |k − k,σ ωkL − σ  ωkL ) k k  |k − k | nml    Ikσ L Ikσ L ×  −  (k,σ ωkL )  (k,σ  ωkL )

×

× δ[σ ωkL − σ  ωkL − (k − k ) · v] fa (p)  4 dk +   (k,σ ωkL ) σ   4ea2 dp × (k − k )2 |  (k − k,σ ωkL − σ  ωkT )| a   kj kn ki (k − k )k δ × χij(2)k (k,σ  ωkT |k − k,σ ωkL − σ  ωkT ) − jn k |k − k | k 2 km (k − k )l (2)∗   T (9.42) × χ (k ,σ ωk |k − k,σ ωkL − σ  ωkT ) k |k − k | mnl    1 Ikσ T Ikσ α × − 2  (k,σ ωkα ) ⊥ (k,σ  ωkT ) × δ[σ ωkα − σ  ωkT − (k − k ) · v] fa (p). Making use of ki kj (k − k )k

χ (2) (k,σ  ωkL |k − k,σ ωkL − σ  ωkL ) k k  |k − k | ij k  (k − k )l (2)   L kn km × χ (k ,σ ωk |k − k,σ ωkL − σ  ωkL ) k k  |k − k | nml e2 (k · k )2 (k − k )2 e = − 2 2 2 [χ (k − k,σ ωkL − σ  ωkL )]2, 4 4me k k ωpe    kj km ki (k − k )k (2)   T • δj m − 2 χ (k ,σ ωk |k − k,σ ωkL − σ  ωkT ) k k |k − k | ij k



262

Electromagnetic Klimontovich Weak Turbulence Theory

kn (k − k )l (2)∗   T χ (k ,σ ωk |k − k,σ ωkL − σ  ωkT ) k |k − k | nml 1 e2 (k × k )2 (k − k )2 × = , 4 4m2e k 2 k 2 ωpe |  (k − k,σ ωkL − σ  ωkT )|2

×

(9.43)

we arrive at    ∂Ikσ L  (k · k )2 σ ωkL   dk dp = − ∂t spont.scatt. 16π 2 n2  k 2 k 2 σ

× δ[σ ωkL − σ  ωkL − (k − k ) · v] 

 × σ  ωkL Ikσ L − σ ωkL Ikσ L [fe (p) + fi (p)]   (k×k )2 σ ωkL   dk dp − δ[σ ωkL − σ  ωkT − (k − k )·v] 16π 2 n2  k 2 k 2 σ  σ T   T σL L Ik  × σ ωk Ik − σ ωk [fe (p) + fi (p)]. (9.44) 2 Next, we take α = S. Among the new terms is the spontaneous emission term for S mode,   μk (ωL )2  Ikσ S ∂  k dp δ(σ ωkS − k · v) fa (p). = ea2 (9.45) 2 ∂t spont.emiss. μk k a This is the counterpart to the induced emission term, given in (8.65). For ion-sound mode, nonlinear scattering processes are generally slow, so we may ignore such processes. The spontaneous emission term in the transverse wave kinetic equation is given by   e2  (k × v)2 Ikσ T ∂  a dp δ(σ ωkT − k · v) fa (p). (9.46) = 2 ∂t spont.emiss. 2 2 k a The spontaneous scattering term for T mode is specified by    ∂  Ikσ T 2e2 T  dk dp = σ ωk  ∂t spont.scatt. 2 (k − k )2 |  (k − k,σ ωkT − σ  ωkL )|2 σ   kj (k − k )k ki km δ χij(2)k (k,σ  ωkL |k − k,σ ωkT −σ  ωkL ) ×  − im k |k − k | k2 k  (k − k )p (2)∗   L × n χ (k ,σ ωk |k − k,σ ωkT − σ  ωkL ) μTk μLk k |k − k | mnp

9.3 Electromagnetic Klimontovich Weak Turbulence Theory: Summary

 × σ ωkT

σ L k

σT I  L Ik − σ ω  k μLk 2μTk



263

δ[σ ωkT − σ  ωkL − (k − k ) · v]

× [fe (p) + fi (p)].

(9.47)

Making use of 

 ki km δim − 2 χij(2)k (k,σ  ωkL |k − k,σ ωkT − σ  ωkL )   k |k − k | k kn (k − k )p (2)∗   L ×  χ (k ,σ ωk |k − k,σ ωkT − σ  ωkL ) k |k − k | mnp e2 (k − k )2 (k × k )2 e = |χ (k − k,σ ωkT − σ  ωkL )|2, 4 4m2e ωpe k 2 k 2

kj (k − k )k

(9.48)

we obtain the desired expression for spontaneous scattering term within the T wave kinetic equation,    Ikσ T (k × k )2 ∂  σ ωkT   dk dp = − ∂t spont.scatt. 2 32π 2 n2  k 2 k 2 σ

− σ  ωkL − (k − k ) · v]  σT  L Ik T σ L [fe (p) + fi (p)]. × σ ωk − σ ω k Ik  2

×

δ[σ ωkT



(9.49)

Gathering all the results we have obtained thus far, we summarize the equations of Klimontovich weak turbulence theory that incorporate the full electromagnetic effects and particle discreteness effects. The following displayed equations list the wave kinetic equations for L, S, and T modes with each wave-particle and wavewave process indicated, as well as the particle equation:   L 2 2  σ ωk ∂Ikσ L ∂fe (p) σ L L 4π e L dp δ(σ ωk − k · v) f (p) + k · • = σ ωk I 2 e ∂t k2 ∂p k &4π '( ) & '( ) spont. & ind. emiss. L ↔ e  2  2 Iσ S L e  μk−k (k · k ) L σ  L k−k dk 2 2 σ ωk Ik + πσ ωk 2Te2   k k |k − k |2 μk−k σ ,σ   Iσ S  L k−k σ L  L σ L σ L S δ(σ ωkL − σ  ωkL − σ  ωk−k − σ ωk I − σ ωk−k Ik Ik ) '( ) & μk−k k 

+ πσ μLk ωkL

e2 2 8 m2e ωpe

 σ ,σ 

decay L ↔ L + S  2  (k × k )2 k k2 2  dk 2 2 + k k |k − k |2 σ  ωkL σ ωkL

264

Electromagnetic Klimontovich Weak Turbulence Theory







σ T Ik−k I σ T   T σ L σ L × Ik Ikσ L − σ  ωkL k−k Ikσ L − σ  ωk−k  Ik 2 2 T × δ(σ ωkL − σ  ωkL − σ  ωk−k ) & '( )



σ ωkL

decay L ↔ L + T     2  Iσ T σ S e2   μk (k × k ) L k−k Ik dk σ ω + πσ ωkL k 2Te2   k 2 k 2 |k − k |2 2 μ k σ ,σ    I σ T Ikσ S σ L T T δ(σ ωkL − σ  ωkS − σ  ωk−k − σ  ωkL k−k Ikσ L − σ  ωk−k I  ) & '( ) 2 μ k k decay L ↔ S + T  k [k · (k − k )]2 L e  dk 1+ + πσ ωk T 2 4m2e   k 2 |k − k |2 (ωkT )2 (ωk−k ) σ ,σ     σ  T I σ  T Iσ T Ikσ T σ L k−k L Ik   T k−k σ L  T × σ ωk − σ ωk I − σ ωk−k I 2 2 2 k 2 k T × δ(σ ωkL − σ  ωkT − σ  ωk−k ) & '( ) 2



2



decay L ↔ T + T   (k · k )2  dk dp 2 2 δ[σ ωkL − σ  ωkL − (k − k ) · v] − k k σ 

 1  L σL L σ L × ω I − σ ω I σ [fe (p) + fi (p)]   k k k k 2 &4π '( ) σ ωkL 4n2



− Ikσ L Ikσ L &

spont. scatt. L ↔ L + p  ∂fi (p)  (k − k ) · ∂p '( )

ind. scatt. L ↔ L + i   (k × k )2 σ ωkL   dk dp − δ[σ ωkL − σ  ωkT − (k − k ) · v] 4n2  k 2 k 2 σ   σ T  1  T σL L Ik  [fe (p) + fi (p)] × σ ωk Ik − σ ωk 4π 2 2 '( ) & 



Ikσ T σ L I 2 k

spont. scatt. L ↔ T + p  ∂fi (p)  ; (k − k ) · ∂p

ind. scatt. L ↔ T + i

(9.50)

9.3 Electromagnetic Klimontovich Weak Turbulence Theory: Summary

265





2 2 ∂ Ikσ S L 4π e dp δ(σ ωkS − k · v) = μ σ ω k k ∂t μSk k2    L  Ikσ S σ ωk ∂ [fe (p) + fi (p)] [fe (p) + fi (p)] + k · × 2 ∂p μk &4π '( ) & '( )

+ πσ ωkL

spont. & ind. emiss. S ↔ p   2    μk [k · (k − k )] σ  L dk σ ωkL Ikσ L Ik−k  2 2  2 k k |k − k |  

e2 4Te2 

σ L − σ  ωkL Ik−k 

σ ,σ

σS Ikσ S L σ  L Ik − σ  ωk−k I  k μk μk



L δ(σ ωkS − σ  ωkL − σ  ωk−k ) & '( )

decay S ↔ L + L + πσ ωkL

   2  Iσ T e2   μk (k × k ) L k−k σ  L dk 2 2 σ ωk I 2Te2   k k |k − k |2 2 k

(9.51)

σ ,σ



− σ  ωkL

σ T σS σS Ik−k  Ik T σ  L Ik − σ  ωk−k I  k 2 μk μk



T δ(σ ωkS − σ  ωkL − σ  ωk−k ) ; '( ) &

decay S ↔ L + T  4π 2 e2  (k × v)2 ∂ Ikσ T a dp • δ(σ ωkT − k · v) = T 2 ∂t 2 k σ ω k a   T σ ωk ∂fa (p) Ikσ T f (p) + k · × 2 a ∂p 2 &8π '( ) & '( ) spont. & ind. emiss. T ↔ p  e2 (k × k )2  dk + π σ ωkT 2 32 m2e ωpe k 2 k 2 |k − k |2 σ ,σ    2 σT k |k − k |2 2 T σ  L σ  L  L σ  L Ik σ ω × − I I − σ ω I     k k k−k k k−k L 2 σ  ωkL σ  ωk−k   σT L σ  L Ik L δ(σ ωkT − σ  ωkL − σ  ωk−k − σ  ωk−k  Ik  ) '( ) & 2

+ π σ ωkT

e2 4Te2

decay T ↔ L + L   2  Iσ S  μk−k (k × k ) T σ  L k−k dk 2 2 σ ωk Ik k k |k − k |2 μk−k  

σ ,σ

266

Electromagnetic Klimontovich Weak Turbulence Theory 

−σ



ωkL

σ S σT σT Ik−k  Ik L σ  L Ik − σ  ωk−k  Ik  μk−k 2 2

+ π σ ωkT 

e2 4m2e

 σ ,σ 

dk

|k − k |2 (ωkT )2 (ωkT )2



S δ(σ ωkT − σ  ωkL − σ  ωk−k ) '( ) &

decay T ↔ L + S  (k · k )2 1 + 2 2 k k





σT Ikσ T σ  L Ik − σ  ωkT Ik−k  2 2 σ T σ T  Ik Ik L δ(σ ωkT − σ  ωkT − σ  ωk−k ) '( ) & 2 2



σ L × σ ωkT Ik−k  L − σ  ωk−k 





decay T ↔ T + L

(k × k )2 σ ωkT  dk dp δ[σ ωkT − σ  ωkL − (k − k ) · v] 8n2  k 2 k 2 σ    σT 1  L Ik T σ L σ ωk [fe (p) + fi (p)] − σ ω k Ik  × 4π 2 2 '( ) &





− Ikσ L &

spont. scatt. T ↔ L + p  Ikσ T ∂fi (p)  ; (k − k ) · 2 ∂p '( )

(9.52)

ind. scatt. T ↔ L + i   k ∂ ∂fe (p) 2 dk · • = πe δ(σ ωkL − k · v) ∂t k ∂p σ =±1   σ ωkL σ L k ∂fe (p) , (p) + I f · × e k 4π 2 k k ∂p   ∂fi (p) k ∂ 2 • dk · = πe μk δ(σ ωkS − k · v) ∂t k ∂p σ =±1   Ikσ S k ∂fi (p) σ ωkL . × fi (p) + · 4π 2 k μk k ∂p • Auxiliary Definitions   3 2 2 L L ωk = ωpe 1 + k λDe , ω−k = −ωkL, 2   1 + 3T /T Ti i e S ωkS = kcS , ω−k = −ωkS , cS = , 2 2 me 1 + k λDe

(9.53)

9.3 Electromagnetic Klimontovich Weak Turbulence Theory: Summary



c2 k 2 T , ω−k = −ωkT , 2 ωpe    me 3Ti 3 3 1+ , μk = |k| λDe mi Te ωkT = ωpe

267

1+

(9.54)

The summary of equations presented in (9.54) constitutes the general weak turbulence theory where all the terms appear in balanced form between spontaneous and induced processes. In Chapter 10, the final chapter, we discuss the applications of the electromagnetic weak turbulence theory.

10 Applications of Electromagnetic Klimontovich Weak Turbulence Theory

In Chapters 5 and 6 we applied the electrostatic Klimontovich weak turbulence theory to two specific problems. First was on the spontaneous emission of electrostatic fluctuations in thermal equilibrium plasma, which includes the emission of plasma eigenmodes as well as non-eignemodes. The discussion also pertained to plasmas that are slightly out of equilibrium. Collisional kinetic theory becomes relevant for such plasma states. It was shown that collisional processes bring the system back to thermal equilibrium. We then made another application to the plasma that can be considered as being far from equilibrium, and one in which collective excitation of plasma eigenmodes via instability dominates the system. Thus, in Chapter 6, we discussed the weak turbulence theory of gentle electron beam-plasma interaction and the excitation of Langmuir instability. The early saturation of Langmuir instability was shown to be well described by quasilinear theory, but for time scales longer than the quasilinear relaxation period, the mode coupling interaction involving decay and scattering enters the picture. It was shown by numerical analysis that the long time scale evolution of Langmuir and ionsound turbulence involves the generation of energetic tail population in the electron velocity distribution function. We subsequently discussed the analytical theory of asymptotically steady-state electron distribution and Langmuir turbulence intensity, and showed that the solution corresponds to the electron kappa distribution. It was also shown that the electron kappa distribution is intimately related to the solar wind electrons measured by artificial spacecraft. In this chapter, we discuss the consequence of retaining the electromagnetic effects. The first example is the generation of electromagnetic fluctuations in thermal plasmas. For plasmas in thermal equilibrium, electrostatic spontaneous emission theory explains the natural emission and absorption of Langmuir and ion-sound waves. Electromagnetic generalization, however, cannot explain the emission of transverse electromagnetic radiation in the high-frequency regime, but instead, such a theory does predict the emission of magnetic field fluctuations in the 268

10.1 Spontaneous Emission of Magnetic Field Fluctuations

269

low-frequency regime. The non-propagating, or aperiodic, magnetic field fluctuations spontaneously generated in thermal unmagnetized plasmas have potentially significant cosmological implications, as such a mechanism may explain the origin of large-scale magnetic field in the early universe, a topic that was explored by Schlickeiser (2012). The emission of high-frequency electromagnetic waves, that is, the radiation, however, requires nonlinear correction, and it will be shown that such a correction indeed constitutes a new radiation emission mechanism. For systems slightly out of equilibrium, we already discussed that the collisional kinetic theory is applicable. Under electromagnetic formalism, the standard Balescu–Lenard–Landau equation (Landau, 1937; Balescu, 1960; Guernsey, 1960; Lenard, 1960) is generalized to Beliaev–Budker equation (Beliaev and Budker, 1956) relevant to fully electromagnetic collisional process. Beliaev–Budker equation becomes important for relativistic plasmas. The final application of fully electromagnetic Klimontovich weak turbulence theory is again on the gentle or weak electron beam-plasma interaction. The inclusion of electromagnetic effects leads to a new physical process, namely, the emission of transverse electromagnetic radiation at the plasma frequency and/or its harmonics. Such a non-thermal radiation emission process, which can be viewed as a process of partially converting the energy associated with the beam-generated Langmuir turbulence into the radiation energy by nonlinear processes, is the basic radiation mechanism responsible for type II and type III solar radio bursts. Such a radio emission process is generally called the “plasma emission,” and it has been extensively discussed in the solar radio physics context. Nevertheless, complete numerical solution starting from the electron beam-plasma induced Langmuir instability, mode coupling, and partial conversion to electromagnetic radiation has not been demonstrated until quite recently (Ziebell et al., 2014a, 2015). We will overview such a solution in the present chapter. 10.1 Spontaneous Emission of Magnetic Field Fluctuations Consider the general spectral balance equation (9.30). For thermal plasmas, we may ignore nonlinear terms such that we may simply balance the linear response terms with the spontaneous emission terms, which, in the non-relativistic limit, is expressed by      (k,ω) δE2 k,ω + ⊥ (k,ω) δE⊥2 k,ω    (4πea )2  1 1 (k · v)2 (k × v)2 dv = + (2π)3 ω2 k 2 ∗ (k,ω) k2 ∗⊥ (k,ω) a × δ(ω − k · v)fa (v),

(10.1)

270

Applications of Electromagnetic Klimontovich Weak Turbulence Theory

where ⊥ (k,ω), defined in (7.32), is given by ⊥ (k,ω) = ⊥ (k,ω) −

c2 k 2 , ω2

(10.2)

and the longitudinal and transverse components of the linear dielectric tensor are defined by  (k,ω) = 1 +

 4πe2  a

ma k 2

a

⊥ (k,ω) = 1 +

 2πe2  a

ma ω

a

dv

k · ∂fa (v)/∂v , ω − k · v + i0

(10.3)

(k×v)×k ∂fa (v) dv · . ω − k · v + i0 k2 ∂v

From (10.1) we readily obtain   2 δE k,ω = 

 2

δE⊥

k,ω

=



a

(4πea )2 (2π)3 ω2 |  (k,ω)|2

a

(4πea )2 (2π)3 ω2 |⊥ (k,ω)|2



dv 

(k · v)2 δ(ω − k · v)fa (v), k2

dv

(k × v)2 δ(ω − k · v)fa (v). k2

(10.4)

For thermal equilibrium we have  4πe2 

ω(k · v) 1 fa (v) Tω ω − k · v + i0 k 2 a   2   2ωpa ω ω 1+ , Z =1+ kvT a kvT a k 2 vT2 a a  2πe2  1 (k×v)2 a dv ⊥ (k,ω) = 1 − fa (v) Tω ω − k · v + i0 k 2 a   2  ωpa ω , Z =1+ ωkvT a kvT a a  (k,ω) = 1 −

a

dv

(10.5)

where vT a = (2T /ma )1/2 and Z(ζ ) is the plasma dispersion function – see (5.5). As a consequence, we have Im  (k,ω) =

 4π 2 e2  a

a

Im ⊥ (k,ω) =



 2π 2 e2  a

a



dv δ(ω − k · v)

(k · v)2 fa (v), k2

dv δ(ω − k · v)

(k×v)2 fa (v), k2

(10.6)

10.1 Spontaneous Emission of Magnetic Field Fluctuations

from which we obtain 

δE2





δE⊥2

k,ω

 k,ω

T Im  (k,ω) , 2π 3 ω |  (k,ω)|2 Im ⊥ (k,ω) T = 3 . π ω | ⊥ (k,ω) − c2 k 2 /ω2 |2

271

=

(10.7)

Note that the magnetic field fluctuations also follow from (10.7), 

δB 2

 k,ω

=

Im ⊥ (k,ω) c2 k 2 T . 2 3 ω π ω | ⊥ (k,ω) − c2 k 2 /ω2 |2

(10.8)

The first expression in (10.7) is the same as (5.5), and it describes the spectrum of spontaneously emitted electrostatic fluctuations. According to (10.7), however, transverse electromagnetic fluctuations satisfying the condition for superluminosity, or fast wave condition, ck/ω < 1,

(10.9)

cannot be spontaneously emitted even for non-relativistic treatment. For relativistic case, the emission is strictly forbidden, since no particle speed can exceed the speed of light in vacuo, c. Even for non-relativistic case, fast mode emission will be virtually non-existent. This is because for thermal equilibrium distribution function,   na ma v 2 , (10.10) exp − fa (v) = (2πT /ma )3/2 2T the superluminal region will be characterized by a negligible fluctuation intensity since the overall emission intensity is proportional to     c2 ω2 ma ω 2 = exp − 2 2 2 1. exp − 2 (10.11) 2k T vT a c k For sub-luminal regime, ck/ω > 1,

(10.12)

on the other hand, thermal plasma may emit electromagnetic fluctuations. Subluminous electromagnetic fluctuations in unmagnetized plasmas have been discussed by Tajima et al. (1992); Yoon (2007); Tautz and Schlickeiser (2007); Schlickeiser and Yoon (2012); Felten et al. (2013); and Lazar et al. (2012). Shown in Figure 10.1 is the contour plot of the sub-luminous magnetic field fluctuations described by (10.8). The spontaneous emission of fluctuating B field is closely associated with the Weibel instability. Recall that the unstable Weibel mode in the presence of temperature anisotropy is a purely growing mode with Re ω = 0. The spontaneously emitted magnetic field fluctuation spectrum has peak intensity for Re ω = 0, which implies that the magnetic field fluctuations can be described as

272

Applications of Electromagnetic Klimontovich Weak Turbulence Theory

being quasistationary. This property is identical to that of Weibel instability. Thus, the spontaneously emitted B field fluctuations for thermal plasmas provide the seed perturbation, which will be amplified as the Weibel instability when excessive perpendicular free energy source becomes available. Here, we should note that the Weibel mode is characterized by purely aperiodic transverse fluctuations with Re ω = 0 and Im ω = finite. Since the basic conjecture of the main body of this book is based upon the assumption of weak wave growth or damping, Re ω |Im ω|, the Weibel mode, whether it is stable or unstable, does not naturally lend itself to the formalism discussed in the main body of this book. The Weibel mode is a classic example of the “reactive” mode – see the discussion in Appendix E, where the weak turbulence theory of reactively unstable mode is considered under electrostatic approximation. The natural question is whether it is possible to extend the electromagnetic weak turbulence formalism for reactively unstable modes, which includes aperiodic Weibel mode. Schlickeiser and Yoon (2012) and Yoon et al. (2014) discussed the re-formulation of quasilinear kinetic theory that includes the effects of spontaneously emitted and reabsorbed Weibel mode. Schlickeiser (2012) applied the theory of aperiodic magnetic field fluctuations in order to put forth a novel theory for origin of cosmic magnetic fields in the early universe. In standard astrophysics, the origin of cosmic magnetic fields is closely associated with the so-called Biermann battery mechanism (Hao et al., 2008), combined with some magnetic dynamo action (Kulsrud and Zweibel, 2008). Schlickeiser (2012), on the other hand, suggested a novel idea based upon the plasma kinetic theory. While the linear formulae for transverse electric and magnetic field fluctuations (10.7) and (10.8) are appropriate for describing the sub-luminous “Weibel” mode branch of the electromagnetic wave spectrum, these linear theoretical formulae cannot describe the emission of fast electromagnetic waves in unmagnetized plasmas. That is, according to the linear theory, unmagnetized plasmas cannot radiate electromagnetic waves spontaneously. However, the situation changes when we include nonlinear terms in the general weak turbulence theoretical formalism. 10.2 Electromagnetic Radiation in Thermal Plasmas To begin the discussion, let us focus our attention on the transverse eigenmode, satisfying the dispersion relation, ω = ωkT . In general, spontaneous emission can take place for all ω including the eigenmode as well as those ω’s that do not satisfy the linear dispersion relation. We focus our attention on the eigenmode solution near the transverse mode only, however, since we are interested in the emission of radiation in thermal plasmas. This means that the equations of electromagnetic Klimontovich weak turbulence theory, (9.50)–(9.53), are applicable. For thermal

10.2 Electromagnetic Radiation in Thermal Plasmas

273

Figure 10.1 Spontaneously emitted magnetic field fluctuation spectrum, (10.8), plotted against normalized frequency ω/ωp and dimensionless wave number kλDe .

plasmas, however, all physical quantities are in stationary state, ∂/∂t = 0. Consequently, we may assume that the particle distribution function and wave intensities are stationary and isotropic in that we may take fa (v) = fa (v), a = e,i, Ikσ α = Ikα , and Ikσ T = IkT . We may further assume that the particle distributions are given by thermal Maxwell–Boltzmann distribution. In the wave kinetic equations for longitudinal electrostatic waves, we may ignore nonlinear mode coupling terms, since the wave intensities for thermal equilibrium plasmas are low so that generally nonlinear terms are inconsequential. However, in the transverse wave equation, since the linear response term is absent owing to the fact that linear wave-particle resonance cannot be satisfied between the particles and the fast radiation mode, nonlinear mode coupling terms become essential. Among the nonlinear mode coupling terms, we ignore decay processes since these terms only involve wave energy and momentum exchanges but do not involve particles. In the steady-state solution that we seek, we pay attention to asymptotic quasi-equilibrium between the electrons and electrostatic and electromagnetic fluctuations. Hence, we may ignore low-frequency ion-sound mode at the outset. The starting point is the transverse wave kinetic equation (9.52) in which spontaneous and induced emission terms as well as various three-wave interaction terms

274

Applications of Electromagnetic Klimontovich Weak Turbulence Theory

depicting various decay processes are ignored. Resorting to the velocity (rather than momentum) representation, the relevant equation is   e2  (k × k )2 ∂ Ikσ T  dk dv = σ ωkT 2 ∂t 2 2m2e ωpe k 2 k 2  σ

× δ[σ ωkT − σ  ωkL − (k − k ) · v]  2 σT  ne T σ L  L Ik [Fe (v) + Fi (v)] × 2 σ ωk Ik − σ ωk ωpe 2  ∂Fi (v) me σ  L Ikσ T . Ik  (k − k ) · +π mi 2 ∂v

(10.13)

Making use of (k × k )2 = k 2 k  2 − (k · k )2 , and assuming Maxwellian particle distributions, we have  k 2 k  2 − (k · k )2 dv δ[ωkT − ωkL + |k − k | vz ] 2 2  k k    3/2  T  mi T T L L Ik −me (vx2 +vy2 +vz2 )/2T −mi (vx2 +vy2 +vz2 )/2T e ω I  − ωk  + e × 4π 2 k k 2 me   3/2 IT mi 2 2 2 (10.14) − IkL k (ωkT − ωkL ) e−mi (vx +vy +vz )/2T , 2 me 

0=

dk

where we have assumed steady state, ∂/∂t = 0, and have assumed that the vector k − k lies along z axis without loss of generality. We have also ignored σ  = −σ at the outset and ignored the σ dependence associated with the wave intensities, since they are assumed to be isotropic, hence independent of the propagation direction. We have then chosen σ = σ  = 1. Integrations along vx and vy are trivial. Upon carrying out the vz integration by virtue of the delta function resonance condition, we have 

0=

k 2 k  2 − (k · k )2 dk 2  2 k k |k − k |     T  me (ωkT − ωkL )2 T T L L Ik exp − × ω I  − ωk  4π 2 k k 2 2T |k − k |2    T   IkT T T T L L L Ik I k  − ωk  − − Ik + ωk 4π 2 2 4π 2 2  1/2   mi mi (ωkT − ωkL )2 × . exp − me 2T |k − k |2

(10.15)

10.2 Electromagnetic Radiation in Thermal Plasmas

275

At this point, we note that the most important contribution to the k integral should come from k  that satisfies ωkT ∼ ωkL , or k ∼ k∗ , where k∗2 =

2 c2 2 k . 3 vT2 e

(10.16)

As a consequence, we may simply consider the quantity within the square bracket associated with the ion thermal distribution, and also the quantities within the square bracket associated with the electron thermal distribution, and set them equal to zero separately. Upon replacing k  by k∗ , we thus have IT ωkT IkL∗ − ωkL∗ k = 0, 2    T T I I T T k T L L L Ik∗ − ωk∗ − − Ik∗ k = 0. ωk 2 2 4π 2 4π 2

(10.17)

Upon making use of 3k∗2 λ2De =

c2 k 2 , 2 ωpe

(10.18)

and making use of (see [6.19]) 1 T , 2 4π 1 + 3k 2 λ2De

(10.19)

2 1 T T ωpe = , 2 2π 2 1 + c2 k 2 /ωpe 2π 2 (ωkT )2

(10.20)

IkL = we immediately obtain IkT =

for both equations in (10.17). This is the electric field spectral intensity for spontaneously emitted radiation in thermal plasmas. In short, thermal plasmas spontaneously emit (and reabsorb) all the eigenmodes naturally excited in plasmas, including Langmuir, ion-sound, and transverse electromagnetic modes, but in order to discuss the fast electromagnetic branch, one must include nonlinear effects. For longitudinal modes (L and S), linear treatment is sufficient, as we have already shown in Section 5.1. The spontaneous emission of electromagnetic radiation discussed in this section was first discovered by Ziebell et al. (2014b), who also confirmed the theoretical findings by numerical particle-in-cell (PIC) simulation. Next, we turn to another problem associated with quasi-equilibrium state. If the plasma is close to but not exactly at thermal equilibrium state, then collisional processes bring such a plasma back to thermal equilibrium state. This was discussed in Chapter 5 under electrostatic formalism. However, when the plasma is relativistic, the simple electrostatic approximation is no longer applicable and one must resort to

276

Applications of Electromagnetic Klimontovich Weak Turbulence Theory

fully electromagnetic formalism for collisional relaxation process. We next discuss the derivation of collisional kinetic equation for relativistic plasma. 10.3 Electromagnetic Collisional Kinetic Equation To discuss the collisional processes for relativistic plasmas slightly out of equilibrium, we return to (9.34), which is repeated here for the sake of convenience:       1 k ∂ Im  (k,ω) ∂fa 2 δ(ω − k · v) = πea dk dω · 3 ∂t k ∂p 2π k |  (k,ω)|2  (k × v)2 ω2 Im ⊥ (k,ω) fa + k |ω2 ⊥ (k,ω) − c2 k 2 |2     2 k ∂fa 1 (k × v)2  2  . (10.21) δE⊥ k,ω + δE k,ω + · 2 ω2 k ∂p If the plasma is slightly out of equilibrium, yet there is no free energy source to excite any collective modes, that is, in the absence of instabilities, then one does not need to solve the wave kinetic equation, but instead, we may employ the electric field spectra taken from the spontaneous emission theory, namely (10.4), which is now represented in momentum formalism,   2   2 2eb2  (k · v ) δE k,ω = dp δ(ω − k · v )fb (p ), 2 | (k,ω)|2 2 πω k  b 

δE⊥2

 k,ω

=

 b

×



2eb2 ω2 π |ω2 ⊥ (k,ω) − c2 k 2 |2 dp

(10.22)

(k × v )2 δ(ω − k · v )fb (p ), k2

where the relativistic momentum-velocity relationship is assumed, p = ma γ v or v = p/(ma γ ) with the relativistic mass factor defined by γ = (1 − v 2 /c2 )−1/2 = [1 + (p/ma c)2 ]1/2 . The dielectric constants and their imaginary parts are given in momentum representation by  4πe2  1 ∂fb (p ) b  dp , k ·  (k,ω) = 1 + k2 ω − k · v + i0 ∂p b  2πe2  (k×v )×k 1 b dp ⊥ (k,ω) = 1 + ω ω − k · v + i0 k2 b    ∂fb (p ) k ∂fb (p )  , (10.23) · + × v × ∂p ω ∂p

10.3 Electromagnetic Collisional Kinetic Equation

and Im  (k,ω) = −

 4π 2 e2  b

b

Im ⊥ (k,ω) = −

k2

 2π 2 e2  b

b

ω2

dp δ(ω − k · v ) k · dp δ(ω − k · v )

277

∂fb (p ) , ∂p

(10.24)

[(k×v )×k]2 ∂fb (p ) k · . k4 ∂p

Substituting (10.22) and (10.24) to (10.21) and carrying out the ω integration, one has     k ∂ ∂fa (p)  2 2  δ(k · v − k · v ) dk ea eb dp = · ∂t k ∂p b   2 (k × v)2 (k × v )2 × 2 + k |  (k,k · v)|2 k 2 |(k · v)2 ⊥ (k,k · v) − c2 k 2 |2   k ∂fb (p ) k ∂fa (p)  fa (p) , (10.25) × · fb (p ) − · k ∂p k ∂p 2 2 2 where we have invoked (k × v) = [(k × v) × k] /k . Under the assumption that dp (k × v) fa (p) = 0, we have a useful identity

(k × v)2 (k × v )2 = 2[(k × v) · (k × v )]2 .

(10.26)

This can easily be checked by direct substitution for k = kˆz. Then one can easily show that (k × v)2 (k × v )2 = k 4 (vx2 v  x + vy2 v  y + vx2 v  y + vy2 v  x ), 2

2

2

2

[(k × v) · (k × v )]2 = k 4 (vx2 v  x + vy2 v  y + 2vx vy vx vy ). 2

2

Upon making use of     2   2   dpx dpy v x fb (p ) = dpx dpy v  y fb (p ),          dpx dpy vx fb (p ) = 0 = dpx dpy vy fb (p ), one can prove the relationship (10.26). Thus, making use of (10.26), we obtain    ∂fa (p) ∂fb (p ) ∂fa (p) ∂   dp Qij fb (p ) − fa (p) , = ∂t ∂pi ∂pj ∂pj   ki kj 2 2 ea eb dk 4 δ(k · v − k · v ) (10.27) Qij = 2 k b   1 [(k × v) · (k × v )]2 × . + |  (k,k · v)|2 |(k · v)2 ⊥ (k,k · v) − c2 k 2 |2

278

Applications of Electromagnetic Klimontovich Weak Turbulence Theory

This is the electromagnetic, or, equivalently, relativistic, generalization of the Balescu–Lenard equation (5.22). One can also derive the generalization to the Landau equation (5.31), which was first derived by Beliaev and Budker (1956). In this approximation, we simply take  → 1,

⊥ → 1.

(10.28)

Then we get Qij = 2

 b

 ea2 eb2

dk

  ki kj [(k × v) · (k × v )]2  . δ(k · v − k · v ) 1 + k4 |(k · v)2 − c2 k 2 |2 (10.29)

As with the Landau equation, we must replace the limit of k integral from 2 kmin = λ−1 De to kmax = T /e . Making use of (k × v) · (k × v ) = k 2 (v · v ) − (k · v)(k · v ), we obtain  ∂ ∂fa (p) ea2 eb2 =2 ∂t ∂pi b





dp



(10.30)

dk [c2 − (v · v )]2 δ(k · v − k · v )

  ki kj ∂ ∂ × 2 2 −  fa (p)fb (p ). [c k − (k · v)2 ]2 ∂pj ∂pj

(10.31)

This is the relativistic version of the Landau collisional kinetic equation, with appropriate cutoffs in the k integral implicitly assumed. 10.4 Plasma Emission by EM Weak Turbulence Process Solar radio burst phenomena in the meter wavelengths were identified and subsequently classified into several types since the decade of 1950s (Wild, 1950, 1951; Wild and McCready, 1950a,b; Wild et al., 1954, 1959). Among the five different types of solar radio bursts, type III radio bursts are the most studied and wellunderstood category in a comparative sense. Most models agree that fast electrons escaping from active regions in the Sun, that is, solar flares, first excite Langmuir instability by electron beam-plasma interaction. Subsequently, nonlinear processes partially convert the electron beam energy into radiation. The details of the conversion process can be described within the framework of electromagnetic weak turbulence theory. A similar process may also be operative for type II emissions (Roberts, 1959; Wild et al., 1959). Type II emissions are associated with the electrons accelerated at the interplanetary shock wave, known as the coronal mass ejection, generated during solar active events.

10.4 Plasma Emission by EM Weak Turbulence Process

279

For general solar and coronal physics, the readers may refer to an excellent monograph by Aschwanden (2005). For the physics of solar wind, an excellent reference may be the monograph by Meyer-Vernet (2007). For general space plasma physics, among excellent books are those by Baumjohann and Treumann (1997); Treumann and Baumjohann (1997); Parks (2004); Cravens (1997); and Kivelson and Russell (1995), just to mention a few. Both type II and III radio bursts are often associated with the fundamental/harmonic two-band structure, although it is more common for type II bursts than for type III emissions. It is commonly believed that the fundamental/harmonic refers to the plasma frequency or its harmonic, hence the process involved in these solar radio bursts is called the “plasma emission.” As the type II or type III radio generating electrons travel outward from the solar coronal source region into the interplanetary space, they encounter decreasing background number density, or, equivalently, decreasing local plasma frequency. As a result, these emissions exhibit characteristic descending tone in frequency-time plot. There are excellent review articles and monographs on the subject of type III radio bursts, as well as some representative research papers. There are too numerous references so that it is practically impossible to cite them all – see, e.g., Wild et al. (1963); Kundu (1965); Zheleznyakov (1970); Zaitsev et al. (1972); Wild and Smerd (1972); Stewart (1974); Fainberg and Stone (1974); Lin et al. (1973, 1981, 1986); Smerd (1976); Rosenberg (1976); Melrose (1980a,b, 1986); Goldman (1983); Suzuki and Dulk (1985); Dulk (1985); Reiner et al. (1992); Ergun et al. (1998); Robinson and Cairns (1998a,b,c); Mann et al. (1999); Cane et al. (2002); Reiner et al. (1992, 2009); Saint-Hilaire et al. (2013); Reid and Ratcliffe (2014), etc. – but perhaps one of the most comprehensive reviews of the progress, which was made on the topic of solar radio emission up to the decade of 1980s, may be the monograph edited by McLean and Labrum (1985). If one is interested in detailed solar radio physics from the perspective of plasma astrophysics, one is referred to the excellent two-volume monograph by Melrose (1980a). The first theoretical model of plasma emission was proposed by Ginzburg and Zheleznyakov (1958), which has undergone considerable improvements ever since, and some alternative models have also been proposed (Tsytovich, 1967; Kaplan and Tsytovich, 1968; Zheleznyakov and Zaitsev, 1970a,b; Papadopoulos et al., 1974; Bardwell and Goldman, 1976; Magelssen and Smith, 1977; Nicholson et al., 1978; Goldstein et al., 1979; Smith et al., 1979; Goldman and Dubois, 1982; Melrose, 1982, 1987; Goldman, 1983; Cairns, 1987a,b,c; Robinson and Cairns, 1998a,b,c; Wu et al., 2002; Li et al., 2008a,b, 2009; Tsiklauri, 2011; Graham et al., 2012; Thejappa et al., 2012; Krafft et al., 2013). In this section the plasma emission will be discussed on the basis of EM weak turbulence theory (Yoon, 2006; Yoon et al., 2012b; Ziebell et al., 2014a). Over the past several decades, the plasma emission

280

Applications of Electromagnetic Klimontovich Weak Turbulence Theory

was discussed within the framework of reduced or partial EM weak turbulence theory (Li et al., 2008a,b, 2009; Schmidt and Cairns, 2012a,b, 2014), but the complete numerical solution of the entire set of EM weak turbulence equations has not been done until quite recently (Ziebell et al., 2014a, 2015). The beam-plasma interaction and plasma emission can also be studied by means of direct EM PIC simulation (Kasaba et al., 2001; Karlick´y and Vandas, 2007; Rhee et al., 2009b,a; Umeda, 2010; Ganse et al., 2012a,b). The analytical EM weak turbulence theory can be employed to interpret the results of PIC simulations. To investigate the problem, we start from the basic equations of EM weak turbulence theory, namely, (9.50)–(9.54). In employing the equations, however, we make note of the fact that the radiation emission is a byproduct of the beam-plasma instability process. The generalized L and S mode wave kinetic equations (9.50) and (9.51) contain various extra terms that describe the back-reaction of L mode in response to the radiation generation, but these effects are generally weak. Consequently, in describing the wave dynamics of L and S modes, let us resort to the simpler electrostatic description as in (6.4) and (6.5). Here, we repeat the wave and particle kinetic equations for the sake of convenience and completeness: 



4πe2 ∂Ikσ L = ∂t me k 2

dvδ(σ ωkL 



+

L dk Vk,k 

σ ,σ  =±1

−σ



L σ L σ L ωk−k  Ik  I k



− k · v) ne Fe (v) + 2





πσ ωkL k



∂Fe (v) σ L · I ∂v k





σ S σ S σL σ ωkL Ikσ L Ik−k σ  ωkL Ik−k   Ik − μk−k μk−k



S δ(σ ωkL − σ  ωkL − σ  ωk−k )

(10.32)





 ne2 L σ  L  L σL I − σ ω I σ ω [Fe (v) + Fi (v)] k k k k 2 ωpe  σ  πme σ  L σ L ∂Fi (v)  δ[σ ωkL − σ  ωkL − (k − k ) · v], + I  I (k − k ) · mi k k ∂v   ∂ Ikσ S 4πμk e2 S dvδ(σ ω = − k · v) ne2 [fe (v) + fi (v)] k ∂t μk me k 2    ∂fe (v) me ∂fi (v) Ikσ S L k· + + πσ ωk k · ∂v mi ∂v μk    σ  L σ S L σ L σ S   σ ωkL Ik−k σ  ωk−k  Ik  Ik  I k  S L σ  L σ  L dk Vk,k σ ωk Ik Ik−k − + − μk μk   +

dk

L dvUk,k 

σ ,σ

L × δ(σ ωkS − σ  ωkL − σ  ωk−k  ),

(10.33)

10.4 Plasma Emission by EM Weak Turbulence Process

∂ Ikσ T = ∂t 2









T LL σ L dk Vk,k σ ωkT Ikσ L Ik−k   −

σ ,σ  

L σ  L k k−k

σ ω I



281

Ikσ T

2

L σ L σT σ  ωk−k  Ik  I k L δ(σ ωkT − σ  ωkL − σ  ωk−k − ) 2   σ  S σ  S σ T  σ ωkT Ikσ L Ik−k σ  ωkL Ik−k   Ik  T LS dk Vk,k + − μk−k 2μk−k σ ,σ   L σ L σ T σ  ωk−k  Ik  I k S δ(σ ωkT − σ  ωkL − σ  ωk−k − (10.34) ) 2   σ  L σ  L σ T  σ ωkT Ikσ T Ik−k σ  ωkT Ik−k   Ik  TTL dk Vk,k + − 2 2 σ ,σ   L σ T σ T σ  ωk−k  Ik  I k L δ(σ ωkT − σ  ωkT − σ  ωk−k − ) 4  2  σT   ne ˆ  T T σ L  L Ik [Fe (v) + Fi (v)] dk dvUk,k 2 σ ωk Ik − σ ωk + ωpe 2 σ  ∂Fi (v) me σ  L Ikσ T  δ[σ ωkT − σ  ωkL − (k − k ) · v], + π Ik  (k − k ) · mi 2 ∂v  k ∂ ∂Fe (v) πe2  dk · = 2 δ(σ ωkL − k · v) ∂t me σ =±1 k ∂v



×

 me σ ωkL σ L k ∂Fe (v) , Fe (v) + Ik · 4π 2 k k ∂v

where L Vk,k  =

πe2 σ ωkL μk−k (k · k )2 , 2Te2 k 2 k  2 |k − k |2

S Vk,k  =

πe2 σ ωkL μk [k · (k − k )]2 . 4Te2 k 2 k  2 |k − k |2

L Uk,k  =

σ ωkL e2 (k · k )2 , 2 ne m2e ωpe k2k2

T LL Vk,k  =

 2  k πe2 σ ωkT (k × k )2 |k − k |2 2 − , L 2 32m2e ωpe σ  ωk−k k 2 k  2 |k − k |2 σ  ωkL 

(10.35)

282

Applications of Electromagnetic Klimontovich Weak Turbulence Theory

πe2 σ ωkT μk−k (k × k )2 , 4Te2 k 2 k  2 |k − k |2   (k · k )2 πe2 σ ωkT |k − k |2 1 + , = 4m2e (ωkT )2 (ωkT )2 k2k2 σ ωkT e2 (k × k )2 = . 2 2nm ˆ 2e ωpe k2k2

T LS Vk,k  = TTL Vk,k  T Uk,k 

(10.36)

Ziebell et al. (2014a, 2015) solved the set of equations (10.32)–(10.36) by numerical means for initial configuration in which the ions are considered stationary and distributed according to thermal equilibrium model with temperature Ti . The electrons are assumed to be initially composed of an isotropic thermal background plus a Gaussian distribution of streaming component, as in Chapter 6. As discussed in Section 10.2, T mode waves are spontaneously emitted via nonlinear spontaneous scattering process. As a consequence, even in the absence of electron beam there is a background level of radiation given by formula (10.19). Ziebell et al. (2014a, 2015) thus initiated their numerical analysis with a finite level of initial T mode intensity, Ikσ T (0) =

1 Te . 2 2 2 2π 1 + c k 2 /ωpe

(10.37)

The input parameter g is related to the particle discreteness, and in their study, Ziebell et al. (2014a, 2015) simply chose g in accordance with the plasma parameter 1/(nλ3D ) = 5 × 10−3 . Other input parameters are the background to total electron density ratio nb /n0 = 10−3 , equal Gaussian beam and background temperatures Te = Tbeam , the ion to electron temperature ratio Ti /Te = 1/7, and the normalized beam speed Vb /vT e , which they varied in their study. Here, we show an example of Vb /vT e = 6. In Figures 10.2–10.5, we show the numerical solution obtained by Ziebell et al. (2015). Figure 10.2 shows snapshots of electron velocity distribution function at two different time intervals corresponding to ωpe t = 100 and 1000. For relatively early time, namely ωpe t = 100, one can still see the drifting electron beam distribution feature, but as time progresses, the positive slope in the parallel velocity is reduced as a result of the well-known quasilinear velocity diffusion process. As a side note, it should be mentioned that quasilinear relaxation of the electron beam associated with the type III radio source has been the focus of intense research in the literature. The challenge was how to explain the persistence of observed beam feature associated with type III events (Lin, 1970; Frank and Gurnett, 1972; Kane, 1972; Lin et al., 1973, 1981, 1986; Ergun et al., 1998; Gosling et al., 2003; Krucker et al., 2007; Wang et al., 2012b) in spite of rapid relaxation by quasilinear plateau formation. This was known in the early literature as Sturrock’s dilemma

10.4 Plasma Emission by EM Weak Turbulence Process

283

(a) ω t = 100 pe

Fe(v)

0 –5 –5

–10 –5

0 0 v /v

|| Te

v /v ^

Te

5

5 10

(b) ω t = 1000 pe

–5

e

F (v)

0

–5

–10 –5

0 0 v||/vTe

v^ /vTe

5

5 10

Figure 10.2 Time evolution of the electron velocity distribution function Fe versus v⊥ /vT e and v /vT e format. Panel (a) shows Fe at ωpe t = 100 and panel (b) is for ωpe t = 1000.

284

Applications of Electromagnetic Klimontovich Weak Turbulence Theory

Figure 10.3 Time evolution of Langmuir wave spectral intensity IL (k) versus k⊥ vT e /ωpe and k vT e /ωpe . Panel (a) shows IL (k) at ωpe t = 100; panel (b) is for ωpe t = 500; panel (c) is for ωpe t = 1000; and the final panel (d) shows IL (k) at ωpe t = 2000 time step.

(Sturrock, 1964). Later, however, the issue was resolved in the context of finite physical dimension of the source size, which leads to the beam reformation via time-of-flight effects. In short, the problem related to the propagation of type III electrons has received extensive discussions in the literature. This includes the study of propagation of finite-sized type III electron source in homogeneous and inhomogeneous media (Sturrock, 1964; Zheleznyakov and Zaitsev, 1970b; Zaitsev et al., 1972; Papadopoulos et al., 1974; Takakura and Shibahashi, 1976; Magelssen and Smith, 1977; Goldstein et al., 1979; Smith et al., 1979; Grognard, 1982; Muschietti, 1990; Mel’nik et al., 1999; Kontar and P´ecseli, 2002; Foroutan et al., 2007; Ratcliffe et al., 2012; Voshchepynets and Krasnoselskikh, 2015). This problem is, however, beyond the scope of the present monograph. Instead, we focus on the detailed radiation generation mechanism. The Langmuir turbulence spectrum is shown in Figure 10.3. The portion of the 2D k space corresponding to k > 0 is for forward-propagating L mode, Ik+L , while the other half space with k < 0 belongs to the backward L mode, Ik−L . For

10.4 Plasma Emission by EM Weak Turbulence Process

285

Figure 10.4 Time evolution of ion-sound wave spectral intensity IS (k) versus k⊥ vT e /ωpe and k vT e /ωpe in the same format as Figure 10.3.

relatively early time, ωpe t = 100, the enhanced forward-propagating component (the primary L) can be seen to be excited, which is the result of initial bump-on-tail instability (panel a). For ωpe t = 500, the backscattering of the primary L mode into the oppositely traveling L mode, via a combined three-wave decay process and nonlinear scattering off ions, becomes visible. For ωpe t = 1000, the primary L mode begins to evolve into a semi-ark shape spectrum. The backscattered L mode was already generated in the form of ring spectrum even at ωpe t = 500. For the final time step ωpe t = 2000, the total L mode spectrum attains more and more ring-like feature, which was first noticed in earlier electrostatic weak turbulence calculations (Ziebell et al., 2008a,b). The numerical solution compares favorably with the simulated spectrum based upon the two-dimensional EM PIC methodology – see, e.g., figure 2 of the paper by Rhee et al. (2009a). Nonlinear decay processes that generate the backscattered L mode is accompanied by the production of ion-sound mode by decay instability, as indicated in Figure 10.4, which shows the time evolution of the S mode spectral intensity in the same format as Figure 10.3.

286

Applications of Electromagnetic Klimontovich Weak Turbulence Theory

Figure 10.5 Time evolution of the radiation spectral intensity IT (k) in the same format as Figure 10.3.

Figure 10.5 plots the transverse EM radiation (T mode) spectrum. In Figure 10.5d, the generation of fundamental/second-harmonic pair of emissions with frequency in the vicinity of ωpe and 2ωpe are clearly visible. The physics of electron-beam, Langmuir, and ion-sound turbulence processes, and also the radiation generation processes based upon reduced methods, have been investigated in detail (Ziebell et al., 2001, 2008a,b; Kontar and P´ecseli, 2002; Foroutan et al., 2007; Li et al., 2008a,b, 2009; Ratcliffe et al., 2012; Schmidt and Cairns, 2012a,b, 2014), but these works employ various simplifying assumptions. In contrast, the complete weak turbulence formalism contains all the terms, and hence is suitable for a detailed analysis (Ziebell et al., 2014a, 2015). According to (10.34), the fundamental emission should be dictated by the threewave process involving L mode decaying into T and S mode and the scattering process involving the beating of L and T mode mediated by the particles. Of the

10.4 Plasma Emission by EM Weak Turbulence Process

287

two processes, the decay mechanism (L + S → T ) is governed by the overall nonlinear coupling coefficient, (k × k )2 ∝ sin2 ϑ. 2 2  k k

(10.38)

If we associate k and ωkL with the Langmuir wave, and k and ωkT with the transverse S radiation (which stems from the resonance condition σ ωkT − σ  ωkL − σ  ωk−k  = 0),  then if we allow k vector to lie along the direction of beam propagation, it can be seen that the fundamental emission along the direction specified by ϑ = 0 should be prohibited. Of course, the L mode spectrum has a broad range of angles in the present case of two-dimensional physics so that the fundamental emission by the decay process (L + S → T ) is allowed even for ϑ = 0. Nevertheless, (10.38) predicts that the overall F emission by the decay mechanism should possess a dipolar radiation angular distribution. Figure 10.5a bears such a feature out. Note that the scattering process is also dictated by the same nonlinear coupling coefficient (10.38), so that the fundamental plasma emission having the dipolar structure with maximum radiation intensity in directions orthogonal to the direction of beam propagation direction is consistent. For the second-harmonic emission, the basic emission mechanism is dictated by the coupling coefficient associated with the decay process L + L → T , 2 2 2 (k × k )2  2  2 2  − |k − k | ∼ sin ϑ k − 2kk cos ϑ ∝ sin2 ϑ cos2 ϑ, k 2 k2k (10.39) where we identify k with the transverse mode and k  with the Langmuir mode, so that one may approximate k k  . According to this picture, the second-harmonic (ω ∼ 2ωpe ) emission should possess a quadrupole angular distribution. Indeed, Figure 10.5 shows that the second-harmonic emission peaks at 45◦ angle. The quadrupole emission is not symmetric with respect to k axis, featuring stronger forward emission, stronger than backward emission. The third-harmonic emission is not apparent for the beam speed Vb /vT e = 6, but for higher beam speed it is possible. The third- or higher-harmonic emission takes place via decay process L + T → T . The coupling coefficient is given by   (k · k )2 ∼ 1 + cos2 ϑ. (10.40) 1 + 2 2 k k As this indicates, the third-harmonic plasma emission should be largely isotropic. In the solar type III bursts third harmonic is only rarely detected (Takakura and Yousef, 1974; Brazhenko et al., 2012). There are reports of occasional third harmonic of

288

Applications of Electromagnetic Klimontovich Weak Turbulence Theory

type II bursts in the literature (Zlotnik et al., 1998). The theory of 3H emission was first developed by Zheleznyakov and Zlotnik (1974), and also discussed in the paper trilogy by Cairns (1987a,b,c) – see also the paper by Kliem et al. (1992). The present theory allows for F , 2H , 3H as well as even higher-harmonic emissions, and as such it generalizes previous theories. In the paper by Ziebell et al. (2015), further analysis is carried out by turning certain terms in the T mode wave kinetic equation on or off artificially, thereby identifying the role of each term. By doing so, they were able to pinpoint the exact radiation generation mechanism for each harmonic mode. This is the advantage of the theoretical approach to study a complicated nonlinear problem in plasmas, as opposed to direct numerical simulation. Simulations are supposed to be more rigorous, but it is not so straightforward to interpret the result since all physical processes take place simultaneously, and it is not always possible to single out certain processes and turn them on or off.

Epilogue

In this book the author attempted to present a detailed discussion regarding the weak turbulence methodology for unmagnetized plasmas, starting from a simple Vlasov– Poisson system, moving on to the Klimontovich–Poisson system of equations. Then, the discussion returned to the Vlasov equation, but this time full Maxwell’s equation is employed. Finally, the fully electromagnetic problem is extended to the Klimontovich framework, thus concluding the mathematical formalism. In all these discussions, however, the effects of ambient magnetic field were not considered, which limited the applicability in an appreciable way. At present, the weak turbulence theory for magnetized plasmas is not at a completely satisfactory state. In the literature, some early works attempted to extend the weak turbulence theory to plasmas immersed in a constant applied magnetic field. Among them are the works by Tsytovich and Shvartsburg (1966), Melrose and Sy (1972), and Pustovalov and Silin (1975), for instance. Among these, the work by Pustovalov and Silin (1975), for which a monumental effort must have been devoted, deserves a close look. However, their result is very formal, and the ramifications thereof are not entirely evident. In short, these early works are preliminary, and further developments are necessary. If one makes simplifying assumptions, such as electrostatic approximation (Ellis and Porkolab, 1968; Porkolab and Chang, 1978; Mikhajlenko and Stepanov, 1981), or drift kinetic approximation, which amounts to ignoring cyclotron interactions (Ganguli et al., 2010; Rudakov et al., 2011; Crabtree et al., 2012; Mithaiwala et al., 2012), then one may find in the literature more concrete works that employ weak turbulence systemology in order to address complex plasma phenomena. The author of this book also made some efforts to extend the existing method of studying the weak plasma turbulence to include the effects of ambient magnetic field, but thus far, the discussions are limited to turbulence propagating along the magnetic field (Yoon, 2015a,b) or exactly perpendicular to it (Yoon, 2015c) – for a discussion of nonlinear

289

290

Epilogue

wave interactions taking place in perpendicular directions, see also the work by Brodin and Stenflo (2015). For turbulent processes taking place in macroscopic dimensions, plasmas immersed in a background magnetic field may be regarded as a magnetized electrofluid so that mathematical and conceptual tools of the magnetohydrodynamic (MHD) turbulence theory may be brought forth (Biskamp, 2003; Bruno and Carbone, 2016). However, for kinetic scale turbulent phenomena, the tools and methodology of weak turbulence theory are called for, but at present, the standard framework of plasma weak turbulence theory has not been fully implemented for magnetized plasmas. It is the author’s wish that not only the readers may find the material presented in this book useful and educational but also the content of this book may form a foundational structure based upon which future fully developed weak turbulence theory for magnetized plasmas may be constructed.

Appendix A Time Irreversible Small Amplitude Perturbations

Linearized Vlasov equation for electrostatic perturbation in unmagnetized plasmas is given by   ∂ ea fa (v) ∂ +v· δfa (r,v,t) + = 0, δE(r,t) · ∂t ∂r ma ∂v   ∂ · δE(r,t) = 4π ea dv δfa (r,v,t). (A.1) ∂r a Applying the spatial Fourier transformation to the perturbed Vlasov equation, we have   ea ∂fa (v) ∂ + ik · v δfka (v,t) + = 0, δEk (t) · ∂t ma ∂v   ik · δEk (t) = 4π ea dv δfka (v,t).

(A.2)

a

The next logical step is to take a similar integral transformation in time. However, physical phenomena do not operate in infinite domain, −∞ < t < ∞, but rather the problem always has an initial time t = 0, so the domain of time is 0 < t < ∞. This means that we are only interested in forward time, or, equivalently, causal processes. Consequently, the type of integral transformation of a temporal function g(t) must be of the form  ∞ dt g(t) eiωt . (A.3) gω = (2π )−1 0

The question is whether this integral transformation is well defined in the mathematical sense. As a concrete example, take, for instance, an exponential function g(t) = eat .

(A.4)

The transformed function is gω =

1 2π



 e(a+iω) t 1 − lim . a + iω t→∞ a + iω

(A.5)

If a < 0, then the limit vanishes, but if a > 0, then the limit diverges; hence, the integral transformation is not defined. Nevertheless, one may salvage the situation if we assume that ω is complex: ω = Re ω + i Im ω,

(A.6)

291

292

Time Irreversible Small Amplitude Perturbations

which is, of course, the Laplace transformation, and this is the approach that Landau took (Landau, 1946) in contrast to Vlasov (1938), who restricted the discussion to only real ω. Returning to the toy problem (A.4), as long as Im ω > a, the integral transformation is well defined, and the result is gω =

1 1 . 2π a + iω

(A.7)

From this, a general condition for the validity of integral transformation is that, for any function g(t), integral transformation (A.3) is defined as long as ω is defined above all singular points associated with gω in complex ω space. This leads to the proper definition of inverse transformation. The naive definition  ∞ dω gω e−iωt g(t) = −∞

is inappropriate since ω must be higher than all singularities associated with gω in complex plane. Consequently, we take the infinite integration along a path L that is taken above the highest singular point associated with gω in complex plane,  g(t) = dω gω e−iωt . (A.8) L

The spectral transformation (A.3) and its inverse (A.8) constitute the Laplace transformation. The path L does not have to be a straight horizontal line – see Figure A.1 – as long as it is above the singular points. For instance, we may take a deformed integration contour as shown on the right of Figure A.1. Applying the Laplace transformation to (A.2) under the assumption that limt→∞ δfka (v,t) = 0, we have ∂fa (v) 1 a (v) + ea δE δf a (v,0) − i (ω − k · v) δfk,ω = 0. k,ω · 2π k ma ∂v

(A.9)

One may also take the Laplace transformation of Poisson’s equation, and take δEk,ω = (k/k) δEk,ω , in order to arrive at  2ea  δf a (v,0) dv k , (A.10) (k,ω) δEk,ω = − k ω−k·v a Im w

Im w L L Re w

X X X

Re w

X X

X

Singularities associated with g(w)

X

X

Singularities associated with g(w)

Figure A.1 Integration paths for inverse Laplace transformation (A.8). The left panel shows a straight horizontal path, but it can be deformed as shown on the right, as long as the path is above the singular points.

Time Irreversible Small Amplitude Perturbations

293

where (k,ω) is the longitudinal dielectric response function (1.55). For the sake of convenience, let us assume that k = zˆ k and let us work with the reduced distribution function Fˆa (vz ) = 2π 0∞ dv⊥ v⊥ Fa (v), where fa = nFa . Then (A.10) becomes δEkω =

S(k,ω) , (k,ω)

(A.11)

where S(k,ω) = −

 2ea  k

a

(k,ω) = 1 +

dv

δfk (v,0) , ω − kvz

2  ∞  ωpa a

k

−∞

dvz

d Fˆa (vz )/dvz . ω − kvz

(A.12)

We are interested in the temporal behavior of perturbed electric field, which can be discussed by considering the inverse Laplace transformation of perturbed electric field:  S(k,ω) , (A.13) dω e−iωt δEk (t) = (k,ω) L where contour L is defined above the highest singularity associated with the integrand S(k,ω)/ (k,ω). The singularity may come from any possible divergences associated with S(k,ω) or from the zeros of denominator (k,ω). Let us focus on the zeros of (k,ω), which are none other than the dispersion relation satisfying (k,ω) = 0, or ω = ωk + iγk . We also assume that the zeros of (k,ω) are made of poles rather than branch cuts. The task is to perform the integral along the path L. The question is how to perform such an integral. In general, Cauchy’s residue theorem cannot be applied in analytically closed form since the general forms of S(k,ω) and (k,ω) do not lend themselves to such a mathematical manipulation, and generally there is no guarantee that integrals converge asymptotically. However, we may carry out the contour integration in an approximate (that is, in an asymptotic) sense, by employing the analytic continuation (Landau, 1946). The idea is to deform the contour L down along the imaginary axis, as shown on the right-hand side of Figure A.2, past the real axis, but avoiding the singularities by encircling around each pole. Suppose that we deform the contour down the imaginary ω axis sufficiently far and close the contour in an asymptotic sense. Let us denote the outer contour  and the closed complete contour C,    S(k,ω) S(k,ω) S(k,ω) = + , (A.14) dω e−iωt dω e−iωt dω e−iωt (k,ω) (k,ω) (k,ω) C L  Im w

Im w L

X X

Re w

X

X

X X

X

X

L

Figure A.2 Analytic continuation of integration path avoiding the singularities.

294

Time Irreversible Small Amplitude Perturbations Im w

Re w

X X X

X

L

Γ

Figure A.3 Deformed contour L, including the encircled poles, cancelled out vertical paths connecting the poles, and outer contour . Complete contour C thus lends itself to Cauchy’s residue theorem. Im w

Re w

X X X

X

Figure A.4 In the time asymptotic sense, only the least damped mode will be important so that the residue contribution from that mode needs to be included in the inverse Laplace transformation.

as shown in Figure A.3. Each circle around the pole makes the residue contribution in the sense of Cauchy’s theorem, but vertical paths cancel out. In the time asymptotic sense, t 1, the contribution from the least damped mode with lowest value of |γk | in magnitude will dominate (here we assume damping with γk ≤ 0), as shown in Figure A.4. Suppose that the most important mode, that is, the least damped mode, is given by

ω = ωk + iγk = ωk − i |γk | = ωs .

(A.15)

Time Irreversible Small Amplitude Perturbations

295

Then, we may employ the Taylor series expansion, and, by invoking the residue theorem, obtain   S(k,ω) S(k,ω) = dω e−iωt dω e−iωt (k,ω) (ω − ωs ) ∂ (k,ωs )/∂ωs C C = −2π i

S(k,ωk + iγk ) e−iωk t+γk t , ∂ (k,ωk + iγk )/∂(ωk + iγk )

(A.16)

where we have used the fact that (k,ωk + iγk ) = (k,ωs ) = 0 by definition. Here, the minus sign is because the contour loops around in the clockwise sense. Making use of (A.16), we have  S(k,ωk + iγk ) e−iωk t+γk t S(k,ω) = −2π i dω e−iωt (k,ω) ∂ (k,ωk + iγk )/∂(ωk + iγk ) L  S(k,ω) . (A.17) − dω e−iωt (k,ω)  As noted already, we generally do not know how to evaluate the integration around the path . However, in the asymptotic sense, if we push the contour down infinitely low along the imaginary axis, then, in the end, we are not concerned with the contribution from  segment, so the result is  S(k,ωk + iγk ) e−iωk t−|γk | t S(k,ω) ≈ −2π i . (A.18) dω e−iωt (k,ω) ∂ (k,ωk + iγk )/∂(ωk + iγk ) L From this it is seen that the initial fluctuation δfk (v,0) exponentially damps away, δfk (v,0) e−|γk | t ,

(A.19)

over time in the asymptotic sense. This is the well known Landau damping, which represents dissipation in collisionless plasmas. In general, dissipation is associated with collisions and irreversibility, and since collisionless plasmas are supposed to be dissipation less, the basic system of equations are time reversible. That is, Vlasov and Maxwell’s equations remain unchanged if we change t to −t. From this, it might appear that the Landau damping is counter-intuitive and paradoxical. However, one should keep in mind that by employing the Laplace transformation, one is only concerned with physically meaningful causal solutions. Had we chosen the anti-causal Laplace transformation (with 0 dt), one would have had an exponentially increasing solution for positive time. In short, integral −∞ Landau damping can be time reversible. It is just that Landau restricted his discussion to physically meaningful causal solution.

Appendix B Resonant Velocity Integral

We pick up where we left off regarding the inverse Laplace transformation,  δEk (t) =

dω e−iωt L

S(k,ω) . (k,ω)

(B.1)

In (B.1), the denominator is made of (k,ω), which is defined with the velocity integral (k,ω) = 0 = 1 −

2  ∞  ωpa a

k2

−∞

dvz

d Fˆa /dvz . vz − ω/k

(B.2)

According to the causal Laplace transformation, the quantity ω that appears in the integrand in (B.1) is defined along the contour L, so that it lies above the highest root. Suppose that all roots have γk ≤ 0. Suppose also that the highest root is the weakly damped mode with γk close to 0. This means that the quantity ω that appears within the definition of (k,ω) must be above that weakly damped root. The implication is that while the velocity integration is along real vz , the pole ω/k lies above imaginary vz axis – Figure B.1. As the contour L is deformed (analytic continuation), however, the pole vz = ω/k, which originally lay above the real vz axis, will now start to move down along imaginary vz axis, until it crosses the real axis. Except for the loop around each pole, ω generally lies below the real axis, and, in fact, in the asymptotic sense, it has a very large imaginary part, implying that vz = ω/k is also associated with large imaginary part. Accordingly, the velocity integral path must be deformed in such a way as to ensure that the pole vz = ω/k always stays above the velocity integration path, as shown in Figure B.2. If γk is infinitesimally small and negative (γk = −0), the velocity integration can be broken into the principal part and the half residue,

1 ωk  1 =P + iπ δ vz − , vz − ωk /k vz − ωk /k k

(B.3)

as shown in Figure B.3. The relationship in (B.3) is known as the Plemelj formula, which is more generally expressed as 1 1 = P ∓ iπ δ(x). x ± i0 x

296

(B.4)

Resonant Velocity Integral Im w

297 Im vz

L

w /k

X Re vz

Re w

X X X

X

Figure B.1 The Laplace contour L, taken above all singularities associated with the integrands, and the corresponding velocity integral path, which lies below the pole ω/k with an infinitesimally positive imaginary part. Im w

Im vz

w /k X Re w

X

Re vz

X X

X

L

Figure B.2 Deformed Laplace contour L and the correspondingly deformed velocity integral path. Im vz

(w k – i½g k½)/k x

Im vz

=

(w k – i½g k½)/k x

Re vz

Im vz

Re vz

+

(w k – i½g k½)/k x

Re vz

Figure B.3 Velocity integration for weakly damped mode (upper panel), which can be broken into the principal part plus one-half residue contribution.

298

Resonant Velocity Integral

In adopting the prescription (B.3) for treating the resonant velocity integration, we should keep in mind that the sign of γk can be either positive or negative depending on whether one is interested in instability or damping, but before one actually computes γk , one must always apply the Plemelj formula for positive sign associated with ω. This is the origin of the factor +i0 in the resonant denominator (1.42), 1 . ω − k · v + i0

(B.5)

Appendix C Nonlinear Dispersion Relations

In this appendix, we discuss the consequence of retaining the nonlinear correction terms in the real part of nonlinear spectral balance equation. Two main results come out as consequences of retaining the nonlinear correction terms. One is the existence of harmonics of plasma frequency, which are nonlinear eigenmodes of the system, and the other is the nonlinear frequency shift of linear eigenmodes.

C.1 Nonlinear Eigenmodes An interesting phenomenon unveiled through computer simulations (Klimas, 1983, 1990; Umeda et al., 2003) is that the Langmuir wave instability driven by a weak electron beam is accompanied by the generation of harmonic modes of fundamental plasma frequency, but the harmonics of ωpe are not the normal modes of a plasma according to the textbook theory of small amplitude plasma waves; hence their existence cannot be discussed on the basis of linear Vlasov theory, nor can they be explained in terms of the plasma emission scenario of Chapter 10, as the excited modes are longitudinal. On the other hand, such a phenomenon can be explained by a theory of nonlinear dispersion relation. Let us focus on the first harmonic Langmuir wave with frequency close to twice the plasma oscillation frequency. Let us denote the new solution by ω = ±ωkN .

(C.1)

Since the nonlinear eigenmode is presupposed to possess a wave frequency near 2ωpe , while ωkL ∼ ωpe , and ωkN − ωkL ∼ ωpe , all three waves possess the fast-wave conditions, |ωkN | kvT e, |ωkL | k  vT e, |ωkN − ωkL | |k − k | vT e .

(C.2)

This means that the second-order susceptibility is dominated by the electron response, which can be approximated by (see 1.128) χ (2) (k, ± ωkL |k − k, ± ωkN ∓ ωkL ) 2 ωpe 1 i e =  N N L L 2 me ω ω  (ω − ω  ) k k |k − k | k k k k  2  |k − k |2 k2  k    × k · (k − k ) + N k · k + N k · (k − k ) ωkL ωk − ωkL ωk ≈

ie 2 2me ωpe

(C.3)

2k 2 k · (k − k ) + 2|k − k |2 k · k + k 2 k · (k − k ) . 4 k k  |k − k |

299

300

Nonlinear Dispersion Relations

Let us introduce the following expression for notational simplicity: {k 2 k · (k − k ) + 2k 2 k · (k − k ) + 2|k − k |2 (k · k )}2 . 16k 2 k 2 |k − k |2

ak,k =

(C.4)

The nonlinear dispersion equation (1.90), upon ignoring the off diagonal terms involving δE 2 k,ω δE 2 k−k,ω−ω and also ignoring the third-order nonlinear response, is given by     0 = Re (k,ω) + 2 dk dω {χ (2) (k,ω |k − k,ω − ω )}2  ×

δE 2 k,ω δE 2 k−k,ω−ω +   (k ,ω ) (k − k,ω − ω )

 δE 2 k,ω .

(C.5)

We treat ω and ω−ω as belonging to the fundamental Langmuir wave branch, while treating ω as the harmonic Langmuir wave. Expressing both L and N mode wave spectrum in the form of eigenmode intensities, δE 2 k,ω = IkL δ(ω − ωkL ), L  L δE 2 k−k,ω−ω = Ik−k  δ(ω − ω − ωk−k ),

δE 2 k,ω = IkN δ(ω − ωkN ),

(C.6)

where only forward propagation is considered without loss of generality, and, making a trivial change of integral variables, we have    ak,k IkL e2 IkN . dk 0 = Re (k,ωkN ) − 2 4 me ωpe (k − k,ωkN − ωkL )

(C.7)

Making use of the following – see (1.115), 2  ωpe 2 λ2 1 + 3k De N2 , ωkN2 ωk L2 ω  Re (k − k,ωkN − ωkL ) = 1 − N k−k L , (ωk − ωk )2

Re (k,ωkN ) = 1 −

2 ωpe



(C.8)

2 (1 + 3k 2 λ2 ), ωL2 = ω2 (1 + 3k 2 λ2 ), and ωL2 = ω2 (1 + 3|k − k |2 λ2 ), where ωkL2 = ωpe pe pe De De De k k−k we obtain

 2  ωpe 2 λ2 1 + 3k De N2 ωkN2 ωk  ak,k (ωkN − ωkL )2 IkL e2 − 2 4 dk . L2 me ωpe (ωkN − ωkL )2 − ωk−k 

0=1−

2 ωpe

(C.9)

If we ignore thermal effects then we have 0 = ωkN + ωpe −

ωkN (ωkN − ωpe ) e2 4 (ωkN − 2ωpe ) m2e ωpe



dk ak,k IkL .

(C.10)

C.1 Nonlinear Eigenmodes

301

Upon approximating ωkN ∼ 2ωpe everywhere except in the denominator, we have 

2e2

ωkN = 2ωpe +

3 3m2e ωpe

dk ak,k IkL .

(C.11)

This is the desired (first) harmonic Langmuir wave dispersion relation, which has a peculiar property of persisting even in the limit of IkL → 0. One can extend the discussion to higher harmonic Langmuir waves. Let us designate the harmonic eigenmodes including the fundamental and first harmonic mode by ω = ωkLn,

n = 1,2,3, . . . .

(C.12)

Then, the wave intensity is given by 

δE 2 kω =

IkLn δ(ω − ωkLn ).

(C.13)

n=1,2,3,...

One may insert (C.13) to (C.5). In direct generalization of (C.3), we may evaluate the second-order susceptibility, χ (2) (k,ωk

L(n−1)

|k − k,ω − ωk

L(n−1)

#2

≈−

)

e2 4 4m2e ωpe

n , ak,k 

1 n ak,k { (n − 1) k 2 [k · (k − k )]  = n4 (n − 1)4 k 2 k 2 |k − k |2 + n k 2 [k · (k − k )] + n(n − 1) |k − k |2 (k · k ) }2 . (C.14) This leads to  0 = Re (k,ω) −



e2 4 m2e ωpe



L(n−1)

dk

n I ak,k  k

L(n−1)

(k − k,ω − ωk

.

(C.15)

)

Approximately writing the quantities Re (k,ωkLn ) and Re (k − k,ωkLn − ωk

L(n−1)

Re (k,ωkLn ) ≈ 1 − Re (k − k,ωkLn − ωk

L(n−1)

)≈



2 ωpe

(ωkLn )2

1 + 3k 2 λ2De

2 ωpe

(ωkLn )2

L(n−1) 2 L1 )2 ) − (ωk−k 

(ωkLn − ωk

L(n−1) 2 )

(ωkLn − ωk

) as

 ,

(C.16)

,

we have 0=1−

2 ωpe

(ωkLn )2

e2 − 2 4 me ωpe



 1 + 3k 2 λ2De dk

2 ωpe



(ωkLn )2

L(n−1) 2 L(n−1) ) Ik . L(n−1) 2 Ln L1 )2 (ωk − ωk ) − (ωk−k 

n (ωLn − ω ak,k  k k

(C.17)

302

Nonlinear Dispersion Relations

Ignoring thermal effects, we have 2 (ωkLn )2 − ωpe

0=

(C.18)

(ωkLn )2 −

[ωkLn − (n − 1)ωpe ]2



e2

2 m2 ω4 [ωkLn − (n − 1)ωpe ]2 − ωpe e pe

n I L(n−1) . dk ak,k  k

Replacing ωkLn by nωpe everywhere except in the denominator associated with the second term on the right-hand side, we readily obtain (Yoon et al., 2003a) ωkLn = nωpe +

e2

n2 4 n2 − 1 2m2e ωpe



n I dk ak,k  k

L(n−1)

,

(C.19)

for n ≥ 2. This is the desired multiple harmonic Langmuir wave dispersion relation when thermal corrections are ignored. For n = 2, we recover (C.11). In (C.19), the nonlinear dispersion relation depends on the harmonic mode intensities, IkLn . This implies that the discussion of harmonic Langmuir waves is incomplete without computing the intensities of these modes. Gaelzer et al. (2003) formulated the quasilinear theory of harmonic Langmuir waves with which the wave intensities can be calculated. Interested readers may refer to that paper.

C.2 Nonlinear Frequency Shift We now discuss the classical problem of nonlinear frequency shift. For this situation, we only consider the linear eigenmodes and the frequency shifts thereof when the underlying plasma is weakly turbulent. We decompose intensity by considering only linear eigenmodes in the manner $ the spectral $ of (1.95), δE 2 kω = σ =±1 α=L,S Ikσ α δ(ω − σ ωkα ). Inserting this to (1.90), we have 0=

  σ =±1 α



Re (k,σ ωkα ) Ikα δ(ω − σ ωkα )

dk dω

+ 2Re





γ   γ 2  { χ (2) (k,σ ωkα − σ  ωk−k  |k − k ,σ ωk−k ) } γ

(k,σ ωkα − σ  ωk−k )

σ,σ  =±1 α,γ σ  γ

× Ikσ α Ik−k δ(ω − σ ωkα ) δ(ω − σ ωkα + σ  ωk−k ) +



γ

 { χ (2) (k,σ  ωβ |k − k,σ ωα − σ  ωβ ) }2 k k k

σ,σ  =±1 α,β σ β

δ(ω − σ ωkα ) δ(ω − σ  ωk ) β

× Ikσ α Ik 



β

(k − k,σ ωkα − σ  ωk )

γ 2  | χ (2) (k,σ  ωkβ |k − k,σ  ωk−k ) |

σ ,σ  =±1 β,γ σ  β σ  γ

γ

β

∗ (k,σ  ωk + σ  ωk−k )

× Ik Ik−k δ(ω − σ  ωk ) δ(ω − σ  ωk − σ  ωk−k )   β β χ¯ (3) (k,σ  ωk | − k, − σ  ωk |k,σ ωkα ) − σ,σ  =±1 α,β

β

β



σ β σ α β Ik δ(ω − σ ωkα ) δ(ω − σ  ωk ) .

× Ik

γ

(C.20)

C.2 Nonlinear Frequency Shift

303

To the leading order, { χ (2) (k,ω |k − k,ω − ω ) }2 ≈ −| χ (2) (k,ω |k − k,ω − ω ) |2 . Consequently, the real part of second-order nonlinear terms comes from the principal parts of inverse dielectric functions. The principal part of the third term on the right-hand side within the second-order nonlinear term becomes unimportant if the condition ω = σ ωkα is satisfied, since the principal part excludes

the point σ  ωk + σ  ωk−k = ω, by definition, when ω is equal to σ ωkα . The first and second terms within the second-order nonlinear correction terms are equivalent, which can be seen by a simple interchange of dummy integral and summation variables. As a consequence, (C.20) reduces to       ) I σ β I σ α, dk Aσ,σ (k,k 0 = Re (k,σ ωkα ) +  k α,β k γ

β

σ  =±1 β  α (2)   β   β 2   ) = 2 P 2{ χ (k ,σ ωk |k − k ,σ ωk − σ ωk ) } (k,k Aσ,σ α,β β (k − k,σ ωkα − σ  ωk )

(C.21)

 β β −χ¯ (3) (k,σ  ωk | − k, − σ  ωk |k,σ ωkα ) .

C.2.1 Nonlinear Frequency Shift of Langmuir Waves For Langmuir mode, α = L, we have 0 = Re (k,σ ωkL )       ) I σ  L + Aσ,σ  (k,k ) I σ  S . + Re dk Aσ,σ (k,k   k L,L L,S k

(C.22)

σ  =±1

Making note of the inequalities, ωkL kvT e , ωkL k  vT e , ωkL kvT i , and ωkL k  vT i , we may write χ¯ (3) (k,σ  ωkL | − k, − σ  ωkL |k,σ ωkL ) 1 e2 (k · k )2 |k − k |2 χe (k − k,σ ωkL − σ  ωkL ), 2 m2e k 2 k 2 (σ ωL )3 (σ  ωL ) k k { χ (2) (k,σ  ωkL |k − k,σ ωkL − σ  ωkL ) }2 ≈−

≈−

1 e2 (k · k )2 |k − k |2 { χe (k − k,σ ωkL − σ  ωkL ) }2 . 4 m2e k 2 k 2 ωL2 ωL2  k k

(C.23)

Note that the electrons dominate nonlinear responses since the lighter and mobile electrons respond to nonlinear perturbations much more rapidly than the sluggish and heavier ions. Denoting (k − k,σ ωkL − σ  ωkL ) = 1 + χe (k − k,σ ωkL − σ  ωkL ) +χi (k − k,σ ωkL − σ  ωkL ),

(C.24)

we have 

 e2 (k · k )2  |k − k |2  σ L Re Aσ,σ L,L (k,k ) Ik = − 2 k 2 k 2 (σ ωkL )3 (ωkL )2 me σ  =±1 σ  =±1    χe2 × Re σ ωkL − σ  ωkL χe Ikσ L . 1 + χe + χi

(C.25)

304

Nonlinear Dispersion Relations

In (C.25), χe and χi are shorthand notations for the full expression given by (C.24). From this, it can be seen that when the enhanced turbulence is primarily of Langmuir waves, the appropriate nonlinear dispersion equation is given by  e2 (k · k )2 |k − k |2 dk 0 = Re (k,σ ωkL ) − 2 4 k 2 k 2 me ωpe    ωL  χe2 × Re − σ σ  kL χe Ikσ L . (C.26) 1 + χ + χ ω e i k σ  =±1 If σ and σ  are of the same sign, then we may approximate Re χe ≈

1 |k − k |2 λ2De

Re χi ≈ −

,

2 ωpi

(ωkL − ωkL )2

≈−

me 1 . mi [(3/2)(k 2 − k 2 ) λ2 ]2 De

(C.27)

We further approximate ωkL ωkL

≈1−

3 2 (k − k 2 ) λ2De . 2

(C.28)

Upon ignoring the ion response, we have  Re

ωL χe2 − kL χe 1 + χe + χi ωk

 ≈

3 k 2 − k 2 − 1. 2 |k − k |2

(C.29)

On the other hand, if σ and σ  are of the opposite sign, then we have 3 Re χe ≈ , 4

Re χi ≈ −

me , 2mi

(C.30)

from which we obtain  Re

ωL χe2 + kL χe 1 + χe + χi ωk

 ≈

15 . 14

(C.31)

As a result, (C.26) reduces to Re (k,σ ωkL ) =



(k · k )2 4π nT ˆ e k 2 k 2  15 L . + |k − k |2 Ik−σ  14 λ2De

dk



 3 2 (k − k 2 ) − |k − k |2 Ikσ L 2

This leads to the modified Langmuir wave dispersion relation   3 ωkL = ωpe 1 + k 2 λ2De + δωkL, 2 2  3ω λ (k · k )2 pe De dk 2 2 δωkL = 16π nˆ e Te k k    2 5 2 2  |2 I −L . |k − k × k − k − |k − k |2 Ik+L +  k 3 7

(C.32)

(C.33)

C.2 Nonlinear Frequency Shift

305

The quantity δωkL is the nonlinear frequency shift of Langmuir wave caused by Langmuir self turbulence. Next, we consider the contribution to the nonlinear frequency shift from ion-sound waves. For this case, the appropriate inequalities are ωkL k vT e , ωkS k  vT e , |ωkL − ωkS | |k − k | vT e , ωkL k vT i , ωkS > k  vT i , and |ωkL − ωkS | |k − k | vT i . This leads to the following evaluation of nonlinear responses: χ (2) (k,σ  ωkS |k − k,σ ωkL − σ  ωkS )

k · (k − k ) i e k χe (k,σ  ωkS ), 2 me σ ωL (σ ωL − σ  ωS ) k |k − k | k k k χ¯ (3) (k,σ  ωkS | − k, − σ  ωkS |k,σ ωkL )   1 k · (k − k ) e2 k · k 2k 2 χe (k,σ  ωkS ). ≈− + 2(σ ωkL )3 m2e k 2 σ ωkL σ ωkL − σ  ωkS ≈−

(C.34)

If we further approximate Re χe (k,σ  ωkS ) ≈

1 k 2 λ2De

(C.35)

,

then the nonlinear dispersion equation for Langmuir waves subject to ion-sound turbulence is   e2 1 dk Re Re (k,σ ωkL ) = 2 4 2 λ2 me ωpe k De σ  =±1  2  2 1 1 k [k · (k − k )] × P k 2 |k − k |2 k 2 λ2De (k − k,σ ωkL − σ  ωkS )   (k · k )(3k 2 − k · k ) − Ikσ S . (C.36) k2 In (C.36), the term associated with the denominator, | (k − k,σ ωkL − σ  ωkS )|2 , is the dominant term, which simplifies (C.36). Upon replacing σ ωkL by a generic ω, we obtain Re (k,ω) =

1 4π nT ˆ e ×P



dk Re

 [k · (k − k )]2 k 2 |k − k |2 

σ =±1



Ikσ S

1

(k − k,ω − σ  ωkS ) k 2 λ2De

.

(C.37)

Further approximating the denominator, Re (k − k,ω − σ  ωkS ) ≈

L2 ω2 − ωk−k 

ω2

,

(C.38)

we have 0=1−

ωkL ω

 −

dk βk,k

ωpe L ω − ωk−k 

, 

βk,k =

 [k · (k − k )]2 I σ S 1 k . 16π nT ˆ e  k 2 |k − k |2 k 2 λ2De σ =±1

(C.39)

306

Nonlinear Dispersion Relations

One may iteratively solve this equation to obtain the Langmuir wave dispersion relation subject to the ion sound turbulence:   3 (C.40) ωkL = ωpe 1 + k 2 λ2De + δωkL, 2  1/2 dk βk,k . δωkL = ωpe The nonlinear frequency shift of Langmuir waves, δωkL , given above, is owing to the ion sound turbulence, which may be compared with (C.33), where it arises due to Langmuir self turbulence.  In the expression for βk,k , the quantity Ikσ S /(k 2 λ2De ) may appear to diverge at first sight. However, if we define “plasmon” number density by – see (2.20) Nkσ α =

Ikσ α , α h¯ (σ ωk ) [∂Re (k,σ ωkα )/∂(σ ωkα )]

(C.41)

where h¯ is the Planck constant, and make use of (C.8), specifically, 1

≈ μk ∂Re (k,σ ωkS )/∂(σ ωkS )

(σ ωkL ) 2

,

(C.42)

where μk is defined in (C.8), then it is more convenient to absorb μk into the definition of the ion-sound wave spectrum, IσS Ikσ S → k . μk

(C.43)

Then in the new scheme, the quantity βk,k is well behaved mathematically.

C.2.2 Nonlinear Frequency Shift of Ion-Sound Waves Let us consider the ion-sound mode, α = S, in (C.21): 0 = Re (k,σ ωkS )       ) I σ  L + Aσ,σ  (k,k ) I σ  S . +Re dk Aσ,σ (k,k   k S,L S,S k

(C.44)

σ  =±1

For this case, we have the following inequalities: ωkS k vT e,

ωkL k  vT e,

ωkS > k vT i ,

ωkL k  vT i ,

|ωkS − ωkL | |k − k | vT e, |ωkS − ωkL | |k − k | vT i ,

ωkS k vT e, ωkS k  vT e, |ωkS − ωkS | |k − k | vT e, ωkS > k vT i , ωkS > k  vT i , |ωkS − ωkS | > |k − k | vT i .

(C.45)

Note that ion-sound waves can exist only in a plasma where Te Ti . Therefore, ωkS ∼ kcS , ωkS ∼ k  cS , ωkS − ωkS ∼ (k − k  ) cS , and cS /vT i ∼ (Te /Ti )1/2 1.

C.2 Nonlinear Frequency Shift

307 σ,σ 

Of the various nonlinear response functions within the coefficient AS,L (k,k ), the second-order electron response is the most important one. Others can be ignored. χe (k,σ  ωkL |k − k,σ ωkS − σ  ωkL ) (2)

≈−

2 ωpe k · (k − k ) 2 i e . 2 me (σ  ωL )(σ ωS − σ  ωL ) k k  |k − k | v 2 Te k k k

(C.46)

Consequently, we have 2   2  1  ) ≈ − e [k · (k − k )] P (k,k Aσ,σ . S,L S 2 2 2  2 Te k k |k − k | (σ ωk − σ  ωkL )

(C.47)



 Among the nonlinear response functions within the coefficient Aσ,σ S,S (k,k ), while it is true that in general the third-order susceptibility can be ignored, the ion second-order response can be of the same order as the electron term,

χe (k,σ  ωkS |k − k,σ ωkS − σ  ωkS ) ≈ (2)

2 ωpe ie 1 , Te k k  |k − k | v 2

Te (2)   S  S  S χi (k ,σ ωk |k − k ,σ ωk − σ ωk ) 2 ωpi i e 1 ≈− 2 mi (σ ωS )(σ  ωS )(σ ωS − σ  ωS ) k k  |k − k | k k k k   2 2 k |k − k |2 k     k · (k − k ) + k · (k − k ) + k·k . × σ ωkS σ  ωkS σ ωkS − σ  ωkS

(C.48)

Combining both electron and ion responses, we have χ (2) (k,σ  ωkS |k − k,σ ωkS − σ  ωkS )  2   |k − k |2 ie me ωpe (k · k ) . 1 + = −σ σ  Te 2Te k 2 k 2 |k − k | (σ k − σ  k  )2

(C.49)

This leads to 2 S2 2 2 ω4 (k · k )2  ωk−k   pe  ) = − e me 1 + Aσ,σ (k,k S,S Te2 Te2 k 4 k 4 |k − k |2 (σ ωkS − σ  ωkS )2 1 ×P . (k − k,σ ωkS − σ  ωkS )

(C.50)

Collecting the results we have obtained thus far, we arrive at Re (k,ω) =

     1 1 [k · (k − k )]2 Ikσ L dk P 2  2 2  2 4π nT ˆ e  k |k − k | k λDe (k − k ,ω − σ  ωkL ) σ =±1

 2 S2 ωk−k  (k · k )2 + 2 2 1 + k k (ω − σ  ωkS )2

 × P , (k − k,ω − σ  ωkS ) k 2 k 2 |k − k |2 λ6De 

Ikσ S

1

(C.51)

308

Nonlinear Dispersion Relations

where we have replaced the argument σ ωkS by a generic variable, ω. Since we are looking for solutions in the ion-sound mode range, ω ∼ ωkS , we approximate various linear responses by Re (k,ω) ≈

2(1 + k 2 λ2De ) k 2 λ2De

 1−

ωkS ω

 ,

2σ  L (ω − σ  ωkL + σ  ωk−k  ), ωpe   S2 ωk−k 1 + |k − k |2 λ2De  Re (k − k,ω − σ  ωkS ) ≈ 1 − , |k − k |2 λ2De (ω − σ  ωkS )2

Re (k − k,ω − σ  ωkL ) ≈ −

(C.52)

where ωkS2 is defined in (C.6). This simplifies (C.52): ωkS 0 = ω − ωkS + 8π nT ˆ e

σ  =±1

me mi

1/2 

  σ  Ikσ L ωpe [k · (k − k )]2  dk 2 k 2 |k − k |2 ω − σ  ωL + σ  ωL  k k−k

   3Ti 1/2 (k · k )2 |k − k |2 2 σ  S 1 + Ik Te k 2 k 2 (k − σ  k  )2  (k − σ  k  )2 cS2 , × (ω − σ  k  cS )2 − (k − k )2 cS2 − k  λDe



 

1+

(C.53)

where we have redefined the ion-sound spectrum by absorbing the factor μk in the denominator. In general, (C.53) requires numerical solution. If we ignore contribution from the ion-sound = Ik−L = Ik , then we obtain spectrum, and assume isotropic Langmuir turbulence, Ik+L   ω = ωkS + δωkS ,  ωkS [k · (k − k )]2 Ik dk 2 , δωkS = 12π nT ˆ e k |k − k |2 (2k · k − k 2 ) λ2De

(C.54)

where we have assumed ωkS ωpe in order to ignore ωkS the denominators. Here, δωkS represents the nonlinear frequency shift of ion-sound waves by background high-frequency Langmuir turbulence.

Appendix D Plasma Dispersion Function

The plasma dispersion or Fried–Conte function (Fried and Conte, 1961), or simply Z function, is defined by ⎧ 2 ∞ ⎪ 1 e−x ⎪ ⎪ , Im ζ > 0 dx ⎪ ⎪ x−ζ ⎪ π 1/2 −∞ ⎪ 2 ⎨ ∞ e−x 1 2 (D.1) Z(ζ ) = + i π 1/2 e−ζ , Im ζ = 0 . P dx ⎪ 1/2 −∞ x − ζ ⎪ π ⎪ ⎪ 2 ∞ ⎪ ⎪ e−x 1 2 ⎪ ⎩ + 2i π 1/2 e−ζ , dx Im ζ < 0 x−ζ π 1/2 −∞







• Alternative Expression For Im ζ > 0, we may express  ∞ 1 =i dτ e−i (x−ζ ) τ , x−ζ 0

(D.2)

 0 1 = −i dτ e−i (x−ζ ) τ . x−ζ −∞

(D.3)

while for Im ζ < 0, we may use

Consequently, we have

Z(ζ ) =

⎧ ⎪ ⎪ ⎨

∞ ∞ dτ dx e−x −i (x−ζ ) τ , −∞ π 1/2 0  0 ∞ i i

2

Im ζ > 0

⎪ 2 2 ⎪ ⎩ − dτ dx e−x −i (x−ζ ) τ + 2i π 1/2 e−ζ , −∞ π 1/2 −∞

.

(D.4)

Im ζ < 0

Carrying out the x integration, we obtain

Z(ζ ) =

⎧ ⎪ ⎪ ⎨ 2i

∞ dτ e−τ +2i ζ τ , 0 0

⎪ ⎪ ⎩ −2i

2

−∞

2 2 dτ e−τ +2i ζ τ + 2i π 1/2 e−ζ ,

Im ζ > 0 ,

(D.5)

Im ζ < 0

309

310

Plasma Dispersion Function

which can be further manipulated as Z(ζ ) = 2i e−ζ

2

 iζ −∞

2 dτ e−τ .

(D.6)

Note that we have only discussed the case of Im ζ > 0 and Im ζ < 0, but not the marginal case Im ζ = 0. This is because for Im ζ = 0, the identity (D.3) is not valid. However, since the alternative expression in (D.6) for the Z function is valid for both signs of Im ζ , and since Im ζ = 0 is a singular case, the expression in (D.6) is extended to the singular case as well. • Series Expansion Since the alternative expression for Z function (D.6) is valid for both signs of Im ζ , it is permissible to consider only the case of Im ζ > 0 without loss of generality. From (D.5), for Im ζ > 0, namely,  ∞ 2 dτ e−τ +2i ζ τ , (D.7) Z(ζ ) = 2i 0

we employ the Taylor series expansion for e2i ζ τ , to obtain Z(ζ ) = 2i

 ∞ n  2 2 n ∞ i dτ e−τ τ n ζ n . n! 0 n=0

Making use of the integral identity   ∞ √ 2 (2a − 1)! ! π/2a+1 dτ e−τ τ n = b! /2 0

n = 2a , n = 2b + 1

(D.8)

(D.9)

we have Z(ζ ) = i

∞ ∞  √  (−ζ 2 )a 2b b! − 2ζ (−2 ζ 2 )b . π a! (2b + 1)! a=0

(D.10)

b=0

The imaginary term is but the Taylor series expansion of the exponential function. Consequently, we have Z(ζ ) = i

∞  √ −ζ 2 πe − 2ζ n=0

=i

2n n! (−2 ζ 2 )n (2n + 1)!

(D.11)

  √ −ζ 2 2 4 4 8 6 ζ − ζ + · · · + an ζ 2n + · · · . πe − 2ζ 1 − ζ2 + 3 15 105

The coefficients an can be automatically generated from the recursion relation: an = −

2 an−1, 2n + 1

n = 1,2,3, . . . ,

a0 = 1.

(D.12)

• Asymptotic Expansion For asymptotic expansion, we start from 2  ∞ 2 e−x + iσ π 1/2 e−ζ , dx Z(ζ ) = π −1/2 P x−ζ −∞ ⎧ ⎨ 0 Im ζ > 0 1 Im ζ = 0 . σ = ⎩ 2 Im ζ < 0

(D.13)

Plasma Dispersion Function

311

For large ζ , we expand the factor 1/(x − ζ ) to obtain  ∞ −∞

dx

 2  2 1 ∞ e−x =− dx e−x 1 + x−ζ ζ −∞

x2 x4 + + ··· + ζ2 ζ4

 x 2n + · · · . ζ 2n

(D.14)

Making use of √ 2 (2n)! π dx e−x x 2n = , n! 22n −∞

 ∞

(D.15)

we have  1 3 an 1 1 + 2 + 4 + · · · + 2n ζ 2ζ 4ζ ζ 

⎧ ⎨  n + 12 (2n)! = , σ = an = 1 2n ⎩ n! 2 π2

Z(ζ ) = −

 + ··· 0 1 2

1 2 + iσ π 2 e−ζ ,

Im ζ > 0 Im ζ = 0 Im ζ < 0

.

(D.16)

In general, one may automatically generate the coefficients using the recursion relation: an =

2n − 1 an−1, 2

n = 1,2,3, . . . ,

a0 = 1.

(D.17)

• Differential Equation If we take the derivative of the dispersion function, we have 2  ∞ e−x d d Z(ζ ) = π −1/2 = −2 − 2ζ Z(ζ ). dx dζ dζ −∞ x−ζ

(D.18)

From this, we obtain the differential equation for Z(ζ ), Z  (ζ ) = −2 [1 + ζ Z(ζ )] .

(D.19)

• Continued Fraction The continued fraction expression for plasma dispersion function can be obtained based upon the asymptotic expansion. The conversion of an infinite series to a continued fraction is standard. For a given series of the form s=

1 1 1 1 1 + + + + + ··· , a0 a1 a2 a3 a4

(D.20)

the continued fraction can be generated by 1

s=

.

a02

a0 +

a12

−(a1 + a0 ) +

a22

(a2 + a1 ) + −(a3 + a2 ) +

a32

(a4 + a3 ) + · · ·

(D.21)

312

Plasma Dispersion Function

From this, it follows that the continued fraction representation of Z function is given by ζ

Z(ζ ) = −

− 12 ζ 2

ζ2 +

− 32 ζ 2

ζ 2 + 12 +

ζ 2 + 32 +

− 52 ζ 2 ζ 2 + 52 + · · ·

− 12 ζ 2 − 32 ζ 2 − 52 ζ 2 2 ζ =− 2 + iσ π 1/2 e−ζ . ζ + ζ2 + 1+ ζ2 + 3+ ζ2 + 5 + ··· 2 2 2

(D.22)

Some computer programs can generate the continued fraction representation by the following algorithm: An , n→∞ Bn

2 ζ + iσ π 1/2 e−ζ , Z(ζ ) = − 2 ζ +P An+1 = bn+1 An + an+1 An−1,

Bn+1 = bn+1 Bn + an+1 Bn−1, A−1 = 1, A0 = 0, B0 = 1, an = −

P = lim

(n = 0,1,2, . . .),

B−1 = 0, (2n − 1) ζ 2 , 2

bn = ζ 2 +

2n − 1 . 2

(D.23)

Appendix E Weak Turbulence Theory for Reactive Instabilities

In the present Appendix, we discuss a possible method to extend the standard weak turbulence theory to a more general case. For the sake of simplicity, we restrict our discussion to one-dimensional situation. In the standard weak turbulence theory, which is the focus of the main part of this monograph, the plasma eigenmode is determined solely by setting the real part of the dielectric response function equal to zero: Re (k,ωk ) = 0.

(E.1)

However, for reactive instability, it is no longer appropriate to consider just the real part of the dielectric response function in order to evaluate the dispersion relation, but rather, both the real and imaginary parts of (k,ω) must be set equal to zero. Moreover, it is no longer valid to consider that ω is real, since the root of the equation (k,ω) = 0, which is generally given by a transcendental function, is complex: (k,ωk + iγk ) = Re (k,ωk + iγk ) + i Im (k,ωk + iγk ) = 0.

(E.2)

Linear and quasilinear theories of waves and instabilities for generally complex ω are well known. However, perturbative nonlinear theory that goes beyond the quasilinear method is not fully developed. In what follows, we discuss one possible way to formulate the weak turbulence theory for reactively unstable plasmas. Let us begin the discourse starting from one-dimensional Vlasov–Poisson system of equations,   ∂ ea ∂ ∂ +v + fa (x,v,t) = 0, E(x,t) ∂t ∂x ma ∂v   ∂E(x,t) = 4π ea dv fa (x,v,t). (E.3) ∂x a Separating the total particle distribution and electric field as is customary, fa (x,v,t) = na Fa (v,t) + δfa (x,v,t) and E(x,t) = δE(x,t), we have  ea ∂ ∂na Fa (v,t) =− dk δEk∗ (t) δfka (v,t), ∂t ma ∂v   ea dv δfka (v,t), ik δEk (t) = 4π 

a

 ea ∂na Fa (v,t) ∂ + ikv δfka (v,t) = − δEk (t) ∂t ma ∂v  ea ∂ a (v,t) − δE  (t) δf a (v,t) ], dk  [ δEk  (t) δfk−k −  k k−k  ma ∂v

(E.4)

313

314

Weak Turbulence Theory for Reactive Instabilities

where we have made use of the spatial Fourier spectral representation,   1 δfka (v,t) = dx δfa (x,v,t)e−ikx , δfa (x,v,t) = dk δfka (v,t) eikx , 2π   1 dx δE(x,t)e−ikx , δEk (t) = δE(x,t) = dk δEk (t) eikx , 2π

(E.5)

and have made use of the fact that δE−k = δEk∗ .

E.1 Generalized Quasi-Linear Theory Let us first review the textbook quasi-linear theory (Sagdeev and Galeev, 1969; Davidson, 1972), which is the lowest-order nonlinear theory. In the purely quasi-linear picture, the spectral amplitudes do not depend on time, and we ignore nonlinear mode coupling, so that we have    ea ∂ ∂na Fa (v,t) ∗ δf a (v) ei (ω −ω) t , =− dk dω dω δEk,ω  k,ω ∂t ma ∂v   a (v), ea dv δfkω ik δEkω = 4π a

∂na Fa (v,t) a (v) = − ea δE . −i (ω − kv) δfk,ω k,ω ma ∂v

(E.6)

In writing (E.6), we have represented the fluctuating quantities in terms of temporal spectral representation (Laplace transformation),  1 a (v,t) = δfk,ω dx δfa (x,v,t)e−ikx+iωt , (2π )2   a (v,t) eikx−iωt , δfa (x,v,t) = dω dk δfk,ω  1 dx δE(x,t)e−ikx+iωt , δEk,ω (t) = (2π )2   δE(x,t) = dω dk δEk,ω (t) eikx−iωt , (E.7) where the adiabatic time dependence of the spectral amplitude is assumed, but,unlike in the main  body of this book, ω is assumed to be fully complex, and the integrals dω and dω are along the usual paths L, defined above all the singularities associated with the integrands. a (v), we have Solving for δfk,ω a (v) = − δfk,ω

iea δEkω ∂na Fa (v,t) . ma ω − kv ∂v

(E.8)

In (E.8), the resonant denominator ω − kv is no longer approximately treated by ωr − kv + i0, but we assume that ω is fully complex, ω = ωr + iγ , except that we assume γ > 0 in the sense of causality requirement. Inserting (E.8) to Poisson equation, we obtain (k,ω) δEk,ω = 0,

(E.9)

where the linear dielectric response function is given by (k,ω) = 1 +

2   ωpa a

k

dv

∂Fa (v,t)/∂v . ω − kv

(E.10)

E.2 Generalized Weak Turbulence Theory

315

We may designate the solution to the dispersion relation (k,ω) = 0 by ω = ωk + iγk ,

(E.11)

and write the spectral electric field amplitude as δEk,ω = δEk δ(ω − ωk − iγk ).

(E.12)

From this, upon inserting (E.12) to the definition of spatial Fourier representation of electric field in (E.5), we may construct the spectral wave energy density:   ∗ (E.13) δEk (t) δEk (t) = Ik e2γk t . , where we have defined δE 2 = Ik . Here, we reiterate that in the lowest-order nonlinear theory, that k is, purely quasi-linear picture, the amplitude Ik does not depend on time. Instead, the time dependence of the wave energy density comes solely from the exponential factor e2γk t . Upon taking the time derivative of (E.13), we obtain the quasilinear wave kinetic equation, ∂ 2γk t  Ik e = 2γk Ik e2γk t . ∂t

(E.14)

a (v) (E.8) into the particle kinetic equation, that is, the Inserting the perturbed distribution δfk,ω first equation in (E.6), we obtain the customary quasilinear particle kinetic equation,

e2 ∂ ∂Fa (v,t) = a2 ∂t ma ∂v

 dk

γk Ik e2γk t (ωk − kv)2 + γk2

∂Fa (v,t) . ∂v

(E.15)

If we take the limit of γk → 0, and make use of lim

γk

γk →0 (ωk − kv)2 + γ 2 k

= π δ(ωk − kv),

(E.16)

then we recover the customary form of quasilinear particle kinetic equation for weakly damped or growing mode,  π ea2 ∂ ∂Fa (v,t) ∂Fa (v,t) = 2 dk Ik δ(ωk − kv) . (E.17) ∂t ∂v ma ∂v However, for reactively unstable plasmas, the quasilinear particle kinetic equation is (E.15), rather than (E.17).

E.2 Generalized Weak Turbulence Theory Moving on to the next order theory, the key aspect concerns the time dependence of wave amplitude. In quasilinear theory, the spectral amplitudes δfk,ω and δEk,ω do not depend on adiabatic time variable. The time dependence of the wave energy density arises purely by virtue of the exponential factor e2iγk t . In contrast, in the next-order weak turbulence scheme, we now allow the slow adiabatic time dependence on these amplitudes, which will be accounted for on the basis of nonlinear mode coupling effects. Thus, if we allow the slow-time dependence on the spectral wave energy density, then the quasi-linear wave kinetic equation (E.14) must be generalized to ∂Ik 2γk t ∂ 2γk t  Ik e e = 2γk Ik e2γk t + . ∂t ∂t

(E.18)

316

Weak Turbulence Theory for Reactive Instabilities

The extra factor, ∂Ik /∂t, is to be determined from the nonlinear terms. In order to accomplish that, we solve the nonlinear perturbed Vlasov equation in the usual manner, by absorbing the slow-time derivative into the complex wave frequency, ω → ω + i ∂/∂t, and reintroducing it later. The spectral form of nonlinear equation for the perturbed distribution function is thus given by   iea 1 ∂ a = − iea ∂na Fa /∂v δE dk  dω δfk,ω k,ω − ma ω − kv ma ω − kv ∂v -

, a a . (E.19) × δEk  ω δfk−k ,ω−ω − δEk  ω δfk−k ,ω−ω The iterative solution of (E.19) may be proceeded in the same manner as expounded in the main part of this book, where solutions up to third order were discussed. If we simplify the matter by restricting to just second order, then it is a straightforward exercise to obtain  ∗  0 = (k,ω) δEkω δEkω     ∗ + dk  dω χ (k ,ω |k − k ,ω − ω ) δEkω (E.20) δEk  ω δEk−k ,ω−ω , where the linear dielectric constant is already defined, and the second-order nonlinear response function is defined in the customary manner, namely, 2 i  ea ωpa 2 a ma k    1 1 ∂ 1 ∂Fa + , × dv     ω − kv ∂v ω − k v ω − ω − (k − k ) v ∂v

χ (k ,ω |k − k ,ω − ω ) = −

(E.21)

except that ω and ω are complex. The third-order cumulant may also be obtained in the same manner:   ∗ δEkω δEk  ω δEk−k ,ω−ω  ∗   -, χ (k ,ω |k − k ,ω − ω ) , ∗ ∗ ,ω δE   δEk−k ,ω−ω δE δE = −2   k k ,ω k−k ,ω−ω ∗ (k,ω)       χ (k,ω| − k + k , − ω + ω ) ∗ δE δEkω δEkω + k−k ,ω−ω δE−k+k ,−ω+ω   (k ,ω )    χ (k,ω| − k , − ω )  ∗ δEkω δEkω δEk ,ω δE−k ,−ω . (E.22) + (k − k ,ω − ω ) Upon reintroducing the adiabatic time derivative, and denoting the complex root of the dispersion relation by k ≡ ωk + iγk ,

(E.23)

one arrives at 4 ∂Ik = Im ∂t ∂ (k,k )/∂k   |χ (k ,k  |k − k ,k−k  )|2  δ(k − k  − k−k  ) Ik  Ik−k  × dk ∗ (k,k  + k−k  ) χ (k ,k − k−k  |k − k ,k−k  ) χ (k,k | − k + k , − k−k  ) Ik Ik−k  (k ,k − k−k  )  χ (k ,k  |k − k ,k − k  ) χ (k,k | − k , − k  ) (E.24) Ik Ik  . + (k − k ,k − k  )

+

E.3 Generalized Second-Order Nonlinear Susceptibility

317

This equation can be substituted to (E.18), which, in principle, completes the derivation of wave kinetic equation. The resulting generalized weak turbulence theory is not so straightforward to analyze, either by analytical means or by direct numerical computation, since the nonlinear susceptibility involves complex velocity integrals with linear and nonlinear resonant denominators. For this reason, the weak turbulence theory of the type discussed in this Appendix has not been fully developed in the existing literature, although Yoon (2010); Yoon and Umeda (2010) applied the method outlined in this Appendix to the problem of Buneman instability. We close the present Appendix by considering the dielectric response functions for thermal distribution.

E.3 Generalized Second-Order Nonlinear Susceptibility Let us consider the Maxwellian velocity distribution function. After a straightforward but somewhat tedious manipulations, it can be shown that %   a 2  ea ωpa 2ζka,ω 2 ζk,ω dx 1 2 + + χ (k ,ω |k − k ,ω − ω ) = −i a a ma k 2 v 4 x − ζk,ω x − ζk,ω π 1/2 k  a Ta  2 (ζka,ω )2 ζka,ω 1 1 + a + a x − ζka,ω x − ζk,ω (x − ζka,ω )2 x − ζk,ω  a a 2ζk−k 2 ζk,ω ,ω−ω 1 + 2 + + a a k − k x − ζk,ω x − ζk,ω  a a 2 2 (ζk−k ζk−k ,ω−ω ) ,ω−ω 2 1 1 e−x , + + a a a 2 x − ζa x − ζk−k (x − ζk−k ) ,ω−ω x − ζk,ω ,ω−ω k,ω (E.25) where ω , ζka,ω =  k vT a

a ζk−k ,ω−ω =

ω − ω . (k − k  ) vT a

(E.26)

Carrying out the x integration by making use of the definition for Z function, one can show that χ (k ,ω |k − k ,ω − ω ) = i

2 !  ea ωpa a ) k Z  (ζkω 4Ta (kω − k  ω)2 a

" a − k  Z  (ζka ω ) − (k − k  ) Z  (ζk−k ,ω−ω ) .

(E.27)

Appendix F On Higher-Order Perturbative Expansion

a beyond the third order truncation, If we employ perturbative solution for δfkω a(1)

a(2)

a(3)

a(4)

a(5)

a = δf δfkω kω + δfkω + δfkω + δfkω + δfkω + · · · ,

(F.1)

then after some tedious but otherwise straightforward algebraic manipulations, it is possible to obtain the iterative solution, which is explicitly written down up to fifth order, a = α(K) n F φ fkω a a K  + α (2) (1|2) na Fa (φ1 φ2 − φ1 φ2 ) 1+2=K



+

α (3) (1|2|3) na Fa (φ1 φ2 φ3 − φ1 φ2 φ3  − φ1 φ2 φ3 )

1+2+3=K



+

α (4) (1|2|3|4) na Fa (φ1 φ2 φ3 φ4 − φ1 φ2 φ3 φ4 

1+2+3+4=K

− φ1 φ2 φ3 φ4  − φ1 φ2 φ3 φ4  + φ1 φ2  φ3 φ4 )  α (5) (1|2|3|4|5) na Fa (φ1 φ2 φ3 φ4 φ5 − φ1 φ2 φ3 φ4 φ5  + 1+2+34+5=K

− φ1 φ2 φ3 φ4 φ5  − φ1 φ2 φ3 φ4 φ5  + φ1 φ2 φ3  φ4 φ5  − φ1 φ2 φ3 φ4 φ5  + φ1 φ2 φ3  φ4 φ5  + φ1 φ2  φ3 φ4 φ5 ) , ··· ,

(F.2)

where short-hand notations, K ≡ (k,ω), 1 ≡ (k1,ω1 ), 2 ≡ (k2,ω2 ), etc., are employed, δ is omitted for δfk,ω and δφk,ω , and α(K) = k · gkω, 1 α (2) (1|2) = [(k1 · gk1 +k2,ω1 +ω2 ) (k2 · gk2 ω2 ) 2 + (k2 · gk1 +k2,ω1 +ω2 ) (k1 · gk1 ω1 )], 1 α (3) (1|2|3) = (k1 · gk1 +k2 +k3,ω1 +ω2 +ω3 ) 2 × [ (k2 · gk2 +k3,ω2 +ω3 ) (k3 · gk3 ω3 ) + (k3 · gk2 +k3,ω2 +ω3 ) (k2 · gk2 ω2 ) ],

318

On Higher-Order Perturbative Expansion α (4) (1|2|3|4) =

319

1 (k1 · gk1 +k2 +k3 +k4,ω1 +ω2 +ω3 +ω4 ) 2 × (k2 · gk2 +k3 +k4,ω2 +ω3 +ω4 ) [ (k3 · gk3 +k4,ω3 +ω4 ) (k4 · gk4,ω4 ) + (k4 · gk3 +k4,ω3 +ω4 ) (k3 · gk3,ω3 ) ],

α (5) (k1,ω1 |k2,ω2 |k3,ω3 |k4,ω4 |k5,ω5 ) =

1 (k1 · gk1 +k2 +k3 +k4 +k5,ω1 +ω2 +ω3 +ω4 +ω5 ) 2 × (k2 · gk2 +k3 +k4 +k5,ω2 +ω3 +ω4 +ω5 )(k3 · gk3 +k4 +k5,ω3 +ω4 +ω5 ) × [ (k4 · gk4 +k5,ω4 +ω5 )(k5 · gk5,ω5 ) + (k5 · gk4 +k5,ω4 +ω5 )(k4 · gk4,ω4 ) ], ··· .

(F.3)

The quantity of relevance is a δφ    = α(K) KK   n F δfkω a a kω  + α (2) (1|2) 12K   na Fa 1+2=K



+

! " α (3) (1|2|3) 123K   − 1K   23 na Fa

1+2+3=K



+

! " α (4) (1|2|3|4) 1234K   − 12K   34 − 1K   234 na Fa

1+2+3+4=K



+

! α (5) (1|2|3|4|5) 12345K   − 123K   45

1+2+3+4+5=K

" − 12K   345 − 1K  2345 + 1K  2345 na Fa ··· ,

(F.4)

where we have made use of even more compact notations, 12 = φ1 φ2 , etc. Some careful considerations show that many-body cumulants of electrostatic potential can be expressed as various sums of two- and three-body cumulants, as follows: 123K   − 1K  23 = 2123K  , 1234K   − 12K  34 − 1K  234 = 2 1234K   + 13K  24 + 23K  14 + 1342K   + 34K  12, 12345K   − 123K  45 − 12K  345 − 1K  2345 + 1K  2345 = 6 12345K   + 13245K   + 14235K   + 14253K   + 14352K   + 2 12453K   + 13452K   + 21K  2435 + 1K  2345 + 2 12435K   + 13425K   + 14K  235 + 12345K   + 14523K   + 13K  245.

(F.5)

320

On Higher-Order Perturbative Expansion

Putting all the results together, after somewhat lengthy but otherwise straightforward manipulations, one arrives at   α (3) (1| − 1|K) φ 2 1 fKa φK   = δ(K + K  ) α(K) + 2 +2

1

 !

3α (5) (1| − 1|2| − 2|K) + 3α (5) (1|2| − 1| − 2|K)

1

2

+ 3α (5) (1|2| − 2| − 1|K) + 3α (5) (1|2|K| − 1| − 2) + 3α (5) (1|K|2| − 1| − 2) + α (5) (1| − 1|K|2| − 2) + α (5) (1|K| − 1|2| − 2) + α (5) (K|1| − 1|2| − 2)  " + 2α (5) (K|1|2| − 1| − 2) φ 2 1 φ 2 2 + · · · φ 2 K na Fa  + α (2) (1|K − 1) φ1 φK−1 φK   na Fa 1

+

 !

2α (4) (1|2|K + K  − 1 − 2| − K  )φ1 φ2 φK+K  −1−2 φ 2 K 

1

2 (4) + 2α (1|2|K − 1| − 2)φ1 φK−1 φK  φ 2 2

+ 2α (4) (1|2|K − 2| − 1)φ2 φK−2 φK  φ 2 1 + α (4) (1| − K  |2|K + K  − 1 − 2)φ1 φ2 φK+K  −1−2 φ 2 K  " + α (4) (1| − 1|2|K − 2)φ2 φK−2 φK  φ 2 1 + · · · na Fa  + α (5) (1|2|3|4|K − 1 − 2 − 3 − 4) !

1

2

3

4

× 2φ1 φ2 φ4 φ3 φK−1−2−3−4 φK   + 2φ1 φ3 φ4 φ2 φK−1−2−3−4 φK   + 2φ1 φ4 φK  φ2 φ3 φK−1−2−3−4  + φ1 φ2 φ3 φ4 φK−1−2−3−4 φK   + φ1 φ4 φK−1−2−3−4 φ2 φ3 φK   " + φ1 φ3 φK  φ2 φ4 φK−1−2−3−4  na Fa + ··· .

(F.6)

In (F.6) all the terms on the right-hand side are expressed in terms of two-body correlations of the type φ 2 , three-body correlations φ 3 , and their products. If we approximately express the three-body cumulants, following the closure scheme (1.78), ! φ1 φ2 φ3  = δ(1 + 2 + 3) β(1|2|3) φ 2 2 φ 2 3 " +β(2|1|3) φ 2 1 φ 2 3 + β(3|1|2) φ 2 1 φ 2 2 , k k χ (2) (−2| − 3) , β(1|2|3) = 2i 2 3 k1 (1)

(F.7)

then after another quite lengthy manipulations, it can be shown that the result simplifies as follows: fKa φ−K  = α(K)φ 2 K na Fa !  α (2) (K  |K − K  ) β(K  |K − K  | − K) φ 2 K−K  φ 2 K + K

On Higher-Order Perturbative Expansion

321

+ β(K − K  |K  | − K) φ 2 K  φ 2 K +β(−K|K  |K − K  ) φ 2 K  φ 2 K−K  # +2α (3) (K  | − K  |K) φ 2 K  φ 2 K na Fa "  ! 2α (4) (K  |K  | − K  − K  |K) + α (4) (K  |K|K  | − K  − K  ) +

"

K  K 

!

× β(K  |K  | − K  − K  )φ 2 K  φ 2 K  +K  + β(K  |K  | − K  − K  )φ 2 K  φ 2 K  +K  " +β(−K  − K  |K  |K  )φ 2 K  φ 2 K  φ 2 K ! + 2α (4) (K  |K  |K − K  | − K  ) + 2α (4) (K  |K  |K − K  | − K  ) "! +α (4) (K  | − K  |K  |K − K  ) β(K  |K − K  | − K)φ 2 K−K  φ 2 K + β(K − K  |K  | − K)φ 2 K  φ 2 K

" # +β(−K|K  |K − K  )φ 2 K  φ 2 K−K  φ 2 K  na Fa ! 3α (5) (K  | − K  |K  | − K  |K) +2 K  K  + 3α (5) (K  |K  | − K  | − K  |K) + 3α (5) (K  |K  | − K  | − K  |K)

+ 3α (5) (K  |K  |K| − K  | − K  ) + 3α (5) (K  |K|K  | − K  | − K  ) + α (5) (K  | − K  |K|K  | − K  ) + α (5) (K  |K| − K  |K  | − K  )

"

+α (5) (K|K  | − K  |K  | − K  ) + 2α (5) (K|K  |K  | − K  | − K  ) # ×φ 2 K  φ 2 K  φ 2 K na Fa ! 2α (5) (K  |K  |K  | − K  − K  |K − K  ) + K  K  K  + 2α (5) (K  |K  |K  | − K  − K  |K − K  )

+ 2α (5) (K  |K  |K  |K − K  | − K  − K  ) + α (5) (K  |K  |K − K  |K  | − K  − K  ) + α (5) (K  |K  |K − K  |K  | − K  − K  ) " +α (5) (K  |K  | − K  − K  |K  |K − K  ) ! × β(K  |K  | − K  − K  )φ 2 K  φ 2 K  +K  + β(K  |K  | − K  − K  )φ 2 K  φ 2 K  +K  " +β(−K  − K  |K  |K  )φ 2 K  φ 2 K  ! × β(K  |K − K  | − K)φ 2 K−K  φ 2 K + β(K − K  |K  | − K)φ 2 K  φ 2 K

" +β(−K|K  |K − K  )φ 2 K  φ 2 K−K  na Fa

+ ··· .

(F.8)

322

On Higher-Order Perturbative Expansion

It is instructive to rewrite (F.8) in terms of the electric field amplitudes via E 2 K = k 2 φ 2 K , and make use of explicit definitions for various α’s: fKa φ−K  =

k · gK E 2 K na Fa k2 1 [k · gK (k − k ) · gK−K  + (k − k ) · gK k · gK  ] + 2  K



β(K − K  |K  | − K) 2 β(K  |K − K  | − K) 2 E K−K  E 2 K + E K  E 2 K 2  2 k |k − k | k 2 k 2  β(−K|K  |K − K  ) 2 2   + E  E K K−K k 2 |k − k |2 2 E 2 K  E 2 K na F a −k · gK (k · gK k · gK−K  − k · gK−K  k · gK  ) k 2 k 2  1  −k · gK k · gK k · gK−K  (k + k ) · gK−K  −K  + 2 k  

×

K K + k · gK k · gK−K  (k + k ) · gK  +K  k · gK−K  −K 

1  k · gK k · gK  k · gK−K  (k + k ) · gK  +K  2  1 + k · gK (k + k ) · gK  k · gK  k · gK−K  2  β(K  |K  | − K  − K  ) 2 × E K  E 2 K  +K  k 2 |k + k |2 β(K  |K  | − K  − K  ) 2 + E K  E 2 K  +K  k 2 |k + k |2  β(−K  − K  |K  |K  ) 2 2   E 2  n F  + E  E K a a K K k 2 k 2   1 k · gK k · gK  k · gK−K  (k − k ) · gK−K  −K  + 2   k −

K K

+ k · gK k · gK−K  (k − k ) · gK−K  k · gK−K  −K  + k · gK k · gK  k · gK−K  (k − k ) · gK−K  −K 

− k · gK (k − k ) · gK−K  k · gK−K  k · gK−K  −K  1 − k · gK k · gK (k − k ) · gK−K  k · gK−K  2  1  − k · gK (k − k ) · gK k · gK  k · gK−K  2  β(K − K  |K  | − K) 2 β(K  |K − K  | − K) 2 × E K−K  E 2 K + E K  E 2 K 2  2 k |k − k | k 2 k 2  β(−K|K  |K − K  ) 2 2 2   + E  E K K−K E K  na Fa k 2 |k − k |2   1 3 k · gK k · gK k · gK k · gK−K  k · gK−K  +2 2 k 2 k 2 2 k   K K

On Higher-Order Perturbative Expansion − + − + − + + − − − −

323

3  k · gK k · gK k · gK  k · gK−K  k · gK−K  2 3 k · gK k · gK k · gK−K  k · gK−K  k · gK−K  −K  2 3  k · gK k · gK  k · gK−K  k · gK−K  k · gK−K  −K  2 3 k · gK k · gK k · gK−K  k · gK−K  k · gK−K  −K  2 3  k · gK k · gK  k · gK−K  k · gK−K  k · gK−K  −K  2 3  k · gK k · gK  k · gK−K  k · gK  +K  k · gK−K  −K  2 3  k · gK k · gK  k · gK−K  k · gK  +K  k · gK−K  −K  2 3  k · gK k · gK  k · gK  k · gK−K  k · gK  +K  2 3  k · gK k · gK  k · gK  k · gK−K  k · gK  +K  2 k · g0 k · gK k · gK  k · gK  k · gK  +K   k · g0 k · gK k · gK  k · gK  k · gK  +K 

× E 2 K  E 2 K  E 2 K na Fa + ··· .

(F.9)

The perturbation series in (F.9) diverges when various resonance conditions are satisfied. For instance, if we are interested in the velocity space that satisfies the linear wave-particle resonance condition, ω − k · v = 0, then, by noting that 1 , (F.10) gK ∝ ω−k·v we may rearrange the various terms in (F.9) as an ascending series in gK . If we keep only those terms that contain this factor, and retain the most important contributions from each term within the perturbation expansion in powers of wave field energy, then the partial series becomes ⎡ 2  E E 2 K  K  · g k · g ⎣1 −  k fKa φ−K  = k · gK K K−K k2 k 2 K ⎤ ⎛ ⎞2 2   E K ⎠ 3 ) + ···⎥n F . + 3⎝ k · gK k · gK−K  + O(gK (F.11) ⎦ a a k 2  K

This shows that the straightforward perturbation series diverges beyond the first couple of terms, and that the entire treatment requires some sort of renormalization. The situation is reminiscent of the mathematical asymptotic series, where retaining more terms does not necessarily improve the approximation, but may actually make the situation worse. Note that (F.11) has a qualitative form of a series given by  , -   , -  2 3

, - 2 2 O E O E2 O E + + + ··· . (F.12) ω−k·v (ω − k · v)2 (ω − k · v)3

324

On Higher-Order Perturbative Expansion

This implies that for velocity space that satisfies the linear wave-particle resonance condition, ω − k · v ≈ 0, the straightforward perturbative expansion appears to fail, but if the series in (F.12) can be summed, then it must be given by the form

, - O E2   . (F.13) ω − k · v − O E2 This points to the need for a renormalization scheme, where the influence of wave intensity must be incorporated into the resonance condition, ω − k · v → ω − k · v + iηk,ω, where iηk,ω ∝ O

,

δE 2

(F.14)



k,ω

.

(F.15)

One particular method to achieve such a renormalization procedure is discussed in Appendix G.

Appendix G On Renormalized Kinetic Turbulence Theory

Appendix F pointed to the fact that partial summation of leading terms in the infinite series implies the necessity for a correction of the resonant denominator where the wave intensity modifies the waveparticle resonance condition. This provides a clue as to how to systematically modify the resonance factor at the outset. The following discussion is based on the work carried out by Rudakov and Tsytovich (1971). We return to the Vlasov–Poisson system of equations 

 ∂ ea ∂δφ ∂ ∂ +v· − · fa = 0, ∂t ∂r ma ∂r ∂v  ∂ 2 δφ = −4π ea 2 ∂r a

 dv δfa .

(G.1)

The relevant equations for averaged and perturbed quantities are given by  4π ea  a , dv δfk,ω k2 a   iea ∂ ∂na Fa a , = δφ−k,−ω δfk,ω dk dω k · ∂t ma ∂v   ∂ ∂na Fa a = − ea δφ ω−k·v+i δfk,ω k,ω k · ∂t ma ∂v   ea dk dω k − ma " ∂ ! a a δφk,ω δfk−k · ,ω−ω − δφk,ω δfk−k,ω−ω  . ∂v δφk,ω =

(G.2)

a is iteratively solved In the straightforward weak turbulence scheme, the perturbed distribution δfk,ω in terms of the field strength δφk,ω as the series expansion parameter.

G.1 Renormalization of Kinetic Equation a δf a In the renormalized kinetic theory by Rudakov and Tsytovich (1971), first, a quantity iηk,ω k,ω a is added to both sides of the equation for δfk,ω , which is to be determined later, resulting in the expression

325

326

On Renormalized Kinetic Turbulence Theory ea ∂na Fa a δf a + iηk,ω δφk,ω k · k,ω ma ∂v   ea dk dω k − ma " ∂ ! a a δφk,ω δfk−k · ,ω−ω − δφk,ω δfk−k,ω−ω  . ∂v

a ) δf a = − (ω − k · v + iηk,ω k,ω

(G.3)

Note that (G.3) is formally identical to the original equation in (G.2). Recall that the form of resonance a , is given in the form of (F.14). Then an operator factor prescribed in (G.3), namely, ω − k · v + iηk,ω a gk,ω (v) is introduced, which satisfies the equation a (v) (ω − k · v + iηa ) F a = F a , gk,ω k,ω k,ω k,ω

(G.4)

a is an arbitrary function, or, equivalently, we may formally write where Fk,ω a (v) = (ω − k · v + iηa )−1 . gk,ω k,ω

(G.5)

a (v) represents the renormalized resonant velocity denominator. Evidently, gk,ω The formal solution to (G.3) is then given by

ea a (v) k · ∂na Fa + i g a (v) ηa δf a δφk,ω gk,ω k,ω k,ω k,ω ma ∂v   ! ea a ∂ a δφk,ω δfk−k − g (v) dk dω k · ,ω−ω ma k,ω ∂v " a − δφk,ω δfk−k ,ω−ω  .

a =− δfk,ω

(G.6)

Changing the spectral argument from (k,ω) to (k − k,ω − ω ), we also have ea ∂na Fa a    g δφ   (v) (k − k ) · ma k−k ,ω−ω k−k ,ω−ω ∂v a a a + i gk−k (v) η δf ,ω−ω k−k,ω−ω k−k,ω−ω   ea a ∂ − gk−k,ω−ω (v) dk dω k · ma ∂v " ! a a × δφk,ω δfk−k  −k,ω−ω −ω − δφk,ω δfk−k −k,ω−ω −ω  .

a δfk−k ,ω−ω = −

Let us insert (G.7) to the right-hand side of (G.6): ea ∂na Fa a δf a + iηk,ω δφk,ω k · k,ω ma ∂v     e2 ∂ δφk,ω δφk−k,ω−ω − δφk,ω δφk−k,ω−ω dk dω k · + a2 ∂v ma ∂na Fa a  × gk−k ,ω−ω (v) (k − k ) · ∂v  

∂ ea a a ga  dk dω k · −i  (v) ηk−k,ω−ω δφk,ω δfk−k,ω−ω ma ∂v k−k ,ω−ω

a ) δf a = − (ω − k · v + iηk,ω k,ω

(G.7)

G.1 Renormalization of Kinetic Equation

327

-

,

a a a − gk−k ,ω−ω (v) ηk−k,ω−ω δφk,ω δfk−k,ω−ω     e2 ∂ a ∂ gk−k,ω−ω (v) k · + a2 dk dω dk dω k · ∂v ∂v ma

, a a × δφk,ω δφk,ω δfk−k −k,ω−ω −ω − δφk,ω δφk,ω δfk−k −k,ω−ω −ω - , a . (G.8) − δφk,ω δφk,ω δfk−k  −k,ω−ω −ω

This equation is exact and no approximation is made. Rudakov  Tsytovich  (1971) make the observation that in the last term, which involves double  and integral, dk dω dk dω , if we choose the singular term where k = −k and ω = −ω , namely, e2 − a2 ma



dk



dω k ·

∂ a  ∂ δφ  δφ  a ,  δf g   (v) k · ∂v k−k ,ω−ω ∂v k ,ω −k ,−ω k,ω

(G.9)

a δf a . Thus the quantity ηa is identified formally as then this term can be equated with iηk,ω k,ω k,ω 2 a δf a = − ea ∂ iηk,ω k,ω m2a ∂vi



dk



a dω ki kj |δφk,ω |2 gk−k ,ω−ω

a ∂δfk,ω

∂vj

.

(G.10)

Note that the quantity |δφk,ω |2 is equivalent to the ensemble average δφ 2 k,ω for stationary and homogeneous turbulence with random phases. When (G.10) is inserted back to (G.8), then the term a δf a no longer appears on the right-hand side: iηk,ω k,ω ea ∂na Fa δφk,ω k · ma ∂v 2  e ∂ + a2 δφk1,ω1 δφk2,ω2 k1 · ∂v ma

a ) δf a = − (ω − k · v + iηk,ω k,ω

k1 +k2 =k ω1 +ω2 =ω

  ∂na Fa − δφk1,ω1 δφk2,ω2 gka2,ω2 k2 · ∂v  ∂ a ea a g k1 · η δφk1,ω1 δfka2,ω2 −i ma ∂v k2,ω2 k2,ω2 k1 +k2 =k ω1 +ω2 =ω

- − gka2,ω2 δηka2,ω2 δφk1,ω1 δfka2,ω2  e2 ∂ a ∂ gk2 +k3,ω2 +ω3 k2 · + a2 k1 · ∂v ∂v ma k +k +k =k ,

1

2

3

ω1 +ω2 +ω3 =ω

,

× δφk1,ω1 δφk2,ω2 δfka3,ω3 − δφk1,ω1 δφk2,ω2 δfka3,ω3 - , (G.11) − δφk1,ω1 δφk2,ω2 δfka3,ω3 . In (G.11), of course, the singular term ∂ a ∂ e2  a gk−k1,ω−ω1 k1 · δφk1,ω1 δφ−k1,−ω1 δfk,ω k1 · − a2 ∂v ∂v ma k ,ω 1

1

328

On Renormalized Kinetic Turbulence Theory

a fa . should be excluded since this term was already singled out in order to eliminate the term iηk,ω k,ω $ However, since this term is but a term of measure zero within the integral, k1 +k2 +k3 =k       $ ω1 +ω2 +ω3 =ω = dk1 dk2 dk3 δ(k1 +k2 +k3 −k) dω1 dω2 dω3 δ(ω1 +ω2 +ω3 −ω), when averaged over the ensemble, the presence or absence of this term does not matter. Besides, in what follows, we will truncate (G.11) by ignoring explicit nonlinear terms. Consequently, in the renormalization scheme by Rudakov and Tsytovich (1971), the nonlinear coupling term serves the a , but in the subsequent application, explicit purpose of providing the means to identify the term iηk,ω nonlinear mode coupling terms are not taken into account.

G.2 Renormalized Quasilinear Theory Let us thus write the truncated solution for perturbed distribution, a =− δfk,ω

ea a ∂na Fa δφk,ω, g k· ma k,ω ∂v

(G.12)

where we have only kept the first term on the right-hand side of (G.11). Inserting this solution to Poisson’s equation, we have (k,ω) δφk,ω = 0,

(G.13)

where (k,ω) is the renormalized dielectric response function, (k,ω) = 1 +

2   ωpa a k · ∂Fa . dv gk,ω 2 ∂v k a

(G.14)

Multiplying φ−k,−ω and taking the ensemble average, we obtain (k,ω) δφ 2 k,ω = 0.

(G.15)

a is determined from (G.4), and, in turn, the quantity iηa is defined by (G.10). The operator gk,ω k,ω Combining the two equations, we have    e2 ∂ a · dk dω k k gk,ω (v) ω − k · v − a2 ma ∂v  , ∂ a a (v) = F a (v). Fk,ω × δφ 2   gk−k (v) · (G.16) ,ω−ω k,ω k ,ω ∂v a (v) is an arbitrary function. We reiterate that Fk,ω Inserting (G.12) to the formal particle kinetic equation in (G.2), we have

∂ ∂Fa ∂Fa = −Im · D(v) · , ∂t ∂v ∂v   , e2 dk dω kk δφ 2 g a (v). D(v) = a2 k,ω k,ω ma

(G.17)

Upon reintroducing the slow-time derivative associated with ω, we also obtain the wave kinetic equation from (G.15), upon taking the imaginary part, (k,ω) , 2 ∂ , 2δφ δφ = 2Im . k,ω k,ω ∂t ∂ (k,ω)/∂ω

(G.18)

G.3 Green’s Function for Resonant Velocity Denominator

329

To sum up, the renormalized quasilinear kinetic theory comprises of the particle kinetic equation (G.17), wave kinetic equation (G.18), dispersion relation (G.15) with the dielectric response a , function defined by (G.14), and the equation for renormalized resonant velocity denominator, gk,ω given by (G.16). The equations of renormalized quasilinear theory are formal, but their solution is not so straightforward to obtain. In what follows, we introduce an approximate analysis. Before we do that, however, let us introduce Green’s function for the resonant velocity denominator.

G.3 Green’s Function for Resonant Velocity Denominator At this point Rudakov and Tsytovich (1971) introduce another quantity, Gak,ω (v,v ), defined by  a (v) H (v) = gk,ω

dv Gak,ω (v,v ) H (v ).

(G.19)

a (v ). dv Gak,ω (v,v ) δfk,ω

(G.20)

a (v), then we have If we take H (v) = δfk,ω

 a (v) δf a (v) = gk,ω k,ω

a (v) = Ga (v,v ) in (G.16), we obtain Upon taking Fk,ω k,ω



  , e2 ∂ · dk dω k k δφ 2   (ω − k · v) − a2 k ,ω ma ∂v  ∂ a Gak,ω (v,v ) = Gak,ω (v,v ). × gk−k ,ω−ω (v) · ∂v

a (v) gk,ω

(G.21)

Upon choosing H (v) = (ω − k · v) Gak,ω (v,v ) in (G.19), it follows that a (v) (ω − k · v) Ga (v,v ) = gk,ω k,ω



dv Gak,ω (v,v ) (ω − k · v ) Gak,ω (v,v ).

(G.22)

a By simply making use of different spectral arguments in the definition (G.19), namely, gk−k ,ω−ω (v)   a H (v) = dv Gk−k,ω−ω (v,v ) H (v ), and making the choice of

H (v) = k ·

∂ a G (v,v ), ∂v k,ω

(G.23)

we obtain a  gk−k ,ω−ω (v) k ·

∂ a G (v,v ) = ∂v k,ω



dv Gak−k,ω−ω (v,v ) k ·

∂ Ga (v,v ). ∂v k,ω

(G.24)

We make use of (G.19) again, but this time we choose H (v) = k ·

∂ a  ∂ Ga (v,v ), g   (v) k · ∂v k−k ,ω−ω ∂v k,ω

(G.25)

330

On Renormalized Kinetic Turbulence Theory

in order to arrive at a (v) k · ∂ g a  ∂ a  gk,ω k−k,ω−ω (v) k · ∂v Gk,ω (v,v ) ∂v  ∂ a ∂   Ga (v,v ) = dv Gak,ω (v,v ) k ·  gk−k ,ω−ω (v ) k · ∂v ∂v k,ω  ∂ = dv Gak,ω (v,v ) k ·  ∂v  ∂ × dv Gak−k,ω−ω (v,v ) k ·  Gak,ω (v,v ), ∂v

where we have made use of (G.24) in the last equality. Making use of (G.22) and (G.26), (G.21) can be rewritten as follows:     , e2 ∂  · dk dω k k δφ 2   dv (ω − k · v) δ(v − v ) − a2 k ,ω ma ∂v  ∂ × Gak−k,ω−ω (v,v ) ·  Gak,ω (v,v ) = δ(v − v ), ∂v

(G.26)

(G.27)

where we have made some trivial replacements of dummy integral variables. This equation constitutes the differential equation for Green’s function Gak,ω (v,v ), which will be coupled to the perturbed a (v) via (G.20). distribution function δfk,ω

G.4 Approximate Analysis To solve (G.27), let us consider the solution near linear wave-particle resonance regime, η ≡ ω − k · v ≈ 0.

(G.28)

It turns out that the solution Gak,ω (v,v ) is proportional to δ(v−v ) for this case. With this a posteriori knowledge, we may expand Gak,ω (v,v ) by Gak,ω (v,v ) = Gk,ω (v) δ(v − v ) + η Gk,ω (v) + · · · . a(0)

Inserting this to (G.27), we have % 

e2 ∂ · dv η δ(v − v ) − a2 ma ∂v

a(1)



dk

× Gk−k,ω−ω (v) δ(v − v ) · a(0)

 

, dω k k δφ 2

(G.29)

k,ω

∂ a(0) Gk,ω (v ) δ(v − v ) ∂v

= δ(v − v ). Upon carrying out the trivial velocity integration, we have    , e2 ∂ · dk dω k k δφ 2   η − a2 k ,ω ma ∂v  ∂ a(0) a(0) Gk,ω (v) δ(v − v ) = δ(v − v ). × Gk−k,ω−ω (v) · ∂v

(G.30)

(G.31)

G.4 Approximate Analysis

331

Making note of the fact that ∂ ∂ = −k , ∂v ∂η we obtain   ∂ ∂ a(0) η+ D(η) Gk,ω (η) = 1, ∂η ∂η e2 D(η) = − a2 ma



k ·

dk



∂ ∂ = −(k · k ) . ∂v ∂η

, dω δφ 2

(G.32)

(k · k )2 Gk−k,ω−ω (η). (G.33) a(0)

k,ω

This is a nonlinear equation, but if we keep in mind that the present scheme is valid for η ≈ 0, then we a(0) a(0) may simply replace Gk−k,ω−ω (η) that appears in D(η) by Gk−k,ω−ω (0). Let us call the resultant diffusion coefficient associated with this approximation, −iD0 ,  ∂D(η)  D(η) = D(0) + η + ··· , ∂η η=0 D(0) ≡ −iD0,   , e2 a(0) D0 = i a2 dk dω δφ 2   (k · k )2 Gk−k,ω−ω (0). k ,ω ma Upon inserting (G.34) to (G.33), we have 

∂2 η − iD0 2 ∂η

(G.34)

 a(0)

Gk,ω (η) = 1.

(G.35)

To solve this equation, let us express the solution as  ∞ 1 a(0) ˆ a(0) (τ ) exp(iητ ), Gk,ω (η) = dτ G k,ω 2π −∞ ˆ a(0) (τ ) = 0, G k,ω

τ < 0.

(G.36)

Then we have  ∞ 1 ∂ 2 a(0) ˆ a(0) (τ ) exp(iητ ), G (η) = − dτ τ 2 G k,ω 2π 0 ∂η2 k,ω  ∞ i ∂ ˆ a(0) a(0) dτ exp(iητ ) (τ ), η Gk,ω (η) = G 2π 0 ∂τ k,ω  ∞ 1 1= dτ 2π δ(τ ). 2π 0

(G.37)

From this we have ˆ a(0) (τ ) ∂G k,ω ∂τ

ˆ a(0) (τ ) = −2π i δ(τ ). + τ 2 D0 G k,ω

For τ = 0, the solution is obtained as follows:   τ3 a(0) ˆ , Gk,ω = g0 exp −D0 3

g0 = const.

(G.38)

(G.39)

332

On Renormalized Kinetic Turbulence Theory

To obtain the coefficient g0 , let us integrate (G.38) from τ = − to τ = , where 1:  −

ˆ ∂G k,ω (τ ) a(0)



∂τ

+

 −

ˆ a(0) (τ ) = −2π i dτ τ 2 D0 G k,ω

 −

(G.40)

dτ δ(τ ),

or ˆ a(0) (τ = − ) = −2π i. ˆ a(0) (τ = ) − G G k,ω k,ω

(G.41)

ˆ a(0) (τ = − ) = 0, or G ˆ a(0) (τ ) = 0, for τ < 0, we have Since G k,ω k,ω ˆ a(0) (τ = ) = g0 = −2π i. lim G k,ω

(G.42)

→0

Inserting the solution (G.39) to (G.36), we arrive at   ∞

a(0) Gk,ω (v) = −i

0

τ3 dτ exp i (ω − k · v) τ − D0 3

 .

(G.43)

Inserting this result to (G.29), and inserting the resultant expression to (G.20), one arrives at    ∞ τ3 a . (G.44) gk,ω (v) = −i dτ exp i (ω − k · v) τ − D0 (k,ω) 3 0 If we set D0 = 0, then we have the customary expression for the resonant velocity denominator that characterizes the weak turbulence theory. The quantity D0 is yet to be determined, but it can be formally solved from (G.33) and (G.43): D0 (k,ω) =

ea2



m2a ×

 ∞ 0

dk



, dω δφ 2 

(k · k )2

k,ω

dτ exp i [ω − ω − (k − k ) · v] τ − D0 (k − k,ω − ω )

τ3 3

 .

(G.45)

This formal equation does not enjoy any closed-form solution. Consequently, we need further approximation.

G.5 Resonance-Broadening Approximation As a further approximation, let us ignore D0 (k − k,ω − ω ) in (G.45) to write D0 (k,ω) =

ea2



m2a ×

 ∞ 0

dk



, dω δφ 2

k,ω

(k · k )2

< = dτ exp i [ω − ω − (k − k ) · v] τ .

(G.46)

Carrying out the τ integration and taking the real part, we have D0 (k,ω) =

π ea2 m2a



dk



, dω δφ 2

k,ω

(k · k )2 δ[ω − ω − (k − k ) · v].

(G.47)

G.5 Resonance-Broadening Approximation

333

Upon carrying out the ω integration by virtue of the delta function, we obtain D0 (k,ω) =

π ea2



m2a

, dk δφ 2

k,ω−k·v+k ·v

(k · k )2 .

(G.48)

Inserting this result to (G.44), we arrive at the renormalized resonant velocity denominator obtained under the resonance-broadening approximation, a (v) = −i gk,ω



 ∞ 0

π ea2

 dτ exp i (ω − k · v) τ



, dk δφ 2

k,ω−k·v+k ·v

m2a

(k · k )2

 τ3 . 3

(G.49)

With this approximate result, we now have a set of formal equations for renormalized quasilinear theory,     , e2 ∂ ∂Fa (v) ∂Fa (v) a (v) δφ 2 = −Im a2 gk,ω , dk dω k · k· k,ω ∂t ∂v ∂v ma , ∂ δφ 2 2 (k,ω) , 2 k,ω = −Im δφ , k,ω ∂t ∂ (k,ω)/∂ω 2   ωpa a (v) k · ∂Fa = 0. dv gk,ω (k,ω) = 1 + 2 ∂v k a

(G.50)

In order to render the formal theory in (G.50) to a more concrete form, we must consider the eigenmode solution, or, equivalently, the dispersion relation. Suppose that the eigenmode ω = ωkα can be approximately obtained from the real part of the renormalized dielectric constant, Re (k,ωkα ) = 0, then we may write the eigenmode amplitude as , k 2 δφ 2

k,ω

=



Ikα δ(ω − ωkα ).

(G.51)

α

Then we have ∂ ∂Fa (v) =− ∂t ∂vi ∂Ikα ∂t (k,ωkα ) gkαa (v)



 ea2 

ki kj



∂Fa (v) , 2 ∂vj ma α 2 Im (k,ωkα ) =− I α, ∂ Re (k,ωkα )/∂ωkα k 2   ωpa ∂Fa = 0. dv gkαa (v) k · =1+ 2 ∂v k a   ∞ = −i dτ exp i (ωkα − k · v) τ 0   π e2  (k · k )2 τ 3 β β . − 2a dk Ik δ[ωkα − ωk − (k − k ) · v] 3 ma β k 2 Im

dk 2 gkαa (v) Ikα k

(G.52)

This result represents the renormalized quasilinear theory. While the result is self-consistent, the final set of equations is not so straightforward to analyze.

334

On Renormalized Kinetic Turbulence Theory 4 D = 0.01

6

– Ri( z, D)

Rr( z, D)

D = 0.01

D = 0.1

2

D = 0.5 0

4 D = 0.1

2

D = 0.5

–2

0 –4 –4

–2

0

2

4

–4

–2

0

2

4

z

z

Figure G.1 Real (left) and imaginary (right) parts of the broadened resonance function R(z,D) for several values of D. We close the present Appendix by considering the resonance function,    ∞ τ3 , Im z > 0, dτ exp izτ − D R(z,D) ≡ −i 3 0

(G.53)

which is related to the function gkαa (v) defined in (G.52). If D → 0, then we simply obtain R(z,D → 0) =

1 1 ≈ P − iπ δ(z), z z

(G.54)

for small Im z. For finite D, the function R(z,D) depicts the broadened resonance function. Numerical plots of real and imaginary parts of R(z,D) are shown in Figure G.1 for D = 0.01, 0.1, and 0.5.

Appendix H One-Dimensional Normalized Equations

In this Appendix, we further consider the normalized equations (6.13)–(6.15) and take the limit of one-dimensional situation. We assume that the ions are stationary. The velocity space diffusion coefficient D is of interest to us,  ∞  dq δ(σ xqL − qu) Eσ L (q) D= −∞

σ =±1

%   ∞ 2 dq δ q − = |u| 0

xqL u



 E+L (q) + δ q +

xqL



u

 E−L (q) .

(H.1)

Consequently, we have    ∂e ∂ ∂e  D , = ∂T diff. ∂u ∂u

2 (u) E+L (q) + (−u) E−L (q) q=x L /|u| , D= q |u| where

1 xqL =

2

1 + 3q /4  − 1 + 3q 2 /4

(q > 0) (q < 0)

.

(H.2)

(H.3)

The velocity drag term is likewise manipulated as  ∂ ∂e  [(u) − (−u)] e (u). = 2g ∂T drag ∂u

(H.4)

Spontaneous and induced emission terms in normalized one-dimensional form are given by    π g ∂e (u) ∂E±L (q)  L e (u) ± xq E±L (q) = 2 ,  ∂T q ∂u q emiss. u=±xqL /q   π μq g ∂E±S (q)  [e (u) + i (u)] =  ∂T q2 q emiss.   ∂e (u) 2u − i (u) , (H.5) ±xqL E±S (q) ∂u τ u=±x S /q q

335

336

One-Dimensional Normalized Equations

where  xqS = A=

Aq/(1 + q 2 /2)1/2 −Aq/(1 + q 2 /2)1/2

(q > 0) , (q < 0)

(1 + 3τ )1/2 . (2M)1/2

(H.6)

The decay terms are considered next. Consider the decay terms for L mode,     ∞ ∂Eσ L (q)  1 + 3τ 1/2  L = σ x dq  |q − q  | q ∂T decay 2M −∞   σ ,σ =±1 !

L  S σ xqL Eσ  L (q  ) Eσ  S (q − q  ) × δ σ xq − σ xqL − σ  xq−q  " L  − σ  xqL Eσ  S (q − q  ) Eσ L (q) − σ  xq−q  Eσ  L (q ) Eσ L (q) .

(H.7)

In order to explicitly carry out the q  integration via three wave resonance condition, we first rewrite the q  integral over positive range by invoking the symmetry property, x−q = −xq , then we write out the double summations over indices σ  = ±1 and σ  = ±1, and evaluate the following three wave resonance delta functions of all possible combinations:   

3 2 2 S (q − q  ) − A (q − q  ) δ xqL − xqL − xq−q  = δ 4       4A 4/3 4A  − q δ q + q − + δ(q  − q) , = |2q − 4A/3| 3 3

 4/3 S δ xqL − xqL + xq−q δ(q  − q),  = |2q + 4A/3|   

3 2 2 S (q + q  ) − A (q − q  ) : no root, δ xqL + xqL − xq−q  = δ 4   

3 2 2 S (q + q  ) + A (q − q  ) : no root, δ xqL + xqL + xq−q  = δ 4   

3 S 2 + q  2 ) − A (q + q  ) : no root, (q = δ δ xqL + xqL − xq+q  4   

3 2 S (q 2 + q  ) + A (q + q  ) : no root, δ xqL + xqL + xq+q  = δ 4     

4A 4/3 S  − q + 4A ,  q − δ q = δ xqL − xqL − xq+q  |2q − 4A/3| 3 3   

4/3 4A S δ q − q − . (H.8) δ xqL − xqL + xq+q  = |2q + 4A/3| 3 Making use of (H.8), (H.7) is now written explicitly as  ∂E±L (q)  = q∗ xqL xqL E∓L (q + q∗ ) E∓S (2q + q∗ )  ∂T decay ! " # L L − xq+q E (2q + q ) − x E (q + q ) E (q) ∗ ∗ ∓S ∓L ±L 2q+q∗ ∗ + q∗ (q − q∗ ) xqL xqL E∓L (q − q∗ ) E±S (2q − q∗ )

One-Dimensional Normalized Equations !

337 "

L L − xq−q E (2q − q∗ ) + x2q−q E (q − q∗ ) E±L (q) ∗ ±S ∗ ∓L

#

+ q∗ (q∗ − q) (2q − q∗ ) xqL xqL E±L (q∗ − q) E±S (2q − q∗ ) " # ! L E (q − q) E±L (q) − xqL∗ −q E±S (2q − q∗ ) + x2q−q ∗ ±L ∗ + q∗ (q∗ − q) (q∗ − 2q) xqL xqL E±L (q∗ − q) E±S (q∗ − 2q) " # ! − xqL∗ −q E±S (q∗ − 2q) − xqL∗ −2q E±L (q∗ − q) E±L (q) , q∗ =

4 3



 1 + 3τ 1/2 . 2M

(H.9)

We may likewise manipulate for the S-mode, where the various possible three wave resonance conditions are given below:    2/3

q + 4A/3 L (q  − q) δ q  − δ xqS − xqL − xq−q  = |q| 2     2/3 q + 4A/3 4A  − q δ q − , = |q| 3 2   

 2/3  4A q + 4A/3 L  q− δ q − , δ xqS − xqL + xq−q  = |q| 3 2   

 2/3  4A q − 4A/3 L  k− δ q − , δ xqS + xqL − xq−q  = |q| 3 2    

 2/3 L  − q)  q − 4A δ q  − q − 4A/3 = 0, (q δ xqS + xqL + xq−q =  |q| 3 2   

 2/3  4A 4A/3 − q L  − q δ q − , δ xqS + xqL − xq+q  = |q| 3 2 

L δ xqS + xqL + xq+q  = 0,

 L δ xqS − xqL − xq+q  = 0,   

3 2 3  S L L (H.10) δ xq − xq  + xq+q  = δ Aq + q + qq = 0. 4 2 In (H.10), those delta functions designated as zeros either have no contributions or they cannot be satisfied. Making use of (H.10), we have       q∗ q + q∗ q − q∗ ∂E±S (q)  L xL E (q − q E = ) x ∗ q ∓L q ±L  ∂T 2 2 2 decay       q − q∗ q + q∗ L L − x E − x(q+q E E (q) ∓L ±L ±S (q−q∗ )/2 ∗ )/2 2 2      q∗ q + q∗ q∗ − q (q∗ − q) xqL xqL E±L E±L + 2 2 2       q∗ − q q + q∗ L L − x E − x(q+q E E (q) . (H.11) ±S (q∗ −q)/2 ±L ∗ )/2 ±L 2 2

338

One-Dimensional Normalized Equations

For induced scattering involving L mode, we have   ∞ ! "   ∞ ∂Eσ L (q)  =− dq  du δ σ xqL − σ  xqL − (q − q  )u  ∂T −∞ −∞ ind.sc. σ  =±1   σ xqL ∂   L  L (σ xq − σ xq  ) e (u) − i (u) . × Eσ  L (q ) Eσ L (q) (q − q ) ∂u M (H.12) Again, re-expressing the q  integral over positive range only by making use of the symmetry property, x−q = −xq , and explicitly writing out the summation over the index σ  = ±1, one may obtain    ∞ ∂E±L (q)   u |q − q  | ∂e (u) = ∓ dq  ∂T ∂u 0 ind.sc.   2 q −q + xqL i (u) τ |q − q  | u=±3(q+q  )/4   ∞ ∂ e (u) ∓ dq  u |q + q  | ∂u 0  2 L q + q i (u) + xq τ |q + q  | u=±3(q−q  )/4

E±L (q  ) E±L (q)

E∓L (q  ) E±L (q).

(H.13)

The spontaneous scattering term may also be evaluated likewise:   ∞ ! "   ∞ ∂Eσ L (q)   = −g dq du δ σ xqL − σ  xqL − (q − q  )u  ∂T −∞ −∞ spt.scatt. σ  =±1 ! "

× σ xqL σ  xqL Eσ L (q) − σ xqL Eσ  L (q  ) e (u) + i (u) ⎧  ∞ ⎨ x L E±L (q) − xqL E±L (q  )

q e (u) + i (u) u=±3(q+q  )/4 dq  = −g xqL | ⎩ |q − q 0 ⎫ L ⎬ xq  E±L (q) − xqL E±L (q  )

e (u) + i (u) u=±3(q−q  )/4 . + (H.14)  ⎭ |q + q | Putting all the terms together, we obtain a set of equations that may be solved readily by numerical finite difference scheme, subject to the initial condition specified in Chapter 6:   ∂e (u) ∂ ∂e (u) = Ae (u) + D , ∂T ∂u ∂u ∂ [(u) − (−u)] , A = 2g ∂u

2 (u) E+L (q) + (−u) E−L (q) q=x L /|u| , D= q |u|   π g ∂e (u) ∂E±L (q) = 2 e (u) ± xqL E±L (q) ∂T q ∂u q u=±x L /q q

+ q∗ xqL xqL E∓L (q + q∗ ) E∓S (2q + q∗ )

One-Dimensional Normalized Equations !

"

L L − xq+q E (2q + q∗ ) − x2q+q E (q + q∗ ) E±L (q) ∗ ∓S ∗ ∓L

339 #

+ q∗ (q − q∗ ) xqL xqL E∓L (q − q∗ ) E±S (2q − q∗ ) ! " # L L − xq−q E (2q − q ) + x E (q − q ) E (q) ∗ ∗ ±S ∓L ±L 2q−q∗ ∗ + q∗ (q∗ − q) (2q − q∗ ) xqL xqL E±L (q∗ − q) E±S (2q − q∗ ) ! " # L − xqL∗ −q E±S (2q − q∗ ) + x2q−q E (q − q) E±L (q) ∗ ±L ∗ + q∗ (q∗ − q) (q∗ − 2q) xqL xqL E±L (q∗ − q) E±S (q∗ − 2q) ! " # − xqL∗ −q E±S (q∗ − 2q) − xqL∗ −2q E±L (q∗ − q) E±L (q)   ∞ ∂e (u) dq  u |q − q  | ∓ ∂u 0   2 q −q i (u) + xqL E±L (q  ) E±L (q) τ |q − q  |  u=±3(q+q )/4   ∞ ∂e (u)   ∓ dq u |q + q | ∂u 0  2 L q + q i (u) + xq E∓L (q  ) E±L (q) τ |q + q  | u=±3(q−q  )/4 ⎧  ∞ ⎨ x L E±L (q) − xqL E±L (q  )

q e (u) + i (u) u=±3(q+q  )/4 dq  − g xqL | ⎩ |q − q 0 ⎫ ⎬ xqL E±L (q) − xqL E±L (q  )

 + (u) +  (u)  e i u=±3(q−q )/4 ⎭ , |q + q  |  π μq g ∂E±S (q) = 2 [e (u) + i (u)] ∂T q q   ∂e (u) 2u − i (u) ± xqL E±S (q) ∂u τ u=±xqS /q      q∗ q + q∗ q − q∗ (q − q∗ ) xqL xqL E±L E∓L + 2 2 2       q − q q + q∗ ∗ L L − x E E E (q) − x(q+q ∓L ±L ±S (q−q∗ )/2 ∗ )/2 2 2      q∗ q + q∗ q∗ − q (q∗ − q) xqL xqL E±L E±L + 2 2 2       − q q q + q∗ ∗ L L − x E E E (q) , − x(q+q ±S (q∗ −q)/2 ±L ∗ )/2 ±L 2 2   4 1 + 3τ 1/2 . (H.15) q∗ = 3 2M

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Index

adiabatic time dependence, 12, 156 aperiodic, 204, 272 Balescu–Lenard equation; electromagnetic, 278 Balescu–Lenard equation; relativistic, 278 Balescu–Lenard–Guernsey equation, 112 Beliaev–Budker equation, 278 bump-on-tail instability, 41, 43, 130, 201 Buneman instability, 317 Cerenkov emission, 101 closure problem, 19 coalescence, 50 collision integral, 112 collisional integral; relativistic, 276 Coulomb logarithm, 115 Debye length, 40, 68 decay, 50, 177, 178, 182, 206, 219–222, 257, 259, 263–266 dispersion relation, 39, 68, 199, 204 dispersion relation; electromagnetic, 202 dispersion relation; electrostatic, 200 dispersion relation; electrostatic dispersion relation, 141 dispersion relation; harmonic, 301, 302 dispersion relation; ion-sound, 40, 142, 201 dispersion relation; Langmuir, 39, 141 dispersion relation; longitudinal, 170 dispersion relation; modified Langmuir, 304 dispersion relation; multiple harmonic, 302 dispersion relation; nonlinear, 299, 304–306 dispersion relation; transverse, 170, 199 eigenmode, 105 electromagnetic instability, 203 electrostatic instability, 200 ensemble, 4 ensemble average, 6

fast-wave condition, 33, 35 fluctuations, 105 Fokker–Planck equation, 99 Fourier–Laplace transformation, 11 Fried–Conte, 34, 309 fully-symmetrized third-order nonlinear susceptibility, 18 fundamental emission, 279, 286 Green’s function, 329 growth rate, 41, 201 H theorem, 118 harmonic emission, 279 higher-order perturbative expansion, 318 homogeneous and stationary, 7, 8 induced decay, 52, 69, 103, 104 induced emission, 48, 68, 69, 103, 104, 177, 181, 205, 219–221, 257, 259, 263, 265 induced scattering, 54, 178, 182, 213, 220, 222, 258, 260, 264, 266 induced scattering; ion sound, 69, 104 induced scattering; off thermal ions, 69, 103 instability; electromagnetic, 203 instability; electrostatic, 200 instability; ion-acoustic, 202 instability; Weibel, 203 ion-acoustic instability, 202 ion-acoustic speed, 40, 68 ion-sound speed, 40, 68 ion-sound wave dispersion relation, 108 Kappa distribution, 122, 135, 139, 150 kinetic temperature, 33 Klimontovich equation, 77, 225 Klimontovich function, 76 Klimontovich function; free particles, 82 Klimontovich function; non-interacting particles, 82 Kramers–Kr¨onig, 111

351

352 Landau damping, 41 Landau damping rate, 43 Landau equation, 114 Landau equation; relativistic, 278 Langmuir condensation, 132 Langmuir wave dispersion relation, 108 linear dielectric conductivity, 16, 18 linear dielectric constant, 18 linear dielectric susceptibility, 16, 33 linear dielectric susceptibility; tensor, 161, 183 linear wave-particle resonance, 49 magnetic field fluctuations, 271 Maxwell’s equation, 75, 155, 225 Maxwell–Boltzmann–Gaussian distribution, 106 non-eigenmode, 105 non-extensive entropy, 123 nonlinear frequency shift, 302 nonlinear frequency shift; ion-acoustic, 306 nonlinear frequency shift; Langmuir, 303 partial third-order nonlinear susceptibility, 18 particle kinetic equation, 70, 125, 128, 170, 218, 238, 239, 280 particle kinetic equation; discrete-particle effects, 99 plasma dispersion function, 34, 203, 270, 309 plasma dispersion function; asymptotic expansion, 310 plasma dispersion function; continued fraction, 311 plasma dispersion function; differential equation, 311 plasma dispersion function; series expansion, 310 plasma emission, 278 plasma frequency, 18, 68 plasma parameter, 115, 127 plasmon, 52 Plemelj formula, 296 Poisson equation, 9 principal value, 28 quasi-thermal noise spectroscopy, 111 quasilinear, 49, 169, 238, 282 quasilinear; diffusion, 61 quasilinear; renormalized, 328, 329, 333 quasilinear; saturation, 130, 131, 149

Index radiation; non-thermal, 278 radiation; thermal, 272, 275 random phase, 8 reactive, 71, 272, 313 renormalization, 323 renormalized, 72, 325, 326 renormalized; quasilinear, 328 resonance broadening, 332 Rosenbluth potentials, 119 second-harmonic emission, 287 second-order nonlinear susceptibility, 16, 35 second-order nonlinear susceptibility; tensor, 161, 186 slow-wave condition, 33, 36 solar radio bursts, 278 solar wind, 150 spectral balance equation, 22, 92, 93, 163, 233, 238 spontaneous decay, 52, 68, 69, 103, 104 spontaneous emission, 101, 103, 104, 257, 259, 260, 262, 263, 265 spontaneous scattering, 102–104, 258–262, 264, 266 Sturrock’s dilemma, 284 symmetry property, 32 thermal speed, 33 thermal speed; electron, 68 thermal speed; proton, 68 third-harmonic emission, 287 third-order nonlinear susceptibility, 17, 37 third-order nonlinear susceptibility; tensor, 161, 190 three-wave resonance, 50 turbulent quasi equilibrium, 133, 148 type II radio bursts, 279 type III radio bursts, 278 velocity diffusion, 61, 99, 240 velocity drag, 99, 240 velocity friction, 99, 240 Vlasov equation, 9, 155 wave intensity, 26, 100 wave intensity; ion-sound, 103, 133 wave intensity; Langmuir, 130, 131 wave intensity; transverse, 221 wave kinetic equation, 68, 125, 128, 280 Weibel instability, 203