The Vlasov equation 1, History and general properties 9781786302618, 1786302616

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The Vlasov Equation 1

Series Editors Pierre-Noël Favennec and Frédérique de Fornel

The Vlasov Equation 1 History and General Properties

Pierre Bertrand Daniele Del Sarto Alain Ghizzo

First published 2019 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.

Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address: ISTE Ltd 27-37 St George’s Road London SW19 4EU UK

John Wiley & Sons, Inc. 111 River Street Hoboken, NJ 07030 USA

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© ISTE Ltd 2019 The rights of Pierre Bertrand, Daniele Del Sarto and Alain Ghizzo to be identified as the author of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988. Library of Congress Control Number: 2019943752 British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library ISBN 978-1-78630-261-8

Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix

Chapter 1. Introduction to a Universal Model: the Vlasov Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.1. A historical point of view . . . . . . . . . . . . . . . . . . . . . 1.2. Individual and collective effects in plasmas . . . . . . . . . . . 1.3. Graininess parameter . . . . . . . . . . . . . . . . . . . . . . . 1.4. The collective description of a Coulomb gas: an intuitive approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5. From N -body to Vlasov . . . . . . . . . . . . . . . . . . . . . 1.6. The graininess parameter and 1D, 2D or 3D models . . . . . . 1.7. The Vlasov equation at the microscopic fluctuations level . . 1.8. The Wigner equation (Vlasov equation for quantum systems) 1.9. The relativistic Vlasov–Maxwell model . . . . . . . . . . . . . 1.10. References . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . .

1 5 7

. . . . . . .

8 12 16 19 21 26 28

Chapter 2. A Paradigm for a Collective Description of a Plasma: the 1D Vlasov–Poisson Equations . . . . . . . . . . .

31

2.1. Introduction . . . . . . . . . . . . . . . . . . . . . . 2.2. The linear Landau problem . . . . . . . . . . . . . . 2.2.1. The Maxwellian case . . . . . . . . . . . . . . . 2.2.2. Landau poles and others . . . . . . . . . . . . . 2.2.3. Unstable plasma: two-stream instability . . . . 2.3. The 1D cold plasma model: nonlinear oscillations . 2.3.1. Hydrodynamic description . . . . . . . . . . . . 2.3.2. Lagrangian formulation through the Von Mises transformation . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

31 33 34 36 38 39 39

. . . . . . .

40

vi

The Vlasov Equation 1

2.3.3. The wave-breaking phenomenon . . . . . . . . . . . . . 2.4. The water bag model . . . . . . . . . . . . . . . . . . . . . 2.4.1. Basic equations . . . . . . . . . . . . . . . . . . . . . . 2.4.2. Linearized theory . . . . . . . . . . . . . . . . . . . . . 2.4.3. Water bag hydrodynamic description . . . . . . . . . . 2.5. Connection between the hydrodynamic, water bag and Vlasov models . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1. A Vlasov hydrodynamic description . . . . . . . . . . . 2.5.2. Vlasov numerical simulations of P n−3 . . . . . . . . . 2.5.3. The fundamental contribution of poles besides Landau 2.6. The multiple water bag model . . . . . . . . . . . . . . . . 2.6.1. A multifluid description . . . . . . . . . . . . . . . . . . 2.6.2. Linearized analysis . . . . . . . . . . . . . . . . . . . . 2.7. Further remarks . . . . . . . . . . . . . . . . . . . . . . . . 2.8. References . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . .

. . . . .

. . . . .

42 44 44 47 48

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

50 50 52 56 58 59 63 66 71

Chapter 3. Electromagnetic Fields in Vlasov Plasmas: General Approach to Small Amplitude Perturbations . . . . . .

75

3.1. Introduction and overview of the chapter . . . . . . . . . . . . 3.2. Linear analysis of the Vlasov–Maxwell system: general approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1. Dispersion relation and response matrix . . . . . . . . . . 3.2.2. The choice of the basis for the response tensor . . . . . . . 3.2.3. About the number of “waves” in plasmas . . . . . . . . . . 3.2.4. Real or complex values of k and ω: steady state and initial value problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Polynomial approximations of the dispersion relation: why and how to use them . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1. Truncated-Vlasov and fluid–plasma descriptions for the linear analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2. Wave dispersion and resonances allowed by inclusion of high-order moments in fluid models . . . . . . . . . . . . . . . . 3.3.3. An example: fluid moments and Finite–Larmor–Radius effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.4. Key points about approximated normal mode analysis . . 3.4. Vlasov plasmas as collisionless conductors with polarization and finite conductivity: meaning of plasma’s “dielectric tensor” . 3.4.1. Polarization charges and wave equation in dielectric materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2. The “equivalent” dielectric tensor and its complex components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

75

. . . .

77 81 83 89

.

92

.

93

.

96

.

99

. 103 . 108 . 109 . 112 . 115

Contents

3.4.3. Temporal and spatial dispersion in plasmas . . . . . . . . . 3.4.4. Conductivity and collisional resistivity in Vlasov plasmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5. Symmetry properties of the complex components of the equivalent dielectric tensor and energy conservation . . . . . . . . 3.5.1. Onsager’s relations . . . . . . . . . . . . . . . . . . . . . . 3.5.2. Poynting’s theorem . . . . . . . . . . . . . . . . . . . . . . 3.5.3. Symmetry of the coefficients of the equivalent dielectric tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.4. More about Onsager’s relations for wave dispersion . . . . 3.5.5. Energy dissipation versus real and imaginary parts of σij and ij . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

vii

. 120 . 122 . 126 . 126 . 129 . 130 . 134 . 138 . 141

Chapter 4. Electromagnetic Fields in Vlasov Plasmas: Characterization of Linear Modes . . . . . . . . . . . . . . . . . . 147 4.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Characterization of electromagnetic waves and of wave-packets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1. Polarization of electromagnetic waves in plasmas . . . . . 4.2.2. Phase velocity, group velocity and refractive index . . . . 4.2.3. Example of propagation in unmagnetized plasmas: underdense and overdense regimes . . . . . . . . . . . . . . . . . 4.2.4. Example of propagation in magnetized plasmas: ion-cyclotron resonances and Faraday’s rotation effect . . . . . . 4.2.5. Wave–particle resonances, Landau damping and wave absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.6. Resonance and cut-off conditions on the refractive index . 4.2.7. Graphical representations of the dispersion relation . . . . 4.3. Instabilities in Vlasov plasmas: some terminology and general features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1. Linear instabilities . . . . . . . . . . . . . . . . . . . . . . . 4.3.2. Absolute and convective instabilities and some other classification criteria . . . . . . . . . . . . . . . . . . . . . . . . . 4.4. On some complementary interpretations of the collisionless damping mechanism in Vlasov plasmas . . . . . . . . . . . . . . . 4.4.1. Landau damping as an inverse Vavilov–Cherenkov radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2. Landau damping in N -body “exact” models . . . . . . . .

. 147 . 148 . 153 . 156 . 161 . 166 . 172 . 176 . 178 . 182 . 184 . 192 . 198 . 199 . 203

viii

The Vlasov Equation 1

4.4.3. Some final remarks about interpretative issues of collisionless damping in Vlasov mean field theory . . . . . . . . . 206 4.5. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 Chapter 5. Nonlinear Properties of Electrostatic Vlasov Plasmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 5.1. The Vlasov–Poisson system . . . . . . . . . . . . . . . . . . . 5.2. Invariants of the Vlasov–Poisson model . . . . . . . . . . . . . 5.3. Stationary solutions: Bernstein–Greene–Kruskal equilibria . . 5.4. Some mathematical properties of the Vlasov equation . . . . . 5.5. The Bernstein–Greene–Kruskal solutions . . . . . . . . . . . . 5.5.1. The case of (electrostatic) two-stream instability . . . . . . 5.5.2. Chain of BGK equilibria . . . . . . . . . . . . . . . . . . . 5.5.3. Stability of the periodic BGK steady states . . . . . . . . . 5.6. Traveling waves of BGK-type solutions . . . . . . . . . . . . . 5.7. Role of minority population of trapped particles . . . . . . . . 5.7.1. Nonlinear Landau damping and the emergence of the nonlinear Langmuir-type wave . . . . . . . . . . . . . . . . . . . 5.7.2. Electron acoustic wave in the nonlinear Landau damping regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7.3. Kinetic electrostatic electron nonlinear waves . . . . . . . 5.7.4. Emergent resonance for KEEN waves . . . . . . . . . . . . 5.8. Nature of KEEN waves and NMI . . . . . . . . . . . . . . . . 5.8.1. Adiabatic model for a single linear wave: the (electrostatic) trapped electron mode model . . . . . . . . . . . . . . . . . . . . 5.8.2. The Dodin and Fisch model connected to the emergence of KEEN waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.9. Electron hole and plasma wave interaction . . . . . . . . . . . 5.10. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . Index

. . . . . . . . . .

215 216 217 220 229 230 235 236 242 245

. 247 . . . .

254 260 268 270

. 270 . 274 . 281 . 291

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297

Preface

Plasma physics is a rather young branch of modern science. The reason is that plasmas are not naturally present on Earth, except during a thunderstorm. But around 1920–1930, the pioneering works of Irving Langmuir and Lewis Tonks on electrical discharges pointed to the peculiar properties of a “new medium”: the plasma state. During this same period, some of these properties were being indirectly and independently observed in the measurements of radio-wave transmission through that part of the atmosphere which at that time was being discovered: the ionosphere, which represents the “gas” in a natural, persistent plasma state that is closest to Earth’s surface. In a neutral gas, the main physical properties can be explained in terms of a statistical description of the mutual interactions of a large number of particles (usually neutrally charged molecules), where each particle interacts (i.e. “collides”) with just a very few other particles at a time. On the contrary, in a plasma, the presence of free charged particles (ions and electrons) points to a different and new physical behavior, since a single charge can interact with a huge number of other charged particles at the same time. These collective properties can be explained as due to the long-range interaction of Coulomb forces, which decays as r−2 , while the number of particles at a distance r from a test particle grows as r+2 . This number can be larger than 108 in astrophysical plasmas (sun, stars, interstellar medium etc.) as well as in laboratory plasmas for thermonuclear fusion experiments (by magnetic or inertial confinement). In the limit in which this number can be considered as infinite in a mathematical sense, a master equation can be derived which makes it possible to describe the collective behavior of plasmas. This is the well-known (at least by plasma physicists!) Vlasov equation, introduced by A. Vlasov in 1945 and L. D. Landau in 1946.

x

The Vlasov Equation 1

However, the modeling of a Vlasov plasma is not that easy, especially because of the intrinsic nonlinearities of the equations involved. As this represents an active field of research, approaching it by relying on modern scientific literature can be a quite challenging task for a newcomer, despite several excellent books that already exists about the physics of Vlasov plasmas. Most books on this subject tend to focus either on the physics of “warm/diluted plasmas”, to which Vlasov formalism applies, or to more strictly mathematical properties of the Vlasov equation, related to issues more relevant to applied mathematics (convergence, integrability, etc.). The present book aims to privilege a discussion of the general properties of the Vlasov equation and of its applications, which are presented to the reader from a historical and – hopefully – pedagogical point of view. Although plasma physics itself is a rather young discipline, being less than 100 years old, one of us (Pierre Bertrand) has been working on this subject for more than 50 years. He has been a modest actor, but he has been involved during more than half of the “history” of plasma physics. Thus, it seemed to us that introducing some of the historical works which were carried out during all these years (a task which is mainly accomplished in the first two chapters of the present book) would help the understanding of more current works, whose discussion is developed in the following chapters: notably, the implications of the Vlasov formalism for describing the propagation of small amplitude electromagnetic waves in a plasma (Chapters 3 and 4), and the nonlinear behavior of the latter, in the electrostatic Vlasov limit (Chapter 5). This book is addressed to students in plasma physics and to young researchers, as well as to whomever wants to get a good understanding of a Vlasov plasma, in order to be able to grasp the essence of its main properties. Pierre B ERTRAND Daniele D EL S ARTO Alain G HIZZO June 2019

1 Introduction to a Universal Model: the Vlasov Equation

1.1. A historical point of view Halfway between the N body model and the usual hydrodynamics, the Vlasov equation, or better the Vlasov model, describes different media going from nuclear matter to the expanding universe (via semiconductors, plasmas and stellar dynamics problems and the introduction of a quantum counterpart, the so-called Wigner equation). The aim of this book is to provide the reader with a good knowledge of the Vlasov model, which offers a specific mathematical description for different parts of physics or astrophysics.

Figure 1.1. Sir James Hopwood Jeans (1877–1946)

The Vlasov Equation 1: History and General Properties, First Edition. Pierre Bertrand; Daniele Del Sarto and Alain Ghizzo. © ISTE Ltd 2019. Published by ISTE Ltd and John Wiley & Sons, Inc.

2

The Vlasov Equation 1

About 100 years ago, Jeans (1915) for the first time (to our knowledge) considered this equation to study the behavior of an infinite number of interacting masses (galaxies)1. But due to analytical difficulties, he considered only time-independent problems. But when Vlasov (1945) and Landau (1946) gave the first time-dependent solution for the plasma case, the plasma physicists took the leadership in the study of this equation and called it .... the Vlasov equation.

Figure 1.2. Anatoly Alexandrovitch Vlasov (1908–1975)

Figure 1.3. Lev Davidovitch Landau (1908–1968) (Nobel prize 1962)

1 See also the classical textbook by Chandrasekhar (1942).

Introduction to a Universal Model: the Vlasov Equation

3

For a historical review on the Vlasov equation, the reader is also referred to an interesting paper by M. Hénon (1982). The aim of this book is to present an overview of the Vlasov model in its various aspects: 1) From a physical point of view, what situations and which physical systems are described by this model? 2) From a mathematical point of view, what problems can be solved analytically? 3) Another concern is the link with other models, especially hydrodynamic or more specifically magnetohydrodynamic (MHD). 4) A further point deals with the numerical aspect of the problem and is connected with the huge field of computer simulation with hundreds of papers published each year. We will concentrate on what, in our opinion, is the central problem, putting aside technical and often non-trivial aspects. The word “plasma” was introduced by I. Langmuir (1881–1957) around 1920.

Figure 1.4. Irving Langmuir (1881–1957) (Nobel prize, Chemistry, 1932)

Together with Lewis Tonks, he discovered the phenomenon of electronic oscillations in electrical discharges and for the first time introduced the idea of collective phenomena. Considering a homogeneous neutral distribution of ions and electron (with equal electron ne and ion density ni ), if an electron (e, me ) is moved from its equilibrium position, the other particles create a net charge

4

The Vlasov Equation 1

and our test electron further oscillates around its equilibrium position with a “frequency” (or rather a pulsation ωp ) given by the well-known formula  ωp =

ne e 2  0 me

This collective behavior of charged particles was also observed at the same time by Peter Debye (1884–1966) together with Erich Huckel in electrolytes.

Figure 1.5. Peter Debye (1884–1966) (Nobel prize, Chemistry, 1936)

Considering a test ion, the canonical equilibrium distribution of surrounding electrons at temperature Te is given by the Laplace equation for the so-called Debye screening (eφ/κTe ) =

1 ne − ni λ2D ne

where λD is the well-known Debye length given by  λD =

0 κTe ne e 2

As a matter of fact, Debye length λD and plasma frequency ωp are fundamental quantities in plasma physics and further details can be found in elementary textbooks. In a usual weakly ionized or even neutral gas, binary interactions involve only a limited number of particles and we have to cope with individual

Introduction to a Universal Model: the Vlasov Equation

5

phenomena. On the other hand, in a high-temperature tokamak plasma (Te = 108 K) particles are quite ionized, and the electromagnetic interactions, involving a large number of charges, point to the idea of collective phenomena. Let us clarify this concept. 1.2. Individual and collective effects in plasmas A more rigorous derivation of long-range versus short-range interactions can be obtained by considering a simplified plasma model consisting of discrete charged particles: electrons and massive (immobile) ions. The electron gas (charge e, mass me , density ne and temperature Te ) is immersed in a fixed neutralizing homogeneous ion background. The discrete (individual) character implies the existence of Coulomb collisions. Suppose a large deflection angle (see Figure1.6).

Figure 1.6. Large angle collision

The impact parameter Pimpact is defined as the distance between the electron and the attractive ion when the thermal energy of the electron κTe and the potential energy are of the same order of magnitude κTe ∼

e2 0 Pimpact

giving an estimate of the mean free path mf p mf p ∼

1 ne



0 κTe e2

2

and the collision frequency νcoll ∼

 κTe /me mf p

6

The Vlasov Equation 1

Obviously, particles experience large angle collisions if the distance between them is smaller than the Debye length. Above λD the Debye screening is the dominant phenomenon. Introducing the interparticle distance inter ∼ n−3 e this condition can be written as λD  inter or equivalently ne λ3D  1

[1.1]

The dimensionless parameter ne λ3D appears to play a fundamental role. It is a measure of the number of electrons in a Debye cube. Actually, in our model, we have to deal with two characteristic lengths: the mean free path mf p and the Debye length λD . In the same way, we have two characteristic frequencies: the collision frequency νcoll and the plasma frequency ωp . Individual phenomena are characterized by a typical length mf p and a typical frequency νcoll . Collective phenomena are characterized by a typical length D and a typical frequency ωp . The relations between these characteristic quantities are summarized in Table 1.1. The parameter ne λ3D appears to play a fundamental role; it gives the scaling between collective and individual phenomena. A thought experiment as suggested by Rostoker and Rosenbluth (1960) (see also papers cited therein) will help to understand the role of this parameter. Let us imagine a dichotomy process in which each electron is cut into two parts, each “half-electron” being cut into two parts again and so on: dichotomy: (e, me ) → 2(e/2, me /2) → 4(e/4, me /4) → · · · giving the mathematical limit e → 0, me → 0, ne → ∞ and Te → 0 but leaving invariant the following quantities e/me = const, Te /me = const and ne e = const.

Introduction to a Universal Model: the Vlasov Equation

Characteristic lengths

Characteristic frequencies

Individual phenomena Mean free path  2 0 κTe /e2 mf p ∼ n−1 e

7

Collective phenomena Debye length  λD = 0 κTe /ne e2

mf p ∼ (ne λ3D )λD Collision frequency Plasma frequency  νcoll ∼ κTe /me /mf p ωp2 = ne2 /0 me νcoll ∼ (ne λ3D )

−1

ωp

Table 1.1. Individual and collective characteristic quantities

Therefore, the collective characteristic quantities remain invariant ωp2 = ε−1 0 (ne e)(e/me ) = const λ2D = (κTe /m)/ωp2 = const while the individual characteristic quantities are modified at each step of the dichotomy to reach the following limits:  −1 νcoll ≈ ne λ3D ωp → 0   mf p = ne λ3D λD → ∞ 1.3. Graininess parameter 3 in We have seen above the important role played by the parameter ne λD allowing the distinction between collective and individual phenomena. Let us consider the dimensionless parameter

g=

1 ne λ3D

[1.2]

The following properties are obvious: – in the dichotomy experiment, g is divided by a factor 2 at each step; – g is clearly a measure of the discrete (or granular) character of the plasma; – collective effects are described in the limit g → 0.

8

The Vlasov Equation 1

In terms of characteristic lengths and times, we have – λD ≈ g mf p with the collective limit coll  λD ; −1 −1 with the collective limit νcoll  ωp−1 . – ωp−1 ≈ g νcoll

Since the Debye length λD in [1.2] depends on the electron density ne and the electron temperature Te , it is interesting to rewrite g in practical units g = 3.10−6

1/2

ne

3/2

Te

,

[1.3]

where ne is expressed in m−3 and Te in kelvin. For some typical plasmas, the corresponding values for the parameter g are summarized in Table 1.2. Discharges Tokamak Solar corona Interstellar plasma

  ne cm−3 1014 1014 106 1

Te [K] 104 108 106 104

mf p [m] 2.10−5 2.10+3 2.10+3 2.10+9

D [m] 7.10−7 7.10−5 7.10−7 7

g 3.10−2 3.10−8 3.10−9 3.10−9

Table 1.2. The graininess parameter for typical plasmas

Both small density and high temperature are needed to get a collective plasma with small g. Such a plasma appears to be a completely different medium than a usual gas (g not small) for which collisions are the dominant process. For fusion or astrophysical plasma, collective effects are so important that a usual fluid description cannot be used. On the other hand, it is interesting to consider the electron gas in a metal like copper. Although a full quantum treatment would be necessary, an estimate for g with our formulas would give g = 2.10+6 , which is largely beyond the scope of a Vlasov description. 1.4. The collective description of a Coulomb gas: an intuitive approach To build a Vlasov model, we must keep in mind the limit g → 0 and derive an equation, which is invariant with respect to g in this limit.

Introduction to a Universal Model: the Vlasov Equation

9

As described above, we shall consider a fully ionized electron gas in a fixed homogeneous neutralizing ion background. To further simplify the problem, we deal with an electron–proton plasma (with an ion density n0 ). No individual atomic physics processes are present and only Coulomb interactions will be taken into account (without any magnetic effects). These simplifying hypotheses are needed to get a comprehensive knowledge of the collective behavior. Generalization to more complex plasmas will be carried out in the following chapters. Due to the huge value of the number of particles N , the motion of each electron is hard to derive even with the most powerful supercomputers available nowadays. Therefore, we introduce the concept of a distribution function f (r, v, t) such as: f dr dv is the probability to find an electron at the spatial coordinate r having a velocity v inside a phase space elementary volume dr dv. Obviously, we have the normalization condition: f (r, v, t) dr dv = 1 The electron charge density is given by ρe = en0 (

f dv − 1)

and it is important to note that both f and ρ are invariant in the limit g → 0. In the Hamiltonian formalism, the distribution function is theoretically a function of canonically conjugated variables (position r and momentum p). But it is a widespread practice to consider position and velocity as variables of the distribution function. In a general way, impulsion and linear momentum are not always proportional, for example in the relativistic regime or in the case of particles in a magnetic field. We will use p when needed. Let us consider the motion of N particles in phase space (r, p), interacting through self-consistent Coulomb forces. To simplify the notations and without any loss of generality, we can deal with particles moving in a one-dimensional (1D) space (and consequently a two-dimensional (2D) phase space). A rigorous demonstration will be given in section 1.5, but here we do not want to cope with too much complicated mathematical details. Thus, at time t0 we draw a closed “contour” (C0 ) in phase space defining a “surface”

dS0 =

dr dp

10

The Vlasov Equation 1

At a later time t > t0 , the particles on the initial contour have moved and define a new contour (C) delineating a new surface

dr dp = dS Since particles evolve through Coulomb forces, the motion is Hamiltonian and obviously dS = dS0

[1.4]

Furthermore, individual collisions are neglected in the g → 0 limit. Consequently, all particles which at time t0 are inside the contour (C0 ) are exactly the same as those which, at time t, are inside the contour (C). No particles have left or entered the moving contour. Now, letting dS0 → 0 and dS → 0 allows us to define the phase space densities f (r0 , p0 , t0 ) and f (r, p, t). From the conservation relation f (r, p, t)dS = f (r0 , v0 , t0 )dS0

[1.5]

together with [1.4] we easily deduce f (r, p, t) = f (r0 , p0 , t0 )

which means

df =0 dt

[1.6]

In [1.6], d/dt stands for the total derivative d ∂ dr ∂ dp ∂ = + · + · dt ∂t dt ∂p dt ∂p Taking into account dr =v dt

(velocity) and

dp =F dt

(force)

equation [1.6] can be explicitly written as ∂f ∂f ∂f +v· +F · =0 ∂t ∂r ∂p

[1.7]

In the non-relativistic case, p = me v. The electrostatic force is F = eE. Equation [1.7] can be reformulated in terms of f (r, v, t) ∂f ∂f ∂f e E· +v· + =0 ∂t ∂r me ∂v

[1.8]

Introduction to a Universal Model: the Vlasov Equation

11

[1.8] is just the Vlasov equation introduced by Vlasov (1945) for a plasma. Now we have to compute the field E in [1.8]. As far as collective phenomena are our concern, in the limit g → 0, Coulomb interaction between charged particles must indeed be replaced by a mean field calculated from Poisson (or Maxwell equations) using charge density (and/or current density) where the microscopic fluctuations are averaged over the Debye length. In other words, it means that we stay at a space scale where fluctuations can be ignored, although the plasma is not a continuum but is formed of discrete particles. In our plasma model, the electric field E is self-consistently computed using Poisson’s equation divE = ρ/0

with ρ = en0

f dv − en0

[1.9]

Remembering that f and ρ are invariant through the dichotomy process described in section 1.2, it is clear that the Vlasov equation (or more precisely the Vlasov–Poisson system [1.8]–[1.9] is the pertinent model to describe an unmagnetized electronic plasma in the limit g → 0. More generally for more complex plasmas, Maxwell’s equations have to be used to compute the self-consistent electromagnetic fields E, B (see section 1.9). In the original work by Jeans in the case of interacting masses m, the Vlasov equation writes ∂f ∂f F ∂f +v· + · =0 ∂t ∂r m ∂v

[1.10]

where the interacting force F = mE (where this time E is the gravitational field) is obtained self-consistently using Poisson’s equation divE = −4πGρ where G is the gravitational constant and ρ is the mass density given by ρ=m

f (r, v, t) dv

[1.11]

12

The Vlasov Equation 1

1.5. From N -body to Vlasov A more rigorous approach using the Hamiltonian formalism for the N -body system of interacting particles will now be outlined. Let us consider again our plasma model consisting of a collection of N electrons embedded in a fixed homogeneous neutralizing ion background. Position and momentum coordinates of the electrons are, respectively 

ri (t)

 i=1,...,N

and

  pi (t) i=1,...,N

Using the Hamiltonian formalism, these conjugated phase space coordinates obey the Hamilton equations ∂H dri = ∂pi dt

[1.12]

dpi ∂H =− ∂ri dt

[1.13]

where H is the Hamiltonian of our system. Since the forces acting on each particle i are supposed to derive from scalar potentials (no magnetic terms), the Hamiltonian can be written as H=

N N      p2i + ϕext (ri ) + φij |ri − rj | 2m e i=1 i=1 i j>i

[1.14]

In [1.14], two different potentials have been introduced: 1) an external potential ϕext (ri ) corresponding to external forces acting on the electron i and forces due to the ion background as well;   2) the interaction Coulomb potential φij |ri −rj | between electrons i and j given by φij =

e2 1 4π0 |ri − rj |

[1.15]

Equations [1.12]–[1.13] form a set of 6N -coupled equations, which are impossible to analytically solve since N is usually huge and statistical methods are necessary. A set of possible realizations must be considered, each of them being represented by a point in the 6N -grand phase space and the statistical set

Introduction to a Universal Model: the Vlasov Equation

13

represented by a cloud. Some configurations can be more probable than others, and it is convenient to introduce the distribution probability DN (r1 , . . . , rn , p1 , . . . , pn , t) Now the Liouville theorem tells us that the cloud behaves like an incompressible fluid in the grand phase space. Consequently, DN obeys Liouville’s conservative equation

dDN ∂DN [1.16] = + DN ; H = 0 dt ∂t where ·; · denotes the usual Poisson brackets. Using the Hamiltonian [1.14] Liouville’s equation [1.16] now writes ∂DN  pi ∂DN  ∂ϕext ∂DN · − · + ∂t me ∂ri ∂ri ∂pi i=1 i=1 N

N

−

N   ∂φij i=1 j>i

[1.17]

∂DN · =0 ∂ri ∂pi

Actually, we have replaced the 6N time differential equations [1.12]–[1.13] by one partial differential equation, but involving 6N phase space variables plus time. The microscopic information is the same in equations [1.12]–[1.13] or equation [1.16]. This microscopic information is clearly oversized. On the contrary, the macroscopic information available in an experiment is not sufficient and involves only some quantities like particle density n(r, t) or pressure (P r, t) at a given point r at time t in the plasma. To reduce the unnecessary information, let us introduce a series of reduced distributions   Dn (r1 , . . . , rn , p1 , . . . , pn , t) n=1,...,N −1

by averaging over particles running from n + 1 to N Dn (r1 , . . . , rn , p1 , . . . , pn , t) 1 DN drn+1 · · · drN dpn+1 · · · dpN = (N − n)!

14

The Vlasov Equation 1

Note that the factor 1/(N − n)! has been introduced to take into account the electrons identity since DN must be symmetrical by exchanging (ri , pi ) and (rj , pj ) with the corresponding normalization condition 1 N!

DN dr1 · · · drN dp1 · · · dpN = 1

The nature of the first reduced distribution D1 (r1 , p1 , t) is particularly interesting: D1 (r1 , p1 , t) dr1 dp1 is a measure of the number of electrons in the ordinary phase space volume dr1 dp1 around (r1 , p1 ). The knowledge of D1 (r1 , p1 , t) allows a straightforward calculation of the macroscopic (fluid) quantities: – the local density of particles: n(r1 , t) =

D1 (r1 , p1 , t) dp1

– the mean fluid velocity: 1 u(r1 , t) = me n(r1 , t)

p1 D1 (r1 , p1 , t) dp1

– the kinetic pressure tensor: P (r1 , t) =

1 me

(p1 − me u) ⊗ (p1 − me u) D1 (r1 , p1 , t) dp1

from which we get the kinetic temperature nκT = tr(P ). Obviously, D1 (r1 , p1 , t) is just the distribution function f (r, p, t) already introduced in section 1.4. It can be interpreted as a one particle distribution function describing a particle (which we have labeled as particle 1), which statistically represents all the particles. Therefore, r is identified as r1 and p as p1 . From the Liouville equation [1.17], it is easy to obtain an equation for D1 (r1 , p1 , t) (and thus for f (r, p, t)): we have to integrate [1.17] over dr2 · · · drN dp2 · · · dpN to get ∂D1 ∂ϕext ∂D1 p1 ∂D1 · − · = + ∂t me ∂r1 ∂r1 ∂p1



∂φ12 ∂D2 · dr2 dp2 ∂r1 ∂p1

[1.18]

Introduction to a Universal Model: the Vlasov Equation

15

Unfortunately, this equation for D1 depends on D2 . Repeating this process gives an equation for D2 , which depends on D3 and so on. Actually, we get a hierarchy of equations that is equivalent to solving the Liouville equation alone and is known as the BBGKY hierarchy (Born, Bogoliubov, Greene, Kirkwood, Yvon). For further explanations about the BBGKY, see, for instance, the book by Montgomery and Tidman (1964). Again we are faced with the same problem of cutting a hierarchy of equations and it seems that no advantage has been obtained. But remember our discussion about collective effects and the limit g → 0. Let us therefore introduce an expansion with respect to the small parameter g. First remember that two random variables x1 and x2 are said uncorrelated if the 2-probability P2 (X1 , X2 ) to have x1 = X1 and x2 = X2 can be factorized in terms of the 1-probability P1 namely P2 (X1 , X2 ) = P1 (X1 ) P1 (X2 ) Coming back to our electrons, the limit g → 0 is equivalent to consider particles i and j (∀i, j, i = j) as uncorrelated. Let us introduce the following ordering in powers of g: D1 (r1 , p1 , t) = O(g 0 ) D2 (r1 , p1 , r2 , p2 , t) = D1 (r1 , p1 , t) D1 (r2 , p2 , t) + O(g 1 ) Reporting this expansion into [1.18] and keeping only first-order terms in g yields ∂f p ∂f ∂ϕext ∂f · + − · = ∂t me ∂r ∂r ∂p



∂φ12 ∂f (r, p, t) · f (r2 , p2 , t) dr2 dp2 ∂r ∂p

where D1 has been identified as f and index 1 has been dropped. This equation is nothing else but the Vlasov equation that can be written in the more simple form as ∂f p ∂f ∂f · + +F · =0 ∂t me ∂r ∂p

[1.19]

where the force term F = Fext + Fsc is split into two parts. Referring to the Hamiltonian [1.14], we can distinguish:

16

The Vlasov Equation 1

– external forces Fext (i.e. external forces applied to the whole plasma, and forces due to the ion background): Fext = −

∂ ϕext (r) ∂r

[1.20]

– self-consistent forces Fsc due to Coulomb interactions between electrons in the mean field approximation (i.e. smoothed using f ): Fsc

∂ =− ∂r



  φ12 |r − r2 | f (r2 , p2 , t) dr2 dp2

[1.21]

Introducing the electric field in the plasma E = F /me and taking the divergence of [1.20] and [1.21] allow us to recover Poisson’s equation [1.9]. 1.6. The graininess parameter and 1D, 2D or 3D models The Vlasov–Poisson system we have derived is a self-consistent model that describes the time evolution of density perturbations. If these perturbations are along a given axis (say the x-axis) and if no external field is applied, then the self-consistent electric field E has only one component along this axis. In that case, the model is often referred to as 1D electrostatic. This 1D plasma model invites to go back to the 1D meaning of the graininess parameter. Let us consider again the microscopic plasma model described in section 1.5, and imagine an initial situation where the charged particles are distributed on parallel infinite sheaths perpendicular to the x-axis. The electric forces take the form of plane wave. Between sheaths the electric field is constant and the sheaths experience a uniformly accelerated motion. To advance the system, we have only to compute the crossing times, find the smallest and rearrange the sheaths (see Figure 1.7). This is the basis of an algorithm to solve exactly the N -body problem in one dimension. It is very easy to handle in 1D because the field computation is straightforward, the motion of each sheath being uniformly accelerated. For higher dimensions, for instance 2D, where charges are now rods as seen in Figure 1.8, or in three-dimensions (3D) (points), the problem becomes much more complicated because the interacting forces now depend on the distance between particles. As introduced by Rostoker and Rosenbluth (1960), the analysis we have developed in section 1.2 can be extended to any dimensionality d of the plasma (d = 1, 2 or 3). The reader is referred to an

Introduction to a Universal Model: the Vlasov Equation

17

interesting paper by Lotte and Feix (1984) for more details we have not developed here. Their analysis is based again on the search of length and time scales and on a dimensionless parameter g used under the “virtual” dichotomy experiment (D) described in sections 1.2 and 1.3: each particle (e,me ) is divided into two (charge e/2, mass me /2, kinetic energy divided by two but the velocity is consequently left unchanged).

Figure 1.7. The 1D N -body model: charged sheaths (above) and corresponding electric field (below)

Now we look for length and time scales invariant under this dichotomy (D) while the parameter g is divided by two (under D). It can be shown that λD and ωp−1 are just the length and time scale invariant under D. Furthermore, the Vlasov equation also remains invariant. The general expression for g becomes g=

1 ne λdD

For d = 3 (respectively, 2,1), i.e. a 3D plasma (respectively, a 2D and a 1D plasma), ne is the density per cubic meter (respectively, square meter and meter), e and me the charge and mass (respectively, charge and mass per meter or per square meter). It is interesting to point out that the graininess factor g = 1/ne λdD decreases with ne in a 3D plasma, it is independent of ne in a 2D plasma and decreases when ne increases in a 1D plasma. In this last case, rather than speaking of collisions it is more convenient to introduce the notion of

18

The Vlasov Equation 1

fluctuations connected to the discrete nature of the phase space fluid while both concepts (collisions and field fluctuations) agree in 3D.

Figure 1.8. The three possible plasma models

In a plasma, the size of the system can be arbitrarily large. In the gravitational case, the quasi-totality of the steady states has a size of order L ∼ λD = vT /ωJ where ωJ is the Jeans frequency defined by ωJ2 = 4πGρ and ρ is the typical mass density and G is the gravitational constant. Consequently, nλdD ∼ nLd ∼ N , where N is the total number of particles. As shown in Table 1.2 (section 1.3), the Vlasov equation is an excellent approximation in many cases. Fusion plasmas exhibit a graininess factor g of order 10−7 to 10−8 and space plasmas can have still smaller values of g. The situation is similar in the gravitational case with g ∼ N −1 (from 10−5 in a cluster to 10−9 -10−10 for galaxies).

Introduction to a Universal Model: the Vlasov Equation

19

1.7. The Vlasov equation at the microscopic fluctuations level The Vlasov model describes the time evolution of a collective plasma; it is obvious that time-dependent solutions are valid up to a time of order (ωp g)−1 (or (ωJ g)−1 ). Now it turns out that for g → 0 but not strictly 0, there is a correcting term for D2 in [1.18], which is of order g and should be evaluated. Before discussing this problem, it should be pointed out that when g is not small, the plasma behaves like a classical gas where binary interactions between particles are the dominant processes. The right-hand side of [1.18] can be evaluated using the Boltzmann formalism without having to cope with D2 . This RHS is usually written in the general form of a collision operator (∂f /∂t)collision and [1.18] becomes Boltzmann’s equation:  ∂f  ∂f p ∂f ∂ϕext ∂f · + − · = ∂t me ∂r ∂r ∂p ∂t collision

[1.22]

Now coming back to the case g → 0, it turns out that the correcting term (of order g) can be computed by extending the application of the Vlasov equation at this microscopic level through one of the most fruitful concepts in plasma physics: the test particle picture, introduced in the early 1960s by Rostoker and Rosenbluth (1960) and Rostoker (1961). The idea is the following: a particle with velocity v0 is launched into a homogeneous stable plasma, and this test particle is considered as a perturbation of the Vlasov distribution function while all the others (the field particles) are treated as a continuum. After some algebra, drag and diffusion coefficients can be calculated. This resembles the Langevin formalism for Brownian motion. Thus, in a similar way, a Fokker–Planck-like equation can be developed. It is interesting to note that the same result has been obtained by Balescu (1960) and Lenard (1960) using complex diagram methods of statistical mechanics. For more details, the reader is referred to the textbook by R. Balescu (1963). This equation is known as the so-called Lenard–Balescu equation (see also Lenard (1960)) and written as ∂fd ∂t

=





∂ −4 k |(k, ω = k · (vt − vd )|−2 ∂vd     ∂ ∂ fd (vd ) ft (vt ) δ k · (vd − vt ) − [1.23] ∂vd ∂vt 16π 3 e4 me

dk

dvt k ·

20

The Vlasov Equation 1

Actually, the RHS of [1.23] gives the first-order O(g) correction to the Vlasov equation. This rather complicated equation is quite subtle. It gives the evolution of a distribution function fd of distinguished particles when the total distribution is ft . Note that the total distribution includes the distinguished distribution and that quantifying the evolution of the distinguished distribution is only possible in a computer experiment. Finally, (k, ω) is the plasma dispersion function we shall discuss in Chapter 2 (section 2.2). We have written the Lenard–Balescu equation using the form [1.23] to exhibit the fundamental differences between 1D and 3D plasmas and between the evolution of the distinguished and global distributions (the global distribution is obtained with fd = ft .) It is clear from equation [1.23] that we just have to consider the kernel     ∂ ∂ δ k · (vd − vt ) fd (vd ) ft (vt ) − [1.24] ∂vd ∂vt Because of the presence of the δ distribution in [1.24], contribution to [1.23] is non-zero only in the three following cases: 1) k = 0; 2) k⊥(vd − vt ); 3) vd = vt . The first case does not give any contribution if we assume global neutrality. For cases 2 and 3, the situation for 1D and 3D plasmas and for distinguished and global distribution functions is quite different as pointed out by Rouet and Feix (1991). Table 1.3 shows the contribution of these terms. Distribution Dimensionality Global Global

1D 3D

k⊥(vd − vt ) Non-existing term Non-zero contribution

Distinguished

1D

Non-existing term

Distinguished

3D

Non-zero contribution

   

∂ ∂vd ∂ ∂vd ∂ ∂vd ∂ ∂vd

vd =vt ∂ fd f t − ∂v t  ∂ − ∂vt fd ft  ∂ fd f t − ∂v t  ∂ fd f t − ∂v t

=0 =0 = 0 = 0

Table 1.3. Contributions to the Lenard–Balescu equation

Consequently, for 3D plasma the contribution is not zero both for fd = ft and fd = ft . Remembering that these calculations are of the first order in

Introduction to a Universal Model: the Vlasov Equation

21

−1 the graininess   parameter g, we can state that for time up to (ωp g) , namely −1 3 ωp ne λD , both distinguished and global distribution functions relax toward a thermodynamic equilibrium.

On the other hand, for 1D plasma the global distribution function does not change while the distinguished distribution relaxes toward this global distribution. This result is a priori strange because this is exactly what happens for very short range potential between two charged sheets, which either pass each other without changing their velocities or bounce off each other by exchanging velocities; in both cases, the total distribution function is not changed whereas for the bounce case, a distinguished particle changes its velocity. The result remains true for a 1D plasma. The above results, very briefly detailed in this section, involve complex and sophisticated treatments. Computer experiments in one dimension were ideally suited to check these results and from the very beginning of computer simulations, “experimental results” have been compared with theory by different groups (see Dawson (1962), Eldridge and Feix (1962), Eldridge and Feix (1963), Dawson (1964)). In all cases, the agreement was well inside error amplitudes. The last result concerning the relaxation of the distinguished distribution toward the global one (which does not change at first order in g) had to wait for the possibility of treating systems with typical values of nλD = 200 and L/λD = 200 (i.e. 4.104 particles) over times of order 1,000 ωp−1 using the exact N -body code sketched in Figure 1.7 (see, for instance, the papers by Rouet and Feix (1996) or Ricci and Lapenta (2002)). This is one of the finest example of checking a theory by computer experiments. These theories are largely based on the use of the Vlasov model to help to understand the behavior of the microscopic fluctuations in a plasma. For further studies, the reader will find a very detailed discussion in the book by Elskens and Escande (2003). 1.8. The Wigner equation (Vlasov equation for quantum systems) Introducing quantum effects in Vlasov plasmas is a complex and difficult task. The quantum treatment of the N -body quantum problem and its domains of validity are beyond the scope of this book. This subject is addressed in the papers by Clérouin et al. (1990), Clérouin et al. (1992), Yalabik et al. (1989) and Remler and Madden (1990), for instance.

22

The Vlasov Equation 1

Nevertheless, it is very interesting to focus on a paper by Wigner (1932) which allows us to discuss similarities and differences with the classical Vlasov equation: discussing the properties of Wigner equation will hopefully shed some light on the more general quantum statistical problem. The simplest way to introduce the Wigner equation is to consider the wave function ψ(x, t) describing the motion of one particle (i.e. an electron in a 1D plasma so as to simplify the problem). Following Wigner (1932), let us introduce the Wigner function fW (x, p, t) in the 2D (x, p) phase space through the relation      +∞  ipΔ Δ 1 Δ ∗ fW (x, p, t) = ψ x− exp − dΔ ψ x+ 2π −∞ 2 2  It is easy to see that the respective integrations of fW with respect to p and x lead to the so-called marginal distributions n(x) and N (p) in agreement with the well-known quantum interpretations n(x) = ψ(x)ψ ∗ (x) and N (p) = −1 Φ(p/)Φ∗ (p/) where Φ is the Fourier transform of ψ. Unfortunately, this Wigner function can exhibit not only positive but also negative values, which prevents fW from being a bona fide distribution in phase space. It is nevertheless a useful mathematical tool allowing us to understand the transition from classical to quantum mechanics. Since ψ obeys the usual Schrödinger equation i

∂ψ 2 ∂ 2 ψ + V ψ, =− ∂t 2me ∂x2

where V (x, t) is the potential seen by our electron, it is easy to find the corresponding Wigner equation for fW . ∂fW p ∂fW + = [1.25] ∂t me ∂x       λ i iλ λ   V x + (x, p , t) exp ) dλdp − V x − f (p − p W 2π2 2 2  Detailed calculations can be found in the works by Wigner (1932) or Moyal (1949).

Introduction to a Universal Model: the Vlasov Equation

23

The form of this equation is very similar to the Vlasov equation. Nevertheless, the field term exhibits a more complicated expression involving convolution products. But it is easy to demonstrate that the Wigner equation [1.25] coincides with the classical Vlasov one in only three cases, as explained in the paper by Bertrand et al. (1980): 1) free particle motion V = 0; 2) uniform field V = −xE0 ; 3) harmonic oscillator V (x) = (A/2)x2 . The first case is obvious: for free particle motion, we have V = 0 and the second member of [1.25] is zero. For the two other cases, it is obvious that     λ λ −V x+ = −λE(x) V x− 2 2 where E(x) = −∂V /∂x is the electric field deriving from the potential V . Then the λ-integration is easily performed which brings in a Dirac delta function δ(p − p ), allowing the p -integration by part. Therefore, the right hand side of [1.25] is just equal to −E(x)

∂fW ∂p

i.e. the classical result. At the first sight, this result is surprising since we know that the quantum treatment of the harmonic oscillator brings quantified levels of energy. As a matter of fact, we must pay attention to the initial conditions introduced in the Wigner equation [1.25]: the conditions corresponding to a given ψ(x, 0) describe what is called a pure state. Following the usual explanation of Von Neumann, the Wigner equation also describes mixtures, i.e. situations described by different ψi with probability Pi and a Wigner distribution given by fW =



Pi f W i

i

where fW i is the Wigner distribution associated with ψi .

24

The Vlasov Equation 1

This very delicate aspect of quantum statistical mechanics needs the introduction of the random phase approximation between the different ψi ’s. Details can be found in the reference textbook by Huang (1963). Coming back to plasma, this mixture interpretation is needed to deal with the quantum treatment of any basic plasma problems. Let us consider the wellknown linear analysis of a small perturbation in a homogeneous plasma. This is the well-known Landau (1946) problem. We shall discuss it more extensively in Chapter 2 (section 2.2). Let us briefly outline the main idea. Starting with the Vlasov–Poisson system [1.8]–[1.9] describing a 1D (say x-axis) electron plasma in a fixed homogeneous ion background ∂f ∂f ∂f e E +v + = 0 and ∂t ∂x me ∂v

∂E en0 = ∂x 0



+∞ −∞

 f dv − 1

[1.26]

we look at a small perturbation around an electronic homogeneous equilibrium solution (with a normalized homogeneous equilibrium distribution function F0 (v)) f (x, v, t) = F0 (v) + εf1 (x, v, t) and

E(x, t) = 0 + εE1 (x, t)

with ε 1. Keeping only first order ε terms, [1.26] becomes linear dF0 ∂f1 ∂f1 e E1 +v + = 0 and ∂t ∂x me dv

∂E1 en0 = ∂x 0



+∞ −∞

f1 dv

[1.27]

allowing us to seek harmonic solutions for f1 and E1 of the form exp (i(kx − ωt)) so that equation [1.27] becomes (k, ω) E1 (k, ω) = 0 where (k, ω) is the plasma dispersion function ωp2 (k, ω) = 1 + k



+∞ −∞

dF0 /dv dv ω − kv

[1.28]

We recognize the plasma frequency ωp2 = ne e2 /0 me . Repeating the same treatment on the Wigner equation [1.25], we get (k, ω) = 1 +

ωp2 k



+∞ −∞

F0 (v + k/2me ) − F0 (v − k/2me ) dv (k/me )(ω − kv)

[1.29]

Introduction to a Universal Model: the Vlasov Equation

25

Comparing relations [1.29] and [1.28], we see that the quantum treatment replaces the derivative dF0 /dv by its centered finite difference using a step k/m. Obviously, long wavelengths are unaffected by the quantum treatment (of course an expected result). More precisely, wavenumbers such that kλ 1 have negligible corrections. Here, λ = /mvT is the characteristic de Broglie wavelength of the plasma and vT is a characteristic velocity of the plasma (for example the thermal velocity). An interesting difference can be pointed out in studying the behavior of a cold plasma with F0 (v) = δ(v). The well-known classical result does not exhibit any k-dispersion (we have ω 2 = ωp2 ). On the other hand, using [1.29] and solving  = 0 yields the corresponding quantum result ω 2 = ωp2 +

2 k 4 4m2e

[1.30]

with a k 4 correction as pointed by Bertrand et al. (1980). A useful model to study the oscillations and the stability properties of a quantum plasma is given by the so-called multistream model introduced by Haas et al. (2000) FW =

N 

δ (v − ai )

[1.31]

i=1

This Wigner distribution corresponds to a mixture of N pure states. The state i is characterized by a wavefunction ψi =

√

ni exp (−iki x) with

ki =

me ai 

[1.32]

Plugging [1.32] into [1.29] and introducing the plasma frequency of beam (i) defined by ωi2 = ni e2 /me 0 yields (k, ω) = 1 −

N  i=1

ωi2 (ω − kai )2 − 2 k 4 /4m2e

[1.33]

Note that [1.33] generalizes to the quantum case the well-known dispersion relation of the classical case. Obviously, with a sufficiently large

26

The Vlasov Equation 1

number of streams we can modelize any homogeneous plasma. The form of [1.33] confirms for this general case the k 4 quantum correction of [1.30]. Details and applications can be found in the paper by Haas et al. (2000). As a final remark, a similarity has to be pointed out between the classical Vlasov and the quantum Wigner equations even for the general nonlinear problem. As noticed in [1.29] for the linear case, the potential term in the Wigner equation [1.25] exhibits a replacement of a derivative by its finite difference counterpart. This similarity is somewhat surprising since the classical equation is entirely based on a trajectory concept and the incompressibility of the phase space fluid. At least, it points out the usefulness of this phase space treatment of quantum problems. But a priori this was not suspected. 1.9. The relativistic Vlasov–Maxwell model Up to now, we have derived the Vlasov equation as the fundamental paradigm for collective effects in the case of an electrostatic model of a one species (electrons) plasma embedded in a fixed ion background taking into account only electrostatic forces in classical mechanics. The generalization to a fully electromagnetic relativistic multispecies plasma needs further discussion but involves at least the following steps. For each species α, we define a distribution function fα (r, pα , t) and add to the force term the Lorentz force (pα /mα γα ) × B, where mα γα is the relativistic mass and γα is the Lorentz factor   γα2 = 1 + p2α /m2α c2 In laser-plasma experiments, some electrons may reach velocities close to the light velocity c. Now for each species, the Vlasov equation is written   ∂fα pα pα ∂fα ∂fα + + qα E + · ×B · =0 ∂t mα γα ∂r mα γ α ∂pα

[1.34]

Introduction to a Universal Model: the Vlasov Equation

27

where the electromagnetic fields obey Maxwell’s equations ∂ ∂B ×E =− ∂r ∂t   ∂E ∂ × B = μ0 J + ε0 ∂r ∂t

[1.35] [1.36]

∂ ρ ·E = ∂r ε0

[1.37]

∂ ·B =0 ∂r

[1.38]

Finally, to ensure self-consistency, charge and current densities are to be expressed in terms of distribution functions fα (r, pα , t) ρ=





J=

fα dpα

qα

α



qα

α

pα fα dpα mα γ α

[1.39] [1.40]

The set of equations [1.34]–[1.40] forms the basis of the relativistic Vlasov–Maxwell model. Two specific points deserve a few comments. First, the formal derivation of the Vlasov equation from the N -body distribution function in the presence of a magnetic field is a little more delicate than the electrostatic example that we have seen in section 1.5. The reason can be ascribed to the complications introduced by the need to extend the mean field theory to B: similarly to the way that the redistribution of charges in a plasma modifies the effective electric field felt by an electron (or ion) in a certain point of space and time, the collective motion of the same charges corresponds to electric currents, which in turn contribute to a mean magnetic field inside of the plasma. The fact that the magnetic induction B appearing in the set of equations above corresponds to the self-consistent field, which is obtained inside of the plasma, requires some further hypotheses, especially in relativistic regimes. These hypotheses are in the end related to those that justify the use of the well-known minimal coupling assumption, Pα = pα + qα A,

28

The Vlasov Equation 1

between the particle kinetic momentum pα and the canonical momentum Pα , which defines the “appropriate” conjugate coordinate to spatial position r in the phase space of a magnetized system. In view of the BBGKY cluster expansion, some approximations are indeed required already at the level of the N -body Hamiltonian of a charged system in order to write it by isolating the kinetic component of the energy from higher order interparticle interactions of magnetic nature. Because of these hypotheses, the relativistic Lorentz factor of each particle in the plasma can be expressed in terms of the canonical momentum by relying on the minimal coupling contribution only, and the subsequent integrations in the velocity space lead to the Vlasov–Maxwell system in the form it is written above. A more detailed discussion on this topic, toghether with some bibliographical references, can be found, for example, in the PhD thesis of Sarrat (2017). The second comment is more like a note, aimed at drawing the attention of the reader to a point: the procedure with which we have obtained the Vlasov equation in the electrostatic regime (see section 1.5) authorized us to write it in terms of the self-consistent electric field E defined by means of equation [1.9]. This definition in terms of the free charges inside of the plasma makes it possible to write the electrostatic forces in terms of the Poisson equation for the electric field E, by adopting for it the same definition used in vacuum, without any need to introduce a “dielectric induction” vector D. For similar reasons, when including the magnetic field, in the end we can couple the Vlasov equation with the set of Maxwell’s equations used for electromagnetic fields in a vacuum. In Chapter 3, we will discuss the response of the plasma to the electromagnetic perturbations and we will see why and in which context in collisionless plasmas we speak of polarization, finite conductivity, dielectric tensors and other features, which are normally related to the physics of electromagnetic fields in materials of dielectric nature. 1.10. References Balescu, R. (1960). Irreversible processes in ionized gases. Phys. Fluids, 3, 52. Balescu, R. (1963). Statistical Mechanics of Charged Particles. Wiley Interscience, New York. Bertrand, P., Nguyen, V.T., Gros, M., Izrar, B., Feix, M.R., Gutierrez, J. (1980). Classical Vlasov plasma description through quantum numerical methods. J. Plasma Phys., 23, 401–422. Chandrasekhar, S. (1942). Principles of Stellar Dynamics. University of Chicago Press, Chicago.

Introduction to a Universal Model: the Vlasov Equation

29

Clérouin, J., Pollock, E., Zerah, G. (1992). Thomas-Fermi molecular dynamics. Phys. Rev. A, 46, 5130. Clérouin, J., Zerah, G., Benisti, D., Hansen, P. (1990). Plasma simulations using the car-parrinello method. Europhys. Lett., 13, 685. Dawson, J.M. (1962). One dimensional plasma model. Phys. Fluids, 5, 445. Dawson, J.M. (1964). Thermal relaxation in a one-species, one-dimensional plasma. Phys. Fluids, 7, 419. Eldridge, O.C., Feix, M.R. (1962). One dimensional plasma model at thermodynamical equilibrium. Phys. Fluids, 5, 1076. Eldridge, O.C., Feix, M.R. (1963). Numerical experiment with a plasma model. Phys. Fluids, 6, 398. Elskens, Y., Escande, D.F. (2003). Microscopic Dynamics of Plasmas and Chaos. IOP Publishing, Bristol. Haas, F., Manfredi, G., Feix, M.R. (2000). Multistream model for quantum plasmas. Phys. Rev. E, 62, 2763. Hénon, M. (1982). Vlasov equation? Astron. Astrophys, 114, 211. Huang, K. (1963). Statistical Mechanics. John Wiley and Sons, New York. Jeans, J.H. (1915). On the theory of star-streaming and the structure of the universe. Mon. Not. Roy. Astron. Soc., 76(71), 799–814. Landau, L.D. (1946). On the vibration of the electronic plasma. J. Phys. USSR, 10, 25. Lenard, A. (1960). On Bogoliubov’s kinetic equation for a spatially homogeneous plasma. Ann. Phys., 3, 390. Lotte, P., Feix, M. (1984). Plasma models and rescaling methods. J. Plasma Phys., 31, 141–151. Montgomery, D.C., Tidman, D.A. (1964). Plasma Kinetic Theory, McGraw Hill, New York. Moyal, E. (1949). Quamtum mechanics as a statistical theory. Proc. Cambridge Phil. Soc., 45, 99. Remler, D.K., Madden, P. (1990). Molecular dynamics without effective potentials via the Car-Parrinello approach. Mol. Phys., 70, 921. Ricci, P., Lapenta, G. (2002). Computer experiments on dynamical cloud and space time fluctuations in one-dimensional meta-equilibrium plasmas. Phys. Plasma, 9, 430. Rostoker, N. (1961). Fluctuations of a plasma. Nucl. Fusion, 1, 130. Rostoker, N., Rosenbluth, M. (1960). Test particles in a completely ionized plasma. Phys. Fluids, 3, 1.

30

The Vlasov Equation 1

Rouet, J.L., Feix, M.R. (1991). Relaxation for one-dimensional plasma: Test particles versus global distribution behavior. Phys. Fluids B, 3, 1830. Rouet, J.L., Feix, M.R. (1996). Computer experiments of dynamical cloud and space time fluctuations in one-dimensional meta-equilibrium plasma. Phys. Plasma, 3, 2538. Sarrat, M. (2017). Physique des instabilités de type Weibel. PhD Thesis, University of Lorraine, Institut Jean Lamour, Nancy, France. Vlasov, A.A. (1945). On the kinetic theory of an assembly of particles with collective interaction. J. Phys. USSR, 9, 25. Wigner, E.P. (1932). On the quantum correction for thermodynamical equilibrium. Phys. Rev., 40, 749. Yalabik, M.C., Neofotistos, G., Diff, K., Guo, H., Gunton, J. (1989). Quantum mechanical simulation of charge transport in very small semiconductor structures. IEEE Trans. Electr. Devices, 36, 1009.

2 A Paradigm for a Collective Description of a Plasma: the 1D Vlasov–Poisson Equations

2.1. Introduction The set of Vlasov–Maxwell equations as written in section 1.9 is a formidable problem to solve. Let us give a short list of the main difficulties: – a Vlasov equation for each species has to be solved; – for each species α, we have to deal with seven variables r, v plus time t; – the different ion plasma frequencies ωi with respect to the electron  plasma frequency are in the ratio mi /me , which can be important for heavy ions introducing multiple time scales; – the magnetic fields, whether applied to the plasma (in tokamaks for example) or self-consistently present, give rise to very different frequencies and again point to other time scales; – finally, the nonlinear character is due to self-consistency: the electromagnetic fields E and B in the Vlasov equation are obtained from Maxwell’s equations and consequently from particle and current densities: ρ=





α

j=

 α

fα dpα

qα qα

pα fα dpα mα γ α

The Vlasov Equation 1: History and General Properties, First Edition. Pierre Bertrand; Daniele Del Sarto and Alain Ghizzo. © ISTE Ltd 2019. Published by ISTE Ltd and John Wiley & Sons, Inc.

32

The Vlasov Equation 1

The field term in the Vlasov equation  qα E +

pα ×B mα γ α



becomes a functional of the fα ’s so that the cross-product with ∂fα /∂pα is a quadratic nonlinear term. To simplify the points mentioned above, we can: – consider a one-species plasma (electrons) with immobile ions forming a neutralizing background; – deal with a one-dimensional (1D) model (i.e. consider plane wave solutions) as already discussed in Chapter 1; – treat only electrostatic modes that allow us to replace Maxwell’s equation with Poisson’s equation. Even with these simplifications the problem remains sufficiently nonlinear in itself. Therefore, in this chapter, we shall discuss the following Vlasov–Poisson model to exhibit basic features of collective plasma properties: ⎧ ∂f ∂f ∂f e ⎪ ⎪ ⎪ ⎨ ∂t + v ∂x + me E ∂v = 0  +∞  ⎪ ∂E en0 ⎪ ⎪ f dv − 1 = ⎩ ∂x 0 −∞

[2.1]

This model is kinetic (since it involves the velocity variable in the distribution function) and nonlinear as well. In this chapter, we shall essentially concentrate on: – the linearized theory, Landau damping and the existence of poles besides the Landau ones; – the full nonlinear properties in the cold plasma limit and the wave breaking phenomenon; – a fundamental question, related to an attempt at a fluid description, in order to further simplify the model and so treat it like a normal neutral gas. The introduction of special models (cold plasma and water bag) will be illuminating for both points of view.

A Paradigm for a Collective Description of a Plasma

33

2.2. The linear Landau problem Before starting the aforementioned program, it is worth coming back to the linear case we have already introduced in Chapter 1 when discussing classical versus quantum plasma. This is the well-known Landau (1946) problem. Let us consider a small perturbation around a homogeneous equilibrium described by the distribution function F0 (v), namely f (x, v, t) = F0 (v) + εf1 (x, v, t)

and E(x, t) = 0 + εE1 (x, t)

with ε 1. We have already shown that if we keep only first order (in ε) terms then [2.1] becomes a linear system. Furthermore, we concentrate on the initial value problem: 1) Perform a Fourier transform in space (k is the conjugated variable of space x) ∂f1 dF0 e E1 − ikvf1 + =0 ∂t me dv en0 +∞ f1 (k, v, t) dv − ikE1 = 0 −∞

[2.2]

2) Perform a Laplace transform in time to introduce the initial conditions (s is the conjugated variable of time t) f1 (k, v, s) =

+∞

f1 (k, v, t) exp (−st) ds

0

[2.3]

Introducing the initial condition f1 (k, v, t = 0) [2.2] becomes (s − ikv) f1 (k, v, s) +

e dF0 = f1 (k, v, t = 0) E1 (k, s) n0 dv

[2.4]

From [2.4] and Poisson’s equation, the perturbed quantities f1 (k, v, s), E1 (k, s) can be easily deduced. The perturbed density is written as n1 (k, s) = n0

f1 (k, v, s)dv

n0 = (k, s)



+∞ −∞

f1 (k, v, t = 0) dv s − ikv

[2.5]

34

The Vlasov Equation 1

where (k, s) is the plasma dispersion function (already introduced in Chapter 1, section 1.8). (k, s) = 1 + i

ωp2 k



+∞ −∞

dF/dv dv s − ikv

[2.6]

2.2.1. The Maxwellian case Consider an equilibrium described by the usual Maxwellian distribution1:   1 v2 [2.7] exp − 2 F0 (v) = √ 2vth 2π vth It is convenient to introduce the complex function Z(z) introduced by Fried and Conte (1960): ⎧ +∞ exp (−x2 ) ⎪ ⎨ √1 dx x−z π −∞ Z(z) = ⎪ ⎩ analytical continuation

if Im(z) > 0 if Im(z) < 0

It is very important to note that the analytical continuation is requested for Z(z) to be an analytical function in the whole complex plane. As a matter of fact, let us consider a complex function defined by the integral (x real) h(z) =

+∞ −∞

f (x) dx x−z

for Im(z) > 0

The Plemelj theorem states that for Im(z) → 0± , h(z) → P P

+∞ −∞

f (x) dx ± iπf (z) x−z

where the symbol P P means “principal value”. Therefore, the analytical continuation of h(z) is defined by ¯ h(z) =

+∞ −∞

f (x) dx + 2iπf (z) x−z

for Im(z) < 0

1 vth is the thermal velocity so that the Debye length λD can be written as λD = vth /ωp .

A Paradigm for a Collective Description of a Plasma

35

Coming back to our calculations and defining the dimensionless variables k¯ = λD ,

s¯ =

s , ωp

i s¯ θ= √ ¯ 2 k

[2.8]

the plasma dispersion function can be simply expressed in terms of the Fried–Conte function: ¯ s¯) = 1 + 1 + 1 θ Z(θ) (k, k¯2 k¯2

[2.9]

Furthermore, let us introduce the initial condition f (x, v, t = 0) = (1 + ε cos k0 x) F0 (v) corresponding to a perturbation on modes ±k0 . Using the dimensionless variables [2.8], the Laplace transformed perturbed density [2.5] becomes n0 ε Z(θ) n1 (k¯0 , s¯) = −i √ ¯ 2 2k0 (k¯0 , s¯)

[2.10]

Now we have to perform the Mellin–Fourier inversion transform to obtain the time dependence of n1 (k¯0 , t¯) (where t¯ = ωp t is the dimensionless time). Since Z(θ) is analytic in the whole complex plane, we have just to cope with the poles s¯j given by the dispersion relation (k¯0 , s¯j ) = 0

[2.11]

and sum over the contributions of each pole s¯j . Using the residue method, equation [2.10] is transformed into n1 (k¯0 , t¯) = n0

2 2 ε  k¯0 (k¯0 + 1) exp (¯ sj t¯) 2 j 1 + k¯0 2 + s¯2 j

where the sum extends over all the poles given by [2.11].

[2.12]

36

The Vlasov Equation 1

2.2.2. Landau poles and others Since the seminal paper by Landau (1946), much work has been devoted to the solution of [2.11] and is available in any textbook on plasma physics. We give the main results here. Looking at the general solution given by [2.12] for k¯0 → 0, i.e. k0 λD → 0 (or in other words long wavelength as compared to the Debye length), two kinds of poles s¯j have to be considered: 1) First the statement “long wavelength as compared to the Debye length” means a collective behavior and we must recover the plasma frequency ωp , i.e. sj = ±i ωp or in dimensionless form s¯2j = −1. From  [2.12], it is clear that the excitation level (the coefficient of exp in the sum j in [2.12] ) of these two poles goes to 1: these two poles are known as the two Landau poles s¯L . Performing an expansion in power of the small parameter k¯0 , we easily get  2 4 γL ± i 1 + 3k¯0 + 6k¯0 + · · · s¯L = −¯ The imaginary part is the real frequency (also known as the Bohm–Gross frequency), which can be written with dimensional quantities using [2.8]   2 ωBG = ωp2 1 + 3k02 λ2D + · · ·

[2.13]

The real part −¯ γL corresponds to a collisionless damping. This is the well-known Landau damping:  γ L = ωp

    1 π 1 3 exp − 2 2 exp − 8 2 k03 λ3D 2k0 λD

[2.14]

In fact, for k0 λD < 0.4 the damping is usually quite negligible. On the other hand, for k0 λD > 0.7 the damping becomes of the same order as the real part of the frequency, which means that no oscillations with wavenumber greater than λ−1 D can exist for a long time in a plasma. 2) For the other poles s¯j → 0 when k¯0 → 0 and from [2.12], their excitation level goes to zero. These poles are easily computed from the numerical solution of 1+

1 1 2 + ¯ 2 θ Z(θ) = 0 ¯ k0 k0

A Paradigm for a Collective Description of a Plasma

37

This equation can be simplified by noticing that Z  (θ) = −2 (1 + θ Z(θ)) and we have to solve Z  (θ) = 2 k¯0

2

For k0 λD = 0.1 (negligible Landau damping), we have computed the first 32 poles (plus the two Landau ones). The results are shown in Figure 2.1.

1

0.6

-1

-0.6

-0.2

0.2

-0.6

-1

Figure 2.1. Poles for a Maxwellian distribution with k0 D = 0.1

As a matter of fact, it is clear that these other poles are strongly damped but can play a role in transient regimes. We will come back on this point later, in section 2.5.

38

The Vlasov Equation 1

2.2.3. Unstable plasma: two-stream instability Using the Maxwellian distribution above and considering the limiting case where the spread of velocities at a given point goes to zero (i.e. vth → 0), we get the cold plasma model. The equilibrium distribution function [2.7] is simply a Dirac delta function F0 (v) = δ(v). The plasma dispersion function becomes (k, s) = 1 + ωp2 /s2 and we recover the plasma oscillations at the frequency ωp . The cold plasma case is a somewhat singular case where all k-dependence has disappeared since the natural length unit (the Debye length λD ) is equal to zero. We shall come back to this model in the next section. Up to now for Maxwellian-like distribution, the poles given by (k, s) = 0 are located in the half-plane Re(s) < 0 giving damped behavior. A question arises if (k, s) can have zeros for Re(s) > 0. The answer is yes. For instance, let us consider the velocity distribution consisting of two counter-streaming cold beams with velocities ±a: F0 (v) =

 1 δ (v + a) + δ (v − a) 2

[2.15]

The dispersion equation is written as (k, s) = 1 + ωp2

s2 − k 2 a 2 (s2 + k 2 a2 )2

and the solutions of  = 0 are   1  −(ωp2 + 2k 2 a2 ) ± ωp4 + 8ωp2 k 2 a2 2 √ The solution with the − sign in front of the · corresponds to s2 < 0 for all k (i.e. a real frequency). On the other hand, the solution with the + sign gives s2 > 0 for ka/ωp < 1 and consequently the charge density (or the electric field) experiences an exponentially increasing behavior with time: such a plasma is said unstable. s2 =

A Paradigm for a Collective Description of a Plasma

39

This problem was investigated for the first time by Penrose (1960) who has given the following criterion: if a distribution function F0 (v) has a local minimum at v = v0 , then it is unstable if +∞ F0 (v) − F0 (v0 ) dv > 0 2 (v − v0 ) −∞ This is obviously the case for our two-stream plasma [2.15]. 2.3. The 1D cold plasma model: nonlinear oscillations 2.3.1. Hydrodynamic description The cold plasma model we have just introduced is in fact a subset of our 1D Vlasov model. More precisely, a cold nonlinear plasma model deals with electrons with velocities as large as we like that experience forces, contrary to the hypothesis adopted in the linear treatment of a Vlasov homogeneous plasma where all particles experience only first-order changes in their velocity. In other words, “cold” means simply that at a given time t all electrons at a given point x have the same velocities. More precisely, f (x, v, t) =

 n(x, t)  δ v − u(x, t) n0

[2.16]

Here, n(x, t) is the electron density n(x, t) = n0

+∞

f (x, v, t) dv

−∞

and u(x, t) is the mean fluid velocity already defined in Chapter 1: u(x, t) =

n0 n(x, t)



+∞ −∞

v f (x, v, t) dv

Of course, this hypothesis is true at initial time but it is not a self-preserving property and we will have to check the results of our calculations with respect to this ansatz. It is interesting to deduce the equation governing n(x, t) and u(x, t). We start from the Vlasov equation [2.1] multiplied by dv and integrated over v. The continuity equation is obtained: ∂n ∂ + (nu) = 0 ∂t ∂x

[2.17]

40

The Vlasov Equation 1

Next we multiply the Vlasov equation by v and integrate over v. A straightforward calculation yields the fluid motion equation: ∂u ∂u e E(x, t) +u = ∂t ∂x me

[2.18]

These equations [2.17] and [2.18] for n(x, t) and u(x, t) must be self-consistently completed by Poisson’s equation  ∂E e  n(x, t) − n0 = ∂x 0

[2.19]

The set of equations [2.17]–[2.19] are just the fluid equations of a selfconsistent hydrodynamic model, which is equivalent to the Vlasov–Poisson one provided the ansatz [2.16] remains true at each time. We shall discuss this question later. 2.3.2. Lagrangian transformation

formulation

through

the

Von

Mises

First used by Konyukov (1960) and Kalman (1960), the nonlinear solution of equations [2.17]–[2.19] can be obtained by introducing a well-known transformation of hydrodynamics, the Von Mises transformation, and a function ξ(x, t) defined by ⎧ ∂ξ n ⎪ ⎪ ⎨ ∂x = n 0 ⎪ ∂ξ nu ⎪ ⎩ =− ∂t n0

[2.20]

The continuity equation [2.17] is automatically satisfied. Moreover, it is obvious that the total time derivative of ξ(x, t) is zero: dξ ∂ξ ∂ξ = +u dt ∂t ∂x =0 This is a fundamental property: for an observer following the motion of particles, ξ is a constant and can be considered as a label attached to a particle

A Paradigm for a Collective Description of a Plasma

41

as explained by Dawson (1959). Now we interchange the roles of x and ξ, i.e we consider x as a function of ξ (and t of course). We have u=

dx ∂x = dt ∂t

du ∂u ∂2x = = 2 dt ∂t ∂t and the motion equation [2.18] becomes ∂2x e = E(ξ, t) 2 ∂t me

[2.21]

Now looking at Poisson’s equation [2.19], we get en0 ∂E = ∂x 0



 ∂ξ −1 ∂x

which can be integrated (assuming no external field) E(ξ, t) =

 en0  ξ − x(ξ, t) 0

[2.22]

Finally, combining [2.21] and [2.22] yields a linear equation ∂2x (ξ, t) + ωp2 x(ξ, t) = ωp2 ξ ∂t2 with a general solution (assuming no drift) x(ξ, t) = ξ + A(ξ) sin ωp t + B(ξ) cosωp t

[2.23]

We have thus obtained the exact solution of the nonlinear hydrodynamic cold plasma equations [2.17]–[2.19] in a parametric form: ⎧  −1 dA dB ⎪ ⎪ n(ξ, t) = n0 1 + sin ωp t + cos ωp t ⎪ ⎪ dξ dξ ⎪ ⎪ ⎨   u(ξ, t) = ωp A(ξ) cos ωp t − B(ξ) sin ωp t ⎪ ⎪ ⎪   ⎪ ⎪ ⎪ ⎩ E(ξ, t) = − en0 A(ξ) sin ωp t + B(ξ) cosωp t 0 where A(ξ) and B(ξ) are determined through initial conditions.

[2.24]

42

The Vlasov Equation 1

2.3.3. The wave-breaking phenomenon The Von Mises transformation [2.20] has completely solved the problem in a parametric form, provided the Jacobian does not vanish:  ∂(x, t)  ∂x =  ∂ξ ∂(ξ, t)  0



∂x  ∂t 

∂x

= 0 = 1  ∂ξ

As a matter of fact, ∂x/∂ξ = 0 gives n/n0 → ∞, an obvious warning that something has gone wrong in the model. Actually, ∂x/∂ξ = 0 means that at a given point and at a given time, particles with different labels ξ converge at the same point. They do not have the same velocity (otherwise their past history would have been the same) and consequently our cold plasma assumption [2.16] breaks down. To illustrate this phenomenon, let us consider the following initial conditions: n(x, t = 0) = n0 u(x, t = 0) = u0 sin k0 x From these initial conditions, the functions A(ξ) and B(ξ) in [2.24] are easily obtained: B(ξ) = 0 u0 A(ξ) = sin k0 ξ ωp and the exact parametric solution [2.24] now writes: ⎧  −1 k u ⎪ ⎨ n(ξ, t) = n0 1 + 0 0 cos k0 ξ sin ωp t ωp ⎪ ⎩ u(ξ, t) = u0 sin k0 ξ cos ωp t where the relation between x and ξ is given by x=ξ+

U0 sin k0 ξ sin ωp t ωp

allowing us to come back from variables (ξ, t) to (x, t).

[2.25]

A Paradigm for a Collective Description of a Plasma

43

Let us now examine the different cases: 1) If k0 u0 /ωp 1, then [2.25] exhibits the well-known oscillatory behavior at the plasma frequency. 2) If k0 u0 /ωp is not small but close to 1, the electron density can reach very high values but the plasma remains cold in the sense that our model [2.16] remains valid. A typical density profile for k0 u0 /ωp = 0.9 is shown in Figure 2.2. At spatial locations given by cos k0 x = −1 (at those points x = ξ) n/n0 reaches very high values. 3) If k0 u0 /ωp → 1, then n/n0 → ∞ in a finite time ωp t = π/2 for x given by cos k0 x = −1. 4) If k0 u0 /ωp > 1, the mean velocity u becomes a multivalued function of x giving rise to an overtaking of particles in phase space, or wave breaking.

Figure 2.2. Electron density at time ωp t = π/2 for k0 u0 /ωp = 0.9

The phase space plots showing u(x, t) for k0 u0 /ωp = 1.4 at different times are presented in Figure 2.3. In this graph corresponding to case 4 above, plot (B) clearly exhibits a quite vertical slope at the spatial locations defined by cos k0 x = −1, where x = ξ.

44

The Vlasov Equation 1

An overtaking of electrons around those points begins to occur, and clearly our cold plasma model is no more valid.

Figure 2.3. Phase space plots for k0 u0 /ωp = 1.4 at different times. (A) ωp t = 0.1 × π/2, (B) ωp t = 0.5 × π/2, (C) ωp t = 0.89 × π/2. For a color version of this figure, see www.iste.co.uk/delsarto/vlasov1.zip

Furthermore, plot (C) becomes a multivalued function. It corresponds to an exact solution of an invalid model and should be discarded, but has been kept here in Figure 2.3 to figure out the beginning of the wave breaking. Indeed, no analytical solutions are available and the recourse to numerical simulation is the only way to study these phenomena. These large nonlinear phenomena with strong charge separations are involved in ultra-intense laser–plasma interaction and will be considered in volume 2 of this book. 2.4. The water bag model 2.4.1. Basic equations Coming back to the phase space properties of the Vlasov equation for our 1D electron plasma, we have already noted that the distribution function f (x, v, t) can be viewed as the density of an incompressible fluid moving in

A Paradigm for a Collective Description of a Plasma

45

the phase space, which in our 1D problem is reduced to a two-dimensional (2D) (x, v)-plane. To simplify further, let us imagine an initial state (t = 0), where all the electrons are located inside a phase space domain D limited by a closed curve (C) (the contour) with a constant density f = A. An infinite plasma (C) will be represented by two single-valued functions v+ (x, t = 0) and v− (x, t = 0) closing together at x → ±∞ (see Figure 2.4).

Figure 2.4. Phase space portrait of the water bag

At any point x, the initial distribution function is a step function of the form (see Figure 2.5):  f (x, v, 0) =

A if

v− (x, t = 0) < v < v+ (x, t = 0)

0 elsewhere

[2.26]

From Liouville’s theorem, three important properties must be pointed out: 1) The distribution function remains constant along a dynamical trajectory in phase space. 2) The value A of the distribution inside D is an absolute constant (both in phase space and time). 3) Only the contour (C) changes this topology. Inside D the fluid behaves like a homogeneous incompressible one, suggesting the name “water bag” given by Depackh (1962).

46

The Vlasov Equation 1

Figure 2.5. Water bag distribution function

Consequently, a point on (C) does remain on the same contour and the state of the system is completely described by the equations governing the motion of the particles (e, me ) on this contour (C): me

dv± (x, t) = eE(x, t) dt

or equivalently ∂v± e ∂v± E(x, t) + v± = ∂t ∂x me

[2.27]

completed by Poisson’equation which in our water bag model takes on the form   ∂E n0   A v+ (x, t) − v− (x, t) − 1 = ∂x 0

[2.28]

The electron density n can obviously be recognized: n(x, t) = n0

+∞ −∞



f (x, v, t) dv

= n0 A v+ (x, t) − v− (x, t)



Equations [2.27] and [2.28] form the basis of our water bag model.

[2.29]

A Paradigm for a Collective Description of a Plasma

47

2.4.2. Linearized theory The first task we have to complete is to find the Landau poles for this particular distribution function. As shown in section 2.2, we have to solve the dispersion equation (k, s) = 0 with (k, s) given by [2.6] 1+i

ωp2 k



+∞ −∞

dF0 /dv dv = 0 s − ikv

where F0 (v) is the initial homogeneous equilibrium distribution function (assuming no drift)  F0 (v) =

A

if − a < v < +a

0

elsewhere

where v± (x, 0) = ±a are the constant equilibrium initial contours.  Since F0 (v) dv = 1, we obviously have 2aA = 1 and the dispersion relation becomes 1+

ωp2 =0 s2 + k 2 a 2

This last equation exhibits two undamped poles (no Landau damping): s = ±iωk

with

ωk2 = ωp2 + k 2 a2

or equivalently2 2 ωk2 = ωp2 + 3 k 2 vth

[2.30]

= ωp2 (1 + 3 k 2 λ2D )

This relation is just the Bohm–Gross relation already derived in section 2.2. Although in the Maxwellian case the Bohm–Gross relation [2.13] is valid up to the second order in kλD (with kλD → 0), this relation is exact whatever the value of kλD in the water bag case.

2 2 The thermal velocity is given by vth = distribution function.



v 2 F0 (v) dv = a2 /3 for a water bag

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The Vlasov Equation 1

2.4.3. Water bag hydrodynamic description In the cold plasma model, we have shown that both electron density3 and average “fluid” velocity obey hydrodynamic equations (continuity and Euler) without kinetic pressure. In the water bag case, density has been already given by [2.29]. The averaged velocity u(x, t) can be easily calculated as well +∞ n0 u(x, t) = vf (x, v, t)dv n(x, t) −∞  1 v+ (x, t) + v− (x, t) = 2 Now adding and subtracting v+ (x, t) and v− (x, t) from the contour equations [2.27] allows us to get the corresponding hydrodynamic equations for n(x, t) and u(x, t) ⎧ ∂n ∂ ⎪ ⎪ + (nu) = 0 ⎪ ⎪ ∂t ∂x ⎪ ⎪ ⎪ ⎪ ⎪ ∂u e 1 ∂P ∂u ⎪ ⎪ ⎨ ∂t + u ∂x = m E(x, t) − mn ∂x [2.31] m ⎪ −3 ⎪ ⎪Pn = 2 2 ⎪ 12 n0 A ⎪ ⎪ ⎪ ⎪   ⎪ ⎪ ⎪ ⎩ ∂E = e n(x, t) − n0 ∂x 0 The set of equations [2.31] are continuity, Euler and state equations supplemented by Poisson’s equation to ensure self-consistency. Since a hydrodynamic description involves particle density, average velocity and pressure, it was clear that we could predict the possibility of casting the water bag into the hydrodynamic frame with, in addition, an automatically provided state equation. This result was demonstrated for the first time by Bertrand and Feix (1968). This state equation takes on the form of an adiabatic law: P n−3 = const.

[2.32]

3 Since we have only one species (electrons), from now on the subscript “e” has been dropped.

A Paradigm for a Collective Description of a Plasma

49

where P is the kinetic pressure defined by P (x, t) = mn0

+∞ −∞



v − u(x, t)

2

f (x, v, t) dv

This adiabatic law P n−γ = const. with γ = 3 (which is different from the value γ = 5/3 for a monoatomic perfect gas) results from the 1D character of particles’ motion: consider a charged particle (or rather its guiding center) moving along the axis of a linear magnetic machine between two parallel magnetic mirrors (see Figure 2.6).

Figure 2.6. 1D motion of a charged particle between moving mirrors

One mirror (say the left one on the figure) is motionless, whereas the second moves with a constant velocity VM . Let v be the velocity of the particle along the machine axis and suppose v  VM . Each reflection of the particle on the moving mirror increases its velocity by 2VM . The number of reflections per unit time is v/2L. Thus we have dv v = 2VM dt 2L v dL =− L dt Integrating this equation gives a velocity v ∝ 1/L and a kinetic energy K ∝ 1/L2 . Consider now a population of particles with density n ∝ 1/L. From the kinetic energy K above, its temperature is T ∝ n2 , so the kinetic pressure P = nκT behaves like P ∝ n3 .

50

The Vlasov Equation 1

This result is nothing but the signature of a 1D adiabatic compression for which the water bag offers a rigorous demonstration4. 2.5. Connection between the hydrodynamic, water bag and Vlasov models 2.5.1. A Vlasov hydrodynamic description As already mentioned, the Vlasov equation is a difficult one mainly because of its high dimensionality. A question arises immediately: can it be reduced to the sole configuration space as in usual hydrodynamics? In the case of a usual gas, the presence of collisions with frequency much greater than the inverse of all characteristic times implies the existence of a local thermodynamic equilibrium usually (e.g. for a perfect gas) described by a local Maxwellian distribution function, which can be replaced by a few macroscopic quantities such as particle density n(r, t), average velocity u(r, t) or pressure P(r, t). On the contrary, a collisionless plasma involves a microscopic description involving a distribution function f (r, v, t), which is an arbitrary function of r, v (and t of course) in phase space. Therefore, a macroscopic description cannot be carried out except for the two 1D models described above, which actually are subsets of the 1D Vlasov model: – the cold plasma; – the water bag. Since particle density n(x, t), average velocity u(x, t) and pressure P (x, t) are given in terms of integrals (the moments) involving the distribution function, it is tempting to derive the equations governing these functions from the 1D Vlasov equation ∂f ∂f e ∂f +v + E =0 ∂t ∂x m ∂v

4 Of course, if particles were randomized in three-dimensional space (due to collisions, for instance), the γ-factor would have taken the usual 5/3 value.

A Paradigm for a Collective Description of a Plasma

51

by multiplying, respectively, by 1, v and v 2 and integrating over v. The set of hydrodynamic equations (of course supplemented by Poisson’s equation) is easily derived and takes the hydrodynamic form ⎧ ∂n ∂ ⎪ + (nu) = 0 ⎪ ⎪ ⎪ ∂t ∂x ⎪ ⎪ ⎨ ∂u ∂u e 1 ∂P +u = E(x, t) − ⎪ ∂t ∂x m mn ∂x ⎪ ⎪ ⎪ ⎪ ⎪ ∂P ∂u ∂Q ∂P ⎩ +u + 3P + =0 ∂t ∂x ∂x ∂x

[2.33]

Actually, three equations for four macroscopic quantities are obtained: the first equation for n involves u, the second equation for u involves P and the third equation for P involves the electron heat flux Q Q = mn0

+∞ −∞

(v − u)3 f (x, v, y) dv

Clearly, the equation for Q involves a quantity defined from some integral of the form v 4 f dv and so on. To stop this infinite hierarchy of equations, some further assumptions have to be made. Usually, heat flux processes are assumed to be sufficiently weak so that ∂Q/∂x can be neglected as compared to ∂P/∂t, and the set of equations [2.33] is now closed and the pressure equation can be transformed into a more compact Eulerian form: ∂ d ∂  (P n−3 ) = +u (P n−3 ) = 0 dt ∂t ∂x

[2.34]

Thus, an approximation (weak heat flux) is compulsory to transform the microscopic Vlasov–Poisson system into a macroscopic hydrodynamic one: ⎧ ∂n ∂ ⎪ ⎪ + (nu) = 0 ⎪ ⎪ ∂t ∂x ⎪ ⎪ ⎪ ⎪ ⎪ ∂u e 1 ∂P ∂u ⎪ ⎪ ⎨ ∂t + u ∂x = m E(x, t) − mn ∂x ⎪ ∂ ∂  ⎪ ⎪ + u (P n−3 ) = 0 ⎪ ⎪ ∂t ∂x ⎪ ⎪ ⎪   ⎪ ⎪ ⎪ ⎩ ∂E = e n(x, t) − n0 ∂x 0

[2.35]

52

The Vlasov Equation 1

On the contrary, the water bag – characterized by a special class of initial conditions – is strictly equivalent to the Vlasov–Poisson system for all wavelengths and all degrees of nonlinearity. Furthermore, it is important to note that the quantity P n−3 is an absolute constant (both in space and time) in the water bag model [2.31], while in [2.35] it is a constant only in the Eulerian form d (P n−3 ) = 0 dt (i.e. following the fluid). In both cases, the physics will clearly be quite different. We shall discuss this topic in section 2.5.2. The linearized version of [2.35] around a no-drift homogeneous equilibrium is interesting to consider: n(x, t) = n0 + εn1 (x, t) u(x, t) = εu1 (x, t) 2 P (x, t) = mn0 vth + εP1 (x, t)

E(x, t) = εE1 (x, t) Looking for harmonic solutions of the form exp i(kx − ωk t) yields the Bohm–Gross relation: 2 ωk2 = ωp2 + 3 k 2 vth

Remembering that the Bohm–Gross relation is valid only in the long wavelength limit (kλD → 0) in the linear theory of the Vlasov–Poisson model, it is easy to understand why it has often been stated that long-wavelength phenomena could be approximated by a hydrodynamical approach, at least for linear perturbations. This claim needs to be corrected. 2.5.2. Vlasov numerical simulations of P n−3 To summarize what has been said above, it clearly appears that the behavior of the adiabatic quantity P n−3 plays a central role. To establish a more rigorous approach in the nonlinear case, the recourse to Vlasov numerical simulations is compulsory. The work by Gros et al. (1978) will help. Carried out in the 1970s, these simulations used the popular

A Paradigm for a Collective Description of a Plasma

53

Fourier–Hermite algorithm. More details on numerical simulations of plasmas will be considered in Volume 3. A first series of simulations is conducted using the usual initial conditions already used in section 2.2.1, namely a Maxwellian distribution with a harmonic perturbation on a single mode k0 : f (x, v, 0) = (1 + ε cos k0 x) F0 (v)

[2.36]

where F0 is a Maxwellian homogeneous distribution F0 (v) = √

  1 v2 exp − 2 2vth 2π vth

The initial first moments of f are easily obtained: n(x, 0) = n0 (1 + ε cos k0 x) u(x, 0) = 0 P (x, 0) =

2 mn0 vth

[2.37] (1 + ε cos k0 x)

so that the initial value of P n−3 writes 2 P n−3 (x, 0) = mn−2 0 vth (1 + ε cos k0 x)

−2

[2.38]

The Vlasov–Poisson system is solved using parameters chosen to match the conditions described above in section 2.5.1 to get a hydrodynamic description from the Vlasov model, namely: – long wavelengths (k0 λD → 0); we choose different values between 0.03 and 0.1 to get a negligible Landau damping; – a small amplitude perturbation ε = 3.10−3 . 2 The behavior of P n−3 (normalized to the equilibrium value mn0 vth ) as a function of time for different values of k0 λD is shown in Figure 2.7.

Initially, the different plots start from the normalized initial value [2.38] given by (1 + ε)−2 = 0.994 at x = 0. For small time, these plots exhibit a horizontal slope as predicted by the hydrodynamic model u

∂ (P n−3 ) ∼ O(ε2 ) ∂x

54

The Vlasov Equation 1

Figure 2.7. Normalized value of P n−3 at spatial location x = 0 for k0 λD (here labeled kD) between 0.03 and 0.10

so that to the first order in ε [2.34] reduces to ∂ (P n−3 ) = 0 ∂t Nevertheless, after a few ωp−1 ’s, the quantity P n−3 (0, t) experiences a −2 2 jump from the initial value mn−2 to its equilibrium value 0 vth (1 + ε) 2 mn−2 0 vth around which it oscillates with a very small amplitude. Moreover, this jump is of order ε and does not depend on k0 λD . Thus, even in the long-wavelength weak nonlinear limit, Vlasov simulations do not verify the predictions of the hydrodynamic model. Furthermore, it must be pointed out that the initial conditions [2.38] above cannot match with a water bag description since P n−3 (x, 0) is not a constant with respect to the space variable x. These initial conditions will be labeled as non-adiabatic. Therefore, let us consider another initial state where the plasma is now prepared with an initial pressure:  3 P (x, 0) = P0 n−3 n(x, 0) 0

[2.39]

A Paradigm for a Collective Description of a Plasma

55

2 where P0 = mn0 vth is the equilibrium pressure. For this initial state, the solution of the Eulerian adiabatic law [2.34] is given as:

 3 P (x, t) = P0 n−3 n(x, t) 0 so that the Vlasov hydrodynamic description is equivalent to the water bag. Let us consider a second series of numerical Vlasov simulations using [2.39], which are now referred as adiabatic initial conditions. The distribution function [2.36] has to be modified along   f (x, v, 0) = (1 + ε cos k0 x) F0 v/(1 + ε cos k0 x

[2.40]

giving an initial kinetic pressure 2 P (x, 0) = mn0 vth (1 + ε cos k0 x)

3

and an initial value for P n−3 2 P n−3 (x, 0) = mn−2 0 vth = const. 2 The behavior of P n−3 (normalized to the equilibrium value mn0 vth ) as a function of time is shown in Figure 2.8 (A: full line) for kλD = 0.1 and ε = 0.003. P n−3 oscillates with a very small amplitude around the predicted equilibrium constant value: 2 (P n−3 )eq = mn−2 0 vth

These fluctuations have a very small amplitude as compared to the non-adiabatic jump −2 2 2 Δna (P n−3 ) = mn−2 0 vth − mn0 vth (1 + ε)

−2

2 ≈ 2 ε mn−2 0 vth

obtained for the non-adiabatic initials conditions [2.36]–[2.38] presented in Figure 2.7 and reproduced here (plot B). The Vlasov model now agrees very well with the hydrodynamic (water bag) description.

56

The Vlasov Equation 1

Figure 2.8. Normalized value of P n−3 at spatial location x = 0. (A) Adiabatic initial conditions. (B) Non-adiabatic initial conditions

For these adiabatic conditions, a water bag can be constructed with initial contours v± (x, 0) and phase space density A given by5 √ v± (x, 0) = ± 3 vth (1 + ε cos k0 x) 1 A= √ 2 3vth To conclude the hydrodynamic approach it is worthwhile to approximate the Vlasov–Poisson model only through the water bag formalism. The choice of pertinent initial conditions is crucial and the question arising now is why the hydrodynamic model does not agree with the Vlasov one for non-adiabatic initial conditions (even in the long wavelength limit). The next section will shed light on the role of the poles besides the Landau ones. 2.5.3. The fundamental contribution of poles besides Landau In Figure 2.7, the time duration of the non-adiabatic jump can be roughly estimated to scale as ωp−1 (k0 λD )−1 . On the other hand, it is known from the

2 5 Remember that for a homogeneous equilibrium, the contours are ±a with a2 = 3vth .

A Paradigm for a Collective Description of a Plasma

57

work by Feix (1964) that the solutions s¯j of the dispersion relation [2.11] besides the two Landau poles (see Figure 2.1) are |sj | ∼ ωp k0 λD . It is a clear indication of the role played by these poles. To be more precise, the calculation of density and pressure to the first order in ε can be carried out using the linear theory developed in section 2.2. The density formula as given by formula [2.12] in the case of non-adiabatic initial conditions can also be extended to the adiabatic initial conditions. Pressure and finally P n−3 are evaluated using the same method (correct to the first order in ε) at the spatial location k0 x = 0. Without any difficulties, we obtain:  Pn

−3

(0, t¯) =

P0 n−3 0

1+ε

 j

 2 1 + 3k¯0 + s¯2j Cj exp s¯j t¯ 2 1 + k¯0 + s¯2

[2.41]

j

using the same normalized quantities as in section 2.2, namely k¯0 = k0 λD and t¯ = ωp t, and s¯j = sj /ωp being the jth pole of the Landau dispersion equation [2.11]. Cj is non-dimensional quantity characterizing the type of initial conditions:  1 + k¯02 for non-adiabatic initial conditions [2.36] Cj = s¯2j for adiabatic initial conditions [2.40] For k0 λD = 0.1, these poles are shown in Figure 2.1. The contributions of the poles (i.e. solutions of equation [2.11]) when k¯0 → 0 are quite different: – The two principal Landau poles −¯ s2j = 1 + 3k¯02 + 6k¯04 + . . . give a 2 ¯ negligible contribution of order εk0 in both cases. – The poles besides Landau |¯ s2j | ∼ k¯02 also give a negligible contribution for adiabatic initial conditions. But for non-adiabatic initial conditions, they have a dominant role and allow us to recover the jump that is indeed of order ε. These results are presented in Figure 2.9 in the non-adiabatic case with k0 λD = 0.1 and ε = 0.003. The summation in [2.41] is extended on, respectively, N = 8 and N = 32 poles besides the two Landau poles. After very few ωp−1 , the agreement is excellent, proving the unique role of those poles. Furthermore, a very high number of poles is clearly required when the time variable is close to zero, where a divergence occurs (below ωp t ≈ 6 with N = 8 and below ωp t ≈ 3 with N = 32). This phenomenon has already been pointed out by Denavit (1968) who has shown the non-uniform convergence of some poles series as t → 0.

58

The Vlasov Equation 1

Figure 2.9. Normalized value of P n−3 for non-adiabatic initial conditions. Comparison between numerical simulation (full line) and analytical formula [2.41] using N = 8 poles (dots) and N = 32 poles (crosses)

To summarize, it has been clearly pointed out that the water bag can be a substitute for the more complex Vlasov model for a special class of initial conditions including not only nonlinear but also thermal effects that the cold plasma cannot offer. It turns out that in this case the hydrodynamic approach is equivalent to the water bag. But for spatially non-adiabatic type initial conditions, the hydrodynamic model becomes rapidly inaccurate and the poles besides Landau, although strongly damped, play a non-negligible role. One should be aware of that when dealing with more recent subjects including trapped electron acoustic waves or KEEN waves, which will be discussed in following chapters. 2.6. The multiple water bag model A drawback to the water bag comes from the lack of Landau resonance (and of course of Landau damping) since the phase velocity falls in a region of

A Paradigm for a Collective Description of a Plasma

59

velocity space for which there are no particles vϕ = ω/k0  = a2 + ωp2 /k02 >a To recover the Landau damping, the water bag has to be generalized into the multiple water bag. This generalization was carried out by Navet and Bertrand (1964) and Bertrand et al. (1972). As a matter of fact, Berk and Roberts (1970) and Finzi (1972) used a double WB model to study two-stream instabilities using computer simulations and exhibiting the phase space filamentation of the water bag contours and their multivalued nature (a highly difficult problem from a programming point of view). 2.6.1. A multifluid description Let us consider 2N contours in phase space labeled vj+ and vj− (where j = 1, · · · , N ). Figure 2.10 shows the phase space contours for a three-bag system (N = 3) where the distribution function takes on three different constant values F1 , F2 and F3 .

Figure 2.10. Multiple water bag: phase space plot for a three-bag model. For a color version of this figure, see www.iste.co.uk/delsarto/vlasov1.zip

60

The Vlasov Equation 1

Figure 2.11. Multiple water bag (MWB): corresponding MWB distribution function

Let us introduce the bag heights A1 , A2 and A3 as shown in Figure 2.11. We obviously have F3 = A3 , F2 = A2 + A3 and F1 = A1 + A2 + A3 . For an N -bag system, the distribution function writes f (x, v, t) =

N 

  Aj Υ(v − vj− (x, t) − Υ(v − vj+ (x, t) ,

[2.42]

j=1

where Υ is the Heaviside unit step function. Note that some of the Aj can be negative. Let us now introduce for each bag j the density nj , average velocity uj and pressure Pj in the same way as we did above for the one-bag case: ⎧ nj = Aj (vj+ − vj− ) ⎪ ⎪ ⎨ uj = (1/2)(vj+ + vj− ) ⎪ ⎪ ⎩ 2 2 Pj n−3 j = m/(12n0 Aj )

[2.43]

Each bag j behaves like a fluid and obeys continuity and Euler’s equations,  the coupling between the bags given by computing the total density j nj in

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61

Poisson’s equation: ⎧ ∂nj ∂ ⎪ ⎪ + (nj uj ) = 0 ⎪ ⎪ ∂t ∂x ⎪ ⎪ ⎪ ⎪ ⎪ ∂uj 1 ∂Pj e ⎨ ∂uj + uj =− + E ∂t ∂x mnj ∂x m ⎪ ⎪ ⎪ ⎪ N  ⎪ ⎪ e  ∂E ⎪ ⎪ n (x, t) − n = j 0 ⎪ ⎩ ∂x 0 j=1

[2.44]

The connection with a multifluid model is more illuminating if we consider the equivalence in the fluid moment sense of a multiple water bag distribution and a continuous distribution. The starting point of the discussion comes from the fact that the distribution function involved in the Vlasov theory (the microscopic theory) contains much more information than any experiment can afford. Indeed, the macroscopic quantities like particle density (respectively, mean velocity, pressure, heat flow, etc.) are computed from the zero-order (respectively, first, second, third, etc. order) moment of the distribution function, the -moment of f being defined by M (f ) =

∞ −∞

v  f (v) dv

[2.45]

As usual, in this chapter, we consider a homogeneous equilibrium distribution function6 f0 (v). For reasons of simplicity, we suppose that f0 is an even function of v (odd moments are zero). In the water bag formalism, it means symmetrical equilibrium contours ±aj so that the -momentum of the corresponding water bag writes M (WB) =

1  2Aj a+1 j +1 j

[2.46]

6 To avoid confusion with phase space bag values Fj , the distribution function is now written as f0 instead of F0 .

62

The Vlasov Equation 1

Let us now sample the v-axis with appropriate aj ’s. Thus, equating [2.45] and [2.46] for  = 0, 2, . . . , 2(N − 1) yields a system of N equations for the N unknown Aj, j=1, ..., N . Integrating by parts we obtain 

2Aj a+1 =− j

j

∞ −∞

v +1

df0 dv, dv

 = 0, 2, . . . , 2(N − 1).

[2.47]

A water bag model with N bags is equivalent to a continuous distribution function for moments up to max = 2(N − 1). Nevertheless, from a computational point of view, [2.47] has the form of a Vandermonde system, which becomes ill-conditioned for higher values of the number of bags N (for instance for N = 15 and a cut-off in velocity space aN = 5vth , the matrix elements vary from 1 to 528 ).

Figure 2.12. Constructing the bags from a continuous distribution

A more convenient solution can be found for a regular sampling aj = (j − 12 )Δa and is explained in Figure 2.12: we consider Fj at the middle of the interval Δa =

2aN 2N − 1

and compute Fj = f0 (aj −

[2.48] Δa 2 ).

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63

From equation [2.47], the solution is straighforward:   Δa  Δa  Aj = f0 aj − − f0 aj + + O(Δa3 ). 2 2

[2.49]

For a Maxwellian distribution, all Aj ’s are positive. On the contrary, in the case of a two-stream distribution for example, some Aj ’s can be negative. 2.6.2. Linearized analysis Linearizing [2.44] for an electronic plasma around a homogeneous (density n0 ) equilibrium, i.e. perturbing the equilibrium contours vj± (x, t) = ±aj + wj± (x, t)

[2.50]

with |wj± | aj and looking for harmonic solutions exp i(kx − ωt), yields the dispersion relation 1 − ωp2

N  j=1

2aj Aj = 0. ω 2 − k 2 a2j

[2.51]

A schematic plot of this dispersion function is shown in Figure 2.13 in the case of a four-bag system (N = 4). We have chosen all Aj ’s positive (single hump distribution function like a Maxwellian). Therefore, the dispersion equation [2.51] has 2N (here 2 × 4) real frequencies ωn located in the intervals [aj , aj+1 ] and [−aj+1 , −aj ]. The last frequency +ωN (respectively, −ωN ) lies in the interval [aN , +∞[ (respectively, −ωN lying in the interval ] − ∞, −aN ]). Since all poles are real, a question arises: how can the Landau damping be recovered? The excitation level of the different poles of the dispersion equation [2.51] depends on the initial perturbation [2.50]. For instance, choosing as usual a harmonic perturbation vj± (x, 0) = ±aj (1 + ε cos k0 x) the electric field is easily obtained: E(k0 , t) = ε

N ien0  Cn cos ωn t 20 k0 j=1

[2.52]

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The Vlasov Equation 1

where the excitation level Cn of the corresponding pole n is given by ⎛ Cn = ⎝ωp4

N  j=1

⎞−1 2aj Aj ⎠ (ωn2 − k02 a2j )2

Figure 2.13. Dispersion function for N = 4 bags

Note that Aj ’s are computed using [2.49] to describe a Maxwellian distribution. Choosing k0 λD = 0.7, the Landau damping value is 0.37ωp as given by [2.14]. The electric field [2.52] is shown in Figure 2.14 for N = 50 bags and in Figure 2.15 for N = 100 bags. In both figures, a damping clearly occurs, and a logarithmic plot gives a damping coefficient γ/ωp = 0.37, which is in accordance with the Landau value for k0 λD = 0.7. The Landau damping is recovered as a phase mixing process of real frequencies as shown by Navet and Bertrand (1964) and Bertrand et al. (1976). This process is reminiscent of the special treatment of the electronic plasma oscillations (different from the Landau one) given by Van Kampen (1955) and Case (1959).

A Paradigm for a Collective Description of a Plasma

Figure 2.14. Mode k0 of the electric field for N = 50 bags. A recurrence appears for ωp t > 200

Figure 2.15. Mode k0 of the electric field for N = 100 bags. A recurrence appears for ωp t > 400

65

66

The Vlasov Equation 1

The Van Kampen–Case treatment is alternate to the Landau treatment we discussed in section 2.2: instead of introducing a Laplace transform to take into account the proper initial conditions, the Vlasov–Poisson equations are considered as an eigenvalue problem and the initial conditions are developed on a special basis. The eigenvalues form an infinite dense spectrum on the real frequency axis. The superposition of all those frequencies allows us to recover the Landau damping through a phase mixing process. This is what the multiple water bag does. Nevertheless, this model has only a finite number of real frequencies with a frequency interval Δω ≈ k0 Δa, where Δa is given by [2.48]. This finite number of frequencies gives rise to a recurrence time (for which the oscillators are again in phase) TR ≈ 2π/k0 Δa i.e. ωp TR is proportional to the number of bags. The Van Kampen–Case theory corresponds to an infinite number of bags N → ∞. With the parameters above, we get TR ≈ 210ωp−1 for N = 50 bags and TR ≈ 420ωp−1 for N = 100 bags. 2.7. Further remarks Throghout this chapter, we have outlined some selected properties of the 1D Vlasov–Poisson model to help with the description of collective effects in plasmas. Some other aspects have long been studied but have not been presented here and would have certainly deserved some attention. From a pedagogical point of view, we hope our choice is pertinent enough to help the reader to cope with this extraordinary rich branch of plasma physics. Among these other topics, one is the quasi-linear theory. This theory has been developed to describe the weak-warm beam–plasma instability and has been the source of many controversies for several decades. Instead of the usual Maxwellian case discussed in section 2.2, let us consider an initial distribution function F0 (v) describing a one-dimensional spatially uniform beam–plasma system. This (weak) beam corresponds to a bump on the tail of the electron velocity distribution function. In that case, the calculation presented in section 2.2.1 must be extended to the general case of any distribution F0 . In the long wavelength limit, the Landau damping formula [2.14] is generalized as follows: π ωp3 γL = − 2 k2



dF0 dv

 [2.53] v=vϕ

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67

Taking for F0 the Maxwellian distribution [2.7] allows us to retrieve the Landau formula. Furthermore, [2.53] shows that γL is strongly related to the slope of F0 around v = vϕ , the phase velocity of the wave. In the Maxwellian case, this slope is negative, giving a γL positive and damped behavior of the form exp (−γL t). On the contrary, the presence of a “bump” on the tail creates a velocity domain where the slope can be positive and the waves that have their phase velocity in that domain are unstable. They first grow linearly, but when the electron dynamics becomes chaotic enough in this range of phase velocities, the bump is eroded and a plateau in the distribution function may build up. Simultaneously these waves get a turbulent spectrum due to the transfer of momentum from particles to waves. This scenario was first theoretically predicted by Drummond and Pines (1962) and Vedenov et al. (1962) by considering wave–particle interactions as a perturbation and neglecting mode couplings due to the nonlinear term E∂f /∂v in the Vlasov equation, except for their effect on the space-averaged distribution function F0 (v, t). Using these assumptions, “quasilinear” equations coupled with the slowly time-dependent F0 (v, t) and the waves power spectra were derived. Unfortunately, the complexity of this situation has long led to controversy about the validity of this quasilinear theory to predict the saturation of the beam–plasma instability (see section 4.4). Due to the lack of powerful computers and accurate numerical schemes, the quasilinear theory could not be properly checked. Nowadays, the situation is quite different, thanks to the powerful computers presently available and to the development of Vlasov codes characterized by both high accuracy and weak numerical diffusion. (see the paper by Besse et al. (2011) on this subject). In the same way, the Landau damping in the linear regime and the associated wave–particle interaction have also led to controversy. Although the Landau damping was described analytically by Landau in 1946, it was fully recognized only after its experimental evidence by Malmberg and Wharton (1964). Furthermore, the behavior of the nonlinear Landau damping for a long time has also been the subjects of many discussions, especially when the linear theory we described in section 2.2 is no more valid. The recent work by Mouhot and Villani (2010) has brought the long-awaited mathematical proof of (nonlinear) Landau damping in infinite time with exponential decay. The reader will find many useful references on the topic in this last paper.

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Last but not least a question arises at the end of this chapter, about the interest of a 1D model to describe the real world. As a matter of fact, let us consider a typical example provided by the so-called gyrokinetic modeling in fusion plasmas. Predicting turbulent transport in nearly collisionless fusion plasmas requires us to solve kinetic (Vlasov) equations. Although more accurate, as explained in section 2.5, the kinetic calculation of turbulent transport is obviously much more demanding in terms of computer resources than fluid simulations. More precisely, low-frequency ion-temperature-gradient-driven (ITG) instabilities are commonly held responsible for turbulence giving rise to anomalous radial energy transport in the core of tokamaks. The computation of turbulent thermal diffusivities in fusion plasmas is of prime importance since the energy confinement time is determined by these transport coefficients. Ion turbulence in tokamaks was intensively studied in the 1990s both with fluid (or rather gyro-fluid) models (see, for instance, the works of Dorland and Hammett (1993), Garbet and Waltz (1996) or Manfredi and Ottaviani (1997)), and with kinetic models using particle-in-cell (PIC) simulations (Parker et al. (1993), Sydora et al. (1996)) or Vlasov simulations (Depret et al. (2000) or Grandgirard et al. (2006)). As a matter of fact, the thermal diffusivity χ computed from fluid simulations exhibits an overestimation as compared to kinetic simulations. The physical model (see, for instance, Hahm (1988) and papers cited above) is based on the Vlasov gyrokinetic equation for the ions with an adiabatic response for the electrons. The Vlasov equation acts on the guiding center ion distribution function f (r, v , μ, t), where r, v , μ are, respectively, the position, the parallel component of the velocity along the confinement 2 magnetic field B0 and the first adiabatic invariant μ = v⊥ /2B0 of the ion guiding centers. The time evolution is given by the gyro-averaged Vlasov equation written here for a uniform B0 ∂f ∂f + r˙ · ∇f + v˙  =0 ∂t ∂v where ⎧ E × B0 ⎪ ⎪ ⎨ r˙ = vGC = v b + B02 ⎪ q ⎪ ⎩ v˙  = i E · b mi

[2.54]

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69

where b is the unit vector along B0 . The subscript ·GC stands for “guiding center”. The gyro-averaged electric field on the ion guiding center r is defined by 1 E(r, μ, t) = 2π



2π 0

E(r + ρL , t) dα

[2.55]

ρL being the Larmor radius. Assuming adiabatic electrons and using the long wavelength limit allows for the replacement of Poisson’s equation for the potential φ by the more simple quasi-neutrality condition   e ¯ = n0 1 + (φ − φ) Te    # " ∂ ∂ ∂φ ∂φ 1 n0 + n0 = ni  + B0 Ωci ∂x ∂x ∂y ∂y

[2.56]

where φ¯ is the potential averaged over a magnetic surface and ni  denotes the gyro-averaged ion density: ni (r, t) =

f (r − ρL , v , μ, t) dv

In [2.56], n0 (x, y) and Te (x, y) are the perpendicular profiles of the initial density and electron temperature. It must be pointed out that μ is actually a continuous label in [2.54]. Only a finite number of discrete values of μ can be considered, thus drastically decreasing the computational effort. Even one single value can be chosen. For a given choice of μ, a distribution function fμ can be defined. Each fμ obeys a Vlasov equation that can be solved on one CPU on a parallel computer. The different fμ are then coupled through the gyro-averaged quasi-neutrality equation [2.56]. The Vlasov gyrokinetic equation [2.54] with gyro-averaging [2.55] is derived according to the usual gyrokinetic ordering: ω/Ωci ∼ k /k⊥ ∼ eφ/Te ∼ ρL /Lgrad ∼ ε

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The Vlasov Equation 1

where Lgrad is the length of the gradient scale (both density and temperature gradient) in the perpendicular direction (x, y) and ε is a small parameter, which is usually of order 10−3 in a tokamak like ITER. Furthermore, this gyrokinetic approximation (ε → 0) is known to preserve the Hamiltonian structure of the original Vlasov–Poisson model even in the non-uniform magnetic field B0 . Now the most important and interesting feature of [2.54] is that f depends on only one velocity component v parallel to B0 . Similar to the case with gyro-fluid models, the topology remains three dimensional in the real space, but only one velocity component has to be added to get a full kinetic description. At this point, it is very interesting to note that a water bag representation (or more generally a multiple water bag) of the distribution function can be carried out for this velocity component. Remember that this is not an approximation but rather a special class of initial conditions allowing us to reduce the full kinetic Vlasov equation into a set of hydrodynamic equations while keeping its kinetic character. For each μ, let us consider a multiple water bag distribution fμ (v ) given by a form similar to [2.42] fμ (r, v , t) =

N 

  − + Aμj Υ(v − vμj (r, t)) − Υ(v − vμj (r, t) )

j=1

Such a model allows us to keep the same complexity as the fluid one, while ± keeping the kinetic character of the problem. For each μ, the contours vμj (r, t) obey the following set of equations: ± ± ∂vμj ∂vμj E × B0 qi ± ± · ∇v + v Ez + = μj μj 2 ∂t B0 ∂z mi

j = 1...,N

The one bag case had already been considered about 40 years ago by Bertrand and Baumann (1976). Nowadays, the multiple gyro-water bag model has been extensively studied for magnetized plasmas (see, for instance, Morel et al. (2007), Gravier et al. (2008), Morel et al. (2011)). Furthermore, its interesting mathematical properties have been pointed out by Besse et al. (2009).

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2.8. References Berk, H.L., Roberts, K.V. (1970). Methods in Computational Physics, vol. 9. Academic Press, New York. Bertrand, P., Baumann, G. (1976). Stability of three dimensional guidingcentre-water bag plasma. Phys. Lett., 57A, 237. Bertrand, P., Feix, M.R. (1968). Non linear plasma oscillation: The “water bag model”. Phys. Lett., 28A, 68. Bertrand, P., Gros, M., Baumann, G. (1976). Nonlinear plasma oscillations in terms of multiple water bag eigenmodes. Phys. Fluids, 19, 1183–1188. Bertrand, P., Dorémus, J.P., Baumann, G., Feix, M.R. (1972). Stability of inhomogeneous two-stream plasma with a water-bag model. Phy. Fluids, 13, 1275–1281. Besse, N., Berthelin, F., Brenier, Y., Bertrand, P. (2009). The multi-water-bag equations for collisionless kinetic modeling. Kinetic Theory Related Models, 2, 39–80. Besse, N., Elskens, Y., Escande, D.F., Bertrand, P. (2011). Validity of quasilinear theory: Refutations and new numerical confirmation. Plasma Phys. Control. Fusion, 53, 025012. Brodin, G., Marklund, M., Manfredi, G. (2008). Quantum plasma effects in the classical regime. Phys. Rev. Lett., 100, 175001. Case, K.M. (1959). Plasma oscillations. Ann. Physics, 7, 349. Dawson, J. (1959). Nonlinear electron oscillations in a cold plasma. Phys. Rev., 113, 383. Denavit, J. (1968). Landau damping in Maxwellian plasmas as t → 0. Phys. Fluids, 2, 680. Depackh, D.C. (1962). The water-bag model of a sheet electron beam. J. Electron. Control, 13, 417. Depret, G., Garbet, X., Bertrand, P., Ghizzo, A. (2000). Trapped-ion driven turbulence in tokamak plasmas. Plasma Phys. Control. Fusion, 42, 949–971. Dorland, W., Hammett, G.W. (1993). Gyrofluid turbulence models with kinetic effects. Phys. Fluids B, 5, 812. Drummond, W.E., Pines, D. (1962). Non-linear stability of plasma oscillations. Nucl. Fusion, 3, 1049–1057. Feix, M.R. (1964). Impedance of RF grids and plasma condensers. Phys. Lett., 12, 316. Finzi, U. (1972). Accessibility of exact nonlinear stationary states in waterbag model computer experiments. Plasma Phys., 14, 327.

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Fried, B.D., Conte, S.D. (1960). The Plasma Dispersion Function. Academic Press, New York. Garbet, X., Waltz, R.E. (1996). Action at distance and Bohm scaling of turbulence in tokamaks. Phys. Plasma, 3, 1898. Grandgirard, V., Brunetti, M., Bertrand, P., Besse, N., Garbe, X., Ghendrih, P., Manfred, G., Sarazin, Y., Sauter, O., Sonnendrücker, E., Vaclavik, J., Villard, L. (2006). A drift-kinetic semi-lagrangian 4D code for ion turbulence simulation. J. Comput. Phys., 217, 395–423. Gravier, E., Klein, R., Morel, P., Besse, N., Bertrand, P. (2008). Gyrokineticwater-bag modeling of low-frequency instabilities in a laboratory magnetized plasma column. Phys. Plasmas, 15, 122103. Gros, M., Bertrand, P., Feix, M.R. (1978). Connection between hydrodynamics, water bag and vlasov models. Plasma Phys., 20, 1075– 1080. Hahm, T.S. (1988). Nonlinear gyrokinetic equations for tokamak microturbulence. Phys. Fluids, 31, 2670–2673. Kalman, G. (1960). Nonlinear oscillations and non stationary flow in a zero temperature plasma : part I. Initial and boundary value problems. Ann. Phys., 10, 1. Konyukov, M.V. (1960). Nonlinear Langmuir electron oscillations in a plasma. Soviet Phys. JETP, 37, 570. Landau, L.D. (1946). On the vibration of the electronic plasma. J. Phys. USSR, 10, 25. Malmberg, J.H., Wharton, C.B. (1964). Collisionless damping of electrostatic plasma waves. Phys. Rev. Lett., 13, 184–186. Manfredi, G., Ottaviani, M. (1997). Gyro-Bohm scaling of ion thermal transport from global numerical simulations of ion-temperature-gradientdriven turbulence. Phys. Rev. Lett., 79, 4190. Marklund, M., Brodin, G. (2007). Dynamics of spin-1/2 quantum plasmas. Phys. Rev. Lett., 98, 025001. Morel, P., Gravier, E., Besse, N., Klein, R., Ghizzo, A., Bertrand, P., Garbet, X., Gendrih, P., Grandgirard, V., Sarazin, Y. (2007). Gyrokinetic modelling: A multi water bag approach. Phys. Plasmas, 14, 112109. Morel, P., Gravier, E., Besse, N., Klein, R., Ghizzo, A., Bertrand, P., Bourdelle, C., Garbet, X. (2011). Water bag modeling of a multispecies plasma. Phys. Plasmas, 18, 032512. Mouhot, C., Villani, C. (2010). Landau damping. J. Math. Phys., 51, 015204.

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Navet, M., Bertrand, P. (1964). Multiple “water-bag” model and Landau damping. Phys. Lett., 34A, 117–118. Parker, S.E., Lee, W.W., Santoro, R.A. (1993). Gyrokinetic simulation of ion temperature gradient driven turbulence in 3D toroidal geometry. Phys. Rev. Lett., 71, 2042. Penrose, O. (1960). Electrostatic instabilities of a uniform non-Maxwellian plasma. Phys. Fluids, 3, 258. Sydora, R.D., Decyk, V.K., Dawson, J.M. (1996). Fluctuation-induced heat transport results from a large global 3D toroidal particle simulation model. Plasma Phys. Control. Fusion, 38, A281. Van Kampen, N.G. (1955). On the theory of stationary waves in plasmas. Physica, 21, 949. Vedenov, A.A., Velikhov, E.P., Sagdeev, R.Z. (1962). Quasi-linear theory of plasma oscillations. Nucl. Fusion, 2, 465–475.

3 Electromagnetic Fields in Vlasov Plasmas: General Approach to Small Amplitude Perturbations

3.1. Introduction and overview of the chapter At the end of Chapter 1, we discussed the possibility of extending the mean field theory to include a magnetic field. At the end of Chapter 2, we mentioned an application of the Vlasov–Maxwell formalism to provide an approximated (i.e., “reduced”) description of low-frequency instabilities and electrostatic turbulence in tokamak plasmas. Although in this volume we mostly focus on the Vlasov–Poisson system as a relatively “simpler” yet paradigmatic example of the distinctive physics of the collisionless plasma state of matter, it is now useful to discuss more formally some general features related to the response of the plasma to harmonic electromagnetic perturbations. We have already seen (Chapter 2) examples of electrostatic oscillations with no group velocity (the Langmuir waves), which become propagating when dispersive effects related to the plasma response are kept into account (the Bohm–Gross corrections to the dispersion relation) and which can be damped in time even in absence of collisions (Landau damping) or which can lead, depending on the initial configuration, to instabilities that exponentially grow in time (e.g. the two-stream instability). Including a magnetic component of course widens the spectrum of modes that can be excited and the physics behind them. There are however some general features, which we are going to discuss, which can help us to identify on a very general basis the properties of electromagnetic waves in plasmas.

The Vlasov Equation 1: History and General Properties, First Edition. Pierre Bertrand; Daniele Del Sarto and Alain Ghizzo. © ISTE Ltd 2019. Published by ISTE Ltd and John Wiley & Sons, Inc.

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The multiscale character of the Vlasov–Maxwell system pointed out at the beginning of Chapter 2 leads indeed to a variety of normal modes, whose classification depends on the plasma regime and on the characteristic considered. Moreover, depending on the initial configuration assumed, these normal modes can give rise to a large number of instabilities. There are several excellent manuals that are devoted to the detailed analysis of normal modes and of instabilities in plasmas, which are not discussed here. For that, we suggest an interested reader look at classical monographs and textbooks (Stix, 1962; Allis et al., 1963; Briggs, 1964; Sitenko, 1967; Mikhailovskii, 1974a,b; Kadomtsev, 1979; Cap, 1976, 1978, 1982; Hasegawa and Uberoi, 1982; Landau et al., 1981; Melrose, 1986; Swanson, 1989; Stix, 1992; Gary, 1993; Brambilla, 1998; Fitzpatrick, 2014). The purpose of this (and the next) chapter is instead to discuss some general properties of the Vlasov plasma as a medium in which electrostatic and electromagnetic waves can be excited, propagate and/or become unstable, and to revise some key elements of the formalism which is normally used to study them. In discussing the response of the plasma to small electromagnetic perturbations, we will also analyze the notion of polarization charge and the meaning which should be attributed to the “dielectric function” , which we have introduced in Chapter 2, and to conductivity and resistivity in plasmas. To this purpose, we will adopt here and in Chapter 4 a slightly different notation than before. This will involve symbols used for the dispersion relation, but also for vector differential operators: while we will keep on using ∂/∂r and ∂/∂v in order to distinguish the two kinds of gradients in kinetic calculations, we will use the more customary ∇ symbol in fluid-type models (e.g. in the so-called magnetohydrodynamic (MHD) regime), in which there is no ambiguity about the variable with respect to which differentiation is performed. Moreover, since we will discuss the general formalism developed to treat the full Vlasov–Maxwell system, quantities which before were mostly referred to electrons only (e.g. the distribution function, the plasma frequency, the particle density and the charge) will appear here with the species index α = e, i. Later, since the remainder of this volume will focus again on applications to the Vlasov–Poisson regime in a limit in which the ion dynamics can be neglected, the notation of previous chapters will be recovered.

Electromagnetic Fields in Vlasov Plasmas

3.2. Linear analysis of the Vlasov–Maxwell system: approach

77

general

In this section, we first summarize the general approach to the solution of the normal mode problem in Vlasov plasmas by discussing its key points. Most statements that are mentioned here and which are related to the properties of the matrix operators involved in the linear analysis will be discussed in detail and refined, when required, in the following sections. Some more detailed elements of discussion as well as some specific examples of application will be discussed in Chapter 4. Like for any continuous medium, the dispersion relation of electromagnetic waves in a Vlasov plasma is obtained after linearization of Maxwell equations in the material, combined with the equations expressing the response of the medium to the electromagnetic fields. We have however seen (section 1.9) how the mean field theory allows us to use, for most purposes, Maxwell’s equation valid in vacuum. Nevertheless, the response of the plasma, being collective in the sense discussed in previous chapters because of the long-range interparticle interactions, is more complex than that of solid dielectric materials even for small-amplitude perturbations. The plasma response is described by the variation of the distribution function induced by the electromagnetic fields: this variation modifies the density charge ρ and the current density J through integrals in the velocity space, which in turn cause a nonlinear response because of the feedback that the particle distributions have on the mean fields. In general, one must solve the linearized Vlasov–Maxwell system [1.34–1.40] or the linearized Vlasov–Poisson system [1.7–1.8], if a restriction to electrostatic perturbations is sought. Perturbations of both the electromagnetic field components and of the distribution functions fe and fi are generally assumed to have a harmonic form dependent on a wave vector k and on a frequency1 ω. For example, for a quantity F we will consider small-amplitude perturbations of the form2 δF ∼ AF ei(k·r−ωt) + c.c.

[3.1]

1 As it is frequently done in plasma physics literature – and as we implicitly did so far – we will keep on not distinguishing between “frequency” (formally defined as the inverse of the oscillation period T −1 ) and pulsation of an oscillation (that is 2π/T ) and we will use the two terms as interchangeable. 2 We recall that the complex conjugate contribution in [3.1], A∗F e−i(k·r−ωt) , can be omitted when studying linear perturbations of an equilibrium. It must be instead retained while looking, for example, at nonlinear processes like three waves coupling.

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The Vlasov Equation 1

Consistently with the Fourier analysis on which this approach is based, the amplitude AF itself can generally depend on the space coordinate(s) along which the vector k has null component(s). Some special cases of a linear harmonic analysis can be however considered, in which AF can in principle depend both on all the spatial coordinates and on time: this is possible provided a sufficiently large-scale separation exists between the gradients of AF and both k and ω, which makes the wave amplitude substantially uniform over an oscillation period of the wave: it is the well-known WKB3, or eikonal approximation, as it was known and used in geometrical optics (see, for example, Born and Wolf (1959, §3.1, 3.2.2 and Appendix I)). For example, a version of the WKB approximation is used to describe MHD modes in toroidal geometry in tokamak plasma physics, where the dependence of fields on the toroidal angle is usually “slow” with respect to the dependence on the other variables, and goes under the name of “ballooning mode” representation (Connor et al., 1978; Pegoraro and Schep, 1986). In general, while k and ω are real for normal modes, one of the two becomes complex when a harmonic perturbation is linearly unstable. The stable or unstable nature of the perturbation follows from the normal mode analysis, which, in absence of source currents, always leads to a linear system for the perturbed electric field components Ei , which is of the kind Λij (ω, k)Ej = 0.

[3.2]

Here, we have adopted for the response tensor of the medium the notation Λij used by Melrose (1986). As we will see later (section 3.4.2 and 3.5), Λij (ω, k) is in general a matrix with complex coefficients and depends on both the polarizability and conductivity of the medium. In a lossless plasma, in which the total energy of both the particles and the electromagnetic field is conserved, the response tensor is Hermitian, that is Λij = Λ∗ji . Deferring to later more technical details regarding these points, we note here that linear analysis make it possible in principle to relate the components of the current density in the plasma to the electric field perturbations. In the most general

3 After Gregor Wentzel (1898–1978), Hendrik Anthony Kramers (1894–1952) and Léon Nicolas Brillouin (1889–1969), who in 1926 developed applications of this approximation method to the solution of wave equations of Schrödinger’s form. The general approach however dates back at least to the early 19th Century as a general technique for finding approximated solutions to differential equations whose highest derivative is weighed by a small parameter (for details, see Bender and Orszag (1999, Ch. 10)).

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case, for small-amplitude perturbations we can assume such a relation to be expressed by a tensorial coefficient, which with “obvious” notation we write as σij . The fact that σij is interpreted in plasmas as a conductivity tensor, so as it is normally meant in materials, is however subtle: we will come back to this in section 3.4.4. The collective response to electromagnetic perturbations, combined with some causality prescriptions which make the mean field in a point in space depend at a certain moment on the way that electromagnetic perturbations have influenced nearby charged particles at previous times, makes a collisionless plasma generally display both temporal dispersion (or dispersion in frequency) and spatial dispersion (or dispersion in wavelength) (see section 3.4.3). As a result, for each Fourier component of the form of [3.1], the relation between the linearized current density (apex (1)) and the small-amplitude electric field (1) is of the kind Ji = σij (ω, k)Ej . Formally speaking, the temporal dispersion (dependence on ω) and the spatial dispersion (dependence on k) of the plasma conductivity come from the time derivative and spatial gradient operators in the linearized Vlasov and Poisson equations. The tensorial character of σij can be understood due to the explicit dependence of the dispersion on the wave vector orientation, which breaks by itself the spherical isotropy of the plasma medium. Using this notation, the linearization of the Vlasov–Maxwell system [1.34-1.40] with no source current easily yields the wave equation in the form "

#   c2  i 2 ki kj − k δij + δij + σij (ω, k) Ej = O(E 2 ), ω2 ε0 ω

[3.3]

where we identify Kij (ω, k) = δij +

Λij (ω, k) =

i σij (ω, k), ε0 ω

 c2  ki kj − k 2 δij + Kij (ω, k). 2 ω

[3.4]

[3.5]

The equations above are quite general, since the whole information about the plasma response is contained in the explicit form of σij (ω, k): technical difficulties have been “postponed” in its evaluation in terms of the plasma parameters and polarization of the perturbation. In some regimes, this may be performed without using the linearized Vlasov equation but by relying instead on approximated models. In specific ranges of frequency and wavelengths,

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The Vlasov Equation 1

the average plasma response can be deduced from the microscopic equation for the single particle dynamics under the effects of external fields, or by using a large-scale fluid approximation, in which the first fluid moments of the Vlasov equation suffice for the purpose. The tensor Kij (ω, k) assumes the name of (equivalent) dielectric tensor of the plasma: in section 3.4.2, we will see its physical meaning and differences with respect to the usual dielectric tensor in materials, such as semiconductors or neutral fluids. In a lossless plasma Kij is Hermitian as Λij is, and these properties both follow (see equation [3.4]) from the fact that energy conservation imposes that σij be instead anti-Hermitian, that is ∗ σij = −σji . In the most general case, however, the tensors above have both a Hermitian and an anti-Hermitian part and the two do not coincide with the real and imaginary parts of the tensor. These features are important when processes of absorption of the electromagnetic perturbation by the plasma are considered, for example when wave–particle resonance conditions are met. We will further discuss all these statements in section 3.5. Since the explicit dependence of σij and then of Kij on k introduces a privileged spatial direction, the components of the equivalent dielectric tensor (and those of σij , too) are not isotropic even if the equilibrium plasma is. Inclusion of further privileged spatial directions, such as those related to a background equilibrium magnetic field, to source current densities or to spatial orientation of non-isotropic velocity distribution functions, increases the complexity of the explicit form of σij and hence of Kij . In the absence of a magnetic field, however, since only k provides a preferred direction that breaks the isotropy of the system, the dielectric tensor Kij gets a diagonal block form and can be split by introducing two scalar functions, K|| (ω, k) and K⊥ (ω, k), which depend on the frequency and modulus of the wave vector (the labels || and ⊥ refer here to the direction of k): Kij (ω, k) = K|| (ω, k)

ki kj + K⊥ (ω, k) k2

 δij −

ki kj k2

 .

[3.6]

In this case, as follows from equation [3.5], K|| (ω, k) = Λ|| (ω, k). Such a matrix element coincides with the dispersion function (ω, k) of the electrostatic example considered in equation [2.6]. However, if a magnetic field is present, even if parallel to k, off-diagonal elements of Kij can appear. This occurrence is related to the asymmetry that

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the magnetic field introduces between the propagation of right- and left-circularly polarized waves, even in absence of spatial dispersion (see section 4.2.4). At a more fundamental level, there are indeed further constraints on the symmetry of the coefficients of σij , Kij and Λij , which follow from the different transformation properties with respect to the axis reversal of k, which is a polar, or “true” vector, and of B, which is an axial, or pseudo-vector. It can be shown that by the definition of the conductivity tensor in the presence of dispersion (see section 3.4.3) and because of consistency with microscopic reversibility (i.e., the so-called Onsager relations, section 3.5), the components of σij must in general satisfy the following symmetries with respect to the change of sign of k and B at fixed orientation:  ∗ σij (k, B) = σji (k, B) [3.7] σij (k, B) = σij (−k, −B) In a plasma with a uniform magnetic field oriented as B = B0 ez this forces some of the out-of diagonal components of Kij (ω, k) to be purely imaginary or purely real. In this geometry, the most general form of Kij (ω, k), regardless of the orientation of k, is (see section 3.5.3): ⎛

Kxx Kij (ω, k) = ⎝ −Kxy Kxz

Kxy Kyy −Kyz

⎞ Kxz Kyz ⎠ , Kzz

[3.8]

where the components Kxy and Kyz are purely imaginary, whereas Kxz is real. 3.2.1. Dispersion relation and response matrix The solution of det[Λij (k, ω)] = 0

[3.9]

gives the dispersion relation of electromagnetic waves in the plasma. In most of the problems we are interested in, the roots of the above equation, which define the dispersion relation of each specific mode, can be written in terms of a dependence of the complex frequency ω from a wave vector k with real components, ω = ω(k).

[3.10]

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The Vlasov Equation 1

In Chapter 2, we have already seen some examples of the typical procedure to be followed in order to obtain such a result, for example with application to the collisionless Landau damping mechanism for electrostatic perturbations in an isotropic, isothermal plasma (section 2.2.1). The integrals of the perturbed distribution functions intervene in the calculation of σij (ω, k). The perturbation on f α is in turn obtained by linearizing Vlasov equation of the species α around the equilibrium. Resonant integrals of the kind ∼

(0)

1 ∂fα 3 d vk F(v) ω−k·v ∂vi

[3.11]

must be therefore dealt with. In the expression above, we have indicated with F a function which can in principle depend on the velocity components that (0) differ from the vi with respect to which fα is differentiated. The specific form of F, which is related to the linearization of the force term in the Vlasov equation4,  qα E +

pα ×B mα γα0

 ·

∂fα , ∂p

[3.12]

depends on the equilibrium field profile and on the way equation [3.12] is related to the electric field components after appropriate integration strategies – the typical one consists of the integration along the characteristics of the Vlasov equation (see e.g. Krall and Trivelpiece (1973, §8) or Brambilla (1998, Ch. 4)), which we will speak of in the second volume. Let us forget for the moment about the problem of the resonant denominator, which we have already seen to be related to the collisionless Landau damping mechanism and about which we will make some more comments later (section 4.2.5). Even when we can neglect the F(v) contribution, integral [3.11] may lead to non-algebraic solutions, which makes the dispersion relation generally non-polynomial. In order to bring equation [3.9] to a polynomial form, some limit cases are typically (0) considered, for example by expanding ∂fα /∂vk in powers of the phase velocity ω/k : depending on the regime of interest, ω/k can be assumed to be

4 For example, in the non-relativistic, electrostatic cases of section 2.2.1, we simply had F = 1 for the integral contribution along the Ei component. However, had we considered a relativistic regime, a dependence on the modulus of the velocity would have appeared through a gamma factor.

Electromagnetic Fields in Vlasov Plasmas

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small or large with respect to other characteristic velocities in the plasma (e.g. the particle average thermal speed). Even in these limits, however, the degree of the polynomial may be such that a numerical integration of it is required to find the real and imaginary parts of each root. We will discuss these approximation techniques in section 3.3. 3.2.2. The choice of the basis for the response tensor Since equation [3.2] is the rewriting of a linear problem, in which all perturbed quantities have been eliminated as a function of the electric field component, nothing prevents us to consider a different basis with respect to which to write the response tensor Λij . From a formal point of view, nothing prevents us either from casting the linear problem in a matrix form with respect to a “vector” with dimensions higher than three. For example, we could consider a “vector” constituted by all the components of field quantities and of scalar quantities for which an evolution equation is available, without having eliminated those which are “redundant” in terms of the others: setting to zero the determinant of the corresponding matrix will still give the dispersion relation obtained from [3.3]. However, if a linear transformation Vi = Mij Ej

with

det[Mij ] = 0

[3.13]

is defined, then a well-known property of determinants5 tells us that the dispersion relation [3.2] remains unchanged if, instead, we evaluate it from det[M ΛM −1 ] = 0.

[3.14]

Albeit trivial, the above statement has important practical consequences. The most evident is the possibility to exploit the possible symmetries under rotation of the original Λij tensor by applying the appropriate rotation matrix so to change the orientation of the relevant axes. For example, when equation [3.6] holds, applying a rotation that aligns k to one of the Cartesian axes, say k = kez , brings Λij to the diagonal form ⎛ ⎞ k 2 c2 K 0 0 (ω, k) − ⎜ ⊥ ⎟ ω2 ⎜ ⎟ k 2 c2 Λij (ω, k) = ⎜ ⎟, [3.15] 0 0 K⊥ (ω, k) − 2 ⎝ ⎠ ω 0 0 K|| (ω, k)

5 That is, det[AB] = det[A] det[B]. In linear algebra, this is known as Binet’s theorem.

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whence we immediately recognize that transverse and longitudinal modes are independent: this corresponds to a linear decoupling between purely electromagnetic and purely electrostatic branches. Analogous simplifications can be obtained, for example, in cylindrical symmetric configurations. In this case, we can rotate just the components of k, which are, for example, perpendicular to a background magnetic field. If we choose the background magnetic field to be oriented along, say, z, we can always rotate the orthogonal Cartesian axes so that the wave vector can be written as k = k sin θex + k cos θez , with θ which is the angle between k and B0 = B0 ez (Figure 3.1). In this geometry, sometimes referred to as “Stix reference frame” (after Stix (1962)) the dispersion relation can be written after some algebra in the general form (Allis et al., 1963) A(ω, k)

 ω 4 k

− B(ω, k)

 ω 2 k

+ C(ω, k) = 0.

[3.16]


   



















Figure 3.1. A convenient choice of the Cartesian axes for wave propagation in a uniformly magnetized plasma

The coefficients A(ω, k), B(ω, k) and C(ω, k) depend on ω and k through the components Kij (we omit below the explicit writing Kij (ω, k) for the sake

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of notation) according to the definitions: A(ω, k) ≡ Kxx sin2 θ + Kxz cos θ sin θ + Kzz cos2 θ, [3.17]   2 2 2 B(ω, k) ≡ Kxx Kzz − Kxz + Kzz Kyy + Kxy sin θ   2 +2 (Kyy Kxz + Kxy Kyz ) sin θ cos θ + Kyy Kzz + Kyz cos2 θ [3.18] C(ω, k) ≡ det[Kij ].

[3.19]

The roots of the dispersion relation [3.19] now read ω = ω(k, θ).

[3.20]

Absence of dispersion means here a linear dependence of ω on k. A particularly simple example, important because of the general relevance of the matrix form it gives, is that obtained in the “cold plasma” limit (see section 2.3.1), in which first order temperature effects are neglected in the calculation of the plasma response. In that case, again, the response tensor Λij assumes a diagonal block form. When written with respect to a Cartesian basis with B aligned with z and by choosing the x, y axes so as to use the angle θ defined as above, the response tensor becomes (Stix, 1962) Λij (x, ω) = ⎛

k 2 c2 2 ⎜ K⊥ − ω 2 cos θ ⎜ ⎜ ⎜ ⎜ ⎜ i|Kxy | ⎜ ⎜ ⎜ ⎝ k 2 c2 cos θ sin θ ω2

−i|Kxy | K⊥ − 0

k 2 c2 ω2

k 2 c2 cos θ sin θ ω2 0 K|| (ω, k) −

k 2 c2 sin2 θ ω2

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟. ⎟ ⎟ ⎟ ⎠

[3.21]

From this, a dispersion relation in the form of [3.16] is easily calculated. Besides these quite evident applications, equation [3.14] also indicates the possibility of evaluating the response tensor of the plasma with respect to the components of a field other than E. In a linear problem, this is always possible, and in some regimes of description this may be more convenient from an algebraic point of view. For example:

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– In the large-scale ideal MHD description, the Vlasov–Maxwell model can be approximated by a set of fluid equations for the plasma density, bulk velocity and pressure, and by the non-relativistic Maxwell equations, which are coupled to the plasma response by an equation for J (the so-called ideal Ohm’s law). In this modeling, it is usually more convenient to reformulate the linear problem by referring all equations to the components of the bulk fluid velocity, that is by writing it in the form Λij uj = 0. – In the still fluid but higher frequency regime of the so-called incompressible electron-magnetohydrodynamics (EMHD), where the fast dynamics of an incompressible electron fluid is considered, which moves rapidly with respect to a neutralizing ion background, all fluid equations can be combined in the so-called equation for the generalized vorticity that just contains the components of the magnetic field (Kingsep et al., 1980). In this case, it can be operationally easier to linearize the system by writing it as a null matrix product between a response matrix Λij and the components of B, that is to write it as Λij Bj = 0. In this regard, it must be however noted that, unless the matrix of basis change M is Hermitian, too, the new response tensor Λ = M ΛM −1 may not be symmetric anymore6, which can be an inconvenient feature, since the symmetry of the components of the response tensor defined with equation [3.9] often represents a useful check for the calculations. Another useful consequence of [3.15] is the possibility of considering a linear combination of the components of E: this may be convenient when we want to describe waves that are not linearly polarized. In this case, the Jones’ vector7 representation is required. For example, supposing again with no loss

6 It is not difficult to verify that this occurs for example in the description of Alfvéntype waves in the Hall-MHD set of equations, in which the response tensor written with respect to the electric field is not symmetric. Instead, it is symmetric when evaluated with respect to the formally equivalent set of two-fluid polytropic equations in the me /mi = 0 limit. The reason is in the fact that the passage from the two-fluid to the Hall-MHD set is performed by substituting the vectors of the electron and ion fluid velocities with their linear combinations, which give the bulk plasma velocity and the current density: as a result, the response tensor corresponding to Λij Ej = 0 is written with respect to two different basis in the two models. 7 The vector representation used to describe the polarization of coherent electromagnetic waves by using the reference frame of coordinates rotating with the electric field of a circularly polarized mode is named after Robert Clark Jones (1916– 2004), who introduced it in a series of papers in between 1941 and 1942 (Jones, 1941a,b,c, 1942).

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of generality to choose a local Cartesian basis so that k is along the z axis, we can consider a new basis for the electric field components, defined by the (unitary) matrix of coordinate change ⎛ 1 −i 1 ⎝ 1 i Uij = √ 2 0 0

⎞ 0 ⎠, √0 2

−1 Uij

⎛ 1 1 ⎝ i =√ 2 0

1 −i 0

⎞ 0 ⎠ √0 2

[3.22]

which transforms the Cartesian electric field components into those rotating with left-hand (EL ) and with right-hand (ER ) circularly polarized waves according to: ⎛

⎞ ⎞ ⎛ EL Ex ⎝ ER ⎠ = [U ] · ⎝ Ey ⎠ . E|| Ez

[3.23]

The new rotating coordinates EL , ER and E|| , for which the evident notation E− = EL and E+ = ER is also sometimes encountered in the literature, are defined as ⎧ 1 ⎪ EL = E− ≡ √ (Ex − iEy ) ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎨ 1 ER = E+ ≡ √ (Ex + iEy ) ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎩ E|| = Ez

[3.24]

This change of coordinate is associated with the change of basis that defines the right-hand (eR ) and left-hand (eL ) “polarization states” (see also Figure 3.2) ⎧ 1 ⎪ eL ≡ √ (ex + iey ) ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎨ 1 eR ≡ √ (ex − iey ) ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎩ e|| = ez

[3.25]

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Figure 3.2. Jones vector respresentation for left-hand and right-hand circularly polarized waves

Applying this change of basis to the Λij defined by equation [3.3] gives a symmetric response matrix with real coefficients (Stix, 1962; Allis et al., 1963) ⎛

Λij

k 2 c2 ⎜ KL − ω 2 ⎜ ⎜ ⎜ = ⎜ KLR ⎜ ⎜ ⎝ KL||

KLR KR −

k 2 c2 ω2

KR||

⎞ KL|| ⎟ ⎟ ⎟ ⎟ ⎟ KR|| ⎟ ⎟ ⎠ K||

[3.26]

where ⎧ Kxx + 2iKxy + Kyy ⎪ KL ≡ ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎨ Kxx − 2iKxy + Kyy , KR ≡ ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎩ K|| ≡ Kzz (cf. Eq.[3.8])

⎧ Kxx − Kyy ⎪ ⎪ KLR = ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ Kxz − iKyz √ KL|| = [3.27] 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Kxz + iKyz ⎪ ⎪ √ ⎩ KR|| ≡ 2

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In the case of validity of the decomposition [3.6], the response tensor defined by [3.26] maintains a diagonal form of the kind in [3.15]. When this happens, the roots (ω/k)2 of equation [3.16] are always real (see e.g. Stix (1962, Ch. 2) or Allis et al. (1963, §2.4)) and the dispersion relation can be cast in an alternative, useful form, in which the propagation angle is written in terms of the diagonal elements KR , KL and K|| and of (kc/ω)2 (Aström, 1951): 

 2 2  k 2 c2 k c − K − K L R ω2 ω2  2 2  tanh2 θ = −  2 2 k c k c KR + KL − K K − K L R || ω2 ω2 2 K||

[3.28]

3.2.3. About the number of “waves” in plasmas Another point deserving some comments concerns the counting of different “waves”, that is, of the roots of the linear problem. As long as the original response tensor Λij is a Hermitian matrix, we know that at most three real independent eigenvalues of it exist, the product of which gives the dispersion equation when set to zero. Each eigenvalue of Λij defines the dispersion equation of an eigenvector with fixed relations between the electric field components of the corresponding perturbation, that is, it defines the polarization of the wave in terms of the corresponding ω and k and as a function of the plasma parameters. However, the number of different combinations of these values depends, for each eigenvalue, on the number of its roots, once it is equated to zero. This can in principle be infinite in a medium with both dispersion in wavelength and frequency, as it can be understood by looking for example at the diagonal expression of Λij , which we have written in absence of a magnetic field: we see in [3.15] that one of the eigenvalues is twice degenerate, but the number of the roots of each eigenvalue depends on the explicit dependence of K⊥,ij and K|| on ω/k. In the example of the “other-than-Landau’s” poles in a Maxwellian plasma, which has been discussed in section 2.2.2 in relation to the roots of K|| = 0, we have seen this number to be infinite. In the case of a plasma with a background magnetic field, equation [3.16] provides a rewriting of the dispersion equation, which is useful when looking at this problem. If dispersion effects in A, B and C were negligible, we should solve a quadratic equation for (kc/ω)2 in terms of coefficients depending on k 2 and

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The Vlasov Equation 1

on the angle of propagation. Assuming (kc/ω)2 to be real, such as in the last example discussed in section 3.2.2, we would then find two independent electromagnetic branches (i.e. polarizations) identified by two different values of (kc/ω)2 . In particular: 1) if (kc/ω)2 < 0, assuming that k is real, the existing branches can differ for both the values of the real and the imaginary part of ω. Complex conjugate solutions typically exist. There are however some steady-state problems in which, instead of considering a solution with a complex ω and a real k, it is meaningful to look for a complex k and real ω. We will discuss an example of this kind in section 3.2.4; 2) when (kc/ω)2 > 0, instead, both k and ω are real, and each branch would then correspond to two modes, oppositely propagating with respect to each other at the same frequency ω. The numerical value of the roots generally depends on the angle of propagation with respect to the background magnetic field (see equation [3.20]). However, in the counting of branches, we usually identify perturbations, which propagate at different angles according to dispersion relations of the kind (kc/ω)2 = f (k, θ), as corresponding to the same kind of wave. That is, a typology of waves in a branch is identified by the dispersion relation that relates its frequency ω to the a spectrum of wave vectors k, which indicate propagation at different wavelengths in a certain cone of directions. There can be however special angles of propagation for which a given branch can assume a particular behavior, for example when two different branches coalesce into one. Such an occurrence is more easily identified when considering the dispersion relations written in the form of equation [3.27]. These limit cases will lead us to later introduce the notions of “resonance” and “cut-off” conditions of a mode (sections 4.2.3, 4.2.4, 4.2.6). For the moment, we can look at this feature from the point of view of a relatively simpler example, borrowed from classical optics in dielectrics: let us consider the problem of enumeration of modes in the probably more familiar example of light propagating through an anisotropic, dielectric crystal displaying no dispersion. In this case, too, a polynomial of the form of [3.16], quadratic in (kc/ω)2 , is obtained. Here, the analogous of the coefficients A, B and C only display quadratic dependence on k 2 and depend on the angle of propagation with respect to the principal axes of the anisotropic dielectric tensor of the medium. A particularly simple form of this

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dispersion relation can be written by choosing the Cartesian basis, which diagonalizes the dielectric tensor εij of the crystal8 (see, for example, Born and Wolf (1959, §15.2.2) for a derivation). In this form, the roots of the dispersion relation formally appear at the resonant denominators  ω 2 k

ky2 kx2 kz2  2  +    2  +    2  = 0. [3.29] c ε0 c ε0 c ε0 ω 2 ω 2 − − − εxx k εyy k εzz

The absence of dispersion in frequency allows us to identify for each arbitrary vector k two independent type of waves, i.e. branches. Their different polarizations are identified by the two different phase velocities expressing the values of the roots of the second-order polynomial in (kc/ω)2 , which equation [3.29] corresponds to. Then, for each polarization, two light rays oppositely propagating along |k| and −|k| can be found due to the ∼ k 2 dependence of the root. There are, however, special angles of propagation along which the two roots coalesce into one. This occurs when propagation along one of the principal axes is chosen, which identifies the unique value of phase velocity with which light propagates in the medium along that direction. Coming back to the plasma case, in which both dispersion in ω and k are present, we have already seen while discussing equation [3.11] why the eigenvalues of Λij (ω, k) can have a non-polynomial form with respect to ω. This explains the rich “zoology” of electromagnetic waves, which can propagate in a warm plasma. From a practical point of view, however, we could say that the effective number of branches relevant to a plasma model, in principle infinite in the full Vlasov–Maxwell or Vlasov–Poisson systems, depends on the considered plasma regime and therefore also on the degree of accuracy which we want to adopt in the description of the kinetic model for quantitative purposes. This, in turn, obviously depends on the approximations done, which leads us to the topic that we will discuss in section 3.3. Before doing so, let us conclude the introductory discussion about the general formalism with something more specific about the real and imaginary parts of the roots of the dispersion relation.

8 This rewriting of the quadratic dispersion relation is known in optics as Fresnel’s equations (of wave normals), after Augustin Jean Fresnel (1788–1827), who obtained it.

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The Vlasov Equation 1

3.2.4. Real or complex values of k and ω: steady state and initial value problems While for normal modes both the components of k and ω are real, we have mentioned before that “instabilities” with a complex ω can occur when (kc/ω)2 < 0, since we have assumed the components of the wave vector to be real numbers. In principle, however, we could solve the same dispersion relation by fixing ω as real and by looking for a complex k. When does this make sense and what does it mean? Let us address these questions by refining the notion of unstable modes related to the (kc/ω)2 < 0 condition. One should more properly speak of damped modes when the sign of the imaginary part of ω corresponds to an exponential decrease in time (it is, e.g. the case of Landau damped waves studied in Chapter 2). The word instability is instead usually restricted to modes displaying an exponential growth of their amplitude in time. Formally speaking, one refers to unstable normal modes or to unstable waves when the wave vector is real and the frequency is complex. The terms amplifying and evanescent wave are instead normally used for the case of an imaginary wave vector and a real frequency. The possibility to solve the dispersion relation for some complex ki components in terms of a real ω or to solve it for a complex ω in terms of real ki can leave ambiguous the interpretation of the nature of the perturbation. This ambiguity is however ruled out by the nature of the problem itself, depending on whether we consider it to be a steady-state or rather an initial value problem. From a mathematical point of view, this assumption is linked to the chosen boundary conditions. – Steady-state problem: in this kind of problem, we formally look to a wave which “has been always existing”. This allows us to perform a Fourier transform with respect to time over the whole interval [−∞, +∞]. This is one of the possible interpretations of equation [3.1]. Solving a steady-state problem fixes ω as real. In this case, the instability of the perturbation [3.1] corresponds to a growth (or damping) in space, which respectively occurs for k(I) · r < 0 and k(I) · r > 0, with k = k(R) + ik(I)

[3.30]

and ω, ki,(R) , ki,(I) ∈ R. Such unstable perturbations correspond, respectively, to the amplifying and evanescent waves mentioned above. We will see an example of the latter kind in section 4.2.4, while discussing the notion of overdense plasma.

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– Initial value problem: in this case, we consider a wave that has been emitted with a frequency ω from a source localized in time (and in space). Mathematically speaking, an initial value problem is treated by means of the Laplace transform, which can be seen as a “version” of the Fourier transform, which is compatible with a causal function, that is a function that vanishes before a given time that we may arbitrarily set to be t = 0. We have already seen the importance of such an initial condition related to Landau-type poles, and we will see later another implication of this, concerning the spectral width of a wave packet in section 3.4. Solving an initial value problem fixes k as real. In this case, the unstable perturbation [3.1] is growing or damped in time for a given wavelength when, respectively, ω(I) > 0 or ω(I) < 0, with ω = ω(R) + iω(I)

[3.31]

and ki , ω(R) , ω(I) ∈ R. More specifically, the ω(I) > 0 case is the one typically termed (linear) instability, whereas the ω(I) < 0 case is simply referred to as a damped mode. It is worth noticing that, since the distinction between the two problems stated above is formally fixed by the boundary conditions, both points of view are compatible with the formulation of the problem of wave propagation posed in terms of the harmonic perturbations of the kind in [3.1]. The correspondence between the variable s used in Laplace transform in equations [2.3–2.6] and the frequency ω of the linear perturbation [3.1] is s → iω,

with

ω = ωR + iωI .

[3.32]

A more detailed discussion about the comparison between spatial and temporal damping can be found, for example, in Briggs (1964), Melrose (1986, §2.5–2.6), or Landau et al. (1981, Ch. VI). 3.3. Polynomial approximations of the dispersion relation: why and how to use them We have seen in section 2.2.2 that, despite the infinite number of roots of the dispersion relation in a collisionless electrostatic plasma, only a few branches are usually physically relevant in the electrostatic linear regime: those related to the poles that are beyond the one considered in Landau’s solution are subject to an increasingly strong collisionless damping. While this is surely true for a linear perturbation, for which it does not make any sense to speak of a “wave”, if it is damped before it can complete an oscillation, we will see in Chapter 5 that nonlinear effects may make the existence of poles beyond Landau’s ones quite relevant: for example,

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The Vlasov Equation 1

nonlinearly driven, self-consistent structures akin to “phase-space” vortices (i.e. BGK-like structures, section 5.5), for which particle-trapping effects oppose collisionless damping by locally flattening the distribution function, may in principle “hook” to these extra modes by acquiring a propagative character related to the real part of their frequency (see section 5.6). If we then consider a magnetized plasma, the number of roots of physical interest increases also in the linear regime, as the electromagnetic (i.e. transverse to B) branches, too, are involved: for example, branches related to resonances at all integer multiples of the particle cyclotron frequencies exist, ∼ ω ± nΩα . Even if these branches are linearly damped as more as n increases because of similar reasons to the electrostatic case (see sections 4.2.5 and 4.2.6), their existence is very important for heating processes in collisionless plasmas: the energy of electromagnetic waves can indeed heat particles through resonant absorption at these frequencies. This, for example, has important practical applications for heating techniques of magnetically confined laboratory plasmas: e.g. the ion cyclotron resonance heating (ICRH) and electron cycloctron resonance heating (ECHR) methods9. Identifying the roots of the dispersion relation beyond the first Landau poles is therefore of practical interest, especially in geometries and configurations that complicate the form of the response tensor. At the same time, for each physical problem, a sort of hierarchy in the relevance of the roots of the dispersion relations can be established, which allows us to focus only on a few of the eigenmode solutions. It is here that approximate solutions to the Vlasov dispersion relation become useful. Besides, and at a more general level, it is worth stressing that looking for approximated analytical forms of the dispersion relation is not even a second rate choice with respect to the search for roots by numerical integration of the “complete form” of the dispersion relation: it is rather a complementary tool for the investigation and identification of the normal modes in a plasma. There are three main reasons for this:

9 In a fusion machine like ITER, for example, ICHR and ECHR represent two of the three external heating systems (the third one consisting in neutral beam injection), which are devised to heat the initial plasma up to burning conditions for self-sustained thermonuclear fusion reactions. ICHR (at frequency of ∼ 40 − 50 MHz) and ECHR (at frequency of ∼ 170 MHz) together are expected to be able to depose up to ∼ 20 − 25 MW heating power out of the ∼ 50 MW needed to raise the core plasma temperature up to the required ∼ 150 × 106  K.

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1) First, a reason of numerical character: identifying the roots by quantifying the values of their real and imaginary parts can be practically difficult in many plasma regimes, because ill conditioning problems can occur in the numerical solution. This is true, in general, also in the simplest case of homogeneous equilibrium configurations. Furthermore, in plasmas that are spatially inhomogeneous at equilibrium, some cases can be encountered in which the amplitude of the eigenmodes varies so much in space that ensuring an accurate convergence of the numerical eigensolver becomes a quite challenging task10. 2) Second, approximated solutions and reduced models usually provide a deeper physical insight. The reasons and ways this occurs may be of different nature, depending on the kind of model simplification that has been performed, whether it is by approximation (i.e. by restriction of the parameter range in a narrower region and/or by restriction of the domain of frequency and wavelength in which the solution is sought), or by reduction to a subclass of exact solutions of the Vlasov–Maxwell system (we have already seen an example of this in sections 2.4–2.6, concerning the “water-bag” model). Whatever the case, the driving idea – and the desired outcome – is to reduce the complexity of the physical problem, not only from a mathematical but also from a conceptual point of view, so that the fundamental ingredients and mechanisms at play can be better recognized: it is in the end the reductionism paradigm on which the whole of classical mechanics and linear physics grounds itself and which can still provide a guide to tackle complex phenomena in plasma physics. 3) Last but not least, an approximated or “reduced” linear analysis is fundamental in connection with approximated nonlinear models it can be related to. Approximated models are of interest especially because of their both theoretical and numerical applications to the nonlinear dynamics. This is indeed the regime in which their usefulness is more evident. The linear analysis of approximated models, that is, the “linear benchmark” of the model, aimed at verifying its consistence with a full Vlasov–Maxwell description is therefore a mandatory step for their use in nonlinear regimes.

10 This kind of problem can be encountered, for example, in the numerical eigenmode analysis of tearing-type instabilities (Furth et al., 1963), especially if an accurate approximation of the eigenfunction, and not only of the eigenvalue, is sought (we recall indeed that in a first-order perturbation theory performed with respect to an expansion parameter ε, the perturbed eigenvalue displays an error ε smaller than the corresponding eigenvector).

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A distinction is then worth making about the different methodologies related to the kinds of simplification of the linear Vlasov–Maxwell (or Vlasov–Poisson) system mentioned in the second point above. We will not consider further in this section the techniques based on reduction by restriction to a class of exact solution of the problem, for example the water bag model already presented. We just note here that, from an operational point of view, the simplification in that case comes from a specific hypothesis about the form of the distribution function, which makes it possible to write in closed form the velocity integral of the perturbations of fα , formally without performing any kind of approximation. We will see in the second volume other kinds of reduced “exact” models of this type, for instance, the “multi-stream” model in which deltas of Dirac are considered instead of Heaviside functions, or the model based on considering distribution functions as given by the sum of Hermite polynomials in the velocity coordinates. It is only when we want to use such exact solutions as a “basis” of functions with respect to which we can represent whatever distribution function is of practical interest, that the notion of approximation intervenes. It is at this level that we can try to put it in the physical terms we are used to, by looking for a physical interpretation of some characteristic parameters that are related to the convergence of the solutions obtained with the reduced model to the solution obtained with the full Vlasov model. We have seen an example of such parameters in the waterbag description: the number of bags and the fluid density or velocity associated with them (see section 2.6). The case of the truncated-Vlasov and fluid-type descriptions, which we are going to discuss next, are different. They represent more “traditional” examples of approximation of the Vlasov–Maxwell system, both from a historical point of view (they were the first techniques developed to tackle the linear mode analyses in collisionless plasmas) and from the point of view of closeness to the reduction methods used in classical mechanics (they rely on approximations well identifiable in terms of some expansion parameter). 3.3.1. Truncated-Vlasov and fluid–plasma descriptions for the linear analysis In most cases, as already mentioned, a polynomial form of the components of Kij (ω, k) = Kij (ω, k, θ) can be obtained in terms of powers of ω/k compared to the amplitude of some characteristic velocity of the plasma. We speak in this case of a truncated Vlasov description.

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We have already used this procedure, although without using this terminology, in section 2.5, while comparing solutions of the linear Vlasov–Poisson problem with those of the waterbag model used to approximate it. We are now going to revise and discuss such a procedure from a more formal point of view. The degree of the polynomial form of Kij (ω, k), and therefore the number of possible branches, typically increases the higher the number of terms retained in the expansion is. At the same time, the effect of spatial dispersion, that is, the exponent with which k enters in dispersion relations [3.16], also increases together with the number of terms of the truncation that are retained: we have already seen an example of this, concerning the calculation of the Bohm–Gross dispersion relation (see equations [2.13]). Making a truncation corresponds to limiting the spectrum of modes considered in the model. The truncation is defined in terms of an expansion parameter that can be expressed in powers of kV /ω, k or ωτ for some V ,  or τ , indicating the characteristic velocity, or length, or time scale of the plasma respectively. The restriction in wavelength or frequency can be macroscopically interpreted as a thermodynamical prescription on the considered phenomena. Typical examples are the two opposite limits of an adiabatic-type assumption, ω/k  vth,α , or of an isothermal-type assumption, ω/k vth,α . Further discussion and examples can be found in Pegoraro (1991). It is worth stressing in this regard that looking at the approximation as a restriction on the phenomenon under investigation is usually more appropriate than considering it as a self-consistent assumption about the plasma itself: since Vlasov plasmas are typically out of equilibrium systems, an a priori thermodynamical closure is seldom justifiable. Instead, it is generally legitimate to assume, for example, incompressibility when the phase velocities considered are much smaller than sound speed in the medium, and so on for other restrictions interpretable in thermodynamical terms. For related reasons, the adiabatic-like kvth /ω 1 limit is also sometimes called the “hydrodynamic limit”, since the adiabatic condition it corresponds to is the one which formally justifies, in a collisional gas, the closure of the infinite chain of fluid moments by assuming almost thermodynamical equilibrium conditions and therefore negligibility of heat fluxes. In the limit in which vth → 0, the cold plasma model, as discussed in section 2.3.1, is recovered.

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The Vlasov Equation 1

Provided the above interpretation about the limitations introduced in the description of the phenomenon, both approximations ω/k  vth,α and ω/k vth,α (as well as further ones corresponding to other kinds of thermodynamical “closures”) can be then reinterpreted within a fluid-type description, in which a limited number of fluid moments of the Vlasov equation are used to compute σij (ω, k) and then Kij (ω, k). In this case, the thermodynamical interpretation of the performed expansion is also the criterion with which the corresponding closure of the infinite chain of fluid moments must be read. In early plasma literature, such a fluid approach was termed “transport theory” (see e.g. Allis et al. (1963)), basically with reference to the derivation of the diffusion and thermal conduction coefficient in gases developed by Chapman and Cowling, which relayed on the evaluation of the fluid moments of the Boltzmann equation (Chapman and Cowling, 1953, Ch. 6). Today, expressions such as fluid modeling or extended11 fluid modeling are instead preferred12. When not needing a finer level of specification, we will generically refer henceforth to these approaches as to a fluid plasma description. Particular care is however required when considering fluid models that allow a description of temporal and spatial scales which approach kinetic ones. Beside coping with problems related to thermodynamical prescriptions (i.e. the fluid closure assumption), a fluid modeling also requires a consistent macroscopic description of the plasma response to electromagnetic fluctuations. In standard “non-extended” fluid models, this is formally granted by means of the ideal or resistive Ohm’s law when spatial scales larger than the ion gyroradius and frequencies smaller than ions gyrofrequency are

11 “Extended” is here meant with respect to the hydrodynamic cold plasma limit of section 3.3, in order to express the inclusion of microscale effects in the model, for example effects related to non-ideal terms of the generalized Ohm’s law, or related to the inclusion of non-isotropic pressure tensor components, or even approximated modeling of Landau damping effects. 12 The term “transport theory” nowadays refers to the same physical problem as in the past (the quantification of transport processes of matter, momentum, energy, etc., in plasmas) but in a much wider sense. It does not necessarily imply a fluid-type description but can acquire instead some specificity related to the phenomenology of interest, and depending on the subdomain of plasma physics it is applied to (e.g. the “physics of transport in tokamaks”).

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considered. Rigorously speaking, however, a fluid MHD modeling can become questionable as spatial scales approach from above the largest of i particle gyroradii, that is, typically that of ions, ρi = vth /Ωi . At this scale, ions become unmagnetized and microscopic effects of kinetic nature, that is, related to the shape and evolution of the particle distribution function, start becoming important. In the single fluid MHD model, in which the center of mass of both species is considered and both electron and ion temperatures contribute to the whole plasma pressure, therelevant gyrotropic scale is the so-called ion-sound Larmor radius, ρs = P/(nmi Ω2i ). In plasmas with comparable magnetic and kinetic pressure, the latter is of the order of the ion skin depth di = cA /Ωi , cA being the Alfvén velocity, since ρ2s = βd2i /2, with β = 2μ0 P/B 2 . In most extended fluid models, microscopic effects are then introduced as low-frequency corrections, occurring at microscopic scales that may be comparable to particle gyroradii. A thorough discussion of the orderings with which kinetic effects can be introduced in magnetized plasmas by following this kind of approach can be found in the book by Hazeltine and Meiss (1992). Among the first extended fluid plasma descriptions so devised, Grad’s “13 moment model” (Grad, 1949) and the Braginskii model (Braginskii, 1965) should be recalled. In order to provide a practical example and to elucidate some delicate points which may arise, in sections 3.3.2–3.3.3 we will consider and compare two extended fluid models in which non-ideal MHD effects are introduced with the purpose of improving the description of dispersive effects of the normal modes: in the one model, such effects are introduced as low-frequency corrections at microscopic scales, and in the other model, the “extension” is realized by allowing inclusion of a high-frequency dynamics (at cyclotron scales). As we will see, both models have their own advantages and delicate issues or drawbacks. These, in general, may not be trivial to identify a priori. 3.3.2. Wave dispersion and resonances allowed by inclusion of high-order moments in fluid models The problem of identifying the number of modes that can be excited in a fluid plasma description was probably first addressed by Stix (1962) and Allis et al. (1963). The latter, in particular, pointed out that when an adiabatic closure for the scalar pressure is assumed in a collisionless electron-ion plasma with m different ion species, a total of m + 3 waves can be found:

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1) two electromagnetic waves (i.e. with transverse polarization); 2) one electron plasma wave (i.e. the Bohm–Gross mode); 3) m further modes, which can be identified as ion-plasma waves and which correspond to out-of-phase oscillations of the different ion species with respect to each other and to the electrons. These fundamental branches can then couple with each other and/or acquire new features as the geometry and conditions on the equilibrium system are modified. In general, changing the kind of closure of the highest fluid moment considered changes the nature and number of branches described: we have mentioned before the case of incompressibility, which can trivially rule out the coupling of the plasma response with compressive modes. More subtle is instead the effect on the dispersion relation, which follows from truncations or expansions that are performed in the fluid moment equations and which affect the number of degrees of freedom that are retained in the evolution of higher order fluid moments. For example, inclusion of the full pressure tensor dynamics in the fluid description of a collisionless plasma model allows, in certain regimes of frequency and of propagation, a reasonable description of further branches, which normally are not included in a polytropic closure. In this way, it is for instance possible to model (up to a certain degree of accuracy) the propagation of electromagnetic waves, which are dominated by the electron dynamics and which become linearly unstable to Weibel-type modes when an anisotropic electron distribution – i.e. an anisotropic pressure tensor – is considered in an unmagnetized plasma (Sarrat et al., 2016). Or, with analogous formalism, we can describe ion electromagnetic modes that propagate perpendicular to a magnetic field, while oscillating at twice the ion cyclotron frequency: they correspond to the ω 2  4Ω2i ion-Bernstein waves (Verheest, 1967; Del Sarto et al., 2017). In order to exemplify the usefulness and limitations of this fluid moment-based approach in comparison with the truncated Vlasov model, let us first discuss the possibility of including, in the formal eigenmode solutions of a fluid model of a magnetized plasma, resonances at the integer multiples of the cyclotron frequency. Here, we focus in particular on the possibility of describing the ω 2  4Ω2i ion-Bersntein branch, of which we can provide an intuitive explanation without entering into the details of calculations.

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It is known indeed that even a single fluid model in which electron inertia is neglected, namely the Hall-MHD model, allows a description of electromagnetic waves that includes the ion-cyclotron resonance ω 2 = Ω2i (see section 4.2.4). Despite the problems of consistency13 of a fluid description when particles (in this case ions) become unmagnetized at ω ∼ Ωi , and with all the caveats about the interpretation of the fluid eigenvalue solutions near such a resonance, the related linear dispersion relations are in reasonable agreement with those obtained from the Vlasov model. The description of the ω 2 = Ω2i resonance is related to the v α × B Lorentz forces in Vlasov equations, which appear in the first-order fluid moments of the two Maxwellian species in the form (uα and Pα below express, respectively, the order 1 and isotropic order 2 velocity moment of f α) qα qα ∂uα E + Ωα + uα · ∇uα = ∂t mα |qα |

  ∇Pα B uα × − . [3.33] B nα mα

Once these two equations for α = e, i are combined with the other set of fluid equations and the me /mi → 0 limit is taken so to get the single fluid model, it is the ion contribution related to the Lorentz force which ends up dominating with a ∼ (ω 2 − Ω2i )−1 resonant term in the dispersion relation. We can then interpret the period 2π/Ωi as the time required by the components of the vectors ui locally perpendicular to B to make a complete rotation around the magnetic field under the effect of the term ui × B/B above14. With the same limitations in mind, the argument can be in principle extrapolated also to the ω 2 = Ω2e electron case, when me /mi is considered to be finite. Analogous is then the interpretation that we can give of the ∼ (ω 2 − 4Ω2i )−1 resonance we were speaking of previously: it comes from the v α × B contribution of the Vlasov equation in the dynamics equation of the components of the second-order fluid moment, Πα . Focusing just on the

13 We will later see (section 4.2.5) this to be related to the need of introducing in the fluid description some appropriate dissipation mechanisms associated with resonant absorption, which in Vlasov formalism are accounted for by Landau’s prescription. 14 We stress once more that, because of the limitations of the fluid model close to the resonance, this interpretation is qualitative. We recall indeed that only at the driftordering ω  Ωi , the second right-hand side term of equation [3.33] dominates over the last one and the perpendicular components of both ions and electrons are equal to E × B/B 2 . As the resonance ω 2 = Ω2i is approached, this of course does not hold anymore but the fluid model itself formally breaks down (see section 4.2.5).

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The Vlasov Equation 1

terms of interest here, this equation can be written, in vector notation, as: ∂ α qα Π + ...... = Ωα ∂t |qα |

  B B Πα × + × Πα − ∇ · Qα , [3.34] B B

where on the left-hand side we have omitted some symmetrized terms related to further contributions of the first-order moment uα and which are due to convection (∼ (uα · ∇)Πα ), compression (∼ (∇ · uα )Πα ) and the action of the strain tensor (∼ (∇uα ) · Πα ). The first right-hand side term of equation [3.34] expresses the “rotation” of the pressure tensor components around the local magnetic field direction. Note, like in equation [3.33], the Ωα “weight” in front of this term: it indicates that the components of Πα , as the components of uα do, rotate at the respective cyclotron frequency. From a geometrical point of view, however, differently from uα , the components of Πα perpendicular to B are represented not by a one-dimensional object like a segment with an orientation in space, but rather by an ellipse, whose semiaxes have length proportional to the values of the perpendicular eigenvalues of Πα . This means that only half the period 2π/Ωα is required for the ellipse to make an oscillation, that is to return to a position indistinguishable with respect to the initial one (see Figure 3.3). Of course, this kind of recurrence cannot be seen in the isotropic limit, in which the components of Πα perpendicular to B are represented by a circle.



















Figure 3.3. The period of congruency of an ellipse (i.e. a quadrupole, Nm = 2; central frame) is half of that of an arrow with orientation (i.e. dipole, Nm = 1; left frame) rotating in the plane at the same frequency Ω. For an octupole (Nm = 4; right frame), the period would be 1/4 shorter than the dipole case. For a color version of this figure, see www.iste.co.uk/delsarto/vlasov1.zip

Going further, one could argue that, in principle, as higher order fluid moments of the distribution functions are retained in the model, then faster oscillations – and then, arguably, resonances at higher frequencies – can be described with this fluid approximation of a Vlasov plasma. Roughly

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th speaking, indeed, any 2Nm -order multipole of a quantity in a plane, say a th “2Nm -pole”, can be imagined like a geometrical shape with 2Nm in-plane axes of symmetry and discrete central symmetries of angles, which are integer th multiples of 2π/Nm (see Figure 3.3): in this sense, the Nm order velocity th moment of a distribution function, which is build as an Nm -order rank th symmetric tensor, can be associated with an 2Nm -order multipole of f α .

However, a problem of consistency with a full Vlasov description can arise, which is related to the hypotheses that have been made to get rid of the higher order moments in this fluid-type normal mode analysis: since higher order fluid moments enter in the dynamic equations of lower ones by means of the components of their spatial gradients15, the specific choice of closure made about them affects the way dispersive effects are retained in the model. In practice, neglecting some features of the dynamical evolution of the fluid moment on which the closure assumption has been done eliminates from the components of the response tensor some terms which are proportional to ki and possibly to further ω-dependent and ki -dependent contributions: these come from the way that the perturbation on the highest order moment is closed in terms of the lower moments. A comparison between the extended fluid model and a truncated Vlasov description within some appropriate limits can help with identifying the problem and fixing the issues. 3.3.3. An example: effects

fluid moments and Finite–Larmor–Radius

In order to fix the ideas without any need to enter in the details of the calculations16, one could think of closing equation [3.34] by neglecting the gradient of the heat flux Q, or, rather, by taking the gyrotropic limit of the same equation in the case in which a low-frequency “double adiabatic” approximation (Chew et al., 1956) is sought, possibly with the inclusion of first-order finite Larmor radius effects (Thompson, 1961; Macmahon, 1965b,a).

15 For example, the fluid velocity u enters in continuity equation for density with a ∼ ∇·u term. The isotropic (P ) or anisotropic pressure tensor (Π) enter in the momentum equation for ∂t u with force terms like ∼ ∇P or ∼ ∇ · Π (see equation [3.33]). Heat fluxes enter into the equation for the pressure, scalar or tensorial, with their gradients (see equation [3.34]), and so on. 16 These can be found in the articles already cited in section 3.3.2 and in those which are going to be cited next.

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The result of both operations may in general lead to quantitative discrepancies or even to spurious dispersive effects with respect to the Vlasov approach, not only for the newly added branches, but also for the description of those that were correctly modeled by simpler fluid closure conditions that neglected such dispersive effects: adding further effects, if not done carefully, does not necessarily correspond to an improvement of the model. The reason is related to the further approximations that have been introduced to justify the closure of the fluid-moment modeling and that are not necessarily equivalent to (i.e. do not necessarily commute with) a truncation-type approximation. In the example we are considering: the gyrotropic assumption quoted above, which allows us to simplify the second-order velocity moments of Vlasov equation by replacing them with two equations for the pressure components (which are, respectively, perpendicular and parallel to the magnetic field), is grounded solely by the ω ∼ k|ui | Ωi hypothesis. The two equations, which can be deduced from equation [3.34] by taking its isotropic part and its projection along the local magnetic field (Chew et al., 1956), read, in tensor notation17: ∂P||α ∂t

+ uα k

∂P⊥α ∂t

∂P||α ∂xk

+ uα k

+ P||α

∂uα ∂uα k = −2P||α bl bk l , ∂xk ∂xk

∂P⊥α ∂uα ∂uα + 2P⊥α k = P⊥α bl bk l , ∂xk ∂xk ∂xk

[3.35]

where bi = Bi /|B| is the unitary vector indicating the local direction of the magnetic field, with respect to which the symbols || and ⊥ are defined. If one then considers the dispersion relation of waves that propagate perpendicular to a uniform background magnetic field B0 as it is obtained from the set of MHD equations closed with equations [3.35], it coincides with that obtained from the full Vlasov–Maxwell model truncated at the lowest order with respect to the expansion parameter ε1 = ω 2 /Ω2i 1. In the kinetic framework, the zeroth order of this expansion corresponds to the gyrokinetic ordering, which is based on the adiabatic conservation (up to the order ε1 ) of the particle magnetic moment (Kruskal, 1962). The mode

17 A re-derivation of the double adiabatic equations in the formulation given in equations [3.35] can be found in (Del Sarto et al., 2017); it differs from the “usual” form often encountered in literature since the further ideal MHD hypothesis has not been used, yet.

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described at this ordering is the magnetosonic branch at purely perpendicular propagation. Here, the resonance ω 2 = Ω2i is of course ruled out by the low-frequency limit, but on the other hand it would not even concern this branch, which can cross it. In any case, no higher frequency modes can be described by the extended MHD fluid model closed with equations [3.35]. Inclusion in the extended model of first-order FLR corrections can be then realized by taking the next order in the ε1 1 expansion combined with the i further hypothesis |u⊥ | ∼ vth,i (Thompson, 1961) (see also Cerri et al. (2013) for a recent rederivation). This introduces the further expansion 2 parameter ε2 = k 2 vth,i /Ω2i = k 2 ρ2i ∼ ε1 1 with respect to which we can truncate both the fluid and Vlasov–Maxwell models. It should be noted that the ordering ε1 ∼ ε2 1 is more restrictive than the gyrokinetic one based on the condition ε1 1 only. The latter condition, under certain hypotheses about the ordering of the parallel and perpendicular gradients, which is relevant for example to the description of Alfvénic turbulence, is indeed compatible with the ordering ε2 ∼ 1 (Schekochihin et al., 2009). However, as first pointed out by Macmahon (1965a), a truncation performed for ε1 ∼ ε2 1 evidences an inconsistency of the extended fluid model based on equations [3.35] with inclusion of FLR corrections obtained without making use of the equation for the heat flux: when corrections at the order ε1 ∼ ε2 are obtained by expansion of equation [3.34] only, without taking account of the equation for ∂t Q, a positive dispersion for the magnetoacoustic mode at perpendicular propagation is obtained, instead of the expected negative dispersion predicted by the truncated Vlasov model (see Table 3.1). The reason can be understood as related to the fact that the magnetoacoustic branch has a compressive part, which makes the components of the fluid velocity perpendicular and parallel to k very different in modulus one with i respect to each other: this does not agree with the condition |u⊥ | ∼ vth,i required by coherence with the assumption of isotropy over a gyration period for the perturbation of the gyrotropic, perpendicular pressure. Such an assumption introduces at the next order of the expansion some errors that correspond to differences not only in modulus but also in the sign of the dispersive corrections. In order to fix these errors one should consider in the calculation of the linear dispersion relation of the extended fluid model also the relevant powers of ε2 , which enter in equation [3.34] via the heat flux components involved: in this case, as first shown by Mikhailowskii and Smolyakov (1985), the limit ω Ωi taken both in equations [3.34] and in the equation of ∂t Q makes it possible to close the relevant heat flux components in terms of the pressure

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tensor in a way, which gives an exact18 description of the magnetosonic branch up to the ε1 ∼ ε2 order. Dispersion Relation

Model

   k2 ρ2i 2 ω 2  k2 c2A + vth 1+ 16

Truncation Hypothesis First ε01 (CGL)

CGL-FLR

then

Full Π evolution

ε1 ∼ ε2 (FLR) First ε1 unordered

with

then

∇·Q=0 Full Π evolution,

ε1 ∼ ε2

     k2 ρ2i k2 ρ2i 2 + vth 1− ω 2  k2 c2A 1 − 8 16

     k2 ρ2i 5k2 ρ2i 2 + vth 1− ω 2  k2 c2A 1 − with contribution 8 16 from ∂t Q = ...

ε1 ∼ ε2

     k2 ρ2i 5k2 ρ2i 2 ω 2  k2 c2A 1 − Truncated Vlasov 1− + vth 8 16

ε1 ∼ ε2

Table 3.1. Dispersion relation of the magnetoacoustic mode at perpendicular propagation, obtained at the order of truncation ε2 ∼ k2 ρ2i  1 from models based on different initial orderings with respect to ε1 ∼ ω 2 /Ω2 . Note the equivalence of the results of the two lowest lines of the table

Quite different would have been the outcome of performing an expansion in powers of ε2 1 if the extended fluid model had been closed by simply neglecting the heat flux gradients in equations [3.34], that is, by initially relaxing the ε1 1 hypothesis in the full pressure tensor equation. In this case also a higher frequency branch would have appeared among the normal modes described by the model (see section 3.3.2), whose dispersion relation at perpendicular propagation is in agreement with the generalized n = 2, i.e. ω 2 ∼ 4Ω2i , non-quasi-electrostatic Bernstein mode obtained within a Vlasov–Maxwell approach (Fredricks and Kennel, 1968). In particular, retaining the further degrees of freedom of the full pressure tensor dynamics lets the perturbations on the component of Πα evolve unbound by the constraint of gyrotropy and comparison between the resulting polynomial dispersion relation and the numerical solution of the untruncated Vlasov equation gives results in qualitative good agreement over a quite large spectrum of wavelengths and frequencies in the parameter space, both for the

18 The notion of exactness is here referred to the Vlasov–Maxwell result, in this case obtained after truncation at the same order of expansion.

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magneotacoustic and Bernstein-like branch at perpendicular propagation (Del Sarto et al., 2017). Performing then the truncation ε2 1 yields for the high-frequency branch a spatial dispersion with opposite sign with respect to the Vlasov limit for small values of k⊥ but, as k⊥ increases, the kinetic result is well approximated and a good quantitative agreement with the truncated Vlasov 2 2 analysis is obtained for ε2 ∼ k⊥ ρi 1 (Del Sarto et al., 2017). If, in the same ε2 1 limit, we just look at the lower frequency, magnetoacoustic branch (which allows us indeed to take the ε1 ∼ ε2 1 limit previously considered), a good agreement is recovered with respect to the truncated Vlasov result, despite the fact of having disregarded the heat flux contribution in equation [3.34]: in this case, the spatial dispersion appears with the correct sign, even if there are quantitative discrepancies with respect to the truncated Vlasov result because of a factor 5 in the dispersion term. The results of the approximation k 2 ρ2i 1 performed on the magnetosonic dispersion relation in the different models compared above can be summarized as in Table 3.1. A more detailed discussion on the combined problems of closure of extended fluid models and truncation, related to the description of linear MHD-type modes with inclusion of FLR and further kinetic effects (e.g. Landau damping), can be found in the works by Passot and Sulem, such as (Passot and Sulem, 2007; Sulem and Passot, 2014), and in the review articles by Hunana et al. (2019a,b). From the discussion and comparison of the two fluid models above, it should be evident that using an extended fluid approach to approximate the Vlasov–Maxwell system is a particularly delicate task, if not a tricky one, even when it is interpreted as a restriction to the range of linear modes that are described. As a general rule of thumb, however, the following criterion can be followed: – In the limit in which a fluid moment-based model and a full Vlasov– Maxwell model can be analytically compared by means of a truncated Vlasov description, the possible discrepancies between the two linear mode analyses can be corrected, even quantitatively, at a given order of the expansion parameter ε, if in the fluid model all the relevant dispersive effects are consistently retained, including those that can come from fluid moments of higher order than that of the highest order moment whose dynamics equation has been explicitly considered in the model. In order to give an explicit example, in the case considered above, the highest order moment equation retained in the model is that for ∂t Πiij , in

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which also the heat flux gradients ∂k Qiijk appear. Quantitative consistency with a truncated Vlasov expansion requires including in the equation for Πiij the contribution from ∂k Qiijk , closed at the relevant expansion order by using the equation for ∂t Qiijk (see Mikhailowskii and Smolyakov (1985)). The appropriate fluid closure criterion must be in this way determined for each specific physical problem: the lowest order fluid moment, which can be neglected, is the one that starts contributing to the dispersion relation of the linear modes involved in the problem with powers of the expansion parameter ε that are negligible at the level of truncation chosen by comparison with the truncated Vlasov dispersion relation. The technical difficulty is in the fact that, in the determinant of the response matrix, contributions at a given order in the expansion parameter, say εn , can result from the product of different powers of ε, which are contained in the diagonal and out-of-diagonal matrix elements. This makes the closure criterion above generally difficult to be recognized a priori. 3.3.4. Key points about approximated normal mode analysis In order to summarize what we have seen in this section: – the complexity related to the solution of the dispersion relation and identification of the different branches that can propagate in a collisionless Vlasov plasma can be eased by the use of reduced models, which yield polynomial approximations of the dispersion equations. We have discussed the approximations based on the expansion of the distribution function (truncated Vlasov models) and on the integration in the velocity space (fluid models); – expanding the Vlasov–Maxwell system with respect to some parameter ε is not equivalent to first closing the infinite chain of fluid moments of the Vlasov equation and then performing the same expansion in powers of ε (obviously!). This is to say that care must be taken with the formal assumptions made to perform the fluid closure and the truncation, respectively, since the two operations may be not equivalent nor commute (even if they may apparently “seem to do so” on an intuitive physical footing: see the ω Ωi and kvth,i Ωi limits of the example considered above); – inclusion of higher order moments in a fluid-type description allows in principle the inclusion of higher order resonances and higher order dispersive effects, but reasonable consistency of the closure assumption must be checked against the full Vlasov model, at least with respect to the propagation of the linear modes in the frequency and length scale of interest: based only on the reasonability of physical assumptions, the agreement is not granted, a priori,

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not even when an analytical comparison can be made with a truncated Vlasov model. Consistency may indeed require the inclusion of contributions from the dynamics of fluid moments whose order is higher than the order of the dynamic equation on which closure has been performed – in the examples of the previous section, depending on the case, different kinds of “corrections” have been considered for perpendicular magnetoacoustic waves in an initially gyrotropic description: in the one case by relaxing and extending the gyrotopic pressure assumption so to include the out-of-diagonal components of the pressure tensor; in the other case by including the components of the heat flux with respect to both the gyrotropic and non-gyrotropic limits. We conclude by noting that, even if the truncated-Vlasov or fluid approximation (or combinations of the two) surely provides a simplification of the problem, useful for an easier identification of the principal normal modes in a given plasma regime, a numerical solution may still be required to find the roots of the polynomial dispersion relation. In this case, too, the problems related to ill conditioning of solutions, already discussed in section 2.6.1, can occur. 3.4. Vlasov plasmas as collisionless conductors with polarization and finite conductivity: meaning of plasma’s “dielectric tensor” Having recalled on several occasions, so far in this chapter, the results previously discussed about the solution of the linear Vlasov–Poisson problem for electrostatic perturbations, it is the moment to come back to symbol , which for historical reasons and coherently with the majority of plasma physics literature, we have used in Chapter 2 for the dispersion function, and which evidently resembles the dielectric permittivity in materials. This is in reality more than a resemblance, considering that the tensor that we have named Kij (and which in some books is indicated with ij ) is called the “dielectric tensor” of the plasma. Let us now discuss this point, as well as why it is legitimate to consider the set of Maxwell equations in a vacuum to describe the electromagnetic waves in a Vlasov plasma, while at the same time we speak of a plasma polarization. Let us first recall Maxwell’s macroscopic equations in materials, which we can a priori expect to be more appropriate to describe electromagnetic waves propagating in a medium, regardless of the “state” of matter it is in: ∂ ∂B ×E =− , ∂r ∂t

[3.36]

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∂ ∂D ×H =J + , ∂r ∂t

[3.37]

∂ · D = ρ, ∂r

[3.38]

∂ · B = 0. ∂r

[3.39]

The writing of the effective electric field D and effective magnetic field H in the medium, in terms of the fields in vacuum, E and B, generally requires some “constitutive equations” D = D(E, B) and H = D(E, B), which close the system and which makes the response of the medium to the external fields intervene. A further constitutive equation J = J (E, B) is then required to relate the current J , which is due to the motion of free charges, to E and B: this typically takes the name of Ohm’s law, when a macroscopic fluid-like scale description is used, or, in a full kinetic Vlasov approach, it is the integraldifferential relation [1.40], which we have in principle used to obtain equation [3.3]. An element of curiosity, and then probably of confusion, could emerge when one thinks of the reason for which D and H have been first introduced, and then compares the notation above with the constitutive equations that we have used for the mean electric and magnetic field in the plasma. The reason of having mostly abandoned the use of D and H in modern plasma physics literature19 is probably due to a combination of elements: on the one hand, a sort of “economy of notation”, at least from a formal point of view even if not from a conceptual one, due to the lack of the need to introduce a further polarization or magnetization vector in the constitutive equation for D and H, since the fields in the plasma are uniquely determined by fe and fi . On the other hand, we can also think of a motivation of an essentially historical character and related to the implicit meaning, other than the general meaning of being the effective electromagnetic fields in a material medium, which is nowadays attributed to D and H in the context of plasma physics (and in several other fields of classical physics).

19 It is interesting to note that, at the dawn of plasma physics, before Vlasov’ paper but also for some years later, most scientific articles and books used D and H for the mean field in plasmas. This usage seems to have been progressively abandoned (with obvious exceptions) since the 1960s–1970s, becoming nowadays essentially a matter of personal taste of the authors.

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We recall indeed that the dielectric induction D and the total magnetic field H are, generally speaking, the effective amplitude of the electric field and magnetic field in the medium, which result from the superposition of the external fields and of the fields generated by the microscopic response of the medium, averaged over macroscopic scales: the meaning they assume for dielectric materials, for which these symbols have been historically introduced so to point out the difference with respect to E and B, often identifies D and H with the constitutive equations that relate them to the polarization and magnetization vectors in dielectrics. This habit may be somehow enforced by the fact that in most classical electrodynamics textbooks the case of dielectrics is the typical example considered for an explicit writing of the constitutive equations of D and H. And in materials which are not in the plasma state, like dielectrics or metals (at least for magnetic properties at non-superconducting conditions), the microscopic response of the medium is meant to be due to polarization charges and magnetic multipoles of atoms and molecules (see, for example, Jackson (1962); Landau and Lifshitz (1960); Feynmann et al. (1964)). Now, a certain amount of further confusion about terminology may occur, since Vlasov plasmas are collisionless conducting media in which polarization charges of atoms and molecules do not contribute, but for which we nevertheless normally speak of “polarization”, “diamagnetism”, “paramagnetism”, “resistivity tensor” and so on. The reason is, obviously, that in plasma physics these terms have a different physical origin (and therefore a subtly different meaning) with respect to the analogous terms usually encountered when one looks at Maxwell equations in solid materials. The most evident point should probably be the fact that the symbol ij (or  for its longitudinal component) had been often used, since the beginning of plasma wave theory, to indicate a plasma “dielectric tensor”, which in fact includes the plasma conductivity (or “resistivity”). In particular, all the mentioned effects in classical plasmas, starting from polarization charges and current polarization densities, are entirely20 related to

20 An exception to this could be represented by the polarization charges and current densities that appear because of “finite size” particle effects in some reduced Vlasov models. The example is provided by gyrokinetic theory, where the difference in position between the center of mass of the particles and that of the “gyrocenters” is at the origin of charge separation effects, which more closely resemble polarization charges due to dipoles, as they are met in dielectrics. These are however model-dependent features, which are related to the specific reduction technique used for the Vlasov Maxwell system in a certain frequency regime. We will speak of this in the second volume.

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The Vlasov Equation 1

the free charges and to their collective response to the external electromagnetic fields. Let us now discuss in more detail these features and some specific properties of the “Vlasov plasma state” with respect to these effects. 3.4.1. Polarization charges and wave equation in dielectric materials Let us first obtain the wave equation in media which also display dielectric properties, and let us compare it with equation [3.3], which we have written for plasmas. This will let us better understand the different sources contributing to the polarization of the medium and the notion of polarization in plasmas. In materials displaying dielectric properties, the dielectric induction D and the total magnetic field H are, respectively, defined in terms of the electric polarization vector P (E, B) and magnetic polarization vector M (E, B) by means of the following constitutive relations: D = ε0 E + P , H=

[3.40]

1 B − M. μ0

[3.41]

The vectors P and M , respectively, correspond to a “polarization charge” density ρ(p) and “polarization current” density J(p) , whose dependence on the external fields E and B can be made explicit by rewriting equations [3.37]– [3.38] in the form ∂ × B = μ 0 J + μ0 ∂r

 ∂ ∂E ∂P + μ0 ε 0 ×M + , ∂r ∂t ∂t & '( ) 

[3.42]

J(p)

1 ∂ ρ ∂ + ·E = ·P . ∂r ε0 ε0 &∂r'( )

[3.43]

ρ(p)

The quantities ρ(p) and J(p) defined above, historically introduced for non-plasma materials, are related to the concentration and oscillation of microscopic polarization charges of the constituents of the medium (atoms,

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molecules, etc.). On average, over mesoscopic time and length scales in solids and neutral fluids these microscopic constituents appear as neutrally charged. Because of this they are kept distinct from the “free” charges (electrons and ions), which are present in metal conductors, electrolytic solutions and plasmas, and which can move inside of the material. In equations [3.42]–[3.43], the concentration and motion of “free” charges accounts for the charge density ρ and for the current density J . However, there is often talk of “polarization” charges and also “polarization” currents for plasmas, even if the contribution of ρ(p) and J(p) , meant in the sense above, is completely negligible with respect to the contribution of ρ and J . This is perfectly legitimate in the sense that the collective response of the plasma to the electric filed causes a complex space- and time-dependent reorganization of the free charges, which gives rise to the generation of charge densities and currents induced by the electromagnetic perturbations. In order to appreciate the different sources of polarization in dielectrics and plasmas, in the following we will keep on distinguishing the contribution of free and polarization charges by continuing to use ρ(p) and J(p) as they are meant in classical materials, while we will denote the free charges and currents with symbols ρ and J . We will nevertheless speak of polarization charges in plasmas, meaning with it the space and time reorganization, which are induced on ρ and J by the electromagnetic perturbations because of the collective response. Then, in order to account for the possibility of injecting charged beams in the medium, as for example in the case of plasmas, it is convenient to separate a source component, say J(s) , from J . The remaining part of the free current density is the one that can be induced by the fields and can be interpreted as a “polarization free current”. However, for the sake of notation, instead of writing J = J(in) + J(s) , from now on we will omit the index “in” for the induced part by simply substituting J → J + J(s) in the equations above. As mentioned previously, the “closure” equation relating the induced current density J to E and possibly to the other macroscopic quantities takes the name of Ohm’s law21. While in the most general case the latter is a nonlinear relation, for sufficiently small amplitudes the (generalized) Ohm’s law gets a linear form with respect to the involved tensor quantities.

21 When other quantities besides E intervene in the constitutive relation J = J (E, J(s) , ...), like in the case of magnetized plasmas, the term “generalized” Ohm’s law is normally used.

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The polarization vectors represent the macroscopic averages of the superposition of the microscopic electric and magnetic multipoles originated by the imposed fields E and B on the material. Therefore, generally P and M are nonlinearly dependent on E and B (see e.g. Jackson (1962, §1.4 and §4)). However, for small electromagnetic perturbations around homogeneous background fields, the first-order term of the multipole expansion (henceforth labeled by an apex “(n)” for n = 1, 2, ..) dominates and a linear – that is, matrix – relation between the components of both P (1) and M (1) and the components of both E and B can be assumed. By combining equations [3.36] and [3.42] and by using [3.1], the wave equation of electromagnetic perturbations in the medium can be written as:    c2  i (1) i 2 + ki kj − k δij Ej + Ei + J Ji = ω2 ε0 ω i,(p) ε0 ω    (n) i Ji,(s) + [3.44] Ji,(p) . − ε0 ω n The small amplitude, linear limit corresponds to the negligibility of the (n) Ji,(p) contributions for n ≥ 2. The expression above is general and its comparison with equation [3.3] is immediate. The specifity of the medium provides indeed the constitutive relations for D, H and J , which lead in the (1) end to the closure condition for the writing of the components of J(p) and of J in terms of those of E. It is at this level that the distinction between plasmas and media in different states of matter becomes manifest. We can now make a comment about the role of the magnetic polarization M in plasmas. In the majority of Vlasov plasma regimes, like those of full ionization of interest here, the contribution of the magnetic polarization vector M is perfectly negligible, since the microscopic magnetic multipole effects are negligible22. Therefore, we can assume B  H with respect to the notation in dielectrics (see section 1.9). It should be however noted that an a priori distinction between the P - and M -related contributions to J(p) is not particularly meaningful in media that, like plasmas, display dispersion in both time and space. The reason is that such a double dispersion would make it possible, in principle, to reciprocally relate without any distinction, in the linear limit, the two contributions of P and M in equation [3.42], since a linear relation between the spatial variation of E and the temporal variation of

22 Unless, of course, quantum plasmas are considered, in which spin effects can be important (see, e.g. (Marklund and Brodin, 2007) and (Brodin et al., 2008)).

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B (or vice versa) is implied by equation [3.36] (or by equation [3.37]). Also because of this, from now on, in this section we will focus only on the plasma response to the electric field. In this regard, some further words are due about the notions of diamagnetic and paramagnetic properties23, when they are referred to a plasma: in this case, of course, we refer once more to the way that free charges re-organize themselves to respond to an applied magnetic field. These properties, such as the diamagnetic plasma response expressed by Alfvén’s theorem in the MHD frequency regime or examples of plasma paramagnetism provided by instabilities that amplify magnetic perturbations (like pressure-driven instabilities), play a crucial role in plasma physics and have a wide range of applications/implications: from magnetic plasma confinement in laboratory, to magnetic generation processes in laser plasma experiments, to a variety of phenomena, related to energy conversion between particle kinetic energy and electromagnetic energy, which are relevant to astrophysics and cosmology. Coming back to our discussion of plasma polarization: being concerned with the plasma response to the electric field only, we can now reintroduce the equivalent dielectric tensor Kij defined by equation [3.4] as a generalization of a combination of both the “true” dielectric tensor related to the polarization currents J(p) and to the conductivity tensor related to the free currents J . 3.4.2. The “equivalent” components

dielectric

tensor

and

its

complex

We now consider the small amplitude limit, which makes it possible to establish linear relations between the involved fields. Let us postulate a linear dependence of D (1) on E only, so to consider P (1) as completely determined by E: (1)

Di

= ij Ej .

[3.45]

The tensor ij is called the dielectric permittivity tensor or simply the dielectric tensor and is dimensionally homogeneous to ε0 . Assuming for the

23 We recall that a material is said to be diamagnetic if the total magnetic field H is smaller than the applied magnetic induction, whereas it said to be paramagnetic in the opposite case.

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The Vlasov Equation 1

moment its coefficients to be complex (we will discuss their physical meaning in section 3.5), we can split ij = Re(ij ) + i Im(ij ).

[3.46]

When the material does not display any intrinsic anisotropy, the tensor ij takes the form of a coefficient  multiplying the identity matrix: D (1) = E. When the dielectric properties of the medium are negligible, the dielectric tensor must then converge to the dielectric coefficient in vacuum, that is ij −−−→ Re(ij )|ω=0  ε0 δij . ω→0

[3.47]

Similarly, in the small amplitude limit, the induced current J can be assumed to linearly depend on the components of the imposed electric field only. This leads to the classical Ohm’s law writing by means of the conductivity tensor σij or by means of the resistivity tensor ηij , Ji = σij Ej

or

Ei = ηij Jj ,

[3.48]

−1 . We will discuss in section 3.4.4 the which are evidently related by σij = ηij relation between conductivity and the absence of binary collisions in Vlasov plasmas. Assuming this time, too, complex coefficients, whose meaning we will discuss in detail in section 3.5, we write

σij = Re(σij ) + i Im(σij ).

[3.49]

Again, the tensor character is here due to the anisotropic response of the medium: as already mentioned in section 3.2.1, the presence of a magnetic field is sufficient to bring in an explicit dependence on the frequency and wavelength of the electromagnetic perturbation, which also breaks the isotropy of σij . In most classical (i.e., non-plasma) materials, conductivity and resistivity are, up to a good approximation, isotropic, that is J = σE and E = ηJ . In wide frequency, ranges σ and η are also weakly dependent on the frequency and on the wave length of the electromagnetic perturbation. In particular, σ = ∞, η = 0 in perfect conductors, whereas σ = 0, η = ∞ in perfect (dielectric) insulators. In plasmas, the isotropic limit provides a reasonable approximation only in a non-Vlasov regime, that is at frequencies sufficiently smaller than the binary collision rate and for sufficiently weak magnetic fields.

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We then recall that the real and imaginary parts of both ij and σij are not independent one from each other but are related by means of the Kramers–Kronig relations24: given a complex analytic function χ(ω) = Re(χ(ω)) + i Im(χ(ω)) of complex argument ω, it can be demonstrated that, under an appropriate hypothesis of integrability25, the following reciprocal relations hold Re(χ(ω)) = P.P.

+∞ −∞

Im(χ(ω)) = − P.P.

+∞ −∞

Im(χ(ν)) dν, ν−ω Re(χ(ν)) dν. ν−ω

[3.50]

These integral relations, which become relevant to our discussion when the functions χ(ω) are the components of the tensors ij or σij , respectively, have much broader applications than optics and are known in mathematics as the Hilbert transform26. In the cases of interest to us, several properties of the medium are related to or can be deduced from equations [3.50] above. To be noted in this regard is the statement of causality for the propagation of electromagnetic waves in the medium, formally equivalent to relations [3.50] applied to the (equivalent) dielectric tensor (Toll, 1956), and the fact that temporal dispersion and absorption in a medium are intrinsically related phenomena. In the following, we will come upon this latter point in section 3.5.5 but we make no further use, here, of these integral relations. The interested reader can delve into the subject by looking at more specific monographies like Lucarini et al. (2005) or at textbooks of mathematical methods, such as Arfken and Weber (1995) and Cicogna (2018) (the subject is also treated in most of textbooks cited at the end of the chapter, such as Stix (1962) or Brambilla (1998) for plasma physics or Born and Wolf (1959), Jackson (1962) or Landau and Lifshitz (1960) for electromagnetism and optics).

24 After Hendrik Anthony “Hans” Kramers (1894–1952) and Ralph de Laer Kronig (1904–1955) (Kramers, 1928; Kronig, 1926), who first established them for the dielectric polarizability. 25 Different procedures can be followed to prove the result, which depends on some hypothesis about the “good behavior” of the function χ(ω) in the Im(ω) > 0 demiplane as |ω| → ∞, but it is in general required that χ(ω) vanishes rapidly enough so to grant its L1 integrability. 26 After David Hilbert (1862–1943), who first formulated them (Hilbert, 1905).

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We have now all the elements for introducing a “generalized” equivalent dielectric tensor, which we write in dimensionless form as     1 1 1 1 Kij ≡ Re(ij ) − Im(σij ) +i Im(ij ) + Re(σij ) . [3.51] ε0 ω ε0 ω '( ) & '( ) & Re(Kij )

Im(Kij )

In the small amplitude limit, this definition of Kij makes it possible to rewrite the second left-hand side parenthesis term of equation [3.44] as Ei +

i (1) i (1) + = Kij Ej . J J ε0 ω i,(p) ε0 ω i

[3.52]

The complex equivalent dielectric tensor Kij encompasses the global response of the medium to the electromagnetic perturbation, which accounts for the redistribution of both polarization charges and free charges. In semiconductors (and in plasmas which are not fully ionized), their relative importance is measured by the ratio (ω||ij ||)/(c2 ||σij ||). This global contribution can been indicated as a generalized “equivalent” ˜ (1) defined as dielectric induction D ˜ (1) ≡ ε0 Kij Ej . D i

[3.53]

The “∼” symbol above and the specification “equivalent” are just to indicate the differences with respect to the constitutive equation [3.40], with which the symbol of dielectric induction D is often identified. It is however worth stressing once more that, despite the different physical origins, both D ˜ have the same physical meaning, as long as they are regarded as and D components of the effective electric field in the medium. This is why the symbol D had been used also in plasma, especially in earlier literature (see e.g. Allis et al. (1963); Kadomtsev (1979)). Here, for utility in forthcoming discussions, we keep on identifying D with that of equation [3.40] and on ˜ henceforth, instead. using D ˜ allows the rewriting of Maxwell’s equation in The introduction of D material media for small amplitude perturbations (which is practically the only strong assumption needed to justify the linear relation [3.53], the condition H  B representing more a simplification) in an even more symmetric form: ∂ ∂B ×E =− , ∂r ∂t

[3.54]

Electromagnetic Fields in Vlasov Plasmas

˜ (1) ∂ ∂D ×H = , ∂r ∂t

where

˜ (1) ∂E ∂D (1) , = J (1) + J(p) + ε0 ∂t '( ∂t) & ∂t D

∂ ˜ (1) = 0, ·D ∂r ∂ · B = 0. ∂r

˜ (1) = ε0 [K] · E, where D

119

[3.55]

of Eq.[3.40]

[3.56] [3.57]

For compactness of notation in the second part of equation [3.56], we have used “[K] · ...” to express the matrix product of the matrix K with the vector E on its right. Note that the null source term in this rewriting of the Gauss equation follows from the definitions given, and it is consistent with the first in equations [3.55]: taking the divergence of the latter, the continuity equation of electric charge,  ∂ρ(1) ∂  (1) (1) + · J + J(p) = 0 ∂t ∂r

[3.58]

is as usual recovered, now being written in the form ˜ (1) ∂ ∂D · = 0. ∂r ∂t

[3.59]

In the case of fully ionized Vlasov plasmas, the “purely dielectric” contribution to the equivalent dielectric tensor is given by the limit expressed using equation [3.47] and the equivalent dielectric tensor of equation [3.51] reduces to Kij = δij + i

1 (Re(σij ) + i Im(σij )) . ε0 ω

[3.60]

That is, as we noted from the beginning, the equivalent dielectric tensor of a plasma, Kij , coincides with what we have named the conductivity tensor (but for the additive identity matrix δij ). As we already remarked, however, while on the one hand this identification is self-consistent with the definition we have adopted and it is functional for understanding the motivation of the varied terminology for this subject in plasma physics, on the other hand, it is not the only one which is possible or legitimate. Indeed, another point to be discussed remains, which we will address in section 3.4.4 and which is related to the collisionless nature of σij in Vlasov plasmas: if one wants to privilege the identification of the conductivity (or resistivity) of a medium as related to

120

The Vlasov Equation 1

collisions among charge carriers, it may be preferable to use the symbol ij in the place of the symbol σij , which we have instead adopted. As we said, both are encountered in literature – compare for example Landau et al. (1981) and Melrose (1986) – and have their sound motivations behind. We also note that, so far, we have not yet addressed from a formal point of view the symmetry properties of Kij , ij and σij . This will be done in section 3.5, once we will have settled all the elements for the discussion. 3.4.3. Temporal and spatial dispersion in plasmas We can now discuss from a more formal point of view the origin of the spatial and temporal dispersion in plasmas by following the general argument exposed, for example, in Landau and Lifshitz (1960, Ch. IX, §77, §82; Ch. XII, §103). ˜ (1) of the medium to a time varying In general, the global response D electric field E(t) is not instantaneous. This makes the complex dielectric tensor Kij dependent on the frequency of the oscillation, which is called temporal dispersion or dispersion in frequency. Causality imposes indeed ˜ (1) (t) to generally depend not only on E(t) but also on E(t ) for any D t < t, which can be expressed with the aid of a time convolution product, ∞ ˜ (1) (t) = ε0 Ei (t) + ε0 D [3.61] f˜ij (τ )Ei (t − τ ) dτ. i 0

Here, f˜ij (t) is a tensor with complex coefficients dependent on time, which are related to both ij and σij . By then decomposing each perturbation in Fourier series, we can specialize equation [3.61] for each frequency component (superscript “(ω)” in vector components below) and identify ∞ Kij (ω) ≡ δij + [3.62] f˜ij (τ )eiωτ dτ, 0

which satisfies27 ˜ (ω) ei(k·r−ωt) = ε0 Kij (ω) E (ω) ei(k·r−ωt) . D i j

[3.63]

27 Relation [3.63] of course applies also to the ∼ e−i(k·r−ωt) , complex conjugate component of [3.1], provided each scalar quantity is substituted by its complex conjugate.

Electromagnetic Fields in Vlasov Plasmas

121

When collective effects are important, such as in plasmas, the global ˜ (1) of the medium to a time varying electric field E(t) is not even response D ˜ (1) (r, t) is linearly related to the values of E(r ˜  , t ) local. That is, D measured not only at previous times (temporal dispersion) but also in some points r  in a volume Vr in the neighborhood of r. Because of this, the complex dielectric tensor Kij gets a dependence on the wavelength (or wave vector) of the perturbation, and one speaks of spatial dispersion or dispersion in wavelength. This can be formalized with an expression similar to [3.61], which includes an integral over space. In the case of instantaneous, non-local response (i.e. of spatial dispersion but no temporal dispersion), we would write (1) ˜ Di (r, t) = ε0 Ei (r, t) + ε0 [3.64] f˜ij (r, r  , t)Ei (r  , t) d3 r  . Vr

˜ (1) and E ˜ therefore keeps The most general linear relation between D account of both kinds of dispersion and can be written as ∞ ˜ (1) (r, t) = ε0 Ei (r, t) + ε0 D dτ f˜ij (r, r  , τ )Ei (r  , t − τ ) d3 r  . [3.65] i 0

Vr

When the medium is sufficiently homogeneous in space over distances much larger than the wavelength of the perturbation, we can write f˜ij (r, r  , τ )  f˜ij (r − r  , τ )

[3.66]

and we are allowed to consider a Fourier transform also with respect to space (provided periodic boundary conditions or an unlimited extension of the spatial domain). In this way, similarly to what has been done for equations [3.62]– [3.63], we can specialize equation [3.65] for each frequency component ∼ ei(k·r−ωt) and, by introducing ρ = r − r  and using d3 r  = d3 ρ, we can write ∞ [3.67] f˜ij (τ, ρ)ei(ωτ −k·ρ) d3 ρ dτ. Kij (ω, k) ≡ δij + 0

Vr

This satisfies ˜ (ω,k) ei(k·r−ωt) = ε0 Kij (ω, k) E (ω,k) ei(k·r−ωt) . D i j

[3.68]

From the definition above, it immediately follows the symmetry property of the coefficients of Kij under simultaneous ω ↔ −ω and k ↔ −k transformation: ∗ Kij (−ω, −k) = Kij (ω, k).

[3.69]

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The Vlasov Equation 1

It is worth noting that equations [3.67]–[3.68] can be meaningful also for materials with M = 0 if the polarization current J(p) is referred uniquely to the electric field, and if the electric polarization vector is redefined (see comment at the end of section 3.4.1). The dispersion functions f˜ij (τ, ρ) are typically negligible over large time ˜ (1) intervals (i.e., for τ → ∞) and for large distances from the point where D is evaluated (i.e., for ρ → ∞). While in dielectrics the characteristic distance over which f˜ij (τ, ρ) becomes negligible is of the order of the atomic radius, in conductors such as metals and plasmas the mobility of free charges makes the spatial dispersion relevant over a wide range of spatial scales, and therefore of electromagnetic frequencies. 3.4.4. Conductivity and collisional resistivity in Vlasov plasmas A last point about terminology may deserve clarification: the plasma conductivity is finite, but the Vlasov plasma is collisionless. What is then the meaning of the conductivity tensor of plasmas? What is the meaning of an equivalent dielectric tensor that includes the contribution of conductivity? Let us discuss these points. Due to the lack of binary collisions, the proportionality constant between the free current and the electric field in plasmas gets a different origin and meaning with respect to what happens in metals, where charges have a finite collision rate. In metals and conducting lattices, indeed, the finite collision frequency νc between charge carriers, that is electrons and other electrons or ions, makes it possible to express the scalar resistivity as η (c) =

νc 1 me  νc = ε0 2 , ne e 2 ωpe σ (c)

[3.70]

where ne is the density of free (i.e. valence) electrons in the medium. We have introduced in the writing above the apex “(c) ” so to stress the collisional origin of the involved quantities. The equation above also applies to plasmas in which interspecies collisions are taken into account, for example because of a partial ionization or of relatively low dilution and/or temperature (depending on the dimensionality of the problem, see equations [1.2]–[1.3] and related discussions in sections 1.3 and 1.6); in that case, of course, ne is the electron plasma density and νc is the electron-ion collision rate (see for example Landau et al. (1981, Ch. IV) or Golant et al. (1980) for an extensive discussion of the role of collisions in plasma collisional resistivity).

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We have seen instead (section 1.1) that the Vlasov limit corresponds to νc = 0, which means that Vlasov plasmas have null collisional resistivity. For these, the finite coefficients of the conductivity tensor σij are indeed due to the collective plasma response to electromagnetic perturbations, which limits the possibility of the electric field to arbitrarily accelerate the charge carriers (which would imply a diverging conductivity, instead). Some confusion may then arise when in the strictly collisionless Vlasov limit one refers to the −1 inverse of the conductivity tensor, σij as a “plasma resistivity tensor”, or when one speaks of “plasma electric conductivity”, meant as the reciprocal of the collisional resistivity, according to equation [3.70]. In this latter case, it must be made clear that a collisional plasma model is considered, which goes beyond the collisionless Vlasov description, or models that account for an “efficient resistivity” have been adopted. In order to make explicit the difference with respect to the collisionless quantities σij and ηij previously (c) (c) introduced, let us use σij and ηij for the corresponding quantities, which are related to collisions. When numerical directions which the (c)

particle collisions are considered in plasmas, an appreciable difference typically exists between the collision rates in the parallel and perpendicular to the magnetic field, with respect to (c) (c) (c) symbols ν|| (and therefore σ|| and η|| ) and ν⊥ (and hence σ⊥

and η⊥ ) are adopted. Different estimations of the collision rates and hence of σ|| and σ⊥ exist, which depend on assumptions based on the rate of ionization and magnitude of the graininess parameter (for a synthetic list of formulae of frequent use see, for example, Huba (2006, pp. 32–40)). In fully ionized plasmas, the usual estimation is that first suggested by Spitzer and Härm (1953), which is based on the assumption (we neglect below the numerical factors that make the difference between parallel and perpendicular resistivity)    ∂fα pα nα qα dpα  −νc J . [3.71] mα γα ∂t collision α It here manifests the connection with the estimation νc ∼ ωpe /(ne λ3D ) already discussed in Chapter 1, from which it follows (c)

ηS =

1 (c) σS

∼

1 0 ∝ Te−3/2 . ωpe ne λ3D

[3.72]

In this way, for small g, Spitzer’s resistivity can be considered as a first ∼ O(g) correction to the Vlasov conductivity tensor. Refinements to this estimation may be introduced, which keep account of spatial anisotropies

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The Vlasov Equation 1

induced by a magnetic field to the collision rates (see Braginskii (1965)), and the Spitzer-type resistivity defined above can be generalized to a tensor form (c) (c) (for example, to give the σ|| and σ⊥ previously mentioned). In this limit, the parallel and perpendicular (with respect to B) components (c) of σS can be added to those of the σij (ω, k) tensor written in equations [3.3] and [3.51]. As already noted at the end of section 3.4.2, this explains and justifies an alternative choice of notation, with respect to the one we have here adopted, (c) which consists of substituting in a plasma σij → ij and σij → σij . We then note that Vlasov plasma nonlinear effects, which lead to the formation of small-scale structures, modify the properties of diffusion and transport of energy and matter. This leads to an “anomalous” diffusion28 and hence to an anomalous resistivity that, although being consistent with an essentially collisionless physics, plays the role of a “collisional resistivity” (roughly speaking, we could say that it acts as filamentation and phase-mixing do in collisionless dissipation). The term “anomalous” (or “non-classical”) is due to the value that is phenomenologically measured for this kind of resistivity, which is typically higher than one would estimate, based on the particle collision rates that are expected and/or measured in those regimes. Several models exist in literature for a quantitative estimation of anomalous resistivity, which depend on (and usually are restricted to) the specific phenomenon that is being considered. Turbulence, often triggered by some eigenmode that has become unstable, is however the most typical source of this further possible contribution to the quantity, which we have (c) here named σij . This anomalous resistivity can then provide the trigger which destabilizes further eigenmodes, such as tearing modes in magnetic reconnection29. Investigating the latter, in particular, has motivated studies on specific models for anomalous resistivity, for example with applications to reconnection in the terrestrial magnetotail or magnetosphere in general (Papadopulos, 1977). Examples are provided by models in which an efficient

28 In laser-plasma interactions an analogous concept is that of “anomalous absorption” (of the incident electromagnetic wave). 29 We recall that magnetic reconnection is a mechanism of energy conversion in plasma, which implies a transfer of the energy associated with inhomogeneous, largescale magnetic fields to the particle kinetic energy. Although locally occurring where strong current densities are developed, magnetic reconnection is associated with a global re-organization of the magnetic field topology (see e.g. Biskamp (2000)).

Electromagnetic Fields in Vlasov Plasmas

125

resistivity is related to particle inertia intervening according to the generalized Ohm’s law (Speiser, 1970); to fluctuations in transport properties of particles, which are induced by the excitation of plasma modes (Coppi and Mazzucato, 1971; Kruer and Dawson, 1972); to lower hybrid turbulence (Coroniti, 1985); or to assuming that the mean free time of charge carriers in a magnetized plasma be related to the averaged transit time of electrons close to an X-point magnetic configuration (Ma et al., 2018). In numerical simulations of collisionless plasmas, an “artifact” source of anomalous resistivity is due to the truncation error of the numerical scheme and becomes manifest when spatial gradients comparable to the inverse mesh spacing are generated. The interested reader can explore the literature about these subjects, which is usually related to the subfield of transport theory in plasmas, and especially to the more specific domain30 of “anomalous” transport theory. Over the years, a large number of articles and books have been published on these topics, due to their relevance to both plasma astrophysics and tokamak plasma confinement. Among the monographies devoted to discussing the delicate issues related to transport theory in collisionless plasmas, we make a special mention of the books by Balescu cited in the footnote, as well as of the further books by Balescu (1997) and Elskens and Escande (2003). A final remark can be made, concerning the dependence of collisional Spitzer’s resistivity on temperature. It must be noted in this regard a further fundamental difference between plasmas on the one side, and both metals and semiconductors on the other side: the fact that η (c) in metals generally increases with temperature, whereas in plasmas it decreases according to [3.72], is a consequence of the different dependence of the collision rate on temperature in the two types of conductors. The fact instead that in typical semiconductors the resistivity decreases with temperature (according to the empirical Steinhart–Hart law, (Steinhart and Hart, 1968)) is only qualitatively

30 Especially in tokamak-related terminology, a distinction is typically made between “classical” (see Balescu (1988a)), “neo-classical” (see Balescu (1988b)) and “anomalous” transport theory (see Balescu (2005)). The terming “classical” usually refers to Boltzmann-type models in which binary collisions dominate in diffusion processes of charged particles in homogeneous and stationary electromagnetic fields. “Neo-classical” specifically refers to the inclusion in the theory of inhomogeneous magnetic fields. “Anomalous”, or “non-classical”, refers to further effects, which usually dominate over classical and neoclassical ones, and which are due to the turbulent and chaotic dynamics mostly ruled by long-range particle interactions.

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The Vlasov Equation 1

similar to the case of plasmas: in seminconductors, indeed, the dominant effect is the fact that increasing the temperature makes more bound electrons excited across the band gap, which this way become conduction electrons; in plasmas, where charges are free and diluted, increasing the temperature reduces the cross-section of binary collisions, since a shorter interaction distance becomes necessary in order for the energy exchanged through electrostatic interaction to appreciably affect the kinetic energy associated with the thermal motion. This further differs from the conductor case, in which an increase in temperature enhances the collision rate of electrons in the conduction layer, whose density is relatively higher than in semiconductors, with other charge carriers. 3.5. Symmetry properties of the complex components of the equivalent dielectric tensor and energy conservation Let us now discuss the real and imaginary part of both ij and σij . As anticipated in section 3.2.1, their symmetry properties as tensors are fixed by consistence with energy conservation and with the second principle of thermodynamics. More specifically, these symmetry constraints follow from what in thermodynamics are called Onsager relations (Casimir, 1945; De Groot, 1951), which are conditions descending from energy conservation at microscopic scales, which make it possible to express the macroscopic entropy increase. Let us briefly discuss their origin and meaning before looking at their application to the matrices ij , σij and Kij . 3.5.1. Onsager’s relations While inviting the reader to look at specific monographies on the subject for details, we recall that Onsager’s theorem (Onsager, 1931a,b) states that, once some phenomenological coefficients Lij have been individuated, which linearly relate some “flows” or “currents” Ji (in a generalized sense) to some “forces” Xi according to Ji = Lij Xj ,

[3.73]

the condition of microscopic reversibility requires Lji = Lij

(Onsager’s reciprocal relations).

[3.74]

Let us recall here the main points of the demonstration, which will let us understand its extensibility to the case of interest to us.

Electromagnetic Fields in Vlasov Plasmas

127

Onsager’s result can be obtained (by following, for example De Groot (1951, Ch. II), to which we direct the reader for details) by using the fact that for each phenomenon under investigation both the “fluxes” Ji and the “forces” Xi represent the deviations from thermodynamical conditions of some “state variable” ai , which identifies the properties of the system related to the phenomenon. Once entropy fluctuations ΔS from equilibrium are considered, we can always define some “force” Xi = ∂ΔS/∂ai with respect to some state variable ai , whose time derivative gives the corresponding flux Ji . Owing to the fact that entropy is maximum at equilibrium, a quadratic relation of the kind 1 ΔS = − gik ai ak 2

[3.75]

for some positive definite form gik can be established. This, after some algebra, makes it possible to show that ai Xj m.c. = −kB δij .

[3.76]

In the equation above, kB is Boltzmann’s constant and ...m.c. expresses the average over a microcanonical ensemble or, thanks to the ergodic theorem, the average with respect to time. Onsager’s result follows then from two further ingredients: 1) The first one is using microscopic reversibility for fluctuations between times t and t + τ , which is expressible as ai (t)aj (t + τ )m.c. = ai (t)aj (t − τ )m.c. .

[3.77]

This can be used to show that    ai (t)aj (t + τ ) − aj (t)ai (t),...,an (t) m.c. =    aj (t)ai (t + τ ) − ai (t)ai (t),...,an (t) m.c. ,

[3.78]

where ...ai (t),...,an (t) means an average over a portion of the microcanonical ensemble, in which all the values ai (t), ...an (t), including the ai (t) and aj (t) under investigation, have been kept fixed. Note that aj (t + τ )ai (t),...,an (t) can be equivalently interpreted as the ensemble average of the values, which the quantity aj has reached a time τ after a configuration in which the system was described by the set of variables ai (t), ..., an (t). The passage from equation [3.77] to [3.78] is obtained by first noting the formal equivalence between ai (t)aj (t − τ )m.c. and ai (t + τ )aj (t)m.c. and by then noting

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The Vlasov Equation 1

the to perform the averages as meant in  latter expression to be equivalent   ai (t + τ )aj (t)ai (t),...,an (t) m.c. because of the alternative interpretation mentioned above. Using these substitutions in equation [3.77] and then subtracting aj (t)ai (t),...,an (t) m.c. from both sides yields equation [3.78]. 2) The second one is to assume that, on average (over time or microcanonical ensemble), the decay of fluctuations follows phenomenological macroscopic laws that linearly relate the time derivatives a˙ i and the state variables ai themselves. To obtain them, we can use the writing of Xi in terms of the derivatives of ΔS. It follows that Xi = −gik ak , whence some coefficient Lij can be introduced to write a˙ i m.c. = Lij Xj = −Lij gjl al .

[3.79]

The “trick” of rewriting the linear relation Ji = a˙ i m.c. ∝ ai by introducing a linear combination of the state variables al for l = i, .., n at the right-hand side is because this last relation is phenomenologically easier to be verified for some suitable choice of the forces and fluxes (e.g. the coefficients Lij can correspond to diffusion coefficients, heat or electrical conductivity, etc.), and the conjugated quantities Xi and Ji can be obtained by the definitions above, once the entropy production rate ΔS˙ = Jj Xj is determined. Then, by noting that a˙ i (t)m.c. =

ai (t + τ ) − ai (t)m.c. ai (t + τ ) − ai (t)ai ,...,an = τ τ

[3.80]

for some suitably small τ , equations [3.74] are obtained after substituting equation [3.80] into equation [3.78] and by using equation [3.79]. A further element that is of crucial interest to us is when the system is constituted by charged particles interacting with magnetic field through Lorentz forces v × B. In this case, reversibility of trajectories requires that B → −B when the transformation t → −t (i.e. v → −v) is performed. The above derivations must be therefore corrected (see De Groot (1951, Ch. II and XI)) by replacing ai (t; B)aj (t + τ ; B)m.c. = ai (t; B)aj (t − τ ; −B)m.c. ,

[3.81]

into equation [3.74] and hence aj (t + τ ; B) − aj (t; B)ai (t),...,an (t) and ai (t + τ ; −B) − ai (t; −B)ai (t),...,an (t) , respectively, into equation [3.74]. By repeating the procedures above, in the end this implies Lji (B) = Lij (−B)

(Onsager’s reciprocal relations (II)).

[3.82]

Electromagnetic Fields in Vlasov Plasmas

129

Relations expressed by equation [3.82] are relevant to the equivalent dielectric tensor in conducting material, since, as we are going to see, the coefficients of both the dielectric and conductivity tensors fulfill the requirements with which the phenomenological coefficients Lij can be identified, according to the definitions related to equation [3.73]. We can, however, anticipate that when Onsager relations are considered for a dielectric or for an equivalent dielectric tensor with application to wave propagation in a dispersive medium, equation [3.82] must be refined in order to take into account the dependence of the dielectric tensor components on k. The result is that equation [3.82] must be specified in this case as Lji (k, B) = Lij (−k, −B)

(Onsager’s reciprocal relations (II) for wave dispersion).

[3.83]

Obtaining such a kind of relation is less straightforward: an early derivation, with an explicit application to conductivity in a dispersive medium, has been provided by Kostantinov and Perel (1960), who evaluated current density fluctuations due to harmonic electromagnetic perturbations in a dielectric medium by using a semiclassical quantum model of radiation31. Such a derivation has been then rediscussed for plasmas (Rukhadze and Silin, 1962) and crystals (Agranovich and Ginzburg, 1962). However, as anticipated in section 3.2.1, equation [3.83] can be deduced from Onsager’s conditions given as in equation [3.82] and from the different transformation properties under axis reversal of B and k, with respect to which the principal axes of the dielectric tensor can be identified (see Landau et al. (1981, §52)). We will discuss this point in detail below (section 3.5.3). Before doing so, in order to identify the components of the equivalent dielectric tensor as Onsager-like phenomenological coefficients and in order to make the connection with the formalism expressed by equation [3.73], it is first useful to recall another important theorem, which is implied by the set of Maxwell’s equations and which expresses the conservation of the energy density transported by a wave: Poynting’s theorem32. 3.5.2. Poynting’s theorem Let us consider Maxwell’s equation in materials, written as in [3.36]–[3.39]. Combining equations [3.36]–[3.37] scalarly multiplied,

31 That is, a model in which charged particles are described with quantum mechanics formalism, whereas the electromagnetic field is not quantized. 32 After John Henry Poynting (1852–1914), who discovered it (Poynting, 1884).

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The Vlasov Equation 1

respectively, by H and E and using a well-known vector identity33, we obtain Poynting’s theorem E·

∂D ∂B +H · + ∇ · P = −E · J , ∂t ∂t

[3.84]

written as usual in terms of what is known as the Poynting vector, P = (E × H) .

[3.85]

In a vacuum, the first two left-hand side terms of equation [3.84] correspond to the time derivative of the electromagnetic energy density, Ee.m. ≡ ε0

E2 B2 , + 2 2μ0

[3.86]

of which E·D B·H + , 2 2

[3.87]

once averaged over an oscillation period, becomes the generalization in a dielectric medium. The result stated by the theorem holds independently from the linear approximation. When applied to the fields associated with a propagating wave, the Poynting vector can be interpreted as the flow of energy density transported by the electromagnetic wave. In particular, it measures the amount of energy that crosses per second a unit area whose normal vector is orthogonal to both E and H (more detailed discussions can be found in practically all books of classical electrodynamics and optics: see the bibliography at the end of the chapter for a few examples). 3.5.3. Symmetry of the coefficients of the equivalent dielectric tensor We can now use Onsager’s relations applied to the tensors σij , ij or Kij , once we identify the “fluxes” Ji and “forces” Xi as the density currents Ji (or

33 ∇ · (A × B) = B · ∇ × A − A · ∇ × B.

Electromagnetic Fields in Vlasov Plasmas

131

displacement currents ∂Di /∂t) and electric fields Ei , respectively. Indeed, once equation [3.85] is integrated over a volume crossed by the wave, the variation of the energy that the wave transports is due to the absorption by the medium, which can be due to the E · J term and to contributions related to the dielectric induction D (and total magnetic field H), which are possibly dephased with respect to E (and to B). Since these quantities contribute to energy dissipation (i.e., entropy increase) in the medium through scalar products of the form Jk Xk , Onsager’s relation applies34 to the coefficients of proportionality between the couples of vector fields involved, that is σij and ij . In particular, the phase relations between the quantities involved in the scalar product make the contributions of the real and imaginary components of ij and σij distinct. The real and imaginary parts intervene indeed in the scalar products of perturbations of the form of equation [3.1] when the combination between a phase ∼ i(k · r − ωt) and its complex conjugate is considered: since quantities involved in Poynting’s theorem are quadratic with respect to the perturbation, and since they must of course be real, their average over a period must be considered, for which the contributions of the “c.c.” terms of equation [3.1] must be kept into account. Therefore, the symmetry properties of the components of the dielectric and conductivity tensors will be not only determined by Onsager’s relations [3.82], but they will also depend on the symmetry properties stated by equation [3.69]. Using the latter and by identifying Kij (ω, k, B) with a phenomenological coefficient Lij (ω, k, B) satisfying [3.82], we have, in general: ⎧ ∗ (−ω, −k; B), ⎨ Kij (ω, k; B) = Kij (Symmetry constraints [3.88] on the Kij components). ⎩ Kij (ω, k; B) = Kji (ω, −k; −B). Owing to definition [3.51], both ij and σij respect of course the same symmetries of Kij that are stated in the equation above, and which we anticipated for the conductivity tensor in plasmas while writing equation [3.7]. Then, non-trivial relations between the real and imaginary parts of these tensors and their Hermitian and anti-Hermitian parts can be deduced by using

34 In reality, while for the case of fully ionized Vlasov plasmas where equation [3.60] holds, the way the medium response is described through σij makes the application of relations [3.82]–[3.83] quite straightforward, some more care should be deserved when different constitutive equations for D and H are considered in materials where dielectric and magnetic polarization properties are important (see, e.g. Lakhtakia and Depine (2005) for a discussion).

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The Vlasov Equation 1

equation [3.51]. In this regard, it is useful to recall that, given a tensor with complex coefficients, such as Kij a priori is, we can distinguish its real and imaginary parts according to Re(Kij ) ≡

∗ Kij + Kij , 2

Im(Kij ) ≡

(H)

∗ Kij − Kij , 2i

[3.89]

(A)

and its Hermitian (Kij ) and anti-Hermitian (Kij ) parts according to (H)

Kij

≡

∗ Kij + Kji , 2

(A)

Kij

≡

∗ Kij − Kji . 2i

(H)

[3.90] (A)

The correspondences Re(Kij ) = Kij and Im(Kij ) = Kij only hold ∗ ∗ if Kij = Kji . In particular, it is worth noticing that a formal definition of ˜ the kind Di ≡ Kij (Ei + Ei∗ )/2 (and analogous ones for D(E) and J (E)), which we will need in order to carry out the calculations, implies Di ≡

 ε0  Ej Kij (ω, k; B)ei(k·r−ωt) + Kij (−ω, −k; B)e−i(k·r−ωt) 2 [3.91]  1  ∗ = Ej Kij (ω, k; B)ei(k·r−ωt) + Kij (ω, k; B)e−i(k·r−ωt) , [3.92] 2

where in the second line we have used the second line of [3.88], that is, equation [3.69]. Let us now specify the constraints stated by [3.88] to the case of a fully ionized Vlasov plasma in which polarization charges are negligible (ij → ε0 δij ) and for which, therefore, equation [3.60] holds. These constraints lead in a Vlasov plasma to equations [3.7]. Let us first demonstrate such symmetry rules for the coefficients Kij by starting from equation [3.88]. Using Onsager relations in the form of [3.83] makes the derivation quite straightforward, as it is shown for example by (Brambilla, 1998, §6.1), whom we are going to follow here. As a second step, in section 3.5.4, we will rely on analogous geometrical arguments so to show how Onsager conditions [3.13] follow from conditions [3.13]. To this purpose, we consider once more a configuration in which the background magnetic field B is along z and the perpendicular component of

Electromagnetic Fields in Vlasov Plasmas

133

k is along x, so that k = k sin θex + k cos θey (so as we assumed in equations [3.16]–[3.19]). In this geometry (see Figure 3.1), the y-axis direction is defined by B × k/|kB0 |. It follows from this that the transformation (k, B) −→ (−k, −B) just flips the axes x and z, as is shown in Figure 3.4, and it is therefore described by the rotation matrix of angle ϕ = +π around the y axis, ⎛ ⎞ −1 0 0 1 0 ⎠. Rij (y; π) = ⎝ 0 [3.93] 0 0 −1 As is evident from Figure 3.4, we can write −1 (y; π). Kij (−ω, k, −B) = Ril (y; π) Klm (ω, k, B) Rmj

The second part of equation [3.88] can in this way be reformulated as −1 Kji (ω, k, B) = Ril (y; π) Klm (ω, k, B) Rmj (y; π),

[3.94]

whence the conditions on the elements of Kij , specific for this geometry, follow:  Kxx = Kxx , Kyy = Kyy , Kzz = Kzz . [3.95] Kxy = −Kyx , Kxz = Kzx , Kzy = −Kyz . For the sake of notation, we have omitted in the writing above the argument (ω, k, B) on which all elements depend. These are the relations that have been used to write equation [3.7] and to express the tensor Kij in the general form of equation [3.8]. Then, once they are combined with the first of equations [3.88] and use of the definitions [3.89–3.90] is made, the following relations between the coefficients of the Hermitian/anti-Hermitian and real/imaginary parts of Kij can be established: ⎛ ⎞ i Im(Kxy ) Re(Kxz ) Re((Kxx ) Re(Kyy ) i Im(Kyz ) ⎠ Kij = ⎝ −i(Im(Kxy ) Re(Kxz ) −i Im(Kyz ) Re(Kzz ) '( ) & (H)

⎛

i Im(Kxx ) + ⎝ − Re(Kxy ) i Im(Kxz ) &

Kij

Re(Kxy ) i Im(Kyy ) − Re(Kyz ) '( (A)

Kij

⎞ i Im(Kxz ) Re(Kyz ) ⎠ . i Im(Kzz ) )

[3.96]

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The Vlasov Equation 1





















 
 






 













Figure 3.4. Once the Cartesian axes are chosen as shown above (see also Figure 3.1), the simultaneous reversal of the magnetic field and wave vector directions, B → −B and k → −k, corresponds to a rotation R(y, φ = π) of angle +π around the y axis

3.5.4. More about Onsager’s relations for wave dispersion As noted in Landau et al. (1981, §54) it is also possible to deduce the symmetry relations [3.95] by using together: i) the Onsager relations in the form of equation [3.82], Kij (B) = Kji (−B); ii) the invariance of the equivalent dielectric tensor under complete axis reversal, r → −r, that is (x, y, z) → (−x, −y, −z); iii) the pseudovector character of B and the true vector character of k and B × k. In order to complement the discussion in section 3.5.3, we will here follow this kind of argument to prove that, using (ii) and (iii) above, it is possible to deduce Onsager relations [3.83] for wave dispersion by starting from the simpler form expressed by [3.82], that is from assumption (i) above. This will make manifest how symmetry relations [3.83] are specific for the phenomeological coefficients related to the propagation of waves, since such relations depend on the way the orientations of the wave vector and

Electromagnetic Fields in Vlasov Plasmas

135

background magnetic field components identify the directions with respect to which the characteristic axes of the dispersion matrix (or of the conductivity or dielectric tensors) are defined. This procedure, too, is quite straightforward. It just requires a little lengthier algebra than the derivation previously shown in order to obtain equations [3.96]. Choosing again the reference axes as in Figures 3.1 and 3.4, we note that the operation of complete axis reversal, once applied to the coordinate components, corresponds to a rotation of the triad B, k⊥ , B × k by an angle ϕ = +π around the magnetic field direction B. This is because of the pseudo-vector nature of B and true vector nature of B × k, which we can also see to be proportional to the vector potential A, since B × k = (k × A) × k ∝ A: according to the choice made about the Cartesian axes, the transformation (x, y, z) → (−x, −y, −z) indeed implies (see Figure 3.5) r→−r

(k⊥ , A, B) −−−−→ (−k⊥ , −A, B), Using the rotation matrix ⎛

−1 Rij (z; π) = ⎝ 0 0

0 −1 0

⎞ 0 0 ⎠, 1

[3.97]

the equivalence summarized in Figure 3.5 can be expressed by the relation: Kij (ω, −k, B) = Ril (z; π) Klm (ω, k, B) Rmj (z; π)−1 . This leads to the set of non-trivial relations for the out-of-diagonal components of Kij , the diagonal components being unaffected by the transformation (k, B) → (−k, B) (we omit below, for the sake of notation, the dependence on ω): Kxy (k, B) = Kxy (−k, B), Kyx (k, B) = Kyx (−k, B),

[3.98]

Kxz (k, B) = −Kxz (−k, B), Kzx (k, B) = −Kzx (−k, B),

[3.99]

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The Vlasov Equation 1

Kyz (k, B) = −Kyz (−k, B), Kzy (k, B) = −Kzy (−k, B).

[3.100]



 
































Figure 3.5. An axis inversion r → −r applied to the components of the tensor Kij (ω, k, B) corresponds to considering its components Kij evaluated with respect to the vectors (−k, B) because of the transformation rules of the triad k⊥ , A, B under axis reversal

In order to use Onsager relations [3.82], we can then consider the outcome of the transformation (k, B) → (k, −B). Since the latter can be made correspond to a rotation of angle ϕ = +π around the x axis (see Figure 3.6), ⎛

1 Rij (x; π) = ⎝ 0 0

0 −1 0

⎞ 0 0 ⎠, −1

[3.101]

we can write: −1 (x; π). Kij (ω, k, −B) = Ril (x; π) Klm (ω, k, B) Rmj

A set of relations analogous to [3.98]–[3.100] can be established: Kxy (k, B) = −Kxy (k, −B), Kyx (k, B) = −Kyx (k, −B),

[3.102]

Electromagnetic Fields in Vlasov Plasmas

137

Kxz (k, B) = −Kxz (k, −B), Kzx (k, B) = −Kzx (k, −B),

[3.103]

Kyz (k, B) = Kyz (k, −B), Kzy (k, B) = Kzy (k, −B).




[3.104]




























 















Figure 3.6. The reversal of the magnetic field only, (k, B) → (k, −B), corresponds to a rotation R(x, φ = π) of angle +π around the x axis

Onsager’s conditions [3.83] can be then obtained by combining equations [3.98-3.100] with equations [3.102]–[3.104] and by using Onsager relations [3.82] written as Kij (k, B) = Kji (k, −B). For example, we deduce from the latter and from the first part of equation [3.102] that Kyx (k, B) = Kxy (k, −B).

[3.105]

On the other hand, we deduce from the first part of equation [3.98] by direct substitution of B with −B in both sides of the equation35 that Kxy (k, −B) = Kxy (−k, −B).

[3.106]

35 The sign of B, that is of the scalar product B · ez , corresponds here just to a convention in the definition of the orientation of the z axis with respect to B.

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The Vlasov Equation 1

Combining [3.105] and [3.106], the condition Kyx (k, B) = Kyx (−k, −B) is then obtained. Repeating the procedure for all the other equations, the conditions Kij (k, B) = Kji (−k, −B) for all components are recovered. 3.5.5. Energy dissipation versus real and imaginary parts of σij and ij In order to conclude the discussion about the equivalent dielectric tensor, we still have to say something about the Hermitian and anti-Hermitian components of ij and σij , and about the way they are related to energy dissipation. In order to explicitly see this, it is useful to reformulate Poynting’s theorem for small amplitude perturbations by means of equations [3.54]–[3.57], so to write (we consider the H  B limit) E·

˜ (1) ∂D ∂B +B· + ∇ · P = 0. ∂t ∂t

[3.107]

In the expression above, the E · J contribution has been re-absorbed in the first left-hand side term. Using [3.91], we write (1)

˜ ∂t D i

  ω ∗ = −i ε0 Ej Kij (ω, k; B)ei(k·r−ωt) − Kij (ω, k; B)e−i(k·r−ωt) . 2 [3.108]

After integration over a period T = 2π/ω, the contributions of ˜ (1) /∂t, which are proportional to ∼ e±2iωt , vanish. We see that the E · ∂D surviving contributions correspond to products between a phase term and its complex conjugate. By noting that in the definition of Im(Kij ) in equation [3.89], division by “i” corresponds to dephasing of −π/2, we obtain that the average electromagnetic energy absorbed by the medium per unit time over a period is E · =

ω π



π ω

0

dt

˜ (1) ∂D ω T ≡ ∂t 2π



2π ω

0

dt E ·

˜ (1) ∂D ∂t

 ω  ∗ (ω, k; B) Ei Ej ε0 Kij (ω, k; B) − Kij 2 ω = ε0 Im(Kij (ω, k; B)) Ei Ej . 2

[3.109]

Electromagnetic Fields in Vlasov Plasmas

139

Only the imaginary part of the equivalent dielectric tensor contributes to the wave absorption by the medium. Using equation [3.51], we then see that the relation above concerns, according to the notation adopted in this chapter, the imaginary part of ij and the real part of σij only: in the most general case and (c) taking into account also the σij contributions to the conductivity discussed in section 3.4.4, we can write E ·

˜ (1) ∂D ω 1 (c) T = Im(ij ) Ei Ej + Re(σij + σij ) Ei Ej . ∂t 2 2

[3.110]

The contribution due to Im(ij ) is due to the absorption and scattering of photons, when intrinsic polarization effects of charge carriers are important. In dielectrics, this absorption is related to the intrinsic electric dipoles of atoms and molecules and it is caused by bound electrons in atoms. In plasmas, it may appear as due to a partial ionization or, for example, if spin effects of free charges are kept into account. Both occurrences go beyond the Vlasov model we are interested in, here. The absorption properties related to what we named conductivity tensor can be split into two contributions, which we did in equation [3.110] by distinguishing, according to the notation introduced in section 3.4.4, σij and (c) σij . The first mechanism of absorption, related to the collisionless component we named σij , is due to resonances between the oscillation of the electric field of the wave and the motion of the charged particles in phase with it: this is the Landau-type resonant absorption mechanism (see sections 4.2.5 and 4.2.6). The other one, related to the collisional component σ (c) , is generally due to the Joule heating of the electrons accelerated by the wave. The origin of such a heating is in the scattering of electrons with other charge carriers through binary collisions, as happens in the case of metals and semiconductors especially at optic frequencies (see e.g. Born and Wolf (1959, §15.6)) or as it occurs in plasmas when Spitzer’s resistivity is accounted for. Otherwise, electrons can be involved in “collisional” scattering with the outcome of some collective plasma response, for example turbulent fluctuations of the magnetic field: in this case, the energy of the wave is (c) dissipated in the plasma because of the contributions to σij which come from some efficient, anomalous resistivity, which has been modeled ad hoc for the phenomenon under consideration, and to which the electrons accelerated by the wave are subject. It must be then noted that the whole discussion above, centered on the absorption of the energy of the wave by the medium, is equally valid, but for the sign of Im(Kij ), for possible sources of amplification of the wave by

140

The Vlasov Equation 1

the medium. This should be evident by simple inspection of [3.110]: while ˜ T > 0 means that a net loss of the electromagnetic energy of the wave E·∂t D occurs inside a certain volume, in which the energy has been transferred to the ˜ T < 0 means that, inside that volume, the medium, the opposite case E · ∂t D energy associated with the wave has increased. This may occur because energy has been injected across the surface of the volume under consideration, or because a perturbation has displaced the system from an equilibrium condition: this is what happens during a plasma instability. More specifically, when the system is at equilibrium and in absence of time-varying external fields, according to equation [3.110] the second principle of thermodynamics requires Im(Kij ) ≥ 0. Then, for perturbations of the form ∼ e−iωt , it follows that ω = ωR + iωI has ωI < 0, since the wave is damped (see e.g. Landau and Lifshitz (1960, Ch. XI, §80) for a more detailed discussion). When, instead, the system is open and energy is injected from external electromagnetic radiation, or when the initial state is not at equilibrium, then Im(Kij ) can be negative, which corresponds to an amplification of the harmonic perturbation, that is to an instability. We conclude by noting that equation [3.110] also implicitly evidences that the real part of Kij , which is related to the real part of ij and to the imaginary part of σij , is responsible for the propagation of the electromagnetic perturbations in the medium. In dielectrics, this is allowed by the redistribution of the polarization charges related to Re(ij ). In conductors, this is related to the imaginary part of the conductivity tensor Im(σij ): in metals, as known, conduction occurs in a limited region, of thickness of the order of the electron skin depth, close to the external surface of the medium36, whereas in plasmas, the collective behavior of charged particles may allow the propagation of the perturbation inside of the whole volume. It also follows, as an important implication of Kramers–Kronig relations [3.50], that when the medium displays dispersion in frequency, which appears as an explicit dependence on ω in ij or σij , electromagnetic absorption by the medium is unavoidable. For further discussions on the properties of the dielectric and conductivity tensors and on their asymptotic behavior, we suggest the reader to look, for example, at Landau and Lifshitz (1960, §77-78 and §124). 36 It is the well-known skin effect, which is discussed in classical electrodynamics textbooks – see also Casimir and Ubbinik (1967) for a concise, practical review on the subject.

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3.6. References Agranovich, V.M., Ginzburg, V.L. (1962). Crystal optics with allowance for spatial dispersion: exciton theory. Sov. Phys. Usp., 5, 323. Allis, W.P., Buchsbaum, S.J., Beers, A. (1963). Waves in Anisotropic Plasmas. MIT Press, Cambridge. Arfken, G.B., Weber, H.J. (1995). Mathematical Methods for Physicists. Academic Press, San Diego. Aström, E. (1951). On waves in a ionized gas. Arkiv Fysik, 2, 443. Balescu, R. (1988a). Transport Processes in Plasmas. 1. Classical Transport Theory. North-Holland Publishing, Amsterdam. Balescu, R. (1988b). Transport Processes in Plasmas. 2. Neoclassical Transport Theory. North-Holland Publishing, Amsterdam. Balescu, R. (1997). Statistical Dynamics. Matter Out of Equilibrium. Imperial College Press, London. Balescu, R. (2005). Aspects of Anomalous Transport in Plasmas. Institute of Physics Publication, Bristol. Bender, C.M., Orszag, S.A. (1999). Advanced Mathematical Methods for Scientist and Engineers I. Asymptotic Methods and Perturbation Theory. Springer, New York. Biskamp, D. (2000). Magnetic Reconnection in Plasmas. Cambridge University Press, Cambridge. Born, M., Wolf, E. (1959). Principles of Optics. Cambridge University Press, Cambridge. Braginskii, S.I. (1965). Transport processes in a plasma. In Reviews of Plasma Physics, Leontovich, M.A. (ed.). Consultants Bureau, New York (NY), 1, 205. Brambilla, M. (1998). Kinetic Theory of Plasma Waves. Clarendon Press, Oxford. Briggs, R.J. (1964). Electron-Stream Interaction with Plasmas. MIT Press, Cambridge. Cap, F.F. (1976). Handbook on Plasma Instabilities, Volume 1. Academic Press, New York. Cap, F.F. (1978). Handbook on Plasma Instabilities, Volume 2. Academic Press, New York. Cap, F.F. (1982). Handbook on Plasma Instabilities, Volume 3. Academic Press, New York.

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Casimir, H.B.G. (1945). On Onsager’s principle of microscopic reversibility. Rev. Mod. Phys., 17, 343. Casimir, H.B.G., Ubbinik, J. (1967). The skin effect. Philips Tech. Rev., 28, 271. Cerri, S., Henri, P., Califano, F., Del Sarto, D., Faganello, M., Pegoraro, F. (2013). Extended fluid models: Pressure tensor effects and equilibria. Phys. Plasmas, 20, 112112. Chapman, S., Cowling, T.G. (1953). The Mathematical Theory of NonUniform Gases. Cambridge University Press, Cambridge. Chew, G.F., Goldberger, M.L., Low, F.E. (1956). The Boltzmann equation and the one-fluid hydromagnetic equations in the absence of particle collisions. Proc. R. Soc. A, 236, 112. Cicogna, G. (2018). Exercises and Problems in Mathematical Methods of Physics. Springer Verlag, Berlin. Connor, J.W., Hastie, R.J., Taylor, J.B. (1978). Shear, periodicity and plasma balloning modes. Phys. Rev. Lett., 40, 396. Coppi, B., Mazzucato, E. (1971). Anomalous resistivity at low electric fields. Phys. Fluids, 14, 134. Coroniti, F.V. (1985). Space plasma turbulent dissipation: Reality or myth? Space Sci. Rev., 42, 399. De Groot, S.R. (1951). Thermodynamics of Irreversible Processes. Interscience, New York. Del Sarto, D., Pegoraro, F., Tenerani, A. (2017). Magnetoelastic waves in an anisotropic magnetised plasma. Plasma Phys. Controll. Fusion, 59, 045002. Elskens, Y., Escande, D.F. (2003). Microscopic Dynamics of Plasmas and Chaos. Institute of Physics Publishing, Bristol. Feynmann, R.P., Leighton, R.B., Sands, M. (1964). The Feynman Lectures on Physics. Volume II. Addison-Wesley, Reading. Fitzpatrick, R. (2014). Plasma Physics: An Introduction. CRC Press, Boca Raton. Fredricks, R.W., Kennel, C.F. (1968). Magnetosonic wave propagating across a magnetic field in a warm collisionless plasma. J. Geophys. Res., 73, 7429. Furth, H.P., Killeen, J., Rosenbluth, M.N. (1963). Finite-resistivity instabilities of a sheet pinch. Phys. Fluids, 6, 459. Gary, S.P. (1993). Theory of Space Plasma Microinstabilties. Cambridge University Press, Cambridge.

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Golant, V.E., Zilinski, A.P., Sakharov, I.E. (1980). Fundamentals of Plasma Physics. John Wiley, New York. Grad, H. (1949). On the kinetic theory of rarefied gases. Comm. Pure Appl. Math., 2, 325. Hasegawa, A., Uberoi, C. (1982). The Alfvén Wave. Technical Information Centre U.S. Department of Energy, Oak Ridge. Hazeltine, R.D., Meiss, J.D. (1992). Plasma Confinement. Addison-Wesley, Reedwood City. Hilbert, D. (1905). Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen. Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen. Math-Phys. Klasse, Band 1905, 307. Huba, J.D. (2006). NRL Plasma Formulary. Naval Research Laboratory. Plasma Physics Division, Washington, D.C. Hunana, P., Tenerani, A., Zank, G.P., Goldstein, M.L., Webb, G.M., Velli, M., Adhikari, L. (2019a). A brief guide to fluid models with anisotropic temperatures part 1-cgl description and collisionless fluid hierarchy. ArXiv. Hunana, P., Tenerani, A., Zank, G.P., Goldstein, M.L., Webb, G.M., Velli, M., Adhikari, L. (2019b). A brief guide to fluid models with anisotropic temperatures part 2-kinetic theory, padé approximants and landau fluid closures. ArXiv. Jackson, J.D. (1962). Classical Electrodynamics. John Wiley & Sons, New York. Jones, R.C. (1941a). A new calculus for the treatment of optical systems i. Description and discussion of the calculus. J. Opt. Soc. Am., 31, 488. Jones, R.C. (1941b). A new calculus for the treatment of optical systems ii. Proof of three general equivalence theorems. J. Opt. Soc. Am., 31, 493. Jones, R.C. (1941c). A new calculus for the treatment of optical systems iii. the sohncke theory of optical activity. J. Opt. Soc. Am., 31, 500. Jones, R.C. (1942). A new calculus for the treatment of optical systems iv. J. Opt. Soc. Am., 32, 486. Kadomtsev, B.B. (1979). Phénomènes collectifs dans les plasmas. Editions MIR, Moscow. Kingsep, A.S., Chukbar, K.V., Yan’kov, V.V. (1980). Electron magnetohydrodynamic. In Reviews of Plasma Physics, Leontovich, M.A. (ed.). Consultants Bureau, New York. Kostantinov, O.V., Perel, V. (1960). Quantum theory of spatial dispersion of electric and magnetic susceptibilities. Sov. Phys. JETP, 37, 560.

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Krall, N.A., Trivelpiece, A.W. (1973). Principles of Plasma Physics. McGraw-Hill, New York. Kramers, M.H.A. (1928). La diffusion de la lumière par les atomes. Atti del Congresso Internazionale de Fisici (Transaction of Volta Centenary Congress), Como, Zanichelli, Bologna, 2, 545. Kronig, R. d.L. (1926). On the theory of dispersion of X-rays. J. Opt. Soc. Am., 12, 547. Kruer, W.L., Dawson, J.M. (1972). Anomalous high-frequency resistivity of a plasma. Phys. Fluids, 15, 446. Kruskal, M. (1962). Asymptotic theory of Hamiltonian and other systems with all solutions near periodic. J. Math. Phys., 3, 806. Lakhtakia, A., Depine, R.A. (2005). On Onsager relations and linear electromagnetic materials. Inst. J. Electron. Commun., 59, 101. Landau, L.D., Lifshitz, E.M., Pitaevskii, L.P. (1981). Course of Theoretical Physics, Volume 10. Physical Kinetics. Pergamon Press, Oxford. Landau, L.D., Lifshitz, E.M. (1960). Course of Theoretical Physics, Volume 8. Electrodynamics of Continuous Media. Pergamon Press, Oxford. Lucarini, V., Saarinen, J.J., Peiponen, K.-E., Viartiainen, E.M. (2005). Kramers-Kroning Relations in Optical Materials Research. Springer Verlag, Berlin. Ma, Z.W., Cen, T., Zhang, H.W., Yu, M.Y. (2018). Effective resistivity in collisionless magnetic reconnection. Sci. Rep., 42, 399. Macmahon, A. (1965a). Finite gyro-radius corrections to the hydromagnetic equations for a vlasov plasma. Phys. Fluids, 8, 1840. Macmahon, A. (1965b). On finite gyro-radius corrections to the hydromagnetic equations for a Vlasov plasma. PhD Thesis, University of California, E. O. Lawrence Radiation Laboratory, Berkley. Melrose, D.B. (1986). Instabilities in Space and Laboratory Plasmas. Cambridge University Press, Cambridge. Mikhailovskii, A.B. (1974a). Theory of Plasma Instabilities, Vol. 1: Instabilities of a Homogeneous Plasma. Springer, New York. Mikhailovskii, A.B. (1974b). Theory of Plasma Instabilities, Vol. 2: Instabilities of an Inhomogeneous Plasma. Springer, New York. Mikhailowskii, A.B., Smolyakov, A.I. (1985). Theory of low frequency magnetosonic solitons. Soviet Physics -JETP, 61, 109. Onsager, L. (1931a). Reciprocal relation in irreversible processes. I. Phys. Rev., 37, 405.

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Onsager, L. (1931b). Reciprocal relation in irreversible processes. II. Phys. Rev., 38, 2265. Papadopulos, K. (1977). A review of anomalous resistivity for the ionosphere. Rev. Geophys. Space Phys., 15, 113. Passot, T., Sulem, P.L. (2007). Collisionless magnetohydrodynamics with gyrokinetic effects. Phys. Plasmas, 14, 08502. Pegoraro, F. (1991). High-pressure equations of state: Theory and applications. In Proceedings of the International School of Physics “Enrico Fermi”. Course CXII, 1989, vol. 1, Part II. Eliezer, S., Ricci, R.A. (eds). NorthHolland Publishing. Varenna, Lake Como. Pegoraro, F., Schep, T.J. (1986). Theory of resistive modes in the balloning representation. Plasma Phys. Control. Fusion, 28, 647. Poynting, J.H. (1884). On the transfer of energy in the electromagnetic field. Philosophical Transactions of the Royal Society of London, 175, 343. Rukhadze, A.A., Silin, V.P. (1962). Electrodynamics of media with spatial dispersion. Sov. Phys. Usp., 4, 459. Sarrat, M., Del Sarto, D., Ghizzo, A. (2016). Fluid description of weibeltype instabilities via full pressure tensor dynamics. EuroPhysics Letters, 115, 45001. Schekochihin, A., Cowley, S.C., Dorland, W., Hammett, G.W., Howes, G.G., Quataert, E.T.T. (2009). Astrophysical gyrokinetics: Kinetic and fluid turbulent cascade in magnetized weakly collisional plasmas. Astro. J. Suppl. Ser., 182, 310. Speiser, T.W. (1970). Conductivity without collisions or noise. Planet. Space Sci., 18, 613. Spitzer, L.J., Härm, R. (1953). Transport phenomena in a completely ionized gas. Phys. Rev., 89, 977. Steinhart, J.S., Hart, R. (1968). Calibration curves for thermistors. Deep Sea Research and Oceanographic Abstracts, 15, 497. Sitenko, A.G. (1967). Electromagnetic Fluctuations in Plasma. Academic Press, New York. Stix, T.H. (1962). Theory of Plasma Waves. McGraw-Hill, New York. Stix, T.H. (1992). Waves in Plasmas. American Institute of Physics, New York. Sulem, P.L., Passot, T. (2014). Landau fluid closures with nonlinear large-scale finite larmor radius corrections for collisionless plasmas. J. Plasma Physics, 81, 3203. Swanson, D.G. (1989). Plasma Waves. Academic Press, Boston.

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Thompson, W.B. (1961). The dynamics of high temperature plasmas. Rep. Prog. Phys., 24, 363. Toll, J.S. (1956). Causality and the dispersion relation: Logical foundations. Phis. Rev., 104, 271. Verheest, F.G. (1967). Waves in a multicomponent Vlasov plasma with anisotropic pressure. Zeit. Natur. A, 22, 1201.

4 Electromagnetic Fields in Vlasov Plasmas: Characterization of Linear Modes

4.1. Introduction We can now look at some application of the general formalism for the linear mode analysis introduced in Chapter 3 to characterize some properties of stable and unstable linear modes in Vlasov plasmas. Like for the previous chapter, the purpose here is not to provide an exhaustive treatment of these subjects, for which several books already exist, often entirely devoted to these topics (see the references at the end of this chapter and previous chapter). The aim of the sections that follow is rather on providing a synthetic and organic presentation, as well as some illustrative examples of application, of the fundamental tools and terminology, which are typically used in literature to identify and characterize some properties of waves and instabilities in plasmas. This will allow us to look at and to revise from a somehow more formal point of view some notions and definitions already used, sometimes implicitly, in Chapters 1–2, as well as to introduce some elements that will be used in Chapter 5 (and in the next volume). In particular, in this chapter we will revise the relation between polarization, phase and group velocities, refractive index, cut-off and resonant conditions of electromagnetic waves propagating in a plasma (section 4.2), and we will discuss some general features and classification criteria of linear instabilities (section 4.3).

The Vlasov Equation 1: History and General Properties, First Edition. Pierre Bertrand; Daniele Del Sarto and Alain Ghizzo. © ISTE Ltd 2019. Published by ISTE Ltd and John Wiley & Sons, Inc.

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In doing so, we will also revise the Landau damping mechanism already analyzed in Chapter 2 by discussing in more detail the notion of wave-particle resonance and of resonant absorption (section 4.2.5). In this regard, we will also consider an alternative interpretation proposed to explain the physical mechanism behind collisionless damping in plasmas and some related, open issues (section 4.4). 4.2. Characterization wave-packets

of

electromagnetic

waves

and

of

Based on the choice of perturbation of equation [3.1], a monochromatic, electromagnetic plane wave propagating in a homogenous plasma with wave vector k is completely determined, for example, by the components of its electric field, as they oscillate in time and space according to the phase of the wave, φ(ω, k) ≡ (k · r − ω(k)t).

[4.1]

We can write this in terms of the polarization vector e(k) , related to the wave vector k, and of the frequency ω(k), which satisfies the dispersion relation for a specific choice of k: E = e(k) Eei(k·r−ω(k)t) + c.c.

[4.2]

In general, we can assume the polarization vector e(k) to have complex components, so to keep account of a possible de-phasing ∼ eiφ0 of angle φ0 between waves which are not synchronous. The components of the polarization vector for non-rectilinear polarization can be written in terms of the Cartesian components (i.e. of rectilinear polarization vectors) according to Jones’ transformation defined in section 3.2.2 However, in most practical cases, both in nature and in a laboratory, the wave described by equation [4.2] is a mathematical idealization. Even in a vacuum, the components of an electromagnetic wave written with respect to the electric field components are functions which solve the d’Alembert equation 

 ∂2 2 2 E = 0, − c ∇ ∂t2

[4.3]

with the appropriate boundary conditions and in the geometry defined by the Laplacian operator ∇2 . The case of plasma is complicated by the presence of

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source terms which, in the linear limit, add to the equation above further integral-differential tensorial operators applied to the components of E. A general electromagnetic wave is therefore identified by a set of functions Ei (r, t) for i = 1, 2, 3, of which equation [4.2] represents a particular case. However, Fourier’s theorem, applied with respect to the time variable to such functions Ei (r, t), grants the possibility to write any component Ei (r, t) as the integral of monochromatic waves over all possible (real) frequencies: Ei (r, t) = Ai (ω, r) ei(ϕ(r)−ωt) dω + c.c. [4.4] The writing of equation [4.2] is then recovered by each component of the frequency spectrum, under further hypotheses about the spatial dependence. In particular, spatial homogeneity of the medium is required, which allows a further Fourier expansion of Ai (ω, r) with respect to the Fourier-conjugate components of r, that is, the components of the wave vector k, and a plane wavefront is assumed1, which also makes the wave periodic in space, so that a finite Fourier series in k can be considered. A case more general than a single plane wave, which is of physical interest for its relevance to a variety of phenomena, is that of a wave packet. A wave described by [4.4] is typically termed “wave packet” of average frequency ω ¯ and spectral width Δω when the envelope of Fourier modes at the right-hand side is centered around the frequency ω0 in an interval of width Δω > 0. That is, when the coefficients Ai (ω, r) appreciably differ from zero only in an interval of frequencies ω ¯−

Δω Δω ≤ ω ≤ ω ¯+ 2 2

with

Δω 1. |¯ ω|

[4.5]

A wave satisfying these conditions is also termed an almost monochromatic wave. It can be shown by using properties of the Fourier transform that the wave packet is more monochromatic, the larger the time interval over which the oscillating electromagnetic field is. Conversely, the finite extension in time of a wave train at a given frequency corresponds in general to a wave packet. An ideal example that is illustrative for this is that of an “almost monochromatic” wave, whose time dependence is described by a harmonic

1 We recall that locally, at sufficient distance from the source, any spherical wave is well approximated by a plane wave.

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The Vlasov Equation 1

electric field oscillating at a frequency ω ¯ , which is “switched on” at a time t = −T and then “switched off” at t = T . It is easy to see how this case could in principle concern all experiments of laser-plasma interaction, as well as cyclotron heating processes or plasma diagnostic measurements by electromagnetic scattering. Assuming we can separate the spatial dependence from the temporal one as Ai (ω, r) = f (r)A˜i (ω), if the processes of switching on and off were sharp, the frequency dependence of the Fourier coefficient A˜i (ω) would be obtained by Fourier transforming the function  ˜i (t) ≡ E

e−i¯ωt 0

for for

|t| ≤ T , |t| > T

[4.6]

which yields sin((ω − ω ¯ )T ) A˜i (ω) = 2 . ω−ω ¯

[4.7]

The behavior of the function A˜i (ω) for two values of T is shown in Figure 4.1.

Figure 4.1. Examples of functions A˜i (ω) (equation [4.7]). For a color version of this figure, see www.iste.co.uk/delsarto/vlasov1.zip

The squared amplitude |A˜i (ω)|2 takes the name of spectral density of the ˜ wave packet. The reason is that, given an arbitrary electromagnetic pulse E(t) with a temporal profile generally different from that of equation [4.6], its

Electromagnetic Fields in Vlasov Plasmas: Characterization of Linear Modes

151

average frequency and spectral width can be formally defined in terms of the spectral density through the integrals: +∞ 1 2 dω ˜ ω |A(ω)| ω ¯ ≡  +∞ , [4.8] dω 2 ˜ 2π |A(ω)| 2π −∞ −∞ +∞ 1 2 dω ˜ (ω − ω ¯ )2 |A(ω)| . [4.9] (Δω)2 ≡  +∞ dω 2 ˜ 2π |A(ω)| −∞ −∞

2π

The example of equation [4.6] is “ideal” in the sense that, even if that specific temporal behavior allows a qualitative and intuitive understanding of the reciprocality between T and Δω, the electromagnetic wave considered is not a wave packet in the sense of [4.5] because, according to [4.9], we find Δω = ∞ for any T < ∞: the reason of this is in the unphysical discontinuity ˜i (t) at t = ±T . However, it generally occurs that for a assumed for E ˜i (t), which goes to zero in a certain time interval, “smooth” function of time E the spectral width is finite and we can speak of a quasi-monochromatic wave packet in the sense discussed above. Moreover, given a general temporal ˜ profile for the function E(t), similarly to equations [4.8]–[4.9], we can define an average “time baricenter” t0 and a “time width” Δt, +∞ 1 ¯ t |E(t)|2 dt, [4.10] t ≡  +∞ 2 dω |E(t)| −∞ 2π −∞ +∞ 1 (t − t¯)2 |E(t)|2 dt, [4.11] (Δt)2 ≡  +∞ 2 dt −∞ |E(t)| −∞ where, owing to Parseval identity, +∞ +∞ 2 2 dω ˜ |E(t)| dt = |A(ω)| . 2π −∞ −∞

[4.12]

It can then be proved as a general result2 that the time and spectral width thus defined are always related by the “uncertainty relation” Δt Δω ≥

1 , 2

[4.13]

2 See, for example, (Born, 1935), Appendix XII, for a demonstration of the result in the Gaussian case, and (Folland and Sitaram, 1997) for a more recent review on the general mathematical approach. The subject is also treated in some recent textbooks of mathematical methods, which can present different derivations of the aforementioned result (see, for example, (Cicogna, 2018) for a demonstration of the general result).

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The Vlasov Equation 1

which quantifies the lower bound of the spectral width of a wave packet in terms of the characteristic length Δt of the electromagnetic pulse. In particular, because of the Fourier transform of a Gaussian function, the strict equality Δt Δω = 1/2 holds only for a profile which is Gaussian in time (and therefore in frequency). Analogous results hold for the components of r and their Fourier-conjugate components of k, if an electromagnetic pulse with finite spatial extension along a direction is considered: for example, a typical laser beam, which obviously has a finite spot, has an intensity which has a Gaussian radial profile with respect to the beam axis; more importantly, as it has on origin in time, any electromagnetic pulse has traveled for a finite distance. The most interesting application of this “space-coordinate” point of view indeed occurs when we consider an electromagnetic wave that satisfies a dispersion relation ω = ω(k, θ), under the hypotheses for which this can be done (homogeneity of the medium, certain symmetry conditions, etc.). In this case, the envelope defined by equation [4.4] in terms of an integral over frequency can be re-expressed as an integral with respect to the vector k, or with respect to the modulus of k and to the angle ϑ between k and a local reference axis: Ei (r, t) = dϑ Ai (k, ϑ) ei(kr cos ϑ−ω(k,ϑ)t) dk + c.c. [4.14] The whole discussion above provided in terms of ω and t can then be reformulated in terms of the components of the space coordinates and wave vector. In particular, an average wave vector length k¯ and a corresponding spectral width Δk can be considered to satisfy a condition equivalent to [4.5] in order to identify an almost monochromatic wave packet in terms of its wavelength. The set of definitions [4.8]–[4.12] can be then adapted to the relevant components of r and k, and the equivalent of [4.13] reads as Δr Δk ≥

1 . 2

[4.15]

With this approach, for example, it is easy to quantify the effect of dispersivity of a medium on the spectral width of a wave packet that propagates in it (see for example Jackson (1962, §7.8–§7.9)). Let us now discuss some more specific features that characterize an electromagnetic wave and let us see some of their applications to the case of plasmas by starting from polarization and the consequences of spatial dispersion on the optical properties of the plasma.

Electromagnetic Fields in Vlasov Plasmas: Characterization of Linear Modes

153

4.2.1. Polarization of electromagnetic waves in plasmas Having previously discussed the meaning of charge polarization that is related to the mean field, we can now look at how the collective response affects the polarization of waves that propagate in the plasma. The wave polarization is usually defined by the components of the electric field perturbations associated with the wave. Regardless of the specific polarization of the mode, something can be said on a very general basis by just considering the orientation of the components of the electromagnetic fields associated with a plane wave corresponding to one of the roots of det(Λij ) = 0. To this purpose, it is sufficient to consider Maxwell equations in the plasma, ˜ rewritten as in materials but by using the equivalent dielectric induction D previously introduced [3.53]. Even if not needed in the discussion that follows, we assume H = B, which, as we have seen, is relevant to most Vlasov plasma regimes. For a plane wave of the form of [4.16], the following relations are obtained after simple substitution ∂/∂t → −iω and ∂/∂r → ik (being clear of the linear approximation, we omit the apex (1) for the sake of notation): We find: * ˜ k × E = ωB, k × B = −ω D, (Vlasov plasmas),[4.16] ˜ = 0, k·D k · B = 0, which should be compared with the respective relations obtained in vacuum k × E = ωB,

˜ k × B = −ω E,

k · E = 0,

k · B = 0,

* (vacuum).

[4.17]

The two cases are schematized in Figure 4.2, which shows how the vectors ˜ B (or H, in case) and k form a Cartesian triad in plasmas, differently D, from E, B and k in vacuum. A longitudinal component of the electric field can appear for waves in plasmas when the projection of Kij on the subspace (plane) orthogonal to k is not diagonal. In general, for waves in a plasma, we can therefore split:  E = Ek + Et ,

where

Ek || k, ˜ Et || D.

[4.18]

This provides a further point of view about the richness of behavior of electromagnetic waves in plasmas and about the reason for which here it is not always possible to distinguish normal modes by separating purely

154

The Vlasov Equation 1

electrostatic (i.e. Et = 0) from purely “electromagnetic” (i.e. Ek = 0) ˜ is branches and instabilities. If one looks at the way the time derivative of D defined in terms of the currents (see the second of equations [3.55]), for a plane wave we obtain the relation   ˜ = i J (1) + J (1) + ε0 E. D [4.19] (p) ω

Figure 4.2. Scheme of the typical orientation of the perturbed field vectors ˜ (1) , E (1) , B (1) and of the wave vector k for a plane wave propagating in D a plasma (left) and in vacuum (right). For a color version of this figure, see www.iste.co.uk/delsarto/vlasov1.zip

Since J(p) = 0 in fully ionized Vlasov plasmas, the relation above makes it possible to interpret the transverse and longitudinal behavior of modes as related to the relative dominance or balancing between the components of the displacement and free currents in the plasma. One of the earliest discussions of this subject, extended to consider the conditions of cut-off and resonance of a wave, which we will speak of next, can be found in Allis et al. (1963). Using the explicit form of Kij (ω, k) given by equation [3.8], the relative amplitudes of the electric field components of the wave in a plasma with a magnetic field B0 along z can be written (E0 is the reference electric field amplitude) as: Ex = E0 Ey = Kxy E0





k 2 c2 − Kyy ω2



k 2 c2 sin2 θ − Kzz ω2

k 2 c2 sin2 θ − Kzz ω2



 + Kyz



2 + Kyz ,

[4.20]

 k 2 c2 sin θ cos θ + Kxz , [4.21] ω2

Electromagnetic Fields in Vlasov Plasmas: Characterization of Linear Modes

Ez = E0



k 2 c2 − Kyy ω2



k 2 c2 sin θ cos θ + Kxz ω2

155

 + Kxz Kyz . [4.22]

Finally, when the response matrix can be cast in the form of equations [3.26]–[3.27], two limit cases of wave polarization can be recognized: the quasi-circular polarization and quasi-linear polarization (Booker, 1934) (see also (Allis et al., 1963, §3.6–3.7 and Ch. 4)). These two cases, respectively, correspond to the limits sin θ → 0 and cos θ → 0 of the dispersion relation [3.16], which in this case reads (KR KL − K|| K⊥ )2 sin4 θ + K||2 (KL − KR )2 cos2 θ = 0.

[4.23]

In particular, we have: – A quasi-circular polarization when K||2 (KL − KR )2 cos2 θ  0,

[4.24]

and the first left-hand side term of equation [4.23] is negligibly small with respect to the second one. In this case, the corresponding modes are quasi-parallel to the ambient magnetic field B0 (k||B0 ) and both left-hand and right-hand circularly polarized waves can propagate (see Figure 3.2).






















   

   





Figure 4.3. Polarization of the principal waves at perpendicular propagation (k ⊥ B0 ): ordinary mode (left), extraordinary mode (right). The electric field perturbation of the extraordinary mode can rotate either left-handed or righthanded with respect to the ambient magnetic field. For a color version of this figure, see www.iste.co.uk/delsarto/vlasov1.zip

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The Vlasov Equation 1

– A quasi-planar polarization when (KR KL − K|| K⊥ ) sin2 θ  0,

[4.25]

and the second left-hand side term of equation [4.23] is negligibly small with respect to the first one. In this case, the corresponding modes propagate nearly across the ambient magnetic field B0 (k ⊥ B0 ), with an electric field perturbation which always remains in a plane containing k (hence it is “quasiplanar”). The two principal waves are named ordinary mode and extraordinary mode: the former is linearly polarized, with electric field perturbation along B0 ; the latter has magnetic perturbations along B0 and can rotate both left and right handed with respect to B0 (see Figure 4.3); this differs from the notion of circular polarization related to Figure 3.2. 4.2.2. Phase velocity, group velocity and refractive index The notions of phase velocity, vφ , and group velocity, vg , which we have already used, are well-known. Their formal definitions in terms of the dispersion relation ω = ω(k) of a wave propagating with wave vector k, ⎧ ω ⎪ ⎪ ⎨ vφ ≡ k ∂ω , vg ≡ , [4.26] ⎪ ∂k ⎪ ⎩ vφ = v φ k k can be of course adapted for a wave packet (with the evident substitutions k → ¯ and ω → ω k ¯ ). Even if typically identified with the scalar value expressing its modulus, the phase velocity, as a vector, is formally defined as the velocity vφ with which the wave front moves in the direction locally orthogonal to itself. Hence, the orientation of vφ is along k. The modulus of the phase velocity can also be expressed in terms of the ratio between the amplitude of the magnetic induction |B| and the amplitude of the transverse component of electric field, Et (Figure 4.2). The following relation, which follows from equations [4.16]– [4.17], is indeed always verified: |vφ | =

ω |Et | = . k |B|

[4.27]

The group velocity represents instead the velocity at which the components of the wave vector itself travel by changing in space and time. This can be seen by considering the definition of the phase of the wave (or wave packet) given by equation [4.14] and by differentiating it twice, once with respect to time and once by calculating its spatial gradient: since, from the definition,

Electromagnetic Fields in Vlasov Plasmas: Characterization of Linear Modes

∂φ = −ω, ∂t

∂φ = k, ∂r

157

[4.28]

by equating ∂(∂φ/∂r)/∂t and ∂(∂φ/∂t)/∂r and by using ∂k ∂ω ∂k ∂ω = · = vg · ∂r ∂k ∂r ∂r

[4.29]

we obtain ∂k ∂k + vg · = 0. ∂t ∂r

[4.30]

It is also well-known that vg represents the velocity with which a wave packet is moving in space. If we consider an envelope of the kind of [4.14], peaked around ω0 and k0 , a local expansion of ω(k) around k0 gives ω  ω0 +

∂ω · (k − k0 ) = ω0 + vg · (k − k0 ), ∂k

which allows us to write the envelope as Ei (r, t) = ei(k0 ·r−ω0 t) Ai (k) ei(k−k0 )·(r−vg t) dk + c.c.

[4.31]

[4.32]

In the above expression, the time dependence is split into a phase, oscillating at the frequency ω0 , and an “amplitude” which propagates at velocity vg . A further relation between the group velocity and the Poynting vector defined by equation [3.85] could then be proved, which shows how the energy of the wave propagates at the speed vg . While addressing the reader to other books for its writing and demonstration − see for instance (Kadomtsev, 1979, Chapter 2, §5) – for our purposes it is sufficient to note in this regard that the intensity of the energy transported by a wave is proportional to its squared amplitude, whose time dependence is given by equation [4.32]. In this sense, for example, electrostatic plasma oscillations at the Langmuir frequency, already encountered in section 2.2.3, do not propagate energy 2 since they have a dispersion relation ω 2 = ωpe . This is the reason for which sometimes one preferably speaks of them as “oscillations”, rather than as “waves”. It is only when thermal effects are included, which yield the Bohm–Gross dispersive corrections to the dispersion relation [2.30], that these electron plasma modes obtain a non-null group velocity (Bohm and Gross, 1949). The spatial dispersion of the medium is then characterized by a different dependence of the phase and group velocity with respect to the wave length.

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The Vlasov Equation 1

Depending on the sign of the derivative of the phase velocity with respect to k, for a wave with dispersion relation ω = ω(k, θ) we can distinguish the two opposite cases of positive dispersion (∂vφ /∂k > 0) and negative dispersion (∂vφ /∂k < 0). When applied to a wave packet, both cases of dispersion make its spectral width change by increasing the time (spectral broadening due to dispersion). This is the case because components of the spectrum with different wavelengths travel with different group velocities: the component of the spectrum at small wavelengths (large k) will spread faster in a case of positive dispersion, whereas it will be the large wavelength component (small k) to do so for a negative dispersion. A limit of interest because of its generality is when the dependence of vφ on dispersion is a quadratic function of k, as one expects it to be for example in an isotropic medium. A dependence of this kind is also frequently encountered in plasmas, despite their intrinsic anisotropy. For small dispersive effects with respect to a critical reference length L0 around a characteristic phase velocity cφ as defined in absence of dispersion, we can summarize the two cases of quadratic dispersion when writing vφ = cφ (1 ± (kL)2 /2), whence it follows |v|g = cφ (1 ± 3(kL)2 ). For more specific examples and more detailed discussions, we suggest the reader goes through the references given in Jackson (1962) and Kadomtsev (1979). The group and phase velocities can then be related by an expression, which is useful for geometrical interpretation. Using ∂k k = , ∂k k

[4.33]

which can be easily verified by rewriting ∂k/∂k in tensor notation, 1

∂(kj kj ) 2 kj ∂kj kj ki ∂k = = = δij = , ∂ki ∂ki k ∂ki k k

[4.34]

we obtain: v g = vφ

k ∂vφ +k = vφ + Δvg,⊥ . k ∂k

[4.35]

The second term Δvg,⊥ = k∂vφ /∂k, which is non-zero only in presence of spatial dispersion, is always directed orthogonally to k, since it expresses the variation of the modulus of the phase velocity with respect to the variation of the wave vector. Instead, when vφ does not depend on the modulus of k, that is, when there is no spatial dispersion and ω(k) ∝ k, the group and phase velocities coincide coherently with their definitions.

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As it is well-known from optics, the notion of dispersion is then related to the dependence of the refractive index Nr ≡

c kc = vφ ω

[4.36]

on the wavelength (i.e. on k) and possibly on the angle of propagation with respect to some preferred axes, which break the isotropy of the medium. As we are going to discuss in section 4.2.6, this allows a useful criterion for the classification of waves in plasmas (Stix, 1962; Allis et al., 1963), since there are some critical values of the refractive index, namely 0, 1 and ∞, which identify transition points of the properties of propagation of the electromagnetic branch in the k parameter space. In principle, ω is a complex number and so it can be Nr , whence it follows that plasmas Nr2 can assume both positive and negative values. In particular, the conditions that make Nr2 change sign separate the regions of the plasma in which a certain wave can propagate or not. We recall, in this regard, that in classical optics the above-mentioned features are related to Fresnel equations3, which describe the behavior of electromagnetic waves, namely their reflection and transmission properties in terms of wave polarization, when the wave crosses a surface separating two media with different refractive indices. Such equations, which we do not need to recall here (we address instead to textbooks like Born and Wolf (1959, section 1.5) or Landau and Lifshitz (1960, Ch. X, §86), for their derivation and for a detailed discussion) follow from the boundary conditions on the electromagnetic field components of the wave, which are deduced by using the relations ⎧ kx,0 = kx,rf l = kx,rf r , ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ kz,rf l = −kz,rf r , [4.37] ⎪ ⎪ ⎪  ⎪ ⎪ ⎩ kz,rf r = ω N 2 − N 2 sin2 ϑ0 r,2 r,1 c which refer to a geometry like that in Figure 4.4, where indices 1 and 2 label the two media and “0”, “rf l” and “rf r” refer to the incident, reflected and refracted wave, respectively. Relations [4.40] follow the hypothesis that the thickness of the interface between the two media, δ, be much smaller than the

3 Discovered by Fresnel (1821).

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The Vlasov Equation 1

wavelength of the incident wave (k0 δ 1), and from the assumption of spatial homogeneity with respect to the coordinates on the surface separating the two media. When both media are transparent, that is, when no reflection occurs at their interface, conditions [4.40] lead to the well-known Snellius–Descartes law Nr,1 sin ϑ0 = Nr,2 sin ϑrf r ,

[4.38]

which are valid also in the opposite limit k0 δ  1 of geometrical optics, in which reflection of the incident light wave can be neglected in the above-mentioned problem.

Figure 4.4. Reflection (angle ϑrf l ) and refraction (angle ϑrf r ) of a wave, incident on the surface of separation between a medium with refractive index Nr,1 and medium with refractive index Nr,2 with an angle of incidence ϑ0 (the example drawn in this figure schematizes a case with Nr,2 > Nr,1 ). For a color version of this figure, see www.iste.co.uk/delsarto/vlasov1.zip

As we are going to see in the following two sections, relations [4.37]– [4.38] are sufficient to let us understand some simple applications relevant to the scattering and propagation of electromagnetic waves in plasmas. The first of which is the spatial attenuation4 of an electromagnetic wave incident from 2 < 0, such as it happens for an incident vacuum in a plasma for which Nr,2

4 It is the damping in space we spoke of for the state-state problem in section 3.2.4.

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laser on an overdense plasma or for the scattering of radio waves on layers of the ionosphere having different densities. We recall in this regard the notion of the critical angle of total internal reflection: the law of Snellius–Descartes [4.38] implies that ϑrf r > ϑ0 when Nr,2 < Nr,1 . Therefore, if the latter condition is met, a critical angle ϑc defined as sin ϑc =

Nr,2 Nr,1

[4.39]

exists, for which ϑrf r = π/2. That is, for ϑ0 > ϑc the incident wave is completely reflected at the interface between the two media, instead of being refracted by crossing it, even in regimes in which standard reflection is negligible (i.e. in regimes where equation [4.38] holds). As already mentioned, in the two sections that follow, we are going to consider two illustrative examples of some optical properties in Vlasov plasmas: – the condition on frequency for the penetration of electromagnetic waves in an unmagnetized plasma in which temperature effects are disregarded, which leads to the distinction between underdense and overdense plasma regimes; – and the dependence on frequency of the refractive index of electromagnetic waves, which is different for right- and left-circular polarized waves propagating parallel to a magnetic field. Both examples are quite simple to consider, since the relevant properties of the refractive index can also be easily derived from reduced models based on single particle dynamics. 4.2.3. Example of propagation in underdense and overdense regimes

unmagnetized

plasmas:

Let us consider a collisionless, unmagnetized plasma, at an order of approximation in which particle temperature can be neglected. We are interested in the propagation of purely electromagnetic waves, that is, transverse waves, with a frequency high enough to allow us to neglect ion dynamics. We therefore consider the electron response only by assuming ions to form a neutralizing background. Due to the absence of preferred directions other than that of k, we can use the response tensor of equation [3.15].

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The Vlasov Equation 1

Transverse waves, having electric field components orthogonal to k, have dispersion relation K⊥ (ω, k) =

k 2 c2 . ω2

[4.40]

Under the hypotheses we have assumed, due to the absence of a magnetic field and due to the negligibility of |kvth,α |/|ω| corrections in the expansion of a Maxwellian distribution function, we should not expect any difference between the components of the equivalent dielectric tensor that are parallel and perpendicular to k. By leaving this point to be verified by the reader as an exercise or by looking at other plasma physics textbooks, we therefore consider K⊥ (ω, k) = K|| (ω, k). We can identify K⊥ (ω, k) with the dispersion function K|| (ω, k) = (k, ω), already evaluated in section 2.2.3 for 2 Langmuir oscillations in the form  = 1 − ωpe /ω 2 . We obtain5 two equivalent writings for the dispersion relation 2 + k 2 c2 , ω 2 = ωpe

Nr2 = 1 −

2 ωpe . ω2

[4.41]

It follows that only electromagnetic waves with a frequency higher than the local plasma frequency can propagate. In particular, we see from the first of [4.41] that for ω < ωpe the squared value k 2 changes sign and therefore the wave number must be imaginary, k = i|k|. Therefore, |k|2 =

2 ωpe − ω2 c2

−−− −−− → 2 2 ω ωpe

1 . d2e

[4.42]

That is, below a threshold value fixed by ωpe , a low-frequency electromagnetic wave incident on a collisionless, unmagnetized plasma is reflected after penetrating for a characteristic distance of the order of the electron skin depth de : for ω ωpe , indeed, a plane wave perturbation of the form [3.1] becomes ∼ (AF eiωt + c.c.) e−x/de . We see from the second of [4.41] that the threshold condition at which k 2 changes sign occurs at 2 Nr2 = 0. Since ωpe ∼ ne , a critical density nc (ω) can be associated with an

5 A simpler model based on the single electrons dynamics (i.e. the Drude–Lorentz model) would suffice to obtain the same result (see, for instance, (Feynmann et al., 1964, §32) or (Jackson, 1962)). To this purpose, the calculations can also be adapated from those in section 4.2.4 by taking the B0 = 0 limit.

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incident electromagnetic wave with frequency ω. In the regime of plasma here considered, the dependence of the refractive index on ω is written as Nr2 = 1 −

ne . nc (ω)

[4.43]

For each electromagnetic frequency, this identifies two regimes, depending on the electron density. We speak of: – overdense plasma when ne > nc (ω) (i.e. Nr2 < 0), in which the light at frequency ω is capable to penetrate only for short distances of the order of ∼ de , before being mostly reflected back; – underdense plasma when ne < nc (ω) (i.e. 0 < Nr2 < 1), in which the wave with frequency ω can propagate, although being refracted at angles typically larger6 than the incident one with respect to the local normal to the plasma surface. These properties of plasmas, which can be understood in terms of the discussion previously provided in equations [4.37]–[4.39], are fundamental for long-range radio communications and for satellite transmissions, since they describe refraction and reflection of radio waves in the ionospheric plasma well, as long as magnetic field effects can be neglected. In radiophysics, equation [4.43] is indeed known as (the unmagnetized limit of the) Altar–Appleton–Hartree–Lassen equation, after the four scientists who obtained it in the second half of 1920s7. A first use of what would have been later8 identified as the transition between underdense and overdense plasmas has indeed been in the estimation of the dependence on altitude of the

6 For comparison, note that most transparent solids have Nr > 1 (and in vacuum obviously Nr = 1). 7 Although the equation is usually only entitled to Edward Victor Appleton (1892– 1965) and Douglas Hartree (1897–1958), the first fundamental contribution to its formulation has been later attributed (Gillmor, 1982) to Wilhelm Altar (1900–1995), coworker of Appleton in those years, and the equation seems to have been established and first published by Hans Lassen (1897–1974) (Lassen, 1926). All these scientists were interested in modeling the radiowave scattering by the ionosphere, whose existence had been hypothesized in 1902 by Olivier Heaviside (1850–1925) and Arthur Edwin Kennelly (1861–1939), and which had been experimentally discovered in 1924 by Appleton, after systematic measurements of radio wave propagation. 8 It is interesting to note that, when these studies about the ionosphere were performed, plasma physics itself was being born (as we recalled in Chapter 1, Langmuir’s first use of the word “plasma” in an article dates back to 1928): the works quoted before as well as that by Chapman, cited next, were not based on the theory of wave propagation

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The Vlasov Equation 1

electron density in the ionosphere. This has been done by measuring the time needed for a radio signal to be reflected back from a layer where ne  nc (ω), according to equations [4.39] and [4.43]. As a practical example of an application of the previous discussion, we can discuss the outcome of these studies and their practical consequences for telecommunications. In this way, the ionospheric density has been understood to increase, on average, from about 100 km to about 250–300 km, and then to progressively decrease again as the altitude increases further. The density profile has also been found to be strongly influenced by the sun’s activity and is well-described by Chapman’s model (Chapman, 1931), dating back to 1931: the day-side ionosphere, being more intensely irradiated by solar electromagnetic waves, has a ionization rate higher9 than the night side, and hence a typically higher density almost at all altitudes. Today, satellite transmissions must therefore be done at frequencies sufficiently high to be scarcely affected by refraction and spatial attenuation (see equations [4.37]) by the ionosphere they cross, whose peak plasma frequency is about ∼ 9 MHz. This corresponds to transmission, which is typically in the ∼ GHz frequency range. Instead, the relatively smaller frequency (∼ 3 − 30 MHz) of the “short wave” band (λ ∼ 10 − 102 m) is used for long-range radio communications, which use the refraction properties of the underdense ionospheric plasma: depending on their frequency and on the local density of the ionospheric layers, the angle of incidence of these “short waves” is regulated in order to trespass the critical angle of equation [4.39], so that the diffracted angle allows a propagation of the signal beyond the curvature of the Earth at distances of thousands of kilometers from the emitter. The even lower frequency “medium wave” band (∼ 0.3 − 3 MHz), which is usually used for radio broadcasting at a national scale, uses the same principles but is diffracted by lower layers of the ionosphere, since these frequencies cannot penetrate in the higher density regions. However, at night time, due to the lower ionization that lowers the value of ωpe , they can penetrate further in the ionosphere and, being refracted by the higher layers, their signal can travel farther and reach distances of thousands of kilometers.

in Vlasov or magnetohydrodynamic plasmas (more than 10 years should have been waited!), but rather on single particle dynamics models, namely, the Drude–Lorentz model for light refraction in materials. 9 Treating the ionosphere as a collisionless, fully ionized plasma is a strong approximation. Nevertheless, equation [4.43] obtained in this simplistic model provides an excellent qualitative and a sufficient quantitative agreement with measurements.

Electromagnetic Fields in Vlasov Plasmas: Characterization of Linear Modes

165

Another implication of equation [4.43] is of course for laser–plasma interactions, where the notions of underdense and overdense plasmas are probably more commonly encountered, especially from the point of view of terminology. In these processes, electrons are often accelerated by the laser light up to relativistic velocities, so that non-negligible corrections to the above formula are required because of the relativistic inertial mass increase: √ due to the√dependence de ∼ me , the penetration length also increases by a factor ∼ Γe and the threshold between underdense and overdense regimes is modified as NR2 = 1 −

ne . Γe nc (ω)

[4.44]

A parameter often used to quantify the importance of these relativistic effects in terms of the laser intensity is the so-called laser strength parameter,

a0 ≡

pquiv eE e = , me c me cω

[4.45]

which measures in “relativistic units” the electron quiver momentum, that is the momentum gained by an electron in a complete oscillation due to the transverse electric field associated with the wave. By keeping into account the light polarization of the wave, a0 can be related to the intensity of the laser. For circular polarization, for example, the practical relation (the relevant units are indicated as subscripts in brackets) a0  0.6 × 10−9 λ[μm]

 I[W/cm2 ]

[4.46]

holds. For typical wavelengths of the order of λ ∼ 1 μm, this indicates that a relativistic regime for electrons (that is, a0 approaching unity, let us say then a0 ∼ 0.1 − 1) is achieved already for intensities I ∼ 1015 − 1018 W/cm2 . For a more thorough reading about laser plasma interactions, specific books (Atzeni and Meyer-Ter-Vehn, 2004; Macchi, 2013) are suggested. Relation [4.44] can provide a picture of the initial stage of propagation of a high-intensity monochromatic wave in an otherwise overdense plasma because of a sort of relativistic “self-induced transparency”. However, because of the large amplitude that perturbations rapidly reach in these regimes, nonlinear effects soon become fundamental for propagation (Cattani et al., 2000; Tushentsov et al., 2001; Berezhiani et al., 2005). Moreover, in these extreme regimes, the kinetic effects we have neglected so far in these

166

The Vlasov Equation 1

examples based on a cold plasma model play a fundamental role, since part of the laser energy is also transferred to the thermal motion. The resulting dynamics becomes quite complex and requires a Vlasov–Maxwell description in which coupling with plasma waves and instabilities mediated by wave–particle trapping effects occur (Ghizzo et al., 2007). Examples of this kind of wave particle effects will be discussed in Chapter 5. 4.2.4. Example of propagation in magnetized plasmas: ion-cyclotron resonances and Faraday’s rotation effect For a second illustrative example, we can add to the plasma considered in the previous section a uniform equilibrium field, let us say along z: B = B0 ez . We then relax the high-frequency hypothesis, so to consider both ions and electrons. For this case, we have not yet evaluated the relevant components of the equivalent dielectric tensor. By addressing once again the reader to other books (e.g. (Stix, 1962; Krall and Trivelpiece, 1973; Melrose, 1986; Swanson, 1989; Brambilla, 1998)) for the evaluation of Kij (ω, k) by taking the appropriate limits of the Vlasov–Maxwell linear system, let us examine how to get the same result by means of the single particle models mentioned previously. This could also be seen as a “historically accurate” approach: it was first developed by Paul Karl Ludwig Drude (1862–1906) (Drude, 1900a, 1900b) to model the transport of electrons in conducting materials with collisions and is based on the idea that, on average, the electrons subject to the oscillating field of the incident wave in a conductive resistive medium (a metal) are all governed by the same dynamic equation, which accounts for the electric and Lorentz forces and for a friction term. In collisionless plasmas, we could formally neglect the latter while we retain the dynamics of ions too (indices α = e, i): me

dvα = qα (E + vα × B) . dt

[4.47]

By noting that in this modeling, the current density is given by J=



q α nα v α

[4.48]

we can look for an equivalent dielectric tensor that satisfies the definitions of equations [3.3]–[3.5]. To this purpose, we assume that the velocity is determined by the small electromagnetic perturbations, so we can linearize the Lorentz-force term in [4.47] as ∼ vα × B0 . The components of vα can be determined assuming a dependence on ω and on k of the kind of [3.1] and by

Electromagnetic Fields in Vlasov Plasmas: Characterization of Linear Modes

167

substituting d/dt → −iω. Notably, as equations [4.47]–[4.48] do not contain any spatial gradient, the equivalent dielectric tensor will not display any spatial dispersion. A resonance in frequency however appears due to the characteristic time scale (i.e. the particle gyration period) introduced by Lorentz force (see section 3.3.2). Writing equations [4.47] by components: qα Ex , mα ω qα −iωvyα + sα Ωα vxα = Ey , mα ω qα vzα = i Ez , mα ω

−iωvxα − sα Ωα vyα =

[4.49] [4.50]

where sα indicates the sign of the charge, that is “+” for α = i and “−” for α = e. After some algebra, by using [3.4] and [4.47], we obtain the equivalent dielectric tensor Kij (ω, k) = ⎞ ⎛ 2 2  ωpα  ωpα Ωα −i 0 ⎟ ⎜ 1− ω 2 − Ω2α ω(ω 2 − Ω2α ) ⎟ ⎜ α α ⎟ ⎜ ⎟ ⎜ ⎟ ⎜  2 2  ωpα ωpα Ωα ⎟ ⎜ ⎟ . [4.51] ⎜ i 0 1− 2 − Ω2 ) 2 − Ω2 ⎟ ⎜ ω(ω ω α α ⎟ ⎜ α α ⎟ ⎜ ⎟ ⎜ ⎜ 2  ωpα ⎟ ⎠ ⎝ 0 0 1− 2 ω α The dispersion relation is then obtained after inserting the tensor above in the definition [3.5] of Λij (ω, k). The different branches corresponding to this dispersion relation are discussed in most books. Here, in order to highlight some specific features, we just focus on the case of parallel propagation, k = kez , that is to the θ = 0 limit in which the response tensor written with respect to the Cartesian basis reads Λij (ω, Nr ) = ⎞ ⎛ 2 2  ωpα  ωpα Ωα 2 − Nr −i 0 ⎟ ⎜1− ω 2 − Ω2α ω(ω 2 − Ω2α ) ⎟ ⎜ α α ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 2 2  ωpα  ωpα Ωα ⎟ ⎜ 2 ⎟ . [4.52] ⎜ i − Nr 0 1− 2 − Ω2 ) −2 Ω2 ⎟ ⎜ ω(ω ω α c ⎟ ⎜ α α ⎟ ⎜ ⎟ ⎜ ⎜ 2 ⎟  ωpα ⎠ ⎝ 0 0 1− ω2 α

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The Vlasov Equation 1

We immediately recognize that, as we set both Ωe = Ωi = 0 for B0 = 0, the results discussed in section 4.2.3 are recovered, with the additional 2 contribution of the ion species that becomes negligible as ωpi ω 2 . Such a correction appears equally both for the electromagnetic, i.e. transverse, and electrostatic, i.e. longitudinal, branches. The presence of the magnetic field introduces however some resonances at the denominator of the transverse branch, at the cyclotron frequency of each of the two species, respectively. The form of the tensor [4.52] suggests the use of Jones transformations to rewrite the electric field with respect to the (EL , ER , E|| ) components defined by equations [3.24]. With respect to this basis Λij (ω, Nr ) becomes diagonal, which leads to the dispersion relations of three completely decoupled modes: 2 Nr,L = 1−

2 2 ωpi ωpe − , ωL (ωL + Ωe ) ωL (ωL − Ωi )

[4.53]

2 = 1− Nr,R

2 2 ωpi ωpe − , ωR (ωR − Ωe ) ωR (ωR + Ωi )

[4.54]

0 = 1−

2 2 ωpi ωpe − . ω||2 ω||2

[4.55]

In particular, the transverse, purely electromagnetic branches differ also because of the polarization (indices L, R above), whereas the electrostatic branch (index ||) also now depends on the ion dynamics. Their oscillations  occur however at a frequency me /mi smaller than the ωpe of the electrons, which is why it was neglected in the calculations of section 4.2.3. Let us comment on a few specific limits of these results, by considering for simplicity a hydrogen plasma, for which |qe | = |qi | = e and therefore, for a quasineutral equilibrium, ne = ni : – Faraday’s rotation effect Equations [4.53] state that the phase (and thus group) velocity of the two electromagnetic waves with parallel propagation and opposite circular polarization are different. The difference is larger, the closer the frequency is to some of the resonant values at the denominator (ω = ±Ωα ). Also, because of the temporal dispersion, it follows that two oppositely circularized waves that propagate at the same frequency have different wavelengths: if we solve each equations [4.53] for k by fixing in both solutions the same numerical value of ω, we get two different values that we name, with coherence of notation, kL and kR . According to definitions [3.24]–[3.25], the electric field vectors EL ∼ eL EL ei(kL z−ωt) + c.c. and ER ∼ eR ER ei(kR z−ωt) + c.c. can be used

Electromagnetic Fields in Vlasov Plasmas: Characterization of Linear Modes

169

to express the two circularly polarized modes. If we now consider a linearly polarized wave as a superposition of two circularly polarized waves with the same electric field amplitude (ER = EL = E0 ), then we can write   E(z, t) ∼ E0 eL ei(kL z−ωt) + eR ei(kR z−ωt) + c.c.

[4.56]

By introducing a “mean wave vector” k0 ≡ (kL + kR )/2 and a difference Δk ≡ (kL − kR )/2, equation [4.56] can be rewritten in terms of the Cartesian components as E(z, t) ∼ E0 cos(k0 z − ωt) (ex cos(Δkz) + ey sin(Δkz)) .

[4.57]

We see that, while the polarization remains linear at any time (the ratio between the components Ex and Ey does not depend on t), the plane of polarization rotates with a phase φ(z, t) = k0 z − ωt as the wave propagates along the magnetic field. This behavior of electromagnetic waves, discovered in 1845 by Michael Faraday (1791–1867) during the experiments with which he first recognized the electromagnetic nature of light (Faraday, 1846), is of fundamental importance in astrophysics for the measurements of magnetic fields in the interstellar plasma through the indirect estimation of wave emission by synchrotron radiation (see, e.g. (Wentzel, 1974)). – Alfvén waves 2 Let us now consider the low-frequency limit ω 2 Ω2i ωpi of the purely electromagnetic branch (waves with transverse polarization). We first note that 2 2 ωpe me ωpi me c2 = = 2 2 Ωe mi Ωi mi c2A

and

2 2 ωpi ωpe = , Ωi Ωi

[4.58]

where cA ≡ √

B0 0 ni mi

[4.59]

2 is the Alfvén speed, which satisfies the condition c2A c2 owing to Ω2i ωpi . Using the second of [4.55] and no further hypothesis, we can rewrite equations

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The Vlasov Equation 1

[4.53]–[4.55] as: 2 Nr,L = 1−

2 2 + ωpi ωpe , (ωL + Ωe )(ωL − Ωi )

[4.60]

2 = 1− Nr,R

2 2 + ωpi ωpe , (ωR − Ωe )(ωR + Ωi )

[4.61]

whence we see that in the chosen frequency regime and neglecting the contribution of electron inertia, i.e. taking me /mi → 0, the two transverse branches with opposite circular polarizations coalesce into a single, double degenerate branch: 2 Nr,L 2 Nr,R

+

2 −→ Nr,⊥ 1−

c2 . c2A

[4.62]

This dispersion relation is formally rewritten as (ω 2 − k 2 c2A )2 = 0, which corresponds to the “magnetohydrodynamic waves” identified by Alfvén (1942). We recognize them to correspond to linearly polarized electromagnetic waves, which propagate parallel to the magnetic field with an extremely small phase velocity (vφ = cA ). Low-frequency electromagnetic waves entering a plasma with a wave vector parallel to B are therefore bound to parallel propagation since Nr,⊥  1 implies refraction angles tending to zero. – Whistler branch Despite the apparent “symmetry” between the dispersion relations of the left-hand and right-hand polarized waves, we see that in the frequency window 2 Ωi  ω Ωe the squared refractive index, Nr,L , becomes negative: we have seen in section 4.2.4 that this means an exponential attenuation of the wave over a short penetration distance. In this frequency range, indeed, the left-hand polarized wave cannot propagate. The right-hand polarized branch instead can take the name of a whistler wave. From equation [4.61], we obtain, after a few simplifications related to the frequency limit stated above, the approximated dispersion relation ω  k 2 d2e Ωe .

[4.63]

– Cyclotron resonances The refractive indices Nr,L and Nr,R tend to diverge as the frequency of the wave approaches the resonances at the denominator of equations [4.52].

Electromagnetic Fields in Vlasov Plasmas: Characterization of Linear Modes

171

The divergences implied by these limit values indicate the limits of the model, which for these critical values would require one to keep account of a full kinetic, Vlasov description. In particular, the Landau prescription for evaluating the resonant phase-space integrals, which gives the electrostatic Landau damping (section 2.2), also applies, with some slight differences, in a magnetized plasma. Here, it leads to the collisionless cyclotron damping (see, e.g. (Landau et al., 1981, Ch. V, §55)), which involves the particles of the species α, moving along the magnetic field with speed vz that satisfy a resonance condition of the kind ω − kz vz = nΩα

[4.64]

for integer numbers n = ±1, ±2, ±3, ... However, even if the divergence can be this way corrected, the resonance in the dispersion relation indicates a peak in the refractive index for which the wave vector itself tends to diverge. To see this in the example previously considered, we can, for instance, take the limit ω 2  Ω2i in equations [4.52] and solve them for k as a function of a real ω. We obtain: kL

ω = c

kR

ω = c



2 ωpe 1− ω(ω + Ωe )



 12

2 ωpe 1− ω(ω − Ωe )

We see that:  ∞ kL → + i∞  ∞ kR → i∞

,

[4.65]

.

[4.66]

 12

as as

(ω + Ωe ) → 0− (ω + Ωe ) → 0+

[4.67]

as as

(ω − Ωe ) → 0+ . (ω − Ωe ) → 0−

[4.68]

The discontinuity in the sign of Nr2 is therefore associated with a divergence in k. That is, close to the Nr2 = 0 condition, the attenuation of the wave in the Nr2 < 0 region diverges over small distances (i.e. e−|k|Δz → 0 even for Δz arbitrarily small). This occurs because of the mechanism of resonant absorption, which leads us to the point which we are going to address in the following section.

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The Vlasov Equation 1

4.2.5. Wave–particle resonances, Landau damping and wave absorption In the latter examples discussed in the previous section, we have seen that, close to a resonance in frequency, the refractive index tends to diverge. On the other hand, in sections 2.2.1–2.2.2 we have seen how these resonances lead to Landau poles and to the collisionless damping. The two are of course related; in particular they are connected to the process of absorption of the wave by the plasma. In order to see how, let us look at these features, which at first may appear as quite technical and related to the mathematics, from a more intuitive physical point of view. To this purpose, let us first consider the definition of resonance, as it is used in classical mechanics: it indicates an increase in the amplitude of an oscillation, which occurs because a periodic force is applied with a frequency close to one of the natural frequencies of oscillation of the system. The classical example is the forced harmonic oscillator. In optics, this notion translates into that of “resonant absorption” encountered in the classical theory of light dispersion in materials. We have already seen its application in section 4.2.4 with Drude’s model: in that case we considered the oscillation of an electron, perturbed by the electric field E = E0 e−iωt + c.c. of an incident electromagnetic wave. If, instead of assuming the electron to be free, we consider it to be bound to an equilibrium position10 r0 in an atom or molecule because of some restoring force of the kind −κ(r − r0 ), its momentum equation that we wrote as in equation [4.47] can be re-cast in the form: me r¨ = −κ(r − r0 ) + e(E0 e−iωt + c.c.).

[4.69]

This admits a trivial oscillatory solution of the form (r − r0 ) = r0 e−iωt + c.c. when the equilibrium position satisfies r0 =

e E0 , 2 m (ω0 − ω 2 )

with

ω02 ≡

κ . m

[4.70]

When the frequency of the incident electromagnetic radiation resonates at a value equal to plus  or minus the characteristic oscillation frequency of the electron, ω0 = κ/m, the amplitude of the oscillation diverges. This

10 It can be the case, for example, of electrons in dielectrics or non-conduction electrons in metals.

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example suits the notion of resonance we are interested in, since if we assume the response of the single electron to be weighed by the electron density in the medium, it becomes possible to relate its collective response, that is the electromagnetic perturbation propagating in it at the frequency ω, to the microscopic behavior of the single electron. This exemplifies the mechanism of wave–particle resonance with which the electromagnetic wave exchanges efficiently the energy it transports with the kinetic motion of particles. Of course, in a system in which the electromagnetic wave propagates because of the collective response of all charges, this exchange of energy between the resonant particles and the “wave” can occur in the two senses, and in which one dominates depending on the configuration of the system and on its boundary conditions (for example, a resonant amplification or instability could occur, rather than a damping of the wave). In a collisionless plasma, wave–particle resonances appear at the denominator of integrals [3.11]. In absence of Landau damping, this would diverge over time intervals inversely proportional to the density in the phase-space of the resonant particles, because of the contribution of those particles whose velocity component along k equals the phase velocity ω/k of the wave. The “critical values” of ω/k are fixed by the roots of Λij (ω, k) = 0. In the electrostatic examples considered in section 2.2, we have seen that Λ(ω, k) also appears at the denominator in front of the integral of the perturbed distribution function (equation [2.5]). In order to make a comparison with the example of the oscillating electron, the unphysical divergence related to the resonant denominator of equation [4.70] can be corrected by an inclusion of a further friction term of the kind of −me ν r˙ at the right-hand side of [4.69]. This is justified by the fact that an accelerated electron radiates and the damping factor ν can be in principle related to the energy lost by photon emission. This modifies the solution [4.70] as r0 =

e E0 , 2 m (ω0 − ω 2 ) − iνω

[4.71]

due to the inclusion of a damping term −iνω. Such a kind of correction appears in analogous form in a Vlasov description of a plasma because of the Landau damping rate γL : by noting that 1 2(ω0 + iε) 1 − = 2 ω − ω0 − iε ω + ω0 + iε ω − ω02 − 2iεω + ε2

[4.72]

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we can indeed rewrite equation [4.71] as e E0 r0  m 2ω0



1 1 − ω − ω0 − iν/2 ω + ω0 + iν/2

 + O(ν).

[4.73]

This is of immediate comparison with the inclusion of the imaginary part of resonant poles in Vlasov problems. Figure 4.5 shows the typical behavior of the real part of resonances with (dashed lines) and without (continuous line) the inclusion of a damping factor.

0

ω





Figure 4.5. Typical dependence in ω of a resonance with (continuous lines) and without (dashed line) inclusion of a damping factor. In the left panel, the real part of ∼ (ω − ω0 )−1 . In right panel, the behavior of ∼ (ω − ω0 )−2

Let us then consider the case of the one-dimensional Vlasov–Poisson plasma studied in section 2. We can refer the s variable of Laplace transform to ωR and ωI according to equation [3.32], by writing s = iω = iωR − ωI . Looking for the Landau damping frequency corresponds then to looking for an ωI = −γL < 0 damping factor, which is small with respect to the oscillation frequency, |ωR |  γL . Using the notation already introduced for the parallel component of the response tensor in the one-dimensional Vlasov–Poisson problem (i.e. Λij (ω, k) → (s, k)), the identification of γL follows from expanding the dispersion relation (s, k) = 0 for (j) (j) γL /|ωR |  1 as (ω, k)  (ω, k)|ω=ω(j) − i γL R

  ∂ . (ω, k) (j) ∂ωR ω=ω R

[4.74]

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We obtain from this:

γL

   Im[(ω, k)]  = −  ∂ Re[(ω, k)]  ∂ω R

,

[4.75]

(j)

ω=ωR

(j)

(j)

(j)

where ωR (k) is the solution, exact up to the order O(γL /|ωR |), of the dispersion relation Re[(ω, k)]|ω=ωR (k) relative to the jth-resonance. The divergence of the latter is then formally removed because of the appearance of the Landau damping term, 1 1 . −→ ωR (k) − kv ωR (k) − kv − iγL

[4.76]

For infinitesimally small values of γL , equation [4.76] agrees with the Plemelj formula used to integrate around the pole: 1 1 ... dv −−−−−→ P P ...dv − iπδ(ωR − kv).[4.77] γL →0 ωR − kv − iγL ωR − kv The Landau damping coefficient, γL , corresponds to half of what is usually called the absorption coefficient of a longitudinal mode (half because the absorption is measured with respect to the energy decay, which is quadratic in the wave amplitude). The notion of absorption coefficient γAbs can be generalized by keeping into account a possible attenuation in space related to an imaginary component kI of the wave vector as (see, e.g. (Melrose, 1986, §2.5) for a discussion and some examples): γAbs ≡ −2(ωI − kI · vg (k))

[4.78]

The meaning, from the point of view of physics, of the role played by the Landau damping factor γL , as the result of the absorption of the electromagnetic wave by the plasma, becomes evident by a comparison of equation [4.76] with equation [4.73]. Instead, the actual physical mechanism with which this damping occurs in a collisionless plasma does not appear evident within the Vlasov mean field theory alone, and it is indeed still a matter of discussion in plasma physics. This is a long standing problem. As already mentioned in section 2.7, the incertitudes about this latter point have made several scientists doubt for a long time the effective relevance of the damping phenomenon predicted by Landau, until it had been experimentally measured by Malmberg and Wharton (1964), first, and later through the indirect experimental proof of its collisionless nature by the laboratory measurement of the plasma echo phenomenon (Gould et al., 1967; Malmberg

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et al., 1968). We have already mentioned in this regard the more recent result by Mouhot and Villani (see section 2.7), who have formalized from a mathematical point of view the role played by filamentation and phase-mixing in the large-scale irreversibility of the otherwise microscopically reversible Landau damping process (Mouhot and Villani, 2011). While this mathematical proof also solves some controversy about physics, related to the formal consistency of the Vlasov model, it still leaves some questions related to the physical origin of the collisionless damping phenomenon unanswered. The answers to these questions must probably be sought beyond the mean field Vlasov formalism, which goes beyond the purposes of the present volume. Nevertheless, we will dedicate a separate discussion to some of these elements in section 4.4. We conclude this discussion about the role of the coefficient γL in the resonant absorption, with a comment on resonances encountered in approximated dispersion relations of collisionless plasmas, in which the procedure to cure the divergence by means of Landau prescription cannot be followed (e.g. in fluid-type models). In these cases, the presence of a divergent resonance is however indicative, even if in a not quantitatively predictive way, of a condition in which the energy transported by the wave can be absorbed by the plasma. It is the example of the ion and electron cyclotron resonance mechanisms previously quoted, whose efficiency as a tool to heat a tokamak plasma can be predicted on the basis of the simplistic cold-plasma model discussed in sections 4.2.3–4.2.4. Then, in numerical simulations, an efficient dissipation mechanism can be introduced by adding some resistivity or fluid viscosity or by relying on numerical dissipation due to the truncation of the scheme and/or to spatial filtering. These fluid models, however, cannot account for fundamental wave–particle resonance effects, such as those that allow trapped particles in a potential well of a wave to sustain autocoherent nonlinear modes (“KEEN waves”) by opposing to Landau damping. Identifying these processes, which will be discussed in detail in Chapter 5, requires a kinetic description. For further reading about resonant absorption in plasmas, we recommend, for example, reading (Stix, 1992, Ch. 13). 4.2.6. Resonance and cut-off conditions on the refractive index We can now summarize the conditions of the resonance and cut-off of a wave on the refractive index based on the elements introduced so far in sections 4.2.1–4.2.5:

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– Cut-off condition: Nr2 = 0 This condition separates between the region of the medium in which Nr2 < 0 and the wave becomes evanescent by penetrating only for a small distance over which it is exponentially damped, and the region at Nr2 > 0 in which it can propagate, being diffracted. The angle of diffraction is larger than the incident but only if, according to Snellius–Descartes law [4.38]–[4.39], the refractive index in the medium is smaller than that of the medium from which the wave arrived. In the case of incidence from vacuum, such a condition can be satisfied by a plasma as long as its refractive index is 0 < Nr2 < 1. Since Nr2 = (c/vφ )2 = (kc/ω)2 , when approaching a cut-off condition the phase velocity of a wave of finite real frequency ω tends to infinity or, equivalently, its wave vector k tends to zero: vφ = ∞

and

k2 = 0

(at Nr2 = 0).

[4.79]

The conditions above have their correspondence on the conditions about the components of the transverse electric field and currents accordingly: because of [4.27], at cut-off conditions |B| → 0; therefore, from equation [4.16] and the second of [3.55], the density current J and the displacement current ∂t E compensate each other along each direction. Finally, based on the dispersion relation written as in equation [4.27], we see that the cut-off condition, that is, the existence of at least one root (ω/k)2 = 0 for at least one among the coefficents A or B, which is non-zero, is realized when det(Kij ) = 0. This means that the cut-off condition does not depend on the direction of propagation, since the determinant of the equivalent dielectric tensor does not depend on θ (Allis et al., 1963). We finally recall that reflection of a wave does not necessarily require Nr2 = 0 but it can also occur if just one component of k passes through zero and the other two keep on being fixed by the boundary conditions (see, e.g. (Stix, 1962)). – Resonance condition: Nr2 = +∞ The condition for which the electromagnetic wave is absorbed by the plasma according to one of the mechanisms discussed in section 4.2.5 is expressed in terms of the phase velocity and wave vector of a wave with finite frequency ω as: vφ = 0

and

k 2 = +∞

(at Nr2 = ∞).

[4.80]

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We see from equation [4.27] that the electric field must be purely longitudinal (|Et | → 0) and hence, from equation [4.16], |J |/|H| → ∞. The behavior of the refractive index close to a resonance is represented in Figure 4.5. Using again the dispersion relation written in the form of [3.16], we see that resonance is obtained when the coefficient of the highest power of the polynomial goes to zero. The resonance condition is therefore satisfied along the surface of a “cone of directions” defined by the angle of resonance that solves equation [3.17] for A = 0 (Allis et al., 1963). However, as k tends to diverge close to resonance conditions, inclusion of spatial dispersive effects can be expected to be important: they can indeed modify the solution of the dispersion relation in the presence of further roots. A discussion of these topics can be found in (Landau and Lifshitz, 1960, Ch. XII, §103). Further properties of resonance conditions, notably related to the polarization of plasma modes, are discussed in (Allis et al., 1963, Ch. 4). For a more thorough reading on cut-off, resonance conditions and absorption in plasmas, we address the reader to (Stix, 1992, Ch. 13). 4.2.7. Graphical representations of the dispersion relation The dispersion relation ω = ω(k) provides information on several quantities of physical interest. Different graphical representations, which provide useful synthetic information on some of these quantities at each time, are therefore possible. In particular, we have seen in section 3.2 that, in the most general case, a single privileged direction beside that of propagation can be identified. In these cases, it is useful to consider the dependence of ω on k and on the angle of propagation with respect to that direction, ω = ω(k, θ), which in the typical example of a magnetized plasma is the angle with respect to the background magnetic field B0 (Figure 3.1). We have, on the other hand, discussed in previous sections how the angle of propagation in an anisotropic plasma influences the number and properties of the existing branches. Even if this is not an issue for most of the electrostatic examples on which the remainder of this volume focuses, it is worth completing the discussion of this chapter, on the characterization of linear modes in plasma, by recalling and

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discussing their graphical representations, most frequently encountered in literature. The most typical representation of the dispersion relation is that of ω = ω(k) at fixed θ (or of ω(θ) at fixed k), which makes it possible to identify the resonances of the different branches existing for a given propagation angle. A well-known example is that represented in Figure 4.6, which shows the case of MHD-type modes.

Figure 4.6. Curves ω = ω(k) of MHD-type modes at fixed propagation angle θ

A complementary kind of representation displays the value of the phase velocity vφ as a function of the angle θ indicated in polar coordinates. Because of this, it is usually called a “polar diagram” or a “polar plot”: each branch is in this way represented by a closed curve, the angle θ varies as usual in the counterclockwise sense, and the value of vφ of a branch at a fixed propagation angle is expressed by the radial distance of the point on the line from the origin of axes. This diagram highlights the anisotropy of propagation in the medium, since a mode propagating isotropically would obviously correspond to circles centered around the origin of axes. Intersection between curves represents the critical values at which the merging (or emergence) of branches occurs. Immediate information about critical plasma parameters intervening in the different branches (e.g. those which concur to the value of Alfvén, or of sound velocity) is provided by the relative shapes of the

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The Vlasov Equation 1

corresponding curves. These curves, which in a 3D representation would be 2D surfaces, are usually called wave normal surfaces or phase velocity surfaces. A classical example is that of the low-frequency Alfvén modes in an isotropic plasma in the MHD regime (Figure 4.7): the eccentricity of the ellipses provides information on the dependence on the plasma β parameter, which is β ≡ 2μ0 P0 /B02 = c2s /c2A . Here, P0 and B0 are the total plasma pressure and magnetic field intensity at equilibrium, cs is the sound velocity and Alfvén velocity cA is defined by equation [4.67].

Figure 4.7. Polar diagram (wave normal surfaces) for Alfvén-type modes. The radial distance of a point on the curve from the origin gives the value of vφ of that mode for the angle of inclination ϑ with respect to the equilibrium magnetic field B0 . For a color version of this figure, see www.iste.co.uk/delsarto/vlasov1.zip

A further graphical polar-type representation, usually called “Friedrichs diagram”, displays the modulus of the group velocity |vg | as a function of the propagation angle (Figure 4.8). The latter allows a rapid visualization of the spatial dispersion of the mode (see, e.g. (Kadomtsev, 1979, Ch. 2, §1) for more detailed examples). We finally note that further diagrams, particularly useful to give a synthetic representation of the fundamental properties of modes existing in different regimes at the varying of the plasma parameters, can be obtained by displaying the value of a quantity of interest (i.e. ω, or k, or vφ ) as a function of two other ones, properly normalized along the Cartesian axes with respect to some characteristic plasma parameters (which depend, e.g. on plasma

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density, background magnetic field amplitude, plasma temperature and so on). The Cartesian coordinates of this kind of representation therefore define a certain parameter space.

Figure 4.8. Friederichs diagram for Alfvén modes: analogous to the polar diagram in Figure 4.7, it represents |vg | at the place of vφ . For a color version of this figure, see www.iste.co.uk/delsarto/vlasov1.zip

2D examples of this kind of diagram were first suggested by Clemmow and Mullaly (1955) and then by Allis et al. (1963), and are therefore usually named “Clemmow–Mullaly–Allis diagrams” (see, for example, (Stix, 1992, §1.7) or (Swanson, 1989, §2.2)), where a detailed discussion of these diagrams can be found. In these kinds of representations, a certain domain of the parameter space is subdivided into subspaces in which common features of the propagating modes can be recognized, for example by looking at common “shapes” of the wave normal surfaces or at the conditions for cut-offs and resonances. It is in this way possible to provide synthetic classifications of the normal modes in the plasma. For example, Allis et al. (1963) did so by representing the values of 1/ω 2 , 2 2 respectively, normalized to ωpe + ωpi on the x-axis and to Ωi Ωe on the y-axis. 2 2 They named (α , β )-diagram this kind of representation, having defined α2 = 2 2 (ωpe +ωpi )/ω 2 and β 2 = Ωe Ωi /ω 2 . Its practical utility is shown in Figure 4.9, where the conditions for cut-off and resonances are shown in the parameter

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The Vlasov Equation 1

space, and in Figure 4.3, in which the polar diagrams of different branches are represented. The two figures have been reproduced and adapted from (Allis et al., 1963) and (Buchsbaum, 1963).

Figure 4.9. “(α2 , β 2 )-diagram” representing the cut-off curves (dashed lines) and resonance curves (solid lines) of the principal waves in a cold magnetized plasma. Figure reproduced from Allis et al. (1963, ibid Figure 3.6, p. 33) and adapted. For a color version of this figure, see www.iste.co.uk/ delsarto/vlasov1.zip

4.3. Instabilities in Vlasov plasmas: general features

some terminology and

If the terminology about waves propagating in a plasma is varied and has undergone some changes over the years, it is nevertheless quite precise and well-established in comparison with the terminology encountered when dealing with instabilities. Many of them are present day research subjects: the nonlinear dynamics, the coupling with other modes as well as inclusion of

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kinetic effects and the extension to more complex geometries (e.g. non-homogeneous equilibria and/or curvilinear coordinate systems) have often required to wait for the progress in computational resources in order to be addressed. It is then easy to understand that some features that have been recognized as a criterion of identification or classification in some context may happen to be revised in another one and/or after a relatively short time, following the progresses of research that may have occurred in a time lapse of just a few years.






  
  


  
 




 
 


 

 
 

  

 

Figure 4.10. “(α2 , β 2 )-diagram” representing wave normal surfaces for waves in a cold magnetized plasma. Figure reproduced from Buchsbaum (1963, ibidem Figure 4, p. 14) and adapted. For a color version of this figure, see www.iste.co.uk/delsarto/vlasov1.zip

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The Vlasov Equation 1

It is also worth mentioning that cultural aspects should be kept into account, which are related to the way a certain instability or effect is termed in a subfield of plasma physics or by communities of scientists working on that instability in a specific context (e.g. in the context of transport theory of magnetically confined fusion plasmas rather than in the context of transport processes in astrophysical plasmas): this contributes to a scattered terminology that may appear as a real babel to a reader who first approaches the study of an instability by looking at the specialized literature represented by research articles. An example among all, which can be easily verified by a quick search on a scientific web search engine, concerns for example the usage of the term “Weibel instability”: nowadays used to indicate a quite large class of kinetic instabilities driven by an anisotropy in the velocity space distribution of charged particles, depending on the circumstances it is sometimes used to identify quite specific features, which in different contexts are recognized by other names (e.g. with different names of instabilities, like the “current filamentation instability” or the “whistler instability”) or with the addition of defining adjectives (e.g. time resonant, non-relativistic, magnetized, etc.). On the other hand, the opposite problem also exists, related to the terminology that has been acquired and is going on to be used for an instability that had first been dubbed with a “new” name, and only later it has been recognized as belonging to a more general class of instabilities with which it can be identified. After this caveat, we can however speak of some “firm ground” to be on. By inviting once more the reader to look for a more thorough discussion within specific books on plasma instabilities and plasma physics textbooks such as those indicated in the references, we discuss in this last section just a few key points concerning the general behavior and some classification criteria of plasma instabilities. 4.3.1. Linear instabilities Considering the linear mode approach which has been characterized in this chapter, the most natural notion to start with is that of “linear instability”. With this term, we mean a harmonic perturbation of the kind of [3.1], in which the imaginary part of the frequency corresponds to an exponential growth. The linear mode analysis is usually based on studying perturbations of an equilibrium configuration. In the case of linear steady instabilities, this “equilibrium” is stable and some instability criterion must be satisfied in order for the normal mode perturbation to start growing.

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The onset of the instability is associated with some threshold condition, which for spontaneous modes is often related to the presence of spatial inhomogeneties and gradients of quantities, but can also be related to geometrical features (e.g. the aspect ratio of a current sheet, which influences the number of unstable modes in tearing-type reconnecting instabilities), to some matching conditions between the parameters (e.g. in the case of parametric instabilities) and so on. As a rule of thumb, in spontaneous instabilities, a source of “free energy” can typically be identified, which provides the reservoir for the growth of the mode: the instability indeed occurs when a “displacement” induced by a perturbation on an unstable equilibrium is amplified by a force associated with a favorable change (i.e. a decrease) of the related potential energy. The growth of the mode then normally implies a conversion of energy from one component to another: for example, from electrostatic or magnetic to kinetic energy of the particles, which can be disordered (thermal motion) or “ordered” (particle beams and currents), etc. In some cases, an energy principle can be used to find conditions on the polarization of the perturbations and/or on the geometry and spatial inhomogeneity of the equilibrium configuration in order to assess its (in)stability with respect to a certain mode. It in general consists of individuating an energy functional for the whole system, let us say W (x), where x is a “vector” related to all the different quantities of the system, which can be varied because of the perturbation (e.g. components of the electromagnetic fields, perturbation of fα , or perturbation of the components of the other fluid moments, if a fluid-type approximation is used). One then evaluates the sign of the dominant variation of W after a perturbation δx. Around an equilibrium point, the dominant variation δW is necessarily quadratic in δx. The instability condition is then met when 1 δW  2



∂2W ∂xi ∂xj

 δxi δxj < 0,

[4.81]

which corresponds to a decrease in the potential energy. A general feature can be further deduced by these arguments: since the unstable mode needs some kind of energy to grow, in a closed system it is more likely that the most unstable configurations are those in which the redistribution of the free energy among the other stable modes is minimal. This is the reason for which in tokamaks, for example, disruptive MHD-type instabilities like reconnecting modes are more likely to occur on resonant

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The Vlasov Equation 1

magnetic surfaces, since it is here that the energy conveyed to the excitation of (stable) Alfvén waves is minimized11. Several examples of (in)stability criteria based on the principle of the minimization of energy [4.81] can be found in plasma physics as well as in classical mechanics and hydrodynamics. It is worth mentioning in this regard the importance of this kind of approach to investigate the stability of magnetic equilibrium profiles in tokamaks and laboratory devices (see (Wesson, 1987)). Based on a fluid-type description of the collisionless plasma, this stability study is a key step for the design of devices for plasma magnetic confinement, in particular with application to thermonuclear fusion experiments. In other cases, combining this search of the extremals of the energy with some constraints related to quantities (typically integral ones), which are expected to remain constant or at least to vary negligibly with respect to the involved potential energy, makes it possible to determine the final states toward which the system relaxes after an instability. An important example of this approach, whose agreement with experimental evidence has been successfully verified, comes once again from magnetic confinement in tokamak plasmas, and is related to the process known as “Taylor’s relaxation” (Taylor, 1974). The modeling based on extremal energy-type principles, however, cannot follow the dynamics of the instability but just provides some information on its initial and final states. If one wants to follow the whole dynamics of the linear instability, from its onset to its “end”, that is its saturation, a numerical approach is normally required. Figure 4.11 summarizes the typical stages of a linear instability in a plasma by visualizing as a function of time the logarithm of the amplitude of one of the growing quantities (all quantities in the figure are normalized to some reference value which defines the “unity” value for the equilibrium system).

11 We recall that, in a tokamak, a resonant surface of radius r = rres is formally identified in toroidal geometry by the condition |q(rres )| = m/n, where m and n are integer numbers and q(r) ≡ Δϕ/(2π) expresses the number of toroidal turns after which a magnetic line laying on a toroidal surface of radius r closes on itself. Due to the magnetic configuration in the torus, it turns out that q(r) is well-approximated by q(r) Bϕ (r)r/(Bθ (r)R), for R being the major radius of the tokamak and Bϕ and Bθ being the toroidal and poloidal magnetic field components, respectively: the condition B(r) · k(r) = 0, which rules out the polarization of Alfvén modes, is then trivially satisfied on a resonant surface r = rres by a perturbation with wave vector k = m/(2πr)eθ + n/(2πR)eϕ , written with respect to the toroidal coordinate vectors er , eθ and eϕ .

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The different blue and green curves correspond to different components of the Fourier spectrum with respect to a single (periodic) spatial coordinate. Since these are harmonic waves oscillating in space along such a coordinate, we can identify them in terms of the number m of spatial oscillations in the corresponding interval of the simulation box, which we can say to have extension L. Each wave number is therefore k=

2πm . L

[4.82]

Figure 4.11. Typical behavior in time of the Fourier modes of an (absolute) linear instability. For a color version of this figure, see www.iste.co.uk/delsarto/vlasov1.zip

The blue curve corresponds in particular to the most unstable mode, which in the example has m = 1 oscillations in the box: having been excited since the beginning of the simulation by adding it as a small amplitude eigenmode to the equilibrium profile, it starts with an exponential growth. Its growth rate γ(k) is given by the slope of the “straight” portion of the line. The green curves represent Fourier components with different values of m. In this particular case, the components with m = 2, 3, 4, 5 are represented; the m = 2 component displays what appears as a “two-step” linear growth. The first part is due to the exponential growth of the m = 2 eigenmode, which in this problem results as linearly unstable, too, even if it has a growth rate

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The Vlasov Equation 1

γ(m = 2) much smaller than γ(m = 1). Then, at a later time (about t  7.5 in the simulation units), the m = 2 Fourier component abruptly changes slope: this is because its spontaneous growth mixes with that of the second harmonic of the m = 1 mode, which, as a nonlinear effect, grows at twice the rate γ(m = 1). In general indeed, given a linear mode of the kind of equation [3.1] with ω = ωR + iγ, which we rewrite here (assuming the small amplitude ε to be a real number) as   ∼ ε ei[k(m)x−ωR (m)t] + e−i[k(m)x−ωR (m)t] eγ(m)t ,

[4.83]

at the N th-order nonlinear coupling with itself it will have also generated an N th-order harmonic, corresponding to a mode oscillating both in time and space N -times faster than the original one. Since N k(m) = k(N m), we can write it as   ∼ εN ei[k(N m)x−N ωR (m)t] + e−i[k(N m)x−N ωR (m)t] eN γ(m)t . [4.84] Despite the much smaller starting amplitude (∼ εN ), this harmonic will grow exponentially, N -times faster than the original mode m: γ(N m) = N γ(m). This explains the appearance, in the graphics of the figure, of the green lines after t  8, as well as the fact that the growth rate 2γ(m = 1) of the first harmonic of the m = 1 eigenmode soon overtakes the spontaneous growth γ(m = 2) γ(m = 1) of the m = 2 eigenmode. The red line represents the logarithmic time evolution of another quantity, which is used here as an alternative measure of the instability growth rate: this quantity, as it typically happens in experimental measurements, depends on the contributions of all the components of the Fourier spectrum. For example, it can be a component of the magnetic field (as in this specific example which concerns a “tearing mode” instability; (see Del Sarto et al. (2011) for details), of which the blue and green lines correspond to some Fourier components along a coordinate. Often, however, the quantity which is more easily measured in order to provide an observational estimate of the growth rate is the energy: having a quadratic dependence on the field amplitudes, it grows at about twice the rate, at least, of the fastest growing mode, which has been excited (in reality, of course, it usually grows faster because of the nonlinear contributions of the other unstable modes or harmonics). In general, the definition of the growth rate of an instability may depend on circumstances and measurement limitations, since the formal definition given above in terms of the imaginary

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part of the frequency may not be of practical use if a Fourier analysis of the perturbation is not feasible. In the case exemplified in the figure, however, the agreement between the growth rate of the dominant mode γ(m = 1) (dashed line) and that estimated by the superposition of the Fourier modes (red curve) are in good agreement until the amplitude of the m > 1 components of the spectrum has become comparable with that of the dominant mode: it is the beginning of the nonlinear stage, which is followed by the saturation, in which the growth rate of the instability tends to zero, the system has achieved a new macroscopic state, and new normal modes may be excited or even become unstable (secondary instabilities). For practical use in the interpretation of nonlinear simulations of instabilities, the different stages of the evolution can be summarized as follows: – Transient It is the initial stage of an instability, which takes place when a mode different from the unstable eigenmode has been excited. An intuitive example is that of the tuning fork: when it is set vibrating by striking it on a surface, quite a large amount of energy can be transferred to it, but the human can perceive, nevertheless, the finite time interval between the strike and the hearing of the typical “A tune”. When, instead, it is brought close to another emitter of sound vibrations at the 440 Hz frequency, one recognizes it starts vibrating at the same frequency almost immediately. This is analogous to the cases we are interested in. The occurrence of a transient time is a problem that especially concerns numerical simulations, since this “error” in the initialization, if unwanted, implies that the unstable mode starts with a much smaller amplitude than was expected – namely, such an effective amplitude is that of the component that the excited perturbation has on the unstable mode, once the former is projected on the basis of the normal modes of the system. Then, the smaller the initial amplitude, the longer the time needed to “see” the effects of the instability on the equilibrium. If we consider the limit case in which the initial perturbation had been entirely put on a mode orthogonal to the unstable one, no instability is formally to be expected. Nevertheless, in a numerical simulation the instability will manifest after a sufficiently long time: it will start with a perturbation with a “zero-machine” amplitude or with an amplitude related to the nonlinear harmonics of the excited mode projected on the unstable wave, depending on which is larger. Of course, letting instabilities grow from a “white noise” may be desirable, since it can more closely resemble that which happens in nature, but in this way the

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length time of simulations, which are typically already very expensive in terms of computational resources, may increase considerably. If one wants to avoid the problem, in the cases in which the analytical profile of the eigenfunction is not available, a practical alternative to a numerical eigensolver is, for example, to use the output of a first explorative evolutive simulation, stored before the linear instability entered the nonlinear stage, to obtain the eigenmode profile to be used to initialize a second simulation.

Figure 4.12. An example of transient for the same instability obtained with two different almost indentical numerical setups: despite the formally equivalent initialization of the perturbation, a slight temporal mismatch is observed between the two curves because of the different transient in the two numerical models (a 1D2V and a 2D2V semi-Lagrangian Vlasov–Maxwell solver), which have been used. The plot displays the time evolution of the magnetic energy density associated with the magnetic field component generated by a Weibeltype instability (Ghizzo et al., 2017). During the nonlinear stage, after saturation (tωp ≥ 60), discrepancies between the two numerical simulations become evident also from the plot of the represented amplitude

A further example of transients is displayed in Figure 4.12, which shows the growth in time of a (squared) magnetic field component, amplified by a Wiebel-type instability (the figure refers to a numerical study of (Ghizzo et al., 2017)). The two curves correspond to two numerical simulations run with a 1D-2V and a 2D-2V version of the same Vlasov–Maxwell solver, initialized with identical perturbations (the considered instability in the linear stage is indeed dominated by 1D dynamics): a slight temporal mismatch for the otherwise almost identical linear evolutions is measured because of the different transient times in the two runs. The different transients are due to the

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fact that the initial perturbation chosen is not an exact unstable eigenmode of the system, and the energy on such an initial perturbation is not equivalently redistributed among the eigenmodes of the 1D-2V and 2D-2V numerical models during the transient time. – Linear stage It is the time interval in which the numerical results are coherent with the linear analysis performed. This means that: - the instantaneous amplitude of the perturbation must still be “small” with respect to that of comparable equilibrium quantities; - the amplitude of the harmonics must be “smaller” than that of the driving mode taken at the same time; - a time evolution of the kind ∼ e−iωt+γt is observed for the unstable mode. The fact that linear analysis is based on the asymptotic limit of formally vanishing perturbation amplitudes poses the problem of how to quantify the adjective “small(er)” in the above statements. As rule of thumb, two orders of magnitudes in the differences of amplitude should be required. Depending on the quantity that is considered, the oscillating part (∼ eiωt ) of the time evolution may not appear as visible in the plot if the time oscillations are averaged to zero because of an integral in space. On the other hand, when amplitudes including the contributions of more than one oscillating Fourier components are plotted, oscillations can manifest as a result of wave beating between different modes. – Nonlinear stage Defined as the stage in which the linear analysis formally breaks down, its distinction from the linear stage becomes effectively “appreciable” when consequences of nonlinearity appear in the evolution of quantities: the first, most evident manifestation is usually the fact that the equilibrium configuration starts being sensibly modified by the perturbation. Another effect, quite easily recognizable, can be the break of the spatial symmetries that were respected by the linear set of equations and by the imposed boundary conditions. Such a symmetry break is allowed by the nonlinear amplification of perturbations that violate the initial conservations. This may possibly be associated with the onset of secondary instabilities and/or with the excitation of further

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normal modes (i.e. waves), which may sometimes be completely “new” (i.e. incompatible with the initial configuration), being allowed by the modification of the equilibrium. The nonlinear stage may also lead to turbulence, selforganization processes and nonlinear and secondary instabilities. In general, however, as soon as nonlinear perturbation has grown enough, the remainder of the evolution of the system can be considered to occur within the nonlinear stage of the primary instability. Of course, the initial amplitude of the perturbation may influence how long the instability will take before entering the nonlinear phase (and before eventually saturating −see below). – Saturation This is the part of the nonlinear stage in which the unstable mode m ends its growth as γ(m) → 0. This is a general occurrence of instabilities that are not externally forced, since the free energy of a closed system is finite. As the growth rate diminishes, periodic oscillations may appear in the time evolution of the amplitude of the unstable modes (Berk et al., 1996). Note that nonlinear oscillations can also appear as a result of the energy exchange between the de-phased components of a multicomponent system. This, too, is a quite general feature that concerns linear instabilities (see, e.g. (Galeotti and Califano, 2005)) but also other unstable processes, for example, the merging of two current bundles advected by a flow (Bergmans and Schep, 2001). With regard to the nonlinear stage of a linear instability, nothing prevents the system, which has reached saturation, from evolving with new instabilities or selforganization processes that can lead to the conversion between other forms of energy. Many recent research studies in plasma physics are devoted to the prediction of the saturation stage of primary instabilities, as they can both represent the final state of a plasma or a “metaequilibrium” from which it can further evolve. 4.3.2. Absolute and convective instabilities and some other classification criteria Let us now discuss just a few other general criteria of identification that can be related to, or go beyond, the class of linear instabilities just considered. What we termed “linear instability”, based on harmonic-type perturbations, represents just a part of the family of plasma instabilities. For example, in this chapter we assumed the unperturbed plasma state to be that of a non-evolving equilibrium (i.e. stationary, or at least steady). In principle,

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however, the instability of small amplitude perturbations can be studied by relaxing this hypothesis. Otherwise, a still steady or stationary equilibrium can be considered, which is perturbed by finite amplitude perturbations. We speak of this latter case of nonlinear instabilities. These can be related to the further notion of subcritical instabilities, which are destabilized only when the amplitude of the perturbation trespasses a certain threshold (for an introductive tutorial on these instabilities in plasmas, the reader could look at (Lesur et al., 2018)). Furthermore, an external drive may put the system in a condition, which is not of equilibrium (e.g. a strong laser radiation generating a plasma by target ablation): this evolution from an intrinsically unstable configuration can allow some quantities to be amplified over time even by growing from zero – this is a key difference with respect to unstable harmonic perturbations, which require an initial finite amplitude to grow, even if it is “infinitesimally” small. Examples of these “growths from zero in unstable configurations” are provided by the Biermann battery mechanism for the generation of magnetic fields (Biermann, 1950) or by the shear-induced anisotropization mechanism for the generation of pressure anisotropy (Del Sarto et al., 2016), which we will speak of in the second volume. Then, we can speak of forced instabilities, which are driven or allowed by some external force such as currents, i.e. charged beams, which would intervene as a source term at right-hand side of equation [3.3], expressed in equation [3.44] as a Ji,s contribution; neutral beams injected in the plasma12, which, even if they do not induce electromagnetic perturbations directly, may do so indirectly by changing the plasma pressure profile; high-intensity light radiation, which can accelerate electrons via ponderomotive effects or lead to absorption–emission processes; and so on. Even the linear instabilities we have considered so far can change behavior if they are externally forced. Moreover, most instabilities, linear or not, can become “spatially resonant” because of spatial inhomogeneities, which we have mostly neglected in the previous discussions on the linear mode analysis. We have touched in passing the topic of the role of spatial inhomogeneities in wave propagation while speaking of the refractive index of the ionosphere by noting that it varies with the altitude because of (among other effects) the dependence on the plasma density. Since the particle density enters in most of the characteristic plasma parameters that can be related to a velocity (e.g.

12 We recall that neutral injection is a technique of plasma heating in tokamaks – see footnote about ICRH and ECHR in section 3.3.

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Alfvén velocity), to a time scale (e.g. plasma frequency), or to a characteristic length (e.g. electron skin depth) that may be involved in a resonant condition [4.72]–[4.73], it follows that such a resonance will be localized in space. An example of such a kind of phenomenon is provided by the current filamentation instability made spatially resonant due to the presence of an initial inhomogeneity in the counterstreaming electron beams (Califano et al., 1997, 1998, 2006). Besides, while spatial inhomogeneities provide the driving mechanism of several instabilities (see (Mikhailovskii, 1974)), the propagation of stable modes can be radically modified in their presence. For example, modes containing compressional components can be generated by the scattering of transverse modes on plasma inhomogeneities (Kaghashvili et al., 2006; Kaghashvili, 2007). We also recall in this regard that, as mentioned at the beginning of Chapter 3 (see comments below equation [3.1]), under certain hypotheses a linear analysis can be performed even for propagation along the inhomogeneous direction. For further information on these topics, we refer the reader to Swanson (1989, Ch. 9). Not having the pretention to be exhaustive about the topic of the classification of the instabilities, we conclude this chapter with two fundamental classification criteria that have quite general application: that of absolute and convective instability and that of macro- and microinstability. Let us start with the distinction between absolute and convective instabilities. Thinking of initial value problems, in which a wave with an imaginary ωI is found, one could have noticed that an ambiguity may arise due to the possibility of changing the reference frame, by choosing one which is co-moving at the wave phase velocity, so as to find a null real frequency of the wave, ω(R) = 0. This type of instability would make sense: it is a perturbation that exponentially grows in amplitude and does not oscillate in time; just in space. Incidentally, it is exactly the behavior of the tearing-mode instability shown in Figure 4.11: had a time oscillation been present, the slope indicating the linear stage would have not been “straight” but would have displayed an oscillatory modulation, instead. On the other hand, we can expect that both ωR and ωI differ from zero. This leads to the distinction between “convective” and “absolute” instabilities – see (Briggs, 1964, §1–2): – A linear instability is termed convective with respect to the reference frame in which it propagates in time while growing, if this occurs in such a way that the perturbation disappears if one stands at a fixed point. This corresponds to have ω = ω(R) + iω(I) with ω(R) = 0 for a real wave vector in section 3.1.

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– A linear instability is termed absolute with respect to a reference frame in which the involved amplitudes grow (or damp) in time at every point in space. This corresponds to the non-propagating case with ω = iω(I) and ω(R) = 0 for a real wave vector.







Figure 4.13. Typical behavior of a convective (left frame) and an absolute instability (right frame). The two plots display the amplitude of a growing envelope with respect to space (x-axis), the different curves corresponding to different times (t1 < t2 < t3 )

Any convective instability appears to be absolute in the reference frame, which is co-moving with the wavefront. However, if we look at the behavior of a wave packet in the chosen reference frame (see Figure 4.13), an a priori fundamental distinction based on physical arguments, can be made between the two types of instability. If we consider an unstable wave packet with a finite spatial spread, it corresponds to a convective instability if it travels so fast that the amplitude of the perturbation tends to zero as t → ∞ in every point of space. Instead, the wave packet can be identified as an absolute instability if, regardless of its propagation, the amplitude of the perturbation grows indefinitely in time as t → ∞ (supposing, of course, that the linear modeling holds indefinitely in time). Given a “preferred” reference system in which the instability is studied, its identification as convective or absolute can then be made by studying the sign of the imaginary part of some “critical pole” of the dispersion relation, ωc , for which the group velocity goes to zero. Details about this procedure and its applications can be found, for example, in (Briggs, 1964) or (Stix, 1962, Ch. 9) and (Landau et al., 1981, Ch. VI, §62). As discussed in (Stix, 1962), a simple formulation of this procedure of identification has been shown by Feix (1963). It was developed to analyze the exponential growth of a Gaussian envelope of the form of equation [4.32] under a two-stream instability. In particular, let us consider the 1D limit (say

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E(r, t) → E(z, t)) of such a wave packet, where k0 is chosen so that ∂ Im(ω)/∂k|k=k0 = 0 in the reference frame moving at the group velocity of the envelope (i.e. ∂(Re(ω))/∂k = 0). This makes it possible to approximate ∂2ω 1 ω(k0 )  ω0 + (k − k0 )2 2 , 2 ∂k by retaining the second-order term of the expansion in |k − k0 | 1. This expression should replace equation [4.31] used to obtain equation [4.32]. After some algebra related to the expansion of [4.32] in powers of |k − k0 | 1, the requirement that the amplitude E(z = −vg t, t) exponentially grow with time leads to the condition: ⎛ ⎞ ⎜ Im (ω(k0 )) > Im ⎝

vg2 ⎟ ⎠. ∂ ω(k0 ) 2 ∂k 2

[4.85]

Equation [4.85] is the condition for an absolute instability in the reference frame co-moving with the Gaussian wave-packet, but it also holds in identic form for ω(k0 ) measured in the laboratory frame due to the difference by a simple Doppler shift, real term k0 vg . Even if the distinction between convective and absolute instabilities may appear as artificial, as far as the choice of the reference frame from a mathematical point of view is concerned, it is not so from an experimentalist perspective: knowing the nature of an instability in the laboratory frame is important for the kind of diagnostics that must be implemented – think, for example, of a detector that is too far from the line of propagation of a convective instability. Concerning, finally, the distinction between macro- and microinstabilities: of course, kinetic effects dominate the dynamics of Vlasov plasmas that we are interested in. Nevertheless, Vlasov plasmas are affected by some instabilities in which kinetic processes, meant as due or related to the “fine-scale” evolution of the distribution function in phase space, play a somehow secondary role. A fluid-type, “large-scale” description based on just a few moments of the Vlasov equation (possibly “extended” to include some microscopic effects) may suffice for them, at least until scales too small are developed during the nonlinear evolution, which makes a kinetic treatment mandatory. There are other instabilities, instead, for which information about the specific shape of the distribution function is needed, since they strongly depend on them. This leads to the two following quite broad categories (see, e.g. (Cap, 1976, 1978,

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1982; Melrose, 1986) for specific examples of instabilities classified in these terms): – Macroinstabilities These are the instabilities which are driven by some features of the medium that are related to its configuration in the physical space and do not require specific information about the details of the particle distribution function. In practice, they are usually related to (or simply driven by) spatial gradients of macroscopic quantities and only need information about the fluid moments of the distribution function. Typical examples are fluid-type or magnetohydrodynamic instabilities. – Microinstabilities (or kinetic instabilities) These are the instabilities that are usually driven by anisotropies in momentum of the particle distribution function, and which we could therefore also term “kinetic instabilities”. The adjective “micro-” is due to the fact that, of course, kinetic effects of this kind are related to microscale phenomena. Regarding this latter distinction, it is however worth recalling that collisionless plasmas are prototypical examples of multiscale systems, so that, even if the distinction between micro- and macro- given above is meaningful as far as the driving mechanism is concerned, it does not hold anymore when we are interested in the nonlinear evolution of the instability. As a general rule, the lack of a dissipation scale makes practically all instabilities in collisionless plasma induce couplings between well-separated spatial scales. Small scales are rapidly generated from the macroscopic scales (i.e. even starting from “small” values of k) because of nonlinearities (see equation [4.84]). In this sense, an unstable mode that can be a priori classified as a macroinstability can turn out to be strongly affected by kinetic dynamics once the latter is taken into account; this occurs especially as far as the saturation mechanisms are concerned. Moreover, it can happen that instabilities driven by gradients of macroscopic quantities also require information on kinetic features, like an accurate description of wave–particle resonances: an example is provided by “trapped ion modes” (TIM), which are kinetic-type microinstabilities which can be regarded as a subclass of ion temperature gradient (ITG) instabilities (Depret et al., 2000): the latter, as long as wave–particle interactions result to be negligible, can be formally classified as macroscopic instabilities of interchange-type, many of which, such as the Rayleigh–Taylor instability, play a fundamental role in fluid models described by MHD and/or Navier–Stokes-type equations.

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On the other hand, changes induced by microinstabilities have an immediate consequences on large-scale plasma dynamics, for example because of inverse cascade processes related to the reorganization of fluid enstrophy and vorticity (Hasegawa, 1985), or otherwise by simply changing the topological configuration of the magnetic field13 in the plasma (examples are provided by Weibel-type instabilities (Weibel, 1959) or by reconnecting instabilities driven by kinetic effects (see, e.g. (Cai and Lee, 1997; Daughton, 1999)). We will consider explicitly in the second volume some of these microinstabilities and the rather rich physics related to the coupling of kinetic effects with the dynamics of plasma particles in the presence of magnetic fields. For the remainder of this volume, particularly in Chapter 5, we will keep on considering electrostatic effects in Vlasov–Poisson systems. We will see how several nonlinear properties of electrostatic collisionless plasmas, like the existence of nonlinear self-coherent states or the onset of nonlinear electrostatic instabilities, can be strongly affected by plasma eigenmodes through wave–particle resonances. 4.4. On some complementary interpretations of the collisionless damping mechanism in Vlasov plasmas We conclude this chapter with a complementary section, in which we discuss some further conclusions on the origins of the Landau damping mechanism, which can be drawn if the phenomenon is investigated by looking at it from perspectives that go beyond the Vlasov mean field formalism. From a formal point of view, the collisionless damping predicted by Landau is indeed a rather universal phenomenon expected to generally occur in Hamiltonian systems with long-range interactions: stars in a galaxy (Lynden-Bell, 1967), neutrinos in dense stellar plasmas (Silva et al., 2000), even fluids and possibly some biological systems (see (Ryutov, 1999) for further references) and, in general, any continuum system in which quasi-particle dynamics can be described by Vlasov-type equations, such as photons (Blaizot and Iancu, 1996; Bingham et al., 1997). 13 We recall that charged particles tend to spiral along magnetic field lines: if the collision rate is low, as in plasmas, magnetic lines constitute “preferred routes” for charged particles; changing their large-scale structure modifies the connection between fluid elements of the plasma.

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Focusing here, for obvious reasons, on the case of electrons in a Vlasov plasma, it is interesting to mention two complementary points of view on the subject of the physical source of the collisionless damping mechanism, which are based on quite different arguments, which go beyond the sheer Vlasov mean field approach and which shed light on it by underlining two features of the damping phenomenon. – The first point of view suggests identifying the Landau damping mechanism, at least for electrostatic modes, as an inverse Vavilov–Cherenkov emission process (Ginzburg and Zheleznyakov, 1958). This interpretation is based on heuristic arguments related to microscopic reversibility and on the formal equivalence, in modulus, between the electrostatic Landau damping rate and the coefficient of emission of Langmuir waves produced by electrons moving faster than their phase velocity (see (Melrose, 2017) for an up-to-date discussion). – The second point of view, based on a self-consistent N -body description of the resonant particles that intervene in the Landau damping and in the inverse Landau damping resonance of the weak beam-plasma instability, demonstrates that collisionless damping corresponds to the exponential relaxation of a superposition of Langmuir oscillation toward their thermal level, because of the synchronization of particles with waves (Elskens and Escande, 2003; Escande and Elskens, 2003). This approach proves some limitations in the description of collisionless damping, as it is obtained from Landau’s prescription in the Vlasov formalism, which has already been mentioned in section 2.7: these are related to the approximations contained in the passage from N -body dynamics of the resonant particles to their statistical continuum description in the mean field N → ∞ limit. 4.4.1. Landau damping as an inverse Vavilov–Cherenkov radiation This explanation of the collisionless damping mechanism for electrostatic modes in plasmas is suggested by the interpretation of the role of Landau damping, which we have previously discussed in terms of equations like [4.73]. In particular, it provides an interpretation of the mechanism of energy loss by photon emission, which we mentioned just before equation [4.73], that is specific for electrostatic modes in plasmas. This interpretation is grounded in consistency arguments with the second principle of thermodynamics, which requires, for any spontaneous emission mechanism, a corresponding absorption mechanism to exist (otherwise an indefinite accumulation of energy in the form of electromagnetic radiation

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would be allowed by violating Clausius’ statement). A well-known form of this correspondence is, for example, encountered at a microscopic level in atomic physics, in which the assumption of detailed balance is done14 in order to grant thermodynamical equilibrium for an atom interacting with an electromagnetic field: the equivalence of the ratio of Einstein’s coefficients of spontaneous absorption and emission with the ratio of the degeneracies of the two energy levels involved in the electron transition follows from this. An analogous argument can be applied at a macroscopic level, in a medium displaying collective properties (namely the existence of waves), to the mechanism known as Vavilov–Cherenkov emission15 or radiation. A general theoretical framework of the phenomenon isolated by Cherenkov had been provided by Tamm and Frank (1937). They pointed out that particles moving faster than the wave front of an eigenmode in a certain medium emit waves in that specific mode. This concerns both material particles, which can excite, for example, surface waves in deep water (e.g. the wake of a ship) as well as sound waves in the air (which led, e.g. to the acoustic “bang” generated by a supersonic aircraft), and charged particles, which this way can excite “electromagnetic” perturbations in a plasma. Cherenkov emission occurs then in the form of excitation of a specific mode ω = ω(k), if the particle velocity overtakes the phase velocity vφ = ω/k = c/Nr . The condition v > vφ , rewritable as v > c/Nr , does not violate any relativistic constraint as long as Nr > 1 for the frequency and wavelength of that specific mode. In this way, energy is transferred from the particles to the wave: as a consequence of Vavilov–Cherenkov emission, particles with v > vφ will slow down by transferring their energy to the normal mode: for typical “bump-on-tail”-like configurations with a small population of particles near a resonance condition, this would mean an increase in the steepening of the distribution function on the right of the resonant value v = vφ (see Figure 4.14 and see also the discussion in section 2.7), which is opposite to the behavior related to the linear Landau damping mechanism. This occurrence

14 This is related to the microscopic reversibility assumption on the basis of Onsager’s relations −see section 3.5.1 15 The name is after Sergey Iovanonvich Vavilov (1891–1951) under whose supervision Pavel Alekseyevich Cherenkov (1904–1990) identified, during his PhD research, the bluish glow of water containing radioactive salts as being unrelated to fluorescence (Cherenkov, 1934; Vavilov, 1934). The observational phenomenon itself had already been noted or predicted by other scientists, even if this went substantially unnoticed by the scientific community before Cherenkov’ work – see e.g. Ginzburg (1996) for a brief historical account.

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has been already discussed in section 2.2.3 for the two-stream instability. In the limit in which the number of particles of one of the two beams is much smaller than the other, so that it can be visualized as a small “bump-on-tail” of the largest distribution – see Figure 4.14, right frame – an inverse Landau damping mechanism can take place. In this case, the two-stream instability is usually termed bump-on-tail instability and its linear growth rate can indeed be shown to correspond to an “inverse Landau damping” coefficient, being the former equal in modulus and opposite in sign to the γL previously introduced (see section 2.7)). Going further with this point of view, it has been noted by Ginzburg and Zheleznyakov (1958) that the absorption coefficient of a Langmuir wave on a plasma due to an inverse Vavilov–Cherenkov emission perfectly matches the collisionless Landau damping coefficient.







Figure 4.14. Typical configuration of a beam plasma instability (left frame). When the second particle population is microscopic with respect to the other one, we speak of weak beam-plasma instability. The latter also takes the name of “bump-on-tail” instability, if the bulk velocities of the two distributions are separated enough (right frame)

This suggests that the collisionless dissipation coefficient γL , obtained by Landau as the imaginary part of the pole of the dispersion relation, be physically identifiable as (half of) the “absorption coefficient” of the inverse Vavilov–Cherenkov-type emission mechanism. In particular, the condition for Vavilov–Cherenkov emission, and therefore of absorption, coincides with the resonant requirement ω − k · v = 0,

[4.86]

which corresponds indeed to the resonant denominator of the Landau problem for electrostatic modes in plasmas. For emission, equation [4.86] defines the

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“Cherenkov angle” Θ at which the modes radiated by this mechanism form a frontwave with respect to the direction of motion of the particle16, cos Θ =

ω . kv

[4.87]

Since Cherenkov resonance is particularly efficient for electrostatic modes, in which electrons are accelerated along k (Θ = 0 above), this point of view would provide a physical interpretation for why collisionless Landau damping corresponding to the resonance of equation [4.86] is particularly important at purely parallel propagation, i.e. for purely electrostatic modes. Moreover, for these modes the condition Nr > 1 required by Cherenkov emission in plasmas is generally satisfied. Resonant condition [4.86] does not concern, instead, the purely electromagnetic branches, and we had previously seen (section 4.2) that these can have Nr < 1, which would invalidate, indeed, the Cherenkov condition v > vφ . For these branches, cyclotron resonance conditions of the kind of [4.64] replace that of Vavilov–Cherenkov emission (and absorption) expressed by equation [4.86]. Notably, it can be shown (see, e.g. Landau and Lifshitz (1960)) that only values of the integer n with a sign such that nqα > 0 make it possible to satisfy the resonance condition [4.64] when Nr < 1. This also allows a simple interpretation of the absorption/emission condition related to cyclotron-type resonances: exchanges of energy (absorption or emission) between charged particles in plasmas and cyclotron-type waves can occur only when the sense of rotation of the circularly polarized wave along the direction of k matches the sense of rotation around B of the particle species with which it resonates, once the rotation sense of the latter is projected in the k direction: in this way, the particles will stay in phase with the rotating electric field of the wave during a whole gyration period. For a more thorough reading of the relation between absorption, Landau damping and Cherenkov radiation, we suggest the reader to look at books by Melrose (1986, §1.3.) and Somov (2000, Ch. 5) and especially to review articles by Ginzburg (1996) and Melrose (2017). A detailed analysis of the estimation of Cherenkov emission in plasmas can be found in several earlier research articles (see, e.g. Cohen (1961) and references therein).

16 An example that is easy to visualize is that of a ship moving in deep water: if the dispersivity of the surface wave was negligible, the Cherenkov angle Θ would correspond to half the angle of opening of the characteristic triangle-shaped wake behind the ship.

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4.4.2. Landau damping in N -body “exact” models Complementary elements related to the physical interpretation of the Landau damping mechanism manifest when the process is investigated beyond the Vlasov–Maxwell formalism. This may be done in some electrostatic problems by considering the exact N -body “Newtonian” dynamics applied to the particles, which are involved in a bump-on-tail-like resonant interaction, which occurs in the velocity space close to the tails of a distribution function (see Figure 4.14). This localization in the velocity-space makes such an approach numerically affordable, because of the fact that the number of particles involved, N , is relatively smaller than that involved in the mean field dynamics, which rules the remainder of the particle distribution. Some elements of this kind of approach have been partially discussed in section 1.6 in relation to the numerical experiment by Lotte and Feix (1984) and, more explicitly, in the conclusive remarks of section 2.7. We can now say something more about this. A Hamiltonian model, essentially corresponding to this kind of description, has been formulated by Antoni et al. (1998). It is based on a self-consistent Hamiltonian describing the 1D dynamics of the N resonant particles (the electrons) coupled with M harmonic oscillators, each representing a Langmuir oscillation in the 1D plasma with a finite, yet large, spatial period (the Bohm–Gross mode of the Vlasov limit corresponds in this framework to the statistical superposition of these M Langmuir waves). Such a Hamiltonian has the form Hsc =

N M N  M     p2i + ω j Aj − νj Aj cos(kj xj − θj ), 2m e i=1 j=1 i=1 j=1

[4.88]

where Aj is the amplitude of the jth-Langmuir oscillation with wave-vector kj , θj is a phase and νi is a coupling coefficient that depends on the features of the bulk. The latter corresponds to the remainder of the particles described by the mean field theory and whose number is assumed to be large with respect to the sum N + M . The coefficient νi therefore contains an explicit dependence on both the bulk plasma density and its dispersion relation through a ∂(ω, k)/∂ωR term like the one we already discussed in section 4.2.5 (for more details, see the references cited above and which are

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going to be cited in the following). The system conserves the total wave–particle momentum P =

N  i=1

pi +

M 

kj Aj ,

[4.89]

j=1

which evidences how, in this modeling, the way a Langmuir oscillation grows or is dampened (i.e. Aj changes in time) symmetrically affects the particle acceleration (i.e. the change in pi ), and vice versa. This highlights the essential difference with respect to the mean field Vlasov approach, in which the growth of an electrostatic eigenmode and the Landau-damped solution are differently treated from a formal point of view and do not appear as complex conjugate solutions. That is, this approach unifies from a formal point of view the collisionless damping and spontaneous emission processes. This model has been then applied to investigate the convergence of the N → ∞ limit to the Vlasov–Poisson description, with a special reference to the Landau damping mechanism, in a series of papers starting from (Firpo and Elskens, 2000). In these studies, among which we note (Escande and Elskens, 2003; Besse et al., 2011) as well as the recent review (Escande et al., 2018) and, especially, the book by Elskens and Escande (2003), some limitations of the Vlasov approach to describe the long-term chaotic dynamics of kinetic turbulence have been evidenced and discussed. In particular, by studying the linear and nonlinear dynamics of the weak beam plasma instability, the following fundamental points of interest for our discussion about the linear Vlasov–Poisson system have been pointed out: – From a mechanics point of view, it is the total momentum conservation [4.89] that makes two oppositely accelerated particles in the potential well of a Langmuir oscillation17 tend to synchronize with the wave. As a result, the amplitude of the wave changes accordingly, which can result in its damping or growth, depending on the total number of particles slower or faster than ωj /kj that is, depending on the slope of the particle “distribution” (taken in the large N limit) with respect to velocity. – The collisionless damping of a Bohm–Gross mode turns out to be a phenomenon that follows from the phase mixing among the “true” eigenmodes of the Vlasov plasma, that is, the Langmuir oscillations: the Bohm–Gross mode corresponds in this sense to a statistical superposition of different

17 Assume the two particles to have equal velocities but symmetric positions with respect to the bottom of the electrostatic potential.

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Langmuir oscillations (i.e. van Kampen modes, see section 2.6.2) interacting with several electrons that have a random thermal motion (i.e. that have a thermal velocity). – By applying a perturbation theory up to the second order in the coupling parameter νj for a superposition of oscillators with random equilibrium position and phases, a quasi-linear equation of the form dEk2  = 2γL (k, t)Ek2  + Sk (t), dt

[4.90]

is obtained, which describes the time evolution of the statistical average of the electric field amplitude Ek corresponding to the wave number kj . The difference with respect to the quasi-linear approximation normally adopted in the Vlasov limit (see (Vedenov et al., 1961)) is in the presence of the Sk (t) term, which expresses here the spontaneous emission of waves by particles. Since Sk (t) results to be inversely proportional to the number of resonant particles, 1/N , it vanishes in the Vlasov limit N → ∞, for which the classic Landau damping result is recovered when γL (k, t) < 0. - The peculiarity of the collisionless Landau damping solution for the single mode turns out to be a formal outcome of the passage from the N -body problem to the mean field approximation. Despite the convergence of the mean field result to the N → ∞ limit of the N -body Hamiltonian treatment, the quantitative and interpretative discrepancies between the two models for a finite number of resonant particles leave open some interesting questions: notably about the appropriateness of the stochastic approach inherent to the mean field Vlasov approximation for the description of the long-term chaotic dynamics of the single particles and, in the end, about the appropriateness of the Vlasov equation to model a turbulent collisionless plasma. These fundamental issues, related to the investigation of the limits of validity of the passage from a Klimontovich-type description of a collisionless plasma to a mean field Vlasov theory (see Chapter 1), rather than to a Fokker–Planck-type approximation (see (Escande and Sattin, 2007)) or to other transport models, are a matter of present day research. We recommend the interested reader to look at the articles and monography quoted above for a deeper discussion of these issues, related to the limits of the mean field Vlasov theory and especially of the quasi-linear modeling of plasma turbulence based on the Vlasov formalism. We also note that applications of the N -body approach have been developed in addition to investigations about these fundamental issues. One of

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these applications is for the modeling of Langmuir wave turbulence, aimed to study the role of density inhomogeneities in the generation of electromagnetic radiation in solar wind plasma during solar bursts (a review on this subject can be found in (Krafft et al., 2019)). Implications of the self-consistent Hamiltonian chaos described by this N -body model for the generation of coherent structures have been also studied by del Castillo-Negrete (2000). 4.4.3. Some final remarks about interpretative collisionless damping in Vlasov mean field theory

issues

of

A couple more words can be spent on interpretative issues that a reader may encounter while delving further into the subjects we have afforded in this section, but which might have already emerged from the brief discussion provided above. Operationally speaking, the Landau damping factor γL follows from a causality prescription on the eigenmode solution of the Vlasov–Maxwell system, whereas its identification in terms of the inverse Vavilov–Cherenkov absorption coefficient follows from consistency with the second principle of thermodynamics: not only is the link between causality end entropy increase not evident, but it can be questioned. Such a debate, related to subjects relevant to the foundations of physics and, in the end, to the origin of time irreversibility in nature (i.e. the problem of the “arrow of time”), would bring us too far and would lead us to cross the border with philosophy. Nevertheless, we can at least note some kind of consistency of the Vlasov–Maxwell model, which should be reassuring: even if the actual origin, in a physical sense, of collisionless damping is not clear from the Vlasov–Maxwell formalism (or better, from the Vlasov–Poisson formalism), the overall picture that comes from different interpretations of it, besides being supported by experimental measurements, shows a sufficient level of coherence from an operational point of view. From the point of view of the philosophy of science, it is indeed interesting to note that, as we pointed out in section 3.2.4, the identification of the eigenmode problem with a causal problem is related to the choice of boundary conditions in time, and it is from this that Landau prescription follows. Similarly, studies about the origin of irreversibility in physics point out that in present physical theories, the time asymmetry has to be ascribed, again, to the “choice” of the boundary conditions, in the end those at the cosmological scale which are related to the origin of the universe, (Davies, 1974). Therefore, as it happens for the H-theorem in Boltzmann microscopic

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theory, in which the loss of information (i.e. entropy increase) is postulated at a mesoscopic level through the molecular chaos hypothesis so as to make the statistical transport model for gases consistent with the second principle of thermodynamics (see, e.g. (Huang, 1963, Ch. 4)), what seems to matter most in the Vlasov–Maxwell formalism we are discussing is that, at some level of description (e.g. boundary conditions of the normal mode analysis), the assumptions made in order to yield results that are in agreement with fundamental thermodynamic principles do not provide evidence for any inner incoherence of the model itself. In this sense, the correspondence between the collisionless damping and inverse Cherenkov radiation (section 4.4.1) is operationally sound and is not contradictory with the interpretation of the mean field formalism, which comes from comparison with results obtained with the N -body Hamiltonian model for wave-particle interactions (section 4.4.2). And this, together with Mouhot and Villani’s mathematical proof of phase-mixing as the ingredient that gives Landau damping the macroscopic effective “irreversibility” (Mouhot and Villani, 2011), adds coherence to the global mean field framework. At the same time, the open questions that these different interpretations raise and the quantitative discrepancies, which emerge in some regimes of the different models for collisionless plasmas, motivate to pursue with these comparative investigations in order to shed light onto the limits of the Vlasov model and onto the most fundamental features of the physics of collisionless plasmas. Even if this topic transcends the purposes of this this volume, and we will not delve in it any further, we can conclude this section by noting that an example relevant to this discussion will be detailed in Chapter 5, section 5.8.2, which will discuss the role played by trapped particles in the transition to meta-stable, non-stationary BGK-like electrostatic structures: as we will see, the interpretation of the process given in terms of the negative mass instability applied to a Vlasov plasma will provide evidence for a kinetic dissipation process related to a re-synchronization of trapped particles with the wave, which is similar to the one discussed above with reference to the beam-plasma instability in the N -body Hamiltonian model, but which is well-described in the framework of a mean field theory. 4.5. References Alfvén, H. (1942). Existence of electromagnetic-hydrodynamic waves. Nature, 150, 405.

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Allis, W.P., Buchsbaum, S.J., Beers, A. (1963). Waves in Anisotropic Plasmas. MIT Press, Cambridge, MA. Antoni, M., Elskens, Y., Escande, D.F. (1998). Explicit reduction of n-body dynamics to self-consistent particle-wave interaction. Phys. Plasmas, 5, 841. Atzeni, S., Meyer-Ter-Vehn, J. (2004). The Physics of Inertial Fusion. Beam Plasma Interaction, Hydrodynamics, Hot Dense Matter. Clarendon Press, Oxford, UK. Berezhiani, V.I., Gacuchava, D.P., Mikeladze, S.V., Sigua, K.I., Tsintdadze, N.L., Mahajan, S.M., Kishimoto, Y., Nishikawa, K. (2005). Fluid-Maxwell simulation of laser pulse dynamics in overdense plasma. Phys. Plasmas, 12, 062308. Bergmans, J., Schep, T.J. (2001). Merging of plasma currents. Phys. Rev. Lett., 87, 195002. Berk, H.L., Breizman, B.N., Pekker, M. (1996). Nonlinear dynamics of a driven mode near marginal stability. Phys. Rev. Lett., 76, 1256. Besse, N., Elskens, Y., Escande, D.F., Bertrand, P. (2011). Validity of quasilinear theory: Refutations and new numerical confirmation. Plasma Phys. Control. Fusion, 53, 025012. Biermann, L. (1950). Über den Ursprung der Magnetfelder auf Sternen und im Interstellaren Raum (mit einem Anhang von A. Schlüter). Zeitschrift Naturforschung, Teil A, 5, 65. Bingham, R., Mendonça, J.T., Dawson, J.M. (1997). Photon Landau damping. Phys. Rev. Lett., 78, 247. Blaizot, J.-P., Iancu, E. (1996). Lifetime of quasiparticle in hot QED plasmas. Phys. Rev. Lett., 76, 3080. Bohm, D., Gross, E.P. (1949). Theory of plasma oscillations. A. Origin of medium-like behavior. Phys. Rev., 75, 1851. Booker, H.G. (1934). Some general properties of the formulae of the magnetoionic theory. Proc. Royal Soc. London A, 150, 267. Born, M. (1935). Atomic Physics. Blackie & Son Ltd., Glasgow, UK (reprinted by Dover Publ. Inc., New York, 1989). Born, M., Wolf, E. (1959). Principles of Optics. Cambridge University Press, Cambridge, UK. Brambilla, M. (1998). Kinetic Theory of Plasma Waves. Clarendon Press, Oxford, UK. Briggs, R.J. (1964). Electron-stream Interaction with Plasmas. MIT Press, Cambridge, MA.

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Buchsbaum, S.J. (1963). Waves in uniform, magnetoactive plasmas. Proceedings of the 7th Lockheed Symposium on Magnetohydrodynamics. Standford University Press, Standford, CA. Cai, H.J., Lee, L.C. (1997). The generalised Ohm’s law in collisionless magnetic reconnection. Phys. Plasmas, 4, 509. Califano, F., Del Sarto, D., Pegoraro, F., Bulanov, S.V. (2006). Threedimensional magnetic structures generated by the development of the filamentation (Weibel) instability in the relativistic regime. Phys. Rev. Lett., 96, 105008. Califano, F., Pegoraro, F., Bulanov, S.V. (1997). Spatial structure and time evolution of the Weibel instability in collisionless inhomogeneous plasmas. Phys. Rev. E, 56, 963. Califano, F., Prandi, R., Pegoraro, F., Bulanov, S.V. (1998). Nonlinear filamentation instability driven by an inhomogeneous current in a collisionless plasma. Phys. Rev. E, 58, 7837. Cap, F.F. (1976). Handbook on Plasma Instabilities. Volume 1. Academic Press, New York, NY. Cap, F.F. (1978). Handbook on Plasma Instabilities. Volume 2. Academic Press, New York, NY. Cap, F.F. (1982). Handbook on Plasma Instabilities. Volume 3. Academic Press, New York, NY. Cattani, F., Kim, A., Anderson, D., Lisak, M. (2000). Threshold of induced transparency in the relativistic interaction of an electromagnetic wave with overdense plasmas. Phys. Rev. E, 62, 1234. Chapman, S. (1931). The absorption and dissociative or ionizing effect of monochromatic radiation in an atmosphere on a rotating earth. Proc. Phys. Soc. London, 43, 26. Cherenkov, P.A. (1934). Visible emission of clean liquids by action of γ radiation. Dokl. Akad. Nauk. SSSR, 2, 451. Cicogna, G. (2018). Exercises and Problems in Mathematical Methods of Physics. Springer Verlag, Berlin. Clemmow, P.C., Mullaly, R.F. (1955). Dependence of the refractive index in magneto-ionic theory on the direction of wave normal. In Physics of the ionosphere: Report of the Physical Society Conference held at Cavendish Laboratory, September 1954, Beckerely, J.G. (ed.). Physical Society, London, UK. Cohen, M.H. (1961). Radiation in a plasma. I. Cerenkov effect. Phys. Review, 123, 711.

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Daughton, W. (1999). The unstable eignemodes of a neutral sheet. Phys. Plasmas, 6, 1329. Davies, P. (1974). The Physics of Time Asymmetry. University of California Press, Berkeley, CA. del Castillo-Negrete, D. (2000). Self-consistent chaotic transport in fluids and plasmas. Chaos, 10, 75. Del Sarto, D., Marchetto, C., Pegoraro, F., Califano, F. (2011). Finite larmor radius effects in the nonlinear dynamics of collisionless magnetic reconnection. Plasma Phys. Control. Fusion, 53, 035008. Del Sarto, D., Pegoraro, F., Califano, F. (2016). Pressure anisotropy and small spatial scales induced by velocity shear. Phys. Rev. E, 93, 053203. Depret, G., Garbet, X., Bertrand, P., Ghizzo, A. (2000). Trapped-ion driven turbulence in tokamak plasmas. Plasma Phys. Control. Fusion, 42, 949. Drude, P. (1900a) Zur Eleketrontheorie der Metalle. Annalen der Physik, 306, 566. Drude, P. (1900b). Zur Eleketrontheorie der Metalle; II Teil. Galvanomagnetische und thermonagnetische Effecte. Annalen der Physik, 308, 11. Elskens, Y., Escande, D.F. (2003). Microscopic Dynamics of Plasmas and Chaos. Institute of Physics Publishing, Bristol, UK. Escande, D.F., Bénisti, D., Elskens, Y., Zarzoso, D., Doveil, F. (2018). Basic microscopic plasma physics from n-body mechanics. A tribute to PierreSimon Laplace. Rev. Mod. Plasma Phys., 2, 9. Escande, D.F., Elskens, Y. (2003). Microscopic dynamics of plasmas and chaos: The wave-particle interaction paradigm. Plasma Phys. Control. Fusion, 45, A115. Escande, D.F., Sattin, F. (2007). When can the fokker-planck equation describe anomalous or chaotic transport? Phys. Rev. Lett., 99, 185005. Faraday, M. (1846). On the magnetization of light and the illumination of magnetic lines of force. Phil. Trans. R. Soc. London, 136, 1. Feix, M. (1963). Propagation of a double-stream instability in a plasma. Nuovo Cimento, 27, 1130. Feynmann, R.P., Leighton, R.B., Sands, M. (1964). The Feynman Lectures on Physics. Volume II. Addison-Wesley, Boston, MA. Firpo, M.-C., Elskens, Y. (2000). Phase transition in the collisionless damping regime for wave-particle interaction. Phys. Rev. Lett., 84, 3318. Folland, G.B., Sitaram, A. (1997). The uncertainty principle. A mathematical survey. Fourier Journal Analysis and Applications, 3, 207.

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Fresnel, A.J. (1821). Notes sur le calcul des teintes que la polarisation développe dans les lames cristallisées. Annales de Chimie et de Physique, 17, 167. Galeotti, L., Califano, F. (2005). Asymptotic evolution of weakly collisional Vlasov-Poisson plasmas. Phys. Rev. Lett., 95, 015002. Ghizzo, A., Del Sarto, D., Réveillé, T., Besse, N., Klein, R. (2007). Selfinduced transparency scenario revisited via beat-wave heating induced by doppler shift in overdense plasma layer. Phys. Plasmas, 14, 062702. Ghizzo, A., Sarrat, M., Del Sarto, D. (2017). Vlasov models for kinetic Weibeltype instabilities. J. Plasma Phys., 83, 705830101. Gillmor, C.S. (1982). Wilhelm Altar, Edward Appleton, and the magneto-ionic theory. Proc. Am. Philos. Soc., 126, 395. Ginzburg, V.L. (1996). Radiation by uniformly moving sources (Vavilov– Cherenkov effect, transition radiation, and other phenomena). Phys. Uspekhi, 39, 973. Ginzburg, V.L., Zheleznyakov, V.V. (1958). On the possible mechanism of sporadic radio emission (radiation in an isotropic plasma). Soy. Astron., 2, 653. Gould, R.W., O’Neil, T.M., Malmberg, J.H. (1967). Plasma wave echo. Phys. Rev. Lett., 19, 219. Hasegawa, A. (1985). Self-organisation processes in continuous media. Adv. Phys., 34, 1. Huang, K. (1963). Statistical Mechanics. John Wiley and Sons, New York, NY. Jackson, J.D. (1962). Classical Electrodynamics. John Wiley & Sons, New York, NY. Kadomtsev, B.B. (1979). Phénomènes collectifs dans les plasmas. Editions MIR, Moscow. Kaghashvili, E.K. (2007). Alfvén wave driven compressional fluctuations in shear flows. Phys. Plasmas, 14, 044502. Kaghashvili, E.K., Raeder, J., Webb, G.M., Zank, G.P. (2006). Propagation of Alfvén waves in shear flows: Nature of driven longitudinal velocity and density fluctuations. Phys. Plasmas, 13, 112107. Krafft, C., Volokitin, A.S., Gauthier, G. (2019). Turbulence and microprocesses in inhomogeneous solar wind plasmas. Fluids, 4, 69. Krall, N.A., Trivelpiece, A.W. (1973). Principles of Plasma Physics. McGrawHill, New York (NY). Landau, L., Lifshitz, E.M. (1960). Course of Theoretical Physics. Volume 8: Electrodynamics of Continuous Media. Pergamon Press, Oxford, UK.

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Landau, L., Lifshitz, E.M., Pitaevskii, L.P. (1981). Course of Theoretical Physics. Volume 10: Physical Kinetics. Pergamon Press, Oxford, UK. Lassen, H. (1926). Ionisation der atmosphäre und ihr einfluß auf die ausbreitung der kurzen elektrischen wellen der drahtlosen telegraphie. Jahrbuch der drahtlosen Telegraphie und Telephonie: Zeitschrift für Hochfrequenztechnik, 28, 109. Lesur, M., Médina, J., Sasaki, M., Shimizu, A. (2018). Subcritical instabilities in neutral fluid and plasmas. Fluids, 3, 89. Lotte, P., Feix, M. (1984). Plasma models and rescaling methods. J. Plasma Physics, 31, 141–151. Lynden-Bell, D. (1967). Statistical mechanics of violent relaxation in stellar systems. Mon. Not. R. Astr. Soc., 136, 101. Macchi, A. (2013). A Superintense Laser-Plasma Ineteraction Theory Primer. Springer Verlag, Dordrecht, The Netherlands. Malmberg, J.H., Wharton, C.B. (1964). Collisionless damping of electrostatic plasma waves. Phys. Rev. Lett., 13, 184. Malmberg, J.H., Wharton, C.B., Gould, R.W., O’Neil, T.M. (1968). Plasma wave echo experiment. Phys. Rev. Lett., 20, 95. Melrose, D.B. (1986). Instabilities in Space and Laboratory Plasmas. Cambridge University Press, Cambridge, UK. Melrose, D.B. (2017). Coherent emission mechanisms in astrophysical plasmas. Rev. Mod. Plasma Phys., 1, 4. Mikhailovskii, A.B. (1974). Theory of Plasma Instabilities. Volume 2: Instabilities of an Inhomogeneous Plasma. Springer, New York, NY. Mouhot, C., Villani, C. (2011). On Landau damping. Acta Mathematica, 207, 29. Ryutov, D. (1999). Landau damping: Half a century with the great discovery. Plasma Phys. Controll. Fusion, 41, A1. Silva, L.O., Bingham, R., Dawson, J., Mendonça, J.T., Shukla, P. (2000). Neutrino Landau damping. Phys. Lett. A, 5, 265. Somov, B. (2000). Cosmic Plasma Physics. Springer Verlag, Dordrecht, The Netherlands. Stix, T.H. (1962). Theory of Plasma Waves. McGraw-Hill, New York, NY. Stix, T.H. (1992). Waves in Plasmas. American Institute of Physics, New York, NY. Swanson, D.G. (1989). Plasma Waves. Academic Press, Inc., Boston, MA.

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5 Nonlinear Properties of Electrostatic Vlasov Plasmas

5.1. The Vlasov–Poisson system The system developed in this chapter is described by equations [2.1] (Chapter 2) and consists of Vlasov’s equation and Poisson’s equation for the  L  +∞ electric field E (x, t). Here, f is normalized so that L1 0 dx −∞ dvf = n0 , and we note by e, m and n0 the electron charge, mass and uniform ion density, respectively. This pair of equations is a highly idealized description of a plasma under the following assumptions and approximations: i) The ion background is taken to be immobile and uniformly distributed. ii) All disturbances of the plasma from the spatial uniform state are one dimensional (1D). iii) By the initial value problem, which we shall mainly be concerned with here, except for the problem of kinetic electrostatic electron nonlinear (KEEN) waves in section 5.7, we mean that we know the initial data of f , f (x, v, t = 0) (often periodic in x) and compute f (x, v, t) and E (x, t) for t > 0. The initial value problem is conceptually simpler than the more realistic boundary value problem (which we get roughly by imposing conditions on x = 0 or x = L) since the Poisson equation determines E (x, t) uniquely from the data of f .

The Vlasov Equation 1: History and General Properties, First Edition. Pierre Bertrand; Daniele Del Sarto and Alain Ghizzo. © ISTE Ltd 2019. Published by ISTE Ltd and John Wiley & Sons, Inc.

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The Vlasov equation is a transport equation and can therefore be solved by the method of characteristics. These characteristics correspond to Newton’s laws, which are written as dX dV eE (X (t) , t) = V (t) and = dt dt m

[5.1]

with (X (0) , V (0)) = (x, v). Thus, the resulting equation on f generally involves the Jacobian determinant of the flow at time t. If the motion is Hamiltonian, the Liouville theorem shows that the distribution function is conserved along their characteristics. Depending on the situation, we may consider the nonlinear Vlasov model either from the Eulerian point of view (compute the data of f (x, v, t)) or from the Lagrangian point of view (compute now the particle trajectories). This also affects the numerical approach, since numerical methods can be Eulerian (look at values of f on a grid), Lagrangian (consider particle motion or Newton’s laws as in particle-in-cell codes for instance) or semi-Lagrangian (make the particles move and interpolate to reconstruct f on a grid). 5.2. Invariants of the Vlasov–Poisson model The Vlasov–Poisson system given by [2.1] preserves the total number of particles N . This is evident from the way we got the Vlasov equation. Moreover, integrating the Vlasov equation over the velocity v and noticing that limv→±∞ f (x, v, t) = 0, we see that ∂ ∂t



+∞ −∞

∂ f dv + ∂x



+∞ −∞

vf dv = 0

[5.2]

 +∞ Now n (x, t) = f dv is the density of particles and −∞  +∞ J (x, t) = −∞ vf dv is the current density of particles. We now integrate [5.2] over the space x (from zero to the length of the system L in the case of a periodic system for instance), noticing that the current density takes the same values at the boundaries in x = 0 and x = L leads to d dt



L 0

n (x, t) dx =

dN =0 dt

[5.3]

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The Vlasov equation also preserves the total energy. Multiplying the Vlasov equation by v 2 and then integrating over v leads to ∂ ∂t



+∞ −∞

∂ f v dv + ∂x 2



+∞ −∞

eE f v dv + m 3



+∞ −∞

v2

∂f dv = 0 ∂v

[5.4]

2 ∂t K (x, t), where The first term in the left-hand-side (LHS) of [5.4] is m K (x, t) is the density of kinetic energy. The last term can be transformed; we  +∞ can integrate by part −∞ v 2 ∂f ∂v dv = −2eJ (x, t), and equation [5.4] can be written as

∂ 2eK (x, t) ∂ + ∂t m ∂x



+∞ −∞

f v 3 dv −

2e E (x, t) J (x, t) = 0 m

[5.5]

Finally, by introducing the total kinetic energy by Ec (t) =  L  +∞ 1 mv 2 f dxdv and by integrating [5.4] over the variable x, we obtain: 0 −∞ 2 dEc + dt



L 0

0 E

∂E d dx = (Ec + Ep ) = 0 ∂t dt

[5.6]

since eJ (x, t) + 0 ∂E ∂t = 0 can be considered as one of the Maxwell equations where, in the electromagnetic regime, the magnetic field is zero. L Here, Ep = 12 0 0 E 2 dx is the electrostatic energy. It must be pointed out that the Vlasov equation preserves all the nonlinear integrals of f , often called the Casimir’s invariants of the equation. They take the form F (f ) dxdv, where F is arbitrary. In other words, they express the fact that the Vlasov equation induces a transport of a Hamiltonian preserving flow. From a mathematical point of view, all Lp −norms  are preserved and in particular it is also the case of the entropy S = − f lnf dxdv. The latter property is in sharp contrast with the Boltzmann equation for which the entropy can only increase in time due to the existence of collisions (the well-known H−theorem), unless it is at equilibrium. Thus, the conservation of the entropy reflects the preservation of information. 5.3. Stationary solutions: Bernstein–Greene–Kruskal equilibria As far as collective phenomena are our concern, it is worth remembering that in a Vlasov plasma, Coulombian interactions between charged particles are actually replaced by a mean field calculated from the Poisson equation using the charge density, where the microscopic fluctuations (due to the fact

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that a plasma is not a continuum) are averaged over the Debye length. This mean field concept is thus the basic idea, which points to the concept of the Vlasov–Poisson model. Thus, while a property of the Boltzmann equation is that it only has Gaussian equilibria, in contrast the Vlasov equation may have many shapes of equilibria. Nearly 50 years ago, Bernstein et al. (1957) discovered a class of dissipation-less nonlinear waves (the so-called Bernstein–Greene–Kruskal (BGK) waves) in collisionless plasmas. Thus, a 1D homogeneous ion neutralizing background can support a BGK equilibrium, consisting of stationary spatially periodic structures exhibiting holes (or vortices) in phase space. Moreover, the appearance of (semi-Lagrangian) Vlasov codes have allowed a precise observation of these structures of the phase space, especially in these regions where the distribution function f is very small. It is interesting to note that, while the importance of phase space holes was well recognized in phenomena like two-stream instability, their importance in other phenomena had to wait for the undertaking of the semi-Lagrangian Vlasov codes. We begin by considering the steady-state solution of the Vlasov equation, i.e. when ∂t = 0. The construction of a class of equilibria is made easy by means of the so-called Jeans’ theorem, which states that the general solution f is an arbitrary function of the invariants of the system. Let us start with a 1D plane geometry with two-dimensional (x, v) phase space. The Vlasov equation is written as v

∂f eE ∂f + =0 ∂x m ∂v

[5.7]

which is solved through the characteristics method leading to dx mdv = v eE (x, t)

[5.8]

Taking into account the expression of the electrostatic field E = − ∂φ ∂x , we get the invariant H=

1 mv 2 + eφ = ε 2

[5.9]

Thus, the general solution of the steady-state Vlasov equation is written as f (x, v, t) = F (ε), where F is an arbitrary function. We then get by, the chain rule,   ∂f ∂φ ∂φ e ∂φ ∂f v − = eF  () v − v =0 [5.10] ∂x m ∂x ∂v ∂x ∂x

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so F (ε) is an equilibrium. The solution f = F (ε) must be included in the Poisson equation, leading to compatibility conditions for the potential φ. For instance, let us assume that the potential φ is small compared to the characteristic kinetic energy of particles at a given point. Thus, it is possible to consider a Taylor’s expansion of the distribution F (ε) in the form  2  F (ε) = F mv + φF (ε) in the Poisson equation. We then have the 2 condition d2 φ e 2 φ + dx2 0



+∞

−∞

dF dv = 0 dε

[5.11]

which can be rewritten in the usual form: d2 φ − k2 φ = 0 dx2

[5.12]

where k2 = −

e2 m0



+∞

−∞

1 dF dv v dv

[5.13]

The solution will depend on the sign of k 2 that implies a dependence on the distribution F (v). Thus, in the case of a Maxwellian distribution of type 2  − v2 1 F (v) = √2πv e 2vth , where vth = kBmTe is the electron thermal velocity, th we get the solution k 2 λ2D = 1 and find the usual localized sheath of a Maxwellian plasma. Now, in the case where F (v) exhibits a depression, k 2 can become negative and in that case the plasma can develop a periodic non-uniform and non-neutral structure (although their exists a neutralizing homogeneous background). Equations [5.12] and [5.13] can be generalized to the case where the potential φ is not very small and it is possible to write, without difficulty, that E verifies the equation d2 E − k 2 (x) E = 0 dx2

[5.14]

where k 2 (x) is given by equation [5.13]. We note that [5.14] is formally identical to [5.12], except that E is the function and not φ, but k 2 (x) is now a function of x. This point will be discussed later in section 5.5.

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The Vlasov Equation 1

5.4. Some mathematical properties of the Vlasov equation The fact that little effort was spent on Vlasov codes during the 1980s may be connected to the limited size of the available computers, and up until very recently these codes could only treat a 2D or 3D phase space. But another explanation must be also discussed. Previous attempts at the numerical solution of the Vlasov equation were not capable of solving, in a practical way, the problem of filamentation in the velocity space. The source of the problem lies in the treatment of the free streaming term of the Vlasov equation: ∂f ∂f +v =0 ∂t ∂x

[5.15]

which has a general solution in the Fourier k-space in the form fˆ (k, v, t) = fˆ (k, v, 0) eiktv

[5.16]

We see in [5.16] that, as a function of v, fˆ (k, v, t) oscillates at the frequency kt and when this frequency reaches the size of the cell, we can no longer follow the exact evolution of f . This is the unavoidable filamentation problem. From a mathematical point of view, the properties of the free transport equation [5.15] differ in the whole real space (open system) and in the confined space (periodic system) (see Villani (2010a,b, 2014)). In the former case, dispersion at infinity dominates the asymptotic behavior, while in the latter case, one observes a homogenization process due to phase mixing as illustrated in Figure 5.1, which shows the time evolution of a double Maxwellian distribution in x and v, at different times, when solving the free transport equation. The free transport equation [5.15] can be solved explicitly in the form f (x, v, t) = f (x − vt, v, 0)

[5.17]

which gives the expression [5.16] in the Fourier space. It can be pointed out that this free transport equation takes the form of an advective equation, but on the other hand, the fact that a simple shift can be expressed in the corresponding Fourier space (by simply changing the phase in [5.16]) suggests the use of fast Fourier transform (FFT) to solve the equation numerically.

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Figure 5.1. Phase space representation of the distribution function in the case of the free streaming problem. We clearly observe the formation of filaments that are becoming thinner and thinner. As expected, we may also observe the and rebuilding of the distribution function at the recurrence time TR = k02π Δv finally the initial data are recovered at 2TR . For a color version of this figure, see www.iste.co.uk/delsarto/vlasov1.zip

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By performing a Fourier transform in the x-space and using the previous solution [5.17], the free particle motion from tn = nt to tn+1 = (n + 1) t is given by f n (x, v)

F T ˆn F T −1 n+1 f (k, v) × eikv t f (x, v) =⇒ =⇒

[5.18]

Here, F T means Fourier transform with respect to x and F T −1 means the inverse Fourier transform, while the symbol ×eikv t indicates of course multiplication by the corresponding factor (i.e. shift of phase). A first series of experiments was conducted with a phase space sampling of Nx Nv = 256×256 points, showing an evolution of a double Gaussian in x and v up to tωp = 600 (with a time step of tωp = 0.10) and consequently strengthening the conjecture of the production of fine filaments until a homogenization of phasespace structures is reached at time tωp = 70. Figure 5.1 shows the time evolution of the distribution function (initially a double Maxwellian at time tωp = 0), the formation of filaments that become finer until the first reconstruction of the distribution at time tωp = 255 (but with a de-phasing). Note that this time corresponds to the recurrence time TR = k02π Δv . Finally, after a second process of filamentation, the initial distribution is recovered at double TR . Thus, the filamentation in the velocity space leads to the occurrence of a fine microstructure in phase space. In simulations, the numerical grid in phase space inevitably becomes too coarse as this fine graining develops. This filamentation problem is well known in Vlasov simulations and has been discussed by Feix et al. (1994). Furthermore, a measure of the loss of information can be obtained by means of the entropy  of2 the system. To do that, we have used two different norms: first L2 =  f dxdv (which favors the higher values of f ) and the entropy S = − f lnf dxdv (which now privileges the smaller values of f ). We recall that any functional C (f ) of the distribution function H (t) = C (f ) dxdv gives an exact invariant of the Vlasov–Poisson system.   The relative variation of L2 − L20 /L20 (respectively, (S − S0 )/S0 ) is shown in Figure 5.2 in the top (and middle) panels, together with the total density (total mass) in the bottom panel, showing a small variation of these quantities due to the numerical integration of the free transport equation. From a physical point of view, information goes toward the small velocity scales and the oscillations of f are amplified when time becomes large. Such oscillations, or filaments in phase space, clearly visible in Figure 5.1, are the fundamental mechanism which preserves the microscopic typical reversibility

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of the equation. Thus, we can represent such a filamentation mechanism in the velocity space as a cascade from the low- to high-velocity modes (the cascade being faster for higher spatial modes), a process already seen in weak turbulence in plasmas. Such temporal variation of the invariants is a numerical artifact connected to an information loss in phase space. By increasing the sampling of the phase space, as shown in Figure 5.3 where we have chosen now to increase the sampling in the velocity space to Nv = 65536 grid points, while keeping Nx at 256, we observe that such a variation becomes weaker. It must be pointed out that now we recover the standard thermodynamic property of an increase in the norm L2 . A key idea, suggested by Mouhot and Villani (2010), consists of concentrating on the Fourier modes that matter for the solution of the free transport equation, and following this cascade over the course of time. This concept was called the sliding regularity1 and comes with the degradation of the regularity bounds in velocity, but simultaneously with the improvement of the regularity in position, when velocity averages have been formed. Such a mechanism is shown in Figure 5.4 in the space (λ, t), where λ is theFourier  space of v. We have here represented the distribution fˆ x = L4 , λ, t . In his work concerning Landau damping, Villani (2010a,b) used this new concept of sliding regularity and gave a new interpretation of the Landau damping as a transfer of regularity away from the velocity space v toward the spatial variable x: the improvement in time of the regularity of the electrostatic field, which implies that its amplitude decreases, leads to the famous Landau damping in time. This gliding regularity is thus the consequence of the cascade: the information shifts in phase space from low to high modes. Although from a mathematical point of view, no Poincare recurrence is possible (since the confined system is of infinite dimension), a numerical recurrence can be however observed at time 2TR = k04π

v where the initial 2π state is recovered. Here, k0 = L is the fundamental wave vector. Such a numerical recurrence is linked to the introduction of an elementary cell of finite size in the velocity space (i.e. v) and to the bounded box in space (through the fundamental vector k0 ). In the numerical simulation context, it may be noted that several reflection processes are produced for subharmonics recurrence times.

1 The regularity of a function being a mathematical expression of how smooth its variations are.

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The Vlasov Equation 1

Figure 5.2. Time evolution of the relative invariants fluctuations of the L2 norm, the entropy S and finally of the electron density, for a sampling Nx Nv in phase space of 2562 grid points. Note that the linear variation, induced by the numerical discretization, remains weak

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Figure 5.3. Similar diagnostics as those previously shown in Figure 5.2 but now for a high sampling of the distribution function of Nv = 65536 in the velocity space, which allows us to reduce the level of fluctuations of the electron density. However, the L2 -norm and the entropy continue to vary in a linear way, showing that the loss of information in the phase space is an inevitable process

226

The Vlasov Equation 1

  Figure 5.4. (λ, t)-representation of the distribution fˆ x = L4 , λ, t , where λ is the Fourier space of the velocity v. The function is chosen at a fixed point x = L4 in the configuration space x. This representation illustrates the “gliding” regularity process, a mechanism proposed by Mouhot and Villani, corresponding to the information shift in phase space from low to high modes, responsible for the formation of strong fluctuations in the velocity space. For a color version of this figure, see www.iste.co.uk/delsarto/vlasov1.zip

It must be pointed out that the free transport equation (or the Vlasov equation from a more general point of view) is time reversible. This time reversibility is quite natural for a system where interactions are taken into account through the force appearing in Newton’s law. In fact, we might distinguish between a purely mathematical point of view and a physically reasonable one. A clear distinction is provided by the following very simple problem. Let us consider a rectangular box, i.e. phase space ( |x| ≤ L0 = L2 and |v| ≤ vmax ), which contains the initial information given by a uniform distribution of particle, the box being limited by two reflecting walls located at x = ±L. After a certain amount of time, the fastest particles strike the wall and appear to the other side (due to the periodicity in space). An example of such a situation is shown in Figure 5.5. We observe the formation of five filaments. Figure 5.5 shows what the system looks like after a long time, when phase mixing takes place. Now, what conclusion do we draw from such an evolution?

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Figure 5.5. Illustration of the filamentation process of the distribution function in phase space for an initial homogeneous distribution. At a given time in the simulation, it is no longer possible to distinguish two adjacent filaments, leading to an inevitable increase in the entropy. For a color version of this figure, see www.iste.co.uk/delsarto/vlasov1.zip

Let us consider the entropy S, which can be put in the following form: S=−

f lnf dxdv = −

nlnndx −

f ln

  f dxdv n

[5.19]

i.e. in the form of the summation of two terms, the former is linked to a smoothed distribution while the latter is a measure of the kinetic entropy. Mathematically the entropy S is conserved but there is also a transfer of information from spatial to kinetic  variables. Initially for a simple example, we have f = 4L0N , where f dxdn = N is the total number of particles, vmav which is of course conserved, and S0 = N ln

4L0 vmax N

[5.20]

Mathematically speaking f is a constant and at any time S = S0 . But what will an experimenter say when faced with the difficult problem of

228

The Vlasov Equation 1

distinguishing the values of v for which f = f0 and those for which f = 0? At a certain moment, the interval δv between two neighboring beams becomes smaller than the accuracy with which v can be measured (since the number of beams increases with time) and we are forced to take f=

N 4L and S = N ln vmax 4Lvmax N

[5.21]

Figure 5.6. Linear increase in the L2 -norm for a sampling Nx Nv of 2562 grid points in the case of a homogeneous distribution in phase space. Because the information is unavoidably lost, the entropy increases in a linear way

Now the entropy variation is S − S0 = N ln LL0 and we observe an increase in entropy of ln LL0 per particle, a consequence of the scrambling of the trajectories and of the filamentation of phase space. It is through such a mechanism that physically (or numerically due to the introduction of a phase space cell of finite size) meaningful increase in entropy takes place. An example is plotted in Figure 5.6, which shows the linear increase in the L2 norm as a function of time for a sampling Nx Nv of 2562 grid points in phase space. We consequently see that statistical treatment of the Vlasov equation will sometimes be necessary. From a mathematical point of view, a simpler conceptual solution is to invoke the notion of weak convergence: time reversibility leads to the conservation of information (which allows us to recover the plasma echoes), but information may be lost, in an irreversible way, in the limit when weak convergence is considered (leading to Landau damping, a relaxation process with entropy conservation). The free transport equation is time reversible, yet, from a numerical point of view, we are

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looking for irreversible behavior. For a sampling of Nx Nv = 256 × 256, the value of the L2 - norm (plotted in Figure 5.6) increases, which means that there is a loss of information in the distribution function, when filaments reach the size of the cell xv. 5.5. The Bernstein–Greene–Kruskal solutions There is extensive literature on the equilibrium solutions of the Vlasov equation, which evidently correspond to stationary flows of the fluid in phase space, and also on the many linearized waves and instabilities that can be superimposed on these equilibria. But, it would clearly be of interest to develop a general kinetic theory of such a type of classical fluid capable of generating nonlinear structures. Starting from initial configurations that are fairly uniform but unstable, the first numerical simulations have demonstrated the formation of large-scale persistent structures in phase space. These have been seen in a variety of problems such as two-stream instability (see Berk et al. (1970); Ghizzo et al. (1988a)), a negative-mass instability (see Landau and Neil (1966); Dodin and Fisch (2013)) and gravitational Jeans instability (see Hohl and Feix (1967)). The problem starts with the study of Landau damping in the linear and nonlinear regimes. Landau damping is the oldest and most famous kinetic effect exhibited by the Vlasov–Poisson system. By performing numerical Vlasov simulations, Bertrand et al. (1989) have shown that an increase in the nonlinearity (i.e. the size of the initial perturbation level) stops the damping of the plasma wave after some oscillations with possible modulation of the amplitude of the oscillations. Finally, the establishment of steady undamped oscillation is observed in the electric field. The distribution function exhibits the formation of phase space holes and the persistence of the oscillation of the electric field is simply connected to the “ballistic” motion of these holes. This hole’s motion at constant velocity vhole associated with the spatial periodicity with wave-vector k0 gives the corresponding frequency ω = k0 vhole , obviously close to the Bohm–Gross frequency since vhole is close to the phase velocity vϕ of the plasma wave. Note that the formation of these holes (or phase space depressions) corresponds to the presence of a small population of trapped particles which here plays a fundamental role in stopping the Landau damping (see, for instance, section 5.7). Another point observed in Vlasov or particle-in-cell (PIC) simulations is the possibility of obtaining holes with a frequency well below the plasma

230

The Vlasov Equation 1

frequency due to the tendency of holes to merge. In particular, the coalescence of two electron holes leads to a wave vector divided by two (due to the vortex merging) and the frequency of the driven oscillation ω = kvϕ is not quite divided by two, but jumps from a value of ω ≥ ωp (given by the Bohm–Gross dispersion relation) to a value well below the plasma frequency. Such phase space structures are usually apparent in the BGK equilibria. 5.5.1. The case of (electrostatic) two-stream instability As our first example, we consider two-stream instability in order to handle the difficult problem of the phase space representation of the distribution function. The most striking advantage of the semi-Lagrangian Vlasov code is the very fine resolution in phase space, capable of describing the vortex fusion of BGK holes. We begin with a 1D plasma of electrons (with a fixed homogeneous neutralizing ion background). Starting with a homogeneous two-stream unstable state, periodic BGK structures appear with a number of holes proportional to the length of the system (the chosen length is four times the marginal box length), followed in a second step by a vortex merging to end up with a large hole BGK-type structure. A first series of numerical simulations showing the appearance of a four-hole structure at time tωp = 40 and its subsequent coalescence is presented in Figure 5.7. These curves represent, at different times, the contour representation of f . This kind of representation shows clearly the “cleaning” of the microstructure, a phenomenon which embodies an increase in the “sliding” regularity principle (i.e. an improvement of regularity in the position space). Here, we have taken an initial equilibrium of type n0 μ 2 − 2ξ F0 (ε) = √ 2πvth 3 − 2ξ

 1+

ε 1−ξ



e−ε

[5.22]

where μ and ξ are parameters which characterize the BGK equilibrium (see 2 Ghizzo et al. (1988a) for more details). Here, ε = v2 + eφ (x) is the total energy normalized to kB Te (here we have used the dimensionless quantities v = vvth and φ = kBeφTe ). In the situation considered here, we have chosen to start with a homogeneous system, i.e. with μ = 1 and no electrostatic potential (φ = 0). For such a homogeneous distribution, it is possible to calculate the corresponding value of the marginal mode (i.e. the maximum wave-vector of the instability) in the form: 2 2 kM λD =

2ξ − 1 3 − 2ξ

[5.23]

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Figure 5.7. Phase space representation of the distribution function, at different times, in the case of two-stream instability for a short plasma: we clearly observe the formation of four holes at time tωp = 25, followed by several successive processes of vortex merging leading finally to the formation of a single stable vortex. Notice the “cleaning” phenomenon of the microstructure in phase space. For a color version of this figure, see www.iste.co.uk/delsarto/vlasov1.zip

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The Vlasov Equation 1

Choosing ξ = 0.95, we have kM λD = 0.904, with a marginal wavelength of λM = 6.946λD . This two-stream system, shown in Figure 5.7, was slightly kM perturbed initially on both the fundamental mode k0 = 2π L = 2 (with a small perturbation amplitude of α = 0.001) and on the mode 4k0 (now with an amplitude of α = 0.1 ). Here, L  55.7570λD is the length of the plasma and is eight times the marginal wavelength. Consequently, we forecast the appearance of a four-hole structure, which shows up rather quickly (at time tωp = 40). In simulations, x, v and t are normalized to the Debye length λD , the thermal velocity vth and the inverse plasma frequency ωp−1 , respectively. Two remarks must be pointed out: i) The first point concerns the rather smooth structure in the final state in phase space. Such a mechanism was observed using different numerical schemes: an examination of the long-time behavior shows that it depends on how the treatment of small wavelengths was performed, but the “cleaning” of the microstructure was always observed, and that after the “cleaning”, the same final state is obtained with a distribution function that is shown to depend explicitly of the sole energy, a feature of BGK equilibria. ii) The second interesting feature is the tendency of electron holes to behave as quasi-particles before coalescence when perfect symmetry is used in the initial condition (i.e. without introducing a small dephasing in the perturbation term); the “forces” acting on the vortex structures balance one another perfectly and no merging is observed (see Ghizzo et al. (1987) for more details). Furthermore, we have also studied the time evolution of the position of the center of mass of each electron hole, confirming the existence of “attractive” and “central” forces between vortices, depending only on the distance. The existence of these attractive forces was mentioned for the first time by Berk et al. (1970) using a waterbag model. Returning to the question of vortex coalescence, we can now envisage larger systems. Let us consider the phase space representation of a distribution function f (x, v, t), a solution of the same problem, but for a longer plasma of length L = 64λM  492.48λD , which exhibits at the beginning of the evolution (at time tωp = 20) about 30 vortices (in this case the sampling of the distribution function has been chosen to Nx Nv equal to 2048 × 256). Figures 5.8 and 5.9 show the time evolution of this distribution function at different times: the time evolution of plasma is characterized by several pair-wise vortex merging processes until a three-hole structure is formed at time tωp = 3000 (in fact, at time tωp = 40000, the system has evolved toward a two-hole structure). It is the inherent phase space representation resolution of the Vlasov code that allows the presentation of

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such detailed structures in phase space, which in the common PIC codes are usually very coarsely represented.

Figure 5.8. A similar problem for a larger plasma of length L 492.5λD in which 30 holes appear at the beginning of the temporal evolution. In these pictures, a gray-shading representation has been used. The temporal evolution of the plasma is characterized by several pair-wise vortex merging processes leading to an intermediate state forming (ten) larger holes at time tωp = 300

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The Vlasov Equation 1

Figure 5.9. Continuing the simulation of the two-stream instability for large plasma, shown in Figure 5.8, over a long time. The pair-wise vortex merging process continues over time, giving rise to a three-hole intermediate state at time tωp = 3, 000. The continuation of the simulation until a time of 40, 000ωp−1 did not allow us to obtain an asymptotic state

There are some physical similarities with the study of 1D gravitational systems. Using a waterbag model, Hohl and Feix (1967) have studied Jeans instability and the stability of an isolated structure. If the initial state is near equilibrium, thin streams of phase fluid are ejected from the boundaries of the structure and rotate about the center of the vortex. There are also some physical similarities (or mathematical duality) between two-stream instability, or the negative-mass instability met in the problem of the particle accelerator, and Jeans instability. Thus, with the help of these “duality principles”, we see

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that the large-scale hole structures that develop in the nonlinear two-stream problem can be interpreted in several different ways. It was found, perhaps somewhat surprisingly, that electron holes (of same charge) attract each other. We can explain this phenomenon first of all by realizing that the boundaries of the holes are determined by negatively charged electrons (although electron holes have positive charge due to the local excess of ions), and that they are indeed attracted toward neighboring positively charged regions. However, numerical simulations have shown that the coalescence of several vortices does not take place immediately but results in several successive pair-wise vortex merging processes. Second, because of the duality principle (with negative mass instability (NMI)), the holes behave as gravitational bodies that attract one another through a Debye-shielded Coulomb force. 5.5.2. Chain of BGK equilibria Following Ghizzo et al. (1988a), we consider now the BGK equilibrium with a distribution function given by equation [5.22]. As previously mentioned, the distribution F0 (ε) corresponds to two-stream plasma in the homogeneous case (for φ = 0), a case previously studied in section 3.5.1. Here, ξ is a parameter that obviously satisfies the inequality 0 ≤ ξ < 1. The second parameter μ is defined as μ = n(0) n0 at the boundary x = 0. The −φ = μ 3−2ξ+2φ . Consequently, the normalized electron density is n(x) n0 3−2ξ e equilibrium potential φ obeys the condition

d2 φ 3 − 2ξ + 2φ −φ +μ e −1=0 dx2 3 − 2ξ

[5.24]

Without loss of generality, we can select the minimum of the density n (x) n(0) at x = 0 with φ (0) = 0 and dφ dx (0) = 0. We obtain the condition n0 = μ. The solution of equation [5.24] is obtained by a Runge–Kutta method and is characterized by the parameters μ and ξ. Thus, μ is the minimum value of the relative electron density and is a measure of the inhomogeneity of the system (μ ≤ 1), and ξ characterizes the depth of the hole in the phase space density around x = 0. Equation [5.24] can be written in terms of the electric field E = − dφ dx to give equation [5.14]. The structure of equation [5.14] makes it possible to get a periodic or aperiodic solution (see Holec (1974) for more details). In that case, we obtain the condition μ>

3 − 2ξ ξ− 1 e 2 2

[5.25]

which gives the limit for periodic and aperiodic waves in the parameter space.

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The Vlasov Equation 1

5.5.3. Stability of the periodic BGK steady states Several methods have been used to study the stability of two-stream-type BGK structures using mode coupling by Goldman (1970), using thermodynamics arguments by Minardi (1973), or as an eigenproblem in Fourier–Hermite basis by Siminos et al. (2011). Here, we present a quit different analysis based on an analogy with the research of unstable state of Kelvin–Helmholtz instability for the guiding center model using the Rayleigh criterion. We first consider the stability of a wave train of N periods with one hole or BGK, each of length LBGK , with a periodic boundary condition over the whole system of length L = N LBGK . The originality of such an approach is that it distinguishes between the case N = 1 and N ≥ 2. Some of the features being already published by Ghizzo et al. (1988a), we indicate here the main steps. For these modes, following the marginal adiabatic mode theory (see also Bertrand et al. (1972)), we must compute the k 2 (x) factor appearing in equation [5.14], i.e. e2 k (x) = − m0 2



+∞

−∞

1 ∂f dv v ∂v

[5.26]

Introducing the equilibrium [5.22] into [5.26], we obtain k 2 (x) λ2D = μ

1 − 2ξ + 2φ −φ e 3 − 2ξ

[5.27]

Let us first recall that for an infinite homogeneous plasma, k 2 as given by [5.26] is of course independent of x, and it is well known from the properties of the linear homogeneous dispersion function that the sign of k 2 indicates stability: k 2 > 0 for a stable state plasma while the condition k 2 < 0 characterizes an unstable state. In the inhomogeneous case, k 2 is now dependent on x and to obtain the d2 stability criterion, we introduce the operator Λ = k 2 (x) − dx 2 . Now the stability properties are given by the sign of the eigenvalues of the operator Λ. To compute the lowest eigenvalue, we calculate the Rayleigh factor  gΛgdx R (g) =  2 [5.28] g dx where the integral in [5.28] is taken over an integer number of periods. We know that the lowest eigenvalue is always smaller than R (g) for any arbitrary

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function g and if we can find a trial function g for which R (g) is negative we can state that the corresponding steady-state equilibrium is unstable. To construct the trial function g, we first note that the equilibrium electric field E as given by equation [5.14] is a solution of ΛE = 0, so that R (E) = 0. Thus, a trial function is given with g equal to either the data of E or the minimum or maximum of E. When g takes a constant value, the contribution to the sign of R (g) is negative since in these regions the function k 2 (x) is negative. Since in these regions, where g is taken as a constant, the sign of k 2 is always negative, the Rayleigh factor is negative, implying instability. Note that we must take a trial function extending on two spatial periods (which indicates that this instability implies interaction between at least two periods and consequently is a long wavelength or a slow process). Moreover, it is impossible to build a trial function g by considering only one-period BGK equilibrium since g would not be continuous, preventing the use of the Rayleigh factor method. However, such a construction can be extended to three or more periods. Nevertheless, in the slightly inhomogeneous BGK equilibria, it was noted that the minimum wave vector that can be fitted in the 2π system, namely LBGK , is just equal to kM ; kM is defined by equation [5.23], which is the maximum wave-vector for which the plasma becomes unstable. In that case, the equilibrium is at the marginal state. It can be conjectured that such limiting behavior will persist for strongly inhomogeneous plasma. Finally, Ghizzo et al. (1988a) have solved equation [5.24] numerically for μ = 0.92 and ξ = 0.90 to obtain a BGK equilibrium. The spatial period LBGK  14.710λD corresponds to a single vortex structure. In order to allow for subharmonic perturbations (sidebands), a plasma of length L = N LBGK has been considered, supporting the periodic potential φ to build an “N −vortex” structure or an “N −cell replica” of the basic cell (a single BGK electron hole). Numerical experiments were carried out in order to study the stability of a chain of N BGK equilibria. For the case N = 1, a single BGK hole, we observe that the system remains unchanged. It can thus be conjectured that the equilibrium is marginally stable. For N ≥ 2, numerical simulations show that the plasma evolves toward a final state, in which all the electron holes finally merge into a single hole, a marginally stable state since it does not evolve in time, when the simulation is continued. It has also been shown that this final state is described by a distribution function that depends only on the total particle energy, a feature of BGK equilibria. An example of such simulation was reported in Figures 5.10 and 5.11 for N = 3. In the first step in Figure 5.10, the holes begin to move through a mutual attraction. Next, they coalesce and we are left with a final equilibrium state shown in Figure 5.11, which again

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The Vlasov Equation 1

consists of a single smooth hole in which the noisy microstructure has disappeared. The phase space representation allows us to have a more detailed insight into the vortex coalescence process. Although this vortex coalescence starts with three-phase space holes, one hole is however ejected and finally we observe the successive pair-wise vortex merging process. The temporal behavior of the first three modes of the electric field is reported in Figure 5.12. The modes k0 and 2k0 (where k0 is the fundamental wave-vector) exhibit unstable behavior from tωp = 0 to tωp = 100 followed by the nonlinear saturation of the instability corresponding to the vortex merging. We see that while a one-hole structure is at the stability limit, the mode 3k0 is now the marginal one and remains constant before the coalescence takes place. Two points must be made at this stage of presentation. i) First, the structure presented in Figures 5.10–5.12 was slightly perturbed initially in the following form: f (x, v, 0) = F0 (ε) (1 + αcos (k0 x + ψ)) ,

[5.29]

where L = N LBGK (with k0 = 2π L ) and ψ is a small phase shift allowing a breaking of the initial symmetry and consequently an acceleration of the evolution of the plasma. Here α = 10−3 . Without such an initial dephasing ψ, the system remains in a state of marginal stability and it is necessary to wait for several hundreds of plasma oscillations to see the start of the merging process. ii) More recently, the study of vortex merging has been revisited by Siminos et al. (2011) as an eigenproblem in a Fourier–Hermite basis of finite dimension. Such an approach leads to a general formulation for the perturbation δf of a distribution function around an equilibrium f0 of type: ∂δf ˆ = Aδf ∂t

[5.30]

where Aˆ is a linear operator that depends on f0 and the eigenvalues of the operator Aˆ determine the stability of the equilibrium. Thus, to study the stability of a chain of N BGK equilibria, the authors have proposed to solve the eigenproblem given by [5.30] by making use of a Galerkin spectral method. They have shown that the combination of spectral deformation and Fourier–Hermite expansion can be used to recover the growth rate of the different modes of the electric field.

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Figure 5.10. Phase space representation using 3D plot and the corresponding f = const contours in the case of the study of an initial plasma state of three BGK structures at four different times in the initial phase of the instability. While a system formed by a single BGK equilibrium remains stable, a system made up of several BGKs is unstable. We observe clearly the mutual attraction of vortices, followed by the beginning of the coalescence of vortices

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The Vlasov Equation 1

Figure 5.11. Simulation of Figure 5.10 continued over time. The pictures show clearly the ejection of one of the vortices followed by fusion. Thus, two successive pair-wise vortex merging are observed leading to the formation of a global large phase space structure

Nonlinear Properties of Electrostatic Vlasov Plasmas

Figure 5.12. Time evolution of the three first modes of the electric field, in a logarithmic scale for N = 3. We observe that the mode k = 3k0 is the marginal mode that remains constant before the coalescence takes place at time tωp ∼ 100

241

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The Vlasov Equation 1

When the contribution of the advection term in the Vlasov equation is significant, the eigenfunctions of the operator Aˆ are expected to invoke high-order velocity modes (or equivalently fine velocity scales due to the filamentation phenomenon in velocity), leading to a slow convergence of the solution. To ensure fast convergence of the eigenvalue calculation, a spectral deformation method has been introduced (originally developed for quantum mechanical problems). The electric field is written in the form  Ek0 (t) = Ek0 (0) j ej eiωj t , where Ek0 (0) ej is the projection of Ek0 (0) onto the eigenspace corresponding to the imaginary eigenvalue iωj , which allows an accurate presentation of the vortex merging scenario. In particular, in the case of a chain constituted of three BGK holes, the method predicts a growth rate of ωγp  0.047, consistent with the time scale of development of the instability tωp = 150 in our simulation. Here, the second harmonic (i.e. 2k0 ) appears to be the most unstable and is found close to ωγp = 0.044, in close agreement with the predicted value. More interesting is the result obtained for a BGK hole, which can be seen as a system of two eigenvalues γ1 γ2 ωp = 0.055 (corresponding to the e1 state) and ωp = 0.040 (for the e2 state). The authors showed that it was possible to excite each mode independently by imposing perturbations of appropriate wavelength, leading to a pair-wise vortex merging process, often observed in semi-Lagrangian Vlasov simulations. For e1 , the fastest growing mode, the plasma evolves in to a two-hole system, whereas at the same time the state e2 corresponds to a more complex situation, in which a symmetric vortex-merging process is occurring, with the subsequent nonlinear evolution toward a single BGK state. 5.6. Traveling waves of BGK-type solutions Because we want to make a connection with the linear theory when the electrostatic potential is assumed to be weak, we now begin here by postulating the existence of solutions of type f (x − vφ t, v − vφ ) and E (x − vφ t) for the corresponding electron distribution function and the electric field. Recently, utilizing the formal approach of BGK, Holloway and Dorning (1989, 1991) have given a rigorous description of spatially periodic nonlinear waves of this type, the existence of which illustrates the incompleteness of the linear theory of longitudinal wave phenomena. For more details, see also Buchanan and Dorning (1993); Lancellotti and Dorning (1998, 1999, 2003). These authors have examined traveling waves with a phase velocity vφ that are small perturbations of a given equilibrium, described by a spatially uniform distribution function F0 (v), and saw what conclusion we can draw about the minimum “amplitude” of these waves. Thus by choosing a class of even distribution, a traveling-wave solution with

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phase velocity vφ can be described by the set of a distribution function of type f (x, v, t) = f (x − vφ , v − vφ ) and an electric field E (x, t) = E (x − vφ t). By transforming into the wave frame, the quantities f and E satisfy the following conditions (with the following variable changes ζ = x − vφ t and u = v − vφ ): u

∂f eE (ζ) ∂f + =0 ∂ζ m ∂u

dE e = dζ 0



+∞ −∞

[5.31] 

f (ζ, u) du − n0

[5.32]

Moreover, for a periodic traveling wave to exist near the homogeneous equilibrium given by F0 (v), the electric field E must be a periodic function, which satisfies the following condition: d2 E + κ2 E + γ (ζ) E = 0 dζ 2

[5.33]

where e2 κ = m0 2



+∞

−∞

1 dF0 e2 du and γ (ζ) = u du m0



+∞

−∞

1 dδf du u du

[5.34]

Note that the integrals appearing in equation [5.34] are not singular because only the distribution function, which is here written in the form of a summation of the equilibrium state plus a fluctuating part (f = F0 (v) + δf ), is assumed to be an even function. To find traveling waves near the equilibrium F0 (v), a necessary condition is κ2 ≥ 0, or equivalently, κ2 =

e2 PP m0



+∞

−∞

1 dF0 (v + vϕ ) dv ≥ 0 v dv

[5.35]

for a given phase velocity vφ . Then, there is a wave-vector k = −κ and a corresponding frequency ω = kvφ , which must satisfy the dispersion relation originally given by Vlasov (1945) (here P P denotes the principal part): e2 1− PP m0 k 2



+∞ −∞

F0 (v) dv = 0 v − ωk

[5.36]

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The Vlasov Equation 1

The principal value integral, which was introduced without justification by Vlasov and in approximations to the linear theory by Landau (1946) or by Van Kampen (1955), arises here, in a natural way, because the part added to F0 (v) does not contribute to κ2 . After a change in integration variable, the dispersion relation [5.36] becomes k2 =

e2 PP m0



+∞ −∞

F0 (v) dv v − ωk

[5.37]

which is just the Vlasov equation relation. It is not equivalent to the Landau dispersion relation, which, for undamped waves, takes the standard form k2 =

e2 PP m0



+∞ −∞

F0 (v) iπe2   ω  dv − F v − ωk m0 0 k

[5.38]

  and which can be satisfied only if F0 vφ = ωk = 0. The results of the nonlinear analysis summarized in this subsection neither support nor contradict the linear analysis concerning Landau damping, but do establish the existence of small-amplitude traveling wave solutions with an undamped electric field in plasma presenting single-humped equilibria. A plot of k versus ω, computed from equation [5.37] for the same thermal equilibrium plasma, is shown in Figure 5.13. This dispersion diagram shows a high-frequency branch usually called (nonlinear) Langmuir waves, but unlike 2 the traditional (linear) Langmuir waves (described by ω 2 = ωp2 + 3k 2 vth ), their nonlinear version does not damp, not even slowly. Another branch of waves, found at low frequency and long wavelength, corresponds to the trapped electron acoustic wave (TEAW). It is sometimes simply referred to as the electron acoustic wave (EAW), having a frequency close to ωEAW = 1.3kvth . It will be observed that these two branches are in fact connected, so there is both a frequency cut-off and a wavenumber cut-off leading to a mechanism of resonance loss. Finally, it must be pointed out that the TEAW (or EAW) branch corresponds to the excitation of a low-frequency pole (different from the Landau pole) as mentioned in section 2.5.3. These developments show that the linear theory does capture some quantitative features of exact undamped traveling waves, but nevertheless its predictions are incomplete because it does not describe the population of trapped electrons.

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5.7. Role of minority population of trapped particles

Figure 5.13. Plot of the nonlinear dispersion diagram showing two different branches: the upper branch corresponds to the high-frequency regime (the Langmuir wave in the linear standard approach), whereas the lower branch is now related to the low-frequency nonlinear “electron acoustic wave”

Landau damping is the oldest – and most famous – kinetic effect exhibited by the Vlasov equation. Landau damping become the paradigm of all wave–particle resonance phenomena. It may be worthwhile to mention that, in the linearized treatment of his equation, Vlasov (1945) missed an aspect connected to analytical continuation in the complex plane (although today such an aspect of the problem received new attention by Buchanan and Dorning (1993, 1994)) for its applicability to obtaining a traveling BGK solution of the Vlasov–Poisson equation in the regime of strong Landau damping. Landau (1946) gave a correct – but somewhat obscure – treatment of the problem. Consequently, a long discussion took place until 1962 (some authors claiming that it was a mathematical artifact with no physical foundation) until Dawson (1961) and independently Eldridge and Feix (1972) showed its existence through computer experiments. It is worth pointing out that they showed the occurrence of Landau damping not on the evolution of a macroscopic perturbation but in the microscopic fluctuation properties of a plasma at thermal equilibrium (i.e. through an extension of the Landau theory at a microscopic level). A more direct check was finally given by the Eulerian codes of Knorr (Fourier–Fourier) and the Hermite Fourier algorithms of Grant and Feix (1967), Armstrong (1967) and then the spline interpolation technique by Shoucri and Gagne (1978). These authors noted that an increase

246

The Vlasov Equation 1

in the nonlinearity (i.e. the amplitude of the perturbation) stops the damping after some oscillations with a possible modulation of the amplitude, and finally, the establishment of a steady undamped oscillation. A clear physical picture of this phenomenon has awaited the appearance of graphical software allowing a representation of the 2D phase space f (x, v). Using these, Bertrand et al. (1989) showed phase space pictures which had not been obtained by particle codes. Landau solved the linearized Vlasov equation by using the separation of modes and the Fourier transform in space and Laplace transform in time. An alternative consists of writing the solution as a combination of eigenfunctions, the so-called Van Kampen modes. Landau damping has been understood at the linearized level for a long time, but the study of the nonlinear Landau problem poses important conceptual and technical problems. Dawson (1961) and O’Neil (1965) have proposed an interpretation of Landau damping as the result of resonant wave–particle interactions on which there is nowadays a quite general agreement. Since Dawson’s interpretation, it is widely accepted that the dissipation of the wave energy is due to the energy continuously gained by “resonant” particles, i.e. particles located in the region in velocity close to the phase velocity of the electron plasma wave (EPW). This phenomenon has been the subject of a great deal of research since it remains of considerable interest today. Recently (see the works of Klimas et al. (2017) and Belmont et al. (2008)), attention has been drawn for the transition that can occur, as the strength of the density perturbation is increased, from this Landau damping scenario to what is sometimes referred as the O’Neil scenario. In such a scenario, the damping is stopped and a non-zero asymptotic wave follows. The O’Neil treatment invokes electron trapping as the main mechanism of saturation of nonlinear Landau damping. Simulations of the arrested Landau damping scenario often end in a quasi-steady state that suggests the possibility of the emergence of a BGK equilibrium state (see Bertrand et al. (1989) and Manfredi (1997) for more details). Sichenko (1997) found that at long times, the longitudinal electric field scales algebraically as t−1 , but Lancellotti and Dorning (1998) pointed out that the analysis is not self-consistent. More recently, Villani (2010a) and Mouhot and Villani (2010) proved that, for any initial data near an analytical linearly stable stationary state, the electric field decays. They proved that the nonlinear Vlasov equation does enjoy the same long-time mixing properties as the linearized Vlasov–Poisson version, when the initial data are a perturbation of a stable equilibrium, and provided an interpretation of the damping property as related to regularity (the so-called “gliding” regularity). Thus, the convergence of the distribution function holds only in the weak

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sense: the norms of velocity derivatives grow quickly in time, which reflects the filamentation process of the distribution function in the velocity space and the transfer of energy (or information) from low to high frequencies (a notion seen in weak turbulence). It must be pointed out that weak convergence preserves only the information of low-frequency modes (such as in a weak turbulence scenario), and there is a loss of information. It is this transfer of information to small scales that allows us to recover the reversibility of the Vlasov equation (as in the echo phenomenon), with the seemingly irreversible long-time behavior. Thus, the convergence seems to be based on a reversible, purely deterministic mechanism, without any Lyapunov function nor variational interpretation. The asymptotic solution may keep the memory of the initial data and the interaction. A last remark concerns the parallel with the classical Kolmogorov–Arnold–Moser (KAM) theorem. There is an analogy between the phenomenon of small divisors in the KAM theory and time resonance, which causes the echo phenomenon in plasma physics. 5.7.1. Nonlinear Landau damping and the emergence of the nonlinear Langmuir-type wave We now come back to nonlinear Landau damping. The problem is the following: the plasma is unidimensional and electrostatic with only electrons moving (ions forming a homogeneous neutralizing background). The problem is given with the following initial condition: f (x, v, 0) = FM (v) (1 + αcos (k0 x)) where FM (v) =

2 2 √ 1 e−v /2vth 2πvth

[5.39]

is the Maxwellian distribution function.

Before studying the nonlinear regime of Landau damping, we first present an example of linear Landau damping. We choose to start with k0 λD = 0.5 using a phase space sampling Nx Nv = 128 × 512 grid points and a time step of Δtωp = 0.1. Numerical simulations have been performed using a semi-Lagrangian version of the electrostatic Vlasov code, using a spline cubic interpolation technique. The linear Landau damping of the electric field energy is clearly visible in Figure 5.14, in which we have used a logarithm scale. A good agreement is obtained between the measured (numerical) damping rate of γnum /ωp  0.152 and the expected theoretical value, which is γL /ωp = 0.153.

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The Vlasov Equation 1

Figure 5.14. Time evolution of the electrostatic energy for the study of linear Landau damping. We clearly observe the damping of the Langmuir wave in good agreement with the theoretical prediction. Here, we have taken k0 λD = 0.5 and α = 10−3

Figure 5.15. Time evolution of the electrostatic energy for the study of nonlinear Landau damping. We clearly see that the amplitude of the oscillation is now modulated at the beginning of the evolution until time tωp 300, where we reach a steady amplitude of the oscillation. The corresponding phase space representation is shown in Figure 5.17. The physical parameters are k0 λD = 0.3 and α = 0.2

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We now consider the nonlinear regime of Landau damping by considering a series of experiments performed with α = 0.2 and k0 λD = 0.3 in a periodic box of length L = 2π/k0 . The Landau damping is then γωLp  0.018 with a phase velocity of vφ  3.9vth . Both theoretical and numerical studies indicate that after an initial Landau damping, the amplitude of the wave changes in time in an oscillatory manner, due to a periodic exchange of energy between the electrons trapped in the potential well of the plasma wave and the wave itself. Figure 5.15 gives the variation with time of the electrostatic energy in a logarithmic scale. The fundamental oscillation is at a frequency of ω  2.3ωp (i.e. ωe = ω2  1.16ωp ), a value somewhat larger than the linear value of  3  1.126ω . The spectrum in frequency is plotted in ωLang = ωp2 + 3k02 vth p Figure 5.16. As time goes on (tωp ≥ 300), the amplitude oscillations tend to a constant level. A rough explanation was given through the use of quasi-linear theory arguments. The help of the high-resolution phase space diagnostics of the Vlasov code allows to give a much more precise explanation. The upper part (v ≥ 0) of the phase space is represented in Figure 5.17, which focuses on the small values of the distribution function in the range of 0.0005 ≤ f ≤ 0.05. We clearly see in the first pictures in Figure 5.17 the overtaking and the subsequent breaking of the plasma wave, followed by the formation of a hole (a BGK-traveling-type structure) with a filamentary structure winding around this hole. The rotation of these filaments inside the hole structure is related to a bounce frequency and is responsible for the oscillations of the amplitude of the electric energy. After a long time, the filaments have disappeared due to the phase mixing, and the persistence of the oscillation is simply related to the “ballistic” motion of the holes (each hole of positive velocity vhole is associated with another hole of negative velocity). This hole’s motion at constant velocity associated with the spatial periodicity with wavenumber k0 gives the corresponding frequency ω = k0 vhole , obviously close to the Bohm–Gross frequency since vhole is close to the phase velocity of the wave vφ = 1.16 0.3 vth = 3.86vth . Note that the formation of holes corresponds to the presence of a small population of trapped particles, which plays the fundamental role in stopping Landau damping. Reducing α with the same k0 λD gives rise to smaller holes and consequently a smaller asymptotic energy level. If we reduce the amplitude perturbation α to a weak level, we then recover the standard linear Landau damping without the presence of trapped particles. Note that the presence of the holes was already obtained 30 years ago by Bertrand et al. (1989) with a modest numerical effort (the treatment of Nx Nv being 64 × 256 grid points). With this number of particles, typical PIC simulations hardly demonstrate the existence of the holes. The phase space sampling can be increased in order to test the validity

250

The Vlasov Equation 1

of the traveling BGK-type solutions proposed by Dorning et al. in the previous section. It must be pointed out that the propagating hole, observed in this semi-Lagrangian Vlasov–Poisson simulation, exhibits high-frequency behavior close to the Bohm–Gross dispersion relation, i.e. largely above the low-frequency KEEN waves. To check the prediction of a limit in the frequency wave-vector diagram close to kλD  0.53, we now increase the parameter k0 , keeping however only one structure in the system.

Figure 5.16. Spectrum in frequency of the electrostatic energy (top graph) and component |Ek0 (ω)|2 , Ek0 being the first Fourier mode (at k0 ) of the electric field (bottom graph). While the spectrum of the global electrostatic energy exhibits a narrow peak close to ω ∼ 2.3ωp , we observe clearly the contribution of the two first harmonics in |Ek0 (ω)|2 . The physical parameters are k0 λD = 0.3 and α = 0.2

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251

Figure 5.17. Phase space representation of the distribution function at eight different instants during the nonlinear evolution of the system in the regime of Landau damping with k0 λD = 0.3 and α = 0.2. We clearly see in the first four pictures the overtaking and the subsequent breaking of the “phase space wave”, followed by the formation of an electron hole. For a color version of this figure, see www.iste.co.uk/delsarto/vlasov1.zip

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The Vlasov Equation 1

For such a high-frequency regime, if we come back to the analogy with the KAM theorem, we now have to deal with a continuum of particles as in the Vlasovian approach, so one might well expect that the system keeps memory of the linearized situation where Landau damping is the dominant process (this state being recovered when the perturbation is weak). Thus, by increasing the perturbation amplitude and by maintaining the same initial condition of the Maxwellian distribution function, one might expect to find without difficulty a new (nonlinear) equilibrium state, now characterized by a trapping process. Note that, in the case of a low-frequency regime, the situation is somewhat more complex, and it may become difficult to reach said nonlinear state without modifying the initial distribution function. Let us now consider a third numerical simulation performed with a value of k0 λD = 0.5 and for a perturbation amplitude of α = 0.2, keeping the other numerical and physical parameters identical. The behavior of the system in phase space is shown in Figure 5.18 at different instants. We clearly observe the formation of a filamentary structure (this structure is quite apparent in the three first pictures, at times tωp = 10, 20 and 40). After tωp  70, a hole is forming and filling the gaps between filaments. A large hole is now apparent in phase space at tωp  450, which does not evolve any more (another being present in the v ≤ 0 part of the phase space not shown in the pictures). Now these holes drive oscillation for the bulk of the plasma at a frequency ω = k0 vφ  1.35ωp (with vφ = 2.7). But since the holes’ velocity is just vφ , the phase velocity of the initial (linear) plasma oscillation, we see that the plasma goes on oscillating at a frequency very close to the initial oscillation. The nature of the nonlinear wave is somewhat modified in comparison to that observed for small kλD values, as indicated by the spectrum in frequency of the electrostatic energy shown in Figure 5.19. Thus, for larger kλD , kinetic effects are more important and the plasma wave now exhibits a low-frequency component. We see up to the time tωp  70 the appearance of the filamentation (or Van Kampen modes) characteristic of the free particles’ motions. Then around this time, the hole appears pushing up and down the neighboring filaments and reaching a sort of asymptotic regime (at time tωp = 1,000). By increasing kλD to 0.6, the plasma wave is strongly affected by the Landau damping and now no hole is forming during the initial phase of the plasma’s evolution, as previously predicted by the linear analysis (the plasma dispersion relation being plotted in Figure 5.13). The size of the resulting electron hole, observed at the asymptotic regime, depends strongly on the parameter kλD . A plot of such phase space structure is shown in Figure 5.20 for four different values of kλD = 0.15, 0.3, 0.5 and 0.6, showing that the hole size strongly decreases at kλD  0.6, although the hole does not disappear completely for this value.

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Figure 5.18. Phase space representation of f for larger kλD : kinetic effects are more important and the wave experiences more significant damping before the appearance of smaller electron holes. The phase space pictures correspond to k0 λD = 0.5 and α = 0.2. For a color version of this figure, see www.iste.co.uk/delsarto/vlasov1.zip

254

The Vlasov Equation 1

Figure 5.19. The corresponding spectrum of the electrostatic energy for k0 λD = 0.5 and α = 0.2. The phase space representation of the distribution function is shown in Figure 5.18. Note the strong contribution of a low-frequency mode

5.7.2. Electron acoustic wave in the nonlinear Landau damping regime The concept of KEEN waves is a particular class of the (non-resonant) general BGK equilibria in 1D plasma, in which there might be a wave train with trapped electrons such that there is no further evolution in some particular frame (the “wave frame”) in which the electron velocity distribution function is stationary. Because these waves are essentially kinetic and electrostatic, because they involve electrons (with fixed ions playing a negligible role) and because they are, as it proves, essentially nonlinear and non-resonant, it was decided that a distinct acronym was required to emphasize this (see also Afeyan et al. (2004); Johnston et al. (2009)). Because the waves are non-resonant, they can be driven in over 2 : 1 range of frequencies from 0.8ωEAW to 1.6ωEAW , and in general they have no linear counterpart, while trapped EAWs in their nonlinear version exhibit an = 1.35vth and can be interpreted as a emergent resonance at vφ = ωEAW k resonant version of KEEN waves, the resonance coming from the amplification, here possible due to the existence of a linear wave (EAW) at small value of the electric field. It must be pointed out that the acronym EAW has been previously used to denote waves in the linear wave frequency gap

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between the electron and ion plasma frequencies in contexts ranging from ionosphere observations by Gary (1987), possibly in plasmas with appropriate amounts of clearly defined populations of cold and hot electrons, or in plasmas with Landau damping set equal to zero (perhaps because of some special adjustment to the electron distribution function at some designated phase velocity, to choose the distribution of such trapped particles appropriately so as to create electrostatic plasma waves with an essentially arbitrary relationship between frequency ω and wave vector k).

Figure 5.20. Phase space representation of the asymptotic regime, at the same time tωp = 960, for four different values of the parameter k0 λD , showing that the size of the electron hole in phase space strongly decreases when the Landau damping becomes stronger. For a color version of this figure, see www.iste.co.uk/delsarto/vlasov1.zip

However for EAWs, in their nonlinear version characterized by the excitation of electron holes in phase space, the relationship between ω and k

256

The Vlasov Equation 1

seems to be less arbitrary, and is given in the limit of zero amplitude by the Vlasov dispersion relation ωp2 1 − 2 PP k

dv

F0 (v) =0 v − ωk

[5.40]

where the function F0 describes the equilibrium plasma state. As already mentioned above, the concept of kinetic waves has however become of wider and wider importance, and one of these reasons is the remarkable set of results by Montgomery et al. (2001, 2002), in a series of laser–plasma interaction experiments. These experiments were aimed at improving our knowledge of stimulated Raman scattering (SRS) from EPWs from a single speckle of laser light, and employed laser scattering as a key diagnostic. In addition to the expected EPW scattering signal, there is also an unexpected signal that was much smaller than that associated with SRS-related EPW (∼ 10−3 lower intensity), but nonetheless well out of the thermal noise. The extremely surprising aspect of this scattering result was that the frequency of the relatively weak electrostatic density wave associated with this signal was only a modest fraction (∼ 0.37) of the plasma frequency, which put it right in the so-called frequency gap where no electrostatic wave thought to be, at least according to linear wave theory for a plasma with Maxwellian electrons. Note that the observed wave-vector was at a modest fraction (∼ 0.26) of the reciprocal Debye length, giving the electrostatic wave a phase velocity not far above the electron thermal velocity. Because we want to analyze the difference between EAWs and KEEN waves, we decide to perform a series of numerical simulations by now building an initial condition allowing us to excite a linear EAW and to observe its transition in the nonlinear regime. The initial condition is given by  f (x, v, 0)

=

2

√

2

− v2 − v2 ncold nhot e 2vcold + √ e 2vhot 2πvcold 2πvhot

× (1 + αcosk0 x)



[5.41]

In equation [5.41], vcold and vhot are the thermal velocity of the cold and hot electron components of respective density ncold and nhot (with ncold + nhot = n0 ). From [5.41], it is possible to choose units such that vcold = 1 (the normalization being made on vth = vcold ) and finally ωp = λD = 1. EAWs are by no means new. They were first identified by

Nonlinear Properties of Electrostatic Vlasov Plasmas

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Fried and Gould (1961) as a strongly damped acoustic-like solution of the dispersion relation for electrostatic waves. By considering a fixed (cold) ion background, we can examine an EAW due solely to the electron dynamics following the work of Gary and Tokar (1985). They showed that the EAW was distinct from the Langmuir wave. Among the more important results were the approximate existence criteria for EAW: Thot ≥ 10 and ncold < 0.80n0 Tcold

[5.42]

Figure 5.21. Temporal evolution of the electrostatic energy, in a logarithmic scale, clearly showing the Landau damping process in the case of an initial condition composed of two (hot and cold) electron populations, describing an electron acoustic wave (EAW). The physical parameters are Thot = 30Tcold , nhot = 0.6n0 , ncold = 0.4n0 , k0 λDhot = 0.15 and α = 10−3 . We recover the expected linear Landau damping until the time tωp = 70, followed by a saturation process

They showed that the dispersion relation for the wave is given by ω = kcse at small wavelength (kλDhot 1, λDhot being defined as the parameter λD by replacing the mean density n0 and the temperature Te by ncold nhot and Thot , respectively) where cse = nhot vhot . As shown by Gary (1987), the EAW can be destabilized by introducing a relative drift between the two components, which is capable of providing the free energy necessary to destabilize the mode. Here, we choose to take Thot = 30Tcold , ncold = 0.4n0 and nhot = 0.6n0 .

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The Vlasov Equation 1

Figure 5.22. Phase space representation of the distribution function in the case of an EAW initially excited in the plasma. The nonlinear dynamics is studied when the level of the initial perturbation is high. The pictures reported here correspond to a numerical simulation performed with k0 λDhot = 0.15 and α = 0.2. Strong filaments are observed in the phase space just before the formation of an electron hole. Note that the size of the hole is strongly reduced in that case. For a color version of this figure, see www.iste.co.uk/delsarto/ vlasov1.zip

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Figure 5.23. The corresponding spectrum in frequency of the electrostatic energy in the case of the initial excitation of an EAW in the nonlinear regime. We observe the formation of a low-frequency electron hole emerging from phase space filaments. The corresponding phase space representation is shown in Figure 5.22. Note that the hybrid nature of the wave also exhibits a highfrequency component

Assuming Thot  Tcold , the Landau damping rate can be approximated by the following relation:  π ncold kvhot γL =  0.075ωp [5.43] 8 nhot (1 + k 2 λ2 )2 Dhot Figure 5.21 shows the time behavior of the electrostatic energy in a logarithmic scale, clearly indicating a Landau damping process with a damping rate of γnum  0.08ωp found in good agreement with the theoretical value obtained in [5.43]. Thus, by increasing the amplitude of the perturbation, we have again been able to reach a nonlinear equilibrium state close to that observed in the linear case but now with the emergence of two small electron holes in phase space. Figure 5.22 exhibits the corresponding dynamics of their formation in the case of an initial EAW case. The corresponding frequency spectrum of the electrostatic energy is shown in Figure 5.23, showing the excitation of a broad spectrum with a strong sideband for weak values of the frequency. Such a characteristic spectrum has

260

The Vlasov Equation 1

previously been found in the case of the high-frequency Langmuir wave in the kinetic regime, where kλD ∼ 0.5. The pictures reported in Figure 5.22 correspond to a numerical simulation carried out with α = 0.2 and k0 λDhot = 0.15. 5.7.3. Kinetic electrostatic electron nonlinear waves To explore the physics of such a low-frequency wave generation and to make contact with a possible laboratory experiment using the ponderomotive force (PF) between two opposing laser beams appropriately separated in frequency, it was decided (beginning in 2001) and now published by Johnston et al. (2009) to undertake a study of the effect of applying a PF drive to a collisionless uniform Maxwellian plasma. More recently, more light has been shed on these nonlinear trapping structures by Ghizzo and Del Sarto (2014), who made Vlasov–Maxwell to deal specifically with the problem of the generation of KEEN waves, but now in the presence of a strong backward SRS. With the aim of obtaining the finest detail of the velocity distribution function, rather than using the commonly used PIC model for the Vlasov plasma with its inevitable noise, we have used the simplest (1D electrostatic, periodic with non-relativistic electrons and stationary ions) semi-Lagrangian Vlasov–Poisson code, as developed initially by Cheng and Knorr (1976) and further developed by Izrar et al. (1989) and Ghizzo et al. (1988b). By varying the carrier frequency of the drive, it has been found that not only can KEEN waves be sustained nonlinearly and self-consistently, but that a wide range of frequencies, hitherto thought to be inaccessible for coherent collective wave excitation (i.e. in a band gap between the linear ion acoustic wave and Langmuir wave), were all legitimately sustained as well. The simulations demonstrated successful generation of indefinitely self-sustained low-frequency waves, produced by using a PF drive which, as it turned out, needed to be applied for a limited time only. Because of the low-noise properties of our SL Vlasov algorithm, some useful diagnostics are readily available, which are not suitable for most PIC codes. For a particular signal, if the frequency spectrum (or a selected portion of it) proves to be dominated by a single slowly varying frequency component, one can obtain at any time the amplitude of the signal envelope and the instantaneous signal phase (and hence the instantaneous frequency) by using the time quadrature signal. Thus, if only a real time signal (say from a probe) is available, one can readily generate the required time quadrature component using the Hilbert transform.

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The signal envelope and signal phase are obtained, respectively, as the square root of the sum of the squares of the real signal and the quadrature signal and as the arc-tangent of the ratio of the real and quadrature signals (provided one arranges for phase continuity as the branch cuts of the arctangent function are encountered). Often one can readily obtain equivalent information concerning the spectrum in frequencies directly from various temporal FFT. However, the use of FFT in that case requires following the plasma diagnostics over a long time in order to obtain an accurate estimation of the signal frequency in the low-frequency regime. Here, in this section, when we show a frequency, it is the frequency of the fundamental spatial Fourier component of the electron charge density, which is here available in complex form (i.e. with real and imaginary components) at each time step and which could be imaged to be also available in a scattering diagnostics in a real experiment. Such a frequency is thus obtained via its temporal derivative ω = ∂ϕ ∂t , where ϕ is the phase of the fundamental spatial component of the electron density n. It is possible to use such a Hilbert method to implement a hybrid version of the Vlasov–Maxwell code, the so-called Vlasov–Hilbert code (see Ghizzo et al. (1995, 1996)). An important application of interest is the beat-wave driven trapping process of electrons, for which the experimental ratios of driver frequency to plasma frequency are very high (ω/ωp ∼ 37 in the experiment described by Clayton et al. (1993) or even ω/ωp ∼ 100 in the experiment of Amiranoff et al. (1992)). Such high ratios impose a prohibitive computer burden on a direct attack via Vlasov–Maxwell simulations. Thus, the full Vlasov apparatus can be used for the longitudinal plasma wave aspect of the problem, but now with the addition of the PF driver, which is itself obtained from complex amplitude (i.e. complex envelope) coupled wave equations for the electromagnetic driver waves. Thus, the complex envelope for the rapidly oscillating phases is obtained by using a spatial Hilbert transform of the mode effected as follows: the signal E is Fourier transformed and the positive k modes are multiplied by +i, while the negative k modes are multiplied by −i. The resulting function is then Fourier transformed back EHilbert , which results in a Hilbert transform. The complex amplitude of the field Ee is then given by the relation: Ee (x, t) = a (x, t) eiϕ(x,t)

[5.44]

with an envelope given by a (x, t) =

 2 E 2 + EHilbert

[5.45]

262

The Vlasov Equation 1

with a phase ϕ defined by the relation tgϕ =

EHilbert E

[5.46]

For the waves of interest here, numerical simulations have shown that, if the PF drive is of sufficient level and if applied long enough, a KEEN wave is produced with distinctively non-sinusoidal waveform, and this wave then persists for very long time after the drive has been turned off, far longer than the linear decay time, as if the linear Landau damping has been destroyed. This striking longevity of the KEEN waves is clearly remarkable and has led to a study of the drive threshold above which the persistent KEEN waves are produced. We have found empirically that, for any particular time program (i.e. particular values of Ttrans and Tplateau 2), that we used for the PF drive, there is a rough threshold for the PF drive strength, above which a KEEN wave can be produced readily and below which no enduring KEEN wave would result. Conversely, for a given PF drive strength, a sufficiently long PF drive envelope effective duration Tef f is required for producing a self-sustained KEEN wave (here we define the effective drive duration as Tef f = Tplateau +Ttrans , which is the effective time during which the drive is at its maximum). Figure 5.24 shows an example of KEEN wave generation in the plasma ekφ0 when the FP drive is high enough (with a0 = 0.05), where a0 = mω is p vth the normalized PF drive amplitude. Because we wished to be able to take spatial Fourier transforms easily, we chose to have Nx = 512 space mesh points in our space box, which was chosen to be exactly two KEEN wavelengths long. For a typical KEEN wave value of 0.26 for kλD , this meant that our space mesh step was thus typically x = 0.0944. The time step was chosen to be tωp = 0.04, so for a typical plasma KEEN wave 1 frequency value of 0.37ωp , the time step was roughly 425 of this KEEN wave period, allowing excellent representation of the highest harmonics of interest. Finally, in our Vlasov simulation, we used Nv = 1024 points in the velocity space, over a range of ±6vth , where vth is the electron thermal velocity of our initial Maxwellian plasma (or equivalently for a velocity interval of v = 6/512 = 0.01172 in vth units.

2 The parameters of the PF are chosen at well, these parameters being the driving frequency and wave-vector, maximum amplitude, and envelope shape as a function of time, including duration, and in particular the temporal envelope of the PF drive consists of two smooth transitions from zero to maximum drive level and back to zero, each with a duration Ttrans and a plateau Tplateau in between.

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Figure 5.24. Successful KEEN wave results with the trapping time τB shown here by the double-headed arrow, being about equal to the drive duration. The result of the external drive is to produce a long-lasting KEEN wave after time tωp = 200

Numerical simulations performed for this problem suggest that a criterion of necessary drive strength and duration for producing a surviving KEEN wave can be established in terms of the characteristic electron bounce (or trapping) time τB relevant to the formation of an electron trapping structure in phase space. When electron trapping is discussed, usually only an electrostatic field is involved. In calculating the electron bounce time τB when both PF drive and the induced electrostatic field coexist, as here, one needs to obtain the result including the dynamics of coexistence in a self-consistent way. A reasonable concept, used by Johnston et al. (2009), is to estimate the response of untrapped particles using the linear theory. A

264

The Vlasov Equation 1

Fourier transform of the linear response to the combination of ponderomotive potential φP F and plasma electrostatic potential φ can be as δn = −k 2 0 χe (ω, k) (φP F (ω, k) + φ (ω, k))

[5.47]

where χe is the electron susceptibility and ω and k are the frequency and the wave vector of the response. Taking a Fourier transform of the Poisson equation leads to the expression k 2 φ (ω, k) = δn(ω,k) . Eliminating φ results 0 in equation [5.48], from which one gets the relationship between δn and φP F , and then, upon taking the inverse Fourier transform, that between φ (x, t) and φP F (x, t) we then obtain δn = −k 2 0

χe χe φP F and φ (x, t) = − φP F (x, t) 1 + χe 1 + χe

[5.48]

The result for the total effective potential for the electrons, i.e. φtotal = φP F φP F + φ, is then given by φtotal = 1+χ . Approximating the total potential e φtotal trough with a parabolic one, we have φtotal

  φ 0 1 − k 2 x2 =− 1 + χe

[5.49]

and near its minimum, the bounce period of an electron in this potential becomes  2π 2π 1 + χe = [5.50] τB = ωB ωp kλD a0 ekφ0 is the normalized ponderomotive drive amplitude used where a0 = mω p vth in the simulation, φ0 the maximum value of the PF drive and χe the electron susceptibility, e and m the usual electron charge and mass and k is the drive wave-vector.

It must be pointed out that for a resonant electron plasma situation, such as the Langmuir electron plasma resonance with low loss, the function 1 + χe is very small, being about Im (χe ), and the bounce period is then determined by the very high field of a high-Q resonance. However, for the response at the low frequencies of interest here, χe is large and complex, so the electric field is almost anti-phase to the PF, leaving a relatively small combined field 1 ) nearly in quadrature (i.e. differing in phase by nearly ± π2 ) (close to 1+χ e with the two large fields (electric and pondemomotive). We have found (see Figure 5.24) that the electron trapping and the associated nonlinearity essential

Nonlinear Properties of Electrostatic Vlasov Plasmas

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for the formation of a self-sustaining KEEN wave, is fairly well established during the plasma to about t = τB . This bounce period was indicated in Figure 5.24 by the length of the double-headed arrow labeled τB between the EF (electrostatic force) and PF frames. For our usual parameters ω  0.37ωp , kλD = 0.26, we have = −1.21 + 9.59i and the normalized trapping time is then  −1 9.59 τB ωp = 2π 0.26a0  38.2a0 2 . As shown directly in Figure 5.24, between τB and τB the waveforms of the plasma field become markedly 2 nonsinusoidal, indicating strong nonlinearity and there is also a very characteristic change in the net force that becomes comparable to the maximum ponderomotive driving force. χe

Although the KEEN waves we discuss are driven up using a spatially purely sinusoidal traveling-wave PF driver, it was noted from the outset that when we obtained a self-sustaining KEEN wave (by which is meant one that survived long after the drive had ended), the waveform of the KEEN wave thus produced always turned out to be very nonsinusoidal, as shown in Figure 5.24, third panel. This nonsinusoidal waveform is a clear indication that essentially nonlinear physics is involved (hence the emphasis on nonlinear in the acronym KEEN). Strong nonlinearity is not surprising since, as we will see below, particle trapping is absolutely central to KEEN wave formation. This behavior is in contrast with the behavior of the usual Langmuir plasma wave of modest wavenumber (k 2 λ2D 1), which can exist at very low amplitudes with little trapping and with quite sinusoidal waveforms even at significant strength. The effects of the nonsinusoidal waveform are seen to be significant if one looks at the dynamics of the electron distribution function in phase space, over a wavelength for the formation of a typical KEEN wave, as shown in Figure 5.25. The evolution of the distribution function is plotted for t 1 τB = 2 , 1 and 10 for a0 = 0.05 with their total force potentials superposed. The development of the electron distribution function up to t = τ2B reflects a very normal early evolution of f . Between times t = 12 τB and t = τB , one sees the formation of a well-organized vortex in phase space (an electron hole), which would seem likely to survive, just as the drive is being terminated. With the drive already turned off, the phase space vortex shows some complicated evolution with fine ripples. By t = 10τB , the distribution function in Figure 5.25 appears to have evolved to relatively smooth states, which resemble BGK-like states, after a cleaning mechanism is operating, a process already observed in previous simulations of BGK coalescence presented in section 5.5.

266

The Vlasov Equation 1

Figure 5.25. Phase space pictures of the electron distribution function during the drive-up period (for t ≤ τB ) and after it (namely at t = τ2B , τB and 10τB ). The total force potential profiles for each time frame are superposed for convenience. Here, a grey shaded representation has been used

The question of whether there is continuing evolution to an ideal BGK state or whether fluctuations about this ideal state do not die out with time must be left to investigations with more sophisticated diagnostics than those employed here. However for this aspect, as well as investigating the KEEN waves driven up ponderomotively, it was judged appropriate to check the validity of the final phase space structure as an ideal BGK state. One should recall that the BGK concept, when reduced to its essentials, is the assertion that steady state depends only on invariants. In the 1D electrostatic case of interest here, this means that the distribution function must be a function only

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of the particle total energy (from which the velocity may be calculated when the electrostatic potential is given).It is often not appreciated that the existence of a BGK solution (see the work of Bernstein et al. (1957)) does not say anything about whether it is stable or not in a particular reference frame. The assertion was only that if such stationary solutions exist, then they must follow the BGK rule. One should therefore always test such a BGK candidate for stability with a proper simulation to see that it will evolve close to the original formulation.

Figure 5.26. Typical phase space representation of a stable nonsinusoidal KEEN wave distribution function snapshot (in banded rainbow presentation) at time tωp = 1, 000. Note that the density contours are indeed parallel to the contours of constant wave-frame total energy as expected from the BGK concept. For a color version of this figure, see www.iste.co.uk/delsarto/vlasov1.zip

What we have found is that, as shown in Figure 5.26, electron distribution density contours in phase space seem to follow quite closely the contours of the total particle energy as they should in a BGK equilibrium. In the phase space snapshot in Figure 5.26, the banded rainbow presentation shows the contours of constant density, and how they lie parallel to the contours of 2 constant total energy H (where H = 12 m (v − vφ ) + eφ). It is useful to note that, to the extent that the KEEN waves resemble perfectly stationary BGK waves in the wave frame, the constant density contours (and in particular the

268

The Vlasov Equation 1

ones that go through the X points) reveal the form of the square root of the electrostatic potential. Note that the PF drive plateau value was a0 = 0.052, Ttrans ωp = 50, Tplateau ωp = 150, for a total duration of 250 with ω = 0.37ωp and kλD = 0.26. 5.7.4. Emergent resonance for KEEN waves We have found that electron trapping and the associated nonlinearity, essential for the formation of a self-sustaining KEEN wave, are fairly well established during the driving phase by t = τ2B , but to reach a well-established KEEN state, one should at least continue to drive the plasma to about t = τB . This is the threshold criterion we adopt here for the PF-driver KEEN waves. As is well-known, at the low frequencies of interest, the linear plasma response has no weakly damped eigenmodes, so one might well expect that there is little selectivity in frequency at a given wavenumber for driving up KEEN waves, and this is more or less true. In Figure 5.13, the (ω, k) results obtained by artificially setting the imaginary part of the linear electron susceptibility equal to zero are shown for reference, thus artificially eliminating Landau damping. As previously mentioned, the lower part of this “thumb” curve is what has been called the EAW branch. While the EAW curve does give some guidance as to the lowest frequency that can be easily excited as a KEEN mode, it is clear that KEEN modes for the wavenumber shown can be driven up very far away (in the ω − k space) from that locus. We see that KEEN waves can be readily produced over a 2:1 range of frequency (80% of the EAW frequency to 160% of the EAW frequency) for the typical drive wavenumber used here. Thus, one of the major points of the work presented here is that the excitation of KEEN waves is a broadband phenomenon, consistent with the heavily damped nature of the linear plasma response at these KEEN frequencies, i.e. well below the plasma frequency. In a result that differs remarkably from this broadband result, Valentini et al. (2006) obtained a result for a particular case with very low-drive amplitude that seems to be in direct contradiction with the typical result of Figure 5.26, in that a response was obtained that seems to indicate a possible resonance from the linear EAW (which may here play the role of a seed necessary to a resonant amplification till the formation of a BGK-type solution). However, these two apparently contradictory situations are in fact quite consistent with each other. In the numerical experiment of Valentini et al. (2006, 2012), the emergence of the trapping structure is the direct

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consequence of the existence of the linear EAW seed, which allows a resonant amplification. To resolve the apparent contradiction, one must examine the time history of the behavior of the amplitude growth under external drive. The temporal history is shown in Figure 5.27. Beyond the trapping period, there emerges (in this low-drive case at least) a narrow resonance behavior that was termed an emergent resonance by Johnston et al. (2009). It is clear from Figure 5.27 that the narrow resonance for low drives is much longer than the bounce trapping time τB and does not appear in the linear phase of the plasma response to the drive but rather emerges after about two trapping periods, in the sense of a frequency-localized (i.e. in a narrow frequency range centered in this case at about ωdriver ∼ 1.10ωEAW ) immunity to the saturation via trapping oscillation of the kind so clearly seen at neighboring drive frequencies (such as ωdriver /ωEAW = 1.0 − 1.3).

Figure 5.27. Extended time history of the KEEN wave energy accumulation from the drive for different driving frequencies, ranging from 0.80ωEAW to 1.4ωEAW . Emergence of the nonlinear resonance is apparent at late times. Long-time behavior of added energy (up to 0.01 for the monotonically growing case) shows how the emergent KEEN resonance at ω = 1.10ωEAW = 0.4015ωp appears at the time where saturation is usually seen at about 2τB . For a color version of this figure, see www.iste.co.uk/delsarto/vlasov1.zip

Understanding how this comes about will require a much more profound study than has been made so far. It is clear that the effect is related to the basic physics of why the normal saturation of the KEEN wave drive occurs and

270

The Vlasov Equation 1

how this does not occur for the right frequency. There is however usually a modification of KEEN frequency (and therefore of its phase velocity) as the drive is removed. Thus, the emergence of the resonance may well be also related to this phenomenon, which may be apparent to an autoresonant mechanism. It must be pointed out that the excitation of the third low-frequency non-Landau pole, shown in Figure 2.1 close to ω = 0.4ωp , is also possible, provided that a large PF is applied. We will see in later sections that the bunching instability, linked to what appears as a trapped particle instability (TPI) coupling, may play a major role in the resonance. 5.8. Nature of KEEN waves and NMI As previously mentioned, KEEN waves are a non-stationary version of BGK equilibria and it was shown in previous sections that the electron distribution density contours in phase space seem to follow quite closely the contours of the total particle energy, as they should in BGK solution. However, on closer examination, deviation from the ideal state was observed that seems to indicate that the BGK state is an ideal state, not necessarily an accessible attractor state; it appeared that it would be quite difficult to produce such KEEN waves at very low field amplitude, so they would not be confused with the result of any linear perturbation or amplification of a linear wave seed. KEEN waves contain multiple robust phase-locked harmonics and differ strongly from EAWs. In a KEEN wave, the trapped electrons play the role that hot electrons play in two-temperature linear EAWs, while the free electrons play the role of the cold population but with trapping eliminating the severe Landau damping. The sensitivity of KEEN waves to the excitation process and the driver amplitude is still an open question. During the growth phase, high spatial harmonics of the plasma field become very noticeable as the KEEN’s net force nonsinusoidal waveform is set up and such a change seems essential to the transition to a free-running KEEN wave. It was recently proposed by Dodin and Fisch (2011, 2014) that such a sensitivity may depend on NMI, a collective property of trapped particles to bunch together with the formation of sideband oscillations of the plasma waves. 5.8.1. Adiabatic model for a single linear wave: the (electrostatic) trapped electron mode model While the basic BGK formalism is easy to understand in the stationary regime and has been discussed previously, the detailed dependence of the nonlinear behavior on NMI and more generally the sideband instability connected to the trapped particle population, is sufficiently complex as to

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271

make physical interpretation difficult. Rather, it is purposed in this section to introduce results on the adiabatic theory for a monochromatic linear wave. First consider the following Hamiltonian: H (x, p) =

p2 p2 + eφ (x) = − eφ0 cos (kx) 2m 2m

[5.51]

where e and m are the particle charge and mass. Governed by equation [5.51], both passing and trapped particles will undergo oscillations that are convenient to describe in terms of the action

1 J= pdx [5.52] 2π C where the integral is taken over one cycle of the motion. Let α be the canonical angle associated with the action J. Following the work of Rosenbluth et al. (1990), we introduce normalized quantities (the Hamiltonian H0 being normalized to kB Te , the momentum variable p to mvth , the time to the inverse of the plasma frequency ωp−1 ), and equation [5.51] becomes H0 P2 2 = [5.53] − Ω0 cosψ k B Te 2  2 2 ωB eφ0 k2 where Ω0 = ω2 k2 λ2 and ωB = is the electron bounce frequency. m H0 =

p

D

Thus, we introduce the normalized action–angle coordinates J = defined for the steady-state Hamiltonian H 0 (with ψ = kx) as:

+π    1 1 2 P dψ = 2 H 0 + Ω0 cosψ dψ J= 2π 2π −π

Jk mvth ,

α

[5.54]

and α=Ω

ψ

dψ    2 2 H 0 + Ω0 cosψ

[5.55]

where ⎛ ⎜ Ω (J) = 2π ⎜ ⎝2



⎞−1 ψ

⎟ dψ ⎟    ⎠ 2 2 H 0 + Ω0 cosψ

=

1 ∂H0 ∂J

[5.56]

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The Vlasov Equation 1

Following the standard Hamilton–Jacobi formalism, the steady-state J Hamiltonian may be expressed as H 0 = Ω (J  ) dJ  . The element of phase space area Γ is then dΓ = dJdα. For a population of trapped particles, we introduce the trapping parameter κ as κ2 =

2

2

H 0 +Ω0 2 2Ω0

(with κ < 1 for trapped

particles and κ > 1 for passing particles). We then have 1 J= 2π



+π −π

  +π ψ 1 2 2 H 0 + Ω0 cosψdψ = 2Ω0 κ2 − sin2 dψ [5.57] 2π −π 2

By introducing the variable change sin

ψ = κsinθ 2

[5.58]

we have, for the action J, J=

4 Ω0 π



π 2

0

4 κ2 cos2 θdθ √ = Ω0 I π 1 − κ2 sin2 θ

[5.59]

The equilibrium trajectories are then expressible in terms of action-angle variables J and α by dα ∂H0 = = Ω (J) dt ∂J

[5.60]

dJ ∂H0 =− =0 dt ∂α

[5.61]

and

In equation [5.59], the quantity I can be rewritten in the following form: I=

π 2

0

 √

 κ2 − 1 dθ

1 − κ2 sin2 θ

+

π 2

0

 1 − κ2 sin2 θdθ

[5.62]

which is given by   I = E (κ) + 1 − κ2 K (κ)

[5.63]

leading to the expression of the invariant (with κ < 1): J=

    4 Ω0 E (κ) + 1 − κ2 K (κ) π

[5.64]

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273

where K and E are the complete elliptic integrals of the first and second kind defined by K (κ) =

π 2

0

√

dθ 1 − κ2 sin2 θ

[5.65]

and E (κ) =

π 2

0

 1 − κ2 sin2 θdθ

[5.66]

In a similar way, the frequency Ω (J) can be calculated from equation [5.56], giving Ω (J) =

πΩ0 2K (κ)

[5.67]

Motivated by applications in laser fusion or laser–plasma interactions, the utility of the adiabatic approximation has recently been confirmed by Dewar and Yap (2009), who visualized the dynamics of a population of electrons trapping in an electrostatic wave of slowly increasing amplitude, illustrating that, despite disordering of particles as they pass close to the X-point of separatrix, the concept of adiabatic invariant is still valid. If the Hamiltonian now depends on the time, i.e. H = H (J, t), the nonlinear frequency is again given by Ω = ∂H ∂J . When the system changes slowly in time, the adiabatic invariant J remains constant to all orders of the parameter measuring the slow variation (see Lenard (1959)). The Hamiltonian of the system is not conserved and the energy of the particles can vary substantially. However, Krapchev and Ram (1980) have shown that it is possible to build a new adiabatic invariant: their theory predicts that from an initial equilibrium with a Maxwellian (and no electric field), the system can evolve in an adiabatic way to only two stable states. One of them corresponds to the standard Langmuir wave that exists even for very small amplitudes of the electric potential. The authors call the other state a trapped particle mode and it exists only at sufficiently large amplitude of the electric field, i.e. when most of the particles become trapped. The corresponding phase velocities of this mode are √ always smaller than those of the Langmuir wave and given by vφ = ωk ∼ φ, where φ is the amplitude of the self-consistent potential (or equivalently ω ∼ kωB ). Some basic phenomena, however, have been left out that correspond to the irreversible processes, which originate in the domain of phase space where the adiabatic theory fails. Thus, the trapped particle mode

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The Vlasov Equation 1

solution seems to be valid nearly everywhere, except for a narrow region of the phase space near the separatrix, i.e. the border line in phase space between trapped and passing particle trajectories. Close to the separatrix one has to keep all the whole irreversible time history of the nonlinear energy transfer, and the theory is essentially non-local. This region in fact corresponds to the non-adiabatic particles, i.e. the resonant particles (in the common language adiabatic particles, or nonresonant particles refer to the condition where ωB  γL , where γL is the Landau damping). Particles near the separatrix are most susceptible to stochastic perturbations. These may ultimately lead to a change in the plasma state by the irreversible transfer of energy and momentum from the wave (the trapped particle mode) to the (resonant) particles. 5.8.2. The Dodin and Fisch model connected to the emergence of KEEN waves It was recently proposed by Dodin and Fisch (2011, 2013, 2014) that KEEN waves can result from NMI. Due to NMI and collective effects, trapped electrons can group together to form a “macro-particle” due to the bunching instability. NMI leads a trapped electron population to “bunch” into rotating macrostructure, which then produces sideband oscillation of the wave field, a signature of the so-called TPI. Before going further, we find it interesting to come back to TPI. Particles trapped by the traveling plasma wave can be considered as a source of plasma wave instability, usually referred as the TPI. Kruer et al. (1969) (see also Kruer and Dawson (1970), Rosenbluth et al. (1990) and more recently Brunner and Valeo (2004), Brunner et al. (2014)) showed that trapped particles led to the growth of sidebands, which is due to the nonlinear resonance between the frequency of the plasma wave and the bounce frequency of trapped particles in the potential trough of the wave. Numerical simulations of these sidebands’ instability (see, for instance, Shoucri (1977)) have shown, in the first phase of the nonlinear evolution, the formation of a stable bump in the distribution function in the velocity space. The evolution of the plasma is then characterized by the formation of a quasi-linear plateau in the distribution function. This stochastization process, driven by the growth of sidebands, tends to flatten the trapped distribution and leads to the formation of a plateau in the region close to the resonant phase velocity. Similar results have been also found in the case of the beam–plasma interaction when several modes become highly nonlinear, so that particle trapping becomes important (see Ghizzo et al. (1988b) for more details). Thus, this interaction leads to an

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effective transfer of energy from the original EPW to the resonant sidebands. More recently, in addition to TPI, NMI can be included in such an approach. What still distinguishes the Dodin and Fisch (2013) model from that of Kruer et al. (1969) is that the former model accounts for the variation in the bounce trapping frequency while the latter does not take into account these effects. It is well known that phase space holes attract one another and it is interesting that the same phenomenon occurs for vortices in a real incompressible fluid in a two-dimensional system. The tendency of electron holes to behave as quasi-particles in phase space comes from the self-bunching instability, a mechanism seen when a fast (keV) bunch of particles oscillates between two electrostatic mirrors, which form a stable trap. This counterintuitive behavior referred as bunching instability belongs to a much larger class of behaviors called NMI (see, for instance, Strasser et al. (2002)), which was first introduced by Nielson et al. (1959) for relativistic circular accelerators or storage rings. In these experiments, NMI occurs when the angular velocity decreases with increasing energy, a situation occurring above the so-called transition energy. An example of NMI, but in a different physical situation, is today offered in particle accelerators or in Saturn’s rings. Under the action of the gravitational field of Saturn, dust particles move along closed orbits and it is the mutual gravitation interaction between particles that leads to the dust’s confinement. The physical conditions in an accelerating device are more rich than those occurring in the mechanism of Saturn’s rings. In such a circular accelerator, a charged particle beam can execute millions or billions of revolutions under the action of electromagnetic forces instead of the planet’s gravitational field. Due to their electric charges, particles repel each other instead of the gravitational mutual attraction among the dust particles. Consequently, the system may be unstable. In particular, when the space charge dominates the beam, particles are paradoxically drawn in the opposite direction toward which the total force is acting, like the behavior of particles having a negative mass. For this reason, the resulting instability is usually referred as the NMI. It was first introduced in particle accelerators by Nielson et al. (1959) and extensively considered in a number of theoretical works by Fedele (1996, 2008) showing the analogy of NMI with quantum mechanics. A nonlinear Schrödinger-like theory for coherent instabilities and in particular for NMI has been developed by Fedele (2008) and shown to be fully similar to the modulational instability that occurs in nonlinear dispersive media such as plasmas. In addition, it has been shown by Fedele (1996) that a deep analogy exists between the physical mechanism used in accelerator physics to describe coherent instabilities for particle bunches and those used in plasma

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The Vlasov Equation 1

physics to describe the oscillating two-stream instability (known also as the filamentation instability in the electromagnetic regime and the parametric decay instability). Coming back to NMI in plasmas, these sidebands may survive in the long run only if they are phase-locked to the main wave, provided that a Langmuir wave was excited initially. One can view such a rotating vortex, resulting from the bunching instability, as an effect akin to the phase-mixing process, allowing us to obtain an ideally BGK-type solution in asymptotic regime. Once phase locked, the wave should be able to tolerate moderate variations of the wave amplitude from the exact resonance, i.e. one can expect that the mechanism is robust enough without strong modification of the amplitude. Such a mechanism resembles the amplification of stimulated Raman backscattering in optical mixing, which exhibits a nonlinear phase locking (i.e. autoresonance) by adiabatic passage through resonance. The observed behavior in SRS corresponds to the combined action of the (nonlinear) shift in frequency of the amplified Langmuir wave induced by nonlinear particle trapping effects (first predicted by Morales and O’Neil (1972) or also by Bertrand and Feix (1976)), compensated for by a retuning of wave number as well, so as to maintain the parametric resonance over a long time (see also Albrecht-Marc et al. (2007) for more details). Another variant of such a mechanism is the chirped beat-wave process as described by Ghizzo et al. (1998), which exploits a robust transition to nonlinear phase locking. Rather than starting on resonance, we chirp from above resonance, slowly sweeping the beat frequency through resonance and below. Thus, under certain conditions, the plasma wave frequency locks into the frequency of the drive (see also Lindberg et al. (2004)). Following the work of Dodin and Fisch (2013), to study such phase locking (or autoresonance) we express the Hamiltonian in canonical action-angle variables: H=

0

J

Ω (J  ) dJ  − eφ (x, t)

[5.68]

The equations of motion are then given by α˙ =

∂H ∂φ = Ω (J) − e ∂J ∂J

∂H ∂φ J˙ = − =e ∂α ∂J

[5.69]

[5.70]

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and the particle distribution f (Γ, t) is governed by the Vlasov equation ∂f ∂f ∂f + α˙ + J˙ =0 ∂t ∂α ∂J

[5.71]

which reads as   ∂φ ∂f ∂φ ∂f ∂f + Ω (J) − e +e =0 ∂t ∂J ∂α ∂J ∂J

[5.72]

which is somewhat unusual as the electrostatic potential is now formally a function of both coordinate and momentum, which obeys the Poisson’s equation: ∂2φ entrapped = 2 ∂x 0

dΓδ (x − X (Γ)) f (Γ, t) −

entrapped 0

[5.73]

where ntrapped is the density of the trapped particle population. We have here   +∞  +π  dΓf (Γ, t) = 1 and Γ dΓ ≡ 0 dJ −π dα. By considering a solution of Γ f in the standard form f=

+∞  F0 (J) δfn (J) ei(nα−ωt) + 2π n=−∞

[5.74]

and the electrostatic potential φ=

+∞ 

δφn (J) ei(nα−ωt)

[5.75]

n=−∞

equation [5.72] leads to −i (ω − nΩ (J)) δfn + i

neδφn dF0 =0 2π dJ

[5.76]

or equivalently δfn =

enδφn (J) dF0 2π (ω − nΩ (J)) dJ

[5.77]

In what  follows, we seek a solution in the linear regime, i.e. using +∞ 2J X (Γ) = mΩ cosα, or by considering that 0 δ (x − X (Γ)) dΓ = 0

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H (cosα) where H is the Heaviside step function. Thus, the Poisson’s  +∞ equation [5.75] writes now (assuming that 0 F0 (J) dJ = 1): ∂ 2 δφ entrapped = 2 ∂x 0

Γ

dΓδ (x − X (Γ)) δf (Γ, t)

[5.78]

Hence, integrating the resulting Poisson’s equation over x from zero to infinity, and assuming that the fluctuations of the electrostatic potential δφ tend to zero when x → +∞, we have: ∂δφ ∂δφ (+∞) − (0) = ∂x ∂x +∞ +∞ + π 2 entrapped −iωt  e einα dJδfn (J) π 0 0 n=−∞ − 2 Using equation [5.77] and assuming that we can write equation [5.79] in the form: δEx (0) = Using Ω20 =

∂δφ ∂x

[5.79]

→ 0 at the limit x → +∞,

+∞ ntrapped e2  nπ +∞ δφn (J) dF0 sin dJ [5.80] π0 2 0 ω − nΩ (J) dJ n=−∞ eφ0 k2 m ,

it is always possible to introduce, in the linear

approximation, a Hamiltonian of type H0 = transformed in the following expression:

p2 2m

p2 p2 mΩ2 mΩ2 H0 = − 2 0 coskx  − 20 2m k 2m k

− eφ0 coskx, which may be



k 2 x2 1− 2

 [5.81]

or equivalently, in a dimensionless form:

H0 =

 2 H0 +

mΩ20 k2 2 mΩ0

 =

p2 + x2 = 2JΩ0 m2 Ω20

[5.82]

allowing us to introduce the expression of x and p in terms of action-angle variables:   2J 2J x= cosα and p= sinα [5.83] mΩ0 mΩ0

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Equation [5.80] represents a coupled set of equations for the Fourier amplitude of one mode δφ (or δEx (0)) in terms of the Fourier amplitudes of other modes. We simplify this set by observing that the plasma does not support wave-type solutions for ω greatly different from the plasma frequency ωp . Thus, we may keep in the summation only two components of n = ±1, 

2J δEx (0) . Here, we replace δφn (J) i.e. δφn (J) = − (δn,1 + δn,−1 ) mΩ 2 0 with the previous expression in the numerator of equation [5.77], which is now written as  √ +∞ 2ω 2pt  J Ω (J) dF0 J0 dJ 2 1+ =0 [5.84] 2 (J) πx0 ω − Ω Ω0 dJ 0

In equation [5.84], x0 is the maximum amplitude of unperturbed oscillations, so the corresponding choice for the action is J0 = 12 mΩ0 x20 . We √ have also introduced in equation [5.84] the quantities ω pt = ωpt / πx0 and e2

2 ωpt = trapped (ntrapped being the density of trapped particles). An m0 integration by parts of equation [5.84] leads to the expression n

√ 2ω 2pt J0 +∞ F0 (J) 1− √ dJ √ Ω0 2 JΩ (ω 2 − Ω2 (J)) 0 ,

ω 2 Ω (1 − β) − Ω2 (3β + 1) = 0 

[5.85]

with β = − JΩ Ω . The Dodin and Fisch model provides a somewhat more detailed description of the trapped particle dynamics than the standard TPI. In addition to TPI, this model also accounts for the so-called NMI, which is a consequence of the decrease in the bounce frequency of less deeply trapped particles compared to deeply trapped ones, potentially leading to the bunching of particles in their phase space rotation (see Hara et al. (2015)). Thus, if we consider two trailing trapped particles on a same phase space orbit (say with the invariant J), Coulomb repulsion will have these trapped particles switch orbits. The leading particle gains energy and then moves to a less deeply trapped orbit (where it sees its bounce frequency Ω (J) decrease), while the trailing particle, losing energy, then moves to a more deeply trapped one with its bounce frequency increased. It is this difference in bounce frequency between the inner and outer orbits that leads to the bunching of particles (a condensation phenomenon) that underlies NMI. This condensation process may or not eventually saturate in the form of a stable quasi-particle (phase space hole) but its key formation constitutes a fundamental instability in itself. Thus, the essential difference between the Dodin and Fisch model and

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The Vlasov Equation 1

TPI is based on the fact that the former accounts for this decrease in the bounce frequency Ω (J), whereas the latter model does not take into account this effect. When formed, such an electron hole may produce sidebands in theory of the main wave field. As the driver continues to feed the instability, these sidebands grow in time and may initiate a phase mixing or a kind of stochastization of the electron orbits, which makes the “ideal” BGK solution more accessible (see also Ghizzo and Del Sarto (2014)). Thus, the asymptotic state is somewhat modified when these sidebands are in an appropriate resonance leading to the formation of an invariant torus in the particle phase space, a solution close to the ideal BGK state. We may also interpret the pair-wise vortex merging as an example of NMI, where a BGK hole can be obtained as the result of the coalescence of two (or several) phase space holes being initially different from BGK states. Thus, in the case of a ring Dirac-type distribution, F0 (J) = δ (J − J0 ), equation [5.85] reads as   ω 4 − ω 2 2Ω20 + ω 2pt (1 − β) + Ω20 ω 2pt (1 + 3β) + Ω40 = 0

[5.86]

For Harmonic bounce oscillations with Ω (J) = const = Ω0 , the ω2

ω2

resolution of equation [5.85] leads to roots ω 2 = Ω20 + 2pt ± 2pt , or equivalently to ω 2 = Ω20 or ω 2 = Ω20 + ω 2pt . For a constant value of β = β0 = β (J0 ), equation [5.86] gives to a solution of type: ω 2 = Ω20 +



1 − β0 2



ω 2pt ±

1 2

 2 ω 4pt (1 − β0 ) − 4β0 Ω20 ω 2pt

[5.87]

If we assume that β0 1 and ω 2pt Ω20 , then we can neglect the term 2 ω 4pt (1 − β0 ) and the solution is written as ω 2 = Ω20 +

ω 2pt  + −β0 Ω20 ω 2pt 2

[5.88]

Hence, having β0 > 0, or equivalently Ω0 = Ω (J0 ) < 0, leads to an instability with ω 2pt ω ω pt  1+ +i |β0 | 2 Ω0 2Ω0 Ω0 which finally corresponds to NMI.

[5.89]

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281

Moreover, the influence of low-frequency nonlinear BGK-type waves induced by particle trapping in backward SRS has been investigated by Albrecht-Marc et al. (2007) and more recently by Ghizzo and Del Sarto (2014) in optical mixing for parabolic plasma profiles in the study of laser–plasma interaction. Numerical experiments performed by an electromagnetic semi-Lagrangian Vlasov–Maxwell scheme have shown that the plasma wave undergoes a frequency decrease according to the basic idea of Morales and O’Neil (1972) regarding the effect of trapped particles on the frequency of the plasma (Langmuir) wave, while the wave-vector increases slightly so as to maintain the SRS resonance. It was originally conjectured by Vu et al. (2001, 2002) that the frequency change would act to detune the interaction and at least substantially limit the effects of the instability but in fact the simultaneous wave-vector retuning allows the instability to continue. What then appears to happen is that, instead of a constant decrease in instability rate due to the increasing energy losses as in early, smaller simulations, the SRS-driven retuned EPW interaction continues easily with well-trapped electrons, until arrested by a new process. This SRS-limiting process begins with a symmetry-breaking instability of pair-wise merging of electron holes in phase space. This vortex merging symmetry breaking behavior also proves to be pertinent even though we are faced with a large parabolic plasma with the persistence of these vortices over several thousands of inverse plasma frequencies. In particular, the electromagnetic spectrum has been analyzed in detail and shown to result from the combined action of the nonlinear shift due to particle trapping and the parametric resonance, which is here rigorously maintained in time in the optical mixing scenario. Thus, the low-frequency system seems to sustain large coherent structures that are stable, and once phase locked, the main wave should be able to tolerate moderate variations of its amplitude from the exact resonance (in spite of the initially parabolic profile), as in a typical autoresonance, i.e. the nonlinear features of the physical process are particularly robust. 5.9. Electron hole and plasma wave interaction The surprising result is that for each realization of an (numerical) experiment of the emergence of an electron phase space hole (TEAW, KEEN hole, “Landau” hole), the coherent structure that persists in time behaves as a quasi-particle and seems to be connected to a given macroscopic wave (EAW, trapped particle mode, Langmuir wave). This occurs when a (nonlinear) autoresonant mechanism takes place: the electron hole and its global nonlinear associated wave forms a symbiotic structure (which resembles a “wave–particle”, duality but note that the associated wave now becomes,

282

The Vlasov Equation 1

nonlinear). Such a “physical picture” provides a means of using one subject or analogy to gain insight into another. For instance in the stimulated Raman backscattering, where the laser pulse now plays the role of the driver and amplifies the Langmuir wave, we can interpret why the Langmuir wave is not Landau damped when the interaction takes place in the so-called kinetic regime of the interaction, i.e. for kλD ≥ 0.3. If the electron hole disappears (by vortex merging for instance or when the external driver is turned off), the initial Langmuir wave is then modified or vanished (by Landau damping). In reverse, if the wave is damped, the energy transfer to resonant particles is stopped and the electron hole cannot be created. Thus, it is the interaction between the quasi-particle (the phase space electron hole) and the nonlinear version of the Langmuir wave that allows the plasma wave to survive. The coupling between the quasi-particle and the plasma “wave” results in a non-local dynamics. The concept of quasi-particle, as the result of energy transfer from the wave with a resonance at the phase velocity, might offer insight into the origins of the quasi-particle formation in space. We see that the large-scale hole structures that develop in plasma as the result of an external PF driver, or due to the modification of the initial distribution function (as seen in the trapped EAWs since the distribution function usually exhibits a plateau in the velocity space), can be interpreted in several different ways. There is an important qualitative difference between the so-called TEAW instability and the KEEN wave formation driven by an external low-frequency PF driver (i.e. well below the plasma frequency and above the EAW signature). In the problem of KEEN waves, the excitation of the mode is rapid and the structure is created on half of the bounce period. For the trapped EAW distribution, however, the system could still support waves at a phase velocity equal to the velocity of the linear EAW, even initially. The emergence of the trapped EAW mode arises with a very narrow resonance for low drives much larger than the trapping/bounce time τB when the initial distribution is a Maxwellian. Thus, an initial deformation of the electron distribution function, exhibiting a “plateau” in velocity around vϕ , is then required in the linear phase of the plasma response (to decrease the contribution of the Landau damping). In the nonlinear state of KEEN wave growth, we saw that when a hole is formed, it tends to move as an independent particle. Since there is little viscosity due to the background fluid, the rest of the plasma seems to play a passive role. However, we found that the electron trapping and the associated nonlinearity, essential for the formation of a self-sustaining KEEN structure, is fairly well established during the driving phase at time t = τ2B , but to reach a well-established KEEN state, one should at least continue to drive the plasma to about t = τB (to produce a KEEN

Nonlinear Properties of Electrostatic Vlasov Plasmas

283

wave, one would have to drive harder in comparison to the EAW fast emergence, or longer or both). Although the KEEN wave, as we discuss here, has been produced using a spatially purely sinusoidal traveling-wave PF driver, it was noted from the outset that when we obtain a self-sustained KEEN wave, the waveform thus produced always turns out to be very nonsinusoidal, as shown in Figure 5.28. It is immediately apparent that it is the energy transfer between the wave (here described by the trapped particle mode) and resonant particles that allows the electron hole to survive and to maintain in a self-sustaining regime. It can be said that most of the distinctively nonlinear effects, resulting in the maintenance of the electron holes, which have emerged so far from the numerical simulations are intimately associated with the region of phase space of strong interaction between bouncing trapped particles and waves. Resonant particles whose velocities are near to the phase velocity can exchange energy with the wave and thus amplify its electrostatic energy. The situation shown in Figure 5.28 indicates that the stability of the self-sustaining electron hole is due to a process by which a collective transfer between the mode and the resonant particles takes place. The resonant particles with velocities a little slower than the phase velocity gain energy and those with velocities a little faster than the phase velocity lose energy. The same basic phenomenon already occurs within the framework of the intuitive physical picture of the Landau damping mechanism given by Dawson (1961) or O’Neil (1965). The situation here is however slightly different from the one familiar in the Landau damping process, because in that case a kind of equilibrium between the quasi-particle and the associated wave is obtained, allowing both structures to exist in a symbiotic way. When the exchange of energy between particles and a longitudinal electrostatic field wave with dependence of type ei(kx−ωt) is considered, resonant particles can be categorized into two general populations: trapped and untrapped. It is found that untrapped particles tended to be dragged toward the wave phase velocity vφ = ωk . Thus, untrapped particles moving slower than vφ can gain energy, while particles moving faster lose energy. Thus, if there are more slow particles than fast particles, the electrostatic wave is Landau damped. However, conversely, if there are more fast than slow particles, now net energy flows from the resonant particles to the associated plasma mode. However, the physical process in question can become amplitude dependent (as in the particle trapping scenario) above some critical amplitude threshold and the nonlinearity now plays a major role. In practice, the breakdown of the superposition principle leads to mode coupling and new instabilities. In situations where the nonlinear coupling is weak, linear theory may be invoked as a reasonable first approximation and then used as a basis

284

The Vlasov Equation 1

for developing the nonlinear model. However, in situations where nonlinear effects become dominant, the linear theory cannot be used in principle. Thus, an equilibrium in the energy transfer between the quasi-particle and the (associated and now nonlinear) “wave” can be found in the self-consistent regime.

Figure 5.28. The waveform is very nonsinusoidal but apparently remarkably very stable with little charge from one cycle to the next. The waveform corresponds to the formation of a KEEN wave. Although the KEEN wave, as we discuss here, has been produced using a spatially purely sinusoidal travelingwave PF driver, it was noted from the outset that when we have obtained a self-sustained KEEN wave, the waveform thus produced always turned out to be very nonsinusoidal

Analysis of a single electron hole shows that it is stable. However, an ensemble of several holes was found to be unstable, because the holes attract one another. Simulations performed by Berk et al. (1970) show that outer contours behave in an adiabatic way and merely shield the positive charges of holes in contrast to the non-adiabatic behavior of inner contours, forming the hole structure. In their simulations based on the multiple water bag model, these authors pointed out that the hole configuration is strictly not a state of maximum total energy (it is always possible to excite positive energy plasma oscillations in the outer contours). Since these have phase velocities that are quite different from those of negative energy waves excited by the hole contours, the two types of waves do not couple effectively. Hence, in their simulation experiment, a simple hole is found to be stable because the system cannot coherently transfer energy from the inner (hole) to the outer contours (wave).

Nonlinear Properties of Electrostatic Vlasov Plasmas

285

However, at this step, two fundamental remarks must be pointed out: i) the BGK solution construction from the Vlasov theory shows that fully nonlinear collisionless electrostatic waves, time-independent in some frame of reference, can in principle exist, but this theory makes no statement as to the physical accessibility of these structures. ii) Vlasov–Poisson simulations show clearly the excitation of a single KEEN wave from an external driver; the simulations also show the possibility of emergence of a KEEN wave train followed by a pair-wise merging process. Such a scenario seems to be inconsistent with Liouville’s theorem, which asserts that two particle trajectories can be infinitely close together but cannot merge. Thus, Liouville’s theorem states that Hamiltonian flows in phase space are incompressible. Incompressibility is central to classical mechanics and can be interpreted as information conservation since the entropy is here conserved where particle trajectories in phase space cannot merge or intersect. Hence, two trajectories never cross in phase space. The work of Dodin and Fisch (2011) shows a possible connection between KEEN waves and NMI. Thus, the asymptotic state is modified when the sidebands are in appropriate resonance, leading to the bunching instability (see Hara et al. (2015); Brunner et al. (2014); Brunner and Valeo (2004)) or to a (dissipative) phase mixing mechanism. Thus, one can interpret the two-vortex merging mechanism as an example of NMI which, in the work of Dodin and Fisch (2013), was suggested to be a necessary ingredient to obtain a self-sustained state. We now return to the modeling of two-stream instability, previously presented in section 5.5. The initial condition is based on the same initial distribution function given by equation [5.22]. A disturbance is introduced on both of the first two Fourier modes k0 and 2k0 to cause the vortex fusion mechanism. The equation can be written as f (x, v, t = 0) = F0 (ε) (1 + α2 cos2k0 x + α1 cosk0 x)

[5.90]

Choosing ξ = 0.95, we have a marginal wavelength of λM = 6.946λD . This two-stream system, shown in Figure 5.29, was slightly perturbed initially kM on both the fundamental mode k0 = 2π L = 2 (with a small perturbation amplitude of α1 = 0.001) and on the mode 2k0 (now with an amplitude of α2 = 0.1). Here, L  27.785λD is the length of the plasma. Two mechanisms can be observed in Figure 5.29: the first concerns the fusion of the two electron holes and the second leads to the well-known mixing phenomenon in phase space. When the two holes are joined together, a process of stretching and folding occurs at time tωp = 100. Then, as predicted by NMI, one observes the appearance of two smaller structures at time tωp = 200 (on the top and

286

The Vlasov Equation 1

right panel in Figure 5.29) in rotation. It is the rotation of the two substructures in the electron hole that leads to a phase mixing mechanism and finally to the emergence of a single and stable electron hole. To study the formation of the electron hole, we choose to take the disturbance amplitude α2 to zero and examine in detail the formation and the growth of a single electron hole in the phase space. The physical and numerical parameters are kept identical in this simulation. Numerical results are shown in Figure 5.30, which exhibits the formation of the vortex in the phase space. During the formation of the central vortex, at time tωp = 50, several filaments appear and become thinner and thinner until they completely disappear. Here, the plasma length is L  27.785λD and corresponds to two equilibrium periods, which explains why the electron hole size is large in this simulation. Figures 5.31–5.32 show an another example when the size of the system is double and there is the possibility of a coupling between the BGK structure and an EPW. By further increasing the size of the system, the formation of the central vortex is now accompanied by a non-negligible disturbance of the surrounding plasma region. This disturbance gives rise to a nonlinear (Langmuir) plasma wave followed by the appearance of two other electron holes at time tωp = 65 in Figure 5.31. The result is a fusion of the two substructures and finally the plasma is now made up of two electron holes that remain stable for a long time. What is surprising is that the formation of a nonlinear Langmuir structure is not observed at a certain phase velocity. Thus, it seems that the interaction between the Langmuir wave and the BGK hole leads to a stationary Langmuir wave. This stationary Langmuir wave is thus formed by two waves propagating in opposite directions, of wave vectors ke ≥ 0 and ke ≤ 0. The interaction resembles a typical instability encountered during the decay of a Langmuir wave by an ion acoustic wave. The spectrum of the Fourier transform of the plasma field is plotted in Figure 5.32, for the fundamental mode ke = k0 . The panel displays an asymmetric broadening toward small wave frequency (here connected to the BGK mode), interpreted as a signature of a secondary EPW produced in the Langmuir decay instability. Note that here the low-frequency ion acoustic wave is replaced by a BGK structure.

Nonlinear Properties of Electrostatic Vlasov Plasmas

287

Figure 5.29. Phase space representation of the distribution function in the case of two-stream instability. Two mechanisms can be observed: the vortex merging and the negative mass instability (NMI). The fusion of the two electron holes is obtained at times tωp = 65 − 100. When the two holes are joined together, a process of stretching and folding occurs at time tωp = 100. Then, as predicted by NMI, one observes the appearance of two smaller rotating structures at time tωp = 200 indicated by arrows. For a color version of this figure, see www.iste.co.uk/delsarto/vlasov1.zip

288

The Vlasov Equation 1

Figure 5.30. Figure obtained when the disturbance in mode 2 is set to zero. The system is identical to that shown in Figure 5.29. During the formation of the central vortex, at time tωp = 50, several filaments appear and become thinner and thinner until they completely disappear. For a color version of this figure, see www.iste.co.uk/delsarto/vlasov1.zip

Nonlinear Properties of Electrostatic Vlasov Plasmas

289

Figure 5.31. By further increasing the size of the system (in comparison with Figure 5.30), the formation of the central vortex is now accompanied by a non-negligible disturbance of the surrounding plasma region. This disturbance gives rise to a nonlinear (Langmuir) plasma wave followed by the appearance of two other electron holes at time tωp = 65. The result is a fusion of the two substructures and finally the plasma is now made up of two electron holes that remain stable for a long time. For a color version of this figure, see www.iste.co.uk/delsarto/vlasov1.zip

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The Vlasov Equation 1

The interaction mechanism can be described by the following resonance: ke = ke + 2kBGK and ωe = ωe + 2ωBGK

[5.91]

where both EPWs have frequency and wave vectors (ωe , ke ) and (ωe , ke ) and the BGK hole is described by (ωBGK , kBGK ). Equation [5.91] describes the beating of the two EPWs of wave vectors ke = k0 and ke = −k0 . The corresponding frequency of Langmuir waves can be estimated by using the linear Bohm–Gross dispersion relation leading to ωe = ωe =  2 ωp2 + 3ke2 vth  1.089ωp for a value of the wave vector given by

2π = 0.113. This beating leads to the growth of a BGK mode with ke λD = 55.57 kBGK = k0 . Note that the corresponding frequency is not equal to zero (but is found to be close to ωBGK  0.1ωp ) as indicated in Figure 5.32, which shows the presence of the mode kBGK = k0 corresponding to the emergence of the first hole when the instability starts up.

Figure 5.32. Spectrum in frequency of the first spatial Fourier mode of the plasma field |Ek0 (ω)|. The panel displays an asymmetric broadening toward small wave frequency (here connected to the BGK mode), interpreted as a signature of a secondary EPW produced in the Langmuir decay instability. We observe that the plasma wave has a nonlinear nature

We have presented several electrostatic problems, mostly one dimensional (2D phase space), where typically 105 grid points was an excellent sampling

Nonlinear Properties of Electrostatic Vlasov Plasmas

291

of the distribution function in phase space, and evolution over hundreds of plasma units of time ωp−1 has been obtained. With a rough and noisy particle code, basically correct results can be obtained with a number of particles as low as 103 while the semi-Lagrangian Vlasov code will certainly blow up with a mesh of 103 points. These codes imply a minimum size of the computation. However, these codes provide unparalleled accuracy for describing distribution function tails where the number of particles remains generally insufficient in PIC code. It is interesting to note that for 1D plasma personal computers today provide the necessary memory size and processor speed to handle these 1D plasma semi-Lagrangian Vlasov codes. Of course, 2D problems imply an effort 104 –105 times bigger but these simulations are now possible on parallel architectures, as evidenced by simulations of the generalized BGK problems in 2D performed by Fijalkow (1999a,b) or even in the electromagnetic regime by Ghizzo et al. (2003). Moreover, for interesting 3D problems, we reach the level of tomorrow’s exascale supercomputers. 5.10. References Afeyan, B., Won, K., Savchenko, V., Johnston, T., Ghizzo, A., Bertrand, P. (2004). Kinetic Electrostatic Electron Nonlinear (KEEN) waves and their interactions driven by the ponderomotive force of crossing laser beams. In Third International Conference on Inertial Fusion Sciences and Applications, Monterey California 7–12 September, Hammel, B., Meyerhofer, D., Meyer-ter-Vehn, J., Azechi, H. (eds). American Nuclear Society, LaGrange Park. Albrecht-Marc, M., Ghizzo, A., Johnston, T., Reveille, T., Del Sarto, D., Bertrand, P. (2007). Saturation process induced by vortexmerging in numerical Vlasov-Maxwell experiments of stimulated Raman backscattering. Phys. Plasmas, 14, 072704. Amiranoff, F., Laberge, M., Marques, J., Moulin, F., Fabre, E., Cros, B., Matthieussent, G., Benkheri, P., Jacquet, F., Meyer, J., Mine, P., Stenz, C. (1992). Observation of modulational instability in Nd-laser-beatwave experiments. Phys. Rev. Lett., 68, 3713. Armstrong, T. (1967). Numerical studies of the nonlinear Vlasov equation. Phys. Fluids, 10, 1269. Belmont, G., Mottez, F., Chust, T., Hess, S. (2008). Existence of non-Landau solutions for Langmuir waves. Phys. Plasmas, 15, 052310. Berk, H., Nielsen, C., Roberts, K. (1970). Phase space hydrodynamics of equivalent nonlinear systems: Experimental and computational observations. Phys. Fluids, 13, 980.

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Index

A, B Alfvén speed, 169 wave, 169 BBGKY hierarchy, 12, 15, 27 BGK equilibria, 218, 229, 230, 235, 267 stability analysis, 237 C central force between vortices, 232 CGL equations, 104 circular polarization, 87 cleaning of microstructure, 232 Clemmow-Mullaly-Allis diagrams, 181 collision as field fluctuation, 18 binary, 10 frequency, 5, 122 cut-off condition, 177 curve of, 182 D

dispersion relation, 81 spatial, 79, 100, 158, 178 temporal, 79 double adiabatic, 104 E electron acoustic wave linear EAW, 254 nonlinear EAW, 244 energy principle, 185 entropy increase, 207 F Faraday’s rotation, 168 filamentation in velocity space, 220, 224 fluid closure problem, 104 moment, 104 free streaming, 220, 222 Fresnel equations for wave normals, 91 for wave transmission, 159 Friederichs diagram, 180

Debye length, 4 dichotomy tought experiment, 6

The Vlasov Equation 1: History and General Properties, First Edition. Pierre Bertrand; Daniele Del Sarto and Alain Ghizzo. © ISTE Ltd 2019. Published by ISTE Ltd and John Wiley & Sons, Inc.

298

The Vlasov Equation 1

G, H

M, N, O

gliding regularity, 246 group velocity, 155 Hamiltonian model N -body resonant particles, 203 adiabatic, 270 charged particles, 16 Hilbert transform, 260

mean free path, 5 minimal coupling, 27 nonlinear stage (instability), 191 Onsager relations, 81

I, J instability absolute, 194 convective, 194 macro-, 197 micro-, 197 stages of a linear, 189 subcritical, 193 two-stream, 38, 195, 230, 231, 234 ionosphere, 163 radio-wave transmission, 163 Jeans frequency, 18 Jones vector representation, 86 K KAM theorem, 247 KEEN wave emergent resonance, 268 negative mass instability, 270, 275 KEEN waves, 176, 250, 254, 260 L Landau linear damping, 33, 174, 175, 204, 224, 229 nonlinear damping, 245, 247 Langmuir decay instability, 290 wave, 287 nonlinear aspect, 247 laser strength parameter, 165 Lenard–Balescu equation, 19 linear stage (instability), 191

P pair-wise vortex merging, 234, 240, 242, 280 phase velocity, 155 polarization of waves circular, 81 extraordinary mode, 155 ordinary mode, 155 quasi-circular, 155 quasi-linear, 155 R Rayleigh criterion, 236 resistivity anomalous, 124 collisional, 122 dependence on temperature, 125 Spitzer’s, 123 resonance absorption, 172, 175, 201 condition, 177 curve of, 182 cyclotron, 170, 202 wave particle, 173 resonant surface (tokamak), 186 S, T saturation (instability), 192 Snellius–Descartes law of optics, 160 stimulated Raman scattering, backward, 281 test particle, 19 transient (instability), 189 trapped-electron mode (TEM) electrostatic trapping, 270 trapped particle instability, 274

Index

V

W

Van Kampen modes, 64, 246 Vavilov-Cherenkov emission, 200 Vlasov–Hilbert code, 261 Von Mises transformation, 42 vortex merging, 285 of two-stream instability, 285

wave normal surfaces, 180 packet, 149 spectral density, 150 spectral width, 151 Whistler waves, 170 Wigner equation, 21 WKB-type approximation, 78 ballooning representation, 78

299

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