Eulerian Codes for the Numerical Solution of the Kinetic Equations of Plasmas [1 ed.] 9781613245613, 9781616684136

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Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved. Eulerian Codes for the Numerical Solution of the Kinetic Equations of Plasmas, Nova Science Publishers, Incorporated, 2010. ProQuest Ebook Central,

Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved. Eulerian Codes for the Numerical Solution of the Kinetic Equations of Plasmas, Nova Science Publishers, Incorporated, 2010. ProQuest Ebook Central,

PHYSICS RESEARCH AND TECHNOLOGY

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EULERIAN CODES FOR THE NUMERICAL SOLUTION OF THE KINETIC EQUATIONS OF PLASMAS

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PHYSICS RESEARCH AND TECHNOLOGY

EULERIAN CODES FOR THE NUMERICAL SOLUTION OF THE KINETIC EQUATIONS OF PLASMAS

MAGDI SHOUCRI Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved.

EDITOR

Nova Science Publishers, Inc. New York

Eulerian Codes for the Numerical Solution of the Kinetic Equations of Plasmas, Nova Science Publishers, Incorporated, 2010. ProQuest Ebook

Copyright © 2011 by Nova Science Publishers, Inc. All rights reserved. No part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic, tape, mechanical photocopying, recording or otherwise without the written permission of the Publisher. For permission to use material from this book please contact us: Telephone 631-231-7269; Fax 631-231-8175 Web Site: http://www.novapublishers.com NOTICE TO THE READER The Publisher has taken reasonable care in the preparation of this book, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained in this book. The Publisher shall not be liable for any special, consequential, or exemplary damages resulting, in whole or in part, from the readers’ use of, or reliance upon, this material. Any parts of this book based on government reports are so indicated and copyright is claimed for those parts to the extent applicable to compilations of such works.

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Independent verification should be sought for any data, advice or recommendations contained in this book. In addition, no responsibility is assumed by the publisher for any injury and/or damage to persons or property arising from any methods, products, instructions, ideas or otherwise contained in this publication. This publication is designed to provide accurate and authoritative information with regard to the subject matter covered herein. It is sold with the clear understanding that the Publisher is not engaged in rendering legal or any other professional services. If legal or any other expert assistance is required, the services of a competent person should be sought. FROM A DECLARATION OF PARTICIPANTS JOINTLY ADOPTED BY A COMMITTEE OF THE AMERICAN BAR ASSOCIATION AND A COMMITTEE OF PUBLISHERS. Additional color graphics may be available in the e-book version of this book. LIBRARY OF CONGRESS CATALOGING-IN-PUBLICATION DATA Eulerian codes for the numerical solution of the kinetic equations of plasmas / editor, Magdi Shoucri. p. cm. Includes index. ISBN H%RRN 1. Kinetic theory of gases--Mathematics. 2. Plasma (Ionized gases) I. Shoucri, Magdi Mounir, 1961QC175.E85 2009 530.4'4--dc22 2010015592

Published by Nova Science Publishers, Inc. † New York

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CONTENTS Editor’s Foreword

vii

Dedication

xi

Chapter 1

Splitting Methods for the Vlasov-Maxwell Equations in Plasma C.Z. Cheng

Chapter 2

A Vlasov Approach to Collisionless Space and Laboratory Plasmas Francesco Califano and André Mangeney

23

Chapter 3

Eulerian Conservative Advection Schemes for Vlasov Solvers T.D. Arber, N.J. Sircombe and R.G.L. Vann

65

Chapter 4

Eulerian-Lagrangian Kinetic Simulations of Laser-Plasma Interactions D. J. Strozzi, A.B. Langdon, E.A. Williams , A. Bers and S. Brunner

89

Chapter 5

Gyrokinetic Vlasov Simulations for Turbulent Transport in Magnetized Plasmas Tomo-Hiko Watanabe and Hideo Sugama

123

Chapter 6

Numerical Solution of the Relativistic Vlasov-Maxwell Equations for the Study of the Interaction of a High Intensity Laser Beam Normally Incident on an Overdense Plasma Magdi Shoucri

163

Chapter 7

Semi-Analytical Adaptive Vlasov - Fokker-Planck - Boltzmann Methods Oleg V. Batishchev

237

Chapter 8

The Bump-on-Tail Instability Magdi Shoucri

317

Index

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1

359

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EDITOR’S FOREWORD Transport of charged particles and collisions are main aspects of fundamental kinetic processes which we need to study in electronic devices and plasmas. Considerable amount of effort is devoted to the application of numerical techniques to the study and comprehension of these kinetic processes, which are generally highly nonlinear problems. There are two important numerical approaches which are generally used in these studies. The first approach includes essentially Lagrangian methods which have become known as the particle-in-cell (PIC) methods. In this case the plasma is approximated by a finite number of macro-particles which are advanced in time in the self-consistent fields computed on a background mesh. This approach has been widely developed, with successful applications to several if not all aspects of plasma phenomena and devices. An important drawback with this approach is that there is a numerical noise inherently associated with PIC methods, which decreases as

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1 / N , where N is the number of macro-particles in the computational cell. This problem becomes important when the physics of interest is in the low density regions of the phasespace where there is only a relatively small fraction of the macro-particles, as for instance in the high energy tail of the distribution function. An alternative approach is to look for the     direct calculation of the particle distribution function f (r , , t ) , where r and  are respectively the position and velocity vectors at time t. The methods which discretize the full phase-space on a multi-dimensional grid and perform the direct solution of the distribution   function on this grid, by solving numerically the partial differential equation for f (r , , t ) in the multi-dimensional phase-space, are called Eulerian methods. Since the original publication of C.Z. Cheng and G. Knorr (J. Comp. Phys. 22, 330, 1976), who used a split Eulerian scheme for the numerical solution of a one-dimensional Vlasov equation by successive advection in the different directions of the phase-space, this approach has become increasingly important, especially for the numerical solution of the Vlasov equation which is used to study the kinetic processes when the collisions between the particles are negligible, as in high temperature and low-density plasmas. A great advantage of the Eulerian codes is their very low noise level, which allows accurate representation of the fine scale structures associated with the particle distribution function, especially in the low density regions of the phase-space. The areas of the kinetic processes in plasmas are broad, and of special interest are the applications to the study of collisionless transport processes using the Vlasov equation for a wide range of phenomena displaying collective behaviour, and in different manyparticles systems in plasmas, charged-particle beams, laser-plasma interactions, inviscid fluid

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Magdi Shoucri

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viii

turbulence, and in a variety of other domains such as astrophysics, the study of galaxies and other self-gravitating systems, and more recently in semiconductors and thin films. The present volume will review the important developments and the present state of the art in the application of the Eulerian methods to the kinetic equations of plasmas. It was intended to show the rich diversity of ideas which have developed in this field, as well as the wealth of results and physics obtained from numerous applications. These methods offer a great potential for further applications in the future, and with the continuous advances in computer technology, the best is still to come. The first three chapters of this book review different variations of the original timesplitting schemes applied to the Vlasov-Poisson system, extend these Eulerian schemes to different Vlasov-Maxwell systems including the relativistic Vlasov equation, and discuss important developments and progress in this field. Some schemes recently developed for the numerical solution of these equations will be compared, together with relevant applications, and adaptation to multi-dimensional simulations will be presented. Chapter 4 will describe a characteristics-based Vlasov-Maxwell solver and apply it to electron trapping in Langmuir waves, with a major study of stimulated Raman scattering in the interaction of a high intensity laser beam with a plasma, for parameters appropriate to inertial confinement fusion. In Chapter 5, the splitting scheme in the two-dimensional phase-space is generalized by the use of symplectic integrators, and applied to the gyrokinetic equations to study plasma turbulence in magnetic confinement fusion devices, where macroscopic transport is simulated from a level of microscopic fluctuations of the distribution function in a five-dimensional phasespace. Chapter 6 presents applications of Eulerian codes for the numerical solution of the relativistic Vlasov-Maxwell equations for the study of the interaction of a high intensity laser beam normally incident on an overdense plasma, with special attention to the problems of ion acceleration in the case of a circularly polarized laser beam and harmonics generation for the case of a linearly polarized laser beam. In Chapter 7 purely kinetic ionized gas descriptions are considered, and also neutral particles collisions. Particle and continuous methods are compared for problems with and without binary collisions. Coulomb collisions are considered and several benchmarking tests are presented along with few problems of practical interest. In Chapter 8, an Eulerian code is used to study the bump-on-tail instability for the case of a strong beam-plasma interaction, when particle trapping effects become important and play an important role in the growth of sidebands excited by a variety of mode coupling events mediated by the trapped particles oscillation. We are grateful to all the authors who have contributed a chapter to the present volume. Their invaluable contributions help make the field of Eulerian codes an important numerical tool for the theoretical studies of nonlinear kinetic processes in plasma physics. We hope that the present volume will find an interested audience.

December, 2009

Magdi M. Shoucri

Eulerian Codes for the Numerical Solution of the Kinetic Equations of Plasmas, Nova Science Publishers, Incorporated, 2010. ProQuest Ebook

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Editor’s Foreword

Professor Georg Knorr

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ix

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DEDICATION Georg Knorr was born in Munich, Germany, in 1929. His formative years were during the Second World War and the difficult years in post-war Germany. The many hardships naturally influenced his adulthood. He learned to be happy in spite of scarcity and to extract useful results with limited resources. That applied to building housing for his rabbits to supplement the meagre rations of the family during the war, as well as to his first published work on the integration of the non-linear Vlasov equation. It was obtained on a computer with no more than 4096 ‘words’, including both storage and program. Georg Knorr was first introduced to physics during the war by a neighbour, a chemist, who taught him calculus and thermodynamics. Schooling at that time had all but stopped due to the destruction of the buildings and almost incessant day-and-night air raids. He did his undergraduate work at the almost completely destroyed Institute of Technology in Munich from 1948 to 1951, and one year at the University of Illinois in Champaign/Urbana in 1951/52 as a Fulbright student. Upon graduating in 1954, he worked a few years in industry. In 1958 Georg Knorr joined the Max-Planck Institute for Physics and Astronomy. Assisted by the mentorship of director Ludwig Biermann and thesis advisor Arnulf Schlueter, and helpful discussions with Dieter Pfirsch, he received his PhD in 1963 for work on the solutions of the one-dimensional non-linear Vlasov equation. G. Knorr spent post-doctoral years with the Princeton Plasma Physics Laboratory in 1964 and the Department of Physics at the University of California in Los Angeles with Professor Burton Fried in 1966. He joined the Department of Physics of the University of Iowa in 1967, where he became a full professor in 1974. Georg Knorr belongs to the generation of the founding fathers of the numerical simulation in plasmas. He pioneered numerical methods to study kinetic processes in plasmas, especially for the solution of the Vlasov equation. Last year was the 50th anniversary of his paper presenting the first numerical solution of the one-dimensional Vlasov-Poisson system, published in 1958 (Z. Naturforsch. 13a, 941). He continued his pioneering work in the field of plasma physics for several years, and published many outstanding papers. The numerical studies were coupled with powerful theoretical studies to gain understanding of the turbulence in guiding-center plasmas and in the Vlasov-Poisson system. In 1976 he published with C.Z. Cheng his ground-breaking paper on the split Eulerian scheme for the numerical solution of the one-dimensional Vlasov-Poisson system (this work was published first as a report in 1975), which became the spark which initiated great interest and applications of Eulerian codes for the numerical solution of the kinetic equations in plasmas, especially the Vlasov equation. The Eulerian Vlasov codes are characterized by a very low noise level,

Eulerian Codes for the Numerical Solution of the Kinetic Equations of Plasmas, Nova Science Publishers, Incorporated, 2010. ProQuest Ebook

Magdi Shoucri

xii

which allows accurate studies of phenomena in the low density regions of the phase-space. The splitting scheme reduces the dimensionality of the Vlasov equation in each fractional step and adapt easily to computers with parallel architecture. A wealth of physics has resulted from many successful applications of these codes to study kinetic processes in plasmas. The present volume, dedicated to Georg Knorr, will present selected topics and reviews showing the rich diversity of ideas developed in this field and of physical results obtained. Georg Knorr has deserved well of the computational physics community. He has always generously shared his ideas and provided much encouragement to his students and collaborators. I would like to end this dedication on a personal note to mention the patient attention and care he used to provide to his students. I remember his patient tutorial sessions when he taught me, as his graduate student, numerical simulations. Georg Knorr is celebrating this year the 80th return of his birthday. On this special occasion and on behalf of all the authors of this volume, we wish him many more happy returns.

The editor, on behalf of the authors.

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December, 2009

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In: Eulerian Codes for the Numerical Solution… Editor: Magdi Shoucri, pp. 1-21

ISBN: 978-1-61668-413-6 © 2010 Nova Science Publishers, Inc.

Chapter 1

SPLITTING METHODS FOR VLASOV-MAXWELL EQUATIONS IN PLASMA SIMULATIONS C.Z. Cheng Plasma And Space Science Center, and Institute of Space, Astrophysical And Plasma Sciences, National Cheng Kung University, Tainan, Taiwan

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Abstract Since the splitting scheme with second order accuracy in time step for efficient computation of numerical solutions of the Vlasov-Poisson equations was published by Cheng and Knorr in 1976, Vlasov simulations have made tremendous progress in the study of nonlinear plasma kinetic physics. The splitting scheme has since been extended to include electromagnetic effects and external magnetic field with complex magnetic field geometry. The splitting scheme splits the Vlasov equation into a series of lower-dimensional hyperbolic partial differential equations (i.e., the free-streaming and accelerating equations) in the spatial and velocity space separately. These lower-dimensional equations have simple constant advection speed in the spatial dimensions and analytically tractable acceleration in the velocity space, and thus exact analytical solutions of these lower-dimensional equations can be obtained. The lower-dimensional equations can be solved numerically by either performing interpolation or finite difference or finite element or finite volume methods. Because the lower-dimensional equations can be solved exactly by following the particle characteristics, the most natural numerical scheme is to perform interpolation. The important features of the splitting scheme are that not only the computation time is quite low, but also excellent results can be obtained with a few number of grid points. The advantage of the Vlasov simulations employing the splitting scheme over the Particle-In-Cell methods has also been demonstrated on the consideration of numerical noise, selective mode resolution, etc. Because the splitting method can be easily adapted to the massively parallel processor (MPP) technology, Vlasov simulations using the splitting scheme should play a greater role in the study of nonlinear plasma kinetic physics. Here, we review the splitting scheme and its applications in the interaction of nonlinear Langmuir solitons with plasma, and electron cyclotron waves.

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2

C Cheng C.Z.

Introductiion The major effort in plaasma physics is to understaand fundamenntal kinetic prrocesses of coollisionless pllasmas, and the t Vlasov-M Maxwell equattions are the fundamentall equations gooverning the behavior b of coollisionless plaasmas. The Vlasov V equation describes thhe behavior G G G G f (xx , v , t ) , wheree ( x , v ) are thhe position off the plasma single s particle distribution functions fu

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annd velocity veectors, of ions and electronss under the inffluence of elecctric and magnnetic fields. C Coupled with the t Maxwell equations forr the electrom magnetic fieldss, which are determined d frrom the plasm ma charge and current dennsities compuuted from the moments of the single paarticle distribu ution functionns of ions and electrons, onee obtains a higghly nonlinearr system of diifferential and d integral equaations. The Vllasov-Maxwelll equations caan be solved analytically a onnly for a few very simple problems. Forr this reason numerical sim mulations of thhe VlasovM Maxwell equatiions have become an imporrtant tool for theoretical t stuudies of nonlinnear kinetic pllasma physiccs processes. Because of the advanceed computer technologies numerical siimulations of the t Vlasov-Maxwell equatioons have beenn instrumental in understandding kinetic pllasma behaviors with stroong nonlineariity, and havee been applieed to a wide variety of prroblems in fussion, space, asstrophysical, and a industrial plasmas. p Basically th here are two approaches a to obtain numerrical solutionss of the Vlasovv equation: thhe particle-in-ccell (PIC) metthod [e.g., 1] and a the Vlasovv simulation method m [e.g., 2]. 2 The PIC m method approximates the plaasma by a finitte number of macroparticles m s which move in the selfcoonsistent electtromagnetic fields fi computeed by taking moments m on a background mesh. The keey drawback with the PIC C approach is that the num merical noise only o decreasees as 1/√N, w where N is thee number of finite-sized particles in anny particular computational c l cell. This G G prroblem is partticularly pronoounced in studdies where thee fine-scale sttructure of f ( x , v , t ) is im mportant or where w the phyysics of intereest is in the high-energy h taail of the disttribution in w which there is only o a relativeely small fractiion of the totaal number of particles. p Evenn with these lim mitations the PIC approachh can producee accurate ressults when thee high-energyy tail is not siignificant, and d a sufficiently high N cann be maintainned to resolve the broad distribution d fuunction. ov simulation method solvves numericallly the Vlasov equation as a partial The Vlaso diifferential equ uation in the multi-dimensiional particle phase space. The Vlasov simulation m methods offer some advanntages over PIC P simulationns, as they do d not sufferr from the coonsequences of o the statisticcal fluctuationns (or noise) inherent in using u a finite number of m macroparticles much smallerr than the actual plasma poppulation. 70s almost alll Vlasov simuulation methoods integrate the t distributioon function Before 197 diirectly in multti–dimensionaal phase spacee either directlly on discretizzed phase spacce Eulerian grrids [3] or by expanding thee distribution in terms of a finite set of analytical a basiis functions off continuous phase p space variables v [e.g., 2,4]. These methods usuaally demand much m more coomputation tim me than the PIC P methods and a are generrally considereed impracticaal for a full siix-dimensionaal particle phase space. Thuus, most Vlasoov simulationss were perform med for the onne-dimensionaal Vlasov-Poissson equations. To overcom me these difficculties a grounnd-breaking m method, a splittting scheme with second-oorder accuracy in time stepp

, was puublished by

Cheng and Kno C orr [5] in 19766 for unmagneetized plasmass and was lateer extended byy Cheng [6] inn 1977 for mag gnetized plasm mas. The methhod splits the Vlasov V equatioon into a seriees of lower-

Eulerian Codes for the Numerical Solution of the Kinetic Equations of Plasmas, Nova Science Publishers, Incorporated, 2010. ProQuest Ebook

Splittting Methodss for Vlasov-M Maxwell Equattions in Plasm ma Simulationss

3

diimensional hy yperbolic partiial differentiall equations (i.ee., the free-strreaming and accelerating a eqquations) in th he spatial andd velocity spaace separatelyy. Moreover, these t lower-ddimensional eqquations havee simple consttant advectionn speed in thee spatial dimeensions and analytically a trractable acceleeration in the velocity v spacee, and thus exaact analytical solutions of thhese lowerdiimensional eq quations can be obtained. Numericallyy these analyytical solutions can be coomputed by employing interpolation i methods succh as the cuubic spline or o Fourier innterpolations. This second-oorder splittingg scheme not only o improvess the time stepp accuracy, buut also greatly y simplifies the numerical procedures to advance a the single particle distribution d inn lower-dimen nsional phase space variables sequentiallly, thus allowiing significannt saving in coomputation time. t Moreovver, advanceed computer technology of parallel--computing arrchitecture or vector operatiions can be easily adopted too obtain the nuumerical soluttions of the loower-dimensio onal equationss to speed up the computattion and makee applicationss of Vlasov siimulations to higher h dimenssional problem ms feasible. As describ bed in the origginal paper of Cheng and Knorr [5], thhe second-order splitting

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sccheme in

can be straaightforwardlyy extended too the Vlasov--Maxwell equuations for

pllasmas with non-periodic n sppatial boundarry conditions, two- and threee-dimensionaal problems w and withou with ut external maagnetic and eleectric fields [66-30], and elecctromagnetic effects e [31477]. Since then n, numerous investigations i on improvingg the solutionn accuracy of the lowerdiimensional eq quations havee been reporteed. Applicatioons of Vlasovv simulation have been suuccessfully peerformed to stuudy various tyypes of highlyy nonlinear eleectrostatic plassma kinetic phhenomena succh as Langmuuir solution dyynamics and turbulence annd large ampliitude BGK w waves, and man ny electromaggnetic problem ms such as parrticle beam-plasma interactiions, X-ray frree electron lasers, laser wakke field acceleeration of partiicles, etc. The lower-dimensional equations off the Vlasov equation resuulting from thhe splitting sccheme can be solved by eithher performinng interpolationn or finite diffference or finite element orr finite volum me methods. Thhe finite difference schemes are relativelly easy to impplement but offten suffer nu umerical instaabilities and produce p non-ppositivity in thhe distributionn function. A Although num merical schem mes that conserve particle number havve been deveeloped, the im mplementation n can be quite complicateed and time consuming. Finite F elemennt or finite voolume method ds also suffer similar s difficuulties. Becausee the lower-dimensional equuations can bee solved exacttly by followinng the particlee characteristiccs, the most natural n numerical scheme iss to locate thee original phaase space locaation of a choosen fixed Euulerian grid annd perform innterpolation to o obtain the distribution d fuunction at thatt phase space location. Theerefore, the acccuracy of fo ollowing the distribution function f in tim me depends on the accuracy of the innterpolation scchemes used. In the originaal paper of Chheng and Knoorr [5] two innterpolation scchemes were used: Fourieer interpolatioon and cubic--spline interpolation. In general, the Foourier interpo olation is the most m accuratee scheme in a periodic systtem, but it is difficult to appply the meth hod for non-peeriodic boundaary conditionss. Thus, the cuubic-spline intterpolation, w which has the property p of miinimum curvatture among all third-order scheme, s has beecome very poopular becausse it is easy to t implement for various boundary b condditions and caan produce reesults with satiisfactory accuuracy. It is also worthwhile w to mention that the semi-Laggragian methood, which hass long been appplied in the field of meeteorology using fixed Euulerian grids with the fluiid velocity chharacteristics obtained iteraatively, has alsso been develooped for the numerical n soluution of the V Vlasov-Maxwe ell equations [48-53]. [ The semi-Lagrangi s ian method iss based on folllowing the

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C.Z. Cheng

characteristics backwards in time step in the original multi-dimensional phase space and interpolate the distribution function at the particle characteristics location at the previous time step. The interpolated value of the distribution function then moves forward to the fixed Eulerian grid points. Because the particle characteristics are usually nonlinear and in high phase dimensions, the particle characteristic curves is usually obtained via iterative processes. Therefore, such methods are very computation time consuming and difficult to implement, and thus become prohibitively expensive and difficult in multi-dimensional simulations. It is important to mention that the symplectic integration scheme which preserves the canonical character of the equations of motion was introduced in early 1990s for the integration of Hamiltonian equations. It can be shown that the second-order splitting scheme for solving the Vlasov-Maxwell equations introduced by Cheng and Knorr [5] is also a symplectic integration scheme accurate to second order in time step [54-57]. Although higher-order symplectic integration schemes have been developed, they are more difficult to program and consume much more computation time than the second order splitting scheme with small improvement in accuracy, and for this reason they have not been very practical for applications. A common problem with all Vlasov simulation methods is that the numerical solutions of the Vlasov equation often involve velocity space filamentation which is fine-scale structure that increases in time. The velocity space filamentation is associated with the amplitude and wave number of the initial perturbed distribution. For example, the solution to the linear Landau damping of a Fourier mode with initial amplitude A and wave number k has a perturbed distribution function which varies as A exp(ikvt), where t is time and v is particle velocity. Due to filamentation of the distribution function in phase space, the velocity space ripples are continuously generated and an exact representation would requires more and more numerical grid points as the computation goes on in time. This makes the integration of the Vlasov equation a lengthy and expensive task in more than one dimension. Fortunately, this filamentation has little influence on the lower moments of the distribution function, and in fact does not contribute to the linearized analytical treatment of the Vlasov equation. Thus, it is desirable to dispose the velocity space ripples even for the simulations of the Landau damping problems where the initial perturbation amplitude dominates over the amplitudes of the nonlinear solutions. For simulations of plasma instabilities the initial perturbation amplitude should in principle be very small in the fluctuation levels, but for practical applications are chosen to be much larger than the fluctuation values to allow for faster increase to large amplitudes to study nonlinear effects. Thus, it is also desirable to remove the velocity space filamentation associated with the initial perturbations. A general convolution integral method of removing the velocity space filamentation was introduced by Cheng and Knorr [5]. Although the smoothing method produces artificial entropy to the Vlasov equation, it was shown to be very effective in removing the velocity filamentation and producing quite accurate results. Fortunately, the velocity space filamentation effect is naturally smoothed out by the phase mixing of many Fourier modes in plasmas with highly nonlinear solitary structures or in turbulence states. This was demonstrated in the study of Langmuir solitons by Lin et al. [13] that the simulation can be carried out accurately for times much longer than the time limit imposed by the velocity space filamentation effect. In this paper we review the second-order time splitting scheme for solving the VlasovPoisson equations with external magnetic field, and show that the scheme can be easily extended to include electromagnetic effects. Then, we review simulation results which justify

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Splitting Methods for Vlasov-Maxwell Equations in Plasma Simulations

5

the discussions made in the Introduction section. Finally, we conclude the paper by discussing the major features and future applications of the Vlasov-Maxwell equations for studying plasma kinetic physics.

Splitting Scheme Here we present the splitting scheme with second order accuracy in time step for integrating the Vlasov-Poisson equations with external electric and magnetic fields. The splitting scheme can be straightforwardly generalized to solve the general Vlasov-Maxwell equations in higher dimensional problems with electromagnetic effects. For simplicity we consider a three-dimensional, electrostatic collisionless electron plasma immersed in a

G

uniform static magnetic field B = Beˆ z . The dimensionless form of the Vlasov-Poisson

G G

G G

equations for the electron distribution function f ( x , v , t ) and the electric field E ( x , t ) are given by

G G G G ∂f G ∂f + v • ∇f − E ( x , t ) + v × ω c • G = 0 , ∂t ∂v

)

(1)

G ∇ • E = 1 − ∫ f d 3v ,

(2)

(

where

ωc = eB / mcω pe is the ratio of the electron cyclotron frequency to the electron

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plasma frequency

ω pe , e is the absolute value of the electron charge, and m is the electron

mass, and the electric field and the electron distribution are normalized with the electronic charge. The electric field in Eq. (1) can also include the external electric field. Following the principles of the splitting scheme used in [5, 6], Eq. (1) is splitted into lower-dimensional equations in the following four-steps: (i) Solve the free streaming equation

∂f G + v • ∇f = 0 ∂t

(3)

for half a time step with the solution

G G G G G f * ( x , v ) = f n ( x − v Δt / 2, v ) n

(4)

G G

where f ( x , v ) denotes the distribution at tn . (ii) Calculate the electric field from the Poisson equation

G G G ∇ • E * = 1 − ∫ f * ( x , v )d 3 v . Eulerian Codes for the Numerical Solution of the Kinetic Equations of Plasmas, Nova Science Publishers, Incorporated, 2010. ProQuest Ebook

(5)

6

C.Z. Cheng

(iii) Integrate the acceleration equation

G G G G ∂f ∂f − E * ( x ) + v × ωc • G = 0 ∂t ∂v

(

)

(6)

for a whole time step with the solution

G G G G f ** ( x , v ) = f * ( x , v * )

(7)

G*

where v is the solution of the characteristics of Eq. (6) for one time step backward and is given by

G G G E *y ( x ) E *y ( x ) G E * (x) v ( x) = + (v x − ) cos ωc Δt + ( v y + x ) sin ωc Δt , * x

G v *y ( x )

ωc

=−

ωc

G E x* ( x )

ωc

+ (v y +

G E x* ( x )

ωc

ωc

) cos ω c Δt − (v x −

G E *y ( x )

ωc

) sin ω c Δt ,

G G v *z ( x ) = v z + E z* ( x ) Δt .

(8)

∂f G + v • ∇f = 0 again for half a time step, and the ∂t solution of the distribution function at t n +1 = t n + Δt is given by

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(iv) Solve the free streaming equation

G G G G G f n +1 ( x , v ) = f ** ( x − v Δt / 2, v ) .

(9)

By substituting Eqs. (7) and (4) into Eq. (9) we obtain

G G G G f n +1 ( x , v ) = f n ( x ' , v ' ) with

(10)

G G G G x ' = x − ( v + v ' ) Δt / 2 , G G G G v ' = v * ( x − v Δt / 2) ,

(11)

G* G

where v ( x ) is given by Eq. (8). The characteristics of Eq. (1) are given by

G G G G G dv / dt = −[ E ( x , t ) + v × ωc ] ,

(12)

G G dx / dt = v .

(13)

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Splittting Methodss for Vlasov-M Maxwell Equattions in Plasm ma Simulationss Integrating g Eqs. (12) andd (13) from , are given by b

to

7

, the solutions, coorrect to seconnd order in

G G G G x n = x n +1 − ( v n + v n +1 ) Δt / 2 ,

E yn +1 / 2 E yn +1 / 2 G E n +1 / 2 v xn ( x ) = + ( v xn +1 − ) cos ωc Δt + ( v ny +1 + x ) sin ωc Δt ,

ωc

ωc

ωc

n +1 / 2

Ey E n +1 / 2 E n +1 / 2 G + ( v ny +1 + x ) cos ωc Δt − ( v xn +1 − v ny ( x ) = − x

ωc

ωc

) sin ωc Δt ,

ωc

v zn = v zn +1 + E zn +1 / 2 Δt , w where

the

time

ceentered

eleectric

fieldd

(14)

G E n +1 / 2

is

evaluuated

at

G G G G G x n +1 / 2 = x n +1 − v n +1Δt / 2 + O ( Δt 2 ) , v n and v n +1 are the vellocity at tn and tn +1 , reespectively. A compariison between Eqs. (11) annd (14) showss that the spllitting schemee (3)-(9) is

G Δt . It is clear that t after we have h shifted f in the x sppace at time G for onee half time steep, we can integrate f in the t v space for f a full timee step, then G coombine the sh hift of f in x space in Eqs. E (4) and (99) for a full tiime step, andd repeat the coorrect to secon nd order in

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prrocedure. It is also easy to show s that as

ωc → 0 the soolutions givenn in Eq. (8) reccover to the

unnmagnetized case. c The abovve procedure can be applied to more com mplicated situations with noonuniform ex xternal magnettic field and self-consistennt electromagnnetic fields byy assigning thhe magnetic field fi in Eq. (112) to be timee-centered at the t n+1/2 tim me step and evvaluated at

G x n +1 / 2 , and thhe splitting sccheme remainns accurate too second ordeer in

. Thhe proof is

sttraightforward d and will not be b presented here. h Note that for f the three-ddimensional free f streamingg equations (44) and (9) we can either peerform three-d dimensional (3D) interpolaation or furtheer split the 3D D interpolationn into three suubsequent onee-dimensional (1D) interpollations in eachh direction of the t ( x, y , z ) rectangular r cooordinate. To obtain the veelocity space solutions s of Eq. E (6) for a fuull time step Δt , Eq. (6) caan be replaced d by the follow wing sequencee of equations

E *y ∂f E * ∂f ∂f − + x = 0, ∂t ωc Δt ∂v x ωc Δt ∂v y

(15a)

∂f ∂f + ωc = 0, ∂t ∂φ

(15b)

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C Cheng C.Z.

E *y ∂f E * ∂f ∂f − x = 0, + ∂t ωc Δt ∂v x ωc Δt ∂v y

(15c)

∂f ∂f − E z* = 0, ∂t ∂v z

(15d)

w where Eqs. (15a), (15c) mustt be integratedd for the time step

. The variable

inn Eq. (15b)

iss the azimuthaal angle in the ( v x , v y ) pllane. The form mal solutions of Eqs. (15aa)-(15d) are giiven by the sequence of shifftings of the distribution d

f * ( v x , v y , v z ) = f b ( v x + E *y / ωc , v y − E x* / ωc , v z ) ,

(16a)

f ** (v x , v y , vz ) = f * (v x coos(ωc Δt ) + v y sin( s ωc Δt ), v y cos(ωc Δt ) − v x sin(ωc Δt ), vz ) , (16b)

f *** ( v x , v y , v z ) = f ** ( v x − E *y / ωc , v y + E x* / ωc , v z ) ,

(16c)

f e ( v x , v y , v z ) = f *** ( v x , v y , v z + E z* Δt ) ,

(16d)

where the sup w perscripts denote thhe initial andd the final diistribution, reespectively. Suubstituting (16 6d), (16c), (166b) into (16a), we obtain

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f e ( v x , v y , v z ) = f b ( v *x , v *y , v *z ) *

*

*

w where b Eq. (8) andd the equivalennce between Eqs. E (15a)-(155d) and Eq. v x , v y , v z are given by (66) is obvious. Note that Eqs. (4), (9),, (15a) and (15c) can be further splitteed into one-ddimensional eqquations, thuss greatly simpplifying the soolutions. The sub-dimensioonal equationss, Eqs. (4), (99), and (15a)-1 15(d), resulting from the spllitting schemee are solved inn general by innterpolation m methods. As prresented in thee paper by Chheng and Knorrr [5] the Fourrier interpolatiion method w employed was d for periodicc boundary condition in the t real space and the cuubic spline innterpolation method m was useed for non-perriodic boundaary conditions in the velocitty space. In geeneral, the Fo ourier interpolation is the most m accurate method m in a periodic p system m, but it is diifficult to app ply the methood for non-perriodic boundaary conditions. Thus, the cuubic spline innterpolation, which w has the property of minimum m currvature amongg all third-ordder scheme, haas become verry popular beccause it is easyy to implemennt for various boundary b connditions and caan produce ressults with highhly satisfactoryy accuracy.

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Splitting Methods for Vlasov-Maxwell Equations in Plasma Simulations

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Langmuir Soliton Nonlinear wave-particle interactions between Langmuir solitons and plasmas have been extensively investigated by theoretical and experimental studies. Electrons can be accelerated by large amplitude Langmuir solitons to form a non-Maxwellian high energy tail velocity distribution function. Langmuir solitons also interact with thermal ions nonlinearly through both wave-particle resonance and coupling between Langmuir solitons and acoustic waves. The nonlinear processes not only slow down the soliton propagation but also damp the amplitude of the solitons. Although analytical results provide some understanding of nonlinear heating and slowing down of soliton propagation, highly nonlinear wave-particle and wave-wave interactions and the dynamical evolution can only be studied through numerical simulations, and we will present highlights of the simulation results. Simulations of the interaction of Langmuir solitons with plasmas and the soliton propagation require a system with large phase space domain in both real space and velocity directions [13]. The Vlasov simulation method has a clear advantage over the particle simulation method for simulating accurately the wave-particle interaction in the high energy tail portion of the distribution function because the high energy tail distribution is well represented. To obtain similar accuracy the particle-in-cell simulation method would require a very large number of particles to reduce the statistical error because only a small fraction of particles are involved. Since the computation time of the particle-in-cell method is proportional to the number of particles, it will take a much longer computation time to simulate the same physical problem with similar accuracy. In addition, the Vlasov simulation does not have statistical noise and can accurately compute the particle distribution function. Below we will present one-dimensional Vlasov simulations of Langmuir solitons and their interaction with plasmas. A rectangular mesh with the computational domain, R={(x,v) | 0 ≤ x < L, |v | ≤ vmax}, is used to represent the x-v phase space. We will consider a periodic system with L being the spatial system length and vmax is the cutoff velocity. Typically, we use 128 grid points along the velocity direction for electrons and 64 grids for ions. The number of mesh points along the x direction is usually 128 except for additional needs in some simulations for electron heating where we used 1024 grid points. To obtain a better spatial resolution of soliton structure, the mesh size Δx is selected to be 2λde, where λde is the electron Debye length, such that the spatial system length is 256λde for 128 spatial grid points. The cutoff velocity of electrons and ions are usually taken to be 8 vte and 6 vti, where vte and vti are the initial electron and ion thermal velocity, respectively. Convergence has been checked with larger system size and more grid points to ensure very high accurate results. The electric field and plasma density profile at the initial time is chosen as the solutions of Zakharov's equations [58]. If we normalize the time with the inverse of electron plasma frequency, the distance by λde, the particle velocity by its initial thermal speed, the electric field by (4πn0Te)1/2, where n0 is the electron density averaged over the system, Te is the initial

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10

C.Z. Cheng

electron temperature, then the normalized dimensionless form of the electric field and perturbed ion density are given by [59] E(x,t) = E0 sech[K0(x-Vgt)] exp[i(K1x-Ωt)],

(17)

N(x,t) = −6 K02{cosh[K0(x-Vgt)]}-2 n0,

(18)

and

2 2 2 where E0 = {24(1+3Ti/Te)K02[1– 9 K1 (mi/me)]}1/2, Vg = 3 K1, and Ω = 1 + 3(K1 –K0 )/2, K0 and K1 are dimensionless parameters related to the wave numbers, and the physical group velocity is given by vg = Vgvte. When the soliton is stationary, the group velocity Vg = 0 and K1 = 0, and the normalized electric field amplitude is proportional to the parameter K0. The

total ion density is given by ni = n0 + δni(x,t), where δni(x,t) = 12 n0K0λde/L + N(x,t). The initial electron and ion velocity distribution functions are chosen to be shifted Maxwellians to produce the soliton solution and they are given by fs(x,v,t=0) =( ns /(2π)1/2vts ) exp[-(vVs)2 /2 vts2], where the subscript s = e, i denotes electrons and ions, respectively, vts is the particle thermal speed, the ion drift velocity is given by Vi = (δni /ni)vg, and the electron drift velocity Ve is determined by the current density equation ∂E ( x, t ) / ∂t = −4πe( niVi − neVe ) . Below we review the simulation results of electron heating by Langmuir solitons and the propagation of Langmuir solitons and its interaction with ions and sound waves [13].

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A. Electron Heating by Langmuir Soliton The soliton amplitude controls the nonlinear wave-particle interaction between Langmuir soliton and plasma. To study the electron heating, an initial soliton is chosen to be stationary (K 1 = 0) at the center of the simulation system for simplicity. The ionto-electron mass ratio mi /me is chosen to be 100. The electron-to-ion temperature ratio Te /Ti is chosen to be 100 to exclude ion thermal effects. From Zakharov's solutions, the soliton wave amplitude E0 is proportional to the characteristic wave number K0. Three different initial wave amplitudes of Langmuir solitons, determined by K 0 = 1/15, K 0 = 2/15, and K 0 = 3/15 are chosen in simulations, which correspond to the initial maximum electric field amplitude E0 of about (1/3)(4 πn 0Te)1/2, (2/3)(4 πn 0Te)1/2 , and (4 πn 0Te)1/2 , respectively. The steady state solution of the average electron velocity distribution function at −1 t = 8000ω pe due to nonlinear interaction between electrons and solitons for these three

initial wave amplitudes are presented in Figs. 1(a), (b) and (c). Both Figs. 1(b) and (c) show super-thermal tails formed in each distribution function which approaches a specified slope while simulation time increases. The temporal evolution of the average electron velocity distribution function changes very slowly after t = 6000ω pe . Thus, the results at −1

−1 t = 8000ω pe which almost keep a constant distribution shape could be interpreted as the

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Splitting Methods for Vlasov-Maxwell Equations in Plasma Simulations

11

asymptotic solution for electron heating to avoid further numerical errors. In the electron resonant regions, where the curve f(v) rises up, the slope of f(v) for K0 =2/15 is –4.6 , i.e., f(v) ∝ v–4.6 , and the velocity slope of f(v) for K =3/15 is –4.2, i.e., f(v) ∝ v–4.2. These 0

results suggest that the heated electron distribution function forms a super-thermal tail which approach some constant slope. Comparing with the quasi-linear theory of Gorev and Kingsip [60], the numerical results in the above examples agree with the theoretical calculation f(v) ∝ v–4. For the case of K =1/15, Fig. 1(a) shows no obvious electron 0

heating. From the quasi-linear diffusion theory, the electron resonant velocity (9.38 vte for K0 =1/15) is very large such that the resonance effect can be neglected. It is to be noted that

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because we adopt a periodic system of 256λde in system length, the electrons have time to traverse across the system and interact with the soliton many times. Thus, the simulation can also be interpreted as the electron interaction with many solitons that have the same wave amplitude and are distributed with equal space in an infinite system. Those fast electrons under continuing wave-particle interactions and random scattering by high frequency electric field cause f(v) diffusion in velocity space with a higher energy tail population. The evolution of the spatial structure of ion density profiles of Langmuir solitons corresponding to Fig. 1 are given in Fig. 2. Figure 2(a) shows that the ion density profile has nearly the same soliton shape during the simulation time, and thus the soliton is in a steady state. However, the amplitudes of ion density depression attenuates faster with larger soliton amplitude, but the ion density depletion maintains the soliton shape as shown in Figs. 2(b) and 2(c). Also, note that a pair of ion acoustic waves are emitted propagating away at ion sound speed. The above results confirm that fluid theory of Langmuir soliton by Zakharov is satisfied very well for small wave amplitude. When the wave amplitude becomes larger, the wave-particle interaction becomes important, and the strong nonlinearity must be considered. B. Propagation of Langmuir Soliton

The slowing down of Langmuir soliton propagation in uniform plasmas is mainly due to nonlinear damping by thermal ions. The propagation of Langmuir solitons is measured by the position of the maximum density depression since the high frequency electric fields are trapped in this region. Taking into account the nonlinear effect of induced scattering by thermal ions which occurs strongly when the soliton group velocity vg ≤ vti, the momentum balance equation for a moving soliton [60] is given by 1/ 2

⎡ miTi ⎤ d d ⎛ 6 E0Te k ⎞⎟ 3 P≡ ⎜ =− ⎢ 3 2 dt dt ⎜⎝ 2πeω pe ⎟⎠ 4( 2π ) ⎣ me (Ti + Te ) ⎥⎦

(kλde )

E04 4πnTe

(18)

where P is the soliton momentum, k is the wave number of the Langmuir wave and is proportional to K1 in Eq. (17) and is related to the soliton group velocity vg, and E0 is the electric field amplitude and is assumed to be undamped. Thus, the soliton momentum or the

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12

C.Z. Cheng

soliton group velocity vg changes as a result of a change in wave number k and the wave amplitude E0. Note that for vg >> vti, the slowing down rate is exponentially small. To study the effect of vg/vti we perform simulations for three temperature ratios (a) Te /Ti = 100, (b) Te /Ti = 10, and (c) Te /Ti = 1, respectively and the evolution of ion density profile is shown in Fig. 3(a)-(c). The mass ratio is chosen to be mi /me = 100 and the initial group velocity is vg = 0.06 vte (or K1 = 0.02), thus we have (a) vg = 6 vti for Te /Ti = 100, (b) vg = 1.9 vti for Te /Ti = 10, and (c) vg = 0.6 vti for Te /Ti = 1. The simulation system length L

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is 256λde and the initial wave amplitude is obtained by choosing K0 = 2/15, for these three temperature ratio cases. Figure 4 shows the position of maximum depression of ion density profile as a function of time for these three cases in Fig. 3, in which the straight dash line shows the initial soliton group velocity and is plotted for comparison. When vg/vti >> 1 such as in the Figs. 3(a) and 4(a) with vg = 6 vti, the resonant interaction between the soliton wave packet and thermal ions is very weak and the soliton maintains its initial group velocity very well. As the ion thermal velocity is comparable to the soliton group velocity, the resonant interaction between the soliton wave packet and thermal ions becomes stronger and the soliton motion slows down and eventually becomes stationary as shown in Figs 3(b) and 4(b) for vg = 1.9 vti. However, as the ion thermal velocity becomes larger than the soliton group velocity, the resonant interaction between the soliton wave packet and thermal ions is very strong and the soliton very quickly slows down to become stationary as shown in Figs 3(c) and 4(c) for vg = 0.6 vti. Also, it should be noted that in cases (a) and (b) ion sound waves are clearly emitted moving with the sound speed Cs = 0.1 vte. But, in case (c) the ion Landau damping of sound waves is so strong due to strong resonant interaction with thermal ions so that the sound wave amplitude is small and becomes nonlinear at the late stage. The results are insensitive if the initial soliton wave amplitude is reduced by a factor of 2.

Electron Cyclotron Wave In this section we will present the 2D Vlasov simulations in the 2-dimensional

G ( x, y , v x , v y ) space with the external magnetic field B = Beˆ z . We use a rectangular mesh in the ( x, y ) plane, and in the velocity space we use the cylindrical coordinate. No mesh point is placed at the origin of the velocity space and the smallest velocities associated with the mesh points in the radial velocity direction is Δv ⊥ / 2 . The computational domain is

G G R = {( x , v ) | 0 ≤ x ≤ Lx , 0 ≤ y ≤ L y , 0 ≤ φ ≤ 2π , v ⊥ ≤ v ⊥ max } , and Nx, Ny, Nφ, and Nr

denote the number of mesh points used in x, y ,

φ , and v ⊥ directions, respectively.

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Splitting Methods for Vlasov-Maxwell Equations in Plasma Simulations

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Figure 1. The steady-state solutions of the average electron velocity distributions for three initial soliton maximum electric field amplitudes: E0 /(4πn0Te)1/2 = 1/3 with K0 = 1/15, (b) E0 /(4πn0Te)1/2 = 2/3 with K0 = 2/15, and (c) E0 /(4πn0Te)1/2 = 1 with K0 = 3/15. The fixed parameters are Te /Ti = 100, and mi/me =

100. Note that in the Figure the electron plasma frequency ωpe is abbreviated as ωp and the electron thermal speed vte is abbreviated as ve. The figure is reprinted from Phys. Plasmas, Vol. 2, C. H. Lin, J. K. Chao, C. Z. Cheng, One Dimensional Vlasov Simulations of Langmuir Solitons, 4195-4203, Copyright 1995, with permission from American Institute of Physics.

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14

C.Z. Cheng

Figure 2. The temporal evolution of the perturbed ion density profile, N(x,t), of the Langmuir soliton for different initial electric field amplitudes corresponding to cases shown in Figs. 1(a) , 1(b), and 1(c). Note that in the Figureλde is abbreviated as λd and the electron plasma frequency ωpe is abbreviated as

ωp. The figure is reprinted from Phys. Plasmas, Vol. 2, C. H. Lin, J. K. Chao, C. Z. Cheng, One

Dimensional Vlasov Simulations of Langmuir Solitons, 4195-4203, Copyright 1995, with permission from American Institute of Physics.

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Splitting Methods for Vlasov-Maxwell Equations in Plasma Simulations

15

Figure 3. The evolution of ion density profiles of the moving soliton with an initial group velocity vg =0.06 vte (or K1 = 0.02) for (a) Te /Ti = 100, (b) Te /Ti = 10, and (c) Te /Ti = 1, respectively. The mass ratio is mi /me = 100, thus vg = 6 vti for (a) Te /Ti = 100, vg = 1.9 vti for (b) Te /Ti = 10, and vg = 0.6 vti for

(c) Te /Ti = 1. The simulation system length L is 256λde and the initial soliton amplitude is determined by choosing K0 = 2/15. Note that in the Figure λde is abbreviated as λd and the electron plasma

frequency ωpe is abbreviated as ωp. The figure is reprinted from Phys. Plasmas, Vol. 2, C. H. Lin, J. K. Chao, C. Z. Cheng, One Dimensional Vlasov Simulations of Langmuir Solitons, 4195-4203, Copyright 1995, with permission from American Institute of Physics.

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16

C.Z. Cheng

Figure 4. The position of the maximum density depression of the soliton is shown as a function of time for the cases shown in Fig. 3. The position of the solitons (solid lines) ais compared with that without slowing down (dashed lines). The figure is reprinted from Phys. Plasmas, Vol. 2, C. H. Lin, J. K. Chao, C. Z. Cheng, One Dimensional Vlasov Simulations of Langmuir Solitons, 4195-4203, Copyright 1995, with permission from American Institute of Physics.

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Splittting Methodss for Vlasov-M Maxwell Equattions in Plasm ma Simulationss

17

Fiigure 5. Powerr spectrum of the t 2D nonlinear electron cycclotron waves in i a magnetizeed plasma is shhown

by

considering

k x = 2π / Lx = 1.02 ρ

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= Ny = 8, N

= Nr = 16,

−1 e ,

ions annd

as

a an

immobile

k y = 2π / L y = 1.02 ρ

−1 e

backgroundd

for

ωce / ω pe = 1

,

. The compuutational parameeters are Nx

ω pe Δt = π / 8 v ⊥ max = 4.5vtee , , and A = B = 0.01. The spectrum peaks

, 1.18875, 2.125, 3.0625, 4.0. The figure f is reprintted from J. Com mput. Phys., arre located at V 24, C. Z. Cheng, Vol. C The integration of thhe Vlasov equaation for a maggnetized plasm ma, 348-360, Copyright 1977, with permission from Elsevierr.

The equilib brium distribuution is a Maxw wellian,

, with w two or thhree velocity components c G G annd the initiaal condition of the distriibution functtion is choseen as f j (x , v , t = 0) = G f 0 j (v )(1 + A cos k x x + B coos k y y ) . The subscript referss to electronss and ions, reespectively. t simulationn results for 2D D electron cycclotron wavess with ions connsidered as We show the ann

immobilee

backgrounnd,

and

ωce / ω pe = 1 ,

k x = 2π / Lx = 1.02 ρ e−1 ,

and

k y = 2π / L y = 1.02 ρ e−1 . The T computattional parametters are Nx = Ny = 8, Nφ = Nr = 16,

ω pe Δt = π / 8 , v ⊥ max = 4.5vte , and A = B = 0.01. Figuures 5 and 6 show the poweer spectrum annd autocorrelaation of the Fourier modee of the elecctric field pottential Φ (k x , k y , t ) for

k x ρ e = 1.02 and k y ρ e = 0 . The autoccorrelation off the electric potential is defined as Eulerian Codes for the Numerical Solution of the Kinetic Equations of Plasmas, Nova Science Publishers, Incorporated, 2010. ProQuest Ebook

188

C Cheng C.Z.

C k (τ ) = (1 / t max m )∫

tmax

0

2556 time stepss with

Φ ( k x , k y , t )Φ * ( k x , k y , t − τ )dτ . The compuutation is carrried out for

ω pe t maax = 32π . Thhe time sequeence of the pootential Φ (k x , k y , t ) is 2

Foourier transfo ormed in timee to obtain thhe power spectrum P (ω ) ∝ Φ ( k x , k y , ω ) . The peeaks in the po ower spectrum m in Fig. 5 havve the values of o 4..0 which agreee with the linear l theory of

ω / ωce = 0, 0 1.1875, 2.1225, 3.0625, , 1.1354, 2.043, 3.0055, etc. The

auutocorrelation of the potenntial shows thee effects of thhe phase mixxing and dampping of the ellectron cyclottron wave. The conservatiion of the total energy haas relative errror 10-3 at . This simulatiion correspondds to the use of o 16384 grid points p in the phase p space

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ass far as the com mputer storage is concernedd.

Fiigure 6. The au utocorrelation of o the potentiall

Φ(k x , k y , t )

versus time is i shown for thhe nonlinear

magnetized plasm m ma as in Fig. 5. 5 The figure is reprinted from m J. Comput. Phhys., Vol. 24, C. C Z. Cheng, Thhe integration of the Vlasov equation for a magnetized plasma, 348-3360, Copyright 1977, with peermission from Elsevier.

We also no ote that the recurrence effecct, which occuurs in the num merical integraation of 1D V Vlasov equatio on for unmagnnetized plasm mas because of the finite reesolution in thhe velocity sppace, disappeaars for a 2D magnetized m plaasma. This is due to the facct that the maggnetic field pllays a dominaant role in the particle motioon. For particlle motion alonng the magnetiic field, we w would still exp pect recurrennce effects for single modde case with large initial amplitude. H However, for most m of the phhysically interresting cases, the wavelength along the field f line is veery large com mpared to the perpendicular p wavelength ( tim me

T the recurrence ). Therefore,

w would be very long for a moodest number of grid pointss in the

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Splitting Methods for Vlasov-Maxwell Equations in Plasma Simulations

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direction. On the other hand, the recurrence effect can be smoothed out by adding a collision term or other artificial entropy producing term to the system.

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Conclusion The Vlasov-Maxwell equations are the fundamental equations describing plasma dynamics. A revolutionary splitting scheme with second order accuracy in time step for obtaining efficiently numerical solutions of the Vlasov-Poisson equations was published by Cheng and Knorr in 1976 [5]. The splitting scheme was later shown to be a second-order symplectic scheme in 1990 [54]. Since the Cheng & Knorr paper was published, the splitting scheme was first extended to include the external magnetic field in 1977 [6], and then to electromagnetic cases with complex geometry by many authors. The splitting scheme lays the foundation for Vlasov simulations and has since been extensively employed to simulate various nonlinear electrostatic and electromagnetic plasma physics problems in higher dimensional phase space. The splitting scheme splits the Vlasov equation into a series of lower-dimensional hyperbolic partial differential equations (i.e., the free-streaming and accelerating equations) in the spatial and velocity space separately. These lower-dimensional equations have simple constant advection speed in the spatial dimensions and analytically tractable acceleration in the velocity space, and thus exact analytical solutions of these lower-dimensional equations can be obtained. The lower-dimensional equations can be solved by either performing interpolation or finite difference or finite element or finite volume methods. Because the lower-dimensional equations can be solved exactly by following the particle characteristics, the most natural numerical scheme is to perform interpolation. The important features are that not only the computation time requirement is quite low, but also excellent results can be obtained with a few number of grid points. Moreover, the method has also been extended to simplified plasma dynamical equations such as the gyrokinetic equations. Advantage of the Vlasov simulations employing the splitting scheme over the PIC methods has also been demonstrated on the consideration of numerical noise, selective mode resolution, etc. Moreover, with the advancement of computer technology such as massively parallel processors (MPP), Vlasov simulations using the splitting scheme should play a greater role in the study of nonlinear plasma kinetic physics in the future.

References [1] Birdsall, C. K.; Langdon, A.B. Plasma Physics via Computer Simulation; McGraw-Hill: New York, NY, 1985. [2] Armstrong, T. P.; Harding, R. C.; Knorr, G.; Montgomery, D. In Methods in Computational Physic; Alder, B., Fernbach, S., Rotenberg, M.; Ed.; Academic Press: New York, NY, 1970; Vol. 9, pp 29. [3] Gazdag, J. J. Comp. Phys., 1973, 13, 1175-1183. [4] Knorr, G. J. Comput. Phys., 1973, 13, 165-180. [5] Cheng, C. Z.; Knorr, G. J. Comput. Phys., 1976, 22, 330-351. [6] Cheng, C. Z. J. Comput. Phys., 1977, 24, 348-360.

Eulerian Codes for the Numerical Solution of the Kinetic Equations of Plasmas, Nova Science Publishers, Incorporated, 2010. ProQuest Ebook

20 [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]

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[25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41]

C.Z. Cheng Shoucri, M.; Gagne, R. A. J. Comput. Phys., 1978, 27, 315-322. Shoucri, M.; Gagné, R. Phys. Fluids, 1978, 21, 1168-1175. Shoucri, M. Phys. Fluids, 1978, 21, 1359-1365. Rickman, J.D. IEEE Trans. Plasma Science, 1982, 10, 45-52. Shoucri, M.; Storey, O. Phys. Fluids, 1986, 29, 262-265. Izar, B.; Ghizzo, A.; Bertrand, P.; Fijalkow, E.; Feix, M. R. Comput. Phys. Comm., 1989, 52, 375-382. Lin, C. H.; Chao, J. K.; Cheng, C. Z. Phys. Plasmas, 1995, 2, 4195-4203. Manfredi, G.; Shoucri, M.; Feix, M. R.; Bertrand, P.; Fijalkow, E.; Ghizzo, A. J. Comput. Phys., 1995,121, 298-313. Manfredi, G. Phys. Rev. Lett., 1997, 79, 2815-2818. Fijalkow, E. Comput. Phys. Comm., 1999, 116, 319-328. Fijalkow, E. Comput. Phys. Comm., 1999, 116, 336-344. Nakamura, T.; Yabe, T. Comput. Phys. Comm., 1999, 120, 122-154. Newman, D. L.; Goldman, M. V.; Ergun, R. E.; Mangeney, A., Phys. Rev. Lett., 2001, 87, 255001. Filbet, F.; Sonnendrucker, E.; Bertrand, P. J. Comput. Phys., 2001, 172, 166-187. Arber, T. D.; Vann, R. G. L. J. Comput. Phys., 2001, 180, 339-357. Filbet, F.; Sonnendrucker, E. Comput. Phys. Comm., 2003, 150, 247-266. Fijalkow, E.; Nocera, L. J. Plasma Phys., 2003, 69, 93-108. M. Gutnic, M.; Haefele, M.; I. Paun, I.; Sonnendrücker, E. Comput. Phys. Comm., 2004,164, 214-219. Pohn, E.; Shoucri, M.; Kamelander, G. Comput. Phys. Comm., 2005,166, 81-93. Pohn, E.; Shoucri, M.; Kamelander, G. J. Plasma Phys, 2006, 72, 1139-1143. Schmitz H.; Grauer R. Comput. Phys. Comm, 2006, 175, 86–92. Umeda, T.; Ashour-Abdalla, M.; Schriver, D. J. Plasma Phys., 2006, 72, 1057-1060. Califano, F.; Galeotti, L.; Mangeney, A. Phys. Plasmas, 2006, 13, 082102; DOI:10.1063/1.2215596. Shoucri, M. Comm. Nonl. Science Numer. Simuln, 2007, 13, 174-182. Yee, K. S., IEEE Trans. Antenn. Propagat., 1996, AP-14, 302–307. Boine-Frankenheim, O.; Hofmann, I.; Rumolo, G. Phys. Rev. Lett., 1999, 82, 32563259. Mangeney, A.; Califano, F.; Cavazzoni, C.; Travnicek, P. J. Comput. Phys., 2002, 179, 495. Venturini, M.; Warnock, R.; Ruth, R.; Ellison, J. A. Phys. Rev. Special Topics - Accel. Beams, 2005, 8, 014202, pp. 15. Schmitz H.; Grauer R. J. Comput. Phys., 2006, 214, 738–756. Schmitz H.; Grauer R. Phys. Plasmas, 2006, 13, 092309. Warnock, R. Methods Phys. Res., Sect. A, 2006, 561, 186. Warnock, R.; Bassi, G.; Ellison, J. Methods Phys. Res., Sect. A, 2006, 558, 85. Besse, E. N.; Mauser, N.; Sonnendrucker, E. Int. J. Appl. Math. Comput. Sci., 2007, 17, 361–374. Strozzi, D. J.; Williams, E. A.; Langdon, A. B.; Bers, A. Phys. Plasmas, 2007, 14, 013104, pp. 13. Venturini, M.; Warnock, R.; Zholents, A. Phys. Rev. Special Topics – Accelerators and Beams, 2007, 10, 054403(1-11).

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[42] Shoucri, M. In Computer Physics Research Trends; Bianco, S.J.; Ed.; Nova Science Publishers: New York, NY, 2007; pp. 1-87. [43] Shoucri, M. Comm. Comp. Phys. 4, 2008, 703-718. [44] Shoucri, M. Numerical Solution of Hyperbolic Differential Equations; Nova Science Publishers: New York, NY, 2008. [45] Shoucri, M.; Afeyan, B.; Charbonneau-Lefort, M. J. Phys. D., 2008, 41, 239901, pp. 68. [46] Umeda, T. Earth Planets Space, 2008, 60, 773–779. [47] Umeda, T.; Togano, K.; Ogino, T. Comput. Phys. Comm., 2009, [48] doi:10.1016/j.cpc.2008.11.001 (in press). [49] Sonnendrucker, E.; Roche, J.; Bertrand, P.; Ghizzo, A. J. Comput. Phys., 1999, 149, 201. [50] Coulaud, O.; Sonnendrucker, E.; Dillon, E.; Bertrand, P.; Ghizzo, A. J. Plasma Phys., 1999, 61, 435-448 [51] Besse, N.; Sonnendrucker, E. J. Comput. Phys., 2003, 191, 341-376. [52] Ghizzo, A.;Huot, F.; Bertrand, P. J. Comput. Phys., 2003, 186, 47–69. [53] Elkina, N.V.; Buchner, J. J. Comput. Phys., 2005, 213, 862–875. [54] Besse, N.; Segré, J.; Sonnendrücker, E. Transp. Theory Statist. Phys., 2005, 34, 311-332. [55] Forest, E.; Ruth, R. D. Physica D, 1990, 43, 105-117. [56] Yoshida, H. Phys. Lett.,1990, 150, 162. [57] Watanabe, T.-H.; Sugama, H.; Sato, T. J. Phys. Soc. Jpn., 2001, 70, 3565. [58] Watanabe, T.-H.; Sugama, H. Transp. Theory Statist. Phys., 2005, 34, 287-309. [59] Zakharov, V. E. Sov. Phys. JETP, 1972, 35, 908-914. [60] Pereira, N. R.; Sudan, R. N.; Denavit, J. Phys. Fluids, 1977, 20, 271-281. [61] Gorev, V. V.; Kingsep, A. S. Sov. Phys. JETP, 1974, 39, 1008-1011.

Eulerian Codes for the Numerical Solution of the Kinetic Equations of Plasmas, Nova Science Publishers, Incorporated, 2010. ProQuest Ebook

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In: Eulerian Codes for the Numerical Solution... Editor: Magdi Shoucri, pp. 23-63

ISBN 978-1-61668-413-6 c 2010 Nova Science Publishers, Inc. °

Chapter 2

A V LASOV A PPROACH TO C OLLISIONLESS S PACE AND L ABORATORY P LASMAS Francesco Califano∗ and André Mangeney† Physics Department, University of Pisa, Italy Observatoire de Paris, France

Abstract

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The study of the collisionless, non-linear electrostatic and electromagnetic plasma dynamics provides a paradigm for the analysis of dynamical systems. Their investigation has fundamental applications to different fields ranging from space plasmas to geophysical and astrophysical systems as well as to the achievement of novel energy sources through the development of magnetic and inertial fusion . In this perspective, kinetic Vlasov simulations are today one of the basic tools in plasma physics research that benefit of the impressive development of super computers.

PACS 52.65.Ff, 52.65.-y, 52.35.Mw, 52.35.Sb Keywords: Vlasov equation, Plasma kinetic simulations

1.

Introduction

Numerical simulations are today a powerful tool for the understanding of the phenomena occuring in laboratory and space plasmas. Since these phenomena cover a very wide range of time and spatial scales, it has been necessary to develop different models, each coping with a specific physical regime. In particular, the relative importance of collisions discrimates between collisional and collisionless regimes depending on wether the collisional time and spatial scale legths, τc and `c , are much smaller or much greater than the time and spatial scales characterizing the constraints applied on the plasma τH , `H , for example the duration of the discharge and the dimensions of the container for a laborary plasma or the scale height and expansion time for the solar wind plasma. ∗ E-mail † E-mail

address: [email protected] address: [email protected]

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Francesco Califano and André Mangeney

In the collisional regime, particle distribution functions remain close to Maxwellian ones so that the physical state of the plasma is well described by a small number of velocity moments, such as the density n, the fluid velocity u, the electron and proton temperatures Te and Tp , . . . , and their gradients. The evolution of these moments is governed by the fluid like equations of MHD, expressing conservation of mass, momentum and energy. On the contrary, in the collisionless regime characterized by τc À τH , `c À `H

(1)

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the particle distribution functions may be so strongly distorted by ballistic effects that they become unstable and excite electrostatic or electromagnetic waves which react back on the particles in order to limit these distortions. Such kinetic effects play a key role in the plasma dynamics and are often crucial to transfer ordered (large scale) energy toward small scale fluctuations which, reacting on the particles, push the system toward some sort of "thermalisation”. This process of wave particle interaction can somewhat replace the role of collisions, at least sufficiently to explain why a hydrodynamic approach for the modeling of collisionless plasmas has been often very successfull in laboratory or space plasmas, while one would expect the MHD model to give good results only for time and scale lengths larger than the collisional scales. For fully ionized plasmas, the collisional time scale τc is proportional to the number of particles in the Debye sphere ω p τc ' 2πnλ3D / ln Λ (2) where ω p = (4πne2 /me )1/2 is the electron plasma frequency, λD = vth,e /ω p the Debye length e the proton electric charge, vth,e the electron thermal speed, and ln Λ = ln(λmin /λD ) 2 the Coulomb logarithm. Here λmin = e2 /me vth,e is the impact parameter for strong collisions. Note that this expression corresponds to the three dimensional case; similar expressions for one and two dimensions are respectively 2nλD , and 2π2 nλ2D . The number of particles in a Debye sphere n λ3D number is very large for high temperature, tenuous plasmas usually found in space and often in the laboratory. For example, high temperature plasmas, Te ∼ a few keV, are routinely obtained by running high magnetic field (B ∼ 10T), high currents and low electron density (< a few 1013 cm−3 ) discharges, resulting in ∼ 107 particles in the Debye sphere. This number reaches values of ∼ 1010 in the solar wind at the earth orbit where typical values of the macroscopic fields are n ∼ 10 cm−3 , Te ∼ 10 eV, B ∼ 10−8 T. As a result, the conditions (1) are satisfied in these plasmas and in first approximation collisions may be neglected. For example in the solar wind the mean free path is of the order of, or greater than, the dimension of the system. Observations show indeed that distribution functions are often non Maxwellian. Here the electron distribution function displays a three component structure with a dominant Maxwellian core, a tenuous suprathermal halo and a beam like "strahl” propagating away from the sun. Significant technical developments in space observations have offered the possibility to obtain measurements of electric and magnetic fields, density, etc. with a sufficient sensitivity and resolution (in space and time) to reach the kinetic scales, such as the Debye length and electron plasma frequency. It is today possible to investigate the fine structuring of space plasmas and its role in converting the large-scale macroscopic motions in thermal energy, in particle acceleration as well as in the heat transport that drive the dynamics and/or

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A Vlasov Approach to Collisionless Space and Laboratory Plasmas

25

the stability of space plasmas. In particular, one can check the well established assumption concerning wave-particle interaction process, responsible for the partial thermalization of collisionless plasmas discussed above, implies an electric (or electromagnetic) field made of the superposition of incoherent weakly nonlinear waves (the so called quasi-linear approximation). Furthermore, the plasma "thermal electric noise”, created by the random motion of charged particles is now routinely measured; it is so close to the (very low) level predicted by a quasilinear type of theoretical model that it can be used as a diagnostic tool. In the context of space observations, the strong surprise has been the discovery by satellite measurements (e.g. Fast, Wind, Cluster) of small-scale coherent structures with corresponding non Maxwellian distribution functions. Coherent electrostatic solitary structures are observed in space in the form of bipolar electric pulses mainly parallel to the ambient magnetic field. These were first observed in strongly non-homogeneous regions of the Earth’s environment as the bow shock, the plasma sheet boundary, the aurora zone and were detected also in the solar wind. Today, they are routinely observed in all planetary environments. The electric field associated to such coherent structures spans over a wide range of values depending on the observed location, of the order or less than 0.1 mV/m in the solar wind and in the plasma sheet boundary layer, while up to 100mV/m or more in the bow shock and in the aurora regions. Among the most spectacular observations, we quote the discovery of ion and electron holes on a typical scale length of a tenth of Debye lengths. More recently, Cluster data have given evidence of magnetized coherent structures together with energetic supra-thermal electrons in the Magnetotail. These structures are embedded in current sheets and are probably the signature of previous strongly energetic events, as for example magnetic reconnection. Magnetic reconnection is a fundamental plasma physics process where kinetic effects manage to mimic some MHD like features due to the "natural” generation of smaller and smaller scales by the system. It is the only process able to reorganize the large scale topology of the magnetic field as well as to dissipate the magnetic energetic of the system; it is considered to be the driver of the most important energetic phenomena observed in thermonuclear fusion machines, in the solar atmosphere, in many regions of the (outer) Earth Magnetosphere. In general, energetic effects are of primary interest in astrophysics, while magnetic field topology is considered as the crucial point in fusion plasmas since it affects the confinement properties. From a physical point of view, magnetic reconnection is able to modify the large scale field topology in plasmas due to a local violation of the so-called frozen-in law which links the evolution of the magnetic field lines to the corresponding fluid elements. The large and small-scale dynamics are strongly linked in a multi-scale process, characterized by two key points. The first one is that the breaking of the linking between the fluid and the magnetic field occurs in a region of microscopic dimensions where electrons decouple from the magnetic field lines. It is therefore a process driven by the large fluid scale evolution, the role of collisions being replaced by the electron microphysics processes such as electron inertia, finite Larmor radius effects, Landau resonances, etc. The second key point is that in a real system the large scale evolution of the full system creates the conditions for reconnection to occur which in turn influences its further dynamics [30]. Therefore, the large and small-scale dynamics are strongly linked in a multi-scale process, so that the study of kinetic microphysics is considered as crucial in the field of reconnection. In other words,

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Francesco Califano and André Mangeney

the process of magnetic reconnection is probably the best example of a process driven by the large fluid scales evolution, but where the role of microscopic kinetic physics replacing collisions has a key role on its macroscopic evolution. Small scales coherent structures, as dipolar electric fields associated to phase space holes similar to those mentionned above, have been observed in recent laboratory experiments along the separatrix [32] or by satellite observations in the magnetotail [40]. These coherent structures could have a basic role in the final energy dissipation and be a local source of anomalous resistivity. The inverse process with respect to the annihilation of magnetic flux inside the reconnection layer is the magnetic field generation due to the presence of temperature anisotropies or of sources of ordered kinetic energy (particle beams). The corresponding instability is known as the Weibel instability. This instability is mostly important in the case of electron velocities close to the speed of light, i.e. for plasmas close to (or in) relativistic regimes. The Weibel instability is a very important process since, on time scales of the order of the inverse of the electron plasma frequency, it creates a quasi-static magnetic field perpendicular to the plane of the beams and of the perturbation wave vector. In general, since the instability develops on electron time scales, ions can be taken to remain at rest until the electrostatic field, that is rapidly generated in the nonlinear phase of the instability due to charge separation, becomes important. In a collisionless plasma, the saturation of the instability does not occur because of dissipative processes but because of particle trapping in the magnetic field generated by the instability, as for the case of the well known two-stream (electrostatic) instability. Therefore, the nonlinear regime of the Weibel (and two-stream) instability can only be studied in a framework of a full kinetic analysis. Presently, strong interest in the Weibel instability has been renewed in the framework of the interaction of intense laser pulses with plasmas and of the accompanying generation of ultraintense magnetic fields as well as in Astrophysics as a mechanism for the generation of a seed magnetic field to be then amplified by some dynamo like effect. In summary, the analysis of kinetic processes in collisionless or weakly collisional plasmas can be today considered as one of the outstanding problems of plasma physics. At such "small” length scales (and high frequencies) the collisionless Vlasov mean field theory is the physical model for the understanding of the processes in which collisions are still neglected, but kinetic effects play now a key role in the plasma dynamics. This analysis is based on the Vlasov equation, self consistently coupled to the Maxwell equations, which describes the plasma as a continuous, Hamiltonian collisionless system where the distribution function is not systematically brought back toward a Maxwellian type equilibrium. Indeed, in the Vlasov approach the description of a plasma as a collection of discrete charged particles is completely lost in favor of a more tractable model where, however, the plasma considered as a continuous medium.

2.

The Vlasov–Maxwell Equations as a Hamiltonian Flow on Phase Space

The Vlasov equation is obtained by integrating the Liouville equation for the N-particles distribution; It is based on a mean field theory in which each particle interacts with an average field generated by all plasma particles, while single particles interactions are completely

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A Vlasov Approach to Collisionless Space and Laboratory Plasmas

27

neglected. It can be considered as a "transport” equation in phase space. The basic quantity at this level of description is the one particle distribution function fa (d.f. in what follows) defined on the one particle phase space (position x, velocity v) and giving the number of particles of species a fa (x, v) dx dv

(3)

in the phase space volume element dx dv. This distribution function satisfies the Vlasov equation d fa ∂ fa ∂ fa ∂ fa +v· +F · =0 (4) = dt ∂t ∂x ∂v where F is the force acting on a particle located at position x with a velocity v. The Vlasov equations may describe electrostatic processes when self-consistently coupled to the appropriate Poisson equation ∇ · E = 4π ρ

(5)

Then the force F reduces to the electric force

F =

Za e E ma

(6)

where ma and Za e are the mass and electric charge of the species a. It can also be used for the study of electromagnetic phenomena in a collisionless, magnetized plasma when coupled to the Maxwell equations

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4π 1 ∂E = ∇×B− j; c ∂t c

1 ∂B = −∇ × E c ∂t

(7)

and F is now the full Lorentz force · ¸ Za e (v × B) F = E+ ma c

(8)

The electric and magnetic fields E(x,t) and B(x,t) are determined by the distribution of the charge densities ρ = e ∑a Za na and current densities j: Z

na =

Z

fa dv ; j = e ∑ Za

v fa d 3 v

a

The Vlasov–Poisson or Vlasov–Maxwell system of equations covers an impressive range of physical regimes separated by many order of magnitude in frequency and scale lengths. It is therefore useful to work in dimensioless variables. Thus, we define a characteristic ¯ −1 , mass m¯ and scale length l¯ and so a characteristic velocity time scale and frequency t¯ = ω ¯ t¯. Furthermore, n¯ is the total particle number density, ρ¯ = en¯ the characteristic charge u¯ = l/ ¯ and density and j¯ = en¯ u¯ a characteristic current density. Finally, we define E¯ = mu¯ω/e ¯ the characteristic electric and magnetic field, where c is the speed of light. As B¯ = mcω/e ¯ B¯ is the associated drift speed. The Vlasov equation becomes in dimensionless a result, cE/

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Francesco Califano and André Mangeney

form (physical and dimensioless quantities are designed by the same symbols, risks of confusion being small): ∂ fa ∂ fa Za ∂ fa + va · + [E + va × B] · =0 ∂t ∂x ma ∂va

(9)

(here ma is the dimensionless mass of particles of species a, ma /m) ¯ while the dimensionless Poisson equation reads: ∇·E =

1 4πne ¯ 2 1 Za na ; α1 = ∑ ¯2 α1 a m¯ ω

(10)

Note that α1 = 1 when the equations are normalized on the electron plasma time scale ¯ = ω pe , where ω pe is the electron plasma frequency. The dimensionless since m¯ = me and ω Ampere and Faraday equations read: ∂E = α2 ∇ × B − α1 j ; ∂t

∂B = −∇ × E ; ∂t

α2 = c2 /u¯2

(11)

The Vlasov–Maxwell or Vlasov–Poisson system is mathematically a very complex non linear system of equations, since the force F depends on the distribution functions fa . A fundamental feature of this model is that the d.f. is subjected to strong topological constraints, provided by the existence of invariants,

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d dt

Z

dxdv H( fa ) = 0

for any function H. This reduces significantly the (infinite) number degrees of freedom of the system. For example, the d.f. can be transported and roll up in a complex way in phase space, but different d.f. isolines can never be broken and reconnect. As a result, transitions from "unconnected states” in phase space are forbidden, as for example from a laminar type state (i.e. free streaming) to a vortex type state (i.e. particle trapping). This situation is similar to that of an ideal plasma where the magnetic field lines frozenin condition prevents any transition from different (not connected) magnetic energy states. Two phase space elements initially connected by an iso-line of f will remain connected by the same isoline in time even if the shape of the isoline can be strongly changed by the system evolution. In other terms, the formation of phase space "holes” or "vortices” during the development of a number of kinetic instabilities, as observed in many published numerical simulations, cannot be correctly described within the Vlasov approximation; actually all numerical schemes are at least weakly dissipative, allowing the observed break up of d.f. isolines. However, the comparison with observations apparently authorizes to believe that this small amount of numerical dissipativity, even if unphysical, allows to follow the essential of the evolution of the systems under study (see Ref. [17] for a dicussion). The Vlasov equation, for given electromagnetic fields, is a hyperbolic partial differential equation with characteristics given by dv dx = v; =F dt dt

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(12)

A Vlasov Approach to Collisionless Space and Laboratory Plasmas

29

which obey the same equations as particle trajectories; note that in general isolines of the distribution function must not be confused with the characteristics. Equations (12) may be put in Hamiltonian form; this is where the Vlasov equation makes contact with Hamiltonian mechanics. The electrostatic limit (equations 5 and 6) is particularly well suited to illustrate this point, since the equations (12) can be written as dq p ∂H dp ∂H = = ; = ma F = − dt ma ∂p dt ∂q

(13)

using canonical variables (q = x, p = ma v) which are proportional to the usual phase space coordinates. The corresponding Hamiltonian reads: H=

p2 − Za Φ(q) 2ma

(14)

Here Φ(x) is the electrostatic potential depending only on time and the space coordinate. The corresponding electric field is given by E = −∇Φ. The trajectories describe thus a "Hamiltonian flow” >t in phase space. If we define z0 = (x0 , v0 ) as a phase space point at time t = 0 and zt = (x, v) the phase space point with coordinates at time t which are the (unique) solutions of equations (12) with initial conditions r0 , >t is given by >t z0 = zt ; >−t zt = z0 The flow >t is reversible and preserves the phase space volume element: >t >−t = identity , dzt = dz0

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The solution of the Vlasov equation may then be expressed formally in terms of a propagating operator P acting on the distribution function. If f0 is the d.f. at t = 0, then f (z,t) = P t f0 = f0 (>−t z)

(15)

This (formal) solution may be termed "Lagrangian” in the sense that the value of f (z,t) at time t depends on the initial coordinates z0 at t = 0 along the characteristic arriving at the phase space point z at time t. The Vlasov equation can be rewritten using the Poisson brackets, ∂f = [H, f ] ∂t

(16)

where, as usual, the brackets are given by: [a, b] = (∂a/∂x)(∂b/∂v) − (∂a/∂v)(∂b/∂x). By defining Λ as the "Poisson bracket operator”, Λ f ≡ [H, f ], the Vlasov propagator can be written as P t = exp(Λt) (17) The fully electromagnetic case is a little more involved, but without significant consequences for what follows.

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30

3.

Francesco Califano and André Mangeney

Commonly Used Numerical Schemes

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In the free streaming case (F = 0), the characteristic equations have the trivial solution x = x0 + vt, v = v0 and the propagator P t takes the form of a displacement operator µ ¶ ∂ t P f (x, v,t) = exp −tv f (x, v,t) = f (x − vt, v,t = 0) = f0 (x0 , v0 ) (18) ∂x as may be checked by a simple Taylor expansion. Except for the free streaming case, one does not know explicit expressions of the propagator P t appearing in Equation (15), even for very simple physical problems. As a consequence, the theoretical study of the Vlasov–Poisson or Vlasov–Maxwell system of equations is based today on (large scale) numerical simulations. These simulations are very challenging, due to the huge size of the problem. The Vlasov–Maxwell system is indeed nonlinear and the d.f.’s depend on seven variables: three configuration space variables, three velocity space variables and time, for each species of particles.This feature makes it essential to choose carefully the numerical schemes to be used. Furthermore, in order to tackle the most realistic possible physical problems, it is important also to use all the modern computing power and techniques, in particular parallelism. Several approaches may be used to replace the infinite number of degrees of freedom a priori necessary to describe the particle distribution functions to only a finite number. The first one is to discretize directly the Vlasov equation on a grid in phase space and the field equations on a spatial grid and solve the resulting system of finite difference equations by standard methods. The main draw back of such a purely "Eulerian” approach is that the time step is severely limited by CFL type conditions, prohibiting very long simulations, and we will not discuss them any further. The other approaches use the hyperbolic character of the Vlasov equation by following the characteristics along which the d.f. are constant. A significant advantage of semi-Lagrangian techniques over traditional Eulerian methods, long recognized in atmosphere-ocean applications, is their ability to run with relatively long time steps, even permitting violation of the well-known Courant–Friedrichs–Lewy criterion (see [59] for meteorological simulations). It is a numerically efficient method for advection problems and is well suited for parallelization.They are widely used in many research fields to solve partial differential equations describing an advection process. Here we will focus on the Vlasov equation.

3.1.

Particle Methods

Among these more or less "Lagrangian” methods, particle simulations have been used for a long time to give a numerical solution of purely convective problems, such as the incompressible Euler equation in fluid mechanics (see Refs. [42], [55]) or the Vlasov equation in plasma physics (see Ref. [5]). They are still the most widely used numerical techniques for computational and theoretical studies of fluids and plasmas. Over many decades of research, they have reached a mature state with the development of a vast number of techniques, which can now be found in classic texts and monographs on the subject. A general introduction of the particle methods and their numerical analysis for general advection equations may be found in Ref. [23], while a study of the convergence rate of these methods

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31

is given in Ref. [22]. Particle methods replace the infinite number of characteristics corresponding to expression (15) by the trajectories of a large set, but finite "computational macro-particles” of finite size. These macro-particles follow the characteristics (12), i. e. obey the classical laws of mechanics and interact with each other through self-consistently generated fields. While in fluid applications the particles are introduced as a numerical artifice to add an appealing lagrangian character to the model, in plasma simulations the computational particles behave as true charged particle interacting with each other through long range forces. In the particle approach, the solution is sought as a linear combination of Dirac distributions: Np ¢ ¡ fN (Z,t) = ∑ αi (t)δ Z − Zip (t) i=1

{Zip (t)}

and coefficients {αi (t)} represent locations and weights of the Np The positions computational particles, respectively. These quantities are chosen to provide a good approximation to the distribution function f (Z), in the sense that Z

I(A) =

Np ¡ ¢ A(Z) f (Z)dZ ∼ ∑ αi A Zip

(19)

i=1

for any function A(Z) of the phase-space coordinates Z. For example, I(A) would be the number density in configuration space if A = 1 and the integral is over the velocity space. As discussed in Ref. [2] this is equivalent to a Monte Carlo evaluation of I(A), a numerical quadrature in a multidimensional phase space using pseudorandom numbers. To see this, let us write the expression of I(A) as the mean value of a quantity g(Z) depending on a random variable Z distributed with a known probability density p(Z) Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved.

Z V

p(Z)dZ = 1

Then, the integral for I(A) can be rewritten ¶ Z µ A(Z) f (Z) p(Z)dZ I(A) = hgi = p(Z) V i.e. as the mean value of the random variable g(Z) =

A(Z) f (Z) p(Z)

Using an independent random sample (Z1 , Z2 , . . . , ZNp ) from the random variable Z, we may obtain the sample mean g¯ as g¯ =

1 N

N

∑

g(Z j )

j=1

which is the unbiased and consistent Monte Carlo evaluation of I(A) = hgi. The standard error p in the Monte Carlo evaluation decreases slowly with the number of particles ∼ O(1/ Np ). Eulerian Codes for the Numerical Solution of the Kinetic Equations of Plasmas, Nova Science Publishers, Incorporated, 2010. ProQuest Ebook

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Francesco Califano and André Mangeney

In this picture, the computational particle (or Lagrangian marker) positions in phase space play the role of the random sample, while the weights αi contain statistical informations about the distribution function f αi =

1 f (Zi ) Np p(Zi

p and to order O(1/ Np ), equation (19) can be written in terms of fN : Z

I(A) =

A(Z) fN (Z)dZ

In particular, the number N(Ω) of physical particles in a given volume Ω of phase space, is given by Z Z Ω

f (Z)dZ ∼

Ω

fN (Z)dZ =

∑

α j = N(Ω)

Z j ∈Ω

p up to discretization error of ∼ O(1/ Np ). The weights αi are therefore constant along the particle trajectory; this shows that moving the markers along the orbits defined by Eq. (12) while letting them carry the information {αi }about the distribution function f allows to determine the time evolution of f and its moments. In particular, these initial positions and weigths, {Zip (0)} and {αi (0)}, are chosen to provide a good approximation to the initial distribution function f 0 (Z), i.e, to accurately approximate the integral equation Np

∑ αi (0)A(Zip (0)) =

Z

dZ f 0 (Z)A(Z)

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i=1

for all functions A(Z). For example, a plain Monte Carlo algorithm would sample points uniformly in the integration region. Since the numerical domain in phase space is finite, of size, say, VT , p(Z) = 1/VT is constant in this case. Now one may cover the numerical domain in phase space with a mesh consisting of Np cells Ci , i = 1, ..., Np of volume Vi and center of mass ZCM i . Then, placing the particles at t = 0 into the centers of mass of the corresponding cells gives the following initial data for the system ¡ ¢ ; αi (0) = Vi f 0 ZCM Zip (0) = ZCM i i There are more elaborate sampling strategies (see e.g. [2]) to reduce the numerical noise. Furthermore, in order to recover a proper approximation of the solution f (Z,t) and the corresponding moments at some time t > 0, one needs to regularize the particle solution fN (Z,t), and hence the performance of the method depends on the quality of the regularization procedures, allowing the recovery of the approximate moments from this particle distribution. It is as if the parameter (electric charge and mass) of interaction of a particle with the fields were spatially distributed with a known law S(r), i.e., instead of pointlike (structureless) particles. In other word, a new object, which is a particle and its charge

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cloud, is introduced [5, 44]. The characteristic space scale h generated by the distribution S is considered a particle "size”. The particle size h must not be too small compared to the distances between the particles and their neighbors, while too large values of h will generate unacceptable smoothing errors. Since collisions come about due to the rapid variation of the force, a finite particle size will reduce these strong variations for close encounters so that the collision rate is also greatly reduced. The corresponding increase in the collision time is described in Ref. [24]. 3.1.1.

Fully Lagrangian Schemes

Grid-free Lagrangian particle simulation methods for collisionless plasmas are the only truly Lagrangian ones. In these simulation methods the force exerted on a given particle by the other ones is explicitly calculated, for example using a tree-code field solver (a O(Np log Np ) method) to efficiently solve Poisson’s equation without using an underlying mesh [21, 46]. One may even insert new particles to maintain resolution in phase space instead of keeping a fixed number of Lagrangian particles in order to avoid loss of resolution as gaps appear in the particle distribution during long simulations [46]. This method has been applied to the two stream instability for cold and warm plasmas. Adaptive particle insertion allows one to capture details that are difficult to resolve with traditional particle simulations.

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3.1.2. Particle in Cell Methods Particle-mesh techniques use a numerical mesh in the physical space to compute more effectively the forces acting on the model particles, i.e., the interactions with self-consistent and external fields (see, e.g., [5] or [23]) . Otherwise known as Particle-In-cell (PIC), particle-mesh codes was originally invented at Los Alamos [42] for application in compressible fluid flows. It was essentially reinvented in the mid sixties in Ref. [44] for application to plasmas. Since then it has been applied most often to plasmas, but also to gravitational N-body systems, in solid state device design, compressible fluid flow, incompressible fluid flow (the vortex method) and MHD. These techniques combine Lagrangian features (the numerical particles advance) with eulerian features (the field solvers) and require four basic steps: 1. Solve continuum equations on an Eulerian grid (Poisson or Maxwell equations, electron equations for the hybrid case) 2. Track the individual particles by solving the equations of motion; 3. Couple the Eulerian to the Lagrangian framework by interpolating the fields to the particle positions, E(x p ), B(x p ); 4. Couple the Lagrangian to Eulerian framework, by evaluating the values of the electric

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Francesco Califano and André Mangeney charge and current densities at the spatial grid points x j : Np

ρ(x j ) =

∑ qi S(|xi − x j |)

(20)

∑ qi vi S(|xi − x j |)

(21)

i=1 Np

j(x j ) =

i=1

Although the solution of the dynamical equations in the second step introduces some error and noise, the "noise in particle simulations” is predominantly associated with the fourth step, where low-order moments of the distribution function are calculated to find the source terms for Poisson’s or Ampere’s equations.

3.2.

Semi-Lagrangian Methods

Instead of tracking numerical particles along the characteristics of the Vlasov equations, methods relying on a discretization of the phase space but following the characteristic curves at each time step have been proposed (see section 4.1.) under the generic appelation of "Semi Lagrangian” schemes, altough the terminology is not well settled. The name originates from the mixed Lagrangian-Eulerian aspect of such method, i.e. the solution of a Lagrangian flow on a fixed Eulerian grid. Semi-Lagrangian methods are today also used in plasma physics research, in particular in the relativistic limit. Two broad classes of discretization techniques may be distinguished.

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3.2.1. Grid Discretizations In the first one, the distribution function is discretized on a fixed Eulerian grid {Zm } in phase space, { fm (t)} being the values of f at the nodes, at time t. Suppose that { fm (tl )} is known at times tl = l∆t, l = 0, . . . , n where ∆t is the time step. Since the d.f. f is conserved following the flow, the computation of the d.f. at the new time step { fm (tn+1 )} is done by following backwards the characteristic curves during the time step ∆t and interpolating the values of { fm (tn )} at the origin of the characteristics. In other terms, the trajectory of each Lagrangian point Zm on the grid at time tn + ∆t is integrated backwards in time to find the "departure point” Zdm , at some earlier time, say tn . Then f (Zm ,tn + ∆t) = f (Zdm ,tn ) = f (Zm − DZm ,tn )

(22)

The "displacement” vector DZm = Zm (tn + ∆t) − Zdm (tn ) is obtained through some numerical approximation of the equations of motion (12), usually at second order, in ∆t. Then an interpolation is used to obtain the values f (Zdm ,tn ) of the distribution function at the departure points from the known values { fm (tn )}. Critical elements of the procedure, therefore, typically involve (a) determination of the departure points Zdk , which requires an interpolation of the force field F to Zd and/or some other point along the characteristics and (b) an interpolation of the distribution function to the departure points, all to sufficient accuracy to preserve the main characteristics of the process. Various procedures involving differing combinations of iteration and interpolation have been suggested and largely discussed in the meteorological context [50, 56] and in the

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plasma communities [58], while element (b) is generally implemented using cubic or higher-order interpolation schemes since linear interpolation is not sufficiently accurate. An important question is the extent to which compromises may be made, especially in the design of element (b), to achieve savings in computer time and storage. Other uncertainties concern (i) the representation of boundary conditions and (ii) the role and significance of conservation properties, since these schemes do not, in general, possess formal conservation properties which are analogous to those of many Eulerian schemes. While the consequences of non-conservation in semi-Lagrangian advection are generally taken to be modest, the absence of such properties is a matter of some concern, especially where such schemes are used for long-term simulations. "Symplectic” integrators, exploiting the hamiltonian character of the phase space flow, have even been proposed to search for the downstream departure point in order to improve the preservation of the physical quantities (e.g. [60]). It is worth to underline that the Vlasov equation does not contain any dissipative effect. Dissipation is only due to numerical errors depending on the accuracy of the interpolation and accuracy of the trajectory estimation. 3.2.2. Finite Elements Discretizations In the second classes of discretization techniques, which will be mostly considered here, one uses a partitioning of the computational domain into M small cells Cm , each one bearing a cell average value f¯m , m = 1, . . . , M of the distribution function 1 f¯m (t) =

Z

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Vm

Cm

dxdv f (x, v,t)

(23)

where Vm is the volume of the cell Cm . From the Vlasov equation one then may obtain an exact equation relating the time variations of f¯m to the fluxes crossing the cell boundaries. These fluxes depend on the value of the distribution function f at the cell boundaries but not on the cell averages { f¯m }. A critical step is then to express the fluxes at the cell interfaces in terms of the cell averages, a procedure implying an interpolation; once this is done, one obtains an explicit discretization scheme, known under the name of Finite Volume Method, which is known to be a mass conservative, robust and computationally "cheap” method for the discretization of conservation laws (see, e.g., [29] and the references therein). Finite volume schemes have already been implemented to approximate the solution of the Vlasov equation coupled with the Poisson equation and, more recently, with the Maxwell system. One may also use the cell averages { f¯m } to obtain a finite element discretization of the distribution function, where f is expressed in terms of a finite elements basis functions on each cell. Mathematically, the finite element discretization of a function f (Z) is given by a linear combination of a finite number S of basis functions χsm (Z), s = 0, .., S for each cell. In other word, it is a projection of f on a finite dimensional functional space having both a finite complete basis χsm (Z) and the corresponding dual basis ξsm : Z

f (Z) = ∑ m,s

am,s (t) χsm (Z)

, am,s =

dZ ξsm (Z) f (Z)

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(24)

36

Francesco Califano and André Mangeney

The dual basis ξsm is defined such that Z

0

dZ ξsm0 (Z)χsm (Z) = δm,m0 δs,s0 . The basis functions may be considered as providing a low order interpolation inside each cell; for example a piecewise linear basis functions leads to a first order interpolation. As discussed in Section 2. (see in particular Eq. 15) the solution of the Vlasov equation can be expressed in terms of a propagator describing the flow in phase space. The same is true for a time step ∆t f (Z,t + ∆t) = P ∆t f (t) = f (>−∆t Z,t) (25) If the coefficients asm (t) are known at time t, f (Z,t + ∆t) at a later time can be written in terms of these known coefficients ¡ ¢ f (Z,t + ∆t) = ∑ asm (0) χsm >−∆t r m,s

However, this expression is not in the form of Eq. (24). One has to project onto the finite element basis. This may be done by making use of the dual basis ξsm . The coefficients asm (t + ∆t) are finally obtained as: r asm (t + ∆t) = ∑ Ψm,s n,r an (t)

(26)

n,r

where the matrix element of the evolution operator >−∆t is given by Z

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Ψm,s n,r =

£¡ ¢¤ dZ ξsm (Z) >−∆t χrn (Z)

Equation (26) provides a numerical scheme for the time advance of the coefficients asm , which we will refer to as the "discontinuous Galerkin method”, as it relies on the projection on a function space spanned by a finite basis of functions, without requiring any condition on the regularity of f and its derivatives at the cell boundaries. Imposing such conditions will naturally decrease the dimension of the space on which f is projected; the key point is that the cell averages { f¯m } may be used to interpolate f in the numerical domain, so that one expects that the coefficients am,s may be related to the cell averages { f¯m }. The requirements imposed on this interpolation define the projection {am,s } to { f¯m }. To illustrate this point while avoiding cumbersome notations, we shall assume for the sake of simplicity that f depends on one variable only, say x, the generalisation to a six dimensionnal phase space being relatively straigthforward. The numerical domain [0, L] is partitionned into M intervals (the cells) of width ∆, centered on x¯m and bounded by the grid points [xm−1 , xm ], xm = m∆, m = 1, . . . , M. The finite element basis functions χsm are also chosen to be polynomials of order s ≤ S : ¶ µ x − x¯m s Ym (x) (27) χm (x) = Ps 2∆ where Ym (x) is the characteristic function of m-th cell. An example which will be developped futher in this paper uses Legendre polynomials, P0 (u) = 1, P1 (u) = u, P2 (u) =

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u2 /2 − 1, etc. In that case ξsm = χsm . Starting with the definition of the primitive function of f, Z H(x) =

x

0

dx0 f (x0 )

we can exactly express its values at the node points xm by means of the cell mean values of the function f [43]: Hm = H(xm ) = ∆ ∑ f¯l l≤m

˜ { f¯m }) be some interpolant for the primitive function over the numerical interval. Let H(x, One may require from this interpolant a certain number of regularity conditions, for example continuity at the nodes, or continuity of the derivatives up to some order n ≤ m − 1 at the nodes. However one must recall that f is a distribution function, which is everywhere, greater or equal to 0, but does neither need to be continuous, nor have continuous derivatives. But its primitive, which is nothing but the total mass in the interval [0, x] should be continuous if the mesh size ∆ is sufficiently small to resolve all important mass variations. In other terms, this condition excludes unresolved δ-like contributions to f . ˜ { f¯m }): For example, consider the third order interpolant H(x, ˜ { f¯m }) = H(x,

∑

αs,m (x − x¯m )s xm−1 ≤ x ≤ xm

s=0,3

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˜ m−1 ) = Hm−1 , H(x ˜ m ) = Hm ; one may also require more elaboContinuity requires that H(x ˜ ˜ m+1 ) = Hm+1 . rate regularity conditions, such as H(xm−2 ) = Hm−2 , H(x Then the coefficients αs,m can be obtained from the cell averages: f¯m+1 − f¯m−1 f¯m+1 + f¯m−1 − 2 f¯m α1,m = f¯m , α2,m = , α3,m = (28) 4∆ 6∆2 A piecewise quadratic polynomial interpolant for f is then obtained by taking the ˜ { f¯m }) and, comparing with (24), one obtains an explicit linear relation derivative of H(x, between the coefficients am,s and the cell averages { f¯m }: am,s = ∑ χsm,n { f¯n }

(29)

f¯m = ∑ ξsm,n asm

(30)

n

and by taking the cell average

s

( f¯m = a0m for the case of Legendre polynomials) Finally, combining Equation (26) with Equation (30) relating the cell averages and the coefficients {asm } one gets s ¯ f¯m = ∑ ∑ Nms Ψm,s n,r Mm,n f m

(31)

n s,r

allowing to advance the approximate distribution function as expressed by Eq. (24). This explicit, one step scheme may be shown to preserve the mass, and will be called the "Van Leer scheme”. One may also require a greater regularity at the cell boundaries, as for

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Francesco Califano and André Mangeney

example, continuity of the distribution function and some of its derivatives, which may be achieved by using splines. Note that since the finite element representation of f is obtained by a projection of f on a finite dimensional functional space which is in general not invariant under the flow operator >t . Therefore, the Hamiltonian character of the Vlasov equation, and so reversibility, is lost. Any scheme will therefore be dissipative (and dispersive), as expected since we are solving a continuum model on a discrete grid. As a consequence, it is necessary to limit oneself to sufficiently small time step ∆t such that the solution can be approximated to order ∆t s .

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3.3.

A Comparison between Numerical Techniques

Following the particle orbits, as it is typically done in particle simulations, or solving the Vlasov equation for the distribution function f along the characteristics, is a priori completely equivalent. But the numerical techniques are very different in many respects. The appeal of particle codes is rather obvious. By modeling a system, such as a plasma using particles, one incorporates automatically some important systems structure in the model. In particular, negative mass or energy densities never appear. They can use a time step significantly larger than it would be possible for an Eulerian discretization scheme on a grid in phase space, which is subject to CFL type stability conditions . They allow for a full coupling between particles and fields. Paricle codes are today able to describe very large systems requiring 6Np dimensional arrays for the particles of a given species at each time step. In general, Particle In Cell (PIC) methods have proven to be a very efficient tool for the numerical simulation of charged particle systems. The main advantage of such methods is that they can adequately give very good results for the low order moments of primary interest in plasma physics. Of course particle codes have limitations of their own. The number of computational macro-particles is limited. As a result, the level of the random noise is relatively large so that the collisionless character of the plasma is lost. Vlasov solvers are usually less noisy, but are very demanding since the distribution function is discretized on a grid in a six dimensional space. In practice, Vlasov codes can be only used by reducing the phase space dimension at least to 5D. On the other hand, due to the deveolpment of numerical resources, the particle methods loose part of their interest when dealing with a reduced phase space since the noise decrerases slowly when the number of computational particle increases. Thus Eulerian or Semi Lagrangian methods which discretize the full phase space on a multi-dimensional grid are more and more attractive since the procedure is intrinsically devoid of statistical noise; numerical errors are only associated with the discretization of the Vlasov equation on the phase-space grid. In particular one may be interested in more detailed collective wave phenomena, for example, when particles in the tail of the distribution play an important physical role, or when the numerical noise due to the finite number of particles becomes too important.

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4. 4.1.

39

The Vlasov Equation A Multi-advection Equation

The calculation of the full Vlasov propagator P ∆t for a time step ∆t is in general complicated, so that one tries to use a succession of free streaming operators which takes into account the fact that the Vlasov equation can be view as a multi-dimensional advection equation in phase space. This technique was introduced in plasma physics many years ago by Cheng and Knorr [20] in the electrostatic limit and extended to the electromagnetic case by Mangeney et al. [48]. It is known as the splitting method; it is most simply explained in the case considered by [20].

4.2.

The Particles Motion, Electrostatic Limit

For the sake of simplicity, we shall consider here the electrons only, a = e, and we shall ¯ = ω pe , u¯ = vth,e in Eq. (9). use the electron characteristic quantities, m¯ = me , ω Then, using the (dimensionless) Hamiltonian describing the motion of the electrons in phase space (in this case q = x, p = v) H= H =H =

p2 2

1 2 v − Φ(x) 2

(32)

− Φ(q), the Vlasov equation can be written as (see Section 2. and Eq. (16)) ∂f = [H, f ] ∂t

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P t = exp(Λt) given in terms of the "Poisson bracket operator”, Λ f ≡ [H, f ]. Actually, H = H1 + H2 is the sum of two terms H1 = v2 /2 and H2 = −Φ(x), corresponding to the two separate advection in space and velocity ∂f ∂f = [H1 , f ] = −v ∂t ∂x ∂f ∂f = [H2 , f ] = E ∂t ∂v The propagators corresponding to equations (33)-(34), are defined as:

Pxt = exp(Λ1t) ;

(33) (34)

Pvt = exp(Λ2t)

where Λi f ≡ [Hi , f ], and simply reduce to a translation either in space or velocity, respectively: f (x, v,t) = f0 (x − vt, v) ≡ ; Pxt [ f0 ] ;

f (x, v,t) = f0 (x, v + Et) ≡ Pvt [ f0 ]

(35)

Since the operators Λ1 and Λ2 do not commute, P ∆t does not reduce to the product Px∆t × Pv∆t but instead may given by a series expansion in ∆t, called the Magnus expansion ∆t/n ∆t/m ( [54]), involving ordered products of Px ,Pv , n and m being integers.

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4.3.

Francesco Califano and André Mangeney

Splitting Scheme

The splitting scheme uses the Magnus expansion truncated at order ∆t 2 leading to a second order approximation of the full Vlasov propagator by the following combination of the two propagators defined in Eqs. (35)): o n (36) P dt [ f (x, v)] = Pxdt/2 Pvdt Pxdt/2 [ f (x, v)] + O(dt 3 ) This formula becomes exact if the two operators Px and Pv commute, which is not the case of the Vlasov equation. Therefore, assuming that the distribution function at time t is known, the solution of the electrostatic, Vlasov equation giving the distribution function at time t + dt reduces to three successive translations of the distribution function at time t: dt/2

f ∗ (x, v) = Px

[ f (x, v,t)]

f˜(x, v) = Pvdt [ f ∗ (x, v)] dt/2

f (x, v,t + dt) = Px

[ f˜(x, v)]

As discussed by Cheng and Knorr [20], in order to maintain the second order time accuracy the electric field must be calculated (using the Poisson equation in the case of periodic boundary conditions) at the intermediate step just after the calculation of f ∗ . Furthermore, except at first time step, the initial and final translation can be put together giving the electrostatic Vlasov time step advancement: 1) f ∗ (x, v) = Pxdt [ f˜(x, v,t)] Z 2

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2) ∇ Φ = ni −

f ∗ (x, v)

3) f˜(x, v,t + dt) = Pvdt [ f ∗ (x, v)] where ni is the ion density taken here as constant for the sake of simplicity (as for example in the case where the study is limited to electron dynamics). The distribution function at dt/2 any time must be obtained by a semi-translation of f˜, i.e. f (x, v,t) = Px f˜(x, v,t). Let us now consider the electromagnetic case and, for the sake of simplicity, we first consider the case of a one directional, uniform magnetic field along the z-axis, so that we can reduce to the 1D-2V phase space (x, vx , vy ). The corresponding dimensionless, electron Vlasov equation reads: ∂f ∂f ∂f ∂f − (Ey − vx Bz ) =0 + v − (Ex + vy Bz ) ∂t ∂x ∂vx ∂vy

(37)

In this case, it is possible to apply the same idea of splitting the full Vlasov propagator into a sum of a propagation in space, as in the electrostatic 1D case, and a propagation in velocity. This last corresponds to a "Lorentz operator” that is in turn splitted into two other operators each one representing a translation plus a semi-rotation. These operators are defined as:

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(38)

A Vlasov Approach to Collisionless Space and Laboratory Plasmas

41

Rvdtx [ f (x, vx , vy )] = f (x, vx + [Ex + vy Bz ] dt, vy )

(39)

Rvdty [ f (x, vx , vy )] = f (x, vx , vy + [Ey − vx Bz ] dt)

(40)

The full advancement of the 1D-2V Vlasov equation is now obtained as: h i dt dt/2 P dt [ f (x, vx , vy )] = Pxdt/2 Rvdt/2 R R Pxdt/2 [ f (x, v)] + O(dt 3 ) v x x vy As in the electrostatic case, the electric and magnetic fields must be calculated after the first dt/2 spatial translation Px . Let us now consider the case of a pure rotation of a particle in a uniform magnetic field, i.e. Ex = 0, Ey = 0, Bz = 1 and ∂ f /∂x = 0. In this case the solution of the Vlasov dt/2 dt/2 equation, P dt [ f ] ≡ Rvx Rvdty Rvx [ f ], reduces to apply two vx semi-translations of the form f (vx , vy ) → f (vx + vy dt/2, vy ), and one vy translation of the form f (vx , vy ) → f (vx , vy − vx dt, vy ), giving at second order: f (vx , vy ) → f (vx (1 + dt 2 ) + vy dt, vy (1 + dt 2 ) − vx dt) The corresponding map is symplectic (product of symplectic operators) and the motion is very close to a circular motion. The energy cannot grow numerically; rather, it decreases at a rate proportional to dv3 , where dv is the velocity discretization interval in phase space [48]. In the most general 3D-3V case, we first split the Vlasov equation into two multiadvection equations, the space advancement part, ∂ f /∂t + v · ∂ f /∂r = 0 and the force advancement part, ∂ f /∂t − [E + v × B] · ∂ f /∂v = 0. Correspondingly, the Vlasov propagator can be splitted in term of a space and a force propagator:

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P dt [ f (r, v)] = Psdt/2 PFdtL Psdt/2 [ f (r, v)]

(41)

Let consider first the space Vlasov term that we further split into three advection equations of the type of Eq. (33). As already discussed, the analytical exact solution is a space advection of the form f (s, . . . ) → f (s−vst, . . . ), where s = x, y, z. By defining Pxt [ f ] ≡ f (x− vxt, . . . ), Pyt [ f ] ≡ f (. . . , y − vyt, . . . ), Pzt [ f ] ≡ f (.., z − vzt, ..) as the three space operators corresponding to each space advection equation, we can write

Pst [ f (r, v)] = Pxt Pyt Pzt [ f (r, v)]

(42)

t since the three space translation operators Px/y/x commute. On the other hand, this is not the case for the force propagators. This is easily seen even in the 1D-2V limit previously discussed. For example, let consider the two operators Rvx and Rvy . Eqs. (39)-(40). We see that Rvdtx Rvdty [ f (x, vx , vy )] 6= Rvdty Rvdtx [ f (x, vx , vy )]

Therefore, it is necessary to make use of the the splitting scheme, Eq. (36). Let define Fvdtx/y/z = f (r, vx/y/z + [Ex + vy/z/x Bz/x/y − vz/x/y By/z/x ] dt, vy/z/x , , vz/x/y ) as the three force operators in the full 3D-3V case. Correspondingly, we need to solve three advection equations, each solution corresponding to one of the F operators. In order to maintain II order symplectic accuracy, we first split the force Vlasov equation into the vx and the vy − vz addt/2 dt/2 vection terms: Fvx PFt y−z Fvx . The vy − vz propagator can be then split into the vy and vz

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42

Francesco Califano and André Mangeney dt/2

dt/2

dt/2

component: PFt y−z = Fvy Fvz Fvy . In summary, the force terms of the Vlasov equation are integrated by the following scheme: h i PFdtL [ f (r, vx , vy , vz )] = Fvdt/2 Fvdt/2 Fvdtz Fvdt/2 Fvdt/2 [ f (r, vx , vy , vz )] (43) x y y x more performant than the first version presented in Ref. [48] where the Force Vlasov equation is splitted one-half step in the vx − vy component and one intermediate step in vz . In that case, the vx − vy propagator is then further split such that the total force propagator reads: dt/4 dt/2 dt/4 PFdtL ≡ [Fvdt/4 Fvdt/2 Fvdt/4 ]Fvdt [Fvx Fvy Fvx ]. x y x z The full splitting scheme for the Vlasov equation is therefore given by Eq. (41) using the results of Eqs. (42)-(43).

4.4.

Discrete Representation of the Distribution Function on Functional Spaces

We now come to the discretization in the (x, v) phase space. As in Section 3.2.2. (see in particular Eq. 24), we assume that the distribution function f can be approximated on the phase space cells by a projection on a finite dimensional functional space where χm (Z) is a complete basis and where ξ is the dual basis, such that ξm (χl ) = δm,n . The coefficients am = ξm ( f ) allow us to express the distribution function as f (Z) = ∑ asm (t) χsm (Z)

(44)

m,s

As discussed in the subsubsection (Finite elements discretizations), the solution of the Vlasov equation can be advanced in time in term of the flow,

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f (Z,t) = f (>−t Z, 0) resulting in

s ¯ f¯m = ∑ ∑ Nms Ψm,s n,r Mm,n f m n s,r

with the matrix elements Z

Ψm,s n,r

4.5.

=

£¡ ¢¤ dZ ξsm (Z) >−∆t χrn (Z)

Discontinuous Galerkin Schemes

To discretize the propagator operator of the form P t [ f (x, 0)] = f (x − αt,t), we use the discontinuous Galerkin methods. As in Section 3., we define [0, L] the numerical domain divided into N cells Ci , i = 1, . . . , N, of the same width ∆. The boundaries of the i-cell are the points xi−1 , xi , corresponding to the N + 1 points xi = i∆ where now i = 0, . . . , N. We also define x¯i = (i − 1/2)∆ as the center of each cell. Note that the name discontinuous Galerkin methods come from the fact that the Galerkin approach does not require explicitly the continuity of the function f (and its derivatives) at the cell boundaries, i.e. on xi and

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A Vlasov Approach to Collisionless Space and Laboratory Plasmas

43

xi+1 for the cell Ci . We define hi (x) the characteristic function of the cell such that hi (x) = 1 if y ∈ Ci while hi (x) = 0 if y ∈ / Ci , and < f > the cell mean value < f >i =

1 ∆

Z

icell

f (x)dx

We take the Legendre polynomials polynomials Pl (x) in the interval [−1, 1] to construct our basis of functions χi,l = hi (x)Pl (2(x − xi )/∆) allowing us to express a generic function belonging to the functional space of our basis as f = ∑i,l ai,l χi,l . As discussed in Ref. [48], the coefficients ai,l are calculated as µ ¶ Z ∆ 2 1 +1 Pl (z) f x¯i + z dz; cl = ai,l = 2 ; z = 2(x − xi )∆ 2 2l + 1 cl −1 Note that ai,0 represents the cell average < f >i . The order of the truncation of the expansion, i.e. the maximum value of l, corresponds to the degree of accuracy of the scheme to approximate the function f (x, 0). Let now consider the initial, approximated, distribution function, f (x, 0) = ∑ ai,l (0) χi,l (x) i,l

The solution f (x,t) at time t can be expressed in terms of the propagator P t and can be approximated (or projected) on the same functional space: f (x,t) ≡ P t [ f (x, 0)] = ∑ ai,l (t) χi,l (x) ;

N

ai,l (t) =

M

∑ ∑ Ai,ln,m (t) an,m (0)

n=1 m=0

i,l

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Since the coefficient ai,l (t) are calculated by the following integration, 2 an,m (t) = 2 ∆cl

Z

χn,m (x) P t f (x, 0)dx

in the limit where the propagator corresponds to the 1D advection equation, i.e. P t [ f (x, 0)] = f (x − vt,t), we get: Ai,l n,m =

Z

2 ∆c2l

χi,l (x)χi,l (x − vt)dx

The velocity coordinate v can be either positive or negative. Therefore, the above coefficients depends on the "signed” Courant number vt/∆. The numerical values of Ai,l n,m for M+1 v > 0 and v < 0 can be found in Ref. [48] in the case of M = 1, 2 of accuracy O (∆t ). We underline that the above numerical scheme is "local”, in the sense that only nearby informations are used to advance f in a given point xi , a properties that makes this approach particularly suited for massively parallel computations. Furthermore, the total density, Z

n(t) =

L

N

f (x)dx = ∑ ai (0) i=1

is conserved in time if periodic boundary conditions are used. However, in practical situation where a limited phase space interval is used, i.e. −vmax ≤ v ≤ vmax , (charge) density conservation can be violated for any physical mechanism accelerating particles at velocities greater than vmax .

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4.6.

Francesco Califano and André Mangeney

Van Leer Interpolation

We finally end this Section by projecting the function from the function space FM N on a smaller one of dimension N. This can be done by considering the cell averaged value of the function f¯ and its derivatives ∂k f¯/∂yk . Following Ref. [48], it is possible to calculate the distribution function at the point xi at the next time step as: M

f¯i (t + ∆t) =

∑

A j (δ) f¯i+ j (t) ,

v>0

j=−(M+1)

f¯i (t + ∆t) =

M+1

∑

A j (δ) f¯i− j (t) ,

v k1 ) are given with a time interval of τp . Both waves are damped in the real space by the phase mixing while generating fine structures of f in the velocity space. Then, a plasma echo with the wavenumber of k2 − k1 appears at techo = τp k2 /(k2 − k1 ) through a nonlinear coupling of the two waves. The plasma echo is also found in the drift wave system in Eqs.(19)–(21) through the E × B nonlinearity. In simulation of drift wave echo, we set ηi = 0 and Ci = 0 and the symmetry condition of fkx ,ky = f−kx ,ky . The initial perturbed distribution function is set to zero, while an electrostatic potential of Φ cos k1 x cos k1 y is externally imposed at t = 0 with a duration time τD . Then, the second pulse of Φ cos k2 x cos k2 y is added at t = τp . The drift wave echo with the wavenumber of k3,x = k1 + k2 and k3,y = k2 − k1 is expected to appear at t = techo . A numerical simulation of the drift wave echo is carried out for the following parameters; τp = 30Ln /vt , k1 ρi = 0.4, k2 ρi = 0.5, Φ = 0.1, τD = 0.2Ln /vt and ∆t = 0.1Ln /vt . Thus, a drift wave echo with k3,x ρi = 0.9 and k3,y ρi = 0.1 appears at t = techo = 150Ln /vt . The velocity space of −5vt ≤ vk ≤ 5vt is discretized by 2049 grid points of which resolution is fine enough to accurately simulate fine velocityspace structures of the distribution function. We employ the implicit midpoint rule for the time-integration.

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134

Tomo-Hiko Watanabe and Hideo Sugama 1.5 ×10−5

Re[φ(k3,x,k3,y)] Im[φ(k3,x,k3,y)]

Potential (φk3)

1.0 ×10−5 5.0 ×10−6 0.0 ×100 -5.0 ×10−6 -1.0 ×10−5 -1.5 ×10−5 0

50

100

150

200

250

300

Time (Ln/vt)

Figure 6. Drift wave echo found in time evolution of the (k3,x , k3,y ) = (k1 + k2 , k2 − k1 ) mode of which potential amplitude peaks at t = 150Ln /vt in agreement with the theoretical estimate [21].

Time-history of complex amplitude of the electrostatic potential for (k3,x , k3,y ) is shown in Fig.6. As is expected, the drift wave echo peaks at t = 150Ln /vt demonstrating that the collisionless damping process with the time reversibility is successfully simulated by the nondissipative time-integration scheme based on the implicit symplectic integrator.

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2.4.3.

Three-Mode Coupling

Time reversibility of the collisionless drift kinetic simulation is also verified by the three-mode coupling problem which was studied as one of the basic benchmark test for the slab ITG simulation [31–35]. Various types of simulation codes have been tested for the three-mode ITG system, such as the PIC, the gyrofluid, and the Vlasov (continuum) ones. A class of exact solutions of the symmetric three-mode equations is derived in Ref. [20,26], and is completely reproduced by the nondissipative drift kinetic simulation. Some of the results are briefly summarized below. Taking the long wavelength limit (kρi ≪ 1), the gyrokinetic equations in Eqs.(22)–(24) reduce to ∂t fk + iΘvk ky fk +

X

k=k′ +k′′

and


ky′ kx′′ − kx′ ky′′ φk′ fk′′

h  i ηi  2 = −iky φk 1 + vk − 1 + Θvk FM (vk ) , (25) 2 Z

fkdvk = φk ,

(26)

where no zonal flow potential is involved in the drift kinetic limit. The collision term is also neglected. We keep only (kx , ky ) = (±k, ±k) and (±2k, 0) modes which are called Eulerian Codes for the Numerical Solution of the Kinetic Equations of Plasmas, Nova Science Publishers, Incorporated, 2010. ProQuest Ebook

Gyrokinetic Vlasov Simulations for Turbulent Transport in Magnetized Plasmas 135 as (±1, ±1) and (±2, 0) modes, respectively. We also impose the symmetry condition of ∗ ∗ ∗ f1,1 = f−1,1 = f1,−1 = f−1,−1 , f2,0 = f−2,0 , and Re(f2,0 ) = φ2,0 = 0, where the asterisk means the complex conjugate. For simplicity, vk , f1,1 , Im(f2,0 ) and φ1,1 are, respectively, denoted by v, f , h and φ in this subsection. Then, Eqs.(25) and (26) are rewritten as (∂t + ikΘv)f (v, t) + 2ik 2 φ(t)h(v, t) = −ikφ(t)G(v) ,

(27)

∂t h(v, t) = 4k2 Im[φ∗ (t)f (v, t)] , Z φ(t) = dv f (v, t) ,

(28)

where f and φ are complex-valued while h is real-valued and G(v) is defined by h i ηi G(v) ≡ 1 + (v 2 − 1) + Θv FM (v) . 2

(29)

(30)

Equations (27)–(29) have a class of exact nonlinear solutions which is represented in terms of the real and imaginary parts of the linear eigenfunction fL and theR real part of eigenfrequency ωr of the ITG instability [26]. By taking the normalization of dvfLr = 1 R and dvfLi = 0 for fL (v) = fLr (v) + ifLi (v), we consider f (v, t) = [a(t)fLr (v) + ib(t)fLi (v)] exp(−iωr t), h(v, t) = c(t)fLi (v), φ(t) = a(t) exp(−iωr t) ,

(31)

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where a(t), b(t), and c(t) are real-valued functions of the time t. The linear solution is represented by a(t) = b(t) ∝ exp(γt) and c(t) = 0, where γ (> 0) is the linear growth rate of the ITG mode with (kx , ky ) = (±k, ±k). Then, a set of nonlinear ordinary differential equations for [a(t), b(t), c(t)] are obtained from Eqs.(27)–(29), da/dt = γb , db/dt = γa − 2k 2 ac ,

dc/dt = 4k 2 ab.

(32)

It is straightforward to find that Eq.(32) has stationary solutions, (a, b, c) = (as , 0, γ/2k 2 ) and (a, b, c) = (0, 0, cs ) with arbitrary real-valued constants, as and cs . Two invariants are easily obtained from Eq.(32), such that c−

2k 2 2 a = C1 , γ

(33)

and

1 γ b2 + c2 − 2 c = C2 . (34) 2 2k Substituting Eq.(33) into Eq.(32), we obtain the Hamiltonian equations of motion [36], da/dt = ∂H/∂b and db/dt = −∂H/∂a where H(a, b) is given by H(a, b) =

k4 γ 2 (b − a2 ) + k 2 C1 a2 + a4 . 2 γ

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Figure 7. Simulations of the three-mode ITG system by means of the implicit midpoint rule (solid), the Runge-Kutta-Gill method (dashed), and the predictor-corrector scheme (dotted) [20].

Since H(a, b) has no explicit time-dependence, and is a constant of motion, the Hamiltonian system is integrable. Analytic solutions of Eq.(32) are given byRthe Jacobi elliptic snu functions of dnu = (1 − κ2 sn2 u)1/2 and cnu = (1 − sn2 u)1/2 where 0 [(1 − x2 )(1 − κ2 x2 )]−1/2 dx = u [26]. The periodic solution of the three-mode ITG system is accurately reproduced by the drift kinetic simulation with the nondissipative integrator in Eq.(16). Figure 7 shows the amplitude of φ1,1 obtained from simulations of the three-mode ITG system, where Eqs.(27)– (29) are numerically integrated by the nondissipative implicit midpoint rule (solid), the fourth-order Runge-Kutta-Gill (RKG) method (dashed), and the predictor-corrector scheme (dotted) [20]. The used parameters are k = 0.1, ηi = 10, and Θ = 1. The initial condition of f is given by the Maxwellian, f (v, t = 0) = εFM (v) with ε = 10−5 , while h(v, t = 0) = 0. In the early stage of the simulation, through the phase mixing process, the potential amplitude approaches the class of exact solution given by Eq.(31). The three simulations give the correct results with respect to the linear growth rate and the first peak level of |φ|. After the first peak, |φ| decreases with the same rate as the linear growth. Then, the nondissipative scheme successfully reproduces the analytical solution with the period of T = 353 for these parameters. The damping process is, however, stopped at earlier times in other two simulations because of the numerical dissipation. The periodic behavior of |φ| is lost in the simulation with the predictor-corrector scheme. Thus, time interval of the first and second peaks of |φ| is shorter (T = 143). The periodic solution is not correctly reproduced by the RKG method even with less numerical dissipation than the predictor-corrector. The above benchmark test demonstrates that the nondissipative scheme successfully reproduces the three-mode ITG system where the results obtained by other methods are

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Gyrokinetic Vlasov Simulations for Turbulent Transport in Magnetized Plasmas 137 influenced by the numerical dissipation. Broader applicability of the simulation method has also been verified by benchmark tests for dissipative or asymmetric three-mode ITG systems [20]. In the next section, we will see application of the nondissipative method to investigation of a quasisteady state of the collisionless slab ITG turbulence [28]. The implicit midpoint rule, however, demands several times larger computational cost than that for the RKG method, since Eq.(16) is solved by iterations. Thus, a weakly dissipative scheme could be more practical, if finite physical dissipation introduced is much larger than the numerical dissipation. To study a statistically steady state of the ITG turbulence with finite collisionality, we employ the RKG method while carefully monitoring the entropy balance [37]. The simulation results will also be shown in the next section.

3.

Turbulent Transport and Fine-Scale Distribution Functions

In collisionless plasmas, fine-scale structures of distribution functions are spontaneously generated by ballistic motions of particles. Phase mixing of the fine structures leads to collisionless damping of density fluctuations. In the slab gyrokinetic system considered in section 2.4.1, ITG turbulence causes anomalous ion heat transport across the magnetic field. The anomalous transport generated by the drift wave turbulence has been one of the main subjects in the magnetic confinement fusion research [2]. Here, we discuss some fundamental aspects of the kinetic plasma turbulent transport focusing on the fine structures of the distribution function in the phase space.

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3.1.

Steady and Quasisteady States of Plasma Turbulence

An anomalous heat transport flux caused by plasma turbulence is defined by ensemble average of a correlation of fluctuating temperature δT and flow δu, that is, δT δu. Here, · · · denotes the ensemble average. Since both δT and δu are usual fluid variables defined on the real space, it might be expected that the conventional fluid equations were enough to describe the anomalous transport phenomena. However, a fundamental question arises regarding to the origin of the irreversible transport in the collisionless plasma turbulence. Fine-scale structures of distribution functions are continuously generated in the collisionless turbulence. Since the collisionless kinetic (Vlasov or gyrokinetic) equation has the time reversibility, to find an irreversible transport process, a coarse-grained form of the fluctuating distribution function should be taken into account. It is also pointed out that a steady transport flux will be observed in a quasisteady state of the collisionless turbulence [36], where the low-order moments are constant in average, but high-order velocity-space moments of the perturbed distribution function continue to grow. To reveal the collisionless transport process, we have conducted nondissipative kinetic simulations of the slab ITG turbulence by using the simulation model shown in section 2.4.1 [28]. First, let us consider an entropy variable δS as a measure for fluctuations of the distribution function in the phase space, δS ≡

Z

d3 v(δf )2 /2FM .

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(36)

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Tomo-Hiko Watanabe and Hideo Sugama

Neglecting third-order term for δf = R the R f − FM , δS is rewritten as δS = SM − Sm where 3 SM = − d vFM ln FM and Sm = − d3 vf ln f represent macroscopic and microscopic entropy per unit volume, respectively. Here, the ensemble average of δf is assumed to vanish. Let us suppose that the ensemble average can be replaced with the spatial average, and that the periodic boundary condition is applied to fluctuations. By multiplying fk∗ /FM to Eq.(19) and taking the velocity-space integral and summation over k, the entropy balance equation is derived: d (δS + W ) = ηi Qi + Di , (37) dt where the entropy variable δS, the ion heat transport flux Qi , the potential energy W , and the collisional dissipation Di are defined by XZ δS = dvk |fk|2 /2FM , (38) k

Qi =

XZ

dvk (−iky e−k

2 /2

φk)vk2 f−k/2 ,

(39)

k

W =

X [1 + (Te /Ti )(1 − Γ0 )]|φk|2 /2,

(40)

dvk f−kCi (fk)/FM ,

(41)

k

Di =

XZ k

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respectively. In the present simulation model of the slab ITG turbulence, it is expected that the density and temperature gradients fixed to constant continuously drive the statistically steady turbulence which produces a constant transport flux. According to the entropy balance equation, a quasisteady state characterized by continuous growth of δS with constant W and Qi should be realized, d(δS) ≈ ηi Qi , dt

(42)

if the constant transport flux is generated in the collisionless ITG turbulence. This scenario is based on the idea that continuous growth of fine-scale structures of δf in the velocity space (or high-order moments of δf ) contributes to monotonic increase of δS, while loworder moments providing W and Qi reach steady values. To confirm the conjecture, the nondissipative simulation method is applied to the collisionless slab ITG turbulence [28]. Employing the model described above, we have performed the kinetic simulation without zonal flow components, where the ky = 0 modes of δf are artificially fixed to zero. This is because, when the zonal flows were included, the ITG turbulence was almost completely suppressed in the slab geometry, where we observed a trivial steady state with no transport, d(δS)/dt ≈ ηi Qi ≈ 0 [28]. Considering the collisionless damping of zonal flows in toroidal systems, we studied the case without the zonal flow components. Physical parameters used are ηi = 10 and Θ = 2.5. To keep enough resolution for fine-scale structures of δf , 8,193 grid points are employed for discretization of the velocity space of −5 ≤ vk ≤ 5. The minimum and maximum values of the wavenumber are set to be kmin = 0.1 and kmax = 3.2, respectively. The time step is set to ∆t = 6.25 × 10−3 .

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Gyrokinetic Vlasov Simulations for Turbulent Transport in Magnetized Plasmas 139

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Figure 8. Time histories of (a) d(δS)/dt, dW/dt, −ηi Qi , (b) entropy variable δS and its low-pass filtered values for ηi = 10 and Θ = 2.5 in collisionless slab ion temperature gradient (ITG) turbulence without zonal flow component [28].

Figure 8(a) shows time history of d(δS)/dt, dW/dt, and −ηi Qi , where the entropy balance is well satisfied with a constant level of d(δS)/dt and ηi Qi in the saturated turbulence for low-order moments (dW/dt ≈ 0). Since Eq.(42) is satisfied with the constant transport flux, δS linearly increases in time [see Fig. 8(b), where dashed and dotted lines indicate values of the entropy variable δScut calculated from low-pass filtered δf at the velocitywavenumber lkcut ]. Also, more rapid growth of δScut is observed for larger lkcut . This means that the high-order moments of δf continue to grow while keeping the low-order moments constant on the average. This is the first confirmation of the quasisteady state in the collisionless plasma turbulence. Here, it should be reminded that the nondissipative time-integration scheme based on the implicit symplectic integrator guarantees conservation of the entropy variable when the inhomogeneous term is absent. This makes the present evaluation of the entropy balance quite reliable. A different scenario on the steady turbulent transport is considered in case with finite collisionality. A Fokker-Planck type collision term leads to dissipation of fine-scale fluctuations of δf in the velocity space. Small velocity-scale fluctuations of δf are continuously generated by the ballistic motions of particles, and the finest velocity-scale becomes shorter after longer time in the collisionless limit. Thus, the collision term definitely affects evolution of the system through dissipation of the fluctuations in a small velocity-space scale Lv with (ν/γ)(vt /Lv )2 ∼ O(1), even if the collision frequency is negligibly smaller than the instability growth rates γ. In case with the finite collisionality, therefore, a statistically

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Tomo-Hiko Watanabe and Hideo Sugama

Figure 9. Time-evolution of entropy variable δS for collisionless (ν = 0) and weakly collisional (ν = 1.25 × 10−4 ) slab ITG turbulence [37]. steady state of kinetic plasma turbulence is expected to be realized: ηi Qi ≈ −Di with

d(δS) ≈0, dt

(43)

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where δf itself is statistically steady. Kinetic simulations of the weakly collisional slab ITG turbulence are performed by the use of the fourth-order RKG method for time integration with careful convergence checks to the time step size. We employ the Lenard-Bernstein model collision operator [38],   ∂ ∂ + vk fk(vk ) (44) Ci (fk) = ν ∂vk ∂vk with the collision frequency ν. Since the collision operator used makes D negative-definite, the heat transport flux for LT = −∂Ti /∂x > 0 is positive in the steady turbulent state. Time history of δS for the weakly collisional case with ν = 1.25 × 10−4 is presented in Fig. 9 where the result of the collisionless simulation is also plotted for comparison. The other parameters used are the same as those for the collisionless ITG turbulence simulation in Fig. 8, except for the time-step size and the velocity-space resolution which is determined so as to accurately satisfy the entropy balance. In the finite collisionality case, growth of δS saturates in the saturated state of turbulence. As δS is represented by the sum of squares of the velocity-space moments such as the density, flow velocity, and temperature, the result shown in Fig. 9 indicates that all orders of moments of δf are statistically steady in the weakly collisional case (see section 3.3 for more detailed discussions). The two scenarios on the slab ITG turbulent transport for the steady and quasisteady states have been confirmed by the kinetic simulations. Regarding to a limiting behavior of the transport, an important question arises: whether the turbulent transport in the limit of ν → 0 approaches to that in the collisionless case. The collision frequency dependence of the ion heat transport coefficient χi has been investigated by changing ν over a wide range from ν = (1/512) × 10−3 to ν = 8 × 10−3 . The thermal diffusivity χi obtained from a series of simulation runs is summarized in Fig. 10. The horizontal dashed line in

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Figure 10. Anomalous ion heat transport coefficient (χi ) obtained from slab ITG turbulence simulations with weak collisionality is plotted for different collision frequency ν. Data is time-averaged from t = 1000 to 3000. A horizontal dashed line represents χi in collisionless case with ν = 0 [37].

the figure represents the collisionless simulation result. For relatively lager values of ν, 1.25 × 10−4 < ν < 8 × 10−3 , χi has a logarithmic dependence on ν. The ν-dependence of χi becomes quite weak for the lower collision frequency (ν ≤ 1.25 × 10−4 ). Then, χi in the limit of ν → 0 approaches the collisionless result. From the results shown above, it is concluded that the finite collisionality is indispensable to realizing the statistically steady turbulence (including δf ), and that the collision term with sufficiently small ν does not influence the low-order moments as well as χi . Thus, it is considered that large velocity-scale structures of δf are not much affected by the small collisionality, while the time history of the entropy variable associated with fine structures of δf show totally different behaviors for the collisionless and weakly collisional cases.

3.2.

Generation of Fine Structures of Distribution Function

In the collisionless or weakly collisional plasma turbulence, fine-scale fluctuations develop in the phase-space distribution function. The parallel advection term in Eq. (19) generates fine structures of δf in the velocity-space because of the ballistic motion of particles. Generation of the fine-structures in the collisionless ITG turbulence are directly recognized in Fig. 11 where the real and imaginary parts of δf normalized by the potential are plotted for the same simulation shown in Fig.8. A smooth profile of the eigenfunction is observed in the linear growth phase of the ITG instability (shown in the top of Fig. 11). In the saturated turbulence, the fine structures develop in the velocity space, as the linear eigenmode is no longer an approximate solution of the nonlinear gyrokinetic equation. Then, the small velocity-scale fluctuations of δf are continuously generated as the time goes, and the finest scale finally reaches the grid size. Detailed processes of entropy production, transfer, and dissipation in the slab ITG turbulence have been investigated [37] in analogy with a passive scalar convection in the homogeneous isotropic turbulence of a neutral fluid [39]. For describing the entropy transfer

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Figure 11. Profiles of the perturbed distribution function of the linearly most unstable ITG mode [kx = 0.1 and ky = 0.3; (1,3) mode] normalized by potential observed at four different time steps [28].

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process from macro to micro velocity scales, let us consider the Hermite-polynomial expansion of fk, fk(vk ) = FM (vk )

∞ X

fˆk,n Hn (vk /vt ) ,

(45)

n=0

2

2

where Hn (x) = (−1)n ex /2 dn e−x /2 /dxn denotes the nth-order Hermite polynomial of x. The expansion coefficient fˆk,n is defined by 1 fˆk,n = n!

Z

∞

d(vk /vt )fk(vk )Hn (vk /vt )

(46)

−∞

for n = 0, 1, 2, .... From Eqs. (19)–(21), and (44), we find the entropy transfer equation in the n-space, # " d 1X |φk|2 {1 − Γ0 (bk)} = Jn−1/2 − Jn+1/2 + δn,2 ηi Qi − 2νnδSn . δSn + δn,1 dt 2 k (47)

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Figure 12. Plots of entropy transfer function Jn−1/2 normalized by σ ≡ ηi Qi for different collision frequencies ν with kmax = 3.2 [37].

where δSn ≡ Jn−1/2 ≡

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Jn+1/2 ≡

X k

X

δSk,n ≡

X1 k

2

n!|fˆk,n |2 ,

(48)

∗ Θky n!Im(fˆk,n−1 fˆk,n ),

(49)

∗ Θky (n + 1)!Im(fˆk,n fˆk,n+1 ),

(50)

k

X k

and δn,m = 1 (for n = m) or 0 (for n 6= m). No zonal flow component is considered here. The entropy transfer from the (n − 1)th to the nth components is caused by the parallel advection term, and is expressed by Jn−1/2 . Collisional entropy dissipation is represented by −2νnδSn . The Hermite-polynomial expansion is convenient for deleting the potential energy and the entropy production terms in the range of n ≥ 3 in Eq.(47). In the real steady state of the weakly collisional ITG turbulence, the entropy transfer balances with the dissipation for n ≥ 3, −2νnδSn = Jn+1/2 − Jn−1/2 ≃

dJn . dn

(51)

In derivation of the right-hand-side of Eq.(51), n is regarded as a continuous variable. The entropy variable δSn is transferred in the n-space where the collisional dissipation is neglin gible, such that, Jn+1/2 − Jn−1/2 ≃ dJ dn = 0. This is analogous to the concept of inertial subrange which is universally found in the well-developed homogeneous neutral fluid turbulence. The entropy transfer function normalized by the production rate, σ ≡ ηi Qi , is plotted for different collision frequency in Fig. 12. A flat profile of Jn−1/2 is clearly found in the n range where no entropy production nor dissipation works. The flat profile of Jn−1/2 spans a wider n-space for the smaller ν. In analogy with the spectral analysis of the passive scalar advection in the large Prandtl number case [39], it is supposed that the E × B flow is steady and statistically independent

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Figure 13. Plot of δSn for ν = 1.25 × 10−4 and kmax = 12.8 (solid). Theoretical spectra given by Eq.(52) [that is Eq.(18) in Ref. [37]] with γ = 0.5 and Eq.(53) [that is Eq.(20) in Ref. [37]] with γM = 15 are also shown by dashed and dotted lines, respectively, where σ = 36 [37].

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of fˆk,n for large n. Strain of the steady flow causes an exponential growth of effective wavenumbers of fˆk,n . Then, the spectrum of the entropy variable is derived as [37],   σ νn δSn = exp − . (52) 2γn γ In actual simulations, the finite resolution limits the exponential growth of the effective wavenumber. In case of the effective wavenumber independent of n, a different spectrum is obtained for δSn ! σ 2 νn3/2 √ exp − , (53) δSn = 3 γM 2γM n where γM is a parameter representing the limit of wavenumber growth. As shown in Fig. 13, spectrum of the entropy variable obtained from the ITG turbulence simulation with kmax = 12.8 is well explained by combining the analytical expressions in Eqs.(52) and (53). The simulation result shows good agreement with Eq.(52) (dashed) in a range of 3 ≤ n . 103 , where the entropy variable is transferred from the low to high n portion. In the dissipation range found on the high-n side (n & 103 ), the spectrum is fitted by Eq.(53) (dotted). From the numerical and theoretical studies above, the following picture on the turbulent transport and the entropy variable can be drawn: On the macro velocity scale with low nvalues, the ITG turbulent transport produces the entropy variable which is mainly carried by the fluid variables (low-order moments of δf ). The entropy variable produced on the low n side is transferred through a ‘inertial subrange’ in the n-space by the ballistic motion of particles as well as in the k-space by the E ×B nonlinearity. Then, it is finally dissipated by the collision term on a high n side. Here, it is reminded that the transport coefficient χi has very weak collisionality dependence for sufficiently small values of ν as shown in Fig.10. For the smaller ν, the wider ‘inertial subrange’ is formed in the spectrum of the entropy

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Gyrokinetic Vlasov Simulations for Turbulent Transport in Magnetized Plasmas 145 variable so that the collisional dissipation balances with the production. This situation is consistent to the conjecture by Krommes and Hu such as ‘flux determines dissipation’ [40].

3.3.

Relation to Kinetic-Fluid Closure Model

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As we have seen in the Vlasov simulations of the nonlinear Landau damping and the collisionless ITG turbulence, the distribution function is stretched and folded by particle motions, and is strongly deformed in the phase space as the time advances, even if the initial condition is given by a smooth function. Thus, initial perturbations of fluid variables given in a stable system are damped by phase mixing of the fine-scale fluctuations, while the distribution function never reaches a steady state in the collisionless case. In this subsection, let us consider what kind of extension in fluid modeling is demanded for application to the quasisteady state of the collisionless plasma turbulence. The fluid equations are obtained by taking velocity-space moments of kinetic equations. For example, the density, the flow velocity, and the temperature are provided by calculating the zeroth-, the first-, and the second-order moments. However, the nth-order moment equation involves the (n+1)th- or higher-order terms. In the conventional derivation of fluid equations, for truncation of the moment hierarchy, the distribution function is approximated by expansions around the Maxwellian in terms of a smallness parameter of ν −1 (Knudsen number). In application to the collisionless or weakly collisional systems, however, the conventional assumption is not valid, and the fluid model should be extended by means of novel moment-closure relations. The fluid equations for the two-dimensional slab ITG model in the long wavelength limit are obtained by taking velocity-space moments of Eq.(19) with bk = 0, eφk c ∂nk + ikk n0 uk − iω∗i n0 − ∂t Ti B

n0 mi

n0

X

[b · (k′ × k′′ )]φk′ nk′′ = 0 ,

(54)

k′ +k′′ =k

∂uk n0 mi c +ikk (Ti nk +n0 Tk +n0 eφk)− ∂t B

∂Tk n0 c +ikk (2n0 Ti uk +qk)−iω∗i ηi n0 eφk − ∂t B

X

[b·(k′ ×k′′ )]φk′ uk′′ = 0 , (55)

X

[b·(k′ ×k′′ )]φk′ Tk′′ = 0 . (56)

k′ +k′′ =k

k′ +k′′ =k

Fluctuations of the density, the parallel flow velocity, the temperature, andRthe paralR lel heatR flux for gyrocenter positions are defined by n = dv f , u dvk fkvk , k k = k k R Tk = dvk fk(mi vk2 − Ti ), and qk = dvk fk(mi vk3 − 3Ti vk ), respectively. The gyrokinetic Poisson equation reduces to nk = n0

eφk , Te

(57)

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obtained from Eqs. (54)–(57), d X n0 dt k

! 1 nk 2 1 uk 2 1 Tk 2 Te eφk 2 + + + 2 n0 2 vt 4 Ti 2Ti Te

X q⊥ = · (−∇ ln Ti ) + Re Ti k



Tk ikk qk∗ 2Ti2



, (58)

P ∗ ) with where the perpendicular ion heat flux is denoted by q⊥ = (n0 /2) k Re(TkvEk vEk = i(c/B)b × kφk. Hence, ηi Qi = q⊥ /(n0 vt Ti ) · (−∇ ln Ti )/|∇ ln n0 |. The entropy balance relation in Eq. (37) derived from the collisionless gyrokinetic equations with bk = 0 can also be represented in a similar form to Eq. (58), that is, d X n0 dt k

1 nk 2 1 uk 2 1 Tk 2 1 qk 2 + + + 2 n0 2 vt 4 Ti 12 n0 Ti vt  2 2 X n! Te eφk  q⊥ · (−∇ ln Ti ) . (59) + = fˆk,n + 2 2Ti Te Ti

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n≥4

The quasi-steady state of the collisionless ITG turbulence is described by continuous growth of high-order moments of n ≥ 4 which corresponds to generation of fine-scale fluctuations of δf . In the derivation of Eq. (59), we used the Hermite polynomial expansion of fk of which each term is related to fluid quantities. Comparison of Eqs. (58) and (59) leads to   X q⊥ Tk ∗ · (−∇ ln Ti ) = − Re ikk qk Ti 2Ti2 k   (60) X X 2 d  n! ˆ  = n0 , fk,n dt 2 k n≥4

when q⊥ and the low-order moments of n ≤ 3 are statistically steady. In the quasisteady state of the collisionless ITG turbulence, the entropy production rate balances with correlation of the temperature and the parallel heat flux. Moreover, the turbulent heat transport drives the growth of fluctuations in the high-n moments (the entropy production) through the correlation term Re[iTkqk∗ ]. To complete the set of fluid equations in Eqs.(54)–(56), the parallel heat flux qk (that is, the third-order moment of fk) should be represented in terms of the lower-order moments. It is called a closure relation. The above discussion highlights importance of choosing the closure model in fluid simulations of the collisionless turbulent transport. A simple kinetic closure model was proposed by Hammett and Perkins [41], where the parallel heat flux is given by qk = −n0 χk ikk Tk , (61) with the parallel heat diffusivity χk = 2(2/π)1/2 vt /|kk |. In this model, a phase angle between the complex variables, Tk and qk, is −π/2, and leads to a dissipative closure relation. A nondissipative closure model (NCM) has also been proposed [36, 42], where real-valued

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Figure 14. Phase angle histograms of Tk and qk for (a) the longest wave length mode [kx = ky = 0.1; (1, 1) mode] and (b) the linearly most unstable mode [kx = 0.1 and ky = 0.3; (1, 3) mode] resulted from collisionless slab ITG turbulence simulation [28] .

coefficients are employed in the closure relation of qk = CT kn0 vt Tk + Cukn0 Ti uk so that the time reversibility of unstable eigenmodes can be preserved. Results of the collisionless gyrokinetic simulation of the slab ITG turbulence shown in Fig. 8 are also utilized to evaluate the phase relation between Tk and qk. Probability distribution of the phase angle is plotted in Fig.14 for the longest wave length and the linearly most unstable modes. The histogram for the former that is responsible for carrying the heat flux peaks around −π/2, suggesting the dissipative phase relation in consistence with the Hammett-Perkins closure model. An oscillatory phase relation is obtained for the latter, where the histogram peaks around ±π, and can not be reproduced by the dissipative closure in Eq. (61). A direct comparison of the gyrokinetic and kinetic-fluid simulations of the slab ITG turbulence has been conducted in Ref. [42]. The simulation results showed that the turbulent ion heat diffusivity χi obtained from the Hammett-Perkins model was significantly larger than that from the nondissipative closure model. The latter value agreed well with the gyrokinetic simulation result. Closure relations for zonal flow components in toroidal plasmas are also proposed [43, 44], and the further extensions of the kinetic-fluid model are currently in progress.

4.

Gyrokinetic Vlasov Simulations of Toroidal Plasmas

In magnetic fusion plasma with high temperature, turbulence causes transport of particles, momentum, and energy, and leads to confinement degradation. The transport levels are often orders of magnitudes higher than those estimated from the classical and neoclassical transport theories based on binary collisions of particles. Thus, the turbulent transport is called ‘anomalous’. The anomalous transport has been one of central subjects in the magnetic fusion research. Since 1990’s, gyrokinetic simulation studies have been conducted worldwide with aims of understanding the anomalous transport mechanism and predicting the transport levels [45, 46]. One of the most important findings obtained from the gyrokinetic simulations concerns zonal flows and their turbulent transport suppression in toroidal

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plasmas [47]. In this section, we describe basic algorithm and numerical methods of a gyrokinetic Vlasov simulation code, GKV [48], and its application to turbulent transport and zonal flows in toroidal plasmas. After a brief review of the GKV code in the next subsection, applications to collisionless damping of zonal flows and geodesic acoustic mode (GAM) are provided in section 4.2. It is followed by a subsection for the entropy balance in toroidal ITG turbulence with zonal flows. The last subsection is devoted for GKV simulations of the ITG turbulent transport and zonal flows in helical systems.

4.1.

Gyrokinetic Vlasov Simulation Code GKV

The gyrokinetic Vlasov simulation code, GKV, is developed for investigating drift wave turbulence in toroidal magnetized plasmas. The gyrokinetic equations for perturbed ion distribution function [49] are numerically solved in the electrostatic limit where the flute reduction for a large-aspect-ratio torus is employed [1]. Thus, we consider a local ITG turbulence under constant magnetic shear and gradients of density and temperature, and assume a circular poloidal cross-section with an effective minor radius r0 . The gyrokinetic equation for perturbed ion gyrocenter distribution function, δf , defined on the five-dimensional phase space (x, y, z, vk , µ) is given by c ∂δf ∂δf + vk b · ∇δf + {Φ, δf } + vd · ∇δf − µ (b · ∇Ωi ) ∂t B0 ∂vk
e∇Φ = v∗ − vd − vk b · FM + Ci (δf ) (62) Ti

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2 /2Ω . Notations in Eq. (62) are standard. Also, we define the magnetic moment µ ≡ v⊥ i The background ion distribution is assumed to be the Maxwellian, FM , with the constant density and temperature gradients. Abbreviations used in Eq. (62) are defined as

1 ∂ ∂Φ ∂δf ∂Φ ∂δf , {Φ, δf } = − , q0 R0 ∂z ∂x ∂y ∂y ∂x    cTi mi v 2 3 ∂ v∗ · ∇ = − − , 1 + ηi eLn B0 2Ti 2 ∂y

b·∇=

where v 2 = vk2 + 2Ωi µ. The term with the Poisson brackets represents the E × B drift motion. The toroidal magnetic drift term is denoted by vd · ∇δf of which detailed expressions for tokamak and helical systems are given in Refs. [48] and [50, 51], respectively. We employ a flux-tube coordinate system [52] defined by x = r − r0 , y = r0 [q(r)θ − ζ], and z = θ which are chosen for accurately reproducing the ballooningq0 type mode structure. The poloidal and toroidal angles are represented by θ and ζ, respectively. The safety factor has a linear dependence on r, q(r) = q0 [1 + sˆ(r − r0 )/r0 ], with q(r = r0 ) = q0 and the magnetic shear sˆ. The flux-tube coordinates defined on a flux surface are illustrated in Fig. 15, where the poloidal angle z = θ is measured along field lines. A squared area (A − B − C − D) in the (θ, ζ) coordinates is mapped onto a parallelogram (A − B − E − F ) in the flux tube coordinates. The use of the flux tube coordinates enables us to reduce number of grid points in the field-aligned (z) direction, because the ballooning-type mode structure of drift waves has much longer parallel scale-length than perpendicular wave lengths.

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Figure 15. Flux-tube coordinates defined on a flux surface.

The local flux-tube model allows us to utilize the periodic boundary condition in the (x, y) directions. Then, δf can be Fourier transformed, fkx ,ky . The quasi-neutrality condition is simply defined in the wavenumber space (kx , ky ). Z eφkx ,ky n0 (1 − Γ0 ) = ne,kx ,ky , (63) J0 (k⊥ v⊥ /Ωi )fkx ,ky d3 v − Ti 2 = (k + s where J0 is the zeroth order Bessel function and k⊥ ˆzky )2 + ky2 . The electrostatic x potential Φ acting on the gyrocenter is related to φ,

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Φkx ,ky = J0 (k⊥ v⊥ /Ωi )φkx ,ky . Here, we assume the adiabatic electron response,


 Ti e φkx ,ky − φkx ,ky    ne,kx ,ky Te Ti =  n0 eφ   Ti kx ,ky Te Ti

(64)

for ky = 0 .

(65)

for ky 6= 0

The toroidal mode number n = 0 for the zonal flow components corresponds to ky = 0. The flux surface average is denoted by h· · · i, such that Z +Nθ π Z +Nθ π A 1 hAi = dz dz (66) B B −Nθ π −Nθ π for the ky = 0 component of A. Magnitude of the confinement magnetic field is modeled as B = B0 [1 − ǫt (r) cos z] , (67)

for tokamak systems with the large-aspect-ratio ǫt ≪ 1. A model for helical confinement field is provided by ) ( l=L+1 X B = B0 1 − ǫ00 (r) − ǫt (r) cos z − ǫl (r) cos [(l − M q0 )z − M α] , (68) l=L−1

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where L and M denote the poloidal and toroidal period numbers of the main component of the helical field. In case with finite collisionality, we employ the Lenard-Bernstein collision model [38],


Ci fkx ,ky = νii



1 ∂ v⊥ ∂v⊥

  2 ∂fkx ,ky v⊥ v⊥ + 2 fkx ,ky ∂v⊥ v  ti   2 vk ∂fkx ,ky k⊥ ∂ + 2 fkx ,ky − 2 fkx ,ky , (69) + ∂vk ∂vk vti Ωi

where νii denotes the ion-ion collision frequency. The last term in the square brackets, 2 /Ω2 )f −(k⊥ i kx ,ky , has a minor effect, and is often neglected. The GKV code numerically solves the gyrokinetic equations, Eqs. (62)–(65), defined on the five-dimensional phase space. Equation (62) is Fourier-transformed for the x and y coordinates, and its time-integration is calculated by the fourth-order RKG method. The nonlinear E × B (Poisson brackets) term is computed in the real space, where the 3/2rule is applied in the Fourier-transformation from the wavenumber space to the real space [53]. The parallel and velocity-space coordinates are discretized by numerical grids, where fourth- or fifth-order finite-difference methods with constant grid-spacing are applied to differential terms for z, vk , and v⊥ . In the limit of v⊥ → 0, we rewrite Eq. (69) into a nonsingular form by using l’Hˆopital’s rule so that coordinate singularity at v⊥ = 0 is avoided in implementation of the collision term. The integral of J0 fkx ,ky in the two-dimensional velocity-space is calculated by the trapezoidal rule with a correction at v⊥ = 0 for high accuracy.

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4.2.

Collisionless Damping of Zonal Flows and Geodesic Acoustic Mode

The first application of the gyrokinetic Vlasov simulation code, GKV, is the linear collisionless damping of zonal flows and GAM in toroidal plasmas. Since the zonal flows play a key role in regulating the ITG turbulent transport, it is important to understand how strongly the zonal flows are generated by a given source term. Because of the magnetic drift and mirror motions in the toroidal confinement field, particle orbits are more complicated than those in the slab case. Thus, this is also a good benchmark test for toroidal gyrokinetic codes. The zonal flow response theory was first considered by Rosenbluth and Hinton for a tokamak plasma [54], and has been extended to helical configurations [55, 56]. The initial zonal flow perturbation of ion gyrocenter distribution function given by the Maxwellian is coupled to GAM oscillations through the toroidicity. After the Landau damping of GAM, a portion of the zonal flow remains with a constant amplitude. The linear response of the zonal flow potential is written as φk⊥ (t) = K(t)φk⊥ (0),

K(t) = KGAM (t)[1 − KL (t)] + KL (t)

(70) (71)

where K(t) represents a response function (or kernel). The short-time response KGAM (t) represents the collisionless damping of GAM oscillations with the real frequency ωG and Eulerian Codes for the Numerical Solution of the Kinetic Equations of Plasmas, Nova Science Publishers, Incorporated, 2010. ProQuest Ebook

Gyrokinetic Vlasov Simulations for Turbulent Transport in Magnetized Plasmas 151 the damping rate |γ| = −γ, such that KGAM (t) = cos(ωG t) exp(γt) .

(72)

Analytical expressions of ωG and γ, which have been extended to include the finite-orbitwidth effects, are provided in Refs. [57] and [56] for tokamak and helical systems, respectively. The long-time response function KL (t) provides the residual (normalized) amplitude of zonal flow in the limit of t → +∞, and is considered to affect a saturation amplitude of the ITG turbulence. For tokamaks, Rosenbluth and Hinton derived [54], KL = lim

t→∞

hφkx ,0 (t)i 1 . = hφkx ,0 (t = 0)i 1 + 1.6q 2 /ǫ1/2

(73)

For helical systems, KL (t) has time-dependence, and is given by KL =

1 − (2/π)h(2ǫH )1/2 {1 − gi1 (t, θ)}i
2 a2 i 1 + G + E(t)/ n0 hk⊥ i

(74)

(see Ref. [55,56] for further details). The response kernel for helical systems has a constant limiting value for t/τc → ∞,

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K> =

hk⊥ a2i i[1

hk⊥ a2i i[1 − (2/π)h(2ǫH )1/2 i] − (2/π)h(2ǫH )1/2 i + G] + (2/π)(1 + Ti /Te )h(2ǫH )1/2 i

(75)

where τc ∼ 1/|kx vdr | denotes a characteristic time of zonal flow damping caused by the radial drift motion of typical helical-ripple-trapped particles. An initial value problem for the zonal flow component with ky = 0 is solved by a linearized version of the GKV code [48]. As shown in Fig. 16, the zonal flow amplitude, hφkx ,0 i, initially given by the Maxwellian perturbation decreases due to the collisionless damping of GAM oscillation. Here, we set ǫt ≡ r0 /R0 = 0.18 and q0 = 1.4. The radial wave number is kx = 0.1715ρ−1 i . The residual level of the zonal flow potential obtained by the GKV simulation agrees well with the theoretical estimate. Velocity-space profiles of a real part of the perturbed ion gyrocenter distribution function at different time steps during the zonal flow damping are plotted in Fig.17, where the boundary of trapped and passing ions is shown by dotted lines. The ballistic motion of passing particles deforms the initial Maxwellian distribution, and generates fine-scale structures of fkx ,0 in the passing-particle region of the velocity space. Phase mixing of the fine structures causes the collisionless damping of GAM. A coherent structure found in the tapped particle region is responsible for keeping the residual zonal flow, and is associated with the neoclassical polarization effect. The coherent profile of the gyrocenter distribution agrees with a bounce-averaged analytical solution of the gyrokinetic equation for the zonal flow damping [48]. Results of the GKV simulation of the collisionless zonal flow damping in a helical system are shown in Figs. 18 and 19 [56]. A simple equilibrium model of the Large Helical Device (LHD) [58] is employed with a single helicity component of the confinement field with L = 2 and M = 10 where we set ǫt = ǫL = 0.1, q0 = 1.5, and kx ρi = 0.131. The solid and dashed lines in Fig.18 show time-history of the zonal flow potential amplitude and

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〈φkx(t)〉/〈φkx(t=0)〉

1 0.8

kxρi=0.1715 (1+1.6q2/ε1/2)−1

0.6 0.4 0.2 0 -0.2 -0.4 0 5 10 15 20 25 30 35 40 45 50 Time (Ln/vti)

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Figure 16. Time-history of zonal flow potential resulted from GKV simulation of collisionless damping of axisymmetric (n = 0) mode [48].

the residual zonal flow level given by K> in Eq.(75), respectively. The zonal flow amplitude, hφk(t)i, oscillating with the GAM frequency, asymptotically approaches the residual level. The observed frequency and the damping rate of the GAM oscillations also agree well with the theoretical analysis of the short-time response function [56]. A contour plot in Fig.19 shows a real part of the perturbed ion gyrocenter distribution function normalized by the zonal flow potential. Fine striation structures along the v⊥ direction are caused by ballistic motions of passing particles going through toroidal and helical ripples. The radial drift motion of helical-ripple-trapped particles produces a hollow part of the distribution function in the trapped region, which influences the long-time behavior of the zonal flow. An averaged profile of the gyrocenter distribution shows a good agreement with the gyrokinetic theory of zonal flows in helical systems [56]. Our theoretical and numerical analyses demonstrate that a higher-level zonal flow response is maintained for a longer time by suppressing the radial drift velocity of the helical-ripple-trapped particles. It means that the inward-shifted LHD plasma optimized for reducing the neoclassical ripple transport should have a large zonal flow response, which has recently been verified by more detailed GKV simulations with multi-helicity components of the confinement field [51]. This idea suggests better confinement in the inward-shifted LHD plasma, and is confirmed by recent GKV simulations of ITG turbulent transport [60, 61], which will be described in section 4.4.

4.3.

Entropy Balance in Toroidal ITG Turbulence

We have performed ITG turbulence simulation for a tokamak plasma with the Cyclone DIII-D base case parameters [59]; R0 /LT = 6.92, ǫt ≡ r0 /R0 = 0.18, r0 /ρi = 80, sˆ = 0.78, q0 = 1.4, and ηi = 3.114. Starting from initial perturbations with small amplitudes, the toroidal ITG instability grows with ballooning-type mode structures. Figure 20 shows a contour plot of the electrostatic potential fluctuations mapped on a flux surface and a poloidal plane observed at t = 90.6Ln /vti in a fully-developed ITG turbulence. Zonal flows with a wide spectrum in the radial wavenumber space are generated in the ITG turbu-

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Gyrokinetic Vlasov Simulations for Turbulent Transport in Magnetized Plasmas 153 t=0; θ=0

5 4 3 2

√2µΩi

(a)

1 0 -5 -4 -3 -2 -1 0 1 2 3 4 5 v|| t=20; θ=0

5 4 3 2

√2µΩi

(b)

1 0 -5 -4 -3 -2 -1 0 1 2 3 4 5 v|| t=50; θ=0

5 4

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3 2

√2µΩi

(c)

1 0 -5 -4 -3 -2 -1 0 1 2 3 4 5 v||

Figure 17. Velocity-space profiles of real part of the perturbed distribution function at z = θ = 0 for different time steps of simulation. Positive and negative parts are shown by solid and dashed lines, respectively. Dotted lines show the boundary between trapped and passing particle regions [48].

lence, and regulate the turbulent transport. In the simulation, we have found a statistically steady turbulent state realized where zonal flows and turbulent eddies of the ITG mode co-exist for the tokamak plasma with mean ion heat transport. The entropy balance considered in section 3.1 is examined for the toroidal ITG turbulent transport in a tokamak configuration with zonal flows. The entropy balance equation

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Figure 18. Time-history of zonal flow potential obtained by GKV simulation for a helical system (solid) and residual zonal flow level (K> ) derived from gyrokinetic theory (dashed). A single helicity configuration with L = 2 and M = 10 is considered with ǫL = 0.1, q0 = 1.5 and kx ρi = 0.131 [55].

Figure 19. Velocity-space structure of perturbed ion distribution function at θ = 8π/13 during collisionless damping of zonal flow [t = 12.5(R0 /vti )] in a helical system. The same parameters as in Fig.18 are used [56].

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Gyrokinetic Vlasov Simulations for Turbulent Transport in Magnetized Plasmas 155

Figure 20. A snapshot of electrostatic potential mapped on a flux surface and a poloidal plane observed at t = 90.6Ln /vti in GKV simulation of toroidal ITG turbulence [62].

derived from Eqs. (62) and (63) has the same form as in the slab ITG case,

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d (δS + W ) = ηi Qi + Di . dt

(76)

The entropy variable, δS, the potential energy, W , the ion heat transport flux, Qi , and the collisional dissipation, Di , are, respectively, given by + *Z X |fkx ,ky |2 3 1 X δSkx ,ky = δS = d v , (77) 2 FM kx ,ky kx ,ky     X 2  2 1 X Ti Ti

Wkx ,ky = φkx ,ky δky ,0 , W = 1 − Γ0 + − φkx ,ky 2 Te Te kx ,ky

kx ,ky

(78)   Z X 1 (79) iky φ−kx ,−ky v 2 J0 fkx ,ky d3 v , Qi = Qi,kx ,ky = 2 kx ,ky kx ,ky    X Z  X
3 f−kx ,−ky J0 φ−kx ,−ky + Di = Di,kx ,ky = C fkx ,ky d v . (80) FM X

kx ,ky

kx ,ky

Here, contributions of the zonal flow components are taken into account. Time-history of each quantity in Eq.(76) is presented in Fig.21, where a residual error defined by ∆ ≡ d (δS + W ) /dt−ηi Qi −Di is also plotted. The entropy balance is accurately satisfied with |∆/Di | less than 5% in the well-developed turbulence, and indicates that the statistically

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Tomo-Hiko Watanabe and Hideo Sugama d(δS)/dt, dW/dt, ηiQi and Di

156

100 80 60 40 20 0 -20 -40 -60

d(δS)/dt dW/dt ηiQi Di ∆

0

50

100 150 Time (Ln/vti)

200

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Figure 21. Time-history of each quantity in the entropy balance equation, such as timederivatives of entropy variable and potential energy, transport flux multiplied by ηi , and collisional dissipation. Residual error in the entropy balance (∆) is also plotted [62].

steady state of the turbulent transport is realized by balance of the two terms, ηi Qi ≈ −Di in the toroidal ITG system with zonal flows. According to the detailed study on production, transfer, and dissipation processes of the entropy variable in section 3.2, we expect development of fluctuations in velocity-space profiles of the perturbed distribution function. Unstable eigenmodes with wavenumbers of ky ρi < 0.6 linearly grow in the early stage of simulation. In the turbulent phase, the fluctuations are transferred to a wavenumber region of linearly stable modes through the nonlinear advection term. Fluctuations of perturbed distribution function in the velocity space are found in Fig.22, where positive and negative real-parts of fkx ,ky (kx ρi = 0.0858 and ky ρi = 0.7) are plotted by solid and dashed contours, respectively. Fine velocityspace fluctuations develop in the profile of Re[fkx ,ky ] due to the entropy transfer, and are dissipated by collisions.

4.4.

Application to ITG Turbulent Transport in Helical Systems

To investigate effects of helical magnetic field on the ITG turbulence and zonal flows, the GKV simulations are extended to the LHD configurations with L = 2 and M = 10 [50, 60–62]. We have employed magnetic configuration models relevant to the LHD experiments with the inward-shifted and standard plasma positions [51]. The ITG turbulence simulations for the LHD plasmas demand fine numerical resolution in the z (θ) direction due to helical ripples of which mode number along the field line is |L − M q0 | ≫ 1. In addition, as we have seen in Fig.19, motion of helical-ripple-trapped particles causes complicated fine structures of the distribution function in the velocity space [51, 56]. Thus, we employ a huge number of grid points over 5 × 1010 so that fluctuations of the gyrocenter distribution is well resolved in the phase space with the entropy balance accurately satisfied. Contours of the electrostatic potential in the statistically steady state of the ITG turbulence are plotted in Fig.23 for the inward-shifted (left) and standard (right) configurations. The ballooning-type mode structure of the ITG instability observed in the linear

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Gyrokinetic Vlasov Simulations for Turbulent Transport in Magnetized Plasmas 157

3 2

√2µΩi0

t=105; θ=0; kx=0.0858, ky=0.7; ν=0.001 5 Re(fkx,ky) 4

1 0 -5 -4 -3 -2 -1 0 1 2 3 4 5 v||

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Figure 22. Contour plot of real part of perturbed ion distribution function for stable ITG mode (kx ρi = 0.0858 and ky ρi = 0.7) observed at t = 105Ln /vti in the saturated toroidal ITG turbulence [62].

growth phase is destroyed in the latter turbulent state by the self-generated zonal flows. For the inward-shifted case, a clear radial structure of poloidal E × B zonal flows is identified in the potential profile mapped on a poloidal plane, while more isotropic vortices are found in the standard case. Time-averaged amplitude of the zonal flow potential, hφky =0 (x)i ≈ 4.5Te ρi /eLn for the inward-shifted plasma is about six times larger than that for the standard case, hφky =0 (x)i ≈ 0.74Te ρi /eLn . The stronger generation of zonal flows in the inward-shifted plasma is attributed to their higher response in the neoclassically optimized helical system as investigated in Ref. [51, 55, 56]. The stronger zonal flows in the inward-shifted case lead to significant reduction of turbulent transport. Time-history of the ion heat diffusivity χi is plotted in Fig.24. In the welldeveloped turbulent state, the time-averaged value of χi ≈ 1.27ρ2ti vti /Ln for the inwardshifted case is about 30% smaller than that of χi ≈ 1.78ρ2ti vti /Ln for the standard one. The result shown here confirms the theoretical prediction [55,56] that helical configurations optimized for reducing neoclassical ripple transport can simultaneously improve the turbulent transport with enhancing zonal flow generation, and also provides a possible explanation to the confinement improvement observed in the LHD experiments with the inward plasma shift [63]. It is concluded that the anomalous transport in helical systems is coupled with the neoclassical transport through zonal flows, and that synergetic studies on both the transport processes should largely contribute to comprehension of the helical plasma confinement.

5.

Conclusions

Our recent researches on Vlasov simulation methods and their applications to turbulent transport in magnetized plasmas are reviewed in this article. The pioneering work by Cheng and Knorr [4] on the splitting technique of the Vlasov equation is generalized by means of the symplectic integrators. The higher-order methods for the Vlasov-Poisson system are constructed by using the explicit symplectic integrators, and demonstrate improvements in the total energy conservation. The drift kinetic simulation method based on the implicit

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Tomo-Hiko Watanabe and Hideo Sugama

Figure 23. Contours of the electrostatic potential φ of the zonal flow and the ion temperature gradient (ITG) turbulence obtained by GKV simulation for inward-shifted (left) and standard (right) model configurations of the Large Helical Device (LHD) [60].

3

Standard Inward-Shifted

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χi (ρ2i vti/Ln)

2.5 2 1.5 1 0.5 0 0

50

100 150 Time (Ln/vti)

200

250

Figure 24. Time-history of the ion heat conductivity χi for the inward-shifted (solid) and the standard (dashed) LHD configurations [60].

symplectic scheme preserves the time reversibility of collisionless kinetic equations. The drift wave echo, three-mode ITG problem, and the entropy conservation are accurately reproduced by the nondissipative drift kinetic simulations. The gyrokinetic simulations applied to slab ITG turbulent transport revealed existence of the steady and quasisteady states of plasma turbulence, where the entropy production, transfer, and dissipation processes are thoroughly investigated in relation to phase-space structures of the distribution function. The kinetic simulation results of the collisionless ITG turbulence are also utilized for construction of a kinetic-fluid closure model. An oscillatory phase relation between temperature and parallel heat flux manifests necessity of the

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Gyrokinetic Vlasov Simulations for Turbulent Transport in Magnetized Plasmas 159 nondissipative closure model. The gyrokinetic Vlasov simulations are employed for studying the plasma turbulent transport and zonal flows in toroidal plasma confinement. A series of studies on zonal flow responses in tokamak and helical configurations demonstrates fairly good agreements between theory and simulations. Being based on the zonal flow response analysis, we have discovered turbulent transport reduction by zonal flow enhancement in helical systems. As predicted by the gyrokinetic theory, the GKV simulation confirms lower ion heat transport with stronger zonal flow generation in helical configurations with less neoclassical transport of helical-ripple-trapped particles. The obtained results also provide a reasonable explanation to the better confinement in the inward-shifted LHD plasma. In the last three decades, the Vlasov simulations have made tremendous progress in various fields of kinetic plasma phenomena. For investigations of the anomalous transport in the magnetic fusion plasma, several gyrokinetic Vlasov simulation codes have been developed [45, 46], where the macroscopic turbulent transport in magnetized plasmas can now be investigated from microscopic fluctuation levels of the distribution function in the five-dimensional phase space. According to rapid growth of the computer technology and the numerical algorithms, the gyrokinetic Vlasov simulations should further develop in the near future, and will also be applied to other areas of plasma physics, such as space and astrophysical phenomena.

Acknowledgments

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This work is supported in part by grants-in-aid of the Ministry of Education, Culture, Sports, Science and Technology (No. 17360445), and in part by the National Institute for Fusion Science (NIFS) Collaborative Research Program (NIFS08KDAD008).

References [1] Hazeltine, R.D., Meiss, J.D. Plasma Confinement, Addison-Wesley:Redwood City, 1992. [2] Horton, W., Rev. Mod. Phys. 1999, 71, 735. [3] Watanabe, T.-H., Sugama, H., Idomura, Y., J. Plasma Fusion Res. 2005, 81, 534-546. [4] Cheng, C.Z., Knorr, G., J. Comput. Phys. 1976, 22, 330. [5] Shoucri, M., Gagn´e, R., J. Comput. Phys. 1978, 27, 315. [6] Ghizzo, A., Izrar, B., Bertrand, P., Fijalkow, E., Feix, M.R., Shoucri, M., Phys. Fluids 1988, 31, 72. [7] Izar, B., Ghizzo, A., Bertrand, P., Fijalkow, E., Feix, M.R., Comput. Phys. Comm. 1989, 52, 375. [8] Klimas, A.J., Farrell, W.M., J. Comput. Phys. 1994, 110, 150. [9] Fijalkow, E., Comput. Phys. Comm. 1999, 116, 319.

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Tomo-Hiko Watanabe and Hideo Sugama

[10] Nakamura, T., Yabe, T., Comput. Phys. Comm. 1999, 120, 122. [11] Filbet, F., Sonnendr¨ucker, E., Bertrand, P., J. Comput. Phys. 2001, 172, 166. [12] Mangeney, A., Califano, F., Cavazzoni, C., Travnicek, P., J. Comput. Phys. 2002, 179, 495. [13] Arber, T.D., Vann, R.G.L., J. Comput. Phys. 2002, 180, 339. [14] Filbet, F., Sonnendr¨ucker, E., Comput. Phys. Comm. 2003, 150, 247. [15] Besse, N., Sonnendr¨ucker, E., J. Comput. Phys. 2003, 191, 341. [16] Umeda, T., Ashour-Abdalla, M., Schriver, D., J. Plasma Phys. 2006, 72, 1057-1060. [17] Umeda, T., Earth, Planets and Space 2008, 60, 773-779. [18] Birdsall, C.K., Langdon, A.B., Plasma Physics via Computer Simulation; McGrawHill:New York, 1985. [19] Yoshida, H., Phys. Lett. 1990, 150, 162. [20] Watanabe, T.-H., Sugama, H., Sato, T. J. Phys. Soc. Jpn 2001, 70, 3565-3576. [21] Watanabe, T.-H., Sugama, H., Transp. Theo. Stat. Phys. 2005, 34, 287-309. [22] Forest, E., Ruth, R.D., Physica D 1990, 43, 105. [23] Watanabe, T.-H., J. Plasma Fusion Res. 2005, 81, 686-697.

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[24] Sanz-Serna, J.M., Physica D 1992, 60, 293. [25] Goldstein, H., Poole, C., Safko, J., Classical Mechanics; Addison-Wesley:San Francisco, 2002, Chapt. 9. [26] Watanabe, T.-H., Sugama, H., Sato, T., Phys. Plasmas 2000, 7, 984. [27] deFrutos, J., Sanz-Serna, J.M., J. Comput. Phys. 1992, 103, 160. [28] Watanabe, T.-H., Sugama, H., Phys. Plasmas 2002, 9, 3659. [29] Dubin, D.H.E., Krommes, J.A., Oberman, C., Lee, W.W., Phys. Fluids 1983, 26, 3524. [30] Lifshitz, E.M., Pitaevskii, L.P., Physical Kinetics; Pergamon Press, New York, 1981, Chapt. III. [31] Lee, W.W., Krommes, J.A., Oberman, C.R., Smith, R.A., Phys. Fluids 1984, 27, 2652. [32] Federici, J.F., Lee, W.W., Tang, W.M., Phys. Fluids 1987, 30, 425. [33] Dimits, A.M., Phys. Fluids B 1990, 2, 1768. [34] Parker, S.E, Dorland, W., Santoro, R.A., Beer, M.A., Liu, Q.P., Lee, W.W., Hammett, G.W., Phys. Plasmas 1994, 1, 1461. Eulerian Codes for the Numerical Solution of the Kinetic Equations of Plasmas, Nova Science Publishers, Incorporated, 2010. ProQuest Ebook

Gyrokinetic Vlasov Simulations for Turbulent Transport in Magnetized Plasmas 161 [35] Mattor, N., Parker, S.E, Phys. Rev. Lett. 1997, 79, 3419. [36] Sugama, H., Watanabe, T.-H., Horton, W., Phys. Plasmas 2001, 8, 2617. [37] Watanabe, T.-H., Sugama, H., Phys. Plasmas 2004, 11, 1476-1483. [38] Clemmow, P.C., Dougherty, J.P., Electrodynamics of Particles and Plasmas; AddisonWesley: Redwood City, 1969. [39] Batchelor, G. K., J. Fluid Mech. 1959, 5, 113. [40] Krommes, J.A., Hu, G., Phys. Plasmas 1994, 1, 3211. [41] Hammett, G. W., Perkins, F. W., Phys. Rev. Lett. 1990, 64, 3019. [42] Sugama, H., Watanabe, T.-H., Horton, W., Phys. Plasmas 2003, 10, 726-736. [43] Beer, A., Hammett, G. W., Proceedings of the Joint Varenna- Lausanne International Workshop on Theory of Fusion Plasmas, Varenna, 1998, edited by J. W. Connor et al.; Societa Italiana de Fisca: Bologna, Italy, 1999. [44] Sugama, H., Watanabe, T.-H., Horton, W., Phys. Plasmas 2007, 14, 022502. [45] Idomura, Y., Watanabe, T.-H., Sugama, H., C. R. Physique 2006, 7, 650. [46] Garbet, X., Idomura, Y., Villard L., Watanabe, T. H. Nucl. Fusion 2010, 50, 043002. [47] Diamond, P.H., Itoh, S.-I., Itoh, K., Hahm, T.S., Plasma Phys. Control. Fusion 2005, 47, R35.

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[48] Watanabe, T.-H., Sugama, H., Nucl. Fusion 2006, 46, 24. [49] Frieman, E.A, Chen, L., Phys. Fluids 1982, 25, 502. [50] Watanabe, T.-H., Sugama, H., Ferrando-Margalet, S., Nucl. Fusion 2007, 47, 1388. [51] Ferrando-Margalet, S., Sugama, H, Watanabe, T.-H., Phys. Plasmas 2007, 14, 122505. [52] Beer, M.A., Cowley, S.C., Hammett, G.W., Phys. Plasmas 1995, 2, 2687. [53] Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A., Spectral Methods in Fluid Dynamics; Springer-Verlag, New York, 1988. [54] Rosenbluth, R.M., Hinton, F.L., Phys. Rev. Lett. 1998, 80, 724. [55] Sugama, H, Watanabe, T.-H., Phys. Rev. Lett. 2005, 94, 115001. [56] Sugama, H, Watanabe, T.-H., Phys. Plasmas 2006, 13, 012501. [57] Sugama, H., Watanabe, T.-H., J. Plasma Phys. 2006, 72, 825. [58] Motojima, O. et al., Nucl. Fusion 2003, 43, 1674. [59] Dimits, A.M et al., Phys. Plasmas 2000, 7, 969. Eulerian Codes for the Numerical Solution of the Kinetic Equations of Plasmas, Nova Science Publishers, Incorporated, 2010. ProQuest Ebook

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[60] Watanabe, T.-H., Sugama, H., Ferrando-Margalet, S., Phys. Rev. Lett. 2008, 100, 19502. [61] Sugama, H., Watanabe, T.-H., Ferrando-Margalet, S., Plasma Fusion Res. 2008, 3, 041. [62] Watanabe, T.-H., Sugama, H., Ferrando i Margalet, S., Proc. Joint Varenna-Lausanne Int. Workshop on Theory of Fusion Plasmas, Varenna, 2006 edited by J. W. Connor, et al.; American Institute of Physics, Melville, New York, 2006, p. 264.

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[63] Yamada, H. et al., Plasma Phys. Control. Fusion 2001, 43, A55.

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In: Eulerian Codes for the Numerical Solution … Editor: Magdi Shoucri, pp. 163-236

ISBN: 978-1-61668-413-6 © 2010 Nova Science Publishers, Inc.

Chapter 6

NUMERICAL SOLUTION OF THE RELATIVISTIC VLASOV-MAXWELL EQUATIONS FOR THE STUDY OF THE INTERACTION OF A HIGH INTENSITY LASER BEAM NORMALLY INCIDENT ON AN OVERDENSE PLASMA Magdi Shoucri Institut de Recherche d’Hydro-Québec(IREQ), Varennes, Qué. J3X1S1,Canada

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Abstract We study the interaction of a high intensity laser beam with an overdense plasma, when the beam is incident normally on the plasma surface. An Eulerian Vlasov code is used for the numerical solution of the one-dimensional (1D) relativistic Vlasov-Maxwell set of equations, for both electrons and ions. The incident high intensity laser radiation is pushing the electrons at the plasma surface through the ponderomotive pressure, producing a sharp density gradient at the plasma surface. There is a build-up of the electron density at this sharp edge which creates a space-charge, giving rise to a longitudinal electric field. The combined effects of this field and the ponderomotive pressure of the wave are responsible for an important ion acceleration. The results obtained differ substantially in several aspects when circular or linear polarization for the incident laser wave is considered. For the case of

a circular polarization, when the laser wave free space wavelength  0 is greater than the scale length of the jump in the plasma density at the plasma edge

Ledge ( 0  Ledge ) and

the ratio of the plasma density to the critical density is such that

n / ncr  1 , the radiation

pressure is pushing the sharp edge in the forward direction and the ions are accelerated and reach a free streaming expansion phase where they are neutralized by the electrons. For the case of a linear polarization, there is a standing structure with a sharp edge which forms at the wave front, and in this case the electrons at the sharp plasma edge oscillate nonlinearly in the field of the wave, which results in an important distorsion in the reflected electromagnetic wave and includes the generation of harmonics. The generation and propagation of collisionless shock waves in these systems are investigated, with proper

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164

Magdi Shoucri attention to the spatial and temporal shocks’ structures evolution and the associated electric fields. The method of characteristics for the numerical solution of hyperbolic type partial differential equations is applied for the numerical solution of the relativistic VlasovMaxwell equations for several selected situations especially pertinent to these problems, with different polarization and with different values of the ratios

n / ncr  1 .

0 / Ledge

and

The results underline the importance of including the ion dynamics in the

interaction of high intensity laser waves with overdense plasmas.

1. Introduction

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15

The field of the interaction of a high intensity femtoseconds ( 10 s.) laser beam incident on foil targets has attracted considerable interest and attention in recent years. Under these conditions a solid target transforms very rapidly, in a few cycles of the light wave, into a plasma. During the short period of the laser pulse, the laser interacts directly with a plasma with density of the order of the density of the solid. This interaction takes place in the relativistic domain because of the high intensity of the laser field. Collisions become negligible since the electrons quiver energy is relativistic, and the interaction at the plasma surface will be dominated by collective effects. Promising applications in a variety of areas in physics and medicine [1-7] have been demonstrated. Because of the production of high energy accelerated ions [1-5], especially in the case of a circular polarization of the laser wave, the phenomenon of energization and expansion of particles at the front of a high intensity wave and the subsequent formation and evolution of collisionless shock waves have been of particular interest since they play a fundamental role not only in laboratory laserplasma interactions, but also in a number of physical situations in astrophysics, semiconductors and nano-technology, where plasma jets of high speed are observed [8,9,10]. It is believed to provide the main acceleration mechanism for cosmic rays at the front of collisionless shock waves generated during supernova explosions [11]. Also of great interest is the generation of coherent high order harmonics when the laser wave is linearly polarized, 18

which has important potential applications as radiation sources with attosecond ( 10 s.) duration and with unprecedented intensities, offering a combination of short wavelengths and very high time resolution [12-15]. In the present chapter, we study the interaction of a high intensity laser beam normally incident on an overdense plasma, using an Eulerian Vlasov code for the numerical solution of the 1D relativistic Vlasov-Maxwell equations for both electrons and ions [16-18]. If the intensity of the wave is sufficiently high to make the oscillation of the electrons relativistic, interesting interactions between the wave and the surface of the plasma take place. Of particular interest for the problem of the acceleration of the ions is the case of circular polarization of the laser wave, when the free space wavelength of the wave  0 is greater than the scale length of the jump in the plasma density at the plasma edge Ledge ( 0  Ledge ), and the plasma density is such that n / ncr  1 , where ncr  1.1x10 21 02 cm 3 (  0 is the laser wavelength expressed in microns). In these regimes the radiation pressure causes a charge separation and an electric field to appear at the plasma edge, and produces a plasma surface whith a steep density profile. There is a build-up of the electron density at the plasma edge.

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The combined effects of the edge electric field and the radiation pressure result in an important acceleration of the ions in the forward direction. In the ongoing acceleration process, the accelerated ions reach a free-streaming expansion phase where the electrons neutralize the charge of the expanding ions, and shock structures are formed. A dense compact bunch of quasineutral plasma is formed and expands. These results are in qualitative agreement with recent particle-in-cell (PIC) simulations which show that the ion acceleration process during the interaction of an intense circularly polarized wave with a thin foil takes place on the front side of the foil [19-21]. Recent theoretical results [22] have considered the effect of a steady longitudinal electric field at the plasma edge, but the self-consistent effect of the ponderomotive pressure and the feedback effect of the electrons and ions can be studied only by numerical simulations. There are substantial differences in the interaction of a circularly polarized or a linearly polarized high intensity laser beam incident on an overdense plasma [19]. In the case of a linear polarization of the incident laser beam, we find that a standing structure with a sharp edge forms at the wave front. The electrons at the sharp edge oscillate nonlinearly in the field of the wave, which results in an important distorsion in the reflected electromagnetic signal which includes the generation of harmonics. We study the generation of coherent high order harmonics in the reflected wave from the laser-generated overcritical plasma surface (see the recent references [12-15]). Numerical simulations are indeed the only alternative to study the kinetic effects in these highly relativistic and nonlinear problems. Kinetic effects (e.g. particles trapping and acceleration) are often simulated numerically using particle-in-cell (PIC) codes [23], as recently applied to study short-pulse laser plasma interactions in [12-15, 20,21] for instance. In the present chapter, we study these problems by using an Eulerian Vlasov code for the numerical solution of the 1D relativistic Vlasov-Maxwell equations. The numerical code integrates these equations along their characteristics, using a two-dimensional advection technique in phase-space, of second order accuracy in time-step, and where the value of the distribution function is advanced in time by interpolating in two dimensions along the characteristics using a tensor product of cubic B-splines. Numerical details have been discussed in the review articles [17,18] for instance. See also Refs. [16,24]. The low noise level of the Eulerian Vlasov code allows an accurate representation of the phase-space, especially for the low density regions where particles are accelerated and expanding. A fully nonlinear 1D relativistic Vlasov-Maxwell model to study the self-consistent interaction of intense laser pulses with plasmas can be found, for instance, in [25,26]. A characteristic parameter of a high power laser beam is the normalized vectror potential or quiver







momentum a   eA / me c  a0 , where A is the vector potential of the wave, e and me are the electronic charge and mass respectively, and c the velocity of light. We are interested in the regime a 0  1 .

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2. The Relevant Equations 2.1. The 1D relativistic Vlasov-Maxwell Model Time t is normalized to the inverse electron plasma frequency  pe , length is normalized 1

to l 0  c pe , velocity and momentum are normalized respectively to the velocity of light c 1

and to me c . The general form of the Vlasov equation is written for the present problem in a 4D phase-space for the electron distribution function Fe ( x, p xe , p ye , p ze , t ) and the ion distribution function Fi ( x, p xi , p yi , p zi , t ) (one spatial dimension) as follows:

Fe,i t



   pe,i x B Fe,i  (E  )   0 x  e ,i pe,i

Fe,i

p xe,i M e ,i  e , i

(1)

with

 e ,i

  1 2 2  1  2 ( p xe  p ye  p ze2 ,i )  , i , i  M  e ,i  

1/ 2

(2)

(the upper sign in Eq.(1) is for electrons and the lower sign for ions, and subscripts e or i denote electrons or ions respectively). In our normalized units M e  1 and M i 

mi . me

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We write the Hamiltonian of a particle in the electromagnetic field of the wave:

H e,i  M e,i ( e,i  1)   ( x, t ) . where

(3)

 (x,t) is the scalar potential. Eq.(1) can be reduced to a two-dimensional phase-space 

Vlasov equation as follows. The canonical momentum Pce,i is connected to the particle









 

momentum pe,i by the relation Pce,i  pe,i  a . a  eA / me c is the normalized vector potential. From Eq.(3), we can write: 1/ 2

H e ,i  M e ,i 

   1  1 ( Pce,i  a ) 2  2  M  e ,i  

 1    ( x, t ) .

(4)



Choosing the Coulomb gauge ( diva  0 ) , we have for our one dimensional problem

a x    0 , hence a x  0 . The vector potential a  a ( x, t ) , and we also have the x following relation along the longitudinal direction:

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dPcxe,i dt



H e,i

167

(5)

x

and since there is no transverse dependance :

 dPc e,i dt

   H e,i  0 .

(6)



This last equation means Pce,i  const. We can choose this constant to be zero without loss of generality, which means that initially all particles at a given (x,t) have the same





perpendicular momentum pe,i  a ( x, t ) . The Hamiltonian now is written:



2 2 2 2 H e,i  M e,i  1  p xe ,i / M e,i  a ( x, t ) / M e,i



1/ 2

 1    ( x, t ) .

(7)



The 4D distribution function Fe,i ( x, p xe,i , pe,i , t ) can now be reduced to a 2D distribution function f e,i ( x, p xe,i , t ) :

   Fe,i ( x, p xe,i , pe,i , t )  f e,i ( x, p xe,i , t ) ( pe,i  a ) .

(8)

f e,i ( x, p xe,i , t ) verify the relation:

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df e,i dt



f e,i t



H e,i f e,i p xe,i x



H e,i f e,i x p xe,i

0

(9)

which gives the following Vlasov equations for the electrons and the ions::

f e,i t where



p xe,i

f e,i

M e,i  e,i x

 ( E x 



2 M e ,i  e ,i

 e,i  1   p xe,i / M e,i 2  a / M e,i 2  Ex   x

a 2 f e,i )  0. x p xe,i

1



1/ 2

and

(10)

.

  a  E   t

and Poisson’s equation is given by:

 2  f e ( x, p xe )dp xe   f i ( x, p xi )dp xi (12) x 2 

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(11)

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An alternative way, used in the present work, is to calculate E x from Ampère’s equation:

E x  J x , t

The transverse electromagnetic fields E y , Bz and E z , B y for the circularly polarized 

wave obey Maxwell’s equations. With E  E y  B z and F



 E z  B y , we write these

equations in the following form:

(

      )E  J y . ; (  )F   J z t x t x

(13)

which are integrated along their vacuum characteristic x=t [23]. In our normalized units we have the following expressions for the normal current densities:

  a J  e ,i   M e ,i

   J   J  e  J i ;



f e ,i





dp xe,i .

(14)

e ,i

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2.2. The Numerical Scheme There have been important advances in the last few decades in the domain of the numerical solution of hyperbolic type partial differential equations using the method of characteristics. We use a method of characteristics for the numerical solution of Eq.(10), as recently discussed in [16,17,18,24] for instance. These methods are Eulerian methods which use a computational mesh to discretize the equations on a fixed grid, and have been successfully applied to different important problems in plasma physics (see for instance the articles [16-18]). The numerical scheme to advance Eq.(10) from time tn to tn+1 necessitates the knowledge of the electromagnetic field E  and F  at time tn+1/2 . This is done using a centered scheme where we integrate Eq.(13) exactly along the characteristics with x  t , to calculate E n1 / 2 and F n1/ 2 as follows:

E  ( x  t , t n1 / 2 )  E  ( x, t n1 / 2 )  tJ y ( x  t / 2, t n ) with J y ( x  t / 2, t n ) 

(15)

J y ( x  x, t n )  J y ( x, t n ) 2

A similar equation can be written for F n1/ 2 with Jz. From Eq.(11) we also have

      an1  an  tEn1 / 2 , from which we calculate a n1 / 2  (a n1  a n ) / 2 . We also have:

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J xn 

We

calculate

1 Mi

E xn1 / 2



p xi





f i n dp xi 

i

from

1 Me

Ampère’s



p xe





169

f en dp xe

e

equation:

E x  J x , t

from

which

Exn1 / 2  Exn1 / 2  tJ xn . Now given f en,i at mesh points (we stress here that the subscript i n 1

denotes the ion distribution function), we calculate the new value f e ,i

at mesh points from

the relation [17,18]:

f en,i1 ( x, p xe,i )  f en,i ( x  2 xe,i , p xe,i - 2 pxe,i ) ;

(17)

 xe,i and  p xe,i are calculated from the solution of the characteristics equations for Eq.(10), which are given by:

dx 1 p xe,i  dt M e,i  e,i dp xe,i dt

 Ex 

1 2 M e ,i  e ,i

a 2 x

.

(18)

This calculation is effected as follows. We rewrite Eq.(17) in the vectorial form:

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f en,i1 ( X e,i )  f en,i ( X e,i  2 X e,i ) ; .

(19)

Xe,i is the two dimensional vector Xe,i  x, p xe,i  , and  X e,i  ( xe,i ,  pxe,i ) is the two dimensional vector calculated from the implicit relation [17,18]:

 X e ,i 

t Ve,i (X e,i -  X e,i , t n1 / 2 ) . 2

(20)

 1 p xe,i  (a ( n 1 / 2) ) 2  1  . Eq.(20) for  X e,i is implicit and Ve,i   ,  E xn 1 / 2   M  2 M   x e , i e , i e , i e , i   t k 1 is solved iteratively:  Xe,i  Ve,i (X e,i - kXe,i , t n1 / 2 ) , where we start the iteration with 2 0  Xe,i  0 for k=0. Usually two or three iterations are sufficient to get a good convergence. n 1

Then f e ,i

n

is calculated from f e ,i in Eq.(17) by calculating the shifted value using a two-

dimensional interpolation in the two dimensional phase-space ( x, p xe,i ) . Similarly in Eq.(20)

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the shifted value was calculated at every iteration using a two-dimensional interpolation. In all cases these two-dimensional interpolations are effected using a tensor product of cubic Bsplines [17,18]. Cubic splines have been tested in several applications and shown to give accurate results [16,27,28]. We note that a linear interpolation has been used sometimes for the solution of Eq.(20), and a cubic spline interpolation used for Eq.(17) [24]. This however has been shown in several applications to enhance the numerical diffusion [28]. In the results presented in this chapter, the cubic spline interpolation is used for both Eq.(17) and Eq.(20) as mentioned.

3. A Circularly Polarized Laser Wave Incident on an Overdense Plasma (n/ncr=25) A. Case with a0  4 The forward propagating circularly polarized laser wave enters the system at the left hand (x=0) boundary, where the forward propagating fields are E   2E0 Pr (t ) cos(t ) and

F   2E0 Pr (t ) sin(t ) . The shape factor

Pr (t )  sin(t /(2 )) for t    50 ,

Pr (t )  1 otherwise. A characteristic parameter of a high power laser beam is the normalized   vectror potential or quiver momentum a   eA / me c  a0 . We are interested in the regime a 0  1 . We choose for the amplitude of the potential vector a 0  4 . For the circularly polarized wave 2a02  I20 / 1.368x1018 , I being the intensity in W / cm ,  0 is 2

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the laser wavelength in microns.

  0.2 pe , which corresponds to n / ncr  25 , where ncr

is the critical density . The Lorentz factor for the transverse oscillation of an electron in the field of the wave is

 0  1  a02  17 .The initial distribution functions for electrons and

ions are Maxwellian with temperature Te=1. keV for the electrons and for the ions Ti=0.1 keV. These parameters are close to what has been recently presented in [29], except that in the present case we have a steeper edge of the plasma slab, and the length of the central flat top density is smaller to allow the laser wave to cross completely the plasma slab. The total length of the system is L  125.66 (in units of c /  pe ). We use a fine resolution in phasespace, with N =7000 grid points in space and 1400 grid points in momentum space for the electrons and the ions (extrema of the electron momentum are  5 , and for the ion momentum  87 , momentum in units of me c ). This fine resolution of the phase-space guaranties

accurate

results.

We use a

time-step

and a

grid size

such

that

t  x  0.01795 . We have a vacuum region of length Lvac  56.20 on each side of the plasma. The steep jump in density at the plasma edge on each side of the flat top density is of length Ledge  1.795 . In free space   k for the electromagnetic wave, and

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0  2 / k  31.415 . It follows that 0  Ledge . The length of the central plasma slab with flat top density of 1 is L p  9.67 .

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Figure 1a. Electron (full curve) and ion (dash curve) density profile at t=134.625.

Figure1b. Electron (full curve) and ion (dash curve) density profile at t=242.325.

Figure 1c. Electron (full curve) and ion (dash curve) density profile at t=394.9.

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Figure 1d. Electron (full curve) and ion (dash curve) density profile at t=550.

Figs.(1) show the plot of the density profiles against distance (full curves for the electrons

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and dash curves for the ions) at times t=134.625, 242.325, 394.9 and 550 (in units of

1  pe ).

Initially the results in Figure (1a) and Fig,(1b) reproduce what has been observed in [29]. Figure (2) presents the same density plots at times (from left to right) t=89.75, 134.625, 242.325, 394.9 and 550, concentrating on the plasma slab region. Note in Figure (2) in the first curve at t=89.75 the electrons (full curve) move first, while the density profile for the ions (dash curve) did not move much. The curve in Figure (1a) and the second curve in Figure (2) at t=134.625 show the ion peak reaching the electron peak at the wave-front plasma-edge interface. Then in the subsequent evolution the ion density peak (dash curve) increases and reaches a much higher value than the electron density peak at the wave-front plasma-edge interface. At the wave-front edge the plasma forms a steep structure, with a solitary like structure for the ions, maintaining a stable profile as the edge moves forward. To the right of this edge the accelerated high velocity ion population is now free streaming towards the right (see the ion phase-space in Figs.(6) below). The density of the ion population in this free streaming region is exactly compensated by the electron density, giving essentially no electric field (see Figs.(2,3) below) in this expansion region at the right in Figure (2). We see in Figs.(1,2) two ion density peaks propagating with different speeds. The high solitary-like left peak at the wave-front plasma-edge interface propagates at a lower speed. The right smaller peak in front of the bunch of the expanding neutral plasma is propagating at a higher speed, and the shape of a small shock-like structure with a steepening process at the right edge of the expanding ions is taking place (see the work in [8] or the experimental results in [9] where collisionless shocks are observed in laser-plasma interaction). In Figure (2) we see that the high peaks at t=242.325 and t=349.9 have to their immediate right a small plateau. At t=550, the edge at the right of the free expansion region has crossed the initial plasma slab location and is now free streaming in the vacuum region, where we had initially no plasma. We note even in Figure (1d) two small peaks in the free expansion region. This planar expansion as it appears to the right edge of the expanding plasma, with the formation of shock-like structures and sharp ion front with ion peaks or spikes in the local ion density (called bunching), has been studied using fluid equations for the ion expansion in vacuum [10,30-32]. We note qualitative similarities with some features we see in the results obtained with the present kinetic code. In Figure (2) a very small density ion population (dash curve) is moving backward to the left of the sharp edge discontinuity (which is not the case for the electrons

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who show a sharp edge at the wave front). The electric field is presented in Figure (3) at times t=89.75 (full curve), 134.625 (dotted curve), 242.325 (dash curve), 394.9 (dash-dot curve) and 550 (dash-3dots curve). The electric field is essentially zero in the free expansion region where the free-streaming plasma is essentially neutral. In Figure (4a) we present again a view of the density profiles in the plasma at t=242.325 and 550, to which we add the electric field profiles at the same times (dash-dot curve, multiplied by a factor of three to make it more visible). Note the stable structure of the electric field edge peak, followed by a rapid decay of the electric field inside the plasma, the penetration of the electric field in the plasma is of the order of c /  pe , as we can verify from Figs.(3) and (4a). The discontinuity at the right edge of the neutral expanding region is propagating supersonically and contributes to the rapid expansion of the plasma towards the right. Figure (4b) is the same as Figure (4a), more concentrated around the peaks to better localize the position of the electric field at the edge. The high ion solitary-like peak at the wave front edge is located at the same position as the electron peak for the curve ‘a’ in Figure (4b), and shows a very slight shift with respect to the electron peak for the curve ‘b’. Note that the high solitary like ion peak builds up very rapidly. We show in Figure (5) the density at the wave-front plasma-edge interface at t=134.625, 143.6 and 152.575. The ion density peak is jumping to a value slightly above 13 in a time of 17.95 , producing a solitary-like structure. In front of the plasma edge in the backward direction, the electric field profiles in Figure (3) show at t=89.75 and t=134.625 a shape decaying to zero away from the plasma. At t=242.325 and at higher times in Figure (3), the electric field shows a negative value in front of the plasma, accelerating the small ion population in front of the edge in the negative direction. However, in Figure (2) for instance, there are only ions in the backward direction in front of the edge position of the plasma. The electrons are pushed in the forward direction by the radiation pressure of the laser pulse. At t=242.325 and higher times in Figure (3), the electric field is monotonic in the immediate vicinity of the plasma, but then changes sign. From Gauss law, the derivative of the electric field is proportional to the charge density that is positive everywhere in the backward direction in front of the plasma edge. We note that down the steep edge in front of the plasma, the cubic spline interpolation used in the code may result in some small negative fluctuations of the distribution functions, as it is well known when interpolating using a cubic spline along a steep gradient. So although the electric field profiles in Figure (3) show at t=89.75 and t=134.625 the correct profiles, the steepening of the edge profile at higher times and the sparse distribution of the low density ions in the phasespace in the backward direction in front of the plasma edge result in the calculation of the electric field in the backward direction, using Ampère’s law, being flawed, building up this negative electric field as the simulation progresses. However, this low density region in front of the plasma edge is completely decoupled from the solution propagating in the forward direction, which is the main interest of this simulation. The solution propagating in the forward direction is verifying with accuracy the relations presented in Eqs.(21-23) below. Figs.(6) present the phase-space contour plots of the ion distribution function at times t= 134.625, 242.325, 394.9 and 550. The loops appearing at the top in Figs.(6b-d) correspond to the free streaming ions, escaping the front edge region. These loops are close to what has been presented in [20] using PIC codes. See also the recent results in [29,33]. The results in Figs.(1,2) and Figure (6c-d) are also close to what is presented in [19]. Figs.(7) present the phase-space contour plot of the electron distribution function. At the left edge the effect of the

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ponderomotive pressure pushing the electrons is apparent. Note the sharp profile at the left at the wave-front plasma-edge interface, and the electronic population being ejected forward. These ejected electrons are bouncing back at the right boundary of the plasma, as in Figs(7b,7c). We note in Figure (7c) another sharp edge appearing at the right of the small plateau, as we previously mentioned in Figure (2). At t=550, the right edge of the free streaming expanding ions has reached around x=77 (see Figs.(6d) and (7e)), beyond the original plasma slab right edge which was located initially around x=69 (see Figure (1a-c)). The shape of the initial plasma slab with a plateau and sharp edges is now completely modified, and replaced by the stable configuration of a bunch traveling in space with shock’s structures to the left and to the right. This bunch is neutral with the exception of a layer at the wave-front plasma-edge interface where a sharp electric field is penetrating the plasma in a layer of the order of c /  pe .

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Figure 2. Plot of the density at the plasma slab (full curve electrons, dash curve ions) at (from left to right) : t=89.75, t= 134.625, t=242.325, t=394.9, t=550.

Figure 3. Electric field at : t=89.75 (full curve), t=134.625 (dotted curve), t=242.325 (dash curve), t=394.9 (dash-dot curve), t=550 (dash-3dots curve).

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Figure 4a. Plot of the density at the plasma slab (full curve electrons, dash curve ions), at: a) t=242.325, b) t=550. The dash-dot curve gives the electric field (multiplied by a factor of 3 to improve visibility) at the same times.

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Figure 4b. Same as in Figure 4a , concentrating the horizontal scale around the steep gradients.

Figure 5. Density profiles at the edge of the plasma for the electrons (full curves) and the ions (dash curves) at: t=134.625, b) t=143.6, c) t=152.575 , showing the rapid build-up of the ion density peak over a time of 17.95.

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Figure 6a. Phase-space plot of the ion distribution function at t=134.625.

Figure 6b. Phase-space plot of the ion distribution function at t=242.325.

Figure 6c. Phase-space plot of the ion distribution function at t=394.9.

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Figure 6d. Phase-space plot of the ion distribution function at t=550.

Figure 7a. Phase-space plot for the electron distribution function at t=89.75.

Figure 7b. Phase-space plot for the electron distribution function at t=134.625.

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Figure 7c. Phase-space plot of the electron distribution function at t=242.32

Figure 7d. Phase-space plot of the electron distribution function at t=394.9.

Figure 7e. Phase-space plot of the electron distribution function at t=550.

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The ions at the left boundary have a solitary like structure, which travels at a speed of about 0.0185 (calculated by following the left edge in Figure (2) or Figure (4)), and maintain an electric field whose penetration in the plasma is of the order of c /  pe . The velocity of the pushing front calculated in [34] by balancing the electromagnetic pressure at the absorber surface with the rate of increase in ion momentum yields the following expression for the velocity of the surface of discontinuity:

n m  u i  a0  c e   n mi  with

a0  4 ,

n / nc  25 and

1/ 2

me / mi  1. / 1836 , we calculate

(21)

ui  0.01867

(normalized to the velocity of light), in good agreement with the measured result previously mentioned. At the right boundary we have a neutral plasma with a smaller peak and with a shock-like structure expanding at a speed of 0.041, which corresponds to the speed of the free streaming ions in Figure (6d). Between these two peaks, a neutral plasma bunch exists (with the exception of the wave- front plasma-edge interface where the penetration of the electric field is of the order of c /  pe ). From Figure (6d) we can check that for the free-streaming ions, we have mi / me c  70 , i.e.

 / c  70me / mi  0.038 , in close agreement with the

calculated value of 0.041. The proton energy is mi c 2  940 MeV. The kinetic energy of the

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ion beam will be mi 2 / 2  mi c 2 ( / c) 2 / 2  0.683 MeV. The entire structure is stable as it propagates to the right. Figure (2d) shows the presence of two small peaks in the neutral free expanding region, corresponding to two vorticity like structures in Figure (7e). Note the difference at the right boundary between Figure (7d) and (7e), where now in Figure (7e) electrons are expanding to the right, neutralizing the free-streaming ions.

Figure 8. A cut taken at x=61.57 in Figure 7c at t=242.325 (just to the right of the electron peak) where

Fe( x  61.57, p xe )

is plotted on a logarithmic scale.

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Figure 9. A cut at the position x=64.26 for Fe( x  64.26, p xe ) at a) t=89.75, b) t=134.625, c) t=394.9.

Figure (8) shows on a logarithmic scale the distribution function Fe( x  61.57, p xe ) obtained by making a cut at x=61.57 in the results presented in Figure (7c) at t=242.325. This cut shows at this position a distribution function with a relatively flat peak and with steep edges. Figs.(9a-c) show on a logarithmic scale cuts at the position x=64.26, obtained from Figs.(7). Figure (9a) shows the cut Fe( x  64.26, p xe ) at t=89.75, which is still close to the initial Maxwellian shape. In Figure (9b) the cut Fe( x  64.26, p xe ) is obtained from Figure (7b), at t=134.625. We note the presence of the ejected electrons forming a bump in tail like structure. Figure (9c) (obtained from Figure (7d) at x=64.26, just behind the steep shock at the right edge at t=394.9) shows a heated distribution with relatively broad top, similar to Figure (8). Figs.(10) present the forward propagating wave E  (full curves) and the backward

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reflected wave E  (dash curves) at t=520.55, 529.525, 538.5, 547.475 and 550. Figs.(11) present the corresponding results for the forward propagating wave F  (full curves) and the backward propagating wave F  (dash-curves). The electromagnetic wave damps in the plasma over a distance of the order of the skin depth c /  pe . The strong increase of the ion and electron densities at the wave-front makes the plasma more opaque, with the steep plasma edge acting as a moving mirror for the incident light. Note the continuous change in phase occurring during reflection. The frequency of the backward reflected wave is slightly down-shifted by the moving reflecting plasma surface. See in Figure (10B) at t=529.525, when the incident and reflected waves are in phase at the reclection point, the reflected wave (dash curve) in vacuum has a wavelength (and a period) slightly bigger than the corresponding ones for the incident wave (full curve). Hence the frequency of the reflected wave is slightly downshifted with respect to the frequency of the incident wave by the Doppler shift due to the motion of the moving reflecting surface. In [35] the following analytical expressions were derived for the reflected wavenumber k r and the reflected frequency

r of the laser wave due to the Doppler shift by the moving reflecting surface:

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 r 1   F2  2 F /    1   F2

(22)

k r 1   F2  2 F   k 1   F2

(23)

 F  ui  0.01867 is the normalized velocity of the discontinuity surface, and    / k =1 for the incident wave, since   k  0.2 for the incident laser where in our units

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wave in free space. We get for the reflected wave in free space from Eqs.(22-23) r /   k r / k  0.963 . This is confirmed for the frequency by a Fourier spectrum of the ingoing and outgoing signals at the entance of the domaim at x=0.



Figure 10. Electromagnetic wave E (full curve) and C) t=538.5, D) t=547.475, E) t=550.

E

(dash curve) at :A) t=520.55, B) t=529.525,

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For the wavenumber, this can be checked from Figs.(10,11), where the wavelength of the reflected signal (dash curve) is r  32.564 , so k r  2 / r  0.193 , and k r / k  0.964 ,in very good agreement with Eq.(23).

B. Case with a0  6 The parameters are the same as in the previous case, except that we now have a0  6 . The interaction is more vigourous, the peaks are higher, and the velocities of the shock structures are bigger. Figure (12a) is the equivalent of Figure (4a). The curve a in Figure (12a) is at t=242.325. The curve and the displacement towards the right is more important than in Figure (4a) at the same time. The curve b in Figure (12a) is at t=439.77. The right expanding edge of the structure has reached the position x=76, which is reached in Figure (4a) only at t=550. The ion peaks in Figure (12a) (dash curves) reach a value around 11.5. The ion peaks seems to be slightly shifted with respect to the electron peaks (full curve) at the wave-front plasma-edge interface at the left of the structure (perhaps due to the higher radiation pressure). The dash-dot curves are the electric fields (multiplied by a factor of 3), and appear more important than in Figure (4a), although the penetration of the electric field at the edge of the plasma is still of the order of c /  pe . Figure (12b) concentrates more in the region of the interface between the wave front and the plasma edge. The same remark concerning the negative electric field in front of the plasma edge discussed in section 3.A applies. Note the steep density profiles at the wave-front plasma- edge interface, and we can check from this figure that the velocity at this interface is about 0.027, in good apreement with Eq.(21) for a 0  6 . We can also calculate from Eqs.(22-23) for the frequency of the

r  0.9474 , and for the wanumber of the reflected wave k r  0.9474k . This is confirmed with good accuracy for r by a Fourier spectrum of the outgoing waves at

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reflected wave

the entance of the domaim at x=0, and for k r by following the peaks of the reflected wave, as we did in Figs.(10,11) in section 3.A. We verify from Figure (12a) the velocity of the expanding right edge at the right of the structure is now 0.0542. From Figure (14b), the free-streaming ions have mi / me c  100 , i.e.

 / c  100me / mi  0.0544 , in good agreement with the calculated value from Figure

(12a). The proton energy is mi c 2  940 MeV. The kinetic energy of the ion beam will be

mi 2 / 2  mi c 2 ( / c) 2 / 2  1.37 MeV.As we mentioned previously, the growth of the ion peak at the wave-front plasma- edge takes place very rapidly. Figure (13) shows a plot of the density profiles at the edge at the wave front (full curves for electrons, dash curves for the ions) at t=125.65, 134.625 and 143.6, showing the rapid growth of the ion peak over a period of 17.95, forming a solitary like structure. The curve c at t=143.6 is reaching a peak of about 16.5. Figure (14a) is the ion phase-space at t=242.325 (to be compared with Figure (6b)), showing a more important deformation and a more important free-streaming ion population (the loop at the top). Figure (14b) at t=439.775 shows a shape close to what we reach at t=550 in Figure (6d).

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

Figure 11. Electromagnetic wave F (full curve) and t=529.525, C) t= 538,5, D) t=547.475, E) t=550.

F

183

(dash curve) at :A) t=520.55, B)

Figure 12a. Plot of the density at the plasma slab (full curve electrons, dashed curve ions), a) t=242.32, b) t=439.77. The dash-dotted curves give the corresponding electric field (multipled by a factor of 3 to improve visibility) at the same times. Note in the expanding region at the right the electron density matching the ion density in the free-streaming expansion region of the ions (case

a0  6

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Figure 12b. Same as Figure 12a, concentrating the horizontal scale around the steep gradients.

Figure 13. Density profiles at the edge for the electrons (full curves) and ions (dash curves) at a) t=125.65, b) t=134.625, c) t=143.6, showing the rapid build-up of the ion density peak over a time of Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved.

17.95 (the peak of the curve c reaches 16.5) (case

a0  6 ).

Figure 14a. Phase-space plot of the ion distribution function at t= 242.32 (case with

a0  6 ).

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Figure 14b. Phase-space plot of the ion distribution function at t= 439.77 (case with

a0  6 ).

Figure 15a. Phase-space plot of the electron distribution function at t= 242.32 ( a 0

 6 ).

Figure 15b. Phase-space plot of the electron distribution function at t= 439.77( a 0

 6)

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Figure (15a) gives the electron phase-space at t=242.325 (to be compared with Figure (7c)). Figure (15b) is the electron phase-space at t=439.77, when the initial plasma slab structure has disappeared and replaced by an expanding bunch with a shock structure having a solitary-like shape at the wave front edge, and another shock structure at the right of the expansion region. We give in Figure (16) the contour plot of the electron phase-space concentrating on the front edge of Figure (15b), showing a solitary-like structure. In Figure (17) we concentrate on the vortical structure in the middle of Figure (15b), and in Figure (18) we concentrate on the shock edge at the right of Figure (15b), where electrons are following the ions and providing a neutral expanding plasma. It is interesting to note that these figures appear more or less symmetric, this symmetry does not appear to be around pxe =0 , but rather around a small value of pxe , reflecting the fact that the entire population is translating to the right.

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Figure 16. Same as Figure 15b, magnifying the region of the wave front edge.

Figure 17. Same as Figure 15b, magnifying the region of the center.

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Figure 18. Same as Figure 15b, magnifying the region of the right edge.

4. A Circularly Polarized Laser Wave Incident on a Moderately Overdense Plasma (n/ncr=1.731) The forward propagating circularly polarized laser wave enters the system at the left hand (x=0) boundary, as previously discussed in section 3. We choose for the amplitude of the potential vector a0 

3 .   0.76 pe , which corresponds to n / ncr  1.731 .The

Lorentz factor for the transverse oscillation of an electron in the field of the wave is

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 0  1  a02  2. The initial distribution functions for electrons and ions are Maxwellian with temperature Te=1. keV for the electrons and for the ions Ti=0.1 keV. The total length of the system is L  67.666 . We use N =6000 grid points in space and 1960 grid points in momentum space for the electrons and for the ions, providing a fine grid for the resolution of the phase-space (extrema of the electron momentum are  7 , and for the ion momentum

 125.88 , momentum in units of me c ). The time-step and the spatial grid size are such that t  x  0.01128 . We have a vacuum region of length Lvac  14.946 on each side of the plasma. The jump in density on each side of the plasma slab is over a length of Ledge  3.948 (in the present case 0  8.267 and Ledge  0 ). The length of the central plasma slab with a flat top density of 1 is L p  29.9 . The plots of the density profiles against distance are presented in Figs.(19) (full curves for the electrons and dash curves for the ions) at times t=78.96, 90.24, 112.8, 124.08, 180.84 and 200. The electron density profile is steepening at the wave-front plasma-edge interface. There is in this case a small electron population from the uderdense region moving in the backward direction, and then pushed back by the incident wave. The beating of the incident and reflected waves results in vortical structures having steep edges (see Figs.(23) below). More attention will be given to these vortical structures in the next section. Also some ions are accelerated backward. There is also an electron population

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accelerated forward with an important modulation of the bulk electron density (see Figure (19b) at t=90.24 and Figure (23b-c) below). Figure (19d) shows a huge jump in the ion density at the plasma edge (reaching a peak of 22 ). This sharp build-up of the ion density peak at the wave front takes place from Figure (19c) to Figure (19d) over a time of 11.28, and is shown in more detailts in Figure (20). In Figs.(19) we see a penetration of a small perturbation along the flat electron density profile. n /  0 ncr  0.865 , however the sharp increase in the plasma density peak at the plasma edge will make the plasma more opaque. Figure (20) shows, concentrating on the edge, the rapid build-up of the ion peak at the edge forming a solitary-like structure ( dash-curves at a) t=112.8, b) t=118.44, c) t=124.08, this last one reaching a peak value of 22), and the corresponding electron density profiles at the same times given respectively by the dotted curve, the dash-dot curve, and the full curve. Figs.(19e-f) shows another ion peak of a shock-like structure propagating to the right, similar but more important compared to what has been observed in the previous section. This shock-like structure at the right corresponds to a streaming ion population (see Figs.(24c-d) below). Figure (21) shows the longitudinal electric field at the edge at t=78.96 (full curve), 124.08 (dotted curve), 146.64 (dash curve), 180.48 (dash-dot curve) and 200 (dash-3dots curve). Figure (22) presents on a magnified scale at the wave front the ion density ( dash curves at a) t=180.48 and b) t=200.), with the corresponding electron density profile (full curves), and the corresponding electric field profiles (dash-dot curve at t=180.48 and dash-3dots curve at t=200). Note the peaks of the electric field located at the position of the steep electron density at the wave-front plasma-edge interface. Also the position of this steep electron density edge and the steep ion density do not coincide as in the previous section. The velocity of this edge taken from Figure (22) is equal 0.02 (calculated following the edge motion), in good agreement again with what is calculated from Eq.(21) with the present parameters. Note that at this stage we might still be evolving towards the steady state. The two peaks at the right of the figure represent the expanding ions at t=180.48 and t=200. The sharp shock-like structure at the right edges of these peaks are propagating at a speed of 0.06, and we note a tendency of the electron density profile to follow the corresponding density profile of the expanding ions. Figs.(23) give the electron distribution function contour plots at: times t=78.96, 90.24, 124.08, 146.64 and 200. Figs.(23a-e) show how electrons are ejected backward, and then under the effect of the incident and reflected waves form vortical structures with steep edges (see Figs.(19a-c)). Their density however, with the present set of parameters, remained low and had little effect on the propagation of the incident or the reflected waves. Also electrons are ejected and rapidly accelerated in the forward direction. Figs.(23b-c) show also an important modulation of the electron bulk propagating forward. Electrons are bouncing back at the right edge (Figure (23d)). Figure (23f) gives the same result at t=200 as in Figure (23e), but concentrating on the region of the bulk. Figs.(24) give the ion distribution function contour plots at times t=78.96, 146.64, 180.48 and 200. Figs.(24a-b) show the acceleration of the ions at the edge of the plasma, and Figs.(24c-d) show the expanding ion population propagating to the right, which spreads on a much wider velocity than in the previous section, and forms the shock-like structures we see at the right in Figure (22).

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Figure 19a. Electron (full curve) and ion (dash curve) density profile at t=78.96.

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Figure 19b Electron (full curve) and ion (dash curve) density profile at t=90.24.

Figure 19c. Electron (full curve) and ion (dash curve) density profile at t=112.8.

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Figure 19d. Electron (full curve) and ion (dash curve) density profile at t=124.08 (the ion peak reaches 22).

Figure 19e. Electron (full curve) and ion (dash curve) density profile at t=180.48 (the ion peak at the left reaches 12).

Figure 19f. Electron (full curve) and ion (dash curve) density profile at t=200.

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Figure 20. Density profiles at the edge for the electrons (dotted curve t=112.8, dash-dot curve t=118.44, full curve t=124.08) and ions (dash – curves, denoted respectively by a, b and c) showing the rapid build-up of the ion density peak over a time of 11.28 (the peak of curve c at t=124.08 reaches 22)

Figure 21. Electric field profile at the edge at t=78.96 (full curve), t=124.08 (dotted curve), t=146.64 (dash curve), t=180.48 (dash-dot curve), t=200 (dash-3dots curve)

Figure 22. Plot of the density at the plasma edge (full curves electrons, dash curves ions), a) t=180.48, b) t=200.The dash-dot and dash-3dots curves give the corresponding electric fields at the same time. Note at the right the steep shock-like structure, the expanding ion and the electron densities getting closer.

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Figure 23a. Phase-space plot of the electron distribution function at t=78.96.

Figure 23b. Phase-space plot of the electron distribution function at t=90.24.

Figure 23c. Phase-space plot of the electron distribution function at t=124.08.

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Figure 23d. Phase-space plot of the electron distribution function at t=146.64.

Figure 23e. Phase-space plot of the electron distribution function at t=200.

Figure 23f. Same as Figure 23e ,concentrating on the center region.

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Figure 24a. Phase-space plot of the ion distribution function at t=78.96.

Figure 24b. Phase-space plot of the ion distribution function at t=146.64.

Figure 24c. Phase-space plot of the ion distribution function at t=180.48.

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Figure 24d. Phase-space plot of the ion distribution function at t=200.



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Figure 25. Incident wave E (full curve) and reflected wave

Figure 26. Incident wave

F

(full curve) and reflected wave

E



F

(dash curve) at t=90.24.

(dash curve) at t=90.24.

Figure (25) gives the incident E  wave (full curve) and the reflected E  wave (dash curve) at t=90.24. The incident wave front is reaching x=20 at the steep edge, decaying first sharply between x=20 and x=22, then more smoothly between x=22 and x=26 (see the edge profile in Figure (19b)). Note the modulation of the incident wave around x=25, corresponding to the density modulation at the edge in Figure (19b). The same behaviour is

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observed on the incident wave F  at t=90.24 in Figure (26) (full curve). In Figs.(27,29) we present the incident wave E  (full curve) and the reflected wave E  (dash curve) at t=112.8 and t=200 respectively, and the corresponding curves for F  and F  are presented in Figure (28,30). Note in Figure (29) the rapid damping of the penetrating wave due to the fact that the plasma is more opaque at the edge due to the sharp increase of the edge density. The penetration is of the order of c /  pe , and the remaining of the wave profile inside the plasma remains essentially flat. A similar behaviour is presented in Figure (30) for the incident wave

F  and the reflected wave F  , where the reflected wave at reflection is in phase with the incident wave. We can verify that the reflected wave (dash curve) has a wavelength (and a period) slightly bigger than the incident wave wavelength (and period). So the frequency of the reflected wave is slightly downshifted with respect to the frequency of the incident wave, due to the Doppler shift by the moving reflecting surface. Using Eqs.(22-23) with  F  ui  0.02 and    / k  1 , we have for the frequency and wavenumber of the reflected wave

r  0.96  0.192 (which is verified from the frequency spectrum of the

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reflected wave),and k r  0.96k  0.192 (which is verified following the peaks of the reflected wave in Figure (30d)).



Figure 27. Incident wave E (full curve) and reflected wave

Figure 28. Incident wave

F

(full curve) and reflected wave

E



F

(dash curve) at t=112.8.

(dash curve) at t=112.8.

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

Figure 29. Incident wave E (full curve) and reflected wave

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Figure 30. Incident wave

F

(full curve) and reflected wave

E

197



F

(dash curve) at t=200.

(dash curve) at t=200.

5. A Circularly Polarized Laser Wave Incident on a Moderately Overdense Plasma (n/ncr=1.6) A. Case with Ledge ~ 0 The forward propagating circularly polarized laser wave enters the system at the left hand (x=0) boundary, as previously discussed in section 3. We choose for the amplitude of the potential vector or quiver momentum a0  2.5 .   0.79 pe , which corresponds to

n / ncr  1.6 , where ncr is the critical density. The Lorentz factor for an oscillating electron in the field of the wave is

 0  1  a02  2.6925 . The initial distribution functions for

electrons and ions are Maxwellian with temperature Te=10. keV for the electrons and for the ions Ti=1. keV. The total length of the system is L  54.54 . We use N =8500 grid points in space and 1500 grid points in momentum space for the electrons and 1200 for the ions (the extrema for the electron momentum are  8 , and for the ion momentum  66.72 , momentum in units of me c ). The time-step and the spatial grid size are such that

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t  x  0.00652 . We have a vacuum region of length Lvac  11.915 on each side of the plasma. The jump in density at the plasma edge on each side of the flat top density takes place over a length Ledge  10.27 (in the present case the free space wavelength

0  2 / 0.79  7.953 , so Ledge ~ 0 ). The length of the central plasma slab with flat top density of 1 is L p  10.18 . So we consider in this section a relatively short slab, and we

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concentrate on the electronic and ionic population which are accelerated in the backward direction. Figs.(31) show the plot of the density profiles against distance (full curves for the electrons and dash curves for the ions) at times t=44.94, 64.2, 89.8, 102.72 and 128.4. Note in Figure (31a) at t=44.94 the electrons (full curve) move first as usual, pushed by the pondromotive force.The density profile for the electrons is steepening at the wave front. We have in the present case an electron population from the uderdense region moving in the backward direction, and then under the combined effect of the incident and reflected waves forms vortical structures having steep edges or spikes (see Figs.(33a-b) below). Figure (31b) indicates that the steepening of the edge electron density profile is accentuated, and the electrons from the underdense region accelerated backward are now forming two vortices, the structure of the first vortex immediately at the left of the steep edge front is showing a sawteeth-like structure. The penetration of the wave or transparency in the present case has little effect on the density profile beyond the point where we have the steep edge, even though n /  0 ncr  0.5937 . The sharp increase in the electron density peak at the plasma edge will make the plasma more opaque. Figs.(31d-e) at t=102.72 and t=128.4 show the complexity of the vortical structures with steep edges which are developing, with a series of over critical density spikes. The wave is now traveling in a region filled with plasma before the wave front reaches the plasma edge. Figure (32) shows the longitudinal electric field at times t=44.94 (full curve), 64.2 (dotted curve), 89.8 (dash curve), 102.72 (dash-dot curve), 128.4 (dash3dots curve). The electric field peak is concentrated at the wave-front plasma-edge interface, and the penetration of the electric field in the plasma is of the order of about 2 c /  pe . Figs.(33) present the contour plots of the electron distribution function in phase-space. In Figure (33a) at t=44.94 electrons are accelerated in the backward direction at the wave-front plasma-edge interface, and then under the cross effect of the incident and reflected waves they form a vortex structure having steep edges, and a cavity-like structure (see Figure (31a)). We note a small bump penetrating in the plasma bulk in the forward direction. (Note that at the extreme right boundary, a small filament of electrons ejected to the right seems independent from the physics at the wave-front plasma-edge interface we are studying). Figure (33b) shows the contour plot at t=54.57 (note the difference in the vertical scale with respect to Figure (33a)). In Figs.(33c-d) at t=57.78 and 64.2 the motion of the ejected electrons in the backward direction is under the cross effect of the incident and reflected electromagnetic waves, and also of the self-consistent longitudinal electric field resulting from the motion of the electrons and the ions in the backward direction. In Figure (33c) the electrons ejected backward travel to the point indicated by the arrow, before reflecting back to the vortex. In a similar way in Figure (33d), there is a tendency for the ejected electrons to form a second vortex behind the first one, before connecting to the first vortex. In Figs.(33e-f) at t=89.8 and 115.56, there is no direct connection between the plasma edge and the vortex

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just in front it. The connection is done through the point indicated by an arrow in Figure (33f) for instance, where the backward ejected electrons are reflected back in the forward direction. Figs.(34) show the contour plots of the ion distribution function at t=44.94, 89.8, 115.56 and 128.4.

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Figure 31a. Electron (full curve) and ion (dash curve) density profile at t=44.94.

Figure 31b. Electron (full curve) and ion (dash curve) density profile at t=64.2.

Figure 31c. Electron (full curve) and ion (dash curve) density profile at t=89.8.

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Figure 31d. Electron (full curve) and ion (dash curve) density profile at t=102.72.

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Figure 31e. Electron (full curve) and ion (dash curve) density profile at t=128.4.

Figure 32. Electric field at t=44.94 (full curve), t=64.2 (dotted curve), t=89.8 (dash curve), t=102.72 (dash-dot curve), t=128.4 (dash-3dots).

Figs.(31) and (34) show a modest ion acceleration at the wave-front plasma-edge interface in both directions, and although the acceleration is relatively modest with the present set of parameters, they do contribute to the formation of the electric field at the plasma edge. Figs.(34c-d) show a modulation on the contour plot of the ion distribution function. When the plasma population becomes important in the underdense region in the backward direction, collective effects associated with the beating of the incident and reflected waves also become important.

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Figure 33a Phase-space plot of the electron distribution function at t=44.94 (plot concentrated on the central region).

Figure 33b. Phase-space plot of the elec tron distribution function at t=54.57.

Figure 33c. Phase-space plot of the electron distribution function at t=57.78.

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Figure 33d. Phase-space plot of the electron distribution function at t=64.2.

Figure 33e. Phase-space plot of the electron distribution function at t=89.8.

Figure 33f. Phase-space plot of the electron distribution function at t=115.56.

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Figure 34a. Phase-space plot of the ion distribution function at t=44.94.

Figure 34b. Phase-space plot of the ion distribution function at t=89.8.

Figure 34c. Phase-space plot of the ion distribution function at t=115.56.

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Figure 34d. Phase-space plot of the ion distribution function at t=128.4.

We present in Figure (35) a plot of the incident wave E  (full curve) and reflected wave E  (dash curve), and a similar plot in Figure (36) for the incident wave F  (full curve) and the reflected wave F  (dash curve), at the time t=128.4. There is a small damped penetration of the wave in the plasma. Figure (37) presents the frequency spectrum for the incident wave E  (full curve) and the reflected wave E  (dash curve), monitored in vacuum at a point very close to the entrance at the left boundary, at x=0.32. Figure (38) presents a similar plot for the frequency spectrum of E  and E  monitored at a point located in the plasma at x=19.9. We see in Figure (37) that close to the left boundary in the vacuum region, the frequency of the incident wave E  is peaking at Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved.

the reflected wave has a frequency peaking at

in    0.79 , and

 re  0.6 . Figure (38) monitored at x=19.9

shows the same frequency peaks as in Figure (37), plus some additional small peaks resulting from the presence of the plasma. There is a Doppler shift in the frequency of the wave reflected at the discontinuity created by the moving wave-front plasma-edge interface. Vortices are formed (see Figs.(33)) due to the beat-wave effect between the incident laser wave and the reflected wave. We can write a three wave parametric interaction in   re   a , (where  a is the frequency of an excited electron plasma wave), and the corresponding relation for the wavenumbers k in  k re  k a . From Figure (38), with

in  0.79 and  re  0.6 , we get  a  0.19 . From Figs.(35) and (36) we see

that close to the left boundary in the vacuum region the wavelength of the incident wave gives k 0  0.79 (corresponding to the incident wave with in  0.79 ), however in the plasma region close to the wave-front plasma-edge interface we notice a deformation of the incident laser wave, with k in  0.628 ( a wavelength of 10 between two consecutive peaks), and for the reflected wave k re  0.42 ( a wavelength of 15 between two consecutive peaks). This will result in a value of k a  0.628+0.42=1.048 in the plasma region, and a wavelength of

a  2 / 1.048  6 , very close to the width of the electron

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vorticies in Figs.(33). Figs.(34) show that a modulation with a similar wavelength appears on the ion distribution function. The low frequency electron wave which seems to be generated by the beating between the incident and the reflected wave is of the acoustic type and seems to verify  a  k a ui , where u i is the velocity of the discontinuity generated at the wave-front plasma-edge interface. With u i  0.17 calculated following the moving discontinuity in Figure (31), we have

 a  k a ui  1.048x0.17 = 0.178, close to the

value of 0.19 calculated above from the three waves parametric interaction.



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Figure 35. Incident wave E (full curve) and reflected wave

Figure 36. Incident wave

F

(full curve) and reflected wave

E



F

(dash curve) at t=128.4.

(dash curve) at t=128.4.

An estimation of the Doppler shift in the frequency of the wave reflected at the wavefront plasma-edge discontinuity can be obtained using Eqs.(22-23). With and  

 F  ui  0.17

in / k in  1.258 , we get from Eqs.(22-23)  re  0.781in  0.61 (close to the

value of 0.6 in the spectrum in Figs.(37,38)), and k re  0.61k in  0.388 (close to the value of 0.42 calculated above). Finally, from the dispersion relation of the incident electromagnetic

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wave in the underdense plasma written in our normalized units as in2  1. /    kin2 , we can calculate 1. /    0.2297 . We can verify for the reflected wave from the relation

re2  1. /    k re2 =0.2297+0.1764=0.406 that re  0.63 , close to the value of 0.6

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calculated above and verified in the spectrum in Figs.(37) and (38).

Figure 37. Frequency spectrum of

E  (full curve) and E 

Figure 38. Frequency spectrum of

E

(dash curve) monitored at x=0.32



(full curve) and E (dash curve)monitored at x=19.9.

B. Case with Ledge  0 The forward propagating circularly polarized laser wave enters the system at the left hand (x=0) boundary, as previously discussed in section 3. We choose for the amplitude of the potential vector a0 

2 . We keep   0.79 pe , which corresponds to n / ncr  1.6 ,

where ncr is the critical density. The Lorentz factor for an oscillating electron in the field of

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 0  1  a02  3 , and n /  0 ncr  0.924 . The initial distribution functions

for electrons and ions are Maxwellian with temperature Te=10. keV for the electrons and for the ions Ti=1. keV. These parameters are close to those used in [36,37]. The total length of the system is L  126.80 . We use N =8500 grid points in space and 1900 grid points in momentum space for the electrons and the ions (the extrema of the electron momentum are  8 , and for the ion momentum  115.25 ). The time-step and the grid size are such that

t  x  0.01492 . We have a vacuum region of length Lvac  21.425 on each side of the plasma. The jump in density at the plasma edge on each side of the flat top density takes place over a distance Ledge  23.872 , (in the present case 0  2 / 0.79  7.953 for the free space wavelength of the electyromagnetic wave and Ledge  0 ). The present case corresponds to the case where the incident laser wave is interacting with a preformed plasma.

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The length of the initial central plasma slab with flat top density of 1 is L p  36.24 . Figs.(39) show the plot of the density profiles against distance (full curves for the electrons and dash curves for the ions) at times t=67.14, 111.9, 179.04, 201.42, 268.56 and 298. The density profile for the electrons is steepening at the wave-front plasma-edge interface. An important electron population from the uderdense region is moving in the backward direction, and then under the effect of the forward and reflected electromagnetic waves are forming vortical structures having steep edges (see Figure (42a) below). We have discussed in the previous section how these vortical structures are formed. We have in the underdense region a series of spikes. Figs.(39b-d) show that electrons are also accelerated in the forward direction (see Figs.(42) below). Figs.(39c-d) show the ions accelerated in the backward direction, and at the wave-front plasma- edge interface (see Figs.(41) below). Figure (39e) shows an important spike of the ion density at the wave-front plasma-edge interface, preceded by another less important spike, with a cavity-like structure between the two spikes. The same picture is also present in Figure (39f). In Figs.(40) we concentrate the density plots around the region of the wave-front plasma-edge interface at t=201.42, 268.56 and 298. We can verify a velocity of the edge of ui =0.0305. The velocity calculated from Eq.(21) with the present set of parameters is 0.026. We note in the present case the incident wave is crossing an important underdense plasma. The dash-dot curves in Figs.(40) give the electric field. Figs.(41a-b) show the contour plots of the ion distribution function, with ions accelerating at the edge. The ions in the underdense region are expanding in the backward direction, and the ion distribution is modulated with a modulation wavelength (about  6) which appears to be the same as the wavelength of the vortical structures in the electron distribution function (see Figs.(42) below). Note in Figs.(41c-d) the steep shocklike edge. Figs.(41e-f) present the same plots as in Figs.(41c-d), concentrating around the location of the steep shock-like structure, where we note at the position of the two peaks appearing in Figs.(40) an X like structure in phase-space, and where the flow seems to be bifurcating or going in opposite direction. When the plasma population becomes important in the underdense region at the left of the edge, its effect on the wave propagation would become important. Figs.(42) give the contour plots of the electron distribution function at the same times as for the curves in Figs.(39). See in Figure (42a) how the second vortex at the left is connected to the first vortex formed in front of the steep density gradients, as

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previously discussed for A-Case in this section. There is no direct connection between the plasma edge and the vortex in front of it. The connection is taking place through the second vortex behind. There is also a strong population of electrons accelerated in the forward direction. They bounce back in Figure (42d), and then fills the phase-space.

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Figure 39a. Electron (full curve) and ion (dash curve) density profile at t=67.14.

Figure 39b Electron (full curve) and ion (dash curve) density profile at t=111.9.

Figure 39c. Electron (full curve) and ion (dash curve) density profile at t=179.04.

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Figure 39d. Electron (full curve) and ion (dash curve) density profile at t=201.4.

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Figure 39e. Electron (full curve) and ion (dash curve) density profile at t=268.56

Figure 39f. Electron (full curve) and ion (dash curve) density profile at t=298.

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Figure 40a. Plot of the density at the plasma edge (full curve electron, dash curve ion), at t=201.42 .The dash-dot curve gives the corresponding electric field.

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Figure 40b. Plot of the density at the plasma edge (full curve electron, dash curve ion) at t=268.56. The dash-dot curve gives the corresponding electric field.

Figure 40c. Plot of the density at the plasma edge (full curve electron, dash curve ion), at t=298 .The dash-dot curve gives the corresponding electric field.

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Figure 41a. Phase-space plot of the ion distribution function at t=111.9.

Figure 41b. Phase-space plot of the ion distribution function at t=201.4.

Figure 41c. Phase-space plot of the ion distribution function at t=268.56

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Figure 41d. Phase-space plot of the ion distribution function at t=298.

Figure 41e. Same as Figure 41c, around the position of the steep edge.

Figure 41f. Same as Figure 41d, around the position of the steep edge.

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Figure 42a. Phase-space plot of the electron distribution function at t=67.14 (note the difference in vertical scale).

Figure 42b. Phase-space plot of the electron distribution function at t=111.9.

Figure 42c. Phase-space plot of the electron distribution function at t=179.04.

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Figure 42d. Phase-space plot of the electron distribution function at t=201.42.

Figure 42e. Phase-space plot of the electron distribution function at t=268.56

Figure 42f. Phase-space plot of the electron distribution function at t=298.

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Figure 43. Incident wave

E

Figure 44. Incident wave

F  (full curve) and reflected wave F  (dash-curve) at t=298.

(full curve) and reflected wave

215

E  (dash-curve) at t=298

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Figure (43) shows a plot of the incident wave E  (full curve) and the reflected E  (dash curve), and a similar plot in Figure (44) for the incident wave F  (full curve) and the reflected F  (dash curve), at the time t=298. Note how the waves damp rapidly as they penetrate the plasma, and the perturbation penetrating seems to attenuate due to the increase at the edge density which makes the plasma more opaque. Figure (45) presents the frequency spectrum for the incident wave E  (full curve) and the reflected wave E  (dash curve) monitored at a point very close to the entrance at the left boundary, at x=0.746. Figure (46) presents a similar plot for the frequency spectrum of E  and E  at a point located in the plasma at x=46.25. Figure (45) shows that close to the left boundary in the vacuum at x=0.746 the frequency of the incident wave E  peaks around   0.79 (Figure (45) full curve), and the reflected wave has a broad spectrum around the peak at

 re  0.77 . However when

moving from x=0.746 to x=46.25 the spectrum of the incident wave is broadening (see full curves in Figure (45) and Figure (46)), and when moving from x=46.25 to x=0.746 the spectrum of the reflected wave is broadening (see dash curves in Figure (46) and Figure (45)). This is due to the interaction when crossing the region of the underdense plasma. We take in  0.79 , and from the observation of the incident wavelength close the wave-front plasma-edge interface in Figs.(43-44) or the Fourier spectrum we have for the incident

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wavenumber kin =0.65. From Eqs.(22,23), with   in / k in =1.185, get

  ui  0.0305 , we F

 re  0.9508in  0.751 , which is within the broad spectrum of the reflected wave in

Figs.(45). Also from Eq.(23) k re  0.9294k in =0.6 , which is what we calculate from Figs.(43,44) where the wavelength of the reflected wave appears slightly bigger than the wavelength of the incident wave close to the wave-front plasma-edge interface. The incident and the reflected waves interact in the plasma region through a three wave parametric interaction in   re   a , (where  a is the frequency of an excited electron acoustic wave), and the corresponding relation for the wavenumbers k in  k re  k a . From the previous relations we deduce

 a  0.039 and k a  1.25 , and a  2 / k a  5. , close to

the width of the vortex structures we see in Figure (42a). We can also recover this value of  a from the relation  a  k a ui  1.25x0.0305 = 0.038. From the realation in the underdense region in2  1. /    kin2 , we get 1. /    =0.2. We see also in the spectrum of the reflected wave in Figure (45) a small peak for a scattered mode at the frequency  s  0.42 . Usually this backward scattered mode is emitted close to cut-off with a frequency  s  1 / 

1/ 2

=0.447, close to the value of 0.42 observed in Figure (45). If we

consider in the spectrum of the reflected wave in Figure (45) the peak at 0.7, we have 0.70.42=0.28 which is close to the peak observed in Figure (45) at 0.26. The two cases presented in this section in A-Case and B-Case illustrate the importance of the factor

0 / Ledge on the results, and hence the importance to treat the evolution of the

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edge self-consistently for accurate results. See also the results in [38]. This is in addition to what we have already found in the previous sections on the important role played by the dynamic of the ions.

Figure 45. Frequency spectrum of

E  (full curve) and E 

(dash curve)monitored at x=0.746.

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Figure 46. Frequency spectrum of

E

217



(full curve) and E (dash curve)monitored at x=46.25.

6. A linearly Polarized Laser Wave Incident on an Overdense Plasma (n/ncr=25): Harmonics Generation The forward propagating linearly polarized laser wave enters the system at the left hand (x=0) boundary, where the forward propagating field is E   2E0 Pr (t ) cos(t ) and

F   0. The laser beam is now linearly polarized. The shape factor Pr (t )  sin(t /(2 )) for t    50 , Pr (t )  1 otherwise. We choose for the amplitude of the potential vector

a0  8 2 , and for the linearly polatized wave we have

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2 0

2 0

microns. We take

18

  0.2 pe , which corresponds to n / ncr  25 , where ncr is the critical

density . The Lorentz factor for an oscillating electron in the field of the wave is

 0  1  a02  129 . The initial distribution functions for electrons and ions are Maxwellian with temperature Te=1. keV for the electrons and for the ions Ti=0.1 keV. The total length of the system is L  125.66 . We use N =7000 grid points in space and 2200 in momentum space for the electrons and the ions (the extrema of the electron momentum are  5 , and for the ion momentum  131.87 ). We use a time-step and a grid size such that

t  x  0.01795 . We have a vacuum region of length Lvac  40.98 on each side of the plasma. The jump in density at the plasma edge on each side of the flat top density is of length Ledge  1.795 , (0  2 / 0.2  31.41  Ledge ) , and the length of the central plasma slab with flat top density of 1 is L p  40.12 . Figs.(47) show the plot of the density profiles against distance (full curves for the electrons and dash curves for the ions) at times t=107.7, 125.65, 143.6, 161.55, 197.55, 215.4, 305.15 and 350. At t=107.7 there is an important build-up of the electron density at the left edge in front of the incident wave. The position of this electron density steep edge remained

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essentially around x  44 during the simulation, and does not propagate, but rather shows a small oscillation around x  44 . Figure (48a) presents the same result at t=107.7, concentrating on the plasma edge region. Initially electrons move first under the influence of the ponderomotive pressure, and there is an important build-up of electron density at the edge with an associated electric field due to the charge separation (dash-dot curve in Figure (48a)).

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Figure 47a. Electron (full curve) and ion (dash curve) density profile at t=107.7

Figure 47b. Electron (full curve) and ion (dash curve) density profile at t=125.65

Figure 47c. Electron (full curve) and ion (dash curve) density profile at t=143.6

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Figure 47d. Electron (full curve) and ion (dash curve) density profile at t=161.55

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Figure 47e. Electron (full curve) and ion (dash curve) density profile at t=197.45.

Figure 47f. Electron (full curve) and ion (dash curve) density profile at t=215.4.

Between t=125.65 ant t= 161.55 in Figs.(47), i.e. over a time of 35.9, the ions show a sharp and steep increase of their density at the edge, under the influence of the longitudinal electric field and the ponderomotive pressure. Figs.(48b,c) present the same density plots concentrating on the edge region at times t= 143.6 and 161.55, to follow the rapid build-up of the ion density peak, showing the rapid build-up of a thin solitary-like structure.

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Figure 47g. Electron (full curve) and ion (dash curve) density profile at t=305.15.

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Figure 47h. Electron (full curve) and ion (dash curve) density profile at t=350.

This thin ion peak moves slowly forward. The dash-dot curves in Figs.(48) are the electric fields at the corresponding times.Then in Figure (47f) and the corresponding one in Figure (48e) three thin ion peaks are appearing. The motion of the ion peak close to x  42.5 is negligible, while the other two peaks move slowly, the right peak forward and the left peak backward. In Figs.(48g-l) we show the profiles at the edge at close time intervals, so that we can follow the oscillations of the steep electron density profile at the edge, which is oscillating around x  44 . We can check from this selection of figures a period of about 18 for this oscillation (a finer resolution would have been desirable, we are plotting every 500 time-steps). This is close to half the period of the incident wave which is 2 / 0.2  31.4 , i.e. the ponderomotive force is acting on the surface, and oscillates at twice the frequency of the laser. We see the same period of oscillation if we follow the peak of the electric field (dash-dot curves) appearing just at the steep electron density edge in Figs.(48g-l). Between the ion peak at x  42.5 and the ion peak next to it, there is a cavity like structure in the ion density, partially filled with an electron density bump showing the steep profile at x  44 . We plot in Figs.(49) the profile of the electric field in the plasma region, at t=296.17 (full curve), 305.15 (dash-curve), and 314.12 (dash-dot curve) (these are the same as the electric field profiles in Figs.(48i-k)). In the peak of the electric field at the position of the steep electron density profile we note the same oscilation we mentioned with a period of about 18 (here again, a finer resolution would have been desirable). We note however in Figure (49)

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that although the electric field profile is oscillating around the position of the edge, it keeps a constant steady-state profile around x  51 , changing rapidly from negative to positive value. This corresponds in Figure (53c) to the position where we note a bifurcation in phase space in the form of X. The negative and positive electric fields at this location are pushing the ions in opposite directions, resulting in the appearance of the X like structure which we see in Figure (53).

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Figure 48a. Electron (full curve) and ion (dash curve) density profile, and electric field (dash-dot curve) at t=107.7.

Figure 48b. Electron (full curve) and ion (dash curve) density profile, and electric field (das-dot curve) at t=143.6.

Figure 48c. Electron (full curve) and ion (dash curve) density profile, and electric field (dash-dot curve) at t=161.55.

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Figure 48d. Electron (full curve) and ion (dash curve) density profile, and electric field (das-dot curve) at t=197.45

Figure 48e. Electron (full curve) and ion (dash curve) density profile, and electric field (dash-dot curve) at t=215.4.

Figure 48f. Electron (full curve) and ion (dash curve) density profile, and electric field (das-dot curve) at t=242.32.

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Figure 48g. Electron (full curve) and ion (dash curve) density profile, and electric field (dash-dot curve) at t=278.22.

Figure 48h. Electron (full curve) and ion (dash curve) density profile, and electric field (das-dot curve) at t=287.2.

Figure 48i. Electron (full curve) and ion (dash curve) density profile, and electric field (dash-dot curve) at t=296.17.

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Figure 48j. Electron (full curve) and ion (dash curve) density profile, and electric field (dash-dot curve) at t=305.15.

Figure 48k. Electron (full curve) and ion (dash curve) density profile, and electric field (dash-dot curve) at t=314.12.

Figure 48l. Electron (full curve) and ion (dash curve) density profile, and electric field (dash-dot curve) at t=350.

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Figure 49. Electric field profile at t=296.17 (full curve), t=305.15 (dash-curve) and t=314.12 (dash-dot curve).



Figure 50a. Incident wave E (full curve) and reflected wave

Figure 50b. Incident wave

E

(full curve) and reflected wave

E



E

(dash curve) at t=107.7.

(dash curve) at t=161.55.

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

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Figure 50c. Incident wave E (full curve) and reflected wave

Figure 50d. Incident wave

E

(full curve) and reflected wave



E



E

(dash curve) at t=305.15.

(dash curve) at t=314.12.

Figure 51. Spectrum of the incident wave E (full curve) and the reflected wave monitored close to the entrance of the domain at x=0.9.

E

(dash curve),

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Figure 52a. Phase-space plot of the ion distribution function at t=107.7.

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Figure 52b. Phase-space plot of the ion distribution function at t=161.55.

Figure 52c. Phase-space plot of the ion distribution function at t=215.4.

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Figure 52d. Phase-space plot of the ion distribution function at t=350.

Figure 53a. Same as Figure 52b, t=161.55.

Figure 53b. Same as Figure 52c, t=215.4.

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Figure 53c. Same as Figure 52d, t=350.

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Figs.(50) present the incident electromagnetic wave E  (full curves) and the reflected wave E  (dash curves) at t=107.7, 161.55, 305.15 and 314.12. In Figure (50a) there is a small penetration of an important perturbation at the plasma edge (full curve for the incident wave). But as the wave penetrates, the arriving wave at the plasma edge are more and more damped, due to the build-up of the plasma density at the edge, which makes the plasma more opaque, and consequently the perturbation penetration is strongly damped. This is clear in Figs.(50c,d), where the incident electromagnetic wave appears to damp strongly at the position of the oscillating steep electron edge at x  44 . Since this edge is not propagating we do not observe a Doppler shift in the fundamental frequency of the reflected wave, but perhaps a modulation of the phase of the reflected wave due to the oscillation of the reflecting surface around x  44 . The reflected wave in Figs.(50) (dash curves) shows a complex periodic structure, which is the signature of the presence of harmonics. The relativistic oscillation of the electron density surface in the field of the high intensity wave is nonlinear, and can thus generate harmonics (similar to the relativistic oscillating mirror ROM [39], see also the review article in [40]). Note in Figure (50c) the incident wave E  is minimum at the plasma edge, which explains the electron density edge relaxing to the left in Figure (48j), because of a minimum in the incident wave pressure. In Figure (50d) the incident wave E  is close to the maximum at the edge, which explains the electron density edge pushed forward to a steep gradient in Figure (48k), with a peak in the longitudinal electric field at the electron density edge, because of a maximum in the incident wave pressure. This is in contrast with the results obtained in the previous sections, where for a circularly polarized wave the steep density edge is continuously pushed in the forward direction. For a circularly polarized wave, when the incident wave E  is minimum at the edge, the other incident wave F  is maximum, and vice-versa (Figs.(10,11)). So there is always an incident wave exerting pressure on the electron density profile at the edge, which causes this edge to move always in the forward direction. Figure (51) presents the frequency spectrum of the incident wave E  (full curve, with a peak at   0.2 ), and the frequency spectrum of the reflected wave (dash curve) (both monitored at x=0.9 at the entrance of the domain). The spectrum of the reflected wave shows

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the peak at   0.2 , and the odd harmonics of this frequency ( 3, 5, 7, 9... ). This presence of odd harmonics at normal incidence of the laser wave is due to relativistic effects as discussed in several publications [40-43]. Figs.(52) present the contour plots of the ion distribution function. It shows at the beginning the ions accelerating at the edge. At t=161.55, Figure (52b) shows a bifurcation in the form of X in the ion motion along the vertical steep line , which we amplify in Figure (53a) (simlar to what we observe in Figs.(41e,f)), corresponding to the huge jump in the ion density peak we see in Figure (48c). Figs.(52c-d) show how this bifurcation increases with time, which is magnified in Figure (53b-c). Figs.(54) show the contour plots of the electron distribution function. In Figure (54a) electrons are accelerated forward, and have a steep profile at the wave-front plasma-edge interface, which is maintained in Figs.(54b-d), with a small oscillation of the surface during the evolution of the system. We amplify in Figure (55) the region of the front edge at t=350 (taken from Figure (54d)), showing the complex vorticity-like structures which have developed. Figure (56) shows on a logarithmic scale the plot of the distribution function Fe( x  53, p xe ) obtained from Figure (55) at t=350 by making a cut at x=53. It shows a

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plot having a relatively flat profile in the central region, with steep edges around. The first vortex in Figure (55) corresponds to the electron bump appearing next to the steep edge at x  44 in Figure (48l), and located between two ion density peaks as in Figs.(48g-l).

Figure 54a. Phase-space plot of the electron distribution function at t=107.7.

Figure 54b. Phase-space plot of the electron distribution function at t=161.55.

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Figure 54c. Phase-space plot of the electron distribution function at t=215.4.

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Figure 54d. Phase-space plot of the electron distribution function at t=350.

7. Conclusion In the present chapter we have used an Eulerian Vlasov code for the numerical solution of the 1D relativistic Vlasov-Maxwell equations to study the interaction of a high intensity laser beam normally incident on the surface of an overdense plasma. In this interaction the electrons are pushed by the ponderomotive pressure of the incident wave, forming a steep gradient at the wave-front plasma-edge interface. This generates a charge separation and an electric field at the plasma edge which accelerates the ions. Depending on the ratio of n / ncr and of 0 / Ledge , electrons and ions can sometimes be accelerated in both the forward and backward directions. All the results presented point to the importance of the ion dynamics at the wave-front plasma-edge interface in the evolution of the system. There is a phase where the ions go through a very rapid build-up of a solitary like structure of the ion density at the plasma edge, which still needs to be further investigated in more details (see [44] for instance, for a discussion on the efficient acceleration of ions in intense laser-plasma interactions). We are far from the picture of heavy immobile ions at the plasma edge which act to prevent the expansion of the electrons [24,36,45].

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Figure 55 Same as in Figure 54d, concentrating on the edge region, t=350.

Figure 56. Electron distribution function

Fe( x  53, p xe ) , obtained from Figure 55 by making a cut

at x=53.

The results obtained differ substantially in several aspects when circular or linear polarization for the incident laser wave is considered (see also [19,46]). Consider for instance a

linearly

polarized

wave:

 E  (0, E y ,0) , we can write a linear analysis

E y  E0 cos( ) ,  (kx  t ) . Faraday’s law is:  E y B  (0,0, ) t x

 B  (0,0, Bz ) with Bz  B0 cos( ) , and B0  E0 k /  . From      E  a / t and p  a , we get p  (0, p y ,0) with p y   p0 sin( ) , and Then

1 p0  E0 /  . The longitudinal Lorentz force is p y Bz   kp02 sin( 2 ) . This drives a 2

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longitudinal response at the 2nd harmonic of the laser wave. In a circular polarization we have



for the laser wave in a linear analysis E  E0 (0, cos( ), sin( )) . From Faraday’s law:

 E y E B  (0, z , ) t x x 



which gives B  B0 (0, sin( ), cos( )) and p  p0 (0, sin( ), cos( )) . Thus we see

 





that pxB is identically zero, p and B being parallel. So in this case there is no 2nd harmonic longitudinal response to the leading order. As discussed in section 6, in the case of a linear polarization, when the incident wave

E  is maximum at the plasma edge, it pushes the edge of the plasma, and when E  is minimum the plasma edge relaxes in the absence of an incident wave pressure. Since there is two minima of the Lorentz force in a wave cycle, then the resulting oscillation of the electron density edge takes place at the harmonic of the wave frequency. With a normal incidence of the laser pulse and with the set of parameters used in section 6, odd harmonics have been generated in the reflected wave [40-43]. The intense linearly polarized laser pulse interacting with a near discontinuous plasma-vacuum interface causes the electron density surface to perform relativistic oscillations, with a frequency equal to twice the laser wave frequency, (see Figs.(48g-l)). Figs.(48), and the phase-space plots in Figs.(52-55) show the complexity of the structures generated at the wave-front plasma-edge interface. For a circularly polarized wave, when the incident wave E  is minimum at the edge, the other incident wave F  is maximum, and vice-versa. So there is always an incident wave exerting pressure on the steep electron density at the edge, which causes this edge to move in Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved.

the forward direction. The case under the condition

0  Ledge and n / ncr  1 is of

particular interest. Figs.(4,7) show a steep density gradient with a shock-like structure for the electrons at the wave front edge, while the ion density profiles in Figs.(4) show the formation of sharp solitary-like structures where the acceleration of the ions takes place. There is a very small density population of ions being left behind to the left of the wave-front plasma-edge interface. The moving steep plasma edge acts as a moving mirror reflecting the incident wave. Following the ion acceleration phase, there is a fraction of the ions which reaches a free streaming expansion state in the forward direction, with the electrons compensating the charge of the expanding ions. This planar free expansion phase of a neutral plasma bunch moving in the forward direction at the right boundary is associated also with the formation of shock-like structures and sharp ion fronts with ion peaks or spikes in the local ion density (called bunching [31]), as observed in the present simulation. The loop at the top of the phasespace in Figs.(6d-f) originates from the ballistic evolution of the expanding ions. There is qualitative agreement between some phase-space features presented in Figure (6e-f) and recent PIC results (see Figure (4) of [20], and also some of the plots in [19]). Laser-driven ion beams have properties which differ in several aspects from beams of comparable energy obtained from conventional acceleration techniques, and have the potential to be applied in a number of innovative applications in the scientific, technological and medical areas.

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Magdi Shoucri The results point to the importance of the ratio

0 / Ledge on the physics involved in the

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interaction of the incident laser pulse with the overdense plasma. For moderately overdense plasma, we have important electron and ion populations coming from the underdense region which are accelerated in the backward direction, in addition to the population accelerated in the forward direction. The backward accelerated electrons are under the influence of the radiation pressure of the incident wave and also of the reflected wave, and show the formation of vortical structures with spikes. When the density of these vortical structures increases, they have important effects on the interaction between the incident and reflected waves, leading to collective modes excited in the plasma created in the backward direction. Discussion of two and three-dimensional effects is beyond the scope of the present work, although the references we have cited contain relevant discussions and references to simulations with higher dimensionality. Recent results [12-15] have shown the generation of high intensities attosecond pulses, combining short wavelengths and very high time resolution, which open applications to new fields in physics. We mention in addition the recent works in [40,47] which contain a review on the problem of harmonics generation, and the work in [48]. Since the original publications in [49], Eulerian Vlasov codes applying the method of characteristics have been successfully applied to study several problems in plasma physics, especially problems associated with wave-particle interactions. Interest in Eulerian grid-based solvers associated with the method of characteristics for the numerical solution of the Vlasov equation arises from the very low noise level associated with these codes, which allows accurate representation of the low density regions of the phase-space [16-18,28]. We have presented results showing detailed representation of the phase-space structures associated with the problem of the interaction of high intensity laser beams with an overdense plasma. For higher dimensionality Eulerian codes we refer the reader to [17,18,27,28] for instance, to the different chapters of this book, to the recent references in [50,51], and to the work in [52] which presents a 2.5D adaptive PIC-Vlasov hybrid method.

Acknowledgments The tutoring, guidance and fruitful discussions with Professor G. Knorr on Eulerian Vlasov codes are gratefully acknowledged. During the course of a career which made this work possible, the constant support and encouragement of the late Dr. Claude Richard, of Dr. Guy Bélanger and Dr. André Besner are also gratefully acknowledged. My gratitude to all of them is as great as it should be. The author is also grateful to the Centre de calcul scientifique de l’IREQ (CASIR) for computer time used to do part of this work.

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[49] [50] [51] [52]

Magdi Shoucri Mora, P. Phys. Rev. Lett. 2003, 90, 185002-(1-4) Robinson, A.P.L., Zepf, M., Kar, S. et al. New J. Phys. 2008, 10, 013021-(1-13) Denavit, J. Phys. Rev. Lett. 1992, 69, 3052-3055 Guérin, S., Mora, P., Adam, J.-C., Héron, A., Laval, G. Phys. Plasmas 1996, 3, 2693 Ghizzo, A., DelSarto, D., Réveillé, T., Besse, N., Klein, R. Phys. Plasmas 2007, 14, 062702-(1-14); DelSarto, D., Ghizzo, A., Réveillé, T., Besse, N., Bertrand, P. Comm. Nonl. Sc. Num. Simul. 2008,13, 59-64 Berezhiani, V.I., Gacuchava, D.P., Mikeladze, S.V., Sigua, K.I., Tsintdadze, N.L., Mahajan, S.M., Kishimoto, Y., Nishikawa, K. Phys. Plasmas 2005, 12, 062308 Charboneau-Lefort, M., Shoucri, M., Afeyan, B. Proc. 35th EPS Conf. Plasma Phys., Hersonissos, Greece, 2008 ECA Vol. 32, P-5.128 Baeva, T., Gordienko, S., Pukhov, A., Phys. Rev. E 2006, 74, 046404 Teubner, U., Gibbon, P. Rev. Mod. Phys. 2009, 81, 445-479 Bulanov, S.V., Naumova, N.M., Pegoraro, F. Phys. Plasmas 1994,1,745 Lichters, R., M. ter Vehn, J.M., Pukhov, A. Phys. Plasmas 1996, 3, 3425 Wilks, S.C., Kruer, W.L., Mori, W.B. IEEE Trans. Plasmas Sc. 1993, 21, 120 Esirkepov, T., Borghesi, M., Bulanov, S.V., Mourou, G., Tajima, T. Phys. Rev. Lett. 2004, 92, 175003-(1-4) Ruhl, H., Mulser P. Phys. Lett. A 1995, 205, 388-392 Liseikina, T.V., Macchi, A. Appl. Phys. Lett. 2007, 91, 171502-(1-4) Lavocat-Dubuis, X., Matte J.P. Phys. Rev. E 2009, 90, 055401-(1-4) Rykovanov, S.G., Geissler, M., Nomura, Y., Hörlein, R., Dromey, B., Schreiber, J., M. ter Vehn, J., Tsakiris, G.D. Proc. 35th EPS Conf. Plasmas Phys., Hersonissos, Greece , 2008, ECA Vol.32, P-1.145 Cheng, C.Z., Knorr, G. J. Comp. Phys. 1976, 22, 330-351; Shoucri, M., Gagné, R. J. Comp. Phys. 1977, 24, 445-449; ibid. 1978, 27, 315-322 Sircombe, N.J., Arber, T.D. J. Comp. Phys. 2009, 228, 4773-4788 Crouseilles, N., Latu, G., Sonnendrücker, E. J. Comp. Phys. 2009, 228, 1429-1446 Batishchev, O.V., Batishcheva, A.A., Zhang, J. ‘ 2.5D Adaptive Mesh PIC-Vlasov Hybrid Method for Laser-Matter interactions in the Presence of Strong Gradients’, Proc. 18th Inter. Conf. Numer. Simul. Plasmas (September 7-10, 2003, Cape Cod, U.S.A.), p.387-390

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In: Eulerian Codes for the Numerical Solution … Editor: Magdi Shoucri, pp. 237-315

ISBN: 978-1-61668-413-6 © 2010 Nova Science Publishers, Inc.

Chapter 7

SEMI-ANALYTICAL ADAPTIVE VLASOV - FOKKERPLANCK - BOLTZMANN METHODS Oleg V. Batishchev Northeastern University, USA / Moscow Institute of Physics and Technology, Russia

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Abstract In this chapter we are considering purely kinetic ionized gas descriptions, which are more general than the fluid models. Particle (PIC) and continuous (Vlasov) methods are compared for problems with and without binary collisions. Coulomb collisions are considered using statistical Monte-Carlo and deterministic Fokker-Planck phase space grid approaches. Where possible the partial analytical solutions are used to simplify numerical integration. Several benchmarking tests are considered along with few problems of practical interest. Extension to many dimensions, adaptivity in time-space, and application to the collisional rarified gas flows are discussed as well.

1. Introduction Ionized matter is a quantum N -body system: atoms and ions with complex electronic structure interact with free electrons, photons and each other, constantly changing energy states. Charged particles dynamics is affected by internal and applied electromagnetic fields. The physical picture can be further complicated by molecular and surface effects. Thus, general description of dense plasmas requires a complex blend of quantum electrodynamics [1] and classical fields theory [2]. However, it is drastically simplified in case of ionized gases, when i) the system's volume X 3 , normalized to the atomic scale, is much bigger than the inter-particle distance X  N 1 / 3 , and ii) the number of particles is large enough, to allow replacing the microscopic discrete N -particle distribution function (DF), f N , with a macroscopic continuum DF, f s , of a finite number of energy states, S , commonly called plasma species:

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Oleg V. Batishchev S   N (t )       f N (t , r , p)    (r  ri (t )) (v  pi (t ))   f s (t , r , p)  0 i 1

s 1

(1.1)

The replacement of a discrete function with a continuous function is justified in the kinetic theory [3] by using a set of coupled statistically averaged integro-differential kinetic equations linking k -particle with k  1 -particle DF, closed at the some K th level. Modeling such partial set is complicated by the highest DF dimension, i.e. 6K . Note that presently accurate mesh covering of the six-dimensional phase space 6  R3  P3 is computationally demanding. Truncating the BBGKY hierarchy at just K  1 yields the Boltzmann kinetic equation [4]: 

S

f s   I sq

(1.2)

q 1

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where the summation is done over partial Boltzmann collisional integrals, I sq , representing binary collisions of s - species with q -species. Eq. (1.2), if accounting for conservation of momentum and energy, is 8-dimensional in velocity sub-space. This makes any "continuous" modeling challenging, and explains why Monte-Carlo (MC) method remains in use for rarefied flows despite the facts that it never gives a converged solution and the computer power increased billions times since 1947 [5]. The Boltzmann integral is an approximation, which does not describe important multibody collisions, such as three-body recombination, and long-range interactions, including Coulomb collisions. The differential cross-section is known with limited certain accuracy from experimental and idealized theoretical computations. Therefore, reasonable models are used in practice for comparable-mass species, e.g. hard sphere model, BGK collisional term [6], or higher-order expansions [7]. For electron collisions with heavy particles the infinitemass ratio approximation is often used [8]. For charged particles, the left part of Eq. (1.2) reads as:

 df s f f v  B  f s  s  v s  es  E   dt t r c  p 

(1.3)

  here velocity v  p / m , mass m  ms ,   (1  v 2 / c 2 ) 1 / 2 is a relativistic factor, ms and es  eZs are the mass at rest and electric charge ( Z s is the charge number) of s - species, e is

the elementary charge, and c is the speed of light in vacuum. The electromagnetic (EM) 



fields in Lorentz force are superposition of applied and internal electric E and magnetic B fields. The latter are calculated from Maxwell equations using plasma charges and currents:

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Semi-Analytical Adaptive Vlasov - Fokker-Planck - Boltzmann Methods    1 E   4π   B   es  f s v dp c t c s      1 B    E ;   B  0 c t      E  4  es  f s dp

.

(1.4)

s

The "simplest" closed set of equations that describes self-consistent evolution of a plasma system is obtained from Eqs. (1.2)-(1.4) assuming i) absence of short-range collisions,    I sq  0 , ii) classical dynamics, p  ms v , and iii) potential approximation, B  0 , neglecting the displacement and internal currents. It is a non-linear system of Vlasov [9] and Poisson equations:   f s  f s es  v  B0 f s v   (E  )  0 t r ms c v (1.5)   2  4 e  Z s  f s dv s

  where Φ is the electric field potential, E  Φ , and B0 is the applied magnetic field. System (1.5) captures important collective phenomena [10] in collisionless plasma: Langmuir waves, beam-plasma instabilities, Landau damping, etc. However, the time step for





1 1 , f ce , a minimum between Eq.(1.5) integration is limited by the "natural" time scale, min f pe



the period of electron plasma oscillations with plasma frequency f pe  n p e2 / me



1/ 2

, and

period of electron gyration with gyrofrequency fce  eB0 / 2mec . The spatial resolution of Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved.

the computational domain is bounded by the intrinsic scale, min D , re  , a minimum between



the Debye length, D  kTe / 4n p e2



1/ 2

, and electron gyroradius, re  2VTe / fce [11]. Here

n p is plasma number density; ne , Te , me and VTe  kTe / me 1/ 2 are the electron density,

temperature, mass and thermal velocity, respectively; k is the Boltzmann constant. In the laboratory plasma devices f ce can reach GHz and f pe PHz level. The electron gyro-radius can be a few microns, while the Debye length can be sub-µm. Modeling of a cubic meter system evolution for a millisecond of physical time will require ~ 10 40 operations, which is presently impossible. The relatively rarified and hot common experimental plasmas at ne  108  1014 cm3 and Te  1 100eV is ND 

4ne 3D

near-ideal

with

large

number

of

charges

per

Debye

sphere,

 10  10  1 . Substantial uncompensated space charge can only occur on 2

5

D scale at the wall sheath [12] and double layers [13] due to exponential Debye shielding. On physically large scale, X  D , the plasma can be considered quasineutral, ne  Z i ni , where ni is ion density and Zi  1 is an average charge number. Indeed, the charge density,

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Oleg V. Batishchev

,

misbalance

is

small,

 / en p  ne  Z i ni / ne  10 5 ,

for

typical

potential

gradients   100V / cm . In such situation the macroscopic electric field has to be calculated through collisional plasma transport rather from the Poisson equation. The Boltzmann integral cannot be applied due to its formal divergence for the coulombic interactions. This problem is resolved with logarithmic accuracy by the Landau integral term [14], Csq , which assumes small momentum transfer in a low-angle   1 scattering collisions. The term removes i) singularity at zero by excluding large-angle scattering and ii) divergence at infinity by applying plasma Debye shielding. The kinetic theory analysis [3] has shown that the reason for the short-distance force regularization, i.e. finite electron deBroglie length,  e  (me kTe ) 1/ 2 (  is the Planck constant) or    / 2 deflections, is unimportant as far as the Coulomb logarithm is sq sq / a min )  1 . For equilibrated electron-ion plasma, the upper sufficiently large,  sq  ln( a max

distance amax  D , while the lower, amin , is the maximum of  e or the classical distance of minimum electron-ion approach,

Z i e 2 / kTe . Typically  ei ~  ee  10 . Thus, Coulomb

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plasma transport coefficients are accurate within O(1/ )  10% . However, the theory fails when  ei ~ 1 , which is not uncommon for dense plasmas, e.g. in inertial fusion. Other case of inapplicability include i) very strong B when electron gyroradius becomes small, e , ii) electron-ion repetitive collisions in oscillating electric field [15], iii) complete re  rmax ionization of high-Z solid-density material by ultrafast relativistic pulses [16]. Thus, mathematically, the system in Eq.(1.5), by assuming for simplicity absence of nonCoulomb collisions and external magnetic field, is replaced with the following system of equations: f s  f s es  f s v   E A    Csq t r ms v q   (1.6) E A  E ( f s1, f s 2 ,...)   es  f s dv  0 s

 Here E A is the averaged macroscopic field, which is calculated from distribution functions of all plasma species. One possibility is to obtain it from the zero current condition,    j   es  v f s dv  0 . The charges are not accumulating inside the plasma, which cancels the s

averaged electrical currents due to plasma ambipolarity. A short comment should be included here on the widely used fluid description of plasma and gas flows. It is valid in the limit of short mean free path (mfp)  :

  L

(1.7)

where L is a spatial scale. For gas flows the neutral particle mfp, N  1/ N NN , is often viewed to be constant; N is gas density and  NN  d 2 a collisional cross-section of neutral

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molecules with effective “hard sphere” diameter d [17]. In case of internal gas flow in a pipe of radius R one can rewrite inequality (1.7) in terms of dimensionless Knudsen number [18], Kn N   N / R  1 . In reality the Knudsen number is never a constant, as the density and cross-section are functions of phase space. The latter is very important for plasma, particularly Coulomb scattering, when  C  v 3 . The spatial scale  of a flow function F (positively defined, as density, temperature or flux):   ln F  Δ   r 

1

(1.8)

is rarely a constant. Moreover, even for a single function several scales can be defined. For instance, for a heat-conduction-radiation problem [19] (parabolic equation with non-linear diffusion and energy sink) one can define different scales for temperature, heat flux, heat diffusion coefficients and the source term. In summary, applicability of a fluid model depends on the particularities of the physical problem it is trying to approximate. There is a common perception that kinetic models are more complex and demanding than fluid counterparts. It is true that the kinetic description adds extra dimensions to a problem that makes the computer memory requirements more stringent, which is the main complication. However, kinetic models have clear benefits: a) Kinetic models operate with one equation per species, instead of several (usually 5) in the fluid models; b) No closure, such as equation of state in fluid models, is required to make the system complete, and the boundary conditions are more clearly defined;

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c) Unlike unknowns in the fluid equations, e.g. flow velocity, that can change sign, the discrete and continuous DF's Eq. (1) are strictly positively defined. In the following sections we are considering different approaches to the purely kinetic ionized gas simulation. Discrete particle [20] and continuous grid [21,22] methods are compared for problems with and without binary interaction. Deterministic treatment of Landau Coulomb collisions are considered and compared in Monte-Carlo [23] and Vlasov approach [24] networks. Where possible the partial analytical solutions are used to simplify numerical integration as discussed. The large-scale algorithm for ambipolar electric field calculation is described. Several benchmarking tests are considered along with few problems of practical interest. Extensions to many dimensions, adaptivity in time and space, application to rarified gas flow are also discussed.

2. Comparison of PIC-DSMC and Vlasov-Boltzmann Methods PIC method [25,26] effectively reduces dimension of a kinetic problem by modeling individual particle trajectories on a spatial grid for fields. It has low diffusivity as particles keep their identity as fixed-shape objects in phase space. This simplification comes with a price: practically irreducible statistical noise and poor resolution of phase space, especially of the regions with low statistical weight, such as energetic tails. This problem can be fixed by

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Oleg V. Batishchev

using particles of different weight, but in PIC simulations particles tend to have the same weight to keep the number of particles per cell limited. Thus, we may conclude, that with Nc particles per cell the distribution function can be resolved up to velocity v/VT  ln Nc . Another weak part of the PIC methods is the inclusion of collisions. These could be either binary, or statistical. Both are very slow converging, and can‟t reproduce situations with low statistical probability. Moreover, some collisions, like impact ionization can produce particles at the same phase space location, which leads to excessive overpopulation and loss of numerical efficiency.

Vlasov node:

PIC particle:

Where I came from?

Where I will be?

x y tor jec tra x

x

x

t = t0+ 

x

x

f x

x

t = t0

tic ris cte ara h c x

x t = t0- 

E

t = t0



x

f x

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Figure 1. a) In PIC method each particle is advanced forward in the interpolated fields, defined on a mesh. New location is re-interpolated onto the grid to find charge distributions on the mesh. b) In Vlasov method each grid node is tracked backwards along the characteristic to find where it originated. The value is interpolated on the grid.

The Vlasov method, on the another hand, is very demanding from the computational standpoint. It simultaneously requires memory for the entire VR region. It also suffers from strong numerical diffusion, which originates from the necessity to constantly interpolate data on computational mesh. High-order interpolations reduce diffusion, but they produce negative values of the distribution function, which is prohibited. But, unlike PIC, Vlasov method guarantees uniform coverage of the entire phase space. Also it allows simulating collisions very efficiently and accurately due to fact that the distribution function is well resolved and represented by a minimal number of grid points possible. The difference between PIC and Vlasov methods is illustrated in Fig.1. In both methods a characteristic or trajectory has to be found. The tasks are different: in PIC the previous particle position is known and a new one along the trajectory has to be found, while in the Vlasov method the new grid position is known and the previous position has to be traced back. The Vlasov method [24] requires one interpolation at the previous time step, while PIC method needs two interpolations, called gather and scatter [26] on old and new steps, respectively. The two methods are different, but show similar performance for the collisionless plasma systems.

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2.1. Collisionless Plasma The kinetic beam instability is a classical problem in plasma physics, and was studied by many authors. The linear growth regimes are well known. Theoretical analysis of the nonlinear cases relies predominantly on small parameters like the beam to plasma density ratio. The non-linear saturation can be studied only numerically. Both, PIC [25] and Vlasov [22,24] methods have been applied in the past. The resolution of the phase space was always limited. Thus, only exponential growth and few system oscillations upon saturation were captured. Due to limitations fine effects were not captured. The simplest beam instability has several scales: the beam size, its density, density of trapped particles, plasma and trapped particles oscillation frequency, growth increments, etc. It is computationally challenging to cover all the parameters. Accordingly, for illustration purposes let's consider the simplest 1D1V problem in a periodical domain [ x]  [v]  [0, L]  [U ,U ] . The DF evolution is described by the following normalized system:  f  f f  1L   E   Edx  0  v  t  x L  0   v  U 1 L U  E   fdv     fdvdx  x L 0 U U 

(2.1)

using the "natural" units. The system physical dimensions are large enough: L  1 and U  1 to minimize the boundary effects. The boundary condition are periodical in space f ( x  0, v, t )  f ( x  L, v, t ), v, t and in velocity space is zero "incoming" particle flux,

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written as f ( x, v  U   , t )  f ( x, v  U   , t )  0, x, t , where   0.2U is a "boundary layer" for the acceleration sub-step. Initially the DF is Maxwellian with a diffuse beam: f M (n  1, u  0, T  1)  n B f B /  f B dv

f ( x, v, t  0) 

U

 { f M  n B f B /  f B dv}dv

U

  v A  , v  [ A, B] sin   fB    B  A  0, v  [ A, B] 

(2.2)

with range U  20 , beam parameters: A  4 and B  8 , and the density nB  0.01  0.3 . The Maxwellian distribution for velocity space dimension Dv  1,2,3 reads as:  (v  U ) 2    s f Ms  exp   Dv 2 1/ 2 VTs    VTs



ns



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(2.3)

244

Oleg V. Batishchev

 here U s is the drift velocity of s - species. Just one unstable harmonic is excited initially and is allowed to grow in the domain: E( x, t  0)  E0 sin( kx)

(2.4)

here wave number k  2 /  for the wavelength   L / N , L  103 , N  31 , E0  10 6 . The number of nodes per period was a free parameter. Dimensionless time step was t  0.1 . The system in Eq.(2.2) was integrated until t  1000 using the conservative method described in Appendix A. Results for weak, nB  0.03 , and moderate, n B  0.1 , cases are discussed here. The evolution of potential energy of the wave, E 2 (t ) 

1 Nx 2  Ei , is presented in Fig.2. The N x i 1

linear growth rate is accurately matching to theoretical [27] of the electrostatic field, 2 L 

3   p f

2 k 2 v v  p / k

(it is doubled for the square), and is strictly exponential until reaching

the saturation that corresponds to the formation of a trapped population at t  110 . The period of oscillations of the trapped beam‟s electrons,  B (E)  (eEk 2 / me )1 / 2 , is proportional to

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 L . This fact was first discovered in [28]. However, we find numerically that the factor of proportionality depends on the shape of the beam in phase space.

Figure 2. a) (left) Potential energy growth in log scale shows linear stage and non-linear saturation (top curve). Below is the same chart in normal scale with four consecutive extremes marked on the saturated part. b) (right) Contours of the 2D DF on the right correspond to these extremes. Bulk electrons modulated by electrostatic wave. Trapped electrons and the void are “rotating” in a concert. At extremes number 18 and 20 trapped electrons have maximum v~7VT, while at extremes 19 and 21 they are at minimum v~3VT.

To further illustrate the dynamics of plasma-wave interaction, the partial evolution of phase space for approximately two wavelengths is given in Fig.2 in a form of the twoEulerian Codes for the Numerical Solution of the Kinetic Equations of Plasmas, Nova Science Publishers, Incorporated, 2010. ProQuest Ebook

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dimensional DF contours for several moments of time. Those are chosen to correspond to the local minima and maxima of the potential energy curve. It is worth mentioning that the trapped electrons and the “hole” on the phase space plane are oscillating in the potential well in a countermotion, thus enhancing the amplitude of the curve pulsations in the saturated regime. The semi-regular pulsations of potential energy shown in Fig.2 are typical for weak beams with n B  1 . We find that once the beam density is increased to nB  0.07 the system experiences bifurcation of oscillations, as shown in Fig.3 for n B  0.1 . As follows from the E 2 (t ) chart, the system transits into a quasi-steady state regime after the linear growth stage

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is saturated at t  75 and followed by a couple of “regular” oscillations. Unlike the weak case, almost complete potential energy transforms back into kinetic at t  105 and for the second time at t  140 . The electrostatic energy grows again afterwards, but the final level of saturation is approximately the same as for the weaker nB  0.03 beam. During resumed growth additional new high-frequency harmonic becomes visible and it is eventually dominating the spectrum as can be seen in from the zoomed portion of the saturated curve presented in Fig.3b. Note that the plasma frequency of the initially uniform system in a nonlinear regime might no longer be the highest, as commonly assumed in kinetic codes.

Figure 3. a) Left chart shows electrostatic wave energy evolution for a moderately strong beam. It includes exponential linear growth terminated at the strongest peak and a complex transition to a highly-modulated quasi-stationary saturated regime. b) Right curve is a zoomed portion of the plateau section with strong high-harmonic modulation. c) Numbered contours of the 2D DF correspond to the extremes as marked on the right insert. Two populations of trapped electrons can be seen in each well along with periodic voids at high velocities.

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To clarify the underlying physical mechanism we plot the phase space at five moments of time, each corresponding to a local extremum of the curve as numbered in the insert. There are two new features in the DF: i) the trapped populations have divided into two “co-rotating” groups in each potential well; ii) a large “hole” per each well is formed at high phase velocity v  8 10VT . To determine which of the two is responsible for the fundamental frequency we examine three consecutive extrema with numbers 55, 56 and 57. The two maxima of wave energy correspond to the “hole” on the phase space being flat at the highest velocity. At the minimum (#56) the void becomes wider and is centered at a lower velocity. During the same time period the trapped populations make just ~one-quarter of a “full rotation”. This observation is in concert with the DF structure at minima 52 and 60. In both cases the large “holes” on the phase space ate at lowest position with maximum kinetic energy in the system. From the same contour plots one can see that the low-frequency modulation is caused by the motion of the “center of gravity” of the trapped electrons. The latter reaches visibly a maximum at the extremum #60. This fact is causing extra sagging of the potential curve minimum at this instance creating an “absolute” minimum.

Figure 4. a) Left panel shows wave energy evolution calculated using PIC method. b) Right panel gives and corresponding result obtained with Vlasov method. The equivalent mesh size was 2048  512 , 4096 1024 and 9192  2048 . In PIC method either 1 particle was placed in the center or 4 particles were placed orderly per rectangular “cell” in phase space, similar to “quiet start” in PIC [26], and also to Vlasov DF on a 1D1V mesh. Vertical arrows mark the moment of the accuracy loss.

We have presented these results to emphasize the richness of kinetic description, which is required in many cases, and to use this simple kinetic problem to illustrate differences between particle PIC and continuum Vlasov methods. Indeed, the high intensity case is characterized with two scales. Due to strong instability, any small inaccuracy in computation immediately transpires in the plot of the electrostatic energy. To expedite such transition and shorten the computational time we compare simulations of the most intensive case. As a benchmark the same problem with n B  0.3 is modeled using PIC and Vlasov methods. We

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start with the coarsest mesh of 2048  512 nodes in X V space, and then double the number of nodes used for each coordinate. Obtained charts of the potential energy are presented in Fig.4 for progressively improved phase space resolution. The evolution of potential energy on coarse grid is very different for two methods with 1M particles and 1M nodes in the phase space, respectively. The two first peaks are very similar, but the rest of the curves have no similarity at all. The PIC curve becomes highly oscillatory while Vlasov chart flattens. Comparison to the higher fidelity simulations indicates that the accuracy in both cases is lost by the third peak in both cases. This means that oscillations revealed by PIC and damping shown by Vlasov method are numerical artifacts. They are the product of high numerical noise and diffusion, respectively. Despite the different nature of the numerical errors, the results obtained by two methods have the same fidelity for three considered meshes. Indeed, for 1M, 4M and 16M nodes both methods lose accuracy at approximately the same physical time. To be more specific the number of nodes in Vlasov has to be compared to the number of particles in PIC. To verify this statement we run PIC with different number of spatial nodes, but the same number of particle. As marked in Fig.4 by arrows, the numerical accuracy is lost at the same moment. Of course, the visual judgment of when this happens is somewhat subjective. It should be mentioned that to make the comparison correct, the particles in PIC method had variable weight. Traditional PIC operates with particles of uniform size, which increases DF fluctuations and makes it impossible to resolve energetic tails of DF due to low statistical weight. Such initial “discrete” particle distribution was in fact identical to the “continuous” mesh DF in Vlasov method. Thereafter, the initial conditions in both methods were completely identical from the numerical standpoint. Despite the fact that both methods used the same initial conditions and linear interpolations for numerical approximations the DFs at any time are different, particularly in the regime of non-linear saturation after the numerical accuracy is no longer maintained. To illustrate, two DFs obtained with PIC and Vlasov method at the same physical time are shown in Fig.5 with the same spectral resolution. The particle DF has high local fluctuations. However, it captures “a void” on the phase space. The DF calculated on a 2D grid appears to be very smooth with low local gradients, but there is no “hole” presence on the phase space. The reason for high numerical noise of the PIC method is in the finite number of particle trajectories that occupy a given element of the phase space. The phase space volume is an integral of motion of a Vlasov system, but not of the PIC method where individual trajectories can intersect due to finite resolution. Individual particles preserve their identity, but the numerical convergence is proportional to the square root of the number of particles per cell, N c . Note that for moments of DF, the cell is a spatial cell. Statistical fluctuations of density are many orders of magnitude higher than in real plasmas, as a macro-particle usually represents millions of real particles. The oscillations of DF are much higher because the cell is an element of phase space. Thus, the instantaneous DF in particle methods is undefined, and the simulations are never converged in numerical sense. On the contrary, in Vlasov method the phase space is evenly covered at any time, but the trajectories are undefined. A “new” particle trajectory crossing each node is re-synthesized every time step. Thus, particles are losing their identity:  - shaped distribution becomes a diffuse spatial spot. This rate, however, varies significantly across velocity space. For instance, in case of a second order linear interpolation and uniform mesh with spacing  the diffusion coefficient behaves as:

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Oleg V. Batishchev vt   Dv   vt  Int    

2

(2.5)

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This quantity varies between 0.252 / t for “particles” finding themselves between the nodes and zero for those moving exactly from node to node. Lastly, a few additional comments regarding the two methods. In the collisionless case PIC requires twice as many variables, coordinate and velocity in 1D1V, compared to just the DF value in Vlasov method. The number of operations is several times higher (see Fig.1) in the Lagrangian PIC than in the Eulerian Vlasov method with the same spectral resolution. The Vlasov method scales better to more dimensions. It also allows more accurate treatment of the boundary conditions, e.g. wall sheaths.

Figure 5. a) Left panel shows instantaneous electron DF calculated using PIC method. It is noisy due to high statistical error. b) The right panel presents contours of the DF obtained for the same problem & physical time using Vlasov method. Gradients in phase space are smoothened by high numerical diffusion.

2.2. PIC-Vlasov Hybrid for Collisionless and Collisional Plasma It is possible to improve accuracy by reducing the statistical noise and numerical diffusion at the same time. One simple way is to combine PIC and Vlasov as shown in Fig. 6.

Figure 6. Scheme of PIC-Vlasov hybrid method for collisionless plasma. After K steps, the discrete DF is converted into “continuous” DF (per Eq.1) defined on a mesh in phase space. Then the system is

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integrated for k steps using Vlasov method. After that new particles are generated, usually by taking one particle per Vlasov cell.

For the same test problem in Eq.(2.2) it was found that the best results are given by choosing K ~ 10 and k  0 . This means periodic re-generation of discrete DF, or in other words repeating “quiet start”. Of course, v must be small to suppress multi-stream instability [26,27]. To interpolate from PIC to 2D mesh we used bi-linear interpolation. The results are given in Fig.7.

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Figure 7. a) Left panel shows evolution of potential energy for different meshes and number of particles per cell in the hybrid PIC-Vlasov method. Arrows mark the moment when the numerical accuracy is lost. b) Right panel shows contours of instantaneous electron DF.

As one can see the electron DF is less noisy than the one obtained by PIC. At the same time the gradients on phase plane are reproduced much better than in Vlasov method. As a result of improved DF calculation, the accuracy of wave energy, E 2 (t ) , is improved as well. The physical time of accurate simulations is approximately tripled. It is calculated for the non-linear part, after the linear instability saturation is achieved. Much higher numerical efficiency can be gained by applying hybrid method for collisional plasmas. One possible scheme is given in Fig.8.

Figure 8. Scheme of PIC-Vlasov hybrid method for collisional plasma. After K steps, the discrete DF is converted into “continuous” DF in velocity space. The collisions are performed in V-space for a cumulative time step. Then new discrete DF is generated, and the process is repeated.

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This scheme is more efficient than PIC-DSMC because the collisions are easier to treat in the continuum limit. This is explained in the following section for Coulomb collisions in Landau approximation. The reason for keeping PIC step makes sense for mixed-collisional regimes, when collisional and collisionless regions coexist in the simulation domain.

2.3. Direct Simulation of Coulomb Collisions in PIC Method The DSMC method is efficiently used in rarified gas dynamics [23] for short-range Boltzmann collisions of neutral particles. Charged particles interact through long-range Coulomb collisions, to which direct application of DSMC is inefficient. The most straight approach is to include those into the PIC method using the very same scheme the PIC equations are derived. Namely, by substituting into Vlasov equation the discrete DF f s   G ( x  x n (t ) ) (v  v n (t ))

(2.6)

n

where particles have shape, or normalized symmetrical form-factor,  G( z)dz  1 , in space. After a couple of integrations, as in Eqs.(18)-(19) below, one obtains a system of equations of motion for individual particles:

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 dx n (t )  v n (t )   dt  n  dv (t )   e  E (t , x)G ( x  x n (t ) )dx   dt  m s 

(2.7)

Let us derive the equations of motion for single-species plasma with Coulomb interactions between particles. The Landau integral describing Coulomb collisions in the Fokker-Planck (FP) form in the Cartesian coordinates can be written as

C ss

L    ss  4 i vi

 2       f s (u )  1     f s  sq    du f s     f s (u ) v  u du  vi v  u 2k    vi vk  v k

  

(2.8)

  here v  u  (v x  u x ) 2  (v y  u y ) 2  (v z  u z ) 2 , Lss is a constant coefficient, which is 2

 4e s eq   , and sq  ms / mq is the mass-ratio factor.   ms 

generally defined as Lsq   sq 

By substituting the discrete DF in Eq.(2.6) into Eq.(2.8), calculating two inner integrals over  -functions in velocity space, and taking their first and second derivatives over Cartesian components of the velocity vector, we obtain the following explicit form of the Coulomb term:

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v  vin (t )  { G( x-x n(t) ) i } fs  3   i vi n v  v n (t )

C ss   Lss 

(v  v n (t ))(vi  vin (t )) f s n )}  { G( x-x (t) )( ik  i i 2   v k k n v  v n (t )

(2.9)

By integrating Landau equation over velocity space we obtain:   G ( x, x n (t ))  v n (t ) G ( x, x n (t ))  t x dx n (t ) G2' ( x, x n (t ))  v n (t )G1' ( x, x n (t ))  0 dt

(2.10)

where subscripts 1 and 2 mean derivatives over first or second argument of the kernel G . Because the form-factor was chosen to be symmetrical and dependant on the difference of arguments one has G1'  G2' . From this we obtain the first equation for the particle motion in Eq.(2.7), which nullifies Eq.(2.10). The equivalent of the second, Newton, equation of motion in Eq.(2.7) is obtained by multiplying both sides of Landau equation by v and integrating over the phase space:  n e v x (t )   G ( x, x n (t )) dx     E ( x) G ( x, x n (t )) dx  t  m s  Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved.

p

v xn  v xp v xn  v xp

3

  G ( x, x n (t )) G ( x, x p (t )) dx  

(2.11)

here X denotes the contribution of the second, diffusion term in Eq.(2.8). To calculate it we multiply Landau equation by v 2 and integrate twice over velocity and space. Retaining just one spatial coordinate for simplicity, we obtain:  n2   f  p v x (t )   G ( x, x n (t )) dx    v x2  G(x,x (t))D xx dv x dx  t v x  p v x  2 p   v x  G(x,x (t))D xx p

f dv x dx  v x

 2 v x2 D xx f f p G(x,x (t)) v D      x xx v x v x p 



 2v x D xx f v x

  2 D xx f  dv x dx  

(2.12)

p n n p   G(x,x (t)) G(x,x (t))  2 D xx (v x , v x ) dx p

Thereafter, the second term describes a space-averaged diffusion in velocity space with the following components of the diffusion tensor D n :

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Oleg V. Batishchev

Dikn

   (vin (t )  vip (t ))(vkn (t )  vkp (t ))     G(x,x (t)) G(x,x (t)) dx    ik   2   n   v n (t )  v p (t )   n

p

(2.13)

It is a well-known fact that the diffusion can be described by a Langevin equation with random force. An exact result is given by the theory of continuous Markovian processes [29]. By combining Eqs.(2.12-2.14) one obtains the final form of the equation for the macroparticle velocities: v xn (t )  e      E ( x) G ( x, x n (t )) dx  t  m s 

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p

v xn  v xp v xn  v xp

3

  G ( x, x n (t )) G ( x, x p (t )) dx 

2D n 

(2.14)

where  is a random Poisson process with zero average and unity dispersion. These “direct summation” models were considered and found to be inefficient [30]. Comparison of Eqs.(2.13-2.14) with the second one in Eq.(2.7) shows the presence of two summations over particles. Unlike collisionless PIC for Vlasov, in the coulombic PIC, each particle has a “cloud” of neighboring particles with dimension comparable to that of G , which defines the collisional dynamics of a given particle. This brings about several obstacles. First of all, the particles must be ordered in space. This can be done efficiently using approaches developed for Molecular Dynamics (MD) [45, 51]. Usually particles are ordered by the cells of the spatial mesh because common kernels have comparable dimensions. To reduce statistical fluctuations, the number of particles per cell has to be sufficiently large. As each particle may have its “cloud particles” located in the same and adjacent cells (2, 8 and 26 in one-, two and 3-D), the total number of pairs is large, N c ( N c  1) / 2  10 3 . The computations become very bulky. The latter are amplified by a need to calculate spatial integrals for each pair to find the drag term and diffusion coefficients, and to calculate the square roots of the diffusion matrix. The biggest difficulty, however, is caused by the singularity v 1 in the drag term. It is feasible to remove it by smoothing or, possibly, by introducing form-factors for particles in velocity space. This will require an introduction of a mesh in velocity space. Equations of motion will be modified and additional calculations of integrals in velocity space will be required, which will further reduce the computational efficiency. Despite these difficulties, we performed initial calculations using direct summation method for two-component plasma [30]. It was found that the DF is very noisy, and accuracy of the method is low. It can be used to obtain qualitative results only.

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2.4. Coulomb Collisions with Averaged Potential Functions An efficient computational strategy is to replace ( N c2 ) binary collisions between particles by ( Nc ) of the individual particle collisions with of the entire ensemble. One can use a similarity with PIC method, which replaces the direct summation of binary interactions with a boundary problem for an electric potential on a fixed mesh in space. The calculated field has fluctuations effectively smoothened on a Debye scale. A similar approach is proposed to reduce fluctuations due to Coulomb scattering. The Debye shielding scale is replaced with the thermal velocity, VT . By introducing two Rosenbluth potential [31] functions  s and  s  1 f s (u )     ( v )     du   s 4 v  u    (v )   1 f (u ) v  u du  s  s 8

(2.15)

which are taken here in Trubnikov‟s form [32], the Landau equation can be written in the compact FP form: df s    Lsq  dt q i vi

2    q   q   sq f s    vi  k  vi v k  

  f s  v   k

   

(2.16)

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If in the first Rosenbluth-Trubnikov potential function,  , velocities of the particles are replaced with their coordinates, it becomes equivalent to electrostatic potential of a system of point charges. Thus, the potentials are calculated via two coupled Poisson equations: 2   v  s  f s  2   v s   s

(2.17)

where subscript v indicates that the Laplace operator acts in velocity space. There is, however, an important difference from the electrostatic potential. The boundary conditions (BC) are undefined because the velocity space in classical formulation is infinite. Usually asymptotical boundary, V / VTs   , BC are imposed: ns     s   4 V  U s     ns  V  U s  s   8 

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(2.18)

254

Oleg V. Batishchev

which implies that the grid has to be large enough, practically reaching to ~ 10  20VTs , as required by the physical problems. Other required parameters: density, drift and thermal velocities are calculated as moments of the DF  n s   f s dv    n sU s   vf s dv Dv n sVTs2

   2(  v f s dv - n sU s2 )

(2.19)

2

The Rosenbluth potentials are calculated at each spatial node on a virtual mesh in Vspace. This is the most expensive operation as in the non-linear case potentials have to be updated every time step. Thereafter, non-linear FP parallelization [33] is very efficient compared to the stochastic method [34]. But if plasma is equilibrated this expensive step can be omitted. The linearized Coulomb operator operates with so-called Maxwell potential functions,  v Ms  f Ms  v Ms   Ms [32]. The numerical schemes using random MC and Langevin methods are widely used to simulate evolution of gas and plasma systems [23,3537]. We find, however, that the stochastical approach has much lower numerical efficiency if compared to the continuous method, per direct comparison.

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2.5. Non-linear Coulomb Collisions in PIC and Fokker-Planck Method There are two approaches to calculate collisional Coulomb integrals: Deterministic [3840] and Statistical [36-37]. Statistical methods as DSMC [23,35] are characterized with high intrinsic statistical noise and slow numerical convergence O( NC1/ 2 ) . This is also typical for PIC when applied to regimes with strong divergence of trajectories, which are characterized by large values of the Lyapunov exponent [41]. Despite drawbacks, these methods, proposed in the late 1940s - mid 1950‟s at Los-Alamos [5,20], made it possible modeling on computers with limited random-access memory (RAM). The deficiency of the stochastical approach could be seen from the following 1V example. The dimensionless spherically-symmetric v  v x2  v 2y  v z2 Coulomb term in FP formulation [31,32] reads as:  q  2 q f s  f s 1  2   2 v  sq fs  t v v v v 2 v  

The units are thermal velocity of q species

2(k T ) [t ]  8



1/ 2 2 mq B q 2  sq nq es2 eq2

(2.20)

[v]  VTq and s  q collisional time

.

The first and second non-linear Rosenbluth potentials are obtained from Poisson equations in spherical coordinates:

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Semi-Analytical Adaptive Vlasov - Fokker-Planck - Boltzmann Methods   2 q 2  q  2   fq v v  v  2    q 2  q  q  2  v v  v

255

(2.21)

For illustration we simulate single test species evolution on field particles with arbitrary DF. For simplicity we assume Maxwellian DF. In this case the potential functions derivatives defining drag and diffusion in v-space are the following analytical functions:   Mq  4 q   v  2 2    Mq  v 2   v  q

where  q  

(2.22)

 erf(v) , subscript q indicates argument normalization, and erf is the Gauss v v

error-function. The corresponding Langevin equation can be written as: dv 2 4  4(λsq  1 )μ q  2 (μ q  2erf(v))  μq ξ dt v v

(2.23)

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The “random force” force   [0,1] is  -correlated with zero mean value: 1,   i (t ) k ( x)    0,

i  k t  x i  k t  x

(2.24)

  i (t )   0

where brackets ... denote ensemble-averaging. This equation describes spherical velocity DF, f vs  v 2 f s . Both stationary solution of the kinetic Eq.(2.20) (with respect to f vs ) and asymptotical distribution of v given by the stochastic Eq.(2.23) can be written as: 2

f vs  v 2 e (v / VTs )  v 2 e

v 2sq

(2.25)

With a proper unconditionally stable scheme [42] the same function can be obtained as a numerical solution. An example of such scheme's numerical convergence is presented in Fig.9a for v -distribution of N  10 8 trajectory points N  1 . Fluctuations are increasing with normalized energy,   v 2 , as N  NC 1/ 4e  / 2  1/ 2 . Here  is the size of energy bin used to calculate the DF per Eq. (2.19). At N  1 the accuracy is completely lost.

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The stochastic data have to be compared with Eulerian simulations results of the same problem presented in Fig.9b. These continuous FP [33,43] simulation included 100 time steps and 100 velocity space bins. Eq.(2.16) was solved using conservative splitting scheme applied to the continuous DF defined on a variable grid in velocity space (see section 3). As one can see the convergence is good through very high energies   400T . This corresponds to numerical values f  10 173 , well exceeding simulation needs. The number of floating-point operations in FP calculations was ~ 2 10 5 , while in Langevin ~ 5 10 9 . Thus, in the collisional case the Deterministic method is much faster than the Stochastic. It also more accurately resolves the energetic population of DF, easily reaching to  ~ 20T and beyond. This feature is vital for accurate plasma modeling of important plasma phenomena.

0

l n f

5

8 N = 1 0 t= 2 

= 0 .0 1  1 0

0

2 2 v e x p { v }

5

2 V 1 0

Figure 9. a) Left chart shows convergence of the discrete spherical velocity DF to the asymptotic equilibrium solution using the stochastic Langevin procedure given by Eq. (31). b) Right chart shows equilibration to Maxwellian of the continuous grid DF using deterministic FP method. DFs are plotted

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in log scale versus normalized energy E / T  v 2 . Asymptotic equilibria are marked by dashed lines.

2.6. Phase Resolution of the Energetic Part of the DF High-energy electron population often called suprathermal "tail" of the DF are characterized with   2Te . These particles account only for ~ 25% of the total number density (Fig.10a). However, this quarter is usually responsible for rates of inelastic collisional processes in plasmas, intensity of plasma-wall interactions including erosion and probe measurements, and plasma heat transport. Indeed, the tail of the electron distribution determines some plasma-neutral interaction processes, particularly in the cold plasma regions. In Fig.10b we show the plasma temperature dependence of the energy E * that maximizes excitation and ionization rates,   ( E) E1.5 exp( E 2 ) for equilibrated plasma. Inelastic cross-sections are characterized with energy thresholds of the corresponding reaction, Eth , and behave as step-functions of the latter

 ( E)   (E  Eth ) . In domains with Te / Eth  1 / 3 the suprathermal electron contribution dominates. Because of the higher energy threshold the effect on the hydrogen ionization is stronger than the effect on the Ly excitation. An elevated electron tail could alter the ionization and recombination balance, shifting it to a lower temperature than its Maxwellian

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value. One can expect an even larger effect on high-Z impurity excitation rates by electron impact, which have higher thresholds, as shown in Fig.10b for C 2 with Eth  39.7eV . 1-FM

100

fM

10-1

10-2

FM

E/T 10

-2

10

-1

100

101

Figure 10. a) Left pane shows normalized partial density of the Maxwellian DF in spherical velocity coordinates, which can be written as f M  E exp {-E} . It is defined as an integral in the energy space E

FM (E)   E exp {-E}d E . Complementary function, 1  FM , characterizes number density in the 0

energetic "tail". b) Right curve† shows normalized energy E * / T , which corresponds to the maximum of electron impact excitation.

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An energetic electron tail can affect the value of the sheath potential. The equilibrated floating potential is V f  3  5Te depending on ion mass [44]. In the presence of an impurity it can be higher. An enhanced non-Maxwellian electron tail increases the sheath potential at the plasma-facing wall. This gives rise to an increase in the sputtering of the wall material due to additional acceleration of heavy ions in the sheath potential. Erosion rate is sensitive to the energy of bombarding particles. It is a serious technological problem for magnetic fusion devices and plasma thrusters [45]. Enhanced or depleted electron tails also affect the interpretation of plasma measurements, which are commonly based on the assumption of a Maxwellian electron distribution. V-I curves are usually taken in the narrow interval around the floating potential to avoid high energy fluxes. If the tail of the electron distribution is not equilibrated with the bulk of the distribution, measured and mean temperatures will differ. Moreover, a small ratio of the parallel transit time with respect to the time necessary to make the temperature isotropic can produce a difference in the parallel and the perpendicular temperatures. This can affect Thomson scattering and neutral beam-based techniques Te measurements.

 Most importantly, the plasma heat conduction flux along the magnetic field lines, l II , is carried predominantly by the suprathermal particles as illustrated by Fig.11a that shows the electron heat flux density in velocity space, q II  v II v 2 f e , here vII  v is the velocity

along magnetic field, and   cos  is a cosine of angle between particle trajectory and magnetic field line. The peak of the parallel heat flux density is located at E  6Te . The curve was obtained using adaptive 1D2V Fokker-Planck code ALLA for highly collisional fusion

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device plasma [46] with KnC  C (Te )

d ln Te  10 3 . It is in a quantitative agreement with dl II

the analytical result, q II  v 5 f1 [47], where f1  f M is a small first correction to  v 2 1

Maxwellian equilibrium. The partial heat flux q pII ( )  2   δq II dμ dv is a measure of 0

1

how much heat is carried by the particles with normalized energies not of  or less. As follows from Fig.11a half of the total heat flux, q II  q pII () , is carried by particles with E  7Te . More than 10% is due to electrons with E  10Te in collisional plasmas. These energetic electrons have much longer mfp, C (  10)  100C (Te ) , due to quadratic dependence on particle energy. Accordingly, the efficient coulombic Knudsen number is  100 larger for these tail electrons than for the thermal population. This explains why it has to be so small in the first place to assure validity of the analytical transport theory results for plasma conductivity obtained in the short mfp limit.

floating potential

10 0 10 -1

fPIC

10 -2

fFP

maximum of heat flux

ionization by tail at low T

10 -3 10 -4 0

1

2

3

4

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V / VT

Figure 11. a) Left pane1 shows calculated normalized heat flux density, qII / qII MAX , (solid curve) and partial flux,

q pII / q II , (dashed) as function of dimensionless energy  . b) Right pane shows

instantaneous discrete DF for a 100 particle ensemble as function of V / VT   . The accuracy is lost for   1 and the suprathermal tail is unresolved.

Summarizing, the energetic tail of the electron DF is responsible for several important physical processes in plasma. Approximately 2-10% of electron population with E  3  5Te is defining the structure of wall Bohm sheath and amplitude of the potential drop. Hence, they affect the intensity of wall bombardment by ions, material erosion rate, plasma potential with respect to wall, plasma probe measurements, wall conductivity, other plasma-wall phenomena. The same fraction of the electrons is responsible for inelastic plasma chemistry processes with Eth  3Te . 

Plasma heat conductivity along B has complex alternative-sign structure in velocity space, but roughly half of the integral is defined by mere 0.2% of particles with E  7Te .

1

Reprinted with permission from Batishchev, O.V., et al, Phys. Plasmas 4 (5), 1672 (1997). Copyright 1997, American Institute of Physics.

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Moreover, a negligible fraction of the number density of 0.01% with E  10Te carries onetenth of the total heat flux . Note that these results are asymptotical KnC   . For finite lengths and gradients the effect can be more profound in some regions and suppressed in others. In special cases, e.g. in a presence of weak high-energy Eb  Eth  Te electron beam, these "kinetic" features would be attributed exclusively to the high- E fraction. There are many instances when plasma DF's are strongly non-equilibrated: i) fast process, such as ultrafast laser-matter interaction, pulsed gas discharges; ii) selective plasma heating, e.g. by electron or ion cyclotron resonances; iii) run-away electrons in strong electric fields, plasma in the oscillating wave, etc. Predictive modeling of all these regimes requires purely kinetic description of plasmas. Kinetic modeling of plasmas is done predominantly by PIC method. Usually PIC simulations are performed with NC  10  100 particles of each species per computational cell. Corresponding statistical fluctuations in number density will be on the order of 10-30%. Fluctuations of plasma density, drift velocity, temperature, pressure will be of the same order. This is a manageable problem if one consider quasi-steady-state regimes. Then time averaging, possible due to system's ergodicity, can be applied to large time intervals t av pe  1 . Special modifications, such as "quiet start" can be used to study linear regimes

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of some plasma instabilities, which is the most prominent scholarly result first obtained by PIC method (Fig.2). The application becomes problematic, however, for non-steady and non-linear regimes. Reducing statistical fluctuations by simply increasing NC is computationally inefficient due to quadratic dependence: to reduce numerical noise to 1% one needs NC  10 4 , which is already impractical. Due to such slow convergence of PIC (and all particle methods including DSMC) none of the discrete simulations is converged fully from the computational standpoint. Moreover, usually particles of the same statistical weight, N C1 , are being used. Such "quantizing" makes accurate resolution of the exponential tail of the Maxwellian DF impossible above energies   ln NC . Hence, approximating an initial distribution with fixed weight particles is very inefficient. Using variable weights is much more efficient and allows simulation of tenuous energetic tails [48]. The problem is that these particles quickly randomize in velocity space on plasma times, as in quiet start. Typical discrete Maxwellian DF for NC  100 is presented in Fig.11b. The resolution in energy space is set to modest   0.1 . Fluctuations of the DF are about 10  30% for   1 , but they rapidly exceed 100 % in the energetic   1 region, indicating complete loss of approximation. This is a fundamental limitation of the statistical method (see Fig.9a), which leads to the following conclusions: i)

The errors in DFs are always on the order of 100% for suprathermal   2 energies. High-energy tail cannot be resolved due to the limited number of particles of finite size.

ii) Kinetic phenomena, such as sheath dynamics, wall potentials, plasma heat fluxes and inelastic processes caused by the energetic particles cannot be resolved adequately; iii) Non-linear dynamics of the fast processes cannot be studied because the instantaneous DF is undefined, and, for instance, non-linear Rosenbluth potentials cannot be calculated;

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iv) Transient and non-steady processes, like turbulence, cannot be simulated accurately; v) Fine processes with amplitude below statistical noise level cannot be accurately resolved; vi) Resonant phenomena are difficult to study as most of the velocity space is not covered by the  -shaped particles. Other important shortcomings are: particle methods are not well compatible with adaptive meshes. If a cell is refined by dividing it into several, this leads to NC reduction and further increase of numerical noise. Particles randomization in space and velocities requires their periodical ordering, for instance, to define a sheath potential. This is an expensive and inefficient procedure. Processes, like ionization, produce low-energy slowly moving secondary electrons. These particles accumulate in the simulations domain, basically at the same phase-space location. This contributes additionally to the problem of the inefficient energetic tail resolution in PIC.

3. Continuous Fokker-Planck Method for Quasineutral Systems

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Any predictive computational method has to be able to resolve the entire phase-space with the required accuracy, which is defined by the physical problem. Mathematically, resolution means reproduction of the gradients of the DF due to the fact that numerical approximations are proportional to their derivatives starting with the first. Because the location of the high gradients is unknown (e.g. of the plasma wake) following the gradients has to be automated. Our basic strategy is the use of moving adaptive meshes. Ideally they have to be used in real and velocity sub-spaces. Here we are considering an intermediate case with i) variable grid in velocity space combined with ii) adaptive mesh in space. Physically we are considering collisional plasma flow along the magnetic field line.

3.1. Splitting Scheme for Electron-Ion Plasma with Coulomb Interaction Let us consider the following model problem. One has to solve a boundary problem for electron and ion DFs, f e,i (t , x, vII , v ) , described by the system of two kinetic equations, which are strongly coupled through self-consistent electric field and Coulomb collisions:      

f e f f e C  vII e  EII e  Cee ( f e , f e )  CeiC ( f e , f i ) t x m vII f i f f e  vII i  EII i  CiiC ( f i , f i )  CieC ( f e , f e ) t x M vII

(3.1)

here m, M ,  e, e are masses and charges of electron and singly charged ion, respectively; C is the Landau EII is the parallel (axial) electric field along the magnetic field x  lII , C

collisional term, vII  vx is velocity along the field, and v  v 2y  vz2 is velocity normal to the axis x .

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The boundary conditions on fe,i are as follows: the distributions of particles being "imported" into the simulation domain X Vx V [0, L][U ,U ][U ,U ] are prescribed and fixed in time: f e,i (t , x  0, vx  0, v )  f e,i (vx  0, v ) f e,i (t , x  L, vx  0, v )  f e,i (vx  0, v )

(3.2)

The domain's span in velocity space, U , is given by the natural boundary condition, no particles at "infinity": f e,i (t , x,U , v )  f e,i (t , x, vII ,U )  0

(3.3)

As we are expecting plasma to be close to equilibrium, it is sufficient to set U  20VT , where VT  2T  / m is the thermal velocity of electrons that corresponds to the boundary DF, f e,i : T 

  2 mv 2   Mv2   ( f e dv   fi dv ) / (  m f e dv   M fi  dv ) 3 2 2

(3.4)

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This choice guarantees that the value of electron DF at the outer boundary will be  exp( U 2 )  10173 , which is close to zero for the calculations performed with 8-byte floating point numbers (double accuracy). Single 4-byte arithmetic is insufficient for two reasons: i) the DFs vary in a broad range, hot electrons and cold ions are characterized with very different scales in velocities, and ii) 6 digits of accuracy is insufficient to maintain the dynamic range required for avoiding f  0 situations, for debugging purposes, particularly benchmarking convergence to actual "computer zero", not to some "small" value. Regarding plasma we assume that the dimension is large, L  D , millions times the Debye length. Hence, the plasma is quasineutral, ne  ni . For a typical plasma density of n p  1012 1014 cm3

and length

L  0.1  10m

the level of statistical fluctuations

n p / n p  10 8 10 7 suggests high charge neutrality and impossibility to obtain electric field EII  4e(ni  ne ) . Accuracy of approximation is limited to 3x 4 digits, while charge density fluctuates in the 7-8th digit. The spatial mesh step has to be of the order of 10D for the applicability of the implicit Poisson equation [49]. Thus, the self-

from the divergence equation,

consistent electric field profile adjusts to precisely maintain plasma neutrality. There is no spatial charge accumulation for the quasineutral flows. This means that on the time scales exceeding charge separation times, which are on the order of plasma times 1011 1010 s for the common plasma devices, the flow is ambipolar. Thus, formally, the electric force compensates all other forces acting on the charged particles. The Coulomb collisional time is

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108 107 s . The ion transport time 105 104 s is the time scale of pressure relaxation in the system. Both scales are much bigger than plasma times. The system in Eq.(3.1) can be normalized in many ways. We chose the following units:

length [ x]  1m , energy [kT]  1eV , elementary electric charge [q]  e  1.6022 10 19C , proton mass [m]  M p  1.6726 1027 kg .

3.1.1. Computational Grid in the Axisymmetrical-Spherical Coordinates We must transform equations (3.1) from Cartesian to a convenient system of coordinates. The convenience is determined by the efficiency of calculations provided the required accuracy is guaranteed. For the sake of Coulomb collisions it is important to accurately resolve DFs in velocity space, including thermal and suprathermal parts. We would like to keep the same mesh in velocity space throughout the simulation domain. We would like to simulate regimes with large temperature variation Tmax / Tmin  102 103 , while resolving details of DF with accuracy v  0.1VT in the thermal core for both electron and ion populations. Because of high temperature range and mass ratio ( VTi / VTe  42.6 ), the ratio of the required velocity mesh step to the maximum speed that must be covered is v / U  105 . Thus, a uniform grid in Cartesian coordinates will have K  1010 nodes per each spatial point, which is well beyond a practical limit of K  10 4 . The solution is to use a variable mesh. The relative grid resolution of the DF's gradients has to be uniform, which means that mesh should vary in accordance with:

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d ln f d ln f dv v   const dv dv dv

(3.5)

In our approach the mesh is variable and fixed in time, not adaptive. If DF is known, then the grid can be selected per this relation. Generally, it is unknown, and, moreover, evolves in time. Neglecting the first term in the middle section of Eq.(3.5) we obtain analytical solution v / V  const  av

(3.6)

which guarantees general uniformity and smoothness of finite-difference approximation of operators for all velocities. The finite solution of this equation is given by the geometrical progression of velocity steps: vn  av vn1 , n  2,..., Nv

(3.7)

This relation gives a required dynamic range to cover velocity space with Nv  100 for av  1.1  1.2 , as 1.12100  10 5 . If directly applied in the Cartesian coordinates, the grid will

be stretched in three directions ( vII , v ), creating extremely elongated cells at low parallel and high perpendicular velocities and vice versa, which further increases the stiffness of the numerical scheme.

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These unwanted features can be removed by switching to the spherical system of coordinates, which in two-dimensional case transforms into axisymmetrical spherical coordinates

 , 

, where

  vII2  v2  [0,U ] is the modulus of velocity and

vII  [1,1] is the cosine of an angle between velocity vector and X-axis. These v coordinates are "natural" for Coulomb collisions as was noticed in [31,32]. An example of computational grid is presented in Fig.12. The mesh is non-uniform in both v and  .

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

Figure 12. a) Variable mesh is

 ,  coordinates easily reaches to high energies,

E  200T p ; zoomed

regions show b) axis v  0 with first grid row v1  0 , and a typical computational cell i, j.

The modulus of velocity interval [0,U ] is divided into Nv non-uniform steps with fixed stretching coefficient av , and the angular range [1,1] is divided into N  uneven part in the following way: v1  0, v2 ,..., vi 1  vi  av vi ,..., vNv  U

vi  vi 1  vi , i  2, Nv  1

 j  cos  j  1 ,





   / N   1 ,

j  1, N  N  2

k

 1 , k  5  7

 j   j 1   j , j  1, N  1

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(3.8)

264

Oleg V. Batishchev

 

The distribution function f ei,,i j is defined in the nodes of this mesh centered quantity. To bound these cells an auxiliary grid of size

vi ,  j  as a cell-

( N v  1)  ( N   1) is

introduced (dashed lines in Fig.12): vi 1  vi , i  1, Nv  1 , vN v  U max 2  j 1   j , j  1, N   1 ,  N   1 0  1 ,  j  2

v0  0 , vi 

(3.9) v

The dimensions of the auxiliary mesh are defined as usual, in velocity, i  vi  vi1 , 

i  1, Nv , and in angular j   j   j 1 , j  1, N  , directions.

For further calculations we are using finite-volume approach. We have to define surface areas of the cells facets in both directions S iv,j and S i, j , and their volumes Vi, j . The cell's volume is defined as a volume of an infinitesimal body of rotation with cross-section vddv and rotation radius v sin  (see Fig.12b) integrated over the cell i, j : 







2 d cos dv 3  vi 1 3

Vi , j   j  vi 2v sin vddv   j vi  j 1 vi 1

 j 1





  3 3 2 2    j  vi ddv 3    j 1   j  vi  vi1  ,   3  j 1 vi 1 3

i  1, Nv , j  1, N 

(3.10a) ,

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In a similar way one can calculate the surfaces of a body of rotation:





S iv,j  2vi  j 1   j , i  1, Nv , j  1, N  S v0,j  0 ,

(3.10b)

j  1, N 

2 2 S i, j   1   j  vi  vi1  , i  1, Nv , j  1, N   1  

S i,0  0 & S i, N  0 ,

i  1, Nv

Thus, the variable mesh in  ,  coordinates is defined.

3.1.2. Kinetic Equation in the Axisymmetrical-Spherical Coordinates The next step is to convert the system Eq.(3.1) into new coordinates, and to write them in a "divergence of a flux" form as required by the finite volume approach. Let us do this operation term-by-term for ion and electron DFs, taking into account that the DFs depend now on new variables f e,i t , x, v x , v   f e,i t , l|| , v,   . It is obvious that plasma free-streaming in space operator can be transformed as (for simplicity omit DF's species sub-script):

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f f f  vf   v cos( )  v  x l|| l|| l||

265 (3.11)

To transform the term describing plasma acceleration in the axial electric field, E x  E|| , we should use the following transformation: 2 2 2 2 f f v f  f  v||  v  f  (v|| / v||  v  )      v|| v v||  v|| v v||  v||

(3.12)

2 f v|| f  1 v||  f 1   2 f   3   v v   v v  v v 

The “divergent” form of the operator can then be written as follows:

E

f f 1   2 f 1  f   E E  2  2vEf  v 2 E   v|| v v  v  v 







 

1 f  1  2 1    2Ef  (1   2 ) E  v Ef  1   2 Ef v   v 2 v v 

(3.13)

Combining (3.11)-(3.13), the system of kinetic equations can be written as:





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

 f e f 1   2 eE x  1    2 eE x  v e  2 fe   f e   CeC v   1   t  x  v m v  m v        f  f eE eE  x  i  v i  1   v 2  x f   1   1   2 f i   CiC i  2 v  t  x M v  M v     



(3.14)

where C eC,i denote the species-specific Coulomb collisional term. The Landau collisional operator can be written in the following form in (v,  ) coordinates (see Appendix B):  2   1   2    1   L /     2  m     f v f      2 v  m v   e, i v v 2 v v 2  v v   

CC  



 1  2 

 mm 







  2  1   f     v v  

 f         

(3.15)

 f   1   1   2  2      f         2  2  2  v  v v    v  v      

The Rosenbluth potentials in Trubnikov's form are calculated from two coupled Poisson equations: (3.16)    f      

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Oleg V. Batishchev

with asymptotic boundary conditions at v   :  U    U    M

(3.17)

 U    M U  



here U  vT , and in the Maxwellian potential functions we use:

M v  

n 

vt

 v      vt 

0 

(3.18)

 v      vt 

 M v   n  vt 0 

where vt  2T / m (note factor 2) is the thermal velocity of  -species. Functions in the right-hand side (rhs) of Eq.(3.18) can be expressed through error-function erf and its derivative erf ' : erf x  

2

x



0

e

t 2

as 0 x   

Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved.

 0 x   

erf ' x  

dt ,

2



e x

2

(3.19)

erf x  4x



(3.20)



erf ' x   2 x  x 1 erf x  16

Note that for the spherically-symmetrical potential functions

SP  SP   0 the  

Coulomb term can be written in a simplified form:   f   /   2 SP  2  m SP  L   C C    2 v f   1 2 2   v  m v v    e,i  v v 



 1v v



SP



 f     

    

(3.21)

The system in Eq.(3.14) with complete Eq.(3.15) or reduced Eq.(3.21) Coulomb terms can be simplified by using the Maxwellian potential functions in Eq.(3.20) everywhere in the domain, and exclude the need of calculating actual potential functions by solving the system of Eqs.(3.14)-(3.18) at each point in space. In this case the derivatives can be obtained explicitly. As an initial conditions for the boundary problem in Eq.(3.14) one can take a stationary isotropic Maxwellian function:

Eulerian Codes for the Numerical Solution of the Kinetic Equations of Plasmas, Nova Science Publishers, Incorporated, 2010. ProQuest Ebook

Semi-Analytical Adaptive Vlasov - Fokker-Planck - Boltzmann Methods  m   f  (t  0, x,  , v)  f M ( x, v)  n p x   2T p x    

3/ 2

 m v 2   exp   2T p x    

267

(3.22)

where n p x and Tp x are the specified profiles of plasma temperature and density.

3.1.3. General Splitting Scheme for a System of Kinetic Equations Let us consider the general numerical scheme of solving the non-linear system (3.14) describing evolution of three-dimensional electron and ion DFs. We are using a variation of the splitting scheme, which was first applied to the 1D2V collisionless kinetic equation in the work [24]. The operator-splitting methods [50] are based on splitting complex multidimensional operators into a series of simplifies terms, applied consecutively to the numerical function. As a result, time-integration of a complex equation is substituted with numerical solution of several simplified equations. A well-known example is the ADI method in 2D [51], where implicit 5-diagonal problem is replaced with two 3-diagonal sub-problems. The choice of operator splitting is also justified by its flexibility, particularly the ease of adding new terms into kinetic equations, and adding new equations into the system, e.g. kinetic equations for multiple ionic species or neutral particles [52]. The system in Eq.(3.14) can be formally written in the following form:

f 14   Ap  f  f in   x  v    [0, Lx ]  [0,U ]  [1,1] t p 1

(3.23)

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f (t  0)  F0 ( x, v,  )

here  is the simulation domain. Individual operators represent different physical process of action in different directions: f , x 1   2 eEx  A2 [ f e ]   2 f , v  m  v v  eEx  1   A3[ f e ]   1  2 f , v   m  A1[ f e ]  v



A4 [ f e ]  A5[ f e ] 



Lee  2   e  2 e f 1   2   e 1  e  f   v f   ,  v 2  v v     v 2 v  v v 2 v    1     

Lee  v

2

e

2



  2 e 1  e  f   f    v v   v  

 1  e 1   2  2 e   e  f    ,  2  2 2  v v     v  v   

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(3.24)

268

Oleg V. Batishchev

A6[ f e ]  A7 [ f e ] 

Lei  2  m i  2 i f 1   2   i 1  i  v f    v 2  v v  v 2 v  M v v 2 v  m   1     M 

Lei  v

2

 f        

  2 i 1  i  f   f    v v   v  

i

2



 1  i 1   2  2 i   ie  f      2  2 2  v v     v  v    f , A8 [ f i ]  v x 1   2 eEx  A9 [ f i ]  2 f, v  M  v v 





A10[ fi ] 

eEx  1   1 2 f ,  v   M 

A11[ fi ] 

Lii  2  i  2 i f 1   2   i 1  i  v f    v 2  v v  v 2 v  v v 2 v

A12[ fi ] 

   1     

Lii  v2

i

2



 f   ,     

  2 i 1  i  f   f    v v   v  

 1  i 1   2  2 i   i  f    ,  2   v v v  2 v 2     

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A13[ f i ]  A14[ f i ] 

Lie  2  M  e  2 e f 1   2   e 1  e  f   v f     v 2  v v     v 2 v  m v v 2 v  M   1     m 

Lie  v2

  2 e 1  e  f   f    v v   v  

e

2



 1  e 1   2  2 e   e  f      2   v v v  2 v 2     

In case of spherically-symmetrical potential functions, for instance when calculating collisions with fixed ionic background, the corresponding collisional operators can be simplified: A6M [ f e ] 

i i Lei  2  m M  2 M f  , v f 2 2 v v  M v v v 

A7M [ f e ]  

 1   1     v v 

Lei  v

2

i M

2



 f  .   

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(3.25)

Semi-Analytical Adaptive Vlasov - Fokker-Planck - Boltzmann Methods

269

The simulation domain   x  v   is covered with logically rectangular, nut nonuniform mesh of N x  N v  N  size. Importantly, the velocity sub-mesh is the same at every spatial cross-section. The spatial sub-space is covered with non-uniform (adaptive) grid. Often we start with a stretched mesh, if left boundary represents a physical wall: x1  0, x2 ,..., xk 1  xk  ax xk ,..., xN x  L ,

xk  xk 1  xk ,

(3.26)

k  1, N x  1

here the fixed factor ax  const  1   , (  1) is the grid stretching coefficient. The spatial mesh can be easily adapted to local gradients, moving with time [33]. The system in Eq.(3.23) can be approximated by the numerical splitting scheme, which can be mathematically written as the following system of n  14 finite-difference equations: f m 1 / n  f m



 



 1 f m f m 1 / n ,

(3.27)

   f

m p / n

f

m  ( p 1) / n









 p f m  ( p 1) / n f m  p / n ,

p  2,..., n  1

   f

m 1

f

m  ( n 1) / n









 n f m  ( n 1) / n f m 1 ,

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where index p denotes the intermediate results due to the "action" of a particular finitedifference operator in the splitting sequence;   tm1  tm is a variable time step, which can be further sub-divided. For example, to improve numerical accuracy or stability, the p -th step is sub-divided into n sub-steps: f

m( p 1 / n ) / n

 f m( p 1(1) / n ) / n



*



 p f

m( p 1( 1) / n ) / n

 f

m( p 1 / n ) / n

, ,

(3.28)

  1,..., n here  *   / n is the reduced time step. Next, all individual terms have to be approximated with proper finite-difference operators, p .

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270

Oleg V. Batishchev

3.2. Semi-analytical and Finite-Volume Approximation of Different Operators in the FP Kinetic Equation The operator splitting method produces a set of simplified equations. In some cases an analytical (or semi-analytical with frozen coefficients) can be obtained. This is our preferred approach, because it usually produces an unconditionally stable numerical scheme, which is better than direct implicit finite-difference approximation. Ideally, one would like to preserve basic properties of the system of kinetic equations in Eq.(3.14) by its numerical model. There are fundamental features, such as DFs never become negative, system's entropy never goes down, macroscopic thermodynamics laws remain valid, etc. There are also important properties of conservation of mass, momentum and total energy of the system, which are obeyed by different terms. For an analytical approach the interpolation procedure has to be chosen accordingly to comply with these physical laws. But if it is impossible to find a semi-analytical solution, we apply a more general finite-volume method. The finite volume approach operates with integral form of physical laws, rather than with commonly used differential form, e.g. Eq.(3.14). Instead of directly approximating derivatives, the equations are integrated over a cell (or bounding contour), followed by application of Gauss (or Stokes) law. To be applicable, the equations must be written in the divergence of a vector (called flux) form. Accordingly, in the finite volume method one has to define volumes of cells, or more generally, compounds of cells [19], and the fluxes crossing individual boundaries between the cells. Mass or number particles conservation is enforced if   the fluxes from cell k to a neighboring cell m , jkm , and from m to k , j m k , are mutually

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canceling out:

  jkm  jmk  0

(3.29)

We apply our approach to all operators listed in Eq.(3.24), and illustrate the accuracy of the numerical schemes on several benchmarking problems with known solution.

3.2.1. Free-Streaming in Space The spatial transport in free space (operators A1 and A8 ) is very simple f f  v 0 t x

(3.30)

and has an exact analytical solution. Because the same axisymmetrical spherical mesh is used at each spatial point, product vμ remains constant along the x -axis for any vi and μ j , vi μ j  v x  const .

Let us consider the continuous function f defined on the interval x0 , xN  of the axis,

which is covered by the non-uniform mesh

xi  :

x0  x1  ...  xi  ...  xN ,

i  0,1,..., N

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271

Semi-Analytical Adaptive Vlasov - Fokker-Planck - Boltzmann Methods

Initial values at moment t are given by a grid function { f i }iN0  0 , positively defined at these nodes. To find solution at the next moment t  t we use the analytical solution of Eq.(3.30): (3.31) f ( x, t   )  f ( x  v x , t ) . The value of the function at the node xi at the new moment is precisely the same as at the point x'i  xi   sh at the previous moment of time. Parameter  sh  v x defines the

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magnitude and direction of the shift of the entire profile of the original function. The main difficulty is that the point x'i is generally falls between the nodes, and the unknown function's value has to be interpolated using known values at the nodes. We are using linear interpolation. The numerical scheme is unconditionally stable. Importantly, it also preserves the positiveness of the (distribution) function for any  , if it was such at t  0 . The reason for this choice is that it conserves mass, momentum and energy (see Appendix A) in a closed system with periodic boundary conditions. In case of open system Dirichlet boundary conditions are used at both ends of the spatial domain. If point xi' is located beyond the boundaries, then the corresponding boundary value is used. The described algorithm is very simple, and was thoroughly benchmarked. For trivial stationary Dirichlet problems the accuracy is absolute. For non-stationary solutions, like propagation of a bell-shaped solitary pulse, the method gives satisfactory accuracy for L is average time step, and sufficiently smooth mean gradients, f  h  1 , here h  N L  x N  x0 . However, the results were poor on uniform and variable grids, when this method was applied to functions with sharp gradients. A possible cure is to use moving adaptive meshes instead of the stationary ones. The following simple 4-step algorithm that traces gradients and keeps the number of nodes fixed was proposed. 1) Using the known grid function  fi  define new monotonically growing function Si : S1  0 ,

i 1

Si   f k 1  f k , k 1

i  2, N

(3.32)

2) Divide the interval of S-values (at first iteration equal S N but can be less later) into equal intervals S  S N / N . Find new nodes

x 

' N j j 0

that correspond to function values

S 'j  jS that correspond to equal increments of the function S . At the same time we

control the maximum step not to exceed xmax  10  L / N : if the j -th step happens to be large, we set it to be equal to xmax , and find the corresponding value. The procedure 2) is then repeated for the remaining N  j  intervals.

3) After the preliminary mesh is constructed, we analyze and correct it's smoothness by applying an additional constraint on adjacent nodes variation:

Eulerian Codes for the Numerical Solution of the Kinetic Equations of Plasmas, Nova Science Publishers, Incorporated, 2010. ProQuest Ebook

272

Oleg V. Batishchev 1 / 1    

xn1

xn

 1   

(3.33)

here we use a small parameter   0.1 . 4) After the adapted mesh is constructed, the new node values are calculated using linear interpolation. As a result of these iterative procedure obtain mesh with node density inversely proportional to the gradient of the evolving function, x ~ f 1 . This approach can be used with spline-function of higher order than liner [53,54].

3.2.2. Acceleration in the Electric Field The effect of electric field on electrons and ions is described by operators A2 , A3 , A9 , A10 , which act independently in each spatial node xk . During the 2-nd and 9-th steps of the

splitting scheme the following equation is solved:





f 1  2  2 v E k f , t v v Ek 

e e ,i E x  x k  m e ,i

(3.34)

. To build a finite-volume approximation we integrate (3.34) over the inner

i, j  cell. For the left-hand side (lhs) we obtain:

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

fi ,j  fi, j f f di, j   2v 2dvi d j  Vi, j t t 

(3.35)

where fi,j denotes unknown DF at the next time layer, t   . Next we apply Gauss law to the integral of the rhs of Eq.(3.34) over di, j , using the following expression jEv  Ek f (see Appendix B) for the "flux" v -component, and omitting some indexes: 







      1  2 v Ek f di , j   divv jEv di , j   jEv nvi, j dS iv, j   jEv nvi 1, j dS iv1, j 2 v v





(3.36)



here nvi , j , nvi 1, j are unity vectors normal to the surfaces S iv, j , S iv1, j , respectively. The first integral in the rhs of Eq.(3.36) and the flux can be approximated as follows: v  v v v  jE nvi, j dS i, j  jEk S i, j 

jEv k 

fi , j  fi 1, j 2

Ek ~ j ,

where ~ j is the average cosine of pitch angle: Eulerian Codes for the Numerical Solution of the Kinetic Equations of Plasmas, Nova Science Publishers, Incorporated, 2010. ProQuest Ebook

(3.37)

Semi-Analytical Adaptive Vlasov - Fokker-Planck - Boltzmann Methods j

j

 j 1

 j 1

~ j   cos d   d sin   1   2j  1   2j 1

273

(3.38)

The v -components of flux are parallel to the normal vectors defined at the centers of surfaces. On the used mesh the following is true: vi 1  vi  vi  vi . Thereafter, the flux

 

through the surface is approximated with the second order,   O v2 , in modulus of velocity, As the coefficient of the velocity grid stretching av  v / v  const is the same throughout the mesh, the relative accuracy of calculations,  / v2 , remains the same. Thus, the fluxes are calculated equally accurately at low and high velocities (thermal bulk and suprathermal tail). The "lower" boundary condition for all i  1 points vanishes as the areas of the 

corresponding surfaces are equal to zero, S 0v, j  0 . The conditions of f  0 at v  0 and mass conservation in the system lead to the following "upper" boundary condition: 

vN v  U .

jEv N  0 , v

(3.39)

By combining Eqs.(3.35)-(3.39), the following 3-point finite-volume approximation of the terms A2 and A9 is obtained: Vi , j  ai, j fi 1, j   ai, j  ai 1, j   

V    fi, j  ai 1, j fi 1, j  i, j fi, j i  2, N v  1 , j  1, N    

(3.40)

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where the a -factors are: ai , j 

Ek ~ j 2



i  1, Nv  1

S iv, j ,

The set of Eq.(3.40) is closed with equations at the boundaries i  1 and i  Nv : V1, j   V1, j   f1, j  a1, j f 2, j   a1, j  f ,    1, j  V    aN 1, j  N v i, j v   

(3.41)

V    f N , j  aN 1, j f N 1, j  N v , j f N , j v v  v v  

The system of linear equations Eqs.(3.40)-(3.42) can be easily solved by the 3-diagonal solver [51]. For the numerical stability of the 3D solver it is sufficient for the modulus of the coefficient in the 3-D recursion  i to be less than unity. From Eq.(3.41) it follows that

1 

a1, j a1, j  Vi, j / 

1

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(3.42)

274

Oleg V. Batishchev

for any time step  . Next we request that the recurrent expression for i 1 (forward) not to be limited as well:

 i 1 

ai, j





 ai, j  ai 1, j  Vi, j /    i ai 1, j

1

(3.43)

which requires that the modulus of denominator is larger than numerator:

ai, j  ai 1, j  Vi, j /  i ai 1, j  ai, j

(3.44)

The latter puts a restriction on the time step:  Vi , j   , i  2, N v  1 .   2ai 1 

  min 

(3.45)

The 3-rd and 10-th steps of the splitting scheme solves the equation:



 

f 1   1   2 Ek f t v 

(3.46)

We use finite-volume interpolation using the same approach as for the previous term:

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







f i,j  fi, j f f di, j   2v 2 dvi d j  Vi , j  t t 

(3.47)



      1  1   2 Ek f di, j   div jE di, j   jE ni , j dS i, j   jE ni 1, j dS i1, j v 

The first integral in the rhs of Eq.(3.47) and the "flux" j E  1   2 Ek f (see Appendix B) in angular direction is approximated as:       jE ni , j dS i , j  jEk S i , j 

f i , j 1  f i , j

k

2

jE  1   j 2





(3.48) Ek 



here ni, j , ni 1, j are unity vectors normal to the "angular" surfaces S i, j , S i1, j , respectively 





Flux jE in orthogonal to surfaces S i, j and S i1, j . The equality  j 1   j   j   j is k valid by definition of how the grid is constructed. Thus, the angular projection of the flux jE



k

 

is approximated with second order, O  2 . The finite-volume 3-node approximation

for inner cells reads as: Eulerian Codes for the Numerical Solution of the Kinetic Equations of Plasmas, Nova Science Publishers, Incorporated, 2010. ProQuest Ebook

Semi-Analytical Adaptive Vlasov - Fokker-Planck - Boltzmann Methods

 V i, j bi, j fi,j 1   bi, j  bi, j 1    

275

    fi, j  bi, j 1 fi ,j 1   V i, j fi, j , i  1, Nv , j  2, N   1 (3.49)   

with b -factors defined as bi , j 

Ek 1   j

2



S ,

i  1, N  .

i, j

2

(3.50)

The “lower” j  1 and "upper" j  N  are vanishing because the corresponding surfaces 



have zero area, S i,1  S i, N  0 . Therefore, the boundary cell at

j  1 and

j  N

approximations couples 2-nodes only:  Vi ,1   Vi ,1 bi ,1 f i ,2   bi ,1  f i ,1   f ,     i,1     bi , N 1  V N vi, j    

(3.51)

    fi, N  bi , N 1 fi,N 1   V N v , j fi, N       

The linear system of Eqs.(3.49)-(3.51) is solved using standard 3-diagonal solver. For numerical stability it is sufficient to have diagonal dominance, or back substitution factor is less than 1. From Eq.(3.51) one has:

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N 

b1, N  b1, N  1  Vi , N  / 

1

(3.52)

for an arbitrary time step   0 . Using recurrence equation for  j 1 (reverse), we obtain the stability condition:

 j 1 



 bi, j 1



 bi, j  bi, j 1  Vi, j /    j bi, j

1

(3.53)

Because the denominator is positive, we find that    bi. j  bi, j 1  Vi, j /     j bi, j  bi,. j 1 ,    

This sets limiting condition on the time step  :  Vi , j   , j  2, N   1 .   min   2b j   

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(3.54)

(3.55)

276

Oleg V. Batishchev

Acceleration in the electric fields terms were benchmarked. The first test was to check the conservation of the number of particles (mass). It was on the order of the computer accuracy, ~14 digits after the floating point.

Figure 13. One-dimensional cross-sections (scaled to fit) of the analytical (solid) and calculated using finite-volume approach (solid line) electron DF. The plots, starting from the top, correspond to crosssections of the DF at different pitch angles 90o(   0 ), 60o(   0.5 ), and 0o (   1 ) after the

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acceleration. Analytical and calculated curves are overlapping on this scale due to very low relative error  10 4 .

Next we calculated acceleration of an arbitrary distribution in uniform electric field. The shape of the distribution in v -space has to remain the same, while drift velocity grow linearly with time. The results for electrons with initially stationary Maxwellian distribution are presented in Fig.13. The final velocity with negative E|| is equal to 4vT . The relative accuracy of the numerical result is ~ 10 4 . Because there are constraints in Eqs.(3.45)-(3.55) on the maximum time step, for strong electric fields the direct shift of the distribution function is often used instead of the finitevolume approximation. Logically it is equivalent to the method used for one-dimensional free-streaming operator, and it is also unconditionally stable. However, in two dimensions automatic conservation of the DF's moments is no longer valid in the curvilinear v,  coordinates, and one has to use a correction procedure, e.g. the iterative one described in [45].

3.2.3. Approximation of the Coulomb Collisional Term

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277

We describe in details the discretization of the Fokker-Planck sub-operators A4  A7 , A11  A14 , which are calculated using Rosenbluth potentials  and  . We include Coulomb

electronic and ionic collisions in all 4 possible combinations. First, four potentials  e,i and  e,i have to be calculated from the system of two coupled Poisson equations: 1   2  v v v 2 v 





 1       1 2  f ,  v 2      

1   2  v v v 2 v 



(3.56)



 1       1 2    v 2      

We integrate the first equation over the volumetric cell i, j  . The integral of the rhs is approximated as: (3.57)  f v, , x  constdi, j  fi, jVi, j , Formally, see Appendix B for details, the lhs of Eq.(3.56) can written in the divergence of a vector form to apply the Gauss law:  4    divv,  jv,  di, j    jv,  nn dSn n 1

(3.58)



here nn in a unity vector normal to the center of the corresponding surface, Sn , which is one 

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of the 4 facets of ring-shaped cell i, j  ; jv,  denotes the diffusive flux of function  (see  divv,  a in Appendix B):

   1  2   jv,   ev  e , v v 

(3.59)

  here ev and e are the unity vectors along v and  directions, respectively. Because on the

used grid (Fig.12) these vectors are orthogonal to the corresponding surfaces, the rhs of Eq.(3.59) can be simplified: 

1   2    1   2     v    dS i , j   dS iv1, j   dS   dS v v v  i , j v  i , j 1

(3.60)

 

The first and second integrals can be approximated with the second order of accuracy, O v2 as the point

vi

is dividing segment [vi , vi 1] in half by the grid's construction: 

    v  dS  S iv, j i 1 i v i , j vi 1  vi

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(3.61)

278

Oleg V. Batishchev 

    v  dS i 1, j  S iv1, j i 1 i v vi  vi 1

The second integral in Eq.(3.61) cancels out for i  1 , as the area of the surface is equal 

to zero S 1v, j  0 . Prior to approximating the two last terms in Eq.(3.60), we simplify the angular fluxes: 1   2  1 2  v  v

 1   2 



 , v

(3.62)

The last term in Eq.(3.62) is approximated :  v



1 2  1 , v  2  1

(3.63)

and because points 1 and 2 correspond to the same value of velocity modulus v , the 3rd term can be approximated as follows:



 1  1 i 1, j  1, j i , j 1  1, j 1   2    , j  1, N   S dS  S  i , j v arccos( i, j v v  i , j    i j 1 )  arccos( j ) i j 1 j

(3.64)

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Similarly, the 4th term approximation is:



 i 1, j  1, j 1   2    1 ,  S dS i , j 1 v arccos( v  i , j 1 i j 1 )  arccos( j )

j  1, N 

(3.65)

Both integrals are nullified at the boundary point in  -direction.. Because the grid is uniform

 

in pitch angle  the approximations have second order O  2 . By combining expressions in Eqs.(3.61), (3.64) and (3.65) and omitting spatial some indexes (operators act in velocity space at each spatial point) we obtain a 5-point finitevolume approximation for both potential functions: ai, ji 1, j  ai 1, ji 1, j  ci, ji, j  bi, ji, j 1  bi, j 1i, j 1  Vi, j fi, j ,

(3.66)

i  2,Nv  1 , j  2, N   1

The expressions for coefficients for the inner cells: 

ai , j  S iv, j

1 , vi 1  vi

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(3.67)

Semi-Analytical Adaptive Vlasov - Fokker-Planck - Boltzmann Methods bi , j  S 



i, j

279

1 1 , ci, j  ai, j  ai 1, j  bi, j  bi, j 1 . vi arccos( j )  arccos( j 1 )

and for the boundary cells: ci, j  ai, j  ai 1, j  bi, j ,

i  2, Nv  1 ,

j 1 .

ci, j  ai, j  ai 1, j  bi, j 1 ,

i  2, Nv  1 ,

j  N .

ci , j  ai, j  bi, j  bi, j 1 ,

i 1,

j  2, N   1 ,

c1,1  a1,1  b1,1 ,

c1, N   a1, N  1  b1, N  1 .

(3.68)

As a boundary condition at i  Nv one can use integral form of the potential functions  and  to calculate it exactly. However this requires calculating integrals with 1/ v singularity. Therefore, we use asymptotical expressions for the potentials. As f U   0 for U   , for U  vT we have the following limits for the Maxwellian potential functions:

 v  

 v =  n p vT

np vT

 v v  T

0 

 v erf '   vT

 n n  =  p erf (v / vT )   p ,  vT 4v / vT 4U 

  v  v   2      vT  vT 16

  

1 

erf  v  v   T

  

(3.69)

 n pU ,

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n p and U in the rhs are the dimensionless plasma density and maximum velocity. Plasma density (and other moments of DF) are calculated as sums over all velocity cells: n p   f i , jVi , j .

(3.70)

i, j

The linear 5-diagonal system in Eqs.(3.66)-(3.68) was inverted using standard fully implicit iterative solver from the NAG library [55]. The numerical calculation of potentials was tested on drifting u||  0 Maxwell distributions by comparing against analytical expressions for Maxwellian potentials in Eq.(3.69). The benchmarks were done for the following plasma and mesh parameters: u||  2  4vT and v1, j  0.01  0.04vT , U  10  20vT . The grid had 129  65 dimensions in v   . Because the second potential depends on the first, to calculate  with 2-3 digits of

accuracy the first potential had to be calculated with 5-6 digits. The calculation of potentials takes the majority of computational time. Potentials themselves are not used in the collisional term, only first and second derivatives. As a further test we checked that collisional drag and diffusion fluxes, which are the complex functions of the potentials' derivatives, cancel each other for the Maxwellian

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Oleg V. Batishchev

distribution. It was found that the total flux amplitude is on the order of 10 3 of each individual component forming the flux. After calculating potentials, the collisions can be calculated. Let us consider finitevolume discretization of the terms in the Coulomb operators. Note, that approximations of non-linear, A6 , A7 and linearized Coulomb terms, A6M , A7M , are very similar. The only difference is that the complete non-linear collisional fluxes are more complicated functions of the potential derivatives and DFs. Here we describe a more compact linearized operator, which can be written in the divergence form: f  Cv,   divv,  jvCM , , t

(3.71)

of the two-dimensional fluxes (see appendix B):  CM   1  2  f   2 f   j v,   L a f 2 ev  L e .  v v v  v v  

(3.72)

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To solve Eq.(3.71) one can use a 5-point approximation for the entire operator. In this case the finite-volume approximation will be very similar to one just described for the potential functions. However, here we consider two 3-point approximations for each of the operators A6 and A7 . In this case the logic of the building finite-difference approximation is repeating the one used to discretize acceleration in the electric field term. Accordingly, we omit the intermediate steps and present here the final form of the finitevolume approximation for operator A6 : fi ,j  fi , j







i  1, Nv , j  1, N 

Vi, j   jvCi S iv, j  jvCi1 S iv1, j ,

(3.73)

The first term of in the rhs of Eq.(3.73) can be approximated as:   f f f f   jvCi S iv, j  S iv, j  a i i 1 i  i i 1 i  , 2 vi 1  vi  

(3.74)

with  - and  - coefficients defined as:

i  i 

1 vi  2  vi

i 1  i vi 1  vi

,

(3.75)

  i  2  i 1  i 1  i    i 1  i  i  i 1  1    .    v v  vi 1  vi  vi 1  vi 1  vi 1  vi vi  vi 1   i  2 i 1

By additionally defining Ci 

a i v  S L, 2 i, j

Bi  

i

vi 1  v



S vi, j L ,

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(3.76)

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281

and uniting two surfaces, after simplifications, arrive to the following 3-point approximation of A6 :



Ci  Bi  fi1, j   Ci  Bi  Bi 1  Ci 1  

Bi 1  Ci 1  fi1, j 

Vi, j    fi, j   

Vi, j



fi, j

(3.77) i  2, Nv  1 , j  1, N  ,

which can be resolved by the 3-diagonal solver. As a boundary condition at the outer boundary v  U we use mass conservation, which implies that there is no incoming and outcoming particle flows trough the surface, jvCU  0 Similar to the procedures used for operators A6M and A3 , and utilizing formulas from Appendix B, the last operator in the splitting scheme Eq.(3.24) is approximated with the following 3-point finite-difference:  V i, j C j fi,j 1    C j  C j 1    

    fi, j  C j 1 fi ,j 1  V i, j fi, j , i  2, N v  1 , j  1, N    

(3.78)

where we used the coefficients 

Cj  S L

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i, j

 i, j

1

2vi arccos( j 1 )  arccos( j ) 2

,

(3.79)

  i 1, j 1  i 1, j 1  i 1, j  i 1, j  .  vi 1  vi 1  

 i. j  

The lower and upper boundary conditions at   1 are excluded as the corresponding 



areas equal to zero, S i,1  S i, N  0 . The order of approximation of operators A6M , A7M is formally first because of vv and  v terms. However it is possible to show that that the order



O av  1v  v 2



. Because the constant coefficient of velocity mesh stretching,

av  1   ,   1 . Thereafter, for small values    the approximation order is quasi-

second. Few comment regarding numerical stability. We find experimentally that for variable meshes with first step in modulus v is in the range v1  0.01  0.04vT , maximum grid velocity U  10  20vT , numbers of nodes Nv  100  200 and N   50  100 , the stability maintained

for time steps up to  ~ 10 C . So, locally the time step was divided in the  / 10 C sub-steps as required by the plasma collisionality n p / T p1.5 , which can vary by several orders of magnitude along the magnetic field line.

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Oleg V. Batishchev

To benchmark the Coulomb solver we performed series of tests [45,53]. First of all, the method preserves the number of particles. Numerically, mass conservation was maintained at the computer round-off level, ~ 10 12 . Next we have tested temperature equilibration for an electron distribution, which was initially close to the Maxwellian. The characteristic time of the exponential TII and T equilibration is known theoretically [11,31]. An example is presented in Fig.14a for plasma with

parameters

n p  7  1013 cm3

and

T p  (TII  2T ) / 3  2.83eV .

The

plasma

equilibration time for such plasma is approximately tee  5 10 9 s

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Figure 14. a) Left chart shows exponential parallel and perpendicular temperatures equilibration of the near-Maxwellian electron distribution versus normalized time. b) Right curves show the relaxation of the normalized temperatures with sporadic (once per 10 time steps) update of the nonlinear Rosenbluth potentials. The dynamics is corrupted, but the same steady-state is achieved.

Numerical experiments showed that all collisional times - slow-down, axial and perpendicular diffusion, energy exchange - in any binary combination of electron and ion species was accurate within 2-3 digits. It was found that the distribution will equilibrate even when the non-linear Rosenbluth potentials are updated sporadically, Fig.14b. The temporal dynamics of equilibration is distorted, but the final equilibrium is the same, while the calculations took 10 times less CPU time. The next test is a self-consistent relaxation of an arbitrary DF to the Maxwellian equilibrium. In Fig.15a we present the results for plasma with n p  1013cm3 and Tp  16.5eV . As one can see the convergence at low energies E happens much faster than at

high energies, as the rate of Coulomb collisions reduces as E 3 / 2 . We find that the continuous Fokker-Planck method allows increasing time step as the system approaches to equilibrium. This helps to drastically (by many orders) accelerate the relaxation of the energetic tail. Electron-ion energy exchange times contain a small parameter, me / M i  0.0005 . In Fig.15b we present the temperature evolution of electrons due to collisions with slightly hotter, 1.1T0 , and colder, 0.9T0 , ions for different mass ratios: M i / me  1 (positrons), 10 3 (~protons), 10 6 (heavy molecules). As one can see the dynamics is the same, but the time

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scale is increased in the direct proportionality to the mass ratio. Note that commonly used ion mass to expedite calculations in PIC codes is M i  100 me due to the fixed time step. Variable time step was used in the calculation of the relaxation to Maxwellian of arbitrary electron DF. But mostly efficient variable time step for the multi-species relaxation.

Figure 15. a) Left chart shows relaxation of an arbitrary electron DF to a Maxwellian, which is marked by thick dashed curve, which is a straight line in the log-linear scale. Cold thermal core relaxes very quickly, t  10 -8 sec , while it takes supernatural tail t ~ 10 3 sec to equilibrate. b) Right chart shows electron-ion thermal equilibration with artificial mass ratios M / m  1 , 10 3 and 10 6 . The curves have the same shape, but linearly shifted by exactly 3 orders in time in the linear-log scale.

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The next benchmarking problem was to calculate 4-tempearture relaxation of electron-ion plasma. Two non-Maxwellian distributions with different temperatures, Ti  TIIi  Te  TIIe we allowed to evolve under mutual Coulomb interaction. The real proton/electron mass ratio was used, M i /me  1836 . The total number of time steps was just a 100. In this calculation we used an adaptive time step τ(t) with large variation  max /  min  10 4 . All three characteristic relaxation times were accurate within 1% , Fig.16a. The relative mass, momentum and kinetic energy conservation was ~ 10 -6 . An iterative correction algorithm [45] was used in this case. All presented tests are important, but all of them are partial. Unfortunately, there are almost no situations with known analytical solutions of the kinetic equations with Coulomb collisions. One special class represents non-equilibrated asymptotic self-similar solutions [43]. Those were successfully simulated within the computer accuracy using early version of the described Fokker-Planck model.

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Figure 16. a) Left plot of 4-temperature relaxation of the electron-proton system with all four combinations of Coulomb pair collisions: e-e, e-i, i-i, and i-e. Initial distributions were far from equilibrium. Electron equilibration is the fastest, followed by ion-ion, and then mutual thermalization. This calculation were done with adaptively increasing time step. b) Right plots show stationary algebraic self-similar solutions, obtained for n pT p1.5  const and n p / T p  const with large spatial variation of collisionality, big simulation domain with stiff boundary conditions, T2  T1 .

We have simulated a few special cases with known asymptotic steady-state self-similar solutions [56, 57]. The latter are formed between half-Maxwellian distributions with large variation in temperature to resolve non-exponential algebraic, f e  v  , supra-thermal tail. The relevant calculations were performed [33, 53] and prove to be very demanding both for computational resources, grid resolution ( E / T  1000 ) and method accuracy. The main problem is that they required huge variation of plasma collisionality, TL1.5n0 / T01.5nL  106 , along the computational domain, L . Only asymptotical analytical prescription exists, there is no solution for finite lengths and temperature variations, making the comparison incomplete. In Fig.16b we present results of simulations for   6 and   1 cases.

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3.2.4. Calculation of the Ambipolar Electric Field Calculation of the ambipolar field is not trivial. Coulombic forces cause particles scattering and they also create electric fields on small,  D , scales, and large,  D , scales. The latter cannot be effectively calculated from the Poisson equation. We design a numerical procedure, which allows relatively simple calculation of the ambipolar fields on the macroscopic scales. It is based on the following physical analysis. The plasma is quasineutral, ne  ni  n p . On characteristic times, t pe1  1 , and scales, lD1  1 , there should be no charge accumulation. This means for the 1D geometry that the

total current is equal to zero:   e jII   e  vII f dv  0

 e,i

 e,i

(3.80)

If the current at t  0 was zero, then its derivative remains zero as well. The latter is  derived by multiplying the system of Eq.(3.1) by vII , integrating over v and summing the equation for electrons and ions (momentum balance):

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 e

 e,i

 jII P  e n E    e ( II    II   vIIC dv )  0 t x m  e,i

285

(3.81)

here jII  n uII is the parallel particle flux and PII is the axial component of pressure tensor divided by the mass of  -species;  vIIC  is their friction against opposite species only (no self-force due to Newton's 3rd law) . Combining Eqs.(3.80-3.81), we arrive to the following expression for the parallel ambipolar field: EIIA ( x) 

Note that





mM  i  PII  PIIe   vIICe    vIICi   en p ( M  m)  x 

(3.82)

mM  1   ,   0.0005 . The mass correction exceeds the method accuracy (we M m

target 2-3 digits of code's accuracy). For approximately equal electron and ion temperatures roughly the same magnitude factor,

m   , enters ionic pressure and friction. This is M

physically sound, because the electric current is predominantly carried by light electrons. Therefore, it is often assumed that eEIIA ( x)  (1   )IITp  TpII ln n p ,

(3.83)

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in the assumption that, PIIp  Tp n p , Te  Tp  Ti ; thermal force factor   0.71 [32,47], a  measure of the electron-ion friction contribution:  vIICedv  n IITp . These results obtained under and assumption that the plasma DFs are close to the Maxwellian equilibrium. In a general case one can try applying Eq.(3.82), but it will require calculating integrals over velocity grid at every spatial point. At the same time we are using operator-splitting scheme in Eq.(3.24), and it seems appealing to utilize benefits that this multi-step method gives. Namely, operator-by-operator integration with discrete time step t gives us knowledge about the DF before and after a particular operator S action on the particle ensemble. Hence, it is easy to calculate the change of momentum caused by this operator action on a given species at every spatial node n:  ps n  m  vx [ f ( xn , t )  f ( xn , t  t )]dv

(3.84)

Furthermore, change of parallel momentum divided by time interval is the axial force, FII  p II / t . It appears that the electric force (ambipolar) must balance out all other forces to prevent particle flow acceleration, creation of currents leading to the prohibited charge misbalance. Therefore, with the accuracy of electron/ion mass ratio (42 times slower than electrons. . Note, that this scheme does not guarantee quasineutrality. To maintain it, an additional step must be included:

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    f i dv f e (t , x, v )  f e (t , x, v )    f e dv

(3.86)

The purpose of this step is to scale electron density to exactly match ion (plasma) density, while preserving the unique shape of the electron DF, which may have elevated or depleted tails, and other features.

3.2.5. Modeling of Non-local Heat Transport in Plasma After testing separate operators from the splitting scheme, a number of test runs that include Coulomb collisions and ambipolar field effects have been performed. The main focus was to study the non-local heat transfer in plasma with steep gradients of temperature and density. One of the first goals was the reproduction of thermal force coefficient,   0.71 , and e electron Spitzer-Harm thermal conductivity. q SH  3.2

Ce T XT p

q efs , which relates it with the

free-streaming electron heat flux, q efs  n p vTe T . Though both results are valid in the longmfp limit, it may confirm validity of the numerical model as such calculations require interaction of many terms of the kinetic equation. Derivation was based on linearized electron collisions with stationary ionic background. This helps to simplify simulations. The only

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important requirement are that the system has to be close to equilibrium, much bigger than the collisional Coulomb mfp, X  Ce , with low temperature variation along the region, T  T . The main controlled parameter is the Knudsen number, Kn C 

Ce L



T p2 npL

, which is

well-defined as plasma parameters variation along the simulation domain [0, L] is small. We assume that the "imported" half of DF, coming from the boundaries, is Maxwellian. Because we want to exclude the force associated with pressure gradient, we set the pressure equal at the boundaries, n pT p  const. Thus, the stationary boundary conditions are as follows: p e, i f eim , i ( x  0, v x  0, v )  FM (n0 , T0 , u x  0)

(3.87)

1 p e, i f eim , i ( x  L, v x  0, v )  FM (n0 , T0 , u x  0)



where  is the second another free parameter controlling temperature gradient, which was fixed   1 in the first series, when only collisionality was varied. Results are shown in Fig.17a. It was found, that the parallel heat flux, qII , is approaching the Spitzer-Harm, qSH , value, and the thermal force coefficient   0.71 , as the Coulomb Knudsen number becomes smaller, KnC  0 . Calculations showed that at K n  10 3 the flux is within 1% from the theoretical value on a mesh of moderate size, N x  N v  N   41x129 x33 . At

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K n  0.1 it is substantially less than the theoretical one, which demonstrates the flux-limit

effect. Note, that the heat flux converges to the theoretical limit slower than the thermal force factor, because it is the highest of all, 3rd moment of the DF in v . Next we varied the parameter   0.1, 0.5, 0.9 , and Knudsen numbers, K n  1 / 16, 1 / 4, 1 (parameter  ). The normalized parallel heat flux is presented in Fig.17b. One can conclude that the described continuous Fokker-Planck method gives the expected short mf. results in the  ,   0 limit. For finite values, however, the discrepancy is big. This supports the need for kinetic modeling in the situations when either the Knudsen number is not small enough, K nC  10 -3 , or the spatial gradients become large, and their characteristic length begins competing with L redefining effective K nC . The kinetic modeling is also supported by the results of comparison with the integral approximation [56], which indicate the accuracy of 30% and more for K nC  0.1 .

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†

Figure 17. a) Left pane shows variation of the i) normalized electron heat flux, qII / qSH , and ii) thermal

force factor,  , as functions of Coulomb Knudsen number,   Kn C . For small Kn C , qII / qSH  1 , and   0.71 . For any finite collisionality the results differ from the theoretical limit. b) Right pane shows normalized flux as a function of parameter   T / T p for different plasma collisionalities

  0.0625 , 0.25, 1 .

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3.2.6. Comparison of Kinetic and Fluid Model of Non-linear Heat Wave A non-linear heat wave propagation is a fundamental physical problem that was studied extensively [59]. The hydrodynamic (fluid) description operates with plasma densities, velocities and temperatures. In the simplest formulation the heat propagates on a fixed background. The one-dimensional equation of a nonlinear heat transfer in the absence of volumetric sinks/sources and with the classical Spitzer-Harm heat conduction coefficient reads as: Te  T (3.88)  aTe5 / 2 e  0 t x x here a is a normalization parameter. The domain is half-space, x  [0, ] . We set the following boundary condition at the x  0 boundary: Te ( x  0, t )  (To5 / 2  t ) 2 / 5

(3.89)

where T0 is the initial temperature, uniform in the domain;  is the normalization parameter, defining the rate of plasma heating at the boundary. Eq. (3.89) can be associated with plasma heating by a fixed power external source. One possibility is laser heating. There is an additional analytical result, so-called DLM distribution [60], which can be used to benchmark relevant Fokker-Planck models [61].

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The system of Eq.(3.88)-(3.89) was integrated numerically. The profiles of the calculated solution at two different times is presented in Fig.18a.

T/T0

9 t=1700 

5 t=360 

1

0

T/T0

9

t=1700 

5 t=360 

x/ 

2000

4000

1

0

2000

x/  4000

Figure 18. a) Left pane shows normalized temperature profiles T /T0 obtained with classical non-linear

heat conduction model. Two profiles correspond t  360 and t  1700 , with  being electron collisional time calculated for the background plasma. A shock-like profile with sharp front is formed. b) Right pane shows similar T /T0 profiles obtained with non-linear kinetic model. The dynamics of the bulk is suppressed (flux-limiting), while there is a fast moving precursor (pre-heating) in front of the main heat wave.

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The profiles resemble a well known self-similar shock-wave-like heat wave propagating on the undisturbed cold background plasma. They must to be compared to the "kinetic profiles", calculated with the continuous Fokker-Planck model described above. The kinetic formulation exactly duplicates the fluid Eq.(3.88). The evolution of the electron DF is described by the kinetic equation: f e f f  v x e  E x e  Ce t x v x

(3.90)

Collisional term in the rhs includes electron Coulomb interaction with electrons and ions. The boundary condition is identical to Eq.(3.89): the incoming DF is a half-Maxwellian, non-drifting, with rising temperature: f e (t , x  0, v x  0, v )  FMe (n0 , T ( x  0, t ), u x  0)

(3.91)

There are two restriction applied on DF, which are dictated by the quasineutrality and ambipolarity constraints: ne  n0  const,

jx  0

(3.92)

which are maintained using procedure used for the ambipolar field calculation. Why the kinetic solution is so different from the fluid one if plasma iv very collisional, characteristic length of heat wave is large,  1000 C , the Coulomb Knudsen number is very small KnC  0.001 ? The answer is in the non-local kinetic effects, which cause the heat

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conduction coefficient to vary in the 0.1  100 range with respect to the theoretical short mfp value, see Fig.19. The assumption of DFs being close to the Maxwellian breaks due to non-local transport. Electrons that carry the majority of the heat flux have high energies E  6  9T . For such particles the effective Knudsen number is  40  80 times bigger than for the thermal electrons. The suprathermal tail of the electron DF can be either enhanced or depleted with respect to equilibrium, depending on actual profiles of plasma parameters. This can be seen from the two-dimensional contours of the electron DF plotted in Fig.20.

q/qSH

10 2

t=1700 

10 1 10 10

T/T0

0

t=360 

x/ 

-1

0

2000

4000

Figure 19. Normalized temperature profiles, T /T0 , (solid lines) obtained in the kinetic calculations

E /T

Fe(x=2000)

T

E /T

heat conductivity inside the heat wave, and large increase on its front, which is responsible for the preheating region preceding the main heat front.

T

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overlapped with corresponding normalized heat fluxes, qII / qSH , (dashed) showing reduction of the

10

0

Fe(x=1000)

10

-10

0

E II /T

10

0

-10

0

E II /T

10

Figure 20. Contours of the f e at time moment t  1700 and different spatial locations, left-to-right:

x  1000  (main heat wave), 2000  (front) and 3000  (precursor).

The big difference between electron DFs at different locations is the fundamental reason for the observed non-local kinetic effects. Inside the heat wave ( x  1000  ) the suprathermal tail is depleted, perpendicular temperature is higher than the parallel, T  TII , and heat conduction coefficient is reduced, K  K SH (flux-limit). In the intermediate section

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( x  2000  ) T  TII and K  K SH . In the pre-heating region the suprathermal tail is enhanced ( x  3000  ) T  TII and K  K SH .

4. Continuous Boltzmann Method for Neutral Particles The fundamental difference between Coulomb interaction of charged particles and Boltzmann collisions of neutral particles is the fact that the first one is predominantly longranged, while the second one is fully defined by short-range interactions. Fully ionized plasma systems can be described with Fokker-Planck equation accounting for low-angle collisions (diffusion in velocity space). Rarified gas can be only described by Boltzmann collisional integral, describing instantaneous "jumps" in velocity space due to binary collisions. As we saw in plasma the Coulomb cross-section is inversely proportional to the energy squared,  p  E 2 . In a gas the cross-section is almost constant,  g  const . Therefore, mean-free-paths in plasma and gas depend differently on particle energy E :

p  E2 ,

 g  const

(4.1)

Accordingly, the collisional times behave very differently:

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 p  E 3/ 2 ,

 g  E 1/ 2

(4.2)

In plasma cold particles are more collisional than energetic. In gas cold particles are less collisional than hot. Thus, kinetic effects are more important for the bulk (majority of the population) rather than to suprathermal tail. Also one may expect less collisional effects in the pure gas than in pure plasma. The situation changes when gas is not completely ionized in case of low-T plasma, gas discharges, etc. With that said, rarefied gases require kinetic description. Traditional approach is to use Monte-Carlo models as DSMC [23,35]. The continuous Boltzmann models for gas are much more scarce [7,62], but they relate to the MC methods the same way continuous Fokker-Planck model relates to the discrete PIC method.

4.1. Splitting Scheme for Gas with Boltzmann Collisions Let‟s now consider the following kinetic equation [62] describing the evolution of a 4dimensional distribution function, f (t, x, y, v x , v y ) , of the inter-colliding gas particles: f f f f f  vx  vy  St  M t x y ( f )

(4.3)

Note that though the collisional operator St at the rhs is taken in the BGK form [6]. Due to the relaxation time   f  dependence on f , it represents a wider range of operators.

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The following comment is appropriate here regarding the BGK approximation. Formally, any arbitrary kinetic equation df  St (4.4) dt can be re-written in the BGK form. Indeed for f  0 (true for Maxwellian distribution), one can make the following transformation: df f f  St  St  * dt f τ (f)

where we have formally introduced the relaxation time τ *(f) 

(4.5)

f . St( f )

To simulate (4.3) we first define the distribution function f on a uniform grid, blockrectangular in space, usually square in velocity space. The only requirement is that it has to be exactly the same at every spatial location. We have to introduce maximum velocity to be resolved, VM  aVT , which is set equal to a few thermal velocities, VT2  kTM / m , where m is

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the mass, and TM is the maximum temperature of the particles in the simulation domain. Because we anticipate the distribution function to be close to the 2V Maxwellian,   2 n0 (v  U 0 ) fM  exp { } , with the magnitude of drift velocity U0 on the order of VT ,  VT2 VT2 the practical value a  5 guarantees a small unresolved fraction n  10 6 n0 of the entire particle population. Next we apply an operator-splitting scheme [50] to the equation Eq.(4.3), similarly to the system (3.23). This yields us the following set:  f  t   f  t   f  t   f  t

f  0, x f  vy  0, y f  ,  vx





fM ( f )



,

(4.6a) (4.6b) (4.6c) (4.6d )

where f M ( f ) in the step Eq.(4.6d) denotes Maxwellian distribution with the same first three moments as the lost distribution in sub-step Eq.(4.6c) at the same x-y location.

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4.2. Use of Analytical Solutions for Sub-steps Let us consider the first step in Eq.(4.6a) describing free transport in x-direction. It has an analytical solution at the time t   , which is expressed via the [known] solution at the time t : f (t  , x)  f (t, x  v x ) . We may now utilize the fact that we have the same velocity grid at every spatial position, and the spatial step, h , is constant. Therefore, the function displacement, d  v x  , within the interpolation interval will stay the same for all points at the line v x  const, y  const . It‟s easy to show that the first 3 moments will be preserved if one uses linear interpolation. Indeed, the linearly interpolated value is f k  (1  s)fi  sfi1 . We find that the total mass over the whole [periodic] interval is preserved: k f k  i {( 1  s)fi  sfi1}  i fi . The conservation of the momentum projection and energy results from the fact that v x is the same for all nodes involved. Unconditional stability results from the observation that 0  s  1 and, hence, max f k  max f i . The same approach is applied to the step in Eq.(4.6b). Similarly, we may find an analytical solution for step (4.6c): f (t  )  f (t ) exp{

 } 0  ( f (t ))

(4.7)

Note that in the   0 limit this expression is equivalent to the explicit scheme. Eq.(4.7) and the linear interpolation guarantee the non-negative values of the distribution function f k

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for any time step   0 provided that f was positive at the previous step. The energy conservation for the consecutive collisional steps in Eqs.(4.6c) and (4.6d) follows from the fact that incoming Maxwellian distribution has exactly the same first moments as the distribution of the lost particles:     v f M dv    v f (t ) (1 - exp{



 ( f (t ))

 } )dv,   0,1,2 .

(4.8)

Note, that the moments have to coincide within computer accuracy to assure the full conservation of the method. This method is efficiently parallelizable on a Massively Parallel platform; in [52] it was coupled with a more complex Fokker-Planck collisional term.

4.3. Benchmarking of the Method The initial benchmark set comprised conservation laws and convergence to the equilibrium state. The accuracy of  10 12 has been demonstrated. More elaborate cases, with known theoretical and experimental results, are described below. i)

Shear wave damping. The v y component of the equilibrated gas in a double-periodical box was modulated with a sinusoidal spatial profile. The decrement of the shear wave,

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Oleg V. Batishchev , for the spatially isothermal and uniform gas with drifting Maxwellian distribution is given by the expression   2 2 2 /  2 , where  is a mean-free path, and l is a wavelength of the disturbance of the gas velocity. The maximum of the shear flow velocity, Vmax, decays exponentially (Fig.21a), in good agreement with the theory [63,64]. Numerical value of the decrement agrees with the theoretical one for the longwave harmonics when the time step is comparable with the fixed collisional time. In Figs.21b we present the ratio of the two decrements for different time steps and spatial resolution, as well as the difference between the proposed scheme and the LaxWendroff scheme [51].

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ii)

Low Re number back-step flow. The experimental data [65] for back-step flow give a certain ratio between the reattachment length and step height as function of the Reynolds number. For low Re number the ratio converges to ~1.2 value. The same ratio could be deduced from Fig.22a, where we present the streamlines for the calculated stationary back-step flow. Note that the non-slippery isotropic reflection boundary condition at all material boundaries has been imposed for this case.

Figure 21. a) (left) Exponential decay of the maximum velocity of a periodical shear flow. b) (right) Ratio of calculated decrement to the analytical value as a function of time step for different numerical methods and grid resolutions.

One practical example. We are interested in rarified gaseous propellant flow in the RFdriven plasma thrusters, e.g. described in [66]. Such devices can utilize light (H2, He), medium (N2, Ar) and heavy (Kr, Xe) atomic weight gases. They typically operate in a mixedcollisional Poiseuille-Knudsen regime due to strongly varying gas density and cross-sections of the flow. Our model shows a good agreement with both experiments and the semianalytical models developed for such conditions. As an example we present in Fig. 22b the contours of gas density and streamlines for the calculated flow in a system of pipes. The Knudsen number varied from 0.1 to 10, while the average Reynolds number was about 5. The

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simulation shows the eddy formation, the gas acceleration to the sound speed at both orifices. The departure of the gas distribution function from the equilibrium was observed. In a concert with expressions (4.1-4.2) the tail of gas DF is more equilibrated than the thermal core. This shape is an opposite to the electron DF presented in Fig.20, which is characterized with equilibrated thermal core population and strongly non-Maxwellian suprathermal tail. 4

0.5

3

fN

2

0.4

vy/V T0

1 0 -1

0.3

-2

y

0.05

-3

Y

0.04

0.2 0.1

-4 -4

-2

0.02

0

2

4

vx/V T0

nN 1225 980 750 520 500 490

0.03

0.01

0

0

0.2

x 0.4

0.6

0

0

0.05

0.1

0.15

X

0.2

Figure 22. a) (left) Structure of the low Re-number flow around back-facing step showing the eddy and flow re-attachment point. b) (right) Contours of (bottom) gas density of the mixed-collisional gas flow in the helicon plasma source, and (top) 2V neutral gas DF at a point near the central axis.

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Conclusion The rapid progress of the parallel computer systems makes it possible to numerically simulate Vlasov, Fokker-Planck and Boltzmann kinetic equations using continuous grid methods. These methods bring many benefits compared to the particle methods such as PIC and DSMC. One of the most prominent features is the ability of the deterministic methods to densely cover the phase space as was demonstrated for charged and neutral particles. An accurate description of the suprathermal population opens doors to modeling of important non-linear transient processes including non-local phenomena and anomalous transport [6769] in plasmas, which are of great importance for magnetic and inertial fusion reactors, gas discharge devices and basic plasma science studies. The next logical steps include the development of truly adaptive grid methods [48, 70] and expanding them to the entire 6dimensional phase space to facilitate kinetic modeling of multi-scale [71] problems. The multi-scaling in time is another fundamental problem to be investigated for the sake of achieving predictive capabilities, which are vital for numerous practical applications and basic physics research.

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Acknowledgments This work was partially supported by the contract N 02.740.11.5047 from the Federal Agency for Science and Innovation of the Russian Federation and by grant 10NE133 from the US Air Force Office of Scientific Research, Physics and Electronics Directorate.

Appendix A. Conservative Kinetic Method for Collisional Gas and Plasma The evolution of electron distribution function f e ( x, v, t ) in Cartesian coordinates {x,v} for a simple collisionless plasma system can be described by the following kinetic equation: f e f eE f e v e  0 t x me v

(A.1)

where E – electric field, e – electron charge and me - electron mass. The self-consistent electric field can be derived from a charge divergence equation and in one dimensional case reads as: E  4e (ne  N I ) x

(A.2)



where ne   f e dv - is electron density, and N I - background ion‟s density. Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved.



To normalize equations (A.1-A.2) we consider the following basic units: [m]  me mass,

[q]  e

-

electric

charge,

[ x]  0D  kT0 / 4n0 e 2 -

length,

and

[t ]  1 /  0pe  me / 4n0 e 2 - time, where n0 and T0 - are referenced density and temperature.

For the derived dimensionless units we have: [E]  4en0 [ x] - electric field, [V ]  [ x] /[t ] - velocity, [ f ]  n0 /[v] - distribution function (DF). The normalized equations (A.1-A.2) will read as: f f  f  t  v x  E v  0   E  n  N  x

(A.3)

Let us consider now the evolution of the equations (A.3) in the following domain [ x]  [v]  [0, L]  [U , U ] with dimensionless L  1 and U  1 , correspondingly.

Let us apply periodic boundary conditions, which is equivalent to the absence of any external forces, only self-consistent force counted: f ( x  0, v, t )  f ( x  L, v, t ), v, t . As a result the total electric charge and mean field for the system are zero.

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297

By integrating the field equation we get: N  n  x

 ( x)    n  n( y )  dy

(A.4)

0

E ( x)   ( x)   

where the bracketed symbol gives the mean value of the corresponding field: L

 a   a( x) dx . 0

Now let us check conservation laws for the system of equations (A.3-A.4). Conservation of mass By integrating Eq. (A.3) over velocity and taking into account the fact that f  0 when v   we have: n nu  0 t x

(A.5)

where u   vf dv /  fdv is the mean velocity. By integrating Eq. (A.3) over space, and taking into account periodic boundary condition, we have conservation of mass condition as follows:

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N ( L) 0, t

(A.6)

x

where we introduced the integral value N ( x)   n( y) dy . Another important quantity is 0 x

I ( x)   N ( y) dy 0

As a result we can proof the following: x

 n(y)N(y)dy 

0

N 2(x) 2

x

 n(y)ydy  xN(x)-I(x)

(A.7)

0

E ( x)  N ( x)  x  E 

I ( L) N ( L)  , L 2

N ( L) L  n 

N ( L) L

Conservation of momentum By multiplying Eq. (A.3) by velocity and then grouping terms under differential operator, we have:

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Oleg V. Batishchev fv fv 2 fEv     fE t x v

(A.8)

Next we integrate Eq.(A.8) over the phase space. The integral on the left side of equation L 

   vfdvdx 0 

. t The integral on the right side of equation (A.8) transforms as follows:

(A.8) gives

L 

L

0  L x

0

  fE dvdx   nE dx   n{  n  n  dy   } dy 

0 L

0

 n({N ( x)  x

0

N ( L) I ( L) N ( L) }{  }) dx  L L 2

(A.9)

N 2 ( L) N ( L)  ( LN ( L)  I ( L))  2 L I N ( L) N ( L)(  )0 L 2

As a result the conservation of impulse equation has the following form: L 

   vfdvdx 0 

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t

0

(A.10)

Total energy conservation By multiplying Eq. (A.3) by velocity squared and transforming to divergence form we have: fv 2 fv3 fEv 2    2vfE t x v

(A.11)

Next by integrating twice both sides of equation (A.11) we have L 

L

0 

0

  vfE dvdx  2  nuE dx

(A.12)

It is easy to prove that the following is correct: E 2 ( x)  ( N ( x)  x

N ( L) 2 ) L

From equation (A.13) and equation (A.5) we obtain:

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(A.13)

Semi-Analytical Adaptive Vlasov - Fokker-Planck - Boltzmann Methods x n( y ) E 2 ( x) N ( x)  2 E ( x)  2 E ( x)  dy  t t 0 t x nu  2 E ( x)  dy  2 E ( x)((nu) x  (nu) 0 ) 0 y

299

(A.14)

The total energy (particle kinetic energy plus potential energy of the electric field) conservation is given below: L   L     d    v 2 fdv  E 2  dx d     2vfdv  E 2  dx 0   0       dt dt L L d {2  nuE dx  2  E[(nu) x  (nu) 0 ]dx}  dt 0 0 L d {2(nu) 0  Edx}  0 dt 0

(A.15)

It is known that collisionless Vlasov system satisfies an infinite number of the conservation laws. But since we are constructing conservative method for the collisional plasma, where only first three moments of the distribution function are conserved, we consider only conservation laws of mass, momentum and energy. Now we move to the construction of a finite difference scheme that will satisfy three conservation laws on the arbitrary numerical solution of the equation system (A.3-A.4). Conservative splitting scheme

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Let‟s split the problem into a set of simple sub steps [45, 50]: f f v 0 t x   n  n  x f f E 0 t v

(A.16)

Next we consider an evolution of the distribution function { fij } , defined at the nodes of the phase space grid. For the simplicity we assume that the grid is equally spaced in velocity v and space x , and t is a variable time step. The first and the last equations in the set (5.17) have exact analytical solutions: f ( x, v, t )  f ( x  vt, v,0), v (A.17) f ( x, v, t )  f ( x, v  Et ,0), x As a result we can approximate the solution at time t by interpolating its value from the previous time step t  t .

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By using linear interpolation for the first and the third equations and integrating the equation for the electric field using trapezoid rule we derive the following discrete analog of the initial splitting scheme (A.16): f1 (t  t , x, v)  f (t , x  vt , v),

v

 ( x  x)   ( x)  0.5x(n1 ( x  x)  n1 ( x)  2  n1 ) f (t  t , x, v)  f1 (t , x, v  E ( x)t , v),

(A.18)

x

Here index 1 marks an intermediate solution. Linear interpolation used in the first and third equations of Eqs.(A.19) gives the following advantages: Linear interpolation keeps DF positively defined. Indeed, for any two values a  b  0 the interpolated value i  sa  (1  s)b  b  s(a  b)  b  0 remains positive for s  [0,1] Linear interpolation automatically gives mass conservation, if corresponding integral is defined as: n(t )    f (x, v, t )xv  const .

(A.19)

x v

To prove Eq.(A.19) we consider the spatial shift first. Because the velocity grid is vt  vt  uniform the whole row is shifted and interpolation coefficient s   Int  is constant. x  x  New mass of all particles in the row of constant velocity will be equal to the initial value: n(t , v)   f ( x, v, t )x  x

 ( sf k 1 ( x  vt , v, t  t )x  (1  s) f k ( x  vt , v, t  t ))x  Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved.

x

s  f k 1 ( x  vt , v, t  t )x  (1 s) f k ( x  vt , v, t  t )x  x

(A.20)

x

 f ( x  vt , v, t  t )x  n(t  t , v) x

 vt  where k  Int  . Because the mass remains constant in each row, the total mass value is  x  constant too. A similar logic is used for the third step of the splitting scheme – acceleration in electric field. The force depends on the spatial value and is not dependent on velocity, the solution is shifted along the columns. If we assume value U  20 , and DF close to the Maxwellian

f  exp{v 2 } (in normalized form), then the values of DF at the velocity boundary of the

simulation domain v  U will be the order of 10 170 , while at the axis ~ 1. Due to energy conservation law the shift Et  1 and electric field energy can not be greater than, the particles kinetic energy. The time step values are of the order of a fraction of a plasma oscillation period ( t  0.1  0.4 ). If we would be able to have an infinite velocity grid, then the number of particles will be preserved exactly (to the computer round-off accuracy in the codes).

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Linear interpolation conserves momentum For the kinetic equation of the first moment of the distribution function we have: fv fv fv v E   fE t x v

(A.21)

If we follow the same procedure for the value fv as we did already for f for the left hand side of Eq.(A.22), then we‟ll get conservation of the corresponding integral: M (t )   vf (x, v, t )xv  const

(A.22)

x v

For the right side of Eq.(A.21) let‟s sum over all cells and use Eq.(A.4) for the field. We‟ll get: N

N

i 1

i 1

  fE( x)xv  x  ni Ei  x  ni ( i    )  x v

 n  nk 1  2  n   x  ni   k x     2 i 1 k 1  N

i

(A.23)

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Nn n k 1  2  n   where      k  - mean field, N – number of nodes in space, and for 2 k 1  kN periodic condition applied. Let‟s introduce the following value:

S i   (nk   n )x    . i

k 1

Using ni 

(A.24)

1 ( Si  Si 1 ) , Eq.(A.23) will transform to: x N

x 

i 1

N



S i  S i 1 ) S i  S i 1 )  x

2



1 2 2  S i  S i 1  2 i 1

S N2

 S 02 0 2

(A.25)

where the last equality is true because of the applied periodic condition. As one can see, it is important to approximate field equation as in Eq.(A.18) for the conservation of M. Note, that for the linear shift the discrete velocity is exactly equal to analytic velocity: f

v t  t  v t (v  Et )  v  f   fE t t

(A.26)

Thus, the scheme in Eq.(A.18) conserves the total momentum of the system. But it is not the same for the total energy. The balance equation for the total energy is: Eulerian Codes for the Numerical Solution of the Kinetic Equations of Plasmas, Nova Science Publishers, Incorporated, 2010. ProQuest Ebook

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Oleg V. Batishchev

fv2 fv2 fv2 v E  2vfE t x v

(A.27)

By summing Eq.(A.27) over all cells, we can show that the total energy is not conserved. The rate of the cell‟s kinetic energy change due to electric field is:





2 2 2 2  f iv (v  Ei t )  v v  2Ei ji t  ni Ei t v

(A.28)

where the particle flux is defined as ji   f iv vv . v

By comparing Eq.(A.28) and Eq.(A.27) we see, that the method is adding kinetic energy to the system and the addition is proportional to t . This means, that the reduction of the time step will not change the error for the any given time interval. Note that the total energy in the system is monotonically increasing, fE 2 t  0 . Also note that the energy change in a given cell is proportional to the value of DF in the cell - the error is local. Thus, the error can be corrected locally. This correction must not change the local number of particles in the cell, fvx , and momentum, vfvx . The correction can be applied after the acceleration step by using the following filter: f (v)  0.5 f ' (v  v)  (1   ) f ' (v)

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 0.5 f ' (v  v),

v  U

(A.29)

where   0 - is a free parameter. The linear filter is used for all velocity nodes excluding boundary, where DF value is negligibly small. This simple manipulation with DF conserves mass and the velocity. The summation over three neighboring layers gives: 0.5  (1   )  0.5  1 (v  v)0.5  (1   )v  0.5 (v  v)  v

(A.30)

At the same time the applied filter is changing the energy value. This energy change must compensate the energy growth in the cell. By equating the energy change to the energy growth value: 0.5 ((v  v) 2  (v  v) 2 )  (1   )v 2  E 2 t 2  v 2

(A.31)

we get always positive solution:



E 2 t 2 v 2

0

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(A.32)

Semi-Analytical Adaptive Vlasov - Fokker-Planck - Boltzmann Methods

303

The system of equations (A.30) for unknown f ' can be solved by use of the threediagonal solver . But when f  0 , usually at the DF “tails” and ”holes” in the phase space, non-physical negative values of f ' may occur. Let‟s now consider the energy change for one time step for the given cell i using Eqs.(A.13-A.15). We have: Ei  Nˆ i   Nˆ k / N



Ei2  Nˆ it   Nˆ kt / N



  Nˆ 2

Nˆ i  ( ˆji  ˆj o )t   {Nˆ k  ( ˆj k

i

  ˆj )t} / N   Nˆ   Nˆ k / N

2

2

o

i

  Nˆ k / N



2

(A.33) 

 (2 Ei  ( ˆji  ˆj o )t   ( ˆj k  ˆj o )t / N )  (( ˆji  ˆj o )t   ( ˆj k  ˆj o )t / N ) 

here symbol

identifies the average value between the given cell and the previous cell, and

summation index runs k  1,.., N . By introducing ji  ( ˆji  ˆjo )t , S   jk / N

(A.34)

The sum (A.34) over all cells i  1,.., N gives: 2 2  Ei   2Ei ji   (ji  S )  S  (2Ei  ji  S )

(A.35)

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The last term in Eq.(A.35) is equal to zero. As a result, we obtain a proof that the electric field energy change in the system is of the order of (t ) and is equal to the kinetic energy change in Eq.(A.28). The discrepancy (the second term in Eq.(A.35)) is representing „‟dispersion'' of the system‟s particle flux squared. It is not local, but it tends to go to zero when the system transitions to a steady-state regime (see Fig.23). 10 0 10 -1

10 -2 10 -3



10 -4 10 -5 10 -6 10 -7 0

50

t w-1pe

100

Figure 23. Evolution of the potential energy and total energy discrepancy, δE/E.

Similar to the particles kinetic energy the potential energy is growing monotonically. The Fig. 23 shows numerically calculated evolution of the total energy  E 2  and  /  as a

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Oleg V. Batishchev

function of time for a periodic plasma-beam system. The total energy in the system is growing. Thus, it appears possible to use the correction procedure [45] of reducing DF's temperature while preserving the mass and the momentum of the particle system. Adding collisions to the conservative splitting scheme Let‟s add to the splitting scheme the collisional operator describing elastic collisions between similar species. The generalized BGK [55] operator describes the “Maxwellization” of distribution for both plasma and gas: f f f  M t 

(A.36)

here relaxation parameter  ( f ) can be non-linear, f M is the Maxwellian distribution function with the same first moments as the initial DF in the case of constant  , or with the same first moments as the outgoing DF in the general case. Let‟s split the BGK operator in Eq.(A.36) into two substeps – outgoing and incoming of the particles:

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f f  t  f f  M t 

(A.37)

The first equation has an analytic solution (for the frozen relaxation time). For the each nth cell we have:  t  (A.38) f n1 (t  t )  f n (t ) exp    0    Here index 1 denotes intermediate DF value. The expression in Eq.(A.38) gives the nonnegative DF value for any time step. After evaluation of the intermediate value f n1 , we now compute the mass change in each spatial section: Nv

ni  v  ( f ij1 (t  t )  f ij (t ))  0 j 1

(A.39)

Using initial distribution mean velocity and temperature in each i-th (spatial) section: Nv

Nv

j 1

j 1

u i   v j f ij (t ) /  v f ij (t ) Nv

Nv 2 Ti   (v j  ui ) 2 f ij (t ) /  v f ij (t ) 3 j 1 j 1

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(A.40)

305

Semi-Analytical Adaptive Vlasov - Fokker-Planck - Boltzmann Methods

we construct Maxwellian DF f ijM using iteration procedure with the same Ti and drifting velocity ui , and unit density. Next we compute DF for each node using local weights for the i-th spatial section: f ij (t  t , x, v)  f ij1 (t  t , x, v)  ni f ijM (t , x, v)  0 ,

(A.41)

j  1,.., N v

It is obvious that all three moments of the distribution function remain constant at the transition to the next time step. We have shown that the whole splitting scheme in Eqs.(A.16,A.38,A.31) is conservative. Similar methods were described before, but the problem of getting a completely conservative scheme in the collisional case was not discussed. The method can be extended to more dimensions and species, and different types of collisions. The only negative factor is a high numerical diffusion. There are ways to control this problem, e.g. by using adaptive grids [48].

Appendix B. Transformation from Cartesian to Axially Symmetric Spherical System of Coordinates





Let‟s transform kinetic equation from Cartesian coordinate system to v x , v y , v z axially symmetric spherical coordinates v,   . First we introduce the following definitions:

  vx / v  v1 / v ; 3k 2 vk2 / v 2  1   2 .

Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved.

v  v x2  v 2y  v z2  v12  3k 2 vk2 ,

So that the expressions for derivatives are: 

k 1 v   , ,  v / v , k  2,3 vk  k

  1   , k 1  v k  vv / v, k  2,3 k 

k 1 vk v,     , ,  v / v , k  2,3 v  k

v k  v  k , k  2,3   2 1   



2

v,

k 1

The following relations is useful:

  vk   0 v  v 

  v k  v 2  vk2 ;   vk  v  v3 

  vk       v  

k 1

1,

v k , k  2,3 2  v 1 





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306

Oleg V. Batishchev

 v x  vk

1  2   v 

2    3 1    v 

(B.1)

v      2 k3 , k  2,3 v v





 Now let‟s transform the components of vector a  a x , a y , a z from Cartesian to v,  

coordinates. av  a x 

a   1   2 a x 

vy

ay 

v vy

v 1 

2

vz az , v

ay 

vz  v 1 2

az .

(B.2)

Gradient Gradient components in Cartesian coordinate system:   1 2     v v  k    v k        v  v v  

k 1 k  2,3

Gradient components in v,   system: 2   v 2 vk a  av  1   a     k   av   a    av   k    v   v v k 1 v v    3

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v a  

2 

 3  vk  2 1  2      a  1    av  a        av  a    2 v v v    k 2   1    



1 2 a v

(B.3)

Divergence of a vector





Let‟s rewrite the divergence of the vector with coordinates a x , a y , a z , which has the



form k  k a , to coordinate system ( v,  ), with coordinates a , a k

a x  a v  1  2 a  ,

v



 . We have:

kx

 vk v k  v   a k   a v  1   2 a v    a  a  , k  1,2   v 1 2 v  1 2  

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(B.4)

307

Semi-Analytical Adaptive Vlasov - Fokker-Planck - Boltzmann Methods   v k  v   v  2   +  a  1   a  a  a      k v k v  v x  1 2  

k ak 

(B.5)

From the right-hand side of the expression in Eq.(B.5) we have: 2  a v  1   a  vk v   a  k v k v v x v x

av

(

1 2

 av +

 

 k

1  2   v



 1   2

(

 avv 

1  2  a v



1  2 v a v

k

 vk v k v

) 

 1 

a = 2

1  2  a  v

(B.6)

 2  vk2 v  vk  2  v  v  a +     a v  a    3 v  v    k   v

)    v

2

  vk   v       a v  a   v  v 1  2  v  

2   1  2 v k  a   .    v     

By grouping the terms in Eq.(B.6) we arrive at an expression for the divergence in v,   coordinate system: 1 2  2 1  2 v 1   a  a = 2 v a  1  2 a  k a k  avv  a v 2 v  v v v v v 1 

(B.7)

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Operator k ak  3

  k a k  

k 1

 v x

2    vk    1       a   a  v      v  vk  v

       =   v v  

    1  2 1  2  v  v   v   a  v    vv   v        v  v  v  v v v  v  vx x x  x x    x

   

    1     v    a v  a    x x v  v   v

 3 v 2  vk2    a   v      3 v k  2 v     vk  v    v           v   v     vv   vk vk v  vk vk  vk v k 2 v    3



3  v   vk     v     (B.8)    av  a vk vk  v  v k 2   The right-hand side of Eq.(B.8) takes the form:

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308

Oleg V. Batishchev

  1  2 1 2  v     vv   v   v v   3

1  2 v

 

2

1  2 v

2     v  1       av v   v  





1  2     2    v      1    vv   v v v v    

 1  2  v v v  v      2 2     a  .    v2  

(B.9)

After grouping the terms, we have:   2 1  2 v a  a vv   v     2 2     . 2 v v     a v v 

1  2 v2

a 

(B.10)

Last expression is equivalent to:  k a k  

1   2   1      a(1   2 )   av  2 2 v  v     v   v 

(B.11)

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Laplace’s operator   

 2

2 i  x, y , z vi



1   2   1      (1   2 )  v  2 2 v   v  v  v  

(B.12)

Operator avx f , a  const

a

    f 1 2 1  2    a f v  f    a f v  f    2 f  2 f  v x v v v v     1  2 1  v af  1   2 af v 2 v v 





After comparing the equations for the divergence, we have:

jav  af

Eulerian Codes for the Numerical Solution of the Kinetic Equations of Plasmas, Nova Science Publishers, Incorporated, 2010. ProQuest Ebook

(B.13)

Semi-Analytical Adaptive Vlasov - Fokker-Planck - Boltzmann Methods j a   1   2 af

309 (B.14)

Coulomb collisional term  i  x, y , z vi

CL 

   2 f  a , f    v  k  x, y , z vi v k v k  i 

(B.15)

here L и a - const,  , - Rosenbluth potential functions , related to the Poisson equations:

using   k

 2 vk2

 v  

1   2   1      (1   2 )  f , v  2 2 v  v     v   v 

 v  

1   2   1     (1   2 ) v  2 2 v  v    v   v 

    

, we can write:  vi

  1  a   f    vi vk 

2      f  .     vi v k  

(B.16)

For the first term in brackets we have:  vi

   1   2   1    2  a   v f   v 2 v  av v f   v 2   a(1   )  f  .   i  

(B.17)

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The second term transforms to:  f    k v v k i  1  f 2 v  v 2 v k v k  vi

     i  k f  i  k    k    1  f    (1   2 )  v v k v 2  k v k v v k

 v k

The right-hand side of Eq.(B.18) transforms to:

Eulerian Codes for the Numerical Solution of the Kinetic Equations of Plasmas, Nova Science Publishers, Incorporated, 2010. ProQuest Ebook

(B.18)

310

Oleg V. Batishchev f    v k vi v k  f v f      v         v v   v  v  v v   v   k k  k k       1 2 1 2 1  2 f    vv   v     k  1  f v  2 v v v          1  2 1 k  2,3  1    v f v  2 f   v  vv  2  v  3    v v v    



(B.19)



After grouping the corresponding terms we arrive at: 1  2  f   1  4   vv    v  3    f v . vk vi vk v  v 

(B.20)

Next: f    f v f      v          vk  vk  v vk  vk    v vk  vk   vk v     vk       fv  v k   v   f          v  v v   v  v   v k    k  k     1 2 1 2 2 f    v   v        k  1  f v  v v v     2    1   1 2 1        1    v f v  2 f   v  v  3 2 2 2 v v 1   1  v     Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved.













(B.21)

k  2,3

after grouping :  k

 f    1  2   v     f v  v k  v k  v 

(B.22)

2 2 1    v  1       1      f  2 2 v  v v  

Finally we arrive to the following form of the Coulomb collisional term: CL

1   2    2 f 1   2 v a f    v 2  v  v v 2 v v2

  2   f      v         

  2   f     (1   2 ) a f         v   v  1  1   2  2    f    v v  v 2  2  v 2       

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(B.23)

311

Semi-Analytical Adaptive Vlasov - Fokker-Planck - Boltzmann Methods By comparing to the divergence operator in expressions for the collisional fluxes:

v,  

coordinate system, we get the

 1 2  1   jCv  La v f   vv f v   v     f   2 v  v    1 2 v

 1   a  f   v     f v  v    1 1      v  2 1   2       f   v v   jC   L





(B.24)



For the case of spherically symmetric potentials in Eq.(B.23) can be reduced (

  0 ), 

and the fluxes take the form: jCv M  La v f   vv f v , 

jC   L

1 2 v2

(B.25)  v f .

Normal diffusion ("cyclotron heating") operator We have the following expression for the velocity diffusion normal to x-direction:  f  f 1  f D  D  Dv v y v y v z v z v v v

(B.26)

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where v  v 2y  v z2 . Let‟s transform Eq.(B.26) to divergence form (omitting D): 1  2 f 1 f   v  v v  2 v 1   2   v 1   2 

  



2 f  v 1   2   

2



 f f  2   2 1      v 1   2  v  1   2 v  1

2

 1 2    2 f  2 1  2  1 2 2 f f      2 2 v v v v v 1   2  1 2

  2    1   2  v v  1 2  

  2 f  1   2 f     v v  1   2   

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(B.27)

312

Oleg V. Batishchev 2

2

 1 2   1 2      2 2 2  1 2 2 f f    f   1   f   2 2 2 2 v v v v v v v 1    1 





2

1  v

2

 f 2

v 1   2

 1 2      v2

4

2 f  1 2

2



2 2 1   2 v

2

f  1 2

After grouping the terms in Eq.(B.27) we arrive to the divergence form of the operator:    1   2 2 f   2 2 f (B.28) v 1   1      v v  1   2   v 2 v    

 1 2

 1 2    v  1   2  

  2 f  1   2 f     v v  1   2    

Finally, expressions for the diffusion fluxes (with the "perpendicular" velocity diffusion coefficient D) take the following form:   f  2 f  Av  D 1   2  1   2  v v  1   2    

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  f  2 f  A  D  1   2  v v  1   2    

(B.29)

References [1] Feynman, R. P., Quantum Electrodynamics, Addison Wesley, 1962. [2] Landau L. D. and Lifshitz E. M., The Classical Theory of Fields, Oxford, Pergamon Press; Reading, Mass., Addison-Wesley Pub. Co., 1962. [3] Bogoliubov, N.N., Kinetic Equations, Journal of Phys. USSR, 1946, 10 (3), 265-274. [4] Boltzmann, L., Über das Wärmegleichgewicht von Gasen, auf welche äussere Kräfte wirken.1875 Sitzungsberichte der Akademie der Wissenschaften Wien, 72, 427-457. [5] Ulam, S., Adventures of a Mathematician, New York, Charles Scribner's Sons, 1983. [6] Gross E.P. and Krook M., Phys. Rev. 102, 593, 1956. [7] Shakhov, E.M., Mathematical Methods for Rarified Gases, Nauka (in Russian), 1974. [8] Mitchener, M., Partially Ionized Gases, John Wiley & Son, 1973. [9] Vlasov, A.A., Many-Particle Theory and Its Application to Plasma, New York, Gordon and Breach, 1961. [10] Kadomtsev, B.B., Cooperative Effects in Plasmas, in Series: Reviews of Plasma Physics, 2001, Vol. 22, pp.1-226, New York: Kluwer Acad./ Consultants Bureau. [11] NRL Plasma Formulary (Ed. J.D. Huba), published by ONR, revised 2007.

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313

[12] Langmuir, I., Positive Ion Currents from the Positive Column of Mercury Arcs, Science, 1923, 58 (1502), pp. 290-291. [13] Alfvén, H., On the theory of magnetic storms and aurorae, Tellus, 1958, 10, 104. [14] Landau, L.D., Die kinetische Gleichung für den Fall Coulombscher Wechselwirkung, Phys. Z. Sowjetunion, 1936, 10 (2), 154–164. [15] Balakin, A. A., Fraiman, G.M., Numerical simulations of electron-ion collisions in UHI plasmas, Comp. Physics Comm. 2004, 164 (1-3), 46-52. [16] Bulanov, S. V., et al., Relativistic interaction of laser pulses with plasma, Reviews of Plasma Physics, 2001 Ed. V. D. Shafranov Vol. 22, pp. 227-335, Kluwer Academic/Plenum, New York. [17] Loeb, L. B., The Kinetic Theory of Gases, Dover Publications; Inc., New York, 1961. [18] Knudsen, M., Kinetic Theory of Gases: some modern aspects; Methuen, London; Wiley, New York, 1950. [19] Batishchev, O.V., Batishcheva, A.A., Kholodov, A.S., Unstructured adaptive grid and grid-free methods for magnetized plasma fluid simulations, J. Plasma Phys. 1999, 61, part V, p.701. [20] Harlow, F. H., Hydrodynamic Problems Involving Large Fluid Distortion, J. Assoc. Comp. Mach., 1957, 4, p. 137. [21] Knorr, G., Shoucri M., J. Comp. Phys., 1974, 14, p.1 and p.84. [22] Cheng, C.Z., Knorr, G., J. Comp. Phys. 1976, 22, 330-351. [23] Bird, G.A., Molecular gas dynamics. Clarendon Press, Oxford, 1976. [24] Shoucri, M., Gagné, R., J. Comp. Phys. 1978, 27, 315-322. [25] Hockney, R. W., Eastwood, J. W., Computer Simulation Using Particles , McGraw-Hill, New York, 1981. [26] Birdsall, C. K., Langdon, A. B., Plasma Physics via Computer Simulation, McGrawHill, New York, 1985. [27] Mikhailovsky, A. B., Theory of Plasma Instabilities, Vol. 1, Consultant Bureau, New York, 1974. [28] Fried, B.D., Liu, C.S., Means, R., Sagdeev, R.Z., Report UCLA, PPG-93, 1971. [29] Gardiner, C.W., Handbook of Stochastic Methods: for Physics, Chemistry and the Natural Sciences, (Springer Series in Synergetics), Springer (3rd ed.), 2004. [30] Batishchev, O.V. et al, Kinetic Modelling of Transport Processes in Tokamak Edge Plasma, Contributions Plasma Physics, 1992, 32 (3/4), 237-242. [31] Rosenbluth, M.N., MacDonald, W.M., Judd, D.L., Phys. Rev. 1957,107, 1. [32] Trubnikov, B. A. Reviews of Plasma Physics, Vol. 1 (ed. M. A. Leontovich), p. 105. Consultants Bureau, New York, 1965. [33] Batishcheva, A.A., et al., Massively Parallel Fokker-Planck Code ALLA, Contributions Plasma Physics, 1996, 36 (2/3) 414. [34] Xu, X.Q. et al, Parallelization of and Results from the Kinetic Edge Plasma Code W1, Contributions Plasma Physics, 1996, 36 (2/3) 424. [35] Takizuka, T. and Abe, H., J. Comp. Phys. 1977, 25, 205. [36] Batishchev, O.V. et al, Kinetic Effects on Particle and Heat Fluxes in Detached Plasmas, Physics of Plasmas, 1996, 3 (9), 3386. [37] Cadjan, M.G., Ivanov, M.F., Langevin approach to plasma kinetics with Coulomb collisions, J. Plasma Phys. 1999, 61 (1) 89.

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[38] Killeen, J. , Kerbel, G.D., McCoy, M.G., Mirin, A.A., Computational Methods for Kinetic Models of Magnetically Confined Plasmas, Springer-Verlag, 1986. [39] Epperlein, E.M., a) Phys. Plasmas 1994, 1, 109; b) J. Comp. Phys. 112, 291, c) Laser and Part. Beams, 1994, 12 (2) 257. [40] Shoucri, M., Shkarofsky, I., A fast Fokker-Planck solver with synertgetic effects, Comp. Phys. Comm. 1994, 82, 287; Shoucri, M., Peysson, Y., Shkarofsky, I., Numerical solution of the Fokker-Planck equation, Report EUR-CEA-FC-1737, CEA/Cadarache, France, 2006. [41] Lyapunov, A.M., The General Problem of the Stability of Motion, Taylor & Francis, London, 1992. [42] Averina, T.A. and Artem‟ev, S.S. , Soviet DAN, 1987, 288 (4) 777-778, [in Russian]. [43] Batishcheva, A.A., et al, A Kinetic Model of Transient Effects in Tokamak Edge Plasmas, Physics Plasmas 1996, 3 (5), 1634. [44] Stangeby, P. C., The Plasma Boundary of Magnetic Fusion Devices (Series on Plasma Physics), Taylor & Francis, 2000. [45] Batishchev, O.V., Kinetic models for collisional plasma in fusion devices and space thrusters, Dr. Habil Diss., MIPT, Moscow, [in Russian] 2001. [46] Batishchev, O.V., et al, Kinetic Effects in Tokamak Scrape-off Layer Plasmas, Phys. Plasmas. 1997, 4 (5), 1672. [47] Braginskii, S. I. Reviews of Plasma Physics, 1965, Vol. 1 (ed. M. A. Leontovich), p. 205., Consultants Bureau, New York. [48] Batischeva, A.A. and Batishchev, O.V., Hybrid Kinetic Method for Laser-Plasma Interaction in the Presence of Strong Spatial Gradients, SBIR Phase I Final Report, AFRL-PR-ED-TR-2004-0006, July 2005. [49] Mason R.J. Implicit moment Particle Simulation of Plasmas, J. Comp. Phys. 1981, 41, p. 233-244, 1981. [50] Marchuk G.I., Splitting Methods, Nauka, [in Russian] 1988. [51] Potter, D., Computational Physics, Wiley & Sons, London 1973. [52] Batishchev, O.V., Shoucri, M.M., Batishcheva, A.A., Shkarofsky, I.P., Fully kinetic simulation of coupled plasma and neutral particles in scrape-off layer plasmas of fusion devices, Plas. Phys., 1999, 61, II, 347. [53] Batishcheva, A.A., et al., Fokker-Planck code ALLA, Report Plasma Fusion Center MIT, PFC/JA-95-23, 1995. [54] Batishchev, O.V., et al, Fokker-Planck Simulation of Electron Parallel Transport in the Scrape-off Layer of TdeV, Report Centre Canadien de Fusion Magnetique, RI 466e, -42 p., 1996. [55] NAG Fortran Library Manual, NAG Central Office, Oxford, 1985. [56] Krasheninnikov, S.I., Soviet ZETPh, vol.1988, 67, p.2483, [in Russian]. [57] Bychenkov V.Yu., et al, Nonlocal Electron Transport in a Plasma, Phys. Rev. Lett., 1995, 75 (24), 4405–4408. [58] Novikov, V.N., J. Comp. Math. and Mat. Phys, 1990, 6, 920, [in Russian]. [59] Zel'dovich, Ya. B. and Raizer, Yu. P. Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena. Mineola, NY: Dover Publications, 2002. [60] Langdon, A.B., Phys. Rev. Lett.1980, 44, 575. [61] Batishchev, O.V., et al, Heat transport and electron distribution function in laser produced plasmas with hot spots, Phys. Plasma, 2002, 9 (5) 2302.

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315

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[62] Batishchev, Oleg and Batishcheva, Alla, An efficient statistical noise-free kinetic method for rarified collisional gas simulations, Proc. 2nd MIT Conf. Comp. Fluid and Solid Mech., Book of Proceedings, 2003 Elsevier Sci. Ltd., -4p, Cambridge, USA. [63] Chapman, S., Cowling, T.G., The Mathematical Theory of Non-Uniform Gases, Univ. Printing House, Cambridge, 1970. [64] Cercignani C., Mathematical Methods in Kinetic Theory, New York, Plenum Press, 1990. [65] Armaly, F., Durst, J.C., Pereira, F., Schonung, B., Experimental and theoretical investigations of backward-facing step flow, J. Fluid Mech. 1983, 127, 473. [66] Batishchev, O.V., Minihelicon Plasma Thruster, IEEE Trans. Plasma Science, 2009, 37 (8) 1563 – 1571. [67] Bohm, D., The characteristics of electrical discharges in magnetic fields (Ed. A. Guthrie and R.K. Wakerling) New York, McGraw-Hill, 1949. [68] Zaslavsky, G.M., Non-universality of anomalous transport, J. Plasma Physics, 1998, 59 (4), 671. [69] Balescu, R., Aspects of Anomalous Transport in Plasmas (Series in Plasma Physics), Taylor & Francis, 2005. [70] Sonnendrucker, E., et al, Vlasov simulation of beams with a moving grid, Comp. Phys. Comm. 2004, 164 (1-3), 390. [71] Cambier, Jean-Luc and Batishchev, Oleg, Kinetic Algorithms for Multi-Scale Plasma Simulations, in Proc. Symp. Computational Multiphysics Applications, ASME IDETCCIE Conf., Montreal, August 15-18, 2010 [accepted].

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In: Eulerian Codes for the Numerical Solution… Editor: Magdi Shoucri, pp. 317-358

ISBN: 978-1-61668-413-6 © 2010 Nova Science Publishers, Inc.

Chapter 8

THE BUMP-ON-TAIL INSTABILITY Magdi Shoucri Institut de Recherche d’Hydro-Québec(IREQ), Varennes, Qué. J3X1S1, Canada

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Abstract We study the nonlinear evolution of the bump-on-tail instability in the case of a strong beamplasma interaction, when the ratio of the beam density to the background density is about 10%. An Eulerian code is used for the numerical solution of the one-dimensional (1D) Vlasov-Poisson system of equations for electrons, with ions forming an immobile background. The code applies a method of fractional step previously reported. We follow the growth and saturation of the instability, and the formation of a Bernstein-Greene-Kruskal (BGK) wave. In the case when the system length L is greater than the wavelength of the unstable mode  , growing sidebands develop and disrupt the BGK wave, with energy flowing to the longest wavelengths (inverse cascade). Using a fine resolution grid to follow accurately the dynamic of particles being trapped in the phase-space, it is shown that the electric energy of the system is reaching in the asymptotic state a steady state with constant amplitude modulated by the persistent oscillation of the trapped particles, and of particles which are trapped, untrapped and retrapped. Analysis of the excited spectrum in space and in time shows a rich variety of mode coupling events between the different modes, mediated by the persistent oscillations of the trapped particles. Oscillations at frequencies below the plasma frequency are associated with the longest wavelengths, with phase velocities above the initial beam velocity, resulting in trapped particles to be accelerated to higher velocities. Localized holes are formed in the phase-space, which translates as cavity-like structures in the density plot. A fine resolution grid is used, since numerical grid size effects and small timesteps can have important consequences on the number and distribution of the trapped particles, on the dynamical transitions of the Vlasov-Poisson system, and on kinetic microscopic processes such as the chaotic trajectories which appear in the resonance region at the separatrix of the vortex structures where particles can make transitions from trapped to untrapped and retrapped motion.

1. Introduction The bump-on-tail instability is one of the most fundamental and basic instabilities in plasma physics. When the bump in the tail of the distribution function presents a positive

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318

Magdi Shoucri

slope, the distribution function becomes unstable to a perturbation whose phase velocity lies along the positive slope of the distribution function. Some early numerical simulations have demonstrated that the dominant stabilization mechanism for the beam-plasma instability appeared to be the particle trapping in coherent phase-space vortices [1-3]. A comprehensive discussion of this problem and the associated formation and dynamics of coherent structures involving phase-space vortices in plasmas has been recently presented in several publications (see for instance [4-6]). Using Eulerian codes for the solution of the Vlasov-Poisson system [7], it has been possible to present a better picture of the nonlinear evolution of the bump-ontail instability [8], where it has been shown that the initial bump in the tail of the distribution is distorted during the instability, and evolves to an asymptotic state having another bump in the tail of the spatially averaged distribution function, with a minimum of zero slope at the phase velocity of the wave (in this way the large amplitude wave can oscillate at constant amplitude without growth or damping). The phase-space in this case shows BGK vortex structures [9,10,11] traveling at the phase-velocity of the wave. These results are also confirmed in several simulations (see [12] for instance). Since the bump-in-tail is usually in the low density region of the distribution function, the Eulerian codes, because of their low noise level, allow an accurate study of the evolution of the bump, and on the transient dynamics for the formation and representation of the traveling BGK structures. The simulations in [8,12] were executed with the object of providing informations while making the code as economic as possible, i.e. with a grid in space and velocity space as economic as possible. During the evolution of the system, once the microstructure in the phase-space is reaching the mesh size, it is smoothed away by numerical diffusion, and is therefore lost. Larger scales appear to be unaffected by the small scale diffusivity and appear to be treated with good accuracy. This however can have important consequences in smoothing out information on trapped particles, and modifying some of the oscillations associated with these trapped particles, and those particles at the separatrix region of the vortex structures which evolve periodically between trapping and untrapping states. Numerical grid size effects and small time-steps can have important consequences on the number and distribution of the trapped particles, on kinetic microscopic processes such as the chaotic trajectories which appear in the resonance region at the separatrix of the vortex structures where particles can make periodic transitions from trapped to untrapped motion. These trapped particles play an important role in the macroscopic nonlinear oscillation and modulation of the asymptotic state, and require a fine resolution phase-space grid to be studied as accurately as possible [13-16]. In addition when the length of the system L is greater than the initially unstable wavelength  , unstable sidebands can grow [17-19] and disrupt the BGK wave. Trapped particles are accelerated to higher velocities, and form a warm tail, and a spectrum is excited which allows mode coupling mediated by resonant particles. The transient dynamics is sensitive to grid size effects.The system usually evolves in such a way that energy flows by inverse cascade to the longest wavelengths available in the system (inverse cascade), an evolution characteristic of 2D systems [20]. The resulting spectrum shows oscillations below the plasma frequency associated with the longest wavelengths available. The fine grid resolution is necessary and essential to follow as accurately as possible the trapped particles oscillation and the growth and saturation of the sidebands, and to study their effect in modifying the vortex structures of the phase-space.

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319

The Bump-on-Tail Instability

Many theoretical and numerical calculations have studied the problem of the bump-ontail instability in the context of a small bump in the tail, using a weak beam density where electrons are partitioned essentially in two populations, the bulk and the tail, and where resonant particles in the beam interact with a single wave representing the longitudinal oscillations of the bulk particles. The collective oscillations of the bulk particles is represented by a slowly varying amplitude in an envelope approximation. In the present work, we follow the nonlinear evolution of the bump-on-tail instability with a fine grid in the phasespace and a sufficiently small time-step, in the case of a strong beam-plasma interaction when the ratio of the beam density is about 10% of the total density, and the instability and trapping oscillations have an important feedback effects on the oscillation of the bulk. We follow in the asymptotic state the details of the oscillation and modulation due to the trapped particles in the case L   and in the case L   .

2. The Relevant Equations The relevant equations are the 1D Vlasov equation for the electron distribution function f ( x, , t ) , coupled to the Poisson equation. Ions form an immobile background. These equations are written in our normalized units:

f f f   Ex 0 t x 

(1)



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 2  (1  ne ( x)) ; ne ( x)   f ( x, )d x 2  Ex  

(2)

 x

Time t is normalized to the inverse electron plasma frequency

(3) 1  pe , velocity is

normalized to the electron thermal velocity  th and length is normalized to the Debye length

D  th /  pe . Periodic boundary conditions are considered. These equations are discretized on a grid in phase-space and are numerically solved with an Eulerian code, by applying a method of fractional step which has been previously presented in the literature [7,8,21].

3. The Bump-on-Tail Instability The distribution function of a homogeneous beam-plasma system, with a small density electron beam drifting with a velocity  d relative to a stationary homogeneous plasma is given by:

Eulerian Codes for the Numerical Solution of the Kinetic Equations of Plasmas, Nova Science Publishers, Incorporated, 2010. ProQuest Ebook

320

Magdi Shoucri

f ( ) 

np 2

e

1  2 2



nb 2  thb



e

1   d  2 2 thb

2

(4)

We take for the plasma density n p  0.9 and for the beam density nb  0.1 for a total density of 1. This high beam density will cause a strong beam-plasma instability to develop. The beam thermal spread is  thb  0.5 and  d  4.5 . We perturb the system initially with a perturbation such that:

f ( x, )  f ( )(1   cos(kx)) with   0.04 and with k  n

(5)

2 , and f ( ) is given in Eq.(4). We consider the case L

where k  0.3 , and the approximate initial frequency response of the system will be

 2  1  3k 2 , or   1.127 (nonlinear solutions can give slightly different results) , with a phase velocity of the wave  / k  3.756 . This phase velocity corresponds to a velocity

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where the initial distribution function in Eq.(4) has a positive slope. Hence the density perturbation in Eq.(5) will lead to an instability. Two cases will be considered for the solution of this initial value problem. A case where the mode k  0.3 is a fundamental mode in the system with n=1, (the length of the system is equal to the wavelength of the excited oscillation). And another case where the mode k  0.3 corresponds to the harmonic n=3, in which case unstable sidebands can grow and modify the complex vortex structures of the phase-space (the length of the system is bigger with respect to the wavelength of the excited oscillation). In both cases an important trapped population exists, which results from the strong beam plasma interaction, and which plays an important role in the evolution of the system.

4. Excitation of a Fundamental Mode with k=0.3 We consider the case where a fundamental mode with n  1 is excited. So the length of the system in this case is L  2 / 0.3 . The first results we show are obtained on a spacevelocity grid of 64x1024. With extrema of the velocity at  7 , the recurrence time is



2  1530.38 . We use t  0.05 . Figure 1a) shows the time evolution of the total k

electric energy, essentially resulting from the initially unstable fundamental mode. The initial growth is followed by saturation and by the oscillation of trapped particles, and finally the energy is oscillating around a constant level. We present the results until t=1600, even though the recurrence effect is at t=1530.38. Figure 1b) shows a plot of the results in Figure 1a) from t=1350 to t=1550. There is a very small modulation superimposed on the observed oscillation. The initial bump on tail has evolved to a shape showing another bump on tail in Figure 2), with a minimum at the phase velocity of the wave, at   3.433 . The spatially averaged distribution function F ( ) in Figure 2) is calculated from the relation:

Eulerian Codes for the Numerical Solution of the Kinetic Equations of Plasmas, Nova Science Publishers, Incorporated, 2010. ProQuest Ebook

The Bump-on-Tail Instability

F ( ) 

321

L

1 f ( x, )dx L 0

(6)

Indeed we can verify by following the oscillation of the fundamental mode, that the nonlinear plasma oscillation has a frequency   1.03 , corresponding to a phase velocity of the wave

 ph   / k  3.44 , which corresponds to the minimum in Figure 2). In the tail of the

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distribution function after the bump maximum in Figure 2 there is a very small persistent (nevertheless negligible) oscillation of the distribution functions, which are plotted in Figure 2 at t=1470 (dash-dot curve), t=1500 (full curve) and t=1530 (dash curve). Figure 3a shows at t=1500 a contour plot concentrating on the region of the vortex which is formed in the asymptotic state. It shows essentially a BGK mode whose center is traveling at the phase velocity of the wave. Figure 3b presents the corresponding 3D plot, at t=1500. Figs. 3a,b show essentially very smooth curves. It is generally accepted that once the microstructure in the phase-space reaches the mesh size, it is smoothed away by numerical diffusion due essentially to the interpolation technique used in the code, and is therefore lost. Larger scales appear to be unaffected by the small scale diffusivity and appears to be treated with good accuracy. This results in the smooth curves we see in Figs.3a,3b. It eliminates however important physics associated with the oscillation of the trapped particles.

Figure 1a. Time evolution of the total electric energy.

Figure 1b. Time evolution of the total electric energy.

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Figure 2. Spatially averaged distribution function

F ( )

at t=1470 (dash-dot curve), t=1500 (full

curve) and t=1530 (dash curve).

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Figure 3a. Contour plot of the distribution function.

Figure 3b. Three-dimensional view of the distribution function.

We repeat the previous problem with a more refined grid and with a smaller time-step, to catch in more details the oscillations and dynamic of the trapped particles, and of the particles which are periodically trapped and untrapped at the separatrix during the oscillations. We use a space-velocity grid of 512x1400, with a time step t  0.002 . With extrema of the

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2  2092.9 . Figure 4a shows the k time evolution of the total electric energy, showing growth, saturation and trapped particle oscillations, and seems to asymptotically oscillate around the same value as in Figure 1a, however with more important modulation due to the trapped particles than what is presented in Figure 1a. Indeed it shows during the time evolution a more important and rich diversity of modulation associated with the oscillations of the trapped population. This is confirmed by looking after saturation at the beginning of the trapped particle oscillations, where the amplitude and the modulation are more important with respect to what is presented in Figure 1a. This is also apparent in the plot in Figure 4b compared to the plot in Figure 1b.

velocity at  7 , the recurrence time in this case is  

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Figure 4a. Time evolution of the total electric energy.

Figure 4b. Time evolution of the total electric energy (same as Figure 4a).

Figure 5. Spatially averaged distribution function

F ( ) at t=0 (full curve), t=1880 (dash curve), and

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Figure 5 shows the spatially averaged distribution function calculated as in Eq.(6). The initial bump-on-tail (full curve in Figure 5), after evolving through complicated structures, is finally reaching a shape with a smooth bump and an elongated tail, shown in Figure 5 at t=1880 (dash curve) and t=1980 (dash-dot curve). These curves show a small oscillation which goes deep in the bulk of the distribution function. Figure 6 (to be compared with Figure 2) concentrates the plot of the spatially averaged distribution function more on the bump region, at t=1970 (dash curve), t=1980 (dash-dot curve) and t=1990 (full curve), and shows more clearly this persistent oscillation. The minimum of the bump shows a small oscillation around   3.44 , and the tail beyond the bump is following also the same oscillation, as well as the part of the distribution function penetrating the bulk. This oscillation period of about 20 is present between the high peaks in Figure 4b. It is persistent and repeats itself indefinitely. Note how this oscillation is more important than what we see in Figure 2. Figure 7a and Figure 8a show the contour plots concentrating on the vortex structure appearing in the tail, at t=1960 and t=1980 respectively. They show in more details the complexity of the trapped population compared to Figure 3a. The corresponding 3D plots of the distribution function at the same times are presented in Figure 7b and Figure 8b. The center of the big vortex structure is traveling at the phase velocity of the wave. We note however in Figs. 7a,8a the presence of a second vortex structure above the big central one, which corresponds to the part of the distribution function which is oscillating just following the local maximum of the bump in Figure 6), with particles oscillating at the separatrix between the two trapping regions. So different macroscopic asymptotic physical states are reached depending only on grid scale effects. We have now two adjacent trapping regions traveling together in the phasespace. The 3D plots in Figure 7b and Figure 8b show also different horizontal modulation or striation on the bulk of the distribution function (which are not present in Figure 3b, corresponding to the phase velocities of different modes excited in the system by wave coupling mediated by the trapped particle oscillations (see the discussion of the spectrum below). Some of these modulations appear at velocities below the thermal velocity of the bulk.

Figure 6. Spatially averaged distribution function

F ( ) at t=1970 (dash curve), t=1980 (dash-dot

curve), and and t=1990 (full curve).

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Figure 7a. Contour plot of the

distribution function at t=1960.

Figure 7b. Three-dimensional view of the distribution function at t=1960.

Figure 8a. Contour plot of the

distribution function at t=1970.

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Figure 8b. Three-dimensional view of the distribution function at t=1970.

Figs. 9a-12a) show the plots for the time evolution of the harmonics n=1,2,3,4 respectively. Figs. 9b-12b) show the same plots between t=1800 and t=2000, to make more apparent the modulations of the different modes. We can see that these modulations differ for the different harmonics. Figs. 13,16) show the frequency spectrum for the different Fourier modes, in the steady state regime from t=700 to t=2000. Figure 13a shows the frequency spectrum for the mode n=1 in Figure 9. It shows the dominant mode at   1.0306 , and then different other small amplitude modes responsible for the modulation of the Fourier mode. We repeat this figure in Figure 13b, limiting the maximum of the vertical scale at 100 to exhibit more clearly the different peaks. The phase velocity of the main wave  / k  1.0306 / 0.3  3.435 and corresponds to the minimum we see in Figure 6. This is where the big vortex is centered in Figs.7,8. The other frequency peaks in Figs. 13 are at 0.738, 1.112, 1.313, 1.788. Figure 14 shows the frequency spectrum of the harmonic n=2, or k  0.6 . It has the harmonic peak at   2.056 , and the additional frequency peaks at, for instance, 0.08628, 1.3, 1.76, 2.349, 2.818, 3.59 (note the difference in the vertical scale between Figure 13 and Figure 14). Figure 15 shows the frequency spectrum of the harmonic n=3. It has the harmonic peak at   3.08 , and the additional frequency peaks at, for instance, 2.32, 2.483, 2.794, 3.379, 3.695, 3.849, 4.141, 4.611. Figure 16 shows the frequency spectrum of the harmonic n=4 (the peaks are small compared to the peak of 400in Figure 13a). There is a rich variety of nonlinear coupling between these modes, causing the modulation we see in Figs. 9,12. For instance, in Figure 13 the modes of frequencies 1.030 and 0.738 couple to give the mode of frequency 1.76 in Figure 14 (1.030+0.738=1.768 and k+k=2k). The modes of frequencies 1.030 and 1.313 in Figure 13 couple with the mode of frequency 2.349 in Figure 14 (1.030+1.313=2.343 and k+k=2k). The mode of frequency 2.056 in Figure 14 couples with the modes of frequencies 1.313 and 0.738 in Figure 13 (2.051=1.313+0.738 and 2k=k+k). The mode at the frequency 0.738 in Figure 13b has a phase velocity 0.738/0.3=2.46. We see indeed a small knee appearing in the distribution function in Figure 6 at   2.46 . The mode at the frequency 1.313 in Figure 13b

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corresponds to a phase velocity 1.313/0.3=4.376. We see a local peak of zero slope appearing at the bump in the distribution function in Figure 6 at   4.376 .

(a) Figure 9. Time evolution of the Fourier mode with

(b)

k  0.3 .

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(a) Figure 10. Time evolution of the Fourier mode with

(b)

k  0.6 .

(a) Figure 11. Time evolution of the Fourier mode with

(b)

k  0.9 .

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(a)

(b)

Figure 12. Time evolution of the Fourier mode with

k  1.2 .

(a)

(b)

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Figure 13. Frequency spectrum of the mode with

Figure 14. Frequency spectrum of the mode

k  0.3 .

k  0.6 .

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Figure 15. Frequency spectrum of the mode

k  0.9 .

Figure 16. Frequency spectrum of the mode

k  1.4 .

5. Excitation of a Harmonic Mode n=3 with k=0.3 We consider the case where the mode with n  3 is excited. Since k  n

2 = 0.3, then L

the length of the system in this case is L  20 . The first results we present are obtained on a space-velocity grid of 128x256. With extrema of the velocity at  7 , the recurrence time is



2  381.48 . We use t  0.05 . We use the same initial perturbation as in Eq.(5). k

Figure 17 shows the time evolution of the electric energy, essentially resulting from the unstable mode with n=3, k  0.3 . The initial growth saturates and is followed by the oscillation of the trapped particles, and finally the energy tends to oscillate around a constant level. We present the results until t=400, although the recurrence effect is at t=381.48. The initial bump on tail (full curve in Figure 18) has evolved to a shape showing another smooth bump on tail (dash curve at t=300, dash-dot curve at t=380), with a minimum at the phase

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velocity of the wave, at   3.435 . Indeed we can verify by following the oscillation of the mode n=3, that the nonlinear plasma oscillation has a frequency   1.03 , and a phase velocity

 ph  1.03 / 0.3  3.43 . Figure 19a shows a contour plot at t=380, concentrating

on the region of the vorticies. It shows essentially a BGK mode with three vorticies, whose center is traveling at the phase velocity of the wave. Figure 19b presents the corresponding 3D plot at t=380. Figs. 19a,b show smooth curves. These results are essentially those obtained in [12], and with a grid of 128x128 in [8]. Although unstable sidebands growing from roundoff errors exist in this case, their level however did not affect the results at t=400.

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Figure 17. Time evolution of the total electric energy.

Figure 18. Spatially averaged distribution function

F ( ) at t=0 (full curve), t=300 (dash curve), and

t=380 (dash-dot curve).

However the evolution is different if a more refined grid is used with a smaller time-step, to catch in more details the oscillations and dynamic of the trapped particles, and for those particles which are periodically trapped, untrapped and retrapped during the oscillations. Two cases will be studied. In the first case, we call Case 1, we use a space-velocity grid of 512x1400, with a time step t  0.002 . With extrema of the velocity at  7 , the recurrence

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2  2092.9 . In the second case, we call Case 2, we use a spacek velocity grid of 512x2400, with a time step t  0.002 . With extrema of the velocity at 2  7 , the recurrence time in this case is    3587 . The unstable mode is initially k

time in this case is



excited as in Eq.(5) with n=3, however unstable sidebands with n=1,2 or n=4,5 for instance, growing from round-off errors, can completely modify the evolution of the system after t=400.

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Figure 19a. Contour plot of the distribution function at t=380.

Figure 19b. Three-dimensional plot of the distribution function at t=380.

Figs. 20 present the time evolution of the electric energy, showing growth, saturation and trapped particle oscillations. Then there is a sudden decrease in the electric energy down to a constant value. This is caused by the fact that growing sidebands have reached a level where there is a rapid fusion of the three vortices into a single vortex, associated with a heating of the distribution function. Figure 20a corresponds to Case1 and Figure 20b corresponds to

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Case 2 (the calculation for Case2 is pushed up to t=3000, but we show the results in Figure 20b up to t=2000 to make a comparison with the result in Figure 20a easier).

Figure 20a. Time evolution of the total electric energy, space-velocity grid 512x1400.

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Figure 20b. Time evolution of the total electric energy, space-velocity grid 512x2400.

Figure 21. Spatially averaged distribution function

F ( ) at t=0 (full curve), t=280 (dash curve), and

t=300 (dash-dot curve).

The evolution in Figure 20a and Figure 20b appears to be the same until t  370 . Although the saturation of the initial instability of the mode n=3 appears to be the same as in Figure 17, the amplitude of the trapped particle oscillations up to about t=370 appears more important in the results in Figure 20 than what is presented in Figure 17. Figure 21 shows the initial bump

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on tail at t=0 (full curve), and the bump on tail which is formed at t=280 (dash curve) and at t=300 (dash-dot curve). The results are identical for both Case 1 and Case 2. Figure 22a shows at t=300 the three BGK vortices formed by the initially unstable mode n=3 (we concentrate the contour plot on the vortices region). On the top of the three vortices we see a small layer of another population of trapped particles. Figure 22b presents the 3D plot of Figs. 22a at t=300. The details of the trapped particles we see in Figure 22a are far from the smooth curves we see in Figure 19. Figure 22c presents a plot of the electric field at t=300, and Figure 22d presents a plot of the density at the same time. These results are so far close for both Case 1 and Case 2 up to about t  370 (with the exception of Figure 22a), as we can see from Figure 20a and Figure 20b for instance, inspite of the important difference in the number of grid points in velocity between these two cases. The subsequent evolution of the system after t  370 is however different for Case 1 and Case 2, due to the difference of the number of grid points on the velocity grid. Figs. 23a-g show for Case 1 the evolution of the phase-space for t  370 , at a time where the sidebands have saturated.

Figure 22a. Contour plot of the distribution function at t=300.

Figure 22b. Three-dimensional plot of the distribution function at t=300.

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Figure 22c. Electric field at t=300.

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Figure 22d. Electron density at t=300.

There is a fusion of the vortices in Figs. 23 which leads finally to a single vortex, a signature of a dominant n=1 mode, as can be verified in Figure 30a below. Figure 24a shows the 3D plot of the vortex in Figure 23g, showing the complex vortical structure in the tail of the distribution function. It is interesting to note going through the complex fusion of the vortices and the complex microstructure appearing in Figs. 23b-c for instance, how a single vortex corresponding to the dominant n=1 mode has emerged in Figs. 23e-23g.

Figure 23a. Contour plot of the distribution function at t=400, grid 512x1400.

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Figure 23b. Contour plot of the distribution function at t=500, grid 512x1400.

Figure 23c. Contour plot of the distribution function at t=1000, grid 512x1400.

Figure 23d. Contour plot of the distribution function at t=1200, grid 512x1400.

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Figure 23e. Contour plot of the distribution function at t=1920, grid 512x1400.

Figure 23f. Contour plot of the distribution function at t=1950, grid 512x1400.

Figure 23g. Contour plot of the distribution function at t=1990, grid 512x1400.

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Statistical mechanics of 2D systems [20] indicates that energy should flow by inverse cascade to the longest wavelength available in the system, which is the dominant fundamental n=1 mode shown in Figure 30a, a process characteristic of 2D systems. The frequency spectrum of the Fourier mode n=1 for Case 1 (given in Figure 37) shows the presence of a frequency at   0.4458 , which corresponds to a phase velocity  ph  4.458 for the mode

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with k  0.1 , close to the center of the vortex in Figs. 23e-23g. Hence the fusion of the vortices resulted in a shift of the velocity of the center of the final vortex to a higher velocity compared to Figure 22, at a frequency below the plasma frequency. Note in Figs. 23e-23g a persistent small vertical oscillation of the center of the vortex between   4. and   4.5 , of a period about 70. This will be studied in more details for the more accurate results of Case 2 presented in Figs. 25. We plot in Figure 24b the spatially averaged distribution function at t=1950 (full curve) and at t=1990 (dash-dot curve), where we concentrate on the region of the vortex structure. We use a logarithmic scale for the spatially averaged distribution function, to emphasize the decay of the function.

Figure 24a. Three-dimensional plot of the result in Figure 23g at t=1990.

Figure 24b. Spatially averaged distribution function

F ( ) at t=1950 (full curve) and t=1990 (dash-dot

curve), grid 512x1400.

The distribution function in Figure 24b shows a small oscillation and seems to decay very slowly, showing periodically a point of zero slope at  ph  4.458 , which as we have

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previously mentioned corresponds to the phase velocity of the dominant k  0.1 mode. It also shows during this oscillation what appears to be an inflexion point of zero slope at   3.7 , which corresponds to the phase velocity of the mode n=3 with k  0.3 , and where the frequency spectrum in Figure 38 shows the presence of a frequency with   1.112 , and

 ph  1.112 / 0.3  3.7 . In order to have a zero slope at the velocities 3.7 and 4.458, we see

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in Figure 24b a persistent small bumpy structure appearing in between at a velocity about 4.1, which allows the second point of zero slope at 4.458 to appear. These results will be discussed in more details for Case 2, where more accurate results have been obtained using the finer velocity grid. We draw attention also to the difference between the result in Figure 38 for k  0.3 for Case 1, ant the corresponding one in Figure 41b obtained for the finer velocity grid in Case 2, which shows a second more important peak at a frequency of 1.404. This will be discussed again with more details with the results of Case 2.

Figure 25a. Contour plot of the distribution function at t=400, grid 512x2400.

Figure 25b. Contour plot of the distribution function at t=500, grid 512x2400.

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Figure 25c. Contour plot of the distribution function at t=1000, grid 512x2400.

Figure 25d. Contour plot of the distribution function at t=1200, grid 512x2400.

Figure 25e. Contour plot of the distribution function at t=2000, grid 512x2400.

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Figure 25f. Contour plot of the distribution function at t=2920, grid 512x2400.

Figure 25g. Contour plot of the distribution function at t=2960, grid 512x2400.

Figure 25h. Contour plot of the distribution function at t=3000, grid 512x2400.

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Figure 26. Three-dimensional plot for the result in Figure 25h.

Figure 27a. Spatially averaged distribution function

F ( ) at t=0 (full curve), t=2960 (dash curve), and

t=3000 (dash-dot curve).

Figure 27b. Spatially averaged distribution function

F ( ) at t=2960 (full curve) and t=3000 (dash-dot

curve).

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Figure 27c. Spatially averaged distribution function

F ( ) at t=2800 (full curve) and t=2840 (dash-dot

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curve).

The time evolution of the Fourier modes n=1 to 6 and n=9 for Case1 are presented in Figs. 30a-36a respectively, and in Figs. 30b-36b for Case 2. Up to around t  370 the time evolution of these modes is essentially identical for the two cases, as we previously mentioned also for Figure 20a and Figure 20b . It is for t>370 that important differences appear. Figs. 25a-h show the evolution of the phase-space for Case2 when t>370, for a spacevelocity grid of 512x2400 at t=400, 500, 1000, 1200, 2000, 2920,2960 and 3000 respectively. The results now differ from what is presented in Figs. 23. For instance at t=1000 Figure 25c shows the appearance of the single vortex structure, which is not the case in the equivalent one of Case 1 in Figure 23c. It is clear that the dynamic of the trapped particles are in this case different from Case 1. Again the merging of the vortices in the presence of the growing sidebands has resulted in an inverse cascade with a transfer of energy to the longest wavelength available, i.e. the mode k  0.1 shown for Case 2 in Figure 28b, plotted until t=3000. Also the phase velocity of the final vortex has shifted to slightly higher values. The frequency spectrum of the mode k  0.1 in Figure 37b, calculated from by taking the frequency spectrum from t=1689 to t=3000, shows a dominant frequency   0.4697 , corresponding to a phase velocity

 ph  4.697 . Figs. 25f-25h at t=2920, t=2960 and t=3000

show the position of the center of the vortex having a small vertical oscillation around this value of 4.697, the period of this vertical oscillation is about 80. Figure 26 shows the 3D plot of the results in Figure 25h. Note the striations of the bulk of the distribution function corresponding to the phase velocities of different modes excited, which will be examined in more details with Figs. 37-42. Figure 27a shows the spatially averaged distribution functions at t=0 (full curve), at t=2960 (dash-curve) and at t=3000 (dash-dot curve). Figure 27a shows the tail of the distribution function curves smoothly decaying. These tails are plotted in more details on a logarithmic scale in Figure 27b, at t=2960 (full curve) and t=3000 (dash-dot curve), which shows the presence of a persistent oscillation in the tail of period about 80. This oscillation corresponds to the small vertical oscillation of the center of the vortex we mentioned in Figs. 25f-25h.

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Figure 28a. Electric field at t=2920.

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Figure 28b. Electric field at t=2960.

Figure 28c. Electric field at t=3000.

The nature of the solution obtained in Figs. 22 after the saturation of the unstable mode with n=3, and the solution in Figs. 25f-h, appears different. See in Figure 22b how the modulation of the n=3 mode is penetrating deep in the bulk. We present in Figs. 28a-c the electric field corresponding to the solution in Figs. 25f-h. It shows the electric field with a peak at the phase-space hole, and a rapid variation from positive to negative values. The phase-space hole in Figure 25f corresponds to the density cavity which appears in the density plot of Figure 29. This density cavity is traveling at the phase velocity of the dominant mode n=1, while the hole performs a small vertical oscillation in the phase-space plot.

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Figure 29. Density at t=2920.

Figure 30a. Time evolution of the Fourier mode with

k  0.1 , grid 512x1400.

Figure 30b. Time evolution of the Fourier mode with

k  0.1 , grid 512x2400.

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Figure 31a. Time evolution of the Fourier mode with

k  0.2 , grid 512x1400.

Figure 31b. Time evolution of the Fourier mode with

k  0.2 ,grid 512x2400.

Figure 32a. Time evolution of the Fourier mode with

k  0.3 , grid 512x1400.

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Figure 32b. Time evolution of the Fourier mode with

k  0.3 , grid 512x2400.

Figure 33a. Time evolution of the Fourier mode with

k  0.4 , grid 512x1400.

Figure 33b. Time evolution of the Fourier mode with

k  0.4 , grid 512x2400.

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Figure 34a. Time evolution of the Fourier mode with

k  0.5 , grid 512x1400.

Figure 34b. Time evolution of the Fourier mode with

k  0.5 , grid 512x2400.

Figure 35a. Time evolution of the Fourier mode with

k  0.6 , grid 512x1400.

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Figure 35b. Time evolution of the Fourier mode with

k  0.6 , grid 512x2400.

Figure 36a. Time evolution of the Fourier mode with

k  0.9 , grid 512x1400.

Figure 36b. Time evolution of the Fourier mode with

k  0.9 , grid 512x2400.

We look to the different Fourier modes for the Case 2, presented in Figs. 30b-36b, and their corresponding frequency spectra presented in Figs. 39-44. Figure 39a gives the frequency spectrum of the dominant mode k  0.1 , presented in Figure 30b, from a time t1 =40 to t2 =695, during the early growth of the mode and the beginning of the fusion of the

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The Bump-on-Tail Instability

vortices. It shows a growing mode at   0.441 , and a plasma mode at   1.102 . In Figure 39b we see the steady state spectrum of the mode k  0.1 , calculated from t1 =1689 to t2 =3000, with a dominant frequency at   0.4697 . This corresponds to a phase velocity  ph  4.697 , which corresponds to the inflexion point of zero slope (or a local minimum with very small positive slope) we see in the full curve in Figure 27b and Figure 27c around

  4.7 . There is another very small plateau of zero slope appearing

around   3.7 , which corresponds to the phase velocity of the mode at   1.112 in Figure

41b,

corresponding

to

the

mode

with

k  0.3

in

Figure

32b,

 ph  1.112 / 0.3  3.7 . It is clear that the oscillation we see in Figure 27b is taking place in

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such a way as to maintain a very small plateau and an inflexion point of zero slope (or a minimum followed by a very small positive slope), corresponding to the phase velocities of the waves. This oscillation of the tail we see in Figure 27b of half-period 40 is due to the trapped particles, and corresponds to the vertical oscillation we see in the vortex center in Figs. 25f,25g,25h. This oscillation of the trapped particles persists indefinitely, as we can see from Figure 27c where we plot the tail of the distribution at t=2800 and t=2840, showing exactly the same oscillation of half-period 40 we observe in Figure 27b at t=2960 and t=3000.

Figure 37. Frequency spectrum of the mode

k  0.1 , grid 512x1400.

Figure 38. Frequency spectrum of the mode

k  0.3 , grid 512x1400.

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Figure 39a. Frequency spectrum of the mode

k  0.1 , grid 512x2400, (from t1 =40

Figure 39b. Frequency spectrum of the mode

k  0.1 , grid 512x2400, (from t1 =1689

Figure 40. Frequency spectrum of the mode

k  0.2 , grid 512x2400, (from

to t2 =695 ).

to t2 =3000).

t1 =1689 to t2 =3000).

Note that the vortex we see in Figs. 25c-25h is located on the decaying side of the spatially averaged distribution function in Figs. 27b,27c, for   3.7 .

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Figure 41a. Frequency spectrum of the mode

k  0.3 , grid 512x2400, (from

t1 =40 to t2 =695 ).

Figure 41b. Frequency spectrum of the mode

k  0.3 , grid 512x2400, (from t1 =1689 to t2 =3000).

Figure 42. Frequency spectrum of the mode

k  0.4 , grid 512x2400, (from t1 =1689

to t2 =3000).

Figure 31b presents the time evolution of the Fourier mode with k  0.2 , and the frequency spectrum of this mode calculated during the steady state oscillation from t1 =1689 to t2 =3000, is presented in Figure 40. It shows two important peaks, one at   0.709 with

 ph  0.709 / 0.2  3.54 corresponds to the small plateau we see in Figures 27b-27c, and the other one at   0.9347 with a phase velocity

a phase velocity

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Magdi Shoucri

 ph  0.9347 / 0.2  4.673 corresponds to the inflexion point at   4.69 (note the exact

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location of this inflexion point is showing a very small oscillation) in Figs. 27b-27c. Other very small peaks like the one at   0.839 are due to nonlinear coupling of modes and contribute to the modulation we see in the oscillation.

Figure 43. Frequency spectrum of the mode

k  0.5 , grid 512x2400, (from t1 =1689 to t2 =3000).

Figure 44a. Frequency spectrum of the mode

k  0.6 , grid 512x2400, (from t1 =40 to t2 =695 ).

Figure 44b. Frequency spectrum of the mode . k

 0.6 , grid 512x2400, (from t1 =1689.to t2 =3000).

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Figure 32b presents the time evolution of the Fourier mode k  0.3 . Initially this mode shows the same evolution as in Figure 32a, but after the fusion of the vortices the mode has a higher level with respect to the result in Figure 32a. The frequency spectrum of the mode k  0.3 calculated from t1 =40 to t2 =695, during the transient period of the growth of the mode and the fusion of the vortices, is shown in Figure 41a, and shows a strong peak at   1.026 , and other peaks such as at   1.112 for instance. During the steady state oscillation from t1 =1689 to t2 =3000 the frequency spectrum of the mode k  0.3 is presented in Figure 41b (note the difference with Figure 38). It shows two peaks, one at

  1.112 with a phase velocity  ph  1.112 / 0.3  3.7 corresponding to the small plateau we see in Figs. 27b-27c, and the other one at   1.404 (the third harmonic of the fundamental mode with   0.4697 and k  0.1 ) with a phase velocity  ph  1.404 / 0.3  4.68 corresponds to the inflexion point in Figs. 27b-27c. Other very small peaks at   1.179 ,   1.308 ,   1.634 are due to nonlinear coupling of modes and contribute to the modulation we see in the oscillation. For instance the mode with   0.4697 and k  0.1 can couple with the mode at   0.709 at k  0.2 and the mode with   1.179 at k  0.3 (0.4697+0.709=1.1787 with wavenumbers 0.1+0.2=0.3). The mode with   0.4697 and k  0.1 can couple with the mode at   0.839 at k  0.2 and the mode with   1.308 at k  0.3 (0.4697+0.839=1.3087 with wavenumbers 0.1+0.2=0.3). Figure 33b presents the time evolution of the Fourier mode k  0.4 (note the difference with Figure 33a), and the frequency spectrum of this mode calculated during the steady state oscillation from t1 =1689 to t2 =3000, is presented in Figure 42. It shows an important peak at   1.874 (the fourth harmonic of the mode with k  0.1 ), with a phase velocity Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved.

 ph  1.874 / 0.4  4.685 which corresponds to the inflexion point of zero slope (or the minimum with a very small positive slope) in Figs. 27b-27c. The other small peaks due to nonlinear coupling of modes contribute to the modulation of the mode k  0.4 (at   0.656 ,   0.9347 ,   1.553 ,   1.644 ,   2.1 etc…). The mode with   0.4697 and k  0.1 can couple with the mode at   1.179 and k  0.3 and the mode with   1.644 at k  0.4 (0.4697+1.179=1.648 with wavenumbers 0.1+0.3=0.4). The mode with   0.4697 and k  0.1 can couple with the mode at   1.634 at k  0.3 and the mode with   2.10 and k  0.4 (0.4697+1.634=2.104 with wavenumbers 0.1+0.3=0.4). The presence of the modes at   0.656 and   0.9347 , is interesting, Their phase velocity is respectively

 ph  1.64 and  ph  2.337 ,

corresponding to the small knees we see in the corresponding velocities in the spatially averaged distribution functions in Figs. 27a,27b,27c, i.e. at velocities where the waves should be damped. They also corresponds to the horizontal striations or modulations we see in the 3D plot of Figure 26. Figure 34b presents the time evolution of the Fourier mode k  0.5 (note the difference with Figure 34a), and the frequency spectrum of this mode calculated from t1 =1689 to t2 =3000, is presented in Figure 43. It shows an important peak at   2.344 (the fifth

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354

Magdi Shoucri

harmonic of the mode with k  0.1 ), with a phase velocity

 ph  2.344 / 0.5  4.688

which corresponds to the inflexion point (or a minimum followed by a very small positive slope) in Figs. 27b-27c. The other small peaks due to nonlinear coupling of modes contribute to the modulation we see in the mode k  0.5 (at   2.114 ,   2.57 for instance). The mode with   0.4697 and k  0.1 can couple with the mode at   2.1 at k  0.4 and the mode with   2.57 and k  0.5 (0.4697+2.1=2.5697 with wavenumbers 0.1+0.4=0.5). The mode with   0.4697 and k  0.1 can couple with the mode at   1.644 and k  0.4 and the mode with   2.114 at k  0.5 (0.4697+1.644=2.1137 with wavenumbers 0.1+0.4=0.5). The mode with   0.709 and k  0.2 can couple with the mode at   1.404 and k  0.3 and the mode with   2.114 at k  0.5 (0.709+1.404=2.114 with wavenumbers 0.2+0.3=0.5). These are few examples of mode coupling mediated by the oscillations of the trapped particles causing the different modulations we see in the different Fourier modes. Figure 35b presents the time evolution of the Fourier mode k  0.6 (see the difference with Figure 35a), and the frequency spectrum of this mode calculated from t1 =40 to t2 =695, during the transient period of the growth of the mode and the fusion of the vortices, is shown in Figure 44a, and shows the harmonic of the peaks in Figure 39a, a strong peak at   2.22 and a peak at   2.5 , and other peaks. During the steady state oscillation from t1 =1689 to t2 =3000 the frequency spectrum of the mode k  0.6 is presented in Figure 44b. It shows an important peak at   2.81 (the sixth harmonic of the fundamental mode with   0.4697

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and k  0.1 ) with a phase velocity  ph  2.81 / 0.6  4.683 corresponding to the inflexion point (or the minimum with very small positive slope) we see in Figs. 27b-27c. Other small peaks, like the one at   2.583 and   3.039 are due to nonlinear coupling of modes and contribute to the small modulation we see in the oscillation of the mode. The mode with   2.81 and k  0.6 can couple with the mode at   0.9347 and k  0.2 and the mode with   1.874 and k  0.4 (0.9347+1.874=2.8087 with wavenumbers 0.2+0.4=0.6), and also with the mode at   0.7094 and k  0.2 and the mode with   2.099 and k  0.4 (0.7094+2.099=2.8084 with wavenumbers 0.2+0.4=0.6). The mode with   0.9347 and k  0.2 can couple with the mode with   2.099 and k  0.4 and the mode with   3.039 and k  0.6 (0.9347+2.099=3.0337 with wavenumbers 0.2+0.4=0.6). The frequency spectrum of the mode with k  0.9 in Figure 36b is negligibly small.

6. Conclusion Wave-particle interaction is among of the most important and extensively studied problems in plasma physics. Langmuir waves and their familiar Landau damping and growth are fundamental examples of wave-particle interaction (see for instance [18,19,22-27] and references therein). Following the nonlinear evolution of a monochromatic wave in [25], it was shown that the wave decays to a constant value continuously modulated by the oscillation of the trapped particles. It was also pointed out in [19] that by increasing the number of grid

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points, low frequency oscillations in the electric field due to the trapped particles are observed. By studying the phase-space trajectories of resonant particles, the results in [13] have shown that the dynamics near the border of the resonant region (separatrix) becomes ergodic and chaotic and the trapped particles can escape from the potential well and perform more or less long flights in the phase-space, and are retrapped in the potential well, and this has important effects on the long time modulation observed in the constant asymptotic state. However the trapping, detrapping and retrapping processes are not sufficiently long to allow energy dissipation in the wave following the loss and gain of energy which takes place during the processes. Analyzing the Lagrangian trajectories of resonant particles in the selfconsistent field calculated from the Vlasov-Poisson system is certainly an interesting way which sheds light in the wave-particle interaction physics of these problems [28]. The importance of the dynamics of trapped particles, especially in the resonant region of the phase-space, has been also pointed out in [10,11]. The bump-on-tail instability has been generally studied analytically and numerically under various approximations, either assuming a cold beam, or the presence of a single wave, or assuming conditions where the beam density is weak so that the unstable waves exhibit negligible growth and can be considered as essentially of slowly varying amplitude in an envelope approximation (see for instance [4,14] and references therein). There are of course situations where a single wave theory and a weak beam density do not apply. In the present work, we have presented a study for the long-time evolution of the Vlasov-Poisson system for the case when the beam density is about 10% of the total density, which provides a vigorous beam-plasma interaction and an important wave-particle interaction, and which results in important trapped particles effects. A warm beam is considered, so that sidebands can grow due to mode coupling mediated by resonant particles. We have seen how a fine resolution grid in the phase-space and a small time-step are necessary to follow accurately the nonlinear dynamic of the trapped particles. In the results presented in section 4 for the time evolution of the fundamental mode n=1, the fine resolution grid showed important differences of the particles trapped in the phase-space plots (see Figs. 3 and Figs. 7), with results showing the importance of the microscopic processes for their possible consequences on the large scale dynamics, with interesting physics associated with the macroscopic solution, and with the persistent oscillations and modulations in the nonlinear asymptotic state of the bump-on-tail system. We have also studied the bump-on-tail instability in the case when the length of the system L is larger than the initially unstable wavelength  , which allows the growth of sidebands. In this case the evolution shows initially the classical behaviour of the growth of the initially unstable wave, followed by the saturation of the instability and the formation of BGK vortices, where the electric field is indeed oscillating around a constant amplitude modulated by the trapped particles oscillation. At this stage we find little difference between the two cases studied, where 1400 grid points and 2400 grid points were used in the velocity grid. However this is now only an intermediate regime, since sidebands are growing in this case where the length of the system L is larger than the initially unstable wavelength  . A spectrum is excited when tail particles form a warm beam which allows mode coupling mediated by the resonant particles, in contrast with the single wave situation when a cold beam is used to study the bump-on-tail instability (see [14] and references therein). When the growing sidebands reach the level of the BGK wave, they disrupt the BGK equilibrium

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Magdi Shoucri

formed in the first phase of the nonlinear evolution. The subsequent evolution is dominated by an inverse cascade where energy is flowing to the longest wavelengths available in the system, a process characteristic of 2D systems [20], and the formation of a single hole in the phase-space, where the trapped particles are accelerated to high velocities. The density plot show a corresponding cavity (Figure 29). Frequencies below the plasma frequency are associated with the longest wavelengths, and in the modulation of the different Fourier modes. At this stage important differences are seen between the two cases studied where 1400 and 2400 grid points have been used in the velocity grid, the second case providing a more accurate calculation of the spectrum, the phase-space and the spatially averaged distribution function. The finer grid has important effects on the microscopic kinetic processes which in turn have consequences on the macroscopic evolution of the large scale dynamics of the system, as recently pointed out in several publications [29-32]. Figs. 27 show curves with a decaying slope, and this slope takes the value of zero at the phase velocities of the excited waves. The low frequencies associated with the dominant longer wavelengths result in higher phase velocities of different modes, which are accelerating trapped particles to values corresponding to the velocities in the tail of the distribution function with kinetic energies above the initial energy of the beam. The accelerated population is adjusting in order to provide the distribution function with a zero slope, allowing the different modes to oscillate at a constant amplitude (modulated by the oscillation of the trapped particles). The increase in the kinetic energy due to the particles acceleration is equivalent to the decrease in the electric energy we see in Figs. 20. Note the difference between the distribution functions in Figs. 27, and the plateau-like shape in Figure 24b. See also the differences in the Fourier modes in Figs. 30a-36a and Figs. 30b-36b. These results point to the importance of a fine phase-space grid and a small time step to study the physics and oscillations of the trapped particles. We note that the formation of phase-space holes in situations of beam-plasma interactions has been recently reported in [33,34]. The problem when several unstable modes are initially excited (instead of a single unstable mode) is certainly of interest. Some results presented in [35] for this problem have shown a strong acceleration of particles in the case nb  0.1 , in which case the tail particles are accelerated considerably to velocities higher than twice the initial beam velocity. The distribution function in this case takes the shape of a two-temperature Maxwellian distribution function with a high energy tail having a smooth negative slope. This result seems to agree with experimental observations from current drive experiments using an electron beam injected into the plasma (Advanced Concept Torus ACT-1 device), where it is observed that a significant fraction of the beam and background electrons are accelerated considerably beyond the initial beam velocity [36]. In none of the ACT-1 discharges is a distinctive feature of a plateau predicted from quasilinear theory [37] apparent in the distribution function. The evolution of the waves amplitude in [35] shows a rapid rise, followed by an abrupt collapse of the waves amplitude, the energy being delivered to the accelerated particles. When nb is decreased, the acceleration of the particles is decreased, and when it is reduced to nb  0.01 , a quasilinear plateau is formed and the waves amplitude saturate at a constant level, showing trapped particles oscillation. Finally, we mention the work in [38], with two spatial dimensions and a magnetic field (a configuration closer to the ACT-1 device previously

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cited), which shows in a bump-on-tail instability a rich variety of physics including also acceleration of particles to high energies.

Acknowledgments M. Shoucri is grateful to the Institut de Recherche d’Hydro-Québec (IREQ) computer center CASIR for computer time used to do the present work.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]

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[12] [13] [14] [15] [16] [17] [18] [19] [20] [21]

[22] [23] [24]

Dawson, J.M., Shanny, R. Phys. Fluids 1968, 11, 1506 Joyce, G., Knorr, G., Burns, T. Phys. Fluids 1971, 14, 797 Denavit, J., Kruer, W.L. Phys. Fluids 1971, 14, 1782 Umeda, T., Omura, Y., Yoon, P.H., Gaelzer, R., Matsumoto, H. Phys. Plasmas 2003, 10, 382-391 Schamel, H., Phys. Plasmas 2000, 7, 4831 Eliasson, B., Shukla, P.K. Phys. Rep. 2006, 422, 225 Cheng, C.Z., Knorr, G. J. Comp. Phys. 1976, 22, 330-351 Shoucri, M. Phys. Fluids 1979, 22, 2038-2039 Bernstein, M., Greene, J.M., Kruskal, M.D. Phys., Rev. 1957, 108, 546 Buchanan, M., Dorning, J. Phys. Rev. E 1995, 52, 3015 Bertrand, P., Ghizzo, A., Feix, M., Fijalkow, E., Mineau, P., Suh, N.D., Shoucri, M., 1988 Computer Simulations of Phase-space Holes Dynamics; Nonlinear Phenomena in Vlasov Plasmas, Ed. F. Doveil, Editions de Physique , Orsay. Nakamura, T., Yabe, T. Comput. Phys. Comm. 1999, 120, 122-154; Crouseilles, N., Respaud, T., Sonnendrücker, E. Comp. Phys. Comm. 2009, 180, 1730-1745 Valentini, F., Carbone, V., Veltri, P., Mangeney, A. Trans. Theory Stat. Phys. 2005, 34, 89 Firpo, M.-C., Doveil, F., Elskens, Y., Bertrand, P., Poleni, M., Guyomarc’h, D. Phys. Rev. E 2001, 64, 026407 Califano, F., Lantano, M. Phys. Rev. Lett. 1999, 83, 96-99 Califano, F., Pegoraro, F., Bulanov, S.V. Phys. Rev. Lett. 2000, 84, 3602-3605 Kruer, W.L., Dawson, J.M., Sudan, R.N. Phys. Rev. Lett. 1969, 23, 838 Shoucri, M. Phys. Fluids 1978, 21, 1359-13654 Brunetti, M., Califano, F., Pegoraro, F. Phys. Rev. E 2000, 62, 4109-4114 Knorr, G. Plasma Phys. 1977, 19, 529-538 Shoucri, M. Numerical Solution of Hyperbolic Differential Equations, Nova Science Publishers,Inc.: New-York, 2008; Shoucri, M. The Application of the Method of Characteristics for the Numerical Solution of Hyperbolic Differential Equations; in Numerical Simulation Research Progress, Simone P. Colombo et al. Eds., p.1-98, Nova Science Publ.:N.Y., 2008. Landau, L.D. J. Phys. (Moscow) 1946, 10,25-34 O’Neil, T. Phys. Fluids 1965, 8, 2255 Lancelotti, C., Dorning, J. Phys. Rev. Lett. 1998, 81, 5137

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Magdi Shoucri

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[25] Manfredi, G. Phys. Rev. Lett. 1997, 79, 2815-2818 [26] Elskens, Y., Escande, D.F. 2003 Microscopic Dynamics of Plasmas and Chaos, Bristol, Institute of Physics Publishing. [27] Doveil, F., Firpo, M.-C., Elskens, Y., Guyomarc’h, D., Poleni, M., Bertrand, P. Phys. Lett. A 2001, 284, 279-285 [28] Ghizzo, A., Shoucri, M.M., Bertrand, P., Johnson, T., Lebas, J. J. Comp. Phys. 1993, 108, 105-111 [29] Califano, F., Pegoraro, F., Bulanov, S.V. Phys. Rev. Lett. 2000, 84, 3602-3605 [30] Califano, F., Lantano, M. Phys. Rev. Lett. 1999, 83, 96-99 [31] Califano, F., Pegoraro, F., Bulanov, S.V., Mangeney, A., Phys. Rev. E 1998, 57, 70487059 [32] Palodhi, L., Califano, F., Pegoraro, F. Plasma Phys. Contol. Fusion 2009, 51, 125006(1-13) [33] Sircombe, N.J., Dieckman, M.E., Shukla, P.K., Arber, T.D. Astron. Astrophys. 2006, 452, 371 [34] Sircombe, N.J., Bingham, R., Sherlock, M., Mendonca, T., Norreys, P. Plasma Phys. Control. Fusion 2008, 50, 065005-(1-10) [35] Ghizzo, A., Shoucri, M.M., Bertrand, P., Feix, M., Fijalkow, E. Phys. Lett. A 1988, 129, 453-458 [36] Okuda, H., Horton, R., Ono, M., Wong, K.L. Phys. Fluids 1985, 28, 3365 [37] Laval, G., Pesme, D. Plasma Phys. Control. Fusion 1999, 41, A239-A246 [38] Manfredi, M., Shoucri, M., Shkarofsky, I., Ghizzo, A., Bertrand, P., Fijalkow, E., Feix, M., Karttunen, S., Pattikangas, T., Salomaa, R. Fusion Tech. 1996, 29, 244-260

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INDEX A

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Air Force, 296 algorithm, 32, 46, 48, 50, 51, 52, 53, 67, 70, 73, 77, 79, 87, 101, 116, 118, 131, 148, 241, 271, 283, 286 ambiguity, 92 amplitude, 3, 4, 9, 10, 11, 12, 15, 18, 44, 45, 47, 71, 72, 85, 86, 102, 103, 105, 106, 109, 110, 111, 115, 118, 134, 136, 150, 151, 152, 157, 170, 187, 197, 206, 217, 245, 258, 260, 280, 317, 318, 319, 323, 326, 332, 355, 356 anisotropy, 46, 52, 53, 55, 57, 124 annihilation, 26 arithmetic, 261 atoms, 237 azimuthal angle, 8

B background, vii, 2, 17, 48, 62, 66, 67, 71, 84, 102, 104, 125, 130, 132, 148, 268, 287, 288, 289, 296, 317, 319, 356 bandwidth, 91, 116, 117, 118 beams, vii, 26, 45, 53, 55, 56, 57, 58, 59, 84, 90, 113, 114, 118, 233, 234, 245, 315 behavior, 2, 47, 90, 102, 136, 140, 152 benchmarking, viii, 237, 241, 261, 270, 283 benchmarks, 279 benign, 86 Boltzmann constant, 239 breakdown, 72 broadband, 89, 108, 118 burning, 86

C calculus, xi Canada, 163, 317 capsule, 90 cast, 68

categorization, 86 cell, vii, 2, 35, 36, 37, 42, 43, 44, 47, 52, 66, 68, 69, 70, 79, 82, 104, 110, 242, 246, 247, 248, 249, 252, 259, 260, 263, 264, 270, 272, 275, 277, 286, 302, 303, 304 charge density, 173, 239, 261 classes, 34, 35 classification, 85 closure, 48, 124, 146, 147, 158, 159, 241 codes, vii, viii, xi, xii, 33, 38, 44, 45, 47, 48, 49, 52, 59, 61, 66, 67, 71, 72, 73, 76, 80, 82, 87, 91, 94, 102, 103, 108, 113, 114, 134, 150, 159, 165, 173, 234, 245, 283, 301, 318 collisions, vii, viii, 23, 24, 25, 26, 33, 58, 84, 113, 114, 147, 156, 237, 238, 239, 240, 241, 242, 249, 250, 253, 260, 262, 263, 268, 277, 280, 283, 284, 286, 287, 291, 304, 305, 312, 313 combined effect, 163, 165, 198 communication, 49, 122 complexity, 198, 233, 324 complications, 77, 81 components, 17, 50, 77, 78, 82, 106, 138, 143, 145, 147, 149, 152, 155, 250, 251, 306 compounds, 270 comprehension, vii, 157 compression, 90 computation, 1, 2, 3, 4, 9, 18, 19, 34, 246 computational grid, 75, 263 computer systems, 295 computer technology, viii, 3, 19, 159 computing, 30, 76, 87, 94 concentrates, 182, 324 conduction, 84, 257, 288, 289, 290, 291 conductivity, 258, 287 configuration, 30, 31, 60, 65, 71, 80, 82, 87, 130, 153, 154, 156, 174, 356 confinement, viii, 25, 47, 84, 86, 89, 90, 124, 137, 147, 149, 150, 151, 152, 157, 159 conservation, 18, 24, 35, 43, 47, 52, 62, 73, 92, 95, 97, 98, 104, 107, 110, 118, 123, 128, 129, 131, 132, 139, 157, 158, 238, 270, 273, 276, 277, 281, 282, 283, 293, 297, 298, 299, 300, 301 construction, 95, 118, 158, 278, 299

Eulerian Codes for the Numerical Solution of the Kinetic Equations of Plasmas, Nova Science Publishers, Incorporated, 2010. ProQuest Ebook

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contour, 127, 152, 173, 186, 188, 198, 199, 200, 207, 230, 246, 270, 321, 324, 330, 333 control, 271, 305 convergence, 30, 140, 169, 247, 254, 255, 256, 259, 261, 282, 293 cosmic rays, 164 Coulomb gauge, 166 Coulomb interaction, 283, 289, 291 couples, 106, 275, 326 coupling, viii, 9, 38, 133, 134, 317, 318, 324, 326, 352, 353, 354, 355 covering, 77, 80, 127, 238 CPU, 104, 282 critical density, 108, 163, 170, 197, 206 cycles, 164

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D damping, 4, 11, 12, 18, 44, 46, 47, 59, 63, 66, 71, 72, 73, 74, 75, 76, 84, 89, 90, 91, 98, 103, 104, 105, 106, 107, 108, 109, 110, 111, 114, 115, 118, 123, 127, 128, 129, 134, 136, 137, 138, 145, 148, 150, 151, 152, 154, 196, 239, 247, 293, 318, 354 decay, 44, 47, 48, 72, 73, 112, 116, 119, 173, 294, 337 decomposition, 49, 80 deficiency, 254 deflation, 111 deformation, 182, 204 degradation, 147 density fluctuations, 137 Department of Energy, 119 depression, 11, 12, 16 derivatives, 36, 37, 38, 42, 44, 47, 95, 156, 250, 251, 255, 260, 266, 270, 280, 305 destruction, xi deviation, 72 differential equations, 1, 164 differentiation, 99 diffraction, 113 diffusion, 11, 45, 49, 170, 241, 242, 247, 248, 251, 252, 255, 280, 282, 291, 305, 311, 312, 318, 321 diffusivity, 140, 146, 147, 157, 241, 318, 321 dimensionality, xii, 119, 234 discharges, 24, 259, 291, 315, 356 discontinuity, 172, 173, 179, 181, 204, 205 discretization, 32, 34, 35, 38, 41, 42, 52, 101, 138, 277, 280 dispersion, 71, 72, 86, 95, 96, 97, 99, 100, 101, 102, 103, 104, 109, 111, 112, 118, 205, 252, 303 displacement, 30, 34, 50, 67, 84, 182, 239, 293 distortions, 24 divergence, 240, 254, 261, 264, 270, 277, 280, 296, 298, 306, 307, 308, 310, 311 diversity, viii, xii, 323 dominance, 275 duration, 23, 75, 133, 164 dynamical systems, 23

E electric charge, 24, 27, 32, 238, 262, 297 electric current, 285 electromagnetic, 1, 2, 3, 4, 5, 7, 19, 23, 24, 25, 27, 28, 29, 39, 40, 44, 45, 46, 49, 52, 54, 55, 56, 58, 59, 65, 76, 77, 79, 82, 90, 91, 112, 116, 124, 163, 165, 166, 168, 170, 179, 180, 198, 205, 207, 229, 237, 238 electromagnetic fields, 2, 7, 28, 91, 124, 168, 237 electromagnetic waves, 24, 198 elucidation, 65 encouragement, xii, 234 energy transfer, 46 entropy, 4, 19, 75, 76, 90, 123, 128, 130, 137, 138, 139, 140, 141, 142, 143, 144, 146, 148, 153, 155, 156, 158, 270 environment, 25 equality, 274, 301 equating, 302 equilibrium, 17, 26, 44, 45, 67, 73, 84, 86, 92, 102, 132, 151, 256, 258, 261, 282, 283, 284, 285, 287, 290, 293, 295, 355 erosion, 256, 258 estimating, 72 etching, 111

F FFT, 67, 127 fidelity, 247 field theory, 26 filament, 198 filters, 49 flexibility, 267 floating, 257, 261, 276 fluctuations, viii, 2, 24, 44, 45, 46, 52, 53, 54, 98, 108, 115, 116, 123, 124, 137, 138, 139, 141, 145, 146, 152, 156, 173, 247, 252, 253, 259, 261 fluid, vii, 3, 11, 24, 25, 26, 30, 31, 33, 46, 52, 53, 56, 66, 67, 68, 77, 87, 89, 90, 91, 92, 95, 119, 124, 137, 141, 143, 144, 145, 146, 172, 237, 240, 241, 288, 289, 290, 313 focusing, 137 France, 23, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 313 freedom, 28, 30 friction, 285 fuel, 86, 90 function values, 271 fusion, viii, 2, 23, 25, 44, 50, 84, 86, 89, 90, 123, 124, 137, 240, 257, 295, 314, 331, 334, 337, 348, 353, 354

Eulerian Codes for the Numerical Solution of the Kinetic Equations of Plasmas, Nova Science Publishers, Incorporated, 2010. ProQuest Ebook

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G gases, 237, 291, 294 generalization, 124, 125, 126, 127 generation, viii, xi, 25, 26, 44, 45, 46, 48, 56, 89, 113, 146, 157, 159, 163, 164, 165, 234 global communications, 49 gravity, 246 Greece, 236 grid resolution, 262, 284, 294 grids, 2, 3, 9, 66, 70, 71, 80, 82, 93, 95, 100, 127, 128, 150, 271, 305 grouping, 298, 307, 308, 310, 311 groups, 246 growth, viii, 48, 55, 75, 76, 103, 110, 116, 135, 136, 138, 139, 140, 141, 144, 146, 157, 159, 182, 243, 244, 245, 302, 317, 318, 320, 323, 329, 331, 348, 353, 354, 355 growth rate, 55, 103, 110, 135, 136, 139, 244 growth time, 116

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H Hamiltonian, 4, 26, 29, 38, 39, 62, 126, 130, 131, 132, 135, 136, 166, 167 heat, 24, 86, 124, 137, 138, 140, 141, 145, 146, 147, 153, 155, 157, 158, 159, 241, 256, 257, 258, 259, 287, 288, 289, 290, 291 heat conductivity, 158, 258, 290 heat transfer, 287, 288 heating, 9, 10, 11, 44, 47, 48, 63, 84, 86, 259, 289, 311, 331 heavy particle, 238 height, 23, 294 helicity, 151, 154 histogram, 147 hot spots, 314 House, 314 housing, xi hybrid, 33, 46, 49, 234, 248, 249 hydrogen, 256

I implementation, 3, 67, 150 incidence, 230, 233 inclusion, 89, 104, 116, 242 industry, xi inelastic, 256, 258, 259 inequality, 241 inertia, 25, 49 infinite, 11, 28, 30, 31, 116, 238, 253, 299, 301 inflation, 89, 90, 103, 108, 110, 114, 115, 116, 118, 119 instability, viii, 26, 33, 44, 45, 46, 47, 52, 53, 54, 55, 56, 57, 59, 60, 87, 90, 96, 110, 111, 119, 132, 135,

139, 141, 152, 156, 243, 246, 249, 317, 318, 319, 320, 332, 355, 357 integration, xi, 4, 17, 18, 32, 43, 52, 60, 62, 63, 87, 88, 140, 237, 239, 241, 285 interaction, viii, 1, 9, 10, 11, 12, 24, 25, 26, 32, 45, 56, 58, 61, 80, 88, 89, 163, 164, 165, 172, 182, 204, 205, 215, 216, 231, 234, 241, 244, 256, 259, 287, 312, 317, 319, 320, 354, 355 interaction process, 25 interactions, vii, 3, 9, 11, 26, 33, 44, 54, 56, 77, 80, 90, 120, 127, 164, 165, 231, 234, 236, 238, 240, 250, 253, 256, 291, 356 interface, 172, 173, 174, 179, 182, 187, 188, 198, 200, 204, 205, 207, 215, 216, 230, 231, 233 interval, 37, 41, 43, 52, 53, 133, 136, 257, 263, 270, 271, 286, 293, 302 invariants, 28, 45, 128, 135 inversion, 47 ion transport, 262 ionization, 240, 242, 256, 258, 260 ions, 2, 9, 10, 11, 12, 17, 26, 48, 52, 60, 67, 71, 74, 77, 80, 86, 108, 119, 124, 125, 132, 151, 163, 164, 165, 166, 167, 170, 172, 173, 174, 175, 179, 182, 183, 184, 186, 187, 188, 191, 197, 198, 206, 207, 216, 217, 219, 221, 230, 231, 233, 237, 257, 258, 261, 272, 283, 285, 286, 289, 317 isotropisation, 55, 58 Italy, 23, 59, 161 iteration, 34, 169, 170, 271, 305

K kinetic equations, viii, xi, 50, 123, 124, 131, 132, 145, 158, 260, 265, 267, 270, 283, 295 kinetic model, 90, 124, 241, 287, 288, 289, 295 kinetics, 313

L laminar, 28, 48 laser radiation, 163 lasers, 3, 80, 108, 113 laws, 31, 35, 62, 270, 293, 297, 299 lens, 113, 114 Lie group, 94 limitation, 259 line, 12, 18, 45, 48, 51, 53, 54, 55, 56, 71, 72, 75, 83, 85, 104, 105, 107, 112, 113, 115, 116, 133, 140, 141, 156, 230, 257, 260, 276, 282, 283, 293 linear dependence, 148 linear function, 69 linear model, 72 links, 25

Eulerian Codes for the Numerical Solution of the Kinetic Equations of Plasmas, Nova Science Publishers, Incorporated, 2010. ProQuest Ebook

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362

Index

M

O

magnetic field, 1, 2, 4, 5, 7, 12, 18, 19, 24, 25, 26, 27, 28, 40, 41, 45, 46, 47, 48, 52, 53, 54, 55, 56, 57, 58, 82, 124, 130, 132, 137, 149, 156, 239, 240, 257, 260, 282, 315, 356 magnetic fusion, 124, 147, 159, 257 magnetic moment, 130, 148 magnetosphere, 48 manipulation, 302 mapping, 126, 127, 128, 131 matrix, 36, 42, 47, 252 Maxwell equations, 2, 27, 33, 164, 238 memory, 77, 79, 241, 242, 254 mentorship, xi Mercury, 312 microstructure, 318, 321, 334 Ministry of Education, 159 mixing, 4, 18, 103, 115, 133, 136, 137, 145, 151 model, 24, 25, 26, 28, 31, 33, 38, 44, 48, 51, 52, 63, 77, 84, 85, 88, 90, 91, 110, 114, 124, 125, 132, 137, 138, 140, 145, 146, 147, 149, 150, 151, 158, 159, 165, 238, 241, 260, 270, 283, 287, 289, 291, 294 model system, 85 modeling, 24, 38, 60, 90, 145, 238, 241, 254, 256, 259, 295 models, 23, 44, 62, 63, 90, 118, 119, 124, 156, 237, 238, 241, 252, 289, 291, 294, 314 modulations, 110, 324, 326, 353, 354, 355 modulus, 53, 263, 273, 274, 278, 282 molecules, 241, 283 momentum, 11, 24, 51, 52, 55, 65, 72, 77, 80, 84, 89, 91, 92, 93, 95, 98, 104, 106, 107, 114, 118, 130, 147, 166, 167, 170, 179, 187, 197, 207, 217, 238, 240, 270, 271, 283, 285, 286, 293, 298, 299, 301, 302, 304 motion, 4, 12, 18, 25, 33, 34, 39, 41, 67, 77, 92, 93, 111, 124, 127, 135, 136, 141, 144, 151, 152, 156, 180, 188, 198, 220, 230, 246, 247, 250, 251, 252, 317, 318 motivation, 71, 90 multidimensional, 31, 62, 65, 80, 267

observations, 24, 25, 26, 28, 48, 88, 356 one dimension, 4, 77, 80, 166, 296 operator, 29, 30, 36, 38, 39, 40, 42, 84, 85, 89, 90, 91, 92, 94, 98, 99, 102, 108, 114, 118, 140, 253, 254, 264, 265, 267, 269, 270, 277, 280, 281, 285, 286, 292, 298, 304, 308, 310, 311 Operators, 270 optimization, 59 orbit, 24, 89, 111, 118 oscillation, viii, 71, 75, 83, 84, 109, 151, 164, 170, 187, 218, 220, 229, 230, 233, 243, 301, 317, 318, 319, 320, 321, 324, 329, 330, 337, 338, 342, 343, 349, 351, 352, 353, 354, 355, 356 overpopulation, 242

N neglect, 99, 101 nodes, 34, 37, 244, 247, 248, 262, 264, 271, 282, 293, 299, 301, 302 noise, vii, xi, 1, 2, 9, 19, 25, 32, 34, 38, 44, 48, 53, 54, 60, 66, 108, 117, 118, 165, 234, 241, 247, 248, 254, 259, 260, 318 nonlinear dynamics, 49 numerical analysis, 30

P parallelism, 30 parallelization, 30, 59, 254 parameter, 10, 24, 32, 53, 62, 67, 86, 144, 145, 165, 170, 244, 272, 283, 287, 288, 289, 302, 304 parameters, viii, 10, 13, 17, 85, 89, 103, 104, 106, 108, 109, 116, 133, 136, 138, 140, 152, 154, 170, 182, 188, 200, 206, 207, 233, 243, 254, 279, 282, 287, 290 partial differential equations, 3, 19, 30, 50, 99, 102, 124, 168 particle collisions, 253 particle physics, 59 passive, 141, 143 periodicity, 97 phenomenology, 88 photons, 237 physics, vii, viii, xi, xii, 1, 2, 5, 19, 23, 25, 26, 30, 34, 38, 39, 45, 48, 49, 65, 66, 67, 87, 88, 90, 91, 102, 104, 119, 159, 164, 168, 198, 234, 243, 295, 317, 321, 354, 355, 356, 357 pitch, 272, 276, 278 plane waves, 48 Poisson equation, 4, 5, 27, 28, 35, 40, 125, 145, 240, 253, 261, 277, 284, 309, 319 polarization, 47, 113, 133, 151, 163, 164, 165, 232, 233 polynomial functions, 47 positron, 92 positrons, 283 power, 17, 18, 30, 54, 87, 111, 112, 116, 117, 165, 170, 238, 289 prediction, 157 predictors, 79 pressure, 49, 92, 163, 164, 165, 173, 179, 182, 218, 219, 229, 231, 233, 234, 259, 262, 285, 286, 287 probability, 31, 116, 242 probability distribution, 116 probe, 256, 258

Eulerian Codes for the Numerical Solution of the Kinetic Equations of Plasmas, Nova Science Publishers, Incorporated, 2010. ProQuest Ebook

Index production, 44, 123, 141, 143, 145, 146, 156, 158, 164 propagation, 9, 10, 11, 40, 45, 51, 66, 114, 163, 188, 271 propagators, 39, 40, 41 proportionality, 244, 283 protons, 53, 57, 283 pulse, 51, 56, 111, 133, 164, 173, 233, 234, 271

Q quantum electrodynamics, 237

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R radiation, 163, 164, 165, 173, 182, 234 radius, 25, 148, 241, 264 range, vii, 23, 25, 27, 31, 96, 119, 140, 143, 144, 243, 261, 262, 263, 282, 290, 292 recombination, 238, 256 reconstruction, 52, 80, 104 recovery, 32 recurrence, 18, 19, 49, 72, 102, 103, 104, 105, 106, 116, 118, 275, 320, 322, 329, 330, 331 reflection, 49, 93, 180, 196, 294 reflectivity, 89, 108, 109, 110, 114, 115, 116, 117, 118, 119 region, 11, 25, 32, 45, 48, 59, 110, 111, 113, 114, 116, 118, 151, 152, 156, 170, 172, 173, 179, 182, 183, 186, 187, 188, 193, 198, 200, 201, 204, 207, 215, 216, 217, 218, 219, 220, 230, 232, 234, 242, 259, 287, 290, 291, 317, 318, 321, 324, 330, 333, 337, 355 regulation, 45 relaxation, 50, 85, 89, 90, 91, 92, 115, 118, 262, 282, 283, 284, 292, 304 relaxation rate, 92 relaxation times, 283 reliability, 80 reproduction, 260, 287 residual error, 155 resolution, 1, 9, 18, 19, 24, 33, 72, 74, 75, 80, 87, 95, 104, 106, 127, 128, 130, 133, 138, 140, 144, 156, 170, 187, 220, 234, 239, 241, 243, 247, 248, 259, 260, 294, 317, 318, 355 resources, xi, 38, 284 Russia, 237

S safety, 148 satellite, 25, 26, 48, 49 saturation, 26, 47, 48, 52, 54, 55, 56, 60, 104, 111, 118, 151, 243, 244, 245, 247, 249, 317, 318, 320, 323, 331, 332, 343, 355 savings, 35, 95

363

scatter, 112, 113, 117, 118, 242 scattering, viii, 11, 48, 59, 61, 88, 89, 90, 107, 108, 111, 118, 119, 240, 241, 253, 257, 284 seed, 26, 52, 89, 91, 108, 109, 110, 111, 112, 115, 116, 117, 118 seeding, 116 semiconductors, viii, 164 sensitivity, 24 separation, 26, 56, 164, 218, 231, 261 shape, 10, 11, 28, 57, 170, 172, 173, 174, 180, 182, 186, 217, 244, 250, 276, 283, 286, 295, 320, 324, 329, 356 shear, 61, 148, 293, 294 shock, 25, 49, 61, 163, 164, 165, 174, 180, 182, 186, 207 shock waves, 163, 164 signals, 113, 116, 181 signs, 72, 75, 96, 103 single test, 255 skin, 52, 53, 180 smoothing, 4, 33, 59, 69, 113, 252, 318 solid state, 33 solitons, 1, 4, 9, 10, 11, 16 sound speed, 11, 12, 295 space-time, 100, 112 spatial location, 286, 290, 292 species, 27, 28, 30, 38, 48, 67, 83, 84, 92, 237, 238, 240, 241, 244, 254, 255, 259, 264, 267, 282, 285, 286, 304, 305 spectrum, 17, 18, 53, 54, 111, 112, 113, 116, 117, 144, 152, 181, 182, 196, 204, 205, 206, 215, 216, 217, 229, 245, 317, 318, 324, 326, 328, 329, 337, 338, 342, 348, 349, 350, 351, 352, 353, 354, 355, 356 speed, 1, 3, 9, 10, 13, 19, 24, 26, 27, 49, 53, 57, 66, 67, 69, 71, 77, 102, 109, 130, 164, 172, 179, 188, 238, 262 speed of light, 26, 27, 53, 57, 130, 238 stability, 25, 38, 85, 86, 88, 93, 95, 269, 273, 275, 282, 286, 293 stabilization, 48, 318 standard error, 31 standards, 91 stars, 115 statistics, 113 storage, xi, 18, 35 storms, 312 strategies, 32 stress, 114, 169, 286 stretching, 127, 129, 263, 269, 273, 281 structuring, 24, 73 substitution, 275 Sudan, 21, 357 supernatural, 283 suppression, 108, 123, 147 susceptibility, 110, 114 Switzerland, 89, 120 symbols, 28 symmetry, 96, 133, 135, 186

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364

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T

U

targets, 113, 114, 164 TEM, 78, 79 temperature, vii, 10, 12, 24, 26, 44, 48, 50, 52, 53, 57, 85, 104, 111, 118, 124, 132, 133, 137, 138, 139, 140, 145, 146, 147, 148, 158, 170, 187, 197, 206, 217, 239, 241, 256, 257, 259, 262, 267, 282, 283, 284, 287, 289, 290, 291, 292, 296, 304 thermal energy, 24 thermalization, 25, 284 thermodynamics, xi, 270 threshold, 85, 89, 90, 110, 111, 114, 115, 118, 256 thresholds, 256, 257 time resolution, 164 time use, 234, 357 tokamak, 44, 123, 124, 148, 149, 150, 151, 152, 153, 159 topology, 25 torus, 148 total energy, 18, 105, 123, 128, 129, 157, 270, 299, 302, 303, 304 trajectory, 32, 34, 35, 242, 247, 255, 257 transformation, 127, 131, 265, 292 transition, 28, 44, 45, 245, 246, 305 transitions, 28, 303, 317, 318 translation, 39, 40, 41 transparency, 198 transport, vii, viii, 24, 27, 47, 60, 70, 87, 103, 123, 124, 132, 137, 138, 139, 140, 141, 144, 146, 147, 148, 150, 152, 153, 155, 156, 157, 158, 159, 240, 256, 258, 270, 290, 293, 295, 314, 315 transport processes, vii trapezium, 68, 69 traveling waves, 102 turbulence, viii, xi, 3, 4, 46, 48, 50, 61, 63, 123, 124, 137, 138, 139, 140, 141, 143, 144, 145, 146, 147, 148, 151, 152, 155, 156, 157, 158, 260

UK, 65, 119 unconditioned, 113 uniform, 5, 11, 40, 41, 67, 70, 71, 86, 93, 95, 96, 102, 103, 113, 115, 116, 125, 130, 132, 242, 245, 247, 262, 271, 276, 278, 289, 292, 294, 300 updating, 82 USSR, 312

V vacuum, 78, 79, 93, 97, 100, 108, 109, 113, 114, 168, 170, 172, 180, 187, 198, 204, 207, 215, 217, 238 validation, 118 variables, 2, 3, 27, 29, 30, 51, 78, 79, 80, 82, 86, 94, 130, 137, 144, 145, 146, 248, 264 variance, 116 vector, 3, 34, 130, 131, 165, 166, 169, 170, 187, 197, 206, 217, 250, 263, 270, 277, 306

W wave number, 4, 10, 11, 12, 151, 244 wave propagation, 207, 288 wave vector, 26, 53, 54, 56, 91 wavelengths, 53, 54, 98, 164, 234, 244, 317, 318, 356 wind, 23, 24, 25

X X-axis, 263

Eulerian Codes for the Numerical Solution of the Kinetic Equations of Plasmas, Nova Science Publishers, Incorporated, 2010. ProQuest Ebook