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Reviews of Plasma Physics: Volume 13 PDF

392 Pages·1987·12.359 MB·English
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Reviews of Plasma Physics VOPROSY TEORII PLAZMY BOnpOChI TEOPMM nJIA3MhI Translated from Russian by J. G. Adashko Reviews of Plasma Physics Edited by Acad. B. B. Kadomtsev Volume ® CONSULTANTS BUREAU· NEW YORK-LONDON The Library of Congress cataloged the first volume of this title as follows: Reviews of plasma physics. v. 1- New York, Consultants Bureau, 1965- Iv. illus. 24 cm. Translation of Voprosy teorii plazmy. Editor: v. 1- M. A. Leontovich. 1. Plasma (Ionized gases)-Collected works. I. Leontovich, M. A., ed. II. Consul tants Bureau Enterprises, Inc., New York. III. Title: Voprosy teorii plazmy. Eng. QC718.V63 64-23244 The original text, published by Energoatomizdat in Moscow in 1984, has been corrected and updated by the authors. ISBN-13: 978-1-4612-8996-8 e-ISBN-13: 978-1-4613-1777-7 DOl: 10.1007/978-1-4613-1777-7 © 1987 Consultants Bureau, New York Softcover reprint of the hardcover lst edition 1987 A Division of Plenum Publishing Corporation 233 Spring Street, New York, N.Y. 10013 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, mechanical, photocopying, microfilming, recording, or otherwise, without written permission from the Publisher CONTENTS PARTICLE DYNAMICS IN MAGNETIC TRAPS B. V. Chirikov 1. Introduction. Budker's Problem. 1 2. Choice of Unperturbed System. . 4 3. A Few Examples. • . . • • . • . 5 4. Adiabatic Perturbation. . . . • • 15 5. Insignificant Effect of Perturbation 18 6. Nonlinear Resonances 19 7. Resonant til..!. • • • • • 22 8. Mapping ...••... 35 9. Standard Mapping. . . 39 10. Limit of Global Stability 41 11. Local Diffusion . . . 50 12. Dynamic Correlations 58 13. Global Diffusion. . . 65 14. Cohen's Mapping• ••••••• 73 15. Remarks on Adiabatic Invariance. 84 References. . . . . . . . . . 88 TRANSPORT PROCESSES IN AXISYMMETRIC OPEN TRAPS D. D. Ryutov and G. V. Stupakov 1. Introduction. . . . . . . .. ..... 93 2. Plasma Equilibrium. . . . .. ..... 100 3. Drift Motion of Charged Particles in Open Traps 118 4. Neoclassical Transport in Open Traps . . . • 145 5. Resonant Stochastic Transport in Open Traps 172 Appendix 1. 192 Appendix 2. 194 Appendix 3. • 195 Appendix 4. 197 Appendix 5. 198 References . 200 v vi CONTENTS CLASSICAL LONGITUDINAL PLASMA LOSSES FROM OPEN ADIABATIC TRAPS V. P. Pastukhov Introduction. . . . . . . . . . • . . . • . . . 203 1. Basic Principles of Plasma Confinement in Simple Mirror Traps . . . . . . . . . . . 205 2. Electron Confinement in Simple Mirror Traps 214 3. Ion Confinement in Simple Mirror Traps. . 228 4. Traps with Improved Longitudinal Plasma Confinement 245 References . . . . . . . . . . . . . . . . . . 257 SPECTRAL-LINE BROADENING IN A PLASMA V. I. Kogan, V. S. Lisitsa, and G. V. ShoUn Introduction. • . . . . . . . . • . . . . . . . . . 261 1. General Equations for the Intensity Distribu- tion in a Line . . . . . . . . . . . . . . 263 2. Plasma Microfield. . . . . . . . . . . . . . 267 3. Line-Broadening Mechanisms in a Plasma. Character of Atom Interaction with a Plasma Microfield . . . • . . . . . . . . . . . . . 280 4. Static Theory and Its Generalizations. . . . 298 5. Impact Theory of Broadening by Particles and Waves . . • . . . . . . . . . . . . . . 313 6. Transition from the Impact to the Static Broadening Mechanism . . . . . . . . . 321 7. Comparison of Theory with Experiment. Conclusions 325 References . . . . . . . . . . . . . . . . . 331 ELECTRON CYCLOTRON PLASMA HEATING IN TOKAMAKS A. D. Piliya and V. I. Fedorov Introduction. • . . . . . . . . . . . . . 335 1. Electromagnetic Waves in the Region of Elec- tron-Cyclotron Resonance Frequencies . . . . 336 2. Electron-Cyclotron Resonance in a Homogeneous Plasma with Allowance for the Thermal Motion of the Electrons . . . . . . . . . . . . . 340 '3. Poynting Vector and Energy Absorption in Cyclotron Resonance . . . . . . • . . • . 347 CONTENTS vli 4. Accessibility of Resonances in a Tokamak. . . 354 5. Cyclotron Damping of Waves in the Mod.el of One-Dimensional Inhomogeneity. . . . . . . 359 6. Reflection of Waves from a Resonance Region. 364 7. Linear Transformation of Waves Near the Upper Hybrid Resonance. . . . . . . . . . . 366 8. Eikonal Equation in the Vicinity of the Cyclo tron Resonance and Ray Trajectories of Waves. . . . . . . . . . . . . . . . . . . . 368 9. Applicability of Geometric Optics in the Vicin- ity of Cyclotron Resonance. . . . . . . . . 373 10. Ray Trajectories and Wave Absorption in Tokamaks . . . . . . . . . . . . . . . . . 377 11. Role of Nonlinear Effects in Electron-Cyclo- tron Heating.. 381 References . . . . . . . . . . . . . . . . . . . 385 PARTICLE DYNAMICS IN MAGNETIC TRAPS B. V. Chirikov 1. Introduction. Budker's Problem [1] The investigation of the dynamics of individual (nonin teracting) charged particles in a magnetic trap is probably the simplest of the problems of prolonged plasma containment for controlled thermonuclear fusion. Nonetheless, even this "simple" problem is quite rich in content and is far from com pletely solved, notwithstanding many years' efforts in this direction (see, e. g., [2-4]). In addition, the dynamics of an individual particle is an integral part of the more compli cated problem of collective processes in a plasma. Finally, the problem of containing a single particle in a magnetic trap must be faced every time when a new scheme or a sub stantial modification of an old method of magnetic confine ment of a plasma appears. * An example is Dimov's ambipolar (tandem mirror) trap [7]. It is one of the so-called open systems of plasma confinement, or traps with "magnetic mir rors," which will be discussed below. We shall refer to them for brevity simply as traps. The dynamics of a particle in a trap can be regarded as a problem of nonlinear oscillations induced possibly by some external perturbation in a system having, generally *Certain very simple particle-interaction effects can be ac counted for also within the scope of the single-particle prob lem, for example the change of the magnetic-field configura tion on account of plasma diamagnetism or particle scatter ing in a plasma. 1 2 CHIRIKOV speaking, three degrees of freedom. The term "oscillations" is used here in a broad sense as a synonym for finite mo tion of a particle in a bounded region of space (particle con tainment). In other words, we assume here that the con struction of the trap (the magnetic-field configuration) en sures "locking" of the particle for a time equal at least to several passes of the particle through the trap. Under these conditions, the main factor that determines the character of a prolonged evolution of the oscillations are the resonances, or the commensurability of the periods of various degrees of freedom of the oscillations, as well as commensurability with the external perturbation. The mechanism whereby the resonances act is particularly clear (and well known) in the simplest case of linear oscillations. Nonlinear dynamics is much more complicated, but in this case resonant processes are unique for the very same reason: small resonant per turbations repeat in time and accumulate, leading to a great er long-time effect than nonresonant perturbations. This important property can be used to define resonant processes in a broad sense. Such a "resonant" approach means, figuratively speak ing, that before some dynamic problem is solved it must be checked for the presence of resonances, even if it seems ob vious at first glance that none exist here. A splendid ex ample of such a situation is the action exerted on a system by an adiabatic (Le., a very-low-frequency) perturbation, and is in fact the subject of this article. We shall accordingly pay principal attention below to a fullest possible analysis of various resonances and to the in teraction between them. The latter means the effect of the joint action of several resonances, which does not add up simply to the sum of the effects due to individual resonances, since the equations of motion are nonlinear and the superpo sition rule is thus invalid. It turns out that, under certain conditions, the interaction of the resonances alters radically the character of the motion and transforms it from the well-known regular or quasiperio dic oscillations into the relatively little-known "random (stochastic) walk" of the particle in its phase space (see, e. g., [8, 9]). The latter is very similar to (and in some cases indistinguishable from) random oscillations, L e., oscil lations caused by some external (relative to the dynamic sys tem) random perturbation or "noise," although such a per- CHIRIKOV 3 turbation need not necessarily be present. Strange as it may seem, this unusual motion regime is quite widespread for nonlinear oscillations in general, and for a particle in a magnetic field in particular. From the standpoint of pro longQd containment of particles in a trap, the onset of ran dom oscillations is harmful, since it leads as a rule to par ticle loss through diffusion in phase space. We confine oursel-v-es in this review to a discussion of the simplest case of nonlinear oscillations with two degrees of freedom, when a random regime of motion in a conserva tive system is still possible. For particle motion in a trap this presupposes some symmetry of the magnetic field, mean ing an additional integral of motion (besides the energy). In the case of axial symmetry, for example, the additional integral is the component of the particle's canonical momen tum along the symmetry axis. The motion is then over the intersection, in phase space, of the constant-energy sur face and the constant-value surface of the additional inte gral. We actually consider below a certain quite special case, when the connection between the two degrees of freedom is adiabatic, i.e., the ratio of the fundamental unperturbed frequencies is very large (or small). This special problem, which stems from an analysis of the operation of a trap with magnetic mirrors (the Budker problem), is nonetheless con nected with one classical problem of mechanics, viz., adia batic invariance of the action variables. It is worthwhile noticing that it is precisely research into particle contain ment in traps which made possible considerable progress to ward the solution of this general problem, too. It became clear, in particular, that, under certain conditions, an adia batic invariant becomes an exact integral of the motion [10]. These and other results are discussed below using as an example particle dynamics in several quite simple yet typical modifications of a magnetic trap. The problem can be solved in some cases completely; i.e., it is possible to obtain in explicit form the particle containment conditions, on the one hand, and the statistical properties of the mo tion in the unstable region, on the other [2]. The author is indebted to G. I. Dimov, D. D. Ryutov, and D. L. Shepelyanskii for interesting discussions of the questions touched upon in the article, and to G. B. Minchen kov for help with the numerical simulation. 4 CHIRIKOV 2. Choice of Unperturbed System As a rule, an analytic investigation is possible only by using some approximate methods or perturbation theory. The first step is therefore to divide the investigated system into an unperturbed part and a perturbed one. This division is not unique and is dictated by the type of physical problem. In the present case we might, for example, choose as the unperturbed system a particle in a uniform magnetic field, and include the spatial inhomogeneity of the field as a whole in the perturbation. Although the unperturbed motion is in this case extremely simple, this is not the best choice. The point is that such an unperturbed motion is infinite and, therefore, differs qualitatively from the actual (perturbed) motion. The unperturbed motion is said in this case to be degenerate (one of the fundamental frequencies of the sys tem is zero), and an arbitrarily small perturbation (field in homogeneity of suitable configuration) leads to bifurcation. To choose an unperturbed system more suitable for in vestigation we recall that particle containment in an open trap is the result of approximate conservation of the par = ticles' magnetic moment ~ mv 12/2B, which is proportional to an adiabatic invariant (the action variable J.l. of the Lar mor rotation). Taking as the basic units the speed of light, = = = the charge, and the particle mass (c e m 1), we get (2.1) The subscript .1 labels here quantities that characterize mo = tion in a plane perpendicular to the magnetic field B rot A ; = w B is the Larmor frequency. We shall be interested be low in the nonrelativistic case (v « 1), but all the relations remain in force at arbitrary velocities if the substitution m -+ ym, y = (1 - V2)-1/2 is made. Expression (2.1) for the action J 1. is exact only in a uniform field. We can, how ever, define the unperturbed system by stipulating just this quantity to be an exact integral: == (2.2) !l. const. The identity sign indicates that the last condition is an arbi trary choice of the unperturbed system rather than the prop erty of the quantity ~. We refer all the changes of ~ to the perturbation, which we assume to be small enough. The last condition is known to be met for a sufficiently strong mag-

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