Reviews of Plasma Physics VOPROSY TEORII PLAZMY BOnPOCbl TEOPJ1l1 nJIA3Mbl Translated from Russian by Dave Parsons Translation Editor Herbert Lashinsky University of Maryland Reviews of Plasma Physics Edited by Acad. M. A. Leontovich Volume ®SPRINGER SCIENCE+BUSINESS MEDIA, LLC The Library of Congress cataloged the first volume of this title as follows: Reviews of plasma physics. v. 1- New York, Consultants Bureau, 1965- /v. illus. 24 em. Translation of Voprosy teorii plasmy. 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 plasmy. Eng. QC718.V63 64-23244 The original text, published by Atomizdat in Moscow in 1974, has been corrected and updated by the authors. Library of Congress Catalog Card Number 64-23244 ISBN 978-1-4615-7816-1 ISBN 978-1-4615-7814-7 (eBook) DOI 10.1007/978-1-4615-7814-7 © 1980 Springer Science+Business Media New York Originally published by Plenum Publishing Corporation, New York in 1980 Softcover reprint of the hardcover 1st edition 1980 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 TRANSLATOR'S PREFACE In the interest of speed and economy the notation of the orig- inal text has been retained so that the cross product of two vectors A and B is denoted by [AB], the dot product by (AB), the Laplacian operator by ~. etc. It might also be worth pointing out that the temperature is frequently expressed in energy units in the Soviet literature so that the Boltzmann constant will be missing in various familiar expressions. In matters of terminology, whenever pos- sible several forms are used when a term is first introduced, e.g., magnetoacoustic and magnetosonic waves, "probkotron" and mirror machine, etc. It is hoped in this way to help the reader to relate the terms used here with those in existing translations and with the conventional nomenclature. In general the system of literature citation used in the bibliographies follows that of the American Institute of Physics "Soviet Physics" series. Except for the cor- rection of some obvious misprints the text is that of the original. We wish to express our gratitude to Academician Leontovich for kindly providing the latest corrections and additions to the Russian text. v CONTENTS Steady-State Plasma Flow in a Magnetic Field A. I. Morozov and L. S. Solov'ev Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Chapter 1. Acceleration Mechanisms , • • • • • • • • • • . 2 §1. Microscopic Picture of Plasma Acceleration . . . . . . . . . . . . . . . . . . . 2 §2. Conditions for the Existence of an Electric Field in a Plasma. , ••• , , , • , . . • • . • 5 §3. Basic Equations; the Replacement Factor ~ . . . . . . . . . . . . . . . . . . . . . 8 §4. Perturbations of the Plasma Stream •.• , , 11 Chapter 2. Axisymmetric Flow of an Ideal Plasma in a Magnetic Field with ~ = 0 , , , • • . • 18 §1. Conservation Laws and Equations for the Stream Functions , • , . , • • . • • • • • 18 §2. Axisymmetric Flow across an Azimuthal Magnetic Field. • • • • • • • • • • • • • • • • • 23 §3. Plasma Flow in a Narrow Flux Tube (Hr!=O)....................... 26 §4. Integrable Flows Which Vary Slowly along the z Axis. • • • • • • • • • • • • • • • • 35 §5. Flow Velocity Equal to Signal Velocity at Some Point ••••• , .•••• , •• , •• , 43 § 6, Current Eddies and Critical Surfaces, , , , • 45 vii viii CONTENTS Chapter 3. Flow in Axisymmetric Channel with a Longitudinal Magnetic Field 54 §1. Integrated Characteristics •••••••••••• 54 §2. Flow in Narrow Axisymmetric Channel .•• 56 §3. Flow of Cold Plasma in Channel of Slowly Varying Cross Section• ••••••••••.• 68 Chapter 4. Axisymmetric Hall Flow of an Ideal Plasma . ...................... 70 §1. Conservation Laws and Integral Parameters . ................... 70 §2. Qualitative Analysis of Equations (4.18) and (4.19) ..................... 74 §3. Flow of a Two-Component Plasma in Narrow Tube with H 1 = 0 . . . . . . . . . 81 §4. Flow in Smooth Channel with H 1 = 0 . . . . . 84 Appendix 1. Two-Parameter Steady-State Hydro- magnetic Flow of Ideally Conducting Medium in Arbitrary Curvilinear Coordinate System • • • • • 85 Appendix 2. Steady-State Symmetric Flow in · Two-Fluid Magnetohydrodynamics . • . • • • • . • • • 94 § 1. Flow of the Electron and Ion Fluids • • . • • • 94 §2. Symmetric Flows of Two-Fluid Magneto- hydrodynamics with Slow Variation along One of the Coordinates . • • • . • • . 100 References ........... ~ . . . . . . . . . . . . . . . . . . 102 C a 1 c u 1 at ion of Two-Dimension a 1 Plasma Flows in Channels K. V. Brushlinskii and A. I. Morozov Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 Chapter 1. Formulation of the Problem and Method of Analysis . • • • • . . • • . . . • • • . . . • • 108 §1.1. Physical Formulation of the Problem. • • • • 108 §1.2. Mathematical Models • • • • . • • . • . . • • • • 122 §1.3. Method for Numerical Solution of the Problem. • . • • . • . • . • . . • . . . . 137 CONTENTS ix Chapter 2. Flow of Fully Ionized Plasma •••••••.•. 145 §2.1. Establishment of Steady-State; Stability of the Flow • • • • • • . • . • • • • • 145 §2.2. Quasi-One-Dimensional Flow •.•••••••• 148 §2.3. Two-Dimensional Flow of Ideal Plasma •.. 152 §2.4. Influence of a Finite Conductivity on the Flow . • • • . • • • • . • • • • • . • • • . 157 §2 ,5. Hall Effect • • • • • • • • • • • • • . . • • • • • . . 161 §2.6. Compressional Plasma Flows • • • . • • • • • 172 Chapter 3. Flow of a Gas Which Is Ionized in the Channel . . . • . . . . . . . . . . . . . . . . . . 182 §3.1. Ionization Process in the Flow Model 182 §3.2. Steady-State Flow with a Conductivity Discontinuity in a Channel of Constant Cross Section • • . . • • . • • . 184 §3.3. Calculations for the Flow of an Ionizing Gas • . . • • • . . . • • . . . . . . • . . • . . . • 188 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 Two-Dimensional Magnetohydrodynamic Model for the Dense Plasma Focus of a Z Pinch V. F. D'yachenko and V. s. Imshennik Introduction. . . • . . • . . . • • . . . • • . . • . . . . • . . . . . 199 Chapter 1. MHD Analysis of a Noncylindrical Z Pinch ....................... . 202 §1. Description of Dissipative Processes ..•.. 202 §2. Two-Dimensional Magnetohydrodynami c Equations (MHD Model) ••.•••••••.• 209 §3. Energy Conservation Law and Kirchhoff Equation . . . . . . . . . . . . . . . . . . . . . . 215 §4. Initial Conditions and Boundary Conditions for the MHD Problem ••....•••.•.. 223 §5. Dimensionless Form of the Equations; Complete Formulation of the MHD Problem. . . . . . . . . . . . . . . . . . . . . . 228 X CONTENTS Chapter 2. Numerical Solution of the Two-Dimensional MHD Problem • • • • • • • • • . • • . • • • • • • 233 §6. General Comments. . • . • . • • • • • • . • . • • • 233 §7. The Free-Point Method •••••.••.•••.•• 236 §8. The Selection of Points •••.••••••••••• 239 §9. Method for Finding the Difference Equations • • . • • • • • • • • . . • • • • . • • • • 244 §10. Calculation Equations •.•.••••.••.•••• 248 §11. Difference Formulation of the Boundary Conditions. • . . • . • . • • . • • • • • . • • • • • 258 §12. Formulation of the Calculation Problem •••• 261 Chapter 3. MHD Analysis of the Z Pinch • . • • • • • . • . • 2 63 §13. A Few Comments. • • • • • • • • . • • . . . • • . • 263 §14. Collapse of the Shock Wave and the Current Sheet to the Axis of the System •••••••• 265 §15. First Plasma Compression •••••••••••. 268 §16. Second Plasma Compression •••••.••••• 273 §17. Instability of the Plasma Interface with the Magnetic Field •••.••.••••.• 280 §18. Energy Balance and Neutron Yield of the Plasma Focus •.••.•..••••••• 287 Conclusion 295 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 97 Plasma Optics A. I. Morozov and S. V. Lebedev Chapter 1. Introduction . ..................... . 301 §1. Basic Concepts of Plasma Optics •••••••• 301 §2. Electric Field in the Plasma for Te = 0 •.•• 308 §3. Analysis of Plasma-Optical Systems ••.••• 322 Chapter 2. Analysis of a Paraxial Beam in the Single-Particle Approximation .••• 325 §1. Specification of the Magnetic Fields •••••• 325 §2. Paraxial Approximation •..••••••.•••• 330 §3. Incorporation of Aberrations in the Grinberg Scheme. • • • • • • • • • • • • • • 342 CONTENTS xi §4. Harmonic Systems with Equipotential Prin- cipal Trajectory• ••••••••••••••••• 345 §5. The "Weak-Field Approximation" for Lenses ..................... 353 §6. Variational Formulation of Particle- Optical Problems ••••••••••••••••• 362 Chapter 3. Plasma Lenses . . . . . . . . . . . . . . . . . . . . 367 §1. Axial Plasma Lens • • • • • • • • • • • • • • • • • • 367 §2. Annular Magnetic Lens ••••••••••••••• 373 §3. Electrostatic Plasma Lens ••••••••••••• 377 §4. First-Order Aberrations of an Annular Plasma Lens . . . . . . . . . . . . . . . . . . . . 382 §5. Multiple-Lens, Multiply Connected Charged-Particle Accelerators •••••••• 386 §6. Toroidal Multiply Connected, Multiple-Lens Accelerator . .................... 394 Chapter 4. Plasma-Optical Systems with Two- Parameter Focusing • • • • • • • • • • • • • • • 400 §1. Systems with Stabilized Focus •••••••••• 400 §2. Focusing in Terms of the Radial and Azimuthal Velocities (Whirler System) ••• 404 §3. Energy Recovery • • • • • • • • • • • • • • • • • • • 410 Chapter 5, Equilibrium Neutralized Ion Beams ••••••• 414 §1. Thermalized Potential• ••••••••••••••• 414 §2. Distributions of the Thermalized Potential and the Electron Temperature in a Steady-State Plasma System •••••••• 418 §3. Ion Dynamics ..................... 427 §4. Electron Sheath of a Neutralized Ion Beam Detached from the Walls ••.••.••.•.. 439 §5. Grazing a Neutralized Ion Beam in an Axial Lens ..................... 448 §6. Magnetic Fields Produced by the Currents Flowing in a Multiply Connected, Multiple-Lens Accelerator ••• 456 References ...............•............... 458