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Magneto-Fluid and Plasma Dynamics http://dx.doi.org/10.1090/psapm/018 PROCEEDINGS OF A SYMPOSIUM IN APPLIED MATHEMATICS OF THE AMERICAN MATHEMATICAL SOCIETY Held in New York City April 13-15,1965 HAROLD GRAD EDITOR PROCEEDINGS OF SYMPOSIA IN APPLIED MATHEMATICS VOLUME XVIII MAGNETO-FLUID AND PLASMA DYNAMICS AMERICAN MATHEMATICAL SOCIETY PROVIDENCE, RHODE ISLAND 1967 Prepared by the American Mathematical Society with the support of the U.S. Army Research Office (Durham) under Contract No. DA-31-124-ARO-D-353 and the Mathematics Division of the Air Force Office of Scientific Research under Grant No. AF-AFOSR-835-65. Library of Congress Catalog Card Number 66-20436 Copyright ©1967 by the American Mathematical Society Printed in the United States of America All rights reserved except those granted to the United States Government. Otherwise, this book, or parts thereof, may not be reproduced in any form without permission of the publishers. PREFACE This volume contains the manuscripts of the invited addresses which were presented at a Symposium on Magneto-fluid and Plasma Dynamics in New York City on April 13-15, 1965. The Invitations Committee con• sisted of Andrew Lenard, Marshall N. Rosenbluth, William R. Sears, Harold Weitzner, and Harold Grad, Chairman. The subject matter ranges widely in viewpoint across mathematical physics and applied mathematics. Some presentations tend to emphasize the physical aspects while others, although physically motivated, are guided somewhat more by a desire to expose the underlying mathematical structure. We can hope and even expect that this subject, which describes the physical interaction between electromagnetic fields and fluids, will contribute as much to the future course of mathematics as did the individual subjects of electromagnetic theory and fluid dynamics in the past to mathematical analysis and even to topology. Harold Grad Courant Institute of Mathematical Sciences New York University CONTENTS Sorrrfe remarks on the stability of hydromagnetic shock waves 1 By C. K. CHU A model of some plasma shock structures 17 By P. GERMAIN Stability of a slightly resistive plasma . 46 By JOHN M. GREENE On the controversy over an aligned magnetic field 57 By G. S. S. LUDFORD Hydromagnetic instability of the sub-Alfven equations 64 By WILLEM V. R. MALKUS Singular eigenfunctions and plasma problems 78 By K. M. CASE A mathematical problem in the " quasi-linear" theory of plasma waves. 100 By MICHEL TROCHERIS Longitudinal plasma oscillations 127 By HAROLD WEITZNER Exchange invariance in fluid systems 152 By WILLIAM A. NEWCOMB The guiding center plasma 162 By HAROLD GRAD Problems in plasma microinstability 249 By M. N. ROSENBLUTH The present state of plasma kinetic theory 257 By EDWARD A. FRIEMAN Quasi-particles in a plasma 270 By NORMAN ROSTOKER On Ohm's law resulting from instability 281 By R. Z. SAGDEEV Author Index 287 Subject Index 289 vii AUTHOR INDEX Roman numbers refer to pages on which a reference is made to the author or a work of the author. Italic numbers refer to pages on which a complete reference to a work by the author is given. Boldface numbers indicate the first page of the articles in the book. Abe, R., 269 Goldberger, M., 163,249 Adlam, J. H., 45 Golden, K. I., 45 Akheizer, A. I., 6, 16 Goldman, R., 269 Allen, J. E., 8 Grad, H., 1, 13, 162, 163, 165, 247, 250, 265, 269 Andreoletti, J., 165, 169, 171, 191, 222, 246, 247 Greene, J. M., 46, 56, 269, 269 Greenspan, 64, 77 Backus, G., 126, 127, 150 Gross, R. A., 1 Baldwin, D. E., 126 Guernsey, R. L., 269, 280 Balescu, R., 269, 280 Bazer, J., 5, 16 Harris, E. G., 279, 280 Berkowitz, J., 247, 248 Hastie,R. I., 243, 244,247 Bernstein, I. B., 161, 269, 269 Hide, R., 67, 77, 77 Bhadra, D., 280 Blank, A. A., 248 Johnson, J. L., 46, 56 Bogoljubov, N., 256, 260, 261, 269, 270 Bohn, D., 271, 280 Kadish, A., 163, 168, 195, 247 Bremerman, H. J., 99 Kadomcev, B. B., ( = Kadomtsev) 247, 248, Brueckner,K. A., 163,247 266, 269, 286 Buneman, O., 286 van Kampen, N. G., 126 Burgers, J. M., 160 Karpman, V., 286 Kaufman, A.N. ,163, 247 Cartan, H., 126 Killeen, J., 56 Case, K. M., 78, 99, 126 Klimontovic, Ju. L., ( = Kilmontovich) 279, 280 Chandrasekhar, S., 163, 247 Krall, N. A., 156, 280 Chew, G., 163, 247, 249 Kruskal, M. D., 46, 47, 50, 56, 85, 99, 161, 163 Chu, C.K., 1,16,247 169, 170, 171, 188, 247, 269, 269 Coppi, Bruno, 46, 56 Kilikovskii, A. G., 6, 16,45 Courant, R., 4 Kulsrud, R. M., 161. 163, 171, 247 Dawson, J.,280 Landau, I., 99, 126, 127, 128, 129, 150 Drummond, W. E., 126, 266, 269, 286 Langmuir, I., 127, 150 Dupree, T. H., 279, 280 Lax, P. D., 8,9, 16 Durand, L., 99 Leibovich, S., 58, 60, 63 Lenard, A., 269 Ericson, W. B., 5,16 Liubimov, G. A., 16 Longmire, C. L., 163, 237, 247 Fried, B. D., 280 Low, F.,163, 247 Friedrichs, K. O., 2, 16, 45, 248 Ludford, G. S. S., 6, 7, 12, 13, 16, 57, 58, 60, 63 Frieman, E. A., 161, 257, 266, 269 Lust, R., 248 Furth, H. P., 56 Galeev, A. A., 280, 286 Malkus, W. V. R., 64, 64, 77 Gardner, C. S., 45, 249 Messiah, A., 126 Germain, P., 5, 6, 16, 17, 45 Michael, D. H., 63 287 288 AUTHOR INDEX Mihailovskii, A. B., 280 Rotenburg, M., 161 Mitropolsky, Y. A., 262, 269 Rubin, H., 228,248 Moiseev, S., 286 Rudakov, L.I. ,247 Morawetz, C.S.,45 Ruelle, D., 259, 269 Morikawa, G. K.,45 Rutherford, P., 266, 269 Morton, K. W., 44, 45, 248 Muskhelishvili, N. I., 99 Sagdeev, R. Z., 247, 266, 280, 283, 286 Sandri,G., 268,269 Newcomb, W. A., 48, 50, 55, 56, 56, 152,156, Sarason, L., 1, 13,16 160, 189 Sears,W.R.,57,63 Northrup, T. G., 156, 247, 248 Silin, V. P., 280 Simon, A., 161,219,280 Oberman, C, 85, 99, 163, 169, 170, 171, 247, 280 Soubbaramayer, 38, 45 O'Neil, T., 259, 269 Stewartson, K., 57, 58, 61, 63, 63, 64, 67, 77 Oraevskii,V.N.,280 Su, C.H.,269 Taussig, R., 1, 15,16 Parker, E. N., 248 Taylor, J. B., 67, 77, 168,171, 221, 243, 244, Penrose, O., 150 246,247 Peyret, R., 45 Todd, L., 1, 12, 13,16 Pines, D., 126, 266, 269, 241, 280, 286 Tonks, L., 127, 150 Polovin, R. V., 6,16 Trocheris, M., 100, 126 Prigogine, I., 258, 269 Trubnikov, B. A., 163, 170, 247 Resler, E. L., 57, 63 Vedenov, A. A., 247, 269, 286 Roberts, P. H., 64, 77 Velikhov, E., 264, 286 Roman, P., 154 Ron, A., 278, 279 Watson, K.M.,163, 247 Rose, D. J., 163, 247 Weitzner,H., 126, 127,150 Rosenbluth, M. N., 56,156,161, 163, 168, 237, Wendroff, B., 9,16 247,249,250,219,280 Whiteman, K. J., 248 Rostoker, N., 156, 163, 168, 247, 261, 270, 280 Wyld,H., 280 SUBJECT INDEX acoustic instability, ion, 285 conductivity action integral, 153, 154 finite, 31 adiabatic infinite (collision-free shock), 26 equations of motion, 169, 190 conservation invariant, second, 171 of energy, 184 adjoint equation, 82 of momentum, 184 admissible constant (s) class, 169, 171, 198 entropy, 24, 34 deformations, 215 of motion, 85 distribution functions, 214 contact discontinuity, 4, 9 magnetic fields, 214 continuous subsonic flow, 43 Alfven continuum modes, 90 discontinuity, 4, 9, 13 convexity inequalities, Weyl's, 18 wave(s),57, 58 coriolis force, 66 of finite amplitude, 60 Coulomb force, 261 longitudinal, 58, 60 coupling, weak, 258 modified, 193 curves, integral, 24, 25 regions, 60 cusped fields, 245 aligned magnetic field controversy, 57 annihilator(s), 53 damping, 133 differential, 54 of plasma oscillations, Landau, 144 asymptotology, 46, 47, 50, 53 Debye length, 179 axially symmetric problem, 201, 205 shielding, plasma limit, 261 deformations, admissible, 215 Balescu-Lenard-Guernsey, 260, 265, 267 diagrams, normal speed, 2 collision operators, 268 differential annihilators, 54 Boltzmann diffuse collision operators, 264, 265, 268 linear pinch, 48 equation, 252 reflection, 96 regime, 260, 261 discontinuity boundary Alfven, 4, 9,13 conditions, 54, 56 contact, 49 value problems, 89, 92 discrete modes, possible, 90 and initial value, 78 dispersion relation, 64, 67 Boussinesq equations, 66 distribution(s), 79,81 functions, 131, 133, 145, 147 Cauchy principal value, 80 admissible, 214 channelled waves, 62 Maxwell's, 282 characteristic equations, 193 disturbance, potential, 63 charge-neutral (neutrality), 177, 180, 210 dominating mode approximation, 102, circulation, 155,157, 158 106,112, 121, 124 collapse, wave, 61, 63 drift equations, 243 collision operators, Fokker-Planck, 268 collisionless plasma, 247 expansions, 259 system, 128 eigenfunctions, instantaneous, 102, 111 commutator product, 154 electromagnetic potentials, 172 compression wave, modified, 193 electron plasma frequency, 284 289 290 SUBJECT INDEX energy, conservation of, 184 plasma (GCP), 163 entropy, 22, 191 equations, linearized, 186 constant, 24, 34 equilibrium, 208 equation, 192 gyroscopic, 157 specific, 24, 28, 40, 41 equilibria, slow and fast, 242 high frequency conductivity, 277 equilibrium Hilbert expansion, 281 GCF, 202 Hydro-magnetic GCP, 208 equations, 65 MH, 197 planetary waves, 77 evolutionary shocks, 1, 6 condition, 6 hyperplane, 40 shock, 6 exchange induced potential disturbance, 61 conditions, 154, 155, 157, 159 infinite conductivity (collision-free shock), 26 invariant, 154, 155, 156, 157, 159 initial value problem(s), 85, 127, 129, 143, 193 expansion in instantaneous modes, 111 and boundary value, 78 initial value-boundary value problem, 129 fast mixed, 130, 135 equilibria, 242 inner shock, 25, 35, 36, 44 normal shocks, 4, 6, 7, 10, 11, 12 instantaneous eigenfunctions, 102 finite and eigenfunctionals, 111 amplitude, Alfven waves of, 60 insulating walls, 214 conductivity, 31 integral curve(s), 24, 25 first variation, 216 interchange, 199 flow particle(s), 242 continuous subsonic, 43 stability, 232, 235, 244 pattern, 61 intermediate shocks,
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