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http://dx.doi.org/10.1090/psapm/018

PROCEEDINGS OF A SYMPOSIUM IN APPLIED MATHEMATICS OF THE AMERICAN MATHEMATICAL SOCIETY

Held in April 13-15,1965

HAROLD GRAD EDITOR PROCEEDINGS OF SYMPOSIA IN APPLIED MATHEMATICS XVIII

MAGNETO- 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 ranges widely in viewpoint across mathematical 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 and , will contribute as much to the future course of mathematics as did the individual subjects of electromagnetic theory and in the past to mathematical analysis and even to topology.

Harold Grad Courant Institute of Mathematical Sciences 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 57 By G. S. S. LUDFORD Hydromagnetic 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 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 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, , 285 conductivity action integral, 153, 154 finite, 31 adiabatic infinite (-free shock), 26 equations of motion, 169, 190 conservation invariant, second, 171 of energy, 184 adjoint equation, 82 of , 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 , 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 , 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 , 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 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, , 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 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, 5, 6, 7,11,12, 13,15 Sears-Resler, 63 invariant potential, 59, 61 exchange, 154, 155, 156, 157, 159 slow, 58, 60, 63 second adiabatic, 171 slug, 58, 59, 61 ion steady, of a fluid, 17 acoustic instability, 285 sub-Alfvenic, 60, 62, 63 frequency, 284 velocity, magnetic, 174 isotropic distribution, 230 fluid guiding center, 163 jump conditions, Rankine-Hugoniot, 3 macroscopic, 191 motion (mean motion), 18 kinetic flux coordinates, 174 equations, 181 Fokker-Planck for unstable plasma, 265 collision operators, 268 plasma theory, 257 equation, 264, 265 force-free field, 197, 214 , 128, 135, 147 free vortex sheets, stability of, 62 of plasma oscillations, 144 Friedrichs diagram, 2 least-action principle, 153 limit, plasma, 260 galilean invariance, 177 Debye shielding, 261 geomagnetic field, 64, 65 linear pinch, diffuse, 48 guiding center linearized GCP equations, 186 fluid (GCF), 163 Liouville equation, 251, 258, 259 equilibrium, 202 local stability, 228, 235, 244 macroscopic, 191 localized disturbances, 164 SUBJECT INDEX 291 longitudinal natural boundary condition, 198 Alfven waves, 58, 60 neutrality, charge, 177, 210 plasma oscillations, 100 magnetic, 180 , 66 neutralizing potential, 169 low 0, 244 Noether's theorem, 153 stability, 238 nonequilibrium processes, 261 Lundquist equations, 1, 8 nonevolutionary shock, 1, 7, 12, 15 normal, 7, 14 nonexistence of solutions, 164 macroscopic guiding center fluid, 191 normal magnetic mode expansion technique, 78 fields, admissible, 214 shocks flow velocity, 174 fast, 4, 6, 7, 10, 11, 12 moment, 176, 191 slow, 5, 7, 10, 12 neutrality, 180 speed diagram, 2 Reynold's number, 51, 55 Nyquist diagram analysis, 254 shielding length, 179 well, 241, 244 Ohm's law, 36, 180 magneto-fluid-dynamics (M.F.D. or MH) one-fluid classical equations, 183 system in, 36 theory, 35, 208 theory, 18 classical M.F.D., 18, 37 one-fluid, 18, 37 MH, 202 two-fluid, 37 orbits, particle, 175 equilibrium, 202 ordering, 51 theory, 36, 37, 38 orthogonality theorems, 82, 85, 86, 95 majorizing equations, 117 oscillations marginal stability, 165 Landau damping of plasma, 144 matching condition, 55 plasma, 128,129, 133, 134, 135, 144, 147,148 Maxwell ('s) distribution, 281 equations, 172, 179 particle(s), 274 set, Vlasov, 247 interchange, 242 microinstability, 164, 194, 234, 251, 283 motion, 172 106,112,121,124 orbits, 175 mixed initial value-boundary value problem, path invariants, 192, 204 130,135 test, 271, 272, 276 mode approximation, dominating, 102, 106, path invariants, particle, 192, 204 112,121,124 perpendicular shocks, 4 modes pessimistic continuum, 90 equations of motion, 222 possible discrete, 90 second variation, 170 modified variation, 168, 217, 227 Alfven wave, 193 planetary waves, hydro-magnetic, 77 compression wave, 193 plasma(s), 237 moment, magnetic, 176, 191 classical, 270, 271, 272 momentum, conservation of, 184 collisionless, 247 monotonicity requirements, 168 domain, 265 motion function, 92 constants of, 85 dispersion, 80, 88 particle, 172 guiding center, 163 symmetric equations of, 170 instability, 249 multiple root, 88 kinetic theory, 255 292 SUBJECT INDEX limit, 260 inner, 25, 35, 36, 44 Debye shielding, 261 intermediate, 5, 6, 7, 11,12,13,15 nonuniform, variational stability of, 196 normal, 5, 6,11, 12 oscillations, 128, 129,133,134,135, 144, nonevolutionary, 1, 7,12,15 147,148 normal, 7, 14 Landau damping of, 144 perpendicular, 4 longitudinal, 100 slow, 4, 5, 6, 7, 10, 11, 12, 43 stability, 128, 129, 130, 136, 139, 144, 165, singular 270, 280 integral equation, 82 uniform, 195 points, 22, 32 unstable, 130 slow equations for, 265 equilibria, 242 waves, 100 flow, 58, 60, 63 Poisson equations, 252 normal shocks, 5, 7,10,12 potential disturbance, 63 shock(s), 4, 5, 6, 7, 10,11, 12, 43 induced, 61 slowly time dependent Hamiltonian potential flow, Sears-Resler, 59, 61 operator, 102 Precession, 64 slug flow, 58 pre-Maxwell system, 179 small-perturbation equations, 57, 63 MH, scalar, 194 specific entropy, 24, 28, 40, 41 stability, 157 quantum plasma, 275 exponential, 165 quasi-linear of free vortex sheets, 62 approximation (Q.L.A.), 281, 282, 284 interchange, 232, 235, 244 theory, 100, 101 low 0, 238 quasi-particle(s), 270, 277 at plasma edge, 237 -scaled distribution function, 239 variational, 167, 213 radiation, 270 of nonuniform plasma, 196 random approximation (RPA), 282 stabilization, 156 Rankine-Hugoniot jump conditions, 3 stable plasma (s), 128, 129, 130,136, 139,144 regime steady Boltzmann, 260 flow of fluid, 17 weak coupling, 258 -state reflection, diffuse, 96 equations, possible solutions of, 62 resistive layer, 50, 51, 52, 54, 55, 56 solution, 143 Reynold's number, magnetic, 51, 55 Stewartson slug flow, 59, 61 stream function, 201 scalar pressure, MH, 194 structural stability, 6 scaling, 55 sub-Alfven Sears-Resler equations, 65, 66 flow, 63 flow, 60, 62, 63 potential, 59, 61 -off shocks, 5, 7, 11, 13 solution, 57, 58, 62 switch-on shocks, 1, 4, 5, 7, 11, 13, 14, 15 second symmetric adiabatic invariant, 171 equations of motion, 170, 187 variation, 219 symmetry, axial, 201, 205 pessimistic, 170 self-adjoin tness, 167 test particle, 271, 272, 276 shielding length, magnetic, 179 tied lines, 214, 235 shock (s) transient problem, 57 evolutionary, 6 transition point, 61 fast, normal, 4, 6, 7, 10, 11, 12 transitional solution, 63 hydromagnetic, 1, 6 transverse variation, 170, 220 SUBJECT INDEX 293 two-fluid theory, 35, 210 vortex sheets, stability of free, 62 classical M.F.D., 37 , 155, 156 two-stream instability, 283, 284 wave(s) uniform plasma, 195 Alfven, 57, 58 ultimate of finite amplitude, 60 flow, 61 channelled, 58, 62 sub-Alfvenic flow, 62 collapse, 61, 63 longitudinal Alfven, 58, 60 variation modified pessimistic, 168, 217, 227 Alfven, 193 second, 219 compression, 193 variational plasma, 100 formulation, 198 regions, Alfven, 60 -single, fluid, 213 weak coupling, 260 stability, 167, 196, 213 behavior, 261 variety, 19 regime, 260 velocity gradient instability, 229 well-posed problems, 193,210 Vlasov, 250 Weyl's convexity inequalities, 18 equation(s), 79,85,163,177,266, 273, 276 Maxwell set, 247