Plasma physics via computer simulation

Plasma Physics via Computer Simulation......Page 1
Contents......Page 3
Foreword......Page 11
Preface......Page 14
Acknowledgments......Page 17
Preface to the Adam Hilger Edition......Page 20
Acknowledgments to the Adam Hilger Edition......Page 21
PART ONE: PRIMER ONE DIMENSIONAL ELECTROSTATIC AND ELECTROMAGNETIC CODES......Page 22
CHAPTER ONE: WHY ATTEMPTING TO DO PLASMA PHYSICS VIA COMPUTER SIMULATION USING PARTICLES MAKES GOOD PHYSICAL SENSE......Page 23
2-1 INTRODUCTION......Page 27
2-2 THE ELECTROSTATIC MODEL: GENERAL REMARKS......Page 28
PROBLEMS......Page 30
2-3 THE COMPUTATIONAL CYCLE: GENERAL REMARKS......Page 31
2-4 INTEGRATION OF THE EQUATIONS OF MOTION......Page 32
2-5 INTEGRATION OF THE FIELD EQUATIONS......Page 36
PROBLEMS......Page 38
2-6 PARTICLE AND FORCE WEIGHTING; CONNECTION BETWEEN GRID AND PARTICLE QUANTITIES......Page 39
2-7 CHOICE OF INITIAL VALUES; GENERAL REMARKS......Page 42
PROBLEMS......Page 43
2-8 CHOICE OF DIAGNOSTICS; GENERAL REMARKS......Page 45
2-9 ARE THE RESULTS CORRECT? TESTS......Page 46
3-2 GENERAL STRUCTURE OF THE PROGRAM ES1......Page 48
3-3 DATA INPUT TO ES1......Page 52
3-4 CHANGE OF INPUT PARAMETERS TO COMPUTER QUANTITIES......Page 53
3-5 NORMALIZATION; COMPUTER VARIABLES......Page 54
3-6 INIT SUBROUTINE; CALCULATION OF INITIAL CHARGE POSITIONS AND VELOCITIES......Page 55
PROBLEMS......Page 56
3-7 SETRHO, INITIALIZATION OF CHARGE DENSITY......Page 58
3-8 FIELDS SUBROUTINE; SOLUTION FOR THE FIELDS FROM THE DENSITIES; FIELD ENERGY......Page 59
PROBLEMS......Page 60
PROBLEMS......Page 62
3-11 ACCEL, SUBROUTINE FOR ADVANCING THE VELOCITY......Page 63
PROBLEMS......Page 64
PROBLEMS......Page 67
3-14 HISTRY SUBROUTINE; PLOTS VERSUS TIME......Page 69
3-15 PLOTTING AND MISCELLANEOUS SUBROUTINES......Page 70
4-2 PARTICLE MOVER ACCURACY; SIMPLE HARMONIC MOTION TEST......Page 74
4-3 NEWTON-LORENTZ FORCE; THREE-DIMENSIONAL v X B INTEGRATOR......Page 77
PROBLEMS......Page 78
4-4 IMPLEMENTATION OF THE v x B ROTATION......Page 80
4-5 APPLICATION TO ONE-DIMENSIONAL PROGRAMS......Page 82
4-6 PARTICLES AS SEEN BY THE GRID; SHAPE FACTORS S(x) S(k)......Page 84
4-7 A WARM PLASMA OF FINITE-SIZE PARTICLES......Page 87
4-8 INTERACTION FORCE WITH FINITE-SIZE PARTICLES IN A GRID......Page 89
4-9 ACCURACY OF THE POISSON SOLVER......Page 91
4-10 FIELD ENERGIES AND KINETIC ENERGIES......Page 92
4-11 BOUNDARY CONDITIONS FOR CHARGE, CURRENT, FIELD, AND POTENTIAL......Page 94
PROBLEM......Page 98
5-2 RELATIONS AMONG INITIAL CONDITIONS; SMALL AMPLITUDE EXCITATION......Page 99
5-3 COLD PLASMA (OR LANGMUIR) OSCILLATIONS; ANALYSIS......Page 104
PROBLEMS......Page 107
5-4 COLD PLASMA OSCILLATIONS; PROJECT......Page 108
Additional suggestions:......Page 109
5-5 HYBRID OSCILLATIONS; PROJECT......Page 110
PROBLEMS......Page 111
5-6 TWO-STREAM INSTABILITY; LINEAR ANALYSIS......Page 112
5-7 TWO-STREAM INSTABILITY; AN APPROXIMATE NONLINEAR ANALYSIS......Page 116
PROBLEMS......Page 121
5-8 TWO-STREAM INSTABILITY; PROJECT......Page 122
5-9 TWO-STREAM INSTABILITY; SELECTED RESULTS......Page 123
PROBLEM......Page 125
5-10 BEAM-PLASMA INSTABILITY; LINEAR ANALYSIS......Page 128
5-11 BEAM-PLASMA INSTABILITY; AN APPROXIMATE NONLINEAR ANALYSIS......Page 132
5-12 BEAM-PLASMA INSTABILITY; PROJECT......Page 137
PROBLEMS......Page 138
5-13 BEAM-CYCLOTRON INSTABILITY; LINEAR ANALYSIS......Page 140
5-14 BEAM-CYCLOTRON INSTABILITY; PROJECT......Page 141
5-15 LANDAU DAMPING......Page 142
5-16 MAGNETIZED RING-VELOCITY DISTRIBUTION; DORY-GUEST-HARRIS INSTABILITY; LINEAR ANALYSIS......Page 145
5-17 MAGNETIZED RING-VELOCITY DISTRIBUTION; PROJECT......Page 148
5-18 RESEARCH APPLICATIONS......Page 149
6-2 THE ONE-DIMENSIONAL MODEL......Page 150
6-3 ONE-DIMENSIONAL FIELD EQUATIONS AND INTEGRATION......Page 151
6-4 STABILITY OF THE METHOD......Page 154
6-5 THE EM1 CODE, FOR PERIODIC SYSTEMS......Page 155
6-6 THE EM1BND CODE, FOR BOUNDED SYSTEMS; LOADING FOR f(x,v)......Page 156
6-7 EMlBND BOUNDARY CONDITIONS......Page 158
6-8 EM1, EM1BND OUTPUT DIAGNOSTICS......Page 159
7-1 INTRODUCTION......Page 162
7-2 BEAT HEATING OF PLASMA......Page 163
7-3 OBSERVATION OF PRECURSOR......Page 166
PART TWO: THEORY PLASMA SIMULATION USING PARTICLES IN SPATIAL GRIDS WITH FINITE TIME STEPS......Page 169
8-1 INTRODUCTION; EARLY USE OF GRIDS AND CELLS WITH PLASMAS......Page 170
8-3 SOME GENERAL REMARKS ON THE EFFECTS OF A PERIODIC SPATIAL NONUNIFORMITY......Page 173
8-4 NOTATION AND CONVENTIONS......Page 176
8-5 PARTICLE TO GRID WEIGHTING; SHAPE FACTORS......Page 177
8-6 MOMENTUM CONSERVATION FOR THE OVERALL SYSTEM......Page 179
8-7 FOURIER TRANSFORMS FOR DEPENDENT VARIABLES; ALIASING DUE TO FINITE FOURIER SERIES......Page 180
8-8 MORE ACCURATE ALGORITHM USING SPLINES FOR S(x)......Page 183
8-9 GENERALIZATION TO TWO AND THREE DIMENSIONS......Page 185
8-10 LINEAR WAVE DISPERSION......Page 186
8-11 APPLICATION TO COLD DRIFTING PLASMA: OSCILLATION FREQUENCIES......Page 187
PROBLEM......Page 189
8-12 COLD BEAM NONPHYSICAL INSTABILITY......Page 190
8-13 SOLUTION FOR THERMAL (MAXWELLIAN) PLASMA; NONPHYSICAL INSTABILITIES CAUSED BY THE GRID......Page 192
PROBLEMS......Page 196
9-1 INTRODUCTION......Page 197
9-2 WARM UNMAGNETIZED PLASMA DISPERSION FUNCTION; LEAPFROG ALGORITHM......Page 198
PROBLEMS......Page 203
9-3 ALTERNATIVE ANALYSIS BY SUMMATION OVER PARTICLE ORBITS......Page 205
PROBLEMS......Page 207
9-4 NUMERICAL INSTABILITY......Page 208
9-5 THE DISPERSION FUNCTION INCLUDING BOTH FINITE Ax AND A t......Page 210
9-6 MAGNETIZED WARM PLASMA DISPERSION AND NONPHYSICAL INSTABILITY......Page 211
(a) Derivation of the Dispersion Function......Page 212
(b) Properties of the Dispersion Relation......Page 214
(c) Numerical Instability......Page 215
PROBLEMS......Page 217
(a) Subcycling......Page 218
(b) Implicit Time Integration......Page 219
9-8 OTHER ALGORITHMS FOR UNMAGNETIZED PLASMA......Page 220
(a) Class C Algorithms......Page 221
(b) Class D Algorithms......Page 224
PROBLEMS......Page 225
10-2 NONEXISTENCE OF A CONSERVED ENERGY IN MOMENTUM CONSERVING CODES......Page 227
10-3 AN ENERGY-CONSERVING ALGORITHM......Page 229
PROBLEMS......Page 231
10-4 ENERGY CONSERVATION......Page 232
10-5 ALGORITHMS DERIVED VIA VARIATIONAL PRINCIPLES......Page 234
PROBLEM......Page 235
10-7 LEWIS’S POISSON DIFFERENCE EQUATION AND THE COULOMB FIELDS......Page 236
10-8 SMALL-AMPLITUDE OSCILLATIONS OF A COLD PLASMA......Page 237
10-9 LACK OF MOMENTUM CONSERVATION......Page 239
10-10 ALIASING AND THE DISPERSION RELATION FOR WARM PLASMA OSCILLATIONS......Page 242
(a) Momentum Conservation and Self-Forces......Page 243
(b) Macroscopic Field Accuracy......Page 245
10-12 THE QUADRATIC SPLINE MODEL......Page 246
PROBLEM......Page 247
11-1 INTRODUCTION......Page 248
11-2 THE MULTIPOLE EXPANSION METHOD......Page 249
PROBLEMS......Page 252
11-3 THE “SUBTRACTED” MULTIPOLE EXPANSION......Page 253
11-4 MULTIPOLE INTERPRETATIONS OF OTHER ALGORITHMS......Page 255
PROBLEMS......Page 257
11-5 RELATIONS BETWEEN FOURIER TRANSFORMS OF PARTICLE AND GRID QUANTITIES......Page 258
11-6 OVERALL ACCURACY OF THE FORCE CALCULATION; DISPERSION RELATION......Page 261
11-7 SUMMARY AND A PERSPECTIVE......Page 264
12-1 INTRODUCTION......Page 267
12-2 TEST CHARGE AND DEBYE SHIELDING......Page 269
(a) The Spectrum......Page 271
(2) Spatial spectrum......Page 273
(4) Ax==O, A t # 0......Page 275
( 5 ) Ax, A t both nonzero......Page 276
12-4 REMARKS ON THE SHIELDING AND FLUCTUATION RESULTS......Page 277
(a) Velocity Diffusion......Page 278
(b) Velocity Drag......Page 280
(c) The Kinetic Equation......Page 281
PROBLEMS......Page 282
12-6 EXACT PROPERTIES OF THE KINETIC EQUATION......Page 283
PROBLEMS......Page 285
12-7 REMARKS ON THE KINETIC EQUATION......Page 286
(a) The Sheet Model......Page 288
(b) The Equilibrium Velocity Distribution is Maxwellian......Page 290
(c) Debye Shielding......Page 291
(d) Velocity Drag......Page 293
(e) Relaxation Time......Page 296
13-3 THERMALIZATION OF A ONE-DIMENSIONAL PLASMA......Page 297
(a) Fast Time-Scale Evolution......Page 298
(b) Slow Time-Scale Evolution......Page 299
PROBLEMS......Page 303
(b) Cooling Due to Damping in the Particle Equations of Motion......Page 304
13-5 COLLISION AND HEATING TIMES FOR TWO-DIMENSIONAL THERMAL PLASMA......Page 306
13-6 UNSTABLE PLASMA......Page 311
PART THREE: PRACTICE PROGRAMS IN TWO AND THREE DIMENSIONS: DESIGN CONSIDERATIONS......Page 313
14-1 INTRODUCTION......Page 314
14-2 AN OVERALL 2D ELECTROSTATIC PROGRAM......Page 317
14-3 POISSON’S EQUATION SOLUTIONS......Page 319
14-4 WEIGHTING AND EFFECTIVE PARTICLE SHAPES IN RECTANGULAR COORDINATES: S (x), S (k), FORCE ANISOTROPY......Page 320
14-5 DOUBLY PERIODIC MODEL AND BOUNDARY CONDITIONS......Page 324
(a) Doubly Periodic Poisson Solver......Page 325
(b) Periodic Boundary Conditions; k = 0 Fields......Page 326
14-6 POISSON'S EQUATION SOLUTIONS FOR SYSTEMS BOUNDED IN x AND PERIODIC IN y......Page 327
14-7 A PERIODIC-OPEN MODEL USING INVERSION SYMMETRY......Page 331
14-8 ACCURACY OF FINITE-DIFFERENCED POISSON’S EQUATION......Page 334
14-9 ACCURACY OF FINITE-DIFFERENCED GRADIENT OPERATOR......Page 336
14-10 POISSON’S EQUATION FINITE-DIFFERENCED IN CYLINDRICAL COORDINATES r , r - Z , r-8......Page 340
(a) r only......Page 341
(b) r-z......Page 342
(c) r - 0......Page 343
14-11 WEIGHTING IN CYLINDRICAL COORDINATES FOR PARTICLES AND FIELDS......Page 345
14-12 POSITION ADVANCE FOR CYLINDRICAL COORDINATES......Page 347
14-13 IMPLICIT METHOD FOR LARGE TIME STEPS......Page 348
(a) Implicit Time Differencing of the Particle Equations of Motion......Page 349
(b) Direct Method with Electrostatic Fields; Solution of the Implicit Equations......Page 350
(c) A One-Dimensional Realization......Page 351
(d) General Electrostatic Case......Page 353
14-14 DIAGNOSTICS......Page 354
14-15 REPRESENTATIVE APPLICATIONS......Page 357
(b) Instabilities......Page 358
(c) Heating......Page 359
15-1 INTRODUCTION......Page 360
15-2 TIME INTEGRATION OF THE FIELDS AND LOCATION OF THE SPATIAL GRIDS......Page 361
15-3 ACCURACY AND STABILITY OF THE TIME INTEGRATION......Page 363
15-4 TIME INTEGRATION OF THE PARTICLE EQUATIONS......Page 365
PROBLEM......Page 366
15-5 COUPLING OF PARTICLE AND FIELD INTEGRATIONS......Page 367
15-6 THE V B AND V E EQUATIONS; ENSURING CONSERVATION OF CHARGE......Page 368
15-7 A-4 FORMULATION......Page 370
15-8 NOISE PROPERTIES OF VARIOUS CURRENT-WEIGHTING METHODS......Page 371
(a) Subcycling of the Maxwell Equations......Page 373
(b) Fourier-Transform Field Integration......Page 374
PROBLEM......Page 375
(a) The Longitudinal Field......Page 376
(b) Absorbing Outgoing Electromagnetic Waves in a Dissipative Region......Page 377
(c) A Simple Closure of the Maxwell Equations at the Open Boundaries......Page 378
(d) Boundary Conditions for Waves Incident at (almost) Any Angle......Page 380
(a) Closure of Maxwell’s Equations at the Walls......Page 382
(b) Electrostatic Solutions in 2d......Page 384
(c) Combined Particle and Field Calculation......Page 385
15-13 INTEGRATING MAXWELL'S EQUATIONS IN CYLINDRICAL COORDINATES......Page 386
15-14 DARWIN, OR MAGNETOINDUCTIVE, APPROXIMATION......Page 388
15-15 HYBRID PARTICLE/FLUID CODES......Page 389
(a) Particles......Page 390
(d) Remarks......Page 391
(a) Interaction of Intense Laser Light with Plasmas......Page 392
15-19 REMARKS ON LARGE-SCALE PLASMA SIMULATION......Page 394
16-1 INTRODUCTION......Page 396
16-2 LOADING NONUNIFORM DISTRIBUTIONS f 0 (v ), n 0 (x); INVERSION OF CUMULATIVE DISTRIBUTION FUNCTIONS......Page 397
16-3 LOADING A COLD PLASMA OR COLD BEAM......Page 398
16-4 LOADING A MAXWELLIAN VELOCITY DISTRIBUTION......Page 399
PROBLEMS......Page 401
16-5 QUIET STARTS: SMOOTH LOADING IN X-v SPACE: USE OF MIXED-RADIX DIGIT-REVERSED NUMBER SETS......Page 402
16-6 QUIET START: MULTIPLE-BEAM AND RING INSTABILITIES AND SATURATION; RECURRENCES......Page 403
16-7 LOADING A MAGNETIZED PLASMA WITH A GIVEN GUIDING CENTER SPATIAL DISTRIBUTION n0(xgc)......Page 411
16-8 PARTICLE INJECTION AND ABSORPTION AT BOUNDARIES; FIELD EMISSION, IONIZATION, AND CHARGE EXCHANGE......Page 414
16-9 PARTICLE AND FIELD BOUNDARY CONDITIONS FOR AXIALLY BOUNDED SYSTEMS; PLASMA DEVICES......Page 417
(a) Charge and Field Boundary Conditions in 1d......Page 418
(b) Solutions with an External Circuit......Page 420
PROBLEMS......Page 425
PART FOUR: APPENDICES......Page 426
(a) Complex Periodic Discrete Fourier Transform......Page 427
(b) Transform of Real-valued Sequences, Two at a Time......Page 429
(c) Sine Transform of Real-valued Sequences, Two at a Time......Page 430
PROBLEMS......Page 431
(d) Listings for CPFT, RPFT2, and RPFTI2......Page 432
APPENDIX B: COMPENSATING AND ATTENUATING FUNCTIONS USED IN ES1......Page 438
APPENDIX C: DIGITAL FILTERING IN ONE AND TWO DIMENSIONS......Page 444
APPENDIX D: DIRECT FINITE-DIFFERENCE EQUATION SOLUTIONS......Page 449
PROBLEMS......Page 452
APPENDIX E: DIFFERENCING OPERATORS; LOCAL AND NONLOCAL (V - ik, V2 - -k2>......Page 453
REFERENCES......Page 459