Cosmic Plasma Physics

By: B.V. Somov

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Hardcover | 30 September 2000

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of the approximation at hand will be outlined from the viewpoint of physics rather than of many possible astronomical applications. After that, as an aid to detailed understanding, please return with pencil and paper to work out the missing steps (if any) in the formal mathematics. On the basis of such an approach the student interested in modern astro- physics, its current practice, will find the answers to two key questions: (1) what approximation is the best one (the simplest but sufficient) for descrip- tion of a phenomenon in cosmic plasma; (2) how tO, build an adequate model for the phenomenon, for example, a solar flare. Practice is really important for understanding the theory of cosmic plasma. Related exercises (problems and answers supplemented to each chapter) to improve skill do not thwart the theory but serve to better understanding of cosmic plasma physics. As for the applications, preference evidently is given to physical processes in the solar plasma. Why? - Much attention to solar plasma physics is con- the possibility of th all-round observational test of theoretical ditioned by models. This statement primarily relates to the processes in the solar atmo- sphere. For instance, flares on the Sun, in contrast to those on other stars as well as a lot of other analogous phenomena in the Universe, can be seen in their development, i. e.

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Preface	p. xvii
Introduction	p. 1
Particles and Fields: Exact Self-Consistent Description	p. 5
Charged Particles in the Electromagnetic Field	p. 5
A general formulation of the problem	p. 5
The continuity equation for electric charge	p. 6
Initial equations and initial conditions	p. 7
Cosmic plasma applications	p. 8
Gravitational systems	p. 9
Liouville's theorem	p. 10
Continuity in phase space	p. 10
The character of particle interactions	p. 12
The Lorentz force, gravity	p. 14
Collisional friction in plasma	p. 14
The exact distribution function	p. 16
Practice: Problems and Answers	p. 17
A Statistical Description of Cosmic Plasma	p. 21
The averaging of Liouville's equation	p. 21
Averaging over phase space	p. 21
Two statistical postulates	p. 23
A statistical mechanism of mixing in phase space	p. 24
The derivation of a general kinetic equation	p. 27
A collisional integral and correlation functions	p. 28
The exact distribution function	p. 28
Binary correlation	p. 30
The collisional integral and binary correlation	p. 31
Equations for correlation functions	p. 33
Approximations for binary collisions	p. 36
Small parameters of kinetic theory	p. 36
The Vlasov kinetic equation	p. 37
The Landau collisional integral	p. 38
The Fokker-Plank equation	p. 40
The correlation function and Debye shielding	p. 41
The Maxwellian distribution function	p. 41
Pair correlations and the Debye radius	p. 42
Gravitational systems	p. 46
Comments on numerical simulations	p. 47
Practice: Problems and Answers	p. 49
Propagation of Accelerated Particles in Cosmic Plasma	p. 51
Derivation of the basic equation	p. 51
Basic approximations	p. 51
Dimensionless equation	p. 53
A kinetic equation at high speeds	p. 55
The classical thick-target model	p. 57
An approximate account of scattering	p. 60
The reverse-current electric-field effect	p. 64
The necessity for a beam-neutralizing current	p. 64
A formulation of a realistic kinetic problem	p. 66
Dimensionless parameters of the problem	p. 68
Coulomb energy losses	p. 70
Basic physical results	p. 72
Practice: Problems and Answers	p. 74
The Motion of a Particle in Given Fields	p. 75
A particle in constant homogeneous fields	p. 75
Constant non-magnetic forces	p. 76
Constant homogeneous magnetic fields	p. 76
Non-magnetic forces in a homogeneous magnetic field	p. 79
Weakly inhomogeneous slowly changing fields	p. 81
Small parameters in the motion equation	p. 81
Expansion in powers of m/e	p. 83
The averaging over gyro-motion	p. 85
Spiral motion of the guiding center	p. 87
Inertial and gradient drifts	p. 88
Adiabatic invariants in cosmic plasmas	p. 92
General definitions	p. 92
Three main invariants	p. 92
Approximation accuracy. Exact solutions	p. 101
What is magnetic reconnection?	p. 101
Neutral points of a magnetic field	p. 101
Reconnection in vacuum	p. 103
Reconnection in plasma	p. 105
Three stages in the reconnection process	p. 107
Acceleration in current sheets, why?	p. 108
The origin of particle acceleration	p. 108
Acceleration in a neutral current sheet	p. 109
Practice: Problems and Answers	p. 113
Wave-Particle Interactions in Cosmic Plasma	p. 117
The basis of kinetic theory	p. 117
The linearized Vlasov equation	p. 117
The Landau resonance and Landau damping	p. 119
Gyroresonance	p. 121
Stochastic acceleration of particles by waves	p. 123
The principles of particle acceleration by waves	p. 123
MHD turbulent cascading	p. 125
Stochastic acceleration of electrons	p. 127
Acceleration of protons and heavy ions	p. 128
Electron-dominated solar flares	p. 130
The relativistic electron-positron plasma	p. 133
Practice: Problems and Answers	p. 134
Coulomb Collisions of Particles in Cosmic Plasma	p. 137
Close and distant collisions	p. 137
The Rutherford formula and collision parameters	p. 137
The test particle concept	p. 139
Particles in a magnetic trap	p. 140
The role of distant collisions	p. 141
Debye shielding and plasma oscillations	p. 143
Collisional relaxations in cosmic plasma	p. 146
Some exact solutions	p. 146
Two-temperature plasma in solar flares	p. 148
An adiabatic model for two-temperature plasma	p. 153
Two-temperature accretion flows	p. 154
Dynamic friction in cosmic plasma	p. 155
The collisional drag force and energy losses	p. 155
Electric runaway	p. 160
Thermal runaway in cosmic plasma	p. 162
Practice: Problems and Answers	p. 163
A Hydrodynamic Description of Cosmic Plasma	p. 167
Transition to macroscopic transfer equations	p. 167
Distribution function moments	p. 168
Equations for moments	p. 169
General properties of the transfer equations	p. 174
Hydrodynamic equations for cosmic plasma	p. 175
The continuity equation	p. 175
The momentum conservation law in cosmic plasma	p. 176
The energy conservation law	p. 178
The equation of state and transfer coefficients	p. 178
Gravitational systems	p. 180
The generalized Ohm's law in cosmic plasma	p. 181
Basic equations	p. 181
The general solution	p. 184
The conductivity of magnetized plasma	p. 184
The physical interpretation	p. 186
Cosmic plasma conductivity	p. 187
Volume charge and quasi-neutrality	p. 188
Practice: Problems and Answers	p. 190
Magnetohydrodynamics of Cosmic Plasma	p. 197
Basic assumptions and the MHD equations	p. 197
Old and new simplifying assumptions	p. 197
Non-relativistic magnetohydrodynamics	p. 201
Relativistic magnetohydrodynamics	p. 203
Magnetic flux conservation. Ideal MHD	p. 204
Integral and differential forms of the law	p. 204
An approximation and the equations of ideal MHD	p. 206
The main approximations in ideal MHD	p. 208
Dimensionless equations	p. 208
Weak magnetic fields in cosmic plasma	p. 210
Strong magnetic fields in cosmic plasma	p. 211
Accretion discs and relativistic jets	p. 214
Angular momentum transfer in binary stars	p. 214
Accretion discs near black holes	p. 216
Jets near black holes	p. 217
Flares in accretion disc coronae	p. 220
Relativistic jets from disc coronae	p. 221
Practice: Problems and Answers	p. 221
Cosmic Plasma Flows in a Strong Magnetic Field	p. 225
The general formulation of the problem	p. 225
The formalism of two-dimensional problems	p. 227
The first type of problems	p. 228
The second type of problems	p. 229
On the existence of continuous flows	p. 234
Flows in the field of a time-dependent dipole	p. 237
Plane magnetic dipole fields	p. 237
Axisymmetric dipole fields in cosmic plasma	p. 241
Practice: Problems and Answers	p. 243
MHD Waves in Cosmic Plasma	p. 247
The general dispersion equation in ideal MHD	p. 247
Small-amplitude waves in ideal MHD	p. 250
Entropy waves	p. 250
Alfven waves	p. 251
Magnetoacoustic waves	p. 253
The phase velocity diagram	p. 254
Dissipative waves	p. 256
Damping of Alfven waves	p. 256
Slightly damped MHD waves	p. 258
Practice: Problems and Answers	p. 259
Discontinuous Flows in a MHD Medium	p. 261
Discontinuity surfaces in hydrodynamics	p. 261
The origin of shocks in ordinary hydrodynamics	p. 261
Boundary conditions and classification	p. 262
Dissipative processes and entropy	p. 264
Magnetohydrodynamic discontinuities	p. 265
Boundary conditions at a discontinuity surface	p. 265
Discontinuities without plasma flows across them	p. 269
Perpendicular shock wave	p. 271
Oblique shock waves	p. 273
Peculiar shock waves	p. 279
The Alfven discontinuity	p. 281
Transitions between discontinuities	p. 282
Shock waves in collisionless plasma	p. 284
Practice: Problems and Answers	p. 285
Evolutionarity of MHD Discontinuities	p. 291
Conditions for evolutionarity	p. 291
The physical meaning and definition	p. 291
Linearized boundary conditions	p. 294
The number of small-amplitude waves	p. 296
Domains of evolutionarity	p. 299
Consequences of evolutionarity conditions	p. 300
The order of wave propagation	p. 300
Continuous transitions between discontinuities	p. 302
Dissipative effects in evolutionarity	p. 303
Discontinuity structure and evolutionarity	p. 306
Perpendicular shock waves	p. 306
Discontinuities with penetrating magnetic field	p. 311
Practice: Problems and Answers	p. 312
Particle Acceleration by Shock Waves	p. 315
Two basic mechanisms	p. 315
Shock diffusive acceleration	p. 316
The canonical model of diffusive mechanism	p. 316
Some properties of diffusive mechanism	p. 319
Nonlinear effects in diffusive acceleration	p. 320
Shock drift acceleration	p. 321
Perpendicular shock waves	p. 321
Quasi-perpendicular shock waves	p. 324
Oblique shock waves	p. 328
The collapsing trap effect in solar flares	p. 329
Fast plasma outflows and shocks	p. 329
Particle acceleration in collapsing trap	p. 331
The upward motion of coronal HXR source	p. 334
Practice: Problems and Answers	p. 336
Cosmic Plasma Equilibrium in Magnetic Field	p. 339
The virial theorem in MHD	p. 339
A brief pre-history	p. 339
Deduction of the scalar virial theorem	p. 340
Some astrophysical applications	p. 343
Force-free fields and Shafranov's theorem	p. 346
The simplest examples of force-free configurations	p. 346
The energy of a force-free field	p. 348
Properties of equilibrium configurations	p. 349
Magnetic surfaces	p. 349
The specific volume of a magnetic tube	p. 351
The flute or convective instability	p. 354
Archimedean forces in MHD	p. 356
A general formulation of the problem	p. 356
An oversimplified consideration of the effect	p. 358
MHD equilibrium in the solar atmosphere	p. 359
Practice: Problems and Answers	p. 360
Stationary Flows in a Magnetic Field	p. 363
Ideal plasma flows	p. 363
Incompressible medium	p. 364
Compressible medium	p. 365
Astrophysical collimated streams (jets)	p. 366
MHD waves of arbitrary amplitude	p. 366
Differential rotation and isorotation	p. 367
Flows at small magnetic Reynolds numbers	p. 369
Stationary flows inside a duct	p. 370
The MHD generator or pump	p. 372
Weakly-ionized plasma in space	p. 374
The [sigma]-dependent force and vortex flows	p. 375
Simplifications and problem formulation	p. 375
The solution for a spherical ball	p. 377
Forces and flows near a spherical ball	p. 378
Large magnetic Reynolds numbers	p. 383
The general formula for the [sigma]-dependent force	p. 383
The [sigma]-dependent force in solar prominences	p. 386
Practice: Problems and Answers	p. 388
Magnetic Reconnection in Current Sheets	p. 389
Small perturbations near a neutral line	p. 389
Historical comments	p. 389
Reconnection of strong magnetic fields	p. 390
A linearized problem in ideal MHD	p. 391
Converging waves and the cumulative effect	p. 393
Large perturbations near the neutral line	p. 395
Magnetic field line deformations	p. 395
Plasma density variations	p. 398
The dynamic dissipation of a magnetic field	p. 400
Conditions of appearance	p. 400
The physical meaning of dynamic dissipation	p. 402
Nonstationary analytical models of the RCS	p. 403
Self-similar 2D MHD solutions	p. 403
Magnetic collapse at the zeroth point	p. 406
From collisional to collisionless reconnection	p. 410
Reconnection in solar flares	p. 411
The role of magnetic fields	p. 411
Three-dimensional reconnection in flares	p. 414
The solar flare of 1980 November 5	p. 418
A current sheet as the source of energy	p. 422
A current sheet as a part of an electric circuit	p. 425
New topological models	p. 427
Stationary Models of Reconnecting Current Sheets	p. 433
Magnetically neutral current sheets	p. 433
The simplest MHD model	p. 433
The current sheet by Syrovatskii	p. 435
Simple scaling laws	p. 438
Magnetically non-neutral RCS's	p. 440
Transversal magnetic fields	p. 440
Longitudinal magnetic fields	p. 441
Basic physics of the HTTCS	p. 443
A general formulation of the problem	p. 443
Problem in the strong field approximation	p. 446
Basic local parameters of the HTTCS	p. 447
The general solution of the problem	p. 448
Plasma turbulence inside the HTTCS	p. 450
Formulae for the basic parameters of the HTTCS	p. 450
HTTCS in solar flares	p. 453
Why are flares so different?	p. 453
Superhot plasma production	p. 456
Concluding comments	p. 458
Practice: Problems and Answers	p. 459
Particle Acceleration in Current Sheets	p. 463
Magnetically non-neutral RCS's	p. 463
An introduction in the problem	p. 463
Dimensionless parameters and equations	p. 464
An iterative solution of the problem	p. 466
The maximum energy of an accelerated particle	p. 469
The non-adiabatic thickness of current sheet	p. 470
Regular versus chaotic acceleration	p. 471
Reasons for chaos	p. 472
The stabilizing effect of the longitudinal field	p. 473
Characteristic times of processes	p. 475
Dynamics of accelerated electrons in solar flares	p. 476
Particle simulations of collisionless reconnection	p. 477
Ion acceleration in current sheets	p. 477
Ions are much heavier than electrons	p. 477
Electrically non-neutral current sheets	p. 479
Maximum particle energy and acceleration rates	p. 481
Early and late acceleration in solar flares	p. 484
Structural Instability of Reconnecting Current Sheets	p. 487
Properties of reconnecting current sheets	p. 487
Current sheet splitting	p. 487
Evolutionarity of reconnecting current sheets	p. 489
Magnetic field near the current sheet	p. 490
Current sheet flows	p. 491
Additional simplifying assumptions	p. 493
Small perturbations outside the RCS	p. 494
Basic assumptions	p. 494
The propagation of perturbations normal to the RCS	p. 494
The inclined propagation of small perturbations	p. 496
Perturbations inside the RCS	p. 500
Linearized dissipative MHD equations	p. 500
Boundary conditions	p. 502
Dimensionless equations and small parameters	p. 503
Solution of the linearized equations	p. 505
Solution on the boundary of the RCS	p. 508
The criterion of evolutionarity	p. 510
One-dimensional boundary conditions	p. 510
Solutions of the boundary equations	p. 511
Evolutionarity and splitting of current sheets	p. 515
The Tearing Instability of a Reconnecting Current Sheet	p. 517
The origin of the tearing instability	p. 517
Two necessary conditions	p. 517
Historical comments	p. 518
The simplest problem and its analytic solution	p. 520
The model and equations for small disturbances	p. 520
The external non-dissipative region	p. 522
The internal dissipative region	p. 523
Matching of the solutions and the dispersion relation	p. 525
The physical interpretation of the instability	p. 527
The stabilizing effect of the transversal field	p. 530
Compressibility and a longitudinal field	p. 533
Neutral current sheets	p. 533
Non-neutral current sheets	p. 534
The kinetic approach	p. 536
The tearing instability of neutral sheet	p. 536
Stabilization by the transversal field	p. 540
The tearing instability of the geomagnetic tail	p. 541
Selected Trends in Cosmic Plasma Physics	p. 545
Reconnection and magnetic helicity	p. 545
General properties of complex MHD systems	p. 545
Helical scaling in turbulence	p. 547
Coronal heating in solar active regions	p. 548
Reconnection and helicity in solar flares	p. 549
Reconnection in weakly-ionized plasma	p. 550
Some observations and classical models	p. 550
Balance equations and their solution	p. 551
Characteristics of the reconnecting current sheet	p. 553
Reconnection under solar prominences	p. 556
Element fractionation by reconnection	p. 558
The photospheric dynamo	p. 560
Current generation mechanisms	p. 560
Physics of thin magnetic flux tubes	p. 560
FIP fractionation theory	p. 563
Mechanisms of coronal heating	p. 565
Heating of the quiet solar corona	p. 565
Coronal heating in active regions	p. 566
Practice: Problems and Answers	p. 568
Magnetic Reconnection of Electric Currents	p. 571
Introductory comments	p. 571
Flare energy storage and release	p. 572
From early models to future investigations	p. 572
Some alternative trends in the flare theory	p. 576
Current sheets at separatrices	p. 577
Current sheet formation mechanisms	p. 578
Magnetic footpoints and their displacements	p. 578
Classical 2D reconnection	p. 580
The creation of current sheets by shearing motions	p. 582
Antisymmetrical shearing motions	p. 584
The third class of displacements	p. 586
The shear and reconnection of currents	p. 586
Physical processes related to shear and reconnection	p. 586
The topological interruption of electric currents	p. 589
The inductive change of energy	p. 590
To the future observations by Solar-B	p. 591
Epilogue	p. 593
Notation	p. 595
Useful Expressions	p. 601
Constants	p. 605
Bibliography	p. 607
Index	p. 639
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Cosmic Plasma Physics

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Hardcover

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