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Fundamentals of Plasma Physics Paul M Bellan to my parents Contents Preface xi Basic concepts 1.1 History of the term “plasma” 1.2 Brief history of plasma physics 1.3 Plasma parameters 1.4 Examples of plasmas 1.5 Logical framework of plasma physics 1.6 Debye shielding 1.7 Quasi-neutrality 1.8 Small v large angle collisions in plasmas 1.9 Electron and ion collision frequencies 1.10 Collisions with neutrals 1.11 Simple transport phenomena 1.12 A quantitative perspective 1.13 Assignments 1 3 11 14 16 17 20 22 Derivation of fluid equations: Vlasov, 2-fluid, MHD 2.1 Phase-space 2.2 Distribution function and Vlasov equation 2.3 Moments of the distribution function 2.4 Two-fluid equations 2.5 Magnetohydrodynamic equations 2.6 Summary of MHD equations 2.7 Sheath physics and Langmuir probe theory 2.8 Assignments 30 30 31 33 36 46 52 53 58 Motion of a single plasma particle 3.1 Motivation 3.2 Hamilton-Lagrange formalism v Lorentz equation 3.3 Adiabatic invariant of a pendulum 3.4 Extension of WKB method to general adiabatic invariant 3.5 Drift equations 3.6 Relation of Drift Equations to the Double Adiabatic MHD Equations 3.7 Non-adiabatic motion in symmetric geometry 3.8 Motion in small-amplitude oscillatory fields 3.9 Wave-particle energy transfer 3.10 Assignments 62 62 62 66 68 73 91 95 108 110 119 viii Elementary plasma waves 4.1 General method for analyzing small amplitude waves 4.2 Two-fluid theory of unmagnetized plasma waves 4.3 Low frequency magnetized plasma: Alfvén waves 4.4 Two-fluid model of Alfvén modes 4.5 Assignments 123 123 124 131 138 147 Streaming instabilities and the Landau problem 5.1 Streaming instabilities 5.2 The Landau problem 5.3 The Penrose criterion 5.4 Assignments 149 149 153 172 175 Cold plasma waves in a magnetized plasma 6.1 Redundancy of Poisson’s equation in electromagnetic mode analysis 6.2 Dielectric tensor 6.3 Dispersion relation expressed as a relation between n2 and n2 x z 6.4 A journey through parameter space 6.5 High frequency waves: Altar-Appleton-Hartree dispersion relation 6.6 Group velocity 6.7 Quasi-electrostatic cold plasma waves 6.8 Resonance cones 6.9 Assignments 178 178 179 193 195 197 201 203 204 208 Waves in inhomogeneous plasmas and wave energy relations 7.1 Wave propagation in inhomogeneous plasmas 7.2 Geometric optics 7.3 Surface waves - the plasma-filled waveguide 7.4 Plasma wave-energy equation 7.5 Cold-plasma wave energy equation 7.6 Finite-temperature plasma wave energy equation 7.7 Negative energy waves 7.8 Assignments 210 210 213 214 219 221 224 225 228 Vlasov theory of warm electrostatic waves in a magnetized plasma 8.1 Uniform plasma 8.2 Analysis of the warm plasma electrostatic dispersion relation 8.3 Bernstein waves 8.4 Warm, magnetized, electrostatic dispersion with small, but finite k 8.5 Analysis of linear mode conversion 8.6 Drift waves 8.7 Assignments 229 229 234 236 239 241 249 263 MHD equilibria 9.1 Why use MHD? 9.2 Vacuum magnetic fields 264 264 265 ix 9.3 9.4 9.5 9.6 9.7 9.8 9.9 9.10 Force-free fields Magnetic pressure and tension Magnetic stress tensor Flux preservation, energy minimization, and inductance Static versus dynamic equilibria Static equilibria Dynamic equilibria: flows Assignments 268 268 271 272 274 275 286 295 10 Stability of static MHD equilibria 10.1 The Rayleigh-Taylor instability of hydrodynamics 10.2 MHD Rayleigh-Taylor instability 10.3 The MHD energy principle 10.4 Discussion of the energy principle 10.5 Current-driven instabilities and helicity 10.6 Magnetic helicity 10.7 Qualitative description of free-boundary instabilities 10.8 Analysis of free-boundary instabilities 10.9 Assignments 298 299 302 306 319 319 320 323 326 334 11 Magnetic helicity interpreted and Woltjer-Taylor relaxation 11.1 Introduction 11.2 Topological interpretation of magnetic helicity 11.3 Woltjer-Taylor relaxation 11.4 Kinking and magnetic helicity 11.5 Assignments 336 336 336 341 345 357 12 Magnetic reconnection 12.1 Introduction 12.2 Water-beading: an analogy to magnetic tearing and reconnection 12.3 Qualitative description of sheet current instability 12.4 Semi-quantitative estimate of the tearing process 12.5 Generalization of tearing to sheared magnetic fields 12.6 Magnetic islands 12.7 Assignments 360 360 361 362 364 371 376 378 13 Fokker-Planck theory of collisions 13.1 Introduction 13.2 Statistical argument for the development of the Fokker-Planck equation 13.3 Electrical resistivity 13.4 Runaway electric field 13.5 Assignments 382 382 384 393 395 395 14 Wave-particle nonlinearities 14.1 Introduction 14.2 Vlasov non-linearity and quasi-linear velocity space diffusion 398 398 399 x 14.3 Echoes 14.4 Assignments 412 426 15 Wave-wave nonlinearities 15.1 Introduction 15.2 Manley-Rowe relations 15.3 Application to waves 15.4 Non-linear dispersion formulation and instability threshold 15.5 Digging a hole in the plasma via ponderomotive force 15.6 Ion acoustic wave soliton 15.7 Assignments 428 428 430 435 444 448 454 457 16 Non-neutral plasmas 16.1 Introduction 16.2 Brillouin flow 16.3 Isomorphism to incompressible 2D hydrodynamics 16.4 Near perfect confinement 16.5 Diocotron modes 16.6 Assignments 460 460 460 463 464 465 476 17 Dusty plasmas 17.1 Introduction 17.2 Electron and ion current flow to a dust grain 17.3 Dust charge 17.4 Dusty plasma parameter space 17.5 Large P limit: dust acoustic waves 17.6 Dust ion acoustic waves 17.7 The strongly coupled regime: crystallization of a dusty plasma 17.8 Assignments Bibliography and suggested reading References Appendix A: Intuitive method for vector calculus identities Appendix B: Vector calculus in orthogonal curvilinear coordinates Appendix C: Frequently used physical constants and formulae Index 483 483 484 486 490 491 494 495 504 507 509 515 518 524 528 Preface This text is based on a course I have taught for many years to first year graduate and senior-level undergraduate students at Caltech One outcome of this teaching has been the realization that although students typically decide to study plasma physics as a means towards some larger goal, they often conclude that this study has an attraction and charm of its own; in a sense the journey becomes as enjoyable as the destination This conclusion is shared by me and I feel that a delightful aspect of plasma physics is the frequent transferability of ideas between extremely different applications so, for example, a concept developed in the context of astrophysics might suddenly become relevant to fusion research or vice versa Applications of plasma physics are many and varied Examples include controlled fusion research, ionospheric physics, magnetospheric physics, solar physics, astrophysics, plasma propulsion, semiconductor processing, and metals processing Because plasma physics is rich in both concepts and regimes, it has also often served as an incubator for new ideas in applied mathematics In recent years there has been an increased dialog regarding plasma physics among the various disciplines listed above and it is my hope that this text will help to promote this trend The prerequisites for this text are a reasonable familiarity with Maxwell’s equations, classical mechanics, vector algebra, vector calculus, differential equations, and complex variables – i.e., the contents of a typical undergraduate physics or engineering curriculum Experience has shown that because of the many different applications for plasma physics, students studying plasma physics have a diversity of preparation and not all are proficient in all prerequisites Brief derivations of many basic concepts are included to accommodate this range of preparation; these derivations are intended to assist those students who may have had little or no exposure to the concept in question and to refresh the memory of other students For example, rather than just invoke Hamilton-Lagrange methods or Laplace transforms, there is a quick derivation and then a considerable discussion showing how these concepts relate to plasma physics issues These additional explanations make the book more self-contained and also provide a close contact with first principles The order of presentation and level of rigor have been chosen to establish a firm foundation and yet avoid unnecessary mathematical formalism or abstraction In particular, the various fluid equations are derived from first principles rather than simply invoked and the consequences of the Hamiltonian nature of particle motion are emphasized early on and shown to lead to the powerful concepts of symmetry-induced constraint and adiabatic invariance Symmetry turns out to be an essential feature of magnetohydrodynamic plasma confinement and adiabatic invariance turns out to be not only essential for understanding many types of particle motion, but also vital to many aspects of wave behavior The mathematical derivations have been presented with intermediate steps shown in as much detail as is reasonably possible This occasionally leads to daunting-looking expressions, but it is my belief that it is preferable to see all the details rather than have them glossed over and then justified by an “it can be shown" statement xi xii Preface The book is organized as follows: Chapters 1-3 lay out the foundation of the subject Chapter provides a brief introduction and overview of applications, discusses the logical framework of plasma physics, and begins the presentation by discussing Debye shielding and then showing that plasmas are quasi-neutral and nearly collisionless Chapter introduces phase-space concepts and derives the Vlasov equation and then, by taking moments of the Vlasov equation, derives the two-fluid and magnetohydrodynamic systems of equations Chapter also introduces the dichotomy between adiabatic and isothermal behavior which is a fundamental and recurrent theme in plasma physics Chapter considers plasmas from the point of view of the behavior of a single particle and develops both exact and approximate descriptions for particle motion In particular, Chapter includes a detailed discussion of the concept of adiabatic invariance with the aim of demonstrating that this important concept is a fundamental property of all nearly periodic Hamiltonian systems and so does not have to be explained anew each time it is encountered in a different situation Chapter also includes a discussion of particle motion in fixed frequency oscillatory fields; this discussion provides a foundation for later analysis of cold plasma waves and wave-particle energy transfer in warm plasma waves Chapters 4-8 discuss plasma waves; these are not only important in many practical situations, but also provide an excellent way for developing insight about plasma dynamics Chapter shows how linear wave dispersion relations can be deduced from systems of partial differential equations characterizing a physical system and then presents derivations for the elementary plasma waves, namely Langmuir waves, electromagnetic plasma waves, ion acoustic waves, and Alfvén waves The beginning of Chapter shows that when a plasma contains groups of particles streaming at different velocities, free energy exists which can drive an instability; the remainder of Chapter then presents Landau damping and instability theory which reveals that surprisingly strong interactions between waves and particles can lead to either wave damping or wave instability depending on the shape of the velocity distribution of the particles Chapter describes cold plasma waves in a background magnetic field and discusses the Clemmow-Mullaly-Allis diagram, an elegant categorization scheme for the large number of qualitatively different types of cold plasma waves that exist in a magnetized plasma Chapter discusses certain additional subtle and practical aspects of wave propagation including propagation in an inhomogeneous plasma and how the energy content of a wave is related to its dispersion relation Chapter begins by showing that the combination of warm plasma effects and a background magnetic field leads to the existence of the Bernstein wave, an altogether different kind of wave which has an infinite number of branches, and shows how a cold plasma wave can ‘mode convert’ into a Bernstein wave in an inhomogeneous plasma Chapter concludes with a discussion of drift waves, ubiquitous low frequency waves which have important deleterious consequences for magnetic confinement Chapters 9-12 provide a description of plasmas from the magnetohydrodynamic point of view Chapter begins by presenting several basic magnetohydrodynamic concepts (vacuum and force-free fields, magnetic pressure and tension, frozen-in flux, and energy minimization) and then uses these concepts to develop an intuitive understanding for dynamic behavior Chapter then discusses magnetohydrodynamic equilibria and derives the Grad-Shafranov equation, an equation which depends on the existence of symmetry and which characterizes three-dimensional magnetohydrodynamic equilibria Chapter ends Preface xiii with a discussion on magnetohydrodynamic flows such as occur in arcs and jets Chapter 10 examines the stability of perfectly conducting (i.e., ideal) magnetohydrodynamic equilibria, derives the ‘energy principle’ method for analyzing stability, discusses kink and sausage instabilities, and introduces the concepts of magnetic helicity and force-free equilibria Chapter 11 examines magnetic helicity from a topological point of view and shows how helicity conservation and energy minimization leads to the Woltjer-Taylor model for magnetohydrodynamic self-organization Chapter 12 departs from the ideal models presented earlier and discusses magnetic reconnection, a non-ideal behavior which permits the magnetohydrodynamic plasma to alter its topology and thereby relax to a minimumenergy state Chapters 13-17 consist of various advanced topics Chapter 13 considers collisions from a Fokker-Planck point of view and is essentially a revisiting of the issues in Chapter using a more sophisticated point of view; the Fokker-Planck model is used to derive a more accurate model for plasma electrical resistivity and also to show the failure of Ohm’s law when the electric field exceeds a critical value called the Dreicer limit Chapter 14 considers two manifestations of wave-particle nonlinearity: (i) quasi-linear velocity space diffusion due to weak turbulence and (ii) echoes, non-linear phenomena which validate the concepts underlying Landau damping Chapter 15 discusses how nonlinear interactions enable energy and momentum to be transferred between waves, categorizes the large number of such wave-wave nonlinear interactions, and shows how these various interactions are all based on a few fundamental concepts Chapter 16 discusses one-component plasmas (pure electron or pure ion plasmas) and shows how these plasmas have behaviors differing from conventional two-component, electron-ion plasmas Chapter 17 discusses dusty plasmas which are three component plasmas (electrons, ions, and dust grains) and shows how the addition of a third component also introduces new behaviors, including the possibility of the dusty plasma condensing into a crystal The analysis of condensation involves revisiting the Debye shielding concept and so corresponds, in a sense to having the book end on the same note it started on I would like to extend my grateful appreciation to Professor Michael Brown at Swarthmore College for providing helpful feedback obtained from using a draft version in a seminar course at Swarthmore and to Professor Roy Gould at Caltech for providing useful suggestions I would also like to thank graduate students Deepak Kumar and Gunsu Yun for carefully scrutinizing the final drafts of the manuscript and pointing out both ambiguities in presentation and typographical errors I would also like to thank the many students who, over the years, provided useful feedback on earlier drafts of this work when it was in the form of lecture notes Finally, I would like to acknowledge and thank my own mentors and colleagues who have introduced me to the many fascinating ideas constituting the discipline of plasma physics and also the many scientists whose hard work over many decades has led to the development of this discipline Paul M Bellan Pasadena, California September 30, 2004 522 Appendix B used to construct the full Laplacian The first calculation gives ∇2 (Vr r) = ˆ = = = ∇ · ∇ (Vr r) ˆ ∇ · ((∇Vr ) r + Vr ∇ˆ) ˆ r r∇ Vr + (∇Vr ) · ∇ˆ + ∇ · (Vr ∇ˆ) ˆ r r 2 r∇ Vr + 2∇Vr · ∇ˆ + Vr ∇ r ˆ r ˆ ˆφ ˆ 2φ Vr ∂ = r ∇2 V r + ˆ ˆ · ∇Vr + 2 r r r ∂φ ˆ 2φ ∂Vr Vr = r ∇2 V r + ˆ − 2r ˆ r ∂φ r (B.33) while the second calculation gives ˆ = ∇ · ∇ Vφ φ ˆ ∇2 Vφ φ ˆ ˆ = ∇ · (∇Vφ ) φ + Vφ ∇φ ˆ ˆ ˆ = φ∇2 Vφ + 2∇Vφ · ∇φ + Vφ ∇2 φ 2ˆ ∂Vφ Vφ ˆ r ˆ − φ = φ∇2 Vφ − r ∂φ r (B.34) Since ∇2 (Vz z) = z∇2 Vz it is seen that the Laplacian of a vector in cylindrical coordiˆ ˆ nates is Vr ∂Vφ − r2 ∂φ r ∂Vr Vφ ˆ +φ ∇2 Vφ + − r ∂φ r ∇2 V = r ∇2 Vr − ˆ +ˆ∇2 Vz z (B.35) Equation (B.28) can also be used to calculate V·∇V giving V·∇V = Vr V ∂ ∂ ∂ + φ + Vz ∂r r ∂φ ∂z = r Vr ˆ ˆ V r r + Vφ φ + V z z ˆ ˆ Vφ ∂Vr ∂Vr Vφ ∂Vr + + Vz − ∂r r ∂φ ∂z r ∂Vφ Vφ ∂Vφ Vφ Vr ∂Vφ + + Vz + ∂r r ∂φ ∂z r ∂Vz ∂Vz Vφ ∂Vz +ˆ Vr z + + Vz ∂r r ∂φ ∂z ˆ +φ Vr (B.36) Appendix B 523 Application to spherical coordinates {x1, x2 , x3 } = {r, θ, φ}; {h1 , h2 , h3 } = {1, r, r sin θ} ˆ ∂ψ ˆ ∂ψ θ φ ∂ψ + + ∂r r ∂θ r sin θ ∂φ ∂ ∂ ∂ r Vr + ∇·V = (sin θ Vθ ) + (Vφ ) r2 ∂r r sin θ ∂θ r sin θ ∂φ ∂ ∂Vθ ∇×V = (Vφ sin θ) − r ˆ r sin θ ∂θ ∂φ ∂Vr ∂ + θ − (Vφ r sin θ) ˆ r sin θ ∂φ ∂r ∂ ∂Vr ˆ + (Vθ r) − φ r ∂r ∂φ ∂ ∂ψ ∂ ∂ψ ∂ 2ψ ∇2 ψ = r2 + sin θ + 2 r2 ∂r ∂r r sin θ ∂θ ∂θ r sin θ ∂φ2 ∇ψ = r ˆ (B.37) (B.38) (B.39) (B.40) Appendix C Frequently used physical constants and formulae Physical constants (to significant figures) electron mass proton mass vacuum permeability vacuum permittivity speed of light electron charge Avogadro’s number Boltzmann constant symbol me mp µ0 ε0 c e κ value 9.11 × 10−31 1.67 × 10−27 4π × 10−7 8.85 × 10−12 3.00 × 108 1.60 × 10−19 6.02 × 1023 1.60 × 10−19 unit kg kg N A−2 F m−1 m s−1 C mol−1 J eV−1 Formulae (all quantities in SI units, temperatures in eV, A is ion atomic mass number) Lengths Debye length (p.7) = λ2 D σ λ2 σ where σ is over all species participating in shielding and λσ = ε0 κTσ = 7.4 × 103 n0σ qσ Tσ m n0 σ Electron Larmor radius (p.234) rLe = Ion Larmor radius (p.234) rLi = √ κTe /me Te = × 10−6 m |ωce | B √ κTi /mi −4 ATi √ m = × 10 |ωci | B Z where A is atomic mass and Z is ion charge Electron collisionless skin depth (p.145) c × 106 = √ m ω pe ne 524 Appendix C 525 Ion collisionless skin depth (assuming quasi-neutral plasma so ni Z = ne c = × 108 ωpi A m Zne Frequencies Electron plasma frequency (p.126) fpe = ωpe = 2π 2π √ ne e2 = ne Hz ε0 me Ion plasma frequency (p.126) fpi = ωpi = 2π ni qi = 0.21 ε0 mi Zne Hz A Electron cyclotron frequency (p.79) eB |ω ce | = = × 1010 B Hz 2π 2πme Ion cyclotron frequency (p.79) fce = ZeB ZB |ωci | = = 55 × 107 Hz 2π 2πmi A Upper hybrid frequency (p.186) fci = fuh = 2 fpe + fce Lower hybrid frequency (p.186) flh = fci + fpi 1+ fpe fce Diocotron frequency (pure electron plasma, p.462) fdioc ≃ fpe /2fce Velocities Electron thermal velocity (p.154) vT e = Ion thermal velocity (p.154) vT i = 2κTe = × 105 me Te m s−1 2κTi = 1.4 × 104 mi Ti m s−1 A 526 Appendix C Ion acoustic velocity (p.(4.33)) κTe = 9.8 × 103 mi cs = Te m s−1 A Electron diamagnetic drift velocity (p.250) ud,e = Te κT ∇n = ∇n m s−1 eB n B n Ion diamagnetic drift velocity (flow in opposite direction from electrons, (p.250) Ti κTi ∇n = ∇n m s−1 qi B n ZB n ud,e = Alfvén velocity (p.132) B = µ0 ni mi Wave phase velocity (p.202) vA = √ Dimensionless Plasma beta (p.319) β= Lundquist number (p.381) B = 2 ì 1016 B à0 ne Amp /Z vph = Z m s−1 ne A ω = f k 026 ì 1025 2à0 nT = nT B2 B2 S= µ0 vA L η Collisions, resistivity, and runaways Electron-electron collision rate (p.21) ν ee = × 10−12 n ln Λ 3/2 TeV s−1 where ln Λ ∼ 10, electron-ion rate same magnitude, ion-ion slower by Spitzer resistivity (p.394) η = 1.03 × 10−4 Z ln Λ 3/2 Te Ohm-m me /mi Appendix C 527 Dreicer runaway electric field (p.395) EDreicer = × 10−18 neZ Typical neutral cross-section (p.16) ln Λ V/m Te σ neut ∼ × 10−20 m2 Warm plasma waves Electrostatic susceptibility (p.165) χσ = where α = ω/kvT σ k2 λ2 Dσ [1 + αZ(α)] Plasma dispersion function (p.165) Z(α) lim Z(α) α1 = π1/2 ∞ −∞ dξ exp(−ξ ) (ξ − α) 2α2 + + iπ1/2 exp(−α2 ) 3 = − + + + + iπ1/2 exp(−α2 ) α 2α 4α = −2α − Index arc, 290 astrophysical jet, 295 axisymmetry, 280 action integral in Lagrangian formalism, 63 in wave-wave coupling, 433 relation to quantum number, 433 action integral, 67 adiabatic invariance breakdown of, 95 pendulum, 66 adiabatic invariant general proof for Hamiltonian system, 70 J, second adiabatic invariant, 88 mu conservation, 81 relation to quantum number, 68 third adiabatic invariant, 89 adiabatic limit of energy equation, 40 Airy equation, 241 Alfven time, 370 Alfven velocity, 183 Alfven wave, 131 compressional (fast), 134, 136 in parameter space, 196 inertial Alfven wave, 187, 207 shear (slow), 137 two-fluid model, 138 compressional mode, 146 inertial Alfven wave, 144 kinetic Alfven wave, 145 alpha particle, 296 Altar-Appleton-Hartree dispersion relation, 197 ambipolar diffusion, 18 analytic continuation, 160 antenna, 193 Appleton-Hartree dispersion relation, 197 bad curvature, 305 ballistic term, 415 beam echo, 416 beat waves, 398 Bennett pinch, 275, 295 Bernstein waves, 236 Bessel function model, 358 Bessel relationships, 233 beta, 319 bilinear function, 311 Bohm-Gross wave, 129 Boltzmann relation difficulties with, 498 in Debye shielding, limitations of, 495 bounded volume, 190 bounding surface, 190 break-even fusion, 297 break-even, fusion, 296 Brillouin backscatter, 438 Brillouin flow, 460 Brillouin limit, 462 Bromwich contour, 156 Calugareanu theorem, 356 canonical angular momentum, 99 in non-neutral plasma, 464 toroidal confinement, 285 canonical momentum, 259 definition of, 64 cathode emission, 27 caviton, 214, 451 center of mass frame dynamics, 382 528 Child-Langmuir space charge limited emission, 29 CMA diagram, 188 coalescence mode, 240 coalescence, mode, 212 cold plasma wave energy equation, 221 collimation, 294 collision frequencies, 14 relations between cross-section, mean-free-path, 27 collisions and quasi-linear diffusion, 426 Coulomb, 11 Fokker-Planck model, 389 qualitative treatment in Vlasov equation, 34 with neutrals, 16 condensation of dusty plasma, 495 confinement time, 296 constant of the motion definition of, 64 convective derivative definition of, 38 coronal loop, 295 correspondence of drift equations with MHD, 91 coupled oscillator, 442 cross-section effective, for dust grain charging, 484 fusion, 297 neutral, 17 small-angle scattering, 13 crystallization, 495 crystallization criterion, dusty plasma, 501 current curvature, 93 diamagnetic, 91, 250 force-free in Grad-Shafranov equation, 281 grad B, 93 parallel, 268 polarization, 93 poloidal, 280 sheet, 362 sheet, in Sweet-Parker reconnection, 380 toroidal, 280 curvature good v bad, 305, 318 magnetic field, 271 curvature current, 93 curvature drift, 80 curved magnetic field, 79 cusp field, 103 cutoff, 131, 185, 190 cyclotron motion sense of rotation, 29 daughter wave, 429 Debye shielding, 129 derivation of, solution of pde for, 24 decay instability, 428, 438 delta W in MHD energy principle, 313 destructive interference, 202 deuterium, 296 diamagnetic, 282 current, 91 drift velocity, 250 dielectric constant, 95 dielectric tensor, 179 dielectric tensor elements, 181 right, left hand polarization, 183 diffusion ambipolar, 18 magnetic, 364 quasi-linear velocity space, 405 diocotron mode, 465 resistive wall, image charge method, 479 distribution function definition of, 31 for collisionless drift wave, 260 moments of, 33 529 double adiabatic laws derivation of, 49 Dreicer electric field, 395 drift equations, 73 curvature drift, 80 derivation of, 75 drift in arbitrary force field, 78 grad B drift, 80 grad B force, 78 polarization drift, 78 drift wave, 249 collisional, 253 collisionless, 258 destabilization of, 257, 262 features, 257 nonlinear pumping of plasma, 257 drifts curvature, 80 E x B, 73, 75 force, 75 generalization to arbitrary frequency, 110 grad B, 80 polarization, 78, 80 dumbell wave normal surface, 191 dust charge on grain, 486 charging of grains, 485 levitation, 505 dust acoustic waves, 491 dust charging, 504 dust ion acoustic waves, 494 dusty plasma, 483 crystalization criterion, 501 crystallization (condensation), 495 parameter space, 490 dynamic equilibria, 274, 286 in inhomogeneous plasma, 211 electromagnetic wave derivation of, 130 in decay instability, 442 electron plasma wave, 129 in decay instability, 440 electrostatic ion cyclotron wave, 263 ellipsoid wave normal surface, 191 energy inductive, 273 per helicity, 344 wave, 219 energy conservation in nonlinear waves, 399 energy equation adiabatic limit, 40 isothermal limit, 40 wave, 219 energy equation, two-fluid, 40 energy principle, 306 energy transfer between waves and particles, 114 entropy collisions, 43 conservation of, 413 of a distribution function, 41 equilibrium impossibility of spherically symmetric MHD, 295 stable v unstable, 298 extraordinary wave, 186 fast mode, 134, 192 Fermi acceleration, 88 first adiabatic invariant, 82 floating potential, 57 flow (MHD accelerated), 286 compressible, 291 incompressible, 286 stagnation of, 294 flux accumulation of toroidal flux in flow, 294 frozen-in, inductance, 272 poloidal, 97 private and public, 379 E cross B drift, 73, 80 echoes, 412 higher order, 426 spatial, 425 effective potential, 99 electric field runaway (Dreicer), 395 electromagnetic plasma wave 530 definition of, 76 relation to vector potential, 97 volume per unit, 334 flux linkage, 346 flux preservation, 272 flux surface, 96 in mirror, 91 flux tube Grad-Shafranov solution for, 295 Fokker-Planck theory, 382 force axial MHD force, 287 between parallel currents, 268 centrifugal, 79 flux-conserving, 273 hoop, 270 non-conservative, 286 pinch, 269 force-free magnetic field, 344 force-free equilibrium, 322, 335 Bessel function model, 358 Lundquist solution, 358 force-free magnetic field, 268 free-energy in sheared non-neutral plasma, 477 frequently used formulae, 524 frictional drag, 391 qualitative derivation of, 37 frozen-in flux proof of, 48 fusion break-even, 297 criteria for, 296 Hall term, 48 Hamilton-Lagrange formalism, 62 Hamiltonian, 64, 65 coupled harmonic oscillator, 430 geometric optics, 213 Helmholtz equation, 358 Hermitian part, 223 hollow profile, 474 hoop force, 270 ignitron, 29 ignorable coordinate, 64 image charge, line, 479 inductance, 272 induction equation, 49 in magnetic reconnection, 366 inhomogeneous plasmas, 210 initial condition Poisson’s equation as, 179 initial value, 158 instability caviton, 451 decay, 428 free-boundary, 323, 326 kink, 325, 331 parametric, 428 positron-electron streaming, 150 Rayleigh-Taylor, 299 resistive wall, 471 sausage, 323, 331 streaming, 149 ion acoustic velocity in sheath, 56 ion acoustic wave, 129 in decay instability, 439 Landau damping of, 170 Landau instability, 176 ion saturation current, 56 ion-ion resonance, 208 isothermal limit of energy equation, 40 Gaussian integrals, 59 geometric optics, 213 good curvature, 305 grad B current, 93 drift, 80 force, 78 Grad-Shafranov equation, 278 Green’s function in diocotron mode, 474 group velocity, 201 guiding center J, second adiabatic invariant, 87 531 magnetic field force-free, 268 minimum energy, 343 most general axisymmetric, 279 pressure, 268 reconnection in sheared field, 371 Solov’ev solution, 282 tension, 268 vacuum, 265 magnetic field curvature, 271 magnetic helicity, 320, 336, 345 association with current-driven instability, 319 conservation equation, 321, 357 conservation of, 341 global, 322 linkage, 336 linked ribbons, 358 twist, 338 magnetic mirror, 84 magnetic moments density of, 92 magnetic pressure, 268 magnetic stress tensor, 271 magnetic tension, 268 magnetization, 92 magnetofluid, definition of, 306 magnetoplasmadynamic thruster, 290 magnetron, 481 Manley-Rowe relations, 430 Maxwellian distribution definition of, 44 MHD equations derivation of, 46 Ohm’s law, derivation of, 47 MHD with neutrals, 60 minimum energy frozen-in flux, 272 minimum energy magnetic field, 343 moments of distribution function, 33 momentum conservation in nonlinear waves, 399 mu conservation, 81 mu, definition of, 82 jet astrophysical, 295 joining, in mode conversion, 249 kappa magnetic curvature, 271 Kelvin vorticity theorem, 463 kink, 323 kink instability, 325, 331 diamagnetic or paramagnetic?, 335 Korteweg-de Vries equation, 455 Kruskal-Shafranov stability criterion, 333 Lagrange multipliers, 58 Lagrange’s equation, 64 Lagrangian electromagnetic, 64 Landau damping, 169 and entropy, 413 Landau problem, 153 Langmuir probe, 56 Langmuir wave, 129 Laplace transform, 155 and ballistic term, 415 Laplace’s equation, 265 Larmor radius poloidal, 285 laser fusion, 458 Lawson criterion, 296 leap-frog numerical integration, 25 lightning bolt, 200 line-averaged density, 131 linear mode conversion, 241 linearization, general method, 123 loss-cone, 87 lower hybrid frequency, 186 Lundquist number, 381 Lundquist solution, 358 magnetic axis, 283 diffusion, 364 islands, 376 reconnection, 360 shear, 305 National Ignition Facility, 459 532 in Alfven wave, 132 relation to MHD equation of motion, 93 stochastic motion, 106 poles, 160 poloidal definition of, 279 poloidal flux, 279 poloidal Larmor radius, 285 poloidal magnetic field, 280 ponderomotive force, 436, 448 Poynting flux, 220, 309 Poynting theorem, 357 pressure magnetic, 268 pressure tensor, definition of, 37 principal resonances, 190 private flux, 379 product rule for oscillating physical quantities, 221 public flux, 379 pump MHD acceleration of fluid, 290 pump depletion, 457 pump wave, 429 pure electron plasma, 460 negative energy wave, 225 neutron, 296 nominal plasma parameters, 21 non-adiabatic motion, 95 non-linear dispersion relation, 444 non-neutral plasma, 460 relation to Landau damping, 476 non-resonant particles in quasilinear velocity space diffusion, 411 numerical integration of particle trajectory, 25 orbital motion limited model, 485, 504 ordinary wave, 186 orthogonal curvilinear coordinates, 518 overdense, 197 parallel current, 268 paramagnetic, 282 parameter space, 195 parametric decay instability, 428, 458 Parker reconnection model, 378 Penrose criterion, 172 phase integral, 232 phase mixing, 202 and echoes, 416 phase space, 30 physical constants, 524 pinch Bennett, 295 pinch force, 269 plasma dispersion function definition of, 165 large argument limit of, 167 small argument limit of, 167 plasma frequency definition of, 126 plateau velocity distribution, 412 Plemelj formula, 171 polarization current, 93 relation to capacitance, 95 polarization drift, 78, 80 change in particle energy, 120 QL modes, 199 QT modes, 199 quantum mechanics, 323 quantum mechanics, correspondence to, 214 quasi-linear velocity space diffusion, 399 quasi-neutrality, quasilinear diffusion and collisions, 426 quiver velocity, 436 radiation pressure, 436 radius of curvature, 79 Raman backscatter, 438 random velocity, 37 reduced mass, 383 redundancy of Poisson’s equation, 178 533 refractive index definition of, 181 relaxation, 341 relaxed state, 344 residues, 160 resistive mode, 360 resistive time, 371 resistive wall instability, 471, 479 resistivity Fokker-Planck model, 393 simple derivation of, 17 resonance, 185, 190 ion cyclotron, 193 ion-ion, 208 resonance cones, 204 resonant particles in quasilinear velocity space diffusion, 407, 409 reversal of magnetic field spatial (cusp), 103 temporal, 102 reversed field pinch, 285, 344 ribbon, gift-wrapping, 358 Richardson-Dushman temperature limited emission, 29 Rosenbluth potentials, 388, 395 runaway electric field, 395, 397 Rutherford scattering derivation, 22 sheath physics, 53 sheet current, 362 short wave radio transmission, 228 slow mode, 192 slow wave, 133 Snell’s law, 211 solar corona loop, 295, 381 soliton interaction, 456 ion acoustic wave, 454 propagating envelope, 453 stationary envelope, 452 Solov’ev solution, 282 orbits in, 295 space-charge-limited current, 27 spheromak, 285, 344 stagnation of flow, 294 static equilibria, 274 steepest-descent contour, 242 stellarator, 285 stimulated Raman scattering, 458 Stirling’s formula, 58 stochastic motion, 106 streaming instability, 149 stress tensor magnetic, 271 strongly-coupled plasma, 495 surface wave, 214 susceptibility definition of, 126 Sweet-Parker reconnection, 378 symmetry assumption in MHD jet, 286 canonical angular momentum & toroidal confinement, 285 in Grad-Shafranov equation, 280 in Lagrangian formalism, 64 in solution of Grad-Shafranov equation, 280 saddle point, 243 safety factor, 333 tokamak, 375 sausage instability, 323, 331 scale factor, 518 scattering energy transfer, 23 large angle, 12 Rutherford, 11, 22 small angle, 13 second adiabatic invariant, 87 self-adjointness, 310 self-confinement impossibility of, 276 self-focusing, 438 separatrix, 284 taxonomy of cold plasma waves, 188 Taylor relaxation, 341 TE mode, 216 tearing, 364 temperature limited current, 27 534 magnetic reconnection, 364 vorticity, 286 source for, 289 tension magnetic, 268 thermal force, 59 third adiabatic invariant, 89 thruster, 290 TM mode, 216 tokamak, 285, 333 topology minimum energy, 273 torque, 286 trapped particle in mirror, 87 tritium, 296 twist and writhe, 351, 356 two-fluid equations derivation of, 36 wall stabilization, 326 warm plasma dispersion relation (electrostatic), 234 water beading, 362 wave Alfven, 131 beating, 398 Bernstein, 236 definition of cold plasma wave, 178 drift, 249 dust acoustic, 491 dust Alfven, 504 dust ion acoustic, 494 dust whistler, 505 electron-plasma, 129 electrostatic ion cyclotron, 263 energy equation, 219, 221 energy, finite temperature wave, 224 extraordinary, 186 frequency dispersion, 201 ion acoustic, 129 magnetized cold plasma dispersion, 181 negative energy, 225 ordinary, 186 quasi-electrostatic cold plasma, 203 surface, 214 whistler, 200 wave energy in diocotron mode, 468 wave normal surface, 188, 191 wave-wave nonlinearity, 428 waveguide, plasma, 214 wheel wave normal surface, 191 whistler wave, 200, 207 Wigner-Seitz radius, 487 WKB approximation, 66 WKB criterion, 212 WKB mode underdense, 197 unperturbed orbit, 232 untrapped particle in mirror, 87 upper hybrid frequency, 186 vacuum magnetic field, 265 vector calculus identities, 515 vector potential, 320 relation to flux, 97 velocity Alfven, 132, 183 bounce-averaged, 90 definitions of ‘average’ velocity, 110 diamagnetic drift, 250 group, 201 in sawtooth wave, 111 in small amplitude oscillatory field, 108 quiver, 436 virial theorem, 276 viscosity, 274, 286 Vlasov equation derivation of, 31 moments of, 34 treatment of collisions, 34 volume per unit flux, 334 vortices 535 correspondence to saddle point, 245 WKB solution, 211 WKB solutions of mode conversion problem, 249 Woltjer-Taylor relaxation, 341 work function, 29 writhe and twist, 351, 356 X-point, 379 Yukawa solution, 9, 24, 495, 498 536 ... xi Basic concepts 1.1 History of the term ? ?plasma? ?? 1.2 Brief history of plasma physics 1.3 Plasma parameters 1.4 Examples of plasmas 1.5 Logical framework of plasma physics 1.6 Debye shielding... versa Applications of plasma physics are many and varied Examples include controlled fusion research, ionospheric physics, magnetospheric physics, solar physics, astrophysics, plasma propulsion,... Most of the astrophysical plasmas that have been investigated have temperatures in the range of 1-100 eV and these plasmas are usually fully ionized 1.5 Logical framework of plasma physics Plasmas