Lecture notes Plasma physics

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Plasma Physics Richard Fitzpatrick Professor of Physics The University of Texas at Austin Contents 1 Introduction 5 1.1 Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 What is Plasma? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Brief History of PlasmaPhysics . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.4 Basic Parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.5 PlasmaFrequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.6 Debye Shielding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.7 PlasmaParameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.8 Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.9 Magnetized Plasmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.10 PlasmaBeta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2 Charged Particle Motion 17 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.2 Motion in Uniform Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.3 Method of Averaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.4 Guiding Centre Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.5 Magnetic Drifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.6 Invariance of Magnetic Moment . . . . . . . . . . . . . . . . . . . . . . . . 25 2.7 Poincare´Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.8 Adiabatic Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.9 Magnetic Mirrors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.10 Van Allen RadiationBelts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.11 Ring Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2.12 Second Adiabatic Invariant . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.13 Third Adiabatic Invariant . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.14 Motion in Oscillating Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3 Plasma Fluid Theory 43 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432 PLASMA PHYSICS 3.2 Moments of the Distribution Function . . . . . . . . . . . . . . . . . . . . . 45 3.3 Moments of the CollisionOperator . . . . . . . . . . . . . . . . . . . . . . . 47 3.4 Moments of the Kinetic Equation . . . . . . . . . . . . . . . . . . . . . . . . 49 3.5 Fluid Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.6 Entropy Production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.7 Fluid Closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.8 BraginskiiEquations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.9 Normalizationof the BraginskiiEquations. . . . . . . . . . . . . . . . . . . 67 3.10 Cold-PlasmaEquations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 3.11 MHD Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 3.12 Drift Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 3.13 Closure in Collisionless Magnetized Plasmas . . . . . . . . . . . . . . . . . 78 3.14 Langmuir Sheaths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 4 Waves in Cold Plasmas 89 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 4.2 Plane Waves in a HomogeneousPlasma . . . . . . . . . . . . . . . . . . . . 89 4.3 Cold-PlasmaDielectric Permittivity . . . . . . . . . . . . . . . . . . . . . . 91 4.4 Cold-PlasmaDispersion Relation . . . . . . . . . . . . . . . . . . . . . . . . 93 4.5 Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 4.6 Cutoff and Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 4.7 Waves in an UnmagnetizedPlasma . . . . . . . . . . . . . . . . . . . . . . 96 4.8 Low-Frequency Wave Propagation . . . . . . . . . . . . . . . . . . . . . . . 97 4.9 Parallel Wave Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 4.10 Perpendicular Wave Propagation . . . . . . . . . . . . . . . . . . . . . . . . 106 4.11 Wave PropagationThrough InhomogeneousPlasmas . . . . . . . . . . . . 107 4.12 Cutoffs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 4.13 Resonances. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 4.14 Resonant Layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 4.15 CollisionalDamping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 4.16 Pulse Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 4.17 Ray Tracing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 4.18 Radio Wave PropagationThrough the Ionosphere . . . . . . . . . . . . . . 124 5 Magnetohydrodynamic Fluids 127 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 5.2 Magnetic Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 5.3 Flux Freezing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 5.4 MHD Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 5.5 The Solar Wind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 5.6 Parker Model of Solar Wind . . . . . . . . . . . . . . . . . . . . . . . . . . 136 5.7 Interplanetary Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . 140 5.8 Mass and Angular Momentum Loss . . . . . . . . . . . . . . . . . . . . . . 144CONTENTS 3 5.9 MHD Dynamo Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 5.10 HomopolarGenerators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 5.11 Slow and Fast Dynamos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 5.12 Cowling Anti-DynamoTheorem . . . . . . . . . . . . . . . . . . . . . . . . 154 5.13 PonomarenkoDynamos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 5.14 Magnetic Reconnection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 5.15 Linear Tearing Mode Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 162 5.16 NonlinearTearing Mode Theory . . . . . . . . . . . . . . . . . . . . . . . . 169 5.17 Fast Magnetic Reconnection . . . . . . . . . . . . . . . . . . . . . . . . . . 171 5.18 MHD Shocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 5.19 Parallel Shocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 5.20 Perpendicular Shocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 5.21 Oblique Shocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 6 Waves in Warm Plasmas 187 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 6.2 LandauDamping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 6.3 Physics of Landau Damping . . . . . . . . . . . . . . . . . . . . . . . . . . 194 6.4 PlasmaDispersion Function . . . . . . . . . . . . . . . . . . . . . . . . . . 197 6.5 Ion Sound Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 6.6 Waves in Magnetized Plasmas . . . . . . . . . . . . . . . . . . . . . . . . . 200 6.7 Parallel Wave Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 6.8 Perpendicular Wave Propagation . . . . . . . . . . . . . . . . . . . . . . . . 2064 PLASMA PHYSICSIntroduction 5 1 Introduction 1.1 Sources The major sources for this course are: The Theory of Plasma Waves: T.H.Stix, 1st Ed. (McGraw-Hill,New York NY, 1962). Plasma Physics: R.A. Cairns (Blackie,Glasgow UK, 1985). The Framework of Plasma Physics: R.D.Hazeltine,andF.L.Waelbroeck(Westview,Boulder CO, 2004). Other sources include: The Mathematical Theory of Non-Uniform Gases: S.Chapman,andT.G.Cowling(Cambridge University Press,Cambridge UK, 1953). Physics of Fully Ionized Gases: L. Spitzer, Jr., 1st Ed. (Interscience, New York NY, 1956). Radio Waves in the Ionosphere: K.G. Budden (Cambridge University Press, Cambridge UK, 1961). The Adiabatic Motion of Charged Particles: T.G.Northrop(Interscience,NewYorkNY,1963). Coronal Expansion and the Solar Wind: A.J. Hundhausen(Springer-Verlag, Berlin, 1972). Solar System Magnetic Fields: E.R.Priest,Ed.(D.ReidelPublishingCo.,Dordrecht,Nether- lands, 1985). Lectures on Solar and Planetary Dynamos: M.R.E.Proctor,andA.D.Gilbert,Eds.(Cambridge University Press,Cambridge UK, 1994). Introduction to Plasma Physics: R.J.Goldston,andP.H.Rutherford(InstituteofPhysicsPub- lishing, Bristol UK, 1995). Basic Space Plasma Physics: W.Baumjohann,andR.A.Treumann(ImperialCollegePress, LondonUK, 1996). 1.2 What is Plasma? The electromagnetic force is generally observed to create structure: e.g., stable atoms and molecules, crystalline solids. In fact, the most widely studied consequences of the elec- tromagneticforce formthe subjectmatter ofChemistry andSolid-State Physics,which are both disciplines developed to understand essentially static structures.6 PLASMA PHYSICS Structured systems have binding energies larger than the ambient thermal energy. Placed in a sufficiently hot environment, they decompose: e.g., crystals melt, molecules disassociate. At temperatures near or exceeding atomic ionization energies, atoms sim- ilarly decompose into negatively charged electrons and positively charged ions. These charged particles are by no means free: in fact, they are strongly affected by each others’ electromagnetic fields. Nevertheless, because the charges are no longer bound, their as- semblage becomes capable of collective motions of great vigor and complexity. Such an assemblageis termed a plasma. Of course, bound systems can display extreme complexity of structure: e.g., a protein molecule. Complexity in a plasma is somewhat different, being expressed temporally as much as spatially. It is predominately characterized by the excitation of an enormous variety of collective dynamical modes. Sincethermaldecompositionbreaksinteratomicbondsbeforeionizing,mostterrestrial plasmas begin as gases. In fact, a plasma is sometimes defined as a gas that is sufficiently ionized to exhibit plasma-like behaviour. Note that plasma-like behaviour ensues after a remarkably small fraction of the gas has undergone ionization. Thus, fractionally ionized gases exhibit most of the exotic phenomenacharacteristic of fully ionized gases. Plasmas resulting from ionization of neutral gases generally contain equal numbers of positive and negative charge carriers. In this situation, the oppositely charged flu- ids are strongly coupled, and tend to electrically neutralize one another on macroscopic length-scales. Suchplasmasaretermedquasi-neutral(“quasi”becausethesmalldeviations from exact neutrality have important dynamical consequences for certain types of plasma mode). Strongly non-neutral plasmas, which may even contain charges of only one sign, occur primarily in laboratory experiments: their equilibrium depends on the existence of intense magnetic fields, aboutwhich the charged fluid rotates. It is sometimes remarked that 95% (or 99%, depending on whom you are trying to impress) of the baryonic content of the Universe consists of plasma. This statement has the double merit of being extremely flattering to Plasma Physics, and quite impossible to disprove (or verify). Nevertheless, it is worth pointing out the prevalence of the plasma state. In earlier epochs of the Universe, everything was plasma. In the present epoch, stars, nebulae, and even interstellar space, are filled with plasma. The Solar System is also permeated with plasma, in the form of the solar wind, and the Earth is completely surrounded by plasmatrapped within its magnetic field. Terrestrialplasmasarealsonothardtofind. Theyoccurinlightning,fluorescentlamps, avarietyoflaboratoryexperiments,andagrowingarrayofindustrialprocesses. Infact,the glow discharge has recently become the mainstay of the micro-circuit fabricationindustry. Liquidandevensolid-statesystemscanoccasionallydisplaythecollective electromagnetic effectsthatcharacterizeplasma: e.g.,liquidmercuryexhibitsmanydynamicalmodes,such as Alfve´n waves, which occur in conventional plasmas.Introduction 7 1.3 Brief History of Plasma Physics When blood is cleared of its various corpuscles there remains a transparent liquid, which was named plasma (after the Greek wordπλασμα, which means “moldable substance” or “jelly”) by the great Czech medical scientist, Johannes Purkinje (1787-1869). The Nobel prize winning American chemist Irving Langmuir first used this term to describe an ion- ized gasin 1927—Langmuir wasreminded of the way bloodplasmacarries red and white corpuscles by the way an electrified fluid carries electrons and ions. Langmuir,along with hiscolleagueLewiTonks,wasinvestigatingthephysicsandchemistryoftungsten-filament light-bulbs, with a view to finding a way to greatly extend the lifetime of the filament (a goal which he eventually achieved). In the process, he developed the theory of plasma sheaths—the boundarylayers which form betweenionized plasmasand solid surfaces. He also discovered that certain regions of a plasma discharge tube exhibit periodic variations of the electron density, which we nowadaysterm Langmuir waves. This was the genesis of PlasmaPhysics. Interestinglyenough,Langmuir’sresearchnowadaysformsthetheoretical basis of most plasma processing techniques for fabricating integrated circuits. After Lang- muir, plasma research gradually spread in other directions, of which five are particularly significant. Firstly, the development of radio broadcasting led to the discovery of the Earth’s iono- sphere,alayerofpartiallyionizedgasintheupperatmospherewhichreflectsradiowaves, and is responsible for the fact that radio signals can be received when the transmitter is over the horizon. Unfortunately, the ionosphere also occasionally absorbs and distorts radio waves. For instance,the Earth’s magnetic field causes waves with different polariza- tions(relative to the orientationofthe magneticfield) to propagateat different velocities, an effect which can give rise to “ghost signals” (i.e., signals which arrive a little before, or a little after, the main signal). In order to understand, and possibly correct, some of the deficiencies in radio communication, various scientists, such as E.V. Appleton and K.G. Budden, systematically developed the theory of electromagnetic wave propagation through non-uniformmagnetized plasmas. Secondly,astrophysicistsquicklyrecognizedthatmuchoftheUniverseconsistsofplasma, and, thus, that a better understandingof astrophysical phenomenarequires a better grasp of plasma physics. The pioneer in this field was Hannes Alfv´en, who around 1940 devel- oped the theory of magnetohydrodyamics, or MHD, in which plasma is treated essentially as a conducting fluid. This theory has been both widely and successfully employed to in- vestigate sunspots, solar flares, the solar wind, star formation, and a host of other topics in astrophysics. Two topics of particular interest in MHD theory are magnetic reconnection and dynamo theory. Magnetic reconnectionis a processby which magnetic field-linessud- denly change their topology: it can give rise to the sudden conversion of a great deal of magneticenergyintothermalenergy,aswellastheaccelerationofsomechargedparticles to extremely high energies, and is generally thought to be the basic mechanism behind solar flares. Dynamo theory studies how the motion of an MHD fluid can give rise to the generation of a macroscopic magnetic field. This process is important because both the terrestrial and solar magnetic fields would decay away comparatively rapidly (in astro-8 PLASMA PHYSICS physical terms) were they not maintainedby dynamo action. The Earth’s magnetic field is maintained by the motion of its molten core, which can be treated as an MHD fluid to a reasonableapproximation. Thirdly, the creation of the hydrogen bomb in 1952 generated a great deal of inter- est in controlled thermonuclear fusion as a possible power source for the future. At first, this research was carried out secretly, and independently, by the United States, the Soviet Union, and Great Britain. However, in 1958 thermonuclear fusion research was declassi- fied,leadingtothepublicationofanumberofimmenselyimportantandinfluentialpapers in the late 1950’s and the early 1960’s. Broadly speaking, theoretical plasma physics first emerged as a mathematically rigorous discipline in these years. Not surprisingly, Fusion physicists are mostly concerned with understanding how a thermonuclear plasma can be trapped—in most cases by a magnetic field—and investigating the many plasma instabili- ties which may allow it to escape. Fourthly, James A. Van Allen’s discovery in 1958 of the Van Allen radiation belts sur- roundingthe Earth, using data transmitted by the U.S. Explorer satellite, marked the start ofthesystematicexplorationoftheEarth’smagnetosphereviasatellite,andopenedupthe field of space plasma physics. Spacescientists borrowedthe theory ofplasmatrappingbya magneticfield fromfusionresearch,the theory ofplasmawavesfromionosphericphysics, and the notion of magnetic reconnection as a mechanism for energy release and particle acceleration from astrophysics. Finally, the development of high powered lasers in the 1960’s opened up the field of laserplasma physics. Whenahighpoweredlaserbeamstrikesasolidtarget,materialisim- mediately ablated, and a plasma forms at the boundary between the beam and the target. Laserplasmastendtohavefairlyextremeproperties(e.g.,densitiescharacteristicofsolids) notfoundinmoreconventionalplasmas. Amajorapplicationoflaserplasmaphysicsisthe approach to fusion energy known as inertial confinement fusion. In this approach, tightly focusedlaserbeamsareusedtoimplodeasmallsolidtargetuntilthedensitiesandtemper- atures characteristic of nuclear fusion (i.e., the centre of a hydrogen bomb) are achieved. Another interesting application of laser plasma physics is the use of the extremely strong electric fields generated when a high intensity laser pulse passes through a plasma to ac- celerate particles. High-energy physicists hope to use plasma acceleration techniques to dramatically reduce the size and cost of particle accelerators. 1.4 Basic Parameters Consider an idealized plasma consisting of an equal number of electrons, with massm e andcharge −e(here,edenotesthemagnitudeoftheelectroncharge),andions,withmass m and charge +e. We do not necessarily demand that the system has attained thermal i equilibrium,but nevertheless use the symbol 1 2 T ≡ m hv i (1.1) s s s 3Introduction 9 to denotea kinetic temperature measuredinenergy units(i.e.,joules). Here,v is aparticle speed, and the angular brackets denote an ensemble average. The kinetic temperature of speciess is essentially the average kinetic energy of particles of this species. In plasma physics, kinetic temperature is invariably measured in electron-volts (1 joule is equivalent 18 to6.24×10 eV). Quasi-neutralitydemands that n ≃n ≡n, (1.2) i e wheren is thenumberdensity(i.e.,thenumberofparticlesper cubicmeter) ofspeciess. s Assuming that both ions and electrons are characterized by the sameT (which is, by no means, always the case in plasmas), we can estimate typical particle speeds via the so-called thermal speed, q v ≡ 2T/m. (1.3) ts s Note that the ion thermal speed is usually far smaller than the electron thermal speed: q v ∼ m/m v . (1.4) ti e i te Of course,n andT are generally functions of position in a plasma. 1.5 Plasma Frequency The plasma frequency, 2 ne 2 ω = , (1.5) p ǫ m 0 is the most fundamental time-scale in plasma physics. Clearly, there is a different plasma frequency for each species. However, the relatively fast electron frequency is, by far, the most important, and references to “the plasma frequency” in text-books invariably mean the electron plasma frequency. It is easily seen thatω corresponds to the typical electrostatic oscillation frequency p of a given species in response to a small charge separation. For instance, consider a one- dimensionalsituationinwhich aslabconsistingentirelyofonecharge speciesisdisplaced fromitsquasi-neutralpositionbyaninfinitesimaldistanceδx. Theresultingchargedensity which develops on the leading face of the slab is σ = enδx. An equal and opposite chargedensitydevelopsontheoppositeface. Thex-directedelectricfieldgeneratedinside the slab is of magnitude E = −σ/ǫ = −enδx/ǫ . Thus, Newton’s law applied to an x 0 0 individual particle inside the slab yields 2 dδx 2 m =eE = −mω δx, (1.6) x p 2 dt givingδx = (δx) cos(ω t). 0 p Notethatplasmaoscillationswillonlybeobservediftheplasmasystemisstudiedover timeperiodsτlongerthantheplasmaperiodτ ≡1/ω ,andifexternalactionschangethe p p10 PLASMA PHYSICS system at a rate no faster thanω . In the opposite case, one is clearly studying something p other than plasma physics (e.g., nuclear reactions), and the system cannot not usefully be considered to be a plasma. Likewise, observations over length-scalesL shorter than the distancev τ traveled by a typical plasma particle during a plasma period will also not t p detect plasma behaviour. In this case, particles will exit the system before completing a plasma oscillation. This distance, which is the spatial equivalent toτ , is called the Debye p length, and takes the form q −1 λ ≡ T/mω . (1.7) D p Note that s ǫ T 0 λ = (1.8) D 2 ne is independent of mass,and therefore generally comparable for different species. Clearly, our idealized system can only usefully be considered to be a plasma provided that λ D ≪1, (1.9) L and τ p ≪1. (1.10) τ Here,τ andL represent the typical time-scale and length-scale of the process under inves- tigation. It should be noted that, despite the conventional requirement (1.9), plasma physics is capable of considering structures on the Debye scale. The most important example of this is the Debye sheath: i.e., the boundary layer which surrounds a plasma confined by a material surface. 1.6 DebyeShielding Plasmas generally do not contain strong electric fields in their rest frames. The shielding of an external electric field from the interior of a plasma can be viewed as a result of high plasmaconductivity: i.e.,plasmacurrentgenerallyflowsfreelyenoughtoshortoutinterior electric fields. However, it is more useful to consider the shielding as a dielectric phenom- ena: i.e., it is the polarization of the plasma medium, and the associated redistribution of space charge, which prevents penetration by an external electric field. Not surprisingly, the length-scale associatedwith such shielding is the Debye length. Let us consider the simplest possible example. Suppose that a quasi-neutral plasma is sufficientlyclosetothermalequilibriumthatitsparticledensitiesaredistributedaccording to the Maxwell-Boltzmannlaw, −e Φ/T s n =n e , (1.11) s 0 whereΦ(r) is the electrostatic potential,andn andT are constant. Frome = −e =e, it 0 i e is clear that quasi-neutrality requires the equilibrium potential to be a constant. SupposeIntroduction 11 thatthisequilibriumpotentialisperturbed,byanamountδΦ,byasmall,localizedcharge densityδρ . The total perturbed charge density is written ext 2 δρ =δρ +e(δn −δn ) =δρ −2e n δΦ/T. (1.12) ext i e ext 0 Thus, Poisson’sequationyields 2 δρ δρ −2e n δΦ/T ext 0 2 ∇δΦ = − = − , (1.13) ǫ ǫ 0 0 which reduces to 2 δρ ext 2 ∇ − δΦ = − . (1.14) 2 λ ǫ 0 D If theperturbingcharge densityactually consistsofapointchargeq,located atthe origin, so thatδρ =qδ(r),then the solution to the above equationis written ext √ q − 2r/λ D δΦ(r) = e . (1.15) 4πǫ r 0 Clearly, the Coulomb potential of the perturbing point charge q is shielded on distance scales longer than the Debye length by a shielding cloud of approximate radiusλ consist- D ing of charge of the opposite sign. Note that the above argument, by treatingn as a continuous function, implicitly as- sumes that there are many particles in the shielding cloud. Actually, Debye shielding remains statistically significant, and physical, in the opposite limit in which the cloud is barely populated. In the latter case, it is the probability of observing charged particles within a Debye length of the perturbing charge which is modified. 1.7 Plasma Parameter Let us define the average distance between particles, −1/3 r ≡n , (1.16) d and the distance of closest approach, 2 e r ≡ . (1.17) c 4πǫ T 0 Recall thatr is the distance at which the Coulomb energy c 2 1 e 2 U(r,v) = mv − (1.18) 2 4πǫ r 0 of one charged particle in the electrostatic field of another vanishes. Thus,U(r,v ) =0. c t12 PLASMA PHYSICS The significance of the ratio r /r is readily understood. When this ratio is small, d c charged particles are dominated by one another’s electrostatic influence more or less con- tinuously, and their kinetic energies are small compared to the interaction potential ener- gies. Suchplasmasaretermedstronglycoupled. Ontheotherhand,whentheratioislarge, strong electrostatic interactions between individual particles are occasional and relatively rare events. A typical particle is electrostatically influenced by all of the other particles within its Debye sphere, but this interaction very rarely causes any sudden change in its motion. Such plasmas are termed weakly coupled. It is possible to describe a weakly cou- pled plasma usinga standardFokker-Planckequation(i.e.,the sametype of equationasis conventionally used to describe a neutral gas). Understanding the strongly coupled limit is far more difficult, and will not be attempted in this course. Actually, a strongly coupled plasma has more in common with a liquid than a conventional weaklycoupled plasma. Let us define the plasma parameter 3 Λ =4πnλ . (1.19) D This dimensionless parameter is obviously equal to the typical number of particles con- tainedinaDebyesphere. However,Eqs.(1.8),(1.16),(1.17),and(1.19)canbecombined to give   3/2 3/2 3/2 λ 1 r 4πǫ T D d 0 √ Λ = = = . (1.20) 3 1/2 r r e n c 4π c ItcanbeseenthatthecaseΛ≪1,inwhichtheDebyesphereissparselypopulated,corre- spondstoastronglycoupledplasma. Likewise,thecaseΛ≫1,inwhichtheDebyesphere is densely populated, correspondsto a weakly coupled plasma. It can alsobe appreciated, fromEq.(1.20),thatstronglycoupledplasmastendtobecoldanddense,whereasweakly coupled plasmas are diffuse and hot. Examples of strongly coupled plasmas include solid- density laser ablation plasmas, the very “cold” (i.e., with kinetic temperatures similar to the ionization energy) plasmas found in “high pressure” arc discharges, and the plasmas which constitute the atmospheres of collapsed objects such as white dwarfs and neutron stars. On the other hand, the hot diffuse plasmas typically encountered in ionospheric physics, astrophysics, nuclear fusion,and space plasmaphysics are invariably weaklycou- pled. Table 1.1 lists the key parameters for some typical weakly coupled plasmas. Inconclusion,characteristiccollective plasmabehaviourisonlyobservedontime-scales longer than the plasma period, and on length-scales larger than the Debye length. The statistical character of this behaviouris controlled by the plasmaparameter. Althoughω , p λ ,andΛarethethreemostfundamentalplasmaparameters,thereareanumberofother D parameters which are worth mentioning. 1.8 Collisions Collisionsbetweenchargedparticlesinaplasmadifferfundamentallyfromthosebetween molecules in a neutral gas because of the long range of the Coulomb force. In fact, it isIntroduction 13 −3 −1 n(m ) T(eV) ω (sec ) λ (m) Λ p D 6 −2 4 6 Interstellar 10 10 6×10 0.7 4×10 18 10 −6 3 Solar Chromosphere 10 2 6×10 5×10 2×10 7 5 10 Solar Wind (1AU) 10 10 2×10 7 5×10 12 7 −3 5 Ionosphere 10 0.1 6×10 2×10 1×10 20 11 −7 2 Arc discharge 10 1 6×10 7×10 5×10 20 4 11 −5 8 Tokamak 10 10 6×10 7×10 4×10 28 4 15 −9 4 Inertial Confinement 10 10 6×10 7×10 5×10 Table 1.1: Key parameters for some typical weakly coupled plasmas. clearfromthediscussioninSect.1.7thatbinarycollisionprocessescanonlybedefinedfor weakly coupled plasmas. Note, however, that binary collisions in weakly coupled plasmas arestillmodifiedbycollectiveeffects—themany-particleprocessofDebyeshieldingenters in a crucial manner. Nevertheless, for large Λ we can speak of binary collisions, and therefore of a collision frequency, denoted byν ′. Here,ν ′ measures the rate at which ss ss ′ particles of speciess are scattered by those of speciess . When specifying only a single subscript, one is generally referring to the total collision rate for that species, including impacts with all other species. Very roughly, X ν ≃ ν ′. (1.21) s ss ′ s The species designations are generally important. For instance, the relatively small elec- tron mass implies that, for unit ionic charge and comparablespecies temperatures,   1/2 m i ν ∼ ν. (1.22) e i m e Note that the collision frequencyν measures the frequency with which a particle trajec- tory undergoes a major angular change due to Coulomb interactions with other particles. Coulomb collisions are, in fact, predominately small angle scattering events, so the colli- sion frequency is not the inverse of the typical time between collisions. Instead, it is the inverseofthetypicaltimeneededforenoughcollisionstooccurthattheparticletrajectory ◦ is deviated through90 . For this reason, the collision frequency is sometimes termed the ◦ “90 scattering rate.” It is conventionalto define the mean-free-path, λ ≡v/ν. (1.23) mfp t Clearly, the mean-free-path measures the typical distance a particle travels between “col- ◦ lisions” (i.e.,90 scattering events). A collision-dominated, or collisional, plasma is simply one in which λ ≪L, (1.24) mfp14 PLASMA PHYSICS whereL is the observation length-scale. The opposite limit of large mean-free-path is said to correspond to a collisionless plasma. Collisions greatly simplify plasma behaviour by driving the system towards statistical equilibrium, characterized by Maxwell-Boltzmann distribution functions. Furthermore, short mean-free-paths generally ensure that plasma transport is local (i.e., diffusive) in nature,which is a considerable simplification. The typical magnitude of the collision frequency is lnΛ ν ∼ ω . (1.25) p Λ Note thatν≪ω in a weakly coupled plasma. It follows that collisions do not seriously p interfere with plasma oscillations in such systems. On the other hand, Eq. (1.25) implies thatν ≫ ω in a strongly coupled plasma, suggesting that collisions effectively prevent p plasma oscillations in such systems. This accords well with our basic picture of a strongly coupled plasma as a system dominated by Coulomb interactions which does not exhibit conventionalplasma dynamics. It follows from Eqs. (1.5) and (1.20) that 4 e lnΛ n ν ∼ . (1.26) 2 1/2 3/2 4πǫ m T 0 Thus, diffuse, high temperature plasmas tend to be collisionless, whereas dense, low tem- perature plasmasare more likely to be collisional. Note that whilst collisions are crucial to the confinement and dynamics (e.g., sound waves) of neutral gases, they play a far less important role in plasmas. In fact, in many plasmas the magnetic field effectively plays the role that collisions play in a neutral gas. In such plasmas,charged particles are constrained from moving perpendicular to the field by their small Larmor orbits, rather than by collisions. Confinement along the field-lines is more difficult to achieve, unless the field-lines form closed loops (or closed surfaces). Thus, it makes sense to talk about a “collisionless plasma,” whereas it makes little sense to talk about a “collisionless neutral gas.” Note that many plasmas are collisionless to a very good approximation, especially those encountered in astrophysics and space plasma physics contexts. 1.9 Magnetized Plasmas A magnetized plasma is one in which the ambient magnetic field B is strong enough to significantly alter particle trajectories. In particular, magnetized plasmas are anisotropic, responding differently to forces which are parallel and perpendicular to the direction of B. Note that a magnetized plasma moving with mean velocity V contains an electric field E = −V×B which is not affected by Debye shielding. Of course, in the rest frame of the plasma the electric field is essentially zero. As is well-known,charged particles respond to the Lorentzforce, F =qv×B, (1.27)Introduction 15 by freely streaming in the direction of B, whilst executing circular Larmor orbits, or gyro- orbits,intheplaneperpendiculartoB. Asthefield-strengthincreases,theresultinghelical orbits become more tightly wound,effectively tying particles to magnetic field-lines. The typical Larmor radius, or gyroradius, of a charged particle gyrating in a magnetic field is given by v t ρ≡ , (1.28) Ω where Ω =eB/m (1.29) is the cyclotron frequency, or gyrofrequency, associated with the gyration. As usual, there is a distinct gyroradius for each species. When species temperatures are comparable, the electron gyroradius is distinctly smaller than the ion gyroradius:   1/2 m e ρ ∼ ρ. (1.30) e i m i A plasma system, or process, is said to be magnetized if its characteristic length-scaleL is large compared to the gyroradius. In the opposite limit,ρ≫L, charged particles have essentially straight-line trajectories. Thus, the ability of the magnetic field to significantly affect particle trajectories is measured by the magnetization parameter ρ δ≡ . (1.31) L Therearesomecasesofinterestinwhichtheelectronsaremagnetized,buttheionsare not. However, a “magnetized” plasma conventionally refers to one in which both species are magnetized. This state is generally achieved when ρ i δ ≡ ≪1. (1.32) i L 1.10 Plasma Beta The fundamental measure of a magnetic field’s effect on a plasma is the magnetization parameterδ. The fundamental measure of the inverse effect is calledβ, and is defined as 2 the ratio of the thermal energy densitynT to the magnetic energy densityB/2μ . It is 0 conventionalto identify the plasmaenergy density with the pressure, p≡nT, (1.33) as in an ideal gas, and to define a separateβ for each plasmaspecies. Thus, s 2μ p 0 s β = . (1.34) s 2 B The totalβ is written X β = β. (1.35) s s16 PLASMA PHYSICSCharged Particle Motion 17 2 Charged Particle Motion 2.1 Introduction All descriptions of plasma behaviour are based, ultimately, on the motions of the con- stituent particles. For the case of an unmagnetized plasma, the motions are fairly trivial, since the constituent particles move essentially in straight lines between collisions. The motions are also trivial in a magnetized plasma where the collision frequencyν greatly exceeds the gyrofrequencyΩ: in this case, the particles are scattered after executing only a small fraction of a gyro-orbit, and, therefore, still move essentially in straight lines be- tween collisions. The situation of primary interest in this section is that of a collisionless (i.e.,ν≪Ω), magnetized plasma,wherethe gyroradiusρismuch smallerthanthe typical −1 variation length-scaleL of the E and B fields, and the gyroperiodΩ is much less than the typical time-scale τ on which these fields change. In such a plasma, we expect the motionoftheconstituentparticlestoconsistofarapidgyrationperpendiculartomagnetic field-lines, combined with free-streamingparallel to the field-lines. We are particularly in- terested incalculating howthis motionis affected bythe spatialandtemporal gradients in the E and B fields. In general, the motion of charged particles in spatially and temporally non-uniformelectromagneticfieldsisextremelycomplicated: however,wehopetoconsid- erablysimplifythismotionbyexploitingtheassumedsmallnessoftheparametersρ/Land −1 (Ωτ) . What we are really trying to understand,in this section, is how the magnetic con- finement of an essentially collisionless plasma works at an individual particle level. Note that the type of collisionless, magnetized plasma considered in this section occurs primar- ilyinmagneticfusionandspaceplasmaphysics contexts. Infact, inthefollowingweshall be studying methods of analysis first developed by fusion physicists, and illustrating these methods primarily by investigating problems of interest in magnetospheric physics. 2.2 Motion in Uniform Fields Let us, first of all, consider the motion of charged particles in spatially and temporally uniform electromagnetic fields. The equation of motion of an individual particle takes the form dv m =e(E+v×B). (2.1) dt The component of this equation parallelto the magnetic field, dv e k = E , (2.2) k dt m predictsuniformaccelerationalongmagneticfield-lines. Consequently,plasmasnearequi- librium generally have either small or vanishingE . k18 PLASMA PHYSICS As can easily be verified by substitution, the perpendicular component of Eq. (2.1) yields E×B v = +ρΩe sin(Ωt+γ )+e cos(Ωt+γ ), (2.3) ⊥ 1 0 2 0 2 B where Ω = eB/m is the gyrofrequency, ρ is the gyroradius, e and e are unit vectors 1 2 such that (e , e , B) form a right-handed, mutually orthogonal set, andγ is the initial 1 2 0 gyrophase of the particle. The motion consists of gyration around the magnetic field at frequencyΩ, superimposed on a steady drift at velocity E×B v = . (2.4) E 2 B Thisdrift,whichistermedtheE-cross-Bdriftbyplasmaphysicists,isidenticalforallplasma species, and can be eliminated entirely by transforming to a new inertial frame in which E = 0. This frame, which moves with velocity v with respect to the old frame, can ⊥ E properly be regarded as the rest frame of the plasma. We complete the solution by integrating the velocity to find the particle position: r(t) =R(t)+ρ(t), (2.5) where ρ(t) =ρ−e cos(Ωt+γ )+e sin(Ωt+γ ), (2.6) 1 0 2 0 and 2 e t E b+v t. (2.7) R(t) = v t+ 0k k E m 2 Here, b≡ B/B. Of course, the trajectory of the particle describes a spiral. The gyrocentre R of this spiral, termed the guiding centre by plasma physicists, drifts across the magnetic field with velocity v , and also accelerates along the field at a rate determined by the E parallel electric field. The concept of a guiding centre gives us a clue as to how to proceed. Perhaps, when analyzingchargedparticlemotioninnon-uniformelectromagneticfields,wecansomehow neglect the rapid, and relatively uninteresting, gyromotion, and focus, instead, on the far slower motion of the guiding centre? Clearly, what we need to do in order to achieve this goal is to somehow average the equation of motion over gyrophase, so as to obtain a reduced equationof motion for the guiding centre. 2.3 Methodof Averaging In many dynamical problems, the motion consists of a rapid oscillation superimposed on a slow secular drift. For such problems, the most efficient approach is to describe the evolution in terms of the average values of the dynamical variables. The method outlined 1 below is adapted from a classic paper by Morozov and Solov’ev. 1 A.I.Morozov,andL.S.Solev’ev,MotionofChargedParticlesinElectromagneticFields,inReviewsofPlasma Physics,Vol. 2 (Consultants Bureau, New YorkNY, 1966).Charged Particle Motion 19 Consider the equation of motion dz =f(z,t,τ), (2.8) dt wheref is a periodic function of its last argument, with period2π, and τ =t/ǫ. (2.9) Here, the small parameter ǫ characterizes the separation between the short oscillation periodτ and the time-scalet for the slow secular evolution of the “position”z. The basic idea of the averaging method is to treat t and τ as distinct independent variables, and to look for solutions of the form z(t,τ) which are periodic inτ. Thus, we replace Eq. (2.8) by ∂z 1∂z + =f(z,t,τ), (2.10) ∂t ǫ∂τ and reserve Eq. (2.9) for substitution in the final result. The indeterminacy introduced by increasingthenumberofvariablesisliftedbytherequirementofperiodicityinτ. Allofthe secular drifts are thereby attributed to thet-variable, whilst the oscillations are described entirely by theτ-variable. Let us denote theτ-average ofz byZ, and seek a change of variables of the form z(t,τ) =Z(t)+ǫζ(Z,t,τ). (2.11) Here,ζ is a periodic function ofτ with vanishing mean. Thus, I 1 hζ(Z,t,τ)i≡ ζ(Z,t,τ)dτ =0, (2.12) 2π H where denotes the integral over a full period inτ. The evolutionofZ is determined by substituting the expansions 2 ζ = ζ (Z,t,τ)+ǫζ (Z,t,τ)+ǫ ζ (Z,t,τ)+···, (2.13) 0 1 2 dZ 2 = F (Z,t)+ǫF (Z,t)+ǫ F (Z,t)+···, (2.14) 0 1 2 dt into the equationof motion (2.10), and solving order by order inǫ. To lowest order, we obtain ∂ζ 0 F (Z,t)+ =f(Z,t,τ). (2.15) 0 ∂τ The solubility condition for this equationis F (Z,t) =hf(Z,t,τ)i. (2.16) 020 PLASMA PHYSICS Integrating the oscillating componentof Eq. (2.15) yields Z τ ′ ζ (Z,t,τ) = (f−hfi)dτ. (2.17) 0 0 To first order, we obtain ∂ζ ∂ζ 0 1 F + +F ·∇ζ + =ζ ·∇f. (2.18) 1 0 0 0 ∂t ∂τ The solubility condition for this equationyields F =hζ ·∇fi. (2.19) 1 0 The final result is obtainedby combining Eqs. (2.16) and (2.19): dZ 2 =hfi+ǫhζ ·∇fi+O(ǫ ). (2.20) 0 dt Note that f = f(Z,t) in the above equation. Evidently, the secular motion of the “guiding centre”positionZisdeterminedtolowestorderbytheaverageofthe“force”f,andtonext order by the correlation between the oscillation in the “position” z and the oscillation in the spatial gradient of the “force.” 2.4 Guiding Centre Motion Consider the motion of a charged particle in the limit in which the electromagnetic fields experienced by the particle do not vary much in a gyroperiod: i.e., ρ∇B ≪ B, (2.21) 1 ∂B ≪ B. (2.22) Ω∂t The electric force is assumed to be comparableto the magnetic force. To keep track of the order of the various quantities, we introduce the parameterǫ as a book-keeping device, −1 and make the substitutionρ→ǫρ, as well as (E,B,Ω)→ǫ (E,B,Ω). The parameterǫ is set to unity in the final answer. In order to make use of the technique described in the previous section, we write the dynamical equationsin first-order differential form, dr = v, (2.23) dt dv e = (E+v×B), (2.24) dt ǫm

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