plasma 2 lecture 1: introductionsites.apam.columbia.edu/courses/apph6102y/plasma2...plasma physics...

Plasma 2 Lecture 1: Introduction

APPH E6102y Columbia University

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Syllabus and Class WebsiteAPPH E6102y Site InformationPlasma Physics 2Prof. Michael MauelEmail: [email protected]

GeneralWelcome to the APPH E6102y class information site.

This is the second semester of a two-semester sequence in plasma physics. Plasma physics is the study of "luminous matter", matter that has been heated sufficiently or prepared specially in order to be ionized. In plasma long-range electromagnetic forces are more important than short range forces. Plasma dynamics is dominated by "collective" motion of large populations of neighboring particles. Electric and electromagnetic waves propagate at speeds that resonate with particles and allow energy and momentum exchange. Plasma motion and the self-consistent electric and magnetic fields exhibit beautiful nonlinear physics.Plasma is studied in the laboratory and in space. Most of the visible universe is in the plasma state. Laboratory generated plasma are used to studied the fundamental properties of high-temperature matter, and they are employed for many valuable applications like surface processing and lighting. Integrated circuits are manufactured using plasma processing, and plasma displays are status symbols of today's world of entertainment. Controlled fusion energy research reflects the remarkable success of plasma physics. The controlled release of more than 10 MW of fusion power has occurred within the strong confining fields of tokamak devices, and the world is now building the first experimental fusion power source, called ITER.

Topics covered include: Motion of charged particles in space- and time-varying electromagnetic fields. Magnetic coordinates. Equilibrium, stability, and transport of torodial plasmas. Ballooning and tearing instabilities. Kinetic theory, including Vlasov equation, Fokker-Planck equation, Landau damping, kinetic transport theory. Drift instabilities. Quasilinear theory. Introduction to drift-kinegtic and gyro-kinetic theory.

APPH 6102 requires prior experience with plasma physics. The formal prerequisites are APPH E6101 Plasma physics. The goal of this course is to provide a working understanding of plasma physics and prepare students for research.

http://sites.apam.columbia.edu/courses/apph6102y/

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mailto:[email protected]://www.iter.org/http://www.columbia.edu/cu/bulletin/uwb/subj/APPH/E6101-20173-001/http://sites.apam.columbia.edu/courses/apph6102y/

(No/Optional) TextbookTextbook There will be no textbook for the course. Instead, we will make frequent use of journal

publications.

For those who want a well-rounded textbook, I recommend Introduction to Plasma Physics (2nd Edition) by Don Gurnett (University of Iowa) and Amitava Bhattacharjee (Princeton University). Don Gurnett is a well known space plasma physicists and plasma wave expert. Bhattacharjee has worked in many areas of magnetized plasma physics, including magnetic fusion, plasma astrophysics, theory, and high-performance computation.

Since Columbia's plasma physics program has a focus on fusion energy, I also recommend an inroductory textbook by Garry McCracken and Peter Stott: Fusion: The Energy of the Universe. McCracken has made pioneering studies of tokamak plasma confinement. He's an expert on plasma wall interactions and worked at Alcator CMOD and JET. Peter Stott is also a leading expert on tokamak fusion confinement, having worked at PPPL, JET, and most recently working on ITER diagnostic systems.

These books are available as ebooks for Columbia University students: see CLIO.Occasionally, I will present numerical illustrations of plasma physics using Mathematica. Mathematica is available to all students through Columbia University.

Most frequently, I will distribute published journal articles that illustrate the scientific progress and discoveries in the field of plasma physics.

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https://www.amazon.com/Introduction-Plasma-Physics-Astrophysical-Applications/dp/1107027373/ref=sr_1_1_twi_har_2?ie=UTF8&qid=1514909454&sr=8-1&keywords=plasma+physics+bhattacharjeehttps://physics.uiowa.edu/people/donald-gurnetthttp://www.apam.columbia.edu/courses/apph6102y/amitava%20bhattarchjee%20princetonhttps://www.amazon.com/Fusion-Second-Universe-Garry-McCracken/dp/0123846560https://courseworks.columbia.edu/http://www.wolfram.com/mathematica/http://cuit.columbia.edu/mathematica

GradingGrading A student's grade for the course will be based on two take-home exams

and two research and writing assignments:

one midterm paper and one final paper

These papers must follow the style used for publication in Physics of Plasmas. You will clearly describe in your own words a plasma physics topic or question and then review the topic or present an answer to the question.

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http://aip.scitation.org/journal/php

July 2016 Volume 23 Number 7

pop.aip.org

Physics ofPlasmas

Radiation from laser-microplasma-waveguide interactions in the ultra-intense regime

by L. Yi, A. Pukhov, and B. Shen

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Learn to search…

Web of science core collection Published [Philadelphia, PA] : Thomson Reuters, [2014-] Online

http://www.columbia.edu/cgi-bin/cul/resolve?clio2054244

Format Online

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https://clio.columbia.edu/databases/2054244http://www.columbia.edu/cgi-bin/cul/resolve?clio2054244

Classic “Plasma” ArticlesINTERACTION OF SOLITONS IN A COLLISIONLESS PLASMA AND RECURRENCE OF INITIAL STATES By: ZABUSKY, NJ; KRUSKAL, MD PHYSICAL REVIEW LETTERS Volume: 15 Issue: 6 Pages: 240-& Published: 1965 (Times Cited: 1,990)

RELAXATION OF TOROIDAL PLASMA AND GENERATION OF REVERSE MAGNETIC-FIELDS By: TAYLOR, JB PHYSICAL REVIEW LETTERS Volume: 33 Issue: 19 Pages: 1139-1141 Published: 1974 (Times Cited: 1,323)

ABSORPTION OF ULTRA-INTENSE LASER-PULSES By: WILKS, SC; KRUER, WL; TABAK, M; et al. PHYSICAL REVIEW LETTERS Volume: 69 Issue: 9 Pages: 1383-1386 Published: AUG 31 1992 (Times Cited: 1,312)

PLASMA CRYSTAL - COULOMB CRYSTALLIZATION IN A DUSTY PLASMA By: THOMAS, H; MORFILL, GE; DEMMEL, V; et al. PHYSICAL REVIEW LETTERS Volume: 73 Issue: 5 Pages: 652-655 Published: AUG 1 1994 (Times Cited: 1,243 )

SPONTANEOUSLY GROWING TRANSVERSE WAVES IN A PLASMA DUE TO AN ANISOTROPIC VELOCITY DISTRIBUTION By: WEIBEL, ES PHYSICAL REVIEW LETTERS Volume: 2 Issue: 3 Pages: 83-84 Published: 1959 (Times Cited: 1,123 )

DIRECT OBSERVATION OF COULOMB CRYSTALS AND LIQUIDS IN STRONGLY COUPLED RF DUSTY PLASMAS By: CHU, JH; I, L PHYSICAL REVIEW LETTERS Volume: 72 Issue: 25 Pages: 4009-4012 Published: JUN 20 1994 (Times Cited: 1,048 )

Plasma expansion into a vacuum By: Mora, P PHYSICAL REVIEW LETTERS Volume: 90 Issue: 18 Article Number: 185002 Published: MAY 9 2003 (Times Cited: 617)

TRANSPORT OF DUST PARTICLES IN GLOW-DISCHARGE PLASMAS By: BARNES, MS; KELLER, JH; FORSTER, JC; et al. PHYSICAL REVIEW LETTERS Volume: 68 Issue: 3 Pages: 313-316 Published: JAN 20 1992 (Times Cited: 582 )

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Classic “Tokamak” ArticlesREGIME OF IMPROVED CONFINEMENT AND HIGH-BETA IN NEUTRAL-BEAM-HEATED DIVERTOR DISCHARGES OF THE ASDEX TOKAMAK By: WAGNER, F; BECKER, G; BEHRINGER, K; et al. PHYSICAL REVIEW LETTERS Volume: 49 Issue: 19 Pages: 1408-1412 Published: 1982 (Times Cited: 1,504)

ELECTRON HEAT-TRANSPORT IN A TOKAMAK WITH DESTROYED MAGNETIC SURFACES By: RECHESTER, AB; ROSENBLUTH, MN PHYSICAL REVIEW LETTERS Volume: 40 Issue: 1 Pages: 38-41 Published: 1978 (Times Cited: 994)

ENHANCED CONFINEMENT AND STABILITY IN DIII-D DISCHARGES WITH REVERSED MAGNETIC SHEAR By: STRAIT, EJ; LAO, LL; MAUEL, ME; et al. PHYSICAL REVIEW LETTERS Volume: 75 Issue: 24 Pages: 4421-4424 Published: DEC 11 1995 (Times Cited: 514)

H-MODE BEHAVIOR INDUCED BY CROSS-FIELD CURRENTS IN A TOKAMAK By: TAYLOR, RJ; BROWN, ML; FRIED, BD; et al. PHYSICAL REVIEW LETTERS Volume: 63 Issue: 21 Pages: 2365-2368 Published: NOV 20 1989 (Times Cited: 491)

CONFINING A TOKAMAK PLASMA WITH RF-DRIVEN CURRENTS By: FISCH, NJ PHYSICAL REVIEW LETTERS Volume: 41 Issue: 13 Pages: 873-876 Published: 1978 (Times Cited: 461 )

Electron temperature gradient turbulence By: Dorland, W; Jenko, F; Kotschenreuther, M; et al. PHYSICAL REVIEW LETTERS Volume: 85 Issue: 26 Pages: 5579-5582 Part: 1 Published: DEC 25 2000 (Times Cited: 391 )

STUDIES OF INTERNAL DISRUPTIONS AND M = 1 OSCILLATIONS IN TOKAMAK DISCHARGES WITH SOFT-X-RAY TECHNIQUES By: VONGOELER, S; STODIEK, W; SAUTHOFF, N PHYSICAL REVIEW LETTERS Volume: 33 Issue: 20 Pages: 1201-1203 Published: 1974 (Times Cited: 462 )

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Recent “Hot” Articles“Hot” articles rank within the top 1% of all articles published in physics.

Active control of type-I edge-localized modes with n=1 perturbation fields in the JET tokamak By: Liang, Y.; Koslowski, H. R.; Thomas, P. R.; et al. PHYSICAL REVIEW LETTERS Volume: 98 Issue: 26 (Times Cited: 301) Published: JUN 29 2007

First Observation of Edge Localized Modes Mitigation with Resonant and Nonresonant Magnetic Perturbations in ASDEX Upgrade By: Suttrop, W.; Eich, T.; Fuchs, J. C.; et al. Group Author(s): ASDEX Upgrade Team PHYSICAL REVIEW LETTERS Volume: 106 Issue: 22 (Times Cited: 218) Published: JUN 2 2011

Observations of plasmons in warm dense matter By: Glenzer, S. H.; Landen, O. L.; Neumayer, P.; et al. PHYSICAL REVIEW LETTERS Volume: 98 Issue: 6 (Times Cited: 301) Published: FEB 9 2007

Radiation-Pressure Acceleration of Ion Beams Driven by Circularly Polarized Laser Pulses By: Henig, A.; Steinke, S.; Schnuerer, M.; et al. PHYSICAL REVIEW LETTERS Volume: 103 Issue: 24 (Times Cited: 291) Published: DEC 11 2009

Generating high-current monoenergetic proton beams by a circularly polarized laser pulse in the phase-stable acceleration regime By: Yan, X. Q.; Lin, C.; Sheng, Z. M.; et al. PHYSICAL REVIEW LETTERS Volume: 100 Issue: 13 (Times Cited: 274) Published: APR 4 2008

Dynamics of spin-1/2quantum plasmas By: Marklund, Mattias; Brodin, Gert PHYSICAL REVIEW LETTERS Volume: 98 Issue: 2 (Times Cited: 268) Published: JAN 12 2007 9

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Most Cited from Nuclear FusionRECONSTRUCTION OF CURRENT PROFILE PARAMETERS AND PLASMA SHAPES IN TOKAMAKSBy: LAO, LL; STJOHN, H; STAMBAUGH, RD; et al.NUCLEAR FUSION Volume: 25 Issue: 11 Pages: 1611-1622 Published: 1985 (Times Cited: 951)

NEOCLASSICAL TRANSPORT OF IMPURITIES IN TOKAMAK PLASMASBy: HIRSHMAN, SP; SIGMAR, DJNUCLEAR FUSION Volume: 21 Issue: 9 Pages: 1079-1201 Published: 1981 (Times Cited: 900)

Plasma-material interactions in current tokamaks and their implications for next step fusion reactorsBy: Federici, G; Skinner, CH; Brooks, JN; et al.NUCLEAR FUSION Volume: 41 Issue: 12R Special Issue: SI Pages: 1967-2137 Published: DEC 2001 (Times Cited: 804)

MEASUREMENTS OF MICROTURBULENCE IN TOKAMAKS AND COMPARISONS WITH THEORIES OF TURBULENCE AND ANOMALOUS TRANSPORTBy: LIEWER, PCNUCLEAR FUSION Volume: 25 Issue: 5 Pages: 543-621 Published: 1985 (Times Cited: 670)

FAST-WAVE HEATING OF A 2-COMPONENT PLASMABy: STIX, THNUCLEAR FUSION Volume: 15 Issue: 5 Pages: 737-754 Published: 1975 (Times Cited: 582)

A NEW LOOK AT DENSITY LIMITS IN TOKAMAKSBy: GREENWALD, M; TERRY, JL; WOLFE, SM; et al.NUCLEAR FUSION Volume: 28 Issue: 12 Pages: 2199-2207 Published: DEC 1988 (Times Cited: 530)

PLASMA BOUNDARY PHENOMENA IN TOKAMAKSBy: STANGEBY, PC; MCCRACKEN, GMNUCLEAR FUSION Volume: 30 Issue: 7 Pages: 1225-1379 Published: JUL 1990 (Times Cited: 512)

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Some “Most Cited” from Physics of Plasmas

IGNITION AND HIGH-GAIN WITH ULTRA-POWERFUL LASERSBy: TABAK, M; HAMMER, J; GLINSKY, ME; et al.PHYSICS OF PLASMAS Volume: 1 Issue: 5 Pages: 1626-1634 Part: 2 Published: MAY 1994 (Times Cited: 2,360)

INFLUENCE OF SHEARED POLOIDAL ROTATION ON EDGE TURBULENCEBy: BIGLARI, H; DIAMOND, PH; TERRY, PWPHYSICS OF FLUIDS B-PLASMA PHYSICS Volume: 2 Issue: 1 Pages: 1-4 Published: JAN 1990 (Times Cited: 1,096)

LABORATORY OBSERVATION OF THE DUST-ACOUSTIC WAVE MODEBy: BARKAN, A; MERLINO, RL; DANGELO, NPHYSICS OF PLASMAS Volume: 2 Issue: 10 Pages: 3563-3565 Published: OCT 1995 (Times Cited: 981)

Effects of ExB velocity shear and magnetic shear on turbulence and transport in magnetic confinement devicesBy: Burrell, KHPHYSICS OF PLASMAS Volume: 4 Issue: 5 Pages: 1499-1518 Part: 2 Published: MAY 1997 (Times Cited: 975)

VELOCITY SPACE DIFFUSION FROM WEAK PLASMA TURBULENCE IN A MAGNETIC FIELDBy: KENNEL, CF; ENGELMANN, FPHYSICS OF FLUIDS Volume: 9 Issue: 12 Pages: 2377-+ Published: 1966 (Times Cited: 719)

PSEUDO-3-DIMENSIONAL TURBULENCE IN MAGNETIZED NONUNIFORM PLASMABy: HASEGAWA, A; MIMA, KPHYSICS OF FLUIDS Volume: 21 Issue: 1 Pages: 87-92 Published: 1978 (Times Cited: 694)

COLLISIONLESS DAMPING OF NONLINEAR PLASMA OSCILLATIONSBy: ONEIL, TPHYSICS OF FLUIDS Volume: 8 Issue: 12 Pages: 2255-& Published: 1965 (Times Cited: 681)

A PERTURBATION THEORY FOR STRONG PLASMA TURBULENCEBy: DUPREE, THPHYSICS OF FLUIDS Volume: 9 Issue: 9 Pages: 1773-& Published: 1966 (Times Cited: 611)

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https://doi.org/10.1063/1.1761193

Contents ix

8.4 Charged Particle Acceleration by MHD Shocks 308References 316Further Reading 318

9 Electrostatic Waves in a Hot Unmagnetized Plasma 3199.1 The Vlasov Approach 3199.2 The Landau Approach 3289.3 The Plasma Dispersion Function 3469.4 The Dispersion Relation for a Multi-component Plasma 3499.5 Stability 356References 377Further Reading 377

10 Waves in a Hot Magnetized Plasma 37810.1 Linearization of the Vlasov Equation 37910.2 Electrostatic Waves 38210.3 Electromagnetic Waves 403References 426Further Reading 427

11 Nonlinear Effects 42811.1 Quasi-linear Theory 42811.2 Wave–Wave Interactions 44211.3 Langmuir Wave Solitons 46211.4 Stationary Nonlinear Electrostatic Potentials 470References 477Further Reading 478

12 Collisional Processes 47912.1 Binary Coulomb Collisions 48012.2 Importance of Small-Angle Collisions 48112.3 The Fokker–Planck Equation 48412.4 Conductivity of a Fully Ionized Plasma 49112.5 Collision Operator for Maxwellian Distributions

of Electrons and Ions 495References 499Further Reading 499

Appendix A: Symbols 500

Appendix B: Useful Trigonometric Identities 509

Appendix C: Vector Differential Operators 510

Appendix D: Vector Calculus Identities 512

Index 513

8C D 9 D8 4 4 45 8 4 D- 64 5C 7:8 C: 6 C8 8C D D- 7 C: , , ,/ 4787 9C D- 64 5C 7:8 C: 6 C8 2846 8CD . 8:8 1 5C4C . 5 4 3 8CD 04 4 - - D 5 86 8 .4 5C 7:8 . C8

INTRODUCTION TOPLASMA PHYSICS

With Space, Laboratoryand Astrophysical Applications

DONALD A. GURNETTUniversity of Iowa

AMITAVA BHATTACHARJEEPrinceton University, New Jersey


INTRODUCTION TOPLASMA PHYSICS

With Space, Laboratoryand Astrophysical Applications

DONALD A. GURNETTUniversity of Iowa

AMITAVA BHATTACHARJEEPrinceton University, New Jersey


Further Reading 377

(d) Show that the instability condition for the current-driven ion acousticmode is

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(− Te

2Ti

)⎤⎥⎥⎥⎥⎥⎦ .

(e) Plot |Ue| as a function of Te/Ti. Where does the approximation fail?

ReferencesArfken, G. 1970. Mathematical Methods for Physicists. New York: Academic Press, pp.

311–315.Armstrong, T. P. 1967. Numerical studies of the nonlinear Vlasov equation. Phys. Fluids

10, 1269–1280.Buneman, O. 1959. Dissipation of currents in ionized media. Phys. Rev. 115, 503–517.Flanigan, F. J. 1983. Complex Variables: Harmonic and Analytic Functions. Mineola, NY:

Dover Publications, pp. 272–275. Originally published in 1972.Fried, B. D., and Conte, S. D. 1961. The Plasma Dispersion Function. New York:

Academic Press, pp. 2–3.Gardner, C. S. 1963. Bound on the energy available from a plasma. Phys. Fluids 6,

839–840.Ginzburg, V. L., and Zhelezniakov, V. V. 1958. On the possible mechanism of sporadic

solar radio emission (radiation in an isotropic plasma). Sov. Astron. AJ 2, 653–666.Gould, R. W., O’Neil, T. M., and Malmberg, J. H. 1967. Plasma wave echo. Phys. Rev.

Lett. 19, 219–222.Gurnett, D. A., Hospodarsky, G. B., Kurth, W. S., Williams, D. J., and Bolton, S. J. 1993.

Fine structure of Langmuir waves produced by a solar electron event. J. Geophys. Res.98, 5631–5637.

Landau, L. 1946. On the vibration of the electron plasma. J. Phys. (USSR) 10 (1), 85–94.Malmberg, J. H., and Wharton, C. B. 1966. Dispersion of electron plasma waves. Phys.

Rev. Lett. 17, 175–178.Nicholson, D. R. 1983. Introduction to Plasma Theory. Malabar, FL: Krieger Publishing,

pp. 87–96.Nyquist, H. 1932. Regeneration theory. Bell System Tech. J. 11, 126–147.O’Neil, T. M. 1965. Collisionless damping of nonlinear plasma oscillations. Phys. Fluids

8, 2255–2262.Penrose, O. 1960. Electrostatic instabilities of a uniform non-Maxwellian plasma. Phys.

Fluids 3, 258–265.Vlasov, A. A. 1945. On the kinetic theory of an assembly of particles with collective

interaction. J. Phys. (USSR) 9, 25–44.

Further ReadingChen, F. F. 1990. Introduction to Plasma Physics and Controlled Fusion. New York:

Plenum Press, Chapter 7.Nicholson, D. R. 1983. Introduction to Plasma Theory. Malabar, FL: Krieger Publishing,

Chapter 6.Stix, T. H. 1992. Waves in Plasmas. New York: American Institute of Physics, Chapter 8.Swanson, D. G. 1989. Plasma Waves. New York: Academic Press, Section 4.2.


13

Linearized Vlasov Equation320 Electrostatic Waves in a Hot Unmagnetized Plasma

later. The equations that must be solved then consist of Vlasov’s equation (withq = −e,E = −∇Φ, and B = 0)

∂ f∂t+v ·∇ f + e

m∇Φ ·∇v f = 0 (9.1.1)

and Poisson’s equation

∇2Φ = −ρqϵ0= − e

ϵ0

[n0 −

∫ ∞

−∞f d3υ

]. (9.1.2)

Following the usual linearization procedure, we assume that the electron velocitydistribution function, f (v), consists of a constant uniform zero-order distribution,f0(v), plus a small first-order perturbation, f1(v):

f (v) = f0(v)+ f1(v). (9.1.3)

Similarly, we write Φ = Φ1. Linearizing Eqs. (9.1.1) and (9.1.2) and noting that∫f0d3υ = n0, we obtain for the first-order equations

∂ f1∂t+v ·∇ f1 +

em∇Φ1 ·∇v f0 = 0 (9.1.4)

and

∇2Φ1 =eϵ0

∫ ∞

−∞f1(v) d3υ. (9.1.5)

Next, we attempt to solve these equations using the usual Fourier transformapproach. Since no preferred direction exists, the z axis can be aligned parallelto k. After dropping the subscript 1 on the first-order terms and making theusual operator substitutions (∂/∂t → −iω and ∇ → ik), it is easy to see that theFourier-transformed equations are

−iω f̃ + ikυz f̃ + iem

kΦ̃∂ f0∂υz= 0 (9.1.6)

and

k2Φ̃ = − eϵ0

∫ ∞

−∞f̃ (v) d3υ. (9.1.7)

Solving for f̃ from Eq. (9.1.6),

f̃ =−1

(kυz −ω)em

kΦ̃∂ f0∂υz, (9.1.8)


320 Electrostatic Waves in a Hot Unmagnetized Plasma


∂ f∂t+v ·∇ f + e

m∇Φ ·∇v f = 0 (9.1.1)


∇2Φ = −ρqϵ0= − e

ϵ0

[n0 −

∫ ∞

−∞f d3υ

]. (9.1.2)


f (v) = f0(v)+ f1(v). (9.1.3)


∂ f1∂t+v ·∇ f1 +

em∇Φ1 ·∇v f0 = 0 (9.1.4)

and

∇2Φ1 =eϵ0

∫ ∞

−∞f1(v) d3υ. (9.1.5)



kΦ̃∂ f0∂υz= 0 (9.1.6)

and

k2Φ̃ = − eϵ0

∫ ∞

−∞f̃ (v) d3υ. (9.1.7)


f̃ =−1

(kυz −ω)em

kΦ̃∂ f0∂υz, (9.1.8)




∂ f∂t+v ·∇ f + e

m∇Φ ·∇v f = 0 (9.1.1)


∇2Φ = −ρqϵ0= − e

ϵ0

[n0 −

∫ ∞

−∞f d3υ

]. (9.1.2)


f (v) = f0(v)+ f1(v). (9.1.3)


∂ f1∂t+v ·∇ f1 +

em∇Φ1 ·∇v f0 = 0 (9.1.4)

and

∇2Φ1 =eϵ0

∫ ∞

−∞f1(v) d3υ. (9.1.5)



kΦ̃∂ f0∂υz= 0 (9.1.6)

and

k2Φ̃ = − eϵ0

∫ ∞

−∞f̃ (v) d3υ. (9.1.7)


f̃ =−1

(kυz −ω)em

kΦ̃∂ f0∂υz, (9.1.8)




∂ f∂t+v ·∇ f + e

m∇Φ ·∇v f = 0 (9.1.1)


∇2Φ = −ρqϵ0= − e

ϵ0

[n0 −

∫ ∞

−∞f d3υ

]. (9.1.2)


f (v) = f0(v)+ f1(v). (9.1.3)


∂ f1∂t+v ·∇ f1 +

em∇Φ1 ·∇v f0 = 0 (9.1.4)

and

∇2Φ1 =eϵ0

∫ ∞

−∞f1(v) d3υ. (9.1.5)



kΦ̃∂ f0∂υz= 0 (9.1.6)

and

k2Φ̃ = − eϵ0

∫ ∞

−∞f̃ (v) d3υ. (9.1.7)


f̃ =−1

(kυz −ω)em

kΦ̃∂ f0∂υz, (9.1.8)




∂ f∂t+v ·∇ f + e

m∇Φ ·∇v f = 0 (9.1.1)


∇2Φ = −ρqϵ0= − e

ϵ0

[n0 −

∫ ∞

−∞f d3υ

]. (9.1.2)


f (v) = f0(v)+ f1(v). (9.1.3)


∂ f1∂t+v ·∇ f1 +

em∇Φ1 ·∇v f0 = 0 (9.1.4)

and

∇2Φ1 =eϵ0

∫ ∞

−∞f1(v) d3υ. (9.1.5)



kΦ̃∂ f0∂υz= 0 (9.1.6)

and

k2Φ̃ = − eϵ0

∫ ∞

−∞f̃ (v) d3υ. (9.1.7)


f̃ =−1

(kυz −ω)em

kΦ̃∂ f0∂υz, (9.1.8)


14

Fourier-Laplace Transform



∂ f∂t+v ·∇ f + e

m∇Φ ·∇v f = 0 (9.1.1)


∇2Φ = −ρqϵ0= − e

ϵ0

[n0 −

∫ ∞

−∞f d3υ

]. (9.1.2)


f (v) = f0(v)+ f1(v). (9.1.3)


∂ f1∂t+v ·∇ f1 +

em∇Φ1 ·∇v f0 = 0 (9.1.4)

and

∇2Φ1 =eϵ0

∫ ∞

−∞f1(v) d3υ. (9.1.5)



kΦ̃∂ f0∂υz= 0 (9.1.6)

and

k2Φ̃ = − eϵ0

∫ ∞

−∞f̃ (v) d3υ. (9.1.7)


f̃ =−1

(kυz −ω)em

kΦ̃∂ f0∂υz, (9.1.8)




∂ f∂t+v ·∇ f + e

m∇Φ ·∇v f = 0 (9.1.1)


∇2Φ = −ρqϵ0= − e

ϵ0

[n0 −

∫ ∞

−∞f d3υ

]. (9.1.2)


f (v) = f0(v)+ f1(v). (9.1.3)


∂ f1∂t+v ·∇ f1 +

em∇Φ1 ·∇v f0 = 0 (9.1.4)

and

∇2Φ1 =eϵ0

∫ ∞

−∞f1(v) d3υ. (9.1.5)



kΦ̃∂ f0∂υz= 0 (9.1.6)

and

k2Φ̃ = − eϵ0

∫ ∞

−∞f̃ (v) d3υ. (9.1.7)


f̃ =−1

(kυz −ω)em

kΦ̃∂ f0∂υz, (9.1.8)




∂ f∂t+v ·∇ f + e

m∇Φ ·∇v f = 0 (9.1.1)


∇2Φ = −ρqϵ0= − e

ϵ0

[n0 −

∫ ∞

−∞f d3υ

]. (9.1.2)


f (v) = f0(v)+ f1(v). (9.1.3)


∂ f1∂t+v ·∇ f1 +

em∇Φ1 ·∇v f0 = 0 (9.1.4)

and

∇2Φ1 =eϵ0

∫ ∞

−∞f1(v) d3υ. (9.1.5)



kΦ̃∂ f0∂υz= 0 (9.1.6)

and

k2Φ̃ = − eϵ0

∫ ∞

−∞f̃ (v) d3υ. (9.1.7)


f̃ =−1

(kυz −ω)em

kΦ̃∂ f0∂υz, (9.1.8)


9.1 The Vlasov Approach 321

and substituting into Eq. (9.1.7) gives

k2Φ̃ =e2

ϵ0mkΦ̃

∫ ∞

−∞

(∂ f0/∂υz)(kυz −ω)

d3υ. (9.1.9)

Rearranging the terms and factoring out Φ̃ then gives the homogeneous equation[1− e

2

ϵ0mk2

∫ ∞

−∞

∂ f0/∂υz(υz −ω/k)

d3υ]Φ̃ = 0. (9.1.10)

For the potential Φ̃ to have a non-trivial solution, the term in the brackets must bezero, which gives the dispersion relation

Ð(k,ω) = 1− e2

ϵ0mk2

∫ ∞

−∞


d3υ = 0. (9.1.11)

At this point, it is useful to define a new normalized one-dimensional distributionfunction:

F0(υz) =1n0

∫ ∞

−∞f0(v)dυx dυy. (9.1.12)

The function F0(υz) is sometimes called the reduced distribution function. Notethat by dividing by n0 the reduced distribution function is normalized such that∫

F0(υz)dυz = 1. Recognizing thatω2p = n0e2/(ϵ0m), the dispersion relation equation

(9.1.11) can be written

Ð(k,ω) = 1−ω2p

k2

∫ ∞

−∞

∂F0/∂υz(υz −ω/k)

dυz = 0. (9.1.13)

In some applications it is useful to put the dispersion relation into a slightlydifferent form by integrating by parts once, which gives

∫ ∞

−∞


dυz =[

F0(υz −ω/k)

]∞

−∞+

∫ ∞

−∞

F0(υz −ω/k)2

dυz. (9.1.14)

Since F0 goes to zero at υz =±∞, the first term in the integration by parts vanishes,so the dispersion relation becomes

Ð(k,ω) = 1−ω2p

k2

∫ ∞

−∞

F0(υz −ω/k)2

dυz = 0. (9.1.15)

Both of the above forms of the dispersion relation suffer from a serious problem.Because the denominator goes to zero at υz = ω/k, the integrals do not convergeunless F0 and ∂F0/∂υz are zero at υz = ω/k. Physically, the dispersion relationexists only if there are no particles moving with a velocity equal to the phase




k2Φ̃ =e2

ϵ0mkΦ̃

∫ ∞

−∞

(∂ f0/∂υz)(kυz −ω)

d3υ. (9.1.9)


2

ϵ0mk2

∫ ∞

−∞


d3υ]Φ̃ = 0. (9.1.10)


Ð(k,ω) = 1− e2

ϵ0mk2

∫ ∞

−∞


d3υ = 0. (9.1.11)


F0(υz) =1n0

∫ ∞

−∞f0(v)dυx dυy. (9.1.12)




Ð(k,ω) = 1−ω2p

k2

∫ ∞

−∞


dυz = 0. (9.1.13)


∫ ∞

−∞


dυz =[

F0(υz −ω/k)

]∞

−∞+

∫ ∞

−∞

F0(υz −ω/k)2

dυz. (9.1.14)


Ð(k,ω) = 1−ω2p

k2

∫ ∞

−∞

F0(υz −ω/k)2

dυz = 0. (9.1.15)





k2Φ̃ =e2

ϵ0mkΦ̃

∫ ∞

−∞

(∂ f0/∂υz)(kυz −ω)

d3υ. (9.1.9)


2

ϵ0mk2

∫ ∞

−∞


d3υ]Φ̃ = 0. (9.1.10)


Ð(k,ω) = 1− e2

ϵ0mk2

∫ ∞

−∞


d3υ = 0. (9.1.11)


F0(υz) =1n0

∫ ∞

−∞f0(v)dυx dυy. (9.1.12)




Ð(k,ω) = 1−ω2p

k2

∫ ∞

−∞


dυz = 0. (9.1.13)


∫ ∞

−∞


dυz =[

F0(υz −ω/k)

]∞

−∞+

∫ ∞

−∞

F0(υz −ω/k)2

dυz. (9.1.14)


Ð(k,ω) = 1−ω2p

k2

∫ ∞

−∞

F0(υz −ω/k)2

dυz = 0. (9.1.15)



15

Electrostatic Dispersion Relation for a Plasma



k2Φ̃ =e2

ϵ0mkΦ̃

∫ ∞

−∞

(∂ f0/∂υz)(kυz −ω)

d3υ. (9.1.9)


2

ϵ0mk2

∫ ∞

−∞


d3υ]Φ̃ = 0. (9.1.10)


Ð(k,ω) = 1− e2

ϵ0mk2

∫ ∞

−∞


d3υ = 0. (9.1.11)


F0(υz) =1n0

∫ ∞

−∞f0(v)dυx dυy. (9.1.12)




Ð(k,ω) = 1−ω2p

k2

∫ ∞

−∞


dυz = 0. (9.1.13)


∫ ∞

−∞


dυz =[

F0(υz −ω/k)

]∞

−∞+

∫ ∞

−∞

F0(υz −ω/k)2

dυz. (9.1.14)


Ð(k,ω) = 1−ω2p

k2

∫ ∞

−∞

F0(υz −ω/k)2

dυz = 0. (9.1.15)





k2Φ̃ =e2

ϵ0mkΦ̃

∫ ∞

−∞

(∂ f0/∂υz)(kυz −ω)

d3υ. (9.1.9)


2

ϵ0mk2

∫ ∞

−∞


d3υ]Φ̃ = 0. (9.1.10)


Ð(k,ω) = 1− e2

ϵ0mk2

∫ ∞

−∞


d3υ = 0. (9.1.11)


F0(υz) =1n0

∫ ∞

−∞f0(v)dυx dυy. (9.1.12)




Ð(k,ω) = 1−ω2p

k2

∫ ∞

−∞


dυz = 0. (9.1.13)


∫ ∞

−∞


dυz =[

F0(υz −ω/k)

]∞

−∞+

∫ ∞

−∞

F0(υz −ω/k)2

dυz. (9.1.14)


Ð(k,ω) = 1−ω2p

k2

∫ ∞

−∞

F0(υz −ω/k)2

dυz = 0. (9.1.15)





k2Φ̃ =e2

ϵ0mkΦ̃

∫ ∞

−∞

(∂ f0/∂υz)(kυz −ω)

d3υ. (9.1.9)


2

ϵ0mk2

∫ ∞

−∞


d3υ]Φ̃ = 0. (9.1.10)


Ð(k,ω) = 1− e2

ϵ0mk2

∫ ∞

−∞


d3υ = 0. (9.1.11)


F0(υz) =1n0

∫ ∞

−∞f0(v)dυx dυy. (9.1.12)




Ð(k,ω) = 1−ω2p

k2

∫ ∞

−∞


dυz = 0. (9.1.13)


∫ ∞

−∞


dυz =[

F0(υz −ω/k)

]∞

−∞+

∫ ∞

−∞

F0(υz −ω/k)2

dυz. (9.1.14)


Ð(k,ω) = 1−ω2p

k2

∫ ∞

−∞

F0(υz −ω/k)2

dυz = 0. (9.1.15)



16

Tom O’Neil (1965)





17

Next Week: Read…


9

Electrostatic Waves in a HotUnmagnetized Plasma

In this chapter we investigate the propagation of small-amplitude waves in a hotunmagnetized plasma. Because of the shortcomings of the moment equations,the approach used is to solve the Vlasov equation directly using a linearizationprocedure similar to that used in the analysis of waves in cold plasmas and MHD.Although both electromagnetic and electrostatic solutions exist, the discussion inthis chapter is limited to solutions that are purely electrostatic, i.e., the electric fieldis derivable from the gradient of a potential, E = −∇Φ. Electromagnetic solutionsare discussed in the next chapter.

From Faraday’s law it is easily verified that electrostatic waves have no magneticcomponent. This greatly simplifies the Vlasov equation by eliminating the v ×Bforce. For electrostatic waves, it is usually easier to solve for the potential ratherthan for the electric field. Therefore, in the following analysis, the electric fieldis replaced by E = −∇Φ and the potential is calculated from Poisson’s equation,∇2Φ = −ρq/ϵ0.

9.1 The Vlasov Approach

In an initial attempt to analyze the problem, we assume that normal modes of theform exp(−iωt) exist and represent them by using Fourier transforms, followingthe same basic procedure used in Chapter 4. This is the approach used by Vlasov(1945), who first considered this problem. As we will see, the Vlasov approachencounters a mathematical difficulty that can only be resolved by reformulatingthe linearized problem not as a normal mode problem using Fourier transform, butas an initial value problem using the technique of Laplace transforms.

As we have done before, we start by only considering the motion of theelectrons. The ions are considered to be immobile, with the same zero-ordernumber density, n0, as the electrons. The effects of ion motions will be considered

319


https://doi.org/10.1063/1.1761193

18

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