Lecture11: Plasma Physics 1. APPH E6101x Columbia University

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1 Lecture11: Plasma Physics 1 APPH E6101x Columbia University 1

2 Last Lecture Introduction to plasma waves Basic review of electromagnetic waves in various media (conducting, dielectric, gyrotropic, ) Basic waves concepts (especially plane waves) Electromagnetic waves in unmagnetized plasma Electrostatic waves in unmagnetized plasma 2

3 This Lecture Wave energy density and its relationship to the dispersion function, D(ω) Measurement of electrostatic plasma waves Waves in a (cold) magnetized plasma 3

4 Review of EM Waves E = B t ( ) E B = µ 0 j + ε 0 t ( E) = B t = t ( B) = µ 0 ε 0 2 E t 2 µ 0 j t 4

5 Review of EM Waves ( E) + 1 c 2 2 E t 2 = µ 0 ik Ê = iω ˆB j t ik ˆB = iωε 0 µ 0 Ê + µ 0 ĵ n, we discuss the wave equat 0 k (k Ê) = (kk k ] 2 I)Ê udes (6.7) can be transforme {kk k 2 I + ω2 c 2 I + iωµ 0σ (ω) } } {kk k 2 I + ω2 c 2 ε(ω) All of the plasma physics here Ê = 0 Ê = 0 5

6 Wave Energy Density (Poynting s Theorem) 6

7 Electrostatic Waves n t + x (nv) = 0 m v = q dφ dx γ n d(nk B T ) dx Electron Pressure Force 7

8 Electrostatic Plasma Waves n t + x (nv) = 0 iω ˆn + ikn 0 ˆv = 0 m v = q dφ dx γ n d(nk B T ) dx iωm ˆv = ikq ˆφ ikγ k B T ˆn Electron Pressure Force 8

9 Electrostatic Plasma Waves ω = ( ω 2 pe k2 v 2 Te) 1/2 = ω pe ( 1 + 3k 2 λ 2 ) 1/2 De Electron Pressure Force Electron Pressure Force 9

10 Electrostatic Ion Sound Waves Ion Pressure Force iωm i ˆv i = eê ik n i0 (γ i k B T i ) ˆn i 0 = eê ik n e0 (k B T e ) ˆn e ˆn i = ˆn e = ek iω 2 m i + ik 2 Ê γ i k B T i e Ê, ikk B T e Look! No electron acceleration Electron Pressure Force ik Ê = ( ni0 e 2 ) ε 0 m i k iω 2 + ik 2 γ i k B T i /m i Ê + ( ne0 e 2 ε 0 k B T e ) 1 ik Ê 10 =

11 Electrostatic Ion Sound Waves ε(k, ω) = 1 ( ω 2 pi ω 2 k 2 γ i k B T i /m i + 1 k 2 λ 2 De ( ) ω 2 = k 2 γ i k B T i + ω2 pi λ2 De ( m i 1 + k 2 λ 2 De ω kc s 1 + k 2 λ 2 De T e T i. d C s = ω pi λ De stic wave. 11

12 Energy Density for Electrostatic Waves 12

13 Damping and Dispersion 13

14 Slowly-Varying Wave 14

15 Slowly-Varying Wave 15

16 Electrostatic Wave Energy Conservation 16

17 Electrostatic Wave Energy Conservation 17

18 Electrostatic Wave Energy Conservation 18

19 John Malmberg and Chuck Wharton The first experimental measurement of Landau Damping 19

20 John Malmberg (obit, Nov 1992) Prof. Malmberg joined UCSD from General Atomics in 1969 as a professor of physics. Much of his work revolved around theoretical and experimental investigations of fully ionized gases or plasmas. The field could offer insights into how stars work and how to ignite and control thermonuclear reactions to produce fusion energy--the power that drives the sun. A plasma is the fourth state of matter, with solids, liquids and gases making up the other three. Most of the matter in the Universe is in the plasma state; for example, the matter of stars is composed of plasmas. In recent years, Prof. Malmberg had been experimenting with pure electron plasmas that were trapped in a magnetic bottle. By contrast with electrically neutral plasmas that contain an equal number of positive and negative electrons, pure electron plasmas are rare in nature. Before joining UCSD, Prof. Malmberg was director of the Plasma Turbulence group at General Atomics, where he carried out some of the first and most important experiments to test the basic principals of plasma physics. Perhaps his most important experiment involved the confirmation of the phenomenon called "Landau damping," where electrons surf on a plasma wave, stealing energy from the wave and causing it to damp (decrease in amplitude). For his pioneering work in testing the basic principals of plasma, and for his more recent work with electron plasmas, Prof. Malmberg was named the recipient of the American Physical Society's James Clerk Maxwell Prize in Plasma Physics in

21 Chuck Wharton (emeritus, Cornell) 21

22 22

23 Description of the Experimental Device 4 Movable Probes Coils Magnetic Mirror K Gauss em ,'-i Charged Plale (Eleclroslalic Eleclron Reflector) [ I Plasma Diameter V Supressor Grid LDuoPlosmolron c V P to acuum ump He Supply Electron Gun FIGURE 10.8,4 Schematic of experiment used to investigate plasma wave echoes. [After j, H. Malmberg, et al., Proceedings of Conference on Phenomenes d'ionization dans les Gaz, 4; 229 (1963).1 23

24 Description of the Experimental Device 548 PRINCiPLES OF PLASMA PHYSICS [ 4 Movable Probes I Coils Magnetic Mirror K Gauss em ,'-i Plasma Diameter V Supressor Grid LDuoPlosmolron c V P to acuum ump He Supply Charged Plale (Eleclroslalic Eleclron Reflector) Electron Gun FIGURE 10.8,4 Schematic of experiment used to investigate plasma wave echoes. [After j, H. Malmberg, et al., Proceedings of Conference on Phenomenes d'ionization dans les Gaz, 4; 229 (1963).1 If L is large compared with the Landau damping length, and if w,/(w, - WI) is of order unity, this third electric field, which is the spatial plasma echo, appears at a position well separated from the first two electric field excitation positions. The experiment used by Malmberg et al. 1 to study the spatial plasma wave echoes is depicted schematically in Fig The plasma column is 180 cm long and 5 cm in diameter, with a central density of 1.5 x 10 8 cm- 3. The axial magnetic field is 300 G and can be regarded as infinite for the purposes of the experiment. The plasma has a temperature of 9.4 ev and a Debye length of 2 mm. The electron mean free path is 10 5 cm for electron-ion collisions and 4 x 10 4 cm for electron-neutral collisions. The plasma column is surrounded by a 5.2-cmradius cylinder that acts as a waveguide beyond cutoff and reduces the stray electromagnetic coupling between the excitation and detection probes. A plasma wave echo obtained with this experiment is shown in the lower trace of Fig The upper trace is the spatial distribution of the 120-MHz signal in the vicinity of the excitation probe at x O. The middle trace is the spatial distribution of the l30-mhz signal in the vicinity of the second probe at 1 J. H. Malmberg, C. B. Wharton, R. W. Gould, and T. M. O'Neil, Phys. Fluids, 11: 1147 (1968). 24

25 25 Look!

26 TG Modes: Low Frequency Surface Waves 26

27 Raw Data 27

28 Waves in Magnetized Plasma v (α) t = q α m α ( E 1 + v (α) B 0 ) α = e, i. q ˆv x = i ωm (Ê x +ˆv y B 0 ), q ˆv y = i ωm (Ê y ˆv x B 0 ) ˆv ± =ˆv x ± i ˆv y, Ê ± = Ê x ± iê y ˆv ± q = i ωm (ʱ i ˆv ± B 0 ) ˆv ± = i q m ʱ 1 ω sω c 28

29 ˆv x ˆv y ˆv z Waves in Magnetized Plasma q = i ωm ε ω = I + ω 2 sωω c ω 2 ωc 2 i ω 2 ωc 2 0 sωω c ω 2 Ê x i ω 2 ωc 2 ω 2 ωc 2 0 Ê y Ê z ωpα 2 α ω 2 ω 2 i ωpα 2 s α cα α ω 2 ωcα 2 σ (ω) = iωε 0 i ωpα 2 ω cα ωpα 2 s α α ω 2 ωcα 2 ω ω α 2 ωcα i and with the aid of (6.14) the dielectric tensor ωε 0 σ ω. ω cα ω α 0 0 ω 2 pα ω 2 29

30 Waves in Magnetized Plasma ε ω = I + ε(ω) = i S id 0 id S P ωε 0 σ ω. S = 1 α D = α s α P = 1 α ω 2 pα ω 2 ω 2 cα ω 2 pα ω 2 ω 2 cα ω 2 pα ω 2. ω cα ω Working both in the laboratory and with theoretical calculations, he found many ways to put waves to work in fusion research in succeeding decades, and his 1962 book, ''The Theory of Plasma Waves,'' codified the subject in mathematical form for the first time. 30

31 Waves in Magnetized Plasma S N 2 cos 2 ψ id N 2 cos ψ sin ψ id S N 2 0 N 2 cos ψ sin ψ 0 P N 2 sin 2 ψ k = (k sin ψ, 0, k cos ψ). N k, which we w Ê x Ê y Ê z = 0 Good places to start: Propagation along B (ψ = 0) Propagation to B (ψ = π/2) S = 1 α D = α s α P = 1 α ω 2 pα ω 2 ω 2 cα ω 2 pα ω 2 ω 2 cα ω 2 pα ω 2. ω cα ω 31

32 Waves k B S N 2 id 0 id S N P Ê x Ê y Ê z = 0 E = E R E + = E L B 32

33 ( ) ( ) Waves k B Extra-Ordinary Mode ( ) S id id (S N 2 ) N X = ( S 2 D 2 S ( ) ( Êx ) 1/2 Ê y ) = 0 Plus: Ordinary Mode ω 2 = ω 2 pe + k2 c 2 ω uh = (ω 2 ce + ω2 pe )1/2 ω lh = ( ω 2 ci + ω2 pi ω2 ce ω 2 pe + ω2 ce ) 1/2 33

34 Extra-Ordinary Mode ( ) S id id (S N 2 ) Waves k B ( ) ( Êx Ê y ) = 0 Plus: Ordinary Mode ω 2 = ω 2 pe + k2 c 2 34

35 Next Lecture Chapter 7: Plasma Boundaries Probes (!) 35

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