Fundamentals of Plasma Physics Waves in plasmas

Transcription

1 Fundamentals of Plasma Physics Waves in plasmas APPLAuSE Instituto Superior Técnico Instituto de Plasmas e Fusão Nuclear Vasco Guerra 1

2 Waves in plasmas What can we study with the complete description of a plasma we already have? Study of the different waves that propagate in a plasma! -Send information out of the plasma -May become unstable, growing as they propagate and eventually destroying the plasma confinement 2

3 Linearization For small oscillations,we consider all the quantities in the form n=n0+n1 - n0: term of order zero (equilibrium values, i.e., without the perturbation) - n1: sinusoidal perturbation n 1 = n 1 exp(k r wt) 3

4 Wave descriprion u = u 0 exp[i( k r t)] k: wave vector; ω: angular speed Phase speed: Group speed: v ' = k v g = d dk In a dispersive medium: -vg v φ -Dispersion relation ω = ω(k) 4

5 Plasma oscillations Non-magnetized, cold, homogeneous, infinite and linear medium B=0; M [ni(r)=n0]; kte = 0; Electron motion along xx (longitudinal oscillations) It is the simplest system then we shall the constraints, one by one :) 5

6 Plasma oscillations System of equations: n e t + (n e v e )=0 m e n e apple ve t +( v e ) v e = en e E (Maxwell s equations) E = 0 6

7 Plasma oscillations We cannot use the plasma approximation (fast electron oscillations) We take n e = n 0 + n 1 n i = n 0 E = E 1 u x v e = v 1 u x 7

8 Plasma oscillations Keeping only the terms of first order, we get (...)! =! pe s n 0 e 2 0 m pe = Oscillations at the plasma frequency :) 8

9 Plasma oscillations dω/dk=0 the group speed is zero! We have only electrostatic oscillations, an initial wave packet does not propagate 9

10 Langmuir waves We now abandon the restriction of cold electrons We need to keep the Pe term due to the thermal agitation T e = T 0 + T 1 Assume (as before) motion along xx (longitudinal waves) 10

11 Langmuir waves Consider the wave compressions as adiabatic: -a typical electron travels only a small fraction of the wavelength in the time corresponding to the wave period ve/ω λ 11

12 Langmuir waves We consider a one-dimensional problem: -the collision frequency is small (the variation of temperature along the direction of wave propagation is not transmitted to the other directions) vc ω Unidimensional adiabatic compression: Υ=3 Pn -3 =cte 12

13 Langmuir waves n s m s apple vs t +( v s ) v s = q s n s E Ps Dispersion relation: (...) 2 = 2 pe +3k 2 v 2 te v 2 te = k BT e m 13

14 Langmuir waves Group speed: v g = d dk =3v2 te v The Langmuir waves are also denoted as electron plasma waves The adiabatic hypothesis means that the dispersion relation is valid for v te k pe k 14

15 Langmuir waves For kλe 1, the dispersion relation can be approximated by = ± pe k2 2 De 15

16 Langmuir waves 16

17 Langmuir waves: Raoul Franklin EscampigHerald_issue3.pdf Plasma Sources Sci. Technol. 18 (2009)

18 Dielectric constant In the absence of free charges, we can replace by E = 0 D =0 D = E 18

19 Dielectric constant For oscillating fields, we try to relate ρ with E, in such a way as to write (at 1 dimension) ik E ike = 0 1 ik E =0 ik 0 Note: in the general case we can write an equation of the form DE=0, where D is a matrix. The dispersion relation is then obtained from det(d)=0 19

20 Dielectric constant For cold waves (plasma oscillations...) (...) ( )=1 2 pe 2 The dispersion relation is obtained from the zeros of ε(ω)! ( )=0 = ± pe 20

21 Dielectric constant For the Langmuir waves (,k)=1 (...) 2 pe 2 3k 2 v 2 te (,k)=0 = ± pe 2 +3k 2 vte 2 1/2 21

22 Ion plasma waves Electron plasma waves: high frequency waves (ω>ωpe), the ion motion can be neglected Now: low frequency waves (ω<ωpi), where the ion motion is dominant We need 5 fluid equations: -2 for the ions -2 for the electrons -Poisson s equation 22

23 Ion plasma waves For longitudinal waves (motion along xx): n e t + x (n ev e )=0 n e m e apple ve t + v e v e x = en ee P e x 23

24 Ion plasma waves n i t + x (n iv i )=0 n i m i apple vi t + v i v i x = en ie P i x E x = e(n i n e ) We write Ps=γkTs ns, expand ns=n0+n1s, and keep only the terms of first order: 0 24

25 Ion plasma waves n e1 t + n 0 v e1 x =0 n 0 m e v e1 t n e1 = en 0 E 1 e k B T e0 x n i1 t + n 0 v i1 x =0 v i1 n i1 n 0 m i =+en 0 E 1 i k B T i0 t x E 1 x = e(n i1 n e1 ) 0 25

26 Ion plasma waves The ions move at low frequencies the electrons adjust almost instantaneously to this perturbation However, since they move very fast, they cannot completely cancel the perturbation in the ion density (they fail for potentials of the order of kte) 26

27 Ion plasma waves Isothermal electrons (Υe=1) Ionic compression: unidimensional adiabatic (Υi=3) Since we have low frequency waves, we use the plasma approximation ne1 ni1 (and avoid Poisson s equation) 27

28 Ion plasma waves (...) Sound speed 2 n e1 t 2 Dispersion relation: = c 2 s c s 2 n e1 x 2 ek B T e + i k B T i m i 2 = k 2 c 2 s 1/2 For a sound wave: c s =( kt/m) 1/2 In this limit, the ion plasma waves are known as ion acoustic waves 28

29 Ion plasma waves The ion thermal motion is responsible for the term proportional to Ti But... when Ti 0 the wave still exists! In the limit Ti Te, the sound speed is (Υe=1) c s ' r kb T e m i 29

30 Ion plasma waves Validity of the plasma approximation (i.e., in which conditions is the dispersion relation ω=csk valid): keep Poisson s equation neglect electron inertia 2 = k 2 (...) ik B T i m i + e k B T e m i 1 1+ e k 2 2 De 30

31 Ion plasma waves In the limit kλde 1 (large wavelengths as compared to the Debye length) we recover the ion acoustic waves The second term must be dominant (otherwise ω/k~kbti/mi~vi, many ions would have speeds of the order of the wave phase speed the fluid description fails and a kinetic description is required - Vlassov, to see later). 31

32 Ion plasma waves In the limit kλde 1 (λ λde) and Ti 0 we have ω ωpi (ion plasma oscillations: the electrons cannot shield the potential, the ions oscillate in a background of approximately uniform negative charge). 32

33 Ion plasma waves 33

34 Ion plasma waves 34

35 Ion plasma waves 35

36 Ion plasma waves 36

37 Ion plasma waves 37