8 Waves in Uniform Magnetized Media

Transcription

1 8 Wave n Unform Magnetzed Meda 81 Suceptblte The frt order current can be wrtten j = j = q d 3 p v f 1 ( r, p, t) = ɛ 0 χ E For Maxwellan dtrbuton Y n (λ) = f 0 (v, v ) = 1 πvth exp (v V ) v th 1 πv th v vth χ = ê ê pv k vth + p e λ Y n (λ) n I n λ A n n(i n I k ni n n)a n Ω λ B n n n(i n I n)a n λ I n + λi n λi n A n k Ω(I n I n)b n k ni n Ω λ B n k Ω(I n I n)b ( nω) n k vth B n A n = 1 I n = I n (λ), λ = k v th Ω = 1 k < ρ L > T T T + 1 ( k V nω)t + nωt T Z 0 (ζ n ) B n = 1 ( nω)t (k V nω)t k T + 1 nω ( k V nω)t + nωt Z 0 (ζ n ) k T ζ n = k V nω For V = 0, T = T A n = 1 k v th Z 0 (ζ n ) B n = 1 k 1 + ζ n Z 0 (ζ n ) 30

2 8 Parallel Propagaton For parallel propagaton k B 0, and n the lmt k 0 (λ 0), e λ Y 0 (λ) = V ê ê k vth B 0 e λ Y ±1 (λ) = 1 1 ± A ± The dperon relaton can be derved a n = 1 + p T T T + ( k V ± Ω)T ΩT T Z 0 k V ± Ω whch correpond to cold plama dperon relaton n = R and n = L, and 0 = 1 + p k v th 1 + k V k V Z 0 whch correpond to cold plama dperon relaton 0 = P For the cae V = 0, T = T, 0 < R Ω e k c 1 p { 1 Ω( Ω) + k T m( Ω) 3 π k v th The dampng rate (aumng k real and r ) gven by { 3 4 } p πω Ω r Ω 3 Ω k 4 c4 k k v th The dampng length (aumng real and k k r ) gven by { k } p πω Ω k Ω k c k v th k v th For Ω n = 1 p Ω + p k v th Z 0 (ζ 1 ) Note that Z 0 (ζ) max = π at ζ = 0 For on cyclotron wave πω k 3 p < c v th and k doe not become nfnte a approache Ω } Ω k v th 31

3 83 Cyclotron Harmonc Dampng Wave aborpton at harmonc of cyclotron frequency ( = nω) can occur when fnte k effect taken nto account Conder the cae of econd harmonc n = wth E = ŷe n(k x x t), = Ω, k x = π/(ρ L ) The rate of perpendcular knetc energy ncreae gven by dw dt = d dt mv = m v d v dt = q v E + v B = q v E Subttutng the unperturbed orbt x = ρ L n(ωt + φ) n the expreon for E gve dw dt = qeωρ L co k ρ L n(ωt + φ) t co(ωt + φ) = qeωρ L R J n (k ρ L )e R n(ωt+φ) t e (Ωt+φ) n= It can be een that when = (n ± 1)Ω, W wll keep ncreang (or decreang) Fg 1 Schematc howng the mechanm of nd harmonc cyclotron dampng A an example, conder the cae of the fat wave at = Ω Aumng r, V = 0, and Ω, n whch cae { π } Ω IA k v th The wave equaton can be wrtten a ( Lh + R n ) L h R Ex L h + R L h + R n = 0 E y 3

4 where L h = L + L and L = (p /)λia The dperon relaton gven by n (R n )(L n ) S n 1 R n S n L = 0 Th can be olved to obtan the dampng rate R n { π p } Ω λ Ω S n k v th c k Ω + k ( + Ω) 84 Trant Tme Dampng When the wave magnetc feld ha a gradent along the confnng magnetc feld, the equaton of moton can be wrtten a m dv dt = µ ˆb B( r, t) where µ = mv /(B 0) the magnetc moment of a charged partcle Dampng decrbed by th equaton called trant tme dampng, whch the magnetc analog of Landau dampng decrbed by m dv dt = q ˆb φ Trant tme dampng and Landau dampng both correpond to the n = 0 cae, and the two procee are coherent The net reult of the total nteracton depend on the relatve phae of E and B (1) It not correct to calculate them eparately and add them together, but cro term mut be condered Fg Schematc howng the mechanm of trant tme dampng A an example, conder the cae of compreonal Alfvén mode wth k v (e) th Ω where ɛ n 0 n n 0 ɛ + ɛ n (ɛ m + ɛ m ) n n (ɛ m + ɛ m ) ɛ + ɛ n ɛ = S 1 + ρ ɛb, ɛ = λ T T δ E x E y E z = 0 33

5 ) ɛ = ( p RZ 0, ɛ = ζ k v 0δ th ɛ m = 1 n δ = n p p T RZ 0 Ω T e π e ζ k v th 0 e e, ɛ m = n n, ζ 0 = Ω e T T δ Dampng decrbed by the ant-hermtan part of the wave equaton whch occur only n the lower rght part of the matrx, o the relevant wave equaton ( ɛ + ɛ n ) (ɛ m + ɛ m ) Ey (ɛ m + ɛ m ) ɛ + ɛ n = 0 E z The real part of the dperon relaton gve and the magnary part gve (ɛ n )ɛ ɛ m 0 ɛ (ɛ n ) ɛ m + ɛ ɛ + ɛ m ɛ ɛ m ɛ m 0 ɛ The lat three term repreent trant tme dampng, Landau dampng, and cro term, repectvely In the cae T = T, the coherence between B z (1) (or equvalently E y ) and E z gven by 85 Power Aborpton E y ɛ + ɛ Ω e = E z ɛ m + ɛ m k k vth Under the aumpton that r, power aborpton can be calculated once the electrc feld E gven, P = E j = q n E < v > In complex notaton, P = P = n q ( E < v > + E 4 < v > ) = ɛ ( 0 E χ χ 4 ) E = ɛ 0 E χ a 4 E 34

6 For the cae of Landau dampng wth E = ẑe, { nq P = k T e λ I 0 (λ) ɛ 0 E } π( k V ) k V ɛ 0 k v th For the cae of trant tme dampng wth E = ˆxE x + ŷe y, E = 0, P = β T e λ I 0 (λ) I T 0(λ) B µ 0 (1) π( k V ) k V k v th If B (1) 0 and E 0, Landau dampng and trant tme dampng are coherent, and cro term ( E B(1) ) mut be ncluded For the cae of cyclotron dampng wth E = 0, P = ɛ 0p { n e λ I n λ E xe x n(i n I n)(e xe y EyE x ) n= ( n ) I n + λ + λi n λi n E ye } y IA () n In the lmt λ 1 P ɛ 0 p n=1 n λ n 1 ( E + IA n + E ) IA n (n 1)!, and for hfted Maxwellan IA n = π ( k V nω)t + nωt k V nω k v th T 35