Unit 4. n 1 = n 1 ex[i(k x ωt + δ n )] n 1 : k: k = π/λ δ n : k = kxˆ, δ n = n 1 = n 1 os(k x x ωt) v = ω k k k Re(ω) > Im(ω) > Im(ω) < Im(k) E 1 = E 1 ex[i( k x - ωt)] E 1 : + v g ω ω ω ω = = xˆ + yˆ + zˆ k k k k x y z E 1 = φ 1 E 1 = B erturbation E erturbation k // E 1 k E 1 = ωb 1 = eletrostati Eletron lasma wave ω/k v te - Vlasov eq. u mn e e t + ( u e )u e = en ee e n e t + ( n e u e )= ε E =e( n i n e ) Z = 1 u =, E = u e = u e xˆ, E = Exˆ 1 iωmn u 1 = en E 1 ik 1 = en E 1 ik3tn 1 1-D adiabati γ = 3 iωn 1 + ikn u 1 = ikε E1 = en1
iω mn1 e n n1 = + ik3tn1 k ikε ω = ω (k) ω = ω e + 3k T m = ω e + 3k v te Bohm-Gross disersion relation ω e = n e e ε m e ω e ω Ion aousti wave u Mn i i t adiabati ion, isothermal eletron v ti << ω/k << v te + ( u i )u i = en ie i iωmn i u i1 = en i ikφ 1 γ i T i ikn i1 Boltzmann n e = n e ex eφ 1 n e 1 + eφ 1 eφ n e1 = n 1 e 1 Poisson ε E 1 = ε k eφ φ 1 =en ( i1 n e1 )= en i1 n 1 e k e n i1 = n i ε φ 1 e n i t + ( n i u i )= iωn i1 =n i iku i1 ω Mn i ωn kn i1 i = n i en e T e i kn ε i1 k e + γ T kn i i i1 ω k T = e /M 1+ k λ + γ i T i D M kλ D << 1 ω k = + γ i T i M = s s : >> T i s = M kλ D >> 1 ω = m M ω e = ω i ω i Ω
E 1 B erturbation k E 1 E 1 = σ e = n e1 = e1 = Eletromagneti wave Maxwell ik B 1 = µ j 1 iω E 1 ik E 1 = iωb 1 k k E 1 = k E 1 kk ( E 1 )= ω j E 1 + i 1 ε ω E =, B =, u = 1 iωmu 1 = ee 1 j 1 = n eu 1 = n e E 1 iωm ( k ω ) E = j = E1 = ωe E1 1 iω ε 1 n e mε E 1 ω = ω e + k ω =k 1+ ω e k ω > ω e n<n = mε ω ω e = ω ω < ω e k= ω ( ω e ) 1/ e 1/ = ±i ( ω e ω ) 1/ 1/ ex(ikx) = ex ω e ω x ( ω e ω ) 1/ ω << ω e /ω e ollisionless skin deth
B = X k B, ω = ω B = L k B, ω = ω B = R k B, ω = ω
Ion Aousti Soliton n i n = n, u ix s = u, x λ D = ξ, ω i t=τ, eφ = ϕ ontinuity eq. ion eq. motion n τ + ξ ( nu)= u τ + u u ξ + ϕ ξ = el. eq. motion Boltzmann n e n = e ϕ Poisson eq. ϕ ξ = eϕ n n = 1 + n s y = ξ τ n τ + ( y u n + n u)= u τ + u u y + ( y ϕ u)= ϕ y = n ϕ ϕ L 1 y ϕ, u, n u = n = ϕ u τ + u u y + 1 3 u 3 = Korteweg-de Vries (K-dV) eq. y nonlinear 3 disersion K-dV eq. u = u + B seh x t δ, = u + B 3, δ = 6 B (ω, k) (ω, k), (3ω, 3k), Soliton ω = k s soliton
Ponderomotive Fore E = E (r) os(ω t) r(t) = r + r(t) E (r) = E (r) + r(t) E (r) m dv dt = qe (r) os(ω t)+ v B(r,t) Ponderomotive π/ω m dv dt = m v q = 4mω E (r ) Ponderomotive fore Caviton and Enveloe Soliton Eletron Plasma Wave (EPW) A os(kx ωt) [ ] A os ( kx ωt)= 1 A os( kx ωt)+ 1 k =, ω = EPW t u ex + ω e u ex 3v te x u ex = (ω e n e ) (v te ) k 3 n n + δn e (x,t) δn e t u ex 3v te x u ex + ω e 1+ δn e (x, t) u ex = n u ex (x,t) = u(x,t) ex( iω e t) + u *(x, t) ex(iω e t) u u/ t i t u(x,t) + 3 v te ω e x u(x, t) ω e δn e (x,t) u(x,t) = n
3 Shrödinger eq. soliton enveloe soliton Enveloe Soliton δn e (x,t) u(x,t) ω e δn e m e u ex t + m e u ex x u ex = 1 e n e x + e φ x 1 m e x u ex = x ( m e u(x,t) ) onderomotive fore 3 δn e n x ( x m e u(x,t) )= δn e eφ x n u(x,t) δn e, φ δn e n = eφ m e u(x,t) φ m i u i t = e φ x 1 n ( x n it i ) eφ = T i δn i n δn i = δn e
δn e n = m e u(x,t) + T i u(x,t) i t u(x,t) + 3 v te ω e x u(x, t) + ω e m e +T i u(x,t) u(x,t) = diffration nonlinear Shrödinger eq. onderomotive fore onderomotive fore (A) (B)
old lasma aroximation ( = T i = ) k B k = k xˆ, B = B ẑ ordinary wave (O-mode) E 1 // B, E 1 k extraordinary wave (X-mode) E 1 B iωmu x1 = ee x1 + u y1 B iωmu y1 = ee y1 u x1 B u x1 = e m iωe x1 + ω E y1 ω ω u y1 = e m iωe y1 ω E x1 ω ω k E 1 kk ( E 1 )= ω j E 1 + i 1 ; j 1 = n eu 1 ε ω ω ω ( + ω )E x1 iω ω ω E y1 = iω ω ω E x1 + 1 k ω ( ω ω )+ ω E y1 = E 1 det = k ω v =1 ω ω ω ω ω ω h = resonane (k ) ω = ω h ; ω h ω + ω ω h : uer hybrid frequeny utoff (k ) ω = ω + 4ω ± ω ω R ω L ω R : right hand utoff ω L : left hand utoff k // B B = B ẑ k E 1 = (k E 1 ) j 1 = n eu 1 iω ω ω E x1 + 1 k ω ( ω ω )+ ω E y1 = 1 k ω ( ω ω )+ ω E x1 iω ω ω E y1 =
det = k ω = v =1 ω ωω± ω + : L wave E y1 = ie x1 : R wave E y1 = +ie x1 L(R) wave E x1 (t) = Re( E x1 e iωt )= E x1 os(ωt) E y1 (t) = Re( mie x1 e iωt )= me x1 sin(ωt) ( ) L wave utoff at ω = ω L ω > ω L R wave utoff at ω = ω R ω > ω R resonane at ω = ω ω < ω v (R) > v (L) Faraday rotation warm lasma - B = B ẑ, k = k x xˆ + k z ẑ = ksinθ xˆ + kosθ ẑ mn u 1 t = qn ( E 1 + u 1 B ) γt n 1 iωmu x1 = qe ( x1 + u y1 B ) ik x γt n 1 n iωmu y1 = qe ( y1 u x1 B ) iωmu z1 = qe z1 ik z γt n 1 n n 1 = k ( n ω u x1sinθ + u z1 osθ) iωmu x1 = qe ( x1 + u y1 B ) i k ω γtu ( x1sin θ + u z1 sinθ osθ) iωmu y1 = qe ( y1 u x1 B ) iωmu z1 = qe z1 i k ω γtu x1sinθ osθ + u z1 os θ
j 1 = n qu 1 σ E 1 σ : k E 1 kk ( E 1 )= ω det = σ I + i E 1 = ω σ µ ε E 1 ε ε I + i : ε ω ε ω ω µ ε I + kk = k Cold Plasma Disersion Relation = T i = k index of refration n = Stix notation ω v ω R 1 ωω ω L 1 ωω+ ω Ω ω ωω+ Ω ωω ω = ω e, ω = ω e Ω = ω i, = ω i S R + L D R L P 1 ω ω Ω ω n x = n sinθ, n z = n osθ S n os θ id n sinθ osθ id S n n sinθ osθ P n sin θ E x1 E y1 E z1 = det = ( tan θ = Pn R)n ( L) ( Sn RL)n ( P) // roagation (θ = ): n = R (R wave) n = L (L wave) roagation (θ = π/): n = P (O mode) n = RL/S (X mode) utoff (n ): PRL= ω = ω, ω R, ω L θ resonane (n ): tan θ = P/S θ θ =: P = ω = ω, S ω = ω (R ), ω = (L ) θ = π/: S = ω = ω h (uer hybrid), ω = ω lh (lower hybrid)
Shear Alfvén wave // roagation k // B, E 1 B, k E 1 // B // k ion aousti wave B = B ẑ E z1 =, u z1 =, j z1 = ω <<ω u xe1 = e m iωe x1 + ω E y1 ω ω e m E y1 ω = e M E y1 u ye1 = e m iωe y1 ω E x1 ω ω e m E x1 ω = e M E x1 u xi1 = e M iωe x1 E y1 ω u yi1 = e M iωe y1 + E x1 ω j 1 = n e M iω Ω ω Ω Ω ω + 1 Ω ω 1 Ω iω Ω ω E x1 E y1 = σ E 1 k σ ω µ ε I + kk =, ε ε I + i ε ω n = k/ω Ω 1 n + I( 1 n σ Ω )+ nn + i ε ω = ω Ω i ω Ω ω 1 n + Ω i ω Ω ω Ω ω = n = + Ω ± ω ± ω + R wave L wave zz Shear Alfvén R wave utoff / resonane R wave ω = ω ω n = 1 + Ω
Alfvén seedv A = Ω = eb M ε M ne = B µ nm v A << v = ω k = n = 1 + Ω = 1+ v A v A Shear Alfvén L wave ω = resonane ω = ω L = + Ω / utoff ω n = 1 + Ω R wave Shear Alfvén R wave R wave L wave ω << R L Faraday rotation ω << E 1 B Shear Alfvén wave u 1 = k u 1 = Shear Alfvén L wave
Magnetosoni wave roagation k B, E 1 B X-mode E 1 // B O-mode ω < ω B = B ẑ, k = k xˆ << ω << ω eletron olarization drift Ω 1+ Ω ω + ω ω Ω i ω Ω ω Ω i ω Ω ω Ω 1 n + Ω ω + ω ω = n Ω ω +Ω Ω ω + ω ω = ω +Ω Ω ω + ω ω Ω ω ( Ω ω ) n n ω = ω lh = ω +Ω Ω + ω Ω >> 1 ω lh = 1 Ω + 1 ω ω n = 1+ Ω Ω + ω ω lh : lower hybrid frequeny ω 1+ Ω Alfvén wave magnetosoni wave (omressional Alfvén wave k u 1 Comressional Alfvén wave