流体プラズマ中の波動 - fusion.k.u-tokyo.ac.jp

1 Unit 4.. n 1 = n1 exp[i(k x t + n )]. n1 : k: k = 2 / n: . k = kx , n = 0 . n 1 = n1 cos(k x x t). k vp = . k k Re( ) > 0 Im( ) > 0 Im( ) < 0 . Im(k) . E1 = E1 exp[i(k x - t)]. E 1 : + .. vg = = x + y + z . k k x k y k z . E1= 1 E1 = 0 B perturbation E perturbation . k // E1 k E1 = B1 = 0 electrostatic Electron plasma wave . /k vte - Vlasov eq.. u . mn e e + (u e )u e = en e E p e t . n e + (ne u e ) = 0. t 0 E = e(n i n e ) Z = 1 . u0 = 0, E0 = 0 u e = u e x , E = Ex 1 . i mn0 u1 = en 0 E1 ikp1 = en 0 E1 ik3Tn1 1-D adiabatic = 3 . i n1 + ikn0 u1 = 0. ik 0 E 1 = en 1.. i 2 mn 1 e 2 n 0 n 1. = + ik3Tn 1. k ik 0. = (k) . T. 2 = pe 2 + 3k 2 = pe 2 + 3k 2 v te 2 Bohm-Gross dispersion relation m ne e 2. pe 2 = pe p . 0m e Ion acoustic wave adiabatic ion, isothermal electron vti << /k << vte.

2 U . Mn i i + (u i )u i = en iE pi t . i Mn i0 ui1 = en i0 ik 1 iTiikn i1. Boltzmann . e e e 1. n e = n e0 exp 1 n e0 1 + 1 n e1 = n e0. Te Te Te 1 Poisson . e 1 k2 . 0 E1 = 0 k 2 1 = e (n i1 n e1 ) = e n i1 n e0. e n i1 = n i0 0 1. Te Te e .. n i + (n i u i ) = 0 i n i1 = n i0 ikui1. t . n i1 en i0 kn i1. Mn i0 = + i Ti kn i1. kn i0 e k2. n i0 0. Te e . T. 2. Te /M. = + i i k 1+ k D. 2 2. M. T + i Ti 2. Te k D << 1 = e = c s2 cs: Te >> Ti c s2 =. k M M. m k D >> 1 2 = pe 2 = pi 2 pi p . M. E1 0 B perturbation . k E1 E1 = 0 e = 0 ne1 = 0 pe1 = 0 . Electromagnetic wave Maxwell . E1. ik B1 = 0 j1 i . c2. ik E 1 = i B1. 2 j1 . k k E1 = k 2 E 1 k(k E1 ) = E1 + i c 2. 0 . E0 = 0, B0 = 0, u0 = 0 . n0 e 2 E1. 1 i mu1 = eE 1 j1 = n 0 eu 1 = . i m (c 2. ). k 2 2 E1 =. i . 0. n e2. j1 = 0 E1 = pe 2 E1.

3 M 0. E1 0 . 1/ 2. pe 2 . = pe + c k 2 2 2 2. = ck 1 + 2 2 . c k . m 0 2. > pe n < nc = pe = . e2. ( 2 pe2 ) ( ). 1/ 2 1/ 2. i pe 2 2. < pe k= =. c c 2 2. ( ) . 1 /2. exp(ikx) = exp x . pe c .. ( pe ). 1/ 2.. c . 2. 2. << pe c/ pe collisionless skin depth . B = 0 X k B, c2 = 2 p2 . B = 0 L k || B, c2 = 2 p2 . B = 0 R k || B, c2 = 2 p2 .. Ion Acoustic Soliton ni u x e . = n, ix = u, = , pi t = , = . n0 cs D Te n . continuity eq. + (nu) = 0.. u u . ion eq. motion +u + =0.. ne el. eq. motion Boltzmann = e . n0. 2 . Poisson eq. =e n 2. n = 1 + n cs y = . n . + (u n + n u) = 0. y u u . +u + ( u) = 0. y y 2 2. = n L. y2 2. 1 |y| , u, n 0 u = n = . 2 . u u 1 3 u +u + =0 Korteweg-de Vries (K-dV) eq. y 2 y3. 2 nonlinear . 3 dispersion . K-dV eq.. x ct B 6. u = u + B sech 2 , c = u + , =.

4 3 B. Soliton ( , k) (2 , 2k), (3 , 3k), .. = kcs . soliton . Ponderomotive Force E = E0(r) cos( 0t) r(t) = r + r(t) . E 0 (r) = E 0 (r ) + r(t) E 0 (r ).. dv m = qE 0 (r) cos( 0 t)+ v B(r, t). dt Ponderomotive . 2 / 0 . dv m 2 q2 2. m = v = 2 E0 ( r ). dt 2 4m 0. Ponderomotive force .. Caviton and Envelope Soliton Electron Plasma Wave (EPW) . A cos(kx t) 2 . A [cos(2kx 2 t) + 1]. 1 2. A2 cos 2 (kx t) =. 2. k = 0, = 0.. EPW . 2 2 . 2. u + 2. u 3v u =0. t 2 ex pe ex te x2 ex ( pe2 ne) (vte2 Te). k 0 3 . n0 n0 + ne(x,t) ne .. 2 2 . 2 ne (x, t) . 2 u ex + pe 1 + u =0. 2. n0 ex u 3v t 2 ex te x . u ex (x, t) = u(x, t) exp( i pe t) + u *(x, t) exp(i pe t) u 2u/ t2 .. 3 v te2 2 pe n e (x, t). i u(x, t) + 2 u(x, t) u(x, t) = 0. t 2 pe x 2. 2 n0. 3 Schr dinger eq.. soliton envelope soliton.

5 Envelope Soliton ne(x,t) u(x,t) pe ne . u ex 1 p e . me + m e uex u ex = +e t x n e x x 1 2 . me T n e . x 2. uex 2 =. x (2. ). m e u(x, t) ponderomotive force 3 e n0 x . T . x ( 2. ). me u(x, t) = e n e e . x n 0 . u(x,t) 0 ne, 0 . ne e m e u( x , t ) 2. =. n0 Te .. u i 1 . mi = e (n T ). t x n 0 x i i . n i e = Ti n0. ni = ne . ne me u( x , t ) 2. =. n0 Te + Ti u(x,t) . 3 v te2 2 pe me 2. i u(x, t) + 2 u(x, t) + u(x, t) u(x, t) = 0. t 2 pe x 2. 2 Te + Ti 2 diffraction . nonlinear Schr dinger eq.. ponderomotive force . ponderomotive force . (A) (B) .. cold plasma approximation (Te = Ti = 0) . k B0 k = k x , B0 = B0 z . ordinary wave (O-mode) E1 // B0, E1 k . extraordinary wave (X-mode) E1 B0. (. i mux1 = e E x1 + u y1B 0 ). i muy1 = e(E y1 u x1B 0 ).. e i E x1 + c Ey1. u x1 =.

6 M c2 2. e i E y1 cE x1. u y1 =. m c2 2.. 2 j1 . k E 1 k(k E1 ) = E1 + i ; j1 = n0eu1. 2. 2. c 0 . ( c2 2 + p 2 )Ex1 i p2 c Ey1 = 0. c c 2 k2 . i p 2.. ( ). E x1 + 1 2 c 2 + p 2 E y1 = 0.. 2.. E1 0 det = 0 . c 2k 2 c2 (. p 2 2 p 2 ) h 2 p 2 + c 2 h: upper hybrid frequency ( ). = = 1 ;. 2. vp 2 2 2 h2. resonance (k ) = h c 2 + 4 p 2 c R R : right hand cutoff cutoff (k 0) = . 2 L L : left hand cutoff k // B0 B0 = B0 z . k E1 = 0 (k E1). j1 = n0eu1 . c c 2 k2 . i p 2.. ( ). E x1 + 1 2 c 2 + p 2 E y1 = 0.. 2.. c 2 k2 .. ( 2. ). 1 2 c 2 + p 2 E x1 i p 2 c E y1 = 0.. det = 0 . c 2k 2 c2 p2 + : L wave E y1 = iE x1. = = 1 . 2. vp 2 ( c ) : R wave E y1 = +iEx1. L(R) wave . ( i t )= E x1 cos( t ). E x1 (t) = Re E x1e E y1 (t) = Re(miE x1e i t ) = mE x1 sin( t). ( ) . L wave cutoff at = L > L.

7 R wave cutoff at = R > R . resonance at = c < c . vp(R) > vp (L) Faraday rotation .. warm plasma - . B0 = B0 z , k = kx x + kz z = ksin x + kcos z .. u1. mn 0 = qn 0 (E1 + u 1 B 0 ) T n1. t ( ). i mux1 = q E x1 + uy1 B0 ik x T. n1. n0. (. i muy1 = q E y1 ux1B0 ). n1. i muz1 = qE z1 ikz T. n0. = (ux1sin + uz1 cos ) . n1 k . n0 . (. i mux1 = q E x1 + uy1 B0 i ) k2.. (. T u x1sin 2 + u z1sin cos ). (. i muy1 = q E y1 ux1B0 ). i muz1 = qE z1 i k2.. (. T ux1sin cos + uz1 cos 2 ). j1 = n0qu1 E1 : .. 2 . k E 1 k(k E1 ) = I + i E = 0 E1 0 I + i 2 2. : . 2. c 0 1 0 . det = 0 . 2 0 k 2 I + kk = 0. Cold Plasma Dispersion Relation Te = Ti = 0 . ck c index of refraction n = Stix notation . vp p2 p 2. R 1 p = pe, c = ce ( c ) ( + c ). p2 p 2. L 1 p = pi, c = ci ( + c ) ( c ). R+L. S . 2.

8 R L. D . 2. p 2 p 2. P 1 2. 2 . nx = n sin , nz = n cos . S n 2 cos2 iD n 2 sin cos Ex1 .. S n Ey1 = 0. 2. iD 0. n 2 sin cos . 0 P n 2 sin 2 E z1 . det = 0 . tan 2 =. ( )( ). P n 2 R n 2 L. (Sn RL)(n2 P). 2. // propagation ( = 0): n2 = R (R wave) n2 = L (L wave). propagation ( = /2): n2 = P (O mode) n2 = RL/S (X mode). cutoff (n2 0): PRL=0 = p, R, L . resonance (n2 ): tan2 = P/S . =0: P = 0 = p, S = c (R ), = c (L ). = /2: S = 0 = h (upper hybrid), = lh (lower hybrid). Shear Alfv n wave // propagation k // B0, E1 B0, k E1 // B0 // k ion acoustic wave . B0 = B0 z Ez1 = 0, uz1 = 0, jz1 = 0 c << c . e i Ex1 + c E y1 e E y1 e E y1. u xe1 = =. m c . 2 2. m c M c e i Ey1 c E x1 e E x1 e E x1. u ye1 = = . m c . 2 2. m c M c e i E x1 c E y1. u xi1 = . M c 2 2. e i E y1 + c E x1. u yi1 =.

9 M c 2 2.. i c 1 . 2 . E x1 . n e c 2 2 c2 c . j1 = 0 . M c 1 i = E1. 2 +. E . c y1 . 2. c 2. c 2.. 2 0 k 2 I + kk = 0 , 0 I + i n = ck/ . 0 . p2 p 2 . 1 n2 + i (. I 1 n 2. ). + nn + i . =. c2 2 (. c c2 2 ) =0. 0 p 2. p2. i 1 n2 +. (. c c2 2 ) c 2 2.. p2. c + . c + R wave n2 = . c L wave zz . Shear Alfv n R wave cutoff / resonance . R wave = c . p2. 0 n = 1 +. c2. c eB 0 M B. Alfv n speed v A = c =c = vA << c . p M ne 2 0 nM. c c c vp = = = = vA. k n p 2. c 2. 1+ 1+ 2. c2 vA. Shear Alfv n L wave = c resonance = L = c + p2/ c cutoff p2. 0 n = 1 + R wave Shear Alfv n R wave c2. R wave L wave << c R L . Faraday rotation . << c E1 B0 . Shear Alfv n wave u1 = 0 . k u1 = 0 . Shear Alfv n L wave Magnetosonic wave propagation k B0, E1 B0 X-mode E1 // B0 O-mode < p . B0 = B0 z , k = k x c << << c electron polarization drift.

10 P2 p2 p 2 . 1+ + i c2 2 c 2 (. c c2 2 ). = 0 . p 2. p2 p2. i 1 n2 + +. (. c c2 2 ) c 2 2 c 2. 2. c 2 2 + p 2 p 2 c 2 2 + p 2 p 2 . 2. p 2 . n . 2. + = + . c 2 2 c 2 c 2 2 c 2 c c2 2 ( ) . n2 0 n . 2 = lh 2 =. (. c c c2 + p 2 ) lh: lower hybrid frequency p + c c 2. 1 1 1. p2 >> c2 = +. lh 2 p 2 c c p 2 p 2 p 2. 0 n 2 = 1+ + 1+ Alfv n wave . c 2 c2 c 2. magnetosonic wave (compressional Alfv n wave k u1 0 .. Compressional Alfv n wav)