67 1 Landau Damping and RF Current Drive Kazuya UEHARA* 1 Abstract The current drive due to the rf travelling wave has been available to sustain the plasma current of tokamaks aiming the stational operation. Simple derivation of Landau damping and radio-frequency current drive is described on the standpoint of particle acceleration and deceleration by the rf potential, whereas the current drive is usually described by the quasi-linear theory. This picture is available to understand the physical picture of Landau damping and the current drive. This report starts from the original explanation of Landau damping and then describes the picture of the Landau damping due to the potential as well as the application to the current drive. Finally the new formation of the current drive theory is tried to given. Keywords: Landau damping, radio-frequency current drive, particle trapping, physical picture 1 10 60 1 On the oscillation of the electron plasma Institute of physical problem 1945 6 2 J. Phys. USSR 1946 2) 3 4) 1987 1 Japan Atomic Energy Agency
68 JAXA-RR-09-003 1. 1 1 2 r v m Ó e ε 0 E = - Ó/ r f f 0 f 1 f = f 0 + f 1 x k f 1 = f k e ikx Ó = Ó k e ikx Ó k 3
2007/2008 69 g = f k (v, t = 0) ε v, e -i t Im ( ) - >- å g f 0 ε = k-iî k Ó k e -i k t-î k t k Î k kx<<1 p Î k. 4 N = ˇå -å f 0dv p = (e 2 N/e 0 m) 1/2 Dawson Jackson C-S Wu 5 7 f Ecos(kz- t) 5 t = 0 v = v 0 z = z 0 z = v 0 t+z 0 v 1 = (ee/mï[sin (kz 0 + Ït)-sin (kz 0 )] 6 Ï = kv 0 - z = z 0 + v 0 t + z 1 5 kz 1 << 1 cos(kz 1 )~1 sin(kz 1 )~kz 1 mv 2 /2 z 0 7 f t å Ë 8 df (v))/dv < 0 df (v))/dv > 0
70 JAXA-RR-09-003 8) Í = Í 0 cos (kz - t) /k = V z = z-vt dz /dt = v, dz/dt = v v = v-v Í = Í 0 coskz 2 Í = 0 v 0 9 v c = (2 eí/m) 1/2 (mv 2 )/2 = m (v 0 2 - v c 2 )/2 v 0 < v c Í(z ) v -v v Í 0 -(v-v)+v v = -2(v-V) = -2 v T T T = mv v = -2 mvv v v > V T v < V 2 z Í (z ) dt/dt = P N v f v /L -v c v c v 10 L s f V f(v + v ) = f(v) + v f (V) s 1 v 0 s 2 s = v 4 c f (V)/2 L = 2e 2 Í 2 0 f (V)/m 2 L W w W E = <E 2 z >/2ε 0 = ε 0 k 2 Í 2 0 /2 W T W T = p = (e 2 N/ε 0 m) 1/2 W W = W E + W T = ε 0 (kí 0 ) 2 T + W w = 0 dt/dt = -dw w /dt 11 dw w /dt = ÎW w Î V = /k Î = 2 mvns/ε 0 (kí 0 ) 2 = 4 e 2 Nf (V)/ε 0 mlk 3 kl = 8/Û 12
2007/2008 71 9) JFT 2 10 JT 60 3.25 MA 11) 3 6 32 TRIAM-1 M 12) Ecos(kz - t) 13 13) F F = m v/ t = -2mv v /L (mv)/ t N f P 14 = E eq = F/-e = 2mv v /el E eq J current = env = ˆE eq ˆ ˆ = e 2 N /m (= 1/v) 15 ˆ 14) 13 F N E 0 = kí 0 P F F = ee eff ˆ/e J current J current P
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