68 JAXA-RR 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

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1 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 On the oscillation of the electron plasma Institute of physical problem J. Phys. USSR ) 3 4) Japan Atomic Energy Agency

2 68 JAXA-RR 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

3 2007/ 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

4 70 JAXA-RR ) 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 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

5 2007/ ) JFT 2 10 JT MA 11) 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

6 72 JAXA-RR F.F. Chen Introduction to plasma physics 15 energy momentum 1 z 0 17 t å Ë d(mv)/dt z0 = ee eff E eff 3 energy momentum 18 Ú ei mdv/dt = ee eff - mvú ei d/dt = 0 19 j = -eˇå vf(v)dv v -å 20 16, 17 1 D. D. Ryutov, Plasma Phys. Contr. Fusion 41 (1999) A 1 2 L. D. Landau, J. Phys., UUSR, 10 (1946) 25 3 L. D. Landau, Phys Z. Sowjet, 10 (1936) 154

7 2007/ J. Dawson, Phys. Fluids, 4 (1961) J. D. Jackson, Plasma Phys. 1 (1960) 71 7 Ching-Sheng Wu, Phys. Rev. 127 (1962) A. Sakharov, Memoirs (Knopf, New York, 1990) p T. Yamamoto et al., Phys. Rev. Lett. 45 (1980) K. Uehara, JAERI-M H. Zushi et al., Nucl Fusion 39 (1999) T. H. Stix, The Theory of Plasma Waves (McGraw-hill, New York, 1962) p ˆ v 3 ˆ = e 2 N /m - > v 3 /V ) F. F. Chen, Introduction to Plasma Physics (Plenum Press, NewYork and London, 1974) p ) K. Uehara, Phys. Fluids B 3 (1991) ) J. H. Malmberg and C. B. Wharton Phys. Rev. Lett. 17 (1966) 175 & K. Yamagiwa et al., J. Phys. Soc. Jpn. 40 (1976) 1157

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