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is 1/h = 1 + 2kz\u3c1i(1 + p)s and \u3a02e (x) \u3a92e = \u3a02M.C. \u3a92e \u2261 p 1 + 2kz\u3c1i(1 + p)s . Accordingly, the mode conversion occurs at the position satisfying \u3c92 \u3a02i = ( 1\u2212 \u3c9 2 |\u3a9i|\u3a9e ) + N\u2016vTe2 \u221a 3 c \u239b\u239dTi Te + 1 4 ( \u3c92 \u3a9i\u3a9e )2\u239e\u23a01/2 157 158 12 Wave Propagation and Wave Heating Fig.12.6 Array of waveguides to excite a lower hybrid wave (slow wave). and the value of k2\u22a5\u3c1 2 i at this position becomes k2\u22a5\u3c1 2 i |M.C. = mi me kz\u3c1i s . If the electron temperature is high enough at the plasma center to satisfy vTe > (1/3)c/N\u2016, the wave is absorbed by electrons due to electron Landau damping. After the mode conversion, the value N\u22a5 becomes large so that c/N\u22a5 becomes comparable to the ion thermal velocity (c/N\u22a5 \u223c vTi). Since \u3c9 \ufffd |\u3a9i|, the ion motion is not a\ufb00ected by the magnetic \ufb01eld within the time scale of \u3c9\u22121. Therefore the wave with phase velocity c/N is absorbed by ions due to ion Landau damping. When ions have velocity vi larger than c/N\u22a5 (vi > c/N\u22a5), the ions are accelerated or decelerated at each time satisfying vi cos(\u3a9it) \u2248 c/N\u22a5 and are subjected to stochastic heating. The wave is excited by the array of waveguides, as shown in \ufb01g.12.6, with an appropriate phase di\ufb00erence to provide the necessary parallel index N\u2016 = kzc/\u3c9 = 2\u3c0c/(\u3bbz\u3c9). In the low-density region at the plasma boundary, the component of the electric \ufb01eld parallel to the magnetic \ufb01eld is larger for the slow wave than for the fast wave. Therefore the direction of wave-guides is arranged to excite the electric \ufb01eld parallel to the line of magnetic force. The coupling of waves to plasmas is discussed in detail in (ref.[11]) and the experiments of LHH are reviewed in (ref.[12]). For the current drive by lower hybrid wave, the accessibility condition (12.30) and c/N\u2016 \ufffd vTe are necessary. If the electron temperature is high and \u3baTe \u223c 10 keV, then vTe/c is already \u223c 1/7. Even if N\u2016 is chosen to be small under the accessibility condition, (12.30), the wave is subjected to absorbtion by electron damping in the outer part of the plasma, and it can not be expected that the wave can propagate into the central part of the plasma. When the value of N\u2016 is chosen to be N\u2016 \u223c (1/3)(c/vTe), electron heating can be expected and has been observed experimentally. Under the condition that the mode conversion can occur, ion heating can be expected. However, the experimental results are less clear than those for electron heating. 12.6 Electron Cyclotron Heating The dispersion relation of waves in the electron cyclotron range of frequency in a cold plasma is given by (10.79). The plus and minus signs in (10.79) correspond to ordinary and extraordinary waves, respectively. The ordinary wave can propagate only when \u3c92 > \u3a02e as is clear from (10.86) (in the case of \u3b8 = \u3c0/2). This wave can be excited by an array of waveguides, like that used for lower hybrid waves (\ufb01g.12.6), which emits an electric \ufb01eld parallel to the magnetic \ufb01eld. The phase of each waveguide is selected to provide the appropriate value of the parallel index N\u2016 = kzc/\u3c9= 2\u3c0c/(\u3c9\u3bbz). The dispersion relation of the extraordinary wave is given by (10.87). When \u3b8 = \u3c0/2, it is given by (10.52). It is necessary to satisfy \u3c92UH > \u3c9 2 > \u3c92L, \u3c9 2 LH. As is seen from the CMA diagram of \ufb01g.10.5, the extraordinary wave can access the plasma center from the high magnetic \ufb01eld side (see \ufb01g.12.7) but can not access from the low \ufb01eld side because of \u3c9 = \u3c9R cuto\ufb00. The extraordinary wave can be excited by the waveguide, which emits an electric \ufb01eld perpendicular to the magnetic \ufb01eld (see sec.10.2a). 158 12.6 Electron Cyclotron Heating 159 Fig.12.7 The locations of electron cyclotron resonance (\u3c9 = \u3a9e), upperhybrid resonance (\u3c9 = \u3c9LH) and R cut o\ufb00 (\u3c9 = \u3c9R) in case of \u3a9e0 > \u3a0e0, where \u3a9e0 and \u3a0e0 are electron cyclotron resonance frequency and plasma frequency at the plasma center respectively (left-hand side \ufb01gure). The right-hand side \ufb01gure is the CMA diagram near electron cyclotron frequency region. The ion\u2019s contribution to the dielectric tensor is negligible. When relations b \ufffd 1, \u3b60 \ufffd 1 are satis\ufb01ed for electron, the dielectric tensor of a hot plasma is Kxx = Kyy = 1 +X\u3b60Z\u22121/2, Kzz = 1\u2212X +N2\u22a5\u3c7zz, Kxy = \u2212iX\u3b60Z\u22121/2, Kxz = N\u22a5\u3c7xz, Kyz = iN\u22a5\u3c7yz, \u3c7xz \u2248 \u3c7yz \u2248 2\u22121/2XY \u22121 vT c \u3b60(1 + \u3b6\u22121Z\u22121), \u3c7zz \u2248 XY \u22122 ( vT c )2 \u3b60\u3b6\u22121(1 + \u3b6\u22121Z\u22121), X \u2261 \u3a0 2 e \u3c92 , Y \u2261 \u3a9e \u3c9 , \u3b6\u22121 = \u3c9 \u2212\u3a9e 21/2kzvT , N\u22a5 = k\u22a5c \u3c9 . The Maxwell equation is (Kxx \u2212N2\u2016 )Ex +KxyEy +N\u22a5(N\u2016 + \u3c7xz)Ez = 0, \u2212KxyEx + (Kyy \u2212N2\u2016 \u2212N2\u22a5)Ey + iN\u22a5\u3c7yzEz = 0, N\u22a5(N\u2016 + \u3c7xz)Ex \u2212 iN\u22a5\u3c7yzEy + (1\u2212X \u2212N2\u22a5(1\u2212 \u3c7zz))Ez = 0. The solution is Ex Ez = \u2212 iN 2 \u22a5\u3c7xz(N\u2016 + \u3c7xz) +Kxy(1\u2212X \u2212N2\u22a5(1\u2212 \u3c7zz)) N\u22a5(i\u3c7xz(Kxx \u2212N2\u2016 ) +Kxy(N\u2016 + \u3c7xz)) , Ey Ez = \u2212 N2\u22a5(N\u2016 + \u3c7xz) 2 \u2212 (Kxx \u2212N2\u2016 )(1\u2212X \u2212N2\u22a5(1\u2212 \u3c7zz)) N\u22a5(i\u3c7xz(Kxx \u2212N2\u2016 ) +Kxy(N\u2016 + \u3c7xz)) . The absorption power P\u22121 per unit volume is given by (12.24) as follows: P\u22121 = \u3c9X\u3b60 \u3c01/2 2 exp ( \u2212(\u3c9 \u2212\u3a9e) 2 2k2zv2Te ) \ufffd0 2 |Ex \u2212 iEy|2. 159 160 12 Wave Propagation and Wave Heating When \u3c9 = \u3a9e, then \u3b6\u22121 = 0, Z\u22121 = i\u3c01/2, Kxx = 1+ ih, Kxy = h, \u3c7yz = \u3c7xz = 21/2X(vTe/c)\u3b60 = X/(2N\u2016), \u3c7zz = 0, h \u2261 \u3c01/2\u3b60X/2. Therefore the dielectric tensor K becomes K = \u23a1\u23a3 1 + ih h N\u22a5\u3c7xz\u2212h 1 + ih iN\u22a5\u3c7xz N\u22a5\u3c7xz \u2212iN\u22a5\u3c7xz 1\u2212X \u23a4\u23a6 . For the ordinary wave (O wave), we have Ex \u2212 iEy Ez = iN2\u22a5(O)N\u2016(N\u2016 + \u3c7xz)\u2212 i(1\u2212N2\u2016 )(1\u2212X \u2212N2\u22a5(O)) N\u22a5(O)(N\u2016h+ i\u3c7xz(1\u2212N2\u2016 )) . When N\u2016 \ufffd 1 and the incident angle is nearly perpendicular, (10.82) gives 1 \u2212 X \u2212 N2\u22a5(O) = (1\u2212X)N2\u2016 . Since \u3c7xz = X/2N\u2016, \u3c7xz \ufffd N\u2016. Therefore the foregoing equation reduces to Ex \u2212 iEy Ez = iN\u22a5(O)N\u2016\u3c7xz N\u2016h+ i\u3c7xz . For extraordinary wave (X wave), we have Ex \u2212 iEy Ey = \u2212 iN2\u22a5(X)N\u2016(N\u2016 + \u3c7xz)\u2212 i(1\u2212N2\u2016 )(1\u2212X \u2212N2\u22a5(X)) N2\u22a5(X)(N\u2016 + \u3c7xz)2 \u2212 (Kxx \u2212N2\u2016 )(1\u2212X \u2212N2\u22a5(X)) . WhenN\u2016 \ufffd 1 and \u3c9 = \u3a9e, (10.83) gives 1\u2212X\u2212N2\u22a5(X) \u2248 \u22121+N2\u2016 . Since \u3c72xz = (2\u3c0)\u22121/2(vTe/cN\u2016)Xh\ufffd h, the foregoing equation reduces to Ex \u2212 iEy Ey = \u2212(1 +N2\u22a5(X)N\u2016(N\u2016 + \u3c7xz)) h\u2212 i(1 +N2\u22a5(X)(N\u2016 + \u3c7xz)2) \u223c \u22121 h . The absorption power per unit volume at \u3c9 = \u3a9e is P\u22121(O) \u2248 \u3c9\ufffd02 |Ez| 2 hN2\u22a5(O)N 2 \u2016\u3c7 2 xz (N\u2016h)2 + \u3c72xz exp(\u2212\u3b62\u22121) \u2248 \u3c9\ufffd0 2 |Ez |2 1(2\u3c0)1/2 ( \u3a0e \u3c9 )2( vTe cN\u2016 ) N2\u22a5(O)N 2 \u2016 N2\u2016 + (vTe/c) 2(2/\u3c0) for ordinary wave and P\u22121(X) \u223c \u3c9\ufffd02 |Ey| 2 1 h = \u3c9\ufffd0 2 |Ey|22 ( 2 \u3c0 )1/2 (\u3a0e \u3c9 )\u22122 (N\u2016vTe c ) . for extraordinary wave (ref.[13]). Since P (O) \u221d neT 1/2e /N\u2016, P (X) \u221d N\u2016T 1/2e /ne, the ordinary wave is absorbed more in the case of higher density and perpendicular incidence, but the extraordinary wave has opposite tendency. The experiments of electron cyclotron heating have been carried out by T-10, ISX-B, JFT-2, D-IIID, and so on, and the good heating e\ufb03ciency of ECH has been demonstrated. Heating and current drive by electron cyclotron waves are reviewed in (ref.[14]). 160 References 161 References [1] T. H. Stix: The Theory of Plasma Waves, McGraw-Hill, New York 1962. T. H. Stix: Waves in Plasmas, American Institute of Physics, New York, 1992. [2] B. D. Fried and S. D. Conte: The Plasma Dispersion Function, Academic Press, New York 1961. [3] B.D.Fried and S. D. Conte: The Plasma Dispersion Function Academic Press, New York 1961 [4] M. Porkolab: Fusion (ed. by E. Teller) Vol.1, Part B, p.151, Academic Press, New York (1981). [5] J. E. Scharer, B. D. McVey and T. K. Mau: Nucl. Fusion 17, 297 (1977). [6] T. H. Stix: Nucl. Fusion 15, 737 (1975) [7] A. Fukuyama, S.Nishiyama, K. Itoh and S.I. Itoh: Nucl. Fusion 23 1005 (1983). [8] M. Ono, T.Watari, R. Ando, J.Fujita et al.: Phys. Rev. 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