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in \ufb01g.12.3. The series expansion of the plasma dispersion function Z(\u3b6) is Z(\u3b6) = i\u3c01/2 kz |kz | exp(\u2212\u3b6 2)\u2212 2\u3b6 ( 1\u2212 2 3 \u3b62 + 4 15 \u3b64 + · · · ) in the case of |\u3b6| \ufffd 1 (hot plasma). The asymptotic expansion of Z(\u3b6) is Z(\u3b6) = i\u3c01/2\u3c3 kz |kz| exp(\u2212\u3b6 2)\u2212 1 \u3b6 ( 1 + 1 2 1 \u3b62 + 3 4 1 \u3b64 + · · · ) \u3c3 = 0 for Im\u3c9 > 0, \u3c3 = 2 for Im\u3c9 < 0, but \u3c3 = 1 for |Im \u3b6||Re \u3b6| <\u223c \u3c0/4, and |\u3b6| \ufffd 1 in the case of |\u3b6| \ufffd 1 (cold plasma) (ref.[1],[2]). Refer to appendix C for the derivation. The part i\u3c01/2(kz/|kz |) exp(\u2212\u3b62) of Z(\u3b6) represents the terms of Landau damping and cyclotron damping as is described later in this section. When T \u2192 0, that is, \u3b6n \u2192 ±\u221e, b \u2192 0, the dielectric tensor of hot plasma is reduced to the dielectric tensor (10.9)\u223c(10.13) of cold plasma. In the case of b = (kx\u3c1\u3a9)2 \ufffd 1 (\u3c1\u3a9 = vT\u22a5/\u3a9 is Larmor radius), it is possible to expand e\u2212bXn by b using In(b) = ( b 2 )n \u221e\u2211 l=0 1 l!(n + l)! ( b 2 )2l = ( b 2 )n ( 1 n! + 1 1!(n + 1)! ( b 2 )2 + 1 2!(n + 2)! ( b 2 )4 + · · · ) . The expansion in b and the inclusion of terms up to the second harmonics in K give Kxx =1 + \u2211 j ( \u3a0j \u3c9 )2 \u3b60 ( (Z1 + Z\u22121) ( 1 2 \u2212 b 2 + · · · ) + (Z2 + Z\u22122) ( b 2 \u2212 b 2 2 + · · · ) + · · · ) j , 150 12.3 Dielectric Tensor of Hot Plasma, Wave Absorption · · · 151 Kyy =1 + \u2211 j ( \u3a0j \u3c9 )2 \u3b60 ( Z0(2b+ · · ·) + (Z1 + Z\u22121) ( 1 2 \u2212 3b 2 + · · · ) + (Z2 + Z\u22122) ( b 2 \u2212 b2 + · · · ) + · · · ) j , Kzz =1\u2212 \u2211 j ( \u3a0j \u3c9 )2 \u3b60 ( 2\u3b60W0(1\u2212 b+ · · ·) + (\u3b61W1 + \u3b6\u22121W\u22121)(b+ · · ·) +(\u3b62W2 + \u3b6\u22122W\u22122) ( b2 4 + · · · ) + · · · ) j , Kxy = i \u2211 j ( \u3a0j \u3c9 )2 \u3b60 ( (Z1 \u2212 Z\u22121) ( 1 2 \u2212 b+ · · · ) + (Z2 \u2212 Z\u22122) ( b 2 + · · · ) + · · · ) j , Kxz = 21/2 \u2211 j ( \u3a0j \u3c9 )2 b1/2\u3b60 ( (W1 \u2212W\u22121) ( 1 2 + · · · ) + (W2 \u2212W\u22122) ( b 4 + · · · ) + · · · ) j , Kyz = \u221221/2i \u2211 j ( \u3a0j \u3c9 )2 b1/2\u3b60 ( W0 ( \u22121 + 3 2 b+ · · · ) + (W1 +W\u22121) ( 1 2 + · · · ) + (W2 \u2212W\u22122) ( b 4 + · · · ) + · · · ) j (12.22) Kyx = \u2212Kxy, Kzx = Kxz, Kzy = \u2212Kzy where Z±n \u2261 Z(\u3b6±n), Wn \u2261 \u2212 (1 + \u3b6nZ(\u3b6n)) , \u3b6n = (\u3c9 + n\u3a9)/(21/2kz(\u3baTz/m)1/2). When x\ufffd 1, ReW (x) is ReW (x) = (1/2)x\u22122(1 + (3/2)x\u22122 + · · ·). The absorbed power by Landau damping (including transit time damping) may be estimated by the terms associated with the imaginary part G0 of \u3b60Z(\u3b60) in (12.22) of Kij : G0 \u2261 Im\u3b60Z(\u3b60) = kz|kz |\u3c0 1/2\u3b60 exp(\u2212\u3b620 ). Since (ImKyy)0 = ( \u3a0j \u3c9 )2 2bG0, (ImKzz)0 = ( \u3a0j \u3c9 )2 2\u3b620G0, 151 152 12 Wave Propagation and Wave Heating (ReKyz)0 = ( \u3a0j \u3c9 )2 21/2b1/2\u3b60G0 the contribution of these terms to the absorption power (12.17) is P ab0 = 2\u3c9 ( \u3a0j \u3c9 )2 G0 ( \ufffd0 2 )( |Ey|2b+ |Ez|2\u3b620 + Im(E\u2217yEz)(2b)1/2\u3b60 ) . (12.23) The 1st term is of transit time damping and is equal to (11.16). The 2nd term is of Landau damping and is equal to (11.10). The 3rd one is the term of the interference of both. The absorption power due to cyclotron damping and the harmonic cyclotron damping is obtained by the contribution from the terms G±n \u2261 Im\u3b60Z±n = kz|kz|\u3c0 1/2\u3b60 exp(\u2212\u3b62±n) and for the case b\ufffd 1, (ImKxx)±n = (ImKyy)±n = ( \u3a0j \u3c9 )2 G±n\u3b1n, (ImKzz)±n = ( \u3a0j \u3c9 )2 2\u3b62±nG±nb\u3b1nn \u22122, (ReKxy)±n = \u2212 ( \u3a0j \u3c9 )2 G±n(±\u3b1n), (ReKyz)±n = \u2212 ( \u3a0j \u3c9 )2 (2b)1/2\u3b6±nG±n\u3b1nn\u22121, (ImKxz)±n = \u2212 ( \u3a0j \u3c9 )2 (2b)1/2\u3b6±nG±n(±\u3b1n)n\u22121, \u3b1n = n2(2 · n!)\u22121 ( b 2 )n\u22121 . The contribution of these terms to the absorbed power (12.17) is P ab±n = \u3c9 ( \u3a0j \u3c9 )2 Gn ( \ufffd0 2 ) \u3b1n|Ex ± iEy|2. (12.24) Since \u3b6n = (\u3c9 + n\u3a9i)/(21/2kzvTi) = (\u3c9 \u2212 n|\u3a9i|)/(21/2kzvTi) the term of +n is dominant for the ion cyclotron damping (\u3c9 > 0), and the term of \u2212n is dominant for electron cyclotron damping (\u3c9 > 0), since \u3b6\u2212n = (\u3c9 \u2212 n\u3a9e)/(21/2kzvTe). The relative ratio of E components can be estimated from the following equations: (Kxx \u2212N2\u2016 )Ex +KxyEy + (Kxz +N\u22a5N\u2016)Ez = 0, \u2212KxyEx + (Kyy \u2212N2\u2016 \u2212N2\u22a5)Ey +KyzEz = 0, (12.25) (Kxz +N\u22a5N\u2016)Ex \u2212KyzEy + (Kzz \u2212N2\u22a5)Ez = 0. 152 12.4 Wave Heating in Ion Cyclotron Range of Frequency 153 For cold plasmas, Kxx \u2192 K\u22a5, Kyy \u2192 K\u22a5, Kzz \u2192 K\u2016, Kxy\u2192 \u2212iK×, Kxz \u2192 0, Kyz \u2192 0 can be substituted into (12.25), and the relative ratio is Ex : Ey : Ez = (K\u22a5 \u2212 N2) ×(K\u2016 \u2212 N2\u22a5) : \u2212iKx(K\u2016 \u2212N2\u22a5) : \u2212N\u2016N\u22a5(K\u22a5 \u2212N2). In order to obtain the magnitude of the electric \ufb01eld, it is necessary to solve the Maxwell equation with the dielectric tensor of (12.19). In this case the density, the temperature, and the magnetic \ufb01eld are functions of the coordinates. Therefore the simpli\ufb01ed model must be used for analytical solutions; otherwise numerical calculations are necessary to derive the wave \ufb01eld. 12.4 Wave Heating in Ion Cyclotron Range of Frequency The dispersion relation of waves in the ion cyclotron range of frequency (ICRF) is given by (10.64); N2\u2016 = N2\u22a5 2[1\u2212 (\u3c9/\u3a9i)2] ( \u2212 ( 1\u2212 ( \u3c9 \u3a9i )2) + 2\u3c92 k2\u22a5v 2 A ± \u23a1\u23a3(1\u2212 ( \u3c9 \u3a9i )2)2 + 4 ( \u3c9 \u3a9i )2 ( \u3c9 k\u22a5vA )4\u23a4\u23a61/2). The plus sign corresponds to the slow wave (L wave, ion cyclotron wave), and the minus sign corresponds to the fast wave (R wave, extraordinary wave). When 1\u2212 \u3c92/\u3a92i \ufffd 2(\u3c9/k\u22a5vA)2, the dispersion relation becomes k2z = 2 ( \u3c92 v2A )( 1\u2212 \u3c9 2 \u3a92i )\u22121 , (for slow wave) k2z = \u2212 k2\u22a5 2 + \u3c92 2v2A . (for fast wave) Since the externally excited waves have propagation vectors with 0 < k2z < (1/a) 2, k2\u22a5 > (\u3c0/a) 2 usually, there are constraints \u3c92 v2A 2 (1\u2212 \u3c92/\u3a92i ) < ( \u3c0 a )2 , n20a 2 < 2.6× 10\u22124B 2(1\u2212 \u3c92/\u3a92i ) A for slow wave and \u3c92 2v2A > ( \u3c0 a )2 , n20a 2 > 0.5× 10\u22122 (\u3a9i/\u3c9) 2 A/Z2 for the fast wave (ref.[3]), where n20 is the ion density in 1020 m\u22123, a is the plasma radius in meters, and A is the atomic number. An ion cyclotron wave (slow wave) can be excited by a Stix coil (ref.[1]) and can propagate and heat ions in a low-density plasma. But it cannot propagate in a high-density plasma like that of a tokamak. The fast wave is an extraordinary wave in this frequency range and can be excited by a loop antenna, which generates a high-frequency electric \ufb01eld perpendicular to the magnetic \ufb01eld (see sec.10.2). The fast wave can propagate in a high-density plasma. The fast wave in a plasma with a single ion species has Ex + iEy = 0 at \u3c9 = |\u3a9i| in cold plasma approximation, so that it is not absorbed by the ion cyclotron damping. However, the electric \ufb01eld of the fast wave in a plasma 153 154 12 Wave Propagation and Wave Heating with two ion species is Ex + iEy = 0, so that the fast wave can be absorbed; that is, the fast wave can heat the ions in this case. Let us consider the heating of a plasma with two ion species, M and m, by a fast wave. The masses, charge numbers, and densities of the M ion and m ion are denoted by mM, ZM, nM and mm, Zm, nm, respectively. When we use \u3b7M \u2261 Z 2 MnM ne , \u3b7m \u2261 Z 2 mnm ne we have \u3b7M/ZM + \u3b7m/Zm = 1 since ne = ZMnM + Zmnm. Since (\u3a0e/\u3c9)2 \ufffd 1 in ICRF wave, the dispersion relation in the cold plasma approximation is given by (10.64) as follows: N2\u22a5 = (R\u2212N2\u2016 )(L\u2212N2\u2016 ) K\u22a5 \u2212N2\u2016 , R = \u2212\u3a0 2 i \u3c92 ( (mM/mm)\u3b7m\u3c9 \u3c9 + |\u3a9m| + \u3b7M\u3c9 \u3c9 + |\u3a9M| \u2212 \u3c9 |\u3a9M|/ZM ) , L = \u2212\u3a0 2 i \u3c92 ( (mM/mm)\u3b7m\u3c9 \u3c9 \u2212 |\u3a9m| + \u3b7M\u3c9 \u3c9 \u2212 |\u3a9M| + \u3c9 |\u3a9M|/ZM ) , K\u22a5 = \u2212\u3a0 2 i \u3c92 ( (mM/mm)\u3b7m\u3c92 \u3c92 \u2212\u3a92m + \u3b7M\u3c9 2 \u3c92 \u2212\u3a92M ) , \u3a02i \u2261 nee 2 \ufffd0mM . Therefore ion-ion hybrid resonance occurs at K\u22a5 \u2212N2\u2016 = 0, that is, \u3b7m(mM/mm)\u3c92 \u3c92 \u2212\u3a92m + \u3b7M\u3c9 2 \u3c92 \u2212\u3a92M \u2248 \u2212 \u3c9 2 \u3a02i N2\u2016 \u2248 0, \u3c92 \u2248 \u3c9IH \u2261 \u3b7M + \u3b7m(\u3bc 2/\u3bc\u2032) \u3b7M + \u3b7m/\u3bc\u2032 \u3a92m,