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B ·B\u2217 \u3bc0 + \ufffd0 2 (\u3c9iRe(E\u2217 ·K ·E) + \u3c9rIm(E\u2217 ·K ·E)) . (12.5) From the relations E\u2217 ·K ·E = \u2211 i E\u2217i \u2211 j KijEj , E ·K\u2217 ·E\u2217 = \u2211 i Ei \u2211 j K\u2217ijE \u2217 j = \u2211 j E\u2217j \u2211 i (KTji) \u2217Ei = \u2211 i E\u2217i \u2211 j (KTij) \u2217Ej we \ufb01nd Re(E\u2217 ·K ·E) = E\u2217 · K + (K T)\u2217 2 ·E, Im(E\u2217 ·K ·E) = E\u2217 · (\u2212i)[K \u2212 (K T)\u2217] 2 ·E. (KT)\u2217 is the complex conjugate of transpose matrix KT (lines and rows of components are ex- changed) of K, i.e., KTij \u2261 Kji. When a matrix M and (MT)\u2217 are equal with each other, this kind of matrix is called Hermite matrix. For the Hermite matrix, (E\u2217 ·M ·E) is always real. The dielectric tensor may be decomposed to K(k, \u3c9) = KH(k, \u3c9) + iKI(k, \u3c9). As is described in sec.12.3, KH and KI are Hermite, when k, \u3c9 are real. It will be proved that the term iKI corresponds to Landau damping and cyclotron damping. When the imaginary part of \u3c9 is much smaller than the real part (\u3c9 = \u3c9r + i\u3c9i, |\u3c9i| \ufffd |\u3c9r|) we may write K(k, \u3c9r + i\u3c9i) \u2248KH(k, \u3c9r) + i\u3c9i \u2202 \u2202\u3c9r KH(k, \u3c9r) + iKI(k, \u3c9r). When the Hermite component of W (the term associated to KH in W ) is denoted by W0, W0 is given by W0 = 1 2 Re ( B\u22170 ·B0 2\u3bc0 + \ufffd0 2 E\u22170 ·KH ·E0 + \ufffd0 2 E\u22170 · ( \u3c9r \u2202 \u2202\u3c9r KH ) ·E0 ) = 1 2 Re ( B\u22170 ·B0 2\u3bc0 + \ufffd0 2 E\u22170 · ( \u2202 \u2202\u3c9 (\u3c9KH) ) ·E0 ) (12.6) and (12.3),(12.5) yield \u2202W0 \u2202t = \u2212\u3c9r12\ufffd0E \u2217 0 ·KI ·E0 \u2212\u2207 · P . (12.7) 146 12.2 Ray Tracing 147 Fig.12.2 F (x, t) and f(k) cos(kx\u2212 w(k)t) The 1st term in (12.6) is the energy density of the magnetic \ufb01eld and the 2nd term is the energy density of electric \ufb01eld which includes the kinetic energy of coherent motion associated with the wave. Equation (12.6) gives the energy density of the wave in a dispersive media. The 1st term in the right-hand side of (12.7) represents the Landau and cyclotron dampings and the 2nd term is the divergence of the \ufb02ow of wave energy. Let us consider the velocity of the wave packet F (r, t) = \u222b \u221e \u2212\u221e f(k) exp i(k · r \u2212 \u3c9(k)t)dk (12.8) when the dispersion equation \u3c9 = \u3c9(k) is given. If f(k) varies slowly, the position of the maximum of F (r, t) is the position of the stationary phase of \u2202 \u2202ki (k · r \u2212 \u3c9(k)t) = 0. (i = x, y, z) (see \ufb01g.12.2). Consequently the velocity of the maximum position is( x t = \u2202\u3c9(k) \u2202kx , y t = \u2202\u3c9(k) \u2202ky , z t = \u2202\u3c9(k) \u2202kz ) that is, vg = ( \u2202\u3c9 \u2202kx , \u2202\u3c9 \u2202ky , \u2202\u3c9 \u2202kz ) . (12.9) This velocity is called group velocity and represents the velocity of energy \ufb02ow. 12.2 Ray Tracing When the wavelength of waves in the plasma is much less than the characteristic length (typically the minor radius a), the WKB approximation (geometrical optical approximation) can be applied. Let the dispersion relation be D(k, \u3c9, r, t) = 0. The direction of wave energy \ufb02ow is given by the group velocity vg = \u2202\u3c9/\u2202k \u2261 (\u2202\u3c9/\u2202kx, \u2202\u3c9/\u2202ky , \u2202\u3c9/\u2202kz), so that the optical ray can be given by dr/dt = vg. Although the quantities (k, \u3c9) change according to the change of r, they always satisfy D = 0. Then the optical ray can be obtained by dr ds = \u2202D \u2202k , dk ds = \u2212\u2202D \u2202r , (12.10) dt ds = \u2212\u2202D \u2202\u3c9 , d\u3c9 ds = \u2202D \u2202t . (12.11) 147 148 12 Wave Propagation and Wave Heating Here s is a measure of the length along the optical ray. Along the optical ray the variation \u3b4D becomes zero, \u3b4D = \u2202D \u2202k · \u3b4k + \u2202D \u2202\u3c9 · \u3b4\u3c9 + \u2202D \u2202r · \u3b4r + \u2202D \u2202t · \u3b4t = 0 (12.12) and D(k, \u3c9, r, t) = 0 is satis\ufb01ed. Equations (12.10),(12.11) reduce to dr dt = dr ds ( dt ds )\u22121 = \u2212\u2202D \u2202k ( \u2202D \u2202\u3c9 )\u22121 = ( \u2202\u3c9 \u2202k ) r,t=const. = vg. Equation (12.10) has the same formula as the equation of motion with Hamiltonian D. When D does not depend on t explicitly, D = const. = 0 corresponds to the energy conservation law. If the plasma medium does not depend on z, kz =const. corresponds to the momentum conservation law and is the same as the Snell law, N\u2016 = const.. When k = kr + iki is a solution of D = 0 for a given real \u3c9 and |ki| \ufffd |kr| is satis\ufb01ed, we have D(kr + iki, \u3c9) = ReD(kr, \u3c9) + \u2202ReD(kr, \u3c9) \u2202kr · iki + iImD(kr, \u3c9) = 0. Therefore this reduces to ReD(kr, \u3c9) = 0, ki · \u2202ReD(kr, \u3c9) \u2202kr = \u2212ImD(kr, \u3c9). (12.13) Then the wave intensity I(r) becomes I(r) = I(r0) exp ( \u22122 \u222b r r0 kidr ) , (12.14) \u222b kidr = \u222b ki · \u2202D \u2202k ds = \u2212 \u222b ImD(kr, \u3c9)ds = \u2212 \u222b ImD(kr, \u3c9) |\u2202D/\u2202k| dl. (12.15) where dl is the length along the optical ray. Therefore the wave absorption can be estimated from (12.14) and (12.15) by tracing many optical rays. The geometrical optical approximation can provide the average wave intensity with a space resolution of, say, two or three times the wavelength. 12.3 Dielectric Tensor of Hot Plasma, Wave Absorption and Heating In the process of wave absorption by hot plasma, Landau damping or cyclotron damping are most important as was described in ch.11. These damping processes are due to the interaction between the wave and so called resonant particles satisfying \u3c9 \u2212 kzvz \u2212 n\u3a9 = 0. n = 0,±1,±2, · · · In the coordinates running with the same velocity, the electric \ufb01eld is static (\u3c9 = 0) or of cyclotron harmonic frequency (\u3c9 = n\u3a9). The case of n = 0 corresponds to Landau damping and the case of n = 1 corresponds to electron cyclotron damping and the case of n = \u22121 corresponds to ion cyclotron damping (\u3c9 > 0 is assumed). Although nonlinear or stochastic processes accompany wave heating in many cases, the experi- mental results of wave heating or absorption can ususally well described by linear or quasi-linear theories. The basis of the linear theory is the dispersion relation with the dielectric tensor K of \ufb01nite-temperature plasma. The absorbed power per unit volume of plasma P ab is given by the 1st term in the right-hand side of (12.7): P ab = \u3c9r ( \ufffd0 2 ) E\u2217 ·KI ·E. 148 12.3 Dielectric Tensor of Hot Plasma, Wave Absorption · · · 149 Since KH, KI is Hermit matrix for real k, \u3c9 as will be shown later in this section, the absorbed power P ab is given by P ab = \u3c9r ( \ufffd0 2 ) Re (E\u2217 · (\u2212i)K ·E)\u3c9=\u3c9r . (12.16) As is clear from the expression (12.19) of K, the absorbed power P ab reduces to P ab = \u3c9 \ufffd0 2 ( |Ex|2ImKxx + |Ey|2ImKyy + |Ez|2ImKzz +2Im(E\u2217xEy)ReKxy + 2Im(E \u2217 yEz)ReKyz + 2Im(E \u2217 xEz)ReKxz ) . (12.17) Since (10.3) gives j = \u2212i\u3c9P = \u2212i\ufffd0\u3c9(K \u2212 I) ·E, (12.16) may be described by P ab = 1 2 Re(E\u2217 · j)\u3c9=\u3c9r . (12.18) The process to drive the dielectric tensor K of \ufb01nite-temperature plasma is described in appendix C. When the plasma is bi-Maxwellian; f0(v\u22a5, vz) = n0F\u22a5(v\u22a5)Fz(vz), F\u22a5(v\u22a5) = m 2\u3c0\u3baT\u22a5 exp ( \u2212mv 2 \u22a5 2\u3baT\u22a5 ) , Fz(vz) = ( m 2\u3c0\u3baTz )1/2 exp ( \u2212m(vz \u2212 V ) 2 2\u3baTz ) the dielectric tensor K is given by K = I + \u2211 i,e \u3a02 \u3c92 [\u2211 n ( \u3b60Z(\u3b6n)\u2212 ( 1\u2212 1 \u3bbT ) (1 + \u3b6nZ(\u3b6n)) ) e\u2212bXn +2\u3b720\u3bbTL ] , (12.19) Xn = \u23a1\u23a2\u23a3 n2In/b in(I \u2032n \u2212 In) \u2212(2\u3bbT)1/2\u3b7n n\u3b1In\u2212in(I \u2032n \u2212 In) (n2/b+ 2b)In \u2212 2bI \u2032n i(2\u3bbT)1/2\u3b7n\u3b1(I \u2032n \u2212 In) \u2212(2\u3bbT)1/2\u3b7n n\u3b1In \u2212i(2\u3bbT)1/2\u3b7n\u3b1(I \u2032n \u2212 In) 2\u3bbT\u3b72nIn \u23a4\u23a5\u23a6 (12.20) Z(\u3b6) \u2261 1 \u3c01/2 \u222b \u221e \u2212\u221e exp(\u2212\u3b22) \u3b2 \u2212 \u3b6 d\u3b2, In(b) is the nth modi\ufb01ed Bessel function \u3b7n \u2261 \u3c9 + n\u3a921/2kzvTz , \u3b6n \u2261 \u3c9 \u2212 kzV + n\u3a921/2kzvTz , \u3bbT \u2261 Tz T\u22a5 , b \u2261 ( kxvT\u22a5 \u3a9 )2 , \u3b1 \u2261 kxvT\u22a5 \u3a9 , v2Tz \u2261 \u3baTz m , v2T\u22a5 \u2261 \u3baT\u22a5 m . The components of L matrix are zero except Lzz = 1. 149 150 12 Wave Propagation and Wave Heating Fig.12.3 Real part ReZ and imaginary part ImZ of Z(x) in the case of real x. When the plasma is isotropic Maxwellian (Tz = T\u22a5) and V = 0, then \u3b7n = \u3b6n, and \u3bbT = 1, and (12.19) reduces to K = I + \u2211 i,e \u3a02 \u3c92 [ \u221e\u2211 n=\u2212\u221e \u3b60Z(\u3b6n)e\u2212bXn + 2\u3b620L ] . (12.21) The real part ReZ(x) and the imaginary part ImZ(x) in the case of real x are shown