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the excitation of \ufb01shbone instability, the necessary condition is \u3a9i > 0, that is, \u3b1h < \u3c0 and \u3008\u2212\u2202\u3b2h \u2202r \u3009rs > rs R \u3c9dh,mx \u3c9A 1 \u3c02 K2b K22 . There is a threshold for \u3008|\u2202\u3b2h/\u2202r|\u3009rs for the instability. Banana orbits of trapped ions drift in toroidal direction as is shown in \ufb01g.14.1. The toroidal precession velocity and frequency are\u2217 (refer to (3.59)) v\u3c6 = mv2\u22a5/2 eBr , \u3c9\u3c6 = mv2\u22a5/2 eBRr . (14.35) Therefore \u3c9dh,mx is equal to the toroidal precession frequency of trapped ions with the initial (maximum) velocity. It seems that the \ufb01shbone instability is due to an interaction between ener- getic particles and m=1, n=1 MHD perturbation. The interaction of resonant type is characterized by Landau damping. The resonance is between the toroidal wave velocity of instability and the toroidal precession of trapped energetic particles. 173 174 14 Instabilities Driven by Energetic Particles (* Note: The toroidal vertical drift velocity is vd = (mv2\u22a5/2eBR), so that the poloidal displace- ment of particles between bounces is r\u3b4\u3b8 \u223c vd\u3c4d, \u3c4d being the bounce period. Since d\u3c6/d\u3b8 = q along the magnetic line of \ufb01eld, the associated toroidal displacement between bounces is Rd\u3c6 = (Rqvd\u3c4d/r), q = 1. Thus toroidal precession velocity is given by (14.35).) 14.2 Toroidal Alfve´n Eigenmode Alfve´n waves in homogeneous magnetic \ufb01eld in in\ufb01nite plasma have been analyzed in sec.5.4. Shear Alfve´n wave, fast and slow magnetosonic waves appear. In the case of incompressible plasma (\u2207 · \u3be = 0 or ratio of speci\ufb01c heat \u3b3 \u2192\u221e), only the shear Alfve´n wave can exists. In the case of cylindrical plasma in the axisymmetric magnetic \ufb01eld, the displacement of MHD perturbation \u3be(r, \u3b8, z) = \u3be(r) exp i(\u2212m\u3b8 + kz \u2212 \u3c9t) is given by Hain-Lu¨st equations (8.114-117) as was discussed in sec.8.4. In the case of incompressible plasma, Hain-Lu¨st equation (8.117) is reduced to [In sec.8.4 perturbation is assumed to be \u3be(r) exp i(+m\u3b8 + kz \u2212 \u3c9t)] d dr ( F 2 \u2212 \u3bc0\u3c1m\u3c92 m2/r2 + k2 ) 1 r d dr (r\u3ber) + ( \u2212(F 2 \u2212 \u3bc0\u3c1m\u3c92) + 2B\u3b8 ddr ( B\u3b8 r ) + 4k2B2\u3b8F 2 r2(m2/r2 + k2)(F 2 \u2212 \u3bc0\u3c1m\u3c92) + 2r d dr ( (m/r)FB\u3b8 r2(m2/r2 + k2) )) \u3ber = 0 (14.36) where F = (k ·B) = (\u2212m r B\u3b8(r) + n R Bz(r) ) = Bz R ( n\u2212 m q(r) ) , q(r) = R r Bz B\u3b8 . The position at which F 2 \u2212 \u3bc0\u3c1m\u3c92 = 0 \u2192 \u3c92 = k2\u2016v2A, v2A \u2261 B2/\u3bc0\u3c1m holds is singular radius. It was shown by Hasegawa and L.Chen (ref.[7]) that at this singular radius (resonant layer) shear Alfve´n wave is mode converted to the kinetic Alfve´n wave and absorbed by Landau damping. Therefore Alfve´n wave is stable in the cylindrical plasma. Alfve´n waves were also treated in sec.10.4a, 10.4b by cold plasma model. The dispersion relation in homogenous in\ufb01nite plasma is given by (10.64b) showing that Alfve´n resonance occurs at \u3c92 \u2248 k2\u2016v 2 A and cuts o\ufb00 of compressional Alfve´n wave and shear Alfve´n wave occur at \u3c9 2 = k2\u2016v 2 A(1+\u3c9/\u3a9i) and \u3c92 = k2\u2016v 2 A(1\u2212 \u3c9/\u3a9i) respectively. 14.2a Toroidicity Induced Alfve´n Eigenmode Let us consider shear Alfve´n waves in toroidal plasma and the perturbation of (-m, n) mode given by \u3c6(r, \u3b8, z, t) = \u3c6(r) exp i(\u2212m\u3b8 + n z R \u2212 \u3c9t) (14.37) where R is major radius of torus and k\u2016 is k\u2016 = k ·B B = 1 R ( n\u2212 m q(r) ) . The resonant conditions of m and m+1 modes in linear cylindrical plasma are \u3c92 v2A \u2212 k2\u2016m = 0 \u3c92 v2A \u2212 k2\u2016m+1 = 0. However wave of m mode can couple with m±1 in toroidal plasma since the magnitude of toroidal 174 14.2 Toroidal Alfve´n Eigenmode 175 Fig.14.2 The Alfve´n resonance frequency \u3c9 of toroidally coupled m and m+1 modes. \ufb01eld changes as Bz = Bz0(1\u2212 (r/R) cos \u3b8), as will be shown in this section later. Then the resonant condition of m and m+1 modes in toroidal plasma becomes\u2223\u2223\u2223\u2223\u2223\u2223 \u3c92 v2 A \u2212 k2\u2016m \u3b1\ufffd\u3c9 2 v2 A \u3b1\ufffd\u3c9 2 v2 A \u3c92 v2 A \u2212 k2\u2016m+1 \u2223\u2223\u2223\u2223\u2223\u2223 = 0 where \ufffd = r/R and \u3b1 is a constant with order of 1. Then the solutions are \u3c92± v2A = k2\u2016m + k 2 \u2016m+1 ± ( (k2\u2016m \u2212 k2\u2016m+1)2 + 4\u3b1\ufffd2k2\u2016mk2\u2016m+1 )1/2 2(1\u2212 \u3b12\ufffd2) . (14.38) The resonant condition (14.38) is plotted in \ufb01g.14.2. At the radius satisfying k2\u2016m = k 2 \u2016m+1, the di\ufb00erence of \u3c9± becomes minimum and the radius is given by 1 R ( n\u2212 m q(r) ) = \u2212 1 R ( n\u2212 m+ 1 q(r) ) , q(r0) = m + 1/2 n , k\u2016m = \u2212k\u2016m+1 = 1 2q(r0)R . (14.39) q(r0) = 1.5 for the case of m=1 and n=1. Therefore Alfve´n resonance does not exist in the frequency gap \u3c9\u2212 < \u3c9 < \u3c9+. The continuum Alfve´n waves correspond to the excitation of shear Alfve´n waves on a given \ufb02ux surface where the mode frequency is resonant \u3c92 = k2\u2016mv 2 A(r) and such a resonance leads wave damping. However frequencies excited within the spectral gaps are not resonant with the continuum and hence will not damp in the gap region. This allows a discrete eigen-frequency of toroidicty-induced Alfve´n eigenmode or toroidal Alfve´n eigenmode (TAE) to be established. This TAE can easily be destabilized by the kinetic e\ufb00ect of energetic particles. The equations of TAE will be described according to Berk, Van Dam, Guo, Lindberg (ref.[8]). The equations of the \ufb01rst order perturbations are \u2207 · j1 = 0, \u3c1 dv1 dt = (j ×B)1, (14.40) E1 = \u2207\u3c61 \u2212 \u2202A1 \u2202t , B1 = \u2207×A1. (14.41) For ideal, low \u3b2 MHD waves, we have following relations: E\u2016 = 0, B\u20161 = 0, A1 = A\u20161b (14.42) 175 176 14 Instabilities Driven by Energetic Particles so that i\u3c9A\u20161 = b · \u2207\u3c61, v1 = E1 × b B . (14.43) From (14.40), we have \u2207 · j\u22a51 +\u2207 · (j\u20161b) = 0, (14.44) and \u2212i\u3c9\u3c1(v1 × b) = (j\u22a51 ×B)× b+ (j ×B1)× b, j\u22a51 = \u2212 i\u3c9\u3c1 B2 E\u22a51 + j\u2016 B B\u22a51. (14.45) Equations (14.41-43) yield B\u22a51 = \u2207× (A\u20161b) = \u2207 ( A\u20161 B ) ×B + A\u20161 B \u2207×B \u2248 \u2212i \u3c9 \u2207 ( b · \u2207\u3c61 B ) ×B (14.46) j\u20161 = b · j1 = 1 \u3bc0 b · \u2207 ×B\u22a51 = \u2212i \u3c9\u3bc0 b · \u2207 × ( B2\u2207\u22a5 ( (B · \u2207)\u3c61 B2 ) × B B2 ) = i \u3c9\u3bc0 ( b · B B2 ) \u2207 · ( B2\u2207\u22a5 ( B · \u2207\u3c61 B2 )) = i \u3c9\u3bc0B \u2207 · ( B2\u2207\u22a5 ( B · \u2207\u3c61 B2 )) . (14.47) Then (14.44-47) yield \u2207 · ( i \u3c9 \u3bc0 1 v2A \u2207\u22a5\u3c61 ) +\u2207 ( j\u2016 B B\u22a51 ) +\u2207 ( j\u20161 B B ) = 0 \u2207 · ( \u3c92 v2A \u2207\u22a5\u3c61 ) + \u3bc0\u2207 ( j\u2016 B ) ·B ×\u2207 ( (B · \u2207)\u3c61 B2 ) + (B · \u2207) ( 1 B2 \u2207 · ( B2\u2207\u22a5 · ( B · \u2207\u3c61 B2 ))) = 0. (14.48) When (R,\u3d5,Z) and (r, \u3b8, \u3b6) coordinates are introduced by, R = R0 + r cos \u3b8, Z = r sin \u3b8, \u3d5 = \u2212 \u3b6 R and following notations are used \u3c61(r, \u3b8, \u3b6, t) = \u2211 m \u3c6m(r) exp i(\u2212m\u3b8 + n\u3d5\u2212 \u3c9t), (b · \u2207)\u3c6m = i R0 ( n\u2212 m q(r) ) \u3c6m = \u2212ik\u2016m\u3c6m, k\u2016m = 1 R0 ( n\u2212 m q(r) ) , Em \u2261 \u3c6m R , (14.48) is reduced to (ref.[8]) d dr ( r3 ( \u3c92 v2A \u2212 k2\u2016m ) dEm dr ) + r2Em d dr ( \u3c9 vA )2 \u2212 (m2 \u2212 1) ( \u3c92 v2A \u2212 k2\u2016m ) rEm + d dr ( r3 ( \u3c9 vA )2 2r R0 ( dEm+1 dr + dEm\u22121 dr )) = 0. (14.49) As is seen in \ufb01g.14.3, mode structure has a sharp transition of m=1 and m=2 components at the gap location. Therefore m and m+1 modes near gap location reduces( \u3c92 v2A \u2212 k2\u2016m ) dEm dr + 2r R0 ( \u3c9 vA )2 dEm+1 dr \u2248 0 176 14.2 Toroidal Alfve´n Eigenmode 177 Fig.14.3 Left-hand side \ufb01gure: The toroidal shear Alfve´n resonance frequencies \u3a9 that corresponds to (n=1, m=1) and (n=1. m=2), q(r) = 1 + (r/a)2, a/R = 0.25, \u3a9 \u2261 \u3c9/(vA(0)/R0). Right-hand side \ufb01gure: The structure of the global mode amplitude as a function of radius. ( \u3c92 v2A \u2212 k2\u2016m+1 ) dEm+1 dr + 2r R0 ( \u3c9 vA )2 dEm dr \u2248 0, so that toroidal shear Alfve´n resonance frequency is given by\u2223\u2223\u2223\u2223\u2223\u2223\u2223\u2223 ( \u3c92 v2A \u2212 k2\u2016m ) 2\ufffd ( \u3c9 vA )2 2\ufffd ( \u3c9 vA )2 ( \u3c92 v2 A \u2212 k2\u2016m+1 ) \u2223\u2223\u2223\u2223\u2223\u2223\u2223\u2223 = 0. (14.50) When Shafranov shift is included in the coordinates of (R,\u3d5,Z) and