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(be \u2248 0, |\u3b60e| \ufffd 1) and the ion is hot. Then the contribution of the electron term is dominant in (13.11) and the ion term can be neglected. Equation (13.11) becomes k2 \u2212\u3a02e k2z \u3c92r = 0. (13.13) i.e., \u3c9r = ±\u3a0ekz k . The substitution of \u3c9r into (13.12) yields \u3b3 = \u3c01/2 2k2 \u2211 j \u3a02j ( m \u3baTz ) j 1 (2\u3baTz/m) 1/2 j |kz| × \u239b\u239d \u221e\u2211 l=\u2212\u221e Ile \u2212b\u3c9r ( \u2212(\u3c9r + l\u3a9) + Tz T\u22a5 l\u3a9 ) exp(\u2212\u3b62l ) \u239e\u23a0 j . (13.14) exp(\u2212\u3b62l ) has meaningful value only near \u3c9r + l\u3a9i = \u3c9r \u2212 l|\u3a9i| \u2248 0. The \ufb01rst term in the braket of (13.14), \u2212(\u3c9r \u2212 l|\u3a9i|)| can be destabilizing term and the 2nd term \u2212(Tz/T\u22a5)l|\u3a9i| is stabilizing term. Accordingly the necessary conditions for instability (\u3b3 > 0) are \u3c9r \u223c l|\u3a9i|, \u3c9r < l|\u3a9i|, Tz T\u22a5 l < 1 2 , that is, \u3c9r = \u3a0e kz k < l|\u3a9i|, (13.15) T\u22a5 Tz > 2l. (13.16) When the density increases to the point that \u3a0e approaches |\u3a9i|, then plasma oscillation couples to ion Larmor motion, causing the instability. When the density increases further, an oblique 165 166 13 Velocity Space Instability (Electrostatic Wave) Langmuir wave couples with an ion cyclotron harmonic wave l|\u3a9i| and an instability with \u3c9r = \u3a0ekz/k \u223c l|\u3a9i| is induced. As is clear from (13.16), the degree of anisotropy must be larger for the instability in the region of higher frequency (l becomes large). In summary, the instability with ion cyclotron harmonic frequencies appear one after another in a cold-electron plasma under the anisotropic condition (13.16) when the electron density satis\ufb01es ne \u223c l2Z2me mi ( B2 \u3bc0mic2 ) k2 k2z . (l = 1, 2, 3, · · ·) This instability is called Harris instability (ref.[2],[3]). Velocity space instabilities in simple cases of homogeneous bi-Maxwellian plasma were described. The distribution function of a plasma con\ufb01ned in a mirror \ufb01eld is zero for loss cone region (v\u22a5/v)2 < 1/RM (RM is mirror ratio). The instability associated with this is called loss-cone instability (ref.[4]). In general plasmas are hot and dense in the center and are cold and low density. The instabilities driven by temperature gradient and density gradient are called drift instability. The electrostatic drift instability (ref.[5],[7]) of inhomogeneous plasma can be analyzed by the more general dispersion equation described in appendix C. In toroidal \ufb01eld, trapped particles always exist in the outside where the magnetic \ufb01eld is weak. The instabilities induced by the trapped particles is called trapped particle instability (ref.[6]). References [1] R. J. Briggs: Electron-Stream Interaction with Plasma, The MIT Press, Cambridge, Mass. 1964. [2] E. G. Harris: Phys. Rev. Lett. 2, 34 (1959) [3] E. G. Harris: Physics of Hot Plasma, p.145 (ed. by B. J. Rye and J. B. Taylor) Oliver & Boyd, Edinburgh (1970). [4] M. N. Rosenbluth and R. F. Post: Phys. Fluids 8, 547 (1965). [5] N. A. Krall and M. N. Rosenbluth: Phys. Fluids 8, 1488 (1965). [6] B. B. Kadomtsev and O. P. Pogutse: Nucl. Fusion 11, 67 (1971). [7] K. Miyamoto: Plasma Physics for Nuclear Fusion (revised edition) Chap.12, The MIT Press, Cambridge, Mass. 1989 166 167 Ch.14 Instabilities Driven by Energetic Particles Sustained ignition of thermonuclear plasma depends on heating by highly energetic alpha particles produced from fusion reactions. Excess loss of the energetic particles may be caused by \ufb01shbone instability and toroidal Alfve´n eigenmodes. Such losses can not only reduce the alpha particle heating e\ufb03ciency, but also lead to excess heat loading and damage to plasma-facing components. These problems have been studied in experiments and analyzed theoretically. In this chapter basic aspects of theories on collective instabilities by energetic particles are described. 14.1 Fishbone Instability Fishbone oscillations were \ufb01rst observed in PDX experiments with nearly perpendicular neutral beam injection. The poloidal magnetic \ufb01eld \ufb02uctuations associated with this instabilities have a characteristic skeletal signature on the Mirnov coils, that has suggested the name of \ufb01shbone oscillations. Particle bursts corresponding to loss of energetic beam ions are correlated with \ufb01shbone events, reducing the beam heating e\ufb03ciency. The structure of the mode was identi\ufb01ed as m=1, n=1 internal kink mode, with a precursor oscillation frequency close to the thermal ion diamagnetic frequency as well as the fast ion magnetic toroidal precessional frequency. 14.1a Formulation Theoretical analysis of \ufb01shbone instability is described mainly according to L. Chen, White and Rosenbluth (ref.[1]). Core plasma is treated by the ideal MHD analysis and the hot component is treated by gyrokinetic description. The \ufb01rst order equation of displacement \u3be is (refer to (8.25)) \u3c1m\u3b3 2\u3be = j × \u3b4B + \u3b4j ×B \u2212\u2207\u3b4pc \u2212\u2207\u3b4ph. (14.1) where \u3b4pc is the \ufb01rst order pressure disturbance of core plasma \u2207\u3b4pc = \u2212\u3be · \u2207pc \u2212 \u3b3sp\u2207 · \u3be. \u3b4ph is the \ufb01rst order pressure disturbance of hot component. The following ideal MHD relations hold: \u3b4E\u22a5 = \u3b3\u3be ×B, \u3b4E\u2016 = 0, \u3b4B = \u2207× (\u3be ×B), \u3b4j = \u2207 · \u3b4B. By multiplying \u222b dr\u3be\u2217 on (14.1) and assuming a \ufb01xed conducting boundary, we have \u3b4WMHD + \u3b4WK + \u3b4I = 0 (14.2) where \u3b4I = \u3b32 2 \u222b \u3c1m|\u3be|2dr (14.3) \u3b4WK = 1 2 \u222b \u3be · \u2207\u3b4phdr (14.4) and \u3b4WMHD is the potential energy of core plasma associated with the displacement \u3be, which was discussed in sec.8.2b and is given by (8.79). \u3b4WK is the contribution from hot component. 14.1b MHD Potential Energy Let us consider the MHD term of \u3b4WMHD, which consists of the contribution of \u3b4W sMHD from sin- gular region near rational surface and the contribution \u3b4W extMHD from the external region. External contribution \u3b4W extMHD of cylindrical circular plasma is already given by (8.92) \u3b4W extMHDcycl 2\u3c0R = \u3c0 2\u3bc0 \u222b a 0 ( f \u2223\u2223\u2223\u2223d\u3berdr \u2223\u2223\u2223\u22232 + g|\u3ber|2 ) dr (14.5) 167 168 14 Instabilities Driven by Energetic Particles where f and g are given by (8.93) and (8.95). When r/R\ufffd 1 is assumed, f and g of (-m, n) mode are f = r3 R2 B2z ( 1 q \u2212 n m )2( 1\u2212 ( nr mR )2) g = r R2 B2z (( 1 q \u2212 n m )2( (m2 \u2212 1) + n 2r2 R2 )( 1\u2212 ( nr mR )2 \u2212 2 ( 1 q2 \u2212 ( n m )2)( nr mR )2)) +2 ( nr mR )2 \u3bc0 dp0 dr where q(r) \u2261 (rBz/RB\u3b8(r)) is the safety factor. Let us consider the m=1 perturbation with the singular radius r = rs (q(rs) = m/n). In this case the displacement is \u3ber =const. for 0 < r < rs and \u3ber = 0 for rs < r < a (refer sec.8.3b). Then \u3b4W extMHDcycl is reduced to (ref.[2]) \u3b4W extMHDcycl 2\u3c0R = \u3c0B2\u3b8s 2\u3bc0 |\u3bes|2 ( rs R )2 ( \u2212\u3b2p \u2212 \u222b 1 0 \u3c13 ( 1 q2 + 2 q \u2212 3 ) d\u3c1 ) (14.6) where \u3c1 = r/rs, \u3b2p \u2261 \u3008p\u3009s/(B2\u3b8s/2\u3bc0) and B\u3b8s \u2261 (rs/Rq(rs))Bz is the poloidal \ufb01eld at r = rs. The pressure \u3008p\u3009s is de\ufb01ned by \u3008p\u3009s = \u2212 \u222b rs 0 ( r rs )2 dp dr dr = 1 r2s \u222b rs 0 (p\u2212 ps)2rdr. (14.7) MHD potential energy \u3b4W extMHDtor/2\u3c0R per unit length of toroidal plasma with circular cross section is given by (ref.[3]) \u3b4W extMHDtor 2\u3c0R = ( 1\u2212 1 n2 ) \u3b4W extMHDcycl 2\u3c0R + \u3c0B2\u3b8s 2\u3bc0 |\u3bes|2\u3b4W\u2c6T, \u3b4W\u2c6T = \u3c0 ( rs R )2 3(1\u2212 q0) ( 13 144 \u2212 \u3b22ps ) . (14.8) In the case of m=1 and n=1, \u3b4W extMHDtor/2\u3c0R is reduced to only the term of \u3b4W\u2c6T. Let us consider the contribution from singular region. In this case we must solve the displacement \u3ber in singular region near rational surface. The equation of motion in singular surface was treated in sec.9.1 of tearing instability. From (9.13) and (9.9), we have (in the limit of x\ufffd 1) \u3bc0\u3c1m\u3b3 2\u2202 2\u3ber \u2202x2 = iF \u22022B1r \u2202x2 (14.9) \u3b3B1r = iF\u3b3\u3ber + \u3b7 \u3bc0r2s \u22022 \u2202x2 B1r (14.10) where F \u2261 (k ·B) = \u2212m r B\u3b8 + n R Bz = B\u3b8 r (\u2212m+nq) = B\u3b8n r dq dr \u394r = B\u3b8ns rs x, x \u2261 r \u2212 rs rs , s \u2261 rs dqdr \u2223\u2223\u2223\u2223 rs . By following normalizations \u3c8 \u2261 iB1rrs B\u3b8,ssn