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M. Brambilla and W.D. Getty : Nucl. Fusion 17, 929 (1977). [13] S. Takamura: Fundamentals of Plasma Heating, Nagoya Univ. Press, 1986 (in Japanese). [14] I. Fidone, G. Granata and G. Ramponi: Phys. Fluids 21, 645 (1978). [15] R. Pratter: Phys. Plasmas 11, 2349 (2004). 161 162 Ch.13 Velocity Space Instabilities (Electrostatic Waves) Besides the magnetohydrodynamic instabilities discussed in ch.8, there is another type of insta- bilities, caused by deviations of the velocity space distribution function from the stable Maxwell form. Instabilities which are dependent on the shape of the velocity distribution function are called velocity space instabilities or microscopic instabilities. However the distinction between microscopic and macroscopic or MHD instabilities is not always clear, and sometimes an instability belongs to both. 13.1 Dispersion Equation of Electrostatic Wave In this chapter, the characteristics of the perturbation of electrostatic wave is described. In this case the electric \ufb01eld can be expressed by E = \u2212\u2207\u3c6 = \u2212ik\u3c6. The dispersion equation of electrostatic wave is give by (sec.10.5) k2xKxx + 2kxkzKxz + k 2 zKzz = 0. The process in derivation of dispersion equation of hot plasma will be described in details in appendix C. When the zeroth-order distribution function is expressed by 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 2n\u3baT\u22a5 )1/2 exp ( \u2212mv 2 z 2\u3baTz ) the dispersion equation is given by (C.36) as follows k2x + k 2 z + \u2211 i,e \u3a02 m \u3baTz ( 1 + \u221e\u2211 n=\u2212\u221e ( 1 + Tz T\u22a5 (\u2212n\u3a9) \u3c9n ) \u3b6nZ(\u3b6n)In(b)e\u2212b ) = 0 (13.1) where \u3b6n \u2261 \u3c9n21/2kzvTz , \u3c9n \u2261 \u3c9 \u2212 kzV + n\u3a9, b = (kxvT\u22a5/\u3a9)2, v2Tz = \u3baTz/m, v 2 T\u22a5 = \u3baT\u22a5/m, In(b) is n-th modi\ufb01ed Bessel function Z(\u3b6) is plasma dispersion function. When the frequency of the wave is much higher than cyclotron frequency (|\u3c9| \ufffd |\u3a9|), then we \ufb01nd \u3b6n \u2192 \u3b60, n\u3a9 \u2192 0, \u2211 In exp(\u2212b) = 1, so that the dispersion equation is reduced to k2x + k 2 z + \u2211 i,e \u3a02 m \u3baTz (1 + \u3b60Z(\u3b60)) = 0 (|\u3c9| \ufffd |\u3a9|). (13.2) The dispersion equation in the case of B = 0 is given by k2 + \u2211 i,e \u3a02 m \u3baT (1 + \u3b6Z(\u3b6)) = 0. ( \u3b6 = \u3c9 \u2212 kV 21/2kvT , B = 0 ) (13.3) When the frequency of wave is much lower than cyclotron frequency (|\u3c9| \ufffd |\u3a9|), then we \ufb01nd \u3b6n \u2192\u221e (n = 0), \u3b6nZn \u2192 \u22121 and \u2211 In(b)e\u2212b = 1. k2x + k 2 z + \u2211 i,e \u3a02 m \u3baTz ( I0e \u2212b (1 + \u3b60Z(\u3b60)) + Tz T\u22a5 ( 1\u2212 I0e\u2212b )) = 0. (|\u3c9| \ufffd |\u3a9|) (13.4) 162 13.1 Dispersion Equation of Electrostatic Wave 163 When the frequency of wave is much higher than cyclotron frequency or the magnetic \ufb01eld is very small, the dispersion equations (13.2),(13.3) are reduced to k2x + k 2 z + \u2211 \u3a02 m \u3baTz \u239b\u239c\u239c\u239dkz \u3baTzm \u222b \u2202(f/n0) \u2202vz \u3c9 \u2212 kzvz dvz \u239e\u239f\u239f\u23a0 = 0. Partial integration gives k2x + k 2 z k2z = \u2211 \u3a02 \u222b (f/n0) (\u3c9 \u2212 kzvz)2 dvz. (13.5) 13.2 Two Streams Instability The interaction between beam and plasma is important. Let us consider an excited wave in the case where the j particles drift with the velocity Vj and the spread of the velocity is zero. The distribution function is given by fi(vz) = nj\u3b4(vz \u2212 Vj). The dispersion equation of the wave propagating in the direction of the magnetic \ufb01eld (kx = 0) is 1 = \u2211 j \u3a02j (\u3c9 \u2212 kVj)2 . In the special case \u3a021 = \u3a0 2 2 (n 2 1q 2 1/m1 = n 2 2q 2 2/m2), the dispersion equation is quadratic: (\u3c9 \u2212 kV¯ )2 = \u3a02t ( 1 + 2x2 ± (1 + 8x2)1/2 2 ) where \u3a02t = \u3a0 2 1 +\u3a0 2 2 , x = k(V1 \u2212 V2) 2\u3a0t , V¯ = V1 + V2 2 . For the negative sign, the dispersion equation is (\u3c9 \u2212 kV¯ )2 = \u3a02t (\u2212x2 + x3 + · · ·) (13.6) and the wave is unstable when x < 1, or k2(V1 \u2212 V2)2 < 4\u3a02t . The energy to excite this instability comes from the zeroth-order kinetic energy of beam motion. When some disturbance occurs in the beam motion, charged particles may be bunched and the electric \ufb01eld is induced. If this electric \ufb01eld acts to amplify the bunching, the disturbance grows. This instability is called two-streams instability. 13.3 Electron Beam Instability Let us consider the interaction of a weak beam of velocity V0 with a plasma which consists of cold ions and hot electrons. The dispersion equation (13.5) of a electrostatic wave with kx = 0 (Ex = Ey = 0, Ez = 0, B1 = 0) is given by Kzz = K\u2016 \u2212 \u3a02b (\u3c9 \u2212 kV0)2 = 1\u2212 \u3a02i \u3c92 \u2212\u3a02e \u222b \u221e \u2212\u221e fe(vz)/n0 (\u3c9 \u2212 kvz)2 dvz\u2212 \u3a02b (\u3c9 \u2212 kV0)2 = 0. (13.7) 163 164 13 Velocity Space Instability (Electrostatic Wave) For the limit of weak beam (\u3a02b \u2192 0), the dispersion equation is reduced to K\u2016(\u3c9, k) \u2248 0, if \u3c9 = kV0. The dispersion equation including the e\ufb00ect of weak beam must be in the form of \u3c9 \u2212 kV0 = \u3b4\u3c9(k). (\u3b4\u3c9(k)\ufffd kV0) Using \u3b42\u3c9 we reduce (13.7) to \u3a02b \u3b42\u3c9 = K\u2016(\u3c9 = kV0, k) + ( \u2202K\u2016 \u2202\u3c9 ) \u3c9=kV0 \u3b4\u3c9. If \u3c9 = kV0 does not satisfy K\u2016 = 0, K\u2016 = 0 holds and the 2nd term in right-hand side of the foregoing equation can be neglected: \u3a02b \u3b42\u3c9 = K\u2016(\u3c9 = kV0, k). The expression for K\u2016(\u3c9 = kV0, k) is K\u2016(\u3c9r) = KR(\u3c9r) + iKI(\u3c9r). The KI term is of the Laudau damping (see sec.12.3). When the condition \u3c9 = kV0 is in a region where Landau damping is ine\ufb00ective, then |KI| \ufffd |KR| and the dispersion equation is reduced to \u3a02b (\u3c9 \u2212 kV0)2 = KR. (13.8) Therefore if the condition KR < 0 (13.9) is satis\ufb01ed, \u3b4\u3c9 is imaginary and the wave is unstable. When the dielectric constant is negative, electric charges are likely to be bunched and we can predict the occurrence of this instability. If \u3c9 = kV0 is in a region where Landau damping is e\ufb00ective, the condition of instability is that the wave energy density W0 in a dispersive medium (12.6) is negative, because the absolute value of W0 increases if \u2202W0/\u2202t is negative; \u2202W0 \u2202t = \u2202 \u2202t ( \ufffd0 2 E\u2217z \u2202 \u2202\u3c9 (\u3c9Kzz)Ez ) = \u2212\u3c9r 2 \ufffd0E \u2217 zKIEz < 0. When energy is lost from the wave by Landau damping, the amplitude of wave increases because the wave energy density is negative. Readers may refer to (ref.[1]) for more detailed analysis of beam-plasma interaction. 13.4 Harris Instability When a plasma is con\ufb01ned in a mirror \ufb01eld, the particles in the loss cone ((v\u22a5/v)2 < 1/Rm Rm being mirror ratio) escape, so that anisotropy appears in the velocity-space distribution function. The temperature perpendicular to the magnetic \ufb01eld in the plasma heated by ion or electron cyclotron range of frequency is higher than the parallel one. Let us consider the case where the distribution function is bi-Maxwellian. It is assumed that the density and temperature are uniform and there is no \ufb02ow of particles (V = 0). In this case the dispersion equation (13.1) is k2x + k 2 z + \u2211 j \u3a02jmj ( 1 \u3baTz + \u221e\u2211 l=\u2212\u221e Il(b)e\u2212b ( 1 \u3baTz + 1 \u3baT\u22a5 (\u2212l\u3a9) \u3c9 + l\u3a9 ) \u3b6lZ(\u3b6l) \u239e\u23a0 j = 0, (13.10) 164 13.4 Harris Instability 165 \u3b6l = \u3c9 + l\u3a9 (2\u3baTz/m)1/2kz . We denote the real and imaginary parts of the right-hand side of (13.10) for real \u3c9 = \u3c9r by K(\u3c9r) = Kr(\u3c9r) + iKi(\u3c9r). When the solution of (13.10) is \u3c9r + i\u3b3, i.e., for K(\u3c9r + i\u3b3) = 0, \u3c9r and \u3b3 are given by Kr(\u3c9r) = 0, \u3b3 = \u2212Ki(\u3c9r)(\u2202Kr(\u3c9r)/\u2202\u3c9r)\u22121 when |\u3b3| \ufffd |\u3c9r| and Taylor expansion is used. Accordingly we \ufb01nd k2x + k 2 z + \u2211 j \u3a02jmj ( 1 \u3baTz + 1 (2\u3baTz/m)1/2kz \u221e\u2211 l=\u2212\u221e Ile \u2212b ( \u3c9 + l\u3a9 \u3baTz \u2212 l\u3a9 \u3baT\u22a5 ) Zr(\u3b6l) ) j = 0, (13.11) \u3b3 = \u2212 1 A \u3c01/2 \u2211 j \u3a02jmj × \u239b\u239d \u221e\u2211 l=\u2212\u221e Ile \u2212b 1 (2\u3baTz/m)1/2|kz | ( \u3c9 + l\u3a9 \u3baTz \u2212 l\u3a9 \u3baT\u22a5 ) exp(\u2212\u3b62l ) ) j , (13.12) A = \u2211 j \u3a02jmj \u221e\u2211 l=\u2212\u221e [ Ile \u2212b ( Zr(\u3b6l)/\u3baTz (2\u3baTz/m)1/2kz + 1 (2\u3baTz/m)k2z ( \u3c9 + l\u3a9 \u3baTz \u2212 l\u3a9 \u3baT\u22a5 ) Z \u2032r(\u3b6l) )] j . Zr(\u3b6l) is real part of Z(\u3b6l) and Z \u2032r(\u3b6l) is the derivatives by \u3b6l. Let us assume that the electron is cold