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< 0 in the case of damping) n \u2329 d dt mv2 2 \u232a z0v0 = \u22122\u3b3W and the growth rate \u3b3 is given by \u3b3 \u3c9 = \u3c0 2 ( \u3a0 \u3c9 )2 ( \u3c9 |k| )( v0 \u2202f(v0) \u2202v0 ) v0=\u3c9/k (11.11) where \u3a02 = nq2/\ufffd0m, W \u2248 2\ufffd0E2/4, \u222b f(v)dv = 1. There is a restriction on the applicability of linear Landau damping. When this phenomenon runs its course before the particle orbit deviates from the linear-approximation solution, the reductions leading to linear Landau damping are justi\ufb01ed. The period of oscillation in the potential well of the electric \ufb01eld of the wave gives the time for the particle orbit to deviate from the linear approximation ( \u3c92 \u223c eEk/m from m\u3c92x = eE). The period of oscillation is \u3c4osc = 1 \u3c9osc \u2248 ( m ekE )1/2 . Consequently the condition for the applicability of linear Landau damping is that the Landau damping time 1/\u3b3 is shorter than \u3c4osc or the collision time 1/\u3bdcoll is shorter than \u3c4osc. |\u3b3\u3c4osc| > 1, (11.12) |\u3bdcoll\u3c4osc| > 1. (11.13) On the other hand, it was assumed that particles are collisionless. The condition that the collision time 1/\u3bdcoll is longer than \u3bb/vrms is necessary for the asymptotic approximation of the integral (11.9) as t \u2192 \u221e, where \u3bb is the wavelength of the wave and vrms is the spread in the velocity distribution; 1 \u3bdcoll > 2\u3c0 kvrms . (11.14) 139 140 11 Landau Damping and Cyclotron Damping 11.2 Transit-Time Damping We have already described the properties of Alfve´n waves in cold plasmas. There are compres- sional and torsional modes. The compressional mode becomes magnetosonic in hot plasmas, as is described in ch.5. In the low-frequency region, the magnetic moment \u3bcm is conserved and the equation of motion along the \ufb01eld lines is m dvz dt = \u2212\u3bcm\u2202B1z \u2202z . (11.15) This equation is the same as that for Landau damping if \u2212\u3bcm and \u2202B1z/\u2202z are replaced by the electric charge and the electric \ufb01eld, respectively. The rate of change of the kinetic energy is derived similarly, and is \u2329 d dt mv2 2 \u232a z0,v0 = \u2212\u3c0\u3bc 2 m|k| 2m |B1z|2 ( \u3c9 k )( \u2202f(v0) \u2202v0 ) v0=\u3c9/k . (11.16) This phenomena is called transit-time damping. 11.3 Cyclotron Damping The mechanism of cyclotron damping is di\ufb00erent from that of Landau damping. Here the electric \ufb01eld of the wave is perpendicular to the direction of the magnetic \ufb01eld and the particle drift and accelerates the particle perpendicularly to the drift direction. Let us consider a simple case in which the thermal energy of particles perpendicular to the magnetic \ufb01eld is zero and the velocity of particles parallel to the magnetic \ufb01eld B0 = B0z\u2c6 is V . The equation of motion is m \u2202v \u2202t +mV \u2202v \u2202z = q(E1 + v × z\u2c6B0 + V z\u2c6 ×B1). (11.17) As our interest is in the perpendicular acceleration we assume (E1 · z\u2c6) = 0. B1 is given by B1 = (k×E)/\u3c9. With the de\ufb01nitions v± = vx ± ivy, E± = Ex ± iEy, the solution for the initial condition v = 0 at t = 0 is v± = iqE±(\u3c9 \u2212 kV ) exp(ikz \u2212 i\u3c9t) m\u3c9 1\u2212 exp(i\u3c9t\u2212 ikV t± i\u3a9t) \u3c9 \u2212 kV ±\u3a9 , (11.18) \u3a9 = \u2212qB0 m . The macroscopic value of v\u22a5 is obtained by taking the average weighted by the distribution function f0(V ) as follows: \u3008v\u22a5\u3009= iq exp(ikz \u2212 i\u3c9t)2m ((c ++c\u2212)E\u22a5+ i(c+\u2212c\u2212)E\u22a5 × z\u2c6), (11.19) c± = \u3b1± \u2212 i\u3b2±, (11.20) \u3b1± = \u222b \u221e \u2212\u221e dV f0(V )(1\u2212 kV/\u3c9)(1\u2212 cos(\u3c9 \u2212 kV ±\u3a9)t) \u3c9 \u2212 kV ±\u3a9 , (11.21) \u3b2± = \u222b \u221e \u2212\u221e dV f0(V )(1\u2212 kV/\u3c9) sin(\u3c9 \u2212 kV ±\u3a9)t \u3c9 \u2212 kV ±\u3a9 . (11.22) As t becomes large we \ufb01nd that \u3b1± \u2192 P \u222b \u221e \u2212\u221e dV f0(V )(1\u2212 kV/\u3c9) \u3c9 \u2212 kV ±\u3a9 , (11.23) \u3b2± \u2192 \u2213\u3c0\u3a9 \u3c9|k| f0 ( \u3c9 ±\u3a9 k ) . (11.24) 140 11.3 Cyclotron Damping 141 When t\ufffd 2\u3c0 kVrms (11.25) where Vrms = \u3008V 2\u30091/2 is the spread of the velocity distribution, the approximations (11.19)\u223c(11.24) are justi\ufb01ed. The absorption of the wave energy by the plasma particles is given by \u3008Re(qE exp(ikz \u2212 i\u3c9t))(Re\u3008v\u22a5\u3009)\u3009z = q2 4m (\u3b2+|Ex + iEy|2 + \u3b2\u2212|Ex \u2212 iEy|2). (11.26) Let us consider the case of electrons (\u3a9e > 0). As was described in sec.10.2, the wave N2 = R propagating in the direction of magnetic \ufb01eld (\u3b8 = 0) satis\ufb01es Ex+ iEy = 0, so that the absorption power becomes Pe = q2 4m \u3b2\u2212|Ex \u2212 iEy|2. When \u3c9 > 0, (11.24) indicates \u3b2\u2212 > 0. In the case of \u3c9 < 0, \u3b2\u2212 is nearly zero since f0 ((\u3c9 \u2212\u3a9e)/k)\ufffd 1. Let us consider the case of ions (\u2212\u3a9i > 0). Similarly we \ufb01nd Pi = q2 4m \u3b2+|Ex + iEy|2. When \u3c9 > 0, (11.24) indicates \u3b2+ > 0. In the case of \u3c9 < 0, \u3b2+ is nearly zero, since f0 (\u3c9 +\u3a9i/k)\ufffd 1. The cyclotron velocity Vc is de\ufb01ned so that the Doppler shifted frequency (the frequency of wave that a particle running with the velocity V feels) is equal to the cyclotron frequency, that is, \u3c9 \u2212 kVc ±\u3a9 = 0, Vc = \u3c9 k ( 1± \u3a9 \u3c9 ) . Accordingly particles absorb the wave energy when the absolute value of cyclotron velocity is smaller than the absolute value of phase velocity of the wave (±\u3a9/\u3c9 < 0)(see (11.24)). This phenomena is called cyclotron damping. Let us consider the change in the kinetic energy of the particles in the case of cyclotron damping. Then the equation of motion is m dv dt \u2212 q(v ×B0) = qE\u22a5 + q(v ×B1). Since B1 = (k ×E)/\u3c9 and Ez = 0, we have m dvz dt = qkz \u3c9 (v\u22a5 ·E\u22a5), m dv\u22a5 dt \u2212 q(v\u22a5 ×B0) = qE\u22a5 ( 1\u2212 kzvz \u3c9 ) so that mv\u22a5 · dv\u22a5dt = q(v\u22a5 ·E\u22a5) ( 1\u2212 kzvz \u3c9 ) . Then d dt ( mv2z 2 ) = kzvz \u3c9 \u2212 kzvz d dt ( mv2\u22a5 2 ) , 141 142 11 Landau Damping and Cyclotron Damping v2\u22a5 + ( vz \u2212 \u3c9 kz )2 = const. In the analysis of cyclotron damping we assumed that vz = V is constant; the condition of the validity of linearized theory is (ref.[3]) k2zq 2E2\u22a5|\u3c9 \u2212 kzvz|t3 24\u3c92m2 < 1. We have discussed the case in which the perpendicular thermal energy is zero. When the per- pendicular thermal energy is larger than the parallel thermal energy, so-called cyclotron instability may occur. The mutual interaction between particles and wave will be discussed again in chs.12 and 13 in relation to heating and instabilities. 11.4 Quasi-Linear Theory of Evolution in the Distribution Function It has been assumed that the perturbation is small and the zeroth-order terms do not change. Under these assumption, the linearized equations on the perturbations are analyzed. However if the perturbations grow, then the zeroth-order quantities may change and the growth rate of the perturbations may change due to the evolution of the zeroth order quantities. Finally the pertur- bations saturate (growth rate becomes zero) and shift to steady state. Let us consider a simple case of B = 0 and one dimensional electrostatic perturbation (B1 = 0). Ions are uniformly distributed. Then the distribution function f(x, v, t) of electrons obeys the following Vlasov equation; \u2202f \u2202t + v \u2202f \u2202x \u2212 e m E \u2202f \u2202v = 0. (11.27) Let the distribution function f be divided into two parts f(x, v, t) = f0(v, t) + f1(x, v, t) (11.28) where f0 is slowly changing zeroth order term and f1 is the oscillatory 1st order term. It is assumed that the time derivatives of f0 is the 2nd order term. When (11.28) is substituted into (11.27), the 1st and the 2nd terms satisfy following equations; \u2202f1 \u2202t + v \u2202f1 \u2202x = e m E \u2202f0 \u2202v , (11.29) \u2202f0 \u2202t = e m E \u2202f1 \u2202v . (11.30) f1 and E may be expressed by Fourier integrals; f1(x, v, t) = 1 (2\u3c0)1/2 \u222b fk(v) exp(i(kx\u2212 \u3c9(k)t))dk, (11.31) E(x, t) = 1 (2\u3c0)1/2 \u222b Ek exp(i(kx\u2212 \u3c9(k)t))dk. (11.32) Since f1 and E are real, f\u2212k = f\u2217k , E\u2212k = E \u2217 k , \u3c9(\u2212k) = \u2212\u3c9\u2217(k) (\u3c9(k) = \u3c9r(k) + i\u3b3(k)). The substitution of (11.31) (11.32) into (11.29) yields fk(v) = e m ( i \u3c9(k)\u2212 kv ) Ek \u2202f0 \u2202v . (11.33) If (11.32) (11.33) are substituted into (11.30), we \ufb01nd \u2202f0(v, t) \u2202t = ( e m )2 \u2202 \u2202v \u2329 1 2\u3c0 \u222b Ek\u2032 exp(i(k\u2032x\u2212 \u3c9(k\u2032)t))dk\u2032 142