= qe cos(kz ωt). ω sin(αt) This is further averaged over the distribution of initial ( ) α + ω. = 2m k P g(α) sin(αt) g(α) = g(0) + αg (0) + α2 2 g +

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1 6 Landau Damping 61 Physical Picture of Landau Damping Consider a 1-dimensional electrostatic (longitudinal) wave with k E in the absence of magnetic field Taking v = ẑv and E = ẑe cos(kz ωt), the singleparticle equation of motion can be written as m dv = qe cos(kz ωt) The th order solution (for E = ) z = v t + z can be substituted into the 1st order equation to give m dv 1 = qe cos(kz + kv t ωt) This equation is solved as an initial value problem with the initial condition v 1 = at t = The solution is given by v 1 = qe m sin(kz + kv t ωt) sin(kz ) kv ω The time rate of change of the kinetic energy, averaged over initial positions z is d mv = z q E [ ω sin(αt) ] ωt cos(αt) m α + t cos(αt) + α where α = kv ω velocities v to give d mv Expanding in the vicinity of α = gives d This is further averaged over the distribution of initial ( ) α + ω f(v ) = f = g(α) k = z, ωq E m k P g(α) sin(αt) dα v α g(α) = g() + αg () + α g + mv z, πωq E [ ] df(v ) mk k dv v v = ω k This expression signifies that resonant particles with velocity close to the wave phase velocity determine absorption of wave power by particles 6 A Simple Kinetic Model Consider a 1-dimensional oscillation along the magnetic field (or in the absence of magnetic field) Vlasov equation can be written as f t + v f z + q E(z, t) f m v =

2 The first order equation is df 1 = f 1 t + v f 1 z = q m E(z, t)df (v) dv The left hand side is called the convective derivative, and signifies time derivative along the particle trajectory Taking E(z, t) = R[E 1 exp(ikz iωt)], the solution can be expressed as [ iqe1 f 1 (z, v, t) = g 1 (z vt, v) R m df (v) dv 1 eikz iωt ei(ω kv)(t t) ω kv where g 1 (z vt, v) is the homogeneous solution (solution for E = ), and is chosen to satisfy the initial condition The last term ], is shown in Fig 1 F (u) = 1 ei(u u)τ u u = 1 ei(ω kv)(t t ) ω kv Fg 1 Real and imaginary parts of the resonance term F (u) It was pointed out in Chap 3 that to satisfy causality the integration contour on the complex ω plane must be taken above the singularity that exists on the real ω axis This is mathematically equivalent to shifting the singularity slightly to the negative imaginary side of the real axis This can be accomplished by introducing randomization by collisions The probability of not suffering collisions since t = t is given by exp[ ν(t t )], where ν is the collision frequency Multiplying this probability and averaging over all particles reaching z and v at time t, [ ] iqe1 df (v) 1 f 1 (z, v, t) = R m dv eikz iωt ω kv + iν For a nearly Maxwellian distribution function, df /dv has a velocity wih of order v th, whereas 1/(ω kv + iν) has a wih of ν/k, as illustrated in Fig 3

3 Fg The functions df dv and 1 ω kv + iν 63 Validity Conditions for Landau Damping The bottom of a sinusoidal potential well can be approximated as qϕ [1 cos(kx)] qϕ k x A charged particle oscillates harmonically in this potential with period τ osc = 1 m = ω osc qke The following conditions must be satisfied for valid Landau damping 1 ω i τ osc > 1 Significant growth (or damping) must occur before v changes substantially (which occurs in an oscillation time in the potential well, τ osc ), since linear theory assumes v = const ν coll τ osc > 1 The collision time τ coll must be shorter than τ osc to ensure the assumption v = const 3 k λ mfp > 1 (v th > ν coll /k ) Mean free path must be longer than a wavelength for particles to recognize the presence of a wave The valid range of collision frequency is shown in Fig 3 Fg 3 The valid range of collision frequency for Landau damping 4

4 64 ES Waves in a Maxwellian Unmagnetized Plasma Assume a drifting Maxwellian plasma, given by f s (v) = n s exp [ (v V s) ], πvths v ths where v ths = T s/m s For real k, Iω > and defining τ = t t f 1 (ω, k, v) = nqe(ω, k) πmvth d dv (v V ) dτe i(ω kv)τ e v th It is useful to define v p moments of the distribution function, Z p (ω, v th, k, V, nω) = ik π In particular, for p = where and dv v p Z (ζ n ) = i π sgn(k )e ζ n S(ζn ) ζ n = ω k V nω k v th ζ S(ζ) = e ζ dz e z = S( ζ) dτ e i(ω nω k v)τ e (v V ) v th The functions S(ζ) and ( π/) exp( ζ ) are illustrated in Fig 4 π Fg 4 The functions S(x) and e x parts of the plasma dispersion function which describe the real and imaginary The dispersion relation for a 1-D electrostatic wave in the absence of magnetic field can be derived from Poisson s equation ike(ω, k) = s q s ϵ f s1 (v, ω, k)dv 5

5 as k = s 1 λ ds Z (ζ (s) ) where and λ ds = n sq s ϵ T s Z (ζ) = [1 + ζz (ζ)] A closely related function is called the plasma dispersion function, which is defined as ) Z(ζ) = i dz exp (iζz z 4 The functions Z and Z are related to each other as { Z(ζ) for k > Z (ζ) = Z( ζ) for k < 6