Notes on Landau damping by Youjun Hu Institute of Plasmas, Chinese Academy of Sciences Email: [email protected] In Sec. 1 of this note, resonant wave particle interactions are demonstrated by a simple test- particle simulation. Section 2 reproduces Stix's derivation of Landau damping from the view of test particles[4]. In Sec. 3, linearized Vlasov-Poison equations are solved numerically to demon- strate the phase mix in velocity space and the resulting Landau damping. 1 Test-particle simulation of Landau damping Consider a longitudinal wave given by E = z^E cos(kz !t): (1) ¡ The equations of motion of a test particle in the wave eld are given by dv m = qE cos(kz !t); (2) dt ¡ and dz = v; (3) dt where v v z^ is the z component of the velocity of the particle. Normalize z by the wavelength , t by the wave period T , v by the phase velocity vp, i.e., z t v z = ; t = ; v = ; (4) T vp where = 2 / k, T = 2 / !, vp = ! / k. Using the normalized quantities, Eqs. (2) and (3) are written, respectively, as d v 2kqE = cos[2(z t)]; (5) d t m!2 ¡ and d z =v: (6) d t The initial distribution function of particles f0(z; v) is taken to be uniform in space and Maxwellian in velocity, 1 v2 f (z; v) = f (v) = exp ; (7) 0 m p ¡2v2 vt 2 t which satises the normalization condition 1 f0(v)dv = 1 (Note that here vt = T /m, which ¡1 is dierent from the usual denition vt = R 2T /m used Sec. 3.4 of this note). pIn my particle simulation code (/home/yj/project_new/pic_code), 4 105 particles are initially loaded p random in z and Maxwellian in v. Then the motion equations of every particle are followed numerically to obtain the location and velocity at later time. In the numerical code, when a par- ticle leaves from the region 0 6 x 6 1, it is shifted by one wavelength to return to this region. This shift does not inuence the force on the particle and it simulates the situation of innite length in z direction, where when a particle leave the region 0 6 x 6 1 from the right boundary, a particle of the same velocity will enter the region from the left boundary, and vice versa. 1 2 Section 1 The velocity distribution at later time is obtained by counting the number of particles in each velocity interval. Figure 1a compares the velocity distribution function at t = 0 and t = 10, which shows that the distribution is atted in the resonant region v /vp = 1, which suggests that the total kinetic energy of particles may be increased. Figure 2 plots the temporal evolution of the total kinetic energy of the particles, which conrms that the kinetic energy is increased by the wave. The conservation of energy tell us that the increased kinetic energy of particles must be drawn from the wave, i.e., the wave encounters damping. 0.6 0.6 (a) t=0 t=0 t/T=10 t/T=1 0.5 0.5 t/T=5 t/T=10 t/T=20 0.4 0.4 0.3 0.3 0.2 0.2 velocity distribution function 0.1 velocity distribution function 0.1 0 0 -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 v/vp v/vp Figure 1. Comparison of the velocity distribution function (spatially averaged) at various time, which shows that the distribution is distorted in the resonant region (v / v 1). Other parameters: v / v = 1, p t p 2kqE /m!2 = 1. 1.14 1.12 1.1 1.08 k E 1.06 1.04 1.02 1 0 5 10 15 20 t/T E = i=N 1mv2 / Figure 2. Temporal evolution of the total kinetic energy of the particles, where k i=1 2 i N 1 mv2 v /v = 1 2kqE /m!2 = 1 2 t . Other parameters: t p , . P Although the above simulation is performed by holding the wave amplitude constant, it takes into account all the nonlinear physics of the particle motion in the wave eld. Therefore this is a nonlinear simulation. 1.1 Linear Landau damping In the early phase of the simulation t p2 /! ; (8) b where !b is the bounce angular frequency of particles in the trough of the wave, the trapped particles eects can be neglected. This phase can be considered as the linear phase where the linear Landau damping theory (discussed later in this note) is valid. The bounce angular fre- quency of particles in the trough of the wave is given by[1] kqE ! = : (9) b m r Test-particle simulation of Landau damping 3 Using this, the condition (8) is written as t m!2 : (10) T 2kqE r This condition reduces to t / T 1 for the case plotted in Fig. 2, where we see that the total kinetic energy of the particles increases monotonously with time during this period. From the data in Fig. 2, the temporal change rate of the total kinetic energy can be computed, which gives d Ek / dt = 0.08. Next, I compare this result with those given by the analytic formula (39) (given later in this note), which is written dE ! (qE)2 df(v ) k = 0 : dt ¡ k k 2m dv0 j j v0=!/k Using Eq. (7) and vt/vp = 1, the above expression is written dE (qE)2 1 1 k = exp dt 2m! p ¡2 2 2 Multiplying by T and then dividing by mvt /2, the above expression is written d E (kqE)2 1 1 k = 42 exp d t m2!4 p ¡2 2 2 In the simulation 2kqE /m!2 = 1. Using this, expression (10) is written as d E 1 1 k = exp = 0.12 d t p ¡2 2 2 The result given by the analytic formula is slightly dierent from that of the simulation (0.12 vs 0.08). Considering the various approximations used in deriving the analytic formula, the two results can be considered to be in agreement with each other. 1.2 Nonlinear Landau damping In the phase after the linear phase, Fig. 2 indicates that the total kinetic energy of the particles oscillates with time, with the saturation level larger than the initial kinetic energy of the parti- cles, i.e., there is nonzero net energy drawn from the wave by the particles. This is the nonlinear Landau damping, which clearly demonstrates that net energy exchange between waves and par- ticles can be nonzero on the long time scale without including any collisional eect. 1.3 Inverse Landau damping Choose a drift-Maxwellian distribution 2 1 (v vb) f0(z; v) = f m(v) = exp ¡ ; (11) p ¡ 2v2 vt 2 t with vb / vp = 2. Then the derivative of the distribution function with respect to the velocity in the resonant region is positive. This is the case where an inverse Landau damping is expected to appear. Figure 3 compares the velocity distribution function at t = 0 and t = 10, which shows that the distribution is atted in the resonant region v /vp = 1. Figure 4 plots the temporal evo- lution of the total kinetic energy of the particles, which confirms that the kinetic energy is reduced by the wave. 4 Section 1 0.4 (a) t=0 0.35 t/T=10 0.3 0.25 0.2 0.15 0.1 velocity distribution function 0.05 0 -2 -1 0 1 2 3 4 5 6 v/vp Figure 3. Comparison of the velocity distribution function (spatially averaged) at various time, which shows that the distribution is distorted in the resonant region (v / v 1). Other parameters: v /v = 1, p t p 2kqE /m!2 = 1. 1.005 1 0.995 k E 0.99 0.985 0.98 0.975 0 5 10 15 20 t/T E = i=N 1mv2 /E Figure 4. Temporal evolution of the total kinetic energy of the particles, where k i=1 2 i k0, 2 E is the initial total kinetic energy. Other parameters: vt/vp = 1, 2kqE /m! = 1. k0 P 1.4 Density uctuation induced by the wave Figure (5) compares the spatial distribution at t = 0 and t = 10T , which shows that the distribu- tion become nonuniform at t = 10T . 1.15 t=0 t/T=10 1.1 1.05 1 0.95 spatial distribution function 0.9 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x/λ Figure 5. Comparison of the spatial distribution function at t = 0 and t = 10, which shows that the dis- 2 tribution seems to become nonuniform at t = 10. Other parameters: vt = vp, 2kqE /m! = 1. Analytical theory of wave particle interaction 5 Figure 6 is a GIF animation, which shows the time evolution of the spatial and velocity dis- tribution (spatially averaged) of the particles. The GIF animation can be viewed only in the HTML version of this document (it does not work in the PDF version). As the animation shows, the distribution function in the resonant region (v/v 1) oscillates with large amplitude p at early stage, and then the amplitude becomes smaller and saturated. The spatial distribution also oscillates with large amplitude at early stage and then become much smaller and saturated. The spatial uctuation of the density induced by the longitude wave may explain the density pump out phenomina induced by low-hybrid waves observed in many tokamaks.