5.15 Methods of identifying resonant particles

Analytically, resonant particles are defined as those particles whose velocity is close to the phase-velocity of the wave. These particles are expected to exchange more energy with the waves compared with the non-resonant particles. Next, we examine the phase-space structure of δF in order to find a general way of identifying the resonant region in the phase-space. The initial phase-space structure of δF is plotted in Fig. 6a, which shows the fluctuation in x direction and Maxwellian distribution in v_x direction. Figure 6b plots the phase-space structure of δF at t = 20∕ω_pe. It is not obvious what kind of useful information can be obtained from the figure. Note that lower velocity particles carry more perturbation than higher velocity particles because of the exp(−v²∕v_t²) dependence in δF. The dominant structure of δF in the lower velocity region may blur the change of δF in the higher velocity region. To make the change of δF obvious, define a new function S(v,x) ≡ δF∕, which eliminate the initial variation of δF in v_x direction. Figure 7 plots the contour of S(v,x) in phase space (x,v) at t = 0 and t = 20∕ω_pe, which shows that there are peaks developing near v ≈±2.44 at t = 20∕ω_pe. The location of the peaks of S in the phase-space, v ≈±2.44, is very near the phase-velocity of the wave excited in the simulation (v_p∕v_t = ω∕kv_t = ±2.44). Therefore, the peaks of S prove to be a good indication of the resonant region.

Figure 6: Contour of δF in phase-space (x,v) at t = 0 (left figure) and t = 20∕ω_pe (right figure) in the δf simulation with uniform sampling of phase-space.. Initial value of δF is set as δF = 0.001cos(kx)exp(−v²∕v_t²)∕ with k = 8 × 2π∕L.

Figure 7: Contour of S(x,v_x) ≡ δF∕ at t = 0 (left figure) and t = 20∕ω _pe (right figure) in the δf simulation with uniform sampling of phase-space. Initial value of δF is set as δF = 0.001cos(kx)exp(−v²∕v_t²)∕ with k = 8 × 2π∕L. The solid lines on the figure indicate the phase-velocity of the wave excited in the simulation (v_p∕v_t = ω∕kv_t = ±2.44). Other parameters used in the simulation are dt = 0.0125, L∕λ_D = 100, dx∕λ_D = 0.25, N = 2 × 10⁵, and [v_min∕v_t,v_max∕v_t] = [−3.5,+3.5].

We select the top 500 markers that have large variation in δf∕ and then compare their velocity with the phase-velocity of the wave. The results are plotted in Fig. 8, which confirms that these velocities are close to the phase-velocity of the wave. Note that, since the wave excited in the simulation is a standing-wave, which has two opposite phase-velocities, the corresponding resonant velocity also have two opposite values.

Figure 8: Velocity of the top 500 particles that have large variation in δf∕ between t = 0 and t = 20∕ω_pe in the δf simulation with Maxwellian velocity sampling (a) and uniform velocity sampling (b). Initial value of the δf is set as δf = 0.001cos(kx)exp(−v²∕v_t²)∕ with k = 8×2π∕L. The markers are ordered according to their magnitude of the variation in δf∕. The first marker has the largest variation. The solid lines on the figure indicate the phase-velocity of the wave excited in the simulation (v_p∕v_t = ω∕kv_t = ±2.44). Other parameters used in the simulation are dt = 0.0125, L∕λ_D = 100, dx∕λ_D = 0.25, N = 2 × 10⁵, and [v_min∕v_t,v_max∕v_t] = [−3.5,+3.5]; spatial sampling is uniform.

There is difference between Fig. 8a and Fig. 8b, which arises from the different sampling of the velocity space. In Fig. 8b, we note that the top 50 resonant particles all have positive velocity, which is nonphysical because there is no preferred direction in the system with a standing wave and symmetric velocity distribution.