Diamonds show the real frequency and the linear damping rate as a function of the wave number for LHWs with and , which corresponds to a fixed and . The solid lines are based on the analytical theory.
Time evolution of the spatial Fourier modes of the electric field (inthe natural logarithm scale) for (a) case 1 with and (electron resonance only), (b) case 2 with and (ion resonance only), and (c) case 3 with and (electron and ion resonance). All three cases have common , and initial electric field . The red dashed lines show the measurement of the damping rate.
Damping rate, , as a function of the initial wave amplitude E 0 in the cases with and , for , 0.066, 0.0404, and 0.001.
Damping rate vs. E 0 in the cases with and , for , 1.0, and 0.33.
Saturated electric field vs. in the cases with and , 0.0404, and 0.001.
(a) Contour plots of the electron distributions in the particle phase space and (b) the corresponding parallel velocity distribution functions averaged over x, shown with the black solid curves, at times , 0.22, 2.41, and 8.78 obtained from case 1. All the distribution functions are plotted in the logarithm scales. The red solid lines show the ion distribution functions, and the black dashed lines show the electron resonant phase velocity based on the theoretical prediction. The ion velocities are normalized to the ion thermal speed.
Time (normalized to ) evolution of the electron phase-space distribution in case 2: contour plots of the (a) electron and (b) ion phase-space distributions and (c) the corresponding electron (black solid lines) and ion (red solid lines) distribution functions. The ion velocities are normalized to the ion thermal speed. The black (red) dashed line shows the theoretical electron (ion) resonant phase-velocity .
Resulting particle distributions obtained from case 3. Plots (a), (b), and (c) are in the same format as those in Fig. 7 .
Time evolution of (normalized to ) obtained from case 1.
Resulting parallel currents as a function of the initial LHW amplitude E 0.
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