Density fluctuation induced by the wave

Figure (5) compares the spatial distribution at $t = 0$ and $t = 10 T$, which shows that the distribution become nonuniform at $t = 10 T$.

Figure 5: Comparison of the spatial distribution function at $\overline {t} = 0$ and $\overline {t} = 10$, which shows that the distribution seems to become nonuniform at $\overline {t} = 10$. Other parameters: $v_t = v_p$, $2 \pi k q E / m \omega ^2 = 1$.

Figure 6 is a GIF animation, which shows the time evolution of the spatial and velocity distribution (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 ( $v / v_p \approx 1$) oscillates with large amplitude 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 fluctuation of the density induced by the longitude wave may explain the density pump out phenomina induced by Low-hybrid waves observed in many tokamaks.

Figure 6: GIF animation of the time evolution of the spatial and velocity distribution functions during $t = [0, 20 T]$, where $T$ is the period of the longitudinal wave. Other parameters: $v_t = v_p$, $2 \pi k q E / m \omega ^2 = 1$. The GIF animation can be viewed only in the HTML version of this document (it does not work in the PDF version).