Spatial Damping of Large Amplitude Waves

Abstract

The spatial damping of a large amplitude plasma wave was studied to permit direct comparison with experimental data. Using both the quasilinear treatment of Al'tshul and Karpman, and following exact particle trajectories of resonant particles as done by O'Neil for the temporal case, it was found that the spatial damping results are analogous to the temporal ones. The spatial damping formula is obtained by replacing the nondimensional time $\mathrm{t/\tau }$ and damping coefficient ${}_{\mathrm{\gamma L}}\tau$ for the temporal case by the distance $\mathrm{x/L}$ and spatial damping coefficient ${\Lambda }_{L}L$ , respectively. Here, ${\mathrm{\tau \hspace{0.17em}=\hspace{0.17em}(m/eEk)}}^{\text{1/2}}{\mathrm{,\gamma }}_L$ is the linear Landau damping coefficient, $${\Lambda }_L\mathrm{\hspace{0.17em}=\hspace{0.17em}}\frac{{\gamma }_L}{\mathrm{\partial \omega /\partial k}}$$ , and $\mathrm{L\hspace{0.17em}=\hspace{0.17em}\tau \omega /k}$ . Damping curves have been plotted for comparison with measured data. The validity condition of the spatial case, namely, ${\Lambda }_L\text{L\hspace{0.17em}≪\hspace{0.17em}1}$ , is far more stringent than for the temporal one.