Electron Modulational Instability in the Strong Turbulent Regime for an Electron Beam Propagating in a Background Plasma

We study collective processes for an electron beam propagating through a background plasma using simulations and analytical theory. A new regime where the instability of a Langmuir wave packet can grow locally much faster than ion frequency is clearly identified. The key feature of this new regime is an electron modulational instability that rapidly creates a local Langmuir wave packet, which in its turn produces local charge separation and strong ion density perturbations because of the action of the ponderomotive force, such that the beam-plasma wave interaction stops being resonant. Three evolution stages of the process and observed periodic burst features are discussed. Different physical regimes in the plasma and beam parameter space are demonstrated for the first time.

Figure 1

Snapshots of strong Langmuir turbulence for case 1 (with EMI) at $t=40\mathrm{ns}$, 60 ns, 160 ns shown for part of the simulation domain $x=(0,10\mathrm{mm})$. (a1)–(a3) show the 2D color plots of the electric field $E_x$. The black dashed lines show the $y=6.8\mathrm{mm}$ location where we plot (c1)–(c3), (d1)–(d3). The blue dashed rectangle outlines the region used to produce the line plots shown in Figs. 2 and 2. The red rectangle shows the region used to calculate the EVDF plotted in Figs. 2 and 2. (b1)–(b3) show the time evolution of the ion density profile, $n_i$. (c1)–(c3) show the $⟨E_x^2⟩$ and density profiles of ions and electrons along the black dashed lines, where the $⟨\dots ⟩$ denotes the time average over the time interval 3.025 ns (10 plasma periods). Note the charge separation in (c1) and (c2). (d1)–(d3) show the electron phase space along the same black dashed lines. The color bar to the right of the figure shows EVDF normalized to unity by the integration of the EVDF.

Figure 2

(a) Shows the periodic burst feature of EMI in $E_x$ (detected by probe at $x=2.8\mathrm{mm}$, $y=4.5\mathrm{mm}$ for case 1) and increase of average bulk electron temperature $T_e$. The three periods are roughly indicated by the red dashed lines. Note only the first burst is in the EMI regime for this case. (b) Presents the time evolution of $|E_x|$ at the initial stage. The red line shows the linear growth rate of two-stream instability while the yellow line gives the EMI growth rate ${\gamma }_{\mathrm{EMI}}\approx 7.4\times {10}^7{\mathrm{s}}^{-1}>{\omega }_{pi}\approx 3\times {10}^7{\mathrm{s}}^{-1}$ calculated by Eq. (19) in our companion paper [46]. (c) and (d) Show the time evolution of the averaged electric field energy ${\epsilon }_{E,\mathrm{mean}}$, averaged kinetic energy for bulk electrons ${\epsilon }_{K,\mathrm{mean}}$, averaged energy transfer rate from wave to beam $\mathit{E}·{\mathit{J}}_{\text{beam}}$ (hence negative) and averaged energy transfer rate from wave to the bulk electrons $\mathit{E}·{\mathit{J}}_{\mathrm{bulk}}$ during $t=0–450\mathrm{ns}$ for cases 1(c) and 2(d), respectively. (e) and (f) show color plots of the electron velocity distribution function (EVDF) at $t=60\mathrm{ns}$ for cases 1 and 2. Both EVDFs are normalized to unity by the integration of EVDF. (g) and (h) Show temporal evolution of the energy spectrum $E^2(k)$ for cases 1 and 2, where $k=\sqrt{k_x^2+k_y^2}$.

Figure 3

Parameter space of ratio of the beam energy to the bulk electron temperature $E_b/T_e$ versus the ratio of the beam density to the plasma density, $n_b/n_p$. The blue line shows the threshold Eq. (4) and the red line shows the threshold Eq. (5). Physical pictures of different regimes are shown ($|\tilde{\mathit{E}}|$ denotes wave packet). The yellow curve shows the threshold of Langmuir parametric decay instability (PDI) (which comes from Ref. [1]). Red and blue markers show the cases with and without strong turbulence, respectively. Red markers are used only if the clear large amplitude standing wave feature and the associated ion density dips are observed. Red plus-over-an-$x$ markers denote the cases with EMI, where fast localization of Langmuir waves faster than ${\omega }_{pi}$ and electrostatic force resulting from charge separation that balances the ponderomotive force are clearly observed. Pink dashed circles mark cases 1 and 2 used for analysis in this Letter.