Physical regimes of electrostatic wave-wave nonlinear interactions generated by an electron beam propagating in a background plasma

Electron-beam plasma interaction has long been a topic of great interest. Despite the success of the quasilinear and weak turbulence theories, their validities are limited by the requirements of a sufficiently dense mode spectrum and a small wave amplitude. In this paper, we extensively study the collective processes of a mono-energetic electron beam emitted from a thermionic cathode propagating through a cold plasma by performing high-resolution two-dimensional particle-in-cell simulations and using analytical theories. We confirm that, during the initial stage of two-stream instability between the beam and background cold electrons, it is saturated due to the well-known wave-trapping mechanism. Further evolution occurs due to strong wave-wave nonlinear processes. We show that the beam-plasma interaction can be classified into four different physical regimes in the parameter space for the plasma and beam parameters. The differences between the regimes are analyzed in detail. We identify a new regime in the strong Langmuir turbulence featured by what we call electron modulational instability (EMI) that could create a local Langmuir wave packet growing faster than the ion plasma frequency. Ions do not have time to respond to EMI in the initial growing stage. On a longer timescale, the action of the ponderomotive force produces very strong ion density perturbations, and eventually, the beam-plasma wave interaction stops being resonant due to the strong ion density perturbations. Consequently, in this EMI regime, electron beam-plasma interaction occurs in a repetitive (intermittent) process. The beam electrons are strongly scattered by waves, and the Langmuir wave spectrum is significantly broadened, which in turn gives rise to strong heating of bulk electrons. Associated energy transfer from the beam to the background plasma electrons has been studied. A resulting kappa (κ) distribution and a wave-energy spectrum $E^2\left(k\right)\sim k^{-5}$ are observed in the strong turbulent regime.

Figure 1

Schematic diagram of the simulation domain.

Figure 2

Schematic diagram of different regimes in the parameter space of the ratio of the beam energy to the bulk electron temperature $E_b/T_e$ vs the ratio of the beam to plasma density $n_b/n_p$. The yellow line shows the threshold of Langmuir parametric decay instability (PDI); the blue line, the threshold of standing wave modulational instability (SWMI); the red curve, the threshold of electron modulational instability (EMI). The analytical relations for the lines are given in the theory part of Sec. 4 and Appendix pp1. In the weak turbulence regime, the wave packet undergoes only slight modulations but does not grow locally. In the strong turbulent regime dominated by SWMI, the wave packet grows locally on the ion time scale, and ion density perturbations grow together with the wave. Above the threshold of EMI, the wave localization is faster than the ion response, and the wave grows locally before ions have time to move. The drawing for the ion and electron density (bottom, where “l” denotes time average) and electric field amplitude (top) schematically shows these processes for PDI, SWMI, and EMI.

Figure 3

(a) Time evolution of averaged bulk electron temperature $T_e$. (b) Evolution of the electric field, which is output of the EDIPIC probe diagnostics at $x=2.8\mathrm{mm}$, $y=4.5\mathrm{mm}$, where there is an intense standing Langmuir wave. The intermittent process can be observed. The three periods are roughly indicated by the red dashed lines. (c) Initial evolution of $|E_x|$. The red line shows the linear growth rate of the two-stream instability, and the yellow line shows the theoretical growth rate of electron modulational instability (EMI) calculated by Eq. (19) in the Sec. 4.

Figure 4

Evolution of turbulence and energy for Case 1. Blue lines in (a) show the evolution of the averaged electric field energy ${\epsilon }_{E,\mathrm{mean}}$ (solid line) and the averaged kinetic energy for bulk electrons ${\epsilon }_{K,\mathrm{mean}}$ (dashed line) during $t=0\text{–}450\mathrm{ns}$. Red lines in (a) show the averaged energy transfer rate from wave to beam $\mathbf{E}·{\mathbf{J}}_{\mathrm{beam}}$ (solid line) and to the bulk electrons $\mathbf{E}·{\mathbf{J}}_{\mathrm{bulk}}$ (dashed line). $\mathbf{E}·{\mathbf{J}}_{\mathrm{beam}}$ has a negative value, indicating that the beam energy is transferred to waves; $\mathbf{E}·{\mathbf{J}}_{\mathrm{bulk}}$ always has a positive value, indicating that waves are transferring energy to the bulk electrons. The averages are taken over $x=2.5\text{–}4.5\mathrm{mm}$ [shown by the blue dashed rectangle in (b1)]. (b)–(d) Color plots of $E_x$, $n_i$, the electron velocity distribution function (EVDF) and the one-dimensional (1D) electron phase space plots at different times. The two-dimensional (2D) EVDFs for $f(w_x,w_y)$ are averaged over region $x=4\text{–}5\mathrm{mm}$, $y=1\text{–}8\mathrm{mm}$. The one-dimensional (1D) EVDFs $f\left(v_x\right)$ are averaged over region $y=1\text{–}8\mathrm{mm}$ and $x$ regions denoted in the figure legend. Each EVDF is normalized to unity by the integration of EVDF. The purple dashed line in (d4) shows the κ distribution with coefficients $\kappa =1.6$ and $T=1.0\mathrm{eV}$. The data for the 1D phase space plot is taken along a fixed $y$ value, the black dashed line $y=6.8\mathrm{mm}$ in (b1).

Figure 5

Properties of strong turbulence produced by the beam for Case 1. (a) Two-dimensional (2D) color plot of the parallel electric field $E_x$ along $y=6.8\mathrm{mm}$ in Fig. 4 as a function of $x$ and $t$ during $t=50\text{–}120\mathrm{ns}$. (b) High-frequency spectrum of plasma waves corresponding to the dispersion relation of Langmuir ($L$) waves. The dispersion relations are obtained by performing three-dimensional (3D) fast Fourier transform (FFT) during $t=50\text{–}80\mathrm{ns}$ and then summing over $k_y$ space. The $k_{\max ,1}\approx 9.1\times {10}^3{\mathrm{m}}^{-1}$ therein denotes the $k$ wave vector with the most wave energy. (c) Spectrum of low-frequency waves corresponding to the dispersion relation of ion acoustic waves $\left(\mathrm{IAW}\mathrm{s}\right)$, which is also obtained from 3D FFT, but during $t=0\text{–}450\mathrm{ns}$. The black dashed lines therein indicate the dispersion relation of IAWs. (d) One-dimensional (1D) energy spectrum $E^2\left(k\right)$ at different times, where $k=\sqrt{{k_x^2+k_y^2}^{\phantom{\int }}}$, and a clear $-5$ power spectrum can be seen at $t=30\mathrm{and}60\mathrm{ns}$.

Figure 6

Evolution of turbulence and energy for Case 2, like Fig. 4. Blue lines in (a) show the time evolution of the averaged electric field energy ${\epsilon }_{E,\mathrm{mean}}$ (solid line) and the averaged kinetic energy for bulk electrons ${\epsilon }_{K,\mathrm{mean}}$ (dashed line) during $t=0\text{–}450\mathrm{ns}$. Red lines in (a) show the averaged energy transfer rate from wave to beam $\mathbf{E}·{\mathbf{J}}_{\mathrm{beam}}$ (solid line) and to the bulk electrons $\mathbf{E}·{\mathbf{J}}_{\mathrm{bulk}}$ (dashed line). $\mathbf{E}·{\mathbf{J}}_{\mathrm{beam}}$ has a negative value, indicating that the beam energy is transferring to waves; $\mathbf{E}·{\mathbf{J}}_{\mathrm{bulk}}$ always has a positive value, indicating that waves are transferring energy to the bulk electrons. The averages are taken over $x=7\text{–}11\mathrm{mm}$ [shown by the blue dashed rectangle in (b1)]. (b)–(d) Snapshots of the system at different times. The two-dimensional (2D) electron velocity distribution functions (EVDFs) for $f(w_x,w_y)$ are averaged over region $x=10.5\text{–}11.5\mathrm{mm}$, $y=1\text{–}8\mathrm{mm}$. The one-dimensional (1D) EVDFs $f\left(v_x\right)$ are averaged over region $y=1\text{–}8\mathrm{mm}$ and $x$ regions denoted in the figure legend. Each EVDF is normalized to unity by the integration of EVDF. The data for the 1D phase space plot is taken along a fixed $y$ value, the black dashed line $y=4.5\mathrm{mm}$ in (b1).

Figure 7

Evolution of turbulence and energy for Case 3, like Fig. 4. Blue lines in (a) show the time evolution of the averaged electric field energy ${\epsilon }_{E,\mathrm{mean}}$ (solid line) and the averaged kinetic energy for bulk electrons ${\epsilon }_{K,\mathrm{mean}}$ (dashed line) during $t=0\text{–}450\mathrm{ns}$. Red lines in (a) show the averaged energy transfer rate from wave to beam $\mathbf{E}·{\mathbf{J}}_{\mathrm{beam}}$ (solid line) and to the bulk electrons $\mathbf{E}·{\mathbf{J}}_{\mathrm{bulk}}$ (dashed line). $\mathbf{E}·{\mathbf{J}}_{\mathrm{beam}}$ has a negative value, indicating that the beam energy is transferring to waves; $\mathbf{E}·{\mathbf{J}}_{\mathrm{bulk}}$ always has a positive value, indicating that waves are transferring energy to the bulk electrons. The averages are taken over $x=4.5\text{–}7.5\mathrm{mm}$ [shown by the blue dashed rectangle in (b1)]. (b)–(d) Snapshots of the system at different times. The two-dimensional (2D) electron velocity distribution functions (EVDFs) for $f(w_x,w_y)$ are averaged over region $x=6.5\text{–}7.5\mathrm{mm}$, $y=1\text{–}8\mathrm{mm}$. The one-dimensional (1D) EVDFs $f\left(v_x\right)$ are averaged over region $y=1\text{–}8\mathrm{mm}$ and $x$ regions denoted in the figure legend. Each EVDF is normalized to unity by the integration of EVDF. The data for the 1D phase space plot is taken along a fixed $y$ value, the black dashed line $y=4.5\mathrm{mm}$ in (b1).

Figure 8

Properties of turbulence produced by the beam for Case 3. (a) Two-dimensional (2D) color plot of the parallel electric field $E_x$ along $y=4.5\mathrm{mm}$ in Fig. 7 as a function of $x$ and $t$ during $t=50\text{–}80\mathrm{ns}$. (b) High-frequency spectrum of plasma waves corresponding to the dispersion relation of Langmuir ($L$) waves. The dispersion relations are obtained by performing three-dimensional (3D) fast Fourier tranform (FFT) during $\mathrm{t}=50\text{–}80\mathrm{ns}$ and then summing over $k_y$ space. The $k_{\max ,2}\approx 5.5\times {10}^3{\mathrm{m}}^{-1}$ therein denotes the $k$ wave vector with the most wave energy. (c) Spectrum of low-frequency waves corresponding to the dispersion relation of ion acoustic waves (IAWs), which is also obtained from 3D FFT, but during $t=0\text{–}450\mathrm{ns}$. (d) One-dimensional (1D) energy spectrum $E^2\left(k\right)$ at different times, where $k=\sqrt{{k_x^2+k_y^2}^{\phantom{\int }}}$.

Figure 9

Evolution of turbulence and energy for Case 4, like Fig. 4. Blue lines in (a) show the time evolution of the averaged electric field energy ${\epsilon }_{E,\mathrm{mean}}$ (solid line) and the averaged kinetic energy for bulk electrons ${\epsilon }_{K,\mathrm{mean}}$ (dashed line) during $t=0\text{–}450\mathrm{ns}$. Red lines in (a) show the averaged energy transfer rate from wave to beam $\mathbf{E}·{\mathbf{J}}_{\mathrm{beam}}$ (solid line) and to the bulk electrons $\mathbf{E}·{\mathbf{J}}_{\mathrm{bulk}}$ (dashed line). $\mathbf{E}·{\mathbf{J}}_{\mathrm{beam}}$ has a negative value, indicating that the beam energy is transferring to waves; $\mathbf{E}·{\mathbf{J}}_{\mathrm{bulk}}$ always has a positive value, indicating that waves are transferring energy to the bulk electrons. The averages are taken over $x=6\text{–}10\mathrm{mm}$ [shown by the blue dashed rectangle in (b1)]. (b)–(d) Snapshots of the system at different times. The two-dimensional (2D) electron velocity distribution functions (EVDFs) for $f(w_x,w_y)$ are averaged over region $x=12\text{–}13\mathrm{mm}$, $y=1\text{–}8\mathrm{mm}$. The one-dimensional (1D) EVDFs $f\left(v_x\right)$ are averaged over region $y=1\text{–}8\mathrm{mm}$ and $x$ regions denoted in the figure legend. Each EVDF is normalized to unity by the integration of EVDF. The data for the 1D phase space plot is taken along a fixed $y$ value, the black dashed line $y=4.5\mathrm{mm}$ in (b1).

Figure 10

Averaged $|v_y\left(x\right)|$ of the beam as a function of $x$, showing beam scattering at different times for (a) Case 1, (b) Case 3, (c) Case 5, and (d) Case 6.

Figure 11

Evolution of turbulence for Case 1 with a narrower beam. The beam now is injected only in the spatial interval $y=3.9\text{–}5.1\mathrm{mm}$. The blue line in (a) shows the probe diagnostics of $E_x$ [which is the sum of the three probes shown in (c)]. The red line in (a) is the electron temperature. (b)–(g) Snapshots of $E_x$ and $n_i$ at $t=60,950,\mathrm{and}2400\mathrm{ns}$.

Figure 12

Different terms of Eq. (11) at initial instability growth for different cases: (a) verifies that the first term in the left-hand side (LHS; blue line) of Eq. (11) nearly balances the term in the right-hand side (RHS; yellow line) for Case 2. The data is taken in Case 2 at $t=128.4\mathrm{ns}$ and $y=4.5\mathrm{mm}$. The time average is taken over the neighboring $15.1\mathrm{ns}$ (50 electron plasma period). (b) shows that the second term in the LHS (red line) of Eq. (11) nearly balances the term in the RHS (yellow line) for Case 1. The data is taken in Case 1 at $t=16.1\mathrm{ns}$ and $y=6.8\mathrm{mm}$. The time average is taken over the neighboring $3.0\mathrm{ns}$ (10 electron plasma period).

Figure 13

The saturated electric field before onset of the modulational instability as a function of beam energy $E_b$ for $n_b/n_p=0.015$; it shows applicability of Eq. (24) in estimating the initial saturated wave packet amplitude before the onset of modulational instability. The blue curve shows the theoretical value given by Eq. (24), and the red dots show the simulation data before strong Langmuir turbulence (SLT) develops. The error bars are obtained by taking data from several different snapshots over one or several cases.

Figure 14

Parameter space of ratio of the beam energy to the bulk electron temperature $E_b/T_e$ vs the ratio of the beam density to the plasma density $n_b/n_p$. The blue line shows the threshold in Eq. (25), and the red line shows the threshold in Eq. (26), respectively. The yellow curve shows the threshold of Langmuir parametric decay instability (PDI) given by Eq. (27). Red and blue markers show the cases with and without strong turbulence, respectively. Red plus-over-an-x markers show the cases with electron modulational instability (EMI). Pink circles mark different cases used for detailed analysis in this paper. The numbers denote the case number shown in Table 3. In total, 57 simulations are shown here (all the simulation parameters are given in Table 2).

Figure 15

Balance of terms in Eq. (A31) for Case 2. Data is taken from the particle simulation at $t=128.4\mathrm{ns}$ and $y=4.5\mathrm{mm}$. The time averaging is over the centered window of $15.1\mathrm{ns}$ (50 electron plasma periods). Since the instability is on ion time scale, the averaging window is of that order. We see that the first term (blue line) nearly balances the third term (yellow line).

Figure 16

Balance of terms in Eq. (A31) for the simulated Case 1. Data is taken from Case 1 at $t=16.1\mathrm{ns}$ and $y=6.8\mathrm{mm}$. The time average is taken over the window of $3.0\mathrm{ns}$ (10 electron plasma periods, i.e., on electron time scale). The convergence with respect to time resolution had been verified. It is seen that the second term (red line) nearly balances the third term (yellow line).

Figure 17

Profiles of $n_l-n_i$ and $n_{bl}$. Data is taken from Case 1 at $t=45.2\mathrm{ns}$ and $y=6.8\mathrm{mm}$, when the Langmuir wave envelope has grown by a large factor. The centered time-averaging window is $3.0\mathrm{ns}$ (10 electron plasma periods, i.e., on the electron time scale).

Figure 18

Adiabatic index for electrons in a sinusoidal profile of electrostatic potential vs $e{\mathrm{\Phi }}_{\max }/2T_e$ and $k{\lambda }_{De}$ (a) at the trough of the potential energy and (b) at the peak position.

Figure 19

(a) and (b) One-dimensional (1D) electron velocity distribution function (EVDF) $f\left(v_x\right)$ with energetic bulk electrons collected for Case 1 at $t=0,100,200,\mathrm{and}300\mathrm{ns}$ and Case 3 at $t=100,200,\mathrm{and}300\mathrm{ns}$. The black dashed line in (a) shows the κ distribution with coefficients $\kappa =1.6$ and $T=1.0\mathrm{eV}$. Here, the calculation of the EVDF used all the particles in the simulation domain plus the collected at anode energetic particles. All EVDFs are normalized to unity by the integration of EVDF. The EVDF in Case 1 resembles a κ distribution, and there is a stronger wave heating than in Case 3.

Figure 20

Energy spectrum plot for Case 30 in standing wave modulational instability (SWMI) regime at $t=130,150$, and 170 ns. A $-5$ spectrum, indicating presence of strong turbulence, is observed. The pump length and the scaled Debye length are also shown by the vertical dashed lines.