### Abstract

- Top of page
- Abstract
- 1. Introduction
- 2. Simulation Model
- 3. One-Dimensional Simulation
- 4. Two-Dimensional Simulation
- 5. Discussion
- 6. Conclusions
- Acknowledgments
- References
- Supporting Information

[1] We study formation process of electrostatic solitary waves (ESW) observed by recent spacecraft via one- and two-dimensional electrostatic particle simulations with open boundaries. The previous simulations have demonstrated that ESW correspond to Bernstein-Greene-Kruskal electron holes formed by electron beam instabilities. However, since the previous simulations were performed in uniform periodic systems, wave-particle interaction of an electron beam instability was taking place uniformly in the systems. In the present study, we inject a weak electron beam from an open boundary into the background plasma to study spatial and temporal development of a bump-on-tail instability from a localized source. In the open system, spatial structures of electron holes vary depending on the distance from the source of the electron beam. In an early phase of the simulation run, electron holes that are initially uniform in the direction perpendicular to the magnetic field become twisted through modulation by oblique electron beam modes. As the electron holes propagate along the magnetic field, they are aligned in the perpendicular direction through coalescence. Spatial structures of electron holes in a distant region from the source become one-dimensional. In a long-time evolution of the instability, ion dynamics becomes important in determining spatial structures of electron holes. A lower hybrid mode is excited locally in the region close to the source of the electron beam through coupling with electron holes at the same parallel phase velocity. The lower hybrid mode modulates electron holes excited in later phases, resulting in formation of modulated one-dimensional potentials. Since the perpendicular electric fields of electron holes are carried by the electron holes at the drift velocity of the electron holes, they can be observed even at a distant place from the source.

### 1. Introduction

- Top of page
- Abstract
- 1. Introduction
- 2. Simulation Model
- 3. One-Dimensional Simulation
- 4. Two-Dimensional Simulation
- 5. Discussion
- 6. Conclusions
- Acknowledgments
- References
- Supporting Information

[2] Electrostatic Solitary Waves (ESW) were first observed by the Geotail spacecraft in the Earth's magnetotail [*Matsumoto et al.* 1994]. ESW are bipolar electric pulses longitudinal to the geomagnetic field. ESW are modeled as electron phase-space density holes which are Bernstein-Greene-Kruskal (BGK) modes, i.e., one-dimensional equilibrium solutions to the time-independent Vlasov-Poisson equations [*Bernstein et al.*, 1957; *Krasovsky et al.*, 1997; *Muschietti et al.*, 1999]. Electron holes were observed much earlier in computer simulations [e.g., *Berk et al.*, 1970] as well as in laboratory plasmas [*Saeki et al.*, 1979] than in the space plasma. Electron holes are formed through nonlinear evolution of electron beam instabilities. In a long-time nonlinear evolution of a instability, electron holes coalesce with adjacent holes and merge into larger, more intense and isolated holes. The previous one- and two-dimensional simulation studies have confirmed that ESW in the magnetotail are one-dimensional potentials generated through nonlinear evolution of an electron bump-on-tail instability [*Omura et al.*, 1994, 1996; *Miyake et al.*, 1998].

[3] The FAST spacecraft also observed bipolar parallel electric fields accompanied by strong perpendicular electric fields in the auroral region [e.g., *Ergun et al.*, 1998]. Formation processes of such multidimensional electron holes were studied by recent computer simulations [e.g., *Goldman et al.*, 1999; *Oppenheim et al.*, 1999; *Muschietti et al.*, 1999, 2000; *Miyake et al.*, 1998, 2000]. Nonlinear evolution of electron holes falls into three categories depending on the amplitude of the external magnetic field. When the electron cyclotron frequency Ω_{e} is much larger than the electron plasma frequency Π_{e}, electron holes drifting in the direction parallel to the external magnetic field involve the emission of electrostatic whistler waves (oblique Langmuir waves) [*Goldman et al.*, 1999; *Oppenheim et al.*, 1999]. The electron holes become multidimensional through modulation by the electrostatic whistler waves. When Ω_{e} is smaller than the bounce frequency of the trapped electrons ω_{b}, electron holes are unstable [*Miyake et al.*, 1998; *Muschietti et al.*, 2000]. For Π_{e} ∼ Ω_{e} > ω_{b}, formation and stability of one-dimensional electron holes are confirmed in the two-stream and bump-on-tail instabilities [*Miyake et al.*, 1998, 2000]. Under this condition, ion dynamics is important in the emission process of oblique electrostatic modes. The electron holes couple with lower hybrid waves at the same parallel phase velocity [*Miyake et al.*, 2000]. Multidimensional electron holes are formed through modulation by the lower hybrid waves.

[4] Simulations of electron beam instabilities were conventionally performed in uniform periodic systems. In simulation runs with periodic boundary conditions, unstable velocity distribution functions are assumed to exist uniformly in space as initial conditions. In the real space plasma, however, wave-particle interaction of electron beam instabilities does not necessarily take place uniformly. Sources of electron beams are localized, when the electron beams result from acceleration by localized electric fields that appear in a magnetic reconnection process [*Omura et al.*, 1999a, 1999b] or a shock transition [*Matsumoto et al.*, 1997].

[5] Recently a number of simulations of electron holes were also performed in nonperiodic systems. *Muschietti et al.* [1999] loaded a velocity space distribution function of an electron hole and confirmed its stability. In the present study we injected a weak electron beam from an open boundary into a homogeneous background plasma. We studied spatial and temporal development of an electron beam instability from a localized source of an electron beam. A similar simulation was performed by *Mandrake et al.* [2000] with injection of a narrow electron beam into a nonuniform background plasma. In the present study we assume a wide electron beam with perpendicular dimension much longer than both electron and ion gyroradii. We focus on the interaction of electron holes with lower hybrid waves that occurs under weak magnetic fields with Π_{e} ∼ Ω_{e} > ω_{b}.

[6] In section 2, we describe the model and parameters. Simulation results of the one- and two-dimensional simulations are presented in section 3 and 4, respectively. Section 5 gives discussion, and conclusions are given in section 6.

### 2. Simulation Model

- Top of page
- Abstract
- 1. Introduction
- 2. Simulation Model
- 3. One-Dimensional Simulation
- 4. Two-Dimensional Simulation
- 5. Discussion
- 6. Conclusions
- Acknowledgments
- References
- Supporting Information

[7] We developed one- and two-dimensional electrostatic particle codes modified from Kyoto University Electromagnetic Particle Code (KEMPO) [*Matsumoto and Omura*, 1984; *Omura and Matsumoto*, 1993]. We used open boundaries at both ends of the system in the *x* direction. In the two-dimensional system we used periodic boundaries in the *y* direction. The external magnetic field is taken in the *x* direction.

[8] All electron and ion components have Maxwellian distributions given by

where *V*_{d}, *V*_{t} and *n* are drift velocity, thermal velocity and density, respectively. Subscripts *s* = b, e, and i represent beam electrons, background electrons, and ions, respectively. In the present study, only the electron beam drifts along the external magnetic field with velocity *V*_{d,b} ≡ *V*_{d} (*V*_{d,e} = *V*_{d,i} = 0). As the initial condition, we assume that the background electrons exist uniformly in the simulation box without the electron beam as shown in Figure 1. In the region outside of the right boundary the velocity distribution function *f*_{−} is equal to that in the simulation box *f*_{0}. On the other hand, in the region outside of the left boundary we assume the bump-on-tail velocity distribution function *f*_{+}. Ions with mass ratio *m*_{i}/*m*_{e} = 100 are assumed to exist uniformly. We assume that there is no perturbation in the regions outside of both left and right boundaries. When a computer simulation is started, the electron beam is continuously injected from the left boundary of the simulation box into the background homogeneous plasma. The injected particles (beam electrons, background electrons, and ions) have flux velocity distribution functions given by *v*_{x}*f*(*v*_{x}) [*Birsall and Langdon*, 1985]. The flux functions at the left and right boundaries are given by *v*_{x}*f*_{+} and *v*_{x}*f*_{−}, respectively. In the present study we inject particles with the constant flux, while exiting particles are removed at the open boundaries as if there is no boundary.

[9] In electrostatic simulations we do not need a special numerical treatment of the open boundaries such as absorbing boundary conditions for electromagnetic waves used in electromagnetic simulations [e.g., *Umeda et al.*, 2001]. The electrostatic fields at the open boundaries are determined by the distribution of charged particles. We first compute the charge density distribution at the open boundaries. Then we solve Poisson's equation to obtain electrostatic potential and electrostatic field as follows:

We used “marching method” [*Buneman*, 1973] for solving Poisson's equation.

[10] We set up the initial total electron plasma frequency in the simulation box (*x* ≥ 0) and in the region outside of the left boundary (*x* < 0) constant as specified below.

where *n*_{0}, *n*_{e}, and *n*_{b} refer to local quantities, and *n*_{0} represents the initial density of the background electrons in the simulation box. The electron density is assumed to be constant for all *x* as the initial condition, namely, *n*_{0} = *n*_{e} + *n*_{b}. However, since the electron beam is injected with the constant flux, the total particle number varies by less than 1%. Thus the charge neutrality is not necessarily satisfied. Since the charge neutrality is not enforced in the present open system, there exists a DC component of parallel electric field in the system. To suppress nonphysical acceleration of the background plasma, we subtract the DC electric field from the parallel electric field so that the following condition is satisfied:

We performed a test run without cancellation of the DC electric field. In this test run we observed acceleration of the background electrons by the DC electric field, which leads to nonphysical Langmuir oscillation of the background electrons. This is a limitation of the present one-dimensional model. However, we also observed formation of electron holes in this test run. The physical process of the electron beam instability is less affected by the DC electric field because both beam electrons and background electrons are accelerated, keeping the same relative velocity between them. By subtracting the DC component, the simulations are performed in the rest frame of reference where the background plasma is free from the nonphysical Langmuir oscillation.

[11] We used 1600 superparticles per cell for electrons and ions, respectively. The density ratio of the electron beam *R* is defined as *R* = *n*_{b}/(*n*_{b} + *n*_{e}). In the present study we assume a weak electron beam with *R* = 0.06. The common parameters for all simulation runs are listed in Table 1. Assuming the electron plasma frequency Π_{e} = 1.0 and the initial background electron thermal velocity *V*_{t,e} = 1.0, we can normalize frequencies and velocities by Π_{e} and *V*_{t,e}, respectively. Length is normalized by the electron Debye length λ_{e} = *V*_{t,e}/Π_{e}. The bump-on-tail velocity distribution function of the present study is based on the previous one- and two-dimensional studies [*Omura et al.*, 1996; *Miyake et al.*, 1998]. Electron bump-on-tail velocity distribution functions are consistent with the recent statistical analyses of the Geotail data [*Omura et al.*, 1999b; *Kojima et al.*, 1999b]. In the particle measurement of the Geotail spacecraft, electron beams are observed as enhanced nonthermal fluxes with flat diffused velocity distributions that result in after saturation of electron beam instabilities.

Table 1. Common Simulation Parameters for All RunsParameter | Equation | Value |
---|

Electron cyclotron frequency | Ω_{e}/Π_{e} | 1.0 |

Drift velocity of electron beam | *V*_{d}/*V*_{t,e} | 2.0 |

Thermal velocity of electron beam | *V*_{t,b}/*V*_{t,e} | 0.1 |

Thermal energy ratio | *T*_{e}/*T*_{i} | 64.0 |

[12] In the present study, we define “bump-on-tail instability” when phase velocities of unstable modes are localized in velocity space. In this case, the instability is a resistive instability. On the other hand, “two-stream instability”, in the present study, refers to a reactive instability driving waves with different phase velocities.

### 3. One-Dimensional Simulation

- Top of page
- Abstract
- 1. Introduction
- 2. Simulation Model
- 3. One-Dimensional Simulation
- 4. Two-Dimensional Simulation
- 5. Discussion
- 6. Conclusions
- Acknowledgments
- References
- Supporting Information

[13] We present simulation results of a very long one-dimensional open system with *L*_{x} = 40960Δ*x* (Δ*x* = 0.1λ_{e}), where *L*_{x} and Δ *x* represent system length and grid spacing, respectively. We show *x* − *v*_{x} phase diagrams at different times in Figure 2. A cold and weak electron beam is injected from the left boundary into the warm background electrons to form the bump-on-tail velocity distribution at *x* = 0. As the electron beam propagates from the left to the right, the bump-on-tail instability evolves in space. Since the positive gradient of the velocity distribution function is localized at the small bump, a coherent electron beam mode, whose phase velocity is slightly smaller than the drift velocity of the bump electron beam, is excited. The beam mode traps the beam electrons and diffuses the bump to a flat distribution, forming electron holes in the *x* − *v*_{x} phase space. As the electron holes propagate, they coalesce with adjacent holes to form larger and more isolated electron holes. The spatial scales of electron holes and the distance between them become larger depending on the distance from the source of the electron beam. As shown in Figure 3, at a distant place from the source of the electron beam (*x*/λ_{e} = 4096), we found larger and more isolated electron holes. We confirmed the formation and coalescence processes of electron holes in the one-dimensional open system.

[14] In the present simulation model the leading edge of the electron beam is associated with the sudden turn-on of the electron beam. However, when electron holes penetrate into the background plasma, they can propagate through the background plasma without significant disturbances of the background plasma. This implies that the electron holes are formed essentially by dynamics of trapped electrons rather than that of untrapped electrons with velocities close to the beam drift velocity that pass through the electron holes. Thus the simulation result is less affected by the sudden injection of the electron beam. Electron holes can propagate stably over a long distance through the background plasma, which is consistent with the statistical analysis of the Geotail data [e.g., *Omura et al.*, 1999a; *Kojima et al.*, 1999b]. It is possible to observe electron holes at a more distant place from the source region where the instability is taking place.

[15] We compared the result of the open system with that of the periodic system [*Omura et al.*, 1996] and confirmed that timescale of evolution of the bump-on-tail instability in both systems are the same. With conversion of time to position by *x* = *V*_{d}*t* the temporal evolution of the instability in the periodic system can be transformed to the spatial evolution of the instability in the open system. This implies that the periodic system simulates a localized region of the open system in a frame of reference moving with *V*_{d}.

[16] We also performed the simulations with immobile ions (*m*_{i}/*m*_{e} = ∞). However, we found little difference between the two cases. Even with cold ions, the electron holes do not lead to nonlinear decay to ion acoustic waves such as observed in an electron two-stream instability [*Omura et al.*, 1994, 1996]. In Figure 4 we show a ω − *k* spectrum obtained by taking Fourier transformation of *E*_{x} data for *x*/λ_{e} = 204.8 ∼ 409.6, Π_{e}*t* = 307.2 ∼ 819.2. We found clear enhancement of the beam mode, while the amplitudes of both Langmuir waves and ion acoustic waves are very small. Since the electron holes propagate very fast, the electron holes are separated from the background ions in the *x* − *v*_{x} phase space. Ions cannot be reflected by the positive potentials of electron holes.

[17] In the magnetotail the Geotail spacecraft observed both tailward and earthward propagating ESW simultaneously [*Omura et al.*, 1999a; *Kojima et al.*, 1999a]. To demonstrate the counterstreaming ESW, we performed another simulation run with counterstreaming two electron beams in a short system (*L*_{x} = 8192Δ*x* (Δ*x* = 0.1λ_{e})). As shown in Figure 5, we injected both forward and backward propagating electron beams from the left and right boundaries, respectively. As time elapses, we found formation of both forward and backward moving electron holes. When the forward and backward traveling electron holes encounter, however, they go through each other without coalescence. Since the relative velocity is much larger than the trapping velocity of the electron holes, the electrons which form a potential of an electron hole are not trapped by the other potential. Namely, there is little interaction between the forward and backward propagating potentials.

### 4. Two-Dimensional Simulation

- Top of page
- Abstract
- 1. Introduction
- 2. Simulation Model
- 3. One-Dimensional Simulation
- 4. Two-Dimensional Simulation
- 5. Discussion
- 6. Conclusions
- Acknowledgments
- References
- Supporting Information

[18] In this section we extend the previous one-dimensional simulations to a two-dimensional simulation with *L*_{x} × *L*_{y} = 1024Δ*x* × 64Δ*y* (Δ*x* = Δ*y* = λ_{e}). Before performing a two-dimensional simulation, we numerically solved linear dispersion relations of the bump-on-tail instability. In Figure 6a we show the maximum growth rate of electrostatic waves as a function of wave numbers *k*_{x} and *k*_{y}. We found that three modes are linearly unstable in this instability. (A) Electron beam modes have the maximum growth rate γ/Π_{e} = 0.1 at θ = 0°, where θ represents wave normal angle. We also show the linear dispersion relation of the electron beam modes in Figure 6b. (B) A quasi-perpendicular mode is unstable at θ = 87°. The maximum growth rate of this mode is γ/Π_{e} = 0.01. Frequency of this mode is ω ≃ ω_{LHR} = 0.07Π_{e}, ω_{LHR} represents the lower hybrid resonance frequency. (C) Oblique electrostatic modes are unstable for θ = 50° ∼ 80°. The maximum growth rate of the oblique modes is γ/Π_{e} = 0.05 at θ = 75°. These modes are unstable only in the presence of cold ions. Frequencies of these modes are ω ∼ ω_{LHR} = 0.07Π_{e}.

[19] To analyze spatial structures of electron holes, we show spatial profiles of potentials ϕ at different times in Figure 7. The potentials are normalized by *m*_{e}*V*_{t,e}^{2}/*e*. The zero-level of potentials corresponds to the spatial average. As the electron beam propagates along the external magnetic field, the bump-on-tail instability develops in space as observed in the previous one-dimensional simulation. The injected electron beam is uniform in the *y* direction. We first observed one-dimensional potentials in the leading edge of the electron beam (Figure 7a, *x*/λ_{e} = 85). Then the potentials excited by the instability are modulated in the *y* direction (*x*/λ_{e} = 45). According to the linear dispersion relation of the electron beam modes (Figure 6b), the parallel beam mode has the maximum growth rate, and oblique beam modes with larger wave normal angle θ have smaller growth rates. Thus we observed the parallel beam mode with *k*_{y} = 0 at first, and then we observed oblique beam mode with *k*_{y} = Δ*k*_{y}.

[20] As time elapses, amplitude of the oblique beam modes in the generation region of electron holes (*x*/λ_{e} = 64 ∼ 128) becomes larger and larger. The electron holes excited in the later phases are more modulated by the oblique beam modes. Spatial structures of these holes are twisted as observed in Figure 7b and 7c. We also observed two-dimensional potentials isolated in both *x* and *y* directions infrequently, as at *x*/λ_{e} = 130 in Figure 7d. The scales of *x* and *y* axes of Figure 7d are the same. The ratio of the parallel to perpendicular characteristic widths of the two-dimensional potential in Figure 7d is 0.5. In the present simulation, however, existence of such two-dimensional potentials is infrequent, and the ratio of parallel to perpendicular characteristic widths of them is <0.5. Namely, the perpendicular dimension of electron holes is much longer than the parallel dimension. In the leading edge of the electron beam we found clear one-dimensional potential structures at all times. The one-dimensional potentials can propagate without changing their characteristics. On the other hand, the twisted potentials and two-dimensional potentials are aligned in the direction perpendicular to the external magnetic field through the coalescence process [*Miyake et al.*, 1998].

[21] To show the difference between potential structures in the leading edge of the electron beam and those in the generation region of electron holes quantitatively, we analyzed Fourier spectra of electric fields in these regions. We specified a region with width *x*/λ_{e} = 64 from the leading edge of the electron beam and computed *k*_{x} − *k*_{y} spectra of ∣*E*_{x}∣ and ∣*E*_{y}∣ shown in Figure 8a. The spectra are the time average at Π_{e}*t* = 102.4, 204.8, 307.2, 409.6. The intensity is normalized by *m*_{e}*V*_{t,e}Π_{e}/*e*. We found enhancement of *E*_{x} components at *k*_{y} ≃ 0, while the intensities of *E*_{y} components are the same as that of the background thermal fluctuations. We also show *k*_{x} − *k*_{y} spectra of ∣*E*_{x}∣ and ∣*E*_{y}∣ in the generation region of electron holes (*x*/λ_{e} = 64 ∼ 128) in Figure 8b. The spectra are the time average at Π_{e}*t* = 102.4, 204.8, 307.2, 409.6. We found clear enhancement of ∣*E*_{y}∣ in Figure 8b. These spectra show that both parallel and oblique electron beam modes exist in the generation region. The present result shows that there is a spatial gap between the leading edge of the bipolar pulses (*E*_{x}) and that of the oblique modes (*E*_{y}).

[22] In Figures 9a and 9d we plotted spatial profiles of potentials in later phases. In a region far from the source of the electron beam (*x*/λ_{e} = 700 ∼ 1000) we found one-dimensional electron holes. The one-dimensional potential structures are also indicated from the clear bipolar waveforms of *E*_{x} in Figure 9b. However, these one-dimensional electron holes are accompanied by *E*_{y} components as shown in Figure 9c. In a long-time evolution of the bump-on-tail instability up to ω_{LHR}*t* > 50, the intensities of *E*_{y} components become larger and larger in a region close to the source of the electron beam. We show a spatial profile of ∣*E*_{y}∣^{2} along the magnetic field in Figure 10a. The energy density of ∣*E*_{y}∣^{2} for *k*_{y}λ_{e} = 0 ∼ π averaged for Π_{e}*t* = 819.2 ∼ 1024.0 is shown by the black line (line A). We also plotted the energy density of ∣*E*_{y}∣^{2} for *k*_{y}λ_{e} = 1.0 ∼ 1.6 by the gray line (line B). We found strong enhancement of ∣*E*_{y}∣^{2} with *k*_{y}λ_{e} = 1.0 ∼ 1.6 for *x*/λ_{e} = 150 ∼ 200. We specified a region *x*/λ_{e} = 128 ∼ 256 where *E*_{y} is enhanced and computed a *k*_{x} − *k*_{y} spectrum of ∣*E*_{y}∣ at Π_{e}*t* = 1024. In Figure 10b we found clear enhancement of a quasi-perpendicular mode propagating at θ = 87° relative to the external magnetic field. We found that the region *x*/λ_{e} = 150 ∼ 200 is the source of the quasi-perpendicular mode. As in Figure 6, this mode is linearly unstable. However, generation mechanism of this mode is not the linear growth because oblique electrostatic modes with θ ≃ 75°, whose growth rate is much larger than that of the quasi-perpendicular mode, are not excited. In Figure 11 we show ω − *k*_{x} and ω − *k*_{y} spectra of ∣*E*_{y}∣ for Π_{e}*t* = 819.2 ∼ 1228.8, *x*/λ_{e} = 128 ∼ 256. The plotted ω − *k*_{x} and ω − *k*_{y} spectra are obtained by integration over *k*_{y} and *k*_{x}, respectively. We obtain the frequency of the quasi-perpendicular modes as ω/Π_{e} ≃ 0.08, which is close to the lower hybrid resonance frequency (ω_{LHR}/Π_{e} ≃ 0.07). The ω − *k*_{x} spectrum shows that the parallel phase velocity of the quasi-perpendicular mode corresponds to the drift velocity of the electron holes *V*_{eH} = 0.8*V*_{d} = 1.6*V*_{t,e}. We conclude that a lower hybrid mode is excited through coupling of the parallel phase velocity [*Miyake et al.*, 2000]. The dispersion relation of the excited lower hybrid mode at θ = 90° is given by

where *V*_{s} represents the ion sound speed [*Lominadze*, 1981]. In the present simulation, ω_{LHR} ≫ Ω_{i}. From Figures 10b and 11b the perpendicular wave number of the lower hybrid mode is *k*_{y}λ_{e} ≃ 1.0. The perpendicular phase velocity of this mode is approximately equal to *V*_{s}/. Therefore the propagation angle is given by θ ≃ tan^{−1}(*V*_{EH}/*V*_{s}).

[23] There are some recent theoretical works regarding emission process of electrostatic whistler waves (oblique Langmuir waves) from the electron holes by bounce motions of the trapped electrons [*Vetoulis and Oppenheim*, 2001] or vibration of the election holes [*Newman et al.*, 2001]. However, it is very difficult to compare the present simulation result with these works because the parameters are very different. In the present study, since the frequency of the lower hybrid mode is much smaller than the bounce frequency ω_{b}, the emission process of the lower hybrid mode is not consistent with that described in the former reference. However, it is unclear whether the present simulation result is consistent with the latter one. In the present study the parallel phase velocity of the lower hybrid mode satisfies Cherenkov's condition, namely ω = *V*_{EH}*k*_{x}. A theoretical work of this emission process is conducted by *Singh et al.* [2001].

[24] In the two-dimensional system with the open boundary, nonlinear evolution of the bump-on-tail instability is different from that in the periodic system. In the periodic system the coupling process takes place uniformly in space, and the lower hybrid mode is excited uniformly. In the open system, on the other hand, the lower hybrid mode is excited locally in a region close to the source of the electron beam. The spatial development of the instability cannot be transformed by taking a frame of reference moving with *V*_{d} as observed in the previous one-dimensional system. In the periodic system, the bump-on-tail instability saturates in an early phase, and then the lower hybrid mode grows through the wave-wave coupling. However, since the electron holes, which are the energy source of the lower hybrid mode, begin to decay through their coalescence process after the saturation, the growth of the lower hybrid mode also saturates in the early phase. The amplitude of the lower hybrid mode is much smaller than that in the open system. In the open system, on the other hand, the lower hybrid mode grows continuously in the region close to the source of the electron beam because the electron holes are continuously generated by the constant flux of the electron beam.

[25] By Π_{e}*t* = 1228.8, the system has reached a kind of a steady state because there is no change in the temporal evolution of potential structures in the frame of reference moving with the electron holes. However, in the rest frame of the simulation the system has not reached a steady state because the lower hybrid mode does not reach saturation and continues to grow. In a much later phase of the simulation run we also found excitation of another wave mode at *x*λ_{e} ∼ 100 in Figure 9d. We specify a region *x*/λ_{e} = 64 ∼ 128 and computed a *k*_{x} − *k*_{y} spectrum of ∣*E*_{y}∣ at Π_{e}*t* = 1228.8 (Figure 12a). The spectrum shows that the propagation angle of this mode is θ = 65° relative to the external magnetic field. We also show ω − *k*_{y} and ω − *k*_{x} spectra of ∣*E*_{y}∣ in Figures 12b and 12c. We obtain the frequency of this mode as ω/Π_{e} ≃ 0.09 ∼ ω_{LHR}/Π_{e}. As shown in Figure 12d, the positive gradient of the velocity distribution function still remains in this region. It is possible that this mode is excited by a very long-time linear evolution of the bump-on-tail instability. We note that this mode is not excited in the run with periodic boundaries. From Figure 6b, oblique electrostatic modes with θ = 50° ∼ 80° are linearly unstable. However, the θ = 75° mode with the maximum growth rate γ/Π_{e} = 0.05 is not excited in the present run, but the θ = 65° mode with γ/Π_{e} = 0.01 is excited. In Figures 8b we found enhancement of the electron beam modes at *k*_{x}λ_{e} ∼ 0.6. There also exists enhancement of the lower hybrid mode at *k*_{y}λ_{e} ∼ 1.0. We expect that both electron beam modes and lower hybrid mode “seed” the linear growth of the θ = 65° mode. It is noted that this mode is excited only in runs with cold ions. Further analysis of this mode will be reported in a future paper.

### 6. Conclusions

- Top of page
- Abstract
- 1. Introduction
- 2. Simulation Model
- 3. One-Dimensional Simulation
- 4. Two-Dimensional Simulation
- 5. Discussion
- 6. Conclusions
- Acknowledgments
- References
- Supporting Information

[34] In the two-dimensional open system, spatial and temporal development of the electron bump-on-tail instability is different from that in the uniform periodic system. Potential structures of electron holes are similar everywhere in the periodic system. On the other hand, spatial structures of electron holes vary depending on the distance from the source of the electron beam.

[35] In the early phase of the two-dimensional simulation runs, spatial structures of electron holes are determined by the oblique electron beam modes. In the generation region of the electron holes, electron holes initially uniform in the *y* direction are twisted through modulation by the oblique beam modes. In regions far from the source of the electron beam, electron holes become one-dimensional through the coalescence/decay process. In the later phase of the two-dimensional simulation run the wave-wave coupling process takes place uniformly in space in the periodic system, and the lower hybrid mode is excited uniformly. In the open system, on the other hand, the coupling process takes place in the localized region, because the parallel group velocity of the lower hybrid mode is much slower than the drift velocity of the electron holes. Electron holes are accompanied by the perpendicular electric fields through modulation by the lower hybrid mode, resulting in formation of modulated one-dimensional potentials. The perpendicular electric fields of electron holes can be observed even at a distant place from the source because they are carried by the electron holes at the drift velocity of the electron holes.

[36] In the open system, since both oblique electron beam mode and lower hybrid mode propagate much slower than electron holes in the parallel direction, electron holes in the leading edge of the electron beam are not affected by these oblique modes. On the other hand, electron holes excited at the later times are more modulated by these oblique modes. In the present simulation study we found that both perpendicular electric fields of electron holes and strong lower hybrid modes are observed in the generation region of the electron holes. We can make use of these characteristics to identify source regions of electron holes from waveform data of plasma wave observations.

[37] In the present study we demonstrated excitation of oblique electrostatic modes via the wave-wave coupling process for parameters of Π_{e} ∼ Ω_{e} > ω_{b}. There are also other instabilities of electron holes under strong magnetic fields with Ω_{e} ≫ Π_{e} [*Goldman et al.*, 1999; *Oppenheim et al.*, 1999] and weak magnetic fields with ω_{b} > Ω_{e} [*Muschietti et al.*, 2000], in which other oblique modes with different parallel group velocities are excited. In the open system, parallel group velocities of oblique modes are especially important because the difference of parallel group velocities results in a different wave-wave interaction region. Simulation studies of these instabilities with open boundaries are left as future works.