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Keywords:

  • electrostatic solitary waves;
  • particle simulation;
  • open boundary condition

Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Simulation Model
  5. 3. One-Dimensional Simulation
  6. 4. Two-Dimensional Simulation
  7. 5. Discussion
  8. 6. Conclusions
  9. Acknowledgments
  10. References
  11. 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

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Simulation Model
  5. 3. One-Dimensional Simulation
  6. 4. Two-Dimensional Simulation
  7. 5. Discussion
  8. 6. Conclusions
  9. Acknowledgments
  10. References
  11. 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 Ωe is much larger than the electron plasma frequency Πe, 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 Ωe is smaller than the bounce frequency of the trapped electrons ωb, electron holes are unstable [Miyake et al., 1998; Muschietti et al., 2000]. For Πe ∼ Ωe > ωb, 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 Πe ∼ Ωe > ωb.

[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

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Simulation Model
  5. 3. One-Dimensional Simulation
  6. 4. Two-Dimensional Simulation
  7. 5. Discussion
  8. 6. Conclusions
  9. Acknowledgments
  10. References
  11. 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

  • equation image

where Vd, Vt 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 Vd,bVd (Vd,e = Vd,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 f0. 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 mi/me = 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 vxf(vx) [Birsall and Langdon, 1985]. The flux functions at the left and right boundaries are given by vxf+ and vxf, 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.

image

Figure 1. Schematic illustration of the initial condition. In the simulation box and the region outside of the right boundary, electrons form a Maxwellian velocity distribution function f0 = f. In the region outside of the left boundary, electrons form a bump-on-tail velocity distribution function f+. The local electron densities given by ∫f(v) dv are assumed to be constant for all x. At the open boundaries, particles are injected with the constant flux vf(v), while the exiting particles are removed.

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[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:

  • equation image
  • equation image

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.

  • equation image

where n0, ne, and nb refer to local quantities, and n0 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, n0 = ne + nb. 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:

  • equation image

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 = nb/(nb + ne). 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 Πe = 1.0 and the initial background electron thermal velocity Vt,e = 1.0, we can normalize frequencies and velocities by Πe and Vt,e, respectively. Length is normalized by the electron Debye length λe = Vt,ee. 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 Runs
ParameterEquationValue
Electron cyclotron frequencyΩee1.0
Drift velocity of electron beamVd/Vt,e2.0
Thermal velocity of electron beamVt,b/Vt,e0.1
Thermal energy ratioTe/Ti64.0

[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

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

[13] We present simulation results of a very long one-dimensional open system with Lx = 40960Δxx = 0.1λe), where Lx and Δ x represent system length and grid spacing, respectively. We show xvx 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 xvx 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 (xe = 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.

image

Figure 2. The xvx phase diagrams at Πet = 100, 200, 400, and 1000.

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image

Figure 3. (a) A xvx phase diagram at Πet = 4000 and (b) the corresponding spatial profile of electric fields Ex. The amplitude of the electric field is normalized by meVt,eΠe/e.

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[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 = Vdt 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 Vd.

[16] We also performed the simulations with immobile ions (mi/me = ∞). 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 Ex data for xe = 204.8 ∼ 409.6, Πet = 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 xvx phase space. Ions cannot be reflected by the positive potentials of electron holes.

image

Figure 4. A ω − k spectrum of ∣Ex∣ for xe = 204.8 ∼ 409.6, Πet = 307.2 ∼ 819.2. The amplitude is normalized by meVt,eΠe/e.

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[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 (Lx = 8192Δxx = 0.1λe)). 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.

image

Figure 5. Simulation results of counterstreaming electron beams. The xvx phase diagrams at Πet = 125, 250, and 500.

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4. Two-Dimensional Simulation

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

[18] In this section we extend the previous one-dimensional simulations to a two-dimensional simulation with Lx × Ly = 1024Δx × 64Δyx = Δy = λe). 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 kx and ky. We found that three modes are linearly unstable in this instability. (A) Electron beam modes have the maximum growth rate γ/Πe = 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/Πe = 0.01. Frequency of this mode is ω ≃ ωLHR = 0.07Πe, ωLHR 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/Πe = 0.05 at θ = 75°. These modes are unstable only in the presence of cold ions. Frequencies of these modes are ω ∼ ωLHR = 0.07Πe.

image

Figure 6. (a) Maximum growth rate γ/Πe as a function of kx and ky and linear dispersion relation of electron beam modes: (b) frequency and (c) growth rate with wave normal angles of 0°, 15°, 30°, 45°, and 60°, respectively.

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[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 meVt,e2/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, xe = 85). Then the potentials excited by the instability are modulated in the y direction (xe = 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 ky = 0 at first, and then we observed oblique beam mode with ky = Δky.

image

Figure 7. Spatial profiles of potentials Φ at different times. The potentials are normalized by meVt,e2/e.

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[20] As time elapses, amplitude of the oblique beam modes in the generation region of electron holes (xe = 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 xe = 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 xe = 64 from the leading edge of the electron beam and computed kxky spectra of ∣Ex∣ and ∣Ey∣ shown in Figure 8a. The spectra are the time average at Πet = 102.4, 204.8, 307.2, 409.6. The intensity is normalized by meVt,eΠe/e. We found enhancement of Ex components at ky ≃ 0, while the intensities of Ey components are the same as that of the background thermal fluctuations. We also show kxky spectra of ∣Ex∣ and ∣Ey∣ in the generation region of electron holes (xe = 64 ∼ 128) in Figure 8b. The spectra are the time average at Πet = 102.4, 204.8, 307.2, 409.6. We found clear enhancement of ∣Ey∣ 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 (Ex) and that of the oblique modes (Ey).

image

Figure 8. (a) The kxky spectra of ∣Ex∣ and ∣Ey∣ in the leading edge of the electron beam. We specified a region with width xe = 64 from the leading edge of the electron beam. (b) The kxky spectra of ∣Ex∣ and ∣Ey∣ for xe = 64 ∼ 128. The spectra are the time average at Πet = 102.4, 204.8, 307.2, 409.6. The amplitudes are normalized by meVt,eΠe/e.

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[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 (xe = 700 ∼ 1000) we found one-dimensional electron holes. The one-dimensional potential structures are also indicated from the clear bipolar waveforms of Ex in Figure 9b. However, these one-dimensional electron holes are accompanied by Ey components as shown in Figure 9c. In a long-time evolution of the bump-on-tail instability up to ωLHRt > 50, the intensities of Ey components become larger and larger in a region close to the source of the electron beam. We show a spatial profile of ∣Ey2 along the magnetic field in Figure 10a. The energy density of ∣Ey2 for kyλe = 0 ∼ π averaged for Πet = 819.2 ∼ 1024.0 is shown by the black line (line A). We also plotted the energy density of ∣Ey2 for kyλe = 1.0 ∼ 1.6 by the gray line (line B). We found strong enhancement of ∣Ey2 with kyλe = 1.0 ∼ 1.6 for xe = 150 ∼ 200. We specified a region xe = 128 ∼ 256 where Ey is enhanced and computed a kxky spectrum of ∣Ey∣ at Πet = 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 xe = 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 − kx and ω − ky spectra of ∣Ey∣ for Πet = 819.2 ∼ 1228.8, xe = 128 ∼ 256. The plotted ω − kx and ω − ky spectra are obtained by integration over ky and kx, respectively. We obtain the frequency of the quasi-perpendicular modes as ω/Πe ≃ 0.08, which is close to the lower hybrid resonance frequency (ωLHRe ≃ 0.07). The ω − kx spectrum shows that the parallel phase velocity of the quasi-perpendicular mode corresponds to the drift velocity of the electron holes VeH = 0.8Vd = 1.6Vt,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

  • equation image

where Vs represents the ion sound speed [Lominadze, 1981]. In the present simulation, ωLHR ≫ Ωi. From Figures 10b and 11b the perpendicular wave number of the lower hybrid mode is kyλe ≃ 1.0. The perpendicular phase velocity of this mode is approximately equal to Vs/equation image. Therefore the propagation angle is given by θ ≃ tan−1(equation imageVEH/Vs).

image

Figure 9. (a, d) Spatial profiles of Φ at Πet = 1024.0 and 1228.8, respectively. (b, c) Spatial profiles of Ex and Ey for xλe = 600 ∼ 1024 at Πet = 1024.0, respectively. The amplitude of potentials and electric fields are normalized by meVt,e2/e and meΠeVt,e/e, respectively.

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image

Figure 10. (a) Spatial profiles of ∣Ey2 along the magnetic field: (line A) Energy density of ∣Ey2 for kyλe = 0 ∼ π, and (line B) energy density of ∣Ey2 for kyλe = 1.0 ∼ 1.6. We took time average of ∣Ey2 for Πet = 819.2 ∼ 1024.0. (b) The kxky spectrum of ∣Ey∣ for xe = 128 ∼ 256 at Πet = 1024. The amplitudes of the electric fields are normalized by meVt,eΠe/e.

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image

Figure 11. (a) The ω − kx and (b) ω − ky spectra of ∣Ey∣ for xe = 128 ∼ 256. These ω − kx and ω − ky spectra are given by integration over all values of ky and kx, respectively. The amplitude is normalized by meVt,eΠe/e.

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[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 ωb, 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 = VEHkx. 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 Vd 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 Πet = 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 xλe ∼ 100 in Figure 9d. We specify a region xe = 64 ∼ 128 and computed a kxky spectrum of ∣Ey∣ at Πet = 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 − ky and ω − kx spectra of ∣Ey∣ in Figures 12b and 12c. We obtain the frequency of this mode as ω/Πe ≃ 0.09 ∼ ωLHRe. 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/Πe = 0.05 is not excited in the present run, but the θ = 65° mode with γ/Πe = 0.01 is excited. In Figures 8b we found enhancement of the electron beam modes at kxλe ∼ 0.6. There also exists enhancement of the lower hybrid mode at kyλe ∼ 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.

image

Figure 12. (a) The kxky, (b) ω − ky, (c) ω − kx spectra of ∣Ey∣, and (d) electron velocity distribution function in the generation region of electron holes (xe = 64 ∼ 128): averaged in the time period of Πet = 1228.8 ∼ 1638.4. The velocity distribution function is integrated over xe = 64 ∼ 128, ye = 0 ∼ 64. The amplitudes of the electric fields are normalized by meVt,eΠe/e.

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5. Discussion

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

5.1. Spatial Profile of Potentials Along the Magnetic Field

[26] In the very early phase of the simulation run (Πet = 0 ∼ 100), potential structures of electron holes formed by the bump-on-tail instability are uniform in the direction perpendicular to the external magnetic field. As time elapses, amplitudes of the oblique beam modes become larger and larger in the generation region of electron holes. The excited oblique beam modes modulate the background electron densities in the y direction. Potential structures that are initially uniform in the y direction become twisted through modulation by the oblique beam mode as schematically illustrated in Figure 13a. We also observed two-dimensional electron holes isolated in both x and y directions infrequently (Figure 7d). As the twisted electron holes and two-dimensional electron holes propagate farther from the generation region, the electron holes are more and more isolated in the x direction through coalescence. The perpendicular dimensions of two-dimensional electron holes also become longer and longer through coalescence of the electron holes. Since the electrons trapped therein are mixed in the y direction at the coalescence because of their cyclotron motions, the twisted potentials and two-dimensional potentials are aligned in the y direction as schematically illustrated in Figure 13b. The perpendicular electric fields of the electron holes decay and potential structures become one-dimensional. The decay process of the perpendicular electric fields through the coalescence is described by Miyake et al. [1998, 2000].

image

Figure 13. Schematic illustration on formation and evolution of potential structures along the magnetic field. (a) In the early phase of the simulation run (Πet = 0 ∼ 100), potential structures are modulated by oblique electron beam modes. (b) In the later phase (Πet = 100 ∼ 400), two-dimensional potentials and one-dimensional potentials are observed in the region close to the source of the electron beam and in distant regions from the source, respectively. (c) For ωLHRt ≫ 1, modulated one-dimensional potentials are formed by ion dynamics. (d) Schematic illustration of a modulated one-dimensional potential.

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[27] One-dimensional potentials in the leading edge of the electron beam can propagate without changing their characteristics. The linear dispersion relation of the electron beam modes in Figure 6b shows that both parallel and oblique beam modes have the same parallel phase velocity V/cosθ = VeH. However, group velocities of the oblique beam modes in the parallel direction given by Vcosθ = VeHcos2θ are much slower than that of the electron holes (i.e., the parallel mode). Therefore in the leading edge of the electron beam, potential structures are free from the oblique beam modes propagating slower than the parallel beam mode in the x direction.

[28] In a long-time evolution up to ωLHRt > 50 a strong lower hybrid mode is excited through coupling with the electron holes [Miyake et al., 2000]. This wave-wave coupling process is localized in a region close to the source of the electron beam because the parallel group velocity of the lower hybrid mode is much slower than the drift velocity of the electron holes. As shown in Figure 10a, intensities of Ey components with kyλe = 1 ∼ 1.6 are especially large for xe = 150 ∼ 200. The electron holes excited in the later phases are more modulated by the lower hybrid mode in the y direction. As a result, the electron holes are accompanied by the perpendicular electric fields to form “modulated” one-dimensional potentials as schematically illustrated in Figure 13c. The perpendicular electric fields of electron holes are carried by the electron holes at the drift velocity of the electron holes. As the strong Ey component is observed at xe ∼ 1000 in Figure 10a, we can observe the perpendicular electric field of electron holes even in distant regions from the source.

5.2. Effect of Mass Ratio

[29] We performed three additional simulation runs with different mass ratios. The kxky spectra of ∣Ey∣ for xe = 128 ∼ 374 are plotted in Figure 14. Times of Figures 14b, 14c, and 14d correspond to ωLHRt = 50.7. In the run with immobile ions (mi/me = ∞), simulation results in early phases of the simulation run are almost the same as those with mobile ions (mi/me = 100). In the later phase of this run, however, the lower hybrid mode is not enhanced as in the kxky spectrum of Figure 14a. From this result we found that formation processes of twisted electron holes and two-dimensional electron holes by the oblique beam modes are due to electron dynamics.

image

Figure 14. The kxky spectra of ∣Ey∣ for xe = 128 ∼ 348 with different mass ratios: (a) mi/me = ∞ at Πet = 1228.8, (b) mi/me = 100 at Πet = 716.8, (c) mi/me = 25 at Πet = 358.4, and (d) mi/me = 400 at Πet = 1433.6. The amplitudes are normalized by meVt,eΠe/e. Times of (b), (c), and (d) correspond to ωLHRt = 50.7.

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[30] In the run with lighter ions (mi/me = 25), we found excitation of the lower hybrid mode in the earlier phase. The lower hybrid mode propagates at θ = 83° relative to B0 as shown in Figure 14c. In the run with heavier ions (mi/me = 400), on the other hand, the lower hybrid mode is not excited in the later phase (Πet ∼ 800), and thus spatial structures of electron holes are similar to those in the run with immobile ions. However, we performed a long run up to ωLHRt > 50 and confirmed that a strong lower hybrid mode can be excited in the region close to the source of the electron beam, as shown in Figure 14d.

[31] We listed the angles θ between the wave vectors and the external magnetic field, frequencies ω, parallel and perpendicular wave numbers kx, ky of the excited lower hybrid modes for different mass ratios in Table 2. Equation (6) shows that the wave normal angle and the parallel wave number are approximately given by θ ≃ tan−1(equation imageVeH/Vs) and kx ≃ ωLHR/VeH, respectively. We obtain the perpendicular wave number as kyequation imageωLHR/Vs. These equations are in agreement with the parameters listed in Table 2. Thus the mass ratio is not essential in the emission process of the lower hybrid mode, but it changes the timescale of growth of the lower hybrid mode and its propagation angle. Even with much heavier ions, we expect that the lower hybrid modes is excited. Since the parallel wave number of the lower hybrid mode becomes much smaller with the real mass ratio, a much longer system in the x direction is needed for observation of the lower hybrid mode in a more realistic system.

Table 2. Angles Between the Wave Vector and the External Magnetic Field, Frequencies, Parallel and Perpendicular Wave Numbers of the Excited Lower Hybrid Modes for Different Mass Ratios
mi/meθω/Πekxλekyλe
2583°0.160.10.8
10087°0.080.051.2
40089°0.040.0261.3

5.3. Perpendicular Scale of Electron Holes

[32] From the POLAR observation, Franz et al. [2000] estimated the perpendicular scale of electron holes by assuming lx/ly = Ey/Ex, where lx and ly represent the parallel and perpendicular characteristic widths of electron holes, respectively. In their estimation the ratio of the parallel to perpendicular characteristic widths of two-dimensional electron holes in a plasma with a parameter with Πe = Ωe is ∼0.7. In the present simulation runs, on the other hand, spatial potential structures of electron holes are almost one-dimensional. Although two-dimensional electron holes isolated in both x and y directions are formed through modulation by the oblique beam modes, existence of such electron holes is infrequent. We note that the perpendicular dimension of electron holes is affected by the system length in the y direction because periodic boundary condition is applied in the y direction. To clarify the ratio of the parallel to perpendicular characteristic widths of two-dimensional electron holes, we must use a much longer system in the y direction. However, from the present simulation the ratio of the parallel to perpendicular characteristic widths of two-dimensional electron holes is expected to be much less than 0.5.

[33] Although potential structures are one-dimensional, the electron holes even in regions far from the source are accompanied by perpendicular electric fields as in Figure 9. From Figures 9b and 9c we obtain the amplitude ratio approximately Ey/Ex ≃ 0.7, which is consistent with the POLAR observation [Franz et al., 2000]. In the region close to the source of the electron beam, electron holes are strongly modulated in the y direction by the lower hybrid mode. The perpendicular wave number of the lower hybrid mode is estimated as

  • equation image

The parallel wave number of the beam mode is given by kx = Πe/Vd = 0.5/λe. Thus the perpendicular wavelength of the lower hybrid mode is shorter than the parallel characteristic width of electron holes. The variation of the potentials in the perpendicular direction is smaller than that in the parallel direction as found in Figure 9. One-dimensional potentials are modulated in the perpendicular direction with a shorter wavelength as schematically illustrated in Figure 13d.

6. Conclusions

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Simulation Model
  5. 3. One-Dimensional Simulation
  6. 4. Two-Dimensional Simulation
  7. 5. Discussion
  8. 6. Conclusions
  9. Acknowledgments
  10. References
  11. 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 Πe ∼ Ωe > ωb. There are also other instabilities of electron holes under strong magnetic fields with Ωe ≫ Πe [Goldman et al., 1999; Oppenheim et al., 1999] and weak magnetic fields with ωb > Ωe [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.

Acknowledgments

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

[38] We thank K. Hashimoto and H. Kojima for discussions. The computer simulations were performed on the KDK computer system at Radio Science Center for Space and Atmosphere, Kyoto University. The present work was supported by grant-in-aid for JSPS research fellows 03821 and a grant from ACT-JST.

[39] Arthur Richmond thanks Meers Oppenheim and another reviewer for their assistance in evaluating this paper.

References

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

Supporting Information

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

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