GEOPHYSICAL RESEARCH LETTERS, VOL. 27, NO. 16, PAGES 2469-2472, AUGUST 15, 2000

Three-dimensional structure of electron holes driven by an electron beam

Nagendra Singh, S. M. Loo, B. Earl Wells, C. Deverapalli Department of electrical and computer Engineering, University of Alabama in Huntsville, Huntsville, Alabama

Abstract. Using 3-D particle-in-cell (PIC) simulations we studied the structure and temporal behavior of electron holes (e-holes) in a magnetized plasma driven by an electron beam. When e-holes are fully evolved from high-frequency waves in a time of about a few tens of electron plasma periods, most of the wave energy in the plasma resides in them. Parallel to the ambient magnetic field Bo, the potential distribution of an e- hole is approximately a Gaussian, and the scalelength gz is only a few Debye lengths when determined by the effective temperature of the beam-modified electron distribution function. Transverse to Bo, the potential distribution tends to have a flat top, which makes it difficult to fit a Gaussian distribution, but the scalelengths at which the potential decays in the transverse directions (gx and gz) are found to be only slightly longer than gz. The passages of electron holes monitored at several points in the simulation volume has the signature of bipolar parallel electric field and unipolar perpendicular electric-field pulses as measured from FAST and POLAR. The eventual decay of e-holes is accompanied by the generation of lower hybrid (gh) waves.

1. Introduction

Recent observations of electron holes (e-holes) as solitary potential pulses [Omura et al., 1996; Mozer et al., 1997; Ergun et al., 1998a,b; Franz et al., 1999] have renewed interest in them despite their known formation in electron- beam plasma system since the early work of Berk and Roberts [1967]. Schamell [1982] studied these structures as BGK modes. One- and two-dimensional double layer simulations revealed their formation as solitary potential pulses [Singh and Schunk, 1984; Singh et al., 1987]. In response to recent observations of electron holes, further studies have been performed [Omura et al., 1996; Miyake et al., 1998; Muschietti et al., 1999; Goldman et al., 1999; Oppenheim et al., 1999; Singh, 2000]. However, most of the theoretical studies are limited in dimensionality. Observations reported by Ergun et al. [1998a] clearly show that e-hole structures in the auroral return current region are three-dimensional with comparable electric field component parallel to the ambient magnetic field B0, say Ej, and the two transverse components Ex and Ey. Even the observations from Geotail sometimes show transverse electric fields of the e-holes [Omura et al., 1999]. The purpose of this paper is to report the formation of the 3-D structure of electron holes and their temporal evolution as seen in 3-D particle-in-cell (PIC) simulations of a magnetized plasma driven by an electron beam.

Copyright 2000 by the American Geophysical Union.

Paper number 2000GL003766. 0094-8276/00/2000GL003766505.00

2. Simulation Model

Simulations were preformed using a fully 3-D electrostatic parallel PIC code [Loo et al., 1999]. Periodic boundary conditions on both particles and fields were used. We performed several simulations by varying the size of the 3-D simulation volume given by LxxLyxLz. The length of the simulation box parallel to the ambient magnetic field Bo is Lz and it was fixed at 2563.ao while the transverse dimensions Lx and Ly were varied from 16 to 64 3.ao. In the simulations performed so far, the beam velocity along z (llBo) was assumed to be Vb=4Vte or 8 Vte, where Vt• is the initial electron thermal velocity. The initial electron and ion temperatures were assumed to be equal, that is, Te =Ti-To and Vte =(kBro/me) •/2 , where kB is the Boltzmann constant and m e is the electron mass. Beam temperature was varied in different simulations, but as long as Vb >> Vte • Vtb, the beam thermal velocity, the actual value of Vt• did not matter much. In the runs we used 36 pairs of electrons and ions in each cell of size •ao 3. The electron beam density nb was 10%. The simulations were performed for ion to electron mass ratio M/me = 1836, and •/COpo - 2 and 5, where • and %o are the electron cyclotron and plasma frequencies, respectively. In our discussions we have used the following normalizations and definitions: distance • = x/•ao, time • = t O)po , velocity V: V / Vte , potential • = e{ / kaTo , electric field E = E/(!caTo/ekao), and kao = Vt•/COpo,. In the figures the normalized quantities appear without the bar on top.

3. Numerical Results

We find that only when the electron beam is relatively slow long lasting e-hole structures form. For very fast electron beams such structures are transitory. Therefore, we present results here for a beam with Vo=4 Vte.

Spatial Distribution: Figures l a and lb show the temporal evolution of the potential q•(t) and parallel electric field Ez(t) at the observation point (32,32,5), respectively. Figure 1 c shows Ey at the observation point (32,48,5). The plots for 4) and Ez show that for •-<_ 120, the temporal evolution is dominated by fast oscillations which are high-frequency (HF) waves. As shown later, the dominant frequency component in the HF oscillations is co-•0.8%o. The most noteworthy features of {(t) (Figure l a) and Ez(t ) (Figure lb) are that after about [-• 120 the fast oscillations nearly disappear rather abruptly and the time evolution is dominated by unipolar positive potential pulses in {(t) and bipolar pulses in E•(t), like solitary pulses measured from FAST [Ergun et al., 1998a] and POLAR [Mozer et al., 1997]. The perpendicular electric-field component (Figures l c) also shows this transformation. Since the sign and magnitude of the perpendicular components depend on the relative location

2469

2470 SINGH ET AL.: 3-D STRUCTURE OF ELECTRON HOLES

1.,5

0.,5

-0.,5

-1.5

0.3 .........

0.0

0.25 ' '

0.00

-0.25

0 i 2 3 4 5 6 7 8 9 10

Figure 1. Temporal evolution of fields at the point (32,32,5): (a) 4•(t) and (b) Ez(t). (c) Ey(t) at the point (32,48,5). Note the emergence of unipolar positive potential pulses and bipolar pulses of Ez from the high-frequency waves after t •-120. For the bipolar pulses of Ez(t), the corresponding pulses in Ey are unipolar.

of the observation point with respect to the 3-D structure of the solitary pulses, we now examine the spatial structure of the solitary pulses before we further elaborate on the perpendicular component.

Figure 2 shows an example of the spatial structure of the waves when the HF waves dominate before the emergence of the solitary waves. In this figure equipotential contours are plotted in the x-z plane at • = 32 and [ = 50. The structure in the y-z plane at • = 32 if found to be qualitatively similar. A Fourier analysis of the 2-D wave structure in Figure 2 reveals that a significant level of wave power occurs for k, =m2rclLz with rn = 8-10, which corresponds to parallel wavelengths Xll • 25-30Xdo. If these waves are resonantly driven by the beam, their frequencies are determined by co "' 2 rc 1/b / Xll and they lie in the range 0.8 Z co <copo. Figure 3a shows the frequency spectrum of Ez(t) for t _< 120 shown in Figure lb. There is a relatively broad peak in the spectrum peaking at co • 0.8copo. The HF waves in this frequency band belong to the oblique resonance cone (RC) waves[Fisher and Gould, 1971; Singh and Gould, 1971]. The interference between the wave modes described here and

their nonlinear evolution leads to localization of wave power in positive potential pulses. Figure 2 shows that already at

64

48

x 32

16

0

0 64 128 192 256 z

Figure 2. Wave structures at • = 50 when high-frequency waves dominate. Equipotential contours are shown in the x-z plane at • = 32. Contour for qb = 0 is labeled; the rest of the contours are at intervals of A• = 0.5. The solid and dotted contours show positive and negative potentials.

15

I0

0

0

' 50 ' (a)

E• ' E u ,

.

25 . SP

I 2 O.Ol 0.10 1.00

Figure 3. Frequency spectra of E•(t). (Figure lb) and Ey(t) (Figure l c). (a) For •_<120 and (b) for Z>_120. Solid line curves are for E• while the dotted curves are for Ey.

• = 50, such pulses are beginning to form, but still at this time the potentials variations are oscillatory in both time and space as revealed by the presence of both positive and negative potentials of comparable magnitudes. Figure 2 also shows that in addition to the localization of the wave power in positive potentials, the regions of positive potentials have begun to coalesce. It is worth noting that the early-time structure in Figure 2 appears to be quite analogous to the early-time structure formed in 2-D simulations of Oppenheim et al. [1999].

The coalescence and localization are accompanied by increasing rectification of the waves; increasingly larger positive potentials develop at the expense of the decreasing negative potentials. We find that for [_< 30, the amplitudes of the positive and negative perturbations are nearly equal while at • = 50 the maximum positive potential is • max • 2.2 while the minimum negative potential is 4• min •--1.5. This process continues until the negative potentials are much weaker than the positive potentials localized in a few pulses in the simulation volume. The rectification is caused by the trapping of electrons in positive potential wells. The coalescence is a consequence of the fact that for the oblique

0 64 128 192

Figure 4. Localization of the wave power into a few PPPs at _

t =400. Equipotential contours in (a) x-z plane at • = 32, and (b) y-z plane at • = 32. The contour levels are as in Figure 2. Pulse #1 and #2 are approximately centered near • = 32 and • = 32 and as they travel they repeatedly pass through the point (32,32,5), producing the time history of the fields shown in Figures 1 a and lb.

SINGH ET AL.: 3-D STRUCTURE OF ELECTRON HOLES 2471

waves for c0 < CO po, ell < 0. In such a medium the Coulomb force between like charges is attractive, unlike the repulsive force for ell > 0 [e.g., see Bekefi, 1966]. Thus the regions with positive potentials merge into a few solitary positive potential pulses (PPPs) in the entire simulation box, as shown in Figures 4a and 4b. The PPPs are the e-holes.

In Figures 4a and 4b equipotential contours are plotted at •=400 in the x-z (,F=32) and y-z plane (•=32), respectively. These figures show that there are a few PPPs in the simulation volume, but there are two approximately centered at (32,32,63) and (32,32,175), and they are numbered as 1 and 2 in Figures 4a and 4b. The PPP #1 has the peak potential 4• max • 2.2 and PPP #2 has 4• max --• 1.5. The other PPPs do not extend to the horizontal line passing through the point (32,32) in the x-y planes and they are relatively weaker. This spatial distribution of PPPs has a direct beating on the time plots in Figures la-lc. The solitary positive pulses in Figure l a are the manifestation of PPPs #1 and #2 repeatedly passing through the point (32,32, 5) with a nearly perfect periodicity. The pulse # 1 is stronger of the two pulses until about •- 700, as mentioned above. We point out that the simulation system is periodic, so when the pulses reach z = Lz they reappear at z -0, and propagate through the system repeatedly. As expected the periodicity of the pulse #1 is •pi •68(-0po -1 which yields a velocity of PPP #1 given by V•j •256)•ao/68O)po-l•3.8Vte, which is slightly smaller than the initial beam velocity Vb = 4 Vte. Likewise the pulse #2 has the same periodicity and nearly the same velocity. Thus the PPPs propagate along the positive z axis with a velocity slightly smaller than the initial electron beam velocity. The periodic traversal of the PPPs through the point (32,32,5) gives the bipolar pulses in the E z time-series plot as shown in Figure lb; first the positive field appears followed by the negative field. The positive field is parallel to the motion of the pulses, which is along the initial beam velocity. The point (32,48,5) falls in the region of the large slope c•4•/• in the potential profile of PPP #1 (see Figure 4b) when it repeatedly passes through • = 5. This generates large positive Ey pulses as shown in Figure 1 c. The point (32,48,5) falls in the outer region of the PPP #2, thus the corresponding Ey pulses are relatively weak. Similar features of the transverse component Ex are also seen at strategically located points.

We point out here a major contrast between our 3-D and 2- D results of Oppenheim et al. [1999]. In 2-D simulations electron holes have essentially Ell as they are long structures in the transverse dimensions. This is the reason why Oppenheim et al. call them "tubes" in their presentation in terms of IEI 2 . Only at late times of several thousand O)po -1 when the tubes decay, Oppenheim et al. show formation of e- holes with significant perpendicular electric fields. During this late time the electron holes are embedded in whistler waves. In contrast, in our 3-D simulations e-holes have El I -El as soon as the HF turbulence subsides and solitary structures emerge (Figures l a-1 c). We point out that in 2-D simulations of Oppenheim et al. •/e / COpo - 5. We find that in our 3-D simulations the formation and dynamics of e-holes are nearly identical for •-•e/O)po=2 and 5. Thus we suspect that the dimensionality of the simulation matters a great deal in the evolution and structure of the e-holes.

2.5

2.0

1.5

1.0

0.5

0.0

-32 -16 0 16 32

S

Figure 5. Structure of PPP #1 at i = 400. The horizontal axis (S) is the distance from the center of the pulse along the x, y, and z directions. The curves marked 'x', 'y', and 'z' give the spatial variations of 4)(x,y,z) along the x, y, and z directions, respectively. The dot-dashed curve is a Gaussian fit to the z-curve as explained in the text.

where the electric field components are about zero. In this figure the distance S = r- rc where r stands for x, y, or z. The solid, dashed, and dotted curves show the variation of 4) with z, y, and x, respectively. For 4•(z) we have fitted a Gaussian distribution given by 4•(s) = 4•o exp[ -S2 / 2•2z ], where 4•o is the amplitude of the pulse and e z is the scalelength for variation with z. The dot-dashed curve straddling the curve for 4•(z) shows the fit with •o= 2.2 and • z = 6.5. For 4•(x) and 4•(y) it is difficult to fit a simple Gaussian shape because the top of the pulse tends to have a flat top and it shows oscillations. The flat top makes the pulse broader in x and y directions, but when the potential decays the scalelengths ex and e y associated with the decay are found to be comparable to •z- Therefore the maximum values of E.•, Ey and Ez are found to be comparable. We performed similar analysis on different PPPs at different times and found that he parallel variation (4)(z)) can always be fitted with a Gaussian shape with e z of the order of few Debye lengths. But the variation in the transverse directions are more complex; the potential pulse is generally broader in these directions but the maximum value of the transverse field components can be comparable to Ezmax as explained above.

The excitation of the h-f waves and the formation of the e-

holes plateau the electron velocity distribution by • • 120. The plateaued distribution function has an average drift of Va =0.4Vte and an effective temperature of Teff---2.2To, which yields an effective Debye length of Xa---1.5Xao, and f z---6.5)•ao _--4.3)•a. For the Gaussian pulse the maximum value of the parallel electric field Ezmax --- 4)o / e z4e. The above value of e z and q• o_-- 2.2 gives Ezmax --- 0.22 as seen from Figure lb near [ • 400. Figure 1 e shows that Ermax is of the similar magnitude near [ • 400. For the above value of q•o, eq)o/kB Teff_--1 for which the value of e z found above is somewhat larger than the minimum size predicted by the BGK model of Muschietti et al., [ 1999].

Parallel and Perpendicular Scalelengths' Figure 5 shows the spatial potential distributions 4)(x- xc, y- yc ,z - z) of PPP #1. The point (Xc, y½, Zc) is the center of the potential pulse

Decay of Electron Holes' The frequency spectra of the data for E_• (Figure lb) and Ey (Figure l c) for [ < 120 and [ > 120 are shown in Figures 3a and 3b, respectively. The wave

2472

power in Ez dominates the power in Ey only at the early times • < 120 and for high frequencies c0 > 0.7COpo as seen from Figure 3a for • < 120. The peak at c0---0.2 COpo, marked by SP (solitary pulse) in Figure 3b, occurs in the spectra of both Ez and Ey, and it corresponds to the periodic traversal of the PPP #1 and 2 through the plane J-5 as shown in Figures l a and l c. The power in Ey dominates the power in Ez at low frequencies c0 < 0.05C0po when • > 120 (Figure 2b). In the simulation the lower hybrid (œh) frequency is c0eh --_ 0.2 COpo. The large wave power in the frequency band

_

0.02 • co/COpo _< 0.05 begins to appear only after about t = 500 when very low-frequency oscillations begin to modulate the time-series plot for Ey shown in Figure l c. As the low frequency oscillations develop, the e-hole structure progressively weakens and decays. Figure 6 shows the

_

structure of the e-holes in the x-z plane at t = 1000; the e-hole centered at • = 192 is the one numbered #1 in Figure 4a. Note the laminar behavior of the e-hole structure at [ = 400, while the structure at [ = 1000 has developed undulations, its peak potential has decayed and it is almost fragmented. From the fragments waves are emanating into the plasma with predominantly perpendicular electric fields. This decay process is quite similar to the decay of planar e-holes in 2-D simulations [Oppenheim et al., 1999]. The exact cause of the decay in 2- and 3-D simulations remains unclear. However, we suggest that the waves emanating from the e-holes are directly related to the motion of an e-hole, which is essentially a moving space charge. A charge moving in a plasma generates waves through the Cerenkov condition when the plasma dispersion relation is also satisfied.

4. Conclusions and Discussion

For the first time we have reported here the 3-D structure of electron holes in a magnetized plasma. The holes evolve from HF waves in the frequency range 0.7 < m/COpo • l, corresponding to the propagation of oblique resonance cone waves. The temporal and spatial behavior of the electron holes seen in the simulations are found to be in agreement with observations from FAST and POLAR. The 3-D structure

has Ellma x --• E•max as highlighted by Ergun et al. [1998a,b] and Mozer et al. [ 1997]. This feature of the holes appears as soon as they emerge from the HF waves, unlike in 2-D simulations in which holes with this feature emerge after several thousand COpo -• when they are fragmented by radiating whistler waves. In 3-d simulations also the holes eventually decay and emit waves near the lower hybrid frequency. Exact reason for the difference between available 2-D and 3-D

48

X 32.

0 ..... .. 0 64 128 192 256

Figure 6. Gray scale plots of potential distribution in the x-z plane at •=1000. The whiter and darker scales show increasing and decreasing potentials. Note the fragmentation of the holes and waves radiating from them into the plasma.

results are not clear. It appears that the dimensionality is a key factor in determining the spatial structure and temporal evolution of e-holes.

The temporal behavior (Figures l a-lc) and spatial distribution of e-holes nicely mimic the temporal behavior measured from satellites. Our 3-D simulations predict a periodic lattice structure of e-holes. The fast parallel motion of e-holes in the lattice combined with the slow satellite

velocity allows us to construct the measured temporal behavior.

Acknowledgements. This work was supported by NSF grant ATM9814571.

References

Bekefi, G., Radiation Processes in Plasmas, pp. 283-310, John Wiley, New York, 1966.

Berk, H. L., and K. V. Roberts, Nonlinear study of Vlasov's equation for a special class of distribution functions, Phys. Fluids, 10, 1595, 1967.

Ergun, R. E., et al., FAST satellite observations of large-amplitude solitary structures, Geophys. Res. Lett., 25, 2041, 1998a.

Ergun, R. E., et al., Debye-scale plasma structures associated with magnetic-field-aligned electric fields, Phys. Rev. Lett., 81, 826, 1998b.

Fisher, R. W. and R. W. Gould, Resonance cones in the field pattern of a radio frequency probe in a warm anisotropic plasma, Phys. Fluids, 14, 857, 1971.

Franz, J. R., et al., On the perpendicular scale of electron phase- space holes, Geophys. Res. Lett., 27, 169, 2000.

Goldman, M. V., M. M. Oppenheim, and D. L. Newman, Nonlinear two-stream instabilities as an explanation for auroral bipolar wave structures, Geophys. Res. Lett., 26, 1821, 1999.

Loo, S. M., B. Earl Wells, Nagendra Singh, and Edith P. Huang, Case Study: A Portable Parallel Particle-in-cell Code simulation, Proceedings of the International Conference on Parallel and Distributed Processing Techniques and Applications (PDPTA' 98), Las Vegas, Nevada, July 13-16, 1999.

Miyake, T., et al., Two-dimensional computer simulations of electrostatic solitary waves observed by Geotail spacecraft, J. Geophys. Res., 103, 11,841, 1998.

Mozer, F. S., et al., New features of time domain electric field structures in the auroral acceleration region, Phys. Rev. Lett., 79, 1281, 1997.

Muschietti, L., et al., Phase-space electron holes along magnetic field lines, Geophys. Res. Lett., 26, 1093, 1999.

Omura, Y., et al., Electron beam instabilities as generation mechanism of electrostatic solitary waves in the magnetotail, J. Geophys. Res., 101, 2685, 1996.

Omura, Y., et al., Two dimensional electrostatic solitary waves observed by Geotail in the magnetotail, Adv. Space Res., 24, 55, 1999.

Oppenheim, M. M., M. V. Goldman, D. L. Newman, Evolution of electron phase-space holes in 2-D magnetized plasma, Phys. Rev. Lett., 73, 2344, 1999.

Schamell, H., Kenetic theory of phase space vortices and double layers, Physica $cripta T2/1, 228, 1982.

Singh, N., Electron holes as a common feature of double-layer-driven plasma waves, Geophys. Res. Lett., 27, 927, 2000.

Singh, N and R. W. Gould, Radiation from a short electric dipole in a uniaxial plasma, Radio Science, 6, 1157, 1971.

Singh, N., and R.W. Schunk, Plasma response to the injection of an electron beam, Plasma Phys. Controlled Fus., 26, 859, 1984.

Singh, N., H. Thiemann, and R.W. Schunk, Simulations of auroral plasma processes: Electric fields, waves, and particles, Planet. Space $ci., 35, 353, 1987.

Nagendra Singh, S. M. Loo, B. Earl Wells, C. Deverapalli, Department of Electrical and Computer Engineering, University of Alabama in Huntsville, Huntsville, AL 35899. (email: smloo, wells, singh•ece.uah.edu)

(Received February 16, 2000; revised June 5, 2000; accepted June 9, 2000.)