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Propagation of electron beams in space
Adv. Space Res. Vol.8, No. 1, pp. (1)137—(1)149. 1988 Printed in Great Britain. All rights reserved.
0273—1177/88 S0.1X~÷.50 Copyright © 1988 COSPAR
PROPAGATION OF ELECTRON BEAMS IN SPACE M. Ashour-Abdalla and H. Okuda*~ Department of Physics and Institute of Geophysics and Planetary Physics, University of California, Los Angeles, CA 90024, U.S.A. ~Plasma Physics Laboratory, Princeton University, Princeton, NJ 08544, U.S.A.
ABSTRACT We have carried out particle simulations in order to study the effects of beam plasma interaction and the propagation of an electron beam in a plasma with a magnetic field. It is found that the beam plasma instability results in the formation of a high energy tail in the electron velocity distribution which enhances the mean free path of the beam electrons. Moreover, the simulations show that when the beam density is much smaller than the ambient plasma density, currents much larger than the thermal return current can be injected into a plasma. Results from the two—dimensional simulation indicate that the beam particles suffer 2 drift. large radial diffusion due to a c~x~JB I.
INTRODUCTION
Although the concept of active experiments in space plasma is not a new one, until recently most space experiments have been passive in nature. In the past few years, however, active experiments have become increasingly common, and this trend is continuing. The main objective of such experiments is to see how particle beams injected into space plasma affects the environment, and also how the background plasma interacts with and modifies the beam. The most recent experiment of this kind was that of Spacelab—2 on board the shuttle Challenger on its July 29, 1985 launch. During this mission, and other earlier ones, it was observed that both the active experiments themselves (such as an electron beam injection) as well as the presence of a relatively large body (i.e., the orbiter) can greatly modify the surrounding environment by generating wave turbulence and plasma heating /1,2,3/. In addition, it has been observed that the injected beams are modified considerably due to their interactions with the background plasma and the neutrals. These observed changes take place through a variety of processes, including beam—plasma and beam—neutral interactions. In fact, given that many of the observed interactions take place within a finite and often inhomogeneous region, special care must be taken to understand the results from active experiments. For example, in studying naturally occurring beam plasma interactions, it is often assumed that both the plasma and the beam are infinite and homogeneous because in many cases this assumption is justified. Clearly, such assumption cannot usually be made when studying active beam experiments where the beam width or length may be of the order of, or less than, the wave length of interest. In addition, boundary conditions play an important role in determining the dynamics of the system. Thus, it is obvious that in order to understand the results of active experiments in space, it is necessary to develop new approaches to modeling and studying such experiments. In this regard, the development of analytic theory as well as numerical simulations are both necessary and, more than ever, complementary. In fact, since the physics involved in many of the processes are nonlinear and unpredictable it will be essential to perform preliminary simulations in order to gain an insight into the problem, thereby allowing us to take appropriate steps in developing analytic models. To this end, we present results and interpretations of one— and two—dimensional particle simulations. Following the introduction in section II, we detail our results from an electrostatic one—dimensional particle simulation. This study was motivated by observations from the Advanced Concepts Torus—i (ACT—i) device /4/. The observations indicated the presence of a smooth high energy tail of electrons which could not be explained either in terms of quasi— linear theory /5/ or in terms of existing nonlinear theory for a single wave mode /6/. In order to understand the physics associated with beam injection into space, such as the beam experiment on board the space shuttle, we examined a spatially nonuniform system using a one— dimensional bounded electrostatic model. Results from this simulation model, for the case where beam electrons are injected into a plasma at a constant rate in time, are discussed in section III. In order to understand the spatial diffusion of a beam across the magnetic (1)137
M. Ashour-Abdalla and i-I. Okuda
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field a two—dimensional study was required. This study indicated (section IV) that due to a 2drift, the beam particles suffer significant spatial diffusion. large c~xBJB A discussion and suggestions for future research topics are presented in section V. II.
RESULTS FROM ONE—DIMENSIONAL PERIODIC SIMULATIONS
As stated earlier, this study was motivated by results from laboratory experiments, specifically the ACT—i device /4/. This device permits one to initiate a detailed study of beam propagation. In the current—driven experiment performed in the Advanced Concept Torus—i (ACT—i) device, a nonrelativistic cold electron beam of a few hundred eV with densit~rib < 1012 cm3 was injected from a cathode into a target plasma of ~iO eV and 1 1 x 10 ~ cm~’3. Measurements of the electron velocity distribution revealed that cold injected beam electrons evolve into a smooth inonotonically decreasing distribution function with increasing energy, and with a high—energy tail whose energy extends to twice the initial beam energy. Moreover, the observed current carried by the beam electrons was larger than predicted from quasilinear theory. This discrepancy may be traced to the fact that the velocity distribution predicted from a quasilinear theory does not agree with experimental observation /7/. Instead, observations showed a velocity distribution with a high—energy tail, significantly exceeding the initial beam velocity. Since existing nonlinear theory for a single mode /6/ is valid only when the beam density is much smaller than the background plasma density, it cannot be applied for a high current experiment such as the ACT—i experiments. While both quasilinear theory (many unstable modes) and a single—wave theory may be useful, numerical simulations yield detailed information on the dynamics of more general conditions where neither theory is appropriate. Consequently one— and two—dimensional electrostatic particle simulations have been performed in order to understand the generation of the high—energy tail and enhanced beam currents. The code used is a standard electrostatic one—dimensional model (or two—dimensional) with finite—size particles in a spatial grid. For the one—dimensional case the typical simulation length is L = 1O24~where ~ Is the grid size, and the number of simulation particles per grid is xi — 80~1. An initial electron Debye length of Ae ~ and an integration time step of woe~t 0.2 are used; c — nb/nt is varied from io—~to 0.2, while vO/vt is varied from 3 to 4b. Charge neutrality is assumed initially so that ~b + 1e nt is taken equal to the uniform ion density. Here nb is the beam electron number densit 3~rie is the background electron 2 is the electron lengthelectron of the perinumber density, v0 is the 2)1 in~tialbeam velocity,Debye VT length, (Te/m)’ and isL is the the ambient therodic system.Xe The(Te/4irnoe beam temperature in the direction of the magnetic field is assumed to be mal speed, zero initially, while the perpendicular temperature of the beam is taken to be equal to Let us study the case where the wave propngation is parallel to the magnetic field in which the longitudinal motion is completely decoupled from the transverse motion in the electrostatic limit. Note in this case that the particle velocity is one—dimensional so that v 1 — V.
Figure 1 shows the time development of the phase—space 1024~e~ distribution Initially we for note a one—dimensional the two component electron plasma: the ambient the cold 10. simulation where nb/nt — 0.1, plasma v0/v~ and 10 and L = electron beam with vO/vt
Electron Phase Space
~iI1Ii~1i1~
~~
~
~
I
~
0
500
x Fig. 1.
I
EI~~1F
1000 0
500
400
1000 0
x
Phase space plots for the beam plasma instability.
500
1000
x Here a
—
0.1, v 0
6
=
0~, and L
—
1024~. Note the generation of a high energy tail.
—
iOv~,
Propagation ot Electron Beams in Space
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1, correspondAccording to linearto theory the growth — ~e/Vo 0.0707A ing approximately the twelfth mode rate of themaximizes system. near The kphase space plot at ixpet — 20, indicates modulation of both the beam and background electrons by the twelfth mode. The relative amplitudes of the Fourier modes’ electric field energy at different times are shown in Figure 2, where we find that there are several large amplitude modes at wpet — 20, including mode 12. Amplitude of Fourier Modes 2~3OJJ\~J
~
020
:~
Fig. 2. Relative amplitude of the electromagnetic field energy for each Fourier mode at four different times. The parameters for the simulation are the same as in Figure 1. As the instability grows further to ~et 30, modes 12 and 14 are the most dominant modes (Figure 2); the trapping of beam electrons as well as background electrons can be seen clearly (Figure 1). It is interesting to note that the motion of a trapped particle is roughly symmetric with respect to the wave phase velocity, thereby accelerating a significant fraction of beam and background electrons to a velocity nearly twice the original beam velocity. At ~pet — 50, a significant diffusion in velocity space has taken place and is associated with the coalescence of the unstable modes. This coalescence results in smaller amplitudes of the electrostatic waves. The diffusion process continues until the end of the run. In fact, little evidence of trapped particles is seen at capet — 100, whereas nearly complete randomization occurs by Wpet — 400. By that time, the wave energy is much smaller than its peak value (Figure 2). This is because the wave energy is transferred back to the particles via Landau damping associated with the presence of a negative slope in the electron distribution. Figure 3 shows the beam, background and the total electron distribution for different times. The initial distribution shown in (a) spreads with increasing time and, concurrently, the background electrons are accelerated to form a high energy tail. Beam electrons (shown in Figures 3c and 3f) spread to both high and low energies, while the average bean velocity decreases with increasing time. The final beam distribution has a triangular shape, peaking at zero velocity and smoothly extending to nearly twice the initial beam velocity. Background electrons are also accelerated to form a high—energy tail as a result of trapping. The final tail distribution is very similar to the beam distribution, suggesting that at the final stage of the simulations diffusion in velocity space is strong enough to smear out beam and background electrons. It is interesting to observe that the largest velocity for both beam and background electrons is almost twice the initial beam velocity. This can be explained in the following manner. For a strong instability, such as in this example, the wave energy grows to an amplitude large enough to trap both beam and background electrons. Those trapped electrons then oscillate back and forth in the potential well, thereby producing a high—energy tail. As the amplitude of the unstable waves grows, the trapping width grows and more background particles are trapped. Since the trapping of the background electrons causes wave damping, the instability must saturate when the number of trapped background electrons approaches that of the beam electrons. Since the motion of a trapped electron is nearly symmetric in the wave frame moving with the phase speed, the maximum trapping velocity is v 0 in the beam frame, which is 2v0 in the laboratory franc. After the saturation of the instability, the mean free path of the beam electrons is determined by the classical Coulomb Colision. Moreover, when the beam density is small compared with the ambient density the beam mean free path may be estimated by neglecting collisions 3 for vtarget. >> v~/5/. between beam electrons and treating the background plasma as a Maxwellian NoteThus that the collision time for an electron with speed v is proportional to v
(1)140
M. Ashour-Abdalla and H. Okuda
Electron Velocity Distributions Total
Backgronnd
Bean,
~I:~L~.~F1400~~~1~F40° ~ ~~
:::~~°~~ ca)~ ~~:.‘0~k’~
-20
0
20 -20
0
20 -20
~/k~..kLT~
0
20
V/Vt
Fig. 3. Electron distribution functions for the total, beam, and background electrons. Initial distribution (a), background distribution at capet = 30 (b), beam distribution at scpet — 30 (c). The final total distribution (d), background (e), and beam (f) at Wpet — 400 are shown in the lower panel. The simulation parameters are the same as in Figure 1. the generation of a high—energy tail of electrons as a result of beam—plasma instabilities could enhance the mean free path and result in the enhanced generation of a toroidal current. The mean free path of the electrons may be estimated by 9. —, where < > denotes averaging over the electron distribution. Here te is the collision time of a beam electron with the background electrons and ions, and v >> v~is given by, Te’
tee1 + rei1
where 23
tee
—
—
2
tei
=
m~v __________ 8ixne logA
Therefore the mean free path is given by 9. — (me2/i2tne1’logA) = . The mean free path for the current is given by 9.’ — . Figure 1v 4 shows the 1v time development of 9. and 9.’, normalized to their initial value using the electron distribution from the 1 simulation. The initial mean free paths have been calculated using a cold beam ~(v 1 — v0) streaming through a Maxwellian background plasma. Figure 4 shows, that for small time values, capet 20, both mean free paths retain their initial values until at capet — 25 they are suddenly reduced by 20% or so. By this time, the unstable waves have grown to large amplitudes so that the beam electrons are decelerated, having lost their energy to the waves. At later times when the trapping of electrons begins, a high energy tail is formed (see 3. As electrons a result, have both a9. long and 9.’ with and their frequency final Figure 3). Such high energy meanincrease free path as time, the collision values are rapidly larger than decreases as v their initial values, suggesting that the mean free path of the electrons and the Current generated are greater than those of the original beams. In this sense, we may conclude that at saturation the beam—plasma instability generates an anomalous conductivity rather than an anomalous resistivity as commonly believed.
1
In actual beam injection experiments the beam radial dimension is finite. Because of this the beam plasma instability occurs at an angle of propagation that is oblique with respect to ambient magnetic fields, even when beam particles are injected along the magnetic fields. Two—dimensional simulations however, show the presence of high energy particles, results similar to those of the one—dimensional simulations. The main difference is that in two— dimensions, the wave amplitude is reduced, which results in a smaller number of accelerated particles /9/. This is because the obliquely propagating waves have a small parallel electric field E 11 along the ambient magnetic field. The final mean free paths, 9. and 9.’, are still much larger than predicted from quasilinear theory and Coulomb Collisions. It should be emphasized that the enhanced beam current generation is in agreement with laboratory experiment /9/.
(1)lJl
Propagation of Electron Beams in Space
Mean Free Path of Electrons
-0
200
Fig. 4. Time development of the mean free paths normalized to the mean free paths for the initial III.
and Current
a and II’ of cold beam.
the
electrons
and current
INJECTION OF AN ELECTRON BEAM INTO A SPACE PLASMA
In the previous section it was found that beam plasma instabilities can generate high energy Such high energy particles are electron populations whose energies exceed the beam energy. This ins taaccelerated by a strong electric field resulting from beam plasma instability. bility effectively energizes both resonant and nonresonant particles by trapping. One of the important consequences of this study is the generation of an enhanced mean free path for the beam electrons, because the electrons suffer little collisional slowing down. While the results of the previous section correctly reproduce the observed electron distribution of laboratory experiments at steady state, the physics associated with beam injection and spatial nonuniformity cannot be studied by using such a model. Spatial amplification of the instability, beam neutralization and the propagation of electrons in space can only be studied using a noaperiodic model. A noaperiodic one-dimensional model was therefore developed to study beam injection into a space plasma in the direction of an ambient magnetic field /lo/. We shall now study the The background plasma is results of beam injection into a plasma at a constant rate in time. - the plasma consists of the ambient electrons and neutralizing ions initially charge neutral The plasma is no - so that the injection of beam electrons produces excess negative charges. longer charge neutral and the electric field generated by the excess electrons pushes the background electrons out of the system at x - 0. Electrons, both ambient and beam, that leave the left boundary at x = 0 are considered lost from the system. leaving the At the right end of the system, at x = L, ambient electrons When the beam electrons system are reflected as if they were entering the system from x > L. reach the right boundary at x = L, the simulation is stopped. E(x=L) - 0 is assumed throughout the simulation. No electric field exists in front of the electron beam. When the beam electrons reach x - L, E(x-L) may no longer equal zero because of the generation of the electric field associated with beam particles. It should be pointed out that the choice of the electric field at the right boundary, E(L) = 0, indicates that no net charge exists for the region for x < L although the plasma region for 0 < x < L is in general not charge neutral. This is because if the charge neutrality is not satisfied for x < L, then the electric field flux should be finite in general at x - L, E(L) f 0. that no net charge exists for x < L. The choice of E(L) = 0 indicates, therefore, One such example corresponds to an electron beam Injection from a spacecraft. As the beam electrons leave the spacecraft, an equivalent amount of positive charges remains on the spacecraft so that the total charge is always zero. Using the model described above we have carried out simulations L = 1024X,, aeXe = 40, vg/vt = 10, nb/nt = l/8. Note that in enbVo is larger than the thermal current, eaevt/dT=.
for the this case
following parameters, the beam current
Figure 5 shows the electric field (a), the target electron density (b), the phase space for the beam electrons (c) and the phase space for the target electrons (d) at %at - 10. In only a quarter of the systam length is order to show the spatial structure in more detail, shown in Figure 5 (L = 256X,), although the simulation system is still L - 1024X,. Note that for w ,t - 10, the electric repelling both the beam and backfield at x - 0 is positive, groua B electrons Panel (b) shows that large electron and forcing them to move to the left. density modulations are induced in the plasma. It is interesting to note that at early times, w ,t = 10, the tip of the beam electrons manages to propagate into a plasma at roughly the iait Pal velocity as shoy in (c). This is because the plasma cannot respond to the beam for the plasm to respond to the incoming particles. As the instantly; it takes t _ I+,~beam electrons are bunched near x - 5OA. electric field builds up to stop the beam electrons,
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M. Ashour-Abdalla and H. Okuda
The phase space of the target electrons, shown in (d), indicates a modulation whose wavelength is given by 2lrvo/cape.
Target Electron Density
Electric Field
___________
~
Beam Phase Space
Target Phase Space
Fig. 5. Electric field profile (a), target electron density profile (b), beam electron phase space (c), and target electron phase space (d) at Woet — 10. Here nb/nt — 1/8, v 0 — i0vt, 6 — 0°, L 1024~. Note only one quarter of the system length is shown here. At later times, Upet — 20, the amplitude of the beam—plasma instability increases. This results in the formation of a vortex structure apparent in the phase space of the beam and the background electrons (Figure 6). Both the electric field (a) and the background electron density (b) indicate a coherent wave structure propagating to the right. The electrons exhibit a coherent vortex structure. The wavelength of the coherent structure corresponds approximately to the most unstable mode given by A 2~tvo/wpeas shown in (a). Note the presence of a very sharp density spike in (b). Target Electron Density
Electric Field
20
Beam Phase Space Ic)
20
Target Phase Space Id)
Fig. 6. Electric field profile (a), target electron density profile (b), beam electron phase space (c), and target electron phase space (d) at Wpet — 20. Here nb/nt = 1/8, — i0vt, ~ — 00, L — iO24~. Note only one quarter of the system length is shown here. At a much later time, 24Ae so capetthat — 100, the computing shown in Figure is stopped 7, the at beam this electrons time. Note fill thethecoalescence entire sysof atemcoherent takes place as the beam electrons propagate into the plasma as shown in length trapping L = lO (c) and (d). The phase space distribution and amplitude of the beam instability at this time are similar to those obtained in the previous section using a periodic model.
Propagation of Electron Beams in Space
Beam Phase Space
Target Phase Space
40
I
(a)
w~t —
-40
0
(b)
50 I
500
1000
0
x
1000
500
x
Fig. 11. Beam phase space (a) and target phase space at Upet parameters are the same as those in Figure 10. IV.
(1)145
—
50.
The simulation
ANOMALOUS SPATIAL DIFFUSION OF BEAM ELECTRONS ACROSS THE MAGNETIC FIELD
In sections II and III, diffusion in velocity space was emphasized, indicating that the formation of a high energy tail of beam electrons leads to an enhanced mean free path. Here we consider a spatial diffusion of beam electrons across the magnetic field, a phenomenon which plays an important role in beam propagation and beam divergence along the beam line. It is well known that an intense electron beam is subject to a number of low frequency magnetohy— drodynamic instabilities such as fire hose, Weibel and filamentation instability. These macroscopic instabilities tend to distort and break up the current channel, thereby prohibiting beam propagation. Here we present a different mechanism responsible for rapidwhich spatial 2 drift takes beam across field. which This are diffusion is due with to the placediffusion exclusively for magnetic beam electrons in resonance the c~xWB electrostatic waves and therefore experience a nearly d.c. electric field. Consider simulations in a uniform magnetic field with doubly periodic boundary conditions in x and y. A localized electron beam is initially confined to a narrow strip in x while uniform in y (Figure 12). The beam electrons are completely neutralized initially in this model, propagating in the direction of a uniform ambient magnetic field ~ — (O,Byo,Bzo).
Two Dimensional Simulation
Model
z
B
0
~L
ambient plasma
Lx
X
electron beam
24Ae
(1)146
Nt. Ashour-Abdalla and H. Okuda
beam electrons experience a nearly d.c. wave electric field. This electric field causes the beam electrons to diffuse not only in velocity apace along the magnetic field but also in real space across the magnetic field. The ambient particles, on the other hand, experience a rapidly oscillating electric field so that no net displacement results.
Particle 1.ocation Beam ~,,i
(a)
25
>‘
50
Ib)
~I
0
Background I-,, c~n,t 25
Ic)
‘
~I SO
(dl
Fig. 13. Location of the beam and ambient electrons at w~t — 25 and 50 in a two— dimensional (x,y) space. Here nb/nt — 1/4, vO/vt — 10 ana wce/wpe — 1.2. In order to determine the particle diffusion coefficient, of beam 2 — <(xj(t) displacement — xj(O))2> are shown and in field Figure 14. particles in x from the original positions (Ax) It is clear that the beam electrons undergo a displacement almost one order of magnitude larger than the field particles. following manner:
The diffusion coefficient Dx can be estimated in the
D~
—
(Ax)2> 2t~
~c2
—
Ey2Bz2tc B2
where t~ is the correlation time and E~is the y (azimuthal) component of the electric field. Using the measured and r~— where ~ is the linear growth rate of the beam plasma
Spatial Diffusion
30-
00
o
0
o
-
0 0
0
beam particles
20
-
-
SC
ambient particles
-
* *
**
-
** **
* I
0
I
~00
200
300
c.Jpat
Fig. 14. Displacement, <(Ax)2> across the magnetic field for beam electrons (open circles) and ambient electrons (crosses) initially located in the same area in the two—dimensional model. The simulation parameters are the same as those in Figure 13.
Propagation of Electron Beams in Space
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2wpe in good agreement instability for an order of magnitude estimate, find of -<(Ax)2> 0.1 Xe in Figure 14. with the diffusion coefficient determined for thewe slope Diffusion in velocity space is shown in Figure 15 for parallel and perpendicular components of the electron distribution. These measurements are obtained by averaging over the entire x—y space, mot just the region where the beam particles reside. It is clear that the parallel distribution is quite similar to the one—dimensional results indicating a presence of a high energy tail. In addition to parallel diffusion in velocity, we note perpendicular diffusion also takes place, indicating the presence of a perpendicular component of the electric field K 1 with respect to the magnetic field /6/.
j~
Parallel Distribution
Perpendicular Distribution lb~
I
—0
—
3~2O0O~-
I000L
IJ
~I
Fig. 15. Parallel, (a), and perpendicular (b), velocity space distributions at capet 250 in a two—dimensional simulation.
=
Figure 16 shows the time development of the potential contours in the (x,y) plane. Initial random noises represented by the small scale structures shown in (a), become coherent with its scale size equal to the wavelength at the most unstable mode A1 — 211v0/titpe When the Contours of Electrostatic Potential —
(a)
25
so
(b)
L, ~
too
-i
‘~-~
a.~
(c) -~
r-
2oo .
(d) .
~ L~0~
o-~
0
--
~H
~ ~-~=~-—-.
.-~
-.~
-
L, 0
x x Fig. 16. The time development of the potential contours in the (x,y) plane. Initial 2iTv~/woe. random noises represented by the small scale structures in (a) become coherent with When largernonlinear amplitudes. saturation The most takes unstable place mode the size has aof wavelength the potential givencontour by A1 —becomes larger.
M. Ashour-Abdalla and H. Okuda
(1)148
wave amplitude becomes large the contour plots indicate that unstable waves spread out beyond the region where the beam was initially located. This is caused partly by the radial diffusion of beam electrons and partly by wave propagation across the magnetic field. In order to study the dependence of beam diffusion on beam density, we look at the results frost a two—dimensional simulation using the earlier parameters except for beam density which is reduced to nb/no — 1/8 are shown in Figure 17. The beam drift speed is kept at VO/Vt = 10. The parallel velocity distribution shown in (a) at u~t — 250 clearly indicates that the main characteristics of velocity space diffusion remain the same. The formation of a high energy tail beyond the beam energy and the smooth monotonically decreasing velocity distribution in energy are observed. Perpendicular heating shown in (b) remains small as the beam density is reduced in agreement with previous simulations /9/. Potential contours, (c), show a coherent structure associated with the beam plasma instability. The amplitude of the potential contours remains small. Parallel Distributions
200
°
Perpendicular Distributions
3000
(a)
100
2000
—250
~2O
0 VII /V~
20
1000
Electrostatic Potential
L,
A
(
I~
0 V~/V~
10 x
Fig. 17. The results of a two—dimensional simulation using the same parameters as in Figure 13 except the beam density is now reduced to nb/nt — 1/8. The beam drift is kept at va/vt — 10. The parallel velocity distribution shown in (a) at (dpet — 250 clearly indicates that the main characteristics of velocity space diffusion remains the same. The formation of a high energy tail occurs beyond the beam energy and the smooth, mono— tonically decreasing velocity distribution in energy. Note that the perpendicular heating shown in (b) remains small as the beam density is reduced. The potential contours in (c) show a coherent structure associated with the beam plasma instability. V.
CONCLUSIONS
We have discussed several fundamental plasma processes inherent to the propagation of electron beams in the plasma with a magnetic field. Using one— and two—dimensional simulation models, we have found that: 1. The beam plasma instability results in the formation of a high energy tail in the electron velocity distribution with energy extending to twice the initial beam energy. 2. The mean free path of the beam electrons with a high tail is greater than that predicted from quasilinear theory, but in agreement with ACT—i observations. 3. When the injection current is increased beyond the thermal return current by increasing the beam density, most of the beam electrons are reflected back to the spacecraft, prohibiting beam electrons from propagating. 4. On the other hand, when the injection current is increased by increasing the beam energy but the beam density is kept small, the beam electrons can propagate freely into the ambient plasma /11/. 5. Two—dimensional simulations using spatially localized beam electrons show the presence of anomalous spatial diffusion of beam electrons across the magnetic field, a diffusion 2 much than that of theresonant ambient with electrons. This diffusion is causedand by takes the cExBJB drift greater of the beam electrons the electrostatic instabilities place on a time scale much faster than the low frequency magnetohydrodynamic instabilities. 6. In addition to velocity space diffusion along the magnetic field, the two—dimensional results show a perpendicular velocity space diffusion caused by obliquely propagating electrostatic waves. While we considered several important plasma processes associated with an electron beam injection into space, much remains to be done. One of the most important remaining problems is the electromagnetic stability of the high current electron beam which may be important in determining the ultimate mean free path of propagation. Another is the effect of spacecraft charging caused by the injection of high current electron beams which can prohibit beam propagation away from the spacecraft /11/. At low ionospheric altitudes where the neutral particles are the majority species, ionization of neutral atoms by beam plasma instabilities and the critical velocity ionization associated with the spacecraft becomes very important problems. These processes should be addressed in the future.
Propagation ofElectron Beams in Space
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ACKNOWLEDGMENTS This work was supported by Air Force contract Fl9628—85—K—0027, National Science Foundation grants AI~~(86—12512 and ATM85—132l5, NASA Solar Terrestrial Theory grant NAGW—78, and U.S. Deparment of Energy Contract DE—ACO2—76—CHO—3O73. REFERENCES 1.
P.M. Banks, K. Baker, N. Kawashima, J. Rait, and K. Williams, Vehicle charging and potential experiment, in Spacelab 2 90—day Post—Mission Science Report, ed. by E. Ubran, NASA Marshall Space Flight Center, 1985.
2.
L.A. Frank, N. D’Angelo, J. Grebowsky, D. Gurnett, D. Reasoner, N. Stone, Ejectable plasma diagnostics package, in Spacelab 2 90—day Post—Mission Science Report, ed. by E. Ubran, NASA Marshall Space Flight Center, 1985.
3.
D.A. Gurnett, W.S. Kurth, J.T. Steinberg, P.M. Banks, R.I. Bush, and W.J. Mitt, Whistler—mode radiation from the Spacelab 2 electron beam, Geophys. Keg. Lett., 13, 225, 1986.
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K.L. Wong, M. Otto, and G.A. Wurden, ACT—I: A steady—state torus for basic plasma physics research, Rev. Sci. Instrum., 53, 409, 1982.
5.
W.E. Drummond and D. Pines, Non—linear stability of plasma oscillations, Nucl. Fusion Suppl. Pt.3, p. 1049, 1962.
6.
T.M. O’Neil, J.M. Winfrey, and J.U. Malmberg, Nonlinear interaction and a plasma, Phys. Fluids, 14, 1204, 1971.
7.
B.B. Kadomtsev, Plasma Turbulence, Academic Press, New York, p. 15, 1965.
8.
D.V. Sivuklin, Reviews of Plasma Physics, ed. by M.A. Leontovich, Consultants Bureau, New York, Vol. 4, p. 142, 1966.
9.
H. Okuda, K. Horton, H. Otto, and K.L. Wong, Effects of beam plasma instability on current drive via injection of an electron beam into a torus, Phys. Fluids, 28, 3365, 1985.
of a small cold beam
10. II. Okuda, K. Horton, M. Ono, and H. Ashour—Abdalla, Propagation of a nonrelativistic electron beam in a plasma in a magnetic field, Phys. Fluids, in press, 1986. ii. H. Okuda and J.R. Kan, Injection of an electron beam and vehicle charging in space, Phys. Fluids, submitted, 1986.
0273—1177/88 S0.1X~÷.50 Copyright © 1988 COSPAR
PROPAGATION OF ELECTRON BEAMS IN SPACE M. Ashour-Abdalla and H. Okuda*~ Department of Physics and Institute of Geophysics and Planetary Physics, University of California, Los Angeles, CA 90024, U.S.A. ~Plasma Physics Laboratory, Princeton University, Princeton, NJ 08544, U.S.A.
ABSTRACT We have carried out particle simulations in order to study the effects of beam plasma interaction and the propagation of an electron beam in a plasma with a magnetic field. It is found that the beam plasma instability results in the formation of a high energy tail in the electron velocity distribution which enhances the mean free path of the beam electrons. Moreover, the simulations show that when the beam density is much smaller than the ambient plasma density, currents much larger than the thermal return current can be injected into a plasma. Results from the two—dimensional simulation indicate that the beam particles suffer 2 drift. large radial diffusion due to a c~x~JB I.
INTRODUCTION
Although the concept of active experiments in space plasma is not a new one, until recently most space experiments have been passive in nature. In the past few years, however, active experiments have become increasingly common, and this trend is continuing. The main objective of such experiments is to see how particle beams injected into space plasma affects the environment, and also how the background plasma interacts with and modifies the beam. The most recent experiment of this kind was that of Spacelab—2 on board the shuttle Challenger on its July 29, 1985 launch. During this mission, and other earlier ones, it was observed that both the active experiments themselves (such as an electron beam injection) as well as the presence of a relatively large body (i.e., the orbiter) can greatly modify the surrounding environment by generating wave turbulence and plasma heating /1,2,3/. In addition, it has been observed that the injected beams are modified considerably due to their interactions with the background plasma and the neutrals. These observed changes take place through a variety of processes, including beam—plasma and beam—neutral interactions. In fact, given that many of the observed interactions take place within a finite and often inhomogeneous region, special care must be taken to understand the results from active experiments. For example, in studying naturally occurring beam plasma interactions, it is often assumed that both the plasma and the beam are infinite and homogeneous because in many cases this assumption is justified. Clearly, such assumption cannot usually be made when studying active beam experiments where the beam width or length may be of the order of, or less than, the wave length of interest. In addition, boundary conditions play an important role in determining the dynamics of the system. Thus, it is obvious that in order to understand the results of active experiments in space, it is necessary to develop new approaches to modeling and studying such experiments. In this regard, the development of analytic theory as well as numerical simulations are both necessary and, more than ever, complementary. In fact, since the physics involved in many of the processes are nonlinear and unpredictable it will be essential to perform preliminary simulations in order to gain an insight into the problem, thereby allowing us to take appropriate steps in developing analytic models. To this end, we present results and interpretations of one— and two—dimensional particle simulations. Following the introduction in section II, we detail our results from an electrostatic one—dimensional particle simulation. This study was motivated by observations from the Advanced Concepts Torus—i (ACT—i) device /4/. The observations indicated the presence of a smooth high energy tail of electrons which could not be explained either in terms of quasi— linear theory /5/ or in terms of existing nonlinear theory for a single wave mode /6/. In order to understand the physics associated with beam injection into space, such as the beam experiment on board the space shuttle, we examined a spatially nonuniform system using a one— dimensional bounded electrostatic model. Results from this simulation model, for the case where beam electrons are injected into a plasma at a constant rate in time, are discussed in section III. In order to understand the spatial diffusion of a beam across the magnetic (1)137
M. Ashour-Abdalla and i-I. Okuda
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field a two—dimensional study was required. This study indicated (section IV) that due to a 2drift, the beam particles suffer significant spatial diffusion. large c~xBJB A discussion and suggestions for future research topics are presented in section V. II.
RESULTS FROM ONE—DIMENSIONAL PERIODIC SIMULATIONS
As stated earlier, this study was motivated by results from laboratory experiments, specifically the ACT—i device /4/. This device permits one to initiate a detailed study of beam propagation. In the current—driven experiment performed in the Advanced Concept Torus—i (ACT—i) device, a nonrelativistic cold electron beam of a few hundred eV with densit~rib < 1012 cm3 was injected from a cathode into a target plasma of ~iO eV and 1 1 x 10 ~ cm~’3. Measurements of the electron velocity distribution revealed that cold injected beam electrons evolve into a smooth inonotonically decreasing distribution function with increasing energy, and with a high—energy tail whose energy extends to twice the initial beam energy. Moreover, the observed current carried by the beam electrons was larger than predicted from quasilinear theory. This discrepancy may be traced to the fact that the velocity distribution predicted from a quasilinear theory does not agree with experimental observation /7/. Instead, observations showed a velocity distribution with a high—energy tail, significantly exceeding the initial beam velocity. Since existing nonlinear theory for a single mode /6/ is valid only when the beam density is much smaller than the background plasma density, it cannot be applied for a high current experiment such as the ACT—i experiments. While both quasilinear theory (many unstable modes) and a single—wave theory may be useful, numerical simulations yield detailed information on the dynamics of more general conditions where neither theory is appropriate. Consequently one— and two—dimensional electrostatic particle simulations have been performed in order to understand the generation of the high—energy tail and enhanced beam currents. The code used is a standard electrostatic one—dimensional model (or two—dimensional) with finite—size particles in a spatial grid. For the one—dimensional case the typical simulation length is L = 1O24~where ~ Is the grid size, and the number of simulation particles per grid is xi — 80~1. An initial electron Debye length of Ae ~ and an integration time step of woe~t 0.2 are used; c — nb/nt is varied from io—~to 0.2, while vO/vt is varied from 3 to 4b. Charge neutrality is assumed initially so that ~b + 1e nt is taken equal to the uniform ion density. Here nb is the beam electron number densit 3~rie is the background electron 2 is the electron lengthelectron of the perinumber density, v0 is the 2)1 in~tialbeam velocity,Debye VT length, (Te/m)’ and isL is the the ambient therodic system.Xe The(Te/4irnoe beam temperature in the direction of the magnetic field is assumed to be mal speed, zero initially, while the perpendicular temperature of the beam is taken to be equal to Let us study the case where the wave propngation is parallel to the magnetic field in which the longitudinal motion is completely decoupled from the transverse motion in the electrostatic limit. Note in this case that the particle velocity is one—dimensional so that v 1 — V.
Figure 1 shows the time development of the phase—space 1024~e~ distribution Initially we for note a one—dimensional the two component electron plasma: the ambient the cold 10. simulation where nb/nt — 0.1, plasma v0/v~ and 10 and L = electron beam with vO/vt
Electron Phase Space
~iI1Ii~1i1~
~~
~
~
I
~
0
500
x Fig. 1.
I
EI~~1F
1000 0
500
400
1000 0
x
Phase space plots for the beam plasma instability.
500
1000
x Here a
—
0.1, v 0
6
=
0~, and L
—
1024~. Note the generation of a high energy tail.
—
iOv~,
Propagation ot Electron Beams in Space
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1, correspondAccording to linearto theory the growth — ~e/Vo 0.0707A ing approximately the twelfth mode rate of themaximizes system. near The kphase space plot at ixpet — 20, indicates modulation of both the beam and background electrons by the twelfth mode. The relative amplitudes of the Fourier modes’ electric field energy at different times are shown in Figure 2, where we find that there are several large amplitude modes at wpet — 20, including mode 12. Amplitude of Fourier Modes 2~3OJJ\~J
~
020
:~
Fig. 2. Relative amplitude of the electromagnetic field energy for each Fourier mode at four different times. The parameters for the simulation are the same as in Figure 1. As the instability grows further to ~et 30, modes 12 and 14 are the most dominant modes (Figure 2); the trapping of beam electrons as well as background electrons can be seen clearly (Figure 1). It is interesting to note that the motion of a trapped particle is roughly symmetric with respect to the wave phase velocity, thereby accelerating a significant fraction of beam and background electrons to a velocity nearly twice the original beam velocity. At ~pet — 50, a significant diffusion in velocity space has taken place and is associated with the coalescence of the unstable modes. This coalescence results in smaller amplitudes of the electrostatic waves. The diffusion process continues until the end of the run. In fact, little evidence of trapped particles is seen at capet — 100, whereas nearly complete randomization occurs by Wpet — 400. By that time, the wave energy is much smaller than its peak value (Figure 2). This is because the wave energy is transferred back to the particles via Landau damping associated with the presence of a negative slope in the electron distribution. Figure 3 shows the beam, background and the total electron distribution for different times. The initial distribution shown in (a) spreads with increasing time and, concurrently, the background electrons are accelerated to form a high energy tail. Beam electrons (shown in Figures 3c and 3f) spread to both high and low energies, while the average bean velocity decreases with increasing time. The final beam distribution has a triangular shape, peaking at zero velocity and smoothly extending to nearly twice the initial beam velocity. Background electrons are also accelerated to form a high—energy tail as a result of trapping. The final tail distribution is very similar to the beam distribution, suggesting that at the final stage of the simulations diffusion in velocity space is strong enough to smear out beam and background electrons. It is interesting to observe that the largest velocity for both beam and background electrons is almost twice the initial beam velocity. This can be explained in the following manner. For a strong instability, such as in this example, the wave energy grows to an amplitude large enough to trap both beam and background electrons. Those trapped electrons then oscillate back and forth in the potential well, thereby producing a high—energy tail. As the amplitude of the unstable waves grows, the trapping width grows and more background particles are trapped. Since the trapping of the background electrons causes wave damping, the instability must saturate when the number of trapped background electrons approaches that of the beam electrons. Since the motion of a trapped electron is nearly symmetric in the wave frame moving with the phase speed, the maximum trapping velocity is v 0 in the beam frame, which is 2v0 in the laboratory franc. After the saturation of the instability, the mean free path of the beam electrons is determined by the classical Coulomb Colision. Moreover, when the beam density is small compared with the ambient density the beam mean free path may be estimated by neglecting collisions 3 for vtarget. >> v~/5/. between beam electrons and treating the background plasma as a Maxwellian NoteThus that the collision time for an electron with speed v is proportional to v
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M. Ashour-Abdalla and H. Okuda
Electron Velocity Distributions Total
Backgronnd
Bean,
~I:~L~.~F1400~~~1~F40° ~ ~~
:::~~°~~ ca)~ ~~:.‘0~k’~
-20
0
20 -20
0
20 -20
~/k~..kLT~
0
20
V/Vt
Fig. 3. Electron distribution functions for the total, beam, and background electrons. Initial distribution (a), background distribution at capet = 30 (b), beam distribution at scpet — 30 (c). The final total distribution (d), background (e), and beam (f) at Wpet — 400 are shown in the lower panel. The simulation parameters are the same as in Figure 1. the generation of a high—energy tail of electrons as a result of beam—plasma instabilities could enhance the mean free path and result in the enhanced generation of a toroidal current. The mean free path of the electrons may be estimated by 9. —
tee1 + rei1
where 23
tee
—
—
2
tei
=
m~v __________ 8ixne logA
Therefore the mean free path is given by 9. — (me2/i2tne1’logA)
1
In actual beam injection experiments the beam radial dimension is finite. Because of this the beam plasma instability occurs at an angle of propagation that is oblique with respect to ambient magnetic fields, even when beam particles are injected along the magnetic fields. Two—dimensional simulations however, show the presence of high energy particles, results similar to those of the one—dimensional simulations. The main difference is that in two— dimensions, the wave amplitude is reduced, which results in a smaller number of accelerated particles /9/. This is because the obliquely propagating waves have a small parallel electric field E 11 along the ambient magnetic field. The final mean free paths, 9. and 9.’, are still much larger than predicted from quasilinear theory and Coulomb Collisions. It should be emphasized that the enhanced beam current generation is in agreement with laboratory experiment /9/.
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Propagation of Electron Beams in Space
Mean Free Path of Electrons
-0
200
Fig. 4. Time development of the mean free paths normalized to the mean free paths for the initial III.
and Current
a and II’ of cold beam.
the
electrons
and current
INJECTION OF AN ELECTRON BEAM INTO A SPACE PLASMA
In the previous section it was found that beam plasma instabilities can generate high energy Such high energy particles are electron populations whose energies exceed the beam energy. This ins taaccelerated by a strong electric field resulting from beam plasma instability. bility effectively energizes both resonant and nonresonant particles by trapping. One of the important consequences of this study is the generation of an enhanced mean free path for the beam electrons, because the electrons suffer little collisional slowing down. While the results of the previous section correctly reproduce the observed electron distribution of laboratory experiments at steady state, the physics associated with beam injection and spatial nonuniformity cannot be studied by using such a model. Spatial amplification of the instability, beam neutralization and the propagation of electrons in space can only be studied using a noaperiodic model. A noaperiodic one-dimensional model was therefore developed to study beam injection into a space plasma in the direction of an ambient magnetic field /lo/. We shall now study the The background plasma is results of beam injection into a plasma at a constant rate in time. - the plasma consists of the ambient electrons and neutralizing ions initially charge neutral The plasma is no - so that the injection of beam electrons produces excess negative charges. longer charge neutral and the electric field generated by the excess electrons pushes the background electrons out of the system at x - 0. Electrons, both ambient and beam, that leave the left boundary at x = 0 are considered lost from the system. leaving the At the right end of the system, at x = L, ambient electrons When the beam electrons system are reflected as if they were entering the system from x > L. reach the right boundary at x = L, the simulation is stopped. E(x=L) - 0 is assumed throughout the simulation. No electric field exists in front of the electron beam. When the beam electrons reach x - L, E(x-L) may no longer equal zero because of the generation of the electric field associated with beam particles. It should be pointed out that the choice of the electric field at the right boundary, E(L) = 0, indicates that no net charge exists for the region for x < L although the plasma region for 0 < x < L is in general not charge neutral. This is because if the charge neutrality is not satisfied for x < L, then the electric field flux should be finite in general at x - L, E(L) f 0. that no net charge exists for x < L. The choice of E(L) = 0 indicates, therefore, One such example corresponds to an electron beam Injection from a spacecraft. As the beam electrons leave the spacecraft, an equivalent amount of positive charges remains on the spacecraft so that the total charge is always zero. Using the model described above we have carried out simulations L = 1024X,, aeXe = 40, vg/vt = 10, nb/nt = l/8. Note that in enbVo is larger than the thermal current, eaevt/dT=.
for the this case
following parameters, the beam current
Figure 5 shows the electric field (a), the target electron density (b), the phase space for the beam electrons (c) and the phase space for the target electrons (d) at %at - 10. In only a quarter of the systam length is order to show the spatial structure in more detail, shown in Figure 5 (L = 256X,), although the simulation system is still L - 1024X,. Note that for w ,t - 10, the electric repelling both the beam and backfield at x - 0 is positive, groua B electrons Panel (b) shows that large electron and forcing them to move to the left. density modulations are induced in the plasma. It is interesting to note that at early times, w ,t = 10, the tip of the beam electrons manages to propagate into a plasma at roughly the iait Pal velocity as shoy in (c). This is because the plasma cannot respond to the beam for the plasm to respond to the incoming particles. As the instantly; it takes t _ I+,~beam electrons are bunched near x - 5OA. electric field builds up to stop the beam electrons,
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M. Ashour-Abdalla and H. Okuda
The phase space of the target electrons, shown in (d), indicates a modulation whose wavelength is given by 2lrvo/cape.
Target Electron Density
Electric Field
___________
~
Beam Phase Space
Target Phase Space
Fig. 5. Electric field profile (a), target electron density profile (b), beam electron phase space (c), and target electron phase space (d) at Woet — 10. Here nb/nt — 1/8, v 0 — i0vt, 6 — 0°, L 1024~. Note only one quarter of the system length is shown here. At later times, Upet — 20, the amplitude of the beam—plasma instability increases. This results in the formation of a vortex structure apparent in the phase space of the beam and the background electrons (Figure 6). Both the electric field (a) and the background electron density (b) indicate a coherent wave structure propagating to the right. The electrons exhibit a coherent vortex structure. The wavelength of the coherent structure corresponds approximately to the most unstable mode given by A 2~tvo/wpeas shown in (a). Note the presence of a very sharp density spike in (b). Target Electron Density
Electric Field
20
Beam Phase Space Ic)
20
Target Phase Space Id)
Fig. 6. Electric field profile (a), target electron density profile (b), beam electron phase space (c), and target electron phase space (d) at Wpet — 20. Here nb/nt = 1/8, — i0vt, ~ — 00, L — iO24~. Note only one quarter of the system length is shown here. At a much later time, 24Ae so capetthat — 100, the computing shown in Figure is stopped 7, the at beam this electrons time. Note fill thethecoalescence entire sysof atemcoherent takes place as the beam electrons propagate into the plasma as shown in length trapping L = lO (c) and (d). The phase space distribution and amplitude of the beam instability at this time are similar to those obtained in the previous section using a periodic model.
Propagation of Electron Beams in Space
Beam Phase Space
Target Phase Space
40
I
(a)
w~t —
-40
0
(b)
50 I
500
1000
0
x
1000
500
x
Fig. 11. Beam phase space (a) and target phase space at Upet parameters are the same as those in Figure 10. IV.
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—
50.
The simulation
ANOMALOUS SPATIAL DIFFUSION OF BEAM ELECTRONS ACROSS THE MAGNETIC FIELD
In sections II and III, diffusion in velocity space was emphasized, indicating that the formation of a high energy tail of beam electrons leads to an enhanced mean free path. Here we consider a spatial diffusion of beam electrons across the magnetic field, a phenomenon which plays an important role in beam propagation and beam divergence along the beam line. It is well known that an intense electron beam is subject to a number of low frequency magnetohy— drodynamic instabilities such as fire hose, Weibel and filamentation instability. These macroscopic instabilities tend to distort and break up the current channel, thereby prohibiting beam propagation. Here we present a different mechanism responsible for rapidwhich spatial 2 drift takes beam across field. which This are diffusion is due with to the placediffusion exclusively for magnetic beam electrons in resonance the c~xWB electrostatic waves and therefore experience a nearly d.c. electric field. Consider simulations in a uniform magnetic field with doubly periodic boundary conditions in x and y. A localized electron beam is initially confined to a narrow strip in x while uniform in y (Figure 12). The beam electrons are completely neutralized initially in this model, propagating in the direction of a uniform ambient magnetic field ~ — (O,Byo,Bzo).
Two Dimensional Simulation
Model
z
B
0
~L
ambient plasma
Lx
X
electron beam
24Ae
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Nt. Ashour-Abdalla and H. Okuda
beam electrons experience a nearly d.c. wave electric field. This electric field causes the beam electrons to diffuse not only in velocity apace along the magnetic field but also in real space across the magnetic field. The ambient particles, on the other hand, experience a rapidly oscillating electric field so that no net displacement results.
Particle 1.ocation Beam ~,,i
(a)
25
>‘
50
Ib)
~I
0
Background I-,, c~n,t 25
Ic)
‘
~I SO
(dl
Fig. 13. Location of the beam and ambient electrons at w~t — 25 and 50 in a two— dimensional (x,y) space. Here nb/nt — 1/4, vO/vt — 10 ana wce/wpe — 1.2. In order to determine the particle diffusion coefficient, of beam 2 — <(xj(t) displacement — xj(O))2> are shown and in field Figure 14. particles in x from the original positions (Ax) It is clear that the beam electrons undergo a displacement almost one order of magnitude larger than the field particles. following manner:
The diffusion coefficient Dx can be estimated in the
D~
—
(Ax)2> 2t~
~c2
—
Ey2Bz2tc B2
where t~ is the correlation time and E~is the y (azimuthal) component of the electric field. Using the measured
Spatial Diffusion
30-
00
o
0
o
-
0 0
0
beam particles
20
-
-
SC
ambient particles
-
* *
**
-
** **
* I
0
I
~00
200
300
c.Jpat
Fig. 14. Displacement, <(Ax)2> across the magnetic field for beam electrons (open circles) and ambient electrons (crosses) initially located in the same area in the two—dimensional model. The simulation parameters are the same as those in Figure 13.
Propagation of Electron Beams in Space
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2wpe in good agreement instability for an order of magnitude estimate, find of -<(Ax)2> 0.1 Xe in Figure 14. with the diffusion coefficient determined for thewe slope Diffusion in velocity space is shown in Figure 15 for parallel and perpendicular components of the electron distribution. These measurements are obtained by averaging over the entire x—y space, mot just the region where the beam particles reside. It is clear that the parallel distribution is quite similar to the one—dimensional results indicating a presence of a high energy tail. In addition to parallel diffusion in velocity, we note perpendicular diffusion also takes place, indicating the presence of a perpendicular component of the electric field K 1 with respect to the magnetic field /6/.
j~
Parallel Distribution
Perpendicular Distribution lb~
I
—0
—
3~2O0O~-
I000L
IJ
~I
Fig. 15. Parallel, (a), and perpendicular (b), velocity space distributions at capet 250 in a two—dimensional simulation.
=
Figure 16 shows the time development of the potential contours in the (x,y) plane. Initial random noises represented by the small scale structures shown in (a), become coherent with its scale size equal to the wavelength at the most unstable mode A1 — 211v0/titpe When the Contours of Electrostatic Potential —
(a)
25
so
(b)
L, ~
too
-i
‘~-~
a.~
(c) -~
r-
2oo .
(d) .
~ L~0~
o-~
0
--
~H
~ ~-~=~-—-.
.-~
-.~
-
L, 0
x x Fig. 16. The time development of the potential contours in the (x,y) plane. Initial 2iTv~/woe. random noises represented by the small scale structures in (a) become coherent with When largernonlinear amplitudes. saturation The most takes unstable place mode the size has aof wavelength the potential givencontour by A1 —becomes larger.
M. Ashour-Abdalla and H. Okuda
(1)148
wave amplitude becomes large the contour plots indicate that unstable waves spread out beyond the region where the beam was initially located. This is caused partly by the radial diffusion of beam electrons and partly by wave propagation across the magnetic field. In order to study the dependence of beam diffusion on beam density, we look at the results frost a two—dimensional simulation using the earlier parameters except for beam density which is reduced to nb/no — 1/8 are shown in Figure 17. The beam drift speed is kept at VO/Vt = 10. The parallel velocity distribution shown in (a) at u~t — 250 clearly indicates that the main characteristics of velocity space diffusion remain the same. The formation of a high energy tail beyond the beam energy and the smooth monotonically decreasing velocity distribution in energy are observed. Perpendicular heating shown in (b) remains small as the beam density is reduced in agreement with previous simulations /9/. Potential contours, (c), show a coherent structure associated with the beam plasma instability. The amplitude of the potential contours remains small. Parallel Distributions
200
°
Perpendicular Distributions
3000
(a)
100
2000
—250
~2O
0 VII /V~
20
1000
Electrostatic Potential
L,
A
(
I~
0 V~/V~
10 x
Fig. 17. The results of a two—dimensional simulation using the same parameters as in Figure 13 except the beam density is now reduced to nb/nt — 1/8. The beam drift is kept at va/vt — 10. The parallel velocity distribution shown in (a) at (dpet — 250 clearly indicates that the main characteristics of velocity space diffusion remains the same. The formation of a high energy tail occurs beyond the beam energy and the smooth, mono— tonically decreasing velocity distribution in energy. Note that the perpendicular heating shown in (b) remains small as the beam density is reduced. The potential contours in (c) show a coherent structure associated with the beam plasma instability. V.
CONCLUSIONS
We have discussed several fundamental plasma processes inherent to the propagation of electron beams in the plasma with a magnetic field. Using one— and two—dimensional simulation models, we have found that: 1. The beam plasma instability results in the formation of a high energy tail in the electron velocity distribution with energy extending to twice the initial beam energy. 2. The mean free path of the beam electrons with a high tail is greater than that predicted from quasilinear theory, but in agreement with ACT—i observations. 3. When the injection current is increased beyond the thermal return current by increasing the beam density, most of the beam electrons are reflected back to the spacecraft, prohibiting beam electrons from propagating. 4. On the other hand, when the injection current is increased by increasing the beam energy but the beam density is kept small, the beam electrons can propagate freely into the ambient plasma /11/. 5. Two—dimensional simulations using spatially localized beam electrons show the presence of anomalous spatial diffusion of beam electrons across the magnetic field, a diffusion 2 much than that of theresonant ambient with electrons. This diffusion is causedand by takes the cExBJB drift greater of the beam electrons the electrostatic instabilities place on a time scale much faster than the low frequency magnetohydrodynamic instabilities. 6. In addition to velocity space diffusion along the magnetic field, the two—dimensional results show a perpendicular velocity space diffusion caused by obliquely propagating electrostatic waves. While we considered several important plasma processes associated with an electron beam injection into space, much remains to be done. One of the most important remaining problems is the electromagnetic stability of the high current electron beam which may be important in determining the ultimate mean free path of propagation. Another is the effect of spacecraft charging caused by the injection of high current electron beams which can prohibit beam propagation away from the spacecraft /11/. At low ionospheric altitudes where the neutral particles are the majority species, ionization of neutral atoms by beam plasma instabilities and the critical velocity ionization associated with the spacecraft becomes very important problems. These processes should be addressed in the future.
Propagation ofElectron Beams in Space
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ACKNOWLEDGMENTS This work was supported by Air Force contract Fl9628—85—K—0027, National Science Foundation grants AI~~(86—12512 and ATM85—132l5, NASA Solar Terrestrial Theory grant NAGW—78, and U.S. Deparment of Energy Contract DE—ACO2—76—CHO—3O73. REFERENCES 1.
P.M. Banks, K. Baker, N. Kawashima, J. Rait, and K. Williams, Vehicle charging and potential experiment, in Spacelab 2 90—day Post—Mission Science Report, ed. by E. Ubran, NASA Marshall Space Flight Center, 1985.
2.
L.A. Frank, N. D’Angelo, J. Grebowsky, D. Gurnett, D. Reasoner, N. Stone, Ejectable plasma diagnostics package, in Spacelab 2 90—day Post—Mission Science Report, ed. by E. Ubran, NASA Marshall Space Flight Center, 1985.
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D.A. Gurnett, W.S. Kurth, J.T. Steinberg, P.M. Banks, R.I. Bush, and W.J. Mitt, Whistler—mode radiation from the Spacelab 2 electron beam, Geophys. Keg. Lett., 13, 225, 1986.
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K.L. Wong, M. Otto, and G.A. Wurden, ACT—I: A steady—state torus for basic plasma physics research, Rev. Sci. Instrum., 53, 409, 1982.
5.
W.E. Drummond and D. Pines, Non—linear stability of plasma oscillations, Nucl. Fusion Suppl. Pt.3, p. 1049, 1962.
6.
T.M. O’Neil, J.M. Winfrey, and J.U. Malmberg, Nonlinear interaction and a plasma, Phys. Fluids, 14, 1204, 1971.
7.
B.B. Kadomtsev, Plasma Turbulence, Academic Press, New York, p. 15, 1965.
8.
D.V. Sivuklin, Reviews of Plasma Physics, ed. by M.A. Leontovich, Consultants Bureau, New York, Vol. 4, p. 142, 1966.
9.
H. Okuda, K. Horton, H. Otto, and K.L. Wong, Effects of beam plasma instability on current drive via injection of an electron beam into a torus, Phys. Fluids, 28, 3365, 1985.
of a small cold beam
10. II. Okuda, K. Horton, M. Ono, and H. Ashour—Abdalla, Propagation of a nonrelativistic electron beam in a plasma in a magnetic field, Phys. Fluids, in press, 1986. ii. H. Okuda and J.R. Kan, Injection of an electron beam and vehicle charging in space, Phys. Fluids, submitted, 1986.