Double layer formation due to current injection

Planer. Spue &I., Vol. 33. No. 7, pp, 853-861. 198s Prmted in Great IMain.

DOUBLE

0

LAYER FORMATION CURRENT INJECTION TAKASHI

Geophysical

Institute,

YAMAMOTO

University

and

DUE TO

J. R. KAN

of Alaska, Fairbanks,

(Receioed 4 February

0032m633/85$3.00+0.00 1985 Pergamon Press Ltd.

AK 99701, U.S.A.

1985)

Abstract-It is shown by numerical simulations that enhanced current density can generate double layers, even when the electron drift speed is significantly below the electron thermal speed. The double layer potential is spontaneously produced by the space charge self-consistently developed inside the simulation domain. The particle influxes from the low-potential boundary of our simulation domain are independent of the outfluxes. Thepotentialdifference$,isshown toincreasewithincreasingnumberdensityoftheinjectioncurrent.Strong double layers with potential energy e$O >> kT, (the electron thermal energy) are stably formed when the injection electron current much exceeds the thermal current of ambient electrons. The backscattered and mirrored electrons are found to have stabilizing effects on the current-driven double layers.

1. INTRODUCTlON

Potential double layers are believed to play an important role in the acceleration of aurora1 electrons. Recent numerical simulations and laboratory experiments showed that double layers can develop either by applying a potential drop or by injecting a current along field-lines in a plasma. For convenience, the former may be called potential-driven double layers while the latter called current-driven double layers. The potential-driven double layer concerns with physical processes responsible for sustaining the applied potential. On the other hand, the current-driven double layer is concerned with processes which lead to spontaneous generation of potential drops associated with the injection current. The potential-driven double layers have been studied extensively in recent years by means of numerical simulations (Joyce and Hubbard, 1978 ; Hubbard and Joyce, 1979; Wagner et al., 1981; Singh and Schunk, 1982a; Borovsky and Joyce, 1983; Wagner and Kan, 1985; Yamamoto and Kan, 1985). Joyce and Hubbard (1978) and Hubbard and Joyce (1979) simulated potential-driven double layers with potential differences e&ranging from 10 kT, to 200 kT, in a one-temperature plasma. The current-driven double layers have been studied by numerical simulations (Goertz and Joyce, 1975; DeGrootetal., 1977;SatoandOkuda, 1980;Singhand Schunk, 1982b, 1983, 1984). In the simulation by DeGroot et al. (1977), the double layers are formed by electron currents with vd = 1.4 v,,_ where vd and ugp are the electron drift velocity and thermal velocity, respectively. Sato and Okuda (1980) showed that an ion-acoustic instability (u, = O.~V,,) results in the 853

formation ofweak double layers (e&, < kT,). Singh and Schunk (1982b, 1983, 1984) simulated double layers driven by relatively high electron drift in the range of 1.5 uoe < ud < 5.5 uoV In the above-mentioned simulations, except Goertz and Joyce (1975), the potentials are set equal to each other at the two boundaries of the simulation domain. In contrast, Sato and Okuda( 1981) simulated a series of current-driven weak ion-acoustic double layers in which the total potential difference is determined by the external circuit. Goertz and Joyce (1975) notably used floating potential boundary conditions in their simulations of current-driven layers. They showed that strong double layers (e& >> kT,) can develop spontaneously when the electron drift velocity exceeds the thermal velocity. The particle boundary conditions in their model are as follows : A particle which leaves the system is replaced by a particle entering into the system at the same initial velocity of the existing one. On this type of particle boundary condition, the following point should be noted : Particle influxes into the simulation domain are correlated with particle outfluxes. As double layers start to develop, acceleration of electrons results in an increase in the electron outflux. This in turn calls for an increase of electron influx. This enhanced electron influx possibly causes the double layer potential to grow. The purpose of the present paper is to show, by particle simulations, that the strong double layers (e& x kT,) can develop spontaneously in a plasma carrying a sufficiently high current density even if the electron drift velocity is less than the electron thermal velocity (i.e., ud cc II,,,), in contrast but complementary with the simulations of Goertz and Joyce (1975). The current density in our simulations is increased by

854

increasing electrons.

T. the number

2.

YAMAM~T~ and

density of the current-carrying

SIMCJLATION MODEL

Our simulations are performed using a onedimensional finite-size particle code. The currents flowing in the x-direction may be regarded as fieldaligned currents because dynamical behavior of particles in the x-dimension is independent ofa uniform magnetic field assumed in the x-direction. Initially, the simulation domain is filled with a Maxwellian plasma of temperature T, and density n,, and the potential is zero everywhere. In this paper, we define the electron thermal flux by F We) = &&,/2)t&!

- 0.4 flouoe

(1)

which is calculated from a half-Maxwellian distribution with the temperature T,(= M,&) and the density n,/2. The corresponding ion thermal flux is where M,/Mi is the electron to FW = ~%&%+ ion mass ratio. Also, the thermal current density is defined by Jth = eFrhcel. As will be seen later, the density of the source plasma at the ionospheric boundary is approximately equal to no. Therefore, the abovedefined thermal fluxes can be referred either to the ionospheric source plasma or to the initial ambient plasma. The potential at the magnetospheric boundary (i.e., x = L) is assumed to be zero as the reference potential. Thepotentialat theionosphericboundary(i.e.,x = 0)is self-consistently determined by the charge distribution inside the simulation domain. Note that the electric field must vanish at both boundaries because of the overall charge neutrality imposed in the simulation domain, which is to be discussed later. These potential boundary conditions are similar to those used in the simulation by Goertz and Joyce (1975). The particle boundary conditions are assumed as follows. (i) The magnetospheric boundary supplies a plasma from a non-drifting half-Maxwellian source at a temperature To and a number density nr(t)/2. The number density nr(t) as a function of time is introduced to regulate the field-aligned current. Consequently, the electron influx from this boundary is &

(n r(f)/2)~oe - 0.4n,Wo,

and the ion influx is $F

~Ki%(n,(r)/2)~o,.

(ii)Theionospheric boundarysuppliesaplasmafrom a non-drifting half-Maxwellian source at a temperature To. The electron influx and ion influx from this

J. R.

KAN

boundary are varied to maintain overall charge neutrality in the simulation system and kept on the order of the electron and ion thermal fluxes as defined in (I). Note that the temperatures of the source plasmas at both boundaries are To which is also the initial plasma temperature of the system. The particle boundary condition in our model is different from those of Goertz and Joyce (1975) in that the particle influxes at the magnetospheric boundary in our model can be controlled independently of the particle outfluxes from the simulation domain. Note that our particle boundary condition ensures that particle fluxes across the boundaries remain uncorrelated with the potential drop to prevent feedback effects and resulting in unrealistically large doublelayer potentials. In our simulations the following parameters and nondimensionalizations are introduced : The ion to electron mass ratio M,/M, = 64, and the average number ofcharge sheets per Debye length /I,is initially 20 where 1, is defined by (kTo/4nnoeZ)“Z. The system length L is taken to be 256 iD- The mesh size A = 1, and the time step At = 0.1 uU,’ where w,_ = (4nn,e2/M,)1’2 is the electron plasma frequency. The dimensionless quantities are specified by T = rw,, for the time, X = x/l, for distance, V = c’Jv,, for velocity and 4* = ecj/kT, for potential. The current density J and the flux Fare normalized, respectively, by the reference current density (en,u,,) and the reference flux (n,u,,). Note that the electron thermal flux Ffhcej is defined earlier in (1) and the thermal current density Jfh is eFl,,(,). 3. SIMULATION

RESULTS

Potential profiles in the simulation results given below are time-averaged over 50 WP,’ to remove highfrequency (-cope) plasma oscillations, unless specified otherwise. (i) Test run To test the simulation code, Run 1 is performed with n,(t) = n,. Both boundaries are expected to supply approximately equal thermal fluxes of electrons and ions so that the field-aligned current is equal to zero. These features ofthe fluxes and the current densities are confirmed by the results of Run 1which was performed for 0 < T < 400. The averaged (over 200 < T < 400) potential difference across the simulation system is found to be very small, $. = 0.19 kTo/e although the potential fluctuations are not so small. Thus, the test run showed that the plasmas in our simulation model are stable. The potential jumps are negligible when the held-aligned current density is much smaller than the thermal current level Jth.

855

Double layer formation due to current injection (ii) Strong clot&e layers When the influxes from the magnetospheric boundary significantly exceed those from the ionospheric boundary, potential jumps develop. This is demonstrated in Run 2, in which the density function n,(T) is specified by n,(T) = no

for T d 100

n,(T) = [1+3(T-100)/200]n, for 100 < T d 300 n,(T) = 4n,

for T > 300

This represents a continuous increase of the influxes at the magnetospheric boundary from the thermal level to four times the thermal level. The results in Run 2 are shown in Figs. l-5. Figure 1 shows the electron fluxes (a) at the magnetospheric boundary and (b) at the ionospheric boundary, and the ion fluxes (c) at the magnetospheric boundary and (d) at the ionospheric boundary. Figures l(a) and l(c) confirm that the electron and ion influxes from the magnetospheric boundary are regulated as specified by the density function n,(T). From Figs. l(b) and l(d), the influxes

(a) ELECTRON FLUX AT MAGNETOSPHERIC BOUNDARY

from the ionospheric boundary are, on the average, at the level of the thermal fluxes (Fthcejand FthciJ.Figure 2 shows (a) the total current density exiting from the magnetospheric boundary, (b) the electron current density exiting from the magnetospheric boundary, (c) the electron current density entering at the ionospheric boundary, and (d) the ion current density exiting from the magnetospheric boundary. It is seen from Fig. 2 that the field-aligned currents are carried predominantly by the electrons and the profile of the electron current density at the magnetospheric boundary is quite similar to that at the ionospheric boundary. This similarity indicates that the change in total number ofelectrons in the simulation domain is relatively small. Figure 3 shows the averaged potential profiles at T = 50, 150, 400, 1100 and 1500, and the temporal variation ofthe averaged potential difference across the simulation domain. Figure 4 shows the temporal evolution of the density profiles for the ions (upper) and electrons (lower), which are also averaged over 50 w&‘. Here, we briefly describe temporal evolution of the double layers in our simulations. During T < 100 only the thermal fluxes are provided at the magnetospheric boundary, the plasma is in a thermal equilibrium and

(C) ION FLUX AT MAGNETOSPHERIC BOUNDARY

_;pi

I-l

(b)ELECTRON IONOSPHERIC

FLUX AT BOUNDARY

(d) ION FLUX AT IONOSPHERIC BOUNDARY

_;il_:::

0

600 TIME

1200

. 0

600 TIME

1200

FIG. I. TEMPORALVARIATION OFTHE PARTICLEFLUXES AT THE ROUNDARIES IN RUN 2. The electron fluxes (a)at the magnetospheric boundary and(b) at the ionospheric boundary. The ion fluxes (c) at the magnetospheric boundary and (d) at the ionospheric boundary. The fluxesare normalized by the reference flux(n,v,,).

856

T. YAMAMOTO CURRENT DENSITY AT MAGNETOSPHERIC BOUNDARY 2.5 (

I

1.25j /--Ol/ ELECTRON CURRENT DENSITY AT MAGNETOSPHERIC BOUNDARY

OI. ELECTRON CURRENT DENSITY AT IONOSPHERIC BOUNDARY 2.511 1.25 iOL ION CURRENT DENSITY AT MAGNETOSPHERIC BOUNDARY o.5/ 0.25.

-:::Ij , , / j 0

600

1200

TIME

F1c.2. TEMPORALVARIATIONOFTHECUKKENTDENSITIESATTHE BOUNDARIESINRUN

2.

(a) The total current density exiting from the magnetospheric boundary, (b) the electron current density exiting from the magnetospheric boundary, (c) the electron current density entering at the ionospheric boundary and (d) the ion current density exiting from the magnetospheric boundary. The current densities are normalized by the reference current density (en,v,,).

only small amplitude electrostatic fluctuations are seen in the averaged potential profiles. When the influxes from the magnetospheric boundary increase, negative space charges tend to accumulate on the magnetospheric side of the simulation domain because the electron influx is greater than the ion influx by the mass ratio. Negative charge excess on the magnetospheric side induces positive charge excess on the ionospheric side. Thecharge separation produces an approximately linear potential profile as seen at T = 150 in Fig. 3(b). Subsequent acceleration of electrons and ions in

and J. R. KAN

opposite directions excites the Buneman instability. After the Buneman instability sets in, the electric field energy is found to increase by more than 10 times. This leads to rising of the potential surge as seen at T = 400 in Fig. 3(c), which finally develops into a double layer. The fully developed double layer with a laminar structure is seen at T = 1100 in Fig. 3(d). Note that the positive current carried by the ionospheric ions are enhanced around T = 1300 [see Fig. l(c) and 2(d)] due to acceleration by the large potential jump. A comparison between Figs. 3 and 4 shows that the potential rises just at the density front of the denser plasma originating from the magnetospheric boundary. The potential jump is also found to move toward the ionospheric boundary at a velocity N OSv,, where u,~ is the ion-acoustic speed defined by (kTo + 3kT,)/M, Such temporal evolution of the current-driven double layers is quite similar to the behavior of the potentialdriven double layers (e.g., Joyce and Hubbard, 1978 ; Hubbard and Joyce, 1979; Singh and Schunk, 1982a, 1983). In the quasi-steady state (T > 700), the net potential difference across the system is concentrated mostly in a single double layer. Figure 3(f) shows that the double layer potential jump settles around 10 kT,. This means that the electrons are substantially accelerated by the double layer in Run 2. Figure 5 shows (a) electron and(b) ion densities in the X-P’ phase space at T = 1500, (c) the velocity distribution (not flux) for the electrons which penetrated the magnetospheric boundary between T = 1450 and 1500, where the positive velocity is toward the magnetospheric boundary. Figure 5(d) shows the temporal variations in the thermal velocity u,,, (upper) and the drift velocity vd (lower) of the electrons at the magnetospheric boundary, where the drift velocity is in the direction of flowing into the simulation domain. The velocities v,,, and v,, are calculated from the electron distribution function at the boundary averaged over 50 CL&‘.As can be seen from the electron phase space, a considerable number of electrons originating from the magnetospheric plasma source are reflected before reaching the double layer. This is consistent with the result in Fig. l(a) which shows that roughly one third of the electrons emitted from the magnetospheric boundary are reflected. It follows that the field-aligned current flowing through the double layers is roughly 30% less than the electron influx at the magnetospheric boundary. These reflected electrons give rise to most of the z) > 0 portion of the electron distribution in Fig. 5(c). This phenomenon can be understood as follows : Before accelerated by the double layer, the electrons supplied by the half-Maxwellian plasma source tend to relax toward a thermal distribution through inter-

Double layer formation due to current injection

0

128

256

0

256

128

128

0

256

(e)

0

128 X

256

T=1500)

1

-20 0

128 X

256

0

600 TIME

1200

PROFILESIN RUN 2 AT (a) T = 50, (b) T = 150, (c) T = 400, (d) T = 1100 AND (e)T= 1500. Temporal variations of the averaged potential difference across the simulation domain are shown in (f).

FIG. 3. AVERAGEDWTENTIAL

action with plasma oscillations excited by the preaccelerated electron beam itself. In Fig. 5(d), an increase in ud and a decrease in uth during T 2 250 are caused by sudden injection of the denser plasma from the magnetospheric boundary. It is found that vd and vthr respectively, approach around T = 250 the drift velocity v, = J2/7tv0,

(iii) Weak double layers In Run 3, the density function n,(T) =

for T < 100

no

n,(T) = [l+

1.75(T-

100)/200]n,

for 100 < T & 300

= O.~V,,, and

the thermal velocity v,,, = J-v,,, = 0.6u,, which are estimated from the half-Maxwellian distribution. However, owing to aforementioned thermalization of the electrons, u,, tends to decrease toward - 0.4v,, in the quasi-steady state and the thermal velocity increases up to -O.~V,,. The velocity ratio v,,/v,,, decreases to approximately 0.4 in the quasi-steady state, after it increases to -0.86 around T = 250. Therefore, it can be concluded that the double layers are formed and stably maintained even when u,, is less than oth for the electron distribution at the magnetospheric boundary. Note that the electron drift velocity presutiably exceeds the electron thermal velocity in the middle of the simulation domain before the formation of the double layer when the potential is linearly distributed as seen at T = 150 in Fig. 3(b). This can lead to the excitation of the Buneman instability which is found to start around T = 200.

n,(T) is given as

n,(T) = 2.75no

for T > 300.

The results of this Run are shown in Fig. 6: (a) The temporal evolution of the averaged potential profiles in the quasi-steady state. (b) The temporal variation in the net potential difference across the simulation domain. The double layer potential is seen to oscillate around - 1.5 kT,. The double layer in this Run is often extended over the whole system length as seen at T = 750. The thickness of the double layer in Run 3 is much greater than that in Run 2, where’the double layer is highly localized. As studied by Yamamoto and Kan (1985), the double layer tends to be extended when the potential energy of the double layer is comparable to the particle kinetic energy. The field-aligned current density averaged over the period between T = 600 and 800 is 0.68 of the reference current density (enouo,). These results indicate that the double-layer potential jumps

858

T. ELECTRON 2.0~

AND ION CHARGE

YAMAMOTO

DENSITIES

T=150

LO-~

- 1.0-w-2.07 0

64

128

192

256

5.0T=400 2.5. 0-J -2.5-5.07

the backscattered and mirrored electrons on the current-driven double layers. In this simulation study, of each three electrons with 101< 2% incident on the ionospheric boundary, two electrons are mirrored and reflected without changing the magnitude of the velocity. Of each three electrons with (VI> 2u,, incident on the ionospheric boundary, one electron is backscattered with 20% reduction in the speed. It is implicitly assumed that most of the accelerated electrons with IuI > 2u,,< are inside the losscone, i.e., precipitate into the ionosphere. In what follows, for simplicity, the backscattered and mirrored electrons shall be called the backscattered electrons. Runs 4 and 5 are performed with the number density function n,(T) given by n,(T) = n0

for T < 100

n,(T) = [1+3S(T-100)/200]n, 0

64

192

128

5.0

256

64

128

192

256

5.0 T=1500

/

2.5. O-\ _*_ JW, -2.5. 0

y\_.

64

128 X

192

256

FIG. 4. TEMPORAL EVOLUTION OF THE AVERAGED DENSITY PROFILES FOR THE IONS (UPPER) AND ELECTRONS (LOWER) IN RUN 2.

tend to decrease density.

with decreasing

field-aligned

for 100 < T < 300 n,(T) = 4.5n,

T=llOOj

-5.0-1 0

-5.o\

and J. R. KAN

current

(iv) Eflects of buckscattered electrons on the currentdriven double layers Some of the precipitating electrons along aurora1 field-lines are known to be backscattered by electronneutral collisions in the ionosphere and wave-particle interactions or reflected by the magnetic mirror (for a review, see Kan, 1982). Here we examine the effects of

for T > 300

For comparison, Run 4 is performed without the backscattered electrons; Run 5 includes the backscattered electrons after T = 200. Figure 7 shows the net potential differences across the simulation domain in Runs 4 and 5. Without the backscattered electrons (in Run 4), the potential jump can grow to be as large as -90 kT,/e and the amplitudes of the potential fluctuations are larger in response to the greater electron influx from the magnetospheric boundary. Note that the density n, ofthe injection current in Run 4 is greater than n, in Run 2. In Run 5, the backscattered electrons are found to suppress the growth of the potential jump, resulting in a more stable double layer by the with e&, - 10 kT,. Double layer stabilization backscattered electrons can be interpreted as follows. As a potential jump grows, the flux of the accelerated electrons at the ionospheric boundary increases. Consequently, more electrons with higher energies are backscattered from the ionospheric boundary. Since the double layer potential is fluctuating, some of the energetic backscattered electrons can penetrate the double layer and reach the magnetospheric side. These electrons can appreciably reduce the electron current density on the magnetospheric side, which suppresses the growth of the potential jump. On the other hand, as the potential jump becomes smaller the electron current on the magnetospheric side is less reduced by the energetic backscattered electrons, which in turn allows the potential jump to grow. Thus, the backscattered electrons play a stabilizing role in the current-driven double layers. In this connection, it may be noted that the backscattered electrons have been

Double

24

ELECTRON

PHASE

due to current

853

injection

ELECTRON DISTRIBUTION T=1500 I(c)

2 o

SPACE

.

T=1500

1 (a) 16

layer formation

[ 1.6

i

Vd = -0.47 Vth = 0.89

1.2 1 -8-.’

0.8

~‘-

0.4

-16-24I

0

64

128

256

192

01 -16

-8

16

1.2 1(d)

0.61 0.8 Vd 0.4 0k 0

6

600 1200 TIME

FIG. 5. PARTICLEDISTRIBUTIONS IN RUN 2. (a) Electron and(b) ion densities in the X-V phase space at T = 1500, (c) the velocity distribution

the electrons

which penetrated

the magnetospheric

boundary

(not flux) for between T = 1450 and 1500, where the positive

velocity is toward the magnetospheric boundary.(d) Temporal variations in the thermal velocity u,,,(upper) and the drift velocity v,(lower) of the electrons at the magnetospheric boundary. The velocities arenormalized by the electron

shown to play a crucial role in determining the fieldaligned scale length of the potential-driven double layers (Yamamoto and Kan, 1984). The above physical interpretation implicitly assumes that the magnitude ofthe potential jump increases with the density of the electron current flowing through the double layers. This is verified in our simulation results as summarized in Fig. 8. Here, the total current density J and the potential difference &, are averaged over the period of the last 200 w&l of each run during which the double layers are in a quasi-steady state.

thermal

velocity

uOr.

Finally, we briefly discuss the critical conditions for the development of the current-driven double layers. From the results of Run 3, the critical condition can be written approximately as J, > 2J,,, - en,,uOe

(2)

where J, is the electron current density flowing through the double layers. Note that the number density no and the thermal velocity uoe in (2) refer to the plasma on the ionospheric side or the ambient plasma before the denser plasma is injected from the magnetospheric

860

T. YAMAMOT~ and J. R. KAN 2

(a) T=700

5 T=750

P x

(b) POTENTIAL

DIFFERENCE

2.5-

I& Q, O5 T=800

2.5-

O, 84

0

(a) Temporal

evolution

I 192

0

258 TIME

FIG. 6. WEAK DOUBLE LAYERS IN RUN 3. of the averaged potential profiles in the quasi-steady state. (b) Temporal the net potential difference across the simulation domain.

Potential

100,

-201

I 128 X

Difference

600 TIME

1200

I

~~j

RL;ytial

Difference

variation

in

~

-‘Ok--z-id TIME

FIG.~.TEMPORALVARIATIONSINTHENETPOTENTIALDIFFERENCESACROSSTHESIM~LATIONDOMAININ (LEFT)AND RUN ~(RIGHT).

RUNS

Double

layer formation

. DATA

861

injection

magnetospheric electrons can be substantially accelerated by the current-driven double layers. The backscattered and mirrored electrons arefound to have stabilizing effects on the current-driven double layers.

050

5

due to current

POINT

l$40f / /

g 3 30(r ii! )$ 20-

Acknowledgements-It

is our pleasure to thank D. W. Swift and L. C. Lee for useful discussions. This work was supported in part by NSF Grants ATM 83-17456 and ATM 83-12515.

i

REFERENCES $ 5

loi:

? i?

/ 0, 0

/ / 0.3 0.6 CURRENT

0.9 1.2 DENSITY

1.5

FIG.~.THE POTENTIAL DIFFERENCE @ PLOTTED AGAINSTTKE TOTAL CURRENT DENSITY J NORMALIZED BY THE REFERENCE CURRENT DENSITY(eta&& J and do are averaged over the period of the last 200 w&’ of

each run. The plotting includes the data of J = 0.88 enDuOeand e&, = 2.7 kT,, which are from the result of an additional computer run (not presented in the text).

boundary. The condition (2) does not require the drift speed u, of the magnetospheric electrons to exceed the thermal speed uDe.In our simulations, (2) is satisfied by increasing the number density n 1of the magnetospheric source plasma above the ambient plasma density n,.

4.CONCLUSIONS

We have demonstrated, by particle simulations, that the double layers are formed when the electron current density J, significantly exceeds the thermal current density Jr,, of the ambient electrons. The potential jumps of the double layers can be much greater than the electronkineticenergywhen(i) thefield-alignedcurrent density is sufficiently higher than the thermal current density Jth of the ambient electrons and (ii) the drift velocity ofthe magnetospheric electrons is significantly less than the thermal velocity uOe. Therefore, the

Borovsky, J. E. and Joyce, G. (1983) Numerically simulated two-dimensional aurora1 double layers. J. geophys. Res. 88, 3116. DeGroot, J. S., Barnes, C., Walstead, A. E. and Buneman, 0. (1977) Localized structures and anamolous dc resistivity. Phys. Rev. L&t. 38, 1283. Goertz,C.K.and Joyce,G.(l975)Numericalsimulation ofthe plasma double layer. Astrophys. Space Sci. 32, 165. Hubbard, R. F. and Joyce, G. (1979) Simulation of aurora1 double layers. J. geophys. Res. 84,4297. Joyce, G. and Hubbard, R. F. (1978) Numerical simulation of plasma double layers. J. Plasma Phys. 20,391. Kan, J. R. (1982) Toward a unified theory of discrete auroras. Space Sci. Rev. 31, 71. Sato, T. and Okuda, H. (1980) Ion-acoustic double layers. Phys. Rev. Lett. 44, 740. Sato, T. and Okuda, H. (1981) Numerical simulations on ion acoustic double layers. J. geophys. Res. 86, 3357. Singh, N. and Schunk, R. W. (1982a) Dynamical features of moving double layers. J. geophys. Res. 87,356l. Singh, N. and Schunk, R. W. (1982b) Current-driven double layers and the aurora1 plasma. Geophys. Rex Lett. 9,1345. Singh, N. and Schunk, R. W. (1983) Comparison of the characteristics of potential drop and current-driven double layers. J. geophys. Res. 88, 10081. Singh, N. and Schunk, R. W. (1984) Plasma response to the injection of an electron beam. Pfasma Phys. and Controlled Fusion 26, 859. Wagner, J. S., Kan, J. R., Akasofu, S.-I., Tajima, T., Leboeuf, J. N. and Dawson, J. M. (1981) A simulation study of Vpotential double layers and aurora1 arc formations, in Physics of Aurora1 Arc Formation (Edited by Akasofu, S.-I. and Kan, J. R.). Geophys. Mono. Series, AGU. Wagner, J. S. and Kan, J. R. (1985) On the field-aligned scale length of the V-shaped aurora1 potential structure. Planet. Space Sci. 33, 89. Yamamoto, T. and Kan, J. R. (1985) The field-aligned scale length of one-dimensional double layer. J. geophys. Res. 90, 1553.