Energy dependent pitch angle distributions of auroral primary electrons

Planet. Space Sci., Vol. 36, No. 2, pp. 117-122, Printed in Great Britain.

ENERGY

1988

CKI324633/88 53.CK1+0.00 Pergamon Press plc

DEPENDENT OF AURORAL

PITCH ANGLE DISTRIBUTIONS PRIMARY ELECTRONS N. SINGH

Department

of Electrical

and Computer

Engineering,

Univeristy

of Alabama,

Huntsville,

AL 35899, U.S.A.

(Received in final form 4 August 1987) Abstract-The acceleration of aurora1 primary electrons by dc parallel electric field is often questioned on the ground that the pitch angle distribution of the primary electrons shows a complex behavior; the relatively low energy electrons are found to be field-aligned while the relatively energetic ones are found to be “isotropized” over all pitch angles directed into the atmosphere. In the central regions of arcs or potential structures the field-aligned and the “isotropized” components are found simultaneously. On the other hand, near the arc edges, where electrons have relatively low energies, the field-aligned distributions prevail. It is shown here that double-layer/parallel-electric field acceleration and the subsequent electronbeam plasma interactions involving Cerenkov and anomalous cyclotron resonances yield pitch angle distributions as observed from rockets and satellites. It is highlighted that the electron acceleration by weak parallel electric fields forming a runaway electron tail is limited to a critical parallel energy determined by the anomalous cyclotron resonance. On the other hand, with acceleration by a localized parallel electric field, like that in a double layer, such a limitation does not occur. The bean-plasma interactions after the acceleration by a strong double layer reshape the electron distribution function. During the early stage of the interaction an extended plateau forms. The electrons with parallel energies above the critical energy undergo anomalous cyclotron resonance and thereby their parallel energy is partly transferred to their perpendicular energy. This causes the plateau to shrink towards the critical energy, and also a pitch angle scattering of the electrons leading towards isotropization. The extent of isotropization depends on the time that the electrons spend in the interaction zone. The electrons having parallel energies less than the critical energy are likely to remain field aligned.

ture such as a V or S shock, which is known to accelerate aurora1 electrons causing arcs (Torbert and Mozer, 1978), the electrons just below the potential drop cannot have the above properties. However, if the electrons travel down after their acceleration from such a structure, it is quite likely that they can acquire the above properties when they undergo beamplasma interactions. The purpose of this paper is to discuss these interactions and to estimate their temporal and spatial scales, and the respective energy ranges over which field-aligned and isotropized electron populations are expected to occur.

1. INTRODUCTION

The acceleration of aurora1 primary electrons by dc parallel electric fields has been questioned by several authors (e.g. see Whalen and Daly, 1979 ; Arnoldy et al., 1985). The objections to the parallel electric field as a mechanism for electron acceleration have been due to the following observed features of the primary electrons. (1) Field-aligned electrons occur at relatively low energies, and at relatively large energies the electrons are isotropied over all pitch angles directed into the atmosphere. In central regions of arcs or aurora1 potential structures these electron populations are simultaneously observed (see Arnoldy et al., 1985). (2) The field-aligned (FA) electrons are observed over a wide range of energies, often extending from the peak in the isotropic population to very low energies (Arnoldy et al., 1985 ; Whalen and Daly, 1979). (3) Electron precipitations tend to be mainly fieldaligned near the arc edges or between active arcs (Lin and Hoffman, 1979 ; Whalen and Daly, 1979 ; Arnoldy et al., 1985).

2. RESONANCES

AND BEAM RELAXATION

The relaxation of runaway electrons in a plasma driven by dc electric fields has been studied by several authors (Kadomtsev and Pogutse, 1968 ; Liu et al., 1977 ; Rowland and Palmadesso, 1983 ; Lin and Rowland, 1985). Haber et al. (1978) and Muschietti et al. (1981) have studied analogous problems, in which they analyze the relaxation of an electron beam in a plasma. In all these studies the basic interactions involve Cerenkov and anomalous-cyclotron resonances. Figure 1 schematically shows the role of these res-

If one considers electron acceleration by a localized parallel potential drop in an aurora1 potential struc117

118

N.

SCHEMATIC:

SINGH

PLASMA PROCESSES IN RESHAPING OF VELOCITY DISTRIBUTION OF AMORAL ELECTRONS

BACKGROUNDPLASMA+ ELECTRONBEAM

CERENKOV INTERACTION

PRECIPITATION

FIG. ~.SCHEMATICDMGRAMSHOWINGTHE

RELAXATION

OFAN ELECTRONBEAM.

At the top electrons of ionospheric and plasmasheet origins being fed into the acceleration region are shown. After acceleration, an electron beam in a background plasma is shown. The beam transforms into a plateau via Cerenkov interaction. The energetic part of the tail undergoes anomalous cyclotron resonance (ACR) and thereby the electron parallel energy is partly transformed into its perpendicular energy. This leads to “isotropization” of the energetic electrons. The relatively low-energy electrons remain unaffected by ACR. In the middle of an arc (or aurora1 potential structure) where the electron energy spectrum extends to high energies, both the field-aligned and “isotropized” electrons are expected as shown in the box marked with central precipitation. On the other hand, near the edges of an arc or a potential structure, where electrons gain comparatively smaller parallel energies, they remain field-aligned (as indicated by edge precipitation) provided the mirror force is not effective on them.

119

Pitch angle distributions of aurora1 primary electrons onances in modifying an electron beam after its acceleration. It is shown here that the electrons in the acceleration region, either of plasma sheet origin or of ionospheric origin, are accelerated by parallel electric fields, which could be extended or localized as supported by double layers. We do not worry here how the cold ionospheric electrons are transported to the aurora1 acceleration region. However, their presence in this region has been established by satellite observations (Temerin et al., 1982; Kintner et al., 1979). The electric field could be supported by the aurora1 potential structures known as electrostatic shocks (Mozer et al., 1980). After the acceleration, the electrons appear as electron beams in a background plasma. However, the electron beams cannot last for long in their original state ; they are quickly transformed into a plateau in the parallel velocity distribution function. The plateau is attached to the background electron distribution at relatively low velocities. The plateau formation occurs because of wave-particle interactions involving Cerenkov resonance (e.g. see Goldstein et al., 1978). The formation of the plateau is a quick process. Numerical simulations of double layers have shown the plateau formation within a distance of about 10 Debye lengths from a double layer (Singh and Thiemann, 1980a, b). Similar observations have been made from laboratory experiments on double layers (Hershkowitz et al., 1981). Starting with an electron beam in a plasma, Muschietti et al. (1981) have shown the plateau formation in a time of about zP < lo3 o&l, where copeis the electron plasma frequency in the background plasma. Taking a nominal density of N, N 10 cmd3, zP < 5 ms. Thus, with the temporal resolutions available for particle measurements aboard satellites it may be difficult to observe electron beams unless the background density in an extended acceleration region is sufficiently low. After the plateau formation in the parallel velocity distribution function the relaxation of the extended tail occurs in a manner analogous to that of an extended runaway electron tail (e.g. see Lin and Rowland, 1985 and references therein), but with an essential difference which is important for the aurora1 electrons ; this difference lies in the magnitude of the parallel energy to which electrons can be accelerated. When weak electric fields accelerate electrons forming runaway tails, the acceleration of electrons stops at a critical energy determined by the anomalous cyclotron resonance as discussed below. On the other hand, when acceleration occurs by a large localized electric field, the electron beams can have energies much greater than the critical energy. Hence, the plateau can extend to energies beyond the critical energy. The

ensuing wave-particle interactions tend to shorten the plateau. The extent to which the plateau shortens depends on the temporal and spatial scales of the wave-particle interactions. Besides the Cerenkov resonance, ion-cyclotron resonances play an important role in wave-particle interactions in magnetized plasmas. The cyclotron resonance condition is given by o-k,,

V,, = nn

(1)

where n is the electron-cyclotron frequency and the integer n can be positive or negative. The parallel cyclotron-resonance velocity V, is given by

v, N

CO-d2 ___

k,, ’

When n = 0, one obtains the Cerenkov resonance, o = k,, V,,. For n > 1, the resonances are called normal cyclotron resonances. On the other hand, when n < - 1, the resonances are called anomalous-cyclotron resonances. It is well known that the Cerenkov resonance drives waves when af/aV,, > 0. The normal cyclotron resonance drives waves when af/av, > 0. Under proper conditions, the anomalous cyclotron resonance can drive waves when neither of the above conditions are met. The dispersion relation and growth rate for the waves are given by E

R

co2 k2 co2k2

El-pellY =

-E,(W,k)i(ae,/aw)l,=,

where eR and &Iare the real and imaginary part of the plasma dielectric function E(k, w), wk is the solution of (3), JJy) is the Bessel function and y = k, VJQ. It can be seen from (3) that as,/& > 0. Thusjnstabilities are possible only when either aflaV,, > 0 or (n aJ/aV,) > 0. The latter condition is met when n > 1 and af/aV, > 0 or when n < - 1 and afpv, <: 0. Thus, the anomalous cyclotron resonance can contribute to instability even if af/aV, < 0 as normally is the case for the precipitating electrons, For Maxwellian perpendicular velocity-distributions, the normal and anomalous cyclotron resonances give rise to damping and growth of the waves, respectively. With the downcoming electrons forming an extended tail to the parallel velocity distribution

N.

120

function, this situation changes as discussed below. If the parallel resonant velocity for the normal cyclotron resonance, V,, can be made sufficiently negative, where there are no electrons, the damping can be eliminated. In that case, the anomalous cyclotron resonance can cause instability because the corresponding resonant velocities V, can fall in the electron-populated plateau region. The above restrictions on the resonant velocities are met if o << R ;

VC”= -Q/k,,

(6)

Ka = w,, .

(7)

From (3) we find that waves with frequencies are excited when CI+,”CCSz ;

o, CCR

The normal cyclotron resonance provides an effective damping for the high frequency waves with w N nR, for which V, ‘v 0, where electrons are heavily populated. The excitation of the low frequency waves via the anomalous resonance has an interesting feature in that it is an effective mechanism for the pitch angle scattering of the resonant electrons. This easily follows from the resonance condition (1). Using quantum mechanical concepts, (1) can be written as (Kadomtsev and Poguste, 1968) ti,, V,, = Pm, - nhR

(8)

where h is Plan&s constant divided by 27~.The term lik,, V is the decrement in the electron parallel energy as a plasmon of energy hw, and momentum tik,, is radiated. In the process of the radiation, a part of the energy goes into the orbital motion of the electrons as given by -nhZ2 when 12is negative. It is interesting to note that when wk CC0, most of the parallel energy loss of the electrons goes into its perpendicular energy. Thus, the anomalous cyclotron resonance acts as an effective mechanism for pitch angle scattering. An alternative explanation for the exchange between parallel and perpendicular energies of the anomalously resonant electrons is as follows. When the electrons interact with the waves, their parallel and perpendicular velocity components obey the relation (Shapiro and Shevchenko, 1968) VT + Vi - 2

F s

du,, = constant

(9)

II

or dV1

w=

-(vc,-wk/k,,)/vL II

=

-vca/vL.

(lo)

SINGH

The latter relation in (10) follows from V, >> w,/k,,. The interesting fact to note from (10) is that as V,, decreases, Vl increases causing the pitch angle scattering.

3. APPLICATION TO AURORAL ELECTRONS

Having reviewed the attributes of the anomalous cyclotron resonance, we now examine its role in the relaxation of aurora1 electron beams. We note that the anomalous cyclotron resonance velocity critically depends on the parallel wavenumber k,,. Since in a plasma the maximum value of k N cd; ‘, where & is the Debye length in the background plasma and c( < 1, the minimum electron parallel velocity above which cyclotron resonance is possible, is given by

(11) In writing this relation we have used & = V&ope when ape and V, are the electron plasma frequency and the thermal velocity in the background plasma. The corresponding minimum parallel energy is

where k, is the Boltzman constant and T, is the background electron temperature. Thus, W,,minfor t( = 1 gives the absolute minimum energy above which the pitch angle scattering by the anomalous cyclotron resonance is possible. It is an absolute minimum in the sense that maximum 1; ‘. Normally the maximum value of k,, is a k,, fraction of 1; ’ and, therefore, pitch angle scattering occurs at energies considerably larger than that given by (12) for LX= 1. The main consequence of the pitch angle scattering is that part of the parallel electron energy is transformed into its perpendicular energy. This leads to the “isotropization” of the electrons having initial parallel energies W,, > W,,,,. As an illustrative example, we assume that the precipitating electrons are accelerated by parallel electric fields at an altitude of h 2: 7000 km, where w = 1068 rad s-‘. If we further assume that the plasma density is 5 cmm3, I+ = 1.28 x lo5 rad s-i and CZ&+ N 8.3 (Lotko and Maggs, 1981). Thus, if the background plasma temperature T, N 10 eV, electrons having energies above about 350 eV are expected to be isotropized. However, such estimates can be highly variable depending on the ratio &I/o,,), factor c( and T,. Since the critical energy strongly depends on a, R, wpe and T, and their parameters vary over a wide range as the electrons travel from the acceleration

121

Pitch angle distributions of aurora1 primary electrons region to the ionosphere, it is important to discuss the temporal and spatial features of the pitch angle scattering, and the consequent “isotropization” of the relatively energetic electrons. First we note that the scattering is effective only when R > wpe, which is possible only at high altitudes. In the topside ionosphere cure >> R,. Thus, scattering by the anomalous cyclotron resonance must be limited to high altitudes. Starting with an electron beam, numerical simulations (Haber et al., 1978; Muschietti et al., 1981) have shown that it takes a time of about At N 2 x lo3 w;* for the pitch angle scattering to isotropize the electrons for W,, > W,,min.Considering the same example as for calculating W,,minabove, At = 16 ms. Over this time period, the accelerated electrons over the typical aurora1 electron energy range lo’--lo4 eV can travel distances from 10 to 1000 km. Thus, it is quite likely that the “isotropization” is completed in the near vicinity of the acceleration region. This is especially true for the relatively slow moving electrons with low energies, but greater than W,,min. We must remark that above estimates for the “isotropization” are rough in the sense that the relevant plasma parameters over the estimated distances can change considerably making the problem nonlocal, in contrast to the simulations which are valid locally. However, nonlocal calculations on the relaxation of the aurora1 electrons have not been carried’out. Figure 1 schematically shows the fate of the electrons after the plateau formation. The relatively lowenergy electrons (W,, < W,,miJ do not undergo the scattering via the anomalous cyclotron resonance and remain field-aligned. On the other hand, the relatively energetic part of the plateau ( W,, > W,,,J undergoes the scattering and the corresponding electron distribution is modified. The electron distribution function shown on the bottom right-hand side of Fig. 1 shows the composite distribution consisting of both the low-energy field-aligned and high-energy isotropized populations. Since in the central region of an arc or a potential structure the parallel energy is likely to extend to several kiloelectronvolts, it is quite possible that precipitating electrons have both the electron populations. On the other hand, near the edges of the arcs or potential structures, where only relatively low energy electrons are possible, only the field-aligned electrons exist. In the above discussion, the effect of the mirror force on the electron velocity distribution function has been ignored. Thus, the results are strictly valid either in the near vicinity of the potential structures where electrons undergo the parallel acceleration or when the perpendicular electron temperature is

sufficiently small. However, we must note that for the relatively energetic electrons undergoing the pitch angle scattering by the anomalous cyclotron resonance, the effect of the mirror force is accentuated as their perpendicular energy increases. 4. CONCLUSION

AND DISCUSSION

The main purpose of this brief report has been to suggest that the relaxation of aurora1 electron beams after their acceleration by parallel electric fields can drastically modify the velocity distribution function of the electrons. The relaxation occurs via Cerenkov and anomalous cyclotron resonances. The former resonance creates a plateau in the electron parallel velocity distribution function extending from the energy of the background plasma to energies beyond the initial beam energy. This may be the main process for creating the field-aligned electrons over a wide range of energies. The latter resonance acts only on the relatively energetic electrons and transforms part of their parallel energies into their perpendicular energies. If plasma conditions on the fields lines are suitable this energy transfer can “isotropize” the relatively energetic electrons with initial energies W,, > W,,min. On the other hand, if the conditions are not suitable, the “isotropization” may not be completed and the entire population may remain field aligned but with a considerable perpendicular heating of the electrons in the plateau. Arnoldy et al. (1985) have reported field-aligned electrons up to 15 keV. The energy spectrum of the electrons show an abrupt cut-off at 11 keV, which is believed to be the parallel acceleration by electric fields. The perpendicular energy distribution was found to be Maxwellian with a temperature of about 2.5 keV. It is likely that the perpendicular energy distribution of the electrons was reshaped by the anomalous cyclotron resonance nullifying any effect of the parallel acceleration on it, but the isotropization process remained uncompleted. It is worth mentioning that in the present paper we have considered the relaxation of an electron beam as it travels from the acceleration region to the topside ionosphere. The mixing of electron populations from different beams is likely to complicate the picture. For example, at times both the field-aligned and isotropized electrons occur simultaneously at the same energy (Arnoldy et al., 1985). Such events are possible when the former electron population permeates an ambient isotropized population (with loss cone), which is locally trapped. Lotko (1986) has formulated a diffusive acceleration model for the FA electrons seen near the edges

122

N. SINGH

of aurorae. However, this model does not account for the simultaneous existence of FA and isotropized electrons in their respective energy ranges. Finally we comment that there is an essential difference between the accelerations of electrons by localized strong electric fields and extended weak electric fields as discussed below. In the latter case the parallel acceleration is limited by the anomalous cyclotron resonance ; the parallel acceleration continues till the runaway electron tail reaches V,,tin. When the electrons are accelerated beyond this velocity, they lose their additional parallel energy to the perpendicular component. This has been shown by the numerical simulations of Rowland and Palmadesso (1983) and Lin and Rowland (1985). Thus the model based on parallel acceleration by weak extended electric fields can produce electrons limited in their parallel energy and it cannot be applicable to field-aligned electrons having energies extending to several tens of kiloelectronvolts (Amoldy et al., 1985). On the other hand, when the parallel acceleration by the electric fields is localized, it is possible to produce field-aligned electrons to several tens of kiloelectronvolts depending on the parallel potential drop. Whether such electrons remain field-aligned or not depends on the temporal and spatial evolutions as they propagate down, along the field lines. It is likely that when the electrons are highly energetic, they move through the magnetospheric plasma so fast that they do not have enough time to be isotropized before they precipitate into the ionosphere. Acknowledgements-The author thanks Dr L. P. Block for bringing this problem to his attention and Dr Torbert for supporting the work under the grant NAGS-639 to the University of Alabama, Huntsville, U.S.A. REFERENCES Arnoldy, R. L., Moore, T. E. and Cahill, L. J. (1985) altitude field-aligned electrons. J. geophys. Res. 90, Goldstein, B., Carr, W., Rosen, B. and Seidl, M. Numerical simulation of the weak beam-plasma action. Physics Fluidr 21, 1569.

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