Naval Research LaboratoryWashington, DC 20375-5000

NRL Memorandum Report 6217

Electrostatic Ion Instabilities in the Presence of'Parallel Currents and Transverse Electric FieldscvV

N G. GANGULI

Science Applications International Corporationa) McLean, VA 22102

P. J. PALMADESSO

Plasmna Physics Division

SLECTESEP 2 ~3 1988K*.-

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NASA - Washington, DC 20546 1(-See pag i) (See p ge ii) (e page ii)ItI TITLE (include Secunity Classification)

Electrostatic Ion Instabilities in the Presence of Paral lel Currents and TransverseElectric Fields12 PERSONAL AUTHOR(S)

GangulI i , G. and PalImadesso, P.,!. 14DT FRPR YaMnhDy AE(~13a TYPE OP REPORT 13b TIME COVERED ' 4DT FRPR Ier ot a) 5PG O;interim FROM _ ___TO!__ 1988 Julv 13 26

16 SUPPLEMENTARY NOTATIONThis work was partially sponsored by NA uinder "Weapons Phenomenology and CodeDeve'Kormn" WokUnt'o & Til,: M 'C01,. P s;

7 COSATI CODES iB SU9 TERMS (Continue on reverse if necessary and identify by block number)FED%, GROP SUB-GROUP Electrostatic Ion Instahil it icS)

Tranisvurse Localized Electric Fields0Ohbliquely Prclpa at ingpElectros t:itic '1cIm Modes . (it I i

'9 ABSTRAQt._font~nue on revere if necessary and identify by block number) Jf .'"

The electrostatic ion instabilities are studied for oblique propagation in the presence of magneticfield-aligned currents and transverse localized electric fields in a weakly collisional plasma. It isshown that the presence of transverse electric fields may result in mode excitation for the magnetic

E field aligned current values that are otherwise stable. The electron collisions enhance the growthwhile ion collisions have a damping effect. These results are discussed in the context of observationsof Io~k frequency ion modes in the auroral ionosphere by radar and rocket experiments.

:0 E)StP Pti'ON AVAIAB t 'Y ()f ABS7RAI' 1 AIIYRAL' SIE P 1R(ASS (ATIIN

Ul 0N.ASFEI ,M1)L SAVE AS opI E1] )TI C SER5 UNCLASS I Fl ED22a NAME OF -ESPi)'.RL NU~'.At 21, 'EF IUI\ (Include Awa' 7,77-7 I I V/i

1 .D. 11111m (2O2 767-3630 Code 4780DID Form 1473, JUN 86 Pre ..... i'd,iiiii erohsoii'te LI F I-A 5 1.IIT','

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4e

CONTENTS

1. INTRODUCTION .................................................................................................... I

2. THEORY .................................................................................................................. 2

3. NUMERICAL RESULTS ............................................................................................. 5

4. DISCUSSION ............................................................................................................ 7

ACKNOW LEDGEMENTS ............................................................................................ 8

REFERENCES .......................................................................................................... 9

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0ELECTROSTATIC ION INSTABILITIES IN THE PRESENCE OFPARALLEL CURRENTS AND TRANSVERSE ELECTRIC FIELDS

I. INTRODUCTION

It is well known that an equilibrium current parallel to the magnetic

field may lead to the excitation of obliquely propagating electrostatic

ion modes in a plasma, such as the ion-cyclotron (EIC) mode in the

collisionless [Drummond and Rosenbluth, 1962; Kindel and Kennel, 19711 and

in collisional limits (Milic, 1972; Chaturvedi and Kay, 19751 and ion

acoustic waves [Chaturvedi et al., 1987]. These instabilities have been

extensively studied in the context of the auroral ionosphere [Kindel and

Kennel, 1971; D'Angelo, 1973; Chaturvedi, 1976; Satyanarayana et al.,

.. . ..... ....s et a. .: C d E' al. 19871. Rada~r

observations [Fejer et al., 1984; Haldoupis et al., 1985; Villain et al.,

1987) and in-situ rocket measurements [Kelley et al., 1975; Ogawa et al.,

1981; Yau et al., 1983; Bering, 1984] have detected these waves in the

auroral ionosphere. It has been shown by Kindel and Kennel 119711 that

the EIC instability has the lowest threshold of excitation among the

various current driven ion instabilities for the topside ionosphere.

Thus, these observations have been interpreted as the modes excited by

field-aligned currents (FAC). Often however, along with the FAC,

transverse d.c. electric fields have also been observed IFejer et al.,

1984; Providakes et al., 1985], and the threshold values for excitation by

FAC alone are likely to be exceeded only during periods of strong

geomagnetic activity [Providakes et al.. 1985; Chaturvedi et al., 1987].

Recently. it has been demcnstrated tha, the presence of transverse

localized electric (TLE) fields can result in the excitation of

electrostatic ion modes [Ganguli et a.. l8'a. !.85b.19881. These modes

have recently been seer, in P:I simula :ons conducted by NIshika.'a e- a:.

1198-]. This nonlocal instability is a result of coupling between the

negative-energy wave in the localized electric field region with the

positive energy wave outside it. These studies were for collisionless

plasmas and largely for k; >> k waves: but the important role of the TLE

fields in a realistic study of the fow-frequency ion-wave excitation in

the auroral ionosphere was demonstrated. In this paper. we study the

combined effects of a TLE field and an FAC on the stability of the ion-

modes. We also include the electron neutral (e ) and the ion neutral

(Vin) collisions. We find that the inclusion of TLE field can have

notable effects on the dispersive characteristics of the ion-modes. For

example, the real frequency of the unstable modes can have values,

Manuscript approved December 14, 1987.

0% %%

depending on parameters such as the TLE field strength, that vary from

below 2 to values greater than Qi' the ion cyclotron frequency. More

Importantly, we find that the mode excitation may occur for sub-threshold

parallel current (electron drift) values when the TLE field is included.

These results may be of interest for the ion wave observations in auroral

ionosphere, especially for those observations where it appears that the

ambient conditions set the threshold value of the parallel current for

excitation of the ion waves above the level that one might expect to be

attainable under normal circumstances. A combination of A raralle!

uz IInt ar.d a TL_ f ,d ca!. lead to exc ?' ;r, of thc io) ver undE

much less severe conditions.

2. THEORY

We wish to study the stability properties of a weakly ionized

collisional plasma which is typical of the auroral ionosphere,

characterized by an equilibrium current flowing parallel to the ambient

magnetic field and a localized transverse d.c. electric field (see figure

I). We describe the particle dynamics by using the Vlasov equation with a

BGK-type of model relaxation collision term. The equilibrium current is

predominantly carried by thermal electrons drifting along the magnetic

field, B0z, with a drift velocity, Vdz. The transverse d.c. electric

field, E(x), is localized along the x-axis to a distance L and is~Vpiecewise continuous. Thus, the ions and electrons both have an

equilibrium cross-field drift (Vo'. V, = -cE, B-,. which is also localized

in the x-direction to the distance Lv. In reality. however, the profile

0 4V 0 is no longer of the square hat shape an' is rather smooth due to the

finite-Larmor-radius (FLR) effects at the boundaries of the electric field

region. For simplicity we shall use a sharp profile for V0 in the present

study, and shall subsequently relax this condition in the future. A

similar approach was taken in developing the collisionless TLE only theory

[Ganguli et al. 1985a, 1985b, 19881, where it was found that the

transition from the idealized square hat profile to a more self consistent

smooth profile resulted in quantitative but not qualitative changes in the

theoretical predictions. The effects associated with the Hall and the

Pederson mobilities are small for weak collisions, especially when the

velocity shear is weak; and are therefore neglected. Our treatment is

general in as much as it includes collisions and thus is appropriate for

2

.. a -,.-.'.. . . ---. -- .'-. , .-4,.- - . e . - . ..%, , ./. , ...,,.: .'- ....,. < "." 5r -i4, . :,N6,.

low-altitude applications; and, at the same time, the collisionless

results (applicable at higher altitudes) can be readily obtained by

setting the collision frequencies to zero.

The dispersion properties of the collisional ion cyclotron modes in

the local limit and in the absence of a d.c. electric field are described

by the relation

I + E. C e = 0 (1)1 e

where expression for rj(j e.i) is given as jClerr,-'." arid Dougherty, ]q 6 9;

Kindel and Kennel, 19711

C. = [ /k x)]l + /k v )Z[(Wj - nQ.)/kiivtj} (2)

1+ i (vj/kivltj Z r (wj)Z + nQj )/k vtj]~n -- -" cot

The notations used here are standard: n refers to the harmonic

number; w. = - iv. where w is the complex eigenfrequency, w = w r iy;

,D j = (Ti/41R n .e2) I$ 2 is the Debye length of the species; and T.j, n. and

m. are respectively temperature (in energy units), number density and mass3 IJ/

of the species j; v. = (2T./m.) is the species thermal velocity; v.J J J J

their collision frequency with neutrals; = v./9., the gyroradius; r =j ] n

I nexp (-b) where n is the rmodified Bessel function of order n; b =22,,, -s i n tcn (o k 9 6k: oj Z: and Z(C) is the plasma dispersion function (Book. 1q86].

We now introduce a localized transverse d.c. electric field (see

figure I). This modifies the dispersion relation. Reg.on I (:x <L,/2).

over whicn the electric fheld :s localized, is characterized by EO=Eox;0XV0.=VoY; W' = 4I 1 1 e = ,A -' kzd .... In reg:on iI we have E - ;

V=0; = W ' e = W +'e - k zVd Here A) w - kyVO

We shall assume that the density gradient is negligible. Strictly

speaking, the parallel current is also of finite width in the x-direction.

The typical scaleleigth of this width. L is assumed to be either L > L

or L Lv . We discuss implications of these inequalities in thec v

discussion section. For a more rigorous description of the equilibrium we

refer to Ganguli et al. (1988).

* p.

We follow the approach adopted by Ganguli et al. 11985a1 in deriving

the nonlocal dispersion relation. We use the quasi-neutrality assumption

(Ee + ti - 0). As a result of the x-dependence of the equilibrium the

perturbed quantities cannot be Fourier transformed in x. Instead we

obtain a differential equation in x. For the specific profile of the

electric field under consideration (see figure 1), the modes are described

by

(I', *1() -- 0 , for Region 1 (4)

and

KII *ii( ) = 0 , for Region II (5)

2where - x/pi and K I = 0 /AI and

* n i%;= C /k + E n n (/ktv i Z l'i '6

n kii )(6

D k ,', i'k ,v

A nT , , ,,'vi

-Ir

+ 'r D~ r, i ,. Ii (7)n i ji i, kl i{ t /C = i ir /k-,v 1Z /k Ve D = )'I (W_ /k, ve 'WO k,• v

i e ,' e ) I e ,

,,--• only the n =0 harmonic for the electrons< is considered and r" a =r /ab. i

I 'II

~4

{ , V . %"w - o I, k I Ile - -- - .- - - --- -

'0 OF,\ e ~ ~ ) . I e

In the above, i = Ti/Te, and KII is obtained by setting V0 - 0 in K.

The nonlocal dispersion relation for the even modes obtained by matching

the logarithmic derivatives of the solutions of (4) and (5) at xL v/2 (for

details see Ganguli et al. 11985a1) is given by,

KI tan (KI/2E) = i K (9)

where E = p/L . A similar relation can also be obtained for the odd

modes.

The well-knovn limits of (9) are straightforward to obtain. In the

local approximation, and for L0 = 0, we recovei the cuirent driven E1C

instability in the collisionless (vj=O) and the collisional (viO) * limits.

Further, for \j = 0 and Vd 0, (9) reduces to the dispersion relation

given by Ganguli et al. [1985a] to yield an electrostatic ion-instability.

Here we study the combined effects of the two aforementioned

processes, i.e.,the simultaneous presence of the FAC and the TLE field.

There is evidence that except for periods of moderate geomagnetic

activity, the observed values of the field-aligned currents fall in the

sub-threshold domain for the collisional EIC and IA instabilities [Fejer

et al., 1984; Chaturvedi et al., 19871. Also, there are several

suggestions that the observations indicate the presence of transverse

electric fields in the auroral regions [Fejer et al., 1984: Basu et al.,

1984; 1985; Prikryl et al., 1986]. We therefore investigate the role of

the TLE field on the current driven ion modes.

3. NUMERICAL RESULTS

we proceed to evaluate the nonlocal dispersion relation (9). e

consider ion and e'ectron temperatures tc be equal (T 1). Other typical

parameters used are e = Ve /2 12 . / - 0. D3, E = 0.1 and u

k k= 0.15. We study an 0' ion-plasma system i.e., w = mn/i 29392.I i e

Figure (2) shows the real and the imaginary parts of the frequency

(w and y) as a function of b(= k 2 for V -V /V -( and Vd /v30.r CYO dd -

The mode is stable ier the k, domain of interest. The real frequency

shows the expected weak-dependence on b. This is in agreement with the

previous studies which suggest a higher threshold value of Vd for EIC wave

eyritation (Satyanarayana et ai. [19S] Providakes et a , :1985]).

5.0 N 1

_N 11%

Next we consider V - 0 and let V0= -2.8, v 0, v - 0. Fromd 0' e ifigure (3) we see that the real frequency of the mode has a roughly linear

dependence on b. We find that the mode is stable for the b-domain which

we have scanned and which is of interest to us. Thus, for d = 30, =-

2.8, u = 0.15, Ve = 12, v 0.033, the nonlocal electrostatic instability

discussed by Ganguli et al. 119851 and the collisional current driven EIC

instability are both stable individually.

Now we combine the two drifts (V and 90) for the values given aboved fo0h aus ie bvand solve (9) numerically. The results (w and - plotted vs. b) are shown

in figure 4. A comparison with figs. (2) - (3) reveals that the mode

frequency follows the approximate pattern of the nonlocal electrostatic

instability (fig. 3) but the growth rate is positive. Thus, we find that

in presence of a parallel electron drift (Vd), the nonlocal electrostatic

instability exhibits growth, even in the presence of collisions. Based

upon this result, we suggest that in the collisional low-altitude auroral

ionosphere, the observed low frequency plasma waves may have been excited

due to the simultaneous presence of both the transverse electric field and

the parallel current. A similar conclusion for the collisionless (high-

altitude) case can also be drawn, indicating that excitation of the ion-

modes jointly by the parallel current and the transverse electric field is

p~ssible for sub-threshold electron current.

We find that the effects of finite channel-width (L ) on the nonlocal

ion instability are relatively small if L > L. In figs. (3) - (4), wec -

corc r'.i* we plot y vs. b in fig. '. for the L.7

case L = L. (dashed curve). We see that the growth rate in this case is

of the same order as before fT > L ) though slightly larger.

)ualitatively the reason for the :ela'ivelv smaller influence of finite L "

on the nonlocal ion instab:lt'y in cases when L > L is that the wavec -V

packet (localized around the electric field) samples the region of near %

peak current in space.

The value of the 1901 = 2.8 used in the numerical computations above

is larger than the typical observed value of iv,- 0.5. In figure (5) wenow consider 0 - -0.5 and V - 30 and 25 with u = 0.15 and 0.17.

0- d-respectively. Thc rest of the parameters remain unchanged from the figlire

(4). Now we see a rather coherent electrostatic wave growth around the

ion cyclotron frequency for sub-critical values of Vd.

6

Ap-'P

4. DISCUSSION

We have shown in this article that the simultaneous existence of a

localized transverse electric field and a parallel current can excite the

oblique ion modes even though the electric field and the current values

are separately stable. The parameters we considered are typical to the

low-altitude auroral ionosphere (the upper E-region). In this region

various experiments have indicated observations of EIC or EIC-like ion-

modes. These observations are frequently attributed to a field-aligned

current-driven EIC generation, but often the current is below the

thresh'> valuE reqjuited for exit->EEt. e.g., Fjer et al. , 198-

And the frequency of the modes may also display variations; it may be

equal to the ion-cyclotron frequency or sometimes different from it

[Prikryl et al., 19871. In absence of detailed data on the ambient

parameters, it is not possible to obtain close comparisons with the

observations. However, for the parameters we have considered, it appearsthat the excitation of the transverse electrostatic modes by a combination

of a TLE field and a parallel FAC can be a possibility. The parallel

currents (corresponding to tens of uA/m 2 , i.e., Vd - 30) and the

transverse electric fields (corresponding to 100 mv/m, i.e., -9o1 - 0.5)

correspond to the typical values that are believed to have been observed

in the regions of auroral wave activity. Further, the range of unstable

mode frequencies varies from belov 9. to values greater than Q. and may1 1

have relevance to the auroral situations where the observed wave

frequenc.es sometimes are different from the ion-:yclotron frequency.

The mechanism of mode excitation in the combined presence of V and0

V is of interes" to o -he s)uatons as e!l. For example. a, higher

auroral altitudes. the obser.'ations of EIC-like modes (and lower hybrid

(LH)-like modes) have been made where the TI.E fields may have been non-

neglig.ble [Kintner et al., 1979]. We are presently conducting a

parameter study for this case. In addition, the profile of electric field

chosen by us in this paper is an idealization. Preliminary investigations

with a smooth profile suggest that our results are not severely modified.

Detailed result- will be provided in a forthcoming publication.

In conclusion, we ha-e shovn that the sub-threshold parallel currentsP

and stable transverse electric fields may combine to result in the growth

of electrostatic ion modes in a collisional plasma. We have applied our

results to the lower altitude auroral ionosphere and find that reasonable

4 7

.5V % -

values of parallel currents and transverse fields may result in the ion

mode excitation under relatively less severe conditions of geomagnetic

activity. Detailed results including the collisionless case applicable to

higher altitudes will be presented in a future publication.

Acknowledgments

Numerous discussions vith Dr. Pradeep Chaturvedi are gratefully

acknowledged. This work was supported by NASA, ONP and DNA.

A.

".%

8-==-

% 8PL0

References

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09

d % . % V~ % ~ % .

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Cattell, P.F. Mizera, and J.F. Fennel, Simultaneous observations of

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Nishikawa, K.I., G. Ganguli, and P. Palmadesso, Simulation of

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Ossako.. Theory of the current-driven ion-cyclotron instability in

the bortomside ioncsphere. J. Geophys. Res. . 90, 12209. 1985.

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acceleration events. J. Geophys. Res.. 88. 341, 1983.

10

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