Role of Coulomb collisions in the equatorial F-region plasma instabilities

0032 0633/X4 $3 00 + 0.00 Peryamn Press Ltd

Phw1p1S,me SC;, Vol. 32, No. 5, pp. 635 640, 1984 Printed in Great Britam.

ROLE

OF COULOMB F-REGION

Indian

COLLISIONS IN THE EQUATORIAL PLASMA INSTABILITIES

A. BHATTACHARYYA Institute of Geomagnetism,

and G. S. LAKHINA Colaba, Bombay-400

005, India

(Received 22 September 1983) Abstract-The effect of collisions on electrostatic instabilities driven by gravity and density gradients perpendicular to the ambient magnetic field is studied. Electron collisions tend to stabilize the short wavelength (k,p, >> 1, where k, is the perpendicular wavenumber of the instability and pi is the ion Larmor radius) kinetic interchange mode. In the presence of weak ion-ion collisions, this mode gets converted into an unmagnetized ion interchange mode which has maximum growth rate one order smaller than that of the collisionless mode. On the other hand, electron collisions can excite a long wavelength resistive interchange mode in a wide wavenumber regime (10m3 < k,p,2 0.3) with growth rates comparable to that of the collisional Rayleigh-Taylor mode. The results may be relevant to some of the spread F irregularities.

1. INTRODUCTION

The phenomenon of equatorial spread F has been extensively studied in recent years through ionosonde, radar back-scatter, scintillations and in situ experiments which have been reviewed by Fejer and Kelley (1980).A number of theories based on various plasma instabilities of wavelength varying from hundreds of kilometers down to tens of centimeters, have been put forward as the mechanism for spread F (Haerendel, 1974; Hudson and Kennel, 1975; Costa and Kelley, 1978a, b; Huba and Ossakow, 1979a, 1981). The night-time equatorial F-region has a steep density gradient at the bottomside due to recombination effects and electrodynamic forces. With gravity acting in a direction opposite to Vn, the collisional Rayleigh-Taylor instability can grow in this region when ion-neutral collisions are considered. According to Haerendel’s “hierarchy” model this instability sets up density variations which give rise to other instabilities and at one stage of the hierarchy, the density gradient scale lengths are small enough that ion-neutral collisions cease to be important and the collisionless interchange instability takes over. These instabilities have wavelengths 2 100 m in a direction perpendicular to the magnetic field. For wavelengths of a few tens of meters or less, drift instabilities have been invoked, which can grow on the density gradients created by the long wavelength instabilities (Costa and Kelley, 1978a, b ; Huba and Ossakow, 1979a). Recently, GaryandThomsen(l982) havestudied thecollisionless interchange instability for short perpendicular wavelengths satisfying k,p, >> 1, where k, is the perpendicular wavenumber of the instability and pi is the ion Larmor radius. This instability treats the ions as magnetized and grows on much weaker density gradients than 635

those required by the lower hybrid drift instability which also operates in the same wavelength regime. Such a short wavelength kinetic interchange mode has earlier been discussed by Mikhailovskii (1967). In this paper, we have studied the effects of electron collisions (with ions and neutrals) on the interchange (Rayleigh-Taylor)modes. We find that the transversely propagating short wavelength interchange mode of Gary and Thomsen (1982), hereafter referred to as GT, can be stabilized by electron collisions in the F-region of the ionosphere. However, electron collisions have a de-stabilizing effect on the obliquely propagating short-wavelength interchange mode. For long wavelength modes having k,p, 5 1, we find that electron collisions can excite a resistive interchange mode instability. Similar modes have been predicted in tokamak plasma having magnetic shear as an intrinsic feature (Coppi and Rosenbluth. 1966; Kadomtsev and Pogutse, 1970). However application of their results to the ionosphere is not feasible because firstly, there is no feature equivalent to magnetic shear, and secondly, geometry of the problem is entirely different. In Section 2 of this paper, we derive the general dispersion relation for electrostatic waves in a magnetized plasma in the presence of a density gradient and gravity. In Section 3 we study the short wavelength modes, whereas in Section 4, the long wavelength resistive interchange mode is investigated and a simplified mechanism for its excitation is given. In Section 5, we discuss the application of our results to the F-region of the equatorial ionosphere. 2. DISPERSION

RELATION

We consider a low /? ([I being the ratio of plasma pressure to magnetic field pressure), two component

636

A.

BHATTACHARYYA

plasma consisting of electrons and ions placed in a uniform magnetic field B = B,i. A density gradient is assumed to be present in the x-direction (vertical direction) with scale length L = n(dn/dx)-’ much greater than the wavelength of the perturbation so that the local approximation can be used. The electrons and ions have uniform temperatures T, and x respectively, the associated thermal speeds being given by ‘/T, = (Tjlmj)“’ (j = e, i). The electrons and ions also have diamagnetic drift velocities V,, = (V$,/Lfi,)g where Qj = (qj B,/mj) are the cyclotron frequencies. The gravitational force acts along the negative x-axis giving rise to an acceleration g = -g?, then each species has a gravitational drift velocity Vgj = (g/Q)j? We consider electrostatic waves with the fields fluctuating as exp [i(k,y + k,z - wt)] with a complex frequency w = w, + iy. Assuming both species to be magnetized, a zeroth order time independent distribution function for each species is given by

and G. S.

where kj = (4nnjoqj/Tj)“” is the Debye wave number, I,(Ij) is the modified Bessel function of order m with the argument lj = kzp;, pi = VTj/fij is the Larmor radius and Z is the plasma dispersion function with the argument cy = (o~+iv~-k,~~-rnR~)/fik~V~,. Also for the ions, vi = vinr the ion-neutral collision frequency. While writing equation (2) terms of order e,/k have been dropped in accordance with the local approximation. For electrostatic waves the linear dispersion relation is l+Xe+Xi

+?)I

(1 -ri)

exp[-OJ

(3)

1

k, f4if-i + (O - k, Vgi+ ivin)

ivi,Ti

(1)

where a, = l/L. We include ion-neutral collisions in our formalism, using the BGK model to represent them. For electrons we include electron-neutral as well as electron--ion collisions in an effective collision frequency v, = vei + ven. Since the effective electronelectron collision frequency is negligible as compared to vei unless kp, >> 1 (Huba and Ossakow, 198 1) we have not included it in the effective collision frequency v, Whereas the BGK model is inadequate for describing electron-electron collisions, its use is justified for describing electron-neutral and electron-ion collisions in the absence of a temperature gradient provided v, >> k,l/,# (Rukhadze and Silin, 1969; Huba and Ossakow, 1979b). The susceptibility xj (j = e, i) of each species, defined by xj+r = -4zqjnj,/k2 where +I and nj, are the perturbations in the electrostatic potential and density respectively, can be derived by the usual procedure of integration of the pertinent Boltzmann equation with BGK collision operator over unperturbed orbits. This yields :

= 0.

For w < Q, only the m = 0 terms need be retained in equation (2). We consider ions to be non-resonant with the modes under consideration, i.e. IU - k, si + iv,1 >>k,l/,<. Then using the large argument expansion for Z(I$) we obtain from equation (2)

njo .fOjtxTv) = (2Rp7+jj3,2

[I+e,(x

LAKHINA

’ -(w-k,i$+ivi,)

I

(4)

where Tj = e-“3 I,(I,).

3. SHORT WAVELENGTH

INSTABILITIES

We shall consider two cases, namely, the case where ion collisions are neglected altogether so that the ions are magnetized and the situation where ions become unmagnetized on account of ion-ion collisions. a. Kinetic interchange mode We consider the ion-ion collision frequency vii = 0 and vin = 0 so that the ions behave as magnetized. We consider the limit k, = 0 and Ik,V,,f <
For ions we consider

kp, >> 1 and use the asymptotic

+

(2)

631

Resistive interchange mode value of Ii in equation Xi=${l

(4) to obtain

+&[-I +&I}-

Now with k2 <
relation

t6) given by

@:;,a 1

’ +;kpi-’ +

7; [( 1 - I,) (w + iv,) + k f&r,] +yp [w + iv,( 1 - r,)] P

= 0.

(7)

If we assume w,? >> y2, vf, the real part of equation yields :

(7)

w, = k G,r,/[ I+ 7J 1- T,)/T,l

b. Unmagnetized ion interchange mode At wavelengthssatisfying(v,&)k~p~ > 1 even when vii <
Vqi- k, Vdi Vi-,

w + ivin- k,V$ - k, Vdi

(8) w +

iv,, - k, Vqi- k, V,,

which is identical to that of GT. From the imaginary part ofequation (7) we obtain the following solution for 1’(when w, N k I$) : y = [-B+(B2+4C)1’2]/2

(9)

where B = v,(l

-r,)(i + 7yqTel/,i)

(10)

and C = k2V;;/[&kpiTJ.

(11)

For v, = 0, we get y = & which is the growth rate obtained by GT. For finite values of v,, the growth rate is reduced from that obtained by GT. In fact, for BZ >>4C equation (9) gives y = ($%or

K

YGT

=

8.

(13) Since we are concerned with a short wavelength instability here with kp, >> 1, the condition vi,& << kpi is easily satisfied for the values of vi& encountered in the ionospheric F-region (vi& 5 lo-‘). Consequently, we have vin << klr,.,. Since pi << L, we have I& << V,;. Also for the density gradients considered, I& > I$. Thus the small argument limit for the plasma dispersion function Z can be used in equation (13). For electrons, we shall assume that v, >> k,Vr,, o so that we can use an asymptotic expansion for Z in the large argument limit 1 Z(<,o) 2 - + - __ e 2(5Y . ‘.

(12)

For the F-region ofthe ionosphere, we have T, = T and 11/9,1/VT, z 10m4. Then for a density gradient such that 1VJ/VT< = 0.01 as considered by GT and for kpi in the range 10-30, the inequality B2 >>4C requires that v, >>(0.6-2) x 10m2 R, which is easily satisfied in this region of the ionosphere where v, > Ri. The kinetic interchange mode is stabilized by electron collisions through the transverse diffusion of electrons which short-circuits the perturbation electric field. A simplified explanation of this process is as follows: since the transverse diffusion coefficient for electrons is given by PAVEfor v, << Sz,, the time taken by electrons to traverse a distance of half the transverse wavelength is proportional to (k,pJ2v;‘. This time period becomes comparable to the growth time of the collisionless kinetic interchange instability since this mode has short wavelength (k,p, 2 0.1). Hence the effect of electron collisions on the kinetic interchange mode is strongly stabilizing.

(14)

Also we consider k:p,Z <
ib]/[oS + ib]

(15)

where rU = (w- k,V,,)

and

b = kf V$,lv,.

The resulting dispersion relation assumes the form of a quadratic equation for w. A solution of this equation yields cl, z k, Ei and a growth rate y given by

Since v,,/kV,.,
A.

638

BHATTA~HARYYA

be driven by a field-aligned current in the presence of a density gradient (Bhattacharyya and Lakhina, 1982) with the current due to gravitational drift playing the role of a field-aligned current, as can be seen by comparing the expressions for the growth rate in the two cases.

4. RESISTIVE INTERCHANGE

From equations

MODES

(4) and (15), with the condition that much longer wavelength, the dispersion relation

kZ << k; which is satisfied for wavelengths

than the Debye becomes

T, T

1

kybiri a +(w--k,l$+iv,,) ivJi ’ -(co--k,,l$+ivJ

1

+ (k, v,,+ib) (W+ ib)

= 0,

(17)

where a = 1 - Ti. For k, = 0 and k,p, -+ 0, equation (17) gives the growth rate

+vi”+(v~+$)llz]

(18)

which is identical to the growth rate for the collisional Rayleigh-Taylor instability (Haerendel, 1974 ; Hudson and Kennel, 1975). For vin = 0, equation (18) gives the well known interchange instability growth rate in the MHD limit (Rosenbluth et al., 1962). We emphasize that this mode is stabilized for finite k, when k,/k, ’ 1 II2 because the charge separation > (meClenlmiRi perpendicular to the field lines that drives the instability is short-circuited by the large electron conductivity in the parallel direction (Hasegawa, 1975). On the other hand when k,p, is linite but small and k, is large enough to satisfy ak,l&+ V,,j cc b and k,l V,,l K w we get the following solution from equation (17):

This is a resistive interchange mode and it arises purely on account of electron collisions. The inequality ktp, cc k;lk: << Q:/v,z, arising from our assumption

and G. S.

LAKHINA

k,ZV”,,/vz, is crucial for the existence of this mode. In fact, this inequality provides a clue to the nature of the instability and the role of electron collisions in it. We know that in the weaklycollisional limit,i.e.(Q/v,) >> 1, which holds in the F-region of the ionosphere, the ratio of the diffusion coefficients for diffusion perpendicular and parallel to the magnetic field is (v,/QJ2. Hence the ratio of the time taken by electrons to diffuse through half a wavelength in the parallel direction to the time taken to diffuse through half a wavelength in the Ydirection is (v,/t2J2k:/k~. By virtue of the above inequality, this ratio is much smaller than unity. Consequently, perpendicular diffusion, which could otherwise lead to collisional damping as the electrons would short-circuit the perturbation electric field by moving in the Y-direction, is negligible compared to parallel diffusion. In the case of parallel diffusion, electron collisions provide resistance to the motion of electrons and it is this resistance which maintains the perturbation electric field provided the electrons undergo many collisions in a distance of half the parallel wavelength, which requires v, >> k, VT.. Further as k, increases, the parallel wavelength decreases until it becomes comparable to the electron mean free path (k,VTe _ v,). Then the electrons can move easily along the field lines to short-circuit the perturbation electric field and stabilize this mode. Hence it is expected that the growth rate would show an increase with increasing v, and decreasing k,. The lower limit on k, is set by the inequality k,,/k, cc Q/v,. When k, falls below this limit, the parallel wavelength becomes so large compared to the perpendicular wavelength that perpendicular diffusion begins to gain importance. At this stage, the role ofelectron collisions changes entirely because the collisions would facilitate perpendicular diffusion thereby bringing in collisional damping. We have also solved equation (17) numerically for certain values of the parameters which are relevant to the night-time equatorial F-region. The dependence of the growth rate and real part ofthe frequency on k,p, for different values of L is illustrated in Fig. 1. The curves 2, 3 and 4 of this figure start at higher values of k,p, as compared to curve 1 because the local approximation breaks down for k,p, < 5 x 10m3 for curve 2, k,p, < 10m2 for curve 3, and k,p, < 2 x 10m2 for curve 4. It isclear from this figure that thegrowth rate increases and the range of unstable wave numbers decreases as the density gradient becomes steeper (i.e. L decreases) as given by the analytical solution (equation (19)). For large values of L (L > 3 km) the growth rate is insensitive to an increase in k,p, until k,p, approaches 0.1 where the growth rate falls sharply due to finite Larmor radius stabilization and ion-neutral collisions.

Resistive interchange

639

mode

0.03

0 06

0.02 t

:”

I% g 0.01

-lo

0 16'

16'

100

kyf’----t

FIG. 1. VARIATIONOFGROWTHKATEY(SOLIDCURVES,-)ANDA, = (w,- k,l/,J WHERECU,ISTHEREALFREQLJENCY (DASHEDCURVES,---)WITH k,p, FORTHEPARAMETERSQ = 2OOs-', k,/k, = 5x 10m4,v, = 800~‘,v, = 0.3 Sal AND vgi= -lO~"v,C.

Forcurves 1,2,3and4,L = 14,4.2,1.4and0.84km respectively.ValuesofA,areshownonlyfor L = 0.84 km, and they follow the scale given in the right hand side of this diagram

The dependence of the growth rate and real frequency on v, is shown in Fig. 2 for different values of the ratio kJk,. Initially the growth rate increases with v,and then tends to saturate for large values of v,. Departure from linear behaviour for large v, occurs when the inequality ak,l Vgi+ V&l -K b which amounts to

breaks down and the analytical solution given by equation (19) is not valid any more. The same situation occurs for curves 2,3 and 4 of Fig. 1, which do not show a linear dependence on Vdi beyond k,pi 2 0.05 as predicted by equation (19).

L = 14kmand as also in Fig. 2.

5. DISCUSSION

Our analysis shows that when ion-ion collisions are absent, the kinetic interchange mode of GT is strongly damped by electron collisions. In the presence of weak ion-ion collisions, on the other hand, their mode is converted into an unmagnetized ion kinetic interchange mode. For 1V,,l = 0.01 V,.>,Vgi= 0.5Vdi and k,p, = 30.0 which are the values considered by Gary and Thomsen (1982), we find that the maximum growth rate for the unmagnetized ion kinetic interchange mode, for k, = 0, to be of the order of 10 -3 Ri which is smaller by a factor of 10 than that of the kinetic interchange mode of

0 0: 1

O-08

--__--__--_ 1-. -. --__ /

o-0;

-

0.04

--__3

NW

t

2

-.__ --__

‘” $001

‘i

-P

7

t 0

-7

*YJ

2

3 -004

0

/, .o 08

-0

006

0

500

1000

ye beE’) FIG. 2. VARIATION OF GROWTH RATE (-) AND A,(- -) WITH ELECTRON COLLISION FREQUENCY v, FOR THE PARAMETERSQ = 200~~', k,p, = 0.1, Vdi = -3 x 10m3 VT,(L = 1.4km), V’! = -lo-“ VT, AND yin= 0.3 s-l.

For the curves 1,2 and 3, k,/k,

= 10M4, 5 x 10e4 and 10e3 respectively.

640

A. BH.~TTACHAKYYA

GT at that wavelength. An ion-ion collision frequency vii = lo-’ C&would be sufficient to convert the CT mode into the unmagnetized ion mode. In the equatorial F-region, an ion diamagnetic drift velocity II&] = 0.01 I& corresponds to a density gradient scale 1engthL = 420mfora lOOOK,O+ plasmain a0.3 gauss magnetic field. For this value of L, the ratio Vgi/Vdi is nearly 0.01, and the growth rate, according to equation (16) is about 6 x lo- 3s- 1for k,,p, = 30. Elowever, these short wavelength irregularities, which would be present along with the long wavelength Rayleigh-Taylor instability, would not be observed in back-scatter experiments because the wave vector associated with them is essentially horizontal on account of the geometry of the problem. They could be detected by rocket experiments. We also find that the presenceofcollisions can lead to a long wavelength resistive interchange mode. The unstable wavenumber regime extends from k,p, = 10m3 to k,p, x 0.3 for the density gradient scale lengths present in the night time equatorial F-region as seen from Fig. 1. For typical values of k,/k, = 5 x 10m4, v, = 800 s-l and vi,, = 0.3 s-i the maximum growth rates for L = 14,4.2,1.4 and 0.84 km, according to Fig. 1, are respectively 0.0019,0.0059,0.018 and 0.029 s-l. The corresponding growth rates for the collisional Rayleigh-Taylor mode given by equation (18) are 0.002, 0.0075, 0.022 and 0.035 s-* respectively. This shows that the growth rates for the resistive interchange mode are comparable to those of the collisional Rayleigh-Taylor instability. For smaller values of kdk,,, growth rates can even exceed the collisional Rayleigh-Taylor mode growth rates [cf. Fig. 2 and equation (1911but for these values of k,/k,, the parallel wavelength & = 2nJk, may become comparable or greater than the length of the field line itself. For instance, for curves 1, 2 and 3 of Fig. 2, the corresponding parallel wavelengths are ii, = 2500,500 and 250 km. Thus curve 1 of Fig. 2 does not correspond to a realistic situation as the parallel wavelength exceeds the length of the field line within the ionosphere. Exclusion of recombination from our formal-

and G. S.

LAKHINA

ism is justified by the fact that the recombination rates found in the equatorial F-region, typically 6 10e3 s- ’ are much smaller than the growth rates obtained for the resistive interchange mode. It is interesting to note that the resistive interchange mode can exist for parallel wavenumbers for which the usual RayieighTaylor mode is stabilized. For instance, for the parameters of the night-time equatorial F-region, and a density gradient scale length of 14 km, the RayleighTaylor

mode

is stabilized

for

k,/k, > 1.6 x IO-(’

whereas the resistive interchange mode exists for much larger values of k,/k, with appreciable growth rate as seen from curve 1 of Fig. 1. This is expected to lead to faster dissipation of the irregularities and may affect the saturated irregularity spectrum.

REFERENCES

Bhattacharyya, A. and Lakhina, G. S. (1982) Planet. Space Sci. 30,581. Coppi, B. and Rosenbluth, M. N. (1966) Proceedings Conference on Plasma Physics and Controlled Nuclear Fusion Research CU~~U~I, Vol. I. IAEA, Vienna. Costa, E. and Kelley, M. C. (1978a) J. geophys. Rex 83,4359. Costa, E. and Kelley, M. C. (1978b). J. geophys. Res. 83,436s. Dougherty, J. P. (1964) Phys. Fluids. 7, 1788. Fejer, B. G. and Kelley, M. C. (1980)Rev. Geophys. Space Phys. 18,401.

Gary,S. P. and Thomsen,M. F.(1982)J. PlasmaPhys.28,551. Haerendel, G. (1974) Preprint. Max Planck Institute, Garching, West Germany. Hasegawa, A. (1975) Plasma Instabilities and Non-Linear riffacts. Springer, Berlin. Huba, J. D. and Ossakow, S. L. (1979a) J. geophys. Res. 84, 6697. Huba, J. D. and Ossakow, S. L. (1979b) Phys. Fluids. 22,1349. Huba, J. D. and Ossakow, S. L. (1981)J. geophys. Rex 86,829. Hudson, M. K. and Kennel, C. F. (1975) J. geophys. Res. SO, 4581. Kadomtsev, B. and Pogutse, 0. (1970) Reviews of Plasma Physics Consultants Bureau, New York, Vol. 5. Mikhailovskii, A. B. (1967) Reviews of Plasma Physics, Consultants Bureau, New York, Vol. 3. Rosenbluth, M. N., Krall, N. A. and Rostoker, N. (1962) Nucl. Fusion Supp. 1, 143. Rukhadze, A. A. and Sihn, V. P. (1969) Son Phys. Usp. 11,659.