Indian Journal of Radio & Space Physics Vol. 28, April 1999, pp. 95-102

Role of field aligned currents and parallel electric field on the generation of whistler mode instability in the earth's magnetosphere

Harsha Maru, Sushi I Kumar & A K Gwal

Space Plasma Laboratory, Department of Physics, Barkatullah University, Bhopal 462 026

Received 11 June 1998; revised 11 November 1998; accepted 15 February 1999

Dispersion relation for the whistler mode waves in an anisotropic bi-Maxwellian magnetoplasma in presence of field aligned currents and parallel electric field has been reformulated to study their effects on whistler mode waves. The results s\1ow that waves grow in amplitude for both parallel electric field and field aligned currents. The effect of electric field is pronounced towards lower frequency spectrum, whereas the effect of field aligned currents is more towards higher frequency spectrum.This study is significant to express their effects on different types of wave phenomena and their amplification .

1 Introduction Among the global problems in the magnetospheric

plasma physics, the study of field aligned currents and parallel electric fields and their effects on wide variety of waves excited in plasma under different conditions is of scientific importance. The parallel electric fields were first proposed by Alfven and Falthammar' from theoretical arguments. The parallel electric fields are closely related to the magnetospheric plasma, current systems in the ionosphere and magnetosphere and to the acceleration of the charged particles. Many workers2

.6 have

reported the presence of electric fields aligned parallel to the magnetic field lines in earth ' s magnetosphere. Temerin and Mozer7

, on the basis of measurements, concluded that the average electric field in an electrostatic shock is not likely to be greater than 25 mV/m. Block and Falthammar8 from the Viking observation reported that weaker parallel electric fields of the order of few mV/m to 20 mV/m are more common with a duration that can be as long as tens of seconds . The existence of parallel electric fields in the magnetosphere is believed to originate from the anomalous or turbulent resistivity along the geomagnetic field lines. Wave-particle interaction in presence of parallel e lectri.:: field may control the morphologi cal features of the whistlers recorded on the ground . The resonant interaction between energetic particles and whi stler mode waves is responsible for the VLF emissions .

Various current systems exist In the magnetosphere.Olson and Pfitzer9 have described

three major magnetospheric current systems (magnetopause, ring and tail) . Another current system that .flows along the magnetospheric field lines, called Birkeland currents, links the interplanetary medium with the earth's upper atmosphere lO

. The presence of Birkeland currents has been confirmed by satellite observ.ations of charged particles and magnetic fields ll

.12

• The field aligned currents are downward along the downside electric field reversal. The downward currents can be carried by upgoing ionospheric electrons . However, upward field aligned currents are from precIpItating magnetospheric electrons. There are simultaneous observations of the Birkeland currents in northern and southern cusp. It is reported that currents flowing in the northern cusp have a counterpart flowing downward in the southern cusp and vice-versa I'. Haerendel et al. 14 have shown fr?m HEOS -I observations that field aligned currents flow in the horns of the plasma sheet during the substorm activity and the currents are strong enough to affect the magnetic field observed at the ground. These currents are also present at the edge of plasma sheet ' 5. ' 6. Therefore, it is speculative to regard these currents as flowing along closed field lines.

Misra and Singh 17 have discussed the amplification of electromagnetic waves in an isotropic bi­Maxwellian weakly ionized plasma by electric fields in the equatorial magnetosphere. Kumar et ai.' s have reported the role of parallel electric fields on electron cyclotron waves in earth's auroral plasma. In the present work, the effects of parallel electric field and field aligned currents have been analyzed on whistler

96 INDIAN J RADIO & SPACE PHYS. APRIL 1999

mode waves in equatorial magnetosphere. Kulkarni and Landage l2 have reporteti the effects of Birkeland currents on the VLF emissions in the equatorial magnetosphere at L=4. The expression for growth rate is obtained after modifying the dispersion relation of Kulkarni and Landage12 in the presence of field aligned currents and parallel electric field.

2 Theoretical considerations

2.1 bistribution function

We consider an uniform. non-collisional, anisotropic weakly ionized low density magnetospheric plasma, embedded into uniform earth's magnetic field Bo and electric field parallel to Bo. The drift velocity of plasma is assumed to be much smaller than phase velocity of waves and, hence, does not introduce any appreciable changes in the distribution function. It is assumed that ions form an immobile neutralizing background and do not contribute significantly to the VLF phenomena and, hence,ion dynamics are neglected completely. In such a plasma the presence of the weak electric field does not lead to the runway of electrons ..

The following bi-Maxwellian distribution for the main body of the plasma has been considered.

. . . (I) This is penetrated by current carrymg electrons

with a distribution

... (2)

where. VII, V 1- are the particle velocity .components; m is the electron mass; Til, T1- are known to be in the electron volt range (The temperatU1'es are given in energy units); and Vb is the beam velocity.

The equilibrium distribution functionfo is given by fo = fom + fOb. where fom is the distribution supporting the wave propagation and fOb is that of the current carrying electrons. The frequency range of VLF waves is given by Q j < (D < Q e, where Q j and Q e are the ion and electron gyro-frequencies. respectively, and the range lies between 100 Hi. and 50 kHz.

The parallel and perpendicular electron thermal velocities denoted by Oil and a1-, respectively, are given by

_ [T. )"2 _ (T1- )1/2 Oil - - , a1- - -m m

Now, Eq. (l) in terms of Oil and a1- is written as

(

2 2 ) 1 -v - Vn fOm = 12 2 exp ~----2

(2n y a 1- . all 2a 1- 2aJ1 ' . . . (3)

The applied electric field parallel to Bo has an effect of modifying the electron thermal velocity in that direction from a, to complex value Oile and is

given by

( jeEo ) llile = all 1---2 mkall

.. . (4)

Here, Eo is applied electric field, j the imaginary quantity and k the wave number. The parameter Oile. in tum, modifies the parallel temperature Til in the direction of Bo to the complex temperature ~Ie (Ref. 19) and is given as

Tile = Til )---(

jeEo ) kKTu

.. . (5)

where, K is the Boltzmann's constant

2.2 Dispersion relation The dispersion relation for the right hand circularly

polarized electromagnetic wave propagating exactly parallel to the external magnetic field in the presence of parallel electric field in plasma under consideration can be written as

C2~2 = 1 + 2W; [WZ(~) + A'(1 + Z(~»] W W ~2kall

where,

A' = T1- _ I ; ~ W - n 'fuc - ~2kall

-4'

MARU et al.: CURRENTS & ELECTRIC FIELD FOR WHISlLER MODE WA YES 97

... (6)

... (7)

Here, A' is the modified temperature anisotropy, % the plasma frequency and Z(;) the usual plasma dispersion function.

The dispersion relation for right handed circularly polarized electromagnetic wave propagating exactly parallel to the external magnetic field in presence of field aligned currents is given by

... (8)

Using Eqs (2), (6) and (8), the dispersion relation for these waves in presence of parallel electric field and field aligned currents becomes

... (9)

The expression for the growth rate are derived by

using [~ ,] of the form ~'= Jii£: for non-drifting 2k~lb

collisionless bi-Maxwellian magnetospheric plasma, where ~ ' is the plasma dispersion function in presence of current carrying electrons.

2.3 Expression for the growth rate To obtain the expression for growth rate, Eq. (9) is

solved by using an asymptotic expression for Z(;) . The real frequency is not affected by the electric field 17

,'8 and only the lowest order terms containing

Eo are taken in the imaginary part. The dispersion relation, i.e. Eq. (9) may be written as

+2Xj[~(1rY(A- wIn )(I_W)e-e ka, 2) I-wIn) n

(eEok)/n2 (1 w/n)~

+ m(l-w/n)2 + I-wIn U

... (10)

where,

plasma density to the main plasma density.

Equation (10) can be written as D( w, k)=O, where D(w, k) is the dispersion relation. The dispersion Eq. (10) can be written in the form D( (J),k)=D where,

D = Re(D) + j 1m (D) .. ' ( 11)

Using the standard formula of the growth rate (n given by

-lm(D) Y--~-----

- n~(Re(D)w=w) aw ' ... (12)

the expression for growth rate can be written as

Y= PIQ ( \3)

where,

98 INDIAN J RADIO & SPACE PHYS, APRIL 1999

s:t (-C I - X) 2 +u - exp _ 2 2k

n(l-x) 4 _ XV, - 2 ...

2k (allb )

Q = [ 1 + -(I-~-:-)-:;-2 2f2][ x] (I- x) A- I _ x

where,

(V - ka ll T - eEo x=_r ' k =--'A =~-I ' KI =---;

n ' n ' ~I ' K~l n

:::: kall b k=-n

3 Results and discussion A wide variety of waves can be excited in plasma

under different conditions . The study of these waves

is very useful in understanding various astrophysical and geophysical phenomena and numerous plasma applications. The existence of field aligned currents and parallel electric fields has significantly changed the electrodynamics, transport, energization, precipitation and acceleration of charged particles in

. the ionosphere and magnetosphere. These two parameters are coupled with each other and the characteristics of that coupling is the fundamental issue of many studies. Electric field existing along geomagnetic field lines in the magnetospheric regions plays a significant role in the wave-particle interaction . In the present paper, dealing with linear wave-particle approach , the effects of parallel electric field and field aligned currents on the growth of whistler waves in equatorial magnetosphere at R= 4RE have been worked out. For wave-particle interaction studied here in presence of parallel electric field and field aligned currents we ignore the variations in electric field and field aligned currents along geomagnetic field and consider the existence of uniform electric field as assumed by Singh and Prasad20 and Kulkarni and Landage12

, respectively. The effect of parallel electric field has been introduced only through the modification of parallel temperature Til to the complex temperature ~Ic ' The growth rate has been computed for various values of parallel electric fields and energy of current carrying electrons using Eq. ( 13) and shown in Figs 1-1 0.

In Fig. I, the growth rate C r In) has been plotted

against the normalized real frequency for ev./n = 0.54

6xIo-"-.,-----------------------.,

KT II '5 klV

4XIO!'> KTllb'6k1V

iii t-< a:: x 3XIO- 5 t-~ 0 a:: CI 2XI05

IXI05

00!5~4~~0~5~5~~0~56~~0~. 5~7~~0~5~i8~::0:. ~~9--0-. '~0--0'~-I--o-J.62 Wr /A

Fig. I- Vari ation of growth rate (yin ) with normali zed frequency (w, In ) for KTjj = 5 keY and KTjj.,=6 keY and for different values of Eo in mV 1m

MARU e/ al.: CURRENTS & ELECfRIC FIELD FOR WHISTLER MODE WAVES 99

- 0.62, for various values of £0(=0, 5, 10, 15 mV/m) and taking energy of current carrying electron as KTIlb = 6 keY. It is clearly seen that as the (4IQ increases, the · growth rate increases slowly and then rises sharply beyond (4IQ =0.6. The electric field parallel to' the wave vector enhances the growth rate. The effect is more pronounced towards the lower frequency spectrum. The frequency of the domain, i.e. the range of w,/Q in which instability exists is also increased toward the lower frequency spectrum as the applied paralle l e lectric field increases. With the ihcrease in growth rate reported here, the parallel electric fi e lds ex isting in the magnetosphere also enhance the charged particle precipitation in the

h 21?2 F' 2 5 h h .. upper atmosp ere '- . Igures - s ow t e varIatIOn of y/Q with w,/Q when KTllb = 0,6,7,8 keY for £ 0=0, 5, 10, I S mY/m, respectively.

Figures 2-5 indicate that increase in energy of current carryi ng e lec tron increases the growth rate. The increase in growth rate is insignificant towards lower frequency spectrum and is quite significant towards hi gher frequency spectrum. The solar wind interacti on with the central dipolar geomagnetic field lines gives rise to a variety of current systems resulting in the di stortion in the geomagnetic field . It is also reported that solar wind variability results into corresponding variability in the distorted geomagnetic field '). Singh and Prasad2J reported that increase in solar wind activity enhances the geomagnetic field di stortions which, in tum, increases influx of precipitated charged particles. Since the upward

flowing field aligned currents are carried by precipitating magnetospheric electrons, the increase in growth rate with increase in Birkeland currents can be accounted for the increase in solar wind activity. A source of free energy in plasma is required for the wave growth. Increase in wave growth in presence of parallel electric field is due to the acceleration of electrons in the electric field so that they get into resonance with the wave and transfer free energy to the wave which leads to the wave growth 17,18 . Larger growth under the field aligned currents as compared to parallel electric' fields indicates that the fie ld aligned currents are better source of free. energy than the parallel electric field s.

Figure 6 shows the variation of the growth rate (y /0.) with temperature ani sotropy parameter

A=(T 1- ITlirl for different values of £0=0,5, 10, 15

mYlm when KTllb=6 keY . It reveals that as the temperature ani sotropy increases, the growth rate also increases linearly. A minimum value of temperature ani sotropy is required for the initial excitation of the instability and the instability to be of any significance ( y = 10-5

). The minimum value of temperature ani sotropy required for the present case is about 1.7. Figures 7- 10 show the variations of growth rate ( y IQ) with temperature anisotropy parameters

A=(T 1- /Tarl for the A= 1.7 - 2.6, when KTII = 5 keY

and KTllb=O, 6, 7, 8 keY and taking £0=0, 5, 10, 15 mV/m, respective ly . The growth rate increases with the increase in temperature anisotropy and the effect

12XIO- 5'-r--------------------------.

B XI05 Eo • 0 mV 1m

LoJ KTII:~k.V ...

<l a:

6XI05 :z: ... ~

~ 4XI05 C)

2X16~

O~~~~~~~~~~~~--~__J 0'~4 O · ~~ O·~ O· 7 0 ·~8 0'~9 0 ·60 0 ·61 0 ·62

W,/A

Fig.2- Variation of growth rate (yIn) with normalized frequency (co, In) for KTII = 5 ke V and EI) = 0 mV 1m and for different values of KTllh in keV

100 INDIAN J RADIO & SPACE PHYS. APRIL 1999

12.16'

10KI6'

8K10~ III h • s.v/. ~ liT • • ShV II:

6KIO-' % ~ • i

.KU,-5 " 2.10~

Fig.3-Same as in Fig.2. but for KTII = 5 keY and Eo = 5 mY/m and for different values of K1ilb in keY

12xlo5~--------------------------------------------------.,

10X10~

o

Eo , 10mv/m

KTn '5keV

054 0 ·55 0 ·56 0 ·58 0 · 59 0 ·60 0·61 0'62

Wrl Ii

Fig.4--Same as in Fig.2. but for KTII = 5 keY and En = 10 mY/m and for different values of KTII b in keY

13110-5

11110-5

9110-5 Eo • 15 mV 1m

~ KTll a5keV ~

7110-5 %

~ i 5 110-5

" 3110- 5

1110-5

0 0 ·54 0 ·55 0 ·56 0 ·57 0 ·58 0-59 0 ·60 0 ·61 0 ·62

Wr/n Fig.5-Same as in Fig.2. but for KTII = 5 keY and En = 15 mY/m and for different values of KTrI b in keY

.,

' ''1(

,

MARU et al.: CURRENTS & ELECTRIC FIELD FOR WHISTLER MODE WA YES 101

2010-5..-_________________ -,

1· 7 1· 8 I · ' 2 ·0 2 ·, 2 ·2 2 · 3 2 ·. 2 ·5 2 ·6

A ' (~r' Til

Fig.6-Yariation of growth rate (yin ) with temperature

ani sotropy (A) for KTu = 5 ke Y, KTuh =6 keY and (co,. In) = 0 .6 and for different values of Ell in mY/m

• 110- '

III ... 31"0-5 4(

'" ;: 21"0'"

~ '1'0" 0

'" " 0

I·' '·e

KT". ·8

Kl U b to

19 2 -0 2· ' 2·2 2 <5 2 · 4 2 ·5 2·6

A ' (T.1fl Til

Fig.7-Yarialion o f growt h rate (yin) with temperature

anisotropy (A) for Ell = 0 mY/m. KTu =5 keY and (OJr In ) =0.6 and for different va lues of KTu h in keY

,·7 ' ·8 I·' 2 ·0 2·' 2·2 2·3 2·' 2 ·5 2·6

Fig.8-Same as in Fig.7. but for Eo = 5 mY/m. KTu = 5 keY and ( co,. In) = 0.6 and fo r different values of KTuh in keY

III ' Ol~

~ 41110'

X 301<11 ... 201O-!I ~

0 1010-5 lit

C) KTll b s O

0 1-7 1·8 I·g 2 ·0 2 · 1 2 ·2 2·3 2 ·4 2~ 2 ·e

A' (~r'

: ig.9-Same as in Fi g.7, but for Ell = 10 mY/m. ( co,. In) = 0 .6, (Til = 5 keY and for differen t values of KTuh in keY

'0165

III • 0165 KTII •••

~ 3010-' lit

KTIlIO .1 X

2 010 5 ... ~ 0 1010-5 lit C)

0 1-7 1·11 1· 9 2 ·0 2 ' 1 2 ·2 2 ·3 2 ·4 2 · 5 2-.

A • (~)-I Til

Fig IO--Same as in Fig.7, but fo r KTII= 5 keY and Eo = 15 mY/m, (co,. In) = 0 .6 and for different values of KTuh in keY

of field aligned currents is to further enhance the growth rate significantly .

4 Conclusions Based on the results presented here one may

conclude that temperature anisotropy is a major factor for initial excitation of instability and the effect of parallel electric field and field aligned currents is to further enhance the growth rate. The effect of parallel electric field is more pronounced towards the lower frequency spectrum, whereas the effect of fi eld aligned currents is more pronounced towards higher frequency side. The effect of field aligned currents is stronger than the effect of parallel electric field.

Acknowledgements One of the authors (AKG) is grateful to Madhya

Pradesh Council of Science and Technology, Bhopal, for fi nancial support . The authors also thank the anonymous referees for their valuable comments and suggestions.

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