Adv. SpaceRes.Vol. 8, No. 8, pp. (8)101—(8)104, 1988 0273—1177/88$0.00 + .50Printed in Great Britain. All rights reserved. Copyright© 1989 COSPAR

ELECTROSTATICWAVE EMISSIONANDDIFFUSION OFCHARGEDPARTICLESINPLASMASPHERICREGION

R. Prasad*andR. N. Singh*** NationalPhysicalLaboratory, NewDelhi—110012,India* * Departmentof Applied Physics, Institute of Technology, Banaras Hindu

University, Varanasi—221005,India

ABSTRACT

Consideringa mixture of cold and denseplasmaof ionosphericorigin and hot andtenuousplasma of plasmasheetorigin, the emission of electrostaticspectrum is studied. Use ismade of the test particle approachin computing the electric field power spectrum. Thegrowth/damping rate of theseelectrostaticwavesis studied. The diffusion coefficientsofchargedparticles interactingresonantlywith emitted electrostaticspectrumare computed.

INTRODUCTION

There havebeenabundantin situ observationsof theelectrostaticspectrum.Thermalfluc-tuations of the ambientbackgroundplasma is reported to account for severalclassesofnaturally occurring plasmaemissions/1-3/. Bandedemissionsat frequenciesw~<w <wi,have beenobservedabundantly. Here w~and w~,are electronangular cyclotron and elec-tron angular plasmafrequencyrespectively. Theseemissionsare found to peak in inten-sity betweenconsecutiveharmonicsin the bandsnw~<w < (n + 1)w~.The wave activityseemsto confinebetween±100geomagneticlatitude of theequator.Extremelystrongwaves(> 100 mV/rn) at w/w

0 1.5 havebeenobservedduringtheexpansionphaseof a substormon the magneticshell of aurora! oval /4/. Using measuredfeaturesof electroncyclotronwavesBelmont ~ /5/ have studied the potentiality of theseelectrostaticwavesfor thepitch anglescatteringand precipitation of chargedparticles into theaurora! ionosphere.

In the present paper,we have studied thegenerationof electrostaticwavesat w 1.5w~in the equatorial region correspondingto L=4. The contributions of thermal fluctuationsof thebackgroundplasma,which is consideredto be a combination of a cold and isotropicMaxwellian and a hot andanisotropicMaxwellian are estimated.The diffusion coefficientsof electronsinteractingresonantlywith the generatedelectrostaticemissionsare computed.

THEORETICAL FORMULATION

Eachparticle constituting thedistribution areconsideredto act independentlyas aseparatetest particle incoherentlycontributing its shareto the total electrostaticpowerspectrum.Solving the linearized Vlasov equationleads to the dielectricresponseof the plasmato thepresenceofthetest particles. The Fourier transformedelectrostaticpowerspectrumis givenas /6/.

<E2(k,w) >= ~2I~.(kw)I~(’) (1)

where ~,, is the permittivity of free space,< p2(k,w) > is thepowerspectrumof the spacechargedensity fluctuations and <> denotesthe ensembleaverage.In the abovederivation~ ~,. is considered.Here c

1 and e, are respectivelythe imaginary and thereal partsof

the dielectric function ~(k,w). Assumingthe plasma as a combination of multispeciesbi-Maxwellians the chargefluctuation spectrumand dielectric function are respectivelygivenas /6/.

e2 n~ Ik~4 \ _____

> 2ir~/~, kiVziis ) (2)

(8)102 R. Prasadand R. N. Singh

where As is the anisotropy of components and Zns Z [(w — nw~)/ki1a1i.)]is plasmadispersionfunction. In deriving (3) the frequencyrangeis confined by a~<<w/k << cwhere a, is the the ion thermal velocity. The plasma dielectric function is separatedintoreal and imaginary partsforIw~/w~I< < 1 Substituting theexpressionsfor imaginary part ofthe dielectric function and chargefluctuation spectruminto equation (1), the electrostaticpowerspectrumis rewritten as

<E2(k,w)>=~~~ ~ Ans e~1[w~ 2~~”(1+A.).• “Ih~n=—oo $ a1s 1k

— ~~W~j) Afl$ e,,~]’irS(c,) (4)

where~ = exp— [(w — nw0~)/k11a11~]2.The growth/dampingrateof linear instability is given

as~,(k,w)

= 8~(k.w)/8w

The generatedelectrostaticspectrum interacts effectively with resonantelectronsand theelectronsconsequentlydiffuse both in pitch angle andenergy. The pitch angle,mixed andvelocity diffusion coefficients for electronsinteracting resonantlywith w ~ 1.5 ~ wavesarerespectivelygiven as /7/.

D~oa = Jk1dk±T~lk ~ — sma]l k11 = k11~ (6)

srnczcosa j

Day = fk±dk1T1I’l,k w~/w— sin2a I i’~’= (~)

srncsCo5a )

~ = Jk.Ldk±’I’i,kIkII= k11~ (8)

where ~ = ~ ~ < E2(k,w) > . Here V and s are total velocity and

pitch angle respectivelyof the interacting electronsand .Jj is first order Besselfunction.

RESULTS AND DISCUSSION

Thermal velocity andnumberdensityof isotropic componentarerespectivelytaken as7 eVand iOT m3. The perpendicularand parallel thermal velocities; and numberdensity of theanisotropiccomponentare respectivelytaken as 70 eV and 20 eV; and 106 m3. The realpart of the dielectric function is separatedfrom equation (3) andequatedto zero to studythedispersioncharacteristicsof normal modeelectrostaticwaves.

Frequenciesand vectors of excited wavesare made dimensionlessrespectivelyby gyrofre-quencyandgyroradiusof cold electrons. Variations of normalmode frequencieswith per-pendicular and parallel componentsof dimensionlessvector are plotted in Figure 1 andFigure 2 respectively. The lower frequencycomponentsof normal modesare found to beexcited at higher k

1. The w 1.5w~wavesare found to be excited at k1p0 = 2 (Figure 1).The frequencyof electrostaticwavesis found to vary with k11 (Figure2). The variation withk11 is slower than that with k~.In this studysmall valuesof are consideredkeepinginview that the excitedwavespropagatealmost perpendicularto themagnetic field andarehighly dampedat smaller anglesto themagneticfield /8/.

ElectrostaticWave Particleand Diffusion (8)103

Useis madeof equations(5) and (1) andgrowth ratesof excitedspectrumare computed.The excited electrostaticspectrum is found to damp. Figure 3 displays the variation ofdamping rateswith k

11p~for different valuesof k±p~.For k~p~= 2, at which w ~ 1.5w0wavesare excited, the damping rate is found minimum at =.1. The damping rateisfound to increasewith increasingk11p~.First the damping rateincreasesfast and saturateslater for higher valuesof k11p~,.For higher kLPC values, sayfor 3 and5, thedamping ratesare found ordersof magnitudehigher and tend to saturateat smaller valuesof k11p0. Thus,electrostaticspectrumexcited at higher k1 dampsat muchhigher ratesevenif propagatingperpendicularto themagneticfield.

With thehelp of equation(4) the electrostaticpowerspectrumis computedand its variationwith is plotted in Figure 4. At k1p0=2, theelectrostaticpowerspectrumis found tohave a constantvalue of 4.5 x 10fl V

2 mHz_i for k11p0 < 0.102. As k11p0 increasesbeyond

0.102 theelectrostaticpower spectraldensitystarts decreasingfrom its constantvalue. At= 0.3, it attains a value of 5.8 x 10_12 V

2 mHz~1.The electrostaticpower spectraldensity saturatesat lower level for still higher valuesof k

11p0. As it is seenfrom Figure 4,the spectraldensityfor kIp0 = 3 and 5 saturatesat low level in thewhole rangeof k11p0.

4.50101 1 _________________________________________

_I_0.1 0.2 0.3 0.4 0.5 0.05 0.1 0.2 03 0,4

5,, PC

Fig. 3. Variation of normalizeddamping Fig. 4. Variationof electrostaticpowerspec-ratesof electrostaticwaveswith k11p0. tral densitywith k11p0.

Computationsfor theelectrostaticpowerspectraldensityandcorrespondingdampingratesare made for varying temperatureanisotropy of thehot componentof compositeplasma.This variationof temperatureanisotropyresultsfromacorrespondingvariationof theparallelthermalvelocityof thehot component.The variationsof spectraldensityand damping ratewith anisotropyare shownin Figure 5. The electrostaticpower spectra!density is foundto increasewith increasingtemperatureanisotropy. The damping rate tends to decreasewith increasingtemperatureanisotropy. This variation is in accordancewith thefact thatanisotropyis free energysourcefor the excitation of electrostaticspectrum. Other freeenergysourcesmayfurther enhancethe level of electrostaticpowerspectrum.

Considering electronsinteracting resonantlywith k11p0 = 0.1 and kjp0 = 2 electrostaticwaves, the pitch angle, mixed and velocity diffusion coefficientshave beencomputedforvarying pitch anglesand plotted in Figure 6. The diffusion coefficientshave beendividedby V

2 to have thedimensionas s1. it is seenfrom thefigure that thepitch angle diffusioncoefficient decreaseswith increasingpitch angles.Thus, particles with smallerpitch anglesareproneto diffuse in pitch angles.The velocity diffusioncoefficient is found first to increasewith increasingpitch angle and peaking at some pitch angle and at pitch angles greaterthan peak value thevelocity diffusion coefficient is found to decrease.The mixed diffusioncoefficientis seento lie betweenpitch angleandvelocity diffusioncoefficients.The pitch anglediffusion coefficient is found higherthan velocity diffusion coefficient in theentire rangeofpitch angle. However,the differencebetweenthesetwo diffusion coefficientsdecreaseswithincreasingpitch angle. Thus, at smaller pitch angles the pitch angle diffusion coefficientdominatesover other diffusion coefficients,whereasat higher pitch anglesall the diffusion

(8)104 R. PrasadandR. N. Singh

5.5oltY~ ., 8.0.10_S 10—li

%1 i~L::~~0~4C 30 1015

1.0 2.0 3.0 4.0 8.0 12.0 16.0 200

ANISOTROPY PITCH ANGLE IN DEGREES

Fig. 5. Variationof electrostaticpowerspec- Fig. 6. Variation of diffusion coefficientsoftral densityandnormalizeddampingratewith electronsin resonancewith k

11p0=O.1compo-temperatureanisotropyof hot component. nent of the spectrumof k±p~=2.0with pitch

angle.

with a portion of distribution function of the energeticelectron and consequentlydiffusethem into pitch angle and velocity. Young ~ /9/ hiaveproposeda model for thespectraldistribution of the electrostaticwave energywhich is convenientin studying the diffusionof interacting chargedparticles. The interaction with electrostaticspectrumis localized ina narrowlatitudinal range around the equator/5/. The chargedparticles in the vicinityof loss cone interact resonantly with the electrostaticspectrumin this narrow region anddiffuse into the loss cone to precipitatedown into the lower ionosphere.

Acknowledgement:The assistancein preparingthecamera-readymanuscriptby theSwedishInstitute of SpacePhysics,UppsalaDivision whereoneofus (RP) workedasVisiting Scientistis acknowledged.

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