Volume 35A, number 2 PHYSICS LETTERS 17 May 1971

VLF WAVE AMPLIFICATION BY WAVE-PARTICLE INTERACTION

R. N. STNGH * and R. P. SINGH **

PhysicsSection, Institute of Technology,Banaras Hindu Unii’ersit~’, Varanasi-5, India

Received19 March 1971

A chargedparticle beamis assumedto move throughthe magnetosphere.It is shownthat the VLF wavespropagatingat non—zeroangles with the beamare amplified.

It is well known that for electromagneticwave casionalinflux of chargedparticleswhich con-propagationalonga streamingplasmabeam(co- stitutesthe secondarypeak in the Maxwellianincidentwave, beamandmagneticfield direc- velocity distribution function. Therefore, thetions), thereis no couplingbetweenthe longitu- amplification of electromagneticwavespropa-dinal modeoscillations in theplasmaandthe gating throughthe ionosphereand the magneto-progapatingtransverseelectromagneticwaves, sphereat an arbitrary angle to theplasmabeamThis is owing to thefact that thedispersionless streamingalongthegeomagneticfield may be ofplasmais not effective in coupling a purely longi- muchimportance. In the caseof electromagnetictudinal wavewith theplasmaoscillations of the wave generationby artificial electronbeamin-streamingbeam. However, in the caseof elec- jection in the ionosphereand in themagneto-tromagneticwavespropagatingat certain angle spherethe TWT amplification mechanismmayto the streamingplasmabeam,thereis an elec- play a decisiverole. However, this is not speci-tric field componentparallel to thestreaming fically accountedin theanalysisof electronbeamplasmabeam. The presenceof parallel electric injection experiment[6] andin theanalysisoffield componentpermits theexchangeof energy simultaneouslymeasuredelectronflux andVLFbetweenthe longitudinalplasmaoscillations and emissionin the ionosphere.thepropagatingelectromagneticwaves. Under The dispersionequationof a magnetoplasmasuitableconditionsthere may be energytransfer with a streamingplasmabeamhasbeenderivedfrom the streamingplasmaparticles to thepro- by Knox [4] andWang [5].pagatingwaves. This energytransferprocessis D(w k) - w~/c2(w2- k U ) = 0 (1)analogousto the travelingwavetube (TWT) me- ‘ ii bchanismandis capableof amplifying the inter- where Ub andwb arethe streamingplasmabeamacting electromagneticwaves[1]. The detailed velocity andthebeamplasmafrequencyrespec-studyof TWT mechanismin the ionosphereand tively. In eq. (1) the first term is characteristicin themagnetospherewasmadeby several of VLF wavepropagation in thewhistler modeworkers [2-5]. Although, theTWT mechanism throughthecold andcollisionlessmagnetoplasmais basedon soundexperimentalandtheoretical andthesecondterm characterisesthecontribu-foundationsbut its importanceandrelevanceto tion of streamingplasmabeam. The generaldis-the electromagneticwave amplification, in lack persion equationappropriateto this situation isof experimentalevidenceof streamingplasma rathercomplicatedanddoesnot permit the ana-beam, wasignored. lytical study of w andk variations. Therefore,

It is moreor establishedthat thequite time we confineourselvesto the frequencyrangeionosphericandmagnetosphericplasmaplays WHi < ~ < andignoretherole of ions in thethe role of backgroundthermal plasmawith Max- dispersionequation. It hasbeenshown[3-5] thatwellian velocity distribution andthereareoc- eq. (1) depictsthe instability of VLF wavespro-

pagatingin thewhistler mode(w complexandreal k). The maximumtemporalgrowth rate

* ESROVisiting Scientist, ESRIN, Frascati (Rome), consistentwith eq. (1) is givenbyItaly. — /

** At present with Groupede Rechercheslonospheriques, amax = 0.86!aD/awI 1 3 (w/c)2/3 (2)CNRS, France. k

11 Ub

105

Voluno’ 3i.\. nuiolwr 2 I’ [‘I E H 8 1 5l~o1971

Substituting for ~D w from eq. (1 and ye ‘md to reflect several times between thenorthern:iI’ranginr, we rewrite en. (2) as mud the southernhemispheres. However, in the

om’esent easewe a’ e onsiclerlimg the propagation~42 C~1 ÷ftsmn

2 at ‘iou - run (I angle to the magnetic field thereD ens Q(1 2 lhcos e “re. thepropagating waveswill refract awa~

utter sometime. The time taken by the VLF‘~vhei 7? ~5b 52 (w

1~0)012 and ~0)He wavesto propagateHorn the equatorial region 0

I’hi~formulation shows thai electromagnetic the magnetosphereto the ground is given byv.,uvespresent in a thermal plasma with a stream rrur(

ing plasmabeam and having proper w and k will t~he amplified. It is well known that such stream-

ing plasmabeam in the ionospheric and magnetspheric plasmaalso satisfy the well known where I ms the group ye] ocit\ 01 the VLF waves(~erenkuvcondition and give rise to VLF hiss and dx, is an element of pathalong the geomag-7,81. Therefore, the VLF wavespropagating in netic lines of force. The time of propagation of

the whistler modebetweenthe northernand the VLF wavesdependsthepath of propagationsouthernhemisphereswheneversatisfy the i.e (I. -values or different latitudes). Choosingsame

quirementsareamplified. We havechosenpa- plasmaparametersas used for thecalculation oframetersappropriateto the ionosphericand ~rnax’ we find that the propagationtime variesnuagnetosphericconditions and have computed from 0.01 to 0.1 second. The crude estimatethe maximum growth rate of VLF wavespropa— shows that this mechanismwhen operative calmgating in thewhistler mode. The computedgrowth give rise to VLF power densiH increaseby aiate is found to vary between10 to 200 per second factor of 50-100. This analysisleadsus to con

The aboveestimateshowsthat at times the eludethat the interpretation of VLF wavesre-amplification of VLF wavespropagatingthrough ceived on theground or by satellites should takethe ionosphereandthe magnetospherecould be into accountthepossibility of this amplificationquite considerable. Therefore, theelectric field mechanism.of the VLF waves goingthrough this amplificationprocesscan be expressedas The presentwork waspartly supportedby the

‘S,A,t’ (leant EOOAR-70-007f1F E

0(w) exp(omaxt) 14and the correspondingtemporal growth rate otthe energy spectrum can be written as

fieterencesF] co F0(w) exp(2~mnaxl) (5~ 1] R.M.Gallet and R.A, Hellmoe.ll, J. of Res., N.B.S.

(RadioProp.) 63D (1957) 21.Sincethe exponentialfactor dependson thenum- [2] C.F. Knox. Proc. Phvs. Soc. 83 ~1964)783.ber of beamparticles andnot on their energy [3] T. F. Bell and O.Buneman.Phys. Rev. 133 (1964)spectrum, it can be arguedthat the total radiated 1 300,power will also be enhancedby this factor. [4) C. F. Knox, J. Plasma Phvs.1 ~l967) 1,

Further eq. (5) shows that the contribution due tm [3) T.N. C.Wang. JEEETrans. \nts, Propn. 11 (i969~

[mite growth rate is controlled by the time for [6] T. F. Bell, .1. Goophys.Res. 73 (1968) 4409,which the amplification mechanismis operative. [7] JF~IcKenzj~ Phys. Fluids 10 (1967) 2680.

The VLF waves in the whistler modeare known H] B. N. Sjnghand B. P. Singh. ~nn. de Geophys.25to be guidedalongthe geomagneticlines of force (11)69) 029,

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