# VLF wave amplification by wave-particle interaction

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Volume 35A, number 2 PHYSICS LETTERS 17 May 1971

VLF WAVE AMPLIFICATION BY WAVE-PARTICLE INTERACTION

R. N. STNGH * and R. P. SINGH **Physics Section, Institute of Technology, Banaras Hindu Uniiersit~,Varanasi-5, India

Received 19 March 1971

A charged particle beam is assumed to move through the magnetosphere. It is shown that the VLF wavespropagating at nonzero angles with the beam are amplified.

It is well known that for electromagnetic wave casional influx of charged particles which con-propagation along a streaming plasma beam (co- stitutes the secondary peak in the Maxwellianincident wave, beam and magnetic field direc- velocity distribution function. Therefore, thetions), there is no coupling between the longitu- amplification of electromagnetic waves propa-dinal mode oscillations in the plasma and the gating through the ionosphere and the magneto-progapating transverse electromagnetic waves, sphere at an arbitrary angle to the plasma beamThis is owing to the fact that the dispersionless streaming along the geomagnetic field may be ofplasma is not effective in coupling a purely longi- much importance. In the case of electromagnetictudinal wave with the plasma oscillations of the wave generation by artificial electron beam in-streaming beam. However, in the case of elec- jection in the ionosphere and in the magneto-tromagnetic waves propagating at certain angle sphere the TWT amplification mechanism mayto the streaming plasma beam, there is an elec- play a decisive role. However, this is not speci-tric field component parallel to the streaming fically accounted in the analysis of electron beamplasma beam. The presence of parallel electric injection experiment [6] and in the analysis offield component permits the exchange of energy simultaneously measured electron flux and VLFbetween the longitudinal plasma oscillations and emission in the ionosphere.the propagating electromagnetic waves. Under The dispersion equation of a magnetoplasmasuitable conditions there may be energy transfer with a streaming plasma beam has been derivedfrom the streaming plasma particles to the pro- by Knox [4] and Wang [5].pagating waves. This energy transfer process is D(w k) - w~/c2(w2- k U ) = 0 (1)analogous to the traveling wave tube (TWT) me- ii bchanism and is capable of amplifying the inter- where Ub and wb are the streaming plasma beamacting electromagnetic waves [1]. The detailed velocity and the beam plasma frequency respec-study of TWT mechanism in the ionosphere and tively. In eq. (1) the first term is characteristicin the magnetosphere was made by several of VLF wave propagation in the whistler modeworkers [2-5]. Although, the TWT mechanism through the cold and collisionless magnetoplasmais based on sound experimental and theoretical and the second term characterises the contribu-foundations but its importance and relevance to tion of streaming plasma beam. The general dis-the electromagnetic wave amplification, in lack persion equation appropriate to this situation isof experimental evidence of streaming plasma rather complicated and does not permit the ana-beam, was ignored. lytical study of w and k variations. Therefore,

It is more or established that the quite time we confine ourselves to the frequency rangeionospheric and magnetospheric plasma plays WHi < ~ < and ignore the role of ions in thethe role of background thermal plasma with Max- dispersion equation. It has been shown [3-5] thatwellian velocity distribution and there are oc- eq. (1) depicts the instability of VLF waves pro-

pagating in the whistler mode (w complex andreal k). The maximum temporal growth rate

* ESRO Visiting Scientist, ESRIN, Frascati (Rome), consistent with eq. (1) is given byItaly. /

** At present with Groupe de Recherches lonospheriques, amax = 0.86!aD/awI 1 3 (w/c)2/3 (2)CNRS, France. k11 Ub

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Voluno 3i.\. nuiolwr 2 I [I E H 8 1 5l~o 1971

Substituting for ~D w from eq. (1 and ye md to reflect several times between the northern:iIranginr, we rewrite en. (2) as mud the southern hemispheres. However, in the

omesent ease we a e onsiclerlimg the propagation~42 C~1 ftsmn2 at iou - run (I angle to the magnetic field thereD ens Q(1 2 lhcos e re. the propagating waves will refract awa~

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

1~0)012 and ~0)He waves to propagate Horn the equatorial region 0

Ihi~formulation shows thai electromagnetic the magnetosphere to the ground is given byv.,uves present in a thermal plasma with a stream rrur(ing plasma beam and having proper w and k will t~he amplified. It is well known that such stream -ing plasma beam in the ionospheric and magnetspheric plasma also satisfy the well known where I ms the group ye] ocit\ 01 the VLF waves(~erenkuv condition and give rise to VLF hiss and dx, is an element of path along the geomag-7,81. Therefore, the VLF waves propagating in netic lines of force. The time of propagation of

the whistler mode between the northern and the VLF waves depends the path of propagationsouthern hemispheres whenever satisfy the i.e (I. -values or different latitudes). Choosing samequirements are amplified. We have chosen pa- plasma parameters as used for the calculation oframeters appropriate to the ionospheric and ~rnax we find that the propagation time variesnuagnetospheric conditions and have computed from 0.01 to 0.1 second. The crude estimatethe maximum growth rate of VLF waves propa shows that this mechanism when operative calmgating in the whistler mode. The computed growth give rise to VLF power densiH increase by aiate is found to vary between 10 to 200 per second factor of 50-100. This analysis leads us to con

The above estimate shows that at times the elude that the interpretation of VLF waves re-amplification of VLF waves propagating through ceived on the ground or by satellites should takethe ionosphere and the magnetosphere could be into account the possibility of this amplificationquite considerable. Therefore, the electric field mechanism.of the VLF waves going through this amplificationprocess can be expressed as The present work was partly supported by the

S,A,t (leant EOOAR-70-007f1F E

0(w) exp(omaxt) 14and the corresponding temporal 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.

(Radio Prop.) 63D (1957) 21.Since the exponential factor depends on the num- [2] C. F. Knox. Proc. Phvs. Soc. 83 ~1964)783.ber of beam particles and not on their energy [3] T. F. Bell and O.Buneman. Phys. Rev. 133 (1964)spectrum, it can be argued that the total radiated 1 300,power will also be enhanced by 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. JEEE Trans. \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 mechanism is operative. [7] JF~IcKenzj~ Phys. Fluids 10 (1967) 2680.The VLF waves in the whistler mode are known H] B. N. Sjngh and B. P. Singh. ~nn. de Geophys. 25to be guided along the geomagnetic lines of force (11)69) 029,

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