Fundamentals of Very Low Frequency Emission Generation Mechanisms

Home , Very Low Frequency, Whistler (radio)

JOURNAL OF GEOPHYSICAL RESEARCH VOL. 69, No. 21 NOVEM•ZR 1, 1964

•N. BRICE 1

Radioscience Laboratory Stanford University, Stanford, California

Abstract. The transfer of energy between whistler-mode signals and energetic charged particles is examined. Resonance conditions are derived, leading to a classification of the mechanismspreviously suggested for the generationof VLF emissions.The relationshipbetween change in energy and change in pitch angle of the particles is derived for the transverse resonance interaction with longitudinal whistler-mode waves. Features of the transverse resonanceplasma instabilities and the anomalousDoppler effect are clarified.

Introduction. A number of mechanisms have force F and incremental distance As, and we been suggestedfor the generationof VLF emis- can write for the distancethe productof velocity sions which are observedin the range 300 to and time 30,000 cps. All assume that the emissionsare generated in the ionosphereor magnetosphere AW = F.As (2) by the interaction between whistler-modeelec- = F.vAt (3) tromagneticwaves and energeticparticles in the medium.In this work, by the applicationof = q(E q- v X B).vAt (4) simplephysical principles, the resonanceconditions for the interactionsare derived,leading to = qE.vAt (5) a systematic classification of the suggested To obtain a significantamount of energy mechanisms. Some of the more important transfer,we requirethe scalarproduct of electric properties of the mechanismsare also simply field and particle velocity to have a constant explained. (zerofrequency) component. Resonance conditions. In this section we are The motionsof chargedparticles in the mag- concernedwith the derivationfrom first principles netosphereare controlledprimarily by the earth's of the conditions necessaryfor the simplest magnetic field, so that we can conveniently resonances, so that the mutual interaction considerthe velocity to be made up of two between electromagneticwaves and particles components,%1 and vx, parallel and perpendicular which satisfy these resonanceconditions can be to the earth's magneticfield, respectively.The studied. (For a more detailed descriptionof the directionof the transverse velocity will, of course, effectsof waveson the motionof particleswhich rotate at a rate correspondingto the gyro- satisfy transverseresonance conditions see, for frequencyof the particle. example, Helliwell and Bell [1960], Bell et al. If the longitudinalvelocity of the particle is [1963],or Dungey[1963a].) matched to the wave phase velocity and the The force F on a charged particle due to wave has a longitudinal componentof electric electric (E) and magnetic(B) fieldsis given by field, we have a longitudinalresonance, since

F = q(E q- v X B) (1) Ell'vii - constant (6) where q is the chargeand v the velocity of the Alternatively, if the particle experiencesan particle. electric field which rotates about the earth's The incremental change in energy of the magnetic field at the same rate and with the particle, AW, is given by the scalar product of same sense as the particle, the transverse resonancecondition is satisfied,since • Now at Faculty of Engineering, Carleton Uni- versity, Ottawa, Ontario, Canada. E•_.v•_= constant (7) 4515 4516 N. BRICE

To satisfy the longitudinal resonancecondi- For frequenciesof interest,we can assume tion, we require v• cos 0 = v• (8) [y > [ >> [, (12) wherevp --- wave phasevelocity, and 0 - angle vp<< c (13) betweenwave normal and earth's magneticfield. For the transverseresonance and electrons,we For the transverse resonance, the Doppler- can then write shifted wave frequency seen by the particle must equal the particle gyrofrequencyin magni- /'a- ! (14) tude and have the same polarization. Since cos0 = --v•, whistler-mode waves are right circularly polar- and for protons(y •,• 1) ized, the electric and magnetic fields of the wave rotate about the earth's magnetic field in the same senseas an electron. The longitudinal cos0 = v•! +! Ji (15) velocity required by energetic electrons to satisfy the transverse resonance condition can • v• (16) then be derived from the classical formula for For a wave frequencyof 5 kc/s and zero Doppler shift [Panofskyand Phillips, 1962] as wave normal angle, the energiesof resonance electronsare plotted in Figure I as a function y cos0vii = v•!- (9) of geomagneticlatitude a along the field line terminatingat 60ø geomagnetic latitude (L - 4). where ]H is the gyrofrequencyof the energetic It was assumedthat in the magnetospherethe electrons and plasmafrequency was given by y = (1 -- v]!c")-•/• (10) /'o= 1000fH1/2 (17) _•_1 for nonrelativistic electrons. To obtain a This model of electron density distribution transverse resonance for whistler mode w•ves was deducedfrom measurementsof whistlersby •nd protons, • reversal of polarization must be Smith [1960] for the geomagneticequatorial effected, since the w•ve as seen by the proton plane. This doesnot necessarilyimply that the must •ppe•r to be left circuitfly polarized. For same model applies to the electron density • given circuitfly polarizedw•ve, •s the velocity variation along a field line. The use of this of the observeris increased,the observedw•ve modelis justifiedhere, since we are onlyinterested frequency decreasesuntil it re•ches zero, when in indicating the order of magnitudeof the the observer'svelocity is m•tched with the w•ve particleenergies required to satisfythe resonance phase velocity. As the velocity is further in- conditionsat differentlocations in the magneto- creased,the observedfrequency increasesfrom sphere.For the ionosphere(a = 60ø) energies zero, •nd the w•ve •ppe•rs to h•ve the opposite were computedfor plasmafrequencies of 3 and polarization from that found by • st•tion•ry 7 Mc/s. It was alsoassumed that one-thirdof observer. Thus when the observer's velocity the kinetic energyof the electronwas contained exceedsthe w•ve phase velocity, the polariza- in the longitudinalvelocity. For protons,for tions in the fixed •nd moving framesof reference the longitudinalresonance, the requiredenergy •re opposite, •nd the Doppler s•ft is referred is alsoshown in Figure 1. The energyrequired to •s •nom•lous [•nzburg, 1960]. for the transverseresonance is not greatly As noted •bove, we require •n •nom•lous different,as is shownby (16). From Figure 1 Doppler shift to obtain • transverse resonance it is seenthat, for the longitudinalresonance, between whistler-mode w•ves •nd protons, so the energiesrequired are smallest in the iono- that the proton longitudinal velocity required sphere. For the transverse resonance and for the resonanceis given by electrons,the energiesrequired are large except near the top of the magnetic field line path •v•cos J+ J' (11) (small valuesof a). In examiningthe suggestedmechanisms for the where[• is the ion (proton)gyrofrequency. generationof VLF emissions,we note that all GENERATION OF VLF EMISSIONS 4517

ratio of proton to electron mass) greater than that for electrons. As a result, the number

+ + densityof suitableparticles that can realistically iOs be postulatedis much less for protons than for + electrons. The other six possible mechanisms + + have all been previously suggested for the • I05 + generationof VLF emissions. + Cerenkov radiation from electrons (L, e, S) was suggestedor consideredby Kolomenskii øo" [1956], Ellis [1957], Eidman [1958], Dowden [1960], Ellis [1960], Gendrin [1960], Ondoh[1961, I 1962, 1963], Gershman and Ugarov [1961], Beneditkovand Eydman [1961], Clemmow[1962], io$ E] PROTONENERGY, LONGITUDINAL RESONANCE and McKenzie [1963]. G ELECTRONENERGY, LONGITUDIN.Z•_ RESONANCE The traveling wave amplificationhypothesis + ELECTRONENERGY, TRANSVERSE RESONANCE (L, e, I) has been examinedby Helliwell [1956], io2 I i I I I •) o I0 20 30 40 50 60 Gallet [1959], Gallet and Helliwell [1959], Bell GEOMAGNETIC LATITUDE (DEGREES) and Helliwell [1960], Barrington [1960], Kimura [1961], Adachi and Mushiake [1962], and Dowden Fig. 1. Particle energies required to satisfy the resonance conditions at different latitudes along a [1962a].Doppler-shifted cyclotron radiation from magnetic field line path. protons (T, p, S) has been treated by Aarons [1960], MacArthur [1959], Murcray and Pope involve either the longitudinalor the transverse [1960a, b], Santirocco [1960], and Knox and resonance,and that in all casesprotons or elec- Rycro•t[1964], while the correspondingradiation trons are suggestedto be the energeticparticles. from electrons(T, e, S) has been postulatedby In addition some authors have suggestedthat Dowden[1962b, c, d, e, •; 1963] and considered individual particles will emit Cerenkov or cyclo- by Brice [1963] and Hansen [1963]. tron radiation and that, if a large number of A transverseresonance plasma instability for particlesradiate, an emissionis observed.These a proton beam (T, p, I) was found by Kimura radiation processesare, in essence,single particle [1961] and Maeda and Kimura [1962, 1963], and effects. Others have considered collective effects the transverseresonance plasma instability for of energetic particles and suggested plasma electrons(T, e,I) wassuggested by Brice [1963] instabilities. We can now classify the various and wasinvestigated by Bell andBuneman [1964]. suggestedmechanisms according to the resonance Energytransfer. It is instructiveto examine condition, longitudinal (L) or transverse(T), the transfer of energy from waves to particles and the type of particles, protons (p) or elec- for the transverse resonance condition. trons (e). For each of these cases,we can suggest Consider a whistler-modesignal propagating singleparticle effects(S) in postulatingCerenkov with zero wave normal angle. The wave vector k or cyclotronradiation or suggestthat collective is then parallel to the earth's magnetic field, effectsgive rise to plasma instabilities (I). Thus and there is no componentof the electric field Cerenkov radiation from electrons can be of the wave in this direction. classified as (L, e, S) and the proton beam From Maxwell's equation amplification of Kimura [1961] as (T, p, I). V X E = --OB/Ot (18) There are eight possible combinations of we obtain letters and hence eight possible mechanisms. The longitudinal resonance mechanisms for k X E - o•B (19) protons (L, p, S and L, p, I) have not been where o•is wave frequencyin radians per second. suggestedin the literature for generationof VLF Since there is no componentof E parallel to k, emissionssince, for this resonance,protons have we can write no advantage over electrons and one serious disadvantage.The kinetic energy of the reso- = 1,,,,I nance protons must be some 1800 times (the 4518 N. BRICE and note that the vectorsE, B, and k (or v•) Thusfor protons,if the total energyof the constitutea right-handedrectangular coordinate particleis decreased(AW negative)there is a system. smallincrease in the transverseenergy of the From (1), the forceon a particle moving with proton (AW• positive). the phasevelocity of the wave is For electrons,however, we notethat • = q(• + • x •) (2•) v,/v• = --(fn- i)/i (34) = q(E- E) (22) and =0 A W•./AW = 1 + (fn -- ])/] (35) Thus the energy of a particle in a frame of = •/] > I (36) referencemoving with the wave phasevelocity is constant.We can write, then, For electronswhich lose energy (AW negative), the transverseenergy decreases by a greater «mv•.• -{- «m(v,i-- v•)• -- constant (23) amount, and so the longitudinalenergy is This result is not new [e.g., Stix, 1962]. The increased. derivationis includedhere for completeness. It is alsoof interestto considerchanges in If the wave effects small changes,Av. and the pitch anglesof the particles.The pitch Avll,in the transverseand longitudinalvelocities, angle/9 of the particle spiral about the earth's respectively,we obtain to first order magnetic field is defined as mv•_Ave. -Jr- mvll Av, -- mv• Av, -- 0 (24) fi = tan-x v•./v, (37) The changein energyof the particle in the rest We cannow obtain the changein pitchangle to frame is first-order from

A W = mv•.Av•.-Jr- mv,Avll (25) A(tan tan -- 2I(AW•_ \ •--• AW,)Wii (38) = mv•Av•l (26) noting that We define the changein transverseenergy of the particles as d(tan•)/d• = 1 q- tan'.• (39) A W•. = mv_•Av_• (27) Wz/W = sin'.• (40) and the changein longitudinalenergy as W,,/W = cos"• (41) we obtain AWii = my, Avl, (28) Then tan fi a w,/a w = •,,/• (29) Aft= 2(1-Jr- tan 2fi) and AWl A W, [ AW_•AW,2 lAW (42) •w - • aw- •-- (so) For electrons,assuming a pitch angleof 45 Let us now examine the transverse resonance degreesand using(29), (34), and (36), we find interactionsbetween whistler-mode signals and (nonrelativistic) protons and electrons. For protons,we seefrom (11) that A• = • -- I • (43)

•,/• = (• + 5)/! (31) whle for protonswith 45-degreepitch angles we obtain Substitutinginto (30) we find that AW AWz/AW- I -- (f + (32) w (44) = (33) In general,we expect GENERATION OF VLF EMISSIONS 4519

TABLE 1. Relationshipsbetween Energy and Pitch Angle Changesfor Transverse Resonance Interaction with Whistler-Mode Waves

Type Total Whistler Particle Particle Particle of Particle Wave Longitudinal Transverse Pitch Particle Effergy Energy Energy Energy Angle

Electron Decreases Increases Increases Decreases Decreases

Proton Decreases Increases Decreases Increases Increases

Electron Increases Decreases Decreases Increases Increases

Proton Increases Decreases Increases Decreases Decreases

•i•/• > 2 h/f (( I (45) Doppler effect. Ginzburg [1960] consideredthe motion of chargedparticles spiraling along field For ½•sesin which f• is much gre•ter th•n f, lines as analogousto a hVarmonicoscillator in we obtain for electrons with 45-degree pitch which the energy of the oscillator was the •ngles transverseenergy of the particle. For anomalous AB -----(]•/1')(A W/W) (46) Doppler shifts, he found that the emissionof a photon of energy was accompanied by an For protons with 45-degree pitch angles we increase in the level of excitation of the oscillator. find that We have shown that consideration of resonance conditionsshows that the polarization reversal requires particle longitudinal velocities greater The qualitative relationshipbetween the changes than the wave phase velocity. It has also been in the total particle energy, the wave energy, simply shown that, for this condition, emission the longitudinal and transverse energies, and of radiation must be accompaniedby an increase the pitch anglesof the particlesare summarized in the transverseenergy of the particle. in Table 1. Returning to considerationof (46), it is of The resultsgiven in Table i provide a simple interest to note that for frequenciesmuch less explanation of a curious feature of the trans- than the electron gyrofrequency a relatively verse resonance.Kimura [1961] found a whistler- small changein the energy of a whistler may mode instability for an ambient plasma with a cause a large transfer of energy from the linear beam of protons; i.e., the protons were longitudinal to the transverse energy of the assumedto have initially no transversevelocity particles (or vice versa) and hence significant (or zero pitch angle).Neufeld and Wright [1963] changesin the pitch anglesof the particles. found no such instability for a linear beam of Dungey[1963b] and Cornwall[1964], to explain electrons,but the instability was found by Bell the loss of energetic electrons from the lower and Buneman [1964] with a gyrating beam for magnetosphere,postulated that whistlerscaused which it was assumed that the electrons in the changesin the pitch angles of these electrons. beam had nonzero initial transverse velocities. They suggestedthat there are no (or very few) It is apparent from the resultsgiven in Table 1 energeticelectrons with very small pitch angles, that zero initial transverse energy for the elec- since these are lost by collisions near theft trons in the beam precludes the possibility of mirror points. Also, if there are more particles obtaining an instability, since the transverse with large pitch angles than small, a 'random energy of these electronsmust decreaseduring walk' of pitch anglescaused by the whistler will the emission. A linear beam of protons may give a net decreasein the pitch angles.For some give a whistler-mode instability, since the of the particleswith (small) pitch anglesclose to transverseenergy of the protonsincreases during the 'loss cone,' a small decreasein pitch angle the endssion. will place them in the loss cone (and they will The analysis above also provides a simple therefore be lost), so that the whistler reduces explanationof another feature of the anomalous the number of trapped particles. 4520 N. BRICE

Noting that the electrons that have their to the resonancecondition, type of particle,and mirror points loweredalso contributeenergy to type of emission.For the transverse resonance the whistler, we may suggestthat a net lowering and whistler-modewaves, the relationships of mirror points will give a net increasein the betweenchanges in energyof the wave and in energy of the whistler. It may be further sug- the longitudinaland transverse energy and pitch gested,then, that the condition for growth of angle of the particle have been derived. The the whistler is that there be more particleswith resultsobtained have a widerange of application. large pitch anglesthan small. Alternative state- ments of this condition are that the average Acknowledgment. I thank Dr. R. A. Helliwell transverse velocity of the particles of interest and Dr. R. L. Smith of StanfordUniversity for exceed the average longitudinal velocity, or helpful advice and criticism. that the transverse temperature exceed the This work was sponsoredby the United States longitudinal temperature. This is the condition Air Force Oflqceof ScientificResearch under grant AF-AFOSR 62-370. found necessaryfor the existenceof the transverse resonance plasma instability by Sudan I•EFERENCES [1963] and the condition suggestedby Bell and Buneman [1964] in order that the growth rate Aarons,J., Naturalbackground noise at verylow of the whistler-mode instability exceed that of frequencies,The RadioNoise Spectrum, edited the longitudinalelectrostatic instability. by DonaldI5. Menzel,chapter 8, pp. 111-122, 15arvardUniversity Press, Cambridge, Mass., The results obtained above apply for any 1960. interaction between a whistler-modesignal and Adachi,S., and Y. Mushtake,On VLF emissions energeticparticles for the transverseresonance. in the exosphere,IRE Trans.,AP10(6), 785-787, If we wish to postulate that whistler-mode November 1962. signalscause dumping of electronsin large num- Barrington,R. E., The interactionof the whirler modewith the spacecharge modes of an elec- bers, we should note that to lower the mirror tron stream, Proc. Symp. Phys. ProcessesSun- pointsof electronswe must require that the total Earth Environ.,20-21 July 1959,Defence Res. electron energy decrease. Acceleration and TelecommunicationsEstab., DRTE Publ.1025, dumpingwould then requirea two-stageprocess. pp. 223-230,Ottawa, Canada,March 1960. An adiabatic compressionof the ambient mag- Bell, T. F., and O. Buneman,Plasma instability in the whistlermode caused by a gyratingelec- netic field could lead to an increase in the tron stream,Phys. Rev., 133(5A),25-26, March transverse energy of the electron, leaving the 2, 1964. longitudinal energy unaffected.This would raise Bell, T. F., and R. A. 15elliwell,Traveling-wave the mirror point of the particle. Emission of amplificationin the ionosphere,in Proc.Syrup. Phys.Processes Sun-Earth Environ., 20-21 July VLF signals could then convert some of the 1959,Defence Res. Board, Dept. Natl. Defence, transverseenergy into longitudinal energy, and DRTE Publ.1025, pp. 215-222, Ottawa, Canada, thus lower the mirror point of the electron.The March 1960. 'efficiency' of the conversionof transverse to Bell,T. F., R. A. Itelliwell,R. F. Mlodnosky,and longitudinalenergy is given by (46) and may be R. L. Smith, Geocyclotronfeasibility study, A. F. Spec. WeaponsCenter, A. F. Systems quite high for large ratios of electron gyro- Command,AFSWC-TDR-63-35, Final Rept., frequency to wave frequency, so that a net contract AF 29(601)-4506,May 1963. increase in energy and a net lowering of the Benediktov,E. A., and V. Ya. Eydman,On the mirror point may be accomplishedby this two- incoherentradio emission of fastmoving charged particlesin the earth'smagnetic field, Radio- stage process.However, these large ratios could fizika, J(2), 253-258,1961. only be obtainedby high energy particles,since Brice, N.M., An explanationof triggeredVLF large longitudinal velocitieswould be required emissions,J. Geophys.Res., 68(15), 4626-4628, to Doppler-shift the gyrofrequencyto the wave August 1, 1963. frequency. Clemmow,P. C., Wave amplificationin a plasma stream in a medium of high refractive index, Summary. Resonances in the interaction Proc. Phys.Soc. London, $0, 1322-1332,1962. between electromagnetic waves and charged Cornwall,J. M., Scatteringof energetictrapped particles are simply related to energy transfer. electronsby very-low-frequencywaves, J. Geo- The varioussuggested mechanisms for generation phys. Res., 69(7), 1251-1258,April 1, 1964. Dowden, R. L., Geomagneticnoise at 230 kc/s, of VLF emissionshave beenclassified according Nature, 187(4738), 677-678, August 1960. GENERATION OF VLF EMISSIONS 4521

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