JOURNAL OF GEOPHYSICAL RESEARCH VOL. 69, No. 21 NOVEM•ZR 1, 1964 Fundamentals of Very Low Frequency Emission Generation Mechanisms •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 relationshipbe- tween 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 resonancecondi- tions 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 longitudinal or 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