Journal of Research of the National Bureau of Standards- D. Radio Propagation Vol. 63D, No.1, July- August 1959
Origin of "Very-Low-Frequency Emissions" 1
R. M. Gallet and R. A. Helliwe1l 2
(Jan uary 16, 1959) SC'lecL ive travclin g-wave nmplificatiol1 in t he outer ionosphere is postulated to explain vpr·.v-l ow-freque ll c.v emissio,)s, fI class of very Jow-frequency (1 to 30 kilocycles per second) nal ural n o i ~e . Byanalogy willl Lhc mechanism of Lravelin g wa ve tubes, low-level ambient nOise In Lho ou Le r lo no phel'e is a mplified in streams of in coming ionized solar particles at f requenclcs for whICh the stream a nd wave veloc it ies arc equal. Required velocities a re in t he r ~ n ge 0.01 to 0.1 c (where c is the velocity of light). Streams with densities of the o rder of one electron per cubic centimeter would provide snfficient energy. Phenomena whi ch can be explained quaiJ tatl vely by th e theo r' y arc the hi ss, quasi-constant tones, dawn chorus and related tranSlCnts, a nd very long trains of whis t ler echoes. A quantitative example shows how t he theory can reproduce t he general form of certain characteristic di s cr~te spectra " hooks" of emission s, and how t his leads to defi nite values of part icle velOC ity and a Jaw for the distribution of electron density in t he outer ionospllerc. 1. Introduction of the origin of vlf emissions, based on selective travcling-wave amplification takinO' place in the 'When a high-gain audio amplifier IS connected outer ionosphere. The inpu t signal is asslmled to to a loop or long-wire an tenna, unusual sounds be provided b.v tbermal radiation, whistler energy, of natural origin arc oJLen heard. The more or possibl:v Cerenkov radiaLion (Ellis [9]). It is impulsive sounds called " atmospherics" arc pro assumed that the energ ~T for amplification of the duced by lightning and reach the receiver by electl'omagn eLic wave is provided b~T the streams traveling between the ground and thc lower edo:e of ioni7.ed solar pal'Licles (boLh positive and negative) of the ionosphere. They arc described by var i o~s which are thought to cause the aurora. These tenns such as "clicks,"" " Lweeks," and "crashes." streams are fI,ssmned to penetrate the ambient Some of thc cn ergy from a ligh tning discharge ionizaLion of the outcr ionosphere and Lo be guided 3;1so enters the ionosphere and is guided by the along Lhe lines of force of the earth's magnetic fi eld. hnes of force of the earth's magnetic field. As The mechanism of ampliGcaLion is essentially similar the energy propagates from one hemi sphere Lo th e to thaL which Lakcs place in an ordinary traveling oLher, it is dispersed and arrives at the receivCl" wave tube (Pierce [10]), excepL Lhat the "slow-wave" as a wh~ stling tone, usually of descending frequency. circuit (provided by the helix: in the tube) is the Such sIgnals arc called " whistlers," and their ambien L ionization of Lh e ouLer ionosphere in the theory has been developed in som e detail (Storey presenee of the earLh's magneti c field. Such a [1] , 3 H elliwell et a1. [2], E llis [3], and :Maeda [4]). m edium is c1ispcl'sive, i.c., the wave velocity is a . There is y?t anotl~ ~ 1" class of a udiofreq u en c~~ signals function of frequency, and at the vcr~T low fre wluch we WIll call very-low-frequency emlSS lons". quencics considered here Ute vc l oc il~~ ma~T bc rcduced These appear Lo be related to magncLic clisLurbance as much as two orders of magnitude below that in and occur most frequen tl.\T neal' t he auroral 7.ones. free space. Furthermore, as shown b~ T Storey [I} (Storey [1], Pope [5], All cock [6], Dinger [7], and in connection with the theory of whisLie rs the energy "Watts [8] .) One of the most common is Lhe "I ti ss" of a vlf wave tends to be guided along Lhe earth 's a band of noise several kilocycles wide which i ~ magnetic field lines rather closel.", and thus may usually heard below 10 kc; it is occasionally ac follow the same path as the postulated particle compained by one or more discrete tones in the same stream. The wave normal, on the other hand, may range of frequencies. Another is the so-called make any angle with the field ; and since the wav·e " dawn chorus", a combination of short (0.1- 0.2 polarization is approximately circular , there will sec) rising whis tIes and warbling tones. Various normally be a substantial component of the electric isolated events, including rising and falling tones and field of the wave which is parallel Lo Lhe magneLi c combinations thereof can be observed. Very-low field. This component will acL on the moving par ticle frequency emissions are frequently associated with stream to produce the required sinu oidal modulation strong whistlers. Sometimes a whistler appears of charge density. For given values of the ambient 1,0 " trigger" short periods of dawn ehorus or hiss. plasma anel gyro frequencies , th e sLream velocity Whistlers an~ :"ometimes preceded or followed b y will approximate the phase velocity at two par one or more n s mg tones. ticular frequencies, and traveling wave amplification It is the purpose of this paper to outline a theory can take place. As the parameters of the m edium change with position along the stream, th e fre quencies of amplification will cb ange. This paper I The basic ideas described in this paper we"e pre
~ 10 and the phase velocity is then simply u~ ~ B ~ --- "- (2) ~6 --- 3 where j = wave frequency ; jH= gyrofrequency; j p= 4 (81N)J.2 = plasma frequency; with N = number of electrons per cubic centimeter whenjp is in kilocycles per second. It is assumed thatjp>j,jH>f. From (1) and (2), oL3~~~~~o 4 6 8 10 12 14 16 18 20 the frequencies of amplification are given by Gy ro Frequency, kc
FIGURE 1. F arial1:on of excited wave frequency with gyro ( 2Vs fP)2 ]X"\ frequency. f= fH { 1± [ 1- cf~ l. (3) 2 1- (vs/e) 2 r , ) EXCITED If we restrict the discussion to cases in which the FREQUENCIES p: ::t.. f : 3 kc kc stream velocity isless than about 0.1 e, then (v s/c)2«1 c P and (3) can be written '
(4) where P - jp(vs/e) and is conveniently measured in kilocycles per second. For interaction to occur, the solution to (4) must be real, and hence the condition for interaction is
jH"i;;.2P. (5) 2 3 4 5 DISTANCE IN EARTH RADII ~he maximu~ value of the wave frequency j is jH, ~ mce.the maxlmum.r~al value of the radical in (4) IS umty. The conditIOns for interaction are shown in figure 1, which is a plot of wave frequency j as a function of gyrofrequency jH for different values of the parameter P. All curves are asymptotic to the line j jH (ignoring the dashed curve for the mo FIGURE 2. Illuslration of the the01'y of hiss, for the auroral line ment). of f 01'ce X=67°. 22 3 . Stream Velocities 10- 5 w/m2• The required efficiency of conversion from kinetic energy to electromagnetic energy is The stream velocity required for interaction l S thus 0.1 percent, a low value compared with traveling found from (4) and is given by wave tube efficiencies, which are of order 10 percent. 5. Comparison of Theory With Observations (6) Using the traveling wave hypothesis the hiss is explained in terms of a continuous stream of particles Values for the stream velocity for appropriate along which amplification occurs only for a restricted values of the m edium parameters and the wave band of frequencies. The amount of gain at any frequency are given in table 1. It is seen that the one frequency will depend upon th e interaction required stream velocities could vary roughly be distance available, which in turn will depend on the tween 0.01 and 0.1 of the velocity of light. These way in which the plasma frequency varies witb dis velocities seem reasonable, since the lower value, tance along the path. In a lilce manner the quasi 0.01 c, is comparable to the lower limit found by constant tones sometimes observed on different fre Meinel [11] in spectroscopic studies of hydrogen in quencies can b e explained by assuming an ionization the aurora. On the other hand, it appears necessary model in which the plasma frequency increases with that electrons which penetrate to the E region must gyrofrequency in sucb a way that j = constant over have velocities of approximately 0.3 c (VegaI'd [12]) . an appreciable length of path. Several different tones Thus the velocities required for the traveling wave observed simultaneously could be explained by mechanism appear to lie in the same range as those differcnt streams flowing along lines of force of ascribed to auroral particles. sightly different geomagnetic latitude, or possibly by streams of different velocities. 4. Stream Energy Requirements Transient phenomena, of ,,,"hich the dawn chorus is an example, are less easily explained. The travel If traveling wave amplification is to be accepted ing wave hypo the is can be adapted to the generation as a possible mechanism, it is necessary that the of transient signals by assuming that the incoming power density of the incoming stream exceed that particles arrive in bunches. As the bunches travel of the observed signals. A measurement of the along the lines of force, amplification takes place in strength of relatively strong hiss was made at the region of the bunch. However, the character Stanford, California (Stanford University [13]) on istic frequ encies are amplified for a length of time March 3, 1956, during a period of magnetic dis which depends upon the length and velocity of the turbance. The power density was of the order of bunch. As th e bunch moves, the frequency which 10 - 10 w/m2. for a signal having a bandwidth of about is generated changes in accordance with changes in 1 kc. Observations of whistlers indicate tbat the the parameters of the medium. The mechanism of total attenuation from the source to the observer is formation of such bunches is not lrnown; however, not likely to exceed 20 db. Thus the power den ity the visual phenomena observed during aurora sug of the signal as it leaves the amplifiying region is of gest that solar streams may be bunched. the order of 10- 8 ·w/m2• The interpretation of observed transient signals Let us now calculate the power density of the in terms of bunches of particles is complicated by incoming stream. Since the particle densities and the fact that the frequency-time relation of the velocities of streams from the sun are not lrnown, received signal will depend on the integrated group we must make a conservative estimate. From the delay from the poin t of origin to the receiver. The work of Martyn [14] the stream is estimated to have group delay, which is a fun ction of frequency, a density of at least 1 electron/cm3 and a velocity of clearly will affec t the frequency-time dependence of 3,000 km/s. The power density in such a stream is the received sjgnal. However, it cannot by itself (!mv2) (Nv), where m is the particle mass, v the explain the appearance of a given frequency at two velo city, and N the number of particles per unit different times, a phenomenon seen frequently in volume. For electrons, m = 9.11 X 10- 31 kg, and the dawn chorus. Such cases might better be ex the power density of the stream is thus about plained in terms of variations in the ambient ioniza-
T ABLE 1. Calculated stl·eam velocities for different lJaramelers of amplifying region
W ave frequency G yrofrcq uency Distance from surface Minimum geomagnetic lat itude Plasma 1---,----,---- frequency 2 ke 5 ke 10ke ------1------1------_ ·_------/ll= 1,000 k c______Regular layers______Middle latitudc3 ______4, 000 k c 0. 011 e O. 018 e O. 025 e 1, 000 k c . 044 c . 071e . l e /1( = 100 kc______~1.0 earth radiL ______44 0 ______1, 000 kc .014 c . 022e . 030e 200 k c . 070 c . 11 c . 15 c /1( = 10 kc______3.6 earth radiL ______610 ______200kc . 02 c . 025 e / 1( = 5 k c______9.0 earth radiL ______65 0 ______200 k c . 014 c
23 tion density in the interaction region. For example, small that we can assume the accelerating signal to consider the hypothetical model, jp= 200 kc+jH 2j be of constant frequency. Thus as the whistler (11m), wherejH is measured in kilocycles per second. travels down the field line, particles of a given A short bunch of charge (say, with dimensions of velocity are literally picked up by the wave at every about 100 km) traveling through such a region point where a wave component of the same velocity would excite the frequencies given by the dashed exists. As the wave continues to propagate, its curve of figure 1. The energy would be limited to phase velocity increases and the particles riding on short groups produced at different times. The the front of the wave are accelerated, as in a linear curve of excited frequency versus gyrofrequency accelerator. With the additional kinetic energy transforms to a somewhat similar shape in the which the stream acquires from the signal it would frequency-time plane. Some distortion of this be more efficient as a traveling wave amplifier and curve occurs at the receiver because group delay is might conceivably produce audible vlf emissions. a function of both frequency and location of source. No completely closed trace of this type has yet been observed, but many of the recorded signals first fa11 6. Conclusions and then rise with time, corresponding to the lower portion of the dashed curve. Thus there is at least The rather close similarity between conditions in a qualitative r esemblance between the theory and the outer ionosphere and those in a traveling wave the experimental data. tube lead to the hypothesis that traveling wave The steady hiss and the sharply defined transients amplification might take place at very low audiofre represent extremes among vlf emission phenomena. quencies. Even though the traveling wave hy In between lie many forms which show varying pothesis may not account successfully for all of the degrees of irregularity. Their character often observed phenomena, it is probably an important changes markedly in the course of a few hours. It factor to be considered in whistler propagation as is suggested that the observed variety of form is well as in the generation of vlf emissions. Confirma simply the result of variations in the degree of bunch tion of the hypothesis must await a more detailed ing of the charge in the streams. study of the traveling wave-gain problem and some The traveling wave hypothesis offers a convenient what more detailed information regarding the dis ·explanation of very long trains of echoes of a tribution of ionization in the outer ionosphere. Such whistler. The echo intensity decreases very slowly data presumably will come from whistler analysis. with order, and may even increase. It is difficult This theory, if correct, should provide a useful new to explain this behavior in terms of the passive tool for a study of the densities, velocities, and loca properties of the medium. Discrete echoes in a tions of streams of charged particles in the outer single train have been detected for as long as 4 min. ionosphere. These rather rare events are usually accompanied by magnetic disturbance and the dawn chorus or 7. Appendix: A Quantitative Comparison hiss. The whistler components which are most prolonged often lie in the 3- to 4-kc range where of the Theory With Observation the hiss is known to predominate. On the basis of these facts it is suggested that the low apparent 7.1. Example Chosen for Comparison With the attenuation in very long whistler trains can be Theory attributed to traveling wave amplification some where along the path. The whistler energy propa The theory presented above has been developed in gates along the same path followed by the incoming extensive numer ical applications by one of us solar stream and is amplified in the regions where the (R. M. G.) to see if the theory is able to account in whistler phase velocity approximates the stream detail for spectra of discrete vlf emissions, using a velocity. minimum of reasonable assumptions. One of the most puzzling phenomena is the inter There is considerable information on the distribu action between loud whistlers and vlf emissions. tion of electron density up to the height of maximum I If during a period of dawn chorus and whistler density in the ionosphere, but very little is lmown activity, an ordinary loud whistler should occur, it about densities above this height. In particular, the is frequently accompanied by one or more rising law governing the distribution of electron density l whistles (Storey [1]) . In attempting to explain this with height is not known for this region. In these connection in terms of the traveling wave concept, investigations several models of the dis tribution we suggest that the loud whistler may be capable of above the main body of the ionosphere were con accelerating an incoming stream of particles. For sidered. They were: Cons tan t electron density; a energy to be transferred from the whistler to the power law r - n where r is the dis tance from the cen tel' stream , the wave must accelerate the particles in of the earth and n was taken from 1 through 6; and its electrical field. Equation (2) shows that in a an exponential distribution e- k T where k is a con region where j p does not incr ease too rapidly with stant. The power law was found to give the best jH, the phase velocity of a fixed frequency wave will representation of certain distinguishing features of incr ease with jH' For our purposes the rate of the vlf emissions and also to be consistent with in change of whistler frequ ency with time is sufficiently formation obtained from parallel studies of whistlers
24 " (dispersion co nstants and analysis of nose whistlers). part of the figure. The latter is a typical member The arne criteria eem to require th e exponent of the "Hook" class of vlf emissions. This class is n~3. Furthermore, a power law distribution wi th one of the best defined, and the reproducibility of n~3 appears to be justifi ed theoretically in other the emissions is remarkably good in general shape. research in progress on t heO"l"avitational equilibrium However, no two hooks, even though separated by of ex tended atmo phere , at large distances from the only a few seconds in time, are exactly the same. earth's surface. It is perhaps significant for the Therefore, the exact matching of any individual tone physics of the exosph ere tILa t n~3 means that the is of no particular importance. In the example electron density is nearly proporLiona1 to the strength shown here, no great accuracy in matching was at of the ear th's magnetic field . tcmpted; the same computations made with a An example of a computed spectrum (frequency lightly lower speed for the cloud of particles (say ver us time at the earth's urface) of waves emitted 9,000 km/sec) would have resulted in a better match. by a discrete cloud of ionized particles moving with I t is more significant to r ealize that with reason constant speed parallel to the lines of force of the able numerical values for the parameters, the theory earth's magnetic fi eld is shown in the upper part of leads very na turally to calculated spectra very figure 3. This example was chosen to approximately similar to those observed in regard to values of match the observed spectrum shown in the lower frequency, duration, and ge neral shape.
12 I I I I 10 I I I I 1 8
~6 '+------~ 4
2
0 .0 0 .5 1.0 1.5 TIME , sec
---itf' I ~1• 1 ' , ,• . ., ,
U . ..x:: '+-
0.0 0 .5 1.0 1.5 2 .0 TIME, sec
FIGURE 3. Comparison of theory and observation (see appendix fOI' details).
25 7.2. Model of the Distribution of Electron Density value of the scale height used was ~ = 200 km. Below hmax, J'& = 100 km was used, The model of electron density versus altitude used Thus the total electron density for the model used is for the numerical calculations is illustrated in figure 4. Eleetron density (Ne/cm3) is given on the right scale, and the corresponding local plasma fre quency f71 in kilocycles can be read on the left scale. In the exosphere, far above the earth's surface (above 1,000 km), rather than r-3 , the distribution chosen was: + {KfH exp {-(h-1,000)2/J't'.2x} for h< l,OOO km} KfH for h> l,OOO km N e/H = constant, or (7) (9) where H is the local magnetic field strength and jH is the local electron gyrofrequency. Below 1,000 km, Note that only a small range of values of the con as a means of joining smoothly with a plausible stant K is possible if the electronic density in the ionospheric distribution, the above expression for exosphere is not to be too large near the earth (smaller than the lowest values of N max ) nor too small to per j~ is multiplied by the factor exp {-(h-1,000)2/ mit the propagation of whistlers at the greater J't';.}, where h is the height in kilometers above the distances. The value chosen here was 4,000 kc. earth's surface. The scale height J'~x is taken as 200 km. The ionospheric distribution is taken to be a 7.3. Geometric Conditions Chapman-like layer The calculations were made for an ionized cloud N.= N max e:A"p {Yz(l-z-e- Z), of particles movu1g along the line of force which in or tersects the earth's surface at the geomagnetic lati j~'lono=n,max exp {Yz(l-z-e-')}, (8) tude A= 55°. The observations were made at Boulder at geomagnetic latitude 49°, but it is known where Z= (h-hmax ) /.JIt: The chosen value of N max (10 6 from studies of whistlers that a signal propagating cm- 3) corresponds closely to the critical frequency in the whistler mode can be received at places far of the F2 layer at the place and time of the observa from the foot of the nominal line of force along which tion to be matched (Boulder, Colorado, October 26, the wave is propagated. For a dipole field, the line 1956, 2230 UT or 1530 local standard time). The corresponding to A= 55° intersects the plane of the value of hmax was taken as 350 km. Above hmax the geomagnetic equator at 2.04 earth radii, or 13,010 km above the earth's surface. (N ote that fig. 4 shows the model only out to the height 3,500 km.) The total length along the line of force from the equator to the earth's surface is 3.17 earth radii, or 20,214 km. At the equator the magnetic field strength along the line of force is a minimum. Its value is 0.011 gauss compared to 0.315 gauss at the surface. The corresponding gyrofrequency is 31.4 kc, and with the exosphere model (7) N .= 1,560 andjp= 354 kc at this height. Using as the origin of time a starting position in the equatorial plane, the calculations were made for points regularly spaced along the trajectory to obtain: (1 ) the frequencies emitted; (2) the time of the emission; (3) the time for propagation of the wave from this point to the earth's surface along the remaining arc, using the full formula for group velocity in the whistler mode; and, (4) the sum of these two times to obtain the arrival tilne at the earth. At 10,000 km/sec the total time of flight for the cloud is nearly 2 sec while the range of arrival tilnes of the emitted waves is about 76 sec. Of course, one observes only the range of arrival times. The "knee" in the "hook" in the present example 106L-_-'-_-'-_-.J.__ -'--_-'-_-'-_--' corresponds to the transition at about 2,500-km o · 1000 2000 3000 height between the electron density characteristic DISTANCE FROM EARTH'S SURFACE. k m of the exosphere and the more rapidly increasing FIGURE 4. Distribution of electron density usedfol' the theoretical density in the ionosphere, which is responsible for calculation shown in figure 3. the high branch of the "hook." 26 7.4. Velocity of the Cloud 8. References I Generally speaking, the shape of the computed [1] L. R . O. Storey, Ph. D. dissertation, Cambridge Uni spectrum will remain approximately the same for versity, England, and Phil. Trans. Roy Soc. [A]246, different velocitie. But the frequency and time 113 (1953). [2] R. A. Helliwell, J. H . Crary, J . I-I. Pope, and R. L. Smith, scales will vary rapidly. A small velocity furnishes The nose whi t Ier, a new high latitude phenomenon, a range of arrival times that arc too long and emitted J . Geophys. R esearch 61, 139 (1956). frequencies that arc too low. The durations and [3] G. R. E llis, J . Atmospheric and T errest. Phys. 8, 33 (1956) . frequen cies are very sensitive to the velociLy. [4] Ie. Maeda and 1. Kimura, A t heoretical in vestigation Velocities of about 1,000 km/sec, as uggested by the on the propagation path of t he whistlin g atmospherics, delay b etween related olar and terrestrial events, Rep. Ionosphere R esearch J apan 10, 105 (1956). give frequencies roughly ten times too small and [5] J . H. Pope, Diurnal variation in t he occurrence of D a wn Chorus, Nature 180, 433 (1957). durations 10 times too long. Satisfactory fits are [6] G. McE:. Allcock, A study of the audio-Frequency radio obtained only for velocities of the order of 10,000 phenomenon known as D awn Chorus, AusLralian km/sec. Such a high velocity seems to indicate that J. Phys. 10, 286 (1957). some process in the interaction between the earth's [7] H . E . Dinger, Whistling atmospherics, Nav. R esearch exosphere and the solar corpuscular radiations gives Lab. R ep . 4825 (September 14, 1956). [8] J. M . Watts, An observation of audio-freq uency elec high local velocities to the small clouds responsible tromagnetic noise during a period of sola r disturbance, for the vIf emissions. These clouds are tentatively J . Geophys. R esearch 62,199 (1957). associated with the flashes of luminosity in active [9] G. R. E llis, Low frequency radio emission from aurorae, J . Atmospheric and Terrest. Phys. 10, 302 (1957). auroras. [10] J. R. Pierce, Traveling-wave Tubes (D . Van Nostrand It is significant that this velocity is of the same Co., New York, N.Y., 1950). order as that required for protons to penetrate to [11 ] A. B. Meillel, Evidence for the entry into the upper the 100-km level in the aurora. atmosp here of high-speed protrons during auroral activity, Science, 112, 590 (1950). [12] L. Vegare! , Terrestrial magnetism and electricity, J . A. T ABLE 2. Summary of the condit'ions assumed for the sample theoretical calculation F leming, cd., p. 609 (McCraw-Hill, New York, 1 .Y., 1937). [J 3] Stanford Univer ity, Low-frequency propagation studies, Distribution of electron denSity I Other condi tions pt. 1, \Vhistlers ane! related phenomena, Final report, Contract AF 19 (604)- 795 (June 15, 1953 to Sep Ionos ph ere Exosphere tember 30, 1956). [1 4] D. F. Mar tyn, The theory of magnetic storms a nd Nmax=106/cm3 h.. = l,OOO km 1- =55° v.=1O' klll /s auroras, Nature, 167, 92 (1951 ). hm AX = 350 km £',= 200 kill JI.f.=200 km [(= 4,000 kc ,;1;;= 100 km
The work reported in this paper was supported partially by the Air Force Office of R esearch under Contract AF18 (603)- 126. The helpful sugges Lions of T . N. Gautier and L. R. O. SLorey arc gratefully acknowledged. BOULDER, COLO. (Paper 63Dl- 4).
27