- Email: [email protected]
On the amplification of VLF hiss
Planet. Sp~lceSci., Vol. 27, pp. 27%2&4. Pergamon Press Ltd., 1979. Printed in Northern Ireland
ON THE
AMPLIFICATION
OF VLF HISS
TAKASHl YAMAMOTO
Geophysics Research Laboratory, University of Tokyo, Tokyo 113, Japan (Received in final form 18 July 1978) Abstract-Based on the model calculation of VLF hiss power flux spectrum resulting from convective beam amplification of incoherent Cerenkov whistler radiation by the beam of precipitating aurora1 electrons, which has been developed by Maggs (1976), we examine the altitude dependence of power flux levels. Their strong altitude dependence leads us to suggest that non-linear processes are important in determining the spectrum of VLF hiss at high altitude. It is also shown that estimated power fluxes inside the electron precipitation region at low altitude might not reach as high levels as observed when the electron beam is weak. In this case, wave propagation outside of the precipitation region will account for the high power flux levels as well as significant magnetic components of VLF hiss observed especially at low altitude. In addition, we show that the transformation of the electron beam in transit to lower altitudes, determined from Liouville’s theorem, may influence appreciably VLF hiss power flux spectrum. Finally, it is pointed out that two types of VLF hiss spectrum observed at the ground level can be accounted for by the difference in strength of the electron beam.
1. JNTRODUCII~N
Estimations of VLF hiss intensity associated with auroras have been made by many authors, based on
the assumption that hiss is generated by incoherent Cerenkov or cyclotron radiation from the precipitating electrons (Jorgensen, 1968; Lim and Laaspere, 1972; James, 1973; Taylor and Shawhan, 1974; Maeda, 1975). Jorgensen (1968) calculated the flux of Cerenkov emission arising from electrons with energies of l-20 keV and obtained a peak amplitude of lo-r4 W/m’Hz at 10 kHz by assuming that all radiation is ducted unattenuated. Lim and Laaspere obtained a peak of the order of intensity of the power lo-r3 W/m’ Hz. However, the computations of Cerenkov radiation taking into account ray path spreading or collisional damping (James, 1973; Taylor and Shawhan, 1974) show that the incoherent mechanism cannot produce observed power fluxes as high as 10-‘3-10-” W/m’I-Iz (Gurnett, 1966; Barrington et al. 1971; Gurnett and Frank, 1972; Mosier and Gurnett, 1972). Therefore, it has been suggested that VLF hiss is generated by a coherent mechanism rather than incoherent Cerenkov radiation (Gurnett and Frank, 1972; James, 1973; Taylor and Shawhan, 1974; Swift and Kan, 1975; Noda and Tamao, 1976). Recently, Maggs (1976) has shown that the observed power fluxes of VLF hiss can be accounted for by convective beam amplification of incoherent Cerenkov whistler radiation by the beam of precipitating aurora1 electrons. It has clearly been demonstrated that the power flux expected from pre-
cipitating electrons without beam amplification cannot possibly account for strong VLF hiss. The beam amplification mechanism was investigated by using the lowest-order WKB wave kinetic equation and linear growth rates. His model and results are briefly summarized in the next section. The purpose of this paper is to develop some consequences resulting from Maggs’ model that have not been considered in his paper. In Section 3, we first examine the variation of temporal and spatial growth rate with altitude. Even if the velocity space distribution of the beam does not vary with altitude, the variation of cold plasma density causes a large change in the growth rate. If the altitude of the beam generation point, where the electrons are supposed to be accelerated downward by a kilovolt potential drop along the geomagnetic field lines (Frank and Ackerson, 1971; Reasoner and Chappell, 1973; Amoldy and Choy, 1973; Arnoldy, 1974; Swift, 1975; Kaufmann et al. 1976; Yamamoto, 1976), is much higher than the observation point, the variation of beam velocity distribution with altitude cannot be neglected. The beam distribution functions at any point along the geomagnetic field line can be determined from Liouville’s theorem and the conservation of the first adiabatic moment, provided that the beam is unaltered by wave-particle interactions. The effects of transformation of the beam on the linear growth rate are also considered. In Section 4, based on the results given in Section 3, we consider the altitude dependence of power flux levels of VLF hiss predicted by convective beam amplification. This leads 273
214
TAKASHI YAMAM~TO
us to suggest that non-linear processes play an important role in determining levels and spectral shapes of VLF hiss power flux at high altitude, if the altitude of the beam generation point is higher than the observation point. It is also shown that power fluxes at an observation point inside an electron precipitation region at low altitude might not rise to as high levels as observed when the electron beam is “weak,” because of the small growth rates and the short length of rays contributing power. The importance of wave propagation outside the precipitation region is pointed out in accounting for high power flux levels in the case of weak beam and significant magnetic components of VLF hiss, observed at low altitude. Finally, we point out that two types of VLF hiss frequency spectrum can be produced by the difference in “strength” of the electron beam. 2. MODEL
OF BEAM AMPLIFICATION VLFHISS
OF
In this section we review the model calculation of the power flux of VLF hiss resulting from beam amplification of incoherent Cerenkov whistler radiation, which has been developed by Maggs (1976). The differential power flux spectrum at an observation point in the middle of an electron precipitation region is calculated by using the lowest-order WKB wave kinetic equation. Integration of this equation gives the power flux per unit frequency at an altitude h
where w whistler frequency k, whistler parallel wave number at the observation point y linear growth rate V, group velocity E power flux per unit frequency of incoherent whistler mode radiation F geometric term resulting from the divergence of ray paths 1 length of a ray path passing through the observation point. The total differential power flux at the frequency w is obtained by calculating (1) for different initial values of k,, and summing the resultant power flux for each kll. In an aurora1 arc the electrons precipitate in a thin curved field-aligned sheet. The au-
Aurora1 arc model
w-T_
_L
FIG. 1. THE MODELELECTRON PRECIPITATION REGION. Noise reaching the satellite at frequencies near o, travels mainly down the magnetic field, while noise near T_ or Q_ propagates almost across the magnetic field (after Maggs, 1976).
roral electron precipitation region is modelled as a straight sheet of fixed north-south thickness aligned along the magnetic field. The effect of arc curvature is simulated by introducing a horizontal cutoff distance which varies with altitude as the radius of a flux tube. The thickness of the aurora1 beam and the horizontal cutoff distance at an altitude of - 1000 km are respectively taken as several kilometers and a few hundred. Figure 1 illustrates the model electron precipitation region (after Maggs, 1976). The electron plasma in the precipitation region is modelled as consisting of three electron populations: cold ionospheric electrons, secondary electrons, and precipitating electron beam. The precipitating electron beam is assumed to be represented by a drifting Maxwellian velocity distribution with a drift energy of -1 keV. The beam number density varies with altitude inversely as the flux tube cross-section, while the beam speed and temperature are taken to be constant with altitude. The secondary electrons are specified by two Maxwellian distributions with temperatures of tens of eV. The parameter values associated with the beam and secondaries, number densities, temperatures and beam drift energy, are chosen so that the observed characteristics of directional differential intensity of precipitating electrons, including a positive slope in the velocity distribution, can be modelled by the combination of beam and secondary electrons. The ionosphere is modelled as a horizontally stratified medium with hydrogen the only ionic constituent. The magnetic field strength varies with
275
On the amplification of VLF hiss
I
50
I
I
100
I
500 1000
Frequency,
KHZ
IO
20
50
100
Frequency,
200
KHz
FIG.~. POWER~UXSPE~PREDI~DBYTHEBEAMAMPL~FICATIONMECHANISMATALT~TUDESOF 1200 km WHEREr_ > fl_ AND (b) 3000 km WHERErr_
(a)
The horizontal cutoff distance is 100 km (after Maggs, 1976).
altitude but the field direction is taken to be vertical. Figure 2 shows the differential power flux spectra resulting from the beam amplification of incoherent whistler noise at altitudes of (a) 1200 and (b) 3000 km, taken from Maggs (1976). We note that the electron gyrofrequency R- is less than the electron plasma frequency r- at 1200 km altitude, while 51_ greater than n_ at 3000 km. Since the amplitude of the power flux is mostly determined by the exponential term in (l), the shape of the power flux spectrum can be explained by investigating the variations of ray path length and spatial growth rate y/V, with frequency. The ray paths at frequencies near oLH, where oLH is the lower hybrid frequency, are almost vertical, because the wave number vector k at o - or_,+is perpendicular to the magnetic field B and the group velocity perpendicular to k. Vertical ray paths have the longest lengths since they are not limited by the horizontal cutoff distance (see Fig. 1). However, the length of ray path along which the growth rate is positive becomes shorter near mLH because of a more rapid change in the parallel phase velocity along the ray path (see Fig. 8). The spatial growth rate near the lower hybrid resonance is proportional to cos 6, where 0 is the angle between B and k. As the frequency increases, the spatial growth rate in-
creases and the parallel phase velocity changes less rapidly, but the path length becomes limited by the horizontal cutoff distance and decreases. Consequently, it can be seen that the peak power fluxes occur near 20, and their amplitudes drop sharply near W- In the case of R_ < n-, the spatial growth rate goes to zero as the frequency approaches a_. Since the path lengths are limited by the horizontal cutoff distance, the power flux near fI_ drops rapidly. On the other hand, when a._> II_, the power flux is large and increasing for frequencies approaching II_ because of the increasing spatial growth rate. 3. VARIATION OF THE GROWI’R WITH ALllTUDE
RATE
We assume that the precipitating electron beam at the beam origin point is represented by a drifting Maxwellian velocity distribution. Though the altitude of the beam generation point (s’) is unknown, it is taken as 10,000 km in the following calculations (Swift, 1975). (In our calculations, the altitude is measured along a field line from the Earth’s surface, s.) Provided that the beam is unaltered by wave-particle interactions, the beam distribution function at an altitude s is given by the solution of the Vlasov equation which satisfies the assumed distribution function at an altitude s’. The
216
TAKASHIYAMAMOTO
result can be written as
F-(v,, u,, 4 = &
exp -
VIZ+ II,=+ u2 a2
[
1
Xexp[$(u,2+(I-~)uL2~]
(2)
where u, and u, are respectively the velocity component perpendicular and parallel to the magnetic field, n,, is the beam number density, a the thermal velocity, u the drift speed, B the magnetic field intensity, and a prime denotes a quantity at an altitude s’. In deriving (2) the conservation of the Iirst adiabatic moment (sin’ a/B = const, where a is the pitch angle) has been assumed. Following the calculation of the linear growth rate given by Maggs (1976)+ the growth rate for the beam distribution (2) can be written as ;=+4’-R’/Z)D,{iV(A*-R”IZ) + GACr + sin’ @C,}-’
(3)
where 4=-2n---_
nb 0 II_’
which is independent of -l.lX106(cm2seV)-‘, altitude. An addition of a suitable secondary electron distribution to the assumed beam distribution would give a typical differential flux spectrum of precipitating electrons observed in aurora1 regions 197 1; Reasoner and and Ackerson, (Frank 1973; Arnoldy and Choy, 1973; Chappell, Arnoldy, 1974; see also Fig. 2 of Maggs, 1976). Taking into account a great deal of variability in beam electron distributions, we later discuss the dependence of -y/o on ratio u/a. The cold electron density and geomagnetic field intensity at various altitudes are determined by the model of Maeda (1975). In his model, the magnetic field has been expressed by the “Mead-Fairfield Model” (Mead and Fairfield, 1975), and a static model of the iono-magnetospheric plasma distribution has been constructed with data accumulated by recent satellites (Alouette-I, -11, ISIS-I, OGO-4, -6 and Explorer 22). Figure 3 shows the electron gyrofrequency, a_ and the plasma frequency, n_ as functions of the altitude, which have been calculated based on this model (Maeda, 1975). These frequencies are plotted against the distance from the ground measured along the magnetic field lines, s
no k, k
X$ F-tat, w/k,, s) and the notations, R, Z, A, N, G, C, and C, are given in the Appendix. In (4) no is the cold plasma density, Jo the Bessel function of order 0, and
where k, is the wave number perpendicular to B. Only the factor DL in the expression for the growth rate given by (3) comes from the contribution from the beam electrons to the imaginary part of the whistler dispersion relation. The contribution from the secondaries is not included in DL. The real part of the dispersion relation has been determined by cold plasmas. For definiteness, we first specify the beam parameters as follows: drift speed u = 2.13 x lo4 km/s thermal velocity a = 1.17 x lo4 km/s
I
IO3
IO' S in km
beam number density n,, = 1.0 cnY3. They give a beam drift energy of 1.3 keV, a ratio of beam temperature to drift energy of 0.3, which are the same values as used in Maggs (1976), and a a=0 of intensity at differential peak
I
FIG.
THE
QUENCYOFELEcTRONS,n_,
DEPENDENCE
OF
GYROFRE-
ANDTHEPLASMAFREQUENCY,
T_, IN THE MAGNETOSPHERE ALONG THE 70” FIELD LINE FROM THE EARTH'S SURFACE, s (IN km) (AFTER MAZDA,
1975).
//-_--_
On the amplification of VLF hiss
lo-’ I
10000
-2 IO
$
.--3000
IO-3
km
km----..
lo-= IO
100 Frequency,
FIG.
MAXIMUMGROWIX
4. 0
AS FUNCTIONS
OF 0
RATES
1000 KHz
IN UNITS
AT VARIOUS
OF FREQUENCY
ALTITUDES.
Dashed curves correspond to the maximum growth rates for the original beam distribution represented by a drifting Maxwellian, and the full and dotted curves correspond to the cutoff values of CL,,,22.5 and 30.0, respectively. (in km), for the geomagnetic latitude at the surface 70” in the night-side. Our calculations are also made along the 70” field line in the night-side. The growth rate at a fixed frequency w is found to be peaked at some value of k, which satisfies ck,/o( = pII)- 18, when the beam distribution function is given by a drifting Maxwellian. However, when the beam distribution is transformed and approaching an isotropic distribution in transit to lower altitudes due to the variation of geomagnetic field with altitude, as shown later, the value of pll corresponding to the maximum growth rate increases. The waves with large values of ~11, however, are absorbed through a collisionless resonant interaction with secondary electrons. Then, we assume that the waves with pll larger than a cutoff value cannot grow. The cutoff values are taken as 22.5 and 30.0 (which correspond to resonant energies of 0.5 and 0.28 keV, respectively,) in order to represent the observed variation in the range of positive slope in the electron velocity distribution. Figure 4 shows the maximum growth rates against w at various altitudes denoted, and the full and
277
dotted curves correspond to the 22.5 and 30.0 cutoff, respectively. To elucidate the effect of beam transformation, we show the growth rates for the original drifting Maxwellian by the dashed curves. It should be noted that they increase with frequency at altitudes of 1000, 3000 and 10,000 km, while the growth rate drops near R- at 300 km altitude. This behavior is due to the fact that the waves are not electrostatic as the frequency approaches fi- in the case of R-
278
TAKASHI YAMAMOTO
/ 300
K7
7 13
I-B’/B=I.O
km
20
I5
30
25
CK,/w
II
14
I,
I 15
I,,,
II,,
,I
16
,,I,
17
3
CK,/w FIG.~. THEGROWIHRATES,$O,ATW= 1043kHz OF B'/BWTHE CASESOF u/a = 1.83 (a)
ASFUNCZTIONSOF~~~(C~,/~)FORDIFFERENTVALIJFS 10.0 (b) (FOR u = 2.13X lo4 km/s). They are calculated for parameter values of the ambient medium at 300-km altitude.
q(B’/B)( = j F_(B’/B) dv) for a given no. But, this is not true when the functional form of beam distribution changes. As was shown in Figs. 4 and 5, as for the present case, the growth rate decreases with B owing to the transformation of beam distribution in spite of an increase in the beam number density and the growth rate for the original beam is considered to be an upper limit to that for the transformed beam. We note that the growth rate without the beam transformation effect is proportional to the ratio of beam density to cold electron density, nb/no (see (3) and (4)). Taking account of the fact that the spatial growth rate y/u, is important in determining the amplitude
ANo
of the power flux rather than the growth rate y, we show in Fig. 8 the maximum spatial growth rates (in km-‘) as functions of frequency at various altitudes. The dashed curves correspond to the cases of the original beam distribution, and the full and dotted curves the 22.5 and 30.0 cutoff, respectively. 4. DISCUSSION
ON POWER
FLUX SPECTRA
1. VLF hiss intensity In estimating the amplitude of the VLF hiss power flux resulting from convective amplification of incoherent Cerenkov whistler noise, based on Maggs’ model, we must examine the length of ray
279
On the amplification of VLF hiss
I:
u/a =lO.O
c
:=5.0
ray paths the path lengths for frequencies near aLH can also be limited by the horizontal cutoff distance. However, the effective length of nearly vertical ray path may be determined from a change in the growth rate because the parallel phase velocity changes rapidly along the ray path. We then examine the change in the component of the refractive index parallel to the magnetic field caused by changes in the ambient medium. In calculating the variation of the parallel component of the refractive index ~11,the gradient in the cold electron density is assumed to be in the girection of the magnetic field. In this case the perpendicular component of the refractive index is constant along a ray path (Snell’s law). Figure 9 shows the distances measured in s (not path lengths), over which the values of ~11change from 14.0 to 22.5, as functions of frequency, where ~11at any frequency is taken as 14.0 at each altitude denoted. The value of ~11,14.0, corresponds to the electron drift speed assumed. Therefore, in other words, Fig. 9 shows the maximum distances (we shall call them the maximum amplification distances) over which the waves could keep being amplified through resonant interactions with the beam electrons, if the beam distribution were not changed from the original drifting Maxwellian and the precipitation region
1: =3.5 \
=I.83
8
FIG. WHOLE
6.
PEAK
GROWTH
pll RANGE
RATE
AT
AS A FUNCTION VALUES
OF
o = OF
1043 kH.z
B’/B
IN
THE
7
FOR DIFFERENT
u/a.
rate -y(B) is normalized to y(B = B’). Variation of cold plasma density n, with altitude (a300 km) is plotted against the corresponding B’/B (dotted line). The growth
paths (we shall call it the effective length of ray paths), passing through an observation point, contributing to the power flux as well as the spatial growth rate which has already been investigated. The effective path length is limited by the geometry of the precipitation region, i.e. the horizontal cutoff distance (only ray paths lying in planes containing the arc sheet are important), and by a change in the growth rate along a ray path. Tracing back along a ray arriving at the observation point, we see the direction of the ray path deviating from the magnetic field. In other words, the angle between the group velocity and the magnetic field increases along the ray path with altitude. This is because the frequency along the ray path, relative to the local value of a_ or II_, increases with altitude. (We note that this does not hold true in the altitude range below 300 km.) Because of the deflection of
6
0.5
I.0
0.0
B’/B FIG.
7.
VARIATION
OF BEAM
(= j F_ to)
NUMBER
AS A FUNCTION
DENSITY OF
B’IB.
n,(B’/B)
280
TAKASHIYAMAMOTO
I I
time being, we consider the growth rate for the beam distribution given by a drifting Maxwellian. We recall that this gives an upper limit to the actual growth rate for the transformed beam in the case of u/a = 1.83.) However, the linear theory could not be applied to estimating the power flux spectrum at high altitudes (roughly, above 3000 km altitude for the beam parameters chosen here) because of the large values of effective path length and spatial growth rate. This suggests that some non-linear mechanism could act to limit power flux levels at high altitudes and it could be important in determining the power flux spectrum. (The importance of a non-linear process in limiting power flux levels near fI_ in the altitude range where II-
i
I
-
I
,o-5,IO
1000
100
Frequency,
KHz
FIG. 8. ~~AXMUM SPATIALGROWTHRATES (km-‘)
AS
FUNfXIONSOFFREQUENcYATVARIOUSALTlTuDES.
Dashed curves correspond to the maximum spatial growth rates for the original beam distribution, and the full and dotted curves the 22.5 and 30.0 cutoff, respectively.
were spatially unlimited. The maximum amplification distance is much shorter at lower altitude due to more rapid changes in the cold electron density and the magnetic field with altitude. Since the horizontal cutoff distance is assumed to vary with altitude as the radius of a flux tube, the effective path length turns out to be much shorter when the observation point is located at lower altitude, whether it may be determined by the horizontal cutoff distance (in the case of higher frequencies) or the maximum amplification distance (lower frequencies). Since the power flux spectrum of VLF hiss is mostly determined by the exponential term in (l), which is roughly proportional to the product of effective path length and average spatial growth rate, the variations of effective path length and spatial growth rate with altitude (see Section 3) leads to a prediction of an anomalously strong altitude dependence of the power flux amplitude within the framework of the linear theory. (For the
sE ‘5; %
Frequency,
kHz
FIG.~. ~~XIMIJMAMPLIRCATIONDISTANCESMEASUREDIN s (km) ASF~N~~IONSOFFF~QUEN~Y. JL , which is 14.0 at each altitude denoted, changes from 24.5 to 14.0 over the maximum amplification distance, during wave propagation.
281
On the amplification of VLF hiss Papadopoulos and Coffey (1974), and Matthews et al. (1976) in relation to non-thermal features of the aurora1 plasma. The non-linear process discussed in their papers is as follows (other possible processes have been suggested by Maggs (1976)): the waves excited by the beam electrons, as an external driver wave, bring about the parametric instability which produces growing ion fluctuations and electron waves with much lower phase velocities. Wave energy is transferred from the driver waves to the new electron waves, and the beam can be stabilized against quasi-linear diffusion. Thus, the power flux of the driver waves is limited, whereas the new waves are eventually absorbed by the ionospheric plasma. The analyses of the effective path length and of the spatial growth rate also indicate that, at low altitudes (roughly, below 1000 km altitude for the assumed beam parameters), the wave may not enfold a sufficient number of times to be observable before it becomes damped by the beam electrons. To explain this situation more clearly, we have investigated to what extent 40 kHz waves emitted as Cerenkov radiation at different altitudes (I”= 1,) can be amplified along a ray path due to a resonant interaction with the beam electrons. This can be represented by amplification factors
40 KHz
S in km
10. A~~PLIFICATION FACTORS FOR 40-kHz WAVES EMITTED AT FIG.
exp(j!0b(2y/ug)dl”) ALTITUDES
DENOTED
(SOURCE ALTITUDES), AS FUNCI'IONS OF THE ALTITUDE OF OBSERVATIONPOINT,S.
e*p as functions of the altitude of observation point. The result is shown in Fig. 10, where the value of ~11at I” = 1, has been taken as 22.5 so that the wave emitted at each altitude can be amplified most. In these calculations the amplification region has been assumed to be spatially unlimited. As apparent from this figure, the power flux amplitude in the altitude range below about 1000 km cannot rise to levels as high as observed. Here, we note that the amplification factor at the low-frequency peak of a power flux spectrum calculated by Maggs (1976) is about 25. It should also be noted that the noise emitted at an altitude much higher than an observation point is eventually absorbed by the beam electrons even if it is amplified sufficiently, and cannot contribute to the power flux at the observation point in the aurora1 arc. However, if the observation point is outside the precipitation region, the situation becomes vastly different. Consider an observation point in planes containing the arc sheet, but outside the arc, and ray paths originating in the precipitation region at an altitude much higher than the observation point and arriv-
The
value of pII at I” = I, is taken as 22.5.
As is illustrated in Fig. 11, we can take a ray path the most part of which is outside the precipitation region, even if the observation point is near the precipitation region. This is due to the fact that the ray direction tends to the magnetic field through downward propagation, as was indicated previously. If the waves are sufficiently amplified in the precipitation region, these intense waves can propagate unattenuated along these ray paths to the observation point at a much lower altitude, since their resonant velocities increase during propagation while the electron beam is assumed to be very weak or non-existent outside the precipitation region. Thus, it is suggested that high power flux levels of VLF hiss can be observed at low altitudes outside the beam rather than inside. However, if the electron beam is stronger than assumed in our numerical calculations, i.e. it has larger number density, wider range of positive slope in its velocity distribution, or steeper positive slope, high power flux levels will also be observed inside the beam at low altitudes. For example, in Fig. 12, we show the peak growth rate y in the ing at the point.
282
TAKASHIYAMAMOTO
i
‘\
,
f
I
I
I
I
E
I I I
i
I ‘ecipi+atio, region I I ?N, & .
Observation paint
FIG. 11. A RAY PATH WHICH ORIGINATES IN THE PRJXIPITATION REGION AND ARRIVES AT AN OBSERVATION POINT OUTSIDE OF THE REGION. The most part of the ray
path is outside of the region.
whole CL,,range at a frequency of 1043 kHz as a function of the ratio u/a. Here, the drifting Maxwellian is assumed as the beam distribution and y is taken as unity when u/a = 1.83. Though the calculation has been done for parameter values of the
ambient medium at 300 km altitude and for the fixed frequency, the result is almost independent of altitudes and frequencies except near oLH and n_(K). The important point to be noted is that the above growth rate ratio is dependent on the parameters n and a, approximately, only through the ratio u/a. Furthermore we note that, when a becomes larger, n,, must be taken as larger, proportional to a3, and then y must be larger so that the beam distribution can give the same peak differential intensity at o! = 0. Combination of Figs. 6 and 12 leads us to conclude that large values of y are retained even at low altitudes in the case of larger u/a ratio (for a fixed nb). In this case power flux levels will be limited by such non-linear processes as discussed earlier in a much wider range of altitudes and waves of observable power flux levels can be emitted at low altitudes. Finally, it should be noted that the whistler mode VLF produced at high altitudes approaches the local lower hybrid resonance frequency as it propagates toward the ionosphere and some of the lower frequency VLF-waves are reflected back into the magnetosphere. Consequently, when the electron beam is weak (i.e. VLF-waves cannot be amplified enough except at high altitudes), VLF hiss at the ground levels will have a comparatively narrow and distinct frequency spectrum having a lowerfrequency limit near the maximum mLH in the altitude range. On the other hand, in the strong beam case VLF hiss with the broad band spectrum
‘O’ti Y
u/a FIG.
12.
PEAK GROWTH RATE y AT o = 1043 kHz IN THE WHOLE U/ll.
CL,, RANGE
AS A FUNCTION OF THE RATIO
The drifting Maxwellian is assumed as the beam distribution and y is taken as unity when u/a = 1.83. It is calculated for parameter values of the ambient medium at 300 km altitude.
On the amplification of VLF hiss
will be observed at the groumj levels since it can be sufficiently amplified even at low altitudes. These two types of VLF hiss observed at the ground stations have already been reported. (Makita and Fukunishi, 1973). 2. A magnetic field component of VLF hiss Though VLF hiss is generated with wave normal angles close to the resonance cone angle because of the large refractive index, it will deviate from the resonance cone angle through propagation. For a constant power as the wave normal angle of a wave approaches the resonance cone angle, the electric field of the wave increases, and the magnetic field decreases. The magnetic component can be substantial for wave number vectors only a small angular distance from the resonance cone angle (Taylor and Shawhan, 1974). Since the waves can propagate longer distances unattenuated outside the precipitation region, the magnetic field associated with VLF hiss will be detected outside the beam. Therefore, the power flux of VLF hiss with a significant magnetic component, measured by satellites, (Jarrgensen, 1968; Mosier and Gurnett, 1972), can be explained by propagation of waves amplified at much higher altitudes. Acknowledgements-I am grateful to Dr. T. Tamao for helping to initiate this work, and Dr. K. Hayashi, Mr. A. Noda and other members at The Geophysics Research laboratory for discussions. Important comments on the present calculation have been given by the referee, to whom I am very grateful. It is a-pleasure for me to thank Mr. K. Makita for his valuable comments on the recent observation of VLF hiss.
R.EmRENm
Arnoldy, R. L. (1974). Auroral particle precipitation and Birkeland currents. Reu. geophys. 12, 217. Amoldy, R. L. and Choy, L. W. (1973). Anroral electrons of energy less than 1 keV observed at rocket altitudes. 1. geophys. Res. 78, 2187. Amoldy, R. L., Lewis, P. B. and Issacson, P. 0. (1974). Field-aligned aurora1 electron fluxes. J. geophys. Res. 79, 4208. Barrington, R. E., Hartz, T. R. and Harvey, R. W. (1971). Diurnal distribution of ELF, VLF, and LF noise at high latitudes as observed by Alouette 2. J. geophys. Res. 76, 5278. Frank, L. A. and Ackerson, K. L. (1971). Observations of charged particle precipitation in the auroral zone. J. geophys. Res. 76, 3612. Gurnett, D. A. (I%@. A satellite study of VLF hiss. 3. geophys. Res. 71, 5599.
283
Gnmett,
D. A. and Frank, L. A. (1972). VLF hiss and related plasma observations in the polar magnetosphere. 1. geophys. Res. 77, 172. James, H. G. (1973). ~stler-me hiss at low and medium frequencies in the dayside-cusp ionosphere. J. geoph.sy. ReS. 78, 4578. _ Jorgensen, T. S. (1968). Intermetation of aurora1 hiss &ea.&d on O&O-2. and at-Byrd station in terms of incoherent Cerenkov radiation. J. geophys. Res. 73, 1055. Kaufmann, R. L., Walker, D. N. and Amoldy, R. L. (1976). Acceleration of anroral electrons in parallel electric fields. J. geo~hys. Res. Sl, 1673. Lim, T. L. and Laaspere, T. (1972). An evaluation of the intensity of Cerenkov radiation from auroral electrons with energies down to 100eV. J. geophys. Res. 77, 4145. Maeda, K. (1975). A calculation of aurora1 hiss with improved models for geoplasma and magnetic field. Planet. Space Sci. 23, 843. Makita, K. and Fukunishi, H. (1973). Observation of VLF einissions at Syowa Station in 1970-1971-I. Relationship between the occurrence of aurora1 hiss emissions and the location of aurora1 arcs. Report of the Japanese Antarctic Research Expedition, 46, 1.
Maggs. J. E. (1976). Coherent generation of VLF hiss. J. geophys. Res. 81, 1707. Matthews, D. L., Pongratz, M. and Papadoponlos, K. (1976). Nonlinear production of suprathermal tails in anroral electrons. J. geophys. Res. 31, 123.
Mead, G. D. and Fairfield, D. H. (1975). Quantitative magnetospheric models derived from spacecraft magnetometer data. 1. geophys. Res. So, 523. Mosier, S. R. and Gnmett, D. A. (1972). Observed correlations between auroral and VLF emissions. J. geophys. Res. 77, 1137. Noda, A. and Tamao, T. (1976). The model dependence of differential power flux spectra of incoherent Cerenkov radiation. J. geophys. Res. 81, 287. Papadopoulos, K. and Coffey, T. (1974). Nonthermal features of the anroral plasma due to precipitating electrons. .I. geophys. Res. 79, 674. Reasoner, D. L. and Chappell, C. R. (1973). Twin payload observations of incident and backscattered au&al electrons. J. geophys. Res. 78, 2176. Swift, D. W. (1975). On the formation of aurora1 arcs and acceleration of &oral electrons. J. geophys. Res. SO, 2096. Swift, D. W. and Kan, J. R. (1975). A theory of aurora1 hiss and implifications on the origin of auroral elec-
trons. J. geophys. Res. 80, 985. Taylor, W. W. L. and Shawhan, S. D. (1974). A test of incoherent Cerenkov radiation for VLF hi and other ma~et~phe~c emissions. J. geophys. Res. 79, 105. Yamamoto, T. (lQ76). On the formation of electrostatic shock waves associated with aurora1 electron precipitation. Planet. Space Sci. 24, 1073.
APPENDIX
Notations used in (3) are as follows:
J+(!!X~
+)’
+
284 A=/.L~(~-Z)-~+Z-R+(~-Z)~R/Z
TAKASHIYAMAMOTO C,=A-2(p2-Z)
S=R+(l-Z)(l-6R/Z) G = S sin’ 0 + (1 - Z){ 1 - (1 + @R/Z} cosz 13 N= -Z+R(6+cos20)/Z
where c is the speed of light and S the electron to ion mass ratio.
ON THE
AMPLIFICATION
OF VLF HISS
TAKASHl YAMAMOTO
Geophysics Research Laboratory, University of Tokyo, Tokyo 113, Japan (Received in final form 18 July 1978) Abstract-Based on the model calculation of VLF hiss power flux spectrum resulting from convective beam amplification of incoherent Cerenkov whistler radiation by the beam of precipitating aurora1 electrons, which has been developed by Maggs (1976), we examine the altitude dependence of power flux levels. Their strong altitude dependence leads us to suggest that non-linear processes are important in determining the spectrum of VLF hiss at high altitude. It is also shown that estimated power fluxes inside the electron precipitation region at low altitude might not reach as high levels as observed when the electron beam is weak. In this case, wave propagation outside of the precipitation region will account for the high power flux levels as well as significant magnetic components of VLF hiss observed especially at low altitude. In addition, we show that the transformation of the electron beam in transit to lower altitudes, determined from Liouville’s theorem, may influence appreciably VLF hiss power flux spectrum. Finally, it is pointed out that two types of VLF hiss spectrum observed at the ground level can be accounted for by the difference in strength of the electron beam.
1. JNTRODUCII~N
Estimations of VLF hiss intensity associated with auroras have been made by many authors, based on
the assumption that hiss is generated by incoherent Cerenkov or cyclotron radiation from the precipitating electrons (Jorgensen, 1968; Lim and Laaspere, 1972; James, 1973; Taylor and Shawhan, 1974; Maeda, 1975). Jorgensen (1968) calculated the flux of Cerenkov emission arising from electrons with energies of l-20 keV and obtained a peak amplitude of lo-r4 W/m’Hz at 10 kHz by assuming that all radiation is ducted unattenuated. Lim and Laaspere obtained a peak of the order of intensity of the power lo-r3 W/m’ Hz. However, the computations of Cerenkov radiation taking into account ray path spreading or collisional damping (James, 1973; Taylor and Shawhan, 1974) show that the incoherent mechanism cannot produce observed power fluxes as high as 10-‘3-10-” W/m’I-Iz (Gurnett, 1966; Barrington et al. 1971; Gurnett and Frank, 1972; Mosier and Gurnett, 1972). Therefore, it has been suggested that VLF hiss is generated by a coherent mechanism rather than incoherent Cerenkov radiation (Gurnett and Frank, 1972; James, 1973; Taylor and Shawhan, 1974; Swift and Kan, 1975; Noda and Tamao, 1976). Recently, Maggs (1976) has shown that the observed power fluxes of VLF hiss can be accounted for by convective beam amplification of incoherent Cerenkov whistler radiation by the beam of precipitating aurora1 electrons. It has clearly been demonstrated that the power flux expected from pre-
cipitating electrons without beam amplification cannot possibly account for strong VLF hiss. The beam amplification mechanism was investigated by using the lowest-order WKB wave kinetic equation and linear growth rates. His model and results are briefly summarized in the next section. The purpose of this paper is to develop some consequences resulting from Maggs’ model that have not been considered in his paper. In Section 3, we first examine the variation of temporal and spatial growth rate with altitude. Even if the velocity space distribution of the beam does not vary with altitude, the variation of cold plasma density causes a large change in the growth rate. If the altitude of the beam generation point, where the electrons are supposed to be accelerated downward by a kilovolt potential drop along the geomagnetic field lines (Frank and Ackerson, 1971; Reasoner and Chappell, 1973; Amoldy and Choy, 1973; Arnoldy, 1974; Swift, 1975; Kaufmann et al. 1976; Yamamoto, 1976), is much higher than the observation point, the variation of beam velocity distribution with altitude cannot be neglected. The beam distribution functions at any point along the geomagnetic field line can be determined from Liouville’s theorem and the conservation of the first adiabatic moment, provided that the beam is unaltered by wave-particle interactions. The effects of transformation of the beam on the linear growth rate are also considered. In Section 4, based on the results given in Section 3, we consider the altitude dependence of power flux levels of VLF hiss predicted by convective beam amplification. This leads 273
214
TAKASHI YAMAM~TO
us to suggest that non-linear processes play an important role in determining levels and spectral shapes of VLF hiss power flux at high altitude, if the altitude of the beam generation point is higher than the observation point. It is also shown that power fluxes at an observation point inside an electron precipitation region at low altitude might not rise to as high levels as observed when the electron beam is “weak,” because of the small growth rates and the short length of rays contributing power. The importance of wave propagation outside the precipitation region is pointed out in accounting for high power flux levels in the case of weak beam and significant magnetic components of VLF hiss, observed at low altitude. Finally, we point out that two types of VLF hiss frequency spectrum can be produced by the difference in “strength” of the electron beam. 2. MODEL
OF BEAM AMPLIFICATION VLFHISS
OF
In this section we review the model calculation of the power flux of VLF hiss resulting from beam amplification of incoherent Cerenkov whistler radiation, which has been developed by Maggs (1976). The differential power flux spectrum at an observation point in the middle of an electron precipitation region is calculated by using the lowest-order WKB wave kinetic equation. Integration of this equation gives the power flux per unit frequency at an altitude h
where w whistler frequency k, whistler parallel wave number at the observation point y linear growth rate V, group velocity E power flux per unit frequency of incoherent whistler mode radiation F geometric term resulting from the divergence of ray paths 1 length of a ray path passing through the observation point. The total differential power flux at the frequency w is obtained by calculating (1) for different initial values of k,, and summing the resultant power flux for each kll. In an aurora1 arc the electrons precipitate in a thin curved field-aligned sheet. The au-
Aurora1 arc model
w-T_
_L
FIG. 1. THE MODELELECTRON PRECIPITATION REGION. Noise reaching the satellite at frequencies near o, travels mainly down the magnetic field, while noise near T_ or Q_ propagates almost across the magnetic field (after Maggs, 1976).
roral electron precipitation region is modelled as a straight sheet of fixed north-south thickness aligned along the magnetic field. The effect of arc curvature is simulated by introducing a horizontal cutoff distance which varies with altitude as the radius of a flux tube. The thickness of the aurora1 beam and the horizontal cutoff distance at an altitude of - 1000 km are respectively taken as several kilometers and a few hundred. Figure 1 illustrates the model electron precipitation region (after Maggs, 1976). The electron plasma in the precipitation region is modelled as consisting of three electron populations: cold ionospheric electrons, secondary electrons, and precipitating electron beam. The precipitating electron beam is assumed to be represented by a drifting Maxwellian velocity distribution with a drift energy of -1 keV. The beam number density varies with altitude inversely as the flux tube cross-section, while the beam speed and temperature are taken to be constant with altitude. The secondary electrons are specified by two Maxwellian distributions with temperatures of tens of eV. The parameter values associated with the beam and secondaries, number densities, temperatures and beam drift energy, are chosen so that the observed characteristics of directional differential intensity of precipitating electrons, including a positive slope in the velocity distribution, can be modelled by the combination of beam and secondary electrons. The ionosphere is modelled as a horizontally stratified medium with hydrogen the only ionic constituent. The magnetic field strength varies with
275
On the amplification of VLF hiss
I
50
I
I
100
I
500 1000
Frequency,
KHZ
IO
20
50
100
Frequency,
200
KHz
FIG.~. POWER~UXSPE~PREDI~DBYTHEBEAMAMPL~FICATIONMECHANISMATALT~TUDESOF 1200 km WHEREr_ > fl_ AND (b) 3000 km WHERErr_
(a)
The horizontal cutoff distance is 100 km (after Maggs, 1976).
altitude but the field direction is taken to be vertical. Figure 2 shows the differential power flux spectra resulting from the beam amplification of incoherent whistler noise at altitudes of (a) 1200 and (b) 3000 km, taken from Maggs (1976). We note that the electron gyrofrequency R- is less than the electron plasma frequency r- at 1200 km altitude, while 51_ greater than n_ at 3000 km. Since the amplitude of the power flux is mostly determined by the exponential term in (l), the shape of the power flux spectrum can be explained by investigating the variations of ray path length and spatial growth rate y/V, with frequency. The ray paths at frequencies near oLH, where oLH is the lower hybrid frequency, are almost vertical, because the wave number vector k at o - or_,+is perpendicular to the magnetic field B and the group velocity perpendicular to k. Vertical ray paths have the longest lengths since they are not limited by the horizontal cutoff distance (see Fig. 1). However, the length of ray path along which the growth rate is positive becomes shorter near mLH because of a more rapid change in the parallel phase velocity along the ray path (see Fig. 8). The spatial growth rate near the lower hybrid resonance is proportional to cos 6, where 0 is the angle between B and k. As the frequency increases, the spatial growth rate in-
creases and the parallel phase velocity changes less rapidly, but the path length becomes limited by the horizontal cutoff distance and decreases. Consequently, it can be seen that the peak power fluxes occur near 20, and their amplitudes drop sharply near W- In the case of R_ < n-, the spatial growth rate goes to zero as the frequency approaches a_. Since the path lengths are limited by the horizontal cutoff distance, the power flux near fI_ drops rapidly. On the other hand, when a._> II_, the power flux is large and increasing for frequencies approaching II_ because of the increasing spatial growth rate. 3. VARIATION OF THE GROWI’R WITH ALllTUDE
RATE
We assume that the precipitating electron beam at the beam origin point is represented by a drifting Maxwellian velocity distribution. Though the altitude of the beam generation point (s’) is unknown, it is taken as 10,000 km in the following calculations (Swift, 1975). (In our calculations, the altitude is measured along a field line from the Earth’s surface, s.) Provided that the beam is unaltered by wave-particle interactions, the beam distribution function at an altitude s is given by the solution of the Vlasov equation which satisfies the assumed distribution function at an altitude s’. The
216
TAKASHIYAMAMOTO
result can be written as
F-(v,, u,, 4 = &
exp -
VIZ+ II,=+ u2 a2
[
1
Xexp[$(u,2+(I-~)uL2~]
(2)
where u, and u, are respectively the velocity component perpendicular and parallel to the magnetic field, n,, is the beam number density, a the thermal velocity, u the drift speed, B the magnetic field intensity, and a prime denotes a quantity at an altitude s’. In deriving (2) the conservation of the Iirst adiabatic moment (sin’ a/B = const, where a is the pitch angle) has been assumed. Following the calculation of the linear growth rate given by Maggs (1976)+ the growth rate for the beam distribution (2) can be written as ;=+4’-R’/Z)D,{iV(A*-R”IZ) + GACr + sin’ @C,}-’
(3)
where 4=-2n---_
nb 0 II_’
which is independent of -l.lX106(cm2seV)-‘, altitude. An addition of a suitable secondary electron distribution to the assumed beam distribution would give a typical differential flux spectrum of precipitating electrons observed in aurora1 regions 197 1; Reasoner and and Ackerson, (Frank 1973; Arnoldy and Choy, 1973; Chappell, Arnoldy, 1974; see also Fig. 2 of Maggs, 1976). Taking into account a great deal of variability in beam electron distributions, we later discuss the dependence of -y/o on ratio u/a. The cold electron density and geomagnetic field intensity at various altitudes are determined by the model of Maeda (1975). In his model, the magnetic field has been expressed by the “Mead-Fairfield Model” (Mead and Fairfield, 1975), and a static model of the iono-magnetospheric plasma distribution has been constructed with data accumulated by recent satellites (Alouette-I, -11, ISIS-I, OGO-4, -6 and Explorer 22). Figure 3 shows the electron gyrofrequency, a_ and the plasma frequency, n_ as functions of the altitude, which have been calculated based on this model (Maeda, 1975). These frequencies are plotted against the distance from the ground measured along the magnetic field lines, s
no k, k
X$ F-tat, w/k,, s) and the notations, R, Z, A, N, G, C, and C, are given in the Appendix. In (4) no is the cold plasma density, Jo the Bessel function of order 0, and
where k, is the wave number perpendicular to B. Only the factor DL in the expression for the growth rate given by (3) comes from the contribution from the beam electrons to the imaginary part of the whistler dispersion relation. The contribution from the secondaries is not included in DL. The real part of the dispersion relation has been determined by cold plasmas. For definiteness, we first specify the beam parameters as follows: drift speed u = 2.13 x lo4 km/s thermal velocity a = 1.17 x lo4 km/s
I
IO3
IO' S in km
beam number density n,, = 1.0 cnY3. They give a beam drift energy of 1.3 keV, a ratio of beam temperature to drift energy of 0.3, which are the same values as used in Maggs (1976), and a a=0 of intensity at differential peak
I
FIG.
THE
QUENCYOFELEcTRONS,n_,
DEPENDENCE
OF
GYROFRE-
ANDTHEPLASMAFREQUENCY,
T_, IN THE MAGNETOSPHERE ALONG THE 70” FIELD LINE FROM THE EARTH'S SURFACE, s (IN km) (AFTER MAZDA,
1975).
//-_--_
On the amplification of VLF hiss
lo-’ I
10000
-2 IO
$
.--3000
IO-3
km
km----..
lo-= IO
100 Frequency,
FIG.
MAXIMUMGROWIX
4. 0
AS FUNCTIONS
OF 0
RATES
1000 KHz
IN UNITS
AT VARIOUS
OF FREQUENCY
ALTITUDES.
Dashed curves correspond to the maximum growth rates for the original beam distribution represented by a drifting Maxwellian, and the full and dotted curves correspond to the cutoff values of CL,,,22.5 and 30.0, respectively. (in km), for the geomagnetic latitude at the surface 70” in the night-side. Our calculations are also made along the 70” field line in the night-side. The growth rate at a fixed frequency w is found to be peaked at some value of k, which satisfies ck,/o( = pII)- 18, when the beam distribution function is given by a drifting Maxwellian. However, when the beam distribution is transformed and approaching an isotropic distribution in transit to lower altitudes due to the variation of geomagnetic field with altitude, as shown later, the value of pll corresponding to the maximum growth rate increases. The waves with large values of ~11, however, are absorbed through a collisionless resonant interaction with secondary electrons. Then, we assume that the waves with pll larger than a cutoff value cannot grow. The cutoff values are taken as 22.5 and 30.0 (which correspond to resonant energies of 0.5 and 0.28 keV, respectively,) in order to represent the observed variation in the range of positive slope in the electron velocity distribution. Figure 4 shows the maximum growth rates against w at various altitudes denoted, and the full and
277
dotted curves correspond to the 22.5 and 30.0 cutoff, respectively. To elucidate the effect of beam transformation, we show the growth rates for the original drifting Maxwellian by the dashed curves. It should be noted that they increase with frequency at altitudes of 1000, 3000 and 10,000 km, while the growth rate drops near R- at 300 km altitude. This behavior is due to the fact that the waves are not electrostatic as the frequency approaches fi- in the case of R-
278
TAKASHI YAMAMOTO
/ 300
K7
7 13
I-B’/B=I.O
km
20
I5
30
25
CK,/w
II
14
I,
I 15
I,,,
II,,
,I
16
,,I,
17
3
CK,/w FIG.~. THEGROWIHRATES,$O,ATW= 1043kHz OF B'/BWTHE CASESOF u/a = 1.83 (a)
ASFUNCZTIONSOF~~~(C~,/~)FORDIFFERENTVALIJFS 10.0 (b) (FOR u = 2.13X lo4 km/s). They are calculated for parameter values of the ambient medium at 300-km altitude.
q(B’/B)( = j F_(B’/B) dv) for a given no. But, this is not true when the functional form of beam distribution changes. As was shown in Figs. 4 and 5, as for the present case, the growth rate decreases with B owing to the transformation of beam distribution in spite of an increase in the beam number density and the growth rate for the original beam is considered to be an upper limit to that for the transformed beam. We note that the growth rate without the beam transformation effect is proportional to the ratio of beam density to cold electron density, nb/no (see (3) and (4)). Taking account of the fact that the spatial growth rate y/u, is important in determining the amplitude
ANo
of the power flux rather than the growth rate y, we show in Fig. 8 the maximum spatial growth rates (in km-‘) as functions of frequency at various altitudes. The dashed curves correspond to the cases of the original beam distribution, and the full and dotted curves the 22.5 and 30.0 cutoff, respectively. 4. DISCUSSION
ON POWER
FLUX SPECTRA
1. VLF hiss intensity In estimating the amplitude of the VLF hiss power flux resulting from convective amplification of incoherent Cerenkov whistler noise, based on Maggs’ model, we must examine the length of ray
279
On the amplification of VLF hiss
I:
u/a =lO.O
c
:=5.0
ray paths the path lengths for frequencies near aLH can also be limited by the horizontal cutoff distance. However, the effective length of nearly vertical ray path may be determined from a change in the growth rate because the parallel phase velocity changes rapidly along the ray path. We then examine the change in the component of the refractive index parallel to the magnetic field caused by changes in the ambient medium. In calculating the variation of the parallel component of the refractive index ~11,the gradient in the cold electron density is assumed to be in the girection of the magnetic field. In this case the perpendicular component of the refractive index is constant along a ray path (Snell’s law). Figure 9 shows the distances measured in s (not path lengths), over which the values of ~11change from 14.0 to 22.5, as functions of frequency, where ~11at any frequency is taken as 14.0 at each altitude denoted. The value of ~11,14.0, corresponds to the electron drift speed assumed. Therefore, in other words, Fig. 9 shows the maximum distances (we shall call them the maximum amplification distances) over which the waves could keep being amplified through resonant interactions with the beam electrons, if the beam distribution were not changed from the original drifting Maxwellian and the precipitation region
1: =3.5 \
=I.83
8
FIG. WHOLE
6.
PEAK
GROWTH
pll RANGE
RATE
AT
AS A FUNCTION VALUES
OF
o = OF
1043 kH.z
B’/B
IN
THE
7
FOR DIFFERENT
u/a.
rate -y(B) is normalized to y(B = B’). Variation of cold plasma density n, with altitude (a300 km) is plotted against the corresponding B’/B (dotted line). The growth
paths (we shall call it the effective length of ray paths), passing through an observation point, contributing to the power flux as well as the spatial growth rate which has already been investigated. The effective path length is limited by the geometry of the precipitation region, i.e. the horizontal cutoff distance (only ray paths lying in planes containing the arc sheet are important), and by a change in the growth rate along a ray path. Tracing back along a ray arriving at the observation point, we see the direction of the ray path deviating from the magnetic field. In other words, the angle between the group velocity and the magnetic field increases along the ray path with altitude. This is because the frequency along the ray path, relative to the local value of a_ or II_, increases with altitude. (We note that this does not hold true in the altitude range below 300 km.) Because of the deflection of
6
0.5
I.0
0.0
B’/B FIG.
7.
VARIATION
OF BEAM
(= j F_ to)
NUMBER
AS A FUNCTION
DENSITY OF
B’IB.
n,(B’/B)
280
TAKASHIYAMAMOTO
I I
time being, we consider the growth rate for the beam distribution given by a drifting Maxwellian. We recall that this gives an upper limit to the actual growth rate for the transformed beam in the case of u/a = 1.83.) However, the linear theory could not be applied to estimating the power flux spectrum at high altitudes (roughly, above 3000 km altitude for the beam parameters chosen here) because of the large values of effective path length and spatial growth rate. This suggests that some non-linear mechanism could act to limit power flux levels at high altitudes and it could be important in determining the power flux spectrum. (The importance of a non-linear process in limiting power flux levels near fI_ in the altitude range where II-
i
I
-
I
,o-5,IO
1000
100
Frequency,
KHz
FIG. 8. ~~AXMUM SPATIALGROWTHRATES (km-‘)
AS
FUNfXIONSOFFREQUENcYATVARIOUSALTlTuDES.
Dashed curves correspond to the maximum spatial growth rates for the original beam distribution, and the full and dotted curves the 22.5 and 30.0 cutoff, respectively.
were spatially unlimited. The maximum amplification distance is much shorter at lower altitude due to more rapid changes in the cold electron density and the magnetic field with altitude. Since the horizontal cutoff distance is assumed to vary with altitude as the radius of a flux tube, the effective path length turns out to be much shorter when the observation point is located at lower altitude, whether it may be determined by the horizontal cutoff distance (in the case of higher frequencies) or the maximum amplification distance (lower frequencies). Since the power flux spectrum of VLF hiss is mostly determined by the exponential term in (l), which is roughly proportional to the product of effective path length and average spatial growth rate, the variations of effective path length and spatial growth rate with altitude (see Section 3) leads to a prediction of an anomalously strong altitude dependence of the power flux amplitude within the framework of the linear theory. (For the
sE ‘5; %
Frequency,
kHz
FIG.~. ~~XIMIJMAMPLIRCATIONDISTANCESMEASUREDIN s (km) ASF~N~~IONSOFFF~QUEN~Y. JL , which is 14.0 at each altitude denoted, changes from 24.5 to 14.0 over the maximum amplification distance, during wave propagation.
281
On the amplification of VLF hiss Papadopoulos and Coffey (1974), and Matthews et al. (1976) in relation to non-thermal features of the aurora1 plasma. The non-linear process discussed in their papers is as follows (other possible processes have been suggested by Maggs (1976)): the waves excited by the beam electrons, as an external driver wave, bring about the parametric instability which produces growing ion fluctuations and electron waves with much lower phase velocities. Wave energy is transferred from the driver waves to the new electron waves, and the beam can be stabilized against quasi-linear diffusion. Thus, the power flux of the driver waves is limited, whereas the new waves are eventually absorbed by the ionospheric plasma. The analyses of the effective path length and of the spatial growth rate also indicate that, at low altitudes (roughly, below 1000 km altitude for the assumed beam parameters), the wave may not enfold a sufficient number of times to be observable before it becomes damped by the beam electrons. To explain this situation more clearly, we have investigated to what extent 40 kHz waves emitted as Cerenkov radiation at different altitudes (I”= 1,) can be amplified along a ray path due to a resonant interaction with the beam electrons. This can be represented by amplification factors
40 KHz
S in km
10. A~~PLIFICATION FACTORS FOR 40-kHz WAVES EMITTED AT FIG.
exp(j!0b(2y/ug)dl”) ALTITUDES
DENOTED
(SOURCE ALTITUDES), AS FUNCI'IONS OF THE ALTITUDE OF OBSERVATIONPOINT,S.
e*p as functions of the altitude of observation point. The result is shown in Fig. 10, where the value of ~11at I” = 1, has been taken as 22.5 so that the wave emitted at each altitude can be amplified most. In these calculations the amplification region has been assumed to be spatially unlimited. As apparent from this figure, the power flux amplitude in the altitude range below about 1000 km cannot rise to levels as high as observed. Here, we note that the amplification factor at the low-frequency peak of a power flux spectrum calculated by Maggs (1976) is about 25. It should also be noted that the noise emitted at an altitude much higher than an observation point is eventually absorbed by the beam electrons even if it is amplified sufficiently, and cannot contribute to the power flux at the observation point in the aurora1 arc. However, if the observation point is outside the precipitation region, the situation becomes vastly different. Consider an observation point in planes containing the arc sheet, but outside the arc, and ray paths originating in the precipitation region at an altitude much higher than the observation point and arriv-
The
value of pII at I” = I, is taken as 22.5.
As is illustrated in Fig. 11, we can take a ray path the most part of which is outside the precipitation region, even if the observation point is near the precipitation region. This is due to the fact that the ray direction tends to the magnetic field through downward propagation, as was indicated previously. If the waves are sufficiently amplified in the precipitation region, these intense waves can propagate unattenuated along these ray paths to the observation point at a much lower altitude, since their resonant velocities increase during propagation while the electron beam is assumed to be very weak or non-existent outside the precipitation region. Thus, it is suggested that high power flux levels of VLF hiss can be observed at low altitudes outside the beam rather than inside. However, if the electron beam is stronger than assumed in our numerical calculations, i.e. it has larger number density, wider range of positive slope in its velocity distribution, or steeper positive slope, high power flux levels will also be observed inside the beam at low altitudes. For example, in Fig. 12, we show the peak growth rate y in the ing at the point.
282
TAKASHIYAMAMOTO
i
‘\
,
f
I
I
I
I
E
I I I
i
I ‘ecipi+atio, region I I ?N, & .
Observation paint
FIG. 11. A RAY PATH WHICH ORIGINATES IN THE PRJXIPITATION REGION AND ARRIVES AT AN OBSERVATION POINT OUTSIDE OF THE REGION. The most part of the ray
path is outside of the region.
whole CL,,range at a frequency of 1043 kHz as a function of the ratio u/a. Here, the drifting Maxwellian is assumed as the beam distribution and y is taken as unity when u/a = 1.83. Though the calculation has been done for parameter values of the
ambient medium at 300 km altitude and for the fixed frequency, the result is almost independent of altitudes and frequencies except near oLH and n_(K). The important point to be noted is that the above growth rate ratio is dependent on the parameters n and a, approximately, only through the ratio u/a. Furthermore we note that, when a becomes larger, n,, must be taken as larger, proportional to a3, and then y must be larger so that the beam distribution can give the same peak differential intensity at o! = 0. Combination of Figs. 6 and 12 leads us to conclude that large values of y are retained even at low altitudes in the case of larger u/a ratio (for a fixed nb). In this case power flux levels will be limited by such non-linear processes as discussed earlier in a much wider range of altitudes and waves of observable power flux levels can be emitted at low altitudes. Finally, it should be noted that the whistler mode VLF produced at high altitudes approaches the local lower hybrid resonance frequency as it propagates toward the ionosphere and some of the lower frequency VLF-waves are reflected back into the magnetosphere. Consequently, when the electron beam is weak (i.e. VLF-waves cannot be amplified enough except at high altitudes), VLF hiss at the ground levels will have a comparatively narrow and distinct frequency spectrum having a lowerfrequency limit near the maximum mLH in the altitude range. On the other hand, in the strong beam case VLF hiss with the broad band spectrum
‘O’ti Y
u/a FIG.
12.
PEAK GROWTH RATE y AT o = 1043 kHz IN THE WHOLE U/ll.
CL,, RANGE
AS A FUNCTION OF THE RATIO
The drifting Maxwellian is assumed as the beam distribution and y is taken as unity when u/a = 1.83. It is calculated for parameter values of the ambient medium at 300 km altitude.
On the amplification of VLF hiss
will be observed at the groumj levels since it can be sufficiently amplified even at low altitudes. These two types of VLF hiss observed at the ground stations have already been reported. (Makita and Fukunishi, 1973). 2. A magnetic field component of VLF hiss Though VLF hiss is generated with wave normal angles close to the resonance cone angle because of the large refractive index, it will deviate from the resonance cone angle through propagation. For a constant power as the wave normal angle of a wave approaches the resonance cone angle, the electric field of the wave increases, and the magnetic field decreases. The magnetic component can be substantial for wave number vectors only a small angular distance from the resonance cone angle (Taylor and Shawhan, 1974). Since the waves can propagate longer distances unattenuated outside the precipitation region, the magnetic field associated with VLF hiss will be detected outside the beam. Therefore, the power flux of VLF hiss with a significant magnetic component, measured by satellites, (Jarrgensen, 1968; Mosier and Gurnett, 1972), can be explained by propagation of waves amplified at much higher altitudes. Acknowledgements-I am grateful to Dr. T. Tamao for helping to initiate this work, and Dr. K. Hayashi, Mr. A. Noda and other members at The Geophysics Research laboratory for discussions. Important comments on the present calculation have been given by the referee, to whom I am very grateful. It is a-pleasure for me to thank Mr. K. Makita for his valuable comments on the recent observation of VLF hiss.
R.EmRENm
Arnoldy, R. L. (1974). Auroral particle precipitation and Birkeland currents. Reu. geophys. 12, 217. Amoldy, R. L. and Choy, L. W. (1973). Anroral electrons of energy less than 1 keV observed at rocket altitudes. 1. geophys. Res. 78, 2187. Amoldy, R. L., Lewis, P. B. and Issacson, P. 0. (1974). Field-aligned aurora1 electron fluxes. J. geophys. Res. 79, 4208. Barrington, R. E., Hartz, T. R. and Harvey, R. W. (1971). Diurnal distribution of ELF, VLF, and LF noise at high latitudes as observed by Alouette 2. J. geophys. Res. 76, 5278. Frank, L. A. and Ackerson, K. L. (1971). Observations of charged particle precipitation in the auroral zone. J. geophys. Res. 76, 3612. Gurnett, D. A. (I%@. A satellite study of VLF hiss. 3. geophys. Res. 71, 5599.
283
Gnmett,
D. A. and Frank, L. A. (1972). VLF hiss and related plasma observations in the polar magnetosphere. 1. geophys. Res. 77, 172. James, H. G. (1973). ~stler-me hiss at low and medium frequencies in the dayside-cusp ionosphere. J. geoph.sy. ReS. 78, 4578. _ Jorgensen, T. S. (1968). Intermetation of aurora1 hiss &ea.&d on O&O-2. and at-Byrd station in terms of incoherent Cerenkov radiation. J. geophys. Res. 73, 1055. Kaufmann, R. L., Walker, D. N. and Amoldy, R. L. (1976). Acceleration of anroral electrons in parallel electric fields. J. geo~hys. Res. Sl, 1673. Lim, T. L. and Laaspere, T. (1972). An evaluation of the intensity of Cerenkov radiation from auroral electrons with energies down to 100eV. J. geophys. Res. 77, 4145. Maeda, K. (1975). A calculation of aurora1 hiss with improved models for geoplasma and magnetic field. Planet. Space Sci. 23, 843. Makita, K. and Fukunishi, H. (1973). Observation of VLF einissions at Syowa Station in 1970-1971-I. Relationship between the occurrence of aurora1 hiss emissions and the location of aurora1 arcs. Report of the Japanese Antarctic Research Expedition, 46, 1.
Maggs. J. E. (1976). Coherent generation of VLF hiss. J. geophys. Res. 81, 1707. Matthews, D. L., Pongratz, M. and Papadoponlos, K. (1976). Nonlinear production of suprathermal tails in anroral electrons. J. geophys. Res. 31, 123.
Mead, G. D. and Fairfield, D. H. (1975). Quantitative magnetospheric models derived from spacecraft magnetometer data. 1. geophys. Res. So, 523. Mosier, S. R. and Gnmett, D. A. (1972). Observed correlations between auroral and VLF emissions. J. geophys. Res. 77, 1137. Noda, A. and Tamao, T. (1976). The model dependence of differential power flux spectra of incoherent Cerenkov radiation. J. geophys. Res. 81, 287. Papadopoulos, K. and Coffey, T. (1974). Nonthermal features of the anroral plasma due to precipitating electrons. .I. geophys. Res. 79, 674. Reasoner, D. L. and Chappell, C. R. (1973). Twin payload observations of incident and backscattered au&al electrons. J. geophys. Res. 78, 2176. Swift, D. W. (1975). On the formation of aurora1 arcs and acceleration of &oral electrons. J. geophys. Res. SO, 2096. Swift, D. W. and Kan, J. R. (1975). A theory of aurora1 hiss and implifications on the origin of auroral elec-
trons. J. geophys. Res. 80, 985. Taylor, W. W. L. and Shawhan, S. D. (1974). A test of incoherent Cerenkov radiation for VLF hi and other ma~et~phe~c emissions. J. geophys. Res. 79, 105. Yamamoto, T. (lQ76). On the formation of electrostatic shock waves associated with aurora1 electron precipitation. Planet. Space Sci. 24, 1073.
APPENDIX
Notations used in (3) are as follows:
J+(!!X~
+)’
+
284 A=/.L~(~-Z)-~+Z-R+(~-Z)~R/Z
TAKASHIYAMAMOTO C,=A-2(p2-Z)
S=R+(l-Z)(l-6R/Z) G = S sin’ 0 + (1 - Z){ 1 - (1 + @R/Z} cosz 13 N= -Z+R(6+cos20)/Z
where c is the speed of light and S the electron to ion mass ratio.