Approximate derivation of self-exciting whistler-mode sideband wave frequencies

Transcription

1 Indian Journal of Radio & Space Physics Vol. 31, June 2002, pp Approximate derivation of self-exciting whistler-mode sideband wave frequencies M Ikeda Musashi University, Toyotamakami , Nerima-ku, Tokyo , Japan Received 14 September 2001 ; revised 4 February 2002; accepted 19 March 2002 Using a new equation system, a possibility of the sideband wave generation in whistler-mode via a non-linear Dopplershifted cyclotron resonant interaction between untrapped electrons and the whistler mode carrier signal is examined. The untrapped electrons resonant with the quasi-monochromatic whistler mode signal are phase-bunched with the trajectory gap, just outside the separatrix, on the phase space in the frame of electron of the Doppler-shifted cyclotron resonance with the carrier. Then, they may be able to radiate the whistler mode sideband waves with frequencies of fundamental, second and third harmonics, whose currents may never be zero because of strong non-linear interaction. It is imagined that, at the same time, two kinds of plasma may interact with the carrier signal to form the broadening, and with the sideband waves showing frequencies of fundamental, second and third harmonics. The fundamental, second and third harmonics sideband wave frequencies may be related to the saturated amplitude of the carrier signal. )r 1 Introduction A quasi-monochromatic carrier whistler wave transmitted to the magnetosphere may self-excitedly generate sideband waves around the frequency of the carrier signal under special conditions. There are many theoretical and numerical studies I -7 on whistler mode sideband generation, but another possibility for the sideband generation mechanism is presented in this paper. Matsumot0 8 and Omura et al. 9 reviewed theoretical and numerical studies of whistler mode non-linear interactions, and the analyses of sideband generation were also examined by them However, the mechanisms of whistler mode sideband generation were introduced there, as unresolved problems. Brinca l considered that the distortion caused by a large-amplitude whistler on the energetic-electron velocity distribution of a predominantly cold magneto-plasma might make the whistler test waves grow largely and consecutively at the early stages of the wave-particle interaction. Karpman et al. 2 considered that the modulational instability of the quasi-monochromatic whistler-mode wave in the equatorial region led to the non-linear frequency shift coitesponding to sideband generation observed experimentally. Denavit and Sudan 3 considered that resonant electrons in a large-amplitude whistler wave acquired a correlation in phase of their perpendicular velocities with respect to the wave magnetic field, which resulted in unstable growth of other whistler waves with frequencies near the main wave frequency. Nunn 4 investigated the non-linear sideband stability of VLF waves excited by non-linear cyclotron resonance, assuming that the treatment was non-selfconsistent, the wave field was specified a priori and the non-linear resonant particle current was directly computed. Tripathi and Patel 5 considered the process that two whistler waves with nearly equal frequency UlI-Ul2 and forward and backward travelling components exerted a parallel ponderomotive force on the electrons and generated a non-linear current to drive the first-order sidebands. Taking account of Canadian power distribution system, Sa,6 considered that single-frequency sidebands were due to the combined effect of a transmitted wave from Siple and radiation present in the duct at multiples of 60 Hz. Sotnikov et al. 7 considered that the non-linear coupling of the VLF transmitter signal to natural ELF emission was invoked to explain the symmetric sidebands. In this paper, a possibility of the sideband wave generation in whistler-mode via a non-linear Dopplershifted cyclotron resonant interaction between untrapped electrons and the whistler mode carrier signal has been examined. The untrapped electrons resonant with the quasi-monochromatic whistler mode signal are phase-bunched with the trajectory gap just outside the separatrix. Even if external magnetic field, ambient plasma density and amplitude of carrier whistler mode signal are all constant, the phasebunched electrons outside the separatrix can generate currents on the plane perpendicular to the external magnetic field.

2 122 INDIAN J RADIO & SPACE PHYS, JUNE Modelling Equations of electron motion in the wave-electron Doppler-shifted cyclotron resonance frame in whistler mode are given as follows (see Ref. 8 for in homogeneous geomagnetic field). have, d~o --=ku 0 dt Z... (4)... (1)... (5) du_ ): --' = -u 1. ro,sin", dl... (2) U 1.0 = constant... (6)... (7)... (3) where I is the time, U1. the velocity perpendicular to the external magnetic field and Uz the velocity relative to the resonant one (VR) and parallel to the external magnetic field. The symbol S represents the phase angle between the wave field vector b and the velocity vector U1., and k the wave number of the monochromatic whistler mode signal. Further no and rol are the cyclotron angular frequencies for external magnetic field and wave field, respectively. The laboratory coordinate system is shown in Fig. 1. The parameter U1. is separated into U1.0 of constant component and U1.1 which is a perturbation. Furthermore, the parameters (S, U Z ) are separated each into two terms of (So,UzO ) and (SI,Uzi ), where So and UzO follow pendulum equations, and both of SI and Uzi are perturbations added to pendulum motions 'o. Thus we... (8)... (9) The current produced by the untrapped electrons is defined as follows. J 1. (/) = -nle(u1.0 +U1.I)COS(~o +~I) X h(u 1.0' U 1.li' ~()i)u 1.0~u 1.0~u zoi ~~O i... (l0) where, h(u1.o, U zo, ~o) is a loss-cone type distribution function. The subscript 'i' means initial conditions. The ~ means the arbitrary increment of each parameter, and nl is an energetic electron number x Fig. I- Laboratory coordinate system [where I is the time, III the velocity amplitude perpendicular to the external magnetic field, <p the phase angle of the velocity vector Ill, ~ is related to the phase angle between the wave field vector b and the velocity vector Ill, k the wave number of the monochromatic whistler mode signal, w the angular frequency of the monochromatic carrier wave. The external magnetic field points towards the z direction, and the wave vector k also points towards the identical direction. while the resonant electrons move towards the z direction.]

3 IKEDA: DERIVATION OF WHISTLER-MODE SIDEBAND WAVE FREQUENClES 123 density, and e the electric charge. It is assumed that Iuzol and IUzd«lnolkl, IU-Ld«lu-Lol and tentatively l~d«1 in order to search for approximate solutions in the initial stage of time. Then Eqs (7)-(9) become very simple as follows: du.0 0. _.1_1::::: (() sm): dt k 1 ':>0... (11)... (12)... (13)... (16) where, y is the function of time t, V R the Dopplershifted cyclotron resonance velocity, ~i the initial value of ~, K( lit;,) the complete elliptic integral of the first kind with modulus lit;" by which trajectories of untrapped electrons for (, > 1 are classified, and F(~/2, 11(,) the elliptic integral of the first kind. The u«(,) means the potential energy of 0 order for the ~I perturbation and expressed as follows: Differentiating Eq. (13) and then substituting Eq. (12) into it, we get:... (17)... (14) Only the untrapped electrons just outside the separatrix for the sideband generation are taken into account. The untrapped electrons are resonant with the monochromatic whistler mode signal, but not trapped in the potential well in the phase space, while the phase angle ~o proceeds conforming to the pendulum equation. Furthermore, the phase angle ~I is added to ~o. Consequently the phase angle (~+~I) changes with time in a complicated manner. Equation (14) for ~I variation has the resonance point located just outside the separatrix on the phase diagram (u z - ~ diagram). Next, the parameters relating to the resonance point are searched. Equations (4)-(6) are solved as zero order solutions of 'Jacobian elliptic functions' corresponding to the pendulum motion with the constant modulus t;,. The zero order solutions are substituted into Eq. (14) and then Fourier-expanded to the lowest order. As a result, a Mathieu's equation with forced oscillation terms is derived as follows:... (15) where, K, is {~) and K, is K[ JI-,I, ), and further r«(,), CI«('), C2«(,) are complicated functions with variable (,. Equation (15) is obtained from Eq. (14) under the condition in which ~I is sufficiently small, but it has a significant meaning in the aim of looking for the phase resonance points. When ~I equalizes to se2(y, r) which is a Mathieu function, and A:; (r) is an eigen value of se2(y, r), then A resonance condition is obtained as follows:... (18)... (19)... (20) As shown in Fig. 2, a solution of Eq. (20) is that ~ ; = This value is an approximate one because of the lowest order approximation, but it is sure that this means the existence of phase resonance. Actually, the exact value should be derived from the future computer simulation. After time elapses

4 124 INmAN J RAmo & SPACE PHYS, JUNE 2002 sufficiently, the SI of trajectories near the S~ may be large enough positively or negatively to result in the trajectory gap on the phase diagram, thus untrapped electrons may get trapped due to the phasesynchronized deceleration by the V Rxb electric field of the carrier signal. Next, the trajectory gap may result in the frequency gap between the sideband wave and the carrier signal. Accordingly, taking account of only the term giving the fundamental sideband frequency, the electric current, as a function of time, is given from Eq. (10) on the base of strong non-linear influence as follows. J 1. (t) = nleu 1.1 (t)sin~o (t). ~I (t)h(u 1.0' u zo' ~o) x U 1.0~u 1.0~u z Oi~~Oi (21) It is Fourier-expanded to the lowest order, then the fundamental frequency is carefully selected. Near the resonance point, U1.I is approximately proportional to cos(2ny) and further both sinso and SI approximately become proportional to sin(2ny). As a result, h also becomes proportional to sin(2ny), and the discrete frequencies of the sideband waves Ii' are approximately given from Eqs (16) and (20) as follows. where, K(1/ t;~. ) z 7.9, and n is a natural number. => " c: zeta... (22) I o I I I. ()()()()()6 I I I I t---~-~-~--~-~-~-~--, OJ w => -' <: >- z w C!> w L- --' Fig. 2--Curves showing the eigen value Az'(r) of Mathieu function sez, and It(~) [Both of the eigen value A2'(r) of Mathieu function sez, and It(~) are functions of parameter ~ (zeta).] Furthermore, trapped electrons for r, < I can also have another phase resonance condition just inside the separatrix in the same manner, but the contribution to the sideband generation formed from them is similar to that of untrapped electrons for r, > 1, because they also get untrapped. The contribution from the electrons, extremely near the phase resonance points, may be negligible due to the reduced electron number density in untrapped condition. 3 Results and discussion According to the results obtained by Park", sideband separations for the carrier signal frequency of 4.44 khz were measured at the intervals of 1 sec. It can be seen that there was no clear relationship between the carrier signal amplitude and the sideband separation, and that the sideband separations ranged from -2 to 100 Hz. However, Fig. 5 of Ref. 11 showed that the sideband separations were plotted for the period hrs UT on 26 July 1977, and they gathered around 30 Hz. Furthermore, Rastan! et ai. 12 showed that the measured sideband spacing were ± (30 ± 2 Hz) for the carrier signal frequency of khz. The values of parameters obtained by Park" were used here to estimate the sideband separation in a rough sense (which were: fir =4.03 khz, L-4.23 and Ne=313/cc). For the comparison between the results obtained from the parameters of Park" and those of Sonwalkar et ai. 13, which are fir =2.4kHz, L-5. 1 and Ne=280/cc in the observation of January 1988, the values of sideband wave frequency are estimated and shown in Tables 1 and 2. According to the paper of Park I I, Table 2 represents that the sideband wave frequency is 32.5 Hz for the transmitted wave frequency fir=4.03 khz, amplitude b= 11 my, resonant perpendicular energy E1.2=8.8 key, and the parameters described above. This value is much similar to the observed frequencies 11 of whistler-mode sideband waves. The parameters of Table 2 correspond to those generally seen in ~=4.23 located inside LI=5.1. In order to express simply the energy changes due to the ExB drift from LI=5.1 to ~=4.23, it is assumed that the magnetic moment 11 conserves along the flow line, even if the periods, places and parallel energies are different, namely E1.2=E1.1 ( i :::: E1. I. Based on this simple assumption, E1.I=2.5 key corresponds to E1.2=4.4 key, E1.I=5.0 key corresponds to E1.2=8.8 key, and E1.I=7.5 key corresponds to E1.2=13.1 key. The results of pitch angles in Table 2 are similar to the ones shown in Table I, but the sideband spectra of.il.

5 IKEDA: DERIVATION OF WHISTLER-MODE SIDEBAND WAVE FREQUENCIES 125 Table I-Fundamental sideband frequency (Hz) fori., =2.4 khz, N c =280.0/cm 3, LI=5.1, and E 1P =0.315 key Sideband freguencr and Eitch angle at Transmitted Eu=2.5 key E.u=5.0 key E1.I=7.5 key wave amplitude 7.0 my 18.7(40.4) 22.3(48.0) 24.6(53.1) 9.0 my 21.2( 45.8) 25.3(54.4) 27.9(60.2) 11.0 my 23.5(50.6) 27.9(60.2) 30.9(66.6) 13.0 my 25.5(55.0) 30.4(65.4) 33.6(72.4) Pitch angle (0) Note: Values in the bracket represent the broadening widths of carrier signal. Table 2-Fundamental sideband frequency (Hz) for 1...=4.03 khz, N c =313.0/cm 3, ~=4. 23, and E II2 =0.97 key Sideband freguencr and Eitch angle at Transmitted wave E1.2=4.4 key E1.2=8.8 key E1.2=13.1 key amplitude 7.0 my 9.0 my 11.0 my 13.0 my 21.8(45.9) 25.9(54.5) 24.7(52.0) 29.4(61.8) 27.3(57.5) 32.5(68.4) 29.7(62.5) 35.3(74.3) 28.6(60.2) 32.5(68.3) 35.9(75.5) 39.0(82.1) Pitch angle (0) 75 Note: Values in the bracket represent the broadening widths of carrier signal. Tables 1 and 2 are different from each other. Generally, the sideband separations of Table 2, which are coincident with the observed ones, are a little greater than the results of Table 1. This is because the geomagnetic field increases in intensity from LI=5.1 to ~=4.23, even if the periods, places and parallel energies of resonant electrons do not connect each other. On the other hand, Table 1 represents that the sideband wave frequency is 27.9 Hz for the transmitted wave frequency Ir,=2.4 khz, amplitude b=11 my, resonant perpendicular energy E1.I=5.0 key, and the parameters described by Sonwalkar et at. 13 and Carpenter et al. 14. According to Sonwalkar et at. 13, the energies and pitch angles for the resonant electrons range from 0.6 key and 40 to 11 key and 80, respectively. The parameters used in Table 1 are within the range of energies and pitch angles obtained by Sonwalkar et at. 13 and Carpenter et al. 14, where the parallel energy of the electrons resonant with the whistler-mode carrier signal is key and the total energies (E1.I+ EIII) are 2.815,5.315, and key, respectively. As a result, the parameters used in Table 1 and the values shown in Table 1 are reasonable for the magnetospheric condition of Ir" Ne and LI at the corresponding period. The perpendicular energies E1.2 in Table 2 are greater than those in Table 1 and further the separation frequencies are greater in Table 2 than those in Table 1. This tendency may be coincident with the perpendicular acceleration of the energetic electrons drifting towards the earth due to the dawn-to-dusk electric field 15, or coincident with the perpendicular energy distribution of electrons in the outer radiation bele 6. These sideband waves may be hidden or intensified by the overlap of the broadening width of the carrier signal 6.f This broadening width is given from the Doppler-shifted cyclotron resonance condition and the extent of the separatrix on the phase space 6.u, = 2~ U 1.0 W I, and - k... (23) where, 0) and O)p are the wave angular frequency and the plasma angular frequency, respectively. In Tables 1 and 2, the values of 6.1 are given in the brackets located on the right of sideband frequency values Ii'. The values of 6.1 are maximum in the case where all the emissions radiated by the whole trapped electrons appear perfectly on the spectrum films, but the observed broadening width is generally smaller than the values of 6.1 described in Tables 1 and 2. According to Sonwalkar et at. 13, the bandwidths of the signal received at Lake Mistissini were about 20 Hz. One of the conclusion in this paper is that the possibility of self-exciting whistler-mode sideband waves generated in the magnetosphere is not negligible. Thus, the possibility of the self-exciting whistler-mode sideband wave generation is examined. Three kinds of the possibility are presented below. Possibility I-Overlapping of the broadening signal with frequency I (fr, -Af < I < Ir, +6.j) and the selfexciting sideband waves with frequency I (f> Ii' + Irr), where Irr means the frequency of transmitted signal. The sideband waves may be intensified inside the broadening signals, in addition to the causes of amplification due to some instabilities. It is imagined that the band widths of whistler-mode sideband waves may be included within the frequency interval between the curves of I { and 6.f Otherwise, the

6 126 INDIAN J RADIO & SPACE PHYS, JUNE 2002 whistler-mode sideband waves may appear outside the broadening width of the transmitted carrier signal and have the gap of frequency on the spectrum film, because the actual broadening, for example of 20 Hz, is generally smaller than!>:..f Possibility 1 is shown in Fig. 3, where the curves of Ii' and!>:..f are described for the perpendicular energies of the Doppler-shifted cyclotron resonant electrons, namely 2.5, 5.0, and 7.5 key, respectively. Possibility 2-The whistler-mode sideband waves may be observed, if the second or third harmonics of whistler-mode sideband are amplified outside the broadening width of the carrier signal and the fundamental of whistler-mode sideband is hidden by the broadening carrier signal on the spectrum fi lm. Possibility 2 is shown in Fig. 4, where the perpendicular energy of 0.75 key is used so as to generate the second and third harmonics of whistlermode sideband signal with a gap above the line of!>:..f Possibility 3-This mechanism include the idea that the electrons related to the Doppler-shifted interaction with the carrier signal may be different from those related to the generation of the whistlermode sideband waves. It is probable that two kinds of perpendicular velocities may be related to the generation and amplificati on of whistler-mode sideband waves at the same time. Ikeda et al. 17 indicated that the two kinds of energetic electrons might overlap to amplify the whistler-mode carrier signal at the same place in the magnetosphere. It is thought that this overlap is possible to explain amplification due to any whistler-mode instabilities and, therefore, to explain various kinds of amplitude spectrum of whistler-mode sideband waves described on the film. Possibility 3 is shown in Fig. 5, where the frequency broadening!>:..f of the whistler-mode carrier signal is described for the perpendicular energy of 0.5 key as an example, and the sideband waves emerging above the broadening carrier signal are described for the perpendicular energy of 12.5 key. Taking account of the observed variety of spectrum forms in whistlermode sideband waves, it is considered that Possibility 3 is most probable among the three. In many studies related to instability theory, several distribution forms of energetic electrons in addition to cold electrons have been employed for calculation of growth rates. Amplitudes of Siple signals observed onboard satellites are presented as a few my by Inan et al. ls, 0.1 m y by Kimura et al. 19 and m y by Rastani et al. 12. All the amplitudes shown in those studies are much smaller than 11 my. These may be due to the fact that the signals were detected before amplification or before exciting sideband waves. If the signals described above are amplified by 30 db near the equatorial plane l8, it is considered that the amplitudes of the carrier signals may become 1-10 my, which are consistent with those predicted from the model described in this paper. Be1l 2D showed 12 my as the amplitude observed onboard after ampli fication. The amplification of 30 db were estimated by many researchers It is also inferred r ~~~ 80.0 ~ N ~ 60.0 >- <..:> :z w => c w 40.0 c:: I.L E.ll =7. 5keV [Fundamental] ~_~ ke V b (PT) Fig. 3- Approxim ati on of wave frequenci es with fundamental o f the sideband current speclra as calculated from Eq. (22) [The'sideband frequencies are g iven for parameters assumed at LI=5. 1,};,=2.4 kh z, and Nc=280/cc. The fundamenjai sideband wave frequencies may be re l:lied to the saturated amplitude b of the carrier signal and E.l I o f Ihe e nergy perpendi cular to th e ex ternal magnetic fi eld. na me ly and 7.5 kc V. T he broadening width /1,./ in the same way, is also shown here from Eq. (23).J

7 IKEDA: DERIVATION OF WHISTLER-MODE SIDEBAND WAVE FREQUENCIES 127 from the gain of the Siple signals observed on the ground 24 that the amplification of 30 db obtained through the equatorial plane in the magnetosphere is very probable for Siple signals propagating along the field line near L=4.1. As the difference of factor is negligible in relation to the amplitude of carrier wave magnetic field, the saturated amplitude of the carrier signal may show the possibility of7-13 my. Furthermore, Draganov and Taranenk0 25 also presented the idea that un trapped particles have some influence on non-linear wave-particle interaction. Although amplitudes of the sideband waves generated "'" N ::I: '-' >- C,,) :z UJ :::l UJ 0:: u Broadening width El. 1 =O.75keV Fundamental Second Ha rmonics b (pn Fig. 4-Approximation of wave frequencies with fundamental, second and third harmonics of the sideband current spectra as calculated from Eq. (22) [The sideband frequencies are given for parameters assumed at LI=5.1, 1..-=2.4 khz, and Ne=280/cc. The sideband wave frequencies may be related to the saturated amplitude b of the carrier signal and E1.1 of the energy perpendicular to the external magnetic field, namely 0.75 key. The broadening width 6./, in the same way, is also shown here from Eq. (23).) , , 80. 0,... N C ~... Jl.l =12. 5keV 1~~~B~ro:a:d:en:i~n:g~w~i:dt:h~E~1.~1~O~.~5;ke;v~~~~~~~~~~~~~~~~~::~ El.l =10. 0keV ffi ~ ~ El.l 7. 5keV 20. 0?-" [Fundamental] 0. 0 L- ~ ~ ~ ~ ~L ~ ~ ~ ~ ~ b (pn Fig. 5-Approximation of wave frequencies with fu ndamental of the sideband current spectra is calculated from Eq. (22) [The sideband frequenc ies are given for parameters assumed at LI=5.1,1..-=2.4 khz, and Nc=280/cc. The fundamental sideband wave frequencies may be related to the saturated amplitude b of the carrier signal and E1.1 of the energy perpendicular to the external magnetic field, for example 7.5,10.0, and 12.5 key. The broadening width 6./, in the same way, is also shown here from Eq. (23) for E1. I=0.5 kev.)

8 128 INDIAN J RADIO & SPACE PHYS, JUNE 2002 by the untrapped electrons are unknown, the sideband waves described in this paper may be regarded as seeds of observed sideband waves. Accordingly, it is noted that some mechanisms like spatial wave growth 13, 14 together with the non-linear amplification described above may be also necessary for the sideband wave amplification. 4 Summary and conclusions The conclusive results are obtained as follows. (i) The untrapped electrons generating sideband waves have an external resonance point just outside the separatrix, which is labelled by = (ii) Using a new analytical formula system, it is imagined that their trajectories in the phase diagram form the trajectory gap near the separatrix, and so form phase-bunched sideband currents due to a strong non-linearity. (iii) One of the conclusion in this paper is that the possibility of self-exciting whistler-mode sideband waves generated in the magnetosphere is not negligible. Thus, the possibility of the self-exciting whistler-mode sideband wave generation is examined. Three kinds of the possibility are presented. (iv) It is most probable that two kinds of perpendicular velocities may be related to the generation and amplification of whistler-mode sideband waves at the same time. It is thought that this overlap of energetic electrons is possible to explain amplification of carrier signal due to any whistler-mode instabilities and, therefore, to explain various kinds of amplitude spectrum of whistler-mode sideband waves described on the film. (v) The presented model includes many problems, but also includes the possibility that the model can explain the frequency spectra of whistlermode sideband waves as a first approximation. In future, by carrying out observations and analyses in detail, it is quite possible that the electron distribution function generating sideband waves in the magnetosphere can be estimated with increased accuracy on the ground. As a result, it is noted that the present method will become one of the new tools for measuring the plasma parameters of the magnetosphere. However, the preliminary results obtained from a possible mechanism of self-exciting whistler-mode sideband waves show that more exact computation is necessary to make sure of this mechanism. Furthermore, the generation problem of whistler-mode sideband wave may develop non-linear plasma physics more. Acknowledgements This research started during the period the author stayed at Stanford University in 1994 and The author would like to appreciates Drs R A Helliwell, D L Carpenter, U S Inan, T F Bell and V S Sonwalkar for their kind help and effective suggestion, and thank the kindness and cooperation of many students of Stanford University. The author also thanks Dr S Machida of Kyoto University, Japan, for programme resources of Jacobian Elliptic Functions, and appreciates much the useful suggestion and advice of Drs H Matsumoto, K Hashimoto and Y Omura of Radio Atmospheric Science Center, Kyoto University. Further, the author wants to express a lot of appreciation to Dr K Tsuruda of Institute of Space and Astronautical Science, Japan, for his support to the author's staying at Stanford University. The author used the computers of Musashi University and National Institute of Polar Research, Tokyo. He is greatly thankful to many colleagues of Musashi University for kind cooperation. References I Brinca A L, J Geophys Res (USA), 77 (1972) Karpman Y I, Istomin J A N & Shklyar D R, Planet & Space Sci (UK), 22 (1974) Denavit J & Sudan R N, Phys Fluids (USA), 18 (1975) Nunn D, Planet & Space Sci (UK), 34 (1986) Tripathi Y K & Patel Y L, Geophys Res Lett (USA), 15 (1988) Sa' LAD, J Appl Phys (USA), 66 (1989) Sotnikov Y I, Fiala Y, Lefeuvre F & Lagoutte D, J Geophys Res (USA), 96 (1991) Malsumoto H, Wave Instabilities ill Space Plasmas (Netherlands), 74 (1979) Omura Y, Nunn D, Matsumoto H & Rycroft M 1, J Atmos & Terr Phys (UK), 53 (1991) Ikeda M, Proceedings of the Fifth International School/Symposium for Space Simulations (Japan), (1997) 21. II Park C G, J Geophys Res (USA), 86 (1981) Rastani K, Inan U S & Helliwell R A, J Geophys Res (USA), 90 (1985) Sonwalkar Y S, Carpenter D L, Helliwe ll R A, Walt M, Inan U S, Caudle D L & Ikeda M, J Geophys Res (USA), 102 (1997) Carpenter 0 L, Sonwalkar Y S, Helliwell R A, Walt M, inan U S, Ikeda M & Caudle D L, J Geophys Res (USA), 102 (1997) Maeda K, Bewtra N K & Smith PH, J Geophys Res (USA), 83 (1978) ).

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