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Damping of the resonant mode in a plasma-maser
6 March 1995 PHYSICS
LETTERS
A
Physics Letters A 198 (1995) 352-356
Damping of the resonant mode in a plasma-maser Bipuljyoti Saikia and S. Bujarbarua Centre of Plasma Physics, Saptaswahid Path, Dispur, Guwahati 781006, India
Received 17 October 1994; revised manuscript received 21 November 1994; accepted for publication 6 January 1995 Communi~ted by M. Porkolab
Abstract The damping rate of resonant waves is evaluated considering
their interaction with nonresonant
waves and plasma particles
due to the plasma-maser effect. It is found that the polarization coupling terms gives the dominant con~bution to the damping rate. On the other hand, the growth rate of the nonresonant waves consists only of the direct coupling term contribution.The
significanceof our result is discussed.
The nonlinear interaction of a resonant mode (which satisfies the Cherenkov resonance condition w-k* u = 0) mode (which neither satisfies the Cherenkov resonance condition .f2- K*v f 0 nor the scattering resonance condition a- o - (K-k) .o # 0)is termed the plasma-maser effect in the literature [ 11. Consideration of this type of nonlinear interaction began with the work of Nambu [ 21 and Tsytovich et al. [ 37. Much research has since been devoted to the study of the nonlinear amplification of the high frequency nonresonant waves by low frequency resonant ones [ 4-41. The simultaneous effect of the resonant and the nomesonant waves on the particle distribution function has also been studied (the inverse plasma-maser effect [ 71). Recent investigation [ 81 shows that the energy and momentum conservation laws are exactly satisfied by the plasma-maser interaction, while the Manley-Rowe relation is violated; as a result an efficient up-conversion of energy takes place from the low frequency resonant mode to the high frequency nonresonant mode without electron population inversion. It is generally agreed that the plasma-maser con~bution comes only from the direct nonlinear coupling term for no~esonant waves in an unmagnetized plasma. The absorption process reverse to the plasma-maser originates from the slow time change of the medium due to the quasilinear interaction between resonant electrons and resonant waves. In an unmagnetized, closed system (where there are no sources and sinks of particles and/or energy (momentum) ) the above two effects are exactly balanced [ 91. But in an open system where particle and/or energy (momentum) sources are available, the reverse absorption effect itself vanishes and the nonresonant wave grows at the expense of the resonant wave. The plasma-maser is also effective in a magnetized plasma with a symmetry breaking factor, i.e., finite Larmor radius effect [ 51. The anomalous high frequency radiation in the presence of low frequency fluctuations is often reported in laboratory and space plasmas. The Langmuir wave excitation from ion sound waves and high frequency whistler mode driven by low frequency ion modes are shown in Refs. [IO] and [ 111, respectively. The Langmuir wave emission from ion sound solitons and the emission of upper-hybrid wave solitons from ion waves are reported in Refs. [ 121 and [ 131, respectively. The ex~aordin~y mode radiation from the electrostatic lower hybrid turbulences is observed in space [ 141 and by computer simulation [ 151. The plasma-maser effect predicts most of the characwith a nonresonant
0375-9601/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved SSDIO375-9601(95)00019-4
6. Saikia, S. Bujarbarua / Physics Letters A 198 (1995) 3S2-356
353
teristics of the observations. Similar high frequency radiation is also reported in laboratory experiments [ 161. Concurrent excitation of Langmuir waves in turbulent heating experiments, in which ion sound wave activity is much enhanced, is a typical phenomenon of the laboratory plasma [ 171. Thus the plasma-maser effect may play an important role in interpreting numerous anomalous radiation phenomena in laboratory and space plasmas. As has been mentioned earlier, in most of the work on the plasma-maser the nonlinear evolution of the nonresonant mode was studied [ 4,5], Only very recently, the simultaneous evolution of the resonant mode was considered [ 181. The consideration of the evolution of the resonant mode shows that the polarization mode coupling term offers a non-zero contribution to the damping of the resonant mode even for an unmagnetized plasma. This is important from the point of view of energy exchange among the waves and particles and the evolution of the particle dist~bution function in the plasma-maser interaction. Here we investigate the evolution of the resonant mode in the plasma-m~er interaction. We obtain an expression for the nonlinear damping rate of the resonant mode, and then its basic difference with the nonlinear growth rate of the nonresonant mode [ 191 is pointed out. We use the Vlasov-Poisson equation to obtain the nonlinear dielectric function of the low frequency resonant (ion sound) wave due to its interaction with the nonresonant (~ngmuir) wave. We assume an unmagnetized plasma with ,$, > k > K, where kc is the electron Debye wave number, and k and K are the wave numbers of the resonant and the nonresonant waves, respectively. After lengthy but straightforward calculations, we obtain the nonlinear dispersion function of the ion sound wave as +(k,
w) = e&k,
w) + e&k,
u) -t e+,(k, w) ,
(1)
where eeo( k, w) is the linear part given by
led( k, w) is the direct mode coupling term given by
1
X
w-ko+iO
1 -a dv co-L?-(k-K)u
and eyp( k, w) is the polarization
lpp(k, 0)
1 --a au ( O-Ku
+ k”-~-i~)~fti
dv
(3)
mode coupling term,
= -
where A=
+
I
and B=A(w++w-0,
k-k-K,
fl*
-f),
Fit, -II).
(6)
In the above equations w and fl are the frequencies of the resonant and the nonresonant waves, respectively; wF ( Wpi) is the electron (ion) plasma frequency, e and m are the electronic charge and mass, E,,( K, 0) is the electric field of the nonresonant wave, and foe (j&i) is the unperturbed electron (ion) distribution function. Thus we see that the nonlinear evolution of the resonant mode due to the plasma-maser interaction is described by two processes: the direct mode coupling and the polarization mode coupling of the waves,
354
B. Saikia, S. Bujarbarua / Physics Letters A 198 (1995) 352-356
3/e(k, 0) = -
Im eed(k, w) +Im l ep(k, w) (a/&~> Re e&k, w) ’
where Re and Im denote the real and imaginary parts. We must mention here that in a closed system, an additional imaginary part of the dielectric function due to the nonstationarity of the system (originating from the quasilinear process) is to be taken into account in evaluating the damping rate. However, here we consider an open system where constant particle distributions are maintained. The imaginary part of the direct nonlinear term can be written as
(8) where M is the mass of the ion. In evaluating Eq. (8) we assume a Maxwellian distribution for the unperturbed distribution function. From Eq. (8) we obtain the damping rate due to the direct coupling term as
yedk, 0)
(9)
w
where W, = I&( K, 0) I2/4?rNT is the normalized energy of the Langmuir mode. Here we make use of the real part of the linear dispersion function of the ion sound wave, Re eeO(k, w) = 1 + (k,/k) 2 - ( O,i/W) 2. Next, we determine the damping rate due to the polarization coupling term. From Eqs. (5) and (6) we obtain
and k-K
B=--
(11)
R2
which give
(12)
Im(AB)=2
Accordingly, we obtain the damping rate due to the polarization term as
Yep(k~1
(13)
w
In deriving Eq. ( 13) we use 1
k, 2
e&k-K)
I-- 0k
’
It is now straightforward to compare the magnitudes of the two damping rates,
I2%=k-2>1 IO ‘yed
k
fork,>k.
(14)
This result is markedly different from the results obtained for the nonresonant waves. For nonresonant waves, the imaginary part of the polarization term of the dielectric function vanishes in an unmagnetized system [ 51. Following Ref. [ 51 we obtain the polarization term of the dielectric function for the nonresonant waves as
B. Saikia, S. Bujarbarua /Physics Letters A 198 (I 99.5) 352-356
355
(15)
(16) and D=C(&-+t,-cow,
K++K-k,
0”
-w,
k*
-k)
.
(17)
Here E,( k, o) is the electric field of the resonant wave. It is straightforward to show that 1 'f,du, o-ku+iOau
1 a--w-(K-k)u
Da-
1 L?-o-(K-k)u
1 O-KU
1 o-kv-i0
Lf&,du= au
-C*.
(If-9
Here an asterisk represents the complex conjugate. Thus Im( CD) = 0, and accordingly, the growth rate of the nonresonant wave due to the polarization coupling term for an unmagnetized system vanishes. We have obtained the nonlinear damping rate of the resonant ion sound wave considering its interaction with nonresonant Langmuir waves and particles due to the plasma-maser effect. It is found that in contrast to the nonresonant mode, whose growth rate comes only from the direct nonlinear coupiing of the waves, the damping rate of the resonant mode comes from both the direct and the polarization coupling terms. Furthermore, the damping rate due to the polarization term is larger than that due to the direct nonlinear coupling of the waves for the resonant mode. This result is signific~t, because the energy exchange between particles and waves in the pl~ma-m~er interaction is impo~nt. In particular, the evolution of the dis~bution function due to the simul~neous interaction of plasma particles with the resonant and nonreson~t waves must take into account this pol~zation con~bution which has so far been ignored. For example, in Ref. [ 71, the authors derive a nonlinear collision integral describing the plasma-maser interaction without considering this polarization contribution. Since strong energy exchange takes place between waves and particles in a turbulent plasma and the changes in particle energy and momentum are described by the corresponding collision integral, they must be derived more accurately. In Ref. [20], it is shown that ye = 0 for a stationary resonant field without any explicit evaluation of ‘yeincluding the polarization effect. Recently, the conservation of energy in closed and in open systems for plasma-maser interaction is studied [ 81. This indicates an effective energy variation of the resonant mode in plasma-maser interaction. From our results, we write dW, L = -/3ewswt, dt where we drop the summation sign for brevity. From a previous result [ 191 we obtain
dK - dt =h,wsw,, where W, = ( 1E&k, w) 1214mT) (k,/k)’ and W, are the normalized wave energy of the ion sound and Langmuir waves, res ectively; and &=2] -yf(k, w) I/ WL=2( 1y&l + I yepI ) /Wi_ (EIqs. (9) and (13)) and ph= 2y,/ mn/MKl k 1/kz (Eq. ( 19) of Ref. [ 191). For k> K, it can be shown that & > ph. Under this condition W, = oF JR--the linear Landau damping of the Langmuir wave is negligibly small. This implies that the total wave energy ( W, f W,J decreases due to the abso~tion of energy by resonant particles in the plasma-maser interaction. It can
356
B. Saikia. S. Bujarbarua /Physics Leners A 198 (1995) 352-356
also be expected that under certain conditions, the plasma-maser can also lead to an increase of energy of the resonant mode. This may occur under conditions similar to those for the acceleration of beam particles [ 211. This and other features of the plasma-maser may be explored in the future. One of the authors (BS) acknowledges the Council of Scientific & Industrial Research (CSIR), India for a research fellowship.
References [ 11M. Nambu, Space Sci. Rev. 44 (1986) 357. [2] M. Nambu, Phys. Rev. L&t. 34 (1975) 387. [3] V.N. Tsytovich, L. Stengo and H. Wilhelmsson, Phys. Ser. 11 (1975) 251. 141 S. Bujarbarua, S.N. Sarma and M. Nambu, Phys. Rev. A 29 (1984) 2171. [5] M. Nambu, S. Bujarbarua and S.N. Sarma, Phys. Rev. A 35 (1987) 798. [6] V.S. Krivitskii, V.N. Tsytovich and S.V. Vladimirov. Phys. Rep. 218 ( 1992) 141. [7] V.S. Krivitskii and S.V. Vladimirov, J. Plasma Phys. 46 (1991) 209. [ 81 M. Nambu and T. Hada, Phys. Fluids B 5 (1993) 742. [9] S.B. Isakov, V.S. Krivitskii and V.N. Tsytovich. Zh. Eksp. Teor. Fiz. 90 (1986) 933 [ Sov. Phys. JETP 63 ( 1986) 5451. [ 101 Y. Amagishi, J. Phys. Sot. Japan 29 (1970) 764. [ 111 H. Fujiyama and M. Nambu, Phys. Lett. A 105 ( 1984) 295. [ 121 R.W. Boswell, Geophys. Res. L&t. 11 (1984) 1015. [ 131 I. Mori and K. Ohya, Phys. Rev. Lett. 59 (1987) 1825. [ 141 M. Malingre, R. Pottelette, N. Dubouloz, P.A. Lindqvist, G. Holmgren and B. Aparicio, Geophys. Res. Lett. 19 (1992) 1339. [ 151 K.G. McClements, R. Bingham, J.J. Su, J.M. Dawson and D.S. Spicer, Astrophys. J. 409 (1993) 465. [ 161 T. Mieno. M. Gertl, R. Hatakeyama and N. Sato, Phys. Lett. A 184 (1994) 445. [ 171 H. de Kluiver, N.F. Perepelkin and A. Hirose, Phys. Rep. 199 (1991) 281. [ 181 S.1. Popel, S.V. Vladimirov and M.Y. Yu, Phys. Ser. 47 (1993) 239. [ 191 M. Nambu, J. Phys. Sot. Japan 53 ( 1984) 1594. [20] V.N. Tsytovich, Zh. Eksp. Tear. Fiz. 89 ( 1985) 842 [Sov. Phys. JETP 62 ( 1985) 4831. [21] S.V. Vladimirov and V.S. Krivitskii, Sov. Phys. Lebedev Inst. Report No. 1 ( 1990) 53.
LETTERS
A
Physics Letters A 198 (1995) 352-356
Damping of the resonant mode in a plasma-maser Bipuljyoti Saikia and S. Bujarbarua Centre of Plasma Physics, Saptaswahid Path, Dispur, Guwahati 781006, India
Received 17 October 1994; revised manuscript received 21 November 1994; accepted for publication 6 January 1995 Communi~ted by M. Porkolab
Abstract The damping rate of resonant waves is evaluated considering
their interaction with nonresonant
waves and plasma particles
due to the plasma-maser effect. It is found that the polarization coupling terms gives the dominant con~bution to the damping rate. On the other hand, the growth rate of the nonresonant waves consists only of the direct coupling term contribution.The
significanceof our result is discussed.
The nonlinear interaction of a resonant mode (which satisfies the Cherenkov resonance condition w-k* u = 0) mode (which neither satisfies the Cherenkov resonance condition .f2- K*v f 0 nor the scattering resonance condition a- o - (K-k) .o # 0)is termed the plasma-maser effect in the literature [ 11. Consideration of this type of nonlinear interaction began with the work of Nambu [ 21 and Tsytovich et al. [ 37. Much research has since been devoted to the study of the nonlinear amplification of the high frequency nonresonant waves by low frequency resonant ones [ 4-41. The simultaneous effect of the resonant and the nomesonant waves on the particle distribution function has also been studied (the inverse plasma-maser effect [ 71). Recent investigation [ 81 shows that the energy and momentum conservation laws are exactly satisfied by the plasma-maser interaction, while the Manley-Rowe relation is violated; as a result an efficient up-conversion of energy takes place from the low frequency resonant mode to the high frequency nonresonant mode without electron population inversion. It is generally agreed that the plasma-maser con~bution comes only from the direct nonlinear coupling term for no~esonant waves in an unmagnetized plasma. The absorption process reverse to the plasma-maser originates from the slow time change of the medium due to the quasilinear interaction between resonant electrons and resonant waves. In an unmagnetized, closed system (where there are no sources and sinks of particles and/or energy (momentum) ) the above two effects are exactly balanced [ 91. But in an open system where particle and/or energy (momentum) sources are available, the reverse absorption effect itself vanishes and the nonresonant wave grows at the expense of the resonant wave. The plasma-maser is also effective in a magnetized plasma with a symmetry breaking factor, i.e., finite Larmor radius effect [ 51. The anomalous high frequency radiation in the presence of low frequency fluctuations is often reported in laboratory and space plasmas. The Langmuir wave excitation from ion sound waves and high frequency whistler mode driven by low frequency ion modes are shown in Refs. [IO] and [ 111, respectively. The Langmuir wave emission from ion sound solitons and the emission of upper-hybrid wave solitons from ion waves are reported in Refs. [ 121 and [ 131, respectively. The ex~aordin~y mode radiation from the electrostatic lower hybrid turbulences is observed in space [ 141 and by computer simulation [ 151. The plasma-maser effect predicts most of the characwith a nonresonant
0375-9601/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved SSDIO375-9601(95)00019-4
6. Saikia, S. Bujarbarua / Physics Letters A 198 (1995) 3S2-356
353
teristics of the observations. Similar high frequency radiation is also reported in laboratory experiments [ 161. Concurrent excitation of Langmuir waves in turbulent heating experiments, in which ion sound wave activity is much enhanced, is a typical phenomenon of the laboratory plasma [ 171. Thus the plasma-maser effect may play an important role in interpreting numerous anomalous radiation phenomena in laboratory and space plasmas. As has been mentioned earlier, in most of the work on the plasma-maser the nonlinear evolution of the nonresonant mode was studied [ 4,5], Only very recently, the simultaneous evolution of the resonant mode was considered [ 181. The consideration of the evolution of the resonant mode shows that the polarization mode coupling term offers a non-zero contribution to the damping of the resonant mode even for an unmagnetized plasma. This is important from the point of view of energy exchange among the waves and particles and the evolution of the particle dist~bution function in the plasma-maser interaction. Here we investigate the evolution of the resonant mode in the plasma-m~er interaction. We obtain an expression for the nonlinear damping rate of the resonant mode, and then its basic difference with the nonlinear growth rate of the nonresonant mode [ 191 is pointed out. We use the Vlasov-Poisson equation to obtain the nonlinear dielectric function of the low frequency resonant (ion sound) wave due to its interaction with the nonresonant (~ngmuir) wave. We assume an unmagnetized plasma with ,$, > k > K, where kc is the electron Debye wave number, and k and K are the wave numbers of the resonant and the nonresonant waves, respectively. After lengthy but straightforward calculations, we obtain the nonlinear dispersion function of the ion sound wave as +(k,
w) = e&k,
w) + e&k,
u) -t e+,(k, w) ,
(1)
where eeo( k, w) is the linear part given by
led( k, w) is the direct mode coupling term given by
1
X
w-ko+iO
1 -a dv co-L?-(k-K)u
and eyp( k, w) is the polarization
lpp(k, 0)
1 --a au ( O-Ku
+ k”-~-i~)~fti
dv
(3)
mode coupling term,
= -
where A=
+
I
and B=A(w++w-0,
k-k-K,
fl*
-f),
Fit, -II).
(6)
In the above equations w and fl are the frequencies of the resonant and the nonresonant waves, respectively; wF ( Wpi) is the electron (ion) plasma frequency, e and m are the electronic charge and mass, E,,( K, 0) is the electric field of the nonresonant wave, and foe (j&i) is the unperturbed electron (ion) distribution function. Thus we see that the nonlinear evolution of the resonant mode due to the plasma-maser interaction is described by two processes: the direct mode coupling and the polarization mode coupling of the waves,
354
B. Saikia, S. Bujarbarua / Physics Letters A 198 (1995) 352-356
3/e(k, 0) = -
Im eed(k, w) +Im l ep(k, w) (a/&~> Re e&k, w) ’
where Re and Im denote the real and imaginary parts. We must mention here that in a closed system, an additional imaginary part of the dielectric function due to the nonstationarity of the system (originating from the quasilinear process) is to be taken into account in evaluating the damping rate. However, here we consider an open system where constant particle distributions are maintained. The imaginary part of the direct nonlinear term can be written as
(8) where M is the mass of the ion. In evaluating Eq. (8) we assume a Maxwellian distribution for the unperturbed distribution function. From Eq. (8) we obtain the damping rate due to the direct coupling term as
yedk, 0)
(9)
w
where W, = I&( K, 0) I2/4?rNT is the normalized energy of the Langmuir mode. Here we make use of the real part of the linear dispersion function of the ion sound wave, Re eeO(k, w) = 1 + (k,/k) 2 - ( O,i/W) 2. Next, we determine the damping rate due to the polarization coupling term. From Eqs. (5) and (6) we obtain
and k-K
B=--
(11)
R2
which give
(12)
Im(AB)=2
Accordingly, we obtain the damping rate due to the polarization term as
Yep(k~1
(13)
w
In deriving Eq. ( 13) we use 1
k, 2
e&k-K)
I-- 0k
’
It is now straightforward to compare the magnitudes of the two damping rates,
I2%=k-2>1 IO ‘yed
k
fork,>k.
(14)
This result is markedly different from the results obtained for the nonresonant waves. For nonresonant waves, the imaginary part of the polarization term of the dielectric function vanishes in an unmagnetized system [ 51. Following Ref. [ 51 we obtain the polarization term of the dielectric function for the nonresonant waves as
B. Saikia, S. Bujarbarua /Physics Letters A 198 (I 99.5) 352-356
355
(15)
(16) and D=C(&-+t,-cow,
K++K-k,
0”
-w,
k*
-k)
.
(17)
Here E,( k, o) is the electric field of the resonant wave. It is straightforward to show that 1 'f,du, o-ku+iOau
1 a--w-(K-k)u
Da-
1 L?-o-(K-k)u
1 O-KU
1 o-kv-i0
Lf&,du= au
-C*.
(If-9
Here an asterisk represents the complex conjugate. Thus Im( CD) = 0, and accordingly, the growth rate of the nonresonant wave due to the polarization coupling term for an unmagnetized system vanishes. We have obtained the nonlinear damping rate of the resonant ion sound wave considering its interaction with nonresonant Langmuir waves and particles due to the plasma-maser effect. It is found that in contrast to the nonresonant mode, whose growth rate comes only from the direct nonlinear coupiing of the waves, the damping rate of the resonant mode comes from both the direct and the polarization coupling terms. Furthermore, the damping rate due to the polarization term is larger than that due to the direct nonlinear coupling of the waves for the resonant mode. This result is signific~t, because the energy exchange between particles and waves in the pl~ma-m~er interaction is impo~nt. In particular, the evolution of the dis~bution function due to the simul~neous interaction of plasma particles with the resonant and nonreson~t waves must take into account this pol~zation con~bution which has so far been ignored. For example, in Ref. [ 71, the authors derive a nonlinear collision integral describing the plasma-maser interaction without considering this polarization contribution. Since strong energy exchange takes place between waves and particles in a turbulent plasma and the changes in particle energy and momentum are described by the corresponding collision integral, they must be derived more accurately. In Ref. [20], it is shown that ye = 0 for a stationary resonant field without any explicit evaluation of ‘yeincluding the polarization effect. Recently, the conservation of energy in closed and in open systems for plasma-maser interaction is studied [ 81. This indicates an effective energy variation of the resonant mode in plasma-maser interaction. From our results, we write dW, L = -/3ewswt, dt where we drop the summation sign for brevity. From a previous result [ 191 we obtain
dK - dt =h,wsw,, where W, = ( 1E&k, w) 1214mT) (k,/k)’ and W, are the normalized wave energy of the ion sound and Langmuir waves, res ectively; and &=2] -yf(k, w) I/ WL=2( 1y&l + I yepI ) /Wi_ (EIqs. (9) and (13)) and ph= 2y,/ mn/MKl k 1/kz (Eq. ( 19) of Ref. [ 191). For k> K, it can be shown that & > ph. Under this condition W, = oF JR--the linear Landau damping of the Langmuir wave is negligibly small. This implies that the total wave energy ( W, f W,J decreases due to the abso~tion of energy by resonant particles in the plasma-maser interaction. It can
356
B. Saikia. S. Bujarbarua /Physics Leners A 198 (1995) 352-356
also be expected that under certain conditions, the plasma-maser can also lead to an increase of energy of the resonant mode. This may occur under conditions similar to those for the acceleration of beam particles [ 211. This and other features of the plasma-maser may be explored in the future. One of the authors (BS) acknowledges the Council of Scientific & Industrial Research (CSIR), India for a research fellowship.
References [ 11M. Nambu, Space Sci. Rev. 44 (1986) 357. [2] M. Nambu, Phys. Rev. L&t. 34 (1975) 387. [3] V.N. Tsytovich, L. Stengo and H. Wilhelmsson, Phys. Ser. 11 (1975) 251. 141 S. Bujarbarua, S.N. Sarma and M. Nambu, Phys. Rev. A 29 (1984) 2171. [5] M. Nambu, S. Bujarbarua and S.N. Sarma, Phys. Rev. A 35 (1987) 798. [6] V.S. Krivitskii, V.N. Tsytovich and S.V. Vladimirov. Phys. Rep. 218 ( 1992) 141. [7] V.S. Krivitskii and S.V. Vladimirov, J. Plasma Phys. 46 (1991) 209. [ 81 M. Nambu and T. Hada, Phys. Fluids B 5 (1993) 742. [9] S.B. Isakov, V.S. Krivitskii and V.N. Tsytovich. Zh. Eksp. Teor. Fiz. 90 (1986) 933 [ Sov. Phys. JETP 63 ( 1986) 5451. [ 101 Y. Amagishi, J. Phys. Sot. Japan 29 (1970) 764. [ 111 H. Fujiyama and M. Nambu, Phys. Lett. A 105 ( 1984) 295. [ 121 R.W. Boswell, Geophys. Res. L&t. 11 (1984) 1015. [ 131 I. Mori and K. Ohya, Phys. Rev. Lett. 59 (1987) 1825. [ 141 M. Malingre, R. Pottelette, N. Dubouloz, P.A. Lindqvist, G. Holmgren and B. Aparicio, Geophys. Res. Lett. 19 (1992) 1339. [ 151 K.G. McClements, R. Bingham, J.J. Su, J.M. Dawson and D.S. Spicer, Astrophys. J. 409 (1993) 465. [ 161 T. Mieno. M. Gertl, R. Hatakeyama and N. Sato, Phys. Lett. A 184 (1994) 445. [ 171 H. de Kluiver, N.F. Perepelkin and A. Hirose, Phys. Rep. 199 (1991) 281. [ 181 S.1. Popel, S.V. Vladimirov and M.Y. Yu, Phys. Ser. 47 (1993) 239. [ 191 M. Nambu, J. Phys. Sot. Japan 53 ( 1984) 1594. [20] V.N. Tsytovich, Zh. Eksp. Tear. Fiz. 89 ( 1985) 842 [Sov. Phys. JETP 62 ( 1985) 4831. [21] S.V. Vladimirov and V.S. Krivitskii, Sov. Phys. Lebedev Inst. Report No. 1 ( 1990) 53.