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VLF wave amplification by wave-particle interaction
Volume 35A, number 2
VLF
WAVE
PHYSICS
AMPLIFICATION
LETTERS
17 May 1971
BY WAVE-PARTICLE
INTERACTION
R. N. STNGH * and R. P. SINGH ** Physics Section, Institute of Technology, Banaras Hindu Unii’ersit~’, Varanasi-5, India Received 19 March 1971
A charged particle beam is assumed to move through the magnetosphere. It is shown that the VLF waves propagating at non—zero angles with the beam are amplified.
It is well known that for electromagnetic wave propagation along a streaming plasma beam (coincident wave, beam and magnetic field directions), there is no coupling between the longitudinal mode oscillations in the plasma and the progapating transverse electromagnetic waves, This is owing to the fact that the dispersionless plasma is not effective in coupling a purely longitudinal wave with the plasma oscillations of the streaming beam. However, in the case of electromagnetic waves propagating at certain angle to the streaming plasma beam, there is an electric field component parallel to the streaming plasma beam. The presence of parallel electric field component permits the exchange of energy between the longitudinal plasma oscillations and the propagating electromagnetic waves. Under suitable conditions there may be energy transfer from the streaming plasma particles to the propagating waves. This energy transfer process is analogous to the traveling wave tube (TWT) mechanism and is capable of amplifying the interacting electromagnetic waves [1]. The detailed study of TWT mechanism in the ionosphere and in the magnetosphere was made by several workers [2-5]. Although, the TWT mechanism is based on sound experimental and theoretical foundations but its importance and relevance to the electromagnetic wave amplification, in lack of experimental evidence of streaming plasma beam, was ignored. It is more or established that the quite time ionospheric and magnetospheric plasma plays the role of background thermal plasma with Maxwellian velocity distribution and there are oc-
* **
ESRO Visiting Scientist, ESRIN, Frascati (Rome), Italy. At present with Groupe de Recherches lonospheriques, CNRS, France.
casional influx of charged particles which constitutes the secondary peak in the Maxwellian velocity distribution function. Therefore, the amplification of electromagnetic waves propagating through the ionosphere and the magnetosphere at an arbitrary angle to the plasma beam streaming along the geomagnetic field may be of much importance. In the case of electromagnetic wave generation by artificial electron beam injection in the ionosphere and in the magnetosphere the TWT amplification mechanism may play a decisive role. However, this is not specifically accounted in the analysis of electron beam injection experiment [6] and in the analysis of simultaneously measured electron flux and VLF emission in the ionosphere. The dispersion equation of a magnetoplasma with a streaming plasma beam has been derived by Knox [4] and Wang [5]. D(w k) - w~/c2(w2- k U ) = 0 (1) ‘ ii b where Ub and wb are the streaming plasma beam velocity and the beam plasma frequency respectively. In eq. (1) the first term is characteristic of VLF wave propagation in the whistler mode through the cold and collisionless magnetoplasma and the second term characterises the contribution of streaming plasma beam. The general dispersion equation appropriate to this situation is rather complicated and does not permit the analytical study of w and k variations. Therefore, we confine ourselves to the frequency range WHi < ~ < and ignore the role of ions in the dispersion equation. It has been shown [3-5] that eq. (1) depicts the instability of VLF waves propagating in the whistler mode (w complex and real k). The maximum temporal growth rate consistent with eq. (1) is given by — / amax
=
0.86!aD/awI
1 3 (w/c)2/3
(2)
k
11 Ub 105
nuiolwr 2
Voluno’ 3i.\.
I’
Substituting for ~D w from eq. (1 and ye :iI’ranginr, we rewrite en. (2) as ~42 D
[‘I
EH
8
2
C~1 ÷ftsmn ens Q(1 2 lhcos e
quirements are amplified. We have chosen parameters appropriate to the ionospheric and nuagnetospheric conditions and have computed the maximum growth rate of VLF waves propa— gating in the whistler mode. The computed growth iate is found to vary between 10 to 200 per second The above estimate shows that at times the amplification of VLF waves propagating through the ionosphere and the magnetosphere could be quite c onsiderable. Therefore, the electric field of the VLF waves going through this amplification process can be expressed as
“re. the propagating waves will refract awa~ utter sometime. The time taken by the VLF waves to propagate Horn the equatorial region 0 the magnetosphere to the ground is given by rrur( t~
where I ms the group ye] ocit\ 01 the VLF waves and dx, is an element of path along the geomagnetic lines of force. The time of propagation of the VLF waves depends the path of propagation (I. -values or different latitudes). Choosing same
plasma parameters as used for the calculation of ~rnax’ we find that the propagation time varies from 0.01 to 0.1 second. The crude estimate shows that this mechanism when operative calm give rise to VLF power densiH increase by a factor of 50-100. This analysis leads us to con elude that the interpretation of VLF waves received on the ground or by satellites should take into account the possibility of this amplification mechanism.
The present work was partly supported by the ‘S,A,t’
F0(w) exp(2~mnaxl)
(5~
Since the exponential factor depends on the number of beam particles and not on their energy spectrum, it can be argued that the total radiated power will also be enhanced by this factor. Further eq. (5) shows that the contribution due tm [mite growth rate is controlled by the time for which the amplification mechanism is operative. The VLF waves in the whistler mode are known to be guided along the geomagnetic lines of force
106
(leant EOOAR-70-007f1
E
0(w) exp(omaxt) 14 and the corresponding temporal growth rate ot the energy spectrum can be written as
F] co
5l~o 1971
‘md to reflect several times between the northern mud the southern hemispheres. However, in the om’esent ease we a’ e onsiclerlimg the propagation at ‘iou - run (I angle to the magnetic field there
‘~vhei 7? ~5b 52 (w 0)He I’hi~formulation shows 1~0)012 thai electromagnetic and ~ v.,uves present in a thermal plasma with a stream ing plasma beam and having proper w and k will he amplified. It is well known that such stream ing plasma beam in the ionospheric and magnet spheric plasma also satisfy the well known (~erenkuv condition and give rise to VLF hiss 7,81. Therefore, the VLF waves propagating in the whistler mode between the northern and southern hemispheres whenever satisfy the i.e
F
1
fieterences R.M.Gallet and R.A, Hellmoe.ll, J. of Res., N.B.S. (Radio Prop.) 63D (1957) 21. [2] C. F. Knox. Proc. Phvs. Soc. 83 ~1964)783. [3] T. F. Bell and O.Buneman. Phys. Rev. 133 (1964) 1]
1 300, [4) C. F. Knox, J. Plasma Phvs. 1 ~l967) 1, [3) T.N. C. Wang. JEEE Trans. \nts, Propn. 11 (i969~ [6] T. F. Bell, .1. Goophys. Res. 73 (1968) 4409, [7] JF~IcKenzj~ Phys. Fluids 10 (1967) 2680. H] B. N. Sjngh and B. P. Singh. ~nn. de Geophys. 25 (11)69) 029,
VLF
WAVE
PHYSICS
AMPLIFICATION
LETTERS
17 May 1971
BY WAVE-PARTICLE
INTERACTION
R. N. STNGH * and R. P. SINGH ** Physics Section, Institute of Technology, Banaras Hindu Unii’ersit~’, Varanasi-5, India Received 19 March 1971
A charged particle beam is assumed to move through the magnetosphere. It is shown that the VLF waves propagating at non—zero angles with the beam are amplified.
It is well known that for electromagnetic wave propagation along a streaming plasma beam (coincident wave, beam and magnetic field directions), there is no coupling between the longitudinal mode oscillations in the plasma and the progapating transverse electromagnetic waves, This is owing to the fact that the dispersionless plasma is not effective in coupling a purely longitudinal wave with the plasma oscillations of the streaming beam. However, in the case of electromagnetic waves propagating at certain angle to the streaming plasma beam, there is an electric field component parallel to the streaming plasma beam. The presence of parallel electric field component permits the exchange of energy between the longitudinal plasma oscillations and the propagating electromagnetic waves. Under suitable conditions there may be energy transfer from the streaming plasma particles to the propagating waves. This energy transfer process is analogous to the traveling wave tube (TWT) mechanism and is capable of amplifying the interacting electromagnetic waves [1]. The detailed study of TWT mechanism in the ionosphere and in the magnetosphere was made by several workers [2-5]. Although, the TWT mechanism is based on sound experimental and theoretical foundations but its importance and relevance to the electromagnetic wave amplification, in lack of experimental evidence of streaming plasma beam, was ignored. It is more or established that the quite time ionospheric and magnetospheric plasma plays the role of background thermal plasma with Maxwellian velocity distribution and there are oc-
* **
ESRO Visiting Scientist, ESRIN, Frascati (Rome), Italy. At present with Groupe de Recherches lonospheriques, CNRS, France.
casional influx of charged particles which constitutes the secondary peak in the Maxwellian velocity distribution function. Therefore, the amplification of electromagnetic waves propagating through the ionosphere and the magnetosphere at an arbitrary angle to the plasma beam streaming along the geomagnetic field may be of much importance. In the case of electromagnetic wave generation by artificial electron beam injection in the ionosphere and in the magnetosphere the TWT amplification mechanism may play a decisive role. However, this is not specifically accounted in the analysis of electron beam injection experiment [6] and in the analysis of simultaneously measured electron flux and VLF emission in the ionosphere. The dispersion equation of a magnetoplasma with a streaming plasma beam has been derived by Knox [4] and Wang [5]. D(w k) - w~/c2(w2- k U ) = 0 (1) ‘ ii b where Ub and wb are the streaming plasma beam velocity and the beam plasma frequency respectively. In eq. (1) the first term is characteristic of VLF wave propagation in the whistler mode through the cold and collisionless magnetoplasma and the second term characterises the contribution of streaming plasma beam. The general dispersion equation appropriate to this situation is rather complicated and does not permit the analytical study of w and k variations. Therefore, we confine ourselves to the frequency range WHi < ~ < and ignore the role of ions in the dispersion equation. It has been shown [3-5] that eq. (1) depicts the instability of VLF waves propagating in the whistler mode (w complex and real k). The maximum temporal growth rate consistent with eq. (1) is given by — / amax
=
0.86!aD/awI
1 3 (w/c)2/3
(2)
k
11 Ub 105
nuiolwr 2
Voluno’ 3i.\.
I’
Substituting for ~D w from eq. (1 and ye :iI’ranginr, we rewrite en. (2) as ~42 D
[‘I
EH
8
2
C~1 ÷ftsmn ens Q(1 2 lhcos e
quirements are amplified. We have chosen parameters appropriate to the ionospheric and nuagnetospheric conditions and have computed the maximum growth rate of VLF waves propa— gating in the whistler mode. The computed growth iate is found to vary between 10 to 200 per second The above estimate shows that at times the amplification of VLF waves propagating through the ionosphere and the magnetosphere could be quite c onsiderable. Therefore, the electric field of the VLF waves going through this amplification process can be expressed as
“re. the propagating waves will refract awa~ utter sometime. The time taken by the VLF waves to propagate Horn the equatorial region 0 the magnetosphere to the ground is given by rrur( t~
where I ms the group ye] ocit\ 01 the VLF waves and dx, is an element of path along the geomagnetic lines of force. The time of propagation of the VLF waves depends the path of propagation (I. -values or different latitudes). Choosing same
plasma parameters as used for the calculation of ~rnax’ we find that the propagation time varies from 0.01 to 0.1 second. The crude estimate shows that this mechanism when operative calm give rise to VLF power densiH increase by a factor of 50-100. This analysis leads us to con elude that the interpretation of VLF waves received on the ground or by satellites should take into account the possibility of this amplification mechanism.
The present work was partly supported by the ‘S,A,t’
F0(w) exp(2~mnaxl)
(5~
Since the exponential factor depends on the number of beam particles and not on their energy spectrum, it can be argued that the total radiated power will also be enhanced by this factor. Further eq. (5) shows that the contribution due tm [mite growth rate is controlled by the time for which the amplification mechanism is operative. The VLF waves in the whistler mode are known to be guided along the geomagnetic lines of force
106
(leant EOOAR-70-007f1
E
0(w) exp(omaxt) 14 and the corresponding temporal growth rate ot the energy spectrum can be written as
F] co
5l~o 1971
‘md to reflect several times between the northern mud the southern hemispheres. However, in the om’esent ease we a’ e onsiclerlimg the propagation at ‘iou - run (I angle to the magnetic field there
‘~vhei 7? ~5b 52 (w 0)He I’hi~formulation shows 1~0)012 thai electromagnetic and ~ v.,uves present in a thermal plasma with a stream ing plasma beam and having proper w and k will he amplified. It is well known that such stream ing plasma beam in the ionospheric and magnet spheric plasma also satisfy the well known (~erenkuv condition and give rise to VLF hiss 7,81. Therefore, the VLF waves propagating in the whistler mode between the northern and southern hemispheres whenever satisfy the i.e
F
1
fieterences R.M.Gallet and R.A, Hellmoe.ll, J. of Res., N.B.S. (Radio Prop.) 63D (1957) 21. [2] C. F. Knox. Proc. Phvs. Soc. 83 ~1964)783. [3] T. F. Bell and O.Buneman. Phys. Rev. 133 (1964) 1]
1 300, [4) C. F. Knox, J. Plasma Phvs. 1 ~l967) 1, [3) T.N. C. Wang. JEEE Trans. \nts, Propn. 11 (i969~ [6] T. F. Bell, .1. Goophys. Res. 73 (1968) 4409, [7] JF~IcKenzj~ Phys. Fluids 10 (1967) 2680. H] B. N. Sjngh and B. P. Singh. ~nn. de Geophys. 25 (11)69) 029,