Amplification of signal by Cerenkov resonance interaction

Planet. Space Sci. 1972, Vol. 20, pp. 2073 to 2080.

AMPLIFICATION

G.R.I.-C.N.E.T.,

Pergamon Press. Printed in Norlhern Ireland

OF SIGNAL BY CERENKOV INTERACTION

RESONANCE

R. P. SINGH 3 Avenue de la Republique, 92, Issy les Moulineaux, France (Received injinalform

6 June

1972)

Abstract-The interaction between VLF waves propagating at an angle to the geomagnetic field and an electron beam in the presence of cold plasma is discussed. It is shown that the Cerenkov signal is amplified during the process of resonance interaction by three orders of magnitude. The variation of the amplification factor with frequency is studied and its application to the explanation of the observed VLF intensities is indicated. INTRODUCTION to explain the experimental observations of VLF broad band noise, known as hiss, in terms of the Cerenkov radiation process (Jorgensen, 1968; Singh and Singh, 1969/70). Using the Cerenkov process, one easily explains the frequency spectrum but the maximum observed amplitude remains unexplained. The theoretical computation of total emitted VLF power is based on certain unrealistic assumptions and generally gives an upper limit to the total radiated power. The scattered power and absorption loss are ignored when compared with the experimental measurements. This shows that the Cerenkov radiation process alone cannot explain the amplitude of VLF emissions. Therefore one must take account of other simultaneous effects, such as wave-particle interactions. Dowden (1962), using the Cerenkov resonance conditions, has studied the frequency spectrum of exospheric VLF noises. Kennel (1966) has shown that the VLF wave is amplified through the Cerenkov process in the presence of an electron beam. Recently, Singh and Singh (1971) have shown that the VLF hiss generated in the innermagnetosphere can be amplified by two orders of magnitude during the process of Cerenkov resonance interaction. In the present paper, we study the longitudinal resonance between VLF hiss propagating at an angle with the geomagnetic field and streaming electron influx. We have analysed the dispersion equation and have computed the growth rate for various frequencies. Using the growth rate, we have studied the amplification coefficient for VLF emission. We have shown that the amplified VLF hiss propagating through the magnetosphere accounts for the enhancement of observed power on the ground by roughly three orders of magnitude. Several

attempts

have been

made

THRORETICAL

CONSIDERATIONS

The VLF emissions propagating in the magnetosphere are known to interact whenever the doppler-shifted frequency seen by the streaming electron satisfies dition w -

k - U,, = SW,

strongly the con(1)

where w is the angular frequency of VLF hiss k is the wave vector U,, is the streaming electron’s velocity parallel to the magnetic field w, is the magnitude of electron gyrofrequency; s = 0, &I, &2, &3, harmonic number. 2073

2074

R. P. SINGH

In Equation (l), s = 0, gives the Cerenkov resonance interaction condition, whereas s # 0 represents gyroresonance interaction. Here, we will be studying VLF hiss amplification by wave-particle interaction using the Cerenkov resonance condition. The VLF wave propagating at an arbitrary angle to the geomagnetic field has a finite longitudinal component of electric field vector which is parallel to the geomagnetic field and the streaming electron influx. When resonance condition (U,, cos 0 = w/k = V,, 0 being wave-vector angle; V, wavephase velocity) is satisfied, the longitudinal component of the electric field vector comes into generative resonance with the stream and thus extracts energy from the streaming electron influx. Using this condition, Wang (1969) derived an expression of the temporal growth rate for VLF-waves. In the following section we have analysed the dispersion equation and have computed growth rate in the frequency range !Ai < w < w,, Qi is the magnitude of ion gyrofrequency. DISPERSION EQUATION,

GROWTH RATE AND GROUP VELOCITY OF VLF-WAVE

Considering the propagation of VLF electromagnetic waves through the cold magnetoplasma and expressing the fluctuation current due to streaming electrons in terms of perturbing electric field, one obtains the wave equation (Wang, 1969):

where v=fl,O

and

&

= (E, f iE,)/d/z,

k,,

= (k, f

=%= l -

j=il,O

ik,>ld,

(w -

E, = E, k, = k,

wD2 YW,)(W + YQJ

w B2 = wpe2(1 + A&/M,)

w, = the plasma frequency of streaming electron

60,= 1, = 0,

v=o v # 0.

Equation (2) can be separated into three homogeneous equations. Setting the determinant of these homogeneous equations equal to zero, the following dispersion equation is obtained. s2 sin2 0[w2a+,a_,/~, - k2c2](w2&, - k2c2) + &ocos2 ~(w~E+~ - k2c2)(w2&_1 - k2c2) cos2 B(w~E+~ - k2c2)(w2&_, - k2c2) + sin2 8(w2&+l&__,/e2 - k2c2)w2Ez -

w, (w - k - U,,)2 = ’

where .52= ((4 F+~+ 2/~_J/2}~. Using this dispersion equation and following the method of Bell and Buneman (1964), the maximum temporal growth rate is written as

AMPLIFICATION

OF SIGNAL

BY CERENKOV

RESONANCE

INTERACTION

2075

where D(k, w, 0) is the first term of the dispersion Equation (3). Substituting the expression for (~D~~~)~=~.~~, into Equation (4) the maximum growth rate in the frequency range Sz, < w < w, < w, is written as

= Ymar 0.87

l/3 II

ws2w[(w, + w)w - w,C$ cos %]sin2 % w 4W,QY,[W& + w cos 0) + w(w, + 2w) cos %] iI(

W=JPU((

(5)

In Equation (20) of Wang’s formulation, w should be replaced by k * U;,. From Equation (5) it is clear that the growth rate vanishes for 8 = 0” and 8 = 90”. This is physi~lIy valid because for 8 = 0 and 90”, there is no electric field parallel to the geomagnetic field and hence there is no first order interaction between the wave and the stream. When % = O”, Equation (3) is the dispersion relation for the two stream system (moving beam and stationary plasma). For %= 90”, using Equation (2) the longitudinal and transverse component of the electric field can be written as

and

From these two equations, it is seen that the longitudinal electric field is not coupied with the transverse electric field. Therefore, any perturbation in the longitudinal electric field due to the stream will not affect the transverse electric field. It can be seen from Equation (6) that for the frequency range considered, E, does not exist; this confirms the result that for %= 90” longitudinal interaction does not exist. Singh and Gendrin (1972) have shown that the refractive index corresponding to the extra-ordinary mode in the frequency range C&-=cw < w, < w, is real and positive, which permits the VLF wave to propagate in the magnetosphere as whistlers. Solving D(w, lkj, %) = 0, and using appropriate approximations, the phase velocity for the whistler mode propagation in the above frequency range is written as y = c = w = cw we2cos2 0 + (w,& - w2) sin2 % 1/Z 9 k * w, ( cos % [w(w, + w) - w, R, cos e] P

(8)

From this equation, the group velocity of the wave is written as

2c(w cos %(w, + w) - f&w, cos2 %}1’2{w2cos2 % -t (w&J - w2) sin2 %)3/z = IV~W@ cos %{w,(2w + w,) cos2 % + (w2 + we& + 2wQ(l - cos %))sin2 %}’

(9)

In the next section we will use this equation for studying the wave propagation and for calculating the interaction time and amplification coefficient. To evaluate the growth rate, one is obliged to know the characteristic frequency #, which is the simultaneous solution of the resonance condition V,, cos 8 = w/k and the cold plasma dispersion relation, D(w, (kl, %) = 0. There are four roots, one root is negative, one root corresponds to ELF waves and two roots correspond to VLF waves which have

2076

R. P. SINGH

been shown in Fig. 1. From the figure it is seen that for a given frequency there are two angles of propagation. In one branch the angles are always greater than in the other branch, but the trend of the curve is the same. Using Equation (5), we have computedithe growth rate for different values of 17and wave-normal angle 6. The results given in Fig. 2 show that the growth rate increases when the characteristic frequency fl or the wavenormal angle 8 increases. For 8 = O”, the growth rate is zero, which confirms the analytica solution of Equation (5). The growth rate will be used for the computation of the amplification factor in the next section.

Characteristic

frequency

(Hz)

FIG. 1. VARIATIONOF CHARACTERISTICPREQUENCYWITHWAVE-NORMALANGLE. The two curves represented in the figure are the two simultaneous solutions of resonance condition and the dispersion relation in the VLF range. The other two roots (one negative and one belonging to ELF wave) have not been shown. E = 1 keV, cc = 20”, fg= 1.8 x lo5 Hz,

fa=

AMPLIFICATION

Considering the conservation action, we can write vg&

3.2 x lo4 Hz (I, = 3, $I = O”).

FACTOR

of input and output energy during the process of inter-

($E2)

=2y

(;I?)

fS’(w,W)

w9

where S’ is the input of energy per second and per cm3. V, is the group velocity of the signal, y(w, 6, VII) is the growth rate, E is the amplitude of instantaneous radiated power in the Cerenkov mode. Equation (10) is a partial hnear differential equation which can be solved by finding the integrating factor [““P

(1

+?)l

’

AMPLIFICATION

FIG. 2. 4C”,

VARIATION

E = 1.0 keV,

OF SIGNAL

OF GROWTH

u = 20”,

BY CERENKOV

RATE WITH

fp= 1.8X

FREQUENCY

=

CHARACTERISTIC

RESONANCE

FREQUENCY

lo&Hz, fa = 3.2 x 10’Hz fb= 4.42 X lo3 Hz,

2077

INTERACTION

FOR AND

6 =

10, 30

BEAM

AND

PLASMA

Solving Equation (10) we have,

+$xexpg)/a’exp(-$z)d; where IE(,,, is the amplitude of the electric field at initial time t = 0. If there is no outside excitation except the spontaneous emission [S(W, 8, U, a)] which is a function of frequency and wave-normal direction of the waves and velocity and pitch angle of the particles, then E,=, = 0, S’ = S and Equation (11) can be written as t PI” = $

Q

[exp (WV,)

/s’exp (--2yz/V,)

dz] .

(12)

If the medium is homogeneous and uniform in the interaction region, y and V, may be taken as independent of z and simple integration of Equation (12) yields t I.!Zj2= $

[exp (2yz/V,) - 11.

(13)

This equation describes the amplified power level of the radiated signal during the process of linear interaction. The behaviour of this equation can be studied analytically under the following approximations. (1) 0 ==c2yz/Vg < 1, the righthand side can be expanded and

2078

the first order approximation

R. P. SINGH

can be written as

where r is the interaction time. This equation is independent of growth rate, which indicates that the interaction is weak and that the signal is coherent during the time of interaction. There is no amplification of the signal. (2) 2Y - z > 1, Equation (13) is written v,

as

This is dependent on growth rate and describes the amplification of the signal. Using an approximate form of this equation Singh and Singh (1971) have studied the amplification of the signal for VLF waves in the inner magnetosphere. (3) If the growth rate is negative and 1(2yZ/V,)I > 1 Equation (13) is simplified to

It is seen that this equation describes the absorption of the signal (1~1> l), which is evident from the negative growth rate. Thus, Equation (13) describes the amplification and absorption of the signal depending upon the sign of the growth rate. Using Equation (13) we can define the amplification factor which is the ratio of the output power to the instantaneous input power and is written as Amplification factor = ; jE12/S(w, 0, U, CC) = & [exp (272/V,) - 11. Thus the amplification factor depends on the growth rate and the interaction time which are functions of the characteristics of the medium. The interaction region is taken to be near the equatorial region along the geomagnetic field line with a L-value of 3. The plasma frequency and the gyro-frequency of the electrons are evaluated using the thermal electron density model of Thorne and Kennel (1967) and the dipole geometry of the geomagnetic field line. The energy of the particle in the beam is taken to be 1 keV because the spontaneously radiated flux of energy is large for the low energy electrons (0.5-5 keV; Jorgensen 1968; Singh and Singh 1969). The particle density of 1 keV electron in the beam has been taken to be 0.25 cm-3 at L = 3, 4 = 0”. The interaction length Z is taken to be 1000 km following Helliwell (1967). Using these parameters, the amplification factor for the branch for which wave-normal angle is smaller has been calculated and the results are shown in Fig. 3. From the figure it is seen that the amplification factor first increases with frequency and then decreases showing a maximum at certain frequency. This maximum amplification factor is of the

AMPLIFICATION

OF SIGNAL

BY CERENKOV

RESONANCE

INTERACTION

FIG. 3. VARIATION OF AMPLIFICATION FACTOR WITH FREQUENCY, fg = 14 X 105Hz, 3.2 x 104Hz, E = l$lkeV, CL= 2O”,f, = 4.42 x 1OaHz, Z = 1 X 106meter.

2079

fa=

order of 103. Figure 3 shows a spike in the flat region. This spike in the flat region is attributed to the linear and exponential dependence of the amplification factor on the growth rate which in turn is a complicated function of characteristic frequency and wave normal angle. The computation of amplification factor for the second branch of the curve, in which the wave-normal angle is very large (near the resonance angle), has not been made because the validity of the linear theory of longitudinal resonance may be questioned. The growth rate for this branch becomes very large because of the large wave-normal angle and the linear theory breaks down. As the linear theory breaks down, one is obliged to consider the non-linear theory by taking into account the energy and pitch angle diffusion of the particle. In this paper, we have presented the amplification for frequencies up to 4 kHz. This upper cut-off frequency depends on the particle energy and the characteristics of the medium. Therefore, the upper cut-off frequency will depend on the location of the interaction region and the energy of the particle in the beam. In order to explain the flat spectrum of the VLF hiss at the higher frequencies (up to 20-40 kHz) the interaction region should be considered at lower L-value or at high latitude along the same geomagnetic field line. Studying the wave-particle interaction as a mechanism of the generation of VLFemissions, Helliwell (1967) has shown that the interaction length depends on the wave frequency and the parameters of the medium. The interaction length z, decreases when the frequency of the wave increases. But for our computations, we have taken a fixed value of interaction length, because 2 does not change significantly in the frequency range of interest.

2080

R. P. SINGH

In our results, for simplicity we have assumed a homogeneous medium. In order to include the effect of inhomogeneity of the medium, the amplification factor should be computed with the help of Equation (12), using numerical integration. CONCLUSION

The wave-particle interaction using the longitudinal resonance condition has been studied and it has been shown that the spontaneously generated power is amplified by three orders of magnitude depending on the interaction length, the plasma parameters and the number of particles participating in the interaction process. This amplification factor may explain the discrepancies between the measured and theoretically evaluated powers (Singh, 1972). Because linear theory does not take into account the reaction of the wave on the particles, i.e. the energy and pitch angle diffusion of particles are ignored, it is necessary to make quasi-linear calculations for detailed interpretations of experimental observations. Acknowledgement--The author wishes to thank Drs. R. Gendrin, R. M. Pellat (France), T. N. C. Wang (U.S.A.) for useful discussions; and is grateful to Prof. R. N. Singh (B.H.U., India) for his keen interest and constant encouragement. The work is supported by the French National Center for Scientific Research (C.N.R.S.). REFERENCES BELL, T. F. and BUNEMAN,0. (1964). Phys. Rev. 133, A.1300. DOWDEN, R. L. (1962). J.geophys. Res. 67,2223. HELLIWELL,R. A. (1967). J.geophys. Res. 72,4773. JORGENSEN,T. S. (1968). J. gee&v. Res. 73, 1055. KENNEL, C. F. (1966). Phys. Flui&, 9,219O. SINGH, R. N. and SINGH, R. P. (1969). Annls. Gkophys. 25,629. SINGH, R. P. and SINGH, R. N. (1970). Nature 22549. SINGH, R. N. and SINGH, R. P. (1971). Phys. Left. 35A, 105. SINGH, R. P. (1972). A&s. Gkophys. 28. SINGH, R. P. and GENDRIN, R. (1972). G.R.I. technical note. THORNE,R. M. and KENNEL, C. F. (1967). J. geophys. Res. 72, 857. WANG, T. N. C. (1969). I.E.E.E. Trans. Antennas and Propag. AP17,690.