L I M I T I N G W A V E G R O W T H F O R E L E C T R O N

C Y C L O T R O N W A V E S

D. P. S I N G H , 1 U. P. S I N G H 2 AND R. P. S I N G H 2

1Physics Department, Government College Adampur, Hisar - 125052, India; 2Atmospheric Research Lab, Physics Department, Banaras Hindu University, Varanasi - 221005, India

(Received 23 November i993)

Abstract. Quasilinear weak diffusion theory presented by Kennel and Petschek (1966) and advanced by Schulz and Davidson (1988) is further extended to put an upper limit on the growth of electron cyclotron waves. It is shown that the power gain of whistler mode electron cyclotron wave (other than plasmaspheric hiss) can not exceed 40 dB. Inside the outer radiation belt, the upper limit of temporal wave growth is ~350 rad s -1 and normalised wave growth is 0.017. The limits are independent of the kind of the electron diffusion and are applicable for on the equator/off the equator locations of wave- particle interactions.

1. Introduction

Kennel and Petschek (1966) put a limit on stably trapped fluxes of energetic particles in the magnetospheric radiation belts which can be computed only if we know in advance the pitch angle anisotropy (A), the perpendicular/parallel temperatures (T±/TII) of the bi-Maxwellian distribution and the exponent of the power law spectrum. Cornwall (1966), Haerendel (1970) and, Brice and Lucas (1971) have shown variations on the general theme of Kennel and Petschek (1966). Cornwall (1966) treated the ULF wave (f~< 3 Hz) amplification by anisotropic pitch angle distribution of ring current protons at flux levels initially exceeding the stably trapped limit. Whereas Cornwall (1966) considered over-simplified initial particle-energy spectrum, Haerendel (1970) considered the same energy spectrum but calculated the distribution function consistent with marginal occurrence of a magnetospheric maser at all wave frequencies below proton gyrofrequency. Brice and Lucas (1971) postulated a pitch angle anisotropy for outer zone electrons and deduced that the whistler mode maser would be band limited for reasonable electron spectra. Roux and Solomon (1971), starting with initial energy spectrum as well as pitch angle anisotropy considered quasilinear pitch angle scattering by electron cyclotron waves and obtained final energy spectrum and anisotropy. Etcheto et al. (1973) obtained a self consistent steady state solution for whistler mode wave spectrum, electron energy spectrum and electron pitch angle distribu- tion. Etcheto et al . 's (1973) theory was basically different from that of Kennel and Petschek (1966) in that whereas Kennel and Petschek (1966) considered interacting wave's spectral density (/~f-) as a measure of pitch angle diffusion, Etcheto et al.

(1973) considered integral flux (J) responsible for pitch angle scattering. Recently, Schulz and Davidson (1988) extended the work of Kennel and Pets-

Earth, Moon, and Planets 64: 145-154, 1994. © 1994 Kluwer Academic Publishers. Printed in the Netherlands.

146 D . p . S INGH ET AL.

chek and put an upper bound on integral flux of stably trapped energetic particles (electrons and protons both). Whereas Kennel and Petschek (1966) could provide a limit on J (>Ell) for energies above a threshold energy (Ell), Schulz and Davidson (1988) gave the expression for J for any energy (E) above threshold energy (Eli).

Like integral flux of anisotropic electrons, the wave growth measurement of cyclotron waves, too, is important to study the Wave Particle Interaction (WPI) inside the radiation belts. Two characteristic measures of wave growth are, growth rate and saturation level. The growth itself can be somewhat irregular but in many cases is exponential with growth rates (intensity of the wave) between 25 dB s -1 and 250 dB s -1. Though wave growth measurements based on experimental evi- dences have been reported by many authors (Helliwell and Katsufrakis, 1974; Paschal and Helliwell, 1984), the satellite data on the unamplified signal strength are limited and no in-situ measurement of the amplitude of a ducted signal has yet been made. The experimental measurements of whistler mode growth rates have produced data on both the amplitude and phase evolution of (an amplified) received signal, though the phase measurement could be given due consideration only after 1978 (Dowden et al., 1978; Rietveld et al., 1978). Phase measurements of signals from VLF transmitters indicate that pulses with temporal growth exhibit a relative phase advance with time (Paschal, 1988; Dowden et al., 1978; Carlson et al., 1990; Paschal and Helliwell, 1984). In this paper we further extend the works of Kennel and Petschek, and Schulz and Davidson and impose an upper limit on temporal wave growth, normalised wave growth and power amplification of electron cyclotron waves.

2. Wave Growth Formulation

Since we will proceed on the lines of Kennel and Petschek (henceforth KP) and Schulz and Davidson (henceforth SD) we refer these papers frequently. Further, SD considered the interaction region Z to be LRo (L is McIlwain parameter and R0 is earth radius (6370 km), we take Z < LRo. For a parallel propagating whistler wave, KP gave following expression for temporal wave growth (3').

7 = ~r{~(1 - x)lA(1 - x ) - x l r / (1)

where A = pitch angle anisotropy, f~ (=qB/mc) is gyroffequency of the particle, B the geomagnetic induction, q charge of the particle, rn mass of the particle and c is the velocity of light, x is normalised wave frequency ~o/f~ where w is wave frequency. ~ is the fraction of resonant particles and is given by the expression (SD)

2wm~ (1 - x) |p,f @ l (2)

~ - N k 30

where N is number density of cold plasma, p • (11) is particle momentum perpendicu-

GROWTH FOR ELECTRON CYCLOTRON WAVES 147

lar (parallel) to geomagnetic field) and y is equatorial phase space density of energetic particles such that

)~(P±, Pit) = (P~-/Pll)zSg(P), (3)

In Equation (3), s is an energy independent index to measure pitch angle anisotropy (A) and g ( p ) is the isotropy factor which defines the energy spectrum of the trapped particle population (SD). Following SD and using p2 = (pa)2 + (pFi) 2, we can write

fi * p ± f - - " c / v r l ) 2 1 n R - l ( ~ " ~ - ¢0) 2

- - 8"rr3 q 2 Z ( s a - s o ) - c o ) " ( 4 )

Substituting Equation (4) into Equation (2), the growth rate (7) is given as

2~-rn~) y = 7rYe(1 - x) [A(1 - x ) - x] --TZ-- (1 - x )

N/¢

(c/riO 2 In R - 1 ( a - o )2 (5) 8"Tr3Q2 Z ( s ~ ~ - - S¢.O - - o 9 ) "

The above equation reduces to a still simpler case when A = s is considered (SD, P.72)

f~3c2(1 - x) 4 In R -1 7 = 2 2 , (6)

rrp Z k vlp

where 7r 2 = square of plasma frequency = 4rrq2N/m. Putting the well known con- dition for first order cyclotron resonance (kVll = 12 - 60) in Equation (6) we obtain

~ 3 C 2 ( 1 - - X) 4 In R -1 " ) / : 2 ~p Zvrl (f~ - w) (7)

which further reduces to

2 ( i - x) 3 in R -z , (8) y = v,~ Zvll

where v~ is Alfvdn velocity for electrons. Since vrl --- v~ (1 - X)3/2/X 1/2 we can write

V= In R -1 (1 - x)3/2xl/2 = (9)

Z

Equation (9) can be reduced to get normalised temporal growth rate (3,/f~) as under

")/ C ( 1 - - X)3/2X 1/2 in R - 1 - ( l o )

Wp Z

148 D . P . SINGH ET AL.

0.35

0 . 3 -

0.25 -

¢q

" × 0.2 -

,"0.15-

0.1

0 . 0 5

0 i i i i

0 0.2 0.4. 0.6 0.8

X

Fig. 1. V a r i a t i o n o f (1 - X)3/2X 112 wi th n o r m a l i s e d f r e q u e n c y x.

3. Ionospheric Model

The low frequency wave propagation parallel to geomagnetic field is in plasma- sphere. We confine the present study to higher L shells lying in the outer radiation belt (2 < L < 8). The cold plasma density (N) for electrons at these L shells is taken from the experimental profiles obtained by Angerami and Carpenter (1966) employing whistler technique during moderate storm periods (Kp = 2 - 4) and values are 4000 (L = 2.4), 1000 (L = 3) and 400 electrons cm -3 (g = 4). These values have earlier been used by Singh (1991, 1992). We have taken N = 182 for L = 5 for inside plasmapause position and N-- 18 for outside plasmapause posi- tion (Angerami, 1966; Church and Thorne, 1983; Singh, 1991). The cold plasma density for L = 6 - 8 is taken to be 1 - 10 electrons cm -3 which have been experimentally observed by Cornilleau-Wehrlin et al. (1985) and Solomon et al.

(1988) aboard GEOS satellites. Singh (1992) used these electron densities in his study of strong diffusion regimes for electrons diffused by ELF waves.

4. Results and Discussions

The variation of (1 -x)3/2x 1/2 with x, as shown in Figure 1, clearly shows that maximum growth will occur at x = 0.25. Almost all the wave spectra (y/f l versus x) in Solomon et al. (1988, 1989) depict this characteristic. Thus Equation (10) can also be written as

(~/) ~_ X)3/2X1/2]max lnR-1 c [(1 - =_ c i n R -1 (0.3248). (11) m a x 7J'p Z m i n "Tgp Z m i n

G R O W T H FOR E L E C T R O N C Y C L O T R O N W A V E S 149

The above equation contains Zmin which is minimum interaction region. Roed- erer (1970) has reported that the interaction region is limited to latitude range (Am) ~< 20 ° and beyond this point wave growth vanishes. Helliwell (1967) and Helliwell and Katsufrakis (1974), using theoretical computations and experimental evidences, have shown that the region of temporal growth is within a few degrees (3 - 5 °) of the geomagnetic equator. To calculate y . . . . and (Y/f~)max we consider that the interaction region (Z minimum) is limited to -+ 1 ° latitudes around the geomagnetic equator. Applying the formulae given in Helliwell (1967) we get

Zmi~ = 0.02 LRo (12)

Recently, Carlson et al. (1990), in computing space-time evolution of whistler mode wave growth, considered interaction region lying between upstream boun- dary (Zup) and downstream boundary (Zd . . . . ). They computed Zup = +250km and Zdow~ = --250 km at L = 4. Thus in their case the total interaction region is

of 500 km at L -- 4. Equation (12) gives Zmin = 510 km at L = 4 i.e. the interac- tion region considered by Equation (12) is still larger than that taken by Carlson et al. (1990, Figure 6) and so our Z is not only minimum but an acceptable minimum. Another important parameter which controls the wave growth is reflec- tion coefficient R.

Etcheto et al. (1973), Sazhin (1984) and Singh (1991) have considered, in their wave spectral density (B}) calculations, imaginary reflection coefficient R to be 0.1 ( lnR -1 = 2.3). On the other hand, Villalon et al. (1989) considered R to be a real quantity which depends on E and F region cold plasma parameters and took R = 0.8 i.e. in their case ln R - 1 = 0.2231. All these R values were theoretical assumptions/calculations without any experimental evidences. Using the wave growth measurements the value of R can be evaluated. Following Solomon et al.

(1989) we can write

Gain (dB) = 10 log e a (13)

where G = In R -1. Based on the wave growth measurements (Stiles and Helliwell, 1977; Chang and Helliwell, 1979; Carlson et al., 1985, 1990) we can have an upper boundary on wave growth of 40 dB. This value produces G = In R -~ = 9.3, which comes out to be the same as computed by Thorne et al. (1979) for quiet time ELF hiss generation. Thus Equation (11) can be written as

(~)m = 3×108×9.3×0.3248 ax 6.28fp(0.02 × 6.37 × 106) L

= 1 1 3 3 / L f p (14)

Since plasma frequency f p = 8.979 N ~/2 (kHz) and electron gyrofrequency in the equatorial region 1~ = 2~r(873.6 kHz)/L 3, we can write

150 D. P, S1NGH I~T AL.

' ~ INSIDE OUTSIDE 300 ~LASMASPHERE PLASMASPHERE ~21 i'150

E I,,"

0 . . . . ~ - " _ _ I x ~ _ _ L 0

2 ~ 4 5 6 7 8

L VALUES

Variation of reduced wave growth (7/~)r~ax and temporal wave growth rate Ymax with L values.

2 0 0

1t 100

Fig. 2.

6.9 x 10 5 rad s -I . (15)

Y m a x L4N1/2

The variations of reduced growth rate (Y/f~)m~x and temporal growth rate ")/max are depicted in Figure 2. The figure shows that the reduced wave growth is minimum at L = 2.2, the lower boundary of the outer radiation belt and then slowly increases up to L = 5 (plasmapause position) inside the plasmasphere. Outside the plasmasphere, it increases sharply till the upper boundary of the outer radiation belt.

The variation of 'Ymax with L-values shows that in this case the slope of the curve is opposite to that of (Y/f~)~ax Curve. The temporal growth rate is highest at the lower boundary of the outer radiation belt. It, then, tends to decrease till L = 5 inside the plasmasphere. Outside the plasmasphere the value of Ymax is highest at L = 5 and remains almost constant at 6 ~< L <~ 8. In the plasmapause there is sudden increase in the temporal growth rate. This shows that plasmapause gives rise to large temporal wave growth and thereby supports intense VLF/ELF spectral densities as is generally observed aboard various satellites (Church and Thorne, 1983; Etcheto et al. 1973). Though we have taken plasmapause position at L = 5 (quiet time position), it may be at any position between L = 3 and L = 6 depending upon the Kp indices, intensity of magnetic storms and the distortion of dipolar geomagnetic field (Sazhin, 1984) and will always produces intense

(Bs). ELF/VLF spectral density 2 In their analyses of ELF/VLF intensity data recorded aboard GEOS satellites

Cornilleau-Wehrlin et al. (1985) and Solomon et al. (1988, 1989) have reported the variation of experimentally determined reduced growth rates with normalised

G R O W T H FOR E L E C T R O N C Y C L O T R O N W A V E S 151

TABLE I

Comparison of computed reduced wave growth (Y/f~)max values with experimentally determined values of Cornilleau-Wehrlin et al. (1985).

Figure L N A,n Experimental Theoretical Case ( c m - 3 ) V a l u e Value

3(b) 6.78 5 0.5 ° 1 × 10 - 3 8.32 x 10 -3 4(b) 6.78 7 0.5 ° 6 × 10 - 4 70.4 x 10 ~ 6(c) 5.79 6 20 ° 2 x 10 .5 890 × 10 -5 8 5.92 6 20 ° 3 x 10 -s 870 x 10 -5 Right 9 6.42 4 23 ° 6x10 -s 983x I0 5

Strong Diffusion Moderate Diffusion Strong Diffusion Strong Diffusion

Weak Diffusion

Comparison of experimental values of

TABLE II

(Y/f~) .... reported by Solomon values.

et aL (1988) with theoretical

Figure L N Am Experimental Theoretical Case (cm -3) Value Value

1 Left 7.1 7.7 t0 ° 4 x 10 -s 641 x 10 5 Weak Diffusion 1 Right 7.1 4.5 l0 ° 8 x 10 s 838 x 10 -5 Strong Diffusion 3 Left 4.5 120 3.5 ° 5 × 10 - 4 25.6 x 10 .4 Weak Diffusion 3 Right 4.5 120 3.0 ° 4 X 10 - 4 25.6 X 10 - 4 Moderate Diffusion 4 Left 5.2 120 11 ° 2.5 X 10 - 4 22.2 X 10 - 4 Weak Diffusion 4 Right 4.7 120 10 ° 5 × 10 .4 24.5 X 10 - 4 Weak Diffusion 5 Left 4.6 500 24 ° 1 x 10 -6 1227 × 10 -6 Weak Diffusion 5 Right 4.9 1100 31 ° 2 × 10 .6 777 × 10 -6 Weak Diffusion

f requency (x). Most of the figures in So lomon et al. (1989) are actually repeats of

the figures in Corn i l l eau -Wehr l in et al. (1985). We have made a compara t ive s tudy

of exper imenta l ly d e t e r m i n e d (Y/f~)max values, given by Corn i l l eau-Wehr l in et al.

(1985) in Table I, with those compu ted f rom E q u a t i o n (14). Tab le II shows same

compar i son for the reduced wave growths compu ted by So lomon et al. (1988). It

is ev ident f rom these tables that the exper imenta l ly de t e rmined values are lower

than our l imit ing values. It is to be emphas ized here that (i) we have fol lowed the

weak diffusion theory of K e n n e l and Petschek (1966) and der ived the expression

for the reduced growth rate max ima and t empora l growth rate max ima and our

values (Tables I, II) are greater than (Y/f~)max de t e rmined exper imenta l ly for

s t rong diffusion cases, (ii) our expressions are der ived for the equa tor ia l values of

e lec t ron gyrof requency (f~) whereas the observa t ions recorded by G E O S satellites

are sl ightly/largely away f rom the equator . Thus our fo rmulae are appl icable for

bo th weak and s t rong diffusions at equa to r /o f f - equa to r locations.

F igure 2 in Corn i l l eau -Wehr l in et al. (1985) shows much higher normal i sed wave

growth of 1.14 × 10 -2 but , for tunate ly , this value, too, is lower than our l imit at

the co r respond ing L-shell . Ac tua l ly in this case the wave may have exper ienced

152 D . P . S I N G H E T A L .

energy enhancement (it is the strongest wave recorded aboard GEOS satellites) due to multiple reflections at the base of the resonance tube or magneto-ionic ducting (Helliwell, 1965). As is clear from Figure 2, a whistler mode cyclotron wave can never have a temporal growth rate above ~350 rad s -1 and normalised growth rate above ~0.017 inside the outer radiation belt. These limits are indepen- dent of strength of diffusion (weak and diffusion) or the location of the interaction between energetic electrons and whistler mode waves. It is well known that wave- particle interaction region is mainly at the equator and most of the waves recorded at various positions away from the equator might have generated at the equator and propagated to these locations in ducted/nonducted mode.

We will further discuss why we have taken 40 dB gain to be maximum for electron cyclotron waves. Solomon et al. (1988, 1989) have computed the gain for plasmaspheric hiss observed aboard GEOS 1/2 satellites to be >100 dB. Thorne et al. (1979), Church and Thorne (1983) and Solomon et al. (1988, 1989) by doing ray tracing computations for such waves have found that these waves exhibit cyclic trajectories with multiple field-aligned transits through the equatorial growth region. Thus the waves are amplified many times and hence the observed gain >100 dB. In the present case, we consider only one transit of the whistler wave through the interaction region (Kennel and Petschek, 1966). Moreover, Solomon et al. (1988, 1989) considered background incoherent emissivity as the source for plasmaspheric hiss and they computed the gain above the incoherent emissive power of 10 -H pT 2 Hz 1 for thermal noise.

It may be argued that we have taken the gain of 38 dB for coherent waves which interact always in nonlinear mode. We have done so for three reasons: Firstly, Kennel and Petschek (1966) have shown that for weak whistler turbulence non- linear interaction reduces into linear/quasilinear interaction. Secondly, Inan (1987) has observed that for weak signals (Bw = 1 - 7 pT) coherent and incoherent signals produce same results. Thirdly, it can be shown easily that wave growth expression of Etcheto et al. (1973) for quasilinear interaction reduces into the expression for wave growth employed by Carlson et al. (1990) for nonlinear interaction between coherent waves and energetic electrons. It may be emphasized here that Dowden et al. (1978) have experimentally shown that linear amplification for 6.6 kHz waves produced a gain of 20 ± 10 dB whereas nonlinear amplification for same transmitted frequency gave the gain of ~40 dB. This explains why the authors have taken the maximum gain to be 40 dB.

We will like to add here that though our formulae are independent of plasma parameters and one can easily compute y, ,//f~ at any position inside the outer radiation belt, one will have to determine imaginary reflection coefficient 'R' first employing the method of Villalon et al. (1989).

5. Summary

Quasilinear theory given by Kennel and Petschek for pitch angle scattering of energetic electrons by whistler mode waves and wave amplification is extended to

G R O W T H FOR E L E C T R O N C Y C L O T R O N W A V E S 153

put an uppe r l imit on e lec t ron cyclotron wave growths. The expressions der ived for

t empora l and norma l i sed wave growths are free of any k ind of part icle d is t r ibut ion

funct ions. The theoret ica l values of these limits are compared with exper imenta l ly

de t e rmined wave growth max ima recorded aboard G E O S satellites and experi-

men ta l values are lower than theoret ical limits. The wave growth 's uppe r limits

are appl icable no t only for weak or s t rong diffusion of energet ic e lect rons bu t also

on the equa to r /o f f the equa to r locations.

Acknowledgements

DPS is grateful to the Univers i ty Gran t s Commiss ion , New Delhi for awarding

him a Minor Research Project , and to the Principal of his college for facilities in

the depa r tmen t . UPS is grateful to CSIR, New Delhi for providing him financial

assistance in the form of Senior Research fellowship.

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