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Simulation of nose whistlers: An application to low latitude whistlers
ARTICLE IN PRESS
Planetary and Space Science 54 (2006) 594–598 www.elsevier.com/locate/pss
Simulation of nose whistlers: An application to low latitude whistlers Kalpana Singha,, R.P. Singhb, Abhay Kumar Singhb, R.N. Singhb a
Department of Astrophysics, Radboud University, 6525 D Nijmegen, The Netherlands b Department of Physics, Banaras Hindu University, Varanasi-221 005, India
Received 5 February 2005; received in revised form 16 December 2005; accepted 20 December 2005 Available online 24 March 2006
Abstract Simulation technique for whistler mode signal propagating through inhomogeneous plasma using WKB approximation has been developed (Singh, K., Singh, R.P., Ferencz, O.E., 2004. Simulation of whistler mode propagation for low latitude stations. Earth Planet Space 56, 979–987). In the present paper, we have used it for the analysis of recorded signals at low latitudes and estimated the nose frequency, which is not observed on the dynamic spectra. At low latitudes nose frequency is 100 kHz or more and therefore it is absent in the dynamic spectra due to attenuation of the signal at higher frequencies. The importance of nose frequency is in determining the exact path of propagation, which is required in probing of ambient medium. It is shown that the method permits to study the nose frequency variation, it can be used to deduce physical parameters as the global electric field. A case study permits to get a reasonable value of the electric field, which up to now could not be done at very low latitude. r 2006 Elsevier Ltd. All rights reserved. Keywords: WKB solutions; Magnetosphere; Noise whistler; Dynamic spectra; Laplace transform; Fourier transform
1. Introduction It is well known that the cloud-to-ground (CG) electrical discharges produce an impulse of electromagnetic fields, which experience dispersion due to the frequency dependence of group velocity of the waves during propagation through the plasma medium, which in the audio frequency range produces whistling sound in the radio receivers. The lightning generated signals from CG or CC (cloud-tocloud) discharges at lower latitudes may not align themselves to geomagnetic field lines and propagate back and forth without attenuation. The exact mechanism of generation of lightning, radiation of electromagnetic waves from lightning and details of geomagnetic field alignment are still not clear. The curvature of geomagnetic field poses an additional problem regarding the field-aligned propagation. Hence the occurrence rate of low latitude whistlers is quite low (Singh, 1993; Hayakawa and Ohta, 1992; Singh Corresponding author. Department of Astrophysics, Radboud University, 6525 ED, Nijmegen, The Netherlands. Tel.: +31 63 389 3837; fax: +31 24 365 2807. E-mail address: [email protected] (K. Singh).
0032-0633/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.pss.2005.12.018
and Hayakawa, 2001). The potential uses of whistlers in deriving medium parameters involve the knowledge of path of propagation and distribution of plasma density/magnetic field along the path of propagation. The path of propagation can be determined with the help of nose frequency. However, the observed dynamic spectra of low latitude whistlers do not contain the nose frequency (Ohtsu and Iwai, 1967), which lies in the high frequency range (100 kHz). Hence some extension technique is needed to determine the nose frequency (Sazhin et al., 1992; Singh et al., 1998a). For the analysis of non-nose whistlers, Smith and Carpenter (1961) used a method that involved evaluation of elliptical integralspinffiffiffi casting the whistler dispersion in a suitable form D t f , where f is frequency of the signal and t is corresponding group time delay. On the other hand, Brice (1965) used a graphical method, which is however, found unsuitable in processing large amount of data when a high precision is required. Further, Dowden and Allcock (1971) used a method in which a linear function of frequency Q is expressed in terms of D as Q 1=D, where D is the dispersion. This method is simple, but requires a large number of scaled data points in the
ARTICLE IN PRESS K. Singh et al. / Planetary and Space Science 54 (2006) 594–598
analysis. In addition, Bernard (1973) developed a method for extrapolating the nose frequency f n and determining the nose frequency arrival time tn by selecting two points on the whistler trace. This method although renders speedy data processing but increases the random scale error as pointed out later by Stuart (1977). In most of these methods, the systematic errors involved in the analyses of whistlers do have different magnitudes that depend primarily on the actual plasma distributions along different L-values and the approximations used in the evaluation of the dispersion integral (Tarcsai et al., 1989; Singh et al., 2000). It is worthwhile to note that above nose extension methods developed are not suitable in the analysis for the low latitude whistlers because nose frequency lies in the range of 100 kHz, where as the observed dynamic spectra mostly lie in the frequency range of 1–10 kHz. Due to errors involved in the estimation of the nose frequency, low latitude whistlers are seldom used in probing the medium (Singh et al., 1998a). Using continuity equation, momentum equation and Maxwell’s equations, Ferencz et al. (2001) and Singh et al. (2004) derived wave equation, which was solved with the help of Laplace transformation technique, using WKB approximations, to obtain analytical expressions for the wave fields of whistler mode signal propagating longitudinally through the inhomogeneous magnetoplasma. However, Omid and Hayakawa (1994) have considered propagation of electromagnetic waves excited from a current sheet in an arbitrary direction. They have studied special and temporal evolution of wave packets in different frequency ranges. The WKB approximation used by Singh et al. (2004), for the calculation purpose is valid only when the variation of geomagnetic field intensity is much bigger than the wavelength, quantitatively this validation condition can be written as (Budden, 1961) 1 3 1 dm 2 1 1 d2 m 51, 2 m3 dz2 k2 4 m2 dz where m ¼ ck=o, k being the wave number. We have numerically evaluated and found that it is valid for the whistler mode propagation in the terrestrial magnetosphere as well as when the frequency range is extended in higher range for computation. Singh et al. (2004) have developed a simulation technique to derive the full dynamic spectra using the frequency–time data of recorded low latitude whistlers. The validity of this technique in reproducing the full dynamic spectra of the whistler wave has been tested (Singh et al., 2004) by comparing the theoretically simulated and experimentally observed spectra of nose whistler recorded at the Eights Station of Antarctica (Guthart, 1965). In this paper, we attempt to use this simulation technique for the determination of the nose frequency of the whistlers recorded at low latitude stations such as Varanasi and Nainital in India. In this method, we first fix the electron
595
density, electron temperature and L-values by comparing the dynamic spectra as obtained by this theoretical method with the observed whistlers in the lower frequency range. These parameters are further used to reproduce the full dynamic spectra of the whistler including nose frequency f n and nose time tn . 2. Computational results and discussion The simulated dynamic spectra of the whistler mode signal is obtained by integrating the expression of wave electric field propagating from the source to the receiving point. Considering the terrestrial lightning discharge in the form of Dirac delta function, the expression of electric field for the whistler mode signal propagating along the magnetic field direction is written as (Singh et al., 2004) sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z Z0 1 k1 ðx0 ; oÞ E w ðx; tÞ ¼ I x0 k1 ðx; oÞ 4p 0 Rx j½ot k1 ðx;oÞdx k0 ðoÞ x0 e do, k0 ðoÞ þ k1 ðx0 ; oÞ where k0 ðoÞ ¼ o=c, o is the wave frequency, c is the velocity of light, k1 ðx; oÞ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 oob ðxÞo2p ðxÞ þ o2 fo2p ðxÞ þ o2b ðxÞ o2 g ¼ , c ðo2b ðxÞ o2 Þ op ðxÞ and ob ðxÞ are space dependent plasma- and gyrofrequencies, Z0 is wave impedance for vacuum, I x0 is the amplitude of the source, and x0 is the boundary where signal is supposed to enter in the terrestrial magnetosphere. In the above expression, the integration is to be carried out along the path of propagation and hence we require electron density and magnetic field distribution along the path of propagation. As the whistler mode wave propagates along the geomagnetic field line, the dipolar variation of magnetic field is considered, however, the electron density distribution along geomagnetic field line is taken into account with diffusive equilibrium model (DE-1) with NðOþ Þ ¼ 90%, NðHeþ Þ ¼ 2% and NðH þ Þ ¼ 8% (Park, 1972). Simpson’s quadrature integration method is used for numerical integration. The electric field is numerically calculated for the whole frequency range ð0oooob Þ. Later on, inverse fast Fourier transform is carried out to get the temporal variation of wave field. To derive the dynamic spectrogram of signal we have used short time Fourier transform (STFT) technique, which involves chopping of signal into short pieces, and the fast Fourier transform is carried out piecewise. The wave electric field as a function of frequency and time is numerically evaluated using the parameters derived from whistlers recorded at ground stations Varanasi (9th March, 1991) and Nainital (25th March, 1971) in India and the results are shown in Figs. 1(a) and (b). In the simulation process, physical parameters are varied to make
ARTICLE IN PRESS K. Singh et al. / Planetary and Space Science 54 (2006) 594–598
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an exact replica of the observed whistler traces using the lower frequency range for computation. Thereafter, keeping the parameters fixed at which perfect matching is found, frequency range is enhanced to obtain the full dynamic spectra including nose- and higher-frequency trace. We have chosen L-values separately for each of the 8000 Varanasi 9th March, 1991
7000
1- 0.147 IST
1
2- 0.142 IST
Frequency [Hz]
6000 2
3- 0.121 IST
5000 3 4000 3000
Observed part of the Spectrum
2000 1000 0
0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
Time [sec]
(a) 8000
Nainital 25th March, 1971 1- 0403 IST 2- 0237 IST 3- 0105 IST
7000
Frequency [Hz]
6000 5000 4000
3 2
Observed part of the Spectrum
3000 1 2000 1000 0
0
0.2
0.4
0.6
(b)
0.8
1.0
1.2
1.4
1.6
1.8
2.0
Time [sec]
Fig. 1. (a) Whistlers recorded at Varanasi (Singh et al., 1998b) and simulated for lower frequency range. (b) Whistlers recorded at Nainital (Singh and Singh, 1997) and simulated for lower frequency range. (For both (a) and (b) refer to Table 1 for input parameters. In both (a) and (b) observed part of the spectrum of whistlers is marked by arrows.)
recorded traces and correspondingly other physical parameters are also taken (Table 1). Dispersion of whistler depends sensitively on the distribution of electron number density and gyrofrequency variation along the path of propagation, i.e., on L-value (Singh et al., 1998a); therefore, we have mainly concentrated on the variation of the L-value and the particle number density at reference height (Nref) which is taken as 750 km (x0) for computations, while the variation in temperature at reference height (Tref), is not considered (Table 1) because we are considering cold plasma approximation where phase velocity of the wave is much larger than the thermal velocity of the plasma electrons. Matching of the experimentally observed signal to that of the simulated one is done with the variation of L-value mainly, however, fine-tuning of matching is obtained with the variation of number density at the reference height (Nref). Fig. 1(a) shows a set of simulated and observed whistlers at Varanasi both are exactly matched, whereas Fig. 1(b) shows the set of simulated whistlers and observed whistlers at Nainital. Figs. 2(a) and (b) are obtained by extending the frequency range above nose frequency during computations for the whistlers shown in Figs. 1(a) and (b). It is clear from Figs. 2(a) and (b) that whistlers are more dispersive in the lower frequency range as compared to higher frequency range. The reported L-value of whistler-3 as marked in Fig. 1(a) recorded at Varanasi on the 9th March 1991 is 2.30 (Singh et al., 1998b); however, the L-value through simulation is 2.39, which differs from the previously reported values by about 5%. In some cases this difference goes up to 15%. Table 1 indicates that nose frequency ðf n Þ decreases as the L-value increases, a result that is usually obtained in whistler analysis (Angerami and Carpenter, 1966). It is expected because the nose frequency depends on the distribution of magnetic field along the path of propagation. In the case of dipolar geometry it is directly dependent on the minimum value of geomagnetic field strength along the path of propagation. In the case of terrestrial magnetosphere, the minimum value of magnetic field corresponds to the equatorial region and f n ffi f He =0:4, where f He is the gyrofrequency of the electrons in the equatorial region, and hence the higher the L-value the lower will be the f He and the lower will be the f n as well. Fig. 2(a) shows that whistlers recorded at very low latitude stations may change their L-values
Table 1 Physical parameters used for simulation and observed nose frequency fn on frequency extension SI No.
Recording station
Recording date
Recording time (IST)
href (km)
Tref (K)
Nref (cm3)
L
fn (kHz)
1 2 3
Varanasi
9th March 1991 || ||
0147 0142 0121
7.5 105 || ||
1.6 104 || ||
1.5 104 1.3 04 8.8 103
2.07 2.15 2.39
38 34 25
1 2 3
Nainital
25th March 1971 || ||
0403 0237 0105
|| || ||
|| || ||
1.6 104 1.8 104 1.9 104
1.45 1.46 1.52
88 87 77
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East to West. We have calculated the change in fn with time for simulated whistlers for Varanasi and found jEj in the range 0.19–0.33 mV/m and positive which is in close agreement with the reported |E|0.2–0.3 mV/m during the post-midnight sector corresponding to L ¼ 2.12–2.76 (Block and Carpenter, 1974; Park, 1978; Singh et al., 1998b). In order to study, the effect of small variation in electron density along the entire path on whistler traces, we have simulated whistlers with variation in particle number density at reference height, therefore, particle density exceeds or deficits constantly throughout the path of propagation. The simulated full dynamic spectra are shown in Fig. 3(a) the third observation at Nainital. Density variation is allowed only in the range of 10%. The curve (ii) is simulated for electron number density N eref ¼ 1:895 104 =cm3 at reference height while curves (i) and (iii) are computed with respective 10% increase and
Fig. 2. (a) Simulated whistlers corresponding to the observed whistlers reported in Fig. 1(a) and extended to higher frequencies including the nose frequency. (b) Simulated whistlers corresponding to the observed whistlers reported in Fig. 2(b) and extended to higher frequencies including the nose frequency (see Table 1).
appreciably from one event to another. Singh et al. (2006) have reported similar results from the fine structure analysis of whistlers recorded at the low latitude stations Varanasi and Jammu. The precise knowledge of nose frequency and its variation with time is used to estimate the large scale electric fields (Sazhin et al., 1992; Singh et al., 1998a). In this analysis successively recorded large number of whistlers are used for the estimation of electric field, which is given by Bernard (1973) dðf 2=3 n Þ ðv=mÞ, dt where a ¼ 2:07 102 , time is measured in minutes and fn is measured in Hz (Block and Carpenter, 1974; Park, 1976). If dfn/dt is positive then the electric field is directed from jEj ¼ a
Fig. 3. (a) Dynamic spectra of simulated whistlers for L ¼ 1:52 (i) electron density increased by 10%, (ii) normal electron density N ref ¼ 1:895 104 =cm3 , (iii) decreased electron density by 10%. (b) Dynamic spectra of simulated whistlers for (i) L ¼ 1:37, (ii) L ¼ 1:52, and (iii) L ¼ 1:67 with N ref ¼ 1:895 104 =cm3 .
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K. Singh et al. / Planetary and Space Science 54 (2006) 594–598
decrease in N eref . Comparing these three curves of Fig. 3(a), we find that the nose frequency fn remains almost constant in all the three cases. This shows that fn depends only to a small extent on the ionization distribution (Smith, 1961), while dispersion is different in these three cases and it appears to be merged at the tail end in the lower frequency range. This is because of the fact that dispersion depends on electron number density. The D-value for the three traces are 26:6; 25:9 and 25.1 s1/2, respectively. Thus, we find that an increase in electron density increases the dispersion of whistlers and do not affect the nose frequency fn. However, with the same N eref ð¼ 1:895 104 =cm3 Þ, the effect of variation in L-value is studied, for the third observation at Nainital, by computing the full dynamic spectra for L ¼ 1:37, 1.52 and 1.67 as shown in Fig. 3(b). It is observed that higher the L-value, the lower is the nose frequency (fn). Further, the higher the L-value, the longer will be the path of propagation and hence the larger the dispersion of the simulated spectra. Comparing the simulated and the observed whistlers, we find that the computational technique discussed in this paper appropriately reproduces the full dispersion curve showing the nose frequency appearance on the dynamic spectra, which is very much important in the interpretation of dynamic spectra, propagation mechanism and probing of the medium. 3. Summary The simulation technique has been used to two series of whistlers recorded at low latitude stations namely at Varanasi and Nainital. In particular, the variation of the nose frequency thus obtained for Varanasi events permits us to estimate the large-scale electric field. Since at low latitudes nose frequency is not observed, this technique would be useful in the interpretation of dynamic spectra and probing of the upper ionosphere by the right hand polarized VLF waves. Acknowledgments One of the authors (K. S.) is thankful to CSIR, New Delhi for providing financial support. The work is partly supported by the Department of Science and Technology, New Delhi. References Angerami, J.J., Carpenter, D.L., 1966. Whistler studies of the plasmapause in the magnetosphere, (2). Electron density and total tube electron content near the knee in magnetospheric ionization. J. Geophys. Res. 71 (3), 711–725. Bernard, L.C., 1973. A new nose extension method for whistlers. J. Atmos. Terr. Phys. 35, 871–880.
Block, L.P., Carpenter, D.L., 1974. Derivation of magnetospheric fields from whistler data in a dynamic geomagnetic field. J. Geophys. Res. 79, 2783–2789. Brice, N., 1965. Electron density and path latitude determination from VLF emissions. J. Atmos. Terr. Phys. 27, 1–6. Budden, K.G., 1961. Radio Waves in the Ionosphere. Cambridge University Press, New York. Dowden, R.L., Allcock, G.M., 1971. Determination of nose frequency of non-whistlers. J. Atmos. Terr. Phys. 33, 1125–1129. Ferencz, Cs., Ferencz, O.E., Hamar, D., Lichtenberger, J., 2001. In: Burton, W.B., et al. (Eds.), Whistler Phenomena. Kluwer, Netherlands. Guthart, H., 1965. Nose dispersion as a measure of Magnetosphere Electron temperature. Radio Sci. J. Res. NBS/UNSC-URSI, 69D, 1417–1424. Hayakawa, M., Ohta, K., 1992. The propagation of low latitude whistlers: a review. Planet. Space Sci. 40, 1339–1351. Ohtsu, J., Iwai, A., 1967. Low-latitude nose whistlers. Proc. Res. Inst. Atmos. Nagoya Univ. 14, 75–123. Omid, M., Hayakawa, M., 1994. Excitation of electromagnetic wave by delta function current sheets in the ionospheric plasma. Radio Sci. 29, 867–877. Park, C.G., 1972. Methods of determining electron concentration in the magnetosphere from nose whistlers. Technical Report No. 3454, Radioscience Laboratory, Stanford University, Stanford, California, pp. 11–17. Park, C.G., 1976. Substorm electric fields in the evening plasmasphere and their effects on the underlying F-layer. J. Geophys. Res. 81, 2283–2288. Park, C.G., 1978. Whistler observations of substorm electric fields in night side plasmasphere. J. Geophys. Res. 83, 3137–3144. Sazhin, S.S., Hayakawa, M., Bullough, K., 1992. Whistler diagnostics of magnetospheric parameters: a review. Ann. Geophys. 10, 293–308. Singh, B., Hayakawa, M., 2001. Propagation modes of low- and very-lowlatitude whistlers. J. Atmos. Solar-terr. Phys. 63, 113–147. Singh, K., Singh, R.P., Ferencz, O.E., 2004. Simulation of whistler mode propagation for low latitude stations. Earth Planet Space 56, 979–987. Singh, R.P., 1993. Whistler studies at low latitudes: a review. Indian J. Radio & Space Phys. 22, 139–155. Singh, R.P., Singh, A.K., Singh, D.K., 1998a. Plasmaspheric parameters as determined from whistler spectrograms. A review. J. Atmos. Terr. Phys. 60, 495–508. Singh, R.P., Singh, U.P., Singh, A.K., Singh, D.P., 1998b. A case study of whistlers recorded at Varanasi. Earth Planets Space 50, 1–8. Singh, R.P., Patel, R.P., Singh, A.K., Hamar, D., Lichtenberger, J., 2000. Matched filtering parameter estimation method and analysis of whistlers recorded at Varanasi. Pramana J. Phys. 55, 685–691. Singh, R.P., Singh, K., Hamar, D., Lichtenberger, J., 2006. Matched filtering analysis of whistlers and its propagation characteristics at low latitudes. J. Atmos. Terr. Phys., in press. Singh, U.P., Singh, R.P., 1997. Study of plasmasphere–ionosphere coupling fluxes. J. Atmos. Terr. Phys. 59, 1321–1327. Smith, R.L., 1961. Properties of the outer ionosphere deduced from nose whistlers. J. Geophys. Res. 66, 3709–3716. Smith, R.L., Carpenter, D.L., 1961. Extension of nose whistler analysis. J. Geophys. Res. 66, 2582–2586. Stuart, G.F., 1977. Systematic errors in whistler extrapolation 1. Linear Q analysis and the extrapolation factor. J. Atmos. Terr. Phys. 39, 415–425. Tarcsai, G., Strangeways, H.J., Rycroft, M.J., 1989. Error sources and travel time residuals in plasmasferics whistlers interpretation. J. Atmos. Terr. Phys. 51, 249–258.
Planetary and Space Science 54 (2006) 594–598 www.elsevier.com/locate/pss
Simulation of nose whistlers: An application to low latitude whistlers Kalpana Singha,, R.P. Singhb, Abhay Kumar Singhb, R.N. Singhb a
Department of Astrophysics, Radboud University, 6525 D Nijmegen, The Netherlands b Department of Physics, Banaras Hindu University, Varanasi-221 005, India
Received 5 February 2005; received in revised form 16 December 2005; accepted 20 December 2005 Available online 24 March 2006
Abstract Simulation technique for whistler mode signal propagating through inhomogeneous plasma using WKB approximation has been developed (Singh, K., Singh, R.P., Ferencz, O.E., 2004. Simulation of whistler mode propagation for low latitude stations. Earth Planet Space 56, 979–987). In the present paper, we have used it for the analysis of recorded signals at low latitudes and estimated the nose frequency, which is not observed on the dynamic spectra. At low latitudes nose frequency is 100 kHz or more and therefore it is absent in the dynamic spectra due to attenuation of the signal at higher frequencies. The importance of nose frequency is in determining the exact path of propagation, which is required in probing of ambient medium. It is shown that the method permits to study the nose frequency variation, it can be used to deduce physical parameters as the global electric field. A case study permits to get a reasonable value of the electric field, which up to now could not be done at very low latitude. r 2006 Elsevier Ltd. All rights reserved. Keywords: WKB solutions; Magnetosphere; Noise whistler; Dynamic spectra; Laplace transform; Fourier transform
1. Introduction It is well known that the cloud-to-ground (CG) electrical discharges produce an impulse of electromagnetic fields, which experience dispersion due to the frequency dependence of group velocity of the waves during propagation through the plasma medium, which in the audio frequency range produces whistling sound in the radio receivers. The lightning generated signals from CG or CC (cloud-tocloud) discharges at lower latitudes may not align themselves to geomagnetic field lines and propagate back and forth without attenuation. The exact mechanism of generation of lightning, radiation of electromagnetic waves from lightning and details of geomagnetic field alignment are still not clear. The curvature of geomagnetic field poses an additional problem regarding the field-aligned propagation. Hence the occurrence rate of low latitude whistlers is quite low (Singh, 1993; Hayakawa and Ohta, 1992; Singh Corresponding author. Department of Astrophysics, Radboud University, 6525 ED, Nijmegen, The Netherlands. Tel.: +31 63 389 3837; fax: +31 24 365 2807. E-mail address: [email protected] (K. Singh).
0032-0633/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.pss.2005.12.018
and Hayakawa, 2001). The potential uses of whistlers in deriving medium parameters involve the knowledge of path of propagation and distribution of plasma density/magnetic field along the path of propagation. The path of propagation can be determined with the help of nose frequency. However, the observed dynamic spectra of low latitude whistlers do not contain the nose frequency (Ohtsu and Iwai, 1967), which lies in the high frequency range (100 kHz). Hence some extension technique is needed to determine the nose frequency (Sazhin et al., 1992; Singh et al., 1998a). For the analysis of non-nose whistlers, Smith and Carpenter (1961) used a method that involved evaluation of elliptical integralspinffiffiffi casting the whistler dispersion in a suitable form D t f , where f is frequency of the signal and t is corresponding group time delay. On the other hand, Brice (1965) used a graphical method, which is however, found unsuitable in processing large amount of data when a high precision is required. Further, Dowden and Allcock (1971) used a method in which a linear function of frequency Q is expressed in terms of D as Q 1=D, where D is the dispersion. This method is simple, but requires a large number of scaled data points in the
ARTICLE IN PRESS K. Singh et al. / Planetary and Space Science 54 (2006) 594–598
analysis. In addition, Bernard (1973) developed a method for extrapolating the nose frequency f n and determining the nose frequency arrival time tn by selecting two points on the whistler trace. This method although renders speedy data processing but increases the random scale error as pointed out later by Stuart (1977). In most of these methods, the systematic errors involved in the analyses of whistlers do have different magnitudes that depend primarily on the actual plasma distributions along different L-values and the approximations used in the evaluation of the dispersion integral (Tarcsai et al., 1989; Singh et al., 2000). It is worthwhile to note that above nose extension methods developed are not suitable in the analysis for the low latitude whistlers because nose frequency lies in the range of 100 kHz, where as the observed dynamic spectra mostly lie in the frequency range of 1–10 kHz. Due to errors involved in the estimation of the nose frequency, low latitude whistlers are seldom used in probing the medium (Singh et al., 1998a). Using continuity equation, momentum equation and Maxwell’s equations, Ferencz et al. (2001) and Singh et al. (2004) derived wave equation, which was solved with the help of Laplace transformation technique, using WKB approximations, to obtain analytical expressions for the wave fields of whistler mode signal propagating longitudinally through the inhomogeneous magnetoplasma. However, Omid and Hayakawa (1994) have considered propagation of electromagnetic waves excited from a current sheet in an arbitrary direction. They have studied special and temporal evolution of wave packets in different frequency ranges. The WKB approximation used by Singh et al. (2004), for the calculation purpose is valid only when the variation of geomagnetic field intensity is much bigger than the wavelength, quantitatively this validation condition can be written as (Budden, 1961) 1 3 1 dm 2 1 1 d2 m 51, 2 m3 dz2 k2 4 m2 dz where m ¼ ck=o, k being the wave number. We have numerically evaluated and found that it is valid for the whistler mode propagation in the terrestrial magnetosphere as well as when the frequency range is extended in higher range for computation. Singh et al. (2004) have developed a simulation technique to derive the full dynamic spectra using the frequency–time data of recorded low latitude whistlers. The validity of this technique in reproducing the full dynamic spectra of the whistler wave has been tested (Singh et al., 2004) by comparing the theoretically simulated and experimentally observed spectra of nose whistler recorded at the Eights Station of Antarctica (Guthart, 1965). In this paper, we attempt to use this simulation technique for the determination of the nose frequency of the whistlers recorded at low latitude stations such as Varanasi and Nainital in India. In this method, we first fix the electron
595
density, electron temperature and L-values by comparing the dynamic spectra as obtained by this theoretical method with the observed whistlers in the lower frequency range. These parameters are further used to reproduce the full dynamic spectra of the whistler including nose frequency f n and nose time tn . 2. Computational results and discussion The simulated dynamic spectra of the whistler mode signal is obtained by integrating the expression of wave electric field propagating from the source to the receiving point. Considering the terrestrial lightning discharge in the form of Dirac delta function, the expression of electric field for the whistler mode signal propagating along the magnetic field direction is written as (Singh et al., 2004) sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z Z0 1 k1 ðx0 ; oÞ E w ðx; tÞ ¼ I x0 k1 ðx; oÞ 4p 0 Rx j½ot k1 ðx;oÞdx k0 ðoÞ x0 e do, k0 ðoÞ þ k1 ðx0 ; oÞ where k0 ðoÞ ¼ o=c, o is the wave frequency, c is the velocity of light, k1 ðx; oÞ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 oob ðxÞo2p ðxÞ þ o2 fo2p ðxÞ þ o2b ðxÞ o2 g ¼ , c ðo2b ðxÞ o2 Þ op ðxÞ and ob ðxÞ are space dependent plasma- and gyrofrequencies, Z0 is wave impedance for vacuum, I x0 is the amplitude of the source, and x0 is the boundary where signal is supposed to enter in the terrestrial magnetosphere. In the above expression, the integration is to be carried out along the path of propagation and hence we require electron density and magnetic field distribution along the path of propagation. As the whistler mode wave propagates along the geomagnetic field line, the dipolar variation of magnetic field is considered, however, the electron density distribution along geomagnetic field line is taken into account with diffusive equilibrium model (DE-1) with NðOþ Þ ¼ 90%, NðHeþ Þ ¼ 2% and NðH þ Þ ¼ 8% (Park, 1972). Simpson’s quadrature integration method is used for numerical integration. The electric field is numerically calculated for the whole frequency range ð0oooob Þ. Later on, inverse fast Fourier transform is carried out to get the temporal variation of wave field. To derive the dynamic spectrogram of signal we have used short time Fourier transform (STFT) technique, which involves chopping of signal into short pieces, and the fast Fourier transform is carried out piecewise. The wave electric field as a function of frequency and time is numerically evaluated using the parameters derived from whistlers recorded at ground stations Varanasi (9th March, 1991) and Nainital (25th March, 1971) in India and the results are shown in Figs. 1(a) and (b). In the simulation process, physical parameters are varied to make
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an exact replica of the observed whistler traces using the lower frequency range for computation. Thereafter, keeping the parameters fixed at which perfect matching is found, frequency range is enhanced to obtain the full dynamic spectra including nose- and higher-frequency trace. We have chosen L-values separately for each of the 8000 Varanasi 9th March, 1991
7000
1- 0.147 IST
1
2- 0.142 IST
Frequency [Hz]
6000 2
3- 0.121 IST
5000 3 4000 3000
Observed part of the Spectrum
2000 1000 0
0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
Time [sec]
(a) 8000
Nainital 25th March, 1971 1- 0403 IST 2- 0237 IST 3- 0105 IST
7000
Frequency [Hz]
6000 5000 4000
3 2
Observed part of the Spectrum
3000 1 2000 1000 0
0
0.2
0.4
0.6
(b)
0.8
1.0
1.2
1.4
1.6
1.8
2.0
Time [sec]
Fig. 1. (a) Whistlers recorded at Varanasi (Singh et al., 1998b) and simulated for lower frequency range. (b) Whistlers recorded at Nainital (Singh and Singh, 1997) and simulated for lower frequency range. (For both (a) and (b) refer to Table 1 for input parameters. In both (a) and (b) observed part of the spectrum of whistlers is marked by arrows.)
recorded traces and correspondingly other physical parameters are also taken (Table 1). Dispersion of whistler depends sensitively on the distribution of electron number density and gyrofrequency variation along the path of propagation, i.e., on L-value (Singh et al., 1998a); therefore, we have mainly concentrated on the variation of the L-value and the particle number density at reference height (Nref) which is taken as 750 km (x0) for computations, while the variation in temperature at reference height (Tref), is not considered (Table 1) because we are considering cold plasma approximation where phase velocity of the wave is much larger than the thermal velocity of the plasma electrons. Matching of the experimentally observed signal to that of the simulated one is done with the variation of L-value mainly, however, fine-tuning of matching is obtained with the variation of number density at the reference height (Nref). Fig. 1(a) shows a set of simulated and observed whistlers at Varanasi both are exactly matched, whereas Fig. 1(b) shows the set of simulated whistlers and observed whistlers at Nainital. Figs. 2(a) and (b) are obtained by extending the frequency range above nose frequency during computations for the whistlers shown in Figs. 1(a) and (b). It is clear from Figs. 2(a) and (b) that whistlers are more dispersive in the lower frequency range as compared to higher frequency range. The reported L-value of whistler-3 as marked in Fig. 1(a) recorded at Varanasi on the 9th March 1991 is 2.30 (Singh et al., 1998b); however, the L-value through simulation is 2.39, which differs from the previously reported values by about 5%. In some cases this difference goes up to 15%. Table 1 indicates that nose frequency ðf n Þ decreases as the L-value increases, a result that is usually obtained in whistler analysis (Angerami and Carpenter, 1966). It is expected because the nose frequency depends on the distribution of magnetic field along the path of propagation. In the case of dipolar geometry it is directly dependent on the minimum value of geomagnetic field strength along the path of propagation. In the case of terrestrial magnetosphere, the minimum value of magnetic field corresponds to the equatorial region and f n ffi f He =0:4, where f He is the gyrofrequency of the electrons in the equatorial region, and hence the higher the L-value the lower will be the f He and the lower will be the f n as well. Fig. 2(a) shows that whistlers recorded at very low latitude stations may change their L-values
Table 1 Physical parameters used for simulation and observed nose frequency fn on frequency extension SI No.
Recording station
Recording date
Recording time (IST)
href (km)
Tref (K)
Nref (cm3)
L
fn (kHz)
1 2 3
Varanasi
9th March 1991 || ||
0147 0142 0121
7.5 105 || ||
1.6 104 || ||
1.5 104 1.3 04 8.8 103
2.07 2.15 2.39
38 34 25
1 2 3
Nainital
25th March 1971 || ||
0403 0237 0105
|| || ||
|| || ||
1.6 104 1.8 104 1.9 104
1.45 1.46 1.52
88 87 77
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East to West. We have calculated the change in fn with time for simulated whistlers for Varanasi and found jEj in the range 0.19–0.33 mV/m and positive which is in close agreement with the reported |E|0.2–0.3 mV/m during the post-midnight sector corresponding to L ¼ 2.12–2.76 (Block and Carpenter, 1974; Park, 1978; Singh et al., 1998b). In order to study, the effect of small variation in electron density along the entire path on whistler traces, we have simulated whistlers with variation in particle number density at reference height, therefore, particle density exceeds or deficits constantly throughout the path of propagation. The simulated full dynamic spectra are shown in Fig. 3(a) the third observation at Nainital. Density variation is allowed only in the range of 10%. The curve (ii) is simulated for electron number density N eref ¼ 1:895 104 =cm3 at reference height while curves (i) and (iii) are computed with respective 10% increase and
Fig. 2. (a) Simulated whistlers corresponding to the observed whistlers reported in Fig. 1(a) and extended to higher frequencies including the nose frequency. (b) Simulated whistlers corresponding to the observed whistlers reported in Fig. 2(b) and extended to higher frequencies including the nose frequency (see Table 1).
appreciably from one event to another. Singh et al. (2006) have reported similar results from the fine structure analysis of whistlers recorded at the low latitude stations Varanasi and Jammu. The precise knowledge of nose frequency and its variation with time is used to estimate the large scale electric fields (Sazhin et al., 1992; Singh et al., 1998a). In this analysis successively recorded large number of whistlers are used for the estimation of electric field, which is given by Bernard (1973) dðf 2=3 n Þ ðv=mÞ, dt where a ¼ 2:07 102 , time is measured in minutes and fn is measured in Hz (Block and Carpenter, 1974; Park, 1976). If dfn/dt is positive then the electric field is directed from jEj ¼ a
Fig. 3. (a) Dynamic spectra of simulated whistlers for L ¼ 1:52 (i) electron density increased by 10%, (ii) normal electron density N ref ¼ 1:895 104 =cm3 , (iii) decreased electron density by 10%. (b) Dynamic spectra of simulated whistlers for (i) L ¼ 1:37, (ii) L ¼ 1:52, and (iii) L ¼ 1:67 with N ref ¼ 1:895 104 =cm3 .
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decrease in N eref . Comparing these three curves of Fig. 3(a), we find that the nose frequency fn remains almost constant in all the three cases. This shows that fn depends only to a small extent on the ionization distribution (Smith, 1961), while dispersion is different in these three cases and it appears to be merged at the tail end in the lower frequency range. This is because of the fact that dispersion depends on electron number density. The D-value for the three traces are 26:6; 25:9 and 25.1 s1/2, respectively. Thus, we find that an increase in electron density increases the dispersion of whistlers and do not affect the nose frequency fn. However, with the same N eref ð¼ 1:895 104 =cm3 Þ, the effect of variation in L-value is studied, for the third observation at Nainital, by computing the full dynamic spectra for L ¼ 1:37, 1.52 and 1.67 as shown in Fig. 3(b). It is observed that higher the L-value, the lower is the nose frequency (fn). Further, the higher the L-value, the longer will be the path of propagation and hence the larger the dispersion of the simulated spectra. Comparing the simulated and the observed whistlers, we find that the computational technique discussed in this paper appropriately reproduces the full dispersion curve showing the nose frequency appearance on the dynamic spectra, which is very much important in the interpretation of dynamic spectra, propagation mechanism and probing of the medium. 3. Summary The simulation technique has been used to two series of whistlers recorded at low latitude stations namely at Varanasi and Nainital. In particular, the variation of the nose frequency thus obtained for Varanasi events permits us to estimate the large-scale electric field. Since at low latitudes nose frequency is not observed, this technique would be useful in the interpretation of dynamic spectra and probing of the upper ionosphere by the right hand polarized VLF waves. Acknowledgments One of the authors (K. S.) is thankful to CSIR, New Delhi for providing financial support. The work is partly supported by the Department of Science and Technology, New Delhi. References Angerami, J.J., Carpenter, D.L., 1966. Whistler studies of the plasmapause in the magnetosphere, (2). Electron density and total tube electron content near the knee in magnetospheric ionization. J. Geophys. Res. 71 (3), 711–725. Bernard, L.C., 1973. A new nose extension method for whistlers. J. Atmos. Terr. Phys. 35, 871–880.
Block, L.P., Carpenter, D.L., 1974. Derivation of magnetospheric fields from whistler data in a dynamic geomagnetic field. J. Geophys. Res. 79, 2783–2789. Brice, N., 1965. Electron density and path latitude determination from VLF emissions. J. Atmos. Terr. Phys. 27, 1–6. Budden, K.G., 1961. Radio Waves in the Ionosphere. Cambridge University Press, New York. Dowden, R.L., Allcock, G.M., 1971. Determination of nose frequency of non-whistlers. J. Atmos. Terr. Phys. 33, 1125–1129. Ferencz, Cs., Ferencz, O.E., Hamar, D., Lichtenberger, J., 2001. In: Burton, W.B., et al. (Eds.), Whistler Phenomena. Kluwer, Netherlands. Guthart, H., 1965. Nose dispersion as a measure of Magnetosphere Electron temperature. Radio Sci. J. Res. NBS/UNSC-URSI, 69D, 1417–1424. Hayakawa, M., Ohta, K., 1992. The propagation of low latitude whistlers: a review. Planet. Space Sci. 40, 1339–1351. Ohtsu, J., Iwai, A., 1967. Low-latitude nose whistlers. Proc. Res. Inst. Atmos. Nagoya Univ. 14, 75–123. Omid, M., Hayakawa, M., 1994. Excitation of electromagnetic wave by delta function current sheets in the ionospheric plasma. Radio Sci. 29, 867–877. Park, C.G., 1972. Methods of determining electron concentration in the magnetosphere from nose whistlers. Technical Report No. 3454, Radioscience Laboratory, Stanford University, Stanford, California, pp. 11–17. Park, C.G., 1976. Substorm electric fields in the evening plasmasphere and their effects on the underlying F-layer. J. Geophys. Res. 81, 2283–2288. Park, C.G., 1978. Whistler observations of substorm electric fields in night side plasmasphere. J. Geophys. Res. 83, 3137–3144. Sazhin, S.S., Hayakawa, M., Bullough, K., 1992. Whistler diagnostics of magnetospheric parameters: a review. Ann. Geophys. 10, 293–308. Singh, B., Hayakawa, M., 2001. Propagation modes of low- and very-lowlatitude whistlers. J. Atmos. Solar-terr. Phys. 63, 113–147. Singh, K., Singh, R.P., Ferencz, O.E., 2004. Simulation of whistler mode propagation for low latitude stations. Earth Planet Space 56, 979–987. Singh, R.P., 1993. Whistler studies at low latitudes: a review. Indian J. Radio & Space Phys. 22, 139–155. Singh, R.P., Singh, A.K., Singh, D.K., 1998a. Plasmaspheric parameters as determined from whistler spectrograms. A review. J. Atmos. Terr. Phys. 60, 495–508. Singh, R.P., Singh, U.P., Singh, A.K., Singh, D.P., 1998b. A case study of whistlers recorded at Varanasi. Earth Planets Space 50, 1–8. Singh, R.P., Patel, R.P., Singh, A.K., Hamar, D., Lichtenberger, J., 2000. Matched filtering parameter estimation method and analysis of whistlers recorded at Varanasi. Pramana J. Phys. 55, 685–691. Singh, R.P., Singh, K., Hamar, D., Lichtenberger, J., 2006. Matched filtering analysis of whistlers and its propagation characteristics at low latitudes. J. Atmos. Terr. Phys., in press. Singh, U.P., Singh, R.P., 1997. Study of plasmasphere–ionosphere coupling fluxes. J. Atmos. Terr. Phys. 59, 1321–1327. Smith, R.L., 1961. Properties of the outer ionosphere deduced from nose whistlers. J. Geophys. Res. 66, 3709–3716. Smith, R.L., Carpenter, D.L., 1961. Extension of nose whistler analysis. J. Geophys. Res. 66, 2582–2586. Stuart, G.F., 1977. Systematic errors in whistler extrapolation 1. Linear Q analysis and the extrapolation factor. J. Atmos. Terr. Phys. 39, 415–425. Tarcsai, G., Strangeways, H.J., Rycroft, M.J., 1989. Error sources and travel time residuals in plasmasferics whistlers interpretation. J. Atmos. Terr. Phys. 51, 249–258.