Vertical profile of atmospheric conductivity that matches Schumann resonance observations
© Nickolaenko et al. 2016
Received: 28 June 2015
Accepted: 18 January 2016
Published: 1 February 2016
We introduce the vertical profile of atmospheric conductivity in the range from 2 to 98 km. The propagation constant of extremely low frequency (ELF) radio waves was computed for this profile by using the full wave solution. A high correspondence is demonstrated of the data thus obtained to the conventional standard heuristic model of ELF propagation constant derived from the Schumann resonance records performed all over the world. We also suggest the conductivity profiles for the ambient day and ambient night conditions. The full wave solution technique was applied for obtaining the corresponding frequency dependence of propagation constant relevant to these profiles. By using these propagation constants, we computed the power spectra of Schumann resonance in the vertical electric field component for the uniform global distribution of thunderstorms and demonstrate their close similarity in all the models. We also demonstrate a strong correspondence between the wave attenuation rate obtained for these conductivity profiles and the measured ones by using the ELF radio transmissions.
A comparison of experimental Schumann resonance data with those computed from Eqs. (1–4) has confirmed the validity of the model by Ishaq and Jones (1977), although some other models are used in the literature suggesting simpler expressions for the ν(f) dependence (Nickolaenko and Hayakawa 2002, 2014). We use relations (1–4) in what follows as the standard or the reference model.
In the field computations and in the interpretation of experimental data, the knowledge is redundant of the vertical profile of atmospheric conductivity σ(h). It is sufficient to use the regular expressions for the electromagnetic fields incorporating the propagation constant, the current moment of the field source, and the ionosphere effective height (see e.g. Nickolaenko and Hayakawa 2002, 2014).
However, information on the vertical profile of atmospheric conductivity σ(h) becomes obligatory when using the direct modeling methods such as finite difference time domain (FDTD) technique or the 2D telegraph equation (2DTE) (Kirillov 1996; Kirillov et al. 1997; Kirillov and Kopeykin 2002; Morente et al. 2003; Pechony and Price 2004; Yang and Pasko 2005). This kind of computations is impossible without knowing a particular vertical profile of air conductivity and the relevant complex permittivity of atmosphere. The range of heights 50–100 km is crucial for the ELF radio propagation, but it is inaccessible by any modern remote sensing. The existing experimental data on the air conductivity within these altitudes are rare and have been usually obtained by the rocket probes. Therefore, one can find only a limited amount of altitude profiles of the air conductivity in the literature. It is significant that none of these profiles provides a realistic frequency dependence of ELF propagation constant as given by Eqs. (1–4).
The objective of our paper is a realistic σ(h) profile consistent with the Schumann resonance observations. Such a model profile is desirable when modeling the sub-ionospheric radio propagation in the real Earth–ionosphere cavity.
The air conductivity as a function of altitude
Logarithm of air conductivity (S/m) as function of altitude above the ground surface
The profiles of atmospheric conductivity are shown in Fig. 1 for the altitudes ranging from 0 to 100 km. The thin curve with points 1 shows the classic profile (Cole and Pierce 1965) and the smooth thick curve 2 depicts the more realistic profile σ(h). As might be seen from the figure, the both curves are rather close to each other, although profile 2 has a more pronounced alteration in the 50–60 km interval (the so-called “knee”). Deviations begin from the 30 km altitude, and the profile 2 becomes “elevated” over the classical plot.
The heuristic “knee model” is popular in the modern Schumann resonance studies proposed in the paper by Mushtak and Williams (2002). It might be applied in computations of the propagation constant instead of formulas (1–4). Similarly to previous works (Kirillov 1996; Kirillov et al. 1997; Kirillov and Kopeykin 2002; Greifinger and Greifinger 1978; Nickolaenko and Rabinowicz 1982, 1987; Sentman 1990a, b; Fullekrug 2000), the knee model postulates a set of parameters allowing computing the two complex characteristic heights (the “electric” and “magnetic” heights) together with the real (i.e., having no imaginary part) scale heights nearby these altitudes. The propagation constant is computed by substituting these parameters into the “standard” equations, while the frequency dependence is postulated for all the model parameters in Mushtak and Williams (2002). After finding the appropriate propagation constant, one can turn to the field computations (Nickolaenko and Hayakawa 2014; Williams et al. (2006)).
Unfortunately, all the works applying the knee model are based on only the verbal description of the relevant σ(h) profile. None of these depicts the conductivity profile nearby the both characteristic heights. Obtaining such a profile is not a simple task, provided that it is possible at all, especially because all the model parameters depend on the signal frequency. Thus, it is not clear in what a way the real function of height σ(h), being independent of frequency, might be constructed from the complex functions of frequency. At any rate, the problem remains currently unresolved.
The simplified conductivity profiles are widely used in the direct methods of field computation. These are typically the lg[σ(h)] plot incorporating the two straight lines that form a twist at the knee altitude due to the change in the scale height (see e.g. Morente et al. 2003; Yang and Pasko 2005; Toledo-Redondo et al. 2013; Molina-Cuberos et al. 2006; Zhou et al. 2013). The vicinity of upper, “magnetic” characteristic height is ignored. The curved height dependence of the air conductivity is in fact the well-known two-scale exponential model. Advantages and drawbacks of such a model are quite well known, and these were comprehensively discussed in the literature (Mushtak and Williams 2002; Sentman 1990a, b; Greifinger et al. 2007). Besides, the two-scale exponential model does not predict any correct values of the peak frequencies and the Q-factors of the Schumann resonance modes when applied in the FDTD technique.
The propagation constant
The ELF propagation constant ν (f) is usually constructed on the assumption that the ionosphere plasma is isotropic and horizontally homogeneous. Then, by using the full wave solution (see Hynninen and Galuk 1972; Bliokh et al. 1997, 1980; Galuk and Ivanov 1978; Galuk et al. 2015), one might compute the ν (f) dependence corresponding to a given profile σ (h). The full wave solution is the rigorous solution of the radio propagation problem within the vertically stratified ionosphere, and it allows us to obtain the sub-ionospheric propagation constant (f). We will mention the major steps in obtaining the solution without reproducing equations here, as these could be found in the above-cited works. The upward and downward waves are taken into account in every plasma layer. Their thickness is much smaller than the wavelength in the medium. The tangential field components are continuous at the layer boundaries. It might be shown then (Hynninen and Galuk 1972; Bliokh et al. 1997; Galuk and Ivanov 1978; Galuk et al. 2015) that the electromagnetic problem is reduced to a nonlinear differential equation of the first order for the surface impedance (the ratio of the tangential components of E and H fields). The surface impedance satisfies boundary conditions at the ground and at the upper boundary in the ionosphere from where the plasma density is supposed to remain constant. The problem is solved numerically by using the iteration procedure, and the desired propagation constant ν (f) is obtained as a result. The method is regarded as the full wave solution, since it strictly accounts for all the fields propagating in the stratified plasma and in the air.
As might be seen, all models give very close values in the real part of the propagation constant (i.e., the phase velocity of radio waves), because deviations are only a few percents. So, the resonant frequencies are almost coincident for all three models. But, deviations in the imaginary part or in the attenuation rate of radio waves are more distinct. The standard or the reference model (Ishaq and Jones 1977) and the conductivity profile 2 provide similar dependences (curves 4 and 6), while the attenuation factor following from the classical conductivity profile (Cole and Pierce 1965) (curve 5) considerably deviates from them.
Plots in Fig. 3 indicate that profile 2 provides a rather good propagation constant being close to the reference model in the entire Schumann resonance band: deviations in the phase velocity do not exceed 1 %, and those in the attenuation rate are still within an interval of ±5 %. Therefore, profile 2 might be used in modeling of the global electromagnetic resonance of the Earth–ionosphere cavity, especially, in direct methods of field computations, such as FDTD and 2DTE (the parameters are listed in Table 1).
Validity of the conductivity profile #2 might be illustrated also by comparing the computed wave attenuation rate with the data of direct measurements, which was based on the monochromatic radio signals from the ELF transmitters (Bannister 1999; Nickolaenko 2008a, b). Data from the paper by Bannister (1999) were based on the amplitude monitoring of the signal arriving from the US Navy transmitter regarded as the Wisconsin Test Facility (WTF), in which the global network was used to receive the signal. Data were obtained at the frequency of 76 Hz, and the average attenuation rate was 0.82 dB/1000 km for the ambient night and 1.33 dB/1000 km in the ambient day conditions. The average attenuation at this frequency was equal to 1.08 dB/1000 km, and the relative standard deviation due to seasonal variations was ±25 %.
The imaginary part of the propagation constant at this frequency for the profile 2 is equal to Im[ν (f)]∣ f=76 = 0.86, and this value corresponds to the attenuation α (76 Hz) = 1.17 dB/1000 km. This attenuation is practically coincident with that by observations, and this fact is certainly in favor of the model.
The imaginary part of the propagation constant was also published in the papers (Nickolaenko 2008a, b), and it was measured at the 82 Hz frequency. It is equal to Im[ν (f)]∣ f=82 = 0.92, which corresponds to the attenuation factor α (76 Hz) = 1.25 dB/1000 km. The attenuation rate in Nickolaenko (2008a) was inferred from the distance dependence of the signal amplitude in the vertical electric field components while the radio wave was emitted from the Kola Transmitter of the Soviet Navy. The model imaginary part of the propagation constant Im[ν (f)]∣ f=82 = 0.92 is equal to the value measured experimentally.
A comparison with observations of the man-made ELF radio transmissions justifies the employment of the conductivity profile #2 in ELF applications.
Comparison of the power spectra
Two resonance spectra are shown in Fig. 4. The smooth line 1 corresponds to the spectrum computed with the reference propagation constant (Ishaq and Jones 1977), and the line with dots 2 is the spectrum relevant to our conductivity profile of the atmosphere. Relative deviations in percents from the reference spectrum are shown by curve 3 relevant to the right ordinate. By comparing Figs. 3 and 4, we observe that deviations in the spectra are more apparent than in the dispersion curves ν (f). Even a difference in the phase velocity of about 1–2 % is clearly visible in the spectra: the peak frequencies of the higher modes noticeably diverge. Curve 3 in Fig. 4 illustrates that relative deviations of the power spectrum occupy the interval from −5 to +15 %, and this is 3–4 times smaller than deviations pertinent to the classical profile (Ishaq and Jones 1977).
Accounting for ambient day and night conditions
The horizontal axis of Fig. 5 depicts the logarithm of air conductivity, and the vertical axis is the altitude above the ground. The smooth curve 2 reproduces the σ(h) profile that was shown by line 2 in Fig. 1. Line 1 in Fig. 5 corresponds to the conductivity at ambient night, i.e. when the ionosphere is known to be higher than by day. The curve 3 corresponds to ambient day condition.
By using the full wave solution, we computed the frequency dependence of the complex propagation constant for the day and the night profiles, and compared these data with the reference model of ν(f). When propagation constant is known, one can compute the power spectra of resonance oscillations in the ambient day and ambient night conditions. We are not going to investigate the effect of the ionosphere day–night asymmetry on the global electromagnetic resonance. Therefore, the term “ambient day condition” means that the horizontally uniform ionosphere is described by the day profile all over the globe. Similarly, the words “ambient night condition” mean in what follows that the night profile of the ionosphere is valid over all points of the Earth’s surface.
Figure 6 shows the computational data for the day and night conductivity profiles. Graphs in the upper panel of Fig. 6 demonstrate that the reference attenuation rate (curve 1) lies between the values obtained for the night (curve 2) and the day (curve 3) conductivity profiles within all Schumann resonance band. The bottom panel of this figure depicts the power spectra of the vertical electric field. As it was expected, the resonance peaks of the “night” power spectrum (curve with stars) are higher than those of the “day” spectrum. The resonance frequencies and the quality factors corresponding to the night conductivity profile are also higher than those corresponding to the daytime ionosphere. The spectrum relevant to the reference propagation constant occupies an intermediate position between the “day” and “night” spectra. Thus, the outline of power spectra confirms the validity of the day and night conductivity profiles that we have shown in Fig. 5 and presented in Table 1.
Discussion and conclusions
Radio wave attenuation at discrete frequencies obtained from conductivity profile and measured experimentally
Table 2 compares values of attenuation rate of ELF radio waves computed for the conductivity profile presented in Table 1 with those published in the literature and presenting the results of measurements of radio signals from the ELF radio transmitters. Data at 76 Hz were taken from the survey (Bannister 1999), which summarized the long-term observations of the signals transmitted by the US Navy Wisconsin Test Facility.
The model values Im(ν) from Table 2 were translated in accordance with this formula to the equivalent attenuation α. As might be seen, the average model attenuation rate at the frequency of 76 Hz is 1.17 dB/1000 km, and the experimentally measured value is 1.08 dB/1000 km. The deviation is about 7 %. Deviation of the model attenuation from that measured in the ambient day and night conditions are equal to 2 % and 24 % correspondingly. The attenuation values at 82 Hz are equal to 1.25 dB/1000 km, and the mutual deviation was less than 1 %. These data lead to the conclusion that the vertical profile 2 of the air conductivity suggested here is justified not only in the frequency band of global electromagnetic resonance, but also at frequencies above it.
We analyzed and compared model results with the literature data available and demonstrated that the suggested vertical profile of the atmospheric conductivity is a rather realistic model. Firstly, it is consistent with the classical concept of the air ionization. Secondly, application of this profile in the full wave solution provides the frequency dependence of the ELF propagation constant close to the reference one in the whole Schumann resonance band. Third, the computed the propagation constant is in good agreement with measurements of the man-made ELF radio signals.
When thinking about areas of future works, we anticipate that our profile will be useful in direct modeling of Schumann resonance: the FDTD algorithms and in the 2DTU approach. In particular, all published FDTD solutions had Schumann resonance frequencies exceeding the observed values. Deviations have arisen from unrealistic conductivity profiles applied in these models. We are sure that profiles presented here will improve the data of direct modeling, and we plan applying the profiles in future investigations of Schumann resonance.
APN and YPG carried out the full wave computation for sub-ionospheric ELF waves, and MH participated in the general discussion of the paper. All authors read and approved the final manuscript.
The authors declare that they have no competing interests.
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- Bannister PR (1999) Further examples of seasonal variations of ELF radio propagation parameters. Radio Sci 34(1):199–208View ArticleGoogle Scholar
- Bliokh PV, Nickolaenko AP, Filippov YuF (1980) Schumann resonances in the Earth-ionosphere cavity. Peter Perigrinus, LondonGoogle Scholar
- Bliokh PV, Galuk YuP, Hynninen EM, Nickolaenko AP, Rabinowicz LM (1997) On the resonance phenomena in the Earth-ionosphere cavity. Izv. VUZOV, Radiofizika 20(14):501–509 (in Russian) Google Scholar
- Cole RK, Pierce ET (1965) Electrification in the Earth’s atmosphere from altitudes between 0 and 100 kilometers. J Geophys Res 70(11):2735–2749View ArticleGoogle Scholar
- Fullekrug M (2000) Dispersion relation for spherical electromagnetic resonances in the atmosphere. Phys Lett A 275(1):80–89View ArticleGoogle Scholar
- Galuk YP, Ivanov VI (1978) Deducing the propagation characteristics of VLF fields in the cavity Earth—non-uniform along the height anisotropic ionosphere. In: Problems of diffraction and radio wave propagation, vol 16. Leningrad State University Press, Leningrad, pp 148–153 (in Russian) Google Scholar
- Galuk YP, Nickolaenko AP, Hayakawa M (2015) Knee model: comparison between heuristic and rigorous solutions for the Schumann resonance problem. J Atmos Sol-Terr Phys 135:85–91View ArticleGoogle Scholar
- Greifinger C, Greifinger P (1978) Approximate method for determining ELF eigenvalues in the Earth-ionopshere waveguide. Radio Sci 13:831–837View ArticleGoogle Scholar
- Greifinger PS, Mushtak VC, Williams ER (2007) On modeling the lower characteristic ELF altitude from aeronomical data. Radio Sci. 42:RS2S12. doi:10.1029/2006RS003500 View ArticleGoogle Scholar
- Hynninen EM, Galuk YP (1972) Field of vertical dipole over the spherical Earth with non-uniform along height ionosphere. In: Problems of diffraction and radio wave propagation, vol 11. Leningrad State University Press, Leningrad, pp 109–120 (in Russian) Google Scholar
- Ishaq M, Jones DL (1977) Method of obtaining radiowave propagation parameters for the Earth–ionosphere duct at ELF. Electron Lett 13(2):254–255View ArticleGoogle Scholar
- Kirillov VV (1996) 2D theory of ELF electromagnetic wave propagation in the Earth–ionosphere cavity. Izv. VUZOV, Radiofizika 39(12):1103–1112 (in Russian) Google Scholar
- Kirillov VV, Kopeykin VN (2002) Solution of 2D telegraph equations with anisotropic parameters. Izv. VUZOV, Radiofizika 45(12):1011–1024 (in Russian) Google Scholar
- Kirillov VV, Kopeykin VN, Mushtak VC (1997) Electromagnetic waves of ELF band in the Earth–ionosphere cavity. Geomagn Aeron 37(3):114–120 (in Russian) Google Scholar
- Molina-Cuberos GJ, Morente JA, Besser BP, Portí J, Lichtenegger H, Schwingenschuh K, Salinas A, Margineda J (2006) Schumann resonances as a tool to study the lower ionospheric structure of Mars. Radio Sci. 41:RS1003View ArticleGoogle Scholar
- Morente JA, Molina-Cuberos GJ, Portí JA, Besser BP, Salinas A, Schwingenschuch K, Lichtenegger H (2003) A numerical simulation of Earth’s electromagnetic cavity with the Transmission Line Matrix method: Schumann resonances. J Geophys Res 108(A5):1195. doi:10.1029/2002JA009779 View ArticleGoogle Scholar
- Mushtak VC, Williams E (2002) Propagation parameters for uniform models of the Earth–ionosphere waveguide. J Atmos Solar Terr Phys 64(6):1989–2001View ArticleGoogle Scholar
- Nickolaenko AP (2008a) Deducing the ELF attenuation rate from the distance dependence of radio wave emitted by man-made source. Radio Phys Electron 13(1):40–44 (in Russian) Google Scholar
- Nickolaenko AP (2008b) ELF attenuation factor derived from distance dependence of radio wave amplitude propagating from an artificial source. Telecommun Radio Eng 67(18):621–629View ArticleGoogle Scholar
- Nickolaenko AP, Hayakawa M (2002) Resonances in the Earth–ionosphere cavity. Kluwer, DordrechtGoogle Scholar
- Nickolaenko A, Hayakawa M (2014) Schumann resonance for tyros (Essentials of global electromagnetic resonance in the earth–ionosphere cavity). Springer, BerlinView ArticleGoogle Scholar
- Nickolaenko AP, Rabinowicz LM (1982) On a possibility of global electromagnetic resonances at the planets of Solar system. Kosm Issled 20(1):82–89 (in Russian) Google Scholar
- Nickolaenko AP, Rabinowicz LM (1987) On applicability of ELF global resonances for studying thunderstorm activity at Venus. Kosm Issled 25(2):301–306 (in Russian) Google Scholar
- Pechony O, Price C (2004) Schumann resonance parameters computed with a partially uniform knee model on Earth, Venus, Mars, and Titan. Radio Sci 39:RS5007. doi:10.1029/2004RS003056 View ArticleGoogle Scholar
- Sentman DD (1990a) Approximate Schumann resonance parameters for two-scale-height ionosphere. J Atmos Terr Phys 52(1):35–46View ArticleGoogle Scholar
- Sentman DD (1990b) Electrical conductivity of Jupiter Shallow interior and the formation of a resonant planetary–ionospheric cavity. Icarus 88:73–86View ArticleGoogle Scholar
- Toledo-Redondo S, Salinas A, Morente-Molinera JA, Mendez A, Fornieles J, Portí J, Morente JA (2013) Parallel 3D-TLM algorithm for simulation of the Earth–ionosphere cavity. J Comput Phys 236:367–379View ArticleGoogle Scholar
- Williams ER, Mushtak VC, Nickolaenko AP (2006) Distinguishing ionospheric models using Schumann resonance spectra. J Geophys Res 111:D16107. doi:10.1029/2005JD006944 View ArticleGoogle Scholar
- Yang H, Pasko VP (2005) Three-dimensional finite-difference time domain modeling of the Earth–ionosphere cavity resonances. Geophys Res Lett 32:L03114. doi:10.1029/2004GL021343 Google Scholar
- Zhou H, Yu H, Cao B, Qiao X (2013) Diurnal and seasonal variations in the Schumann resonance parameters observed at Chinese observatories. J Atmos Solar Terr Phys 98(1):86–96View ArticleGoogle Scholar