Amplitude variations of ELF radio waves in the Earth–ionosphere cavity with the day–night non-uniformity

Journal of Atmospheric and Solar-Terrestrial Physics xxx (2017) 1–14

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Amplitude variations of ELF radio waves in the Earth–ionosphere cavity with the day–night non-uniformity Yu P. Galuk a, A.P. Nickolaenko b, M. Hayakawa c, * a

Saint-Petersburg State University, Saint-Petersburg, Peterhof, 198504, Russia A.Ya. Usikov Institute for Radio-Physics and Electronics, National Academy of Sciences of the Ukraine, Kharkov, 61085, Ukraine Hayakawa Institute of Seismo Electromagnetics Co. Ltd.(Hi-SEM), The University of Electro-Communications (UEC) Alliance Center #521, Advanced & Wireless and Communications Research Center, UEC, Chofu, Tokyo, 182-8585, Japan

b c

A R T I C L E I N F O

A B S T R A C T

Keywords: Schumann resonance Mesosphere conductivity profile Day–night non-uniformity Diurnal amplitude variations

The real structure of lower ionosphere should be taken into account when modeling the sub–ionospheric radio propagation in the extremely low frequency (ELF) band and studying the global electromagnetic (Schumann) resonance of the Earth–ionosphere cavity. In the present work we use the 2D (two dimensional) telegraph equations (2DTE) for evaluating the effect of the ionosphere day-night non-uniformity on the electromagnetic field amplitude at the Schumann resonance and higher frequencies. Properties of the cavity upper boundary were taken into account by the full wave solution technique for realistic vertical profiles of atmosphere conductivity in the ambient day and ambient night conditions. We solved the electromagnetic problem in a cavity with the day–night non-uniformity by using the 2DTE technique. Initially, the testing of the 2DTE solution was performed in the model of the sharp day–night interface. The further computations were carried out in the model of the smooth day–night transition. The major attention was directed to the effects at propagation paths "perpendicular" or "parallel" to the solar terminator line. Data were computed for a series of frequencies, the comparison of the results was made and interpretation was given to the observed effects.

1. Introduction Influence of the ionosphere day-night non-uniformity on the parameters of extremely low frequency (ELF: 3 Hz < f < 3 kHz) radio propagation and the global electromagnetic (Schumann) resonance is among the most popular topics (Madden and Thompson, 1965; Bliokh et al., 1968, 1977, 1980; Rabinowicz, 1986; Nickolaenko, 1986; Kirillov, 1993a, 1993b; Kirillov et al., 1997; Kirillov and Kopeykin, 2002; Nickolaenko and Hayakawa, 2002; Kulak et al., 2003, Pechony and Price, 2004; Pechony, 2007; Pechony et al., 2007, Yang and Pasko, 2005; Yang et al., 2006, Kulak and Mlynarczyk, 2013; Toledo-Redondo et al., 2013, 2016, 2017; Nickolaenko and Hayakawa, 2014; Zhou et al., 2016, Mlynarczyk et al., 2017; Nickolaenko et al., 2017). Usually, the model solutions of the electromagnetic problem are used in the framework of various difference schemes, such as the 2D (two dimensional) telegraph equations (2DTE) (Madden and Thompson, 1965; Kirillov, 1993a, 1993b; Kirillov et al., 1997, Kirillov and Kopeykin, 2002; Pechony and Price, 2004; Pechony, 2007; Pechony et al., 2007, Kulak et al., 2003, Kulak and Mlynarczyk, 2013; Nickolaenko et al., 2017), the (Finite Difference in Time Domain -

FDTD) (Hayakawa and Otsuyama, 2002; Otsuyama et al., 2003, Yang and Pasko, 2005; Yang et al., 2006, Zhou et al., 2016), the 3D (three dimensional) transmission line model (3DTLM) (Toledo-Redondo et al., 2013, 2016, 2017). Formal solutions with the method of moments (Rabinowicz, 1986) and the Stratton – Chu integral equation (Nickolaenko, 1986) were also used. We evaluate in the present work the influence of the day-night nonuniformity on the diurnal amplitude variations of the vertical electric field component of the ELF radio wave on fixed propagation paths and several frequencies. The classical solution is based on the 2DTE approach which was described in detail in Kirillov (1993a, 1993b), Kirillov et al. (1997), and Kirillov and Kopeykin (2002), and it was also utilized in several publications (Kulak et al., 2003, Pechony and Price, 2004; Pechony, 2007; Pechony et al., 2007, Kulak and Mlynarczyk, 2013; Nickolaenko et al., 2017). The improved model of the vertical conductivity profile of the atmosphere is adopted in our computations for obtaining the propagation constant in the frequency band of 4–200 Hz. Then, diurnal changes were computed in the field amplitude for a propagation path at several frequencies.

* Corresponding author. E-mail addresses: [email protected] (Y.P. Galuk), [email protected] (A.P. Nickolaenko), [email protected] (M. Hayakawa). https://doi.org/10.1016/j.jastp.2018.01.001 Received 10 October 2017; Received in revised form 7 December 2017; Accepted 1 January 2018 Available online xxxx 1364-6826/© 2018 Published by Elsevier Ltd.

Please cite this article in press as: Galuk, Y.P., et al., Amplitude variations of ELF radio waves in the Earth–ionosphere cavity with the day–night nonuniformity, Journal of Atmospheric and Solar-Terrestrial Physics (2017), https://doi.org/10.1016/j.jastp.2018.01.001

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2. Conductivity profiles of mesosphere

The attention will be directed to the radio propagation paths so-called “parallel” or “perpendicular to the terminator line”. The first of these corresponds to the source and receiver positioned at the same meridian at equal distance from the equator to the north and to the south. The computations are performed in the equinox conditions when the center of the day and the night hemisphere lies on the equator line. Therefore, distances from the correspondent points to the solar terminator (daynight boundary) vary in time, but these distances remain coincident (the propagation path is parallel to the terminator line). The perpendicular path corresponds to the equatorial source and receiver; it is perpendicular to the terminator line in equinox conditions. Owing to the daily rotation of the Earth, the propagation path moves along the great circle arc being perpendicular to the day-night interface. We consider in our computations the sharp day-night transition first, which allows for verifying the computational technique and the consistency of the model results. Afterwards, the major data set is obtained in the model of the smooth day-night transition, which is more relevant to the reality.

The vertical conductivity profiles σ (h) for the ambient day and ambient night conditions are given for the altitude interval h 2 [0; 110] km (see Fig. 1a and Table A1 as Appendex 1). The logarithm of air conductivity σ (h) in S/m is plotted along the abscissa in Fig. 1a and the height h is plotted along the ordinate in km above the ground surface. The night profile is shown by the curve with asterisks, and the day profile is a curve with dots. We improved the conductivity profiles so that these remain consistent, but not exactly coincident with those published in Kudintseva et al. (2016) and Nickolaenko et al. (2016). Two major modifications were made. We extended the altitude interval of the profile from 100 to 110 km since the magnetic characteristic height HL(f) exceeds the 100 km level at the basic Schumann resonance frequency in the ambient night condition (see Table 1). To improve correspondence of model to the global electric circuit data, we slightly elevated the profile (by ~2 km) thus reducing the conductivity near the ground and obtaining a more realistic fair weather field amplitude of ~130 V/m for the Earth–ionosphere leakage current of 1 pA/m2. As follows from the graphs in Fig. 1a, the conductivities of the day and the night hemispheres depart from the Fig. 1. Parameters of the Earth–ionosphere cavity with the day – night non-uniformity: a) – conductivity profiles of model atmosphere in ambient day and night conditions; b) – frequency variations of complex magnetic HL and electric HC heights in the ionosphere; c) – ELF propagation constant ν versus frequency f, an inset in the panel illustrates the wave attenuation factor in the Schumann resonance band for ambient day and ambient night conditions.

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near the light and shadow boundary (the solar terminator). Here, we use the models of sharp or of smooth transition from characteristic daytime heights to the nighttime ones (see Fig. 2). In the sharp terminator model, the profile changes abruptly over the light–shadow boundary at the ground surface, as it is schematically shown in the left part of Fig. 2. The coordinates of the sharp terminator are coincident with the light–shadow line. In a more realistic model of smooth terminator, the horizontal changes of conductivity occur linearly from the nighttime to the daytime parameters. In this model, the ambient day conductivity profile is valid up to the distance of 875 km from the solar terminator line toward the shadow. This occurs due to the location of ionosphere in the shadow thrown by the planet and its lower boundary of definite height. The strip of 875 km is obtained when the ionosphere lower edge is found around the 50 km altitude, as seen in the right diagram of Fig. 2. The height of ionosphere lower boundary grows to approximately 90 km at the night side of the globe, which corresponds to a “dusk” area lying over the dark side of the planet at distances 875–1070 km from the solar terminator line. In the 875–1070 km interval (from the terminator line along the surface of the Earth), parameters of ionospheric plasma vary, and the smooth linear transition takes place from the day characteristic heights to the night ones. Thus, the illuminated region of the ionosphere in the smooth terminator model encompasses the daytime hemisphere plus the strip 875 km wide from the day–night boundary toward the shadow. Then, the transition region follows by the linear changes of characteristic heights, and the “night comes” starting from the distance of 1070 km from the terminator line where the ambient night conductivity profile is valid. The geometry described does not account for the finite size of the solar disk. The size of smooth terminator model suggests that the night in the ionosphere comes in ~40 min after the sunset at an equatorial observatory. It is no doubt that such a model is highly idealized, but its application is justified by the giant wavelength of Schumann resonance reducing its sensitivity to many delicate plasma processes occurring at the real day–night interface. We prefer using purely geometric considerations here thus avoiding very complicated physics with minor relevance to the sub-ionospheric ELF radio propagation. The following geometry of the problem is considered. The light–shadow line on the ground surface corresponds to the equinox conditions, i.e., the center of the day and the night hemisphere is located at the equator. The solar terminator line is coincident with the meridians of the geographic coordinate system. To be specific, we assume that the Universal Time (UT) is equal to tU ¼ 0 h (GMT midnight), so that the center of the night hemisphere is located at the zero latitude and the zero longitude. The center of the day hemisphere lies at the zero latitude and the 180 longitude. The meridian φ ¼ 90 (i.e., 90 E) corresponds to the morning terminator tU ¼ 6 h, and the meridian φ ¼ ¡90 (90 W) corresponds to the evening terminator tU ¼ 18 h. Consider an observer with geographic coordinates 0 N and 0 E positioned at the center of nocturnal hemisphere when tU ¼ 0 h. With the passage of time, the latitude of observer remains constant, but the longitude increases by 15 per hour. At the time moment tU ¼ 6 h, the observer crosses the morning terminator line, when tU ¼ 12 h it arrives at the center of the day hemisphere, and when tU ¼ 18 h it crosses the evening terminator and then returns to the night side of the globe. The analytical solution of a radio propagation problem in the Earth–ionosphere waveguide might be constructed only in a few particular cases. The field computations in any arbitrary ionosphere model depending on the angular spherical coordinates θ and φ are possible only by using numerical methods. The most popular and commonly used WKB (Wentzel-Kramers-Brillouin) approximation is not valid at our frequencies, while the updated and very popular FDTD method (Yang and Pasko, 2005; Yang et al., 2006) is extremely time-consuming. This is why we turn to the upgraded method of a spherical 2D transmission line, which leads to a system of two telegraph equations. For the first time, this model was applied by Madden and Thompson (1965) in the Schumann resonance problem. The technique requires modest

Table 1 Propagation constant and characteristic heights and the day and night side of the globe in the non-uniform cavity at a set of frequencies.

Day

Night

f, Hz

8

32

80

120

180

Re[ν] Im[ν] Re[HL], km Im[HL],km Re[HC],km Im[HC], km Re[ν] Im[ν] Re[HL], km Im[HL],km Re[HC],km Im[HC], km

1.039 0.159 97.104 3.4 49.418 10.145 1.009 0.163 103.051 4.093 54.547 11.708

4.93 0.424 93.954 4.308 57.799 6.53 4.836 0.402 99.832 4.365 63.664 6.981

12.539 0.913 91.686 5.333 61.368 5.05 12.357 0.846 97.818 5.315 67.454 5.243

18.739 1.325 91.016 5.849 63.428 4.673 18.495 1.226 97.355 5.877 69.793 4.806

27.889 1.954 91.214 6.52 66.765 4.406 27.561 1.804 97.973 6.629 73.794 4.481

27 km altitude where air conductivity reaches the 1011 S/m level (see Table A1). Below 27 km, the day and the night conductivities are coincident and smoothly decrease toward the 7.6⋅1015 S/m value near the ground surface. Thus, diurnal variations of the fair weather field do not depend on the local time and reflect the daily variations in the current of the global electric circuit. Frequency variations of propagation characteristics of ELF radio waves are collected in Fig. 1b and c. These quantities were estimated by using the full wave solution in the form of Riccati equation (Hynninen and Galuk, 1972; Galuk et al., 2015) for the improved ambient day and ambient night profiles. The propagation constant ν(f) and the complex characteristic heights HC(f) and HL(f) were found as functions of signal frequency f for the day (curves with dots) and the night (curves with stars) conductivity profiles. The smaller characteristic altitude is regarded as the electric height HC(f) since the electric field of the radio wave incident on the ionosphere penetrates to this height, and it rapidly attenuates into the plasma afterwards. The magnetic field reaches a higher altitude HL(f) > HC(f), which is called the magnetic height (Nickolaenko et al., 2015, 2016, 2017). Fig. 1b depicts the complex characteristic heights versus frequency in the range 4–200 Hz. One may observe that the real part of electric height increases with frequency while the magnetic height tends to decrease with signal frequency. The values found in the ambient night systematically and noticeably exceed those in the day conditions. The magnetic height may surpass the 100 km altitude during the night. The imaginary parts of characteristic heights have opposite signs (as it must be in accordance with their physical meaning), and these are much smaller than the real parts. They exhibit a wide spectral maximum at lowest frequencies, and deviations between the day and the night values are noticeable only around the basic Schumann resonance frequency of 8 Hz. The propagation constant ν(f) is shown in Fig. 1c. Its real parts are nearly coincident in ambient day (curves with dots) and in night (curves with stars) conditions, thus indicating that the phase velocity of ELF radio waves remains almost the same on the day and the night sides of the globe. The lower plot in Fig. 1c shows frequency variations of the wave attenuation rate: it is not high, and the losses are smaller under the night ionosphere. The inset shows the wave attenuation in more detail within the Schumann resonance band. One may notice that here the losses at the sunlit and at the shadow sides of the globe are practically coincident in the vicinity of the first three resonance modes. The greater dayside attenuation rate becomes visible at higher frequencies. The imaginary part of the propagation constant is an important characteristic. It was shown (Galuk et al., 2015; Kudintseva et al., 2016, Nickolaenko et al., 2015, 2016, 2017) that the attenuation rate thus obtained remain very close to the commonly used standard model by Ishaq and Jones (1977) within the entire Schumann resonance frequency band. We assume in the model of the Earth–ionosphere cavity with the daynight non-uniformity that properties of the middle atmosphere in the day and the night hemispheres are independent of the angular coordinates and vary only along the altitude. An exception is the transition region 3

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Fig. 2. Diagram of the sharp and the smooth day-night interface.

in the models of the uniform Earth–ionosphere cavity formed either by the ambient day (line “suns”) or by ambient night ionosphere (line with stars). Diurnal variations of the field amplitude are absent in these “whole day” and “whole night” uniform resonators. One may observe that the field amplitude is by 1.3 dB higher in the “whole day” cavity than in the purely nocturnal cavity. This difference arises from the reduction of ionopshere height at the dayside of the globe. The line 1 and line 5 in Fig. 3 demonstrate diurnal variations at two “mirror” traces No.1 and No.5, when the equatorial observer is shifted to the west or to the east from the field source. We must remark that owing to the Earth's rotation, the longitudinal separation of 5000 km along the equator (45 ) is covered by the terminator line in 3 h. Therefore, the abrupt increase of the field amplitude occurs on the path No.1 at tU ¼ 6 h caused by the reduction of ionosphere height above the source. The further sharp increase at tU ¼ 9 h takes place due to transition of the observer from the night to the dayside of the globe. For the propagation path No.5, the observer is by 45 closer to the morning terminator line than the source. Therefore, the first discontinuity of the field amplitude at tU ¼ 3 h is confined to the observer transition toward the illuminated side of the Earth. The further increase at tU ¼ 6 h is relevant to alteration in the local ionosphere height above the source when the latter crosses the sharp terminator and also enters the day hemisphere. One may note that daily variations caused by the day–night nonuniformity reach the level of 1.8 dB exceeding the 1.3 dB difference pertinent to the “whole day” and “whole night” cavities. This excess arises from the interference of the direct source–receiver wave and the wave “reflected” from the day–night interface. Plot 3 (diamonds) and the dashed line 7 in Fig. 3 depict diurnal amplitude variations of the field when the observer occupies the same meridian as the source being separated from it along the latitude to the south or to the north correspondingly (propagation paths No.3 and No.7). Completely coincident graphs 3 and 7 clearly illustrate the validity of reciprocity theorem in the non-uniform isotropic cavity. Diurnal variations for the propagation paths 2, 4, 6, and 8 are shown by the relevant plots in Fig. 3. One may conclude that the reciprocity theorem also holds here, so that the pair of graphs 4 (open circles) and 6 (dash-dot line) and the pair 2 (the squares) and 8 (dashed line) are coincident. This result was expected since these paths are symmetric to each other with respect to the equatorial plane. The amplitude of diurnal variations remains unchanged for all orientations, only the field discontinuities shift along the time axis in accordance with the transition time from the night to the day hemisphere. The propagation path turns into the “mixed” or “intermediate” one when the terminator line intersects the source–observer arc, and the relevant field amplitude acquires the relatively stable intermediate value. The tests have confirmed the conclusion of paper by Nickolaenko et al. (2017) that the 2DTE solution remains robust even in the model of the sharp day–night interface. The obtained diurnal variations satisfy the reciprocity theorem, and their value is about 0.9 dB relative to the daily averaged field level, which is in accordance with the published model data. Successful testing allowed us to proceed to investigating the terminator impact on the field amplitude at different frequencies in the

computing resources due to reduction of the original 3D problem into the 2D one. The reduction is possible owing to the relatively small height of the lower ionosphere boundary over the ground surface (~100 km) in comparison with the Earth's radius (~6400 km), while the wavelength of resonant ELF radio signal matches the 40000 km length of the Earth's equator. The approach used by Madden and Thompson (1965) was later elaborated and generalized by Kirillov (1993a, 1993b), Kirillov et al. (1997) and Kirillov and Kopeykin (2002) where the relationships were shown connecting the particular conductivity profile of atmosphere and parameters of geomagnetic field with the equivalent transmission line characteristics (the complex line inductance L and the line capacitance C) combined with the updated algorithm of solving the 2DTE. We apply in what follows the specific version of this approach, which is valid for the isotropic non-uniform ionosphere. The relevant description is given in Appendix 2. 3. Tests The resonator field was computed in the tests for the sharp day–night transition, the signal frequency was chosen to be equal to 8 Hz, and the source–receiver distance was D ¼ 5000 km or 45 . The complex characteristic heights were derived by the full wave solution for the conductivity profiles, and the propagation constants were computed at this frequency presented in Table 1. The UT tU was assumed to be equal to 0 h, so that the field source occupied the center of the night hemisphere (0 N and 0 E). Eight propagation paths were oriented in such a way that the observer went around the source along the circle of the 5000 km radius. Relevant coordinates of the observer are collected in Table 2. Plots in Fig. 3 illustrate the computational results. Time tU is plotted along the abscissa in hours, and the relative variations of the field amplitude in arbitrary units are shown along the ordinate. The relative field amplitude is shown in dB, which means that we use the logarithmic scale along the ordinate. The numbers denoting different curves correspond to the path number from Table 2. The source–observer propagation path is drawn in the upper right corner of figure. The letter S in the central circle of this diagram indicates the equatorial source, and the black numbered dots denote position of the observer relevant to the specific path number. Two horizontal lines in Fig. 3 refer to the field amplitudes computed

Table 2 Coordinates of the observer in the field test calculations. Path N1 N2 N3 N4 N5 N6 N7 N8

Observer coordinates 



0 E; 45 W 30 S; 35.26 W. 45 S; 0 W 30 S; 35.26 E 0 N; 45 E 30 N; 35.26 E 45 N; 0 E 30 N; 35.26 W

Shift from the field source to the west to the south-west to the south to the south-east to the east to the north-east to the north To the north-west

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Fig. 3. Results of test computations for the sharp terminator model in equinox conditions. The vertical electric dipole source is positioned at the equator on Greenwich meridian. The observer occupies one of eight possible positions around the source at a distance of 5 Mm of 45 . The source – observer geometry is drawn in the upper right corner, and the propagation path (from 1 to 8) moves in time around the globe with the source always positioned at the equator. This motion alters the distance of propagation path from the day-night interface thus causing dirnal variations in the signal amplitude shown by curves 1–8 relevant to the observer positions. Horizontal lines depict the invariant field levels computed in the uniform cavity formed by either the day or the night ionopshere.

Earth–ionosphere cavity is shown for the time moment tU ¼ 0. The center of the night hemisphere occupies the point 0 N and 0 E being the intersection of the Equator and Greenwich meridian. The points marked by S indicate the source positions and those marked by O correspond to the observer. The line S1 – O1 is the perpendicular propagation path, and the line S2 – O2 is the parallel one. With the growing time of day tU, both

framework of a more realistic smooth day–night transition model. 4. Model results for parallel and perpendicular paths We show in Fig. 4 the diagram explaining positions of parallel and perpendicular propagation paths. Here the night side of the

Fig. 4. Parallel and perpendicular propagation paths used in computations.

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these paths move from the midnight meridian φ ¼ 0 to the right, toward the morning terminator line of φ ¼ 90 E. The impact of the smooth day–night interface on the amplitude diurnal variations of the vertical electric field component was analyzed at the propagation distance D ¼ 5000 km for the path perpendicular to the terminator (S1 – O1) or parallel to it (S2 – O2). Computational data are shown for frequencies 8, 32, 80, and 120 Hz. The corresponding plots are shown in Fig. 5 a–d. Fig. 5a illustrates daily variations of the field amplitude at 8 Hz for the parallel and the perpendicular propagation paths. The initial (for the time moment tU ¼ 0) geographic coordinates of the source and the receiver are S1 (0 S; 22.5 W) and O1 (0 N; 22.5 E) at the perpendicular path, and S2 (22.5 S; 0 E) and O2 (22.5 N; 0 E) for the parallel trace. Thus, the middle of both propagation paths is initially positioned at the center of the night hemisphere. With the passage of time, it moves to the east with the velocity of 15 per hour. The graphs in Fig. 5a resemble those of Fig. 3. The distinctions have raised either from the symmetry of the propagation path with respect to the equator (parallel path) or the median meridian (perpendicular path) or by the transition to a smooth terminator model, in which the size of illuminated area of the Earth's ionosphere exceeds that of the dark one. Even in the smooth terminator model the abrupt change in the field amplitude on the parallel path occurs when the source and the receiver simultaneously enter the night hemisphere. The discontinuity is observed at all frequencies of Fig. 5. The amplitude of 8 Hz field changes in two steps for the perpendicular propagation path, as seen in Fig. 5 a. Initially, there occurs a jump associated with the crossing of the night–day interface by the observer, and the propagation path acquires the “mixed” nature. No temporal variation is observed at the path of this kind in the field amplitude at f ¼ 8 Hz. When the field source also crosses the night–day boundary and enters the sunlit area, the second amplitude jump takes place, which is approximately equal to the first one. The similar transient and exactly “mirror” process is observed when the path crosses the evening terminator. Amplitude of vertical electric field amplitude gradually varies in time when the source–observer arc is found in the day or in the night hemisphere. These smooth changes with local maximum around the midnight and minimum at the UT noon are explained by the interference

of the direct wave arriving from the source to the observer and the wave scattered by day–night non-uniformity. The phase shift in the waves reflected from the inhomogeneity deviates by 180 at the different sides of terminator, therefore, the interference being constructive at the night side of the Earth (the waves meet in phase) becomes the destructive one at the dayside (the reflected waves arrives in anti-phase). Fig. 5b contains similar “parallel” and “perpendicular” graphs computed for the fifth resonance mode frequency of Schumann resonance f ¼ 32 Hz. In distinction from the 8 Hz when the wavelength is equal to the Earth's circumference, the signal of 32 Hz has the wavelength smaller by the factor of five. This explains why interference at the pure day and the night propagation paths becomes faster than in Fig. 5a. One may observe several periods of sinusoidal oscillations now. It is interesting to note that oscillations are also present at the mixed paths. Remarkably, the wave beating caused by the reflections from the day–night non-uniformity changes its sign after crossing the terminator line (e.g. around 4 and 7 h). This indicates that the sign of reflection coefficient changes when the source moves from one side of terminator to the other. Graphs of Fig. 5 b demonstrate that magnitude of the interference pattern decreases with an increase of signal frequency, while the abrupt amplitude change on the parallel path has practically remained the same. Diurnal patterns of the field amplitude are shown in Fig. 5c in and Fig. 5d for the same propagation paths, but at different signal frequencies of 80 and 120 Hz. These are frequencies where the global electromagnetic resonance is absent owing to an increase in radio wave attenuation rate. Obviously, the total change in the field amplitude at the smooth terminator is still close to 2 dB for parallel paths. The interference pattern at higher frequency is characterized by progressively faster changes, while the amplitude of these oscillations decreases. At higher frequencies, the amplitude variations become linear in time at the mixed perpendicular propagation paths. 5. Discussion Plots in Fig. 5 illustrate an important physical feature of electromagnetic field in the non-uniform Earth–ionosphere cavity mentioned by Fraser-Smith and Bannister (1998) and Nickolaenko (2008). There are Fig. 5. Diurnal variations of the field amplitude in the smooth terminator model at the frequencies of 8, 32, 80, and 120 Hz. The parallel propagation paths are shown by curves with points and the curves with pluses correspond to the perpendicular propagation paths.

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two competing mechanisms in a cavity with varying effective height. A decrease in the effective electric height in the daytime hemisphere increases the efficiency of the vertical electric dipole source: it creates the field of greater amplitude when positioned at the dayside. However, a reduction in ionosphere height increases the attenuation factor of ELF radio waves, and the field attenuates faster with the distance at the dayside of the Earth–ionosphere cavity. The opposite situation is observed at the night side: the source creates the field of smaller amplitude, however, this radio wave slowly attenuates with the distance. Obviously, one may specify such a source–observer distance where the field amplitude at a certain frequency in the day conditions is equal to that in ambient night conditions. In other words, there is a distance at which the gain in the excitation factor is compensated by the wave attenuation. It was stated by Fraser-Smith and Bannister (1998) that such a distance would be 3000–4000 km, provided that the signal frequency is 70–80 Hz. Calculations in Nickolaenko (2008) showed this distance close to 7000 km for the signal frequency of 80 Hz, and its exact determination is impeded by oscillations of the field amplitude jE (D)j along the distance D. The solution of 2DTE (see Fig. 5c and d) shows that for the source–receiver distance D ¼ 5000 km and f ¼ 80 Hz, the radio wave attenuation does not compensate yet the change in the excitation factor: so the average field amplitude is higher in the daytime than at night. The situation is closer to the compensation at 120 Hz. The daytime and nighttime levels will become approximately equal at D ¼ 5000 km when 80 < f 180 Hz. We must remark that the amplitude discontinuity of 0.9 dB relative to the median value is retained in this case at the mixed paths, i.e. the localized field modifications are preserved in the vicinity of the solar terminator, although the median day and median night amplitudes become close to each other. Thus, the model computations are physically consistent with the expected features of radio waves propagation in the real Earth–ionosphere cavity, and we will return to this issue below. The information is missing in the Schumann resonance literature on how the resonance pattern changes under the impact of the day–night non-uniformity. We show in Fig. 6 the computed spectra for the source–observer distance D ¼ 5000 km. The four characteristic orientations were modeled of propagation paths relative to the smooth day–night interface (see Table 3). The broad shaded strip in Fig. 6 outlines the zone having the upper boundary coincident with the spectrum in the uniform Earth–ionosphere cavity bounded by the daytime ionosphere. The lower boundary of this strip is the spectrum in the uniform cavity formed by the night ionosphere. The thick black line shows the results of computations in the nonuniform cavity when the center of equatorial path No.1 is found at the center of the night hemisphere. By taking into account the plots of Fig. 5a, one may expect a minute impact of the day–night non-uniformity on the observed field amplitude: rotation of this particular propagation path by 90 around its center does not alter the field level. A comparison of curve 1 with the shaded zone outlining the fields in the “whole day” and the “whole night” cavities shows noticeable departures. Still, it is convenient to use the data of path No.1 as a standard curve facilitating the comparison of spectra obtained for the other path orientations. Curve 2 (the dashed line with dots) in Fig. 6 depicts the amplitude spectrum when the center of the equatorial path is positioned exactly at the light–shadow boundary on the ground (0 N; 90 E). In this case, the field source remains in the night hemisphere while the observer has passed into the dayside. One may observe that spectra 1 and 2 are close to each other in the vicinity of the first resonance maximum, but curve 2 lies above curve 1 by 0.5–1 dB starting from the second spectral peak. The spectrum outline did not change much, except that the second and the third peaks acquired the equal in height. Curves 3 and 4 (dashed and the dashed-dotted line with pluses) correspond to the signals received on the meridional paths near the morning day–night interface at the night side (curve 3, tU ¼ 5 h) and the dayside (curve 4, tU ¼ 7 h). Deviations of spectrum 3 from curve 1 are

Fig. 6. Schumann resonance spectra at 5000 km source–observer distance for different orientations of propagation paths: 1 – center of the path is coincident with the center of the night hemisphere, 2 – equatorial propagation path centered at the morning terminator, 3 – meridian path at tU ¼ 5 h centered at the equator, 4 – meridian path at tU ¼ 7 h centered at the equator.

Table 3 Source and observer coordinates used in spectral computations. Path

Source coordinates

Observer coordinates

Comment

1

0 N; 22.5 W

0 N; 22.5 E

tU ¼ 0 h, symmetric perpendicular path tU ¼ 6 h, symmetric perpendicular path tU ¼ 5 h, symmetric parallel path tU ¼ 7 h, symmetric parallel path









2

0 N; 67.5 E.

0 N; 112.5 E

3 4

22.5 S; 75 E 22.5 S; 105 E.

22.5 N; 75 E. 22.5 N; 105 E.

negligible, whereas departures of curves 4 and 1 are substantial and may reach 2 dB. It is interesting to note that an increase in the field level at the frequency of 8 Hz was present in Fig. 5 a, but now we conclude that an increase in amplitudes is observed at all resonant modes. Thus, the influence of the day–night non-uniformity on paths 2 and 4 tends to raise the spectrum as a whole. Deviations of the resonance spectrum on path 3 are more complicated: the field amplitude drops almost by 1 dB in the neighborhood of the first, fourth, and the fifth spectral maxima, while curves 1 and 4 are practically coincident in the vicinity of the second, third and above the fifth mode. Data depicted in Fig. 6 suggest that accounting for the day–night nonuniformity in the Schumann resonance problem does not result in dramatic changes: the general outline is easily recognized in the Schumann resonance pattern for a given source–observer distance. Typical changes in the spectral amplitude can reach 1.5–2 dB, however, modifications tend to shift the spectra as a whole. Such a “vertical” transfer of the resonance spectrum along the ordinate simply leads to corresponding

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propagation path are shown in Fig. 7 by smooth lines with no markers. Distance dependences of the field amplitude in Fig. 7 at frequencies of Schumann resonance f ¼ 8 and f ¼ 32 Hz resemble the well-known results (Nickolaenko and Hayakawa, 2002, 2014; Price, 2016). One may observe on the upper panel of Fig. 7 that the field amplitude under the day conditions exceeds the nighttime values at all source–observer distances. The field node at the 8 Hz frequency is found close to D ¼ 10 Mm, and a deeper minimum is observed in ambient night conditions lying to the right from the daytime minimum. The displacement appears due to deviations in amplitude of the waves propagating across the day and the night hemispheres. One observes five amplitude minima in the distance dependence at the peak frequency of the fifth Schumann resonance mode (f ¼ 32 Hz). These are associated with the interference of the direct and antipodal waves (Nickolaenko and Hayakawa, 2002, 2014; Price, 2016). Again, the daytime amplitude lies above the night values, except the vicinity of 15–18 Mm where curves are violated by the interference of direct and antipodal waves. As before, the nighttime minima in the field amplitude are positioned at larger source–observer distances than the daytime minima. In general, one observes higher field amplitudes during the day hours in the whole Schumann resonance band. This indicates that characteristic electric height HC dominates at these frequencies. A situation when the daytime gain in the field excitation factor is compensated by the increase in the attenuation rate of ELF radio waves is illustrated by two lower graphs in Fig. 7. The daytime amplitude exceeds the nighttime one at short source–observer distances. Distance curves intersect in the area marked by vertical arrows. With the further increases of distance from the source, the field amplitude in the ambient night propagation conditions exceeds the daytime level. Compensation occurs in the vicinity of 12 Mm at 80 Hz, and it shifts to the distance of about 8 Mm at 120 Hz frequency. We emphasize that the compensation of diurnal amplitude variations in the electric field might be observed at frequencies much higher than those of global electromagnetic resonance. The daily level of the vertical electric field proves to be higher than the nighttime level in the Schumann resonance band. The idea of compensation of diurnal amplitude variations in the ELF field was spelled by Fraser-Smith and Bannister (1998). Model computations by Nickolaenko (2008) have confirmed the feasibility of such counterbalance in observations of a monochromatic signal at a certain distance from the transmitter. An experiment of this kind could be used for validating the model of the vertical conductivity profile of the mesosphere. It would be desirable to use the frequencies substantially exceeding those of Schumann resonance. Measurements at 120 Hz seem preferable since diurnal amplitude variations vanish at a distance of 8–9 Mm where the interference of direct and antipodal waves is small. Of course, the day and night amplitude equality is also present at 80 Hz, but it is displaced to the distance of 12 Mm where an unambiguous conclusion is impeded by interference in the field amplitude. Vanishing of diurnal amplitude variations should not be understood literally. The term means that amplitude measured at the day propagation path is practically equal to that of the night path. Important details are missing in Fig. 7 like the amplitude oscillations in the purely day and night conditions or the serious modulations caused by the terminator passage over the source or the observer (see Fig. 4). To clarify the situation, we made additional computations at 120 Hz and show the data in Fig. 8. The optimal source–observer distance D ¼ 7.3 Mm was chosen, which slightly deviates from 8 Mm marked by the vertical arrow in the lowest frame of Fig. 7. The UT is shown in hours on the abscissa in Fig. 8, and the field amplitude is plotted on the ordinate in dB. For the perpendicular path, the source and observer were placed on the equator and their longitudes were equal to 32.85 W and 32.85 E correspondingly for the initial time moment tU ¼ 0 h. For the parallel propagation path, the correspondents were positioned symmetrically with respect to the equator having the latitudes of 32.85 S and 32.85 N, and the longitudes equal to 0 when

alteration in the estimated magnitude of the source current moment when interpreting observational data. In particular, an increase in the field amplitude by 1.5 dB leads to relevant enhancement in the source current moment (lightning discharge) by 19%. The technique remains applicable for establishing the source–observer distance in the uniform cavity from the general outline of the broadband Schumann resonance spectrum (Nickolaenko and Hayakawa, 2002, 2014; Price, 2016), since the characteristic spectrum outline for the given source–observer distance does not crucially change. We show the distance variations in Fig. 7 of vertical electric field amplitude at frequencies 8, 32, 80 and 120 Hz. The source-receiver distance is shown along the abscissa in Mm (1 Mm ¼ 1000 km). The field amplitude is plotted along the ordinates in dB (relative arbitrary units). Panels in the figure depict data for frequencies of 8, 32, 80, and 120 Hz (from top to bottom). The curves with dots show the distance dependence along the day path, and the smooth lines correspond to the night propagation conditions. Since we use the smooth terminator model, the illuminated region in the lower ionosphere exceeds in size the dark one (see Fig. 2). In computations for the day propagation paths, the equatorial field source was placed at the morning terminator (tU ¼ 6 h or 90 E) while the observer moved from the source along the equator to the east remaining in the day hemisphere up to the distance of 18 Mm. Data for this propagation path are shown in Fig. 7 by lines with dots. The nocturnal propagation was modeled in a different way. The equatorial field source was positioned under the night ionosphere nearby the morning terminator (tU ¼ 5.3 h or 79.5 E), while the observer moved away from the source along the equator to the west remaining under the night ionosphere up to the distance of 18 Mm. Data relevant to the night

Fig. 7. Distance variations of fields at different frequencies on the day (curves with dots) and night (smooth curves) sides of the Earth. 8

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corresponds to the path difference between the direct and the reflected waves altered by λ/2 or about 1 Mm at the frequency of 120 Hz. The path difference for the perpendicular path are readily evaluated: the necessary shift is equal to λ/4 in case of the backscatter, so the relevant oscillation period must be smaller than an hour (about 40 min). On the parallel path, the corresponding estimate becomes more complicated and, by using the flat geometry, one can find that oscillation period should be close to 2 h (~2.3 h). These elementary estimates are in excellent agreement with the computational data of Fig. 8 thus confirming that ELF field perturbations are equivalent to reflections from the day–night interface. We observe that equality of the median day and median night amplitude in the distance dependence of Fig. 7 does not mean that diurnal variations completely vanish at these particular distances. The small regular oscillations remain for the paths within the two hemispheres, and large periodic modulations are present at times of crossing the solar terminator. The 2DTE computations might be used for estimating the field disturbance caused by the day–night non-uniformity. Let us return to the upper panel in Fig. 7 relevant to the signal frequency of 8 Hz. Here, the curve with dots shows the E1(D) distance dependence of the field amplitude in the ambient day conditions, and the curve without markers corresponds to the E2(D) field on the night path. We remember that the gradual (smooth) day–night model is used. Therefore, the equatorial field source is placed at the morning terminator line (tU ¼ 6 h or 90 E) for analyzing the day propagation path while the observer moves eastward from the source along the equator. The plots for this path start from the distance D ¼ 2 Mm or from the 108 E longitude. The maximum source–observer distance is 18 Mm. The night propagation conditions are realized when the field source is positioned in the night hemisphere at the longitude of 79.5 E (tU ¼ 5.3 h). The ambient night propagation path goes along the equator westward from the source, and the data are built starting from the 2 Mm and ending at 18 Mm distance from the source. The day–night interface perturbs the field amplitude relative to its median value and this latter might be found from the following equation:

Fig. 8. Daily changes of field amplitude at 120 Hz frequency and 7.3 Mm distance where no diurnal variations are expected according to Fig. 7.

tU ¼ 0 h. Variations for the parallel path are shown by smooth line in Fig. 8, and those for the perpendicular path are depicted by dashed lines. To be exact, the median day levels for the both orientations are lower by approximately 0.05 dB relative to the median night level. In other words, the average night amplitude exceeds the daytime value by 0.6%. These amplitudes are very close, and we consider them as coincident. In spite of equality between the median day and night amplitudes, serious modulations are visible in the both plots of Fig. 8 reaching 1 dB for the parallel path in the vicinity of terminator (~6 and 18 h). Modulations might be explained in terms of Fresnel zones and/or reflections from the day–night interface. The concept of reflections is not rigorous at ELF, but still it successfully explains amplitude oscillations by interference of the direct source–observer wave and the wave reflected from the non-uniformity. Interaction of these two waves provides a characteristic amplitude “beating”, especially when the path approaches the terminator. A pronounced increase in the beating amplitude at the parallel path in the vicinity of terminator line is readily explained by an increase of reflections at “gliding incidence angles”. The extent of modulations suggests that maximum of the reflection coefficient is about 0.1 for the smooth terminator model. Reflections are not so pronounced at the perpendicular path (dashed line). Instead, we observe interaction of spatial field distribution in the non-uniform cavity with the Fresnel zones moving in time together with the source and observer. The motion causes variations small in amplitude and regular in time when the correspondents are positioned in the same hemisphere. The field amplitude varies almost linearly with time at the mixed path when the terminator line crosses the source–observer arc. In the both cases reflections become clearly visible in the backscattering geometry. This feature was noted in solutions constructed with the help of Stratton-Chu integral equation (Nickolaenko, 1986, 2008; Nickolaenko and Hayakawa, 2002). Time intervals of the wave interference noticeably deviate owing to a different geometry of the parallel and perpendicular propagation paths. The maximum–minimum distance in the field amplitude pattern

E0 ðDÞ ¼

E1 ðDÞ þ E2 ðDÞ 2

(1)

The dashed line in the upper panel of Fig. 7 shows the distance dependence of the median field E0(D) at the frequency of 8 Hz. Obviously, the average field E0(D) might be considered as a field with compensated impact of the day–night interface. The field disturbance by the non-uniformity is estimated by the following relation: EB ðDÞ ¼

E1 ðDÞ  E2 ðDÞ 2

(2)

The disturbance EB(D) corresponds to the half-width of the vertical distance between the nighttime and the daytime curves in Fig. 7. One may introduce the normalized field perturbation in the following way: BðDÞ ¼

EB ðDÞ E0 ðDÞ

(3)

The obvious relations are valid: E1 ðDÞ ¼ E0 ðDÞ½1 þ BðDÞ

(4)

and E2 ðDÞ ¼ E0 ðDÞ½1  BðDÞ

(5)

One can find the relative perturbations relevant to the non-uniformity by using the distance dependence of field amplitudes along the day and night propagation paths:

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B1 ðDÞ ¼

E1 ðDÞ 1 E0 ðDÞ

B2 ðDÞ ¼

E2 ðDÞ  1 ¼ B1 ðDÞ E0 ðDÞ

(6)

-day path

-night path

(7)

The relevant field disturbances are shown in Fig. 9 computed at a set of signal frequencies. The abscissa depicts the source–observer distance in Mm, and the normalized deviations of the field amplitude are shown along the ordinate on linear scale. The pairs of plots correspond to the signal frequencies of 8, 32, 80, and 120 Hz (from top to the bottom). Data for the ambient day path (propagation to the east) are shown by curves with dots. Field disturbances in the night conditions (the westward propagation) are depicted by lines with stars. Four paired graphs in Fig. 9 contain pronounced characteristic oscillations of the normalized disturbance. As might be expected from the energy conservation, maxima of curves with dots correspond to the minima of the curve with stars and vise versa. Physically this means that the non-uniformity “redistributes” the field between the hemispheres, depending on the signal frequency and the source–observer distance. At the fundamental Schumann resonance frequency of 8 Hz, the disturbance is positive for the field in the sunlit hemisphere: the amplitude increases by 0.13% here, while at the dark hemisphere, the amplitude is reduced by the same amount. At the fifth Schumann resonance mode (32 Hz), the sign of field disturbance varies with the source–observer distance, relevant to a more complicated spatial distribution of the resonant field. Data in Fig. 9 demonstrating changes in the impact of terminator indicate that reflections increase with the source distance. The growth might be attributed to two factors. The first implies an extension of the Fresnel zones with the increasing propagation path. The second factor is associated with the waves arriving from the source antipode. Since the spatial periods of interference patterns in the day and the night hemispheres are slightly different (owing to deviations in the wave phase velocities), these departures can alter the magnitude of functions E0 and EB, especially, at higher frequencies and distances exceeding 10 Mm. As might be seen in Fig. 9, the day–night interface is able to alter Schumann resonance field amplitude by approximately 1 dB. This result was also present in Figs. 4, 5 and 8. Such alterations indicate that the normalized disturbance relevant to the day–night interface is close to 10% in magnitude. It should be noted though that exact separation of the field disturbances caused by the non-uniformity is a complex and delicate problem containing may factors, therefore, its comprehensive examination requires further investigation.

Fig. 9. Estimated field disturbances by the smooth terminator against the source–observer distance at different signal frequencies.

counterweighed by alterations in the wave attenuation rate. Therefore, one can find the source–receiver distance where the median day amplitude of a monochromatic signal is equal to the median night amplitude. Though, the temporal field modulations do not completely vanish, especially, in the close vicinity of terminator.  Impact of day–nigh non-uniformity is observed for the day and the night propagation paths in form of interference between the direct source–observer wave and the wave scattered by terminator. The “reflection coefficient” of the day–night interface may reach 0.1, and its sign depends on the orientation of source–observer arc relative to the terminator line and on its distance from the day–nigh interface.

6. Conclusions Data shown above allow formulating the following conclusions.  Workability of specific 2DTE technique was demonstrated. Modeling of ELF radio propagation was performed for the Earth–ionosphere cavity resonator with the realistic model of day–night non-uniformity  Characteristics of particular diurnal variations in the field amplitude at a given frequency depend on the orientation of the propagation path relative to the solar terminator. When one of the corresponding points crosses the day–night interface, the abrupt amplitude change is observed that may reach 1 dB in magnitude.  Discontinuities in amplitude associated with the crossing of the solar terminator show minor dependence on the ELF signal frequency.  The day–night non-uniformity affects the Schumann resonance spectrum by elevating or lowering it as a whole by an amount reaching 1.5 dB. Only the level of the field tends to change, while the overall resonance pattern is easily recognized thus allowing for the successful location of sources of the ELF transients in the non-uniform cavity.  At frequencies above Schumann resonance, changes in the field amplitude related to the source excitation factor might be

Areas of future work We investigated in the present work the influence of the day–night non-uniformity on the amplitude of ELF radio wave and estimated magnitude of field reflections from the terminator. A similar treatment might be performed for the horizontal magnetic field components. In this case, the reflections from terminator will cause deviations in the apparent source bearing, and the results obtained could be compared with observations by Mlynarczyk et al. (2017). We plan to conduct such a study in the nearest future. Acknowledgements The authors express their sincere gratitude to the reviewers of this work, for their valuable comments and suggestions, which allowed us to substantially improve this paper. 10

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Appendix A. Supplementary data Supplementary data related to this article can be found at https://doi.org/10.1016/j.jastp.2018.01.001. Appendix 1 Table A1. Logarithm of air conductivity (S/m) against the altitude above the ground surface z, km

lg(σ ) Day

lg(σ ) Night

z, km

lg(σ ) Day

lg(σ ) Night

z, km

lg(σ ) Day

lg(σ ) Night

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36

14.12 13.97 13.82 13.67 13.40 13.17 12.99 12.84 12.71 12.58 12.46 12.35 12.24 12.13 12.03 11.93 11.84 11.74 11.65 11.57 11.48 11.40 11.32 11.24 11.17 11.10 11.03 10.96 10.89 10.82 10.74 10.65 10.58 10.51 10.44 10.35 10.24

14.12 13.97 13.82 13.67 13.40 13.17 12.99 12.84 12.71 12.58 12.46 12.35 12.24 12.13 12.03 11.93 11.84 11.74 11.65 11.57 11.48 11.40 11.32 11.24 11.17 11.10 11.03 10.96 10.89 10.82 10.76 10.69 10.63 10.57 10.51 10.45 10.39

37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73

10.16 10.09 9.97 9.92 9.84 9.75 9.69 9.63 9.59 9.56 9.53 9.51 9.48 9.46 9.44 9.40 9.38 9.29 9.22 9.10 9.01 8.86 8.75 8.57 8.45 8.24 8.10 7.87 7.73 7.50 7.35 7.17 7.02 6.85 6.72 6.55 6.37

10.32 10.25 10.18 10.11 10.04 9.99 9.93 9,87 9.81 9.75 9.68 9.64 9.62 9.58 9.57 9.56 9.53 9.51 9.48 9.46 9.44 9.40 9.38 9.29 9.22 9.10 9.01 8.86 8.75 8.57 8.45 8.24 8.10 7.90 7.73 7.50 7.35

74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110

6.25 6.12 6.02 5.93 5.83 5.76 5.66 5,58 5.49 5.41 5.29 5.19 5.05 4.94 4.77 4.64 4.43 4.29 4.04 3.89 3.58 3.40 3.01 2.81 2.61 2.41 2.21 2.00 1.87 1.72 1.48 1.29 1.13 0.96 0.81 0.67 0.54

7.17 7.02 6.85 6.72 6.55 6.37 6.25 6,12 6.02 5.93 5.83 5.76 5.66 5.58 5.49 5.41 5.29 5.19 5.05 4.94 4.77 4.64 4.43 4.29 4.04 3.89 3.58 3.35 3.15 3.05 2.96 2.88 2.81 2.76 2.71 2.67 2.64

Appendix 2 2D telegraph equations. Computations of the electromagnetic field by the 2DTE method deal with the partial differential equation for the scalar function u(θ, φ) describing the voltage between the ground surface and the lower boundary of the ionosphere at the current point having the angular coordinates (θ, φ):

 HL ðθ; ϕÞ ∂ sin θ ∂u HL ðθ; ϕÞ ∂ 1 ∂u þ þ k2 a2 S2 ðu þ uст Þ ¼ 0; sin θ ∂θ HL ðθ; ϕÞ ∂θ sin2 θ ∂ϕ HL ðθ; ϕÞ ∂ϕ

(A1)

where a is the Earth's radius and k is the wave number. The parameter S is the complex sine of the incidence angle of a plane monochromatic radio wave on the ionosphere. It is expressed in terms of magnetic HL and electric HC characteristic heights: S2 ¼ HL/HC The spherical coordinate system {r, θ, φ} is used with the polar axis θ ¼ 0 directed to the point vertical electric dipole source located at the ground surface. In this system, the source function uст is defined in the following way: uст ¼ P0

δðθÞ 2πε0 a2 sin θ

(A2)

where P0 is the dipole moment of the vertical source and δ(θ) is the Dirak's delta function. The sought vertical component of the electric field is expressed at the ground surface in terms of u(θ, φ) function: Er ðθ; ϕÞ ¼

uðθ; ϕÞ : HC ðθ; ϕÞ

11

(A3)

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The results obtained with the help of the 2DTE correspond to the sub-ionospheric single-mode radio propagation, which is valid at extremely low frequencies. The solution of equation (A1) might be constructed in different ways. Expansion over the spherical functions was used in the papers (Kirillov, 1993a, 1993b; Kirillov et al., 1997, Kirillov and Kopeykin, 2002). Such an approach requires, however, modifications in the computational code after each alteration of the form of the non-uniformity. This is why we exploit a more universal grid technique and solve the partial differential equation (A1) on the sphere. The grid computational methods solving such equations were developed for the real functions, but it turned out that these are applicable to the telegraph equations in complex variables. The essence of the grid method (or the method of finite differences) consists in replacing the region of continuous variations of arguments of the original problem by a finite discrete set of points called a grid. Then, one obtains the finite-difference equation instead of a partial differential equation. The major difficulty associated with the spherical coordinates (θ, φ) is the presence of singularities of the coefficients in equation (A1) at the poles (θ ¼ 0 and θ ¼ π ). The obstacle might be avoided by an appropriate selection of grid sampling points. A rectangular uniform grid with the number of nodes N and M correspondingly is superimposed on the domain of variables (θ, φ). The grid spacing along the variable θ (co-latitude) is equal to hθ ¼ π /N. The step along the φ variable (longitude) is correspondingly equal to hφ ¼ π /M. The nodal coordinates of the grid are related to their numbers {i, j} by relations: θi ¼ (i þ ½) hθ, i ¼ 2 [0, N–1], φj ¼ j hφ, j ¼ 2 [0, M–1]. With such a choice of the grid nodes along the θ coordinate, the boundary points i ¼ 0 and i ¼ N – 1 are separated from the poles by exactly one-half the grid step hθ. Thus, it becomes unnecessary to calculate coefficients of the equation at the poles. The first derivatives of the sought function at the nodal points are approximated by the finite differences of the first order. As a result of discretization of equation (A1), one obtains a closed system of equations for the grid functions ui,j: h2ϕ

) )  (  (   HL θi ; ϕj HL θi ; ϕj sinðθi þ hθ =2Þ  sinðθ  hθ =2Þ  ui;jþ1  ui;j ui;j  ui;j1   uiþ1;j  ui;j   i  ui;j  ui1j       þ h2θ  sin θi HL θi ; ϕj þ hϕ 2 HL θi ; ϕj  hϕ 2 HL θi þ hθ =2; ϕj HL θi  hθ =2; ϕj sin2 θi   H ; ϕ θ L i j ui;j þ h2θ h2ϕ k 2 a2  HC θ i ; ϕ j ¼ 0;

(A4)

where i ¼ f1; :::; N  2g; ϕj ¼ j hϕ ; and j ¼ f0; :::; M  1g The grid equation is constructed in the pole vicinity θ ¼ 0 (i ¼ 0, θ0 ¼ hθ,/2), j ¼ 2 [0, M–1] with an account of the finiteness (equal to zero) of the expression sin θ ∂∂uθ ¼ 0 at the poles (θ ¼ 0 and θ ¼ π ). These points are separated from the poles by the hθ/2 distance: 2h2ϕ

( )        HL hθ =2; ϕj  HL θ0 ; ϕj u  u0;j u  u0;j1   u1;j  u0;j þ 4HL θ0 ; ϕj  0;jþ1    j   þ h2θ h2ϕ k2 a2  u0;j ¼ 0; j ¼ 0; :::; M  1: HL h θ ; ϕ j HC θ0 ; ϕj HL θ0 ; ϕj þ hϕ 2 HL θ0 ; ϕj  hϕ 2

(A5)

Since we assume that there is a source described by equation (A2) at the other pole (θ ¼ π ), the grid equations acquire the following form in the neighborhood of this pole (i ¼ N – 1, θN–1 ¼ π hθ,/2, j ¼ 2 [0, M–1]): 2h2ϕ

( )        HL θN1 ; ϕj  H θN1 ; ϕj uN1;jþ1  uN1;j u  uN1;j1 k 2 HL ðπ ; 0Þ 2 2 2 2 L   uN2;j  uN1;j þ 4HL θN1 ; ϕj     N1;j   uN1;j ¼ P0 : (A6) þ hθ hϕ k a πε0 h2θ HC ðπ ; 0Þ HL π  hθ ; ϕj HC θN1 ; ϕj HL θN1 ; ϕj þ hϕ 2 HL ðθN1 ; ϕj  hϕ 2

We imply in equation (A4)–(A6) that owing to the cyclicity of variable φ its index should be replaced by M – 1 when it reaches the value of 1, and it must be replaced by 0 when arriving at M. Thus, we have obtained the closed system N  M of linear algebraic equation (A4)–(A6) for the N  M unknowns ui,j. We use in computations N ¼ 299 and M ¼ 80. Equations of this type are often met in problems of electrostatics and hydrodynamics. There are many methods of constructing the solution, including the iterative techniques. We have chosen the direct method of solution called block tri-diagonal matrix algorithm (which is regarded in Russian ! literature “matrix sweep technique” (Samarskyj, 2001)): The matrix sweep method is used in solution of the following problem: the vectors Y i ; ði ¼ 0; :::; NÞ are sought that satisfy the following equation: ! ! ! ! Ai Y iþ1  Bi Y i þ Ci Y i1 ¼  F i ; 1  i  N  1

(A7)

! ! ! A0 Y 1  B0 Y 0 ¼  F 0 ; ! ! ! BN Y N þ CN Y N1 ¼  F N ;

(A8)

and the boundary conditions:

where Ai, Bi, and Ci, ði ¼ f0; :::; N  1gÞ are the square matrices. The solution of the problem has the following form: ! ! ! Y i1 ¼ Xi Y i þ Z i ; i ¼ fN; N  1; :::; 1g ; ! where Xi and Z i are the matrices and vectors sought. Initially, the direct sweep of matrix is performed:

12

(A9)

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Xiþ1 ¼ ðBi  Ci Xi Þ1 Ai ; X1 ¼ B1 A;  ! !0  0 ! Z iþ1 ¼ ðBi  Ci Xi Þ1 Ci Z i þ F i ; ! ! Z 1 ¼ B1 0 F 0 ; i ¼ 1; 2; ::: ; N  1

(A10)

Afterwards, the inverse sweep is made that provides the solution:  ! !  ! Y N ¼ ðBN  CN XN Þ1 CN Z N þ F N ; ! ! ! Y i1 ¼ Xi Y i þ Z i ; i ¼ fN; N  1; ::: ; 1g :

(A11)

A more detailed description of the matrix sweep algorithm might be found in a book by Samarskyj (2001). The major feature of this method is a substansional gain in the computational resources necessary for the implementation when the grid dimensions vary. The number of calculations increases as the third degree (the amount of memory allocated to arrays is proportional to the second degree) of the M parameter being the number of partitions along the φ variable. The growth arises due to conversion of the complex matrix of this dimension. The number of arithmetic operations and the necessary memory increase in proportion to the first degree of N being the partition number along the variable θ. This fact agrees well with the physical picture of ELF radio propagation: the field depends only on the θ variable and is independent of φ in the uniform waveguide. In the actual computations, one can choose the M parameter (the transverse dimension of the grid) several times smaller than the number N (the longitudinal dimension). ! ! It should also be noted that, owing to the selected numbering of grid nodes, all the arrays of the right-hand parts F i (except the last one F N ) are ! equal to zero. This allows us to ignore the intermediate arrays Z i . One may observe from Eq. (A4) that the arrays of coefficients Ai and Bi have the simple diagonal form, and the Ci array is of a tri-diagonal form. This feature substantially simplifies the corresponding algebraic operations in formula (A10). As it was already mentioned, the most laborious operation in implementation of the matrix sweep algorithm is the computation of the inverse matrix during the direct sweep (A10). There are many algorithms realizing this operation. We would recommend using for this purpose the sub-routins zgetrf and zgetri from the BLAS package. This package is an internal part of such well-known scientific software packages as IMSL, MKL, and GSL. The BLAS package might be also downloaded as FORTRAN listing from the NETLIB website (http://netlib.org/blas/). The most efficient way would be to install the OpenBLAS package from (http://www.openblas.net/) site, since it generates the library accounting for the peculiarities of a particular processor and capable to exploit all the processor cores of your computer.

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