Modifications of Middle Atmosphere Conductivity During Sudden Ionospheric Disturbances Deduced From Changes of Schumann Resonance Peak Frequencies

Abstract

This paper presents a novel technique to reveal modifications of the Schumann resonance peak frequencies during sudden ionospheric disturbances (SIDs) caused by the solar X-ray flares. An SID is associated with the abrupt reduction in ionospheric height over the day hemisphere during solar flares. Reduction of the daytime ionosphere lowers the average height of the Earth-ionosphere cavity and leads to an increase in its resonant frequencies. An improved profile is used describing the middle atmosphere conductivity and its modification. The resonance frequencies of the Earth-ionosphere cavity and the characteristic heights of the conductivity profile are computed by the full wave technique for various intensities of ionosphere modifications. Power spectra of the vertical electric field component are calculated using the 2-dimensional telegraph equations for the uniform spatial distribution of global thunderstorms in the cavity with the day-night nonuniformity for several observer positions relative to the solar terminator. Changes in the peak frequencies are obtained as a function of magnitude of ionosphere modifications. The weighted average frequency of Schumann resonance is calculated over the first three modes, and the linear fitting is performed for the dependence of this weighted average frequency on the SID intensity and/or on changes in the magnetic characteristic height of the conductivity profile. The data obtained might be used in further interpretations of observational data and in estimating the lower ionosphere modification by using Schumann resonance records.

Key Point

• Modification of lower ionospheric electron density profile during an SID has been studied with changes of Schuman resonance peak frequencies

1 Introduction

A great attention is directed to the studies of global electromagnetic (Schumann) resonance of the Earth-ionosphere cavity in order to detect various perturbations in the lower ionosphere. The major cause of natural ionosphere modifications is the solar activity, which depends on the space weather in the near geocosmos. Solar activity manifests itself in the form of X-ray flares when the sharply increased flux comes from the solar chromosphere to Earth being the hard electromagnetic radiation ranging from the gamma to ultraviolet wave bands. An X-ray flare is usually followed by a geomagnetic storm. Increases in the corpuscular flux are observed less frequently, and these are either the solar proton or solar electron events. The charged particles in these solar events are deflected by the geomagnetic field, and they modify the polar and subpolar ionosphere of the Northern and Southern Hemispheres. In the wide radio band, the sharp fading of signals is observed at the polar propagation paths, and the relevant phenomenon is regarded as polar cap absorption.

X-ray flares dramatically increase ionization rate in the middle and upper atmosphere on the dayside of the Earth affecting the radio communications in various wave bands. The phenomenon is called a sudden ionospheric disturbance (SID). Studies of links between the solar activity and Schumann resonance have a long history, and extended literature was devoted to SIDs and their role in extremely low frequency (ELF) radio propagation (see e.g., Cannon & Rycroft, 1982; Madden & Thompson, 1965; Sao et al., 1973; Sentman et al., 1996). Unfortunately, convincing evidence of the SID impacts on Schumann resonance frequency is seriously masked by the random nature of Schumann resonance signal: The local noise and fluctuations relevant to the global thunderstorms radiation mask the simultaneous abrupt modification driven by the solar activity; see, for example, Sátori et al. (2005, 2016).

Both the corpuscular and electromagnetic solar radiation are unambiguously observed in radio waves from the very low frequencies (3 to 30 kHz) to the short wavelengths (3–10 MHz) facilitating detection of changes in the ionospheric plasma at the altitude 120 km and above. As for the radio waves of ELFs (ELF = 3 Hz–3 kHz) and for the Schumann resonance range (4–40 Hz) in particular, which is sensitive to the ionosphere characteristics below 100 km, the SID influence on the Schumann resonance remained dubious until recently specially for the median solar X-ray flares not followed by solar proton event or solar electron event (Dyrda et al., 2015; Roldugin et al., 2004; Sátori et al., 2005, 2016).

A special technique was proposed by Shvets et al. (2017) for processing Schumann resonance records facilitating detection of sharp changes of resonance frequencies caused by SIDs. To identify these alterations, the weighted average frequencies (WAFs) were used instead of the customary peak frequencies of the Schumann resonance. The WAF is equal to the average of normalized centered frequencies of several Schumann resonance modes.

As the first step in data processing, the slow trends are subtracted from the records of individual peak frequencies being associated with the daily movement of the global thunderstorms around the globe. This procedure results in obtaining “centered” peak frequencies close to 8, 14, 20 Hz, etc. The impact of the source-observer distance is reduced in centered frequencies, and these are further transformed (weighted) into the “unified” first mode frequency of 8 Hz. We address this operation in more detail below. The weighted frequencies retain fluctuations associated with the noisy nature of radio signal arriving from thunderstorms and carrying a replica of a random distribution of lightning discharges over the globe. These fluctuations are mutually independent at different resonance modes owing to departures in spatial field distributions of individual modes. This independence is preserved in the weighting procedure. Averaging of the weighted frequencies over the ensemble of resonance modes provides a WAF characterized by noticeably reduced fluctuations.

The effect of an SID on the Schumann resonance record is based on the fact that a reduction of the lower ionosphere over the day hemisphere causes a synchronous fast increase in all resonance frequencies, and the temporal outline of these increases is practically coincident at every resonance mode. Reduction of independent fluctuations during averaging of the weighted frequencies makes the WAF more sensitive to a solar X-flare in comparison with the resonance frequencies of individual modes. This feature was demonstrated by Shvets et al. (2017), which allowed them to reveal modifications in the lower ionosphere during several moderate SIDs and to also estimate the vertical displacement of the model knee profile of the middle atmosphere conductivity. The heuristic knee model was used for interpretations of experimental data in this work.

In the present study, we use the improved conductivity profile model corresponding to observations of Schumann resonance (Galuk et al., 2017; Kudintseva et al., 2016; Nickolaenko et al., 2015, 2016) instead of the conventional heuristic knee model. We postulate profile modifications caused by SIDs with different magnitudes and afterward estimate the sensitivity of the WAF to ionosphere modifications. The results of linear fitting of model computations might be applied to observational data for estimating the degree of ionosphere disturbances during various SIDs.

2 Vertical Conductivity Profiles

Parameters observed experimentally of the global electromagnetic (Schumann) resonance in the Earth-ionosphere cavity depend on the vertical conductivity profile of the middle atmosphere. Kudintseva et al. (2016), Nickolaenko et al. (2015, 2016), and Galuk et al. (2017) suggested the model vertical conductivity profile based on the observations both of the Schumann resonance in the uniform Earth-ionosphere cavity and in the cavity nonuniform along the angular coordinates corresponding to the ionosphere heterogeneity of the day-night type.

Those papers addressed the conductivity of the middle atmosphere in regular conditions of the quiet Sun. When the X-ray flash occurs at the Sun, the flux sharply increases of the hard ionizing radiation coming to the dayside hemisphere of the globe. The concentration of the ionospheric plasma increases as a result, and the conductivity profile is modified so that the upper strata of atmosphere with higher conductivity shift downward during the flare. Such perturbations cover the illuminated hemisphere exclusively, and since these occur very fast, they are regarded as SIDs.

The greatest modifications during an SID event concentrate in the upper ionosphere and in the vicinity of the ionization maximum (250–300 km above the ground surface). Modifications reduce when the altitude decreases, and perturbations affecting the observed Schumann resonance signals disappear at altitudes below 50–60 km. Therefore, one may say that an SID reduces the upper (magnetic) characteristic height of the conductivity profile, while the lower (electric) height remains unchanged.

We model these features in the following way. Conductivity profile in the day hemisphere in the absence of solar flares is described by the improved regular ambient day conductivity profile. This profile is shown in Figure 1 by the leftmost smooth curve without markers (curve 0). The height in km above the ground surface is shown along the ordinate, and the logarithm of middle atmosphere conductivity (measured in S/m) is plotted along the abscissa. A set of conductivity profiles is shown in Figure 1, and each curve is numbered from 0 to 10. The curve number 0 depicts the conductivity of the middle atmosphere for a quiet Sun. During the SID, the lower part of this curve (the region from 0 to ~55 km) remains unchanged, while the upper part of profile at altitudes from 55 to 110 km alters “in a linear way.” The conductivity profile is transformed into curve 10 in Figure 1 when the hard radiation from the Sun reaches its maximum flux during the X-ray flare. The data are given in Table 1 corresponding to the quiet and the maximum modification conditions.

Figure 1

Vertical conductivity profiles of the quiet (B = 0) and perturbed (B = 1–9) ionosphere models.

Table 1. Logarithm of Air Conductivity [S/m] Against Altitude Above the Ground Surface

z, km	Quiet day	Max SID	z, km	Quiet day	Max SID	z, km	Quiet day	Max SID
0	−14.12	−11.48	37	−10.16	−9.01	74	−6.25	−5.14
1	−13.97	−11.4	38	−10.09	−8.86	75	−6.12	−5.0
2	−13.82	−11.32	39	−9.97	−8.75	76	−6.02	−4.66
3	−13.67	−11.24	40	−9.92	−8.57	77	−5.93	−4.33
4	−13.40	−11.17	41	−9.84	−8.45	78	−5.83	−4.0
5	−13.17	−11.1	42	−9.75	−8.24	79	−5.76	−3.5
6	−12.99	−11.03	43	−9.69	−8.1	80	−5.66	−3.0
7	−12.84	−10.96	44	−9.63	−7.87	81	−5,58	−2.66
8	−12.71	−10.89	45	−9.59	−7.73	82	−5.49	−2.33
9	−12.58	−10.82	46	−9.56	−7.5	83	−5.41	−2.0
10	−12.46	−10.74	47	−9.53	−7.35	84	−5.29	−1.75
11	−12.35	−10.65	48	−9.51	−7.17	85	−5.19	−1.5
12	−12.24	−10.58	49	−9.48	−7.02	86	−5.05	−1.25
13	−12.13	−10.51	50	−9.46	−6.9	87	−4.94	−1.0
14	−12.03	−10.44	51	−9.44	−6.83	88	−4.77	−0.74
15	−11.93	−10.35	52	−9.40	−6.8	89	−4.64	−0.54
16	−11.84	−10.24	53	−9.38	−6.78	90	−4.43	−0.34
17	−11.74	−10.16	54	−9.29	−6.76	91	−4.29	−0.17
18	−11.65	−10.09	55	−9.22	−6.74	92	−4.04	−0.04
19	−11.57	−9.97	56	−9.10	−9.0	93	−3.89	0.1
20	−11.48	−9.92	57	−9.01	−8.75	94	−3.58	0.25
21	−11.40	−9.84	58	−8.86	−8.5	95	−3.40	0.35
22	−11.32	−9.75	59	−8.75	−8.25	96	−3.01	0.5
23	−11.24	−9.69	60	−8.57	−8.0	97	−2.81	0.56
24	−11.17	−9.63	61	−8.45	−7.75	98	−2.61	0.75
25	−11.10	−9.59	62	−8.24	−7.5	99	−2.41	0.88
26	−11.03	−9.56	63	−8.10	−7.25	100	−2.21	1.0
27	−10.96	−9.53	64	−7.87	−7.0	101	−2.00	1.06
28	−10.89	−9.51	65	−7.73	−6.75	102	−1.87	1.16
29	−10.82	−9.48	66	−7.50	−6.5	103	−1.72	1.25
30	−10.74	−9.46	67	−7.35	−6.25	104	−1.48	1.36
31	−10.65	−9.44	68	−7.17	−6.0	105	−1.29	1.43
32	−10.58	−9.4	69	−7.02	−5.85	106	−1.13	1.5
33	−10.51	−9.38	70	−6.85	−5.7	107	−0.96	1.59
34	−10.44	−9.29	71	−6.72	−5.56	108	−0.81	1.65
35	−10.35	−9.22	72	−6.55	−5.42	109	−0.67	1.74
36	−10.24	−9.1	73	−6.37	−5.28	110	−0.54	1.8

• Note. SID, sudden ionospheric disturbance.

The maximally disturbed profile 10 in Figure 1 and in Table 1 was obtained from the quiet profile 0 in the following way. Modifications are absent at altitudes below 55 km. The regular day conductivity profile of ionosphere is shifted downward by Δ = 21 km at the 110 km altitude, which is indicated by the vertical arrow in Figure 1. Owing to this height reduction, the value of the conductivity logarithm lg(σ) = −0.54 shifts from the altitude of z = 110 km (in quiet conditions) to 89 km (during the maximum SID; see Table 1).

The vertical displacement dh in km at the intermediate height h is found from the following linear equation:

(1)

where Δ = 21 km, z₁ = 55 km, and z₂ = 110 km.

It should be noted that the abscissa in Figure 1 shows the logarithm of conductivity, so that linear changes in the height correspond to the exponential changes in the air conductivity. These model variations are physically justified by the exponential reduction in ionizing radiation during its penetration downward into the middle atmosphere.

Thus, the two ultimate dependencies were obtained in Figure 1 (Table 1) shown by a smooth curve B = 0 (the quiet conditions) and the curve with the wedges B = 10 (the maximum perturbation). All other conductivity profiles are found between these curves, each corresponding to an intermediate magnitude of perturbation B rating from 1 to 9 points. These profiles were obtained by the linear interpolation of the two ultimate dependencies. Thus, we obtain the corresponding conductivity profile in the middle atmosphere for an SID with the intensity measured in relative units 0 < B < 10.

For each profile we calculated the frequency dependence of the complex characteristic heights of conductivity profiles and the propagation constant of ELF radio waves by using the full wave method (Galuk et al., 2015, 2017, 2018; Hynninen & Galuk, 1972; Kudintseva et al., 2016; Nickolaenko et al., 2015, 2016). The data are shown in Figure 2.

Figure 2

(a) Characteristic heights (h_L and h_C) and (b) the propagation constant ν(f) of extremely low frequency radio wave for the quiet and active Sun (sudden ionospheric disturbance) periods.

When constructing the propagation constant of ELF radio waves ν (f) by using the full wave method, one considers the plane vertically stratified conducting medium where the rigorous solution is sought for the electromagnetic problem. The radio waves are considered to propagate from one horizontal layer to another. The thickness of these layers is much smaller than the wavelength in the medium, so that properties of the medium inside a stratum do not change. The tangential components of the fields remain continuous at the boundaries of the layers. Thus, one obtained for M layers an algebraic system of 2 M linear equations for the reflection and the transition coefficients at each boundary, and it is possible to calculate the dependence ν (f) corresponding to a given conductivity profile σ (h) (Galuk et al., 2015, 2017, 2018; Hynninen & Galuk, 1972; Kudintseva et al., 2016; Nickolaenko et al., 2015, 2016).

However, this classical approach encounters difficulties when the air conductivity becomes too small and slowly varies with altitude. The algebraic system of equations may become degenerate in this case. Fortunately, it can be shown that the full wave problem is reduced to the first-order nonlinear differential equation (the Riccati equation), provided that the problem is reformulated from the field amplitudes to the surface impedance at the boundaries of the layers (the ratio of the tangential components of the E [electric] and H [magnetic] fields). The Riccati equation is solved numerically with the help of iterations. The sought complex characteristic heights of the conductivity profile and the propagation constant ν (f) are obtained as a result. The electric h_C and the magnetic h_L(f) characteristic heights (h_C < h_L[f]) are interpreted physically as the effective interfaces in the ionospheric plasma above which the amplitude of the electric or the magnetic field starts to rapidly decrease with altitude when a monochromatic electromagnetic wave is incident on the ionosphere from the ground. The method is regarded as the full wave technique since the formal solution of the problem is rigorously constructed and the changes are obtained for both fields within the whole atmosphere.

Figure 2a illustrates frequency variations of the real parts of the characteristic heights in the ionosphere obtained for particular conductivity profiles of atmosphere. The abscissa shows frequency ranging from 3 to 50 Hz. The upper panel of this figure depicts the real parts of the magnetic (the upper) characteristic height h_L(f) computed for the regular day (curve with points) and the night (curve with asterisks) model profiles. The curve with diamonds shows data obtained for the day profile during the SID of the highest 10-point magnitude. One may observe that magnetic characteristic heights decrease with frequency from ~100 km, and the dependence corresponding to the SID is located by ~15 km below the curve for the quiet ionosphere. It is worth recalling here that in accordance with Figure 1, the actual conductivity profile of middle atmosphere has moved downward by 21 km from 110 km.

The lower panel of Figure 2a shows the frequency dependence of real parts of the electrical (lower) characteristic height h_C(f) computed for the same profiles by using the full wave technique. These heights increase with increasing frequency from 40–45 to 60–65 km. Dependence for the disturbed ionosphere during an SID initially coincides with the curve for the regular daytime ionosphere, and then it is located below the curve for unperturbed case. Such behavior seems quite natural, since the postulated conductivity profile remained unchanged in the altitude region below 55 km during the SID.

Plots in Figure 2b illustrate the frequency dependence of the complex propagation constant ν(f) of ELF radio waves computed for the Earth-ionosphere cavity model with the upper boundary formed by the regular dayside and the nightside ionosphere (curves with points and stars correspondingly). Data for the ambient day ionosphere during the SID event of the 10-point magnitude are shown by curves with diamonds.

The upper panel shows the imaginary part of propagation constant Im(ν[f]) as a function of frequency which characterizes the radio wave attenuation rate. As one may see, a reduction of the upper part of the conductivity profile leads to a significant decrease of losses in the Earth-ionosphere cavity. Such changes must lead to an increase in the observed peak frequencies of the global electromagnetic resonance. This feature manifests itself at the lower panel of Figure 2b by leftward shift of the frequency dependence of the real part of the propagation constant pertinent to the perturbed cavity. Thus, the resonance condition Re(ν[f]) = n, where n = 1, 2, 3, etc. takes place at higher frequencies in the presence of ionosphere modification. The effect is explained physically by the fact that conductivity variations in the lower ionosphere become steeper during an SID, and this raises the peak frequencies and the Q-factors of the resonance oscillations.

To conclude this section, we note that fitting of computational data for the unperturbed imaginary part of the propagation constant by a power function provides the frequency dependence of the form −Im(ν[f]) ∝ f^0.69. This result is extremely close to the widely used standard frequency dependence of the ELF attenuation factor by Ishaq and Jones (1977) with the form of −Im(ν[f]) ∝ f^0.64. The agreement of the model computations with the standard ELF model attenuation rate validates the conductivity profile used here. Deviation from the standard model in the index of frequency dependence by 0.05 is easily explained by the fact that we use the ambient day conductivity profile when the attenuation factor somewhat exceeds the night (and the average) values.

3 Changes of Resonance Frequencies Caused by an SID

To reveal a connection between the different mode resonance frequencies and modifications of the vertical conductivity profile in the middle atmosphere during SIDs, we apply the computed frequency dependence ν(f). This approach is based on the arguments listed in several books and papers on Schumann resonance and verified by various computations (see e.g., Nickolaenko & Hayakawa (2002, 2014); Tanaka et al. (2011); Nickolaenko et al. (2010, 2012)).

The model Earth-ionosphere cavity with the day-night nonuniformity is obtained from a uniform resonator by introducing the antisymmetric perturbation, in which parameters on the dayside and the nightside of the planet deviate from the median values “in opposite directions by the same quantity.” Such a disturbance is regarded as the day-night asymmetry. For example, let the average ionosphere height be 75 km and the day-night asymmetry be 15 km. Then, the night ionosphere height will be equal to 90 km and the day ionosphere height will be equal to 60 km. Obviously, the asymmetry magnitude cannot exceed the median value; otherwise, the model parameter on the dayside may reach zero or even become negative.

Formal analysis and computations (Nickolaenko et al., 2010, 2012; Nickolaenko & Hayakawa, 2002, 2014; Tanaka et al., 2011) showed that the day-night asymmetry provides a minor impact on the eigen-frequencies of the Earth-ionosphere cavity and on the outline of the power spectra for an arbitrary reasonable modification. Thus, it is desirable to explain in how an SID caused by the solar or galactic flare is able to modify the Schumann resonance parameters.

An SID alters the conductivity on the dayside of the globe without affecting the nightside of the Earth. Therefore, the SID modifies both the asymmetry and the median parameters of the cavity, and this latter noticeably changes the peak frequencies in the spectral pattern. Let the regular cavity have the average height of 75 km with the nighttime ionosphere height of 90 km and the daytime height of 60 km. The SID event reduces the dayside ionosphere altitude by 20 km (to 40 km). Hence, the average height of the Earth-ionosphere cavity decreases to 65 km, while the asymmetry increases to 25 km. The increase of asymmetry will not noticeably affect the power spectra, while the lowering of the average ionosphere will perceptibly alter the resonance parameters in observational data. Thus, perturbations localized at the daytime ionosphere provide a noticeable effect via modification of average global ionosphere.

The noisy nature of Schumann resonance signals being a composition of pulsed radio emissions from global thunderstorms hides the spectral modifications caused by SID events. A short time ago, a technique has been suggested for processing the records of Schumann resonance frequencies allowing singling out the sharp synchronous frequency modulations caused by the solar X-ray flares (Shvets et al., 2017). The WAF is used for this purpose instead of the modal peak frequencies. As the first step of processing, the recorded diurnal variations of peak frequencies of three or four resonant modes are “centered”: The slow trends are removed relevant to the drift of global thunderstorms around the Earth during the day. To do this, the recorded diurnal frequency variations are averaged over an ensemble of days covering several days before and several days after the X-ray event date. Thus, the standard daily patterns are obtained for each resonance mode over the observation period. These standard changes are subtracted from the original records, thus eliminating the slow daily trends while the fast frequency fluctuations remain still present in such “centered” records.

One obtains the weighted frequencies, that is, frequencies reduced to 8 Hz by using the centered peak frequencies of each of the resonant modes. Here some explanations are necessary. It is well known that according to Schumann formula the eigen-frequencies of the perfect Earth-ionosphere cavity are proportional to , where n is the resonance mode number. The quantity λ = n (n + 1) is regarded as the separation constant of the spherical coordinate system. The peak frequencies are equal to 8, 14, and 20 Hz, etc. in the actual Earth-ionosphere cavity with losses, and they are smaller than the eigen-values of the ideal cavity but still remain proportional to (Bliokh & Nickloaenko, 1986). By using this property, one can reduce the peak frequencies of the higher modes to the value pertinent to the fundamental resonance frequency 8 Hz. To do this, one must multiply the centered frequencies of the higher modes by , where n is the mode number. The peak frequencies 14, 20, and 26 Hz are reduced to 8 Hz as a result of such normalization. The weighted frequencies thus obtained can be averaged over the ensemble of resonant modes providing the WAF characterized by the reduced random fluctuations. The averaging procedure preserves the coordinated changes of the resonance frequencies caused by the sharp modification of the global ionosphere during an SID, and Shvets et al. (2017) demonstrated that it was possible to identify the sharp increase in experimentally observed resonance frequencies during SIDs. For evaluating the magnitude of modifications of the magnetic characteristic height of the ionosphere, they applied the calibration dependence of the WAF derived for the heuristic knee model. This allowed them to estimate the vertical displacement of the conductivity profile during particular SID events.

Figure 3 demonstrates practical implication of weighted frequencies. Temporal variations of modal frequencies observed in the Karymshino in Russia (geographic coordinates: 52.94°N and 158.25° E) records on 3 March 2015 are shown in Figure 3a (see Shvets et al., 2017 for more details). The abscissa depicts the universal time in hr. The left ordinates are the frequency deviations in Hz shown by black lines, and the right ordinate shows the solar X-ray flux shown by the dashed gray line. The upper frame (Figure 3a) illustrates the frequency fluctuations observed at the first four Schumann resonance modes. The plots are shifted vertically for convenience. One may observe that some kind of abrupt changes is present in all modal frequencies, but its relevance to X-ray flare remains dubious.

Figure 3

Schumann resonance modal frequencies recorded during a sudden ionospheric disturbance on 3 March 2015 at Karymshino observatory, Russia (Shvets et al., 2017). (a) Variations of modal frequencies around their centered values and (b) temporal variations of weighted average frequency in comparison with the concurrent solar X-ray flux. The black triangle and wedge are the moments of local sunrise and sunset, respectively.

The lower frame (Figure 3b) demonstrates the final result of processing. Here the black line depicts alterations of the WAF plotted against the left ordinate. The black triangle and wedge mark the moments of local sunrise and sunset at the Karymshino (Kamchatka) observatory. The gray dashed line illustrates the flux of solar X-rays shown against the right ordinate. Obviously, transition to WAF has clarified the modulation of Schumann resonance frequencies by the SID, and even relatively weak enhancements in X-ray flux became visible in Figure 3b. One may note that the X-ray flare modified the dayside of the globe when the observer was positioned on the nightside. Nevertheless, the effect is clearly seen in the record vividly confirming the global nature of Schumann resonance for one more time.

Let us return to dispersion curves ν(f) of Figure 2 computed for the conductivity profiles of the daytime atmosphere shown in Figure 1. The resonance frequencies satisfy the dispersion equation of the form Re(ν[f]) = n. We computed the Re(ν[f]) functions similar to those shown in Figure 2b corresponding to each SID profile. By fitting them for a particular magnitude of ionosphere disturbance B ∈ [0; 10], we obtained the following expressions for the resonance frequencies f^(B) (in Hz) as a function of the mode number n:

It is easy to obtain from these relations the resonance frequencies of the particular Schumann resonance mode by varying n with a fixed SID magnitude B. Relevant values are listed in Table 2 for the first three modes. The first column in this table indicates the magnitude of the disturbance. The next three columns refer to the resonance frequencies f_n of the first three modes in Hz. The fifth and the sixth columns contain the weighted (normalized) frequencies of the second and third modes in Hz computed from the formula , n = 2 and 3. The last column in Table 2 indicates the values of WAF relevant to a particular SID intensity B.

Table 2. Resonance Frequencies for the Fixed Sudden Ionospheric Disturbance (SID) Intensity

B	Resonance frequencies			Weighted frequency		Weighted average frequency <F(B)>, Hz
	f1, Hz	f2, Hz	f3, Hz	F2, Hz	F3, Hz
1	2	3	4	5	6	7
0	7.733	13.914	20.095	8.033	8.204	7.990
1	7.803	14.051	20.299	8.113	8.287	8.068
2	7.892	14.214	20.537	8.207	8.384	8.161
3	7.99	14.376	20.762	8.30	8.476	8.255
4	8.077	14.518	20.955	8.382	8.555	8.338
5	8.159	14.638	21.117	8.451	8.621	8.411
6	8.226	14.738	21.250	8.509	8.675	8.470
7	8.282	14.818	21.355	8.555	8.718	8.518
8	8.327	14.884	21.440	8.593	8.753	8.558
9	8.364	14.937	21.510	8.624	8.781	8.590
10	8.396	14.982	21.569	8.650	8.806	8.617

After constructing the <F(B)> dependence of the WAF on the disturbance magnitude (B), we performed the linear fitting and obtained the following relations:

(2)

for the WAF in Hz and

(3)

for its deviation (in Hz) caused by the ionosphere modification during the SID.

The following inverse calibration dependence is valid:

(4)

which is very helpful for inferring the magnitude of disturbance (in relative points) from the observed abrupt deviation of WAF (in Hz). Equation 4 indicates that an increase in the WAF by 0.2 Hz corresponds to an SID intensity of 3-point magnitude (see the corresponding profile in Figure 1).

A similar treatment is possible relating the WAF variations with the changes of the magnetic characteristic height h_L of atmosphere conductivity profile (changes of the electrical characteristic height h_C are negligibly small in our model). We obtain data shown in Figure 4 and listed in Table 3 in this case.

Figure 4

Calibration curves for deviations of weighted average frequency associated with the conductivity disturbance in the middle atmosphere during a sudden ionospheric disturbance. (a) Ionospheric magnetic height (h_L) versus <δF>. The broken curve with dots refers to the observation, and a straight line, to the linear fitting as a calibration line. (b) The same as (a), but in a different form of h_L versus (b).

Table 3. Connection of Sudden Ionospheric Disturbance (SID) Magnitude B With the Magnetic Characteristic Height h_L and With Variations of Weighted Average Frequency <δF>

B	Magnetic height hL, km	<δF>, Hz
0	97.104	0
1	95.328	0.078
2	93.256	0.171
3	91.004	0.265
4	88.917	0.348
5	87.136	0.421
6	85.633	0.48
7	84.377	0.528
8	83.347	0.568
9	82.509	0.6
10	81.817	0.627

The abscissa in Figure 4a indicates deviations of the WAF during the disturbance, and the abscissa of Figure 4b is the SID magnitude. The ordinates of the both panels of Figure 4 depict the computed characteristic magnetic height of the ionosphere h_L (in km) averaged over the first three Schumann resonance modes. The computational results are displayed by dots in both panels, and the straight lines show the linear fitting. The following equations are valid:

(6)

and

(7)

relating the characteristic magnetic height h_L (in km) with deviations of WAF <δF> (in Hz) and/or with the SID magnitude B (relative units). It follows from equation 6 that an increase in the WAF <δF> by 0.1 Hz is associated with the decrease in the magnetic characteristic height of the ionosphere <δh_L> by 2.5 km.

An explanation would be appropriate here. We obtained in our computation the data listed in Table 3, which presents the disturbance magnitude B ∈ [0; 10], the relevant magnetic characteristic height h_L, and the WAF deviations <δF> in its three columns. Figure 4 demonstrates two of these functions, namely, h_L(<δF>) and h_L(B) shown by dashed lines with dots. Equations 6 and 7 are obtained as a linear fit of these curves, which are shown by the straight lines in Figures 4a and 4b. Similarly, the dependence δF(B) of equation 3 was obtained from the same set of data.

4 Schumann Resonance Spectra for a Uniform Distribution of Sources

The above estimates were obtained from the dispersion relation Re(ν[f]) = n. Since actual observations exploit the power spectra of Schumann resonance, it is desirable to obtain the calibration curves from the model resonance spectra. One can compute the power spectra of the vertical electric field component by using the characteristic heights of the ionosphere conductivity profiles and the ELF propagation constant. We use the numerical solution of the problem for this purpose with the 2DTE. The specific algorithm for 2DTE was described by Hynninen and Galuk (1972) and Galuk et al. (2015, 2018) with detailed references.

The major difficulty encountered in the field modeling in a nonuniform cavity is related to a necessity of evaluating deviations in the observed peak frequencies caused by ionosphere disturbance positioned on the dayside of the globe when the spatial distribution of thunderstorms (the field sources) remains arbitrary. It would be desirable to compensate somehow the diurnal variations of peak frequencies caused by the movement of global thunderstorms around the Earth during the day. One can obtain the “average” calibration dependence in this case connecting modifications of the conductivity profile with observed abrupt changes in the peak frequencies of Schumann resonance. To eliminate an impact of the source-observer distance, we assume that the lightning activity is uniformly distributed over the surface of the Earth.

An additional problem arises in a cavity with the day-night nonuniformity even for a uniform distribution of sources: possible influence of the observer position relative to the terminator on the results of observations. We performed three series of computations of power spectra: when the observer is located at the center of the day hemisphere, when it occupies the center of the night hemisphere, and when it is positioned at the day-night interface. The model of a smooth day-night transition was used (Galuk et al., 2017, 2018). It turned out that for the other fixed conditions, a displacement of the observer leads to a predominantly vertical shift of the power spectrum above the frequency axis, while the peak frequencies do not change (see Figure 5). Since we are interested in peak frequencies, the choice of observer position becomes unimportant and the particular quantitative results listed below were obtained for the observer located at the light-shadow boundary, that is, at the solar terminator line.

Figure 5

Dynamic spectra of the Schumann resonance intensity <|E|²> (as given on the right side in different degree of black and white) obtained for the uniform distribution of lightning sources. The abscissa is radio frequency in Hz, and the ordinate is the sudden ionospheric disturbance disturbance magnitude, B. Three different observer positions are considered.

Figure 5 demonstrates dynamic spectra of the relative Schumann resonance intensity computed for the uniform distribution of thunderstorms over the globe. The horizontal axes here show the frequency of radio signal in Hz, and the vertical axes depict the magnitude of disturbance of the mesosphere conductivity B ranging from 0 (the cavity formed by regular ionosphere) to 10 (SID of maximum intensity). The relative spectral density of resonance oscillations <|E|²> is shown above this plane by dark inking. Three frames are shown in this figure, and each of them covers three Schumann resonance modes computed for the uniform spatial distribution of field sources in the nonuniform cavity. The upper panel corresponds to an observer located at the center of the day hemisphere; the lower frame was obtained for the observer at the center of the night hemisphere. The middle panel demonstrates data computed for the observer positions at the solar terminator line. One may observe that all three field distributions are very similar, even though there are some deviations in details.

We notice three resonant peaks in Figure 5 corresponding to the first, second, and third modes, and the peak frequencies lie around 8, 14, and 20 Hz. Changes in the conductivity profile caused by an SID lead to complex changes in the shape of the resonance spectra: the height, the position relative frequency axis, and the width of resonance peak. It is important for us that the peak frequencies of all resonance modes monotonically increase when the magnitude of SID, B, increases due to the increase in intensity of solar ionizing radiation.

Such changes are physically explained by a faster increase in atmospheric conductivity with altitude, and this magnifies the ionosphere reflectivity and reduces the losses of electromagnetic energy. Therefore, with the growth in B, the spectral maxima become narrower, and the peak frequencies increase, coming closer to the eigen-values of the perfectly conducting Earth-ionosphere cavity.

Figure 5 illustrates a monotonic increase in the Schumann resonance peak frequencies with increasing B. One can plot the peak frequencies against the magnitude of disturbance B for all three Schumann resonance modes by using computed power spectra. These modal frequencies are shown in Figure 6 after appropriate weighting . Three curves are shown in the vicinity of 8 Hz by the dashed lines with dots, diamonds, and asterisks relevant to the peak frequency f₁ of the first mode, and to the weighted frequencies of the second F₂, and the third F₃ resonance modes. One may notice that all curves in Figure 6 increase similarly with the disturbance magnitude B.

Figure 6

Deviations of weighted average frequency (WAF) associated with the conductivity disturbance during a sudden ionospheric disturbance. Three curves in the vicinity of 8 Hz by dashed lines with dots, diamonds, and asterisks are relevant to the peak frequency f₁ and the weighted frequencies of the second F₂ and the third F₃ resonance modes, respectively. The curve marked by the four-ray stars is the variation of the WAF f_p(B), and the straight line is the linear fitting of this curve as a calibration.

When processing experimental records, one applies averaging of the weighted resonance frequencies, thus reducing random fluctuations and revealing a synchronous change hidden in the record. The dependence of WAF f_p(B) is plotted in Figure 6 by the curve marked by the four-ray stars. The following dependence is obtained after a linear fitting of this curve:

(8)

This function is shown in Figure 6 by the straight line, and it can be used as the calibration relationship when processing the observational data.

5 Link of WAF With SID Magnitude and Magnetic Characteristic Height of Conductivity Profile

We derived the calibration equation 8 that might be used in processing of experimental data. An inverse relationship might be readily obtained having the form:

(9)

According to the calibration equation 9 derived from the power spectra of the vertical electric field component in the cavity with the day-night nonuniformity, an increase in the weighted average resonant frequency of 0.1 Hz corresponds to the 2.5-point intensification of the SID magnitude. As might be expected, this estimate (disturbances on the dayside only) is approximately by a factor of two lower than the sensitivity of Schumann resonance frequencies to the modification of entire ionosphere: 2.5 versus 1.5 points.

The model data are collected in Table 4 obtained from the power spectra of Schumann resonance in the nonuniform Earth-ionosphere cavity during SIDs of different magnitudes B for the uniform distribution of thunderstorms and an observer located at the solar terminator line.

Table 4. Peak Frequencies of Power Spectra and Characteristic Heights of the Dayside Ionosphere in Nonuniform Cavity for Sudden Ionospheric Disturbance (SID) of Different Magnitude B

	Peak frequencies, Hz			Weighted frequencies, Hz		WAF, Hz		Magnetic heights hL, km			Median magnetic height, km	Deviations of median magnetic height, km
B	f1	f2	f3	F2	F3	fp	δfp	n = 1	n = 2	n = 3	<hL> day	<δhL> day
1	2	3	4	5	6	7	8	9	10	11	12	13
0	7.73	14.01	20.22	8.09	8.25	8.02	0	99.595	96.845	95.707	97.382	0
1	7.76	14.08	20.33	8.13	8.30	8.06	0.04	97.979	95.035	93.705	95.573	−1.809
2	7.81	14.17	20.46	8.18	8.35	8.11	0.09	96.227	92.918	91.411	93.518	−3.864
3	7.87	14.26	20.59	8.23	8.41	8.17	0.15	94.248	90.656	89.184	91.363	−6.019
4	7.92	14.34	20.68	8.28	8.44	8.21	0.19	92.084	88.600	87.263	89.315	−8.067
5	7.96	14.41	20.77	8.32	8.48	8.25	0.23	90.005	86.850	85.655	87.503	−9.879
6	8	14.47	20.84	8.35	8.51	8.29	0.27	88.210	85.378	84.325	85.971	−11.411
7	8.04	14.52	20.9	8.38	8.53	8.32	0.30	86.680	84.154	83.244	84.693	−12.689
8	8.06	14.56	20.95	8.40	8.55	8.34	0.32	85.370	83.155	82.372	83.632	−13.75
9	8.08	14.58	20.99	8.42	8.57	8.36	0.34	84.262	82.343	81.656	82.753	−14.629
10	8.1	14.62	21.02	8.44	8.58	8.37	0.35	83.345	81.670	81.051	82.022	−15.36

The first column of Table 4 lists the SID magnitude, B. The next three columns contain the peak frequencies f_n of the first, second, and third modes. In distinction from the data presented in Table 2 which is relevant to the dispersion equation Re[ν] = n, Table 4 is based on the peak spectral frequencies. These are the frequency values over which the maximum is observed in the power spectrum of the vertical electric field component. The fifth and sixth columns refer to the weighted peak frequencies of the second F₂ and the third F₃ modes. The WAF f_p = (f₁ + F₂ + F₃)/3 is located in the seventh column. Deviations of WAF δf_p(B) in Hz are shown as a function of the SID magnitude B in the eighth column. Since the peak frequencies of the first three modes of the Schumann resonance deviate from each other, the corresponding characteristic magnetic heights listed in columns 9, 10, and 11 are also slightly different.

We use the average value of characteristic height <h_L> presented in the 12th column of Table 4. The last column in this table indicates deviations from the unperturbed level of the average magnetic altitude <δh_L> caused by an increasing perturbation magnitude. As we have already noted, alterations of characteristic magnetic height are close to that of the profile modification but do not coincide with it. For example, it is postulated in the model that an SID of maximum 10-point magnitude moves the ambient day conductivity profile of quiet ionosphere from the 110-km altitude to 89 km (Δ = 21 km), whereas the corresponding change in magnetic altitude is <δh_L> = 15.36 km. Though this offset is counted from the height of 97.382 km instead of 110 km, the presented model data allowed us to plot the magnetic characteristic height h_L(B) of the conductivity profile as a function of SID magnitude and fit the data by a linear function. The result has the following form:

(10)

Here the median characteristic magnetic height of the dayside ionosphere <h_L> is measured in km, and the magnitude of disturbance in relative B points.

Using data of Table 4, we plotted in Figure 7 deviations of the median magnetic characteristic height <δh_L> at the illuminated side of the planet caused by an SID (column 13 of Table 4) against deviations of the WAF δf_p from the unperturbed values (column 8 of Table 4). The abscissa in Figure 7 shows the WAF deviations in Hz, and the ordinate depicts the disturbance of median height in km. The tabulated calculated data are plotted on the graph by the dots connected by dashed line. The linear fitting of this dependence yields the following equation:

(11)

Figure 7

Changes of the magnetic height δh_L and the weighted average frequency derived for sudden ionospheric disturbances with different magnitudes. The broken line with asterisks refers to the observation, and the straight line is the corresponding calibration fitting.

which is shown in Figure 7 by a straight line.

Formula 11 is convenient when we want to estimate the impact of solar X-ray flare on the characteristic magnetic height of the mesosphere conductivity profile. Obviously, a sharp 0.1-Hz increase of the WAF induced by the SID indicates a similarly sharp 4.3-km decrease in the height of conductivity profile in the vicinity of the magnetic altitude of the ionosphere. The experimental data presented in Figure 3b correspond to 10 to 11-km reduction in the magnetic characteristic height of the daytime ionosphere during the SID on 11 March 2015.

6 Conclusions

We constructed the Schumann resonance model based on a realistic conductivity profile that enables us to estimate modifications of the middle atmosphere conductivity profiles during solar X-ray flares. Solar flare increases the air conductivity at altitudes above 55 km, and the magnitude of SIDs depends on the X-ray flux ranging from 0 (no perturbation) to 10 points (maximum perturbation).

Changes were modeled of characteristic magnetic heights in the ionosphere, the resonance frequencies of the Earth-ionosphere cavity, and the power Schumann resonance spectra of vertical electric field component with the help of the full wave technique and 2DTE. Computations were performed for SIDs with various intensities, and this allowed us to relate the SID magnitude (B) with deviations of both magnetic characteristic height and the peak frequencies of the Schumann resonance modes.

Processing of model data provided the linearized calibration equations appropriate for interpreting the observational data. One equation relates the increase of the WAF with the SID magnitude, and the second one—with the decrease in ionosphere magnetic characteristic height. It is shown that an increase in the WAF of Schumann resonance by 0.1 Hz is associated with the 2.5-point increase of the SID intensity; the same increase in the WAF also indicates the downward displacement of 4.3 km in conductivity profile of the mesosphere around the magnetic characteristic height.

The obtained calibration equations are suitable for processing and interpreting the results of Schumann resonance observational data.