Line splitting in the Schumann resonance oscillations



[1] We discuss detection of line splitting in the global electromagnetic (Schumann) resonances. The lifting of resonance degeneracy is usually not visible in the ordinary power spectrum of either the electric or magnetic field components since splitting is small in comparison with the natural width of the resonance lines. Splitting may be detected by exploiting the spatial structure of the fields and/or the elliptical polarization of the magnetic field. The spatial properties were utilized in synchronous and coherent measurements of the vertical electric field at two longitudinally separated observatories. The results were attributed to line splitting. An alternative interpretation was also advanced that takes into account the source-receiver separation. The lifting of degeneracy also appears as a frequency-dependent elliptical polarization of the horizontal magnetic field vector, which has been found experimentally. We compare measurement and computational data, and their reciprocity proves the detection of Schumann resonance line splitting.

1. Introduction

[2] Global electromagnetic resonances were predicted by Schumann [1952] and are referred to as the Schumann resonances. Radio waves of extremely low frequencies (ELF) ranging from a few Hz to a few tens of Hz originate from global lightning strokes. An electromagnetic pulse radiated by a discharge circles the planet a few times, and resonance peaks appear in the power spectra at frequencies of nominally 8, 14, 20, 26 Hz, etc. Oscillations in a spherically uniform and isotropic Earth-ionosphere cavity are associated with a degeneracy of the eigenvalues: there is a set of different eigenfunctions corresponding to a single eigenvalue or resonance frequency [Wait, 1962, 1965; Bliokh et al., 1968, 1971, 1977, 1980; Sentman, 1995]. The eigenfunctions have different azimuthal indices m∈[−n, +n], where n denotes the resonance mode number, so that there are (2n + 1) different eigenfunctions corresponding to a single resonance frequency, and we speak about a (2n + 1)–fold degeneracy. This frequency degeneracy is similar in many respects to the energy degeneracy that occurs among the electron states of atomic and molecular systems, so the same concepts and language may be employed.

[3] The presence of the geomagnetic field turns the ionospheric plasma into an anisotropic (locally gyrotropic) medium characterized by a tensor dielectric constant. The superposition of a geocentrically centered geomagnetic dipole magnetic field onto an otherwise spherically symmetric scalar conductivity reduces the symmetry to that of a cylinder with symmetry axis parallel to the geomagnetic dipole axis. The presence of the magnetic field violates the equivalence of the eastward and westward propagation (in magnetic coordinates), and the waves corresponding to distinct eigenfunctions acquire different phase speeds. The degeneracy is therefore removed, the resonance frequencies are split, and elliptical polarization appears [Wait, 1962; Large and Wait, 1966, 1967a, 1967b, 1968; Bliokh et al., 1968, 1971, 1977, 1980; Sentman, 1987, 1989, 1995; Nickolaenko and Hayakawa, 2002].

[4] Attempts to detect the line splitting have been undertaken in many experiments. As far as we know, the first systematic investigation utilized the vertical electric field component [Bliokh et al., 1971; Bormotov et al., 1971]. An attempt was undertaken by Tanahashi [1976] to observe the fine structure in the power spectra of horizontal magnetic field using a Z transform analysis method. Polarization of horizontal magnetic field was addressed in the works of Sentman [1987, 1989] and Labendz [1998].

[5] Sentman [1989] reported on the different polarization characteristics of the magnetic component of the Schumann resonances computed separately for the two orthogonal horizontal magnetic axes. It was shown that large Q bursts often exhibited strong elliptical polarization, with a gradual rotation of the major axis of the polarization ellipse occurring over the duration of the event. This behavior is consistent with the presence of eastward and westward traveling waves of different frequencies. As mentioned above, the differences in the frequencies of eastward and westward traveling waves is an expected signature of the normal modes associated with the side multiplets (m = ±1) in the presence of a magnetized ionosphere, which led Sentman [1989] to conclude that the observed behavior was in fact due to such line splitting. While this interpretation appears to have been correct, the observations of only two components of the magnetic field at a single site did not permit unambiguous resolving of the full set of multiplets, three in the case of the lowest n = 1 mode at 8 Hz.

[6] The basic computational difficulty in detecting the line splitting by using the fine structure of Schumann resonance peaks when computed from either the electric field or a single axis of magnetic field is a consequence of the considerable width of the resonance lines. It is about 2 Hz at the 8 Hz basic mode, so that the Q factor is typically equal to 4. Estimates indicate that the highest splitting may reach a few tenths of a Hz [Bliokh et al., 1968, 1977, 1980], which is much smaller than the natural width of the peak; therefore the resonance pattern is barely perturbed by splitting. For this reason, Tanahashi [1976] turned to the Z transform [see Marple, 1987] instead of FFT in an attempt to sharpen the sublevels of the resonance line. The Z transform method tries to compensate the attenuation by applying the basic functions growing in time. However, it is highly susceptible to noise, and tends to create spectral structure that depends on the order of the model, making it difficult to distinguish real peaks from model-dependent artifacts. There exist several additional natural effects that create fine structure unrelated to the removal of degeneracy. First, a scalloped profile is always observed owing to the phase interference arising among modes excited by randomly occurring global lightning strokes, which result in a highly nonstationary spectrum for spectra computed over short timescales. Averaging raw FFT spectra reduces these fluctuations, but these never vanish completely. Additional mechanisms leading to spectral structure are: Alfven ionospheric resonance observed just below the Schumann resonance [Belyaev et al., 1990] and possible impact of magnetospheric ULF signals [Bliokh et al., 1969].

[7] To completely resolve the line splitting problem requires that the number of independent measurements equal or exceed the order of the degeneracy. Since there are only three components in the TM0 fields (two magnetic and one electric), one must therefore sample the fields at different spatial locations to unambiguously resolve the various subfields. The most effective locations for such measurements would be at a combination of locations corresponding to the various peaks and nodes of the underlying eigenfunctions. However, observations at sites separated by distances that are an appreciable fraction of the characteristic spatial scales of the modes should be adequate to obtain unambiguous results, and form the basis of the present work.

2. Coordinates and Fields in Uniform and Nonuniform Cavities

[8] Studies of the Schumann resonances often employ a model of the Earth-ionosphere cavity that is isotropic and spherically symmetric. The spherical coordinate system {r, θ, ϕ} is used with the origin at the Earth's center and the polar axis θ = 0 directed to the source MS(a, 0, 0) (vertical lightning stroke) placed on the surface of the Earth with radius r = a, so that the observer is found at the point MO(a, θH, 0). We can always place the source (or the observer) at the pole of the spherical coordinate system and the observer (or the source) at the zero meridian in a uniform isotropic cavity, as the cavity is spherically symmetric. As a result, the solution becomes a function of two variables: the frequency ω and the source-observer distance θH. Only TEM (TM0) waves propagate at ELF [Wait, 1962; Galejs, 1972] so there are only a vertical (radial) electric field Er and a single component of the horizontal magnetic field component Hϕ in the case of a perfectly conducting ground. The field Hϕ is linearly polarized perpendicular to the great circle arc connecting the source and the receiver.

[9] When the Earth-ionosphere cavity is not spherically symmetric, the fields are not simply expressible in terms of the linearly polarized modes of the spherical cavity, and account must be taken of the effects of the asymmetries on the normal modes of the system. For a system in which the only departure from spherical symmetry is a magnetized ionosphere from a centered magnetic dipole, the symmetry of the system is with respect to the dipole axis. It is convenient to use geographic coordinates with the polar axis θ = 0 directed to the North Pole. A particular lightning discharge (elementary source) and the observer occupy positions MS(a, θS, ϕS) and MO(a, θO, ϕO) correspondingly. Thunderstorms concentrate in the tropics over the continents, with maximum of activity occurring at local time close to 1600–1700 UT, so that sources move around the globe during the day, and source distance and the wave arrival angle varies correspondingly. A single vertical electric and two crossed horizontal magnetic antennas are the standard complement of detectors used in Schumann resonance studies. Usually, magnetic antennas are oriented along local parallel and meridian, and field components HX = HEW and HY = HNS are projections of the full horizontal magnetic field vector. Temporal variations of HX and HY fields occur “in phase” in an isotropic and uniform cavity, and the tip of horizontal vector Hf draws a straight line: we speak of linear magnetic polarization in this case. The term “polarization” has been used with regard to TEM polarized electromagnetic waves [Wait, 1962; Galejs, 1972]. However, in our description of the polarization of the horizontal magnetic components of the Schumann resonances we adopt the standard conventions used in ULF research [Fowler et al., 1967], which are in turn based on the Stokes parameterization of Born and Wolf [1964].

[10] In the presence of geomagnetic field, the θ = 0 axis must be directed along the line of symmetry, i.e., toward the magnetic pole. We ignore the tilt of the geomagnetic dipole and its eccentricity, so the point θ = 0 is the North Pole, and again we return to the ordinary geographic coordinates. Three nonzero fields Er, Hθ, and Hϕ exist at the perfectly conducting ground in an anisotropic cavity [Wait, 1962; Large and Wait, 1966, 1967a, 1967b, 1968; Galejs, 1972; Bliokh et al., 1968, 1977, 1980; Sentman, 1995; Nickolaenko and Hayakawa, 2002]. The magnetic field components Hθ = −HY (directed along the meridian) and Hϕ = −HX (along the parallel) become now the elements of the solution rather than simple projections of a horizontal vector. The lifting of degeneracy, or splitting of the resonance frequencies, causes a frequency-dependent phase shift between these fields, and elliptical polarization must appear [Sentman, 1987]. Thus the presence of elliptical polarization may indicate line splitting, even though the power spectrum may not reveal visible fine structure corresponding to individual modes.

[11] The experimental study by Sentman [1987] found strong elliptical polarization, especially in the vicinity of the day-night terminators, which appears not to be associated with removal of degeneracy but is rather due to a perturbation in the local fields due to effects of the altitude discontinuity. However, the observations of Sentman [1989] related to polarization characteristic of individual Q bursts were interpreted to be in terms of the removal of degeneracy, as discussed in detail in that report. We discuss below the line splitting at the first mode where the triplet of eigenvalues corresponds to three eigenfunctions, each having different azimuthal numbers: m = −1, m = 0, or m = +1 [Bliokh et al., 1968, 1980; Sentman, 1987, 1989; Nickolaenko and Hayakawa, 2002]. For the lowest n = 1 mode near 8 Hz, the central frequency of the triplet corresponds to the standing wave (m = 0), which is linearly polarized. The two other solutions are related to waves traveling to the west and to the east (m = ±1), and the waves are elliptically polarized with an ellipticity that depends on the angular distance from the source [Sentman, 1989]. A small frequency change around the first Schumann resonance significantly modifies the character of propagation: the standing wave transforms into traveling waves, and the linear polarization turns into the elliptical one (see below).

[12] It should be noted that the sense of elliptical polarization (left or right handed) is with respect to the radius vector, or local vertical, at the point of measurement and is a physical property of the wave. The assignment of which particular azimuthal mode index corresponds to left- and right-handed polarization, respectively, and the corresponding sign of the ellipticity, depends on the sign convention adopted for the harmonic behavior. With an assumed harmonic convention of e+iwt, right hand polarization corresponds to positive ellipticity.

3. Physics of Line Splitting

[13] We only outline the consequences of ionosphere anisotropy since the problem has been addressed in many works [Wait, 1962; Large and Wait, 1966, 1967a, 1967b, 1968; Galejs, 1972; Bliokh et al., 1968, 1969, 1977, 1980; Nickolaenko and Rabinowicz, 1974, 2003; Rabinowicz, 1979; Sentman, 1987; Nickolaenko et al., 2002; Nickolaenko and Hayakawa, 2002]. By applying the method of moments (MOM), one obtains a linear system of algebraic equations for the field expansion coefficients αnm and βnm [see, e.g., Large and Wait, 1967a, 1967b, 1968; Galejs, 1972; Bliokh et al., 1968, 1969, 1980; Nickolaenko and Hayakawa, 2002; Wait, 1965]. Expressions for m = 0, m = 1, and m = −1 eigenfunctions are listed in Table 1. The source of the electromagnetic field of angular frequency ω is a vertical electric dipole placed at MS(a, θS, ϕS). The Earth is assumed to be a perfectly conducting sphere, and the tensor boundary conditions are formulated at the lower ionosphere. The “hedgehog” model of the geomagnetic field was applied in computations: the constant magnetic field is directed along the radius, and its direction abruptly changes when crossing the equator. The magnetization is therefore antisymmetric between the northern and southern hemispheres, and it is further assumed there is no additional day-night asymmetry in the system [e.g., Bliokh et al., 1977, 1980; Nickolaenko and Hayakawa, 2002; Nickolaenko et al., 2002; Nickolaenko and Rabinowicz, 2003]. The impact of anisotropy is specified by parameter ρ = ωr/νe where ωr is the electron gyrofrequency and νe is the effective electron collision frequency.

Table 1. Particular Eigenfunctions of the First Schumann Resonance Mode n = 1
Field ComponentAzimuthal IndexEigenfunction/Resonance Characteristic
ErStanding wave m = 0equation image
Westward traveling wave m = 1equation image
Wave traveling to the east m = −1equation image
Hequation imageStanding wave m = 0equation image
Westward traveling wave m = 1equation image
Wave traveling to the east m = −1equation image
Hequation imageStanding wave m = 00
Westward traveling wave m = 1equation image
Wave traveling to the east m = −1equation image
Rnmm = 0equation image
m = 1equation image
m = −1equation image
R1nmm = 0equation image
m = 1equation image
m = −1equation image

[14] Expressions in Table 1 allow us to interpret the experimental and computational results. The first column here lists the field components and relevant elements of the frequency characteristics. The second column contains the azimuthal index m. The third column presents the contracted expressions for individual field components and their frequency characteristics. Since the first resonance mode n = 1 splits into three submodes, we list the quantities m = 0, m = 1, and m = −1.

[15] It is easy to see from Table 1 that frequency characteristics of different sublevels (separate indices m) ∣Rnm1 (w)/Rnm (w)∣ and ∣1/Rnm (w)∣ reach their maxima at various frequencies. Diagrams in Figure 1 illustrate expected effects of the line splitting. The top diagram shows the angular field distributions for three sublevels of the first Schumann resonance mode, and broadly correspond to the polarization characteristics for the various subfields presented by Sentman [1989]. The lowest plot schematically depicts the idealized amplitude characteristics of split individual sublevels. The second (from the top) plot shows the relative phase shift of the vertical electric field component when observed at two longitudinally separated observatories (see below). The third plot depicts expected polarization of the field, polarization arising from the line splitting.

Figure 1.

Speculative models of impact of the anisotropic ionosphere plasma on the Schumann resonance field.

[16] Consider higher frequencies where the eigenfunction dominates with n = 1 and m = 1. This wave traveling from east to west has the highest amplitude. In contrast, the eigenfunction with n = 1 and m = −1 travels from west to east and dominates at the lower frequencies. The phenomenon, being essentially a Zeeman effect, is caused by the outer geomagnetic field, which forces the plasma electrons to rotate around the field lines with the gyrofrequency ωr. Waves corresponding to eigenfunctions with m = 1 and m = −1 circle the globe in opposite directions and interact with the ionosphere electrons in different ways. The m = −1 wave, whose electric vector rotates in the same sense as electron gyromotion, experiences greater damping in comparison with the m = 0 standing wave or with the wave traveling in the opposite direction. The geomagnetic field modifies the losses and the phase velocity, these becoming functions of propagation direction, and the splitting of resonance frequencies takes place. The splitting expected is visualized in the bottom part of Figure 1, we have three distinct maxima, and each of them corresponds to a separate index m.

4. Line Splitting in Vertical Electric Field Component

[17] The vertical electric field component was used in the first attempt to detect the line splitting. Natural ELF signals were recorded simultaneously and coherently at two longitudinally separated field sites at Kharkov (50°N, 37°E) and Ulan-Ude (50°N, 105°E) in 1969 [Bliokh et al., 1971; Bormotov et al., 1971]. After restoring the synchronization and coherence, magnetic tape with records was replayed with the tenfold speed, which facilitated the analog spectral analysis. Power spectra of individual signals were obtained in the processing together with the complex cross spectra (amplitude and phase). The equipment and the processing were described by Bliokh et al. [1971, 1977, 1980] and Bormotov et al. [1971].

[18] The driving idea behind the experiment was simple (see Figure 1). The standing wave (m = 0) plays the major role at the central 8 Hz frequency, and the mutual phase shift is equal to zero at 8 Hz when signal is detected at longitudinally separated observatories. The sideband m = +1 dominates at higher frequencies, and the westward traveling wave prevails. Consequently, the phase must lead at the eastern site (Ulan-Ude) relative to the western observatory (Kharkov). Similarly, the sublevel m = −1 determines the field at lower frequencies, the wave travels eastward, so the Ulan-Ude signal is delayed in phase, as indicated in the diagram of Figure 1. The line splitting reveals itself as a regular increase with frequency of the complex cross spectrum with phase arg {〈E1(ω) · E2(ω)*〉} where the angular brackets denote averaging, E1(ω) and E2(ω) are the complex spectra of the signals at Kharkov and Ulan-Ude respectively, and the asterisk denotes complex conjugate.

[19] Figure 2 demonstrates a typical experimental result [Bormotov et al., 1971; Bliokh et al., 1971]. The frequency varying from 6.5 to 9.5 Hz is plotted along the abscissa on linear scale. The phase of cross spectrum is shown in degrees by the points. The solid line depicts the average frequency dependence of mutual phase shift. It was concluded that the line splitting was successfully detected since the experiment showed the expected (see Figure 1) behavior of phase [Bormotov et al., 1971; Bliokh et al., 1971; Nickolaenko and Hayakawa, 2002].

Figure 2.

Frequency dependence of the phase shift measured at the base Kharkov–Ulan-Ude in 1969.

[20] No model computations of the cross spectra were available at that time. Later, the phase variations similar to experiment were obtained in the model of an isotropic and uniform cavity. Computations demonstrated the weakness of the assumption that no traveling wave exists in the isotropic and uniform cavity. Such wave is always present in a cavity with losses. Indeed, the direct and antipodal waves arrive to an arbitrary observer, which have different amplitudes owing to different attenuation along propagation paths. There appears “uncompensated remnant”: the wave propagating to the source antipode, since the direct wave (arriving along the shorter path) has the higher amplitude. Residual traveling wave provides a phase shift between two signals at separate positions, the shift sometimes similar to that observed experimentally. By varying the source position, one can obtain different frequency variations of the mutual phase. Therefore, to prove the detection of Schumann resonance splitting, one has to know the actual locations of the sources. Experimental results of Figure 2, so similar to expectations, are not an explicit proof of the line splitting. We must note that necessary measurements are feasible nowadays with the source position found for the synchronous and coherent records of a Q burst at longitudinally separated observatories.

[21] It is worth noting here that results obtained in the gyrotropic cavity model [Nickolaenko and Rabinowicz, 1974; Bezrodny et al., 1977; Bliokh et al., 1977, 1980; Rabinowicz, 1979; Nickolaenko and Hayakawa, 2002] showed the prevalence of the single wave traveling from west to east. The m = −1 sideband has the highest amplitude, and the wave exp[i(ωt − ϕ)] determines the field behavior almost everywhere in the cavity when frequency is in the vicinity of the first Schumann resonance mode.

5. Expected Line Splitting in the Polarization of Horizontal Magnetic Field

[22] Detection of line splitting in the polarization of horizontal magnetic field was addressed in the works by Sentman [1987, 1989] and Labendz [1998]. Preliminary considerations are based on the ±90° phase shift between the Hϕ and Hθ fields, which has to appear at the first Schumann resonance frequency when line splitting takes place (see Table 1). As a result, the complete horizontal magnetic vector rotates in opposite directions when we vary frequency around the first resonance peak. The vector tip draws a circle when the amplitudes of two orthogonal field components are equal and the phase difference is 90°. The trajectory is a straight line when the phase shift is zero. When the phase shift is not zero, the complete horizontal vector outlines an ellipse.

[23] Ellipticity appears owing to the factor im present in the Hθ field (cf. Table 1). As splitting separates resonance characteristics of different m, the polarization depends on frequency. Let the eigenfunction with n = 1 and m = −1 have the highest amplitude at frequencies below 8 Hz (see Figure 1, bottom plot). In this case the component HX = Hϕ leads in phase relative to HY = −Hθ, and the complete horizontal H vector rotates counterclockwise (right-hand polarization (RHP)). Rotation becomes clockwise at frequencies above 8 Hz, where the subfunction n = 1, m = 1 is of the highest amplitude and the mutual phase shift changes its sign. Polarization becomes linear at intermediate frequencies where contributions from m = −1 and m = 1 waves are equal, or where the subfunction n = 1, m = 0 dominates. These arguments lead to the idea that a line splitting may be found from frequency variations of the polarization sense around the first Schumann resonance frequency [Sentman, 1989]. Elementary considerations predict rapid frequency variations of the polarization sense from positive to negative values shown in the third plot of Figure 1.

[24] Experiments have confirmed that elliptical polarization depends on frequency [Sentman, 1987, 1989; Labendz, 1998]. The report by Sentman [1987] introduced the ellipticity spectrum as a tool for studying Schumann resonance polarizations. The elliptical polarization spectrum was computed from observations obtained at Table Mountain, California. Measurements were made in the frequency band from 3 to 60 Hz. The focus in that report was on diurnal properties of the ellipticity, and it was shown that elliptical polarization reached its maximum after local sunset or local sunrise (see Figure 3). It was positive (RHP) several hours after local sunrise reaching the value of 0.7 near the first Schumann resonance. The ellipticity decreased with frequency reaching negative values (LHP) at frequencies between the resonance peaks. It became maximum negative near the 8 Hz resonance immediately after local sunset (0400–0500 UT) reaching the −0.8 level. The limited dynamic range used to plot this data (reproduced here in Figure 3) did not permit showing the expected rapid variations of polarization sense in going across the resonance. However, unpublished ellipticity spectra plotted with greater dynamic range than presented in Sentman's [1987] report, as well as data subsequently obtained in California and in Alaska, quite clearly show the expected switch in polarization in going across the eigenfrequency exactly as described here. In his second work, Sentman [1989] presented observations of the time domain signature of the polarization of discrete ELF events (Q bursts). Transient pulses from the rare powerful strokes also demonstrated RHP and LHP over different frequency bands. No fast frequency variations in the polarization were evident, but this is to be expected because of the limited frequency resolution of the short-length events (<1 s).

Figure 3.

Polarization of horizontal magnetic field in the time-frequency domain measured in 1985 [from Sentman, 1987]. Horizontal bars denote local night condition. The sign of polarization tends to remain unchanged around the first Schumann resonance mode in contrast to expectations shown in Figure 1.

[25] The experiment by Labendz [1998] was carried out in Germany (51°N and 9.5°E) in the frequency band from 0.1 to 20 Hz. These results were in general correspondence to works by Sentman [1987, 1989]. The ellipticity was a function of time of day. Negative sense was detected all through the day with a short period of positive sense during the late evening. Special processing was made to separate components of different polarization. The following split frequencies were announced: f(m = −1) = 7.82 Hz, f(m = 0) = 8.05 Hz, and f(m = 1) = 7.98 Hz. The “central” frequency 8.05 Hz corresponded to the most frequent linearly polarized events. The sidebands were connected with the LHP and the RHP respectively. Unfortunately, we cannot consider these results as robust: the signal fragments processed were extremely short, about a few tenths of a second. Such brief parts cannot provide the frequency resolution of 10−2 Hz mentioned in the paper. One might suspect that results of signal processing were volatile, and the superresolution was an artifact.

[26] It was concluded that, while the magnetic elliptical polarization detected in Q bursts strongly suggested the presence of line splitting [Sentman, 1989], fully resolved lines would require spatially separated measurements analyzed according to the expected spectral pattern shown in Figure 1. The computational results obtained later [Nickolaenko and Hayakawa, 2002; Nickolaenko et al., 2002, 2004] confirmed an absence of fast frequency variations in the polarization sense. Comparison of the model data with measurements indicates that the line splitting was really detected in measurements, but its pattern does not look like that one following from the approximate preliminary considerations.

6. What Does the Line Splitting Look Like?

[27] The polarization of a magnetic field is formally described in a manner similar to that of conventional optical polarization, i.e., by introducing the coherence matrix [e.g., Born and Wolf, 1964; Fowler et al., 1967], but instead of electric field, the magnetic component is used. We omit details of obtaining the polarization, which are described by Sentman [1987, 1989], Nickolaenko et al. [2002, 2004], and Nickolaenko and Hayakawa [2002]. Our model exploits the “hedgehog” geomagnetic field and an equatorial point vertical electric dipole source placed at 1700 local time [Nickolaenko et al., 2002, 2004]. For the n = 1 mode near 8 Hz, such a source excites only the waves with m = ±1. Absence of standing wave m = 0 guarantees the greatest possible separation of individual subpeaks in the power spectrum. The source current moment is a constant. Computations covered a few Schumann resonance modes.

[28] Figure 4 shows the results of model computations. The abscissa depicts the frequency in Hz, and the ordinate shows the characteristics of individual submodes on a logarithmic scale. Marked thin lines depict the frequency response of individual submodes ∣R1−1 (f)∣−2, ∣R10 (f)}∣−2, and ∣R11 (f)∣−2. The peak of the R1−1 (f)}∣−2 eigenfunction is the highest one. We must note that the highest level (and the lowest attenuation) of the curve m = −1 corresponds to the lowest resonance frequency. This property follows from the way the nondiagonal tensor component of the surface impedance enters the resonance characteristics in Table 1. As a result, the smallest imaginary part in the eigenvalue is combined with its highest real part, which is impossible in an isotropic cavity.

Figure 4.

Model amplitude spectra of individual sublevels and of the field component Hϕ in the vicinity of the first Schumann resonance in the hedgehog geomagnetic field model.

[29] The thick black line in Figure 4 shows the resulting power spectrum of the complete horizontal east-west magnetic field. We show the spectrum for the closest distance between the source and observer when the propagation path coincides with the meridian. Despite the largest separation of individual submodes, no fine structure appears in the resulting resonance curve as the natural line width substantially exceeds the splitting. The amplitudes of individual frequency characteristics are completely different, and a “competition” between the sublevels is impossible, which contradicts expectations based on elementary considerations. The polarization pattern depicted in Figure 1 therefore appears to be unrealistic. A single submode dominates around the resonance peak, and this mode determines an invariable polarization sense.

[30] Figure 5 shows the ellipticity spectra computed for the parameter of gyrotropy ρ = 0.5. Particular curves correspond to 0400 UT (dots), 1000 UT (the smooth curve), and 1600 UT (stars). In the first case, the observer is close to the source antipode, it is in the source vicinity in the last one, and 1000 UT corresponds to an intermediate distance. The top plot in Figure 5 shows data for the middle latitude observer (50°N and 0°E). The bottom plot corresponds to the tropical position of the site (10°N and 0°E).

Figure 5.

Computed spectra of ellipticity in the “hedgehog” geomagnetic field for the equatorial source at 1700 LT. The observer is located at (top) 50°N and 0°E and (bottom) 10°N and 0°E.

[31] For the observer in the Northern hemisphere, the horizontal magnetic field vector rotates counterclockwise (right-hand polarization) at all frequencies around the Schumann resonance peaks. The reason is that m = −1 eigenfunction amplitude substantially exceeds that of the m = +1 function, as Figure 4 demonstrated. The single traveling wave regime is realized in a major part of the cavity [see Nickolaenko and Rabinowicz, 1974; Bliokh et al., 1977, 1980; Bezrodny et al., 1977; Rabinowicz, 1979], and we observe the RHP polarization. The standing wave regime occurs only in a close vicinity of equatorial source, and here, the sense of polarization may vary with frequency.

[32] The bottom plot in Figure 5 shows the “derivative-shape” pattern appearing at short source-observer distances. This function is closer to elementary considerations demonstrated in Figure 1, as the direction of vector rotation abruptly changes. The pattern appears when traveling in opposite direction waves have comparable amplitudes. The negative polarization (left-hand polarization) appears just above the resonance, provided that the observer is close to the equatorial source.

[33] The stability of the direction of the field rotation obtained in the major part of the Earth-ionosphere cavity contradicts expectations based on elementary analyses that do not take into account the magnetization of the ionosphere. However, when the magnetization is taken into account, the experimental results may be naturally explained in terms of the concepts outlined above. Below, this is shown by using a reciprocity approach.

7. Modern Experimental Data on Polarization

[34] Monitoring of Schumann resonances was carried out at Lehta, Karelia, Kola Peninsula (64°N and 34°E) and at Karymshino, Kamchatka (53°N and 157°E). The details of experimental setup and data acquisition were described by Belyaev et al. [1999] and Nickolaenko et al. [2002, 2004].

[35] The sample records from these observatories are shown in Figure 6. The date (UT) is shown on the horizontal axes, and the frequency is plotted along ordinate ranging from 4 to 12 Hz. The top contour map presents the Lehta record, it covers the interval from 18 to 24 September 2000. The bottom frame depicts the ellipticity at Karymshino in the interval from 7 to 14 July 2000. The outstanding stability and similarity is obvious between the data recorded at distant observatories and at different times. The polarization sense at the sites is of opposite sign owing to different antenna calibration. At Lehta, the geographic coordinate system is used. At Karymshino, the geomagnetic coordinates are applied with the z axis directed downward. Therefore the same rotating horizontal vector acquired the opposite sense of polarization. Thus modern experimental records at middle latitudes confirm the prediction of theory: the polarization persists at the first Schumann resonance frequency during the day. This fact implies that a single sideband dominates around the peak frequency.

Figure 6.

Similarity of experimental dynamic spectra of polarization at the first Schumann resonance. Shown are the records performed at observatories Lehta (18–24 September 2000) and Karymshino (7–14 July 2000), each 1 week long.

[36] A close correspondence to experimental results is clear when data are depicted in detail, see Figure 7. We use a wider frequency band 4–24 Hz here. The model data are compared with experiment at Lehta during 48 hours of continuous record from 18 to 20 September 2000. The top frame presents the raw dynamic spectrum based on the hourly averaged ellipticity spectra. High-frequency resolution (∼0.1 Hz) is conditioned by the duration of elementary data sample in the FFT procedure (12 s) that computes the complex spectra of individual field components. The resolution seems to be excessive, as the image is fragmented along the ordinate so that general structure of the spectrum is lost in details. The regular patterns emerge when we smooth the spectra and reduce the frequency resolution to 0.5 Hz (the middle frame). Characteristic horizontal dark strips correspond to the positive polarization (RHP) of the field. The trek is permanent around the first Schumann resonance mode and tends to be present around the second mode. Simultaneously, the crescent elements of negative ellipticity are found over the intermediate frequencies relevant to short source-observer distances, as it was predicted by model. The bottom plot depicts the results of computations. An optimist will find some more coincident details in the smoothed experimental and the model dynamic spectra of Figure 7, especially around the first and the second mode frequencies.

Figure 7.

Detailed comparison of dynamic spectra of ellipticity at Lehta: experiment and the model.

[37] The simplest global thunderstorm distribution was used in computations: a point equatorial source of constant intensity moving around the globe. However, correspondence among computations and the experiment is high. Reciprocity is probably explained by the independence of polarization of the signal amplitude. Ellipticity rather depends on the phase relations in the cavity, and these are connected with the ionosphere properties.

8. Conclusion

[38] We have demonstrated how the difference in spatial properties of individual eigenfunctions facilitates the detection of the Schumann resonance line splitting. Standing waves correspond to the central eigenvalues, while traveling waves correspond to the eigenfrequencies of the side multiplets. Splitting separates individual peaks in the spectral line and causes noticeable modifications of the field distribution, which depends on the frequency. Consequently, the line splitting not unambiguously detectable in single-station observations may be discerned by exploiting spatial characteristics of the fields, which differ according to mode.

[39] The lifting of resonance degeneracy has been sought using two kinds of experiments. Measurements of the vertical electric field were made of the mutual phase shift at two longitudinally separated points, and the expected frequency dependence was found in the phase shift. It was later found that there is a source-receiver distance dependence in the phase, which needs to be taken into account. Measurements of polarizations of the horizontal magnetic field vector confirmed that this occurs, and exhibit the expected elliptical polarization. However, its frequency dependence was different from that following from elementary considerations. Modeling of the Earth-ionosphere cavity with anisotropic upper boundary showed that waves travel from the west to east in the major part of the cavity for frequencies in the vicinity of resonance peaks. The computed polarization taking into account the source-observer geometry behaves similarly to that observed experimentally. Thus the Schumann resonance line splitting of the lowest, n = 1 mode near 8 Hz has been resolved. A preliminary estimate 0.5 < ρ < 1 is valid for the effective parameter of gyrotropy, and further detailed research may permit an improved value of this parameter.


[40] We thank M. Füllekrug and A. Shvets, the reviewers of this paper, for their attention and helpful comments.