In the case of spherical systems, assuming no dependence on the ϕ coordinate, the system of six Maxwell equations in 3D spherical coordinates (r, θ, ϕ) can be reduced to 2D spherical systems (r, θ) with three equations for Er, Eθ, and Hϕ field components (Holland, 1983; Inan and Marshall, 2011): 1ε0∂Er∂t+σEr+Jr=1rsin⁡θ∂∂θHϕsin⁡θ,2ε0∂Eθ∂t+σEθ=-1r∂∂rrHϕ,3μ0∂Hϕ∂t=-1r∂∂rrEθ-∂Er∂θ, where σ=σ(r) is the conductivity profile of Earth–ionosphere cavity, and ε0 and μ0 are the vacuum permittivity and permeability, respectively.

These equations were discretized using central-difference approximations to the space and time partial derivatives. The resulting finite-difference equations are solved in a leapfrog manner: the electric field vector components in a volume of space are solved at a given instant in time; then the magnetic field vector components in the same spatial volume are solved at the next instant in time and the process is repeated over and over again until the desired transient or steady-state electromagnetic field behavior is fully evolved (Inan and Marshall, 2011). So the update equations for Er, Eθ, and Hϕ are the following (Holland, 1983): 4Er|i,jn+1/2=ε0/Δt-σi,j/2ε0/Δt+σi,j/2Er|i,jn-1/2+1ε0/Δt+σi,j/2-Jr|i,jn+1ri+1/2sin⁡θj⋅sin⁡θj+1/2Hϕ|i,jn-sin⁡θj-1/2Hϕ|i,j-1nΔθ,5Eθ|i,jn+1/2=ε0/Δt-σi,j/2ε0/Δt+σi,j/2Eθ|i,jn-1/2+1ε0/Δt+σi,j/2ri⋅ri+1/2Hϕ|i,jn-ri-1/2Hϕ|i-1,jnΔr,6Hϕ|i,jn+1=Hϕ|i,jn-Δtμ0ri+1/2⋅ri+1Eθ|i+1,jn+1/2-riEθ|i,jn+1/2Δr-Er|i,j+1n+1/2-Er|i,jn+1/2Δθ, where Δt is the time step, Δr and Δθ are the sizes of grid cell in r and θ coordinates, respectively, ri=R0+iΔr, ri±1/2=R+(i±1/2)Δr, θj=jΔθ, θj±1/2=(j±1/2)Δθ, and R0 is the mean of Earth's radius. Superscript n signifies that the quantities are to be evaluated at time t=nΔt, and i,j represent the point (r=iΔr, θ=jΔθ) in the spherical grid. The half time steps indicate that the electric and magnetic fields are calculated alternately.

The updated equation for Er cannot be applied for poles (where θ=0 and θ=π) because of sin⁡θ in the denominator. To solve this singularity problem the Holland's approach was used (Holland, 1983). Namely, for the poles the integral form of the curl equation with a small contour around the poles was applied, which leads to updated equations for poles without singularities. After that, the updated equation for θ=0∘ takes the following form (Holland, 1983): 7Er|i,jn+1/2=ε0/Δt-σi,j/2ε0/Δt+σi,j/2Er|i,jn-1/2+1ε0/Δt+σi,j/2⋅-Jr|i,jn+sin⁡θ1/2Hϕ|i,0nri+1/2(1-cos⁡θ1/2). And for θ=180∘: 8Er|i,jn+1/2=ε0/Δt-σi,j/2ε0/Δt+σi,j/2Er|i,jn-0.5+1ε0/Δt+σi,j/2⋅-Jr|i,jn-sin⁡θNθ-1/2Hϕ|i,Nθ-1nri+1/2(1+cos⁡θNθ-1/2), where θ1/2=Δθ/2, θNθ-1/2=(Nθ-1/2)Δθ, and Nθ is the number of grid cells in the θ direction.

The size of grid cell (in the r and θ directions) is defined by the desired range of frequencies, which are going to be analyzed. For the present analysis we used Δr=1 km and Δθ=0.1∘, which lets us analyze frequencies up to 1 kHz. The ground was modeled as a perfect electric conductor. Also, the upper boundary Rmax⁡ for the model is a perfect conductor, but it was placed at high enough altitude to make sure that there would be no reflection of the waves from this boundary.

The total simulation time is defined by the desired frequency resolution of FFT (Δf=1/tmax⁡ Hz). The time step is defined by the stability of Courant's criterion Δt≤1/c(Δr)-2+(aΔθ)-2, where c is the speed of light and a is Earth's radius (Inan and Marshall, 2011). In order to get sufficient frequency resolution (in our case it is 0.01 Hz, to be able to study the differences between different models), we have conducted the FDTD simulation up to 100 s.

In order to analyze the propagation of electromagnetic waves originating from lightning discharges, one has to use the source model with similar characteristics to real lightning discharge. We used the time profile of lightning discharge proposed in Kulak et al. (2010) and Rakov (2007), which has a flat spectrum in a wide frequency range. But taking into account the restrictions connected with the FDTD computational grid, we had to modify the source: the highest frequency (or smallest wavelength) is defined by the cell size, and according to the generally accepted rule the smallest wavelength should be at least 10–20 times larger than the cell size. Therefore, the source has to be filtered with a loss-pass filter in order to remove frequency components that do not fit the size of the computational cell.

Also, it should be noted that if the source contains the direct current (DC) in its spectrum, it can introduce artifacts, which are not physical (Li et al., 2013). Therefore, additional filtering has to be used for the lowest frequencies by a high-pass filter. For such filtering we have used a fifth-order Butterworth low-pass filter with a cutoff frequency of 300 Hz and a third-order Butterworth high-pass filter with a cutoff frequency of 2 Hz. The final spectrum of this modified source is almost flat in the range 5–100 Hz but decreases rapidly for f<3 and f>1000 Hz.

We placed the source in the FDTD grid at the pole (θ=0∘) in radial direction in nodes with i = 0–6, which corresponds to the source length L=6 km. Also, for the implementation of the source into the FDTD grid we took into account the lightning stroke speed, and for our analysis we used v=108 m s-1 (Rakov, 2007). We assumed a cloud-to-ground (CG) discharge and implemented it according to the time delay between adjacent nodes in the source.

The first model for testing was a lossless spherical waveguide with perfect electric conductors at the ground and the upper boundary. The distance between the conductors was set to h=74 km. The precise theoretical resonance frequencies for such a waveguide for a given height are presented by the equation fn=(c/2πa)n(n+1)1-h/a, where c is the speed of light, a is Earth's radius, and n is the mode of resonance frequency (Bliokh et al., 1977). A comparison of analytical and numerical results for such a model is presented in Table 1. The results of the FDTD analysis for this model are shown at Fig. 1.

A more complicated model for testing had a perfect conductor at the ground and a constant conductivity of 4×10-6 S m-1 above 70 km. The analytical solutions for this model were obtained by Kulak and Mlynarczyk (2013). A comparison of analytical and numerical results for such a model is presented in Table 2. The results of the FDTD analysis for this model are shown at Fig. 1.

The third model for testing was a two-layered model with a perfect conductor at the ground, a constant conductivity of 5×10-7 S m-1 at altitudes between 70 and 110 km, and a constant conductivity of 5×10-5 S m-1 above 110 km. The analytical results for such a model can be found in Kulak et al. (2013), which deals with multi-layer waveguides. A comparison of analytical and numerical results for this model is presented in Table 3. The results of the FDTD analysis for this model are shown at Fig. 1.

To be able to compare the numerical results with the analytical solutions, we have extracted the complex characteristic electric and magnetic altitudes from the FDTD model. The corresponding altitudes can be extracted using their definition (Mushtak and Williams, 2002; Kirillov, 1993): 9h‾e(f)=1E‾r0∑i=0NE‾riΔr,10h‾m(f)=1H‾ϕ0∑i=0NH‾ϕiΔr, where N is the number of radial nodes, E‾ri and H‾ϕi are the complex field values at the radial node i for a given frequency, and E‾r0, H‾ϕ0 are the values of these fields at the ground level.

The complex electric altitude can also be expressed using the conductivity profile by the analytical equation in a normalized form (Mushtak and Williams, 2002): 11h‾e=∫0∞dr1-iσe, where σe=σ/σe∗ and the characteristic conductivity value for electric altitude is σe∗=ωε0, and ω=2πf.

Assuming a similar dependence for complex magnetic altitude and taking into account that the characteristic conductivity value for magnetic altitude is (Greifinger and Greifinger, 1978) σm∗=14μ0ωζ2(r), where the scale height ζ(r)=σ(r)/(dσ/dr), we can write a similar equation for the complex magnetic altitude: 12h‾m=∫0∞dr1+iσm, where σm=σ/σm∗. We compared the characteristic electric and magnetic altitudes calculated analytically from Eqs. (11) and (12) with the FDTD results calculated from Eqs. (9) and (10). The results are presented in Fig. 4.

We calculated the phase velocity Vph and the attenuation rate α using two different methods: numerical and analytical. Analytically the propagation parameters can be calculated using the conductivity profile and electric and magnetic altitudes given by Eqs. (11) and (12), through the following relationship (Kulak and Mlynarczyk, 2011): 13Vph=cReS‾,14α=ωcImS‾, where S‾2=h‾m/h‾e.

In the case of numerical calculations those parameters were obtained by two approaches: (a) similarly to analytical as described above but using electric and magnetic altitudes from FDTD model by Eqs. (9) and (10); and (b) directly from the spectra of electric and magnetic field components. Assuming that in our coordinate system the wave propagates in the θ direction, the following relationship can be written in the frequency domain: 15H‾ϕ1H‾ϕ2sin⁡(ρ1/a)sin⁡(ρ2/a)=exp⁡(γ‾(ρ2-ρ1)), where the ratio of magnetic field complex amplitudes H‾ϕ(f) are calculated for two probes “1” and “2” which are located at distances ρ1 and ρ2 from the source, respectively. In this equation the propagation constant γ‾=α+iβ, where α is the attenuation rate (in units Np m-1) and β is the phase constant, so the phase velocity is Vph=ω/β. For further convenient usage we convert the units of attenuation rate to dB Mm-1. A similar relationship to Eq. (15) but for electric field components E‾r(f) can be written as well.

We should note that Eq. (15) for E‾r and H‾ϕ spectra calculated in a closed cavity gives non-monotonic behavior of propagation parameters. This is caused by superposition of wave attenuation along meridian and different amplitudes of the Schumann resonances for different source–observer distances. One possible solution for removing the influence of Schumann resonances is to implement a perfectly matched layer (PML) along the θ direction. However, we solved this problem in a different way, transforming the Maxwell equations at angle θ>90∘ from spherical coordinates to plane equations, changing the behavior of equations from “closed” to “open”. In this case the waves are unable to propagate around Earth and therefore the Schumann resonance does not occur. The calculated propagation parameters from all the methods listed above are presented in Fig. 5.

As an additional validation we applied the spectral decomposition technique to Schumann resonance power spectra. Figure 6 presents the resonance frequencies obtained from the spectra measured at different distance from the source, and the decomposed frequencies that do not depend on the source–observer distance. Following the decomposition algorithm (Kulak et al., 2006; Dyrda et al., 2014), the spectrum is approximated by the function 16W(f)=s+zfm+∑k=13pk1+ekf-fkf-fk2+gk/22, where W(f) is the signal power spectrum, s is the white noise component, z/fm is the color noise term, pk is the power parameter of the kth resonance peak, ek is the peak asymmetry parameter, fk is the resonance frequency, and gk is the resonant mode half-width parameter. Fitting this function to the FDTD spectra allows us to extract the resonance frequencies fk of the cavity, which are not equal to Schumann resonance frequencies. The Schumann resonance frequencies obtained from the spectra depend on the distance from the source and represent the superposition of standing and traveling waves (Kulak et al., 2006). To analyze the standing waves separately and reveal the properties of the resonant cavity, we use the spectral decomposition method described in Kulak et al. (2006) and Dyrda et al. (2015). After applying this method, the resonance frequencies become independent of the source–observer distance (see details in Kulak et al., 2006 and Dyrda et al., 2015). The decomposition method shows that the solutions for the electric field are symmetric at θ=180∘ because the traveling waves cancel each other out and only the standing waves remain. Therefore, the resonant peaks measured from the spectrum at θ=180∘ for the electric field component (Er180) represent the resonance frequency of the cavity, and they are in agreement with the decomposed frequencies.

The resonance frequencies of the Earth–ionosphere cavity for a given conductivity profile can be calculated by different approaches, and the consistency between them can be considered as an additional validation of the model. We have considered the followings ways to calculate the resonant frequencies:

Analytically using the conductivity profile. The resonance frequencies in this approach are calculated by solving the following equation (Mushtak and Williams, 2002; Galejs, 1972):17fcn=c2πan(n+1)ReS‾(fcn)|S‾(fcn)|,where S‾2=h‾m/h‾e depending on the characteristic complex altitudes, which are discussed in Sect. 4.1;