A numerical modeling of Earth's atmosphere is carried out by means of the Transmission Line Matrix (TLM) numerical method with the aim of calculating the Schumann resonance frequencies. The numerical results obtained are very close to the experimental ones and those obtained with the widely accepted two-scale-height ionospheric model, which allows us to affirm that this is a valid numerical tool for predicting the Schumann frequencies in the atmospheres of other planets and moons. The great flexibility of the TLM numerical method also allows the study of slight shifts in the Schumann resonance frequencies due to an increase in electrical conductivity at the Earth's poles, originating from high-energy particle precipitation emitted from the Sun in conjunction with solar flares. A slight increase in the first Schumann resonance frequency is observed during these events, which is associated with a reduction in the dimensions of the electromagnetic cavity.
 The Earth and other celestial bodies with electric conductivity starting from low values near the surface and increasing with height, form an electromagnetic cavity for waves propagating in the extremely low frequency (ELF) range between the surface and the ionized layer. If the atmosphere between both conductive layers is not excessively dissipative, global lightning activity excites the resonant frequencies for the cavity, also called Schumann's resonances. These resonance frequencies were predicted by Schumann  and detected by Balser and Wagner . Nowadays, there is renewed interest for the phenomenon of Schumann resonances mainly for two fundamental reasons: first, a clear correlation between annual variations of the first Schumann resonance intensity and the tropical temperature over a 6-year span, which suggests the possibility of using these frequencies as a global thermometer for the tropics of the Earth [Williams, 1992]. Second, to explore terrestrial-like electrical activity in other celestial bodies such as Mars [Schwingenschuh et al., 2001] and the Saturnian moon, Titan [Fulchignoni et al., 1997].
 Schumann resonance frequencies are lower than expected for an ideal cavity, i.e., two concentric layers of infinite conductivity filled by a lossless dielectric medium [Jackson, 1999], since Joule dissipation associated with the finite conductivity of the atmosphere makes the real system resonate at lower frequencies than a lossless system with an almost perfectly dielectric atmosphere. Therefore in order to obtain results close to experimental values, this loss of ideality in the dielectric behavior must be accounted for. As a consequence of considering ground and atmospheric conductivity, the exact analytical study of the electromagnetic system becomes unfeasible. Thus the use of strong mathematical approximations or the numerical analysis of the system is required.
 In the last few decades, a growing interest in numerical analysis and the design of complex systems has emerged, since it offers greater flexibility, easier diagnostics, lower costs, and rapid evaluation of alternatives when compared to conventional analytical or experimental studies. Once an adequate numerical model is available, it can be used to complement experimental work by numerical simulation of the complex interactions that take place in physical experiments and systems. Transmission Line Matrix (TLM) is a numerical method that models electromagnetic field phenomena by means of circuit concepts and has been successfully used in the simulation of especially complex electromagnetic systems [Christopoulos, 1997; Portí et al., 1998a].
 In this paper, a numerical modeling of the natural cavity formed by the Earth's surface and the lower ionosphere is carried out by means of the TLM method. The results obtained are compared to the widely accepted two-scale-height ionospheric model [Greifinger and Greifinger, 1978; Sentman, 1990], a good correlation between them being observed. Numerical values for Schumann resonance frequencies are very close to the experimental ones, enabling us to affirm that this numerical tool is useful in predicting the Schumann frequencies of other planets or moons without making it necessary to fit the conductivity profile to a two-layer model. Finally, shifts in the Schumann resonance frequencies due to a sudden increase in the conductivity at a pole, caused by high-energy particle precipitation emitted from the Sun in conjunction with solar flares, have also been numerically modeled and compared with experimental resonances [Schlegel and Füllekrug, 1999]. A slight increase in the first Schumann resonance frequency is observed, which is related to a reduction in the electromagnetic cavity dimensions, since an increase in the electric conductivity is equivalent to a descent in the upper boundary of the Earth-ionosphere cavity.
2. The Transmission Line Matrix (TLM) Numerical Method
 The TLM method is a powerful simulation tool widely used in the analysis and design of antennas, microwave, and electromagnetic compatibility structures with arbitrary geometry. The TLM method sets up a temporal and spatial iterative process, which allows the temporal evolution of the six electromagnetic field components to be obtained. It is based on the construction of a three-dimensional transmission line network where voltages and currents behave similar to the electromagnetic fields in the original system [Christopoulos, 1995]. This three-dimensional mesh is constructed by interconnecting unitary circuits, termed nodes or cells. Different particular nodes can be used depending on the specific problem to model, but the most usual ones are variations of the symmetrical condensed node [Johns, 1987], shown in Figure 1 for spherical geometry. A TLM node models the electromagnetic wave propagation through a small block of space, whose dimension must be equal to or less than one tenth of the wavelength at the highest frequency of interest, in order to avoid errors due to numerical dispersion [Morente et al., 1995]. The details on the node geometry and characteristic parameters will be discussed later for the particular case to be used.
 Independent of the geometry of the specific node used, the basis of the TLM algorithm is quite simple. At each time step, a set of incident voltage pulses, represented by the column matrix, Vi, are incident at each node in the TLM network, and after scattering, produce a set of reflected voltage pulses, represented by the column matrix, Vr. The incident and reflected voltage pulses are related by the scattering matrix, S, by the equation
The reflected pulses become incident to adjacent nodes at the next time step. Time synchronism must be imposed for the nodes so that all the voltage pulses in the mesh are simultaneously incident at all node centers at each discrete time nΔt, where Δt is the time needed by any pulse in passing from one node center to an adjacent node center.
 Despite the usual Cartesian shape of the TLM nodes, a modified spherical TLM condensed node will be used in this paper in order to better fit the Earth surface and its atmosphere. Each node must model an elementary spherically shaped volume ΔV = (Δr)(riΔϑ)(ri sin ϑjΔφ), where Δr, Δϑ, and Δφ are constant across the entire space, and ri, ϑj, and φk are the coordinates describing the center of the (i, j, k) node, given by the equations
with r0 being the radius of the inner conducting sphere.
 In electromagnetic terms, each portion ΔV can be considered as a total capacitance, inductance, and electric conductance for each direction, which for the spherical volume under study are
where ε, μ, and σ are the electric permittivity, magnetic permeability, and electric conductivity, respectively.
 In terms of TLM, an analogous spherical TLM node must define the electromagnetic properties of this volume, while the voltage and current pulses propagating through the node must define the propagation of the electromagnetic field in the original medium. Twelve main or link lines form the basis of the node, as shown in Figure 1. These lines are connected to link lines in adjacent nodes and are the lines in charge of the propagation. Each line has associated a component of the electric field and a component of the magnetic field, propagating along a certain direction. For instance, line 1 defines components Er and Hφ propagating along the ϑ direction. Table 1 shows these associations for all the link lines. The characteristic impedance of the link lines, Z0, is chosen to be identical for all the link lines
where Li and Ci are the inductance and capacitance per unit length of the ith link line, respectively. It is worth noting that these lines may present different length, Δli/2, but for synchronism to be met, voltage pulses must cover their length in a constant time, Δt/2, independent of the particular line length. Therefore the speed of the voltage pulse at the ith line is related to its length by Δli/2 = viΔt/2, which means that the total capacitance and inductance introduced by the line are given by
which is independent of the actual line length.
Table 1. Node Lines and Related Quantities
Cr = ()Y0Δt
Lφ = ()Z0Δt
Cr = ()Y0Δt
Lϑ = ()Z0Δt
Cϑ = ()Y0Δt
Lφ = ()Z0Δt
Cϑ = ()Y0Δt
Lr = ()Z0Δt
Cφ = ()Y0Δt
Lr = ()Z0Δt
Cφ = ()Y0Δt
Lϑ = ()Z0Δt
Cr = ()Y0YrΔt
Cϑ = ()Y0YϑΔt
Cφ = ()Y0YφΔt
Lr = ()Z0ZrΔt
Lϑ = ()Z0ZϑΔt
Lφ = ()Z0ZφΔt
 In addition, and not represented in Figure 1, some extra lines or stubs are connected at the node center. These lines are not connected to adjacent nodes and are present for independent control of ε, μ, and σ. Three lines, 13–15, are open-circuit or capacitive stubs. Each capacitive stub is associated with a component of the electric field, with no particular direction of propagation or magnetic field component, and introduces a total capacitance to the node. Thus line 13, for example, is associated with Er, with characteristic admittance, Yr (normalized to Y0), for reasons of synchronism, must be covered by pulses in half the time step, and adds a capacity C13 = (1/2)Y0YrΔt to the node. Three other extra lines, 16–18, are short-circuited or inductive stubs, each one defining a component of the magnetic field and introducing inductance to the node. Thus line 16 is associated with the r-component of the magnetic field, has characteristic impedance Zr (normalized to Z0), must be covered in half the time step, and adds a total inductance of L16 = (1/2)Z0ZrΔt to the node. Pulses entering lines 13–18 at time nΔt, reach the end of the stubs at time (n + 1/2)Δt, and after reflection, go back to the same node center, being incident pulses at time (n + 1)Δt. This means that these lines do not directly contribute to propagation, but instead control the group velocity by storing in the stubs part of the charge and current at each time step. Finally, three infinitely long stubs are also connected at the node center, lines 19–21. The pulse entering these lines never come back again to the node, so they are a means of modeling Joule losses associated with each component of the electric field. Thus line 19, for example, is associated with Er and has a normalized admittance, Gr, to model losses due to σ along the r direction. Table 1 also includes the fields defined by each stub line and other related quantities.
 Although conceptually intuitive, the global three-dimensional node is difficult to deal with because there is no actual circuit containing all the lines in the node. The node must be understood as a conceptual circuit in which connections between lines are not real but are formal describing Ampère's and Faraday's laws. For simplicity sake, these Maxwell equations are usually split into six coupled scalar equations. In a similar manner and to simplify the situation, the node can be regarded as six coupled two-dimensional transmission line circuits or two-dimensional nodes. Three of them are termed parallel or shunt nodes and describe a component of Ampère's law, including Joule losses, while the other three are series nodes, each one describing a component of Faraday's law.
 Regarding the parallel nodes, they are constructed by the parallel connection of four link lines, a capacitive stub, and a lossy stub. The circuit defines a common voltage, has a total capacitance associated with this common voltage, and also considers Joule losses for that voltage. Figure 2a is a plot of the parallel node corresponding to Er, including the link lines associated with Er, lines 1, 2, 9, and 12, the capacitive stub, line 13, and the lossy stub, line 19. It can be shown that the transmission line equations governing the parallel node are analogous to the r component of Ampère's law [Christopoulos, 1995], with the result that the voltage at the center of the node is analogous to ErΔr and the total capacitance of the circuit is analogous to the medium capacitance, Cr, defined by equation (3a). Finally, the circuit also has an admittance of GrY0 defined by line 19 that models the term corresponding to Joule losses in Ampère's law, or alternatively, the conductance term, gr, in equation (3a).
 The specific value of Yr is obtained by considering the total capacitance of the circuit in Figure 2a,
In a similar manner, for the circuit to model the medium conductance, the following equations must be met:
 A similar procedure for parallel nodes corresponding to the ϑ and φ directions yields:
 As regards the series nodes, each one is designed by a series connection of four link lines and a short-circuit or inductive stub that defines a common current and a total inductance that are analogous to a component of the magnetic field and to the magnetic permeability for a certain direction. Figure 2b shows the series node for Hφ including the link lines associated with Hφ, lines 1, 3, 11, and 12, and the inductive stub, line 18. It can be shown that the equations governing this circuit are analogous to the φ component of Faraday's law [Christopoulos, 1995], with the result that the common current Iφ is analogous to Hφ times the node length along the φ direction, ri sin ϑjΔφ, and that the total inductance of the circuit is the medium inductance, Lφ, defined in equation (3c).
 For determining the normalized impedance of stub 18, let us consider the total inductance of circuit in Figure 2b,
which must be equal to Lφ in equation (3c), which gives us the following equation:
 Using a similar procedure for the other series nodes, the following values for the r and ϑ directions are obtained:
 The stub parameters defined by equations (7) and (9) depend on the node location and must always be positive or zero because they represent the characteristic admittances or impedances of transmission lines. Thus Δt must be chosen so that the admittances and impedances mentioned above are nonnegative for all the points in the mesh; that is to say,
 Once the maximum value of Δt compatible with equations (10a)–(10c) is determined, the characteristic normalized admittance or impedance of the stub lines for all nodes in the mesh are simply determined by applying equations (7) and (9).
 The derivation of the scattering spherical coordinate's S matrix can be directly done by imposing Maxwell equations to the node, together with energy and charge conservation laws, and conditions of electric potential continuity as described by Johns , with modifications for considering a non-Cartesian node. This is an elegant but complicated task to do because of the great number of equations to be dealt with. Fortunately, this process can be greatly simplified by applying the concept of common and uncommon lines introduced by Portí et al. [1998b] to the series and parallel nodes described above. In this manner, simple circuital considerations together with only two simple conditions on charge conservation and continuity of electric potential provide the corresponding scattering matrix. The final result is a matrix which resembles that of a standard lossy Cartesian node [Naylor and Desai, 1990] by simply changing the quantities related to x, y, z in the Cartesian node by those related to r, θ, φ in the spherical node.
 Additional information provided by the parallel and series circuits allows the determination of expressions for the electromagnetic field in terms of the incident pulses. So Thevenin's theorem applied to the parallel node in Figure 2a leads to a simple parallel circuit in which each line in the node is substituted by a series connection of a voltage source of value twice the incident voltage and an impedance equal to the characteristic impedance of the transmission line. The common voltage at the node center can be simply derived. As mentioned above, this voltage is analogous to ErΔr, which yields
 In a similar manner, the ϑ, and φ components of the electric field are given by
 By applying Thevenin's theorem to the series nodes in Figure 2b and by identifying current at the node with Hφri sin ϑj Δφ, the following expression for Hφ can be derived
 The other series nodes provide Hr and Hϑ given by
These expressions not only allow the output field corresponding to a set of incident pulses to be obtained, but also define appropriate incident pulses to impose a desired excitation.
 Finally, it is important to note that the TLM must take into account the presence of system boundaries. The solution is straightforward for the case of limits that can be represented by an equivalent load impedance, ZL. The link lines reaching the actual boundary are connected to ZL and any pulse reaching the load through these lines is partially reflected toward the node center, with a reflection coefficient given by
Thus a perfectly conducting boundary corresponds to ZL = 0, so the boundary inverts pulses maintaining amplitude. Other situations such as nonperfectly conducting boundaries can be represented by a more elaborate load, which turns into a reflection coefficient less than unity, i.e., into an energy loss.
3. Earth's Electromagnetic Resonant Cavity
 The simplest model to simulate wave propagation through the space between the terrestrial surface and the ionosphere in the ELF range is by assuming the system as a cavity formed by two spherical conducting surfaces. This approximation is based on the relatively high values of conductivity at the surface, around 4 S/m in the case of seawater and from 10−3 to 10−5 S/m in the case of a solid surface, and the strong increase in conductivity in the atmosphere around 60 km, which changes from 10−10 S/m to values higher than 10−7 S/m in a few kilometers. Figure 3 shows different conductivity profiles for Earth's lower atmosphere used in this paper. The quiet conductivity profile was estimated using an electron density profile from the international reference ionosphere for afternoon equinox conditions [Schlegel and Füllekrug, 1999] and the disturbed conductivity profile corresponds to the solar protons event which was observed in Tromsö (69.6°N, 19.5°E), northern Scandinavia, on 20 October 1989, during the daytime [Collis and Rietveld, 1990]. As shown in the quiet atmosphere conductivity profile of Figure 3, the atmospheric conductivity increases from 10−14 S/m near the surface to 10−3 S/m just below 100 km. So the relation between the conduction current and the displacement one, σ/ωε0, changes from 7 × 10−6 to 7 × 105 at 25 Hz, i.e., the atmosphere behaves as a good dielectric for low altitudes, as a conductor for high altitudes, and can be considered as a dielectric with nonnegligible losses in the intermediate zone.
 In the simplest model, electric losses are excluded, while the terrestrial surface and the ionosphere are considered as perfectly conducting. This configuration produces a simple resonant cavity formed by two concentric spherical surfaces with high conductivity. The model is simple enough to be analytically studied [Jackson, 1999; Balanis, 1989]. According to Balanis , the electric and magnetic fields, E and H, respectively, for spherical transmission lines and cavities can be derived from potentials, F and A, in the form
with ε and μ being the electrical permittivity and magnetic permeability, respectively.
 In source-free regions and using spherical coordinates, the transverse electric to r modes, TEr, can be constructed by letting the vector potential, A, equal zero and
where jn and yn are the spherical Bessel functions, and Pnm(cos ϑ) with m < n, are the associated Legendre functions. A, B, C, and D are constants to be calculated from the boundary conditions, and n is an integer that takes nonzero values because P0m(cos ϑ) = 0.
 Similarly, the transverse magnetic to r modes, TMr, can be constructed canceling out the F potential and obtaining the electric and magnetic fields from the potential
 For both polarization modes, the boundary conditions impose zero tangential components for the electric field at the conducting surfaces. If r0 denotes the Earth's radius and h is the height above the surface at which the ionosphere is located, the boundary conditions to impose are
For TEr modes the boundary conditions lead to the following expression:
 For each n value, the above equation is a transcendent one with multiple roots. A simple numerical analysis of this equation shows that for our particular geometry in which r0 ≫ h the roots kn,p and the frequencies fn,p are approximately given by
This solution does not depend on n or m; therefore a great number of degenerated modes is observed as n increases.
 Following a procedure similar to the one outlined for the TEr modes, it can be shown that the TMr modes between two concentric conducting spheres must satisfy the equation
where the prime denotes the derivative with respect to the function argument, kr. A simple numerical analysis of the above transcendent equation provides the roots shown in equation (19), together with a new set of roots or resonances with a much lower value than those of equation (19). Table 2 summarizes the lowest modes for the case under study, r0 = 6370 km and h = 60 km, together with lossless TLM results obtained later.
Table 2. Theoretical and Numerical Resonance Frequencies for the Electromagnetic Cavity of the Atmosphere Between the Terrestrial Surface and the Ionosphere
Theoretical resonances (f, Hz)
TLM resonances (f, Hz)
Error between theoretical and numerical, %
Experimental resonances (f, Hz)
 The theoretical resonance frequencies shown in Table 2 are the lowest resonance frequencies for the lossless electromagnetic cavity modeling Earth. These depend both on Earth's radius and also on the ionospheric height, and should correspond to Schumann frequencies, but theoretical frequencies are noticeably larger than experimental values. The differences between the theoretical and experimental frequencies are partially due to neglecting the effect of electric losses on the resonance frequencies. The fact that losses tend to reduce the resonance frequencies can be understood by considering a simple system such as a damped oscillator with a dissipative part proportional to the velocity. The system behavior is described by the differential equation:
whose solution is
where ωd2 = ω02 − β2 and δ is the initial phase. Therefore as indicated above, the inclusion of damping effects not only reduces the amplitude as time proceeds, but also reduces the resonance frequency, ωd, the larger the dissipative constant, the larger the reduction. The effect of losses in a more complicated system such as the atmosphere can be predicted by considering the lossy wave equation for a damped medium
Similarities between equations (21) and (23) are evident; so it is reasonable to expect that inclusion of loss terms in the wave equation must be considered to reduce frequencies in order to achieve a more accurate prediction of the Schumann resonance frequencies for the Earth.
 But in addition to losses due to propagation through an atmosphere with an electric conductivity that increases with height, the effect of extra losses due to the fact that the ionosphere and the terrestrial surface are not perfect conductors must also be considered. In conclusion, both type of losses must be included in order to obtain realistic resonance frequencies of the damped system, but unfortunately, these corrections make an exact analytical study impossible and therefore a numerical simulation is required.
4. TLM Numerical Model for Earth
4.1. A Simple Lossless Cavity Model
 In order to predict the Schumann resonance frequencies for the Earth, a full three-dimensional TLM model seems to be required. Although this is the most direct way, excessive memory demand for storage and calculation time requirements make this approach unfeasible. Fortunately, a close analysis of transcendent equations (18) and (20) provides an alternative and a more efficient solution. Effectively, the roots of these equations yield the resonance frequencies, which, according to these equations, do not depend on m. This means that the roots provided by these equations are the same if, for instance, the particular value m = 0 is chosen. It is true that some modes will disappear using this choice, i.e., the field distribution will not be the same, but we are not interested in the field distribution, but in the value of the different resonance frequencies. It is clear from equations (15) and (16) that the vector potentials F and A for the particular case m = 0 are not φ dependent. In other words, the resonance frequencies can only be dependent on the radial and zenithal variables, and the information we are interested in can be obtained by considering the whole spherical system as a superposition of meridian wedges, i.e., associated with a certain φ, which present identical behavior whatever the value for φ. Therefore the system to be modeled by means of the TLM method can be simulated by using a two-dimensional mesh, which turns into an important computational saving, with a variable number of nodes along the r and ϑ directions, but only one node along the φ direction. Of course, added symmetry conditions that ensure nondependence on the φ variable must be imposed both in the excitation and also in the boundary conditions.
 In the first situation, a lossless resonant electromagnetic cavity constituted by two perfectly conducting concentric spheres, with radius r0 = 6370 km and r0 + h = 6370 + 60 km, separated by a vacuum, is considered. The two-dimensional TLM mesh is 12 × 18 × 1 nodes wide in the r, ϑ, and φ directions. This means that Δr = 5 km and Δϑ = 10°. As the number of meridian wedges forming the sphere is variable, a specific value for Δφ is undetermined. In order to reduce numerical dispersion, a node as isotropic as possible is advisable because this implies lower stub values to be used [Morente et al., 1995]. In a spherical system, fully identical size for all directions and nodes is impossible. For this reason, Δφ has been chosen to be comparable to Δϑ in the sense that the minimum value for Δt equals that provided for the ϑ coordinate. According to expressions (10b)–(10c), this value is
which yields an optimum time step of Δt = 2.46 × 10−6 s.
 As regards excitation and output point choice, a lossless system produces a high-amplitude response at resonance frequencies, so this choice is not relevant except for excessively symmetrical points where a number of modes vanish, because in this case, these modes are neither excited nor numerically detected. Whatever the mode, the electric field takes nonzero normal values or the magnetic field takes nonzero tangential values at a perfect conductor and therefore a good choice for all the modes to be excited and detected is to locate output points on one conducting surface and the excitation on the opposite conducting surface. For the practical situation modeled, the tangential magnetic field, Hφ, is obtained near the Earth's surface, point (1, 8, 1), for the reasons discussed above together with the fact that most of the measurements are carried out by antennas at ground level. A wide band excitation has been introduced at point (3, 9, 1) by imposing an impulse signal formed by an initial positive unitary voltage pulse for lines 1–6 at this node and opposite voltage pulses for lines 7–12. In this manner, all components of electric and magnetic fields are excited and the continuous level due to the static component is minimized. The center of the excitation node is at 12.5 km altitude, which approximately corresponds to the region where electrical activity occurs. A total of 2.4 × 106 time step calculations have been carried out, which allows the signal to cover the Earth's perimeter more than 44 times and therefore a quasi-permanent regime to be established. Figure 4 shows the discrete Fourier transform of the time-dependent magnetic field, Hφ, at the output point near the Earth's surface. This Fourier transform is carried out using the Cooley-Tukey algorithm with the entire time series as periodic input for the Fourier transform. The high-frequency aliasing error is negligible because the bandwidth represented in Figure 4 is very far from the Nyquist critical frequency, (2Δt)−1 ≈ 0.2 MHz. The system resonances are clearly observed, their values being included in Table 2 for comparison with theoretical values previously obtained. The numerical results are close to the theoretically expected ones, with differences below 2.5%, which proves the suitability of the TLM method for dealing with these kinds of electromagnetic systems.
4.2. Damped Cavity Model
 Now a more realistic modeling is implemented by including the conductivity profile in Figure 3 for a quiet Earth's atmosphere [Schlegel and Füllekrug, 1999] and allowing a small part of the energy to escape from the outer layer, due to a large but finite conductivity. As mentioned in section 2, this energy loss is evaluated by considering that the outer layer defining the ionosphere is substituted by an equivalent load impedance for the nonperfect conductor, . This equivalent load is connected to lines 10 and 11 (see Figure 1) of the most external nodes in the TLM mesh. As these lines have characteristic impedance Z0, equal to the vacuum impedance, the voltage pulses reaching the ionosphere are partially reflected. According to equation (13), the reflection coefficient for these pulses is
which, after simple manipulations, becomes
 The conductivity of the atmosphere significantly increases above 60 km, reaching similar values to the conductivity at the ground level around 100 km high. With the average conductivity of the Earth's surface, σ = 10−3 S/m, and for a frequency of 30 Hz, in the middle of the low-frequency band to be studied, the real and imaginary parts of the reflection coefficient are ΓR ≈ −0.998 and ΓI ≈ 0.002, ∣ΓR∣≫∣ΓI∣ and Γ ≈ −0.998. This indicates that, for each time calculation, a very small fraction, approximately equal to 4 × 10−6 times the incident energy, is lost in each conducting spherical border. Despite the small value of the lost energy, it is important to take this factor into account due to the large number of time calculations involved in the total simulation. Figure 4 includes a plot of the φ component of the magnetic field versus frequency for the Earth's conductivity profile of a quiet atmosphere. This plot is the Fourier transform of the time domain results obtained for an atmosphere with a height of 100 km, bounded by two good, but nonperfectly, conducting spheres. The TLM mesh size is 20 × 18 × 1 in node units; that is Δr = 5 km, Δθ = 10°, and Δφ is given by equation (24). According to equations (10a)–(10c), the optimum time step is Δt = 2.46 × 10−6 s. A total of 2.4 × 106 time step calculations has been carried out, which implies that the signal completes 44 turns around the Earth, thus ensuring that the quasi-permanent regime is reached. The excitation point is located at node (3, 9, 1), and the output point is taken to be near the Earth's surface, at point (1, 8, 1). The details of the excitation signal are the same as that for the nondamped ideal case. As shown in Figure 4, the resonance frequencies of the magnetic field obtained via discrete Fourier transform are lower than in the lossless case because of the damping due to both losses taken into account in the model: nonzero electric conductivity for the dielectric material and a reflection coefficient at the outer layers that absorb part of the incident energy. The first six numerical Schumann resonances are compared with experimental values in Table 3. Errors range from 6.4% for the fundamental mode to 13.0% for the worst case.
Table 3. Experimental, TLM Numerical, and Two-layer Model Values for the Schumann Resonance Frequencies
Experimental resonances (f, Hz)
TLM resonances (f, Hz)
TLM resonance error,%
Two-layer resonances (f, Hz)
Two-layer resonance error,%
4.3. Comparison With Two-Scale-Height Ionospheric Model
 Observing the conductivity profile of Earth's quiet atmosphere shown in Figure 3, two regions can be detected. The first one is located below 60 km and the second one is defined by points above 60 km. Conductivity increases with height in both regions but the increasing rate is considerably higher for the second region. This characteristic conductivity shape is the basis of the widely accepted two-scale-height ionospheric model proposed by Greifinger and Greifinger  and Sentman . In this two-layer model, two contiguous exponential sections with different scale heights approximate the conductivity height profile; that is
where r is the height above the Earth's surface, ξ1 and ξ2 are local scale heights that range between 3 and 5 km, depending on the authors, h1 is the altitude for which σ1 = ε0ω, and h2 is chosen as the altitude for which 4 μ0ωσ2ξ22 = 1.Values for h1 range between 40 and 50 km and those of h2 between 75 and 90 km. Figure 3, discussed above, also includes a two-scale-height ionospheric conductivity profile which matches the experimental measurements at 45 km in region I and at 75 km in region II.
 This two-layer model, using typical values h1 = 50 km, h2 = 90 km, and ξ1 = ξ2 = 5 km yields Schumann resonance frequencies in good agreement with experimental observations. These values for the Schumann resonances are also included in Table 3. Errors provided by the TLM method are larger than those corresponding to the two-scale-height ionospheric model, but it must be noted that a single conductivity profile of the atmosphere has been introduced in the TLM model, provided by Schegel and Füllekrug  and that no parameters are available to be specifically adjusted for achieving a good agreement with experimental values, as occurs with the semianalytical two-scale-height ionospheric model, where h1 and h2 can be adapted in order to agree with the Schumann resonances.
 By hypothesis, two-layer model assumes that there is no altitude dependence of the magnetic field in region I and the electric field is purely transverse to r in region II. Despite these simplifications, the electric and magnetic field expressions provided by this method are quite complex and are usually evaluated by means of an average over the unit sphere. In order to compare the results provided by the TLM method with published results obtained with the two-scale-height ionospheric model, a numerical simulation of the first 120 km of the atmosphere has been carried out by using a TLM mesh 24 × 18 × 1 nodes wide. In this case, link lines 1 and 12 have been excited for all the nodes in contact with the Earth's surface in order to generate only the Er component of the electric field, as the two-layer model does. After the signal evolves all over the spherical lossy cavity, the tangential component of the electric field, Et, is activated throughout the entire ionosphere. Figure 5 shows the electric and magnetic profiles frequency-averaged around the first Schumann resonance frequency and normalized to ∣Z0Hφ∣ = 1 at ground level, and Figure 6 is a sketch of the corresponding Joule dissipation profiles. TLM results are very similar to that reported by Greifinger and Greifinger  and Sentman , both in shape and also in order of magnitude. Nevertheless, some differences between both models may be noted:
 (1) As regards the TLM results, the transverse component of the electric field, Et, at ground level is nonzero, since the numerical model for the Earth's surface is a good conductor but not a perfect one. The Et field amplitude at the Earth's surface is very similar to the maximum amplitude through the whole atmosphere.
 (2) This component presents a sharp increase around a height of 90 km, which is not observed in the two-layer model.
 (3) In the TLM numerical scheme, the drastic decay of the electric and magnetic fields is predicted for an altitude between 15 and 20 km higher.
 (4) In the two-scale-height ionospheric model, the dissipation profile exhibits two relative maximums at heights corresponding to parameters h1 and h2. In the TLM model the behavior is very similar, the maximum at lower altitude almost entirely arises from vertical currents, while the upper one is mainly due to horizontal currents, but maximum dissipation altitudes are located between 15 and 20 km higher than in the two-layer model. In addition, due to the maximum value of Et between 90 and 100 km, the dissipation corresponding to the transverse field, Pt, is greater than the Joule dissipation from the radial field, Pr.
4.4. Schumann Resonance Changes During a High-Energy Particle Precipitation
 High-energy protons and electrons are often emitted from the Sun in conjunction with solar flares. These solar particle events cause additional ionization in the D region over large circular-shaped areas centered on the geomagnetic poles. Changes in the D region ionization lead to conductivity profile changes and may thereby modify the Schumann resonance frequencies. In Figure 3, a plot of a conductivity profile for a strong solar proton event, reported by Schlegel and Füllekrug  from an experimental paper of Collis and Rietveld , is included. Comparing this profile with the undisturbed one reveals that the conductivity level usually present at heights above 50 km is downshifted by ∼10–12 km during proton precipitation. The solar proton events imply an increase of conducting particles, which produces two simultaneous effects in the ionosphere: an increase in the electric conductivity and a decrease of the ionosphere altitude. These two effects produce opposite actions on the Schumann resonances: the increase in the conductivity implies a decrease in the resonance frequencies, but the lower value for the ionosphere altitude produces higher values for these frequencies. Therefore depending on the particular modification of the conductivity and ionosphere altitude, a solar proton event may cause an increase [Schegel and Füllekrug, 1999] or a decrease [Roldugin et al., 1999, 2001] in the first Schumann resonance.
 Nonhomogeneous problems like this are easy to solve by using the TLM method, since a different value for the electric and magnetic constants can be assigned for each node in the TLM mesh. A numerical modeling of a polar high-energy particle precipitation has been carried out by means of a TLM mesh with size 20 × 18 × 1 nodes, using for the column of nodes near one of the Earth's poles the disturbed ionosphere profile included in Figure 3 and the quiet atmosphere profile for the rest of the nodes. The excitation shape and location are the same as those used when obtaining the Schumann resonance frequencies for a nonperturbed damped cavity. In this case, the magnetic field first resonance shifts from 8.3 to 8.5 Hz, i.e., an increase of 0.2 Hz on the order of the experimental shift observed during solar proton events by Schlegel and Füllekrug , which were never >0.14 Hz. One of the reasons for the difference between the experimental and numerical values may reside in the dimensions of the polar region that is affected by these phenomena. Figure 4 includes the frequency response for Hφ at point (1, 8, 1) in the TLM mesh, i.e., in a vertical column with ϑ = 75°, near the equatorial line. This magnetic field spectrum is clearly disturbed by the increase in the conductivity of the column of nodes near the pole.
 The Schumann frequencies on Earth have been evaluated by using the TLM numerical method. These frequencies are lower than the expected ones for an ideal, lossless electromagnetic cavity with the dimensions of the Earth's atmosphere due to the strong electromagnetic field damping produced by atmospheric conductivity. The numerical results obtained are very close to the experimental ones, indicating that this is a valid numerical tool for predicting the Schumann frequencies of other planets and moons. The numerical method provides results in good agreement with those obtained with the widely accepted two-scale-height ionospheric model, but is also capable of modeling systems with more general conductivity profiles. In addition, the method is useful in analyzing nonhomogeneous systems or situations, such as those occurring during periods of high-energy particle precipitation over the Earth's poles.
 This work was supported in part by the project between Austria and Spain, HU2001-0017, and project 15/2002 of the WTZ-Programme of the ÖAD. G. J. Molina-Cuberos thanks Caja Murcia for a research grant, and B. P. Besser acknowledges financial support by a Friedrich-Schmiedl-Stipendium.
 Arthur Richmond thanks Yu. P. Maltsev and Kristian Schlegel for their assistance in evaluating this manuscript.