A partially uniform knee (PUK) model is a combination of two two-dimensional telegraph equation (TDTE)-based techniques: the “knee” model, which addresses the problem of approximating the knee-like conductivity profile (on a semilogarithmic scale) of the Earth's ionosphere, and the global partially uniform day-night model, which allows a convenient treatment of the day-night asymmetry. Incorporation of the “knee” conductivity profile allows to overcome the shortcoming of the two-exponential technique, widely used in extremely low frequency (ELF) work, too flat frequency dependence of quality factor in the Schumann resonance (SR) range (5–40 Hz). The PUK model predictions for Schumann resonance parameters reasonably represent observations in the SR frequency range. Propagation parameters for other planets were calculated on the basis of existing ionospheric models of the planets. To allow the approximation of structured conductivity profiles of Venus and Mars, the “knee” model was upgraded to a double-“knee” by inserting an additional “knee” to the profile. In general, this technique allows to approximate very structured profiles by adding as many “knees” as necessary. Calculations show that the detection of Schumann resonances on Venus, Mars, and Titan is possible, though low-quality factors on Mars and Titan imply that pronounced peaks are not to be presumed on these planets.
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 There are several reasons for interest in SR. From the very beginning of SR studies, they were used to evaluate characteristics of global thunderstorm activity [Nickolaenko, 1996]. It has also been suggested that SR may be used to monitor planetary temperature [Williams, 1992]. The link between these two parameters is the lightning flash rate, which increases nonlinearly with temperature, providing a natural amplifier of the temperature changes. Recently yet another application has been found. Water vapor and global lightning activity are closely linked through continental deep-convective thunderstorms. SR provides continuous, long-term global lightning record and thus can serve as a tool for monitoring global upper-tropospheric water vapor changes [Price, 2000]. SR can be also used in exploration of the electrical activity and lower ionosphere parameters on celestial bodies [Nickolaenko and Rabinowicz, 1982].
 The Earth-ionosphere waveguide is not an ideal electromagnetic cavity. Losses due to finite ionosphere conductivity make the system resonate at lower frequencies than would be expected in an ideal case. In addition there are a number of horizontal asymmetries: day-night transition, latitudinal changes in the Earth magnetic field, sudden ionospheric disturbances, polar cap absorption, etc.
 Using a full-wave technique Madden and Thompson  identified two characteristic layers within the lower ionosphere with the lower layer being responsible for the behavior of the vertical electric component, and the upper one, for the horizontal magnetic component. The ionospheric properties external to these characteristic layers were found to be of minor importance for ELF wave propagation. On the basis of these findings Greifinger and Greifinger  proposed an analytical procedure for determining propagation parameters. The procedure is governed by Maxwell's equations and generalized Ohm's law. Analytical expressions are obtained assuming exponential conductivity profiles within both lower (electric) and the upper (magnetic) characteristic layers. For more general profiles a numerical solution is required [Greifinger and Greifinger, 1986]. The two-exponential version of Greifinger and Greifinger , based on approximating the conductivity profile within the lower and the upper characteristic layers by exponential functions, is widely used in ELF research. While successfully predicting resonance frequencies the two-exponential model fails to simulate the frequency dependence of the quality factors.
 Another, more general approach is based on the analogy between ELF propagation within a waveguide and wave propagation in a transmission line, two-dimensional telegraph equation (TDTE) technique suggested by Madden and Thompson , and developed by Kirillov  (this method is used in this work and will be described in more detail in following sections). TDTE allows incorporation of any conductivity profile. The “knee” approximation developed by Mushtak and Williams  (hereinafter referred to as M&W) on the basis of the TDTE formalism allows to obtain both realistic resonance frequencies and quality factors, thus overcoming the shortcoming of the two-exponential approach.
 The present work combines the partially uniform global model developed by Kirillov et al.  to treat the day-night asymmetry, and the “knee” model by M&W into a partially uniform knee (PUK) model. In order to allow incorporation of different ionospheric profiles when modeling SR on other planets, the “knee” model was extended to a “multiknee” approximation by introducing additional “knees” to the profile. The resulting model is very flexible, allowing incorporation of structured conductivity profiles and a convenient treatment of the day-night asymmetry with an easy switch from “uniform” to “day-night” model.
 The telegraph equation is obtained from the equation of charge conservation on the Earth surface: divi + ∂ρ/∂t = 0, where i is the current density and ρ is the charge density. Adopting a temporal time dependence of the form exp(−iωt) and integrating the equation over the altitude we arrive at the equation of conservation of surface charge: divj − iωq = 0 where q = ρdz is the surface charge density and j is the surface current density: jx = ixdz, jy = iydz. The vertical component of the current is absent since the Earth is electrically insulated from the ionosphere [Kirillov and Kopeykin, 2002]. Dependency between the surface charge density q and voltage u is described by the relation q = Cu, u ≡ Erdr where C is the surface capacitance density and Er is the vertical component of the electric field. The relationship between the surface current density j and the surface gradient of voltage is iωLj = gradu where L is the local inductance. The above relations transform the equation of conservation of surface charge into a second-order differential equation, the two-dimensional telegraph equation:
 In the general case L has a structure of a 2 × 2 tensor. Kirillov et al.  suggested a partially uniform global model for treating the day-night asymmetry. In the ELF range the day-night transition is smaller than the wavelength, thus the waveguide parameters can be approximated by their average values for day and for night [Kirillov et al., 1997]. With such an approach the inductance L becomes a scalar. The frequency dependence for inductance and surface capacitance has a form of [Kirillov, 1993]:
HE and HM are the lower and the upper characteristic altitudes, respectively. The real parts of these altitudes can be considered as the effective “electric” and “magnetic” ELF heights of the waveguide [Mushtak and Williams, 2002]. The lower characteristic altitude HE (f) is computed as an integral
where σ is the conductivity. It is convenient to approximate the conductivity profile with a function which allows the analytical solution of the integral in equation (4), yet a characteristic knee-like change [Cole, 1965] in the conductivity profile (representing a transition from ion-dominated to electron-dominated conductivity) on a semilogarithmic scale makes it hard to make a good fit with one exponential function. M&W developed a “knee” model which overcomes this problem. This model seems preferable over the widely used two-exponential approach, while both models give rather accurate modal frequencies, the two-exponential model fails to simulate frequency dependence of quality factors in the SR frequency range whereas the “knee” model simulates this dependence rather well. In the “knee” model the approximation for the conductivity profile is formulated as
where σkn is the conductivity at a symbolically defined “knee” altitude hkn and ζb, ζa are the scale heights of the exponential functions approximating the conductivity profile below and above hkn, respectively. With such approximation HE (f) becomes:
where fkn is related to σkn as fkn = σkn/(2πɛ0). For the upper characteristic altitude HM (f) M&W suggested the following phenomenological approximation:
where k0 is the free space wave number, ω0 is the electron plasma frequency, ωHz is the vertical projection of the electron gyrofrequency, ν is the electron collision frequency, ζNe and ζν are the scale heights of the exponential approximations of the electron density and electron collision frequency profiles, respectively.
 In terms of the global partially uniform day-night model the resonance frequencies are computed taking into account the reflection and transmission at the terminator. Reflection and transmission coefficients, Rm and Dm, are computed demanding continuity for the voltage (the analogy of the electric field in transmission line) and the normal component of the current (the analogy of the magnetic field) at the day-night boundary. Then the coefficients are expressed by [Kirillov et al., 1997]
where θT defines the boundary between day and night hemispheres, the terminator (in spherical coordinate system with polar axis pointing at nadir). Here θT was taken to be 81°, as suggested by Kirillov et al. . Note that the coefficients are independent of the source position. Turning the real part of the denominator to zero, viewed as a function of frequency (the real part of the corresponding eigenfrequency) and utilizing equations (2), (8), (9), and (10), gives frequencies of the Schumann resonances in a nonuniform waveguide [Kirillov et al., 1997]. The Legendre functions were calculated using convergence acceleration method developed by Nickolaenko and Rabinowicz  (corrected formulas can be found in the work of Connor and Mackay ).
 In the equation (12), Pνm (cos θ) are associated Legendre functions and complex parameter ν is calculated via the relationship
 The propagation parameters are formulated as the real and imaginary parts of the complex sine S or, in physical terms, as the phase velocity Vph (f) and the attenuation rate α (f)
The quality factor of the resonant cavity may be determined as a ratio between the stored energy and the energy loss per cycle. Considering only the electrically stored energy it can be shown that [Galejs, 1972]:
Q−1 values computed under daytime and nighttime conditions can be averaged to give quality factors of the waveguide [Galejs, 1972].
 The partially uniform knee (PUK) model which combines partially uniform day-night model by Kirillov et al.  and “knee” model by Mushtak and Williams , is characterized by fifteen parameters: fkn, , and for the lower characteristic altitude, f*m, h, ζ* and b for the upper characteristic altitude, and the terminator position, θT.
 As a first test the PUK model was set to a uniform mode (no difference between day and night variables). In such a setting, the PUK model is similar, but not identical to the “knee” model by M&W, the procedure of resonance frequency calculation is different for the two models. The model variables were given the same values as the (uniform) “knee” model in the work of M&W; see Table 1. Table 2 presents resonance frequencies, Q factors and propagation parameters calculated by M&W and those computed with PUK model. Here and throughout the paper the results are presented for the azimuthal index m = 0. The results are almost identical for both models and are very close to experimental values shown in Table 3.
 The criterion for choosing variable values for the day-night PUK model (shown in Table 1) was based on obtaining the best possible agreement between the modeled and experimental values for the resonant frequencies and quality factors while maintaining a reasonable fit of conductivity profile (Figure 1). Model output is shown in Table 4 (the column labeled “f (Hz)” shows the resonant frequency, and the column labeled “frequency (Hz),” frequencies at which the propagation parameters c/Vph and α were calculated). The obtained frequencies and Q factors are in very good agreement with the experimental values (Table 3). The propagation parameters c/Vph and α are compared in Table 5 with values presented in the work of Galejs [1972, Figure 7.12]. The output of the PUK model is in reasonable agreement with the previously published values.
Table 4. Resonant Frequencies, Q Factors and Propagation Parameters for Earth at the Given Frequenciesa
4. Model Results for Schumann Resonances on Other Planets
 Existence of Schumann resonances depends generally on two factors, presence of a substantial ionosphere with electric conductivity increasing with height from low values near the surface to form an ELF waveguide, and a source of excitation of electromagnetic waves in the ELF range. The detection of lightning activity, which is an ELF source, on other planets of the Solar system leads to the possibility of detecting global resonances on celestial bodies. In the present study we will concentrate on three celestial bodies, Venus (were lightning activity may have been detected), Mars (where electrical discharges in dust storms are likely) and Titan (were lightning activity is considered possible). Jupiter, where lightning activity is well established, was left out of the scope of this work. Schumann resonances on this planet were studied by Sentman , and due to the uncertainties in the waveguide structure, we decided not to investigate it further.
 The speculations that lightning occurs on Venus first arose about 30 years ago and led to the installation of instruments capable of detecting optical emissions and electromagnetic radiation associated with lightning on Venus landers and orbiters [Russell and Scarf, 1990]. The strongest evidence for lightning on Venus comes from the impulsive electromagnetic waves seen by the Venera 11 and 12 landers [Ksanfomaliti, 1979, 1983a, 1983b, 1985] and the Pioneer Venus Orbiter [Taylor et al., 1979; Scarf and Russell, 1983].
 Although the Venusian atmosphere has been extensively studied, the electron density profiles were measured in the upper ionosphere, while SR frequencies depend on the structure of the lower part of the ionosphere. Therefore in this study height profiles are adopted from the model of Venusian lower ionosphere by Borucki et al. . This model does not distinguish between day and night therefore the computer simulation is performed in a uniform mode.
 Venusian modeled and fitted conductivity profiles are shown on Figure 2. The highly structured profile around 50km altitude is due to cloud and haze particles which attach the ions, lower ion concentrations and reduce the conductivity [Borucki et al., 1982]. Although clouds have a regional nature in the terrestrial atmosphere, on Venus the ubiquity of the cloud cover make it a global feature. Such a conductivity profile can not be adequately fitted with a “knee” profile. Instead, a “double-knee” (K-2) model was introduced.
 The approximation for the K-2 conductivity profile is formulated as
where σ1, σ2 are conductivities at a first and second “knee” altitudes h1 and h2, and ζ1, ζ2 are the scale heights of the exponential function approximating the conductivity profile below the first, and above the second “knee” respectively. Between the two “knee” altitudes, the profile is defined by two points, (h1, σ1) and (h2, σ2) as ln σ = a1z + a2 when a1 and a2 are found from
which leads to:
With this “multiknee” approach, even highly structured profiles can be approximated by introducing as many “knees” as desired.
 Solving the integral in equation (4) for the K-2 profile, the lower characteristic altitude HE (f) becomes:
where f1,2 are related to σ1,2 as f1,2 = σ1,2/(2πɛ0).
 This model is characterized by ten variables: h1,2, f1,2 and ζ1,2 for the lower characteristic layer, hm*, fm*, ζm*, and bm for the upper characteristic layer. The model variables for Venus, derived from ionospheric profiles of Borucki et al.  are shown in Table 6 (the very small f1 and f2 values are a direct consequence of the conductivity values at the knee heights). In Table 7 the results of the present model (PUK-2) are compared to the values obtained by Nickolaenko and Rabinowicz  (hereinafter referred to as N&R) and to fn(0), the case of perfect conductivity for both the planet and its ionosphere:
The obtained resonant frequencies for the first three modes are very close to those calculated by N&R. Owing to the ionospheric losses the resonant frequencies are about 20% lower than would be expected for the ideal cavity. The quality factors are almost twice higher than predicted by N&R and show much more pronounced frequency dependence. N&R estimated that the predicted Q factors may deviate from measured by a factor of two. The Q factors obtained here are in agreement with these estimates.
 The conductivity profile of the Martian ionosphere was calculated from electron collision frequency and day and night electron density profiles presented by Cummer and Farrell  as:
where ne, e and me are the number density, charge and mass of the electrons, respectively, and ν is the electron-neutral collision frequency. Lack of significant magnetic field on Mars allows us to neglect the effect of ionospheric ions on the wave propagation [Cummer and Farrell, 1999]. Figure 3 shows the resulting conductivity profile and its fit. As in case of Venus, a “double-knee” approximation was used here to achieve a better fit.
 Model variables for Mars are shown in Table 6 (the variables were chosen to provide the best fit to Cummer and Farrell  ionospheric profiles). The resultant resonant frequencies and quality factors for the present model (PUK-2) are presented in Table 8 along with fn(0) values and values obtained by Sukhorukov . The SR frequencies and Q factors predicted by the present model are lower than those obtained by Sukhorukov. The main reason for the disparity is the difference between electron density profile models adopted for the study. The low-quality factors imply that pronounced sharp peaks at resonance frequencies should not be expected for the Marian ELF waveguide.
Table 8. Resonant Frequencies and Q Factors for Marsa
 Although no lightning was observed during Voyager flybys of Titan in 1980 and 1981, it is suggested that lightning dischargers do take place on this moon of Saturn. Recent theoretical papers suggest mechanisms for lightning to occur in Titan's troposphere [Tokano et al., 2001; Lammer et al., 2001]. A thundercloud model by Tokano et al.  showed that cloud-to-ground discharges are possible on Titan.
 The ionosphere of Titan is perhaps the most thoroughly modeled today. The recent interest in the largest satellite of Saturn is associated with the Cassini/Huygens Mission which includes Cassini/Huygens orbiter and the descent of Huygens probe on Titan. In this study the conductivity profile is adopted from a work by Morente et al.  which is based on the model of Titan's ionosphere developed by Molina-Cuberos et al. . The profile is shown on Figure 4. As it can be seen, this model of Titan's conductivity is well approximated by a “single-knee” fit. Since there is no discern between day and night, the PUK model is set to the uniform mode.
 The model variables (chosen to give a best fit to the predicted ionospheric profiles) are shown in Table 1. Table 9 shows computed resonant frequencies and Q factors for the PUK model, fn(0) values, SR frequencies predicted by TLM model [Morente et al., 2003] and frequencies and quality factors estimated by Nickolaenko et al. . SR frequencies obtained in this work are somewhat lower than the values calculated by Nickolaenko et al., and very close to those predicted by the TLM model. The quality factors are within the range estimated by Nickolaenko et al. , but show stronger frequency dependence, as it is expected from a “knee”-type model.
Table 9. Resonant Frequencies and Q Factors for Titana
 The PUK model unites the “knee” method developed by Mushtak and Williams  for approximating conductivity profile, and the global partially uniform day-night model developed by Kirillov et al. . This combination allows a convenient analytical treatment of the day-night asymmetry and of the knee-like Earth ionospheric conductivity profile. Usage of the “knee” model allows to obtain a realistic frequency dependence of Q factors overcoming the limitation of the widely used two-exponential approach in the SR frequency range.
 The PUK model was used to calculate possible values of the resonant frequencies and quality factors of Venus, Mars and Titan. Calculations were based on modeled ionosphere profiles. In order to allow a more precise approximation of the structured conductivity profiles on Venus and Mars, the “knee” method was upgraded to a “double-knee” (K-2) method by introducing an additional “knee”-height into the model. In general, this multiknee technique allows to approximate very structured profiles by simply adding as many “knees” as desired.
 On Venus the basic resonant frequency is around 9 Hz with quality factor approximately equal to 10. This rather high Q factor presumes well pronounced peaks around the resonant frequencies in the ELF spectra. On Mars the first mode frequency is approximately 8.5 Hz with Q slightly over 2. On Titan the first resonance is near 12 Hz with the quality factor of about 2. On both planets resonant frequencies are almost twice lower than it would be expected for an ideal cavity due to losses in the ionosphere. Low Q factors on Mars and Titan imply that sharp, pronounced peaks should not be presumed on these planets.
 The values of the resonance frequencies and quality factors are very dependant on the ionospheric profiles. The accuracy of the modeled conductivity profiles used in this work is limited, and a deeper knowledge of planetary ionospheres would allow more precise predictions of Schumann resonance parameters. Additional inaccuracies in the SR frequencies and Q factors are introduced when considering the real part of the eigenfrequencies, when minimizing the denominator in equation (12), but they are much smaller then those introduced by the uncertainties in the ionospheric profiles, especially on other planets. Nevertheless, the results obtained with the present study help defining the frequency interval of the expected spectral maxima.
 On Earth, improved modeling of the global resonances can be used for global studies of lightning activity. Recent findings show that SR can be used as an index of global temperature changes [Williams, 1992] and of atmospheric water vapor [Price, 2000]. Perhaps these methods can be of use not only on Earth, but also on celestial bodies with terrestrial-like lightning activity. Experimental evaluation of SR parameters can also aid in the elaboration of the effective model of the ionospheric conductivity profile, and contribute substantially to the knowledge of lower ionospheres on planets of the Solar system.
 We wish to sincerely thank V. C. Mushtak and A. P. Nickolaenko for their priceless advice.