Abstract
 Top of page
 Abstract
 1. Introduction
 2. Model
 3. Results
 4. Model Results for Schumann Resonances on Other Planets
 5. Discussion and Conclusions
 Acknowledgments
 References
 Supporting Information
[1] A partially uniform knee (PUK) model is a combination of two twodimensional telegraph equation (TDTE)based techniques: the “knee” model, which addresses the problem of approximating the kneelike conductivity profile (on a semilogarithmic scale) of the Earth's ionosphere, and the global partially uniform daynight model, which allows a convenient treatment of the daynight asymmetry. Incorporation of the “knee” conductivity profile allows to overcome the shortcoming of the twoexponential 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 lowquality factors on Mars and Titan imply that pronounced peaks are not to be presumed on these planets.
1. Introduction
 Top of page
 Abstract
 1. Introduction
 2. Model
 3. Results
 4. Model Results for Schumann Resonances on Other Planets
 5. Discussion and Conclusions
 Acknowledgments
 References
 Supporting Information
[3] 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 deepconvective thunderstorms. SR provides continuous, longterm global lightning record and thus can serve as a tool for monitoring global uppertropospheric 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].
[4] The Earthionosphere 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: daynight transition, latitudinal changes in the Earth magnetic field, sudden ionospheric disturbances, polar cap absorption, etc.
[5] There are two general approaches in determining ELF wave propagation parameters: numerical [Madden and Thompson, 1965; Jones, 1967; Hynninen and Galyuck, 1972; Pappert and Moler, 1974; Ishaq and Jones, 1977] and analytical [Greifinger and Greifinger, 1978, 1979, 1986; Sentman, 1995, 1996; Mushtak and Williams, 2002]. The numerical fullwave technique allows to incorporate detailed ionosphere structure, but does not provide tangible connection between the propagation parameters and the ionospheric properties. The analytical approach, though still less accurate than the fullwave technique, does not have such a shortcoming.
[6] Using a fullwave technique Madden and Thompson [1965] 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 [1978] 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 twoexponential version of Greifinger and Greifinger [1978], 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 twoexponential model fails to simulate the frequency dependence of the quality factors.
[7] Another, more general approach is based on the analogy between ELF propagation within a waveguide and wave propagation in a transmission line, twodimensional telegraph equation (TDTE) technique suggested by Madden and Thompson [1965], and developed by Kirillov [1993] (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 [2002] (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 twoexponential approach.
[8] The present work combines the partially uniform global model developed by Kirillov et al. [1997] to treat the daynight 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 daynight asymmetry with an easy switch from “uniform” to “daynight” model.
2. Model
 Top of page
 Abstract
 1. Introduction
 2. Model
 3. Results
 4. Model Results for Schumann Resonances on Other Planets
 5. Discussion and Conclusions
 Acknowledgments
 References
 Supporting Information
[11] In the general case L has a structure of a 2 × 2 tensor. Kirillov et al. [1997] suggested a partially uniform global model for treating the daynight asymmetry. In the ELF range the daynight 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]:
H_{E} and H_{M} 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 H_{E} (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 kneelike change [Cole, 1965] in the conductivity profile (representing a transition from iondominated to electrondominated 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 twoexponential approach, while both models give rather accurate modal frequencies, the twoexponential 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 σ_{kn} is the conductivity at a symbolically defined “knee” altitude h_{kn} and ζ_{b}, ζ_{a} are the scale heights of the exponential functions approximating the conductivity profile below and above h_{kn}, respectively. With such approximation H_{E} (f) becomes:
where f_{kn} is related to σ_{kn} as f_{kn} = σ_{kn}/(2πɛ_{0}). For the upper characteristic altitude H_{M} (f) M&W suggested the following phenomenological approximation:
where the scale height ζ_{m} depends on frequency as
The real altitude h_{m} can be determined from the equation [Kirillov, 1993; Mushtak and Williams, 2002]
where k_{0} is the free space wave number, ω_{0} is the electron plasma frequency, ω_{Hz} is the vertical projection of the electron gyrofrequency, ν is the electron collision frequency, ζ_{Ne} and ζ_{ν} are the scale heights of the exponential approximations of the electron density and electron collision frequency profiles, respectively.
[12] In terms of the global partially uniform daynight model the resonance frequencies are computed taking into account the reflection and transmission at the terminator. Reflection and transmission coefficients, R_{m} and D_{m}, 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 daynight boundary. Then the coefficients are expressed by [Kirillov et al., 1997]
where θ_{T} defines the boundary between day and night hemispheres, the terminator (in spherical coordinate system with polar axis pointing at nadir). Here θ_{T} was taken to be 81°, as suggested by Kirillov et al. [1997]. 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 [1974] (corrected formulas can be found in the work of Connor and Mackay [1978]).
[13] In the equation (12), P_{ν}^{m} (cos θ) are associated Legendre functions and complex parameter ν is calculated via the relationship
where S is the sine of the wave incidence angle. The relationship between the parameter S and the complex characteristic altitudes is [Kirillov, 1993; Kirillov et al., 1997]:
[14] The propagation parameters are formulated as the real and imaginary parts of the complex sine S or, in physical terms, as the phase velocity V_{ph} (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].
4. Model Results for Schumann Resonances on Other Planets
 Top of page
 Abstract
 1. Introduction
 2. Model
 3. Results
 4. Model Results for Schumann Resonances on Other Planets
 5. Discussion and Conclusions
 Acknowledgments
 References
 Supporting Information
[18] 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 [1990], and due to the uncertainties in the waveguide structure, we decided not to investigate it further.
4.1. Venus
[19] 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].
[20] 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. [1982]. This model does not distinguish between day and night therefore the computer simulation is performed in a uniform mode.
[21] 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 “doubleknee” (K2) model was introduced.
[22] The approximation for the K2 conductivity profile is formulated as
where σ_{1}, σ_{2} are conductivities at a first and second “knee” altitudes h_{1} and h_{2}, and ζ_{1}, ζ_{2} 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, (h_{1}, σ_{1}) and (h_{2}, σ_{2}) as ln σ = a_{1}z + a_{2} when a_{1} and a_{2} are found from
which leads to:
With this “multiknee” approach, even highly structured profiles can be approximated by introducing as many “knees” as desired.
[23] Solving the integral in equation (4) for the K2 profile, the lower characteristic altitude H_{E} (f) becomes:
where f_{1,2} are related to σ_{1,2} as f_{1,2} = σ_{1,2}/(2πɛ_{0}).
[24] This model is characterized by ten variables: h_{1,2}, f_{1,2} and ζ_{1,2} for the lower characteristic layer, h_{m}*, f_{m}*, ζ_{m}*, and b_{m} for the upper characteristic layer. The model variables for Venus, derived from ionospheric profiles of Borucki et al. [1982] are shown in Table 6 (the very small f_{1} and f_{2} values are a direct consequence of the conductivity values at the knee heights). In Table 7 the results of the present model (PUK2) are compared to the values obtained by Nickolaenko and Rabinowicz [1982] (hereinafter referred to as N&R) and to f_{n}^{(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.
Table 6. Model Variables for Venus and Mars^{a}f_{1}, Hz  f_{2}, Hz  h_{1}, km  h_{2}, km  ζ_{1}, km  ζ_{2}, km  f_{m}*, Hz  h_{m}*, km  ζ_{m}*, km  b_{m}, km 


Venus 
6e4  1e4  47.0  49.0  11.0  2.9  10  120.0  2.0  10.0 

Mars 
150.0  1000.0  28.0, 30.0  53.0, 58.0  3.5, 3.5  4.6, 6.1  150.0  78.0, 86.0  7.0, 7.3  10.0 
         10.0 
4.2. Mars
[26] 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 [1999] as:
where n_{e}, e and m_{e} are the number density, charge and mass of the electrons, respectively, and ν is the electronneutral 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 “doubleknee” approximation was used here to achieve a better fit.
[27] Model variables for Mars are shown in Table 6 (the variables were chosen to provide the best fit to Cummer and Farrell [1999] ionospheric profiles). The resultant resonant frequencies and quality factors for the present model (PUK2) are presented in Table 8 along with f_{n}^{(0)} values and values obtained by Sukhorukov [1991]. 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 lowquality 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 Mars^{a}Mode  f_{n}^{(0)}, Hz  Sukhorukov [1991]  PUK2 

f, Hz  Q  f, Hz  Q 


1  20.1  13–14  3.3–3.7  8.6  2.3 
2  34.8  25–26  3.5–3.8  16.3  2.4 
3  49.4  37–38  3.7–4.0  24.4  2.4 
4.3. Titan
[28] 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. [2001] showed that cloudtoground discharges are possible on Titan.
[29] 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. [2003] which is based on the model of Titan's ionosphere developed by MolinaCuberos et al. [1999]. The profile is shown on Figure 4. As it can be seen, this model of Titan's conductivity is well approximated by a “singleknee” fit. Since there is no discern between day and night, the PUK model is set to the uniform mode.
[30] 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, f_{n}^{(0)} values, SR frequencies predicted by TLM model [Morente et al., 2003] and frequencies and quality factors estimated by Nickolaenko et al. [2003]. 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. [2003], but show stronger frequency dependence, as it is expected from a “knee”type model.
Table 9. Resonant Frequencies and Q Factors for Titan^{a}Mode  f_{n}^{(0)}, Hz  TLM  Nickolaenko et al. [2003]  PUK 

f, Hz  f, Hz  Q  f, Hz  Q 


1  26.2  12.8  19.9  0.96–2.6  11.8  1.8 
2  45.4  24.4  35.8  0.97–2.7  22.5  1.9 
3  64.2  38.1  51.8  1.01–2.8  34.1  2.0 
5. Discussion and Conclusions
 Top of page
 Abstract
 1. Introduction
 2. Model
 3. Results
 4. Model Results for Schumann Resonances on Other Planets
 5. Discussion and Conclusions
 Acknowledgments
 References
 Supporting Information
[31] The PUK model unites the “knee” method developed by Mushtak and Williams [2002] for approximating conductivity profile, and the global partially uniform daynight model developed by Kirillov et al. [1997]. This combination allows a convenient analytical treatment of the daynight asymmetry and of the kneelike 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 twoexponential approach in the SR frequency range.
[32] 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 “doubleknee” (K2) 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.
[33] 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.
[34] 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.
[35] 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 terrestriallike 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.