The contribution of sprites to the global atmospheric electric circuit

The global static electric field in the global atmospheric electric circuit resulting from mesospheric sprite discharges is inferred from a coupled model for the global static and dynamic electric fields derived from Maxwell’s equations. It is found that the global atmospheric electric field from individual sprites is ∼ 44 mV/m, which can be measured with conventional ULF/ELF radio wave antennas at frequencies ∼ 4 Hz.


Introduction
Sprites are transient luminous events in the mesosphere (Lyons, 1996;Sentman et al., 1995;Boeck et al., 1995;Franz et al., 1990), which constitute a new element in the global atmospheric electric circuit (Su et al., 2003;Sato and Fukunishi, 2003;Pasko et al., 2002;Rycroft et al., 2000). The impact of sprites on the global circuit has not yet been quantified, even though the quasi-static (DC) atmospheric electric field plays an important role in the global climate system (Carslaw et al., 2002, and references therein). The global DC atmospheric electric field ∼ 150 V/m is mainly maintained by thunderstorm electric fields (Bering et al., 1998;Hays and Roble, 1979;Roble and Hays, 1979). These electric fields exhibit a ∼20% diurnal variation with Universal Time, which is denoted the Carnegie curve (Füllekrug et al., 1999;Holzer and Deal, 1956;Torreson et al., 1946;Hoffmann, 1923). The contribution of sprites to the global DC atmospheric electric field may be similar to the contribution from particularly intense lightning discharges, ∼5-120 mV/m (Füllekrug, 2004). The main difficulty in measuring the contribution of sprites to the global DC atmospheric electric field is the inadequate sensitivity of ordinary electric field mills, > ∼ 1 V/m, such that other measurement technologies need to be considered. This study proposes a new methodology to infer the global DC atmospheric electric field of individual sprites from conventional global dynamic (AC) electric field measurements in the Ultra-Low and Extremely-Low Frequency (ULF/ELF) range made with radio wave antennas.

A Coupled Model for the Global DC and AC Electric Field
The global DC atmospheric electric field is derived from a solution of Maxwell's equations in a spherical geometry (Uman, 1974). For any charge Q in the atmosphere, specifically a sprite, the resulting global DC atmospheric electric field E z points towards the centre of the Earth where a = 6371 km is the Earth's equivolumetric radius and ε 0 is the electric permittivity. In this approach, the Earth and the ionosphere are considered to be concentric spherical shells. If a charge is deposited on the Earth's surface (e.g., by a lightning discharge) or delivered to the ionosphere (e.g., by a sprite), the potential difference between the Earth and the ionosphere (V e − V i ) changes and the electric field adjusts to the new charge configuration according to Eq. (1). The sprite charge is created instantaneously through quasistatic heating of the mesosphere by the causative lighting discharge. The charge is subsequently delivered to the ionosphere such that no mesospheric charge configuration prior to the sprite needs to be considered. The global AC electric field is derived from a solution of Maxwell's equations in a spherical geometry (Sentman, 1996, Bliokh et al., 1980, but it requires a conductivity model of the ionosphere (Füllekrug, 2005;Füllekrug et al., 2002;Füllekrug, 2000;Sentman, 1990). The description of the global AC electric field with a weighted sum of spherical harmonic functions results in the short pulse approximation of the normal mode expansion with frequency dependent ionospheric heights .
(2) In this approach, the electric field spectrum E AC (ω, ϑ) is related to the intensity of the sprite, the geometric spreading of the radio wave and the ionospheric transfer function. The intensity of the sprite is characterised by the charge moment change Ql (Cummer and Füllekrug, 2001;Füllekrug et al., 2001;Pasko et al., 1998;Cummer et al., 1998), which describes the amount of charge Q flowing within the body of a sprite of length l/2, and therefore includes the image current in the conductive ionosphere. The geometric spreading of the radio wave is described by the Legendre polynomials P n (cos ϑ) of degree n (an integer) at an angular distance ϑ from the sprite on a spheroidal Earth with radius a. The ionospheric transfer function is characterised by the frequency dependent conduction boundary h 1 (ω) ≈ 50 km, where the displacement and conduction currents become equal, and the complex modal frequency (3) (Füllekrug, 2000;Sentman, 1990;Greifinger and Greifinger, 1978), where h 2 (ω) ≈ 100 km is the ionospheric height where the radio waves are reflected, s 1 ≈ s 2 ≈ 2.5 km are scale heights, which determine the exponential increase of the ionospheric conductivity in the atmosphere, ω * n is the complex conjugate of ω n , and c ≈ 3 · 10 8 m/s is the speed of light.
The global AC electric field can be expressed in terms of the global DC atmospheric electric field (4) by use of Eqs. (1) and (2). The uniform global AC electric field spectrum (Sentman, 1996, Eq. (38)) is calculated for a sprite with a charge moment change of Ql = 1 kC·km, e.g., a vertical charge transport of Q = 20 C in a sprite of l/2 = 25 km vertical extent (say 60-85 km), by integration along all source-receiver distances with the Gauss quadrature formula (Kautzleben, 1965, p. 21-24). The resulting electric field spectrum exhibits a surprising increase at frequencies < ∼ 4 Hz, which indicates a quasi-static component of the AC electric field (Fig. 1).
This static term results from the Legendre polynomial P 0 of degree n = 0 in Eq. (4) (P 0 ≡ 1 for all source reciever distances ϑ), which corresponds to a constant electric field all around the globe. This peculiar property can readily be verified by calculating the global AC electric field spectrum without the Legendre polynomial P 0 , i.e., extending the summation from n = 1 . . . ∞ (Fig. 1). The resulting electric field spectrum now exhibits a decrease at frequencies < ∼ 4 Hz. It therefore seems possible to infer the global DC atmospheric electric field from an ultra-low frequency approximation of the global AC atmospheric electric field (Wait, 1962, p. 165). The straightforward analytic calculation of the asymptotic expansion of Eq. (2) for ω → 0 requires the treatment of the frequency dependence of the ionospheric heights h 1 (ω) and h 2 (ω) (Füllekrug, 2000) in the normal mode frequency ω n (Eq. (3)), which is beyond the scope of this paper. Since neighbouring modal frequencies exhibit little interference with each other, we calculate the asymptotic expansion from the mode n = 0, where the Legendre polynomial P 0 and the complex modal frequency ω 0 = 0 become the dominant terms such that Eq. (4) reduces to a basic scaling law for ultra-low frequencies < ∼ 4 Hz, i.e., ω ω 1 . In this way, the models for the global AC and DC electric fields are coupled. The major scientific advance of the new AC/DC electric field model is that it is now possible to infer the global DC atmospheric field of sprites from conventional AC electric field measurements by use of the scaling law. Figure 1 illustrates the convergence of the approximated DC atmo-  spheric electric field E DC (Eq. (5)) toward the exact value of the global DC atmospheric electric field of E z = 4.4 mV/m (Eq. (1)) at ultra-low frequencies < ∼ 4 Hz for a sprite discharge with a charge moment change Ql = 1 kC·km. The ratio of the conduction boundary h 1 (ω) ≈ 50 km and the vertical extent of the sprite l/2 = 25 km cancel such that the scaling law in Eq. (5) may be more roughly approximated with E DC ≈ ω E AC (ω, ϑ).

The Accuracy of the Coupled AC/DC Electric Field Model
The accuracy of the coupled global AC/DC electric field model is frequency dependent (Fig. 1) as a result of the frequency dependent conduction boundary h 1 (ω). The relative deviation of the approximated global DC atmospheric electric field (Eq. (5)) from the exact value (Eq. (1)) is ∼1% at 2.1 Hz, ∼10% at 3.6 Hz, and ∼50% at 5.0 Hz (Fig. 1, inset  fig.). A deviation of 50% may seem to be large, but it is comparable to the uncertainty of the vertical extent of the sprite l/2, which exhibits a similar variability. In addition, the quoted accuracies are calculated from the uniform electric field spectrum, i.e., the integration of individual electric field spectra over all source-receiver distances. For an individual sprite event, the electric field spectrum needs to be calculated for one individual source-receiver distance. The source-receiver distance dependence of the approximated global DC atmospheric electric field is displayed in Fig. 2. At distances from 8-10 Mm, the approximated DC atmospheric electric fields are very close to the exact value of the DC atmospheric electric field E z = 4.4 mV/m inferred from Eq. (1) for a vertical charge transport of Q = 20 C within a sprite of l/2 = 25 km vertical extent. The inset figure in Fig. 2 shows in more detail the relative error of the approximation which is <1% at 0.1 Hz for all source-receiver distances, <5% at 2.1 Hz from 8-10 Mm, <10% at 3.6 Hz from 8-10 Mm and <20% at 5.0 Hz from 8.5-9.5 Mm. It is evident from these results that the accuracy of the approximation decreases with increasing frequency (compare to Fig. 1, inset fig.). However, this effect can be compensated for by choosing a suitable location for the ULF/ELF radio wave antenna, at 8-10 Mm from the sprites.

Summary
The global DC atmospheric electric field of sprites can be determined from calibrated AC atmospheric electric field measurements at frequencies < ∼ 4 Hz with an error < ∼ 10% at source-receiver distances from 8-10 Mm. The largest charge moment changes observed on planet Earth are ∼ 10 kC·km (Füllekrug and Constable, 2000). This observation places an upper bound on the DC atmospheric electric field resulting from an individual sprite < ∼ 44 mV/m, or ∼ 3 · 10 −4 E z , where E z ≈ 150 V/m is the total global DC atmospheric electric field.