The mean lightning current waveform of particularly intense lightning discharges is inferred from 52,510 radio wave recordings in the frequency range 1–200 Hz. The current waveform decays initially with a time constant of ∼2 ms, and the current lowers ∼60 C from cloud to ground within the first ∼10 ms of the discharge. The subsequent continuing current exhibits a decay time constant of ∼40 ms and lowers ∼170 C from cloud to ground within the next ∼100 ms of the discharge. The total charge transfer ∼230 C from cloud to ground deposits electrical energy into the stratosphere resulting from quasi-static (Joule) heating. The energy deposition is dominated by the lightning continuing current, and it is ∼10−5 J/m3 at 30 km height. It is speculated that the initiation of blue jets and gigantic jets in the stratosphere may result from lightning continuing current ≳100 ms which can be observed with radio waves at frequencies ≲10 Hz.
 The recent discovery of an electrical discharge from a thundercloud top to the lower ionosphere [Pasko et al., 2002] and gigantic jets [Su et al., 2003] provides direct evidence for an electrodynamic coupling between the troposphere and the ionosphere within the global atmospheric electric circuit [e.g., Rycroft et al., 2000], which may play a critical role in the Earth's climate system [e.g., Carslaw et al., 2002; Markson and Muir, 1980]. These cloud to ionosphere discharges are thought to be initiated by a charge accumulation in the upper part of the thundercloud [Tong et al., 2005; Pasko and George, 2002] since no electromagnetic signatures from the causative lightning discharges have been detected to date [Su et al., 2003; Pasko et al., 2002; Wescott et al., 1998]. On the other hand, it was suggested that blue starters [Wescott et al., 1996] and blue jets [Wescott et al., 1995, 2001] may be initiated by lightning continuing current [Sukhorukov et al., 1996] or horizontal intracloud discharges [Sukhorukov and Stubbe, 1998], which both result in subsequent quasi-static (Joule) heating of the atmosphere [Pasko et al., 1996] and/or relativistic runaway breakdown in the stratosphere [Yukhimuk et al., 1998]. All proposed initiation mechanisms of cloud to ionosphere discharges (cloud charge accumulation, lightning continuing current and horizontal intracloud lightning) share the unique property that they cannot easily be detected with ordinary radio measurements. For example, substantial continuing current may flow in a lightning channel established by a preceding lightning discharge which was too small to be detected in the natural radio noise environment. These electromagnetic signatures have indeed been detected in the early days of radio research [Ogawa et al., 1966, 1967] and they may be associated with the initiation of the newly discovered cloud to ionosphere discharges. This contribution lends further support to this mechanism through quantifying the stratospheric Joule heating by lightning continuing current inferred from measurements of radio waves in the frequency range 1–200 Hz.
2. Lightning Current Waveform
 Particularly intense lightning discharges transmit electromagnetic waves in the frequency range 1–200 Hz, which propagate over long distances within the Earth-ionosphere cavity [Füllekrug, 2005; Sentman, 1995]. The remote signals can be measured simultaneously around the globe with networks of sensitive radio wave magnetometers [Sato and Fukunishi, 2003; Füllekrug and Constable, 2000]. The recorded magnetic field B is linearly related to the lightning current I through the transfer function T [Burke and Jones, 1996]
where ω is the frequency of the radio wave and ϑ is the angular distance between the lightning discharge and the recording magnetometer. The transfer function T(ω, ϑ) can be calculated for a radially symmetric atmosphere from the normal mode expansion with frequency-dependent ionospheric heights [Füllekrug, 2000; Sentman, 1996]
which describes the radio wave propagation by taking into account the ionospheric conductivity [Füllekrug, 2005]. In this approach, the geometric spreading of the radio wave is described with the associated Legendre polynomials Pnm(cos ϑ) of degree n and order m = 1 on a spheroidal earth with radius a and l ≈ 4 km is the length of the lightning channel [Lyons et al., 2003]. The ionospheric conductivity is characterized by the frequency-dependent conduction boundary h1(ω) ≈ 50 km and the ionospheric reflection height h2(ω) ≈ 100 km, which are both included in the complex modal frequency ωn [Füllekrug, 2000; Sentman, 1990; Greifinger and Greifinger, 1978].
 To determine the lightning current waveform from magnetic field recordings, we make use of radio waves of 52,510 particularly intense positive and negative lightning discharges recorded in Silberborn, Germany, during April 1998 [Füllekrug and Constable, 2000]. The angular distances between the lightning discharges and the magnetometer in Silberborn are determined by triangulation of the source locations with a global network of three magnetometers in Silberborn, Germany, Hollister, California, and Lameroo, Australia. Note that this study does not discriminate the polarity of the intense lightning discharges since the Joule heating is an absolute quantity and the differences between positive and negative lightning continuing current are rather small in this frequency range [Füllekrug et al., 2002; Burke and Jones, 1996]. One second long time intervals of the broadband waveforms in the frequency range 1–200 Hz are extracted and averaged into one time series for each source receiver distance from 2,000–18,000 km with a spatial resolution of 100 km which results in 160 mean time series. Figure 1 illustrates for example the mean magnetic field waveform recorded at 10,000 km distance from the lightning discharges. The lightning current spectrum is subsequently determined for each frequency bin from all 160 source receiver distances by use of the Fourier transform and subsequent application of the Gaussian method of least squares for a complex univariate scatter problem [Füllekrug, 1996]. The lightning current waveform is determined from the inverse Fourier transform of the complex lightning current spectrum. The recorded magnetic field waveform can finally be described by use of the inferred lightning current waveform calculated from equation (1). The forward modeling with an assumed impulsive current moment of the lightning discharge, i.e., the short pulse approximation [Huang et al., 1999; Burke and Jones, 1996; Sentman, 1996], fits the observations less well since only one parameter, i.e., the charge moment change, is used for the modeling (Figure 1).
 The inferred lightning current waveform exhibits a fast decaying current ∼10 ms with an exponential decay time constant τf ≈ 2 ms and a subsequent slow continuing current ∼100 ms with an exponential decay time constant τs ≈ 40 ms (Figure 2). The lightning current transports charge from cloud to ground, which can be calculated from the time integral of the inferred lightning current [Cummer and Inan, 2000]. The total charge transfer Qt ≈ 230 C can thus be described with the superposition of the fast charge transfer (Qf ≈ 60 C) and the slow charge transfer (Qs ≈ 170 C)
The fast decaying current dominates the charge transfer from cloud to ground during the first ∼15 ms after the initiation of the lightning discharge, while the continuing current dominates the charge transfer during the subsequent ∼100 ms of the discharge (Figure 2, inset).
3. Stratospheric Joule Heating
 The charge removal from the thundercloud results in an excess electric field from a charge with opposite sign until the charge distribution in the thundercloud adjusts to the new charge configuration [Pasko, 2006]. The electric field above the thundercloud is dominated by the electric (Coulomb) field of a unipolar monopole charge because the higher-order multipole fields of the charge distribution in the thundercloud, if any, fall off more quickly with height. This Coulomb field Ec results from the total charge deposition described in equation (3),
In this first-order, quasi-static description, the electric field varies spatially only with height (in one dimension), where r is the vertical distance from the charge. The electric field will decay in the conducting atmosphere with the local dielectric relaxation constant τr = ɛ0/σ, which is described with a scalar, inhomogeneous, linear differential equation of first-order for the resulting electric field E,
derived from Ampère's law by neglecting magnetic fields. The right hand side of equation (5) results from the driving external displacement current
Note that the electric field at the highly conducting boundaries of the model (the Earth's surface and in the lower ionosphere) need to vanish, which is achieved by placing appropriate mirror charges inside the Earth and into the ionosphere. In a more general approach, it is possible to include a time-dependent ionosphere, i.e., the moving capacitor model, which can lead to a slowing down of the relaxation and even a growth of the electric field below the moving boundary [Pasko et al., 1997]. The electrical (Joule) heating from the resulting electric fields is calculated from the time integral of the quasi-electrostatic power
where W is the energy density deposited by the electric fields in the atmosphere and σ is the atmospheric conductivity. This atmospheric conductivity depends on the height z and it may be approximated with σ(z) = σ0exp((z − z0)/s), where σ0 ≈ 6.4 × 10−10 S/m is a scaling conductivity at z0 = 50 km height and s = 3.2 km characterizes the conductivity gradient in the atmosphere. It is evident that the total electrical energy deposition in the stratosphere is dominated by the energy density resulting from the slow continuing current (Figure 3).
 It is concluded that the lightning continuing current mainly contributes to the electrical (Joule) heating of the stratosphere. The energy deposition at 30 km height ∼10−5 J/m3 is comparable to the impact of solar UV radiation on the ozone layer [Makarova et al., 2004]. Yet, the energy deposition by particularly intense lightning discharges is highly localized in space and time. Blue jets and gigantic jets are initiated right above thunderclouds in the stratosphere as inferred from optical video recordings. If the initiation of the jets is associated with lightning discharges at all, then lightning continuing current is a prime initiation mechanism. For example, a lightning discharge with a very small fast decaying current may establish a lightning channel in which a long continuing current τs ≳ 100 ms flows to ground. This continuing current would mainly appear in the electromagnetic spectrum at frequencies ≲10 Hz, where the natural radio noise background [Fraser-Smith et al., 1991] may inhibit the detection of these signals, which undoubtedly do exist [Ogawa et al., 1966, 1967]. This mechanism may explain why no electromagnetic signals of lightning discharges preceding jets have been reported in the scientific literature so far. In summary, the search for electromagnetic signals associated with blue jets and gigantic jets is a major challenge to future research in the transition region between the electromagnetic spectrum and quasi-static electric fields, which holds lots of promise for new discoveries.
 This research was supported by the Particle Physics and Astronomy Research Council under contract PP/C502430/1. The second author acknowledges support from the European Commission through the Research Training Network Connecting Atmospheric Layers (CAL) under contract HPRN-CT-2002-00216. Special thanks to Ulrich Schmucker for his strong support toward the inversion of magnetic field signals and their use for the advancement of science.