Manoj Kumar Paras and Jagdish Rai

Department of Physics, Indian Institute of Technology Roorkee, Roorkee, Uttarakhand, Pin-247667, India. Corresponding author: M. K. Paras; e-mail: mkparas.iitr@gmail.com

Received July 29, 2011; accepted July 1, 2012

RESUMEN

Los duendes rojos son una clase exótica de rayos que surgen por arriba de las tormentas eléctricas. Se han obtenido expresiones de la velocidad y la corriente de estos fenómenos. La primera expresión es gaussiana. La corriente que fluye en el cuerpo del duende rojo se comporta de manera similar a una corriente de retorno típica. Las variaciones en el tiempo del momento de corriente y del cambio en el momento de carga se han calculado con ayuda de las expresiones de velocidad y corriente. También se ha obtenido el campo de radiación eléctrica generado por el momento de corriente del duende. Este campo alcanza su máximo alrededor de los 40 Hz con una amplitud del orden de 10^–5 V/m a 200 km del canal del duende. La energía total disipada en el cuerpo de los duendes rojos es del orden de 10⁹ J.

ABSTRACT

Red sprites are the exotic kind of lightning above thunderstorms. Expressions for the velocity and current of sprites have been obtained. The velocity expression comes out to be Gaussian. The calculated current flowing in the sprite body behaves just like a typical lightning return stroke current. The variations of current moment and charge moment change with time have been calculated with the help of velocity and current expressions. The approximate radiation electric field generated from the current moment of sprite also has been obtained. The radiation electric field peaks at about 40 Hz with an amplitude of the order of 10^–5 V/m at 200 km from the sprite channel. The total energy dissipated in the body of red sprites is of the order of 10⁹ J.

Keywords: Sprites, extremely low frequency (ELF), current moment, charge moment change.

1. Introduction

Sprites are upper atmospheric lightning that occur above thunderstorms at altitudes ranging from clouds top to as high as 90 km. Probably Franz et al. (1990) were the first authors to detect sprites from ground based imaging systems. Sentman et al. (1995) gave the strong evidence of occurrence of sprites through an aircraft campaign. Sprites are initiated by the strong positive cloud-to-ground (CG) discharges (Boccippio et al., 1995; Reising et al., 1996; Huang et al., 1999; Price et al., 2002; Williams et al., 2007). The proposed generation mechanisms include ambient electron heating by quasi-electrostatic thundercloud fields, resulting in ionization and excitation of optical emissions (Pasko et al., 1997), and the secondary cosmic ray electrons of energy ~1 MeV that become runaway electrons (Bell et al., 1995; Roussel-Dupré and Gurevich, 1996) in the presence of large quasi-electrostatic fields which collide inelastically with the neutral O₂ and N₂ molecules, resulting in loss of energy and therefore in atomic and molecular excitations and subsequent optical emissions. Sprites can be generated by acoustic waves originated in the lower atmosphere due to natural cataclysms like hurricanes, thunderstorm lightning and tornados (Aramyan and Galechyan, 2009), and propagate upwards as well as downwards from their points of origin (Sentman et al., 1995). They endure for several milliseconds, and are typically delayed in time ranging from ~3 ms (Winckler, 1995) to several tens of ms (Fukunishi et al., 1995) with respect to the onset of the causative CG lightning. Generally, sprites occur during the dying stage of thunderstorms, and may appear singly as well as in clusters of two or more. The brightest red region of a unit sprite lies between altitudes of 66 and 75 km, below which faint tendrils extend downward up to 40-50 km (Pasko et al., 1997). The lateral extent of unit sprites is typically 5-10 km and sprite clusters may occupy a large volume greater than 10⁴ km³ (Sentman et al., 1995). The energy dissipated in the sprite channels is of the order of 10¹⁰ J (Dowden et al., 1996). This energy excitates the various chemical constituents and scales the luminosity of sprites, which can have different shapes and sizes, depending upon the causative CG lightning. The most common types are columnar, carrot, angels, jellyfish, and A-bomb sprites (Williams, 2001). The main reason for the development of lightning discharges between a thundercloud and the ionosphere is the exponential reduction of pressure with height (Petrov and Petrova, 1999).

The purpose of this paper is to calculate the electrical parameters of red sprites like current moment, charge moment change, radiated electric field from sprites and total energy dissipated in their body.

1.1 Physical mechanism of red sprites

The breakdown ionization of upper atmosphere by thundercloud electric fields was first mentioned by Wilson (1925). He calculated that the thunder cloud electric field decreases with altitude h as ~h^-3 and the critical breakdown electric field E_k decreases exponentially, i.e. much faster than the thundercloud electric field, so that there would be a height where the thundercloud electric field exceeds E_k and the discharge takes place. The fundamental reasons behind the electro-dynamical coupling between the cloud tops and the ionosphere are the heating of the ambient electrons by large quasi-electrostatic fields E_Q-E generated by the strong CG lightning (Pasko et al., 1995) and the runaway electron (Bell et al., 1995; Roussel-Dupré and Gurevich, 1996; Taranenko and Roussel-Dupré, 1996). Raizer et al. (1998) calculated E_Q-E using a simple dipole model with cloud charge Q left at altitude z above the earth. They assumed the earth and ionosphere as perfect conducting boundaries. The E_Q-E generated in the upper atmosphere (h >> z) is given by

where h_i ~90 km is the height of ionosphere. It can be shown with the help of Eq. (1) that only the large charge moment change from conventional lightning discharge can generate the sufficient amount of electric field to produce a breakdown in the atmosphere (60-90 km). However, it has been found experimentally that sprites can propagate up to the thundercloud tops (40 km), so another mechanism, the "runaway breakdown", is very effective to explain the breakdown of lower atmosphere (40-50 km). Runaway breakdown involves the cosmic ray seed electrons with energy ɛ_c ~ 0.1-1 MeV, which accelerate in the presence of large E_Q-E and bombard the atmosphere (Bell et al., 1995). The electric field E_RB required to accelerate the cosmic seed electrons is one-tenth of the conventional critical breakdown electric field E_k. The calculated value of E_RB (Gurevich and Zybin, 2001) is given by

where H = 7.2 km, the scale height of the atmosphere. Figure 1 shows the heating of the upper atmosphere and the runaway mechanism by the E_Q-E generated in the upper atmosphere as an effect of the strong +ve CG lightning discharge, resulting in red sprites.

2. Velocity of red sprites

Stanley et al. (1999) adopted a high speed imaging technique and measured initial sprite velocities in excess of 10⁷ m/s. McHarg et al. (2002) reported the velocity of propagation of downward and upward luminosity of the order of 10⁷-10⁸ m/s. Moudry et al. (2002) reported the fast and slow expansion of sprites with an average velocity of 10⁷ m/s in the bright region and 10⁶ m/s in the bottom part of the dim sprite. Li and Cummer (2009) measured the downward sprite peak velocities in the range of 3-10% of the speed of light. The sprites consist of streamers which propagate in the E_Q-E produced by the strong causative positive CG lightning. Raizer et al. (1998) described the propagation of red sprites as the cluster of streamers originated from the plasma patches. Their calculations can be fitted by a Gaussian distribution of the form

3. Current moment and charge moment change of red sprites

Earlier studies of extremely low frequency (ELF) electric and magnetic field signals have shown some unusual pulses during sprite events (Cummer, 2003). It has been observed that these signals do not contain energy at frequencies higher than 1 kHz. Cummer (2003) suggested that these pulses were associated with high altitude electric current in the sprite itself and not with a lightning process. The first experimental evidence that electrical currents may be flowing in the body of sprites was given by Cummer et al. (1998). This current generates ELF electromagnetic energy at levels comparable to that produced by the parent CG lightning. Cummer et al. (1998) described that this vertical current flowing in a narrow region (about 10%) of the sprites is the main source of ELF radiation. Cummer et al. (2006) found that the sprite current pulses begin 2.5 ms after the onset of the return stroke. The peak vertical sprite current ranges from 1.6 to 3.3 kA (Cummer et al., 1998), and 5-10 kA (Rycroft and Odzimek, 2010). It can be as high as 25 kA (Cummer, 2003) for typical sprites. According to Cummer et al. (1998), the measured values of the current moment during a sprite have a very large initial peak, which may be due to the continuing current from the causative CG positive discharge. The second peak is due to the sprite current. The current flowing through the body of the sprite at the beginning must be zero and it must have a later peak to account for the peak in current moment. The current flowing through the sprite channel is shown in Figure 3. The sprite current i(t) can be given by

where A ≈ 400 km² is the average cross section area of the sprite in which current flows and it is assumed to be constant with time; σ (t), the ambient conductivity ahead of the propagating front of the sprite channel; and E (t) the electric field at the tip of the channel. Using the simple point charge model (Fullekrug, 2006), the sprite that generates the electric field E(t) is given by

where Q₀ = 200 C, the charge removed by the +ve CG lightning; r = 10 km, the radial distance from the charge point; τ_q = 10 ms, the duration of lightning current; and τ_r = ɛ₀/σ(z), the relaxation time of the electric field. The ambient conductivity ahead of the sprite front (Fullekrug, 2006) is given by

where σ₀ = 6.4 × 10^–10 S/m; z₀ = 50 km; and s = 3.2 km describe the conductivity gradient in the earth's atmosphere. The altitude z of the propagating front of the sprite channel can be written as

where z₀ is the altitude of sprite initiation. The approximate sprite current has been calculated from Eqs. (4), (5), (6), and (7). The calculated value of the current has been shown in Figure 3, where a double exponential current expression similar to the return stroke has also been plotted. As the difference between them is very small, we take the sprite current to be

where i₀, γ and δ are constants. The estimated values of γ and δ are 1.10 × 10² s^–1 and 7 × 10² s^–1 respectively, and i₀ is the constant which can be taken into account for the peak value of the sprite current. The value of i₀ is taken to be 8.35 × 10³ A for the peak current of 5 kA.

The current moment M_c is the multiplication of current and the channel length through which it has propagated. The mathematical expression is given by

It has been assumed here that the sprite channel is perfectly conducting and the current generated on its tip is immediately connected to all parts of the channel. The calculated peak value of M_c from Eq. (9) is around 260 kA/km (Fig. 4). Cummer et al. (1998) measured the M_c of a few sprites and obtained values ranging from 150 to 240 kA/km. The charge moment change M_q defined by has also been calculated. The maximum value is around 2050 C/km and it is shown in Figure 4.

4. ELF emissions from red sprites

Cummer et al. (1998) observed that the lightning sprites produced ELF signals and very low frequency (VLF) signals were absent. Cummer (2003) observed that the lightning sprites radiate ELF pulses. The source of these ELF pulses is the vertical current flowing through the body of sprites. A vertical electric field is developed due to the current moment of the sprites. The peak radiation frequency of the vertical electric field can be obtained with the help of velocity and current expressions. Leise and Taylor (1977) and Rai (1978) showed that the Fourier component of the radiation field E_r(ω) is related to the Fourier component of current i(ω) and velocity V(ω) by

where , ω is the angular frequency of radiation, and R is the distance of observation from the source.

The Fourier component E_r(ω) of the vertical electric field must have a peak. Figure 5 shows the calculated electric field, which lies mainly in the ELF region (0-300 Hz). Radiated electric field peaks at around 40 Hz with an amplitude of the order of 10^–5 V/m at a distance of about 200 km from the sprite channel.

5. Energy dissipated in the red sprites

As we know, the energy loss in the upper atmosphere due to the quasi-electrostatic heating and inelastic collisions between runaway electrons and neutral N₂ and O₂ molecules comes in the form of sprites. The released energy dissociates into ionization, excitation, expansion, radiation and kinetic motion of the particles in the body of sprites. The total energy loss in sprites is given below:

where U_k is the potential difference between 40 km and 90 km altitudes.

The value of U_k can be calculated with the help of conventional breakdown electric field E_k.

The calculated value of potential difference U_k for the propagation of sprites between 40 km and 90 km is around 60 MV. With the help of Eqs. (11) and (12), the value of total energy loss in the body of sprites comes out to be of the order of 10⁹ J, which is one order of magnitude less than the previous estimated value of 10¹⁰ J (Dowden et al., 1996).

The sprite electrical parameters, namely current moment, charge moment change and radiation electric field, have been calculated. The peak value of the current moment is 260 kA/km for a peak current of 5 kA. Cummer et al. (1998) measured the current moment of three sprites and reported values ranging from 150 to 240 kA/km. The total charge transferred from stratosphere to ionosphere by the sprites is around 63 C, corresponding to the peak current of 5 kA. Cummer et al. (1998) observed three sprites having peak currents ranging from 1.3 to 3.3 kA. If we take the peak current in the sprite channel to be 3.3 kA, the calculated value of total charge transfer comes to be around 41 C, which is consistent with the 42 C charge transfer for the largest of the three sprites reported by Cummer et al. (1998). The calculated maximum value obtained for the charge moment change is around 2050 C/km. The value of the radiated electric field in the frequency domain has also been calculated. The vertical electric field radiated by sprites lies within the ELF (0-300 Hz) range. The vertical electric field peaks at around 40 Hz with an amplitude of 9.3 × 10^–5 V/m at 200 km from the sprite channel (Fig. 5), which is in conformity with the experimental values reported by different researchers (Cummer et al., 1998; Cummer, 2003). These ELF waves generated from the sprite propagate in the earth-ionosphere waveguide and may contribute to the Schumann resonances. It is suspected that the sprites play a crucial role in enhancing the ionospheric temperature reported by Sharma et al. (2003). Finally, we calculated the energy dissipated in the sprite body due to the quasi-electrostatic heating and runaway mechanism, which was of the order of 10⁹ J, one order of magnitude less than the previously reported value of 10¹⁰ J (Dowden et al., 1996).

Acknowledgments

Author Manoj Kumar Paras is thankful to CSIR, New Delhi, India, for their financial support. Also, we are thankful to the reviewers for their valuable comments and suggestions.

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