Lightning preceded the presence of life on Earth, and according to some theories was instrumental in causing it. It has been present in the human culture since antiquity. In ancient Greece, Zeus would create lightning from the top of mount Olympus when he was in a bad mood. North American Indian tribes attribute lightning to the thunderbird. The thunderbird would produce lightning by flashing its feathers and thunder by flapping its wings.
Benjamin Franklin was the first to realize, as late as the mid 18th century, that lightning was an electrical phenomenon, by noticing the similarity of lightning with the sparks produced by rubbing a dielectric material. He conducted experiments to show that lightning was electrical and in 1750 invented the lightning rod as a protection against lightning. Figure 1.1 shows the famous kite experiment which demonstrated the electrical nature of lightning. This experiment, if conducted correctly, produces sparks from the hanging key to the ground.
We can distinguish two types of lightning discharges, cloud-to-ground and intracloud. Most discharges observed from the ground are cloud-to-ground discharges. An example of a cloud-to-ground discharge, or stroke, is displayed in Fig. 1.2 which is taken form Uman . Lightning discharges are ultimately caused by the air motion around the clouds, inducing charge separation. This stored energy is dissipated in the form of a lightning discharge or stroke. The stroke starts with a stepped leader in a series of 1 steps of length of about 50 m. Before the stepped discharge reaches the ground, a second discharge of the opposite charge starts from the ground. The two discharges meet shorting the circuit and a return stroke is formed which propagates upwards along the ionized channel lowering the charge.
The velocity of propagation of the return stroke is a fraction of the velocity of light [Uman 1987]. A cloud-to-ground discharge acts as a vertical electric dipole antenna and radiates predominantly in the horizontal direction. An intracloud discharge occurs mainly inside the cloud or between clouds in the horizontal direction. It acts as a horizontal electric dipole antenna radiating its energy predominantly upwards as well as downwards. Uman  reports measurements of intracloud discharges suggesting a propagation speed equal to a fraction of the speed of light. The length of the discharge may reach tens, and sometimes hundreds, of kms in the so called spider lightning mode [Lyons, 1994], hence generating a large dipole moment.
A lightning flash lasts for about 0.5 sec [Uman 1987] and is composed of a few strokes, which last for a few msec. A flash may discharge a charge as large as Q=100 C which was originally separated by a distance, on average, of 5 km, dissipating about J of energy and generating an average power of about 109-1010 W. Since the global flash rate, which is distributed preferentially near the equator, is about 100 flashes/sec, the total mean power generated globally by lightning is about 1011-1012 W. As a reference, the averaged power consumption of the United States is about W.
Most of the initial energy stored in the separated charge is dissipated in the form of heat and radio waves [Uman 1987]. In fact only a very small amount of energy is accessible, or released, at the ground in the case of a cloud-to-ground stroke. Lightning generates two types of electric fields: static electric fields are produced by the separated charge, while electromagnetic pulses (EMP) are produced by the moving charges as they are accelerated during the lightning stroke.
a review of lightning fractal antenna and hal
Over the years lightning was thought to be a predominantly low altitude (<10 km) phenomenon with little of its energy coupling to the mesosphere and ionosphere. As a result, recent observations relating lightning to energy dissipation at altitudes between 30 and 90 km came as a major surprise. These phenomena, grouped under the name of high altitude lightning (HAL), include (a) Red sprites, (b) Blue jets, (c) Gamma ray bursts, (d) Radio bursts (e) more? A diagram referring to the different phenomena and the location of their occurrence is shown in Fig 1.3.
Besides the intrinsic scientific interest, HAL phenomena are important in that they can couple large amounts of energy to the upper atmosphere and provide a direct transient coupling between the ionosphere and the stratosphere, with implications to the global electric circuit of the earth and its atmosphere. The high altitude phenomena may also have important influence in the general chemistry composition of the atmosphere, e.g. blue jets occur around the ozone layer peak. Some of these phenomena are characterized, and in fact named, by their optical signatures or their emission spectrum (some of the relevant optical bands and their properties are listed in Table 1.1):
a review of HAL
The observations associated with HAL phenomena came as a major surprise to the community. A number of models were proposed to account for the startling HAL observations. All the models considered electric fields - quasistatic (QS) or electromagnetic pulses (EMP) - as the agents responsible for transferring the energy form the lightning discharge site upwards. The coupling of this power to the upper atmosphere and lower ionosphere generates the observables discussed in section 1.2 (Fig. 1.3).
Lightning generates two types of fields. During the charging state - when the positive and negative charges are separated in the cloud - static fields are slowly generated in the atmosphere and ionosphere which are quickly (with time scale sec at h=80 km) neutralized in the weakly conducting atmosphere. When the lightning stroke occurs, the static fields are suddenly reduced in the ionosphere leaving for a short time an unbalanced electric field. Such a field could be responsible for the energy deposition and emissions. On the other hand, electromagnetic pulses (EMPs) are generated as the moving charges are accelerated during the lightning stroke. The EMPs transfer the energy to the ionosphere by energizing the ionospheric electrons. This energy transfer is then responsible for the emissions.
Red sprites are a relatively new phenomenon, and only few attempts have been made to describe it. Early papers examined the EM radiation from lightning, using dipole models, and ignored the aspects of the energy coupling to the atmosphere. Farrell and Desch  discussed the radio emission spectrum due to the upward propagation of hypothetical return current pulses with 10 msec duration. Their model was intended to explain the absence of VLF emissions between 0.3-15 kHz associated with long-lasting discharges [ Nemzek and Winckler, 1989; Franz et al., 1990]. Hale and Baginski  suggested that an electric field in the ionosphere can be induced by the monopole that remains in the cloud following the discharge. As indicated by Farrell and Desch  the latter model has several difficulties, since it occurs on a temporal scale much longer than that of the observed events. Krider [1992; 1994] considered the electric field radiated by a return lightning stroke, and concluded that considerable fields can be produced in the lower ionosphere if the velocity of the stroke is close to the speed of light. However the physical mechanism which could produce the drastic increase of the lightning stroke speed remains unclear.
The first published theoretical model of red sprites [Milikh et al., 1995], including energy deposition, associated the red sprite generation with transient electric fields induced by large intracloud lightning discharges, which were modeled as horizontal electric dipoles. Heating, ionization and emissions were computed using the results of a model developed by Papadopoulos et al. [1993a] for EMP atmospheric breakdown. They found the overall emissions consistent with a horizontal cloud discharge moment in excess of C-km to produce the required emissions at heights of about 60-70 km. The paper demonstrated that the energization of the ionospheric electrons by the transient fields could account for several of the observed features.
Two subsequent publications reached the same conclusion following a similar approach, but emphasizing different sources for the lightning generated electric fields. The first [Pasko et al., 1995], following an earlier suggestion of Hale and Baginski , assumed that the dominant electric fields were laminar static fields established in the atmosphere after the lowering of a positive charge to the ground by a cloud-to-ground discharge, i.e. an electric monopole at an altitude of 10 km. They then proceeded to compute electron heating, ionization and emissions using the Papadopoulos et al. [1993a] model. They found that the overall emissions were consistent with discharges lowering in excess of 100 C of charge. The analysis emphasized the importance of dielectric relaxation on the field timescale. The second one [ Rowland et al., 1995] assumed that the fields were due to the transient far field of a vertical electric dipole generated by cloud-to-ground discharges and followed a similar analysis as Milikh et al., .
In comparing the laminar [ Pasko et al., 1995] and EMP [ Milikh et al., 1995; Rowland et al., 1995] models against the observations we note that: all models can account for the color and the altitude of the maximum emission. To account for the total optical emission of 10-100 kR, both models require unrealistically large values in the discharge parameters. A reduction by a factor of 5-10 in the required charge, or cloud dipole moment, will be more in line with the statistics of the observations. Furthermore, the laminar model fails to account for the appearance of displaced pairs of red sprites. Displaced pairs occur naturally in the EMP model by considering energy deposition due to the part of the EM pulse reflected from the ground.
All of the above models, while successful in explaining some observed characteristics of the ''red sprites', such as the color and the generation altitude of the emissions, suffer from two important drawbacks. First, dipole or monopole distributions generate electric fields smoothly distributed at ionospheric heights, thereby failing to account for the persistent fine structure of the red sprites which show vertical striations with horizontal size of 1 km or smaller, often limited by the instrumental resolution [Winckler et al.,1996]. Second, the threshold current, or charge, and dipole moment requirements of all three models have been criticized as unrealistically large [Uman, unpublished comment 1995].
Interest in the runaway discharge was recently renewed by the unexpected observations of -ray flashes detected by the Compton Gamma-ray Observatory [CGRO] overflying massive thunderstorms in the equatorial regions [Fishman et al., 1994]. It was shown that the observed -ray intensity and spectrum is consistent with bremsstrahlung, due to a beam of relativistic electrons with MeV average energy, generated at altitudes higher than 30 km. The generation altitude is a key requirement since -rays generated below 30 km will be absorbed by the atmosphere and will not reach satellite altitudes. Therefore, early speculations centered on the runaway discharge driven by the quasi-static fields induced by lightning [Bell et al., 1995; Roussel-Dupre and Gurevich, 1996]. Such runaway air breakdown is an avalanche of electrons, started by a cosmic ray secondary, that get accelerated to relativistic energies in the presence of a sufficiently large electric field. Taranenko and Roussel-Dupre  have proposed that ''gamma ray flashes of atmospheric origin as well as blue jets and red sprites are naturally explained by high-discharges produced by runaway air breakdown''. Such a relativistic electron discharge interacting with the ambient atmosphere would be responsible for the emissions observed in HAL.
Even though the source of gamma ray flashes and blue jets is probably connected with runaway air breakdown, such connection is not so clear for red sprites. The main objection is that all the standard runaway electric field threshold are radically increased in the presence of the Earth's magnetic field [Gurevich et al., 1996; Papadopoulos et al., 1996] which becomes relevant at heights over 20 kms at low latitudes. Therefore, for the later part of the blue jets, and certainly for any participation in the red sprite evolution, the Earth's magnetic field must be included in such models. This increase in the field threshold due to the presence of the Earth's magnetic field becomes the most important constraint in the modeling of these HAL phenomena, as produced by a runaway air breakdown. An additional problem with the runaway discharge at altitudes exceeding 60 km is that the ionization mean free path of the runaway electrons is longer than the scale height of the density gradient in the neutral ionosphere. Hence, the runaway electrons are free to escape the atmosphere.
The main objective of this work is to explore the physical processes related to the newly discovered phenomena of high altitude lightning (HAL). We are interested in determining the essential features of lightning discharges and thunderstorms responsible for the observed HAL phenomena. In particular we will address the issues of red sprites and gamma ray flashes.
As we mentioned in the previous section red sprite models based on the simple dipole lightning discharges suffer from a number of deficiencies. They don't account for the sprite's fine spatial structure and require large charge or current thresholds. To avoid these problems, in this thesis we considered and found that the self-similar structure of the lightning discharge introduces profound modifications in the intensity and structure of the electromagnetic field pattern at high altitudes.
It is well known that lightning discharges follow a tortuous path [ LeVine and Meneghini, 1978]. Williams  showed that intracloud discharges (Lyons  suggested the name spider lightning) resemble the well known Lichtenberg patterns observed in dielectric breakdown. These patterns have been recently identified as fractal structures of the Diffusion Limited Aggregate (DLA) type with a fractal dimension [Sander, 1986; Niemeyer et al., 1984].
Existing observations show that red sprites are generated by some of positive cloud-to-ground discharges (+CG). As suggested by Lyons  sprite generating +CGs are associated with intracloud spider or dendritic lightning known to accompany many +CG events. He presents a qualitative model which is based on the fact that horizontal discharges of the order of 100 km have been observed in connection with +CG events. The model starts with the initial spider lightning followed by the positive leader toward ground, which in turn is followed by the positive return stroke. The latter generates an intracloud lightning discharge which propagates along the (fractal) spider channel.
The observed delays between +CG events and associated sprites, which ranges from a few ms to tens of ms, favor this model, since they correspond to the time needed to develop the intracloud fractal discharge, while if sprites are caused directly by +CG events they will be delayed by less than a msec. In addition, such model explains why red sprites occur during thunderstorms having a dimension in excess of 100 km, since long horizontal spider discharges impose a large size requirement for the thundercloud extension.
The similarity between lightning discharges and dielectric breakdown helps to understand the role played by +CG discharges in the red sprite generation. In fact, surface dielectric breakdown develops much intense structure if caused by the immersed positive needle than by the negative one, as revealed by Lichtenberg figures Atten and Saker, 1993]. A simple physical explanation of this effect is that the positively charged needle pulls electrons from the media, which is more energetically efficient than pushing electrons ejected by the negatively charged needle. Similarly, the intracloud lightning will acquire better developed fractal structure if caused by a positive rather than negative return stroke.
By incorporating the dendritic fractal structure of the lightning channel as described by Williams  and by Lyons  in the calculation of the lighting induced fields, results in a natural explanation of the observed fine structure of the red sprites and in a significant reduction of the required threshold charge or equivalent dipole moment.
Therefore, in this thesis we study the properties of fractal antennae in general.
Besides its application to the sprite problem, fractal antennae are interesting in their own right. For example, fractal antennae are broadband and due to their spatially structured gain could be of particular interest in ionospheric modification and in other settings [Werner and Werner , 1995; Jaggard 1990].
In particular, careful consideration will be given here to the dependence of the antenna gain on the fractal description of the antenna, e.g. its fractal dimension. Among the most relevant issues concerning fractal antennae, specially for lightning, is that their radiation pattern exhibit an increase in the radiated power density, at least at certain positions, as compared with non-fractal (dipole) models. It is this power density increase that will be responsible for reducing the required threshold current and charge to produce the red sprites. At the same time the spatial structure of red sprites can be understood in terms of the spatio-temporal radiation pattern of the fractal antenna. Due to the fractality, and its inherent power law distribution, the radiation pattern from the fractal antenna will be broadband, meaning that it will be relatively insensitive to the type of current pattern that produces the radiation.
The lightning discharge will radiate as a fractal antenna that, unlike a dipole antenna, generates a spatially non-uniform radiation pattern with regions of high field intensity and regions of low field intensity. The non-uniform radiation pattern can cause the observed fine structure of red sprites. Furthermore, such a fractal antenna naturally leads to an increase in the radiated power density, as compared with dipole-type models.
We apply the ideas from fractal antennae to the generation of red sprites. The concept of antenna gain and spatial structure will be given special consideration for specific fractal models of lightning discharges. Here we will include the propagation, with self absorption, of the lightning induced radiation fields through the lower ionosphere, the heating of the ambient electrons, and the subsequent optical emissions. The required current (and charge) threshold and the emission pattern depend critically on the type of discharge, i.e. its dimension. The entire energy transfer process is modeled with the help of a Fokker-Planck code. This code was developed to study the modification of the electron distribution function [Tsang et al., 1991] in the presence of field energization and inelastic loses.
The modeling of lightning as a fractal antennae reduces the required lightning current (and charge) amplitude for the production of sprites as compared with dipole models. Furthermore, such a fractal antenna naturally generates a spatially inhomogeneous emission pattern.
While the gross phenomenology of the emissions, termed red sprites, has been known for some time now, their spectroscopic structure is only currently emerging [Mende et al., 1995; Hampton et al., 1996]. We generated the first model of the red sprite spectrum, with the help of a Fokker-Planck code [Tsang et al., 1991], that is based on the energization of ionospheric electrons by lightning induced fields. Comparison of the modeled spectrum with the measured ones [ Mende et al., 1995; Hampton et al., 1996], constraints the local power density absorbed by the plasma and the electron energy spectrum in the emission region. This last result may help us discriminate among different models.
For heights of relevance to HAL phenomena, i.e. h>20 km, the standard electron runaway process under a static electric field is strongly influenced by the presence of the magnetic field, specially close to the equator, where most of the thunderstorms occur and the -ray flashes observed. We have therefore, developed the theory of the runaway process under the influence of both electric and magnetic fields The conditions for the electron runaway are different from those described for a pure static electric field. In fact, the electric field threshold is critically increased in the presence of a magnetic field suggesting that the runaway process may be an improbable candidate as a source of HAL phenomena, at least for red sprites. We will estimate the size of the runaway basin in momentum space as a function of the electric and magnetic fields. Such determination will be relevant in understanding the feasibility of the runaway process in each situation. Furthermore, we will compute the diffusion coefficient in the presence of the magnetic field for the case of the E and B parallel.
The thesis is organized as following:
Chapter 2: Fractal antennae
Chapter 3: Red Sprites
Chapter 4: Spectrum of Red Sprites
Chapter 5: Runaway Discharges in the Presence of the Magnetic Field