Conclusions

The objective of this thesis is to provide the physics framework within which some of the observed high altitude lightning phenomena can be studied and quantitatively understood and modeled. These HAL phenomena represent clear evidence of the lightning induced energy dissipation in the lower ionosphere during a low altitude thunderstorm. The lightning energy can be coupled to the upper atmosphere by a multitude of processes such as electromagnetic pulses and runaway beams. In this thesis we studied the properties of these processes as related to HAL.

Electromagnetic Pulses

Electromagnetic pulses (EMP), as generated from our fractal lightning models, can energize the electrons in the lower ionosphere, inducing electronic transitions, as they collide with the molecules in the lower ionosphere. These electronic transitions are then followed by emissions, which are termed red sprites. The theoretical understanding of the red sprites including energy deposition was the subject of three recent publications which considered the lightning induced electric fields as produced by horizontal [Milikh et al., 1995] or vertical [Pasko et al., 1995, Rowland et al., 1995] dipole electric models. The models were able to account for the energetics of the sprites, but produced highly homogeneous and smooth electron heating in the lower ionosphere resulting in the absence of internal structure in the optical emissions. At the same time, Winckler et al. [1996] conducted a detailed study of the spatial structure of sprites with the conclusion that an extremely important characteristic of red sprites is their fine spatial structure, sometimes even down to the detector resolution. Furthermore, the threshold current and dipole moment requirements of all three models have been criticized as unrealistically large [Uman, unpublished comment, 1995].

We presented a novel model of red sprites, the first model to account for the fine structure of the sprites, that is based on the fact that the low altitude lightning has a fractal structure which is reflected in the subsequent spatially dependent optical emission pattern.

We conducted an extensive analysis of fractal antennae to study their properties as compared with dipole type of models. The most important results seems to suggest that by having a power law distribution of phases a fractal antennae can give a considerable increase in the radiated power density, sometimes by a factor of 10, as compared with equivalent dipole models. Such increase in the power density can be of extreme relevance to the modeling of red sprites. Furthermore, fractal antennae can naturally give a spatially structured radiation pattern. The radiation pattern depends on the structure of the discharge, but we expect, as seen from simple fractal models, that the most relevant parameter in determining the spatially dependent radiation pattern is the dimension of the self-similar fractal.

We applied the concept of fractal antennae to the sprite phenomena. Lightning was modeled as a self-similar fractal discharge from which the fields in the lower ionosphere were computed, including self-absorption. The kinetic treatment of the energy deposition and the optical emissions were computed with the help of a Fokker-Planck code [Tsang, 1991]. Beside the trivial parameter I_o and Q and $\beta$ , we also found that the electric field power density and the structure of the optical emissions is critically dependent on the dimension D of the discharge. In fact, the power density scales as $S(W/m^2)\sim \beta ^2I_o^2\,f(D,L,\omega )\,g(\theta ,D,\beta ,\omega )$ , where $\,f(D,L,\omega )$ is the efficiency function of the different fractals. $g(\theta ,D,\beta ,\omega )$ represent the spatial structure of the radiation pattern with $\theta$ as the angular position. These models suggest that we can obtain realistic emission intensities with a current threshold of $I_o\sim 100$ kA (hence a $Q\sim 100$ C) for a particular discharge model, e.g. $\eta =3$ and for n $_f\geq 50$ . Fractal discharges of different dimensions have varying discharge parameter thresholds, but in general a sprite can be generated with I $_o\geq 100$ kA.

Statistics of intracloud lightning are in the best cases incomplete, but information about some independent measurements can be found in the book by Uman [1987]. It seems to suggest that in extreme cases the intracloud discharge can reach $Q\sim 100$ C or 100 kA, with a length of 10s of kms. Some rough estimates can be made of the relevance of the model discharge parameters by comparing with the statistics of cloud-to-ground discharges given in Uman [1987].

Such estimates can be made more precise if we assume the qualitative model suggested by Lyons [1996] where the 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. Therefore, in this model, the statistics of +CG current amplitudes could be related to the particular intracloud events responsible for the sprite generation. In cloud-to-ground discharges, a current peak of $I_o\sim 100$ kA, consistent with a charge transfer of Q=100 C, occurs between 1-5 % of the time. The speed of propagation of a cloud-to-ground is about $\beta \sim 0.5$ , but intracloud discharges seem to propagate with speeds of an order of magnitude smaller [Uman, 1987]. Consequently, our model discharge parameters seem to agree with the sprite occurrence [Lyons, 1994] and the statistics of lightning discharge parameters [Uman, 1987].

We also constructed a model of the red sprite spectrum due to molecular excitation by ionospheric electrons accelerated by the lightning induced electric field. A valuable output of the model is the scaling of the relative intensities of the emissions with the value of the electric field and/or power density. Such scaling can provide additional constraints to the required energy deposition in the red sprite region by comparing with spectrum measurements. Proper account was done of the wavelength dependent atmospheric attenuation. In principle, the model could yield the spatial profile of the amplitude of the electric field in the emission region from spatially resolved measurements of the spectrum. The model also reveals some differences between the aurora and sprite spectra: in the aurora both permitted and forbidden transitions play a noticeable role, while in sprites only permitted transitions are important. Finally, it seems that sprites are produced by electrons of much lesser energy than that of auroral electrons.

Runaway Beams

A new type of electrical air breakdown, called runaway breakdown or runaway discharge, was discussed recently by Gurevich et al. [1992] and applied to the preliminary breakdown phase of a lightning discharge. If the local electric field is large enough an electron breakdown, or runaway, discharge can be created. It is often assumed that these energetic electron beams may be related to some of the other phenomena related to HAL: blue jets, gamma ray burst, radio burst pairs. The biggest issue related with runaway discharges is the fact that these phenomena seem to be occurring at heights where the magnetic field effects must be included. But when the magnetic field is incorporated in the equations, the field threshold conditions for the creation of the electron beams is substantially changed.

Therefore, we have developed the theory of the runaway beam in the presence of static electric and magnetic fields. The role played by the geomagnetic field in the runaway process for heights less than 20 km is negligible. Nevertheless the geomagnetic field plays a noticeable role at heights which ranges from 20 to 30 km. In fact, it significantly changes the threshold electric field E $_\eta$ for $\theta _o\geq 45^o$ , where $\theta _o$ is the angle between the $\mathbf{E}$ and $\mathbf{B}$ fields. At the height above 40 km the effect of the geomagnetic field dominates at large angles $\theta _o$ and the conditions for runaway breakdown becomes even more hindered.

Thus taking into consideration that the static electric field due to thunderclouds is directed almost vertically, one can expect a significant difference in the parameters of high altitude discharges as they occur in the equatorial and midlatitude regions. Close to the equator, $\theta _o\sim \pi /2$ , and for high altitudes, (z>40 km), the runaway breakdown is hindered. While at midlatitudes, for $\theta _o\leq 45,$ the runaway process can proceed freely. Such observations put strong constraints on the type of models needed to explain these high altitude phenomena. At least close to the equator extremely large field may be required since the runaway process is substantially hindered at the relevant heights.

We have computed the basin of acceleration in the electron momentum phase space as the relative importance of the electric and magnetic fields are varied, giving an idea of the relative feasibility in producing the runaway avalanche. To get an idea of the importance of the diffusion process in the runaway discharge, we have computed the diffusion coefficient in the presence of the magnetic field in terms of quadratures for the specific case that E and B are parallel.

It seems that the runaway process is not energetically efficient to produce red sprites due to the height in which they occur, where the B field of the Earth would hindered the process. Even though we expect that the runaway process is ultimately responsible for the blue jets and gamma ray bursts, it is not clear yet what are the specifics of the phenomena. The work must continue.

Implications to Future Work

From the results developed in this thesis we propose a few directions in which to continue the work.

The latest observations of sprites reveal filaments that can be described as streamers propagating down from the main body of the sprite (see Fig. 6.1) with a cross-sectional diameter of 100 m or less. Given the nucleated spatial structure in the conductivity produced by the fractal lightning discharge, the streamers would start naturally in the presence of a laminar field. Therefore, a more comprehensive model of red sprites that includes both the laminar and electromagnetic lightning induced fields and their effects in the lower ionosphere can be developed from a model that solves the nonlinear wave equation, Eq. (B.2), and includes ionization and charge separation. Furthermore, the streamers will strongly influence the red sprite spectra.

**Figure 6.1:** A sprite. The picture shows clear streamers, or filaments, that reach downward from the main body of the sprite.
$\begin{figure} \center \includegraphics [width=3in,height=4in]{images/streamers.eps}\end{figure}$

There is still no comprehensive theory for blue jets, gamma ray flashes and radio bursts. We expect that these phenomena are ultimately related to runaway beams, but the analysis must include the magnetic field. We should develop the proper kinetic theory of runaway discharges in the presence of a magnetic field by solving selfconsistently the Boltzmann equation, Eq. (5.5), in 3D momentum space. Besides its general scientific interest, it is expected that such analysis will be extremely relevant for the understanding of blue jets, gamma ray flashes and radio bursts.