The origin of thunderstorm electrification has long been an unsolved problem in atmospheric physics. Despite a number of simulated laboratory experiments, together with the vast amount of field data collected over the past few decades, our knowledge of how these convective cloud masses get charged still remains sparse at the microphysical level. Sir John Mason in the Bakerian Lecture identified thunderstorm electrification as one of the three leading unsolved problems in cloud physics. He had this to say about the problem: This is, for me, the most intriguing and challenging problem in cloud physics, with a strong incentive to understand one of the most spectacular of natural phenomena, but made all the more interesting by the fact that the search for a continuing solution has led us into a number of rather difficult areas of classical physics, and to a deeper study of the fundamental properties of water and ice. A satisfactory theory must be able to explain all of the observed electrical characteristics of a typical thunderstorm.

Experimental observations of the optical and electromagnetic fields generated by lightning flashes during the last 50 years have significantly advanced our knowledge concerning the mechanism of the lightning flash. Nevertheless, this knowledge is not as exhaustive as that of long laboratory sparks due to our inability to observe lightning flashes under controlled conditions. Thus, the mathematical description of the mechanism of a lightning flash is relatively poor at present even though the main features of lightning flashes themselves are well known. The main goal of this chapter is to provide the reader with the important features of the mechanism of the lightning flash. No attempt is made to provide an exhaustive list of the literature since this can be found elsewhere. Nomenclature: in this chapter a positive discharge is defined in such a way that the direction of motion of electrons in such a discharge is opposite to that of the discharge itself; a negative discharge is defined as one in the opposite sense. According to this definition a negative return stroke is a positive discharge and a positive return stroke is a negative discharge. A positive field is defined in terms of a negative charge being lowered to ground or positive charge being raised. According to this definition a lightning flash that transports negative charge to ground gives rise to a positive field change.

A lightning flash is initiated by electrical breakdown of the air in a cloud. This process, commonly known as the preliminary breakdown, signifies the initiation of a stepped leader. Such a stepped leader propagates towards the earth, in a succession of nearly discontinuous surges or steps. The stepped leader leaves a charged, conducting channel in its wake. When the leader reaches the ground, the current flowing in the channel increases abruptly, marking the beginning of the return stroke. After the first return stroke, several subsequent return strokes may occur, each of which is preceded by a fast, continuously moving leader the dart leader which propagates from cloud to earth down the channel made by the stepped leader. This chapter is concerned with the mathematical modelling of return strokes.

In this chapter, we will present the theory describing the interaction of lightning electromagnetic fields with overhead lines, with particular reference to power systems. In the first part of the chapter, we will present the different approaches and formulations that can be used to describe the coupling between an external electromagnetic field and a transmission line. Then, we will extend the selected field-to-transmission line coupling model to include the effects of a lossy earth serving as a return conductor and to deal with the case of multiconductor lines. The time-domain representation of coupling equations, useful for analysing nonlinearities, will also be dealt with.

Thunderstorms are natural weather phenomena and there are no devices and methods capable of preventing lightning discharges. Direct and nearby cloud-to-ground discharges can be hazardous to structures, persons, installations and other things in or on them, so that the application of lightning protection measures must be considered.