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MODELS OF THE DEVELOPMENT OF THE ELECTRICAL STRUCTURE OF CLOUDS 133 sign of charge, fall faster and farther down with respect to the convected air, leaving the oppositely charged smaller particles above. As the charge centers build up, discharging mechanisms become more effective, and these should be included in any complete description of cloud electrification. Two kinds of discharge are possible: (1) discharge by collisions between drops and ions of opposite polarity and (2) discharge by collision and coalescence with cloud particles of opposite charge (see Colgate et al., 1977). The first mechanism depends on the electrical conductivity of the small ions (Î» = nek, where n is the ion number density, e is the electronic charge, and k is the electrical mobility) and on the ion diffusivity. Often ions of different polarities have different diffusivities; those with higher diffusivity attach preferentially to cloud and free aerosol particles. However, after some charge is built up on a cloud particle, further ion diffusion to it will be limited. Another factor affecting ion attachment to cloud particles is the electric field in the cloud. Strong fields will move free ions from one region to another. In so doing, they also increase the conduction currents and, hence, the discharge of cloud particles. In addition, when the electric field exceeds about 50 kV/m, corona discharge begins near the corners of ice crystals and high concentrations of ions are generated. These ions increase the electrical conductivity and help to prevent or slow down the further buildup of the space charge. The other discharge mechanism (collection of oppositely charged cloud particles) takes place at all stages of particle growth. This mechanism is enhanced if the interacting particles are highly charged, have opposite polarity, and when the ambient field is strong. In this case the collection efficiency increases by increasing the collision efficiency (Coulomb attraction changes the trajectories of particles relative to each other) or by increasing coalescence efficiency (not allowing bouncing, and hence no charge separation, to occur) or by both. Therefore, for a charge mechanism to be effective, it has to separate sufficient charge at a rate sufficiently high to overcome these discharge processes. The many charge-separation mechanisms and their complexity require a detailed discussion that is beyond the scope of this chapter. We recommend that the interested reader refer to Chapter 9 (this volume) by Beard and Ochs and to Mason (1971). The various charging mechanisms that have been proposed as possible major contributors to electrification of thunderstorms can be divided into two major classifications: (1) precipitation mechanisms requiring particle interactions with subsequent space-charge separation by gravitational sedimentation and (2) ion attachment to cloud or precipitation particles and then charge separation by either gravitational settling or by atmospheric convection (updrafts or downdrafts). Mechanisms from group (1) above are divided into two major subprocessesâinductive and noninductive. Most models to date treat these mechanisms with various degrees of detail. On the other hand, only a few models are available that treat the ion convective process. Consequently, since the intention here is to review the present state of knowledge in modeling electrical development in clouds, most of the emphasis is placed on the models dealing with the precipitation mechanisms. As discussed later, there are still a great many questions that these models cannot answer. Inductive Process Charge can be separated by the inductive process during rebounding collisions of particles embedded in an electric field. This mechanism, which is relatively simple to formulate, was treated intensively in cloud electrification models. According to Sartor (1967) and Scott and Levin (1975) the amount of charge that is separated per collision by this process is In this equation âQ represents the charge transfer to the large particle as a smaller particle of radius r collides and rebounds in an electric field E (defined as positive when a positive charge is overhead), making an angle Î¸ between the field and the line connecting the centers of the particles at the point of separation; Î¦ and Ï are constants that depend on the ratio of sizes of the colliding particles (Ziv and Levin, 1974); Q and q represent the initial charge on the particles before the interaction; t c is the contact time of the colliding particles and Ï the relaxation time of the charge carrier (=ÎµÎµ0/K, where Îµ and K are the dielectric constant and the electrical conductivity, respectively; and Îµ0 is the permittivity of free space). The first term on the right-hand side of Eq. (10.1) represents the charge that is transferred from the small to the large particle because of the inductive polarization effect. One can see that the stronger the field or the larger the size of the smaller particle, the larger is the charge separated. The constant Î¦ represents the enhancement of the electric field around the colliding particles, as compared with the the ambient field. Particles may collide at the head-on position but will skid or roll on each other and finally separate at the angle Î¸. For water drops, Î¸ may vary between 50Â° and 90Â° (Levin and Machnes, 1977). Large liquid particles sometimes