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MODELS OF THE DEVELOPMENT OF THE ELECTRICAL STRUCTURE OF CLOUDS 142 agreement with the other two-dimensional models, this model demonstrates that both strong horizontal and vertical fields can be produced by charge separation mechanisms that depend on precipitation. The horizontal fields are generated by horizontal displacement of the charged particles by the air circulation. Even under very weak shear conditions the space-charge centers were found to be displaced horizontally and produce very strong horizontal fields even close to the cloud base. The presence of the shear was found to smooth the development of the charge centers by limiting mixing of precipitation particles of opposite charges. Takahashi (1979) developed a two-dimensional, time-dependent model of a small warm cloud, which treats the microphysics and the macroscale dynamics in detail. Electrification due to the inductive mechanism and to ion attachment by diffusion and conduction is considered in a way that seems to explain the weak electrification of warm maritime clouds. The most important mechanism responsible for charging in such clouds, according to this model, is the attachment of ions to cloud and precipitation drops. This attachment is significantly enhanced during condensation and evaporation. During the former, positive ions are preferentially incorporated into the growing drops, whereas during evaporation negative ions are preferentially attached. In an attempt to evaluate the effectiveness of the convection electrification process, Ruhnke (1972) and Chiu and Klett (1976) developed two-dimensional, axisymmetric steady-state models. Ruhnke calculated the electric fields and charges that arise from ion attachment to cloud water in solenoidal flow, intended to represent the flow in an isolated convective cloud. The actual cloud volume (where liquid water exists) was assumed to be spherical and to be entirely within the updraft. Space charge arises owing to local differences in electrical conductivity. These differences stem from attachment of ions to cloud particles (assumed to depend only on liquid-water content), which form ion currents consisting of both conduction and convection currents. By assigning a specific liquid-water content and a relation between it and ion conductivity, Ruhnke avoided dealing with interactions between ions and cloud droplets. The steady-state assumption precludes any detail of the initial development of the convective electrification. His results show that only very small fields can be developed by this process. Chiu and Klett (1976) improved on this model by using a more realistic convective circulation in which the updraft was within the cloud and the downdraft was at its edges. They also considered the effect of turbulent diffusion in addition to conduction and convection currents on the transport of ions. Attachment of ions to cloud drops was affected by the liquid-water content and by the ambient electric field. Chiu and Klett's results show that convective electrification by itself cannot explain the strong electrification in thunderclouds. One should bear in mind that the terms that are highly variable with time such as the rate of charge buildup, especially at the later stages of thunderstorm development, are ignored in these steady-state models. In a fully developed time-dependent model, such terms may modify the above conclusions. In addition, the dynamics used in the convective models is parameterized and may not be realistic enough to simulate the real convective charging process that is highly dependent on cloud dynamics. Three-Dimensional Models To date, only one three-dimensional, time-dependent model of an electrical cloud has been developed (Rawlins, 1982). This model uses pressure as the vertical coordinate with grid spacings of 50 mbar vertically and 1 km horizontally. The microphysical parameterization of Kessler (1969) is used to describe the growth of cloud particles into precipitation size. Ice, initiated by ice nuclei that freeze the supercooled water drops, is represented by three size classes: 0-100 Âµm, 100-200 Âµm, and 200-300 Âµm in radius. Hail is designated as ice greater than 300 Âµm in radius, and it is forced to be distributed exponentially in size (Marshall and Palmer, 1948). With this model Rawlins tested the effectiveness of various electrical processes, such as the inductive and the contact surface potential (noninductive) mechanisms. He assumed that ion attachment to cloud particles can be ignored altogether. He concluded that the noninductive process is able to produce fields of high enough intensity to initiate lightning within about 20 min after precipitation begins. This process produced a ''classical" dipole [see Figure 10.7 (a)] but without the small positive charge center closer to the cloud base. The inductive process, involving ice-ice collisions, was capable of producing strong fields only in the presence of high concentrations of ice particles and only 30 min after precipitation particles appeared. With this process, a very complex space-charge structure emerged [see Figure 10.7 (b)] as was also found by Kuettner et al. (1981). Allowing ice crystals to rebound more than once from ice pellets reduced the calculated maximum field to below that needed for lightning [compare the values of E z in Figures 10.7(a) and 10.7(b)]. However, one should note that the same restriction of multiple collisions was not applied to Rawlins's calculations of the noninductive process. Multiple collisions, if