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CHARGING MECHANISMS IN CLOUDS AND THUNDERSTORMS 116 stage (10 Âµm, 100 cm-3), the depletion time constant, t = (4ÏRDiN)-1, is 10 sec. Thus equilibrium is achieved rapidly, and the local concentration in clouds can be approximated by a steady-state balance between production by cosmic rays and depletion by recombination and attachment to cloud droplets (Chiu and Klett, 1976). The amount of charge on a cloud droplet from kinetic theory is a Gaussian-like (Boltzmann) distribution centered on zero charge for equal concentration of positive and negative ions. As is evident by the positive space charge near the ground, the ion mixture is not always neutral. In such cases a net charge is collected by droplets. For the zero- centered distribution the rms Boltzmann charge can provide an estimate for the magnitude attained in diffusion charging: where Îµ0 is the permittivity of air (8.85 Ã 10-12 F/m), k is the Boltzmann constant (1.38 Ã 10-23 J/K), and T is the temperature (Gunn, 1957). This equation can be interpreted as a balance between the stored electric energy on the droplet (1/2 Q2/4ÏÎµ0R) and the thermal motion energy of the ions (kT). When the Boltzmann charge is evaluated for a typical droplet size in the cloud stage (R = 10 Âµm), the rms charge in number of electrons is ne = 6RÂ½(Âµm) = 19. This result is consistent with the spread in droplet charge measured for nonconvective clouds with low electric fields (Gunn, 1957). Drift Charging Larger-scale transport of ions is characterized by currents from bulk and eddy transport of ions along with the field-driven drift. The charge-flux equation is the larger-scale version of Eq. (9.1) where K is the eddy diffusivity (m2/sec). If we consider just the drift of ions in the ambient field we find that the vertical drift current (ÏiBiE) at cloud top and cloud base must result in an accumulation of positive and negative space charge, respectively (Figure 9.2). An electric balance is achieved fairly rapidly as a screening layer forms with the capture of incoming ions by cloud droplets. The field within the cloud is increased by the charge that accumulates at the boundary. The amount of charge on droplets in this region can be estimated by considering diffusion capture by ions of only one sign. The maximum charge captured is found from the amount needed to neutralize the induced charge (of opposite sign) from polarization in the electric field: Figure 9.2 Electrification of a model cumulus cloud (after Chiu and Klett, 1976): a, vertical drift currents reflecting the ion deficits with the cloud; b, resultant charge accumulation and field enhancement from ion drift; c, effect of convective transport on charges and field. The capture of ions from the drift current during the cloud stage, for example at cloud base (Figure 9.2b), results in ne = 0.002 R 2 (Âµm) E (V/m) = 20 for R = 10 Âµm and E = 100 V/m. This charge is comparable with the Boltzmann charge given by Eq. (9.3). However charge generation from drift into cloud edges increases with R 2 in contrast to the dependence of Eq. (9.3). Thus charges of several hundred electrons are readily attained for somewhat larger cloud droplets by diffusion of ions of one sign to polarized drops. Somewhat later in this section we shall find out that the size and field dependence given by Eq. (9.5) also applies within clouds for ion capture by polarized drops and for the breakup of these drops. In addition, for the cloud stage we must also consider the role of the bulk and eddy transport terms in Eq. (9.4). For diffusion charging and the simple convective pattern, illustrated in Figure 9.2c, convection transports negative charge upward within the core and carries positive charge downward along the edges (Chiu and Klett, 1976). The effect of eddy diffusion in this simple model is to smooth the charge distribution produced by the bulk transport and ion drift. When all three terms in Eq. (9.4) are included, the field is enhanced within the cloud but is not so strong as the pure drift case. For this early stage of cloud electrification, drift charging of droplets with negative charge at cloud base and positive charge at cloud top is apparently the dominant mechanism. The current into cloud base from drift