Electrical screening layers around charged clouds

Grenet and Vonnegut have shown that an electrified cloud surrounded by conducting clear air acquires an electrical screening layer around its outer boundaries by droplet capture of ions. The process has been modeled for conditions in which diffusion is assumed to be negligible. Numerical solutions have been obtained for the charge accumulations, for the distribution of the electric field, and for the forces acting on cloud droplets in terms of the length l0 of the ionic mean free path in the cloud. The maximum charge concentration is found to be about 0.5417 ∈0E0/l0 with 63% of the maximum being acquired in a time of ∈0/2 λ± where λ± is the appropriate polar conductivity and ∈0 the effective permittivity of the clear air. The layer's effective thickness for the first 63.2% of the charge is 1.34 l0 and 2.41 l0 for the first 86.5% of the charge. The computed maximum electric force per unit volume is 0.21 ∈0E02/l0 with a maximum divergence of 0.29 ∈0E02/l02 above a maximum convergence of 0.099 ∈0E02/l02. The time constant for a Naperian e-fold increase in cloud droplet concentration by electrostatic convergence in Stokes law flow is estimated as about 20 μl0/(ā∈0E02) where μ is the molecular viscosity of the air and ā is the average droplet radius.