Charge screening layers around electrified clouds

Analytical solutions to the cloud charge screening-layer model of Brown et al. are obtained for planar, cylindrical, and spherical cloud geometries. Two specifications of the negative ion concentration n− at the surface of a positively charged cloud, representing upper and lower approximations to the actual supply of ions, are employed: (1) n− = n0 = constant, and (2) n− decays with time t approximately as (1 + αt)−1, where α is constant. The latter form for n− comes from a model that allows for space charge divergence outside the cloud but ignores ionization and recombination. In mksa units the maximum (saturation) space charge is found to be (4/e2) (ε0E0/λ0), where e = 2.718 …, ε0 is the permittivity of air, E0 is the initial cloud electric field, and λ0 is the mean free path of ions in an uncharged cloud (typically, λ0 = 10 meters). The surface electric field when the cloud particles first become saturated is E0/e2, and the final depth of the saturated layer is λ0/4. If the relaxation time of the surface electric field for the first case is τ (typically 40 sec), then the relaxation time for the second case is 1.94 τ. The screening-layer formation time (i.e., the time required for the surface screening-layer charge density to be within e−1 of its saturation value) is 0.49 τ in the first case and 0.67 τ in the second. Brown et al. obtained numerical solutions for the field and screening charge density as a function of time and depth for the special case of a flat layer cloud with n− = n0 at the surface. The agreement between their results and ours for these same conditions is excellent.