A SIMPLE MODEL TO ESTIMATE ELECTRICAL DECAY TIMES IN ANVIL CLOUDS

A model is developed that uses measured particle-size distributions to show that pre-existing electrification can persist for well over an hour in realistic anvil clouds if the initial electric-field intensities are high enough. INTRODUCTION Thunderstorm anvil clouds are a cause for concern to the space-launch community because they are known occasionally to be strongly electrified and because they are suspected to be capable of storing charge for long periods of time [e.g., Marshall et al., 1989]. Therefore, during 2000 and 2001 an experiment was conducted in the vicinity of the NASA Kennedy Space Center, Florida, to measure both the ambient electrostatic fields and the size/shape distributions of the cloud and precipitation particles in anvils [Dye, et al., 2002]. The purpose of the present modeling is to predict the time required for the decay of electrification within such clouds from the measured microphysical properties. The decay modeled here is that caused by electrical conduction currents flowing in the reduced conductivity within these clouds. A companion paper [Dye, et al., this conference] discusses these predictions in relation to the observed radar, electrical, and microphysical structure of the same clouds. MODEL DESCRIPTION Based on a suggestion by Paul Krehbiel [NASA/USAF Lightning Advisory Panel (LAP) meeting, Tucson, AZ, January, 1998], a simple model has been developed to calculate the temporal decay of the vertical electric field, E(t), within a previously charged, horizontally stratified anvil cloud, given a measured particlesize distribution, N(d). This model envisions a microphysically uniform and constant, motionless cloud that contains a thin layer of positive charge between two thin, negative screening layers. (For simplicity each screening layer is assumed to contain half the charge area density of the internal positive layer.) The positive and negative ions within this cloud are assumed to have equal concentrations and identical properties. Because of the simple geometry, all variables are uniform in magnitude throughout the bulk of the cloud (between the charge layers), so the volume charge density there remains zero. Thus, dE/dt = -J(t)/ε0 = -2ekn(t)E(t)/ε0 (1) where J(t) is the vertical conduction-current density, e is the electronic charge, k is the small-ion mobility, n(t) is the polar small-ion density, and ε0 is the dielectric permittivity of free space. (The sign convention is such that the vertical vector components, E and J, are positive above the internal positive layer and negative below it.) Krehbiel [1967] had shown that J(t) becomes constant -independent of both k and E(t) -when the electric field is strong enough. In this limit the field decay is linear and can be quite slow. Our analysis begins with the steady-state, small-ion budget equation in a population of stationary, mono-disperse, spherical cloud particles, from Pruppacher and Klett [1978, Eq. 17-40]. After neglect of smallion recombination and aerosol attachment, simplification to uncharged cloud particles, generalization to allow non-spherical shapes, and use of the "Einstein relation" [e.g., Pruppacher and Klett, 1978, Eq. 12-21], this equation becomes, q ≈ Ae(d)kN(d)n(t)E(t) + [C(d)/ε0][kKT/e]N(d)n(t) (2) where q is the ionization rate, Ae(d) is the effective electrical cross section of a particle of long dimension, d, C(d) is the electrical capacitance of that particle, K is Boltzmann's constant, and T is absolute temperature. The first term on the right represents the small-ion loss rate due to field-driven attachment of ions to cloud particles, which dominates at high enough electric-field intensity, producing an ion density that is inversely proportional to field. The second term is the diffusive loss rate, which dominates at low fields, resulting in an Ohmic conductivity, independent of E. When N(d) is a particle-size spectrum, each term on the right-hand side of (2) must be regarded as an integral over the size distribution. Equation 2 can be solved for n(t) and inserted into (1) to give a first-order, non-linear differential equation for dE/dt: dE/dt = -2eq{ε0N(d)[Ae(d) + C(d)KT/(ε0eE(t))]} (3) where again each term in the denominator on the right is to be considered an integral over the size distribution. Notice that, although both loss terms in (2) depend on the small-ion mobility, (3) is independent of k; thus no results below (except those in Figure 2 itself) depend on k. This equation has been solved numerically to obtain E(t) for various observed particle-size spectra. In general Ae and C are functions of the shape, as well as the long dimension, of the cloud particles. For all numerical calculations herein, however, we have approximated particles of all sizes by spheres of diameter, d. Thus, Ae(d) = 3πd/4 and C(d) = 2πε0d. This turns out to be the conservative approach, as it predicts the slowest possible electrical decay for a given size distribution. RESULTS Here we present one example of this numerical solution for a dense anvil cloud that was penetrated at 210800 UT on 13 June 2000 at a flight altitude of 10.5 km. This case was chosen for illustration because it is typical of the high particle concentrations, field intensities, and radar returns that were encountered at the windward ends of well-developed anvils, just downstream from the convective cores of their parent thunderstorms. (The spatial and temporal structure of this same anvil are discussed in detail by Dye, et al. [this conference].) Figure 1 gives the measured size distribution -a composite of data from the FSSP, 2-DC, and HVPS instruments [Dye, et al., 2002] -integrated over about 3.5 km of flight track. From this N(d) it is easy to calculate size spectra of the small-ion loss rates that are represented by the two terms on the right-hand side of Equation 2. Per unit small-ion density -that is, with n(t) cancelled out -the field-driven-attachment rate is shown in black and the diffusive-loss rate is shown in gray in Figure 2, where we have taken k = 3.6 X 10 m/(Vs) and T = 225 K. To compute the former rate, it is necessary to assume an electric-field intensity. For that purpose we have computed the "transition field," Eγ, at which the two loss rates, integrated over particle size, are equal -551 V/m in this case. (The only effect of changing the ambient field is to shift the black curve vertically in proportion to E.) In plotting the figure, each of these loss terms has been multiplied by the particle size, d, to compensate for the effect of the logarithmic horizontal axis. This weighting is convenient because larger magnitudes on the graph make larger contributions to the total ion-loss rate (the integral over particle size). 1.μ 10-6 0.00001 0.0001 0.001 0.01 Long Dimension HmL 1