Effective dielectric constants of mixed-phase hydrometeors

Melting snow, graupel, and hail are often modeled as uniform mixtures of ai r-ice-water or ice-water. Two-layered models have also been proposed in whi ch the particle consists of a dry snow or ice core surrounded by water or a wet snow mixture. For bath types of particle models, the mixtures are char acterized by effective dielectric constants. This information, along with p article shape, size, and orientation, provides the necessary data for calcu lating the scattering characteristics of the particles. The most commonly u sed formulas for the effective dielectric constant, epsilon(eff), are those of Maxwell Garnett and Bruggeman. To understand the applicability and limi tations of these formulas, an expression for epsilon(eff) is derived that d epends on the mean internal electric fields within each component of the mi xture. Using a conjugate gradient numerical method, the calculations are ca rried out for ice-water mixtures. Parameterization of the results in terms of the fractional water volume and the electromagnetic wavelength provides an expression for epsilon(eff) for wavelengths between 3 and 28 mm. To circ umvent the laborious task of parameterizing epsilon(eff) with wavelength fo r air-ice-water mixtures, several approximate formulations are proposed. Te sts of the accuracy of the formulas are made by calculating the mean and va riance from different particle realizations and by comparison to a previous method. Tests of the applicability of the formulas for epsilon(eff) are ma de by changing the shape, size, and orientations of the inclusions. While t he formulas are adequate over a certain range of inclusion sizes and for a change in shape from cubic to spherical, they are not applicable to highly eccentric, aligned inclusions such as rods or plates.