Effective dielectric constants of mixed-phase hydrometeors

Melting snow, graupel, and hail are often modeled as uniform mixtures of air-ice-water or ice-water. Two-layered models have also been proposed in which 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 characterized by effective dielectric constants. This information, along with particle shape, size, and orientation, provides the necessary data for calculating the scattering characteristics of the particles. The most commonly used formulas for the effective dielectric constant, epsilon(eff), are those of Maxwell Garnett and Bruggeman. To understand the applicability and limitations of these formulas, an expression for epsilon(eff) is derived that depends on the mean internal electric fields within each component of the mixture. Using a conjugate gradient numerical method, the calculations are carried 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 circumvent the laborious task of parameterizing epsilon(eff) with wavelength for air-ice-water mixtures, several approximate formulations are proposed. Tests of the accuracy of the formulas are made by calculating the mean and variance from different particle realizations and by comparison to a previous method. Tests of the applicability of the formulas for epsilon(eff) are made by changing the shape, size, and orientations of the inclusions. While the 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.