Comparisons of Cross Sections for Melting Hydrometeors as Derived from Dielectric Mixing Formulas and a Numerical Method

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  • a Goddard Space Flight Center, Greenbelt, Maryland
  • | b Caelum Research Corporation, Silver Spring, Maryland
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Abstract

One of the impediments to the interpretation of radar signatures from the melting layer is the uncertainty over the dielectric mixing formula for ice-water mixtures. In the commonly used Maxwell Garnett mixing formula, the dielectric constant for ice inclusions in a water matrix differs from that for water inclusions in an ice matrix for the same fraction of meltwater. While the choice of materials for the matrix and inclusion is clear for either small or large fractions of meltwater, it is not obvious how these are to be chosen in the intermediate ranges of melting. In this paper, cross sections derived from the various mixing formulas are compared to a conjugate gradient-fast Fourier transform numerical method. In the numerical method the particle is divided into equi-volume subcells in which the composition of the particle is controlled by assigning a probability of water to each subcell. For a uniform distribution of water and ice, where the probability of water in a subcell is independent of its location within the particle, the numerical results for fractional water contents of less than about 0.7 indicate that the scattering coefficients are closest to those predicted by the Maxwell Garnett mixing formula if an ice matrix with water inclusions is assumed. However, if the meltwater is highly concentrated near the boundary of the particle or if the fractional volume of water is greater than about 0.8, the Maxwell Garnett formula is in fair agreement with the numerical results, if the roles of ice and water are interchanged. A discussion of the relevance of these results to the modeling of melting snow aggregates and the interpretation of radar signatures of the bright band is given in the final section of the paper.

Abstract

One of the impediments to the interpretation of radar signatures from the melting layer is the uncertainty over the dielectric mixing formula for ice-water mixtures. In the commonly used Maxwell Garnett mixing formula, the dielectric constant for ice inclusions in a water matrix differs from that for water inclusions in an ice matrix for the same fraction of meltwater. While the choice of materials for the matrix and inclusion is clear for either small or large fractions of meltwater, it is not obvious how these are to be chosen in the intermediate ranges of melting. In this paper, cross sections derived from the various mixing formulas are compared to a conjugate gradient-fast Fourier transform numerical method. In the numerical method the particle is divided into equi-volume subcells in which the composition of the particle is controlled by assigning a probability of water to each subcell. For a uniform distribution of water and ice, where the probability of water in a subcell is independent of its location within the particle, the numerical results for fractional water contents of less than about 0.7 indicate that the scattering coefficients are closest to those predicted by the Maxwell Garnett mixing formula if an ice matrix with water inclusions is assumed. However, if the meltwater is highly concentrated near the boundary of the particle or if the fractional volume of water is greater than about 0.8, the Maxwell Garnett formula is in fair agreement with the numerical results, if the roles of ice and water are interchanged. A discussion of the relevance of these results to the modeling of melting snow aggregates and the interpretation of radar signatures of the bright band is given in the final section of the paper.

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