Effective Diameter in Radiation Transfer: General Definition, Applications, and Limitations

Abstract

Although the use of an effective radius for radiation transfer calculations in water clouds has been common for many years, the export of this concept to ice clouds has been fraught with uncertainty, due to the nonspherical shapes of ice particles. More recently, a consensus appears to be building that a general definition of effective diameter Deff should involve the ratio of the size distribution volume (at bulk density) to projected area. This work further endorses this concept, describes its physical basis in terms of an effective photon path, and demonstrates the equivalency of a derived Deff definition for both water and ice clouds. Effective photon path is the unifying underlying principle behind this universal definition of Deff. Simple equations are formulated in terms of Deff, wavelength, and refractive index, giving monochromatic coefficients for absorption and extinction, ?abs and ?ext, throughout the geometric optics, Mie, and Rayleigh regimes. These expressions are tested against Mie theory, showing the limitations of the use of Deff as well as its usefulness. For water clouds, the size distribution N(D) exhibits relatively little dispersion around the mean diameter in comparison with ice clouds. For this reason, a single particle approximation for ?abs based on Deff compares well with ?abs predicted from Mie theory, providing a new and efficient means of treating radiation transfer at terrestrial wavelengths. The Deff expression for ?ext agrees well with Mie theory only under specific conditions: 1) absorption is substantial or 2) absorption occurs in the Rayleigh regime, or 3) size parameter xe ? 50, where xe = πDeff/?. Since the Deff expressions for ?abs and ?ext are single particle solutions, it is not surprising that agreement with Mie theory is best when the size distribution dispersion is reduced, approaching the single particle limit. For ice clouds, it is demonstrated that the Deff expressions for ?abs and ?ext are probably inadequate for most applications, at least at terrestrial wavelengths. This is due to the bimodal nature of ice particle size spectra N(D) with relatively high concentrations of small (D < 100 ?m) ice crystals. These small crystals have relatively low absorption efficiencies, causing the N(D)-integrated ?abs to be lower than ?abs based on Deff. This difference in N(D) dispersion between water and ice clouds makes it desirable to use an explicit solution to the absorption and extinction coefficients when calculating the radiative properties of ice clouds. Analytical solutions to the integral definitions of ?abs and ?ext are provided in the appendix, which may not be too computationally expensive for many applications. Most schemes for predicting ice cloud radiative properties are founded on the assumption that the dependence of ?abs and ?ext on the size distribution can be described solely in terms of Deff and ice water content (IWC). This assumption was tested by comparing the N(D) area-weighted efficiencies for absorption and extinction, Qabs and Qext, for three N(D) that have the same IWC and Deff, but for which N(D) shape differs. Analytical solutions for ?abs and ?ext were used, which explicitly treat N(D) shape, over a wavelength range of 1.0 to 1000 ?m. For a chosen Deff value, uncertainties (percent differences) resulting only from N(D) shape differences reached 44% for Qabs, 100% for Qext, and 48% for the single scattering albedo ?o for terrestrial radiation. This sensitivity to N(D) shape has implications relating to the formulation of schemes predicting ice cloud radiative properties, as well as satellite remote sensing of cloud properties.