The Use of Equivalent Spheres to Model Scattering and Absorption of Radiation by Ice Clouds and Snow

The use of "equivalent" spheres to represent nonspherical particles has been unsatisfactory in the past because the sphere of equal volume has too little surface area and thus too little scattering, whereas the sphere of equal area has too much volume giving too much absorption. Their asymmetry factors are also too large. These problems can largely be avoided if the real cloud of nonspherical particles is represented by a model cloud of spheres such that the model cloud contains the same total surface area as well as the same total volume. Each nonspherical particle is then represented not by just one sphere but rather by a collection of independent spheres, each of which has the same volume-to-surface-area ratio (V/A) as the nonspherical particle.

To demonstrate the broad utility of this approach, we show results for ice, whose absorption coefficient varies with wavelength by 8 orders of magnitude. Randomly-oriented infinitely long circular cylinders were used as the first test case because an exact solution is available for all size parameters. The spheres mimicked the cylinders very well, so the study has now been extended to hexagonal columns and plates. The extinction efficiency and single-scattering co-albedo for these prisms are closely approximated by the values for equal-V/A spheres across the ultraviolet, visible, and infrared from 0.2 to 25 micrometers wavelength. Errors in the asymmetry factor can be significant where ice absorptance is weak, at visible wavelengths for example. These errors are greatest for prisms with aspect ratios close to 1. Errors in hemispheric reflectance, absorptance, and transmittance are calculated for horizontally-homogeneous clouds with ice water paths from 0.4 to 200,000 grams per square meter and crystal thicknesses of 1 to 400 micrometers, to cover the range of crystal sizes and optical depths from polar stratospheric clouds (PSCs) through cirrus clouds to surface snow. The errors are less than 0.05 over most of these ranges at all wavelengths, but can be larger at visible wavelengths because of the ideal shapes of the prisms. The errors may therefore be smaller for the more irregular crystals typical of cirrus clouds.

References:

Grenfell, T.C., and S.G. Warren, 1999: Representation of a nonspherical ice particle by a collection of independent spheres for scattering and absorption of radiation. J. Geophys. Res., 104, 31697-31709.

Neshyba, S.P., T.C. Grenfell, and S.G. Warren, 2002: Representation of a nonspherical ice particle by a collection of independent spheres for scattering and absorption of radiation: II. Hexagonal columns and plates. Submitted to J. Geophys. Res.