Indented and hollow ice crystals: representation by collections of independent spheres

The ability of an assembly of spheres to represent scattering and absorption by a nonspherical ice crystal of the same volume-to-area (V/A) ratio was previously evaluated for convex shapes (circular cylinders and hexagonal prisms). Here we extend the comparison to indented and hollow prisms, which are common in ice clouds. Estimates of hollowing and fluffiness are obtained from photomicrographs of natural atmospheric ice crystals. Estimates of micron-scale surface roughness, as well as the degree of hollowing, are obtained from scanning electron micrographs of laboratory-grown ice hexagons.

In the equivalent-sphere representation, the crystal mass and surface area are both conserved. Internal surfaces as well as external surfaces contribute to the total surface area; in the model representation both become external surfaces of spheres. The optical depth (tau) of the model cloud is thus greater than that of the real cloud by the ratio A/4P, where A is the total area of the nonspherical particle and P is the orientation-averaged projected area. This ratio, which we call "fluffiness," is unity for convex particle shapes but may exceed 2 for clusters of hollow bullets. In effect, the scattering at interior surfaces of a hollow crystal becomes classified as multiple scattering in the model of ice spheres. Therefore, rather than directly comparing the asymmetry factor (g) and single-scattering albedo (ssa) of the hollow crystal to those of the equal-V/A sphere, it is more appropriate to compare the product [tau(1-g)ssa], because this quantity largely determines the bulk radiative properties of the cloud.

For multiple-scattering results, the albedo, absorptance, and transmittance of horizontally homogeneous ice clouds are used as criteria for comparison between the equal-V/A representation and the exact results in the geometric-optics limit, the latter obtained by a ray-tracing algorithm. The errors caused by the equal-V/A representation are presented for aspect ratios ranging from thin plates to long columns, for various indentation depths, and for ice-water paths covering the range from thin cirrus clouds to surface snowpacks, at wavelengths in the visible (0.5 microns, non-absorbing) and near-infrared (1.6 microns, moderately absorbing). The errors are less than 0.15 over most of these ranges at both wavelengths and are largest for extreme aspect ratios (long columns or thin plates) rarely found in nature. Errors are smaller when the cloud contains a distribution of aspect ratios, as found in samples of atmospheric ice crystals.

The single-scattering quantities combine with the fluffiness to form a similarity-parameter ratio, which can predict the ability of the equal-V/A formulation to provide an accurate approximation to multiple-scattering results. However, good agreement is also obtained in the limits of large and small optical depth, regardless of the value of the ratio.