The Backscattered Fraction in two-stream Approximations

New formulas for the backscattered fraction in two-stream theory are derived. They express this fraction, for either isotropically or monodirectionally incident radiation, as a single integral over the scattering phase function, thereby effecting a substantial simplification over the customary multiple-integral definitions. From these formulas the globally averaged backscatter of the earth due to typical aerosols is shown to depend primarily on the forward part (0° to 90°) of the scattering phase function, where the disagreement between spherical-and nonspherical-particle scattering is smallest. The new formulas also lead to connections, in terms of standard elliptic integrals, between the backscatter and the phase function asymmetry factor; while rigorously correct only for the Henyey-Greenstein phase function, these relations are shown to be remarkably accurate for all spherical-particle phase functions. The detailed relationship between backscatter and asymmetry factor is shown to be multi-valued; thus two-stream and Eddington approximations cannot be uniquely related. The common approximation of the globally averaged backscatter, or Bond albedo, by the backscatter for radiation incident at solar zenith angles of O° or 60° is shown to lead, for a wide range of particle sizes and optical properties, to systematic and often large underestimates. The solar-spectrum-integrated enhancement of the Bond albedo due to a uniform, optically thin aerosol layer is examined, holding the total mass of aerosol fixed and varying the particle radii and optical properties over wide ranges. The particle radius at which maximum albedo enhancement occurs decreases from 0.3 µm down to about 0.08 µm as the particle absorptivity increases. Also, increasing the absorption of particles smaller than 0.1 µm actually raises the albedo in contrast to the usual situation where absorption suppresses backscattering.