Two-Stream Approximations to Radiative Transfer in Planetary Atmospheres: A Unified Description of Existing Methods and a New Improvement

Existing two-stream approximations to radiative transfer theory for particulate media are shown to be represented by identical forms of coupled differential equations if the intensity is replaced by integrals of the intensity over hemispheres. Thus, one set of solutions suffices for all methods and provides convenient analytical comparisons. The equations also suggest modifications of the standard techniques in order to duplicate exact solutions for thin atmospheres and permit accurate determinations of the effects of typical aerosol layers. Numerical results for the plane albedos of plane-parallel atmospheres (single-scattering albedo = 0.8, 1.0; optical thickness = 0.25, 1, 4, 16; the Henyey-Greenstein phase function with an asymmetry factor of 0.75) are given for conventional and modified Eddington approximations, conventional and modified two-point quadrature schemes, the hemispheric-constant method, and the delta-function method, all for comparison with accurate discrete-ordinate solutions. A new, two-stream approximation is introduced that reduces to the modified Eddington approximation in the limit of isotropic phase functions and to the exact solution in the limit of extreme anisotropic scattering. Comparisons of plane albedos and transmittances show the new method to be generally superior over a wide range of atmospheric conditions (including cloud and aerosol layers), especially in the case of nonconservative scattering.