Improved Delta-Eddington Approximation for Optically Thin Clouds

The δ-Eddington simulations of broadband shortwave net radiation fluxes at the top of the atmosphere (F_TOA) and the surface (F_SURF) are evaluated with different parameterizations of the forward fraction of scattering (f), including the square of the asymmetry factor (f = g²), the fraction of the forward single-scattered intensity over the total single-scattered intensity (f = f_p), and the cube of the asymmetry factor (f = g³). g² and g³ are respectively the 2nd and 3rd moments of the Henyey-Greenstein (HG) phase function and hence approximate the variance and skewness of the phase function. The factor f_p for spherical droplets is estimated by considering a truncation angle that separates the forward peak and diffusive portions of a highly anisotropic phase function. The results show that the simulations of F_TOA and F_SURF are not improved if the conventional approach f = g² is replaced by f = f_p in the δ-Eddington approximation for an atmosphere in the presence of liquid clouds. For the optically thick conditions, multiple scattering plays a dominant role in determining the reflectance (R) and transmittance (T) of the cloud layer, and the conventional parameterization f = g² is the most accurate among the three parameterizations. For the optically thin conditions, single scattering dominates over multiple scattering and thus f = g² results in biased F_TOA and F_SURF calculations, particularly at low solar elevations. For such cases, f = g³ shows most accurate F_TOA and F_SURF results for both liquid and ice clouds, though f = g³ also results in lower cloud layer heating rates in general.