Estimation of Errors in Two-Stream Approximations of the Solar Radiative Transfer Equation for Cloudy-Sky Conditions

Abstract Solar flux densities and heating rates predicted by a broadband, multilayer δ-Eddington two-stream approximation are compared to estimates from a Monte Carlo model that uses detailed descriptions of cloud particle phase functions and facilitates locally nonzero net horizontal flux densities. Results are presented as domain averages for 256-km sections of cloudy atmospheres inferred from A-Train satellite data: 32 632 samples for January 2007 between 70°S and 70°N with total cloud fraction C > 0.05. The domains are meant to represent grid cells of a conventional global climate model and consist of columns of infinite width across track and Δx ≈ 1 km along track. The δ-Eddington was applied in independent column approximation (ICA) mode, while the Monte Carlo was applied using both Δx → ∞ (i.e., ICA) and Δx ≈ 1 km. Mean-bias errors due to the δ-Eddington’s neglect of phase function details and horizontal transfer, as functions of cosine of solar zenith angle μ0, are comparable in magnitude and have the same signs. With minor dependence on cloud particle sizes, the δ-Eddington over- and underestimates top-of-atmosphere reflected flux density for the cloudy portion of domains by ~10 W m−2 for μ0 > 0.9 and −3 W m−2 for μ0 < 0.2; full domain averages are ~8 and −2 W m−2, respectively, given mean C > 0.75 for all μ0. These errors are reversed in sign, but slightly larger, for net surface flux densities. The δ-Eddington underestimates total atmospheric absorption by ~2.5 W m−2 on average. Hence, δ-Eddington mean-bias errors for domain-averaged layer heating rates are usually negative but can be positive. Rarely do they exceed ±10% of the mean heating rate; the largest errors are when the sides of liquid clouds are irradiated by direct beams.