## Estimation of Errors in Two-Stream Approximations of the Solar Radiative...

*J. Atmos. Sci., 72*, 4053-4074, doi:10.1175/JAS-D-15-0033.1.

Solar flux densities and heating rates predicted by a broadband, multilayer d-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 708S and 708N 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 Dx ’ 1 km along track. The d-Eddington was applied in independent column approximation (ICA) mode, while the Monte Carlo was applied using both Dx / ‘ (i.e., ICA) and Dx ’ 1 km. Mean-bias errors due to the d-Eddington’s neglect of phase function details and horizontal transfer, as functions of cosine of solar zenith angle m0, are comparable in magnitude and have the same signs.

With minor dependence on cloud particle sizes, the d-Eddington over- and underestimates top-of-atmosphere reflected flux density for the cloudy portion of domains by ;10 W m22 for m0 . 0.9 and 23 W m22 for m0 , 0.2; full domain averages are ;8 and 22 W m22, respectively, given mean C . 0.75 for all m0. These errors are reversed in sign, but slightly larger, for net surface flux densities. The d-Eddington underestimates total atmospheric absorption by ;2.5 W m22 on average. Hence, d-Eddington mean-bias errors for domain-averaged layer heating rates are usually negative but can be positive. Rarely do they exceed 610% of the mean heating rate; the largest errors are when the sides of liquid clouds are irradiated by direct beams.