Double-delta-function adjustment in thermal radiative transfer

For atmospheric scattering, the weak backward scattering peak is always ignored, in contrast with the strong forward scattering peak. In this paper, the backward peak contribution is incorporated in multiple scattering, along with the forward peak contribution. Thus, a new parameterization, the double-δ-function adjustment, is proposed and its application in a two-stream approximation for infrared radiative transfer is shown. The accuracy of the adding method for the double-δ-two-stream discrete-ordinates approximation (2δ-2DDA) is evaluated by the emissivities in a single layer of atmosphere and the fluxes and heating rate in a multi-layer atmosphere with a realistic atmospheric profile. The results show that 2δ-2DDA produces less bias than the δ-two-stream approximation (δ-2DDA) for thick optical depths, such as water cloud conditions. For thin optical depths, such as ice cloud, δ-2DDA and 2δ-2DDA produce similar errors. Generally, 2δ-2DDA is more accurate than δ-2DDA, and it can be easily applied in climate models.