Representing 3-D cloud radiation effects in two-stream schemes: 1. Longwave considerations and effective cloud edge length

Current weather and climate models neglect 3-D radiative transfer through cloud sides, which can change the cloud radiative effect (CRE) significantly. This two-part paper describes the development of the SPeedy Algorithm for Radiative TrAnsfer through CloUd Sides (SPARTACUS) to capture these effects efficiently in a two-stream radiation scheme for use in global models. The present paper concerns the longwave spectral region, where not much work has been done previously, although the limited previous work has suggested that radiative transfer through cloud sides increases the longwave surface CRE of shallow cumulus by around 30%. To assist the development of a longwave capability for SPARTACUS, we use a reference case of an isolated, isothermal, optically thick, cubic cloud in vacuum, for which 3-D effects increase CRE by exactly 200%. It is shown that for any cloud shape, the 3-D effect can be represented in SPARTACUS provided that correct account is made for (1) the effective zenith angle of diffuse radiation emitted from a cloud, (2) the spatial distribution of fluxes in the cloud, (3) cloud clustering that enhances the interception of emitted radiation by neighboring clouds, and (4) radiative smoothing leading to the effective cloud edge length being less than the measured value. We find empirically that the circumference of an ellipse fitted to a horizontal cross section through a cumulus cloud provides a good estimate of the radiatively effective cloud edge length, which provides some guidance to how cloud observations could be analyzed to extract their most important properties for radiation.