## A Missing Solution to the Transport Equation and Its Effect on Estimation of...

*J. Atmos. Sci., 59*, 3572-3585.

Most of the existing cloud radiation models treat liquid water drops of a variety of sizes as an ensemble of particles. The ensemble approach assumes that all drop sizes are well represented in an elementary volume, and its scattering and absorbing properties can be accurately specified through the use of the drop size probability density distribution function. The concentration of large drops, however, can be so low that a chance to capture them in the elementary volume is rare. Thus the drop ensemble assumption is not always true, though classical radiative transfer theory uses this assumption to simplify the radiative transfer process, as if scattering takes place from an ‘‘average drop’’ rather than from a particular drop. The theoretical analysis presented in this paper demonstrates that if a cumulative distribution function is used to describe drop size variability with jumps accounting for the probability of finding large drops in the elementary volume, one obtains an extra term, the Green’s function, in the solution of the radiative transfer equation. The analysis of data on cloud drop size distribution acquired during the First International Satellite Cloud Climatology Project (ISCCP) Research Experiment (FIRE) field campaign clearly shows jumps in the cumulative drop size distribution; the magnitudes of the jumps are related to the frequencies of large drop occurrence. This discontinuity is primarily responsible for the additional terms that must be added to the solution to properly account for the photon interaction with the large drops. The enhancement of cloud absorption due to accounting for the ‘‘missing solution’’ exhibits a jump-like response to continuous variation in the concentration of large drops and may exceed the enhancement predicted by the ensemble-based models. The results presented here indicate that the missing term might be plausible to explain the enhanced value of the ratio of the shortwave cloud forcing at the surface to the forcing at top of the atmosphere.