A new method is proposed to predict the optical thickness, effective radius, and concentration of cloud droplets in water layer clouds by using the spectrum of cloud condensation nuclei (CCN), ascent velocity at cloud base, and liquid water path (LWP). A retrieval method is also proposed to predict CCN number concentration by using independent observational data of ascent velocity at the cloud base, the optical thickness and LWP of clouds. For this purpose, a newly developed cloud microphysical model that relates cloud droplet size distributions to the updraft velocity, and to CCN constituents and size distribution is used. Cloud droplet growth is calculated, with special care being taken to avoid non-physical numerical diffusion of the droplet spectrum. Near the cloud base, CCN activation and subsequent cloud droplet growth are calculated in a Lagrangian framework to model the effect of CCN on growth by condensation more accurately. In the middle and upper parts of the cloud, an Eulerian framework is used to estimate growth by coalescence for cloud droplet size distributions. Simulated vertical profiles of droplet size distributions, and a solution to the radiative transfer equation using the discrete ordinate method, with no parameterizations, yield the optical properties of the cloud for short-wavelength radiation. Using the approximation equation proposed in this study, the maximum value of supersaturation in a cloud (Smax) is predicted by observing cumulative activated CCN number concentration at 0.075% supersaturation (Nc(0.075%)), and the ascent velocity at the cloud base. Using this Smax, we can estimate Nc(Smax), which is the cumulative number concentration of CCN, whose critical supersaturations are lower than Smax. Nc(Smax) is used to predict the concentration of cloud droplets at the middle altitude of layer cloud (Nd). Conventionally Nd is assumed to be Nc(Smax). However, our parameterization can show that Nd is smaller than Nc(Smax), when Nc(Smax) is large. For the fixed liquid water path, optical thickness and effective radius can be expressed as a function of Nd, unless drizzle is falling from the cloud. The parameterizations developed in this paper are based on the U.S. Standard Atmosphere 1976. If necessary, the parameterizations for the extremely different atmosphere from that used here can be developed in the same way.
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