Supersaturation and diffusional droplet growth in liquid clouds: Polydisperse spectra

Evolution of droplet size distribution (DSD) due to the water vapor diffusion in a vertically moving adiabatic parcels is investigated. Analytical expressions for height dependences of the main DSD parameters and DSD moments are obtained. The asymptotic behavior of the DSD parameters at large heights above cloud base is determined. It is shown that during diffusion growth, the width and the relative dispersion of the DSD decrease with height as z^{- 1/3} and z^{- 2/3}, respectively. The paper presents examples of DSD evolution in cases DSD forms on aerosols with a three-mode lognormal distribution. The aerosol distribution parameters used in the study correspond to four aerosol types: "Marine," "Clean continental," "Background," and extremely polluted "Urban." The vertical profiles of DSD parameters are compared with the asymptotic profiles. It is shown that in case of polydisperse DSD evolution, the vertical profile of supersaturation within several hundred meters above the cloud base can be approximated by a supersaturation profile corresponding to the "equivalent" monodisperse DSD. The initial radius of this equivalent DSD is equal to the mean radius of polydisperse DSD (haze size distribution) at cloud base, which is estimated using the Kohler theory. This result of the relation between the polydisperse and monodisperse solutions is universal. A new equation for estimation of supersaturation maximum for polydisperse case is obtained. The obtained analytical expressions and numerical results are useful for understanding the mechanisms of DSD formation in clouds and for parameterization of warm microphysical processes in cloud models.