Broadening of Cloud Droplet Size Distributions by Condensation in Turbulence

To consider the growth of cloud droplets by condensation in turbulence, the Fokker-Planck equation is derived for the droplet size distribution (droplet spectrum). This is an extension of the statistical theory proposed by Chandrakar and coauthors in 2016 for explaining the broadening of the droplet spectrum obtained from the "Pi-chamber", a laboratory cloud chamber. In this Fokker-Planck equation, the diffusion term represents the broadening effect of the supersaturation fluctuation on the droplet spectrum. The aerosol (curvature and solute) effects are introduced into the Fokker-Planck equation as the zero flux boundary condition at R-2 = 0, where R is the droplet radius, which is mathematically equivalent to the case of Brownian motion in the presence of a wall. The analytical expression for the droplet spectrum in the steady state is obtained and shown to be proportional to R exp (-cR(2)), where c is a constant. We conduct direct numerical simulations of cloud droplets in turbulence and show that the results agree closely with the theoretical predictions and, when the computational domain is large enough to be comparable to the Pi-chamber, agree with the results from the Pi-chamber as well. We also show that the diffusion coefficient in the Fokker-Planck equation should be expressed in terms of the Lagrangian auto-correlation time of the supersaturation fluctuation in turbulent flow.