Another look at stochastic condensation in clouds: Exact solutions, Fokker-Planck approximations and adiabatic evolution

The eddy-diffusivity model of turbulent mixing, in which a Fokker-Planck operator represents the ensemble effect of unresolved, independent, Gaussian velocity fluctuations, is a standard feature of atmospheric models from LES to GCM scales. Forty years ago, Levin-Mazin-Sedunov (LMS) derived the corresponding Fokker-Planck operator for the cloud droplet size distribution, f(r). This operator diffuses mean-square radius and represents the ensemble impact of unresolved, independent, Gaussian supersaturation fluctuations on droplet growth. Despite obvious analogy to the ubiquitous eddy-diffusivity approximation, the implications of the LMS Fokker-Planck operator and its implementation in a cloud physics model have not been fully pursued in cloud physics research.

In this talk, we re-visit the classical problem posed by LMS--the evolution of a population of droplets in a closed, adiabatic volume with independent Gaussian supersaturation fluctuations. We first present the exact analytic solution to this problem and show how it differs from (and in what limit converges to) the LMS Fokker-Planck approximation. We then consider the coupled evolution of the mean supersaturation, S, and f(r) for the present model, a problem treated previously only by Voloshchuk and Sedunov (1977). Our analysis reveals a single non-dimensional number, Nd, that determines the primary impact of fluctuations on {S,f} and which is a function of the time-integrated LMS diffusivity, parcel height, accommodation length and thermodynamic functions in the S-equation. Using the quasi-stationary approximation, we find a critical value Nd=6.5, above which S can become negative in a rising parcel. Finally, we assess the net impact of the LMS operator on the evolution of the droplet spectra. This assessment differs from all past studies (modulo Voloshchuk and Sedonov) in that our approach exactly conserves total water. We find that the LMS operator decreases the parcel's shortwave optical depth, and hence decreases cloud reflectivity, while simultaneously increasing the parcel's precipitation efficiency.