11.6 1 ANOTHER LOOK AT STOCHASTIC CONDENSATION IN CLOUDS: EXACT SOLUTIONS, FOKKER-PLANCK APPROXIMATIONS AND ADIABATIC EVOLUTION Christopher A. Jeffery * Space and Remote Sensing Sciences (ISR-2), LANL, Los Alamos, NM Jon M. Reisner & Miroslaw Andrejczuk Atmospheric, Climate and Environmental Dynamics (EES-2), LANL, Los Alamos, NM 1. INTRODUCTION One of the most important theoretical constructs in at- mospheric science is the gradient transport model for the flux u ′ ζ ′ , u ′ ζ ′ = −K∇ ζ which enables the derivation of logarithmic profiles of velocity (ζ ′ = w ′ ) and temperature (ζ ′ = T ′ ) in the at- mospheric surface layer and provides a subgrid closure (eddy viscosity and diffusivity) for atmospheric models. In modern atmospheric texts and treatments, the gra- dient model is usually postulated a priori. But under- lying this model is a rich stochastic framework that, in the earlier years of atmospheric science, was given due consideration. The interested reader is referred to Sut- ton (1953) for the historical flavor of this discussion. In fact, the stochastic Langevin equation (Risken 1989) underlying the gradient model is simply dX dt = u ′ where X is Lagrangian position and u ′ is indepen- dently distributed with the following constraint: either (i) u ′ is assumed Gaussian with a renewal time, τ , that is smaller—but not vanishingly smaller—than the macro- scopic evolution time, or (ii) u ′ is arbitrarily distributed with assumed renewal time τ → 0, the former being the better assumption for turbulent transport. Seen in this light, the gradient model is, of course, a Fokker-Planck operator (Risken 1989). It is natural, then, to inquire whether this stochas- tic formulation may have applicability to other subgrid problems, and in the 1960s a group of Russian scien- tists (Belyaev 1961; Sedunov 1965; Mazin 1965; Levin and Sedunov 1966a,b) postulated dr dt ∼ S ′ r * Corresponding author address: Christopher A. Jeffery, Los Alamos National Laboratory (ISR-2), PO Box 1663, Mail Stop D-436, Los Alamos, NM 87545, USA. Tel.: (505) 665-9169; fax: (505) 664- 0362. Email: cjeffery@lanl.gov where r is droplet radius and S ′ is a random supersat- uration fluctuation with the correlation S ′ f ′ analogous to the flux u ′ ζ ′ where f (r) is the droplet-size density. Appropriately, this theoretical approach has inherited the name “stochastic condensation”. Despite the ob- vious analogy to the ubiquitous eddy-diffusivity param- eterization, the investigation of the correlation S ′ f ′ and its corresponding gradient model, has largely been ig- nored outside of the Russian community with few ex- ceptions (e.g. Manton (1979); Khvorostyanov and Curry (1999a,b)). This extended manuscript summarizes the results of Jeffery et al. (2006). In Jeffery et al. and here we take another look at the Langevin equation for droplet growth and its corresponding Fokker-Planck equation, with the overarching goal of clarifying and elucidating stochas- tic condensation and its applicability to subgrid atmo- spheric modeling. To this end we begin with a stochas- tic model, described in Sec. 2, that assumes S ′ is in- dependently and normally distributed with given time- dependent variance, σ 2 (t) and fixed renewal time. We do not construct a model for σ 2 (t), nor do we relate S ′ to vertical velocity; rather, we treat σ 2 (t) as externally provided. A discussion of the validity of these model- ing assumptions is postponed until their consequences have been deduced. The exact analytic solution to the present model is derived in Sec. 3, and the correspond- ing Fokker-Planck equation presented in Sec. 4. In Sec. 5, the impact of S ′ -fluctuations on the mean su- persaturation S is derived and the coupled evolution of { S,f } in a closed, adiabatic volume is assessed. We search for evidence of the stochastic condensation mechanism in cloud droplet spectra from cumulus cloud fields in Sec. 7; Sec. 8 contains a summary. 2. THE MODEL Following the proceeding discussion, we introduce the following exactly solvable model of stochastic conden- sation and evaporation. The local supersaturation field experienced by the i-th droplet is decomposed