## 1. Introduction

The governing equations of most cloud models are obtained using the closure of the first or 1.5 order, according to which the fluxes of subgrid quantities are assumed to be proportional to gradients of mean values (the Prandtl mixing length concept). For subgrid scales corresponding to turbulent ones, the influx of the subgrid fluxes represents the rate of turbulent mixing. Using the Prandtl mixing length concept and the K-theory based on this concept (*K* being the turbulent coefficient), the mixing rate of a certain quantity *a* is described by applying operators like in the conservation equations. Strictly speaking, the application of this operator suggests that *ā* is a conservative value, e.g. the potential temperature in a dry adiabatic process or the liquid water potential temperature in a moist adiabatic process. With this assumption, mixing takes into account the effect of subgrid processes in the models that have the resolution of a few tens to a few hundred metres (depending on the model resolution). This approach represents a parametrization of the net *results* of a complicated real mixing accompanied by formation of turbulent filaments of different scales and by other dynamic and thermodynamic processes. The purpose of this parametrization is to represent the general effects of small-scale motions on the averaged (model-resolved) fields. For instance, in the case of dry adiabatic processes, the application of such a mixing operator in the vertical direction should lead to a potential temperature that is constant with height.

The evolution of droplet size distribution (DSD) in Large Eddy Simulation (LES) models of stratocumulus clouds, as well as in models of convective clouds with explicit microphysics, is calculated by solving averaged kinetic equations. The turbulent mixing of DSD in such models is also represented by operator (e.g. Takahashi, 1974; Hall, 1980; Khvorostyanov *et al*, 1989; Kogan, 1991; Feingold *et al*, 1994, 1996; Kogan *et al*, 1995; Khairoutdinov and Kogan, 1999; Yin *et al*, 2000, Stevens *et al*, 2005; etc.). This approach (that will be referred to hereafter as the standard one) does not take into account the non-conservative nature of DSD and may lead to significant physical errors. For instance, the 2 µm radius droplets near a cloud base at 40–50 m aloft may grow by diffusion to, say, 5 µm radius. If the drop size changes occurring during the turbulent mixing between these two layers are neglected, large drops (coming from above) will artificially appear near the cloud base, while small drops will appear at the distance of several tens of metres above cloud base (coming from below). Such an approach to mixing leads to an artificial broadening of the droplet spectrum and to an artificial acceleration of raindrop formation (see e.g. Kogan and Mazin, 1981; Khain *et al*, 2004). In fact, if the small asymmetry between the rates of drop growth and drop evaporation is neglected (the role of this symmetry was investigated in detail by Korolev (1995)), the vertical mixing between moist parcels that are adiabatic in all other respects, undiluted by environmental air, should not change the DSDs in these parcels.

Note that the non-conservativity of different quantities in the atmosphere manifests itself largely during movements (or mixing) in the vertical direction because temperature as well as pressure changes mostly in the vertical direction, and this is why condensation occurs during ascent (saturated vapour pressure is a function of temperature only). The mixing of a non-conservative quantity like temperature or a conservative quantity like potential temperature in the horizontal direction can be described by one and the same operator .

The necessity to take into account the non-conservativity of DSD while describing turbulent mixing was well recognized in the theory of stochastic condensation, which attributes DSD broadening to turbulent fluctuations of supersaturation (Buikov, 1961, 1963; Levin and Sedunov, 1966; Sedunov, 1974; Mazin and Merkulovich, 2008). Studies by Cooper (1989) and Srivastava (1989) which analyse effects of fluctuating supersaturation on the development of DSD also belong to this line of investigation. Detailed derivations of the equations of stochastic condensation using Reynolds averaging and a careful evaluation of covariances (subgrid fluxes) were performed by Stepanov (1975), Clark (1976), Voloshchuk and Sedunov (1977), Manton (1979), Merkulovich and Stepanov (1977), and Khvorostyanov and Curry (1999a, 1999b). Usually, these covariances are described using the Prandtl mixing length concept and the K-theory based on this concept. Supersaturation and droplet growth rate are treated as stochastic variables and the operator of turbulent diffusion is replaced by the operator where *A* reflects the non-conservativity of DSD and depends on fluctuations of vertical velocity and supersaturation, as well as on drop size (e.g. Khvorostyanov and Curry, 1999a; Mazin and Merkulovich, 2008).

In most studies dedicated to the theory of stochastic condensation the non-conservativity of DSD leads to the fact that vertical mixing of cloud volumes containing adiabatic liquid water content (LWC) does change the DSD in the volumes. This conclusion is in obvious contradiction with the results that can be obtained by applying the operators for turbulent fluxes or for the influxes.

Using this approach, Khvorostyanov and Curry (1999b) offered an analytical solution of a somewhat simplified kinetic equation for DSD and obtained solutions of the gamma-distribution type.

Most conclusions of the theory of stochastic condensation are obtained under different simplification assumptions and have not been applied in the cloud models (with the exception of the study by Khvorostyanov and Curry (1999b), where a one-dimensional (1-D) model was used). Besides, the studies treating DSD as a non-conservative quantity do not take into account any droplet nucleation/denucleation processes that may occur during subgrid turbulent mixing. Here we refer to denucleation as the process of transition of droplets to haze particles during their evaporation. This simplification is also used in all cloud models with explicit microphysics.

While the Eulerian LES models use the K-theory for parametrization of turbulent mixing (even with the errors mentioned above), no attempts to mix Lagrangian parcels in the Trajectory Ensemble Models (TEM) (where a set of Lagrangian parcels move along air stream lines) have ever been performed (Stevens *et al*, 1996; Feingold *et al*, 1998; Harrington *et al*, 2000; Erlick *et al*, 2005; Khain *et al*, 2008; Pinsky *et al*, 2008; Magaritz *et al*, 2009). In the Lagrangian parcel model of the cloud-capped boundary layer (Magaritz *et al*, 2009), there are more than 1300 Lagrangian parcels moving within a turbulent–like flow that interact via droplets settling from one parcel onto adjacent parcels. In other TEM models, there is no interaction between parcels at all, which limits the utilization of the models by the analysis of the droplet diffusion growth alone.

In this study, we propose an approach to the turbulent mixing of DSDs that is conceptually close to the stochastic condensation theory, based on the Prandtl mixing length concept extended to the case of mixing of non-conservative values like DSD. Among the specific features of the proposed approach is (1) taking into account droplet nucleation/denucleation in the course of the turbulent mixing, and (2) taking into account the asymmetry in the rates of the diffusion-induced growth and evaporation of small droplets. This approach is applicable for describing in-cloud mixing, mixing between cloudy volumes and drop-free air volumes, as well as mixing between droplet-free volumes. The method is specially designed to be used in both Eulerian and Lagrangian cloud models.