## 1. Introduction

[2] In recent years, increasing attention has been paid to treating the non-linear effects of radiation and precipitation microphysics in large-scale models. The traditional assumption that a cloud is horizontally homogeneous on the scale of a large-scale model grid box for purposes of radiation and precipitation leads to substantial biases due to the non-linear nature of these processes (e.g., *Cahalan et al.* [1994] for radiation; *Larson et al.* [2001] or *Pincus and Klein* [2000] for microphysics).

[3] Statistical cloud schemes provide an attractive framework to self-consistently predict the horizontal inhomogeneity because the probability distribution function (PDF) of total water contained in the scheme can be used to calculate the PDF of cloud condensate, from which the non-linear effects of radiation and precipitation may be self-consistently estimated. (While “assumed-PDF scheme” might be a better term than “statistical cloud scheme,” the term “statistical cloud scheme” is ingrained in the literature and will be retained in this paper.) Statistical cloud schemes were originally developed in the context of boundary layer studies, so their extension to the full atmosphere is non-trivial [*Mellor*, 1977; *Sommeria and Deardorff*, 1977]. For example, in their seminal paper, *Sommeria and Deardorff* [1977, p. 345] state: “For large grid volumes, such as in a global circulation model, the assumption of Gaussian distributions on the subgrid scale, for even θ_{l} [liquid water potential temperature] and *q*_{t} [total water specific humidity, a quantity closely related to total water mixing ratio] would presumably be poor.”

[4] Indeed, one problem not envisioned by the pioneers is that one would want to have a statistical cloud scheme in a model that also contained a separate mass-flux formulation for atmospheric convection. Originally, it was envisioned that the turbulence scheme when formulated with Reynolds averaged equations would be responsible for all the subgrid-scale transport within a grid box. However, it was quickly learned that Gaussian PDFs do not well represent the trade cumulus boundary layer, where intermittent convection leads to highly skewed distributions of total water [*Bougeault*, 1982]. While one can overcome these difficulties for shallow convection within a self-consistent statistical approach that treats all subgrid-scale transport [*Golaz et al.*, 2002], one may choose to represent subgrid-scale vertical transports with both a turbulence and a mass-flux convection scheme. As this represents the case for virtually all large-scale models which must treat both shallow and deep convection, it is worth improving the consistency between a statistical cloud scheme and a mass-flux convection scheme.

[5] If, within a large-scale model, a statistical cloud scheme is to co-exist with a mass-flux convection scheme, how should they be coupled? First attempts at using a statistical cloud scheme in a global model just ignored the coupling altogether. For example, models at the UK Met Office have essentially assumed a triangle distribution to a variable that is essentially the difference of *r*_{t} and *r*_{s}, the total water and saturation mixing ratios, respectively [*Smith*, 1990]. The width of the distribution when normalized by *r*_{s}, however, was either a global constant or a fixed function of pressure. With a time-invariant width to the distribution, this scheme is equivalent to a relative humidity threshold scheme [*Smith*, 1990]. Without any explicit connection to convection, imagine what happens to the parameterized clouds when updrafts detrain cloud condensate into clear air. According to this cloud scheme, clouds can only begin to occur when the relative humidity in the grid box exceeds a threshold. Thus, if the initial relative humidity is less than the threshold value, all cloud water detrained from convection instantaneously evaporates until the relative humidity of the grid box rises to the threshold value. How one might remedy this unnatural behavior in the context of a statistical cloud scheme is the subject of this paper.

[6] Two issues must be dealt with if one wishes to treat clouds in the environment of convection with a statistical approach. The first and relatively straightforward issue is to remove the conventional assumption of a symmetric shape to the *r*_{t} PDF, as PDFs tend to be highly skewed [*Xu and Randall*, 1996; *Bony and Emanuel*, 2001; *Tompkins*, 2002; *Larson et al.*, 2002]. The second and more difficult issue is the prediction of the shape of the *r*_{t} PDF, both its width and asymmetry. *Bony and Emanuel* [2001] use a simple coupling whereby the shape of the PDF is altered so that at every time step, the in-cloud value of cloud condensate equals the sum of that diagnosed by traditional large-scale saturation and that diagnosed from the Emanuel convection scheme. In the formulation of *Tompkins* [2002], prognostic equations for essentially the variance and skewness of the *r*_{t} PDF are added to the large-scale model, with ad-hoc source terms from the mass-flux convection scheme.

[7] In this work, physically based source terms for the impact of convection on the variance and skewness of the *r*_{t} PDF are proposed in section 2. These source terms are intended to replace the ad-hoc source terms given by *Tompkins* [2002]. In section 3, the suitability of these terms for parameterization is tested with output from a cloud-resolving model (CRM) simulation of deep atmospheric convection. The ultimate goal of this work is successful incorporation of these terms into a large-scale model. While this is not accomplished herein, practical suggestions to this end are offered in section 4.