## 1. Introduction

[2] Clouds are of fundamental importance to the Earth's climate, in particular for its radiative energy budget. However, the representation of clouds in general circulation models (GCMs) poses one of the main challenges in large-scale climate modelling due to the subgrid-scale nature of cloud-related processes [e.g., *Randall et al.*, 2007; *Quaas et al.*, 2009]. A first-order challenge is to simulate fractional cloudiness, or the fraction of a GCM grid box which is covered by clouds. Usually, clouds are considered as “boxes” that fill a GCM grid box entirely in the vertical, and to a fraction, *f* ∈ [0, 1], in the horizontal. An assumption which holds well for liquid water clouds is that clouds exist wherever the specific humidity, *q*_{v}, exceeds the saturation specific humidity, *q*_{s}(*T*), which is a function of temperature *T*, and also slighly dependent on pressure. The ratio of specific humidity and saturation specific humidity is called relative humidity, *r* = *q*_{v}/*q*_{s}. A GCM grid box with a typical scale of 20 to 200 km in the horizontal, could be considered entirely cloudy when the grid-box mean relative humidity exceeds 100%, and entirely clear otherwise. A presumably better approach is to consider subgrid-scale variability of humidity, and perhaps of temperature [*Sommeria and Deardorff*, 1997; *Mellor*, 1977]. Advanced GCM cloud schemes thus simulate prognostically the probability distribution function (PDF) of total water specific humidity, *q*_{t}, the sum of *q*_{v} and condensed (liquid and ice) water [e.g., *Bony and Emanuel*, 2001; *Tompkins*, 2002]. An alternative to this is to introduce cloud cover as a prognostic model variable, effectively simulating one more moment of the distribution [e.g., *Sundqvist*, 1978; *Tiedtke*, 1993]. The probably simplest choice for a PDF would be a uniform distribution of *q*_{t} (see Figure 1), and for the simplest assumption, the width of it could be expressed as a fraction *γ* of *q*_{t} [e.g., *Le Treut and Li*, 1991], or as a fraction of *q*_{s}. In this case, *T* and thus *q*_{s} are usually assumed constant throughout the grid box. Aircraft measurements show that these assumptions are simplifications compared to reality [*Larson et al.*, 2001]. The uniform PDF with a width related to *q*_{s} can be formulated in terms of a threshold in relative humidity – the “critical relative humidity”, *r*_{c} [see Appendix A for more details]. Fractional cloudiness, *f*, in such a formulation is expressed in terms of the grid-box mean relative humidity, , as:

with *f* = 0 for ≤ *r*_{c} and *f* = 1 for ≥ 1 [*Sundqvist et al.*, 1989]. When such schemes have been introduced as parameterizations in general circulation models, observations from field campaigns [*Slingo*, 1980], theoretical considerations [*Sundqvist et al.*, 1989] or cloud-resolving simulations [*Xu and Krueger*, 1991; *Lohmann and Roeckner*, 1996] have been applied to estimate *r*_{c}. Now satellite retrievals of and *f* exist for example from the Atmospheric Infrared Sounder (AIRS). So, the parameter *r*_{c} is indeed an observable and can be inferred from equation (1) as

for 0 < *f* < 1. In this study, two different observationally-based data sets are used (described in section 2) to estimate profiles of *r*_{c}, and compared to different versions of the model parameterization for fractional cloudiness (section 3). In section 4, the influence of this choice on the simulated cloud-climate feedback parameter is assessed.