Corresponding author: J. Quaas, Institute for Meteorology, University of Leipzig, Stephanstr. 3, D-04146 Leipzig, Germany. (firstname.lastname@example.org)
 A simple way to diagnose fractional cloud cover in general circulation models is to relate it to the simulated relative humidity, and allowing for fractional cloud cover above a “critical relative humidity” of less than 100%. In the formulation chosen here, this is equivalent to assuming a uniform “top-hat” distribution of subgrid-scale total water content with a variance related to saturation. Critical relative humidity has frequently been treated as a “tunable” constant, yet it is an observable. Here, this parameter, and its spatial distribution, is examined from Atmospheric Infrared Sounder (AIRS) satellite retrievals, and from a combination of relative humidity from the ECMWF Re-Analyses (ERA-Interim) and cloud fraction obtained from CALIPSO lidar satellite data. These observational data are used to evaluate results from different simulations with the ECHAM general circulation model (GCM). In sensitivity studies, a cloud feedback parameter is analyzed from simulations applying the original parameter choice, and applying parameter choices guided by the satellite data. Model sensitivity studies applying parameters adjusted to match the observations show larger positive cloud-climate feedbacks, increasing by up to 30% compared to the standard simulation.
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 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, qv, exceeds the saturation specific humidity, qs(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 = qv/qs. 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, qt, the sum of qv 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 qt (see Figure 1), and for the simplest assumption, the width of it could be expressed as a fraction γ of qt [e.g., Le Treut and Li, 1991], or as a fraction of qs. In this case, T and thus qs 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 qs can be formulated in terms of a threshold in relative humidity – the “critical relative humidity”, rc [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:
for 0 < f < 1. In this study, two different observationally-based data sets are used (described in section 2) to estimate profiles of rc, 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.
 In order to evaluate the GCM parameterization, critical relative humidity, rc is computed from relative humidity, , and cloud fraction, f, derived from different observationally-based data sets. In the first approach, both and f are from a passive satellite infrared sounder, in the second approach, f is from a satellite lidar combined with from meteorological re-analyses. First retrievals from the Atmospheric Infrared Sounder (AIRS) on board the Aqua platform (equator-crossing overpass at 1.30 p.m. local time in the ascending orbit) are used for both relative humidity and fractional cloudiness [Susskind et al., 2003] as reported on a 1° × 1° grid in the AIRX3STD product, bi-linearly regridded to a T63 spectral grid as used in the GCM studies. Daily data - i.e., the instantaneous retrievals - for both ascending (daytime) and descending (nighttime) orbits are used for the year 2003. The retrieval for temperature and specific humidity - allowing to compute relative humidity - is done in clear and partly cloudy scenes with up to 90% cloud fraction for 12 vertical levels at a horizontal resolution of approximately 40 km. Only the partly cloudy scenes are of interest in this study. In terms of accuracy, here it is of main importance that the retrieval error for relative humidity does not depend very much on the cloud fraction in terms of its bias [Susskind et al., 2006]. The statistical error in retrieved will average out since rc is linear in . A statistical error in retrieved f, though, may bias rc which is non-linear in f, and this problem will be investigated in the discussion of the results. The limitation to partly cloudy skies is acceptable for the purpose of this study, since it is only for fractional cloudiness (f < 1) that the rc parameter is useful. Within the AIRS version 5 (V5) retrieval, relative humidity is computed with respect to liquid water for layer-mean temperatures above 0°C, and with respect to ice for temperatures below this freezing temperature.
 As a second data set, relative humidity is used from the European Centre for Medium-Range Weather Forecasts (ECMWF) re-analysis (ERA-Interim) [Dee et al., 2011], generated at a horizontal resolution of T255 and 6-hourly temporal resolution. In ERA-Interim AIRS data are assimilated, so the two data sets are not completely independent. ERA-Interim relative humidity is further constrained by retrievals from TOVS on board of the NOAA operational satellites, by observations from the global radiosonde network, and by data from the GPS network. Relative humidity is defined with respect to liquid water where the grid-box mean temperature is above 0°C, and with respect to ice for temperatures below −23°C. Between these two temperatures, a mixed function is used following Matveev . ERA-Interim data for relative humidity is combined with cloud fraction retrieved from daily daytime spaceborne lidar observations in the GCM-Oriented Cloud-Aerosol Lidar and Infrared Pathfinder Satellite Observations (CALIPSO) Cloud Observations product (GOCCP) [Chepfer et al., 2010]. The GOCCP data set provides a cloud fraction retrieved from the attenuated backscatter at the full 330 m horizontal resolution of the CALIPSO lidar, but at a GCM-like coarser vertical resolution of 480 m. In contrast to the standard CALIPSO product, thus, the signal-to-noise ratio is increased by vertical rather than horizontal averaging for the threshold-based cloud detection. GOCCP data are gridded from the original 1° × 1° grid onto a horizontal T63 grid, at which the GCM data is available, and vertically projected to the coarser vertical grid of the ERA-Interim data. In the vertical projection, the cloud fraction in the layer with mean height closest to the one in the ERA-Interim layer is chosen so that no interpolation is carried out. CALIPSO data are not assimilated in ERA-Interim. The ERA-Interim data are gridded onto the common T63 horizontal grid, and from the six daily time steps, for each longitude, the one with local time nearest to the approximate 1.30 p.m. local time of overpass of the CALIPSO satellite is taken. ERA-Interim and GOCCP data are used for the year 2007. It should be noted that the CALIPSO lidar attenuates below clouds of an optical thickness of about 3 [Comstock et al., 2002; Chepfer et al., 2010], so that layers below optically thick clouds cannot be taken into account in the analysis. Such situations, where no lidar data for cloud fraction is available, are discarded from the analysis. In consequence, the ERA/GOCCP results in particular for the lowest layers of the atmosphere have to be treated with caution.
 For all data sets, rc is first computed from the instantaneous data and then averaged temporally. The observational data sets, both for cloud cover and for relative humidity, are interpolated to the T63 horizontal grid as used for the model. This is necessary since in equation (2), the grid-box mean values are used, and these have to be applied at the grid scale of interest for the model evaluation. The grid-box mean relative humidity, , is composed of the observationally-based humidity from the AIRS retrievals in the clear part of the grid-box, rclr, and an assumed relative humidity of r = 1 in the cloudy part: = f · 1 + (1 − f)rclr. For the ERA-Interim data, the relative humidity is reported as grid-box mean .
 The atmospheric general circulation model used in this study is based on the ECHAM5 GCM [Roeckner et al., 2003]. In this model, two optional cloud fraction parameterizations can be chosen: A scheme applying a “critical relative humidity” [Sundqvist et al., 1989], and a statistical scheme with a prognostic PDF of total water content [Tompkins, 2002]. We use the model in a horizontal T63 spectral resolution with 47 levels with prescribed climatological observed sea-surface temperature (SST) and sea-ice distributions. For the sensitivity studies analyzing different model resolutions, a single month (defined as January by the SST patterns) is simulated; and for estimating the cloud radiative feedback parameter, two simulations of ten years each are run for each parameter setting, one using the climatological SST, and one where SSTs are increased by a uniform 4 K [Taylor et al., 2009]. In ECHAM, relative humidity is computed with respect to liquid water for grid-box-mean temperatures above 0°C, and with respect to ice for temperatures below −35°C. Between these temperatures, the Bergeron-Findeisen process is parameterized by using saturation with respect to ice when some cloud ice already is present in the grid-box, and with respect to liquid water otherwise [Lohmann and Roeckner, 1996].
 From the combination of cloud fraction for grid boxes which are partially cloudy (0 < f < 1, so neither completely clear nor overcast) and grid-box mean relative humidity, equation (2) allows one to compute the critical relative humidity, rc. The geographical distribution of the annual mean distribution of the observed critical relative humidity is shown in Figure 2 for selected vertical levels. For the analysis of these results, remember that low values of rc indicate large subgrid-scale variability of humidity (see section 1 and Figure 1). Quantitatively, there are several differences between the results for AIRS and ERA/GOCCP. In general, rc estimated from AIRS is lower than the one inferred from ERA/GOCCP. A qualitative difference between the two data sets is observed over the Tropical warm pool. This difference can be traced back to the differences in the two quantities which go into the computation of rc, namely and f. Over much of the globe the two data sets show differences more in cloud fraction than in relative humidity. However, over the Tropical warm pool, cloud cover does not vary a lot over the year, and in this region the differences in rc are due to an offset of several percent in the relative humidity, which is larger in ERA-Interim than in AIRS retrievals. A further marked difference is also found over Northern Africa, but this might not be very relevant since few clouds are present over the desert. It is not intended here to judge on the relative quality of the AIRS and GOCCP retrievals. Comparisons between the two retrievals can be found elsewhere [e.g., Kahn et al., 2008]. Here, the diversity of the two is used as a proxy for the observational uncertainty.
 An interesting result for both AIRS and ERA/GOCCP are the homogeneously high values of rc in the 925 hPa level, particularly above the oceans. In this level, both data sets show relatively little cloud cover, implying that rc is close to the mean in many cases, and the value of rc found in this level is a characteristic value of . In the upper troposphere (around 200 hPa) the patterns of rc are rather different in the extra-Tropics. A likely reason for this is that for ice clouds, the assumption of sub-saturation in clear, and saturation in cloudy parts of a grid box is not always valid. Furthermore, retrievals of relative humidity become less accurate for the low specific humidities in the upper troposphere [Gettelman et al., 2004, 2006; Read et al., 2007; Fetzer et al., 2008]. In the lower troposphere, both observation methods may frequently be attenuated, so the results in the lowest layers are sampled from situations without overlying thick clouds only. These limitations will be kept in mind when analyzing the results.
 Qualitatively, several features clearly emerge.
 1. A vertical profile in rc is found (by comparison of the different layers in Figure 2; see also Figure 3), with on average larger subgrid-scale variability (lower rc) in the free/middle troposphere, very little subgrid-scale variability near the surface and in the lower planetary boundary layer (over ocean), and less variability as well in the upper troposphere. This vertical profile is consistent with previous studies investigating cloud-resolving simulations [Xu and Krueger, 1991].
 2. A distinct geographical distribution is observed, with large subgrid-scale variability in the subtropics, less in the extra-tropical strom tracks, and another minimum in the inner Tropics. The subgrid-scale variability tends to be lower above continents than over oceans.
 These findings are consistent with the results of Kahn and Teixeira , who investigated variability in temperature and specific humidity as retrieved from AIRS data at scales between 150 and 1200 km. They also find a clear vertical profile in subgrid-scale variability, with larger variability in the mid-troposphere; a distinct difference in variability between inner Tropics and sub-Tropics, as well as some indication for a land-sea contrast. A land-sea contrast in the parameterized critical relative humidity has also been applied by Rotstayn , who adjusted rc in their model in order to match the observed cloud radiative effects.
with pressure p, surface pressure ps, and the values of rc at the surface, cs, and in the free troposphere, ct. The shape of the vertical profile depends on a parameter nx. In the standard implementation, the parameter choice is constant at cs = 0.9, ct = 0.7, and nx = 4.
Figure 3 shows the global mean profiles for rc as found in the two observational data sets and in various model versions. The observational data sets, from the daytime and nighttime AIRS data, and from the ERA/GOCCP data, qualitatively agree well with each other. A difference between the two consists of slightly larger rc values from ERA/GOCCP over much of the troposphere, especially between 500 and 900 hPa and above 400 hPa. Comparing to these observational data the model results, it is found that the basic formulation of the Sundqvist et al.  scheme (equation (3)) qualitatively captures the vertical profile rather well (blue line in Figure 3). However, for the standard parameter setting, the subgrid-scale variability is much under-estimated compared to the observational data sets (compare the blue to the red/orange and black lines in Figure 3).
 A sensitivity simulation with the ECHAM model applying the statistical cloud scheme of Tompkins  is carried out. This scheme predicts the skewness and variance of an assumed beta-shaped PDF of the subgrid-scale variability of the total water mixing ratio. For the purpose of this study, nevertheless we compare it to the observations and the other parameterizations using the same framework of a critical relative humidity, implicitly assuming a simpler PDF shape. As for the observations, rc is computed from the model-simulated f and , by using equation (2) (purple line in Figure 3). The geographical distribution of rc at selected levels is shown in Figure 4 in a similar way as for the observations in Figure 2. The Tompkins  parameterization unfortunately does not reproduce the observed profile in subgrid-scale variability, and it grossly underestimates this variability. A similar result, namely too little variance virtually everywhere on the globe, is also found when evaluating the subgrid-scale distribution of the vertically integrated total water path simulated by the Tompkins  scheme with satellite data [Weber et al., 2011].
Figure 3 also shows the profile of rc from ERA-Interim alone (i.e., also cloud fraction as provided from ERA-Interim). This thus allows to evaluate the cloud cover parameterization in the underlying model, the Integrated Forecasting System (IFS). rc again is diagnosed from the data for cloud fraction and grid-box mean relative humidity. Note that in the prognostic cloud scheme of the IFS [Tiedtke, 1993], cloud cover is not diagnosed but predicted based on source processes, including new formation of stratiform clouds, where a critical relative humidity threshold is applied, and formation from convective detrainment as well as from boundary layer processes, and a sink term due to evaporation. This scheme does better than the ones applied in ECHAM, both in terms of the vertical profile, and in terms of subgrid-scale variability of humidity, but still slightly underestimates the latter. Also the geographical distribution of rc is much better captured by this scheme compared to the ECHAM GCM simulations with either the Tompkins  or the Sundqvist et al.  schemes (Figure 4). Previously, Teixeira  also found a relatively good agreement of the Tiedtke  cloud scheme to observations of the relationship between f and .
 The observations-based results for the critical relative humidity are also qualitatively consistent with results by large-eddy simulations (T. Heus, Max Planck Institute for Meteorology, personal communication, 2012), which are shown in the auxiliary material.
 Not only a uniform PDF, but also a triangular PDF of the subgrid-scale variability of the total water mixing ratio is consistent with the concept of a critical relative humidity [Smith, 1990, Appendix C]. Assuming a triangular shape of the PDF, the cloud fraction is [Smith, 1990]
The result for the ERA/GOCCP data set is shown by the olive green line to be compared to the black line in Figure 3. It yields a slightly different profile of rc, with a less pronounced vertical gradient. Much more variability in the boundary layer is found assuming a triangular-shaped PDF of total water than for the uniform PDF, but less in the upper troposphere. In absolute values, the diagnosed rc is within the range of the other observational-based estimates throughout the troposphere.
 The resolution-dependency of rc has been investigated from both observational data sets, computing it from f and gridded to a very coarse T21 grid (approximately 5.6°) and a finer T127 grid (approximately 0.9°). However, very little difference between the resolutions have been observed (indeed, the difference is so low that the curves largely overlap, which are thus not shown). In contrast, Kahn and Teixeira  found for a similar range of scales (their αS) a relatively weak, but distinct, scaling of water vapor mixing ratios in cloudy skies (power-law scaling exponents of about 0.2 to 0.4 for the standard deviation). This lack of resolution-dependency might indicate problems with the assumptions underlying this framework and need further investigations.
 In a further test, the temporal averaging has been performed not as a regular averaging, but by applying a non-linear regression in order to find the rc which best fits the time-series of f and at each grid-point (compare the plain and long-dashed black curves in Figure 3). This test also allows to qualitatively assess the effect of statistical errors in the observations of f on rc. The result for ERA/GOCCP shows a profile with lower rc than for the regular averaging (for AIRS, a similar result is found). The reason for this is that rc is non-linear in f, so that the temporal variability in f results in different average profiles of rc for different ways of averaging. The shape of the geographical and vertical distribution of rc, however, remains unaffected, so that the remainder of this study will focus on the results obtained by the “standard” averaging.
 The parameter choices approximately fitting the AIRS and ERA/GOCCP rc profiles using the Sundqvist et al.  formulation (dotted lines in Figure 3) are ct = 0.35, cs = 0.8, nx = 5 and ct = 0.4, cs = 0.85, nx = 3.5, respectively. This approach seems to work well for the global mean profiles for AIRS, except for the upper troposphere. For the global mean ERA/GOCCP profiles, it is not possible with this formulation to capture the slight decrease in rc in the lower boundary layer (Figure 3, black solid curve), and the parameter choice does not completely follow the shape of the curve in the free and upper troposphere. For a sensitivity test, though, this is considered acceptable here. It should be noted that the particular shape of the ERA/GOCCP rc in the lower boundary layer is subject to uncertainties due to frequent attenuation of the lidar cloud observation by overlying thick clouds.
 It might be useful not to prescribe a constant global-mean parameter setting, but to allow for a link to the meteorological regime. Such a link might be particularly useful for realistic climate change simulations. Inspired just by the shape of the geographical distribution, a very simple parameterization which takes into account dynamical influences is tried out here. This is to be seen as an illustration only, a realistic parameterization would need a more careful evaluation. A link to the resolved dynamics can be expressed in terms of the estimated inversion strength (EIS), derived from lower-tropospheric stability (LTS) [Klein and Hartmann, 1993] in combination with an estimate of the lifting condensation level and the average moist adiabat in the boundary layer [Wood and Bretherton, 2006]. Figure 5 shows the geographical distribution of EIS from ERA-Interim. EIS is a good predictor for cloudiness at least in low altitudes [Wood and Bretherton, 2006]. Its geographical distribution resembles the distribution of the critical relative humidity as seen in the observational data sets (Figure 2). Less stable regimes tend to favor turbulent mixing, driving humidity to a more well-mixed state with less variability in the middle troposphere at a large scale – consistent with the geographical distribution found in Figure 2. Thus, two different profiles of rc are used, rather than a single one, depending on the atmospheric stability, where the parameters ct = 0.34, cs = 0.8, nx = 2.5 and ct = 0.37, cs = 0.88, nx = 5 for EIS > 5 K (stable) and EIS ≤ 5 K (less stable), respectively. These fits to the ERA/GOCCP data are shown in Figure 6.
4. Implications for Climate Simulations
 Since the choice of rc is an essential part of the cloud cover scheme, it is expected to change the simulated climate. Here, first the simulated base state (present-day climate) is investigated, and then an idealized warmer climate.
Table 1 summarizes simulation results for a present-day configuration from model experiments applying the standard parameters of ECHAM's implementation of the Sundqvist et al.  parameterization, and of the two fits to the global mean rc profiles to the AIRS and to the ERA/GOCCP observations. The global annual mean model results are shown for a ten-year simulation with a prescribed climatological seasonal cycle of monthly-mean observed sea-surface temperature and sea-ice distributions. For total cloud cover, little sensitivity to the parameter choice is found (f ∈ [65, 67%]). In particular, the high-level and mid-level clouds are not sensitive to the modifications in this parameterization. Some variation is found, however, in the low-level cloudiness, although also for low clouds, the differences remain small. Little sensitivity is also found for the global annual mean top-of-atmosphere cloud radiative effects (only up to 4 Wm−2 in the solar, 0.5 Wm−2 in the terrestrial spectrum). A possible explanation for this rather small effect in the ECHAM model might be that in this model, relative humidities between rc and saturation tend to occur relatively rarely, with completely clear or overcast skies occurring much more frequently than partially cloudy skies (in this model version, which resolves the stratosphere, at 6-hourly output, 90.79% of all 3D grid-boxes in the 10-year simulation are completely clear, 5.25% overcast, and just 3.96% partly cloudy). A more detailed study on this model behavior is ongoing. The fact that there are no large changes in the present-day climate does not necessarily imply that climate change feedbacks are similar, too [Pincus et al., 2008; Klocke et al., 2011]. This is also found here, as described next.
Table 1. Global 10-Year Mean Results for the Model Sensitivity Studies for Total Cloud Cover (TCC), Low-Level Cloud Cover (LCC), Mid-Level Cloud Cover (MCC), High-Level Cloud Cover (HCC), Solar Cloud Radiative Effect (SCRE), Estimated From the Difference in Top-of-Atmosphere (TOA) Radiative Flux in the Solar Spectrum for All-Sky and Assumed Clear-Sky Conditions, Terrestrial CRE (TCRE)a
SCRE (W m−2)
TCRE (W m−2)
TCC: between the surface and 750 hPa; computed from the 3D cloud distribution using the random-maximum overlap hypothesis. MCC: between 750 hPa and 440 hPa. HCC: from 440 hPa to the top of the atmosphere. Geographical distributions of these quantities are shown as auxiliary material.
 In Table 2, the differences in cloudiness and cloud radiative effects are presented between these simulations and an idealized climate change experiment, in which the sea surface temperatures have been increased by a globally constant 4 K [Taylor et al., 2009]. The different model versions react to this perturbation in a similar way. Noticeable is a slightly stronger perturbation of high-level cloudiness and subsequently, of the terrestrial cloud radiative effect, in the model version in which the parameters have been adjusted to the AIRS-derived rc profile. The most substantial change in low-level cloudiness, and subsequently, in the solar cloud radiative effect, is found in the model version in which the rc profile is dependent on atmospheric static stability. Also listed are estimates of the simulated cloud-climate feedback from the difference in net top-of-atmosphere cloud radiative effect from a simulation where sea-surface temperatures are increased by 4 K and the control simulation, normalized by the change in net radiation imbalance at the top of the atmosphere, following Cess et al. . Note that this metric for the cloud-climate feedback is an indication of model diversity, not necessarily a rigorous estimate of realistic cloud-climate feedbacks. It is computed from simulations that are idealized, and the change in cloud radiative effect contains besides the cloud feedback partly also the water vapor feedback [Ringer et al., 2006]. The range of model realizations shown here is at the upper end of the results of the models investigated by Ringer et al. , which covered the range ΔNCRE/G ∈ [−0.05, 0.37]. While the effect of the parameter choice on the unperturbed climate was relatively weak, it nevertheless implies a considerable change in cloud-climate feedbacks (from 0.29 to 0.34 and 0.37, for the fits to AIRS and ERA/GOCCP, respectively). In a previous similar study, Rotstayn  found a stronger effect on the simulated present-day climate, and - in agreement with the present study - a considerable effect on the simulated cloud-climate feedback.
Table 2. Global 10-Year Mean Changes in the Quantities Listed in Table 1, and the Cloud Feedback Parameter (ΔNCRE/G) Defined as the Change in Net CRE (ΔNCRE = ΔSCRE + ΔTCRE) Between an Experiment Where SST is Increased by a Uniform 4 K and the Control Experiment, Normalized by the Change in TOA Radiation Imbalance, G
ΔSCRE (W m−2)
ΔTCRE (W m−2)
 The choice of a constant “critical relative humidity” may limit a GCM's capability in simulating climate feedbacks. Such a parameterization tightly couples cloud fraction, condensation, and relative humidity. It has been shown that relative humidity is approximately constant in GCM simulations of climate change, consistent with observations after the Mt. Pinatubo eruption [Soden et al., 2002]. Analyzing AIRS data, Gambacorta et al.  broadly support this finding on average, but with a large spatial variability. However, GCMs with schemes where relative humidity is limited by a fixed threshold might not be flexible enough to simulate anything different from approximately constant relative humidity, and are also very limited in the simulated responses of cloudiness to climate. For example, Senior and Mitchell  found less (negative) feedbacks acting in a relative-humidity based scheme compared to more flexible cloud parameterizations. As seen from Table 1, the results for the sensitivity study in which the parameters were tied to atmospheric stability in terms of average cloud cover, or cloud radiative effects are similar to the other simulations. However, linking the cloud cover to dynamics by this very simple method enhances the cloud feedback (Table 2). The cloud feedback parameter increases by 30% compared to the standard configuration. It is interesting to note that this larger positive cloud-climate feedback for a presumably more realistic parameter choice is in contrast to findings from idealized climate change simulations with a global cloud-resolving model [Miura et al., 2005] and a GCM with “super-parameterized” clouds [Wyant et al., 2006], which both find smaller cloud feedbacks and climate sensitivities than conventional GCMs.
 In this study, simple parameterizations of fractional cloudiness as implemented in general circulation models are evaluated using satellite data. In such parameterizations, subgrid-scale variability of humidity is accounted for by assuming that a grid box becomes (partially) cloudy where grid-box mean relative humidity exceeds a threshold, the “critical relative humidity”. Since it relates the observable parameters grid-box mean relative humidity and cloud cover, rc itself is an observable. It may serve to analyse subgrid-scale variability of humidity and to evaluate cloud cover parameterizations.
 From AIRS satellite data, and from a combination of ERA-Interim reanalyses and GOCCP lidar satellite retrievals, a consistent picture of vertically varying rc is found, with larger subgrid-scale variability in the middle troposphere. Also found are distinct geographical distributions, with less variability in the inner Tropics and mid-latitude storm-tracks than in the sub-Tropics, and more above continents than above oceans. The profile of rc as parameterized by Sundqvist et al.  captures the global-mean observed profile shape well, but the absolute amount of variability is grossly underestimated. To an even larger degree, the prognostic parameterization of subgrid-scale variability of humidity by Tompkins  largely underestimates the variability, and here, also the shape of the vertical profile is wrong. Among the cloud parameterizations evaluated, the Tiedtke  scheme as applied for the ERA-Interim reanalysis performs best. It is found that for a large range of horizontal grid resolutions (approximately 5° to 1°), the average rc profiles are virtually unchanged. Varying the parameters in the Sundqvist et al.  scheme, the profiles as observed can be better captured.
 Choosing the parameters fitted to the observed profiles of rc does not change the simulated global annual mean cloud cover or cloud radiative effects very much. However, some sensitivity of the cloud feedback to the parameter choice is found. In order to allow for a certain feedback of changes in dynamics to cloud cover, a simple modification to the Sundqvist et al.  parameterization is tested, where different rc profiles are used depending on estimated inversion strength. Less stable regimes are considered to allow for more mixing and thus less subgrid-scale variability in humidity, consistent with the geographical distribution found in the observations of rc. With this slightly modified parameterization, the cloud feedback is increased by 30% compared to the standard parameter setting, despite the finding that the present-day simulated climatological cloud cover and cloud radiative effects are not very different from the control simulation. This indicates that cloud cover parameterizations with fixed variance, or a fixed critical relative humidity, might under-estimate climate sensitivity.
 There are a couple of limitations to the present study. Future studies should investigate the subgrid-scale PDF of humidity in more detail, allowing also for more complex distribution shapes. For this, however, vertically resolved observations of specific humidity at high horizontal resolution would be necessary, which are presently not available from satellites. Ground-based remote sensing, or high-resolved model simulations as “virtual reality” might help. An important axis of further research would be to develop a better understanding, and a more comprehensive parameterization of subgrid-scale variability of relative humidity as a function of atmospheric processes. While this study demonstrates that in its present implementation, the Tompkins  parameterization fails to reproduce observed subgrid-scale humidity variability, this approach might serve as a basis for future studies.
 It has been shown earlier (e.g. by A. Tompkins in his lectures at the ECMWF) that a cloud cover scheme assuming a uniform PDF of total water mixing ratio subgrid-scale variability is equivalent to the Sundqvist et al.  scheme, if the variance is assumed a constant fraction of the saturation water vapor mixing ratio. For completeness, this is repeated here.
 From a scheme assuming a uniform PDF, cloud cover is defined as
as shown in Figure 1, where Δq = γqs and is used in the second step (qs, or, equivalently, temperature, is assumed constant in the grid-box). The grid-box mean relative humidity, is composed of a clear-sky and a cloudy-sky component, = f · 1 + (1 − f)rclr, where within the cloud saturation (r = 1) is assumed. The clear-sky relative humidity rclr can be written as
With this, the grid-box mean relative humidity is
where γ = 1 − rc has been used in the second step. Equation (1) follows from this.
 AIRS data were obtained through the Goddard Earth Sciences Data and Information Services Center (online at http://daac.gsfc.nasa.gov). ERA-Interim data used in in this study have been obtained from the ECMWF data server (http://www.ecmwf.int/products/data). GOCCP data were used as provided by the Laboratoire de Météorologie Dynamique/IPSL/CNRS (http://climserv.ipsl.polytechnique.fr). I would like to thank the groups generating and distributing these data for their very valuable support of this kind of research. Computing time was provided by the German High Performance Computing Centre for Climate- and Earth System Research (Deutsches Klimarechenzentrum, DKRZ). This study has been funded by the German Research Foundation (Deutsche Forschungsgemeinschaft, DFG) in an “Emmy Noether” grant. I am grateful to the Max Planck Institute for Meteorology (MPI-M) for hosting me during this research. I would like to thank Robert Pincus and four anonymous reviewers for helping improving this study considerably.