## 1. Introduction

Uncertainty in the representation of clouds in general circulation models (GCMs) is one of the major causes of the broad spread of predicted future climate change (Mitchell 2000; Stocker 2001), mostly via the impact on radiative heat fluxes and the resultant cloud-climate feedbacks. Many climate and weather forecasting GCMs use two variables to represent the cloud properties in each grid box. Typically one represents the mean mass concentration of condensate across the grid box and the other represents the fraction of the grid box that contains cloud. This cloud fraction is defined either by area (*C _{a}*) or by volume (

*C*) with the distinction shown schematically in Fig. 1. By definition,

_{υ}*C*will always be greater than or equal to

_{a}*C*, and we can expect this difference to increase with the vertical dimension of the grid box.

_{υ}Most schemes for determining the cloud fraction yield *C _{υ}*, which is more easily related to the mass of condensate. For example, in the European Centre for Medium-Range Weather Forecasts (ECMWF) cloud scheme of Tiedtke (1993) and the Met Office Unified Model (Wilson and Ballard 1999; Smith 1990), large-scale cloud fraction is defined to be the proportion of the humidity distribution across the grid box that exceeds saturation, thus giving

*C*directly. However,

_{υ}*C*is more appropriate for calculating the radiative effect of cloud (Stephens 1984; Edwards and Slingo 1996) and the representation of precipitation (Jakob and Klein 1999), but most GCMs assume that the cloudy area of a grid box fills the entire grid box in the vertical, thus setting

_{a}*C*equal to

_{a}*C*. This discrepancy may go some way in explaining discrepancies between the observed and modeled radiative fluxes discussed by Webb et al. (2001).

_{υ}The only previous reference to the distinction between *C _{a}* and

*C*appears to be that in Del Genio et al. (1996), which gives

_{υ}*C*=

_{a}*C*

^{2/3}

_{υ}by assuming that cloud within a grid box takes the form of a regular cube, which does not fill the grid box in the vertical. However when considering real clouds it is apparent that they are not regular cubes, and the real reason that

*C*and

_{a}*C*differ is that clouds have irregular geometries both in the horizontal and vertical, on scales much smaller than the grid box.

_{υ}It is also worthy of note that the difference between *C _{a}* and

*C*is not dealt with by the cloud overlap assumptions used in the models, as these describe the arrangement of cloud between different grid boxes in the vertical (see Morcrette and Fouquart 1986; Tian and Curry 1989). In contrast, the difference between

_{υ}*C*and

_{a}*C*is more a representation of subgrid cloud geometry, and the combination of the parameterization detailed in this paper with a realistic overlap assumption such as Hogan and Illingworth (2000), would enable an unbiased total cloud cover to be determined from a profile of individual

_{υ}*C*values.

_{υ}In this paper we use radar and lidar to directly measure the values of *C _{a}* and

*C*that would be simulated in a model, and develop a parameterization to enable models to determine

_{υ}*C*from

_{a}*C*as a function of grid-box size, cloud phase, and wind shear. The ability of radar and lidar to measure the profile of cloud occurrence with high resolution was demonstrated by Mace et al. (1998), Clothiaux et al. (2000), and Hogan et al. (2001). Indeed Hogan et al. (2001) compared three months of observed

_{υ}*C*with the values held in the ECMWF model and found that although the model simulated the frequency of cloud occurrence well, the amount of cloud when present was overestimated above 6 km and underestimated below 6 km. The underestimate below 6 km would become even more severe if observed

_{υ}*C*were compared with the values of

_{a}*C*currently used within such GCMs for the partitioning of radiative fluxes.

_{υ}In section 2 of this paper, we outline the observational methods used to derive values of *C _{a}* and

*C*, for comparison in section 3. In section 4, a parameterization is developed for use within a GCM to obtain the

_{υ}*C*for cloud from the currently held

_{a}*C*, which is evaluated in section 5.

_{υ}## 2. Observational methods and data

The primary observations are made from the 94-GHz Galileo cloud radar and Vaisala CT75K lidar ceilometer, at Chilbolton in southern England. At this site, stratiform and frontal cloud are predominantly observed. The radar and lidar are vertically pointing and operated near-continuously from 1 May 1999 until 26 May 2000. Examples of the radar and lidar observations for 9 May 1999 are shown in Figs. 2a and 2b.

### a. The 94-GHz cloud radar

The 94-GHz cloud radar records vertical profiles of reflectivity factor (*Z*), with a vertical resolution of 60 m, averaged over a 30-s time period, and is particularly sensitive to larger cloud particles. The cloud radar was calibrated by comparison with the Chilbolton 3-GHz weather radar in drizzle, where the droplets are small enough to Rayleigh scatter at both 3 and 94 GHz whereas larger hydrometeors will Mie scatter at 94 GHz. The 3-GHz weather radar at Chilbolton was absolutely calibrated using the redundancy of the polarization parameters in heavy rain (Goddard et al. 1994; Hogan et al. 2003a). Heavy rainfall results in significant attenuation of the 94-GHz signal, and so all occasions when the rainfall rate exceeds 0.5 mm h^{−1}, as measured by a rapid response drop counting rain gauge, were excluded from the analysis.

### b. Lidar ceilometer

To complement the radar observations, a 905-nm Vaisala CT75K lidar ceilometer was used. This records profiles of lidar backscatter (*β*), which is approximately proportional to particle diameter to the second power. In comparison to the *Z* returned from a radar, the lidar return is much more sensitive to high concentrations of small droplets, so it is therefore able to distinguish the liquid cloud base from any precipitation falling from the cloud, as well as detect the cloud base of even thin liquid water clouds that may be undetectable by the radar alone. The disadvantage of the lidar instrument is that the lidar beam is rapidly attenuated by the presence of any liquid water clouds, so producing lidar information only at or slightly above cloud base.

### c. Model fields

The model wind and temperature data used in this study were taken from daily operational ECMWF forecasts and Met Office operational mesoscale forecasts. The data were extracted for the model grid box closest to Chilbolton and a long time series was created by concatenating consecutive model forecasts, taken daily at 12 to 36 h after the analysis time (termed T+12 to T+36) for the ECMWF and every 6 h at T+4 to T+10 for the Met Office.

During the period of this study the ECMWF spectral model used T_{L}319 truncation, corresponding to a horizontal resolution of around 60 km, and with 25 model levels below 15 km. The Met Office model operated with a horizontal resolution of 0.11° × 0.11°, which is equivalent to approximately 11 km, and with 32 model levels below 15 km.

### d. Defining cloud fraction

To derive the cloud fraction from the radar and lidar observations, the radar and lidar data are analyzed in grid boxes, which contain a large number of the radar pixels, of height 60 m and a time of 30 s, corresponding to the sampling of the instrument. The radar and lidar returns from each pixel are then analyzed and each pixel is determined as either cloudy or cloud free, producing a cloud mask. Then *C _{υ}* is simply the fraction of the pixels within the grid box that are deemed to be cloudy, while

*C*is the fraction of the area of the grid box that contains cloud when viewed from above or below.

_{a}To determine whether an individual pixel is cloudy, we adopt the methodology of Hogan et al. (2001). A radar return is determined to be from cloud when it is located either above the freezing level (defined as the height at which the model wet-bulb temperature is 0°C), or above the lidar cloud base. This excludes any rain or drizzle falling from the base of the cloud, or insects below cloud base, but retains the ice. Mittermaier and Illingworth (2003) compared the freezing levels in the operational Met Office and ECMWF models and found them to be in close agreement with the radar observations.

Thin, nonprecipitating liquid water clouds are also observed by the lidar that the radar may not be able to detect. The bases of these clouds are clearly indicated by high values of *β* (>6 × 10^{−4} sr^{−1} m^{−1}) observed by the lidar, and are included in the cloud mask with an assumed thickness of 180 m (3 radar pixels). Clouds thicker than this typically develop sufficient drizzle droplets to become observable by the radar.

The resultant array of cloudy or noncloudy pixels, hereafter referred to as the cloud mask and shown in Fig. 2c, forms a time–height section of cloud at the radar’s high temporal (30 s) and spatial (60 m) resolution, from which *C _{a}* and

*C*are calculated as described above. The example cloud fractions shown in Fig. 2 are estimated over an hour with a vertical resolution of 720 m, and so are based on 1440 individually cloudy or noncloudy radar pixels. Although the cloud mask is a two-dimensional slice, given sufficient samples it is believed to be representative of the three-dimensional cloud field.

_{υ}To investigate the degree of difference between cloud fraction by volume and cloud fraction by area, *C _{a}* and

*C*were calculated from the cloud masks on a variety of regular grids, with time intervals of 10, 20, 60, 180, and 360 min and 120, 360, 720, 1080, and 1440 m in the vertical. The time-averaging periods are converted to horizontal resolutions using the ECMWF wind speeds, and on average, the selected time intervals are equivalent to horizontal resolutions of approximately 10, 20, 65, 200, and 390 km. Initially only odd numbered months will be analyzed, so that the even numbered months can be used as an independent dataset upon which to test the emerging parameterization.

_{υ}## 3. Evaluating cloud fraction by area and volume

To gauge the significance of the difference between *C _{a}* and

*C*, Fig. 3 shows the mean underestimate of

_{υ}*C*by

_{a}*C*as function of vertical dimension of the grid box, for a variety of horizontal resolutions, for all events in the odd numbered months in the yearlong dataset. It is significant that the mean underestimate is all but independent of the horizontal grid-box dimension (

_{υ}*H*), and strongly dependent on the model vertical grid-box dimension (

*V*). This property is based on an average of 6 months of radar and lidar observations. Examining a single month’s data shows similar results, but with rather more noise.

For a typical numerical weather prediction (NWP) model, vertical resolution is currently of the order of 500 to 1000 m in the midtroposphere, which corresponds to an underestimate of *C _{a}* by

*C*of 30% to 50%. This underestimate is reduced with the vertical grid-box dimension, but even with a vertical resolution of 120 m, which is a typical vertical resolution of an NWP model within the boundary layer,

_{υ}*C*is still a 7% underestimate of the observed

_{υ}*C*. This would indicate that to eliminate the problem solely through increasing vertical resolution would require vertical resolutions throughout the troposphere that may take many decades to be achievable in operational NWP or climate prediction models.

_{a}Now that the significance of the underestimate of *C _{a}* by

*C*has been evaluated, we will examine this bias in more detail. In doing so it is necessary to examine only partially cloudy grid boxes, where 0 <

_{υ}*C*< 1, as by definition,

_{υ}*C*and

_{a}*C*are equal for either fully clear or cloudy grid boxes. Figure 4 shows the mean underestimate of

_{υ}*C*by

_{a}*C*for partially cloudy grid boxes with

_{υ}*V*and

*H*.

When looking only at partially cloudy grid boxes, the effect of increasing *V* is still apparent, but the mean underestimate decreases as *H* increases. This is due to the subdivision of a given cloud field producing more grid boxes that are either completely clear or completely cloudy, in which *C _{a}* and

*C*are equal. In the remaining partially cloudy grid boxes, the underestimate of

_{υ}*C*by

_{a}*C*is correspondingly greater. When averaging over all grid boxes, as in Fig. 3, these two effects of varying

_{υ}*H*cancel out, but in developing a relationship between

*C*and

_{υ}*C*we must examine only partially cloudy grid boxes, where the effect of varying

_{a}*H*is significant.

With a view to examining the physical processes that lead to the generation of this bias, in Fig. 5 the ECMWF model outputs were used to classify each grid box according to temperature (*T*) and vertical wind shear (*s*), which was derived by examining the vector change in the horizontal wind velocity with height. This approach has been successfully used by Hogan and Illingworth (2003) when examining ice-fall streaks and cloud inhomogeneities and is an approach often used in GCMs themselves. For example, it is an intrinsic part of the calculation of turbulent fluxes, and the local Richardson number used in the cloud and boundary layer schemes in the Met Office Unified Model (Smith 1990; Martin et al. 2000). In this analysis it is assumed that within a grid box with *T* < −15°C, only a very small proportion of any cloud is still in liquid phase, so it is classified as ice cloud. These temperature boundaries are selected to remove the majority of the mixed phase clouds (Hogan et al. 2003b; Hogan et al. 2004), clouds with a temperature between 0° and −15°C are ignored.

Figure 5 demonstrates that the mean underestimate of *C _{a}* by

*C*is most significant for liquid water cloud and increases with wind shear. This arises because liquid water clouds typically form in thinner layers than ice clouds; Mace et al. (1997) found that mean cirrus thickness is 1.6 km, whereas for stratocumulus and supercooled altocumulus clouds, 100 to 200 m is typical, so liquid water clouds are less likely to fill a grid box in the vertical.

_{υ}As an investigation of the sensitivity of these results to the assumed thickness of those liquid water clouds detected only by lidar, this thickness was raised from 180 to 360 m. The effect of this was to reduce the observed underestimate of *C _{a}* by

*C*by only a fraction of a percent at the resolutions considered above.

_{υ}The only reported parameterization for obtaining *C _{a}* is that of Del Genio et al. (1996), which uses

*C*to mimic

_{υ}*C*by raising it to the power ⅔. Given the resolution dependence of the observed underestimate of

_{a}*C*by

_{a}*C*, it is to be expected that the Del Genio et al. (1996) correction will also be highly resolution dependent. Figure 6 shows the mean bias in

_{υ}*C*against the observed

_{υ}*C*after correction by the Del Genio et al. (1996) method, as a percentage of the mean

_{a}*C*. For the larger vertical resolutions there is still a significant underestimate in

_{a}*C*of up to 40%, although this is a reduction from almost 60% for unmodified

_{a}*C*in Fig. 3. As the vertical grid separation decreases, so the Del Genio et al. (1996) becomes an overestimate of

_{υ}*C*by up to 40%. There is hence a need to seek a more appropriate parametric from relating

_{a}*C*and

_{υ}*C*.

_{a}## 4. Parameterizing cloud fraction by area

In section 3 we examined the underestimate of *C _{a}* by the use of

*C*and found that, for partially cloudy grid boxes, the underestimate increased with vertical grid separation (

_{υ}*V*), and decreased with horizontal grid separation (

*H*). Also this underestimate was more significant for liquid water clouds than for ice, and increased with wind shear. In this section we aim to produce a simple parameterization that captures these observed trends. To determine an appropriate functional form for a parameterization of

*C*from

_{a}*C*, we will consider the observed relationship between the two cloud properties. Figure 7 shows scatterplots of

_{υ}*C*versus

_{a}*C*for two example resolutions. Note that only a specific range of horizontal grid-box dimensions are shown. The discretization of the

_{υ}*C*values in Fig. 7a is due to the smaller sample size when averaging over a smaller period of time. Overlaid are the mean values

_{a}*C*and

_{a}*C*in bins of

_{υ}*C*, which a successful parameterization should be able to reproduce.

_{υ}*C*allowed to vary, which we will refer to as the power law parameterization, as shown in Eq. (1):A second possible parameterization is formulated based on the observation that the line produced by the mean

_{υ}*C*versus

_{a}*C*in Fig. 7 is roughly symmetrical on reflection in the line

_{υ}*C*= 1 −

_{a}*C*. One suitable equation for such a relationship is given in Eq. (2), and will be referred to as the symmetric parameterization:where the

_{υ}*f*parameter controls, for

*f*> 0, the extent to which

*C*is greater than

_{a}*C*, with

_{υ}*f*= 0 giving

*C*=

_{a}*C*, as shown in Fig. 8.

_{υ}The solid lines overplotted in Fig. 8 show the relationship between *C _{υ}* and

*C*using Eq. (1) and the value of

_{a}*D*that best fits observed mean values of

*C*in the bins of

_{a}*C*, in terms of least squares. Similarly, the dashed lines show the relationship between

_{υ}*C*and

_{a}*C*, using Eq. (2) with a value of

_{υ}*f*that best fits the mean values of

*C*in the

_{a}*C*bins.

_{υ}For cases where *C _{υ}* is small (less than 0.2 and 0.3 in the examples presented), the

*C*produced by the power law parameterization is significantly overpredicted to the extent that, for a

_{a}*C*of 0.05, the

_{υ}*C*predicted by (1) is approximately double the mean observed value of

_{a}*C*. Conversely, for higher values of

_{a}*C*is underestimated by the power law parameterization. Neither of these problems is evident when comparing the symmetric parameterization. Although only two example resolutions have been shown here, the problems with the power law (solid lines) are apparent in all of the resolutions examined. If the value of

_{υ}, C_{a}*D*in Eq. (1) is selected by minimizing the rms error against the raw data points, rather than the mean values of

*C*and

_{a}*C*in

_{υ}*C*bins, then the value of

_{υ}*D*is altered to reflect the distribution of

*C*, to the extent that the overestimate of low cloud fractions is slightly reduced, at the expense of drastically increasing the underestimate of

_{υ}*C*for larger

_{a}*C*. We therefore conclude that the symmetric parameterization, given in Eq. (2) best describes the mean dependence of

_{υ}*C*on

_{a}*C*, and we will continue in this section to develop this parameterization further.

_{υ}To determine the appropriate values of *f*, the cloud fraction data were reanalyzed, splitting the observed values of *C _{a}* and

*C*according to their vertical (

_{υ}*V*) and horizontal (

*H*) grid sizes, where the ECMWF wind speed, at the specific time and height, was used determine the actual horizontal distance which a given averaging period represents. The variability of wind speeds with height is such that the number distribution of cases with a given

*H*will vary with height, but the variability of the wind speed with time ensures that all

*H*bins examined are well populated at all heights. Such an approach significantly improves the fits of

*f*with

*H*compared to using a single wind speed with height to convert to a horizontal distance. The resultant variation of

*f*with

*H*and

*V*is shown in Fig. 9.

*f*increases with

*V*, and decreases with

*H*, which is consistent with the relationship between the mean underestimate of

*C*by

_{a}*C*, and

_{υ}*H*and

*V*shown in Fig. 4. To determine the exact relationship between

*f*and the model resolution, we assume that as

*V*tends to zero, or

*H*tends to infinity, so

*C*will tend to

_{a}*C*, and therefore

_{υ}*f*tend to zero. We therefore assume that

*f*may be given in (3),where

*A*,

*α*, and

*β*are parameters to be fitted to the observations.

Based on the observation in section 3 that the underestimate of *C _{a}* by

*C*was greater for liquid water clouds than for ice, the coefficients in Eq. (3) need to be determined separately. To determine the appropriate values for the coefficients in (3),

_{υ}*α*and

*β*were allowed to vary, iteratively selecting the pair of values for which (3) gave the best fit to the observed values of

*f*, in terms of least squares. The results of this analysis are shown in Fig. 10, where the analysis has been performed separately for ice and liquid water clouds, using the model temperature (

*T*) profiles as in Fig. 5.

Given the dependence of *s*) observed in Fig. 5, a logical extension to this work is to categorize clouds by wind shear as well as by cloud phase. For simplicity we assume that as wind shear varies, Eq. (3) still applies and that *α* and *β* remain constant while we allow *A* to vary.

The analysis shown in Fig. 10 was repeated for a number of wind shear bins; within each wind shear bin a shear-dependent value of *A* was computed as the gradient of *f* with *V ^{α}H^{−β}*. The variation of the mean observed

*A*(

*s*) with wind shear is shown in Fig. 11a, and in order to represent this data, we define the limit of

*A*(

*s*) as

*s*tends to zero [

*A*(

*s =*0)]. For the purposes of fitting a power law, Fig. 11b shows the variation of

*A*(

*s*) −

*A*(

*s*= 0) against wind shear on logarithmic axis, and the resultant power laws are overplotted on Figs. 11a and 11b.

*f*parameter in Eq. (2) can be parameterized as follows, with all terms in the equations in SI units. We propose that

*A*can be determined either with or without a wind shear dependence as appropriate to the user requirements:

With this parameterization, the correction of *C _{υ}* to

*C*is increased with wind shear for both ice and liquid water cloud. However, ice cloud is less sensitive to wind shear, both in that the overall effect of wind shear is less, and that the shear term reaches an almost asymptotic value at lower wind shears than for liquid cloud (consistent with Fig. 5). One physical explanation for this would be that the liquid water cloud droplets fall more slowly than ice particles so that a given wind shear gives a larger difference in horizontal transport for the more slowly falling particles.

_{a}## 5. Evaluation of the *C*_{υ} to *C*_{a} parameterizations

_{υ}

_{a}

To evaluate the parameterizations given above, a test period consisting of the even numbered months over the period 1 May 1999 to 26 May 2000, was examined analyzed to produce an independent dataset of cloud fractions, *C _{a}* and

*C*, using the same time and height resolutions as used in sections 3 and 4, and converting time to

_{υ}*H*using the ECMWF wind speed. The values of

*C*were corrected using the symmetric parameterization, (2), and values of

_{υ}*f*obtained, using Eqs. (4)–(9), by first ignoring and then including the effects of wind shear and compared to the observed

*C*.

_{a}Figure 12 shows, for an example resolution of 65 km in the horizontal by 720 m in the vertical, a scatterplot of *C _{a}* and

*C*, and the mean relationship between

_{υ}*C*and

_{a}*C*as observed and predicted by the symmetric parameterization, (2), obtaining

_{υ}*f*both with and without wind shear. It is clear that the use of

*C*to represent

_{υ}*C*leads to a significant underestimate in

_{a}*C*for partially cloudy grid boxes, and that averaging over a long period both methods of obtaining

_{a}*f*give mean values of

*C*that are in good agreement with the observed

_{a}*C*across the full range of

_{a}*C*.

_{υ}Tables 1, 2 and 3 give the mean bias and rms error in the prediction of *C _{a}* by

*C*during the test period, for uncorrected

_{υ}*C*and

_{υ}*C*and then corrected both without and with wind shear information, respectively. These statistics are reweighted to give equal weight to each grid resolution examined. For the uncorrected

_{υ}*C*, the mean bias is considerable, with the mean

_{υ}*C*during this period, and range of resolutions, is 0.126, and the mean

_{a}*C*0.079, which corresponds to an underestimate of 0.046, or 37% of the mean

_{υ}*C*. This is consistent with the mean underestimate from the odd numbered months as in Fig. 3. Ice clouds exhibit a less significant bias between

_{a}*C*and

_{a}*C*than liquid water clouds, with an underestimate of 27% compared to 48%, respectively. Both methods of correcting

_{υ}*C*(Tables 2 and 3) give good results, with the mean bias in

_{υ}*C*predicted from

_{a}*C*being less than 0.2% of the mean

_{υ}*C*, for both ice and liquid clouds.

_{a}In more detail, when correcting without wind shear, using Eqs. (4), (5), (7), (8) shown in Table 2, the parameterization performs well for all resolutions considered with remaining biases varying between +3% where *H* is less than 20 km, to a bias of −3% for grid boxes with *H* between 20 and 100 km although there remain biases of +5.7% in instances of low wind shear, and −9.0% for high wind shear. These are much less than the biases in the uncorrected *C _{υ}* of −33% and −46%.

Correcting *C _{υ}* using phase and wind shear (Table 3) performs well at all the vertical and horizontal resolutions examined, reducing underestimates ranging from 15%–50% to an underestimate of 2.7% in the very worst case. All of the resolutions considered show improved performance compared to excluding wind shear effects. Clouds in conditions of negligible wind shear exhibit a less significant bias than clouds in a more sheared environment, and when correcting

*C*with wind shear the parameterization accounts for this effect.

_{υ}Both parameterizations reduce the rms errors of the corrected *C _{υ}* by a factor of 2 to 3, relative to the uncorrected

*C*, but are still significant. This is to be expected given that the scheme provides an average correction to account for the random arrangement of cloud within the grid box, however by introducing the ability to both over- and underestimate

_{υ}*C*, the bias is all but eliminated. The inclusion of the effect of wind shear reduces the rms error by 2% (i.e., from 35% to 33%) relative to the parameterization that excludes the effect of wind shear. Combining the 15% variation in the bias of the parameterization between high and low wind shear environments, when wind shear is ignored, with the small improvements across the range of resolutions, when wind shear effects are included, indicates that including the effect of wind shear gives a discernable improvement to the performance of the parameterization, although the effect is small relative to magnitude of the correction that is being applied.

_{a}To quantify the impact of the *C _{a}* parameterization with a realistic GCM resolution, the radar and lidar data were analyzed as described in section 2 to obtain

*C*and

_{a}*C*on the same grid as the Met Office mesoscale model outputs in the vertical and where the time-averaging period was determined as the time taken to advect a 12-km grid box using the model wind speed. This is opposite approach to averaging for a set period of time and using the model winds to calculate

_{υ}*H*, which is the approach adopted in the previous sections (in practice, the results from the two approaches are indistinguishable). The observed mean profile of

*C*is shown in Fig. 13 and is broadly comparable with those output by Met Office mesoscale model, although a detailed evaluation of the differences between the observed cloud fractions and the model outputs is beyond the scope of this paper. Also shown is the mean profile of the observed

_{υ}*C*and the

_{υ}*C*produced by correcting the observed

_{a}*C*using Eq. (2), with the

_{υ}*f*parameter determined using appropriate values of

*H*and

*V*, and values of wind shear taken directly from the model winds.

The mean profiles of *C _{a}* are approximately 30% higher than the mean profile of

*C*, which is consistent with Fig. 3, while the profile of parameterized

_{υ}*C*agrees with the observed mean

_{a}*C*to within 2% at all heights. There is no noticeable difference between the parameterization with and without wind shear when examining averages over long periods of time.

_{a}Last, we calculate the total cloud cover from the profiles of *C _{a}* and

*C*, by applying the most commonly used maximum-random overlap assumption as defined in Geleyn and Hollingsworth (1979). The mean total cloud cover is found to be 53% when

_{υ}*C*is used, and 63% when using

_{υ}*C*(observed or parameterized). This increase is comparable to the 11% increase found by Morcrette and Jakob (2000) when changing from the maximum to the random overlap assumption in the ECMWF model, which in terms of global mean radiative fluxes at the top of the atmosphere reduced the absorbed shortwave radiation by 3.3 W m

_{a}^{−2}and the outgoing longwave radiation by 8.3 W m

^{−2}.

## 6. Conclusions

A quasi-continuous dataset of radar reflectivity and lidar backscatter over Chilbolton, in southern England, has been used to produce a climatology of cloud fractions, at a variety of horizontal and vertical resolutions appropriate for comparison with GCMs. By calculating cloud fractions by area and by volume on regular grid, a systematic underestimate has been found in which cloud fractions calculated by volume (*C _{υ}*) are always lower than the corresponding cloud fraction by area (

*C*), which is of more interest in the calculation of fluxes of radiation and precipitation. For example, at typical model vertical grid-box spacings of 500 to 1000 m, we observe that

_{a}*C*is underestimated by

_{a}*C*by 30% to 50%. This is caused by the stratification of cloud layers occurring with vertical scales that cannot be fully resolved at these resolutions, and by a degree of randomness with which cloud forms, so spreading cloud in the horizontal, rather than filling a grid box in the vertical. The underestimate of

_{υ}*C*by

_{a}*C*has been shown to be approximately twice as large for liquid water clouds than for ice and this underestimate also increases with wind shear.

_{υ}In the interest of expressing this effect in a manner that is easily applicable to most GCM cloud schemes, a parameterization has been developed that gives a correction to the *C _{υ}*, for frontal/stratiform cloud. This parameterization is based on the observation that the mean relationship between

*C*versus

_{a}*C*is roughly symmetric about the line

_{υ}*C*= 1 −

_{a}*C*, and varies with the horizontal and vertical grid-box dimensions (

_{υ}*H*and

*V*, respectively), cloud phase, and to a lesser extent, wind shear. This approach performed better than the formulation of Del Genio et al. (1996), where

*C*=

_{a}*C*

_{υ}^{2/3}, even when the exponent was allowed to vary.

The resultant parameterizations, both excluding and including the effect of wind shear, have been tested against an independent dataset of Chilbolton radar and lidar data, and were found to give good agreement across the range *C _{υ}*. On average, both correction schemes were found to reduce the mean underestimate of

*C*by

_{a}*C*from 37% (at the range of horizontal and vertical resolutions used) to a mean bias of less than 0.2% both with and without the use of wind shear information. The parameterizations cope with the variation in this underestimate with grid-box resolution, as well as phase. However when wind shear is ignored, the parameterization produces a variation of approximately 15% of the mean

_{υ}*C*, between the low and high wind shear environments. When wind shear is included, this variation is accounted for, and the overall performance of the parameterization is improved, although this effect is small relative to the magnitude of the correction that is applied.

_{a}When examining cloud fractions on a grid equivalent to that of the Met Office mesoscale model outputs, the mean profile of *C _{a}* is well captured by the parameterization. When using

*C*rather than

_{a}*C*, the mean total cloudiness of the column is increased from 53% to 63%. We would therefore expect the implementation of this parameterization of

_{υ}*C*would have significant radiative impacts, although the issue is complicated by the representation of the inhomogeneity of water content that could accompany a

_{a}*C*parameterization.

_{a}Before applying the findings of this study it would be useful to study the subgrid geometry statistics from cloud observed in other locations, since the properties of convective cloud may differ significantly from the properties of stratiform and frontal cloud predominantly observed at Chilbolton, although we would expect *C _{a}* to be much closer to

*C*for upright convective clouds.

_{υ}## Acknowledgments

We thank the Radio Communications Research Unit at the Rutherford Appleton Laboratory for providing the 94-GHz Galileo radar and lidar ceilometer data. This research received funding from the European Union CloudNET project (Grant EVK2-CT-2000-00065) and NERC Grant NER/T/S/1999/00105. We are also grateful to Pete Clark and Richard Forbes at the Met Office for their help and for providing the Unified Model output and to Christian Jakob for providing the ECMWF model data. The first author was also supported by an NERC studentship.

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Biases (both absolute, and in terms of the percentage of mean *C _{a}*), and rms errors, in the representation of

*C*by

_{a}*C*. The analysis is based on cloud masks of radar and lidar data over Chilbolton during even numbered months between 1 May 1999 and 26 May 2000.

_{υ}Biases (both absolute, and in terms of the percentage of mean *C _{a}*), and rms errors, in the representation of

*C*by

_{a}*C*after correction using the symmetric parameterization (2), with

_{υ}*f*found without wind shear [(4), (5), (7), (8)]. The analysis is based on cloud masks of radar and lidar data over Chilbolton during even numbered months between 1 May 1999 and 26 May 2000.

Biases (both absolute, and in terms of the percentage of mean *C _{a}*), and rms errors, in the representation of

*C*by

_{a}*C*after correction using the symmetric parameterization (2), with

_{υ}*f*found including the effect of wind shear [(4), (6), (7), (9)]. The analysis is based on cloud masks of radar and lidar data over Chilbolton during even numbered months between 1 May 1999 and 26 May 2000.