## 1. Introduction

In order to produce accurate radiative fluxes in large-scale atmospheric models (e.g., general circulation models; GCM), it is necessary to parameterize subgrid-scale (i.e., on unresolved horizontal scales) cloud distributions. This is due to the nonlinear dependence of radiative fluxes on cloud properties, which makes cloud variability on small spatial scales important to radiative fluxes on large scales (e.g., Wielicki and Welch 1986; Boer and Ramanathan 1997; Stubenrauch et al. 1997). Parameterizations of cloud microphysical processes are also beginning to use detailed information of subgrid cloud distributions (e.g., Jakob and Klein 1999, 2000).

Subgrid cloud distributions are typically constructed from grid-scale values of cloud fraction and cloud water density at each vertical level together with an overlap assumption. Bulk properties of the cloud distribution such as “total cloud fraction” (i.e., the fractional area of a grid-scale column that is cloudy) are determined by the construction and can vary widely depending on the overlap assumption. The overlap assumption is traditionally described in terms of three asymptotic realizations: “maximum overlap,” for which cloud overlap is as large as possible (cf. Geleyn and Hollingsworth 1979); “minimum overlap,”^{1} for which cloud overlap is as small as possible (cf. Tian and Curry 1989); and “random overlap,” which assumes that horizontal cloud distributions from different vertical levels are independent (cf. Manabe and Strickler 1964). However, actual cloud overlap properties are much more complex (e.g., Tian and Curry 1989; Ramsey and Vincent 1995; Dudek et al. 1996). In fact, many GCMs [e.g., those developed at the European Centre for Medium Range Forecasts (ECMWF), the Goddard Institute for Space Studies (GISS), and the next generation GCM from the National Center for Atmospheric Research (NCAR)] use a hybrid overlap assumption, called “maximum-random,” in which clouds from adjacent levels are considered to be maximally overlapped and clouds separated by clear levels are randomly overlapped (e.g., Tian and Curry 1989; Stubenrauch et al. 1997; Morcrette and Jakob 2000; Collins 2001).

Each of these overlap assumptions has significant limitations. Used individually, they are inflexible; that is, there is no adjustable parameter with which cloud distributions can be altered. Each overlap assumption has other deficiencies that can cause a model to produce unrealistic cloud distributions. For example, maximum overlap only allows one to simulate perfectly stacked clouds. That might be reasonable for models with very high horizontal resolutions, such as cloud resolving models, but certainly not for most GCMs.

The random and maximum–random assumptions recognize that clouds are not, in general, perfectly stacked and for many cases, they perform well at the spatial resolution of an atmospheric GCM. However, those assumptions derive cloud overlap fraction (i.e., the fractional area of a grid in which clouds at two different elevations overlap) according to which model level the clouds are in and not according to physical parameters. This causes the subgrid cloud distribution to have certain undesirable properties, and even display pathological behavior. For example, Fig. 1a shows the total cloud fraction under random (short-dashed line) and maximum–random (long-dashed line) overlap assumptions as a function of the vertical resolution (i.e., the number of vertical levels) for the vertical profile of cloud fraction shown in Fig. 1b. Total cloud fraction, under the random overlap assumption, asymptotes to 100% as vertical resolution increases (cf. Raisanen 1999). This is because random overlap neglects the vertical coherence of cloud fields and never allows complete overlap, even at arbitrarily small vertical separations.

Maximum–random overlap is an ingenious construction that, to a large degree, circumvents the convergence problems of random overlap because it allows clouds in adjacent levels to be maximally overlapped. However, it can display erratic behavior with changes of resolution. Maximum-random overlap produces large jumps of total cloud fraction in Fig. 1a for small changes of vertical resolution. This occurs because the determination that two cloudy layers are adjacent depends on details of the vertical discretization. Furthermore, with maximum–random overlap, a small change in the parameterizations that determine cloud fraction in each level can cause a large change in total cloud fraction. Consider, for example, a region depicted by a GCM column (on the order of 1–200 km wide) that contains active convection, with at least some cloudiness in all levels from the boundary layer to near the tropopause. In this example, no one level has more than 30% cloud fraction. However, the total cloud fraction is large, say 75%, because the cloud distribution is complex, containing many individual clouds. If the cloud parameterization places clouds in vertical regions that are separated by clear regions (e.g., the parameterization only produces cirrus anvils together with some midlevel and low-level clouds), then maximum–random overlap can produce a realistic total cloud fraction because there are three randomly overlapped cloudy regions. Now, consider an improvement to the cloud parameterizations that produces more cloud types (e.g., convective towers) and, thus, clouds at more vertical levels. Each of the intermediate clear regions will now have at least a small amount of cloud. The overlap parameterization detects cloudiness in all those levels and produces maximally overlapped clouds throughout, with a total cloud fraction of only 30%. For this case, an improvement to the physical parameterizations that produce clouds acts to degrade the simulated total cloud fraction as well as radiative fluxes.

*α*such that

*α*decreases exponentially with the separation distance of the two cloudy levels in question. A similar exponential decrease has been found in data from cloud resolving models (H. Barker 1998, personal communication).

Utilizing these results, we first recognize the overlap parameter *α* as the correlation between cloudiness at two levels and then parameterize overlap in terms of a decorrelation depth *L*_{c}. In principle, *L*_{c} can be specified as a function of space and time, even as a function of cloud type within the same grid-scale column. Thus, the parameterization is very flexible, allowing a wide range of subgrid cloud distributions to be determined from any given vertical profile of cloud fraction by altering a single parameter that has an obvious physical interpretation.

Section 2 discusses the conceptual framework in which the parameterization is formulated. Details of the parameterization are developed in section 3. It is found that merely specifying the overlap probabilities between pairs of levels does not guarantee a consistent subgrid cloud distribution. In fact, the specification of those overlap probabilities can overconstrain the subgrid cloud distribution. Here, the analogy between vertically coherent cloud distributions and a Markov process is exploited in order to produce consistent subgrid cloud distributions with desirable properties. The utility of the parameterization is demonstrated in section 4, which assesses the sensitivity of radiative fluxes to changes of *L*_{c}. To do so, the parameterization is used in conjunction with two radiative transfer formulations, both utilizing the radiative transfer model from the community climate model (CCM) developed the National Center for Atmospheric Research (NCAR). The first is the independent column approximation (ICA), which divides the grid-scale column into subgrid columns, one for each permutation of clear and cloudy conditions in different levels. The second is a more efficient formulation based on modifications to the NCAR radiative transfer model by Bergman and Hendon (1998). The modified model closely approximates ICA fluxes at greatly reduced computational cost by using less detailed information of the cloud distribution. Sensitivity calculations using idealized cloud distributions reveal fundamental properties of the relationship between the correlation depth and radiative fluxes. Calculations using cloud distributions produced by CCM examine that sensitivity in a more practical application. Concluding remarks are found in section 5.

## 2. Conceptual framework

### a. Cloud overlap matrix

*C*(

*k*) represents the fractional area of the column that has clouds at model level

*k*and can be interpreted as the probability that, at any given horizontal location within the column, clouds are present in that level. Total cloud fraction

*C*

*C*(

*k*) represents the normalized area integral of a cloudiness function

*A*is the area of the column. The cloudiness function

*c*(

*x,*

*y,*

*k*) has a value of 1 if there is a cloud present at the point (

*x,*

*y*) in level

*k,*and is zero otherwise.

*k*and

*l*overlap if, for some point (

*x,*

*y*) in the column,

*c*(

*x,*

*y,*

*k*) and

*c*(

*x,*

*y,*

*l*) are both equal to one. The fractional area over which clouds in the two levels overlap, or the probability that clouds in the two levels overlap, is

*K*×

*K*matrix (

*K*is the number of cloudy vertical levels in the GCM) with elements

*O*

_{kl}. The overlap matrix only describes

*first-order*overlap probabilities (i.e., overlap for clouds from two different levels), but not higher-order probabilities (i.e., the simultaneous overlap for clouds from three or more levels). For example, consider three levels

*k,*

*l,*

*m*containing overlapping clouds. The overlap matrix quantifies the probabilities that clouds in levels

*k*and

*l*overlap, that clouds in levels

*l*and

*m*overlap, and that clouds in levels

*m*and

*k*overlap. It constrains, but does not explicitly give, the second order overlap probability that at some horizontal location within the column there are clouds at all three levels.

*O*

_{kl}is related to a correlation coefficient

*R*

_{kl}that relates the horizontal distributions of cloudiness in levels

*k*and

*l,*or

*c*is as small as possible (i.e., approaches −1.0). For maximum overlap, the cloudiness correlation is as large as possible (i.e., approaches 1.0). Random overlap assumes that clouds from different levels are independent and, thus, the cloudiness correlation is zero.

### b. The cloud matrix

*k*and clear conditions between levels

*k*and

*l,*and cumulative cloud fractions over multiple model levels. We therefore define the cloud matrix 𝗰, whose elements

*C*

_{kl}give the probability that at a location (

*x,*

*y*) there are clouds in level

*k*and clear conditions from level

*k*through level

*l.*Matrix 𝗰 can be calculated from the overlap matrix 𝗼 if assumptions are made regarding the higher-order overlaps. For example, given random overlap, it is consistent to assume that all of the higher-order overlaps are also random. In that case,

*l*>

*k*and the negative sign for

*l*<

*k.*

*C*

_{above}(

*k*), which is the cumulative cloud fraction for all levels above level

*k*(i.e., the probability that clouds exist somewhere in the levels above

*k*), and

*C*

_{below}(

*k*), which is the cumulative cloud fraction for all levels below the top of level

*k*(i.e., including level

*k*). These quantities are readily calculated from the cloud matrix:

*k*= 1 is the top level and

*k*=

*K*is the lowest level. These quantities are explicitly used in modifications to the NCAR radiative transfer model that are described in the appendix.

### c. The independent column approximation

*F*

*F*

_{n}from those calculations weighted by the “column probability”:

*A*is the grid-scale area,

*A*

_{n}is the area represented by the

*n*th subgrid column, and

*N*is the total number of subgrid columns. The column probability

*p*

_{n}is the fractional area of the grid-scale column that is represented by the

*n*th subgrid column. The ICA, because it uses plane-parallel calculations, neglects horizontal fluxes out of the subgrid columns. Hence, as the area represented by individual subgrid columns becomes small, the accuracy of the ICA decreases. On space scales typical of a GCM, ICA has been found to provide a good estimate of radiative fluxes for shallow cloud distributions and for high solar angles but has some difficulty with deep cloud distributions at low solar angles (e.g., Cahalan et al. 1994; Zuidema and Evans 1998; Barker et al. 1998, 1999). Nevertheless, the ICA is a reproducible standard that can be tested against other more realistic calculations. Hereafter, a subgrid cloud distribution is considered to be fully specified if the subgrid column probabilities

*p*

_{n}have all been determined.

## 3. The parameterization

*R*

^{max}

_{kl}

*z*

_{k}is the height of level

*k.*This provides a compact parameterization of

*α*

_{kl}that has a meaningful interpretation. It allows the subgrid cloud distribution to be parameterized in terms of a decorrelation depth

*L*

_{c}. The normalized correlation

*α*

_{kl}is identical to the overlap parameter of Hogan and Illingworth and has the desired property that it becomes 1.0 (i.e., maximum overlap) with no separation of the levels and zero, indicating that cloudiness in two levels becomes independent, as the level separation becomes large. The decorrelation depth can, in principle, be specified as a function of space, time, and even cloud type, allowing the flexibility to generate a wide range of cloud distributions. In the interest of clarity, we ignore that functionality, with no loss of generality, for the remaining discussion.

This exponential form neglects minimum overlap as a feature of cloud distributions in a single GCM column. Minimum overlap is important for describing cloud distributions on large spatial scales (e.g., Tian and Curry 1989), and would be expected to accompany two regions of different climatological conditions. On GCM grid scales, combinations of maximum and random overlap are generally accepted as the most realistic (e.g., Geleyn and Hollingsworth 1979; Morcrette and Jakob 2000). This assertion has, for the most part, been supported by data from cloud resolving models (Barker et al. 1999), radar (Hogan and Illingworth 2000), and studies that combine surface and satellite observations (Tian and Curry 1989; Weare 1999). Furthermore, this parameterization could be generalized further to include negative correlations. This is left for future refinements of the parameterization.

*α*

_{kl}specified, the overlap matrix is uniquely specified:

*K*levels is populated by only one cloud type,

^{2}then there are 2

^{K}probabilities to determine the full cloud distribution for

*K*cloudy levels. By specifying cloud fractions and pair-wise overlap probabilities, we have specified only [

*K*(

*K*+ 1)]/2 constraints. Together with the additional constraint that the sum of column probabilities is 1.0, the total number of constraints is always less than or equal to 2

^{K}. However, there are also the implied constraints that none of the column probabilities can exceed 1.0 or be less than zero. While these are not rigid constraints, there are 2

^{K}of them, allowing the specification of pair-wise overlap probabilities to overconstrain the cloud distribution.

*n*from an atmosphere with

*K*cloudy levels and cloud fraction

*C*(

*k*) in a given cloudy level

*k.*This column has cloud fractions

*c*

^{n}

_{k}

*c*

^{n}

_{k}

*P*{

*c*

^{n}

_{1}

*c*

^{n}

_{1}

*P*{

*c*

^{n}

_{k}

*c*

^{n}

_{k−1}

*c*

^{n}

_{k}

*c*

^{n}

_{k−1}

*C*(

*k*) and

*C*(

*k*− 1) and the overlap probability

*O*

_{kk−1}:

*K,*computationally prohibitive. Fortunately, we can calculate the elements of the cloud matrix 𝗰, by-passing the individual column probabilities with a simple recursive technique:

*l*>

*k*and the negative sign for

*l*<

*k.*

To illustrate the salient features of cloud distributions produced by this parameterization, Fig. 2 shows the total cloud fraction as a function of model resolution for different values of the decorrelation depth *L*_{c}. For these calculations, we consider a cloudy layer of fixed thickness (12 km) and fixed level cloud fraction *C*(*k*) = 0.1. The total cloud fraction *C**C*(*k*), the total cloud fraction decreases with increasing *L*_{c}. For each value of *L*_{c}, the total cloud fraction becomes less sensitive to changes of resolution as the resolution increases (i.e., further subdivision of the cloud layer into more levels). The resolution at which this insensitivity is achieved (i.e., when the curves level off) occurs when the thickness of the levels is half of the decorrelation depth. That is, two levels per decorrelation depth adequately resolves the cloud distribution, so strongly correlated cloud distributions (large *L*_{c}) are resolved with fewer levels than less correlated cloud distributions.

Figure 2 suggests that for nonzero values of *L*_{c}, there is a meaningful asymptotic value of cloud fraction in the limit of high resolution. Further analysis with as many as 10^{4} levels indicates that total cloud fraction *C**K*) approaches the asymptotic value *C**C**K*) − *C**K,* where *K* is the number of levels.

*C*(

*k*) = (0, 0.1, 0.2, 0.3, 0.4) and

*L*

_{c}= (750, 1200, 2000, 3000, 6000 m). The resolution is systematically increased by factors of two. With each doubling of resolution, the difference

*C*

*K*

*C*

*K*

*C*

*K*

*C*

*K*= 2048), is calculated and plotted in Fig. 3 as a function of resolution. A slope of −1.0 with respect to logarithmic axes is obtained for most cases at low resolution (

*K*< 32) and for all but two cases for

*K*greater than 128. The reason for the discrepancy in those two cases is not completely understood, but in those cases convergence is more rapid; the difference |

*C*

*K*) −

*C*

*K*

^{2}.

One can estimate a priori the properties of the cloud distribution based on the number *K*_{eff} of “effective independent” cloud layers, which is calculated as the total thickness of the entire cloud field (12 km in Fig. 2) divided by the decorrelation depth. Table 1 demonstrates this with further analysis of the cloud distributions used for Fig. 2. Shown in Table 1 are the values of *K*_{eff}, total cloud fraction for 16 and 64 levels, asymptotic values *C*^{2}, 10^{3}, and 10^{4} levels), and a random overlap calculation for *K*_{eff} levels (each with cloud fraction 0.1). The asymptotic values of total cloud fraction are approximately equal to the random overlap calculation for *K*_{eff} levels. The equality becomes more precise for smaller decorrelation depths (more independent layers).

In a final demonstration, Fig. 4 repeats Fig. 1a, with an additional calculation of cloud fraction using the new parameterization (solid line). The vertical profile of cloud fraction from Fig. 1b is used together with *L*_{c} = 3 km (i.e., four independent cloud layers) to approximate maximum–random overlap. At the highest resolutions, total cloud fractions from the new parameterization and from maximum–random overlap have similar values. However, the new parameterization produces total cloud fractions that are a smoother function of resolution and that attain a much better estimate of the asymptotic value at lower resolutions than maximum–random overlap does.

## 4. Radiative transfer calculations

The parameterization is now combined with radiative transfer calculations to test the sensitivity of radiative fluxes to changes of the decorrelation depth. The independent column approximation provides a reasonably accurate estimate of fluxes from a detailed description of the cloud distribution. However, the ICA is an expensive calculation; the number of independent columns for which radiative transfer calculations must be performed for the ICA increases exponentially with the number of cloudy levels. To provide a more efficient formulation, we enhance modifications to the NCAR radiative transfer model made by Bergman and Hendon (1998) that use the cloud matrix 𝗰 and the bulk cloud fraction arrays *C*_{above} and *C*_{below} to describe the cloud distribution. The shortwave (SW) modifications (detailed in the appendix) are a compromise between ICA and the unmodified plane-parallel model in which the distinction between clear and cloudy components is maintained but inhomogeneities within the cloud field are ignored. The longwave (LW) modification is a simple replacement of analytical representations of the probabilities represented in the cloud matrix with values calculated in the cloud parameterization.

The modifications to the NCAR radiative transfer model reduce errors with respect to the ICA by an order of magnitude to less than 5 W m^{−2} in tests using a wide range of cloud distributions, while only doubling the time of a single plane-parallel calculation. The improvements in accuracy of fluxes and heating rates to the CCM radiative transfer algorithm make the modifications extremely useful for diagnostic studies with these parameterizations. However, the modifications are not optimal for use in long GCM simulations because they introduce small errors in conservation of photons that can influence the simulation. Other variations on the CCM radiation calculations (which are conservative, but are more costly and require much larger code revisions) are also possible (e.g., Collins 2001).

### a. Idealized calculations

Here, we perform idealized calculations with elementary cloud distributions to examine general features of the “overlap sensitivity” of radiative fluxes in terms of the relationships between decorrelation depth, cloud fraction, and radiative fluxes. For these calculations, clouds are distributed uniformly among eight levels spanning 250–900 mb. Cloud water content in each level is specified to be 25 g m^{−2}, the solar angle is specified to be representative of a diurnal mean in the Tropics cos(*θ*_{s}) = 1/*π,* and temperature and humidity distributions are representative of tropical conditions.

As a diagnostic aid for the overlap sensitivity of radiative fluxes, Fig. 5 demonstrates the sensitivity of total cloud fraction. Values in Fig. 5 are contoured as a function of the cloud fraction in individual levels and decorrelation depth, which has been normalized by the level thickness (i.e., if the normalized decorrelation depth is 2.0, *L*_{c} is twice the thickness of a level). As one might expect, total cloud fraction increases (Fig. 5a) with increasing level cloud fraction and for decreasing *L*_{c} (i.e., as overlap becomes more random).

One measure of the overlap sensitivity is the difference between the calculated total cloud fraction and the corresponding value for random overlap (Fig. 5b). The sensitivity is largest for intermediate values of level cloud fraction (approximately 0.2 for these calculations). For very small values of the level cloud fraction, the sensitivity is small because we are dealing with differences of small values. At very large values, the sensitivity is small because total cloud fraction cannot exceed 1.0 and values “saturate.” Total cloud fraction is most sensitive to changes of decorrelation depth at small values of *L*_{c} because for those values, the cloud distribution is not yet resolved. This is demonstrated by a second measure of overlap sensitivity, the derivative of total cloud fraction with respect to decorrelation depth (Fig. 5c). This aspect is important because estimates from radar data (Hogan and Illingworth 2000) and from cloud resolving models (H. Barker 1998, personal communication) of that depth are on the order of 1–4 km. That is, decorrelation depths are not much larger than the level thickness for a typical GCM and, thus, total cloud fractions in the GCM will be sensitive to uncertainties of the decorrelation depth at realistic values of that parameter.

Figure 6 examines the overlap sensitivity of SW surface fluxes. Results for fluxes at other levels (e.g., at the top of the atmosphere) have the same qualitative features, differing only in magnitude. Here, we examine “cloud radiative forcing” (CRF), which is the difference between two radiative transfer calculations: one including clouds and one without. As with total cloud fraction, the absolute value of SW CRF (Fig. 6a) increases with increasing level cloud fraction and decreasing decorrelation depth. Also like total cloud fraction, SW surface fluxes exhibit the strongest overlap sensitivity (Fig. 6b) for level cloud fractions between 0.2 and 0.4. The choice of decorrelation depth can alter SW surface fluxes, in this idealized case, by as much as 60 W m^{−2}, consistent with values obtained by Barker et al. (1999) and Morcrette and Jakob (2000). Similar results are found for LW fluxes (Fig. 7). Whether the modified NCAR model is used or an ICA calculation is performed makes little difference to the results; the qualitative conclusions drawn from both methods are identical and quantitative differences are less than 5 W m^{−2} (Figs. 6c and 7c).

There are competing factors that contribute to the overlap sensitivity of radiative fluxes. The total cloud water amount in a grid-scale column is not a function of overlap, however the cloud water amount within sub-grid columns is. As cloud overlap increases (i.e., becomes closer to maximum overlap), columns with large cloud water amounts become more probable relative to those with small cloud water amounts. Making clouds thicker has the effect of increasing atmospheric SW reflectivity and LW emissivity. On the other hand, as cloud overlap increases, the total cloud fraction decreases, which decreases atmospheric reflectivity and emissivity. The contour patterns in Figs. 6b, 7b, and 5b are quite similar. In addition, the sign of the sensitivity (not shown in these figures) is consistent with cloud fraction being the dominant influence on radiative sensitivities; downward SW fluxes are smallest and downward LW fluxes are largest for the largest values of total cloud fraction. From basic radiative properties of cloud water (e.g., Stephens 1978), we could have anticipated the total cloud fraction to be the dominant source of overlap sensitivity in radiative fluxes. Reflectivity and emissivity are nonlinear functions of cloud water path, becoming less sensitive to changes of water path as water path increases, but are linear functions of the total cloud fraction under the independent column approximation. So, increasing the water path of clouds will always have less of an impact on radiation than the accompanying decrease of total cloud fraction if the total cloud water amount is held fixed.

In these idealized experiments, the column-integrated atmospheric heating rates (not shown) have a “relative sensitivity” (i.e., the change in flux divided by the flux) on the order of 10%–20% for LW and a smaller sensitivity for SW. However, local (i.e., at individual levels) effects can be larger. Figure 8 shows the distribution of heating rates for both LW (left side or negative values) and SW (right side) for values of normalized *L*_{c} = (0, 1.3, 4.0, ∞). The ICA was used for these calculations, but results with the modified radiative transfer model are nearly indistinguishable. Above the cloudy levels, which are shaded, the overlap sensitivity of heating rates is small. However, within and below the cloudy levels, the relative sensitivity is as large as 50%. Both LW and SW heating rates display a vertical shift of atmospheric heating in addition to the change of amplitude as overlap is changed. This effect is more pronounced for LW cooling than for SW heating. It is indicated by the change of sign of the overlap sensitivity at different levels. For example, below 500 mb and near 300 mb, LW cooling is larger for maximum overlap (*L*_{c} = ∞) than for random overlap (*L*_{c} = 0). However, in the region near 400 mb, the reverse is true. This shift indicates that cloud overlap is more important for atmospheric heating than is indicated by the column-integrated value, for which the contributions from regions with opposite sign cancel.

### b. CCM cloud distributions

The sensitivity of radiative fluxes to cloud overlap in the previous section are consistent with values from previous work (e.g., Barker et al. 1999; Morcrette and Jakob 2000). The Barker et al. results are particularly compelling because they arise from a sophisticated cloud resolving model and apply directly to subgrid variability in a GCM. However, this does not necessarily mean that cloud overlap will make a large difference for all GCMs. To investigate this issue further and to put the overlap parameterization to practical use, we examine the overlap sensitivity of radiative fluxes in a version of NCAR's Community Climate Model. For this purpose, we use hourly instantaneous cloud fields from a 7-day run of a recent version of CCM to perform off-line radiative transfer calculations. Note that, since cloud overlap is not altered in the GCM itself, these calculations measure only the sensitivity of radiative fluxes to changes of overlap and do not include dynamical feedbacks that alter the cloud fields themselves. This version of CCM is a T42 spectral Eulerian model with 30 vertical levels. Convection is parameterized with the combined algorithms of Zhang and McFarlane (1995) for deep convection and Hack (1994) for shallow convection. Cloud fraction and condensate amounts are determined as described by Rasch and Kristjansson (1998).

For brevity, we discuss only the sensitivity of surface fluxes to cloud overlap. Analysis of other fluxes (not shown) offers no further insight, as one might expect from the results of idealized calculations in the previous section. Table 2 and 3 show area-weighted statistical comparisons for grid-scale surface fluxes calculated with different values of *L*_{c} in terms of the bias (Table 2), which is the global 7-day mean of the difference, and the rms difference (Table 3), which is the root mean square of individual hourly gridpoint differences. For this GCM run, the impact of cloud overlap on radiative fluxes is small, even negligible, compared to other sources of GCM error. Biases are typically less than 1 W m^{−2} and rms differences are typically less than 5 W m^{−2}. Most of the sensitivity is due to random overlap, which is clearly unrealistic for a 30-level model. If our attention is restricted only to depth scales of 1.0 km or larger, biases are reduced to less than 0.5 W m^{−2} and rms differences are reduced to 1–2 W m^{−2}.

Figure 9 shows the geographical distribution of the 7-day mean sensitivity in terms of the difference between surface fluxes for *L*_{c} = 10 km and those for *L*_{c} = 1 km. Shortwave fluxes (Fig. 9a) show their strongest sensitivity over land regions, particularly in the Tropics where solar angles, and thus the incident solar fluxes, are largest. Values are reduced as expected at high latitudes, and also over the oceans, where there is essentially no SW sensitivity. Many factors contribute to the magnitude of the SW sensitivity, including the local surface albedo and solar angle. However, from the idealized calculations, we expect that the sensitivity of total cloud fraction to strongly influence the radiative sensitivities. If that is case, the patterns of SW and LW (Fig. 9b) sensitivity should be similar. Indeed, the geographical distribution of LW sensitivity shows some similarity to the SW distribution, with maxima over the same land regions. However, the LW and SW results show some striking differences: some that are easily understood, such as stronger LW sensitivity over high latitudes, and other that are not as easily understood such as stronger LW sensitivity over the oceans and weaker LW sensitivity over some land regions.

The impact of solar angle on SW fluxes is certain to have a large impact on the SW sensitivity. The obvious effect is the smaller SW sensitivity at high latitudes. There is also a diurnal effect because SW sensitivity only occurs during daylight hours and overlap sensitivity of cloud distributions during nighttime could differ from those during daytime. These effects are examined in Fig. 9c, which displays the LW sensitivity that has been weighted by the cosine of the solar zenith angle, just as occurs for SW fluxes. The “solar weighting” enhances the similarities of SW and LW sensitivities. There is reduced sensitivity of the solar-weighted LW fluxes at high latitudes compared to pure LW fluxes. In addition, the maxima over land of the solar-weighted LW sensitivity are very similar to those of the SW sensitivity. Since these maxima are found throughout the Tropics, diurnal effects must be important for these locations. Solar-weighted LW sensitivity is still stronger over the oceans than for SW, indicating that other factors that differentiate SW and LW radiative transfer are important for the different SW and LW patterns.

Another important question is, why is the sensitivity so small? Certainly it is not for lack of clouds produced in the GCM as illustrated by the 7-day mean cloud fraction (Fig. 10a). Total cloud fractions show reasonable spatial variations, with intermediate values of cloud fraction (i.e., not clear or overcast) over much of the globe. Yet, despite these 7-day mean total cloud amounts, the sensitivity of total cloud fraction to overlap (Fig. 10b) is quite small. This explains the small sensitivity of radiative fluxes. But why is the sensitivity of cloud fraction so small?

To explain the small overlap sensitivity of CCM cloud fields, we examine instantaneous cloud fraction at each level. Figure 11a displays a histogram of cloud fraction within individual model levels. Over 35% of cloudy levels have instantaneous cloud fractions greater than 0.95, which, as discussed earlier, reduces the sensitivity to cloud overlap. In fact, 64% of all cloudy columns in the 7-day dataset have either only one cloudy level, in which case overlap is truly irrelevant, or have at least one level in which cloud fraction is greater than 0.95. In contrast, if we examine cloudy columns that have a LW sensitivity greater than 1 W m^{−2}, then the percentage of cloudy levels with level cloud fraction greater than 0.95 is reduced to about 18% (Fig. 11b). It seems, therefore, the propensity for CCMs to produce completely cloudy levels is primarily responsible for that GCM's insensitivity to cloud overlap. While cloud distributions on GCM grid scales have not been widely studied, clouds that extend for 2.8° are not nearly as prevalent as those simulated by CCM (cf. Boer and Ramanathan 1997). As CCM attains more realism, cloud overlap will undoubtedly become a larger factor.

## 5. Conclusions

This manuscript develops a parameterization of subgrid cloud distributions for use in the physical parameterizations of atmospheric models. The assumption that underlies this work is that the resolved horizontal scales of the model are much larger than individual clouds. However, the parameterization will be useful for higher horizontal resolutions as well because it can help produce the appropriate cloud statistics over arbitrary timescales. In this discussion, cloud distributions are described by a cloudiness function, and overlap is described in terms of the correlations between cloudiness functions from different levels. That correlation is assumed to decay exponentially with the separation distance between the levels, allowing the full subgrid cloud distribution to be parameterized in terms of the cloud fraction at each level and a decorrelation depth.

For the overlap parameterization developed here, pair-wise overlap fractions among levels are calculated from a specified decorrelation depth. However, the specification of pair-wise overlap probabilities can overconstrain the subgrid cloud distribution. Considering only the nearest-neighbor overlap probabilities circumvents this problem. A unique cloud distribution is obtained via the multiplication of a series of conditional probabilities. This method also allows fast calculations of the bulk cloud distribution properties without resorting to the calculation of subgrid column probabilities in the ICA.

The parameterization has features with practical contributions to atmospheric modeling in addition to enhancing the representation and interpretation of simulated cloud distributions. It is flexible: in principle, the decorrelation depth could be specified as a function of space and time as well as other quantities such as cloud type. It is related explicitly to physical coordinates, depending on model levels only as a discretization and not to determine the nature of the cloud distribution. As a result, the parameterized cloud fields are consistent with a meaningful continuous cloud distributions at arbitrary vertical resolution, unlike random overlap. Parameterized cloud distributions approach asymptotic distribution smoother and at lower resolution than maximum–random overlap. Bulk properties of the distribution, such as total cloud fraction, calculated with the parameterization become insensitive to changes of vertical resolution once the resolution is adequate to resolve the cloud distribution, that is, if there are at least two vertical levels per decorrelation depth. Using this representation of cloud fields, bulk properties can be estimated by considering the cloud distribution to contain *K*_{eff} randomly overlapped layers, where *K*_{eff} is equal to the total thickness of the cloudy region divided by the decorrelation depth.

The overlap parameterization is combined with two radiative transfer formulations to examine the overlap sensitivity of radiative fluxes. Calculations with idealized cloud distributions reveal strong overlap sensitivities for radiative fluxes, several tens of watts per squares meters. That sensitivity is traced to the overlap sensitivity of total cloud fraction. Cloud distributions from a 7-day run of CCM are not very sensitive at all to cloud overlap. Even fluxes from random and maximum overlaps have less than 5 W m^{−2} rms differences. This small sensitivity occurs because CCM cloud parameterizations have a propensity to produce overcast conditions within individual model levels. However, as CCM attains more realism, cloud overlap is expected to become a larger factor.

The parameterization could be quite useful for radiative transfer calculations in GCMs, particularly as vertical resolution increases and cloud parameterizations produced increasingly complex cloud distributions. The formulation presented here could benefit other components of a GCM as well. For example, to model physical interactions that involve precipitation falling through a given level requires knowledge of cloud overlap at the detail presented in the cloud matrix (e.g., Jakob and Klein 1999). Nevertheless, indiscriminant use of the overlap parameterization is not advised and might be an unnecessary expense in a GCM with unrealistic vertical profiles of cloud fraction. In addition, it is worth asking at what point detailed information about cloud fields becomes unnecessary for the purpose of radiative transfer calculations. Perhaps at the typical resolution of GCMs, 18–30 levels, we have already exceeded the vertical resolution needed for radiative transfer calculations and those calculations could be performed at much lower vertical resolution than is used for the model's dynamical and convective parameterizations. Furthermore, it might be that future increases of horizontal resolution will be adequate to resolve the important features of cloud distributions (cf. Boer and Ramanathan 1997). In that case, cloud fields can be described purely in terms of clear and overcast conditions and cloud overlap is not a concern.

## Acknowledgments

The authors gratefully acknowledge helpful discussions with and contributions from H. Barker and colleagues at CDC and NCAR, notably J. Barsugli and W. Collins. The comments of anonymous reviewers were also very helpful. This study was funded in part by the National Science Foundation, Grant #ATM-9903908.

## REFERENCES

Barker, H. W., J. -J. Morcrette, and G. D. Alexander, 1998: Broadband solar fluxes and heating rates for atmospheres with 3D broken clouds.

,*Quart. J. Roy. Meteor. Soc.***124****,**1245–1271.Barker, H. W., G. L. Stephens, and Q. Fu, 1999: The sensitivity of domain-averaged solar fluxes to assumptions about cloud geometry.

,*Quart. J. Roy. Meteor. Soc.***125****,**2127–2152.Bergman, J. W., and H. H. Hendon, 1998: Calculating monthly radiative fluxes and heating rates from monthly cloud observations.

,*J. Atmos. Sci.***55****,**3471–3491.Boer, E. R., and V. Ramanathan, 1997: Lagrangian approach for deriving cloud characteristics from satellite observations and its implications to cloud parameterization.

,*J. Geophys. Res.***102****,**21383–21399.Cahalan, R. F., W. Ridgway, and W. J. Wiscombe, 1994: Independent pixel and Monte Carlo estimates of stratocumulus albedo.

,*J. Atmos. Sci.***51****,**3776–3790.Collins, W. D., 2001: Parameterization of generalized cloud overlap for radiative calculations in general circulation models.

,*J. Atmos. Sci.***58****,**3224–3242.Dudek, M. P., X-Z. Liang, and W-C. Wang, 1996: A regional climate model study of the scale-dependence of cloud radiative interactions.

,*J. Climate***9****,**1221–1234.Geleyn, J. F., and A. Hollingsworth, 1979: An economical analytical method for the computation of the interaction between scattering and line absorption of radiation.

,*Contrib. Atmos. Phys.***52****,**1–16.Hack, J. J., 1994: Parameterization of moist convection in the NCAR Community Climate Model, CCM2.

,*J. Geophys. Res.***99****,**5551–5568.Hogan, R. J., and A. J. Illingworth, 2000: Deriving cloud overlap statistics from radar.

,*Quart. J. Roy. Meteor. Soc.***126****,**2903–2909.Jakob, Co, and S. A. Klein, 1999: The role of vertically varying cloud fraction in the parameterization of microphysical processes in the ECMWF model.

,*Quart. J. Roy. Meteor. Soc.***125****,**941–965.Jakob, Co, and S. A. Klein, 2000: A parameterization of the effects of cloud and precipitation overlap for use in a general-circulation model.

,*Quart. J. Roy. Meteor. Soc.***126****,**2525–2544.Kiehl, J. T., J. J. Hack, G. B. Bonan, B. A. Boville, B. P. Briegleb, D. L. Williamson, and P. J. Rasch, 1996: Description of the NCAR Community Climate Model (CCM3). NCAR Tech. Note NCAR/TN-420+STR, National Center for Atmospheric Research, 152 pp.

Manabe, S., and R. Strickler, 1964: Thermal equilibrium of the atmosphere with convective adjustment.

,*J. Atmos. Sci.***21****,**361–385.Morcrette, J-J., and C. Jakob, 2000: The response of the ECMWF model to changes in the cloud overlap assumption.

,*Mon. Wea. Rev.***128****,**1707–1732.Raisanen, P., 1999: Effect of vertical resolution on cloudy-sky radiation calculations: Tests with two schemes.

,*J. Geophys. Res.***104****,**27407–27419.Ramsey, P. G., and D. G. Vincent, 1995: Computation of vertical profiles of longwave radiative cooling over the equatorial Pacific.

,*J. Atmos. Sci.***52****,**1555–1572.Rasch, P. J., and J. E. Kristjansson, 1998: A comparison of the CCM3 model climate using diagnosed and predicted condensate parameterizations.

,*J. Climate***11****,**1587–1614.Stephens, G. L., 1978: Radiation profiles in extended water clouds. II: Parameterization schemes.

,*J. Atmos. Sci.***35****,**2123–2132.Stubenrauch, C. J., A. D. Del Genio, and W. B. Rossow, 1997: Implementation of subgrid cloud vertical structure inside a GCM and its effect on the radiation budget.

,*J. Climate***10****,**273–287.Tian, L., and J. A. Curry, 1989: Cloud overlap statistics.

,*J. Geophys. Res.***94****,**9925–9935.Weare, B. C., 1999: Combined satellite- and surface-based observations of clouds.

,*J. Climate***12****,**897–913.Wielicki, B. A., and R. M. Welch, 1986: Cumulus cloud properties using Landsat satellite data.

,*J. Climate Appl. Meteor.***25****,**261–276.Zhang, G. J., and N. A. MacFarlane, 1995: Sensitivity of climate simulations to the parameterization of cumulus convection in the Canadian Climate Centre general circulation model.

,*Atmos.-Ocean***33****,**407–446.Zuidema, P., and K. F. Evans, 1998: On the validity of the independent pixel approximation for boundary layer clouds observed during ASTEX.

,*J. Geophys. Res.***103****,**6059–6074.

## APPENDIX

### Modifications to the SW Radiative Transfer Model

We first present a brief discussion of the salient features of the existing model and then discuss how it was modified. A more detailed description of the NCAR radiative transfer model is available in Kiehl et al. (1996). The NCAR SW radiative transfer model is a plane-parallel model that uses the “adding” method. In this method, the optical properties of each model level are calculated in terms of reflection *R* and transmission *T* functions, or “radiative functions.” Multilevel radiative functions are then calculated so that, at each level interface, there are radiative functions for the combined effects of all levels below and all levels above that interface. From those combined-level functions, SW fluxes at each interface are calculated.

*R*

_{1}and

*T*

_{1}be radiative functions for a given layer (i.e., layer 1, which might be composed of several model levels) and

*R*

_{2}and

*T*

_{2}be the radiative functions for the level that is to be combined with layer 1. The combined radiative functions are functions of the individual layer radiative functions:

*R*

_{12}and

*T*

_{12}become the functions

*R*

_{1}and

*T*

_{1}for the new layer 1, and so on.

*R*

^{cld}and

*T*

^{cld}) and clear (

*R*

^{clr}and

*T*

^{clr}) values. Now, when combining the radiative functions for the two layers, there are additional calculations. The clear-sky value is obtained by combining clear-sky radiative functions for the two layers:

*R*

^{clr}

_{12}

*R*

_{12}

*R*

^{clr}

_{1}

*T*

^{clr}

_{1}

*R*

^{clr}

_{2}

*T*

^{clr}

_{2}

*f*

_{1}is the area over which there are clouds in layer 1 and clear-sky conditions in layer 2 divided by the total cloudy area of the two layers:

*C*

_{1}and

*C*

_{2}and the cloud fractions for the two layers and

*O*

_{12}is the probability that those clouds overlap. Similarly,

*C*

_{above}(

*k*) and

*C*

_{below}(

*k*). For example, consider adding level

*k*(layer 2) to the combined levels above

*k*(layer 1). In this case,

*C*

_{below}when combining levels below an interface.

Total cloud fraction calculated for different decorrelation depths at 16-level, 64-level, and infinite resolution. Also shown are random overlap calculations for *K _{eff}* independent layers

The sensitivity of SW fluxes at the surface to changes of cloud overlap for CCM. Shown are comparisons for different values of the decorrelation depth in terms of bias (fluxes for large *L ^{c}* minus small) and the rms difference. Statistics are area weighted

The sensitivity of LW fluxes at the surface to changes of cloud overlap for CCM. Shown are comparisons for different values of the decorrelation depth in terms of bias (fluxes for large *L _{c}* minus small) and the rms difference. Statistics are area weighted

^{*}

The National Center for Atmospheric Research is sponsored by the National Science Foundation.

^{1}

Minimum overlap is not, to our knowledge, actually used in atmospheric models. We mention it here for completeness.

^{2}

This argument can be generalized to any number of cloud types per level. If there are *T*_{k} cloud types in level *k,* then there are ^{K}_{k=1}*T*_{k} + 1) possible ICA columns and ^{K}_{k=1}^{K}_{l=k}*T*_{k}*T*_{l} constraints made by specifying cloud fraction in each level and the pair-wise overlap probabilities.