## 1. Introduction

The specification of the gaseous transmission, cloud/aerosol optical properties, and radiative transfer are the three main tasks for an atmospheric radiation algorithm. Over the past two decades, there has been a trend in radiative transfer schemes to replace traditional band models for gaseous transmittance with the correlated *k*-distribution (CKD) method (Lacis and Oinas 1991; Fu and Liou 1992; Hollweg 1993; Kratz 1995; Edwards and Slingo 1996; Mlawer et al. 1997; Chou and Suarez 1999; Kato et al. 1999; Chou et al. 2001; Zhang et al. 2003; Li and Barker 2005; and others). In the CKD method, sorting the absorption coefficient in ascending order transforms the problem of integrating a tortuously variable absorption coefficient over frequency space into integrating a smooth sorted absorption coefficient using relatively few quadrature points. In principle, CKD can be applied to a single absorption line while band models utilize mean values for entire bands. Thus, CKD shows considerable promise in simulating atmospheric radiation accurately. However, there are still unresolved issues surrounding the CKD method in climate models. Recently, Li et al. (2010) pointed out that the partitioning of solar versus infrared spectral energy is an unsolved problem in most CKD algorithms; also, the shortwave radiative forcing due to CH_{4} has to be addressed in current climate models. In this work, we would like to raise another issue, namely the inconsistency in specifying gaseous absorption coefficient and cloud optical properties in all current CKD radiation models.

It is realized that cloud plays an important role in the radiation process. Generally, the cloud radiative forcing is much larger than that of the major greenhouse gases (Ramanathan et al. 1989). The interaction between the gas absorption and cloud optical properties has been previously studied (e.g., Ramaswamy and Freidenreich 1992; Espinoza and Harshvardhan 1996; Räisänen 1999; and others). However, most of the studies focus on the water vapor absorption and the cloud absorption by using the band-mean schemes. In Fomin and Correa (2005), the cloud and gas interaction has been considered by mapping the sorted gaseous absorption coefficients into the frequency space where the spectral distributions of cloud extinction coefficient are presented. It is correctly pointed out by Fomin and Correa (2005) that the strong water vapor absorption can mask strong cloud absorption due to the overlapping between the gaseous absorption line and cloud extinction coefficient.

In a CKD algorithm, the gaseous absorption coefficients are sorted in ascending order, but most of the cloud optical property parameterizations (Slingo 1989; Chylek et al. 1992; Hu and Stamnes 1993; Chou et al. 1998; Dobbie et al. 1999; Lindner and Li 2000) are calculated based on the band-mean average. Therefore, the spectral treatments of gaseous transmission and cloud absorption are not the same. In a CKD algorithm, usually the variation of the gaseous absorption coefficients is very large, often spanning several orders to over 10 orders of magnitude in each band. However, the corresponding cloud optical properties are assumed constant. The spectral correspondence between gas and cloud is crucial for cloud radiative heating rates. For example, in the shortwave, if the gaseous absorption coefficient in a spectral region is very large, the downward solar energy distributed in such region is largely attenuated and so the corresponding cloud heating rate is very small regardless of the cloud absorptance. On the other hand, if the gaseous absorption coefficient is relatively small in a spectral region, the cloud heating rate becomes more sensitive to the cloud absorption coefficient in such a region. However, in a band-mean cloud optical property scheme (hereafter called a band-mean cloud scheme), the cloud heating rate is not dependent on the distribution of gaseous absorption coefficient. Thus, the spectral correlation between gas and cloud is lost. The study of the radiative impact of this spectral inconsistency is the purpose of this paper.

In section 2, we begin with an overview of the relevant physics pertaining to the CKD gaseous transmission. In section 3, a simple CKD cloud scheme is proposed to solve the inconsistent treatment of cloud in CKD algorithms. In section 4, the radiative impact of using the CKD cloud scheme rather than uniform cloud optical properties is quantified using a one-dimensional CKD radiation model. Finally, the conclusions are given in section 5.

## 2. Basic theory of CKD gaseous transmission

*ν*iswhere

*w*is the absorber amount (mol cm

^{−2}) and

*k*(

*ν*) is the gaseous absorption coefficient in frequency space. In general,

*k*(

*ν*) depends on temperature

*T*and pressure

*p.*The transmission function can also be written in

*k*space aswhere

*f*(

*k*) is the

*k*-distribution function, withwhere

*δ*[

*k*−

*k*(

*ν*)] is the delta function. The quantity

*f*(

*k*) is the Dirac comb function, which sums up all absorption lines of the same strength. The definition of the

*k*-distribution function by Goody and Yung (1989) is only an approximation of (3) [see the detailed discussion in Li and Barker (2005)]. The

*k*space and frequency space are related to each other by inserting (3) into (2):Following the definition of

*f*(

*k*) by Goody and Yung (1989), the transform of (4) between the frequency space to the

*k*space cannot be obtained. The cumulative probability function is introduced as

^{1}with

*h*(0) = 0 and

*h*(∞) = 1. From (5),

*dh*(

*k*)

*= f*(

*k*)

*dk*, by inserting it into (2), we haveThe function

*h*forms a cumulative probability space (CPS) in the range of (0 ≤

*h*≤ 1), and so

*k*(

*h*) is now the absorption coefficient in CPS. In a CKD algorithm, the gaseous absorption coefficient is sorted over each band, and each band corresponds to a CPS.

*N*points

*H*(

_{i}*i =*0, 1, 2, … ,

*N*), with

*H*

_{0}

*=*0,

*H*1, and letting

_{N}=*h*=

_{i}*H*−

_{i}*H*

_{i}_{−1}, such that

*k*is the absorption coefficient in the

_{i}*h*interval obtained asLi and Barker (2005) proved that the scaling factor

_{i}*α*≤ 1, with the equal being for constant

_{i}*k*(

*h*) in the range

*H*

_{i}_{−1}≤

*h*≤

*H*In each CPS interval

_{i}.*h*,

_{i}*α*can be obtained by a fitting procedure to make the heating rate and flux give the best approximation to the line-by-line results in the same CPS range. Also, the fitting procedure is applied to different atmospheric profiles (McClatchey et al. 1972) to make sure the accuracy in heating rate and flux under different conditions. It is found that the larger the variation of

_{i}*k*(

*h*) in a CPS interval the smaller the value of

*α*. Generally,

_{i}*α*is close to one in a region of small

_{i}*h*, since

*k*(

*h*) is relatively flat in such a region.

## 3. CKD cloud scheme for cloud optical properties

There are three relevant cloud optical properties that are needed in the radiative transfer process. They are the specific extinction coefficient *ψ _{ν}*, the single scattering albedo

*ϖ*, and the asymmetry factor

_{ν}*g*, each defined at frequency

_{ν}*ν.*

*r*is the droplet size,

*ρ*is water density,

*n*(

*r*) is the distribution of droplet sizes, and

*Q*

_{ext}is the efficiency for the droplet to scatter plus absorb radiation. The size distribution of cloud droplets in the atmosphere tends to resemble a gamma distribution, which has the formwhere

*A*is a constant

*α*and

*β*are determined when the effective radius

*r*= (

_{e}*α +*3)/

*β*and the effective variance

*υ*= 1/(

_{e}*α +*3) are given (Chylek et al. 1992). Observations showed that

*υ*varies little and that the observed variations do not change the calculated optical properties significantly (Chylek and Ramaswamy 1982). Therefore, a constant value of

_{e}*υ*= 0.171 is set in the Mie calculations (Dobbie et al. 1999), and the cloud optical properties depend only on the effective radius. In this work, the optical properties are computed over the integral range of 0.05–45

_{e}*μ*m with resolution of 0.05

*μ*m. The refractive index of water is from Segelstein (1981).

*ν*is the frequency range of the band and

*F*

_{s}_{,ν}is the global mean downward solar flux at 500 hPa and solar zenith angle

*θ*

_{0}= 53°, which is close to the daily mean value. The spectral resolution in (10) is 0.1 cm

^{−1}for the shortwave.

For the infrared, the downward radiative flux at a cloud top depends on the emission and attenuation of radiation from all layers above the cloud and the upward radiative flux at the bottom of the cloud depends on the emission and attenuation of radiation from all layers below the cloud. However, in both cases, the layers adjacent to the cloud layer have the biggest influence because the cloud receives the radiative emission from these layers with minimum gaseous attenuation. The emission from neighboring layers is mostly determined by the Planck function. Therefore, we average the band-mean results with respect to the Planck function at a temperature of 270 K with spectral resolution of 0.01 cm^{−1}.

Figure 1 shows the water cloud optical properties and the band-mean results with *r _{e}* = 5.89

*μ*m and

*υ*= 0.171. The cloud optical properties vary dramatically with wavelength in the longwave range. However, the variations of cloud optical properties in the shortwave are relatively small, especially in the visible and ultraviolet range.

_{e}As pointed above, the spectral treatments of the band-mean cloud optical properties and gaseous transmission are not the same in the current CKD algorithm. Therefore, it is necessary to create a parameterization of cloud optical properties in which the spectral correlation between gas and cloud can be addressed. We call this parameterization of cloud optical property the CKD cloud scheme.

The Beijing Climate Center radiation model (BCC_RAD) is used in this study. The algorithm of cloud and aerosol is based on the BSTAR5C and MSTRNX models (Nakajima et al. 2000). The CKD algorithm is adopted by Zhang et al. (2003, 2006a,b). The 10–49 000 cm^{−1} spectral range was divided into 17 bands (8 longwave and 9 shortwave). Six major greenhouse gases—H_{2}O, CO_{2}, O_{3}, N_{2}O, CH_{4}, and chlorofluorocarbons (CFCs)—are considered. Table 1 lists the numbers of intervals in CPS and the absorbers considered in each band. The Rayleigh and cloud scattering is considered in the model by using the two-stream radiative transfer scheme.

The band spectrum ranges, absorber, and the number of intervals in CPS.

The spectral variations of gas and cloud optical properties match in frequency space. To maintain such consistency in CPS, we use the same sorting for both gas and cloud. This guarantees that the correspondence in frequency space between gas and cloud is maintained in CPS.

The availability of frequency-dependent data for water refractive index is limited (Segelstein 1981) compared with that for gaseous absorption coefficient. Therefore, the refractive indices are linearly interpolated to a resolution of 0.1 cm^{−1} for shortwave and a resolution of 0.01 cm^{−1} for longwave. The gas absorption coefficients are calculated using the high-resolution transmission (HITRAN) 2008 molecular spectroscopic database (Rothman et al. 2009).

In each band (each CPS), the gaseous absorption coefficients are sorted from high to low. In Fig. 2, the left column shows the sorted H_{2}O gaseous absorption coefficients *k*(*h*) in ascending order in CPS for bands 9, 10, and 11 at a pressure of 500 hPa and a temperature of 270 K. It is seen that the variation of gaseous absorption coefficients in CPS is usually very large. By applying the same sorting to cloud optical property coefficients, we obtain the sorted cloud optical properties of *ψ*(*h*), *ϖ*(*h*), and *g*(*h*), shown as the light gray lines in the rest of the columns. We also present the absorption coefficient, *ψ*_{abs}(*h*) = *ψ*(*h*)[1 − *w*(*h*)], which will be used to help the understanding of results in the next section.

Generally, there is more than one gas in a band. However, water vapor is the main gas in the near-infrared range (bands 9–11) and ozone is the main gas in the visible and ultraviolet range (bands 12–17). Only the sorting of the main gas is applied to the cloud optical properties. Since cloud optical properties are independent of pressure and temperature, sorting always follows that of the main gas at pressure of 500 hPa and at a temperature of 270 K.

*H*, as shown in (7), and the mean results for each CPS interval must be obtained in a similar way to that of (8) for gas. Since the variations of cloud optical properties are much smaller than those of gaseous absorption coefficients, the scaling factor

_{i}*α*should be close to 1. Therefore, (8) becomes simple average mean. In a manner similar to the band-mean cloud scheme, the Chandrasekhar mean principle can be applied to each CPS interval by weighting the line-by-line result for the solar downward flux at 500 hPa. To maintain spectral consistency, the downward solar flux is also sorted in the same way as those for gaseous absorption coefficients. For a CPS interval

_{i}*i*where

*F*(

_{s}*h*) is the sorted solar downward flux in CPS. It is found that the Chandrasekhar mean produces more accurate in heating rates compared to those computed using the simple average mean, since the solar energy distribution in each CPS interval is addressed in (11). It is worthwhile to point out that Mlawer et al. (1997) and Mlawer and Clough (1998) also provide a formalism on the sorting of the Planck function and the solar source function in each quadrature interval in their radiation algorithm of the Rapid Radiative Transfer Model (RRTM).

In Fig. 2, the results of CPS interval mean are shown as solid lines. Also, the band-mean results are shown as dotted lines.

In Fig. 3, the results of sorted cloud optical properties in the longwave bands 3, 4, and 5 are shown. In the infrared, there is no gaseous scattering and cloud scattering is very weak; thus, the scattering is usually neglected in most of radiation algorithms. We therefore do not present the result of *g*(*h*) in Fig. 3. Instead, we present the sorted Planck function for each band, which aids in the understanding of the results presented in the next section.

In the infrared, water vapor is the main gas. However, other gases also play an important role in some bands, especially in bands 3, 5, and 6. The application of the CKD principle under multiple gases conditions is an interesting and challenging problem. In band 6, H_{2}O, N_{2}O, and CH_{4} are the dominant gases. However, the large cooling rates of N_{2}O and CH_{4} are generally located in the regions above 500 hPa (Li and Barker 2005). Therefore, the main gas in the water cloud regions is H_{2}O. A similar condition exists for band 5, in that O_{3} plays an important role only in the upper atmosphere. Band 3 (550–780 cm^{−1}) is more problematic, since the radiative interaction between H_{2}O and CO_{2} is strong in the lower atmosphere. Here strong interaction means that both gases play important roles in the same region.

_{2}O and CO

_{2}are

_{2}O and CO

_{2},

_{2}O and CO

_{2}, and Δ

*z*is the vertical integral interval in height. The total optical depth for the two gases iswhere

*k*(

*ν*) is arranged to represent the absorption coefficient of a “single” gas. However, since

*η*varies with atmospheric profile and location, sorting cannot be done in a general way. Usually in a CKD algorithm, the gaseous sorting results corresponding to several reference values of the ratio

*η*are precalculated. For a given

*η*, linear interpretation between two neighboring reference results is applied. This in principle can be applied to the sorting process of cloud optical properties. However, we find this is not necessary, since the sorting results of cloud optical properties are not sensitive to

*η*. In Table 2, the sensitivity of

*η*to the sorting results of cloud optical properties is shown for midlatitude summer (MLS) (McClatchey et al. 1972). We set the concentration of CO

_{2}at 385.2 ppmv. With changing cloud height,

*η*changes because of the change of water vapor concentration. We choose three different heights—1, 3, and 5 km—for a sensitivity study, which correspond to

*η*equal to 17.3, 8.3, and 3.2.

CKD cloud optical properties of band 3 for three values of *η* (*η* is the mass ratio of H_{2}O and CO_{2}).

It is found in Table 2 that the differences in cloud optical properties for each CPS interval are very small for three different values of *η*. The relative difference is generally less than a few percent. In section 4, it will be shown that the longwave effect of the CKD cloud scheme is much weaker than that of the shortwave. Therefore, we can simply choose *η* to be 8.3, which determines the gaseous combination of H_{2}O and CO_{2} and finally the cloud optical properties are sorted based on this combined “single gas.”

*B*(

*h*) is the sorted Planck function. Note that

*ϖ*and

_{i}*g*can be obtained in the same way as (11) by replacing

_{i}*F*(

_{s}*h*) with

*B*(

*h*).

## 4. Results in a one-dimensional radiation model

The one-dimensional radiative transfer model is used to quantify the radiative impact of the CKD treatment of cloud optical properties. The one-dimensional radiative transfer model used was discussed in section 3. The atmospheric profile is divided into 280 vertical layers from the surface to 70 km, each layer with a thickness of 0.25 km. The surface albedos are set as 0.0967 and 0.265 for bands 9 and 10; all other bands are set at 0.3. Two water cloud cases are considered. The low cloud is positioned from 1.0 to 2.0 km with liquid water content (LWC) of 0.22 g m^{−3} (optical depth of 60 at 0.55 *μ*m) and *r _{e}* = 5.89

*μ*m. The middle cloud is positioned from 4.0 to 5.0 km with LWC of 0.28 g m

^{3}(optical depth of 72 at 0.55

*μ*m) and

*r*= 6.2

_{e}*μ*m.

The left panels of Fig. 4 present the shortwave heating rates of low and middle clouds for MLS, based on line-by-line calculations, with the solar zenith angle *θ*_{0} = 0° and *θ*_{0} = 60°. The line-by-line model is described in Zhang et al. (2006a) with the HITRAN 2008 molecular spectroscopic database. In line-by-line calculations, the resolution of cloud optical properties is 0.1 cm^{−1}.

The errors in the heating rate of the band-mean cloud scheme and the CKD cloud scheme are shown in the middle and right columns for two solar zenith angles. It is found that the errors of the band-mean cloud scheme can be considerable, with relative errors over 30% in general. The band-mean cloud scheme strongly overestimates the cloud-top solar heating rate for both low and middle clouds. In contrast, the errors of the CKD cloud scheme are considerably smaller. The errors of the CKD cloud scheme generally arise in two ways. First, the CKD radiation algorithm has its own error under clear-sky conditions [see Fig. 4 of Zhang et al. (2006a)], and this kind of error can affect the cloudy-sky results. Second, in a CKD cloud scheme, even though the cloud optical properties are sorted in the same way as those of the gaseous absorption lines, the spectral correlation is only addressed for each of the CPS interval mean. This can cause difference to the results of line-by-line calculations, in which the spectral correlation between the gaseous absorption lines and cloud optical properties is fully captured at fine resolution (0.1 cm^{−1}).

In Table 3, the corresponding results are shown for the upward flux at the top of the atmosphere (TOA) and the downward flux at surface. It is found that the band-mean cloud scheme generally produces lower upward flux in comparison with the line-by-line result. The error can be 20.7 W m^{−2} at *θ*_{0} *=* 0°. This is consistent with the heating rate results because the enhanced solar energy absorption leads less reflected flux to the TOA. Generally, the CKD cloud scheme can considerably improve the accuracy of flux at the TOA and surface.

In the case of low cloud, radiative fluxes by the line-by-line model and the errors by the band-mean cloud scheme and the CKD cloud scheme. [F^{↑} and F^{↓} are the upward and downward fluxes (W m^{−2}).]

It is found that the differences in solar heating rate caused by the band-mean cloud scheme occur mostly in the near-infrared range. In Fig. 5 the heating rate of bands 9–11 are presented for the low and middle clouds in MLS. It is found that the relative errors in heating rate between the band-mean cloud scheme and line-by-line model become larger in bands 9 and 10 compared to the broadband results. Also, the errors of the CKD cloud scheme are relatively small. In band 11, both the band-mean cloud scheme and the CKD cloud scheme are relatively accurate.

To prove that the results in Figs. 4 and 5 are reliable, we perform the same line-by-line calculations but using the band-mean cloud scheme. We called this scheme the LBL (band-mean cloud) model. The differences between LBL (band-mean cloud) and the line-by-line model, which uses the cloud optical property in fine resolution of 0.1 cm^{−1}, are shown in Fig. 6. It is found that the cloud-top heating rates considerably increase by the LBL (band-mean cloud) model. The errors shown in Fig. 6 are similar to those of the band-mean cloud scheme shown in Fig. 4. We also have done the same calculations by using LBLRTM (Clough et al. 2005) and obtained results similar to those in Fig. 6, although LBLRTM does not include multiple-scattering capability. These line-by-line calculations support the results found in Figs. 4 and 5. The strong overestimation of cloud absorption in the near-infrared range by the band model was also found in Espinoza and Harshvardhan (1996).

We need to understand the physics of why the band-mean cloud scheme produces larger cloud heating compared to the CKD cloud scheme. To illustrate the problem clearly, we simply consider the cloud absorption by the direct solar flux without multiple scattering. This is a first-order approximation, since the solar direct beam dominates the cloud-top heating. However, we emphasize that the exact radiative transfer, including multiple scattering, is used in all numerical calculations as shown in Figs. 4–6.

*j*is between levels

*j*and

*j*+ 1, we specify a cloud located in several adjacent layers

*m*,

*m*+ 1, … . In a certain band, the downward direct solar flux at the cloud top is approximatelywhere

*i*= 1, 2, … ,

*N*is the interval number of the CPS in such band,

*D*is the geometric length from the TOA to the cloud top, and

*q*, and

^{j}*d*being the gaseous absorption coefficient, mass concentration, and geometric thickness of layer

^{j}*j*, respectively. Generally, the

*k*〉,

_{i}q*k*and

_{i}*q*can be considered as the result of the vertical mean from the TOA to the cloud-top level. The

*F*

_{0i}is the incoming solar flux at the TOA for interval

*i.*The variation of

*F*in the CPS is much smaller than that of the absorption coefficient

_{0i}*k*as shown in Fig. 2. Therefore, the variation of

_{i}*k*. As a result of

_{i}*k*increasing with

_{i}*i*,

*i.*This is shown in Fig. 7 by the sorted downward solar fluxes at the cloud-top levels of 2 and 5 km.

*Q*is the heating rate,

*ψ*

_{abs i}is the specific absorptance of interval

*i*, and

*d*is the geometric thickness of layer

^{m}*m.*

*ψ*

_{abs i}also increases with

*i*for bands 9 and 10, hence

*i*for these two bands. Using the Chebyshev inequality (see the appendix), we havewhere

*a*

_{1}

*= a*

_{2}= · · · = a

_{N}. Under the band-mean condition, the total input solar energy flux is

*k*and

_{i}*ψ*

_{abs i}are positively correlated in CPS, then the band-mean cloud scheme produces larger cloud-top layer heating than the CKD cloud scheme. On the other hand, if

*k*and

_{i}*ψ*

_{abs i}are anticorrelated, then the band-mean cloud scheme produces a smaller cloud heating rate.

The physics can be explained intuitively. In bands 9 and 10, *k _{i}* is positively correlated with

*ψ*

_{abs i}in the CPS. For an interval of large

*k*, the incoming solar energy is considerably reduced at the cloud top, and the reduced solar energy limits the cloud heating. On the other hand, for an interval of small

_{i}*k*, the small

_{i}*ψ*

_{abs i}also limits the cloud heating. Consequently, the CKD cloud scheme produces less cloud heating. Therefore, to a certain extent the relationship between the gaseous absorption coefficient and the cloud absorption coefficient determines the bias of the cloud solar heating rate of the CKD cloud scheme.

Figure 2 shows that the cloud single scattering albedo in band 11 is very close to one, and hence the cloud absorption coefficient is close to zero, and its correlation to the gaseous absorption coefficient is very weak. It can be seen from Fig. 5 that the difference in heating rate between two cloud optical property schemes is very small (relative error < 0.8%). This supports the above correlation argument. Similar to band 11, the results of other bands 12–17 in the visible and ultraviolet are also not sensitive to the choice of cloud optical property schemes, since the cloud absorption is very close to zero in these ranges as shown in Fig. 1.

For the diffused solar photons, the result is much more complicated, and we cannot simply write down the formula of heating rate like that of (13). However, the positive (or negative) correlation between *k _{i}* and

*ψ*

_{abs i}should also exist and play a similar role in the solar cloud heating.

In the CKD cloud scheme, the spectral correlation between gas and cloud is considered, but the spectral correlation is only addressed for each of the CPS interval mean. The CKD model we used contains relatively small number of CPS interval in each band, and only three bands cover the near-infrared range. Therefore the spectral correlation cannot be fully captured as that of the line-by-line model with very fine spectral resolution. To obtain more accurate results, the slightly adjustment of the CPS interval mean result of the cloud optical property is necessary. This is similar to the case of gas as discussed in (8), in that there is a factor *α* to make the heating rate and flux give the best approximation to the line-by-line result. In the CKD cloud case, the corresponding physical quantity *k*(*h*) in (8) is the cloud absorption coefficient, *ψ*_{abs}(*h*) = *ψ*(*h*)[1 − *ϖ*(*h*)]. Therefore, the factor *α* ≤ 0 indicates a decrease (or no change) of *ψ _{i}* and an increase (or no change) of

*ϖ*from that of (11). By line-by-line calculations, the accurate heating rates and fluxes can be obtained for each of the CPS intervals. Then the fitting procedure is used to ensure that the corresponding result for the CKD cloud scheme is close to that of the line-by-line model result in each CPS interval. For example, in band 9, in order to agree with the line-by-line result,

_{i}*ϖ*in CPS intervals 3–7 (see Fig. 2) has to be slightly increased by a few percent from that of (11). The results of the adjusted CKD cloud scheme (called the CKD* cloud scheme) are also shown in Figs. 4 and 5 and Table 3. It is found that the accuracy in heating rate and flux is generally further improved by using the CKD* cloud scheme.

_{i}If a CKD algorithm contains fine band structure and each band has sufficient CPS intervals, the CKD cloud scheme should be able to fully resolve the spectral correlation between the gas and cloud. Hence, the CKD* is not needed. For example, in the RRTM algorithm (Mlawer et al. 1997), there are 10 bands in the near-infrared range and each band generally contains over 10 CPS intervals. We therefore expect that the CKD cloud scheme can produce very accurate result when applied to RRTM.

In Fig. 8 we present the results for a wide range of effective radius and liquid water content, in order to show that the CKD cloud scheme works for general cloud microphysical condition. The low cloud case is considered with solar zenith angle *θ*_{0} *=* 60°. In the left column, the top panel shows the line-by-line results of shortwave flux versus *r _{e}* and the bottom panels show the errors of the band-mean cloud scheme, the CKD cloud scheme, and the CKD* cloud scheme. It is found that the corresponding error of the band-mean cloud scheme increases with decreasing

*r*The error is up to 8 W m

_{e}.^{−2}for the upward flux at TOA. It is found that the errors of the CKD and CKD* cloud schemes are considerably reduced compared to those of the band-mean cloud scheme. The CKD* cloud scheme is slightly more accurate than the CKD cloud scheme. In the right column of Fig. 8, the top panel shows the line-by-line results of shortwave flux versus LWC. It is found that both the upward flux and the downward flux are saturated when LWC becomes large enough. The corresponding errors of the band-mean cloud scheme, the CKD and CKD* cloud schemes are shown in the bottom panels. Also, the errors of the CKD and CKD* cloud schemes are considerably smaller compared to those of the band-mean cloud scheme.

It is worth pointing out that the overestimation of the cloud-top heating rate due to the band-mean cloud scheme exists in non-CKD radiation models as well such as the band model for gaseous transmission, since in a non-CKD radiation model the correlation between cloud and gas can never be addressed. Therefore, the overestimation of cloud solar heating rate has been a problem in climate models for some time. The situation might not be substantially improved by using a narrowband radiation model. The variation of gaseous absorption coefficients in the CPS is very large and can still span several orders of magnitude in each narrow band. Therefore, the spectral correlation between gaseous absorption coefficients and cloud optical properties cannot be completely solved if the band-mean cloud scheme is used. The result in Fig. 5 supports this opinion. It is seen that the error of the CKD cloud scheme is larger in band 10 than in band 9. This is because band 9 contains more CPS intervals, which can be considered as subbands.

The overestimation of the solar heating rate can enhance cloud evaporation and limit the growth of water clouds. Zhang et al. (2005) had compared 10 general circulation models (GCMs) with observations from the International Satellite Cloud Climatology Project (ISCCP) and Clouds and the Earth’s Radiant Energy System (CERES). Figure 5 of Zhang et al. (2005) clearly showed that half of the models underestimated low and middle cloud amounts, while none overestimated them at a statistical significant level. We believe one cause for the underestimation of water clouds in GCMs is the overestimation of the solar heating rate due to the band-mean cloud scheme. This can be verified by testing the new CKD cloud scheme in a GCM.

The left column of Fig. 9 shows the cloud longwave cooling rates by the CKD cloud scheme for the same cloud case and atmospheric profiles in Fig. 4. The differences in cooling rate caused by the two cloud schemes are shown in the left column. It is found the differences are very small. We do not show the line-by-line calculations for the infrared case.

Since the small difference between the CKD and band-mean schemes indicates that the effect of addressing the vertical spectral correlation is small, the line-by-line calculation fully addresses the spectral correlation at very fine resolution.

As shown in Figs. 2 and 3, the variations of cloud optical properties in the infrared are of the same order of magnitude as those of the shortwave. Why is the longwave cooling rate insensitive to the two different cloud schemes? To understand this, we present in Fig. 10 the results of the cooling rate of low cloud for bands 3, 4, and 5. These three bands dominate the infrared energy.

*h*, the upward flux at level

_{i}*j*iswhere

*i =*1, 2, … ,

*N*is the interval number in CPS;

*j*;

*d*is the geometric thickness of layer

^{j}*j*;

*μ*

_{1}= 1/1.648 72 is the diffusivity factor (Li 2002);

*j*; and

*q*is the gaseous mass concentration at layer

^{j}*j.*In a layer containing cloud, the cloud absorption is generally much larger than the gaseous absorption, so

*j*+ 1 isFor a layer

*m*with a water cloud, the corresponding cooling rate of a considered band isThe physics of (17) is very clear. The cooling rate is determined by the net incoming energy flux

In the longwave case, the thermal source appears everywhere, which can affect the spectral distribution of flux in the CPS. In (15), the upward flux at level *j* is determined by the flux transmission from level *j* + 1 (the first term on the right-hand side) and the thermal emission from layer *j* (the second term). Therefore, the distribution of the upward flux in the CPS is not solely determined by the distribution of

In Figs. 9 and 10, it is seen that the difference in the cooling rate is very small between the two cloud optical property schemes. Also, the sign of the difference is not determined. We do not present the corresponding results in the longwave flux, as in Table 3 for the shortwave, since the difference in flux is very small between the two different cloud schemes.

## 5. Summary and conclusions

This work is an effort to complete the idea of correlated-*k* distribution in a radiation algorithm by extending the CKD method from gaseous absorption to cloud optical properties.

In the CKD cloud scheme, the water cloud optical properties are treated in the same way as the gaseous absorption (i.e., by using the same sorting in the CPS for each band). Since there is generally more than one gas in a band, we sort the cloud optical properties based on the main gas. In the solar spectrum, H_{2}O is the main gas in the near infrared range and O_{3} is the main gas in the visible and ultraviolet range.

It is interesting to find that the commonly used band-mean cloud scheme generally produces large errors in the solar heating rate at the top layer of a cloud in comparison with the line-by-line calculations. The relative error can be over 30%. The physical explanation of the overestimation of the solar heating rate is given as the absence of the spectral vertical correlation between the gaseous absorption coefficient and the cloud optical properties. The corresponding error is considerably reduced by applying the CKD cloud scheme. However, even the CKD cloud scheme cannot completely resolve the spectral consistency between gas and cloud if the band structure is not fine enough. To produce accurate results, a slight adjustment of cloud optical property in a CKD cloud scheme is needed.

This large overestimated solar heating rate at the cloud-top layer can strongly affect the moisture circulation and limit the cloud growth. This might be one of the main causes for the lower bias of water cloud amount in GCMs (Zhang et al. 2005).

It was emphasized that the overestimation of the cloud-top heating rate due to band-mean cloud scheme exists in the non-CKD radiation models as well, since in a non-CKD radiation model the correlation between cloud and gas can never be addressed.

It is interesting to note that the errors in the longwave cooling rate caused by the band-mean cloud scheme are generally very small. In the infrared, the distribution of flux in a CPS is continuously modulated by the distribution of thermal emission, which results in a more random distribution of the fluxes. Therefore, the CKD cloud scheme, which emphasizes the vertical spectral correlation between gas and cloud, does not make an obvious difference in cloud cooling rate and flux.

We do not consider the CKD cloud scheme for high cloud (ice cloud), since the solar heating rate of high cloud is usually much smaller compared to that of the water cloud, due to the thin optical depth. Therefore, the difference between the CKD cloud scheme and the band-mean cloud scheme is expected to be small. However, future work might be needed to clarify if the overestimation of the cloud heating rate is true for ice cloud.

This work should be extended to include aerosol shortwave radiative calculations, since all current aerosol optical parameterizations are based on the band-mean result (e.g., Li et al. 2001).

This study shows that the main radiative impact of the CKD cloud scheme occurs in the near-infrared range. In the visible, ultraviolet, and the infrared ranges, the current band-mean cloud scheme can be used without an obvious loss of accuracy.

In Cess et al. (1995), collocated satellite observations and surface measurements of solar radiation at five geophysically diverse locations showed significantly more solar absorption by clouds compared to that predicated by the theoretical models. Our work shows opposite results in that the cloud solar absorption is strongly overestimated by the current theoretical models in comparison with the benchmark line-by-line calculations.

## Acknowledgments

The authors thank Dr. Jason Cole, Dr. Steve Lambert, and three anonymous reviewers for their constructive comments. This work is partly supported by the National Natural Science Foundation of China (Grant 41075056), and National Basic Research Program of China (Grant 2011CB403405).

## APPENDIX

### Chebyshev Inequality

*a*

_{1}≤

*a*

_{2}≤, … , ≤

*a*and

_{N}*b*

_{1}≥

*b*

_{2}≥, … , ≥

*b*or

_{N}*a*

_{1}≥

*a*

_{2}≥, … , ≥

*a*and

_{N}*b*

_{1}≤

*b*

_{2}≤, … , ≤

*b*, thenIf

_{N}*a*1 ≤

*a*2 ≤, … , ≤

*a*and

_{N}*b*

_{1}≤

*b*

_{2}≤, … , ≤

*b*or

_{N}*a*

_{1}≥

*a*

_{2}≥, … , ≥

*a*and

_{N}*b*

_{1}≥

*b*

_{2}≥, … , ≥

*b*, then

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^{1}

In most CKD papers, *g* is used for the cumulative probability function. We choose *h* since *g* is more commonly used for the cloud asymmetry factor.