1. Introduction

Clouds have a large effect on the radiative heating–cooling of the atmosphere and the earth’s surface. Difficulties in modeling clouds involve the parameterizations of cloud generation, development, and dissipation, as well as cloud microphysical properties (water content, phase, particle shape, and size distribution) and optical properties (extinction coefficient, single-scattering albedo, scattering phase function). The microphysical properties are determined by the thermal and dynamicalproperties of the environment, whereas the cloud optical properties are determined by cloud microphysical properties. In atmospheric models, cloud optical properties are commonly parameterized as functions of particle size and water content.

Difficulties involved in the parameterization for cloud single-scattering properties are many. The single-scattering coalbedo of ice crystals is significantly larger than that of water droplets; thus, ice clouds have a larger absorption than water clouds for a given optical thickness. The optical properties of ice clouds are functions of the size, shape, and orientation of ice crystals, which vary over a large range and cannot be determined in current atmospheric models. The single-scattering coalbedo of ice and water particles vary rapidly with wavelength. Parameterization for the mean effective single- scattering coalbedo for a wide spectral interval dependsnot only on the absorption by clouds but also on the water vapor within and above cloud layers (Ramaswamy and Freidenreich 1992). All these problems make the parameterizations for cloud optical properties very difficult.

Calculations of radiative heating are further complicated by the fact that clouds are not horizontally homogeneous and do not always cover the entire sky. The solar (or shortwave, SW) radiation algorithms used in atmospheric models are developed under the assumption that atmospheric layers are plane parallel and can only be applied to the case where an atmospheric layer is either totally cloud covered or cloud free. The simplest approach to dealing with the partially cloudy case is to assume that clouds in various layers overlap either randomly or maximally. The sky is then divided into sections. Within each section, clouds are homogeneous in a layer with cloud amount equal to either 0 or 1. Shortwave fluxes are calculated for each section and then weighted by the cloud amount to derive total fluxes. This approach requires a huge amount of computations. For the case of random cloud overlapping, it requires 2ⁿ sets of calculations, where n is the number of cloud layers. For the case of maximum cloud overlapping, the computing time required is less. To reduce the computational burden, clouds in a partially covered layer are smeared to cover the entire layer, and the optical thickness is scaled by a function of the cloud amount (e.g., Kiehl et al. 1994; Sud et al. 1993). In addition to being a function of the cloud amount, scaling of the cloud optical thickness should also be a function of the optical thickness itself, the solar zenith angle, as well as the assumption made for cloud overlapping.

In this study, we present parameterizations for cloud single-scattering properties, for the treatment of cloud overlapping, and for the scaling of cloud optical thickness. These parameterizations have been implemented in the GEOS (Goddard Earth Observing System) general circulation model (GCM) (Schubert et al. 1993) and the Goddard cloud ensemble model (Tao et al. 1996). The parameterizations for cloud single-scattering properties are extensions of those of Slingo (1989) for water clouds and Fu (1996) for ice clouds. The scaling of cloud optical thickness is based on the assumption of maximum cloud overlapping. Uncertainties in these cloud-radiation schemes are investigated by comparing the results with more detailed calculations.

2. Radiative transfer calculations

In our radiation routine, the solar spectrum is divided into an ultraviolet (UV) and visible region (λ < 0.7 μm) and an IR region (0.7 μm ≤ λ < 10 μm). In the UV and visible region, we include absorption due to O₃ and aerosols and scattering due to gases, clouds, and aerosols. The UV and visible region is further divided into eight spectral intervals so that fluxes in the PAR (photosynthetically active radiation) (0.4 < λ < 0.7μm), UV-A (0.328 < λ < 0.4 μm), UV-B (0.28 < λ < 0.328 μm), and UV-C (0.18 < λ < 0.28 μm) regions can be separately computed. An effective absorption coefficient for O₃, and an effective extinction coefficient for Rayleigh scattering are derived for each spectral interval. It is noted that, in this study, a single set of parameterizations for cloud optical properties is applied to the eight spectral intervals in the UV and visible region (Band 1 shown in Table 1).

In the IR region, we include absorption due to water vapor, O₂, CO₂, clouds, and aerosols, and scattering due to clouds and aerosols. Calculations of the absorption due to water vapor and O₃ follow Chou and Lee (1996), and those of the absorption due to O₂ and CO₂ follow Chou (1990). This spectral region is divided into three bands (Table 1). To avoid oscillation in the computed heating profile caused by the use of a small number of k intervals (Chou and Lee 1996), each band is grouped into 10 k intervals, where k is the water vapor absorption coefficient. A one-parameter scaling approximation is applied to take into account the effect of temperature and pressure on k.

To reduce computing time, we avoid applying a single multiple-scattering algorithm to the entire inhomogeneous atmosphere for flux calculations. Instead, we first compute the transmission and reflection functions for each layer and then use the two-stream adding method to compute fluxes in the atmosphere and at the surface (cf. Lacis and Hansen 1974). In this approach, the transmission and reflection functions for each layer are separately computed for direct and diffuse incident radiation [cf. (4) and (5) of Chou 1992]. In our radiation model, the reflection and transmission functions of a scattering layer are computed using the δ-Eddington approximation of Joseph et al. (1976) for direct radiation and the two-stream approximation of Sagan and Pollack (1967) for diffuse radiation.

3. Parameterizations for cloud single-scattering properties

The cloud single-scattering properties that affect radiative transfer are the extinction coefficient, single- scattering albedo, and scattering phase function (or asymmetry factor). The extinction coefficient per unit mass of polydispersion cloud particles is given by

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where σ(r) is the extinction cross section of a particle with a size r, the subscript λ is the wavelength, n(r)dr is the number of particles within the size range dr per unit volume, and C is the cloud water mass concentration. The single-scattering albedo is defined as the ratio of the scattering coefficient to the total extinction coefficient

ω_λβ^s_λβ_λβ^s_λβ^s_λβ^a_λ(2)

where the superscripts s and a denote scattering and absorption, respectively. The asymmetry factor is given by

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where P_λ is the scattering phase function of polydispersion cloud particles and μ is the cosine of the scattering angle.

Radiation routines for atmospheric models usually use only a few broad spectral bands with effective optical properties parameterized for each band. The single- scattering coalbedo, 1 − ω_λ, varies by several orders of magnitude in the solar spectrum. It is, therefore, difficult to derive an effective ω for a broad spectral band that can be applied to both strong and weak absorption situations. There are a number of approaches to deriving ω. Slingo and Schrecker (1982) derived ω by weightingω_λ with the solar insolation and linearly averaging over a spectral band. Espinoza and Harshvardhan (1996) used a square-root approximation to derive ω. Fu (1996) used a mix of linear and logarithmic averaging depending on the strength of absorption in a given spectral band. The absorption of SW radiation by clouds depends not only on ω_λ but also on water vapor and cloud ice/water amounts. It is clear that there is no unique method for deriving ω over a broad band. It can only be empirically determined based on the amounts of cloud particles and water vapor encountered in the earth’s atmosphere. Let us define the linear and logarithmic averaging over a band as

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and

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where s_λ is the solar insolation at the top of the atmosphere (TOA) and Δλ is a narrow spectral interval, where optical properties can be treated as constants. The effective mean single-scattering coalbedo of a band is then computed from

ωhωhω(6)

where h is the weight ranging from 0 to 1. The weighth should be close to 1 for the spectral bands with weak absorption and should decrease as absorption increases. It is to be determined empirically.

Compared to ω_λ, the extinction coefficient (β_λ) and the asymmetry factor (g_λ) vary rather smoothly with λ. Their effective mean values over a wide spectral band can be accurately approximated by

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Theoretical considerations and radiative transfer calculations have shown that cloud single-scattering properties are not significantly affected by details of the particle size distribution and can be adequately parameterized as functions of the effective particle size (Fu 1996; Hu and Stamnes 1993; Tsay et al. 1989). Following Slingo and Schrecker (1982), we parameterize the single-scattering properties for a broad spectral band by

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where a, b, and c are regression coefficients, and r_e is the effective particle size defined to be proportional to the ratio of the total volume of cloud particles to thetotal cross-sectional area, A_c, of cloud particles. For spherical water droplets, the effective size is given by

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where ρ_w is the density of water. For hexagonal ice crystals randomly oriented in space, the effective size is shown by Fu (1996) to be

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where ρ_i is the density of ice. The cloud optical thickness, τ, is then given by

τβCz,(14)

where z is the geometric thickness of a cloud layer.

The spectral data ω_λ, β_λ, and g_λ calculated by Fu (1996) for ice clouds and by Tsay et al. (1989) for water clouds are used to derive ω, β, and g from (4)–(8). By assuming hexagonal ice crystals randomly oriented in space, Fu (1996) computed the single-scattering parameters of ice clouds using the improved ray-tracing method of Yang and Liou (1995). A total of 28 size distributions derived from in situ aircraft measurements were used in the calculations, which included samples from the First ISCCP Regional Experiment (FIRE) and Central Equatorial Pacific Experiment (CEPEX) field campaigns. The mean effective size of ice crystals, r_i, ranges from 20 to 130 μm. For water clouds, the single-scattering parameters of five water clouds were computed by Tsay et al. (1989) from the Mie theory assuming spherical droplets and a log-normal size distribution. The effective radius of water droplets, r_w, ranges from 4 to 20 μm.

The size, shape, and refractive indices are different for ice crystals and water droplets. The extinction coefficient of ice clouds is smaller than that of water clouds, because ice crystals are much larger than water droplets. The single-scattering albedo and asymmetry factor of ice clouds and water clouds are also different, as shown in Figs. 1 and 2. The ice cloud optical properties shown in the figures were computed by Fu (1996) for a FIRE I (22 October 1986) case and a CEPEX case, which have a mean effective size, r_i, of 45 μm and 97 μm, respectively. For the water cloud, two log-normal size distributions with effective droplet radii, r_w, of 8 and 16 μm are used to compute the cloud optical properties (Tsay et al. 1989). It can be seen in the figures that the single-scattering coalbedo, 1 − ω_λ, of ice crystals is larger than that of water droplets by nearly a factor of 10, while the asymmetry factor of ice crystals is smaller than that of water droplets for λ < 1.4 μm. Because of these differences, we parameterize the single-scattering properties separately for ice and water clouds.

As in Slingo (1989), we divide the solar spectrum into four wide bands and parameterize the single-scattering properties for these bands. The spectral ranges of these four bands are given in Table 1. There is one band in the ultraviolet and visible spectral regions and three in the IR region. Referring to Fig. 1, the single-scattering coalbedo is small (<0.001, weak absorption) in the first two bands but is large (>0.03, strong absorption) in Band 4. Within individual bands, it varies by two orders of magnitude. Figure 2 shows that the asymmetry factor varies more strongly with wavelength for ice clouds than for water clouds. The extinction coefficient varies with wavelength weakly for both ice and water clouds, and the detailed spectral distributions are not shown in the figures.

The single-scattering parameters ω′, ω", β, and g are computed, respectively, from (4), (5), (7), and (8) for the four spectral bands and all 33 cloud cases (28 ice clouds and 5 water clouds). By specifying various values of the weight h, the effective mean single-scattering albedo of a band are computed from (6), also for the four spectral bands and 33 cloud cases. These band- averaged single-scattering parameters are then fit by (9)–(11) separately for ice and water clouds. Fluxes are computed using these band-averaged single-scattering properties for a midlatitude summer atmosphere taken form McClatchey et al. (1972) with clouds located at various heights in the atmosphere. Optimal values of h are then determined empirically (trial and error) such that the difference in the fluxes at the TOA and at the surface between the parameterization (four bands, Table 1) and high-spectral resolution (100 cm⁻¹) calculations is minimized. For the high-spectral resolution calculations, the cloud single-scattering parameters are interpolated from that of Fu (1996) and Tsay et al. (1989), which have a spectral resolution of ≈1000 cm⁻¹ in the UV spectral region and ≈50 cm⁻¹ in the near IR. Table 1 shows the optimal values of h for the four spectral bands, separately for ice and water clouds. The weighth is 1 for weak absorption bands and 0 for strong absorption bands.

Figures 3–5 show the results of regression using (9)–(11). The coefficients a, b, and c are shown in Tables 2–4, respectively. It can be seen in Fig. 3 that the extinction coefficient varies weakly with spectral band but strongly with particle size. For large particles, it is independent of wavelength for both water and ice clouds. The results shown in Fig. 3 for ice clouds are indistinguishable among the four bands. Due to a large particle size, the extinction coefficient of ice clouds is significantly smaller than that of water clouds. It is interesting to note that in spite of a large difference in the particle size distribution, the ice particle extinction coefficients for the 28 clouds nearly fall onto a single curve of (9). Figure 4 shows the single-scattering coalbedo of Bands 2–4. The single-scattering coalbedo of Band 1 is very small (>10⁻⁵) and is not shown in the figure. For a given particle size, the single-scattering coalbedo varies by three orders of magnitude among the three IR bands. For a given spectral band, it is ≈8 times larger for ice clouds than for water clouds. The asymmetry factor, shown in Fig. 5, varies between 0.78 and 0.94 for different bands and particle sizes. Since the shape of size distribution has little effect on the single-scattering properties (Hu and Stamnes 1993; Slingo and Schrecker 1982), the results shown in Figs. 3–5 should be representative for a wide range of clouds with various particle-size distributions.

Using the radiation model discussed in section 2, the SW heating of the cloud and the atmosphere computed with the parameterization is compared to high spectral- resolution calculations. The temperature and humidity profiles used are typical of a midlatitude summer (McClatchey et al. 1972). The surface albedo is set to 0.2. In addition to clouds, the absorption of SW radiation due to water vapor, O₃, CO₂, and O₂ are also included.The sky is set to be overcast (i.e., fractional cover = 1) with either a high (200–300 mb), middle (500–600 mb), or low (800–900 mb) cloud. The high cloud is assumed to contain ice particles, and the others are water clouds. For the high cloud, two samples of the ice particle size distribution measured during CEPEX and FIRE field campaigns are used. The equivalent particle size, r_i, is 45 μm for the CEPEX case and 97 μm for the FIRE case. For water clouds, two log-normal size distributions are used. The equivalent radii of the droplets, r_w, are 8 and 16 μm.

Fluxes are calculated for a range of 0.1–1.0 in the cosine of the solar zenith angle, μ_o, and 0.1–60 in the cloud optical thickness in Band 1, τ₁. For other spectral bands, i, the optical thickness is scaled from τ_i = τ₁(β_i/β₁), where β_i is the cloud extinction coefficient in band i. The percentage difference in the total (0.18–10 μm) net downward flux at the TOA between the parameterization and the high spectral-resolution calculations is shown in Fig. 6 for the high, middle, and low clouds. It can be seen in the figure that the parameterization introduces a very small error (≤1%) in all cases. Figure 7 shows the percentage error in the SW heating of the atmosphere. For the middle and low clouds, shown in the middle and bottom panels, the error is small (≤1.5%). It increases to 4% for an optically thick cirrus cloud (upper panels). This relatively large error is related to the height of the clouds. Compared to water clouds, the absorption by water vapor above the cirrus cloud top is weak, which makes the heating of the atmosphere more sensitive to the parameterization for cirrus optical properties than the parameterization for water clouds. In addition to the heating of the entire atmospheric column, it is also important to understand the effect of the parameterization on the heating of the cloud layers. Figure 8 shows that the error in the heating of cirrus clouds reaches −14% (or −5.2 W m⁻²) for r_i = 45 μm, τ₁ =1.0, and μ_o = 0.5. This reduced cloud heating causes an increased heating of 1.6 W m⁻² in the atmosphere, both above and below the cloud, leading to a reduced SW heating of 3.6 W m⁻² for the total atmospheric column. This reduced heating is equivalent to −3% of the total atmospheric SW heating (including the cloud), as shown in the upper left panel of Fig. 7. The percentage error in the heating of the earth’s surface is shown in Fig. 9. It is small even in the presence of high cirrus clouds. It is noted that the parameterization for g shown in the upper panel of Fig. 5 has a very small error at r_i = 45 and 97 μm but has a relatively large error (≈0.01) at r_i = 80 μm. When r_i is set to 80 μm, the results are similar to that shown in the upper panels of Figs. 6–9.

4. Cloud overlapping

Clouds could occur at various heights with fractional cover. Nearly all radiative transfer algorithms used inatmospheric models apply only to a plane-parallel (i.e., horizontally homogeneous) atmosphere. Horizontal inhomogeneity is not allowed. A straightforward approach to dealing with a partial cloudiness situation is to divide the sky into sections. Within each section, an atmospheric layer is either free of clouds or filled totally with a homogeneous cloud. Radiative fluxes are then computed for each section, and the total SW heating is the sum of all sections weighted by the fractional cover of individual sections. Depending upon the number of cloud layers and the way these clouds overlap, computational costs of this approach could be huge. To simplify the computations, a cloud that partially fills a layer is usually smeared over the entire layer, and the optical thickness τ is adjusted by a factor dependent on the fractional cloud cover. As demonstrated by a number of studies (e.g., Harshvardhan and Randall 1985; Cahalan et al. 1994), the effect of τ on radiation is highly nonlinear, and it is not appropriate to scale τ (or cloud waterpath) linearly by cloud cover. In the National Center for Atmospheric Research (NCAR) community climate model CCM2, the optical thickness is scaled by f^1.5, where f is the fractional cloud cover (Kiehl et al. 1994). In the GEOS GCM, the optical thickness was previously scaled by inverting a simple reflection function for clouds when illuminated by diffuse radiation and assuming no absorption by clouds (Sud et al. 1993). As can be expected, the effective optical thickness corresponding to a cloud smeared to cover the entire sky is a function of τ, f, and the solar zenith angle, as well as whether scaling is based on reflection or absorption. Scaling of τ is further complicated by a large range of the way clouds in various layers overlap.

b. Scaling of optical thickness

Scaling of cloud optical thickness in the GEOS GCM is based on the maximum-random cloud overlapping assumption shown in Fig. 10. To simplify the scaling of τ, we assume that reflection is more important to climate than absorption, since the former affects the heating of the earth–atmosphere system, but the latter primarily redistributes the heating within the atmosphere as well as between the atmosphere and the surface. As stated in section 2, our radiation algorithmrequires calculations of reflection/transmission of a cloud layer, separately for direct and diffuse radiation. By smearing cloud cover over the extent of the maximum cloud cover of respective height group, the optical thickness of a cloud layer is scaled by factors χ_s and χ_s,

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so that albedos of the layer remain the same before and after scaling,

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and

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where μ_o is the cosine of the solar zenith angle, R is the albedo averaged over the entire solar spectrum fordirect incident radiation, R is the albedo for diffuse incident radiation given by

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f′ is the normalized cloud cover given by

fff_m(20)

and μ is the cosine of the zenith angle of an incident beam. Using the scaling of (15) and (16), a layer with a fractional cloud cover, f, and an optical thickness, τ, is reduced to a layer with a fractional cloud cover, f_m, and equivalent optical thicknesses τ_b for direct radiation and τ_f for diffuse radiation. It is noted that the scaling is a function of the incident angle, and the optical thickness is scaled separately for the direct radiation and the diffuse radiation.

Using values of r_e representative of the water and ice cloud samples used in section 3, ω and g are computed from (10) and (11) for the individual bands given in Table 1. Cloud albedos R and R averaged over the solar spectrum are computed using the radiation model discussed in section 2. The scaling functions χ_s and χ_s are then derived from (15)–(18) by giving f′, R(τ, r_e, μ_o), and R(τ, r_e). Sample calculations show that χ_s and χ_s depend only weakly on r_e and the phase of cloud particles. Therefore, they are reduced to χ_s(f′, τ, μ_o) and χ_s(f′, τ) by taking the mean values of the scaling functions for various particle sizes and phases.

The above scaling ensures that the reflection of a cloud layer is the same before and after the scaling. However, errors are expected in the calculations of atmospheric and surface SW heating when there are other cloud layers present. In addition to the solar zenith angle, cloud cover, and optical thickness, an optimal cloud scaling should, in principle, also depend on the fractional cover and optical thickness of other cloud layers. This situation makes scaling of the optical thickness extremely difficult. To simplify the problem, we made simple adjustments to the scaling functions χ_s and χ_s, so that errors in the fluxes at the TOA and the surface are minimized for the simplest cloud situation, which has two cloud layers in only one of the three height groups. For the two-cloud situation, scaling is applied to the cloud with a smaller cloud cover (see right panel of Fig. 10). It is noted that scaling is not required when there is only one cloud layer in a given height group. The scaling functions are adjusted according to

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where δ and δ are constants empirically (trial and error) chosen to minimize errors in the fluxes at the TOA and at the surface. The errors are defined as the differences between the flux computed with the scaling of τ and the flux computed with the sky divided into horizontally homogeneous sections (three sections in the two-cloud situation). The adjusted scaling functions preserve theshapes of χ_s and χ_s. The results for the scaling functions are saved as tables for later use. It is found that optimal values of δ and δ are 0.2 and 0.3, respectively. The results for the scaling functions χ and χ are shown in Fig. 11. It can be seen that χ(f′, τ, μ_o) < f′, and χ → f′ as τ → 0. For an optically thick cloud, χ and χ are not sensitive to τ.

The error induced in flux calculations due to the scaling of τ is calculated for the case of maximum overlapping of two contiguous cloud layers within a given height group. The scaling uses (15) and (16), except χ_s and χ_s are replaced by χ and χ. The percentage errors in the net downward flux at the TOA and at the surface are shown in Figs. 12–14 for pairs of cloud layers in the high, middle, and low height groups, respectively. The midlatitude summer atmosphere of McClatchey et al. (1972) is used. The high clouds are set to be in the 200–250-mb and 250–300-mb layers and contain ice particles with r_i = 97 μm and τ₁ = 2 for each cloud layer, where τ₁ is the optical thickness in Band 1. Themiddle clouds are set to be in the 400–450-mb and 450–500-mb layers and contain water droplets with r_w = 10 μm and τ₁ = 5 for each cloud layer. The low clouds are set to be in the 800–850-mb and 850–900-mb layers and contain water droplets with r_w = 10 μm and τ₁ = 10 for each cloud layer. The single-scattering albedo and asymmetry factor are derived from (10) and (11) for the four bands shown in Table 1. The optical thickness in Bands 2–4 are scaled from that in Band 1 (τ₁) by the ratio β_i/β₁ derived using (9), where the subscripts denote spectral bands, and i = 2, 3, or 4. The results shown in the figures are for three solar zenith angles, 30°, 60°, and 75°. The ordinate is the fractional cover of the upper cloud layer, f_u, and the abscissa is the fractional cover of the lower cloud layer, f_l.

It can be seen in Figs. 12–14 that, generally, the percentage error increases with increasing solar zenith angle. The errors are smaller when the upper-cloud amount is larger than the lower-cloud amount. In this situation, scaling of the optical thickness is applied to the lower- cloud layer. When the cloud amounts of the two layers are equal, that is, f_m = f_u = f_l, there is no need to scale the optical thickness, and there are no errors, as shown by the diagonal lines in the figures. Except for cases with large solar zenith angles, the error is less than a few percent. In all cases, the absolute errors are small both at the TOA and at the surface.

In the GEOS GCM, the scaling functions derived for the two-cloud situation is applied universally to all situations even if there are more than two cloud layers in a given height group. It implicitly assumes that, in a given height group, accurate scaling of τ is not critical if there are two cloud layers above.

5. Conclusions

We have developed parameterizations for computing cloud single-scattering properties and for scaling theoptical thickness in a partial cloudiness situation. Due primarily to large differences in particle size and shape, differences in the single-scattering properties of ice and water clouds are large. Therefore, the extinction coefficient, single-scattering albedo, and asymmetry factor are parameterized separately for ice and water clouds as functions of the effective mean particle size. These parameterizations are variations from that of Slingo (1989) for water clouds and that of Fu (1996) for ice clouds. The shortwave spectrum is divided into four broad bands nearly identical to those used by Slingo (1989), and the parameterizations for ice clouds are based on the high spectral-resolution single-scattering data from Fu (1996). The single scattering coalbedo varies greatly with wavelength, and the approach to averaging the single-scattering coalbedo over a broad spectral band is empirically determined so that errors in flux calculations are minimized. Depending upon the strength of absorption, the averaging ranges between linear and logarithmic.

Compared to high spectral-resolution calculations, the parameterization for the cloud single-scattering properties introduces a small relative error of ≤1% at TOA. The error exceeds 10% in the SW heating by ice clouds. Due to a substantial compensation in the absorption within clouds and the regions above and below clouds, the relative errors in the absorption of SW radiation in the atmosphere and at the surface are less than a few percent.

Multiple-scattering radiative transfer models used in atmospheric models are developed for plane-parallel atmospheres, and scaling of cloud optical thickness in an atmosphere with multiple cloud layers is necessary to make computations affordable. The scaling of optical thickness depends on the assumption applied to the overlapping of clouds at various heights. In the GEOS GCM, a maximum-random assumption for cloud overlapping is applied. Clouds are identified as high, middle, and low, separated approximately by the 400- and 700- mb levels. Clouds are assumed to be maximally overlapped within each height group but randomly overlapped among the three height groups. Scaling of the optical thickness is separately applied to each height group. Fluxes are then computed by dividing the atmosphere into a maximum of eight horizontally homogeneous sections. The fractional cloud amount of a layer in each section equals either 0 or 1.

Scaling of the optical thickness is based on the case with two maximally overlapped cloud layers. Thus, it implies that flux calculations are not sensitive to the scaling of the optical thickness of a cloud layer with more than two cloud layers lying above. The scaling functions are calculated using a radiative transfer model and then empirically adjusted so that errors in the fluxes at the TOA and at the surface introduced by the scaling are minimized. They are separately applied to direct and diffuse radiation. Tables for these scaling functions were precomputed as functions of the cloud optical thickness, cloud amount, and, for the direct radiation case, the solar zenith angle. Flux errors introduced by the scaling are small, both at the TOA and at the surface. Except for cases with large solar zenith angles, the error is only a few percent. In terms of absolute error, it is within a few watts per meter squared.