## 1. Introduction

It has been demonstrated that accurate estimates of domain-averaged solar irradiances for marine boundary layer clouds can be obtained by applying a two-stream approximation to each subgrid cell and averaging the resulting irradiances (the independent pixel or column approximation; Stephens et al. 1991; Cahalan et al. 1994). Barker et al. (1998) showed that the same can be done using entire columns for large domains containing convective clouds. Since this approximation neglects net horizontal fluxes, these results suggest that if the probability density function of cloud optical thickness is known, domain-averaged irradiances can be computed as a function of a few parameters expressing the probability density function.

Satellite observations (Barker et al. 1996) and cloud-resolving models (e.g., Barker et al. 1998; Oreopoulos and Barker 1999) indicate that distributions of cloud optical thickness for domains the size of cells in typical weather and climate models can be approximated by gamma distributions. Barker (1996) used a two-stream approximation with boundary conditions of no diffuse irradiance from above and below the cloud layer to derive an analytical solution for domain-averaged reflectance and transmittance for an inhomogeneous cloud layer. Oreopoulos and Barker (1999) used the principles of invariance (Chandrasekhar 1960, 161â€“166) combined with Barker's (1996) analytical solution to compute irradiance at each level of a multilayered atmosphere. They used the adding method, which differs from discrete ordinate two-stream multilayered algorithms that solve a matrix equation set up by irradiance boundary conditions (e.g., Shettle and Weinman 1970; Liou 1975; Stamnes and Swanson 1981; Stamnes et al. 1988; Toon et al. 1989, Fu and Liou 1993). Since discrete ordinate two-stream algorithms are widely used to compute irradiance for multilayered atmospheres, it is worthwhile incorporating the gamma-weighted two-stream approximation into the discrete ordinate framework.

The main purpose of this paper is to derive analytical solutions for reflected and transmitted irradiance given gamma distributed cloud optical thickness that can be used with a discrete ordinate two-stream model. This provides an alternative to the adding method and entails solving a tridiagonal matrix built from boundary conditions for radiative transfer computations for multilayer atmospheres.

## 2. Algorithm development

### a. Homogeneous cloud layer

*F*

^{+}and

*F*

^{+}are upward and downward irradiance, respectively;

*Î¾*is the fraction of cloud depth measured from

*Î¾*= 0 at cloud top to

*Î¾*= 1 at cloud base,

*Ï„*

_{m}is optical thickness of the layer at

*Î¾*= 1,

*Ï€F*

_{s}

*Î¼*

_{0}is the solar irradiance incident at cloud top;

*Î¼*

_{0}is the cosine of the solar zenith angle;

*Ï‰*

_{0}is the single-scattering albedo; and

*Î³*

_{1},

*Î³*

_{2},

*Î³*

_{3}, and

*Î³*

_{4}are coefficients that depend on the form of the two-stream approximation (Meador and Weaver 1980).

### b. Inhomogeneous cloud layer

*Î½*and

*Ï„*

_{m}are the shape parameter and mean optical thickness of the layer at

*Î¾*= 1, respectively (Wilks 1995, p. 86). The shape parameter can be estimated from moments such as

*Î½*= (

*Ï„*

_{m}/

*Ïƒ*)

^{2}where

*Ïƒ*is the standard deviation of

*Ï„*

_{m}. For

*Î½*< 10, however, it is recommended that the maximum likelihood method be used to estimate

*Î½*(Wilks 1995, 86â€“90). Domain-averaged irradiances are then computed via the independent column approximation as

*Î½*is independent of

*Î¾.*

*k*

_{1}and

*k*

_{2}, are determined by boundary conditions,

*F*

^{âˆ’}(0) =

*F*

^{+}(

*Ï„*

_{m}) = 0] and expanding the denominators of

*k*

_{1}and

*k*

_{2}by the binomial theorem yields

*Î¾*are

*G*

_{1}and

*G*

_{2}at layer boundaries are

*Î¾*is given by

*Î½*goes to infinity (i.e.,

*Ïƒ*goes to zero),

Since the single-scattering albedo *Ï‰*_{0} and asymmetry parameter *g* are mean values weighted by the optical thickness of constituents, they depend on cloud optical thickness. For the integration of (8), we assume that *Ï‰*_{0} and *g,* as well as *Î»,* which is a function of *Ï‰*_{0} and *g,* are constant throughout the layer.

### c. Multilayer atmosphere

*F*

^{âˆ’}(0) and

*F*

^{+}(

*Ï„*

_{m}) â‰ 0 in (11)]. Thus, assuming that when

*G*

_{1}and

*G*

_{2}are factored out of the unknowns, that are determined by boundary conditions, the remainder is independent of

*Ï„*

_{m}, and domain-averaged upward and downward irradiance for nonzero

*F*

^{âˆ’}(0) and

*F*

^{+}(

*Ï„*

_{m}) are

*k*

^{â€²}

_{1}

*k*

^{â€²}

_{2}

*Î½*goes to infinity, (25) and (26) revert to (1) and (2), except that

*e*

^{Î»Ï„m}

*k*

_{1}.

When an inhomogeneous cloud layer is placed in a vertically inhomogeneous atmosphere consisting of *n* homogeneous layers, the boundary conditions are set up as a matrix equation that can be solved for *n* âˆ’ 1 unknowns of *k*_{1} and *k*_{2} in (1) and (2) and *k*^{â€²}_{1}*k*^{â€²}_{2}

## 3. Results

Broadband, domain-averaged irradiances computed by (25) and (26) are compared to corresponding values obtained by numerically integrating (1) and (2) weighted by the appropriate optical thickness distribution [i.e., Eq. (7)]. For the test, a cloud layer of *Ï„*_{m} = 15 was placed between 1 and 2 km in the midlatitude summer standard atmosphere (McClatchey et al. 1972). We used a spectrally invariant surface albedo of 0.2 and a solar zenith angle of 60Â°. The *k*-distribution tables used here in estimating gaseous absorptions are those given by Kato et al. (1999). For numerical integration, lower and upper limits of optical thickness were 0.001 and 500, respectively. The population outside this range is negligible, less than 0.1% for the cases considered here.

Figure 1 shows the top-of-the-atmosphere reflectance and top-of-the-atmosphere to surface transmittance for the analytic model and the numerical integration. Differences between broadband reflectances and transmittances are less than 0.015, 0.01, and 0.007 for *Î½* = 1.0, 1.5, and 2.0, respectively. As mentioned by Barker et al. (1996), since the algorithm's values result from integration over *Ï„*_{m} from 0 to infinity, differences between algorithmic and numerically integrated irradiances can be caused, in part, by numerical integration over a finite range. As a consequence, differences between estimated irradiances increase as *Î½* and *Ï„*_{m} decrease; the population outside the range of integration increases to more than 1% when *Î½* = 0.5 and *Ï„*_{m} = 5. In this case, differences in broadband reflectance and transmittance are 0.006 and 0.023, respectively.

This algorithm runs into difficulty when *Ï‰*_{0} approaches unity because of the binomial expansion of the denominator in (9) and because of the singularity in the two-stream approximation. Therefore, a maximum single-scattering albedo is set to avoid the problem (Wiscombe 1977). For a maximum single-scattering albedo of 0.99999 and use of 100 terms to compute *G*_{1} and *G*_{2} by (14) and (15), the algorithm reproduces conservative-scattering reflectances given by Barker et al. (1996). For example, for four overcast scenes in Barker et al. (1996), the sums of transmittance and reflectance are greater than 0.9995 for all solar zenith angles.

## 4. Discussion and summary

The result shown in the previous section demonstrates that this algorithm can compute accurate domain-averaged irradiances for multilayered atmospheres with a single-layer inhomogeneous cloud. In such cases, trivial boundary conditions are sufficient for straightforward computation of domain-averaged irradiance. If the cloud layer were to be divided into many layers, trivial boundary conditions are no longer appropriate because irradiance is horizontally variable; columns locating below columns that are optically thinner (thicker) than the mean optical thickness receive more (less) radiation than the domain-averaged irradiance at the top boundary (Stephens 1988; Oreopoulos and Barker 1999). One could, however, resort to an adjustment of optical properties similar to that proposed by Oreopoulos and Barker (1999). When overcast cloud layers are separated by a clear layer, the trivial boundary conditions can be applied once again if one is content to accept the assumption that for multilayered systems, fluctuations in cloud optical properties and irradiance are uncorrelated. Although the discrete ordinate formulation allows for overcast clouds only, the algorithm presented here could be used in a cloud overlap scheme such as that developed by Geleyn and Hollingsworth (1979).

When *Ï„*_{m}/*Î½* is small, (14) and (15) approach (10) and (11) because the second term in the denominators of (14) and (15) are small and the denominators can be approximated by exponential functions. Therefore, the inhomogeneous solutions (25) and (26) approach the homogeneous solutions (1) and (2), respectively. Two cases that give small *Ï„*_{m}/*Î½* are *Ï„*_{m} â‰ª 1 and *Î½* â‰« 1. When the layer is optically thin, the transmittance and reflectance are linear functions of the optical thickness so that the mean optical thickness provides the domain averaged transmittance and reflectance. As the mean optical thickness increases, the difference of transmittance and reflectance increases. When *Î½* is large, the layer is close to homogeneous; hence, the inhomogeneous solution should approach the homogeneous solution. The absorptance of the atmosphere with a homogeneous cloud layer is always greater than that with an inhomogeneous cloud layer for these cases (Fig. 2). As the mean optical thickness or *Î½* of the cloud layer increases, the absorptance of the atmosphere with an inhomogeneous cloud layer approaches to that with a homogeneous cloud layer. However, the absorption vertical profile can be different.

## Acknowledgments

We thank S. Weckmann for useful discussions. The work for the clouds and the Clouds and the Earth's Radiant Energy System (CERES) project was supported by the NASA Langley Research Center Grant NAG-1-1963 to Hampton University and Cooperative Agreement NCC-1-405 to Virginia Polytechnic Institute and State University.

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## APPENDIX

### Matrix Elements of the Domain-Averaged Irradiance for a Multilayer Atmosphere

*j*th layer, upward and downward irradiance for the layer are

*n*equations in 2

*n*unknowns. Each of these equations has four unknowns. Eliminating either

*Y*

_{1}and

*Y*

_{2}for one of the layers results in three unknowns in each equations and, thus, leads to a tridiagonal matrix similar to that given by Toon et al. (1989). Following their notation, the tridiagonal matrix is in the form

*l*= 1 are

*l*from 3 to 2

*N*âˆ’ 1 are

*l*from 2 to 2

*N*âˆ’ 2 are

*l*= 2

*N*the coefficients are

*R*

_{sfc}is surface albedo.

*j*th layer), they are

*C*

^{Â±}

_{j}

*Ï„*

_{j}) are replaced by (16) and (17).

Difference of the atmospheric absorptance for homogeneous cloud and inhomogeneous cloud shown as a function of mean optical thickness for distributions shown in Fig. 1a

Citation: Journal of the Atmospheric Sciences 58, 24; 10.1175/1520-0469(2001)058<3797:GWDOTS>2.0.CO;2

Difference of the atmospheric absorptance for homogeneous cloud and inhomogeneous cloud shown as a function of mean optical thickness for distributions shown in Fig. 1a

Citation: Journal of the Atmospheric Sciences 58, 24; 10.1175/1520-0469(2001)058<3797:GWDOTS>2.0.CO;2

Difference of the atmospheric absorptance for homogeneous cloud and inhomogeneous cloud shown as a function of mean optical thickness for distributions shown in Fig. 1a

Citation: Journal of the Atmospheric Sciences 58, 24; 10.1175/1520-0469(2001)058<3797:GWDOTS>2.0.CO;2