## 1. Introduction

For infrared radiation the scattering process is often neglected in climate models, since it is commonly believed that the scattering effect is small for the infrared. However, recent studies show that the infrared scattering by the cloud droplets could not be neglected in cloud cooling rates and surface radiation fluxes (Edwards and Slingo 1996). The reason for neglecting infrared scattering in climate models is that the scattering would make the calculation process much more complicated and dramatically increase the model’s computing time. This problem recently has been solved by Li and Fu (2000). Since scattering of infrared radiation does not occur in cloud-free layers, it would be beneficial to find a way to account for scattering in cloud layers only. The method proposed by Li and Fu (2000) avoids the scattering calculation in cloud free layers and is therefore computationally efficient.

If scattering is neglected in the infrared (referred to as the absorption approximation), the only cloud optical property needed in the radiative transfer calculation is the absorption coefficient. To avoid the complicated Mie calculation, a parameterization of the absorption coefficient in terms of physical quantities that can be generated in climate models is needed. Chýlek and Ramaswamy (1982) proposed a simple parameterization of the emittance for water clouds in the infrared. A more sophisticated scheme was proposed by Chýlek et al. (1992). If scattering is considered in the infrared, more optical properties, such as the single-scattering albedo and the asymmetry factor, are needed. Also, only the parameterization for single wavelengths is considered in the previous works. The conversion to band results has not been discussed.

Hu and Stamnes (1993) is the only work that gives a parameterization of cloud optical properties in the infrared with scattering effects considered. The differences between this work and that of Hu and Stamnes (1993) are threefold. First, the wavelengths chosen match with those of the parameterization for ice cloud in Fu et al. (1998). This makes it simple to generate a complete parameterization for both water and ice clouds. Second in Hu and Stamnes (1993) exponential functions are used to parameterize the cloud’s optical properties in terms of their effective radius. Exponential functions are computationally expensive and it is therefore preferable to avoid their use in GCMs. We will show that our parameterizations, based on rational functions, approximate well the exact Mie results while maintaining computational efficiency. Finally, in Hu and Stamnes (1993), the parameterizations are for three separate ranges of effective radius. In this work, a single parameterization covers effective radius from 2 to 40 *μ*m.

## 2. Theory and parameterization

*k*

^{λ}

_{ext}

*λ*is defined aswhere

*r*is the droplet size,

*n*(

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

*Q*

_{ext}is the efficiency for the droplet to scatter plus absorb radiation.

*λ, ω*

_{λ}, is the probability that a photon will survive a scattering event as opposed to being absorbed and is defined aswhere

*Q*

_{sca}is the scattering efficiency.

*P,*describes the angular distribution of scattered photons for scattering events. The normalized phase function for wavelength

*λ*is given bywhere

*θ*is the scattering angle, and

*i*

_{1}and

*i*

_{2}are the squares of the vertical and horizontal scattering amplitudes (Bohren and Huffman 1983), respectively. Most radiative transfer methods require various moments of the Legendre expansion of the phase function. For example, the two-stream approximation requires the asymmetry factor

*g*

_{λ}, which is one-third of the first moment of the Legendre expansion of the phase function. All of the optical properties parameterized here (

*Q*

_{ext},

*Q*

_{sca},

*P,*etc.) are obtained from Mie calculations.

*n*

*r*

*Ar*

^{α}

*e*

^{−βr}

*A, α,*and

*β*are constants, and

*r*is the radius of the cloud particle. The effective radius and the effective variance for a gamma distribution are given by

There will not be any dependence on effective variance in our parameterization. Observations showed that *υ*_{e} varies little (Chýlek and Ramaswamy 1982) and that the variations that do occur are not very significant when calculating the optical properties. It is therefore reasonable to use a constant value of *υ*_{e} = 0.172 in the Mie calculations (Dobbie et al. 1999). The constants of *α* and *β* in Eq. (4) can be defined by *r*_{e} and *υ*_{e} through Eqs. (5) and (6).

Usually, the extinction coefficient is normalized by dividing by the liquid water content. This gives *ψ*_{λ} = *k*^{λ}_{ext}*ψ*_{λ} is the specific extinction, and LWC is the liquid water content for the cloud. Since LWC is proportional to *A* in the gamma distribution, *A* is canceled out in the calculation of *ψ*_{λ}.

In Mie calculations for cloud optical properties, the refractive indices of water are taken from Hale and Querry (1984) and Downing and Williams (1975), and supplemented by Kou et al. (1993).

*g*

_{λ}is performed at an angular resolution of 0.5°. A resolution of 0.05

*μ*m is used for the droplet radius in calculations in Eqs. (1)–(3). We use simple rational functions to parameterize the cloud optical properties:

The coefficients are derived from least squares fitting to the Mie calculation values. Given the nature of the least squares fitting it should be noted that our parameterization is meant to be valid for effective radius in the region 2–40 *μ*m. The single-scattering coalbedo, 1 − *ω*_{λ}, is parameterized instead of the single-scattering albedo.

*y*

_{a}(

*x*;

*c*

_{1},

*c*

_{2}, . . . ,

*c*

_{M}) to fit a curve

*y*(

*x*), where

*c*

_{1},

*c*

_{2}, . . . ,

*c*

_{M}are

*M*adjustable parameters, and we fit

*N*data points (

*x*

_{i},

*y*(

*x*

_{i})),

*i*= 1, 2, . . . ,

*N,*the least square fitting process is to minimize the value ofby choosing the suitable parameters

*c*

_{1},

*c*

_{2}, . . . ,

*c*

_{M}(Press et al. 1992). Consequently, we define percentage error asThis is an averaged result for relative errors at

*N*fitting points.

In our calculations, the variable *x* is the effective radius *r*_{e} and *y*(*r*_{e}) represents the exact Mie calculated values. Here *N* is equal to 39, at an interval of 1 *μ*m for *r*_{e} from 2 to 40 *μ*m.

We present the parameterization for 36 individual wavelengths, the same as Fu et al. (1988). Coefficients for Eqs. (7)–(9) are presented in Tables 1–3 along with the percentage error for each wavelength.

We also present the band-averaged results, and fitting is done for 12 bands. The calculations are based on 203 wavelengths ranging from 4 to 1000 *μ*m. The parameterization formulas are the same as Eqs. (7)–(9).

*B*

_{λ}(

*T*) is the Planck function,

*T*is the temperature, and Δ

*λ*

_{i}is the spectral interval for band

*i.*Equation (10) is physically similar to the Chandraskhar mean (Liou 1992). The Chandraskhar mean requires a weighting by incoming radiative flux to the cloud. However, for the infrared the incoming radiative flux is not available till the radiative transfer process is performed.

We find that the band results of cloud infrared optical properties are not sensitive to the temperature for the Planck function. Figures 1a and 1b show the extent of the error that could be introduced by weighting the averages for each band by the Planck function with *T* at 180 and 320 K. The results for two bands are shown. Band 1 (4.54–5.26 *μ*m) is at the beginning of the infrared spectrum with a small band weight in radiative calculations. Band 11 (25.000–35.714 *μ*m) is at the center of the infrared spectrum with a large band weight in radiative calculations. The error introduced by weighting at different temperatures is found to be very small for all bands. Given these small errors it appears that there is little dependence on temperature in our weighting. All band results are derived by weighting the average with the Planck function at 275 K.

Figures 2a–c show the differences between exact values for the single-scattering properties and the curves that are obtained from the parameterization. The results of bands 1, 5, and 11 are presented. It is found that *ψ*^{i}_{band}*ω*^{i}_{band}*g*^{i}_{band}*r*_{e}. The behavior of cloud optical properties are particularly complicated for effective radii less than 5 *μ*m. However, the proposed simple parameterization scheme always approximates well the exact Mie results. Tables 1, 2, and 3 list the percentage error for the parameterizations for band-averaged results. It is found that errors are seldom greater than 5%.

*z*is the cloud layer thickness and (1 −

*ω*

_{λ})

*k*

^{λ}

_{ext}

*z.*Using our parameterization the absorptance for band

*i,*

^{i}

_{band}

*ω*

^{i}

_{band}

*ψ*

^{i}

_{band}

*i.*The precise method would be to calculate the absorption for each wavelength first and then calculate the weighted averages of the absorption over the band, in the following way:

The formulation for absorption in Eq. (13) is significantly more complicated than the one in Eg. (12). Figures 3a, 3b, and 3c (for bands 1, 5, and 11) show a comparison between absorption curves calculated using our parameterization and those using the formulation of Eq. (13) based on exact Mie calculation. The absorption curves are plotted versus the LWP. Here *r*_{e} = 2, 10, and 25 *μ*m are considered. The figures demonstrate that the band parameterization always gives an accurate approximation of the absorption curve. The worst case occurs for band 1 with *r*_{e} = 2 *μ*m, since in this case the errors for both *ψ*^{i}_{band}*ω*^{i}_{band}

## 3. Summary

This paper presents a parameterization of the optical properties of liquid water clouds for radiation in the infrared spectrum. The properties are parameterized in terms of the effective radius of their droplet size distribution and liquid water content. Parameterizations for 36 individual wavelengths in the infrared region are presented. Also, the results from the Mie calculations were averaged into 12 spectral bands. The parameterized curves approximate well the exact Mie values, with percentage errors mostly less than 5% for all the optical properties. Since Fu et al. (1998) has proposed a parameterization for ice cloud in the infrared, a corresponding parameterization for water clouds is necessary to define a complete parameterization for cloud optical properties in the infrared.

## Acknowledgments

The authors would thank Drs. N. A. McFarlane and J. Scinoca and anonymous reviewers for their helpful comments.

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Values of coefficients for the *ψ*_{λ} and *ψ*^{i}_{band} parameterizations.

Values of coefficients for the (1 − *ω*_{λ}) and (1 − *ω*^{i}_{band} parameterizations.

Values of coefficients for the *g*_{λ} and *g*^{i}_{band} parameterizations.