1. Introduction
Cirrus clouds, primarily present in the upper troposphere and lower stratosphere, are globally distributed and are composed almost exclusively of nonspherical ice crystals. Through their radiative effects, these clouds may modulate climate systems on all scales (e.g., Randall et al. 1989; Ramanathan and Collins 1991; Ramanathan et al. 1983; Ramaswamy and Ramanathan 1989;Fu et al. 1995; Starr and Cox 1985; Stephens 1983). Recent studies have shown that variations in assumed cirrus radiative properties can significantly alter the results of climate models (e.g., Ramanathan et al. 1983; Mitchel et al. 1988; Stephens et al. 1990). In his review article, Liou (1986) articulates that cirrus clouds are one of the most important yet least understood components in the climate system. The importance of understanding the radiative properties of cirrus clouds in weather and climate research has been recognized in view of several major field campaigns such as ICE (Raschke 1994), FIRE I (see Monthly Weather Review, Vol. 118, No. 11, 1990), and FIRE II (see Journal of the Atmospheric Sciences, Vol. 52, No. 23, 1995). One of the main conclusions drawn from these field experiments is that the asymmetry factor, which directly affects the reflection and transmission of cirrus clouds, is significantly smaller than that predicted from Mie theory in the solar spectral region (e.g., Francis 1995). However, it is still not clear if Mie theory, with its assumption of equivalent ice spheres, is adequate to describe the absorption and scattering associated with nonspherical particles in thermal infrared wavelengths.
Modeling and prediction of climate require that the parameterization of the radiative effects of cirrus clouds be as accurate as possible. However, to date, there is no exact theoretical solution for scattering and absorption by nonspherical ice particles (Liou and Takano 1994). In the solar spectrum, the geometric optics method (GOM) (e.g., Takano and Liou 1989; Macke et al. 1996), including an improved version (Yang and Liou 1996a), has been widely used to evaluate single-scattering properties of cirrus clouds. This method is an asymptotic approach that becomes accurate as the size parameter increases. In view of the observed ice crystal sizes in cirrus clouds, the geometric ray-tracing technique should be valid for solar wavelengths (Takano and Liou 1989). Based on optical parameters derived from the improved geometric ray-tracing method, Fu (1996) has developed an accurate parameterization of the solar radiative properties of cirrus clouds. However, this method may not be appropriate for thermal infrared wavelengths where ice crystals with small size parameters can be important in determining cirrus cloud radiative properties.
It is common practice to use Mie theory to approximate the absorption and scattering by nonspherical ice crystals in the infrared (e.g., Stephens et al. 1990; Ebert and Curry 1992; Sun and Shine 1995). The radiative properties of cirrus clouds in the infrared are dominated by absorption although the effects of scattering cannot be neglected (Fu et al. 1997). For this reason, the use of Mie theory to evaluate the scattering coefficient and asymmetry factor for ice crystals may not present a serious problem (Fu et al. 1998, manuscript submitted to J. Atmos. Sci.). However, using Mie theory to specify the absorption coefficient, which is the key parameter describing the infrared radiative properties of cirrus clouds, is questionable. Mitchell (1995) argued that not all of the physics that Mie theory embraces is applicable to nonspherical particles.
An alternative method of calculating extinction and absorption coefficients for ice crystals is the anomalous diffraction theory (ADT) (e.g., Ackerman and Stephens 1987; Chylek and Klett 1991; Mitchell and Arnott 1994). Chylek and Videen (1994) found that the accuracy of the ADT depends on the real part of the ice refractive index. Mitchell et al. (1996a) claimed that the ADT is a much better approximation than Mie theory for nonspherical particles. However, by comparing results from the ADT with those from the discrete dipole approximation, Maslowska et al. (1994) concluded that the ADT cannot be used for scattering and absorption by nonspherical particles without verification with rigorous methods. Although the ADT can be improved by considering the effects of refraction and reflection (Ackerman and Stephens 1987; Mitchell et al. 1996a), the resulting absorption coefficients should not be better than those calculated from the geometric ray-tracing method (Fu et al. 1998, manuscript submitted to J. Atmos. Sci.).
Fu and Liou (1993) developed a parameterization of cirrus cloud radiative properties. In the infrared spectrum, the optical parameters for nonspherical ice crystals with size parameters greater than 30 were computed using the geometric ray-tracing technique. For size parameters less than 30, Mie-type solution for spheroids was used. Although spheroids may be a better approach than the spherical approximation of nonspherical ice crystals, further inquiry into this problem appears necessary.
Recently, Yang and Liou (1996b) employed the finite-difference time domain (FDTD) method for light scattering by small ice crystals. The FDTD technique solves the Maxwell equations numerically in the time domain by using the finite-difference analog, which can be applied to nonspherical particles accurately. This numerical approach, however, can be used only for particles with small size parameters in practice because of significant computational efforts. Yang and Liou (1996b) showed that the FDTD can achieve an accuracy of ∼1% for size parameters smaller than 10.
In the present study, a composite method to determine single-scattering properties of hexagonal ice crystals in the infrared is developed by comparing results from the Mie, ADT, GOM, and FDTD. Following Fu (1996), an accurate parameterization for the infrared radiative properties of cirrus clouds is developed based on optical parameters derived from this composite method. The present parameterization complements the scheme for the solar radiative properties of cirrus clouds developed by Fu (1996) for use in climate models. In section 2, the calculation of single-scattering properties of ice crystals, including the development of the composite scheme, is discussed. The parameterization of these properties following Fu (1996) is shown in section 3. In section 4, the present parameterization is validated by examining bulk radiative properties. Summary and conclusions are given in section 5.
2. Determination of single-scattering properties of ice crystals
The radiative properties of cirrus clouds required for radiative transfer calculations in climate models are the scattering and absorption coefficients and the asymmetry factor. The computation of these single-scattering properties involves detailed information of ice particle shapes and their size distributions, knowledge of the refractive indices of ice, and a light-scattering program.
a. Ice crystal size distributions
The 28 ice crystal size distributions adopted in Fu (1996) are used in the present study, which were based on in situ aircraft observations from both midlatitude and tropical regions.
b. A composite light-scattering program
For calculations of the scattering and absorption properties of nonspherical ice crystals, a composite scheme is developed by comparing results based on the Mie, ADT, GOM, and FDTD.
The single-scattering calculations are performed for randomly oriented hexagonal ice crystals. The Mie code is from Wiscombe (1979). The geometric ray tracing program and finite-difference time domain program are based on Yang and Liou (1996a and 1996b). We have developed the anomalous diffraction scheme for randomly oriented hexagonal ice crystals (Fu et al. 1998b, manuscript submitted to J. Atmos. Sci.). In the single-scattering calculations, the refractive indices for ice from Warren (1984) and Gosse et al. (1995) are used.
Figure 1 shows the comparison of absorption efficiency from different methodologies as a function of the size parameter for the wavelengths of 3.7 μm in (a) and 12.99 μm in (b). The refractive indices are 1.40 + 0.0072i and 1.47 + 0.389i for the wavelengths of 3.7 and 12.99 μm, respectively. The different dependence of absorption efficiency on size parameters between Fig. 1a and 1b is due to the difference in the refractive indices. In Fig. 1a, the aspect ratio D/L used is ⅓ and the FDTD results are taken from Yang and Liou (1996a). In Fig. 1b, the aspect ratio used follows Eq. (2.1); the hexagonal particle with a length of 300 μm corresponds to the size parameter of ∼28. From Fig. 1, we see that the absorption efficiency is not sensitive to the particle shape for large size parameters for which the Mie results using different equivalent spheres converge to GOM solutions. The absorption efficiency from ADT approaches one for large size parameters. This is because ADT does not consider the external reflection. The difference between ADT and GOM for smaller size parameters is evident because ADT neglects refraction and internal reflection. At 3.7 μm where ice absorption is quite weak (Fig. 1a), the absorption efficiency is overestimated by using spheres with the equivalent projected area or volume; in both cases, the ratio of V/P is overestimated. Figure 1a also shows that GOM and Mie theory using rvp obtain the best results compared with reference calculations from FDTD. Significant errors are produced by ADT because of neglecting reflection and refraction. At 12.99 μm, where the absorption is quite strong (Fig. 1b), the results from FDTD show that the absorption efficiency of nonspherical ice crystals can be significantly larger than one. This is the so-called tunneling effect, that is, the incident radiation outside the projected area of particles can be absorbed (Guimaraes and Nussenzveig 1992). The tunneling effect can be regarded as the difference between exact results and those from GOM. The Mie results seem to overestimate the tunneling effect associated with nonspherical particles while the ADT and GOM totally ignore this physics. Numerical results demonstrate that the tunneling effect strongly depends on refractive indices including both the real part, mr, and imaginary part, mi. The tunneling effect increases as mr or mi increases. It vanishes when mi approaches zero. However, the tunneling effect does not necessarily vanish when mr = 1.0, which is different from the result predicted using complex angular momentum theory (Guimaraes and Nussenzveig 1992).
For hexagonal ice crystals with size parameters smaller than 10, the asymmetry factor can be obtained from Mie theory for volume equivalent spheres. For size parameters larger than 10, we use the asymmetry factor from GOM. The extinction efficiency is simply derived as the mean value of results from ADT for hexagonal ice crystals and Mie in which both the projected area and volume are preserved.
Figure 2 shows the absorption efficiency for randomly oriented hexagonal ice crystals determined from the composite scheme as a function of size parameter for eight different wavelengths. Also shown is the absorption efficiency predicted from FDTD. The relative differences are generally smaller than ∼3%. There are two points at the wavelength of 18.18 μm and one point at 12.2 μm having the relative differences larger than 5% when the absorption efficiency is small. On the basis of these comparisons and considering the wide range of refractive indices used, we conclude that the composite scheme is generally valid for calculations of the absorption efficiency of hexagonal ice crystals.
Figures 3 and 4 provide comparison of the composite method and FDTD for calculations of asymmetry factor and extinction efficiency, respectively. The results are shown for three wavelengths. For the asymmetry factor, the relative differences are generally smaller than 2% when the asymmetry factor is larger than 0.5. The relative differences in extinction efficiency are usually smaller than 15%.
c. Single-scattering properties of ice clouds
Using the composite light-scattering and absorption scheme, the single-scattering properties of cirrus clouds are derived at 36 wavelengths from 4 to 100 μm (see Table 1). This spectral resolution is fine enough for the calculation of broadband radiative fluxes and heating rates. The 28 ice crystal size distributions are incorporated into the computation of the single-scattering parameters.
Figure 5 shows the single-scattering properties as a function of wavelength for six particle size distributions. The single-scattering properties have strong variation with both wavelength and cloud microphysical properties.
3. Parameterization of the single-scattering properties of ice clouds
The single-scattering properties of cirrus clouds are parameterized at 36 wavelengths in the infrared spectrum. The infrared spectrum is also divided into 12 bands as in the previous study (Fu and Liou 1993). In this case, the cirrus single-scattering properties obtained from the composite scheme are weighted with the Planck function (T = −40°C) to obtain average values in each spectral band (Fu 1991). Tables 1a–c list the wavelengths and band limits and values of the coefficients in Eqs. (3.1)–(3.3), respectively. In the present parameterization, the units for IWC and Dge are in grams per cubic meter and microns, respectively, and the units for β are inverse meters.
Figure 6 shows the fitting for β/IWC, βa/IWC × Dge, and g in the spectral intervals 1250–1100 cm−1 (upper panel), 980–800 cm−1 (middle panel), and 670–540 cm−1 (lower panel). The averaged relative errors shown in the figure are defined as (100%/n) Σn |(xf − xe)/xe|, where n = 28 and xe and xf are “exact” values and those from the numerical fitting, respectively. The maximum absolute value of relative error is also shown in the figure. The averaged relative differences between the fitted and reference calculations are less than ∼5%, ∼5%, and 1% for β/IWC, βa/IWC × Dge, and g, respectively. It should be noted that the linear relationship between β/IWC and 1/Dge, which is a very good approximation in the solar spectrum (Fu 1996), is no longer valid for the infrared.
The scattering of thermal infrared radiation by clouds is often neglected in climate studies. This would introduce an increase of the outgoing longwave fluxes at the top of the atmosphere by ∼6 W m−2 in cloudy atmospheres (Fu et al. 1997). The present parameterization can be directly incorporated into radiation models to consider multiple scattering processes associated with cirrus clouds.
4. Accuracy of bulk radiative properties
The comparisons have been performed between the bulk broadband radiative properties obtained from the present parameterization (both for 36 wavelengths and 12 bands) of single-scattering parameters and those from the composite method at 36 wavelengths. Here 28 types of ice clouds are used with different thicknesses (0.5, 1.0, 2.0, and 4.0 km) to cover a wide range of atmospheric conditions.
Figure 7 shows the absolute values of relative errors due to the 36-wavelength parameterization versus the reference broadband absorptivity, transmissivity, and reflectivity. Each figure contains 112 points, representing 28 types of ice clouds with four different thicknesses. The averaged relative errors in absorptivity, transmissivity, and reflectivity are 2.2%, 3.3%, and 8.8%, respectively. Since thermal emission from the cloud layer is the dominating radiative process, the magnitude of absorptivity, that is, emissivity, must be determined accurately. Figure 7 shows that the relative errors in the absorptivity are less than about 5%. For transmissivity, the large relative errors occur when the transmissivity is small so that the errors in radiative energy may not be large. Since the reflectivity, as shown in Fig. 7c, is always smaller than ∼0.06, the accuracy of ∼10% is satisfactory.
Figure 8 is the same as Fig. 7 except for the 12-band parameterization of single-scattering parameters. From Figs. 7 and 8, it is clear that the accuracy of the 12-band parameterization is similar to that obtained with the 36-wavelength parameterization. This suggests that the accuracy of the parameterization is insensitive to the spectral resolution.
The errors in bulk cloud radiative properties due to the present parameterization are small compared with those produced by other error sources (Fu 1996). These error sources include radiative transfer approximations and uncertainties in refractive indices and atmospheric input parameters. The accuracy of this parameterization guarantees its reliability in the applications to climate studies. The present parameterization complements the scheme for the solar radiative properties of ice clouds developed by Fu (1996) for use in numerical models.
5. Summary and conclusions
In the present study, a composite methodology to determine single-scattering properties of hexagonal ice crystals in the infrared is developed by comparing results from the Mie theory, anomalous diffraction theory, geometric optics method, and the finite-difference time domain technique. The absorption efficiency from this scheme yields relative accuracies within ∼3%. It is found that for size parameters smaller than ∼30, Mie theory using equivalent ice spheres tends to overestimate the absorption efficiency of nonspherical ice crystals, while the ADT and GOM underestimate it.
An accurate parameterization of the infrared radiative properties of cirrus clouds is developed based on improved light-scattering calculations. Here 28 ice crystal size distributions from in situ aircraft observations in both tropical and midlatitude regions are employed. These size distributions are extrapolated to 1.0 μm to include the effects of small ice crystals. Thirty-eight size bins are used to resolve the detailed structure of the observed particle size distributions. The single-scattering properties of hexagonal ice crystals are calculated by using the composite scheme that can produce accurate results for a wide range of size parameters.
Following Fu (1996), a generalized effective size, Dge, is defined to account for the ice crystal size distribution in the radiative calculations. Based on physical principles, the single-scattering properties have been parameterized in terms of ice water content and Dge. The generalized effective size can be related to the total cross-sectional area of ice particles per unit volume, a quantity directly measured by the 2D optical probe in in situ microphysical observations of cirrus clouds.
The present parameterization of single-scattering properties of cirrus clouds is evaluated by examining the bulk radiative properties. Here 28 types of cirrus clouds with four different thicknesses are used to cover a wide range of atmospheric conditions. Compared with reference results, the typical relative error due to the parameterization is ∼2.2% in cirrus cloud emissivity. This error is small when compared with those produced by other error sources. The accuracy of this parameterization guarantees its reliability in the applications to climate studies. The present parameterization complements the scheme for the solar radiative properties of ice clouds developed by Fu (1996) for use in numerical models.
Acknowledgments
The authors wish to thank Drs. P. Chylek, G. Videen, and G. Lesins for useful discussions. Many thanks to M. Cribb for help in preparing the manuscript. The research work contained herein has been supported by DOE Grant DE-FG02-97ER62363, and in part by an NSERC operating grant, an AES Science Subvention grant, and an NSERC collaborative grant.
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Comparison of absorption efficiencies for randomly oriented hexagonal ice crystals derived from different scattering programs: Mie theory for sphere with equal projected area (········),equal volume (—), both equal area and volume (____), anomalous diffraction theory(–·–), geometric optics method (–··–), and finite-difference time domain technique (×). The results are shown as a function of size parameter for wavelengths of 3.7 μm (a) and 12.99 μm (b). In (a) the aspect ratio, D/L, used is ⅓; in (b) the aspect ratio follows Eq. (2.1).
Citation: Journal of Climate 11, 9; 10.1175/1520-0442(1998)011<2223:AAPOTI>2.0.CO;2
Comparison of the absorption efficiency derived from the composite method with the result from the finite-difference time domain technique for randomly oriented hexagonal ice crystals at different wavelengths. The results are shown as a function of size parameter and the aspect ratio used follows Eq. (2.1).
Citation: Journal of Climate 11, 9; 10.1175/1520-0442(1998)011<2223:AAPOTI>2.0.CO;2
As in Fig. 2 but for the asymmetry factor and results are only shown for three wavelengths.
Citation: Journal of Climate 11, 9; 10.1175/1520-0442(1998)011<2223:AAPOTI>2.0.CO;2
As in Fig. 3 but for the extinction efficiency.
Citation: Journal of Climate 11, 9; 10.1175/1520-0442(1998)011<2223:AAPOTI>2.0.CO;2
Single-scattering properties including the extinction coefficient, absorption coefficient, and asymmetry factor as a function of wavelength for six cirrus clouds.
Citation: Journal of Climate 11, 9; 10.1175/1520-0442(1998)011<2223:AAPOTI>2.0.CO;2
Extinction coefficient/IWC, absorption coefficient/IWC × Dge, and asymmetry factor vs the generalized mean effective size for the spectral intervals 1250–1100 cm−1 (upper panels), 980–800 cm−1 (middle panels), and 670–540 cm−1 (lower panels). The “○” points represent reference results from the light-scattering program for 28 measured ice crystal size distributions, and the curves are from the parameterization.
Citation: Journal of Climate 11, 9; 10.1175/1520-0442(1998)011<2223:AAPOTI>2.0.CO;2
Absolute values of relative errors due to the 36-wavelength parameterization of single-scattering parameters in absorptivity (a), transmissivity (b), and reflectivity (c) versus reference calculations. The 112 points in each figure represent results of 28 cirrus clouds with four thicknesses (0.5, 1.0, 2.0, 4.0 km). The reference absorptivity, transmissivity, and reflectivity are based on single-scattering properties from the composite method.
Citation: Journal of Climate 11, 9; 10.1175/1520-0442(1998)011<2223:AAPOTI>2.0.CO;2
As in Fig. 7 but for absolute values of relative errors due to the 12-band parameterization of single-scattering parameters.
Citation: Journal of Climate 11, 9; 10.1175/1520-0442(1998)011<2223:AAPOTI>2.0.CO;2
Table 1a. Values of coefficients in Eq. (3.1) for the parameterization of the extinction coefficient (m−1). (The units for Dge and IWC are in μm and gm−3, respectively.)
Table 1b. Values of coefficients in Eq. (3.2) for the parameterization of the absorption coefficient (m−1). (The units for Dge and IWC are in μm and gm−3, respectively.)
Table 1c. Values of coefficients in Eq. (3.3) for the parameterization of the asymmetry factor. (The units for Dge are in μm.)