The aspect ratio (AR) of a nonspherical ice particle is identified as the key microphysical parameter to determine its asymmetry factor for solar radiation. The mean effective AR is defined for cirrus clouds containing various nonspherical ice particles. A new parameterization of the asymmetry factor of cirrus clouds in terms of AR and mean effective size, Dge, is developed for solar radiation. It is based on geometric ray-tracing calculations for hexagonal ice crystals with a simple representation of particle surface roughness. The present parameterization well reproduces the asymmetry factors of complicated ice particles such as bullet rosettes, aggregates with rough surfaces, and fractal crystals and agrees well with observations. It thus can be properly applied to cirrus clouds containing various nonspherical ice particles. The asymmetry factor from this parameterization in the visible spectrum ranges from about 0.73 to more than 0.85.
Radiative transfer calculations show that for a cirrus cloud with an optical depth of 4 and a solar zenith angle of 60°, changes in AR from 1.0 to 0.5 or from 1.0 to 0.1 result in differences in reflected solar fluxes of about −30 or −70 W m−2, respectively. For the same cloudy conditions, the effect of ice particle surface roughness on the reflected solar flux is found to be about 20 W m−2.
A parameterization of the solar radiative properties of cirrus clouds was developed by Fu (1996) based on light scattering calculations for hexagonal ice particles. Using ice water content (IWC) and generalized effective size (Dge), which conserve both volume and projected area of nonspherical ice particles, the parameterization for the extinction coefficient and single-scattering albedo can be properly applied to cirrus clouds containing various nonspherical particles such as plates, columns, bullet rosettes, and aggregates (Fu 1996; Grenfell and Warren 1999; Edwards et al. 2007). However, the parameterization of the asymmetry factor in terms of Dge is still sensitive to the habit and shapes of ice crystals assumed in the light scattering calculations (Fu 1996; Edwards et al. 2007).
The asymmetry factor, g, is the first moment of the scattering phase function. The quantities (1 − g)/2 and (1 + g)/2 can be interpreted as the integrated fraction of energy backscattered and forward scattered, respectively. The global albedo of clouds in the visible, based on a two-stream radiative transfer equation (Liou 2002), is
where τ is the cloud optical depth. The change in the albedo, ΔR, due to a change in the asymmetry factor, Δg, can then be derived in the following form:
Thus, for a typical cirrus asymmetry factor of 0.8 and Δg of ±0.05, the differences in albedo are about ∓0.03, ∓0.06, and ∓0.06 for the optical depth of 1, 4, and 10, respectively. Modeling and prediction of climate therefore require that the parameterization of the asymmetry factor of cirrus clouds be as accurate as possible.
The asymmetry factors of cirrus clouds derived from observations are 0.7 from Stephens et al. (1990), 0.75 from Garrett et al. (2001), and 0.8 from Francis et al. (1994). Theoretical calculations of asymmetry factors for ice particles in the solar radiation are usually based on the geometric ray tracing technique (e.g., Takano and Liou 1989; Macke 1993; Yang and Liou 1995). Different parameterizations of asymmetry factors in terms of mean effective size of ice particles have been developed and used in climate models (e.g., Ebert and Curry 1992; Fu 1996; Macke et al. 1996; Chou et al. 2002; McFarquhar et al. 2002; Edwards et al. 2007). Shown in Fig. 1 are the asymmetry factors in the visible spectrum from these parameterizations, which assumed different habits and shapes and ice particle size distributions. We see different dependences of g on Dge, and for a given Dge the asymmetry factor ranges from 0.74 to more than 0.81. This indicates that the mean effective size may not be an appropriate microphysical parameter being used for the parameterization of the asymmetry factor. In this study, we identify a different microphysical parameter, the aspect ratio (AR) of ice particles, to determine the asymmetry factor in the solar radiation. We show that g also depends on Dge in the near-IR when the absorption becomes significant.
In section 2 we define a mean effective AR to represent its impact on the asymmetry factor of cirrus clouds. In section 3 we develop a new parameterization of the asymmetry factor in terms of the effective aspect ratio and Dge based on light scattering by hexagonal ice particles. The new parameterization is tested in section 4 to show that it can be properly applied to complicated ice particle shapes. More discussions on this parameterization are given in section 5. Section 6 presents the effect of mean effective aspect ratio and ice particle surface roughness on the cirrus radiative energy budget. A summary and conclusions are given in section 7.
2. Definition of mean effective aspect ratio
The asymmetry factor of hexagonal ice particles in the solar spectrum is dominated by its aspect ratio, D/L, where D and L are the particle width and length, respectively. Figure 2 shows the g of hexagonal ice crystals and delta-transmission fraction associated with light beams passing through the parallel planes of the ice crystals (Takano and Liou 1989) versus D/L based on the geometric optics method at a wavelength of 0.56 μm. We see that the asymmetry factor is very sensitive to the particle aspect ratio, ranging from 0.77 to above 0.9 with the minimum value when D/L is about one.
Figure 3 shows the asymmetry factor and the delta-transmission fraction versus ice particle length L for a D/L of 0.5 at wavelengths of 0.56 and 1.50 μm. In the visible, both quantities are independent of the ice particle size for given D/L. This is actually true for all wavelengths shorter than about 1.4 μm (about 85% of solar energy is contained in this interval) where the ice absorption is small (Fu 1996). Dependence of the asymmetry factor on the Dge in the visible, as shown in Fig. 1, can be largely explained by the dependence of the aspect ratio on the ice particle size assumed in the parameterization.
For hexagonal ice particles in the visible, the asymmetry factor is uniquely determined by the aspect ratio (see Fig. 2), but the aspect ratio (thus the asymmetry factor) is not uniquely determined by the ice particle size. Thus it is more suitable to parameterize the asymmetry factor of nonspherical ice particles in terms of aspect ratio instead of Dge. Here we define the mean effective aspect ratio for given ice particle size distributions in the form
where D and L are the width and length of the hexagonal component, Ac is its projected area, and n(L) is the ice crystal size distribution. The mean aspect ratio can also be defined for more complicated shapes such as bullet rosettes and aggregates by considering their individual components.
Asymmetry factors for more complicated shapes, such as hollow columns, bullet rosettes, and aggregates (Takano and Liou 1995; Yang and Liou 1998), are within the same range shown in Fig. 2, and these particle types essentially behave like their single components (Iaquinta et al. 1995; Macke et al. 1996). Thus, the parameterization of the asymmetry factor in terms of AR based on light scattering by hexagonal ice particles is expected to be generally applied to cirrus clouds containing various nonspherical ice particles.
The natural ice particles in cirrus clouds may have rough surfaces, which should also be taken into account (e.g., Macke et al. 1996; Yang and Liou 1998). The ice particles with rough surfaces should not have the delta transmission associated with light beams passing through the parallel planes of ice crystals. Herein we simply represent the effect of ice particle surface roughness on g by removing the contribution of the delta transmission to g and assuming that the removed part has the same asymmetry factor as the remaining part.
For wavelengths longer than 1.4 μm, the asymmetry factor may depend not only on D/L but also on the ice particle size. Figure 3 shows that at 1.5 μm, where ice exhibits an absorption peak, the asymmetry factor increases with the ice particle size for a given D/L. This phenomenon can be explained as follows. In the geometric optics limit, the scattering phase function is composed of the diffraction peak and geometric reflection and refraction including the delta transmission. The fraction of diffracted rays is ½ω̃ (Fu 1996) where ω̃ is the single-scattering albedo. Since the diffracted rays are largely confined in the forward direction, their asymmetry factor can be approximated as one. Thus the total asymmetry factor, g, can be generally expressed as
where g′ is the asymmetry factor associated with the geometric reflection and refraction. At wavelengths with strong absorption, as the single-scattering albedo decreases with the increase of ice particle size (Fu 1996), the asymmetry factor increases following Eq. (2.2). We derive the asymmetry factor at the wavelength of 1.5 μm based on Eq. (2.2) using a constant g′ of 0.68, which is shown as a dashed line in Fig. 3. It indicates that the dependence of g on L is largely through the dependence of ω̃ on L and the dependence of g′ on L has small impact on g. Note that ½ω̃ is 0.50 for wavelengths shorter than about 1.4 μm (see Fig. 3b in Fu 1996).
3. New parameterization of the asymmetry factor
To provide a reliable parameterization of the asymmetry factor of cirrus clouds, we use 28 ice crystal size distributions based on in situ aircraft observations compiled by Fu (1996). For light scattering calculations, the size distributions are discretized into 30 bins ranging from 8.75 to 3100 μm to resolve the structure of the size spectra. For hexagonal columns, the relationship between L and D follows Eq. (2.1) of Fu (1996). We thus have the mean effective aspect ratios for the 28 size distributions ranging from 0.34 to 0.99, ice water content from 5.5 × 10−4 to 0.28 g m−3, and Dge from 18.6 to 139.2 μm. For hexagonal plates, we use the relationship between L and D following Pruppachar and Klett (1997), given by L = 1.7915D0.474. The resulting ARs are from 3.14 to 16.68, ice water content from 2.4 × 10−4 to 0.28 g m−3, and Dge from 11.6 to 77.4 μm.
The single-scattering properties of hexagonal ice crystals are calculated using a geometric ray-tracing program (Takano and Liou 1989; Yang and Liou 1995). The ice crystals are assumed to be randomly oriented in space. We use the refractive indices of ice following Fu (1996). The single-scattering calculations are performed at 200 wavelengths. The solar spectra are divided into six bands (Fu 1996) and the data are weighted with the solar irradiance to obtain average values in each spectral band. The single-scattering properties for cirrus clouds containing hexagonal columns are the same as in Fu (1996) and we only need to calculate those for hexagonal plates. Following Fu (1996), the contribution of delta-function transmission through parallel planes at the zero-scattering angle is derived based on the conventional ray-tracing calculations (Takano and Liou 1989). Note that using the improved ray-tracing technique we cannot separate the contributions of the delta transmission as well as diffracted rays, but such separations are important in the radiative flux transfer calculations involving cirrus clouds (Fu 1996).
The asymmetry factor in the visible ranges from 0.766 to 0.836 for the 28 cirrus clouds containing ice columns and from 0.833 to 0.938 for ice plates. We derive asymmetry factors for cirrus clouds containing ice particles with rough surfaces, gr, using gr = (gs − fd)/(1 − fd), where gs is the asymmetry factor for ice particles with smooth surfaces and fd is the delta-transmission fraction. The gr ranges from 0.734 to 0.806 for the 28 cirrus clouds containing ice columns and from 0.783 to 0.898 for ice plates. After obtaining gs and gr along with the single-scattering albedo for given ice particle size distributions, we can derive g′s and g′r using Eq. (2.2).
We parameterize g′ in terms of AR in the forms
for 0.1 ≤ AR ≤ 1.0 and
for 1.0 < AR ≤ 20, where the coefficients are obtained by numerical fitting to the data for the 28 cirrus clouds containing ice columns and plates, respectively. Thus, for given AR and Dge, we can derive the asymmetry factor using Eqs. (3.1)–(3.4) and Eq. (2.2) where ω̃ is parameterized in terms of Dge from Fu (1996). Note that the obtained asymmetry factor is independent of Dge for wavelengths shorter than 1.4 μm.
We parameterize the delta-transmission fraction, fd, in the following form:
for 0.1 ≤ AR ≤ 1.0 and
for 1.0 < AR ≤ 20, where the coefficients are again obtained by numerical fitting to the data for the 28 cirrus clouds. The coefficients in Eqs. (3.1)–(3.6) are provided in Tables 1 and 2 for 0.1 ≤ AR ≤ 1.0 and 1.0 < AR ≤ 20, respectively. Note that dc1 and dp1 are zero for the bands 0.2–0.7 and 0.7–1.41 μm.
Comparisons of the parameterizations and the reference calculations are shown for gs and gr versus AR in Fig. 4 and for fd versus AR in Fig. 5, in the band of 0.2–0.7 μm, where g and fd are only functions of AR. The agreements are similar for the other five bands. Figure 6 shows gs, gr, and fd from parameterizations versus those from reference calculations for 28 ice particle size distributions using both columns and plates for all six bands in the solar radiation. The parameterization in terms of AR along with Dge well represents the variation of asymmetry factor and the delta-transmission fraction, with an averaged absolute relative error of about 0.4% for gs and gr and 3% for fd. The parameterizations can be used for AR ranging from 0.1 to 20.
In the radiative transfer calculations, the fraction of scattered energy residing in the forward peak, f, needs to be removed from the scattering parameters to incorporate the strong forward peak contribution in multiple scattering. We use
for ice particles with smooth surfaces (Fu 1996) but
for ice particles with rough surfaces. The single-scattering properties are then adjusted following the similarity principle for radiative transfer calculations (e.g., Liou 2002).
4. Test of new parameterizations
The discussion in section 2 indicates that the parameterization of asymmetry factor in terms of AR developed in section 3 can be generally applied to nonspherical ice particles with more complicated shapes. Herein we test this parameterization for fractal particles (Macke et al. 1996), aggregates, and spatial bullet rosettes (Yang et al. 2000), and with observations (Garrett et al. 2001), focusing on the visible radiation.
Considering randomized Koch fractals for irregular shaped ice particles, Macke et al. (1996) obtained an asymmetry factor of 0.74. Such particles have rough surfaces and their individual components have an aspect ratio of about one.
Yang et al. (2000) constructed the aggregates using eight hexagonal elements with different given aspect ratios (Yang and Liou 1998). In their numerical computation, the number of hexagonal elements of which an aggregate consists is randomly selected from two to eight to average the effect of aggregate orientations. The selected hexagonal elements are attached together without overlap. The mean effective aspect ratio is 0.67 for their aggregates. Yang et al. (2000) considered the effect of rough surface by using a two-dimensional Gaussian probability function. They obtained an asymmetry factor of 0.76.
Figure 7 shows the comparison of asymmetry factors of fractal ice particles and aggregates with those for ice particles with rough surfaces from our parameterization. The differences are smaller than 0.01. Such an agreement justifies our simple representation of ice particle surface roughness for its effect on the asymmetry factor.
Yang et al. (2000) also gave the asymmetry factor for spatial bullet rosettes composed of multiple (randomly selected from two to six) bullet branches with smooth surfaces (Yang and Liou 1998). The typical values of their asymmetry factors are also compared with our parameterizations in Fig. 7, which shows differences of less than about 0.01.
Garrett et al. (2001) analyzed direct airborne measurements of the asymmetry factor of cirrus clouds. With and without delta-function transmission, the derived values of g were 0.76 ± 0.03 and 0.74 ± 0.03, respectively. The majority of ice particles from the Cloud Particle Imager (CPI) were bullet rosettes (the average number of branches were six) and the mean aspect ratio was 0.4 (Garrett et al. 2001). The gs and gr for an AR of 0.4 from our parameterizations are 0.828 and 0.797, which are significantly larger than those from observations. [The asymmetry factor of the bullet rosette with six branches and an aspect ratio of 0.4 is 0.837 based on the geometric ray-tracing method from Iaquinta et al. (1995).] It should be noted that the mean aspect ratio in Garrett et al. (2001) was underestimated since the CPI missed small ice particles. In the cases described in Garrett et al. these small crystals were responsible for ∼38% of the extinction coefficient. Assuming a mean effective aspect ratio of 1.0 for small ice particles, the AR with the consideration of small ice particles should be 0.63 (i.e., 1.0 × 0.38 + 0.4 × 0.62). Thus, we have a gs of 0.798 and a gr of 0.766, which agree with observed asymmetry factors within observational uncertainty.
It can be concluded that the parameterization of the asymmetry factor from this study along with the parameterizations of the extinction coefficient and single-scattering albedo from Fu (1996) can be generally applied to cirrus clouds containing various nonspherical ice particles. New research efforts need to be initiated in theoretical and modeling and observational studies to parameterize/diagnose AR for climate models. The observational studies require both in situ observational data analysis and the development of satellite remote sensing of ice crystal shape information in terms of AR that is not a parameter currently available from operational satellite retrieval products.
5. More discussions on the new parameterization
The asymmetry factor parameterization developed in this study is more general than previous parameterizations because of the following aspects. First, the new parameterization is generally valid for solid hexagonal ice particles including columns, plates, and their assemblages. Second, the new parameterization reproduces the asymmetry factor of nonhexagonal fractal crystals (Macke et al. 1996), which has also been widely used in climate and remote sensing studies. Third, the new parameterization can reproduce various previous parameterizations (e.g., those shown in Fig. 1) by considering the assumed AR–Dge relationship and surface roughness. In addition, the new parameterization agrees well with in situ observations.
However, it should be noted that we did not consider the hollow ice crystals that may have different asymmetry factors for a given aspect ratio (Takano and Liou 1995; Yang et al. 2000; Schmitt et al. 2006). Schmitt et al. showed that for a given hollowness of 0.5 the asymmetry factors of bullet rosettes range from ∼0.79 to ∼0.86 due to the aspect ratio changes, while for a given aspect ratio of 0.5 the asymmetry factors are from ∼0.79 to ∼0.83 due to the change of hollowness. The impact of hollow ice crystals on cirrus asymmetry factor also depends on their percentage in total ice particles. Further research effort is required to consider such impact in the asymmetry factor parameterization.
It should also be noted that the ice particle surface roughness is another independent parameter to determine the asymmetry factor (Macke et al. 1996; Yang and Liou 1998). We have developed the new parameterization separately for ice particles with smooth and rough surfaces. Our simple consideration of ice particle surface roughness is justified by comparing the asymmetry factors of ice particles with rough surfaces recommended in Macke et al. (1996) and Yang et al. (2000).
There are several advantages for incorporating our new parameterization of the cirrus asymmetry factor in climate models over those shown in Fig. 1. First, the new parameterization is more general and physically based. Second, the new parameterization can go back to various previous parameterizations by specifying an assumed AR–Dge relationship [e.g., AR = 1.236 − 1.526 × 10−2Dge + 1.004 × 10−4D2ge − 2.750 × 10−7D3ge from Fu (1996) and AR = 0.67 with rough surfaces from Edwards et al. (2007)]. (Edwards et al. showed that the differences in cirrus reflected solar radiation using different cirrus radiative properties parameterization are mainly due to the differences in the asymmetry factor parameterization.) Third, the AR is a critically important parameter to consistently quantify the cirrus radiative forcing since it only has impact on the cirrus shortwave radiative forcing but has little impact on the longwave forcing (different from Dge and IWC that have an impact on both). (Note that cirrus longwave radiative forcing is dominated by the absorption and emission processes.)
6. Application of the new parameterization to the cirrus radiative energy budget
By incorporating the g parameterization in terms of AR and Dge into the Fu–Liou radiation model (Fu and Liou 1992, 1993; Fu 1996), this section presents the impact of ice particle mean aspect ratio and surface roughness on the radiative energy balance through their effects on the symmetry factor. We will also compare results using the new parameterization with those based on g from Fu (1996).
The U.S. Standard Atmosphere, 1976 profiles (COESA 1976) are used in the solar radiative transfer calculations. We assume a surface albedo of 0.1 and the solar constant of 1365 W m−2. A cirrus cloud is inserted between 10 and 12 km with a Dge of 45 μm. We vary the cirrus optical depth from 0 to 10 by changing the ice water content. Figure 8 shows the effect of AR on the reflected solar radiation as a function of cirrus optical depth for solar zenith angles of 30°, 60°, and 75°. We can see that the magnitude of the differences increases with the increase of optical depth (within the range of optical depth considered) but decreases with the increase of solar zenith angles. For a solar zenith angle of 60° and a cirrus optical depth of 4, the differences in reflected solar radiation are about −30 or −70 W m−2, respectively, due to changes in AR from 1.0 to 0.5 or from 1.0 to 0.1. Figure 9 is similar to Fig. 8 but shows the effect of ice particle surface roughness. Such an effect is larger for smaller AR. For a solar zenith angle of 60° and a cirrus optical depth of 4, the differences in reflected solar radiation due to the consideration of ice particle surface roughness are from 15 to 21 W m−2, depending on the AR used.
For the comparison of the Fu (1996) ,g parameterization in terms of Dge with g in terms of AR and Dge, we still consider the cirrus cloud between 10 and 12 km but with a fixed optical depth of 4. A solar zenith angle of 60° is used. We change the Dge from 10 to 100 μm (the ice water content is also changed to keep the optical depth constant). For the new parameterization we use the ARs of 0.5 and 1.0 for ice particles with both smooth and rough surfaces. The reflected solar fluxes are shown in Fig. 10 as functions of mean effective size. For given AR the reflected solar fluxes decrease with the increase of Dge. This is because the coalbedo, 1 − ω̃, increases with Dge, which enhances cloud absorption (for given optical depth), and increases g in the bands with significant absorption. Both processes contribute to the decrease of reflected solar fluxes but the former dominates (the impact of Dge through g for given AR is only about 3 W m−2 based on our sensitivity calculations). The large decrease of the reflected fluxes with Dge using Fu (1996) is mainly due to the increase of g with Dge (Fig. 1). The results of Fu are about between those using ARs of 0.5 and 1.0 for ice particles with smooth surfaces. However, there are two issues in the g parameterization of Fu (1996). First, the dependence of g on Dge is not generic but depends on the assumed relationship between the aspect ratio and ice particle size built into the parameterization. This shortcoming applies to all parameterizations shown in Fig. 1. Second, it does not consider the effect of ice particle surface roughness on the light scattering, which seems to be a common feature of natural ice particles in cirrus clouds (e.g., Macke et al. 1996; Yang and Liou 1998; Garrett et al. 2001; Baran and Francis 2004; Edwards et al. 2007). We recommend that the parameterization of g in terms of AR and Dge for ice particles with rough surfaces should be used in the global climate models.
7. Summary and conclusions
We have identified the aspect ratio of a nonspherical ice particle as the key microphysical parameter to determine its asymmetry factor for solar radiation. We define the mean effective aspect ratio (AR) for cirrus clouds containing various nonspherical ice particles. We find that for wavelengths longer than 1.4 μm where the absorption becomes significant, the asymmetry factor also depends on ice particle size. This is largely because the change of single-scattering albedo modifies the relative contribution of diffracted rays to the scattered light.
We have developed a new parameterization of asymmetry factor of cirrus clouds in terms of AR and Dge for solar radiation based on geometric ray-tracing calculations for hexagonal ice crystals with a simple representation of particle surface roughness. Twenty-eight ice crystal size distributions based on in situ aircraft observations are used for both columns and plates, with the AR ranging from about 0.3 to 17. The new parameterization is independent of Dge for wavelengths shorter than 1.4 μm.
We show that the new parameterization reproduces the asymmetry factors of complicated ice particles such as spatial bullet rosettes, aggregates with rough surfaces, and fractal crystals and agrees well with observations. We have thus provided a more generalized parameterization of the asymmetry factor for nonspherical ice particles that can be properly applied to cirrus clouds containing various nonspherical ice particles.
The radiative transfer calculations are performed to test the new parameterization of the asymmetry factor. We show that, for a cirrus cloud with an optical depth of 4 and a solar zenith angle of 60°, changes in AR from 1.0 to 0.5 or from 1.0 to 0.1 result in differences in reflected solar fluxes of about −30 or −70 W m−2, respectively. For the same cloudy conditions, the effect of ice particle surface roughness on the reflected solar flux is found to be about 20 W m−2.
The author wishes to thank Prof. T.C. Grenfell for his comments and suggestions on the manuscript. This work was supported by NASA Grants NNG04GM23G and NNG05GA19G, a Boeing contract, and LANL IGPP Award.
Corresponding author address: Dr. Qiang Fu, Department of Atmospheric Sciences, University of Washington, Box 351640, Seattle, WA 98195. Email: firstname.lastname@example.org