Editorial Type:
Article
Article Type:
Research Article

A Flexible Parameterization for Shortwave and Longwave Optical Properties of Ice Crystals and Derived Bulk Optical Properties for Climate Models

Bastiaan van Diedenhoven Columbia University, and NASA Goddard Institute for Space Studies, New York, New York

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https://orcid.org/0000-0001-5622-8619
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Brian Cairns NASA Goddard Institute for Space Studies, New York, New York

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Online Publication:
01 Mar 2020
Print Publication:
01 Apr 2020
DOI:
https://doi.org/10.1175/JAS-D-19-0193.1
Page(s):
1245–1260
Free access

Abstract

We provide a parameterization of the extinction efficiency, single-scattering albedo, and asymmetry parameter of single ice crystals with any combination of particle volume, projected area, component aspect ratio, and crystal distortion at any wavelength between 0.2 and 100 μm. The parameterization is an extension of the one previously published by van Diedenhoven et al. In addition, the parameterized optical properties are integrated over size distributions yielding bulk extinction efficiencies, single-scattering albedos, and asymmetry parameters for large ranges of effective radii, particle component aspect ratios, and crystal distortion values. The parameterization of single-particle optical properties is evaluated with a reference database. The bulk optical properties are evaluated against the ice model selected for the Moderate Resolution Imaging Spectroradiometer (MODIS) collection 6 products, for which accurate optical properties are available. Mean absolute errors in parameterized extinction efficiency, asymmetry parameter, and single-scattering albedo are shown to be 0.0272, 0.008 90, and 0.004 68, respectively, for shortwave wavelengths, while they are 0.0641, 0.0368, and 0.0200 in the longwave. Shortwave and longwave asymmetry parameters and single-scattering albedos are shown to vary strongly with particle component aspect ratio and distortion, resulting in substantial variation in shortwave fluxes, but relatively small variations in longwave cloud emissivity. The parameterization and bulk optical properties are made publicly available.

© 2020 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Bastiaan van Diedenhoven, bv2154@columbia.edu

Abstract

We provide a parameterization of the extinction efficiency, single-scattering albedo, and asymmetry parameter of single ice crystals with any combination of particle volume, projected area, component aspect ratio, and crystal distortion at any wavelength between 0.2 and 100 μm. The parameterization is an extension of the one previously published by van Diedenhoven et al. In addition, the parameterized optical properties are integrated over size distributions yielding bulk extinction efficiencies, single-scattering albedos, and asymmetry parameters for large ranges of effective radii, particle component aspect ratios, and crystal distortion values. The parameterization of single-particle optical properties is evaluated with a reference database. The bulk optical properties are evaluated against the ice model selected for the Moderate Resolution Imaging Spectroradiometer (MODIS) collection 6 products, for which accurate optical properties are available. Mean absolute errors in parameterized extinction efficiency, asymmetry parameter, and single-scattering albedo are shown to be 0.0272, 0.008 90, and 0.004 68, respectively, for shortwave wavelengths, while they are 0.0641, 0.0368, and 0.0200 in the longwave. Shortwave and longwave asymmetry parameters and single-scattering albedos are shown to vary strongly with particle component aspect ratio and distortion, resulting in substantial variation in shortwave fluxes, but relatively small variations in longwave cloud emissivity. The parameterization and bulk optical properties are made publicly available.

© 2020 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Bastiaan van Diedenhoven, bv2154@columbia.edu
Keywords: Cirrus clouds; Cloud radiative effects; Optical properties

1. Introduction

To calculate the effect of ice clouds on Earth’s radiation budget, accurate optical properties are needed that are consistent with the range of ice crystal shapes and sizes present in natural ice clouds (Stephens et al. 1990; Vogelmann and Ackerman 1995). Distributions of ice crystal shapes and sizes in clouds depend on many processes, such as depositional vapor growth, sublimation, crystal aggregation, riming, and crystal deposition, which in turn may depend on dynamics, pressure, temperature, and humidity as well as crystal number concentration and nucleation mode and rate (Bailey and Hallett 2009; Harrington et al. 2013; Heymsfield et al. 2017). The range of processes that are involved generally leads to complex mixtures of ice crystal sizes and shapes varying with cloud type, altitude, and other variables (Bailey and Hallett 2009; Lawson et al. 2010; Heymsfield et al. 2017). In situ, laboratory and remote sensing measurements provide an increasingly detailed view of the variation of ice crystal size and shape characteristics at various scales (van Diedenhoven et al. 2014b, 2020; Schmitt et al. 2016; Schnaiter et al. 2016; Järvinen et al. 2018; Lawson et al. 2019).

The fundamental optical properties needed to calculate the effects of ice clouds on radiative fluxes are the extinction coefficient, single-scattering albedo, and scattering asymmetry parameter. For climate model simulations, generally parameterizations are used to approximate such ice optical properties (e.g., Fu et al. 1998; Fu 2007; Mitchell 2002; Hong et al. 2009). Such parameterizations generally aim to represent the variation of ice optical properties within selected wavelength and size bins, but have limited capabilities to account for variations in ice crystal shapes. Databases of calculated shortwave and longwave optical properties for various ice crystal shapes are available (e.g., Yang et al. 2013). However, owing to the rather extreme computational costs of these calculations, only a limited selection of ice crystal habits are included, which may not be consistent with the ice properties assumed or predicted by a model. Although the macroscale shape (i.e., habit) of ice crystals may partly influence ice optical properties, the extinction coefficient, and single-scattering albedo of randomly oriented ice crystal ensembles mainly depend on the average projected areas and volumes, while asymmetry parameters are primarily determined by the average aspect ratio (defined throughout as the ratio of hexagonal prism height to width) of the ice crystal hexagonal components and their microscale surface roughness or crystal distortion (Fu 1996; Fu et al. 1998; Iaquinta et al. 1995; Macke et al. 1996; Fu 2007; Baran 2009). It has been shown that optical properties of ensembles of complex ice crystals with hexagonal components can be well represented by those of simple proxy hexagonal prisms as long as the ensemble-averaged projected area, volume, component aspect ratio, and crystal distortion are conserved (Iaquinta et al. 1995; Fu 2007; Um and McFarquhar 2007, 2009; van Diedenhoven et al. 2016). However, Smith et al. (2016) concluded that mass–dimensional and area–dimensional relationships of the complex ice crystals cannot be conserved using single hexagonal prisms while maintaining realistic aspect ratios. To overcome this limitation, van Diedenhoven et al. (2014a, VD14 hereafter) developed a flexible parameterization of shortwave optical properties of single ice crystals that approximates the single-scattering albedo and scattering asymmetry parameter of complex ice crystals with any combination of projected area, volume, component aspect ratio, and crystal distortion for any wavelength in the shortwave. Although this parameterization is based on optical properties of single hexagonal prisms with a discrete selection of aspect ratios, sizes, and distortion values, it allows the particle projected area, volume, aspect ratio, and crystal distortion to be varied independently and continuously. As demonstrated by VD14, any mass–dimensional and area ratio–dimensional relationships assumed in (or predicted by) a cloud microphysical scheme can be conserved by our parameterization, while component aspect ratio and crystal distortion can be adjusted independently to be consistent with assumed or predicted values.

The parameterization of VD14 is based on geometric optics principles and provides the scattering asymmetry parameter and single-scattering albedo of single ice crystals. Absolute differences with reference geometric optics calculations of both asymmetry parameter and single-scattering albedo were found to be generally below 0.015. Consistent with the geometric optics approximation, VD14 assume the extinction coefficient to be equal to 2. However, the parameterization of VD14 is not suitable for thermal infrared wavelengths, where ice size parameters decrease outside of the range where geometric optics assumptions may be assumed to be sufficiently accurate. In this paper, we extend the parameterization of ice optical properties of VD14 to infrared wavelengths and include a parameterization of the extinction coefficient, adding just a few lines of simple code and one more coefficient to the 88 coefficients of the original scheme. Furthermore, we use the parameterization to compute bulk optical properties that can be readily used in climate model radiation schemes. Results are provided for a wide range of effective particle sizes, component aspect ratios, and crystal distortion values. The updated parameterization is reevaluated using reference calculations from the database of Yang et al. (2013). The latest version of this database uses the improved geometric optics method (IGOM) for large size parameters, while optical properties for small size parameters are calculated using invariant imbedding T-matrix (II-TM) technique (Bi and Yang 2017).

In section 2 the parameterization of single-particle optical properties is presented and in section 3 the method to integrate the properties over size distributions is given. An evaluation of the parameterized bulk optical properties against those of the Moderate Resolution Imaging Spectroradiometer (MODIS) collection 6 ice model is shown in section 5. Examples of the variation of single-scattering albedo and asymmetry parameter with particle shape and distortion are shown in section 6, before concluding the paper in section 7.

2. Single-particle optical properties

Single-particle asymmetry parameters and single-scattering albedos are calculated using the parameterization of VD14, which is based on geometric optics calculations (e.g., Macke et al. 1996). This parameterization approximates asymmetry parameters and single-scattering albedos for any combination of particle volume, projected area, aspect ratio, and crystal distortion. Evaluation of the parameterization with geometric optics calculations show absolute errors in both single-scattering albedo and asymmetry parameter that are generally below 0.015 for wavelengths from 0.2 to 4 μm. While the original parameterization assumed an extinction efficiency equal to 2, independent of wavelength or size, here we extend the parameterization of extinction efficiency to account for its size and wavelength dependence and include edge effects, as discussed below.

Although the parameterization was developed to be applied to solar wavelengths less than 4 μm, here we present some minor modifications so that it can also be applied to infrared wavelengths up to 100 μm. Comparisons to the MODIS collection 6 ice model, as presented in section 5, show reasonable results for infrared wavelengths justifying this extension.

In this section, first the parameterization of extinction efficiency is given, followed by minor corrections to the parameterization of single-scattering albedo and asymmetry parameter.

a. Extinction efficiency

The geometric optics approximation QGO for the extinction efficiency Qe was implemented in the parameterization of VD14; that is,
QGO=2.

Here, we apply two additional corrections to this approximation.

First, the geometric optics approximation of Eq. (1) is valid for particle sizes that are large compared to the wavelength. For particle sizes that are considerably smaller than the wavelength, Qe is smaller than 2. As previously described by Fridlind et al. (2016), we account for this size dependence by applying a correction factor derived from the simplified anomalous diffraction theory of Bryant and Latimer (1969). Namely,
CR=1−e−miχscat cos⁡[χscat⁡(mr−1)]when χscat<π2⁡(mr−1),
where mr and mi are the real and imaginary parts, respectively, of the ice refractive index (Warren and Brandt 2008) with the scattering size parameter χscat being given by
χscat=2π VλA,
where V and A are the volume and the projected area of the particle, respectively; CR is set to unity for χscat ≥ π/[2(mr − 1)]. Furthermore, the lower limit of CR is set to (1 − ωGO), where ωGO is the single-scattering albedo assuming geometric optics as originally parameterized by VD14 (see section 2b). Hence, a corrected extinction efficiency is given by
QR=CRQGO.
Note that any oscillations in the extinction efficiency typically occurring at small size parameters are not accounted for in this parameterization.
Exact methods for the calculation of extinction coefficients of spherical and nonspherical particles show that edge effects may substantially increase the extinction coefficients at small size parameters (Jones 1957; Nussenzveig and Wiscombe 1980; Guimarães and Nussenzveig 1992; Bi and Yang 2016). These contributions are sometimes termed above- and below-edge and/or tunneling effects, although the terminology seems to be not well defined. Generally, the contribution of edge effects to the total extinction coefficient Qsurf is approximated by (e.g., Zhao and Hu 2003; Yang et al. 2013)
Qedge=ηx2/3,
where η is a parameter depending on particle shape and refractive index, further discussed below, and x is a particle size parameter that we define here as x = (3/4)χscat (Zhao and Hu 2003). Note that Qedge tends to infinity as x goes to zero. To account for this, we follow Zhao and Hu (2003) and calculate the total extinction coefficient Qe as
Qe=QR+QR2/Qedge+1/⁡[|m−1|⁡(QR+1)].
For spheres, a value for η of 0.996 is generally used (Nussenzveig and Wiscombe 1980), but its value is not well determined for nonspherical particles. Bi and Yang (2016) showed that edge effect contributions are considerable smaller for most nonspherical particles. Yang et al. (2013) determined η values separately for all habits in their database and as a function of wavelength by matching exact II-TM calculations and IGOM calculations at overlapping size parameters, although the derived η values are not reported (P. Yang 2020, private communication). To estimate a reasonable value for η, we calculate Qe using Eq. (6) for all habits, sizes and wavelengths included in the database of Yang et al. (2013), applying values of η between 0 and 1 in steps of 0.05. The mean and median value of η that leads to the lowest absolute differences between Qe calculated using Eq. (6) and those of Yang et al. (2013) is 0.5, which is consistent with the finding of Bi and Yang (2016) that edge effect contributions for hexagonal prisms are about half that of spheres. The coefficient η is the only extra coefficient we add to the original scheme.

Although exact calculations of smooth nonspherical particles include edge effect contributions (Bi and Yang 2016), it is not determined how edge effects are affected by surface roughness or crystal complexity. Furthermore, Mitchell et al. (2001) discussed the different processes affecting surface wave edge effects and tunneling effects, which may have different dependencies on size parameter, refractive index, and particle shape. Yang et al. (2013) adjust the IGOM results for edge effects using an equation equivalent to Eq. (5). As we use the Yang et al. (2013) database as a reference, the edge effect contribution is included in our parameterization as described above. However, the software we provide includes an option to change η and possibly set it to zero to remove the parameterized edge effect contribution. The parameterization of Qe is summarized in Fig. 1.

Fig. 1.
Fig. 1.

Summary of the parameterization of extinction efficiency Qe (box 5). The geometric optics approximation for the extinction cross section is given in box 1 with correction for edge effects in box 3 and the correction for small size parameters in box 4. The parameterization for single-scattering albedo ωGO is given in Fig. A1.

Citation: Journal of the Atmospheric Sciences 77, 4; 10.1175/JAS-D-19-0193.1

b. Single-scattering albedo

The parameterization of single-scattering albedo derived by VD14 is based on geometric optics calculations, which, as discussed above, are less accurate for small size parameters. Consistent with the correction CR applied to the extinction efficiency (see section 2a), we apply the following correction to the original single-scattering albedo ωGO of VD14:
ω=1−1−ωGOCR,
where CR is given by Eq. (2). Since the lower limit of CR is set to (1 − ωGO) (see section 2a) and the upper limit is unity, the parameterized ω is appropriately limited between zero and ωGO.

Note that edge or tunneling effects may affect absorption efficiencies Qabs differently to extinction efficiencies Qe and thus may affect the single-scattering albedo (ω = 1 − Qabs/Qe) at small size parameters (Yang et al. 2013; Mitchell et al. 2001, 2010). However, such contributions to extinction and absorption efficiencies are often small and may partly cancel out, hence the effects on the single-scattering albedo are ignored here.

The updated parameterization for the single-scattering albedo is summarized in Fig. A1 in the appendix.

c. Asymmetry parameter

The parameterization of asymmetry parameter described by VD14 consists of separate parameterizations of the diffractive contributions gdif and the refractive plus reflective (or ray tracing) contributions gRT to the total asymmetry parameter. For the extension of the parameterization to longer wavelengths, minor modification of the parameterizations of gRT and gdif are needed.

As described by VD14, the refractive plus reflective part of the asymmetry parameter is first parameterized at a wavelength of 0.862 μm (gRT,λ=862nm). Subsequently, the correction factor Cmr [defined by Eq. (16) in VD14] is applied to account for the ice refractive index varying with wavelength. Here, a minor modification to Cmr is made to address a discontinuity near the strongly absorbing Christiansen bands at wavelengths of 2.9 and 11 μm, where Cmr as originally defined becomes negative, leading to nonphysical negative total asymmetry parameters when using the parameterization described by VD14. This situation is avoided by using the absolute value of correction factor Cmr in all cases by replacing Eq. (16) of VD14 with
Cmr=|⁡(mr,1−ε)⁡(mr,1+ε)⁡(mr,2+ε)⁡(mr,2−ε)|.
See Fig. A2 and VD14 for the definitions of the parameters appearing in Eq. (8). Using Eq. (8) leads to parameterized refraction-plus-reflection asymmetry parameters gRT close to unity at the Christiansen bands, consistent with the value expected on physical grounds for a strongly absorbing particle.

Furthermore, we found that when scattering size parameters are smaller than about unity, the diffraction asymmetry parameter gdif parameterized by VD14 is erroneously negative. Theoretically gdif should approach zero, so here we apply a lower limit of zero to the parameterized gdif.

The original parameterization also applies correction factors Cω,1 and Cω,2 to correct gRT,λ=862nm for the single-scattering albedo varying with wavelength. These correction factors are determined using the single-scattering albedo originally parameterized by VD14 (i.e., ωGO), without applying the correction for small size parameters given by Eq. (7). To avoid these correction factors becoming negative or too large, we calculate Cω,1 and Cω,2 using ωGO instead of the corrected single-scattering albedo given by Eq. (7). Furthermore, a single-scattering albedo is used to weight the gdif and gRT terms to compute the total asymmetry parameter according to geometric optics approximations (Macke et al. 1996; also see Fig. A2). To be consistent with this geometric optics approximation, ωGO is also used for this weighting.

The updated parameterization for the asymmetry parameter is summarized in Fig. A2 in the appendix.

3. Bulk optical properties

To obtain bulk ice optical properties, we numerically integrate the single-particle optical properties over particle size distributions according to equations given by Baum et al. (2005). We aim to calculate bulk ice optical properties for ensembles with a given effective radius re, which is defined as
re=34∫0∞V⁡(s)N⁡(s) ds∫0∞Ap⁡(s)N⁡(s) ds=34VtotAtot,
where N(s) represents the number size distribution as a function of particle size s. Generally, ice crystal sizes are represented by their maximum dimensions Dmax. Here, instead, we opt to represent the sizes of single hexagonal prisms by the side length of their hexagonal planes ahex, which are equal to Dmax/2 for plates, and Dmax/(2α) for columns, where α is the particle aspect ratio. The total volume Vtot of an ensemble of hexagonal ice crystals with equal aspect ratios is then given by
Vtot=33αM3,
where Mn is the nth moment of the size distribution in terms of ahex (Petty and Huang 2011). Similarly, the total projected area Atot of the ensemble is given by (Vouk 1948)
Atot=33+12α4 M2.
When an exponential size distribution is assumed (Field et al. 2005), that is,
N⁡(ahex)=N0e−γahex,
where N0 is a normalization parameter, the moments of the size distribution are given by
Mn=n!γn.
Combining Eqs. (9)–(13) gives an equation for the exponential size distribution parameter γ for any given re and α pair; namely,
γ=9αre⁡(1+43 α).
The above equations together with the single-particle parameterizations described in section 2 are used to derive bulk extinction efficiencies Qe¯, single-scattering albedos ω¯ and asymmetry parameters g¯. Note that the mass extinction coefficient ke that is commonly used in climate model applications (Platt 1997) can be defined as
ke=3Qe¯4ρire,
where ρi is the density of solid ice.

4. Evaluation of single-particle optical properties

To evaluate the updated parameterization of single-particle optical properties, we compare them to optical properties provided by Yang et al. (2013). This database uses IGOM for large size parameters, and II-TM method for small size parameters (Bi and Yang 2017). Crystal distortion or surface roughness is parameterized in the improved geometric optics method similarly as is done by Macke et al. (1996). Note that the definition of the distortion parameter σ applied by Yang et al. (2013) differs from the one applied here δ (Macke et al. 1996), but these definitions yield comparable results (Neshyba et al. 2013; Geogdzhayev and van Diedenhoven 2016). Also note that no crystal distortion is assumed for any of the particles for the small size parameters where the II-TM approach is applied.

For this evaluation, we use the calculations for the smooth solid column and the severely roughened aggregate of columns. The geometry of the solid column needed for our parameterization is given by Yang et al. (2013) and the distortion parameter is set to zero. Complex refractive indices are taken from Warren and Brandt (2008), consistent with Yang et al. calculations. As discussed in the introduction, we approximate the optical properties of the complex aggregate with our parameterization, using the volume and area of the whole aggregate and an aspect ratio equal to the average value of the components of the complex particle, that is, 1.5 (Yang et al. 2000; Fu 2007; Yang et al. 2013). To determine an appropriate effective distortion parameter δ, we aim to closely match the asymmetry parameter given by our parameterization with the calculation of Yang et al. at a wavelength of 0.67 μm for large re, that is, gC6 = 0.7536, yielding δ = 0.59. As the asymmetry parameter of the aggregate model as well as the performance of our parameterization does not vary substantially over visible wavelengths, equivalent results are obtained if the effective distortion parameter is determined at other visible wavelengths.

Figure 2 shows extinction efficiencies, single-scattering albedos and asymmetry parameters of smooth solid columns as a function of scattering size parameter [Eq. (3)]. Results are shown for five different wavelengths selected to span a large range of wavelengths and imaginary part of the refractive index. Generally, the increase of extinction efficiency at small size parameters, as well as the asymptotic behavior to 2 at large size parameters is well represented by our parameterization. As expected, the oscillatory behavior at intermediate size parameters is not represented by our parameterization. Calculated single-scattering albedos are well matched by our parameterization, although performance degrades with increasing imaginary part of the refractive index. The decrease of single-scattering albedo at small size parameters is well represented by applying Eq. (7) with the correction factor given by Eq. (2). As expected, oscillations in ω at intermediate size parameters are not present in our parameterization, but the parameterization follows the overall behavior as a function of size parameter well. Asymmetry parameters at large size parameters increase with size parameter even for the nonabsorbing wavelength (0.86 μm) because assumed aspect ratios increase with size. This behavior is well represented by our parameterization. Interestingly, the calculations show relatively large asymmetry parameters at size parameters around 10 that are not represented by our parameterization since they are not simulated by geometric optics approximations in general. The oscillations are shown to stop rather abruptly at particle sizes where the database transitions from II-TM to IGOM. For the long wavelengths, the decrease of asymmetry parameter with decreasing size parameter is reasonably well represented by the parameterization, although parameterized values do not tend to zero as is the case for the calculations.

Fig. 2.
Fig. 2.

(top) Extinction efficiencies, (middle) single-scattering albedos, and (bottom) asymmetry parameters of (left) smooth solid columns and (right) rough aggregates of columns as a function of scattering size parameter. Solid and dashed lines show results from Yang et al. and from our parameterization, respectively. Colors indicate different wavelengths.

Citation: Journal of the Atmospheric Sciences 77, 4; 10.1175/JAS-D-19-0193.1

Comparisons between calculated and parameterized optical properties of the rough aggregate of columns, shown in Fig. 2b, are qualitatively similar to those obtained for the smooth solid column. Note that II-TM calculations do not take crystal distortion or surface roughness into account, which may lead to overestimation of asymmetry parameters at intermediate size parameters around 10.

Table 1 lists mean absolute differences of calculated and parameterized optical properties for the solar (0.2 < λ < 4 μm) and infrared (4 < λ < 100 μm) wavelengths. Values for smooth columns and aggregates of columns are similar. Mean absolute differences for ω and g at solar wavelengths are below 0.015, agreeing with the conclusions of VD14. In the infrared mean absolute differences for ω and g are about 0.025. Mean absolute differences for Qe are below 0.04 and 0.05 at solar and infrared wavelengths, respectively. Note that assuming Qe = 2 would lead to maximum mean absolute differences with calculations of Yang et al. of 0.07 and 0.2 at solar and infrared wavelengths, respectively.

Table 1.

Mean absolute differences between calculated and parameterized optical properties.

Table 1.

5. Evaluation against MODIS collection 6 model

We evaluate the bulk optical properties with the ice model that is used for the MODIS collection 6 (C6) ice cloud retrieval products (Platnick et al. 2017), which is based on the severely roughened aggregates of columns discussed in section 4. We choose this model for the evaluation because it provides accurate, realistic optical properties for global ice clouds (e.g., Holz et al. 2016; Järvinen et al. 2018), the geometry of the particle is well-documented and the size distribution-integrated properties for a large range of effective radii and wavelengths are available (Yang et al. 2013).

Bulk optical properties are parameterized as explained in sections 2 and 3 for re ranging from 5 to 90 μm in steps of 5 μm and at wavelengths ranging from 0.2 to 100 μm at intervals for which the MODIS C6 values are given (Yang et al. 2013). As discussed in section 4, aspect ratio and distortion parameters are set to 1.5 and 0.59, respectively.

Figures 3, 4, and 5 compare extinction efficiencies, single-scattering albedos, and asymmetry parameters, respectively, of the MODIS C6 model with those calculated using our parameterization as a function of wavelength and re. Largest differences are seen for small size parameters, that is, small re and long wavelengths, which may be expected as geometric optics approximations are less accurate at small size parameters. In the infrared, the parameterization tends to underestimate Qe¯ and g¯, while errors in ω¯ are more variable with wavelength and size. Mean absolute differences of calculated and parameterized optical properties for solar and infrared wavelengths are given in Table 1. At solar wavelengths, mean absolute differences are generally smaller than those obtained for single particles, presumably because oscillations of the optical properties with size parameter that are not captured by the parameterization are partly averaged out by application of a size distribution. In the infrared, however, mean absolute differences in the case of Qe¯ and g¯ are somewhat larger than values obtained for single particle, presumably because of greater weight of smaller size parameters.

Fig. 3.
Fig. 3.

Bulk extinction efficiency as a function of effective radius and wavelength using (top) the MODIS C6 model and (middle) our parameterization. (bottom) The difference between the top and middle panels.

Citation: Journal of the Atmospheric Sciences 77, 4; 10.1175/JAS-D-19-0193.1

Fig. 4.
Fig. 4.

As in Fig. 3, but for the single-scattering albedo.

Citation: Journal of the Atmospheric Sciences 77, 4; 10.1175/JAS-D-19-0193.1

Fig. 5.
Fig. 5.

As in Fig. 3, but for the asymmetry parameter.

Citation: Journal of the Atmospheric Sciences 77, 4; 10.1175/JAS-D-19-0193.1

Although errors in shortwave asymmetry parameters calculated with our parameterization can be relatively large at wavelengths at which the single-scattering albedo is low, the cloud radiative properties at these wavelengths are expected to be dominated by absorption rather than scattering and the effect of errors in asymmetry parameters on flux calculations may be expected to be small. To illustrate this, Figs. 6–8 show shortwave upward and downward fluxes and cloud absorptance, respectively, for an atmosphere containing a uniform cloud layer consisting of ice crystals with optical properties derived with the MODIS C6 model for a range of optical thicknesses and effective radii. In addition, the differences between the calculations with the MODIS C6 model and with our parameterization are shown. To calculate these fluxes, two-stream calculations are made as described by VD14 assuming a cloud layer between 10 and 11 km embedded in a tropical atmosphere over a black surface with a solar zenith angle of 60°. The optical thickness τ at a wavelength λ = 0.67 μm is varied between 0.1 and 10 and is scaled to values at other wavelengths according to τλ=τ0.67⁡(Qe,λ¯/Qe,0.67¯).

Fig. 6.
Fig. 6.

(top) Upward flux calculated using MODIS C6 and (bottom) the difference between results obtained with MODIS C6 and our parameterization.

Citation: Journal of the Atmospheric Sciences 77, 4; 10.1175/JAS-D-19-0193.1

Fig. 7.
Fig. 7.

As in Fig. 6, but for downward fluxes.

Citation: Journal of the Atmospheric Sciences 77, 4; 10.1175/JAS-D-19-0193.1

Fig. 8.
Fig. 8.

As in Fig. 6, but for absorptance.

Citation: Journal of the Atmospheric Sciences 77, 4; 10.1175/JAS-D-19-0193.1

As expected, the upward and downward fluxes shown in Figs. 6 and 7 strongly vary with optical thickness, but are less sensitive to effective radius, while absorptance, shown in Fig. 8, does vary substantially with effective radius, in addition to optical thickness. Absolute errors caused by our parameterization are generally well below 4 W m−2 and largest errors are found for small effective radii. Specifically, mean absolute errors in upward and downward fluxes and absorptance are 1.36, 0.867, and 1.46 W m−2, respectively, for re ≤ 10 μm, while they are 0.633, 0.442, and 0.612 W m−2, respectively, for re > 10 μm. To put these results in perspective, we note that VD14 showed that, for a cloud optical thickness of 4, upwelling and downwelling fluxes range between about 140 and 730 W m−2 when particle aspect ratios are varied between 0.02 and 50 and distortion parameters between 0 and 0.8, while cloud absorptance ranges from 36 to 132 W m−2 for re between 5 and 90 μm. Thus, the parameterization errors are small compared with variations expected as a result of varying aspect ratios and crystal distortion. Similar relative errors in fluxes and absorptance are obtained for other solar geometries.

To further evaluate the parameterization applied to infrared wavelengths, we compare spectrally integrated longwave emissivity calculated using our parameterization with those calculated using MODIS C6 optical properties. At a given wavelength λ in the longwave, the cloud emissivity εc is approximated by (Liou 2002; Tang et al. 2018)
εc=1−e−βτλ⁡(1−ω¯cscat),
where β is the diffusivity factor, which is taken to be 1.66 (Liou 2002), and cscat is a factor correcting for scattering effects. We follow Tang et al. (2018) and set cscat to (1+g¯)/2. Values of εc are calculated using Eq. (16) at all wavelengths from 4 to 100 μm for which the MODIS C6 model is available and subsequently spectrally integrated weighted by the Planck function assuming a cloud temperature of 210 K to obtain broadband infrared emissivity. Calculations are made for cloud optical thickness values ranging from 0.1 to 10, where the cloud optical thickness is specified at a reference wavelength of 0.67 μm and scaled to other wavelengths using the wavelength-dependent extinction efficiencies, as described above. Broadband emissivities are calculated according to this procedure using MODIS C6 optical properties and using optical properties generated by our parameterization and results are subsequently compared.

Figure 9 shows that the broadband emissivity calculated using the MODIS C6 ice model generally increases with cloud optical thicknesses and is less sensitive to effective radius. Figure 9 also shows that errors in broadband emissivity from using our parameterization are largest for small effective radii and optical thickness values around 1. Specifically, the mean absolute parameterization errors in broadband emissivity are 9.64 × 10−3 for re ≤ 10 μm and 8.56 × 10−4 for larger sizes. Note that when broadband emissivity values are calculated only for the atmospheric window region between 8- and 11-μm mean absolute parameterization errors are increased by factors of about 2.

Fig. 9.
Fig. 9.

As in Fig. 6, but for longwave broadband emissivity.

Citation: Journal of the Atmospheric Sciences 77, 4; 10.1175/JAS-D-19-0193.1

In summary, the presented parameterization generally compares well to the MODIS C6 ice model at shortwave wavelengths. In the longwave, larger parameterization errors are obtained for the bulk optical properties, but the resulting errors in broadband emissivity can be considered small.

6. Variation of optical properties with wavelength and crystal properties

The parameterization presented in this paper is used to generate a database of extinction efficiencies, asymmetry parameters and single-scattering albedos for ice crystal ensembles with re values ranging from 5 to 123 μm in steps of 2 μm and wavelengths ranging from 0.2 to 100 μm at spectral intervals for which Warren and Brandt (2008) provided refractive indices. Aspect ratios for plates range from 0.02 to 0.98 in steps of 0.04, while the inverse of these values is used for columns. In addition, a particle geometry with unity aspect ratio is included. Distortion parameters δ range from 0 to 0.8 in steps of 0.02. Here, we illustrate the variation of asymmetry parameters and single-scattering albedos as a function of aspect ratio, crystal distortion, and wavelength. Note that the parameterized extinction efficiencies only depend on the scattering size parameter and our parameterization does not explicitly account for any possible variation of extinction efficiencies with particle shape. The variation in shortwave fluxes and longwave emissivity with aspect ratio, crystal distortion, and size are discussed.

For an effective radius of 31 μm, the variation of single-scattering albedo with wavelength and aspect ratio is shown in Fig. 10. For the shortwave spectral domain, the sensitivity to aspect ratio is minimal, while in the infrared single-scattering albedos are shown to be substantially smaller for ensembles of particles with aspect ratio near unity compared to those of thin plates and long columns. Qualitatively similar variations of the single-scattering albedo with aspect ratio are seen for other sizes. Note that the variation of single-scattering albedo with effective radius is shown in Fig. 4.

Fig. 10.
Fig. 10.

(top) Parameterized single-scattering albedo and (middle),(bottom) asymmetry parameter as a function of particle aspect ratio and wavelength for an ensemble of crystals with an effective radius of 31 μm. Asymmetry parameters shown in the middle panel assume pristine particles (δ = 0), while the those shown in the bottom panel assume distorted crystals (δ = 0.6).

Citation: Journal of the Atmospheric Sciences 77, 4; 10.1175/JAS-D-19-0193.1

The variation of asymmetry parameter with wavelength and aspect ratio is shown in Fig. 10 for an ensemble of smooth particles (δ = 0) with an effective radius of 31 μm. Similarly, the results for distorted particles (δ = 0.6) are shown in Fig. 10. Consistent with results shown by VD14, asymmetry parameters increase as aspect ratios increasingly deviate from unity. Asymmetry parameters increase with wavelengths up to about 1.4 μm owing to the increasing real part of the refractive index. At longer wavelengths, asymmetry parameters are affected by spectral variations in single-scattering albedos. Furthermore, asymmetry parameters at all wavelengths and aspect ratio values are generally smaller for distorted particles compared to smooth particles, as previously shown by VD14 and references therein.

The dependencies of shortwave fluxes on aspect ratio and distortion have been shown by VD14. As can be expected from the variations in asymmetry parameter shown in Fig. 10, upward shortwave fluxes generally decrease as aspect ratios increasingly deviate from unity, while they increase with increasing particle distortion. Absorptance is not substantially affected by particle aspect ratio and distortion.

To illustrate the sensitivity of longwave cloud emissivity to particle aspect ratio and distortion, Fig. 11 shows the broadband emissivity (spectrally integrated weighted by the Planck function as described in section 5) as a function of aspect ratio and effective radius for smooth (δ = 0) and distorted (δ = 0.6) particles. Here, a cloud with a temperature of 210 K and an optical thickness of 1 at 0.67 μm is assumed, but qualitatively similar results are obtained for other temperatures and optical thickness values. The variation of broadband emissivity with optical thickness is shown in Fig. 9. Figure 11 shows that emissivity for a cloud at constant optical thickness is primarily affected by changes in effective radius, while the impacts of varying aspect ratio and crystal distortion are small but may be nonnegligible for some applications. For example, the broadband emissivity for a cloud with an optical thickness of 1 consisting of smooth crystal plates with an effective radius of 31 μm and aspect ratio of 0.02 is 0.4718, while it is 0.070 larger when assuming an aspect ratio of 1 and just 0.0085 larger when assuming distorted particles.

Fig. 11.
Fig. 11.

Broadband cloud emissivity as a function of particle aspect ratio and effective radius, assuming (top) pristine particles (δ = 0) and (bottom) distorted crystals (δ = 0.6). The cloud-top temperature is 210 K, cloud optical thickness at 0.67 μm is assumed to be 1, and a solar zenith angle of 60° is used.

Citation: Journal of the Atmospheric Sciences 77, 4; 10.1175/JAS-D-19-0193.1

7. Conclusions

We extend the parameterization of VD14 to be able to provide extinction efficiency, single-scattering albedos, and asymmetry parameters of single ice crystals with any combination of particle volume, projected area, component aspect ratio, and crystal distortion at any wavelength in the shortwave and longwave (0.2 < λ < 100 μm) spectral domain. Furthermore, the optical properties are integrated over size distributions to obtain bulk shortwave and longwave optical properties as a function of effective radius, aspect ratio, and distortion.

The parameterized single-particle optical properties are evaluated by comparing them to the latest version of the database provided by Yang et al. (Yang et al. 2013; Bi and Yang 2017). In general, dependencies of the optical properties with size parameter and refractive index are well represented by the parameterization. Mean absolute differences for both ω and g at solar and infrared wavelengths are below 0.015 and 0.025, respectively, while mean absolute differences for Qe are below 0.04 and 0.05 at solar and infrared wavelengths, respectively.

Using MODIS C6 ice model (Yang et al. 2013; Platnick et al. 2017) as a reference, we conclude that the uncertainties in bulk optical properties provided by the presented parameterization in the shortwave are small with respect to the variations with particle size, aspect ratio, and distortion. Resulting mean absolute errors in shortwave upward and downward fluxes and cloud absorptance for optical thicknesses ranging from 0.1 to 10 are generally well below 1 W m−2 for re > 10 μm and below 1.5 W m−2 for smaller sizes. In the longwave, uncertainties in the parameterized single-particle optical properties are larger, but resulting mean absolute errors in broadband emissivity are small at about 1 × 10−3 for re > 10 μm and about 1 × 10−2 for smaller sizes.

Parameterized bulk extinction efficiencies are close to 2 for most sizes in the shortwave, but are substantially smaller than 2 at wavelengths longer than 10 μm, especially for re < 20 μm. Single-scattering albedos are only minimally varying with particle aspect ratio in the shortwave, but vary more substantially in the longwave. As shown before by VD14 and others, the asymmetry parameter in the shortwave generally decreases as the aspect ratio increasingly deviates from unity. Here, we show that the dependence is similar in the longwave. Furthermore, asymmetry parameters in the shortwave and longwave generally decrease with increasing distortion. These variations of optical properties with component aspect ratio and crystal distortion, as well as with size, lead to substantial variations in shortwave fluxes and absorptance for a given cloud optical thickness, as shown by VD14. Here, we show that the sensitivity of broadband emissivity in the longwave is mainly dependent on size and optical thickness, while it has a weak sensitivity to particle component aspect and crystal distortion that may nonetheless be significant for specific applications.

Acknowledgments

This material is based upon work supported by the NASA ROSES program under Grant NNX11AG81G. We thank Professor Ping Yang and his team at Texas A&M for providing the ice particles optical properties database. We thank three anonymous reviewers for their valuable input.

Data availability statement: The bulk optical properties and a Python code of the parameterization for single-particle optical properties are publicly available online (https://doi.org/10.17605/OSF.IO/BR8DH).

APPENDIX

Summary and Corrections to the Parameterization of Single-Particle Optical Properties

The parameterization of the asymmetry parameter of single ice crystals for a given wavelength, size, aspect ratio, and crystal distortion parameter is given in Fig. A1. Coefficients needed in box 4 are given in Table A1. This figure is updated and corrected from Fig. 7 in VD14. The minor adjustments as discussed in section 2 is included in Fig. A1. We also corrected an error that appears in Fig. 7 as published in VD14. Namely, equations in boxes 3 and 4, ga=1 and Δgα were erroneously written as a function of logarithm of δ instead of δ itself as is described in the text of VD14. Figure A1 contains the correct equations. The single-scattering albedo is parameterized as summarized in Fig. A2. This figure is updated and corrected from Fig. 4 in VD14 and includes the correction for small size parameters as discussed in section 2. The coefficients required for calculations in box 3 are provided in Table A2. See VD14 for further details.

Fig. A1.
Fig. A1.

Summary of the parameterization of single-scattering albedo ω (box 5), with the absorption size parameter in box 1, parameterization of single-scattering albedo for particles with aspect ratio of unity (ωα=1) in box 2, and the parameterization of the difference between ωα=1 and scattering albedo for particles with other aspect ratios in box 3. The adjustment factor for small size parameters CR is given in box 4.

Citation: Journal of the Atmospheric Sciences 77, 4; 10.1175/JAS-D-19-0193.1

Table A1.

Coefficients ci,j used to compute coefficients li in box 3 of Fig. A1.

Table A1.
Fig. A2.
Fig. A2.

Summary of the parameterization of asymmetry parameter gtot (box 9), with parameterization of the diffraction asymmetry parameter in boxes 1 and 2, parameterization of ray-tracing asymmetry parameter at 862 nm in boxes 3–5, and correction factors for the real refractive index and absorption in boxes 6–8. Single-scattering albedo ωGO is parameterized according to Fig. A1.

Citation: Journal of the Atmospheric Sciences 77, 4; 10.1175/JAS-D-19-0193.1

Table A2.

Coefficients qi,j of the sixth-order polynomial to compute coefficients pΔ,i in box 4 of Fig. A2. This table is reproduced from Table 3 of VD14.

Table A2.

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  • Bailey, M. P., and J. Hallett, 2009: A comprehensive habit diagram for atmospheric ice crystals: Confirmation from the laboratory, AIRS II, and other field studies. J. Atmos. Sci., 66, 2888–2899, https://doi.org/10.1175/2009JAS2883.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Baran, A. J., 2009: A review of the light scattering properties of cirrus. J. Quant. Spectrosc. Radiat. Transfer, 110, 1239–1260, https://doi.org/10.1016/j.jqsrt.2009.02.026.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Baum, B. A., P. Yang, A. J. Heymsfield, S. Platnick, M. D. King, Y. X. Hu, and S. M. Bedka, 2005: Bulk scattering properties for the remote sensing of ice clouds. Part II: Narrowband models. J. Appl. Meteor., 44, 1896–1911, https://doi.org/10.1175/JAM2309.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Bi, L., and P. Yang, 2016: Tunneling effects in electromagnetic wave scattering by nonspherical particles: A comparison of the Debye series and physical-geometric optics approximations. J. Quant. Spectrosc. Radiat. Transfer, 178, 93–107, https://doi.org/10.1016/j.jqsrt.2015.11.022.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Bi, L., and P. Yang, 2017: Improved ice particle optical property simulations in the ultraviolet to far-infrared regime. J. Quant. Spectrosc. Radiat. Transfer, 189, 228–237, https://doi.org/10.1016/j.jqsrt.2016.12.007.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Bryant, F., and P. Latimer, 1969: Optical efficiencies of large particles of arbitrary shape and orientation. J. Colloid Interface Sci., 30, 291–304, https://doi.org/10.1016/0021-9797(69)90396-8.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Field, P. R., R. J. Hogan, P. R. A. Brown, A. J. Illingworth, T. W. Choularton, and R. J. Cotton, 2005: Parametrization of ice-particle size distributions for mid-latitude stratiform cloud. Quart. J. Roy. Meteor. Soc., 131, 1997–2017, https://doi.org/10.1256/qj.04.134.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Fridlind, A. M., R. Atlas, B. van Diedenhoven, J. Um, G. M. McFarquhar, A. S. Ackerman, E. J. Moyer, and R. P. Lawson, 2016: Derivation of physical and optical properties of mid-latitude cirrus ice crystals for a size-resolved cloud microphysics model. Atmos. Chem. Phys., 16, 7251–7283, https://doi.org/10.5194/acp-16-7251-2016.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Fu, Q., 1996: An accurate parameterization of the solar radiative properties of cirrus clouds for climate models. J. Climate, 9, 2058–2082, https://doi.org/10.1175/1520-0442(1996)009<2058:AAPOTS>2.0.CO;2.

    • Crossref
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  • <sc>Fig</sc>. 1.
    View in gallery
    Fig. 1.

    Summary of the parameterization of extinction efficiency Qe (box 5). The geometric optics approximation for the extinction cross section is given in box 1 with correction for edge effects in box 3 and the correction for small size parameters in box 4. The parameterization for single-scattering albedo ωGO is given in Fig. A1.

  • <sc>Fig</sc>. 2.
    View in gallery
    Fig. 2.

    (top) Extinction efficiencies, (middle) single-scattering albedos, and (bottom) asymmetry parameters of (left) smooth solid columns and (right) rough aggregates of columns as a function of scattering size parameter. Solid and dashed lines show results from Yang et al. and from our parameterization, respectively. Colors indicate different wavelengths.

  • <sc>Fig</sc>. 3.
    View in gallery
    Fig. 3.

    Bulk extinction efficiency as a function of effective radius and wavelength using (top) the MODIS C6 model and (middle) our parameterization. (bottom) The difference between the top and middle panels.

  • <sc>Fig</sc>. 4.
    View in gallery
    Fig. 4.

    As in Fig. 3, but for the single-scattering albedo.

  • <sc>Fig</sc>. 5.
    View in gallery
    Fig. 5.

    As in Fig. 3, but for the asymmetry parameter.

  • <sc>Fig</sc>. 6.
    View in gallery
    Fig. 6.

    (top) Upward flux calculated using MODIS C6 and (bottom) the difference between results obtained with MODIS C6 and our parameterization.

  • <sc>Fig</sc>. 7.
    View in gallery
    Fig. 7.

    As in Fig. 6, but for downward fluxes.

  • <sc>Fig</sc>. 8.
    View in gallery
    Fig. 8.

    As in Fig. 6, but for absorptance.

  • <sc>Fig</sc>. 9.
    View in gallery
    Fig. 9.

    As in Fig. 6, but for longwave broadband emissivity.

  • <sc>Fig</sc>. 10.
    View in gallery
    Fig. 10.

    (top) Parameterized single-scattering albedo and (middle),(bottom) asymmetry parameter as a function of particle aspect ratio and wavelength for an ensemble of crystals with an effective radius of 31 μm. Asymmetry parameters shown in the middle panel assume pristine particles (δ = 0), while the those shown in the bottom panel assume distorted crystals (δ = 0.6).

  • <sc>Fig</sc>. 11.
    View in gallery
    Fig. 11.

    Broadband cloud emissivity as a function of particle aspect ratio and effective radius, assuming (top) pristine particles (δ = 0) and (bottom) distorted crystals (δ = 0.6). The cloud-top temperature is 210 K, cloud optical thickness at 0.67 μm is assumed to be 1, and a solar zenith angle of 60° is used.

  • <sc>Fig</sc>. A1.
    View in gallery
    Fig. A1.

    Summary of the parameterization of single-scattering albedo ω (box 5), with the absorption size parameter in box 1, parameterization of single-scattering albedo for particles with aspect ratio of unity (ωα=1) in box 2, and the parameterization of the difference between ωα=1 and scattering albedo for particles with other aspect ratios in box 3. The adjustment factor for small size parameters CR is given in box 4.

  • <sc>Fig</sc>. A2.
    View in gallery
    Fig. A2.

    Summary of the parameterization of asymmetry parameter gtot (box 9), with parameterization of the diffraction asymmetry parameter in boxes 1 and 2, parameterization of ray-tracing asymmetry parameter at 862 nm in boxes 3–5, and correction factors for the real refractive index and absorption in boxes 6–8. Single-scattering albedo ωGO is parameterized according to Fig. A1.

Fig. 1.

Summary of the parameterization of extinction efficiency Qe (box 5). The geometric optics approximation for the extinction cross section is given in box 1 with correction for edge effects in box 3 and the correction for small size parameters in box 4. The parameterization for single-scattering albedo ωGO is given in Fig. A1.
Fig. 2.

(top) Extinction efficiencies, (middle) single-scattering albedos, and (bottom) asymmetry parameters of (left) smooth solid columns and (right) rough aggregates of columns as a function of scattering size parameter. Solid and dashed lines show results from Yang et al. and from our parameterization, respectively. Colors indicate different wavelengths.
Fig. 3.

Bulk extinction efficiency as a function of effective radius and wavelength using (top) the MODIS C6 model and (middle) our parameterization. (bottom) The difference between the top and middle panels.
Fig. 4.

As in Fig. 3, but for the single-scattering albedo.
Fig. 5.

As in Fig. 3, but for the asymmetry parameter.
Fig. 6.

(top) Upward flux calculated using MODIS C6 and (bottom) the difference between results obtained with MODIS C6 and our parameterization.
Fig. 7.

As in Fig. 6, but for downward fluxes.
Fig. 8.

As in Fig. 6, but for absorptance.
Fig. 9.

As in Fig. 6, but for longwave broadband emissivity.
Fig. 10.

(top) Parameterized single-scattering albedo and (middle),(bottom) asymmetry parameter as a function of particle aspect ratio and wavelength for an ensemble of crystals with an effective radius of 31 μm. Asymmetry parameters shown in the middle panel assume pristine p