Abstract

Although the use of an effective radius for radiation transfer calculations in water clouds has been common for many years, the export of this concept to ice clouds has been fraught with uncertainty, due to the nonspherical shapes of ice particles. More recently, a consensus appears to be building that a general definition of effective diameter Deff should involve the ratio of the size distribution volume (at bulk density) to projected area. This work further endorses this concept, describes its physical basis in terms of an effective photon path, and demonstrates the equivalency of a derived Deff definition for both water and ice clouds. Effective photon path is the unifying underlying principle behind this universal definition of Deff.

Simple equations are formulated in terms of Deff, wavelength, and refractive index, giving monochromatic coefficients for absorption and extinction, βabs and βext, throughout the geometric optics, Mie, and Rayleigh regimes. These expressions are tested against Mie theory, showing the limitations of the use of Deff as well as its usefulness.

For water clouds, the size distribution N(D) exhibits relatively little dispersion around the mean diameter in comparison with ice clouds. For this reason, a single particle approximation for βabs based on Deff compares well with βabs predicted from Mie theory, providing a new and efficient means of treating radiation transfer at terrestrial wavelengths. The Deff expression for βext agrees well with Mie theory only under specific conditions: 1) absorption is substantial or 2) absorption occurs in the Rayleigh regime, or 3) size parameter xe ≳ 50, where xe = πDeff/λ. Since the Deff expressions for βabs and βext are single particle solutions, it is not surprising that agreement with Mie theory is best when the size distribution dispersion is reduced, approaching the single particle limit.

For ice clouds, it is demonstrated that the Deff expressions for βabs and βext are probably inadequate for most applications, at least at terrestrial wavelengths. This is due to the bimodal nature of ice particle size spectra N(D) with relatively high concentrations of small (D < 100 μm) ice crystals. These small crystals have relatively low absorption efficiencies, causing the N(D)-integrated βabs to be lower than βabs based on Deff. This difference in N(D) dispersion between water and ice clouds makes it desirable to use an explicit solution to the absorption and extinction coefficients when calculating the radiative properties of ice clouds. Analytical solutions to the integral definitions of βabs and βext are provided in the a1appendix, which may not be too computationally expensive for many applications.

Most schemes for predicting ice cloud radiative properties are founded on the assumption that the dependence of βabs and βext on the size distribution can be described solely in terms of Deff and ice water content (IWC). This assumption was tested by comparing the N(D) area-weighted efficiencies for absorption and extinction, Qabs and Qext, for three N(D) that have the same IWC and Deff, but for which N(D) shape differs. Analytical solutions for βabs and βext were used, which explicitly treat N(D) shape, over a wavelength range of 1.0 to 1000 μm. For a chosen Deff value, uncertainties (percent differences) resulting only from N(D) shape differences reached 44% for Qabs, 100% for Qext, and 48% for the single scattering albedo ωo for terrestrial radiation. This sensitivity to N(D) shape has implications relating to the formulation of schemes predicting ice cloud radiative properties, as well as satellite remote sensing of cloud properties.

1. Introduction

The use of an effective particle size to represent the size dependence of scattering and absorption processes in radiation transfer in both water and ice clouds has found broad acceptance throughout the atmospheric sciences community. In combination with the cloud liquid water content (LWC) or ice water content (IWC), the effective size enables cloud radiation interactions to be quantified for water clouds, although this is less clear for ice clouds. The effective particle size, usually referred to as an effective radius (reff) or diameter (Deff), thus forms the basis for many parameterizations of radiative properties for both water (e.g., Slingo 1989) and ice clouds (e.g., Ebert and Curry 1992; Wyser and Yang 1998; Fu 1996; Fu et al. 1998; Yang et al. 2001). Because of its apparent usefulness, present and planned environmental satellite instrumentation and algorithms are designed to retrieve reff or Deff with global coverage. An intended use of these retrievals is to describe cloud–radiation interactions in global climate models (GCMs) to forecast future climate.

As discussed in McFarquhar and Heymsfield (1998) and Wyser (1998), the definition of Deff for ice clouds is not well understood. Even for the same size distribution and ice crystal shape, the Fu definition of Deff differs from the Ebert and Curry definition by about 60% (Wyser 1998; Fu 1996). Some definitions for Deff assume circular cylinders or hexagonal columns, while others are based on equivalent area spheres, equivalent volume spheres, or the IWC-projected area ratio of the size distribution (i.e., volume at bulk ice density–projected area). A number of studies suggest Deff definitions incorporating the IWC-projected area ratio show promise in describing the radiative properties of ice clouds (Foot 1988; Francis et al. 1994; Francis et al. 1999; Fu 1996; Fu et al. 1998; Wyser and Yang 1998).

In section 2 of this paper, the physical basis of Deff is described for the first time, and its definition is shown to be equivalent for both water and ice clouds. In section 3, simple expressions using Deff are provided for calculating the absorption and extinction coefficients βabs and βext at any wavelength. These expressions are tested against Mie theory using size distributions of water and ice spheres. Using size distributions appropriate for ice clouds, the practice of using only Deff and IWC to calculate βabs and βext is critically evaluated in sections 3 and 4. A summary and concluding remarks are given in section 5. A new scheme for calculating βabs and βext in ice clouds, which treats size distribution shape effects, is described in the appendix.

2. Concept of an effective diameter

The concept of an effective distance or photon path de as being a particle volume-to-area ratio was first suggested by Bryant and Latimer (1969) and further developed in Mitchell and Arnott (1994), Mitchell et al. (1996b), and Yang et al. (2000) to treat absorption and extinction in ice particles. The last three citations defined de for ice particles as the volume V defined at bulk ice density (0.92 g m−3) divided by the particle's projected area at random orientation:

 
formula

where m is the particle's mass and ρi = 0.92 g m−3. This value of ρi must be used since ice refractive indices are referenced to bulk ice density. This concept of de is borne out of the anomalous diffraction approximation (ADA), a simplification of Mie theory (van de Hulst 1981). ADA approximates the absorption efficiency as

 
Qabs = 1 − exp(−4πnide/λ),
(2)

where ni is the imaginary part of the refractive index and λ is the wavelength. As defined in (2), de is the representative distance a photon travels through a particle without internal reflections or refraction occurring. In Mitchell (2000, henceforth M00), it is shown that relevant processes not included in ADA can be parameterized into ADA such that this modified ADA yields absorption efficiencies with errors ≲10%, relative to Mie theory. Absorption processes represented in this modified ADA are based on the principle of effective photon path, indicating that de is the relevant dimension for single particle–radiation interactions. We can take this a step further, and relate the diameter of a sphere D to its effective distance de. Using ice spheres as an example in (1), m = ρi(πD3/6) and P = πD2/4, giving

 
de = 2/3D.
(3)

If there is an effective photon path for a single particle, it can be asked if there is also an effective photon path for the entire size distribution N(D). Based on the formalism in (1), such a photon path should be defined for ice size spectra as

 
formula

where Pt is the total projected area of the size distribution. Based on (3), the effective diameter of the size distribution should then be

 
formula

where IWC is the ice water content of the size distribution.

The same formalism applies to water clouds. That is, the standard definition of effective radius used for water clouds reff is equivalent to (5). Defining reff as ½Deff, then

 
formula

where LWC is the liquid water content, and ρw is the density of liquid water. Defining the LWC and Pt as

 
formula

where N(r)dr is the size distribution with respect to radius, then substituting (7) and (8) into (6) yields the traditional definition of effective radius, as defined in Hansen and Travis (1974) and Slingo (1989):

 
formula

This illustrates how there is simply one general definition for effective radius or diameter for all clouds, regardless of phase, and that this definition can be understood physically as the representative photon path for all particles in the size distribution. Since reff is simply ½Deff, (5) has been used to treat cirrus radiative properties for some time (Foot 1988; Francis et al. 1994).

3. Use of Deff in solar and terrestrial radiation transfer

a. Formulating the absorption and extinction efficiencies using Deff

The coefficients for absorption and extinction are defined as follows:

 
formula

where N(D) is the size distribution. If Deff is the appropriate dimension for describing particle–radiation interactions for a size distribution N(D), then it is natural to ask what the consequences might be if Qabs and Qext were to be taken outside the integrals of (10) and (11), and solved for in terms of Deff. This results in simple analytical equations appropriate for both water and ice clouds:

 
formula

where Qabs and Qext are efficiences representing the entire N(D). While it is not being suggested that (12) and (13) are mathematically consistent with their definitions, what is being suggested in the following hypothesis. When N(D) are sufficiently narrow, βabs and/or βext may be approximated by single particle solutions given by Deff in all or some spectral regions. In such cases, Qabs and Qext can be determined from Deff, and βabs and/or βext can be estimated from (12) and (13). Expressions for Qabs, Qext, and Pt are given below, and (12) and (13) are tested against Mie theory in the next section.

Assuming a gamma size distribution of the form

 
N(D) = NoDν exp(−ΛD),
(14)

and mass and projected area dimensional power-law expressions for m and P,

 
formula

a general expression for the size distribution projected area is given as

 
formula

where D is the maximum particle dimension. Expressions (15) and (16) are defined for various ice particle shapes in Mitchell (1996) and Mitchell et al. (1996b). The parameters ν, Λ, and No can be obtained from measured N(D) properties as described in Mitchell (1991):

 
formula

where D is the N(D) mean D, and Dm is the D that divides the N(D) mass into equal parts.

Substituting (4) into (2) but in terms of spheres having the same De value, we have

 
Q
abs,ADA = 1 − exp(−8πniDeff/3λ),
(21)

where Qabs,ADA is the absorption efficiency representing the entire size distribution based on ADA. For a general solution for water and ice clouds, the complete expression for Qabs as parameterized in M00 can be used, which includes the processes of internal reflection/refraction and photon tunneling. These processes were also expressed in terms of an “effective photon path,” or de. Consistent with section 2, Qabs may be formulated for all absorption processes in terms of Deff:

 
Q
abs = (1 + C1 + C2)
Q
abs,ADA.
(22)

The term C1 accounts for absorption contributions from internal reflection and refraction:

 
C1 = a1 exp(−8;piniDeff/3λ),
(23)

where a1 was found to be about 0.25 in M00. The photon tunneling term C2 in M00 is expressed in terms of Deff as

 
formula

where

 
ta = 0.7393nr − 0.6069,
(25)

εo and kmax are defined in M00, and nr is the real component of the refractive index. The term tf is the tunneling factor, and varies from 0 to 1.0, depending on ice crystal habit and aspect ratio (Baran et al. 2001). A tf of 1.0 gives the tunneling contribution for a sphere, as predicted by Mie theory (within 10% accuracy), while tf = 0 assumes tunneling is negligible. Tunneling is a process by which radiation beyond a particle's physical cross section is absorbed if the particle is a “black body” (i.e., all incident radiation is absorbed). The term ta predicts the maximum potential absorption tunneling contribution for a given wavelength, while εo determines the Deff/λ value kmax, where this maximum occurs.

Equation (12) is now solved using (17) and (22) for size parameter xe > 1, where xe = πDeff/λ. When xe < 0.3, the Rayleigh Qabs is used. In terms of Deff, Qabs is

 
Q
abs = −(4πDeff/λ) Im[(n2 − 1)/(n2 + 2)],
(26)

where n is the complex index of refraction and Im indicates only the imaginary part is taken. When 0.3 < xe < 1, the “bridging functions” described in M00 are used to define Qabs, based on xe.

Like absorption, Qext in M00 may also be formulated in terms of Deff. For xe > 3, Qext is given as

 
Q
ext = (1 + 0.5C2)
Q
ext,ADA +
Q
edge,
(27)

where

 
Q
ext,ADA = 4Re[K(t)].
(28)

Here, Re indicates only the real part of K(t) is used:

 
formula

n is the complex refractive index, and i = (−1)1/2. The term Qedge in (27) gives the contribution of surface waves, often called edge effects, and is parameterized in terms of Deff as

 
Q
edge = a6[1 − exp(−0.06xe)]x−2/3e,
(31)

where a6 may range from about 1 to 2, and can be well approximated for water clouds as 1.0 (M00). For ice clouds, comparisons of laboratory measurements of Qext with this radiation scheme and T-matrix calculations reveal a6 = 0 for ice clouds (Mitchell et al. 2001). For xe < 3, Qext is described in M00 by the functions joining the Rayleigh and Mie regions and by substituting xe for x in the Rayleigh scattering efficiency, making use of the fact that Qext = Qsca + Qabs.

Two minor improvements have been made to the M00 scheme for ice spheres, which are implemented in this work. First, it was found that the dependence of tunneling on nr changed when nr > 1.55 or ni > 0.40. When nr or ni exceeded these values, ta was given by

 
ta = 0.3109nr + 0.0572.
(32)

Second, when xenr < 2.15, the Deff solution for βext was used, since a single particle solution yielded better agreement with Mie theory in this region.

b. Testing with Mie theory: Water clouds

The above expressions for βabs and βext will now be compared with numerical Mie theory integrations over size distributions of water droplets. Since Pt in (12) has no dependence on radiation, only Qabs needs to be compared with Mie theory. However, Mie theory values of Qabs must correspond to the size distribution, and hence βabs and βext from Mie theory is divided by Pt to yield Qabs,Mie and Qext,Mie:

 
formula

Values of Qabs and Qext based on M00, referred to as Qabs,M00 and Qext,M00, were determined similarly.

Size distributions were described by (14), where D and ν are specified and LWC is arbitary. Size spectra N(D) in water clouds often have relatively little dispersion around D in comparison with ice clouds, resulting in relatively high ν values [see Eq. (18)]. In water clouds, ν values typically range from 2 to 40, with values of 4 to 20 most common (M00). This is illustrated in Fig. 1, where two N(D) having ν values of 4 and 20 are contrasted. The D for both N(D) is 15 μm. These N(D) will be used to test the above formulation of Qabs and Qext, since they roughly encompass the range of ν in most water clouds. As will be shown later, it is ν that determines how accurately Deff can be used to calculate Qabs. The Deff corresponding to the ν values of 4 and 20 (D = 15 μm) is 21.0 and 16.4 μm, respectively.

Fig. 1.

Two size distributions illustrating the typical range of dispersion about the mean for water clouds. These are used to test the performance of the Deff expressions for βabs and βext [Eqs. (12) and (13)] relative to Mie theory and modified ADA, as shown in Figs. 4–8 

Fig. 1.

Two size distributions illustrating the typical range of dispersion about the mean for water clouds. These are used to test the performance of the Deff expressions for βabs and βext [Eqs. (12) and (13)] relative to Mie theory and modified ADA, as shown in Figs. 4–8 

First, let us test the hypothesis that a single particle solution for βabs based on Deff is feasible when the N(D) is sufficiently narrow. Using the narrow N(D) in Fig. 1 (ν = 20), βabs was calculated from (10) via numerical integration, where Qabs is determined from Mie theory for each size bin. Then Qabs was determined from (33), and is shown by the solid curve in Fig. 2. The dashed curve in Fig. 2 gives Qabs based on Deff, also predicted by Mie theory. Clearly Deff as defined in (5) [or reff defined in (6)] provides a good estimate of Qabs when spectra are narrow. Also shown in Fig. 2, by the dotted curve, is Qabs predicted from Mie theory when the “generalized effective size,” or Dge, is used in Mie code to represent the N(D) instead of Deff. Fu (1996) and Fu et al. (1998) recommended using Dge as the characteristic radiative dimension for N(D) in ice clouds. But since hexagonal columns were assumed in deriving Dge: Dge = 0.7698 Deff. Therefore the photon path implied by Dge is less than predicted by Deff by this factor, causing Qabs to be lower for Dge in Fig. 2. This is not to say that radiation schemes using Dge are flawed, but one could argue that Deff is a more meaningful physical quantity than Dge, since Deff gives the appropriate N(D) photon path for Mie calculations. Mie calculations based on Deff provide good estimates for βabs when N(D) are sufficiently narrow.

Fig. 2.

Absorption efficiencies for the low dispersion (ν = 20) size distribution in Fig. 1, based on a numerical integration of Eq. (10) using Mie theory (solid curve), Deff using Mie theory (dashed curve), and the generalized effective size Dge using Mie theory (dotted curve)

Fig. 2.

Absorption efficiencies for the low dispersion (ν = 20) size distribution in Fig. 1, based on a numerical integration of Eq. (10) using Mie theory (solid curve), Deff using Mie theory (dashed curve), and the generalized effective size Dge using Mie theory (dotted curve)

As in Fig. 2, explicit Mie integral solutions (solid curve) are compared with Mie solutions based on Deff in Fig. 3, except this time, ν = 4 when calculating Qabs. The greater N(D) dispersion about D renders the Deff solution less accurate, as will be demonstrated further. This is a general finding, as demonstrated by results from other combinations of D and ν (not shown).

Fig. 3.

Absorption efficiencies for the high dispersion (ν = 4) size distribution in Fig. 1, based on a numerical integration of Eq. (10) using Mie theory (solid curve), and Deff using Mie theory (dashed curve)

Fig. 3.

Absorption efficiencies for the high dispersion (ν = 4) size distribution in Fig. 1, based on a numerical integration of Eq. (10) using Mie theory (solid curve), and Deff using Mie theory (dashed curve)

Testing of the Deff expressions from the previous section is described in Figs. 4–6 for the case of ν = 20, D = 15 μm; and in Figs. 7 and 8 for the case of ν = 4, D = 15 μm. In Fig. 4, for Qabs where ν = 20, the integral Mie solution is represented by the solid curve, the dashed curve is the parameterization of M00, and the dotted curve is from (22) and (26) based on Deff. Lower in the figure, the long-dashed curve gives the contribution to Qabs from photon tunneling based on M00, while the dotted–dashed curve gives the contribution from internal reflection/refraction. In Fig. 5, for Qext where ν = 20, the same curve convention applies except the dotted–dashed curve now gives the contribution from edge effects. It is important to note that the Deff scheme gives single particle solutions, and that as the N(D) dispersion decreases (i.e., ν becomes larger), Qabs,Mie and Qabs,M00 calculated via (33) will converge toward the single particle solution of Qabs, given by (22), and Qext,Mie and Qext,M00 calculated via (34) will converge toward the single particle solution of Qext, given by (27). Both the Mie and M00 curves are calculated explicitly from N(D). Since the Deff scheme is based on the parameterizations in M00, differences between the Deff and M00 schemes are only due to N(D) effects (which M00 accounts for but Deff does not). Errors relative to Qabs,Mie are shown in Fig. 6 for Qabs from (22) (solid curve), and for Qabs,M00 (dotted curve). Due to the low N(D) dispersion, errors relative to Qabs,Mie are low (generally within 10%) for both methods. The error “spikes” for λ < 1.6 μm, where absorption is weak, are due to resonance effects where specific frequencies resonate within the droplet, greatly extending the photon path and absorption. Errors relative to Qext,Mie (not shown) were ≤16% for Qext from (27) and were ≤7% for Qext,M00. It is seen in Fig. 5 that for λ > 5 μm, the DeffQext, Qext, behaves similar to Qext,Mie and Qext,M00. This is a manifestation of the dispersion principle noted above, since ν is relatively large. Also, when absorption becomes sufficiently strong, wave interference effects, which cause the large oscillations in Qext over xe dampen and tend to blend together—another reason why relative agreement is found for λ > 5 μm. Moreover, wave interference oscillations broaden and ultimately vanish as xe decreases. When absorption is weaker and xe > 1, which is often true when λ ≲ 5 μm (see Fig. 5), then interference oscillations manifest for Qext, whereas N(D) effects otherwise dampen these oscillations.

Fig. 4.

Comparison of Mie theory (solid), the modified ADA (dashed), and the Deff (dotted) calculation of Qabs using the low-dispersion size distribution in Fig. 1. The long-dashed curve gives the tunneling contribution and the dotted–dashed curve gives the contribution of internal reflection/refraction to Qabs, based on the modified ADA

Fig. 4.

Comparison of Mie theory (solid), the modified ADA (dashed), and the Deff (dotted) calculation of Qabs using the low-dispersion size distribution in Fig. 1. The long-dashed curve gives the tunneling contribution and the dotted–dashed curve gives the contribution of internal reflection/refraction to Qabs, based on the modified ADA

Fig. 7.

Comparison of Mie theory (solid), the modified ADA (dashed), and the Deff (dotted) calculation of Qabs using the high-dispersion size distribution for water clouds in Fig. 1. The long-dashed curve gives the tunneling contribution and the dotted–dashed curve gives the contribution of internal reflection/refraction to Qabs, based on the modified ADA

Fig. 7.

Comparison of Mie theory (solid), the modified ADA (dashed), and the Deff (dotted) calculation of Qabs using the high-dispersion size distribution for water clouds in Fig. 1. The long-dashed curve gives the tunneling contribution and the dotted–dashed curve gives the contribution of internal reflection/refraction to Qabs, based on the modified ADA

Fig. 8.

The Qabs errors in Fig. 7 relative to Mie theory, for the modified ADA (dotted) and the Deff parameterization (solid). Errors correspond to the high dispersion size distribution

Fig. 8.

The Qabs errors in Fig. 7 relative to Mie theory, for the modified ADA (dotted) and the Deff parameterization (solid). Errors correspond to the high dispersion size distribution

Fig. 5.

As in Fig. 4, but for extinction efficiency Qext using the same curve labeling convention as Fig. 4 except the dotted–dashed curve gives the contribution of edge effects to Qext

Fig. 5.

As in Fig. 4, but for extinction efficiency Qext using the same curve labeling convention as Fig. 4 except the dotted–dashed curve gives the contribution of edge effects to Qext

Fig. 6.

The Qabs errors in Fig. 4 relative to Mie theory, for the modified ADA (dotted) and the Deff parameterization (solid)

Fig. 6.

The Qabs errors in Fig. 4 relative to Mie theory, for the modified ADA (dotted) and the Deff parameterization (solid)

The same analysis is repeated for absorption in Figs. 7 and 8 for ν = 4, D = 15 μm. It is seen here that errors for Qabs are greater than for Qabs,M00, due to the failure of Qabs to account for the greater N(D) dispersion. Nonetheless, Qabs errors in Fig. 8 are within 12% (excepting a few resonance spikes), which is probably acceptable for most purposes. These Qabs errors are typical for ν = 4, and represent the highest likely to occur for most water clouds, due to the high N(D) dispersion. Therefore, the above Deff parameterization for Qabs and βabs may find useful application as a rapid, yet reasonably accurate, treatment of radiative properties for water clouds. On the other hand, the above Deff parameterization for Qext and βext may not be sufficiently accurate for some applications, as shown in Fig. 5, especially at near and midinfrared wavelengths. When ν = 4, the errors in Qext were ≤16%, as with the ν = 20 case, except for λ near 5.3 μm, where errors reached 25%. The Qext parameterization based on Deff may be useful for broadband calculations.

Since the zero scattering approximation (Paltridge and Platt 1976), which requires only βabs, is usually sufficiently accurate for terrestrial radiation transfer, the Deff parameterization of βabs may satisfy most needs for radiation transfer in water clouds at terrestrial wavelengths.

The close agreement between Qabs from Mie theory based on Deff and Qabs from numerical Mie integrations over the N(D), and between Qabs from (22) and Qabs from numerical Mie integrations, strongly indicate that Deff as defined here is a physically meaningful radiative parameter, based on the concept of an effective photon path. This also indicates that when formulating the anomalous diffraction approximation, the proper dimension to apply in (2) is de as defined in (1). The radiative significance of de and Deff supports the use of this formulation of ADA, based on the V/P ratio.

c. Testing with Mie theory and modified ADA: Ice clouds

In this section, the above Deff parameterization for Qabs or βabs, and Qext or βext, will be tested with Mie theory for an exponential N(D) of ice spheres, ν = 0 and D = 15 μm. It is also shown that the M00 scheme accuracy is similar for ice spheres as for water droplets having similar N(D).

A means of applying the M00 scheme to ice clouds is presented. To evaluate the error introduced by the absence of N(D) effects in the Deff parameterization, Deff parameterization results will be compared against M00 results [note the only difference between approaches is that M00 includes N(D) effects]. Such an evaluation will be performed for an N(D) typical of tropical cirrus, which are bimodal, with relatively high concentrations of small ice crystals for D < 100 μm.

1) Testing with Mie theory

Until recently, N(D) in ice clouds were often assumed to be monomodal and exponential, where ν = 0 (e.g., Lo and Passarelli 1982; Mitchell 1988). While this is often true for D > Do, where Do ≈ 1 mm for frontal clouds (Herzegh and Hobbs 1985) and Do ≈ 100 μm in cirrus clouds (e.g., McFarquhar and Heymsfield 1996; Mitchell et al. 1996a), ice cloud N(D) are bimodal in the sense that a small particle mode exists for D < Do, containing high ice crystal concentrations relative to the large particle mode (e.g., Heymsfield and Platt 1984; McFarquhar and Heymsfield 1996, 1997; Ryan 1996, 2000; Platt 1997).

To begin, a simple exponential N(D) of ice spheres (not shown) is used to compare Mie theory with the M00 and Deff parameterizations, as shown in Figs. 9–12, where D = 15 μm, ν = 0, and Deff = 45 μm. This D would correspond to an ice particle maximum dimension of 20 to 40 μm approximately. The curve labeling convention is the same as for water clouds, described above. Figures 9 and 10 show how the greater dispersion of exponential spectra introduce larger errors for Qabs regarding the Deff parameterization. Such errors could be troublesome for remote sensing based on relative brightness temperature differences in the window region (8–13 μm), where the Deff parameterization errors fluctuate between −7% and +10%. This smooths out the true variation of Qabs in this region, as shown in Fig. 9. In contrast, the M00 parameterization matches Qabs,Mie rather well. The results for Qext are shown in Figs. 11 and 12, with errors slightly less than obtained for water clouds when λ < 20 μm, due to the larger Deff. The greatest errors occur between the Mie and Rayleigh regimes, where empirical bridging functions are used. Comparisons based on many other N(D) varying D and ν yielded results similar to these.

Fig. 9.

Comparison of Mie theory (solid), the modified ADA (dashed), and the Deff (dotted) calculation of Qabs for an exponential size distribution of ice spheres, where D = 15 μm, ν = 0, and Deff = 45 μm. The long-dashed and dotted–dashed curves show contributions from tunneling and internal reflection/refraction to Qabs, respectively

Fig. 9.

Comparison of Mie theory (solid), the modified ADA (dashed), and the Deff (dotted) calculation of Qabs for an exponential size distribution of ice spheres, where D = 15 μm, ν = 0, and Deff = 45 μm. The long-dashed and dotted–dashed curves show contributions from tunneling and internal reflection/refraction to Qabs, respectively

Fig. 10.

The Qabs errors in Fig. 9 relative to Mie theory, for the modified ADA (dotted) and the Deff parameterization (solid)

Fig. 10.

The Qabs errors in Fig. 9 relative to Mie theory, for the modified ADA (dotted) and the Deff parameterization (solid)

Fig. 11.

Same as Fig. 9, but for Qext, and the dotted–dashed curve shows edge effect contributions

Fig. 11.

Same as Fig. 9, but for Qext, and the dotted–dashed curve shows edge effect contributions

Fig. 12.

The Qext errors in Fig. 11 relative to Mie theory, for the modified ADA (dotted) and the Deff parameterization (solid)

Fig. 12.

The Qext errors in Fig. 11 relative to Mie theory, for the modified ADA (dotted) and the Deff parameterization (solid)

2) Testing using realistic N(D) and ice crystal shape

In this section, we will determine whether Deff can be used in the manner described above to accurately calculate the radiative properties of ice clouds. This will be done by comparing Qabs as determined from Deff with Qabs as determined from the explicit scheme of Mitchell et al. (1996b, hereafter M96).

Actually, the M96 scheme used here is a synthesis of M96 and M00, with ice size spectra transformed into an N(D) of ice spheres. The N(D) of ice particles can be transformed into an N(D) of ice spheres having the same photon path as the parent ice crystals, while at the same time preserving the projected area of the N(D). Since the absorption and extinction properties of the N(D) only depend on the photon paths and projected areas of the ice crystals, a transformed N(D) of ice spheres that preserves these properties will possess the same absorption/extinction properties as the parent N(D) of ice crystals (M96). This is the same principle used in Grenfell and Warren (1999), who also use a population of ice spheres that conserves a crystal's photon path and projected area. From the measured mean ice particle length D and median mass length Dm, their corresponding photon path equivalent spheres De and Dme are defined as 1.5(V/P), where V is the volume at bulk ice density. The V (i.e., mass) and P expressions are given by (15) and (16), which depend on crystal habit. Equations giving the parameters for the transformed N(D) are given as

 
formula

where e denotes de sphere values. When dealing with bimodal size spectra, the small and large particle modes are transformed separately, and consideration must be given to the fact that the mass and area dimensional relationships may change around D = 100 μm (M96). Details of this modified M96 scheme are given in the appendix.

The above N(D) parameters for the N(D) of de-equivalent spheres can now be used in the radiation treatment of M00. Since this scheme has been validated against Mie theory for N(D) of water droplets and ice spheres, the primary remaining uncertainty is the degree of photon tunneling for various ice crystal shapes. Recently, the degree of tunneling was determined for hexagonal columns (about 60% relative to ice spheres), and the percent differences between laboratory measurements of Qext and Qext predicted by the updated M96 were within 3% on average for wavelengths between 2 and 17 μm (Mitchell et al. 2001). For the purpose of intercomparing the updated M96 and Deff schemes, we will assume no tunneling.

A typical example of N(D) found in tropical cirrus is shown in Fig. 13 in log–linear space, based on the N(D) parameterization of Mitchell et al. (2000). This N(D) parameterization for tropical anvil cirrus was based on in situ microphysical and radiometric measurements taken during the CEPEX experiment in the central equatorial Pacific (McFarquhar and Heymsfield 1996), and on microphysical measurements made near anvil tops in the western equatorial Pacific (Knollenberg et al. 1993) and in tropopause cirrus (Heymsfield 1986). The parameterization predicts N(D) similar to those predicted by the anvil cirrus parameterization of McFarquhar and Heymsfield (1997). In Mitchell et al. (2000), 2DC probe measurements revealed ν = 0 for the large particle mode, whereas it was assumed ν = 0 for the small particle mode (which was inferred from radiometric measurements; see Mitchell et al. 1998). The Deff for this N(D) is 39 μm, and the mean size of the large particle mode D1 is 100 μm. From Fig. 13, it is clear that the N(D) for ice clouds are dramatically different in form than for water clouds, as described in Fig. 1.

Fig. 13.

Example size distribution characteristic of those sampled in anvil cirrus during CEPEX, as predicted by the scheme of Mitchell et al. 2000. For the large particle mode, D = 100 μm, and for the total distribution, Deff = 39 μm

Fig. 13.

Example size distribution characteristic of those sampled in anvil cirrus during CEPEX, as predicted by the scheme of Mitchell et al. 2000. For the large particle mode, D = 100 μm, and for the total distribution, Deff = 39 μm

The Qabs based on Deff (referred to as Qabs) is now compared with the Qabs determined from the updated M96, referred to as Qabs,M96. The N(D) of Fig. 13 is used for this intercomparison. In Fig. 14, it is seen that Qabs (dashed) and Qabs,M96 (solid) differ appreciably. For the purpose of error evaluation due to neglect of N(D) shape, we can view Qabs,M96 as “ground truth,” since both approaches are the same except that Qabs,M96 is an analytical solution of (10). Note how Qabs,M96 in the window region (8 μm < λ < 13 μm) is considerably lower than Qabs. Errors for Qabs relative to Qabs,M96 are given in Fig. 15. It is seen that errors can be as high as 24%, and fluctuate widely in the window region between 3% and 22%. When tunneling as predicted for spheres was assumed, the errors were significantly greater.

Fig. 14.

Based on the N(D) described in Fig. 13, Qabs is predicted by the revised M96 scheme (see appendix) and the Deff parameterization (dashed curve). Differences are due solely to N(D) shape effects

Fig. 14.

Based on the N(D) described in Fig. 13, Qabs is predicted by the revised M96 scheme (see appendix) and the Deff parameterization (dashed curve). Differences are due solely to N(D) shape effects

Fig. 15.

The Qabs errors in Fig. 14 for the Deff parameterization, relative to the revised M96 scheme

Fig. 15.

The Qabs errors in Fig. 14 for the Deff parameterization, relative to the revised M96 scheme

Extinction efficiencies for the N(D) in Fig. 13 are given in Fig. 16, where Qext,M96 (solid curve) is contrasted with Qext via the Deff method. Errors relative to Qext,M96 can exceed 50% when λ < 100 μm. In the near IR, errors are generally within 10%, and could be much less for band calculations.

Fig. 16.

As in Fig. 14, but for Qext (i.e., the revised M96 scheme corresponds to the solid curve; the Deff parameterization corresponds to the dashed curve)

Fig. 16.

As in Fig. 14, but for Qext (i.e., the revised M96 scheme corresponds to the solid curve; the Deff parameterization corresponds to the dashed curve)

Results similar to this were obtained for other tropical anvil N(D) obtained at various D values. The more abrupt the transition between N(D) modes (i.e., the Deff difference between modes increases), the greater the error via the Deff method. When ν for the small mode was increased to 3 while conserving small mode Deff, Deff method errors were virtually unchanged. A new N(D) parameterization for midlatitude cirrus (Ivanova et al. 2001), based on 966 N(D) between about −20° and −60°C, also predicts bimodal N(D) similar to the N(D) in Fig. 13. Errors associated with the Deff method for these midlatitude N(D) were usually similar to those found here.

These analyses indicate that Eqs. (12) and (13) are not sufficient for predicting the absorption and scattering properties of ice clouds at terrestrial wavelengths. The reason that Deff overestimates Qabs in ice clouds is due to the much higher relative concentration of small particles, producing greater dispersion, in contrast to water cloud N(D). The smaller crystals have relatively low values of Qabs, which, when integrated over N(D), result in lower overall Qabs values than this Deff approach predicts. This Deff parameterization for Qext, being a single particle solution, fails to smooth out the oscillations due to wave interference effects (M00). In nature, the greater the N(D) dispersion, the more diverse the Qext contributions from individual crystals become, thus smoothing out these oscillations. These findings make it desirable to adopt a different approach for determining terrestrial radiative properties in ice clouds, an approach not using a single particle in explicit solutions [e.g., Mie theory or Eqs. (12) and (13)].

One should note that this Deff approach would be satisfactory for ice clouds if ice cloud N(D) were similar to water clouds, and did not contain relatively high concentrations of small crystals. What is true for water clouds is also true for ice clouds, for a given N(D), and the analysis shown in Figs. 2 and 3 can also be made for ice clouds.

4. Uncertainty in ice cloud radiation schemes using an effective particle size

Most schemes in use today that parameterize ice cloud radiative properties for solar and terrestrial radiation use an “effective particle size” and IWC to represent the size distribution (e.g., Ebert and Curry 1992; Fu 1996; Wyser and Yang 1998; Fu et al. 1998; Yang et al. 2001). The first and last two of these studies treat terrestrial radiation, and all assume that ice cloud radiative properties can be described in terms of only IWC and effective size. Results from the preceding section give cause to reconsider these claims, especially for terrestrial radiation. While Ebert and Curry defined their effective size in terms of an area-equivalent sphere, effective size in the latter four studies was similar in concept to Deff in this study, involving the ratio IWC/Pt.

In this section, the modified M96 scheme described here will be evaluated over a wavelength range of 1 to 1000 μm for N(D) having the same IWC and Deff, but having different dispersion or shape. The findings reveal the uncertainties associated with radiation schemes that describe ice cloud N(D) solely in terms of IWC and effective size. Three N(D) are considered here, referred to as N(D) no. 1, N(D) no. 2, and N(D) no. 3. These are illustrated in Fig. 17. N(D) no. 1 is based on the tropical cirrus bimodal parameterization of Mitchell et al. (2000), with Deff = 25.7 μm, D1 = 74 μm, Dsm = 12.8 μm, and ν = 0 for both large and small N(D) modes. Here, D1 and Dsm refer to the mean size of the large and small particle N(D) modes, respectively. N(D) no. 2 also has a Deff of 25.7 μm, and Dsm = 13.7 μm. However, ν for the small mode (νsm) is now 3.0, which narrows the small N(D) to exclude larger particles. This requires that D1 = 100 μm to maintain a Deff of 25.7 μm, and ν1 = 0 as before. Lastly, N(D) no. 3 is a monomodal exponential (ν = 0) N(D), where D = 22.8 μm and Deff = 25.6 μm. As with the analyses above, βabs and βext will be determined from each scheme and divided by Pt to yield Qabs and Qext values for each N(D). Planar polycrystals were assumed. Ice clouds are primarily comprised of such complex shapes (Mitchell 1996a; M96; Heymsfield and Iaquinta 2000; Korolev et al. 1999, 2000). Since tunneling is shape-dependent with contributions for planar crystals lower than for hexagonal columns (Baran et al. 2001), it was assumed that tf = 0.5 for the small mode N(D) and that tf = 0.3 for the large mode N(D).

Fig. 17.

Three size distributions having the same Deff value of 25.7 μm and IWC of 10 mg m−3, but having different shapes

Fig. 17.

Three size distributions having the same Deff value of 25.7 μm and IWC of 10 mg m−3, but having different shapes

Results for absorption are given in Fig. 18, where Qabs,M96 for different N(D) are indicated by the different line patterns. Here, Qabs,M96 is lowest for N(D) no. 2, since this N(D) is characterized by high concentrations of crystals having relatively low values of Qabs. Crystal sizes in the small mode for N(D) no. 1 are somewhat larger in the “tail” of this distribution, due to its broader dispersion, and such crystals have larger Qabs values. For instance, Deff for the small mode in N(D) no. 1 is 15.2 μm, while Deff for the small mode in N(D) no. 2 is only 8.4 μm. This results in a lower Qabs for the small mode of N(D) no. 2, which impacts the overall Qabs value. This phenomena is further illustrated when the small mode is totally missing, as shown for exponential spectra via N(D) no. 3, which exhibits the highest Qabs,M96 values. It is noteworthy that even though comparable values of Dsm are found for N(D) no. 1 and N(D) no. 2 (12.8 and 13.7 μm, respectively), subtle differences in small mode dispersion give rise to large differences in Qabs,M96 values. Moreover, these Dsm values appear characteristic of those measured in tropical cirrus (McFarquhar and Heymsfield 1997; Puechel et al. 1997; McFarquhar et al. 2000).

Fig. 18.

Based on the N(D) described in Fig. 17, Qabs (i.e., βabs/Pt) is predicted by the updated M96 scheme. Planar polycrystals were assumed, and tunneling factors of 0.5 and 0.3 were used in the updated M96 scheme for the small and large N(D) modes, respectively

Fig. 18.

Based on the N(D) described in Fig. 17, Qabs (i.e., βabs/Pt) is predicted by the updated M96 scheme. Planar polycrystals were assumed, and tunneling factors of 0.5 and 0.3 were used in the updated M96 scheme for the small and large N(D) modes, respectively

Since the value of ν characterizing the small mode in tropical cirrus is not well known, either ν characterizing N(D) no. 1 or N(D) no. 2 could be common. Therefore the differences in Qabs,M96 between N(D) no. 1 and N(D) no. 2 may be representative of our uncertainty regarding complete bimodal N(D), while the Qabs,M96 differences between N(D) no. 2 and N(D) no. 3 may be viewed as the maximum difference likely to be obtained between different parameterizations of ice cloud radiative properties. This is because the simple exponential N(D) is still used to represent N(D) in ice clouds, and because measurements of N(D) from which parameterizations of ice cloud single scattering properties are based often consider primarily the large mode (e.g., D ≳ 60 μm), which tends to be approximately exponential.

Results for extinction are given in Fig. 19. N(D) no. 2, where the small mode N(D) had the greatest impact, is characterized by the highest Qext,M96 values at shorter wavelengths, approaching the geometric optics regime, and generally by lower Qext,M96 values at longer wavelengths. The behavior exhibited for all N(D) is primarily a complex function of refractive index and wave interference phenomena, the latter depending strongly on the small mode Deff value. The variability in the window region and at longer wavelengths is somewhat remarkable.

Fig. 19.

Same as Fig. 18, except Qext is compared

Fig. 19.

Same as Fig. 18, except Qext is compared

Percent differences between N(D) no. 2 and N(D) no. 3 regarding Qabs, Qext and the single scattering albedo ωo are shown in Fig. 20, where ωo = 1 − Qabs/Qext. Percent differences for λ < 100 μm reach 44% for Qabs, 64% for Qext, and 18% for ωo. For λ ≈ 100 μm, differences reach 100% for Qext, and 22% for ωo. In the calculation of ωo, differences tend to cancel to a first approximation throughout much of the near IR. Estimates of possible differences among radiation schemes that attempt to estimate the contribution of small ice crystals (e.g., 3 μm < D < 60 μm), such as radiation schemes based on the N(D) parameterizations of McFarquhar and Heymsfield (1997), Wyser (1998), Ryan (2000), Ivanova et al. (2001), and Mitchell et al. (2000), are given in Fig. 21, where percent differences between N(D) no. 1 and N(D) no. 2 are shown. Percent differences in the thermal IR and beyond reach 26% for Qabs, 96% for Qext, and 48% for ωo. These differences clearly demonstrate that ice cloud radiative properties depend on the assumed N(D) shape as well as Deff and IWC.

Fig. 20.

Percent differences between the Qabs, Qext, and ωo predicted from N(D) no. 3 and N(D) no. 2 (in Fig. 17), based on the revised M96 scheme

Fig. 20.

Percent differences between the Qabs, Qext, and ωo predicted from N(D) no. 3 and N(D) no. 2 (in Fig. 17), based on the revised M96 scheme

Fig. 21.

Same as Fig. 20, except the percent differences correspond to N(D) no. 1 and N(D) no. 2 in Fig. 17 

Fig. 21.

Same as Fig. 20, except the percent differences correspond to N(D) no. 1 and N(D) no. 2 in Fig. 17 

These differences are based on a single value of Deff. While Deff ≈ 25 μm is common for tropical cirrus, larger Deff values will be associated with lower uncertainties as Qabs and Qext approach their limiting values. Smaller Deff values may be associated with uncertainties similar to those shown here, or possibly greater uncertainties.

a. Comparisons using different size distribution schemes

Next, differences in Qabs,M96 and Qext,M96 are evaluated for a single wavelength as a function of Deff, using two N(D) parameterizations: one for tropical cirrus (Mitchell et al. 2000) and one for midlatitude cirrus (Ivanova et al. 2001). Both parameterizations estimate the concentrations of small ice crystals (3 μm < D < 100 μm). Hexagonal columns with a tunneling factor of 0.60 (60% tunneling relative to ice spheres) are assumed, based on Mitchell et al. (2001). Efficiency differences between these N(D) parameterizations are estimates of how tropical and midlatitude cirrus differ radiatively for a given Deff. This is shown in Fig. 22 for Qabs and Qext as a function of Deff, evaluated at wavelengths of 3.73, 8.48, and 11.1 μm. Channels corresponding to these wavelengths are used on polar orbiting and Geostationary Operational Environmental Satellites (GOES), and may be used to retrieve cirrus physical properties (e.g., Stubenrauch et al. 1999; Mitchell and d'Entremont 2000). Both N(D) schemes predict Deff as a function of temperature. For both the midlatitude and tropical schemes, Deff is predicted for the temperature range −20° to −100°C. The unrealistically cold temperature of −100°C was used to provide a realistic range of Deff values. Note that assuming an ice crystal shape other than hexagonal columns would decrease Deff considerably, and this combined with natural variability makes the range of Deff reasonable for each scheme. It is seen that the range for Deff is narrower for the midlatitudes than for the Tropics. In the midlatitude scheme, the small particle mode intensifies with increasing temperature, eventually causing a reversal in Deff at the warmest temperatures in spite of a broadening of the large mode. This creates the “hook” at the end of the solid curves. When λ = 3.73 or 8.48 μm, differences in Qabs for a given Deff are around 10%.

Fig. 22.

Prediction of Qabs and Qext for selected wavelengths as a function of Deff, using two size distribution schemes (tropical and midlatitude) with the updated M96 scheme, and using the Fu (1996) and Fu et al. (1998) schemes based on 28 N(D).

Fig. 22.

Prediction of Qabs and Qext for selected wavelengths as a function of Deff, using two size distribution schemes (tropical and midlatitude) with the updated M96 scheme, and using the Fu (1996) and Fu et al. (1998) schemes based on 28 N(D).

b. Comparisons with the Fu radiation schemes

The results presented suggest that differences between ice cloud radiation schemes can be largely due to the choice of N(D) used to parameterize them. Hence it makes sense to compare results from the above N(D) schemes with results from a radiation scheme based on a priori N(D) information. In the schemes of Fu (1996), Fu et al. (1998), and Yang et al. (2001), 28 or 30 N(D) from midlatitude and tropical cirrus were used to parameterize the single scattering results. Radiative properties from these schemes are parameterized solely in terms of Deff (or Dge; Dge = 0.7698Deff) and IWC. Since these schemes are intended for all cirrus, and since the Fu schemes have high spectral resolution, Qabs and Qext predicted by the Fu (1996) and Fu et al. (1998) schemes have been plotted against Deff given by the tropical N(D) scheme in Fig. 22 (short-dashed curves). Knowing the N(D) projected areas, Qabs and Qext were determined from the Fu scheme as described in (33) and (34), and by noting that βabs = βext(1 − ωo). The use of Deff eliminates the dependence of Qabs and Qext on particle shape to a large degree (Fu 1996), although shape-dependent differences may occur due to different treatments or assumptions regarding the degree of photon tunneling. Therefore, to compare the updated M96 scheme with the Fu scheme most directly, a tunneling factor of 0.6, corresponding to hexagonal columns (Mitchell et al. 2001) was used. If the ice particle–radiation interactions are represented accurately in both schemes, then differences in Qabs and Qext may be due to N(D) effects.

Looking at Fig. 22, it appears that Qabs and Qext from the tropical and/or midlatitude N(D) schemes tend to be in general agreement with Qabs and Qext predicted from the Fu schemes. This could be interpreted that the N(D) used to create the Fu scheme are not too dissimilar in bimodality to those predicted by the tropical and midlatitude N(D) schemes. However, significant differences do exist between the Fu and Mitchell approaches, which may not be due to N(D) assumptions.

The most obvious difference concerns Qabs at 3.7 μm. The wavelength resolution of the Fu scheme could contribute to this discrepancy. This band ranges from 3.4 to 4.0 μm in Fu (1996), and an absorption minimum is located at 3.85 μm. Hence, the integrated mean value of Qabs in this band should be different than the discrete value at 3.7 μm. Moreover, a finite difference time domain (FDTD) calculation taken from Fu et al. (1998) for Deff = 5.6 μm, λ = 3.7 μm, is given by the “o” in Fig. 22, and a T-matrix calculation for a N(D) of hexagonal columns measured in Mitchell et al. (2001) is given by the “X” in Fig. 22 for Deff = 14 μm (courtesy of Anthony Baran). These calculations appear consistent with the Mitchell schemes, and suggest spectral resolution as a factor contributing to the discrepancy.

Another reason for discrepancy between the Fu and Mitchell approaches may lie in the range of Deff used to parameterize the Fu schemes. Large discrepancies exist at the smallest Deff values for both Qabs and Qext. Since the range of Deff in Fu (1996) and Fu et al. (1998) was 24.2 to 169.2 μm, and 14.3 to 168.4 μm, respectively, the curve fits relating Deff to the single scattering calculations may sometimes be weak for Deff < 15 μm.

It is noteworthy that the Wyser and Yang (1998) results for solar radiation also suggested that ice cloud radiative properties depend on N(D) shape, although their conclusions assert that radiative properties only depend on Deff and IWC. Of the four N(D) forms they considered, their power-law N(D) was closest in form to the bimodal N(D) used here. Significant differences in single scattering albedo were observed between their power-law N(D) and the other N(D) for a given value of Deff. Their power-law N(D) exhibited by far the greatest dispersion, with relatively high concentrations of small crystals. Their power-law N(D) results were excluded from their parameterization on the grounds that numerical integrations over such N(D) are less accurate and that such N(D) overestimate the concentrations of ice crystals having D < 20 μm.

5. Summary and conclusions

This study has shown how the concept of effective photon path can be used to understand the physical basis of an “effective diameter,” or Deff. Beginning with this photon path concept, a general definition for Deff was derived for both ice and water clouds. Moreover, the Deff expression for water clouds was twice the value of the “traditional” effective radius definition. Therefore a single definition of Deff is advocated for water and ice clouds, as has been advocated by others (e.g., Foot 1988; Francis et al. 1994) for different reasons.

Simple expressions for the absorption and extinction coefficients, βabs and βext, were derived based on Deff, wavelength and refractive index, and were tested against Mie theory using size distributions [N(D)] of water and ice spheres. For water clouds, the expression for βabs was generally accurate within 12%, while the βext expression was generally accurate within 20% for any wavelength. Using βabs and the zero scattering approximation, this provides a simple means of determining the thermal properties of water clouds. For ice clouds, it was shown that errors in βabs and βext were probably unacceptable for many applications, due to the bimodal nature of ice cloud N(D), with relatively high concentrations of small ice crystals.

It was further demonstrated that the cloud ice water content (IWC) and Deff were not sufficient for describing the radiative properties of ice clouds at thermal wavelengths, and that, in addition, information on the N(D) shape was needed (e.g., degree of bimodality or dispersion about the mean size). For a given Deff and IWC, variations in N(D) shape were shown to produce differences in the N(D) area-weighted efficiencies for absorption (Qabs) and extinction (Qext) by up to 44% and 100%, respectively, at terrestrial wavelengths. In the window region (8–12 μm), these differences reach 30% for Qabs, 48% for Qext, and 18% for the single scattering albedo ωo. Hence, differences in a priori N(D) information could contribute to differences between schemes predicting ice cloud radiative properties. It was also suggested that significant differences between such schemes may arise due to the range of Deff used to parameterize these schemes over a given wavelength band.

To summarize these two primary and independent findings for ice clouds at terrestrial wavelengths, we can say that 1) substantial errors may arise if Deff is used to represent the N(D) in Mie theory or our single particle solutions for βabs and βext, and 2) a single Deff and IWC can apply to multiple N(D)s, with each N(D) having different radiative properties.

One of the implications of this finding is that for satellite retrievals of Deff to be viable for ice clouds, the retrieval algorithms must include implicit assumptions of N(D) shape that are realistic, or N(D) shape parameters must be independently retrieved such that they are not incestuous with retrievals of Deff or other properties. If N(D) shape assumptions are made in these algorithms, then these same assumptions should be adopted in radiation transfer work using the Deff retrievals. With recent and future improvements in measuring the complete N(D) in ice clouds, in situ measurements may provide the needed N(D) shape information that Deff retrievals may require. In fact, considerable progress has already been made in this regard (Heymsfield and Platt 1984; McFarquhar and Heymsfield 1997; Platt 1997; Ryan 2000; Mitchell et al. 2000; Ivanova et al. 2001). One should note that N(D) shape may be a function of cloud type, such as anvil cirrus versus frontal cirrus.

It follows that the treatment of N(D) shape effects would be a desirable feature in an ice cloud radiation scheme, especially in regard to the concentrations of ice crystals having D < 100 μm. Such a scheme is offered here for calculating βabs and βext, described in detail in the appendix.

Acknowledgments

This research was funded entirely by the U.S. Department of Energy, Environmental Sciences Division, Atmospheric Radiation and Measurement (ARM) program, which is gratefully thanked for its support. The findings herein do not necessarily reflect the views of this agency. Dr. Anthony Baran provided the T-matrix calculations in Fig. 22, and is thanked for his contribution. The two reviewers of this paper are gratefully acknowledged for their constructive comments.

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APPENDIX

Analytical Solutions to the Definitions of βabs and βext for Ice Clouds

This appendix describes analytical solutions for the integrals defining the absorption and extinction coefficients, Eqs. (10) and (11), and hence explicitly accounts for size distribution [N(D)] shape. These solutions are given in M00, but are listed below for clarity and completeness. Based on a gamma size distribution as defined in (14), the absorption coefficient βabs is given as

 
formula

while the extinction coefficient βext is given as

 
formula

where

 
formula

and where g = 8πni/3λ, q = i2π(n − 1)/λ is a complex variable (n = complex index of refraction, i = −11/2), Re indicates only the real part of the term is used, Γ is the gamma function, m = 0.50, a6 = 0 for ice clouds and 1.0 for water clouds (Mitchell et al. 2001; M00), σ = π/4 and δ = 2. Note that (A3)–(A7) are only functions of λ, nr, and ni. The computation time required for (A1) and (A2) are orders of magnitude less than Mie theory requires.

The first two terms in (A1) and (A2) are the ADA solutions for βabs and βext. The next two terms in (A1) estimate the contribution of internal reflection/refraction, while the last two terms estimate the contribution of tunneling to βabs. The next two terms in (A2) estimate the contribution of tunneling to βext, while the last term estimates the contribution of edge effects.

The above equations for βabs and βext are accurate to within about 10% relative to Mie theory for ice spheres and xe > 1, where x for a size distribution is given as

 
xe = πDeff/λ.
(A8)

The treatment of βabs and βext for xe < 1 is described in M00.

The size distribution used in (A1) and (A2) is not the “physical” or measured N(D) with D being maximum dimension, but is the N(D) that has been transformed into equivalent de spheres, as described in section 3c(2). This N(D) must account for the fact that the projected area and mass dimensional power-law relations often change near D ≈ 100 μm for a given crystal habit. Regarding section 3c(2), De and Dme must be calculated in accordance with the size regime they fall under, since they are determined by (15) and (16), which may depend on size regime. For example,

 
Dme = 3/2[(αDβm/ρi)/(σDδm)].
(A9)

This approach eliminates much of the error associated with the parameterized Dde relationships used in M96, as documented in Yang et al. (2000), since de is no longer approximated.

The parameter Noe of the transformed N(D), given by (37), requires knowledge of Pt, and Pt requires knowledge of No of the physical size distribution, referred to in (14) and (20). However, the No solution given by (20) is valid only when α and β in (15) are constant. Since they are not (M96), the calculation of No must take this into account:

 
formula

where Do ≈ 100 μm, α1, and β1 correspond to D ≲ 100 μm, and α2 and β2 correspond to D ≳ 100 μm. Equation (A10) can now be solved using the incomplete gamma function (e.g., Mitchell and Arnott 1994) by scaling D with λ such that x = :

 
formula

where xo = Doλ, γ denotes the incomplete gamma function, and Γ denotes the complete gamma function. Since the coefficients in (16) are not necessarily constant either, (17) is not sufficient to determine Pt, and a similar analysis using the incomplete gamma function is required for Pt:

 
formula

Now that Pt is known via (A12), Noe can be solved for via (37), and all of the parameters for the transformed N(D) are now known.

In the Rayleigh regime, the physical N(D) was also transformed. But since βabs and βext depend only on particle volume at bulk density in the Rayleigh regime, the physical N(D) was transformed into a N(D) of equivalent volume or mass spheres. For example,

 
Dme = [6αDβm/(πρi)]1/3.
(A13)

The transformed N(D) parameters νe and Λe were calculated from (35) and (36) as before but using mass (i.e., volume) equivalent spheres, while No for the transformed N(D) was given as

 
formula

The above methodology addresses monomodal size spectra. The same methodology is applied to bimodal size spectra, but each mode of the bimodal N(D) is treated separately to obtain a βabs and βext for each mode. Hence, for the small particle mode (D ≲ 100 μm), denoted N(D)sm, βabs,sm and βext,sm are calculated, and for the large particle mode N(D)1, βabs,1 and βext,1 are calculated, such that the total values are the sum of the mode values: βabs = βabs,sm + βabs,1 and βext = βext,sm + βext,1. The question remains of how to determine Λ, ν, and the IWC corresponding to N(D)sm and N(D)1. This can be estimated by the user, or one could use the relationships in Ryan (2000), or in McFarquhar and Heymsfield (1997) or Mitchell et al. (2000) for tropical anvil cirrus, or in Heymsfield and Platt (1984), Platt (1997), or Ivanova et al. (2001) for midlatitude cirrus. Unfortunately, only the Ivanova et al. and Mitchell et al. schemes describe both N(D) modes as gamma functions.

Footnotes

Corresponding author address: Dr. David L. Mitchell, Atmospheric Sciences Division, Desert Research Institute, 2215 Raggio Parkway, Reno, NV 89512-1095. Email: mitch@dri.edu