## 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 *D*_{eff} 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 *D*_{eff} definition for both water and ice clouds. Effective photon path is the unifying underlying principle behind this universal definition of *D*_{eff}.

Simple equations are formulated in terms of *D*_{eff}, wavelength, and refractive index, giving monochromatic coefficients for absorption and extinction, *β*_{abs} and *β*_{ext}, throughout the geometric optics, Mie, and Rayleigh regimes. These expressions are tested against Mie theory, showing the limitations of the use of *D*_{eff} 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 *β*_{abs} based on *D*_{eff} compares well with *β*_{abs} predicted from Mie theory, providing a new and efficient means of treating radiation transfer at terrestrial wavelengths. The *D*_{eff} expression for *β*_{ext} 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 *x*_{e} ≳ 50, where *x*_{e} = *πD*_{eff}/*λ.* Since the *D*_{eff} expressions for *β*_{abs} and *β*_{ext} 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 *D*_{eff} expressions for *β*_{abs} and *β*_{ext} 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 *μ*m) ice crystals. These small crystals have relatively low absorption efficiencies, causing the *N*(*D*)-integrated *β*_{abs} to be lower than *β*_{abs} based on *D*_{eff}. 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 *β*_{abs} and *β*_{ext} are provided in the a1^{appendix}, 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 *β*_{abs} and *β*_{ext} on the size distribution can be described solely in terms of *D*_{eff} and ice water content (IWC). This assumption was tested by comparing the *N*(*D*) area-weighted efficiencies for absorption and extinction, *Q*_{abs} and *Q*_{ext}, for three *N*(*D*) that have the same IWC and *D*_{eff}, but for which *N*(*D*) shape differs. Analytical solutions for *β*_{abs} and *β*_{ext} were used, which explicitly treat *N*(*D*) shape, over a wavelength range of 1.0 to 1000 *μ*m. For a chosen *D*_{eff} value, uncertainties (percent differences) resulting only from *N*(*D*) shape differences reached 44% for *Q*_{abs}, 100% for *Q*_{ext}, and 48% for the single scattering albedo *ω*_{o} 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 (*r*_{eff}) or diameter (*D*_{eff}), 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 *r*_{eff} or *D*_{eff} 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 *D*_{eff} for ice clouds is not well understood. Even for the same size distribution and ice crystal shape, the Fu definition of *D*_{eff} differs from the Ebert and Curry definition by about 60% (Wyser 1998; Fu 1996). Some definitions for *D*_{eff} 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 *D*_{eff} 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 *D*_{eff} 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 *D*_{eff} are provided for calculating the absorption and extinction coefficients *β*_{abs} and *β*_{ext} 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 *D*_{eff} and IWC to calculate *β*_{abs} and *β*_{ext} is critically evaluated in sections 3 and 4. A summary and concluding remarks are given in section 5. A new scheme for calculating *β*_{abs} and *β*_{ext} 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 *d*_{e} 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 *d*_{e} 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:

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

where *n*_{i} is the imaginary part of the refractive index and *λ* is the wavelength. As defined in (2), *d*_{e} 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 *d*_{e} 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 *d*_{e}. Using ice spheres as an example in (1), *m* = *ρ*_{i}(*πD*^{3}/6) and *P* = *πD*^{2}/4, giving

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

where *P*_{t} is the total projected area of the size distribution. Based on (3), the effective diameter of the size distribution should then be

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 *r*_{eff} is equivalent to (5). Defining *r*_{eff} as ½*D*_{eff}, then

where LWC is the liquid water content, and *ρ*_{w} is the density of liquid water. Defining the LWC and *P*_{t} as

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):

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 *r*_{eff} is simply ½*D*_{eff}, (5) has been used to treat cirrus radiative properties for some time (Foot 1988; Francis et al. 1994).

## 3. Use of *D*_{eff} in solar and terrestrial radiation transfer

### a. Formulating the absorption and extinction efficiencies using D_{eff}

The coefficients for absorption and extinction are defined as follows:

where *N*(*D*) is the size distribution. If *D*_{eff} 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 *Q*_{abs} and *Q*_{ext} were to be taken outside the integrals of (10) and (11), and solved for in terms of *D*_{eff}. This results in simple analytical equations appropriate for both water and ice clouds:

where *Q*_{abs} and *Q*_{ext} 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, *β*_{abs} and/or *β*_{ext} may be approximated by single particle solutions given by *D*_{eff} in all or some spectral regions. In such cases, *Q*_{abs} and *Q*_{ext} can be determined from *D*_{eff}, and *β*_{abs} and/or *β*_{ext} can be estimated from (12) and (13). Expressions for *Q*_{abs}, *Q*_{ext}, and *P*_{t} are given below, and (12) and (13) are tested against Mie theory in the next section.

Assuming a gamma size distribution of the form

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

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

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 *N*_{o} can be obtained from measured *N*(*D*) properties as described in Mitchell (1991):

where *D* is the *N*(*D*) mean *D,* and *D*_{m} is the *D* that divides the *N*(*D*) mass into equal parts.

where *Q*_{abs,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 *Q*_{abs} 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 *d*_{e}. Consistent with section 2, *Q*_{abs} may be formulated for all absorption processes in terms of *D*_{eff}:

The term *C*_{1} accounts for absorption contributions from internal reflection and refraction:

where *a*_{1} was found to be about 0.25 in M00. The photon tunneling term *C*_{2} in M00 is expressed in terms of *D*_{eff} as

where

*ε*_{o} and *k*_{max} are defined in M00, and *n*_{r} is the real component of the refractive index. The term *t*_{f} is the tunneling factor, and varies from 0 to 1.0, depending on ice crystal habit and aspect ratio (Baran et al. 2001). A *t*_{f} of 1.0 gives the tunneling contribution for a sphere, as predicted by Mie theory (within 10% accuracy), while *t*_{f} = 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 *t*_{a} predicts the maximum potential absorption tunneling contribution for a given wavelength, while *ε*_{o} determines the *D*_{eff}/*λ* value *k*_{max}, where this maximum occurs.

Equation (12) is now solved using (17) and (22) for size parameter *x*_{e} > 1, where *x*_{e} = *πD*_{eff}/*λ.* When *x*_{e} < 0.3, the Rayleigh *Q*_{abs} is used. In terms of *D*_{eff}, *Q*_{abs} is

where *n* is the complex index of refraction and Im indicates only the imaginary part is taken. When 0.3 < *x*_{e} < 1, the “bridging functions” described in M00 are used to define *Q*_{abs}, based on *x*_{e}.

Like absorption, *Q*_{ext} in M00 may also be formulated in terms of *D*_{eff}. For *x*_{e} > 3, *Q*_{ext} is given as

where

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

*n* is the complex refractive index, and *i* = (−1)^{1/2}. The term *Q*_{edge} in (27) gives the contribution of surface waves, often called edge effects, and is parameterized in terms of *D*_{eff} as

where *a*_{6} 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 *Q*_{ext} with this radiation scheme and T-matrix calculations reveal *a*_{6} = 0 for ice clouds (Mitchell et al. 2001). For *x*_{e} < 3, *Q*_{ext} is described in M00 by the functions joining the Rayleigh and Mie regions and by substituting *x*_{e} for *x* in the Rayleigh scattering efficiency, making use of the fact that *Q*_{ext} = *Q*_{sca} + *Q*_{abs}.

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 *n*_{r} changed when *n*_{r} > 1.55 or *n*_{i} > 0.40. When *n*_{r} or *n*_{i} exceeded these values, *t*_{a} was given by

Second, when *x*_{e}*n*_{r} < 2.15, the *D*_{eff} solution for *β*_{ext} 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 *β*_{abs} and *β*_{ext} will now be compared with numerical Mie theory integrations over size distributions of water droplets. Since *P*_{t} in (12) has no dependence on radiation, only *Q*_{abs} needs to be compared with Mie theory. However, Mie theory values of *Q*_{abs} must correspond to the size distribution, and hence *β*_{abs} and *β*_{ext} from Mie theory is divided by *P*_{t} to yield *Q*_{abs,Mie} and *Q*_{ext,Mie}:

Values of *Q*_{abs} and *Q*_{ext} based on M00, referred to as *Q*_{abs,M00} and *Q*_{ext,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 *μ*m. These *N*(*D*) will be used to test the above formulation of *Q*_{abs} and *Q*_{ext}, since they roughly encompass the range of *ν* in most water clouds. As will be shown later, it is *ν* that determines how accurately *D*_{eff} can be used to calculate *Q*_{abs}. The *D*_{eff} corresponding to the *ν* values of 4 and 20 (*D* = 15 *μ*m) is 21.0 and 16.4 *μ*m, respectively.

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

As in Fig. 2, explicit Mie integral solutions (solid curve) are compared with Mie solutions based on *D*_{eff} in Fig. 3, except this time, *ν* = 4 when calculating *Q*_{abs}. The greater *N*(*D*) dispersion about *D* renders the *D*_{eff} 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).

Testing of the *D*_{eff} expressions from the previous section is described in Figs. 4–6 for the case of *ν* = 20, *D* = 15 *μ*m; and in Figs. 7 and 8 for the case of *ν* = 4, *D* = 15 *μ*m. In Fig. 4, for *Q*_{abs} 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 *D*_{eff}. Lower in the figure, the long-dashed curve gives the contribution to *Q*_{abs} from photon tunneling based on M00, while the dotted–dashed curve gives the contribution from internal reflection/refraction. In Fig. 5, for *Q*_{ext} 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 *D*_{eff} scheme gives single particle solutions, and that as the *N*(*D*) dispersion *decreases* (i.e., *ν* becomes *larger*), *Q*_{abs,Mie }*and **Q*_{abs,M00} calculated via (33) *will converge toward the single particle solution of **Q*_{abs}, given by (22), and *Q*_{ext,Mie} and *Q*_{ext,M00} calculated via (34) *will converge toward the single particle solution of **Q*_{ext}, given by (27). Both the Mie and M00 curves are calculated explicitly from *N*(*D*). Since the *D*_{eff} scheme is based on the parameterizations in M00, differences between the *D*_{eff }*and *M00 schemes are only *due to **N*(*D*) *effects* (which M00 accounts for but *D*_{eff} does not). Errors relative to *Q*_{abs,Mie} are shown in Fig. 6 for *Q*_{abs} from (22) (solid curve), and for *Q*_{abs,M00} (dotted curve). Due to the low *N*(*D*) dispersion, errors relative to *Q*_{abs,Mie} are low (generally within 10%) for both methods. The error “spikes” for *λ* < 1.6 *μ*m, 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 *Q*_{ext,Mie} (not shown) were ≤16% for *Q*_{ext} from (27) and were ≤7% for *Q*_{ext,M00}. It is seen in Fig. 5 that for *λ* > 5 *μ*m, the *D*_{eff}*Q*_{ext}, *Q*_{ext}, behaves similar to *Q*_{ext,Mie} and *Q*_{ext,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 *Q*_{ext} over *x*_{e} dampen and tend to blend together—another reason why relative agreement is found for *λ* > 5 *μ*m. Moreover, wave interference oscillations broaden and ultimately vanish as *x*_{e} decreases. When absorption is weaker and *x*_{e} > 1, which is often true when *λ* ≲ 5 *μ*m (see Fig. 5), then interference oscillations manifest for *Q*_{ext}, whereas *N*(*D*) effects otherwise dampen these oscillations.

The same analysis is repeated for absorption in Figs. 7 and 8 for *ν* = 4, *D* = 15 *μ*m. It is seen here that errors for *Q*_{abs} are greater than for *Q*_{abs,M00}, due to the failure of *Q*_{abs} to account for the greater *N*(*D*) dispersion. Nonetheless, *Q*_{abs} errors in Fig. 8 are within 12% (excepting a few resonance spikes), which is probably acceptable for most purposes. These *Q*_{abs} 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 *D*_{eff} parameterization for *Q*_{abs} and *β*_{abs} may find useful application as a rapid, yet reasonably accurate, treatment of radiative properties for water clouds. On the other hand, the above *D*_{eff} parameterization for *Q*_{ext} and *β*_{ext} 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 *Q*_{ext} were ≤16%, as with the *ν* = 20 case, except for *λ* near 5.3 *μ*m, where errors reached 25%. The *Q*_{ext} parameterization based on *D*_{eff} may be useful for broadband calculations.

Since the zero scattering approximation (Paltridge and Platt 1976), which requires only *β*_{abs}, is usually sufficiently accurate for terrestrial radiation transfer, the *D*_{eff} parameterization of *β*_{abs} may satisfy most needs for radiation transfer in water clouds at terrestrial wavelengths.

The close agreement between *Q*_{abs} from Mie theory based on *D*_{eff} and *Q*_{abs} from numerical Mie integrations over the *N*(*D*), and between *Q*_{abs} from (22) and *Q*_{abs} from numerical Mie integrations, strongly indicate that *D*_{eff} 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 *d*_{e} as defined in (1). The radiative significance of *d*_{e} and *D*_{eff} 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 *D*_{eff} parameterization for *Q*_{abs} or *β*_{abs}, and *Q*_{ext} or *β*_{ext}, will be tested with Mie theory for an exponential *N*(*D*) of ice spheres, *ν* = 0 and *D* = 15 *μ*m. 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 *D*_{eff} parameterization, *D*_{eff} 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 *μ*m.

#### 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* > *D*_{o}, where *D*_{o} ≈ 1 mm for frontal clouds (Herzegh and Hobbs 1985) and *D*_{o} ≈ 100 *μ*m 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* < *D*_{o}, 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 *D*_{eff} parameterizations, as shown in Figs. 9–12, where *D* = 15 *μ*m, *ν* = 0, and *D*_{eff} = 45 *μ*m. This *D* would correspond to an ice particle maximum dimension of 20 to 40 *μ*m 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 *Q*_{abs} regarding the *D*_{eff} parameterization. Such errors could be troublesome for remote sensing based on relative brightness temperature differences in the window region (8–13 *μ*m), where the *D*_{eff} parameterization errors fluctuate between −7% and +10%. This smooths out the true variation of *Q*_{abs} in this region, as shown in Fig. 9. In contrast, the M00 parameterization matches *Q*_{abs,Mie} rather well. The results for *Q*_{ext} are shown in Figs. 11 and 12, with errors slightly less than obtained for water clouds when *λ* < 20 *μ*m, due to the larger *D*_{eff}. 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.

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

In this section, we will determine whether *D*_{eff} can be used in the manner described above to accurately calculate the radiative properties of ice clouds. This will be done by comparing *Q*_{abs} as determined from *D*_{eff} with *Q*_{abs} 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 *D*_{m}, their corresponding photon path equivalent spheres *D*_{e} and *D*_{me} 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

where *e* denotes *d*_{e} 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 *μ*m (M96). Details of this modified M96 scheme are given in the appendix.

The above *N*(*D*) parameters for the *N*(*D*) of *d*_{e}-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 *Q*_{ext} and *Q*_{ext} predicted by the updated M96 were within 3% on average for wavelengths between 2 and 17 *μ*m (Mitchell et al. 2001). For the purpose of intercomparing the updated M96 and *D*_{eff} 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 *D*_{eff} for this *N*(*D*) is 39 *μ*m, and the mean size of the large particle mode *D*_{1} is 100 *μ*m. 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.

The *Q*_{abs} based on *D*_{eff} (referred to as *Q*_{abs}) is now compared with the *Q*_{abs} determined from the updated M96, referred to as *Q*_{abs,M96}. The *N*(*D*) of Fig. 13 is used for this intercomparison. In Fig. 14, it is seen that *Q*_{abs} (dashed) and *Q*_{abs,M96} (solid) differ appreciably. For the purpose of error evaluation due to neglect of *N*(*D*) shape, we can view *Q*_{abs,M96} as “ground truth,” since both approaches are the same except that *Q*_{abs,M96} is an analytical solution of (10). Note how *Q*_{abs,M96} in the window region (8 *μ*m < *λ* < 13 *μ*m) is considerably lower than *Q*_{abs}. Errors for *Q*_{abs} relative to *Q*_{abs,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.

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

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 *D*_{eff} difference between modes increases), the greater the error via the *D*_{eff} method. When *ν* for the small mode was increased to 3 while conserving small mode *D*_{eff}, *D*_{eff} 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 *D*_{eff} 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 *D*_{eff} overestimates *Q*_{abs} 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 *Q*_{abs}, which, when integrated over *N*(*D*), result in lower overall *Q*_{abs} values than this *D*_{eff} approach predicts. This *D*_{eff} parameterization for *Q*_{ext}, 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 *Q*_{ext} 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 *D*_{eff} 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 *D*_{eff} in this study, involving the ratio IWC/*P*_{t}.

In this section, the modified M96 scheme described here will be evaluated over a wavelength range of 1 to 1000 *μ*m for *N*(*D*) having the same IWC and *D*_{eff}, 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 *D*_{eff} = 25.7 *μ*m, *D*_{1} = 74 *μ*m, *D*_{sm} = 12.8 *μ*m, and *ν* = 0 for both large and small *N*(*D*) modes. Here, *D*_{1} and *D*_{sm} refer to the mean size of the large and small particle *N*(*D*) modes, respectively. *N*(*D*) no. 2 also has a *D*_{eff} of 25.7 *μ*m, and *D*_{sm} = 13.7 *μ*m. However, *ν* for the small mode (*ν*_{sm}) is now 3.0, which narrows the small *N*(*D*) to exclude larger particles. This requires that *D*_{1} = 100 *μ*m to maintain a *D*_{eff} of 25.7 *μ*m, and *ν*_{1} = 0 as before. Lastly, *N*(*D*) no. 3 is a monomodal exponential (*ν* = 0) *N*(*D*), where *D* = 22.8 *μ*m and *D*_{eff} = 25.6 *μ*m. As with the analyses above, *β*_{abs} and *β*_{ext} will be determined from each scheme and divided by *P*_{t} to yield *Q*_{abs} and *Q*_{ext} 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 *t*_{f} = 0.5 for the small mode *N*(*D*) and that *t*_{f} = 0.3 for the large mode *N*(*D*).

Results for absorption are given in Fig. 18, where *Q*_{abs,M96} for different *N*(*D*) are indicated by the different line patterns. Here, *Q*_{abs,M96} is lowest for *N*(*D*) no. 2, since this *N*(*D*) is characterized by high concentrations of crystals having relatively low values of *Q*_{abs}. 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 *Q*_{abs} values. For instance, *D*_{eff} for the small mode in *N*(*D*) no. 1 is 15.2 *μ*m, while *D*_{eff} for the small mode in *N*(*D*) no. 2 is only 8.4 *μ*m. This results in a lower *Q*_{abs} for the small mode of *N*(*D*) no. 2, which impacts the overall *Q*_{abs} 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 *Q*_{abs,M96} values. It is noteworthy that even though comparable values of *D*_{sm} are found for *N*(*D*) no. 1 and *N*(*D*) no. 2 (12.8 and 13.7 *μ*m, respectively), subtle differences in small mode dispersion give rise to large differences in *Q*_{abs,M96} values. Moreover, these *D*_{sm} values appear characteristic of those measured in tropical cirrus (McFarquhar and Heymsfield 1997; Puechel et al. 1997; McFarquhar et al. 2000).

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 *Q*_{abs,M96} between *N*(*D*) no. 1 and *N*(*D*) no. 2 may be representative of our uncertainty regarding complete bimodal *N*(*D*), while the *Q*_{abs,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 *μ*m), 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 *Q*_{ext,M96} values at shorter wavelengths, approaching the geometric optics regime, and generally by lower *Q*_{ext,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 *D*_{eff} value. The variability in the window region and at longer wavelengths is somewhat remarkable.

Percent differences between *N*(*D*) no. 2 and *N*(*D*) no. 3 regarding *Q*_{abs}, *Q*_{ext} and the single scattering albedo *ω*_{o} are shown in Fig. 20, where *ω*_{o} = 1 − *Q*_{abs}/*Q*_{ext}. Percent differences for *λ* < 100 *μ*m reach 44% for *Q*_{abs}, 64% for *Q*_{ext}, and 18% for *ω*_{o}. For *λ* ≈ 100 *μ*m, differences reach 100% for *Q*_{ext}, and 22% for *ω*_{o}. In the calculation of *ω*_{o}, 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 *μ*m < *D* < 60 *μ*m), 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 *Q*_{abs}, 96% for *Q*_{ext}, and 48% for *ω*_{o}. These differences clearly demonstrate that ice cloud radiative properties depend on the assumed *N*(*D*) shape as well as *D*_{eff} and IWC.

These differences are based on a single value of *D*_{eff}. While *D*_{eff} ≈ 25 *μ*m is common for tropical cirrus, larger *D*_{eff} values will be associated with lower uncertainties as *Q*_{abs} and *Q*_{ext} approach their limiting values. Smaller *D*_{eff} 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 *Q*_{abs,M96} and *Q*_{ext,M96} are evaluated for a single wavelength as a function of *D*_{eff}, 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 *μ*m < *D* < 100 *μ*m). 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 **D*_{eff}. This is shown in Fig. 22 for *Q*_{abs} and *Q*_{ext} as a function of *D*_{eff}, evaluated at wavelengths of 3.73, 8.48, and 11.1 *μ*m. 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 *D*_{eff} as a function of temperature. For both the midlatitude and tropical schemes, *D*_{eff} 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 *D*_{eff} values. Note that assuming an ice crystal shape other than hexagonal columns would decrease *D*_{eff} considerably, and this combined with natural variability makes the range of *D*_{eff} reasonable for each scheme. It is seen that the range for *D*_{eff} 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 *D*_{eff} 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 *μ*m, differences in *Q*_{abs} for a given *D*_{eff} are around 10%.

### 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 *D*_{eff} (or *D*_{ge}; *D*_{ge} = 0.7698*D*_{eff}) and IWC. Since these schemes are intended for all cirrus, and since the Fu schemes have high spectral resolution, *Q*_{abs} and *Q*_{ext} predicted by the Fu (1996) and Fu et al. (1998) schemes have been plotted against *D*_{eff} given by the tropical *N*(*D*) scheme in Fig. 22 (short-dashed curves). Knowing the *N*(*D*) projected areas, *Q*_{abs} and *Q*_{ext} were determined from the Fu scheme as described in (33) and (34), and by noting that *β*_{abs} = *β*_{ext}(1 − *ω*_{o}). The use of *D*_{eff} eliminates the dependence of *Q*_{abs} and *Q*_{ext} 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 *Q*_{abs} and *Q*_{ext} may be due to *N*(*D*) effects.

Looking at Fig. 22, it appears that *Q*_{abs} and *Q*_{ext} from the tropical and/or midlatitude *N*(*D*) schemes tend to be in general agreement with *Q*_{abs} and *Q*_{ext} 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 *Q*_{abs} at 3.7 *μ*m. The wavelength resolution of the Fu scheme could contribute to this discrepancy. This band ranges from 3.4 to 4.0 *μ*m in Fu (1996), and an absorption minimum is located at 3.85 *μ*m. Hence, the integrated mean value of *Q*_{abs} in this band should be different than the discrete value at 3.7 *μ*m. Moreover, a finite difference time domain (FDTD) calculation taken from Fu et al. (1998) for *D*_{eff} = 5.6 *μ*m, *λ* = 3.7 *μ*m, 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 *D*_{eff} = 14 *μ*m (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 *D*_{eff} used to parameterize the Fu schemes. Large discrepancies exist at the smallest *D*_{eff} values for both *Q*_{abs} and *Q*_{ext}. Since the range of *D*_{eff} in Fu (1996) and Fu et al. (1998) was 24.2 to 169.2 *μ*m, and 14.3 to 168.4 *μ*m, respectively, the curve fits relating *D*_{eff} to the single scattering calculations may sometimes be weak for *D*_{eff} < 15 *μ*m.

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 *D*_{eff} 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 *D*_{eff}. 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 *μ*m.

## 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 *D*_{eff}. Beginning with this photon path concept, a general definition for *D*_{eff} was derived for both ice and water clouds. Moreover, the *D*_{eff} expression for water clouds was twice the value of the “traditional” effective radius definition. Therefore a single definition of *D*_{eff} 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, *β*_{abs} and *β*_{ext}, were derived based on *D*_{eff}, 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 *β*_{abs} was generally accurate within 12%, while the *β*_{ext} expression was generally accurate within 20% for any wavelength. Using *β*_{abs} 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 *β*_{abs} and *β*_{ext} 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 *D*_{eff} 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 *D*_{eff} and IWC, variations in *N*(*D*) shape were shown to produce differences in the *N*(*D*) area-weighted efficiencies for absorption (*Q*_{abs}) and extinction (*Q*_{ext}) by up to 44% and 100%, respectively, at terrestrial wavelengths. In the window region (8–12 *μ*m), these differences reach 30% for *Q*_{abs}, 48% for *Q*_{ext}, and 18% for the single scattering albedo *ω*_{o}. 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 *D*_{eff} 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 *D*_{eff} is used to represent the *N*(*D*) in Mie theory or our single particle solutions for *β*_{abs} and *β*_{ext}, and 2) a single *D*_{eff} 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 *D*_{eff} 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 *D*_{eff} 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 *D*_{eff} 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 *D*_{eff} 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 *μ*m. Such a scheme is offered here for calculating *β*_{abs} and *β*_{ext}, 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 *β*_{abs} and *β*_{ext} 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 *β*_{abs} is given as

while the extinction coefficient *β*_{ext} is given as

where

and where *g* = 8*πn*_{i}/3*λ, **q* = *i*2*π*(*n* − 1)/*λ* is a complex variable (*n* = complex index of refraction, *i* = −1^{1/2}), Re indicates only the real part of the term is used, Γ is the gamma function, *m* = 0.50, *a*_{6} = 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, **n*_{r}, and *n*_{i}. 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 *β*_{abs} and *β*_{ext}. The next two terms in (A1) estimate the contribution of internal reflection/refraction, while the last two terms estimate the contribution of tunneling to *β*_{abs}. The next two terms in (A2) estimate the contribution of tunneling to *β*_{ext}, while the last term estimates the contribution of edge effects.

The above equations for *β*_{abs} and *β*_{ext} are accurate to within about 10% relative to Mie theory for ice spheres and *x*_{e} > 1, where *x* for a size distribution is given as

The treatment of *β*_{abs} and *β*_{ext} for *x*_{e} < 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 *d*_{e} 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 *μ*m for a given crystal habit. Regarding section 3c(2), *D*_{e} and *D*_{me} 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,

This approach eliminates much of the error associated with the parameterized *D*–*d*_{e} relationships used in M96, as documented in Yang et al. (2000), since *d*_{e} is no longer approximated.

The parameter *N*_{oe} of the transformed *N*(*D*), given by (37), requires knowledge of *P*_{t}, and *P*_{t} requires knowledge of *N*_{o} of the physical size distribution, referred to in (14) and (20). However, the *N*_{o} solution given by (20) is valid only when *α* and *β* in (15) are constant. Since they are not (M96), the calculation of *N*_{o} must take this into account:

where *D*_{o} ≈ 100 *μ*m, *α*_{1}, and *β*_{1} correspond to *D* ≲ 100 *μ*m, and *α*_{2} and *β*_{2} correspond to *D* ≳ 100 *μ*m. 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* = *Dλ*:

where *x*_{o} = *D*_{o}*λ, **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 *P*_{t}, and a similar analysis using the incomplete gamma function is required for *P*_{t}:

Now that *P*_{t} is known via (A12), *N*_{oe} 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 *β*_{abs} and *β*_{ext} 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,

The transformed *N*(*D*) parameters *ν*_{e} and Λ_{e} were calculated from (35) and (36) as before but using mass (i.e., volume) equivalent spheres, while *N*_{o} for the transformed *N*(*D*) was given as

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 *β*_{abs} and *β*_{ext} for each mode. Hence, for the small particle mode (*D* ≲ 100 *μ*m), denoted *N*(*D*)_{sm}, *β*_{abs,sm} and *β*_{ext,sm} are calculated, and for the large particle mode *N*(*D*)_{1}, *β*_{abs,1} and *β*_{ext,1} are calculated, such that the total values are the sum of the mode values: *β*_{abs} = *β*_{abs,sm} + *β*_{abs,1} and *β*_{ext} = *β*_{ext,sm} + *β*_{ext,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