Effective Diameter in Radiation Transfer: General Definition, Applications, and Limitations

Transcription

1 2330 JOURNAL OF THE ATMOSPHERIC SCIENCES Effective Diameter in Radiation Transfer: General Definition, Applications, and Limitations DAVID L. MITCHELL Desert Research Institute, Reno, Nevada (Manuscript received 18 July 2000, in final form 21 February 2002) 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, abs and 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 abs based on D eff compares well with abs predicted from Mie theory, providing a new and efficient means of treating radiation transfer at terrestrial wavelengths. The D eff expression for 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 D eff /. Since the D eff expressions for abs and 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 abs and 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 m) ice crystals. These small crystals have relatively low absorption efficiencies, causing the N(D)-integrated abs to be lower than 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 abs and ext are provided in the 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 abs and 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, Qabs and Qext, for three N(D) that have the same IWC and D eff, but for which N(D) shape differs. Analytical solutions for abs and ext were used, which explicitly treat N(D) shape, over a wavelength range of 1.0 to 1000 m. For a chosen D eff 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 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 sci- Corresponding author address: Dr. David L. Mitchell, Atmospheric Sciences Division, Desert Research Institute, 2215 Raggio Parkway, Reno, NV mitch@dri.edu ences 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 2002 American Meteorological Society

2 1AUGUST 2002 MITCHELL 2331 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 abs and 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 abs and ext is critically evaluated in sections 3 and 4. A summary and concluding remarks are given in section 5. A new scheme for calculating abs and 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: m de, (1) i P where m is the particle s mass and i 0.92 g m 3. This value of 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 Qabs 1 exp(4nid/), e (2) where n i is the imaginary part of the refractive index and 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 i (D 3 /6) and P D 2 /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 IWC De, (4) i P t 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 3 IWC Deff, (5) 2 i P t 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 ½D eff, then 3 LWC reff, (6) 4 w P t where LWC is the liquid water content, and w is the density of liquid water. Defining the LWC and P t as w 3 LWC (4/3)r N(r) dr, (7) 2 Pt r N(r) dr, (8) 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): 3 rn(r) dr reff. (9) 2 rn(r) dr

3 2332 JOURNAL OF THE ATMOSPHERIC SCIENCES 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 ½D 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: abs ext Q (D, )P(D)N(D) dd, (10) abs Q (D, )P(D)N(D) dd, (11) ext 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: abs QabsP, t (12) ext QextP, t (13) 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, abs and/or 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 abs and/or ext can be estimated from (12) and (13). Expressions for Qabs, Qext, 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 N(D) ND o exp(d), (14) and mass and projected area dimensional power-law expressions for m and P, m D, (15) P D, (16) a general expression for the size distribution projected area is given as No( 1) Pt, (17) 1 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,, and N o can be obtained from measured N(D) properties as described in Mitchell (1991): ( 0.67)D D m, (18) D D m ( 1)/D, (19) IWC 1 No, (20) ( 1) where D is the N(D) mean D, and D m is the D that divides the N(D) mass into equal parts. Substituting (4) into (2) but in terms of spheres having the same D e value, we have Q abs,ada 1 exp(8nid eff /3), (21) 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 : Q abs (1 C1 C 2)Q abs,ada. (22) The term C 1 accounts for absorption contributions from internal reflection and refraction: C1 a1 exp(8nid eff /3), (23) 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 m (D eff/) exp(od eff/) 2 f a m k max exp(m) C tt, (24) where ta nr , (25) 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 o determines the D eff / value k max, where this maximum occurs.

4 1AUGUST 2002 MITCHELL 2333 Equation (12) is now solved using (17) and (22) for size parameter x e 1, where x e D eff /. When x e 0.3, the Rayleigh Q abs is used. In terms of D eff, Q abs is 2 2 Q abs (4D eff /) Im[(n 1)/(n 2)], (26) 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 Q ext (1 0.5C 2)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: 2 K(t) 1/2 exp(t)/t [exp(t) 1]/t, (29) t i2d eff(n 1)/, (30) 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 2/3 Q edge a 6[1 exp(0.06x e)]x e, (31) 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 Qext 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 When n r or n i exceeded these values, t a was given by ta nr (32) Second, when x e n r 2.15, the D eff solution for 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 abs and 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 abs and ext from Mie theory is divided by P t to yield Q Q abs,mie and ext,mie: 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 D eff expressions for abs and ext [Eqs. (12) and (13)] relative to Mie theory and modified ADA, as shown in Figs Q /P, (33) abs,mie abs,mie t Q /P. (34) ext,mie ext,mie t Values of Qabs and Q ext based on M00, referred to as Qabs,M00 and Q ext,m00, were determined similarly. Size distributions were described by (14), where D and 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 values [see Eq. (18)]. In water clouds, 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 values of 4 and 20 are contrasted. The D for both N(D) is15m. These N(D) will be used to test the above formulation of Q abs and Q ext, since they roughly encompass the range of in most water clouds. As will be shown later, it is that determines how accurately D eff can be used to calculate Q abs. The D eff corresponding to the values of 4 and 20 ( D 15 m) is 21.0 and 16.4 m, respectively. First, let us test the hypothesis that a single particle solution for abs based on D eff is feasible when the N(D) is sufficiently narrow. Using the narrow N(D) in Fig. 1 ( 20), 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

5 2334 JOURNAL OF THE ATMOSPHERIC SCIENCES FIG. 2. Absorption efficiencies for the low dispersion ( 20) size distribution in Fig. 1, based on a numerical integration of Eq. (10) using Mie theory (solid curve), D eff using Mie theory (dashed curve), and the generalized effective size D ge using Mie theory (dotted curve) 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 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, 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 (not shown). Testing of the D eff expressions from the previous section is described in Figs. 4 6 for the case of 20, D 15 m; and in Figs. 7 and 8 for the case of 4, D 15 m. In Fig. 4, for Q abs where 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 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., becomes larger), Qabs,Mie and Qabs,M00 calculated via (33) will converge toward the single particle solution of Q abs, given by (22), and Qext,Mie and Qext,M00 calculated via (34) will FIG. 3. Absorption efficiencies for the high dispersion ( 4) size distribution in Fig. 1, based on a numerical integration of Eq. (10) using Mie theory (solid curve), and D eff using Mie theory (dashed curve). 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 Qabs,Mie are shown in Fig. 6 for Qabs 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 1.6 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 Qext,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 5 m, the D eff Q ext, Qext, behaves similar to Qext,Mie and Qext,M00. This is a manifestation of the dispersion principle noted above, since 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 5 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 5 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 4, D 15 m. It is seen here that errors for Qabs are greater than for Qabs,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

6 1AUGUST 2002 MITCHELL 2335 FIG. 4. Comparison of Mie theory (solid), the modified ADA (dashed), and the D eff (dotted) calculation of Q abs 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 Q abs, based on the modified ADA. for 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 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 ext may not be sufficiently accurate for some applications, as shown in Fig. 5, especially at near and midinfrared wavelengths. When 4, the errors in Q ext were 16%, as with the 20 case, FIG. 6. The Q abs errors in Fig. 4 relative to Mie theory, for the modified ADA (dotted) and the D eff parameterization (solid). except for near 5.3 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 abs, is usually sufficiently accurate for terrestrial radiation transfer, the D eff parameterization of 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 Qabs from (22) and Qabs 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. FIG. 5. As in Fig. 4, but for extinction efficiency Q ext using the same curve labeling convention as Fig. 4 except the dotted dashed curve gives the contribution of edge effects to Q ext. c. Testing with Mie theory and modified ADA: Ice clouds In this section, the above D eff parameterization for Qabs or abs, and Qext or ext, will be tested with Mie theory for an exponential N(D) of ice spheres, 0 and D 15 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

7 2336 JOURNAL OF THE ATMOSPHERIC SCIENCES FIG. 7. Comparison of Mie theory (solid), the modified ADA (dashed), and the D eff (dotted) calculation of Q abs using the highdispersion size distribution for water clouds in Fig. 1. The longdashed curve gives the tunneling contribution and the dotted dashed curve gives the contribution of internal reflection/refraction to Q abs, based on the modified ADA. 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 m. 1) TESTING WITH MIE THEORY Until recently, N(D) in ice clouds were often assumed to be monomodal and exponential, where 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 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 m, 0, and D eff 45 m. This D would correspond to an ice particle maximum dimension of 20 to 40 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 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. FIG. 8.TheQ abs errors in Fig. 7 relative to Mie theory, for the modified ADA (dotted) and the D eff parameterization (solid). Errors correspond to the high dispersion size distribution. 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 20 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 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

8 1AUGUST 2002 MITCHELL 2337 FIG. 9. Comparison of Mie theory (solid), the modified ADA (dashed), and the D eff (dotted) calculation of Q abs for an exponential size distribution of ice spheres, where D 15 m, 0, and D eff 45 m. The long-dashed and dotted dashed curves show contributions from tunneling and internal reflection/refraction to Q abs, respectively. 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 3.67D e Dme e, (35) D D me e ( 1)/D, (36) e e e 3 4P N e t e oe, (37) (e 3) 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 m (M96). Details of this modified M96 scheme are given in the appendix. The above N(D) parameters for the N(D)ofd 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 Qext and Qext predicted by the updated M96 were within 3% on average for wavelengths between 2 and 17 m (Mitchell et al. 2001). For the purpose of intercomparing the updated M96 and D eff schemes, we will assume no tunneling. FIG. 10. The Q abs errors in Fig. 9 relative to Mie theory, for the modified ADA (dotted) and the D eff parameterization (solid). 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 0 for the large particle mode, whereas it was assumed 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 m, and the mean size of the large particle mode D 1 is 100 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 Qabs based on D eff (referred to as Qabs) isnow 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 m 13 m) 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.

9 2338 JOURNAL OF THE ATMOSPHERIC SCIENCES FIG. 11. Same as Fig. 9, but for Q ext, and the dotted dashed curve shows edge effect contributions. 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 100 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 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 FIG. 13. Example size distribution characteristic of those sampled in anvil cirrus during CEPEX, as predicted by the scheme of Mitchell et al For the large particle mode, D 100 m, and for the total distribution, D eff 39 m. 60C, 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 FIG. 12. The Q ext errors in Fig. 11 relative to Mie theory, for the modified ADA (dotted) and the D eff parameterization (solid). FIG. 14. Based on the N(D) described in Fig. 13, Q abs is predicted by the revised M96 scheme (see appendix) and the D eff parameterization (dashed curve). Differences are due solely to N(D) shape effects.

10 1AUGUST 2002 MITCHELL 2339 FIG. 15. The Q abs errors in Fig. 14 for the D eff parameterization, relative to the revised M96 scheme. FIG. 16. As in Fig. 14, but for Q ext (i.e., the revised M96 scheme corresponds to the solid curve; the D eff parameterization corresponds to the dashed curve). 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. 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 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 m, D1 74 m, Dsm 12.8 m, and 0 for both large and small N(D) 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 FIG. 17. Three size distributions having the same D eff value of 25.7 m and IWC of 10 mg m 3, but having different shapes.

11 2340 JOURNAL OF THE ATMOSPHERIC SCIENCES FIG. 18. Based on the N(D) described in Fig. 17, Q abs (i.e., abs / P t ) 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. 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 D eff of 25.7 m, and D sm 13.7 m. However, for the small mode ( sm ) is now 3.0, which narrows the small N(D) to exclude larger particles. This requires that D m to maintain a D eff of 25.7 m, and 1 0 as before. Lastly, N(D) no.3isa monomodal exponential ( 0) N(D), where D 22.8 m and D eff 25.6 m. As with the analyses above, abs and 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 m, while D eff for the small mode in N(D) no. 2 is only 8.4 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 FIG. 19. Same as Fig. 18, except Q ext is compared. N(D) no. 2 (12.8 and 13.7 m, 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). Since the value of characterizing the small mode in tropical cirrus is not well known, either 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) inice 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 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 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 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 Qabs, Qext and the single scattering albedo o are shown in Fig. 20, where o 1 Q abs / Qext. Percent differences for 100 m reach 44% for Qabs, 64% for Q ext, and 18% for o. For 100 m, differences reach 100% for Qext, and 22% for o. In the calculation of o,

12 1AUGUST 2002 MITCHELL 2341 FIG. 20. Percent differences between the Q abs, Q ext, and o predicted from N(D) no. 3 and N(D) no. 2 (in Fig. 17), based on the revised M96 scheme. 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 m D 60 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 Qabs, 96% for Qext, and 48% for 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 m is common for tropical cirrus, larger D eff values will be associated with lower uncertainties as Qabs and Qext 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 Qabs,M96 and Qext,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 m D 100 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 Qabs and Qext as a function of FIG. 21. Same as Fig. 20, except the percent differences correspond to N(D) no. 1 and N(D) no. 2 in Fig. 17. D eff, evaluated at wavelengths of 3.73, 8.48, and 11.1 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 100C. The unrealistically cold temperature of 100C 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 3.73 or 8.48 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

13 2342 JOURNAL OF THE ATMOSPHERIC SCIENCES FIG. 22. Prediction of Q abs and Q ext for selected wavelengths as a function of D eff, 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). terms of D eff (or D ge ; D ge D eff ) 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 D eff 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 abs ext (1 o ). The use of D eff 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 mid-

14 1AUGUST 2002 MITCHELL 2343 latitude 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 m. The wavelength resolution of the Fu scheme could contribute to this discrepancy. This band ranges from 3.4 to 4.0 m in Fu (1996), and an absorption minimum is located at 3.85 m. Hence, the integrated mean value of Q abs in this band should be different than the discrete value at 3.7 m. Moreover, a finite difference time domain (FDTD) calculation taken from Fu et al. (1998) for D eff 5.6 m, 3.7 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 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 Qabs and Qext. Since the range of D eff in Fu (1996) and Fu et al. (1998) was 24.2 to m, and 14.3 to m, respectively, the curve fits relating D eff to the single scattering calculations may sometimes be weak for D eff 15 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 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, abs and 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 abs was generally accurate within 12%, while the ext expression was generally accurate within 20% for any wavelength. Using 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 abs and 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 ( Qabs) and extinction ( Qext) by up to 44% and 100%, respectively, at terrestrial wavelengths. In the window region (8 12 m), these differences reach 30% for Qabs, 48% for Qext, and 18% for the single scattering albedo 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 abs and 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 m. Such a scheme is