Parameterization of the Mie Extinction and Absorption ... - AMS Journals

1 MAY 2000

MITCHELL

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Parameterization of the Mie Extinction and Absorption Coefficients for Water Clouds DAVID L. MITCHELL Desert Research Institute, Reno, Nevada (Manuscript received 20 January 1998, in final form 4 May 1999) ABSTRACT It was found that the anomalous diffraction approximation (ADA) could be made to approximate Mie theory for absorption and extinction in water clouds by parameterizing the missing physics: 1) internal reflection/ refraction, 2) photon tunneling, and 3) edge diffraction. Tunneling here refers to processes by which tangential or grazing photons beyond the physical cross section of a spherical particle may be absorbed. Contributions of the above processes to extinction and/or absorption were approximated in terms of particle size, index of refraction, and wavelength. It was found that tunneling can explain most of the difference between ADA and Mie theory for water clouds in the thermal IR. The modified ADA yielded analytical expressions for the absorption and extinction efficiencies, Qabs and Qext , which were integrated over a gamma size distribution to yield expressions for the absorption and extinction coefficients, babs and bext . These coefficients were expressed in terms of the three gamma distribution parameters, which were related to measured properties of the size distribution: liquid water content, mean, and mass-median diameter. Errors relative to Mie theory for babs and bext were generally #10% for the effective radius range in water clouds of 5–30 mm, for any wavelength in the solar or terrestrial spectrum. For broadband emissivities and absorptivities regarding terrestrial and solar radiation, the errors were less than 1.2% and 4%, respectively. The modified ADA dramatically reduces computation times relative to Mie theory while yielding reasonably accurate results.

1. Introduction Two-stream radiation transfer models have become popular in regional and global climate modeling, due to their simplicity and computational efficiency. These models require knowledge of the extinction and absorption coefficients, and the asymmetry parameter, which is a measure of the fraction of incident radiation scattered into a particle’s forward and rear hemispheres. These coefficients for water clouds depend on the droplet size distribution and the extinction and absorption efficiencies as predicted by Mie theory. Unfortunately, it is often not practical to apply exact Mie theory solutions for obtaining these efficiencies, due to the large amount of computation time required. Thus, there is a need for computationally efficient approximations to Mie theory, for use in radiation transfer models, which maximize physical realism and accuracy. A number of investigators have developed approximations to Mie theory that are relatively efficient computationally. Two general approaches have been pursued: 1) theoretical or process oriented and 2) curve

Corresponding author address: Dr. David L. Mitchell, Atmospheric Sciences Center, Desert Research Institute, 2215 Raggio Parkway, Reno, NV 89512-1095. E-mail: [email protected]

q 2000 American Meteorological Society

fitting. An example of the curve fitting approach is Chylek et al. (1995), who formulated polynomial approximations to the Mie absorption, scattering, and extinction coefficients, as well as asymmetry parameters. In the thermal infrared, tenth-order polynomials were accurate within 3% and utilized lookup tables. Process-oriented approaches have the potential advantage of yielding physical insight into radiation–particle interactions, which is difficult to extract from Mie theory. An example of a theoretical approach is Nussenzveig and Wiscombe (1980), who used complex-angular-momentum theory to derive asymptotic approximations for Mie extinction and absorption efficiencies. For size parameter x (pD/l, where D 5 diameter, l 5 wavelength of incident radiation) ranging from 10 to 1000, relative errors decreased from 1%–10% to 1022%–1023%, with computation time reduced by an order of x relative to Mie theory. However, this method does not lend itself to analytical expressions for the extinction and absorption coefficients. Ackerman and Stephens (1987) adopted a process-oriented approach yielding analytical expressions for these coefficients by modifying the anomalous diffraction approximation (ADA) by parameterizing refraction and edge effects. However, there has been no process-oriented treatment to date that, in addition to refraction and edge effects, parameterizes the processes of internal reflection and photon tunneling to yield an-

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alytical expressions for the absorption and extinction coefficients, which is the objective of this study. In this work, the phase function and associated asymmetry parameter are not addressed. However, the asymmetry parameter in two-stream models can be treated as constant (Hansen and Pollack 1970) or may be parameterized (Slingo 1989; Chylek et al. 1995). The approach here is described in section 2, where Mie theory results are used to parameterize some of the physical processes contributing to extinction and absorption. These parameterizations were used to modify ADA, leading to equations for the extinction and absorption coefficients in terms of wavelength, refractive index, and size distribution parameters. Different terms in the equations represent different physical processes. Results are given in section 3 and are discussed in section 4. A summary is given in section 5. 2. Methodology The basic approach pursued here was to 1) identify the primary physical processes contributing to absorption, 2) parameterize these processes for absorption, and 3) parameterize these same processes for extinction, as well as other processes that apply. Since fewer processes apply to absorption than extinction, it was possible to isolate and parameterize these processes. What was learned from this exercise then allowed the contribution of a remaining extinction process (edge effects) to be isolated and parameterized, as well as the generalization of the absorption processes to extinction. a. Processes contributing to absorption In this section, the primary processes contributing and not contributing to the absorption efficiency, Qabs , are identified, with justification provided. It is generally accepted that absorption processes include 1) geometrical blocking (analogous to Beer’s law absorption) and 2) internal reflection and refraction, as this extends the photon path through the particle (e.g., van de Hulst 1981). Another process, photon tunneling, may also contribute in the vicinity of the ‘‘resonance region’’ of the extinction efficiency (Qext ) with respect to x, contained mostly within the first major peak of the interference structure. The following discussion provides a physical understanding of the Qabs dependence on tunneling, and the resulting conceptual model provides a basis for the tunneling parameterization developed in this paper. While tunneling in the resonance region is not well understood, Nussenzveig and Wiscombe (1980, 1987) show that the tunneling responsible for large angle diffraction allows Qabs to exceed 1.0 (the geometric optics limit) for large imaginary refractive index, n i , and 10 , x & 50, attaining a maximum value of 1.23 for real index of refraction n r 5 1.33. Photons beyond the physical cross section undergo tunneling in the edge domain

FIG. 1. Depiction of possible trajectories of an incident grazing ray after tunneling to the drop surface. Edge effects are illustrated by ray 1, while rays 2 and 3 are examples of large-angle diffraction whereby the ray travels through the drop and may be absorbed.

between the illuminated and nonilluminated regions of a sphere, launching surface waves that are continually dampened as they propagate, as radiation is scattered along the path of the surface wave (a surface wave propagates along the interface between two mediums). This is physically portrayed in Fig. 1 (similar to Fig. 10 in Nussenzveig 1977), where a ray beyond the physical cross section tunnels through an inertial barrier (Guimares and Nussenzveig 1992), becoming a surface wave, which is then scattered as ray number 1. Also, as described by Nussenzveig (1977), ‘‘At each point along the wave’s circumferential path it also penetrates the sphere at the critical angle for total internal reflection, reemerging as a surface wave after taking one or more such shortcuts.’’ Such surface waves can then be scattered, accounting for the largest scattering angles, as shown by rays 2 and 3 in Fig. 1. Ray 3 contributes to what is known as the glory phenomena (Nussenzveig 1979). The point of this physical model is that rays that penetrate through the interior of the sphere are candidates for absorption, providing a conceptual understanding for the absorption dependence on tunneling via large-angle diffraction. Large-angle diffraction is further discussed in section 2c(2). This large-angle diffraction phenomena is quite similar to wave resonance phenomena. However, with resonance, the incident ray in Fig. 1, after tunneling and penetrating inside the sphere near the critical angle, is internally reflected many times in closed or nearly closed orbits (Guimares and Nussenzveig 1992; Nussenzveig 1979). Such orbits are supported by only certain combinations of wavelength and droplet size, ex-

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FIG. 2. Extinction and absorption efficiencies comparing ADA (dashed) and Mie theory (solid) for a wavelength (l ) of 20 mm, real and imaginary refractive index (n r and n i ) of 1.90 and 0.05, respectively. Regarding Mie theory, note the 1:1 correspondence between the peaks in the ripple structure for Qabs and Qext , which are independent of the larger interference structure in Qext .

plaining why resonance peaks in Qext (i.e., the ripple structure superimposed over the interference structure) occur only at certain x values. When x approaches 1, this ray model of resonance becomes murkier, and resonance may be viewed as x dependent dipole and multipole oscillations (van de Hulst 1981, pp. 151–155). Evidence suggesting that resonance contributes to Qabs in the resonance region is seen in Fig. 2, where there is a 1:1 correspondence between the ripple structure in Qext and the ripple structure in Qabs . Guimaraes and Nussenzveig (1992), using complex angular momentum (CAM) theory, show the extinction ripple structure results from tunneling. The 1:1 correspondence suggests that the same physics is responsible for the ripple structure in Qext and Qabs . This is a general observation, not unique to Fig. 2. The overlapping of the ripple structure in Qabs , especially for 1 , x , 20 approximately, accounts for an enhanced region of absorption relative to ADA, as shown in Fig. 2. As n i increases, the ripples in Qabs broaden, forming a continuum without the ripple pattern. This continuum of enhanced absorption has been shown to result primarily from ‘‘above edge contributions’’ predicted by CAM, which are due to tunneling (Baran et al. 1998). The tunneling parameterization in this paper will further demonstrate that the enhancement of Qabs relative to ADA in the resonance region is due to tunneling. Interference phenomena due to the phase lag of the electromagnetic wave passing through the medium produce the dominant, large undulations in the extinction efficiency, Qext , which are absent in Qabs , as shown in Fig. 2 for both the ADA and Mie curves. This interference is set up with diffracted radiation after the trans-

mitted wave exits the sphere, and thus should not contribute to Qabs . However, the interference in Qext is influenced by Qabs , since as Qabs increases, less radiation is transmitted, causing the interference amplitude to decrease (van de Hulst 1981, p. 182). Classical diffraction, which accounts for roughly half of Qext at large x as well as the strong forward scattering, does not appear to contribute to Qabs either, since these diffracted waves should not pass through the sphere (van de Hulst 1981). One process not addressed is external reflection, where incident radiation is reflected from the particle’s surface, and thus is not absorbed. As shown in section 3, neglecting this process did not appear to compromise the accuracy of this parameterization much for observed cloud droplet sizes. In summary, the following four processes were addressed to parameterize Qabs : 1) geometrical blocking, 2) internal reflection and refraction, 3) large-angle diffraction, and 4) wave resonance. Processes 3 and 4 are similar and are tunneling phenomena, allowing Qabs to exceed 1.0. Hence, they were parameterized collectively as tunneling phenomena. A physical description for Qabs . 1.0 is that most or all photons incident on a sphere’s geometrical cross section are being absorbed and, in addition, some photons beyond this cross section are also being absorbed. b. Parameterization of absorption efficiency The anomalous diffraction approximation (ADA), an approximation of Mie theory (van de Hulst 1981), was used as the foundation of this parameterization. ADA has been used to approximate the radiative properties of water clouds (Ackerman and Stephens 1987) and ice clouds (Mitchell et al. 1996). ADA captures the fundamental behavior of Mie theory, including the phase of the interference patterns apparent in extinction efficiencies, even for nonspherical particles (Asano and Sato 1980). However, it predicts that radiation passes directly through a particle, without refracting or experiencing internal reflections. Moreover, ADA does not account for tunneling processes. To make ADA ‘‘act’’ like Mie theory, the task is then to parameterize the processes of 1) internal reflection/refraction and 2) photon tunneling. An empirical but process-oriented approach, based on Mie theory and other results, is adopted here. 1) INTERNAL

REFLECTION/REFRACTION

We will first address internal reflection/refraction phenomena regarding absorption. For solar wavelengths and large x, where tunneling contributions are small, ADA was used to formulate the absorption efficiency, Qabs , for ice crystals (Mitchell et al. 1996). Internal reflection/refraction phenomena were parameterized. Comparisons between the parameterized Qabs and ray tracing results indicated errors were ,10% for hexag-

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FIG. 3. Extinction and absorption efficiencies predicted by ADA (long-dashed curve) and Mie theory (solid curve) for ice spheres for l 5 13.0 mm (n r 5 1.472, n i 5 0.389). The vertical dashed line indicates the diameter where Mie-to-ADA efficiency ratios are maximum.

onal columns and planar polycrystals, and generally ,15% for hexagonal plates. For spheres, Qabs can be formulated as Qabs 5 (1 1 C1 )Qabs,ADA ,

(1)

where Qabs,ADA is determined by ADA as described in Mitchell and Arnott (1994): Qabs,ADA 5 1 2 exp[24pn i (V/P)/l ],

(2)

where V 5 sphere volume at bulk water density, P 5 projected area of sphere, and l is wavelength. For spheres, V/P is simply (2/3)D, where D 5 sphere diameter. The term C1 corrects for internal reflection and refraction and was formulated as C1 5 a1 (1 2 Q abs,ADA ),

or

(3)

C1 5 a1 exp(28p n i D/3l),

(4)

where n i is the imaginary part of the refractive index and a1 is a1 5 0.25 1 0.25 exp(21167n i ).

(5)

In the thermal infrared, only the first term in (5) is significant. The second term accounts for narrowband resonance effects (causing more absorption), where x . 15, n i , 1023 . Equation (3) was formulated by noting that the contribution of internal reflection and refraction to absorption diminishes as Qabs,ADA increases to its limiting value of 1. Naturally, there can be no such contribution when no radiation can pass through the particle (i.e., absorption depends only on projected area). The coefficient a1 can be viewed as determining the upper limit that internal reflection/refraction can contribute to Qabs , while

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FIG. 4. Dependence of the resonance fraction of maximum Mieto-ADA absorption efficiency ratios on the real index of refraction, where internal reflection/refraction contributions to Qabs are negligible.

(1 2 Qabs,ADA ) estimates the fraction of radiation available for internal reflections after passing once through the sphere. The constants in (5) were determined empirically by minimizing errors relative to Mie theory in x regions where internal reflection/refraction dominated over tunneling effects. 2) TUNNELING

PARAMETERIZATION

To parameterize tunneling for absorption, highly absorbing wavelengths for water or ice spheres were selected in the thermal infrared (IR), where internal reflection/refraction effects were negligible near the strongest resonance peak (where D and l are comparable). These IR wavelengths were used to isolate the tunneling contribution to absorption. Figure 3 compares Qext and Qabs predicted by Mie theory and ADA for ice spheres and a l of 13.0 mm. The maximum difference between ADA and Mie theory occurs near D 5 10 mm, where tunneling contributions are highest. This difference maximum is determined via Rmax , which is the maximum value of the ratio R 5 Qabs,Mie /Qabs,ADA . When Qabs,ADA is near 1.0, the absorption contribution from internal reflection/refraction is negligible. For these conditions, Rmax was estimated. For example, Rmax in Fig. 3 is about 1.5, indicating about ⅓ of Qabs is due to tunneling effects. The component of Rmax due to tunneling, r a , was then estimated. For instance, r a in Fig. 3 is 0.5. In some cases, (4) was used to subtract the contribution of internal reflection/refraction from Rmax to obtain r a . These r a values are plotted against nr in Fig. 4. The linear regression estimating ra is r a 5 0.7393n r 2 0.6069.

(6)

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It is seen that the maximum tunneling contribution is estimated only via n r . This is not always the case but is viable when refractive indeces corresponding to water and ice are used regarding solar and terrestrial radiation. In fact, tunneling also depends on n i , especially when n i . 0.1. Although Mie theory shows the maximum tunneling contribution in water and ice spheres depends on both n r and n i , (6) fortuitously represents this dual dependence only through n r . Hence, the parameterization here is not universal but serves a practical need for parameterizing radiative properties in water and ice clouds. The variable r a indicates the maximum resonancedependent tunneling contribution, but not how tunneling contributions are spread out over a range of size parameters, or x. A Gaussian distribution would be most appropriate for this, but it is desirable to have an expression for Qabs that is integratable, thus yielding an equation for the absorption coefficient. A gamma distribution is well suited for this purpose, and the following ratio of gamma distributions yields the tunneling contribution for any combination of D and l: C2 5 ra

k m exp(2e 0 k) , m k max exp(2m)

(7)

where variable k 5 D/l and m is the dispersion parameter of the gamma distribution. The parameter e 0 determines the k value (kmax ) where Rmax occurs and was found to be estimated reasonably well as a function of the imaginary refractive index, n i :

e 0 5 0.25 1 0.6[1 2 exp(28pn i /3)] 2 .

(8)

This k value is given as kmax 5 m/e 0

(9)

and indicates the droplet size having the maximum tunneling contribution for a given l. Again, (8) is not universal and is valid only for refractive indexes corresponding to water and ice for solar and terrestrial wavelengths. Equation (7) is plotted in Fig. 5 for r a 5 0.4, m 5 e 0 5 0.5, and l 5 50 mm. Since kmax 5 1, C 2 peaks at D 5 50 mm. In general, tunneling contributions peak when D and l are comparable, but often kmax is slightly less than 1.0. An m value of 0.5 was found to best produce the dispersion of tunneling effects predicted by Mie theory. Having parameterized the contributions to Qabs for internal reflection/refraction and tunneling effects, Qabs can be expressed as Qabs 5 (1 1 C1 1 C 2 )Qabs,ADA .

(10)

It is important to note that tunneling was formulated here simply as the absorption of photons beyond a particle’s area cross section under conditions where almost all radiation incident over this cross section was absorbed, as evaluated through comparisons between ADA and Mie theory.

FIG. 5. Gamma function form used to estimate the dispersion of resonance effects over a range of particle diameters or size parameters. Here, Eq. (6) is plotted for l 5 50 mm, r a 5 0.4, and m 5 e 0 5 0.5.

c. Parameterization of extinction due to tunneling 1) TUNNELING

IN THE RESONANCE REGION

The tunneling addressed in this section generally reaches a maximum in the vicinity of the first (largest) interference maximun (typically, 1 , x , 10). The tunneling process for Qabs must also contribute to Qext , since Qext 5 Qabs 1 Qsca (Qsca 5 scattering efficiency). Tunneling effects for Qext were parameterized in the same way as for Qabs , except r a is divided by 2 for the following reason: The tunneling contribution to Qabs , C 2 Qabs,ADA , is maximum for a given n r and k when Qabs,ADA 5 1.0. This fraction of energy is predicted to always undergo tunneling, although it is completely absorbed only when n i is sufficiently large. For smaller n i , the energy fraction not absorbed should be scattered, such that the fraction absorbed and scattered equals C 2 . Since Qext ø 2 on average when tunneling effects are not included (e.g., see ADA curves in Figs. 2 and 3), r a should be divided by 2: rext 5 ra /2, C3 5 rext

k m exp(2e 0 k) . m k max exp(2m)

(11) (12)

Since scattering related to particle cross section is already included in ADA, tunneling effects can be incorporated into Qext as Qext 5 (1 1 C 3 )Qext,ADA ,

(13)

where Qext,ADA is given as (van de Hulst 1981) Qext,ADA 5 4 Re[K(t)], where Re indicates only the real part is used, and

(14)

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FIG. 6. Extinction and absorption efficiencies predicted by ADA (long-dashed curve) and Mie theory (solid curve) for ice spheres at l 5 13.0 mm (n r 5 1.472, n i 5 0.389). The short-dashed curves were predicted by the modified ADA via Eqs. (9) and (12), where Qext does not include edge effects.

1 e 2t e 2t 2 1 K(t) 5 1 1 , 2 t t2

(15)

where t 5 i2pD(n 2 1)/l, n 5 complex index of refraction and i 5 (21)1/2 . From (13), it is evident that the maximum potential tunneling contribution is approximately the same for Qabs and Qext , although it is modified by interference effects for Qext . 2) PARAMETERIZATION

OF

‘‘EDGE

EFFECTS’’

Comparisons of Qext predicted by (13) (short-dashed curve) with Mie theory (solid curve) are shown in Fig. 6 for ice spheres, l 5 13 mm. ADA curves (long-dashed curves) are given for reference. While ADA and tunneling account for Qabs in the resonance region, there is clearly a missing contribution for Qext . It is possible that ‘‘creeping’’ or surface waves, which ray number 1 in Fig. 1 is attributed to, undergo little absorption and hence would not be accounted for in the tunneling parameterization for Qabs . Conveniently, Wu (1956) has isolated the scattering contribution of such surface waves to dielectric spheres as an asymptotic expansion, although a good degree of accuracy was obtained by using only the first term. Subsequently, van de Hulst (1981, p. 364) and later Ackerman and Stephens (1987) have parameterized these ‘‘edge effects’’ as Qedge 5 hx22/3 ,

(16)

where h was 1.8 or 2.0, respectively. Such edge effects are not the diffraction described by the classic Huygens–Fresnel–Kirchhoff theory, where diffraction is strongly forward scattering and pictured as a disturbance of wave propagation due to the removal

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FIG. 7. Same as Fig. 6, but including edge effects for Qext via Eq. (15). The short-dashed curve for Qext is for a 6 5 2 in Eq. 14, while the dot-dashed curve is for a 6 5 1.

of a portion of the wave fronts by the blocking effect of the obstacle. Rather, (16) accounts for the larger scattering angles of diffracted radiation around a curved edge (e.g., ray 1 in Fig. 1), which is due to the tunneling of grazing photons to form surface waves, which are then scattered (Wu 1956; Nussenzveig and Wiscombe 1987; Nussenzveig 1988). The last two references relate to the uniform approximation, which evidently includes, but is not restricted to, contributions from (16). Thus, the term edge effects used here really refers to a subset of the large-angle diffraction phenomena, as described in the Nussensveig references. The absence of a Qedge dependence on n r or n i suggests surface waves are not entering the particle and are not subject to absorption. Although edge effects are also a tunneling phenomena, throughout the text the term tunneling is used to denote tunneling that can potentially result in absorption, as formulated in section 2b and 2c (1), while the tunneling described here will be referred to as edge effects. Unfortunately, (16) overestimates Qext considerably for x , 15. To parameterize edge effects for lower x values, the formulation must attenuate Qedge , since edge effects should be nonexistent for x & 1. This can be done easily for single particles, but the resulting parameterization was not integratable over analytical particlesize distributions. The parameterization below is integratable and, while attenuating Qedge , the attenuation is not rapid enough, giving finite values of Qedge for x , 1: Qedge 5 a 6 [1 2 exp(20.06x)]x22/3 ,

(17)

where a 6 may range from about 1 to 2. Nonetheless, errors in (17) are not too great for describing Qext for x . 1, and the final expression for Qext is given as Qext 5 (1 1 C 3 )Qext,ADA 1 Qedge .

(18)

1 MAY 2000

It is instructive to revisit Fig. 6, but this time with edge effects included, as shown in Fig. 7. The shortdashed curve for Qext is for a 6 5 2 in (17), while the dot-dashed curve is for a 6 5 1. It was found that for most wavelengths regarding water and ice spheres, a 6 5 1 gives better agreement, and henceforth it is assumed that a 6 5 1. This result, as well as many similar comparisons, clearly implicate edge effects as the physics missing in (13).

N(D) 5 N 0 D n exp(2LD),

Cloud droplet size distributions can be parameterized using the gamma distribution,

bext 5 babs 5

E E

(p /4)D 2 Q ext N(D) dD and

(20)

(p /4)D 2 Q abs N(D) dD,

(21)

(10), (18), and (19) can be used to solve for bext and babs , which yields

sN0 G(d 1 n 1 1) sN0 G(d 1 n 1 1) a sN G(d 1 n 1 1) a sN G(d 1 n 1 1) 2 1 1 0 2 1 0 Ld1n 11 (L 1 g) d1n 11 (L 1 g) d1n 11 (L 1 2g) d1n 11 1

bext 5

(19)

where L is the size distribution slope parameter, n determines the concentration of relatively small droplets, and N 0 relates L and n to the liquid water content and droplet number concentration of the cloud. Since the extinction and absorption coefficients, bext and babs , are defined as

d. Parameterization of the extinction and absorption coefficients

babs 5

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MITCHELL

a 2sN0 G(d 1 n 1 m 1 1) a sN G(d 1 n 1 m 1 1) 2 2 0 , d1n 1m11 (L 1 e) (L 1 g 1 e) d1n 1m11

5

(22)

]6

[

pN0 G(n 1 3) G(n 1 2) G(n 1 1) 1 Re pN0 1 ((L 1 q)2(n 11) 2 L2(n 11) ) n 13 n 12 2L q(L 1 q) q2

5

1 Re p a3 N0

1

p a 3 N0 G(n 1 m 1 3) 2(L 1 e) n 1m13

]6

[

G(n 1 m 1 2) G(n 1 m 1 1) 1 ((L 1 e 1 q)2(n 1m11) 2 (L 1 e)2(n 1m11) ) q(L 1 e 1 q) n 1m12 q2

1 a 6sa 5 N0 G(n 1 7/3)[L2(n 17/3) 2 (L 1 a4 )2(n 17/3) ],

where a2 5 a3 5

k

m max

ra , exp(2m)lm

(24)

k

m max

rext , exp(2m)lm

(25)

a4 5 0.06p /l,

(26)

a 5 5 (p /l)22 /3 ,

(27)

e 5 e 0 /l,

(28)

and where g 5 8pn i /3l, q 5 i2p(n 2 1)/l is a complex variable (n 5 complex index of refraction, i 5 211/2 ), Re indicates only the real part of the term is used, G is the gamma function, s 5 p/4, and d 5 2. Note that (24)–(28) are only functions of l, n r , and n i . The computation time required for (22) and (23) are orders of magnitude less than Mie theory requires. The first pair of terms in (22) and (23) are the ADA solutions for babs and bext . The second pair of terms in (22) estimate the contribution of internal reflection/refraction, while the last pair of terms estimate the con-

(23)

tribution of tunneling to babs . The next two terms in (23) estimate the contribution of tunneling to bext , while the last term estimates the contribution of edge effects. The above equations for babs and bext are accurate to within about 10% relative to Mie theory for x e . 1, where x for a size distribution is given as x e 5 2 p r e /l,

(29)

where r e 5 effective radius (Slingo 1989). Similar accuracy can be achieved for x * 0.5 for absorption, x * 0.7 for extinction, if the terms for internal reflection/refraction, tunneling, and edge effects are attenuated to zero as x e approaches the Rayleigh regime (where these processes should not exist). This was done by multiplying each term by its corresponding empirical correction for low x, C x . For edge effects, Qedge depends only on x and hence C x was formulated as C x 5 1 2 exp(2kx e8 ),

(30)

where k 5 0.035. For internal reflection/refraction and for tunneling, the photon travels through the medium, having a refractive index dependence (Guimares and Nussenzveig 1992). The following expression for tun-

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neling reduced errors by up to 6% relative to formulating C x in terms of x e only:

where the bridging functions for extinction and absorption are of the form

C x 5 1 2 exp[2k(x e /n r ) 5 ],

Cray 5 exp(2kx em).

(31)

where k 5 37.52. The x e /n r dependence may relate to the fact that the transition from resonance to Rayleigh region becomes more abrupt as n r increases. For internal reflection/refraction, C x is given as C x 5 1 2 exp[2k(x e n r ) 3 ],

(32)

where k 5 0.014. These three formula are empirical, based on error minimization and physical intuition.

(41)

For babs , k 5 41.2 and m 5 5. For bext , it was found that errors where best minimized by using two Cray expressions, depending on size distribution mean diameter, D, and n. When D , 10 mm and n , 4, then k 5 59.5 and m 5 8.00. Otherwise, k 5 2.28 and m 5 5.60. This limited errors to ,14% in the bridging region for r e $ 5 mm and generally to #10% for r e $ 5.5 mm. f. Size distributions

e. Extension to the Rayleigh regime When the above contributions have attenuated to zero, one is left with the ADA solution for bext and babs . If nothing is done, errors increase rapidly with decreasing x e . However, one can link the modified ADA solution where x e is near unity with the beginning of the Rayleigh regime using a ‘‘bridging’’ function, for example,

bext 5 (1 2 Cray )bext,Mod 1 Cray bext,Ray ,

(33)

where Cray is the bridging function and bext,Ray is the Rayleigh solution for bext . Because errors are small in the vicinity of x e ø 0.7 to 1.1 (or 0.5 for babs ), which is close to the Rayleigh regime (x e ø 0.3), such an empirical approach is feasible. In the Rayleigh regime, Qabs is given as Qabs 5 24x Im[(n 2 2 1)/(n 2 1 2)],

(34)

where n is the complex index of refraction and Im indicates only the imaginary part is taken. Solving for babs as in (21),

babs 5

pK 2 N0 G(n 1 4) , 4Ln 14

(35)

where K 2 5 2(4p /l) Im[(n 2 2 1)/(n 2 1 2)].

(36)

Similarly, the Rayleigh scattering coefficient, bsca, is calculated, noting Q sca 5 (8/3)x |(n 2 1)/(n 1 2)|, 4

bsca 5

2

2

pK1 N0 G(n 1 7) , 4Ln 17

(37) (38)

where K1 5 (8p 4 /3l4 )|(n 2 2 1)/(n 2 1 2)|.

(39)

Extinction is then simply the sum of bsca and babs . However, bsca increases rapidly above Mie theory values with increasing x e outside the Rayleigh regime, producing large errors in the bridging region. To reduce these errors, the Rayleigh extinction coefficient was formulated as

bext,Ray 5 babs,Ray 1 CRay bsca,Ray ,

(40)

This treatment of cloud radiative properties has been formulated through the size distribution parameters L, N 0 , and n, which are related to the measured properties of cloud liquid water content (LWC), mean diameter (D), and median mass diameter (D m ), as described in Mitchell (1991):

n 5 (3.67D 2 Dm )/(Dm 2 D ),

(42)

L 5 (n 1 1)/ D,

(43)

N0 5 6LWCL

41n

/pr w G(4 1 n),

(44)

where D m divides the size distribution into equal masses, r w is the density of water, and G denotes the gamma function. Since this treatment also utilizes x e and hence the effective radius, r e [as commonly defined, e.g., Slingo (1989)] it is useful to note that r e 5 (n 1 3)/2L,

(45)

which, in combination with (43), provides a link between this scheme and others: D 5 2r e (n 1 1)/(n 1 3).

(46)

Another helpful relationship when using this scheme is the number concentration: N5

1

2

6G(n 1 1)LWC n 1 3.67 , pr w G(n 1 4) Dm

Dm 5 (n 1 3.67)/L.

3

(47) (48)

While radiative properties for water clouds are often parameterized in terms of r e and LWC, this scheme requires an additional variable, n. The question arises as to what values of n are appropriate for water clouds. Unlike N, LWC, r e and D, measured values of n are rare in the literature but can be calculated from other properties. In a study of maritime cumuli clouds near Hawaii, Pontikis and Hicks (1993) give a frequency distribution of the size distribution dispersion parameter, d, for one cloud, which can be related to n as

n 5 (1/d) 2 2 1.

(49)

The vast majority of n values ranged from 10 to 43, with 4 being the lowest. Hudson and Li (1995) report

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MITCHELL

droplet size standard deviation and mean, the ratio of which is d, for a clean and a polluted midAtlantic stratus cloud. For both clouds, n was between 4 and 5. Similar data from Hudson and Svensson (1995) for California marine stratus show n ø 24 on average for 26 horizontal cloud penetrations of stratus without drizzle, and n ø 7 on average for 22 other penetrations of stratus exhibiting drizzle production, based on four days of sampling. Analyzing the data in Hoffmann and Roth (1989) with Eq. (42), n was 3.5 6 1.7 and 11.2 6 6.1 for a nonprecipitating stratus and stratocumulus over land, respectively. Finally, Duda et al. (1991) report tethered balloon microphysical measurements through marine stratocumulus off the California coast. For one balloon flight, the average n was 1, while n was 5 for the other two flights. These studies suggest that a reasonable range of n for the above cloud types might be 2 # n # 40. This differs from earlier work by Carrier et al. (1967), where n ranged from 1 to 6 (Chylek at al. 1992). However, measurements of droplet spectra were much fewer for that study, and made no earlier than 1960 [prior to the forward scattering spectrometer probe (FSSP)]. Nonetheless, more recent radiation parameterizations (e.g., Kneizys et al. 1988) still assume model clouds where n ranges from 1 to 6 (Chylek et al. 1992). With a reasonable range for n, it is now possible to determine whether this parameterization will be applicable to most cloud types, since errors in bext can exceed 20% in the thermal IR when r e # 3.0 mm. To determine a lower bound for r e , Slingo and Schrecker (1982, henceforth SS) list typical r e values for eight different cloud types. The lowest r e was for stratus clouds, being about 5.0 mm, which gives D 5 8.5 mm for n 5 10. For stratocumulus clouds, SS show r e averaged 7.65 mm, giving D 5 12.9 mm for n 5 10. Other studies regarding marine arctic stratus, stratocumulus, and cumulus clouds (Herman and Curry 1984; Tsay and Jayaweera 1984; Duda et al. 1991; Pontikis and Hicks 1993; Blyth and Latham 1991) show that r e is rarely less than 5 mm, such that 5 mm may be viewed as a practical lower limit for r e . Parameterization errors are discussed in the next section. 3. Results a. Single particles Figure 7 compares Qabs predicted by (10) (shortdashed curve) with Qabs,Mie (solid) and Qabs,ADA (longdashed curve) for ice spheres, where l 5 13.0 mm, n r 5 1.4717, and n i 5 0.389. Contributions from internal reflection/refraction are not included, as these processes should be diminished or absent in the resonance-Rayleigh regime (xn r & 3). Agreement appears reasonable for D $ 4 mm or x $ 1. Overestimates of Qabs for D . 200 mm are due to the absence of external reflection, a process not parameterized. Extensive comparisons with Mie theory revealed that Qabs errors via (10) were

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FIG. 8. Estimated contributions of physical processes to Qabs and Qext , for l 5 1.93 mm, n r 5 1.309, n i 5 0.0019. Resonance tunneling (two long-dashed curves) contributes to Qext and Qabs , edge diffraction (dot-dash curve) contributes to Qext , and internal reflection/refraction (lower short-dashed curve) contributes to Qabs . Edge effects assume a 6 5 1. The upper two solid curves give Qabs and Qext predicted by Mie theory, which are compared with the modified ADA (shortdashed curve).

generally ,10% for x . 1. The upper short-dashed and dot-dashed curves (a 6 5 2 and 1, respectively) compares Qext via (18) with Qext,Mie (solid) and Qext,ADA (longdashed curve). Errors in Qext via (18) were generally ,10% for x . 3, based on extensive comparisons with Mie theory. The estimated contribution of various physical processes to Qabs and Qext are shown in Fig. 8 for water drops, at an absorption peak in the near-infrared where all processes are active (l 5 1.93 mm, n r 5 1.309, n i 5 0.0019). Solid curves show Mie theory for Qabs and Qext , while short-dashed curves were predicted by the modified ADA, with a 6 5 1 regarding Qext . Interference effects, producing the larger oscillations in Qext , are well represented by ADA. The lowest short-dashed curve (visible between 20- and 300-mm diameter before merging into other curves) gives the contribution of internal reflection/refraction to Qabs . The two long-dashed curves give the contribution of tunneling to Qext and Qabs (the curve for Qabs being barely visible near D 5 10 mm). In the thermal IR, these two curves are almost identical for x . 2, n i $ 0.35, where tunneling contributes to Qabs and Qext about equally. The dot-dashed curve in Fig. 8 gives the edge effect contribution to Qext . Note that this curve makes significant contributions to Qext for x , 1, which is physically unrealistic. b. Extinction and absorption coefficients Comparisons of babs and bext via (22) and (23) with Mie theory were done in terms of efficiencies pertaining to the size distribution, based on the definitions

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FIG. 9. Cloud droplet size distribution shapes found in natural clouds, with gamma distribution parameter n 5 2 (dashed curve) and 20 (solid curve). FIG. 11. Same as Fig. 10 except n 5 2.

Q abs 5 babs /P,

(50)

Q ext 5 bext /P,

(51)

where P 5 size distribution projected area. Refractive indexes used were from Downing and Williams (1975). The accuracy of this scheme will be demonstrated for two reasonable values of n, values of 2 and 20, assuming D 5 10 mm. When n 5 20, r e 5 5.5 mm, which is about the lower limit observed for r e . To better appreciate the influence of n on size spectra, two droplet size distributions are shown in Fig. 9 having the same D and LWC, but having n values of 2 and 20. Increasing n has the effect of narrowing the spectrum, so that the spectrum behaves radiatively more like a single particle.

FIG. 10. Extinction efficiencies predicted via (51) for a cloud droplet size distribution having D 5 10 mm and n 5 20. Compared are Mie theory (solid curve) and modified ADA (short-dashed curve). The contribution of resonance-dependent tunneling (long-dashed curve) and edge effects (dot-dashed curve) to Qext is also shown.

This can be seen in Figs. 10 and 11, where bext via (23) is compared against Mie theory for n 5 20 and 2, respectively, for wavelengths ranging from 1.0 to 100 mm. The solid curve is from Mie theory while the shortdashed curve is the modified ADA. The broader spectra (n 5 2) of Fig. 11 dampen out the peaks and valleys (in the resonance region) of Fig. 10. Also shown in Figs. 10 and 11 are resonance-dependent tunneling contributions to Qext via the long-dashed curve, while edge effects are shown by the dot-dashed-curve. Absorption results are shown in Figs. 12 and 13 for n 5 20 and 2, respectively, where the solid curve is

FIG. 12. Absorption efficiencies predicted via (50) for a cloud droplet size distribution having D 5 10 mm and n 5 20. Compared are Mie theory (solid curve) and modified ADA (short-dashed curve). The contribution of resonance-dependent tunneling (long-dashed curve) and internal reflection/refraction (dot-dashed curve) to Qabs is also shown.

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MITCHELL

FIG. 13. Same as Fig. 12 except n 5 2.

from Mie theory. Broader spectra have a similar effect as above, tending to fill in the ‘‘valleys’’ in Qabs relative to n 5 20. The contribution of tunneling to Qabs is indicated by the long-dashed curve, while the dot-dashed curve gives the contribution from internal reflection/ refraction. Tunneling contributions vary but typically account for 25% of Qabs in the thermal IR. Although difficult to discern, internal reflection/refraction contributes more than tunneling for l & 2.6 mm (see Fig. 8). This is because 1) droplet sizes are kl in this region (tunneling is greatest when l ø D) and 2) lower n i for l , 2.6 mm allows for more internal reflections. Single-scatter albedo (v 0 ) is shown in Figs. 14 and 15 for D 5 10 mm and n 5 20 and 2, respectively.

FIG. 14. Single scattering albedo (v 0 ) predicted via (50) and (51) for a cloud droplet size distribution having D 5 10 mm and n 5 20. Compared are Mie theory (solid curve) and modified ADA (shortdashed curve).

FIG. 15. Same as Fig. 14 except n 5 2.

Since Qext includes the processes determining Qabs , this physically based approach has the tendency for errors in both Qext and Qabs to be of the same sign over the region the parameterized processes are applicable (xe $ 1). This results in relatively low errors in v 0 over this region. However, in the bridging region between the Mie and Rayleigh regimes (xe , 1), where a purely empirical approach was taken, there is no tendency for Qext and Qabs errors to be of the same sign, and errors can be quite large. The way v 0 is defined (v 0 5 1 2 Qabs /Qext ) also contributes to large errors in this region. The percent error of modified ADA relative to Mie theory is plotted against wavelength for Qext , assuming D 5 10 mm and n 5 20 and 2, in Figs. 16 and 17, respectively. Errors are less than 10% for almost all

FIG. 16. Percent error of modified ADA relative to Mie theory for Qext , assuming D 5 10 mm and n 5 20.

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FIG. 17. Same as Fig. 14, except n 5 2.


values of l and converge to zero after entering the Rayleigh regime at x e ø 0.3. A similar analysis for Qabs is shown in Figs. 18 and 19, and for v 0 in Figs. 20 and 21. For both n 5 20 and 2, errors for Qabs are less than 10% for almost all values of l, converging to zero within the Rayleigh regime. For xe $ 1, v 0 errors were generally #4%, while errors can reach 100% or more (not shown) in the bridging region for xe , 1. Errors were generally ,10% for bext and babs at any solar or terrestrial wavelength, provided 5 mm # r e # 30 mm. At smaller r e , errors over the l range of 1 to 100 mm will exceed 10%, with bext error reaching 20% for r e 5 3.0 mm and babs error reaching 17% for r e 5 1.1 mm. At larger r e , bext errors remain within 10% while babs errors for l . 60 mm can be as high as 20%. This

appears due to the behavior of tunneling deviating from a linear n r dependence at large n r values (n r ø 2).

FIG. 18. Percent error of modified ADA relative to Mie theory for Qabs , assuming D 5 10 mm and n 5 20.

FIG. 20. Percent error of modified ADA relative to Mie theory for v 0 , assuming D 5 10 mm and n 5 20.

c. Application to cloud emissivities The accuracy of modified ADA for determining cloud broadband absorption and emissivity in the thermal infrared may be better than at a specific wavelength, since errors cancel when integrating over wavelength. For example, Table 1 lists absorption optical depths (t a ) and emissivities (E) for the thermal IR, obtained by integrating from l 5 4.0 to 100 mm as

t5

E

l2

l1

@E

l2

B(T, l)t (l) dl

B(T, l) dl,

(52)

l1

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MITCHELL


where t refers to either t a or E, and B(T, l ) is Planck’s blackbody radiation function for temperature T 5 278 K. These broadband values of t a and E were calculated for various D values with n 5 20 for a cloud depth of 100 m and a LWC of 0.05 g m23 . Emissivities were calculated in two ways: using the zero-scattering approximation (Paltridge and Platt 1976) and by using the two-stream Delta–Eddington approximation given in Slingo (1989), which uses v 0 , t ext and g (asymmetry parameter) to obtain E as E 5 1 2 T 2 R,

(53)

where T and R are total transmission and reflection, and a single g value of 0.90 was assumed to represent the entire terrestrial spectrum. For purposes of comparison, this g assumption will not compromise the results, nor are the results sensitive to the value of g. These calculations ignore the effect of water vapor absorption. It is seen in Table 1 that E errors are less than 1.2% relative to Mie theory. Moreover, E errors associated with v 0 and (53) are no worse than those based on babs (zeroscattering approximation). This is due to little energy in the terrestrial spectrum for x , 1, making the impact of the above mentioned v 0 errors negligible.

4. Discussion a. Radiation–particle interactions The main development making this parameterization possible was the parameterization of the process of photon tunneling, where tunneling was formulated through evaluating the absorption of photons beyond a particle’s area cross section. Hence, this formulation only addresses tunneling that can potentially result in absorption. This tunneling formulation appears to contribute to large-angle diffraction and resonance, but does not appear to include the tunneling of photons which remain as surface waves without entering the drop and are then scattered. This latter type of tunneling was also parameterized and is referred to as edge effects. Tunneling was formulated in terms of n r , n i , and size parameter x, although the maximum tunneling contribution depended only on n r , provided that only n r values pertaining to water or ice are used. The ability of this simple parameterization to approximate Mie theory with reasonable accuracy is consistent with the assumption that the following physical processes dominate the absorption of radiation by water drops: 1) absorption of incident radiation penetrating into or through the water drop, 2) photon tunneling, 3) internal reflections, and 4) refraction. For scattering, the dominate processes appear to be 1) external reflections, 2) internal reflections, 3) refraction, 4) classic diffraction, 5) tunneling, 6) edge effects, and 7) wave interference phenomena. A process not parameterized is the reduction in absorption due to external reflections. As n i and n r increase, more radiation is externally reflected and less is available for absorption. This modified ADA may also be adapted to treating the radiative properties of nonspherical ice, as evident from the ADA-based scheme of Mitchell et al. (1996). The breakdown of physical processes may be particularly useful in this respect, as it is unclear whether all the processes represented in Mie theory are applicable to ice crystals (Mitchell et al. 1996; Mitchell and Macke 1997). b. Comparison with other schemes The parameterization of babs and bext described in section 2 is based on three size distribution parameters,

TABLE 1. Broadband (4–100 mm) absorption optical depths (t a ) and emissivities (E) calculated for a 100-m thick cloud having a mean temperature of 58C, LWC 5 0.05 g m23, n 5 20, and various values of D. Values for modified ADA (denoted mod) and Mie theory (denoted Mie) are shown, as well as percent error relative to Mie theory values. Emissivities were calculated using the zero scattering approximation and by treating the IR the same as the solar, using the Slingo (1989) two-stream Delta–Eddington approximation and assuming g 5 0.90. E512R2T

Zero scatt. approx. D (mm)

re (mm)

t a,mod

t a,Mie

% error

Emod

EMie

% error

Emod

EMie

% error

5 10 15 20 30

2.7 5.4 8.2 11.0 16.4

0.806 0.642 0.500 0.398 0.272

0.814 0.648 0.504 0.402 0.276

21.01 20.81 20.77 20.82 21.39

0.690 0.637 0.555 0.479 0.361

0.691 0.636 0.556 0.481 0.365

20.06 0.12 20.20 20.45 21.12

0.705 0.653 0.573 0.497 0.376

0.705 0.652 0.574 0.499 0.381

20.03 0.15 20.17 20.43 21.11

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TABLE 2. Percent error relative to Mie theory for broadband (1–4 mm) extinction and absorption coefficients, and the co-albedo, calculated from modified ADA and the Slingo–Schrecker scheme, based on n 5 20 and various values of D . Values in parenthesis were obtained using a constant value of a1 5 0.45 regarding (4). Absorptivity (A) errors using the SS g values and two-stream equations are also given for an idealized cloud of LWC 5 0.15 g m23 , 100-m depth. Percent error Modified ADA

SS scheme

D (mm)

re (mm)

bext

babs

1 2 vo

A

bext

babs

1 2 vo

A

5

2.7

2.2

15.5

13.8

1.4

27.4

29.8

22.5

4.8

15

8.2

20.4

20.9

25.7

24.8

2.3

20

11.0

21.5

4.5

0.9

23.4

5.4

30

16.4

22.1

2.1 (4.5) 21.1 (1.4) 23.7 (21.1) 23.1 (20.4) 22.8 (20.03)

27.7

5.5

22.1 (1.2) 23.9 (20.9) 24.0 (21.0) 22.7 (0.4) 21.6 (1.6)

220.1

10

0.1 (3.4) 22.5 (0.5) 24.4 (21.4) 24.2 (21.1) 23.7 (20.6)

14.3

14.0

20.3

13.2

corresponding to D, D m , and LWC, and thus may differ somewhat from many radiation schemes in use, such as those for solar radiation based on r e and liquid water path (LWP, product of LWC and cloud depth), and those for terrestrial radiation based only on LWP. Parameterizations of cloud emissivity for the thermal IR sometimes assume an LWP dependence only (e.g., Stephens 1978), although Chylek and Ramaswamy (1982) have noted that emissivities between 11.5 and 14 mm also depend on the size distribution. Throughout some of the terrestrial radiation spectrum, absorption is moderate such that radiation can penetrate through droplets, resulting in a mass dependence for absorption (Pollack and Cuzzi 1980; Mitchell and Arnott 1994), and an LWP parameterization is justified. In other regions, such as between 11 and 17 mm, absorption is strong and depends on projected area. Hence, some dependence on the droplet size distribution is expected. Table 1 indicates a substantial dependence on D for thin clouds of low LWC. For instance, decreasing D from 20 to 10 mm increased broadband absorption optical depth and emissivity by 61% and 33%, respectively, for the same LWP. The parameterization of Chylek et al. (1995, 1992) gives bext , babs , bsca , and the asymmetry factor, g, all within 3% accuracy for l between 3 and 25 mm, using tenth-order polynomial curve fits. Hence, extensive lookup tables of fitting coefficients are used. Similar to this scheme, the Chylek et al. scheme was formulated in terms of r e and n, although error tests assumed n ranged from 2 to 6. While the Chylek et al. scheme is more accurate than this one at a particular l between 3 and 25 mm, both schemes appear to have similar accuracy for broadband emissivity calculations. For solar radiation (l , 3 mm), rapid oscillation of bsca due to interference phenomena renders curve fitting impractical, and the Chylek et al. scheme depends on broad size spectra (e.g., n 5 2) which permit sufficient error cancellation for estimates of bext . No information on the scheme’s performance was given for the near-infrared

regarding bext , babs , or the single-scattering albedo, v 0 . The modified ADA scheme does not suffer from these limitations and may be used at any solar or terrestrial wavelength with #10% error for an r e range of 5 to 30 mm. For solar radiation, this scheme was contrasted with the popular 24-band scheme of Slingo and Schrecker (i.e., SS) as discussed in Slingo (1989), as shown in Table 2. In Chylek et al. (1995), the co-albedos of five schemes were compared, of which the SS scheme was superior for l $ 1 mm. Table 2 assumes a model cloud with LWC 5 0.15 g m23 , depth 5 100 m, and n 5 20. Broadband values of bext , babs , and the co-albedo (babs /bext ) were calculated for various D values, obtained by integrating over l between 1 and 4 mm as described by Eq. (52), where B(T, l ) assumes T 5 5780 K at the sun’s surface. These values were calculated from Mie theory, the modified ADA, and the SS scheme, and percent errors relative to Mie theory are given in Table 2 for the modified ADA and SS schemes. In the SS scheme, babs was calculated as

babs 5 bext (1 2 v 0 ),

(54)

where v 0 was given in SS. Corresponding broadband absorptivities (A) were also calculated using the twostream radiation transfer equations in Slingo (1989) and the SS parameterization for g, with Table 2 contrasting A errors resulting from modified ADA and SS. Values in parenthesis under modified ADA were obtained by setting a1 in (4) to 0.45, which was found to yield greater accuracy if only the solar spectrum was of interest. Errors in A for the SS scheme are higher at the extreme D values, but for most applications are no worse than about 5%. The SS errors in the co-albedo are usually ,5%, in general agreement with the error analysis in Chylek et al. (1995). Negative values are underestimates. One should note that the SS scheme was developed from a limited number of size distributions with r e ranging from 4.2 to 16.6 mm, while entries in Table 2 for D 5 5 mm correspond to r e 5 2.7 mm, rarely

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MITCHELL

observed in nature. The modified ADA scheme makes use of 220 wavelengths in this analysis, while the SS scheme uses only 10 bands to cover the l range 1–4 mm. Given the coarser resolution, the SS scheme performs rather well. The main advantage of the modified ADA is greater accuracy over a broader range of size spectra, while the main disadvantage is the increased computational expense. 5. Summary The physical processes contributing to the efficiencies for absorption and extinction in water drops, Qabs and Qext , were either parameterized or represented by the anomalous diffracton approximation (ADA). This resulted in a modified ADA, which predicted Qabs and Qext within 10% of Mie theory values for x . 1 and x . 3, respectively. The parameterized physical processes not included in ADA were 1) internal reflection/refraction, 2) photon tunneling, and 3) edge effects. The second and third categories describe processes by which tangential photons beyond the physical cross section of a particle can be either scattered or absorbed. It appears that tunneling can explain most of the difference between ADA and Mie theory for water clouds in the thermal IR. Analytical expressions for Qabs and Qext were integrated over a gamma size distribution to yield expressions for the absorption and extinction coefficients, babs and bext . By attenuating processes 1–3 above as the Rayleigh regime is approached, and bridging the modified ADA solutions to the Rayleigh solutions for babs and bext , accuracy within 10% of Mie theory was generally obtained for babs and bext for any wavelength in the solar or terrestrial spectrum, for 5 mm # r e # 30 mm. For broadband emissivities regarding terrestrial radiation, the errors were #1.1%. Similar to the parameterization of Chylek et al. (1995, 1992), this parameterization is based on the three parameters of a gamma size distribution, which have been related to the three measured cloud properties of liquid water content, mean, and median mass diameter. It offers improvements over other schemes in both the near- and thermal infrared, and provides for physical insight. This scheme may be adapted to treat the radiative properties of ice clouds. Acknowledgments. This work was funded by the Environmental Sciences Division of the U.S. Department of Energy under Grant DE-FGO3-94ER61775, as part of the Atmospheric Radiation Measurement Program. Financial support does not constitute an endorsement by this agency of the views expressed in this article. Partial funding was also provided by the National Weather Service under NOAA Agreement NA67RJ0146 with the Cooperative Institute for Atmospheric Sciences and Terrestrial Applications (CIASTA). Mr. Yangang Liu provided Eq. (46), and his contribution is much

appreciated. Mrs. Olga Garro is thanked for her computer graphics support. Additional testing of this scheme was performed at CSU by Drs. Paul Stackhouse and Graeme Stephens, and at the Hadley Centre for Climate Prediction by Dr. John Edwards. Their constructive comments improved the quality of this paper and is gratefully acknowledged. Constructive comments from Qianq Fu and two anonymous reviewers are also appreciated. The computer code describing this treatment is freely available; those interested should e-mail the author. REFERENCES Ackerman, S. A., and G. L. Stephens, 1987: The absorption of solar radiation by cloud droplets: An application of anomalous diffraction theory. J. Atmos. Sci., 44, 1574–1588. Asano, S., and M. Sato, 1980: Light scattering by randomly oriented spheroidal particles. Appl. Opt., 19, 962–974. Baran, A. J., J. S. Foot, and D. L. Mitchell, 1998: Ice-crystal absorption: A comparison between theory and implications for remote sensing. Appl. Opt., 37, 2207–2215. Blyth, A. M., and J. Latham, 1991: A climatological parameterization for cumulus clouds. J. Atmos. Sci., 48, 2367–2371. Carrier, L. W., G. A. Cate, and K. J. von Essen, 1967: The backscattering and extinction of visible and IR radiation by selected major cloud models. Appl. Opt., 12, 555–563. Chylek, P., and V. Ramaswamy, 1982: Simple approximation for infrared emissivity of water clouds. J. Atmos. Sci., 39, 171–177. , P. Damiano, and E. P. Shettle, 1992: Infrared emittance of water clouds. J. Atmos. Sci., 49, 1459–1472. , , N. Kalyaniwalla, and E. P. Shettle, 1995: Radiative properties of water clouds: Simple approximations. Atmos. Res., 35, 139–156. Downing, H. D., and D. Williams, 1975: Optical constants of water in the infrared. J. Geophys. Res., 80, 1656–1661. Duda, D. P., G. L. Stevens, and S. K. Cox, 1991: Microphysical and radiative properties of marine stratocumulus from tethered balloon measurements. J. Appl. Meteor., 30, 170–186. Guimaraes, L. G., and H. M. Nussenzveig, 1992: Theory of Mie resonances and the ripple fluctuations. Opt. Commun., 89, 363– 369. Hansen, J. E., and J. B. Pollack, 1970: Near-infrared light scattering by terrestrial clouds. J. Atmos. Sci., 27, 265–281. Herman, G. F., and J. A. Curry, 1984: Observational and theoretical studies of solar radiation in Arctic stratus clouds. J. Climate Appl. Meteor., 23, 5–24. Hoffmann, H.-E., and R. Roth, 1989: Cloudphysical parameters in dependence on height above cloud base in different clouds. Meteor. Atmos. Phys., 41, 247–254. Hudson, J. G., and H. Li, 1995: Microphysical contrasts in Atlantic stratus. J. Atmos. Sci., 52, 3031–3040. , and G. Svensson, 1995: Cloud microphysical relationships in California marine stratus. J. Appl. Meteor., 34, 2655–2666. Kneizys, F. X., E. P. Shettle, L. W. Abreu, G. P. Anderson, J. H. Chetwynd, W. O. Gallery, J. E. A. Selby, and S. A. Clough, 1988: Users Guide to Lowtran 7. AFGL-TR-88-0177. [NTIS AD A206733.] Mitchell, D. L., 1991: Evolution of snow-size spectra in cyclonic storms. Part II: Deviations from the exponential form. J. Atmos. Sci., 48, 1885–1899. , and W. P. Arnott, 1994: A model predicting the evolution of ice particle size spectra and radiative properties of cirrus clouds. Part II: Dependence of absorption and extinction on ice crystal morphology. J. Atmos. Sci., 51, 817–832. , A. Macke, and Y. Liu, 1996: Modeling cirrus clouds. Part II: Treatment of radiative properties. J. Atmos. Sci., 53, 2967–2988.

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pirical theory and its application to tropospheric aerosols. J. Atmos. Sci., 37, 868–881. Pontikis, C. A., and E. M. Hicks, 1993: The influence of clear air entrainment on the droplet effective radius of warm maritime convective clouds. J. Atmos. Sci., 50, 2889–2900. Slingo, A., 1989: A GCM parameterization for the shortwave radiative properties of water clouds. J. Atmos. Sci., 46, 1419–1427. , and H. M. Schrecker, 1982: On the shortwave radiative properties of stratiform water clouds. Quart. J. Roy. Meteor. Soc., 108, 407–426. Stephens, G. L., 1978: Radiation profiles in extended water clouds. II: Parameterization schemes. J. Atmos. Sci., 35, 2123–2132. Tsay, S.-C., and K. Jayaweera, 1984: Physical characteristics of Arctic stratus clouds. J. Climate Appl. Meteor., 23, 584–596. van de Hulst, H. C., 1981: Light Scattering by Small Particles. Dover, 470 pp. Wu, T. T., 1956: High-frequency scattering. Phys. Rev., 104, 1201– 1212.