Extinction efficiency and single-scattering albedo for ...atmos.ucla.edu/~liougst/Group_Papers/Yang_JGR_102_1997a.pdfthe extinction and absorption of light beam by a particle, is the

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 102, NO. D18, PAGES 21,825-21,835, SEPTEMBER 27, 1997

Extinction efficiency and single-scattering albedo for laboratory and natural cirrus clouds

Ping Yang and K. N. Liou Department of Meteorology/CARSS, University of Utah, Salt Lake City

W. P. Arnott

Atmospheric Sciences Center, Desert Research Institute, Reno, Nevada

Abstract. A combination of the finite-difference time domain technique and a ray-by-ray integration method has been applied to compute the extinction efficiency and single- scattering albedo for various size distributions associated with nonspherical ice crystals in laboratory and natural cirrus clouds. The two methods are applicable to small and large size parameters, respectively. The results obtained by the two methods converge when effective size parameters are larger than about 6. For laboratory ice crystals the overall features of the computed extinction efficiency are in general agreement with those determined from measurements. In particular, significant extinction windows at 2.85 and 10.5/•m, associated with the Christiansen effect, are observed in both theoretical and experimental results. These extinction minima appear because the real part of the refractive index approaches unity, so that absorption dominates light attenuation. The single-scattering albedos at the two Christiansen spectral regions are found to be smaller than 0.5 for the laboratory ice crystals. The contours of extinction efficiency and single- scattering albedo versus wavelength and particle size show that the magnitude of the Christiansen effect is dependent on particle size. For large ice crystals, the extinction windows are not significant because the extinction efficiency converges to its asymptotic value of 2, regardless of size parameters. For a number of size distributions observed during FIRE II IFO, the Christiansen effect is small. However, for cold cirrus, the extinction efficiencies in the Christiansen bands are approximately one half of the values at nearby wavelengths due to a significant number of small ice crystals that are present in cold cirrus clouds. It is concluded that the Christiansen effect must be accounted for in

the determination of the extinction efficiency and the single-scattering albedo for small ice particles in order to obtain a reliable optical depth and emissivity for cirrus clouds at infrared wavelengths. Finally, we show that using spherical particles with Mie theory is inadequate to explain the extinction measurements.

1. Introduction

Cirrus clouds reflect solar radiation back to space but also contribute to the greenhouse effect by trapping the infrared emission from the lower atmosphere and the surface. Their effects on the Earth's climate system is dependent on such factors as the cloud position, ice water path, and ice crystal size and shape [Liou, 1986]. Efficient solutions for the single- scattering properties of nonspherical ice crystals covering the whole spectrum of size parameters are not available at this point. For this reason, our understanding of the radiative properties of cirrus clouds is quite limited in comparison with that of warm water clouds consisting of spherical water droplets whose optical properties can be solved by the exact Mie theory.

During the past two decades the geometric ray-tracing approximation (hereinafter referred to as GOM1) has been used to determined the single-scattering properties of nonspherical ice crystals [Wendling, 1979; Cai and Liou, 1982; Takano and Liou, 1989; Macke, 1993]. In GOM1 a constant extinction efficiency of 2 is employed, which is assumed to be contributed

Copyright 1997 by the American Geophysical Union.

Paper number 97JD0 1768. 0148-0227/97/97JD-01768509.00

equally by diffraction and Fresnelian rays. Yang and Liou [1995, 1996a, b] have shown that GOM1 produces substantial errors in the computation of extinction and single-scattering albedo for small and moderate size parameters (<40). These errors are a result of the shortcomings inherent to the conven- tional ray-tracing technique for these size parameters. Another approximate approach known as anomalous diffraction theory (ADT), originally developed by van de Hulst [1957], has also been extensively used to compute the extinction and absorption properties of ice crystals. In ADT the wave propagation inside a dielectric scattering particle is assumed to occur along the incident direction. The variation of field magnitude associated with the external and internal reflections at the particle surface is not accounted for. In addition, the vector properties or polarization configurations of the electromagnetic waves are neglected in ADT. Because of the errors caused by these ap- proximations, ADT is applicable only when the refractive index of the scatterer is close to unity (optically tenuous object). Chylek and Klett [1991] have shown that the errors associated with ADT can reach 25% for a refractive index with a real part of 1.4.

A recent experiment conducted by Arnott et al. [1995] has shown that two significant extinction minima exist at wave-

21,825

21,826 YANG ET AL.: SCATTERING PROPERTIES OF ICE CRYSTALS

lengths around 2.85 and 10.5/xm (known as the Christiansen bands) for ice crystals generated in a cloud chamber. These minima, which cannot be explained by GOM1, are essentially associated with the extinction properties of ice crystals with small size parameters at these wavelengths. Cirrus observations reveal evidence for a large number of small ice particles with sizes of the order of 10/xm [Platt et al., 1989]. It has been shown that small ice crystals can significantly contribute to, and possibly dominate, both solar albedo and infrared emission [Arnott et al., 1994]. Therefore a reliable method must be used to obtain the extinction efficiency and single-scattering albedo of ice crystals. These single-scattering parameters are essential to the determination of the optical depth and emissivity of cirrus clouds at infrared wavelengths for applications to remote sensing and radiative transfer modeling. In particular, accurate evaluation of the single-scattering properties of ice crystals in a Christiansen band is critical to the development of a reliable scheme for retrieving the optical and microphysical properties of cirrus clouds, using, for example, channel 4 of the AVHRR (advanced very high resolution radiometer) at 10.6 •m and channel 31 of MODIS (moderate resolution imaging spectrometer) at 11.0/xm.

In the present study, we use a combination of the finite- difference time domain (FDTD) technique [Yang and Liou, 1995, 1996a] and an improved geometric optics method in terms of ray-by-ray integration (RBRI) algorithm [Yang and Liou, 1997] to compute the extinction and single-scattering albedo for laboratory and natural ice clouds. The FDTD and RBRI methods, in practice, are applicable to effective ice crystal size parameters smaller and larger than about 6, respectively. In section 2, we outline the conceptual basis for FDTD and RBRI. In section 3, we present the size distributions and habits for laboratory and natural ice crystals used in the present computations. Discussed in section 4 are the computed extinction efficiency and single-scattering albedo for ice crystals along with the interpretation of the spectral extinction efficiency determined for laboratory ice clouds. Finally, conclusions are given in section 5.

2. Conceptual Basis of Computational Models

2.1. Finite-Difference Time Domain Technique

Pioneered by Yee [1966], FDTD has been recognized as an efficient method to account for the interaction of electromagnetic waves with a scattering object of arbitrary shape and composition. Yang and Liou [1995, 1996a] have applied this method to the solution of light scattering by nonspherical ice crystals. In practice, the FDTD technique solves the temporal variation of electromagnetic waves within a finite space con- taining the scatterer by a direct implementation of the Maxwell curl equations given by

/x OH(r, t) V x E(r, t)= -- (la) c Ot '

• 0E(r, t) 4rr V x a(r, t) = •- 0• + o-E(r, t) (lb) C '

where E(r, t) and H(r, t) are the magnetic and electric field, respectively; c is the speed of light in a vacuum; and/x, g, and o- are the permeability, permittivity, and conductivity of the medium consisting of the scatterer, respectively. Yang and Liou [1996a] have found that the absorption of a dielectric scatterer

can be efficiently accounted for by introducing an effective conductivity in numerical computations. In the practical implementation of FDTD the unbounded or open space, within which the scattering process occurs, must be truncated by im- posing an absorbing and transmitting boundary condition [Mur, 1981; Liao et al., 1984] at the boundary of the truncated computational domain. Subsequently, the truncated space is discretized by a number of grid meshes, and the presence of the scattering object is replaced by properly assigning the electromagnetic constants in terms of permittivity, permeability, and conductivity over the grid meshes. Furthermore, (la) and (lb) are discretized by using the leap-frog finite-difference scheme in a staggered form for both temporal and spatial differentials. At the initial time, a plane wave which does not require the harmonic condition is turned on. This wave, excited by the source, then propagates toward the scatterer and even- tually interacts with it. The propagation and scattering of the excited wave in the time domain can be simulated by using the finite-difference analogs of (la) and (lb) in a manner of time- marching iteration. The fields solved by the aforementioned finite-difference analogs of the Maxwell equations are in the time domain. In order to obtain the single-scattering properties of the scatterer, one requires the far field in the frequency domain. As pointed out in our previous study, one can input a Gaussian pulse as an initial excitation and apply the discrete Fourier transform technique to transform the near field from the time domain to the frequency domain. The far field in the frequency domain associated with the near field can be obtained by invoking a surface-integration [Yang and Liou, 1995] or a volume-integration transformation [Yang and Liou, 1996a] on the basis of rigorous electromagnetic relationships.

The FDTD technique is applicable to size parameters smaller than about 15 in practice, because of the computer memory and speed limitations even with the present super- computers. Errors in FDTD are mainly produced by the stair- casing effect, imperfect absorbing condition, and finite- difference approximation, all of which essentially depend on the resolution of grid meshes used. In the computation of extinction and absorption efficiencies, Yang and Liou [1995, 1996a] have shown that errors in FDTD are normally of the order of a few percent for some canonical problems in which the exact solutions are available for comparisons.

2.2. Ray-by-Ray Integration Method

On the basis of the conservation principle of the electromagnetic energy associated with the Poynting vector [Jackson, 1975], the extinction and absorption cross sections of a dielectric particle can be exactly expressed as follows:

o- e : Im i-•0[2(g - 1) E(r'). E•(r')d3r ' , (2a)

fff O' a : [-•012 lgi E(r')E*(r')d3r ', (2b)

where the asterisks denote the complex conjugate, E is the electric field inside the particle, Eo is the electric field associated with the incident or initial wave, k = 2 rr/X is the wave- number, g is the complex permittivity of the scatterer, and 8i is the imaginary part of the complex permittivity. The domain of integration in (2a) and (2b), which provide a physical rationale for constructing various approximate approaches to solve for

YANG ET AL.: SCATTERING PROPERTIES OF ICE CRYSTALS 21,827

140

120

100

eo

40

2O

0

• T=-7øC

i I • Measured Size

Ii[ • Distribution I.fl ............ Gamma !' Distribution

0 10 20 30 40 50 60

Maximum Dimension (tJm)

T:'17øC Measured Size Distribution BDii•dbualtion

0 10 20 30 40 50 60

Maximum Dimension (tJrn)

Figure 1. Measured size distributions for laboratory ice crystals produced at two cloud chamber temperatures, along with gamma and bimodal distributions that fit the observed data.

the extinction and absorption of light beam by a particle, is the region inside the scattering particle. By approximating the internal field in the integrand of the preceding equations, a number of methods have been developed: the Rayleigh-Gans approximation, also known as the first-order Born approximation in quantum mechanics [Acquista, 1976; Behren and Huff- man, 1983]; the Wentzel-Kramers-Brillouin approximation and its close ally known as the high energy or eikonal approximation [Klett and Sutherland, 1992; Chen, 1989; Pertin and Chiappetta, 1985; Pertin and Lamy, 1986]; the small perturba- tion expansion [Shiffrin, 1951]; and the anomalous diffraction theory [van de Hulst, 1957]. These methods, however, do not account for the effect of a particle boundary associated with the wave reflection and the deviation of wave propagation at the particle surface. Thus they are usually applicable to cases involving the refractive index close to unity so that the boundary effect is insignificant.

To circumvent the limitations of the aforementioned approaches, Yang and Lieu [1997] have developed the RBRI method in which the geometric ray tracing is employed to calculate the electric field inside the particle by accounting for complete phase and polarization configurations. The integra- tions involved in (2a) and (2b) can be carried out along each individual ray path inside the particle. The potential errors in RBRI are produced by the localization principle for geometric rays in calculating the near field, which requires that the particle size be much larger than the incident wavelength. Yang and Lieu [1997] have further shown that RBRI reduces to ADT when the refractive index approaches unity. The RBRI approach is essentially a proper combination of the geometric optics approximation and the electromagnetic wave theory. We illustrate that the RBRI solutions converge to the FDTD results in the computation of the efficiencies and single- scattering albedo, provided that the effective size parameter of the particle Xcff is larger than about 6 where

rr( 2a L ) •/ 2 )•: x ' (3)

and a and L are the semiwidth and length of a hexagonal ice crystal, respectively.

3. Size Distributions and Habits of Ice Crystals In conjunction with the spectral extinction coefficient mea-

surements, Arnett et al. [1995] generated ice crystals in cloud chamber temperatures ranging from -21øC to -5øC. The size distributions of these ice crystals are observed by two cloud probes: a continuous-running formvar replicator and a newly developed cloud scope. Two representative experimental size distributions were selected for the present theoretical computations. Figure 1 shows the two ice crystal size distributions generated at the cloud chamber temperatures of -17øC and -7øC. The number concentration of ice crystals is plotted versus the ice crystal maximum dimension. Table 1 lists the temperature in the cloud chamber, ice crystal habit, mean length {L}, mean semiwidth {a}, total number concentration, and mean aspect ratio for the two size distributions. Although the ice crystals at a temperature of -17øC cover a larger size spectrum than those produced at a temperature of -7øC, the computed mean effective size parameters are about the same for the two cases. The ratio of the mean effective size parameters for the two size distributions is 1.148.

The laboratory ice crystals were predominately plates and hollow columns at the lower and higher temperatures, respectively. For the hollow column case, a significant peak of ice particle number concentration is noted at the particle size of -12 •m, as is evident in Figure 1. The spikes in the size

Table 1. Mean Size, Mean Aspect Ratio, and Total Number Concentration for Two Laboratory Ice Crystal Size Distributions

Crystal Habit

Temperature, øC

-7

Hollow Columns

-17

Plates

(a}, gm (L), btm Nt, cm -3 (a/L)

4.9

14.8

1582

0.35 _+ 0.15

11.7

8.2

656

1.4


Model Cirrus Cloud: D -< 100 IJm; 60% • , 40% •

D > 100 IJm; 50%+, 30%•, 20% • 10 5

104'

:=- 03 . • 1

o ,m

• 101

o 10 o

o 04 • 1

10-2.

10 '3

-- Cs

- - Ci Uncinus

........ Warm Ci

- - - Cold Ci

ß ,.% .,. t

x, '"...... x '"...

',, ,, ß

ß

% ß

\ : ß

ß

% ß

........ i ........ i ........ i ........

10 ø 10 • 102 103 104 Maximum Dimension (IJm)

FIRE II IFO, 1991 løø 1 • 10_•

o

ß • 10-2-

o

._•

• 10 '3'

10 .4 0

• :..•., _ Dec. 5 ""'•..-. • Nov. 25

.' ,. ,•"...'., ................ Nov. 26 ,,, '•;..' x ..... Nov. 29 ! x x"..: ,%

x ".: ' %

x x ".. i ß ...... '". '\

¾:'" '!

x x i .... 200 400 600 800 1000

Maximum Dimension (IJm)

Figure 2. Ice crystal size distributions for midlatitude cirrus ice clouds. Also shown are proposed ice crystal shapes for radiation calculations.

distribution are associated with undersampling in counting the particles [Arnott et al., 1995]. To smooth the effect of undersampling in the measurement of these size distributions, ana- lytic distributions are used to fit the observed particle number concentration. For the hollow column case, we use a gamma distribution given by

n(D) = NoD • exp (-AD), (4)

where D is the ice crystal maximum dimension; and No, 3/, and A are parameters determined by fitting the observed size distribution. The size distribution of ice plates displays two peaks at D = 16 /•m and 25.6 /•m. Thus we use a bimodal size distribution obtained by selecting two sets of (No, 7, and A) in (4) for D -< 20 and D > 20 to fit the observed data.

The maximum dimensions of ice crystals in natural ice clouds are usually larger than those in laboratory ice clouds. Figure 2 shows eight observed ice crystal size distributions for midlatitude cirrus clouds. The cirrostratus (Cs) and cirrus uncinus (Ci) were observed by Heymsfield [1975], while warm and cold cirrus clouds are modified size distributions reported by Heymsfield and Platt [1984]. These four size distributions have been used by Takano and Liou [1989] for light-scattering calculations using the geometric ray-tracing technique. The right- hand diagram in Figure 2 shows the four size distributions obtained during the First ISCCP Regional Experiment (FIRE) Cirrus Intensive Field Observations (IFOs) in November- December 1991 (phase II). The FIRE II IFO size distributions displayed in the diagram are obtained by smoothing the raw observed data. The geometric shapes of naturally occurring ice crystals are often very complicated [Arnott et al., 1994], though their basic features have a hexagonal structure. For the size distributions displayed in Figure 2, we estimate from the observations that cirrus clouds are composed of plates, hollow columns, and bullet rosettes. The geometry and percentage for each kind of ice crystals in the model cirrus clouds are denoted in the figure.

To carry out the single-scattering computations, the aspect

ratio must be defined for ice crystals with a given shape. For plates, the following relationship given by Pruppacher and Klett [1980] is used:

L = 2.4883a ø'474 5 /•m < a < 1500 /•m. (5)

Equation (5) is the same as (4) in the work of Arnott et al. [1995], but the units used in the present equation are microns.

The aspect ratio for columns has been defined by Mitchell and Arnott [1994] by fitting the observed data presented by Auer and Veal [1970]. For L -> 100/•m the aspect ratio can be defined by

a = 3.48L ø's L > 100 /•m, (6)

where the units are in microns. Note that Mitchell and Arnott

[1994] used units of centimeters in their equations. For small columns the relation is given by

a = 0.35L, L < 100 /•m. (7)

Although (7) indicates that the aspect ratio for small columns is constant, a random variation of the aspect ratio is assumed for small columns in the scattering calculations. This is because the measured mean aspect ratio associated with the ice crystals generated in cloud chamber at a temperature of -7øC is 0.35 +_ 0.15 [Arnott et al., 1995]. The random variation of the aspect ratio is defined by

a = [0.35 + 0.15(1 - 2•)]L, L < 100 t•m, (8)

where • is a random number which is uniformly distributed in the region of (0, 1). Note that a new value of • is selected for each orientation of ice crystal in the single-scattering calculations. The depth of the hollow structure, d, is also assumed to vary randomly and is given by

d = 2/jd, (9)

where • is the mean depth of the hollow structure. We assume d = L/4 in the present study.

YANG ET AL.' SCATTERING PROPERTIES OF ICE CRYSTALS 21,829

1.5-

1

0.5

4- 2.5-

• 2 •....%' .• 3.5 -• -- Experiment •) ,, ,,,h,• ' /4 .

__- Measured. Ice •1.5. Crystal Size ......... Gamma Di.rib•ion

--- Mean Size • ;• 2.5J

.... [l•/ ,. ¾, ,.i

' I , , , •, , , •, , , • , , , • , , , •, , , •, , , •, , 0.5' 4 6 14 16

2.5-

Experiment .-e o ••.,• $ '..7"; Measured Ice 1 5 '"' \

Crystal S,ze .=o \'• ......... Bimodal .• \

Distribution LU 1 -- - -- Mean Size

0.5 .... 2.6 2.8 3

Wavelength (pm)

'"•..,,,

'1'

, 8 10 12

Wavelength (pm) Wavelength (pm)

Figure 3. Comparison of the theoretical and experimental extinction efficiency for laboratory ice crystals. The solid lines are the extinction efficiency measured by using a laser diode and a Fourier transform infrared spectrometer. The other curves are theoretical results computed by using the size distributions displayed in Figure 1 and the mean sizes defined in Table 1.

According to the data obtained during FIRE II [Arnott et al., 1994; Mitchell and Arnott, 1994], bullet rosettes are the pre- dominate ice crystal shape along with columns and plates. Bullet rosette ice crystals in cirrus clouds may consist of a number of branches. In the scattering calculations, we use four typical branches with the geometry defined in Figure 2. For the bullet rosette shape, the aspect ratio is defined by Mitchell and Amott, 1994

a = 1.1552L ø'63, (10)

where a and L are the width and length of the hexagonal columnar part of individual bullet elements. The units are in microns. The length of individual bullet elements is given by

t + L = D/2, (11)

where t is the length of the pyramidal tip of bullet elements, and D is the maximum dimension determined by the two- dimensional optical probe. The length of the pyramidal tip can be specified by the inclination angle a of the pyramidal faces with respect to the major axis of the bullet element as follows:

t = 2 tan•' (12)

Although reliable experimental data are not available for a, 25 ø and 28 ø appear to be reasonable values because the angular position of the halos associated with the ice crystals with pyramidal tips can be explained by the ray-tracing calculation if these values are employed [Greenlet, 1980, p. 62]. We select a = 28 ø in the present scattering and absorption calculations.

4. Extinction Efficiency and Single-Scattering Albedo

Extinction efficiencies have been measured for the labora-

tory ice crystals associated with the two size distributions displayed in Figure 1 by using a laser diode at the visible wave-

length of 0.685 •m and a Fourier transform infrared spectrometer for the spectral region 2-18 •m [Arnott et al., 1995]. In the measurement of extinction efficiency the measured quantities are the transmittance at the visible and infrared wavelengths. Since the particle size is much larger than the visible wavelength, the extinction efficiency can be expected to be 2. Thus the extinction efficiency at the infrared wavelengths can be obtained by the following expression:

Qe,IR -- 2 In (F IR)/ln (Fvis), (13)

where FiR : exp (--q'IR) and ['VlS : exp (--TViS) are the transmittance at infrared and visible wavelengths, respectively,

1.8

(• 1.6

._> 1.4

'• 1. I 0.8/,,,,..,,..,,,,,,,,,,.,,,,,,,.,.,,,,,.,.

x 10 0 'o 10 -1

(D 10 '2 ._> .• 10 -3 ,(0 10_ 4 rr 10-5-

•', 10-6. '5) 10 -7- (o 10.8_; E .

-- 10.9' ,,,•...•.,,• ..................... 0 2 4 6 8 10 12 14 16 18 20

Wavelength ( IJm )

Figure 4. Real and imaginary parts of the refractive index of bulk ice [after Warren, 1984].


2.5 t 2 ',

1.5

I i

II

.

0.5-

• Experiment

- _ Warren's m r

ß O.9m r

ß 0.95m r

Figure 5.

0 1.1m r

2 o•o o ß 1.8•

o

1.4- •

o

(• 1.2-

LU 1 o

o

w• 0.8- LLI • I:xperiment

0.6- - - Warren's m r

0.4- ß 0'9mr

• ß 0.95m r O.P-

0 1.1rn r

Wavelength (pro) Wavelength (pro)

Comparison of the experimental extinction data with the theoretical results using a number of the real refractive indices.

and ZiR and Zvis are the corresponding optical depths at these wavelengths.

To carry out the computations of extinction efficiency and single-scattering albedo for ice crystals, we use the refractive index of ice compiled by Warren [1984] based on the measurements conducted during 1950s and 1970s [Ockman, 1957; Red- ing, 1951; Schaafand William, 1973]. Recent measurements of the imaginary part of ice refractive index differ significantly from Warren's data [Kou et al., 1993; Gosse et al., 1995]. As pointed out by Arnott et al. [1995], it is a practical necessity to assess the accuracy of the current measured refractive-index values for ice in the infrared region under various tempera-

tures and growth conditions for interpretation of the observed radiation signatures of ice clouds. The ice crystals are assumed to be randomly oriented in space. The extinction and absorption efficiencies with a given size distribution are given by

• Q•Pn (D ) dD 1

(Qe) - , (14a)

r>• Pn (D ) dD 1

1-,

0.9-

0.8-

0.7- 0.6-

0.5-

&o.4-

0.3-

0.2-

0.1

0

Wavelength (pm)

1

:< o.•

$ 0.5

•sn

2 4 6 8 10 12 14 16

Wavelength (pm)

Figure 6. Single-scattering albedo associated with the extinction efficiency displayed in Figure 3.


4

3.51 Experiment

- - - Mean Size

.......... Equivalent-S 1 ' :?t Sphere '1 ! I / i ..... -. - Equivalent-V

Ij '! [ i • ....... Sphere 3]i •.: v', \ ........ I : : -" "

_ tl ..-"- ',,.

• t •! i:. ." . w t•.•-'•;:..(V/,,. • :'- /' ' õ l=:a; ,\

"' ' 'i .'", ''

] I \ ' '"•'/ '

'd I

0.5 , •,, , •,, , • , 2 4 6 8 10 12 14 16

Wavelength (pm)

Experiment

Q - - - Mean Size ......... Equivalent-S Sphere

/ø }2: - . - Equivalent-V . '...'"•'"' /' / • Sphere ,'.."' ß • ,\ //

, ',, / '• //

' / '.! .'.;'"" !.• •;:: ./ •i /.. .... ,.;] • L. r../ •. / li• ,• E' :. • ,:. ' "'

-. ,, ;%/, ','11 ',,' \', '/

2 4 e 8 1• 12 1•4 1•6 Wavelength (pm)

Figure 7. Comparison of the extinction efficiency for nonspherical ice crystals and their equivalent spherical counterparts.

fr> •2 Q•,P, (D ) dD 1

(Qa) = , (14b)

fr> •2 Pn (D ) dD 1

where n(D) is the number concentration of ice crystals expressed with respect to the maximum particle dimension; D • and D 2 are the lower and upper cutoff for the size distribution, respectively; P is the geometric projected area of the ice crystal on a screen perpendicular to the incident plane; and Q• and Q, are the extinction and absorption efficiencies, receptively, for individual particles. Note that the effect of the random

orientations of ice crystals has been accounted for in the computation of Q•, Q,, and P. The single-scattering albedo for ice crystals with a given size distribution is defined as follows:

(tOo) = ((Qe)- (Q•,))/(Qe). (15)

Figure 3 shows comparison of the experimental extinction efficiencies and the theoretical results computed from the measured and analytical size distributions displayed in Figure 1. The results corresponding to the mean size distribution are computed by using the mean semiwidths and lengths defined in Table 1. The two inserted subdiagrams show the enlargement of the extinction minimum around 2.85 /•m. In the spectral regions located at 3.5, 4.25, 6.75, and 14/•m, absorption due to gaseous and the cold-box window of cloud chamber is too

0.9-

0.8-

o 0.7-

<r 0.6-

.c_

(• 0.5-

(• 0.4-

03 0.3

0.2

0.1

Figure 8.

1

, [•.. 0.8 ': ...Z ,•

......t

.. :•., 0.7

•? • '""'-'"' 0.6 /'}' ;• 0.5 ...•

0.4

0.3

• • Mean Size 0 2 t Q - - Equivalent-S Sphere Equ ale t V Sphe e

0 ''' i',' I'' ' i ..... , ......

Wavelength (pm) Wavelength (pm)

Comparison of s•n•]c-sca•cdn• albedo fo• nonspherical •cc cwsm]s and


OES• L/a = 5, ill = 0.25

3.5

2.5

Extinction Efficiency

ß

0.4

5 15 25 35 45 55 65

Maximum Dimension (pm)

Single-Scattering Albedo 4

3.5

5 15 25 35 45 55 65


• L/a =0.5

Extinction Efficiency Single-Scattering Albedo

3.5

5 15 25 35 45 55 65


Figure 9. Contours of the extinction efficiency and single-scattering albedo as functions of particle size and wavelength for hollow columns and plates in the spectral region 2-4/•m.

strong to produce sufficient signal to noise. For this reason, the experimental extinction data are subject to substantial errors in these spectral regions [Arnott et al., 1995].

From Figure 3 it is evident that the theoretical results in the plate case are less sensitive to size distribution than in the column case. Both experimental and theoretical results show two pronounced extinction minima at 2.85 and 10.5/•m, which are produced by the strong absorption of ice coupled with the real part of the refractive index close to unity in these wavelengths (Figure 4). Thus the extinction is predominately associated with absorption rather than scattering, known as the Christiansen effect, which has been explained by Arnott et al. [1995] by using an informative model based on ADT. The ratio of the scattered energy to the absorbed energy, that is the Christiansen effect, decreases with the decrease of the size parameter for sufficiently small particles. Because of this mechanism the extinction window at 10.5 /•m is more trans- parent than that at 2.85 /•m, as evident from Figure 3, even though the real part of the ice refractive index is smaller for the latter wavelength. The window at 10.5/•m is also much wider than that at 2.85/•m, corresponding to the variation pattern of

the real part of the ice refractive index versus wavelength. Although overall agreement is illustrated in the comparison of the experimental and theoretical extinction efficiencies, dis- crepancies are noted in the spectral regions near 2.25 and 4/•m.

We have carried out a number of sensitivity studies concern- ing the extinction minima in theoretical results around 2.25 and 4/•m. It is noted that these minima are not sensitive to the variation of particle size. However, the theoretical results can match the experimental data in these wavelength spectral regions by adjusting the refractive index. As noted by Warren [1984], in the near-infrared region (1.4-2.08/•m), uncertainties of a factor of 2 exist for the imaginary refractive index. In the middle infrared (3.5-4.3 /•m), uncertainties can reach 50%. The real refractive index is obtained on the basis of the

imaginary refractive index based on the Kramers-Kroning relationship [Warren, 1984]. Thus the former is also subject to uncertainties. We find that the effect of the imaginary refractive index on the extinction efficiency for the wavelengths round 2.25 and 4/•m is insignificant in comparison with that of the real refractive index if the same percentages of uncertainties are assumed. Figure 5 shows comparison of the experi-

YANG ET AL.: SCATTERING PROPERTIES OF ICE CRYSTALS 21 833

• L/a=5, [/L=0.25

----'- 13

--•'• 12

o)11

(3) 1o

•9


2.8

2.6

::• 1.8

': 1.4 .2

0.8

0.6

•'• 0.4

, ,

s 15 25 3s 45 ss 6s

Maximum Dimension (pro)

•-t3

>e •0

•6 Sing le:s...c..•ttering.: Albedø 15 i I 1 14 ! i 1•i i

{i •' 0.6

ß !i!ti: o.5 ;.i• ßß 0.4

:i O.3 !• 0.2

7 ..•....:::•:•::,•:•,::•:•:.,•:..:.:..::..:..::?::: •::...::....::::..:...... ::.:........ ' 6

5 15 25 35 45 55 65


• L/a = 0.5


---, 13

ß '-•'• 12

0)11

5 •5 •5 35 •5 55 65 Maximum Dimension (pm)

:• aJ 3

':!'?•! 24

"•'•11 2 !

'-.; 1.6

1.4

1.2

ß 0.8

•:: 0.6 i 0.4 • 0.2

0 ,

16

15

14

7

Single-Scattering Albedo

-•5 2s 35 4• 5s 6s


Figure 10. Same as in Figure 9 except for the spectral region 6-16/•m.

mental extinction data and the theoretical results by varying the real refractive index by -10, -5, and 10%. For efficient computations we have used mean sizes associated with the measured size distributions presented in Figure 1, although similar results can be obtained if the detailed size distributions

are accounted for. Agreement between the experimental data and the theoretical results can be achieved by adjusting the refractive index, as demonstrated in this figure. The real refractive indices for 2-2.25 /•m that can be used to fit the experimental data are 10% and -10% of the original Warren's data for hollow columns and plates, respectively. Ice crystals often show some inhomegeous inner skeletons [Nakaya, 19954] and are not completely bubble free associated with the growth condition. In the cloud chamber, plates and columns are generated under different growth conditions. For this reason, the bulk value of refractive index may be different for the two cases.

Figure 6 shows the single-scattering albedo associated with the extinction efficienc7 displayed in Figure 3. Note that the experimental data for the single-scattering albedo are not available at this point. Similar to the extinction efficiency, the single-scattering albedo is less sensitive to size distributions in

the plate case than the column case. Five scattering minima are shown for both the hollow column and the plate cases located at 2.85, 3.26, 4.56, 6.31, and 10.64/•m. The minimum at 2.85/•m corresponds to the minimum in the real part of the ice refractive index at this wavelength, where ice is still strongly absorptive. At 3.26 /•m the imaginary refractive index reaches its maximum, although the real refractive index deviates significantly from unity. Thus extinction is primarily due to absorption leading to a minimum in the single-scattering albedo. The scattering minima at 4.56 and 6.31 /•m correspond to the peaks of the imaginary refractive index at these two wavelengths. The minimum of the single-scattering albedo at the wavelength 10.64/•m corresponds to the minimum of the real ice refractive index in this wavelength region where significant absorption takes place. From Figure 6 it is interesting to note that the single-scattering albedo in the Chris- tiansen bands are much less than 0.5; that is, more than half of incident energy is absorbed by ice crystals. Such a phenomenon cannot be produced by GOM1, because the lowest asymptotic values of the single-scattering albedo computed by this method is 0.5.

Using the equivalent-surface or equivalent-volume ice spheres to approximate nonspherical ice crystals is physically


2.5

:1• 1.5- LM

2.5

FIRE II tFO, 1991

Dec.5

.... Nov.25

............. Nov.26

.... Nov.29

Wavelength (pm)

.... Ci Uncinus

.............. Warm Ci

•, ,^, - '.-- Cold Ci C: 2..•.• ........ '..' .-..• ._O , • •w '

,; ,; ,;

.... i .... i, ! .... i ....

0 4 8 12 16 20

Wavelength (pm)

O 0.8-

030.6. .c_

• 0.4.

._c O3 0.2-

FIRE II IFO, 1991

Dec.5

--- Noy.25

............ N0v.26

.... Nov.29

O 0.8

030.6. .m

•.• 0.4.

O3 0.2.

0

Wavelength (p•m)

Cs

- - - Ci Uncinus

............ Warm Ci

.... Cold Ci

0 0 .... ,• .... • .... 1'2 .... 1'6 .... 20

Wavelength (pm)

Figure 11. Extinction efficiencies and sin. gle-scattering albedos for eight observed ice crystal size distributions displayed in Figure 2.

Figure 11 shows the efficiency and single-scattering albedo for the eight ice crystal size distributions displayed in Figure 2. The Christiansen effect is not significant for the four size distributions observed during FIRE II IFO, because of the lack of the presence of sufficient small ice crystals. For large ice crystals, the extinction efficiency converges to two regardless of the refractive index. For an ice crystal size distribution involving complex shapes, such as bullet rosettes and hollow columns, we define a general mean effective size D e as follows:

2 Vn(D) dD • IWC D = = (16)

e i; 2 pA c , Pn (D ) dD 1

where V is the volume of an ice crystal, P is the projected area defined in (14), p is the mass density of bulk ice, IWC is the ice water content, and A c is the total cross-sectional area of ice crystals per unit volume, which can be measured by the 2-D optical probe [McFarquhar and Heymsfield, 1996]. The mean effective size so defined is now directly related to IWC, a prognostic variable in GCMs and A c, a measured quantity. Table 2 lists the generalized mean effective sizes for the ice crystal size distributions presented in Figures 1 and 2. The generalized mean effective size for laboratory ice crystals is quite small, as are the cases of cirrostratus and cold cirrus. Thus we expect that the Christiansen effect would be pronounced for these clouds. From Table 2 and Figure 11 we note that the extinction window is not pronounced if the generalized mean effective size is larger than 50 approximately.

inadequate for the computations of their single-scattering properties and can lead to significant errors in the applications to remote sensing and climate modeling [Liou and Takano, 1994; Kinne and Liou, 1989]. Figures 7 and 8 show the extinction efficiency and single-scattering albedo for nonspherical ice crystals, using mean sizes and their equivalent-surface and volume spherical counterparts. Significant overestimation of the extinction efficiency is produced by using the equivalent ice spheres in agreement with the results presented by Yang and Liou [1996a]. Variation of the single-scattering albedo with respect to wavelength is similar for spherical ice and nonspherical ice crystals. However, significant differences are produced for X > 11/am, particularly for the plate case. Although we use the mean sizes, a similar conclusion can be made if the detailed ice crystal size distribution is accounted for.

Figures 9 and 10 show the contours of the extinction efficiency and single-scattering albedo as a function of the wavelength and ice crystal size. The oscillation of the extinction efficiency is evident by looking at a fixed wavelength and various ice crystal sizes. The Christiansen effect can be noted from the pronounced change of the extinction character at 2.8 and 10.5 /am, where the real refractive index for bulk ice approaches unity. Comparing the extinction efficiencies shown in Figures 9 and 10, we note that the extinction window is much wider for the spectral region around 10.5/am than 2.8/am; an effect that is also evident from Figures 3 and 6. The dominance of absorption over scattering is clearly evident from the single- scattering albedo diagrams. From Figure 10 we also note that the Christiansen effect is much stronger for smaller ice crystals than for larger ice crystals.

5. Conclusions

The extinction efficiency computed by a combination of FDTD and RBRI methods is compared with experimental data. Overall agreement has been found for the two results, in particular, both show two strong extinction windows at the Christiansen bands. It is also shown that the single-scattering albedos for the laboratory ice crystals are smaller than 0.5 at the Christiansen bands, which cannot be explained by the con- ventional geometric optics method in terms of the ray-tracing technique. Furthermore, we have demonstrated that the Chris- tiansen effect is not critical for most of the ice crystal size distributions observed during FIRE II IFO, because the num-

Table 2. Generalized Mean Effective Size for Laboratory and Observed Ice Crystal Size Distribution

Cloud Type De, /am

Laboratory ice clouds T = -7øC 12.47

T = - 17øC 13.51

Typical cirrus clouds Cs 33.72

Ci uncinus 104.96 warm Ci 48.61

cold Ci 16.63

FIRE II IFO cirrus clouds December 5 57.62 November 25 68.24 November 26 63.62 November 29 75.28


ber of small ice crystals is not substantial. However, for cold and warm cirrus as well as cirrostratus the extinction minimum

at 10.5 btm is pronounced. The Christiansen effect therefore must be accounted for in the determination of optical depth and emissivity for these types of ice clouds. This effect can also be important in the paramerization of the radiative properties of cirrus clouds for applications to remote sensing and climate modeling in view of the fact that a large number of small ice crystals has been observed in tropical cirrus and contrails. We have also shown that using equivalent spheres to approximate nonspherical ice crystals can lead to a significant overestimation of extinction efficiency. Finally, we have presented the extinction coefficients and single-scattering albedos for a number of ice crystal size distributions representative of the midlatitude cirrus cloud systems covering the wavelengths from 0.2 to 20 btm.

Acknowledgments. P. Yang and K. N. Liou have been supported by NSF grant ATM-93-1521, NASA grants NAGl-1719 and NAG5- 2678, and DOE grant DE-FG03-95ER61991. W. P. Arnott has been supported by NSF grant ATM-9413437 and NASA grant NAG 1-1707. Some of the computational results presented in this paper were obtained using CRAY Y-MP 8/864 at the National Center for Atmo- spheric Research, sponsored by NSF. We thank the two anonymous reviewers for constructive comments.

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(Received December 18, 1996; revised June 6, 1997; accepted June 16, 1997.)

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