15 AUGUST 2001

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The Effective Cloud Fraction of Broken Clouds Obtained by Multistream Radiative
Transfer. Part I: Longwave Radiation
HIROHIKO MASUNAGA
Earth Observation Research Center, Roppongi, Minato-ku, Tokyo, JapanTERUYUKI NAKAJIMA
Center for Climate System Research, University of Tokyo, Komaba, Meguro-ku, Tokyo, Japan(Manuscript received 3 March 2000, in final form 12 January 2001)
ABSTRACTThe influence of broken clouds on radiative flux has provided a major source of uncertainty in radiativetransfer models of the atmosphere because plane-parallel approximations are assumed in most of the current
atmospheric models, where horizontal inhomogeneity cannot be adequately taken into account. In this paper,
effects by cloud inhomogeneity on longwave radiation fields are investigated, using a simple model of a cloudarray that consists of identical cuboids following some past studies. In contrast to past work that adopted simplified
formulations of radiative transfer, multistream radiative transfer is considered to obtain the exact solutions ofradiative flux, which enable us to consider semitransparent clouds as well as optically thick clouds in desirable
accuracy. Applicability to semitransparent clouds is important because cirrus clouds, which are considered toplay significant roles for longwave radiation, are often semitransparent to infrared radiation.
The computational results show that the empirical formula previously derived by Harshvardhan and Weinman
systematically underestimates the effective cloud fraction. An alternative formula is proposed for the effective
cloud fraction to supply a better fit to the exact solution of radiative flux. Furthermore, new formulas are derived
to approximate the exact solutions including the dependence on the optical thickness of clouds. They are useful
to convert plane-parallel flux to 3D flux passing through broken clouds, either for optically thick or thin clouds.

1. Introduction
Although clouds are considered to play significant
roles in the earth’s radiation budget, the impacts ofclouds to atmospheric radiative transfer are difficult toevaluate precisely. A main uncertainty lies in the planeparallel approximations, usually assumed in calculationsof atmospheric radiative transfer, because it cannot appropriately take into account the horizontal inhomogeneity (or brokenness) of clouds. The effects on theradiation fields by brokenness of clouds have been studied for more than 20 years ago by authors such as Aida
(1977) for solar radiation and Harshvardhan and Weinman (1982, hereafter HW82) and Ellingson (1982) for
longwave radiation. More recently, Killen and Ellingson(1994), Takara and Ellingson (1996), and Han and Ellingson (1999) have made contributions to this field.
Since the exact treatment of three-dimensional radiativetransfer (3D RT) requires significant computational effort, their estimations of the brokenness effects were
Corresponding author address: Dr. Hirohiko Masunaga, Earth Observation Research Center, 1-9-9 Roppongi, Minato-ku, Tokyo 1060032, Japan.
E-mail: masunaga@eorc.nasda.go.jp

q 2001 American Meteorological Society

based on approximate methods. Aida (1977) considered
the transmittance and reflectance of a cloud array as
statistical average of Monte Carlo results using onecloud and nine-cloud models. HW82 proposed an empirical formula for the effective cloud fraction based on
experimental data and compared it to computational results assuming the variable-azimuth two-stream (VATS)
approximation. The effective cloud fraction for longwave radiation takes account of the emissions and shading by the side faces of broken cloud blocks, which
enhance the emergent flux in comparison with the planeparallel flux (we hereafter call this effect the ‘‘3D effect’’ of clouds.) Validity of the empirical formula of
HW82, however, remains to be examined by more precise calculations since it was derived to fit only six
experimental data. The results of HW82, furthermore,are applicable only to optically thick clouds because
both their experiments using black blocks and the VATS
approximation do not yield correct solutions for optically thin clouds. The effective cloud fraction for longwave radiation was investigated on the basis of detailed
studies on effect of shape and spatial distribution of
black clouds (Ellingson 1982; Killen and Ellingson
1994; Han and Ellingson 1999) or of optically thick2456

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clouds with scattering (Takara and Ellingson 1996),
while dependence on optical thickness, which possibly
attenuates the geometrical effects, remained unexamined in their work. On the other hand, cirrus clouds,
which are considered to play important roles for longwave radiation (Roewe and Liou 1978), are often semitransparent to infrared radiation. Therefore it is important to investigate the 3D effect of clouds with arbitrary
optical thickness.
A number of sophisticated numerical schemes havebeen proposed to date for solving 3D RT problems in
the cloudy atmosphere (e.g., Gabriel et al. 1993 andreferences therein). Although elaborate 3D RT schemes
are powerful tools to solve a certain specific problem,
they are not suitable for general circulation models(GCMs) and algorithms for satellite data analysis because of huge computational cost. In contrast, the strategy of HW82 and of this study is to supply simple
formulas that are available for converting plane-parallel flux to 3D flux depending upon some parametersthat characterize the cloud fields. We follow HW82 forthe model of broken clouds, where an infinitely extended cloud array consists of identical cuboids. This
cloud model is quite simplified compared to complicated geometry of actual clouds, but it has an advantage that the properties of cloud field can be easily
characterized by well-defined parameters such as thecloud fraction and the aspect ratio.
Applicability of the effective cloud fraction may belimited because it does not adequately represent brokenness of clouds if clouds have layer-dependent cloud
fraction, finite horizontal extent, and extreme internal
inhomogeneity as is often the case in actual clouds.
Moreover, vertically asymmetric shape of clouds would
require separate effective cloud fraction for each of upward and downward fluxes. However, the effective
cloud fraction is still a useful tool to approximate radiation fields in broken clouds in GCMs since elaborate
treatment for 3D RT in complicated cloud structure is
too time-consuming.
In contrast to HW82, we solve multistream radiativetransfer to obtain the exact solutions of 3D RT, whichassures reliable estimation of radiative flux regardlesswith optical thickness of clouds. Hence this study does
not only reexamine the empirical formula of HW82 but
also extends it to be applicable to clouds with any
optical thickness.
In section 2 we describe the model characteristics
and the equations to be solved. The computational results are presented in section 3 and discussions are
made in section 4. Finally we summarize our findings
in section 5.
2. Model descriptions
In this section we describe our model of broken
cloud fields and the formulation for calculating theradiative properties. The main target in application of

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our model is cirrus in the longwave window regions
because cloud scattering and gas absorption are notincluded in the present study.
Following HW82, we adopt a regular array of cuboidal clouds for the cloud model. The cloud array ischaracterized by the cloud fraction, f , and the aspectratio, a, defined by
f 5

W2(I 1 W ) 2

a5

H
,
W

and

(1)
(2)

where W, I, and H are the width of a cloud cuboid, theinterval between adjacent cloud blocks, and the cloudheight, respectively. The array that consists of identical
cloud blocks of W 3 W 3 H is assumed to be isothermal
and to be extended infinitely in a horizontal plane. In
contrast to HW82, who assumed each of the cloud
blocks is a black body, we specify arbitrary optical
thickness as a free parameter in the cloud model. The
vertical optical thickness inside a cloud block is denotedby t H .
In the present paper, we concentrate our study on
longwave radiative transfer and consider only the absorption and thermal emission without scattering. The
cloud array with the source function of S c is irradiatedby the upward flux, pIinc , incident through the cloudbase. In terms of the cloud temperature, T c ,S c 5 B n (T c )for monochromatic radiation and
Sc 5

E

n2

Bn (Tc )n1

for integrated radiation over the frequency range between n1 and n 2 , where B n (T) is the Planck function fortemperature T. For the gray radiation, we have Eq. (3)setting n1 5 0 and n 2 5 `,S c 5 sT 4c .

(3)

The incident radiation Iinc is similarly related with theground temperature if the atmospheric absorption between the ground and the cloud base is negligible. However, since the atmospheric absorption below the cloud
can be significant for longwave radiation, the interpretation of Iinc in terms of temperature is complicated ingeneral. The gas absorption inside the cloud is not considered, and thus the resultant fitting formulas presented
in section 3 below are applicable to the infrared window
regions. Inclusion of the gas absorption inside clouds
would attenuate the 3D effects of clouds since horizontal
inhomogeneity is much less significant for gases thanfor clouds.
For an isothermal cloud, the upward radiance emergent through the cloud top is expressed as

15 AUGUST 2001

I ↑ (x t , V) 5 I inc exp[2t (x t , V)]
1 Sc{1 2 exp[2t (x t , V)]},(4)

where x t 5 (x t , y t ) represents the position in the top
plane and V 5 (u, f) is the unit direction vector. Cloud
optical thickness along a given path parallel to V measured from the point x t is represented by t (x t , V). The
downward radiance emergent through the cloud base isI ↓ (x b , V) 5 S c [1 2 exp(2t (x b , V))],(5)

where x b 5 (x b , y b ) is the position in the base plane.
The upward (through the cloud top) and downward(through the cloud base) fluxes are calculated from Eqs.
(4) and (5) as
F↑ 5
F↓ 5

E
E

dx t dytDADA
dx b dyb
DA
DA

E
E

V dVI ↑ (x t , V)

and

12p

12p

V dVI ↓ (x b , V),

E
E

22p

(7)

E
E

2p

1

m dm

df,

0

0

m dm

21

2p

df,

0

where m is cosine of the zenith angle.
In numerical calculations, Eqs. (6) and (7) are discretized as

1O dA 3 O w cosu I 2@DA and (8)F 5 1O dA 3 O w cosu I 2@DA,
(9)
N

F↑ 5

J

n

n51

↑
j nj

j

↓
j nj

J

n

n51

j

j51

N

↓

(10)

↓
FPP
5 pSc [1 2 2E 3 (t PP )],(11)

E n (y) 5E

10

V dV 5↑FPP
5 pSc 1 2p (I inc 2 Sc )E 3 (t PP ),

where

22p

V dV 5symmetric S 8 quadrature formula, that is, J 5 40 withina hemisphere (Lewis and Miller 1984).
Since our interest in this paper is not to obtain theabsolute values of radiative flux but is limited on thegeometrical effects caused by broken clouds, the computational flux presented in this paper is normalized by
the plane-parallel flux, F ↑PP or F ↓PP. This strategy has theadvantage that the results do not suffer from the uncertainties in our knowledge on the optical propertiesof clouds because information of the extinction coefficients of water droplets and ice crystals tends to becanceled out in the ratios of F ↑ /F ↑PP and F ↓ /F ↓PP. The
plane-parallel flux for an isothermal cloud sheet is(6)

respectively. The spatial average of the fluxes is takenover the area DA, which is taken as the (W 1 I) 3 (W1 I) area at the cloud top or base, that is, a unit of thecloud array including both a cloud block and intercloudsky. The integrals over solid angle range over the upper
hemisphere in Eq. (6) and the lower hemisphere in Eq.
(7), that is,

E
E

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MASUNAGA AND NAKAJIMA

j51

where N is the total number of discretized cells insideDA, dA n the area of the n-th cell, J the total number ofthe discretized direction angle, w j the weight for thenumerical integration over solid angle, and u j the zenith
angle. For numerical integration of I ↑ni and I ↓nj according
to Eqs. (4) and (5), the integral path is traced in the ith
direction from the cloud top to the base (or from baseto top) under the cyclic boundary condition on the horizontal boundaries. Note that upward radiance incident
through the side faces and the photon exchange between
clouds are correctly taken into account in computations
due to the cyclic boundary condition. In our calculations,
N is 50 3 50 and wj is determined according to a level`dx
exp(2xy)
xn

is the exponential integral and tPP is the vertical optical
thickness inside the plane-parallel cloud sheet. The
height of the plane-parallel cloud is equal to that of a
block of the broken cloud array, H.
We consider two different cases for determining tPP ,that is, tPP 5 t H and tPP 5 ft H . The first case (tPP 5
t H ) provides a plane-parallel cloud sheet that is filledwith cloud material of the same vertical optical thickness
as t H , while another plane-parallel cloud sheet consistsof spatially averaged cloud material conserving its total
amount. For the sake of convenience, we call the formercase as the maximum plane-parallel case and the latterthe average plane-parallel case. These cases represent
two different ways in which the broken cloud fields are
related to the plane-parallel approximation. The maximum plane-parallel model has the same liquid water
path (LWP) as a block of the broken cloud system buta larger mass of water/ice in total, whereas the averageplane-parallel model contains the equivalent total massof water/ice but has a lower LWP. Actual plane-parallelmodel would lie between these two limiting cases when
one applies the plane-parallel approximation to radiativetransfer models used in GCMs or satellite data analysis.
We will comment on the applicability of these twoplane-parallel models in section 3e.
Although we use only the spatially averaged flux
[Eqs. (6) and (7)] in this work, it may be helpful toshow the spatial distribution of the flux before averaging
to demonstrate how the broken cloud field influences
radiative flux. The spatial distribution of the upward anddownward fluxes normalized by the plane-parallel value(in the maximum plane-parallel model) is drawn in Fig.
1. A unit of the cloud array is presented that contains
a cloud block (0 # x, y # W) and clear-sky region(elsewhere). In this case, we assume f 5 0.25 (andtherefore W 5 I), a 5 1, t H 5 10, and S c /Iinc 5 0.5.
The upward flux shows a complicated spatial structure

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FIG. 1. Spatial distribution of radiative flux for t H 5 10. Upward flux emergent through the cloud top (top) anddownward flux through the cloud base (bottom).

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FIG. 2. Spatial distribution of radiative flux for t H 5 1. Upward flux emergent through the cloud top (top) anddownward flux through the cloud base (bottom).

as a result of combined effects of the illumination by
the ground radiation and the emission from the cloud
surfaces, partially shaded by the cloud itself. While theupward flux is substantially enhanced beyond the plane-parallel value (ø1) in the clear-sky region since Iinc .
S c , the flux just above a cloud block is very close to
the plane-parallel value because of shading by theclouds. In the downward flux, an external component

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L5

1
2p

E E
dA
DA
DA

VOLUME 58

V dV exp(2t (x b , V)).

(14)

22p

Equation (13) shows that f e depends only on geometryof the cloud field and on optical thickness, independently of S c . Under given f e , the downward flux isrelated with plane-parallel flux through a rearranged
form of Eq. (12), that is,F ↓ 5 f e F ↓PP ,(15)

and the upward flux is throughF ↑ 5 f e F ↑PP 1 (1 2 f e )pIinc ,(16)

derived from Eqs. (4)–(7) and (10)–(13). Equation (16)reduces to Eq. (10) of HW82 when the cloud is opticallythick. Equations (15) and (16) are equivalent to Eq. (1)in Han and Ellingson (1999) if downward clear-sky flux
multiplied by (1 2 f e ) is added to the rhs of Eq. (15).
Downward clear-sky flux is necessary at the surface
where water vapor emission is not negligible in clearsky flux, but is omitted here for brevity. However, theeffective cloud fraction defined by Eq. (12) is readily
applied to estimate downward flux in the presence ofthe clear-sky flux using a generalized formulation suchas Eq. (1) in Han and Ellingson (1999).
Partially covered cloud field in GCMs or in algorithms for observational data analysis may be taken into
account according to

FIG. 3. Effective cloud fraction for t H 5 50 with the empiricalformula by HW82 [Eq. (19)] is shown (line). Symbols without error
bars denote the exact solution given by numerical calculations in thepresent study, while the other symbols with error bars show the experimental data obtained by HW82.

such as the ground radiation is not included and therefore the cloud acts only as an emitter, not as an absorber,
which produces an upside-down structure of the upward
flux. Figure 2 is the same as Fig. 1 except that t H equals
1 instead of 10. The spatial structure is much moresmoothed here since an optically thin cloud no longer
absorbs nor emits effectively.
We define the effective cloud fraction as follows:
fe 5

F↓
.
↓
FPP

(12)

Equation (12) is identical to the definition by HW82
when the cloud is optically thick enough to be a blackbody.
Combining Eqs. (5), (7), (11) and (12), one finds
fe 5
where

1 2 2L
,1 2 2E 3 (t PP )(13)↓
F ↓ 5 f FPP
and

(17)

↑
F ↑ 5 f FPP
1 (1 2 f )pI inc ,(18)

where the 3D effect of clouds is not considered. Onceobtaining f e as a function of parameters characterizing
the brokenness of clouds, one can incorporate the 3D
effect of clouds into plane-parallel models through Eqs.
(15) and (16) instead of Eqs. (17) and (18) with noadditional effort for computation.
In the subsequent section, we present the numericalsolutions for f e and provide some formulas approximating the solutions. These formulas, combined with
Eqs. (15) and (16), are intended to be used as correction
factors, with which the 3D effect of clouds is approximately taken into account in plane-parallel calculations
performed in GCMs and remote sensing data analysis.
3. Computational results
In this section, we show the computed radiative flux
for a broken cloud array based on the model describedin section 2. According to HW82, the effective cloud
fraction f e defined by Eq. (12) is presented as a functionof the (true) cloud fraction f given by Eq. (1). In addition to the exact solutions for f e obtained by multistream calculations for radiative transfer, we present
some approximate formulas for more useful applicationof f e .
We have two different forms of f e in Eq. (12) de-15 AUGUST 2001

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MASUNAGA AND NAKAJIMA

One can see that f e as a function of f traces convex
curves due to the 3D effect of clouds. Although theempirical formula of Eq. (19) and the exact solutions
by our calculations exhibit a qualitatively similar trend
in Fig. 3, the empirical formula of Eq. (19) provides
systematically smaller values of f e than the exact solutions by our calculations. The discrepancy is as much
as ;10% in f e for large a.
As a more precise function to fit the exact solution,
we propose another approximate formula on f e forblackbody clouds as follows:[f eBB 5 1 2 exp 2

FIG. 4. Effective cloud fraction for t H 5 50 with the approximate
formula newly proposed by Eq. (20) is shown (line). Symbols withand without error bars are the same as in Fig. 3.

pending upon the definition of F ↑PP, that is, the maximum
plane-parallel or the average plane-parallel case. In thefollowing sections, we first present the optically thickcase on the basis of the maximum plane-parallel model
to compare our results with HW82. Second, the dependence on optical thickness is examined for the maximum
plane-parallel case, and finally f e for the average planeparallel case is shown.
a. Optically thick case in comparison with HW82
First, we consider the cloud array of t H 5 50, which
is so optically thick that the cloud system can be considered as blackbody as was the case in HW82. Figure
3 depicts f e as a function of f for a 5 0.1, 0.5, 1, 2,
and 5. Besides our computational results (plots without
error bars), Fig. 3 shows the experimental results by
HW82 (plots with error bars) and HW82’s formula forblackbody clouds (lines) defined as
f eBB 5

[1 1 2a(1 1 0.15 f )] f.
1 1 2a f (1 1 0.15 f )

(19)

]

(1 1 2a) f,
1 2 f2(20)

which is delineated in Fig. 4 with the same plots as inFig. 3. Equation (20) provides a better fit to the exact
solution than Eq. (19) shown in Fig. 3. In comparison
with the experimental data by HW82, Eq. (20) successfully reproduces two of the six data points but fails
in the other four, all of which are considerably smallerin f e than predicted by Eq. (20). The reason is unclear
why the experimental data do not agree with our exact
solution. Nevertheless, it seems reasonable that the two
experimental data in excellent agreement with the exact
solutions have shorter error bars, thus higher accuracy,
than the other four.
In the limit of low cloud fraction ( f → 0), f BBe shouldbe (1 1 2a) f because the radiation emitted from cloud
surfaces is escaped away without disturbed by the otherclouds, where f BB
e / f equals the ratio of the total emittingarea of a cloud block to that of a planiform cloud piece,
that is,f eBB
W 2 1 2WH
5
5 1 1 2a,
f
W2

(21)

as also noted by HW82. This condition is satisfied in
Eq. (21) for f K 1 as well as in Eq. (19). Anothercondition is required that f BB
should accord with f fore
a 5 0, which is not satisfied in Eq. (20) in the exact
sense. This, however, causes no practical inconveniencesince the error is small.
b. Effective cloud fraction for general optical
thickness based on the maximum plane-parallel
model
HW82 derived Eq. (19) assuming the cloud array isblack, and then applied it to a more realistic cloud modelthat is partially transparent using the VATS approximation (Harshvardhan et al. 1981). The VATS approximation is, however, not exact for optically thin clouds
as also implied by HW82. In contrast, the multistream
treatment of radiative transfer in this paper yields accurate solutions for any optical thickness. We extend
the black (t H 5 50) cloud model considered above tomore general models with variable optical thickness. We
employ the maximum plane-parallel model for calcu-

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FIG. 5. Effective cloud fraction for t H 5 10, 3, 1 as well as t H 5 50 with the approximate formula given by Eq. (22) is shown (line).
Symbols denote the exact solutions as in Fig. 3. The maximum plane-parallel model is used for the normalizing flux.

lating the normalizing flux, that is, tPP 5 t H in Eq. (11),throughout the present section.
Figure 5 presents the f e versus f diagrams for t H 5
10, 3, and 1 as well as t H 5 50. The figure shows thata smaller optical thickness results in smaller excess off BBe over f . In other words, the 3D effect of semitransparent clouds causes less significant influences on f ecompared to opaque clouds. This tendency is easily understood because semitransparent clouds do not effectively shade the surrounding radiation, which attenuates
the 3D effect on the radiation fields. We will, however,show in section 3d that the result is totally different
when the average plane-parallel model is adopted instead of the maximum plane-parallel model for calculating the normalizing flux.
The formula of f e for general optical thickness on thebasis of the maximum plane-parallel case is proposed as[

1

f emx 5 c1 1 2 exp 2(1 1 2c2 a) f
12 f 2]2 1 (1 2 c ) f , (22)1wherec1 5 1 2 exp(2t H ),1 1 1 a2.

c2 5 1 2 exp 2tH

In the optically thick limit, c1 and c 2 become unity andBBthen f mx
e is identical to f e . Equation (22) is delineatedin Fig. 5 for each t H and each a in addition to the exact
solutions indicated by plots. Accordance of the approximate formula with the exact solutions is not always
excellent in particular for small t H , but the error would
not be serious for practical use.
c. Dependence on the normalizing plane-parallelmodelsWe discuss here the upward and downward fluxes
emergent from the broken cloud fields for each of the
maximum plane-parallel and average plane-parallel cases. Figure 6 shows F ↑ /F ↑PP and F ↓ /F ↓PP calculated by Eq.
(8) with Eq. (10), and by Eq. (9) with Eq. (11), respectively, as a function of t H using the maximum planeparallel model. Two cases for the cloud fraction, f 5
0.25 and 0.75, are considered. We assume S c /Iinc 5 0.5
for computing the upward flux. (The downward flux is
independent of this value.) The normalized downward
flux, F ↓ /F ↓PP, is equivalent to f e by its definition.
The downward flux increases with t H but saturates in
the optically thick limit, where the saturated value cor-

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2463

FIG. 6. The upward flux emergent through the cloud top and the downward flux through thebase for the maximum plane-parallel model, i.e., tPP 5 t H .

↓responds to f BB
with t H for mode . The increase of F
erately transparent clouds is more rapid for larger a,where the the side faces of clouds enhance their relative
importance. The optically thin limit of F ↓ /F ↓PP converges
to f for all values of a. The upward and downward
fluxes satisfy the condition of (F ↑ /F ↑PP 1 F ↓ /F ↓PP)/2 5
1 for opaque clouds, because the side faces behave as
emitters for the downward flux while as absorbers forthe upward flux, and these effects compensate for eachother. However, this condition breaks down for semitransparent clouds, where the upward flux tends to ap-

proach the plane-parallel value in the optically thin limit
independently of a. Semitransparent clouds are not effective absorbers and hence the brokenness of clouds
no longer has significant effect on the radiation fields.
Figure 7 illustrates the emergent flux similarly to Fig.
6 but using the average plane-parallel model. A moststriking difference from the maximum plane-parallelcase is that F ↓ /F ↓PP, as well as F ↑ /F ↑PP, approaches tounity in the optically thin limit. This trend results fromthe fact that the emission from semitransparent clouds
directly reflects the total amount of cloud material, and2464

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therefore F ↓ is close to F ↓PP for t H K 1 if the total massof water/ice is common as is the case in the averageplane-parallel case. On the other hand, the behavior in
the optically thick limit shows no difference for the bothcases because the both plane-parallel clouds fall to an
equivalent cloud sheet of blackbody when t H is sufficiently large.
d. Effective cloud fraction for general optical
thickness based on the average plane-parallelmodel
In this section we show f e as a function of f for the
average plane-parallel case. Figure 8 draws f e as a function of f for t H 5 50, 10, 3, and 1, similar to Fig. 5.
As expected, an opaque cloud (t H 5 50) yields
essentially the same dependence of f e on f , while
semitransparent clouds give totally different features.
In contrast to the maximum plane-parallel case, f edoes not converge to zero when f → 0 and insteadf e tends to increase up to unity as t H decrease. Thistendency directly reflects the trend of F ↓ /F ↓PP (or f e )→ 1 for t H → 0 in Fig. 7 as mentioned in section 3c.
The approximate formula of f e for the average planeparallel model, drawn by lines in Fig. 8, is given by

5[

f eav 5 c 1 2 exp 2(1 1 2ca) f
1 2 f2]6

1 1 2 c,

(23)

where

1 1 1 2a2.

c 5 1 2 exp 2

parallel model for determining f e should be chosen appropriately corresponding to the situation.
The independent pixel approximation (IPA) is known
to be a useful tool for calculating radiation flux in thepresence of horizontally inhomogeneous clouds. In the
IPA the cloud inhomogeneity is represented by a set ofplane-parallel columns with different optical properties,
where horizontal radiative interactions are neglected.
Although the IPA reduces computational time for 3D
RT, it can fail to produce some essential physical processes where the horizontal photon transport is important, such as the radiative smoothing in fractal clouds(Marshak et al. 1995). Derivation of the effective cloud
fraction requires a precise calculation of the horizontal
energy transfer by radiation since it involves the emission and absorption by the side faces of broken clouds.
The IPA is, therefore, not able to take into account the3D effect of clouds adequately and fully three-dimensional calculations are inevitable for the purpose of ourpresent study.
4. Discussion
In the previous section we presented how the 3D
effect influences the radiation field in the cloudy atmosphere. We discuss below on longwave radiationbudget of the earth in the presence of broken clouds.
The cooling or heating rate inside a cloud can be
described in terms of the flux divergence in the cloud.
The volume-averaged flux divergence istH

Equation (23) is equivalent to Eq. (20) for t H → `.
e. Applications
In actual applications, either of the maximum or average plane-parallel case, or their combination, should
be chosen depending on purpose. When one applies f e
to correction of flux in fields of view covered partiallywith clouds in infrared or microwave remote sensors,
the average plane-parallel case would be appropriate
because the observed flux is smoothed out within a fieldof view. In contrast, the maximum plane-parallel casemight be preferable in GCMs in case that a plane-parallel cloud is assumed within a grid cell instead of broken clouds of the same liquid water density.
As shown above, f mx
and f avee are identical for optically thick clouds, but they exhibit essentially different
behaviors for semitransparent clouds. Even if t H is as
av
much as 10, f mx
e (in Fig. 5) is distinct from f e (in Fig.
8) for high aspect ratios, a 5 2 and 5, for which thehorizontal optical thickness is not large enough. Considering that clouds typically have optical thickness of;10 or less, one finds that the assumption that clouds
are black is generally invalid and the normalizing plane-

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divF 5E(= · F ) dV

V

V

5

E

F dS

S

V

,

(24)

where V denotes the total volume of the cloud and S
the total surface area. In the limit that the cloud is extended infinitely in a horizontal plane, Eq. (24) leads to

divF 5

E

F dS

S

AH

ø

F ↑ A 2 (pI inc 2 F ↓ )A
,
AH

(25)

where A is the horizontal surface area at the cloud top
or base. Equation (25) shows that the mean heating rate
inside an isothermal and nonscattering cloud is givenby
2divF 5 2

F ↑ 2 (pI inc 2 F ↓ )
.
H

(26)

With Eqs. (4)–(7) and (14), Eq. (26) leads to
2divF 5 p (I inc 2 2Sc )1 2 2L.
H(27)

Substituting E 3 (tPP ) for L as the plane-parallel value,
divFPP , we have

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2465

FIG. 7. The upward flux emergent through the cloud top and the downward flux through thebase for the average plane-parallel model, i.e., tPP 5 ft H .

divF
1 2 2L5
5 fe,
divFPP
1 2 2E 3 (t PP )(28)

which leads to(30)

After the same procedure to derive Eq. (28), one has

using Eq. (13). Therefore the longwave flux divergenceinside an isothermal cloud normalized by the planeparallel value is equal to the effective cloud fraction.
The longwave radiative forcing of an isothermal andnonscattering cloud at its top is given by
Frad 5 F ↑ 2 pIinc ,Frad 5 p (S c 2 Iinc )(1 2 2L).

(29)

Frad
1 2 2L55 fe.
Frad,PP
1 2 2E 3 (t PP )(31)

Thus the longwave forcing of an isothermal cloud normalized by the plane-parallel value as well accords withthe effective cloud fraction.
Equations (28) and (31) indicate that f e shown in
Figs. 5 and 8 is applicable also to examine the clouds’

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VOLUME 58

FIG. 8. Effective cloud fraction for t H 5 10, 3, 1 as well as t H 5 50 with the approximate formula given by Eq. (22) is shown (line).
Symbols denote the exact solutions as in Fig. 3. The average plane-parallel model is used for the normalizing flux.

3D effects on the flux divergence and the radiative forcing for longwave radiation. This indicates that the planeparallel quantities, (= · F)PP and Frad,PP , without any correction for the brokenness of cloud fields are overestimated by a factor of f 21
from the true values. On theeother hand, one finds that modified plane-parallel flux
using the cloud fraction f through Eqs. (17) and (18)
would yield the flux divergence of f (= · F)PP and theradiative forcing of fFrad,PP , replacing f e with f in Eqs.
(28) and (31). In this case, the flux divergence and theradiative forcing for longwave radiation are underestimated by a factor of f / f e since f is always smaller thanf e . The errors can cause substantial misestimation in
the radiation budget of the cloudy atmosphere in any
case when the 3D effect of clouds is not taken into
account adequately.
In this work we assume a simple cloud model of acloud array made of identical cuboids, which is not thecase in actual cloud fields. Several authors have done
statistical analysis of the cloud size distributions employing Landsat Multispectral Scanner (MSS) and Thematic Mapper (TM) data (e.g., Wielicki and Welch 1986;
Cahalan and Joseph 1989; Davis et al. 1997; and references therein). Wielicki and Welch (1986) investi-

gated the cloud size distribution in MSS images, andfound that the cloud size distributions show encouraging
similarities among the cloud fields examined, compilingas well some previous studies based on aircraft photographs by Plank (1969) and Hozumi et al. (1982).
Cahalan and Joseph (1989) also derived the cloud size
distribution using TM data. They reported that the size
distribution as a function of log(Ïcloud area), or cloud
effective diameter, is excellently approximated by a double power-law fit with the power of 20.89 for smallerscales than ;2 km and of 22.76 for larger scales.
Davis et al. (1997) obtained the energy spectrum of
the radiance field of clouds in a TM image, showing
the power-law distribution in the Fourier space with two
scale breaks at 0.2–0.4 and ø20 km. Although directcomparison is impossible between the energy spectrumof cloud fields and the size distribution for ensembles
of individual clouds as described above, cloud fields
seem to be characterized by power-law distributions ingeneral over a wide range of size scales.
Straightforward application of the results in this work
to cloud fields with power-law size distribution is difficult. The effective cloud fraction involves radiativetransfer problems, which do not linearly response to15 AUGUST 2001

2467

MASUNAGA AND NAKAJIMA

cloud size distribution. Moreover, radiative interaction
between clouds with different sizes should be taken into
account. A more sophisticated 3D RT scheme is required
to solve such problems.

the scattering dominates the absorption in radiative processes, arises in a somewhat different way. Our subsequent work will be dedicated to the shortwave versionof the present study.

5. Summary and conclusions

Acknowledgments. The authors greatly appreciate detailed comments of anonymous reviewers, which have
improved the manuscript.

We summarize the conclusions in this paper as follows.
1) We investigate the effects of broken clouds on longwave radiation on the basis of a simple cloud modelthat consists of a cuboidal cloud array. In order toextend the study of HW82 to be applicable to semitransparent clouds as well as optically thick clouds,
multistream radiative transfer is solved to obtain theexact solution of radiative flux passing through thebroken clouds. Since we concentrate the present
study on the geometrical effects of broken clouds to
avoid uncertainties lying in optical properties ofcloud water droplets and ice crystals, the calculated
flux is normalized by plane-parallel flux.
2) We find that the empirical formula for the effective
cloud fraction previously derived by HW82 systematically underestimates the exact solution. Anotherformula [Eq. (20)] is proposed to provide a better fit
to the exact solution.
3) In contrast to HW82, who considered only blackbody
clouds, we also examine the dependence of radiativeflux and the effective cloud fraction on optical thickness. Two different models are considered for computing plane-parallel flux used for the normalization:
the maximum model (tPP 5 t H ) and the average model (tPP 5 ft H ). Approximate formulas of the effective
cloud fraction for general optical thickness are proposed in Eq. (22) for the maximum plane-parallelcase and in Eq. (23) for the average plane-parallelcase.
4) The 3D effect of clouds can affect the thermal structure in the atmosphere. Neglecting the 3D effectcauses overestimation by f e of the longwave flux
divergence and the longwave forcing, while it leads
to underestimation by f / f e if the cloud fraction f isused instead of f e for modifying plane-parallel flux.
The 3D effect of cloud for shortwave radiation, where

REFERENCES
Aida, M., 1977: Reflection of solar radiation from an array cumuli.
J. Meteor. Soc. Japan, 55, 174–181.
Cahalan, R. F., and J. H. Joseph, 1989: Fractal statistics of cloudfields. Mon. Wea. Rev., 117, 261–272.
Davis, A., A. Marshak, R. Cahalan, and W. Wiscombe, 1997: The
Landsat scale break in stratocumulus as a three-dimensional radiative transfer effect: Implications for cloud remote sensing. J.
Atmos. Sci., 54, 241–260.
Ellingson, R. G., 1982: On the effects of cumulus dimensions on
longwave irradiance and heating rates. J. Atmos. Sci., 39, 886–
896.
Gabriel, P. M., S.-C. Tsay, and G. L. Stephens, 1993: A Fourier–
Riccati approach to radiative transfer. Part I: Foundations. J.
Atmos. Sci., 50, 3125–3147.
Han, D., and R. G. Ellingson, 1999: Cumulus cloud formulations forlongwave radiation calculations. J. Atmos. Sci., 56, 837–851.
Harshvardhan, and J. A. Weinman, 1982: Infrared radiative transfer
through a regular array of cuboidal clouds. J. Atmos. Sci., 39,
431–439.
——, ——, and R. Davis, 1981: Transport of infrared radiation in
cuboidal clouds. J. Atmos. Sci., 38, 2500–2513.
Hozumi, K., T. Harimaya, and C. Magono, 1982: The size distributionof cumulus clouds as a function of cloud amount. J. Meteor.
Soc. Japan, 60, 691–699.
Killen, R., and R. G. Ellingson, 1994: The effect of shape and spatial
distribution of cumulus clouds on longwave irradiance. J. Atmos.
Sci., 51, 2123–2136.
Lewis, E. E., and W. F. Miller, 1984: Computational Methods of
Neutron Transport. John Wiley and Sons, 401 pp.
Marshak, A., A. Davis., W. Wiscombe, and R. Cahalan, 1995: Radiative smoothing in fractal clouds. J. Geophys. Res., 100, 26
247–26 261.
Plank, V. G., 1969: The size distribution of cumulus clouds in representative Florida populations. J. Appl. Meteor., 8, 46–67.
Roewe, D., and K.-N. Liou, 1978: Influence of cirrus clouds on theinfrared cooling rate in the troposphere and lower stratosphere.
J. Appl. Meteor., 17, 92–107.
Takara, E., and R. G. Ellingson, 1996: Scattering effects on longwave
fluxes in broken cloud fields. J. Atmos. Sci., 53, 1464–1476.
Wielicki, B. A., and R. M. Welch, 1986: Cumulus cloud properties
derived using Landsat satellite data. J. Climate Appl. Meteor.,
25, 261–276.