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www.elsevier.comrlocateratmos

Geometrically effective cloud fraction for solar

radiation

Michael Batey

1

_{, Harshvardhan}

)

_{, Robert Green}

Department of Earth and Atmospheric Sciences, Purdue UniÕersity, West Lafayette, IN 47907-1397, USA

Received 10 April 2000; accepted 27 June 2000

Abstract

It has been suggested in the past that anAeffectiveB cloud fraction can be used to take into account cloud geometry effects in solar radiation parameterizations. All such models have been based on the reflected flux from non-plane-parallel cloud fields. In this study, it is shown that the

Ž .

AeffectiveB cloud fraction based on absorption or transmission could differ considerably from that based on reflection and, moreover, is not constant throughout the solar spectrum even for fixed cloud geometry. Furthermore, the representation of cloud radiative forcing at the surface and at the top of the atmosphere in terms of cloud fractions are based on differentAeffectiveB cloud fractions. Therefore, a simple interpretation of measured values of the forcing ratio is not possible when geometric effects are present.q_{2000 Elsevier Science B.V. All rights reserved.}

Keywords: Albedo; Atmosphere; Climate; Clouds

1. Introduction

Numerical weather prediction and climate models have always had to grapple with the issue of representing the cloud coverage of grid boxes for purposes of the radiation parameterization. In general, in order to compute the radiation field, one requires information on the physical coverage of the grid box by clouds and the optical properties of the clouds. This is true irrespective of the technique used to generate the simulated

)_{Corresponding author.}

Ž .

E-mail address: [email protected] Harshvardhan . 1

Current affliation: TRW, Aurora, CO, USA

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( ) M. Batey et al.rAtmospheric Research 55 2000 115–129

116

Ž .

clouds in the model Slingo, 1987; Tiedtke, 1993 . The coverage of a grid box by clouds

Ž .

can be termed the Aextrinsic fractional cloudinessB Randall, 1989 and all current models use a weighting of overcast and clear radiation computation to calculate the grid averaged radiation field. The weighting is determined by the cloud fraction and assumptions regarding the overlap between cloud layers in fractionally cloudy grid

Ž .

boxes Harshvardhan et al., 1987; Ridout et al., 1994 .

The separation of a model grid box into clear and cloudy regions does not solve the problem completely since some assumptions need to be made regarding the distribution of optical thickness in the cloudy region. Since the typical climate model is incapable of generating these distributions as yet, some methods have been proposed to calculate

Ž

effective optical thickness to represent such cloud inhomogeneities Cahalan et al.,

.

1994a,b; Barker, 1996 .

There is yet another problem associated with the transfer function that relates simulated clouds and the radiation field. This is the issue of geometrical effects. All radiation parameterizations rely on plane parallel computations as their basis. The diagnosed cloud fraction in a numerical model grid box, of course, does not incorporate any geometric effects. Traditionally, it has been accepted that the cloud fraction provided to the radiation parameterization is an AeffectiveB cloud fraction, which

Ž .

includes geometric effects. Loeb et al. 1998 have modeled conservatively scattering cloud fields that have both horizontal variations in extinction coefficient and structure at cloud top. However, they did not attempt to develop a parameterization that could be used in atmospheric models.

In this note, we revisit the concept of effective cloud fraction because past studies have always considered outgoing fluxes in both the reflected shortwave and emitted longwave when defining the effective cloud fraction for radiation parameterizations. The recent renewed interest in atmospheric solar absorption, in particular, the role of

Ž .

geometrical effects in interpreting measurements Valero et al., 1997 , and computing

Ž .

solar absorption O’Hirok and Gautier, 1998 has brought the issue to the fore again after a long period of dormancy.

2. Effective cloud fraction

The effective cloud fraction, N , is defined for an array of finite clouds but thee arguments presented here can be applied as well to any edge effects that may be

Ž .

observed. As defined by Weinman and Harshvardhan 1982 and Welch and Wielicki

Ž1984 , N is the equivalent cloud fraction of a planiform field of clouds with the same. _e

vertical optical thickness required to give the same flux as that reflected from the finite cloud array, i.e.,

F≠sN Fe pp≠

Ž .

t

Ž .

1

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layer would be embedded in a multi-layer atmospheric model. The concept of an

Ž

effective cloud fraction has also been used for longwave radiation Harshvardhan, 1982;

.

Ellingson, 1982; Takara and Ellingson, 1996 but here we will restrict ourselves to shortwave radiation only.

Ž .

Note from Eq. 1 that N is defined in terms of the reflected flux. There is no reason_e to believe that the same effective cloud fraction will be applicable to the computation of layer absorption or solar transmission to the surface. For the purpose of this study we therefore define the following. The normal cloud fraction, or earth cover, N, is the fraction of earth covered by clouds when the clouds are projected vertically. However, this in not the cloud fraction observed from the surface which is the fractional sky cover

ŽWarren et al., 1988 . We can further refine the definition of N given in Eq. 1 as. e Ž .

Where R and A are the actual reflectance and absorptance of the partly cloudy layer;

R_pp and A_pp are the corresponding quantities when the cloud in the layer is considered to be plane-parallel and homogeneous. All quantities refer to the properties of the entire layer including the clear portions.

3. Cloud model

The above definitions are applied to a very simple model of a partly cloudy layer, shown in Fig. 1. The model consists of rectangular bar clouds where the cloud system is split between a homogeneous cloudy portion where all optical properties are constant for

Ž

an infinite distance into the plane, and a clear portion Harshvardhan and Thomas,

.

1984 . Incident sunlight strikes the array in the plane shown at a prescribed zenith angle. For illustrative purposes, we have restricted the study to a zenith angle, us608. A similar analysis could, in principle, be done for a three-dimensional cloud array made up

Ž .

of cubes or other shapes Welch and Wielicki, 1984 and at different solar zenith angles. The basic non-linearities between field radiative properties and normal cloud fraction remain qualitatively the same. The finite cloud geometry is represented by the aspect ratio, a, of the cloud elements defined as the ratio of the height to the width of each element. Although the cloudy layer is embedded in an atmosphere, we restrict ourselves to the solar near-infrared where there is absorption by both liquid drops and water vapor. Therefore, the insolation is not diffuse and all radiative properties refer to the layer alone.

We have chosen to illustrate cases for two representative portions of the near-infrared

Ž .

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118

Fig. 1. Schematic diagram of the cloud and illumination geometry. The clouds extend to infinity perpendicular to the plane.

spectral characteristics of liquid and vapor absorption in the near-infrared. Note that there are several water vapor windows in which droplet absorption dominates. Closer to the visible wavelengths there are regions of very weak liquid absorption but still significant vapor absorption in the band centers. These are the two regions selected for study here. We have excluded the stronger bands of vapor absorption because there is little insolation incident on low-level clouds at these wavelengths. The two cases are therefore, liquid absorption only and vapor absorption only.

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Fig. 2. Spectral characteristics of water vapor and liquid water absorption. The outer envelope represents the insolation at the top of the atmosphere for a solar zenith angle of 308, while the dotted line represents the total absorption of the clear sky midlatitude summer atmosphere. The solid lines indicate the absorption by a semi-infinite cloud of effective radius 8 and 20 mm, respectively, when there is no water vapor in the

Ž .

atmosphere, after Espinoza and Harshvardhan 1996 .

possibilities that could be encountered, we feel they are sufficient to illustrate the properties of effective cloud fraction.

4. Results

Ž .

The Monte Carlo method described in Marshak et al. 1995 was used to simulate photon transfer in the cloud field. Absorption in each vertical cell was computed by attaching a weight to each photon and modifying the weight at each scattering event. The geometrical cases for which computations were made were seven different normal cloud fractions, N, for each of four aspect ratios of the cloudy element. The cloud fraction was altered by changing the properties of the individual pixels along the field. The model was tested for the plane parallel, homogeneous case by comparing results

Ž .

presented in King and Harshvardhan 1986 who used the doubling method. All results correspond to the entire field which is assumed to be cyclic consisting of alternating rows of cloudy and clear areas.

4.1. Vapor absorption, conserÕatiÕe scattering

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120

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ratio of the scattering optical depth, 20.0, and the extinction optical depth which is the scattering optical depth plus the vapor absorption optical depth in the cloudy portion. We assume here that the vapor column amounts are identical in the clear and cloudy portion. This simplifying assumption does not detract from any of the conclusions reached in this study. Fig. 4 shows the corresponding panels for system absorptance.

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122

Ž .

Energy conservation requires that the residual is the total direct plus diffuse transmit-tance through the system.

Several features stand out. The top left panel of Fig. 3 shows the non-linear relationship between reflectance and normal cloud fraction that has been presented in

Ž .

prior finite cloud studies Welch and Wielicki, 1984 . The actual relationship depends

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quite strongly on the incident solar zenith angle for small cloud fractions. Fig. 4, however, shows that the absorption by the field for conservatively scattering cloud droplets is a fairly linear function of cloud cover. Moreover, both reflectance and absorptance show very weak or non-existent dependence on cloud aspect ratio when

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124

vapor absorption is strong. This is not surprising since most of the absorption occurs in the top portion of the clear and cloudy layers and one may think in terms of the photon aspect ratio of finite clouds which is much less than the geometrical aspect ratio when absorption is strong. In the limit of very strong absorption, finite clouds behave like plane parallel clouds as is evident from the bottom right panel of both Figs. 3 and 4.

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However, this case is of academic interest only because most of the incident solar radiation would have been absorbed by the water vapor above the cloudy layer.

The significance of the results shown in Figs. 3 and 4 is clearer when the concept of

Ž . Ž .

effective cloud fraction introduced earlier is applied. The ratios in Eqs. 2 and 3 are plotted in Fig. 5. The top left panel shows the typical enhancement of effective cloud fraction for moderate to high zenith angle and small cloud fraction that is primarily a

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126

result of the interception of solar radiation by the sides of the cloud elements. The sequence of panels shows results for increasing absorption with the bottom right panel indicating near-plane parallel behavior of cloud field properties.

Although finite cloud effects dominate the reflectance, the absorptance of the entire cloudy layer can be modeled quite well with the simple linear weighting of cloud and clear absorption that is standard practice in model parameterizations. The non-linearity appears, of course, in the transmittance term. The point to note here is that the effective

Ž .

fraction defined by Eq. 2 in terms of the reflectance can be quite different from that

Ž .

defined by Eq. 3 in terms of the absorptance. This has important implications for extracting cloud forcing from measurements, as we shall show later.

4.2. Liquid absorption, waterÕapor window

In this second set of computations, we assume that there is no vapor absorption in the clear or cloudy portions, i.e., t2s0.0 in addition to √2s0.0. The vertical extinction optical depth of the cloud elements is fixed at 20.0 and several values of the single scattering albedo are chosen in turn. The aspect ratios and cloud fractions are as in the earlier runs. Here again, we assume a Henyey–Greenstein phase function with gs0.843. Fig. 6 shows the system reflectance and Fig. 7 the system absorptance for four values of the droplet single scattering albedo. Since there is no vapor absorption, the clear portions are completely transparent and in addition do not scatter either because we have confined ourselves to the near-infrared portion of the spectrum. Although there are similarities between Figs. 3 and 6, there are some striking differences. For example, even for the highly absorbing case, there are non-linearities in the behavior of system reflectance vs. cloud fraction. This is because, unlike the vapor absorption case, the photon aspect ratio is identical to the geometrical aspect ratio. Radiation impinges on the portion of the sides of the cloud element that are not shaded by neighboring clouds since there is no gaseous absorption between cloud elements. The incident radiation will also be significantly greater than in the previous case because there is no absorption by vapor above the cloud layer. These wavelengths will contribute the most to cloudy layer absorption of total energy.

The absorptance, shown in Fig. 7, shows a pronounced non-linear behavior and simple linear weighting will be grossly inadequate. This point is more explicitly made

Ž . Ž .

by Fig. 8, which shows the effective cloud fraction based on Eqs. 2 and 3 . The substantial variation of Ne R and Ne A with single scattering albedo and, more impor-tantly, the difference between Ne R and Ne A point out the inadequacy of effective cloud

Ž

parameterizations based on geometric considerations Harshvardhan and Thomas, 1984;

.

Welch and Wielicki, 1984 . There is simply no unique relationship that can encompass both reflection and absorption, and all wavelengths.

5. Cloud forcing

Some of the recent discussion regarding solar absorption in the atmosphere has

Ž

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.

1995; Ramanathan et al., 1995 . In the context of finite cloud fields such as the bar cloud array discussed here, the cloud forcing at the top of the atmosphere, C , is thest difference in the net downward solar flux at the top between a clear atmosphere and one in which the finite cloud array has been embedded. Likewise, the forcing on the surface,

C , is the difference between surface solar absorption with and without the cloudy layer._ss

Here we can address issues related to the near-infrared solar cloud forcing in the presence of broken clouds, a situation that was fairly common in some of the reported

Ž . Ž .

studies Pilewskie and Valero, 1995 . Following Lubin et al. 1996 , for a partially cloudy atmosphere with normal cloud fraction, N, the mean reflectance R and absorp-tance A of the atmospheric column is usually written as

RsR 1c

Ž

yN

.

qR No

Ž .

4

and

AsA 1c

Ž

yN

.

qA No

Ž .

5

where R and A are the clear sky reflectance and absorptance, respectively, R and Ac c o o

Ž . Ž .

refer to the corresponding overcast quantities. Now we can modify Eqs. 4 and 5 by introducing the respective effective cloud fractions, such that the system reflectance and absorptance are

Note that the effective cloud fraction defined in terms of reflectance is different from that defined by absorptance following the results presented earlier.

A parameter used widely to characterize atmospheric absorption is the ratio of the cloud forcing at the surface and top of the atmosphere, i.e.,

fsCssrCst

Ž .

8

where C_ss and C_st have been introduced earlier. Strictly speaking, our results cannot provide f because all computations have been performed for an isolated cloudy field, but as long as the incident radiation is not diffuse, our estimates of Ne A and Ne R would still be valid and this is the case in the near-infrared as long as there is no overlying cloud layer.

Ž . Ž . Ž . Ž .

Substitution of Eqs. 6 and 7 instead of the usual practice of using Eqs. 4 and 5 in the definition of f yields

A yA N

which reduces to Eq. 3 of Lubin et al. 1996

A_oyA_c

fs1q

Ž

10

.

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128

when Ne AsN . Since overcast skies usually result in enhanced column absorption ande R

Ž .

Ro)R , f is typically greater than 1.0, perhaps as large as 1.5 Cess et al., 1995c

although modeling studies generally obtain somewhat lower values. Inspection of Eq.

Ž ._{9 shows that f is not simply a function of the difference in clear and overcast}

properties but also dependent on geometrical effects. The ratio N_{e A}rN_{e R} can amplify or dampen the positive departure of f from unity. The albeit limited simulations presented here indicate that N_{e A}rN_{e R} is neither a universal ratio nor easily parameterizable. It is a function of wavelength since it depends on whether there is liquid or vapor absorption and also on the geometrical aspect ratio. Therefore, a simple interpretation of measured

Ž .

values of f as in Eq. 10 is not possible when geometric effects are present.

6. Conclusion

It has been shown using a very simple model of heterogeneous clouds that geometric effects introduce uncertainties in the inference of cloud absorption based on measure-ments made above and below cloud layers. Further, any attempts at representing cloud geometrical effects in a parameterization by using an effective cloud fraction are fraught with danger because there is no unique effective cloud fraction for the absorbing regions of the solar spectrum. Nevertheless, the original arguments made by Welch and Wielicki

Ž1984 and others regarding the inappropriateness of using the normal cloud cover to.

compute mean fluxes for partly cloudy regions still holds. Geometric effects are quite pronounced for intermediate cloud coverage. A possible solution is to include geometric effects as an integral part of models of cloud heterogeneity when developing effective thickness and optical depth distribution models.

Acknowledgements

This study was supported by NASA grant NAG5-3992.

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