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 Fig. 3 shows four panels depicting the variation of cloud field reflectance as a
function of normal cloud fraction, N. The individual points are for each of four aspect ratios. The single scattering albedo of the cloudy portion marked on each panel is the
Fig. 3. System reflectance as a function of normal cloud fraction, N, for four different aspect ratios, as 2.0, 1.0, 0.5 and 0.25 and a solar zenith angle of 608. Results are for conservative droplet scattering with the
addition of water vapor yielding single scattering albedos, √ s 0.995, 0.976, 0.952 and 0.909 in the cloudy
1
portion of the field.
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.
Fig. 4. As in Fig. 3 but for the system absorption.
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
Fig. 5. Ratio of the reflected and absorbed fluxes to the corresponding fluxes for plane-parallel clouds with the same normal cloud fraction N. Cloud properties as in Fig. 3.
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
Fig. 6. System reflectance as a function of normal cloud fraction, N, for four different aspect ratios, as 2.0, 1.0, 0.5 and 0.25. Results are for droplet absorption only, no vapor absorption. Single scattering albedos are
√ s 0.995, 0.99, 0.95 and 0.9 in the cloudy portion of the field.
1
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.
Fig. 7. As in Fig. 6 but for the system absorptance.
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
Fig. 8. As in Fig. 5 but for the cloud properties in Fig. 6.
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, water Õapor window In this second set of computations, we assume that there is no vapor absorption in the
clear or cloudy portions, i.e., t s 0.0 in addition to √ s 0.0. The vertical extinction
2 2
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 g s 0.843. 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 N
and N with single scattering albedo and, more impor-
e R e A
tantly, the difference between N and N
point out the inadequacy of effective cloud
e R e A
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
