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 Ž
. Ž .
Warren et al., 1988 . We can further refine the definition of N given in Eq. 1 as
e
follows N s R N rR
N 2
Ž .
Ž .
Ž .
e R pp
and N s A N rA
N 3
Ž .
Ž .
Ž .
e A pp
Ž .
Where R and A are the actual reflectance and absorptance of the partly cloudy layer; R
and A are the corresponding quantities when the cloud in the layer is considered
pp pp
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, u s 608. 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
. spectrum. Shown in Fig. 2 from Espinoza and Harshvardhan 1996 is a schematic of the
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.
The optical properties of the cloudy region are optical depth, t , and single scattering
1
albedo, √ . For all cases, the optical depth of the cloud droplets is set at 20.0 and we
1
assume a Henyey–Greenstein phase function with asymmetry parameter, g s 0.843. For the vapor absorption only case, increasing the vapor optical depth in both clear and
cloudy portions, decreases the single scattering albedo and increases the total cloud optical depth, t . The optical depth of the clear portion, t , is equal to the vapor
1 2
absorption optical depth; √ is always set at 0.0. When there is liquid absorption only,
2
the single scattering albedo, √ , is changed in steps for the cloudy portion. The total
1
optical depth is maintained at 20.0. In the clear portion, both t and √
are set to 0.0
2 2
since there is no vapor absorption. Although the cases shown are a limited set of the
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
