Q or v as these fits can be quickly recomputed. However, this is a problem for the
ext
computation of g, which requires the use of Lorenz–Mie theory. Ž
. We circumvent this problem by adopting the approach of Hu and Stamnes 1993 .
This work showed that when optical properties for varying n were plotted against r ,
e
the resultant curves were fairly independent of n . This result is found to hold in Fig. 5 and suggests that one can compute the optical properties as functions of r for a given n
e
and then use that information for any gamma distribution. Ž
. Of course, there are errors associated with this method when size ; l . For example,
solar bands show the greatest errors at small r , below those typically observed in
e
Ž clouds. However, errors of the same magnitude occur in the infrared near r s 10 not
e
. Ž
. shown , which is within the range typical of marine stratocumulus Stephens, 1978 .
Since g has a smaller influence on diabatic processes than Q or v, this error should
ext
be tolerable. In fact, calculations using this method for ASC with differing n are only slightly affected by using this method for computing g.
3. Bin model parameterization
Bin microphysical models in use today vary in their complexity and structural details Ž
. e.g., Feingold et al., 1994; Kogan et al., 1995 . The liquid-phase bin microphysical
Ž .
scheme used in the RAMS model is essentially that of Feingold et al. 1994 as modified Ž
. by Stevens et al. 1996 while the mixed-phase bin microphysics is that of Reisin et al.
Ž .
Ž .
Ž .
1996 . For both schemes condensation deposition , evaporation sublimation , and collision–coalescence are solved on a discrete grid using the method-of-moments
Ž .
Tzivion et al., 1989; Stevens et al., 1996 . In our work, the grid is defined by size boundaries covering the space from 3.125 to 1008 mm. These diameter boundaries are
defined by mass-doubling between bin edges where edge k q 1 is related to edge k by m
s 2 m . This translates into the following formula for the diameter edges, D s
kq 1 k
k
2
Ž ky1.r3
D , where D is 3.125 mm. A definition of this parameter space requires
1 1
Ž .
Ž .
Ž .
specification of 25 bins 26 edges with both number concentration N and mass M
k k
for each bin varying in both time and space. To most accurately represent cloud optical properties, the radiative transfer model should make use of the bin information as it
evolves.
3.1. Bin optical properties With a bin microphysical representation, the size spectrum can vary significantly
Ž .
during a model run. While it is possible to use the method of Hu and Stamnes 1993 for bin models, there exists a more accurate method of computing the optical properties
from the bin microphysical model information during model integration that is still computationally expedient. Additionally, this approach is necessary for coupling radia-
Ž tion into the vapor growth equations in bin models see Harrington et al., 2000; Wu et
. al., 1998 , where bin-dependent optical properties are required.
Ž . In order to use Eq. 2 , the continuous integral over size must now be broken into a
discrete sum over individual bins and be formulated as a function of the mean diameter Ž
. D
of each bin. However, since the mean diameter in each bin varies with time, and
k
since the wavelength and size dependence are intimately connected through Ž
. Q
D , m, l ,
ext k
N
bins
b s
E A D
Q D , m, l N d l r
E d l, 13
Ž .
Ž . Ž
. Ý
H H
ext l
k ext
k k
l
D l
D l
ks1
is not efficient to solve during model integration. Again, this is because the wavelength integral must be numerically computed.
Ž .
To solve this problem we can replace D in Eq. 13 with the average of the bin edge
k
diameters, D , for each bin. This is possible because bin resolution is finest at the
e, k
small drop end of the spectrum where the optical properties vary the most. By using this
Fig. 6. Relative errors associated with the bin optical property method for 4.6–8.3 mm band. Errors for the Ž
. various optical properties identified in the panel are plotted for a gamma distribution with n s6.
method, the integral over wavelength may be computed before-hand since the D are
e, k
fixed in time. Thus, the above integral is approximated as,
N
bins
b ,
A D N Q
, 14
Ž .
Ž . Ý
ext k
k ext , k
ks1
where Q
, E Q
D , m, l d l r
E d l. 15
Ž .
Ž .
H H
ext , k l
ext e , k
l
D l
D l
Similar forms are easily derived for v and g by thin and thick averaging. For all optical Ž
. properties, solutions to Eq. 15 are computed and stored as model input. During RAMS
Ž .
integration, Eq. 14 is used to compute the optical properties of drops and ice crystals. To test the accuracy of this method, computations were done using gamma distribu-
tion functions for which accurate analytical solutions are known. Tests were then conducted by breaking a gamma distribution of a given shape, n s 6, into 25 discrete
bins and then applying the above method to compute the optical properties. Fig. 6 shows the relative error associated with using the bin method to derive the optical properties
Ž .
for a range of distribution mean sizes 1 to 400 mm and for a particular radiation band. Note that errors in the bin method are, in general, quite small. In particular, the bin
method over-estimates the optical properties by about 1 for extremely narrow distribu- Ž
. tions D
; 1 . For distributions with D
R 4 errors are negligible. This was also
mean mean
found to be true for the ice-phase parameterization.
4. Radiative impacts of ASC
