Three cloud types defined by the ISCCP C product are used: deep cloud combined with CsCc (cloud type 1), cirrus (cloud type 2), and stratus (cloud type 3). CsCc linearly depends on deep cloud, so we combine them to be one cloud type. Cloud type 1 linearly depends on large-scale precipitation with the coefficient:
For more details, see Section 2.2 in Zeng et al. [2].
In version 2.3, a cap is implemented to ensure cloud 1 fraction 
does not exceed 1 (note in theoretical studies this nonlinearity can be
neglected but not in the numerical version especially with higher
variability as in v2.3). Additional conditions are applied to other
cloud types to account for overlap.
Let 
be the total cloud fraction of cloud type k before
taking account of overlap and 
be the cloud fraction
that enters the radiation code after taking overlap into account
(code variables cldtot(k,i,j) and cld(k,i,j)). The
sum of is required to be less than or equal to 1 with
the clear sky fraction ( 
) equal to the remainder.
Since overlapping clouds
are not typically observed by satellite, the fractions are
more directly comparable to ISCCP data as well as being the
radiatively active cloud fraction in the model. When cloud type 1 overlaps
any other cloud type, its radiative properties dominate so
with
In version 2.3, cirrus cloud fraction is parameterized on Cloud Type 1 as
with
The proportionality factor cl2fac=1.5, based on ISCCP C2 data (Chou
1997, Fig. 5.15 and taking into account the combination of deep and CsCc in
cloud type 1) although we note there is considerable scatter about this
regression in the data. This cloud fraction of Cloud 2 is further modified by a
cloud overlap condition to ensure total cloud fraction less than 1. The
physical assumption is that where overlap occurs the deep cloud effect on
radiation dominates so is used in the radiation. The 
factor is equivalent to a random overlap assumption. The cloud 2 fraction
that results from this parameterization in v2.3 results has slightly larger
maximum cloud fraction in convergence zones than ISCCP and
smaller fraction in the subtropics. Spatial averages within the tropics are
similar to observed for 10S-10N but somewhat smaller than observed
for 30S-30N.
For Cloud 3 and Cloud 4, we use
such that
when Cloud 3 or Cloud 4 is shielded by Cloud 1 or Cloud 2, the Cloud 1 or
Cloud 2 properties are used. For stratus, a spatial constant or
observed seasonal climatology from the ISCCP C2 data can be used for
. Note that if observed stratus is used it is modified by
the overlap factor of 
if precipitation occurs in a grid box with stratus.
If using a constant reference value 
,
helps keep spatial mean similar to 
. For Cloud 4,
we choose not to read in an observed climatology since in the ISCCP data
these cloud types do not have an obvious spatial and seasonal dependence.
In previous versions, cloud type 4 was strictly constant and its
cloud fraction was absorbed into the radiation code parameters to
reduce computation. Here cloud type 4 is variable due to overlap.
In addition to overlap with cloud types 1 and 2, cloud type 4 is
reduced if cloud type 3 is large:
The condition 
with these parameterizations
factorizes to 
, 
, and
, which all have been enforced
by the min function conditions in the parameterizations.
With this parameterization, the modeled type 1 cloud fraction is about 0.5-0.6 for intense storms. The ISCCP climatology for deep cloud plus CsCc has maximum around 0.5 as well.