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Þ ð
r
=
r
Þ
2
Q
sc
Q
S
Z
;
e
;
N
ð
Þ
¼
cos
Z
Tr Z
; ðÞ
FN
;
Z
ð
ð
3
:
11
Þ
uence of cloudiness (
1
/
3
≤
F
≤
where F gives the in
Sun
distance, also known as the astronomical unit AU (see Annex 1). The solar constant is
Q
sc
= 1,367 W m
−
2
, and due to the ellipticity of the Earth
fl
1), and
r
is the mean Earth
-
'
s orbit, the Earth
Sun distance
-
varies with time and causes the solar radiation above the Earth
'
s atmosphere to range from
1,294 W m
−
2
22) to 1,483 W m
−
2
22).
The atmospheric transmittance is close to 0.9 except for very large solar zenith angles
(Z > 80
(in June 20
(in December 20
-
-
). Zillman (1972) formula for the clear sky transmittance, found good in many
applications (e.g., Curry and Webster 1999; Lepp
°
ä
ranta and Myrberg 2009), is written as
cos
Z
T
tr
Z
; ðÞ
¼
ð
3
:
12
Þ
Þ
10
3
þ
0
:
01
1
:
085
cos
Z
þ
e
2
:
7
þ
ð
cos
Z
where the water vapour pressure is given in millibars. The maximum of T
tr
(Z = 0 and
e = 0) is 0.913. In cold climate we can assume that e < 10 mbar. At this upper limit, for the
direct radiation we have T
tr
= 0.879 at Z =30
°
, T
tr
= 0.855 at Z =60
°
and T
tr
= 0.764 at
Z =80
°
. At the limit Z
90
°
, we have T
tr
→
0 that is not exactly true since the thickness
→
of the atmosphere is
finite.
For clear sky irradiance at the lake surface we have F = 0 in Eq. (
3.11
). By integration,
it is seen that for horizontal planar irradiance, diffuse radiation corresponds to direct
radiation at Z
D
= 50.5
°
. Therefore, the total clear sky irradiance at the surface can be
estimated as
, where
m
is the fraction of
direct solar radiation. For larger zenith angles the diffuse radiation becomes dominant.
Cloudiness can reduce the level of radiation at the Earth
Q
s
Z
;
e
;
0
ð
Þ
m
Q
s
Z
;
e
;
0
ð
Þþ
1
m
ð
Þ
Q
s
Z
D
;
e
;
0
ð
Þ
s surface to about one-third of
the clear sky level. The cloudiness correction shows much more variability and uncer-
tainty than the transmittance, since it depends on the quality of clouds. Lumb (1964)
examined the hourly incoming radiation at different cloudiness levels and cloud types.
However, usually only the total cloudiness data are available. From the results of Lumb
(1964) we can formulate the cloud correction function as
'
FN
;
Z
ð
Þ
¼1
a
b
ð
cos
Z
Þ
N
ð
3
:
13a
Þ
in the absence of precipitation. Here a
0.1 are empirical parameters. Lumb
(1964) also reported that the radiation reaching the Earth
≈
0.6 and b
≈
is surface is much smaller in the
presence of rain. Several formulae have been designed for estimation of daily solar
'
fl
uxes.
A widely used one is by Reed (1977):
Þ
¼1
c
N
N
þ
0
:
0019
90
Z
FN
;
Z
ð
ð
Þ
ð
3
:
13b
Þ
where c
N
= 0.62 is a cloudiness coef
is the noon solar zenith angle in
degrees. This is, however, inconsistent in that F(0, 0) = 1.17 that would give Q
s
> Q
sc
. The
cient, and Z
°
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