The radiation absorption within the cloud layer is determined by the radiative
divergence Eq. (1.5). It is computed with the obvious equation:
R ¼ 1 À F
"
ð0; m 0 Þ À ð1 À AÞF
#
ðt 0 ; m 0 Þ
¼ 1 À aðm 0 Þ þ
nKðm 0 Þme
Àkt 0
1 À l
le À2kt0
le
Àkt0
À
1 À A
1 À Aa 1
:
(11.5)
Mention that the term “asymptotic” specifies the light regime installed within the
cloud and it does not point out any approximation. Equations 11.1, 11.4 and 11.5
are strict within the diffusion domain. Their accuracy will be studied below
depending on the optical thickness.
11.2 Weak True Absorption of Solar Radiation
In clouds, the absorption is extremely weak compared to scattering (1 o 0 < <1)
within the short-wavelength range. As it has been shown in books in this case both
functions r 1 (m,m 0 ) and K(m) and constants m, l, k are expressed with the expansions
over powers of small parameter (1 À o 0 ). We consider here that parameter s, where
s
2 ¼ (1 À o 0 )/[3(1 À g)], is more convenient for the problem in question than
parameter (1 À o 0 ). Value g is a mean cosine of the scattering angle or, here, the
parameter of Henyey-Greenstein function (1.15). Then, these expansions over
the powers of s (in case of keeping the terms with the power equal to two) for the
constants in Eqs. 11.1–11.5 are looking as follows:
k ¼ 3ð1 À gÞs 1 þ s
2 1:5g À
1:2
1 þ g
m ¼ 8s 1 þ 6 À 7:5g þ
3:6
1 þ g
s
2
l ¼ 1 À 6q
0 s þ 18q
0 2 s
2
a
1
¼ 1 À 4s þ 12q
0 s
2
À 36q
0
À 6g À
1:608
1 þ g
s
3
n ¼ 1 À 3q
0 s þ 9q
0 2 À 3ð1 À gÞ À
2
1 þ g
s
2
:
(11.6)
For the functions in Eqs. 11.1–11.5 the followings expansions are correct:
KðmÞ ¼ K 0 ðmÞð1 À 3q
0 sÞ þ K 2 ðmÞs
2
aðmÞ ¼ 1 À 4K 0 ðmÞs þ a 2 ðmÞs
2
þ a 3 ðmÞs
3
r 1 ðm; m 0 Þ ¼ r 0 ðm; m 0 Þ À 4K 0 ðmÞK 0 ðm 0 Þs þ r 2 ðm; m 0 Þs
2
þ r 3 ðm; m 0 Þs
3
;
(11.7)
108
11 Calculating Solar Radiative Characteristics in Clouds with Asymptotic
divergence Eq. (1.5). It is computed with the obvious equation:
R ¼ 1 À F
"
ð0; m 0 Þ À ð1 À AÞF
#
ðt 0 ; m 0 Þ
¼ 1 À aðm 0 Þ þ
nKðm 0 Þme
Àkt 0
1 À l
le À2kt0
le
Àkt0
À
1 À A
1 À Aa 1
:
(11.5)
Mention that the term “asymptotic” specifies the light regime installed within the
cloud and it does not point out any approximation. Equations 11.1, 11.4 and 11.5
are strict within the diffusion domain. Their accuracy will be studied below
depending on the optical thickness.
11.2 Weak True Absorption of Solar Radiation
In clouds, the absorption is extremely weak compared to scattering (1 o 0 < <1)
within the short-wavelength range. As it has been shown in books in this case both
functions r 1 (m,m 0 ) and K(m) and constants m, l, k are expressed with the expansions
over powers of small parameter (1 À o 0 ). We consider here that parameter s, where
s
2 ¼ (1 À o 0 )/[3(1 À g)], is more convenient for the problem in question than
parameter (1 À o 0 ). Value g is a mean cosine of the scattering angle or, here, the
parameter of Henyey-Greenstein function (1.15). Then, these expansions over
the powers of s (in case of keeping the terms with the power equal to two) for the
constants in Eqs. 11.1–11.5 are looking as follows:
k ¼ 3ð1 À gÞs 1 þ s
2 1:5g À
1:2
1 þ g
m ¼ 8s 1 þ 6 À 7:5g þ
3:6
1 þ g
s
2
l ¼ 1 À 6q
0 s þ 18q
0 2 s
2
a
1
¼ 1 À 4s þ 12q
0 s
2
À 36q
0
À 6g À
1:608
1 þ g
s
3
n ¼ 1 À 3q
0 s þ 9q
0 2 À 3ð1 À gÞ À
2
1 þ g
s
2
:
(11.6)
For the functions in Eqs. 11.1–11.5 the followings expansions are correct:
KðmÞ ¼ K 0 ðmÞð1 À 3q
0 sÞ þ K 2 ðmÞs
2
aðmÞ ¼ 1 À 4K 0 ðmÞs þ a 2 ðmÞs
2
þ a 3 ðmÞs
3
r 1 ðm; m 0 Þ ¼ r 0 ðm; m 0 Þ À 4K 0 ðmÞK 0 ðm 0 Þs þ r 2 ðm; m 0 Þs
2
þ r 3 ðm; m 0 Þs
3
;
(11.7)
108
11 Calculating Solar Radiative Characteristics in Clouds with Asymptotic