F
"
ð0; m 0 Þ ¼ 1 À
4K 0 ðm 0 Þ
3 ð1 À gÞt 0 þ 2q 0 þ
4A
3ð1ÀAÞ
h
i ;
F
#
ðt 0 ; m 0 Þ ¼
4AK 0 ðm 0 Þ
3ð1 À AÞ ð1 À gÞt 0 þ 2q 0 þ
4A
3ð1ÀAÞ
h
i ;
(11.22)
and, finally, the simple expression for the net flux that summarizes both Eq. 11.22 is
feasible at any level in the conservative medium because the net flux is constant
without absorption
Fðt; m 0 Þ ¼
4K 0 ðm 0 Þð1 À AÞ
3ð1 À AÞ ð1 À gÞt 0 þ d
½
þ 4A
:
(11.23)
It should be emphasized that the equality F
"
(t,m 0 ) ¼ F
# (t,m 0 ) ¼ K 0 (m 0 ) is correct
in the semi-infinite conservatively scattered atmosphere at the big optical depth,
where the sense of the escape function K 0 (m), frequently met in our consideration,
is clear from. The case of the conservative scattering comes true in a certain cloud
layer at the some wavelengths within the visual spectral range. Equations 11.21–11.22
are correct in the wider interval of the optical depth (t 0 ! 3) than Eqs. 11.1 and 11.4
derived with taking into account the absorption. Corresponding relations of the
characteristics of the inner radiation field are written as:
for the radiance:
Iðt; mÞ ¼
F 0 m 0 K 0 ðm 0 Þ ð1 À AÞ 3ð1 À gÞðt 0 À tÞ þ 3q
0
þ 3m
½
þ 4A
f
g
p ð1 À AÞ 3ð1 À gÞt 0 þ 6q 0
½
þ 4A
f
g
; (11.24)
for the upward and downward semispherical solar fluxes:
F
"
ðt; m 0 Þ ¼ K 0 ðm 0 Þ
ð1 À AÞ 3ð1 À gÞðt 0 À tÞ þ 3q
0
À 2
½
þ 4A
ð1 À AÞ 3ð1 À gÞt 0 þ 6q 0
½
þ 4A
F
#
ðt; m 0 Þ ¼ K 0 ðm 0 Þ
ð1 À AÞ 3ð1 À gÞðt 0 À tÞ þ 3q
0
þ 2
½
þ 4A
ð1 À AÞ 3ð1 À gÞt 0 þ 6q 0
½
þ 4A
:
(11.25)
The formulas of the radiation characteristics in case of conservative scattering
are possible to apply for the rough estimation even for the very weak absorption but
the computational errors increase fast when the absorption grows and it is necessary
to use the equations for the absorption medium to reach certain accuracy.
11.6 Error Analysis
Asymptotic formulas of the radiation transfer theory presented in this chapter are
obtained strictly. It is necessary to take into consideration that they are describing
the radiation field within the cloud layer and at the cloud top and base boundaries
the more exact, the bigger the optical thickness and the weaker true absorption.
114
11 Calculating Solar Radiative Characteristics in Clouds with Asymptotic
"
ð0; m 0 Þ ¼ 1 À
4K 0 ðm 0 Þ
3 ð1 À gÞt 0 þ 2q 0 þ
4A
3ð1ÀAÞ
h
i ;
F
#
ðt 0 ; m 0 Þ ¼
4AK 0 ðm 0 Þ
3ð1 À AÞ ð1 À gÞt 0 þ 2q 0 þ
4A
3ð1ÀAÞ
h
i ;
(11.22)
and, finally, the simple expression for the net flux that summarizes both Eq. 11.22 is
feasible at any level in the conservative medium because the net flux is constant
without absorption
Fðt; m 0 Þ ¼
4K 0 ðm 0 Þð1 À AÞ
3ð1 À AÞ ð1 À gÞt 0 þ d
½
þ 4A
:
(11.23)
It should be emphasized that the equality F
"
(t,m 0 ) ¼ F
# (t,m 0 ) ¼ K 0 (m 0 ) is correct
in the semi-infinite conservatively scattered atmosphere at the big optical depth,
where the sense of the escape function K 0 (m), frequently met in our consideration,
is clear from. The case of the conservative scattering comes true in a certain cloud
layer at the some wavelengths within the visual spectral range. Equations 11.21–11.22
are correct in the wider interval of the optical depth (t 0 ! 3) than Eqs. 11.1 and 11.4
derived with taking into account the absorption. Corresponding relations of the
characteristics of the inner radiation field are written as:
for the radiance:
Iðt; mÞ ¼
F 0 m 0 K 0 ðm 0 Þ ð1 À AÞ 3ð1 À gÞðt 0 À tÞ þ 3q
0
þ 3m
½
þ 4A
f
g
p ð1 À AÞ 3ð1 À gÞt 0 þ 6q 0
½
þ 4A
f
g
; (11.24)
for the upward and downward semispherical solar fluxes:
F
"
ðt; m 0 Þ ¼ K 0 ðm 0 Þ
ð1 À AÞ 3ð1 À gÞðt 0 À tÞ þ 3q
0
À 2
½
þ 4A
ð1 À AÞ 3ð1 À gÞt 0 þ 6q 0
½
þ 4A
F
#
ðt; m 0 Þ ¼ K 0 ðm 0 Þ
ð1 À AÞ 3ð1 À gÞðt 0 À tÞ þ 3q
0
þ 2
½
þ 4A
ð1 À AÞ 3ð1 À gÞt 0 þ 6q 0
½
þ 4A
:
(11.25)
The formulas of the radiation characteristics in case of conservative scattering
are possible to apply for the rough estimation even for the very weak absorption but
the computational errors increase fast when the absorption grows and it is necessary
to use the equations for the absorption medium to reach certain accuracy.
11.6 Error Analysis
Asymptotic formulas of the radiation transfer theory presented in this chapter are
obtained strictly. It is necessary to take into consideration that they are describing
the radiation field within the cloud layer and at the cloud top and base boundaries
the more exact, the bigger the optical thickness and the weaker true absorption.
114
11 Calculating Solar Radiative Characteristics in Clouds with Asymptotic