It is to be pointed out that the function Pðt; #Þ ¼ expðÀt 0 = cos # 0 Þ is called the
transmission function and used in the calculation of the heat radiation.
Return to the general case of the transfer equation and taking into account
scattering (1.19). Accomplish the transformation to the dimensionless parameters
in the transfer equation for convenience of further analysis. In accordance with
optical thickness definition (1.21) the function t(z) is monotonically decreasing
with altitude that follows from condition a(z
0 ) > 0. In this case there is an inverse
function z(t) that is also decreasing monotonically. Using the formal substitution of
function z(t) rewrite the transfer equation and pass from vertical coordinate t to
coordinate z, moreover, the boundary condition is at the top of the atmosphere
t ¼ 0 and at the bottom t ¼ t 0 , and the direction of axis t is opposite to axis z.
It follows from the definition (1.21): dt ¼ Àa(z)dz. Specify m ¼ cosϑ and pass
from the zenith angle to its cosine (the formal substitution ϑ ¼ arccosm with taking
into account sinϑdϑ ¼ Àdm). Finally, divide both parts of the equation to value
a(t), and obtain instead Eq. 1.19 the following equation:
m
dIðt; m; ’Þ
dt
¼ ÀIðt; m; ’Þ þ
o 0 ðtÞ
4p
ð
2p
0
d’
0
ð 1
À1
xðt;gÞIðt; m
0
; ’
0
Þdm
0
;
(1.24)
where
o 0 ðtÞ ¼
sðtÞ
aðtÞ ¼
sðtÞ
sðtÞþkðtÞ , and the scattering angle cosine cos g ¼ mm
0
þ
ffiffiffiffiffiffiffiffiffiffiffiffiffi
1 À m 2
p
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 À m 02
p
cosð’ À ’
0
Þ.
Dimensionless value o 0 is called the single scattering albedo or otherwise the
probability of the quantum surviving per the single scattering event. If there is no
absorption (k ¼ 0) then the case is called conservative scattering, o 0 ¼ 1. If the
scattering is absent then the extinction is caused only by absorption, s ¼ 0, o 0 ¼ 0
and the solution of the transfer equation is reduced to Beer’s law. After consideration of these cases, the sense of value o 0 is following: it defines the part of
scattered radiation relatively to the total extinction, and corresponds to the probability of the quantum to survive and accepts the quantum absorption as its “death”.
It is necessary to specify the boundary conditions at the top and bottom of the
atmosphere. As it has been mentioned above, solar radiation characterizing with
values F 0 , ϑ 0 , ’ 0 incomes to the top. Usually it is assumed ’ 0 ¼ 0 i.e. all azimuths
are counted off the solar azimuth and specified m 0 ¼ cos# 0 .
As it has been mentioned above solar radiation in the Earth atmosphere consists
of direct and scattered radiation. It is accepted not to include the direct radiation to
the transfer equation and to write the equation only for the scattered one. The
calculation of the direct radiation is elementary accomplished using Beer’s Law
(1.23). Therefore, present the radiance as a sum of direct and scattered radiance
I(t,m,’) ¼ I
0 (t,m,’) + I
00 (t,m,’). From the expression for the direct radiance of the
parallel beam (1.10) the following is correct I
0 (0,m,’) ¼ F 0 d(m – m 0 )d(’ – 0), and
it leads to I
0 (0,m,’) ¼ F 0 d(m – m 0 )d(’)exp(Àt/m 0 ) for Beer’s Law. Substitute the
above sum to Eq. 1.24, with introducing the dependence upon value m 0 and omitting
primes I
00 (t,m,m 0 ,’), we are obtaining the transfer equation for diffuse radiation.
16
1 Radiation in the Earth Atmosphere
transmission function and used in the calculation of the heat radiation.
Return to the general case of the transfer equation and taking into account
scattering (1.19). Accomplish the transformation to the dimensionless parameters
in the transfer equation for convenience of further analysis. In accordance with
optical thickness definition (1.21) the function t(z) is monotonically decreasing
with altitude that follows from condition a(z
0 ) > 0. In this case there is an inverse
function z(t) that is also decreasing monotonically. Using the formal substitution of
function z(t) rewrite the transfer equation and pass from vertical coordinate t to
coordinate z, moreover, the boundary condition is at the top of the atmosphere
t ¼ 0 and at the bottom t ¼ t 0 , and the direction of axis t is opposite to axis z.
It follows from the definition (1.21): dt ¼ Àa(z)dz. Specify m ¼ cosϑ and pass
from the zenith angle to its cosine (the formal substitution ϑ ¼ arccosm with taking
into account sinϑdϑ ¼ Àdm). Finally, divide both parts of the equation to value
a(t), and obtain instead Eq. 1.19 the following equation:
m
dIðt; m; ’Þ
dt
¼ ÀIðt; m; ’Þ þ
o 0 ðtÞ
4p
ð
2p
0
d’
0
ð 1
À1
xðt;gÞIðt; m
0
; ’
0
Þdm
0
;
(1.24)
where
o 0 ðtÞ ¼
sðtÞ
aðtÞ ¼
sðtÞ
sðtÞþkðtÞ , and the scattering angle cosine cos g ¼ mm
0
þ
ffiffiffiffiffiffiffiffiffiffiffiffiffi
1 À m 2
p
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 À m 02
p
cosð’ À ’
0
Þ.
Dimensionless value o 0 is called the single scattering albedo or otherwise the
probability of the quantum surviving per the single scattering event. If there is no
absorption (k ¼ 0) then the case is called conservative scattering, o 0 ¼ 1. If the
scattering is absent then the extinction is caused only by absorption, s ¼ 0, o 0 ¼ 0
and the solution of the transfer equation is reduced to Beer’s law. After consideration of these cases, the sense of value o 0 is following: it defines the part of
scattered radiation relatively to the total extinction, and corresponds to the probability of the quantum to survive and accepts the quantum absorption as its “death”.
It is necessary to specify the boundary conditions at the top and bottom of the
atmosphere. As it has been mentioned above, solar radiation characterizing with
values F 0 , ϑ 0 , ’ 0 incomes to the top. Usually it is assumed ’ 0 ¼ 0 i.e. all azimuths
are counted off the solar azimuth and specified m 0 ¼ cos# 0 .
As it has been mentioned above solar radiation in the Earth atmosphere consists
of direct and scattered radiation. It is accepted not to include the direct radiation to
the transfer equation and to write the equation only for the scattered one. The
calculation of the direct radiation is elementary accomplished using Beer’s Law
(1.23). Therefore, present the radiance as a sum of direct and scattered radiance
I(t,m,’) ¼ I
0 (t,m,’) + I
00 (t,m,’). From the expression for the direct radiance of the
parallel beam (1.10) the following is correct I
0 (0,m,’) ¼ F 0 d(m – m 0 )d(’ – 0), and
it leads to I
0 (0,m,’) ¼ F 0 d(m – m 0 )d(’)exp(Àt/m 0 ) for Beer’s Law. Substitute the
above sum to Eq. 1.24, with introducing the dependence upon value m 0 and omitting
primes I
00 (t,m,m 0 ,’), we are obtaining the transfer equation for diffuse radiation.
16
1 Radiation in the Earth Atmosphere