Zdunkowski W., Trautmann T., Bott A. Radiation in the atmosphere: A course in Theoretical Meteorology
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2.6 Solution of RTE for a horizontally homogeneous atmosphere 55
z
τ
s
d
τ ′
u
g
u′
u′
du′
u
d
τ ′ du′
Top of the atmosphere
Reference level
Ground
τ ′
τ ′
τ
g
Fig. 2.4 Correspondences between the optical depth τ and the absorbing mass u.
of the medium and transmitted to level τ . The second term on the right-hand side of
each equation expresses the total contribution of all elementary layers of optical
thickness dτ
. The black body radiations B(τ
) are multiplied by the transmittance
from level τ
to the level of observation τ .
According to (1.40) and (1.41) for an absorbing medium the differential of the
optical depth dτ and the mass absorption coefficient κ
abs
are related by
dτ = k
abs
ds = κ
abs
du
=⇒ τ =
u
0
κ
abs
du
with du
= ρ
abs
ds =−ρ
abs
dz
(2.118)
Here, u refers to the absorbing mass between the top of the atmosphere and the
reference level, see Figure 2.4. This figure also depicts some other correspondences
between the optical depth and the absorbing mass which will be used below. Sub-
stituting (2.118) into (2.117) yields
I
+
(u,µ) = B
g
exp
−
1
µ
u
g
u
κ
abs
(u
)du
+
1
µ
u
g
u
B(u
)exp
−
1
µ
u
u
κ
abs
(t)dt
κ
abs
(u
)du
I
−
(u,µ) = I
−
(0,µ)exp
−
1
µ
u
0
κ
abs
(u
)du
+
1
µ
u
0
B(u
)exp
−
1
µ
u
u
κ
abs
(t)dt
κ
abs
(u
)du
(2.119)
56 The radiative transfer equation
These equations can also be reformulated as
I
+
(u,µ) = B
g
exp
−
1
µ
u
g
u
κ
abs
(u
)du
−
u
g
u
B(u
)
∂
∂u
exp
−
1
µ
u
u
κ
abs
(t)dt
du
I
−
(u,µ) = I
−
(0,µ)exp
−
1
µ
u
0
κ
abs
(u
)du
+
u
0
B(u
)
∂
∂u
exp
−
1
µ
u
u
κ
abs
(t)dt
du
(2.120)
For ease of notation we introduce the transmission function T via
T (u, u
,µ) = exp
−
1
µ
u
u
κ
abs
(t)dt
(2.121)
Now (2.120) assumes the form
I
+
(u,µ) = B
g
T (u, u
g
,µ) −
u
g
u
B(u
)
∂
∂u
T (u, u
,µ)du
I
−
(u,µ) = I
−
(0,µ)T (0, u,µ) +
u
0
B(u
)
∂
∂u
T (u
, u,µ)du
(2.122)
Partial integration of these equations with respect to u
yields the final result
I
+
(u,µ) =[B
g
− B(u
g
)]T (u, u
g
,µ)+B(u)+
u
g
u
dB
du
T (u, u
,µ)du
I
−
(u,µ) =[I
−
(0,µ) − B(0)]T (0, u,µ)+B(u)−
u
0
dB
du
T (u
, u,µ)du
(2.123)
Here, B(u
g
) describes the Planck radiation of the air temperature existing directly
above the ground. This means that the Earth’s surface itself may have a temperature
T
g
which is different from the air temperature T (u
g
). A similar situation applies to
the upper boundary of the atmosphere where the black body radiation B must be
evaluated at the temperature existing at u = 0. A radiation source from outside the
atmosphere may radiate with the intensity I
−
(0,µ).
An interesting interpretation can be found for the integral terms in (2.123).
Depending on the sign of dB/du
, positive or negative contributions are registered at
the reference level u. Consider, for example, the situation of a temperature inversion
at the reference level as depicted in Figure 2.5. In this case both integral terms will
make a positive contribution to the radiances at the level u.
2.7 Radiative flux densities and heating rates 57
u
T
du
′
du′
dB
du′
> 0
dB
du′
< 0
Reference level
Fig. 2.5 Contributions of the integral terms in (2.123) to the upward and downward
radiances.
In contrast to the formal solution of the RTE for the scattering atmosphere,
see (2.110), equation (2.123) describes the analytical solution for a nonscattering
atmosphere, provided the vertical distributions of the temperature and the absorption
coefficient are known. This is certainly a great advantage of (2.123) compared to
(2.110). However, as will be seen in later chapters, the solution of the RTE for the
purely absorbing atmosphere is still rather complicated. This is due to the fact that
the absorption coefficient strongly depends on the wavelength so that the integration
of (2.123) over a certain wavelength interval causes some problems.
2.7 Radiative flux densities and heating rates
For the computation of the radiative energy balance and the calculation of radiative
heating rates it is sufficient to determine the net radiative flux densities, see Sections
1.2 and 2.3.1. In a horizontally homogeneous atmosphere the radiance depends only
on the variables (z,µ,ϕ) or alternatively on (τ, µ,ϕ) but not on x and y. Hence,
only the vertical component of the net flux density needs to be considered. Since
for a purely absorbing atmosphere we have the analytical solution of the RTE, see
(2.123), it is interesting to determine the radiative flux densities and heating rates
for this particular case. However, before doing so we will again start the discussion
with the general situation of a scattering atmosphere.
2.7.1 The scattering atmosphere
It is convenient to rotate the Cartesian coordinate system in such a way that the
solar azimuth vanishes, that is ϕ
0
= 0, see Figure 2.3. According to (1.37c) and
58 The radiative transfer equation
(2.70) the z-component of the radiative net flux density is then given by
E
net,z
(τ ) =
2π
0
1
−1
∞
m=0
(2 − δ
0m
)I
m
(τ,µ)µ cos mϕ dµ dϕ (2.124)
Since
2π
0
cos mϕ dϕ = 2πδ
0m
(2.125)
we find that only the zeroth Fourier mode m = 0 of the radiance is required for
evaluating E
net,z
, i.e.
E
net,z
(τ ) = 2π
1
−1
I
0
(τ,µ)µdµ
(2.126)
This is a very important result since it shows that for heating rate computations
knowledge of the full directional dependence of the radiation field is not necessary.
Analogously to the radiance field, the term E
net,z
will be split into an upwelling
and a downwelling component by defining
E
net,z
= 2π
1
0
I
0
(τ,µ)µdµ − 2π
−1
0
I
0
(τ,µ)µdµ
= 2π
1
0
I
0
+
(τ,µ)µdµ − 2π
1
0
I
0
−
(τ,µ)µdµ = E
+,z
− E
−,z
(2.127)
Obviously, the upward and downward directed flux densities E
±,z
are positive
definite.
In a similar manner the net radiative flux densities for the x- and y-direction can
be computed. Utilizing (1.37a) one finds for E
net,x
E
net,x
(τ ) =
1
−1
∞
m=0
(2 − δ
0m
)I
m
(τ,µ)(1 − µ
2
)
1/2
dµ
2π
0
cos ϕ cos mϕ dϕ
(2.128)
Since
2π
0
cos ϕ cos mϕ dϕ = (1 + δ
01
)δ
1m
π = πδ
1m
(2.129)
we obtain
E
net,x
(τ ) = 2π
1
−1
(1 − µ
2
)
1/2
I
1
(τ,µ)dµ
(2.130)
Owing to the particular choice of the coordinate system with ϕ
0
= 0 and due
to the orthogonality of the trigonometric functions the y-component of the net
2.7 Radiative flux densities and heating rates 59
radiative flux density vanishes
E
net,y
(τ ) =
1
−1
∞
m=0
(2 − δ
0m
)I
m
(τ,µ)(1 − µ
2
)
1/2
dµ
2π
0
sin ϕ cos mϕ dϕ = 0
(2.131)
where now use was made of (1.37b). If the coordinate system is rotated about the
z-axis so that ϕ
0
= 0, then we obtain a nonvanishing value for E
net,y
. However, in
this case E
net,x
will assume a different value as well.
Obviously, for a horizontally homogeneous medium the net flux density does
not depend on the horizontal variables x and y. Hence we may write
∇·E
net
=
∂ E
net,x
∂x
+
∂ E
net,y
∂y
+
∂ E
net,z
∂z
=
∂ E
net,z
∂z
(2.132)
According to (2.46) for the horizontally homogeneous medium the radiatively
induced temperature change due to the diffuse radiation at height z is then given
by
∂T
∂t
rad,dif
=−
1
c
p
ρ(z)
∂ E
net,z
∂z
=
k
ext
c
p
ρ(z)
∂ E
net,z
∂τ
=
k
ext
c
p
ρ(z)
∂ E
+,z
∂τ
−
∂ E
−,z
∂τ
(2.133)
where k
ext
is the volume extinction coefficient as defined in (1.48) and ρ is the
density of the air. Recall, however, that owing to the splitting of the radiance field
into diffuse and direct radiation, see (2.29), (2.133) represents only the contribution
of the diffuse radiation. The total temperature change by radiative effects is obtained
by adding to ∂ T/∂t|
rad,dif
the contribution by the direct solar radiation. Substituting
(2.29) together with (2.52) into (1.37c) yields the direct solar net flux density
E
net,z,dir
=
2π
0
1
−1
S(τ )δ(Ω − Ω
0
)µ dµ dϕ =−µ
0
S
0
exp
−
τ
µ
0
(2.134)
where δ(Ω − Ω
0
) = δ(µ − µ
0
)δ(ϕ − ϕ
0
). Hence, analogously to (2.133), the tem-
perature change due to the extinction of solar radiation is given by
∂T
∂t
rad,dir
=
k
ext
c
p
ρ(z)
∂ E
net,z,dir
∂τ
=
k
ext
c
p
ρ(z)
S
0
exp
−
τ
µ
0
(2.135)
Combining (2.133) and (2.135) we finally obtain
∂T
∂t
rad,ν
=
∂T
∂t
rad,dif,ν
+
∂T
∂t
rad,dir,ν
=
k
ext,ν
c
p
ρ(z)
∂ E
+,z,ν
∂τ
−
∂ E
−,z,ν
∂τ
+ S
0,ν
exp
−
τ
µ
0
(2.136)
60 The radiative transfer equation
Here, we have again included the subscript ν in order to remind the reader that
this is a spectral equation. Hence, the total radiative heating rate is obtained by
integrating (2.136) over the entire spectral region. If the vertical net flux density of
the total radiation including the parallel solar radiation is constant with height then
the radiative temperature change vanishes.
2.7.2 The nonscattering atmosphere
For the nonscattering atmosphere the upward and downward directed flux densities
are obtained by integrating (2.123) over the upper and lower hemisphere. This gives
E
+
(u) =
2π
0
1
0
I
+
(u,µ)µdµ dϕ
= 2π [B
g
− B(u
g
)]
1
0
T (u, u
g
,µ)µ dµ
+ π B(u) + 2π
u
g
u
dB
du
1
0
T (u, u
,µ)µdµdu
E
−
(u) =
2π
0
1
0
I
−
(u,µ)µdµdϕ
= 2π [I
−
(0) − B(0)]
1
0
T (0, u,µ)µdµ
+ π B(u) − 2π
u
0
dB
du
1
0
T (u
, u,µ)µdµ du
(2.137)
where for simplicity we have assumed that the radiation field incident at the top of
the atmosphere is isotropic. It is convenient to define the so-called flux-transmission
function T
f
via
T
f
(u, u
) = 2
1
0
T (u, u
,µ)µdµ (2.138)
Substituting (2.121) into this equation yields
T
f
(u, u
)=2
1
0
exp
−
1
µ
u
u
κ
abs
(t)dt
µdµ =2
∞
1
ξ
−3
exp(−ξ x)dξ (2.139)
where the following substitutions have been utilized
ξ =
1
µ
, dξ =−
1
µ
2
dµ, x =
u
u
κ
abs
(t)dt (2.140)
2.8 Appendix 61
The last integral expression in (2.139) allows for the introduction of the exponential
integral of third order
E
3
(x) =
∞
1
ξ
−3
exp(−ξ x)dξ (2.141)
Thus we obtain
T
f
(u, u
) = 2E
3
u
u
κ
abs
(t)dt
(2.142)
Substitution of the flux-transmission function into (2.137); gives finally
E
+
(u) = π[B
g
− B(u
g
)]T
f
(u, u
g
) + π B(u) + π
u
g
u
dB
du
T
f
(u, u
)du
E
−
(u) = π[I
−
(0) − B(0)]T
f
(0, u) + π B(u) − π
u
0
dB
du
T
f
(u
, u)du
(2.143)
These expressions may be used in (2.133) to obtain the infrared contributions of the
radiative heating rates whereby the substitution du =−ρ
abs
dz has to be applied.
2.8 Appendix
2.8.1 Local thermodynamic equilibrium
Assuming conditions of local thermodynamic equilibrium, the source function J
e
ν
for infrared emission should be proportional to the Planck black body function
B
ν
. We will now attempt to determine the proportionality factor between the source
function and the Planck function for a simple physical situation. The result, however,
is more general. From (2.36) we obtain
Ω·∇I
ν
=
d
ds
I
ν
(r, Ω) =−k
ext,ν
I
ν
(r, Ω) +
k
sca,ν
4π
4π
P
ν
(Ω
· Ω)I
ν
(r, Ω
)d
+ J
e
ν
(r)
(2.144)
Here, we have omitted the primary scattering term of the solar radiation since in
the range of infrared emission the solar radiation is practically zero for atmospheric
conditions in the troposphere and lower stratosphere. Integration of (2.144) over
the unit sphere, assuming isotropic and homogeneous radiation, gives
4π
dI
ν
(r)
ds
d =−k
ext,ν
4π
I
ν
(r)d +
k
sca,ν
4π
4π
I
ν
(r)
4π
P
ν
d
d +
4π
J
e
ν
(r)d
=−k
abs,ν
4π
I
ν
(r)d +
4π
J
e
ν
(r)d = 0
(2.145)
62 The radiative transfer equation
where the integral of the phase function over unit sphere has been evaluated accord-
ing to (1.46). From (2.145) we obtain for the source function
J
e
ν
(r) = k
abs,ν
I
ν
(r) = k
abs,ν
B
ν
(r)
(2.146)
Since J
e
ν
(r) is the source function for true emission of heat radiation we set I
ν
= B
ν
.
(2.146) is known as Kirchhoff’s law. In case that the conditions of local thermody-
namic equilibrium are not met, the emission of radiative energy becomes a function
of the energy states in the gas. Equation (2.146) is not an exact derivation but a
plausible discussion.
2.9 Problems
2.1: Find the normalization constant C for the phase function P(cos ) = C(1 +
cos
2
). Plot your result in polar coordinates and versus P(cos ).
2.2: Obtain the solution to the RTE in the form (2.76) for radiative transfer within
a plane–parallel cloud of total optical thickness τ
c
. Assume that no diffuse
radiation is incident on the cloud either from above or from below. Ignore
the multiple scattering term and the Planckian emission.
2.3: Consider the following special cases for the solution you obtained from
Problem 2.2
(a) Find I
m
−
(τ,µ = µ
0
).
(b) Find I
m
+
(τ,0) and I
−
(τ,0) for τ = 0 and τ = τ
c
.
(c) Find I
m
+
(τ,µ) and I
m
−
(τ,µ) for τ = 0 and µ = 0.
2.4: In the radiative transfer theory we have to deal with integrals of the type
τ
c
τ
exp
−
τ
− τ
µ
A(τ
)
dτ
µ
and
τ
0
exp
−
τ − τ
µ
A(τ
)
dτ
µ
Evaluate these integrals for µ = 0 where A(τ
) is an arbitrary function.
2.5: Assume isothermal conditions for the system Earth–atmosphere.
(a) From (2.117) find I
+
(τ,µ).
(b) Which additional condition must apply to obtain the same result for I
−
(τ,µ)?
(c) Show that the identical results follow from (2.123).
2.6: The exponential integral of order n is defined by
E
n
(x) =
∞
1
exp(−ξ x)
ξ
n
dξ
(a) Find E
n
(0) for n = 1, 2,...
(b) Find dE
n
/dx.
2.9 Problems 63
(c) Verify
nE
n+1
(x) = exp(−x) − xE
n
(x) = exp(−x) + x
dE
n+1
(x)
dx
, n = 1, 2,...
(d) Verify E
n
(x) =
1
0
exp
−
x
µ
µ
n−2
dµ.
2.7: According to (2.28) for a plane–parallel atmosphere the RTE can be written
in the form
µ
d
dτ
I (τ,µ,ϕ) = I (τ,µ,ϕ) −
1
4π
2π
0
1
−1
P(µ, ϕ, µ
,ϕ
)dµ
dϕ
if ω
0
= 1 and J
e
= 0. In this case the net flux E
net
is a constant.
Show that the solution to this equation can be written as
K (τ ) =
E
net
4π
1 −
p
1
3
τ + C
where C is a constant and K is given by
K (τ ) =
1
4π
2π
0
1
−1
I (τ,µ,ϕ)µ
2
dµdϕ
Hint: Expand the phase function according to (2.55) and make use of the
addition theorem (2.66).
2.8: Consider the RTE in the form stated in Problem 2.7. Assume that the phase
function is given by P = 3/4(1 +cos
2
). Find the source function for the
azimuthally averaged RTE.
2.9: Consider the RTE as given in Problem 2.7. For a semi-infinite plane–parallel
atmosphere and the boundary condition I (0, −µ) = 0.
(a) Find the solution to the azimuthally averaged RTE in case of isotropic scattering.
(b) Introduce the solution found under (a) into the integral I
n
(τ ) =
1
−1
I (τ,µ)µ
n
dµ.
Finally, introduce the exponential integral defined in Problem 2.6 into the latter
integral.
(c) Find the net flux density in terms of the exponential integral.
3
Principles of invariance
The principle of invariance in the original form was stated by Ambartsumian (1942)
expressing the invariance of the diffusely reflected radiation emerging from a semi-
infinite atmosphere to the addition or subtraction of an infinitely thin atmospheric
layer. Chandrasekhar (1960) advanced the original form and stated four general
principles of invariance which apply to finite atmospheric layers. These principles
are not based on the radiative transfer equation, but they are of equal physical valid-
ity. We accept Goody’s (1964a) statement that the principles of invariance may be
viewed as a series of common-sense relations between the scattering and transmis-
sion functions with the radiances emerging from the upper and lower boundaries
of an atmospheric layer and at some intermediate variable level.
3.1 Definitions of the scattering and transmission functions
Let us consider a plane–parallel atmospheric layer of vertical optical thickness τ
1
bounded on both sides by a vacuum, see Figure 3.1. The upper boundary of this
layer is illuminated by a beam of parallel downward directed radiation S
0
, while
at τ = τ
1
no radiation is incident in the upward direction. For simplicity, only
short-wave radiation will be considered. However, inclusion of infrared radiation
causes no particular difficulties. We call this situation the restricted or standard
problem. We define the scattering function S(τ
1
,µ,ϕ,µ
0
,ϕ
0
) and the transmission
function T (τ
1
,µ,ϕ,µ
0
,ϕ
0
) by formulating the reflected and transmitted radiances
I (0,µ,ϕ) and I (τ
1
, −µ, ϕ)as
I (0,µ,ϕ) =
S
0
4πµ
S(τ
1
,µ,ϕ,µ
0
,ϕ
0
)
I (τ
1
, −µ, ϕ) =
S
0
4πµ
T (τ
1
,µ,ϕ,µ
0
,ϕ
0
)
(3.1)
64