Zdunkowski W., Trautmann T., Bott A. Radiation in the atmosphere: A course in Theoretical Meteorology
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2.2 The direct–diffuse splitting of the radiance field 35
Using the boundary condition S
ν
(s = 0) = S
0,ν
, where S
0,ν
is the solar radiation at
the top of the atmosphere, the solution of (2.31) is
S
ν
(s) = S
0,ν
exp
−
s
0
k
ext,ν
(s
)ds
, Ω = Ω
0
(2.32)
S
0,ν
is also called the solar constant or the extraterrestrial solar flux. The expo-
nential function occurring in (2.32) is also known as the transmission function
since it determines the fraction of radiation which is transmitted through the path
s. Obviously, the transmission function is bounded by 0 (no transmission) and 1
(total transmission).
The integral expression on the right-hand side of (2.28) contains the contributions
due to multiple scattering processes. Introducing into this term the definition (2.29)
yields
k
sca,ν
4π
4π
P
ν
(Ω
· Ω)I
ν
(Ω
)d
=
k
sca,ν
4π
4π
P
ν
(Ω
· Ω)I
d,ν
(Ω
)d
+
k
sca,ν
4π
4π
P
ν
(Ω
· Ω)S
ν
δ(Ω
− Ω
0
)d
(2.33)
The last expression in this equation represents the direct solar radiation scattered
from Ω
0
into the direction Ω. Hence, we may also write
4π
P
ν
(Ω
· Ω)S
ν
δ(Ω
− Ω
0
)d
= P
ν
(Ω
0
· Ω)S
ν
(2.34)
We are now ready to obtain the RTE for the diffuse radiation. Substituting (2.29),
(2.31), and (2.33) together with (2.34) into (2.28) gives
Ω ·∇I
d,ν
− k
ext,ν
S
ν
δ(Ω − Ω
0
) = J
e
ν
−k
ext,ν
I
d,ν
− k
ext,ν
S
ν
δ(Ω − Ω
0
)
+
k
sca,ν
4π
4π
P
ν
(Ω
· Ω)I
d,ν
(Ω
)d
+
k
sca,ν
4π
P
ν
(Ω
0
· Ω)S
ν
(2.35)
It can been seen that the two terms involving the direct solar light cancel. Thus,
after introducing the direct–diffuse splitting (2.29), the final version of the RTE for
the stationary diffuse radiation field of a three-dimensional medium reads
Ω ·∇I
d,ν
=−k
ext,ν
I
d,ν
+
k
sca,ν
4π
4π
P
ν
(Ω
· Ω)I
d,ν
(Ω
)d
+
k
sca,ν
4π
P
ν
(Ω
0
· Ω)S
ν
+ J
e
ν
(2.36)
The third term on the right-hand side describes the generation of diffuse light from
the unscattered part of the direct sunlight.
36 The radiative transfer equation
2.3 The radiatively induced temperature change
According to (1.34) the net flux density vector is obtained by taking the first moment
of the radiance with respect to Ω
E
net,ν
=
4π
ΩI
ν
d (2.37)
The total net flux density vector follows from an integration of this equation over
the entire electromagnetic spectrum
E
net
=
∞
0
E
net,ν
dν =
∞
0
4π
ΩI
ν
d dν (2.38)
Since ∇·Ω = 0 the RTE for the total (diffuse plus direct) radiation field (2.28) can
be written as
∇·(ΩI
ν
) + k
ext,ν
I
ν
=
k
sca,ν
4π
4π
P
ν
(Ω
· Ω)I
ν
(Ω
)d
+ J
e
ν
(2.39)
Employing the integral operator
4π
d... to the above equation we find in view
of (2.37)
∇·E
net,ν
+ k
ext,ν
4π
I
ν
d =
k
sca,ν
4π
4π
4π
P
ν
(Ω
· Ω)I
ν
(Ω
)d d
+ 4π J
e
ν
(2.40)
since the source function J
ν
was assumed to be isotropic. Applying the normaliza-
tion condition for the scattering phase function
1
4π
4π
P
ν
(Ω
· Ω)d = 1 (2.41)
to the integral on the right-hand side, we obtain for the divergence of the net flux
density vector
∇·E
net,ν
= k
sca,ν
4π
I
ν
d − k
ext,ν
4π
I
ν
d + 4π J
e
ν
=−k
abs,ν
4π
I
ν
d + 4π J
e
ν
(2.42)
If the medium does not contain any true interior sources (J
e
ν
= 0), then the vector
of the net flux density is given by
∇·E
net,ν
=−k
abs,ν
4π
I
ν
d<0 (2.43)
Whenever ∇·E
net,ν
= 0 the medium is said to be in radiative equilibrium for the
frequency ν. The concept of radiative equilibrium will be discussed in more detail
in a later chapter.
2.4 The RTE for a horizontally homogeneous atmosphere 37
In the absence of frictional effects the first law of thermodynamics can be written
as
de
dt
+ p
dv
dt
=
dh
dt
−
1
ρ
dp
dt
=−
1
ρ
∇·(J
q
+ E
net
) (2.44)
see e.g. Chapter 3 of THD (2004). Here, v = 1/ρ is the specific volume of the air
with density ρ. The quantities e, h, p and J
q
stand for the specific internal energy,
the specific enthalpy, the total pressure, and the heat flux, respectively. Here we
are interested only in the contribution of radiative processes to the atmospheric
temperature change. For an isobaric process (dp = 0) enthalpy and temperature
changes are related by
dh
dt
= c
p
dT
dt
=−
1
ρ
∇·E
net
(2.45)
where c
p
is the specific heat at constant pressure. The local time rate of change of
the temperature caused by radiative processes alone is then given by
∂T
∂t
rad,ν
=−
1
ρc
p
∇·E
net,ν
=−
1
ρc
p
−k
abs,ν
4π
I
ν
d + 4π J
e
ν
(2.46)
From this equation the following conclusions are drawn.
(i) The first term on the right-hand side describes the absorption of photons. Since this
term is never negative it causes local warming.
(ii) The second term on the right-hand side describing the emission of photons is never
positive, thus resulting in local cooling.
(iii) In the absence of absorption and emission no radiatively induced temperature changes
take place.
2.4 The radiative transfer equation for a horizontally
homogeneous atmosphere
The simplest geometry for a scattering and absorbing medium is the so-called plane–
parallel approximation, where in the horizontal direction the medium stretches
to infinity. In such a homogeneous plane–parallel slab all optical properties are
independent of the horizontal position. Moreover, the incident radiation, including
the parallel solar beam, is assumed to be independent of the horizontal coordinates
along the upper and lower boundaries of the atmosphere. In many cases the plane–
parallel assumption represents a good approximation to a planetary atmosphere.
It is important to note that the plane–parallel approximation to the RTE is best
whenever the vertical variations of all radiative quantities dominate over the hori-
zontal variability which is often the case. Two specific examples for which the
plane–parallel theory is inadequate are: (i) radiative transfer in finite clouds located
38 The radiative transfer equation
S
0
s =0
s
Reference level
ϑ
ϑ
ϑ
0
ϑ
x
y
z
µ =1
µ = −1
µ =0
µ > 0
µ <
0
z
dz
ds
ϕ
0
Fig. 2.3 Illustration of the plane–parallel geometry. Upwelling radiation: µ>0.
Downwelling radiation: µ<0. The zenith (µ = 1), nadir (µ =−1), and horizon-
tal (µ = 0) directions are indicated. Solar radiation is incident from (ϑ
0
,ϕ
0
).
over a heterogeneously reflecting ground; and (ii) radiative transfer in a spherical
atmosphere for solar positions near or below the horizon.
In the following we derive the RTE for a plane–parallel medium. For ease of
notation the subscript ν will henceforth be omitted from all radiative quantities.
We start with the RTE for the diffuse radiation in the form (2.36) (omitting for
simplicity the subscript d) and using Ω ·∇I = dI/ds,
1
k
ext
dI(s)
ds
=−I (s) +
ω
0
4π
4π
P(Ω
· Ω)I (s, Ω
)d
+
ω
0
4π
P(Ω·Ω
0
)S(s) +
1
k
ext
J
e
(s) (2.47)
For a black body the emission source function is given by Kirchhoff’s law
J
e
(s) = k
abs
B(s) with
k
abs
k
ext
= 1 − ω
0
(2.48)
where B(s) = B(T (s)) is the Planck function which depends on frequency and
local temperature. The form of (2.48) will be motivated in the Appendix to this
chapter.
In a plane–parallel medium the only spatial variable is the altitude z. The direction
of the radiation is defined by the angle ϑ with respect to the z-axis and by the
azimuth angle ϕ counted from an arbitrary origin. As illustrated in Figure 2.3, the
path element ds is related to dz by
ds =
dz
µ
with µ = cos ϑ (2.49)
2.4 The RTE for a horizontally homogeneous atmosphere 39
From the figure it is also seen that for upward directed radiance µ>0 while for
downward directed radiance µ<0. At an arbitrary reference level the horizontal
direction is characterized by µ = 0. Radiation propagating in the upward and down-
ward vertical direction is specified by µ = 1 and µ =−1, respectively. However,
in order to simplify the notation, for the downward directed radiance with ϑ>90
◦
the cosine of the zenith angle will henceforth be denoted by −µ so that for arbitrary
directions of the radiance µ is positive definite.
The direct solar radiation is expressed by
S(s) = S
0
exp
−
s
0
k
ext
(s
)ds
= S
0
exp
−
1
µ
0
∞
z
k
ext
(z
)dz
(2.50)
where according to (2.49) and due to the fact that θ
0
> 90
◦
, the path increment
ds
has been replaced by −dz
/µ
0
. The integration of the direct beam starts at
the upper boundary s = 0 of the medium and proceeds along a straight path of
length s.
The optical depth τ is defined as the integral of the extinction coefficient over
height along a path perpendicular to the horizontal plane, i.e.
τ =
∞
z
k
ext
(z
)dz
or dτ =−k
ext
dz
(2.51)
Since according to (2.49) and (2.51) k
ext
ds = k
ext
dz/µ =−dτ/µ we obtain for
the solar radiation
S(τ ) = S
0
exp
−
τ
µ
0
with µ
0
> 0 (2.52)
With these results one finds for the plane–parallel approximation of the RTE the
integro-differential equation
µ
d
dτ
I (τ,µ,ϕ) = I (τ,µ,ϕ) −
ω
0
4π
2π
0
1
−1
P(cos )I (τ,µ
,ϕ
)dµ
dϕ
−
ω
0
4π
P(cos
0
)S
0
exp
−
τ
µ
0
− (1 − ω
0
)B(τ )
(2.53)
We repeat that the cosine of the scattering angle is given by cos = Ω
· Ω, which
is a function of the four angles ϑ
,ϕ
,ϑ,ϕ. An expression for cos will be derived
later.
40 The radiative transfer equation
The scattering phase function P(cos ) is assumed to be rotationally symmetric
along the direction of the incident light, an assumption that is exact for spherical
scattering particles. The scattering of light by homogeneous spheres is rigorously
described by the so-called Mie theory which will be thoroughly discussed in a
later chapter. The scattering properties of a single particle are solely a function of
two physical parameters; the complex index of refraction N = n + i κ,
1
and the
so-called Mie size parameter
x =
2πr
λ
(2.54)
where r is the radius of the scattering sphere and λ is the wavelength of the incident
electromagnetic wave. The schematic illustration of the rotationally symmetric form
of P(cos ) shown in Figure 1.18 indicates a strong forward peak of the scattering
phase function. This is typical of cloud droplets and spherical aerosol particles when
they are illuminated by solar radiation. Having specified the size parameter and the
index of refraction, the Mie theory provides a table of numbers of P versus .
In order to give a mathematical description of P, the scattering phase function
will be expanded as an infinite series of Legendre polynomials P
l
(cos )
P(cos ) =
∞
l=0
p
l
P
l
(cos )
(2.55)
For practical purposes the phase function will be truncated after a finite number of
terms.
The Legendre polynomials for argument x, with −1 ≤ x ≤ 1, are defined by
P
l
(x) =
1
2
l
l!
d
l
dx
l
x
2
− 1
l
(2.56)
It will be readily seen that the first three polynomials are given by
P
0
(x) = 1, P
1
(x) = x, P
2
(x) =
3
2
x
2
−
1
2
(2.57)
The Legendre polynomials are a special case of the more general associated
Legendre polynomials P
m
l
(x) as defined by
P
m
l
(x) =
(1 − x
2
)
m/2
2
l
l!
d
l+m
dx
l+m
(x
2
− 1)
l
= (1 − x
2
)
m/2
d
m
dx
m
P
l
(x)
(2.58)
1
n and κ are, respectively, the real and the imaginary part of the index of refraction.
2.4 The RTE for a horizontally homogeneous atmosphere 41
They have the following properties
(a) P
m
l
(x) = 0, m > l, P
m=0
l
(x) = P
l
(x), P
m
l
(−x) = (−1)
l+m
P
m
l
(x)
(b)
1
−1
P
m
n
(x)P
m
l
(x)dx =
2
2l + 1
(l + m)!
(l − m)!
δ
nl
(2.59)
Equation (2.59b) expresses the orthogonality relation of the associated Legendre
polynomials on the interval [−1, 1].
In order to obtain an analytical expression for the phase function P we need to
evaluate the expansion coefficients p
l
occurring in (2.55). This is done by multi-
plying both sides of (2.55) by P
n
(cos ) and then integrating over the unit sphere
1
4π
4π
P(cos )P
n
(cos )d =
1
4π
4π
∞
l=0
p
l
P
l
(cos )P
n
(cos )d (2.60)
In this equation the integration
4π
...d may be replaced by
2π
0
dϕ
1
−1
...d cos . Application of the orthogonality relations for P
l
, which are
given by (2.59b) if we set m = 0, finally yields
p
l
=
2l + 1
2
1
−1
P
l
(cos )P(cos )d cos
(2.61)
In addition to specifying P with the help of the Mie theory, mathematical models
or experimentally determined values of the phase function may be employed. Due
to the normalization of P, i.e.
1
4π
4π
P(cos )d =
1
4π
2π
0
1
−1
P(cos )d cos dϕ
=
1
2
1
−1
P(cos )d cos = 1
(2.62)
the first expansion coefficient p
0
is always given by
p
0
=
1
2
1
−1
P(cos )d cos = 1 (2.63)
Thus the coefficient p
0
expresses conservation of energy in case of pure scattering.
In the following we require an explicit expression for the cosine of the scattering
angle . From Figure 1.16 and equation (1.36) one easily obtains for Ω
· Ω =
cos the expression
cos =(i cos ϕ
sin ϑ
+ j sin ϕ
sin ϑ
+ k cos ϑ
)
· (i cos ϕ sin ϑ +j sin ϕ sin ϑ + k cos ϑ)
(2.64)
42 The radiative transfer equation
With µ = cos ϑ, µ
= cos ϑ
we find
cos = µµ
+ (1 − µ
2
)
1/2
(1 − µ
2
)
1/2
cos(ϕ − ϕ
)
(2.65)
This relation allows us to express the scattering angle with respect to the angles
ϑ
,ϕ
and ϑ, ϕ. From (2.53) it may be seen that the angles ϑ, ϕ are fixed while
ϑ
,ϕ
vary over the unit sphere.
To continue the mathematical development we also need to state the addition
theorem for the associated Legendre polynomials which is given by
P
n
(cos ) =P
n
(µ)P
n
(µ
) + 2
n
m=1
(n − m)!
(n + m)!
P
m
n
(µ)P
m
n
(µ
) cos m(ϕ − ϕ
)
(2.66)
(cf. Jackson, 1975). This theorem will now be used to separate the angles of the
scattering phase function. Substituting (2.66) into (2.55) gives
P(cos ) =
∞
l=0
p
l
P
l
(µ)P
l
(µ
) + 2
l
m=1
(l − m)!
(l + m)!
P
m
l
(µ)P
m
l
(µ
) cos m(ϕ − ϕ
)
=
∞
l=0
l
m=0
(2 − δ
0m
)p
m
l
P
m
l
(µ)P
m
l
(µ
) cos m(ϕ − ϕ
) (2.67)
with p
m
l
= p
l
(l − m)!
(l + m)!
Since P
m
l
(x) = 0 for m > l, P(cos ) can be reformulated as
P(cos ) =
∞
l=0
∞
m=0
(2 − δ
0m
)p
m
l
P
m
l
(µ)P
m
l
(µ
) cos m(ϕ − ϕ
)
=
∞
m=0
(2 − δ
0m
)
∞
l=m
p
m
l
P
m
l
(µ)P
m
l
(µ
) cos m(ϕ − ϕ
)
= P(µ, ϕ, µ
,ϕ
)
(2.68)
It can now be seen that in the scattering phase function the (µ, µ
)-dependence has
been completely separated from the (ϕ, ϕ
)-dependence.
In order to remove the azimuthal dependence of the radiance field we assume
that the radiance can be expressed by means of a Fourier expansion for an even
function of the azimuth angle ϕ
I (τ,µ,ϕ) =
∞
m=0
(2 − δ
0m
)I
m
(τ,µ) cos m(ϕ − ϕ
0
) (2.69)
2.4 The RTE for a horizontally homogeneous atmosphere 43
where ϕ
0
is the azimuth angle of the direct solar beam. It is customary to orient the
Cartesian (x, y, z)-coordinate system in such a way that ϕ
0
= 0, see Figure 2.3, so
that
I (τ,µ,ϕ) =
∞
m=0
(2 − δ
0m
)I
m
(τ,µ) cos mϕ
(2.70)
The expansion (2.70) may be used whenever the phase function is expressed as a
series of Legendre polynomials.
Substituting (2.68) and (2.70) into the RTE (2.53) yields
µ
∞
m=0
(2 − δ
0m
)
d
dτ
I
m
(τ,µ) cos mϕ
=
∞
m=0
(2 − δ
0m
)I
m
(τ,µ) cos mϕ
−
ω
0
4π
S
0
exp
−
τ
µ
0
∞
m=0
(2 − δ
0m
)
∞
l=m
p
m
l
P
m
l
(µ)P
m
l
(−µ
0
) cos mϕ
−
ω
0
4π
2π
0
1
−1
∞
m=0
(2 − δ
0m
)
∞
l=m
p
m
l
P
m
l
(µ)P
m
l
(µ
) cos m(ϕ − ϕ
)
×
∞
i=0
(2 − δ
0i
)I
i
(τ,µ
) cos i ϕ
dµ
dϕ
− (1 − ω
0
)
∞
m=0
δ
0m
B(τ )
(2.71)
The above equation can be simplified by employing the orthogonality relations of
the trigonometric functions
2π
0
cos m(ϕ − ϕ
) cos lϕ dϕ = (1 + δ
0l
)δ
lm
π cos lϕ
(2.72)
An expression for I
m
(τ,µ) can be isolated from (2.71) by carrying out the following
two steps: (i) Multiply (2.71) by cos kϕ; (ii) Carry out the operation
2π
0
...dϕ.
As a consequence, the following relations are needed
2π
0
cos mϕ cos kϕ dϕ = (1 + δ
0k
)δ
mk
π
2π
0
cos kϕ dϕ = (1 +δ
0k
)δ
0k
π
2π
0
cos m(ϕ − ϕ
) cos kϕ dϕ = (1 + δ
0k
)δ
mk
π cos kϕ
(2.73)
44 The radiative transfer equation
From (2.71) it is seen that for the multiple scattering term an integration over ϕ
remains to be done for which we obtain the result
2π
0
cos iϕ
cos kϕ
dϕ
= (1 + δ
0i
)δ
ik
π (2.74)
Using the above intermediate steps we find from (2.71)
µ
∞
m=0
(2 − δ
0m
)
d
dτ
I
m
(τ,µ)(1 + δ
0k
)δ
mk
π
=
∞
m=0
(2 − δ
0m
)(1 + δ
0k
)δ
mk
π I
m
(τ,µ)
−
ω
0
4π
S
0
exp
−
τ
µ
0
∞
m=0
(2 − δ
0m
)
∞
l=m
p
m
l
P
m
l
(µ)P
m
l
(−µ
0
)(1 + δ
0k
)δ
mk
π
−
ω
0
4π
1
−1
∞
m=0
(2 − δ
0m
)
∞
l=m
p
m
l
P
m
l
(µ)P
m
l
(µ
)(1 + δ
0k
)δ
mk
π
×
∞
i=0
(2 − δ
0i
)I
i
(τ,µ
)(1 + δ
0i
)δ
ik
π
dµ
− (1 − ω
0
)
∞
m=0
δ
0m
B(τ )(1 + δ
0k
)δ
0k
π
(2.75)
We recognize immediately that, with the exception of m = k, all terms vanish.
Therefore, after separation of the azimuthal dependence, for the m-th Fourier mode
of the radiance the RTE is given by
(a) µ
d
dτ
I
m
(τ,µ) = I
m
(τ,µ) −
ω
0
2
1
−1
∞
l=m
p
m
l
P
m
l
(µ)P
m
l
(µ
)I
m
(τ,µ
)dµ
−
ω
0
4π
S
0
exp
−
τ
µ
0
∞
l=m
p
m
l
P
m
l
(µ)P
m
l
(−µ
0
) − (1 − ω
0
)B(τ )δ
0m
(b) µ
d
dτ
I
m
(τ,µ) = I
m
(τ,µ) − J(τ, µ)
(2.76)
(2.76b) is the standard form of the RTE for plane–parallel atmospheres. The source
function J (τ,µ) summarizes the terms describing multiple scattering, primary scat-
tering of solar radiation and thermal emissions.
For the special case m = 0 the same result is obtained by performing an azimuthal
average of (2.53). Consider the averaging process for each individual term separ-
ately.